Initial mass function

Edwin E. Salpeter is the first astrophysicist who attempted to quantify IMF by applying power law into his equations.^[9] 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 IMF is $${\displaystyle \xi (m)\Delta m=\xi _{0}\left({\frac {m}{M_{\odot }}}\right)^{-2.35}\left({\frac {\Delta m}{M_{\odot }}}\right).}$$ where ${\displaystyle \xi _{0}}$ is a constant relating to the local stellar density.

Glenn E. Miller and John M. Scalo extended the work of Salpeter, by introducing the log-normal formulation and suggesting that the IMF "flattened" (${\displaystyle \alpha \rightarrow 0}$) when stellar masses fell below 1 M_☉.^[10]

After introducing with Christopher Tout and Gerard Gilmore the corrections for unresolved binary stars in the Galactic field in 1991, Pavel Kroupa found ${\displaystyle \alpha =2.3}$ between 0.5–150 M_☉, but introduced ${\displaystyle \alpha =1.3}$ between 0.08–0.5 M_☉ and ${\displaystyle \alpha =0.3}$ below 0.08 M_☉. Based on Galactic-field star counts analysed by Scalo from 1986, the Galactic-field IMF above 1 M_☉ has ${\displaystyle \alpha =2.7}$, where binary-star corrections are not significant.^[8] The difference above 1 M_☉ between star forming regions (${\displaystyle \alpha =2.3}$) and the Galactic field (${\displaystyle \alpha =2.7}$) is explained by the latter being a composite IMF, or integrated galaxy IMF, made from the sum of many star forming regions. ^[11]

Gilles Chabrier gave the following expression for the density of individual stars in the Galactic disk, in units of pc⁻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 1 M_☉.

For stellar systems (namely binaries) in the Galactic field, he gave: $${\displaystyle \xi (m)={\frac {0.086}{m\ln(10)}}\exp \left[-{\frac {(\log(m)-\log(0.22))^{2}}{2\times 0.57^{2}}}\right]\quad {\text{ for }}m<1.}$$ The binary fraction differs in different regions.^[11]

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 for substellar objects—one IMF covering brown dwarfs and very-low-mass stars, and another ranging from the higher-mass brown dwarfs to the most massive stars. This leads to an overlap region approximately between 0.05–0.2 M_☉ where both formation modes may account for bodies in this mass range.^[13]

The possible variation of the IMF affects our interpretation of the galaxy signals and the estimation of cosmic star formation history^[14] 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.^{[15][16][11][17]}

Measurements of the local universe where single stars can be resolved are consistent with an invariant IMF^{[18][19][20][11][21]} 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.^[22]

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^[23][24][25] and galaxies^{[26][27][28][29][30][31][32][33][34]} 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.^[a]

Recent studies have suggested that filamentary structures in molecular clouds play a crucial role in the initial conditions of star formation and the origin of the stellar IMF. Herschel observations of the California giant molecular cloud show that both the prestellar core mass function (CMF) and the filament line mass function (FLMF) follow power-law distributions at the high-mass end, consistent with the Salpeter power-law IMF. Specifically, the CMF follows ${\displaystyle \Delta N/\Delta \log M\propto M^{-1.4\pm 0.2}}$ for masses greater than ${\displaystyle 1\,M_{\odot }}$, and the FLMF follows ${\displaystyle \Delta N/\Delta \log M_{\text{line}}\propto M_{\text{line}}^{-1.5\pm 0.2}}$ for filament line masses greater than ${\displaystyle 10\,M_{\odot }{\text{pc}}^{-1}}$. Recent research suggests that the global prestellar CMF in molecular clouds is the result of the integration of CMFs generated by individual thermally supercritical filaments, which indicates a tight connection between the FLMF and the CMF/IMF, supporting the idea that filamentary structures are a critical evolutionary step in establishing a Salpeter-like mass function.^[35]