Isotropic Power-Law DF

class galpy.df.isotropicPowerLawdf(pot=None, denspot=None, rmax=10000.0, rmin=None, ro=None, vo=None)

Class that implements the isotropic spherical distribution function for a power-law tracer density \(\nu(r) = \nu_0\,r^{-\gamma}\) in a power-law potential \(\Phi(r) = -\Psi_0\,r^{2-\alpha}\) generated by \(\rho(r) \propto r^{-\alpha}\) with \(\alpha > 2\), where \(\Psi_0 = -\Phi(1)\) and \(\nu_0 = \nu(1)\).

The distribution function is

\[f(\mathcal{E}) = \frac{\nu_0\,\Gamma\!\left(\frac{\gamma}{\alpha-2}+1\right)}{2\sqrt{2}\,\pi^{3/2}\,\Gamma\!\left(\frac{\gamma}{\alpha-2}-\frac{1}{2}\right)\,\Psi_0^{\gamma/(\alpha-2)}}\,\mathcal{E}^{\gamma/(\alpha-2) - 3/2}\]

where \(\mathcal{E} = -E \geq 0\) is the relative energy.

__init__(pot=None, denspot=None, rmax=10000.0, rmin=None, ro=None, vo=None)

Initialize an isotropic power-law distribution function.

Parameters:

• pot (PowerSphericalPotential) – Power-law potential with alpha > 2.

• denspot (PowerSphericalPotential, optional) – Power-law potential representing the tracer density. If None, self-consistent (tracer = potential-generating density).

• rmax (float or Quantity, optional) – Maximum radius for sampling (required because the total mass is infinite). Default: 1e4.

• rmin (float or Quantity, optional) – Minimum radius. Auto-detected for divergent potentials if not specified.

• ro (float or Quantity, optional) – Distance scale for translation into internal units (default from configuration file).

• vo (float or Quantity, optional) – Velocity scale for translation into internal units (default from configuration file).

Notes

• 2025-03-27 - Written - Bovy (UofT)