Erratum Closed-form solutions of spinning eccentric binary black holes at 1.5 post-Newtonian order Phys. Rev. D 108 124039 2023 arXiv 2210.01605 v3 Tom Colin1Rickmoy Samanta 23Sashwat Tanay 1and Leo C. Stein4
2025-04-29
0
0
849.85KB
17 页
10玖币
侵权投诉
Erratum: Closed-form solutions of spinning, eccentric binary black holes at 1.5
post-Newtonian order [Phys. Rev. D 108, 124039 (2023), arXiv: 2210.01605 (v3)]
Tom Colin,
1
Rickmoy Samanta ,
2, 3
Sashwat Tanay ,
1
and Leo C. Stein
4
1
LUX, Observatoire de Paris, Université PSL, Sorbonne Université, CNRS, 92190 Meudon, France
2
Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
3
Birla Institute of Technology and Science Pilani, Hyderabad 500078, India
4
Department of Physics and Astronomy, The University of Mississippi, University, MS 38677, USA
This erratum aims to correct three errors in the previous
arXiv preprint version (v3). The fully revised standalone
article (with the errors fixed) follows this erratum. In this
erratum, first, we fix Eq. (56). Then, we address a missing
±
sign in Eq. (67), and the missing steps to determine
the correct sign. Finally, we also correct Eqs. (73) and
(74). These equation numbers all correspond to the arXiv
preprint version v3 of this article.
Correction of Eq. (56): Eq. (56) should be replaced
with
d⃗r
dt ={⃗r, H}GM. (1)
Correction of Eq. (67): Eq. (67) should be replaced
with:
⃗p ·ˆr=±(2h−ϵ(3ν−1)h2) + 2(1 + ϵ(4 −ν)h)
r+(−l2+ϵ(6 + ν))
r2+−ϵν(l2+ 2seff ·l/ν)
r31/2
≡ ±pQ(r).(2)
r3Q
(
r
)and
Q
(
r
)each have three roots:
r0, r1
, and
r2
(in
ascending order), where
r0→
0in the PN limit
ϵ→
0.
Since
dr/dt′
=
r, H1.5PN
= (
⃗p ·ˆr
)
Q21
, where
Q2̸
= 0
and
H1.5PN
is the 1.5PN accurate Hamiltonian, the two
roots of
˙r
(as a function of
r
) must coincide with two of
the roots r1and r2of Q(r).
From the quasi-Keplerian parametric (QKP) solution
of
r
(Sec. III-D of the paper), the two roots
r−
and
r+
of
˙r
are
ar
(1
∓er
). However, in general, we find
that
r1
and
r2
lie outside the interval
[r−, r+]
. This
non-coincidence of the roots ((
r1, r2
)
̸
= (
r−, r+
)) can be
attributed to the perturbative, and approximate nature
of our PN calculations. However, this leads to a certain
problem of discontinuity when we try to determine
⃗p ·ˆr
(
t
),
and is detailed further below. So, we need to force the
coincidence of the roots (
r1, r2
)=(
r−, r+
)by hand to
overcome this problem.
To do so, we propose the use of the QKP solution
for
r
(Sec. III-D of the paper) with modified
ar
, and
er
parameters, in Eq. (2) above. The modified parameters
(
˜ar,˜er
)are defined as
˜ar
= (
r1
+
r2
)
/
2and
˜er
= (
r2−
r1
)
/
(
r1
+
r2
). With this, the roots of
˙r
derived from the
correspondingly modified QKP solution, which happen to
be ˜ar(1 ∓˜er), coincide with r1, and r2.
The QKP solution of
r
implies
˙r
=
arersin u˙u
, and
n
=
˙u
(1
−etcos u
), which further implies that for an eccentric
1
Recall that
t′
and
t
represent the physical and scaled times, respec-
tively. The dots represent derivatives taken with respect to the
latter.
orbit, the
t
-epochs where
˙r
vanishes are separated by the
interval ∆
t
=
π/n
. At this point, we denote by
T0
, the
unique
t
-epoch such that
˙r
(
t
=
T0
) = 0, and 0
≤T0<
π/n
. We now determine
T0
for it lets us determine the
correct sign in Eq.
(2)
above. Given the initial values
of the phase-space variables (
⃗
R, ⃗
P , ⃗
S1,⃗
S2
), it is a simple
matter to assign
u⋆≡u
(
t
= 0) such that
−π≤u⋆≤π
.
Plugging these initial values in
n
(
t−t0
) =
u−etsin u
gives
us t0. If t0≥0, then T0=t0, otherwise T0=t0+π/n.
Now finally, to determine the sign of
⃗p ·ˆr
at a time
tf
,
we follow the following easily justifiable algorithm:
1.
Calculate
NP/2
, the number of complete half-
periods between
T0
and
tf
. Note that one half
period in tis π/n.
2. Evaluate ⃗p ·ˆr(t= 0) with the initial conditions. If
•⃗p ·ˆr(t= 0) ≤0and NP/2is even, or
•⃗p ·ˆr(t= 0) ≥0and NP/2is odd,
then
⃗p ·ˆr
(
tf
)
>
0, and we set
⃗p ·ˆr
(
tf
) =
pQ(r)
.
Otherwise,
⃗p ·ˆr
(
tf
)
<
0, and we set
⃗p ·ˆr
(
tf
) =
−pQ(r).
Now we summarize the above discussion with the help
of the figure below. The broken-red curve displays the
numerical solution for
⃗p ·ˆr
that we want to model. The
green curve corresponds to
√Q
, and it does not touch the
x-axis because of the above-discussed non-coincidence of
the roots. The use of the modified QKP solution (with
˜ar
and
˜er
) in
pQ(r)
of Eq.
(2)
gives us the blue curve
arXiv:2210.01605v4 [gr-qc] 18 Feb 2025
2
which does touch the x-axis due to the forced coincidence
of the roots. Thereafter, setting
⃗p ·ˆr
(
tf
) =
±|Q
(
r
)
1/2|
as
per the above bullet points, basically amounts to flipping
the blue curve about the x-axis, but only in alternate
intervals of size
π/n
. The result is the orange curve which
is our final emulation of the numerical red curve. Finally
note that had we not used the modified QKP solution in
Eq.
(2)
above, we would have ended up with the broken
black curve as our final model, which is obtained by
flipping the green curve instead. As is evident from the
figure, this broken black curve suffers from the problem
of discontinuity that we alluded to in the beginning. This
concludes our discussion of determining the sign of ⃗p ·ˆr.
-2π
n-π
n
π
n
2π
n
0
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
t
p
·r
(1)Numerical (2)Q[r] (3)Modified Q[r](4)p
·r
model without roots coincidence (5)p
·r
final model
FIG. 1: Comparison of different implementations of ⃗p ·ˆr. Curve (1) shows ⃗p ·ˆrobtained via numerical integration.
Curves (2) and (3) display |⃗p ·ˆr|as per the standard QKP solution, and the modified QKP solution, respectively.
Curves (4), and (5) display
⃗p ·ˆr
obtained by flipping Curves (2) and (3) about the x-axis within alternate time intervals.
Curve (5) is the final analytical model that we propose, and also the one which is closer to the numerical Curve (1).
Correction of Eqs. (73) and (74): To have the
Rj
’s up to the desired accuracy, we need 1.5PN accurate solution
for r, and not just the Newtonian one. Hence, Eqs. (73) and (74) should be replaced with:
r=ar(1 −ercos u),
n(t−t0) = u−etsin u,
where ar,er,n, and etare defined in Eqs. (49), (50), (51), and (52) of the arXiv preprint version v3 of this article.
ACKNOWLEDGMENTS
We thank Laura Bernard for her impetus to initiate
the investigation that led to this erratum, as well as for
her feedback on this article.
Closed-form solutions of spinning, eccentric binary black holes at 1.5 post-Newtonian
order
Rickmoy Samanta ,
1, 2
Sashwat Tanay ,
3, 4, ∗
and Leo C. Stein
4
1
Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
2
Birla Institute of Technology and Science Pilani, Hyderabad 500078, India
3
LUX, Observatoire de Paris, Université PSL, Sorbonne Université, CNRS, 92190 Meudon, France
4
Department of Physics and Astronomy, The University of Mississippi, University, MS 38677, USA
The closed-form solution of the 1.5 post-Newtonian (PN) accurate binary black hole (BBH)
Hamiltonian system has proven to be evasive for a long time since the introduction of the system
in 1966. Solutions of the PN BBH systems with arbitrary parameters (masses, spins, eccentricity)
are required for modeling the gravitational waves (GWs) emitted by them. Accurate models of
GWs are crucial for their detection by LIGO/Virgo and LISA. Only recently, two solution methods
for solving the BBH dynamics were proposed in Ref. [Phys. Rev. D 100, 044046 (2019)] (without
using action-angle variables), and Refs. [Phys. Rev. D 103, 064066 (2021), Phys. Rev. D 107, 103040
(2023)] (action-angle based). This paper combines the ideas laid out in the above articles, fills the
missing gaps and compiles the two solutions which are fully 1.5PN accurate. We also present a
public Mathematica package
BBHpnToolkit
which implements these two solutions and compares
them with the result of numerical integration of the evolution equations. The level of agreement
between these solutions provides a numerical verification for all the five action variables constructed
in Refs. [Phys. Rev. D 103, 064066 (2021), Phys. Rev. D 107, 103040 (2023)]. This paper hence
serves as a stepping stone for pushing the action-angle-based solution to 2PN order via canonical
perturbation theory.
I. INTRODUCTION
Construction of accurate gravitational wave (GW) tem-
plates (or models) has been crucial to the the GW detec-
tions that have taken place so far since 2015 [
1
–
3
]. This
is so because the method of matched filtering for GW
detection requires as one of the inputs, the theoretical tem-
plates of the GW signal to be detected. Post-Newtonian
(PN) theory serves as a useful framework within which
GWs from binary black holes (BBHs) are modeled when
the system is in its initial and longest-lived inspiral stage
[
4
]. At this stage, the two black holes (BHs) of the BBH
are far apart and move slowly around a common center.
This is also referred to as the PN regime. In the PN
framework, quantities of interest are presented in a PN
power series in the small PN paramter (ratio of the typical
speed of the system and that of light). As is typical of
power series, higher-order corrections are of smaller mag-
nitudes and carry higher PN orders. Since GWs from a
BBH are functions of the positions-momenta of the source,
modeling the positions-momenta of the BBH system is
crucial for constructing the GW templates. This paper
deals with the construction of 1.5PN accurate closed-form
solutions of the BBH system.
Since we restrict ourselves to 1.5PN order, the dissipa-
tive effects on the BBH dynamics due to GW emission
don’t enter the picture; they show up at 2.5PN. The
conservative dynamics of the system can be described
with the PN Hamiltonian framework, wherein the system
possesses a Hamiltonian that is a function of the system’s
∗sashwat.tanay@obspm.fr
canonical coordinates [
5
]. The leading PN order effect is
simply that of two point masses moving under mutual
Newtonian gravitational attraction whose Hamiltonian
treatment is a textbook subject matter. Such systems,
move on a closed ellipse if they are bound. The next level
of sophistication is at 1PN order wherein 1PN effects are
added to the above Newtonian system. At this level, spin
effects are ignored (they enter at 1.5PN). Ref. [
6
] provided
the quasi-Keplerian parametric solution for this system;
the trajectory is no longer a closed ellipse and the system
features the advance of periastron. The orbit still remains
confined in a plane due to the constancy of the angular
momentum vector.
Moving further up the PN ladder, we encounter the
1.5PN system whose Hamiltonian was proposed in Ref. [
7
].
At 1.5PN order, spin effects come into play for the first
time via a spin-orbit interaction (linear-in-spin), while the
spin-spin interaction terms enter at 2PN. Via numerical
integration of the resulting equations of motion (EOMs),
it is seen that the orbital plane precesses; the orbital
angular momentum is constant only in magnitude, but
not in direction. This system displays the rich interplay
of non-zero BH spins, periastron precession, along with
spin and orbital-plane precession. We concern ourselves
with this system in this paper.
Over the past decades, solutions to the dynamics of the
spinning BBH system (at 1.5PN order or higher) have
been constructed by many groups [
8
–
16
], but most of
them worked under some simplifying specialization like
only one BH spinning, equal masses, small eccentricity,
orbit-averaging, etc. Two recent breakthroughs have oc-
curred on the front of finding solutions to the most general
1.5PN BBH system without any simplifying assumptions
where the qualifier “most general” indicates a system with
arXiv:2210.01605v4 [gr-qc] 18 Feb 2025
2
arbitrary values of masses, spins, and eccentricity, while
still falling under the PN regime
1
. The first breakthrough
was made by Cho and Lee in Ref. [
17
] where they suc-
ceeded in integrating the EOMs of the system; the 1PN
term in the Hamiltonian was ignored throughout for sim-
plicity. The second breakthrough came in the form of
Refs. [
18
,
19
], where the authors evaluated all the action
variables (actions as in action-angle (AA) variables) and
laid out a scheme on how to construct the AA-based so-
lution of the system
2
. Against this backdrop, this paper
aims to target the following objectives
1.
Re-present the solution of Ref. [
17
] but with the
1PN terms included. We will call it the Standard
Solution.
2.
Present a systematic procedure to construct the AA-
based solution. As we will see later, the construction
of the AA-based solution requires the Standard
Solution as one of the inputs.
3.
Make available
BBHpnToolkit
, a public Mathemat-
ica package [
21
] which (1) implements the Standard
Solution, the AA-based solution, and the numerical
solution (2) gives the numerical values of all five
frequencies (rate of increase of the angle variables)
of a given BBH system, (3) computes the Poisson
brackets (PBs) between any two functions of the
phase-space variables.
Let us mention that the AA-based solution provides a
significant advantage over the Standard solution. This
is so because while extending the Standard Solution to
2PN appears quite difficult, the AA-based solution should
be extendable to 2PN via canonical perturbation theory.
This paper assembles the ideas put forth in Refs. [
18
,
19
,
22
], and provides a solid platform from where one can
springboard to push the 1.5PN solutions to 2PN order
using canonical perturbation theory.
The organization of the paper is as follows. In Sec. II,
we introduce the phase space, the Hamiltonian, the EOMs
of the system as well as the concept of AA variables. In
Sec. III, we re-present the solution of Ref. [
22
] with the
1PN terms included (Standard Solution). In Sec. IV,
we lay out an in a step-by-step fashion, the strategy to
construct the alternative AA-based solution. In Sec. V, we
introduce the Mathematica package
BBHpnToolkit
that
we release with this paper. We also make comparisons of
our analytical solutions with the numerical one, before
summarizing in Sec. VI.
1
In the PN approximation, the spin magnitude of a maximally
spinning BH is 0.5PN order smaller than the BH’s orbital angular
momentum.
2
See Ref. [
20
] for a pedagogical introduction to the action-angle
method of solution.
II. THE SETUP
This paper is the culmination of the research initiated
in Refs. [
17
], [
18
] and [
19
]. Since the notations and con-
ventions used in the first article differs from those in the
other two articles, the notations and conventions of this
paper are a mix of the two types.
The system in consideration is a BBH system in the PN
approximation. It consists of two BHs of masses
m1
and
m2
such that the relative separation vector of the first
BH from the second one is
⃗
R
. We choose to work in the
center-of-mass frame throughout wherein the momenta
of the first BH is equal to the negative of the other and
is represented by
⃗
P
. The total angular momenta of the
BBH system is
⃗
J
=
⃗
L
+
⃗
S1
+
⃗
S2
, where
⃗
L≡⃗
R×⃗
P
, and
⃗
S1
and
⃗
S2
are the spin angular momenta of the two BHs.
We also define the total mass
M
=
m1
+
m2
, the reduced
mass
µ≡m1m2/M
, the symmetric mass ratio
ν≡µ/M
,
and
σ1≡
1+3
m2/
4
m1
and
σ2≡
1+3
m1/
4
m2
, along
with
⃗
Seff ≡σ1⃗
S1+σ2⃗
S2.(1)
As usual,
G
and
c
denote the gravitational constant and
the speed of light, respectively. Finally, note that we
represent the physical time with
t′
, whereas
t≡t′/
(
GM
)
is reserved for the scaled time. Dots represent a derivative
taken with respect to
t
. The norm of any 3D vector will be
denoted by the same letter as the vector but without the
arrow, i.e.
V≡ |⃗
V|
. The unit vector
⃗
V /V
corresponding
to a vector ⃗
Vis denoted by the use of a hat ˆ
V≡⃗
V /V .
We now present the 1.5PN Hamiltonian of the system
using scaled variables ⃗r ≡⃗
R/(GM),⃗p ≡⃗
P /µ,
H=HN+H1PN +H1.5PN +Oc−4,(2)
where the various PN components are
HN=µp2
2−1
r,(3)
H1PN =µ
c21
8(3ν−1)p4+1
2r2(4)
−1
2r(3 + ν)p2+ν(ˆr·⃗p)2,(5)
H1.5PN =2G
c2R3⃗
Seff ·⃗
L. (6)
The evolution of any function
g
is given by its Poisson
bracket (PB) with H
dg
dt′={g, H}.(7)
A few basic rules are needed to evaluate the PB between
any two functions of the phase space variables
⃗
R, ⃗
P , ⃗
S1
and
⃗
S2
, the first one being the PB between the phase
space variables. The only non-vanishing PBs between the
摘要：
展开>>
收起<<
Erratum:Closed-formsolutionsofspinning,eccentricbinaryblackholesat1.5post-Newtonianorder[Phys.Rev.D108,124039(2023),arXiv:2210.01605(v3)]TomColin,1RickmoySamanta,2,3SashwatTanay,1andLeoC.Stein41LUX,ObservatoiredeParis,UniversitéPSL,SorbonneUniversité,CNRS,92190Meudon,France2IndianStatisticalInstitut...
声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！
分类：图书资源
价格：10玖币
属性：17 页
大小：849.85KB
格式：PDF
时间：2025-04-29
作者详情
-
Optimal Persistent Monitoring of Mobile Targets in One Dimension Jonas Hall1 Sean Andersson12 Christos G. Cassandras13 Abstract This work shows the existence of opti-10 玖币0人下载
-
Optimal metabolic strategies for microbial growth in stationary random environments Anna Paola Muntoni and Andrea De Martino10 玖币0人下载
相关内容
-
行政事业单位内部控制报告-关于印发编外聘用人员管理制度和编外聘用人员年度考核制度
分类：办公文档
时间：2025-03-03
标签：无
格式：DOC
价格：5.9 玖币
-
行政事业单位内部控制报告-风险评估管理制度
分类：办公文档
时间：2025-03-03
标签：无
格式：DOCX
价格：5.9 玖币
-
行政事业单位内部控制报告-采购管理内部控制制度
分类：办公文档
时间：2025-03-03
标签：无
格式：DOCX
价格：5.9 玖币
-
行政事业单位内部控制报告-部署单位内部控制专题培训和风险评估工作会议纪要
分类：办公文档
时间：2025-03-03
标签：无
格式：DOC
价格：5.9 玖币
-
行政事业单位内部控制报告-关键岗位轮岗及专项审计制度
分类：办公文档
时间：2025-03-03
标签：关键
格式：DOCX
价格：5.9 玖币
渝公网安备50010702506394
