schechter_vdf

skypy.galaxies.velocity_dispersion.schechter_vdf(alpha, beta, vd_star, vd_min, vd_max, size=None, resolution=1000)

Sample velocity dispersion of elliptical galaxies in the local universe following a Schecter function.

Parameters:

alpha: int

The alpha parameter in the modified Schechter equation.

beta: int

The beta parameter in the modified Schechter equation.

vd_star: int

The characteristic velocity dispersion.

vd_min, vd_max: int

Lower and upper bounds of random variable x. Samples are drawn uniformly from bounds.

resolution: int

Resolution of the inverse transform sampling spline. Default is 100.

size: int

Number of samples returned. Default is 1.

Returns:

velocity_dispersion: array_like

Velocity dispersion drawn from Schechter function.

Notes

The probability distribution function \(p(\sigma)\) for velocity dispersion \(\sigma\) can be described by a Schechter function (see eq. (4) in [1])

\[\phi = \phi_* \left(\frac{\sigma}{\sigma_*}\right)^\alpha \exp\left[-\left( \frac{\sigma}{\sigma_*} \right)^\beta\right] \frac{\beta}{\Gamma(\alpha/\beta)} \frac{1}{\sigma} \mathrm{d}\sigma \;.\]

where \(\Gamma\) is the gamma function, \(\sigma_*\) is the characteristic velocity dispersion, \(\phi_*\) is number density of all spiral galaxies and \(\alpha\) and \(\beta\) are free parameters.

References

[1]

Choi, Park and Vogeley, (2007), astro-ph/0611607, doi:10.1086/511060