Measuring Black Hole Spin in OJ287
M. Valtonen, S.Mikkola, H.J.Lehto, T.Hyvönen, K.Nilsson, D.Merritt, A.Gopakumar, H.Rampadarath, R.Hudec, M.Basta, R. Saunders
aa r X i v : . [ a s t r o - ph . C O ] J a n Measuring Black Hole Spin in OJ287
Valtonen , , S. Mikkola , H. J. Lehto , T. Hyv¨onen , K. Nilsson ,D. Merritt , A. Gopakumar , H. Rampadarath , , R. Hudec , ,M. Basta and R. Saunders Tuorla Observatory, Department of Physics and Astronomy, University of Turku,21500 Piikki¨o, Finland Centre for Computational Relativity and Gravitation, Rochester Institute ofTechnology, 78 Lomb Memorial Drive, Rochester, NY 14623, USA Tata Institute of Fundamental Research, Mumbai 400005, India School of Physics and Astronomy, University of Manchester, Alan TuringBuilding, Oxford Road, Manchester M13 9PL, UK Department of Physics, University of the West Indies, St. Augustine, Trinidad &Tobago Astronomical Institute, Academy of Sciences, Fricova 298, 25165 Ondrejov,Czech Republic Czech Technical University in Prague, Faculty of Electrical Engineering,Technick 2, 166 27 Praha 6, Czech Republic Helsinki Institute of Physics, PL 64, FIN-00014 Helsingin yliopisto, Finland
Abstract.
We model the binary black hole system OJ287 as a spinning primary and a non-spinning secondary. It is assumed that the primary has an accretion disk which isimpacted by the secondary at specific times. These times are identified as majoroutbursts in the light curve of OJ287. This identification allows an exact solutionof the orbit, with very tight error limits. Nine outbursts from both the historicalphotographic records as well as from recent photometric measurements have beenused as fixed points of the solution: 1913, 1947, 1957, 1973, 1983, 1984, 1995, 2005and 2007 outbursts. This allows the determination of eight parameters of the orbit.Most interesting of these are the primary mass of 1 . · M ⊙ , the secondary mass1 . · M ⊙ , major axis precession rate 39 ◦ . .
70. The dimensionless spin parameter is 0 . ± .
01 (1 sigma). The lastparameter will be more tightly constrained in 2015 when the next outburst is due.The outburst should begin on 15 December 2015 if the spin value is in the middleof this range, on 3 January 2016 if the spin is 0.25, and on 26 November 2015 ifthe spin is 0.31. We have also tested the possibility that the quadrupole term inthe Post Newtonian equations of motion does not exactly follow Einstein’s theory:a parameter q is introduced as one of the 8 parameters. Its value is within 30% (1sigma) of the Einstein’s value q = 1. This supports the no − hair theorem of blackholes within the achievable precision. We have also measured the loss of orbitalenergy due to gravitational waves. The loss rate is found to agree with Einstein’svalue with the accuracy of 2% (1 sigma). There is a possibility of improving theaccuracy of both quantities using the exact timing of the outburst on 21 July 2019.Because of closeness of OJ287 to the Sun (8 − ◦ ), the observations would be bestcarried out by a telescope in space. Keywords: gravity — relativity — stellar systems — black hole physicsc (cid:13)
Kluwer Academic Publishers. Printed in the Netherlands.
IAU261Valtonen.tex; 4/11/2018; 19:08; p.1
Valtonen et al. V - band f l u x / m Jy JDyear
Figure 1.
The observations of the brightness of OJ287 from late 1800’s until today.
1. Introduction
The range of celestial mechanics has been expanding from purely Solarsystem studies to binary stars and planetary systems around otherstars, and in recent years also to binary pulsars. The most recent appli-cation of celestial mechanics is in the binary system of two supermassiveblack holes in the quasar OJ287. The identification of this system asa likely binary was made as early as 1982 (Sillanp¨a¨a et al. 1988), butsince the mean period of the system is as long as 12 yr, it has taken aquarter of century to find convincing proof that we are indeed dealingwith a binary system (Valtonen et al. 2008). The primary evidence fora binary system comes from the optical light curve. By good fortune,the quasar OJ287 has been photographed accidentally since 1890’s, wellbefore its discover in 1968 as an extragalactic object. The light curveof over hundred years (Figure 1) shows a pair of outbursts at 11 - 14yr intervals and the two brightness peaks are separated by 1 - 3 yrs.The system is not strictly periodic, but there is a simple mathemat-ical rule which gives all major outbursts of the historical record. Todefine the rule, one assumes that a companion orbits the primary ina Keplerian orbit. Then demand that an outburst is produced everytime the companion passes through a constant phase angle relative tothe primary, and that also another outburst takes place at the oppositephase angle. Due to the nature of Keplerian orbits, this rule cannot bewritten in a closed mathematical form, but its consequences are easily
IAU261Valtonen.tex; 4/11/2018; 19:08; p.2 lack hole spin ◦ per period), the whole historical outburst record of OJ287 iswell reproduced. What is more important, the model is able to predictfuture outbursts. The prediction of the latest outburst on 13 September2007 (Valtonen 2007,2008) was accurate to one day, leaving little doubtabout the capability of the model (see Figure 2; the predicted timerefers in this case to the start of the rapid flux rise, i.e. the beginningof the phase 2 of the outburst, as described in the next section).The more advanced versions of the model make use of the PostNewtonian terms in the two-body orbit calculation as well as likelyastrophysical details of the radiation mechanism of the outbursts. Anatural way to induce two outbursts per orbit is to consider an accretiondisk around the primary body, and to associate the two opposite phaseangles with impacts of the secondary on this disk. By using the standard α disk theory of Shakura and Sunyaev (1973) and its extension tomagnetic disks by Sakimoto and Coroniti (1981), the problem remainsmathematically well defined. In fact, using only 6 outbursts as fixedpoints in the orbit, it is possible to solve five orbital parameters in thefirst approximation (Valtonen 2007). The success of this model in pre-dicting the 2007 outburst has encouraged us to add the remaining welldefined outbursts in the model (9 in all), and to solve the problem for 8parameters. In this paper we first describe the method of solution, andthen list the orbital parameters. Finally, we show that the parametersfit very well with what we would expect of binary black holes, and showthat the likelihood of finding an OJ287-like object in the survey of thewhole sky is of the order of unity, i.e. we have not been particularlylucky in the discovery of this binary system in the nearly whole skysearch of periodic variability of quasars which was initiated by Tuorlaobservatory in 1980.As a side result, we may test the idea that the central body isactually a black hole. One of the most important characteristics ofa black hole is that it must satisfy the so called no-hair theorem ortheorems (Israel 1967, 1968, Carter 1970, Hawking 1971, 1972; seeMisner, Thorne and Wheeler 1973). A practical test was suggested byThorne and Hartle (1985) and Thorne, Price and Macdonald (1986). Inthis test the quadrupole moment Q of the spinning body is measured.If the spin of the body is S and its mass is M , we determine the valueof q in Q = − q S M c . (1) IAU261Valtonen.tex; 4/11/2018; 19:08; p.3
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For black holes q = 1, for neutron stars and other possible bosonicstructures q >
2. OJ287 Outburst Structure
Besides the quasiperiodic pattern of outbursts at about 12 year in-tervals, there is another crucial piece of information which helps inaccurate orbit determination. The radiation at the outburst peaks ismost likely thermal bremsstrahlung radiation from gas at about 10 K temperature. This is evident from the optical-UV spectral energydistribution (Ciprini & Rizzi 2008) as well as from the unpolarizednature of the excess radiation at the outbursts. In contrast, the opticalradiation of OJ287 is non-thermal highly polarized synchrotron emis-sion at ’normal’ times, with the optical-UV spectral index of about-1.5. The reason for the thermal bremsstrahlung is thought to be animpact of a secondary black hole on the accretion disk of the primary(Lehto & Valtonen 1996, Sundelius et al. 1997). An outburst beginswhen a bubble of gas torn off the accretion disk expands, cools andat some point becomes transparent at optical wavelengths. When sucha bubble is viewed from a distance, the emission is seen to grow ina specific way as the observational front advances into the bubble.The size of the bubble, and thus also the rate of development of theoutburst light curve, is a known function of distance from the centerof the accretion disk. The process has three stages: (1) a sphericalbubble becomes transparent at its forward side which causes the initialrise in the light curve. (2) When the bubble is seen approximatelyhalf-way through, it is fully transparent, and the flux rises sharply inthe light-crossing time of the bubble. (3) The fully transparent bubblecontinues its expansion adiabatically, and consequently the radiationfalls in power law manner. Figure 2 illustrates the three stages for theobservations of the 2007 outburst, with the dash-dot and dashed linesoutlining the first and the third stages of the outburst, respectively.From the point of view of timing, it is important to identify the initialmoment of transparency when stage (1) begins. For the 2007 outburstthis moment was at late hours of 10 September GMT; expressed asfractional years the moment is 2007.692 (the line drawn earlier thanthis has no physical significance in Figure 2). Similar timing fits havebeen carried for all 9 outbursts, and the corresponding timings arelisted in Table 1, including the measurement uncertainties. Note thatthe timing ranges are slightly narrower than in Valtonen et al. (2010)where the spin determination was first carried out. It makes the findingof solutions harder but not impossible.
IAU261Valtonen.tex; 4/11/2018; 19:08; p.4 lack hole spin m Jy day, Sept 2007OJ287 outburst at 2007.692 Figure 2.
Observations of the 2007 outburst (crosses) together with the lines mod-elling the phase (1) and phase (3) evolution of the radiating bubble. Phase (2)evolution happens on 13 September when the brightness increases sharply.
3. PN-accurate orbital description
We calculate the binary orbit using PN-accurate orbital dynamics thatincludes the 3PN-accurate conservative non-spinning contributions, theleading order general relativistic, classical spin-orbit and radiation re-action effects (Barker and O’Connell 1975, Damour 1982, Kidder 1995,Mora and Will 2004). The relevant PN-accurate equations of motioncan be written schematically as¨ x ≡ d x dt = ¨ x + ¨ x P N + ¨ x SO + ¨ x Q +¨ x P N + ¨ x . P N + ¨ x P N , (2)where x = x − x stands for the center-of-mass relative separationvector between the black holes with masses m and m and ¨ x repre-sents the Newtonian acceleration given by ¨ x = − G mr x ; m = m + m and r = | x | . The PN contributions occurring at the conservative 1PN,2PN, 3PN and the reactive 2.5PN orders, denoted by ¨ x P N , ¨ x P N ,¨ x P N and ¨ x . P N respectively, are non-spin by nature, while ¨ x SO is thespin-orbit term of the order 1.5PN.The employed PN-accurate equations of motion are in harmoniccoordinates and satify covariant spin supplementary condition. In thepresent analysis, we also included the precessional motion for the spin IAU261Valtonen.tex; 4/11/2018; 19:08; p.5
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Table I. Outbursttimes with estimateduncertainties. The1982/3 outburst is usedas a reference time.1912.970 ± ± ± ± ± ± ± ± ± of the primary black hole due to leading order general relativistic spin-orbit coupling.The quadrupole-monopole interaction term ¨ x Q , entering at the 2PNorder, reads¨ x Q = q χ G m m c r (cid:26)(cid:20) n · s ) − (cid:21) n − ( n · s ) s (cid:27) , where the parameter q , whose value is 1 in general relativity, is intro-duced to test the black hole ‘no-hair’ theorem. The Kerr parameter χ and the unit vector s define the spin of the primary black hole by therelation S = G m χ s /c and χ is allowed to take values between 0and 1 in general relativity. The unit vector n lies along the direction of x . The initial orientation of the binary orbit is perpendicular to theaccretion disk of the primary. The spin axis of the primary black holeis tilted 6 degrees relative to the spin axis of the accretion disk, andremains at this angle while the black hole spin axis precesses around thedisk spin axis, with a period of 1300 yrs. The disk itself also wobbles,but with a smaller amplitude and with a period of 120 yrs. This showsup as modulation in the long term evolution of the light curve. Fordetails, see Valtonen et al. (2009, 2010).
4. Timing experiments
We now require that the correct orbit must reproduce the nine outbursttimes of Table 1 within the stated error limits. An automatic search
IAU261Valtonen.tex; 4/11/2018; 19:08; p.6 lack hole spin
Table II. Solution parameters.∆ φ ◦ . ± ◦ . m (1 . ± . · M ⊙ m (1 . ± . · M ⊙ χ . ± . φ ◦ . ± ◦ . e . ± . q . ± . t d . ± . We will now discuss the physical significance of some of the param-eters. The precession rate 39 ◦ . m = 1 . · M ⊙ which is quite consistent with the values of quasarblack hole masses in a large sample of SDSS quasars; the masses extendwell beyond 2 · M ⊙ (Vestergaard et al. 2008).As to the expected frequency of such high mass values in a mag-nitude limited quasar sample, we may make the following estimate.OJ287 is highly variable mostly due to strong beaming. At its faintestit goes to m B = 18 which may be taken as its intrinsic (unbeamed)brightness. There are about 2 · quasars in the sky brighter thanthis magnitude limit (Arp 1981). Many of these quasars host binaryblack holes, perhaps as many as 50% (Comerford et al. 2009). Thuspotentially there are about 10 bright binary quasars in the sky to bediscovered. Vestergaard and Kelly (private communication) estimatethat there is a 25% probability that about one hundred of them aremore massive than 10 M ⊙ . It is not unlikely that one of them would IAU261Valtonen.tex; 4/11/2018; 19:08; p.7
Valtonen et al. possess a jet which points more a less directly toward us, and thattherefore the quasar would appear as a blazar. The brightness of thequasar is magnified by orders of magnitude by the jet, and so is theprobability of its detection.On the other hand, most of the binary black hole systems shouldhave longer periods than OJ287 which is in the last stages of inspiral,with only 10 yr left out of its potential 10 yr lifetime (Volonteri etal. 2009). This compensates at least partly for the increased discoveryrate due to beaming. Thus the number of bright short period binaries inthe 10 M ⊙ category which could be discovered by techniques similarto the discovery of OJ287 is of the order of unity. We may have beenlucky to discover one such example, OJ287, but anyhow this is onlyan order of magnitude calculation, and more OJ287’s may turn up infuture searches of periodically varying quasars.What about the secondary mass? Its value was calculated by Lehtoand Valtonen (1996) as m ≈ . · M ⊙ , updated to today’s Hubbleconstant and to the outburst peak brightness observed in 2007. Thisestimate is based on the astrophysics of the disk impacts and on theamount of radiation produced in these events, and thus it is totallyindependent of the orbit model. This agrees well with our more exactvalue m = 1 . · M ⊙ . The fact that the mass ratio is as high as 126is quite natural since equal mass binaries do not easily merge within theHubble time (Makino & Funato 2004) and also because the accretiondisk would be unstable if the mass ratio were less than 100.The initial apocenter eccentricity at the beginning of our simulationis e = 0 .
66. It refers to the osculating Keplerian orbit and the oscu-lating eccentricity value varies considerably over the orbit. If definedby using pericenter and apocenter distances as for a Keplerian orbit,the eccentricity is e = 0 .
7. The initial eccentricity at the beginning ofthe final inspiral, say, 10 yr prior to merger, must have been e ≈ . t d gives the ratio of the viscosity parameter α g tothe accretion rate in Eddington units. The value t d = 0 .
74 implies14 ± . .
005 ofthe Eddington rate (Bassani et al. 1983) which gives α g = 0 . ± . χ = 0 .
28. Some of the outbursttimings are particularly sensitive to χ , among them the 1973 outburst.Figure 3 illustrates how the outburst light curves depend on χ in thiscase. Thus we infer that the primary black hole spins approximately IAU261Valtonen.tex; 4/11/2018; 19:08; p.8 lack hole spin
12 14 16 18 20 22 24 26 28 1972.94 1972.96 1972.98 1973 1973.02 1973.04 1973.06 1973.08 1973.1 V f l u x m Jy Year0.24 0.28 0.32 OJ287 in December 1972spin=
Figure 3.
The brightness measurements of OJ287 in 1972/3 (dots with error bars).The error bars represent short-time variability, to lesser extent measurement errors.The three lines show theoretical light curves for three values of χ . at one quarter of the maximum spin rate allowed in general relativ-ity. There is an additional observation which supports this spin value.OJ287 has a basic 46 ± m = 2 mode wave disturbance in the disk, this is likelyto refer to one half of the period. Considering also the redshift of thesystem, and the primary mass value given in Table 2, this correspondsto the spin of χ = 0 . ± .
06 (McClintock et al. 2006). For comparison,it has been estimated that χ ∼ . χ = 0 .
28 the nextoutburst begins on 15 December, 2015. The range of possible outbursttimes extends from 26 November if χ = 0 .
31 to 3 January 2016 if χ = 0 . q cluster around q = 1 . .
3. Figure 4 illustrates the 1995 outburst and how its timing dependson q . The error limits of q may be further narrowed down in 2019, tothe level of 20% (1 sigma) if the timing is carried out accurately. Thefact that q = 1 with the current accuracy indicates that the ’no-hair’theorem is valid. It is interesting that our result converges at the propervalue for a general relativistic black hole. IAU261Valtonen.tex; 4/11/2018; 19:08; p.9 Valtonen et al.
Figure 4.
Observations of brightness of OJ287 in 1995 (dots with error bars). Thethree lines are theoretical curves for the rising part of the flux (for q = 0 . , . , . We have also tested the sensitivity of the solution to the 2.5PN(radiation reaction) term. One may multiply the 2.5PN term by 1 . ± .
06 and still find a solution. Thus we confirm general relativity with6% (3 sigma) accuracy. It is possible to improve this accuracy also by afactor of 2 in the timing of the 2019 outburst. The main observationalproblem with this outburst is that it occurs close to the conjunction ofOJ287 with the Sun. The separation in the sky is only 12 ◦ at the startof the outburst on 21 July 2019, and decreases to 8 ◦ during the week ofthe main burst. It is an observational challenge, probably too much ofa challenge for ground based observations. Thus a space born telescopemeasurement would be desirable. This is a unique opportunity to testthe ’no-hair’ theorem and the radiation reaction term which will notbe repeated for many decades. References
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