stellar astrophysics - How exactly is the Initial Mass Function (IMF) calculated?

What is it?

An IMF, $Phi(m)$, is defined such as $Phi(m){rm d}m$ gives the fractions of stars with a mass between $m - {rm d}m/2$ and $m + {rm d}m/2$, and with a normalized distribution

$$int_{m_{rm min}}^{m_{rm max}}mPhi(m){rm d}m = 1 M_{odot}.$$

Note that these boundaries ($m_{rm min}$ and $m_{rm max}$) are ill-defined, but typically of the order of 0.1 $M_{odot}$ and 100 $M_{odot}$, respectively.

IMFs

The various IMF used are the following, with their main characteristics:

• the Salpeter's IMF, that is a parametrization of the IMF by a simple power-law, of the form
  $$Phi(m){rm d}m propto m^{-alpha}{rm d}m;$$


• the Miller & Scalo's IMF, that is a parametrization of the IMF by a log-normal distribution of the form
  $$xileft(log(m)right) = A_0 + A_1 log(m) + A_2left(log(m)right)^2;$$


• the Kroupa's IMF, that is a parametrization of the IMF by a broken power-law;


• the Chabrier's IMF and Chabrier's system IMF, that is a combination of log-normal distribution (for low mass stars with masses less than 1 $M_{odot}$) and and a power-law distribution (for larger masses). The difference between the IMF and the system IMF is to merge resolved objects into multiple systems to compute the magnitude of systems instead of individual stars.

Determination

As you see, all these IMF are parametrization, deduced from observations. In general, the observations used to infer these mass functions come from star clusters in our galaxy. All you need to do it is to find a mass-magnitude relationship to deduce, from an observed luminosity, a mass function. In general, the number density distribution per mess interval, ${rm d}n/{rm d}m$, takes the following form
$$frac{{rm d}n}{{rm d}m}left(mright)_tau = left(frac{{rm d}n}{{rm d}M_lambda(m)}right) times left(frac{{rm d}m}{{rm d}M_lambda(m)}right)^{-1}_tau,$$
for a given age $tau$ and an observed magnitude $M_lambda$. Then, it is just a matter of parametrization, but also of how well it can arise from a proper theory.

For this matter, the Chabrier's IMF is probably the one that is best back up by theoretical arguments. It relies on a gravo-turbulent theory, taking into account all the possible supports (thermal support, turbulent support and magnetic support) plus the dual nature of turbulence, that both favors star formation by compressing the gas, and impedes star formation, by dispersing the fluid. All the dirty details are given in Hennebelle & Chabrier (2008) and Hennebelle & Chabrier (2009), showing how you can analytically deduce an IMF from these theoretical considerations.

Applications

As far as I know, these IMF are more or less used for every type of population. However, you won't favor Salpeter's IMF if you have enough resolution to resolve low-mass objects, that are not at all well-taken into account with this IMF. You should also favor the Chabrier's system IMF in case of unresolved objects.

To know if all these IMFs are really well-suited to any kind of population is an open and difficult question (the so-called question of the universality of the IMF), in particular because you need to resolve individual stars in clearly identified clusters to deduce an IMF. There are some papers investigating the question (for example, you could have a look at Cappellari et al. (2012) for a recent discussion of the problem).