Black Hole Binaries Separation Growth Rate

Class providing models of the rate of change of the binary separation (in Mpc Gyr\(^{-1}\)) for a black hole binary during its inspiral toward coalescence. After the initial dynamical friction phase brings the two black holes close together, the hardening of the binary is driven by three-body stellar scattering, viscous gas torques, or gravitational wave emission depending on the separation. The separation growth rate (which is negative for inspiral) determines the merger timescale.

Default implementation: blackHoleBinarySeparationGrowthRateZero

Methods

growthRate → double precision

Computes the rate of growth of the separation of the given black hole and its binary companion in units of Mpc/Gyr.

  • class(nodeComponentBlackHole), intent(inout) :: blackHole

blackHoleBinarySeparationGrowthRateStandard

A black hole binary separation growth class that computes the separation growth rate of the binaries following a modified version of Volonteri et al. (2003) which include terms for dynamical friction, hardening due to scattering of stars and gravitational wave emission.

\[\dot{a} = \hbox{min} \left( - \frac{\mathrm{G}\rho _{*}a^2 H}{\sigma}, +\frac{2 \dot{v}_\mathrm{DF} a}{v_c} \right) - \frac{256 G^3 M_{\bullet, 1} M_{\bullet, 2} (M_{\bullet, 1} +M_{\bullet, 2})}{5 c^5 a^3}\]

where \(a\) is the black hole binary separation, \(H\) is a dimensionless hardening parameter \(H\approx 15\) in the limit of a very hard, equal mass binary, \(\rho _\star\) is the density of stars, \(\dot{v}_\mathrm{DF}\) is the acceleration (negative) due to dynamical friction, \(v_\mathrm{c}\) is the circular velocity, \(\sigma\) is the velocity dispersion of stars. Here the first factor represents hardening due to strong scattering of stars, the second results from dynamical friction with distant stars, gas and dark matter and the last results from the emission of gravitational waves Peters (1964).

The acceleration due to dynamical friction is computed using Chandrasekhar’s formula:

\[\dot{v}_\mathrm{DF}=- {2 \pi \mathrm{G}^2 M_\bullet \over V_\mathrm{C}^2} \sum_{i} \rho_i \log(1+\Lambda_i^2) \left[ \hbox{erf}(X_i)-\left\{ {2 X_i \over \sqrt{\pi}} \exp\left(-X_i^2\right) \right\} \right],\]

where the sum is taken over the spheroid (gaseous plus stellar mass) and dark matter halo componentsfootnoteThe disk is ignored as the black hole is assumed to be orbiting in a circular orbit in the disk.. Here,

\[\Lambda_i = {a \sigma^2 \over \mathrm{G}(M_{\bullet, 1}+M_{\bullet, 2})},\]

is the Coulomb logarithm and

\[X_i = V_\mathrm{c} / \sqrt{2} \sigma.\]

In all of the above equations, the velocity dispersion \(\sigma_i\) is computed from the spherical Jeans equation assuming an isotropic velocity dispersion if [computeVelocityDispersion]\(=\)true. Otherwise, \(\sigma_i\) is set to the halo virial velocity for dark matter and to the spheroid characteristic velocity for the spheroid.

In calculating the rate of hardening due to scattering of stars, the stellar density is reduced by a factor (Volonteri et al., 2003)

\[f_\rho = \hbox{min}\left\{ \left[ { 4 a \sigma_\mathrm{spheroid}^2 \over 3 \mathrm{G} (M_{\bullet, 1}+M_{\bullet, 2})} \log\left({\mathrm{G} M_{\bullet, 2} \over 4 \sigma_\mathrm{spheroid}^2 a }\right) \right]^2 , 1 \right\},\]

if [stellarDensityChangeBinaryMotion]\(=\)true to account for the ejection of stars from the loss cone.

Methods

  • factors — Compute factors needed for tidal tensor calculation.

  • tidalTensorGet — Get the tidal tensor.

Parameters

  • [stellarDensityChangeBinaryMotion] (boolean; default .true.) — If true, account for the change in stellar density caused by the black hole binary’s motion through the stellar background when computing the hardening rate of the binary via stellar scattering.

  • [computeVelocityDispersion] (boolean; default .false.) — Specifies whether or not the velocity dispersion of dark matter and stars should be computed using Jeans equation in black hole binary hardening calculations. If false, then the velocity dispersions are assumed to equal the characteristic velocity of dark matter and spheroid.

blackHoleBinarySeparationGrowthRateZero

A black hole binary separation growth class in which the separation does not grow.

(Default implementation)