scrrpy

class scrrpy.DRR(sma, gamma=1.75, mbh_mass=4000000.0, star_mass=1.0, j_grid_size=128, rh=2.0, seed=None)

Bases: scrrpy.cusp.Cusp

Resonant relaxation diffusion coefficient (DRR). Assuming a power law stellar cusp around a massive black hole (MBH). The cusp is assumed to have an isotropic distribution function \(f(E) \propto |E|^p\) corresponding ro a stellar density \(n(r) \propto r^{-\gamma}\) where \(\gamma = \tfrac{3}{2} + p\)

Parameters:

• sma (float) – The semi-mahor axis along which DRR will be computed
• gamma (float, int, optional) – The slope of the density profile. Default: 7/4 (Bahcall wolf cusp)
• mbh_mass (float, int, optional) – Mass of the MBH [solar mass]. Default: \(4.3 \times 10^6\) (Milky Way MBH)
• star_mass (float, int, optional) – Mass of individual stars [solar mass]. Default: 1.0
• rh (float, int, optional) – Radius of influence [pc]. Define as the radius in which the velocity dispersion of the stellar cusp \(\sigma\) is equal to the Keplerian velocity due to the MBH \(\sigma(r_\mathrm{h})^2 = G M_{\bullet} / r_\mathrm{h}\). Default: 2.0

Methods Summary

__call__(l_max[, neval, threads, …])	Returns the RR diffusion coefficient \(D_{JJ}/J_{\mathrm{c}}^2\) [1/yr].
save(file_name)	Save the current instance to an hdf5 file.
from_file(file_name)	Load from file and return an instance

Methods Documentation

__call__(l_max, neval=1000.0, threads=1, progress_bar=True, seed=None)

Returns the RR diffusion coefficient \(D_{JJ}/J_{\mathrm{c}}^2\) [1/yr].

Parameters:

• l_max (int) – Maximal order of spherical harmonics to compute
• neval (int) – The maximum number of integrand evaluations in each iteration of the vegas algorithm. Default: 1000
• threads (int) – Number of parallel threads to use. Default: 1 (no parallelization)
• progress_bar (bool) – Show progress bar. Default: True

save(file_name)

Save the current instance to an hdf5 file.

Example

>>> drr = DRR(0.1, j_grid_size=32)
>>> d, d_err = drr(l_max=3)
>>> drr.save('example.hdf5')
>>> drr = DRR.from_file('example.hdf5')
>>> d, d_err = drr(l_max=drr.l_max, neval=drr.neval)

classmethod from_file(file_name)

Load from file and return an instance

Example

>>> drr = DRR(0.1, j_grid_size=32)
>>> d, d_err = drr(l_max=3)
>>> drr.save('example.hdf5')
>>> drr = DRR.from_file('example.hdf5')
>>> d, d_err = drr(l_max=drr.l_max, neval=drr.neval)

class scrrpy.Cusp(gamma=1.75, mbh_mass=4000000.0, star_mass=1.0, rh=2.0)

A power law stellar cusp around a massive black hole (MBH). The cusp is assumed to have an isotropic distribution function \(f(E) \propto |E|^p\) corresponding ro a stellar density \(n(r) \propto r^{-\gamma}\) where \(\gamma = \tfrac{3}{2} + p\)

TODO - Implement normalization Total mass at \(r_{\mathrm{h}}\)

TODO - Implement normalization \(N(a)\) vs \(N(r)\)

Parameters:

• gamma (float, int, optional) – The slope of the density profile. Default: 7/4 (Bahcall-Wolf cusp)
• mbh_mass (float, int) – Mass of the MBH [solar mass]. Default: \(4.3 \times 10^6\) (Milky Way MBH)
• star_mass (float, int) – Mass of individual stars [solar mass]. Default: 1.0
• rh (float, int) – Radius of influence [pc]. Define as the radius in which the velocity dispersion of the stellar cusp \(\sigma\) is equal to the Keplerian velocity due to the MBH \(\sigma(r_{\mathrm{h}})^2 = G M_{\bullet} / r_{\mathrm{h}}\). Default: 2.0

a_gr1

The sma below which \(\nu_\mathrm{p}\) is only positive, that is \(\nu_\mathrm{p} (a,j=1) = 0\)

d_nu_p(a, j)

The derivative of \(\nu_\mathrm{p}\) with respect to \(j\), defined to be positive

inverse_cumulative_a(x)

The inverse of \(N(a)\). Useful to generate a random sample of semi-major axis.

Parameters:x (float, array) – x in [0, 1]

Example

>>> cusp = Cusp(gamma=1.75)
>>> np.random.seed(1234)
>>> sma = cusp.inverse_cumulative_a(np.random.rand(100))
>>> print("{:0.10}, {:0.10}, {:0.10}".format(sma.min(), sma.mean(), sma.max()))
0.03430996478, 1.147418232, 1.987320281

jlc(a)

Relativistic loss cone

Minimal normalized angular momentum on which orbits are stable.

\(j_{\mathrm{lc}} = J_{\mathrm{lc}} / J_{\mathrm{c}}\), where \(J_{\mathrm{lc}} = 4GM_{\bullet}/c\) is the last stable orbit in the parabolic limit and \(J_{\mathrm{c}} = \sqrt{GM_{\bullet} a}\) is the maximal (circular) stable orbit.

This is an approximation which works when the orbital binding energy \(E\) is much smaller than rest energy of the MBH \(Mc^2\).

Parameters:a (float, array) – Semi-major axis [pc].

mass_ratio

MBH to star mass ratio

nu_gr(a, j)

Precession frequency [rad/year] due to general relativity (first PN term)

Parameters:

• a (float, array) – Semi-major axis [pc].
• j (float, array) – Normalized angular momentum \(j = J/J_\mathrm{c} = \sqrt{1-e^2}\).

nu_mass(a, j)

Precession frequency [rad/year] due to stellar mass.

Parameters:

• a (float, array) – Semi-major axis [pc].
• j (float, array) – Normalized angular momentum \(j = J/J_{\mathrm{c}} = \sqrt{1-e^2}\).

nu_p(a, j)

Precession frequency [rad/year]

\(\nu_{\mathrm{p}} (a, j) = \nu_{\mathrm{gr}} (a, j) + \nu_{\mathrm{mass}} (a, j)\)

Parameters:

• a (float, array) – Semi-major axis [pc].
• j (float, array) – Normalized angular momentum \(j = J/J_\mathrm{c} = \sqrt{1-e^2}\).

nu_p1(a)

Precession frequency at \(j=1\)

nu_r(a)

The orbital frequency in rad/year at \(a\) [pc]

Parameters:a (float, array) – Semi-major axis [pc].

number_of_stars(a)

Number of stars with semi-major axis smaller than \(a\) [pc]

Parameters:a (float, array) – Semi-major axis [pc].

period(a)

The orbital period in years at \(a\) [pc]

Parameters:a (float, array) – Semi-major axis [pc].

rg

Gravitational radius of the MBH [pc]

stellar_mass(a)

Enclosed mass within \(r = a\) [pc].

TODO - check \(M(r)\) vs \(M(a)\)

Parameters:a (float, array) – Semi-major axis [pc].

tg

Light crossing time of the MBH [sec]

total_number_of_stars

Number of stars within the radius of influence \(r_{\mathrm{h}}\)

total_stellar_mass

Total mass within the radius of influence \(r_\mathrm{h}\) [solar mass]

TODO - Implement normalization