Amplitude factors \(Z^\infty_{\ell m}(r)\)

Functions \(Z^\infty_{\ell m}(r)\) giving the amplitude of the gravitational radiation emitted by a particle on a circular orbit at radius \(r\) about a Kerr black hole.

REFERENCES:

  • S. A. Teukolsky, Astrophys. J. 185, 635 (1973)

  • S. Detweiler, Astrophys. J. 225, 687 (1978)

  • M. Shibata, Phys. Rev. D 50, 6297 (1994)

  • D. Kennefick, Phys. Rev. D 58, 064012 (1998)

  • S. A. Hughes, Phys. Rev. D 61, 084004 (2000) [doi:10.1103/PhysRevD.61.084004]

  • E. Gourgoulhon, A. Le Tiec, F. Vincent, N. Warburton, Astron. Astrophys. 627, A92 (2019) [doi:10.1051/0004-6361/201935406]; Arxiv 1903.02049

kerrgeodesic_gw.zinf.Zinf(a, l, m, r, algorithm='spline')

Amplitude factor of the mode \((\ell,m)\).

The factor \(Z^\infty_{\ell m}(r)\) is obtained by spline interpolation of tabulated numerical solutions of the radial component of the Teukolsky equation.

INPUT:

  • a – BH angular momentum parameter (in units of \(M\), the BH mass)

  • l – integer >= 2; the harmonic degree \(\ell\)

  • m – integer within the range [-l, l]; the azimuthal number \(m\)

  • r – areal radius of the orbit (in units of \(M\))

  • algorithm – (default: 'spline') string describing the computational method; allowed values are

    • 'spline': spline interpolation of tabulated data

    • '1.5PN' (only for a=0): 1.5-post-Newtonian expansion following E. Poisson, Phys. Rev. D 47, 1497 (1993) [doi:10.1103/PhysRevD.47.1497], with a minus one factor accounting for a different convention for the metric signature.

OUTPUT:

  • coefficient \(Z^\infty_{\ell m}(r)\) (in units of \(M^{-2}\))

EXAMPLES:

sage: from kerrgeodesic_gw import Zinf
sage: Zinf(0.98, 2, 2, 1.7)  # tol 1.0e-13
-0.04302234478778856 + 0.28535368610053824*I
sage: Zinf(0., 2, 2, 10.)  # tol 1.0e-13
0.0011206407919254163 - 0.0003057608384581628*I
sage: Zinf(0., 2, 2, 10., algorithm='1.5PN')  # tol 1.0e-13
0.0011971529546749354 - 0.0003551610880408921*I
kerrgeodesic_gw.zinf.Zinf_Schwarzchild_PN(l, m, r)

Amplitude factor of the mode \((\ell,m)\) for a Schwarzschild BH at the 1.5PN level.

The 1.5PN formulas are taken from E. Poisson, Phys. Rev. D 47, 1497 (1993), doi:10.1103/PhysRevD.47.1497.

INPUT:

  • l – integer >= 2; the harmonic degree \(\ell\)

  • m – integer within the range [-l, l]; the azimuthal number \(m\)

  • r – areal radius of the orbit (in units of \(M\), the BH mass)

OUTPUT:

  • coefficient \(Z^\infty_{\ell m}(r)\) (in units of \(M^{-2}\))

EXAMPLES:

sage: from kerrgeodesic_gw import Zinf_Schwarzchild_PN
sage: Zinf_Schwarzchild_PN(2, 2, 6.)  # tol 1.0e-13
-0.00981450418730346 + 0.003855681972781947*I
sage: Zinf_Schwarzchild_PN(5, 3, 6.)  # tol 1.0e-13
-6.958527913913504e-05*I