On the function describing the stellar initial mass function

TL;DR

This paper introduces the L_3 IMF, a new functional form for the stellar initial mass function that smoothly combines low- and high-mass power laws, closely matching standard IMFs with analytical advantages.

Contribution

The L_3 IMF provides a natural heavy-tailed approximation with a closed-form cumulative distribution, simplifying analysis and generation of stellar masses.

Findings

L_3 IMF closely matches standard IMFs like Kroupa and Chabrier

It has a closed-form cumulative distribution function

The mass generating function can be explicitly written down

Abstract

We propose a functional form for the IMF, the L_3 IMF, which is a natural heavy-tailed approximation to the log-normal distribution. It is composed of a low-mass power law and a high mass power-law which are smoothly joined together. Three parameters are needed to achieve this. The standard IMFs of Kroupa-2001a, Kroupa-2002a and Chabrier-2003b (single stars or systems) are essentially indistinguishable from this form. Compared to other 3-parameter functions of the IMF, the L_3 IMF has the advantage that the cumulative distribution function and many other characteristic quantities have a closed form, the mass generating function, for example, can be written down explicitly.