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On solving post-Newtonian accurate KeplerEquation
Yannick Boetzel
Physik-Institut der Universitat Zurich
In collaboration with: A. Susobhanan, A. Gopakumar, A. Klein, P. Jetzer
La Thuile, 30.03.2017
Yannick Boetzel Rencontres de Moriond On solving PN accurate Kepler Equation
Eccentric GW templates
We work on providing accurate & efficient prescriptions for h+,×(t)associated with compact binaries merging along eccentric orbits
Yannick Boetzel Rencontres de Moriond On solving PN accurate Kepler Equation
Eccentric GW templates
We work on providing accurate & efficient prescriptions for h+,×(t)associated with compact binaries merging along eccentric orbits
h+,×(t) = −G m η
c2 R ′x∞∑p=0
[Cp+,× cos(pl) + Sp+,× sin(pl)
]
C1+ = s2i
(−e +
e3
8− e5
192+
e7
9216
)+ c2β(1 + c2i )
(−3e
2+
2e3
3− 37e5
768+
11e7
7680
)S1+ = s2β(1 + c2i )
(−3e
2+
23e3
24+
19e5
256+
371e7
5120
)
Yannick Boetzel Rencontres de Moriond On solving PN accurate Kepler Equation
Eccentric GW templates
h+,×(r , r , φ, φ)
Invoke quasi-Keplerian solutionto the conservative dynamics ⇓
(r , r , φ, φ) ⇒ κi (x , et , u(t),m, η)
Explicit expression for u(t) bysolving Kepler equation ⇓
h+,× (κi (x , et , u(t),m, η)) ⇒ h+,×(t) for conservative dynamics
Impose GW emission inducedvariations in x and et
⇓h+,× (κi (x(t), et(t), u(t),m, η)) ⇒ h+,×(t) with radiation reaction
⇓h+,×(t) for binaries inspiralling in PN-accurate eccentric orbits
Yannick Boetzel Rencontres de Moriond On solving PN accurate Kepler Equation
Eccentric GW templates
h+,×(r , r , φ, φ)
Invoke quasi-Keplerian solutionto the conservative dynamics ⇓
(r , r , φ, φ) ⇒ κi (x , et , u(t),m, η)
Explicit expression for u(t) bysolving Kepler equation ⇓
h+,× (κi (x , et , u(t),m, η)) ⇒ h+,×(t) for conservative dynamics
Impose GW emission inducedvariations in x and et
⇓h+,× (κi (x(t), et(t), u(t),m, η)) ⇒ h+,×(t) with radiation reaction
⇓h+,×(t) for binaries inspiralling in PN-accurate eccentric orbits
Yannick Boetzel Rencontres de Moriond On solving PN accurate Kepler Equation
Classical Kepler problem
Equation of motion
d2~r
dt2= −Gm
r2r
Yannick Boetzel Rencontres de Moriond On solving PN accurate Kepler Equation
Classical Kepler problem
Keplerian parametrization
r = a (1− e cos u)
φ− φ0 = v(u)
tanv
2=
√1 + e
1− etan
u
2
where v is called true anomaly and u iscalled eccentric anomaly
Yannick Boetzel Rencontres de Moriond On solving PN accurate Kepler Equation
Classical Kepler problem
Keplerian parametrization
r = a (1− e cos u)
φ− φ0 = v(u)
Classical Kepler Equation
l = u − e sin u
where l = n (t − t0) is called mean anomaly
Yannick Boetzel Rencontres de Moriond On solving PN accurate Kepler Equation
Solving the classical Kepler Equation
Fourier series
u(l)− l =∞∑s=1
As sin(sl)
As =2
π
∫ π
0
(u(l)− l) sin(sl)dl
=2
sπ
∫ π
0
cos(sl)du
=2
s
{1
π
∫ π
0
cos(su − se sin u)du
}=
2
sJs(se)
Yannick Boetzel Rencontres de Moriond On solving PN accurate Kepler Equation
Solving the classical Kepler Equation
Fourier series
u(l)− l =∞∑s=1
As sin(sl)
As =2
π
∫ π
0
(u(l)− l) sin(sl)dl
=2
sπ
∫ π
0
cos(sl)du
=2
s
{1
π
∫ π
0
cos(su − se sin u)du
}=
2
sJs(se)
Yannick Boetzel Rencontres de Moriond On solving PN accurate Kepler Equation
Solving the classical Kepler Equation
Solution to the classical Kepler Equation
u = l +∞∑s=1
2
sJs(se) sin(sl)
Yannick Boetzel Rencontres de Moriond On solving PN accurate Kepler Equation
Relativistic Kepler problem
Equation of motion
d2~r
dt2= −Gm
r2r + A r + B v
A = A1 + A2 + A2.5 + A3 + A3.5 + ...
B = B1 + B2 + B2.5 + B3 + B3.5 + ...
A’s & B’s are functions of r , r , φ, φ, m and η
Radiation reaction terms (2.5PN & 3.5PN) cause orbit to shrink
There exists a quasi-Keplerian parametrization to the 3PNconservative dynamics
Yannick Boetzel Rencontres de Moriond On solving PN accurate Kepler Equation
Relativistic Kepler problem
Quasi-Keplerian parametrization
r = ar (1− er cos u)
φ− φ0 = (1 + k)v + (f4φ + f6φ) sin(2v) + (g4φ + g6φ) sin(3v)
+ i6φ sin(4v) + h6φ sin(5v)
tanv
2=
√1 + eφ1− eφ
tanu
2
The coefficients a, er , eφ, k , f , g , i , h are PN-accurate functions of x ,et and η = µ/m
k represents precession of periastron
Yannick Boetzel Rencontres de Moriond On solving PN accurate Kepler Equation
Relativistic Kepler problem
3PN-accurate Kepler Equation
l = u − et sin u + (g4t + g6t) (v − u)
+ (f4t + f6t) sin(v) + i6t sin(2v) + h6t sin(3v)
Yannick Boetzel Rencontres de Moriond On solving PN accurate Kepler Equation
Relativistic Kepler problem
3PN-accurate Kepler Equation
l = u − et sin u + (g4t + g6t) (v − u)
+ (f4t + f6t) sin(v) + i6t sin(2v) + h6t sin(3v)
Objective
Invert the 3PN-accurate Kepler Equation to get u(l) accurate to 3PNorder
Yannick Boetzel Rencontres de Moriond On solving PN accurate Kepler Equation
Solving the 3PN-accurate Kepler Equation
v − u = 2∞∑j=1
βj
jsin(ju)
sin(v) =2√
1− e2φ
eφ
∞∑j=1
βj sin(ju)
sin(2v) =4√
1− e2φ
e2φ
∞∑j=1
βj(j√
1− e2φ − 1)
sin(ju)
sin(3v) =2√
1− e2φ
e3φ
∞∑j=1
βj(
4− e2φ − 6j√
1− e2φ + 2j2(1− e2φ))
sin(ju)
where β =1−√
1−e2φ
eφ
Yannick Boetzel Rencontres de Moriond On solving PN accurate Kepler Equation
Solving the 3PN-accurate Kepler Equation
l = u − et sin u +∞∑j=1
αj(x , et , η) sin(ju)
αj(x , et , η) =2βj
√1− e2φ
e3φ
((f4t + f6t)e
2φ +
(g4t + g6t)e3φ
j√
1− e2φ
+ 2i6teφ[j√
1− e2φ − 1]
+ h6t[4− e2φ − 6j
√1− e2φ + 2j2(1− e2φ)
])
Yannick Boetzel Rencontres de Moriond On solving PN accurate Kepler Equation
Solving the 3PN-accurate Kepler Equation
Fourier series
u − l =∞∑s=1
As sin(sl)
As =2
π
∫ π
0
(u − l) sin(sl)dl
=2
sπ
∫ π
0
cos(sl)du
=2
s
{1
π
∫ π
0
cos(su − se sin u + s
∞∑j=1
αj sin(ju))du
}
Yannick Boetzel Rencontres de Moriond On solving PN accurate Kepler Equation
Solving the 3PN-accurate Kepler Equation
As =2
sπ
∫ π
0
cos (su − set sin u) du −2
π
∞∑j=1
αj
∫ π
0
sin (su − set sin u) sin(ju)du
=2
sJs (set) +
1
π
∞∑j=1
αj
∫ π
0
{cos ((s + j)u − set sin u)− cos ((s − j)u − set sin u)
}du
=2
sJs (set) +
∞∑j=1
αj {Js+j (set)− Js−j (set)}
Yannick Boetzel Rencontres de Moriond On solving PN accurate Kepler Equation
Solving the 3PN-accurate Kepler Equation
Solution to the 3PN-accurate Kepler Equation
u = l +∞∑s=1
As sin(sl)
As =2
sJs(set) +
∞∑j=1
αj {Js+j(set)− Js−j(set)}
Yannick Boetzel Rencontres de Moriond On solving PN accurate Kepler Equation
Numerical comparison
Numerical solution using Newton’s method
We plot integrated error 12π
(∫ 2π
0
(unum−uanlunum−l
)2dl
)1/2
Yannick Boetzel Rencontres de Moriond On solving PN accurate Kepler Equation
Numerical comparison
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9et
10−7 10−7
10−6 10−6
10−5 10−5
10−4 10−4
10−3 10−3
10−2 10−2
10−1 10−1
100 100
Inte
grat
eder
ror
x = 0.01
x = 0.02
x = 0.03
x = 0.05
x = 0.07
x = 0.10
Yannick Boetzel Rencontres de Moriond On solving PN accurate Kepler Equation
Brief summary
We derived an elegant series solution to the 3PN accurate Keplerequation
We use this solution to calculate eccentric GW templates
Yannick Boetzel Rencontres de Moriond On solving PN accurate Kepler Equation