An Etymological Dictionary of Astronomy and Astrophysics

A mathematical expression describing the relative number of stars found in different ranges of mass for a cluster of stars at the time of its formation. It is defined as φ(log M) = dN / dlog M ∝ M ^-Γ, where M is the mass of a star and N is the number of stars in a logarithmic mass interval. The value of the slope found by Salpeter (1955) for → low-mass and → intermediate-mass stars in the → solar neighborhood is Γ = 1.35. The IMF can be expressed also in linear mass units: χ(M) = dN / DM ∝ M ^-α. Note that χ(M) = (1 / M lm 10) φ(log M), and α = Γ + 1. In this formalism the Salpeter slope is α = 2.35. There is a third way for representing the IMF, in which the exponent is x = -α.
The IMF is not a single power law over all masses, from → brown dwarfs to → very massive stars (Kroupa, 2002, Science 295, 82). Different slopes have been found for different mass segments, as follows: α = 1.3 for 0.08 ≤ M_solar < 0.5; α = 2.3 for 0.5 ≤ M_solar < 1; α = 2.3 for 1 ≤ M_solar.
The IMF at low masses can be fitted by a → lognormal distribution (See Bastian et al., 2010, ARAA 48, 339 and references therein). See also → canonical IMF.

See also: → initial; → mass; → function.