# Discrete Orbit Effect
Lengthens Merger Times for
Inspiraling Binary Black Holes

Don N. Page
Department of Physics
4-183 CCIS
University of Alberta
Edmonton, Alberta T6G 2E1
Canada Internet address<EMAIL_ADDRESS>
(2024 May 21)

###### Abstract

The inspiral merger time for two black holes captured into a nonrelativistic
bound orbit by gravitational radiation emission has been often calculated by a
formula of Peters that assumes the adiabatic approximation that the changes
per orbit are small. However, initially this is not true for the semimajor
axis and period of most of the initially highly eccentric orbits, which change
significantly during closest approach and much less elsewhere along the orbit.
This effect can make the merger time much longer (using other formulas from
Peters that do not assume the adiabatic approximation) than that calculated by
the adiabatic formula of Peters.

##  1 Introduction

The motivation for studying the inspiraling of two black holes has increased
greatly with the enormous success that LIGO and Virgo have had in detecting
the gravitational waves from astrophysical black holes that coalesce to form a
larger black hole [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
18, 19, 20, 21, 22, 23, 24].

In order to learn about the possible origin and evolution of the coalescing
black holes, many scenarios have been investigated, far too many for me to
cite here, though a sample relevant to my considerations is [25, 26, 27, 28,
29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47,
48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60]. Instead, I wish to focus
on scenarios in which two isolated black holes (i.e., not originally a bound
binary pair) become bound through the emission of gravitational radiation and
then inspiral primarily though the process of emitting more gravitational
radiation. That is, I shall ignore the effects of any third-party masses on
the formation and evolution of the bound pair.

To set the notation, I shall adopt the center-of-momentum frame and start with
two initially unbound black holes of masses $M_{1}$ and $M_{2}$ far away from
each other,

$\mathrm{total\ mass\ }\equiv M\equiv M_{1}+M_{2},$ (1) $\mathrm{reduced\
mass\ }\equiv\mu\equiv\frac{M_{1}M_{2}}{M_{1}+M_{2}},$ (2) $\mathrm{symmetric\
mass\ ratio\
}\equiv\eta\equiv\frac{\mu}{M}\equiv\frac{M_{1}M_{2}}{(M_{1}+M_{2})^{2}}\leq\frac{1}{4},$
(3) $\mathrm{initial\ relative\ velocity\ }\equiv v\ll\eta^{1/2}c,$ (4)
$\mathrm{initial\ dimensionless\ relative\ velocity\
}\equiv\beta\equiv\frac{v}{c}\ll\eta^{1/2},$ (5) $\mathrm{initial\
nonrelativistic\ energy\ }\equiv E_{0}=\frac{1}{2}\mu
v^{2}=\frac{1}{2}Mc^{2}\eta\beta^{2}\ll\eta\mu c^{2},$ (6) $\mathrm{relative\
impact\ parameter\ }\equiv b\gg\frac{4GM}{cv},$ (7) $\mathrm{initial\ angular\
momentum\ }L_{0}=\mu bv=Mcb\eta\beta,$ (8)

and

$\mathrm{convenient\ timescale\ }\equiv T_{0}\equiv\frac{2\pi
GM}{v^{3}}=(0.264\
\mathrm{year})\left(\frac{M}{10M_{\odot}}\right)\left(\frac{100\
\mathrm{km/s}}{v}\right)^{3}.$ (9)

I assume that the impact parameter is below the critical value $b_{c}$ (given
below), so that the two black holes are captured into a bound orbit by
emitting gravitational radiation, in the approximation of otherwise empty
spacetime, and I assume that the bound orbit is initially nonrelativistic,
which requires the inequality given above that $b\gg 4GM/(cv)$. (Note below
that even in the formulas for the Keplerian orbits, $v$ or $\beta$ without a
subscript is always the initial relative velocity or its dimensionless form
when the black holes were unbound and not any velocity during an orbit; I use
instead $v_{p}$ for the velocity, and $\beta_{p}=v_{p}/c$ as its dimensionless
form, at periapsis, the point of smallest separation distance, $r_{p}$, during
an orbit.)

I further assume that the two black holes are approaching each other with a
uniform probability distribution in the transverse plane (probability per
small range of impact parameter proportional to the impact parameter). There
is then a critical impact parameter $b_{c}$, proportional to the Schwarzschild
radius of the total mass $M=M_{1}+M_{2}$, multiplied by the 7th root of the
symmetric mass ratio $\eta$, and divided by the 9/7 power of the dimensionless
initial velocity $\beta$, such that for impact parameters $b$ larger than
$b_{c}$, the two black holes scatter without becoming bound, but for
$b<b_{c}$, the black holes emit enough radiation in their initial close
encounter that they become bound. Conditionalizing on $b<b_{c}$, the
cumulative probability for an impact parameter smaller than $b$ is then

$P=\left(\frac{b}{b_{c}}\right)^{2},$ (10)

the fraction $(b/b_{c})^{2}$ of the total area inside the capture disk of
radius $b_{c}$ that also has the impact parameter vector of magnitude smaller
than $b$. This cumulative conditional probability $P$ then serves as a
convenient dimensionless parameter characterizing the impact parameter $b$.

As a result, the coordinate-independent properties of the $n$th orbit (taken
to begin at the $n$th periapsis and end at the $(n+1)$st one) are determined
by the total mass $M$, the symmetric mass ratio $\eta$, the initial
dimensionless velocity $\beta=v/c$, the dimensionless cumulative conditional
probability $P=(b/b_{c})^{2}$, the number $n$ of the orbit, and whatever
combination of $G$ and $c$ that are needed to give the dimensions of the
dimensional quantities. One such quantity is

$\mathrm{the\ {\it{n}}th\ orbit\ time\ period\
}\equiv\tau_{n}=2\pi\sqrt{\frac{a_{n}^{3}}{GM}},$ (11)

with $a_{n}$ the semimajor axis of the $n$th nonrelativistic Keplerian orbit.
Then

$\mathrm{total\ merger\ inspiral\ time\ }\equiv T\equiv\sum_{n}\tau_{n}$ (12)

where the sum is taken over all the orbits until coalescence. (For impact
parameter $b\gg 4GM/(cv)$ as I assume, the number of nonrelativistic orbits
will be large, but with decreasing periods $\tau_{n}$, so an approximate
formula for this decrease of the periods can be summed as if it were an
infinite series, since the error of the terms with large $n$ will be small in
comparison with the contribution from the first several $n$. But the absolute
error will presumably go as $(2GM/c^{3})/\beta^{s}$ with some positive
exponent $s$ that is smaller than the $s=3$ exponent that applies for the
actual timescale that will be of the order of $T_{0}=(2\pi
GM/c^{3})/\beta^{3}$ multiplied by some dimensionless function of $P$.)

Calculations of the capture and subsequent inspiraling of the black holes
usually start with the formulas of Philip Carl Peters and of Peters and Jon
Mathews [25, 26, 27], who show that for a nonrelativistic Keplerian orbit of
semimajor axis $a$ and eccentricity $e$, the power radiated into gravitational
waves, averaged over one orbit, is (Eq. (16) in [25], the same copied on page
92 of [26], and Eq. (5.4) of [27])

$\displaystyle\left\langle-\frac{dE}{dt}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{32}{5}\frac{G^{4}}{c^{5}}\frac{M_{1}^{2}M_{2}^{2}(M_{1}+M_{2})}{a^{5}(1-e^{2})^{7/2}}\left(1+\frac{73}{24}e^{2}+\frac{37}{96}e^{4}\right)$
(13) $\displaystyle=$
$\displaystyle\frac{32}{5}\frac{G^{4}}{c^{5}}\frac{M^{5}\eta^{2}}{a^{5}(1-e^{2})^{7/2}}\left(1+\frac{73}{24}e^{2}+\frac{37}{96}e^{4}\right).$

Similarly, the angular momentum radiated into gravitational waves, averaged
over one orbit, is (Eq. (5.32) on page 99 of [26], and Eq. (5.5) of [27])

$\displaystyle\left\langle-\frac{dL}{dt}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{32}{5}\frac{G^{7/2}}{c^{5}}\frac{M_{1}^{2}M_{2}^{2}(M_{1}+M_{2})^{1/2}}{a^{7/2}(1-e^{2})^{2}}\left(1+\frac{7}{8}e^{2}\right)$
(14) $\displaystyle=$
$\displaystyle\frac{32}{5}\frac{G^{7/2}}{c^{5}}\frac{M^{9/2}\eta^{2}}{a^{7/2}(1-e^{2})^{2}}\left(1+\frac{7}{8}e^{2}\right).$

Since a Keplerian elliptical orbit has energy
$E=-GM_{1}M_{2}/(2a)=-GM^{2}\eta/(2a)$, angular momentum
$L=\sqrt{GM^{3}\eta^{2}a(1-e^{2})}$, and orbital period
$\tau=2\pi\sqrt{a^{3}/(GM)}$, the energy and angular momentum radiated during
one orbit are

$\displaystyle-\Delta E$ $\displaystyle=$
$\displaystyle\tau\left\langle-\frac{dE}{dt}\right\rangle=\frac{64\pi}{5}\frac{G^{7/2}M^{9/2}\eta^{2}}{c^{5}a^{7/2}(1-e^{2})^{7/2}}\left(1+\frac{73}{24}e^{2}+\frac{37}{96}e^{4}\right),$
(15) $\displaystyle-\Delta L$ $\displaystyle=$
$\displaystyle\tau\left\langle-\frac{dL}{dt}\right\rangle=\frac{64\pi}{5}\frac{G^{3}M^{4}\eta^{2}}{c^{5}a^{2}(1-e^{2})^{2}}\left(1+\frac{7}{8}e^{2}\right).$
(16)

Then for simplicity defining

$f\equiv 1-e^{2}=\frac{-2EL^{2}}{G^{2}M^{5}\eta^{3}}$ (17)

(the square of the ratio of the semiminor axis $a(1-e^{2})^{1/2}$ to the the
semimajor axis $a$ of the Keplerian elliptical orbit), one gets the
dimensionless ratios

$-\frac{\Delta
E}{E_{0}}=\frac{340\pi}{3}\left(\frac{GM^{2}\eta}{cL}\right)^{7}\beta^{-2}\eta\left(1-\frac{366}{425}f+\frac{37}{425}f^{2}\right),$
(18) $-\frac{\Delta
L}{L}=24\pi\left(\frac{GM^{2}\eta}{cL}\right)^{5}\eta\left(1-\frac{7}{15}f\right),$
(19)

which are now in a form that can be applied even for the initially hyperbolic
orbit.

For the two initially unbound black holes with positive initial
nonrelativistic energy $E_{0}=(1/2)\mu v^{2}$ (initially very far apart, with
negligible potential energy) to become bound into a nonrelativistic Keplerian
orbit with negative nonrelativistic energy
$E_{1}=-GM_{1}M_{2}/(2a_{1})=-GM^{2}\eta/(2a_{1})$ (including the Newtonian
potential energy $-GM_{1}M_{2}/r=-GM^{2}\eta/r$), the value of $-\Delta
E/E_{0}$ during the transition between the initial hyperbolic orbit and the
first elliptical orbit (taken to begin at the initial periapsis) must be
greater than 1, so the initial angular momentum $L_{0}=\mu bv$ must have the
impact parameter $b$ be less than a critical impact parameter $b_{c}$
(depending linearly on the Schwarzschild radius $2GM/c^{2}$ of the total mass
with a coefficient that is a pure number multiplying powers of the symmetric
mass ratio, $\eta$, and of the dimensionless relative initial velocity,
$\beta=v/c$, to be given shortly).

One can readily show that this requirement of $-\Delta E/E_{0}>1$ with
$\beta\ll 1$ implies that the initial hyperbolic orbit must have its
eccentricity $e$ just slightly larger than 1 (a necessary but not sufficient
condition for capture when $v\ll c$ or $\beta\ll 1$), so the initial
$f=1-e^{2}$ is just slightly negative. Conversely, for the initial elliptical
orbit after $E$ becomes negative to be nonrelativistic (periapsis distance
much larger than the Schwarzschild radius corresponding to the total mass
$M$), one needs the eccentricity $e$ right after binding to be just slightly
smaller than 1, so for the initial orbits at least, $f$ is just slightly
negative. These facts allow one to drop the terms with nonzero $f$ in the
expressions above for the initial orbit and easily to get from them a good
approximation for the critical impact parameter,

$b_{c}=\left(\frac{85\pi}{384}\right)^{1/7}\left(4\eta\right)^{1/7}\beta^{-9/7}\left(\frac{2GM}{c^{2}}\right)\approx
0.949\,429\left(4\eta\right)^{1/7}\beta^{-9/7}\left(\frac{2GM}{c^{2}}\right).$
(20)

Here I have used $4\eta=4M_{1}M_{2}/(M_{1}+M_{2})^{2}$ with the factor of 4 so
that its upper bound is 1, and I have included the 2 in $2GM/c^{2}$ so that it
is the Schwarzschild radius of the total mass $M$, which coincidentally makes
the leading numerical factor only about 5% less than of unity.

After I independently derived Eq. (20) above (see [61] for more details) from
the equations of Peters and Mathews [25, 26, 27], an editor pointed out the
paper of R. O. Hansen [28], which gave two complicated formulas relating the
initial energy, initial eccentricity, and initial impact parameter for the
critical cases (with fixed masses), but it did not use the fact that one can
neglect $f=1-e^{2}$ to get the explicit formula above for the critical impact
parameter in terms of the initial relative velocity $v$ (and hence initial
energy $E_{0}=(1/2)\mu v^{2}$, for fixed given masses). However, I found that
essentially the formula above was given by Gerald D. Quinlan and Stuart L.
Shapiro [30]. Hideaki Mouri and Yoshiaki Taniguchi [34] also got the same
formula but from the starting point of calculations of gravitational radiation
from parabolic orbits by Michael Turner [29] rather than from Peters and
Mathews [25, 26, 27]. A later paper giving the formula (presumably
independently, since not mentioning that Quinlan and Shapiro had given it,
though they were cited for other reasons) is [38].

All of the papers I found [31, 38, 41, 42, 44, 45, 50, 51, 47, 53, 55, 56, 57,
60] using the formula for the critical impact parameter $b_{c}$, as a starting
point for calculating the inspiral merger time, used Eq. (5.14) of [27], or
the unnumbered last equation of that paper by Peters that is the approximation
for initial eccentricity $e_{0}$ near 1 (equivalent to my $f_{1}\equiv
1-e_{1}^{2}\ll 1$ for the first bound orbit),

$T(a_{0},e_{0})\approx(768/425)T_{c}(a_{0})(1-e_{0}^{2})^{7/2},$ (21)

where Eq. (5.10) of that paper gave the decay time for a circular orbit as

$T_{c}(a_{0})=a_{0}^{4}/(4\beta_{P}),$ (22)

with the preceding unnumbered equation giving

$\beta_{P}=\frac{64}{5}\frac{G^{3}m_{1}m_{2}(m_{1}+m_{2})}{c^{5}}$ (23)

(and with Peters’ $m_{1}$ and $m_{2}$ being the same as my $M_{1}$ and
$M_{2}$; here I have added the subscript $P$ to Peters’ $\beta$, which has the
dimension of volume times velocity, so that it would not be confused with my
dimensionless $\beta$ for the initial velocity or $\beta_{p}$ for the
dimensionless velocity at periapsis below).

However, this Eq. (21) assumes an adiabatic approximation for the evolution of
the semimajor axis $a$ and resulting orbit period that I shall call $\tau$,

$\tau=2\pi\sqrt{\frac{a^{3}}{GM}},$ (24)

namely, that the fractional change in $a$ and $\tau$ from one orbit to the
next is small. This approximation is not valid for black hole binaries that
become bound by the emission of gravitational radiation (a situation not
discussed by Peters, so he can hardly be faulted for not pointing out that his
adiabatic approximation would not apply in such cases). Instead, both the
length (major axis) $2a$ and the period $\tau$ of several of the initial
orbits after the black holes become bound change significantly from one orbit
to the next, and Eq. (21) severely underestimates the total merger inspiral
time, especially when the impact parameter $b$ is just slightly smaller than
the critical impact parameter given by Eq. (20).

##  2 Properties of the Orbits and Total Merger Time

We have noted above that in principle the dimensionless properties of the
$n$th orbit, in our idealized situation of two black holes coming in
nonrelativistically from far away with no other perturbing masses around, are
determined by the orbit number $n$, the symmetric mass ratio
$\eta=M_{1}M_{2}/(M_{1}+M_{2})^{2}$, the dimensionless initial relative
velocity $\beta=v/c\ll\eta^{1/2}$, and the cumulative conditional probability
$P=(b/b_{c})^{2}<1$ (the square of the ratio of the actual impact parameter
$b$ to the critical impact parameter $b_{c}$ for capture into a bound orbit by
the emission of gravitational radiation). However, to get a quantative
analysis of the orbits, it is helpful also to define some other dimensionless
constants and variables. After completing this section I found that some, but
certainly not all, of the first part of this section has similarities with the
analysis of [45], though that paper does not discuss the merger time.

First are two dimensionless constants that only depend on the symmetric mass
ratio $\eta$ and on the dimensionless initial relative velocity $\beta$:

$\beta_{c}\equiv\left(\frac{96}{85\pi}\beta^{2}\frac{1}{\eta}\right)^{1/7}=\frac{2GM}{c^{2}\beta
b_{c}}=\frac{2GM/(cv)}{b_{c}},$ (25)
$\delta\equiv\frac{2\beta}{\beta_{c}}\equiv\left(\frac{340\pi}{3}\beta^{5}\eta\right)^{1/7}=\frac{b_{c}\beta^{2}}{GM/c^{2}}=\frac{b_{c}}{GM/v^{2}}=\left(\frac{85\pi}{24}\eta\right)^{1/2}\beta_{c}^{5/2}.$
(26)

The constant $\beta_{c}$ is the dimensionless velocity $\beta_{p}=v_{p}/c$ at
the periapsis of the first orbit in the limit that the impact parameter $b$
approaches the critical impact parameter $b_{c}$ from below. The constant
$\delta$ is the ratio of the critical impact parameter $b_{c}$ to the
lengthscale $GM/v^{2}=T_{0}v/(2\pi)$. Since $\eta\leq 1/4$ and we have assumed
$\beta\ll 1$, we always have $\delta\ll 1$. However, to satisfy our assumption
of nonrelativistic orbits after the two black holes become bound, we need also
to have $\beta_{c}\ll 1$, which is equivalent to the left-hand inequality of

$\beta\ll\eta^{1/2}\leq 1/2.$ (27)

(The right-hand inequality, $\eta^{1/2}\leq 1/2$, follows immediately from the
definition of the symmetric mass ratio $\eta\equiv
M_{1}M_{2}/(M_{1}+M_{2})^{2}$.)

Next, we can define dimensionless constants that depend not only on $\eta$ and
$\beta$ but also on the dimensionless squared impact parameter
$P=(b/b_{c})^{2}$:

$\beta_{p}\equiv
P^{-1/2}\beta_{c}\equiv\left(\frac{96}{85\pi}\beta^{2}\frac{1}{\eta}\right)^{1/7}P^{-1/2}=\frac{2GM}{c^{2}\beta
b_{c}P^{1/2}}=\frac{2GM/(cv)}{b},$ (28) $\epsilon\equiv
P^{-5/4}\delta\equiv\left(\frac{340\pi}{3}\beta^{5}\eta\right)^{1/7}P^{-5/4}=\frac{\beta^{2}b_{c}P^{-5/4}}{GM/c^{2}}=\frac{bP^{-7/4}}{GM/v^{2}}=\left(\frac{85\pi}{24}\eta\right)^{1/2}\beta_{p}^{5/2}.$
(29)

The constant $\beta_{p}$ is, when it is small, a very good approximation for
the dimensionless velocity maximum during the first bound orbit (i.e., at
periapsis, the minimum separation distance $r_{p}$), namely $v_{p}/c$ for the
initial bound orbit with impact parameter $b=b_{c}P^{1/2}\gg 4GM/(cv)$, with
$4GM/(cv)$ being roughly the impact parameter below which there is direct
coalescence without a period of nonrelativistic inspiral. (The actual maximum
impact parameter for direct coalescence is $4GM/(cv)$ multiplied by an order-
one function of $\eta$ and $\beta$ that goes to 1 as both of those constants
are taken to 0.) By our general assumption that there are nonrelativistic
inspiraling orbits after the two black holes become bound, we are requiring
that $\beta_{p}\ll 1$, which is equivalent to $4GM/(cv)\ll b<b_{c}$, or

$\frac{\beta^{4/7}}{\eta^{2/7}}\equiv\frac{(v/c)^{4/7}}{[M_{1}M_{2}/(M_{1}+M_{2})^{2}]^{2/7}}\ll
P\equiv\left(\frac{b}{b_{c}}\right)^{2}<1.$ (30)

(Remember that $v$ is always the initial relative velocity of the two black
holes when they are far apart and unbound, and $\beta=v/c$.) Note that the
nonrelativistic condition $\beta_{p}\ll 1$ implies that
$\epsilon=(85\pi\eta/24)^{1/2}\beta_{p}^{5/2}\approx
1.667\,818(4\eta)^{1/2}\beta_{p}^{5/2}\ll 1$.

As we shall see shortly, the fractional change per orbit of the angular
momentum given by Eq. (19) is proportional to $\epsilon^{2}$, as is the
relative change per orbit of the eccentricity $e=\sqrt{1-f}$ and the absolute
change of $f\equiv 1-e^{2}$ (though not the relative change of $f$, $E$, $a$,
or $\tau$, which can be large per orbit; they are orbit parameters that do not
change adiabatically when the orbit number $n$ is not large). Therefore, for
the angular momentum to change adiabatically, one needs $\epsilon\ll 1$, which
is implied by the condition being assumed of nonrelativistic velocities at
periapsis, $\beta_{p}\ll 1$, which is equivalent to $(\beta^{2}/\eta)^{2/7}\ll
P=(b/b_{c})^{2}$.

After defining these small constants characterizing the initial conditions, it
is also convenient to define dimensionless measures of the time-dependent
energy $E$ and angular momentum $L$, each of which are nearly constant for the
major part of each orbit in which the separation distance $r$ between the
black holes is much larger than the periapsis distance $r_{p}$ for that orbit,
but which change mostly during the relatively short time when $r$ is near
$r_{p}$. That is, it is a good approximation to take the energy $E$ to have
the constant value $E_{n}$ during most of the $n$th orbit and then rather
suddenly change to $E_{n+1}$ as the next periapsis is passed, and similarly
for the angular momentum $L$ to change rather suddenly from $L_{n}$ to
$L_{n+1}$.

In particular, define the orbit-dependent (i.e., depending on the orbit number
$n$) dimensionless variables

$\lambda(n)\equiv\frac{L_{n}}{L_{0}}=\frac{L}{M\eta vb_{c}P^{1/2}},$ (31)

which has the initial value $\lambda(0)=1$, and

$\nu(n)\equiv
P^{7/2}\left(\frac{-E_{n}}{E_{0}}\right)=P^{7/2}\left(\frac{-2E_{n}}{M\eta
v^{2}}\right).$ (32)

which has the initial value $\nu(0)=-P^{7/2}$.

In terms of the small constant $\epsilon^{2}$ and the variables $\nu(n)$ and
$\lambda(n)$, the eccentricity factor for the $n$th orbit is

$f(n)\equiv 1-e(n)^{2}=\epsilon^{2}\lambda(n)^{2}\nu(n).$ (33)

Although $\nu(n)$ has relatively large jumps as $n$ is increased, $\lambda(n)$
and $e(n)$ change relatively slowly (adiabatically), so Eq. (5.11) of [27]
applies, which in my variables (suppressing the $n$ arguments for brevity
henceforth for awhile) gives

$\lambda=\left(1-f\right)^{3/19}\left(1-\frac{121}{425}f\right)^{435/2299}=1-\frac{18}{85}f+O(f^{2}).$
(34)

Then one gets for the changes in $\lambda$ and $\nu$ for one orbit to be

$\Delta\lambda=-\frac{18}{85}\epsilon^{2}\left(1-f\right)^{-12/19}\left(1-\frac{121}{425}f\right)^{1740/2299}\left(1-\frac{7}{15}f\right),$
(35)
$\Delta\nu=\left(1-f\right)^{-21/19}\left(1-\frac{121}{425}f\right)^{3045/2299}\left(1-\frac{366}{425}f+\frac{31}{425}f^{2}\right)=1+\frac{264}{425}f+O(f^{2}),$
(36) $\Delta
T=\tau_{n}=T_{0}P^{21/4}\nu^{-3/2}\left[1+O(\beta_{p}^{2})\right].$ (37)

For $\epsilon^{2}\ll 1$, $\lambda$ stays near 1 for many orbits, but $\nu$
increases by very slightly more than 1 for each orbit, giving
$f\approx\epsilon^{2}(n-P^{7/2})$ for $n\ll
1/\epsilon^{2}\sim\eta^{-2/7}\beta^{-5/7}P^{5/2}$. Then Eq. (36) implies that
$\Delta\nu(n)\approx 1+(264/425)\epsilon^{2}(n-P^{7/2})$, which, along with
the initial condition $\nu(0)=-P^{7/2}$, leads to

$\nu(n)=-P^{7/2}+n+\frac{132}{425}\epsilon^{2}(n^{2}+n-2nP^{7/2})+O(\epsilon^{4}n^{3})+O(\beta_{p}^{2}n).$
(38)

Inserting this back into Eq. (37) and summing over $n$ gives an estimate of
the total inspiraling merger time as

$\displaystyle T\\!\\!$ $\displaystyle\approx$
$\displaystyle\\!\\!\sum_{n=1}^{\infty}\tau_{n}=T_{0}P^{21/4}\sum_{n=1}^{\infty}\nu(n)^{-3/2}\approx
T_{0}P^{21/4}\sum_{n=1}^{\infty}\left(-P^{7/4}+n+\frac{132}{425}\epsilon^{2}n^{2}\right)^{-3/2}$
(39) $\displaystyle\approx$ $\displaystyle\\!\\!\frac{2\pi
GM}{v^{3}}P^{21/4}\left[\zeta\left(\frac{3}{2},1-P^{7/2}\right)-\frac{8}{5}\sqrt{\frac{33}{17}}\left(\frac{340\pi}{3}\beta^{5}\eta\right)^{1/7}P^{-5/4}+O(\beta_{p}^{2})\right]\\!\\!.$

If $\beta_{p}^{2}\equiv[96/85\pi)]^{2/7}(\beta^{2}/\eta)^{2/7}P^{-1}$ and
hence also $\epsilon^{2}\equiv[(340\pi/3)\beta^{5}\eta]^{2/7}P^{-5/2}$ are
small enough that we can neglect them, a highly precise simple elementary
function approximation for the zeta function, using $z\equiv\zeta(3/2)\approx
2.612\,375$, gives

$T\approx\frac{2\pi
GM}{v^{3}}\left(\frac{P^{7/2}}{1-P^{7/2}}\right)^{3/2}\left[1+(1-P^{7/2})^{3/2}\left(z-\frac{1-P^{7/2}}{1-0.5P^{7/2}}\right)\right],$
(40)

which has a relative error always less than 0.07% (assuming $\beta_{p}^{2}$ is
negligible).

The appearance is a bit simpler in terms of

$x\equiv 1-P^{7/2}\equiv 1-(b/b_{c})^{7}$ (41)

and a dimensionless time

$t\equiv T/T_{0}=t/(2\pi GM/v^{3})\approx P^{21/4}\zeta(3/2,1-P^{7/2})$ (42)

with the timescale $T_{0}\equiv 2\pi GM/v^{3}$ from Eq. (9). Then, again
neglecting terms proportional to $\beta_{p}^{2}$ and $\epsilon^{2}$ and higher
powers of these quantities, and using

$z\equiv\zeta(3/2)\approx 2.612\,375,$ (43)

one gets

$T\equiv T_{0}\,t(P)\approx
T_{0}(1-x)^{3/2}\zeta(3/2,x)\approx\left(\frac{1-x}{x}\right)^{3/2}\left[1+x^{3/2}\left(z-\frac{2x}{1+x}\right)\right].$
(44)

##  3 Probability Distribution of the Inspiral Time

From the inspiral merger time $T=T_{0}\,t(P)$ with $T_{0}\equiv 2\pi GM/v^{3}$
when
$\beta_{p}^{2}\equiv[96/85\pi)]^{2/7}(\beta^{2}/\eta)^{2/7}P^{-1}$ can be
neglected, one can get the probability density per logarithmic interval
$d\ln{T}=dT/T$ of the inspiral time $T$ as

$\rho\equiv\frac{dP}{d\ln{T}}=t\frac{dP}{dt}\approx\frac{4}{21}P\left[1+\frac{P^{7/2}\zeta(5/2,1-P^{7/2})}{\zeta(3/2,1-P^{7/2})}\right]^{-1}.$
(45)

The following table gives numerical data, with the first column giving the
cumulative conditional probability $P=(b/b_{c})^{2}$ where $b_{c}$, given by
Eq. (20), is the critical impact parameter for capture, the second column
gives the dimensionless inspiral merger time $t=T/T_{0}=T/(2\pi
GM/v^{3})\approx P^{21/4}\zeta(3/2,1-P^{7/2})$, the third column giving the
inspiral time $T$ for $M_{1}=M_{2}=5M_{\odot}$ (so that $M=10M_{\odot}$ and
$\eta=1/4$) and for $v=100$ km/s $\approx(1/3)10^{-3}c$ (as well as estimates
for $\beta_{p}^{2}=(v_{p}/c)^{2}\sim 2GM/r_{p}$ at periapsis), and the fourth
column gives the probability density per logarithmic interval of the inspiral
time $T$, namely $\rho=dP/d\ln{T}=dP/d\ln{t}$. Note that with $\eta=1/4$ and
this value of $v$, one gets $\beta_{p}^{2}\sim 0.01/P$, which is not very
small when $P$ is small, giving uncalculated relative errors for $T$ of the
order of $\beta_{p}^{2}$. Thus the full 7 digits given for $t=T/T_{0}$ in the
second column and for $\rho=dP/d\ln{T}$ in the fourth column are only all
relevant for
$\beta_{p}^{2}\sim[\beta^{2}/(4\eta)]^{2/7}/P\stackrel{{\scriptstyle<}}{{\sim}}10^{-6}$
or
$\beta\sim(4\eta)^{1/2}P^{7/4}\beta_{p}^{7/2}\stackrel{{\scriptstyle<}}{{\sim}}10^{-11}(4\eta)^{1/2}P^{7/4}$,
but they are given for completeness.

$P=\left(\frac{b}{b_{c}}\right)^{2}$ | $t\equiv T/(2\pi GM/v^{3})$ |  $T$ for $10\,M_{\odot}$, 100 km/s | $\rho=\frac{dP}{d\ln{T}}=T\frac{dP}{dT}$
---|---|---|---
0 | 0 |  $\sim 2\pi GM/c^{3}\sim 10^{-11}$ yr | 0
0.01 | $8.261\,057\times 10^{-11}$ | direct coalescence | 0.001 904 762
0.02 | $3.143\,717\times 10^{-9}$ | direct coalescence | 0.003 809 522
0.05 | $3.860\,446\times 10^{-7}$ | nearly direct coalescence | 0.009 523 673
0.10 | $1.469\,405\times 10^{-5}$ |  $4\times 10^{-6}$ yr, $\beta_{p}^{2}\sim 1/10$ | 0.019 044 525
0.15 | $1.235\,812\times 10^{-4}$ |  $3\times 10^{-5}$ yr, $\beta_{p}^{2}\sim 1/15$ | 0.028 552 230
0.20 | $5.605\,866\times 10^{-4}$ |  $1.5\times 10^{-4}$ yr, $\beta_{p}^{2}\sim 1/20$ | 0.038 025 045
0.25 | $1.814\,879\times 10^{-3}$ |  $4.8\times 10^{-4}$ yr, $\beta_{p}^{2}\sim 1/25$ | 0.047 426 785
0.30 | $4.752\,460\times 10^{-3}$ |  $1.3\times 10^{-3}$ yr, $\beta_{p}^{2}\sim 1/30$ | 0.056 703 625
0.35 | $1.076\,525\times 10^{-2}$ |  $2.8\times 10^{-3}$ yr, $\beta_{p}^{2}\sim 1/35$ | 0.065 780 052
0.40 | $2.196\,685\times 10^{-2}$ |  $5.8\times 10^{-3}$ yr, $\beta_{p}^{2}\sim 1/40$ | 0.074 553 181
0.45 | $4.146\,907\times 10^{-2}$ |  $1.10\times 10^{-2}$ yr, $\beta_{p}^{2}\sim 1/45$ | 0.082 884 298
0.50 | $7.380\,184\times 10^{-2}$ |  $1.95\times 10^{-2}$ yr, $\beta_{p}^{2}\sim 1/50$ | 0.090 586 038
0.55 | $1.255\,964\times 10^{-1}$ |  $3.32\times 10^{-2}$ yr, $\beta_{p}^{2}\sim 1/55$ | 0.097 402 690
0.60 | $2.067\,776\times 10^{-1}$ |  $5.46\times 10^{-2}$ yr, $\beta_{p}^{2}\sim 1/60$ | 0.102 979 918
0.65 | $3.328\,098\times 10^{-1}$ |  $8.79\times 10^{-2}$ yr, $\beta_{p}^{2}\sim 1/65$ | 0.106 818 463
0.70 | $5.293\,246\times 10^{-1}$ |  $1.40\times 10^{-1}$ yr, $\beta_{p}^{2}\sim 1/70$ | 0.108 205 199
0.75 | $8.427\,883\times 10^{-1}$ |  $2.23\times 10^{-1}$ yr, $\beta_{p}^{2}\sim 1/75$ | 0.106 118 525
0.80 | 1.369 021 |  $3.62\times 10^{-1}$ yr, $\beta_{p}^{2}\sim 1/80$ | 0.099 129 628
0.85 | 2.347 172 |  $6.20\times 10^{-1}$ yr, $\beta_{p}^{2}\sim 1/85$ | 0.085 418 810
0.90 | 4.590 029 | 1.21 years, $\beta_{p}^{2}\sim 1/90$ | 0.063 315 302
0.95 | $1.324\,718\times 10^{1}$ | 3.50 years, $\beta_{p}^{2}\sim 1/95$ | 0.033 208 069
0.98 | $5.265\,574\times 10^{1}$ |  $1.39\times 10^{1}$ years, $\beta_{p}^{2}\sim 0.01$ | 0.013 250 261
0.99 | $1.500\,313\times 10^{2}$ |  $3.96\times 10^{1}$ years, $\beta_{p}^{2}\sim 0.01$ | 0.006 619 004
0.999 | $4.815\,761\times 10^{3}$ |  $1.27\times 10^{3}$ years, $\beta_{p}^{2}\sim 0.01$ | $6.655\,242\times 10^{-4}$
0.999 9 | $1.526\,718\times 10^{5}$ |  $4.03\times 10^{4}$ years, $\beta_{p}^{2}\sim 0.01$ | $6.665\,281\times 10^{-5}$
0.999 99 | $4.829\,293\times 10^{6}$ |  $1.28\times 10^{6}$ years, $\beta_{p}^{2}\sim 0.01$ | $6.666\,520\times 10^{-6}$
0.999 999 | $1.527\,202\times 10^{8}$ |  $4.04\times 10^{7}$ years, $\beta_{p}^{2}\sim 0.01$ | $6.666\,652\times 10^{-7}$
0.999 999 9 | $4.829\,451\times 10^{9}$ |  $1.28\times 10^{9}$ years, $\beta_{p}^{2}\sim 0.01$ | $6.666\,665\times 10^{-8}$
0.999 999 99 | $1.527\,207\times 10^{11}$ |  $4.04\times 10^{10}$ years, $\beta_{p}^{2}\sim 0.01$ | $6.666\,666\times 10^{-9}$
0.999 999 999 | $4.829\,452\times 10^{12}$ |  $1.28\times 10^{12}$ years, $\beta_{p}^{2}\sim 0.01$ | $6.666\,667\times 10^{-10}$

Figure 1: The top graph gives the cumulative conditional probability
$P=(b/b_{c})^{2}$ for impact parameters smaller than $b$, given that they are
smaller than the critical impact parameter $b_{c}=[340\pi
G^{7}M^{7}\eta/(3c^{14}\beta^{9})]^{1/7}$ for capture into bound orbits, and
the bottom graph gives $(\ln{10})*\rho=dP/d\log{t}$, both as functions of
$\log{t}$, the common logarithm of the dimensionless inspiral time $t=T/(2\pi
GM/v^{3})$ with $T$ the inspiral time for two black holes of total mass $M$,
symmetric mass ratio $\eta=M_{1}M_{2}/(M_{1}+M_{2})^{2}$, and dimensionless
incident relative velocity $\beta=v/c$.

As a function of the rescaled time $t=T/(2\pi GM/v^{3})\approx
P^{21/4}\zeta(3/2,1-P^{7/2})$, the probability distribution tabulated in the
Table and shown in Figure 1 is very broad, with $t$ varying by a factor of
over a trillion ($1.816\times 10^{12}$) between the 1st and 99th percentiles
(between $P=0.01$ and $P=0.99$), by a factor over 16 billion ($1.675\times
10^{10}$) between the 2nd and 98th percentile, by a factor over 34 million
($3.432\times 10^{7}$) between the 5th and the 95 percentile, by a factor over
312 thousand ($3.127\times 10^{5}$) between the 10th and 90th percentile, and
by a factor slightly over 464 between the 25th and 75 percentiles (the first
and third quantiles, $P=0.25$ and $P=0.75$).

The tails of the probability distribution per range of $\ln{t}$,
$\rho=tdP/dt$, also decay slowly, asymptotically having the following form for
very small and very large $t$ respectively, with $z\equiv\zeta(3/2)\approx
2.612\,375$:

$\rho(t)=\frac{4}{21}\left(\frac{t}{z}\right)^{4/21}\left[1-\frac{9}{7}\left(\frac{t}{z}\right)^{2/3}+O(t)\right],$
(46)
$\rho(t)=\frac{4}{21}t^{-2/3}\left[1-\frac{9}{7}t^{-2/3}+O(t^{-4/3})\right].$
(47)

These tails imply that the mean value of $t^{s}$ converges only for
$-4/21<s<2/3$, though the first (left-hand) inequality in (30),
$P\gg(\beta^{2}/\eta)^{2/7}$, implies that the divergence at infinitesimal
$t=T/T_{0}$ when $s<-4/21$ is not physical. If one defines the inspiral time
$T=T_{0}\,t=(2\pi GM/v^{3})t$ so that it is always greater than $2GM/c^{3}$,
the light travel time for a distance equal to the Schwarzschild radius
corresponding to the total mass $M$, this would imply that one always has $\pi
t>\beta^{3}$, which can be very small, but not infinitesimal, thus cutting off
the divergence arising from the approximation $t=P^{21/4}\zeta(3/2,1-P^{7/2})$
that only applies for $t\gg(v/c)^{3}$.

A quantity both of whose mean and standard deviation is finite is the
logarithm of $t$, which has a mean value of $-3.899\,276$ and a standard
deviation of $\ln{t}$ of 5.636 468. The exponential of the mean of $\ln{t}$ is
0.020 0247, which is small, even smaller than the median value of $t$ of 0.073
802 (at $P=0.5$), but the exponential of the standard deviation of $\ln{t}$ is
280.470, another consequence of the breadth of the probability distribution
for $t$ and $\ln{t}$.

Although Eq. (40) gives an excellent closed-form approximation for $t(P)$ when
$\epsilon$ is negligible (using $z\equiv\zeta(3/2)\approx 2.612\,375$),

$t(P)\approx
T\approx\left(\frac{P^{7/2}}{1-P^{7/2}}\right)^{3/2}\left[1+(1-P^{7/2})^{3/2}\left(z-\frac{1-P^{7/2}}{1-0.5P^{7/2}}\right)\right],$
(48)

it is harder to find a good closed-form approximation for the inverse function
$P(t)$, but one that always has relative error less than 1% (again, neglecting
$\beta_{p}^{2}$) is

$P(t)\approx\left[1+\left(\frac{z^{2}+(z-1)t}{zt+(z-1)t^{2}}\right)^{2/3}\right]^{-2/7}.$
(49)

##  4 Adiabatic Approximation for the Merger Time

Now I shall give the adiabatic approximation for the merger time, $T_{a}$, as
given by Eq. (21) of [27] from the values of the angular momentum
$L_{1}\approx\mu bv=\mu b_{c}P^{1/2}v$ and energy $E_{1}\approx(1/2)\mu
v^{2}(1-P^{-7/2})$ during the first orbit, after the loss of energy and
angular momentum to gravitational radiation during the first close encounter:

$\displaystyle\mathrm{adiabatic\ merger\ time\ }\equiv T_{a}=T_{0}t_{a}(P)$
$\displaystyle\approx$
$\displaystyle\left(\frac{3}{85}\right)\frac{c^{5}a_{1}^{4}(1-e_{1}^{2})^{7/2}}{G^{3}M^{3}\eta}$
(50) $\displaystyle=$
$\displaystyle\left(\frac{3}{85}\right)\frac{c^{5}L_{1}^{7}}{G^{6}M^{25/2}\eta^{15/2}(-2E_{1})^{1/2}}$
$\displaystyle=$
$\displaystyle\left(\frac{3}{85}\right)\frac{c^{5}v^{6}P^{7/2}b_{c}^{7}}{G^{6}M^{6}\eta(P^{-7/2}-1)^{1/2}}$
$\displaystyle=$ $\displaystyle\frac{4\pi
GM}{v^{3}}\frac{P^{7/2}}{(P^{-7/2}-1)^{1/2}}$ $\displaystyle=$ $\displaystyle
2T_{0}P^{21/4}(1-P^{7/2})^{-1/2}$ $\displaystyle=$ $\displaystyle
2T_{0}(1-x)^{3/2}x^{-1/2}$ $\displaystyle=$ $\displaystyle
2x^{-1/2}[\zeta(3/2,x)]^{-1}T$ $\displaystyle\approx$ $\displaystyle
2x\left[1+x^{3/2}\left(z-\frac{2x}{1+x}\right)\right]^{-1}T,$

where $T_{0}\equiv 2\pi GM/v^{3}$, $P\equiv(b/b_{c})^{2}$, $x\equiv
1-P^{7/2}$, $z\equiv\zeta(3/2)$, and the actual inspiral time is $T\approx
T_{0}P^{21/4}\zeta(3/2,1-P^{3/2})=T_{0}(1-x)^{3/2}\zeta(3/2,x)$ for
$(\beta^{2}/\eta)^{2/7}\ll P<1$.

The ratio $R_{a}$ of the adiabatic approximation for the inspiral merger time
$T_{a}$ to the actual inspiral merger time $T$ (for $(\beta^{2}/\eta)^{2/7}\ll
P<1$ and neglecting third-body interactions) is

$R_{a}\equiv\frac{T_{a}}{T}\approx 2x^{-1/2}[\zeta(3/2,x)]^{-1}\approx
2x\left[1+x^{3/2}\left(z-\frac{2x}{1+x}\right)\right]^{-1}.$ (51)

One can see that for $P=0$ or $x=1$, $R_{a}(0)=2/\zeta(3/2)\approx
0.765\,587$, and as $P$ is increased to 1, $R_{a}(P)$ decreases monotonically
to $R_{a}(0)=0$. Thus $R_{a}$ has a relative error a bit more than 23% at
$P=0$ but increasing to 100% at $P=1$.

Figure 2: This graph shows, as a function of the cumulative conditional
probability $P=(b/b_{c})^{2}$, the ratio $R_{a}=T_{a}/T$ of the adiabatic
approximation $T_{a}$ for the inspiral merger time to the actual time $T$ for
the inspiral merger time, using the discreteness of the orbits and the large
relative changes in the periods of the first several orbits, which dominate
the total inspiral time. The adiabatic approximation is about 23% too small
for small $P$ but then becomes 100% too small as $P$ is increased to 1. Figure
3: The solid curve gives the common logarithm (base 10) of the dimensionless
time $t=T/T_{0}\approx P^{21/4}\zeta(3/2,1-P^{7/2})$ for the inspiral time as
the sum of the times of the discrete orbits as a function of the cumulative
conditional probability $P=(b/b_{c})^{2}$, whereas the dashed curve gives the
common logarithm of the adiabatic approximation $t_{a}=T_{a}/T_{0}\approx
2P^{21/4}(1-P^{7/2})^{-1/2}$ for the inspiral time using the adiabatic
continuum approximation of Peters. Figure 4: The solid curve, the same as the
lower curve of Figure 1, though here with an expanded vertical scale, gives
$(\ln{10})*\rho=dP/d\log{t}$ as a function of $\log{t}$, the common logarithm
of the dimensionless inspiral time $t=T/(2\pi GM/v^{3})$ with $T$ the actual
predicted inspiral time for two black holes of total mass $M$, symmetric mass
ratio $\eta=M_{1}M_{2}/(M_{1}+M_{2})^{2}$, and dimensionless incident relative
velocity $\beta=v/c$. The dashed curve gives
$(\ln{10})*\rho_{a}=dP/d\log{t_{a}}$ as a function of $\log{t_{a}}$, the
common logarithm of the dimensionless inspiral time $t_{a}=T_{a}/(2\pi
GM/v^{3})$ with $T_{a}$ the inspiral time as predicted by the invalid
adiabatic approximation. Note that points on the two graphs at the same
horizontal value of $\log{t}=\log{t_{a}}$ do not have the same value of $P$
and hence do not have the same impact parameter $b=b_{c}\sqrt{P}$.

Unlike the inspiral merger time $T(P)$ from the set of discrete orbits in
which the orbit time $\tau_{n}$ jumps significantly as $n$ increases by 1 from
one orbit to the next (until $n$ gets large, since the fractional change in
$\tau_{n}$ is approximately $-3/(2n)$ for large $n$), which has a divergent
mean, the adiabatic approximation giving $T_{a}(P)$ for the merger time
incorrectly gives a finite mean value for it:

$\langle
T_{a}\rangle\approx\frac{11\sqrt{\pi}}{9}\frac{\Gamma(11/14)}{\Gamma(2/7)}\,T_{0}\approx
0.812\,204\,\frac{2\pi GM}{v^{3}}.$ (52)

However, the standard deviation of the adiabatic approximation $T_{a}$ does
diverge as it also does for the correct $T$. Whereas the mean value of $T^{s}$
converges only for $-4/21<s<2/3$, the mean value of $T_{a}^{s}$ converges for
$-4/21<s<2$, a broader range, but not quite broad enough to include $s=2$ as
would be necessary for the standard deviation to converge.

Both $T$ and $T_{a}$ have finite mean values for powers of their logarithms,
so they give one way to get finite comparisons of the adiabatic approximation
$T_{a}$ with the actual $T$ from the discrete orbits. For example, subtracting
out $\ln{T_{0}}$, whereas $\ln{t}$ has a mean value of $-3.899\,276$, whose
exponential is 0.020 025, $\ln{t_{a}}$ has a mean value of $-4.361\,004$,
whose exponential is 0.012 766, which is a bit less than 64% as large as that
for $\ln{t}$. Furthermore, whereas $\ln{t}$ has a standard deviation of 5.636
468, whose exponential is 280.470, $\ln{t_{a}}$ has a standard deviation of
5.433 399, whose exponential is 226.648, a bit less than 81% as large as that
of $\ln{t}$. Also, the median value of $t_{a}$, its value for $P=1/2$, is
0.055 045, which is somewhat less than 75% as large as that of $t$. Thus the
adiabatic approximation is not a very good approximation for the inspiral
merger time, especially for impact parameters $b$ near the critical impact
parameter $b_{c}$, where $P=(b/b_{c})^{2}$ is near 1 and $R_{a}=t_{a}/t\sim
2(1-P^{7/2})=2[1-(b/b_{c})^{7}]$ approaches 0, so that the actual inspiral
time becomes arbitrarily larger than what the adiabatic approximation gives.

The main reason that the adiabatic approximation gives an inspiral merger time
$T_{a}=T_{0}t_{a}$ significantly less than the more accurate calculation
giving an inspiral merger time $T=T_{0}t$ from discrete orbits is that the
period $\tau$ of each orbit is proportional to the 3/2 power of the semimajor
axis $a$, and this changes almost entirely only near the periapsis at the
beginning of each orbit, where most of the gravitational radiation is emitted.
However, the adiabatic approximation assumes that the semimajor axis is
decreasing at a smooth monotonic rate, decreasing the period below what
actually occurs with the semimajor axis decreasing mainly just near the
beginning of each orbit.

In particular, when one can neglect
$\beta_{p}^{2}\equiv[96/85\pi)]^{2/7}(\beta^{2}/\eta)^{2/7}P^{-1}=2GM/(cvb)$,
one can write $T$ and $T_{a}$ in similar forms for comparison (again with
$T_{0}\equiv 2\pi GM/v^{3}$ and $P\equiv(b/b_{c})^{2}$):

$T\approx
T_{0}P^{21/4}\sum_{n=1}^{\infty}(n-P^{7/2})^{-3/2}=T_{0}P^{21/4}\zeta(3/2,1-P^{7/2}),$
(53) $T_{a}\approx
T_{0}P^{21/4}\int_{1}^{\infty}d\nu(\nu-P^{7/2})^{-3/2}=2T_{0}P^{21/4}(1-P^{7/2})^{-1/2}.$
(54)

Thus the fact that the adiabatic approximation $T_{a}$ is significantly
smaller than the valid calculation with discrete orbits can be traced to the
fact that in the part of the integral from $\nu=n$ to $\nu=n+1$, the average
value of $(\nu-P^{7/2})^{-3/2}$ is less than the $(n-P^{7/2})^{-3/2}$ it would
need to be for each $n$ for the adiabatic integral for $T_{a}$ to equal the
discrete orbit sum for $T$. The difference,

$T-T_{a}\approx
T_{0}P^{21/4}\sum_{n=1}^{\infty}\int_{n}^{n+1}d\nu[(n-P^{7/2})^{-3/2}-(\nu-P^{7/2})^{-3/2}],$
(55)

is analogous to the Euler constant

$\gamma=\sum_{n=1}^{\infty}\int_{n}^{n+1}d\nu[n^{-1}-\nu^{-1}].$ (56)

The sum in Eq. (55) for $T-T_{a}$ is dominated by the $n=1$ term, which
corresponds to the first bound orbit. This difference, and also the ratio
$T/T_{a}$, diverges in the limit $P\equiv(b/b_{c})^{2}\rightarrow 1$, which
corresponds to the impact parameter $b$ approaching the critical impact
parameter $b_{c}$ from below, leading to a divergence in the semimajor axis
$a_{1}$ and in the period $\tau_{1}=2\pi\sqrt{a_{1}^{3}/(GM)}$ of the first
orbit in this limit.

##  5 Conclusions

Although the formulas of Peters and of Peters and Mathews [25, 26, 27] work
well for calculating the critical impact parameter
$b_{c}\approx[340\pi\eta/(3\beta^{9})]^{1/7}(GM/c^{2})$ [30, 34, 38, 61] for
two black holes of total mass $M=M_{1}+M_{2}$, symmetric mass ratio
$\eta=M_{1}M_{2}/(M_{1}+M_{2})^{2}$, nonrelativistic initial relative velocity
$v=\beta c\ll\eta^{1/2}c$, impact parameter $b=P^{1/2}b_{c}$ and
nonrelativistic periapsis velocity $v_{p}=\beta_{p}c\approx 2GM/(bv)\ll c$,
the adiabatic approximation of [27] (assuming that the fractional changes of
the semimajor axis and period are small from one orbit to the next) that has
been widely used [31, 38, 41, 42, 44, 45, 50, 51, 47, 53, 55, 56, 57, 60]
significantly underestimates the inspiral merger time, with relative errors
ranging from approximately $1-2/\zeta(3/2)$ (a bit over 23%) for $b\ll b_{c}$
to 100% in the limit as $b$ approaches $b_{c}$. However, for $2GM/(cv)\ll
b\leq b_{c}$, most of the inspiral time is spent in nonrelativistic Keplerian
orbits with eccentricity $e$ near 1, so that most of the gravitational
radiation occurs near periapsis, and one can approximate the dominant sequence
of orbits as having the angular momentum and minimum (periapsis) separation
distance $r_{p}$ stay nearly constant for this sequence of orbits, but the
nonrelativistic energy $E=-GM^{2}\eta/(2a)$ decreasing by a nearly constant
amount $-\Delta E\approx(1/2)\mu
v^{2}P^{-7/2}=(170\pi/3)[GM/(cvb)]^{7}\eta^{2}Mc^{2}$ from one orbit to the
next, and hence the semimajor axis $a=GM^{2}\eta/(-2E)$ and corresponding
period $\tau=2\pi\sqrt{a^{3}/(GM)}$ decreasing by calculable amounts from one
orbit to the next. In this way one can use the formulas of Peters and of
Peters and Mathews [25, 26, 27] to calculate the total inspiral merger time as
$T\approx(2\pi GM/v^{3})P^{21/4}\zeta(3/2,1-P^{7/2})$, where
$P=(b/b_{c})^{2}$.

## Acknowledgments

This work was motivated by research by Ashaduzzaman Joy on the number of
gravitons emitted by black-hole coalescences observed by LIGO and Virgo.
Helpful suggestions by the editor have led to many significant improvements in
the manuscript. Financial support was provided by the Natural Sciences and
Engineering Research Council of Canada.

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