Generalised Gamma (Wise+ 2012)

Generalised Gamma (Wise+ 2012)

Danger

This IMF used to be called ExponentialCutoffPowerLawIMF. It’s possible some old references to this still exist.

Form

Also referred to as the generalized Rosin-Rammler function by Chabrier (2003), the Generalised Gamma Distribution can be expressed as

\[\xi(m) = \xi_0 M^\alpha\exp\left[-\left(\frac{M}{M_c}\right)^\beta\right].\]
This looks like a power law with an exponential cutoff at the low-mass end, which makes it tempting for studies of Population III stars, but is also used for population II.

The default values in our implementation are

\[\alpha = -2.3, \beta=-1.6, M_c = 10M_\odot.\]

Wise+ (2012) provide the values

\[\alpha = -2.3, \beta=-1.6,\]
but suggested a characteristic mass \(M_c\sim100M_\odot\). However, recent studies take values closer to \(10M_\odot\) (e.g. Brauer+ (2025)).

de Bennassuti+ (2017), also refer to a form with \(\beta = -1\) and \(M_c=20M_\odot\) as a “Larson-style” IMF, after Larson (1998).

Implemented in

This is implemented in the GeneralisedGammaIMF class, where the user is free to pick whatever parameters they like or use the default (same as above).

Integrals

Under the hood we will be using scipy’s gamma and gammainc to compute the lower incomplete gamma function

\[\mathtt{gamma}(n+1)\mathtt{gammainc}(n+1, u) = \Gamma(n+1, u) = \int_0^u t^n e^{-t}\mathrm{d}t.\]

We can change the lower limits using the fact \(\int^{b}_{a}f(x)\mathrm{d}x = \int^{b}_{0}f(x)\mathrm{d}x - \int^{a}_{0}f(x)\mathrm{d}x\):

\[ \int_\ell^u t^n e^{-t}\mathrm{d}t = \Gamma(n+1, u) - \Gamma(n+1, \ell) \]

We will also use the substitution

\[u = \left(\frac{M}{M_c}\right)^\beta \Rightarrow\mathrm{d}M = \frac{M_c}{\beta}u^{\frac{1-\beta}{\beta}}\mathrm{d}u.\]

Total number of stars

\[\begin{split}\begin{align} \int^{M_\textrm{max}}_{M_\textrm{min}}\xi(M)\mathrm{d}M &= \xi_0\int^{M_\textrm{max}}_{M_\textrm{min}}M^\alpha\exp\left\{-\left(\frac{M}{M_c}\right)^\beta\right\}\mathrm{d}M \\ &= \frac{\xi_0M_c^{\alpha+1}}{\beta}\int^{u(M_\textrm{max})}_{u(M_\textrm{min})}u^{\frac{\alpha+1}{\beta}-1}e^{-u}\mathrm{d}u \\ &= \frac{\xi_0M_c^{\alpha+1}}{\beta}\left\{ \Gamma\left(\frac{\alpha+1}{\beta}, \left[\frac{M_\mathrm{max}}{M_c}\right]^\beta\right) - \Gamma\left(\frac{\alpha+1}{\beta}, \left[\frac{M_\mathrm{min}}{M_c}\right]^\beta\right) \right\}. \end{align}\end{split}\]

Total mass of stars

\[\begin{split}\begin{align} \int^{M_\textrm{max}}_{M_\textrm{min}}\xi(M)\mathrm{d}M &= \xi_0\int^{M_\textrm{max}}_{M_\textrm{min}}M^{\alpha+1}\exp\left\{-\left(\frac{M}{M_c}\right)^\beta\right\}\mathrm{d}M \\ &= \frac{\xi_0M_c^{\alpha+2}}{\beta}\int^{u(M_\textrm{max})}_{u(M_\textrm{min})}u^{\frac{\alpha+2}{\beta}-1}e^{-u}\mathrm{d}u \\ &= \frac{\xi_0M_c^{\alpha+2}}{\beta}\left\{ \Gamma\left(\frac{\alpha+2}{\beta}, \left[\frac{M_\mathrm{max}}{M_c}\right]^\beta\right) - \Gamma\left(\frac{\alpha+2}{\beta}, \left[\frac{M_\mathrm{min}}{M_c}\right]^\beta\right) \right\}. \end{align}\end{split}\]

See also