Isaac Scientific Publishing

Advances in Astrophysics

A Left and Right Truncated Lognormal Distribution for the Stars

Download PDF (834.2 KB) PP. 197 - 213 Pub. Date: August 18, 2017

DOI: 10.22606/adap.2017.23005


  • L. Zaninetti*
    Physics Department, via P.Giuria 1, I-10125 Turin, Italy


The initial mass function for the stars is often modeled by a lognormal distribution. This paper is devoted to demonstrating the advantage of introducing a left and right truncated lognormal probability density function, which is characterized by four parameters. Its normalization constant, mean, the variance, second moment about the origin and distribution function are calculated. The chi-square test and the Kolmogorov–Smirnov test are performed on four samples of stars.


Stars, characteristics and properties of Stars, normal


[1] E. E. Salpeter, The Luminosity Function and Stellar Evolution., ApJ 121 (1955) 161–167.

[2] J. M. Scalo, The stellar initial mass function, Fundamentals of Cosmic Physics 11 (1986) 1–278.

[3] P. Kroupa, C. A. Tout, G. Gilmore, The distribution of low-mass stars in the Galactic disc, MNRAS 262 (1993) 545–587.

[4] J. Binney, M. Merrifield, Galactic astronomy, Princeton University Press, Princeton, NJ, 1998.

[5] P. Kroupa, On the variation of the initial mass function, MNRAS 322 (2001) 231–246.

[6] N. Bastian, K. R. Covey, M. R. Meyer, A Universal Stellar Initial Mass Function? A Critical Look at Variations, ARA&A 48 (2010) 339–389. arXiv:1001.2965, doi:10.1146/annurev-astro-082708-101642.

[7] P. Kroupa, C. Weidner, J. Pflamm-Altenburg, I. Thies, J. Dabringhausen, M. Marks, T. Maschberger, The Stellar and Sub-Stellar Initial Mass Function of Simple and Composite Populations, Springer Netherlands, 2013, p. 115.

[8] L. Zaninetti, The initial mass function modeled by a left truncated beta distribution , ApJ 765 (2013) 128–135.

[9] M. Evans, N. Hastings, B. Peacock, Statistical Distributions - third edition, John Wiley & Sons Inc, New York, 2000.

[10] R. B. Larson, A simple probabilistic theory of fragmentation, MNRAS 161 (1973) 133. doi:10.1093/mnras/ 161.2.133.

[11] G. E. Miller, J. M. Scalo, The initial mass function and stellar birthrate in the solar neighborhood, ApJS 41 (1979) 513–547. doi:10.1086/190629.

[12] H. Zinnecker, Star formation from hierarchical cloud fragmentation - A statistical theory of the log-normal Initial Mass Function, MNRAS 210 (1984) 43–56. doi:10.1093/mnras/210.1.43.

[13] G. Chabrier, Galactic Stellar and Substellar Initial Mass Function, PASP 115 (2003) 763–795. arXiv:arXiv: astro-ph/0304382, doi:10.1086/376392.

[14] A. N. Cox, Allen’s astrophysical quantities, Springer, New York, 2000.

[15] N. L. Johnson, S. Kotz, N. Balakrishnan, Continuous univariate distributions. Vol. 1. 2nd ed., Wiley , New York, 1994.

[16] F. W. J. e. Olver, D. W. e. Lozier, R. F. e. Boisvert, C. W. e. Clark, NIST handbook of mathematical functions., Cambridge University Press. , Cambridge, 2010.

[17] D. Kahaner, C. Moler, S. Nash, Numerical Methods and Software, Prentice Hall Publishers, Englewood Cliffs, New Jersey, 1989.

[18] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in FORTRAN. The Art of Scientific Computing, Cambridge University Press, Cambridge, UK, 1992.

[19] H. Akaike, A new look at the statistical model identification, IEEE Transactions on Automatic Control 19 (1974) 716–723.

[20] A. R. Liddle, How many cosmological parameters?, MNRAS 351 (2004) L49–L53.

[21] W. Godlowski, M. Szydowski, Constraints on Dark Energy Models from Supernovae, in: M. Turatto, S. Benetti, L. Zampieri, W. Shea (Eds.), 1604-2004: Supernovae as Cosmological Lighthouses, Vol. 342 of Astronomical Society of the Pacific Conference Series, 2005, pp. 508–516.

[22] A. Kolmogoroff, Confidence limits for an unknown distribution function, The Annals of Mathematical Statistics 12 (4) (1941) 461–463.

[23] N. Smirnov, Table for estimating the goodness of fit of empirical distributions, The Annals of Mathematical Statistics 19 (2) (1948) 279–281.

[24] J. Massey, Frank J., The kolmogorov-smirnov test for goodness of fit, Journal of the American Statistical Association 46 (253) (1951) 68–78.

[25] J. M. Oliveira, R. D. Jeffries, J. T. van Loon, The low-mass initial mass function in the young cluster NGC 6611 , MNRAS 392 (2009) 1034–1050. arXiv:0810.4444, doi:10.1111/j.1365-2966.2008.14140.x.

[26] J. Irwin, S. Hodgkin, S. Aigrain, J. Bouvier, L. Hebb, M. Irwin, E. Moraux, The Monitor project: rotation of low-mass stars in NGC 2362 - testing the disc regulation paradigm at 5 Myr, MNRAS 384 (2008) 675–686. arXiv:0711.2398, doi:10.1111/j.1365-2966.2007.12725.x.

[27] J. K. Hill, J. E. Isensee, R. H. Cornett, R. C. Bohlin, R. W. O’Connell, M. S. Roberts, A. M. Smith, T. P. Stecher, Initial mass functions from ultraviolet stellar photometry: A comparison of Lucke and Hodge OB associations near 30 Doradus with the nearby field, ApJ 425 (1994) 122–126. doi:10.1086/173968.

[28] L. Prisinzano, F. Damiani, G. Micela, R. D. Jeffries, E. Franciosini, G. G. Sacco, A. Frasca, A. Klutsch, A. Lanzafame, E. J. Alfaro, K. Biazzo, R. Bonito, A. Bragaglia, M. Caramazza, A. Vallenari, G. Carraro, M. T. Costado, E. Flaccomio, P. Jofré, C. Lardo, L. Monaco, L. Morbidelli, N. Mowlavi, E. Pancino, S. Randich, S. Zaggia, The Gaia-ESO Survey: membership and initial mass function of the Velorum cluster, A&A 589 (2016) A70. arXiv:1601.06513, doi:10.1051/0004-6361/201527875.

[29] W. J. Reed, M. Jorgensen, The double pareto-lognormal distributiona new parametric model for size distributions, Communications in Statistics - Theory and Methods 33 (8) (2004) 1733–1753. arXiv:http: //, doi:10.1081/STA-120037438. URL

[30] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1965.

[31] L. Zaninetti, A right and left truncated gamma distribution with application to the stars , Advanced Studies in Theoretical Physics 23 (2013) 1139–1147.

[32] A. Hald, On the history of maximum likelihood in relation to inverse probability and least squares, Statist. Sci. 14 (2) (1999) 214–222. doi:10.1214/ss/1009212248. URL