Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Stellar initial mass function varies with metallicity and time


Most structural and evolutionary properties of galaxies strongly rely on the stellar initial mass function (IMF), namely the distribution of the stellar mass formed in each episode of star formation1,2,3,4. The IMF shapes the stellar population in all stellar systems, and so has become one of the most fundamental concepts of modern astronomy. Both constant and variable IMFs across different environments have been claimed despite a large number of theoretical5,6,7 and observational efforts8,9,10,11,12,13,14,15. However, the measurement of the IMF in Galactic stellar populations has been limited by the relatively small number of photometrically observed stars, leading to high uncertainties12,13,14,15,16. Here we report a star-counting result based on approximately 93,000 spectroscopically observed M-dwarf stars, an order of magnitude more than previous studies, in the 100–300 parsec solar neighbourhood. We find unambiguous evidence of a variable IMF that depends on both metallicity and stellar age. Specifically, the stellar population formed at early times contains fewer low-mass stars compared with the canonical IMF, independent of stellar metallicities. In more recent times, however, the proportion of low-mass stars increases with stellar metallicity. The variable abundance of low-mass stars in our Milky Way establishes a powerful benchmark for models of star formation and can heavily affect results in Galactic chemical-enrichment modelling, mass estimation of galaxies and planet-formation efficiency.

This is a preview of subscription content, access via your institution

Access options

Buy this article

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: The M-dwarf star sample in the solar neighbourhood.
Fig. 2: Stellar IMF variation revealed by our M-dwarf star sample.
Fig. 3: The IMF power exponent as a function of stellar metallicity for different groups.

Similar content being viewed by others

Data availability

The raw dataset that supports the findings of this study is publicly available at the National Astronomical Data Center ( The data generated and/or analysed during the study are available at the National Astronomical Data Center (

Code availability

The code used to determine the stellar mass of M-dwarf stars and model fitting is publicly available on GitHub at


  1. Kroupa, P. On the variation of the initial mass function. Mon. Not. R. Astron. Soc. 322, 231–246 (2001).

    Article  ADS  Google Scholar 

  2. Kroupa, P. The initial mass function of stars: evidence for uniformity in variable systems. Science 295, 82–91 (2002).

    Article  ADS  CAS  Google Scholar 

  3. Chabrier, G. Galactic stellar and substellar initial mass function. Publ. Astron. Soc. Pacif. 115, 763–795 (2003).

    Article  ADS  Google Scholar 

  4. Bastian, N., Covey, K. R. & Meyer, M. R. A universal stellar initial mass function? A critical look at variations. Annu. Rev. Astron. Astrophys. 48, 339–389 (2010).

    Article  ADS  Google Scholar 

  5. Adams, F. C. A theory of the initial mass function for star formation in molecular clouds. Astrophys. J. 464, 256 (1996).

    Article  ADS  Google Scholar 

  6. Hopkins, P. F. The stellar initial mass function, core mass function and the last-crossing distribution. Mon. Not. R. Astron. Soc. 423, 2037–2044 (2012).

    Article  ADS  Google Scholar 

  7. Hennebelle, P. & Chabrier, G. Analytical theory for the initial mass function. III. Time dependence and star formation rate. Astrophys. J. 770, 150 (2013).

    Article  ADS  Google Scholar 

  8. van Dokkum, P. G. & Conroy, C. A substantial population of low-mass stars in luminous elliptical galaxies. Nature 468, 940–942 (2010).

    Article  ADS  CAS  Google Scholar 

  9. Treu, T. et al. The initial mass function of early-type galaxies. Astrophys. J. 709, 1195–1202 (2010).

    Article  ADS  Google Scholar 

  10. Cappellari, M. et al. Systematic variation of the stellar initial mass function in early-type galaxies. Nature 484, 485–488 (2012).

    Article  ADS  CAS  Google Scholar 

  11. Martín-Navarro, I. et al. IMF–metallicity: a tight local relation revealed by the CALIFA survey. Astrophys. J. Let. 806, L31 (2015).

    Article  ADS  Google Scholar 

  12. Zhang, Z.-Y. et al. Stellar populations dominated by massive stars in dusty starburst galaxies across cosmic time. Nature 558, 260–263 (2018).

    Article  ADS  CAS  Google Scholar 

  13. Bartko, H. et al. An extremely top-heavy initial mass function in the Galactic center stellar disks. Astrophys. J. 708, 834–840 (2010).

    Article  ADS  CAS  Google Scholar 

  14. Smith, R. J. Evidence for initial mass function variation in massive early-type galaxies. Annu. Rev. Astron. Astrophys. 58, 577–615 (2020).

    Article  ADS  CAS  Google Scholar 

  15. Conroy, C. Modeling the panchromatic spectral energy distributions of galaxies. Annu. Rev. Astron. Astrophys. 51, 393–455 (2013).

    Article  ADS  CAS  Google Scholar 

  16. Offner, S. S. R. et al. The origin and universality of the stellar initial mass function. In Protostars and Planets VI (eds Reipurth, B. et al.) 53 (Univ. Arizona Press, 2014).

  17. Kroupa, P. et al. The distribution of low-mass stars in the Galactic disc. Mon. Not. R. Astron. Soc. 262, 545–587 (1993).

    Article  ADS  Google Scholar 

  18. Kroupa, P. & Tout, C. A. The theoretical mass–magnitude relation of low mass stars and its metallicity dependence. Mon. Not. R. Astron. Soc. 287, 402–414 (1997).

    Article  ADS  CAS  Google Scholar 

  19. Li, J. et al. Stellar parameterization of LAMOST M dwarf stars. Astrophys. J. Suppl. Ser. 253, 45 (2021).

    Article  ADS  CAS  Google Scholar 

  20. Liu, C. et al. Mapping the Milky Way with LAMOST I: method and overview. Res. Astron. Astrophys. 17, 096 (2017).

    Article  ADS  CAS  Google Scholar 

  21. Liu, C. Smoking gun of the dynamical processing of solar-type field binary stars. Mon. Not. R. Astron. Soc. 490, 550–565 (2019).

    ADS  CAS  Google Scholar 

  22. Moe, M. et al. The close binary fraction of solar-type stars is strongly anticorrelated with metallicity. Astrophys. J. 875, 61 (2019).

    Article  ADS  CAS  Google Scholar 

  23. Salpeter, E. E. The luminosity function and stellar evolution. Astrophys. J. 121, 161 (1955).

  24. Yan, Z. et al. Chemical evolution of ultra-faint dwarf galaxies in the self-consistently calculated integrated galactic IMF theory. Astron. Astrophys. 637, A68 (2020).

    Article  ADS  CAS  Google Scholar 

  25. Reylé, C. & Robin, A. C. Early galaxy evolution from deep wide field star counts. II. First estimate of the thick disc mass function. Astron. Astrophys. 373, 886–894 (2001).

    Article  ADS  Google Scholar 

  26. Geha, M. et al. The stellar initial mass function of ultra-faint dwarf galaxies: evidence for IMF variations with galactic environment. Astrophys. J. 771, 29 (2013).

    Article  ADS  Google Scholar 

  27. Kordopatis, G. et al. The rich are different: evidence from the RAVE survey for stellar radial migration. Mon. Not. R. Astron. Soc. 447, 3526–3535 (2015).

    Article  ADS  CAS  Google Scholar 

  28. Jeřábková, T. Impact of metallicity and star formation rate on the time-dependent, galaxy-wide stellar initial mass function. Astron. Astrophys. 620, A39 (2018).

    Article  Google Scholar 

  29. Ting, Y.-S. & Rix, H.-W. The vertical motion history of disk stars throughout the Galaxy. Astrophys. J. 878, 21 (2019).

    Article  ADS  CAS  Google Scholar 

  30. Larson, R. B. Early star formation and the evolution of the stellar initial mass function in galaxies. Mon. Not. R. Astron. Soc. 301, 569–581 (1998).

    Article  ADS  Google Scholar 

  31. Padoan, P. & Nordlund, Å.The stellar initial mass function from turbulent fragmentation. Astrophys. J. 576, 870–879 (2002).

    Article  ADS  CAS  Google Scholar 

  32. Papadopoulos, P. P. et al. Extreme cosmic ray dominated regions: a new paradigm for high star formation density events in the Universe. Mon. Not. R. Astron. Soc. 414, 1705–1714 (2011).

    Article  ADS  CAS  Google Scholar 

  33. Zhang, Z.-Y. et al. Gone with the heat: a fundamental constraint on the imaging of dust and molecular gas in the early Universe. Royal Society Open Science 3, 160025 (2016).

    Article  ADS  Google Scholar 

  34. Zhao, G. et al. LAMOST spectral survey — an overview. Res. Astron. Astrophys. 12, 723–734 (2012).

    Article  ADS  Google Scholar 

  35. Deng, L.-C. et al. LAMOST Experiment for Galactic Understanding and Exploration (LEGUE) — the survey’s science plan. Res. Astron. Astrophys. 12, 735–754 (2012).

    Article  ADS  Google Scholar 

  36. Majewski, S. R. et al. The Apache Point Observatory Galactic Evolution Experiment (APOGEE). Astron. J. 154, 94 (2017).

    Article  ADS  Google Scholar 

  37. Jönsson, H. et al. APOGEE data and spectral analysis from SDSS Data Release 16: seven years of observations including first results from APOGEE-South. Astron. J. 160, 120 (2020).

    Article  ADS  Google Scholar 

  38. Zhang, B. et al. Deriving the stellar labels of LAMOST spectra with the Stellar LAbel Machine (SLAM). Astrophys. J. Suppl. Ser. 246, 9 (2020).

    Article  ADS  CAS  Google Scholar 

  39. Yi, Z. et al. M dwarf catalog of the LAMOST pilot survey. Astrophys. J. 147, 33 (2014).

    Google Scholar 

  40. Gaia Collaboration. et al. Gaia Data Release 2: summary of the contents and survey properties. Astron. Astrophys. 616, A1 (2018).

    Article  Google Scholar 

  41. Skrutskie, M. F. et al. The Two Micron All Sky Survey (2MASS). Astrophys. J. 131, 1163–1183 (2006).

    Google Scholar 

  42. Green, G. M. et al. A 3D dust map based on Gaia, Pan-STARRS 1, and 2MASS. Astrophys. J. 887, 93 (2019).

    Article  ADS  CAS  Google Scholar 

  43. Wang, S. & Chen, X. The optical to mid-infrared extinction law based on the APOGEE, Gaia DR2, Pan-STARRS1, SDSS, APASS, 2MASS, and WISE Surveys. Astrophys. J. 877, 116 (2019).

    Article  ADS  CAS  Google Scholar 

  44. Bailer-Jones, C. A. L. et al. Estimating distance from parallaxes. IV. Distances to 1.33 billion stars in Gaia Data Release 2. Astron. J. 156, 58 (2018).

    Article  ADS  Google Scholar 

  45. Bressan, A. et al. PARSEC: stellar tracks and isochrones with the PAdova and TRieste Stellar Evolution Code. Mon. Not. R. Astron. Soc. 427, 127–145 (2012).

    Article  ADS  CAS  Google Scholar 

  46. Chen, Y. et al. Improving PARSEC models for very low mass stars. Mon. Not. R. Astron. Soc. 444, 2525–2543 (2014).

    Article  ADS  Google Scholar 

  47. Chen, T. & Guestrin, C. XGBoost: a scalable tree boosting system. In KDD ’16: Proc. 22nd ACM SIGKDD Int. Conf. Knowledge Discovery And Data Mining, (eds Krishnapuram, B. et al.) 785–794 (ACM, 2016).

  48. Mann, A. W. et al. How to constrain your M dwarf. II. The mass–luminosity–metallicity relation from 0.075 to 0.70 solar masses. Astrophys. J. 871, 63 (2019).

    Article  ADS  CAS  Google Scholar 

  49. Miller, G. E. & Scalo, J. M. The initial mass function and stellar birthrate in the solar neighborhood. Astrophys. J. Suppl. Ser. 41, 513–547 (1979).

    Article  Google Scholar 

  50. El-Badry, K., Weisz, D. R. & Quataert, E. The statistical challenge of constraining the low-mass IMF in Local Group dwarf galaxies. Mon. Not. R. Astron. Soc. 468, 319–332 (2017).

    Article  ADS  Google Scholar 

  51. Xu, Y. et al. Mapping the Milky Way with LAMOST—II. The stellar halo. Mon. Not. R. Astron. Soc. 473, 1244–1257 (2018).

  52. Wang, H.-F., Liu, C., Xu, Y., Wan, J.-C. & Deng, L. Mapping the Milky Way with LAMOST—III. Complicated spatial structure in the outer disc. Mon. Not. R. Astron. Soc. 478, 3367–3379 (2018).

    Article  ADS  Google Scholar 

  53. Jurić, M. et al. The Milky Way Tomography with SDSS. I. Stellar number density distribution. Astrophys. J. 673, 864–914 (2008).

    Article  ADS  Google Scholar 

  54. Salvatier, J., Wiecki, T. V. & Fonnesbeck, C. Probabilistic programming in Python using PyMC3. PeerJ Comput. Sci. 2, e55 (2016).

  55. Sharma, S. et al. Galaxia: a code to generate a synthetic survey of the Milky Way. Astrophys. J. 730, 3 (2011).

    Article  ADS  Google Scholar 

  56. Dotter, A. et al. The Dartmouth Stellar Evolution Database. Astrophys. J. Suppl. Ser. 178, 89–101 (2008).

    Article  ADS  CAS  Google Scholar 

  57. Bovy, J. et al. galpy: a Python library for galactic dynamics. Astrophys. J. Suppl. Ser. 216, 29 (2015).

    Article  ADS  Google Scholar 

  58. The GRAVITY Collaboration. A geometric distance measurement to the Galactic center black hole with 0.3% uncertainty. Astron. Astrophys. 625, L10 (2019).

    Article  ADS  Google Scholar 

  59. Bovy, J. et al. The Milky Way’s circular-velocity curve between 4 and 14 kpc from APOGEE data. Astrophys. J. 759, 131 (2012).

    Article  ADS  Google Scholar 

  60. Schönrich, R. et al. Local kinematics and the local standard of rest. Mon. Not. R. Astron. Soc. 403, 1829–1833 (2010).

    Article  ADS  Google Scholar 

  61. Binney, J. & Tremaine, S. Galactic Dynamics 2nd edn (Princeton Univ. Press, 2008).

  62. Binney, J. Actions for axisymmetric potentials. Mon. Not. R. Astron. Soc. 426, 1324–1327 (2012).

    Article  ADS  Google Scholar 

  63. Sanders, J. L. & Binney, J. A review of action estimation methods for galactic dynamics. Mon. Not. R. Astron. Soc. 457, 2107–2121 (2016).

    Article  ADS  CAS  Google Scholar 

  64. Jenkins, A. & Binney, J. Spiral heating of galactic discs. Mon. Not. R. Astron. Soc. 245, 305–317 (1990).

    ADS  Google Scholar 

  65. Wu, Y. et al. Mass and age of red giant branch stars observed with LAMOST and Kepler. Mon. Not. R. Astron. Soc. 475, 3633–3643 (2018).

    Article  ADS  CAS  Google Scholar 

  66. Delfosse, X. et al. M dwarfs binaries: results from accurate radial velocities and high angular resolution observations. In Spectroscopically and Spatially Resolving the Components of the Close Binary Stars (eds Hilditch R. W. et al.) 166–174 (Astronomical Society of the Pacific, 2004).

Download references


We thank L. Deng and R. de Grijs for their contributions in the very early stage of the project. We thank J. Liu for discussions. This work is supported by the National Key R&D Program of China no. 2019YFA0405500, the China Manned Space Project with no. CMS-CSST-2021-A07 and CMS-CSST-2021-A08. C.L. thanks the National Natural Science Foundation of China (NSFC) for grant no. 11835057. Z.-Y.Z. and Z.-Q.Y. acknowledge the support of NSFC grants no. 12041305 and no. 12173016, and the Program for Innovative Talents, Entrepreneur in Jiangsu. H.T. acknowledges the support of NSFC grant no. 12103062. X.F. acknowledges the support of China Postdoctoral Science Foundation no. 2020M670023, the NSFC grants no. 12203100, no. 11973001 and no. 12090044, and the National Key R&D Program of China no. 2019YFA0405504. Jiao Li acknowledges the science research grants from the China Manned Space Project with no. CMS-CSST-2021-A10 and no. CMS-CSST-2021-B05, and the NSFC grants no. 12090043 and no. 11873016. Z.-Q.Y. acknowledges support from NSFC grants no. 12203021, the Jiangsu Funding Program for Excellent Postdoctoral Talent under grant no. 2022ZB54, the Fundamental Research Funds for the Central Universities under grant no. 0201/14380049. Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope, LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences. The National Development and Reform Commission has provided funding for the project. LAMOST is operated and managed by the National Astronomical Observatories, Chinese Academy of Sciences. This work has made use of data from the European Space Agency (ESA) mission Gaia (, processed by the Gaia Data Processing and Analysis Consortium (DPAC; Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This work also benefited from the International Space Science Institute (ISSI/ISSI-BJ) in Bern and Beijing, thanks to the funding of the team ‘Chemical abundances in the ISM: the litmus test of stellar IMF variations in galaxies across cosmic time’.

Author information

Authors and Affiliations



Jiadong Li contributed most of the modelling and calculations and wrote the initial manuscript. C.L. provided the ideas to initialize the project, supervised Jiadong Li on the modelling, and revised the manuscript. Z.-Y.Z., X.F. and Z.-Q.Y. compared the results with other theoretical and observational work and helped write the manuscript. H.T. ran the calculations to derive vertical actions. Jiao Li helped with discussions on the effect of the binary stars. All authors discussed and commented on the manuscript.

Corresponding author

Correspondence to Chao Liu.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature thanks Charlie Conroy, Antonio Sollima and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data figures and tables

Extended Data Fig. 1 The Sigmoid function fitting result of IMF power-law slope as a function of [M/H].

The yellow line is the best-fit result. The red squares are results in our study using to fit. The blue lines are 100 fitting results selected randomly from the MCMC chains. Error bars represent 1σ uncertainty.

Extended Data Fig. 2 The derived stellar mass by PARSEC model as a function of absolute magnitude of Ks-band (\({M}_{{K}_{s}}\)).

The colours of each pixel represent the median [M/H] in each colour-magnitude bin.

Extended Data Fig. 3 The IMF power-law index as a function of [M/H] measured with different methods and different stellar models.

a, The red filled rectangles are derived by all stars with the hierarchical Bayesian model considering the exponential Galactic disk profile. The filled stars denote the values of α derived directly from the observed densities of stars in the |z| range from 100 to 250 pc. b, The IMF power-law slope as a function of [M/H] based on various stellar models and empirical MLR, respectively. The red filled rectangles are derived by PARSEC, and the black rectangles are derived by Dartmouth56. The blue rectangles denote the values of α from the empirical mass–luminosity relation48. Error bars represent 1σ uncertainty.

Extended Data Fig. 4 Red giant branch stars65 reveals vertical actions Jz increase with stellar age.

a, Age–metallicity distributions of red giant branch stars within 500 pc in the solar vicinity. The red and blue contours display the age–metallicity distribution of dynamically hot (Jz > 20 kpc km s−1) and dynamically cold (Jz < 10 kpc km s−1) stars, respectively. Their contour densities are smoothed by the kernel density estimation method. The dotted lines indicate the separation of [M/H] slices used in the M dwarf samples. b, Stellar ages as a function of Jz in logarithmic form. The left side of the vertical dashed line denotes the dynamically cold stars, and the right side of the vertical dash-dotted line represents the dynamically hot stars. c, The ratio of the normalized number of dynamically hot stars to that of the dynamically cold stars as a function of [M/H]. is the proportion of the number of dynamically hot stars in each [M/H] bin to all dynamically hot stars of M dwarf star sample. fdyn.cold denotes the similar proportion, but for dynamically cold stars. + fdyn.cold) indicate the normalized number ratio between dynamically hot and the sum of dynamically hot and dynamically cold ratios. The vertical shaded region represents the metallicity range of −0.5 < [M/H] < 0.2, corresponding to the area of α variation in Fig. 2b and Fig. 3. Error bars represent 1σ uncertainty.

Extended Data Fig. 5 The test results of the effect of the binary stars on the IMF.

a, The results of simulations to verify the effect of the binary stars in the IMF. The results show the difference of the estimated α from the true values versus binary fraction with the numbers of mock stars equal to 1,000, 10,000 and 100,000. The vertical dotted line represents the binary fraction is 30%, which is approximately the observed mean binary fraction for solar metallicity stars. The vertical dashed line denotes the binary fraction is 60%. b, The test results by setting different slopes of binary fraction as a function of [M/H]. The blue, orange, green and red solid lines show the trend of α with [M/H] by adopting dfb/[M/H] ≈ −0.12, −0.20, −0.50 and −1.00, respectively. The blue dashed line indicates the IMF formula in Yan et al.24. The annotations on the right of the vertical line denote the binary fractions of [M/H] = −0.8. Error bars represent 1σ uncertainty.

Extended Data Fig. 6 The ΔRVmax distribution of the dynamically hot M-dwarf stars and dynamically cold stars.

The red solid line and blue dashed line represented the dynamically hot and dynamically blue stars, respectively. The vertical dotted line denotes the typical 1σ uncertainty of radial velocity uncertainty of LAMOST M-dwarf stars.

Extended Data Table 1 The sigmoid function fitting parameters of IMF power-law slope as a function and corresponding errors
Extended Data Table 2 Binary corrections to IMF slopes in different [M/H]

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, J., Liu, C., Zhang, ZY. et al. Stellar initial mass function varies with metallicity and time. Nature 613, 460–462 (2023).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing