RNDr. Pavel Bakala, PhD

Pavel Bakala
K některým aspektům optických efektů
v blı́zkosti černých děr a neutronových hvězd
(komentovaný soubor článků)
DIZERTAČNÍ PRÁCE
Slezská univerzita v Opavě, Filozoficko-přı́rodovědecká fakulta
Ústav fyziky
K některým aspektům optických
efektů v blı́zkosti černých děr
a neutronových hvězd
(komentovaný soubor článků)
DIZERTAČNÍ PRÁCE
Vedoucı́: prof. RNDr. Zdeněk Stuchlı́k, CSc.
Konzultant: RNDr. Stanislav Hledı́k, Ph.D.
Opava 2010
Pavel Bakala
Poděkovánı́
Na tomto mı́stě bych v prvé řadě rád poděkoval svým rodičům. Můj otec,
který se bohužel obhajoby mé dizertace již nedožije, mi již od dětstvı́
vždy svou přı́tomnostı́ vytvářel intelektuálně inspirujı́cı́ prostředı́. Mé matce
vděčı́m za důvěru a podporu poskytovanou mi kdykoli během mého studia, ač
si nedokáži představit jı́ vzdálenějšı́ vědnı́ obor, než je teoretická fyzika a astrofyzika. Za mnohé vděčı́m také svým přátelům a blı́zkým pro mne zcela nezbytným pro udrženı́ mé duševnı́ rovnováhy, které jenom z nedostatku mı́sta
zde nemohu vyjmenovat. Zvláštnı́ poděkovánı́ však patřı́ těm přátelům, kteřı́
jsou mi zároveň blı́zkými spolupracovnı́ky, Evě Šrámkové, Gabrielu Törökovi
a Martinu Urbancovi. Poděkovánı́ samozřejmě patřı́ mému školiteli Zdeňku
Stuchlı́kovi za vedenı́ mého doktorského studia, inspirativnı́ náměty i diskuze a v neposlednı́ řadě za přátelský přı́stup. Podobné poděkovánı́ náležı́
i mému konzultantu Stanislavu Hledı́kovi. Závěrem bych rád poděkoval mé
drahé dceři Alence za jejı́ nekonečnou trpělivost s otcem trávı́cı́m své večery
studiem.
Obsah
Úvod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Kapitola 1. Virtuálnı́ výlet k horizontu černé dı́ry a povrchu
neutronové hvězdy . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.1. Kontext a motivace . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2. Optické zobrazenı́ ve sféricky symetrických prostoročasech . . . . .
1.2.1. Pohyb fotonů ve sféricky symetrických prostoročasech . . .
1.2.2. Konstrukce optického zobrazenı́ . . . . . . . . . . . . . . .
1.2.3. Geometrie optického zobrazenı́ . . . . . . . . . . . . . . . .
1.3. Softwarová implementace . . . . . . . . . . . . . . . . . . . . . . .
1.3.1. Modul vstupů . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2. Modul relativistického raytracingu . . . . . . . . . . . . . .
1.3.3. Modul zpracovánı́ výstupů . . . . . . . . . . . . . . . . . .
1.4. Vizualizačnı́ výstupy simulacı́ . . . . . . . . . . . . . . . . . . . . .
1.4.1. Statický pozorovatel v blı́zkosti Schwarzschildovy černé dı́ry
1.4.2. Pozorovatel radiálně volně padajı́cı́ do Schwarzschildovy
černé dı́ry . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.3. Povrch rotujı́cı́ superkompaktnı́ neutronové či kvarkové
hvězdy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5. Shrnutı́ a perspektiva . . . . . . . . . . . . . . . . . . . . . . . . .
3
5
5
7
9
12
12
13
13
14
14
15
20
22
Kapitola 2. QPOs: Pohled z nekonečna . . . . . . . . . . . . . . . . . 25
2.1. Fenomén kvaziperiodických oscilacı́ (QPOs) . . . . . . . . . .
2.1.1. kHz QPOs v systémech s neutronovou hvězdou . . . .
2.1.2. Frekvenčnı́ korelace hornı́ch a dolnı́ch QPOs . . . . .
2.1.3. Klastrovánı́ twin-peak QPOs v okolı́ poměrů malých
celých čı́sel. . . . . . . . . . . . . . . . . . . . . . . . .
2.2. Orbitálnı́ modely vzniku QPOs . . . . . . . . . . . . . . . . .
2.2.1. Relativistický precesnı́ model . . . . . . . . . . . . . .
2.2.2. Preferované kruhové orbity . . . . . . . . . . . . . . .
2.2.3. Odhady hmotnosti a spinu s použitı́m relativistického
precesnı́ho modelu: Circinus X-1 . . . . . . . . . . . .
2.3. Shrnutı́ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 25
. . . 28
. . . 29
.
.
.
.
.
.
.
.
.
.
.
.
31
34
36
38
. . . 38
. . . 41
Kapitola 3. Magnetická pole . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1. Observačnı́ motivace . . . . . . . . . . . . . . . . . . . . . . . . . . 45
vii
3.2. Perturbovaný kruhový orbitálnı́ pohyb nabitých testovacı́ch
částic v dipólovém magnetickém poli na schwarzschildovském
pozadı́ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1. Dipólové magnetické pole na pozadı́ Schwarzschildovy
prostoročasové geometrie . . . . . . . . . . . . . . . . . . .
3.2.2. Frekvence perturbovaného kruhového orbitálnı́ho pohybu .
3.2.3. Chovánı́ negeodeticky korigovaných frekvencı́, existence
a stabilita kruhových orbit . . . . . . . . . . . . . . . . . .
3.2.4. Aplikace na relativistický precesnı́ model . . . . . . . . . .
3.3. Perspektiva dalšı́ho výzkumu: magnetické pole pomalu rotujı́cı́
neutronové hvězdy . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1. Lenseova–Thirringova metrika . . . . . . . . . . . . . . . .
3.3.2. Geodetický kvazikruhový orbitálnı́ pohyb . . . . . . . . . .
3.3.3. Dipólové magnetické pole na Lenseově–Thirringově pozadı́
46
46
48
50
52
54
55
56
57
Literatura . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Přı́lohy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
viii
Úvod
Pro svou dizertačnı́ práci jsem zvolil formu komentovaného souboru článků
shrnujı́cı́ch výsledky výzkumu dosažené v rámci mého doktorského studia
pod vedenı́m školitele prof. RNDr. Zdeňka Stuchlı́ka, CSc. Tématem dizertačnı́ práce jsou optické efekty v blı́zkosti černých děr a relativisticky
kompaktnı́ch hvězd. V práci zahrnuté publikace se tématu dotýkajı́ ze třı́
odlišných úhlů, což je reflektováno volbou formy komentovaného souboru
článků i rozčleněnı́m úvodnı́ho textu do třı́ téměř autonomnı́ch kapitol.
Ve shodě s tématickým rozčleněnı́m jsou publikace seřazeny v přı́lohách a
očı́slovány následujı́cı́m způsobem:
1. P. Bakala, P. Čermák, S. Hledı́k, Z. Stuchlı́k, K. Truparová
(2007): Extreme gravitational lensing in vicinity of
Schwarzschild–de Sitter black holes, Central European Journal
of Physics, 5/4, 599
2. G. Török, M. A. Abramowicz, P. Bakala, M. Bursa, J. Horák, W.
Kluźniak, P. Rebusco, Z. Stuchlı́k (2008): Distribution of Kilohertz QPO Frequencies and Their Ratios in the Atoll
Source 4U 1636-53, Acta Astronomica, 58, 15
3. G. Török, M. A. Abramowicz, P. Bakala, M. Bursa, J. Horák, P.
Rebusco, Z. Stuchlı́k (2008): On the origin of clustering of
frequency ratios in the atoll source 4U 1636-53, Acta Astronomica, 58, 113
4. G. Török, P. Bakala, Z. Stuchlı́k, P. Čech (2008): Modelling the twin
peak QPO distribution in the atoll source 4U 1636-53, Acta
Astronomica, 58, 1
5. G. Török, Z. Stuchlı́k, P. Bakala (2007): A remark about possible
unity of the neutron star and black hole high frequency
QPOs, Central European Journal of Physics, 5/4, 457
1
2
Úvod
6. G. Török, P. Bakala, E. Šrámková, Z. Stuchlı́k (2010): On Mass Constraints Implied by the Relativistic Precession Model of
Twin-peak Quasi-periodic Oscillations in Circinus X-1, Astrophysical Journal, 714/1, 748
7. P. Bakala, E. Šrámková, Z. Stuchlı́k, G. Török (2008): On
magnetic-field induced non-geodesic corrections to the relativistic precession QPO model, Cool Disc, Hot Flows: The Varying Faces of Accreting Compact Objects. AIP Conference Proceedings,
1054, 123
8. P. Bakala, E. Šrámková, Z. Stuchlı́k, G. Török (2010): On
magnetic-field-induced non-geodesic corrections to relativistic orbital and epicyclic frequencies, Classical and Quantum
Gravity, 27/4, 045001
V prvnı́ kapitole je diskutována simulace a vizualizace optického zobrazovánı́ vzdáleného vesmı́ru pro pozorovatele nacházejı́cı́ se v těsné blı́zkosti
sféricky symetrických černých děr. Po krátkém přehledu teorie pohybu
fotonů ve sféricky symetrických prostoročasech se zřetelem na vliv elektrického a hypotetického slapového náboje černých děr i repulzivnı́ kosmologické konstanty je dále popisována geometrie optického zobrazovánı́ a
softwarové řešenı́ vyvinutého simulačnı́ho kódu. Závěr kapitoly je doplněn
obrazovými výstupy simulacı́.
Druhá kapitola je po stručném úvodu do fenomenologie kvaziperiodických oscilacı́ (QPOs) a některých teoretických východisek jejich popisu
pomocı́ orbitálnı́ho pohybu v silném gravitačnı́m poli věnována analýze
observačnı́ch dat rentgenových LMXB zdrojů 4U 1636-53 a Circinus X-1.
Zkoumány jsou distribuce dolnı́ch, hornı́ch i twin-peak kHz QPOs a jejich
klastrovánı́ v okolı́ význačných poměrů frekvencı́ kHz QPOs. Na pozadı́ relativisticky precesnı́ho modelu je analyzována souvislost klastrovánı́ detekcı́
s existencı́ preferovaných kruhových orbit a diskutovány možnosti odhadu
hmotnostı́ a spinu neutronových hvězd.
S optickými efekty již relativně volně spojená třetı́ kapitola je věnována
analýze kruhového orbitálnı́ho pohybu nabitých testovacı́ch částic v okolı́
zmagnetizovaných neutronových hvězd. Právě negeodetické korekce orbitálnı́ho pohybu však mohou hrát nezanedbatelnou úlohu při vysvětlenı́ detailů QPO modulace rentgenových toků přicházejı́cı́ch z blı́zkosti akreujı́cı́ch
kompaktnı́ch objektů, tak jak jsou diskutovány v předcházejı́cı́ kapitole.
Kapitola 1
Virtuálnı́ výlet k horizontu černé dı́ry
a povrchu neutronové hvězdy
. . . Kolem Země pluje svět, připomı́ná tér
čtvrtý rozměr bránou je, zmatek černejch děr
Podı́vám se zblı́zka . . .
— Zuzana Michnová & Marsyas
1.1. Kontext a motivace
Článek Extreme gravitational lensing in vicinity of
Schwarzschild–de Sitter black holes, který je obsahem přı́lohy 1, je
věnován analýze výsledků počı́tačové simulace vzhledu vzdáleného vesmı́ru
pro pozorovatele nacházejı́cı́ho se v těsné blı́zkosti Schwarzschildovy–de Sitterovy černé dı́ry, tedy ve výrazně zakřiveném sféricky symetrickém
prostoročase a za přı́tomnosti repulzivnı́ kosmologické konstanty.
Ohyb světla v gravitačnı́m poli (gravitačnı́ lensing) byl jednou z prvnı́ch
astrofyzikálnı́ch predikcı́ obecné teorie relativity a také byl předmětem
jejı́ho prvnı́ho Eddingtonova experimentálnı́ho testu. Přı́pad slabého gravitačnı́ho lensingu (weak lensing), efekt ohybu světla vzdáleného vesmı́rného
objektu mezilehlou hvězdou (galaxiı́) vystupujı́cı́ v roli gravitačnı́ čočky
byl v roce 1936 poprvé analyzován A. Einsteinem, který ovšem zůstával
skeptický k možnostem jeho potvrzenı́ pozorovánı́mi [29]. V současné době
jsou však právě efekty slabého gravitačnı́ho lensingu běžným observačnı́m
nástrojem při zkoumánı́ objektů v hlubokém vesmı́ru, při hledánı́ exoplanet,
i při detekci temné hmoty, hrajı́cı́ klı́čovou úlohu v současných kosmologických modelech [69].
Naproti tomu vizualizace a simulace vzhledu oblohy pro pozorovatele v blı́zkosti černých děr jistě nepatřı́ k fenoménům v současnosti
a pravděpodobně i v blı́zké budoucnosti experimentálně testovatelným,
3
4
Kapitola 1. Virtuálnı́ výlet k horizontu černé dı́ry. . .
nicméně poskytuje působivou ilustraci výrazných rozdı́lů mezi optikou
v prostředı́ silného gravitačnı́ho pole a optikou v plochém prostoročase,
kterou známe z každodennı́ho života a jejı́ž zdánlivě samozřejmé vlastnosti
formovaly naše vnı́mánı́ prostoru a tı́m i naši intuici a představivost. Takto
formulovaná úloha také zcela jistě naplňuje význam termı́nu teoretická
”
astrofyzika“. Teoretický rozbor optického zobrazovánı́ vzdáleného vesmı́ru
pro pozorovatele v blı́zkosti Schwarzschildovy černé dı́ry byl poprvé publikován C. T. Cunnighamem [25]. Odpovı́dajı́cı́ výsledky počı́tačové simulace
vzhledu oblohy pro pozorovatele v blı́zkosti Schwarzschildovy černé dı́ry
nebo neutronové hvězdy lze nalézt ve studii R. J. Nemiroffa [58].
Současné kosmologické testy ukazujı́, že expanze vesmı́ru je v současnosti
urychlována tzv. temnou energiı́, která může být popsána v Einsteinových
rovnicı́ch gravitačnı́ho pole efektivnı́ repulzivnı́ kosmologickou konstantou [42, 61]. V článku, který je obsahem přı́lohy 1, je proto analýza optické
projekce pro statické i radiálně volně padajı́cı́ pozorovatele v silném sféricky
symetrickém gravitačnı́m poli o vliv repulzivnı́ kosmologickém konstanty
rozšı́řena.
Dalšı́m teoreticky možným parametrem sféricky symetrické metriky je
kvadrát elektrického náboje spojeného s centrálnı́ hmotnostı́. Pokud se
jedná o asymptoticky ploché řešenı́, hovořı́me o Reissnerově–Nordströmově
prostoročase, v přı́padě přı́tomnosti repulzivnı́ kosmologické konstanty
pak o prostoročase Reissnerově–Nordströmově–de Sitterově [80]. Vzhledem
k celkové elektrické neutralitě hmoty ve vesmı́ru, ze které gravitačnı́m kolapsem stelárnı́ či supermasivnı́ černé dı́ry vznikajı́, nejsou nejspı́še elektricky nabité černé dı́ry astrofyzikálně realistické. I pokud připustı́me
vznik nabitých černých děr, jejich náboj by byl pravděpodobně velmi
rychle neutralizován atrakcı́ částic s nábojem opačným. Nicméně nabitá
řešenı́ v současné době zažı́vajı́ určitou astrofyzikálnı́ renesanci v souvislosti s novými vı́cedimenzionálnı́mi bránovými kosmologickými modely, ve
kterých existuje třı́da řešenı́ Einsteinových rovnic popisujı́cı́ prostoročasy
v okolı́ relativisticky sféricky symetrických kompaktnı́ch objektů vázaných
na náš tzv. bránový svět formálně právě Reissnerovou–Nordströmovou
metrikou [26, 30]. V takovém přı́padě je ovšem kvadrát elektrického
náboje nahrazen novým parametrem metriky, nábojem slapovým. Tento
parametr oproti kvadrátu elektrického náboje může nabývat kladných
i záporných hodnot, a zdá se dokonce, že jeho záporná hodnota je fyzikálně
přirozenějšı́ [30].
Simulačnı́ kód BHimpaCt použitý ke generovánı́ publikovaných výsledků
(přı́loha 1) v současné verzi proto umožňuje modelovat jak vliv kosmolo-
1.2. Optické zobrazenı́ ve sféricky symetrických prostoročasech
5
gické konstanty tak i elektrického či slapového náboje na vlastnosti optického zobrazovánı́ v sféricky symetrických černoděrových prostoročasech.
Kód umožňuje simulovat přı́spěvky světelných geodetik vyššı́ch řádů (s orbitami ve tvaru vı́cenásobných smyček kolem gravitačnı́ho centra) k optické
projekci, zohledňuje efekty dopplerovského i gravitačnı́ho frekvenčnı́ho posunu (blueshift, redshift) a gravitačnı́m polem indukované amplifikace intenzity. Následujı́cı́ kapitola je po stručném přehledu teorie pohybu fotonů ve
sféricky symetrických prostoročasech věnována diskuzi vlastnostı́ optického
zobrazovánı́ ve sféricky symetrických prostoročasech, popisu softwarové architektury kódu BHimpaCt a demonstraci jeho vizualizačnı́ch výstupů.
1.2. Optické zobrazenı́ ve sféricky symetrických
prostoročasech
1.2.1. Pohyb fotonů ve sféricky symetrických prostoročasech
Prostoročas v okolı́ sféricky symetrické černé dı́ry nebo relativisticky kompaktnı́ (neutronové či podivné) hvězdy lze reprezentovat metrikou s elementem prostoročasového intervalu zapsaným ve standardnı́ch Schwarzschildových souřadnicı́ch zapsaným s použitı́m geometrických jednotek (M =
G = c = 1) ve tvaru
ds2 = −B(r, β, Λ)dt2 + B(r, β, Λ)−1dr 2 + r 2 (dθ2 + sin2 θ dφ2 ) ,
(1.1)
kde funkce B(r, β, Λ) je dána vztahem
B(r, β, Λ) ≡ 1 −
β
Λ
2
+ 2 − r2 .
r r
3
(1.2)
Parametr β v přı́padě bránových řešenı́ značı́ slapový náboj, v přı́padě
nabité černé dı́ry má pak význam kvadrátu elektrického náboje spojeného
s centrálnı́m objektem ( β = Q2 ) a konečně Λ je kosmologická konstanta.
Přı́slušným nastavenı́m parametrů tak metrika ve tvaru 1.1 může reprezentovat čistě Schwarzschildův prostoročas, Schwarzschildovo–de Sitterovo
řešenı́ analyzované v přı́loze 1 anebo elektricky popřı́padě slapově nabitá řešenı́ Reissnerova–Nordströmova typu. Ve sféricky symetrických prostoročasech je pohyb fotonů určen impaktnı́m parametrem definovaným
jako
b≡
Φ
,
E
(1.3)
6
Kapitola 1. Virtuálnı́ výlet k horizontu černé dı́ry. . .
kde Φ a E jsou pohybové konstanty odpovı́dajı́cı́ Killingovým vektorům
ξ(t) and ξ(φ) generovaným přı́slušnými symetriemi prostoročasu [56]. Symetrie prostoročasu implikuje zachovánı́ celkového momentu hybnosti a
tı́m i centrálnı́ roviny pohybu fotonů a lze tedy bez újmy na obecnosti
předpokládat, že pohyb fotonů probı́há v ekvatoriálnı́ rovině. Kovariantnı́
komponenty čtyřhybnosti fotonů lze pak zapsat ve tvaru
pt = −E,
pr =
A(r, β, Λ)
,
B(r, β, Λ)
pφ = Φ,
pθ = 0,
(1.4)
kde
A(r, β, Λ) ≡ sA
r
1 − B(r, β, Λ)
b2
.
r2
(1.5)
Znaménko sA nabývá hodnoty + pro fotony vzdalujı́cı́ se od centrálnı́ho
kompaktnı́ho objektu, − pro přibližujı́cı́ se fotony.
Diskuse vlastnostı́ trajektoriı́ fotonů přicházejı́cı́ch z nekonečna nebo
v přı́padě přı́tomnosti repulzivnı́ kosmologické konstanty z blı́zkosti statického poloměru a tedy reprezentujı́cı́ch zářenı́ objektů vzdáleného vesmı́ru
je pro Schwarzschildův–de Sitterův prostoročas podrobně provedena v [10,
79, 81] a v přı́loze 1 aplikována na konstrukci optického zobrazenı́. Kvalitativnı́ rysy chovánı́ takových geodetik zůstávajı́ zachovány i v přı́padě
nabitých černoděrových prostoročasů [80].
Připomeňme, že existujı́ principálně dva druhy nulových geodetik
přicházejı́cı́ch ze vzdáleného vesmı́ru, které můžeme snadno odlišit dle
hodnoty jejich impaktnı́ho parametru b. Pokud označı́me bcrit impaktnı́
parametr fotonů, které, přicházejı́ce ze vzdáleného vesmı́ru nebo naopak
z blı́zkosti černoděrového horizontu, budou zachyceny na nestabilnı́ kruhové fotonové orbitě, pak všechny geodetiky s b < bcrit odpovı́dajı́ fotonům
finálně dopadajı́cı́m na horizont černé dı́ry a naopak fotony emitované ve
vzdáleném vesmı́ru s b > bcrit se po dosaženı́ svého bodu obratu od černé
dı́ry opět vzdalujı́ a unikajı́ zpět do nekonečna.
Pohyb fotonů je determinován Binetovým vzorcem nabývajı́cı́m v prostoročase s metrikou 1.1 tvaru
1
dφ
= ±q
du
b−2 − u2 + 2u3 − β 2 u4 +
Λ
3
,
u = r −1 .
(1.6)
1.2. Optické zobrazenı́ ve sféricky symetrických prostoročasech
7
Tvar Binetova vzorce přirozeně implikuje podmı́nku existence pohybu fotonů ve tvaru
Λ
−2
2
3
2 4
C (b, u, β, Λ) ≡ b − u + 2u + β u +
≥ 0.
(1.7)
3
Bod obratu geodetik přicházejı́cı́ch ze vzdáleného vesmı́ru rturn = 1/uturn
s b > bcrit je určen kořenem rovnice
C (b, u, β, Λ) = 0
(1.8)
ležı́cı́m v intervalu mezi hodnotou radiálnı́ souřadnice nestabilnı́ kruhové
fotonové orbity rph a statickým poloměrem (pro Λ = 0 jdoucı́m do nekonečna). Fotony na takových geodetikách tak nikdy nedosáhnou polohy
pod nestabilnı́ kruhovou fotonovou orbitou. Naopak pro nulové geodetiky s
b < bcrit je podmı́nka existence splněna na každé hodnotě radiálnı́ souřadnice
a takové fotony přicházejı́cı́ ze vzdáleného vesmı́ru svou trajektoriı́ nestabilnı́ kruhovou fotonovou orbitu protı́najı́ a dopadajı́ na černoděrový horizont. Meznı́m přı́padem jsou pak geodetiky s b = bcrit , které odpovı́dajı́
záchytu fotonů právě na nestabilnı́ kruhové orbitě, jejı́ž poloha rph = 1/uph
odpovı́dá minimu funkce C (b, u, β, Λ), které nezávisı́ na hodnotě kosmologické konstanty a je dáno vztahem1
rph =
p
1
3 + 3 − 8β .
2
(1.9)
Odpovı́dajı́cı́ kritickou hodnotu impaktnı́ho parametru bcrit lze pak snadno
zı́skat dosazenı́m rph do rovnice 1.8 .
1.2.2. Konstrukce optického zobrazenı́
Uvažujme nynı́ nulové geodetiky spojujı́cı́ zdroj se souřadnicemi
(rsource , π/2, φsource ) a pozorovatele se souřadnicemi (robs , π/2, 0), tedy ležı́cı́
v ekvatoriálnı́ rovině. Pak integrálnı́ rovnici vyjadřujı́cı́ impaktnı́ parametr
b jako implicitnı́ funkci okrajových podmı́nek (souřadnic zdroje a pozorovatele) a parametrů metriky můžeme zapsat ve tvaru
∆φ (b, robs , rsource , β, Λ) + φsource + 2kπ = 0 ,
1
(1.10)
Podrobnějšı́ diskuzi o existenci a charakteru kruhových fotonových orbit i statických
poloměrů v souvislosti s černoděrovým nebo nahosingulárnı́m charakterem sféricky symetrických prostoročasů lze nalézt v [79, 80].
8
Kapitola 1. Virtuálnı́ výlet k horizontu černé dı́ry. . .
kde ∆φ je změna souřadnice φ podél přı́slušné geodetiky a lze ji zı́skat
integracı́ Binetova vzorce 1.6. Parametr k značı́ řád obrazu a udává počet
fotonem opsaných smyček kolem gravitačnı́ho centra. Pro geodetiky orbitujı́cı́ pravotočivě a generujı́cı́ tzv. přı́mé obrazy parametr k nabývá hodnot
0, 1, 2, . . . , +∞ , zatı́mco pro geodetiky orbitujı́cı́ levotočivě a generujı́cı́
tzv. obrazy nepřı́mé nabývá k hodnot −1, −2, . . . , −∞. Nekonečné hodnoty k = ±∞ pak korespondujı́ se záchytem fotonů na nestabilnı́ kruhové
fotonové orbitě [10].
Diskuze řešenı́ rovnice 1.10 a integrace Binetova vzorce vzhledem k orientaci geodetiky, existenci a poloze bodu obratu je podrobně provedena
v přı́loze 1 a v [10]. Poznamenejme zde pouze, že tvar podmı́nky existence
pohybu fotonů 1.7 implikuje pro pozorovatele umı́stěného nad nestabilnı́
kruhovou fotonovou orbitou (robs = 1/uobs > rph ) existenci maximálnı́ hodnoty impaktnı́ho parametru dané vztahem
1/2
Λ
2
3
2 4
.
bmax (robs ) = uobs − 2uobs + β u −
3
(1.11)
Fotony přicházejı́cı́ ze vzdáleného vesmı́ru s b > bmax nikdy nedosáhnou
pozice pozorovatele a procházejı́ svým bodem obratu již na rturn > robs .
Maximálnı́ impaktnı́ parametr bmax (robs ) tak odpovı́dá vcházejı́cı́m nulovým
geodetikám s bodem obratu právě na radiálnı́ souřadnici pozorovatele. Pro
pozorovatele umı́stěné pod nestabilnı́ kruhovou fotonovou orbitou jsou relevantnı́ pouze vcházejı́cı́ geodetiky s b < bcrit a maximálnı́ impaktnı́ parametr
pro takové pozorovatele je přirozeně totožný s kritickým impaktnı́m parametrem bcrit .
Hodnoty impaktnı́ho parametru b a znaménka sA , zı́skané řešenı́m rovnice 1.10, postačujı́ k určenı́ komponent čtyřhybnosti fotonu na souřadnicı́ch
pozorovatele a ty po nezbytné transformaci do lokálnı́ho referenčnı́ho
systému spojeného s pozorovatelem jednoznačně určujı́ směrový úhel, pod
kterým danou geodetikou generovaný obraz pozorovatel uvidı́. Směrový úhel
α a odpovı́dajı́cı́ frekvenčnı́ posuv g fotonů (poměr pozorované a emitované
energie) lze pomocı́ lokálnı́ch komponent čtyřhybnosti fotonu zapsat vztahy
(t)
(r)
cos α = −
pobs
(t)
pobs
,
g=
pobs
(t)
psource
.
(1.12)
Gravitačnı́ pole provádı́ časovou, energetickou i prostorovou redistribuci
toku fotonů – intenzity zářenı́ – ze vzdáleného vesmı́ru na oblohu pozorovatele. Časovou redistribuci lze chápat jako změnu dopadajı́cı́ho počtu
1.2. Optické zobrazenı́ ve sféricky symetrických prostoročasech
9
fotonů za časovou jednotku dı́ky rozdı́lnému tempu plynutı́ času v lokálnı́ch
systémech spojených s pozorovatelem a zdrojem zářenı́, energetická je pak
způsobena frekvenčnı́m posuven g (blueshift, redshift) Celkovou amplifikaci
bolometrické intenzity zdroje lze zapsat vztahem
Atotal = Atime · Aangular ,
(1.13)
kde faktor Atime = g 4 reprezentuje právě časovou a energetickou redistribuci
intenzity zdroje a prostorová část amplifikace Aangular souvisı́ s fokusacı́
fotonových svazků gravitačnı́m polem. Uvažujeme-li malý izotropně zářı́cı́
zdroj ve vzdáleném vesmı́ru, pak faktor Aangular pro konkrétnı́ obraz zdroje
je dán poměrem prostorového úhlu, který obraz vytı́ná na pozorovatelově
obloze a prostorového úhlu, který by zdroj vytı́nal na obloze bez přı́tomnosti
gravitačnı́ho pole a může být vyjádřen vztahem
sin γ dγ
Aangular =
,
(1.14)
sin β dβ
kde γ je úhlová vzdálenost gravitačnı́ho centra a zdroje a β úhlová
vzdálenost gravitačnı́ho centra a obrazu [58].
1.2.3. Geometrie optického zobrazenı́
Sférická symetrie gravitačnı́ho pole se manifestuje i v odpovı́dajı́cı́ geometrii optického zobrazenı́. Klı́čovou roli v transformaci optické projekce
hraje optická osa definovaná jako přı́mka spojujı́cı́ pozici pozorovatele a
gravitačnı́ho centra. V nepřı́tomnosti gravitačnı́ho pole na obloze pozorovaná kružnice se středem právě na optické ose a s poloměrem určujı́cı́m
tak souřadnici zdrojů φsource na nı́ ležı́cı́ch2 je efekty extrémnı́ho gravitačnı́ho lensingu projektována na sérii koncentrických kružnic s poloměry
klesajı́cı́mi spolu se vzrůstajı́cı́ absolutnı́ hodnotou přı́slušného řádu obrazu k, přičemž přı́mé obrazy o řádu k+ jsou bezprostředně následovány
nepřı́mými o řádu k− = −(k+ + 1). Chovánı́ směrových úhlů α(φsource , k)
pro libovolné obrazy odlišných řádů lze pak charakterizovat relacemi
α(φsource 1 , k+ ) > α(φsource 2 , k− = −(k+ + 1) ) ,
(1.15)
α(φsource 1 , |k + 1|) > α(φsource 2 , |k|) ,
α(φsource , k) < π .
2
Připomeňme, že ekvatoriálnı́ rovinu je možno dı́ky sférické symetrii metriky (1.1)
volit arbitrárně.
10
Kapitola 1. Virtuálnı́ výlet k horizontu černé dı́ry. . .
Na oblohu pozorovatele v silném gravitačnı́m poli jsou tedy projektována
koncentrická zobrazenı́ celého zářı́cı́ho vzdáleného vesmı́ru tak, že přı́mé
obrazy jsou vždy následovány nepřı́mými a směrový úhel α(φsource , k) klesá
spolu s |k|. Je však nutno podotknout, že celkový úhlový rozměr a tedy
i pozorovaná intenzita zářenı́ jednotlivých obrazů se vzrůstajı́cı́m |k| klesá
exponenciálně [60], a proto lze na vizualizačnı́ch výstupech simulacı́ rozumně modelovat i rozlišit pouze prvnı́ trojici obrazů (nultý přı́mý a prvnı́
přı́mý i nepřı́mý obraz). Obrazy vyššı́ch řádů splývajı́ do jasného prstence
s poloměrem odpovı́dajı́cı́m maximálnı́ hodnotě směrového úhlu αmax (robs )
pro danou hodnotu radiálnı́ souřadnice pozorovatele robs .
V přı́padě pozorovatele umı́stěného nad nestabilnı́ kruhovou fotonovou
orbitou (robs > rph ) úhel αmax (robs ) koresponduje s trajektoriemi fotonů
s impaktnı́m parametrem b → b+
crit , tedy fotonů mnohokrát spirálujı́cı́ch
v těsné blı́zkosti kruhové fotonové orbity, avšak finálně dosahujı́cı́ch bodu
obratu a poté unikajı́cı́ch směrem do vzdáleného vesmı́ru. Naopak v přı́padě
pozorovatele s robs ≤ rph maximálnı́ hodnota směrového úhlu αmax (robs ) koresponduje s trajektoriemi fotonů obdobně mnohokrát spirálujı́cı́ch, avšak
s impaktnı́m parametrem b → b−
crit , a tedy finálně dopadajı́cı́ch na
černoděrový horizont. Maximálnı́ směrový úhel αmax tak vymezuje na pozorovatelově obloze černý region, do kterého nejsou již projektovány žádné
obrazy objektů vzdáleného vesmı́ru a přı́padné zářenı́ pozorované v této
oblasti musı́ nutně přicházet ze zdrojů v těsné blı́zkosti černoděrového horizontu [25,81]. Hranice tohoto regionu je tedy možno interpretovat jako zobrazenı́ nestabilnı́ kruhové fotonové orbity (fotosféry obalujı́cı́ černou dı́ru) a
jeho úhlová velikost může být nazývána zdánlivou úhlovou velikostı́ černé
dı́ry
S(robs ) = 2 αmax (robs ).
(1.16)
Směrový úhel α je na dané hodnotě radiálnı́ souřadnice robs obecně závislý
na parametrech metriky 1.1 i volbě lokálnı́ho referenčnı́ho systému spojeného s pozorovatelem. Kvalitativnı́ a i kvantitativnı́ vlastnosti αmax (robs ) a
tedy i S(robs ) jsou pro statické i radiálně volně padajı́cı́ pozorovatele ve Schwarzschildově–de Sitterově prostoročase podrobně diskutovány v přı́loze 1.
Pro statického pozorovatele umı́stěného nad nestabilnı́ kruhovou fotonovou
orbitou (robs > rph ) je S(robs ) antikorelována k Λ, zatı́mco pro pozorovatele
s robs < rph S(robs ) spolu s Λ roste. Vliv elektrického a slapového náboje
černé dı́ry byl s pomocı́ kódu BHimpaCt analyzován v diplomové práci
M. Vindyše vedené autorem [87]. Přı́tomnost elektrického náboje zdánlivou
1.2. Optické zobrazenı́ ve sféricky symetrických prostoročasech
11
Obrázek 1.1. Zdánlivá úhlová velikost černé dı́ry S(robs ) pro statického pozorovatele jako funkce robs v prostoročasech s rozdı́lnou velikostı́ kvadrátu náboje
Q2 = β a kosmologické konstanty Λ. Plné křivky označujı́ chovánı́ v prostoročasech s Λ = 0, přerušované křivky odpovı́dajı́ prostoročasům s Λ =
5 × 10−3 M −2 . Zdánlivá úhlová velikost klesá k nule na kosmologickém horizontu
a naopak dosahuje maxima S(robs ) = 2π na horizontu černoděrovém.
úhlovou velikost černé dı́ry S(robs ) pro statického pozorovatele na daném
robs zvětšuje, zatı́mco přı́tomnost náboje slapového má opačný efekt. Ve
významném přı́padě statického pozorovatele umı́stěného právě na nestabilnı́
kruhové fotonové orbitě je však zdánlivá úhlová velikost černé dı́ry S(robs )
invariantně rovna π nezávisle na hodnotách Q2 = β i Λ a černý region
pro takové pozorovatele vyplňuje vždy celou hemisféru oblohy orientovanou směrem k černé dı́ře. Obrázek 1.1 ilustruje chovánı́ S(robs ) v elektricky
i slapově nabitých prostoročasech včetně vlivu repulzivnı́ kosmologické konstanty [87]. Zajı́mavou a charakteristickou vlastnostı́ optického zobrazenı́
extrémně silným sféricky symetrickým gravitačnı́m polem je odlišný charakter přı́mých a nepřı́mých obrazů. Zatı́mco přı́mé obrazy zářenı́ vzdáleného
vesmı́ru je pouze úhlově deformovány – komprimovány, transformace zobrazenı́ pro nepřı́mé obrazy je poněkud dramatičtějšı́. Nepřı́mé obrazy jsou
nejenom úhlově komprimovány, ale i úhlově invertovány a navı́c otočeny
o úhel π kolem optické osy zobrazenı́.
12
Kapitola 1. Virtuálnı́ výlet k horizontu černé dı́ry. . .
Dalšı́m dobře známým a charakteristickým efektem gravitačnı́ho
lensingu sféricky symetrickým gravitačnı́m polem jsou Einsteinovy
kroužky [25, 29, 58, 69], zobrazenı́ světelných zdrojů ležı́cı́ch na optické ose
do koncentrických zářı́cı́ch prstenců. V přı́padě lensingu zářenı́ vzdáleného
vesmı́ru existujı́ dvě sady Einsteinových kroužků. Prvnı́ sada, zobrazujı́cı́
zdroj zářenı́ ve vzdáleném vesmı́ru s φsource = π (tedy z hlediska pozorovatele za černou dı́rou) tvořı́ hranice přı́mých řádu k+ s nepřı́mými o řádu
k− = −(k+ + 1). Druhou sadu, zobrazujı́cı́ protilehlý zdroj s φsource = 0
(za zády pozorovatele hledı́cı́ho směrem k černé dı́ře) lze pak lokalizovat
na hranicı́ch přı́mých obrazů s nepřı́mými opačného řádu. Ačkoli intenzita obrazů globálně klesá s ∆φ (b, robs , rsource , β, Λ) a tedy i s |k|, v těsném
okolı́ Einsteinových kroužků naopak lokálně prudce vzrůstá a v použitém
přiblı́ženı́ geometrické optiky v Einsteinových kroužcı́ch diverguje do nekonečna [60]. Prvnı́ Einsteinův kroužek se tak stává nutně velmi signifikantnı́m observačnı́m fenoménem [69].
1.3. Softwarová implementace
Současná verze kódu BHimpaCt je vyvinuta na bázi programovacı́ho jazyka
C++ a je rozdělena na tři základnı́ softwarové moduly implementované jako
C++ objektové třı́dy.
1.3.1. Modul vstupů
Modul vstupů definuje zdroje zářenı́ vstupujı́cı́ do simulace. Na povrchu
zářı́cı́ch objektů zavedené souřadnice umožňujı́ precizovat intenzitu, RGB
barevné komponenty i časovou variabilitu pixelem emitovaného zářenı́
na úrovni buňky mřı́žky (pixelu gridu) s volitelným rozlišenı́m. Zvolené
softwarové řešenı́ tak mimo jiné dovoluje namapovat na povrch objektů
obrázkové textury, což je právě mechanismus využitý při simulacı́ch vzhledu
vzdáleného vesmı́ru. V současné verzi jsou v kódu BHimpaCt implementovány dvě třı́dy zářı́cı́ch objektů, vzdálené rovinné stı́nı́tko a sféra se
středem v počátku souřadnic metriky. Objektová architektura ovšem dovoluje snadno pomocı́ mechanismů dědičnosti a zapouzdřenı́ v programovacı́m
jazyce C++ vytvořit třı́dy pro reprezentaci dalšı́ch a odlišných zářı́cı́ch objektů.
1.3. Softwarová implementace
13
1.3.2. Modul relativistického raytracingu
Modul relativistického raytracingu ve sféricky symetrických prostoročasech
definuje metriku časoprostoru, zvolený lokálnı́ souřadný systém pozorovatele a na základě zadaných parametrů metriky 1.1 vypočı́tává polohu
fotonových orbit, černoděrových i kosmologických horizontů a statických
poloměrů. Jeho hlavnı́ funkcı́ je však řešenı́ problému emitor–observer, tedy
nalezenı́ nulových geodetik spojujı́cı́ch zdroj a detektor zářenı́. Navazujı́cı́
funkcı́ modulu je pak výpočet veličin spojených s nalezenými nulovými
geodetikami a nezbytných pro konstrukci optického zobrazenı́ v lokálnı́m
referenčnı́m systému pozorovatele, jmenovitě směrového úhlu α, hodnoty
frekvenčnı́ho posuvu g, amplifikace intenzity Atotal a také časového zpožděnı́
paprsku. Použitá implementace ovšem použı́vá metodu přenosové funkce a
neprovádı́ raytracing v technickém slova smyslu, tak jak je obvykle chápán.
Pro úspěšné modelovánı́ obrazů vyššı́ch řádů je klı́čová přesnost výpočtu impaktnı́ho parametru b, který se v přı́padě nulových geodetik odpovı́dajı́cı́ch
obrazům vyššı́ch řádů měnı́ spolu s velkým ∆φ (b, robs , rsource , β, Λ) jen velmi
pomalu a také se jen velmi málo lišı́ od limitnı́ hodnoty bcrit . Obdobně
i body obratu se pro takové geodetiky nacházejı́ v těsné blı́zkosti nestabilnı́ kruhové fotonové orbity. Modul relativistického raytracingu počı́tá
hodnotu impaktnı́ho parametru i polohu bodů obratu s přesnostı́ 10−15 M,
což ve většině situacı́ postačuje k modelovánı́ obrazů prvnı́ch čtyř řádů
pro k ∈ (−2, −1, 0, 1). K akceleraci výpočtů optické projekce je využı́vána
sférická symetrie metriky, ale objektové rozhranı́ je navrženo takovým
způsobem, aby modul bylo možno snadno nahradit softwarově kompatibilnı́m modulem raytracingu v odlišných, napřı́klad axiálně symetrických
prostoročasech.
1.3.3. Modul zpracovánı́ výstupů
Konečně modul zpracovánı́ výstupů upravuje výstupy simulacı́ do
požadovaného datového formátu. Vizualizačnı́m výstupem je dvojice bitmapových obrázků pozorovatelovy oblohy, hemisféra oblohy orientovaná
směrem ke gravitačnı́mu centru a projektovaná na virtuálnı́ stı́nı́tko kolmé
na optickou osu spolu s obdobnou projekcı́ hemisféry opačné. Nezbytnou
součástı́ softwarového řešenı́ je proto rutina pro transformaci RGB barevných komponent na základě hodnoty frekvenčnı́ho posuvu g, převzatá
z kódu LightSpeed! [28]. Dalšı́m možným výstupem generovaným v přı́padě
dynamické simulace je světelná křivka zářı́cı́ho objektu, tedy tabulka
14
Kapitola 1. Virtuálnı́ výlet k horizontu černé dı́ry. . .
časového průběhu světelného toku detekovaného pozorovatelem a integrovaného přes celý zářı́cı́ objekt. Tento druh výstupu je možno použı́t pro modelovánı́ časově proměnného zářenı́ horkých skvrn na povrchu kompaktnı́ch
hvězd popřı́padě akrečnı́ch disků.
1.4. Vizualizačnı́ výstupy simulacı́
1.4.1. Statický pozorovatel v blı́zkosti Schwarzschildovy černé dı́ry
Obrázek 1.2. Gravitačnı́m polem nedeformovaný snı́mek Galaxie v Andromedě
M31 spolu se satelitnı́mi galaxiemi M32 (nahoře) a M110 (dole).
Vizualizačnı́ výstupy prvnı́ simulace demonstrujı́ základnı́ rysy geometrie optické projekce pro statického pozorovatele v blı́zkosti sféricky symetrické černé dı́ry, tedy optickou deformaci odpovı́dajı́cı́ sférické symetrii gravitačnı́ho pole, invertovaný charakter nepřı́mých obrazů i formovánı́
Einsteinových kroužků. V simulaci byly modelovány přı́spěvky nulových
geodetik pro prvnı́ čtyři obrazy, k ∈ (−2, −1, 0, 1). Dı́ky zvolenému relativně vysokému rozlišenı́ výstupů jsou detaily obrazů vyššı́ch řádů dobře
1.4. Vizualizačnı́ výstupy simulacı́
15
Obrázek 1.3. Simulace optického zkreslenı́ gravitačnı́m polem černé dı́ry ležı́cı́
mezi pozorovatelem a vzdálenou Galaxie v Andromedě M31. Statický pozorovatel
je vzdálen od virtuálnı́ černé dı́ry o robs = 27M .
rozlišitelné, a výstupy simulace tak dokumentujı́ dosahovanou přesnost integračnı́ho jádra kódu BHimpaCt. Objekty vzdáleného vesmı́ru v podobě
nezkreslené gravitačnı́m polem jsou reprezentovány snı́mkem Galaxie v Andromedě M31 spolu se dvěma satelity M32 a M110 v rozlišenı́ 6000×4800 pixelů [46], viz Obr. 1.2. Výstupy simulace na obrázcı́ch 1.3, 1.4 a 1.5 odpovı́dajı́ optické projekci pro statického pozorovatele umı́stěného nad kruhovou fotonovou orbitou, obrázek 1.6 naopak ilustruje přı́pad statického
pozorovatele pod rph .
1.4.2. Pozorovatel radiálně volně padajı́cı́ do Schwarzschildovy černé
dı́ry
Analyzujeme-li vzhled vzdáleného vesmı́ru pro pozorovatele z nekonečna
radiálně volně padajı́cı́ho do Schwarzschildovy černé dı́ry, pak k efektům
způsobeným silným gravitačnı́m pole přistupujı́ dı́ky pohybu pozorovatele
16
Kapitola 1. Virtuálnı́ výlet k horizontu černé dı́ry. . .
Obrázek 1.4. Detail oblasti maximálnı́ optické deformace obr. 1.3. Zřetelně
lze rozlišit prvnı́ Einsteinův kroužek, invertovaný charakter prvnı́ho nepřı́mého
obrazu i obrazy vyššı́ch řádů splývajı́cı́ spolu s odpovı́dajı́cı́mi Einsteinovými
kroužky v jasný prstenec ohraničujı́cı́ černý region.
1.4. Vizualizačnı́ výstupy simulacı́
17
Obrázek 1.5. Dalšı́ zvětšenı́ části oblasti formovánı́ obrazů vyššı́ch řádů Galaxie
M31 pro statického pozorovatele na robs = 27M . V detailech druhého přı́mého obrazu je rozlišitelný (nahoře vlevo) obraz satelitnı́ galaxie M110. Sekundárnı́ Einsteinův kroužek ohraničujı́cı́ druhý přı́mý obraz a druhý nepřı́mý obraz splývajı́
opět v jasný prstenec ohraničujı́cı́ černý region.
18
Kapitola 1. Virtuálnı́ výlet k horizontu černé dı́ry. . .
Obrázek 1.6. Změna charakteru optické projekce pro statického pozorovatele
robs = 2.7M . Pozorovatel je nynı́ umı́stěn pod kruhovou fotonovou orbitou a
tedy zobrazenı́ celého vzdáleného vesmı́ru je přesunuto na hemisféru oblohy
odvrácenou od černé dı́ry. Vnitřnı́ okraj projekce je tak vnějšı́ hranicı́ obrazu
prvnı́ho řádu a naopak hranice černého regionu zabı́rajı́cı́ho pro takového pozorovatele vı́ce než polovinu oblohy je vnějšı́ hranicı́ zobrazenı́. Výrazný modrý posuv
je na výstupu simulace jasně rozlišitelný, část viditelného zářenı́ je přesunuta do
neviditelného spektrálnı́ho oboru.
19
1.4. Vizualizačnı́ výstupy simulacı́
Nezkreslený obrázek
robs = 200M
robs = 50M
robs = 30M
robs = 15M
robs = 10M
robs = 5M
robs = 1.5M
Obrázek 1.7. Simulace vzhledu galaxie M104 Sombrero pro radiálně volně padajı́cı́ho pozorovatele do Schwarzschildovy černé dı́ry na rozdı́lných hodnotách
radiálnı́ souřadnice.
20
Kapitola 1. Virtuálnı́ výlet k horizontu černé dı́ry. . .
i efekty speciálně relativistické. Chovánı́ zdánlivé úhlové velikosti černé dı́ry
S(robs ) jako funkce radiálnı́ souřadnice volně padajı́cı́ho pozorovatele robs
je oproti statickému přı́padu výrazně kvalitativně odlišné. Hodnota S(robs )
pro takového pozorovatele je vždy menšı́ než odpovı́dajı́cı́ hodnota pro pozorovatele statického a radiálnı́ souřadnice kruhové fotonové orbity zde oproti
statickému přı́padu nevystupuje v nikterak význačné roli. Volně padajı́cı́
pozorovatel uvidı́ černý region okupujı́cı́ polovinu oblohy S(robs ) = π až
při dosaženı́ centrálnı́ singularity. Také chovánı́ frekvenčnı́ho posuvu g je
kvalitativně odlišné dı́ky kombinaci gravitačnı́ho blueshiftu a speciálně relativistického dopplerovského posuvu způsobeného pohybem pozorovatele.
Pozorovaný frekvenčnı́ posuv zářenı́ vzdáleného vesmı́ru tak nutně závisı́
na úhlových souřadnicı́ch zdrojů. Série vizualizačnı́ch výstupů simulace na
obr. 1.7 ilustruje tyto efekty pro padajı́cı́ho pozorovatele v relativně velké
vzdálenosti od černé dı́ry i pro pozorovatele, jenž svým volným pádem
dospěl již pod černoděrový horizont. Objekty vzdáleného vesmı́ru v podobě
nezkreslené gravitačnı́m polem jsou reprezentovány snı́mkem Galaxie M104
Sombrero v rozlišenı́ 1200 × 655 pixelů [21].
1.4.3. Povrch rotujı́cı́ superkompaktnı́ neutronové či kvarkové hvězdy
Simulace vzhledu povrchu rotujı́cı́ relativisticky kompaktnı́ hvězdy (s poloměrem Rstar ∼ rph ) pro vzdáleného pozorovatele v nekonečnu je úlohou
částečně komplementárnı́ k úloze, které byly věnovány předešlé simulace.
Pokud prostoročas v okolı́ takového hvězdného objektu aproximativně popisujeme metrikou ve tvaru 1.1, lze dı́ky stacionaritě tohoto řešenı́ rovnici 1.10
snadno adaptovat pro přı́pad určenı́ impaktnı́ch parametrů nulových geodetik spojujı́cı́ch povrch hvězdy a vzdáleného pozorovatele pouhou vzájemnou
záměnou robs a rsource = Rstar . Analogicky i modrý posuv zářenı́ vzdáleného
vesmı́ru pozorovaný v blı́zkosti černé dı́ry je v přı́padě vzdáleného pozorovatele kompaktnı́ hvězdy nahrazen posuvem rudým.
Přirozený předpoklad neprůhlednosti hvězdy ovšem omezuje obor
možných řešenı́ rovnice 1.10 na vycházejı́cı́ geodetiky s b < bmax (Rstar ).
Proto maximálnı́mu impaktnı́mu parametru bmax (Rstar ) odpovı́dá maximálnı́ možná změna úhlové souřadnice podél geodetiky a tedy i maximálnı́
hodnotě souřadnice φ = φmax
source (Rstar ) na které je izotropně zářı́cı́ bodový
zdroj na rovnı́ku hvězdy (θsource = π/2) ještě viditelný.3 V přı́padě plochého
3
Připomeňme opět, že ekvatoriálnı́ rovinu je možno dı́ky sférické symetrii metriky 1.1
volit arbitrárně.
1.4. Vizualizačnı́ výstupy simulacı́
21
prostoročasu a tedy přı́močarého šı́řenı́ světla je φmax
source roven π/2 a vzdálený
pozorovatel vždy vidı́ pouze přivrácenou hemisféru povrchu hvězdy. Za
přı́tomnosti sféricky symetrického gravitačnı́ho pole a při fixované hmotnosti hvězdy však φmax
source (Rstar ) s klesajı́cı́m poloměrem hvězdy Rstar roste a
tak narůstá i viditelná část povrchu hvězdy. Pokud poloměr hvězdy dosáhne
kritické hodnoty Rc ≃ 3.445M, je φmax
source (Rc ) roven π a viditelným se stává
celý povrch. Pro zobrazenı́ povrchu hvězd s Rstar ≤ Rc se tı́m stávajı́ relevantnı́mi i přı́spěvky obrazů povrchu vyššı́ch řádů. Pro hvězdy s Rstar ≤ rph
max
pak φmax
source (Rstar ) diverguje k nekonečnu. Závislost φsource (Rstar ) na poloměru
hvězdy Rstar ilustruje obrázek 1.8.
Obrázek 1.8. Úhel φmax
source (Rstar ) jako funkce poloměru hvězdy Rstar .
K efektům způsobeným gravitačnı́m polem přistupujı́ obdobně jako
v přı́padě padajı́cı́ho pozorovatele i efekty speciálnı́ relativity zapřı́činěné
rychlostı́ rotace povrchu a vzhled povrchu hvězdy je ovlivněn relativistickým
Dopplerovým jevem i aberacı́. K ilustraci výše diskutovaných efektů byl
použit hračkový model superkompaktnı́ hvězdy (Rstar = 2.4M) s červeně
vyzařujı́cı́m povrchem (λ = 700 nm) a dvojicı́ protilehlých bı́le vyzařujı́cı́ch
skvrn o úhlovém rozměru 0.6π. Hvězda rotuje s vysokou frekvencı́ νstar =
Ωstar /2π = 2000 Hz a osa rotace svı́rá s optickou osou úhel ǫ = 0.57π.
22
Kapitola 1. Virtuálnı́ výlet k horizontu černé dı́ry. . .
Obrázek 1.9. Simulace vzhledu povrchu superkompaktnı́ hvězdy s vyloučenı́m
vlivu gravitačnı́ho rudého posuvu. Dı́ky vysoké superkompaktnosti hvězdy
(Rstar = 2.4M ) je pro vzdáleného pozorovatele viditelný přı́mý i nepřı́mý obraz
úplného povrchu hvězdy. Vlevo: Procesován je pouze speciálně relativistický Dopplerův frekvenčnı́ posuv. Poměrně komplikovaná barevná mapa povrchu vzniká
kompozicı́ klasického a transverzálnı́ho Dopplerova jevu. Zřetelně rozeznatelné je
umı́stěnı́ pólů osy rotace hvězdy i přesun vyzařovánı́ části povrchu pohybujı́cı́ho
se směrem od pozorovatele do infračervené oblasti.Vpravo: Stejná situace jako na
levém panelu, avšak se zahrnutı́m vlivu amplifikace zářenı́ povrchu.
Oba panely Obr. 1.9 ilustrujı́ všechny výše zmiňované efekty s výjimkou
gravitačnı́ho rudého posuvu, který by pouze přesunul zářenı́ povrchu do
neviditelné oblasti elektromagnetického spektra.
1.5. Shrnutı́ a perspektiva
Optické zobrazovánı́ ovlivněné extrémně silným gravitačnı́m polem kompaktnı́ch objektů, at’ již diskutované pro přı́pad pozorovatelů v blı́zkosti
zdroje gravitačnı́ho pole nebo naopak pro pozorovatele vzdálené, je
v současné době již teoreticky velmi dobře popsáno analytickými i numerickými metodami [25, 58, 60, 86]. Existujı́cı́ analytická řešenı́ i softwarové
kódy umožňujı́ detailnı́ analýzu optických efektů ve sféricky i axiálně symetrických prostoročasech. Zde jsou popisovány výsledky zı́skané pomocı́ kódu
BHimpaCt, který je pokusem o vytvořenı́ relativně (i relativisticky) univerzálnı́ softwarové platformy pro modelovánı́ optických efektů v blı́zkosti
1.5. Shrnutı́ a perspektiva
23
hvězdných kompaktnı́ch objektů. Počı́tačové simulace toků elektromagnetického zářenı́ ovlivněného silnou gravitacı́ jsou často použı́vány k modelovánı́ observačnı́ch astrofyzikálnı́ch fenoménů, jako jsou např. kvaziperiodické oscilace v rentgenovém zářenı́ akrečnı́ch disků binárnı́ch systémů
s černými děrami či neutronovými hvězdami nebo také zářenı́ horkých skvrn
na povrchu neutronových a podivných hvězd [11, 24, 67, 78]. Vizualizace optického zobrazenı́ v silném gravitačnı́m poli se v kontextu takových aplikacı́
relativistické optiky jistě může zdát pouhou hřı́čkou, avšak kromě názorného
přiblı́ženı́ vlastnostı́ silně zakřivených prostoročasů může skýtat i estetický
požitek, který snad i v současné době může hrát úlohu inspirace a motivace
vědeckého poznánı́.
Předmětem dalšı́ho plánovaného vývoje kódu BHimpaCt je vytvořenı́
interaktivnı́ho uživatelsky přı́větivého rozhranı́, implementace raytracingu
v jiných než sféricky symetrických černoděrových prostoročasech a podpora
přı́mého generovanı́ videoklipů jakožto výstupů dynamických simulacı́.
Kapitola 2
QPOs: Pohled z nekonečna
Šedivá je teorie, ale věčně zelený je strom života.
— Johann Wolfgang von Goethe
2.1. Fenomén kvaziperiodických oscilacı́ (QPOs)
Předešlá kapitola byla věnována analýze observacı́ virtuálnı́ch pozorovatelů v blı́zkosti černých děr či neutronových hvězd. Skutečné možnosti
pozorovánı́ optických efektů způsobených silným gravitačnı́m polem relativisticky kompaktnı́ch hvězdných objektů jsou však omezeny nejen mezihvězdnými či intergalaktickými vzdálenostmi, ale i vlastnostmi zemské atmosféry, která bohužel bránı́ průchodu a tı́m i detekci elektromagnetického
zářenı́ o vlnových délkách, které jsou relevantnı́ právě pro pozorovánı́
binárnı́ch hvězdných systémů obsahujı́cı́ch černé dı́ry nebo neutronové
hvězdy. Až otevřenı́ nového rentgenového pozorovacı́ho okna za pomoci satelitnı́ch observatořı́ uvedlo na scénu relativistické astrofyziky neočekávaný
optický observačnı́ fenomén, výskyt ostrých pı́ků ve výkonovém spektru
(power density spectrum – dále jen PDS) zı́skávaném Fourierovou transformacı́ rentgenových světelných křivek nı́zkohmotnostnı́ch binárnı́ch systémů
s černou dı́rou či neutronovou hvězdou (LMXB).1
Je všeobecně přijı́máno, že pozorované rentgenové zářenı́ LMXB
systémů je generováno akrecı́ hmoty na černou dı́ru či neutronovou hvězdu,
přesněji řečeno disipacı́ potenciálnı́ a kinetické energie akreovaného materiálu. Neutronovou hvězdou je v tomto kontextu mı́něn relativisticky kompaktnı́ produkt stelárnı́ho kolapsu s parametry neumožňujı́cı́mi dosáhnout
finálnı́ho stadia černé dı́ry. Může se tedy jednat skutečně o neutronové hvězdy, kvarkové hvězdy nebo i dalšı́ kompaktnı́ hvězdné objekty
1
Low Mass X-ray Binaries: Binárnı́ systémy, ve kterých je hmotnost průvodce nižšı́
než hmotnost černé dı́ry či neutronové hvězdy.
25
26
Kapitola 2. QPOs: Pohled z nekonečna
složené z hypotetických bizarnı́ch forem hmoty.2 V LMXB systémech je
předpokládána existence akrečnı́ch disků orbitujı́cı́ch kolem centrálnı́ho
kompaktnı́ho objektu a zářı́cı́ch v rentgenovém oboru a přı́padně také existence obdobně zářı́cı́ch horkých skvrn na povrchu neutronových ohřı́vaných
dopadajı́cı́m akreovaným materiálem. Na pozadı́ relativně dobře zmapovaných spektrálnı́ stavů akreujı́cı́ch LMXB systémů je v PDS pozorován
aperiodický výskyt prozatı́m s obtı́žemi vysvětlovaných výrazných ostrých
pı́ků s charakteristickými vlastnostmi, kvaziperiodických oscilacı́ neboli
QPOs [38].
Slovnı́ spojenı́ ostré pı́ky je ovšem nutno interpretovat v kontextu
současných observačnı́ch možnostı́. Analyzovaná PDS jsou zı́skávána sofistikovanými softwarovými metodami zpracovávajı́cı́mi satelitnı́ observačnı́
data pro relativně dlouhé časové úseky, přičemž identifikace jednotlivých
pı́ků je na hranicı́ch možnostı́ použı́vaných detekčnı́ch i softwarových technologiı́ [13, 52, 53]. Obvykle uváděný rozsah pozorovaných QPO frekvencı́
LMXB zdrojů je 10−2 Hz → 103 Hz. Na základě pozorované frekvence i dalšı́
fenomenologie jsou QPOs obvykle třı́děny do následujı́cı́ch skupin:
• Nı́zkofrekvenčnı́ QPOs (low frequency QPOs, LFQPOs). Skupina s relativně komplexnı́m chovánı́m je shora obvykle ohraničována frekvencı́
100 Hz
• Hektohertz QPOs. Tato jsou pro daný zdroj pozorována na vı́ceméně
konstantnı́ frekvenci v intervalu 100–200 Hz
• Vysokofrekvenčnı́ QPOs (high frequency QPOs, HFQPOs) nebo také
kHz QPOs. Běžně uváděný frekvenčnı́ rozsah pro LMXB zdroje je
200–1200 Hz. Skupina kHz QPOs, vykazujı́cı́ pravděpodobně nejbohatšı́
spektrum vlastnostı́, se těšı́ největšı́ pozornosti teoretických astrofyziků
i pozorovatelů.
Zdůrazněme však, že hodnota pozorované frekvence zde nenı́ jediným
klasifikačnı́m znakem. Pro některé zdroje s neutronovou hvězdou může
napřı́klad dolnı́ hranice frekvenčnı́ho rozsahu kHz QPO dosahovat až 50 Hz.
2
V současnosti je diskutována napřı́klad možnost existence tzv. elektroslabých hvězd,
v jejichž jádru by mělo dı́ky v něm panujı́cı́m extrémnı́m podmı́nkám docházet k elektroslabému sjednocenı́. Mechanismus tzv. elektroslabého hořenı́ by pravděpodobně mohl
působit proti gravitačnı́mu kolapsu podobně jako termonukleárnı́ reakce u běžných
hvězd [27]. Jinými často diskutovanými superkompaktnı́mi objekty jsou hypotetické gravastary [34, 51].
2.1. Fenomén kvaziperiodických oscilacı́ (QPOs)
27
Obrázek 2.1. Výkonové spektrum LMXB zdroje 4U 1608-52 s jasně zřetelnou
přı́tomnostı́ dvojice QPO pı́ků (fitovaných lorentziány). Převzato z [38] a graficky
upraveno.
Dalšı́m a neméně důležitým kritériem je chovánı́ pı́ku odpovı́dajı́cı́ dané skupině; pro systémy s neutronovou hvězdou je kritériem existence vlastnostı́
pı́ku korespondujı́cı́ch s vlastnostmi charakteristických kHz QPO módů,
které jsou podrobněji diskutovány nı́že.
Tato kapitola si ovšem v žádném přı́padě neklade za cı́l poskytnout úplný
přehled rozsáhlé QPOs fenomenologie a teoretických modelů, který lze najı́t
např v [38]. Následujı́cı́ text je věnován kHz QPOs v LMXB systémech s neutronovou hvězdou, předmětu publikacı́ prezentovaných v přı́lohách 2–5.
Termı́nem QPOs (kHz QPOs) bez dalšı́ch přı́vlastků budou dále mı́něny
právě QPOs pozorované v rentgenových tocı́ch přicházejı́cı́ch z LMXB
systémů obsahujı́cı́ch neutronovou hvězdu.
28
Kapitola 2. QPOs: Pohled z nekonečna
2.1.1. kHz QPOs v systémech s neutronovou hvězdou
V systémech s neutronovou hvězdou lze pro kHz QPOs rozlišit dva odlišné
módy pozorované v širokém rozsahu časově proměnných frekvencı́.3 Módy
jsou charakterizované odlišnou korelacı́ nı́že podrobně diskutovaných parametrů pı́ku s jeho frekvencı́ [12, 13, 54].
Pokud jsou oba QPO módy detekovány současně, hodnoty frekvencı́
obou pı́ků splňujı́ vždy shodnou nerovnost a proto hovořı́me o hornı́m (upper ) a dolnı́m (lower ) QPO módu s odpovı́dajı́cı́mi frekvencemi svázanými
relacı́ νu > νl . Zdůrazněme však, že tato relace je relevantnı́ pouze pro
současnou detekci a ve frekvenčnı́m pásmu konkrétnı́ho zdroje mohou hornı́
QPO být detekována na frekvenci menšı́ než nesimultánně pozorovaná
dolnı́ QPO. Částečně zavádějı́cı́ terminologie tak může snadno vést k principiálnı́m nedorozuměnı́m, kterých byl autor svědkem i na mezinárodnı́ch
prestižně obsazených astrofyzikálnı́ch konferencı́ch. Pro simultánnı́ detekci
obou módů je často použı́váno označenı́ twin-peak kHz QPO.4
Tvar frekvenčnı́ch pı́ků v zı́skaných PDS je při dalšı́m procesovánı́ fitován lorentziánem a popisován parametry Q a S.
Faktor kvality Q je definován jako centrálnı́ frekvence pı́ku dělená
jeho šı́řkou (frekvenčnı́m pásmem ∆ν) v polovině maximálnı́ výšky (amplitudy). Za předpokladu QPO pı́ku generovaného jednı́m harmonickým
oscilátorem faktor kvality Q charakterizuje tlumenı́ oscilacı́, vyššı́m hodnotám Q (ostřejšı́m pı́kům) tak odpovı́dajı́ méně tlumené kmity. V přı́padě
vı́ce oscilátorů s blı́zkými frekvencemi Q navı́c charakterizuje i distribuci
frekvencı́.
Signifikance S popisujı́cı́ relaci mezi integrálem plochy pod lorentziánovským fitem pı́ku a chybou měřenı́ může být vyjádřena formulı́
S = k(t)A2
p
Q/ν ,
(2.1)
kde A je rms amplituda pı́ku o frekvenci ν, definovaná jako podı́l energie
pı́ku (dané integrálem plochy pod lorentziánovským fitem) √
k celkové energii
vyzařované zdrojem. Časově proměnný faktor k(t) = I(t) T (count rate)
3
V binárnı́ch systémech s černou dı́rou jsou pozorována kHz QPOs na konstantnı́ch
frekvencı́ch charakteristických pro daný zdroj. Pokud jsou detekovaná v párech, dvojice
frekvencı́ obvykle odpovı́dajı́ poměrům malých celých čı́sel, typicky 3:2 [1, 50].
4
53].
Simultánnı́ detekcı́ je zde ovšem nutno chápat v termı́nech integračnı́ doby měřenı́ [52,
2.1. Fenomén kvaziperiodických oscilacı́ (QPOs)
29
je úměrný počtu detekcı́ fotonů během doby observace časového pozorovacı́ho okna T a a tedy i intenzitě zdroje v okamžiku detekce I(t). Obvykle
použı́vaný dolnı́ limit pro zpracovánı́ pı́ku jako QPO je S ≥ 2–4. Vzhledem
k citlivosti současných detektorů může právě nastavenı́ hodnoty dolnı́ho
limitu signifikance QPO pı́ků výrazně ovlivňovat vlastnosti zkoumaných
setů observačnı́ch dat a tı́m i interpretace pozorovánı́. Kompletnı́ a detailnı́
popis procedury klasifikace QPO pı́ků lze nalézt v [37].
Pro hornı́ QPOs je faktor kvality obvykle malý a zůstává v celém
frekvenčnı́m rozsahu téměř konstantnı́ na hodnotě Q ∼ 10, naproti tomu
pro dolnı́ QPOs Q roste spolu s frekvencı́ a dosahuje maximálnı́ hodnoty
Q ∼ 200. Amplitudy hornı́ch QPO monotónně klesajı́ s frekvencı́, zatı́mco
pro dolnı́ QPOs vykazujı́ zpočátku strmý růst a po dosaženı́ maxima obdobně strmě klesajı́ [12–15]. Typické chovánı́ ukazuje obrázek 2.2.
Obrázek 2.2. Chovánı́ faktoru kvality Q (vlevo), rms amplitudy zde značené
r (uprostřed) a z předchozı́ch parametrů odvozené signifikance S v observačnı́ch
datech zdroje 4U 1636-53. Observačnı́ data v prvnı́ch dvou panelech jsou převzata
z [13], červenými body jsou značeny dolnı́ QPOs, modrými hornı́ QPOs. Spojité
fitujı́cı́ křivky jsou konstruovány sumou exponenciál, korelace stupnic frekvencı́
na hornı́ a dolnı́ frekvenčnı́
p ose odpovı́dá relaci 2.2. Profil signifikance S je modelován jako úměrný r 2 Q/ν , tedy s použitı́m předpokladů konstantnı́ svı́tivosti
zdroje a fixované observačnı́ doby.
2.1.2. Frekvenčnı́ korelace hornı́ch a dolnı́ch QPOs
Na obrázku 2.3 ukazujı́cı́m distribuci detekcı́ twin-peak QPO v rovině νu -νl
je existence korelace přı́liš se neodlišujı́cı́ od lineárnı́ závislosti mezi frekvencemi dolnı́ch o hornı́ch pı́ku vı́ce než zřetelná. Pás detekcı́ twin-peak QPOs
protı́ná přı́mku odpovı́dajı́cı́ poměru frekvencı́ νu : νl = 3 : 2 v blı́zkosti
hodnot νu = 900 Hz , νl = 600 Hz a masivnı́ klastrovánı́ detekcı́ poblı́ž tohoto bodu je ilustrováno i histogramem v pravé části obrázku. Rozlišitelné
jsou však i méně výrazné klastry detekcı́ v okolı́ poměrů frekvencı́ 4:3 a 5:4.
30
Kapitola 2. QPOs: Pohled z nekonečna
Globálně lineárnı́ charakter korelace frekvencı́ lze aproximovat empirickou
relacı́ [6, 17, 93]
νU ≈ 0.7νL + 520 Hz.
(2.2)
Obrázek 2.3. Nahoře: Detekce twin peak kHz QPOs v rovině νl - νu pro odlišné
skupiny LMXB zdrojů s neutronovou hvězdou, data jsou převzata z [18,22,49,90].
Přerušovaná čára odpovı́dá poměru νu : νl = 3 : 2 a šedě je vyznačeno pásmo
odchylky ±5%. Dole: Odchylka od poměru νu : νl = 3 : 2 jako funkce νl .
2.1. Fenomén kvaziperiodických oscilacı́ (QPOs)
31
Obrázek 2.4. Vlevo: Frekvenčnı́ histogram detekcı́ dolnı́ch QPOs zdroje
4U 1636-53. Tmavšı́m odstı́nem (v legendě lower in twin) jsou značeny detekce
odpovı́dajı́cı́ twin-peak QPOs. Vpravo: Analogický histogram pro hornı́ QPOs.
2.1.3. Klastrovánı́ twin-peak QPOs v okolı́ poměrů malých celých
čı́sel.
Detekce twin-peak QPOs vykazujı́ výrazné klastrovánı́ v okolı́ poměrů frekvencı́ odpovı́dajı́cı́ch podı́lům malých celých čı́sel s výraznou dominancı́
hodnoty νu : νl = 3 : 2 [3]. Obrázek 2.4 ukazuje distribuci separátnı́ch
hornı́ch, dolnı́ch i twin-peak QPOs v observačnı́ch datech LXMB zdroje
4U 1636-53 [13]. Existence preferovaných poměrů frekvencı́ twin-peak kHz
QPO naznačujı́cı́ i možnou přı́tomnost fyzikálnı́ho mechanismu ležı́cı́ho
v pozadı́ pozorovaných distribucı́ byla předmětem řady studiı́ a také
i určitých kontroverzı́ [3, 12–15, 17, 18]. Oproti některým publikovaným
tvrzenı́m o závislosti distribucı́ frekvencı́ hornı́ch a dolnı́ch QPOs [17, 18]
výsledky analýzy skutečně pozorovaných distribucı́ publikované v článcı́ch,
které jsou obsahem přı́lohy 2 a 3, ukazujı́ odlišný obraz. Skutečně pozorované distribuce hornı́ i dolnı́ch QPOs zdroje 4U 1636-53 neodpovı́dajı́
předpokládaným distribucı́m odvozeným pomocı́ relace 2.2 z distribucı́
jejich partnerů. Co se týče kvantitativnı́ho zhodnocenı́ odlišnosti takto
predikovaných a pozorovaných partnerských distribucı́ frekvencı́, po porovnánı́ pomocı́ odpovı́dajı́cı́ch kumulativnı́ch distribucı́ zı́skáváme velmi
nı́zké Kolmogorovovy–Smirnovovy (dále jen K-S) pravděpodobnosti pro
shodu predikcı́ a observacı́ pL,KS = 2.35 × 10−5 a pU,KS = 2.24 × 10−3 .
Navı́c je zřejmé z histogramů na obrázku 2.4, že hornı́ QPO jsou detekována
převážně ve spodnı́ části frekvenčnı́ho rozsahu zdroje a naopak. V tomto
smyslu lze distribuce hornı́ch a dolnı́ch pı́ků považovat za komplementárnı́.
Distribuce poměrů frekvencı́ νu a νl v přı́padě detekcı́ twin-peak QPOs
ukazuje levý panel obrázku 2.5. Histogram detekcı́ naznačuje přı́tomnost
32
Kapitola 2. QPOs: Pohled z nekonečna
dvou význačným pı́ků distribuce v okolı́ poměrů 3:2 a 5:4. Observačnı́ data
byla fitována modelovou distribucı́ p2 (r) konstruovanou jako suma dvou
lorentziánů formulı́
p2 (r) = f
λ1 /π
λ2 /π
+ (1 − f )
,
2
2
(r − r1 ) + λ1
(r − r2 )2 + λ22
(2.3)
kde r = νU /νL je poměr frekvencı́ a r1 , r2 , λ1 , λ2 a f jsou volné parametry. Při
dosaženı́ maximálnı́ K-S pravděpodobnosti shody pozorované a fitujı́cı́ distribuce p2,KS = 0.918 volné parametry nabývajı́ hodnot r1 = 1.52, r2 = 1.28,
λ1 = 0.0327, λ2 = 0.0913 a f = 0.722. Naproti tomu nejlepšı́ fit standardnı́
Lorentzovou distribucı́ s jednı́m pı́kem a parametry r0 = 1.50, λ0 = 0.0597
dosahuje K-S pravděpodobnosti pouze p1,KS = 0.340. Lze tedy konstatovat, že pozorované distribuci poměrů frekvencı́ twin-peak QPOs zdroje
4U 1636-53 odpovı́dá s velkou pravděpodobnostı́ výrazná preference dvou
hodnot poměru frekvencı́, 3:2 a 5:4. Pozorovanou distribuci frekvenčnı́ch
poměrů a obě modelové distribuce porovnává pravý panel obrázku 2.5.
Souvislost závislosti amplitud pı́ků na jejich frekvenci a pozorovaného
klastrovánı́ detekcı́ v okolı́ význačných poměrů frekvencı́ byla analyzována
v článku, který je obsahem přı́lohy 3. Článek je věnována softwarovým
simulacı́m pozorovaných frekvenčnı́ch distribucı́ twin-kHz QPOs. Simulace
vycházejı́ z předpokladu, že dolnı́ a hornı́ QPO jsou vždy zdrojem produkovány v párech, jejichž frekvence jsou spojeny relacı́ 2.2, avšak dı́ky
poměrně nı́zké citlivosti družicových detektorů jsou ne vždy detekovány
oba partnerské pı́ky. V simulacı́ch použité aproximativnı́ průběhy rms amplitudy A a faktoru kvality Q konfrontuje s reálnými observačnı́mi daty levý
a střednı́ panel obrázku 2.2.
Pokud byla simulována náhodná distribuce twin-peak QPOs v pozorovaném frekvenčnı́m rozsahu zdroje 4U 1636-53 s výše uvedenými vlastnostmi a za dodatečného předpokladu konstantnı́ luminozity zdroje (k = 1),
podařilo se v obdržené distribuci reprodukovat klastr detekcı́ v oblasti
poměrů frekvencı́ 3:2, nikoli však sekundárnı́ klastr v okolı́ poměru 5:4.
Pokud však byla simulace provedena takovým způsobem, že parametr k
zůstával konstantnı́ (k = 1) až do νl = 700 Hz a poté narůstal lineárně
s frekvencı́ tak aby na νl = 950 Hz nabýval hodnoty k = 2.5, obdržená distribuce nápadně připomı́nala distribuci skutečných observačnı́ch dat. Takový nárůst k nebyl ovšem zvolen zcela arbitrárně, naopak je v souladu
v observačnı́mi daty [13, 53]. V simulaci byl nastaven práh detekce pı́ku pro
signifikanci S na úrovnı́ 3σ a při takovém nastavenı́ se podařilo reproduko-
2.1. Fenomén kvaziperiodických oscilacı́ (QPOs)
33
vat i vlastnosti distribucı́ nesimultánnı́ch detekcı́ hornı́ch a dolnı́ch QPOs.
Výsledky jsou ilustrovány obrázkem 2.6. Lze tedy konstatovat, že s použitı́m
předpokladu korelace luminozity zdroje s frekvencı́ kHz QPO pı́ků je možno
velmi dobře modelovat pozorovaná observačnı́ data. Pozitivnı́ výsledky výše
diskutovaných simulacı́ ovšem nikterak dále nekonkretizujı́ fyzikálnı́ mechanismy stojı́cı́ v pozadı́ pozorovaných distribucı́ frekvencı́ i poměrů frekvencı́
kHz QPOs.
Obrázek 2.5. Vlevo: Detekce twin-peak QPOs zdroje 4U 1636-53 v rovině νl -νu .
Klastrovánı́ detekcı́ v okolı́ poměrů malých celých čı́sel je poměrně zřetelné.
Vpravo: Kumulativnı́ distribuce frekvenčnı́ch poměrů pozorovaných twin-peak
QPOs (stupňovitá křivka), modelová kumulativnı́ distribuce p2 (r) konstruovaná
sumou dvou lorentziánů (silná plná křivka) a modelová kumulativnı́ standardnı́
lorentzovská distribuce p1 (r) (slabá přerušovaná křivka).
Obrázek 2.6. Pozorované a simulované distribuce frekvenčnı́ch poměrů
kHz QPOs zdroje 4U 1636-53. Vlevo: Pozorovaná distribuce frekvenčnı́ch poměrů
twin-peak kHz QPOs. Uprostřed: Simulovaná distribuce (oranžově) twin-peak
kHz QPOs v porovnánı́ s distribucı́ pozorovanou (šedě). Vpravo: Simulovaná
distribuce nesimultánnı́ch detekcı́ hornı́ch a dolnı́ch kHz QPOs.
34
Kapitola 2. QPOs: Pohled z nekonečna
2.2. Orbitálnı́ modely vzniku QPOs
V současné astrofyzikálnı́ komunitě věnujı́cı́ se QPO fenoménu bohužel
neexistuje konsensuálně sdı́lená teoretická báze. Detekované frekvence
kHz QPO se pohybujı́ na škále odpovı́dajı́cı́ frekvencı́m orbitálnı́ho pohybu
testovacı́ch částic v těsné blı́zkosti kompaktnı́ch hvězdných objektů a proto
je nezanedbatelná skupina QPO modelů založena na ztotožněnı́ pozorovaných QPO frekvencı́ právě s frekvencemi charakterizujı́cı́mi relativistický
orbitálnı́ pohyb [38, 44] .
Relativistický popis orbitálnı́ho pohybu se v řadě důležitých aspektů
kvalitativně odlišuje od newtonovského přı́padu [56]. Eliptické orbitálnı́
trajektorie v newtonovském centrálnı́m poli bodové hmotnosti jsou
uzavřeny [45], což v je přı́padě perturbovaného kvazikruhového orbitálnı́ho
pohybu ekvivalentnı́ rovnosti orbitálnı́ frekvence νK , radiálnı́ epicyklické
frekvence νr a vertikálnı́ epicyklické frekvence νθ . Poměrně rozsáhlou
analýzu newtonovského i relativistického orbitálnı́ho perturbovaného pohybu neovlivněného jinými než gravitačnı́mi vlivy i explicitnı́ formule pro
přı́mý výpočet epicyklických frekvencı́ z koeficientů metricky lze nalézt
v [5]. Podrobnějšı́ diskuze relativistického přı́padu včetně zahrnutı́ negeodetických vlivů je předmětem třetı́ kapitoly této práce, na tomto mı́stě
pouze připomeňme, že epicyklické frekvence popisujı́ časovou závislost
malé perturbace polohy částice na stabilnı́ kruhové orbitě a existence oscilačnı́ho charakteru perturbovaného pohybu testovacı́ částice je ekvivalentnı́ tvrzenı́ o stabilitě kruhové orbity vůči radiálnı́m či vertikálnı́m perturbacı́m [5, 7, 8, 59].
Rovnost frekvencı́ νK a νr je však porušena již v přı́padě relativistického
sféricky symetrického gravitačnı́ho pole popsaného Schwarzschildovou metrikou [56]. Nutným důsledkem nerovnosti frekvencı́ je vznik nového efektu
precese periastra5 radiálně perturbované kruhové orbity s frekvencı́ [74].
νP = νK − νr .
(2.4)
Porušenı́ sférické symetrie metriky rotacı́ centrálnı́ hvězdy způsobujı́cı́ efekt
strhávánı́ souřadných systémů (frame-dragging) [82] vede dále i k porušenı́
5
Newtonovský efekt precese periastra obdržı́me v přı́padě porušenı́ sférické symetrie
potenciálu, tedy v přı́padě polárnı́ho zploštěnı́ centrálnı́ho hvězdného objektu [45, 55].
35
2.2. Orbitálnı́ modely vzniku QPOs
rovnosti νK a νθ manifestujı́cı́ se precesı́ orbitálnı́ roviny (často nazývané
Lenseovou–Thirringovou či nodálnı́) s frekvencı́ [73]
νnodal = νK − νθ .
(2.5)
Konečně lze také hovořit o totálnı́ precesi s frekvencı́
νT = νP − νnodal = νθ − νr
(2.6)
a s periodou odpovı́dajı́cı́ časovému intervalu, po kterém periastron i inklinace orbity zaujmou původnı́ pozice. Zdůrazněme však, že i simultánnı́ opakovánı́ pozice periastra a orbitálnı́ roviny neznamená bez dalšı́ch požadavků
kladených na hodnoty frekvencı́ uzavřenost orbitálnı́ trajektorie.
Dobře známé formule pro úhlové frekvence orbitálnı́ho pohybu
v černoděrovém Kerrově prostoročase s použitı́m Boyerových–
–Lindquistových souřadnic a soustavy jednotek, kde G = c = 1 nabývajı́
tvarů [5, 7, 59]
−1
1 3/2
r +a
,
M
= Ω2K , 1 − 4 a r −3/2 + 3a2 r −2 ,
(2.7)
ΩK =
ωθ2
ωr2
=
Ω2K
1− 6r
−1
+ 8ar
−3/2
2
− 3a r
(2.8)
−2
,
(2.9)
kde spin a je definován pomocı́ vnitřnı́ho momentu hybnosti černé dı́ry
(nahé singularity) J a jejı́ hmotnosti M relacı́ a = J/M 2 , r má význam
radiálnı́ souřadnice škálované v jednotkách hmotnosti M. Explicitnı́ tvar
vztahů pro frekvence orbitálnı́ho pohybu pro prostoročas v okolı́ rotujı́cı́ch
neutronových hvězd aproximovaný Hartleovou–Thornovou metrikou [33]
lze nalézt v [2] i v přı́loze 4. V přı́padě geodetického orbitálnı́ho pohybu
v axiálně symetrických prostoročasech orbitálnı́ a epicyklické frekvence
splňujı́ nerovnost νK ≥ νθ > νr . Podrobná diskuze chovánı́ všech třı́ frekvencı́ 2.9, 2.9 a 2.9 je pro přı́pad černých děr i hypotetických nahých singularit předmětem studie [83]. Zde pouze připomeňme, že radiálnı́ epicyklická frekvence νr po dosaženı́ maxima na r > rISCO klesá spolu radiálnı́
souřadnicı́ až nulové hodnotě, kterou dosahuje právě na meznı́ stabilnı́ orbitě rISCO , kde již jakákoli radiálnı́ perturbace vede k pádu testovacı́ částice
na centrálnı́ hvězdný objekt.
Relativistické škálovánı́ 1/M je jednı́m z hlavnı́ch argumentů podporujı́cı́ch ztotožněnı́ pozorovaných kHz QPO frekvencı́ s frekvencemi orbitálnı́ho pohybu. Předpoklad oblasti vyzařovánı́ akrečnı́ch disků v blı́zkosti
36
Kapitola 2. QPOs: Pohled z nekonečna
jejich vnitřnı́ch okrajů přirozeně vede k možnému ztotožněnı́ maximálnı́
pozorované QPO frekvence daného zdroje s orbitálnı́ frekvencı́ právě na
vnitřnı́m okraji akrečnı́ho disku. V přı́padě tenkých disků modelovaných geodetickým pohybem volných částic je vnitřnı́ okraj disku vymezen radiálnı́
souřadnicı́ kruhové meznı́ stabilnı́ orbity rISCO s odpovı́dajı́cı́ orbitálnı́
frekvencı́ νK (rISCO ) [39]. V přı́padě disků stabilizovaných tlakovými gradienty je pak vnitřnı́ okraj disku vymezen meznı́ vázanou orbitou s radiálnı́
souřadnicı́ rmb a odpovı́dajı́cı́ orbitálnı́ rychlost je vyššı́ než νK (rISCO ) [41].
Pozorované rozsahy kHz QPO frekvencı́ u galaktických LMXB zdrojů spolu
s přı́slušnými odhady hmotnostı́ takovému ztotožněnı́ odpovı́dajı́.
Sama identifikace kHz QPO frekvencı́ s orbitálnı́mi frekvencemi jak
geodetického pohybu testovacı́ch částic tak i frekvencemi vztahujı́cı́mi se
k chovánı́ orbitujı́cı́ch diskových struktur je uvažována mnoha způsoby,
jmenujme zde bez nároků na úplnost alespoň beat frequency model
předpokládajı́cı́ interakci orbitálnı́ frekvence s frekvencı́ rotace povrchu neutronových hvězd [9,43], model nelineárnı́ rezonance [1,3,4] a různé varianty
diskoseizmologie [36, 47, 48, 57, 65, 66, 72, 88, 89, 91, 92]. Poněkud podrobnějšı́
přehled orbitálnı́ch QPO modelů lze nalézt v úvodu přı́lohy 7. Následujı́cı́
text bude věnován aplikacı́m často diskutovaného [18, 35, 44, 93] relativisticky precesnı́ho modelu uvedeného na astrofyzikálnı́ scénu L. Stellou a
M. Vietrim [74].
2.2.1. Relativistický precesnı́ model
Relativistický precesnı́ model (dále jen RP model) ve své původnı́ verzi
ztotožňuje pozorované frekvence kHz QPOs s geodetickými orbitálnı́mi frekvencemi zářı́cı́ skvrny (blobu) v tenkém disku relacemi [74]
νL = νK (r) − νr (r),
νU = νK (r).
(2.10)
Frekvence dolnı́ho kHz pı́ku je interpretována jako frekvence precese periastra zářı́cı́ skvrny a frekvence hornı́ho kHz pı́ku jako odpovı́dajı́cı́ orbitálnı́
frekvence. Navı́c jsou ztotožňovány frekvence nı́zkofrekvenčnı́ch QPO νlf
s frekvencı́ nodálnı́ precese [73] relacı́
νlf = νnodal = νK − νθ .
(2.11)
I přes překvapujı́cı́ kvalitativnı́ i kvantitativnı́ shodu observačnı́ch dat
s výše uvedenými frekvenčnı́mi relacemi, ilustrovanou obrázkem 2.7
2.2. Orbitálnı́ modely vzniku QPOs
37
Obrázek 2.7. Rozdı́l ve frekvencı́ch hornı́ch a dolnı́ch kHz QPO ∆ν jako funkce
νK (zde značena νphi ) pro deset LMXB zdrojů. Křivky jsou vykresleny pro nerotujı́cı́ neutronové hvězdy o hmotnostech 2.2, 2.0, 1.8M⊙ . Převzato z [74].
převzatým z původnı́ studie [74], RP model založený na orbitálnı́m pohybu
horkých skvrn ve své původnı́ verzi nedisponuje dostatečně přesvědčivým
vysvětlenı́m mechanismu modulace pozorovaných rentgenových toků [70,
71]. Také kvalita fitů twin-peak QPO dat konkrétnı́ch zdrojů a přı́slušné
odhady hmotnostı́ zůstávajı́ mı́rně diskutabilnı́ [18].
Frekvence orbitálnı́ho pohybu a jejich kombinace nicméně formálně odpovı́dajı́ frekvencı́m oscilačnı́ch módů toroidálnı́ch diskových struktur orbitujı́cı́ch kolem centrálnı́ch kompaktnı́ch objektů a stabilizovaných tlakovými
gradienty. Nabı́zı́ se tak odlišná interpretace alespoň vysokofrekvenčnı́ch relacı́ RP modelu, kdy frekvence hornı́ho a dolnı́ho pı́ku odpovı́dajı́ vlastnı́m
frekvencı́ch oscilačnı́ch módů nebo jejich kombinaci. Frekvence oscilačnı́ch
módů jsou však již negeodeticky ovlivňovány tlakovými gradienty a tedy
závislé na parametrech diskových struktur. Proto takový druh interpretace
může alespoň principiálně korigovat fity twin-peak QPO observačnı́ch dat
žádoucı́m způsobem [20, 68, 77].
38
Kapitola 2. QPOs: Pohled z nekonečna
2.2.2. Preferované kruhové orbity
Přı́loha 4 rozšiřuje simulace distribucı́, které jsou předmětem publikacı́
v přı́lohách 2 a 3, o aplikace vysokofrekvenčnı́ch relacı́ RP modelu. Relace
umožňujı́ přiřadit pozorované hornı́ či dolnı́ QPO frekvenci odpovı́dajı́cı́ orbitálnı́ radius a tı́m i předpokládanou frekvenci partnerského pı́ku. Takovým
způsobem konstruovaná závislost νu (νl ) fitujı́cı́ twin-peak kHz QPO data
je parametrizována radiálnı́ souřadnicı́ přı́slušných kvazikruhových orbit.
Otázka, zda pozorované preferované poměry frekvencı́ hornı́ho a dolnı́ho
pı́ku neodpovı́dajı́ v této interpretaci preferovaným orbitám, je vı́ce než
přirozená.
Pozitivnı́ odpověd’ je podporována výsledky počı́tačových simulacı́. Simulace distribuce twin-peak kHz QPOs založená na frekvenčnı́ch relacı́ch
RP modelu konstruovaných v Hartleově–Thornově prostoročase a pouhém
náhodném výběru orbit z intervalu odpovı́dajı́cı́ho frekvenčnı́mu rozsahu
zdroje 4U 1636-53 je vysoce nekompatibilnı́ s observačnı́mi daty. Naproti
tomu obdobná simulace, avšak s implementacı́ předpokladu preference orbit
odpovı́dajı́cı́ch poměrům frekvencı́ pı́ků 3:2 a 5:4 reprodukuje přesvědčivě
vlastnosti distribucı́ skutečně pozorovaných twin-peak kHz QPOs. Existence preferovaných hodnot radiálnı́ch souřadnice kruhových orbit spojených s význačnými poměry frekvencı́ naznačuje možnou přı́tomnost rezonančnı́ch fenoménů. Závěry o preferovaných hodnotách radiálnı́ souřadnice
mohou být však také interpretovány v kontextu diskových oscilačnı́ch módů.
2.2.3. Odhady hmotnosti a spinu s použitı́m relativistického
precesnı́ho modelu: Circinus X-1
Vysokofrekvenčnı́mi relacemi RP modelu definovaná závislost νu (νl (r)) je
konkretizovaná vztahy pro frekvence orbitálnı́ho pohybu a tedy v geodetickém přı́padě přı́mo volnými parametry metriky. Pokud aproximujeme
prostoročas v okolı́ neutronových hvězd Hartleovou–Thornovou metrikou,
relevantnı́mi parametry jsou hmotnost hvězdy M, vnitřnı́ moment hybnosti j a kvadrupólový moment q. RP model tak principálně umožňuje
zı́skánı́ informacı́ o parametrech časoprostoru i neutronové hvězdy fitovánı́m
twin-peak kHz QPO dat. Je však ukázáno, že ačkoli frekvenčnı́ relace RP
modelu kvalitativně velmi dobře postihujı́ trendy obsažené v observačnı́ch
datech, charakteristická hmotnosti neutronových hvězd M ∼ 2M⊙ je přı́liš
vysoká v porovnánı́ s kanonickou hodnotou M ∼ 1.4M⊙ [18]. Nicméně je
třeba poznamenat, že na rozdı́l od raných studiı́ [55, 75] většina publiko-
39
2.2. Orbitálnı́ modely vzniku QPOs
vaných výsledků pro konkrétnı́ zdroje zanedbává vliv rotace neutronové
hvězdy.
Článek, který je obsahem přı́lohy 6 je věnován právě vlivu rotace na
odhad hmotnosti pekuliárnı́ho zdroje Circinus X-1. Současná odhadovaná
hmotnost zdroje právě s použitı́m frekvenčnı́ch relacı́ RP modelu je M0 =
2.2 ± 0.3M⊙ [22, 23]. Na rozdı́l od většiny LMXB zdrojů, kHz QPOs zdroje
Circinus X-1 vykazujı́ klastrovánı́ v okolı́ poměrů frekvencı́ νu : νl = 3 : 1,
tedy v rámci interpretace RP modelem odpovı́dajı́cı́ kruhovým orbitám již
poměrně vzdáleným od meznı́ stabilnı́ orbity, na kterých již rozdı́ly mezi
Hartleovým–Thornovým a Kerrovým řešenı́m nejsou přı́liš významné. Dodejme dále, že Hartleovo–Thornovo řešenı́ v přı́padě nastavenı́ q = j 2 splývá
s řešenı́m Kerrovým disponujı́cı́m pouze dvěma volnými parametry, hmotnostı́ centrálnı́ho objektu M a jeho spinem a. Relace mezi kvadrupólovým
momentem q a vnitřnı́m momentem hybnosti hvězdy j je určena stavovou
rovnicı́ hvězdného materiálu a právě stavové rovnice dovolujı́cı́ vysoké hmotnosti M0 = 2M⊙ nastavujı́ q do hodnot blı́zkých Kerrově geometrii. Z těchto
důvodů byla pro fitovánı́ twin-peak QPO dat použita frekvenčnı́ relace odpovı́dajı́cı́ Kerrově prostoročasu, která s použitı́m formulı́ pro epicyklické
frekvence (vyjádřené v Hz) 2.7 a 2.9 nabývá tvaru
νL = νU



"
1− 1+
8jνU
−6
F − jνU
νU
F − jνU
2/3
− 3j 2
νU
F − jνU

4/3 #1/2 
,

(2.12)
kde relativistický faktor F je dán vztahem F ≡ c3 /(2πGM).
Pro přı́pad Schwarzschildovy geometrie (j = 0) se vztah (2.12) zjednodušuje na tvar
νL = νU
(
1− 1−6
ν 2/3 1/2
U
F
)
,
(2.13)
vedoucı́ k formuli pro rozdı́l frekvencı́ hornı́ho a dolnı́ho pı́ku
∆ν = νU
q
1 − 6 (2πGMνU )2/3 /c2 ,
(2.14)
která byla použita k výše zmiňovanému odhadu hmotnosti [22, 23]. V principu tak každé dvojici parametrů M a j odpovı́dá unikátnı́ tvar relace
νu (νl (r)).
40
Kapitola 2. QPOs: Pohled z nekonečna
Frekvence hornı́ch a dolnı́ch pı́ků klesajı́ s rostoucı́m M a se zvyšujı́cı́m
se j. Dı́ky tomu lze pro nı́zké hodnoty j do hodnoty ∼ 0.3 nalézt třı́dy téměř
identických křivek, pro které M, j and M0 jsou přibližně svázány relacı́
M = [1 + k(j + j 2 )]M0 ,
(2.15)
kde konstanta k v přı́padě požadavku na aproximativnı́ shodu celého
průběhu frekvenčnı́ch relacı́ nabývá hodnoty k = 0.7. Kvalita fitů observačnı́ch dat je tedy prakticky shodná v přı́padě čistě Schwarzschildova
řešenı́ a hmotnostı́ M0 i při použitı́ Kerrovy metriky s relacı́ 2.15 svázanými
parametry M a j. Pokud je požadována shoda relacı́ pouze v hornı́ části
křivek odpovı́dajı́cı́ kruhovým orbitám v blı́zkosti meznı́ stabilnı́ orbity,
konstanta nabývá hodnoty k = 0.7. Pro dolnı́ části křivek relevantnı́ pro
nı́zkofrekvenčnı́ zdroje, včetně diskutovaného Circinus X-1, je nejlepšı́ho
výsledku dosaženo pro hodnotu k = 0.65 (0.55, 0.5) odpovı́dajı́cı́ okolı́ frekvenčnı́ho poměru νU /νL ∼ 2 (3, 4) . Průběh frekvenčnı́ch relacı́ a existence
třı́d obdobných křivek na pozadı́ observačnı́ch dat lze nalézt na obrázku 2.8.
Výše zmı́něné vlastnosti frekvenčnı́ch relacı́ RP modelu ovšem znamenajı́
nemožnost nezávislého odhadu hmotnosti a spinu fitovánı́m observačnı́ch
twin-peak QPOs dat a frekvenčnı́ relace RP modelu tak mohou poskytnout pouze informaci o třı́dě téměř identických fitů a jim odpovı́dajı́cı́ch
párů hodnot M a j. Přı́má analýza dostupných observačnı́ch twin-peak
kHz QPO dat zdroje Circinus X-1 v kontextu výše diskutovaných vlastnostı́
frekvenčnı́ch relacı́ RP modelu vede tak stanovenı́ relace pro M a j zdroje
Circinus X-1 ve tvaru
M = 2.2M⊙ [1 + k(j + j 2 )],
k = 0.55.
(2.16)
Konstanta M0 odpovı́dajı́cı́ čistě schwarzschildovskému fitu nabývá hodnoty
2.2[+0.3; −0.1]M⊙ s chybou danou jednotkovou variacı́ χ2 a je tedy v dobré
shodě s odhadem hmotnosti v [22, 23].
Výsledky analýzy však vykazujı́ poměrně překvapujı́cı́ trend, pozvolné
zvyšovánı́ kvality fitů spolu s rostoucı́m j a naznačujı́ určitou nekonzistenci použitých geodetických frekvenčnı́ch relacı́ s reálnými observačnı́mi
daty. Na datech vysokofrekvenčnı́ho zdroje 4U 1636-53 již bylo ukázáno,
že modifikace frekvenčnı́ch relacı́ RP modelu zavedenı́m efektivnı́ radiálnı́
epicyklické frekvence vztahem
ν̃t = νt (1 − β) ,
β ∈ (0 ∼ 0.2)
(2.17)
2.3. Shrnutı́
41
Obrázek 2.8. Průběh frekvenčnı́ch relacı́ RP modelu spolu s twin-peak
kHz QPOs observačnı́mi daty pro Circinus X-1 (žlutě), 4U 1636-53 (purpurově) a
jiné LMXB zdroje (černě). Skupiny téměř splývajı́cı́ch křivek ilustrujı́ podobnost
průběhu relacı́ pro M a j svázané relacı́ 2.15.
který poněkud spekulativně aproximuje negeodetické korekce frekvencı́ odpovı́dajı́cı́ napřı́klad interakci zářı́cı́ skvrny s diskem nebo magnetickým
polem neutronové hvězdy, může výrazně zlepšit kvalitu fitů twin-peak
kHz QPOs dat [84]. Geodetické i negeodetické fity dat zdroje 4U 1636-53
i průběh korigovaných frekvencı́ ilustruje obrázek 2.9. Použitı́ takto korigovaných frekvenčnı́ch relacı́ na rozdı́l od čistě geodetického přı́padu generuje
minima χ2 v blı́zkosti j = 0 a tak naznačuje pravděpodobnou přı́tomnost
negeodetických korekcı́ frekvencı́ orbitálnı́ho pohybu. Výsledky analýzy dat
zdrojů Circinus X-1 a 4U 1636-53 přehledně shrnuje obrázek 2.10.
2.3. Shrnutı́
Přes všechny observačnı́ potı́že je dnes známa a popsána poměrně rozsáhlá
fenomenologie QPOs, nicméně komplexnı́ model vysvětlujı́cı́ všechny
aspekty tohoto fenoménu v současné době nenı́ znám. Relativistický pre-
42
Kapitola 2. QPOs: Pohled z nekonečna
Obrázek 2.9. Fitovánı́ twin-peak kHz QPOs dat zdroje 4U 1636-53 frekvenčnı́mi
relacemi RP modelu. Vlevo: Čistě geodetický fit. Vpravo: Fit se zahrnutı́m negeodetické korekce spolu s průběhy geodetických i korigovaných frekvencı́.
Obrázek 2.10. Kvalita fitů twin-peak kHz QPOs dat relacemi RP modelu (charakterizovaná hodnotou χ2 ) jako funkce odhadu hmotnosti M . Pro každou
hodnotu M byla vyhledáno odpovı́dajı́cı́ hodnota j dosahujı́cı́ nejmenšı́ho χ2 .
Vlevo: Výsledky fitovánı́ simulovaných twin-peak kHz QPO dat s přı́tomnostı́
negeodetické korekce a fitovaných relacemi s (přerušovaná křivka) i bez korekčnı́ho faktoru (plná křivka). Výsledky fitovánı́ geodetickými relacemi vykazujı́ nápadnou podobnost s výsledky zpracovánı́ skutečných observačnı́ch dat.
Uprostřed: Výsledky fitovánı́ data zdroje Circinus X-1 pro různé hodnoty korekčnı́ konstanty β. Vpravo: Totéž pro vysokofrekvenčnı́ zdroj 4U 1636-53.
cesnı́ model kvalitativně dobře vysvětluje trendy přı́tomné v observačnı́ch
datech, avšak detaily fyzikálnı́ch mechanismů ležı́cı́ za pozorovanými distribuce observačnı́ch dat jsou stále nejasné. Binárnı́ systémy s relativisticky kompaktnı́m objektem jsou pokládány za přı́rodnı́ relativistické laboratoře a proto se zdá být přirozené vkládat naděje do možnosti testovat
predikce obecné relativity právě pomocı́ observačnı́ch fenoménů spojených
s chovánı́m zářenı́ a hmoty v extrémnı́ch podmı́nkách v blı́zkosti těchto ob-
2.3. Shrnutı́
43
jektů či naopak určovat parametry kompaktnı́ch hvězdných objektů pomocı́
nástrojů relativistické astrofyziky. I přes nesporné úspěchy dosažené na na
tomto poli nám však přı́roda prozatı́m nastavuje poněkud rozporuplnou
tvář.
Kapitola 3
Magnetická pole
3.1. Observačnı́ motivace
Jak bylo v předešlé kapitole ukázáno, ad hoc zavedená modifikace frekvenčnı́ch relacı́ relativistického precesnı́ho modelu spočı́vajı́cı́ v poměrně
malém snı́ženı́ radiálnı́ epicyklické frekvence při současném zanedbatelném
ovlivněnı́ ostatnı́ch frekvencı́ orbitálnı́ho pohybu může významně zlepšit
kvalitu fitovánı́ twin-peak QPOs v observačnı́ch datech. Otázkou samozřejmě zůstává původ a fyzikálnı́ mechanismus této modifikace. Zachováme-li představu orbitujı́cı́ch horkých skvrn v disku a uvažujeme-li
tedy skutečně přı́mé frekvence orbitálnı́ho pohybu testovacı́ch částic, jak
jsou použity původnı́ verzı́ relativistického precesnı́ho modelu, zdá se být
intuitivně zřejmé, že radiálně působı́cı́ negeodetická sı́la by mohla právě
tı́mto způsobem původně geodetickou frekvenčnı́ relaci ovlivnit. Přijmeme-li
dále dodatečný předpoklad, že orbitujı́cı́ hmota v tenkém disku je velmi
slabě elektricky nabita, pak Lorentzova sı́la vznikajı́cı́ interakcı́ náboje nabitých orbitujı́cı́ch částic a magnetického pole neutronové hvězdy aproximativně popsaného magnetickým dipólem kolmým na ekvatoriálnı́ orbitálnı́
rovinu může mı́t požadované vlastnosti.1 V tomto přı́padě se však otevřenou
otázkou nutně stává původ elektrického náboje orbitujı́cı́ hmoty v akrečnı́m
disku.
1
Je nutno podotknout, že se nejedná o jediný možný mechanismus ovlivněnı́ geodetických frekvenčnı́ch relacı́. V přı́padě ztotožněnı́ pozorovaných frekvencı́ s frekvencemi
oscilačnı́ch módů tlustých disků formálně odpovı́dajı́cı́ch epicyklickým frekvencı́m mohou mı́t tlakové gradienty závislé na tloušt’ce disku analogický vliv [19, 20, 68, 76, 77].
Dále je možno uvažovat o přı́padném vlivu viskozity materiálu tenkého disku. Existujı́ také některé odlišné, avšak velmi sofistikované přı́stupy k vysvětlenı́ vzniku vysokofrekvenčnı́ch QPOs pomocı́ modelů oscilujı́cı́ch toroidálnı́ch disků [47, 48, 57, 65, 66, 72,
91, 92], pomocı́ tzv. diskoseizmologie [88, 89] a konečně modely warped“ disků [36].
”
45
46
Kapitola 3. Magnetická pole
3.2. Perturbovaný kruhový orbitálnı́ pohyb nabitých
testovacı́ch částic v dipólovém magnetickém poli
na schwarzschildovském pozadı́
Článek On magnetic-field-induced non-geodesic corrections
to relativistic orbital and epicyclic frequencies, který je obsahem přı́lohy 8 spolu s recenzovaným sbornı́kovým přı́spěvkem On
magnetic-field induced non-geodesic corrections to the relativistic precession QPO model obsaženém v přı́loze 7 je věnován
plně relativistické analýze kruhového orbitálnı́ho pohybu nabitých testovacı́ch částic ovlivněného Lorentzovou silou v silném gravitačnı́m a magnetickém poli neutronových hvězd. V použité aproximaci jsou zanedbávány
efekty strhávánı́ souřadných systémů způsobené rotacı́ (spinem) centrálnı́ho
hvězdného objektu i vliv tenzoru energie-hybnosti magnetického pole
hvězdy na geometrii prostoročasu. Zatı́mco vliv spinu hvězdy je bezesporu
klı́čovým pro modelovánı́ realistických astrofyzikálnı́ situacı́ a jeho zahrnutı́
by mělo být a je předmětem dalšı́ho výzkumu, použitı́ vakuového řešenı́
Einsteinových rovnic lze považovat za velmi dobrou aproximaci dı́ky zanedbatelné hustotě energie magnetického pole i u silně zmagnetizovaných
neutronových hvězd oproti hustotě energie jejich pole gravitačnı́ho [63]. Jako
aproximace kompozice gravitačnı́ho a magnetického pole v okolı́ neutronové
či kvarkové hvězdy je tedy uvažován relativistický magnetický dipól, jehož
osa symetrie je kolmá na ekvatoriálnı́ rovinu statického prostoročasového
pozadı́ reprezentovaného Schwarzschildovou vakuovou metrikou.
3.2.1. Dipólové magnetické pole na pozadı́ Schwarzschildovy
prostoročasové geometrie
Čtyřpotenciál dipólového magnetického pole na uvažovaném schwarzschildovském pozadı́ lze zapsat ve tvaru [62]
Aα = − δαφ f (r)
µ sin2 θ
,
r
(3.1)
tedy ve formě magnetického dipólového řešenı́ Maxwellových rovnic
v plochém prostoročase, avšak násobenému funkcı́ f (r, M) danou formulı́
2M
M
2M
3r 3
+
1+
.
ln 1 −
f (r) =
8M 3
r
r
r
(3.2)
3.2. Dipólové magnetické pole na schwarzschildovském pozadı́
47
Odpovı́dajı́cı́ Maxwellův tenzor elektromagnetického pole Fµν , který lze
zı́skat ze čtyřpotenciálu Aµ definičnı́ relacı́
Fµν =
∂Aµ
∂Aν
−
,
µ
∂x
∂xν
(3.3)
má pouze dvě nezávislé nenulové komponenty,
Frφ
µ sin2 θ
=B =
r2
θ
∂f (r)
f (r) − r
∂r
,
(3.4)
a
Fθφ = −B r = −
µ sin 2θ
f (r),
r
(3.5)
přı́mo svázané s přı́slušnými komponentami vektoru magnetické indukce B.
Obecně relativistickou pohybovou rovnici pro nabitou testovacı́ částici
se specifickým nábojem q̃ ≡ q/m lze zapsat ve tvaru
dU µ
+ Γµαβ U α U β = q̃ Fνµ U ν
dτ
(3.6)
a výše uvedené vztahy tedy ukazujı́, že pohyb nabitých částic bude determinován dvěma volnými parametry neutronové hvězdy, vnitřnı́m magnetickým dipólovým momentem µ a hmotnostı́ M. Nepřı́mé metody založené
na analýze observačnı́ch dat ovšem umožňujı́ určit pouze velikost vektoru magnetické indukce na povrchu hvězdy, která se pro LMXB zdroje
předpokládá v intervalu B ∈ 106 ÷ 109 Gaussů [38, 39]. Lineárnı́ relaci
mezi vnitřnı́m dipólovým magnetickým momentem hvězdy µ a magnetickou indukcı́ Blocal měřenou pozorovatelem umı́stěným na rovnı́ku hvězdy
poloměru R a hmotnosti M však zı́skáme snadno projekcı́ Maxwellova tensoru F µν do lokálnı́ho referenčnı́ho systému takového pozorovatele. Tato
relace pro Schwarzschildův prostoročas a dipólové magnetické pole diskutované výše nabývá tvaru
√
4M 3 R3/2 R − 2M
µ=
6M(R − M) + 3R(R − 2M) ln 1 −
2M
R
Blocal .
(3.7)
Relativně slabé magnetické indukci na povrchu, Blocal = 107 Gauss ≃
2.875 × 10−16 m−1 tak pro hvězdu s hmotnostı́ M = 1.5M⊙ a poloměrem
R = 4M odpovı́dá hodnota dipólového momentu µ = 1.06 × 10−4 m2 .
Hvězdný objekt s výše uvedenými parametry je v dalšı́ analýze použit jako
testovacı́.
48
Kapitola 3. Magnetická pole
3.2.2. Frekvence perturbovaného kruhového orbitálnı́ho pohybu
Standardnı́m nástrojem pro analýzu orbitálnı́ho pohybu je studium chovánı́
efektivnı́ho potenciálu testovacı́ch částic. V přı́padě perturbovaného kruhového orbitálnı́ho pohybu lze jako alternativu s výhodou použı́t analýzu
vlastnostı́ epicyklických frekvencı́ velmi úzce svázaných právě s chovánı́m
efektivnı́ho potenciálu Veff . V okolı́ kruhové orbity dané podmı́nkou
dVeff /dr = 0 je radiálně či vertikálně perturbovaný kruhový orbitálnı́ pohyb
testovacı́ částice determinován přı́slušnými druhými derivacemi efektivnı́ho
potenciálu. V přı́padě stabilnı́ kruhové orbity pak chovánı́ perturbacı́ polohy testovacı́ částice odpovı́dá chovánı́ lineárnı́ho harmonického oscilátoru
a druhé mocniny epicyklických frekvencı́ jsou přı́mo úměrné druhým derivacı́m efektivnı́ho potenciálu,
ωr2
∂ 2 Veff
,
∼
∂r 2
ωθ2
∂ 2 Veff
∼
.
∂θ2
(3.8)
Existence reálné hodnoty radiálnı́ či vertikálnı́ epicyklické frekvence tedy
znamená i existence minima efektivnı́ho potenciálu Veff a současně stabilitu kruhové orbity vůči radiálnı́m či vertikálnı́m perturbacı́m. Oblast
ekvatoriálnı́ch stabilnı́ch kruhových orbit je pak přirozeně omezena hodnotou radiálnı́ souřadnice, kde jedna z epicyklických frekvencı́ch (ve většině
situacı́ radiálnı́) nabývá nulové hodnoty při současné existenci reálné hodnoty druhé epicyklické frekvence. Nulové hodnoty epicyklických frekvencı́
tak definujı́ polohy astrofyzikálně zajı́mavých meznı́ch stabilnı́ch orbit.
Epicyklické frekvence perturbovaného kruhového orbitálnı́ho pohybu lze
přı́mým výpočtem zı́skat ze známých komponent čtyřrychlosti testovacı́
částice na kruhové orbitě [7,8]. Obě nenulové komponenty čtyřrychlosti nabité testovacı́ částice na ekvatoriálnı́ kruhové orbitě (U µ = (U t , 0 , 0 , U φ ) )
snadno obdržı́me řešenı́m soustavy dvou rovnic, radiálnı́ složky relativistické pohybové rovnice 3.6 spolu s normalizačnı́ podmı́nkou pro čtyřrychlost
hmotné částice, U µ Uµ = −1. V diskutovaném přı́padě statické Schwarzschildovy prostoročasové geometrie a potenciálu magnetického pole ve tvaru 3.1
existujı́ dvě odlišná řešenı́ daná páry odpovı́dajı́cı́ch si nenulových komponent čtyřrychlosti testovacı́ch částic. Nicméně tato řešenı́ jsou při fixované
orientaci magnetického dipólu navzájem symetrická vůči simultánnı́ záměně
znaménka specifického náboje q̃ a orientace orbitálnı́ho pohybu a proto je
možno dalšı́ analýzu věnovat bez újmy na obecnosti pouze prvnı́mu z nich.
Relativně komplikovaný explicitnı́ tvar U t , U φ a orbitálnı́ úhlové frekvence
Ω = U φ /U t je uveden v přı́lohách 6 a 7. Mı́ra elektromagnetické interakce
3.2. Dipólové magnetické pole na schwarzschildovském pozadı́
49
Obrázek 3.1. Orbitálnı́ frekvence ν = Ω/2π jako funkce specifického náboje q̃ a
radiálnı́ souřadnice pro neutronovou hvězdu s M = 1.5M⊙ a µ = 1.06 × 10−4 m2 .
je dána součinem q̃ µ a proto lze v analýze opět bez újmy na obecnosti
zafixovat také velikost momentu µ a variovat pouze velikost a znaménko
specifického náboje orbitujı́cı́ hmoty q̃.
Formule pro epicyklické frekvence zı́skáme perturbovánı́m pozice testovacı́ částice na dané ekvatoriálnı́ kruhové orbitě (r, θ) = (r0 , π/2), vyjádřené
jako xµ (τ ) = z µ (τ ) + ξ µ (τ ), kde ξ µ (τ ) je malá perturbace souřadnic. Po
substituci takto vyjádřené polohy částice do rovnice 3.6 a při současném
omezenı́ se na členy prvnı́ho řádu v ξ µ je výsledkem relace pro perturbaci
souřadnic ξ µ ve formě rovnice lineárnı́ho harmonického oscilátoru
d2 ξ a
+ ωa2 ξ a = 0,
dt2
a ∈ (r, θ)
(3.9)
s epicyklickými úhlovými frekvencemi danými jako [8]
ωr
ωθ
∂V r
=
− γAr γrA
∂r
θ 1/2
∂V
.
=
∂θ
1/2
,
A ∈ (t, φ)
(3.10)
(3.11)
50
Kapitola 3. Magnetická pole
Výrazy γαµ a V µ zde nabývajı́ formy
q̃
γαµ = 2Γµαβ U β (U t )−1 − t Fαµ ,
U
q̃ µ α t −1
1 µ α t −1
µ
γ U (U ) − t Fα U (U )
.
V
=
2 α
U
(3.12)
(3.13)
Závěrem zdůrazněme, že derivace v rovnicı́ch (3.10) a (3.11) musı́ být provedeny při fixovaných hodnotách U t and U φ , jinými slovy výše diskutovaná
perturbačnı́ procedura předpokládá zachovávajı́cı́ se energii i moment hybnosti testovacı́ částice na dané orbitě .
Přesné výrazy pro orbitálnı́ i epicyklické frekvence zı́skané výše uvedeným způsobem lze najı́t v přı́loze 7. Omezı́me-li se na korekce prvnı́ho
řádu v q̃ µ vůči geodetickému pohybu v Schwarzschildově řešenı́, lze frekvence zapsat vztahy
Ω± = ±Ωs −
√
r − 3M
f (r) − r ∂f∂r(r)
r 7/2
q̃ µ + O(q̃µ2 ) ,
(3.14)
√
3 r − 3M
q̃ µ + O(q̃µ2 ) ,
(3.15)
ωθ± = Ωs ∓ 5/2
2r (r − 2M)
p
√
M(r − 6M)
3 r − 3M
± 2
q̃ µ + O(q̃µ2 ) .
(3.16)
ωr± =
r2
2r (r − 2M)
p
kde zde a dále Ωs = M/r 3 značı́ scharzschildovskou keplerovskou orbitálnı́
frekvenci.
3.2.3. Chovánı́ negeodeticky korigovaných frekvencı́, existence
a stabilita kruhových orbit
Lorentzova sı́la vznikajı́cı́ dı́ky elektromagnetické interakci elektrického
náboje orbitujı́cı́ hmoty a dipólového magnetického pole hvězdy může
v závislosti na znaménku elektrického náboje a orientace orbitálnı́ úhlové
rychlosti při fixovaném vektoru magnetického dipólového momentu mı́t repulzivnı́ či atraktivnı́ charakter. Obrázek 3.1 ukazuje, že v regionu repulzivnı́ho i atraktivnı́ho působenı́ Lorentzovy sı́ly přı́tomnost elektromagnetické interakce umožňuje existenci kruhových orbit nabitých částic i v těsné
blı́zkosti Schwarzschildova poloměru. Pouze v oblasti malých hodnot specifického náboje (symetrické vůči záměně jeho znaménka) podmı́nka existence reálných hodnot U t a U φ podrobně diskutovaná v přı́loze 7 přirozeně
3.2. Dipólové magnetické pole na schwarzschildovském pozadı́
51
existenci ekvatoriálnı́ch kruhových orbit vylučuje. Tato oblast v rovině
q̃ µ-r začı́ná pro q̃ µ = 0 na r = 3M, dosahuje své maximálnı́ šı́řky pro
q̃ µ = ± 1.971 M 2 na r = 2.441M a pro r → 2M je ohraničena hodnotami q̃ µ = ± 1.333 M 2 . Přı́tomnost elektromagnetické interakce dále
výrazně ovlivňuje polohu meznı́ stabilnı́ orbity, odpovı́dajı́cı́ vnitřnı́mu
okraji tenkých akrečnı́ch disků v okolı́ kompaktnı́ch hvězdných objektů. Pro
takové meznı́ stabilnı́ orbity je nově zavedena zkratka MISCO (Magnetic
Innermost Stable Circular Orbit), zatı́mco geodetické meznı́ stabilnı́ orbity
jsou dále označovány jako GISCO (Geodesic Innermost Stable Circular Orbit). Ve Schwarzschildově prostoročasové geometrie pak samozřejmě rGISCO
nabývá dobře známé hodnoty 6M. Dalšı́ důsledky působenı́ Lorentzovy sı́ly
jsou pro atraktivnı́ a repulzivnı́ region kvalitativně odlišné.
V atraktivnı́m regionu působenı́ Lorentzovy sı́ly vzniká nová třı́da nestabilnı́ch kruhových orbit nabitých částic umı́stěných pod kruhovou fotonovou orbitou (r < rph = 3M) s opačnou orientacı́ orbitálnı́ úhlové rychlosti
oproti orbitám s radiálnı́ souřadnicı́ r > rph . Oblast globálně stabilnı́ch
kruhových orbit je však v atraktivnı́m regionu zdola omezena MISCO orbitami s nulovou hodnotou radiálnı́ epicyklické frekvence, které se spolu
s rostoucı́ velikostı́ specifického náboje q̃ orbitujı́cı́ hmoty vzdalujı́ od geodetické meznı́ stabilnı́ orbity na (rms = 6M). Radiálnı́ epicyklická frekvence
zde klesá spolu se vzrůstajı́cı́ velikostı́ specifického náboje, orbitálnı́ a vertikálnı́ epicyklická frekvence naopak (rozdı́lným tempem) rostou.
V repulzivnı́m regionu vykazuje vliv Lorentzovy sı́ly poněkud komplexnějšı́ charakter. Odpovı́dajı́cı́m způsobem nastavené parametry elektromagnetické interakce (specifický náboj orbitujı́cı́ hmoty q̃ spolu s velikostı́ dipólového magnetického momentu hvězdy µ) mohou překvapivě stabilizovat kruhové orbity nabitých částic i v oblasti pod kruhovou fotonoMISCO
vou orbitou až do hodnoty radiálnı́ souřadnice rmin
= 2.73M spojené
max
s nejvyššı́ možnou orbitálnı́ frekvencı́ ν
= 3284 Hz (1.5M⊙ /M) a tzv.
kritickým nábojem q̃crit při fixovaném magnetickém dipólovém momentu µ.
Kritický náboj orbitujı́cı́ hmoty je svázán s magnetickým dipólovým momentem hvězdy relacı́
µ q̃crit = 1.869M 2 .
(3.17)
Astrofyzikálnı́ relevance takto extrémnı́ch orbit je však diskutabilnı́ v souvislosti se stále otevřenou otázkou možnosti existence relativisticky superkompaktnı́ch neutronových či kvarkových hvězd, t.j. kompaktnı́ch
hvězdných objektů s R < 3M, nebo obecněji kompaktnı́ch hvězdných ob-
52
Kapitola 3. Magnetická pole
jektů s poloměrem menšı́m než je radiálnı́ souřadnice kruhových fotonových
orbit v odpovı́dajı́cı́ch časoprostorových geometriı́ch. V repulzivnı́m regionu je region stabilnı́ch orbit pro q̃ < q̃crit omezen zdola nulovou hodnotou radiálnı́ epicyklické frekvence a radiálnı́ souřadnice MISCO orbity
klesá spolu s rostoucı́ velikostı́ specifického náboje q̃ orbitujı́cı́ hmoty až
MISCO
na rmin
= 2.73M. Pro q̃ > q̃crit meznı́ stabilnı́ orbitu definuje již nulová hodnota vertikálnı́ epicyklické frekvence a radiálnı́ souřadnice takových
MISCO orbit začı́ná spolu s velikostı́ specifického náboje opět růst. Hodnoty
kritického náboje q̃crit a radiálnı́ souřadnice nejnižšı́ meznı́ stabilnı́ orbity
MISCO
rmin
tak přirozeně odpovı́dajı́ simultánnı́mu splněnı́ podmı́nek ωr = 0 a
ωθ =0.
Zkoumané frekvence vykazujı́ v repulzivnı́m regionu opačné chovánı́
než v regionu atraktivnı́m, radiálnı́ epicyklická frekvence zde roste spolu
se vzrůstajı́cı́ velikostı́ specifického náboje a obě zbývajı́cı́ frekvence naopak (rozdı́lným tempem) klesajı́. Nicméně v obou regionech přı́tomnost
elektromagnetické interakce umožňuje vznik exotických ostrůvků existence částečně stabilnı́ch či nestabilnı́ch kruhových orbit v těsné blı́zkosti
černoděrového horizontu oddělených od oblasti globálnı́ stability kruhového
orbitálnı́ho pohybu. Podrobný rozbor lze nalézt opět v přı́loze 7.
Přı́tomnost Lorentzovy sı́ly dále zjevně porušuje sférickou symetrii geometrie časoprostorového pozadı́ manifestujı́cı́ se shodou vertikálnı́ epicyklické a orbitálnı́ frekvence a umožňuje vznik nového efektu, nodálnı́ precese
roviny orbitálnı́ho pohybu s frekvencı́
νn (r) = ν(r) − νθ (r),
(3.18)
kvalitativně podobné Lenseově-Thirringově precesi přı́tomné v rotujı́cı́ch
axiálně symetrických prostoročasech. Fáze této nodálnı́ precese je však
opačná v atraktivnı́ a repulzivnı́ oblasti působenı́ Lorentzovy sı́ly.
3.2.4. Aplikace na relativistický precesnı́ model
V obou kvalitativně odlišných regionech je na astrofyzikálně relevantnı́ch
hodnotách radiálnı́ souřadnice citlivost radiálnı́ epicyklické frekvence vůči
působenı́ Lorenzovy sı́ly významně většı́ než citlivost orbitálnı́ a vertikálnı́
epicyklické frekvence. Existence Lorentzovy sı́ly a jejı́ chovánı́ v atraktivnı́m regionu tedy skutečně modifikuje frekvenčnı́ relace RP modelu
požadovaným způsobem. Pro detailnı́ analýzu observačnı́ch dat konkrétnı́ch
zdrojů v použité aproximaci chybı́ vliv spinu neutronové hvězdy, avšak
3.2. Dipólové magnetické pole na schwarzschildovském pozadı́
53
Obrázek 3.2. Vlevo: Oblast stabilnı́ch kruhových orbit vyplněná vrstevnicovým
grafem orbitálnı́ frekvence ν = Ω/2π. Vpravo: Totožná oblast, avšak vyplněná
vrstevnicovým grafem nodálnı́ precesnı́ frekvence νn . Konstruováno pro for M =
1.5M⊙ a µ = 1.06 × 10−4 m2 .
již hromadný fit observačnı́ch dat rozsáhlé skupiny LMXB zdrojů ukazuje dalšı́ efekt aplikace negeodetických frekvenčnı́ch relacı́ modifikovaných
přı́tomnostı́ Lorentzovy sı́ly, možnost výrazného snı́ženı́ odhadu hmotnosti
neutronových hvězd až ke kanonické hodnotě M = 1.4M⊙ (viz obrázek
3.2.4).2 I alternativnı́ metoda odhadu hmotnosti, založená na předpokladu,
že nejvyššı́ pozorovaná QPO frekvence daného zdroje odpovı́dá orbitálnı́
frekvenci ISCO (zde MISCO) orbity dává obdobný výsledek.
Vlastnosti repulzivnı́ho regionu umožňujı́ stabilizovat kruhové orbity
i pod GISCO orbitou s orbitálnı́mi frekvencemi vyššı́mi než možné frekvence geodetického pohybu. V této souvislosti nenı́ nezajı́mavé, že analýza
observačnı́ch dat známého LMXB zdroje 4U 1636-53 ukazuje ojedinělou
existenci QPO frekvence 1860 Hz, ačkoli běžný rozsah pozorovaných frekvencı́ zdroje je 200–1200 Hz. Geodetické orbitálnı́ modely i s aplikacı́ vlivu
spinu neutronové hvězdy velmi obtı́žně nalézajı́ relaci mezi takto vysokou
pozorovanou frekvencı́ a astrofyzikálně realistickými parametry (hmotnostı́
a spinem) hvězdného objektu [16]. Předpoklad slabě nabité orbitujı́cı́ hmoty
v repulzivnı́m režimu však může přı́tomnost vysokých frekvencı́ snadno
vysvětlit a navı́c na základě nezávislými metodami určené velikosti vek2
Data převzata z [18, 22, 49, 90].
54
Kapitola 3. Magnetická pole
Obrázek 3.3. Hromadné fity observačnı́ch twin-peak kHz QPO“ dat pro širokou
”
skupinu LMXB zdrojů pomocı́ frekvenčnı́ch relacı́ RP modelu. Silná plná křivka
odpovı́dá negeodetické frekvenčnı́ relaci pro M = 1.4M⊙ a Lorentzovu sı́lu indukovanou dipólovým momentem hvězdy µ = 1.06×10−4 m2 a specifickým nábojem
orbitujı́cı́ hmoty q̃ = 5.0 × 1010 . Jako ilustrace jsou také zobrazeny dva čistě schwarzschildovské geodetické fity (tenké přerušované křivky), fit pro M = 2M⊙
diskutovaný v [18] a pro porovnánı́ s negeodetickou frekvenčnı́ relacı́ geodetický
fit pro shodnou hmotnost neutronové hvězdy M = 1.4M⊙ .
toru indukce magnetického pole hvězdy také umožňuje odhad specifického
náboje akreujı́cı́ho materiálu.
3.3. Perspektiva dalšı́ho výzkumu: magnetické pole
pomalu rotujı́cı́ neutronové hvězdy
Pro rozšı́řenı́ v předešlé kapitole diskutovaných výsledků i o popis vlivu
rotace (spinu) neutronové či podivné hvězdy je nutno reprezentovat
časoprostorové pozadı́ axiálně symetrickou geometriı́ popisujı́cı́ i efekty
strhávánı́ inerciálnı́ch systému (frame dragging) způsobené právě rotacı́
3.3. Perspektiva dalšı́ho výzkumu
55
centrálnı́ho hvězdného objektu. Dalšı́m nezbytným krokem pak je aplikace
odpovı́dajı́cı́ho řešenı́ Maxwellových rovnic s charakterem magnetického
dipólu pevně spojeného s rotujı́cı́ neutronovou či podivnou hvězdou.
3.3.1. Lenseova–Thirringova metrika
Velmi často použı́vanou aproximacı́ gravitačnı́ho pole v okolı́ pomalu rotujı́cı́ch neutronových či podivných hvězd je Lenseova–Thirringova metrika3 [31, 32, 82], jejı́ž časoprostorový element v přı́padě vakuového řešenı́
vně centrálnı́ho rotujı́cı́ho objektu lze zapsat ve tvaru
ds2 = −η(r)2 dt2 +
dr 2
+ r 2 dθ2 + sin2 θ dφ2 − 2ω (r) dtdφ , (3.19)
2
η(r)
kde funkce η(r) je dána vztahem
1/2
2M
η(r) ≡ 1 −
.
r
(3.20)
Funkce ω(r) může být interpretována jako úhlová rychlost volně padajı́cı́ch
inerciálnı́ch pozorovatelů a je také známa jako Lenseova–Thirringova úhlová
rychlost. V přı́padě vakuového řešenı́ vně hvězdy je dána jednoduchou formulı́
ω(r) =
2J
,
r3
(3.21)
kde J je celkový moment hybnosti neutronové hvězdy o hmotnosti M a
poloměru R, který je také možno vyjádřit pomoci momentu setrvačnosti
I(M, R) a úhlové rychlosti rotace hvězdy Ωstar jako J = I(M, R)Ωstar .
Všechny uvedené veličiny jsou uvažovány jako měřené statickým pozorovatelem v nekonečnu a metrika 3.19 je zapsána v systému geometrických jednotek, kde c = G = 1. V následujı́cı́ch výpočtech se dále jevı́ výhodné použı́t
rotačnı́ parametr (spin) neutronové hvězdy definovaný jako a = J/M.
Lenseova–Thirringova metrika aproximuje s přesnosti do prvnı́ho řádu
v J jak Kerrovu metriku popisujı́cı́ prostoročas rotujı́cı́ch černých děr, tak
i vnějšı́ (vakuovou) Hartleovu–Thornovu metriku popisujı́cı́ prostoročas
v okolı́ neutronových hvězd včetně vlivu zploštěnı́ neutronové hvězdy
3
Často je namı́sto pojmenovánı́ Lenseova–Thirringova metrika“ použı́ván termı́n
”
aproximace pomalé rotace“ (viz např. [40]).
”
56
Kapitola 3. Magnetická pole
(kvadrupólového momentu distribuce hmotnosti) [33]. Je tedy zřejmé, že
astrofyzikálně relevantnı́ použitı́ Lenseovy–Thirringovy metriky je omezeno
na modely neutronových hvězd s malým rotačnı́m parametrem a, avšak
oproti realističtějšı́mu Hartleovu–Thornovu řešenı́ je jejı́ nezanedbatelnou
výhodou analytická jednoduchost při zachovánı́ všech nezbytných vlastnostı́
pro řešenı́ relativistických Maxwellových rovnic elektromagnetického pole
vně i uvnitř rotujı́cı́ hvězdy [63, 64].
3.3.2. Geodetický kvazikruhový orbitálnı́ pohyb
Řešenı́m radiálnı́ složky rovnice geodetiky DU µ /dτ = 0 spolu s normalizačnı́ podmı́nkou pro hmotné částice, U µ Uµ = −1, a s použitı́m metriky
(3.19) zı́skáváme dva páry nenulových komponent U µ pro korotujı́cı́ (+) a
protirotujı́cı́ (−) ekvatoriálnı́ kruhové orbity ve tvaru
φ
U0±
"
r2
(r − 3M) + 2a a ±
=±
M
t
U0±
=
a±
r
r3
a2 +
M
!
φ
U0±
.
r
r3
a2 +
M
!#−1/2
(3.22)
(3.23)
Odpovı́dajı́cı́ úhlové rychlosti kruhového orbitálnı́ho pohybu Ω0±
φ
t
U0±
/U0±
pak lze zapsat ve tvaru
Ω0± =
a±
r
r3
a2 +
M
!−1
.
=
(3.24)
V přı́padě omezenı́ se na přesnost prvého řádu ve spinu lze úhlovou rychlost
kruhového orbitálnı́ho pohybu zapsat vztahem
Ω0± = ±Ωs − Ω2s a + O(a2 ) .
(3.25)
Dolnı́ index 0 zde a dále značı́ veličiny vztahujı́cı́ se ke geodetickému orbitálnı́mi pohybu neovlivněnému Lorentzovou silou.
Epicyklické frekvence vztažené ke geodetickému orbitálnı́mu pohybu lze
snadno zı́skat metodou popsanou v předešlé kapitole. Navı́c nepřı́tomnost
57
3.3. Perspektiva dalšı́ho výzkumu
elektromagnetické interakce přı́slušné výrazy výrazně zjednodušı́. Při omezenı́ se na členy prvého řádu v a lze geodetické epicyklické frekvence zapsat
vztahy
ωθ0± = Ωs ∓
ωr0±
3 2
Ωs a + O(a2 ) ,
2
p
M(r − 6M)
3M(2M + r)
p
a + O(a2) .
±
=
7
r2
2 r (r − 6M)
(3.26)
(3.27)
Formule pro frekvence orbitálnı́ho pohybu s přesnostı́ prvého řádu ve spinu
a je samozřejmě možno alternativně zı́skat i linearizacı́ přı́slušných vztahů
pro Kerrovo nebo Hartleovo–Thornovo řešenı́.
3.3.3. Dipólové magnetické pole na Lenseově–Thirringově pozadı́
V [63] je odvozováno a analyzováno řešenı́ Maxwellových rovnic pro obecně
orientované dipólové magnetické pole na pozadı́ Lenseovy–Thirringovy metriky 3.19 do prvnı́ho řádu přesnosti v J, včetně podmı́nek napojenı́ na
vnitřnı́ řešenı́ jak pro nekonečně vodivý vnitřek hvězdy tak i pro hvězdu
s konečnou vodivostı́. Dále však předpokládejme, že osa symetrie magnetického dipólu je totožná s osou rotace hvězdy (dipól má nulovou deklinaci), vnitřek hvězdy je nekonečně vodivý, magnetické siločáry jsou tedy
do hvězdy zamrzlé a jsou unášeny jejı́ rotacı́. V takovém přı́padě relativně
komplikované obecné dipólové řešenı́ přecházı́ do jednoduššı́ho tvaru, kdy
komponenta čtyřpotenciálu Aφ odpovı́dá schwarzschildovskému přı́padu 3.1
a časovou komponentu pak lze zapsat ve tvaru
At (r, θ) = at0 (r) + at2 (r)P2 (cos θ) ,
(3.28)
kde P2 je Legendrův polynom 2. stupně [40]. Toto řešenı́ oproti v předešlé
subkapitole diskutovanému dipólovému řešenı́ na sféricky symetrickém
časoprostorovém pozadı́ obsahuje navı́c elektrickou složku čtyřpotenciálu.
58
Kapitola 3. Magnetická pole
Členy at0 a at2 lze zı́skat z Maxwellových rovnic analyticky ve tvaru
at0 =
at2 =
+
−
+
c0
Jµ
Jµ
+
(3r
−
M)
+
(3r − 4M) ln η 2 (r) ,
(3.29)
r
2M 3 r 2
4M 4 r
c1
(r − M)(r − 2M)
M2
3
2
2
2
2
2
2
3r − 6Mr + M + 2 r − 3Mr + 2M ln η (r)
c2
Mr
M
Jµ 4
9r − 3Mr 3 − 30M 2 r 2 + 8M 3 r + 2M 4
6
2
2M r
12r 4 − 36Mr 3 + 24M 2 r 2 + M 3 r ln η 2 (r) ,
(3.30)
kde c0 , c1 a c2 jsou integračnı́ konstanty [40]. Protože konstantu c0 lze snadno
interpretovat jako elektrický náboj hvězdy, je astrofyzikálně přirozené nastavit ji nulovou (c0 = 0). Z podmı́nky regularity v nekonečnu dále zı́skáme
snadno hodnotu konstanty c1 ve tvaru [40]
c1 =
9Jµ
.
2M 4
(3.31)
Konečně zbývajı́cı́ konstanta c2 může být fixována napojovacı́mi
podmı́nkami na povrchu hvězdy. Za předpokladu dokonale vodivého vnitřku
hvězdy rotujı́cı́ho úhlovou rychlostı́ Ωstar a tedy do materiálu hvězdy zamrzlého magnetického pole (uµ Fµν = 0, uµ = (ut , 0, 0, Ωstar ut )) zı́skáme
vztah [40]
c2 =
µJ
12R3 − 24MR2 + 4M 2 R + M 3
5
2
M R
µJ
+
12R3 − 36MR2 + 24M 2 R + M 3 ln η 2 (r)
6
2M R
µΩstar 2
2
2
−
2MR + 2M + R ln η (r)
4M 3
. 2
3R2 − 6MR + M 2
MR
3
2
2
2
+ 2 R − 3MR + 2M ln η (r) .
(3.32)
M
Elektrická složka čtyřpotenciálu je tedy v astrofyzikálně relevantnı́m
přı́padě elektricky nenabité hvězdy indukována pouze rotacı́ hvězdy a efektem strhávánı́ lokálnı́ch inerciálnı́ch systémů, jak je zřejmé z jejı́ závislosti
3.3. Perspektiva dalšı́ho výzkumu
59
na úhlové rychlosti rotace Ωstar a momentu hybnosti J. Je také zřejmé, že
vztah mezi těmito dvěma veličinami lze zı́skat pouze z vlastnostı́ distribuce
hustoty hmoty ve hvězdě, které je určena stavovou rovnicı́ hvězdného materiálu. Rozbor chovánı́ vnitřnı́ho Hartleova–Thornova řešenı́ však ukazuje,
že moment setrvačnosti neutronových hvězd I je možné vyjádřit pouze jako
funkci hmotnosti a poloměru neutronové hvězdy bez ohledu na stavovou
rovnici. Tvar závislosti je podobný jako v Newtonovské mechanice a je dán
vztahem
M
MR2 ,
(3.33)
I=k
R
kde konstanta úměrnosti k(M/R) je funkcı́ pouze kompaktnosti neutronové
hvězdy M/R [85]. Vnitřnı́ moment hybnosti J poté zı́skáme standardnı́m
způsobem jako J = IΩstar .4 Vlastnosti extrémnı́ch forem hmoty vyskytujı́cı́
se v nitru neutronových či podivných hvězd se tak stávajı́ dalšı́ nezbytnou
ingrediencı́ astrofyzikálně realističtějšı́ho modelovánı́ magnetického pole neutronových či podivných hvězd i odpovı́dajı́cı́ch orbit nabitých testovacı́ch
částici a mohou tak přispět k zpřesněnı́ interpretace QPO modulačnı́ch
frekvencı́ rentgenového zářenı́ LMXB zdrojů pomocı́ modelů založených na
frekvencı́ch orbitálnı́ho pohybu ovlivněných elektromagnetickou interakcı́.
4
Obdobně i kvadrupólový moment Q je možné vyjádřit vztahem Q = l(M/R)J 2 /M ,
ve kterém konstanta úměrnosti l(M/R) je opět funkcı́ pouze kompaktnosti hvězdy.
Všechny relevantnı́ parametry vnějšı́ho Hartleova–Thornova řešenı́ je tak možno zı́skat
ze znalosti hmotnosti neutronové hvězdy M , jejı́ho poloměru R a rotačnı́ frekvence Ωstar [85].
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Přı́loha 1
DOI: 10.2478/s11534-007-0033-6
Research article
Extreme gravitational lensing in vicinity of
Schwarzschild–de Sitter black holes
Pavel Bakala1∗ , Petr Čermák2, Stanislav Hledı́k1
Zdeněk Stuchlı́k1, Kamila Truparová1
1
Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava
Bezručovo nám. 13, CZ-746 01 Opava, Czech Republic
2
Institute of Computer Science, Faculty of Philosophy and Science, Silesian University in Opava
Bezručovo nám. 13, CZ-746 01 Opava, Czech Republic
Received 03 February 2007; accepted 19 May 2007
Abstract: We have developed a realistic, fully general relativistic computer code to simulate
optical projection in a strong, spherically symmetric gravitational field. The standard theoretical
analysis of optical projection for an observer in the vicinity of a Schwarzschild black hole is
extended to black hole spacetimes with a repulsive cosmological constant, i.e, Schwarzschild–
de Sitterspacetimes. Influence of the cosmological constant is investigated for static observers and
observers radially free-falling from the static radius. Simulations include effects of the gravitational
lensing, multiple images, Doppler and gravitational frequency shift, as well as the intensity
amplification. The code generates images of the sky for the static observer and a movie simulations
of the changing sky for the radially free-falling observer. Techniques of parallel programming are
applied to get a high performance and a fast run of the BHC simulation code.
c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.
Keywords: black holes, cosmological constant, gravitational lensing, visualization, numerical
relativity
PACS (2006): 04.70.-s, 04.25.Dm, 95.36.+x, 07.05.Tp
1
Introduction
General relativistic deflection of light and lensing effects in gravitational field of stars
were firstly investigated by Einstein [1]. In the vicinity of relativistic compact objects
(black holes or neutron stars) these effects have strong influence on properties of the
∗
E-mail: [email protected]
P. Bakala et al. / Central European Journal of Physics
optical projection which become different than those of the optics in the flat spacetime
as we experience it in our everyday life [2]. Several authors have developed ray-tracing
or simulation computer codes for modeling general relativistic optical projection in the
vicinity of rotating or non-rotating black holes and neutron stars without presence of a
cosmological constant, see, e.g., [3–9].
Recent observations indicate the cosmic expansion accelerated by a dark energy that
can be described by a repulsive cosmological constant, Λ > 0 [10, 11]. We investigate the
influence of Λ > 0 on the appearance of distant universe for observers in close vicinity of
nonrotating Schwarzschild–de Sitter(SdS) black holes. In order to obtain a good qualitative picture of the Λ influence, our simulations have been performed with unrealistically
high values of Λ.
2
Schwarzschild–de Sitter geometry
The line element of the SdS spacetime has in the standard Schwarzschild coordinates and
geometric units (c = G = 1) the form
−1
2M
Λ 2
2M
Λ 2
2
2
ds = − 1 −
− r dt + 1 −
− r
dr 2 + r 2 (dθ 2 + sin2 θ dφ2 ), (1)
r
3
r
3
where M is mass of the central black hole, and Λ ∼ 10−56 cm−2 is the repulsive cosmological constant. It is advantageous to introduce a dimensionless cosmological parameter
y by the relation y = 31 ΛM 2 . The location of horizons is given by the condition gtt = 0.
Two event horizons exist for y ∈ (0, ycrit ), where ycrit = 1/27. The black hole and the
cosmological horizons are located at
2
π+ξ
rh = √ cos
,
3
3y
2
π−ξ
rc = √ cos
,
3
3y
(2)
respectively, where
p
ξ = cos−1 3 3y.
(3)
The spacetime is dynamic at r < rh and r > rc . The static radius, (a hypersurface
where the gravitational attraction of the black hole is balanced by the cosmic repulsion)
is located at
1
rs = y − 3 .
(4)
With increasing value of y, the horizons aproach to each other. In the critical case of
y = ycrit = 1/27, the horizons and the static radius coincide at rh = 3. For of y > 1/27,
the spacetime is dynamic at r > 0, and describes a naked singularity [12]. We consider
only spacetimes admitting existence of static observers that have y < 1/27.
3
Optical projection in Schwarzschild–de Sitterspacetimes
Construction of relativistic optical projection consists of finding all null geodesics connecting the source and the observer, i.e., solving the so-called emitter-observer problem.
P. Bakala et al. / Central European Journal of Physics
An observer will see the image generated by the concrete geodesic in direction tangent to
the photon trajectory in observer’s local frame, therefore given by space part of locally
(µ)
measured 4-momentum of photons pobs . Directional angle α related to the outward radial direction and the frequency shift g of the photon (the ratio of observed and emitted
energy) are given by the general relations
(r)
cos α = −
pobs
(t)
pobs
(t)
,
g=
pobs
(t)
psource
.
(5)
The indeces “obs” (observer) and “source” denote the components locally measured by
an observer or a source. In the SdS spacetimes null geodesics are characterised by the
impact parameter b defined as the ratio of constants of motion, b ≡ Φ/E [12, 13]. For an
observer located at robs , α and g are functions of b only, as shown in Appendix.
Due to spherical symmetry of the SdS geometry (1), it is sufficient to consider only
sources and observers located in the equatorial plane. Considering observers located at
φ = 0, ∆φ along geodesics connecting the source and the observer reads
∆φ = −φsource − 2kπ,
(6)
where φsource is an angular coordinate of the source. The image order k takes values of
0, 1, 2, . . . , +∞ for geodesics orbiting the central black hole clockwise, and −1, −2, . . . , −∞
for geodesics orbiting the central black hole counter-clockwise. The first direct and indirect images correspond to k = 0 and k = −1, respectively.
Photon motion in the SdS spacetimes is governed by the Binet formula [13, 14]
dφ
1
= ±p
,
du
b−2 − u2 + 2u3 + y
(7)
p
where u = r −1 . The critical impact parameter, bc = 27/ (1 − 27y), corresponds to the
circular photon geodesic, which is located at rph = 3M for arbitrary value of Λ [12, 13].
Photons coming from distant universe with b < bc end up in the central singularity,
while photons with b > bc return back towards the cosmological horizont [13, 14]. Using
the term under the square root in (7) as a motion reality condition, a straightforward
calculation yields relation for a turning point rt of geodesics with b > bc :
p
2
1
rt = p
cos
arccos −3 3 (y + b−2 ) ,
(8)
3
3(y + b−2 )
and a relation for the maximum impact parameter bmax for an observer located at given
robs ,
1
bmax = p 2
.
(9)
uobs − 2u3obs − y
Therefore, ∆φ can be expressed as
∆φ(usource , uobs , b) = ∓
Z
uobs
usource
du
p
,
−2
b − u2 + 2u3 + y
(10)
P. Bakala et al. / Central European Journal of Physics
for geodesics with b < bc , or for geodesics passing the observer position ahead of the
turning point. For geodesics passing observer position beyond the turning point ∆φ can
be expressed as
Z uobs
Z uturn
du
du
p
p
∆φ(uobs ) = ±
∓
.
(11)
b−2 − u2 + 2u3 + y
b−2 − u2 + 2u3 + y
uturn
usource
In (10) and (11), the upper (lower) sign corresponds to geodesics orbiting clockwise
(counter-clockwise). These integrals express ∆φ along the photon path as a function
F (b, uobs , usource , y). Equation (6) can then be rewriten in the following way:
F (b, uobs , usource , y) + φsource + k2π = 0.
(12)
This equation, that can be understood as an integral equation with an eigenvalue b determines b as an implicit function of the source and observer coordinates, the image
order and Λ. Unfortunately, F (b, uobs , usource , y) can be expressed in terms of elliptic integrals only, and do not allow to obtain an explicit formula for b (φsource , k, uobs , usource , y).
Consequently, the simulation code uses standard numerical integration and root finding
methods.
4
Numerical solution and code outputs
Final equation (12) was numerically solved using the BHC code for optical projection
of observers located in the static region between the horizons, as well as for observers
located in the dynamic region under the black hole horizon. In the static region, static
observers and observers radially free-falling from the static radius have been considered.
In the dymamic region under the black hole horizon, where static observers cannot exist,
the optical projection was constructed for observers radially free-falling only. We consider
Λ = 5 × 10−3 , Λ = 10−2 and pure Schwarzschild case with Λ = 0.
Left (top) panels of Figs. 1, 2, and 3 show b as a function of |∆φ| along the appropriate
geodesic connecting distant source and observer located at robs . The right (bottom) panel
of these figures show α for static and radially free-falling observers. If we consider spherical
symmetry of the problem, equation (6) implies, that |∆φ| ≤ π corresponds to the first
direct images of distant sources, whereas larger |∆φ| > π corresponds to images with
higher order.
As shown in Fig. 1, for observers located above the circular photon orbit b increases
up to a maximum value bmax given by equation (9), then decreases, and asymptoticaly
aproaches bc from above. Values |∆φ| ≤ |∆φ(bmax )| correspond to ingoing geodesics with
b < bc , or to geodesics passing the observer position ahead the turning point. Values
|∆φ| > |∆φ(bmax )| correspond to geodesics passing the observer position beyond the
turning point. Situation is different for observers located under the circular photon orbit
where only geodesics with b < bc can exist. In this case b increases monotonously and
asymptoticaly aproaches to bc from below.
P. Bakala et al. / Central European Journal of Physics
In all the cases, α monotonously increases up to a maximum value, which determines
the black region on the observer’s sky. We determine the size of the black region as a
function of robs and Λ in the next section.
Fig. 4 shows samples of visualisation outputs of the BHC code, simulated optical
projection of a well-known galaxy M104 “Sombrero”, virtually located behind the black
hole on the optical axis. Images show typical lensing effects as the first and second Einstein
rings, first direct image, first indirect inverted image and mergence of higher order images
with the second Einstein ring around the black region. Another static images, as well as
dynamic simulations, may be downloaded from our web site [15].
9
8
Impact parameter
7
6
5
4
3
2
Pure Schwarzschild
Λ=5.10−3
Λ =10−2
1
0
0.00
1.57
3.14
4.71
6.28
7.85
∆φ
3.14
9.42
11.00
12.57
3.14
Free-falling observer
Static observer
2.36
Directional angle
Directional angle
2.36
1.57
0.79
1.57
0.79
Pure Schwarzschild
Λ=5.10−3
−2
Λ =10
0.00
0.00
1.57
3.14
4.71
6.28
∆φ
7.85
9.42
11.00
Pure Schwarzschild
Λ=5.10−3
−2
Λ =10
12.57
0.00
0.00
1.57
3.14
4.71
6.28
∆φ
7.85
9.42
11.00
12.57
Fig. 1 Optical projection for observers located at the static region above the circular
photon orbit at robs = 6M . Top panel: Impact parameter b as a function of ∆φ.
Bottom panels: Directional angles for static and radially free-falling observers.
P. Bakala et al. / Central European Journal of Physics
6
Impact parameter
5
4
3
2
Pure Schwarzschild
−3
Λ=5.10
Λ =10−2
1
0
0.00
1.57
3.14
4.71
6.28
∆φ
7.85
9.42
11.00
12.57
3.14
Free-falling observer
Directional angle
2.36
1.57
Static observer
0.79
Pure Schwarzschild
Λ=5.10−3
Λ =10−2
0.00
0.00
1.57
3.14
4.71
6.28
∆φ
7.85
9.42
11.00
12.57
Fig. 2 Optical projection for observers located in the static region under the circular
photon orbit at robs = 2.4M . Top panel: Impact parameter b as a function of ∆φ.
Bottom panel: Directional angles for static and radially free-falling observers.
5
Apparent angular size of the black hole
The apparent angular size S of the black hole can be naturally defined as the observed
angular size of the circular black region on the observer sky, in which no images of distant
objects can exist, and only radiation originated under the circular photon orbit can be
observed [2, 13, 14]. For observers located above the circular photon orbit the boundary
of the black region corresponds to outgoing geodesics with b approaching bc from above,
while for observers located under the circular photon orbit the boundary corresponds to
ingoing geodesics with b approaching bc from below.
In the case of static observers we have [14]
s
b2
2
2
S = 2 arccos A(robs , y; b),
A(r, y; b) ≡ ± 1 − 2 1 − − yr .
(13)
r
r
P. Bakala et al. / Central European Journal of Physics
6
Impact parameter
5
4
3
2
Pure Schwarzschild
−3
Λ=5.10
Λ =10−2
1
0
0.00
1.57
3.14
4.71
6.28
∆φ
7.85
9.42
11.00
12.57
3.14
Directional angle
2.36
Free-falling observer
1.57
0.79
Pure Schwarzschild
Λ=5.10−3
Λ =10−2
0.00
0.00
1.57
3.14
4.71
6.28
∆φ
7.85
9.42
11.00
12.57
Fig. 3 Optical projection for observers located under the black hole horizon at robs =
0.7M . Top panel: Impact parameter b as a function of ∆φ. Bottom panels: Directional angle for a radially free-falling observer.
Here + (−) sign corresponds to observers located above (under) the circular photon orbit. Above the circular photon orbit increasing Λ causes downsizing of the black region,
whereas under the circular photon orbit the black region grows with increasing Λ. In the
limit case of observers located just on the circular photon orbit, S is independent of Λ.
It is invariably π, i.e., the black region always occupies just one half of the observer sky.
In the case of observers radially free-falling from the static radius [12–14], the apparent
angular size of the black hole reads [14]
where
p
Z(robs , y) + 1 − 3y 1/3 A(robs , y; b)
S = 2 arccos p
,
1 − 3y 1/3 + Z(robs , y)A(robs , y; b)
Z(r, y) ≡
r
2
+ yr 2 − 3y 1/3 .
r
(14)
(15)
P. Bakala et al. / Central European Journal of Physics
Fig. 4 Simulated appearance of M104 “Sombrero” located behind the black hole. Left
panel: for a radially free-falling observer at robs = 20M in a pure Schwarzschild case
(with nondistorted image in the right-bottom corner). Right panel: for a static observer
at robs = 5M with Λ = 10−3 .
Fig. 5 Apparent angular size of the black hole as function of observer’s radial coordinate.
Left panel: static observers. Right panel: radially free-falling observers.
The Λ dependency is qualitatively different. For radially free-falling observers S grows
P. Bakala et al. / Central European Journal of Physics
with increasing cosmological constant at all values of the radial coordinate except the
central singularity, where S is invariably π, similarly to the case of static observers located
on the circular photon orbit. Consequently, the radially free-falling observer will always
observe smaller S then the static observer at the same radial coordinate.
6
Conclusions
In this paper we discussed the influence of Λ > 0 on the optical projection in strong,
spherically symmetric gravitational field. The influence depends on the value of the dimensionless cosmological parameter y. In the present universe with Λ ∼ 10−56 cm−2
values of y are y ∼ 10−40 for stellar black holes and y ∼ 10−25 for supermassive black
holes in galactic nuclei. Observable effects can be expected for y ≥ 10−15 which corresponds to supergiant black holes with masses M ≥ 1015 M [16]. In the case of primordial
black holes in the very early universe, with assumed high values of repulsive cosmological
constant, one can expect even stronger effects. Considering the electroweak phase transition at Tew ∼ 100 GeV, we obtain an estimate of the primordial effective cosmological
constant Λew ∼ 0.028 cm−2 , while considering the quark confinement at Tqc ∼ 1 GeV we
obtain Λqc ∼ 2.8 × 10−10 cm−2 and consequently higher values of y [16].
BHC code generates numerical solutions of the governing equation of the projection
and static as well dynamic visualization outputs. Results show peculiar influence of Λ
on the apparent angular size of the black hole for observers in different local frames.
This influence vanishes for static observers located at the circular photon orbit. For
future studies we plan to extend our method and the BHC code in order to study axially
symmetric spacetimes with repulsive cosmological constant.
Acknowledgement
The present work was supported by the Czech grants MSM 4781305903 and LC06014
(P. B.). One author (P. B.) would like to thank Eva Šrámková for useful discussions.
A
Appendix: Tetrads and directly measured quantities
It follows from the central symmetry of the geometry (1) that the geodetical motion of
test particles and photons is allowed in the central planes only. The existence of Killing
vector fields ξ(t) and ξ(φ) of the SdS spacetime implies the existence of two constants of
motion
pt = gtµ pµ = −E,
pφ = gφµ pµ = Φ,
(A.1)
and the photon motion is determined by the impact parameter
b≡
Φ
.
E
(A.2)
P. Bakala et al. / Central European Journal of Physics
The 4-momentum of the photon reads [13]
pt = −E,
pr =
A(robs , y, b)
E,
B 2 (robs , y)
pφ = bE = Φ,
(A.3)
where we introduce new variables
2
B 2 (r, y) ≡ 1 − − yr 2 ,
r
A(r, y, b) = ±
r
1 − B 2 (r, y)
b2
.
r2
(A.4)
The + sign corresponds to photons receding from the black hole, while − sign corresponds
to photons infalling into the black hole.
In order to calculate directly measured quantities, one has to transform the 4-momentum of the photon into the local frame of the observer. The local components of 4momentum for the observer at given robs can be obtained using the appropriate tetrad of
(α)
(α)
base 4-vectors eµ , 1-forms ωµ and transformation formulas
µ
ω (α) = e(α)
µ dx ,
µ
p(α) = e(α)
µ p .
(A.5)
A.1 Static observers
The static observers located at rest at r = const, θ = const, φ = const are endowed by a
local frame with an orthonormal tetrad of 1-forms [13]
ω (t) = B(r, y) dt,
ω (r) =
1
dr,
B(r, y)
ω (θ) = r dθ,
ω (φ) = r sin θ dφ.
(A.6)
The local components of 4-momentum of the photon moving in the equatorial plane are
given by the relations [13]
(t)
pobs =
E
B(robs , y)
,
(r)
pobs =
A(robs , y; b)
E,
B(robs , y)
(φ)
pobs =
lE
Φ
=
.
r
robs
(A.7)
Using general formulas (5), the directional angle and the frequency shift are given as [13]
cos αstat = −A(robs , y, b),
gstat =
B(rsource , y)
.
B(robs , y)
(A.8)
A.2 Observers radially free-falling from the static radius
Local components and tetrads for free-falling observers can be obtained using Lorentz
boost between the local frames of the static observer and a moving one at given r obs . The
orthonormal tetrad of 1-forms of appropriate local frame has the form [13]
p
ω (t̃) = 1 − 3y 1/3 dt + Z(r, y)B −2 (r, y) dr,
(A.9)
p
ω (r̃) = Z(r, y) dt + 1 − 3y 1/3 B −2 (r, y) dr,
(A.10)
ω (θ̃) = r dθ,
(A.11)
ω (φ̃) = r sin θ dφ,
(A.12)
P. Bakala et al. / Central European Journal of Physics
where we introduced a new variable
Z(r, y) ≡
r
2
+ yr 2 − 3y 1/3 .
r
(A.13)
The components of 4-momentum of the photon measured by a observer radially freefalling from the static radius at a given robs are given by the relations [13]
p
E
(t̃)
1 − 3y 1/3 + Z(robs , y)A(robs , y; b) ,
(A.14)
pobs = 2
B (robs , y)
p
E
(r̃)
1/3
pobs = 2
(A.15)
Z(r, y) + 1 − 3y A(robs , y; b) ,
B (robs , y)
Φ
Eb
(φ̃)
pobs =
=
.
(A.16)
robs
robs
The directional angle and the frequency shift are given by the formulas [13]
p
1/3
Z(robs , y) + 1 − 3y A(robs , y; b)
,
cos α̃fall = − p
1/3
1 − 3y + Z(robs , y)A(robs , y; b)
(t̃)
g̃fall ≡
References
pobs
(t)
psource
=
B(rsource , y)
p
.
Z(robs , y) cos α̃ + 1 − 3y 1/3
(A.17)
(A.18)
[1] A. Einstein: “Lens-like action of a star by the deviation of light in the gravitational
field”, Science, Vol. 84, (1936), pp. 506–507.
[2] C.T. Cunningham: “Optical Appearance of Distant Observers near and inside a
Schwarzschild Black Hole”, Phys. Rev. D, Vol. 12, (1975), pp. 323–328.
[3] W. Benger: “Simulation of a Black Hole by Raytracing”, In: Relativity and Scientific Computing: Computer Algebra, Numerics, Visualization, Springer-Verlag Telos,
1996.
[4] A. J. S. Hamilton: “Black Hole Flight Simulator”, Bulletin of the American Astronomical Society, Vol. 36, (2004), pp. 810.
[5] D. Kobras, D. Weiskopf and H. Ruder: “Image-based rendering and general relativity”, In: WSCG 2001 Conference Proceedings, University of West Bohemia, Pilsen,
2001, pp. 130–137.
[6] R.J. Nemiroff: “Visual distortion near a neutron star a and black hole”, Am. J. Phys.,
Vol. 61, (1993), pp. 619–631.
[7] H.P. Nollert, H. Ruder, H. Herold and U. Kraus: “The relativistic looks of a neutron
star”, Astron. Astrophys., Vol. 208, (1989), pp. 153—156.
[8] H.C. Ohanian: “The black hole as a gravitational lens”, Am. J. Phys., Vol. 55, (1987),
pp. 428–432.
[9] S.U. Viergutz: “Image generation in Kerr geometry. I. Analytical investigations on
the stationary emitter-observer problem”, Astron. Astrophys., Vol. 272, (1993), pp.
355–377.
P. Bakala et al. / Central European Journal of Physics
[10] L.M. Krauss and M.S. Turner: “The Cosmological constant is back”, Gen. Relat.
Gravit., Vol. 27, (1995), pp. 1137–1144.
[11] J.P. Ostriker and P.J. Steinhardt: “The Observational case for a low density universe
with a nonzero cosmological constant”, Nature, Vol. 377, (1995), pp. 600–602.
[12] Z. Stuchlı́k and. S. Hledı́k: “Some properties of the Schwarzschild–de Sitter and
Schwarzschild–anti–de Sitter spacetimes”, Phys. Rev. D, Vol. 60, (1999), art. 044006.
[13] Z. Stuchlı́k and K. Plšková: “Optical apperance of isotropically radiating sphere
in the Schwarzschild–de Sitter spacetime”, In: Proceedings of RAGtime 4/5, eds.
S. Hledı́k and Z. Stuchlı́k, Silesian University in Opava, Opava, 2004, pp. 167–185.
[14] P. Bakala, P. Čermák, S. Hledı́k, Z. Stuchlı́k and K. Truparová Plšková: “A virtual
trip to the Schwarzschild–de Sitter black hole”, In: Proceedings of RAGtime 6/7,
eds. S. Hledı́k and Z. Stuchlı́k, Silesian University in Opava, Opava, 2005, pp. 11–28.
[15] “Relativistic and particle physics and its astrophysical aplications, Czech research
project MSM 4781305903”, http://www.physics.cz/research/.
[16] Z. Stuchlı́k, P. Slaný and S. Hledı́k: “Equilibrium configurations of perfect fluid
orbiting Schwarzschild—de Sitter black holes”, Astron. Astrophys., Vol. 363, (2000),
pp. 425–439.
Přı́loha 2
ACTA ASTRONOMICA
Vol. 58 (2008) pp. 15–21
Distribution of Kilohertz QPO Frequencies and Their Ratios in the
Atoll Source 4U 1636–53
G. T ö r ö k 1 , M. A. A b r a m o w i c z 1,2,3 , P. B a k a l a 1 , M. B u r s a 4 ,
J. H o r á k 4 , W. K l u ź n i a k 3,5 , P. R e b u s c o 6,7 and Z. S t u c h l í k 1
1
Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava,
Bezručovo nám. 13, 746-01 Opava, Czech Republic
e-mail: [email protected], (pavel.bakala,zdenek.stuchlik)@fpf.slu.cz
2 Department of Physics, Göteborg University, S-412 96 Göteborg, Sweden
e-mail: [email protected]
3 Copernicus Astronomical Centre PAN, Bartycka 18, 00-716 Warsaw, Poland
e-mail: [email protected]
4 Astronomical Institute of the Academy of Sciences, Boční II 1401/1a, 141-31 Praha 4,
Czech Republic
e-mail: (bursa,horak)@astro.cas.cz
5 Johannes Kepler Institute of Astronomy, Zielona Gora University, ul. Lubuska 2,
65-265 Zielona Góra, Poland
6 MIT Kavli Institute for Astrophysics and Space Research, 77 Massachusetts Avenue,
37, Cambridge, MA 02139, USA
e-mail: [email protected]
7 Max-Planck-Institute for Astrophysics, Karl-Schwarzschild-Str. 1, D-85741 Garching,
Germany
Received February 28, 2008
ABSTRACT
A recently published study on long term evolution of the frequencies of the kilohertz quasiperiodic oscillations (QPOs) in the atoll source 4U 1636–53 concluded that there is no preferred
frequency ratio in a distribution of twin QPOs that was inferred from the distribution of a single
frequency alone. However, we find that the distribution of the ratio of actually observed pairs of kHz
QPO frequencies is peaked close to the 3/2 value, and possibly also close to the 5/4 ratio. To resolve
the apparent contradiction between the two studies, we examine in detail the frequency distributions
of the lower kHz QPO and the upper kHz QPO detected in our data set. We demonstrate that for
each of the two kHz QPOs (the lower or the upper), the frequency distribution fina all detections o
QPO differs from the distribution of frequency of the same QPO in the subset of observations where
both the kHz QPOs are detected. We conclude that detections of individual QPOs alone should not
be used for calculation of the distribution of the frequency ratios.
Key words: X-rays: binaries – Stars: neutron – Accretion, accretion disks
16
A. A.
1. Introduction
Kluźniak and Abramowicz (2001) suggested that the kHz twin peak QPOs,
observed in the Fourier power spectra (PDS) from accreting neutron stars, originate
in a non-linear resonance that is possible only in strong gravity. It was reported
later (Abramowicz et al. 2003) that the ratio νL /νU of the upper and lower QPO
frequency in neutron stars usually clusters close to the rational ratio 2/3, with some
frequency pairs possibly clustering close to other ratios, such as 0.78.
Belloni et al. (2005) re-examined the study of Abramowicz et al. (2003) for a
larger set of detections of a single kHz QPO and, on the assumption of a correlation
between the observed QPO and the unobserved second QPO, confirmed that their
(inverse) frequency ratio νU /νL would cluster most often close to the 3/2 value and
less often close to other rational numbers (e.g., 5/4 and 4/3). Because a distribution
of the ratios of two correlated quantities is largely determined by the distribution
of either one of them, Belloni et al. (2005) argued that the peaks in the distribution
of kHz frequency ratios reported by Abramowicz et al. (2003) reflect peaks of
unknown origin in the distribution of a single (upper or lower) kHz QPO. Further,
they argued that such clustering does not provide any useful information about a
possible underlying physical mechanism.
A more recent study of Belloni et al. (2007) is based on a long term evolution
of the QPO frequencies in the atoll source 4U 1636–53 over an eighteen month
period and on the results of their previous research. The authors now conclude that
in fact there are no peaks in the frequency distribution of the lower kHz QPO in
this source. In keeping with their previous argument, they conclude that there are
no peaks in the frequency ratio distribution either.
While Abramowicz et al. (2003) examine the frequency ratios in pairs of observed frequencies, both of the cited papers of Belloni et al. (2005, 2007) study
primarily distributions of the frequencies and focus mainly on the lower QPO. Note
that in most observations of Belloni et al. (2007) only single QPO frequencies have
been detected.
2. Ratio vs. Frequency Distribution in the Atoll Source 4U 1636–53
Using 4U 1636–53 data from the analysis of Barret et al. (2005b), we prepare
a histogram of detections of the lower QPO over nine years (from 1996 until 2005)
of monitoring by RXTE (Fig. 1, left), as well as of the upper QPO (Fig. 1, right).
We restrict our study to frequencies > 400 Hz for the lower QPO, and > 800 Hz
for the upper QPO. The inaccuracies caused by the long-term frequency drift inside
of continuous data segments are not important for the purposes of our paper.
The data of Barret et al. (2005b) have been obtained through a shift-add procedure carried out on individual continuous segments of observation. In this approach
each continuous data segment (corresponding to a few tens minutes of an effective
Vol. 58
17
subset from 1.5 hour RXTE orbital period) is divided into N intervals, and searched
for a QPO. The shortest usable integration time is estimated such that the QPO is
detected above a certain significance in at least 80% of the N intervals; a linear
interpolation is used to estimate the QPO frequency in the remaining intervals. The
N PDS are frequency-shifted to the mean QPO frequency over the segment and
averaged (Méndez et al. 1998). The resulting averaged PDS representing the complete continuous segment is then fitted with one or two Lorentzians plus a constant
corresponding to the counting-statistics noise level (Barret et al. 2005b).
Fig. 1. Left: Frequency histogram for the lower QPO. The subset of lower QPOs detected simultaneously with the upper QPO is denoted by darker bars, which are labeled “lower in twin”. Right:
Analogous histograms for the upper QPO.
When only one significant peak is detected, the QPO is identified as upper or
lower from the parameters of the Lorentzian and we refer to such peaks as single
QPOs. We stress that the value of the QPO frequency itself is not used to distinguish
between the upper and lower QPOs – in principle, a QPO of a given frequency could
be either the upper or the lower QPO. For instance, the quality factor for the lower
kHz QPO is a well determined function of the frequency, and a different function
of frequency for the upper QPO, and we can use this and other relationships to
identify a QPO of given frequency (see Barret et al. 2005a,b,c, 2006, for details).
This way of QPO identification differs from the method based on hardness diagram
applied in Belloni et al. (2005, 2007). The two methods have a different range of
applicability but give comparable results (see e.g., Barret et al. 2005b,c). Of course,
when two significant kHz QPOs are detected, the upper QPO is the one with the
larger frequency, by definition.
Consequently, the frequency values are averaged through intervals of predetermined length ≈ 2000 s. Belloni et al. (2005, 2007) analyzed segments of different
lengths, resulting in a larger number of detections. Hence the histograms we use
here are not comparable in details to those of Belloni et al. (2005), even for the
same RXTE data.
We take into account only the detections of oscillations with quality factor (defined as the QPO centroid frequency over the full-width of the peak at its halfmaximum) Q ≥ 2 and significance (defined as the integral of the Lorentzian fitting
the peak in PDS divided by its error) S ≥ 3.
18
A. A.
2.1.
Different Distributions
Each of the histograms in the Fig. 1 clearly reveals an accumulation of frequencies in the ν ≈ 900 Hz vicinity of the power spectrum. However, for the lower
kHz QPO this range (νL ≈ 900 Hz) is in the high-frequency part of the frequency
distribution of this QPO, while most detections of νU are accumulated in the lowfrequency part of its own range of variation.
If one is interested (for whatever reason) in the distribution of the frequency
ratio νU /νL , then those observations in which both QPO peaks are simultaneously
detected should be considered. Accordingly, we apply our selection criteria to simultaneous significant detections of both QPO frequencies as well. The histograms
of the upper QPO frequency in this sample (darker bars in Fig. 1, right panel) are
strikingly different from the previous histogram of significant detections of the upper QPO (bars of lighter shade in Fig. 1). A new cluster of frequencies appears, in
the range ≈ 1100 Hz to ≈ 1200 Hz, at the expense of frequencies below 900 Hz,
whose occurrence is greatly diminished.
While there is a positive correlation between the QPO frequencies (e.g., Abramowicz et al. 2005, see also Belloni et al. 2005, Zhang et al. 2006),
νU ≈ 0.7νL + 520 Hz
(1)
very clearly the examined data do not support the assumption of Belloni et al.
(2005, 2007) that the distribution of the ratio of two linearly correlated frequencies
is determined by the distribution of one of the frequencies even when the second
frequency remains undetected. There is apparently no direct link between the histogram of all the lower QPO detections (Fig. 1, left, lighter) and the histogram of
the same QPO taken from the subset of twin peak QPO detections (Fig. 1, left,
darker). This result should have an impact on the theory of QPOs. Although a
full discussion is beyond the scope of this paper, we note that the change in the frequency distribution when a second QPO is detected may be suggestive of a physical
mechanism, such as mode-coupling.
To quantify this effect we plot the cumulative distributions of the lower and of
the upper QPO, which are shown in Fig. 2. Using the Kolmogorov-Smirnov (KS) test we compare the frequency distributions of each (the upper and the lower)
QPO measured in all detections, with those measured for the same QPO when
both the upper and the lower QPO are detected. We obtained the K-S probabilities
pL,K−S = 2.35 × 10−5 and pU,K−S = 2.24 × 10−3 in the case of the lower and
upper QPO respectively. Indeed, the two distributions are different in both cases.
We directly conclude that detections of individual QPOs alone cannot be used for
calculation of the distribution of the frequency ratios.
It is interesting to note that the single upper QPOs are mostly detected at relatively low frequencies, while the single lower QPOs are detected at relatively high
frequencies. Taking into account the linear correlation among QPO frequencies,
the distributions of single lower and upper QPOs appear to be complementary, in
Vol. 58
19
Fig. 2. The cumulative distributions for the kHz QPOs corresponding to Fig. 1. Left: Solid curves
correspond to the detected lower QPOs (see left panel of Fig. 1), the dotted line labeled “inferred
lower” indicates the lower QPO frequency calculated from Eq. (1) using all detections of the upper
QPO. The dashed vertical line shows the greatest difference Dmax = 0.515 between the distributions
of “all lower” and “lower in twin”. Right: Analogous lines for the upper QPO. The dashed vertical
line on right panel corresponds to the maximal difference Dmax = 0.4364 between the “all upper”
and “upper in twin” distributions.
the following sense. The lowest-frequency detection of the single lower QPO is at
νL = 651 Hz, which in the linear correlation corresponds to νU = 976 Hz while
the highest-frequency detection of the single upper QPO is at νU = 961 Hz, which
corresponds to νL = 628 Hz. In other words, if one assumed that each of the single upper QPOs is accompanied by a lower QPO of frequency determined from
Eq. (1), the resulting points would all fall to the left of the 3:2 line in Fig. 4 (left
panel), and if the same procedure were applied to the single lower QPOs, the resulting points would fall to the right of the 3:2 line in Fig. 4. This is illustrated in Fig. 3.
Given this fact, it is not surprising that the distribution of actually detected upper
(or lower) kHz QPOs is completely different from the distribution that would be
predicted on Eq. (1) from the distribution of the other kHz QPO, when detections
of single QPOs dominate the data set (Fig. 2).
Fig. 3. Distribution of single QPOs. Frequency axes are aligned according to the correlation of
Eq. (1). The shadow denotes a 50 Hz scatter about the lower QPO frequency of 650 Hz, corresponding
to a 3:2 ratio.
20
A. A.
Fig. 4. Left: The frequencies of detected twin QPOs. The inset shows a corresponding histogram
of the frequency ratio. Right: Cumulative distribution of the frequency ratios, the thick solid line
denotes the best fit by a sum of two Lorentzians. Fit by a single Lorenzian is marked by dotted line.
2.2.
Possible Peaks in the Ratio Distribution
The left panel of Fig. 4 depicts the mutual dependence of frequencies of the
lower and upper QPO when both were significantly detected. It also displays a
corresponding histogram of the frequency ratio. As for Sco X-1 (Abramowicz et
al. 2003), this histogram is peaked close to the 3/2 value, and is suggestive of the
existence of a second peak.
We have fitted the distribution of the frequency ratios by the sum of two suitably
normalized Lorentzians,
p2 (r) = f
λ2 /π
λ1 /π
+ (1 − f )
(r − r1 )2 + λ21
(r − r2 )2 + λ22
(2)
where r = νU /νL is the frequency ratio and r1 , r2 , λ1 , λ2 and f are free parameters. Their values obtained by the maximum likelihood method are r1 = 1.52,
r2 = 1.28, λ1 = 0.0327, λ2 = 0.0913 and f = 0.722, reaching K-S probability p2,K−S = 0.918. The best fit by a single Lorentzian, with r0 = 1.50 and
λ0 = 0.0597 (dotted line in Fig. 4) gives the K-S probability p1,K−S = 0.340. Both
fits are acceptable. In the right panel of Fig. 4 we compare cumulative distributions
of the observed frequencies with both double and single Lorentzians.
3. Conclusions
We have demonstrated for a set of uniform data (Barret et al. 2005b) that the
frequency distribution of a single kHz QPO is not equivalent to the distribution of
the corresponding frequency when a pair of kHz QPOs have been detected.
We stress that if there is a one-to-one correspondence between the frequencies
and their ratio, as is the case for linear functions with a non-vanishing intercept,
the question whether to consider the QPO frequency distribution or the ratio distribution as fundamental is one of theoretical assumptions, as the two distributions
Vol. 58
21
are mathematically equivalent. However, the distribution of a single kHz QPO frequency is not predictive of the distribution of two frequencies detected simultaneously, nor of the distribution of their ratio, even if these frequencies are correlated
when both are actually detected. Thus, the study of Belloni et al. (2007), who conclude that “there is no preferred frequency or frequency ratio in 4U 1636–53” is
based on an invalid assumption, and cannot be accepted as applying to the distribution of ratios, as long as it is based on the detection of a single frequency.
The finding that the frequency distribution of a QPO depends on whether or not
a second QPO can be detected as well should restrict models of the physical origin
of the QPO and of X-ray flux modulation, regardless of whether or not the value of
the frequency ratio is clustered about the specific value of 3/2.
Acknowledgements. We thank Didier Barret for providing the data and software on which this paper builds and for several discussions. We have also benefited
from helpful comments by Tomek Bulik. We thank the referee for very useful suggestions. The authors are supported by the Czech grants MSM 4781305903 and
LC06014, by the Polish grants KBN N203 009 31/1466 and 1P03D 005 30.
REFERENCES
Abramowicz, M.A., Bulik, T., Bursa, M., and Kluźniak, W. 2003, A&A, 404, L21.
Abramowicz, M.A., Barret, D., Bursa, M., Horák, J., Kluźniak, W., Rebusco, P., and Török, G. 2005,
in: Proceedings of RAGtime 6/7, Eds. S. Hledík, Z. Stuchlík, Opava.
Barret, D., Kluzniak, W., Olive, J.F., Paltani, S., and Skinner, G.K. 2005a, MNRAS, 357, 1288.
Barret, D., Olive, J.F., and Miller, M.C. 2005b, MNRAS, 361, 855.
Barret, D., Olive, J.F., and Miller, M.C. 2005c, Astron. Nachr., 326, 808.
Barret, D., Olive, J.F., and Miller, M.C. 2006, MNRAS, 370, 1140.
Belloni, T., Méndez, M., and Homan, J. 2005, A&A, 437, 209.
Belloni, T., Méndez, M., and Homan, J. 2007, MNRAS, 379, 247.
Kluźniak, W., and Abramowicz, M.A. 2000, preprint; astro-ph/0105057.
Kluźniak, W., and Abramowicz, M.A. 2001, Acta Physica Polonica B, 32, 3605.
Méndez, M., van der Klis, M., Wijnands, R., Ford, E.C., van Paradijs, J., and Vaughan, B.A. 1998,
ApJ, 505, L23.
Zhang, C.M., Yin H.X., Zhao, Y.H., Song, L.M., and Zhang, F. 2006, MNRAS, 366, 1373.
Přı́loha 3
ACTA ASTRONOMICA
Vol. 58 (2008) pp. 113–119
On the Origin of Clustering of Frequency Ratios in the Atoll Source
4U 1636–53
G. T ö r ö k 1 , M. A. A b r a m o w i c z 1,2,3 , P. B a k a l a 1 ,
M. B u r s a 4 , J. H o r á k 4 , P. R e b u s c o 5 and Z. S t u c h l í k 1
1
Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava,
Bezručovo nám. 13, 746-01 Opava, Czech Republic
e-mail:[email protected] (pavel.bakala, zdenek.stuchlik)@fpf.slu.cz
2 Department of Physics, Göteborg University, S-412 96 Göteborg, Sweden
e-mail: [email protected]
3 Copernicus Astronomical Centre PAN, Bartycka 18, 00-716 Warsaw, Poland
4 Astronomical Institute of the Academy of Sciences, Boční II 1401/1a, 141-31 Praha 4,
Czech Republic
e-mail: (bursa, horak)@astro.cas.cz
5 MIT Kavli Institute for Astrophysics and Space Research, 77 Massachusetts Avenue,
37, Cambridge, MA 02139, USA
e-mail: [email protected]
Received February 28, 2008
ABSTRACT
A long discussion has been devoted to the issue of clustering of the kHz quasi periodic oscillation
(QPO) frequency ratios in neutron star sources. While the distribution of ratios inferred from an
occurrence of a single QPO seems to be consistent with a random walk, the distribution based on
simultaneous detections of both peaks indicates a preference of ratios of small integers. Based on the
public RXTE data we further investigate this issue for the source 4U 1636–53. Quality factors and
rms amplitudes of both the QPOs nearly equal at the points where the frequencies are commensurable,
and where the twin QPO detections cluster. We discuss a connection of the clustering with the varying
properties of the two QPO modes. Assuming approximate relations for the observed correlations of
the QPO properties, we attempt to reproduce the frequency and ratio distributions using a simple
model of a random-walk evolution along the observed frequency-frequency correlation. We obtain
results which are in qualitative agreement with the observed distributions.
Key words: X-rays: binaries – Stars: neutron – Accretion, accretion disks
1. Introduction
Since the paper of Abramowicz et al. (2003), the issue of distribution of kHz
quasi periodic oscillations (QPOs) in neutron-star low mass X-ray binaries has been
114
A. A.
discussed extensively. In their work, Abramowicz et al. (2003) examined simultaneous detections of the upper and lower QPOs in the Z-source Sco X-1. The authors
show that the ratios of the lower and upper QPO frequencies cluster are most often
close to the value νL /νU = 2/3. They also find an evidence for the second peak in
a distribution of frequency ratios at νL /νU ≈ 0.78. This value is remarkably close
to another ratio of small integers, 4/5 = 0.8. In the most recent paper, Török et
al. (2008) examined occurrences of the twin QPOs in the atoll source 4U 1636–53
applying the same methodology as Abramowicz et al. (2003). They find that the
distribution of the (inverse) frequency ratios νU /νL of two simultaneously detected
QPOs peaks near 3/2 and 5/4.
A preference of the commensurable frequency ratios in kHz QPO data of various sources has been systematically checked by a group of Belloni and his collaborators. Belloni et al. (2005) have re-examined the ratio distribution in Sco X-1
and later also in a larger sample comprising four atoll sources including 4U 1636–
53 (Belloni et al. 2005). They argue that such clustering does not provide any
useful information because frequencies of the two QPOs are correlated and the distribution of the ratio of two correlated quantities is completely determined by the
distribution of one of them. Keeping this argument, a recent study of Belloni et al.
(2007) based on a systematic long term observation of 4U 1636–53 concludes that
there is no preferred frequency ratio.
The apparent disagreement in conclusions of the two groups comes from a confusion between the observed frequency distribution (the one which can be recovered from observed data) and the intrinsic distribution (the “invisible” one really
produced by the source). While Abramowicz et al. (2003) and Török et al. (2008)
examined frequency ratios of the actually observed QPO pairs (twin peaks) only,
the analysis of Belloni et al. (2005, 2007) studies primarily distributions of frequencies of a single QPO and makes implications for the distribution of the other,
often invisible, QPO from the empirical correlation between frequencies.
In this paper we argue (and illustrate) that the observed distributions are affected by the way the signal from a source is being detected and analyzed. We
show that in 4U 1636–53 the observed clustering can be understood in terms of
rms amplitude and quality factor correlations with QPO frequency. Taking these
correlations into account, we simulate the ratio distribution using a random walk
model of QPO frequency evolution and we find that results of the simulation agree
with empirical data.
2. Properties of Oscillation Modes on Large Frequency Range
In the process of data reduction and searching for QPOs, an important quantity
is the significance S of the peak in PDS, which measures the peak prominence.
Shape of a peak in the PDS is most often fitted by a Lorentzian. Usually, S ≥ 2 − 4
is being used as the low threshold limit for detections and only peaks that have
Vol. 58
115
their significances greater than this limit are considered as QPOs. Thus, this imposes a certain selection criterion which could consequently affect the distribution
of detections.
The significance S is given by the relation between the integral area of a Lorentzian in PDS and its error. For a particular detection, it depends on observational conditions, on the quality factor Q of the peak (defined as the QPO centroid frequency
over the peak full-width at its half-maximum) and on the fractional root-meansquared amplitude r (a measure for the signal amplitude given as a fraction of the
total source flux that is proportional to the root mean
p square of the peak power
contribution to the√total power spectrum), S = kr2 Q/ν, where the time-varying
factor k(t) = I(t) T depends on the total length of observation T and the instantaneous source intensity I , which at a given time is the same for both upper and
lower peak. The standard process of the QPO determination is in detail described
in van der Klis (1989).
Barret et al. (2005a,b,c, 2006) have shown that both quality factors and rms
amplitudes are determined by frequency and moreover that their profiles greatly
differ between lower and upper QPO modes. The quality factor of the upper QPO
is usually small and tends to stay at an almost constant level around QU ≈ 10. In
contrast, the lower QPO quality factor improves with frequency and can reach up
to QL ≈ 200 before a sharp drop of coherence at high frequencies. Amplitudes of
upper QPOs generally decrease with frequency, while the lower QPO amplitudes
show first an increase and then they start to decay too.
Fig. 1. The quality factor (left), rms amplitude (middle) and inferred significance (right) behavior in
atoll source 4U 1636–53. Grey points represent lower QPO data, black points are for upper QPO
data. Data in first two panels come from the study of Barret et al. (2005b) and cover large range
of frequencies available via shift-add method through all segments of RXTE observations. Continuous curves are obtained from interpolation by several exponentials (see e.g., Török 2007). The
prospected course of the QPO significance
in the right panel is determined by the rms amplitude
p
and quality factor profiles ( S ∝ r2 Q/ν ). Frequency axes are related using frequency correlation
( νU = 0.701νL + 520 Hz, Abramowicz et al. 2005).
Fig. 1 shows the behavior of amplitudes and quality factors of individual QPO
modes in 4U 1636–53 and how they change with frequencies. In Fig. 1 we use a
correlation νU = 0.701νL + 520 Hz from Abramowicz et al. 2005 (see also Belloni
et al. 2005, Zhang et al. 2006). The displayed data of Barret et al. (2005b) cover
116
A. A.
large frequency range available through the shift-add technique over all RXTE observations (see Méndez et al. 1998, 1999, Barret et al. 2005a,b,c for details).
Note that both of the two properties are becoming similar as the frequency approaches points corresponding to 3/2 or 5/4 ratio (the equality of amplitudes have
been reported by Török 2007). In the right panel of Fig. 1 we then plot the significances of the two oscillation modes, inferred from the combination of the two plots,
while we keep the intensity I and observing time t constant (for simplicity). It is
clearly visible that there is a similar equality of QPO significances close to points,
where the frequencies are close to the 3/2 or 5/4 ratio (as a result of comparable
Q and r at those points), while they are much different elsewhere. We will hereafter call the points of equal significances as the “3/2” and “5/4” points. We may
also observe that the upper QPO mode is usually strong (much more significant)
left from the 3/2 point (at lower frequencies), while right from 3/2 the lower QPO
mode dominates.
3. Clustering of Frequency Ratios
It is likely that if QPOs are produced in a source, they are always produced in
pairs. Because the strength of oscillations is usually around the sensitivity threshold
of measurements, often only one (the stronger) QPO is detected. Around the special
points 3/2 and 5/4, where significances are comparable, there is a good chance that
if one mode can be detected the other could be detected as well, because both peaks
have nearly the same properties. Indeed, this agrees with what is observed and has
been labored or challenged many times (Abramowicz et al. 2005, Belloni et al.
2005, Bulik 2005, Yin and Zhao 2007, Belloni et al. 2007) that pairs of QPOs
cluster close to the 3/2 and some other small rational number ratios.
From time to time, the conditions at the source become such that both QPOs
can be detected simultaneously regardless of their frequency, only because of their
actual high brightness (as the observational sensitivity is relatively low). These
events allows us not only to see QPO pairs close to the critical points, but sporadically also all the way along the frequency-frequency correlation line, even far from
3/2.
The clustering of frequency ratios close to 3/2 is in this view significantly affected by the behavior of rms amplitudes and quality factors and namely by the fact
that these quantities become equal close to that frequency ratio. This is demonstrated in Fig. 2 (left), where we show a fraction of number of observations, in
which both QPOs have been detected simultaneously, to a number of those, in
which at least one QPO has been detected. Fig. 2 is based on data used in Török et
al. (2008). Clearly, the positions of maxima remarkably well correlate with points,
where the two significances equal. Moreover, these positions coincide with peaks
in the distribution of frequency ratios found in Török et al. (2008) which justify a
hypothesis that there is a link between QPO properties and the ratio clustering.
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117
Fig. 2. The distribution of observed frequency ratios. Left: The fraction of the number of observations
with simultaneous detections nUL to the number of observations in which at least one QPO has
been detected nUL + nU + nL (where nU and nL are respectively the numbers of observations with
detections of the upper or lower QPO only). Middle: Simulated ratio distribution assuming a randomwalk in frequency and variable count rate (see text). The gray underlying histogram in the first two
panels shows the actual observed ratio distribution of twin QPO peaks (the data in both panels are
those discussed in Török et al. 2008). Right: The individual distributions of lower and upper QPO
frequencies from the random walk simulation. Black-shaded portions of bars represent simultaneous
occurrences of both modes (twin QPOs) as shown in the middle panel.
4. Random Walk Distribution Model
As previously noticed in several works and further suggested by Belloni et al.
(2005), the observed time evolution of QPO frequency appears consistent with a
series of random walks.
This has been later critised by Bulik (2005) who pointed out that contrary to
the distributions of QPOs that appear qualitatively similar at different times, distributions arising from random walk differ significantly among different realizations
(with different seeds). Nevertheless, using a simple model of random walk we
attempt to at least roughly reproduce the frequency ratio distribution.
Starting with νL = 700 Hz, we model a long-term evolution of QPO frequencies over 10 000 consequent segments. Each segment consists of 50 steps, where
one step is assumed to represent 32 seconds of a real observation. An independent
random variation of ±2 Hz in νL is assigned to each step. This setup roughly
corresponds to the documented frequency drifting through 32 sec integration intervals (Barret et al. 2004, Paltani et al. 2004) and each segment then mimics 1.6
kiloseconds of QPO evolution.
The QPO frequencies are averaged over each segment, and a linear correlation
(νU = 0.701νL + 520 Hz, Abramowicz et al. 2005) between νL and νU is considered, so that finally we obtain 10 000 frequency pairs. To start with, we assume
constant observational conditions (i.e., count rate and observing time), adjusting
k = 1. For each point we calculate its significance based on observed profiles of Q
and rms, which are based merely on datapoints corresponding to twin peak QPO
observations. Only such points are considered in the simulation, where both upper and lower QPOs have significance above 3σ level. The resulting histogram
of frequency ratios shows strong clustering around 3/2 ratio, however, it does not
reproduce the second peak around 5/4, which indicates that the assumption of constant count rate may not be sufficient.
118
A. A.
As a second step, we adopt an additional (still very simplifying) assumption to
the simulation that count rate is varying with frequency. The motivation here comes
from a known fact that for a given source there is not a global correlation between
source luminosity and QPO frequency, but the two quantities stay correlated during
individual (temporary) observational events (so-called parallel-track phenomenon,
e.g., Méndez et al. 1999). In the case of 4U 1636–53, the maximal count rates
related to the highest observed lower QPO frequencies (up to 950 Hz) are 2–3
times higher than the highest count rates at νL ≈ 500−700 Hz (see Fig. 2 in Barret
et al. 2005b). Thus, we keep count rate constant up to νL ≈ 700 Hz and then it
is linearly increased with frequency, being about 2.5 times higher at νL ≈ 950 Hz
than at ≈ 500−700 Hz.
In the middle panel of Fig. 2, we show first the histogram of simultaneous
occurrences of both QPO modes from our simulation on the background of the
observed distribution, and in the right panel of Fig. 2 we also plot individual simulated distributions of lower and upper QPOs. Focusing on twin QPO occurrences,
we have a broad peak around 3/2 and also we obtain a more narrow peak near
5/4. While the presence of the 3/2 clustering seems to be very solid and can be
reproduced with any setup, the second 5/4 peak is more subtle feature and depends
much on assumed behavior of count rate. Apparently, in a real observation, its presence would rely on actual source conditions (and how they would change during
the observation) as well as on how the consequent analysis is done. For instance
the data examined in Belloni et al. (2007) do not exhibit QPO detections above
νU ≈ 1000 Hz while the data used in Török et al. (2008) do. Similarly, if we put
e.g., more stiff limit on significance or consider lower count rates, we would loose
the 5/4 peak.
5. Conclusions
Focused on the atoll source 4U 1636–53 we demonstrate that at frequencies,
where the both QPO modes have comparable properties, there is a high probability
of detecting both peaks of a twin pair simultaneously. We have found a precise
match comparing the observed twin QPO distribution with our simulation based
on the observed correlations between QPO frequencies and their properties. The
simulation not only reproduces the observed clustering, but it also shows the “complementarity” between upper and lower QPO distributions that has been noticed by
Török et al. (2008). This suggests that the ratio clustering may have origin in the
exchange of dominance between the two modes when one mode fades in while the
other one fades out.
Even if the intrinsic distributions of both the mode frequencies were uniform,
there would be a non-trivial profile of the observed distributions and clustering
of the twin peak detections around certain points (narrow regions) prominent due
to behavior of the QPO amplitudes and coherence times determined by the QPO
mechanism. It will require a further detailed analysis to investigate whether the
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119
above influence of the QPO properties can fully explain the ratio clustering observed in 4U 1636–53. For a further understanding of the ratio clustering mechanism (and importance) it is also highly needed to perform a similar analysis for
the other sources. For instance, a very recent study of the atoll source 4U 1820–30
(Barret and Boutelier 2008) found that a point close to the 4/3 value, where the
ratio distribution clusters in that source, and where the amplitudes and quality factors are comparable, is most likely prominent in the intrinsic distribution. (Note
also that in contrary to the case of 4U 1636–53 they reported a lack of the twin
QPO detections close to the 3/2 value, while the amplitudes and quality factors are
comparable there as well.)
Acknowledgements. We thank M. Méndez for several discussions on the subject and, especially, we are thankful to D. Barret for ideas, comments and for providing the data and software on which this paper builds. We are grateful to W. Kluźniak for several comments and suggestions. We also thank the Yukawa Institute for
Theoretical Physics at Kyoto University, where this work was initiated during the
YITP-W-07-14 on "Quasi-Periodic Oscillations and Time Variabilities of Accretion Flows". The authors are supported by the Czech grants MSM 478130590384
and LC06014, and by Polish KBN grants N203 009 31/1466 and 1P03D 005 30.
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Abramowicz, M.A., and Kluźniak, W. 2001, A&A, 374, L19.
Barret, D., Kluźniak, W., Olive, J.F., Paltani, S., and Skinner, G.K. 2004, in: Proc. of the Annual
meeting of the French Astronomical Society (SF2A).
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Barret D., Olive, J.F., and Miller, M.C. 2005b, MNRAS, 361, 855.
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Barret, D., and Boutelier, M. 2008, in: The Proceedings of Jean Piere Lasota Conference.
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Török, G. 2007, A&A.submitted
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Stuchlík, Z. 2008, Acta Astron., 58, 15.
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Yin, H.X., and Zhao, Y.H. 2007, Advances in Space Research, 40, 1522.
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Přı́loha 4
ACTA ASTRONOMICA
Vol. 58 (2008) pp. 1–14
Modeling the Twin Peak QPO Distribution in the Atoll Source
4U 1636–53
G. T ö r ö k, P. B a k a l a, Z. S t u c h l í k and P. Č e c h
Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava,
Bezručovo nám. 13, CZ-746 01 Opava, Czech Republic
e-mail: [email protected], (pavel.bakala,zdenek.stuchlik,petr.cech)@fpf.slu.cz
Received October 5, 2007
ABSTRACT
Relation between the lower and upper frequency mode of the twin peak quasi-periodic oscillations observed in the neutron star X-ray binaries is qualitatively well fitted by the frequency relation
following from the relativistic precession model. Assuming this model with no preferred radius and
the probability of an observable twin QPO excitation being uniform across the inner edge of an accretion disk we compare the expected and observed twin peak QPO distribution in the case of atoll
source 4U 1636–53. We find these two distributions highly incompatible. We argue that the observed
distribution roughly corresponds to the expected one if an additional consideration of preferred resonant orbits is included. We notice that our findings are relevant for some disk-oscillation QPO models
as well.
Key words: X-rays: binaries – Accretion, accretion disks – Stars: neutron
1. Introduction
Several models have been outlined to explain observations of the kHz quasiperiodic oscillations (QPOs) in the X-ray fluxes from neutron-star binary systems,
and it is mostly preferred that their origin is related to orbital motion near the inner
edge of an accretion disk (see van der Klis 2006, Lamb 2003, Lamb and Boutlokous
2007, for a recent review). It is often argued that relation between the lower and
upper QPO frequency mode ( νL , νU ) is qualitatively well fitted by the frequency
relation implied by the particular relativistic precession model (Stella and Vietri
1999). Sources roughly follow the relation given by the model for a central compact
object mass M ≈ 2M⊙ (Belloni, Méndez and Homan 2007a, see Zhang et al. 2006
for a discussion of other possibilities).
In this paper we examine the twin QPO distribution given by the relativistic
precession model (hereafter the RP model) and compare it with the one observed
in the case of atoll source 4U 1636–53. We also discuss a model including preferred
orbits.
2
A. A.
2. Observational Data and their Parametrization
The data we examine are taken from the studies of Barret, Olive and Miller
(2005), Abramowicz et al. (2005) and follow from the shift-add procedure through
continuous segments of observation, see Méndez et al. (1998), Barret et al. (2005)
for details. We seek over data corresponding to nine years of 4U 1636–53 monitoring by RXTE for all detected twin peak QPOs, i.e., for simultaneous detections of
the lower and upper kHz QPO oscillations.
Note that we choose the twin peak QPO occurrences as there is no apparent
link between distributions of the individual QPO modes – see Bulik (2005), Török,
Stuchlík and Bakala (2007).
We take into account only detections of oscillations with quality factor (defined
as the QPO centroid frequency over the full-width of the peak at its half-maximum)
Q ≥ 2 and significance (defined as the integral of the Lorentzian fitting the peak in
PDS divided by its error) S ≥ 3.
In this context we stress that any examined set of QPO detection carries exclusively an information on the distribution of QPOs strong enough to be detected.
The entire information on the QPO excitation is therefore potentially hidden and it
remains as open question whether not-detected oscillations exist or not. Independently of the (yet not known) answer to this question, “unseen” oscillations are at
least suppressed in comparison to those detected.
For the purposes of our study we parametrize the twin peak QPO occurrences
by their frequency ratio
R ≡ νU /νL .
(1)
As discussed in Section 3 this choice makes our discussion less dependent on the
concrete properties of the central compact object in 4U 1636–53. It also avoids
possible confusion with a parametrization of an individual QPO mode distribution.
Further advantage of this choice in the relation to resonant QPO models is discussed
in Section 4.
3.
Distribution Model I
In the RP model (Stella and Vietri 1999) the kHz QPOs represent a manifestation of the modes of a relativistic epicyclic motion of blobs in the inner parts of
accretion disk. The motion of a hot spot (radiating blob) is assumed to be nearly
geodesic. Observed lower QPO oscillation frequency is then related to the relativistic precession of the orbiting hot spot, while the upper QPO oscillation is associated
directly to its Keplerian frequency
νl (r) = νP = νK (r) − νr (r),
νU (r) = νK (r)
(2)
where νK , νr , νθ are Keplerian and radial or vertical epicyclic frequencies of the
geodesic motion and νP is the periastron precession frequency.
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3
In a given axially symmetric spacetime, the relevant angular velocities of the
azimuthal, radial and vertical “quasi-elliptic” orbital motion reads in the spherical
coordinates r, θ, φ (e.g., Abramowicz et al. 2003a). Henceforth, we use the geometrical units: c = 1 = G, M = GM ∗ /c2 , r = r∗ , t = ct ∗ , where the asterisk denotes
the standard units. Thus
ΩK = uφ /ut ,
(gtt + ΩK + gtφ )2 2 2 ∂ Ui ℓ
ω2i =
2gii
(3)
U(r, θ, ℓ) := gtt − 2ℓgtφ + ℓ2 gφφ
(5)
(4)
with gµν being components of the metric tensor, and U being the effective potential
for the equatorial geodetic motion given by the standard relation
where ℓ denotes the specific angular momentum of the orbiting test particle
ℓ = −uφ /ut
(6)
for a Keplerian motion ℓ = ℓK (r, θ). In the following we suppose the external
neutron star spacetime described by the Hartle–Thorne metric (Hartle and Thorne
1968) which represents the solution of vacuum Einstein field equations for the exterior of rigidly and relatively slowly rotating, stationary and axially symmetric body
represented by mass M and two dimensionless parameters – spin j and quadrupole
moment q. The components gµν , µ, ν ∈ {t, r, θ, φ} of the relevant metric tensor together with explicit Eqs. (3) and (4) derived in Abramowicz et al. (2003a) are given
in Appendix I.
In Appendix II we derive the relation between the ratio R of the observed frequencies and radius r of the QPO excitation in RP model for a Schwarzschild case
j = 0, q = 0,
6MR2
(7)
r=
2R − 1
and a few related formulae. Later we calculate all the relevant corrections numerically for Hartle–Thorne spacetimes with non-zero j and Q.
3.1. Modeling the Distribution
Let us assume that there is no preferred radius in the RP model and the probability of a QPO excitation is uniform across the inner part of the accretion disk.
Then, after a sufficient integration time, the number of QPO excitations (and detections) dn(r) should be equal for any given radius r when related to the unit length
in the radial direction
dn
= const,
dr̃
dr̃ =
√
grr dr
(8)
4
A. A.
where r̃ denotes a proper distance in the radial direction in the equatorial plane of
the disk.
Considering mass M , angular momentum j and quadrupole moment q of the
central compact object, one may find from Eqs. (1), (2) and (8) a QPO distribution
dn/dR as illustrated in Fig. 1. (Strictly speaking, the density dn/dR is a function of
j and q, independent of M as frequencies (Eq. 2) scale with 1/M . Nevertheless,
the mass still plays a role for finite distributions if observational restrictions are
connected to the QPO frequency.) In Appendix II we derive analytic formulae for
both differential and cumulative distribution function for the Schwarzschild limit
of the Hartle–Thorne spacetime ( q = 0, j = 0). Generally, for nonzero j and q,
they have to be calculated numerically.
Fig. 1. a) The distribution function dn/dR for two representative values of the angular momentum
j and Kerr limit of quadrupole moment q = j 2 . Shadow in the inserted figure roughly indicates
the relevant part of the accretion disk. While the frequency ratio R = 1 always corresponds to the
Innermost–Stable–Circular–Orbit, R = 2 corresponds to the position of maximum of radial epicyclic
frequency depending on value of j and q with inaccuracy of a few percent. b) Corresponding cumulative distribution functions ñ (normalized to R = 3 ). Note that it is rather difficult to distinguish on
this scale between the two functions j = 0 and j = 0.3 (q = j 2 ) . Two more curves corresponding
to angular momentum (and q = j 2 ) too high for a neutron star are shown for comparison.
3.2.
Properties of Central Compact Object and Differences in Implied Distribution
Because the parameters M , j and q of the central compact object in 4U 163653 are not known, similarly to the other QPO sources, we produce distribution
dn/dR for a large family of sources having the properties consistent with present
standard equations of state, (see e.g., Rikovska et al. 2003). The family we consider
is characterized by an uniform distribution of parameters M ∈ (1.2M⊙ , 2.2M⊙ ),
j ∈ (0, 0.14) and q ∈ ( j 2 , 10 j 2 ) with 100 × 100 × 100 representants (sources).
We produce ≈ 200 datapoints per each source corresponding to a constant density
dn/dr̃ in the region located between the radius corresponding to the maximum of
the radial epicyclic frequency and a marginally stable orbit. The considered radial
interval agrees with rough observational constraints to the model, see e.g., Belloni
et al. (2007a). For the highest neutron star mass we consider here, the lowest peri-
Vol. 58
5
astron precession frequency corresponding to the considered radial interval is close
to 350 Hz. The observational data we use here are restricted above νL = 500 Hz.
However, we have searched through several works, namely Barret et al. (2005),
Barret, Olive and Miller (2006), and there are most likely no twin peak QPO detections for νL ∈ (300, 500) Hz above thresholds corresponding to our dataset. The
500 Hz limit is therefore not really involved here. See also Török et al. (2007).
To mimic an observational error we blur the implied frequencies with the 3%
Gaussian error on a 2σ level of confidence. In this way we produced 106 distributions and also a distribution averaged per all sources (hereafter mean distribution).
We find that the variations of the individual distributions to the mean are rather
small which follows from the partial 1/M scaling of the orbital frequencies (Eq. 2)
and from a small influence of the low neutron star angular momentum to the frequency ratio R which we discuss in Appendix I.II (see also Fig. 1, especially its
right panel). Within the considered radial range the ratio R is a monotonic and
decreasing function of the radial coordinate, changing from R = 1 to R ≈ 2. The
value R = 1 represents rather asymptotic number corresponding to the marginally
stable circular orbit. Notice also that for the Schwarzschild spacetime the value at
the maximum of the radial epicyclic frequency reads exactly R = 2 and slowly decreases with the increasing angular momentum j (see, Török and Stuchlík 2005).
Maximal variations ∆R( j, q) = R(rmax , 0, 0) − R(rmax , j, q) at the maximum of radial epicyclic frequency within the examined interval of the angular momentum do
not exceed ∆R ≈ 0.02.
It follows from the above that considered combinations of j and q imply distributions rather similar to those for j = 0 and q = 0. The mean distribution is
shown in Fig. 2a together with the “Schwarzschild” distribution ( j = 0, and q = 0).
Fig. 2b provides a comparison to the observation which is further discussed in Section 4.
Fig. 2. a) The twin peak QPO distribution implied by RP model with no preferred orbits (model I).
Lines correspond to: dotted – averaged distribution, solid (black) – “Schwarzschild” distribution ( j =
0 , q = 0 ), grey – ( j = 0.3 , q = 0.09 ). Vertical axes scale is arbitrary for a given distribution. b) The
observed twin peak QPO distribution. c) Histogram of K-S test results relevant to the comparison
between observation and each of individual combinations of the considered spacetime parameters.
6
A. A.
4. Discussion and Distribution Model II
One can easily recognize from Fig. 2a,b that the constructed distribution (hereafter the model I) significantly differs from the empirical one. We apply the Kolmogorov-Smirnov (K-S) test (Press et al. 2007) to quantify this statement.
Using the test we compare each of individual distributions to the observation.
Fig. 2c shows a histogram of the results. In terms of the test, the probability that the
constructed and the observed distributions come from the same parent distribution
is pK−S ≈ 10−5 for any parameters from the considered intervals.
The unsatisfactory result presented above is connected to the conclusions of the
studies of kHz QPO ratio distribution in the neutron star sources (Abramowicz et al.
2003b, Belloni, Méndez and Homan 2005, Belloni et al. 2007b, Török et al. 2008)
– the ratio distribution tends to cluster close to ratio of small natural numbers.
It was proposed that the clustering can be connected to different instances of
one orbital resonance (Török, Stuchlík and Bakala 2007) involving modes formally
identical or similar to the modes of Eq. (2). In such a case it is impossible to model
the underlying distribution without a precise knowledge of the physical mechanism.
Nevertheless, in Fig. 2c we show a modified version (model II) of the simulated
distributional model I, satisfying the following restrictions:
• The datapoints are created only close to the “resonant” radii with the ratio
R = k/l , where k, l ∈ {1, 2, 3, 4, 5, 6}.
• The distribution of datapoints around the resonant radii is implied by the
Cauchy–Lorentz distribution in the ratio R
p(R) = wk/l
λk/l /π
.
(R − k/l)2 + λ2k/l
(9)
• The weights wk/l of individual Lorentzians are normalized as
∑ wk/l = 1, wk/l ≈ 1/ j2
(10)
where j is the higher number from the two k, l . The width of the Lorentzians
is arbitrarily given as λ = 0.013R so there is ≈ 97% of datapoints relevant
to the Lorentzians in the interval R ∈ (1, 2).
• All the other properties are the same as in the case of model I.
The distribution guess (model II) given above includes preference of orbits
with the Keplerian and periastron frequency being in resonant ratios, and its detailed properties are rather arbitrary. Its comparison with observation gives the K-S
probability pK−S ≈ 40% within the considered range of the central compact object
parameters. In Fig. 3 we show a comparison of cumulative distributions of models I
and II to observation. Fig. 3 suggests that K-S probability would be much improved
for both of them.
Vol. 58
7
Fig. 3. Comparison of cumulative distributions given by observation (thick curve), model I and model
II. The grey curve corresponds to the best fit (Török et al. 2008) by superposition of two Lorentzians.
It is visible that considerations of number of preferred orbits being located (Eq. 7) between r ≈ 6.2M
(R ≈ 1.2) and r ≈ 7.5M (R ≈ 1.8) can improve the unsatisfactory result of model I.
5.
Conclusions
The observational data from studies of Barret and collaborators (Barret et al.
2005, Abramowicz et al. 2005) which we use correspond to all the RXTE observations of 4U 1636+53 till 2005 proceeded by the shift-add technique through continuous segments of observation. The part of data displaying significant twin peak
QPOs is restricted to about 20 hours of observation represented in our study by 23
datapoints corresponding to the individual continuous observations. It is needed
for a further study to proceed these data by other methods in order to obtain a more
detailed view of the distribution. However, in terms of the RP model the 23 significant datapoints we use represent (under the assumption of the hot spot lifetime
being equal to a few orbits) the statistics of ≈ 107 individual hot spots averaged in
a well defined way which allows us to conclude that:
• The twin peak QPO distribution obtained from the relativistic precession
(RP) model under the consideration of the (observable) QPO excitation probability being uniform across the inner part of the accretion disk is highly incompatible with that given by observational data. This result is independent
of the choice of reasonable sample of intervals of parameters M , j , q. For
completeness, we also check for “extreme values” like j ≈ 0.3. Because of
the shape of resulting histograms (Fig. 1 and Fig. 2a) the result is also independent of the exact delimitation of the radial disk region (which we consider
between a maximum of the radial epicyclic frequency and the marginally stable circular orbit).
• On the other hand the arbitrary consideration of preferred “resonant” radii
implies a twin peak QPO distribution showing similarities to the observational one. (It has been recently noticed (Török et al. 2008) that the distribution can be well described (K-S probability ≈ 98%) by a sum of two
Lorentzians having the centroids at R = 1.51 and 1.28. Notice that from
8
A. A.
.
Eq. (7) this would correspond to radii r = 6.8M and 6.3M . Nevertheless the
eventual relevance of (exactly) these frequency ratios to a QPO model is not
clear at present.)
In principle one can not exclude that non-observed oscillations are produced.
Our findings are, however, relevant even in such a case as the model should explain
why only pairs of oscillations coming from the vicinity of preferred orbits are well
observable.
Finally we notice that several QPO models (hot spot- or disk oscillations-like)
introduce frequency relations which are qualitatively and also quantitatively similar to those implied by the relativistic precession model. Moreover, in the limit of
the Schwarzschild spacetime these relations coincide (Horák et al. 2008 in preparation, Török et al. 2007, Stuchlík, Török and Bakala 2007). Our discussion of
the quantitative distribution of observations is, thus, roughly relevant also for those
models, including the model considering the radial m = 1 and vertical m = 2 diskoscillation modes.
Acknowledgements. We thank to Didier Barret for an important notice that for
usual count-rates the non-observed QPOs can be still involved in the X-ray modulation being in such a case too weak for simultaneous significant detections with
present instruments. We also thank to John Miller and Martin Urbanec for discussion of quadrupole momentum range and to Jiří Horák and Michal Bursa for useful
debates on subject of connection between the RP model and the QPO observability.
The last but not least are our thanks to anonymous referee for several comments and
suggestions which helped to improve the paper. The authors are supported by the
Czech grants MSM 4781305903, LC06014, and GAČR 202/06/0041.
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Belloni, T., Homan, J., Motta, S., Ratti, E., and Méndez, M., 2007b, MNRAS, 379, 247.
Bulik, T. 2005, Astron. Nachr., 326, 861.
Hartle, J.B., and Thorne, K.S. 1968, ApJ, 153, 807.
Kato, S., Fukue, J., and Mineshige, S. 1998, “Black hole accretion disks”, Kyoto University Press.
Lamb, F.K. 2003, ASP Conference Series, 308, 221.
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Results”, Eds. E.F. Milone, D.A. Leahy, and D. Hobill, Dordrecht: Springer.
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10
A. A.
Appendix I
I.I Formulae for Orbital Geodesic Frequencies in the Hartle–Thorne Metric
(after Abramowicz et al. 2003a)
Components of the metric tensor,
ds2 = gtt dt 2 + grr dr2 + gθθ dθ2 + gφφ dφ2 + gφt dφdt + gtφ dtdφ
(11)
are given as
gtt = + (1 − 2M/r) 1 + j 2 F1 t + qF2 t ,
−1
grr = − (1 − 2M/r)
1 + j 2 F1 r − qF2
gθθ = −r2 1 + j 2 F1 θ + qF2 θ ,
φ
φ
gφφ = −r2 sin2 θ 1 + j 2 F1 + qF2 ,
(12)
r
,
gtφ = −2(M 2 /r) j sin2 θ,
(13)
(14)
(15)
(16)
where
F1 t = [8Mr4 (r − 2M)]−1 [u2 (48M 6 − 8M 5 r − 24M 4 r2 − 30M 3 r3 − 60M 2 r4
+ 135Mr5 − 45r6 ) + (r − M)(16M 5 + 8M 4 r − 10M 2 r3 − 30Mr4 + 15r5 )]
+ A1 (r),
F2 t = [8Mr (r − 2M)]−1 [5(3u2 − 1)(r − M)(2M 2 + 6Mr − 3r2 )] − G1 (r),
F1 r = [8Mr4 (r − 2M)]−1 [(G2 − 72M 5 r) − 3u2 (G2 − 56M 5 r)] − G1 (r),
F2 r = F2 t ,
F1 θ = (8Mr4 )−1 (1 − 3u2 )(16M 5 + 8M 4 r − 10M 2 r3 + 15Mr4 + 15r5 ) + G3 (r),
F2 θ = (8Mr)−1 [5(1 − 3u2 )(2M 2 − 3Mr − 3r2 )] − G3 (r),
φ
φ
φ
φ
F1 = F1 ,
F2 = F2 ,
and
15r(r − 2M)(1 − 3u2 )
r
ln
,
2
16M
r − 2M
15(r2 − 2M 2 )(3u2 − 1)
r
G2 =
ln
,
16M 2
r − 2M
G3 = 80M 6 + 8M 4 r2 + 10M 3 r3 + 20M 2 r4 − 45Mr5 + 15r6 ,
G1 =
u = cos θ.
The angular velocity for corotating circular particle orbits reads
#
"
uφ M 1/2
M 3/2
2
Ω
Ω
ΩK = t = 3/2 1 − j 3/2 + j F1 (r) + qF2 (r)
u
r
r
(17)
Vol. 58
11
where
F1 Ω (r) = (48M 7 − 80M 6 r + 4M 5 r2 − 18M 4 r3 + 40M 3 r4 + 10M 2 r5
+ 15Mr6 − 15r7 )[16M 2 (r − 2M)r4 ]−1 + H(r),
5(6M 4 − 8M 3 r − 2M 2 r2 − 3Mr3 + 3r4 )
− H(r),
F2 Ω (r) =
16M 2 (r − 2M)r
15(r3 − 2M 3 )
r
H(r) =
ln
.
3
32M
r − 2M
The epicyclic frequencies of circular geodesic motion are given by formulae
M(r − 6M) ω2r =
1 + jH1 (r) − j 2 H2 (r) − qH3 (r) ,
(18)
4
r
M
ω2θ = 3 1 − jI1 (r) + j 2 I2 (r) + qI3 (r) ,
(19)
r
where
H1 (r) =
6M 3/2 (r + 2M)
,
r3/2 (r − 6M)
H2 (r) = [8M 2 r4 (r − 2M)(r − 6M)]−1 (384M 8 − 720M 7 r − 112M 6 r2 − 76M 5 r3
− 138M 4 r4 − 130M 3 r5 + 635M 2 r6 − 375Mr7 + 60r8 ) + J(r),
H3 (r) =
5(48M 5 + 30M 4 r + 26M 3 r2 − 127M 2 r3 + 75Mr4 − 12r5 )
− J(r),
8M 2 r(r − 2M)(r − 6M)
6M 3/2
,
r3/2
I2 (r) = [8M 2 r4 (r − 2M)]−1 (48M 7 − 224M 6 r + 28M 5 r2
I1 (r) =
+ 6M 4 r3 − 170M 3 r4 + 295M 2 r5 − 165Mr6 + 30r7 ) − K(r),
I3 (r) =
with
5(6M 4 + 34M 3 r − 59M 2 r2 + 33Mr3 − 6r4 )
+ K(r),
8M 2 r(r − 2M)
15r(r − 2M)(2M 2 + 13Mr − 4r2 )
r
ln
,
3
16M (r − 6M)
r − 2M
15(2r − M)(r − 2M)2
r
K(r) =
ln
.
3
16M
r − 2M
For completeness, the relation determining the marginally stable circular geodesic
reads
"
r
3
2 2
2 251647
(20)
+j
− 240 ln
rms = 6M 1 − j
3 3
2592
2
9325
3
+q −
.
+ 240 ln
96
2
J(r) =
12
A. A.
I.II Relation between Keplerian and Precession Frequency Inside the Inner Part
of Accretion Disk
From its definition, function R = νU /νL = νK /(νK − νr ) equals 1 at the marginally stable circular orbit for any angular momentum j . It is then increasing with
increasing radial coordinate. In Fig. 4a,b we show change of the position of maximum of the radial epicyclic frequency rmax [νr ] as well as the related change of the
function R(rmax ) within the range of j and q discussed in this paper.
One should notice that while coordinate position of rmax [νr ] is rather sensitive
to j and q, changing of nearly 10% from rmax = 8 to rmax = 7.4, the corresponding change of frequency reads only about 1%. This suggests that there should not
be a substantial difference between individual distributions related to different j, q
within the discussed range. This is then confirmed by the calculation further discussed in the paper.
Fig. 4. Maximum of the radial epicyclic frequency as depends on parameters j and q . a) Its position
as a function of parameters j and q within intervals discussed in the paper. The three labelled dashed
curves correspond to q/ j 2 = 1, 5 and 10. b) Corresponding frequency ratio. c) Direct comparison of
relative change of radial coordinate and relative change of frequency ratio νU /νL for the extended
range of j .
Vol. 58
13
I.III Relation between Frequency Ratio and Radial Position of QPO Excitation
In Fig. 4c we show the comparison between relative change of rmax [νr ] and
related relative change of R(rmax [νr ]) for the extended range of j and outlines of
q = j 2 and q = 10 j 2 . It is visible from Fig. 4c that up to j = 0.25 (representing
already rapidly rotating neutron star) the relative change of R(rmax [νr ]) is smaller
than 2%.
The above fact has potentially interesting consequences for observational astrophysics. As noticed in several works, (e.g., Nowak and Wagoner 1992, Kato,
Fukue and Mineshige 1998), the physics of accretion disk oscillations is governed
by the behavior of the epicyclic frequencies. In this relation the physically important characteristics for (quantitatively) different spacetimes are locations of ISCO
(where R = 1) and location of rmax . For comparing of radial positions belonging to
different compact objects one can relate “physical” position inside of the accretion
disk to these two points.
As shown above, within the range of j and q expected even for rapidly rotating
neutron star the ratio R(rmax ) is nearly constant. One may therefore use the ratio of
observed QPO frequencies when comparing the radial positions of QPO excitations
from compact object having different M , j , and q using as a reference Eq. (23)
which we derive in Appendix II.
14
A. A.
Appendix II
Distribution Function for the RP Model in the Schwarzschild Spacetime
For j = 0 and q = 0 orbital and epicyclic frequencies given by Eqs. (17–19)
can be expressed in familiar form
q
1 p
1
M(r − 6M).
(21)
M/r3 ,
νr =
νK = νθ =
2π
2πr2
In the RP model, the observable frequencies are identified as
νU
νK
R=
=
.
(22)
νL
νK − νr
The above equations imply relation between the observed frequency ratio and radius of QPO excitation which reads
6MR2
.
(23)
2R − 1
Proper distance r̃ in the radial direction in the equatorial plane of the disk measured
between radii r0 and r is given as
Zr r
Zr
h
ir
p
p
r
√
grr dr =
dr = ln(r − 1 + r2 − 2r) + r2 − 2r .
(24)
r−2
r0
r=
r0
r0
Using Eq. (23), a proper radial distance from radius corresponding to the frequency ratio R0 to radius corresponding to the frequency ratio R can be written
as
s
"
!
12R2
6R2
36R4
r̃ = ln
−
+
− 1)
(2R − 1)2 2R − 1 2R − 1
s
#R
12R2
36R4
.
(25)
−
+
(2R − 1)2 2R − 1
R0
Let us assume the probability of a QPO excitation being uniform across the
inner part of the accretion disk. Then after a sufficient integration time the number
of QPO excitations dn(r) should be equal for any given radius r when related to
the unit length in the radial direction
dn
= const.
(26)
dr̃
Eq. (25) therefore directly determines (cumulative) distribution of QPO excitations
with respect to the frequency ratio R. Relevant differential distribution then reads
(both the cumulative and differential distributions for j = 0 are illustrated in Fig. 1.)
36(R − 1)R2
dn
√
.
=
dR (1 − 2R)2 3 − 6R + 9R2
(27)
Přı́loha 5
DOI: 10.2478/s11534-007-0039-0
Rapid Communication
A remark about possible unity of the neutron star
and black hole high frequency QPOs
Gabriel Török∗, Zdeněk Stuchlı́k, Pavel Bakala
Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava,
Bezručovo nám. 13, CZ-746 01 Opava, Czech Republic
Received 09 May 2007; accepted 15 June 2007
Abstract: In a series of papers it was discussed, on the basis of phenomenological arguments,
whether the high frequency quasiperiodic oscillations (kHz QPOs) observed in the neutron-star
and black-hole X-ray sources originate in the same physical mechanism. Recently it was suggested
that a general trend seen in neutron star kHz QPOs instead excludes such a uniform origin. Using
the example of the atoll source 4U 1636-53 we illustrate that this is not neccesarily true.
c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.
Keywords: X-rays, binaries, accretion, accretion discs, stars, neutron
PACS (2006): 97.60.Lf, 97.10.Gz, 97.10.Sj, 97.60.Jd, 97.80.Jp
1
Introduction
A series of papers discuss, among other things, the possibility of the resonant origin of
the high frequency QPOs observed in bright neutron-star low-mass X-ray binaries and in
Galactic microquasars with a similar/the same mechanism operating in both classes of
sources.
Kluźniak and Abramowicz [1] suggested that the kHz QPOs come from a strong nonlinear gravity resonance and later Abramowicz et al. [2] noticed that the observed ratio
between the lower and upper frequencies νL and νU of a kHz QPO mode usually clusters
close to ratios of small natural numbers, most often close to the value νU /νL = 3/2 which
is also clearly observed in the Galactic microquasars [3, 4] supporting rather strongly the
resonant origin hypothesis.
∗
E-mail: [email protected]
G. Török et al. / Central European Journal of Physics
Belloni et al. [5] re-examined this study and concluded that the frequency ratio clusters most often close to the 3/2 ratio and less often close to the other rational numbers
(e.g., 5/4 and 4/3) and argued that such clustering does not provide any useful information about a possible underlying resonance mechanism in the sources since the distribution
of the ratio of two correlated quantities is completely determined by the distribution of
one of them.†
Abramowicz et al. [6, 7] realized that the slope and shift of lines well approximating the data of individual neutron star frequency-frequency relations are anticorrelated.
Belloni et al. [8] related this anticorrelation with the previously examined frequency
clustering and the general trend seen in the neutron star data — the sources roughly
follow frequency-frequency relation relevant to the relativistic precession model [9] constructed under consideration of the Schwarzschild metric and central compact object mass
M ∼ 2M .
Belloni et al. [8] also argued that this general trend rather excludes the possibility of
the same origin of both black-hole and neutron-star kHz QPOs.
2
A unified QPO model — could it exist or not?
2.1 Microquasars
In the case of Galactic microquasars showing fixed twin peak QPO frequencies, the implied mass and spin relation was discussed for several resonance models based on a resonance between orbital frequencies of geodesic motion [3, 10]. Bursa [11] proposed a
so-called vertical precession resonance model in order to match the spin estimated from
fits of the X-ray spectral continua for the microquasar GRO J1655-40. Note that to date
the observational data do not exclude this model for any of four microquasars displaying
clear twin peak QPOs.
In the vertical precession resonance model, the observed QPO frequencies νL and νU
are for a given black hole spin identified with frequencies
νl (r) = νK (r) − νr ,
νu (r) = νθ (r)
(1)
for a particular choice of r defined by the condition νu = 3/2νl . The frequencies νK , νr
and νθ denote Keplerian, radial and vertical frequencies of orbital motion in the Kerr
spacetime [12, 13]. We note that for the Schwarzschild metric the frequencies νl (r) and
νu (r) merge with the relation predicted by the relativistic precession model mentioned
above.
G. Török et al. / Central European Journal of Physics
Fig. 1 The frequency correlation in the atoll source 4U 1636-53. The νK curve determines
the upper QPO frequency following from the relativistic precession model [9] under consideration of the gravitational field described by the Schwarzschild metric with a central
mass M = 1.84M , the grey curve denotes the same relation but for M = 2M , i.e., the
trend reported by [8]. Intersections of this curve with the 3:2 and 5:4 line correspond to
the relevant vertical precession resonance. Note that the actual (observed) frequencies
of the resonance are allowed to differ from the given resonant eigenfrequencies [6]. The
secondary vertical axes indicate the dimensionless radius related to M = 1.84M . Note
that the assumed 5:4 resonance occurs very close (0.25M) to the innermost stable circular
geodesic orbit, i.e., near the expected inner edge of the accretion disc.
2.2 The atoll source 4U 1636-53
Abramowicz et al. [6], profiting from the studies [14–16], examined frequency correlations
in several neutron star sources.
In Figure 1 we show the correlation corresponding to the occurences of twin peaks for
the atoll source 4U 1636-53 taken from [6], method A in the paper. This correlation was
obtained by the shift-add [17] fitting of continuous segments of observations from all of
the RXTE data available at the time.‡
We stress that in contrast to the studies considering separated single QPO distributions, e.g., the recent paper of Belloni et al. [18], the twin peak QPO distributions
examined in this way consider only simultaneous significant detections of both QPO frequencies (i.e., here the detection of both the peaks above 2.5σ significance having quality
factor higher than 3).
The two distinct clusters of datapoints are easy to recognize on the figure. In the
framework of resonance QPO models, the fact that these two clusters correspond to the
3/2 and the 5/4 frequency ratio may suggest their connection to different instances of a
particular orbital resonance.
†
The goal of our short paper is not in continuing the discussion of this questionable argument which
is given in the different paper [19]. Nevertheless, we at least note that the frequency vs. ratio dilemma
represents rather the question of the choice of the quantity which depends on the assumed model.
‡
See [6, 14–16] for details.
G. Török et al. / Central European Journal of Physics
2.3 Comparison
The 3:2 vertical precession resonance model [11] has been introduced for microquasars
whereas the implied black hole spins seem to be in agreement with the independent
estimates [11, 20].
According to the study of Belloni et al. [8], the datapoint clusters in 4U 1636-53
are very close to the relation predicted by the model of Stella and Vietri and therefore
to the frequencies (1) following from the same vertical precession resonance model as
in microquasars, but for a particular choice of r defined by conditions νu = 3/2 νl and
νu = 5/4 νl corresponding to the 3/2 and 5/4 resonance, respectively.§
2.4 Conclusions
The above clear argument that one can not exlude in 4U 1636-53 a QPO mechanism
similar to that for Galactic microquasars is obviously also applicable to the other neutron
star sources.
Because the neutron star kHz twin peak QPOs typically cluster close to the ratios of
small natural numbers [2, 5] and because the oscillation modes of the vertical precession
resonance and possibly of other similar resonances are close or coincide with some modes
predicted by the relativistic precession model, this type of resonance can be considered
to explain neutron star QPOs as well as the relativistic precession model.
Therefore, an overall agreement of the kHz QPO data in neutron stars with the trend
predicted by the precession model [9] does not rule out resonances in the accretion disk
[1] as the direct cause for the observed oscillations, and, contrary to the statement of
Belloni et al. [5], the same origin for neutron-star and black-hole high frequency QPOs
is not excluded.
Acknowledgments
We would like to thank to the anonymous referees for several suggestions which helped to
improve the paper. The authors have been supported by the Czech grants MSM 4781305903
(GT, ZS), LC06014 (PB) and GAČR 202/06/0041 (ZS).
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§
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Přı́loha 6
The Astrophysical Journal, 714:748–757, 2010 May 1
C 2010.
doi:10.1088/0004-637X/714/1/748
The American Astronomical Society. All rights reserved. Printed in the U.S.A.
ON MASS CONSTRAINTS IMPLIED BY THE RELATIVISTIC PRECESSION MODEL OF TWIN-PEAK
QUASI-PERIODIC OSCILLATIONS IN CIRCINUS X-1
Gabriel Török, Pavel Bakala, Eva Šrámková, Zdeněk Stuchlı́k, and Martin Urbanec
Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava Bezručovo nám. 13, CZ-746 01 Opava, Czech Republic; [email protected],
[email protected], [email protected], [email protected], [email protected]
Received 2009 October 8; accepted 2010 March 15; published 2010 April 14
ABSTRACT
Boutloukos et al. discovered twin-peak quasi-periodic oscillations (QPOs) in 11 observations of the peculiar
Z-source Circinus X-1. Among several other conjunctions the authors briefly discussed the related estimate of
the compact object mass following from the geodesic relativistic precession model for kHz QPOs. Neglecting the
neutron star rotation they reported the inferred mass M0 = 2.2 ± 0.3 M . We present a more detailed analysis
of the estimate which involves the frame-dragging effects associated with rotating spacetimes. For a free mass
we find acceptable fits of the model to data for (any) small dimensionless compact object angular momentum
j = cJ /GM 2 . Moreover, quality of the fit tends to increase very gently with rising j. Good fits are reached when
M ∼ M0 [1 + 0.55(j + j 2 )]. It is therefore impossible to estimate the mass without independent knowledge of the
angular momentum and vice versa. Considering j up to 0.3 the range of the feasible values of mass extends up to 3 M .
We suggest that similar increase of estimated mass due to rotational effects can be relevant for several other sources.
Key words: stars: neutron – X-rays: binaries
Online-only material: color figure
2005; Zhang et al. 2007a, 2007b), oscillations that arise due
to comptonization of the disk–corona (Lee & Miller 1998) or
oscillations excited in toroidal disk (Rezzolla et al. 2003; Rezzolla 2004; Šrámková 2005; Schnittman & Rezzolla 2006; Blaes
et al. 2007; Šrámková et al. 2007; Straub & Šrámková 2009) are
considered as well. At last but not least, already the kinematics
of the orbital motion itself provides space for consideration of
“hot-spot-like” models identifying the observed variability with
orbital frequencies. For instance, recent works of Čadež et al.
(2008) and Kostić et al. (2009) deal with tidal disruption of
large accreted inhomogenities. Among the same class of (kinematic) models belongs also the often quoted “relativistic precession” (RP) kHz QPO model that is the focus of our attention
here.
The RP model has been proposed in a series of papers by
Stella & Vietri (1998, 1999, 2002). It explains the kHz QPOs as a
direct manifestation of modes of relativistic epicyclic motion of
blobs arising at various radii r in the inner parts of the accretion
disk. The model identifies the lower and upper kHz QPOs with
the periastron precession νp and Keplerian νK frequency,
1. INTRODUCTION
Quasi-periodic oscillations (QPOs) appear in variabilities
of several low-mass X-ray binaries (LMXBs) including those
which contain a neutron star (NS). A certain type of these
oscillations, the so-called kHz (or high-frequency) QPOs, often come in pairs with frequencies νL and νU typically in
the range ∼50–1300 Hz. This is of the same order as the range
of frequencies characteristic for orbital motion close to a compact object. Accordingly, most kHz QPO models involve orbital
motion in the inner regions of an accretion disk (see van der
Klis 2006; Lamb & Boutloukos 2007, for a recent review).
There is a large variety of QPO models related to NS sources
(in some but not all cases they are applied to black hole (BH)
sources too). Concrete models involve miscellaneous mechanisms of producing the observed rapid variability. One of the first
possibilities proposed represents the “beat frequency” model assuming interactions between the accretion disk and spinning
stellar surface (Alpar & Shaham 1985; Lamb et al. 1985).
Many other models primarily assume accretion disk oscillations. For instance, non-linear resonance scenarios suggested
by Abramowicz, Kluźniak and Collaborators (Abramowicz &
Kluźniak 2001; Abramowicz et al. 2003b, 2003c; Horák 2008;
Horák et al. 2009) are often debated. A set of the later models join the beat frequency idea, magnetic field influence, and
presence of the sonic point (Miller at al. 1998b; Psaltis et al.
1999; Lamb & Miller 2001). Some of the numerous versions of
non-linear oscillation models and the late beat frequency models rather fade into the same concept that commonly assumes
the NS spin to be important for excitation of the resonant effects
(Kluźniak et al. 2004; Pétri 2005a, 2005b, 2005c; Miller 2006;
Kluźniak 2008; Stuchlı́k et al. 2008; Mukhopadhyay 2009).
Resonance, influence of the spin, and magnetic field also play
a role in the ideas discussed by Titarchuk & Kent (2002) and
Titarchuk (2002). Other resonances are accommodated in models assuming deformed disks (Kato 2007, 2008, 2009a, 2009b;
Meheut & Tagger 2009). Further effects induced in the accreted
plasma by the NS magnetic field (Alphén wave model, Zhang
νL (r) = νp (r) = νK (r) − νr (r),
νU (r) = νK (r),
(1)
where νr is the radial epicyclic frequency of the Keplerian motion. (Note that, on a formal side, for Schwarzschild spacetime
where νK equals a vertical epicyclic frequency this identification
merges with a model assuming m = −1 radial and m = −2
vertical disk-oscillation modes).
In the past years, the RP model has been considered among
the candidates for explaining the twin-peak QPOs in several
LMXBs and related constraints on the sources have been
discussed (see, e.g., Karas 1999; Zhang et al. 2006; Belloni et al.
2007a; Lamb & Boutloukos 2007; Barret & Boutelier 2008a;
Yan et al. 2009). While some of the early works discuss these
constraints in terms of both NS mass and spin and include also
the NS oblateness (Morsink & Stella 1999; Stella et al. 1999),
most of the published implications for individual sources focus
on the NS mass and neglect its rotation.
748
No. 1, 2010
ON MASS OF CIRCINUS X-1
Two simultaneous kHz QPOs with centroid frequencies
of up to 225 (500) Hz have also recently been found by
Boutloukos et al. (2006a, 2006b) in 11 different epochs of
RXTE/Proportional Counter Array observations of the peculiar
Z-source Circinus X-1. Considering the RP model they reported
the implied NS mass to be M ∼ 2.2 M . The estimate was
obtained assuming the non-rotating Schwarzschild spacetime
and was based on fitting the observed correlation between the
upper QPO frequency and the frequency difference Δν = νU −νL .
In this paper, we improve the analysis of mass estimate carried
out by Boutloukos et al. In particular, we consider rotating
spacetimes that comprehend the effects of frame-dragging and
fit directly the correlation between the twin QPO frequencies.
We show that good fits can be reached for the mass–angular
momentum relation rather than for the preferred combination of
mass and spin.
2. DETERMINATION OF MASS
Spacetimes around rotating NSs can be approximated with
a high precision by the three-parametric Hartle–Thorne (HT)
solution of Einstein field equations (Hartle & Thorne 1968;
see Berti et al. 2005). The solution considers mass M, angular
momentum J, and quadrupole moment Q (supposed to reflect
the rotationally induced oblateness of the star). It is known that
in most situations modeled with the present NS equations of
state (EoS) the NS external geometry is very different from
the Kerr geometry (representing the “limit” of HT geometry
for q̃ ≡ QM/J 2 → 1). However, the situation changes
when the NS mass approaches maximum for a given EoS.
For high masses the quadrupole moment does not induce large
differences from the Kerr geometry since q̃ takes values close
to unity (Appendix A.1).
The previous application of the RP model mostly implied
rather large masses (e.g., Belloni et al. 2007a). These large
masses are only marginally allowed by standard EoS. Also
the mass inferred by Boutloukos et al. (2006a, 2006b) takes
values above 2 M . Motivated by this we use the limit of twoparametric Kerr geometry to estimate the influence of the spin
of the central star in Circinus X-1 (see Appendix A.1 where we
pay a more detailed attention to rationalization and discussion of
this choice allowing usage of simple and elegant Kerr formulae).
2.1. Frequency Relations
Assuming a compact object of mass MCGS = GM/c and
dimensionless angular momentum j = cJ /GM 2 described by
the Kerr geometry, the explicit formulae for angular velocities
related to Keplerian and radial frequencies are given by the
following relations (see Aliev & Galtsov 1981; Kato et al. 1998,
or Török & Stuchlı́k 2005):
6
8j
3j 2
ΩK = F (x 3/2 + j )−1 , ωr2 = Ω2K 1 − + 3/2 − 2 , (2)
x x
x
2
where F ≡ c3 /(2π GM) is the “relativistic factor” and x ≡
r/MCGS . Considering Equations (1) and (2), we can write for νL
and νU , both expressed in Hertz (see also Appendix A.1.2 where
we discuss a linear expansion of this formula),
2/3
8j νU
νU
νL = νU 1 − 1 +
−6
F − j νU
F − j νU
4/3 1/2 νU
.
(3)
− 3j 2
F − j νU
749
In the Schwarzschild geometry, where j = 0, Equation (3)
simplifies to
νL = νU
ν 2/3 1/2
U
1− 1−6
F
(4)
leading to the relation
Δν = νU 1 − 6 (2π GMνU )2/3/c2
(5)
that was used by Boutloukos et al. for the mass determination.
2.2. “Ambiguity” in M
There is a unique curve given by Equation (3) for each
different combination of M and j (see Appendix A.2 for the
proof). The frequencies νL and νU scale as 1/M and, as illustrated
in the left panel of Figure 1, they increase with growing j.
Naturally, one may ask an interesting question whether for
different values of M and j there exist some curves that are
similar to each other. We investigate and quantify this task in
Appendix A.2.
There we infer1 that for j up to ∼0.3 one gets a set of nearly
identical integral curves where M, j, and M0 roughly relate as
follows:
(6)
M = [1 + k(j + j 2 )]M0
with
k = 0.7.
This result is illustrated in the right panel of Figure 1. Clearly,
when using relation (6), any curve plotted for a rotating star
of a certain mass can be well approximated by those plotted
for a non-rotating star with a smaller mass, and vice versa.
Furthermore, we find that (see Appendix A.2) when the top parts
of the curves (corresponding to νU /νL ∼ 1–1.5) are considered
only, the best similarity is reached for
k = 0.75.
These parts of the curves are potentially relevant to most of
the atoll and high-frequency Z-sources data. On the other hand,
for the (bottom) parts of the curves that are potentially relevant
to low-frequency Z-sources including Circinus X-1, the best
similarity is achieved for
k = 0.65 (0.55, 0.5)
when
νU /νL ∼ 2 (3, 4).
Taking into account the above consideration we can expect that
the single-parameter best fit to the data by relation (4) roughly
determines a set of mass–angular-momentum combinations (6)
with similar χ 2 . The result of Boutloukos et al. then implies that
good fits to their data, displaying νU /νL ∼ 3, should be reached
for M ∼ 2.2 M [1 + 0.55(j + j 2 )]. In what follows we fit the
data and check this expectation.
1
We first consider a special set of apparently similar curves sharing the
terminal points. The set is (numerically) given by the particular choice of M,
for any j implying the same orbital frequency at the marginally stable circular
orbit. The curves then only slightly differ in their concavity that increases with
growing j.
750
TÖRÖK ET AL.
Vol. 714
Figure 1. Left: relation between the upper and lower QPO frequency following from the RP model for the mass M = 2.5 M . The consecutive curves differ in
j ∈ (0, 0.3) by 0.05. Right: relations predicted by the RP model vs. data of several NS sources. The curves are plotted for various combinations of M and j given by
Equation (6) with k = 0.7. The datapoints belong to Circinus X-1 (red/yellow color), 4U 1636-53 (purple color) and most of other Z- and atoll-sources (black color)
exhibiting large population of twin-peak QPOs.
Figure 2. Left: χ 2 dependence on the parameters M and j assuming Kerr solution of Einstein field equations. The continuous white curve indicates the mass–angular
momentum relation (7). The continuous thin green curve denotes j giving the best χ 2 for a fixed M. The dashed and thick green curve indicates the same dependence but
calculated using formulae (A2) and (A6) linear in j, respectively. The reasons restricting the calculation of the thick curve up to j = 0.4 are discussed in Section A.1.2.
Right: related profile of the best χ 2 for a fixed M. The arrow indicates increasing j.
2.3. Data Matching
In the right panel of Figure 1, we show the twin-peak frequencies measured in the several atoll and Z-sources2 together
with the observations of Circinus X-1. For the Circinus X-1
data we search for the best fit of the one-parametric relation (4).
Already from Figure 1, where these data are emphasized by the
red/yellow points, one may estimate that the best fit should arise
for M0 ∈ 2–2.5 M . Using the standard least squares method
.
.
(Press et al. 2007) we find the lowest χ 2 = 15 = 2 dof for
.
the mass M0 = 2.2 M which is consistent with the value reported by Boutloukos et al. The symmetrized error corresponding to the unit variation of χ 2 is ±0.3 M . The asymmetric
evaluation of M0 reads 2.2[+0.3; −0.1] M . The white curve in
Figure 2 indicates the mass–angular momentum relation implied
by Equation (6),
M = 2.2 M [1 + k(j + j 2 )],
k = 0.55.
(7)
For the exact fits in Kerr spacetime we calculate the relevant
frequency relations for the range of M ∈ 1–4 M and j ∈ 0–0.5.
These relations are compared to the data in order to calculate
2
After Barret et al. (2005a, 2005b), Boirin et al. (2000), Belloni et al.
(2007a), di Salvo et al. (2003), Homan et al. (2002), Jonker et al. (2002a,
2002b), Méndez & van der Klis (2000), Méndez et al. (2001), van Straaten
et al. (2000, 2002), Zhang et al. (1998).
the map of χ 2 . We use the step equivalent to a thousand points
in both parameters and obtain a two-dimensional map of 106
points. This color-coded map is included in the left panel of
Figure 2. One can see in the map that the acceptable χ 2 is
rather broadly distributed. The thin solid green curve indicates
j corresponding to the best χ 2 for a fixed M. It agrees well with
the expected relation (7) denoted by the white curve. The right
panel of Figure 2 then shows in detail the dependence of the
best χ 2 for the fixed M. It is clearly visible that the quality of
the fit tends to very gently, monotonically increase with rising j
and it is roughly χ 2 ∼ 15 for any considered j.
3. DISCUSSION AND CONCLUSIONS
The quality of the fit tends to very gently, monotonically
increase with rising j and it is roughly
χ 2 ∼ 2 dof ⇔ M ∼ 2.2[+0.3, −0.1] M × [1 + 0.55(j + j 2 )].
(8)
Therefore, one cannot estimate the mass without independent
knowledge of the spin or vice versa, and the above relation
provides the only related information implied by the geodesic
RP model.
To obtain relation (8), the exact Kerr solution of Einstein field
equations was considered. The choice of this two-parametric
spacetime description and related formulae (2) is justified by
No. 1, 2010
ON MASS OF CIRCINUS X-1
a large value of the expected mass M0 (see Appendix A.1 for
details). In Appendix A.1.2 we discuss the utilization of the
linearized frame-dragging description. Figure 2 includes the
mass–spin dependence giving best χ 2 resulting when the fitting
of datapoints is based on the associated formulae (A2) and (A6),
respectively. Considering that νL (νU ) formula (3) merge up to the
first order in j with the νL (νU ) relation (A6) linear in j one can
expect that the associated M(j ) relations obtained from fitting
of data should roughly coincide up to j ∼ 0.1–0.2. From the
figure we can find that there is not a big difference between the
resulting M(j ) relations even up to much higher j. The extended
coincidence can be clearly explained in terms of the kHz QPO
frequency ratio R ≡ νU /νL .3
Observations of Circinus X-1 result to R ∼ 2.5–4.5 while
usually it is R ∼ 1.2–3 (and most often R ∼ 1.5; Abramowicz
et al. 2003b; Török et al. 2008a; Yan et al. 2009). Assuming the
RP model along with any j ∈ (0, 1), the ratio R = 2 corresponds
with good accuracy to radii where the radial epicyclic frequency
reaches its maximum (Török et al. 2008c). Only values lower
than R ∼ 2 are then associated with the proximity of the
innermost stable circular orbit (ISCO) where the effects of
frame dragging come to be highly non-linear in both j and r.
Accordingly, for a given j, in the case when R ∼ 3, the individual
formulae restricted up to certain orders in j are already close to
their common linear expansion in j and differ much less than
for R ∼ 1.5 (see Appendix A.1).
The rarely large R and associated high radial distance (both
already remarked by Boutloukos et al. 2006a, 2006b, although in
a different context) in addition to large M0 warrant the relevance
of relation (8) for rather high values of the angular momentum.
Consequently, we can firmly conclude that the upper constrained
limit of the mass changes from the value 2.5 M to 3 M for
j = 0.3 and even to 3.5 M for j = 0.5. The value of M0
that is above 2 M and the increase of M with growing j for
corotating orbits elaborated here are challenging for the adopted
physical model. Further detailed investigation involving realistic
calculations of the NS structure can therefore be effective in
relation to EoS selection or even falsifying the RP model.
Finally, we note that the discussed trend of increase of
estimated mass arising due to rotational effects should be
relevant also for several other sources. Of course, many systems
display mostly low values of R. These low values of R are
in context of the RP model suggestive of proximity of ISCO.
Török (2009) and Zhang et al. (2009) pointed that under
the consideration of the RP model and j = 0, most of the
high-frequency sources data are associated with radii close to
r = 6.75M. Possible signature of ISCO in high frequency
sources data has been also reported in a series of works by
Barret et al. (2005a, 2005b, 2006) based on a sharp drop in
the frequency behavior of the kHz QPO quality factors (for
instance the atoll source 4U 1636-53 denoted by “blueberry”
points in Figure 1 clearly exhibits both low R and a drop of
QPO coherence, see Boutelier et al. 2010). Considering the
proximity of ISCO, high-order non-linearities in both j and r are
important and even small differences between the actual NS and
Kerr metric could have certain relevance. For this reason some
caution is needed when applying our results to high frequency
sources.
3
Orbital frequencies scale with 1/M. For any model considering νL and νU
given by their certain combination, the ratio R represents the measure of radial
position of the QPO excitation (provided that the NS spin and EoS are fixed).
751
This work has been supported by the Czech grants MSM
4781305903, LC 06014, and GAČR 202/09/0772. The authors
thank the anonymous referee for his objections and comments
which helped to greatly improve the paper. We also appreciate
useful discussions with Milan Šenkýř.
APPENDIX
APPROXIMATIONS, FORMULAE, AND EXPECTATIONS
A.1. Matching Influence of Neutron Star Spin
Rotation and the related frame-dragging effects strongly
influence the processes in the vicinity of compact objects and
there is a need of their reflection in the appropriate spacetime
description. External metric coefficients related to up-to-date
sophisticated models of rotating NS are taken out of the model
in two distinct ways. In the first way, the coefficients are obtained
“directly” from differential equations solved inside the numeric
NS model, while in the second (more usual) way, they are
inferred from the main parameters of the numeric model (mass,
angular momentum, etc.) through an approximative analytic
prescription. Several commonly used numerical codes related
to rotating NS have been developed and discussed (see, RNS,
Stergioulas & Morsink 1997; LORENE: Gourgoulhon et al.
2000; and also Nozawa et al. 1998; Stergioulas & Friedman
1995; Cook et al. 1994; Komatsu et al. 1989).
A.1.1. Analytical Approximations and High-mass Neutron Stars
In the context of a simplified analysis of NS frame-dragging
consequences, an approximation through two solutions of
Einstein field equations is usually recalled: Lense–Thirring metric also named linear-Hartle metric (Thirring & Lense 1918;
Hartle & Sharp 1967; Hartle 1967) and Kerr-black-hole metric
together with related formulae (Kerr 1963; Boyer & Lindquist
1967; Carter 1971; Bardeen et al. 1972). It is expected that
the Lense–Thirring metric fits well the most important changes
(compared to the static case) in the external spacetime structure
of a slowly rotating NS. This expectation is usually assumed
for j < 0.1–0.2.4 Due to asymptotical flatness constraints the
formulae related to Lense–Thirring, Kerr and some other solutions considered for rotating NS merge when truncated to the
first order in j. Accordingly, for astrophysical purposes there
is a widespread usage of the approximate terms derived with
the accuracy of the first order in j. While these approximations
are two-parametric, the more realistic approximations—for instance, those given by the HT metric (Hartle & Thorne 1968) and
related terms (Abramowicz et al. 2003a), relations of Shibata
& Sasaki (1998) or the solution of Pachón et al. (2006)—deal
with more parameters and provide less straightforward formulae. Perhaps also because of that they are not often considered
in discussions of concrete astrophysical compact objects.
Astrophysical applicability of the above analytical approaches has been extensively tested in the past 10 years. Criteria
based on the comparison of miscellaneous useful quantities have
The interval 0 < j < 2 × 10−1 is often assumed as one of the several
possible definitions of “slow rotation”. However, in relation to implications of
the frame-dragging effects, the effective size of this interval depends on the
radial coordinate. For x close or below xms the interval in j rather reduces to
low values. On the other hand for x above the radius of the maximum of νr the
interval can be extended to j higher than j = 0.2. The term slow rotation is
also frequently considered in another context. For instance, when using the Ht
metric in NS models the slow rotation is usually associated with the
applicability of the metric and consequently to spins up to ∼800 Hz for most
EOS and NS masses. For these reasons we do not use the term elsewhere in the
paper.
4
752
TÖRÖK ET AL.
Vol. 714
Figure 3. Left: parameter q̃ for several EoS. Shaded areas denote q̃ = 6 and q̃ = 3. Right: ISCO frequencies for the same EoS as used in the left panel. The curves are
calculated for mass 1.4 M and a relevant maximal allowed mass. The curves following from the exact Kerr solution and linear relation (A4) are displayed as well.
The quadratic relation denoted by the black-dashed curve is discussed later in Section A.2.1.
Figure 4. Frequencies of the perturbed circular geodesic motion. Relations for the Kerr metric given by Equation (A2) are denoted by blue and dashed-blue curves.
Relations (A2) are indicated by red curves, while relation (A5) is plotted using the green color. Dotted relations denote the Kerr- and linearized-vertical frequencies
that are not discussed here (see Morsink & Stella 1999; Stella et al. 1999). Inset emphasizes a difference between the radii fulfilling the ISCO condition νr = 0 for the
relations ((2), explicitly given by Equation (A1)), Equation (A2), and the ISCO-radius given by (A3).
Figure 5. Left: the RP model frequency relations given by Equation (3), blue curves; formulae (A2), red curves; relations (A6), green curves. Relation (A7) roughly
determining the applicability of Equation (A6) is denoted by the dashed black/yellow curve. Right: related differences Δν between the lower QPO frequency implied by
the Kerr formulae (3) and those following from Equation (A2) and (A6), respectively indicated by continuous respectively dashed curves. Different colors correspond
to different frequency ratio R. Shaded areas indicate Δν < 5% and Δν < 2%.
been considered for these tests (e.g., Miller et al. 1998a; Berti
et al. 2005). It has been found that spacetimes induced by most
up-to-date NS EoS without inclusion of magnetic field effects
are well approximated with the HT solution of the Einstein field
equations (see Berti et al. 2005, for details). The solution re-
flects three parameters: NS mass M, angular momentum J, and
quadrupole moment Q. Note that Kerr geometry represents the
“limit” of the HT geometry for q̃ ≡ Q/J 2 → 1. The parameter
q̃ then can be used to characterize the diversity between the NS
and Kerr metric.
No. 1, 2010
ON MASS OF CIRCINUS X-1
753
The left panel of Figure 3 displays a dependence of q̃ on
the NS mass. This illustrative figure was calculated following
Hartle (1967), Hartle & Thorne (1968), Chandrasekhar & Miller
(1974), and Miller (1977). The considered EoS are denoted as
follows (see Lattimer & Prakash (2001, 2007) for details):
[EoS1] SLy 4, Rikovska Stone et al. (2003).
[EoS2] APR, Akmal et al. (1998).
[EoS3] AU (WFF1), Wiringa et al. (1988); Stergioulas &
Friedman (1995).
[EoS4] UU (WFF2), Wiringa et al. (1988); Stergioulas &
Friedman (1995).
[EoS5] WS (WFF3), Wiringa et al. (1988); Stergioulas &
Friedman (1995).
Inspecting the left panel of Figure 3 we can see that for
EoS configurations resulting in low or medium mass of the
central star (M up to 0.8Mmax , i.e., roughly up to 1.4M–1.8 M )
depending on EOS, the implied HT geometry is rather different
from the Kerr geometry. More specifically, for a fixed central
density, q̃ strongly depends on the given EoS and substantially
differs from unity. On the contrary, for high mass configurations
q̃ approaches unity implying that the actual NS geometry is
close to Kerr geometry. One can expect that in such cases
formulae related to the Kerr geometry should provide better
approximation than for low values of M. Next, focusing on
high-mass NS, we briefly elaborate some points connected to
the applicability of the Kerr formulae and related linearized
terms.
Note that the root of the expression for ωr2 from Equation (A2)
is of higher order in j so that the exact radius where ωr vanishes
agrees with the solution (A3) only in the first order of j.
The related ISCO frequency can be evaluated as (Kluźniak &
Wagoner 1985; Kluźniak et al. 1990)
A.1.2. Kerr and Linearized Kerr Formulae: Comparison, Utilization,
and Restrictions
When the expression for the radial epicyclic frequency given
by Equation (2) or (A2) is fully linearized in j, it leads to
The radial epicyclic frequency goes to zero on a particular,
so-called marginally stable circular orbit xms (e.g., Bardeen et al.
1972). In Kerr spacetimes it is given by the relation (Bardeen
et al. 1972)
xms = 3 + Z 2 − (3 − Z 1 )(3 + Z 1 + 2Z 2 ),
(A1)
where
Z 1 = 1 + (1 − j 2 )1/3 [(1 + j )1/3 + (1 − j )1/3 ],
Z 2 = 3j 2 + Z 21 .
Below xms there is no circular geodesic motion stable with
respect to radial perturbations. The orbit is often named ISCO
and determines the inner edge of a thin accretion disk. The
corresponding ISCO orbital frequency νK (xms ) represents the
highest possible orbital frequency of the thin disk and the related
“spiraling” inhomogenities (Kluźniak et al. 1990). Dependence
of ISCO frequency on j following from Equation (A1) is shown
in the right panel of Figure 3.
Assuming the description of geodesic motion accurate in the
first order of j, using Taylor expansion around j = 0, one may
rewrite the explicit terms in Equation (2) as
1
6
8j
j
2
2
ΩK = F
,
ω
1
−
.
+
−
=
Ω
r
K
x 3/2
x3
x x 3/2
(A2)
Consequently, linearized formula for the ISCO radius can be
expressed as
2
xms = 6 − 4
j.
(A3)
3
νK (xms ) = (M /M) × (1 + 0.749j ) × νK (xms , M = M , j = 0)
.
= (M /M) × (1 + 0.749j ) × 2197 Hz.
(A4)
This frequently considered relation is included in the right panel
of Figure 3.
In the right panel of Figure 3 we integrate the ISCO frequencies plotted for several EoS (the same as in the left panel).
We choose two groups of models—one calculated for the set
of five different EoS and “canonic” mass 1.4 M , the other
one for the same set of EoS but considering a maximal mass
allowed by each individual EoS. This choice allows for the illustration of medium and high mass behavior of ISCO relations
and comparison of their simple approximations. Clearly, for
medium mass configurations Equation (A4) provides better approximation than using the Kerr-spacetime formulae (see also
Miller et al. 1998a). On the other hand, when the high mass
configurations are considered, the Kerr solution provides better approximation than Equation (A4). Moreover, its accuracy is
higher than the accuracy of both approximations for middle mass
configurations.
A.1.3. Geodesic Frequencies and RP Model
ωr =
(x − 6)x 3/2 + 3j (x + 2)
.
(x − 6)x 7
(A5)
Relation (A5) provides a good approximation except for the
vicinity of xms (j ) as it diverges at x = 6. Note that this
divergence arises only for corotating but not counterrotating
orbits (which we however do not discuss in this paper). For any
positive j < 0.5 the fully linearized frequency (Equation (A5))
does not differ from ωr given by Equation (A2) for more than
about 5% when x 6 + 4j . The left panel of Figure 4 compares
the frequencies of geodesic motion associated directly with Kerr
metric to formulae (A2) and (A6), respectively.
Assuming linearized Keplerian frequency given by
Equation (A2) and the radial epicyclic frequency
(Equation (A5)), we can write for the RP model the relation
between νL and νU as
ν 2/3
√
2j νU (α − 2)
U
νL = νU 1 − 1 − 6α + √
,
, α=
F
F 1 − 6α
(A6)
which equals the first-order expansion of Kerr spacetime
Equation (3) and also to the first-order expansion of the same
relation if it would be derived for Lense–Thirring or HT metric. Similarly to relation (A5), relation (A6) loses its physical
meaning for frequencies close to νK (ISCO) since it reaches a
maximum at frequencies that can be expressed with a small
inaccuracy as
M
νU
(Hz).
(A7)
νL =
12 − νU /200 M
The left panel of Figure 5 compares the frequency relations (A6)
to relations (3) and those following from formulae (A2). It is
754
TÖRÖK ET AL.
useful to discuss their differences in terms of the frequency
ratio R = νU /νL . For a fixed j the frequencies νL and νU scale
with 1/M. The ratio R then represents a “measure” of the radial
position of the QPO excitation. It always reaches R = 1 at
ISCO where the non-linear j terms are important and R = 0
at infinity where the spacetime is flat. Note that R = 2 almost
exactly corresponds to the maximum of νr for any j (Török et al.
2008c).
The right panel of Figure 5 quantifies differences between the
QPO frequency implied by the Kerr formulae (3), relations (A2),
and relation (A6). We can see that differences between the
Kerr relations (3) and those implied by formulae (A2) become
small when R 2 (Δν 5% for j 0.5). For R ∼ 3
and higher, relations (3) and those implied by Equation (A2)
are almost equivalent nearly merging to their common linear
expansion (A5). Note that taking into account relation (A7) the
linear expansion (Equation (A6)) provides reasonable physical
approximation for spins and frequency ratios roughly related as
j 0.3(R − 1).
(A8)
A.1.4. Applications
Several values of NS mass previously reported to be required
by the RP model, including the estimate of Boutloukos et al.,
belong to the upper part of the interval allowed by standard
EoS. We can therefore expect low q̃ and take advantage of the
exact Kerr solution for most of the practical calculations needed
through the paper. Unlike formulae truncated to certain order, all
the formulae derived from the exact Kerr solution are from the
mathematical point of view fully self-consistent for any j. This
allows us to present the content of Appendix A.2 in a compact
and demonstrable form.
In Section 3, we finally compare the results of QPO frequency
relation fits for Circinus X-1 using the Kerr solution and those
done assuming Equations (A2) and (A6), respectively. From the
previous discussion it can be expected that for Circinus X-1,
due to its exceptionally high R, the fits obtained with the Kerr
formulae (3) and “linear” formulae (A2) should nearly merge
with the fits obtained assuming the common linear expansion
(Equation (A6)). Note also that, on a technical side, the linear
expansion can be used up to j ∼ 0.3–0.4 since the highest R in
the Circinus X-1 data is R ∼ 2–2.5 (Equation (A8)).
A.2. Uniqueness of Predicted Curves and “Ambiguity” in M
The radial epicyclic frequency vanishes at xms . In the RP
model it is then νUmax = νLmax = νK (xms ). Obviously, if there are
two different combinations of M and j which, based on the RP
model, imply the same curve νU (νL ), such combinations must
also imply the same ISCO frequency.
In the left panel of Figure 6 we show a set of curves
constructed as follows. We choose M0∗ = 2.5 M and j ∈
(0, 0.5) and for each different j we numerically find M such
that the corresponding ISCO frequency is equal to those for M0∗
and j = 0. Then we plot the νU (νL ) curve for each combination
of M and j. We can see that except for the terminal points the
curves split. The frequencies in the figure can be rescaled for any
“Schwarzschild” mass M0 as M0∗ /M0 . Thus, the scatter between
the curves provides the proof that one cannot obtain the same
curve for two different combinations of M and j.
On the other hand, the discussed scatter is apparently small
and the curves differ only slightly in the concavity that grows
with increasing j. This has an important consequence. The
curves are very similar with respect to the typical inaccuracy
Vol. 714
of the measured NS twin-peak data and there arises a possible
mass–angular momentum ambiguity in the process of fitting the
datapoints. Next, we derive a simple relation approximating this
ambiguity.
A.2.1. Formulae for ISCO Frequency
The ambiguity recognized in the previous section is implicitly
given by the dependence of the ISCO frequency on the NS
angular momentum which for the Kerr metric follows from
relations (2) and (A1). In principle we can try to describe the
ambiguity starting with these exact relations. The other option
is to assume an approximative formula for the ISCO frequency.
One can expect that this formula should be at least of the second
order in j if consideration of spin up to j = 0.5 is required. We
check an arbitrarily simple form
νK (xms ) = (M /M) × [1 + k(j + j 2 )] × 2197 Hz. (A9)
The right panel of Figure 6 indicates the square of difference
between the exact ISCO frequency in Kerr spacetimes following
from Equation (A1) and the value following from Equation (A9).
Inspecting the figure we can find that the particular choice of
k = 0.75 provides a very good approximation.
Figure 7 then directly compares the exact relation and relation (A9) with k = 0.75. For comparison, the first-order
Taylor expansion formula (A4) is indicated. Clearly, using
Equation (A9) one may well approximate the Kerr-ISCO frequency up to j ∼ 0.4 and describe the discussed ambiguity in
terms of Schwarzschild mass M0 as
M ∼ [1 + k(j + j 2 )]M0 ,
(A10)
where k = 0.75. In further discussion we therefore assume this
formula.
A.2.2. Comparison Between Curves
The curves given by Equation (A10) with k = 0.75 are
illustrated in the left panel of Figure 8. Here we quantify their
(apparent) conformity and investigate its dependence on k. It
is natural to consider the integrated area S between the curve
for M0 , j = 0 and the others as the relevant measure. The
right panel of Figure 8 shows this area as the function of k in
Equation (A10) for several values of j. The same panel also
indicates the values related to the set of curves for mass found
numerically from the exact Equations (A1), i.e., curves in the left
panel of Figure 6. We can see that values of S for k = 0.75 are
comparable to those related to Figure 6. Moreover, for a slightly
different choice of k = 0.7, all the values are smaller. The
ambiguity in mass with relation (A10) is therefore best described
for k ∼ 0.7 when the data uniformly cover the whole predicted
curves.
The available data are restricted to certain frequency ranges
and often exhibit clustering around some frequency ratios νU /νL
(see Abramowicz et al. 2003b; Belloni et al. 2007b; Török et al.
2008c, 2008a, 2008b; Barret & Boutelier 2008b; Török 2009;
Boutelier et al. 2010; Bhattacharyya 2009). It is then useful
to separately examine the mass ambiguity for related segments
of the curves. Such investigation is straightforward for small
segments. Let us focus on a single point [νL , νU ] representing
a certain frequency ratio for a non-rotating star (j = 0) of
mass M0 . Assuming relation (A10) one may easily calculate
the value of k which rescales the mass to M = M0 for a fixed
No. 1, 2010
ON MASS OF CIRCINUS X-1
755
Figure 6. Left: set of curves plotted for various combinations of M and j giving identical ISCO frequency. Right: the square of difference between the exact ISCO
frequency and the frequency given by Equation (A9).
Figure 7. Left: ISCO frequency calculated from Equation (A9) vs. exact relation implied by the Kerr solution (dashed vs. thick curve). The linear relation (A4) is
shown as well for comparison (dotted curve). Right: the related relative difference from ISCO frequency in Kerr spacetime.
Figure 8. Left: the set of curves plotted for combinations of M and j given by Equation (A10) with k = 0.75. Right: the integrated area S related to Equation (A10).
Different values of j are color-coded. The same color code is relevant for horizontal lines. These lines denote the values of S arising for the set of curves numerically
found from Equation (A1) and plotted in the left panel of Figure 6. The two red vertical lines denote the case of k = 0.75 (curves νU (νL ) shown in the left panel of this
figure) respectively k = 0.7 (see the text for explanation).
Table 1
The Coefficient k Representing Mass–Angular Momentum Ambiguity (A10)
Segment
νL /νU
νL /νU
νL /νU
νL /νU
νL /νU
νL /νU
∼ 1.5
∼2
∼3
∼4
∼5
∼6
Whole curve
k in M ∼ [1 + k(j + j 2 )]M0
l (%)
Distance from ISCO × M /M (km)
0.75
0.65
0.55
0.50
0.45
0.40
25
50
70
80
83
85
1
3
7
12
16
20
0.7
756
TÖRÖK ET AL.
Figure 9. Values of k approximating the M – j ambiguity for the individual
segments. The upper axes indicate the length of the curve νU (νL ) integrated from
the ISCO point to the relevant frequency ratio.
(A color version of this figure is available in the online journal.)
non-zero j in order to get exactly the same point [νL , νU ]. We
applied this calculation for νU /νL ∈ (1, 10) and j ∈ (0, 0.5).
The output is shown in Figure 9. From the figure, it is possible
to find k that should best describe the ambiguity for a given
frequency ratio (and thus for a small segment of data close
to the ratio). It also indicates the length of the curve νU (νL )
integrated from the terminal (i.e., ISCO) point to the relevant
frequency ratio (assuming j = 0). This length is given in terms
of the percentage share l on the total length L of the curve νU (νL ),
whereas the absolute numbers scale with 1/M.
Apparently, the segments with νU /νL ∈ (1, 2) cover about
50% of the total length L while k only slightly differs from the
value of 0.7. We recall that this top part corresponds to most of
the atoll and Z-sources data. For the segments of curves related
to the sources exhibiting high frequency ratios such as Circinus
X-1, there is an increasing deviation from the 0.7 value and the
coefficient reaches k ∼ 0.6–0.5. More detailed information is
listed up to νU /νL = 5 in Table 1 providing the summary of this
section.
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L171
Přı́loha 7
On magnetic-field induced non-geodesic
corrections to the relativistic precession QPO
model
Pavel Bakala, Eva Šrámková, Zdeněk Stuchlík, Gabriel Török
Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava
Bezručovo nám. 13, CZ-746 01 Opava, Czech Republic
Abstract. Fitting the observational data of the twin peak kHz quasiperiodic oscillations (QPO)
from low mass X-ray binaries (LMXBs) by the relativistic precession model gives a substantially
higher neutron star mass estimate, M ∼ 2M⊙ , than the "canonical value", M ∼ 1.4M⊙ . Using a fully
general relativistic approach we discuss the non-geodesic corrections to the orbital and epicyclic
frequencies of slightly charged circularly orbiting test particles caused by the presence of a neutron
star magnetic field. We show that consideration of such non-geodesic corrections can bring down the
neutron star mass estimate and improve the quality of twin peak QPO data fits based on relativistic
precession frequency relations.
Keywords: stars: neutron — X-rays: binaries — stars: magnetic fields
PACS: 95.30.Sf, 97.10.Gz, 97.10.Ld, 97.80.Jp
INTRODUCTION
X-ray timing measurements provided by the RXTE satellite have revealed existence of
nearly periodic modulation of X-ray flux detected from several low-mass X-ray binaries
(LMXBs), so called quasi-periodic oscillations (QPOs).
Particular, so called high frequency (kHz) QPOs often come in pairs consisting of the
so called lower and upper QPO mode with frequencies νL , νU . Notably, the frequencies
νL , νU roughly correspond to Keplerian periods in the close vicinity of the binary
compact object; see [11] for a review. Miscellaneous orbital QPO models have been
proposed [see, e.g., 14, 3, 16]. In particular, relativistic precession (in next RP) model
relates the upper and lower kHz QPOs to the Keplerian and periastron precession
frequency on an orbit located in the inner part of the accretion disc 1 . Generally, for
neutron star sources correlation between νU (νL) is qualitatively well fitted by the RP
model prediction [see, e.g., 18, 19, 4].
Nevertheless, there are difficulties when modelling QPO frequency relations from
the RP model for individual sources. The mass and angular momentum relevant to the
best fits are questionably high (M ∼ 2 ÷ 3M⊙ , j ∼ 0.2 ÷ 0.4); [see, e.g., 19, 8, 4, 20].
Also the quality of the fits is not satisfactory with chi-square indicating a systematic
deviation between the expected and empirical trend. It has been discussed that the
1 The same model relates another particular so called low frequency QPOs to the “Lense–Thirring” orbit
precession.
above mentioned discrepancies could be connected to non-geodesic corrections to the
orbital and epicyclic frequencies, most likely originating in the presence of a neutron
star magnetic field [18, 19, 20].
In the present paper we discuss in detail non-geodesic perturbative corrections implied by a Lorentz force acting on a slightly charged circularly orbiting matter in the
approximation of a spherically symmetric spacetime and intrinsic dipole magnetic field
of the neutron star2 .
CIRCULAR ORBITAL MOTION IN A DIPOLE MAGNETIC
FIELD ON THE SCHWARZSCHILD BACKGROUND
The line element in the Schwarzschild spacetime using geometric units, c = G = 1, has
the familiar form
dr2
2M 1/2
2
2
2
2
2 2
2
. (1)
+ r (dθ + sin θ dφ ) ,
ds = −η (r) dt +
η (r) ≡ 1 −
η (r)2
r
Solving the vacuum Maxwell equations on the background of the spacetime geometry
(1) for a static magnetic dipole moment µ , parallel to the rotational axis of the star, one
obtains formula for an exterior (r > R, where R is the neutron star radius) four-potential
Aµ [e.g., 22, 7],
M
3r3
2M
µ sin2 θ
φ
2
1+
.
(2)
log η (r) +
,
f (r) =
Aµ = − δµ f (r)
r
8M 3
r
r
In case of potential (2), the Maxwell tensor Fµν has only two independent nonvanishing
components,
µ sin2 θ ( f (r) − r f ′ (r))
µ f (r) sin 2θ
,
−Fθ φ = Br =
.
(3)
2
r
r
Throughout this paper we confine ourselves to studying only circular equatorial motion
with appropriate four-velocity U µ = (U t , 0 , 0 ,U φ )3 . Solving the radial component of
equation of motion (q̃ ≡ q/m is the specific charge of the particle)
Frφ = Bθ =
dU µ
µ
µ
+ Γαβ U α U β = q̃ Fν U ν
(4)
dτ
together with the normalization condition U µ Uµ = −1 for metric (1) and potential (2)
we obtain the nonzero components of U µ in the form
s
r − q̃ µ Φ(r)U φ
ϒ(r , q̃ , µ )
,
Uφ = 3
,
(5)
Ut =
(r − 3M)
2r (r − 3M)
2
We restrict here ourselves to the following assumptions:
the frame-dragging effects are not considered; the neutron star magnetic field is fully dominant over the
magnetic field generated by the currents in the disc.
3 See [13] for a discussion of the existence of nonequatorial, so called "halo", orbits.
and the angular velocity defined as Ω = U φ /U t then reads
Ω=
ϒ(r , q̃ , µ )
r3/2
p
4r4 (r − 3M) − 2q̃ µ Φ (r) ϒ(r , q̃ , µ )
.
(6)
Here Φ(r), χ (r), Ψ(r) and ϒ(r , q̃ , µ ) are given by
Φ(r) ≡ f (r) − r f ′ (r) ,
q
Ψ(r) ≡ 4Mr4 (r − 3M) + (q̃µ χ (r))2 ,
χ (r) ≡ (r − 2M) Φ(r) ,
ϒ(r , q̃ , µ ) ≡ Ψ (r) − q̃ µ χ (r) .
One may obtain the formulae for epicyclic frequencies by perturbing the particle’s
position around the stable circular orbit (r, θ ) = (r0 , π /2), i.e., by presuming that
xµ (τ ) = zµ (τ ) + ξ µ (τ ) where ξ µ (τ ) is a small perturbation [1, 2]. In the spacetime
geometry (1) and magnetic field (2) the appropriate explicit expressions are given by
2
2
2
−7
t −2
ωr = r
U
U φ r6 (3r − 8M) + 2M(M − r)r3 U t
h
io
+ q̃ µ Φ(r) 2U φ r3 (3r − 7M) + q̃ µ χ (r) +U φ r5 (r − 2M) f ′′ (r) ,
(7)
U φ U φ r3 − 2q̃ µ f (r)
2
ωθ =
.
(8)
(U t )2 r3
MAGNETIC FIELD CORRECTIONS TO ORBITAL AND
EPICYCLIC FREQUENCIES
We restrict our consideration to the approach of slowly rotating neutron star that posseses
a dipole magnetic field and a thin accretion disc that is assumed to consist of test particles
moving along nearly circular geodesics in the equatorial plane. As the Maxwell tensor
projected into an orthonormal basis of observer located at the equator on the surface of
the star with radius R has only Fr̂φ̂ non-zero component, one may write
√
4M 3 R3/2 R − 2M
µ=
Bsur f ace .
(9)
6M(R − M) + 3R(R − 2M) log η (R)2
For a neutron star with a rather weak magnetic field strength, Bsur f ace = 107 Gauss =
2.875 x 10−16 m−1 , mass M = 1.5M⊙ and radius R = 4M, we have µ = 1.06 x 10−4 m−2 .
We present here the resulted frequencies for the above value of µ and two different
values of q̃, q̃ = 5.555 x 1010 and q̃ = 1.111 x 1012. Both of these values are still very low
in comparison with the value q̃ = 1.111 x 1018 corresponding to matter purely consisting
of ions of hydrogen. The left panel of Fig. 1, made for q̃ = 5.555 x 1010 , shows a high
sensitivity of the radial epicyclic frequency keeping qualitatively the same profile that is
however shifted to lower values and away from the central object.
The presence of the dipole magnetic field also violates the νK = νθ equality corresponding to spherical symmetry of the background Schwarzschild geometry. However
this corrections are much less significant.
FIGURE 1. Left: An illustration of the radial epicyclic, νr0 = ωr0 /(2π ), vertical epicyclic, νθ0 =
ωθ0 /(2π ), and orbital, νK0 = ΩK /(2π ) = νθ0 , frequency behaviour in the Schwarzschild geometry in a pure
geodesic case compared to case with a presence of an intrinsic external dipole magnetic field B = 107
Gauss on the surface of the star with M = 1.5 M⊙ and R = 4M (quantities νK , νθ and νr without a
superscript). Right: The same comparison but for a higher value of the specific charge q̃.
Effective innermost stable circular orbit (EISCO)
The Lorentz force naturally alters the location of a charged test particle’s effective
innermost stable circular orbit (in next EISCO) given by the condition ωr (rEISCO ) =
0 . With growing values of q̃ it rapidly draws apart from the well-known radius of
ISCO in the Schwarzschild geometry, rISCO = 6M. In case of µ = 1.06 x 10−4 m−2
corresponding to Fig. 1 we find that for q̃ = 5.555 x 1010 there is rEISCO = 7.39M,
while for q̃ = 1.111 x 1012 we obtain rEISCO = 22.16M. For the extremal specific charge
q̃ = 1.111 x 1018 the location of EISCO orbit flies away onto rEISCO = 177864.76M.
IMPLICATIONS FOR THE RELATIVISTIC PRECESSION QPO
MODEL AND DISCUSSION
The widely discussed RP QPO model identifies the frequencies of the lower and upper
QPO peaks (νL and νU , respectively ) as
νL (r) = νK (r) − νr (r),
νU (r) = νK (r),
(10)
where νK (r) and νr (r) are the orbital and radial epicyclic frequencies [17]4 . It has been
shown by [4] that these relations qualitatively well describe the trends presented in the
observational data, but the characteristic mass of neutron stars in LMXBs obtained by
such fits, M ∼ 2M⊙ , is high in comparison with the canonical value. Considering in
the RP model the the corrected frequencies introduced above, the new fits can provide
4
The orbital and epicyclic frequencies also play a significant role in the QPO models dealing with warped
disc [e.g., 9] and tori [e.g., 5] oscillations. Our conclusion is therefore touching directly not only the hot
spot kinematic QPO models, like the RP model, but also the "disc or torus oscillation - like" QPO models.
FIGURE 2. Inspired by [4]. The RP model rough fits of the observational twin peak kHz QPO data for a
wide set of LMXBs. The thick solid curve refers to the case with M = 1.4M⊙ and the orbital and epicyclic
frequencies being corrected by the presence of the Lorentz force induced by the specific charge of orbiting
matter, q̃ = 5 x 1010, and the star intrinsic magnetic dipole moment, µ = 1.06 x 10−4 m−2 . We also present
fits corresponding to a pure geodesic case (thin dashed curves) for M = 2M⊙ that was discussed by [4]
including data from [8, 23, 15, 4].
the characteristic neutron star mass close to the canonical value, M ∼ 1.4M⊙. We
illustrate this finding in Fig. 2 for the intrinsic magnetic dipole moment of the star,
µ = 1.06 x 10−4 m−2 , and the specific charge of the orbiting matter, q̃ = 5 x 1010 , when
the effective innermost stable circular orbit is shifted to rEISCO ∼ 7M. Such a rough
fit for a wide set of LMXBs is shown together with the fits for a pure Schwarzschild
geodesic cases with M = 2M⊙ [4] and M = 1.4M⊙.
A natural implication of the RP model (and several other models) identifies the
highest observed frequency of a particular source with the orbital frequency at ISCO.
It is then possible to derive the mass of source using this direct identification [see, e.g,
10, 14]. Even here straightforward replacing the geodesic ISCO orbital frequency by the
corrected EISCO one provides a significant decrease of the estimated mass. Moreover,
it was shown by [20] that the lowering of the radial epicyclic frequency corresponding
above discussed corections may in general significantly improve the quality of the fits
based on the RP model.
It is widely expected [e.g, 12, 11] that magnetic field of the central compact objects
in LMXBs should be given by an intrinsic exterior magnetic field, B ∈ 106 ÷ 109 Gauss.
There are also several indices supporting evidence of matter being accreted in the region
with r ≤ 10M [see, e.g., 11]. Our results then imply that the specific charge related to the
accreted plasma should not exceed q̃ ∼ 1.86 x 1012 (1.87 x 1011, 1.90 x 1010, 1.91 x 109)
for B = 106 Gauss (107 , 108 , 109 Gauss).
Discussed values of the specific charge are small in comparison to the charge of a
fully ionized matter. Here we do not touch a problem of the (considerable) magnetic
field induced by such a rotating charge. The full discussion of its role exceeds the
framework of the paper. We however note that in principle its external exposure can
be supressed by an influence of a corotating charge in a corona if the total assumed
charge is approximately zero.
Finally we stress that also the diamagnetic effects should be considered in order
to obtain coherent formulae describing approximately motion of a slightly charged
accreted matter. We plan to include relevant corrections within a fully general relativistic
approach in our consequent work.
ACKNOWLEDGMENTS
This work has been supported by the Czech grants LC 06014 (PB, ES) and MSM
4781305903 (ZS, GT). We thank to W. Kluzniak and D. Psaltis for comments.
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Přı́loha 8
IOP PUBLISHING
CLASSICAL AND QUANTUM GRAVITY
Class. Quantum Grav. 27 (2010) 045001 (19pp)
doi:10.1088/0264-9381/27/4/045001
On magnetic-field-induced non-geodesic corrections to
relativistic orbital and epicyclic frequencies
Pavel Bakala, Eva Šrámková, Zdeněk Stuchlı́k and Gabriel Török
Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava,
Bezručovo nám. 13, CZ-746 01 Opava, Czech Republic
E-mail: [email protected]
Received 5 May 2009, in final form 17 December 2009
Published 20 January 2010
Online at stacks.iop.org/CQG/27/045001
Abstract
We discuss non-geodesic corrections to orbital and epicyclic frequencies of
charged test particles orbiting a non-rotating neutron star with a dipole magnetic
field. Using a fully relativistic approach we consider the influence of both the
magnetic attraction and repulsion on the orbital and epicyclic motion. The
magnetic repulsion introduces a rather complex and unusual behaviour of the
circular orbital motion that is well defined down to the radius where the vertical
epicyclic frequency loses its meaning. We demonstrate that for the intensity
of the magnetic interaction appropriately restricted, the stable circular orbits
extend down to the magnetic innermost stable circular orbit (MISCO) that is
located well under the geodetic innermost stable circular orbit (GISCO) and
even can reach the region under the photon circular orbit. The lowest stable
MISCO
= 2.73M, associated with the highest possible orbital
circular orbit at rmin
max
frequency νK = 3284 Hz(1.5 M /M), corresponds to the critical value of the
particle-specific charge and the neutron star magnetic dipole moment product
(q̃μ)crit = 1.87M 2 . For the magnetic attraction acting above the GISCO, the
situation is much more simple and we demonstrate that the most significant
correction arises for the radial epicyclic frequency and consequently for the
location of the MISCO when the strong magnetic attraction pushes its location
far behind the location of GISCO. We show that the Lorentz force also naturally
violates the equality of the orbital and vertical epicyclic frequencies implied
by the spherical symmetry of the background Schwarzschild geometry giving
rise to the new effect of nodal precession of the orbital motion plane. Finally, we
apply the magnetic attraction corrections on the relativistic precession model of
the twin-peak high-frequency quasiperiodic oscillations observed in the galactic
low mass x-ray binaries, showing possible high relevance of the modified radial
epicyclic frequency.
(Some figures in this article are in colour only in the electronic version)
0264-9381/10/045001+19$30.00 © 2010 IOP Publishing Ltd Printed in the UK
1
Class. Quantum Grav. 27 (2010) 045001
P Bakala et al
1. Introduction
The study of charged particles motion in strong gravitational and electromagnetic fields related
to black holes and neutron stars enables us to understand the nature of the objects as well as
the structure of the force fields and their role in astrophysical phenomena. The motion has
been investigated both for Kerr–Newman black holes having intrinsically coupled gravitational
and electromagnetic fields and for strong gravitating objects (black holes and neutron stars)
with a test electromagnetic field influenced by gravity (see, e.g., Johnston and Ruffini (1974),
Prasanna and Vishveshwara (1978), Prasanna (1980), Calvani et al (1982), Balek et al (1989),
Bičák et al (1989), Vokrouhlický and Karas (1991), Stuchlı́k and Hledı́k (1998), Stuchlı́k
et al (1999), Abdujabbarov and Ahmedov (2009)).
It has been shown that magnetic fields around a rotating black hole could be related to
the extraction of the rotation energy from the black hole through the so-called Blandford–
Znajek process enabling formation of the relativistic jets along the black hole rotation axis
(Blandford and Znajek 1977). Motion of charged particles in the magnetic field generated
by accretion discs orbiting black holes was discussed in Znajek (1976) and Mobarry and
Lovelace (1986). On the other hand, the magnetic field tied to a neutron star could
substantially influence the structure of an equatorial accretion disc orbiting the neutron star.
Here we focus our attention on the equatorial orbital and epicyclic motion in the combined
gravitational and dipole magnetic fields related to a slowly rotating neutron star. Its spacetime is
represented by the Schwarzschild geometry that influences the structure of the dipole magnetic
field.
In the case of motion in test fields on strong gravity backgrounds, the equations of
motion are complex and have to be integrated numerically (Prasanna and Vishveshwara 1978,
Prasanna and Sengupta 1994, Preti 2004). Quite recently, off-equatorial circular orbits were
discussed in astrophysically relevant situations (Kovář et al 2008, Stuchlı́k et al 2009b). Of
high interest is the equatorial motion, especially the circular and quasi-circular orbits that
seem to be crucial from the point of view of accretion processes. Numerical integration of
the motion equations gives a number of interesting results but is not sufficient for a complete
classification and understanding of the motion in the equatorial plane. In order to extend the
understanding of the charged particle motion, we consider for the first time its very important
aspect, namely the quasi-circular equatorial epicyclic motion corresponding to oscillations of
particles around stable circular orbits. It is quite interesting that such epicyclic motion can be
excited in the innermost parts of the accretion discs orbiting a neutron star by inhomogeneities
(mountains) on its surface (Stuchlı́k et al 2008).
The epicyclic motion could be relevant in modelling the high-frequency quasi-periodic
oscillations (QPOs) that have been detected during the past two decades from a number of
low-mass x-ray binaries (LMXBs)1 containing a neutron star. These oscillations occur at
frequencies lying in the kHz range and often come in pairs of the lower and upper QPO mode
with frequencies νL , νU , forming the so-called twin-peak QPOs. Notably, νL , νU roughly
correspond to Keplerian periods in the close vicinity of the binary compact object (see, e.g.,
van der Klis (2006)). Moreover, there are indications that the twin-peak frequencies are
clustered near rational ratios that are mostly around 3:2, but also 4:3 and 5:4 (see, e.g., Török
et al (2008a), (2008b), (2008c)).
Miscellaneous orbital QPO models have been proposed (see, e.g., Lamb et al (2007),
Aschenbach (2007) and Miller (2007)). In particular, the relativistic precession (RP) model
(Stella and Vietri 1999) relates the upper and lower kHz QPOs to the Keplerian and periastron
1
2
Binary systems containing a neutron star where the companion mass is smaller than the mass of the neutron star.
Class. Quantum Grav. 27 (2010) 045001
P Bakala et al
precession frequency on a geodesic orbit located in the inner part of the accretion disc2 . It
has been noted that, in general, correlation νU (νL ) is qualitatively well fitted by the RP model
prediction (see, e.g., Stella and Vietri (1999), (1999) and Belloni et al (2007)). There are,
however, other QPO models based on the oscillations of toroidal discs (Šrámková et al 2005,
Straub and Šrámková 2009) or ‘discoseismology’ (Kato et al 1998, Pétri 2005) and so far no
definite agreement on the validity of the QPO models has been established. Let us stress that
the orbital and epicyclic frequencies, that will be discussed in this paper, play an important
role in all of the mentioned QPO models.
When modelling individual frequency relations from the RP model, mass and angular
momentum relevant to the best fits are questionably high (M ∼ 2–3M , j ∼ 0.2–0.4) (see,
e.g., Stella and Vietri (2002), Boutloukos et al (2006), Belloni et al (2007) and Török et al
(2007a)) in comparison with the ‘canonical value’, M ∼ 1.4M , which has been estimated
for a variety of well-studied pulsars (e.g. Glendenning (1997) and Weber (1999). Also, quality
of the fits is not satisfactory with chi-square indicating a systematic deviation between the
expected and empirical trend (Belloni et al 2007, Török et al 2007a, 2007b). In fact, we
show that both discrepancies can be corrected by non-geodesic corrections of the orbital
and epicyclic frequencies using the magnetic attraction introduced in the present paper. On
the other hand, the magnetic repulsion makes the situation worse due to the shift to higher
frequencies (neutron star masses).
In this paper we discuss in detail the non-geodesic, magnetic corrections to the epicyclic
motion using a fully general relativistic approach. These corrections are assumed to be implied
by the Lorentz force acting on a slightly charged matter in the approximation of a spherically
symmetric spacetime. We focus consideration to the case of motion in the field of magnetized
neutron stars. We use the approximation of a dipole magnetic field whose axis of symmetry
coincides with the axis of neutron star’s rotation. The spacetime outside the neutron star is
described by the Schwarzschild geometry and the effects of frame-dragging and contribution of
the electromagnetic field to the stress–energy tensor are thus neglected3 . Such approximation
is suitable for describing the charged particles motion around slowly rotating neutron stars with
a relatively weak magnetic field which does not affect the spacetime curvature in the vicinity
of the neutron star, but its structure is governed by the neutron star spacetime structure4 .
Epicyclic motion and the related frequencies have so far been extensively discussed for
the quasi-circular geodesic motion (see, e.g., Aliev and Galtsov (1981), Abramowicz and
Kluźniak (2005) and Török and Stuchlı́k (2005)). Using the approach of Aliev (2008), we
turn our attention for the first time to a detailed study of magnetically influenced perturbative
epicyclic motion around equatorial circular orbits. We calculate the relevant frequencies of
the non-geodesic charged test particles motion and the corresponding shift of the position
of the innermost stable circular orbit that is governed by vanishing of the radial (vertical)
epicyclic frequency. We consider both the cases of magnetic attraction when the innermost
stable orbit is shifted above the GISCO and magnetic repulsion when it is shifted below the
geodesic orbit. We find a variety of interesting new phenomena of the epicyclic motion, with
unusual behaviour of the epicyclic frequencies and their relation to the orbital (Keplerian)
frequency. We also discuss some implications of the magnetic attraction case for the RP
2
A similar model relates the low-frequency QPO branch to the ‘Lense–Thirring’ orbit precession; see, e.g., Stella
and Vietri (1998).
3 More general and accurate approximation which takes into account the effects of frame dragging and declination
of the dipole magnetic field symmetry axis can be found in Rezzolla et al (2001a), (2001a).
4 The neutron star magnetic field is however fully dominant over the magnetic field generated by the currents in the
disc.
3
Class. Quantum Grav. 27 (2010) 045001
P Bakala et al
model, in particular the remarkable lowering of the neutron star mass estimation obtained by
fitting the QPO observational data.
2. Circular orbital motion in a dipole magnetic field on the Schwarzschild background
The line element in the Schwarzschild spacetime has the familiar form
ds 2 = −η(r)2 dt 2 +
where η(r) is given by
dr 2
+ r 2 (dθ 2 + sin2 θ dφ 2 ),
η(r)2
(1)
2M 1/2
.
(2)
η(r) ≡ 1 −
r
We have adopted here geometric units, c = G = 1, that we will use throughout the paper, if
not stated otherwise.
Solving the vacuum Maxwell equations
∗ μν
μν
F ;μ = 0
F ;μ = 0
(3)
on the background of the spacetime geometry (1) for a static magnetic dipole moment μ,
parallel to the rotational axis of the star, one obtains the formula for an exterior (r > R, where
R is the neutron star radius) 4-potential Aμ (e.g. Wasserman and Shapiro (1983) and Braje and
Romani (2001)):
μ sin2 θ
,
(4)
r
which has the form of the flat space result, multiplied by a function f (r) given by
2M
M
3r 3
2
1+
.
(5)
In η(r) +
f (r) =
8M 3
r
r
In the case of potential (4), the Maxwell tensor Fμν , connected to the 4-potential Aμ through
the relation
∂Aν
∂Aμ
−
,
(6)
Fμν =
μ
∂x
∂x ν
has only two independent non-vanishing components
Aα = −δαφ f (r)
Frφ =
μ sin2 θ (f (r) − rf (r))
r2
(7)
and
μf (r) sin 2θ
,
r
which are related to the components of a magnetic field three-vector B as follows:
Fθφ = −
Frφ = B θ ,
Fθφ = −B r .
(8)
(9)
Note that the symbol ‘ ’ in equation (7) denotes partial derivative with respect to the radial
coordinate r.
In a curved spacetime with the presence of an electromagnetic field, the equation of
motion for a charged test particle of mass m and charge q reads
dU μ
μ
(10)
+ αβ U α U β = q̃ Fνμ U ν ,
dτ
where U μ is the 4-velocity and q̃ ≡ q/m is the specific charge of the particle.
4
Class. Quantum Grav. 27 (2010) 045001
P Bakala et al
We shall study the epicyclic, near circular motion in the equatorial plane of a neutron star
with a dipole magnetic field. In order to obtain maximal information on the epicyclic motion,
we shall consider its properties down to the minimal radius R = 2.25M allowed for internal
Schwarzschild geometry with uniform energy density distribution (Stuchlı́k 2000). On the
other hand, we put a limit of validity of our result in the field of astrophysically plausible
neutron stars using the minimal radius R ∼ 3.5M allowed for a variety of realistic equations
of state (see the appendix for details). The Lorentz force in the equation of motion, and
consequently the described effects on the orbital motion, depends on the product of μ and
q̃ determining the magnitude of the magnetic interaction. Therefore, instead of changing
the magnitude and orientation of μ we can, without any loss of generality, study only the
influence of changes of the specific charge q̃. We shall focus on a typical LMXB neutron star
with a relatively weak magnetic field strength B = 107 Gauss, mass M = 1.5M and radius
R = 4M. Then the magnetic dipole moment μ = 1.06 × 10−4 m2 and it changes linearly with
the field strength B (see the appendix). In order to keep the magnetic force fixed, the specific
charge q̃ must be changed inversely to changes of B, if the neutron star parameters R and M
remain fixed.
2.1. Orbital angular velocity of equatorial circular orbits
The symmetry properties of the spacetime geometry (1) and electromagnetic field (4) allow
for charged test particles motion restricted to the equatorial plane θ = π/2. Throughout this
paper we confine ourselves to studying only circular equatorial motion5 . The 4-velocity then
has only two non-vanishing components, U μ = (U t , 0, 0, U φ ). Solving the radial component
of equation (10) together with the normalization condition U μ Uμ = −1 for metric (1) and
potential (4), we obtain two pairs of the nonzero components of U μ in the form
−q̃μχ(r) ± (r)
,
2r 3 (r − 3M)
φ
r − q̃μ(r)U±
U±t =
,
(r − 3M)
φ
U± =
φ
and the appropriate angular velocities ± = U± U±t then read
−q̃μχ(r) ± (r)
± =
.
r 3/2 4r 4 (r − 3M) − 2q̃μ(r)[−q̃μχ(r) ± (r)]
Here (r), χ (r) and (r) are given by
(11)
(12)
(13)
(r) ≡ f (r) − rf (r),
(14)
χ (r) ≡ (r − 2M)(r),
(r) ≡ 4Mr 4 (r − 3M) + (q̃μχ(r))2 .
(15)
(16)
For uncharged particles we
arrive at the Keplerian geodesic limit with orbital angular velocity
± (q̃ = 0) = ±K = ± M/r 3 .
The constants of motion of charged particles at the equatorial circular orbits are given by
the relations
E = −Ut = η(r)2 U t ,
5
(17)
See Kovář et al (2008) for discussion of the existence of non-equatorial, so-called halo, orbits.
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Class. Quantum Grav. 27 (2010) 045001
P Bakala et al
Figure 1. The orbital frequency ν = /2π as a function of the specific charge q̃ and radial
coordinate for the test neutron star with M = 1.5M and μ = 1.06 × 10−4 m2 .
L = Uφ + q̃Aφ = r 2 U φ − q̃μ
f (r)
,
r
(18)
with E being the specific energy and L being the generalized specific angular momentum.
It is apparent from the form of equations (11)–(13) that for a fixed magnetic dipole
φ
moment of the neutron star, the 4-velocity components U± and U±t are symmetric with respect
to simultaneous interchange of their sign (orientation of the orbital angular velocity ) and the
sign of the specific charge q̃. It is therefore sufficient to analyse only one of these solutions—in
φ
the following we choose U+ , U+t and + .
φ
The existence of the circular orbits is limited by the condition that both U+t and U+ , defined
by equations (11) and (12), take real values. The reality conditions related to the magnitude
of the magnetic interaction given by q̃μ are given by the relations
4r 4 (r − 3M) − 2q̃μ(r)[−q̃μχ(r) ± (r)] > 0
(19)
4Mr 4 (r − 3M) + (q̃μχ(r))2 > 0.
(20)
and
The first of these conditions is satisfied for all values of q̃μ at all radii r > 2M. The second
condition puts limit on the allowed values of q̃μ at radii 2M < r < 3M. The limit region
starts for q̃μ = 0 at r = 3M reaches its maximum of q̃μ = ±1.971 M 2 at r = 2.441M and
takes the value of q̃μ = ±1.333 M 2 for r → 2M.
In figure 1 we illustrate the behaviour of the orbital angular velocity (related frequency
ν = /2π) in dependence on the specific charge q̃ for a fixed magnetic dipole moment
μ = 1.06 × 10−4 m2 and neutron star mass M = 1.5M . For such a value of μ, the
critical values of the specific charge are given by q̃ = ±8.986 × 1010 at r = 2.441M and by
q̃ = ±6.8×1010 for r → 2M. From figure 1 it follows that for positively charged particles the
Lorentz force has a repulsive character and lowers the orbital frequency with respect to the
Keplerian frequency K corresponding to the geodesic motion (q̃ = 0), while for negatively
charged particles the force is attractive and grows with respect to the Keplerian frequency
K .
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Considering both attractive and repulsive character of the Lorentz force, there exist
three qualitatively different types of the orbital angular velocity profile behaviour. For a
sufficiently small charge, corresponding to orbital motion that is not very far from the geodesic
motion, there is a minimum possible value of r for which the circular orbits may exist. With
increasing magnitude of the specific charge (both positive and negative), there appears a second
region of the existence of circular orbits close to the horizon with a certain maximal value of
the radial coordinate. With growing charge both regions merge and circular orbits exist for all
r > Rg . A surprising behaviour of arises for negatively charged particles under the circular
photon orbit at rph = 3M since they orbit in the opposite direction as compared to those
orbiting above rph. (A similar effect was investigated by Balek et al (1989) for ultrarelativistic
charged particles orbiting in the field of Kerr–Newman black holes.) We should stress,
however, that for astrophysically plausible situations, with neutron stars modelled by using
realistic equations of state, validity of our results, being restricted to exterior regions of the
neutron stars, is limited to r > 3.5M, or r > 2.8M, in the most exotic case of the so-called
Q-stars (see the appendix).
The test particle motion in combined gravitational and magnetic fields can be described by
the effective potential Veff (r, θ ) generally determining 3D motion that is reduced to 2D motion
in the symmetry (equatorial) plane. Examples of such effective potential corresponding to our
discussion can be found in Aliev and Galtsov (1981) and Kovář et al (2008). The epicyclic
motion along a stable circular orbit, given by the condition dVeff /dr = 0, is governed by the
second derivatives of the effective potential. In such approximation the effective potential
takes the form corresponding to the linear harmonic oscillation; therefore, the radial and
vertical epicyclic frequencies are related to the effective potential by
∂ 2 Veff
∂ 2 Veff
2
,
ω
∼
.
(21)
ωr2 ∼
θ
∂r 2
∂θ 2
Clearly, vanishing of the radial ωr and vertical ωθ epicyclic frequencies generally determines
the marginally stable circular orbits that are defined by vanishing of the second derivatives of
the effective potential, putting thus limits on the existence of astrophysically important stable
circular orbits.
A detailed analysis of the effective potential that can give an overview of the stability
for the charged particle circular motion can be found, e.g., in Kovář et al (2008), even for
off-equatorial circular orbits. Here we use a more straightforward and simple perturbative
analysis of the epicyclic motion along equatorial stable circular orbits.
3. Epicyclic frequencies and stability of circular motion
Formulae for the radial and vertical epicyclic frequencies of a charged test particle in the
presence of a general electromagnetic field have been derived by Aliev and Galtsov (1981)
and Aliev (2008). One may obtain the formulae by perturbing the particle’s position around
the equatorial circular orbit (r, θ ) = (r0 , π/2), i.e. by assuming that x μ (τ ) = zμ (τ ) + ξ μ (τ ),
where ξ μ (τ ) is a small perturbation. Substituting this into the equation of motion (10) and
restricting to first-order terms in ξ μ one arrives at the relation for ξ μ that takes the form of
equation for a linear harmonic oscillator:
d2 ξ a
+ ωa2 ξ a = 0,
a ∈ (r, θ ),
(22)
dt 2
with the appropriate epicyclic angular frequencies defined as (Aliev 2008)
r
1/2
∂V
r A
ωr =
,
A ∈ (t, φ)
(23)
− γA γr
∂r
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Class. Quantum Grav. 27 (2010) 045001
ωθ =
∂V
∂θ
P Bakala et al
θ 1/2
,
(24)
where γαμ and V μ have the form
q̃
μ
U β (U t )−1 − t Fαμ ,
γαμ = 2
αβ
U
1
q̃
Vμ =
γαμ U α (U t )−1 − t Fαμ U α (U t )−1 .
2
U
(25)
(26)
Note that the derivatives in equations (23) and (24) must be taken at the appropriate equatorial
circular orbit (r, θ ) = (r0 , π/2) with Ut and U φ given by equations (12) and (11).
In the spacetime geometry (1) and the magnetic field (4), the explicit expressions for the
epicyclic angular frequencies are given by
ωr2 = {(U φ )2 r 6 (3r − 8M) + 2M(M − r)r 3 (U t )2
+ q̃μ[(r)(2U φ r 3 (3r − 7M) + q̃μχ (r)) + U φ r 5 (r − 2M)f (r)]}/r 7 (U t )2 ,
(27)
U φ (U φ r 3 − 2q̃μf (r))
.
(28)
(U t )2 r 3
One can easily check that in the absence of the Lorentz force (μ = 0 or q̃ = 0) the expressions
for the orbital (13) and epicyclic (27), (28) frequencies merge into the well-known formulae
for geodesic motion in the Schwarzschild geometry:
ωr = M(r − 6M)/r 2 .
(29)
= ωθ = K = M/r 3 ,
ωθ2 =
3.1. Radial epicyclic frequency
In the context of the perturbation analysis, the existence of the real values of the radial epicyclic
frequency ωr implies the stability of the circular orbit with respect to small radial perturbations
(which lead to oscillation behaviour of the perturbed radial coordinate of the orbiting particle).
In the left panel of figure 2, the line of ωr = 0 in the q̃–r plane denotes the boundary of region
where the stable circular orbits exist. Outside this region the appropriate (a = r) solution
of equation (22) loses its oscillatory character. In the region corresponding to the attractive
Lorentz force ωr decreases with growing q̃ and the marginally stable orbit with respect to radial
oscillations moves away from the analogous orbit in the purely geodesic case, rms = 6M. On
the other hand, in the region of the repulsive Lorentz force ωr grows as the negative charge
q̃ increases and the boundary of region with ‘radially stable’ orbits approaches the horizon
where ωr diverges to infinity.
3.2. Vertical epicyclic frequency
Analogically, the existence of the vertical epicyclic frequency implies the stability of the
circular orbit with respect to small vertical perturbations. The region where such stable
circular orbits may exist is shown in the right panel of figure 2. As seen from the figure,
the behaviour of ωθ exhibits a bit more complicated features than those in the case of ωr .
There are two separate curves of ωθ = 0 defining a part of the boundary of region with circular
orbits that are stable with respect to vertical perturbations. One of the curves lies in the area of
the repulsive Lorentz force, while the other one corresponds to an area with attractive character
of the Lorentz force. Contrary to the radial case, this region of ‘vertical stability’ never reaches
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Class. Quantum Grav. 27 (2010) 045001
P Bakala et al
Figure 2. Left: contour plot of the radial epicyclic frequency νr = ωr /2π as a function of the
specific charge q̃ and the radial coordinate. Right: same as the left panel, but for the vertical
epicyclic frequency νθ = ωθ /2π . Plots are constructed for the test neutron star with M = 1.5M
and μ = 1.06 × 10−4 m2 .
Figure 3. Left: the region of stable circular orbits filled up by the contour plot of the orbital
frequency ν = /2π . Right: same as the left panel, but filled up by the contour plot of the nodal
precession frequency νn . Constructed for M = 1.5M and μ = 1.06 × 10−4 m2 .
the horizon. For relatively small values of both positive and negative charge corresponding
to near geodesic motion, the rest of the boundary of the region where the vertically stable
circular orbits exist coincides with the boundary of region defining the existence of circular
orbits itself (see figure 1).
3.3. Stable orbits and magnetic innermost stable circular orbit (MISCO)
Clearly, stable orbits have to be stable to both radial and vertical perturbations simultaneously.
From the above discussion of the behaviour of the radial and vertical epicyclic frequency, it
is apparent that the region of circular orbits which are stable with respect to both radial and
vertical perturbations is defined by the intersection of regions where the radial and vertical
epicyclic frequencies are defined. As shown in the left panel of figure 3, there exists a critical
value of the specific charge, q̃crit , inside the area of the repulsive Lorentz force, such that for
q̃ > q̃crit the boundary of the region of stable orbits in the q̃–r plane is defined by the ωθ = 0
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Class. Quantum Grav. 27 (2010) 045001
P Bakala et al
curve. For q̃ < q̃crit , the boundary of stable orbits region is formed by the curve of ωr = 0.
These curves thus define the location of the marginally stable orbit for particles of a given
q̃ with a fixed μ. For such orbits we introduce the term MISCO (magnetic innermost stable
circular orbit) to distinguish them from the corresponding geodesic innermost stable circular
orbits that we will refer to as GISCO. In the Schwarzschild spacetimes rGISCO = 6M. It is,
therefore, clear that the repulsive Lorentz force gives rise to a new class of stable circular
orbits with r < rGISCO = 6M that extends below the circular photon orbit. The critical charge
MISCO
= 2.73M and
q̃crit corresponds to the lowest MISCO orbit with the radial coordinate rmin
max
for a given mass of the neutron star. The
the highest possible orbital angular frequency location of the MISCOmin orbit is given by the condition that equations ωr = 0 and ωθ = 0
are fulfilled simultaneously, the critical value of the product of the particle specific charge
and the neutron star magnetic dipole moment is thus given by (q̃μ)crit = 1.869M 2 . For the
test neutron star of M = 1.5M and μ = 1.06 × 10−4 m2 , we have q̃crit = 8.76 × 1010 and
ν max = max /2π = 3124 Hz.
4. Relations of the non-geodesic orbital and epicyclic frequencies
The orbital and epicyclic frequencies exhibit a qualitatively different behaviour in regions of
attractive and repulsive magnetic interaction that strongly depends on the particular value of q̃.
For the test neutron star, we present in figure 4 non-geodesic orbital and epicyclic frequency
profiles in typical situations representing both the repulsive and attractive magnetic interaction.
In the region of magnetic repulsion (q̃ > 0), two qualitatively different types of the frequency
profile behaviour are given by the condition q̃ > q̃crit (q̃ < q̃crit ) when the region of stable
orbits is given by ωθ = 0 (ωr = 0).
The resulted frequency profiles are given for four representative values of q̃ lying in both
attractive and repulsive regions. Namely we choose q̃ = 1.0 × 1011 , q̃ = 8.7 × 1010 , q̃ =
−6.0 × 1010 and q̃ = −1.5 × 1011 . Absolute values of all used specific charge values are very
low in comparison with q̃ = 1.111 × 1018 corresponding to matter consisting purely of ions
of hydrogen.
4.1. Magnetic repulsion
The top-left panel of figure 4 displays the behaviour of the investigated frequencies in the
repulsive region for q̃ = 1.0 × 1011 (q̃ > q̃crit ), whereas and ωr are defined for all radii
above the horizon. exhibits a maximum and converges to 0 at the horizon, while ωr
monotonically grows diverging to infinity at the horizon. On the other hand, ωθ exhibits a
maximum and falls to zero. Therefore, the region of stability of the circular orbits is defined
by the radial coordinate rMISCO where ωθ = 0.
Different features are shown in the top-right panel of figure 4 which illustrates the situation
for q̃ = 8.7 × 1010 (still in the repulsive region but for q̃ < q̃crit ). Contrary to the previous
case there exists an interval of radial coordinate values over which is not defined and where
no circular orbits exist. Similarly, ωr is discontinuous and the boundary of the stability region
rMISCO is now defined by the radial coordinate satisfying ωr = 0. Close to the horizon a new
separate region appears where the circular orbits may again exist, although they are stable
only with respect to radial perturbations. For the value of q̃ used here, the upper boundary
of such a region slightly outreaches the minimal possible size of the stellar compact object,
R = 2.25M.
Generally, for stable circular orbits in the repulsive region, ωr increases with growing
charge, while both and ωθ exhibit opposite behaviour. Both the orbital and vertical epicyclic
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Class. Quantum Grav. 27 (2010) 045001
P Bakala et al
Figure 4. Illustration of the radial epicyclic, νr = ωr /(2π ), vertical epicyclic, νθ = ωθ /(2π ),
and orbital, ν = /(2π ), frequency behaviour in the case of the intrinsic external dipole magnetic
field B = 107 Gauss on the surface of the star with M = 1.5 M and R = 4M compared to
the pure Schwarzschild geodesic case (quantities νK = νθ0 and νr0 ). The top panels illustrate the
situation in the repulsive region, for q̃ = 1.0 × 1011 (left) and for q̃ = 8.7 × 1010 (right). Bottom
panels show the behaviour of frequencies from the attractive region for q̃ = −6.0 × 1010 (left) and
q̃ = −1.5 × 1011 (right).
frequencies are lower than the Keplerian frequency K , and the orbital frequency exceeds the
epicyclic one. The influence of the Lorentz force enables extension of the region with stable
circular orbits deep below the Schwarzschild rGISCO = 6M and, surprisingly, even below the
radius of the circular photon orbit rph = 3M.
4.2. Magnetic attraction
The bottom-left panel of figure 4 illustrates the behaviour of the orbital and epicyclic frequency
profiles in the attractive region for q̃ = −6.0 × 1010 . displays a discontinuity that
is characteristic for the whole attractive region and changes its sign at radius r = 3M
corresponding to the circular photon orbit. The region of inversely orbiting radially unstable
circular non-geodesic orbits does not reach the horizon for the chosen value of q̃. The boundary
of the stability region rMISCO is again defined by the radial coordinate where ωr = 0.
The frequency profiles constructed for q̃ = −1.5 × 1011 , shown in the bottom-right panel
of figure 4, are qualitatively somewhat different when compared with the previous magnetic
attraction case. Even in the attractive region, sufficiently large values of negative q̃ enable
extension of the region of existence of the circular orbits down to the horizon; however, such
orbits are, contrary to the case of magnetic repulsion, unstable with respect to both radial and
vertical perturbations. The region of vertical stability is restricted from below by the radial
coordinate for which ωθ = 0, while the region of both radial and vertical stabilities is again
limited by rMISCO such that ωr (rMISCO ) = 0.
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Class. Quantum Grav. 27 (2010) 045001
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Generally, in the attractive region, the orbital and epicyclic frequency profiles exhibit
opposite behaviour from that in the repulsive region. With increasing negative q̃, the frequency
ωr decreases, while both and ωθ grow. Now both orbital and vertical epicyclic frequencies
exceed the Keplerian one K and, moreover, ωθ > . However, at the circular photon
orbit radius rph = 3M, both and ωθ coincide with the Keplerian angular velocity 0
independently of q̃. In the case of magnetic attraction, the MISCO radius strongly draws apart
from the Schwarzschild rGISCO = 6M with growing q̃.
Finally, we can conclude that at astrophysically relevant values of the radial coordinate
(r > 3.5M) the sensitivity of ωr to q̃ is significantly higher than the sensitivity of the remaining
two frequencies for both attractive and repulsive magnetic interactions. This is qualitatively
in accordance with what one would expect, as the Lorentz force acting on charged particles
moving in the equatorial plane has only the radial non-zero component.
5. Nodal precession
The presence of the Lorentz force violates the ν = νθ equality implied by the spherical
symmetry of the background Schwarzschild geometry. In the repulsive region, both ν and νθ
decrease as the specific charge q̃ grows, while in the attractive region these frequencies increase
with rising negative specific charge. However, νθ is changing faster than ν which gives rise
to the nodal precession of the plane of the orbital motion. The nodal precession is present in
addition to the relativistic precession of periastron having frequency νp (r) = ν(r) − νr (r).
The nodal precession frequency is given by the formula
νn (r) = ν(r) − νθ (r).
(30)
This nodal precession of frequency νn is qualitatively similar to the Lense–Thirring precession
(LTP) occurring in rotating, axially symmetric spacetimes. For attractive magnetic interaction,
some of its features differ from those of the repulsive interaction. It is, however, common for
both attractive and repulsive magnetic interactions that for a fixed value of q̃ the frequency
νn (r) exhibits a maximum at the MISCO orbit and decreases with increasing r.
As follows from the definition of νn given by equation (30), for the attractive interaction,
the nodal precession induced by the Lorentz force has an opposite phase as compared to
the LTP, reflected by its negative values in the right panel of figure 3. It is interesting to
plot νn (rMISCO ) versus negative q̃ for fixed μ and M (see the right panel of figure 5). For the
attractive magnetic interaction the nodal precession frequency νn (rMISCO ) is small (νn 1 Hz)
except for a relatively narrow range of q̃ (about q̃ ∼ −1.8 × 1011 ) where it demonstrates a
sharp maximum νn (rMISCO ) = 0.106ν(rMISCO ).
For the repulsive magnetic interaction, the nodal precession phase is consistent with
the LTP phase. When q̃ < q̃crit , the frequency νn (rMISCO ) grows along with growing q̃.
For MISCO orbits with q̃ > q̃crit , there is νθ (rMISCO ) = 0; the nodal precession frequency
νn (rMISCO ) = νK (rMISCO ) and it decreases with growing q̃. It is evident that for the repulsive
interaction νn (rMISCO ) there exhibits a sharp maximum at q̃ = q̃crit which is identical with the
ν max of the lowest stable circular orbit (see the left panel of figure 5).
6. Implications for the relativistic precession QPO model
The widely discussed relativistic precession QPO model identifies the frequencies of the lower
and upper QPO peaks (νL and νU , respectively) as
νL (r) = ν(r) − νr (r),
12
νU (r) = ν(r).
(31)
Class. Quantum Grav. 27 (2010) 045001
P Bakala et al
Figure 5. The behaviour of νn at the innermost stable circular orbit in the presence of the intrinsic
external dipole magnetic field with B = 107 Gauss on the surface of the test neutron star with
M = 1.5 M and R = 4M as a function of q̃. Left: in the repulsive region the frequency has
its maximum νn = ν max = 3124 Hz for q̃ = q̃crit at the lowest stable circular orbit. Right: in
the attractive region the frequency has its maximum νn = 78.3 Hz for q̃ = −1.8 × 1011 which
corresponds to the shift of MISCO to 9.9 M.
It has been shown by Belloni et al (2007) that these relations qualitatively well describe
the trends presented in the observational data, but the characteristic mass of neutron stars in
LMXBs obtained by such fits, M ∼ 2M , is too high in comparison with the canonical value,
M ∼ 1.4M . Moreover, it was demonstrated by Török et al (2007a) that decreasing the radial
epicyclic frequency may in general notably improve the quality of fits based on the RP model.
The significant reduction of νr (r) along with keeping the other frequencies more or less the
same well corresponds to the above-discussed features of the frequencies in the region of the
attractive Lorentz force.
Consider an astrophysically relevant situation of a rather slowly rotating neutron star that
possesses a dipole magnetic field and is orbited by a thin accretion disc consisting of charged
test particles moving along nearly circular geodesics in the equatorial plane. In addition we
assume the dipole magnetic field to be fully dominant in the total electromagnetic field in the
vicinity of the star, so that the influence of the magnetic field generated by the currents in the
disc and the influence of the total disc charge are both negligible. This criterion is fulfilled if
the specific charge of the material in the disc is very low. Further, such a configuration allows
us to use the test particle approximation, and this is in agreement with the assumed rather
small non-geodesic corrections to geodesic orbital motion.
Considering the RP model in line with the corrected frequencies introduced above, the
new fits can provide the characteristic neutron star mass close to M ∼ 1.4M . In figure 6 we
illustrate this finding for μ = 1.06 × 10−4 m2 and q̃ = 5 × 1010 when the innermost stable
circular orbit is shifted to rMISCO ∼ 7M. Such a rough fit for a wide set of LMXBs6 is shown
together with the fits for the pure Schwarzschild geodesic cases with M = 2M (Belloni et al
2007) and M = 1.4M . However, a detailed analysis for the particular LXMB sources should
be carried out taking into account the above-derived formulae7 .
Natural and simple implication of the RP model (and several other orbital models)
identifies the highest observed frequency of the particular source with the orbital frequency at
the appropriate ISCO, and thus allows for the estimation of the mass of the source (see, e.g.,
6
Data from Boutloukos et al (2006), Wijnands et al (2003), Linares et al (2005) and Belloni et al (2007).
Influence of the neutron star rotation (spin j ) is shown to be relatively weak for both radial and vertical epicyclic
frequencies, and it is quite negligible for small values of the spin (j < 0.1) (Török et al 2008c).
7
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Class. Quantum Grav. 27 (2010) 045001
P Bakala et al
Figure 6. The RP model rough fits of the observational twin-peak kHz QPO data for a wide set of
LMXBs. The thick solid curve refers to the case with M = 1.4M and with the orbital and epicyclic
frequencies being corrected by the presence of the Lorentz force induced by q̃ = 5.0 × 1010 and
μ = 1.06 × 10−4 m2 . For illustration we also present fits corresponding to the pure Schwarzschild
geodesic case (thin dashed curves), namely for M = 2M that was discussed by Belloni et al
(2007), and for M = 1.4M for the comparison with the non-geodesic case.
Figure 7. Left: the location of MISCO in the attractive region as a function of the test particle’s
specific charge q̃ and the intrinsic magnetic dipole moment μ of the star. The curves at the 3D plot
surface and their projections into the μ–q̃ plane denote rMISCO = 10M, 100M, 1000M. Right:
projection of the astrophysically relevant region from the left panel to the μ–q̃ plane with distinctive
values of rMISCO .
van der Klis (2005) and Lamb et al (2007)). A straightforward replacement of the GISCO
orbital frequency by the corrected MISCO orbital frequency provides a significant decrease
of the estimated mass.
Figure 7 illustrates high sensitivity of the MISCO orbit location on the intensity of the
attractive magnetic interaction. With growing values of q̃ or μ, rMISCO rapidly draws apart
from the radius of GISCO. In the case of the test neutron star with fixed μ = 1.06 × 10−4 m2 ,
we find that for q̃ = 6.0 × 1010 corresponding to the bottom-left panel of figure 4, there is
rMISCO = 7.48M, while for q̃ = 1.5 × 1011 corresponding to the bottom-right panel of figure 4
we obtain rMISCO = 9.32M. For the extremal specific charge q̃ = 1.111 × 1018 corresponding
14
Class. Quantum Grav. 27 (2010) 045001
P Bakala et al
to the case of matter purely consisting of ions of hydrogen, the location of the MISCO orbit
flies away to rMISCO = 177 864.76M.
It is widely expected (e.g. Kluźniak et al (1990) and van der Klis (2006)) that the magnetic
field of the central compact objects in LMXBs should be given by the intrinsic exterior magnetic
field, B ∈ 106 –109 Gauss. There are also several indices supporting the evidence of matter
being accreted in the region with r 10M (see, e.g., van der Klis (2006)). Our results then
imply that the specific charge related to the accreting matter should not exceed q̃ ∼ 1.86×1012
(1.87 × 1011 , 1.90 × 1010 , 1.91 × 109 ) for B = 106 Gauss (107 , 108 , 109 Gauss).
7. Conclusions
The aim of this paper is to study the influence of the Lorentz force generated by a magnetic
field of a neutron star on the quasi-circular, epicyclic orbital motion. In particular, we focus
on the behaviour of the non-geodesic orbital and epicyclic frequencies in dependence on the
neutron star magnetic dipole moment and the specific charge of the orbiting matter.
In general, the Lorentz force may be of attractive or repulsive character depending on
the sign of orbiting particle’s specific charge, and the magnetic dipole moment and orbital
velocity orientations. When the specific charge is large enough, the influence of both types
of the force allows for the existence of circular orbits for all radii above the horizon. In the
attractive region, a discontinuity appears, only unstable circular orbits exist under the circular
photon orbit at rph = 3M being oppositely oriented to those located above rph. Surprisingly, in
the repulsive region, the stable circular orbits associated with the radial and vertical epicyclic
oscillations can extend below the circular photon orbit radius rph. A critical charge q̃crit exists
MISCO
= 2.73M with the
for given μ, corresponding to the lowest stable circular orbit at rmin
highest possible orbital frequency max of the stable circular motion8 . In contrast, inside the
attractive region, the MISCO orbits always appear above rGISCO = 6M and the rMISCO can be
substantially shifted above r = 6M. We can conclude that the presence of the Lorentz force
strongly affects the location of the inner edge of the thin accretion disc.
In both repulsive and attractive regions of the magnetic interaction, the behaviour of the
orbital and epicyclic frequency profiles is quite complicated, giving rise to two separated
regions of the circular orbital motion for certain values of q̃. Generally, for stable circular
orbits in the repulsive region, ωr increases with growing specific charge, while both and
ωθ decrease. In the attractive region, on the other hand, the frequencies exhibit opposite
behaviour. For both regions of the magnetic interaction and astrophysically relevant values
of the radial coordinate (r > 3.5M) sensitivity of the radial epicyclic frequency ωr to q̃ is
significantly higher than the sensitivity of the two remaining frequencies.
The presence of the dipole magnetic field also violates the ν = νθ equality corresponding
to the spherical symmetry of the background Schwarzschild geometry. As a result, nodal
precession of the orbital motion plane arises, having an opposite phase for attractive and
repulsive magnetic interaction.
Orbital motion and related epicyclic frequencies have been considered by several authors
as a key agent in their models of the high-frequency QPOs (Kato et al 1998, Stella and Vietri
1999, Kluźniak and Abramowicz 2001, Török et al 2005, Stuchlı́k and Kotrlová 2009); in
this paper, we focused our attention on the relativistic precession QPO model. The models
mostly assume geodesic motion although some non-geodesic corrections have been studied
in the past, e.g., due to pressure-gradient forces (Blaes et al 2006, 2007, Šrámková et al
8
However, it should be stressed that the repulsive magnetic interaction applicability of the stable orbits region has
to be confronted with the location of the neutron star surface (see the appendix).
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Class. Quantum Grav. 27 (2010) 045001
P Bakala et al
2005, Straub and Šrámková 2009), or due to diamagnetic forces in hot plasma interacting
with the central compact object magnetic field (e.g. Vietri and Stella (1998)). However, nongeodesic corrections that arise from the interaction of dipole magnetic field with test particle’s
specific charge (i.e. the Lorentz force) have not been considered in this context yet. The
formulae derived in this work therefore represent the first attempt to describe the appropriate
problem within the scope of general relativity. We have shown that such effects are of high
importance and for attractive magnetic interaction they can improve significantly the fitting of
high-frequency QPOs data for some LMXB sources by the RP model.
Recently, sophisticated attempts appeared that are able to explain the high frequency
QPOs by the models of the oscillating toroidal disc (Rezzolla et al 2003a, 2003b, Lee et al
2004, Li and Narayan 2004, Montero et al 2004, Zhang 2004, Zanotti et al 2005, Schnittman
and Rezzolla 2006) or by discoseismology of (warped) discs (Wagoner 1999, Wagoner et al
2001, Kato 2004, Blaes et al 2007). In all of these models, the orbital and epicyclic frequencies
of the geodetical motion have an important role. It would be interesting to check, if the orbital
and epicyclic frequencies of magnetic non-geodesic motion of slightly charged particles could
be relevant for oscillations of slightly charged toroidal and warped discs.
In the present work we considered the dipole magnetic field on the background of the
spherically symmetric Schwarzschild geometry. Generalization of our results to axially
symmetric spacetimes (e.g. Hartle–Thorne or Lense–Thirring solutions) that describe that
the influence of the neutron star rotation is the subject of our future study.
Acknowledgments
This work has been supported by the Czech grants LC 06014 (PB, ES), MSM 4781305903 and
GACR 202/09/0772 (ZS, GT). The authors (PB, ES, ZS) would like to thank the Copacabana
Rio Hotel in Rio de Janeiro for great hospitality. They would also like to thank Dr J Kovář for
useful discussions.
Appendix. Radius and magnetic dipole moment of neutron stars
The presented analysis of circular and epicyclic motion is relevant in the exterior of the neutron
star only. Therefore, it is very important to fix the neutron star radius R. In order to obtain a
complete view of the motion, we study its properties down to R = 2.25M that represents an
innermost limit on the neutron star radius, being given by the limit on the existence of internal
(but unrealistic) Schwarzschild spacetime with uniformly distributed energy density—for such
configuration the central pressure diverges (Stuchlı́k 2000, Stuchlı́k et al 2001)9 . On the other
hand, realistic equations of state for both neutron and quark stars put the neutron (quark) star
radius into the interval (3–5)M; we take the intermediate value of R = 4M for our test neutron
star. Most of the realistic equations of state put the lower limit on the neutron star radius
at the value of R = 3.5M (Glendenning 1997) which is considered here as a limit radius
of astrophysically plausible neutron stars. Nevertheless, the existence of extremely compact
neutron stars with R < 3M is still discussed and is not excluded; for example, realistic models
of the so-called Q-stars allow R ∼ 2.8M (Bahcall et al 1989, Miller et al 1998, Stuchlı́k et al
2009a). Clearly, in vicinity of extremely compact neutron stars, the exotic phenomena related
to the magnetic repulsion under the photon circular orbit could be observed, giving thus the
signature of the existence of these extreme objects.
9
Admitting the existence of hypothetical gravastars (Mazur and Mottola 2004, Chirenti and Rezzolla 2007) we can
extend our analysis down to the gravitational radius Rg = 2M.
16
Class. Quantum Grav. 27 (2010) 045001
P Bakala et al
Figure A1. Intrinsic magnetic dipole moment μ of the star as a function of the star radius R
and mass M for a fixed magnetic field strength B at the star surface. The z-axis is scaled in
relative units of μ/B while the colour scaling at the 3D plot surface shows the values of μ for
B = 107 Gauss = 2.875 × 10−16 m−1 .
The intrinsic magnetic dipole moment of a neutron star can be obtained from the presumed
magnetic field strength at the neutron star surface. The orthonormal basis of local static
observers in the Schwarzschild spacetime reads
1
, 0, 0, 0 ,
er̂ = 0, η(r), 0, 0 ,
et̂ =
η(r)
(A.1)
1
1
eθ̂ = 0, 0, , 0 ,
eφ̂ = 0, 0, 0,
.
r
r sin θ
Locally measured magnetic field strength is given by the projection of the Maxwell tensor into
μ
the orthonormal basis of a static observer Fα̂β̂ = eα̂ eβ̂ν Fμν , at the surface of the star. For such
an observer located at the equator of the star with radius R, the magnetic field 3-vector has
only one nonzero component:
η(R)
Frφ .
R
Therefore, using equations (5) and (7), one may write
√
4M 3 R 3/2 R − 2M
μ=
B θ̂ .
6M(R − M) + 3R(R − 2M)In η (R)2
B θ̂ = Fr̂ φ̂ =
(A.2)
(A.3)
For a neutron star with a rather weak magnetic field strength, B = 107 Gauss 2.875 × 10−16 m−1 , mass M = 1.5M and radius R = 4M, we have μ = 1.06 × 10−4 m2
(B [cm−1 ] = (G1/2 /c2 )B [Gauss] 2.875 × 10−25 B [Gauss]). We have used neutron stars
of such parameters as the test model for our analysis. The dependence of the magnetic dipole
moment μ (expressed in terms of the surface value of the magnetic field strength B) on the
neutron star mass M and its radius R is illustrated in figure A1.
The electromagnetic 4-potential (4) used here corresponds to the case of magnetic dipole
moment connected to the central compact object (neutron star). In the case of Schwarzschild
black holes with the magnetic field generated by a current loop in the accretion disc (Petterson
17
Class. Quantum Grav. 27 (2010) 045001
P Bakala et al
1974), the discussed solution is valid only for particles orbiting at the radius higher than the
loop radius. Discussion on the so-called internal solution of the 4-potential and orbits of the
particles below the loop radius can be found in Preti (2004).
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