Conservative corrections to the innermost stable circular orbit (ISCO) of a Kerr black hole: a new gauge-invariant post-Newtonian ISCO condition, and the ISCO shift due to test-particle spin and the gravitational self-force

0.48 MB·English

by  Marc Favata

Conservative corrections to the innermost stable circular orbit (ISCO) of a Kerr black
hole: a new gauge-invariant post-Newtonian ISCO condition, and the ISCO shift due
to test-particle spin and the gravitational self-force
Marc Favata∗
Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, California 91109, USA† and
Theoretical Astrophysics, 350-17, California Institute of Technology, Pasadena, California 91125, USA
(Dated: 12 October2010)
The innermost stable circular orbit (ISCO) delimits the transition from circular orbits to those
that plunge into a black hole. In the test-mass limit, well-defined ISCO conditions exist for the
Kerr and Schwarzschild spacetimes. In the finite-mass case, there are a large variety of ways to
define an ISCO in a post-Newtonian (PN) context. Here I generalize the gauge-invariant ISCO
condition of Blanchet and Iyer[Classical Quantum Gravity 20, 755 (2003)] to the case of spinning
1 (nonprecessing) binaries. The Blanchet-Iyer ISCO condition has two desirable and unexpected
1 properties: (1) it exactly reproduces the Schwarzschild ISCO in the test-mass limit, and (2) it
0 accurately approximates the recently calculated shift in the Schwarzschild ISCO frequency due to
2 theconservative-pieceofthegravitationalself-force[BarackandSago,Phys.Rev.Lett.102,191101
(2009)]. Thegeneralization ofthisISCOconditiontospinningbinarieshasthepropertythatitalso
n
a exactlyreproducestheKerrISCOinthetest-masslimit(uptotheorderatwhichPNspincorrections
J are currently known). The shift in the ISCO due to the spin of the test-particle is also calculated.
Remarkably,thegauge-invariantPNISCOconditionexactlyreproducestheISCOshiftpredictedby
4
2 thePapapetrouequationsforafully-relativisticspinningparticle. Itissurprisingthatananalysisof
the stability of the standard PN equations of motion is able (without any form of “resummation”)
] to accurately describe strong-field effects of the Kerr spacetime. The ISCO frequency shift due to
c the conservative self-force in Kerr is also calculated from this new ISCO condition, as well as from
q the effective-one-body Hamiltonian of Barausse and Buonanno [Phys. Rev. D 81, 084024 (2010)].
-
r Theseresultsserveas ausefulpoint ofcomparison for futuregravitational self-force calculations in
g theKerrspacetime.
[

PACSnumbers: 04.25.Nx,04.25.-g,04.25.D-,04.30.-w
2
v
3
I. INTRODUCTION, MOTIVATION, AND plates. The ISCO is also important because its location
5
SUMMARY encodes (potentially observable) information about the
5
2 strong-gravityregion of the BH spacetime.
. The innermost stable circular orbit (ISCO) is a point What happens if we no longer have a geodesic or-
0
of dynamical instability in black hole (BH) spacetimes bit? When dissipation (i.e., radiation-reaction) is in-
1
0 that separates stable, circular, and bound geodesic or- cluded, the location of the ISCO is no longer precisely
1 bits from those that “plunge” into the BH event hori- quantifiable—it becomes “blurred” into a transition re-
: zon. The location of the ISCO can be quantified in a gion(in orbitalradius orfrequency)separatingthe adia-
v
i gauge-invariantmanner by specifying its orbital angular batic inspiralfromthe plunge [1, 2]. However,if we con-
X frequency as measuredby a distant observer. For a test- sider only conservative corrections to geodesic motion, a
r particle in the Schwarzschild spacetime, this frequency precise ISCO can (in some cases) continue to exist. In
a occursatm Ω=6−3/2,wherem isthemassoftheBH.1 particular here we will consider two types of conserva-
2 2
The location of the ISCO is important in the context tive corrections to geodesic motion: (i) the gravitational
of quasicircular, inspiralling compact binaries (an im- self-force (GSF; a force arising from the point-particle’s
portantsourceforgroundandspace-basedgravitational- finite mass which causes it to deviate from geodesic mo-
wavedetectors)becauseitrepresentsthepointwherethe tion) and (ii) the force due to the spin of the test-body.2
characteroftheorbit(andhencethegravitationalwaves) Calculations of the GSF are motivated by the need
abruptly changes. Because of this, the ISCO frequency to model extreme-mass-ratio inspirals (EMRIs), an im-
is often taken as the termination point of inspiral tem- portant source for the planned Laser Interferometer
Space Antenna (LISA) [3] consisting of a compact ob-
ject (m1 1–100M⊙) inspiralling into a massive BH
(m2 104∼–107M⊙)withmassratiosq .10−4. Comput-
∼
∗ NASAPostdoctoral Fellow;[email protected] ingtheGSFischallenging(see[4–7]forreviewsandrefer-
† Copyright2010 CaliforniaInstitute ofTechnology. Government ences),butseveralgroupshavehadrecentsuccess[8–13].
sponsorshipacknowledged.
1 Throughout this article m1 < m2 denote the binary masses,
q = m1/m2 ≤ 1 is the mass ratio, M = m1+m2 is the total
mass, and η = m1m2/M2 = q/(1+q)2 ≤ 1/4 is the reduced 2 The quadrupole and higher-order multipole moments of an ex-
massratio(denoted ν bysomeauthors). tended bodycouldalsocauseashiftintheISCOlocation.
2
In particular, one of the concrete results to emerge from which differs from the exact BS result by 14.7%.
theself-forceprogramhasbeenthecalculationbyBarack TheabovePNISCOconditionisespeciallyinteresting
and Sago (BS) of the shift in the ISCO frequency due because it exactly reproduces the Schwarzschild ISCO
to the conservative GSF in the Schwarzschild spacetime (x = 1/6 or m Ω = 6−3/2) in the test-particle limit.
2
[14, 15]. This result is especially interesting because it It is surprising that a condition derived from the PN
supplies a gauge-invariant, exact strong-field result that equations of motion can reproduce a strong-field result
is only computable using the full self-force formalism. liketheISCO.4 Forexample,astandardwaytocompute
(This is in contrast to standard BH perturbation the- the ISCO in a PN context is by finding the minimum of
ory calculations, which only provide access to the time- thecircular-orbitenergy.5 Inthetest-masslimit,thePN
averageddissipative pieces of the self-force.) The result- expansion of the circular-orbitenergy,
ing conservative GSF ISCO frequency shift can be ex-
E (Ω) (1 2x) 1 3 27
pressed in the form circ = − 1= x 1 x x2
ηM (1 3x)1/2 − −2 − 4 − 8
MΩ=6−3/2[1+ηcGSF(0)+O(η2)], (1.1) − (cid:20)
675 3969 45927
x3 x4 x5+O(x6) , (1.4)
where BS calculated the value cGSF(0) = − 64 − 128 − 512
(cid:21)
1.2512( 0.0004). This value can be used to com- converges slowly: to get within 8% of the exact result
±
pare different GSF codes, and to set constraints on (x=1/6) one needs to truncate the above expression at
the effective-one-body (EOB) [2, 16–18] formalism (see 4PN order or higher.
[19–21]). Part of the motivation for developing “resummation”
In Ref. [21] I compared the above GSF ISCO shift methods was to cure this problemwhile also providing a
with 15 distinct post-Newtonian (PN) or EOB meth- means to compute the ISCO for finite mass-ratio bina-
∼
odsforcomputingtheISCO.Amongthosemethods,two ries. For example, Kidder, Will, and Wiseman [26, 27]
approaches—basedon the EOB formalism and the stan- modified the PN equations of motion by replacing the
dard PN equations of motion—have especially desirable O(η0) terms with the corresponding terms derived from
features. Inparticular,thebestagreement( 10%error) the Schwarzschild geodesic equations (in the appropri-
∼
withthe BSresultwasfoundusinga versionofthe EOB ate coordinate system). This enforced the Schwarzschild
formalism in which a pseudo-4PN term is added to the ISCO in the test-particle limit, but caused deviations
effective metric and calibrated with the Caltech/Cornell from this value for finite-η. Similarly, Ref. [28] intro-
numerical relativity simulations [22]. This method also duced Pad´eapproximantsto improve the convergenceof
adequately predicted (with 16% error) the ISCO fre- PN-based templates (in part by again enforcing agree-
∼
quency for equal-mass binaries as computed from se- ment with the test-particle limit). The EOB formalism
quencesofquasicircularinitialdata[23]. However,inthe providesthe mostsuccessfulversionofthis idea bymod-
absenceofcalibration,themethodwhichmostaccurately eling the two-body dynamics in terms of a Hamiltonian
reproduced the BS result was the gauge-invariant ISCO that is based on a particle with reduced mass µ = ηM
conditionofBlanchetandIyer[24].3 Thisconditionisde- moving in the “η-deformed” Schwarzschild background
rived from a stability analysis of the 3PN (nonspinning) of a central mass M. It is in light of these resummation
equations of motion; it takes the form approaches that the ability of the Blanchet-Iyer ISCO
condition to predict the Schwarzschild ISCO is surpris-
Cˆ0 1 6x+14ηx2 ing (and perhaps not widely appreciated).
≡ −
397 123
+ π2 η 14η2 x3+O(x4), (1.2)
2 − 16 −
(cid:20)(cid:18) (cid:19) (cid:21) A. Summary of results
where x (MΩ)2/3, and Cˆ 0 is required for stable,
0
circular o≡rbits to exist. The ≥ISCO is found by solving ItispossiblethattheabilityoftheBlanchet-IyerISCO
Cˆ = 0 for x (or Ω). The resulting value for the conser- condition to predict the Schwarzschild ISCO is coinci-
0
vative GSF ISCO shift was found to be [21] dental. One of the primary objectives of this study is to
565 41π2
cGSF(0) =1.434912612..., (1.3)
C0 ≡ 288 − 768 4 Indeed, onecanseefromEq.(1.2)thattheSchwarzschildISCO
frequencyarisesonlyfromthe1PNequationsofmotion;the2PN
and 3PN termsaffect onlytheO(η) corrections. Note alsothat
in deriving this result, it was crucial to express Cˆ0 in terms of
3 Inaddition to the above-mentioned reasons, these two methods thegauge-invariantobservablexratherthanagauge-dependent
for computing the ISCO are also preferred over the other ap- radialcoordinate[24].
proaches examined in [21] because: (i) the error in the ISCO 5 The critical point defined in this way is sometimes called an
computedviathesemethodsdecreasesmonotonicallyasthePN ICO (innermost circular orbit). See Sec. IIB of [21] (as well as
orderisincreased,and(ii)theyeacharederivedfromequations Sec. IVA2 of [25]) for a discussion of the difference and rela-
ofmotionthatallowforacompletedescriptionofthetwo-body tionship between the ISCO and ICO. In the rest of this article,
dynamics. IwillrefertobothtermsasanISCO.
3
test this by extending the Blanchet-Iyer ISCO condition
[Eq. (1.2)] to the case of spinning (nonprecessing) bina- 0.40 1
10
ries. This calculation is performed in Sec. II. The result 0
is given by [see also Eq. (2.29) below] 0.35 -1|1Kerr100--21 E3PN
0.30 10
Cˆ0 ≡1−6x+x3/2 14MSℓc2 +6δMmMΣcℓ2 o0.25 |/1100--43 C0
(cid:18) Sc 2 (cid:19) isc 10-5
+x2 14η 3 0,ℓ M 0.20 10-6 Kerr
" − (cid:18)M2(cid:19) # -1.0 -0.5 0.0 2 0.5 1.0
0.15
+x5/2 Sℓc (22+32η) δm Σcℓ (18+15η) E3PN C0
−M2 − M M2 0.10
(cid:20) (cid:21)
397 123
+x3 π2 η 14η2 , (1.5) 0.05
2 − 16 −
(cid:20)(cid:18) (cid:19) (cid:21)
where Sc ℓ Sc, Σc ℓ Σc, Sc ℓ Sc, ℓ is the -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
ℓ ≡ · ℓ ≡ · 0,ℓ ≡ · 0 2
unit vector along the direction of the Newtonian orbital
angular momentum, Sc Sc +Sc, Σc M(Sc/m FIG.1. (coloronline). Comparisonofthreedifferentmethods
Sc/m ), Sc = (1+m /m≡)S1c +(12+m≡/m )Sc2, δm2 −= for computingtheISCOof a nonspinningtest-particlein the
1 1 0 2 1 1 1 2 2
m m , and Sc = χcm2ˆsc are the individual spin Kerrspacetime. Thesolid (black)curve(labeled Kerr)refers
1 − 2 A A A A to the exact result for the Kerr ISCO [Ref. [29] or Eq. (3.7)
angular momenta for body A = 1,2 with dimensionless
spin parameters χc and unit direction vectors ˆsc. This here]. Thedashed(red)curve(labeledC0)istheη→0limit
A A ofthegauge-invariantISCOconditionderivedhere[Eq.(1.5)
condition is derived from the 3PN equations of motion,
or(2.29)]. Thedash-dotted(blue)curve(labeledE3PN)isthe
including all explicitly known spin terms up to 2.5PN
ISCOcomputedbyminimizingthe3PNcircular-orbitenergy
order.
[Eq.(A1)]. TheinsetshowsthefractionalerrorsoftheE3PN
Inthetest-particlelimit(η →0),theISCOdetermined or C0 curveswith respect to theKerr curve.
from Eq. (1.5) can be compared with the ISCO of the
Kerr spacetime [29]. This comparison can be performed
by deriving a condition analogous to Cˆ from the Kerr ToquantifytheseconservativeISCOshifts,weexpand
0
metric, expanding the result in powers of the BH spin the ISCO frequency as [Eq. (4.3) below]
(χK),andcomparingtoEq.(1.5)(seeSec.IIIfordetails).
2
The resulting comparison shows that the two conditions MΩ=m2ΩK(χ2)[1+ηcGSF(χ2)+ηχ1cCOspin(χ2)
agree up to the order to which the PN spin corrections +O(η2)+O(χ η2)+O(χ2η2)], (1.6)
1 1
are known. This comparison is also shown graphically
in Fig. 1. Note the large improvement in comparison whereΩK(χ )istheKerrISCOfrequency[29]. Theshift
2
with the 3PN energy function [which includes spin cor- intheISCOduetotheconservativeGSFisparametrized
rections; see Eq. (A1)]. Presumably, if higher-order spin by the function cGSF(χ ). In Sec. IVB this function is
2
correctionsinthePNequationsofmotionwereincluded, calculated via the EOB and Cˆ approaches; the results
0
the error in comparison with the Kerr ISCO for large
arepresentedgraphicallyinFig.2andtabulatedinTable
values of χ would improve. This excellent agreement
| 2| I. It will be interesting to compare these numbers with
suggests that the standard PN equations of motion are
future GSF calculations in Kerr.
able to exactly recover some strong-field results. In Sec. IVC the function cCOspin(χ ) is also calcu-
2
A second objective of this article is to calculate the lated via the EOB and Cˆ approaches.6 However, in
shift in the ISCO frequency due to conservative effects 0
thiscasetheEOBcalculationviatheHamiltonianin[20]
(Sec.IV). Inparticular,twotypes ofconservativeeffects
yieldstheexact(fullyrelativistic)result. Thisisbecause
are considered: the first due to the GSF, and the second
this Hamiltonian reproducesthe Papapetrou-Mathisson-
due to the spin of the test-particle. As discussed above,
Dixon equations of motion [33–40] in the small-η limit.
theconservativeGSFISCOshiftwascomputedin[14,15]
An analysis of the ISCO shift directly using the Papa-
foraSchwarzschildbackground,andcomparedwithvar-
petrou equations is also presented in Appendix B; the
ious PN calculations in [21]. Here we focus on the ISCO
results are identical to those obtained from the EOB
shiftintheKerrbackground,forwhichGSFcalculations
Hamiltonian(providingfurther confirmationofthe work
arenotcurrentlyavailable. Instead,wemakepredictions
forwhatthatISCO shiftmightbe accordingtotwo ana-
lytic approaches: the ISCO condition in Eq. (1.5) above
and the recently developed spinning-EOB formalism of 6 TheISCOforaspinningtest-particleinKerrwaspreviouslycon-
Barausse and Buonanno [20] (Sec. IVA). (In the latter sidered in [30], but those authors focused on unphysically large
case, the ISCO shift was calibrated to match the exact values of the test-particle spin and did not explicitly compute
Schwarzschildresult [14, 15].) theshiftparametercCOspin (seealso[31,32]).
4
in[20,40]). TheseresultsareshowninFig.2andTableI. A. Equations of motion and the relationship
In the Schwarzschildcase a fully analytic analysis of the between spin variables
Papapetrou equations is straightforward and presented
in Appendix B3. It shows that the Boyer-Lindquist ra- The spin-orbit contributions to the equations of mo-
dial coordinate of the Schwarzschild ISCO is shifted by tion given in [44] are expressed in terms of spin vectors
O(χ1m1): Snc (A = 1,2) whose magnitudes χncm2 do not remain
A A A
constant with time. (Note that Refs. [44, 47] do not use
2 the superscripts “nc”.) An alternative set of spin vari-
risco =6m2−2r3χ1m1+O(χ21m21), (1.7) ables ScA are defined in Eq. (7.4) of [47] [also Eq. (2.21)
of [46]] and have the property that their magnitudes are
andfrequencyshiftoftheISCOduetothepoint-particle constant.7 This choice of spin variables causes the spin-
spin is given by precession equations to take a convenient form and is
generallypreferredin computations. Here we denote the
cCOspin(0)= √6 =0.306186.... (1.8) annodnctohnestcaonnts-tmanatg-nmitaugdneitsupdinesvpeicntovrescotofresacbhybSocd.yWbyeSalnAsco,
8 A
define the spin combinations
Interestingly, this ISCO frequency shift is exactly repro-
ducedbytheCˆ0 ISCOcondition,againshowingthatthe Sc ≡Sc1+Sc2, (2.2a)
standard PN equations of motion are able to exactly re- Σc M Sc2 Sc1 , (2.2b)
produce a strong-fieldresult. (If χ2 6=0, the exactresult ≡ (cid:18)m2 − m1(cid:19)
is only approximately reproduced by the Cˆ condition
0 and analogous relations for Snc and Σnc.
because the PN spin terms are explicitly computed only
The relationship between (Snc,Σnc) and (Sc,Σc) is
to 2.5PN order; see Fig. 2.)
given by Eqs. (2.22) of [46],
SectionVdiscussessomeconclusionsofthisstudy. Ap-
pendix A compares the test-mass limits of several PN
1 M δm
quantities (the orbital energy, angular momentum, and Sc =Snc+ η 2Snc+ Σnc
c2 r M
Keplerian relation) with the analogous quantities com- (cid:26) (cid:20) (cid:21)
puted from the Kerr metric. η v Snc+ δmv Σnc v +O(c−4), (2.3a)
−2 · M ·
(cid:20) (cid:21) (cid:27)
II. GAUGE-INVARIANT ISCO CONDITION
1 M δm
FOR SPINNING BINARIES Σc =Σnc+ Snc+(1 2η)Σnc
c2 r M −
(cid:26) (cid:20) (cid:21)
1 δm
Following the stability analysis of the PNequations of v Snc+(1 3η)v Σnc v +O(c−4),
−2 M · − ·
motion in [24, 26], we can generalize the gauge-invariant (cid:20) (cid:21) (cid:27)
(2.3b)
ISCO condition derived by Blanchet and Iyer [24] to the
case of spinning, nonprecessing binaries.
whereristheorbitalseparationinharmoniccoordinates.
We begin by writing the conservative PN equations of
The inverse relationship is given by
motion for two spinning point-masses as
1 M δm
Snc =Sc+ η 2Sc+ Σc
dv = BN+ B1PN+ B2PN+ B3PN c2 (cid:26)− r (cid:20) M (cid:21)
dt NS NS NS NS η δm
+ B1.5PN+ B2.5PN+ B2PN. (2.1) +2 v·Sc+ M v·Σc v +O(c−4), (2.4a)
SO SO SS+QM (cid:20) (cid:21) (cid:27)
On the first line we list the nonspin terms to 3PN order
1 M δm
(see [41] for references); note that the radiation-reaction Σnc =Σc+ Sc+(1 2η)Σc
c2 − r M −
terms at 2.5PN and 3.5PN order are not present since (cid:26) (cid:20) (cid:21)
we are only concerned with conservative corrections to +1 δmv Sc+(1 3η)v Σc v +O(c−4), (2.4b)
the ISCO. The spin-orbit (SO) term at 1.5PNorder and 2 M · − ·
(cid:20) (cid:21) (cid:27)
the spin-spin (SS) term at 2PN order were first derived
in [42]. The SO term at 2.5PNorder was first derivedin
[43]. HereIusethe formsgiveninEqs.(5.7)of[44]. The
7 Throughoutthissectionallofourspinvariablesarecontravariant
2PNorderquadrupole-monopole(QM)termwasderived
vectors. In[46]thesearedenotedwithanoverbar. Notethatthe
in [45]; Ref. [46] shows how to concisely combine this spin variables used in Kidder [48] are the constant-magnitude,
term (when specialized to black holes) with the 2PN or- contravariant spin vectors denoted ScA here. Note also that we
der spin-spin term [see their Eq. (3.8)]. usethenotationΣforthequantities denoted∆in[46,48].
5
wherethepowersofcwereaddedtoshowthatthecorrec- (aside from the replacements Snc Sc), but the 2.5PN
A ↔ A
tionstothespinsarearelative1PNordereffect. Wealso and higher-order spin terms will differ depending on the
notethe relationshipbetweenthe individualspinvectors choice of spin variables. Throughout this paper the su-
[44, 46, 47], perscripts “c” and “nc” are sometimes dropped where
either index would be appropriate.
m 1 m 2 The2.5PNspin-orbitcorrectionstoEq.(2.1)aregiven
ScA = 1+ c2Br SnAc− 2c2 MB (v·SnAc)v+O(c−4), in Eq. (5.7) of [44] in terms of the variables Snc and
(cid:16) (cid:17) (cid:16) (cid:17) (2.5a) Σnc. Theequivalentexpressionsintermsoftheconstant-
magnitude spin variables are found by substituting the
m 1 m 2
Snc = 1 B Sc + B (v Sc)v+O(c−4). relations(2.4)intothe1.5PNSOterm[Eq.(5.7a)of[44]],
A − c2r A 2c2 M · A and combining the result with the 2.5PN SO term in
(cid:16) (cid:17) (cid:16) (cid:17) (2.5b) Eq. (5.7b) of [44] (into which the substitutions S Sc
andΣ Σc canbemadesinceweonlyrequireaccu→racy
→
Since the spin variables differ at 1PN order, the equa- to relative2.5PNorderin the spin terms). The resulting
tionsofmotion(butnottheequationsofprecession)will SO contributions to Eq. (2.1) in terms of the “c” spin
have the same form for the 1.5PN and 2PN spin terms variables are
1 δm δm δm
B1.5PN = n 12(Sc,n,v)+6 (Σc,n,v) +9(nv)n Sc+3 (nv)n Σc 7v Sc 3 v Σc , (2.6a)
SO r3 M × M × − × − M ×
(cid:26) (cid:20) (cid:21) (cid:27)
1 M δm M 29
B2.5PN = n (Sc,n,v) 30η(nv)2+24ηv2 (44+25η) + (Σc,n,v) 15η(nv)2+12ηv2 24+ η
SO r3 − − r M − − r 2
(cid:26) (cid:20) (cid:18) (cid:19) (cid:18) (cid:18) (cid:19)(cid:19)(cid:21)
δm 3 M
+(nv)v (Sc,n,v)( 9+9η)+ (Σc,n,v)( 3+6η) +n v (nv)(vSc)(1 η) 8 η(nSc)
− M − × − 2 − − r
(cid:20) (cid:21) (cid:20)
δm M 3 45 M
4 η(nΣc)+ (nv)(vΣc) +(nv)n Sc η(nv)2+21ηv2 7 (4+3η)
− M r 2 × − 2 − r
(cid:18) (cid:19)(cid:21) (cid:20) (cid:21)
δm M 23 33 M
+ (nv)n Σc 15η(nv)2+12ηv2 12+ η +v Sc η(nv)2+ (24+11η) 14ηv2
M × − − r 2 × 2 r −
(cid:20) (cid:18) (cid:19)(cid:21) (cid:20) (cid:21)
δm M 11
+ v Σc 9η(nv)2 7ηv2+ (12+ η) . (2.6b)
M × − r 2
(cid:20) (cid:21)(cid:27)
In the aboveequations we define additional notationfol- computed in Refs. [49–59], but the explicit equations of
lowing [44]: the unit vector n=x/r points in the direc- motion have not yet been derived.
tion of the relative separationvector x=y y ; v=x˙
1 2
−
denotes the relative orbital velocity; scalar products of
vectorsare denoted by (ab) a b; and the mixed prod-
B. Restriction to the nonprecessing case
≡ ·
uct of three vectors is denoted by (a,b,c) a (b c).
≡ · ×
The sum of the spin-spin and quadrupole-monopole
Now we restrict to nonprecessing orbits in which the
terms is given in Eq. (3.8) of [46],
individualspinvectorsS arealignedorantialignedwith
A
the direction of the Newtonian orbital angular momen-
3
SS+BQ2MPN =−2Mr4 (Sc0)2−5(nS0c)2 n+2(nS0c)Sc0 , tvuemctovreλctorℓℓ≡nL.NV/e|cLtoNr|s. cWaneathdednitbioendaellcyodmepfionseedthoenutnhiet
(cid:8)(cid:2) (cid:3) (2(cid:9).7) orthonorm≡al×basis n,λ,ℓ as in S = S n+S λ+S ℓ;
where { } n λ ℓ
similar relations hold for Σ and S (in either spin repre-
0
sentation),aswellasforv. Therestrictiontononprecess-
δm m m
Sc 2Sc+ Σc = 1+ 2 Sc + 1+ 1 Sc. ing orbits having a fixed orbitalplane in the direction of
0 ≡ M (cid:18) m1(cid:19) 1 (cid:18) m2(cid:19) 2 ℓ then implies the following relations:
(2.8)
Note that Eq. (2.7) has the same form in terms of the
v=r˙n+rϕ˙λ, v2 =r˙2+r2ϕ˙2, (2.9a)
“nc” spin variables; it is also only valid for Kerr BHs
(nv)=r˙, n v=rϕ˙ℓ, (2.9b)
as the value for the Kerr quadrupole moment was used.
×
Higher-order spin-spin corrections have recently been S=S ℓ, Σ=Σ ℓ, S =S ℓ, (2.9c)
ℓ ℓ 0 0,ℓ
6
(S,n,v)=rϕ˙S , (Σ,n,v)=rϕ˙Σ , (2.9d) Eqs.(181)-(196)ofBlanchet’sreviewarticle[41]. Denot-
ℓ ℓ
n S= S λ, n Σ= Σ λ, (2.9e) ing Blanchet’s expressionsby B,NS and B,NS, ignoring
ℓ ℓ A B
× − × − thedissipativetermsat2.5PNand3.5PNorders,andus-
v S=S (rϕ˙n r˙λ), v Σ=Σ (rϕ˙n r˙λ),
× ℓ − × ℓ − ing the form of the equations without the 3PN logarith-
(2.9f)
mic terms, the nonspin terms in Eqs. (2.11) are related
(nS)=(nΣ)=(vS)=(vΣ)=(nS0)=0. (2.9g) to Blanchet’s by
The above relations allow the conservative PN two- NS = +r˙ , (2.12a)
B,NS B,NS
bodyequationsofmotiontobeputinthefollowingform: A A B
NS =rϕ˙ . (2.12b)
B,NS
B B
dv M
= (1+ tot)n+ totλ , (2.10) The 1.5PN spin-orbit terms are found to be
dt −r2 A B
where (cid:2) (cid:3) M Sc δm Σc
SO = (rϕ˙) 5 ℓ +3 ℓ , (2.13a)
A1.5PN − r M2 M M2
tot = NS+ SO + SO + SS+QM (2.11a) (cid:20) (cid:21)
ABtot =ABNS+BA1S1.O.55PPNN+BA2S2.O5.5PPNN+BA2SPS2P+NNQM. (2.11b) B1S.O5PN =2Mr r˙(cid:18)MSℓc2(cid:19), (2.13b)
Thenonspinterms NSand NShavebeenexplicitlycal- and have the same form in terms of the “nc” variables.
A B
culated by various authors. The results can be found in The 2.5PN spin-orbit terms in both spin variables are
M M 11 Sc M δm Σc
SO,c = (rϕ˙) (20+14η)+ 9 η r˙2 10η(rϕ˙)2 ℓ + (12+9η)+(3 5η)r˙2 5η(rϕ˙)2 ℓ ,
A2.5PN r r − 2 − M2 r − − M M2
(cid:26)(cid:20) (cid:18) (cid:19) (cid:21) (cid:20) (cid:21) (cid:27)
(2.14a)
M M 11 Snc M δmΣnc
SO,nc = (rϕ˙) (17+16η)+ 9 η r˙2 10η(rϕ˙)2 ℓ + (9+10η)+(3 5η)r˙2 5η(rϕ˙)2 ℓ ,
A2.5PN r r − 2 − M2 r − − M M2
(cid:26)(cid:20) (cid:18) (cid:19) (cid:21) (cid:20) (cid:21) (cid:27)
(2.14b)
M M Sc M δm Σc
SO,c = r˙ 2 (2+5η)+ηr˙2+(9 2η)(rϕ˙)2 ℓ + 6 η ηr˙2+(3 η)(rϕ˙)2 ℓ , (2.15a)
B2.5PN r − r − M2 − r − − M M2
(cid:26)(cid:20) (cid:21) (cid:20) (cid:21) (cid:27)
M M Snc M δmΣnc
SO,nc = r˙ 2 (2+3η)+ηr˙2+(9 2η)(rϕ˙)2 ℓ + 4 η ηr˙2+(3 η)(rϕ˙)2 ℓ . (2.15b)
B2.5PN r − r − M2 − r − − M M2
(cid:26)(cid:20) (cid:21) (cid:20) (cid:21) (cid:27)
Finally, the spin-spin + quadrupole-monopole pieces are nents of Eq. (2.10) along n and λ are given by
M
SS+QM = 3 M 2 S0c,ℓ 2, (2.16a) r¨=−r2(1+Atot)+rϕ˙2, (2.17a)
A2PN 2 r M2 1 M
(cid:18) (cid:19) (cid:18) (cid:19) ϕ¨= tot+2r˙ϕ˙ . (2.17b)
SS+QM =0, (2.16b) −r r2B
B2PN (cid:18) (cid:19)
This system can be reexpressed in first-order form by
and have the same form in terms of the “nc” variables. defining u r˙ and ω ϕ˙, resulting in three first-order
≡ ≡
equations in the variables (r,u,ω).
Circular orbits correspond to the conditions r˙ = u˙ =
ω˙ =0. Inparticular,theconditionu˙ =0andEq.(2.17a)
C. Perturbing the equations of motion
imply the following implicit relationship for the circular
orbital frequency:
Havingsimplifiedtheequationsofmotion,wenowwish
M
tostudyperturbationsaboutthecircularorbitsolutions. ω02 = r3[1+At0ot(r0,ω0)], (2.18)
Wefirstreexpresstheequationsexplicitlyintermsofthe 0
polar coordinates (r,ϕ) of the relative position vector. or, in terms of the PN parameter x (Mω )2/3,
0
Differentiating the expression for the velocity vector in ≡
Eq. (2.9a) and using n˙ = ϕ˙λ and λ˙ = ϕ˙n, the compo- x=γ[1+ tot(γ,x)]1/3, (2.19)
− A0
7
whereasubscript0referstoquantitiesevaluatedalonga δ˙u=α δr+β δω, (2.26b)
0 0
circularorbitandwehavedefinedanotherPNexpansion δ˙ω =γ δu, with (2.26c)
parameter γ M/r. 0
≡
Equation (2.19) provides an implicit relationship be-
tweenthe two PNexpansionparametersγ andx. Later,
weshallneedanexplicitPNexpansionforγintermsofx.
M ∂ tot
Tao3PdeNrivseertiehsiserxeplaatnisoinosnhwipitfrhoumn(d2e.t1e9r)m,iwneedfircsotesffiubcisetnittust,e α0 =3ω02− r02 A∂r (cid:12)0, (2.27a)
γ =x(1+c1x+c1.5x3/2+c2x2+c2.5x5/2+c3x3), (2.20) β0 =2r0ω0− Mr2 ∂A∂ωto(cid:12)(cid:12)(cid:12)t , (2.27b)
0 (cid:12)0
into the right-hand-side of Eq. (2.19). Next we series 1 M ∂(cid:12)(cid:12) tot
expand the result in x to 3PN order, and equate the co- γ0 =−r 2ω0+ r2 ∂B(cid:12)u , (2.27c)
efficientsoflikepowersofxonbothsidesoftheequation. 0 (cid:18) 0 (cid:12)0(cid:19)
(cid:12)
This results in a linear system of 5 equations for the 5 (cid:12)
unknownsinEq.(2.20). Solvingthissystemeasilyyields where γ0 is not related to the γ M(cid:12)/r defined earlier.
≡
Nowweassumeaperturbationoftheformδq =E eiλt
q
η 5 Sc δm Σc [where q = (r,u,ω)] and substitute into Eqs. (2.26),
γ =x 1+x 1 +x3/2 ℓ + ℓ
− 3 3M2 M M2 resulting in a linear algebraic system for the Eq and
(cid:26) (cid:16) (cid:17) (cid:18) (cid:19) the eigenvalue λ. A trivial solution corresponding to
65 1 Sc 2
+x2 1 η 0,ℓ λ = 0 is Eu = 0 and Er = (β0/α0)Eω; this rep-
" − 12 − 2(cid:18)M2(cid:19) # resents a nonoscillatory displace−ment from one circu-
10 8 Sc δm Σc lar orbit to another. The remaining eigenvalues are
+x5/2 + η ℓ +2 ℓ λ = [ (α +β γ )]1/2. If the argument of the square-
3 9 M2 M M2 ±− 0 0 0
(cid:20)(cid:18) (cid:19) (cid:21) root is positive, then the resulting solutions are stable.
+x3 1+ 2203 41 π2 η+ 229η2+ η3 . The condition for the existence of stable circular orbits
−2520 − 192 36 81 can therefore be expressed as
(cid:20) (cid:18) (cid:19) (cid:21)(cid:27)
(2.21)
C α β γ >0, (2.28)
In terms of the nonconstant spin-magnitude variables, 0 ≡− 0− 0 0
the 2.5PN order term in the above equation should be
replaced with [see Eq. (6.3) of [47]] and the equality C0 =0 defines the ISCO.
Using Eqs. (2.27) and (2.11), eliminating r via (2.21),
13 2 Snc η δmΣnc
+x5/2 + η ℓ + 3 ℓ , (2.22) and expanding to the appropriate PN order, one can ex-
3 9 M2 − 3 M M2 pressthestabilityconditionexplicitlyintermsofx,yield-
(cid:20)(cid:18) (cid:19) (cid:16) (cid:17) (cid:21)
ingthefollowinggauge-invariantconditionfortheISCO:
whilethe1.5PNand2PNorderspintermshavethesame
form with “c” replaced by “nc”.
Now we examine linearperturbationsto the equations M2 Sc δm Σc
Cˆ C =1 6x+x3/2 14 ℓ +6 ℓ
of motion (2.17) about circular orbits parametrized by 0 ≡ x3 0 − M2 M M2
(r ,ω ). Introducing a small expansion parameter ǫ, we (cid:18) (cid:19)
0 0 Sc 2
substitute the following expansions into Eqs. (2.17): +x2 14η 3 0,ℓ
r =r +ǫδr, (2.23a) " − (cid:18)M2(cid:19) #
0
Sc δm Σc
u=0+ǫδu, (2.23b) +x5/2 ℓ (22+32η) ℓ (18+15η)
−M2 − M M2
ω =ω0+ǫδω, (2.23c) (cid:20) (cid:21)
397 123
and linearize. In doing so we expand tot as +x3 π2 η 14η2 . (2.29)
A 2 − 16 −
(cid:20)(cid:18) (cid:19) (cid:21)
∂ tot ∂ tot ∂ tot
tot = tot+ǫ A δr+ǫ A δu+ǫ A δω,
A A0 ∂r ∂u ∂ω In terms of the nonconstant-magnitude spin variables,
(cid:12)0 (cid:12)0 (cid:12)0 the 2.5PN spin-orbit term is replaced with
(cid:12) (cid:12) (cid:12)(2.24)
(cid:12) (cid:12) (cid:12)
andlikewise for tot. F(cid:12)romthe expli(cid:12)cit formof to(cid:12)t and
Btot, one can verBify that A +x5/2 Sℓnc(13+30η) δmΣnℓc(9+14η) , (2.30)
−M2 − M M2
∂ tot ∂ tot ∂ tot (cid:20) (cid:21)
A = B = B =0. (2.25)
∂u ∂r ∂ω
(cid:12)0 (cid:12)0 (cid:12)0 whilethe1.5PNand2PNspintermshavethesameform
(cid:12) (cid:12) (cid:12)
Then, at O(ǫ(cid:12)0), the eq(cid:12)uations of(cid:12) motion reduce to with“c”relabeledto“nc”. Notethatinthenonspinning
(cid:12) (cid:12) (cid:12)
Eq. (2.18) and tot =0. At O(ǫ1), we have the system case, Eq. (2.29) reduces to the 3PN gauge-invariant sta-
B0 bilityconditionofBlanchetandIyer[24][theirEq.(6.41)
δ˙r =δu, (2.26a) or Eq. (1.2) here].
8
III. COMPARISON WITH THE KERR ISCO Substituting this result into Eq. (3.4), we arrive at the
gauge-invariantrelation
Inthenonspinningcase,Eq.(2.29)reducesinthetest-
X X1/2 X1/2
mass limit to CˆK 1 6 χK 8 3χK . (3.7)
0 ≡ − β2/3 − 2 β1/3 − 2 β1/3
Cˆ =1 6x. (3.1) (cid:20) (cid:18) (cid:19)(cid:21)
0
− For χK = 0 we easily obtain the Schwarzschild value for
2
ThePNISCOcriterionCˆ0 =0inthiscaseclearlyrepro- the ISCO frequency (X = 1/6). One can verify numeri-
duces the exact Schwarzschild ISCO, x = 1/6. We wish callythatsolvingCˆK =0asafunctionofχK reproduces
0 2
to determine if Eq. (2.29) similarly reproduces the Kerr the ISCO frequency computed from Eqs. (3.2) and (3.5)
ISCO. for all values of χK [ 1,1].
Recall that the Kerr ISCO radius in Boyer-Lindquist Now we wish t2o∈com−pare the test-mass limit of the
coordinates is given by [29] ISCO condition derived in Eq. (2.29) with the gauge-
invariant Kerr ISCO expression in Eq. (3.7). Note that
rK
isco =3+Z sign(χK)[(3 Z )(3+Z +2Z )]1/2, Eq. (3.7) is valid for arbitrary spin, while Eq. (2.29) is
m 2− 2 − 1 1 2
2 limited by the PN order to which spin terms have been
Z =1+[1 (χK)2]1/3[(1+χK)1/3+(1 χK)1/3], computed in the equations of motion (currently 2.5PN
1 − 2 2 − 2
order). To allow a meaningful comparison, we must ex-
Z =[3(χK)2+Z2]1/2, (3.2)
2 2 1 pand Eq. (3.7) in the spin parameter χK, yielding
2
where the mass of the Kerr BH is denoted m , and its
2
dimensionless spin is χK [ 1,1] (with negative values CˆK =1 6X+χK(8X3/2 4X5/2)
2 ∈ − 0 − 2 −
corresponding to point-particles with retrograde orbital +(χK)2 3X2+8X3 10X4/3 +O[(χK)3]. (3.8)
motion).8 An expression equivalent to Eq. (3.2) can be 2 − − 2
found by differentiating the reduced particle energy [29] We can also(cid:0)perform a PN expansio(cid:1)n of Eq. (3.7) in X,
which results in
E 1 2w +χKw3/2
E˜ = − BL 2 BL , (3.3)
≡ m1 1−3wBL+2χK2wB3/L2 Cˆ0K =1−6X+8χK2X3/2−3(χK2)2X2
q 4χKX5/2+8(χK)2X3+O[(χK)3X7/2]. (3.9)
where w m /r and r is the Boyer-Lindquist − 2 2 2
BL 2 BL BL
≡
radial coordinate. Some simple algebraic manipulation Note that both expansions give consistent results at the
of dE˜/drBL =0 yields appropriateordersinχK2 andX. This isespeciallyinter-
esting because in Eq. (3.8), no PN expansion has been
Cˆ0K ≡1−6wBL+8χK2wB3/L2−3(χK2)2wB2L =0. (3.4) made. Italsosuggeststhepresenceofadditionalself-spin
termsat3PNand4PNordersinthe equationsofmotion
Solving this equation for r produces results identical
BL (in addition to the currently known 2PN-order terms).
to Eq. (3.2). But note that since w depends on a co-
BL Equation (3.9) suggests that cubic self-spin interaction
ordinateradius,Eq.(3.4)isclearlynotagauge-invariant
terms will not appear until 3.5PN order.
expression.
The above expansions can now be compared with the
Toderiveagauge-invariantversionofEq.(3.4),wefirst
test-mass limit of Eqs. (2.29) and (2.30). Taking η 0,
dtoefitnhee tPhNe pvaarriaambleeteXr x≡(i|nmt2hΩeKt|e2s/t3-,mwahssiclhimisit,anxalogXou)s. δm/M → −1, and (Sℓ/M2,Σℓ/M2,S0,ℓ/M2) → χ2→, the
The frequency ΩK dϕ/dt refers to the circula→r-orbit result is
≡
angularfrequency seen by a distant observerandfollows Cˆ =1 6x+8χ x3/2 3χ2x2 4χ x5/2+O(x3). (3.10)
from the Kerr geodesic equations [Eq. (2.16) of [29]]: 0 − 2 − 2 − 2
Thisisvalidforeitherchoiceofspinvariable(χnc orχc).
m ΩK =sign(χ ) wB3/L2 . (3.5) ComparingwithEq.(3.8)(andidentifyingX w2ithxa2nd
2 2 1+χK2wB3/L2 χK2 with χ2), we see that the PN gauge-invariant ISCO
condition(3.10)agreeswiththe KerrISCOconditionup
Defining β 1 χKX3/2, we invert Eq. (3.5) to obtain tothePNorder(2.5PN)towhichweknowthespinterms
≡ − 2
in the PN equations of motion.
X
w = . (3.6) Figure 1 compares different methods for computing
BL β2/3 the ISCO frequency (in the test-mass limit): (i) the ex-
act Kerr expression [computed from solving Eq. (3.7) or
plugging Eq. (3.2) into Eq. (3.5)]; (ii) solving the gauge-
8 In the notation of the previous section, the BH spin angular invariantISCOconditioninEq.(2.29)or(3.10);and(iii)
momentum is SK ≡χKm2ˆsK, where, in our restriction to non- findingtheminimumofthePNcircular-orbitenergywith
2 2 2 2
precessingcircularorbits,theorbitalangularmomentum points nonspinterms to3PNorderandspintermsto 2.5PNor-
intheℓ=zˆdirectionandwechooseˆsK2 =ˆz. der [Eq. (A1)]. [The ISCO frequencies for approaches
9
(i) and (ii) are also listed in Table I.] The method us- In the remainder of this section, we shall concern our-
ingthegauge-invariantconditionCˆ0 agreesexceptionally selves with the calculation of the coefficients cGSF(χ2)
wellforallspinsup toχ2 .0.5. Inthe nonspinningcase and cCOspin(χ2) via the improved spinning-EOB Hamil-
(χ = 0), the agreement is exact. For nonzero spins, tonianof[20]andthenewgauge-invariantPNISCOcon-
2
agreement with the exact Kerr result is limited by the dition in Eqs. (2.29). In particular, we note that the im-
fact that we only know the spin terms in the equations proved EOB Hamiltonian is constructed such that the
of motion to 2.5PN order. Note also that for small |χ2|, coefficients cGEOSFB(0) and cCEOOBspin(χ2) are exact.
the error is symmetric about χ =0. This is in contrast
2
with the ISCO computed from the 3PNenergy function,
forwhichtheerrorincreases(nearly)monotonicallywith A. The improved effective-one-body Hamiltonian
increasing ISCO frequency (or decreasing radius). This for spinning binaries
indicates that the ISCO computed via Cˆ is limited not
0
by finite-PN corrections but by finite-spin corrections. Recently, Barausse and Buonanno [20] have con-
In Appendix A we examine how other PN expressions structeda new EOBHamiltonianwiththe followingfea-
agree with their Kerr-spacetime counterparts. We find tures: (i) In the test-particle limit, the Hamiltonian re-
that test-masslimits of the circular-orbitenergyandthe duces to the exact Hamiltonian of a spinning test-body
Keplerian relation γ(x) agree with their Kerr analogs if in the Kerr spacetime [40] (to linear order in the test-
we identify χK2 with either choice of spin variable. How- particle’s spin; this limit of the EOB Hamiltonian pro-
ever,thePNorbitalangularmomentumonlyagreeswith ducesequationsofmotionandprecessionthatareequiv-
its Kerr analog if we identify χK2 with χc2. alenttothePapapetrou-Mathisson-Dixonequations[33–
39]). (ii) When PN-expanded, the EOB Hamiltonian re-
produces the 2PN spin-spin and 1.5PN and 2.5PN spin-
IV. CONSERVATIVE SHIFTS IN THE ISCO orbitcouplingsforarbitrarymassratios. (iii)TheHamil-
tonianincludesanadjustablefunctionK(η)thatappears
Consider the general behavior of the ISCO frequency inthespinninggeneralizationoftheeffectivemetricfunc-
whenthetest-particlehasanon-negligiblemassandspin tion A(r) [see Eqs. (6.9)–(6.11) of [20]]; this function is
(butassumethatallspinsarealignedorantialignedwith adjustedtoenforceagreementwiththeBarack-Sagocon-
the orbital angular momentum). The ISCO frequency servative GSF shift in the Schwarzschild ISCO [14, 15].
can be split into the following pieces:9 [But note that this adjustment does not guarantee good
agreementwith the (yet uncalculated) conservativeGSF
m2Ω=ΩˆK(χ2)+δΩˆGSF(χ2,q) shift in the Kerr ISCO.] (iv) For arbitrary mass ratios,
+δΩˆCOspin(χ ,q,χ )+δΩˆGSF+COspin(χ ,q,χ ), (4.1) this improved EOB Hamiltonian provides a well-defined
2 1 2 1
prescription to compute the conservative two-body dy-
where ΩˆK is the Kerr ISCO frequency in units of m namics and spin precession. (v) Finally, in the case of
2
[given by Eqs. (3.2) and (3.5), or Eq. (3.7)], δΩˆGSF and aligned or antialigned spins, this conservative dynamics
produces a well-behaved ISCO for any mass ratio.
δΩˆCOspin are corrections to this frequency (also in units
The improved EOB Hamiltonian of [20] is compli-
of m ) due to the conservative GSF and the spin of the
2 cated to write out explicitly. For the case of equa-
smallercompactobject,andδΩˆGSF+COspinisacorrection
torial (nonprecessing) orbits with spins aligned or an-
that results fromcross-termsbetween both effects. If we
tialigned with the orbital angular momentum, one can
assumethatthemassratioq m /m 1issmall,then
≡ 1 2 ≤ construct the Hamiltonian by starting with Eq. (6.1) of
we can rewrite Eq. (4.1) as
[20] and carefully following their paper for the subse-
quent chain of definitions (see Appendix C of [60] for
m2Ω=ΩˆK(χ2)[1+qc′GSF(χ2)+qχ1cCOspin(χ2) an alternate presentation). Once the EOB Hamilto-
+O(q2)+O(χ q2)+O(χ2q2)]. (4.2) nian is constructed, the ISCO angular frequency can
1 1
be computed from Eqs. (6.6)–(6.8) of [20]. Choosing
MultiplyingbyM/m andusingη =q+O(q2)[19]yields units in which the total mass M = 1, I constructed
2
a numerical code which computes the ISCO frequency
Ω˜ MΩ=ΩˆK(χ2)[1+ηcGSF(χ2)+ηχ1cCOspin(χ2) Ω˜EOB(η,χ1,χ2) giventhe reduced mass-ratioη, the spin
≡
of the test-particle χ , and the BH spin χ . By con-
+O(η2)+O(χ η2)+O(χ2η2)], (4.3) 1 2
1 1 struction, the resulting EOB ISCO has three important
where cGSF = 1+c′GSF was labeled cren in [19, 21] for properties: (i) in the test-particle limit it reduces to the
the χ2 =0 case. Ω Kerr ISCO [Ω˜EOB(0,0,χ2) = ΩˆK(χ2)]; (ii) in the non-
spinning case it reproduces the exact conservative GSF
ISCOshift[cGSF(0)=cren 1.251];and(iii)itcorrectly
EOB Ω ≈
accounts for the conservative ISCO shift due to the test-
9 Intheremainderofthispaperandunlessstatedotherwise,allof particle’sspin(thiswasexplicitlyverifiedinAppendixB
thespinvariablesrefertotheconstant-magnitude spins. by directly analyzing the Papapetrouequations).
10
TABLEI.ISCOquantitiesasafunctionofthedimensionless BHspinparameterχ2. Thesecondcolumndenotesthestandard
Kerr ISCO angular frequency in units of m2 [Eqs. (3.2) and (3.5)]. The third column is the test-particle limit of the ISCO
frequencycomputed from thegauge-invariant ISCO condition Cˆ0 [Eq. (2.29) or(3.10)]. Thefourth column istheconservative
self-forceISCOshiftparametercomputedfromtheEOBISCOfrequency[Eq.(4.4)]. Thefifthcolumnistheanalogousquantity
computedfromtheCˆ0 ISCOcondition[Eq.(4.5)]. ThesixthcolumncomputestheISCOshiftparameterduetothespinofthe
test-particle(computedviathespinning-EOBISCOfrequency[Eq.(4.6)],ordirectlyfromthePapapetrouequations[Appendix
B]). TheseventhcolumnistheanalogousquantitycomputedviatheCˆ0ISCOcondition[Eq.(4.7)]. Notetheperfectagreement
of several of these quantities in theχ2 =0 case, and thecloseness in theirvalues for small χ2.0.6 (see also Figs. 1 and 2).
χ2 ΩKisceorr ΩiCsc0o cGEOSFB cGC0SF cCEOOBspin cCPNOspin
-0.99 0.038635 0.038015 0.9486 1.1903 0.2313 0.1945
-0.9 0.040261 0.039681 0.9449 1.1961 0.2364 0.2020
-0.8 0.042223 0.041694 0.9423 1.2043 0.2424 0.2110
-0.7 0.044372 0.043901 0.9422 1.2148 0.2487 0.2205
-0.6 0.046736 0.046331 0.9458 1.2282 0.2553 0.2308
-0.5 0.049348 0.049016 0.9550 1.2453 0.2625 0.2417
-0.4 0.052251 0.051998 0.9726 1.2670 0.2700 0.2534
-0.3 0.055496 0.055325 1.0027 1.2948 0.2782 0.2657
-0.2 0.059149 0.059057 1.0517 1.3303 0.2868 0.2788
-0.1 0.063295 0.063266 1.1295 1.3759 0.2962 0.2923
0.0 0.068041 0.068041 1.2513 1.4349 0.3062 0.3062
0.1 0.073536 0.073492 1.4418 1.5116 0.3170 0.3199
0.2 0.079979 0.079750 1.7400 1.6118 0.3287 0.3328
0.3 0.087652 0.086978 2.2072 1.7434 0.3414 0.3435
0.4 0.096974 0.095365 2.9338 1.9167 0.3551 0.3502
0.5 0.108588 0.105125 4.0204 2.1441 0.3699 0.3499
0.6 0.123568 0.116470 5.4310 2.4388 0.3856 0.3382
0.7 0.143879 0.129564 6.3967 2.8098 0.4014 0.3097
0.8 0.173747 0.144421 4.2785 3.2524 0.4150 0.2590
0.9 0.225442 0.160767 -3.3671 3.7337 0.4152 0.1837
0.99 0.364410 0.176197 -23.763 4.1440 0.2937 0.0983
B. EOB and PN predictions for the conservative ISCO shift parameter can be computed via
self-force ISCO shift in Kerr
1 Ω˜ (η,0,χ )
cGSF(χ )= lim EOB 2 1 . (4.4)
The conservative self-force ISCO shift parameter de- EOB 2 η→0η " ΩˆK(χ2) − #
notedcGSF inEq.(4.3)is anespeciallyinteresting quan-
titybecauseitisagauge-invariantthatcanbecalculated In the PN case a function Ω˜ (η,χ ,χ ) is computed by
C0 1 2
from self-force calculations. Barack and Sago [14, 15] solvingfortherootofEq.(2.29)numerically. Theresult-
havecomputedthisquantityinthecaseofSchwarzschild, ingconservativeGSFISCOshiftparameterisdefinedby
and in Ref. [21] this result was compared with multiple
PN-based computations of the ISCO shift.10 Gravita- 1 Ω˜ (η,0,χ )
cGSF(χ )= lim C0 2 1 . (4.5)
tsipoancaeltismelef-,fobructe rheesrueltwseareexnpolotreyetthaevapirlaedbilcetfioornsthfeorKtehrer C0 2 η→0η "ΩˆC0(0,0,χ2) − #
conservative GSF ISCO shift in Kerr given by two PN-
Note that in this equation the denominator contains the
based calculations: the spinning-EOB approach[20] and function Ωˆ (0,0,χ ) rather than ΩˆK(χ ). This is be-
the ISCO computed via the gauge-invariant PN ISCO C0 2 2
causethe gauge-invariantPNISCO Ω˜ doesnotreduce
conditionCˆ [Eq.(2.29)]. Basedonthecomparisonstudy C0
0 precisely to the Kerr ISCO (although it is very close for
in[21],thesetwomethodsarethemostviableapproaches
small to moderate values of χ ; see Fig. 1 and Sec. III).
for computing the ISCO in the small-mass-ratiolimit. 2
The resulting values for cGSF(χ ) and cGSF(χ ) are
EOB 2 C0 2
Using the EOB ISCO frequency calculated from [20] listedinTableIandplottedintheleft-halfofFig.2. Note
as described above, the corresponding conservative GSF thatwhiletheEOBcurveiscalibratedtotheexactresult
in the nonspinning case, there is no expectation that it
will also predict the correct ISCO shift in the spinning
case. The function K(η) will presumably need to be re-
calibratedwhen GSF results for the Kerr ISCO shift are
available. To further explore the behaviorofcGSF(χ ), I
EOB 2
10 ForothercomparisonsofPNandGSFresults,see[9,19,61–63]. have varied the value of K from 0 to 4. Figure 2 shows