Initial Mass Function - Form of The IMF

Form of The IMF

The IMF is often stated in terms of a series of power laws, where (sometimes also represented as ), the number of stars with masses in the range to within a specified volume of space, is proportional to, where 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.

The IMF of stars more massive than our sun was first quantified by Edwin Salpeter in 1955. His work favoured an exponent of . 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

Later authors extended the work below one solar mass. Glenn E. Miller and John M. Scalo suggested that the IMF "flattened" (approached ) below one solar mass. Pavel Kroupa kept above half a solar mass, but introduced between 0.08-0.5 solar masses and below 0.08 solar masses.

Commonly used forms of the IMF are the Kroupa (2001) broken power law and the Chabrier (2003) log-normal.

Chabrier 2003 for individual stars:

for ,

for

Chabrier 2003 for stellar systems:

for ,

for

Kroupa 2001:

for ,

for ,

for

There are large uncertainties concerning the substellar region. In particular, the classical assumption of a single IMF covering the whole substellar and stellar mass range is being questioned in favour of a two-component IMF to account for possible different formation modes of substellar objects. I.e. one IMF covering brown dwarfs and very-low-mass stars on the one hand, and another ranging from the higher-mass brown dwarfs to the most massive stars on the other. Note that this leads to an overlap region between about 0.05 and 0.2 solar masses where both formation modes may account for bodies in this mass range.