A Method of Osculating Orbits in Schwarzschild Adam Pound University of Guelph
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Transcript of A Method of Osculating Orbits in Schwarzschild Adam Pound University of Guelph
A Method of Osculating Orbits in Schwarzschild
Adam PoundUniversity of Guelph
Motivation: Self-force Problems• self-force is calculated on a geodesic; true orbit never follows a geodesic path
• we need to determine true path using force on a geodesic
Method: Osculating Orbits• we assume the true orbit x () is tangent to a geodesic z () at each , allowing us to use the force f on that geodesic
True orbit
Osculation with ellipse 1 at 1
Osculation with ellipse 2 at 2
Mathematics of Osculating Orbits
• specify a geodesic z with orbital elements IA and parametrize it with parameter . The osculation conditions state
• insert these conditions into the equations of motion to find
• invert to find evolution equations for IA
),()(
),()(
A
A
Iz
d
dx
Izx
fd
dIz
I
d
dI
I
z
A
A
A
A
0
Bound Eccentric Geodesic Orbits in Schwarzschild
• similar to precessing elliptical orbits
• can be parametrized with a parameter running from 0 to 2 over one period of radial motion:
• can be characterized by analogues of traditional orbital elements in celestial mechanics: IA = (p, e, w, T, )
• principal elements: p = semi-latus rectum, e = eccentricity
• positional elements: w = at periapsis, T = t at periapis, = at periapsis ( = w in Keplerian orbits)
)cos(1),(
we
pMIr A
0
2
2/12
2/12/12
0
)cos(26),( ,),('),(
)cos(1
)sin(),( ,
)cos(1),(
))cos(26))(cos(22())cos(1(
)22()22(),(
),('),(
wep
pI
d
ddssII
we
wpMeI
d
dr
we
pMIr
wepwepwe
epepMpI
d
dt
dssItTIt
AAA
AA
A
AA
The Parametrization in Full (in Schwarzschild coordinates)
22 3 and
)3(
)22)(22(
ep
pML
epp
epepE
• p and e are related to the orbital energy and angular momentum:
Choice of Phase Space• restricting orbits to a plane, we have a 5D phase space corresponding to initial coordinates and velocities (minus one due to normalization)
• rather than using {IA} as our phase space, we use {p(), e(), w(), t(), ()}
• T and can be recovered from t and if we need initial conditions on the tangential geodesic
• using our geodesic parametrization, we invert
to find equations for p, e, and w:
f
d
dIz
Id
dI
I
z A
A
A
A
and 0
f
rf
tf
tf
f
tf
f
wepweepepe
Mpwepppweepp
wepweepepe
Mpeweppep
wepweepepe
pMwpepepep
d
dw
wewepepepe
pMepepepwepep
wewepepepe
Mpepppee
d
de
wewepepep
Mpepepepwep
wewepepep
Mpepp
d
dp
2/14
22/52222
2
2222
2/12
2/12/12
22/1
2/12/122
42/1
22/52222
22/1
22/12/12
2/1
22/722
))cos(26())cos(1)(26)(26(
)sin()3(24162)cos()4812(
))cos(22())cos(1)(26)(26(
8)cos()4128()3(
))cos(26())cos(1)(26)(26(
)sin()6()22()22)(3(2
))cos(1())cos(26)(26)(26(
)22()22)(3))(cos(22)(26(
))cos(1())cos(26)(26)(26(
)3)(8124)(1(
))cos(1())cos(26)(26)(26(
)22()22)(3))(cos(22(2
))cos(1())cos(26)(26)(26(
)3()4(2
Sample Problem:a massive particle orbiting a BH in the post-Newtonian regime
• the particle’s mass causes a gravitational self-force
• we use the hybrid equations of motion presented in Kidder, Will & Wiseman ’93:
• these equations reduce to geodesic motion for = 0
• the self-force is derived from the finite- terms
...)()()(1 542
22
2
vOvOvOM
ildSchwarzschr
M
dt
xd
Radiation-Reaction Approximation
• the self-force has conservative corrections at 1PN and 2PN, and a dissipative correction at 2.5PN
• radiation-reaction approximation uses only 2.5PN correction
• we have shown this approximation fails for electromagnetic self-force (gr-qc/0509122)
• its accuracy has been studied by Ajith et al. (gr-qc/0503124) in post-Newtonian gravitational case
• we test it here using our method of osculating orbits
Comparison of orbits with and without conservative corrections in self-force
p0 = 100e0 = 0.9/M = 0.1
true orbit radiation-
reaction approximation
Dephasing of True and Approximate Orbits(same initial conditions as above)
1.5x1061.0x1060.5x106
• phase difference ~30 rad after p has decreased by 0.4%
0.5x106 1.0x106 1.5x106
1.5x1061.0x1060.5x106
• change in principal elements is roughly correct in radiation-
reaction approximation
• no change in positional elements in radiation-reaction approximation
• slight improvement if we match initial χ-averaged elements (e.g. ) • phase difference ~30 rad after p has decreased by 0.8%
1000 p
• more improvement if we match initial t-averaged elements (e.g. )
• phase difference ~30 rad after p has decreased by 5%
1000 t
p
• total dephasing after p0 0.955p0 (matching non-averaged initial conditions)
2/3~ p
Conclusion
• osculating orbits are ideal for analyzing self-force problems
• our method has been successful in a simple problem
• we have verified the importance of conservative terms in the gravitational self-force
• future applications:- orbits in the spacetime of a tidally-distorted black hole- orbits of a self-accelerated charge or mass in the fully relativistic case in Schwarzschild- generalization of our method to orbits in Kerr?