Radiation Backreaction in Spinning Binaries

3 Pages·0.07 MB·English

by  László Á. Gergely

RADIATION BACKREACTION IN SPINNING BINARIES
La´szl´o A´. Gergely, Zolt´an Perj´es and Ma´ty´as Vasu´th
KFKI Research Institute for Particle and Nuclear Physics,
Budapest 114, P.O.Box 49, H-1525 Hungary
e-mail: [email protected] [email protected] [email protected]
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The evolution under radiation backreaction of a binary system consisting of a black
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hole and a companion is studied in the limiting case when the spin of the companion
1 isnegligiblecompared withthe spin S of theblack hole. Tofirstorder inthespin, the
n motionofthereduced-massparticleexcludingradiationeffects,ischaracterizedbythree
a constants: theenergyE,themagnitudeLoftheangularmomentumandtheprojection
J LS of the angular momentum along the spin S. This motion is quasiperiodic with a
perioddeterminedbyrmin andrmax. Weintroduceanewparametrization,makingthe
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integrationoveraperiodofagenericorbitespeciallysimple. Wegivetheaveragedlosses

intermsofthe’constantsofmotion’duringoneperiodforgenericorbits,tolinearorder
1
inspin.
v
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1 Introduction
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1
0 We describe here the gravitational radiation backreaction on two kinds of binary
8 systems: a black hole accompanied either by another black hole with comparable
9 mass (compact binary, CB) or by a neutron star viewed as a test particle in the
/
c Lense-Thirring (LT) picture.
q
Our recently developed method can be applied whenever the Newtonian evo-
-
r lution of the binary system is perturbed in such a way that an uncoupled radial
g
equation exists. Then for any bounded orbit the turning points rmax are at r˙ =0.
: min
v The half period is defined as the time elapsed between consecutive turning points.
i We introduce the true anomaly parametrization r = r(χ) for the evaluation of
X
integrands of the type: F/r2+n, where n is a positive integer, defined as:
r
a
dr
=−(γ0+Sγ1)r2 , r(0)=rmin and r(π)=rmax (1)
d(cosχ)
whereγ0,γ1 areconstants. Theintegralsoveroneperiodareconvenientlyevaluated
by computing the residues enclosed in the circle ζ =eiχ. This parametrization has
the nice feature that there is only one pole, at ζ = 0. We replace the usual polar
angles by new, monotously changing angle variables.
We employ a post-Newtonian (PN) and additional expansions in both cases:
In the LT limit an additional expansion over the small parameter η = m2/m1 is
necessary. ForaCB,thespinS isofPN1/2 ordersmaller1 thantheorbitalangular
momentum L.
Our main result is that we obtain the leading spin terms in the averaged losses
of the constants of motion on generic orbits.
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2 The orbit in the absence of radiation
The equationsofmotioninthe presenceofthe spincanbe derivedfromthe second
order Lagrangian2,3,4:
µv2 gmµ 2(1+η)gµ ηµ
L= + + v(r×S)+ v(a×S) (2)
2 r c2r3 2c2m
where r=|r| is the relative distance, v the relative velocity,µ=m1m2/(m1+m2)
the reduced mass and m= m1+m2 the total mass of the system. The parameter
η vanishes in the LT case.
Up to linear terms in the spin there are three constants of motion: the energy
E, the magnitude L and the spin projection L of the orbital angular momentum:
S
µv2 gmµ gL S µ gmµ gL S
E = − +η S = [r˙2+r2(θ˙2+sin2θ ϕ˙2)]− +η S
2 r c2r3 2 r c2r3
gµL S 2η
L2 = µ2r4(θ˙2+sin2θ ϕ˙2)−4 S + EL S (3)
c2r c2m S
S
L = L· =Lcosκ .
S
S
From these a pure radial equation follows:
E gm L2 EL S gL S
r˙2 =2 +2 − +2η S −2(2+η) S . (4)
µ r µ2r2 c2mµ2r2 c2µr3
The true anomaly parametrization,by (1), is:
L2 2ηLSS gmµ2A0+(g2m2µ3+EL2)cosχ
r = − (5)
µ(gmµ+A0cosχ) c2mµ2A0 (gmµ+A0cosχ)2
2(2+η)gLSS A0(2g2m2µ3+EL2)+gmµ(2g2m2µ3+3EL2)cosχ
+
c2L2A0 (gmµ+A0cosχ)2
whereA0 =(g2m2µ2+2EL2/µ)1/2 . Byintroducingthree Euleranglevariables,Ψ
(theargumentofthelatitude),ι (theinclinationoftheorbit)andΦ(thelongitude
N
of the node), the equations of motion became simpler. From among these angles
onlythezerothorderpart(inthespin)oftheangleΨ=Ψ0+χenterstheradiative
losses.
3 Averaged radiation losses
Theinstantaneousradiationlossesoftheconstantsofmotionareevaluatedemploy-
ing the radiative multipole tensors of Kidder, Will and Wiseman4, which originate
in the Blanchet-Damour-Iyer formalism 5. The instantaneous losses for the energy
E and total angular momentum J were given by Kidder 6. From these we derive
both in the LT case and in the CB case the instantaneous losses of the constants
E,L and L .
S
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Now in the averagingprocesswe replace the time integrationby the parameter
integration, then we use the residue theorem to find the averaged radiative losses
in the constants of motion. For instance, the power is
dE g2m(−2µE)3/2
=− (148E2L4+732g2m2µ3EL2+425g4m4µ6)
D dt E 15c5L7
S(−2µE)3/2g2cosκ
+ × (6)
10c7L10
520E3L6+10740g2m2µ3E2L4+24990g4m4µ6EL2+12579g6m6µ9
h
+η(256E3L6+6660g2m2µ3E2L4+16660g4m4µ6EL2+8673g6m6µ9) .
i
The LT limit, η → 0, includes the results by Peters and Mathews 7,8 for the
special case of a nonspinning black hole; by Shibata 9 for an equatorialorbitabout
a spinning black hole; and by Ryan for a circular 10 and generic orbit 11 about a
spinning black hole, respectively. With the proper definitions of orbitalparameters
we find perfect agreement.
Complete details of our computations including some subtleties and the radia-
tive losses in E, L and L both in terms of the constants of motion and orbital
S
parameters are described elsewhere 12,13.
Acknowledgments
ThisworkhasbeensupportedbyOTKAno. T17176andD23744grants. Financial
support for participation at the MG8 Conference for L.A´.G. from the Unesco and
for M.V. from EPS is acknowledged.
References
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3. K.S.Thorne and J.Hartle, Phys. Rev. D 31, 1815 (1985).
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5. L.Blanchet, T.Damour and B.Iyer, Phys. Rev. D 51, 5360 (1995).
6. L.Kidder, Phys. Rev. D 52, 821 (1995).
7. P.C.Peters and S.Mathews, Phys. Rev. 131, 435 (1963).
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11. F.Ryan, Phys. Rev. D 53, 3064 (1996).
12. L.Gergely, Z.Perj´es and M.Vasu´th, Phys. Rev. D , (to appear).
13. L.Gergely, Z.Perj´es and M.Vasu´th, submitted
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