Skip to main content
SpringerLink
Log in

The Stellar and Sub-Stellar Initial Mass Function of Simple and Composite Populations

  • Reference work entry
Planets, Stars and Stellar Systems

Abstract

The current knowledge on the stellar IMF is documented. It is usuallydescribed as being invariant, but evidence to the contrary has emerged: it appears to become top-heavy when the star-formation rate density surpasses about \(0.1\,M_{\odot}/(\mathrm{year}\,{\mathrm{pc}}^{3})\) ona pc scale and it may become increasingly bottom-heavy withincreasing metallicity and in increasingly massive elliptical galaxies. Itdeclines quite steeply below about \(0.07\,M_{\odot }\) with brown dwarfs (BDs) and very low mass stars having their own IMF. The most massive star of mass m max formed in an embedded cluster with stellar mass M ecl correlates strongly with M ecl being a result of gravitation-driven but resource-limited growth and fragmentation-induced starvation. There is no convincing evidence whatsoeverthat massive stars do form in isolation. Massive stars form above a density threshold in embedded clusters which become saturated when \(m_{\mathrm{max}} = m_{\mathrm{max{\ast}}}\approx 150\,M_{\odot }\) which appears to be the canonical physical upper mass limit of stars. Super-canonical massive stars arise naturally due to stellar mergers induced bystellar-dynamical encounters in binary-rich very young dense clusters.

Various methods of discretising a stellar population are introduced: optimal sampling leads to a mass distribution that perfectly represents the exact form of the desired IMF and the m max − M ecl relation, while random sampling results in statistical variations of the shape of the IMF. The observed m max − M ecl correlation and the small spread of IMF power-law indices together suggest that optimally sampling the IMF may be the more realistic description of star formation than random sampling from a universal IMF with a constant upper mass limit.

Composite populations on galaxy scales, which are formed from many pc scale star formatiom events, need to be described by the integrated galactic IMF. This IGIMF varies systematically from top-light to top-heavy in dependence of galaxy type and star formation rate, with dramatic implications for theories of galaxy formation and evolution.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Notes

  1. 1.

    As noted by Zinnecker (2011), Salpeter used an age of 6 Gyr for the MW disk; had he used the now adopted age of 12 Gyr, he would have arrived at a “Salpeter index” α ≈ 2. 05 instead of 2.35.

  2. 2.

    The latest version of the catalogue can be found at http://www.ari.uni-heidelberg.de/datenbanken/aricns/, while http://www.nstars.nau.edu/ contains the Nearby Stars (NStars) data base.

  3. 3.

    Here, it should be emphasized and acknowledged that the intensive and highly fruitful discourse between Guenther Wuchterl and the Lyon group has led to the important understanding that the classical evolution tracks computed in Lyon and by others are unreliable for ages less than a few Myr (Tout et al. 1999; Wuchterl and Tscharnuter 2003). This comes about because the emerging star’s structure retains a memory of its accretion history. In particular, Wuchterl and Klessen (2001) present a SPH computation of the gravitational collapse and early evolution of a solar-type star documenting the significant difference to a pre-main sequence track if the star is instead classically assumed to form in isolation.

  4. 4.

    The dynamical properties of a stellar system are its mass and, if it is a multiple star, its orbital parameters (semimajor axis, mass ratio, eccentricity, both inner and outer if it is a higher-order multiple system). See also Sect. 2.6.

  5. 5.

    Note that Scalo (1998) emphasizes that the IMF remains poorly constrained owing to the small number of massive stars in any one sample. This is a true albeit conservative standpoint, and the present authors prefer to accept Massey’s result as a working IMF Universality Hypothesis.

  6. 6.

    Citing from Basu and Jones (2004), “According to the central limit theorem of statistics, if the mass of a protostellar condensation \(M_{c} = f_{1} \times f_{2} \times \ldots \times f_{N}\), then the distribution of M c tends to a lognormal regardless of the distributions of the individual physical parameters f i (i = 1, …N), if N is large. Depending on the specific distributions of the f i , a convergence to a lognormal may even occur for moderate N.” The central limit theorem was invoked for the first time by Zinnecker (1984) to study the form of the IMF from hierarchical fragmentation of collapsing cloud cores.

  7. 7.

    Note that Scalo (1986) calls ξ L(m) the mass function and ξ(m) the mass spectrum.

  8. 8.

    The publicly available C program Optimf allowing the generation of a stellar population of mass M ecl is available at http://www.astro.uni-bonn.de/en/download/software/

  9. 9.

    homepages.physik.uni-muenchen.de/˜Winitzki/erf-approx.pdf

  10. 10.

    More recent work on the physical upper mass limit can be found in Koen (2006) and Maíz Apellániz et al. (2007, 2008).

  11. 11.

    A study by Maschberger and Clarke (2008) of the most-massive star data in young star clusters concluded that “the data are not indicating any striking deviation from the expectations of random drawing.” This statement has been frequently misinterpreted that other sampling mechanisms are ruled out. However, the Maschberger and Clarke analysis focuses on low-mass clusters were the data were insufficient to decide whether star clusters are populated purely randomly from an IMF with constant upper mass limit or, for example, in a sorted fashion. The differences appear clearly at higher cluster masses, not included in their analysis but in Weidner and Kroupa (2006) and Weidner et al. (2010). Maschberger & Clarke (2008) adapt their data set according to the requested result and so their study does not constitute an acceptable scientific standard.

  12. 12.

    Choosing stars randomly from the IMF is, however, a good first approximation for many purposes of study.

  13. 13.

    Owing to the poor statistical definition of Ψnear(M V ) for m ≲ 0. 5 M ⊙, M V ≳ 10, it is important to increase the sample of nearby stars, but controversy exists as to the maximum distance to which the LMS census is complete. Using spectroscopic parallax, it has been suggested that the local census of LMSs is complete to within about 15% to distances of 8 pc and beyond (Reid and Gizis 1997). However, Malmquist bias allows stars and unresolved binaries to enter such a flux-limited sample from much larger distances (Kroupa 2001c). The increase of the number of stars with distance using trigonometric distance measurements shows that the nearby sample becomes incomplete for distances larger than 5 pc and for M V > 12 (Henry et al. 1997; Jahreiss 1994). The incompleteness in the northern stellar census beyond 5 pc and within 10 pc amounts to about 35% (Jao et al. 2003), and further discovered companions (e.g., Beuzit et al. 2004; Delfosse et al. 1999) to known primaries in the distance range 5 < d < 12 pc indeed suggest that the extended sample may not yet be complete. Based on the work by Reid et al. (2003a,b), Luhman (2004), however, argues that the incompleteness is only about 15%.

  14. 14.

    This controversy achieved a maxiumum in 1995, as documented in Kroupa (1995a). The discrepancy evident in Fig.  4-9 between the nearby LF, Ψnear, and the photometric LF, Ψphot, invoked a significant dispute as to the nature of this discrepancy. On the one hand (Kroupa 1995a), the difference is thought to be due to unseen companions in the deep but low-resolution surveys used to construct Ψphot, with the possibility that photometric calibration for VLMSs may remain problematical so that the exact shape of Ψphot for M V ≳ 14 is probably uncertain. On the other hand (Reid and Gizis 1997), the difference is thought to come from nonlinearities in the V − I, M V color–magnitude relation used for photometric parallax. Taking into account such structure, it can be shown that the photometric surveys underestimate stellar space densities so that Ψphot moves closer to the extended estimate of Ψnear using a sample of stars within 8 pc or further. While this is an important point, the extended Ψnear is incomplete (see footnote Sect. 13 on p. 160) and theoretical color-magnitude relations do not have the required degree of nonlinearity (e.g., Fig. 7 in Bochanski et al. 2010). The observational color–magnitude data also do not conclusively suggest a feature with the required strength (Baraffe et al. 1998). Furthermore, Ψphot agrees almost perfectly with the LFs measured for star clusters of solar and population II metallicity for which the color-magnitude relation is not required (Fig.  4-10 ) so that nonlinearities in the color–magnitude relation cannot be the dominant source of the discrepancy

  15. 15.

    When a cloud collapses, its density increases, but its temperature remains constant as long as the opacity remains low enough to enable the contraction work to be radiated away. The Jeans mass ( 4.50) consequently decreases and further fragments with smaller masses form. When, however, the density increases to a level such that the cloud core becomes optically thick, then the temperature increases, and the Jeans mass follows suit. Thus, an opacity-limited minimum fragmentation mass of about 0. 01 M ⊙ is arrived at (Bate 2005; Boss 1986; Kumar 2003; Low and Lynden-Bell 1976).

  16. 16.

    Note that this does not constitute a proof of the stellar IMF being a probabilistic density distribution!

  17. 17.

    Adiabatic contraction (Blumenthal et al. 1986) in UCDs may, however, alleviate this problem but unlikely sufficiently so (Murray 2009).

References

Download references

Acknowledgements

PK is thankful to Christopher Tout and Gerry Gilmore for very stimulatingand important contributions without which much of this material wouldnot have become available. PK is especially indebted to Sverre Aarsethwhose friendly tutoring (against “payments” in the form of many bottles oflieblichen German white wine) eased the numerical dynamics work in1993/1994. Douglas Heggie be thanked for fruitful discussions with PK onoptimal sampling in Heidelberg in September, 2011. We thank SambaranBanerjee for very helpful comments on the manuscript. PK would also like tothank M. R. S. Hawkins who had introduced him to this field in 1987while PK visited the Siding-Spring Observatory as a summer vacationscholar at the ANU. Mike gave PK a delightful lecture on the low-massLF one night when visiting his observing run to learn about modern,state-of-the-art Schmidt-telescope surveying with photographic platesbefore PK embarked on postgraduate work. This research was much latersupported through DFG grants KR1635/2 and KR1635/3 and a Heisenbergfellowship, KR1635/4, KR1635/12, KR1635/13, and currently KR1635/25. MMacknowledges the Bonn/Cologne International Max-Planck Research School forsupport.

Author information

Authors and Affiliations

  1. Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, Bonn, Germany

    Prof. Pavel Kroupa, Dr. Jan Pflamm-Altenburg, Dr. Ingo Thies, Dipl.-Phys. Jörg Dabringhausen & Dr. Michael Marks

  2. Scottish Universities Physics Alliance (SUPA), School of Physics and Astronomy, University of St. Andrews, North Haugh, St. Andrews, UK

    Dr. Carsten Weidner

  3. Instituto de Astrofsica de Canarias, C/Via Lactea s/n, La Laguna (Tenerife), Spain

    Dr. Carsten Weidner

  4. Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, Bonn, Germany

    Dr. Thomas Maschberger

  5. Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, Bonn, Germany

    Dr. Thomas Maschberger

Authors
  1. Prof. Pavel Kroupa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dr. Carsten Weidner
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dr. Jan Pflamm-Altenburg
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Dr. Ingo Thies
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Dipl.-Phys. Jörg Dabringhausen
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Dr. Michael Marks
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Dr. Thomas Maschberger
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Physics & Space Sciences, Florida Institute of Technology, University Boulevard, Melbourne, FL, USA

    Terry D. Oswalt

  2. Institute of Astronomy, Cambridge University, Cambridge, UK

    Gerard Gilmore

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media Dordrecht

About this entry

Cite this entry

Kroupa, P. et al. (2013). The Stellar and Sub-Stellar Initial Mass Function of Simple and Composite Populations. In: Oswalt, T.D., Gilmore, G. (eds) Planets, Stars and Stellar Systems. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5612-0_4

Download citation

Publish with us

Policies and ethics

Search

Navigation