Fast coalescence of post-Newtonian Supermassive Black Hole Binaries in real galaxies
We present the results of theoretical modeling of supermassive black hole binary (SMBHB) mergers using direct 2 -body simulations with a Hermite integration scheme. The BH’s gravitational interaction is described based on the post-Newtonian (PN-terms) approximation up to the 3.5 PN-terms. We carry o...
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irk-123456789-1496342019-03-02T01:23:07Z Fast coalescence of post-Newtonian Supermassive Black Hole Binaries in real galaxies Sobolenko, M. Berczik, P. Spurzem, R. Kupi, G. Внегалактическая астрономия We present the results of theoretical modeling of supermassive black hole binary (SMBHB) mergers using direct 2 -body simulations with a Hermite integration scheme. The BH’s gravitational interaction is described based on the post-Newtonian (PN-terms) approximation up to the 3.5 PN-terms. We carry out a large set of runs using a parametric description of SMBHB orbits. Представлены результаты теоретического моделирования слияния двойных сверхмассивных чёрных дыр с помощью прямого 2-тельного моделирования с эрмитовской схемой интегрирования. Гравитационное взаимодействие черных дыр описывается постньютоновским приближением до 3.5 PN-терма. На основе параметрического описания орбит ДСМЧД получен большой набор моделей. Приводяться результати теоретичного моделювання злиття подвійних надмасивних чорних дір за допомогою прямого 2-тільного моделювання з ермітівською схемою інтегрування. Гравітаційна взаємодія чорних дір описується постньютонівським наближенням до 3.5 PN-терму. На основі параметричного опису орбіт ПНЧД отримано великий набір моделей. 2017 Article Fast coalescence of post-Newtonian Supermassive Black Hole Binaries in real galaxies / M. Sobolenko, P. Berczik, R. Spurzem, G. Kupi // Кинематика и физика небесных тел. — 2017. — Т. 33, № 1. — С. 22-37. — Бібліогр.: 71 назв. — англ. 0233-7665 http://dspace.nbuv.gov.ua/handle/123456789/149634 524.882 en Кинематика и физика небесных тел Головна астрономічна обсерваторія НАН України |
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English |
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Внегалактическая астрономия Внегалактическая астрономия |
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Внегалактическая астрономия Внегалактическая астрономия Sobolenko, M. Berczik, P. Spurzem, R. Kupi, G. Fast coalescence of post-Newtonian Supermassive Black Hole Binaries in real galaxies Кинематика и физика небесных тел |
| description |
We present the results of theoretical modeling of supermassive black hole binary (SMBHB) mergers using direct 2 -body simulations with a Hermite integration scheme. The BH’s gravitational interaction is described based on the post-Newtonian (PN-terms) approximation up to the 3.5 PN-terms. We carry out a large set of runs using a parametric description of SMBHB orbits. |
| format |
Article |
| author |
Sobolenko, M. Berczik, P. Spurzem, R. Kupi, G. |
| author_facet |
Sobolenko, M. Berczik, P. Spurzem, R. Kupi, G. |
| author_sort |
Sobolenko, M. |
| title |
Fast coalescence of post-Newtonian Supermassive Black Hole Binaries in real galaxies |
| title_short |
Fast coalescence of post-Newtonian Supermassive Black Hole Binaries in real galaxies |
| title_full |
Fast coalescence of post-Newtonian Supermassive Black Hole Binaries in real galaxies |
| title_fullStr |
Fast coalescence of post-Newtonian Supermassive Black Hole Binaries in real galaxies |
| title_full_unstemmed |
Fast coalescence of post-Newtonian Supermassive Black Hole Binaries in real galaxies |
| title_sort |
fast coalescence of post-newtonian supermassive black hole binaries in real galaxies |
| publisher |
Головна астрономічна обсерваторія НАН України |
| publishDate |
2017 |
| topic_facet |
Внегалактическая астрономия |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/149634 |
| citation_txt |
Fast coalescence of post-Newtonian Supermassive Black Hole Binaries in real galaxies / M. Sobolenko, P. Berczik, R. Spurzem, G. Kupi // Кинематика и физика небесных тел. — 2017. — Т. 33, № 1. — С. 22-37. — Бібліогр.: 71 назв. — англ. |
| series |
Кинематика и физика небесных тел |
| work_keys_str_mv |
AT sobolenkom fastcoalescenceofpostnewtoniansupermassiveblackholebinariesinrealgalaxies AT berczikp fastcoalescenceofpostnewtoniansupermassiveblackholebinariesinrealgalaxies AT spurzemr fastcoalescenceofpostnewtoniansupermassiveblackholebinariesinrealgalaxies AT kupig fastcoalescenceofpostnewtoniansupermassiveblackholebinariesinrealgalaxies |
| first_indexed |
2025-07-12T22:34:34Z |
| last_indexed |
2025-07-12T22:34:34Z |
| _version_ |
1837482306014543872 |
| fulltext |
UDK 524.882
M. Sobolenko1, P. Berczik1,2,4, R. Spurzem2,3,4, G. Kupi5
1Main As tro nom i cal Ob ser va tory Na tional Acad emy of Sci ences of Ukraine
27 Akademika Zabolotnoho St., 03680, Kyiv, Ukraine
sobolenko@mao.kiev.ua,
2National As tro nom i cal Ob ser va to ries of China and Key Lab o ra tory of Com pu ta tional Astrophysics
Chi nese Acad emy of Sci ences, 20A Datun Rd., Chaoyang Dis trict, 100012, Beijing, China
3Kavli In sti tute for As tron omy and As tro phys ics
Pe king Uni ver sity, Beijing 100871, China
4Astronomisches Rechen-Institut, Zentrum fhr Astronomie, Uni ver sity of Hei del berg
M`nchhofstrasse 12-14, 69120, Hei del berg, Ger many
5Rochester In sti tute of Tech nol ogy
Roch es ter, NY 14623, USA
Fast co ales cence of post-New to nian Supermassive Black
Hole Bi na ries in real gal ax ies
We pres ent the re sults of the o ret i cal mod el ing of supermassive black hole
bi nary (SMBHB) merg ers us ing di rect 2 -body sim u la tions with a Hermite
in te gra tion scheme. The BH’s grav i ta tional in ter ac tion is de scribed based
on the post-New to nian (PN terms) ap prox i ma tion up to the 3.5PN terms. We
carry out a large set of runs us ing a para met ric de scrip tion of SMBHB or -
bits. The fi nal time of the SMBHs grav i ta tional co ales cence is parametrized
as a func tion of ini tial ec cen tric ity e0 and mass ra tio q of the bi nary. We
carry out de tailed tests of our cod ing. We tested our PN terms against the
an a lytic pre scrip tion de scribed at the the o ret i cal works in mid dle 60th. The
grav i ta tional ra di a tion po lar iza tion am pli tudes h+ and h´ from the
SMBHBs merg ing pro cess are also an a lyzed. Based on our nu mer i cal work
we es ti mate the ex pected merg ing time for a list of se lected po ten tial SDSS
SMBHBs. Our re sults show that the merg ing time is a strong func tion of the
as sumed ini tial ec cen tric i ties and fall within the range of thou sands years.
ØÂÈÄÊÅ ÇËÈÒÒß ÏÎÑÒÍÜÞÒÎͲÂÑÜÊÈÕ ÏÎIJÉÍÈÕ ÍÀÄÌÀ -
ÑÈ ÍÈÕ ×ÎÐÍÈÕ Ä²Ð Ó ÐÅÀËÜÍÈÕ ÃÀËÀÊÒÈÊÀÕ, Ñîáî ëåí -
êî Ì. Î., Áåðöèê Ï. Ï., Øïóðçåì Ð., Êóï³ Ã. — Ïðèâîäÿòüñÿ ðå ç óëü -
òàòè òåîðåòè÷íîãî ìîäåëþâàííÿ çëèòòÿ ïîäâ³éíèõ íàäìà ñèâíèõ
÷îð íèõ ä³ð çà äîïîìîãîþ ïðÿìîãî 2-ò³ëüíîãî ìîäå ëþâàííÿ ç åðì³ò³â -
ñüêîþ ñõåìîþ ³íòåãðóâàííÿ. Ãðàâ³òàö³éíà âçàºìîä³ÿ ÷îðíèõ ä³ð îïèñó -
ºòüñÿ ïîñòíüþòîí³âñüêèì íàáëèæåííÿì äî 3.5PN-òåðìó. Íà îñíîâ³
ïàðàìåòðè÷íîãî îïèñó îðá³ò ÏÍ×Ä îòðèìàíî âå ëè êèé íàá³ð ìîäåëåé.
21
ISSN 0233-7665. Êèíåìàòèêà è ôèçèêà íåáåñ. òåë. 2017. Ò. 33, ¹ 1
© M. SOBOLENKO, P. BERCZIK, R. SPURZEM, G. KUPI, 2017
22
M. SOBOLENKO, P. BERCZIK, R. SPURZEM, G. KUPI
ʳíöåâèé ÷àñ ãðàâ³òàö³éíîãî çëèòòÿ ÏÍ×Ä ïà ðà ìåòðèçîâàíî ÿê ôóí -
ê ö³þ ïî÷àòêîâîãî åêñöåíòðèñèòåòó e0 òà â³ä íî øåííÿ ìàñ q ïîäâ³é -
íî¿. Ïðîâåäåíî äåòàëüíå òåñòóâàííÿ íàøîãî êîäó. Ìè ïîð³âíþâàëè
PN-òåðìè ç àíàë³òè÷íèì îïèñîì ó òåîðåòè÷íèõ äîñë³äæåííÿõ ñåðå -
äèíè 1960-õ ðð. Ïðîàíàë³çîâàíî àìïë³òóäó ïîëÿ ðèçîâàíîãî ãðàâ³òà -
ö³éíîãî âèïðîì³íþâàííÿ h+ òà h´ ï³ä ÷àñ çëèòòÿ ÏÍ×Ä. Ç âèêîðèñ -
òàííÿì íàøîãî ÷èñëîâîãî êîäó îö³ íå íî î÷³êóâàíèé ÷àñ çëèòòÿ äëÿ
ñïèñ êó âèáðàíèõ ïîòåíö³éíèõ SDSS ÏÍ×Ä. Íàø³ ðåçóëüòàòè ïîêà -
çóþòü, ùî ÷àñ çëèòòÿ äîñÿãຠòèñÿ÷ ðî ê³â òà º ñòðîãîþ ôóíêö³ºþ
îáðàíîãî ïî÷àò êîâîãî åêñöåíò ðè ñè òåòó.
ÁÛÑÒÐÎÅ ÑËÈßÍÈÅ ÏÎÑÒÍÜÞÒÎÍÎÂÑÊÈÕ ÄÂÎÉÍÛÕ ÑÂÅÐÕ -
ÌÀÑÑÈÂÍÛÕ ×ÎÐÍÛÕ ÄÛÐ Â ÐÅÀËÜÍÛÕ ÃÀËÀÊÒÈÊÀÕ, Ñîáî -
ëåí êî Ì. À., Áåðöèê Ï. Ï., Øïóðçåì Ð., Êóïè Ã. — Ïðåä ñòàâëåíû ðå -
çóëüòàòû òåîðåòè÷åñêîãî ìîäåëèðîâàíèÿ ñëèÿíèÿ äâîéíûõ ñâåðõ ìàñ -
ñèâíûõ ÷¸ðíûõ äûð ñ ïîìîùüþ ïðÿìîãî 2-òåëüíîãî ìîäåëèðîâàíèÿ ñ
ýðìèòîâñêîé ñõåìîé èíòåãðèðîâàíèÿ. Ãðàâèòàöèîííîå âçàèìî äåé ñò -
âèå ÷îðíûõ äûð îïèñûâàåòñÿ ïîñòíüþòîíîâñêèì ïðèáëè æåíèåì äî
3.5PN-òåðìà. Íà îñíîâå ïàðàìåòðè÷åñêîãî îïèñàíèÿ îð áèò ÄÑÌ×Ä
ïîëó÷åí áîëüøîé íàáîð ìîäåëåé. Êîíå÷íîå âðåìÿ ãðàâèòà öèîííîãî
ñëè ÿíèÿ ÄÑÌ×Ä ïàðàìåòðèçîâàíî êàê ôóíêöèÿ íà÷àëüíîãî ýêñöåí ò -
ðèñåòà e0 è îòíîøåíèÿ ìàññ q äâîéíîé. Ïðîâåäåíî äåòàëüíîå òåñ -
òèðîâàíèå íàøåãî êîäà. Ñðàâíèâàëèñü PN-òåðìû ñ àíàëèòè÷åñêèì
îïè ñàíèåì â òåîðåòè÷åñêèõ èññëåäîâàíèÿõ ñðåäèíû 1960-õ ãã. Ïðî -
àíà ëèçèðîâàíà àìïëèòóäà ïîëÿðè çîâàííîãî ãðàâèòà öèîííîãî èçëó ÷å -
íèÿ h+ è h´ âî âðåìÿ ñëèÿíèÿ ÄÑÌ×Ä. Ñ èñïîëüçîâàíèåì íàøåãî ÷èñ -
ëåííîãî êîäà îöåíåíî îæèäàå ìîå âðåìÿ ñëèÿíèÿ äëÿ ñïèñêà âûáðàííûõ
ïîòåí öèàëüíûõ SDSS ÄÑÌ×Ä. Íàøè ðåçóëüòàòû ïîêàçûâàþò, ÷òî
âðåìÿ ñëèÿíèÿ äîñòè ãàåò òûñÿ÷ ëåò è ÿâëÿåòñÿ ñòðîãîé ôóíêöèåé
âûáðàí íîãî íà÷àëüíîãî ýêñöåíòðè ñè òå òà.
IN TRO DUC TION
The for ma tion and evo lu tion of gal ax ies and their SMBHs are con nected in
sev eral ways. This re la tion can be found al ready at the early phases of
protogalaxies for ma tion [64], also at the later stages of hi er ar chi cal LCDM
cos mol ogy [15, 31, 63] and also dur ing the stages of dif fer ent gal axy merg -
ers [36, 42, 51]. One of the most sim ple and plau si ble chan nel of the SMBH
mass growth is an ac cu mu la tion of the BH’s mass dur ing host-gal axy merg -
ers. Gas ac cre tion can sig nif i cantly in crease the mass of BHs dur ing “wet”
merg ing that trig gers star for ma tion [3, 14, 25, 45, 57, 61]. Stel lar ac cre tion
can also in crease BH masses even in “dry” merg ing dur ing the for ma tion
the gi ant el lip ti cal gal ax ies [5, 44, 46, 67, 71]. The M — s re la tion, that
shows a con nec tion be tween the mass of the SMBH and the mass of the cen -
tral bulge of their host gal ax ies [30], we as sume is ev i dence for such a sce -
nario. The fact that the dis tri bu tion of the most lu mi nous and mas sive ac tive
ga lac tic nu clei peaks at higher redshifts also sup port this idea [34].
SMBHBs in side merg ing gal ax ies could be one of the most pow er ful
sources of grav i ta tional waves (GW), which can be de tected by the Pul sar
Tim ing Ar ray (PTA) or fu ture space-based mis sions, such as LISA/eLISA,
DESIGO/BBO [1, 33, 69]. The dy nam i cal evo lu tion of SMBHBs in the
cen ter of a merged stel lar sys tem can be tra di tion ally di vided in three phases
[4].
(I) Two BHs can form a pair in side the merg ing host gal axy due to dy -
nam i cal fric tion in the stel lar back ground. Then these com po nents sink into
the cen tre of the stel lar dis tri bu tion. SMBHBs start to be “hard” when the
length of the semimajor axis of the bi nary reaches the value:
a a
G
q
m
M
h£ º »
+
æ
è
ç
ö
ø
÷
æ
è
çç
ö
ø
÷÷
æ
è
m
s
s
4
2 7
1 10 2002
2
8
. pc
km/s8
çç
ö
ø
÷÷
-2
, (1)
where G is a grav i ta tional con stant, mass of the BH’s is m2 £ m1 , mass ra tio
is q m m= 2 1/ , m = +m m m m1 2 1 2/( ) is a re duced mass, to tal mass is M tot =
m m1 2+ . This means that the bind ing en ergy per unit mass | |/E M tot = G am /2
ex ceeds ~ s 2 (the am bi ent stel lar ve loc ity dis per sion) [50].
(II) Due to the sling shot in ter ac tion mech a nism the bi nary can con tinue
to harden via three-body scat ter ing of sin gle stars. If star’s or bit in ter sects
with the SMBHB or bit, a com plex three-body in ter ac tion can even tu ally
lead to the “ejec tion” of the star. This “ejected” star car ries away en ergy and
an gu lar mo men tum from the binary (see ref er ences in [49, 68, 70]).
But if we as sume spher i cal sym me try, the loss cone of the bi nary BH
sys tem can be de pleted by the sling shot mech a nism be fore this [2, 27].
There fore the sys tem hard en ing time can be more than the Hub ble time
[52]. This is the so called “fi nal parsec prob lem” which can be solved in
N-body sim u la tions as sum ing a more re al is tic stel lar par ti cle dis tri bu tion in
a ro tat ing sys tem [6, 39], ob late / triaxial potential [29, 38, 60] or some com -
bi na tion of these con fig u ra tions.
(III) At the third stage the com po nents sink to ward to the sep a ra tion
when GW emis sion be gins to be ef fi cient. Fi nally, the bi nary inspirals down
to the co ales cence, emit ting a strong GW sig nal. For such a merger the two
SMBHs have to reach a crit i cal sep a ra tion in a time shorter than the Hub ble
time (few Gyr):
a f e
q
q
M
M
GW
tot» ´
+
æ
è
çç
ö
ø
÷÷
-2 10
1 10
3 1 4
1 4
1 2 6
3 4
( )
( )
/
/
/
/
8
pc, (2)
where f e e e e( ) [ ( / ) ( / ) ]( ) /= + + - -1 73 24 37 96 12 4 2 7 2 is a func tion of the bi nary
ec cen tric ity e [58, 59].
To es ti mate the SMBHBs real merg ing times, we need to make our cal -
cu la tions with the real speed of light val ues. Such N-body sim u la tions are
al ready avail able in the lit er a ture (for ex am ple [38, 40]). But on the real
merg ing gal axy scale such sim u la tions re quire a lot of com put ing re sources.
23
FAST COALESCENCE OF POST-NEWTONIAN SUPERMASSIVE BLACK HOLE
In this pa per we pro pose a slightly dif fer ent ap proach. We per form sim u la -
tions for dif fer ent sets of pa ram e ters with var i ous “para met ric” val ues of
light speed (for ex am ple see [9]). To ex plore the con nec tion be tween the
real merg ing time Tmrg , the to tal mass M tot of the SMBHB and ini tial sep a -
ra tion R be tween the BHs we es ti mate a scal ing be tween the merg ing time
Tmrg and the speed of the light c as sum ing the de pend ence be tween these pa -
ram e ters.
NU MER I CAL METH ODS AND INI TIAL CON DI TIONS
Some nu mer i cal de tails. For the two BHB dy nam i cal or bit in te gra tion, we
use the pub licly avail able jGPU* [7, 8] with a 4th or der Hermite in te gra tor
and block hi er ar chi cal in di vid ual time step scheme. This Hermite scheme
re quires us to know the ac cel er a tion and its first time-de riv a tive, called jerk.
Be cause we use this Hermite scheme for our PN runs, we need to in clude the
PN cor rec tions also to the ac cel er a tion and jerk terms. In the jGPU code we
use the gen er al ized “Aarseth” type cri te ria for the time step def i ni tion [53]:
Dt
A
A
p p
p
=
æ
è
çç
ö
ø
÷÷-
-
h
( )
( )
/ ( )1
2
1 3
, (3)
where
A k k k k( ) ( ) ( ) ( )| || | | |= +- +a a a1 1 2 . (4)
Here, a ( )k is the kth de riv a tive of ac cel er a tion, p is the or der of the in te gra tor,
hp is the ac cu racy pa ram e ter. For a 4th-or der Hermite scheme the timestep
looks like:
Dt
A
A
= h4
1
2
( )
( )
, (5)
where
A ( ) ( ) ( ) ( )| || | | |1 0 2 1 2= +a a a , A ( ) ( ) ( ) ( )| || | | |2 1 3 2 2= +a a a . (6)
For all our runs we use the h4 = 0.018 pa ram e ter.
Post-New to nian for mal ism. We use a post-New to nian for mal ism in
the 2-body code for cal cu lat ing the rel a tiv is tic bi nary sys tems dy nam ics.
The re sults for up to 2PN and even up to 2.5PN equa tions of bi nary mo tion
in har monic co or di nates were ob tained by Damour and Deruelle [17—20,
24]. For the 3PN and 3.5PN terms we can use two dif fer ent ways of com pu -
ta tion. One of the pos si bil i ties is to use the ADM-Hamiltonian for mal ism of
gen eral rel a tiv ity [22, 54—56]. Phys i cally equiv a lent re sults [21, 23] can be
ob tained from the post-New to nian it er a tion [11], when we com pute the
equa tion of mo tion di rectly (in stead of via a Hamiltonian) in har monic co -
or di nates.
The equa tion of mo tion is a power se ries of 1/c, where n-PN is pro por -
tional to ( / )v c n2 . Sche mat i cally, one can write the cor rec tion for ac cel er a -
tion dur ing the mo tion of ob ject in bi nary sys tem as [19, 65]:
24
M. SOBOLENKO, P. BERCZIK, R. SPURZEM, G. KUPI
* ftp://ftp.mao.kiev.ua/pub/berczik/phi-GPU/
a a a a a a aNoSpin N
c c c c c
= + + + + +
1 1 1 1 1
2 1 4 2 5 2 5 6 3 7 3PN PN PN PN. .5 8
1
PN +
æ
è
ç
ö
ø
÷O
c
, (7)
where a N is the clas si cal New to nian ac cel er a tion; a1PN , a 2PN , a 3PN are the
non dissipative terms which “con serve” the en ergy of the sys tem. The
a 2 5. PN , a 3 5. PN are the dissipative terms which “carry out” en ergy from the
sys tem due to GW emis sion. We ap ply all PN cor rec tions up to or der
O c( / )1 8 , so the 3.5PN cor rec tion is the high est or der that we take into ac -
count. To com pare our re sults with the an a lyt i cal so lu tion from clas si cal ar -
ti cles [58, 59] we use the code just with the sin gle 2.5PN term.
Sim i lar to the equa tion of mo tion in the cen tre of mass frame [10] the
ac cel er a tion for one par ti cle can be writ ten in the fol low ing form:
a
v
n vNoSpin
d
dt
GM
r
= = - + +
2
1[( ) ]A B , (8)
where r =| |r is the sep a ra tion be tween par ti cles, r r r= -1 2 is the po si tion of
the par ti cles, n r= /r is the nor mal ized rel a tive po si tion vec tor, v v v= -1 2 is
the rel a tive ve loc ity. The func tions A and B con tain dif fer ent or ders of the
PN ap prox i ma tion (sim i lar to Eq. (7)).
For ex am ple the first PN cor rec tion term is given by:
A
Gm
r
Gm
r
1
1 2
2
2
1
2
1 2 2
25 4 3
2
4 2PN = + + × - + × -
é
ëê
ù
ûú
( ) ( )n v v v v v , (9)
B1 1 24 3PN = × - ×( ) ( )n v n v . (10)
De tailed ref er ences and the com plete de scrip tion of the prob lem can be
found in works such as [9, 10, 43]. The com plete equa tions in post-New to -
nian for mal ism up to 3.5PN are given also in [10].
Add ing the spin terms into the equa tion of mo tion we can de scribe as:
a a a a aSpin NoSpin SO SS SO
c c c
= + + +
1 1 1
3 1 5 4 2 5 2 5. , , . ,PN PN PN , (11)
where a1 5. ,PN SO and a 2 5. ,PN SO are the spin-or bit cou pling terms, a 2PN ,SS is the
spin-spin cou pling term (for ex am ple [26]). Now one can write the full
equa tion (like Eq. (7)):
a a a a a aSpin N SO SS
c c c
= + + + + +
1 1 1
2 1 3 1 5 4 2 2PN PN PN PN. , ,( )
+ + + + +
æ
è
ç
ö
ø
1 1 1 1
5 2 5 2 5 6 3 7 3 5 8c c c
O
c
SO( ). . , .a a a aPN PN PN PN ÷, (12)
where the full ex pres sion for a1 5. ,PN SO and a 2 5. ,PN SO can be found in [26], for
a 2PN ,SS can be found in [66]. The value of the phys i cal spin is cho sen from
the the next ex pres sion:
S
Gm
c
true =c
2
, (13)
where the value of c is [0, 1]. At the cen tre of the bi nary mass frame we have
the spin S S Sº +1 2 . We use two body dy nam ics and spin-spin and spin-or bit
25
FAST COALESCENCE OF POST-NEWTONIAN SUPERMASSIVE BLACK HOLE
cou pling just for cal cu la tion of the first or der of the grav i ta tional wave form
con straint (e.g. [41]):
h
G
Dc
v v
GM
r
n nij i j i j» -
é
ëê
ù
ûú
4
4
m
, (14)
where Q v v GMn n rij i j i j= -2( / ) is the usual quadrupole term (sec ond time
de riv a tives of the mass quadrupole mo ment ten sor) and D is the lu mi nos ity
dis tance. Choos ing the vir tual de tec tor ori en ta tion so that as the co or di nate
axes co in cide with the source frame, we can de scribe the two-di men sional
ma trix with only two in de pend ent el e ments:
h
h h
h hij =
-
æ
è
çç
ö
ø
÷÷
+ ´
´ +
. (15)
From hij we can ob tain the am pli tude of po lar iza tion h+ and h´ [12, 16,
62].
Ini tial con di tions and de scrip tion of model. We as sume that the two
point masses which rep re sents our BHs with masses m1 and m2 are placed at
po si tions Y1 and Y2 on the Y axis (see Fig. 1). For our anal y ses we choose the
nat u ral co or di nate sys tem of the two bod ies, con nected by the cen tre of
mass of the sys tem. The ini tial or bital ve loc ity of the two point masses we
chose so that the XY plane con tains the full or bit. The ini tial sep a ra tion be -
tween the com po nents we de fined as R Y Y= +| | | |1 2 . We also set the BH’s mass
ra tio q m m= 1 2/ . We as sume that m m1 2£ . We also fix the to tal BH sys tem
mass M m mtot = +1 2 . The Keplerian mo tion of the two bod ies can be fully
de scribed by two main or bital pa ram e ters: the semimajor axis a and ec cen -
tric ity e. We can write the bind ing en ergy of the bi nary sys tem:
| |E
Gm m
a
G M
a
tot= =1 2
2 2
m
, (16)
where m = m m M tot1 2 / is the re duced mass. We also fix as a pa ram e ter the bi -
nary ini tial or bital ec cen tric ity e0 . The ini tial setup of the par ti cles we show
in Fig ure 1. For fur ther cal cu la tion we as sume the nor mal iza tion R = 1 and
M m mtot = +1 2 = 1.
26
M. SOBOLENKO, P. BERCZIK, R. SPURZEM, G. KUPI
Fig . 1. Con fig u ra tion of the sys tem with two
BHs
We use the N-body (NB) or called HJnon units [32] where we also ac -
cept G = 1 and set the mass units M and length units R to unity*. There fore
the phys i cal val ues of mass, length, en ergy, ve loc ity and time will be in the
form:
[M] = M, [L] = R, (17)
[E] =
GM
R
2
, (18)
[ ] , [ ]
/ /
V
GM
R
T
R
GM
=
æ
è
ç
ö
ø
÷ =
æ
è
çç
ö
ø
÷÷
1 2 3 1 2
. (19)
Con se quently the light speed c in N-body units is:
c
c
V
c
GM
R
M
M
R
= =
æ
è
ç
ö
ø
÷ = ×
æ
è
çç
ö
ø
÷÷
- -
0
0
1 2
8
1 2
3
14213
10 10
/ /
8 pc
æ
è
ç
ö
ø
÷
1 2/
, (20)
where c0 is the light speed in phys i cal units.
DIS CUS SION
Scal ing rou tine be tween merg ing time Tmrg and “para met ric” speed of the
light was made for all mod els from Ta ble 1 (for ex am ple see Fig. 2 for sys -
tem with pa ram e ters M tot = 1 [NB], q = 0.5, R = 1 [NB], e0 = 0.25). Based on
our post-New to nian for mal ism (Eqs (7)-(12)) we can the o ret i cally ex pect
the re la tion ship be tween merg ing time (which is di rectly pro por tional to the
en ergy losses in our post-New to nian for mal ism) and the light speed:
T b c T d c p cmrg mrg5
5
5 7
5 7µ × µ × + ×+, , (21)
where b, d and p are the co ef fi cients of the scal ing. As we can see from Fig.
2 the dif fer ence be tween the two merg ing times are neg li gi ble. So, in this
pa per we use the Tmrg 5 as a ba sic ap prox i ma tion for the bi nary merg ing time
Tmrg .
27
FAST COALESCENCE OF POST-NEWTONIAN SUPERMASSIVE BLACK HOLE
e0
b
q = 1 q = 0.5 q = 0.333 q = 0.25 q = 0.2 q = 0.02
0.00 7.863E-02 8.827E-02 1.043E-01 1.218E-01 1.397E-01 8.611E-01
0.25 2.578E-02 2.900E-02 3.437E-02 4.027E-02 4.639E-02 3.375E-01
0.50 5.584E-03 6.280E-03 7.440E-03 8.716E-03 1.004E-02 7.244E-02
0.75 4.648E-04 5.225E-04 6.186E-04 7.243E-04 8.339E-04 6.003E-03
0.95 1.893E-06 2.126E-06 2.514E-06 2.938E-06 3.383E-06 2.425E-05
0.99 8.146E-09 9.123E-09 1.076E-08 1.255E-08 1.441E-08 1.023E-07
Ta ble 1. The scale fac tor b from Eq. (21) for var i ous mass ra tio q and ini tial ec cen tric ity e0
(sep a ra tion for each sys tem R = 1 [NB] and to tal mass Mtot =1 [NB])
* http://en.wikipedia.org/wiki/N-body_units
We study the evo lu tion of sys tems with var i ous mass ra tios and ini tial
ec cen tric i ties, i. e. with var i ous or bits. We use the fol low ing sets of the pa -
ram e ters: q = 1, 0.5, 0.333, 0.25, 0.2, 0.02 and e = 0.00, 0.25, 0.50, 0.75,
0.95, 0.99.
We ap ply the scal ing fac tors from Ta ble 1 to find the real merg ing times
Tmrg (in phys i cal units) where the phys i cal light speed is c = 2.99792458 ´
´ 108 m/s.
We ap ply the above de scribed “c-scal ing pro ce dure” for a wide range of
phys i cal pa ram e ters for masses (10 5106 9M M Mtot8 8£ £ × ) and the ini tial
sep a ra tion be tween the BHs (10 3- £ £R 102 pc). For each in di vid ual model
we es ti mate the re la tion be tween the merg ing time Tmrg , sep a ra tion be tween
the BHs R and to tal mass M tot of the SMBHB (Fig. 3, 4). For ex am ple us ing
Fig. 3 for sys tem with M tot = 109M8, q = 0.5, R = 10 pc, e0 = 0.5 merg ing
time Tmrg » 1700 years.
In a real cos mo log i cal merg ing sce nario we ex pect that the SMBHBs
merger does not evolve in iso la tion. High res o lu tion cos mo log i cal nu mer i -
cal sim u la tions (see ref er ences in [28, 37, 48]) show us that SMBHB merg -
ers typ i cally need to meet the next large gal axy in a time scale of 1-2 Gyr. If
we as sume the ex is tence of a SMBH in this third gal axy too, in this case our
bi nary BH is trans formed to a tri ple BH sys tem. Ex ten sive di rect N-body
sim u la tions of sys tem with three BHs show that such a con fig u ra tion is
highly un sta ble [1, 13]. So, we as sume that if in a time scale of 1-2 Gyr our
orig i nal BHB sys tem does not merge, the pos si bil ity of such a merger be -
comes very un likely. In Fig ures 3, 4 we show the 1 Gyr merg ing time as the
solid black lines for the dif fer ent ini tial ec cen tric i ties.
For some fixed time this re la tion can be writ ten in the form:
R M SMBHB=10 6
3 4b( ),
/ .
We found that with the rise of ini tial ec cen tric ity e0 the merg ing time
Tmrg of the sys tem de creases. This be hav ior is valid for mass ra tios from q =
= 1 to q = 0.2 and even for extremal q = 0.02. The gen eral con clu sion from
our set of runs is that the lower ini tial ec cen tric ity (cir cu lar) or bits gen er ally
28
M. SOBOLENKO, P. BERCZIK, R. SPURZEM, G. KUPI
Fig . 2. Re la tion be tween the merg ing time Tmrg
and light speed c for sys tem from Ta ble 1 (line
1 — Tmrg = bc5, 2 — Tmrg = dc5 + pc7, stars —
simulation). Ini tial ec cen tric ity e0 = 0.25 and
mass ra tio q = 0.5
have a lon ger merg ing time. For higher mass ra tios even the ec cen tric or bits
be come more sta ble.
Com par i son of the sim u la tion re sults and the o ret i cal work [58, 59]
(which in cludes in the ex pres sions only for the 2.5PN term) is shown in
(Fig. 5, 6). For this nu mer i cal test we use the pa ram e ters M tot = 2 [NB], q =
= 1, R = 1 [NB], e0 = 0.7, c = 15 [NB] and we also in clude only the 2.5PN
term. Our test sim u la tions show that the nu mer i cal model be haves very sim -
i lar to the the o ret i cal curve.
29
FAST COALESCENCE OF POST-NEWTONIAN SUPERMASSIVE BLACK HOLE
Fig . 3. The color coded fi nal merg ing time Tmrg of SMBHB as a func tion of to tal mass and ini tial
sep a ra tion of the bi nary. Each sep a rate plot shows the merg ing time evo lu tion for the spe cific mass
ra tio of the bi nary: q = 1 (a), 0.5 (b), 0.333 (c), 0.25 (d), 0.2 (e), 0.02 (f). On each plots we in di cate the
1 Gyr merg ing time line as a func tion of the ini tial ec cen tric ity e0 of the bi nary. Col ored gamma for
value e0 = 0.00
For ob tain ing the GW con straints, for the se lected test case (M tot =
108 M8, q = 0.5, R = 0.01 pc, e0 = 0.95, S1 = [0, 0, 1], S2 = [0, 0, 1] ), we use
the spin-spin and spin-or bit cou pling which was de scribed above [12]. In
Fig. 7 we show the first periastron passes for h+ and h´. In Fig. 8 we see the
wave form dur ing inspiraling just for h+ po lar iza tion (the h´ looks sim i lar).
In Ta ble 2 we pres ent the GW fre quen cies for BHs with typ i cal masses and
bi nary sys tem or bital pa ram e ters.
30
M. SOBOLENKO, P. BERCZIK, R. SPURZEM, G. KUPI
Fig . 4. The color coded fi nal merg ing time Tmrgof SMBHB as a func tion of to tal mass and ini tial
sep a ra tion of the bi nary. Each sep a rate plot shows the merg ing time evo lu tion for the spe cific mass
ra tio of the bi nary: q = 1 (a), 0.5 (b), 0.333 (c), 0.25 (d), 0.2 (e), 0.02 (f). On each plots we in di cate the
1 Gyr merg ing time line as a func tion of the ini tial ec cen tric ity e0 of the bi nary. Col ored gamma for
value e0 = 0.95
Us ing our well tested PN-rou tine we es ti mate the pos si ble BHBs merg -
ing time for the set of SDSS ob jects [35]. The main pa ram e ters of the bi nary
BHs we pres ent in Ta ble 3. We es ti mate bi nary BH ex pected merg ing times
as sum ing dif fer ent ec cen tric i ties (e0 = 0.00 — 0.99) of the or bits ex cept
J1201, for which we e0 = 0.3. Also we cal cu lated the merg ing time for the
serendipitously dis cov ered SDSS J120136.02 + 3000305.5 (z = 0.146) with
sys tem pa ram e ters M tot = 1.08´107M8, q = 0.08, rmax = 1.3 mpc, e0 = 0.3
[47].
31
FAST COALESCENCE OF POST-NEWTONIAN SUPERMASSIVE BLACK HOLE
Fig . 5. Com par i son the sim u la tion’s evo lu tion
(dots) of the semimajor axis a with an a lyt i cal
re sults (line) for a sys tem with fol low ing ini tial
pa ram e ters: M tot = 2 [NB], q = 1, R = 1 [NB], e0 =
0.6, c = 15 [NB] with just turn ing on 2.5PN
Fig . 6. Com par i son the sim u la tion’s evo lu tion
(dots) of the ec cen tric ity e with an a lyt i cal
re sults (line) for a sys tem with fol low ing ini tial
pa ram e ters: M tot = 2 [NB], q = 1, R = 1 [NB], e0 =
0.6, c = 15 [NB] with just turn ing on 2.5PN
M tot / M8 R (rS) Tmrg , yr Torb , s n, mHz
109 104 0.6 866925 1.15
108 1045 631.5 55175 18.1
107 10451 655148.3 5933 169
Ta ble 2. The GW fre quency for BHs with typ i cal masses Mtot and sys tem pa ram e ters q = 0.5,
e0 = 0.95, R = 0.01 pc, S 1=[0, 0, 1], S 2 =[0, 0, 1]
SDSS ID z log (M tot /M8) rmax, mpc
J075700.70+424814.5 1.17 9.1311 20
J002444.11+003221.4 0.40 9.5618 102
J004918.98+002609.4 1.94 9.3148 96
J161609.50+434146.8 0.49 8.1696 21
J093502.54+433110.7 0.46 9.3425 181
J032223.02-000803.5 0.62 8.2827 32
J095656.42+535023.2 0.61 8.2944 127
Ta ble 3. Con fig u ra tions of the sys tems for SDSS ob jects from [35] (q = 1)
As we can see from Ta ble 4 some of the se lected SDSS ob jects have a
quite short merg ing time even for mod er ately large ec cen tric i ties e0 ³ 0.75.
Al most all of the se lected ob jects (ex cept one J0956) have ex pected merg -
ing times only a few years for ini tial ec cen tric i ties e0 = 0. How ever J1201
has an es ti mated Tmrg = 3.27 Myr, that is not such a grat i fy ing re sult. Hope -
fully our merg ing time pre dic tions can be tested with the larger SDSS4 ob -
ser va tional cat a logues, which are right now in a phase of ob ser va tion.
32
M. SOBOLENKO, P. BERCZIK, R. SPURZEM, G. KUPI
Fig . 8. Strain for a sys tem with
pa ram e ters M tot = 108M
8
(q = 0.5),
e0 = 0.95, R = 0.01 pc, S1 = [0, 0, 1],
S 2 = [0, 0, 1]
Fig . 7. Sim u lated strain from a GW
dur ing the first periastron for a
sys tem with M tot = 108M
8
(q =
0.5), e0 = 0.95, R = 0.01 pc, S1 = [0,
0, 1], S 2 = [0, 0, 1]
e0
Tmrg , yr
J0757 J0024 J0049 J1616 J0935 J0322 J0956
0.00 2.139E+04 6.906E+05 2.916E+06 1.767E+07 2.997E+07 4.390E+07 9.929E+09
0.25 6.796E+03 2.213E+05 9.385E+05 5.714E+06 9.672E+06 1.420E+07 3.214E+09
0.50 1.439E+03 4.738E+04 2.023E+05 1.241E+06 2.095E+06 3.086E+06 7.004E+08
0.75 1.221E+02 3.873E+03 1.668E+04 1.040E+05 1.745E+05 2.591E+05 5.920E+07
0.90 9.528E-01 6.635E+00 5.014E+02 4.195E+02 7.413E+02 1.046E+03 2.453E+05
0.99 9.255E-01 6.435E+00 7.780E+00 2.960E+00 1.946E+01 4.893E+00 1.007E+03
Ta ble 4. Ex pected merg ing time Tmrg for SMBHBs for the se lected SDSS ob jects as the func tion
of the ec cen tric i ties
CON CLU SION
In our study we an a lyze the dy nam i cal be hav ior of SMBHBs. We use a
highly ac cu rate di rect 2-body code where we ap ply the ad di tional PN terms
up to 3.5PN for cal cu la tion of the grav i ta tional forces which act on the BHs
and spin-spin and spin-or bit cou pling for cal cu la tion of GW con straints. As
the main re sult we ob tain the re sult ing merg ing time Tmrg for a large set of
ini tial mass ra tios q of the BBH, ini tial masses, ini tial sep a ra tions and or -
bital ec cen tric i ties e0 . This data we pres ent as a set of color coded 3-D plots.
We also make the orig i nal re sults pre sented on these plots for dif fer ent mass
ra tios q and ini tial ec cen tric i ties e0 pub licly avail able*. Our PN treat ment
was ex ten sively tested and the PN rou tines it self we also make pub licly
avail able via the same link above. In our high or der di rect 2-body im ple -
men ta tion we use not only the PN ac cel er a tions but also the first de riv a tives
of this ac cel er a tions. Our BHBs test cal cu la tions show that for BH masses
in range M tot = (106 — 109)M8 with a fixed ini tial sep a ra tion R = 0.01 pc
and ini tial ec cen tric ity e0 = 0.95 the GW fre quen cies are well in side the
LISA sen si tiv ity band (Ta ble 2) [9]. We use our PN rou tines to ap prox i mate
the ex pected merg ing time for the se lected sam ple SDSS SMBHBs [35].
Our re sults show that for sig nif i cantly large ec cen tric i ties the ex pected
merg ing time for these ob jects are in the range of years.
AC KNOWL EDGE MENTS
MS ac knowl edge the fi nan cial sup port by the NASU un der the Grant for
young re search ers. MS and PB ac knowl edge also the spe cial sup port by the
NASU un der the Main As tro nom i cal Ob ser va tory GRID/GPU com put ing
clus ter golowood pro ject.
RS and PB ac knowl edge sup port by Chi nese Acad emy of Sci ences
through the Silk Road Pro ject at NAOC, through the “Qianren” spe cial for -
eign ex perts pro gram of China. MS grate fully ac knowl edges sup port for
col lab o ra tion vis its in Beijing un der the same program.
Sup ported by the Stra te gic Pri or ity Re search Pro gram “The Emer gence
of Cos mo log i cal Struc tures” of the Chi nese Acad emy of Sci ences, Grant
N XDB09000000.
The main part of the sim u la tions pre sented here was per formed on the
ded i cated GPU clus ters kep ler at the ARI, funded un der the grants I/80
041-043 and I/81 396 of the Volks wagen Foun da tion and the grants
823.219-439/30 and /36 of the Min is try of Sci ence, Re search and the Arts
of Baden-Whrttemberg, Ger many.
The au thors are ac knowl edges the sup port of the Volks wagen Foun da -
tion un der the Trilateral Part ner ships grant No. 90411.
Part of the code de vel op ment work was con ducted us ing the re sources
of the GPU clus ter laohu at the Cen ter of In for ma tion and Com put ing at the
Na tional As tro nom i cal Ob ser va to ries, Chi nese Acad emy of Sci ences,
33
FAST COALESCENCE OF POST-NEWTONIAN SUPERMASSIVE BLACK HOLE
* ftp://ftp.mao.kiev.ua/pub/sobolenko/
funded by the Min is try of Fi nance of Peo ple’s Re pub lic of China un der the
grant ZDYZ2008-2.
We also ac knowl edge the help ful com ments made by Ste phen Justham.
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