Initial mass function

Edwin E. Salpeter is the first astrophysicist who attempted to quantify IMF by applying power law into his equations.[8] His work is based upon the sun-like stars that can be easily observed with great accuracy.[2] Salpeter defined the mass function as the number of stars in a volume of space observed at a time as per logarithmic mass interval.[2] His work enabled a large number of theoretical parameters to be included in the equation while converging all these parameters into an exponent of ${\displaystyle \alpha =2.35}$.[1]

The Salpeter Initial Mass Function is

$${\displaystyle \xi (m)\Delta m=\xi _{0}\left({\frac {m}{M_{\odot }}}\right)^{-2.35}\left({\frac {\Delta m}{M_{\odot }}}\right).}$$

where ${\displaystyle M_{\odot }}$ is the solar mass, and ${\displaystyle \xi _{0}}$ is a constant relating to the local stellar density.

Glenn E. Miller and John M. Scalo extended the work of Salpeter, by suggesting that the IMF "flattened" (approaching ${\displaystyle \alpha =0}$) when stellar masses fell below one solar mass (M_☉).[9]

Pavel Kroupa kept ${\displaystyle \alpha =2.3}$ above half a solar mass, but introduced ${\displaystyle \alpha =1.3}$ between 0.08-0.5 M_☉ and ${\displaystyle \alpha =0.3}$ below 0.08 M_☉. Between 0.5 M_☉ and 1.0 M_☉, correcting for unresolved binary stars also adds a fourth domain with ${\displaystyle \alpha =2.7}$.[7]

Chabrier gave the following expression for the density of individual stars in the Galactic disk, in units of parsec⁻³:[2]

$${\displaystyle \xi (m)={\frac {0.158}{m\ln(10)}}\exp \left[-{\frac {(\log(m)-\log(0.08))^{2}}{2\times 0.69^{2}}}\right]\quad {\text{ for }}m<1,}$$

This expression is log-normal, meaning that the logarithm of the mass follows a Gaussian distribution (up to one solar mass).

For stellar systems (e.g. binaries), he gave:

$${\displaystyle \xi (m)={\frac {0.086}{m\ln(10)}}\exp \left[-{\frac {(\log(m)-\log(0.22))^{2}}{2\times 0.57^{2}}}\right]\quad {\text{ for }}m<1}$$

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 favor 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 M_☉ where both formation modes may account for bodies in this mass range.[11]

The possible variation of the IMF affects our interpretation of the galaxy signals and the estimation of cosmic star formation history[12] thus is important to consider.

In theory, the IMF should vary with different star-forming conditions. Higher ambient temperature increases the mass of collapsing gas clouds (Jeans mass); lower gas metallicity reduces the radiation pressure thus make the accretion of the gas easier, both lead to more massive stars being formed in a star cluster. The galaxy-wide IMF can be different from the star-cluster scale IMF and may systematically change with the galaxy star formation history.[13]

Measurements of the local Universe where single stars can be resolved are consistent with an invariant IMF[14] but the conclusion suffers from large measurement uncertainty due to the small number of massive stars and difficulties in distinguishing binary systems from the single stars. Thus IMF variation effect is not prominent enough to be observed in the local Universe. However, recent photometric survey across cosmic time does suggest a potentially systematic variation of the IMF at high redshift.[15]

Systems formed at much earlier times or further from the Galactic neighborhood, where star formation activity can be hundreds or even thousands time stronger than the current Milky Way, may give a better understanding. It has been consistently reported both for star clusters[16] and galaxies[17] that there seems to be a systematic variation of the IMF. However, the measurements are less direct. For star clusters the IMF may change over time due to complicated dynamical evolution.