Post-Newtonian accurate parametric solution to the dynamics of spinning compact binaries in eccentric orbits: The leading order spin-orbit interaction

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by  Christian Koenigsdoerffer

Post(cid:21)Newtonian a

urate parametri
solution to the dynami
s of spinning
ompa
t
binaries in e

entri
orbits: The leading order spin(cid:21)orbit intera
tion
∗ †
Christian Königsdör(cid:27)er and A
hamveedu Gopakumar
Theoretis
h(cid:21)Physikalis
hes Institut, Friedri
h(cid:21)S
hiller(cid:21)Universität Jena, Max(cid:21)Wien(cid:21)Platz 1, 07743 Jena, Germany
(Dated: 5th February 2008)
We derive Keplerian(cid:21)type parametrization for the solution of post(cid:21)Newtonian (PN) a

urate

onservative dynami
s of spinning
ompa
t binaries moving in e

entri
orbits. The PN a

urate
dynami
s that we
onsider
onsists of the third post(cid:21)Newtonian a

urate
onservative orbital dy-
nami
sin(cid:29)uen
edbytheleadingorderspine(cid:27)e
ts,namelytheleadingorderspin(cid:21)orbitintera
tions.
Theorbital elementsof therepresentation are expli
itly givenin terms of the
onservedorbital en-
ergy,angularmomentumandaquantitythat
hara
terizestheleadingorderspin(cid:21)orbitintera
tions
in Arnowitt, Deser, and Misner(cid:21)type
oordinates. Our parametri
solution is appli
able in the fol-
lowingtwodistin
t
ases: (i)thebinary
onsistsofequalmass
ompa
tobje
ts,havingtwoarbitrary
5 spins, and (ii) the binary
onsists of
ompa
t obje
ts of arbitrary mass, where only one of them is
0 spinningwithanarbitraryspin. Asanappli
ationofourparametrization, we presentgravitational
0 wave polarizations, whose amplitudes are restri
ted to the leading quadrupolar order, suitable to
2 des
ribe gravitational radiation from spinning
ompa
t binaries moving in e

entri
orbits. The
presentparametrization will berequiredto
onstru
t`readyto use'referen
etemplatesforgravita-
n
tionalwavesfromspinning
ompa
tbinariesininspirallinge

entri
orbits. Ourparametri
solution
a
for the post(cid:21)Newtonian a

urate
onservative dynami
s of spinning
ompa
t binaries
learly indi-
J

ates, for the
ases
onsidered, the absen
e of
haos in these systems. Finally, we note that our
5
parametrization provides the (cid:28)rst step in deriving a fully se
ond post(cid:21)Newtonian a

urate `timing

formula',thatmaybeusefulfortheradioobservationsofrelativisti
binarypulsarslikeJ0737(cid:21)3039.
1
v
PACSnumbers: 04.25.Nx,04.30.Db,97.60.Jd,97.60.Lf
1
1
0 (v/c)2 GM/(c2R) v,M,
I. INTRODUCTION of motion in powers of ∼ , where
1 R
0 and are the
hara
teristi
orbital velo
ity, the total
5 Inspiralling
ompa
t binaries of arbitrary mass ratio, massandthe typi
alorbitalseparationofthebinary. At
0 present, the orbital dynami
s of non(cid:21)spinning
ompa
t

onsistingof bla
k holes and neutron stars, arethe most
c/ plausible sour
es of gravitational radiation for the
om- b(vin/ac)r7iesareexpli
itlyknownto3.5PNorder,whi
hgives
q missioned ground based and proposed spa
e based laser
orre
tions to Newtonian equations of motion.
- TheselengthyhighPN
orre
tionswereobtained,forthe
r interferometri
gravitational(cid:21)wave dete
tors [1℄. The
g possibilityofdete
tinggravitationalradiationfromthese (cid:28)rst time, in Arnowitt, Deser, and Misner (ADM)(cid:21)type
:
oordinates [7(cid:21)10℄. Independently, they were later
om-
v sour
esarebasedonthefollowingastrophysi
alanddata
puted in harmoni

oordinates and, as expe
ted, found
i analysis
onsiderations. The detailed astrophysi
al in-
X to be in perfe
t agreement [11(cid:21)16℄. Further, we point
vestigations indi
ate that the inspiral of
ompa
t bina-
r ries should be observed by gravitational(cid:21)wave dete
tors out that the very re
ent determination of the highly de-
a
sirable 3.5PN (relative) a

urate phase evolution for the
[2, 3℄. The temporallyevolvinggravitationalwavepolar-
h (t) h (t)
+ × gravitationalwavepolarizationswaspossible,mainlybe-
izations, and ,asso
iatedwiththeinspiralling

ause of employing te
hniques used in the
omputations

ompa
tbinariesin quasi(cid:21)
ir
ularorbitsareknownvery
of 3.5PN a

urate equations of motion for the dynami
s
pre
isely [4, 5℄. This allows data analysts to employ
of
ompa
t binaries [9℄.
the optimalmethod ofmat
hed (cid:28)lteringto extra
tthese
weak gravitationalwavesignalsfrom the noisy interfero- However,the e(cid:27)e
tofspinsandorbitale

entri
ihty(otn)
+
metri
data [6℄. the
hom(pt)a
t binary dynami
s and the asso
iated
×
It should be noted that the
onstru
tion of the `ready and are not so a

urately determined. The dy-
h (t)
+ nami
s of spinning
ompa
t binaries are determined not
to use sear
htemplates', namely above mentioned
h (t)
× onlybytheorbitalequationsofmotionfortheseobje
ts,
and , is possible mainly due to the fa
t that the
but also by the pre
ession equations for the spin ve
tors
orbital dynami
s, during the binary inspiral, is well de-
themselves [17℄. In fa
t, for the dynami
s of spinning
s
ribed by the post(cid:21)Newtonian (PN) approximation to

ompa
t binaries only the leading order
ontributionsto
generalrelativity. The PN approximationto generalrel-
spin(cid:21)orbitandspin(cid:21)spinintera
tionsarewellunderstood
ativity allows one to express the equations of motion for
[18, 19℄. It is therefore quite
ustomary, in the existing
a
ompa
t binary as
orre
tionsto Newtonian equations
literature, to
onsider only the leading order
ontribu-
tions to spin(cid:21)orbit and spin(cid:21)spin intera
tions, while ex-
ploring the dynami
s of spinning
ompa
t binaries [20℄
∗
†Ele
troni
address: C.Koenigsdoerfferuni-jena.de or the in(cid:29)uen
e of spins on the phase evolution and am-
Ele
troni
address: A.Gopakumaruni-jena.de plitude modulations of the gravitational waveforms [21℄.
2
However, note that the equations of motion for spinning binaries moving even in inspiralling e

entri
orbits.

ompa
t binaries, whi
h in
lude (cid:28)rst post(cid:21)Newtonian There is an ongoing debate about whether or not the

orre
tions to the leading order spin(cid:21)orbit
ontributions spinning
ompa
t binaries, of arbitrarymass ratio,mov-
arealsoavailable[22℄. The leading orderspins e(cid:27)e
ts on ing in
ir
ular or e

entri
orbits under PN a

urate dy-
the gravitationalwavepolarizations,mainlyfor
ompa
t nami
s will exhibit
haoti
behavior [20℄. The question
binaries in quasi(cid:21)
ir
ular orbits, were also explored in of
haoswasoriginallymotivatedbytheobservationthat
greatdetail [23℄. Asfarastheorbhita(lt)e

enthri
(itt)yis
on- theequationsdes
ribingthemotionofspinningtestpar-
+ ×
sidered, the temporally evolving and , whose ti
le in S
hwarzs
hild spa
etime allow
haoti
solutions
amplitudes are Newtonian a

urate and the orbital evo- [29℄. The issue is usually addressed by solving numer-
lution is exa
tly 2.5PN a

urate, was re
ently obtained i
ally PN a

urate equations des
ribing the dynami
s
[24℄. However, the inputs required to
ompute the
ru- of spinning
ompa
t binaries to
ompute
ertain gauge

ialse
ularphaseevolutionfor
ompa
tbinariesininspi- dependent quantities like fra
tal basin boundaries and
rallinge

entri
orbitsarealsoavailableto2PNorderbe- Lyapunovexponents. Thesemi(cid:21)analyti
parametrization
yond the leading quadrupole
ontributions [24, 25℄. Fur- presented here, along with the future extensions, should
ther,veryre
ently,aKeplerian(cid:21)typeparametrizationfor be very e(cid:27)e
tive in analyzing the PN a

urate dynam-
thesolutionof3PNa

urateequationsofmotionfortwo i
s of spinning
ompa
t binaries and to explore if astro-
non(cid:21)spinning
ompa
tobje
tsmovingin e

entri
orbits physi
ally realisti
binary inspiral will be
haoti
or not.
was obtained [26℄. It is therefore desirable to obtain a Further,theparametrizationforthe
ases
onsideredim-
Keplerian(cid:21)type parametri
solution to the dynami
s of pliesthattheasso
iatedPNa

urate
onservativebinary

ompa
t binaries, when spin e(cid:27)e
ts are not negle
ted. dynami
s
an not exhibit
haos.
In this paper, we obtain a Keplerian(cid:21)type parametri
Finally, wenote that ourparametrizationmay be use-
solution to the PN a

urate dynami
s of spinning
om- ful to analyze the high pre
ision radio(cid:21)wave observa-
pa
t binaries moving in e

entri
orbits, when the dy- tions of relativisti
binary pulsars like J0737(cid:21)3039 [30℄.
nami
s in
ludes the leading spin(cid:21)orbit intera
tion. Our These radio(cid:21)wave observations employ a `timing for-
parametri
solutionisappli
ableinthefollowingtwodis- mula', whi
h gives the arrival times at the bary
enter of
tin
t
ases: (i) the binary is
omposed of two
ompa
t the solar system for the ele
tromagneti
pulses emitted
obje
ts of equal mass and two arbitrary spins, and (ii) by a binary pulsar. The `timing formula' is important
thetwoobje
tshaveunequalmassesbut onlyoneobje
t for obtaining astrophysi
al informations from the
om-
has a spin. The present parametrization
an easily be pa
t binary as well as to test generalrelativity in strong
addedtothere
entlyobtained3PNa

urategeneralized (cid:28)eld regimes [31, 32℄. The heavily employed `timing for-
quasi(cid:21)Keplerian parametrization for the solution of the mula',usuallyin
orporates1PNa

urateorbitalmotion,
3PN a

urate equations of motion for two non(cid:21)spinning leading order se
ular e(cid:27)e
ts due to gravitational radia-

ompa
tobje
tsmovingine

entri
orbits[26℄. Thiswill tion rea
tion, spin(cid:21)orbit
oupling as well as 2PN (se
u-
provide,forthe(cid:28)rsttime, almostanalyti
,3PNa

urate lar)
orre
tions to the advan
e of periastron [31, 33(cid:21)35℄.
parametri
solution to the dynami
s of spinning
om- Theparametri
solutionobtainedherewillberequiredto
pa
t binaries moving in e

entri
orbits, when the spin obtain a 2PN a

urate `timing formula', where all spin(cid:21)
e(cid:27)e
ts are restri
ted, for the time being, to the leading orbit and 2PN
orre
tions are fully in
luded. This im-
order spin(cid:21)orbit
ontributions [27℄. As an appli
ation of plies a `timing formula', whi
h in
ludes not only 2PN
the parametrization, we phres(etn)t exphli
i(tt)expressions for order se
ular e(cid:27)e
ts, but all 2PN order periodi
terms,
+ ×
thespin(cid:21)orbitmodulated and , whose ampli- whi
h may leave some potentially observable signature
tudes are restri
ted, for
onvenien
e, to the Newtonian [28, 36, 37℄.
order and the orbital motion is restri
ted to the above
The paper is stru
tured in the following manner. In
mentioned
onservativePN order. Ourresultsgeneralize
Se
.II,weexhibitandexplain,indetail,thetotalHamil-
and improve, for the two
ases
onsidered, a restri
ted
toniandes
ribingthe
onservativedynami
s,whosepara-
parametrization for the orbital motion of
ompa
t bina-
metri
solution we are seeking in this paper. In Se
. III,
ries with leading order spin(cid:21)orbit
oupling [28℄.
we summarize the Keplerian(cid:21)type parametrization that
Our parametrization will have the following possible des
ribes the solution to the 3PN a

urate equations of
appli
ations. The
onstru
tion of `ready to use' sear
h motion for
ompa
t binaries, in e

entri
orbits, when
h (t) h (t)
+ ×
templates, namely and , for
ompa
tbinaries spine(cid:27)e
tsareignored. Se
.IVdes
ribesthedetermina-
in inspirallingorbitsrequiresadetailedknowledgeofthe tion of a Keplerian(cid:21)type parametrization for the
onser-
PNa

uhrTatTeorbitaldynami
sandaPNa

uratedes
rip- vativedynami
sofspinning
ompa
tbinaries,whenspin
tionfor ij ,thetransverse(cid:21)tra
eless(TT)partofthera- e(cid:27)e
ts are restri
ted to the leading order spin(cid:21)orbit in-
h (t)
+
diation (cid:28)eld. As mentioned earlier, time evolving tera
tion. Theparametri
solutions,presentedinSe
.III
h (t)
×
and fornon(cid:21)spinning
ompa
tbinariesofarbitrary and Se
. IV, are
ombined in Se
. V, to obtain PN a
-
masses moving in inspiralling e

entri
orbits were ob-
urate Keplerian(cid:21)type parametrization for the
onserva-
tained in Ref. [24℄. The parametrization presented here, tive dynami
s of spinning
ompa
t binaries moving in
alongwith[26℄,willberequiredto
onstru
tPNa

urate e

entri
orbits. As an appli
ation to our PN a

urate
`ready to use' referen
e templates for spinning
ompa
t parametri
solution, we present expli
it analyti
expres-
3
h (t)
+
sions for the gravitational wave polarizations, and naries, when the spin e(cid:27)e
ts are negle
ted. The leading
h (t)
×
, for spinning
ompa
t binaries in e

entri
orbits orderspin(cid:21)orbit
ouplingtothebinarydynami
sisgiven
h (t)
+ SO
in Se
. VI. The temporal evolution of above and byH . Wedeterminetheparametri
solutiontothedy-
h (t)
×
, whoseamplitudesareNewtoniana

urate,willbe nami
s,givenbyEq.(2.1),inthefollowingway. First,we
governed,forthe timebeing,by the
onservativePN dy- obtain a parametri
solution to the PN a

urate
onser-
nami
s dis
ussed in this paper. Finally, in Se
. VII, we vative dynami
s, negle
ting the e(cid:27)e
ts due to spin(cid:21)orbit
summarize our results and dis
uss its future extensions
oupling. We then
ompute a parametri
solution to the
+
N SO
and appli
ations.
onservativedynami
s, given by H H . The se
ond
step is
onsistent in a post(cid:21)Newtonian framework as the
spin(cid:21)orbit intera
tions are restri
ted to the leading or-
II. DYNAMICS OF COMPACT BINARIES der. In the (cid:28)nal step, we
onsistently
ombine these two
WITH LEADING ORDER SPIN(cid:21)ORBIT parametrizations,toobtainaPNa

urateparametri
so-
INTERACTION
lution to the
onservative
ompa
t binary dynami
s, as
spe
i(cid:28)ed by H.
In this paper, as mentioned earlier, we are interested
in the 3PN a

urate
onservative dynami
s of spinning In the present work, we employ the following 3PN a
-

ompa
t binaries, when spin e(cid:27)e
ts are, for the time be-
urate
onservativeHamiltonian,inArnowitt,Deser,and
ing,restri
tedtotheleadingorderspin(cid:21)orbitintera
tion. Misner (ADM)(cid:21)type
oordinates [38, 39℄, supplemented
The dynami
s is fully spe
i(cid:28)ed by a post(cid:21)Newtonian a
- with a
ontribution des
ribing the leading order spin(cid:21)

urate Hamiltonian H, whi
h may be symboli
ally writ- orbit
oupling. We work, as is usual in the literature
[26℄, with the following 3PN a

urate
onservative re-
ten as H = /µ
du
ed Hamiltonian, H , where the redu
ed mass
H=HN+H1PN+H2PN+H3PN+HSO, (2.1) of the binary is given by µ = mM1m=2/Mm, +m1mand m2
1 2
being the individual masses and is the
N 1PN 2PN 3PN
where H , H , H and H respe
tively are the totalmass. The3PNa

urate(redu
ed)Hamiltonian,in
Newtonian, (cid:28)rst, se
ond and third post(cid:21)Newtonian
on- ADM(cid:21)type
oordinatesandinthe
enter(cid:21)of(cid:21)massframe,
tributions to the
onservative dynami
s of
ompa
t bi- with the leading order spin(cid:21)orbit
ontributions reads
H(r,p,S ,S )=H (r,p)+H (r,p)+H (r,p)+H (r,p)+H (r,p,S ,S ),
1 2 N 1PN 2PN 3PN SO 1 2
(2.2)
where the Newtonian, post(cid:21)Newtonian and spin(cid:21)orbit
ontributions are given by
p2 1
H (r,p)= ,
N
2 − r (2.3a)
H (r,p)= 1 1(3η 1) p2 2 1 (3+η)p2+η(n p)2 1 + 1 ,
1PN c2 8 − − 2 · r 2r2 (2.3b)
(cid:26) (cid:27)
H (r,p)= 1 1 1 5η(cid:0)+5(cid:1)η2 p(cid:2)2 3+ 1 5 20η 3η(cid:3)2 p2 2 2η2(n p)2p2 3η2(n p)4 1
2PN c4 16 − 8 − − − · − · r
1(cid:26) (cid:0) (cid:1)(cid:0) (cid:1) 1 h(cid:0)1 1 (cid:1)(cid:0) (cid:1) i
+ (5+8η)p2+3η(n p)2 (1+3η) ,
2 · r2 − 4 r3 (2.3
)
(cid:27)
H (r,p)= 1 (cid:2) 1 5+35η 70η2+(cid:3)35η3 p2 4+ 1 7+42η 53η2 5η3 p2 3
3PN c6 128 − − 16 − − −
+(2(cid:18) 3η)(cid:0)η2(n p)2 p2 2+3(1 (cid:1)η(cid:0))η2((cid:1)n p)4ph2(cid:0) 5η3(n p)6 1 (cid:1)(cid:0) (cid:1)
− · − · − · r
+ 1 27+136η+(cid:0) 10(cid:1)9η2 p2 2+ 1 (17+30η)η(n p)2p2i+ 1 (5+43η)η(n p)4 1
16 − 16 · 12 · r2
(cid:20) (cid:21)
1(cid:0) (cid:1)(cid:0) (cid:1) 1 1
+ [ 600+ 3π2 1340 η 552η2]p2 (340+3π2+112η)η(n p)2
192 − − − − 64 · r3
(cid:26) (cid:27)
1 (cid:0) 1(cid:1)
+ 12+ 872 63π2 η ,
96 − r4 (2.3d)
(cid:19)
1 (cid:2) (cid:0) (cid:1) (cid:3)
H (r,p,S ,S )= (r p) S .
SO 1 2 c2r3 × · e(cid:27) (2.3e)
4
r = R/(GM) r = r n = r/r
In thpe=abPov/eµequationRs P , | |, e(cid:27)e
ts enter the dynami
s formally at 1PN order. How-
and ,where and aretherelativeseparation ever,mfor
romvpspai
nt obGje
mts2, tvhspeins/pci2n is roughmly ,griven by
ve
torand its
onjugaηte momentumηv=e
µto/rM, respe
tively. Svsp∼in co co ∼ co , where co co and
The (cid:28)nite mass ratio is given by . The spin(cid:21) are the typi
al mass, size and rotationalvelo
ity of
S
orbit
oupling involves the e(cid:27)e
tive spin e(cid:27), given by tvhspeinspinning
ompa
t obje
t. This implies, for moderate
S =δ1S1+δ2S2, values, that the leading order spin orbi(t1/
co4u)pling
e(cid:27) (2.4) will enter the
ompa
t binary dynami
s at O vspin, i.ec.
at 2PN order. It is interesting to note that if ∼ ,
where then theleading orderspin(cid:21)orbitintera
tionwill numer-
3m η 3
2 i
allymanifestat1.5PNorder. However,inthat
asethe
δ =2η 1+ = + 1 1 4η ,
1
(cid:18) 43mm1(cid:19) η2 43(cid:16) −p − (cid:17) (2.5a) bniengaler
yteddynPaNm
i
osrrseh
otiuolndsbteo dHoSmOi.nIantetdhibsypathpeer
,uwrerewntillly,
1
δ =2η 1+ = + 1+ 1 4η .
2 for the sake of
larity and simpli
ity, assume that the
4m 2 4 − (2.5b)
(cid:18) 2(cid:19) (cid:16) p (cid:17) spin(cid:21)orbit intera
tion in(cid:29)uen
e the
ompa
t binary dy-
S S nami
sat2PNorder. Wewouldliketopointoutthatthe
1 2
The redu
ed spinSve
torsS andS =arSe r/e(laµtGedMt)o the presentderivationoftheKeplerian(cid:21)typeparametrization
1 2 1 1
Sindiv=idSual/(sµpGinMs ) and by R aPnd forthe PN a

uratedynami
s ofspinning
ompa
tbina-
2 2
S S , respe
tively. We re
all that , , ries does not really
are if the spin(cid:21)orbit
oupling mani-
1 2
and are
anoni
al variables, su
h that the or- fests at 1.5PN or 2PN order.
bital variables
ommute with the spin variables, e.g. see
In the next se
tion, we will display and explain the
Ref. [17, 19℄.
S re
ently obtained Keplerian(cid:21)type parametri
solution to
The above de(cid:28)nitioζn of the e(cid:27)e
tive spin e(cid:27) is iden- the3PNa

urate
onservativeorbitalmotionoftwo
om-
ti
al to the quantity , introdu
ed in Ref. [28℄, and re-
S 2S = pa
t obje
ts, of negligible proper rotations, moving in
lGaMted2Sto e(cid:27)|TD, introdu
ed in Ref. [19℄, by e(cid:27)|TD e

entri
orbits [26℄.
e(cid:27).
The PN
orre
tions, asso
iated with the motion
of non(cid:21)spinning
ompa
t binaries, are
ompiled using
H III. KEPLERIAN(cid:21)TYPE PARAMETRIZATION
Refs. [9, 40℄. The spin(cid:21)orbit
ontribution to , avail-
FOR THE ORBITAL MOTION OF COMPACT
able in Ref. [35℄, employsthe spin(cid:21)supplementary
ondi-
BINARIES WITH NEGLIGIBLE PROPER
tion (SSC) of Pry
e, Newton and Wigner [41, 42℄. The ROTATION
SSC,whi
hde(cid:28)nesthe
entralworldlineofea
hspinning
body, adopted in this paper is given by
Using the fully determined 3PN a

urate
onserva-
1
2 + Uj =0 (i,j =1,2,3), tive Hamiltonian, in ADM(cid:21)type
oordinates and in the
i0 ij
S cS (2.6)
enter(cid:21)of(cid:21)mass frame, available in Refs. [9, 40℄, very re-
Uµ 4
ently a Keplerian(cid:21)type parametrization for the orbital
µν
where S is the spin(cid:21)tensor and the4 (cid:21)velo
ity of motion of
ompa
t binaries in e

entri
orbits was de-
α
the
enter of mass of the body. The spin (cid:21)ve
tor S is rived [26℄. The derivation of the above parametri
so-
de(cid:28)ned by lution, usually referred to as the 3PN a

urate general-
1 ized quasi(cid:21)Keplerian parametrization,
ru
ially depends
= ε Uβ µν (α,β,µ,ν =0,1,2,3),
Sα −2c αβµν S (2.7) on the following important points. First, the 3PN a
-

urate
onservative Hamiltonian is invariant under time
ε ε = +1
αβµν 0123 translation and spatial rotations, implying the existen
e
where is the Levi(cid:21)Civit(aUtµen=socrδµw)ith . E =H
In the rest frame of the body 0 of 3PN a

urate (redu
ed) eLne=rgyr p and (redu
ed)
orbital angular momentum × , for the binary in
=(0, 23, 31, 12)=(0,S)
Sα S S S (2.8) theL
enter(cid:21)of(cid:21)massframe. Inparti
ular,the
onservation
of indi
ates that the motion is restri
ted to a plane,
S S
1 2
holds. The spins, and , of the
ompa
t obje
ts namely the orbital planre,=anr(dcowseϕ,msianyϕi)ntrodu
e polar
may be given in terms of their moments of ineΩrtiaaan=d
oordinates su
h that . Finally, the
a a
angular velo
ities of proper rotations, I and ( Hamiltonian equations of motion governing the relative
1,2
) respe
tively, as motion of the
ompa
t binary, namely the 3PN a

urate
r˙ = dr/dt ϕ˙ = dϕ/dt
S1 = 1Ω1, expressions for and GM (cid:22) where t de-
I (2.9a) notes the
oordinate time s
aled by (cid:22) in terms of
S = Ω . E,L= L r 1/r
2 I2 2 (2.9b) | |and , arepolynomialsofdegreesevenin .
Thethirdpost(cid:21)Newtoniana

urategeneralizedquasi(cid:21)
Usingtheabovede(cid:28)nitions,itispossibletodetermineat Keplerian parametrization for
ompa
t binaries moving
what PN order the leading order spin(cid:21)orbit intera
tion in e

entri
orbits, in ADM(cid:21)type
oordinates and in the
will manifest. It is easyto see, using Eq. (2.3), that spin
enter(cid:21)of(cid:21)mass frame, is given by
5
r =a (1 e cosu) ,
r r
− (3.1a)
g g f f i h
4t 6t 4t 6t 6t 6t
l n(t t )=u e sinu+ + (v u)+ + sinv+ sin2v+ sin3v,
≡ − 0 − t c4 c6 − c4 c6 c6 c6 (3.1b)
(cid:16) (cid:17) (cid:18) (cid:19)
f f g g i h
4ϕ 6ϕ 4ϕ 6ϕ 6ϕ 6ϕ
ϕ ϕ =(1+k)v+ + sin2v+ + sin3v+ sin4v+ sin5v,
− 0 c4 c6 c4 c6 c6 c6 (3.1
)
(cid:18) (cid:19) (cid:16) (cid:17)
1/2
1+e u
ϕ
v =2arctan tan .
where "(cid:18)1−eϕ(cid:19) 2# (3.1d)
a e2 n e2 k
Te2he PN a

urate orbital elements gr, gr, ,f t ,f , aind by the leading order spin(cid:21)orbit intera
tion. Let us (cid:28)rst
ϕ, and the PN orbital fun
tions 4t, 6t, 4t, 6t, 6t, des
ribe, in the Hamiltonian framework, the binary dy-
h f f g g i h
6t 4ϕ 6ϕ 4ϕ 6ϕ 6ϕ 6ϕ
, , , , , , and expressible in terms nami
s we are interested in.
E,L η
of and are obtainable from Ref. [26℄ and will be
expli
itly displayed in Se
. V, after in
luding the e(cid:27)e
ts
due to the leading order spin(cid:21)orbit intera
tions. A. The Newtonian binary dynami
s with leading
Let us make a few
omments about the parametriza- order spin e(cid:27)e
ts
tion. The radial motion is uniquely parametrized by
Eq.(3.1a)intermsofsomePNa

uratesemi(cid:21)majoraxis We investigate the binary dynami
s, given by the re-
a e u H H =H +H
r r
, radial e

entri
ity and the e

entri
anomaly . du
ed Hamiltonian NSO, where NSO N SO:
The angular motion,vdes
ribed by Eq. 3.1
,eis spe
i(cid:28)ed p2 1 1
by the true anomaly , angular e

entri
ity ϕ, advan
e H = + L S .
k NSO 2 − r c2r3 · e(cid:27) (4.1)
oftheperiastron and some2PNand 3PNorderorbital
fun
tions. The expli
it time dependen
e is provided by
TheaboveHamiltonianpres
ribes,viathePoissonbra
k-
the 3PN a

urate `Kepler equation', namely Eq. 3.1b,
l ets,thefollowingevolutionequationsfortheredu
edan-
whi
h
onne
ts the mean anomaly , and hen
e the
o- L = r p
u v gular momentum ve
tor × , and the individual
ordinate time, to e

entri
and true anomalies and , S1 S2
redu
ed spin ve
tors and
respe
tively. The PN a

urate `Kepler equation' also in-
n dL 1

ludeessomePNa

uratemeanmotion ,timee

entri
- = L,H = S L,
ity t and some2PN and 3PNorderorbitalfun
tions. It dt { NSO} c2r3 e(cid:27)× (4.2a)
e ,e e
shouldbenotedthatallthesee

entri
ities, r ϕ and t dS1 = S ,H = δ1 L S ,
are
onne
tEed,Lto ea
ηh other by PN a

urate expressions dt { 1 NSO} c2r3 × 1 (4.2b)
intermsof and ,givenbyEq.(21)inRef.[26℄. This dS2 = S ,H = δ2 L S ,
implies thatit ispossibleto spe
ify, asin the Newtonian dt { 2 NSO} c2r3 × 2 (4.2
)

ase,aPNa

uratee

entri
orbitintermsof,forexam-
a e
r r ...,...
ple, and . Finally,were
allthattheKeplerian(cid:21)type
where { } denotes the Poisson bra
kets. Note that
dL/dt
parametrization for
ompa
t binaries in e

entri
orbits
, given by Eq. (4.2a), gives only the pre
essional
at (cid:28)rst post(cid:21)Newtonian order was obtained in Ref. [43℄.
motion of the orbital plane. We still need to obtain the
Its extension to 2PN order was derived in Refs. [35, 44℄.
equations des
ribing the orbital motion. Before we go
In the next se
tion, we will improve the above
ontoderivethoseequations,letus
onsiderthe
onserved
H
parametrizationby in
luding the e(cid:27)e
ts due to the lead-
quantities asso
iated wiEth= NHSO.
ing order spin(cid:21)orbit intera
tions.
T∂hHe redu=
ed0 energy L NSO is
onserved be
ause
t
of NSO . Though is noLt
=onLserved, it is fairly
easy to show that its magnitude | | is
onserved:
IV. QUASI(cid:21)KEPLERIAN PARAMETRIZATION
dL2 d 2
OF THE CONSERVATIVE DYNAMICS OF A = (L L)= L (S L)=0.
SPINNING COMPACT BINARY (cid:22) dt dt · c2r3 · e(cid:27)× (4.3)
NEWTONIAN MOTION AUGMENTED BY THE
FIRST(cid:21)ORDER SPIN(cid:21)ORBIT INTERACTION Similarly,belowweshowthatthemagnitudesofthespins
are also
onserved:
In this se
tion, we in
orporate in to the parametriza- dS12 = d (S S )= 2δ1 S (L S )=0,
tion, obtained in the previous se
tion, the e(cid:27)e
ts due to dt dt 1· 1 c2r3 1· × 1 (4.4a)
the leading order spin(cid:21)orbit
oupling. The straightfor- dS2 d 2δ
ward way of implementing that is to analyze the orbital dt2 = dt(S2·S2)= c2r23S2·(L×S2)=0. (4.4b)
dynami
s,whi
hissimplytheNewtonianoneaugmented
6
S˙ = L˙ S = r˙
NotethatEqs.(4.2)indi
atethat − ,where The
omponents of in the spheri
al polar
oordi-
S +S µGM
1 2
is the total spin redu
ed by , the redu
ed nates, whi
h also de(cid:28)ne the orbital equations of motion
H
total spin. We now de(cid:28)ne the
onserved (redu
ed) total asso
iated with NSO, read
angular momentum ve
tor r˙ =n r˙ =p ,
r
· (4.11a)
J =L+S. e S
(4.5) rθ˙=eθ·r˙ =pθ+ φc2·r2e(cid:27) , (4.11b)
µGM
e S
mThoemaebnotuvemexJpr,esnsaiomneltyimJes = R ×givPes+theSt.otaMloarnegouvlearr, rsinθφ˙ =eφ·r˙ =pφ− θc2·r2e(cid:27) . (4.11
)
J
is
onserved both in its magnitude and dire
tion as
dJ/dt=0 dJ/dt=0 J = J
and , where | |. Intheaboveequations,wehaveusedHamilton'sequation
dJ/dt = 0 r˙ =∂H /∂p n e =e n e = e
θ φ φ θ
The fa
t that (eall,oews,ues t)o introdu
e aeref- NSO , × and × − .
X Y Z Z
eren
e orthonormal triad , su
h that is Summarizing, we note that the dynami
s of the bi-
J H
along the (cid:28)xed ve
tor . This gives narysystem,des
ribedbytheHamiltonian NSO,isalso
J =Je . uniquelydeterminedbytheevolutionequationsgivenby
Z
(4.6) Eqs. (4.2) and (4.11). Before trying to (cid:28)nd parametri
solutions to these equations, let us point out severalfea-
Further, for later use, we
hoose the line(cid:21)of(cid:21)sight unit
N e e tures of the above binary dynami
s.
Y Z
ve
tor to lie in the (cid:21) (cid:21)plane (see Fig. 1). This is
It is easy to write down the pre
essionalequations for
S S
always possible and reasonable be
ause of the degree of
eX eY the (redu
ed) total spin and the e(cid:27)e
tive spin e(cid:27) as
freedom asso
iatedwith the
hoi
eof and . These
e e 1
unit ve
tors X and Y spanJthe invariable plane whi
h S˙ =S˙1+S˙2 = c2r3L×Se(cid:27) =−L˙ , (4.12a)
is the plane perpendi
ular to .
1
We now introdu
e highly advantageous spheri
al (po- S˙ =δ S˙ +δ S˙ = L (δ2S +δ2S ).
lar)
oordinates (r,θ,φ) and the asso
iated orthonormal e(cid:27) 1 1 2 2 c2r3 × 1 1 2 2 (4.12b)
(n,e ,e ) θ e
θ φ Z
triad , where is the angle between and S S
n φ
and is the azimuthal angle de(cid:28)ned in the invariable It is
lear that the motiSons of Sand e(cid:27) are more
om-
e e 1 2
X Y pli
ated than those for and . We dedu
e that the
(cid:21) (cid:21)plane (see Fig. 1), su
h that S S
magnitudes of and e(cid:27) are not preserved in general
n=sinθcosφe +sinθsinφe +cosθe .
X Y Z (4.7) and satisfy following equations
dS2 3√1 4η
r = − L (S S ),
rT˙h=e drerl/adtirve separation ve
tor and its time(dne,reiva,teive) dt − c2r3 · 1× 2 (4.13a)
θ φ
, written in the orthonormal triad dS2 3√1 4η(12η+η2)
reads dte(cid:27) =− − 4c2r3 L·(S1×S2). (4.13b)
r =rn,
(4.8a) (L S )
r˙ =r˙n+rθ˙e +rsinθφ˙e , Wenotethatthequantity · e(cid:27) isexa
tlypreserved
θ φ
(4.8b) sin
e
p d dL dS
In spheri
al
oordinates,the linear momentum and its (L S )= S +L e(cid:27) =0.
dt · e(cid:27) dt · e(cid:27) · dt (4.14)
magnitude may be expressed as
L
p=prn+pθeθ+pφeφ, (4.9a) LF=inJaellZy, leSt us
onsider the evolution of Z. Using
p2 =p2+p2+p2 =(n p)2+(n p)2 − on the r. h. s. of Eq. (4.2a) we obtain
r θ φ · ×
dL 3√1 4η
L2 Z = − e (S S ).
=p2r+ r2 . (4.9b) dt 2c2r3 Z · 1× 2 (4.15)
L
E = H The above equations imply that, in general, Z is not
With the help of these relations andp NSO, the
onserved. However LZ along with S and Se(cid:27) are
on-

omponents of the linear momentum
an be expressed
E L (L S ) served for the following two
ases. In the (cid:28)rst instan
e,
using the quantities , and · e(cid:27) as

ase (i), we require the binary to haveequal masses, but
S S m =m η =1/4
p2r =2E+ 2r − Lr22 − 2(Lc2·rS3e(cid:27)), (4.10a) δw1it=hδa2r)b.itIrnartyhespsein
osnd1inasntdan
e2,(
as1e(ii),2th⇔ebinarym⇔ay
pφ = rLsiZnθ , (4.10b) h(mav1e6=armbi2t,raSr1y6=ma0ssoers,Sb2u6=t o0n)l.y one of them is spinning
L2 1 L2 Inthispaper,whenwesolvethedi(cid:27)erentialequations,
p2 = p2 = L2 Z ,
θ r2 − φ r2 − sin2θ (4.10
) Eqs. (4.2) and (4.11), we restri
t ourselves, for the time
(cid:18) (cid:19) being,totheabovetwo
ases. Inthese
ases,thee(cid:27)e
tive
S S
L = e L L spin e(cid:27) and the redu
ed total spin are related by
Z Z Z
where · . Note that in general is not
S =δ S +δ S =χ S,
1 1 2 2

onserved. e(cid:27) so (4.16)
7
L S E
whereχthe newly de(cid:28)ned mass dependent
oupling
on- e2r =1+2EL2+8(1+EL2) ·L2e(cid:27)c2 , (4.21b)
stant so is given by n=( 2E)3/2,
δ =δ =7/8 , − (4.21
)
χso :=(δ11 orδ22 ffoorr ((ii)i),,tthheeesqinugalle(cid:21)(cid:21)mspasins(cid:21)(cid:21)

aassee. e2t =1+2EL2+4L·LS2e(cid:27)cE2 . (4.21d)
We would like to emphasize that in the
ase (i) the re- Eq. (4.20b) gives the `Kepler equation', modi(cid:28)ed by
S S
1
du
ed total spin is the sum of two arbitrary spins the spin(cid:21)orbit intera
tion, whi
h
onne
ts the e

entri
S S S S u
2 1 2
and . However, in the
ase (ii) is either or . anomaly to the
oordinate time. The
ru
ial require-
Theseobservationsallowustopresentthepre
essional ments to determine the above parametrization are the
L S E L L S
equations for and in the following
ompa
tform for
(dorn/sdertv)2ationof , and 1·/re(cid:27). Thisallowsus to treat
the spe
ial
ases as asapolynomialin with
onstant
oe(cid:30)
ients.
dL χ Let us now (cid:28)nd parametri
solution to the angular
= so J L,
dt c2r3 × (4.17a) partsoftheorbitalequationsofmotion,Eqs.(4.11)
om-
dS χ bined with Eq. (4.16), written in the form
= so J S, χ
dt c2r3 × (4.17b) rθ˙ =pθ 1− c2sro , (4.22a)
(cid:16) χ (cid:17) χ Jsinθ
TmhoemEenqtsu.m(4.1L7)ainmdptlyheth(artedthue
e(dr)edtuo
teadl)soprinbitSal,afnorgutlhaer rsinθφ˙ =pφ 1− c2sro + soc2r2 . (4.22b)
J =
(cid:16) (cid:17)
tJweoZ
aatsetshe
osnasmideerraedte, pwriet
heassna(binosuttanthtaen(cid:28)exoeuds)vper
et
oerssion Forthe
asesofinterest,wΘhereLZ isae
onstaLnt,wemay
Z
frequen
y given by Lintro=duL
ecoas
Θonstantangle between and su
hthat
Z
χ J (see Fig. 1) and Eqs. (4.10b) and (4.10
)
ωp = c2sor3 . (4.18) simplify to
LcosΘ
Note that ωp
an only be de(cid:28)ned in a physi
ally mean- pφ = r sinθ , (4.23a)
ingful way when spin e(cid:27)e
ts are in
luded in the binary L2 cos2Θ
p2 = 1 .
dynami
s. In the non(cid:21)spinning limit, the
ross prod- θ r2 − sin2θ (4.23b)
(cid:18) (cid:19)
u
ts appearing on the r. h. s. of Eqs. (4.17) vanish,
L
leaving the (redu
ed) orbital angular momentum as a
The e(cid:30)
ient way of determining solution to the
r

onstant ve
tor. It is interesting to observe that in the
ω χ Eqs. (4.22) begins by expressing the radial ve
tor in
p
non(cid:21)spinning limit, neither nor so go to zero. terms of an orbitaltriad. This is a
hieved by
onne
ting
(e ,e ,e )
In the next subse
tion, we will (cid:28)nd a parametri
so- X Y Z
via two rotations the referen
e triad to an
(i,j,k)
lution to the dynami
s of spinning
ompa
t binaries in
orbitaltriadde(cid:28)ned by . Inourorbitalorthonor-
k k = L/L
e

entri
orbits, given by Eqs. (4.11) and (4.17).
mal triad, the unit ve
tor , given by , and it
e
Z
along with de(cid:28)nes
e k
B. The Keplerian(cid:21)type parametrization asso
iated i= Z ×
H e k (4.24)
with the Hamiltonian NSO | Z × |
e k =sinΘ
Z
Note that | × | , whi
h is indeed always posi-
We start by
onsidering the radial motion, governed 0 Θ<π
tive be
ause of ≤ . We note that the unit ve
tor
by Eq. (4.11a), whi
h reads i
2 L2 2(L S ) givesthe line ofnodes, asso
iatedwith the int(eerse,
etio)n
r˙2 =2E+ r − r2 − c2·r3e(cid:27) (4.19) of the orbital plane with the invariabΘle plane X Y .
The
orrespondingin
linationangleis . Sin
etheabove
mentioned line of nodes vanishes in the non(cid:21)spinning
We obtain parametri
solution to the above equation i
limit, the unit ve
tor is only de(cid:28)ned when spin e(cid:27)e
ts
by followingexa
tlythesamepro
eduredes
ribedin de-
are in
luded in the binary dynami
s.
tailinSe
.IIIofRef.[26℄. Theradialmotionisdes
ribed
The two rotations arise from the observation that the
by
(instantaneous) position and orientation of the orbital
r =ar(1−ercosu), (4.20a) planewithrespe
ttothereferen
etriad(eX,eY,eZ)are
l ≡n(t−t0)=u−etsinu, (4.20b) dΥe(cid:28)(n0ed bΥy<tw2oπa)ngles: the longitude of the line of nodes
, ≤ and the in
lination of the orbital plane
u l eX eY Θ (0 Θ<
where and arethee

entri
andmeananomaEliesL. The wπ)ithrespe
tto theinvariable (cid:21) (cid:21)plane , ≤
orbital elements, expli
itly given in terms of , and . In terms of rotational matri
es, we have
L S
· e(cid:27), are i 1 0 0 cosΥ sinΥ 0 eX
1 L S E j = 0 cosΘ sinΘ sinΥ cosΥ 0 e .
ar =−2E 1−2 ·L2e(cid:27)c2 , (4.21a) k 0 sinΘ cosΘ− 0 0 1eYZ
(cid:18) (cid:19) −
     
8
r
The
omparison of , given by Eqs. (4.8a) and (4.7),
withtheoneinthenewangularvariables,Eq.(4.25)with
(i,j,k)
In the new orbital orthonormal triad , the rel- Eqs.(4.26), implies thefollowingtransformationsforthe
r
ative separation ve
tor is given by angular variables:
r =rcosϕi+rsinϕj, cosθ =sinϕsinΘ
(4.25)
(θ,φ) (Υ,ϕ): sin(φ Υ)sinθ =sinϕcosΘ

where the unit ve
tors (i,j,k) in terms of the referen
e −→ cos(φ− Υ)sinθ =cosϕ
(e ,e ,e ) −
X Y Z
triad are expli
itly given by
 (4.27)
i=cosΥeX +sinΥeY , A
lever
ombination of the above angular transforma-
(4.26a)
j = cosΘsinΥe +cosΘcosΥe +sinΘe , tion equations and their
orresponding time derivatives
X Y Z
−
leadsto thefollowingrelationsforthe angularvelo
ities:
(4.26b)
k=sinΘsinΥe sinΘcosΥe +cosΘe ,
X − Y Z (4.26
) θ˙2 = 1 cos2Θ ϕ˙2,
− sin2θ (4.28a)
(cid:18) (cid:19)
The geometri
al interpretations of the newly introdu
ed cosΘ
angles and basi
ve
tors are illustrated in Fig. 1. φ˙ =Υ˙ + ϕ˙,
sin2θ (4.28b)
J =JeZ
eZ where the over dot again means the derivative with re-
t Θ
k=L/L
spe
t to time and is tθ˙r2eated as a
onstant angle.
Using Eq. (4.28a) for in Eq. (4.22a) with (4.23b)
N Line of sight
leads to
L χ
ϕ˙ = 1 so .
r2 − c2r (4.29)
PSfrag repla
ements (cid:16) (cid:17)
Θ = 0
Note that the above derivation requires 6 , whi
h is
j S = 0
Θ always satis(cid:28)ed for 6 , and hen
e Eq. (4.29) is not
Θ = 0
de(cid:28)ned when . We again invoke the pro
edure
θ
employed in Se
. III of Ref. [26℄ to obtain the following
ϕ
parametri
solution for :
n=r/r
Invariable plane φ ϕ ϕ =(1+k)v,
0
Υ ϕ eY − 1+e 1/2 u (4.30a)
ϕ
v =2arctan tan ,
eX "(cid:18)1−eϕ(cid:19) 2# (4.30b)
i ϕ ϕ t
0 0
Orbital plane where is the value of at the time . The quantity
k e
ϕ
is ameasureof the advan
eof the periastronand is
the so
alled angular e

entri
ity. They are expressible
E,L L S
in terms of and · e(cid:27) as
1 L S
k = χ 3 · e(cid:27) ,
c2L2 so− L2 (4.31a)
(cid:18) (cid:19)
Figure 1: The binary geometry and interpretations of vari-
E
ous angles appearing in this se
tion. Our referen
e frame is e2 =1+2EL2 4χ (1+2EL2)
(eX,eY,eZ), where the basi
ve
tor eZ is aligned with the ϕ − so c2
J = L+S L S E
(cid:28)Jxe=d JtoetZal angular momentum ve
teoXr eY , su
h that +4(3+4EL2) · e(cid:27) .
. The invariable plane ( , ) is perpendi
ular L2 c2 (4.31b)
J
to . Important for the observation is the line(cid:21)of(cid:21)sight unit
N r l ϕ
ve
tor from the observer to the seoXur
e (
oemYpa
t binary). The parametrization for , and
an also be ob-
We may have, byNa
lever
heoYi
eeoZf andk =, tLhe/Lline(cid:21)of(cid:21) tainedusingthe
on
hoidal transformationsemployedin
sight unit ve
tor in the (cid:21) (cid:21)plane. is the Ref. [33℄ and found to be in total agreement.
Υ
unitve
torinthedire
tion oftheorbitalangularmomentum
L Wenowmoveontoderivethetimeevolutionof ,the
, whi
h is perpendi
ular to the orbital plane. The
onstant
longitude of the line of interse
tion (denoted by the line
in
lination of theorbital plane with respe
t to the invariable i
JplaneisΘ,whi
hisalsothepre
ession
oneangleofLaround of nodes unit ve
tor φ˙in Fig. 1), in a parametri
way.
. Theorbitalplaneinterse
tstheinvariableplaneattheline Using Eq. (4.28b) for with Eq. (4.29) in Eq. (4.22b)
i Υ
of nodes ,ewXithΥthe longitude measured in the invariable withΥEq.(4.23a)wegetthefollowingdi(cid:27)erentialequation
planefrom . isalsothephaseoftheorbitalplanepre
es- for :
i,j
sion. Tjh=eokrb×itial planeisspannedbythebasi
ve
tors( ), dΥ = χsoJ ,
where . dt c2r3 (4.32)
9
(Θ = 0) +(J LcosΘ)e ,
Z
whi
h is also restri
ted to the spinning
ase 6 , − (4.39)
ϕ˙
sin
e we made use of Eq. (4.29) for .
Nωotethatther. h. s. oftheaboveequationisidenti
al TheΥabove equation, along with the parametri
solution
to p, given by Eq. (4.18). This is not surpkris=inLg/aLs forS , des
ribes, in a parametri
way, the time evolution
the frequen
ies of the pre
essional motion for of .
i
and should be identi
al due to Eqs. (4.24) and (4.26). We (cid:28)nally
olle
t all the relevant equations, namely
Further,a
loseinspe
tionofFig.1revealsthatthephase Eqs. (4.20), (4.30), and (4.36), and display below our
k (e ,e )
oftheproje
tionofe ontotheinvaΥria+b2le7p0◦lane X Y , Hparametri
solution for the binary dynami
s given by
as measured from X, is given by . NSO:
To solve the ddiϕ(cid:27)/edretntial equation (4.32) analyti
ally, r =rcosϕi+rsinϕj,
we divide it by , given by Eq. (4.29), and dedu
e (4.40)
L=Lk,
at the leading order (4.41)
S =Je Lk,
dΥ χ J 1 Z
− (4.42)
= so .
dϕ L c2r (4.33) (i,j,k)
where the basi
ve
tors are expli
itly given by
r
The radial separation appearing on the r. h. s. of i=cosΥe +sinΥe ,
X Y
(4.43a)
above equatvion may be repla
ed by following expression j = cosΘsinΥeX +cosΘcosΥeY +sinΘeZ,
in terms of as shown below −
a(1 e2) L2 (4.43b)
r =a(1 ecosu)= − = , k=sinΘsinΥe sinΘcosΥe +cosΘe ,
− 1+ecosv 1+ecosv (4.34) X − Y Z (4.43
)
a e
where and maybetreatedastheNewtoniana

urate The time evolution for the radial and angular variables
a e
expressions for r and r. is given by
This allows us to write Eq. (4.33) (cid:22) within the a

u- r=a (1 e cosu),
r r
− (4.44a)
ra
y needed (cid:22) as
n(t t )=u e sinu,
0 t
dΥ χ J − − (4.44b)
dv = c2sLo3(1+ecosv), (4.35) ϕ−ϕ0 =(1+k)v, (4.44
)
χ J
ϕ = v Υ−Υ0 = c2sLo3(v+esinv), (4.44d)
where we also used the fa
t that at the leading
1/2
order. The Eq. (4.35) is easily integrated to obtain the 1+e u
Υ v =2arctan ϕ tan ,
parametri
solution to χasJ "(cid:18)1−eϕ(cid:19) 2# (4.44e)
Υ−Υ0 = c2sLo3(v+esinv), (4.36)
where the orbital elements are given by
v(t )=0
0 1 S E
whTerheerweeepxuistts an alter.nate way to obtain dΥ/dt, given ar =−2E 1−2χsocosαLc2 , (4.45a)
(cid:18) (cid:19)
byEq.(4.32). Notingthatthetimeevolutionfortheunit S E
k e2 =1+2EL2+8(1+EL2)χ cosα ,
ve
tΥor , de(cid:28)ned by Eq. (4.26
), is solelykgiven by that r so Lc2 (4.45b)
for , we
ompute the time derivative of and it reads n=( 2E)3/2,
k˙ =Υ˙ sinΘ(cosΥe +sinΥe ). − (4.45
)
X Y (4.37) e2 =1+2EL2+4χ cosαS E ,
L = Lk L˙ = Lk˙ t so Lc2 (4.45d)
Using the fa
t that and in Eq. (4.17a), 1 S
dΥ/dt
k = χ 3χ cosα ,
wearrive(again)at ,asgivΥenbyEq.(4.32). There- c2L2 so− so L (4.45e)
fore,theparametri
solutionfor readilyde(cid:28)nesasimi- (cid:18) (cid:19)
L=Lk k E
larsolutionfor , where isgivenby Eqs.(4.26
) e2 =1+2EL2 4(1+2EL2)χ
with (4.36). The parametri
solution to L reads ϕ − soc2
S E
L=L(sinΘsinΥe sinΘcosΥe +cosΘe ). +4(3+4EL2)χ cosα .
X − Y Z so Lc2 (4.45f)
(4.38)
e
As noted earlier, is given by the Newtonian
ontribu-
e L S
Finally, let us deriveSa parametri
solution to the tχionLStocosr.αIn Eqs. (α4.45), we have repla
ed · e(cid:27) bLy
pre
essional motion of , given by Eq. (4.17b). This
so S χ, where is the
onstant angle between
parametri
solution follows immediately by noting that
S = J L J = JeZ L = Lk and and so is given by
− , where and . Using
k
Eq. (4.26
) for , we dedu
e that χ := δ1 =δ2 =7/8 for (i), the equal(cid:21)mass(cid:21)
ase,
S =LsinΘ(−sinΥ)eX +LsinΘcosΥeY so (δ1 orδ2 for (ii), the single(cid:21)spin(cid:21)
ase.
10
Θ
Further, we note that the
onstant angle is not a free V. THIRD POST(cid:21)NEWTONIAN ACCURATE
variable and is given by one of the following relations: GENERALIZED QUASI(cid:21)KEPLERIAN
Ssinα PARAMETRIZATION FOR COMPACT
sinΘ= , BINARIES IN ECCENTRIC ORBITS WITH THE
J (4.46a)
FIRST(cid:21)ORDER SPIN(cid:21)ORBIT INTERACTION
L+Scosα
cosΘ= ,
J (4.46b)
We are now in a position to have parametri
solution
H
to the dynami
s, de(cid:28)ned bythe Hamiltonian , asgiven
where the magnitudeJo=f t(hLe2(+reSdu2
+ed2)LtSotcaolsaαn)g1u/2lar mo- by Eq. (2.2). This is done by
ombining
onsistently
mentum is given by .
the parametrizations presented in the previous two se
-
Weareawarethattheaboveparametrizationdoesnot
tions,Se
. IIIandSe
. IV. Thetwoparametri
solutions
lead to the
lassi
Keplerian parametrization simply by
S = 0 of Se
. III and Se
. IV
an be added linearly to obtain
putting . There are several arguments for this ap-
the PN a

urate binary dynami
s, given by Eqs. (2.2)
parent la
k of a simple non(cid:21)spinning limit. Noti
e that
Υ ϕ and (2.3), for the following reasons. Re
all that in the
and are de(cid:28)ned with respe
t to the line of inter-
i
Hamiltonian, given by Eqs. (2.2) and (2.3), the spin(cid:21)
se
tion,
hara
terized by the line of nodes unit ve
tor .
orbit
ontribution is added linearly to the PN a

urate
When the spin e(cid:27)e
ts are negle
ted, the above line of
non(cid:21)spinning
ontributions. Moreover, as we are
on-
interse
tion disappears, sin
e the orbital plane be
omes
sidering only leading orderspin(cid:21)orbitintera
tions, spin(cid:21)
the invariable plane and the asso
iated in
lination an-
Θ Υ ϕ orbit
ontributions only
ross with the Newtonian terms
gle vanishes. In this
ase, the related angles and
in the Hamiltonian. This is why we
an treat the spin(cid:21)
are not individually de(cid:28)ned, though Eq. (4.27) gives us
ϕ+Υ = φ orbit
ontributions and the non(cid:21)spinning PN
ontribu-
, as required. We emphasize that the related
tions separately and later add the two parametri
solu-
di(cid:27)erentialequations,givenbyEqs.(4.29)and(4.32),are
Θ = 0 S = 0 tions linearly. However, as a
autionary note, we state
obtained using 6 ,
orresponding to 6 . There-
Θ = 0 that
are should be taken while merging the above two
fore, Eqs. (4.29) and (4.32) are not de(cid:28)ned if .
parametri
solutions to avoid adding similar
ontribu-
Further, the apparent non(cid:21)vanishing of the time evolu-
Υ tions twi
e. Finally, we display below, in its entirety,
tion of in the non(cid:21)spinning limit is also attributable
ωp the parametri
solution to the
onservative third post(cid:21)
to the similar behavior for . However, a
lose s
rutiny
Newtonian dynami
s of spinning
ompa
t binaries mov-
of Eqs. (4.19), (4.22) with (4.23), (4.27) and (4.28) for
S 0 Θ 0 θ π/2 inginane

entri
orbit,inADM(cid:21)type
oordinates,with
→ , whi
h implies → and → , reveals that
spin e(cid:27)e
ts restri
ted to the leading order spin(cid:21)orbit in-
the binarydynami
sin the non(cid:21)spinning
ase isdes
rib-
r φ φ=ϕ+Υ tera
tions:
able in terms of and , where , as explained
earlier.
r =rcosϕi+rsinϕj,
Our parametri
solution generalizes a restri
ted anal- (5.1)
ysis
onsidered in Ref. [28℄. The analysis, given in L=Lk,
(5.2)
Ref. [28℄, negle
ts the pre
essional motion of the spin S =Je Lk,
S ζ Z
− (5.3)
ve
tors aned restri
ts e(cid:27), denoted by in Ref. [28℄, to
Z
lie along . Due to these restri
tions, that analysis
r (i,j,k)
provides only a parametri
solution to . We have, in where the basi
ve
tors are expli
itly given by
the other hand,
onsistently taken into
onsideration all
the leading order spin(cid:21)orbit intera
tions and obtained a i=cosΥe +sinΥe ,
X Y
parametri
solution to the entire binary dynami
s. In (5.4a)
j = cosΘsinΥe +cosΘcosΥe +sinΘe ,
a future
ommuni
ation we will
onne
t the above pre- − X Y Z
sentedparametri
solutiontoquantitiesrelatedtoobser- (5.4b)
vation, espe
ially asso
iated with binary pulsars [45℄. k=sinΘsinΥeX sinΘcosΥeY +cosΘeZ,
− (5.4
)
In the next se
tion, we will
ombine the Keplerian(cid:21)
type parametrization developed above with the one pre-
sented in Se
. III. and
r =a (1 e cosu) ,
r r
− (5.5a)
g g f f i h
4t 6t 4t 6t 6t 6t
l n(t t )=u e sinu+ + (v u)+ + sinv+ sin2v+ sin3v,
≡ − 0 − t c4 c6 − c4 c6 c6 c6 (5.5b)
(cid:16) (cid:17) (cid:18) (cid:19)
f f g g i h
4ϕ 6ϕ 4ϕ 6ϕ 6ϕ 6ϕ
ϕ ϕ =(1+k)v+ + sin2v+ + sin3v+ sin4v+ sin5v,
− 0 c4 c6 c4 c6 c6 c6 (5.5
)
(cid:18) (cid:19) (cid:16) (cid:17)