The MLP Distribution: A Modified Lognormal Power-Law Model for the Stellar Initial Mass Function

TL;DR

This paper introduces the MLP distribution, a modified lognormal power-law model derived from exponential growth processes, and demonstrates its effectiveness in modeling the stellar initial mass function with analytical properties and practical applications.

Contribution

It presents the MLP distribution, derived from a simple growth model, and applies it to accurately fit the stellar initial mass function, providing useful analytical tools.

Findings

MLP distribution fits the stellar IMF well

Provides analytical expressions for moments and distribution properties

Enables quick calculation of statistical quantities in IMF modeling

Abstract

This work explores the mathematical properties of a distribution introduced by Basu & Jones (2004), and applies it to model the stellar initial mass function (IMF). The distribution arises simply from an initial lognormal distribution, requiring that each object in it subsequently undergoes exponential growth but with an exponential distribution of growth lifetimes. This leads to a modified lognormal with a power-law tail (MLP) distribution, which can in fact be applied to a wide range of fields where distributions are observed to have a lognormal-like body and a power-law tail. We derive important properties of the MLP distribution, like the cumulative distribution, the mean, variance, arbitrary raw moments, and a random number generator. These analytic properties of the distribution can be used to facilitate application to modeling the IMF. We demonstrate how the MLP function provides an excellent fit to the IMF compiled by Chabrier (2005) and how this fit can be used to quickly identify quantities like the mean, median, and mode, as well as number and mass fractions in different mass intervals.