Frequency Evolution of the Gravitational Waves for Compact Binaries

ASPConferenceSeries,Vol.999,2006C.Sterken&C.Aerts

FrequencyEvolutionoftheGravitationalWavesforCompactBinaries

B.Mik´oczi

DepartmentsofTheoreticalandExperimentalPhysics,UniversityofSzeged,Szeged6720,Hungary

Abstract.Wepresenthereacompletelistofcontributionstothegravita-tionalwavefrequencyevolutionfromcompactbinariesoncircularorbitsuptothesecondpost-Newtonianorderbyincludingtheinteractionofmagneticdipolemoments,quadrupole-monopoleinteractionstogetherwiththespin-orbit,thespin-spinandtheself-interactionspincontributions.WeapplyourresultstotheManchesteretal.modeloftheJ0737-3039A-Bdoublepulsar.

1.Introduction

Inspirallingcompactbinaries(likedouble-pulsarsystems)emitgravitationalradiation.Thesebinarysystemsarethemostpromisingsourceofthegravi-tationalwavesfortheEarth-basedinterferometricdetectors(LIGO,VIRGO,GEO,TAMAandAIGO).Thecoalescenceofsuchcompactbinariesisprecededbyamilderinspiralphase,forwhichthepost-Newtonianapproachprovidesareliabledescription.Thisdescriptionisgenerallyconsideredvaliduntilthesys-temreachestheinnermoststablecircularorbit(ISCO).TheISCOisnotevenpreciselydefined.Forthespecialcaseofanon-rotatingblackholetheISCOisr=6Mandatr=Mforamaximallyrotatingblackhole(throughoutthepaperweuseG=c=1).Forneutronstarsthegravitationalwavefrequencyis800-1230Hz(Oechslinetal.2004)atISCO.TheLIGOsensitivityis1000Hz.

Thedescriptionofmotionwasgivento3.5post-Newtonian(3.5PN)order

accuracy,withtheinclusionofspin-orbiteffects(SO)(Wex1995;Rieth&Schafer1997;Gergelyetal.1998)andtheirfirstPNcorrection(Tagoshietal.2001).Spin-spin(SS)(Kidder1995;Gergely2000),quadrupole-monopole(QM)(Poisson1998;Gergely&Keresztes2003)andmagneticdipole-magneticdipole(DD)(Ioka&Taniguchi2000;Vas´uthetal.2003)andself-interactionspinterm(SS-self)(Gergely2000;Mik´oczietal.2005)contributionstotheaccelerationswerealsodiscussed.Onthelongrunduetotheemissionofgravitationalwavestheorbittendstocircularize.Thereforeweconsidercircularorbits,forwhichthegravitationalwavefrequencyistwicetheorbitalfrequency.Weevaluatetherateofincreaseoff.Thisisgivenbytherateofchangeoftheorbitalangularfrequencyω=πfunderradiationreaction.WepresenttheaccumulatednumberofcyclesleftuntiltheISCOandthespinparametersintheManchesteretal.modeloftheJ0737-3039A-Bdoublepulsar.

1

22.

Mik´oczi

FrequencyEvolutionandAccumulatedNumberofCycles

Inthissectionwesummarizesomeoftheresultsof(Mik´oczietal.2005)on

thefrequencyevolutionandaccumulatednumberofcyclesforcompactbinariesoncircularorbit.Theradialprojectionoftheaccelerationdefinestheorbitalangularvelocityω.Weexpressedboththeenergyandtheenergylosswithω.Thentheevolutionoftheradiativeorbitalangularfrequencyis:

󰀈

dω

󰀂

534103

󰀄

1−

󰀋

743

459

η(mω)2/3+(4π−β)mω

󰀍

+

2016

η+

12

σS1S2=

S1S2

2󰀅Sii=1

m2

+75ηcosκi,

󰀃

96m2

2󰀋󰀅Sii=1

2

2󰀅i=1

pi3cosκi−1,

5

󰀇

2

󰀉

σDD=−

πη+

󰀂

󰀌

τ5/8+

󰀋

3715

96258048

ητ3/8+

1855

󰀍

3

4

−πτ1/4

󰀃

󰀍

9275495

η+

τ1/8.

󰀁

(4)

FrequencyEvolutionoftheGravitationalWavesforCompactBinaries

3

wherethedimensionlesstimevariableτ=η(tc−t)/5misrelatedtothetime(tc−t)leftuntilthefinalcoalescence.Thetailand2PNcontributionsagreewiththosegivenin(Blanchetetal.2002).

Table1.

Accumulatednumberofgravitationalwavecycles.

fin(Hz)10104.199×10−4ffin(Hz)10001000

3.997×10−2

Table2.Thespinparametersforthetwosolutions(JR1&JR2)oftheJenet-Ransommodelandfortwosolutions(M1andM2)ofthemodeloftheManchesteretal.representingthebinarypulsarJ0737-3039.

β

σS4−0.1660.0010.1680.0871S2(10−)−0.37200.3710.191σSS−self(10−4)0.2980.3450.2980.333

4Mik´oczi

(JR2).Theotherangleκ2forpulsarBinprinciplecanbedeterminedbysolving

numericallythespinprecessionequations.Accordingto(Kaspietal.2004)itislikelythatwind-torquesfromtheenergeticallydominantcomponenthavedriventhespinaxisoftheothercomponenttoalignwiththedirectionofL,causingκ2=0.Thereforeγ=κ1.WegivetheestimatesforthespinparametersinTable2.

BesidestheJR1andtheJR2solutionswepresenthereforthefirsttimeanmodel(Manchesteretal.2004).Accordingtothismodelthemeasurementsthefavorizemaximallyallowedtheangleκ1=60◦.Thebest-fitsolution(M1)islocatedatκ1=14◦.Wealsopresentresultsforthemaximallyallowedvalueofκ1(M2).Asitcanbeseeninthismodeltheproperspin-spincontributioniscomparablewiththeself-interactionspincontribution.4.

Conclusions

Wehavepresentedthecompletesetofcontributionsuptosecondpost-Newtonianorder(PN,SO,SS,QM,DD,tailand2PN)totheevolutionofgravitationalwavefrequencyandtotheaccumulatednumberofgravitationalwavecyclesleftuntilthefinalcoalescence,withtheinclusiontheself-interactionspinterms(SS-self).

WehaveshownthatontheManchesteretal.modeloftheJ0737-30309A-Bdoublepulsartheself-interactionspincontributionsarecomparablewithspin-spincontributions.

ThisworkwassupportedbyOTKAno.TS044665andT046939grants.

References

Blanchet,L.,FayeG.,Iyer,B.R.,Joguet,B.2002,Phys.Rev.D65,061501Burgay,M.,etal.2003,Nature426,531

´2000,Phys.Rev.D62024007;ibid.D61,024035Gergely,L.A.

´Keresztes,Z.2003,Phys.Rev.D67,024020Gergely,L.&A.,´Perj´Gergely,L.A.,es,Z.I.,Vas´uth,M.1998Phys.Rev.D57,876;ibid.D57,3423;ibid.D58,124001

Ioka,K.&Taniguchi,T.2000,Phys.Rev.537,327Jenet,F.A.&Ransom,S.M.2004,Nature428,919Kaspi,V.M.,etal.2004,ApJ.613,137Kidder,L.1995,Phys.Rev.D52,821Lyne,A.G.etal.2004,Science303,1153Manchester,R.N.,etal.2005,ApJ.621L49

´2005,Phys.Rev.D71124043Mik´oczi,B.,Vas´uth,M.,Gergely,L.A.

Oechslin,R.,etal.2004,Mon.Not.Roy.AstronSoc.349,1469Poisson,E.1998,Phys.Rev.D57,5287

Poisson,E.,Will,C.M.1995,Phys.Rev.D52,848

Rieth,R.&Schafer,G.1997,Class.QuantumGrav,14,2357

Tagoshi,H.,Ohashi,A.,Owen,B.J.2001,Phys.Rev.D63044006

´2003,Phys.Rev.D68,124006Vas´uth,M.,Keresztes,Z.,Mih´aly,A.,Gergely,L.A.

Wex,N.1995,Class.QuantumGrav.12,983