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?