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 during star formation. IMF not only describes the formation and evolution of individual stars, it also serves as an important link that describes the formation and evolution of galaxies.

The IMF is often given as a probability density function (PDF) that describes the probability for a star to have a certain mass during its formation. It differs from the present-day mass function (PDMF), which describes the current distribution of masses of stars, such as red giants, white dwarfs, neutron stars, and black holes, after some time of evolution away from the main sequence stars and after a certain amount of mass loss.{{cite web |title=Astronomy 112: Physics of Stars -n Class 19 Notes: The Stellar Life Cycle |url=https://websites.pmc.ucsc.edu/~glatz/astr_112/lectures/notes19.pdf |publisher=University of Carlifornia, Santa Cruz |access-date=23 December 2023 |archive-url=https://web.archive.org/web/20230406220216/https://websites.pmc.ucsc.edu/~glatz/astr_112/lectures/notes19.pdf |archive-date=6 April 2023}} Since there are not enough young clusters of stars available for the calculation of IMF, PDMF is used instead and the results are extrapolated back to IMF. IMF and PDMF can be linked through the "stellar creation function". Stellar creation function is defined as the number of stars per unit volume of space in a mass range and a time interval. In the case that all the main sequence stars have greater lifetimes than the galaxy, IMF and PDMF are equivalent. Similarly, IMF and PDMF are equivalent in brown dwarfs due to their unlimited lifetimes.

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 of determining its colour, luminosity, radius, radiation spectrum, and quantity of materials and energy it emitted into interstellar space during its lifetime.{{cite book |last1=Scalo |first1=JM |title=Fundamentals of Cosmic Physics |date=1986 |publisher=Gordon and Breach, Science Publishers, Inc |location=United Kingdom |pages=3 |url=https://web.archive.org/web/20220420024218if_/http://www.as.utexas.edu/astronomy/people/scalo/Scalo1986.IMF.FundCosPhys.pdf |access-date=28 February 2023}} 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,{{cite journal | bibcode = 2012ApJ...760...71C | title=The Stellar Initial Mass Function in Early-type Galaxies From Absorption Line Spectroscopy. II. Results | journal=The Astrophysical Journal | volume=760 | issue=1 | pages=71 | year=2012 | author1=Conroy, Charlie |author2=van Dokkum, Pieter G.|arxiv = 1205.6473 | doi = 10.1088/0004-637X/760/1/71 | s2cid=119109509 }}{{cite journal | bibcode = 2013ApJ...763..110K | title=Ultra-Deep Hubble Space Telescope Imaging of the Small Magellanic Cloud: The Initial Mass Function of Stars with M < 1 Msun | journal=The Astrophysical Journal | volume=763 | issue=2 | pages=110 | year=2013 | author1=Kalirai, Jason S. |author2=Anderson, Jay |author3=Dotter, Aaron |author4=Richer, Harvey B. |author5=Fahlman, Gregory G. |author6=Hansen, Brad M.S. |author7=Hurley, Jarrod |author8=Reid, I. Neill |author9=Rich, R. Michael |author10=Shara, Michael M.|arxiv = 1212.1159 | doi = 10.1088/0004-637X/763/2/110 | s2cid=54724031 }}{{cite journal | bibcode = 2013ApJ...771...29G | title=The Stellar Initial Mass Function of Ultra-faint Dwarf Galaxies: Evidence for IMF Variations with Galactic Environment | journal=The Astrophysical Journal | volume=771 | issue=1 | pages=29 | year=2013 | author1=Geha, Marla|author1-link= Marla Geha |author2=Brown, Thomas M. |author3=Tumlinson, Jason |author4=Kalirai, Jason S. |author5=Simon, Joshua D. |author6=Kirby, Evan N. |author7=VandenBerg, Don A. |author8=Muñoz, Ricardo R. |author9=Avila, Roberto J. |author10=Guhathakurta, Puragra |author11=Ferguson, Henry C.|arxiv = 1304.7769 |doi = 10.1088/0004-637X/771/1/29 | s2cid=119290783 }} and potentially dramatically different in early galaxies.{{Cite journal |last1=Sneppen |first1=Albert |last2=Steinhardt |first2=Charles L. |last3=Hensley |first3=Hagan |last4=Jermyn |first4=Adam S. |last5=Mostafa |first5=Basel |last6=Weaver |first6=John R. |date=2022-05-01 |title=Implications of a Temperature-dependent Initial Mass Function. I. Photometric Template Fitting |journal=The Astrophysical Journal |volume=931 |issue=1 |pages=57 |doi=10.3847/1538-4357/ac695e |arxiv=2205.11536 |bibcode=2022ApJ...931...57S |s2cid=249017733 |issn=0004-637X |doi-access=free }}

Development

File:Plot of various initial mass functions.svg

The mass of a star can only be directly determined by applying Kepler's third law to a binary star system. However, the number of binary systems that can be directly observed is low, thus not enough samples to estimate the initial mass function. Therefore, the stellar luminosity function is used to derive a mass function (a present-day mass function, PDMF) by applying mass–luminosity relation.{{cite journal | first = Gilles | last = Chabrier | title = Galactic stellar and substellar initial mass function | journal = Publications of the Astronomical Society of the Pacific | volume = 115 | issue = 809 | pages = 763–795 | date = 2003 | doi=10.1086/376392|arxiv = astro-ph/0304382 |bibcode = 2003PASP..115..763C | s2cid = 4676258 }} The luminosity function requires accurate determination of distances, and the most straightforward way is by measuring stellar parallax within 20 parsecs from the earth. Although short distances yield a smaller number of samples with greater uncertainty of distances for stars with faint magnitudes (with a magnitude > 12 in the visual band), it reduces the error of distances for nearby stars, and allows accurate determination of binary star systems. Since the magnitude of a star varies with its age, the determination of mass-luminosity relation should also take into account its age. For stars with masses above {{Solar mass|0.7|link=y}}, it takes more than 10 billion years for their magnitude to increase substantially. For low-mass stars with below {{Solar mass|0.13}}, it takes 5 × 10⁸ years to reach main sequence stars.

The IMF is often stated in terms of a series of power laws, where $N(m) \mathrm{d}m$ (sometimes also represented as $\xi (m) \Delta m$ ), the number of stars with masses in the range $m$ to $m + \mathrm{d}m$ within a specified volume of space, is proportional to $m^{-\alpha}$ , where $\alpha$ is a dimensionless exponent.

Commonly used forms of the IMF are the Kroupa (2001) broken power law{{cite journal | first = Pavel | last = Kroupa | title = The Initial Mass Function of Stars: Evidence for Uniformity in Variable Systems | journal = Science| volume = 295 | pages = 82–91 | date = 2002 | issue = 5552 | doi=10.1126/science.1067524| pmid = 11778039 |arxiv = astro-ph/0201098|bibcode = 2002Sci...295...82K | s2cid = 15276163 }} and the Chabrier (2003) log-normal.

= Salpeter (1955) =

Edwin E. Salpeter is the first astrophysicist who attempted to quantify IMF by applying power law into his equations.{{cite journal | first = Edwin | last = Salpeter | title = The luminosity function and stellar evolution | journal = Astrophysical Journal | volume = 121 | pages = 161 | date = 1955 | doi=10.1086/145971 | bibcode = 1955ApJ...121..161S }} His work is based upon the sun-like stars that can be easily observed with great accuracy. Salpeter defined the mass function as the number of stars in a volume of space observed at a time as per logarithmic mass interval. His work enabled a large number of theoretical parameters to be included in the equation while converging all these parameters into an exponent of $\alpha = 2.35$ . The Salpeter IMF is

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

where $\xi_{0}$ is a constant relating to the local stellar density.

= Miller–Scalo (1979) =

Glenn E. Miller and John M. Scalo extended the work of Salpeter, by introducing the log-normal formulation and suggesting that the IMF "flattened" ( $\alpha \rightarrow 0$ ) when stellar masses fell below {{Solar mass|1}}.{{cite journal | last1 = Miller | first1 = Glenn | last2 = Scalo | first2 = John | title = The initial mass function and stellar birthrate in the solar neighborhood | journal = Astrophysical Journal Supplement Series | volume = 41 | pages = 513 | date = 1979 | doi=10.1086/190629|bibcode = 1979ApJS...41..513M }}

= Kroupa (2002) =

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

{{Citation |last1=Kroupa |first1=Pavel |title=The Stellar and Sub-Stellar Initial Mass Function of Simple and Composite Populations |date=2013 |url=https://doi.org/10.1007/978-94-007-5612-0_4 |work=Planets, Stars and Stellar Systems: Volume 5: Galactic Structure and Stellar Populations |pages=115–242 |editor-last=Oswalt |editor-first=Terry D. |access-date=2023-11-02 |place=Dordrecht |publisher=Springer Netherlands |language=en |doi=10.1007/978-94-007-5612-0_4 |isbn=978-94-007-5612-0 |last2=Weidner |first2=Carsten |last3=Pflamm-Altenburg |first3=Jan |last4=Thies |first4=Ingo |last5=Dabringhausen |first5=Jörg |last6=Marks |first6=Michael |last7=Maschberger |first7=Thomas |editor2-last=Gilmore |editor2-first=Gerard|arxiv=1112.3340 |bibcode=2013pss5.book..115K }}

= Chabrier (2003) =

Gilles Chabrier gave the following expression for the density of individual stars in the Galactic disk, in units of pc{{sup|−3}}:

$\xi (m) = \frac{0.158}{m\ln(10)} \exp\left[- \frac{(\log(m)-\log(0.08))^2}{2 \times 0.69^2}\right] \quad \text{ for } m < 1,$ This expression is log-normal, meaning that the logarithm of the mass follows a Gaussian distribution up to {{Solar mass|1}}.

For stellar systems (namely binaries) in the Galactic field, he gave:

$\xi (m) = \frac{0.086}{m \ln(10)} \exp \left[- \frac{(\log(m)-\log(0.22))^2}{2 \times 0.57^2} \right] \quad \text{ for } m < 1.$ The binary fraction differs in different regions.{{Citation |last1=Kroupa |first1=Pavel |title=The Stellar and Sub-Stellar Initial Mass Function of Simple and Composite Populations |date=2013 |url=https://doi.org/10.1007/978-94-007-5612-0_4 |work=Planets, Stars and Stellar Systems: Volume 5: Galactic Structure and Stellar Populations |pages=115–242 |editor-last=Oswalt |editor-first=Terry D. |access-date=2023-11-02 |place=Dordrecht |publisher=Springer Netherlands |language=en |doi=10.1007/978-94-007-5612-0_4 |isbn=978-94-007-5612-0 |last2=Weidner |first2=Carsten |last3=Pflamm-Altenburg |first3=Jan |last4=Thies |first4=Ingo |last5=Dabringhausen |first5=Jörg |last6=Marks |first6=Michael |last7=Maschberger |first7=Thomas |editor2-last=Gilmore |editor2-first=Gerard|arxiv=1112.3340 |bibcode=2013pss5.book..115K }}

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.{{cite journal|bibcode=1998ASPC..142...17M|title=The Initial Mass Function of Massive Stars in the Local Group|journal=The Stellar Initial Mass Function (38Th Herstmonceux Conference)|volume=142|pages=17|last1=Massey|first1=Philip|year=1998}}

= 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 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 {{Solar mass|0.05–0.2}} where both formation modes may account for bodies in this mass range.{{cite conference |arxiv= 1112.3340| last1 = Kroupa | first1 = Pavel | display-authors = etal | book-title = Stellar Systems and Galactic Structure, Vol. V | title = The stellar and sub-stellar IMF of simple and composite populations | date = 2013|bibcode = 2013pss5.book..115K |doi = 10.1007/978-94-007-5612-0_4 }}

= Variation =

The possible variation of the IMF affects our interpretation of the galaxy signals and the estimation of cosmic star formation history{{Cite journal |last1=Wilkins |first1=Stephen M. |last2=Trentham |first2=Neil |last3=Hopkins |first3=Andrew M. |date=April 2008 |title=The evolution of stellar mass and the implied star formation history |journal=Monthly Notices of the Royal Astronomical Society |language=en |volume=385 |issue=2 |pages=687–694 |doi=10.1111/j.1365-2966.2008.12885.x |doi-access=free |issn=0035-8711|arxiv=0801.1594 |bibcode=2008MNRAS.385..687W }} 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.{{Cite journal |last1=Kroupa |first1=Pavel |last2=Weidner |first2=Carsten |date=December 2003 |title=Galactic-Field Initial Mass Functions of Massive Stars |journal=The Astrophysical Journal |volume=598 |issue=2 |pages=1076–1078 |doi=10.1086/379105 |issn=0004-637X|doi-access=free |arxiv=astro-ph/0308356 |bibcode=2003ApJ...598.1076K }}{{Cite journal |last1=Weidner |first1=C. |last2=Kroupa |first2=P. |last3=Larsen |first3=S. S. |date=June 2004 |title=Implications for the formation of star clusters from extragalactic star formation rates |journal=Monthly Notices of the Royal Astronomical Society |volume=350 |issue=4 |pages=1503–1510 |doi=10.1111/j.1365-2966.2004.07758.x |issn=0035-8711|doi-access=free |arxiv=astro-ph/0402631 |bibcode=2004MNRAS.350.1503W }}{{Citation |last1=Kroupa |first1=Pavel |title=The Stellar and Sub-Stellar Initial Mass Function of Simple and Composite Populations |date=2013 |url=https://doi.org/10.1007/978-94-007-5612-0_4 |work=Planets, Stars and Stellar Systems: Volume 5: Galactic Structure and Stellar Populations |pages=115–242 |editor-last=Oswalt |editor-first=Terry D. |access-date=2023-11-02 |place=Dordrecht |publisher=Springer Netherlands |language=en |doi=10.1007/978-94-007-5612-0_4 |isbn=978-94-007-5612-0 |last2=Weidner |first2=Carsten |last3=Pflamm-Altenburg |first3=Jan |last4=Thies |first4=Ingo |last5=Dabringhausen |first5=Jörg |last6=Marks |first6=Michael |last7=Maschberger |first7=Thomas |editor2-last=Gilmore |editor2-first=Gerard|arxiv=1112.3340 |bibcode=2013pss5.book..115K }}{{Cite journal |last1=Jeřábková |first1=T. |last2=Zonoozi |first2=A. Hasani |last3=Kroupa |first3=P. |last4=Beccari |first4=G. |last5=Yan |first5=Z. |last6=Vazdekis |first6=A. |last7=Zhang |first7=Z.-Y. |date=2018-12-01 |title=Impact of metallicity and star formation rate on the time-dependent, galaxy-wide stellar initial mass function |url=https://www.aanda.org/articles/aa/abs/2018/12/aa33055-18/aa33055-18.html |journal=Astronomy & Astrophysics |language=en |volume=620 |pages=A39 |doi=10.1051/0004-6361/201833055 |issn=0004-6361|doi-access=free |arxiv=1809.04603 |bibcode=2018A&A...620A..39J }}

Measurements of the local universe where single stars can be resolved are consistent with an invariant IMF{{Cite journal |last=Kroupa |first=P. |date=2001-04-01 |title=On the variation of the initial mass function |journal=Monthly Notices of the Royal Astronomical Society |volume=322 |issue=2 |pages=231–246 |doi=10.1046/j.1365-8711.2001.04022.x |doi-access=free |issn=0035-8711|arxiv=astro-ph/0009005 |bibcode=2001MNRAS.322..231K }}{{Cite journal |last=Kroupa |first=Pavel |date=2002-01-04 |title=The Initial Mass Function of Stars: Evidence for Uniformity in Variable Systems |url=https://www.science.org/doi/10.1126/science.1067524 |journal=Science |language=en |volume=295 |issue=5552 |pages=82–91 |doi=10.1126/science.1067524 |pmid=11778039 |arxiv=astro-ph/0201098 |bibcode=2002Sci...295...82K |issn=0036-8075}}{{Cite journal |last1=Bastian |first1=Nate |last2=Covey |first2=Kevin R. |last3=Meyer |first3=Michael R. |date=2010-08-01 |title=A Universal Stellar Initial Mass Function? A Critical Look at Variations |url=https://www.annualreviews.org/doi/10.1146/annurev-astro-082708-101642 |journal=Annual Review of Astronomy and Astrophysics |language=en |volume=48 |issue=1 |pages=339–389 |doi=10.1146/annurev-astro-082708-101642 |issn=0066-4146|arxiv=1001.2965 |bibcode=2010ARA&A..48..339B }}{{Cite journal |last=Hopkins |first=A. M. |date=January 2018 |title=The Dawes Review 8: Measuring the Stellar Initial Mass Function |url=https://www.cambridge.org/core/journals/publications-of-the-astronomical-society-of-australia/article/dawes-review-8-measuring-the-stellar-initial-mass-function/30FD7936B4C37131AB71C52BBE21B246 |journal=Publications of the Astronomical Society of Australia |language=en |volume=35 |pages=e039 |doi=10.1017/pasa.2018.29 |issn=1323-3580|arxiv=1807.09949 |bibcode=2018PASA...35...39H }} 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.{{Cite journal |last1=Sneppen |first1=Albert |last2=Steinhardt |first2=Charles L. |last3=Hensley |first3=Hagan |last4=Jermyn |first4=Adam S. |last5=Mostafa |first5=Basel |last6=Weaver |first6=John R. |date=2022-05-01 |title=Implications of a Temperature-dependent Initial Mass Function. I. Photometric Template Fitting |journal=The Astrophysical Journal |language=en |volume=931 |issue=1 |pages=57 |doi=10.3847/1538-4357/ac695e |arxiv=2205.11536 |bibcode=2022ApJ...931...57S |s2cid=249017733 |issn=0004-637X |doi-access=free }}

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. 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P. |last5=Ivison |first5=R. J. |date=September 2017 |title=The evolution of CNO isotopes: a new window on cosmic star formation history and the stellar IMF in the age of ALMA |journal=Monthly Notices of the Royal Astronomical Society |language=en |volume=470 |issue=1 |pages=401–415 |doi=10.1093/mnras/stx1197 |doi-access=free |issn=0035-8711|arxiv=1704.06701 }}{{Cite journal |last1=Zhang |first1=Zhi-Yu |last2=Romano |first2=D. |last3=Ivison |first3=R. J. |last4=Papadopoulos |first4=Padelis P. |last5=Matteucci |first5=F. |date=June 2018 |title=Stellar populations dominated by massive stars in dusty starburst galaxies across cosmic time |url=https://www.nature.com/articles/s41586-018-0196-x |journal=Nature |language=en |volume=558 |issue=7709 |pages=260–263 |doi=10.1038/s41586-018-0196-x |pmid=29867162 |issn=1476-4687|arxiv=1806.01280 |bibcode=2018Natur.558..260Z }} 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.{{efn|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.}}

= Origin of the Stellar IMF =

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 $\Delta N/\Delta \log M \propto M^{-1.4 \pm 0.2}$ for masses greater than $1\, M_\odot$ , and the FLMF follows $\Delta N/\Delta \log M_{\text{line}} \propto M_{\text{line}}^{-1.5 \pm 0.2}$ for filament line masses greater than $10\, M_\odot \text{pc}^{-1}$ . 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.{{cite journal | last1 = Zhang | first1 = Guo-Yin | last2 = Andre | first2 = Philippe | last3 = Menshchikov | first3 = Alexander | last4 = Li | first4 = Jin-Zeng | title = Probing the filamentary nature of star formation in the California giant molecular cloud | journal = Astronomy & Astrophysics | volume = 689 | page = A3 | year = 2024 | doi = 10.1051/0004-6361/202449853 | url = https://ui.adsabs.harvard.edu/link_gateway/2024A&A...689A...3Z/doi:10.1051/0004-6361/202449853 | arxiv = 2406.08004 | bibcode = 2024A&A...689A...3Z }}

References

Notes

External links

{{cite web|title=Pavel Kroupa (Prague and Bonn): The stellar initial mass function: the IMF|date=April 8, 2022|website=YouTube|url=https://www.youtube.com/watch?v=Sxtk2cIlU14}}

Category:Stellar astronomy

Category:Equations of astronomy

Category:Mass

Category:Concepts in stellar astronomy