# Initial mass function

In astronomy, the initial mass function (IMF) is an empirical function that describes the initial distribution of masses for a population of stars. The IMF is an output of the process of star formation. The IMF is often given as a probability distribution function (PDF) for the mass at which a star enters the main sequence (begins hydrogen fusion). The distribution function can then be used to construct the mass distribution (the histogram of stellar masses) of a population of stars. It differs from the present day mass function (PDMF), the current distribution of masses of stars, due to the evolution and death of stars which occurs at different rates for different masses as well as dynamical mixing in some populations.

The properties and evolution of a star are closely related to its mass, so the IMF is an important diagnostic tool for astronomers studying large quantities of stars. For example, the initial mass of a star is the primary factor determining its colour, luminosity, and lifetime. At low masses, the IMF sets the Milky Way Galaxy mass budget and the number of substellar objects that form. At intermediate masses, the IMF controls chemical enrichment of the interstellar medium. At high masses, the IMF sets the number of core collapse supernovae that occur and therefore the kinetic energy feedback.

The IMF is relatively invariant from one group of stars to another, though some observations suggest that the IMF is different in different environments.[1][2][3]

## Form of the IMF

Initial mass function

The IMF is often stated in terms of a series of power laws, where ${\displaystyle N(m)\mathrm {d} m}$ (sometimes also represented as ${\displaystyle \xi (m)\Delta m}$), the number of stars with masses in the range ${\displaystyle m}$ to ${\displaystyle m+\mathrm {d} m}$ within a specified volume of space, is proportional to ${\displaystyle m^{-\alpha }}$, where ${\displaystyle \alpha }$ 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. Commonly used forms of the IMF are the Kroupa (2001) broken power law[4] and the Chabrier (2003) log-normal.[5]

### Salpeter (1955)

The IMF of stars more massive than our sun was first quantified by Edwin Salpeter in 1955.[6] His work favoured an exponent of ${\displaystyle \alpha =2.35}$. 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

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

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

### Miller-Scalo (1979)

Later authors extended the work below one solar mass (M). Glenn E. Miller and John M. Scalo suggested that the IMF "flattened" (approached ${\displaystyle \alpha =0}$) below one solar mass.[7]

### Kroupa (2001)

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.

${\displaystyle \xi (m)=m^{-\alpha },}$
${\displaystyle \alpha =0.3}$ for ${\displaystyle m<0.08,}$
${\displaystyle \alpha =1.3}$ for ${\displaystyle 0.08
${\displaystyle \alpha =2.3}$ for ${\displaystyle m>0.5}$

### Chabrier (2003)

Chabrier 2003 for individual stars:

${\displaystyle \xi (m)=0.158(1/(\ln(10)m))\exp[-(\log(m)-\log(0.08))^{2}/(2\times 0.69^{2})]}$ for ${\displaystyle m<1,}$
${\displaystyle \xi (m)=km^{-\alpha }}$ for ${\displaystyle m>1,\alpha =2.3\pm 0.3}$

Chabrier 2003 for stellar systems (e.g. binaries):

${\displaystyle \xi (m)=0.086(1/(\ln(10)m))\exp[-(\log(m)-\log(0.22))^{2}/(2\times 0.57^{2})]}$ for ${\displaystyle m<1,}$
${\displaystyle \xi (m)=km^{-\alpha }}$ for ${\displaystyle m>1,\alpha =2.3\pm 0.3}$

## Slope

The initial mass function is typically graphed on a logarithm scale of log(N) vs log(m). Such plots give approximately straight lines with a slope Γ equal to 1-α. Hence Γ is often called the slope of the initial mass function. The present-day mass function, for coeval formation, has the same slope except that it rolls off at higher masses which have evolved away from the main sequence.[8]

### Uncertainties

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.[9]

### Variation

The possible variation of the IMF affects our interpretation of the galaxy signals and the estimation of cosmic star formation history[10] 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.[11]

Measurements of the local Universe where single stars can be resolved are consistent with an invariant IMF[12] 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.

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[13] and galaxies[14] 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.

## References

1. ^ Conroy, Charlie; van Dokkum, Pieter G. (2012). "The Stellar Initial Mass Function in Early-type Galaxies From Absorption Line Spectroscopy. II. Results". The Astrophysical Journal. 760 (1): 71. arXiv:1205.6473. Bibcode:2012ApJ...760...71C. doi:10.1088/0004-637X/760/1/71.
2. ^ Kalirai, Jason S.; Anderson, Jay; Dotter, Aaron; Richer, Harvey B.; Fahlman, Gregory G.; Hansen, Brad M.S.; Hurley, Jarrod; Reid, I. Neill; Rich, R. Michael; Shara, Michael M. (2013). "Ultra-Deep Hubble Space Telescope Imaging of the Small Magellanic Cloud: The Initial Mass Function of Stars with M < 1 Msun". The Astrophysical Journal. 763 (2): 110. arXiv:1212.1159. Bibcode:2013ApJ...763..110K. doi:10.1088/0004-637X/763/2/110.
3. ^ Geha, Marla; Brown, Thomas M.; Tumlinson, Jason; Kalirai, Jason S.; Simon, Joshua D.; Kirby, Evan N.; VandenBerg, Don A.; Muñoz, Ricardo R.; Avila, Roberto J.; Guhathakurta, Puragra; Ferguson, Henry C. (2013). "The Stellar Initial Mass Function of Ultra-faint Dwarf Galaxies: Evidence for IMF Variations with Galactic Environment". The Astrophysical Journal. 771 (1): 29. arXiv:1304.7769. Bibcode:2013ApJ...771...29G. doi:10.1088/0004-637X/771/1/29.
4. ^ Kroupa, Pavel (2001). "On the variation of the initial mass function". MNRAS. 322 (2): 231–246. arXiv:astro-ph/0009005. Bibcode:2001MNRAS.322..231K. doi:10.1046/j.1365-8711.2001.04022.x.
5. ^ Chabrier, Gilles (2003). "Galactic stellar and substellar initial mass function". Publications of the Astronomical Society of the Pacific. 115 (809): 763–795. arXiv:astro-ph/0304382. Bibcode:2003PASP..115..763C. doi:10.1086/376392.
6. ^ Salpeter, Edwin (1955). "The luminosity function and stellar evolution". Astrophysical Journal. 121: 161. Bibcode:1955ApJ...121..161S. doi:10.1086/145971.
7. ^ Miller, Glenn; Scalo, John (1979). "The initial mass function and stellar birthrate in the solar neighborhood". Astrophysical Journal Supplement Series. 41: 513. Bibcode:1979ApJS...41..513M. doi:10.1086/190629.
8. ^ Massey, Philip (1998). "The Initial Mass Function of Massive Stars in the Local Group". The Stellar Initial Mass Function (38Th Herstmonceux Conference). 142: 17. Bibcode:1998ASPC..142...17M.
9. ^ Kroupa, Pavel; et al. (2013). "The stellar and sub-stellar IMF of simple and composite populations". Stellar Systems and Galactic Structure, Vol. V. arXiv:1112.3340. Bibcode:2013pss5.book..115K. doi:10.1007/978-94-007-5612-0_4.
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## Notes

1.^ Different mass of stars have different ages, thus modifying the star formation history would modify the present-day mass function, which mimics the effect of modifying the IMF.