Initial mass function

In astronomy, the initial mass function (IMF) is an empirical function that describes the initial distribution of masses for a population of stars. The IMF is an output of the process of star formation. The IMF is often given as a probability distribution function (PDF) for the mass at which a star enters the main sequence (begins hydrogen fusion). The distribution function can then be used to construct the mass distribution (the histogram of stellar masses) of a population of stars. It differs from the present day mass function (PDMF), the current distribution of masses of stars, due to the evolution and death of stars which occurs at different rates for different masses as well as dynamical mixing in some populations. In astronomy, the initial mass function (IMF) is an empirical function that describes the initial distribution of masses for a population of stars. The IMF is an output of the process of star formation. The IMF is often given as a probability distribution function (PDF) for the mass at which a star enters the main sequence (begins hydrogen fusion). The distribution function can then be used to construct the mass distribution (the histogram of stellar masses) of a population of stars. It differs from the present day mass function (PDMF), the current distribution of masses of stars, due to the evolution and death of stars which occurs at different rates for different masses as well as dynamical mixing in some populations. The properties and evolution of a star are closely related to its mass, so the IMF is an important diagnostic tool for astronomers studying large quantities of stars. For example, the initial mass of a star is the primary factor determining its colour, luminosity, and lifetime. At low masses, the IMF sets the Milky Way Galaxy mass budget and the number of substellar objects that form. At intermediate masses, the IMF controls chemical enrichment of the interstellar medium. At high masses, the IMF sets the number of core collapse supernovae that occur and therefore the kinetic energy feedback. The IMF is relatively invariant from one group of stars to another, though some observations suggest that the IMF is different in different environments. The IMF is often stated in terms of a series of power laws, where N ( m ) d m {displaystyle N(m)mathrm {d} m} (sometimes also represented as ξ ( m ) Δ m {displaystyle xi (m)Delta m} ), the number of stars with masses in the range m {displaystyle m} to m + d m {displaystyle m+mathrm {d} m} within a specified volume of space, is proportional to m − α {displaystyle m^{-alpha }} , where α {displaystyle alpha } is a dimensionless exponent. The IMF can be inferred from the present day stellar luminosity function by using the stellar mass-luminosity relation together with a model of how the star formation rate varies with time. Commonly used forms of the IMF are the Kroupa (2001) broken power law and the Chabrier (2003) log-normal. The IMF of stars more massive than our sun was first quantified by Edwin Salpeter in 1955. His work favoured an exponent of α = 2.35 {displaystyle alpha =2.35} . This form of the IMF is called the Salpeter function or a Salpeter IMF. It shows that the number of stars in each mass range decreases rapidly with increasing mass. The Salpeter Initial Mass Function is where M ⊙ {displaystyle M_{odot }} is the solar mass, and ξ 0 {displaystyle xi _{0}} is a constant relating to the local stellar density. Later authors extended the work below one solar mass (M☉). Glenn E. Miller and John M. Scalo suggested that the IMF 'flattened' (approached α = 1 {displaystyle alpha =1} ) below one solar mass. Pavel Kroupa kept α = 2.3 {displaystyle alpha =2.3} above half a solar mass, but introduced α = 1.3 {displaystyle alpha =1.3} between 0.08-0.5 M☉ and α = 0.3 {displaystyle alpha =0.3} below 0.08 M☉.