Initial mass function

The initial mass function (IMF) is usually expressed as a power law, ξ(m) ∝ m^−α, where ξ(m) is the number of stars per unit mass interval. The classical form, proposed by Edwin Salpeter in 1955, has a single slope α = 2.35 for masses above about 0.1 M☉.¹ Modern determinations, such as the Kroupa and Chabrier IMFs, find a flattening or turnover at lower masses, with α ≈ 1.3 for 0.08–0.5 M☉ and α ≈ 2.3 for higher masses.²

The IMF is defined for a single stellar population, i.e., stars born at the same time from the same molecular cloud. It is assumed to be invariant across different environments, though this universality is debated. The IMF sets the mass-to-light ratio of galaxies and the rate of supernovae, making it crucial for interpreting observations.

Measuring the IMF directly is challenging because we only observe stars at their current masses, which have evolved from their birth masses. For low-mass stars (≤1 M☉), which live longer than the age of the Universe, the present-day mass function in the solar neighborhood approximates the IMF. For higher masses, corrections for stellar evolution are applied.³

Resolved stellar populations in nearby galaxies, such as the Magellanic Clouds, allow direct counting of stars down to the hydrogen-burning limit. However, for distant galaxies, the IMF is inferred indirectly from integrated light, stellar population synthesis, and the ratio of Hα to ultraviolet flux, which is sensitive to the number of massive stars. These indirect methods suggest a possible variation of the IMF with galactic environment, particularly a top-heavy IMF in starburst galaxies.⁴

The origin of the IMF lies in the process of star formation, which involves the fragmentation of molecular clouds, turbulence, and feedback. Competing theories include the turbulent fragmentation model, which predicts a IMF shaped by the spectrum of turbulence, and the competitive accretion model, where stars grow by accreting gas from a shared reservoir.⁵

Numerical simulations of star formation, such as those by Krumholz and collaborators, reproduce the observed IMF shape, suggesting that the IMF is a natural outcome of supersonic turbulence and gravity. The characteristic mass of the IMF, around 0.5 M☉, is thought to be set by the thermal physics of the gas, specifically the Jeans mass at the typical density and temperature of star-forming regions.⁶

The IMF is a cornerstone of many astrophysical calculations. It determines the chemical enrichment of the interstellar medium through stellar winds and supernovae, the production of heavy elements, and the rate of compact object formation (white dwarfs, neutron stars, black holes).⁷

In galaxy evolution, the IMF affects the stellar mass function, the star formation rate indicators, and the interpretation of galaxy spectra. Variations in the IMF, if real, would have profound implications for our understanding of galaxy formation and the cosmic star formation history. For example, a top-heavy IMF in the early Universe could explain the abundance of elements in the most metal-poor stars.⁸

One lesser-known aspect is the existence of a possible 'bottom-heavy' IMF in some massive elliptical galaxies, inferred from gravitational lensing and stellar dynamics, which would imply an excess of low-mass stars.⁴ Another is the 'IMF mismatch problem' in globular clusters, where the observed mass-to-light ratio is lower than expected, suggesting a deficit of low-mass stars or a different IMF.

Also, the IMF may extend to sub-stellar objects, such as brown dwarfs, but the boundary is fuzzy. The IMF of the first stars (Population III) is predicted to be top-heavy, with masses of tens to hundreds of solar masses, but this has not been directly observed. Finally, the IMF is used in estimating the number of planets via microlensing, as the mass function of the lensing objects is related to the IMF.⁹

Stellar population

A group of stars with a common origin and similar age and metallicity.

Power law

A mathematical relationship where one quantity varies as a power of another.

Jeans mass

The critical mass above which a cloud of gas will collapse under its own gravity.

Top-heavy IMF

An IMF with a higher proportion of massive stars than the standard IMF.

Bottom-heavy IMF

An IMF with a higher proportion of low-mass stars than the standard IMF.