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Authenticating the Presence of a Relativistic Massive Black
Hole Binary in OJ 287 Using Its General Relativity Centenary
Flare : Improved Orbital Parameters
Dey, Lankeswar
2018-10-10
Dey , L , Valtonen , M J , Gopakumar , A , Zola , S , Hudec , R , Pihajoki , P , Ciprini , S ,
Matsumoto , K , Sadakane , K , Kidger , M , Nilsson , K , Mikkola , S , Sillanpaa , A , Takalo ,
L O , Lehto , H J , Berdyugin , A , Piirola , V , Jermak , H , Baliyan , K S , Pursimo , T , Caton
, D B , Alicavus , F , Baransky , A , Blay , P , Boumis , P , Boyd , D , Campas Torrent , M ,
Campos , F , Carrillo Gomez , J , Chandra , S , Chavushyan , V , Dalessio , J , Debski , B ,
Drozdz , M , Er , H , Erdem , A , Escartin Perez , A , Ramazani , V F , Filippenko , A V ,
Gafton , E , Ganesh , S , Garcia , F , Gazeas , K , Godunova , V , Gomez Pinilla , F ,
Gopinathan , M , Haislip , J B , Harmanen , J , Hurst , G , Janik , J , Jelinek , M , Joshi , A ,
Kagitani , M , Karjalainen , R , Kaur , N , Keel , W C , Kouprianov , V V , Kundera , T ,
Kurowski , S , Kvammen , A , LaCluyze , A P , Lee , B C , Liakos , A , Lindfors , E , Lozano
de Haro , J , Mugrauer , M , Naves Nogues , R , Neely , A W , Nelson , R H , Ogloza , W ,
Okano , S , Pajdosz-Smierciak , U , Pandey , J C , Perri , M , Poyner , G , Provencal , J , Raj
, A , Reichart , D E , Reinthal , R , Reynolds , T , Saario , J , Sadegi , S , Sakanoi , T , Salto
Gonzalez , J -L , Sameer , Schweyer , T , Simon , A , Siwak , M , Soldan Alfaro , F C ,
Sonbas , E , Steele , I , Stocke , J T , Strobl , J , Tomov , T , Tremosa Espasa , L , Valdes , J
R , Valero Perez , J , Verrecchia , F , Vasylenko , V , Webb , J R , Yoneda , M , Zejmo , M ,
Zheng , W & Zielinski , P 2018 , ' Authenticating the Presence of a Relativistic Massive Black
Hole Binary in OJ 287 Using Its General Relativity Centenary Flare : Improved Orbital
Parameters ' , Astrophysical Journal , vol. 866 , no. 1 , 11 . https://doi.org/10.3847/1538-4357/aadd95
http://hdl.handle.net/10138/255283
https://doi.org/10.3847/1538-4357/aadd95
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Authenticating the Presence of a Relativistic Massive Black Hole Binary in OJ 287 Using
Its General Relativity Centenary Flare: Improved Orbital Parameters
Lankeswar Dey
1
, M. J. Valtonen
2,3
, A. Gopakumar
1
, S. Zola
4,5
, R. Hudec
6,7
, P. Pihajoki
8
, S. Ciprini
9,10
, K. Matsumoto
11
,
K. Sadakane
11
, M. Kidger
12
, K. Nilsson
2
, S. Mikkola
3
, A. Sillanpää
3
, L. O. Takalo
3,78
, H. J. Lehto
3
, A. Berdyugin
3
, V. Piirola
2,3
,
H. Jermak
13
, K. S. Baliyan
14
, T. Pursimo
15
, D. B. Caton
16
, F. Alicavus
17,18
, A. Baransky
19
, P. Blay
20
, P. Boumis
21
, D. Boyd
22
,
M. Campas Torrent
23
, F. Campos
24
, J. Carrillo Gómez
25
, S. Chandra
26
, V. Chavushyan
27
, J. Dalessio
28
, B. Debski
29
,
M. Drozdz
30
,H.Er
31
, A. Erdem
17,18
, A. Escartin Pérez
32
, V. Fallah Ramazani
3
, A. V. Filippenko
33,34,79
, E. Gafton
35
,
S. Ganesh
14
, F. Garcia
36
, K. Gazeas
37
, V. Godunova
38
, F. Gómez Pinilla
39
, M. Gopinathan
40
, J. B. Haislip
41
, J. Harmanen
3
,
G. Hurst
42
, J. Janík
43
, M. Jelinek
44,45
, A. Joshi
40
, M. Kagitani
46
, R. Karjalainen
47
, N. Kaur
14
, W. C. Keel
48
,
V. V. Kouprianov
41,49
, T. Kundera
29
, S. Kurowski
29
, A. Kvammen
50
, A. P. LaCluyze
41
,B.C.Lee
51,52
, A. Liakos
21
,
E. Lindfors
3
, J. Lozano de Haro
53
, M. Mugrauer
54
, R. Naves Nogues
55
, A. W. Neely
56
, R. H. Nelson
57
, W. Ogloza
5
, S. Okano
46
,
U. Pajdosz-Śmierciak
29
, J. C. Pandey
40
, M. Perri
9,58
, G. Poyner
59
, J. Provencal
28
, A. Raj
60
, D. E. Reichart
41
, R. Reinthal
3
,
T. Reynolds
15
, J. Saario
61
, S. Sadegi
62
, T. Sakanoi
46
, J.-L. Salto González
63
, Sameer
64
, T. Schweyer
65,66
, A. Simon
67
, M. Siwak
5
,
F. C. Soldán Alfaro
68
, E. Sonbas
69
, I. Steele
13
, J. T. Stocke
70
, J. Strobl
44,45
, T. Tomov
71
, L. Tremosa Espasa
72
, J. R. Valdes
27
,
J. Valero Pérez
73
, F. Verrecchia
9,58
, V. Vasylenko
67
, J. R. Webb
74
, M. Yoneda
75
,
M. Zejmo
76
, W. Zheng
33
, and P. Zielinski
77
1
Department of Astronomy and Astrophysics, Tata Institute of Fundamental Research, Mumbai 400005, India; lankeswar.dey@tifr.res.in
2
Finnish Centre for Astronomy with ESO, University of Turku, Finland
3
Tuorla Observatory, Department of Physics and Astronomy, University of Turku, Finland
4
Astronomical Observatory, Jagiellonian University, ul. Orla 171, Cracow PL-30-244, Poland
5
Mt. Suhora Astronomical Observatory, Pedagogical University, ul. Podchorazych 2, PL30-084 Cracow, Poland
6
Czech Technical University in Prague, Faculty of Electrical Engineering, Technicka 2, Prague 166 27, Czech Republic
7
Engelhardt Astronomical observatory, Kazan Federal University, Kremlyovskaya street 18, 420008 Kazan, Russian Federation
8
Department of Physics, University of Helsinki, Gustaf Hällströmin katu 2a, FI-00560, Helsinki, Finland
9
Space Science Data Center—Agenzia Spaziale Italiana, via del Politecnico, snc, I-00133, Roma, Italy
10
Instituto Nazionale di Fisica Nucleare, Sezione di Perugia, Perugia I-06123, Italy
11
Astronomical Institute, Osaka Kyoiku University, 4-698 Asahigaoka, Kashiwara, Osaka 582-8582, Japan
12
Herschel Science Centre, ESAC, European Space Agency, E-28691 Villanueva de la Cañada, Madrid, Spain
13
Astrophysics Research Institute, Liverpool John Moores University, IC2, Liverpool Science Park, Brownlow Hill, L3 5RF, UK
14
Physical Research Laboratory, Ahmedabad 380009, India
15
Nordic Optical Telescope, Apartado 474, E-38700 Santa Cruz de La Palma, Spain
16
Dark Sky Observatory, Department of Physics and Astronomy, Appalachian State University, Boone, NC 28608, USA
17
Department of Physics, Faculty of Arts and Sciences, Canakkale Onsekiz Mart University, TR-17100 Canakkale, Turkey
18
Astrophysics Research Center and Ulupinar Observatory, Canakkale Onsekiz Mart University, TR-17100, Canakkale, Turkey
19
20 Astronomical Observatory of Taras Shevshenko National University of Kyiv, Observatorna str. 3, 04053 Kyiv, Ukraine
20
Valencian International University, E-46002 Valencia, Spain
21
Institute for Astronomy, Astrophysics, Space Applications and Remote Sensing, National Observatory of Athens,
Metaxa & Vas. Pavlou St., Penteli, Athens GR-15236, Greece
22
BAA Variable Star Section, 5 Silver Lane, West Challow, Wantage, OX12 9TX, UK
23
C/ Jaume Balmes No 24 E-08348 Cabrils, Barcelona, Spain
24
C/.Riera, 1, 1
o
3
a
Barcelona, Spain
25
Carretera de Martos 28 primero Fuensanta, Jaen, Spain
26
Centre for Space Research Private Bag X6001, North-West University, Potchefstroom Campus, Potchefstroom, 2520, South Africa
27
Instituto Nacional de Astrofisica, Óptica y Electrónica, Apartado Postal 51-216, 72000 Puebla, México
28
University of Delaware, Department of Physics and Astronomy, Newark, DE 19716, USA
29
Astronomical Observatory, Jagiellonian University, ul. Orla 171, PL-30-244 Krakow, Poland
30
Mt Suhora Observatory, Pedagogical University, ul. Podchorazych 2, PL-30-084 Krakow, Poland
31
Department of Astronomy and Astrophysics, Ataturk University, Erzurum, 25240, Turkey
32
Aritz Bidea No 8 4B (E-48100) Mungia Bizkaia, Spain
33
Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA
34
Miller Institute for Basic Research in Science, University of California, Berkeley, CA 94720, USA
35
Department of Astronomy and Oskar Klein Centre, Stockholm University, AlbaNova, SE-10691, Stockholm, Sweden
36
Muñas de Arriba La Vara, Valdés (MPC J38) E-33780 Valdés, Asturias, Spain
37
Department of Astrophysics, Astronomy and Mechanics, National & Kapodistrian University of Athens, Zografos GR-15784, Athens, Greece
38
ICAMER Observatory of NASU, 27, Acad. Zabolotnoho str., 03143 Kyiv, Ukraine
39
C/Concejo de Teverga 9, 1C E-28053 Madrid, Spain
40
Aryabhatta Research Institute of Observational Sciences (ARIES), Nainital, 263002 India
41
University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA
42
16 Westminster Close Basingstoke Hampshire RG22 4PP, UK
43
Department of Theoretical Physics and Astrophysics, Masaryk University, Kotlár
ská 2, 611 37 Brno, Czech Republic
44
Astronomical Institute, The Czech Academy of Sciences, 25165 Ondr
ejov, Czech Republic
45
Czech Technical University in Prague, Faculty of Electrical Engineering, Prague, Czech Republic
46
Planetary Plasma and Atmospheric Research Center, Tohoku University, Sendai, Japan
47
Isaac Newton Group of Telescopes, Apartado de Correos 321, Santa Cruz de La Palma, E-38700, Spain
48
Department of Physics and Astronomy and SARA Observatory, University of Alabama, Box 870324, Tuscaloosa, AL 35487, USA
49
Central (Pulkovo) Astronomical Observatory of Russian Academy of Sciences, Pulkovskoye Chaussee 65/1, 196140, Saint Petersburg, Russia
50
Department of Physics and Technology, University of Tromsö, Tromsö NO-9019, Norway
The Astrophysical Journal, 866:11 (20pp), 2018 October 10 https://doi.org/10.3847/1538-4357/aadd95
© 2018. The American Astronomical Society. All rights reserved.
1
51
Korea Astronomy and Space Science Institute, 776, Daedeokdae-Ro, Youseong-Gu, 305-348 Daejeon, Republic of Korea
52
Korea University of Science and Technology, Gajeong-Ro Yuseong-Gu, 305-333 Daejeon, Republic of Korea
53
Partida de Maitino, pol. 2 num. 163 (E-03206) Elche, Alicante, Spain
54
Astrophysikalisches Institut und Universitäts-Sternwarte, Schillergäßchen 2-3, D-07745 Jena, Germany
55
52 Observatory Montcabrer , C/Jaume Balmes No 24, Cabrils, Barcelona E-08348, Spain
56
NF/Observatory, Silver City, NM 88041, USA
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1393 Garvin Street, Prince George, BC V2M 3Z1, Canada
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INAF—Osservatorio Astronomico di Roma, via Frascati 33, I-00040 Monteporzio Catone, Italy
59
BAA Variable Star Section, 67 Ellerton Road, Kingstanding, Birmingham B44 0QE, UK
60
Indian Institute of Astrophysics, II Block Koramangala, Bangalore 560034, India
61
Instituut voor Sterrenkunde, Celestijnenlaan. 200D, bus 2401, B-3001 Leuven, Belgium
62
Zentrum fur Astronomie der Universität Heidelberg, Landessternwarte, Knigstuhl 12, D-69117, Heidelberg, Germany
63
Observatori Cal Maciarol mòdul 8. Masia Cal Maciarol, camí de l’Observatori s/n E-25691 Àger, Spain
64
Department of Astronomy & Astrophysics, 525 Davey Lab, The Pennsylvania State University, University Park, PA 16802, USA
65
Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse, D-85748 Garching, Germany
66
Technische Universität München, Physik Department, James-Franck-Str., D-85748 Garching, Germany
67
Astronomy and Space Physics Department, Taras Shevshenko National University of Kyiv, Volodymyrska str. 60, 01033 Kyiv, Ukraine
68
C/Petrarca 6 1
a
E-41006 Sevilla, Spain
69
Department of Physics, University of Adiyaman, Adiyaman 02040, Turkey
70
Center for Astrophysics and Space Astronomy, Department of Astrophysical and Planetary Sciences, Box 389, University of Colorado, Boulder, CO 80309, USA
71
Centre for Astronomy, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, ul. Grudziadzka 5, 87-100 Torun, Poland
72
C/Cardenal Vidal i Barraquee No 3 E-43850 Cambrils, Tarragona, Spain
73
C/Matarrasa, 16 E-24411 Ponferrada, León, Spain
74
Florida International University and SARA Observatory, University Park Campus, Miami, FL 33199, USA
75
Kiepenheuer-Institut fur Sonnenphysic, D-79104, Freiburg, Germany
76
Janusz Gil Institute of Astronomy, University of Zielona Góra, Szafrana 2, PL-65-516 Zielona Góra, Poland
77
Warsaw University Astronomical Observatory, Al. Ujazdowskie 4, PL00-478 Warsaw, Poland
Received 2018 June 15; revised 2018 August 23; accepted 2018 August 27; published 2018 October 5
Abstract
Results from regular monitoring of relativistic compact binaries like PSR 1913+16 are consistent with the
dominant (quadrupole) order emission of gravitational waves (GWs). We show that observations associated with
the binary black hole (BBH) central engine of blazar OJ287 demand the inclusion of gravitational radiation
reaction effects beyond the quadrupolar order. It turns out that even the effects of certain hereditary contributions to
GW emission are required to predict impact flare timings of OJ287. We develop an approach that incorporates this
effect into the BBH model for OJ287. This allows us to demonstrate an excellent agreement between the observed
impact flare timings and those predicted from ten orbital cycles of the BBH central engine model. The deduced rate
of orbital period decay is nine orders of magnitude higher than the observed rate in PSR 1913+16, demonstrating
again the relativistic nature of OJ287ʼs central engine. Finally, we argue that precise timing of the predicted 2019
impact flare should allow a test of the celebrated black hole “no-hair theorem” at the 10% level.
Key words: black hole physics – gravitation – quasars: general – quasars: individual (OJ 287)
1. Introduction
OJ287 (R.A.: 08:54:48.87 and decl.:+20:06:30.6) is a
bright blazar, a class of active galactic nuclei, situated near the
ecliptic in the constellation of Cancer. This part of the sky has
been frequently photographed for other purposes since the late
1800s and therefore it has been possible to construct an
exceptionally long and detailed light curve for this blazar using
historical plate material. It is at a redshift (z) of 0.306
corresponding to a luminosity distance of 1.6 Gpc in the
standard ΛCDM cosmology, which makes it a relatively nearby
object as blazars go. The optical light curve, extending over
120 years (Sillanpää et al. 1988; Hudec et al. 2013 ), exhibits
repeated high-brightness flares (see Figure 1). A visual
inspection reveals the presence of two periodic variations with
approximate timescales of 12 and 60 years, which have been
confirmed through quantitative analysis (Valtonen et al. 2006
).
We mark the ∼60 year periodicity by a red curve in the left
panel of Figure 1 and many observed outbursts/flares are
separated by ∼12 years. The regular monitoring of OJ287,
pursued only in the recent past, reveals that these outbursts
come in pairs and are separated by a few years. The doubly-
peaked structure is shown in the right panel of Figure 1. The
presence of double periodicity in the optical light curve
provided early evidence for the occurrence of quasi-Keplerian
orbital motion in the blazar, where the 12 year periodicity
corresponds to the orbital period timescale and the 60 year
timescale is related to the orbital precession. The ratio of the
two deduced periods gave an early estimate for the total mass
of the system to be ∼18×10
9
M
e
, provided we invoke
general relativity to explain the orbital precession (Pietilä 1998).
It is important to note that this estimate is quite independent of
the detailed central engine properties of OJ287. The host
galaxy is hard to detect because of the bright nucleus; however,
during the recent fading of the nucleus by more than two
magnitudes below the high level state it has been possible to
get a reliable magnitude of the host galaxy. It turns out to be
similar to NGC4889 in the Coma cluster of galaxies, i.e.,
among the brightest in the universe. These results will be
reported elsewhere (M. J. Valtonen et al. 2018, in preparation).
These considerations eventually led to the development of the
binary black hole (BBH) central engine model for OJ287
(Lehto & Valtonen 1996; Valtonen 2008a) .
According to the BBH model, the central engine of OJ287
contains a BBH system where a supermassive secondary black
78
Deceased.
79
Miller Senior Fellow.
2
The Astrophysical Journal, 866:11 (20pp), 2018 October 10 Dey et al.
hole is orbiting an ultra-massive primary black hole in a
precessing eccentric orbit with a redshifted orbital period of
∼12 years (see Figure 2). The primary cause of certain
observed flares (also called outbursts) in this model is the
impact of the secondary black hole on the accretion disk of the
primary (Lehto & Valtonen 1996; Pihajoki 2016). The impact
forces the release of two hot bubbles of gas on both sides of the
accretion disk that radiate strongly after becoming optically
transparent, leading to a sharp rise in the apparent brightness of
OJ287. The less massive secondary BH impacts the accretion
disk twice every orbit while traveling along a precessing
eccentric orbit (Figure 2). This results in double-peaked quasi-
periodic high-brightness (thermal) flares from OJ287. Further-
more, large amounts of matter get ejected from the accretion
disk during the impact and are subsequently accreted to the
disk center. This ensures that part of the unbound accretion-
disk material ends up in the twin jets. The matter accretion
leads to non-thermal flares via relativistic shocks in the jets,
which produce the secondary flares in OJ287, lasting more
than a year after the first thermal flare (Valtonen et al. 2009).
The BBH model of OJ287 can be used to predict the flare
timings (Sundelius et al. 1997; Valtonen et al. 2008b, 2011a)
and the latest prediction was successfully verified in 2015
November. The optical brightness of OJ287 rose above the
levels of its normal variations on November 25, and it achieved
peak brightness on December 5. On that date, OJ 287 was
brighter than at any time since 1984 (Valtonen et al. 2016).
Owing to the coincidence of the start of the flare with the date
of completion of general relativity (GR) by Albert Einstein one
hundred years earlier, it was termed the GR centenary flare.
Detailed monitoring of the 2015 impact flare allowed us to
estimate the spin of the primary BH to be ∼1/3 of the
maximum value allowed in GR. This was the fourth instance
when multiwavelength observational campaigns were launched
to observe predicted impact flares from the BBH central engine
of OJ287 (Valtonen et al. 2008b, 2016). The latest observa-
tional campaigns confirmed the presence of a spinning massive
BH binary inspiraling due to the emission of nano-Hertz
gravitational waves (GWs) in OJ287. These developments
influenced the Event Horizon Telescope consortium to launch
observational campaigns in 2017 and 2018 to resolve the
Figure 1. Left panel displays the optical light curve of OJ287 from 1886 to 2017. We draw a fiducial curve for easy visualization of the inherent long-term variations.
The right panel shows the observed double-peaked structure of the high-brightness flares. The positions of the two peaks are indicated by downward arrows from the
top of the panel.
Figure 2. Artistic illustration of the binary black hole system in OJ287. The present analysis provides an improved estimate for the spin of the primary black hole.
3
The Astrophysical Journal, 866:11 (20pp), 2018 October 10 Dey et al.
presence of two BHs in OJ287 via the millimeter wavelength
Very Long Baseline Interferometry.
Predictions of impact flare timings are made by solving post-
Newtonian (PN) equations of motion to determine the
secondary BH orbit around the primary while using the
observed outburst times as fixed points of the orbit. The PN
approximation provides general relativistic corrections to
Newtonian dynamics in powers of (v/c)
2
, where v and c are
the characteristic orbital velocity and the speed of light,
respectively. The GR centenary flare was predicted using 3PN-
accurate (i.e., third PN order) BBH dynamics that employed
GR corrections to Newtonian dynamics accurate to order (v/c)
6
(Valtonen et al. 2010a, 2010b, 2011b). Additionally, earlier
investigations invoked nine fixed points in the BBH orbit,
which allowed the unique determination of eight parameters
of the OJ287 BBH central engine model (Valtonen
et al. 2010b, 2011b). The GR centenary flare provided
the tenth fixed point of the BBH orbit, which opens up the
possibility of constraining an additional parameter of the
central engine. Moreover, the GW emission-induced rate of
orbital period decay of the BBH in OJ287, estimated to be
∼10
−3
, makes it an interesting candidate for probing the
radiative sector of relativistic gravity (Wex 2014).
These considerations influenced us to explore the observational
consequences of incorporating even higher-order PN contributions
to the BBH dynamics. Therefore, we introduce the effects of GW
emission beyond the quadrupolar order on the dynamics of the
BBH in OJ287 while additionally incorporating next-to-leading-
order spin effects (Faye et al. 2006; Blanchet 2014; Will &
Maitra 2017). Moreover, we incorporate the effects of dominant-
order hereditary contributions to GW emission, detailed in
Blanchet & Schäfer (1993), on to the binary BH orbital dynamics.
It turns out that these improvements to BBH orbital dynamics
cause non-negligible changes to our earlier estimates for the BBH
parameters, especially for the dimensionless angular momentum
parameter of the primary BH in OJ287, and the inclusion of
present improvements to BBH orbital dynamics should allow the
test of the black hole “no-hair theorem” during the present decade.
This is essentially due to our current ability to accurately predict
the time of the next impact flare from OJ287, influenced by the
present investigation.
This paper is structured as follows. In Section 2, we discuss
briefly the improved BBH orbital dynamics. The details of our
approach to obtain the parameters of the BBH system from
optical observation of OJ287 are presented in Section 3. How
we incorporate the effects of dominant-order “hereditary”
contributions to GW emission into BBH dynamics is detailed
in Section 4. Implications of our improved BBH model on
historic and future observations are outlined in Section 5. In the
Appendix, we display PN-accurate expressions used to
incorporate “hereditary” contributions to BBH dynamics.
2. PN-accurate BBH Dynamics
The PN approach, as noted earlier, provides general
relativistic corrections to Newtonian dynamics in powers of
(v/c)
2
. In this paper, we deploy a PN-accurate expression for
the relative acceleration in the center-of-mass frame, appro-
priate for compact binaries of arbitrary masses and spins.
Influenced by Blanchet (2014) and Will & Maitra (2017),we
schematically write
º=++ +
+++ +
++
++
-
()
()
()
x
x
xx x x
xxx x
xx
xx
d
dt
¨¨¨¨¨
¨¨¨ ¨
¨¨
¨¨
,1
2
2
0 1PN 2PN 3PN
2.5PN 3.5PN 4PN tail 4.5PN
SO SS
Q 4PN SO RR
where
=-xx x
12
gives the center-of-mass relative separation
vector between the black holes with masses m
1
and m
2
. The
familiar Newtonian contribution, denoted by
x
¨
0
, is given by
=-xx
¨
Gm
r
0
3
, where m=m
1
+m
2
,
= ∣∣x
r
. Additionally,
below we use
º
ˆ
nxr
,
=
˙
x
v
and η=m
1
m
2
/m
2
. The PN
contributions occurring at 1PN, 2PN, and 3PN orders, denoted
by
x
¨
1PN
,
x
¨
2PN
,
x
¨
3PN
, are conservative in nature and result in a
precessing eccentric orbit. The explicit expressions for these
contributions can easily be adapted from Equations (219)–
(222) in Blanchet (2014) and therefore are in the modified
harmonic gauge. The second line contributions enter the
x
¨
expression at 2.5PN, 3.5PN, 4PN, and 4.5PN orders and are,
respectively, denoted by
x
¨
2.5PN
,
x
¨
3.5PN
,
()
x
¨
4PN tail
, and
x
¨
4.5PN
.
These are reactive terms in the orbital dynamics and cause the
shrinking of the BBH orbit due to the emission of GWs, and
their explicit expressions are available in Equations (219) and
(220) of Blanchet (2014). Later, we will provide explanations
for the
()
x
¨
4PN tail
term in detail.
The conservative spin contributions enter the equations of
motion via spin–orbit and spin–spin couplings and are listed in
the third line of Equation (1). These are denoted by
x
¨
SO
and
x
¨
SS
, while the
x
¨
Q
term stands for a classical spin–orbit coupling
that brings in the quadrupole deformation of a rotating BH. The
term
-()
x
¨
4PN SO RR
stands for the spin–orbit contribution to the
gravitational radiation reaction (RR), extractable from Equation
(8) in Zeng & Will (2007). We adapted Equations 5.7(a) and
5.7(b) of Faye et al. (2006) to incorporate spin–orbit
contributions that enter the dynamics at 1.5PN and 2.5PN
orders, and these equations generalize the classic result of
Barker & O’Connell (1975). The dominant-order general
relativistic spin–spin and classic spin – orbit contributions,
entering the
x
¨
expression at 2PN order, are extracted from
Valtonen et al. (2010b), and we have verified that our explicit
expressions are consistent with Equation (2.3) of Will &
Maitra (2017).
The spin of the primary black hole precesses owing to
general relativistic spin–orbit, spin–spin, and classical spin–
orbit couplings, and the relevant equation for
s
1
, the unit vector
along the direction of primary BH spin, may be symbolically
written as
W=´ ()
s
s
d
dt
,2a
1
1
WW W W=++ (),2b
SO SS Q
where the spin of the primary black hole in terms of its Kerr
parameter (
c
1
) is given by
c=
S
sGm
c
1
1
2
1
1
(χ
1
is allowed to
take values between 0 and 1 in GR ). For the general relativistic
spin–orbit contributions to
W
, we have adapted Equations (6.2)
and (6.3) of Faye et al. (2006), while spin–spin and classical
spin–orbit contributions are listed by Valtonen et al. (2011b).
Let us turn our attention on the RR terms, listed in the
second line of Equation (1). The RR contributions to
x
¨
4
The Astrophysical Journal, 866:11 (20pp), 2018 October 10 Dey et al.
