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

Self-consistent orbital evolution of a particle around a Schwarzschild black hole Barry Wardell University College Dublin Collaborators: Peter Diener, Ian Vega, Steven Detweiler BritGrav Conference, Southampton, 4th April 2012

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)

Motion of a point particle ✤ Solve the coupled system of equations for the motion of the particle and its retarded field. Z δ 4 ( x � z ( τ )) ⇤ Φ ret = � 4 π q d τ p� g ✤ Self-interaction of the particle with Du α its retarded field, 횽 ret . ¯ q = a α = m ( τ )( g αβ + u α u β ) r β Φ ret d τ dm ✤ 횽 ret diverges like 1/r on the world- qu β r β Φ ret d τ = � ¯ line. ✤ “Unphysical” divergence removed by appropriate regularization.

Effective source regularization ✤ Split retarded field into locally Φ ret = Φ S + Φ R constructed field and “regularized” remainder. ✤ Derive an equation for 횽 R . ⇤ Φ R = ⇤ Φ ret − ⇤ Φ S ✤ Always work with 횽 R instead of 횽 ret . Du α ¯ q = a α = m ( τ )( g αβ + u α u β ) r β Φ R d τ dm ✤ If 횽 S is chosen appropriately, then qu β r β Φ R d τ = � ¯ we can just replace 횽 ret with 횽 R in the equations of motion.

Effective source regularization ✤ If 횽 S is exactly the Detweiler- Whiting singular field, 횽 R is a solution of the homogeneous wave equation. ✤ If 횽 S is only approximately the Detweiler-Whiting singular field, then the equation for 횽 R . has an e ff ective source, S. ✤ S is typically finite, but of limited di ff erentiability on the world line.

Self-consistent Evolution ✤ Solve the coupled system of equations for the motion of the ⇤ Φ R = S ( x | z ( τ ) , u ( τ )) particle and its regularized field. Du α ¯ q = a α = m ( τ )( g αβ + u α u β ) r β Φ R d τ ✤ 횽 R = 횽 ret in the wave zone dm qu β r β Φ R d τ = � ¯ ✤ 횽 R finite and (typically) twice di ff erentiable on the world-line

횽 R (t) q = M/32 m = M p 0 = 7.2 e 0 = 0.5

횽 R (t) q = M/32 m = M p 0 = 7.2 e 0 = 0.5

Orbital motion q=0 q=1/32 10 5 4 5 2 6 1 y(M) 0 -5 3 -10 -15 -10 -5 0 5 10 15 -10 -5 0 5 10 x(M) x(M)

Orbital motion ✤ Parametrize orbits in terms of a dimensionless semilatus rectum p and eccentricity e, such that Mp r ± = Mp /(1 ∓ e ). r ( t ) = 1 + e cos( χ − w ) ✤ Separatrix, p = 6 + 2 e ,  2 Mr 0 � d φ dt = 1 − × r − 2 M corresponds to unstable circular [ p − 2 − 2 e cos( χ − w )][1 + e cos( χ − w )] 2 orbits and represents the p p 3 [( p − 2) 2 − 4 e 2 ] M boundary in p – e space separating bound from plunging orbits.

Orbital evolution 5e-07 M 2 a φ Ma r dm/d τ 3e-07 Acceleration 1e-07 -1e-07 -3e-07 -5e-07 Radius (M) 16 12 1 6 4 8 3 4 5 2 0 0 500 1000 1500 2000 2500 Time (M)

Orbital evolution - “dissipative” 0.512 q=1/8 0.51 q=1/16 Eccentricity 0.508 q=1/32 q=1/64 0.506 6 separatrix 0.504 0.502 5 4 0.5 3 2 1 0.498 7 7.05 7.1 7.15 7.2 Semilatus rectum (M) Mp r ( t ) = 1 + e cos( χ − w )

Orbital evolution - “conservative” 16 6 q=1/8 q=1/16 12 q=1/32 8 q=1/64 w / q 2 4 4 5 3 0 1 -4 2 0 500 1000 1500 2000 Time (M) Mp r ( t ) = 1 + e cos( χ − w )

Waveforms at ℐ + 0.015 q=1/8 q=1/16 q=1/32 q=1/64 0.011 � / q 0.007 0.003 -0.001 0 500 1000 1500 2000 2500 Time (M)