Self-consistent orbital evolution of a particle around a - - PowerPoint PPT Presentation

BritGrav Conference, Southampton, 4th April 2012

Self-consistent orbital evolution of a particle around a Schwarzschild black hole

Barry Wardell University College Dublin Collaborators: Peter Diener, Ian Vega, Steven Detweiler

EMRIs and eLISA/NGO

✤ Extreme Mass Ratio Inspirals

have long been a promising source of gravitational waves for the LISA, the space based gravitational wave detector.

✤ Accurate models are a critical

component of any observation.

✤ Even more true now that LISA

is no more and there are proposals for eLISA/NGO which will have less sensitivity.

Image credit: eLISA/NGO Yellow book (ftp://ftp.rssd.esa.int/pub/ojennric/NGO_YB/NGO_YB.pdf)

✤ Solve the coupled system of

equations for the motion of the particle and its retarded field.

✤ Self-interaction of the particle with

its retarded field, 횽ret.

✤ 횽ret diverges like 1/r on the world-

line.

✤ “Unphysical” divergence removed

by appropriate regularization.

Motion of a point particle

⇤Φret = 4πq Z δ4(x z(τ)) pg dτ Duα dτ = aα = ¯ q m(τ)(gαβ + uαuβ)rβΦret dm dτ = ¯ quβrβΦret

✤ Split retarded field into locally

constructed field and “regularized” remainder.

✤ Derive an equation for 횽R. ✤ Always work with 횽R instead of

횽ret.

✤ If 횽S is chosen appropriately, then

we can just replace 횽ret with 횽R in the equations of motion.

Effective source regularization

Φret = ΦS + ΦR

⇤ΦR = ⇤Φret − ⇤ΦS

Duα dτ = aα = ¯ q m(τ)(gαβ + uαuβ)rβΦR dm dτ = ¯ quβrβΦR

✤ If 횽S is exactly the Detweiler-

Whiting singular field, 횽R is a solution of the homogeneous wave equation.

✤ If 횽S is only approximately the

Detweiler-Whiting singular field, then the equation for 횽R. has an effective source, S.

✤ S is typically finite, but of limited

differentiability on the world line.

Effective source regularization

✤ Solve the coupled system of

equations for the motion of the particle and its regularized field.

✤ 횽R = 횽ret in the wave zone ✤ 횽R finite and (typically) twice

differentiable on the world-line

⇤ΦR = S(x|z(τ), u(τ)) Duα dτ = aα = ¯ q m(τ)(gαβ + uαuβ)rβΦR dm dτ = ¯ quβrβΦR

Self-consistent Evolution

q = M/32 m = M p0 = 7.2 e0 = 0.5

횽R(t)

q = M/32 m = M p0 = 7.2 e0 = 0.5

횽R(t)

Orbital motion

• 10
• 5

5 10

• 15
• 10
• 5

5 10 15 y(M) x(M) q=0

• 10
• 5

5 10 x(M) q=1/32

1 2 3 4 5 6

Orbital motion

✤ Parametrize orbits in terms of a

dimensionless semilatus rectum p and eccentricity e, such that r± = Mp/(1 ∓ e).

✤ Separatrix, p = 6 + 2e,

corresponds to unstable circular

• rbits and represents the

boundary in p–e space separating bound from plunging orbits.

r(t) = Mp 1 + e cos(χ − w)

dφ dt =  1 − 2Mr0 r − 2M

• ×

[p − 2 − 2e cos(χ − w)][1 + e cos(χ − w)]2 M p p3[(p − 2)2 − 4e2]

Orbital evolution

4 8 12 16 500 1000 1500 2000 2500 Radius (M) Time (M)

1 2 3 4 5 6

• 5e-07
• 3e-07
• 1e-07

1e-07 3e-07 5e-07 Acceleration dm/dτ M2aφ Mar

Orbital evolution - “dissipative”

0.498 0.5 0.502 0.504 0.506 0.508 0.51 0.512 7 7.05 7.1 7.15 7.2 Eccentricity Semilatus rectum (M) q=1/8 q=1/16 q=1/32 q=1/64 separatrix

1 2 3 4 5 6

r(t) = Mp 1 + e cos(χ − w)

Orbital evolution - “conservative”

• 4

4 8 12 16 500 1000 1500 2000 w / q2 Time (M) q=1/8 q=1/16 q=1/32 q=1/64

1 2 3 4 5 6

r(t) = Mp 1 + e cos(χ − w)

Waveforms at ℐ+

• 0.001

0.003 0.007 0.011 0.015 500 1000 1500 2000 2500 / q Time (M) q=1/8 q=1/16 q=1/32 q=1/64