Initial Mass Function

The initial mass function (IMF) is a term commonly used to describe the statistical distribution function for the initial masses of stars. Its physical interpretation is that the number of stars with masses in the range $\left[ M, M + dM \right]$ is $\Psi \left( M \right) dM$.

If the primary mechanism through which stars accumulate mass is Bondi accretion, then this function can be estimated analytically. We recall that in Bondi accretion the mass accretion rate scales as the mass squared

$\dot{M} \propto M^2$

Solving this ODE yields

$M \left( t \right) = \frac{M_0}{1 - \beta M_0 t}$

where $M_0$ is the initial seed mass at the beginning of the accretion, and $\beta$ is some constant that is independent of $M_0$. The above relation can also be written in the following way

$\frac{1}{M} = \frac{1}{M_0} - \beta t$

Hence at a constant time

$\frac{d M}{M^2} = \frac{d M_0}{M_0^2}$

Assuming that the number of stars is conserved, we can use

$\Psi \left( M \right) dM = \Psi \left( M_0 \right) dM_0$

Combining the two relations yields

$\Psi \left( M \right) = \Psi \left( M_0 \right) \frac{M_0^2}{M^2} \propto M^{-2}$

This value is similar to the Salpeter IMF $\Psi \left( M \right) \propto M^{-2.35}$ obtained from observations.