ABSTRACT (original article): We derive an exact formula
F(e)=2ℏ
N
J
which provides a concrete estimate for the total number and angular momentum of gravitons emitted during the nonrelativistic inspiral of two black holes. We show that the function
F(e)
is a slowly growing monotonic function of the eccentricity
0≤e≤1
and
F(1)=1.0128⋯
. We confirm and extend the results of Page for the function
F(e)
. We also get an exact result for the ratio
ν(
e
i
)=
2ℏN(
L
i
,
e
i
)
L
i
of the sum of the spin angular momentum magnitudes of gravitons emitted to the magnitude
L
i
of the total angular momentum emitted in the graviational waves by inspiraling binary black holes. If the orbit starts off with unit eccentricity
e
i
=1
, we get the exact value
ν(1)=1.0022686662±
-10
10
which confirms the Page conjecture. We also show that the formula
F(e)
for gravitons emitted can be determined by a single function. CITATION (original article): Youngjoo Chung, Hyun Seok Yang (2024), Exact Results On the Number of Gravitons Radiated During Binary Inspiral, arXiv:2409.14808. https://doi.org/10.48550/arXiv.2409.14808
Keywords: Gravitational Wave, Black Hole Binary, Graviton Number​
​Please see paper for the introduction
In[]:=
<<InitPackages`

1. Exact results for graviton number emitted

The average gravitational power per orbit radiated into the nth harmonic (angular frequency
ω
n
=nω=n
GM/
3
a
) is [2, 3]
(2.1)
P(n)=
32
4
G
5
M
2
η
5
5
c
5
a
g(n,e),
where
(2.2)
g(n,e)=
4
n
32
2
J
n-2
(ne)-2e
J
n-1
(ne)+
2
n
J
n
(ne)+2e
J
n+1
(ne)-
J
n+2
(ne)
+
4
n
32
(1-
2
e
)
2
(
J
n-2
(ne)-2
J
n
(ne)+
J
n+2
(ne))
+
2
n
24
2
J
n
(ne)
.
There exists a closed formula for the orbit-averaged gravitational wave power in all harmonics and it is given by [2]
(2.3)
In[]:=
P==
∞
∑
n=1
P[n]​​SCMAF%,RA,At[2],P[n]
32
4
G
5
M
2
η
5
5
c
5
a
g[n,e],{SCFactorSum},​​RA,At[2],
∞
∑
n=1
g[n,e]
1+
73
2
e
24
+
37
4
e
96
7/2
(1-
2
e
)

Out[]=
P
∞
∑
n=1
P[n]
P
32
4
G
5
M
2
η
5
5
a
5
c
∞
∑
n=1
g[n,e]
Out[]=
P
321+
73
2
e
24
+
37
4
e
96
4
G
5
M
2
η
5
5
a
5
c
7/2
(1-
2
e
)
See Appendix 1 for derivation of the relation
∞
∑
n=1
g(n,e)=
1+
73
2
e
24
+
37
4
e
96
7/2
(1-
2
e
)
.
Since each graviton has energy
ℏ
ω
n
=nℏ
GM/
3
a
, the orbit-averaged number rate of graviton emission is
(2.4)
N
t
==
∞
∑
n=1
P[n]
ℏ
ω
n
​​SCMAF%,RR,At[2],P[n]
32
4
G
5
M
2
η
5
5
c
5
a
g[n,e],
ω
n
==nω,ω==
GM
3
a
,{SCFactorSum,PowerExpand}
Out[]=
N
t

∞
∑
n=1
P[n]
ℏ
ω
n
Out[]=
N
t

32
7/2
G
9/2
M
2
η
5
7/2
a
5
c
ℏ
∞
∑
n=1
g[n,e]
n
The angular momentum J per time radiated into gravitational waves averaged over one orbit is also known as [3]
(2.5)
J
t
=
32
7/2
G
9/2
M
2
η
5
7/2
a
5
c
1+
7
2
e
8

2
(1-
2
e
)
.
Therefore, the ratio of the sum of magnitudes of the spin angular momentum
2ℏ
to the magnitude of the total orbital angular momentum vector emitted per orbit is
(2.6)
In[]:=
F[e]≡2ℏ
N
J
​​SCMAF%,SCTransDeriv,{At[2],TransVar->{J,t}},SCRecipDeriv,
t
J
,RA->
N
t

32
7/2
G
9/2
M
2
η
5
7/2
a
5
c
ℏ
∞
∑
n=1
g[n,e]
n
,
J
t

32
7/2
G
9/2
M
2
η
5
7/2
a
5
c
1+
7
2
e
8
2
(1-
2
e
)
,Head->Equal
Out[]=
F[e]≡2ℏ
N
J
Out[]=
F[e]
2
2
(1-
2
e
)
1+
7
2
e
8
∞
∑
n=1
g[n,e]
n
Unlike the total power (2.3) derived from the summation of
g(n,e)
, a closed form for the function
F(e)
is not known so far. But there are many known properties for the function
F(e)
[6]:
(2.7)
F(e=0)=1,
(2.8)
F(e)=1+
1
48
2
e
-
3
128
4
e
+O(
6
e
),forsmalle<1,
(2.9)
F(e≈1)=
248
45
3
π
≈1.012811600479,fore≈1.
These results are obtained by using the asymptotic expansions of the Bessel functions. We will show that
F(e)
is a monotonic function for
0≤e≤1
. This means that the function
F(e)
is remarkably close to the unity for
0≤e≤1
. This implies that the number of gravitons emitted during binary inspiral is equal (within 1% or perhaps about 0.2%) to the magnitude of the total angular momentum emitted divided by the spin angular momentum of a single graviton (
2ℏ
).
The conservation of the total angular momentum of the binary black holes and gravitons emitted implies that the decrease of the angular momentum of the orbiting black holes is equal to the sum of the spin and orbital angular momentum vectors of gravitons emitted:
(2.10)
-Δ

L
BH
=

L
+
S
=

J
.
Some of the spins of gravitons emitted can be parallel to the orbital angular momentum, and some can be opposite. Thus the sum of magnitudes of the spin angular momenta would generically be greater than the magnitude of the sum of the spin angular momentum vectors averaged over one orbit, i.e.,
(2.11)
S=
N
∑
i=1

S
i
<=
N
∑
i=1

S
i
=2ℏN.
It is known [2, 3] that, in the quadrupole approximation, the dominant type of radiation is “magnetic quadrupole” with
J=2
. In this limit, the z-component of angular momentum M can take values
M=−2,0,2
. Emission of an
M=2
graviton would decrease the orbital angular momentum of the black holes by
2ℏ
, an
M=0
graviton would not change it at all and an
M=−2
graviton would increase the orbital angular momentum of the black holes by
2ℏ
. Therefore, in the quadrupole approximation where there is a non-zero amplitude for
M=−2
(when
e>0
), the decrease (2.10) in the black hole orbital angular momentum will be less than the number of gravitons multiplied by
2ℏ
(the magnitude of the spin of each graviton). It may explain why
F(e)>1
when
e>0
.
1
1
We thank Don Page for bringing this discussion to our attention.
This suggests that the formula
F(e)=2ℏN/J
provides a useful information on the distribution of the angular momentum in the gravitational waves emitted from inspiraling binary black holes. A deeper understanding of this formula could enhance our models of binary black hole inspirals and lead to more precise predictions of gravitational wave events.
Therefore we will compute the function
F(e)
exactly although there does not seem to be an explicit closed-form elementary function of the eccentricity e. Later we will illuminate why finding a closed form for
F(e)
is difficult. We will use Mathematica and the add-on package MathSymbolica for the calculation of the summation in (2.6). We find the result
(2.12)
In[]:=
G[e]≡
∞
∑
n=1
g[n,e]
n
​​SCMAF%,RA,At[2],g[n,e]
2
n
(3-3
2
e
+
4
e
+3
3
(1-
2
e
)
2
n
)
2
J
n
[en]
6
4
e
-
(4-3
2
e
)(1-
2
e
)
3
n
J
n
[en]
′
J
n
[en]
2
3
e
+
(1-
2
e
)
2
n
(1+(1-
2
e
)
2
n
)
2
′
J
n
[en]
2
2
e
,Head->Equal,​​SCDivNum,{At[2],n},​​SCExpandSumAll,At[2],Hold->{3-3
2
e
+
4
e
,_+_
2
e
},{SCFactorSum},​​RA,At[2],
∞
∑
n=1
n
2
J
n
[nx_]

1
2
(1-
2
x
)
∞
∑
n=1
c
2n
2n
x
,
∞
∑
n=1
n
2
′
J
n
[nx_]

1
1-
2
x
∞
∑
n=1
d
2n
2n-2
x
,
∞
∑
n=1
3
n
2
′
J
n
[nx_]

2
2
x
4
(1-
2
x
)
∞
∑
n=1
c
2n
2n
x
(4n
2
x
(1-
2
x
)+
2
n
2
(1-
2
x
)
+2(
2
x
+2
4
x
))+1-
1
2
x
∞
∑
n=1
3
n
2
J
n
[nx]
,
∞
∑
n=1
2
n
J
n
[nx_]
′
J
n
[nx_]
∞
∑
n=1
c
2n
-1+2n
x
(2
2
x
+n(1-
2
x
))
3
(1-
2
x
)
,{Expand},​​SCMergeSums,{At[2],Post->Collect->{{
c
2n
,
d
2n
},Simplify}},{SCExpandSum,SCFactorSum},​​Collect,{3+_,n,SCFactorPoly[#,
2
e
,e]&},At[2],Expand,43_,SCFactorGCD,-
4
11
+
2
e
,ChangeSign->(-1|-4)+_,​​SCFactor,At[2],
1
2
2
(1-
2
e
)
,SCMergeSums->Post->SCFactor->
2n-4
e
,SCDivPlus,{3+_,3}
Out[]=
G[e]≡
∞
∑
n=1
g[n,e]
n
G[e]
1
6
4
e
2
(1-
2
e
)
∞
∑
n=1
2n
e
(3-15
2
e
+43
4
e
-3(4-11
2
e
)(1-
2
e
)n+6
2
(1-
2
e
)
2
n
)
c
2n
+
1
2
4
e
∞
∑
n=1
2n
e
d
2n
Out[]=
G[e]
1
2
2
(1-
2
e
)
∞
∑
n=1
-4+2n
e
1-5
2
e
+
43
4
e
3
-(4-11
2
e
)(1-
2
e
)n+2
2
(1-
2
e
)
2
n
c
2n
+
2
(1-
2
e
)
d
2n
where
(2.13)
c
2n
=
5-2n
2
(2n-5)!
(n-3)!(n-2)!
n-3
∑
s=0
s
(-1)
2n-3
(n-s-2)
(2n-s-4)!s!
-
2-n
4
(2n-3)!
(n-2)!(n-1)!
n-2
∑
s=0
s
(-1)
2n-1
(n-s-1)
(2n-s-2)!s!
+
-n
4
(2n)!
2
(n!)
n-1
∑
s=0
s
(-1)
2n+1
(n-s)
(2n-s)!s!
​​=
n-
5
2
!
π
(n-2)!
n-3
∑
s=0
s
(-1)
2n-3
(n-s-2)
(2n-s-4)!s!
-
2n-
3
2
!
π
(n-1)!
n-2
∑
s=0
s
(-1)
2n-1
(n-s-1)
(2n-s-2)!s!
+
n-
1
2
!
π
n!
n-1
∑
s=0
s
(-1)
2n+1
(n-s)
(2n-s)!s!
,
(2.14)
The coefficients satisfy the property
(2.15)
See Appendices 2 and 3 for derivation of Eqs. (2.13), (2.14), and (2.15).
(2.16)
and
(2.17)
(2.18)
(2.19)
In order to calculate the total number of gravitons
(2.20)
(2.21)
We can derive the exact result given by
(2.22)
(3.1)
(3.2)
where
(3.3)
(3.4)
(3.5)
and substitute (3.5) in (3.4) to get the differential equation [9]
(3.6)
Then we have
(3.7)
(3.8)
(3.9)
The summation in (3.9) is known as a Kapteyn series. See Chap. XVII in [7]. If the Kapteyn series
(3.10)
is known, then the series
(3.11)
is given by two simple integrations because
(3.12)
by direct differentiation of (3.10). See 17.33 in [7] and [10] for more details.
Assume that, in Eq. (3.10),
(3.13)
Then we have
(3.14)
where
(3.15)
(3.16)
(3.17)
and
(3.18)
(3.19)
(3.20)

3. Discussion

(See paper for the text.)

4. Note Added

(5.1)
We proved his conjecture by directly computing the integral in this formula:
(5.2)
where we used the result
The result (5.2) is precisely equal to Eq. (3.20).

Acknowledgments

Appendix

References

[1] LIGO Scientific and VIRGO Collaborations, The basic physics of the binary black hole merger GW150914, Ann. Phys. (Berlin) 529, 1600209 (2017), [arXiv:1608.01940].
[2] P. C. Peters and J. Mathews, Gravitational radiation from point masses in a Keplerian orbit, Phys. Rev. 131, 435 (1963).
[3] P. C. Peters, Gravitational Radiation and the Motion of Two Point Masses, Phys. Rev. 136, B1224 (1964).
[4] D. N. Page, Inspiral Time Probability Distribution for Two Black Holes Captured by Emitting Gravitational Radiation, Phys. Rev. D 110, 064062 (2024), [arXiv: 2403.10804].
[5] D. N. Page, Discrete Orbit Effect Lengthens Merger Times for Inspiraling Binary Black Holes, JCAP 7, 017 (2024), [arXiv:2405.13673].
[6] D. N. Page, Graviton Number Radiated During Binary Inspiral, [arXiv:2409.00305].
[7] G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd edition (Cambridge Univ. Press, Cambridge, 1966).
[8] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th edition, eds. A. Jeffrey and D. Zwillinger (Academic Press, San Diego, 2000).
[9] I. Lerche and R. C. Tautz, A Note on Summation of Kapteyn Series in Astrophysical Problems, Astrophys. J. 665, 1288 (2007).
[10] A. I. Nikishov, On Summation of Kapteyn Series, [arXiv:1302.0978].
[11] G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic Press, San Diego, 1985).
[12] R. V. Wagoner and C. M. Will, Post-Newtonian Gravitational Radiation from Orbiting Point Masses, Astrophys. J. 210, 764 (1976).
[13] L. Blanchet, T. Damour, B. R. Iyer, C. M. Will, and A. G. Wiseman, Gravitational-Radiation Damping of Compact Binary Systems to Second Post-Newtonian Order, Phys. Rev. Lett. 74, 3515 (1995), [arXiv:gr-qc/9501027].
[14] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, National Bureau of Standards (1964).

CITE THIS NOTEBOOK

Exact results on the number of gravitons radiated during binary inspiral​
by Youngjoo Chung & Hyun Seok Yang
Wolfram Community, STAFF PICKS, October 25, 2024
​https://community.wolfram.com/groups/-/m/t/3307979