Circular and non-circular nearly horizon-skimming orbits in Kerr spacetimes
Enrico Barausse∗
SISSA, International School for Advanced Studies and INFN, Via Beirut 2, 34014 Trieste, Italy
Scott A. Hughes
Department of Physics and MIT Kavli Institute, MIT, 77 Massachusetts Ave., Cambridge, MA 02139 USA
Luciano Rezzolla
Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, 14476 Potsdam, Germany and
Department of Physics, Louisiana State University, Baton Rouge, LA 70803 USA
(Dated: November 1, 2018)
We have performed a detailed analysis of orbital motion in the vicinity of a nearly extremal Kerr black hole.
For very rapidly rotating black holes — spin parameter a ≡ J/M > 0.9524M — we have found a class of
very strong field eccentric orbits whose orbital angular momentum Lz increases with the orbit’s inclination with
respect to the equatorial plane, while keeping latus rectum and eccentricity fixed. This behavior is in contrast
with Newtonian intuition, and is in fact opposite to the “normal” behavior of black hole orbits. Such behavior
was noted previously for circular orbits; since it only applies to orbits very close to the black hole, they were
named “nearly horizon-skimming orbits”. Our current analysis generalizes this result, mapping out the full
generic (inclined and eccentric) family of nearly horizon-skimming orbits. The earlier work on circular orbits
reported that, under gravitational radiation emission, nearly horizon-skimming orbits exhibit unusual inspiral,
tending to evolve to smaller orbit inclination, toward prograde equatorial configuration. Normal orbits, by
contrast, always demonstrate slowly growing orbit inclination — orbits evolve toward the retrograde equatorial
configuration. Using up-to-date Teukolsky-based fluxes, we have concluded that the earlier result was incorrect
— all circular orbits, including nearly horizon-skimming ones, exhibit growing orbit inclination under radiative
backreaction. Using kludge fluxes based on a Post-Newtonian expansion corrected with fits to circular and
to equatorial Teukolsky-based fluxes, we argue that the inclination grows also for eccentric nearly horizon-
skimming orbits. We also find that the inclination change is, in any case, very small. As such, we conclude that
these orbits are not likely to have a clear and peculiar imprint on the gravitational waveforms expected to be
measured by the space-based detector LISA.
PACS numbers: 04.30.-w
I. INTRODUCTION
The space-based gravitational-wave detector LISA [1] will
be a unique tool to probe the nature of supermassive black
holes (SMBHs), making it possible to map in detail their
spacetimes. This goal is expected to be achieved by observing
gravitational waves emitted by compact stars or black holes
with masses µ ≈ 1 − 100M⊙ spiraling into the SMBHs
which reside in the center of galaxies [2] (particularly the low
end of the galactic center black hole mass function, M ≈
105− 107M⊙). Such events are known as extreme mass ratio
inspirals (EMRIs). Current wisdom suggests that several tens
to perhaps of order a thousand such events could be measured
per year by LISA [3].
Though the distribution of spins for observed astrophysi-
cal black holes is not very well understood at present, very
rapid spin is certainly plausible, as accretion tends to spin-
up SMBHs [4]. Most models for quasi-periodic oscillations
(QPOs) suggest this is indeed the case in all low-mass x-ray
binaries for which data is available [5]. On the other hand,
continuum spectral fitting of some high-mass x-ray binaries
indicates that modest spins (spin parameter a/M ≡ J/M2 ∼
∗Electronic address: barausse@sissa.it
0.6− 0.8) are likewise plausible [6]. The continuum-fit tech-
nique does find an extremely high spin of a/M & 0.98 for the
galactic “microquasar” GRS1915+105 [7]. This argues for a
wide variety of possible spins, depending on the detailed birth
and growth history of a given black hole.
In the mass range corresponding to black holes in galactic
centers, measurements of the broad iron Kα emission line in
active galactic nuclei suggest that SMBHs can be very rapidly
rotating (see Ref. [8] for a recent review). For instance, in
the case of MCG-6-30-15, for which highly accurate observa-
tions are available, a has been found to be larger than 0.987M
at 90% confidence [9]. Because gravitational waves from EM-
RIs are expected to yield a very precise determination of the
spins of SMBHs [10], it is interesting to investigate whether
EMRIs around very rapidly rotating black holes may possess
peculiar features which would be observable by LISA. Should
such features exist, they would provide unambiguous infor-
mation on the spin of SMBHs and thus on the mechanisms
leading to their formation [11].
For extremal Kerr black holes (a = M ), the existence of
a special class of “circular” orbits was pointed out long ago
by Wilkins [12], who named them “horizon-skimming” or-
bits. (“Circular” here means that the orbits are of constant
Boyer-Lindquist coordinate radius r.) These orbits have vary-
ing inclination angle with respect to the equatorial plane and
have the same coordinate radius as the horizon, r = M . De-
http://arxiv.org/abs/0704.0138v2
mailto:barausse@sissa.it
spite this seemingly hazardous location, it can be shown that
all these r = M orbits have finite separation from one another
and from the event horizon [13]. Their somewhat pathological
description is due to a singularity in the Boyer-Lindquist co-
ordinates, which collapses a finite span of the spacetime into
r = M .
Besides being circular and “horizon-skimming”, these or-
bits also show peculiar behavior in their relation of angular
momentum to inclination. In Newtonian gravity, a generic or-
bit has Lz = |L| cos ι, where ι is the inclination angle relative
to the equatorial plane (going from ι = 0 for equatorial pro-
grade orbits to ι = π for equatorial retrograde orbits, passing
through ι = π/2 for polar orbits), and L is the orbital angular
momentum vector. As a result, ∂Lz(r, ι)/∂ι < 0, and the an-
gular momentum in the z-direction always decreases with in-
creasing inclination if the orbit’s radius is kept constant. This
intuitively reasonable decrease of Lz with ι is seen for almost
all black hole orbits as well. Horizon-skimming orbits, by
contrast, exhibit exactly the opposite behavior: Lz increases
with inclination angle.
Reference [14] asked whether the behavior ∂Lz/∂ι > 0
could be extended to a broader class of circular orbits than just
those at the radius r = M for the spin value a = M . It was
found that this condition is indeed more general, and extended
over a range of radius from the “innermost stable circular or-
bit” to r ≃ 1.8M for black holes with a > 0.9524M . Or-
bits that show this property have been named “nearly horizon-
skimming”. The Newtonian behavior ∂Lz(r, ι)/∂ι < 0 is
recovered for all orbits at r & 1.8M [14].
A qualitative understanding of this behavior comes from
recalling that very close to the black hole all physical pro-
cesses become “locked” to the hole’s event horizon [15], with
the orbital motion of point particles coupling to the horizon’s
spin. This locking dominates the “Keplerian” tendency of an
orbit to move more quickly at smaller radii, forcing an or-
biting particle to slow down in the innermost orbits. Lock-
ing is particularly strong for the most-bound (equatorial) or-
bits; the least-bound orbits (which have the largest inclination)
do not strongly lock to the black hole’s spin until they have
very nearly reached the innermost orbit [14]. The property
∂Lz(r, ι)/∂ι > 0 reflects the different efficiency of nearly
horizon-skimming orbits to lock with the horizon.
Reference [14] argued that this behavior could have ob-
servational consequences. It is well-known that the incli-
nation angle of an inspiraling body generally increases due
to gravitational-wave emission [16, 17]. Since dLz/dt <
0 because of the positive angular momentum carried away
by the gravitational waves, and since “normal” orbits have
∂Lz/∂ι < 0, one would expect dι/dt > 0. However, if
during an evolution ∂Lz/∂ι switches sign, then dι/dt might
switch sign as well: An inspiraling body could evolve towards
an equatorial orbit, signalling the presence of an “almost-
extremal” Kerr black hole.
It should be emphasized that this argument is not rigorous.
In particular, one needs to consider the joint evolution of or-
bital radius and inclination angle; and, one must include the
dependence of these two quantities on orbital energy as well
as angular momentum1. As such, dι/dt depends not only on
dLz/dt and ∂Lz/∂ι, but also on dE/dt, ∂E/∂ι, ∂E/∂r and
∂Lz/∂r.
In this sense, the argument made in Ref. [14] should be
seen as claiming that the contribution coming from dLz/dt
and ∂Lz/∂ι are simply the dominant ones. Using the nu-
merical code described in [17] to compute the fluxes dLz/dt
and dE/dt, it was then found that a test-particle on a circu-
lar orbit passing through the nearly horizon-skimming region
of a Kerr black hole with a = 0.998M (the value at which
a hole’s spin tends to be buffered due to photon capture from
thin disk accretion [19]) had its inclination angle decreased
by δι ≈ 1◦ − 2◦ [14] in the adiabatic approximation [20].
It should be noted at this point that the rate of change of in-
clination angle, dι/dt, appears as the difference of two rel-
atively small and expensive to compute rates of change [cf.
Eq. (3.8) of Ref. [17]]. As such, small relative errors in those
rates of change can lead to large relative errors in dι/dt. Fi-
nally, in Ref. [14] it was speculated that the decrease could
be even larger for eccentric orbits satisfying the condition
∂Lz/∂ι > 0, possibly leading to an observable imprint on
EMRI gravitational waveforms.
The main purpose of this paper is to extend Ref. [14]’s anal-
ysis of nearly horizon-skimming orbits to include the effect
of orbital eccentricity, and to thereby test the speculation that
there may be an observable imprint on EMRI waveforms of
nearly horizon-skimming behavior. In doing so, we have re-
visited all the calculations of Ref. [14] using a more accurate
Teukolsky solver which serves as the engine for the analysis
presented in Ref. [21].
We have found that the critical spin value for circular
nearly horizon-skimming orbits, a > 0.9524M , also delin-
eates a family of eccentric orbits for which the condition
∂Lz(p, e, ι)/∂ι > 0 holds. (More precisely, we consider vari-
ation with respect to an angle θinc that is easier to work with
in the extreme strong field, but that is easily related to ι.) The
parameters p and e are the orbit’s latus rectum and eccentric-
ity, defined precisely in Sec. II. These generic nearly horizon-
skimming orbits all have p . 2M , deep in the black hole’s
extreme strong field.
We next study the evolution of these orbits under
gravitational-wave emission in the adiabatic approximation.
We first revisited the evolution of circular, nearly horizon-
skimming orbits using the improved Teukolsky solver which
was used for the analysis of Ref. [21]. The results of this anal-
ysis were somewhat surprising: Just as for “normal” orbits,
we found that orbital inclination always increases during in-
spiral, even in the nearly horizon-skimming regime. This is in
stark contrast to the claims of Ref. [14]. As noted above, the
inclination’s rate of change depends on the difference of two
expensive and difficult to compute numbers, and thus can be
strongly impacted by small relative errors in those numbers.
1 In the general case, one must also include the dependence on “Carter’s
constant” Q [18], the third integral of black hole orbits (described more
carefully in Sec. II). For circular orbits, Q = Q(E,Lz): knowledge of E
and Lz completely determines Q.
A primary result of this paper is thus to retract the claim of
Ref. [14] that an important dynamic signature of the nearly
horizon-skimming region is a reversal in the sign of incli-
nation angle evolution: The inclination always grows under
gravitational radiation emission.
We next extended this analysis to study the evolution of
generic nearly horizon-skimming orbits. The Teukolsky code
to which we have direct access can, at this point, only com-
pute the radiated fluxes of energy E and angular momentum
Lz; results for the evolution of the Carter constant Q are just
now beginning to be understood [22], and have not yet been
incorporated into this code. We instead use “kludge” expres-
sions for dE/dt, dLz/dt, and dQ/dt which were inspired by
Refs. [23, 24]. These expression are based on post-Newtonian
flux formulas, modified in such a way that they fit strong-field
radiation reaction results obtained from a Teukolsky integra-
tor; see Ref. [24] for further discussion. Our analysis indicates
that, just as in the circular limit, the result dι/dt > 0 holds
for generic nearly horizon-skimming orbits. Furthermore, and
contrary to the speculation of Ref. [14], we do not find a large
amplification of dι/dt as orbits are made more eccentric.
Our conclusion is that the nearly horizon-skimming regime,
though an interesting curiosity of strong-field orbits of nearly
extremal black holes, will not imprint any peculiar observa-
tional signature on EMRI waveforms.
The remainder of this paper is organized as follows. In Sec.
II, we review the properties of bound stable orbits in Kerr, pro-
viding expressions for the constants of motion which we will
use in Sec. III to generalize nearly horizon-skimming orbits to
the non-circular case. In Sec. IV, we study the evolution of the
inclination angle for circular nearly horizon-skimming orbits
using Teukolsky-based fluxes; in Sec. V we do the same for
non-circular orbits and using kludge fluxes. We present and
discuss our detailed conclusions in Sec. VI. The fits and post-
Newtonian fluxes used for the kludge fluxes are presented in
the Appendix. Throughout the paper we have used units in
which G = c = 1.
II. BOUND STABLE ORBITS IN KERR SPACETIMES
The line element of a Kerr spacetime, written in Boyer-
Lindquist coordinates reads [25]
ds2 = −
1− 2Mr
dt2 +
dr2 +Σ dθ2
r2 + a2 +
2Ma2r
sin2 θ
sin2 θ dφ2
−4Mar
sin2 θ dt dφ, (1)
where
Σ ≡ r2 + a2 cos2 θ, ∆ ≡ r2 − 2Mr + a2. (2)
Up to initial conditions, geodesics can then be labelled by four
constants of motion: the mass µ of the test particle, its energy
E and angular momentum Lz as measured by an observer at
infinity and the Carter constant Q [18]. The presence of these
four conserved quantities makes the geodesic equations sepa-
rable in Boyer-Lindquist coordinates. Introducing the Carter
time λ, defined by
≡ Σ , (3)
the geodesic equations become
= Vr(r), µ
= Vt(r, θ),
= Vθ(θ), µ
= Vφ(r, θ) , (4)
Vt(r, θ) ≡ E
− a2 sin2 θ
+ aLz
Vr(r) ≡
E̟2 − aLz
)2 −∆
µ2r2 + (Lz − aE)2 +Q
Vθ(θ) ≡ Q− L2z cot2 θ − a2(µ2 − E2) cos2 θ, (5c)
Vφ(r, θ) ≡ Lz csc2 θ + aE
, (5d)
where we have defined
̟2 ≡ r2 + a2 . (6)
The conserved parameters E, Lz , and Q can be remapped
to other parameters that describe the geometry of the orbit. We
have found it useful to describe the orbit in terms of an angle
θmin — the minimum polar angle reached by the orbit — as
well as the latus rectum p and the eccentricity e. In the weak-
field limit, p and e correspond exactly to the latus rectum and
eccentricity used to describe orbits in Newtonian gravity; in
the strong field, they are essentially just a convenient remap-
ping of the orbit’s apoastron and periastron:
rap ≡
, rperi ≡
1 + e
. (7)
Finally, in much of our analysis, it is useful to refer to
z− ≡ cos2 θmin , (8)
rather than to θmin directly.
To map (E,Lz, Q) to (p, e, z−), use Eq. (4) to impose
dr/dλ = 0 at r = rap and r = rperi, and to impose
dθ/dλ = 0 at θ = θmin. (Note that for a circular orbit,
rap = rperi = r0. In this case, one must apply the rules
dr/dλ = 0 and d2r/dλ2 = 0 at r = r0.) Following this ap-
proach, Schmidt [26] was able to derive explicit expressions
for E, Lz and Q in terms of p, e and z−. We now briefly
review Schmidt’s results.
Let us first introduce the dimensionless quantities
Ẽ ≡ E/µ , L̃z ≡ Lz/(µM) , Q̃ ≡ Q/(µM)2 , (9)
ã ≡ a/M , r̃ ≡ r/M , ∆̃ ≡ ∆/M2 , (10)
Figure 1: Left panel: Inclination angles θinc for which bound stable orbits exist for a black hole with spin a = 0.998M . The allowed range
for θinc goes from θinc = 0 to the curve corresponding to the eccentricity under consideration, θinc = θ
inc . Right panel: Same as left but for
an extremal black hole, a = M . Note that in this case θmaxinc never reaches zero.
and the functions
f(r̃) ≡ r̃4 + ã2
r̃(r̃ + 2) + z−∆̃
, (11)
g(r̃) ≡ 2 ã r̃ , (12)
h(r̃) ≡ r̃(r̃ − 2) +
1− z−
∆̃ , (13)
d(r̃) ≡ (r̃2 + ã2z−)∆̃ . (14)
Let us further define the set of functions
(f1, g1, h1, d1) ≡{
(f(r̃p), g(r̃p), h(r̃p), d(r̃p)) if e > 0 ,
(f(r̃0), g(r̃0), h(r̃0), d(r̃0)) if e = 0 ,
(f2, g2, h2, d2) ≡{
(f(r̃a), g(r̃a), h(r̃a), d(r̃a)) if e > 0 ,
(f ′(r̃0), g
′(r̃0), h
′(r̃0), d
′(r̃0)) if e = 0 ,
and the determinants
κ ≡ d1h2 − d2h1 , (17)
ε ≡ d1g2 − d2g1 , (18)
ρ ≡ f1h2 − f2h1 , (19)
η ≡ f1g2 − f2g1 , (20)
σ ≡ g1h2 − g2h1 . (21)
The energy of the particle can then be written
κρ+ 2ǫσ − 2D
σ(σǫ2 + ρǫκ− ηκ2)
ρ2 + 4ησ
. (22)
The parameter D takes the values ±1. The angular momen-
tum is a solution of the system
2 − 2g1ẼL̃z − h1L̃2z − d1 = 0 , (23)
2 − 2g2ẼL̃z − h2L̃2z − d2 = 0 . (24)
By eliminating the L̃2z terms in these equations, one finds the
solution
L̃z =
ρẼ2 − κ
for the angular momentum. Using dθ/dλ = 0 at θ = θmin,
the Carter constant can be written
Q̃ = z−
ã2(1− Ẽ2) +
1− z−
. (26)
Additional constraints on p, e, z− are needed for the orbits
to be stable. Inspection of Eq. (4) shows that an eccentric orbit
is stable only if
(rperi) > 0 . (27)
It is marginally stable if ∂Vr/∂r = 0 at r = rperi. Similarly,
the stability condition for circular orbits is
(r0) < 0 ; (28)
marginally stable orbits are set by ∂2Vr/∂r
2 = 0 at r = r0.
Finally, we note that one can massage the above solutions
for the conserved orbital quantities of bound stable orbits to
rewrite the solution for L̃z as
L̃z = −
g21Ẽ
2 + (f1Ẽ2 − d1)h1 . (29)
From this solution, we see that it is quite natural to refer to
orbits with D = 1 as prograde and to orbits with D = −1
as retrograde. Note also that Eq. (29) is a more useful form
than the corresponding expression, Eq. (A4), of Ref. [21]. In
that expression, the factor 1/h1 has been squared and moved
inside the square root. This obscures the fact that h1 changes
sign for very strong field orbits. Differences between Eq. (29)
and Eq. (A4) of [21] are apparent for a & 0.835, although
only for orbits close to the separatrix (i.e., the surface in the
space of parameters (p, e, ι) where marginally stable bound
orbits lie).
III. NON-CIRCULAR NEARLY HORIZON-SKIMMING
ORBITS
With explicit expressions for E, Lz and Q as functions of
p, e and z−, we now examine how to generalize the condi-
tion ∂Lz(r, ι)/∂ι > 0, which defined circular nearly horizon-
skimming orbits in Ref. [14], to encompass the non-circular
case. We recall that the inclination angle ι is defined as [14]
cos ι =
Q+ L2z
. (30)
Such a definition is not always easy to handle in the case of
eccentric orbits. In addition, ι does not have an obvious phys-
ical interpretation (even in the circular limit), but rather was
introduced essentially to generalize (at least formally) the def-
inition of inclination for Schwarzschild black hole orbits. In
that case, one has Q = L2x+L
y and therefore Lz = |L| cos ι.
A more useful definition for the inclination angle in Kerr
was introduced in Ref. [21]:
θinc =
−D θmin , (31)
where θmin is the minimum reached by θ during the orbital
motion. This angle is trivially related to z− (z− = sin
2 θinc)
and ranges from 0 to π/2 for prograde orbits and from π/2 to
π for retrograde orbits. It is a simple numerical calculation to
convert between ι and θinc; doing so shows that the differences
between ι and θinc are very small, with the two coinciding for
a = 0, and with a difference that is less than 2.6◦ for a = M
and circular orbits with r = M .
Bearing all this in mind, the condition we have adopted to
generalize nearly horizon-skimming orbits is
∂Lz(p, e, θinc)
∂θinc
> 0 . (32)
We have found that certain parts of this calculation, particular
the analysis of strong-field geodesic orbits, are best done us-
ing the angle θinc; other parts are more simply done using the
angle ι, particularly the “kludge” computation of fluxes de-
scribed in Sec. V. (This is because the kludge fluxes are based
on an extension of post-Newtonian formulas to the strong-
field regime, and these formulas use ι for inclination angle.)
Accordingly, we often switch back and forth between these
two notions of inclination, and in fact present our final results
for inclination evolution using both dι/dt and dθinc/dt.
Before mapping out the region corresponding to nearly
horizon-skimming orbits, it is useful to examine stable or-
bits more generally in the strong field of rapidly rotating black
holes. We first fix a value for a, and then discretize the space
of parameters (p, e, θinc). We next identify the points in this
space corresponding to bound stable geodesic orbits. Suffi-
ciently close to the horizon, the bound stable orbits with spec-
ified values of p and e have an inclination angle θinc ranging
from 0 (equatorial orbit) to a maximum value θmaxinc . For given
p and e, θmaxinc defines the separatrix between stable and unsta-
ble orbits.
Example separatrices are shown in Fig. 1 for a = 0.998M
and a = M . This figure shows the behavior of θmaxinc as a
function of the latus rectum for the different values of the ec-
centricity indicated by the labels. Note that for a = 0.998M
the angle θmaxinc eventually goes to zero. This is the general
behavior for a < M . On the other hand, for an extremal black
hole, a = M , θmaxinc never goes to zero. The orbits which re-
side at r = M (the circular limit) are the “horizon-skimming
orbits” identified by Wilkins [12]; the a = M separatrix has
a similar shape even for eccentric orbits. As expected, we
find that for given latus rectum and eccentricity the orbit with
θinc = 0 is the one with the lowest energy E (and hence is the
most-bound orbit), whereas the orbit with θinc = θ
inc has the
highest E (and is least bound).
Having mapped out stable orbits in (p, e, θinc) space, we
then computed the partial derivative ∂Lz(p, e, θinc)/∂θinc and
identified the following three overlapping regions:
• Region A: The portion of the (p, e) plane for which
∂Lz(p, e, θinc)/∂θinc > 0 for 0 ≤ θinc ≤ θmaxinc . This
region is illustrated in Fig. 2 as the area under the heavy
solid line and to the left of the dot-dashed line (green in
the color version).
• Region B: The portion of the (p, e) plane
for which (Lz)most bound(p, e) is smaller than
(Lz)least bound(p, e). In other words,
Lz(p, e, 0) < Lz(p, e, θ
inc ) (33)
in Region B. Note that Region B contains Region A. It
is illustrated in Fig. 2 as the area under the heavy solid
line and to the left of the dotted line (red in the color
version).
• Region C: The portion of the (p, e) plane for which
∂Lz(p, e, θinc)/∂θinc > 0 for at least one angle θinc
between 0 and θmaxinc . Region C contains Region B, and
is illustrated in Fig. 2 as the area under the heavy solid
line and to the left of the dashed line (blue in the color
version).
Figure 2: Left panel: Non-circular nearly horizon-skimming orbits for a = 0.998M . The heavy solid line indicates the separatrix between
stable and unstable orbits for equatorial orbits (ι = θinc = 0). All orbits above and to the left of this line are unstable. The dot-dashed line
(green in the color version) bounds the region of the (p, e)-plane where ∂Lz/∂θinc > 0 for all allowed inclination angles (“Region A”). All
orbits between this line and the separatrix belong to Region A. The dotted line (red in the color version) bounds the region (Lz)most bound <
(Lz)least bound (“Region B”). Note that B includes A. The dashed line (blue in the color version) bounds the region where ∂Lz/∂θinc > 0 for
at least one inclination angle (“Region C”); note that C includes B. All three of these regions are candidate generalizations of the notion of
nearly horizon-skimming orbits. Right panel: Same as the left panel, but for the extreme spin case, a = M . In this case the separatrix between
stable and unstable equatorial orbits is given by the line p/M = 1 + e.
Orbits in any of these three regions give possible general-
izations of the nearly horizon-skimming circular orbits pre-
sented in Ref. [14]. Notice, as illustrated in Fig. 2, that the
size of these regions depends rather strongly on the spin of
the black hole. All three regions disappear altogether for
a < 0.9524M (in agreement with [14]); their sizes grow with
a, reaching maximal extent for a = M . These regions never
extend beyond p ≃ 2M .
As we shall see, the difference between these three regions
is not terribly important for assessing whether there is a strong
signature of the nearly horizon-skimming regime on the inspi-
ral dynamics. As such, it is perhaps most useful to use Region
C as our definition, since it is the most inclusive.
IV. EVOLUTION OF θinc: CIRCULAR ORBITS
To ascertain whether nearly horizon-skimming orbits can
affect an EMRI in such a way as to leave a clear imprint in the
gravitational-wave signal, we have studied the time evolution
of the inclination angle θinc. To this purpose we have used the
so-called adiabatic approximation [20], in which the infalling
body moves along a geodesic with slowly changing parame-
ters. The evolution of the orbital parameters is computed us-
ing the time-averaged fluxes dE/dt, dLz/dt and dQ/dt due
to gravitational-wave emission (“radiation reaction”). As dis-
cussed in Sec. II, E, Lz and Q can be expressed in terms of p,
e, and θinc. Given rates of change of E, Lz and Q, it is then
straightforward [23] to calculate dp/dt, de/dt, and dθinc/dt
(or dι/dt).
We should note that although perfectly well-behaved for all
bound stable geodesics, the adiabatic approximation breaks
down in a small region of the orbital parameters space very
close to the separatrix, where the transition from an inspiral to
a plunging orbit takes place [27]. However, since this region
is expected to be very small2 and its impact on LISA wave-
forms rather hard to detect [27], we expect our results to be
at least qualitatively correct also in this region of the space of
parameters.
Accurate calculation of dE/dt and dLz/dt in the adiabatic
approximation involves solving the Teukolsky and Sasaki-
Nakamura equations [28, 29]. For generic orbits this has been
done for the first time in Ref. [21]. The calculation of dQ/dt
for generic orbits is more involved. A formula for dQ/dt has
been recently derived [22], but has not yet been implemented
(at least in a code to which we have access).
On the other hand, it is well-known that a circular orbit will
remain circular under radiation reaction [30, 31, 32]. This
constraint means that Teukolsky-based fluxes for E and Lz
2 Its width in p/M is expected to be of the order of ∆p/M ∼ (µ/M)2/5 ,
where µ is the mass of the infalling body [27].
are sufficient to compute dQ/dt. Considering this limit, the
rate of change dQ/dt can be expressed in terms of dE/dt and
dL/dt as
= −N1(p, ι)
N5(p, ι)
− N4(p, ι)
N5(p, ι)
where
N1(p, ι) ≡ E(p, ι) p4 + a2 E(p, ι) p2
− 2 aM (Lz(p, ι)− aE(p, ι)) p , (35)
N4(p, ι) ≡ (2M p− p2)Lz(p, ι)− 2M aE(p, ι) p , (36)
N5(p, ι) ≡ (2M p− p2 − a2)/2 . (37)
(These quantities are for a circular orbit of radius p.) Using
this, it is simple to compute dθinc/dt (or dι/dt).
This procedure was followed in Ref. [14], using the code
presented in Ref. [17], to determine the evolution of ι; this
analysis indicated that dι/dt < 0 for circular nearly horizon-
skimming orbits. As a first step to our more general analy-
sis, we have repeated this calculation but using the improved
Sasaki-Nakamura-Teukolsky code presented in Ref. [21]; we
focused on the case a = 0.998M .
Rather to our surprise, we discovered that the fluxes dE/dt
and dLz/dt computed with this more accurate code indicate
that dι/dt > 0 (and dθinc/dt > 0) for all circular nearly
horizon-skimming orbits — in stark contrast with what was
found in Ref. [14]. As mentioned in the introduction, the rate
of change of inclination angle appears as the difference of two
quantities. These quantities nearly cancel (and indeed cancel
exactly in the limit a = 0); as such, small relative errors in
their values can lead to large relative error in the inferred in-
clination evolution. Values for dE/dt, dLz/dt, dι/dt, and
dθinc/dt computed using the present code are shown in Ta-
ble I in the columns with the header “Teukolsky”.
V. EVOLUTION OF θinc: NON-CIRCULAR ORBITS
The corrected behavior of circular nearly horizon-
skimming orbits has naturally led us to investigate the evo-
lution of non-circular nearly horizon-skimming orbits. Since
our code cannot be used to compute dQ/dt, we have resorted
to a “kludge” approach, based on those described in Refs.
[23, 24]. In particular, we mostly follow the procedure de-
veloped by Gair & Glampedakis [24], though (as described
below) importantly modified.
The basic idea of the “kludge” procedure is to use the func-
tional form of 2PN fluxes E, Lz and Q, but to correct the
circular part of these fluxes using fits to circular Teukolsky
data. As developed in Ref. [24], the fluxes are written
= (1− e2)3/2
(1− e2)−3/2
(p, e, ι)
(p, 0, ι) +
fit circ
(p, ι)
, (38)
= (1 − e2)3/2
(1 − e2)−3/2
(p, e, ι)
(p, 0, ι) +
fit circ
(p, ι)
, (39)
= (1− e2)3/2
Q(p, e, ι) ×
(1− e2)−3/2
dQ/dt√
(p, e, ι)−
dQ/dt√
(p, 0, ι)
dQ/dt√
fit circ
(p, ι)
. (40)
The post-Newtonian fluxes (dE/dt)2PN, (dLz/dt)2PN and
(dQ/dt)2PN are given in the Appendix [particularly Eqs.
(A.1), (A.2), and (A.3)].
Since for circular orbits the fluxes dE/dt, dLz/dt and
dQ/dt are related through Eq. (34), only two fits to circu-
lar Teukolsky data are needed. One possible choice is to fit
dLz/dt and dι/dt, and then use the circularity constraint to
obtain3 [24]
dQ/dt√
fit circ
(p, ι) =
2 tan ι
fit circ
Q(p, 0, ι)
sin2 ι
fit circ
, (41)
fit circ
(p, ι) = −
N4(p, ι)
N1(p, ι)
fit circ
(p, ι)
− N5(p, ι)
N1(p, ι)
Q(p, 0, ι)
dQ/dt√
fit circ
(p, ι) . (42)
As stressed in Ref. [24], one does not expect these fluxes to
work well in the strong field, both because the post-Newtonian
approximation breaks down close to the black hole, and be-
cause the circular Teukolsky data used for the fits in Ref. [24]
was computed for 3M ≤ p ≤ 30M . As a first attempt to
improve their behavior in the nearly horizon-skimming re-
gion, we have made fits using circular Teukolsky data for
orbits with M < p ≤ 2M . In particular, for a black hole
with a = 0.998M , we computed the circular Teukolsky-based
fluxes dLz/dt and dι/dt listed in Table I (columns 8 and 10).
These results were fit (with error . 0.2%); see Eqs. (A.4) and
(A.6) in the Appendix.
3 This choice might seem more involved than fitting directly dLz/dt and
dQ/dt, but, as noted by Gair & Glampedakis, ensures more sensible re-
sults for the evolution of the inclination angle. This generates more physi-
cally realistic inspirals [24].
Despite using strong-field Teukolsky fluxes for our fit, we
found fairly poor behavior of these rates of change, particu-
larly as a function of eccentricity. To compensate for this, we
introduced a kludge-type fit to correct the equatorial part of
the flux, in addition to the circular part. We fit, as a function
of p and e, Teukolsky-based fluxes for dE/dt and dLz/dt for
orbits in the equatorial plane, and then introduce the following
kludge fluxes for E and Lz:
(p, e, ι) =
(p, e, ι)
(p, e, 0) +
fit eq
(p, e) (43)
(p, e, ι) =
(p, e, ι)
(p, e, 0) +
fit eq
(p, e) . (44)
[Note that Eq. (40) for dQ/dt is not modified by this proce-
dure since dQ/dt = 0 for equatorial orbits.] Using equatorial
non-circular Teukolsky data provided by Drasco [21, 33] for
a = 0.998 and M < p ≤ 2M (the ι = 0 “Teukolsky” data in
Tables II, III and IV), we found fits (with error . 1.5%); see
Eqs. (A.9) and (A.10). Note that the fits for equatorial fluxes
are significantly less accurate than the fits for circular fluxes.
This appears to be due to the fact that, close to the black hole,
many harmonics are needed in order for the Teukolsky-based
fluxes to converge, especially for eccentric orbits (cf. Figs. 2
and 3 of Ref. [21], noting the number of radial harmonics that
have significant contribution to the flux). Truncation of these
sums is likely a source of some error in the fluxes themselves,
making it difficult to make a fit of as high quality as we could
in the circular case.
These fits were then finally used in Eqs. (43) and (44) to
calculate the kludge fluxes dE/dt and dLz/dt for generic or-
bits. This kludge reproduces to high accuracy our fits to the
Teukolsky-based fluxes for circular orbits (e = 0) or equato-
rial orbits (ι = 0). Some residual error remains because the
ι = 0 limit of the circular fits do not precisely equal the e = 0
limit of the equatorial fits.
Table I compares our kludge to Teukolsky-based fluxes for
circular orbits; the two methods agree to several digits. Tables
II, III and IV compare our kludge to the generic Teukolsky-
based fluxes for dE/dt and dLz/dt provided by Drasco
[21, 33]. In all cases, the kludge fluxes dE/dt and dLz/dt
have the correct qualitative behavior, being negative for all
the orbital parameters under consideration (a = 0.998M ,
1 < p/M ≤ 2, 0 ≤ e ≤ 0.5 and 0◦ ≤ ι ≤ 41◦). The
relative difference between the kludge and Teukolsky fluxes
is always less than 25% for e = 0 and e = 0.1 (even for
orbits very close to separatrix). The accuracy remains good
at larger eccentricity, though it degrades somewhat as orbits
come close to the separatrix.
Tables I, II, III and IV also present the kludge values of the
fluxes dι/dt and dθinc/dt as computed using Eqs. (43) and
(44) for dE/dt and dLz/dt, plus Eq. (40) for dQ/dt. Though
certainly not the last word on inclination evolution (pending
rigorous computation of dQ/dt), these rates of change proba-
bly represent a better approximation than results published to
date in the literature. (Indeed, prior work has often used the
crude approximation dι/dt = 0 [21] to estimate dQ/dt given
dE/dt and dLz/dt.)
Most significantly, we find that (dι/dt)kludge > 0 and
(dθinc/dt)kludge > 0 for all of the orbital parameters we con-
sider. In other words, we find that dι/dt and dθinc/dt do not
ever change sign.
Finally, in Table V we compute the changes in θinc and ι
for the inspiral with mass ratio µ/M = 10−6. In all cases,
we start at p/M = 1.9. The small body then inspirals through
the nearly horizon-skimming region until it reaches the sep-
aratrix; at this point, the small body will fall into the large
black hole on a dynamical timescale ∼ M , so we terminate
the calculation. The evolution of circular orbits is computed
using our fits to the circular-Teukolsky fluxes of E and Lz;
for eccentric orbits we use the kludge fluxes (40), (43) and
(44). As this exercise demonstrates, the change in inclination
during inspiral is never larger than a few degrees. Not only is
there no unique sign change in the nearly horizon-skimming
region, but the magnitude of the inclination change remains
puny. This leaves little room for the possibility that this class
of orbits may have a clear observational imprint on the EMRI-
waveforms to be detected by LISA.
VI. CONCLUSIONS
We have performed a detailed analysis of the orbital mo-
tion near the horizon of near-extremal Kerr black holes. We
have demonstrated the existence of a class of orbits, which we
have named “non-circular nearly horizon-skimming orbits”,
for which the angular momentum Lz increases with the or-
bit’s inclination, while keeping latus rectum and eccentricity
fixed. This behavior, in stark contrast to that of Newtonian
orbits, generalizes earlier results for circular orbits [14].
Furthermore, to assess whether this class of orbits can pro-
duce a unique imprint on EMRI waveforms (an important
source for future LISA observations), we have studied, in
the adiabatic approximation, the radiative evolution of incli-
nation angle for a small body orbiting in the nearly horizon-
skimming region. For circular orbits, we have re-examined
the analysis of Ref. [14] using an improved code for comput-
ing Teukolsky-based fluxes of the energy and angular momen-
tum. Significantly correcting Ref. [14]’s results, we found no
decrease in the orbit’s inclination angle. Inclination always
increases during inspiral.
We next carried out such an analysis for eccentric nearly
horizon-skimming orbits. In this case, we used “kludge”
fluxes to evolve the constants of motion E, Lz and Q [24].
We find that these fluxes are fairly accurate when compared
with the available Teukolsky-based fluxes, indicating that they
should provide at least qualitatively correct information re-
garding inclination evolution. As for circular orbits, we find
that the orbit’s inclination never decreases. For both circular
and non-circular configurations, we find that the magnitude of
the inclination change is quite paltry — only a few degrees at
most.
Quite generically, therefore, we found that the inclination
angle of both circular and eccentric nearly horizon-skimming
orbits never decreases during the inspiral. Revising the results
obtained in Ref. [14], we thus conclude that such orbits are
not likely to yield a peculiar, unique imprint on the EMRI-
waveforms detectable by LISA.
Acknowledgments
It is a pleasure to thank Kostas Glampedakis for enlight-
ening comments and advice, and Steve Drasco for useful
discussions and for also providing the non-circular Teukol-
sky data that we used in this paper. The supercomputers
used in this investigation were provided by funding from the
JPL Office of the Chief Information Officer. This work was
supported in part by the DFG grant SFB TR/7, by NASA
Grant NNG05G105G, and by NSF Grant PHY-0449884. SAH
gratefully acknowledges support from the MIT Class of 1956
Career Development Fund.
Appendix
In this Appendix we report the expressions for the post-
Newtonian fluxes and the fits to the Teukolsky data necessary
to compute the kludge fluxes introduced in Sec. V. In partic-
ular the 2PN fluxes are given by [24]
= −32
(1− e2)3/2
g1(e)− ã
g2(e) cos ι−
g3(e) + π
g4(e)
g5(e) + ã
g6(e)−
sin2 ι
, (A.1)
= −32
(1− e2)3/2
g9(e) cos ι+ ã
(ga10(e)− cos2 ιgb10(e)) −
g11(e) cos ι
g12(e) cos ι−
g13(e) cos ι+ ã
cos ι
g14(e)−
sin2 ι
, (A.2)
)7/2 √
Q sin ι (1− e2)3/2
g9(e)− ã
cos ιgb10(e)−
g11(e)
g12(e)−
g13(e) + ã
g14(e)−
sin2 ι
, (A.3)
where µ is the mass of the infalling body and where the various e-dependent coefficients are
g1(e) ≡ 1 +
e4 , g2(e) ≡
e6 , g3(e) ≡
g4(e) ≡ 4 +
e2 , g5(e) ≡
44711
172157
e2 , g6(e) ≡
e2 , g9(e) ≡ 1 +
ga10(e) ≡
e4 , gb10(e) ≡
e4 , g11(e) ≡
g12(e) ≡ 4 +
e2 , g13(e) ≡
44711
302893
e2 , g14(e) ≡
The fits to the circular-Teukolsky data of Table I are instead given by
fit circ
(p, ι ) = −32
)7/2 {
cos ι+
)3/2 (
cos2 ι+ 4π cos ι
− 1247
cos ι
cos ι
−1625
sin2 ι
d̃1(p/M) + d̃2(p/M) cos ι + d̃3(p/M) cos
+ d̃4(p/M) cos
3 ι + d̃5(p/M) cos
4 ι+ d̃6(p/M) cos
5 ι+ cos ι
)3/2 (
A+B cos2 ι
, (A.4)
(A.5)
fit circ
(p, ι ) =
sin2 ι√
Q(p, 0, ι)
d̃1(p/M) + cos ι
a7d + b
+ c7d
)3/2]
+ cos2 ι
d̃8(p/M) + cos ι
)5/2 [
h̃1(p/M) + cos
2 ι h̃2(p/M)
, (A.6)
where
d̃i(x) ≡ aid + bid x−1/2 + cid x−1 , i = 1, . . . , 8, h̃i(x) ≡ aih + bih x−1/2 , i = 1, 2 (A.7)
and the numerical coefficients are given by
a1h = −278.9387 , b1h = 84.1414 , a2h = 8.6679 , b2h = −9.2401 , A = −18.3362 , B = 24.9034 , (A.8)
and by the following table
i 1 2 3 4 5 6 7 8
aid 15.8363 445.4418 −2027.7797 3089.1709 −2045.2248 498.6411 −8.7220 50.8345
bid −55.6777 −1333.2461 5940.4831 −9103.4472 6113.1165 −1515.8506 −50.8950 −131.6422
cid 38.6405 1049.5637 −4513.0879 6926.3191 -4714.9633 1183.5875 251.4025 83.0834
Note that the functional form of these fits was obtained from Eqs. (57) and (58) of Ref. [24] by setting ã (i.e., q in their
notation) to 1. Finally, we give expressions for the fits to the equatorial Teukolsky data of tables II, III and IV (data with ι = 0,
columns with header “Teukolsky”):
fit eq
(p, e) =
(p, e, 0)− 32
)2 (M
(1− e2)3/2
g̃1(e) + g̃2(e)
+ g̃3(e)
+ g̃4(e)
+ g̃5(e)
, (A.9)
fit eq
(p, e) =
(p, e, 0)− 32
(1− e2)3/2
f̃1(e) + f̃2(e)
+ f̃3(e)
+ f̃4(e)
+ f̃5(e)
, (A.10)
g̃i(e) ≡ aig + big e2 + cig e4 + dig e6 , f̃i(e) ≡ aif + bif e2 + cif e4 + dif e6 , i = 1, . . . , 5 (A.11)
where the numerical coefficients are given by the following table
i aig b
1 6.4590 −2038.7301 6639.9843 227709.2187 5.4577 −3116.4034 4711.7065 214332.2907
2 -31.2215 10390.6778 −27505.7295 −1224376.5294 −26.6519 15958.6191 −16390.4868 −1147201.4687
3 57.1208 −19800.4891 39527.8397 2463977.3622 50.4374 -30579.3129 15749.9411 2296989.5466
4 -49.7051 16684.4629 −21714.7941 −2199231.9494 −46.7816 25968.8743 656.3460 −2038650.9838
5 16.4697 −5234.2077 2936.2391 734454.5696 15.6660 −8226.3892 −4903.9260 676553.2755
e θinc ι
dθinc
dθinc
(deg.) (deg.) (kludge) (Teukolsky) (kludge) (Teukolsky) (kludge) (Teukolsky) (kludge) (Teukolsky)
1.3 0 0 0 −9.108×10−2 −9.109×10−2 −2.258×10−1 −2.259×10−1 0 0 0 0
1.3 0 10.4870 11.6773 −9.328×10−2 −9.332×10−2 −2.304×10−1 −2.306×10−1 1.837×10−2 1.839×10−2 6.462×10−3 6.475×10−3
1.3 0 14.6406 16.1303 −9.588×10−2 −9.588×10−2 −2.359×10−1 −2.360×10−1 2.397×10−2 2.400×10−2 8.645×10−3 8.667×10−3
1.3 0 17.7000 19.3172 −9.875×10−2 −9.876×10−2 −2.420×10−1 −2.421×10−1 2.728×10−2 2.731×10−2 1.007×10−2 1.010×10−2
1.3 0 20.1636 21.8210 −1.019×10−1 −1.019×10−1 −2.486×10−1 −2.488×10−1 2.943×10−2 2.950×10−2 1.111×10−2 1.117×10−2
1.4 0 0 0 −8.700×10−2 −8.709×10−2 −2.311×10−1 −2.312×10−1 0 0 0 0
1.4 0 14.5992 16.0005 −9.062×10−2 −9.070×10−2 −2.386×10−1 −2.386×10−1 2.316×10−2 2.319×10−2 8.823×10−3 8.848×10−3
1.4 0 20.1756 21.7815 −9.520×10−2 −9.526×10−2 −2.482×10−1 −2.482×10−1 2.875×10−2 2.877×10−2 1.141×10−2 1.143×10−2
1.4 0 24.1503 25.7517 −1.006×10−1 −1.007×10−1 −2.595×10−1 −2.596×10−1 3.140×10−2 3.141×10−2 1.289×10−2 1.288×10−2
1.4 0 27.2489 28.7604 −1.067×10−1 −1.068×10−1 −2.725×10−1 −2.725×10−1 3.274×10−2 3.275×10−2 1.378×10−2 1.377×10−2
1.5 0 0 0 −8.009×10−2 −7.989×10−2 −2.270×10−1 −2.265×10−1 0 0 0 0
1.5 0 16.7836 18.1857 −8.401×10−2 −8.383×10−2 −2.348×10−1 −2.343×10−1 2.360×10−2 2.351×10−2 9.602×10−3 9.545×10−3
1.5 0 23.0755 24.6167 −8.917×10−2 −8.897×10−2 −2.454×10−1 −2.449×10−1 2.872×10−2 2.863×10−2 1.228×10−2 1.222×10−2
1.5 0 27.4892 28.9670 −9.537×10−2 −9.516×10−2 −2.583×10−1 −2.579×10−1 3.091×10−2 3.082×10−2 1.372×10−2 1.367×10−2
1.5 0 30.8795 32.2231 −1.025×10−1 −1.023×10−1 −2.733×10−1 −2.728×10−1 3.184×10−2 3.173×10−2 1.452×10−2 1.443×10−2
1.6 0 0 0 −7.181×10−2 −7.156×10−2 −2.168×10−1 −2.162×10−1 0 0 0 0
1.6 0 18.3669 19.7220 −7.568×10−2 −7.545×10−2 −2.242×10−1 −2.237×10−1 2.240×10−2 2.229×10−2 9.600×10−3 9.515×10−3
1.6 0 25.1720 26.6245 −8.084×10−2 −8.062×10−2 −2.346×10−1 −2.341×10−1 2.701×10−2 2.685×10−2 1.223×10−2 1.210×10−2
1.6 0 29.9014 31.2625 −8.708×10−2 −8.687×10−2 −2.474×10−1 −2.470×10−1 2.889×10−2 2.872×10−2 1.363×10−2 1.349×10−2
1.6 0 33.5053 34.7164 −9.425×10−2 −9.399×10−2 −2.622×10−1 −2.616×10−1 2.964×10−2 2.951×10−2 1.441×10−2 1.432×10−2
1.7 0 0 0 −6.332×10−2 −6.317×10−2 −2.034×10−1 −2.031×10−1 0 0 0 0
1.7 0 19.6910 20.9859 −6.702×10−2 −6.687×10−2 −2.101×10−1 −2.098×10−1 2.057×10−2 2.052×10−2 9.202×10−3 9.171×10−3
1.7 0 26.9252 28.2884 −7.197×10−2 −7.184×10−2 −2.199×10−1 −2.196×10−1 2.467×10−2 2.456×10−2 1.170×10−2 1.162×10−2
1.7 0 31.9218 33.1786 −7.794×10−2 −7.782×10−2 −2.319×10−1 −2.316×10−1 2.632×10−2 2.620×10−2 1.306×10−2 1.296×10−2
1.7 0 35.7100 36.8118 −8.475×10−2 −8.465×10−2 −2.457×10−1 −2.455×10−1 2.698×10−2 2.686×10−2 1.384×10−2 1.373×10−2
1.8 0 0 0 −5.531×10−2 −5.528×10−2 −1.888×10−1 −1.887×10−1 0 0 0 0
1.8 0 20.8804 22.1128 −5.879×10−2 −5.874×10−2 −1.948×10−1 −1.946×10−1 1.858×10−2 1.858×10−2 8.635×10−3 8.639×10−3
1.8 0 28.5007 29.7791 −6.343×10−2 −6.336×10−2 −2.036×10−1 −2.035×10−1 2.221×10−2 2.223×10−2 1.098×10−2 1.101×10−2
1.8 0 33.7400 34.9034 −6.901×10−2 −6.894×10−2 −2.146×10−1 −2.144×10−1 2.368×10−2 2.371×10−2 1.228×10−2 1.232×10−2
1.8 0 37.6985 38.7065 −7.533×10−2 −7.533×10−2 −2.271×10−1 −2.271×10−1 2.429×10−2 2.427×10−2 1.306×10−2 1.303×10−2
1.9 0 0 0 −4.809×10−2 −4.811×10−2 −1.740×10−1 −1.740×10−1 0 0 0 0
1.9 0 21.9900 23.1615 −5.132×10−2 −5.134×10−2 −1.792×10−1 −1.793×10−1 1.666×10−2 1.664×10−2 8.022×10−3 8.007×10−3
1.9 0 29.9708 31.1702 −5.562×10−2 −5.564×10−2 −1.872×10−1 −1.872×10−1 1.986×10−2 1.987×10−2 1.019×10−2 1.020×10−2
1.9 0 35.4385 36.5176 −6.078×10−2 −6.077×10−2 −1.971×10−1 −1.970×10−1 2.118×10−2 2.122×10−2 1.143×10−2 1.148×10−2
1.9 0 39.5592 40.4847 −6.659×10−2 −6.658×10−2 −2.082×10−1 −2.082×10−1 2.177×10−2 2.182×10−2 1.222×10−2 1.228×10−2
2.0 0 0 0 −4.174×10−2 −4.175×10−2 −1.598×10−1 −1.598×10−1 0 0 0 0
2.0 0 23.0471 24.1605 −4.471×10−2 −4.472×10−2 −1.643×10−1 −1.643×10−1 1.489×10−2 1.489×10−2 7.425×10−3 7.424×10−3
2.0 0 31.3715 32.4978 −4.867×10−2 −4.871×10−2 −1.713×10−1 −1.714×10−1 1.773×10−2 1.770×10−2 9.436×10−3 9.411×10−3
2.0 0 37.0583 38.0608 −5.341×10−2 −5.345×10−2 −1.801×10−1 −1.801×10−1 1.893×10−2 1.889×10−2 1.062×10−2 1.057×10−2
2.0 0 41.3358 42.1876 −5.873×10−2 −5.875×10−2 −1.900×10−1 −1.900×10−1 1.950×10−2 1.948×10−2 1.141×10−2 1.138×10−2
Table I: Teukolsky-based fluxes and kludge fluxes [computed using Eqs. (40), (43) and (44)] for circular orbits about a hole with a = 0.998M ;
µ represents the mass of the infalling body. The Teukolsky-based fluxes have an accuracy of 10−6.
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e θinc ι
dθinc
(deg.) (deg.) (kludge) (Teukolsky) (kludge) (Teukolsky) (kludge) (kludge)
1.3 0.1 0 0 −8.804×10−2 −8.804×10−2 −2.098×10−1 −2.098×10−1 0 0
1.4 0.1 0 0 −8.728×10−2 −8.719×10−2 −2.274×10−1 −2.275×10−1 0 0
1.4 0.1 8 8.8664 −9.110×10−2 −8.736×10−2 −2.355×10−1 −2.273×10−1 4.066×10−2 2.938×10−2
1.4 0.1 16 17.4519 −1.030×10−1 −8.958×10−2 −2.602×10−1 −2.309×10−1 7.428×10−2 5.475×10−2
1.4 0.1 24 25.5784 −1.243×10−1 −9.771×10−2 −3.037×10−1 −2.415×10−1 9.663×10−2 7.316×10−2
1.5 0.1 0 0 −8.069×10−2 −8.095×10−2 −2.255×10−1 −2.260×10−1 0 0
1.5 0.1 8 8.7910 −8.323×10−2 −8.133×10−2 −2.310×10−1 −2.264×10−1 2.996×10−2 2.070×10−2
1.5 0.1 16 17.3490 −9.121×10−2 −8.395×10−2 −2.483×10−1 −2.314×10−1 5.512×10−2 3.888×10−2
1.5 0.1 24 25.5197 −1.059×10−1 −8.980×10−2 −2.792×10−1 −2.423×10−1 7.255×10−2 5.264×10−2
1.6 0.1 0 0 −7.255×10−2 −7.281×10−2 −2.161×10−1 −2.168×10−1 0 0
1.6 0.1 8 8.7195 −7.430×10−2 −7.321×10−2 −2.201×10−1 −2.173×10−1 2.258×10−2 1.502×10−2
1.6 0.1 16 17.2437 −7.986×10−2 −7.533×10−2 −2.323×10−1 −2.212×10−1 4.179×10−2 2.839×10−2
1.6 0.1 24 25.4388 −9.025×10−2 −8.040×10−2 −2.547×10−1 −2.309×10−1 5.554×10−2 3.886×10−2
1.6 0.1 32 33.2683 −1.082×10−1 −9.435×10−2 −2.920×10−1 −2.551×10−1 6.316×10−2 4.559×10−2
1.7 0.1 0 0 −6.427×10−2 −6.440×10−2 −2.036×10−1 −2.040×10−1 0 0
1.7 0.1 8 8.6555 −6.552×10−2 −6.478×10−2 −2.065×10−1 −2.045×10−1 1.742×10−2 1.124×10−2
1.7 0.1 16 17.1454 −6.953×10−2 −6.651×10−2 −2.154×10−1 −2.075×10−1 3.240×10−2 2.134×10−2
1.7 0.1 24 25.3531 −7.707×10−2 −7.052×10−2 −2.317×10−1 −2.150×10−1 4.342×10−2 2.948×10−2
1.7 0.1 32 33.2416 −9.009×10−2 −7.959×10−2 −2.590×10−1 −2.324×10−1 4.998×10−2 3.512×10−2
1.8 0.1 0 0 −5.640×10−2 −5.640×10−2 −1.897×10−1 −1.897×10−1 0 0
1.8 0.1 8 8.5991 −5.732×10−2 −5.676×10−2 −1.918×10−1 −1.902×10−1 1.371×10−2 8.640×10−3
1.8 0.1 16 17.0562 −6.028×10−2 −5.817×10−2 −1.984×10−1 −1.925×10−1 2.562×10−2 1.647×10−2
1.8 0.1 24 25.2693 −6.588×10−2 −6.139×10−2 −2.105×10−1 −1.983×10−1 3.456×10−2 2.291×10−2
1.8 0.1 32 33.2018 −7.555×10−2 −6.849×10−2 −2.307×10−1 −2.120×10−1 4.020×10−2 2.765×10−2
1.9 0.1 0 0 −4.915×10−2 −4.911×10−2 −1.753×10−1 −1.751×10−1 0 0
1.9 0.1 8 8.5494 −4.985×10−2 −4.945×10−2 −1.768×10−1 −1.755×10−1 1.097×10−2 6.791×10−3
1.9 0.1 16 16.9760 −5.208×10−2 −5.064×10−2 −1.817×10−1 −1.774×10−1 2.055×10−2 1.298×10−2
1.9 0.1 24 25.1898 −5.633×10−2 −5.328×10−2 −1.908×10−1 −1.819×10−1 2.788×10−2 1.816×10−2
1.9 0.1 32 33.1555 −6.364×10−2 −5.870×10−2 −2.059×10−1 −1.920×10−1 3.272×10−2 2.214×10−2
2.0 0.1 0 0 −4.263×10−2 −4.264×10−2 −1.607×10−1 −1.608×10−1 0 0
2.0 0.1 8 8.5057 −4.316×10−2 −4.292×10−2 −1.619×10−1 −1.611×10−1 8.862×10−3 5.424×10−3
2.0 0.1 16 16.9042 −4.488×10−2 −4.390×10−2 −1.656×10−1 −1.625×10−1 1.666×10−2 1.039×10−2
2.0 0.1 24 25.1156 −4.815×10−2 −4.604×10−2 −1.724×10−1 −1.660×10−1 2.271×10−2 1.459×10−2
2.0 0.1 32 33.1064 −5.376×10−2 −5.031×10−2 −1.838×10−1 −1.736×10−1 2.684×10−2 1.793×10−2
2.0 0.1 40 40.8954 −6.339×10−2 −6.236×10−2 −2.027×10−1 −1.967×10−1 2.917×10−2 2.036×10−2
Table II: As in Table I but for non-circular orbits; the Teukolsky-based fluxes for E and Lz have an accuracy of 10
−3. Note that our code, as
all the Teukolsky-based code that we are aware of, presently does not have the capability to compute inclination angle evolution for generic
orbits.
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e θinc ι
dθinc
(deg.) (deg.) (kludge) (Teukolsky ) (kludge) (Teukolsky) (kludge) (kludge)
1.4 0.2 0 0 −8.636×10−2 −8.642×10−2 −2.119×10−1 −2.121×10−1 0 0
1.4 0.2 8 8.8215 −9.853×10−2 −8.240×10−2 −2.374×10−1 −2.015×10−1 1.148×10−1 9.714×10−2
1.5 0.2 0 0 −8.362×10−2 −8.349×10−2 −2.236×10−1 −2.230×10−1 0 0
1.5 0.2 8 8.7595 −9.141×10−2 −8.276×10−2 −2.410×10−1 −2.206×10−1 7.893×10−2 6.549×10−2
1.5 0.2 16 17.2957 −1.145×10−1 −8.394×10−2 −2.915×10−1 −2.215×10−1 1.466×10−1 1.230×10−1
1.5 0.2 24 25.4608 −1.524×10−1 −9.230×10−2 −3.712×10−1 −2.357×10−1 1.952×10−1 1.661×10−1
1.6 0.2 0 0 −7.596×10−2 −7.616×10−2 −2.171×10−1 −2.176×10−1 0 0
1.6 0.2 8 8.6935 −8.111×10−2 −7.641×10−2 −2.292×10−1 −2.177×10−1 5.520×10−2 4.502×10−2
1.6 0.2 16 17.1994 −9.649×10−2 −7.798×10−2 −2.647×10−1 −2.198×10−1 1.032×10−1 8.500×10−2
1.6 0.2 24 25.3891 −1.221×10−1 −8.314×10−2 −3.212×10−1 −2.288×10−1 1.388×10−1 1.160×10−1
1.7 0.2 0 0 −6.765×10−2 −6.799×10−2 −2.057×10−1 −2.068×10−1 0 0
1.7 0.2 8 8.6329 −7.116×10−2 −6.813×10−2 −2.144×10−1 −2.066×10−1 3.963×10−2 3.176×10−2
1.7 0.2 16 17.1064 −8.171×10−2 −6.995×10−2 −2.398×10−1 −2.096×10−1 7.441×10−2 6.024×10−2
1.7 0.2 24 25.3085 −9.948×10−2 −7.443×10−2 −2.806×10−1 −2.178×10−1 1.009×10−1 8.290×10−2
1.7 0.2 32 33.2037 −1.257×10−1 −8.558×10−2 −3.371×10−1 −2.366×10−1 1.175×10−1 9.806×10−2
1.8 0.2 0 0 −5.965×10−2 −5.962×10−2 −1.927×10−1 −1.926×10−1 0 0
1.8 0.2 8 8.5789 −6.211×10−2 −5.997×10−2 −1.990×10−1 −1.930×10−1 2.919×10−2 2.300×10−2
1.8 0.2 16 17.0211 −6.953×10−2 −6.147×10−2 −2.175×10−1 −1.954×10−1 5.504×10−2 4.380×10−2
1.8 0.2 24 25.2283 −8.216×10−2 −6.502×10−2 −2.474×10−1 −2.016×10−1 7.515×10−2 6.068×10−2
1.8 0.2 32 33.1656 −1.009×10−1 −7.410×10−2 −2.890×10−1 −2.190×10−1 8.839×10−2 7.258×10−2
1.9 0.2 0 0 −5.218×10−2 −5.210×10−2 −1.786×10−1 −1.783×10−1 0 0
1.9 0.2 8 8.5312 −5.394×10−2 −5.244×10−2 −1.833×10−1 −1.787×10−1 2.197×10−2 1.704×10−2
1.9 0.2 16 16.9441 −5.928×10−2 −5.373×10−2 −1.970×10−1 −1.807×10−1 4.156×10−2 3.254×10−2
1.9 0.2 24 25.1518 −6.843×10−2 −5.669×10−2 −2.192×10−1 −1.858×10−1 5.706×10−2 4.535×10−2
1.9 0.2 32 33.1207 −8.213×10−2 −6.277×10−2 −2.502×10−1 −1.966×10−1 6.767×10−2 5.475×10−2
2.0 0.2 0 0 −4.528×10−2 −4.530×10−2 −1.637×10−1 −1.638×10−1 0 0
2.0 0.2 8 8.4891 −4.657×10−2 −4.557×10−2 −1.671×10−1 −1.641×10−1 1.679×10−2 1.283×10−2
2.0 0.2 16 16.8749 −5.049×10−2 −4.664×10−2 −1.774×10−1 −1.657×10−1 3.184×10−2 2.457×10−2
2.0 0.2 24 25.0802 −5.725×10−2 −4.904×10−2 −1.941×10−1 −1.696×10−1 4.391×10−2 3.440×10−2
2.0 0.2 32 33.0730 −6.743×10−2 −5.427×10−2 −2.175×10−1 −1.793×10−1 5.243×10−2 4.184×10−2
1.5 0.3 0 0 −8.481×10−2 −8.478×10−2 −2.094×10−1 −2.094×10−1 0 0
1.5 0.3 8 8.7037 −1.006×10−1 −7.824×10−2 −2.442×10−1 −1.934×10−1 1.484×10−1 1.301×10−1
1.5 0.3 16 17.2003 −1.469×10−1 −7.811×10−2 −3.435×10−1 −1.864×10−1 2.766×10−1 2.440×10−1
1.6 0.3 0 0 −8.144×10−2 −8.123×10−2 −2.183×10−1 −2.178×10−1 0 0
1.6 0.3 8 8.6498 −9.182×10−2 −7.807×10−2 −2.426×10−1 −2.095×10−1 1.028×10−1 8.918×10−2
1.6 0.3 16 17.1246 −1.223×10−1 −8.089×10−2 −3.122×10−1 −2.144×10−1 1.928×10−1 1.683×10−1
1.6 0.3 24 25.3046 −1.716×10−1 −8.666×10−2 −4.197×10−1 −2.229×10−1 2.607×10−1 2.295×10−1
1.7 0.3 0 0 −7.362×10−2 −7.314×10−2 −2.104×10−1 −2.095×10−1 0 0
1.7 0.3 8 8.5953 −8.060×10−2 −7.224×10−2 −2.277×10−1 −2.065×10−1 7.240×10−2 6.224×10−2
1.7 0.3 16 17.0415 −1.013×10−1 −7.369×10−2 −2.774×10−1 −2.084×10−1 1.365×10−1 1.180×10−1
1.7 0.3 24 25.2339 −1.349×10−1 −7.800×10−2 −3.547×10−1 −2.153×10−1 1.861×10−1 1.622×10−1
1.8 0.3 0 0 −6.488×10−2 −6.484×10−2 −1.973×10−1 −1.972×10−1 0 0
1.8 0.3 8 8.5454 −6.970×10−2 −6.480×10−2 −2.099×10−1 −1.966×10−1 5.206×10−2 4.436×10−2
1.8 0.3 16 16.9628 −8.402×10−2 −6.671×10−2 −2.461×10−1 −1.998×10−1 9.857×10−2 8.445×10−2
1.8 0.3 24 25.1601 −1.075×10−1 −7.030×10−2 −3.026×10−1 −2.056×10−1 1.353×10−1 1.169×10−1
1.8 0.3 32 33.1047 −1.404×10−1 −8.153×10−2 −3.762×10−1 −2.255×10−1 1.600×10−1 1.394×10−1
1.9 0.3 0 0 −5.669×10−2 −5.690×10−2 −1.829×10−1 −1.832×10−1 0 0
1.9 0.3 8 8.5010 −6.010×10−2 −5.683×10−2 −1.922×10−1 −1.824×10−1 3.823×10−2 3.229×10−2
1.9 0.3 16 16.8911 −7.025×10−2 −5.818×10−2 −2.189×10−1 −1.844×10−1 7.263×10−2 6.165×10−2
1.9 0.3 24 25.0887 −8.701×10−2 −6.054×10−2 −2.609×10−1 −1.874×10−1 1.003×10−1 8.579×10−2
1.9 0.3 32 33.0624 −1.106×10−1 −6.912×10−2 −3.157×10−1 −2.034×10−1 1.195×10−1 1.032×10−1
2.0 0.3 0 0 −4.953×10−2 −4.946×10−2 −1.683×10−1 −1.683×10−1 0 0
2.0 0.3 8 8.4616 −5.199×10−2 −4.970×10−2 −1.753×10−1 −1.685×10−1 2.862×10−2 2.395×10−2
2.0 0.3 16 16.8262 −5.932×10−2 −5.079×10−2 −1.954×10−1 −1.699×10−1 5.452×10−2 4.585×10−2
2.0 0.3 24 25.0215 −7.150×10−2 −5.328×10−2 −2.269×10−1 −1.737×10−1 7.564×10−2 6.411×10−2
2.0 0.3 32 33.0172 −8.878×10−2 −6.003×10−2 −2.682×10−1 −1.864×10−1 9.077×10−2 7.771×10−2
Table III: As in Table II, but for additional values of eccentricity e; the Teukolsky-based fluxes for E and Lz have an accuracy of 10
e θinc ι
dθinc
(deg.) (deg.) (kludge) (Teukolsky ) (kludge) (Teukolsky) (kludge) (kludge)
1.6 0.4 0 0 −7.766×10−2 −7.772×10−2 −1.918×10−1 −1.919×10−1 0 0
1.6 0.4 8 8.5863 −9.433×10−2 −7.645×10−2 −2.297×10−1 −1.881×10−1 1.528×10−1 1.370×10−1
1.6 0.4 16 17.0151 −1.432×10−1 −7.651×10−2 −3.382×10−1 −1.837×10−1 2.873×10−1 2.584×10−1
1.7 0.4 0 0 −7.882×10−2 −7.953×10−2 −2.097×10−1 −2.115×10−1 0 0
1.7 0.4 8 8.5426 −9.002×10−2 −7.408×10−2 −2.367×10−1 −1.978×10−1 1.087×10−1 9.656×10−2
1.7 0.4 16 16.9502 −1.229×10−1 −7.682×10−2 −3.143×10−1 −2.025×10−1 2.054×10−1 1.830×10−1
1.7 0.4 24 25.1282 −1.760×10−1 −8.090×10−2 −4.336×10−1 −2.075×10−1 2.809×10−1 2.514×10−1
1.8 0.4 0 0 −7.107×10−2 −7.007×10−2 −2.013×10−1 −1.988×10−1 0 0
1.8 0.4 8 8.4989 −7.877×10−2 −7.001×10−2 −2.209×10−1 −1.981×10−1 7.788×10−2 6.879×10−2
1.8 0.4 16 16.8817 −1.015×10−1 −7.009×10−2 −2.774×10−1 −1.965×10−1 1.478×10−1 1.309×10−1
1.8 0.4 24 25.0646 −1.383×10−1 −7.314×10−2 −3.646×10−1 −2.003×10−1 2.036×10−1 1.810×10−1
1.8 0.4 32 33.0184 −1.887×10−1 −9.193×10−2 −4.755×10−1 −2.319×10−1 2.414×10−1 2.156×10−1
1.9 0.4 0 0 −6.187×10−2 −6.267×10−2 −1.861×10−1 −1.881×10−1 0 0
1.9 0.4 8 8.4591 −6.728×10−2 −6.216×10−2 −2.006×10−1 −1.861×10−1 5.666×10−2 4.980×10−2
1.9 0.4 16 16.8173 −8.328×10−2 −6.222×10−2 −2.424×10−1 −1.844×10−1 1.079×10−1 9.506×10−2
1.9 0.4 24 25.0006 −1.094×10−1 −6.486×10−2 −3.071×10−1 −1.878×10−1 1.495×10−1 1.322×10−1
1.9 0.4 32 32.9804 −1.452×10−1 −7.884×10−2 −3.896×10−1 −2.158×10−1 1.787×10−1 1.588×10−1
2.0 0.4 0 0 −5.483×10−2 −5.457×10−2 −1.735×10−1 −1.729×10−1 0 0
2.0 0.4 8 8.4235 −5.871×10−2 −5.445×10−2 −1.844×10−1 −1.720×10−1 4.222×10−2 3.686×10−2
2.0 0.4 16 16.7586 −7.020×10−2 −5.555×10−2 −2.158×10−1 −1.733×10−1 8.064×10−2 7.057×10−2
2.0 0.4 24 24.9396 −8.902×10−2 −5.844×10−2 −2.645×10−1 −1.778×10−1 1.122×10−1 9.860×10−2
2.0 0.4 32 32.9389 −1.150×10−1 −6.536×10−2 −3.267×10−1 −1.896×10−1 1.351×10−1 1.193×10−1
1.7 0.5 0 0 −7.421×10−2 −7.401×10−2 −1.815×10−1 −1.810×10−1 0 0
1.7 0.5 8 8.4736 −8.957×10−2 −7.168×10−2 −2.173×10−1 −1.750×10−1 1.379×10−1 1.256×10−1
1.7 0.5 16 16.8300 −1.347×10−1 −6.999×10−2 −3.201×10−1 −1.676×10−1 2.611×10−1 2.378×10−1
1.8 0.5 0 0 −7.589×10−2 −7.620×10−2 −1.993×10−1 −2.000×10−1 0 0
1.8 0.5 8 8.4395 −8.644×10−2 −6.929×10−2 −2.254×10−1 −1.829×10−1 1.005×10−1 9.076×10−2
1.8 0.5 16 16.7776 −1.175×10−1 −7.210×10−2 −3.004×10−1 −1.880×10−1 1.911×10−1 1.726×10−1
1.8 0.5 24 24.9413 −1.678×10−1 −7.395×10−2 −4.158×10−1 −1.881×10−1 2.638×10−1 2.385×10−1
1.9 0.5 0 0 −6.646×10−2 −6.620×10−2 −1.855×10−1 −1.849×10−1 0 0
1.9 0.5 8 8.4059 −7.386×10−2 −6.320×10−2 −2.048×10−1 −1.768×10−1 7.312×10−2 6.579×10−2
1.9 0.5 16 16.7233 −9.572×10−2 −6.551×10−2 −2.603×10−1 −1.809×10−1 1.395×10−1 1.255×10−1
1.9 0.5 24 24.8877 −1.312×10−1 −7.087×10−2 −3.461×10−1 −1.909×10−1 1.937×10−1 1.744×10−1
1.9 0.5 32 32.8741 −1.795×10−1 −8.247×10−2 −4.544×10−1 −2.091×10−1 2.320×10−1 2.092×10−1
2.0 0.5 0 0 −5.987×10−2 −5.995×10−2 −1.761×10−1 −1.763×10−1 0 0
2.0 0.5 8 8.3750 −6.516×10−2 −5.918×10−2 −1.906×10−1 −1.738×10−1 5.456×10−2 4.882×10−2
2.0 0.5 16 16.6725 −8.081×10−2 −5.817×10−2 −2.324×10−1 −1.694×10−1 1.044×10−1 9.343×10−2
2.0 0.5 24 24.8347 −1.063×10−1 −6.254×10−2 −2.970×10−1 −1.776×10−1 1.456×10−1 1.304×10−1
2.0 0.5 32 32.8378 −1.412×10−1 −6.993×10−2 −3.787×10−1 −1.893×10−1 1.756×10−1 1.576×10−1
Table IV: As in Tables II and III, but for different values of eccentricity e; the Teukolsky-based fluxes for E and Lz have an accuracy of 10
e θinc ι ∆t/M ∆θinc ∆ι
(deg.) (deg.) (deg.) (deg.)
0 0 0 1.250×106 0 0
0 5 5.355510 1.217×106 1.949×10−1 4.954×10−1
0 10 10.679331 1.118×106 3.468×10−1 8.631×10−1
0 15 15.943192 9.574×105 4.236×10−1 1.019
0 20 21.125167 7.446×105 4.109×10−1 9.440×10−1
0 25 26.211779 4.981×105 3.158×10−1 6.860×10−1
0 30 31.199048 2.528×105 1.732×10−1 3.527×10−1
0 35 36.092514 6.584×104 4.636×10−2 8.806×10−2
0.1 0 0 1.228×106 0 0
0.1 5 5.351602 1.198×106 4.517×10−1 7.766×10−1
0.1 10 10.671900 1.103×106 6.900×10−1 1.236
0.1 15 15.932962 9.426×105 7.283×10−1 1.344
0.1 20 21.113129 7.315×105 6.433×10−1 1.187
0.1 25 26.199088 4.900×105 4.780×10−1 8.547×10−1
0.1 30 31.186915 2.513×105 2.730×10−1 4.585×10−1
0.1 35 36.082095 6.589×104 8.385×10−2 1.279×10−1
0.2 0 0 1.173×106 0 0
0.2 5 5.339916 1.150×106 1.204 1.598
0.2 10 10.649670 1.064×106 1.698 2.331
0.2 15 15.902348 9.043×105 1.618 2.293
0.2 20 21.077081 6.980×105 1.324 1.900
0.2 25 26.161046 4.693×105 9.545×10−1 1.351
0.2 30 31.150481 2.486×105 5.674×10−1 7.711×10−1
0.2 35 36.050712 7.562×104 2.070×10−1 2.648×10−1
0.3 0 0 1.087×106 0 0
0.3 5 5.320559 1.069×106 2.307 2.788
0.3 10 10.612831 1.001×106 3.256 4.007
0.3 15 15.851572 8.454×105 2.984 3.741
0.3 20 21.017212 6.483×105 2.375 2.998
0.3 25 26.097732 4.408×105 1.700 2.129
0.3 30 31.089639 2.493×105 1.040 1.276
0.3 35 35.997987 1.108×105 4.626×10−1 5.569×10−1
Table V: Variation in the inclination angles ι and θinc as well as time needed to reach the separatrix for several inspirals through the nearly
horizon-skimming regime. In all of these cases, the binary’s mass ratio was fixed to µ/M = 10−6, the large black hole’s spin was fixed to
a = 0.998M , and the orbits were begun at p = 1.9M . The time interval ∆t is the total accumulated time it takes for the inspiralling body
to reach the separatrix (at which time it rapidly plunges into the black hole). The angles ∆θinc and ∆ι are the total integrated change in these
inclination angles that we compute. For the e = 0 cases, inspirals are computed using fits to the circular-Teukolsky fluxes of E and Lz ; for
eccentric orbits we use the kludge fluxes (40), (43) and (44). Notice that ∆θinc and ∆ι are always positive — the inclination angle always
increases during the inspiral through the nearly horizon-skimming region. The magnitude of this increase never exceeds a few degrees.

  We have performed a detailed analysis of orbital motion in the vicinity of a
nearly extremal Kerr black hole. For very rapidly rotating black holes (spin
a=J/M>0.9524M) we have found a class of very strong field eccentric orbits
whose angular momentum L_z increases with the orbit's inclination with respect
to the equatorial plane, while keeping latus rectum and eccentricity fixed.
This behavior is in contrast with Newtonian intuition, and is in fact opposite
to the "normal" behavior of black hole orbits. Such behavior was noted
previously for circular orbits; since it only applies to orbits very close to
the black hole, they were named "nearly horizon-skimming orbits". Our analysis
generalizes this result, mapping out the full generic (inclined and eccentric)
family of nearly horizon-skimming orbits. The earlier work on circular orbits
reported that, under gravitational radiation emission, nearly horizon-skimming
orbits tend to evolve to smaller orbit inclination, toward prograde equatorial
configuration. Normal orbits, by contrast, always demonstrate slowly growing
orbit inclination (orbits evolve toward the retrograde equatorial
configuration). Using up-to-date Teukolsky-fluxes, we have concluded that the
earlier result was incorrect: all circular orbits, including nearly
horizon-skimming ones, exhibit growing orbit inclination. Using kludge fluxes
based on a Post-Newtonian expansion corrected with fits to circular and to
equatorial Teukolsky-fluxes, we argue that the inclination grows also for
eccentric nearly horizon-skimming orbits. We also find that the inclination
change is, in any case, very small. As such, we conclude that these orbits are
not likely to have a clear and peculiar imprint on the gravitational waveforms
expected to be measured by the space-based detector LISA.

Circular and non-circular nearly horizon-skimming orbits in Kerr spacetimes
Enrico Barausse∗
SISSA, International School for Advanced Studies and INFN, Via Beirut 2, 34014 Trieste, Italy
Scott A. Hughes
Department of Physics and MIT Kavli Institute, MIT, 77 Massachusetts Ave., Cambridge, MA 02139 USA
Luciano Rezzolla
Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, 14476 Potsdam, Germany and
Department of Physics, Louisiana State University, Baton Rouge, LA 70803 USA
(Dated: November 1, 2018)
We have performed a detailed analysis of orbital motion in the vicinity of a nearly extremal Kerr black hole.
For very rapidly rotating black holes — spin parameter a ≡ J/M > 0.9524M — we have found a class of
very strong field eccentric orbits whose orbital angular momentum Lz increases with the orbit’s inclination with
respect to the equatorial plane, while keeping latus rectum and eccentricity fixed. This behavior is in contrast
with Newtonian intuition, and is in fact opposite to the “normal” behavior of black hole orbits. Such behavior
was noted previously for circular orbits; since it only applies to orbits very close to the black hole, they were
named “nearly horizon-skimming orbits”. Our current analysis generalizes this result, mapping out the full
generic (inclined and eccentric) family of nearly horizon-skimming orbits. The earlier work on circular orbits
reported that, under gravitational radiation emission, nearly horizon-skimming orbits exhibit unusual inspiral,
tending to evolve to smaller orbit inclination, toward prograde equatorial configuration. Normal orbits, by
contrast, always demonstrate slowly growing orbit inclination — orbits evolve toward the retrograde equatorial
configuration. Using up-to-date Teukolsky-based fluxes, we have concluded that the earlier result was incorrect
— all circular orbits, including nearly horizon-skimming ones, exhibit growing orbit inclination under radiative
backreaction. Using kludge fluxes based on a Post-Newtonian expansion corrected with fits to circular and
to equatorial Teukolsky-based fluxes, we argue that the inclination grows also for eccentric nearly horizon-
skimming orbits. We also find that the inclination change is, in any case, very small. As such, we conclude that
these orbits are not likely to have a clear and peculiar imprint on the gravitational waveforms expected to be
measured by the space-based detector LISA.
PACS numbers: 04.30.-w
I. INTRODUCTION
The space-based gravitational-wave detector LISA [1] will
be a unique tool to probe the nature of supermassive black
holes (SMBHs), making it possible to map in detail their
spacetimes. This goal is expected to be achieved by observing
gravitational waves emitted by compact stars or black holes
with masses µ ≈ 1 − 100M⊙ spiraling into the SMBHs
which reside in the center of galaxies [2] (particularly the low
end of the galactic center black hole mass function, M ≈
105− 107M⊙). Such events are known as extreme mass ratio
inspirals (EMRIs). Current wisdom suggests that several tens
to perhaps of order a thousand such events could be measured
per year by LISA [3].
Though the distribution of spins for observed astrophysi-
cal black holes is not very well understood at present, very
rapid spin is certainly plausible, as accretion tends to spin-
up SMBHs [4]. Most models for quasi-periodic oscillations
(QPOs) suggest this is indeed the case in all low-mass x-ray
binaries for which data is available [5]. On the other hand,
continuum spectral fitting of some high-mass x-ray binaries
indicates that modest spins (spin parameter a/M ≡ J/M2 ∼
∗Electronic address: barausse@sissa.it
0.6− 0.8) are likewise plausible [6]. The continuum-fit tech-
nique does find an extremely high spin of a/M & 0.98 for the
galactic “microquasar” GRS1915+105 [7]. This argues for a
wide variety of possible spins, depending on the detailed birth
and growth history of a given black hole.
In the mass range corresponding to black holes in galactic
centers, measurements of the broad iron Kα emission line in
active galactic nuclei suggest that SMBHs can be very rapidly
rotating (see Ref. [8] for a recent review). For instance, in
the case of MCG-6-30-15, for which highly accurate observa-
tions are available, a has been found to be larger than 0.987M
at 90% confidence [9]. Because gravitational waves from EM-
RIs are expected to yield a very precise determination of the
spins of SMBHs [10], it is interesting to investigate whether
EMRIs around very rapidly rotating black holes may possess
peculiar features which would be observable by LISA. Should
such features exist, they would provide unambiguous infor-
mation on the spin of SMBHs and thus on the mechanisms
leading to their formation [11].
For extremal Kerr black holes (a = M ), the existence of
a special class of “circular” orbits was pointed out long ago
by Wilkins [12], who named them “horizon-skimming” or-
bits. (“Circular” here means that the orbits are of constant
Boyer-Lindquist coordinate radius r.) These orbits have vary-
ing inclination angle with respect to the equatorial plane and
have the same coordinate radius as the horizon, r = M . De-
http://arxiv.org/abs/0704.0138v2
mailto:barausse@sissa.it
spite this seemingly hazardous location, it can be shown that
all these r = M orbits have finite separation from one another
and from the event horizon [13]. Their somewhat pathological
description is due to a singularity in the Boyer-Lindquist co-
ordinates, which collapses a finite span of the spacetime into
r = M .
Besides being circular and “horizon-skimming”, these or-
bits also show peculiar behavior in their relation of angular
momentum to inclination. In Newtonian gravity, a generic or-
bit has Lz = |L| cos ι, where ι is the inclination angle relative
to the equatorial plane (going from ι = 0 for equatorial pro-
grade orbits to ι = π for equatorial retrograde orbits, passing
through ι = π/2 for polar orbits), and L is the orbital angular
momentum vector. As a result, ∂Lz(r, ι)/∂ι < 0, and the an-
gular momentum in the z-direction always decreases with in-
creasing inclination if the orbit’s radius is kept constant. This
intuitively reasonable decrease of Lz with ι is seen for almost
all black hole orbits as well. Horizon-skimming orbits, by
contrast, exhibit exactly the opposite behavior: Lz increases
with inclination angle.
Reference [14] asked whether the behavior ∂Lz/∂ι > 0
could be extended to a broader class of circular orbits than just
those at the radius r = M for the spin value a = M . It was
found that this condition is indeed more general, and extended
over a range of radius from the “innermost stable circular or-
bit” to r ≃ 1.8M for black holes with a > 0.9524M . Or-
bits that show this property have been named “nearly horizon-
skimming”. The Newtonian behavior ∂Lz(r, ι)/∂ι < 0 is
recovered for all orbits at r & 1.8M [14].
A qualitative understanding of this behavior comes from
recalling that very close to the black hole all physical pro-
cesses become “locked” to the hole’s event horizon [15], with
the orbital motion of point particles coupling to the horizon’s
spin. This locking dominates the “Keplerian” tendency of an
orbit to move more quickly at smaller radii, forcing an or-
biting particle to slow down in the innermost orbits. Lock-
ing is particularly strong for the most-bound (equatorial) or-
bits; the least-bound orbits (which have the largest inclination)
do not strongly lock to the black hole’s spin until they have
very nearly reached the innermost orbit [14]. The property
∂Lz(r, ι)/∂ι > 0 reflects the different efficiency of nearly
horizon-skimming orbits to lock with the horizon.
Reference [14] argued that this behavior could have ob-
servational consequences. It is well-known that the incli-
nation angle of an inspiraling body generally increases due
to gravitational-wave emission [16, 17]. Since dLz/dt <
0 because of the positive angular momentum carried away
by the gravitational waves, and since “normal” orbits have
∂Lz/∂ι < 0, one would expect dι/dt > 0. However, if
during an evolution ∂Lz/∂ι switches sign, then dι/dt might
switch sign as well: An inspiraling body could evolve towards
an equatorial orbit, signalling the presence of an “almost-
extremal” Kerr black hole.
It should be emphasized that this argument is not rigorous.
In particular, one needs to consider the joint evolution of or-
bital radius and inclination angle; and, one must include the
dependence of these two quantities on orbital energy as well
as angular momentum1. As such, dι/dt depends not only on
dLz/dt and ∂Lz/∂ι, but also on dE/dt, ∂E/∂ι, ∂E/∂r and
∂Lz/∂r.
In this sense, the argument made in Ref. [14] should be
seen as claiming that the contribution coming from dLz/dt
and ∂Lz/∂ι are simply the dominant ones. Using the nu-
merical code described in [17] to compute the fluxes dLz/dt
and dE/dt, it was then found that a test-particle on a circu-
lar orbit passing through the nearly horizon-skimming region
of a Kerr black hole with a = 0.998M (the value at which
a hole’s spin tends to be buffered due to photon capture from
thin disk accretion [19]) had its inclination angle decreased
by δι ≈ 1◦ − 2◦ [14] in the adiabatic approximation [20].
It should be noted at this point that the rate of change of in-
clination angle, dι/dt, appears as the difference of two rel-
atively small and expensive to compute rates of change [cf.
Eq. (3.8) of Ref. [17]]. As such, small relative errors in those
rates of change can lead to large relative errors in dι/dt. Fi-
nally, in Ref. [14] it was speculated that the decrease could
be even larger for eccentric orbits satisfying the condition
∂Lz/∂ι > 0, possibly leading to an observable imprint on
EMRI gravitational waveforms.
The main purpose of this paper is to extend Ref. [14]’s anal-
ysis of nearly horizon-skimming orbits to include the effect
of orbital eccentricity, and to thereby test the speculation that
there may be an observable imprint on EMRI waveforms of
nearly horizon-skimming behavior. In doing so, we have re-
visited all the calculations of Ref. [14] using a more accurate
Teukolsky solver which serves as the engine for the analysis
presented in Ref. [21].
We have found that the critical spin value for circular
nearly horizon-skimming orbits, a > 0.9524M , also delin-
eates a family of eccentric orbits for which the condition
∂Lz(p, e, ι)/∂ι > 0 holds. (More precisely, we consider vari-
ation with respect to an angle θinc that is easier to work with
in the extreme strong field, but that is easily related to ι.) The
parameters p and e are the orbit’s latus rectum and eccentric-
ity, defined precisely in Sec. II. These generic nearly horizon-
skimming orbits all have p . 2M , deep in the black hole’s
extreme strong field.
We next study the evolution of these orbits under
gravitational-wave emission in the adiabatic approximation.
We first revisited the evolution of circular, nearly horizon-
skimming orbits using the improved Teukolsky solver which
was used for the analysis of Ref. [21]. The results of this anal-
ysis were somewhat surprising: Just as for “normal” orbits,
we found that orbital inclination always increases during in-
spiral, even in the nearly horizon-skimming regime. This is in
stark contrast to the claims of Ref. [14]. As noted above, the
inclination’s rate of change depends on the difference of two
expensive and difficult to compute numbers, and thus can be
strongly impacted by small relative errors in those numbers.
1 In the general case, one must also include the dependence on “Carter’s
constant” Q [18], the third integral of black hole orbits (described more
carefully in Sec. II). For circular orbits, Q = Q(E,Lz): knowledge of E
and Lz completely determines Q.
A primary result of this paper is thus to retract the claim of
Ref. [14] that an important dynamic signature of the nearly
horizon-skimming region is a reversal in the sign of incli-
nation angle evolution: The inclination always grows under
gravitational radiation emission.
We next extended this analysis to study the evolution of
generic nearly horizon-skimming orbits. The Teukolsky code
to which we have direct access can, at this point, only com-
pute the radiated fluxes of energy E and angular momentum
Lz; results for the evolution of the Carter constant Q are just
now beginning to be understood [22], and have not yet been
incorporated into this code. We instead use “kludge” expres-
sions for dE/dt, dLz/dt, and dQ/dt which were inspired by
Refs. [23, 24]. These expression are based on post-Newtonian
flux formulas, modified in such a way that they fit strong-field
radiation reaction results obtained from a Teukolsky integra-
tor; see Ref. [24] for further discussion. Our analysis indicates
that, just as in the circular limit, the result dι/dt > 0 holds
for generic nearly horizon-skimming orbits. Furthermore, and
contrary to the speculation of Ref. [14], we do not find a large
amplification of dι/dt as orbits are made more eccentric.
Our conclusion is that the nearly horizon-skimming regime,
though an interesting curiosity of strong-field orbits of nearly
extremal black holes, will not imprint any peculiar observa-
tional signature on EMRI waveforms.
The remainder of this paper is organized as follows. In Sec.
II, we review the properties of bound stable orbits in Kerr, pro-
viding expressions for the constants of motion which we will
use in Sec. III to generalize nearly horizon-skimming orbits to
the non-circular case. In Sec. IV, we study the evolution of the
inclination angle for circular nearly horizon-skimming orbits
using Teukolsky-based fluxes; in Sec. V we do the same for
non-circular orbits and using kludge fluxes. We present and
discuss our detailed conclusions in Sec. VI. The fits and post-
Newtonian fluxes used for the kludge fluxes are presented in
the Appendix. Throughout the paper we have used units in
which G = c = 1.
II. BOUND STABLE ORBITS IN KERR SPACETIMES
The line element of a Kerr spacetime, written in Boyer-
Lindquist coordinates reads [25]
ds2 = −
1− 2Mr
dt2 +
dr2 +Σ dθ2
r2 + a2 +
2Ma2r
sin2 θ
sin2 θ dφ2
−4Mar
sin2 θ dt dφ, (1)
where
Σ ≡ r2 + a2 cos2 θ, ∆ ≡ r2 − 2Mr + a2. (2)
Up to initial conditions, geodesics can then be labelled by four
constants of motion: the mass µ of the test particle, its energy
E and angular momentum Lz as measured by an observer at
infinity and the Carter constant Q [18]. The presence of these
four conserved quantities makes the geodesic equations sepa-
rable in Boyer-Lindquist coordinates. Introducing the Carter
time λ, defined by
≡ Σ , (3)
the geodesic equations become
= Vr(r), µ
= Vt(r, θ),
= Vθ(θ), µ
= Vφ(r, θ) , (4)
Vt(r, θ) ≡ E
− a2 sin2 θ
+ aLz
Vr(r) ≡
E̟2 − aLz
)2 −∆
µ2r2 + (Lz − aE)2 +Q
Vθ(θ) ≡ Q− L2z cot2 θ − a2(µ2 − E2) cos2 θ, (5c)
Vφ(r, θ) ≡ Lz csc2 θ + aE
, (5d)
where we have defined
̟2 ≡ r2 + a2 . (6)
The conserved parameters E, Lz , and Q can be remapped
to other parameters that describe the geometry of the orbit. We
have found it useful to describe the orbit in terms of an angle
θmin — the minimum polar angle reached by the orbit — as
well as the latus rectum p and the eccentricity e. In the weak-
field limit, p and e correspond exactly to the latus rectum and
eccentricity used to describe orbits in Newtonian gravity; in
the strong field, they are essentially just a convenient remap-
ping of the orbit’s apoastron and periastron:
rap ≡
, rperi ≡
1 + e
. (7)
Finally, in much of our analysis, it is useful to refer to
z− ≡ cos2 θmin , (8)
rather than to θmin directly.
To map (E,Lz, Q) to (p, e, z−), use Eq. (4) to impose
dr/dλ = 0 at r = rap and r = rperi, and to impose
dθ/dλ = 0 at θ = θmin. (Note that for a circular orbit,
rap = rperi = r0. In this case, one must apply the rules
dr/dλ = 0 and d2r/dλ2 = 0 at r = r0.) Following this ap-
proach, Schmidt [26] was able to derive explicit expressions
for E, Lz and Q in terms of p, e and z−. We now briefly
review Schmidt’s results.
Let us first introduce the dimensionless quantities
Ẽ ≡ E/µ , L̃z ≡ Lz/(µM) , Q̃ ≡ Q/(µM)2 , (9)
ã ≡ a/M , r̃ ≡ r/M , ∆̃ ≡ ∆/M2 , (10)
Figure 1: Left panel: Inclination angles θinc for which bound stable orbits exist for a black hole with spin a = 0.998M . The allowed range
for θinc goes from θinc = 0 to the curve corresponding to the eccentricity under consideration, θinc = θ
inc . Right panel: Same as left but for
an extremal black hole, a = M . Note that in this case θmaxinc never reaches zero.
and the functions
f(r̃) ≡ r̃4 + ã2
r̃(r̃ + 2) + z−∆̃
, (11)
g(r̃) ≡ 2 ã r̃ , (12)
h(r̃) ≡ r̃(r̃ − 2) +
1− z−
∆̃ , (13)
d(r̃) ≡ (r̃2 + ã2z−)∆̃ . (14)
Let us further define the set of functions
(f1, g1, h1, d1) ≡{
(f(r̃p), g(r̃p), h(r̃p), d(r̃p)) if e > 0 ,
(f(r̃0), g(r̃0), h(r̃0), d(r̃0)) if e = 0 ,
(f2, g2, h2, d2) ≡{
(f(r̃a), g(r̃a), h(r̃a), d(r̃a)) if e > 0 ,
(f ′(r̃0), g
′(r̃0), h
′(r̃0), d
′(r̃0)) if e = 0 ,
and the determinants
κ ≡ d1h2 − d2h1 , (17)
ε ≡ d1g2 − d2g1 , (18)
ρ ≡ f1h2 − f2h1 , (19)
η ≡ f1g2 − f2g1 , (20)
σ ≡ g1h2 − g2h1 . (21)
The energy of the particle can then be written
κρ+ 2ǫσ − 2D
σ(σǫ2 + ρǫκ− ηκ2)
ρ2 + 4ησ
. (22)
The parameter D takes the values ±1. The angular momen-
tum is a solution of the system
2 − 2g1ẼL̃z − h1L̃2z − d1 = 0 , (23)
2 − 2g2ẼL̃z − h2L̃2z − d2 = 0 . (24)
By eliminating the L̃2z terms in these equations, one finds the
solution
L̃z =
ρẼ2 − κ
for the angular momentum. Using dθ/dλ = 0 at θ = θmin,
the Carter constant can be written
Q̃ = z−
ã2(1− Ẽ2) +
1− z−
. (26)
Additional constraints on p, e, z− are needed for the orbits
to be stable. Inspection of Eq. (4) shows that an eccentric orbit
is stable only if
(rperi) > 0 . (27)
It is marginally stable if ∂Vr/∂r = 0 at r = rperi. Similarly,
the stability condition for circular orbits is
(r0) < 0 ; (28)
marginally stable orbits are set by ∂2Vr/∂r
2 = 0 at r = r0.
Finally, we note that one can massage the above solutions
for the conserved orbital quantities of bound stable orbits to
rewrite the solution for L̃z as
L̃z = −
g21Ẽ
2 + (f1Ẽ2 − d1)h1 . (29)
From this solution, we see that it is quite natural to refer to
orbits with D = 1 as prograde and to orbits with D = −1
as retrograde. Note also that Eq. (29) is a more useful form
than the corresponding expression, Eq. (A4), of Ref. [21]. In
that expression, the factor 1/h1 has been squared and moved
inside the square root. This obscures the fact that h1 changes
sign for very strong field orbits. Differences between Eq. (29)
and Eq. (A4) of [21] are apparent for a & 0.835, although
only for orbits close to the separatrix (i.e., the surface in the
space of parameters (p, e, ι) where marginally stable bound
orbits lie).
III. NON-CIRCULAR NEARLY HORIZON-SKIMMING
ORBITS
With explicit expressions for E, Lz and Q as functions of
p, e and z−, we now examine how to generalize the condi-
tion ∂Lz(r, ι)/∂ι > 0, which defined circular nearly horizon-
skimming orbits in Ref. [14], to encompass the non-circular
case. We recall that the inclination angle ι is defined as [14]
cos ι =
Q+ L2z
. (30)
Such a definition is not always easy to handle in the case of
eccentric orbits. In addition, ι does not have an obvious phys-
ical interpretation (even in the circular limit), but rather was
introduced essentially to generalize (at least formally) the def-
inition of inclination for Schwarzschild black hole orbits. In
that case, one has Q = L2x+L
y and therefore Lz = |L| cos ι.
A more useful definition for the inclination angle in Kerr
was introduced in Ref. [21]:
θinc =
−D θmin , (31)
where θmin is the minimum reached by θ during the orbital
motion. This angle is trivially related to z− (z− = sin
2 θinc)
and ranges from 0 to π/2 for prograde orbits and from π/2 to
π for retrograde orbits. It is a simple numerical calculation to
convert between ι and θinc; doing so shows that the differences
between ι and θinc are very small, with the two coinciding for
a = 0, and with a difference that is less than 2.6◦ for a = M
and circular orbits with r = M .
Bearing all this in mind, the condition we have adopted to
generalize nearly horizon-skimming orbits is
∂Lz(p, e, θinc)
∂θinc
> 0 . (32)
We have found that certain parts of this calculation, particular
the analysis of strong-field geodesic orbits, are best done us-
ing the angle θinc; other parts are more simply done using the
angle ι, particularly the “kludge” computation of fluxes de-
scribed in Sec. V. (This is because the kludge fluxes are based
on an extension of post-Newtonian formulas to the strong-
field regime, and these formulas use ι for inclination angle.)
Accordingly, we often switch back and forth between these
two notions of inclination, and in fact present our final results
for inclination evolution using both dι/dt and dθinc/dt.
Before mapping out the region corresponding to nearly
horizon-skimming orbits, it is useful to examine stable or-
bits more generally in the strong field of rapidly rotating black
holes. We first fix a value for a, and then discretize the space
of parameters (p, e, θinc). We next identify the points in this
space corresponding to bound stable geodesic orbits. Suffi-
ciently close to the horizon, the bound stable orbits with spec-
ified values of p and e have an inclination angle θinc ranging
from 0 (equatorial orbit) to a maximum value θmaxinc . For given
p and e, θmaxinc defines the separatrix between stable and unsta-
ble orbits.
Example separatrices are shown in Fig. 1 for a = 0.998M
and a = M . This figure shows the behavior of θmaxinc as a
function of the latus rectum for the different values of the ec-
centricity indicated by the labels. Note that for a = 0.998M
the angle θmaxinc eventually goes to zero. This is the general
behavior for a < M . On the other hand, for an extremal black
hole, a = M , θmaxinc never goes to zero. The orbits which re-
side at r = M (the circular limit) are the “horizon-skimming
orbits” identified by Wilkins [12]; the a = M separatrix has
a similar shape even for eccentric orbits. As expected, we
find that for given latus rectum and eccentricity the orbit with
θinc = 0 is the one with the lowest energy E (and hence is the
most-bound orbit), whereas the orbit with θinc = θ
inc has the
highest E (and is least bound).
Having mapped out stable orbits in (p, e, θinc) space, we
then computed the partial derivative ∂Lz(p, e, θinc)/∂θinc and
identified the following three overlapping regions:
• Region A: The portion of the (p, e) plane for which
∂Lz(p, e, θinc)/∂θinc > 0 for 0 ≤ θinc ≤ θmaxinc . This
region is illustrated in Fig. 2 as the area under the heavy
solid line and to the left of the dot-dashed line (green in
the color version).
• Region B: The portion of the (p, e) plane
for which (Lz)most bound(p, e) is smaller than
(Lz)least bound(p, e). In other words,
Lz(p, e, 0) < Lz(p, e, θ
inc ) (33)
in Region B. Note that Region B contains Region A. It
is illustrated in Fig. 2 as the area under the heavy solid
line and to the left of the dotted line (red in the color
version).
• Region C: The portion of the (p, e) plane for which
∂Lz(p, e, θinc)/∂θinc > 0 for at least one angle θinc
between 0 and θmaxinc . Region C contains Region B, and
is illustrated in Fig. 2 as the area under the heavy solid
line and to the left of the dashed line (blue in the color
version).
Figure 2: Left panel: Non-circular nearly horizon-skimming orbits for a = 0.998M . The heavy solid line indicates the separatrix between
stable and unstable orbits for equatorial orbits (ι = θinc = 0). All orbits above and to the left of this line are unstable. The dot-dashed line
(green in the color version) bounds the region of the (p, e)-plane where ∂Lz/∂θinc > 0 for all allowed inclination angles (“Region A”). All
orbits between this line and the separatrix belong to Region A. The dotted line (red in the color version) bounds the region (Lz)most bound <
(Lz)least bound (“Region B”). Note that B includes A. The dashed line (blue in the color version) bounds the region where ∂Lz/∂θinc > 0 for
at least one inclination angle (“Region C”); note that C includes B. All three of these regions are candidate generalizations of the notion of
nearly horizon-skimming orbits. Right panel: Same as the left panel, but for the extreme spin case, a = M . In this case the separatrix between
stable and unstable equatorial orbits is given by the line p/M = 1 + e.
Orbits in any of these three regions give possible general-
izations of the nearly horizon-skimming circular orbits pre-
sented in Ref. [14]. Notice, as illustrated in Fig. 2, that the
size of these regions depends rather strongly on the spin of
the black hole. All three regions disappear altogether for
a < 0.9524M (in agreement with [14]); their sizes grow with
a, reaching maximal extent for a = M . These regions never
extend beyond p ≃ 2M .
As we shall see, the difference between these three regions
is not terribly important for assessing whether there is a strong
signature of the nearly horizon-skimming regime on the inspi-
ral dynamics. As such, it is perhaps most useful to use Region
C as our definition, since it is the most inclusive.
IV. EVOLUTION OF θinc: CIRCULAR ORBITS
To ascertain whether nearly horizon-skimming orbits can
affect an EMRI in such a way as to leave a clear imprint in the
gravitational-wave signal, we have studied the time evolution
of the inclination angle θinc. To this purpose we have used the
so-called adiabatic approximation [20], in which the infalling
body moves along a geodesic with slowly changing parame-
ters. The evolution of the orbital parameters is computed us-
ing the time-averaged fluxes dE/dt, dLz/dt and dQ/dt due
to gravitational-wave emission (“radiation reaction”). As dis-
cussed in Sec. II, E, Lz and Q can be expressed in terms of p,
e, and θinc. Given rates of change of E, Lz and Q, it is then
straightforward [23] to calculate dp/dt, de/dt, and dθinc/dt
(or dι/dt).
We should note that although perfectly well-behaved for all
bound stable geodesics, the adiabatic approximation breaks
down in a small region of the orbital parameters space very
close to the separatrix, where the transition from an inspiral to
a plunging orbit takes place [27]. However, since this region
is expected to be very small2 and its impact on LISA wave-
forms rather hard to detect [27], we expect our results to be
at least qualitatively correct also in this region of the space of
parameters.
Accurate calculation of dE/dt and dLz/dt in the adiabatic
approximation involves solving the Teukolsky and Sasaki-
Nakamura equations [28, 29]. For generic orbits this has been
done for the first time in Ref. [21]. The calculation of dQ/dt
for generic orbits is more involved. A formula for dQ/dt has
been recently derived [22], but has not yet been implemented
(at least in a code to which we have access).
On the other hand, it is well-known that a circular orbit will
remain circular under radiation reaction [30, 31, 32]. This
constraint means that Teukolsky-based fluxes for E and Lz
2 Its width in p/M is expected to be of the order of ∆p/M ∼ (µ/M)2/5 ,
where µ is the mass of the infalling body [27].
are sufficient to compute dQ/dt. Considering this limit, the
rate of change dQ/dt can be expressed in terms of dE/dt and
dL/dt as
= −N1(p, ι)
N5(p, ι)
− N4(p, ι)
N5(p, ι)
where
N1(p, ι) ≡ E(p, ι) p4 + a2 E(p, ι) p2
− 2 aM (Lz(p, ι)− aE(p, ι)) p , (35)
N4(p, ι) ≡ (2M p− p2)Lz(p, ι)− 2M aE(p, ι) p , (36)
N5(p, ι) ≡ (2M p− p2 − a2)/2 . (37)
(These quantities are for a circular orbit of radius p.) Using
this, it is simple to compute dθinc/dt (or dι/dt).
This procedure was followed in Ref. [14], using the code
presented in Ref. [17], to determine the evolution of ι; this
analysis indicated that dι/dt < 0 for circular nearly horizon-
skimming orbits. As a first step to our more general analy-
sis, we have repeated this calculation but using the improved
Sasaki-Nakamura-Teukolsky code presented in Ref. [21]; we
focused on the case a = 0.998M .
Rather to our surprise, we discovered that the fluxes dE/dt
and dLz/dt computed with this more accurate code indicate
that dι/dt > 0 (and dθinc/dt > 0) for all circular nearly
horizon-skimming orbits — in stark contrast with what was
found in Ref. [14]. As mentioned in the introduction, the rate
of change of inclination angle appears as the difference of two
quantities. These quantities nearly cancel (and indeed cancel
exactly in the limit a = 0); as such, small relative errors in
their values can lead to large relative error in the inferred in-
clination evolution. Values for dE/dt, dLz/dt, dι/dt, and
dθinc/dt computed using the present code are shown in Ta-
ble I in the columns with the header “Teukolsky”.
V. EVOLUTION OF θinc: NON-CIRCULAR ORBITS
The corrected behavior of circular nearly horizon-
skimming orbits has naturally led us to investigate the evo-
lution of non-circular nearly horizon-skimming orbits. Since
our code cannot be used to compute dQ/dt, we have resorted
to a “kludge” approach, based on those described in Refs.
[23, 24]. In particular, we mostly follow the procedure de-
veloped by Gair & Glampedakis [24], though (as described
below) importantly modified.
The basic idea of the “kludge” procedure is to use the func-
tional form of 2PN fluxes E, Lz and Q, but to correct the
circular part of these fluxes using fits to circular Teukolsky
data. As developed in Ref. [24], the fluxes are written
= (1− e2)3/2
(1− e2)−3/2
(p, e, ι)
(p, 0, ι) +
fit circ
(p, ι)
, (38)
= (1 − e2)3/2
(1 − e2)−3/2
(p, e, ι)
(p, 0, ι) +
fit circ
(p, ι)
, (39)
= (1− e2)3/2
Q(p, e, ι) ×
(1− e2)−3/2
dQ/dt√
(p, e, ι)−
dQ/dt√
(p, 0, ι)
dQ/dt√
fit circ
(p, ι)
. (40)
The post-Newtonian fluxes (dE/dt)2PN, (dLz/dt)2PN and
(dQ/dt)2PN are given in the Appendix [particularly Eqs.
(A.1), (A.2), and (A.3)].
Since for circular orbits the fluxes dE/dt, dLz/dt and
dQ/dt are related through Eq. (34), only two fits to circu-
lar Teukolsky data are needed. One possible choice is to fit
dLz/dt and dι/dt, and then use the circularity constraint to
obtain3 [24]
dQ/dt√
fit circ
(p, ι) =
2 tan ι
fit circ
Q(p, 0, ι)
sin2 ι
fit circ
, (41)
fit circ
(p, ι) = −
N4(p, ι)
N1(p, ι)
fit circ
(p, ι)
− N5(p, ι)
N1(p, ι)
Q(p, 0, ι)
dQ/dt√
fit circ
(p, ι) . (42)
As stressed in Ref. [24], one does not expect these fluxes to
work well in the strong field, both because the post-Newtonian
approximation breaks down close to the black hole, and be-
cause the circular Teukolsky data used for the fits in Ref. [24]
was computed for 3M ≤ p ≤ 30M . As a first attempt to
improve their behavior in the nearly horizon-skimming re-
gion, we have made fits using circular Teukolsky data for
orbits with M < p ≤ 2M . In particular, for a black hole
with a = 0.998M , we computed the circular Teukolsky-based
fluxes dLz/dt and dι/dt listed in Table I (columns 8 and 10).
These results were fit (with error . 0.2%); see Eqs. (A.4) and
(A.6) in the Appendix.
3 This choice might seem more involved than fitting directly dLz/dt and
dQ/dt, but, as noted by Gair & Glampedakis, ensures more sensible re-
sults for the evolution of the inclination angle. This generates more physi-
cally realistic inspirals [24].
Despite using strong-field Teukolsky fluxes for our fit, we
found fairly poor behavior of these rates of change, particu-
larly as a function of eccentricity. To compensate for this, we
introduced a kludge-type fit to correct the equatorial part of
the flux, in addition to the circular part. We fit, as a function
of p and e, Teukolsky-based fluxes for dE/dt and dLz/dt for
orbits in the equatorial plane, and then introduce the following
kludge fluxes for E and Lz:
(p, e, ι) =
(p, e, ι)
(p, e, 0) +
fit eq
(p, e) (43)
(p, e, ι) =
(p, e, ι)
(p, e, 0) +
fit eq
(p, e) . (44)
[Note that Eq. (40) for dQ/dt is not modified by this proce-
dure since dQ/dt = 0 for equatorial orbits.] Using equatorial
non-circular Teukolsky data provided by Drasco [21, 33] for
a = 0.998 and M < p ≤ 2M (the ι = 0 “Teukolsky” data in
Tables II, III and IV), we found fits (with error . 1.5%); see
Eqs. (A.9) and (A.10). Note that the fits for equatorial fluxes
are significantly less accurate than the fits for circular fluxes.
This appears to be due to the fact that, close to the black hole,
many harmonics are needed in order for the Teukolsky-based
fluxes to converge, especially for eccentric orbits (cf. Figs. 2
and 3 of Ref. [21], noting the number of radial harmonics that
have significant contribution to the flux). Truncation of these
sums is likely a source of some error in the fluxes themselves,
making it difficult to make a fit of as high quality as we could
in the circular case.
These fits were then finally used in Eqs. (43) and (44) to
calculate the kludge fluxes dE/dt and dLz/dt for generic or-
bits. This kludge reproduces to high accuracy our fits to the
Teukolsky-based fluxes for circular orbits (e = 0) or equato-
rial orbits (ι = 0). Some residual error remains because the
ι = 0 limit of the circular fits do not precisely equal the e = 0
limit of the equatorial fits.
Table I compares our kludge to Teukolsky-based fluxes for
circular orbits; the two methods agree to several digits. Tables
II, III and IV compare our kludge to the generic Teukolsky-
based fluxes for dE/dt and dLz/dt provided by Drasco
[21, 33]. In all cases, the kludge fluxes dE/dt and dLz/dt
have the correct qualitative behavior, being negative for all
the orbital parameters under consideration (a = 0.998M ,
1 < p/M ≤ 2, 0 ≤ e ≤ 0.5 and 0◦ ≤ ι ≤ 41◦). The
relative difference between the kludge and Teukolsky fluxes
is always less than 25% for e = 0 and e = 0.1 (even for
orbits very close to separatrix). The accuracy remains good
at larger eccentricity, though it degrades somewhat as orbits
come close to the separatrix.
Tables I, II, III and IV also present the kludge values of the
fluxes dι/dt and dθinc/dt as computed using Eqs. (43) and
(44) for dE/dt and dLz/dt, plus Eq. (40) for dQ/dt. Though
certainly not the last word on inclination evolution (pending
rigorous computation of dQ/dt), these rates of change proba-
bly represent a better approximation than results published to
date in the literature. (Indeed, prior work has often used the
crude approximation dι/dt = 0 [21] to estimate dQ/dt given
dE/dt and dLz/dt.)
Most significantly, we find that (dι/dt)kludge > 0 and
(dθinc/dt)kludge > 0 for all of the orbital parameters we con-
sider. In other words, we find that dι/dt and dθinc/dt do not
ever change sign.
Finally, in Table V we compute the changes in θinc and ι
for the inspiral with mass ratio µ/M = 10−6. In all cases,
we start at p/M = 1.9. The small body then inspirals through
the nearly horizon-skimming region until it reaches the sep-
aratrix; at this point, the small body will fall into the large
black hole on a dynamical timescale ∼ M , so we terminate
the calculation. The evolution of circular orbits is computed
using our fits to the circular-Teukolsky fluxes of E and Lz;
for eccentric orbits we use the kludge fluxes (40), (43) and
(44). As this exercise demonstrates, the change in inclination
during inspiral is never larger than a few degrees. Not only is
there no unique sign change in the nearly horizon-skimming
region, but the magnitude of the inclination change remains
puny. This leaves little room for the possibility that this class
of orbits may have a clear observational imprint on the EMRI-
waveforms to be detected by LISA.
VI. CONCLUSIONS
We have performed a detailed analysis of the orbital mo-
tion near the horizon of near-extremal Kerr black holes. We
have demonstrated the existence of a class of orbits, which we
have named “non-circular nearly horizon-skimming orbits”,
for which the angular momentum Lz increases with the or-
bit’s inclination, while keeping latus rectum and eccentricity
fixed. This behavior, in stark contrast to that of Newtonian
orbits, generalizes earlier results for circular orbits [14].
Furthermore, to assess whether this class of orbits can pro-
duce a unique imprint on EMRI waveforms (an important
source for future LISA observations), we have studied, in
the adiabatic approximation, the radiative evolution of incli-
nation angle for a small body orbiting in the nearly horizon-
skimming region. For circular orbits, we have re-examined
the analysis of Ref. [14] using an improved code for comput-
ing Teukolsky-based fluxes of the energy and angular momen-
tum. Significantly correcting Ref. [14]’s results, we found no
decrease in the orbit’s inclination angle. Inclination always
increases during inspiral.
We next carried out such an analysis for eccentric nearly
horizon-skimming orbits. In this case, we used “kludge”
fluxes to evolve the constants of motion E, Lz and Q [24].
We find that these fluxes are fairly accurate when compared
with the available Teukolsky-based fluxes, indicating that they
should provide at least qualitatively correct information re-
garding inclination evolution. As for circular orbits, we find
that the orbit’s inclination never decreases. For both circular
and non-circular configurations, we find that the magnitude of
the inclination change is quite paltry — only a few degrees at
most.
Quite generically, therefore, we found that the inclination
angle of both circular and eccentric nearly horizon-skimming
orbits never decreases during the inspiral. Revising the results
obtained in Ref. [14], we thus conclude that such orbits are
not likely to yield a peculiar, unique imprint on the EMRI-
waveforms detectable by LISA.
Acknowledgments
It is a pleasure to thank Kostas Glampedakis for enlight-
ening comments and advice, and Steve Drasco for useful
discussions and for also providing the non-circular Teukol-
sky data that we used in this paper. The supercomputers
used in this investigation were provided by funding from the
JPL Office of the Chief Information Officer. This work was
supported in part by the DFG grant SFB TR/7, by NASA
Grant NNG05G105G, and by NSF Grant PHY-0449884. SAH
gratefully acknowledges support from the MIT Class of 1956
Career Development Fund.
Appendix
In this Appendix we report the expressions for the post-
Newtonian fluxes and the fits to the Teukolsky data necessary
to compute the kludge fluxes introduced in Sec. V. In partic-
ular the 2PN fluxes are given by [24]
= −32
(1− e2)3/2
g1(e)− ã
g2(e) cos ι−
g3(e) + π
g4(e)
g5(e) + ã
g6(e)−
sin2 ι
, (A.1)
= −32
(1− e2)3/2
g9(e) cos ι+ ã
(ga10(e)− cos2 ιgb10(e)) −
g11(e) cos ι
g12(e) cos ι−
g13(e) cos ι+ ã
cos ι
g14(e)−
sin2 ι
, (A.2)
)7/2 √
Q sin ι (1− e2)3/2
g9(e)− ã
cos ιgb10(e)−
g11(e)
g12(e)−
g13(e) + ã
g14(e)−
sin2 ι
, (A.3)
where µ is the mass of the infalling body and where the various e-dependent coefficients are
g1(e) ≡ 1 +
e4 , g2(e) ≡
e6 , g3(e) ≡
g4(e) ≡ 4 +
e2 , g5(e) ≡
44711
172157
e2 , g6(e) ≡
e2 , g9(e) ≡ 1 +
ga10(e) ≡
e4 , gb10(e) ≡
e4 , g11(e) ≡
g12(e) ≡ 4 +
e2 , g13(e) ≡
44711
302893
e2 , g14(e) ≡
The fits to the circular-Teukolsky data of Table I are instead given by
fit circ
(p, ι ) = −32
)7/2 {
cos ι+
)3/2 (
cos2 ι+ 4π cos ι
− 1247
cos ι
cos ι
−1625
sin2 ι
d̃1(p/M) + d̃2(p/M) cos ι + d̃3(p/M) cos
+ d̃4(p/M) cos
3 ι + d̃5(p/M) cos
4 ι+ d̃6(p/M) cos
5 ι+ cos ι
)3/2 (
A+B cos2 ι
, (A.4)
(A.5)
fit circ
(p, ι ) =
sin2 ι√
Q(p, 0, ι)
d̃1(p/M) + cos ι
a7d + b
+ c7d
)3/2]
+ cos2 ι
d̃8(p/M) + cos ι
)5/2 [
h̃1(p/M) + cos
2 ι h̃2(p/M)
, (A.6)
where
d̃i(x) ≡ aid + bid x−1/2 + cid x−1 , i = 1, . . . , 8, h̃i(x) ≡ aih + bih x−1/2 , i = 1, 2 (A.7)
and the numerical coefficients are given by
a1h = −278.9387 , b1h = 84.1414 , a2h = 8.6679 , b2h = −9.2401 , A = −18.3362 , B = 24.9034 , (A.8)
and by the following table
i 1 2 3 4 5 6 7 8
aid 15.8363 445.4418 −2027.7797 3089.1709 −2045.2248 498.6411 −8.7220 50.8345
bid −55.6777 −1333.2461 5940.4831 −9103.4472 6113.1165 −1515.8506 −50.8950 −131.6422
cid 38.6405 1049.5637 −4513.0879 6926.3191 -4714.9633 1183.5875 251.4025 83.0834
Note that the functional form of these fits was obtained from Eqs. (57) and (58) of Ref. [24] by setting ã (i.e., q in their
notation) to 1. Finally, we give expressions for the fits to the equatorial Teukolsky data of tables II, III and IV (data with ι = 0,
columns with header “Teukolsky”):
fit eq
(p, e) =
(p, e, 0)− 32
)2 (M
(1− e2)3/2
g̃1(e) + g̃2(e)
+ g̃3(e)
+ g̃4(e)
+ g̃5(e)
, (A.9)
fit eq
(p, e) =
(p, e, 0)− 32
(1− e2)3/2
f̃1(e) + f̃2(e)
+ f̃3(e)
+ f̃4(e)
+ f̃5(e)
, (A.10)
g̃i(e) ≡ aig + big e2 + cig e4 + dig e6 , f̃i(e) ≡ aif + bif e2 + cif e4 + dif e6 , i = 1, . . . , 5 (A.11)
where the numerical coefficients are given by the following table
i aig b
1 6.4590 −2038.7301 6639.9843 227709.2187 5.4577 −3116.4034 4711.7065 214332.2907
2 -31.2215 10390.6778 −27505.7295 −1224376.5294 −26.6519 15958.6191 −16390.4868 −1147201.4687
3 57.1208 −19800.4891 39527.8397 2463977.3622 50.4374 -30579.3129 15749.9411 2296989.5466
4 -49.7051 16684.4629 −21714.7941 −2199231.9494 −46.7816 25968.8743 656.3460 −2038650.9838
5 16.4697 −5234.2077 2936.2391 734454.5696 15.6660 −8226.3892 −4903.9260 676553.2755
e θinc ι
dθinc
dθinc
(deg.) (deg.) (kludge) (Teukolsky) (kludge) (Teukolsky) (kludge) (Teukolsky) (kludge) (Teukolsky)
1.3 0 0 0 −9.108×10−2 −9.109×10−2 −2.258×10−1 −2.259×10−1 0 0 0 0
1.3 0 10.4870 11.6773 −9.328×10−2 −9.332×10−2 −2.304×10−1 −2.306×10−1 1.837×10−2 1.839×10−2 6.462×10−3 6.475×10−3
1.3 0 14.6406 16.1303 −9.588×10−2 −9.588×10−2 −2.359×10−1 −2.360×10−1 2.397×10−2 2.400×10−2 8.645×10−3 8.667×10−3
1.3 0 17.7000 19.3172 −9.875×10−2 −9.876×10−2 −2.420×10−1 −2.421×10−1 2.728×10−2 2.731×10−2 1.007×10−2 1.010×10−2
1.3 0 20.1636 21.8210 −1.019×10−1 −1.019×10−1 −2.486×10−1 −2.488×10−1 2.943×10−2 2.950×10−2 1.111×10−2 1.117×10−2
1.4 0 0 0 −8.700×10−2 −8.709×10−2 −2.311×10−1 −2.312×10−1 0 0 0 0
1.4 0 14.5992 16.0005 −9.062×10−2 −9.070×10−2 −2.386×10−1 −2.386×10−1 2.316×10−2 2.319×10−2 8.823×10−3 8.848×10−3
1.4 0 20.1756 21.7815 −9.520×10−2 −9.526×10−2 −2.482×10−1 −2.482×10−1 2.875×10−2 2.877×10−2 1.141×10−2 1.143×10−2
1.4 0 24.1503 25.7517 −1.006×10−1 −1.007×10−1 −2.595×10−1 −2.596×10−1 3.140×10−2 3.141×10−2 1.289×10−2 1.288×10−2
1.4 0 27.2489 28.7604 −1.067×10−1 −1.068×10−1 −2.725×10−1 −2.725×10−1 3.274×10−2 3.275×10−2 1.378×10−2 1.377×10−2
1.5 0 0 0 −8.009×10−2 −7.989×10−2 −2.270×10−1 −2.265×10−1 0 0 0 0
1.5 0 16.7836 18.1857 −8.401×10−2 −8.383×10−2 −2.348×10−1 −2.343×10−1 2.360×10−2 2.351×10−2 9.602×10−3 9.545×10−3
1.5 0 23.0755 24.6167 −8.917×10−2 −8.897×10−2 −2.454×10−1 −2.449×10−1 2.872×10−2 2.863×10−2 1.228×10−2 1.222×10−2
1.5 0 27.4892 28.9670 −9.537×10−2 −9.516×10−2 −2.583×10−1 −2.579×10−1 3.091×10−2 3.082×10−2 1.372×10−2 1.367×10−2
1.5 0 30.8795 32.2231 −1.025×10−1 −1.023×10−1 −2.733×10−1 −2.728×10−1 3.184×10−2 3.173×10−2 1.452×10−2 1.443×10−2
1.6 0 0 0 −7.181×10−2 −7.156×10−2 −2.168×10−1 −2.162×10−1 0 0 0 0
1.6 0 18.3669 19.7220 −7.568×10−2 −7.545×10−2 −2.242×10−1 −2.237×10−1 2.240×10−2 2.229×10−2 9.600×10−3 9.515×10−3
1.6 0 25.1720 26.6245 −8.084×10−2 −8.062×10−2 −2.346×10−1 −2.341×10−1 2.701×10−2 2.685×10−2 1.223×10−2 1.210×10−2
1.6 0 29.9014 31.2625 −8.708×10−2 −8.687×10−2 −2.474×10−1 −2.470×10−1 2.889×10−2 2.872×10−2 1.363×10−2 1.349×10−2
1.6 0 33.5053 34.7164 −9.425×10−2 −9.399×10−2 −2.622×10−1 −2.616×10−1 2.964×10−2 2.951×10−2 1.441×10−2 1.432×10−2
1.7 0 0 0 −6.332×10−2 −6.317×10−2 −2.034×10−1 −2.031×10−1 0 0 0 0
1.7 0 19.6910 20.9859 −6.702×10−2 −6.687×10−2 −2.101×10−1 −2.098×10−1 2.057×10−2 2.052×10−2 9.202×10−3 9.171×10−3
1.7 0 26.9252 28.2884 −7.197×10−2 −7.184×10−2 −2.199×10−1 −2.196×10−1 2.467×10−2 2.456×10−2 1.170×10−2 1.162×10−2
1.7 0 31.9218 33.1786 −7.794×10−2 −7.782×10−2 −2.319×10−1 −2.316×10−1 2.632×10−2 2.620×10−2 1.306×10−2 1.296×10−2
1.7 0 35.7100 36.8118 −8.475×10−2 −8.465×10−2 −2.457×10−1 −2.455×10−1 2.698×10−2 2.686×10−2 1.384×10−2 1.373×10−2
1.8 0 0 0 −5.531×10−2 −5.528×10−2 −1.888×10−1 −1.887×10−1 0 0 0 0
1.8 0 20.8804 22.1128 −5.879×10−2 −5.874×10−2 −1.948×10−1 −1.946×10−1 1.858×10−2 1.858×10−2 8.635×10−3 8.639×10−3
1.8 0 28.5007 29.7791 −6.343×10−2 −6.336×10−2 −2.036×10−1 −2.035×10−1 2.221×10−2 2.223×10−2 1.098×10−2 1.101×10−2
1.8 0 33.7400 34.9034 −6.901×10−2 −6.894×10−2 −2.146×10−1 −2.144×10−1 2.368×10−2 2.371×10−2 1.228×10−2 1.232×10−2
1.8 0 37.6985 38.7065 −7.533×10−2 −7.533×10−2 −2.271×10−1 −2.271×10−1 2.429×10−2 2.427×10−2 1.306×10−2 1.303×10−2
1.9 0 0 0 −4.809×10−2 −4.811×10−2 −1.740×10−1 −1.740×10−1 0 0 0 0
1.9 0 21.9900 23.1615 −5.132×10−2 −5.134×10−2 −1.792×10−1 −1.793×10−1 1.666×10−2 1.664×10−2 8.022×10−3 8.007×10−3
1.9 0 29.9708 31.1702 −5.562×10−2 −5.564×10−2 −1.872×10−1 −1.872×10−1 1.986×10−2 1.987×10−2 1.019×10−2 1.020×10−2
1.9 0 35.4385 36.5176 −6.078×10−2 −6.077×10−2 −1.971×10−1 −1.970×10−1 2.118×10−2 2.122×10−2 1.143×10−2 1.148×10−2
1.9 0 39.5592 40.4847 −6.659×10−2 −6.658×10−2 −2.082×10−1 −2.082×10−1 2.177×10−2 2.182×10−2 1.222×10−2 1.228×10−2
2.0 0 0 0 −4.174×10−2 −4.175×10−2 −1.598×10−1 −1.598×10−1 0 0 0 0
2.0 0 23.0471 24.1605 −4.471×10−2 −4.472×10−2 −1.643×10−1 −1.643×10−1 1.489×10−2 1.489×10−2 7.425×10−3 7.424×10−3
2.0 0 31.3715 32.4978 −4.867×10−2 −4.871×10−2 −1.713×10−1 −1.714×10−1 1.773×10−2 1.770×10−2 9.436×10−3 9.411×10−3
2.0 0 37.0583 38.0608 −5.341×10−2 −5.345×10−2 −1.801×10−1 −1.801×10−1 1.893×10−2 1.889×10−2 1.062×10−2 1.057×10−2
2.0 0 41.3358 42.1876 −5.873×10−2 −5.875×10−2 −1.900×10−1 −1.900×10−1 1.950×10−2 1.948×10−2 1.141×10−2 1.138×10−2
Table I: Teukolsky-based fluxes and kludge fluxes [computed using Eqs. (40), (43) and (44)] for circular orbits about a hole with a = 0.998M ;
µ represents the mass of the infalling body. The Teukolsky-based fluxes have an accuracy of 10−6.
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e θinc ι
dθinc
(deg.) (deg.) (kludge) (Teukolsky) (kludge) (Teukolsky) (kludge) (kludge)
1.3 0.1 0 0 −8.804×10−2 −8.804×10−2 −2.098×10−1 −2.098×10−1 0 0
1.4 0.1 0 0 −8.728×10−2 −8.719×10−2 −2.274×10−1 −2.275×10−1 0 0
1.4 0.1 8 8.8664 −9.110×10−2 −8.736×10−2 −2.355×10−1 −2.273×10−1 4.066×10−2 2.938×10−2
1.4 0.1 16 17.4519 −1.030×10−1 −8.958×10−2 −2.602×10−1 −2.309×10−1 7.428×10−2 5.475×10−2
1.4 0.1 24 25.5784 −1.243×10−1 −9.771×10−2 −3.037×10−1 −2.415×10−1 9.663×10−2 7.316×10−2
1.5 0.1 0 0 −8.069×10−2 −8.095×10−2 −2.255×10−1 −2.260×10−1 0 0
1.5 0.1 8 8.7910 −8.323×10−2 −8.133×10−2 −2.310×10−1 −2.264×10−1 2.996×10−2 2.070×10−2
1.5 0.1 16 17.3490 −9.121×10−2 −8.395×10−2 −2.483×10−1 −2.314×10−1 5.512×10−2 3.888×10−2
1.5 0.1 24 25.5197 −1.059×10−1 −8.980×10−2 −2.792×10−1 −2.423×10−1 7.255×10−2 5.264×10−2
1.6 0.1 0 0 −7.255×10−2 −7.281×10−2 −2.161×10−1 −2.168×10−1 0 0
1.6 0.1 8 8.7195 −7.430×10−2 −7.321×10−2 −2.201×10−1 −2.173×10−1 2.258×10−2 1.502×10−2
1.6 0.1 16 17.2437 −7.986×10−2 −7.533×10−2 −2.323×10−1 −2.212×10−1 4.179×10−2 2.839×10−2
1.6 0.1 24 25.4388 −9.025×10−2 −8.040×10−2 −2.547×10−1 −2.309×10−1 5.554×10−2 3.886×10−2
1.6 0.1 32 33.2683 −1.082×10−1 −9.435×10−2 −2.920×10−1 −2.551×10−1 6.316×10−2 4.559×10−2
1.7 0.1 0 0 −6.427×10−2 −6.440×10−2 −2.036×10−1 −2.040×10−1 0 0
1.7 0.1 8 8.6555 −6.552×10−2 −6.478×10−2 −2.065×10−1 −2.045×10−1 1.742×10−2 1.124×10−2
1.7 0.1 16 17.1454 −6.953×10−2 −6.651×10−2 −2.154×10−1 −2.075×10−1 3.240×10−2 2.134×10−2
1.7 0.1 24 25.3531 −7.707×10−2 −7.052×10−2 −2.317×10−1 −2.150×10−1 4.342×10−2 2.948×10−2
1.7 0.1 32 33.2416 −9.009×10−2 −7.959×10−2 −2.590×10−1 −2.324×10−1 4.998×10−2 3.512×10−2
1.8 0.1 0 0 −5.640×10−2 −5.640×10−2 −1.897×10−1 −1.897×10−1 0 0
1.8 0.1 8 8.5991 −5.732×10−2 −5.676×10−2 −1.918×10−1 −1.902×10−1 1.371×10−2 8.640×10−3
1.8 0.1 16 17.0562 −6.028×10−2 −5.817×10−2 −1.984×10−1 −1.925×10−1 2.562×10−2 1.647×10−2
1.8 0.1 24 25.2693 −6.588×10−2 −6.139×10−2 −2.105×10−1 −1.983×10−1 3.456×10−2 2.291×10−2
1.8 0.1 32 33.2018 −7.555×10−2 −6.849×10−2 −2.307×10−1 −2.120×10−1 4.020×10−2 2.765×10−2
1.9 0.1 0 0 −4.915×10−2 −4.911×10−2 −1.753×10−1 −1.751×10−1 0 0
1.9 0.1 8 8.5494 −4.985×10−2 −4.945×10−2 −1.768×10−1 −1.755×10−1 1.097×10−2 6.791×10−3
1.9 0.1 16 16.9760 −5.208×10−2 −5.064×10−2 −1.817×10−1 −1.774×10−1 2.055×10−2 1.298×10−2
1.9 0.1 24 25.1898 −5.633×10−2 −5.328×10−2 −1.908×10−1 −1.819×10−1 2.788×10−2 1.816×10−2
1.9 0.1 32 33.1555 −6.364×10−2 −5.870×10−2 −2.059×10−1 −1.920×10−1 3.272×10−2 2.214×10−2
2.0 0.1 0 0 −4.263×10−2 −4.264×10−2 −1.607×10−1 −1.608×10−1 0 0
2.0 0.1 8 8.5057 −4.316×10−2 −4.292×10−2 −1.619×10−1 −1.611×10−1 8.862×10−3 5.424×10−3
2.0 0.1 16 16.9042 −4.488×10−2 −4.390×10−2 −1.656×10−1 −1.625×10−1 1.666×10−2 1.039×10−2
2.0 0.1 24 25.1156 −4.815×10−2 −4.604×10−2 −1.724×10−1 −1.660×10−1 2.271×10−2 1.459×10−2
2.0 0.1 32 33.1064 −5.376×10−2 −5.031×10−2 −1.838×10−1 −1.736×10−1 2.684×10−2 1.793×10−2
2.0 0.1 40 40.8954 −6.339×10−2 −6.236×10−2 −2.027×10−1 −1.967×10−1 2.917×10−2 2.036×10−2
Table II: As in Table I but for non-circular orbits; the Teukolsky-based fluxes for E and Lz have an accuracy of 10
−3. Note that our code, as
all the Teukolsky-based code that we are aware of, presently does not have the capability to compute inclination angle evolution for generic
orbits.
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e θinc ι
dθinc
(deg.) (deg.) (kludge) (Teukolsky ) (kludge) (Teukolsky) (kludge) (kludge)
1.4 0.2 0 0 −8.636×10−2 −8.642×10−2 −2.119×10−1 −2.121×10−1 0 0
1.4 0.2 8 8.8215 −9.853×10−2 −8.240×10−2 −2.374×10−1 −2.015×10−1 1.148×10−1 9.714×10−2
1.5 0.2 0 0 −8.362×10−2 −8.349×10−2 −2.236×10−1 −2.230×10−1 0 0
1.5 0.2 8 8.7595 −9.141×10−2 −8.276×10−2 −2.410×10−1 −2.206×10−1 7.893×10−2 6.549×10−2
1.5 0.2 16 17.2957 −1.145×10−1 −8.394×10−2 −2.915×10−1 −2.215×10−1 1.466×10−1 1.230×10−1
1.5 0.2 24 25.4608 −1.524×10−1 −9.230×10−2 −3.712×10−1 −2.357×10−1 1.952×10−1 1.661×10−1
1.6 0.2 0 0 −7.596×10−2 −7.616×10−2 −2.171×10−1 −2.176×10−1 0 0
1.6 0.2 8 8.6935 −8.111×10−2 −7.641×10−2 −2.292×10−1 −2.177×10−1 5.520×10−2 4.502×10−2
1.6 0.2 16 17.1994 −9.649×10−2 −7.798×10−2 −2.647×10−1 −2.198×10−1 1.032×10−1 8.500×10−2
1.6 0.2 24 25.3891 −1.221×10−1 −8.314×10−2 −3.212×10−1 −2.288×10−1 1.388×10−1 1.160×10−1
1.7 0.2 0 0 −6.765×10−2 −6.799×10−2 −2.057×10−1 −2.068×10−1 0 0
1.7 0.2 8 8.6329 −7.116×10−2 −6.813×10−2 −2.144×10−1 −2.066×10−1 3.963×10−2 3.176×10−2
1.7 0.2 16 17.1064 −8.171×10−2 −6.995×10−2 −2.398×10−1 −2.096×10−1 7.441×10−2 6.024×10−2
1.7 0.2 24 25.3085 −9.948×10−2 −7.443×10−2 −2.806×10−1 −2.178×10−1 1.009×10−1 8.290×10−2
1.7 0.2 32 33.2037 −1.257×10−1 −8.558×10−2 −3.371×10−1 −2.366×10−1 1.175×10−1 9.806×10−2
1.8 0.2 0 0 −5.965×10−2 −5.962×10−2 −1.927×10−1 −1.926×10−1 0 0
1.8 0.2 8 8.5789 −6.211×10−2 −5.997×10−2 −1.990×10−1 −1.930×10−1 2.919×10−2 2.300×10−2
1.8 0.2 16 17.0211 −6.953×10−2 −6.147×10−2 −2.175×10−1 −1.954×10−1 5.504×10−2 4.380×10−2
1.8 0.2 24 25.2283 −8.216×10−2 −6.502×10−2 −2.474×10−1 −2.016×10−1 7.515×10−2 6.068×10−2
1.8 0.2 32 33.1656 −1.009×10−1 −7.410×10−2 −2.890×10−1 −2.190×10−1 8.839×10−2 7.258×10−2
1.9 0.2 0 0 −5.218×10−2 −5.210×10−2 −1.786×10−1 −1.783×10−1 0 0
1.9 0.2 8 8.5312 −5.394×10−2 −5.244×10−2 −1.833×10−1 −1.787×10−1 2.197×10−2 1.704×10−2
1.9 0.2 16 16.9441 −5.928×10−2 −5.373×10−2 −1.970×10−1 −1.807×10−1 4.156×10−2 3.254×10−2
1.9 0.2 24 25.1518 −6.843×10−2 −5.669×10−2 −2.192×10−1 −1.858×10−1 5.706×10−2 4.535×10−2
1.9 0.2 32 33.1207 −8.213×10−2 −6.277×10−2 −2.502×10−1 −1.966×10−1 6.767×10−2 5.475×10−2
2.0 0.2 0 0 −4.528×10−2 −4.530×10−2 −1.637×10−1 −1.638×10−1 0 0
2.0 0.2 8 8.4891 −4.657×10−2 −4.557×10−2 −1.671×10−1 −1.641×10−1 1.679×10−2 1.283×10−2
2.0 0.2 16 16.8749 −5.049×10−2 −4.664×10−2 −1.774×10−1 −1.657×10−1 3.184×10−2 2.457×10−2
2.0 0.2 24 25.0802 −5.725×10−2 −4.904×10−2 −1.941×10−1 −1.696×10−1 4.391×10−2 3.440×10−2
2.0 0.2 32 33.0730 −6.743×10−2 −5.427×10−2 −2.175×10−1 −1.793×10−1 5.243×10−2 4.184×10−2
1.5 0.3 0 0 −8.481×10−2 −8.478×10−2 −2.094×10−1 −2.094×10−1 0 0
1.5 0.3 8 8.7037 −1.006×10−1 −7.824×10−2 −2.442×10−1 −1.934×10−1 1.484×10−1 1.301×10−1
1.5 0.3 16 17.2003 −1.469×10−1 −7.811×10−2 −3.435×10−1 −1.864×10−1 2.766×10−1 2.440×10−1
1.6 0.3 0 0 −8.144×10−2 −8.123×10−2 −2.183×10−1 −2.178×10−1 0 0
1.6 0.3 8 8.6498 −9.182×10−2 −7.807×10−2 −2.426×10−1 −2.095×10−1 1.028×10−1 8.918×10−2
1.6 0.3 16 17.1246 −1.223×10−1 −8.089×10−2 −3.122×10−1 −2.144×10−1 1.928×10−1 1.683×10−1
1.6 0.3 24 25.3046 −1.716×10−1 −8.666×10−2 −4.197×10−1 −2.229×10−1 2.607×10−1 2.295×10−1
1.7 0.3 0 0 −7.362×10−2 −7.314×10−2 −2.104×10−1 −2.095×10−1 0 0
1.7 0.3 8 8.5953 −8.060×10−2 −7.224×10−2 −2.277×10−1 −2.065×10−1 7.240×10−2 6.224×10−2
1.7 0.3 16 17.0415 −1.013×10−1 −7.369×10−2 −2.774×10−1 −2.084×10−1 1.365×10−1 1.180×10−1
1.7 0.3 24 25.2339 −1.349×10−1 −7.800×10−2 −3.547×10−1 −2.153×10−1 1.861×10−1 1.622×10−1
1.8 0.3 0 0 −6.488×10−2 −6.484×10−2 −1.973×10−1 −1.972×10−1 0 0
1.8 0.3 8 8.5454 −6.970×10−2 −6.480×10−2 −2.099×10−1 −1.966×10−1 5.206×10−2 4.436×10−2
1.8 0.3 16 16.9628 −8.402×10−2 −6.671×10−2 −2.461×10−1 −1.998×10−1 9.857×10−2 8.445×10−2
1.8 0.3 24 25.1601 −1.075×10−1 −7.030×10−2 −3.026×10−1 −2.056×10−1 1.353×10−1 1.169×10−1
1.8 0.3 32 33.1047 −1.404×10−1 −8.153×10−2 −3.762×10−1 −2.255×10−1 1.600×10−1 1.394×10−1
1.9 0.3 0 0 −5.669×10−2 −5.690×10−2 −1.829×10−1 −1.832×10−1 0 0
1.9 0.3 8 8.5010 −6.010×10−2 −5.683×10−2 −1.922×10−1 −1.824×10−1 3.823×10−2 3.229×10−2
1.9 0.3 16 16.8911 −7.025×10−2 −5.818×10−2 −2.189×10−1 −1.844×10−1 7.263×10−2 6.165×10−2
1.9 0.3 24 25.0887 −8.701×10−2 −6.054×10−2 −2.609×10−1 −1.874×10−1 1.003×10−1 8.579×10−2
1.9 0.3 32 33.0624 −1.106×10−1 −6.912×10−2 −3.157×10−1 −2.034×10−1 1.195×10−1 1.032×10−1
2.0 0.3 0 0 −4.953×10−2 −4.946×10−2 −1.683×10−1 −1.683×10−1 0 0
2.0 0.3 8 8.4616 −5.199×10−2 −4.970×10−2 −1.753×10−1 −1.685×10−1 2.862×10−2 2.395×10−2
2.0 0.3 16 16.8262 −5.932×10−2 −5.079×10−2 −1.954×10−1 −1.699×10−1 5.452×10−2 4.585×10−2
2.0 0.3 24 25.0215 −7.150×10−2 −5.328×10−2 −2.269×10−1 −1.737×10−1 7.564×10−2 6.411×10−2
2.0 0.3 32 33.0172 −8.878×10−2 −6.003×10−2 −2.682×10−1 −1.864×10−1 9.077×10−2 7.771×10−2
Table III: As in Table II, but for additional values of eccentricity e; the Teukolsky-based fluxes for E and Lz have an accuracy of 10
e θinc ι
dθinc
(deg.) (deg.) (kludge) (Teukolsky ) (kludge) (Teukolsky) (kludge) (kludge)
1.6 0.4 0 0 −7.766×10−2 −7.772×10−2 −1.918×10−1 −1.919×10−1 0 0
1.6 0.4 8 8.5863 −9.433×10−2 −7.645×10−2 −2.297×10−1 −1.881×10−1 1.528×10−1 1.370×10−1
1.6 0.4 16 17.0151 −1.432×10−1 −7.651×10−2 −3.382×10−1 −1.837×10−1 2.873×10−1 2.584×10−1
1.7 0.4 0 0 −7.882×10−2 −7.953×10−2 −2.097×10−1 −2.115×10−1 0 0
1.7 0.4 8 8.5426 −9.002×10−2 −7.408×10−2 −2.367×10−1 −1.978×10−1 1.087×10−1 9.656×10−2
1.7 0.4 16 16.9502 −1.229×10−1 −7.682×10−2 −3.143×10−1 −2.025×10−1 2.054×10−1 1.830×10−1
1.7 0.4 24 25.1282 −1.760×10−1 −8.090×10−2 −4.336×10−1 −2.075×10−1 2.809×10−1 2.514×10−1
1.8 0.4 0 0 −7.107×10−2 −7.007×10−2 −2.013×10−1 −1.988×10−1 0 0
1.8 0.4 8 8.4989 −7.877×10−2 −7.001×10−2 −2.209×10−1 −1.981×10−1 7.788×10−2 6.879×10−2
1.8 0.4 16 16.8817 −1.015×10−1 −7.009×10−2 −2.774×10−1 −1.965×10−1 1.478×10−1 1.309×10−1
1.8 0.4 24 25.0646 −1.383×10−1 −7.314×10−2 −3.646×10−1 −2.003×10−1 2.036×10−1 1.810×10−1
1.8 0.4 32 33.0184 −1.887×10−1 −9.193×10−2 −4.755×10−1 −2.319×10−1 2.414×10−1 2.156×10−1
1.9 0.4 0 0 −6.187×10−2 −6.267×10−2 −1.861×10−1 −1.881×10−1 0 0
1.9 0.4 8 8.4591 −6.728×10−2 −6.216×10−2 −2.006×10−1 −1.861×10−1 5.666×10−2 4.980×10−2
1.9 0.4 16 16.8173 −8.328×10−2 −6.222×10−2 −2.424×10−1 −1.844×10−1 1.079×10−1 9.506×10−2
1.9 0.4 24 25.0006 −1.094×10−1 −6.486×10−2 −3.071×10−1 −1.878×10−1 1.495×10−1 1.322×10−1
1.9 0.4 32 32.9804 −1.452×10−1 −7.884×10−2 −3.896×10−1 −2.158×10−1 1.787×10−1 1.588×10−1
2.0 0.4 0 0 −5.483×10−2 −5.457×10−2 −1.735×10−1 −1.729×10−1 0 0
2.0 0.4 8 8.4235 −5.871×10−2 −5.445×10−2 −1.844×10−1 −1.720×10−1 4.222×10−2 3.686×10−2
2.0 0.4 16 16.7586 −7.020×10−2 −5.555×10−2 −2.158×10−1 −1.733×10−1 8.064×10−2 7.057×10−2
2.0 0.4 24 24.9396 −8.902×10−2 −5.844×10−2 −2.645×10−1 −1.778×10−1 1.122×10−1 9.860×10−2
2.0 0.4 32 32.9389 −1.150×10−1 −6.536×10−2 −3.267×10−1 −1.896×10−1 1.351×10−1 1.193×10−1
1.7 0.5 0 0 −7.421×10−2 −7.401×10−2 −1.815×10−1 −1.810×10−1 0 0
1.7 0.5 8 8.4736 −8.957×10−2 −7.168×10−2 −2.173×10−1 −1.750×10−1 1.379×10−1 1.256×10−1
1.7 0.5 16 16.8300 −1.347×10−1 −6.999×10−2 −3.201×10−1 −1.676×10−1 2.611×10−1 2.378×10−1
1.8 0.5 0 0 −7.589×10−2 −7.620×10−2 −1.993×10−1 −2.000×10−1 0 0
1.8 0.5 8 8.4395 −8.644×10−2 −6.929×10−2 −2.254×10−1 −1.829×10−1 1.005×10−1 9.076×10−2
1.8 0.5 16 16.7776 −1.175×10−1 −7.210×10−2 −3.004×10−1 −1.880×10−1 1.911×10−1 1.726×10−1
1.8 0.5 24 24.9413 −1.678×10−1 −7.395×10−2 −4.158×10−1 −1.881×10−1 2.638×10−1 2.385×10−1
1.9 0.5 0 0 −6.646×10−2 −6.620×10−2 −1.855×10−1 −1.849×10−1 0 0
1.9 0.5 8 8.4059 −7.386×10−2 −6.320×10−2 −2.048×10−1 −1.768×10−1 7.312×10−2 6.579×10−2
1.9 0.5 16 16.7233 −9.572×10−2 −6.551×10−2 −2.603×10−1 −1.809×10−1 1.395×10−1 1.255×10−1
1.9 0.5 24 24.8877 −1.312×10−1 −7.087×10−2 −3.461×10−1 −1.909×10−1 1.937×10−1 1.744×10−1
1.9 0.5 32 32.8741 −1.795×10−1 −8.247×10−2 −4.544×10−1 −2.091×10−1 2.320×10−1 2.092×10−1
2.0 0.5 0 0 −5.987×10−2 −5.995×10−2 −1.761×10−1 −1.763×10−1 0 0
2.0 0.5 8 8.3750 −6.516×10−2 −5.918×10−2 −1.906×10−1 −1.738×10−1 5.456×10−2 4.882×10−2
2.0 0.5 16 16.6725 −8.081×10−2 −5.817×10−2 −2.324×10−1 −1.694×10−1 1.044×10−1 9.343×10−2
2.0 0.5 24 24.8347 −1.063×10−1 −6.254×10−2 −2.970×10−1 −1.776×10−1 1.456×10−1 1.304×10−1
2.0 0.5 32 32.8378 −1.412×10−1 −6.993×10−2 −3.787×10−1 −1.893×10−1 1.756×10−1 1.576×10−1
Table IV: As in Tables II and III, but for different values of eccentricity e; the Teukolsky-based fluxes for E and Lz have an accuracy of 10
e θinc ι ∆t/M ∆θinc ∆ι
(deg.) (deg.) (deg.) (deg.)
0 0 0 1.250×106 0 0
0 5 5.355510 1.217×106 1.949×10−1 4.954×10−1
0 10 10.679331 1.118×106 3.468×10−1 8.631×10−1
0 15 15.943192 9.574×105 4.236×10−1 1.019
0 20 21.125167 7.446×105 4.109×10−1 9.440×10−1
0 25 26.211779 4.981×105 3.158×10−1 6.860×10−1
0 30 31.199048 2.528×105 1.732×10−1 3.527×10−1
0 35 36.092514 6.584×104 4.636×10−2 8.806×10−2
0.1 0 0 1.228×106 0 0
0.1 5 5.351602 1.198×106 4.517×10−1 7.766×10−1
0.1 10 10.671900 1.103×106 6.900×10−1 1.236
0.1 15 15.932962 9.426×105 7.283×10−1 1.344
0.1 20 21.113129 7.315×105 6.433×10−1 1.187
0.1 25 26.199088 4.900×105 4.780×10−1 8.547×10−1
0.1 30 31.186915 2.513×105 2.730×10−1 4.585×10−1
0.1 35 36.082095 6.589×104 8.385×10−2 1.279×10−1
0.2 0 0 1.173×106 0 0
0.2 5 5.339916 1.150×106 1.204 1.598
0.2 10 10.649670 1.064×106 1.698 2.331
0.2 15 15.902348 9.043×105 1.618 2.293
0.2 20 21.077081 6.980×105 1.324 1.900
0.2 25 26.161046 4.693×105 9.545×10−1 1.351
0.2 30 31.150481 2.486×105 5.674×10−1 7.711×10−1
0.2 35 36.050712 7.562×104 2.070×10−1 2.648×10−1
0.3 0 0 1.087×106 0 0
0.3 5 5.320559 1.069×106 2.307 2.788
0.3 10 10.612831 1.001×106 3.256 4.007
0.3 15 15.851572 8.454×105 2.984 3.741
0.3 20 21.017212 6.483×105 2.375 2.998
0.3 25 26.097732 4.408×105 1.700 2.129
0.3 30 31.089639 2.493×105 1.040 1.276
0.3 35 35.997987 1.108×105 4.626×10−1 5.569×10−1
Table V: Variation in the inclination angles ι and θinc as well as time needed to reach the separatrix for several inspirals through the nearly
horizon-skimming regime. In all of these cases, the binary’s mass ratio was fixed to µ/M = 10−6, the large black hole’s spin was fixed to
a = 0.998M , and the orbits were begun at p = 1.9M . The time interval ∆t is the total accumulated time it takes for the inspiralling body
to reach the separatrix (at which time it rapidly plunges into the black hole). The angles ∆θinc and ∆ι are the total integrated change in these
inclination angles that we compute. For the e = 0 cases, inspirals are computed using fits to the circular-Teukolsky fluxes of E and Lz ; for
eccentric orbits we use the kludge fluxes (40), (43) and (44). Notice that ∆θinc and ∆ι are always positive — the inclination angle always
increases during the inspiral through the nearly horizon-skimming region. The magnitude of this increase never exceeds a few degrees.
