The gamma function is a good approximation to the luminosity function (LF) of galaxies, and an exponentiated gamma distribution would permit a more rigorous analysis. This paper examines the exponentiated gamma distribution and its double truncation. The new results are applied to five clusters of stars. The magnitude version of the truncated LF is derived in order to fit the observed LF for galaxies. The regular and truncated LFs are applied to the five bands of SDSS galaxies.
References
[1]
Mudholkar, G.S. and Srivastava, D.K. (1993) Exponentiated Weibull Family for Analyzing Bathtub Failure-Rate Data. IEEETransactionsonReliability, 42, 299-302. https://doi.org/10.1109/24.229504
[2]
Gupta, R.C., Gupta, P.L. and Gupta, R.D. (1998) Modeling Failure Time Data by Lehman Alternatives. CommunicationsinStatistics—TheoryandMethods, 27, 887-904. https://doi.org/10.1080/03610929808832134
[3]
Nadarajah, S. and Gupta, A.K. (2007) The Exponentiated Gamma Distribution with Application to Drought Data. CalcuttaStatisticalAssociationBulletin, 59, 29-54. https://doi.org/10.1177/0008068320070103
[4]
Shawky, A.I. and Bakoban, R.A. (2012) Exponentiated Gamma Distribution: Different Methods of Estimations. Journal of Applied Mathematics, 2012, Article ID: 284296. https://doi.org/10.1155/2012/284296
[5]
Evans, M., Hastings, N. and Peacock, B. (2000) Statistical Distributions. 3rd Edition, Wiley.
[6]
Johnson, N.L., Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions. Vol. 1, 2nd Edition, Wiley.
[7]
Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P. (1992) Numerical Recipes in FORTRAN. The Art of Scientific Computing. Cambridge University Press.
[8]
Akaike, H. (1974) A New Look at the Statistical Model Identification. IEEETransactionsonAutomaticControl, 19, 716-723. https://doi.org/10.1109/tac.1974.1100705
[9]
Liddle, A.R. (2004) How Many Cosmological Parameters? MonthlyNoticesoftheRoyalAstronomicalSociety, 351, L49-L53. https://doi.org/10.1111/j.1365-2966.2004.08033.x
[10]
Godlowski, W. and Szydowski, M. (2005) Constraints on Dark Energy Models from Supernovae. In: Turatto, M., Benetti, S., Zampieri, L. and Shea, W., Eds., 1604-2004: Supernovae as Cosmological Lighthouses, Astronomical Society of the Pacific, 508-516.
[11]
Olver, F.W.J., Lozier, D.W., Boisvert, R.F. and Clark, C.W. (2010) NIST Handbook of Mathematical Functions. Cambridge University Press.
[12]
Abramowitz, M. and Stegun, I.A. (1965) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover.
[13]
Zaninetti, L. (2013) A Right and Left Truncated Gamma Distribution with Application to the Stars. AdvancedStudiesinTheoreticalPhysics, 7, 1139-1147. https://doi.org/10.12988/astp.2013.310125
[14]
Kahaner, D., Moler, C. and Nash, S. (1989) Numerical Methods and Software. Prentice Hall Publishers.
[15]
Schechter, P. (1976) An Analytic Expression for the Luminosity Function for Galaxies. TheAstrophysicalJournal, 203, 297-306. https://doi.org/10.1086/154079
[16]
Felten, J.E. (1976) On Schmidt’s Vm Estimator and Other Estimators of Luminosity Functions. TheAstrophysicalJournal, 207, 700-709. https://doi.org/10.1086/154538
[17]
Irwin, J., Hodgkin, S., Aigrain, S., Bouvier, J., Hebb, L., Irwin, M., et al. (2008) The Monitor Project: Rotation of Low-Mass Stars in NGC 2362—Testing the Disc Regulation Paradigm at 5 Myr. MonthlyNoticesoftheRoyalAstronomicalSociety, 384, 675-686. https://doi.org/10.1111/j.1365-2966.2007.12725.x
[18]
Moitinho, A., Alves, J., Huélamo, N. and Lada, C.J. (2001) NGC 2362: A Template for Early Stellar Evolution. TheAstrophysicalJournal, 563, L73-L76. https://doi.org/10.1086/338503
[19]
Oliveira, J.M., Jeffries, R.D. and van Loon, J.T. (2009) The Low-Mass Initial Mass Function in the Young Cluster NGC 6611. MonthlyNoticesoftheRoyalAstronomicalSociety, 392, 1034-1050. https://doi.org/10.1111/j.1365-2966.2008.14140.x
[20]
Prisinzano, L., Damiani, F., Micela, G., Jeffries, R.D., Franciosini, E., Sacco, G.G., et al. (2016) The Gaia-ESO Survey: Membership and Initial Mass Function of the Velorum Cluster. Astronomy&Astrophysics, 589, A70. https://doi.org/10.1051/0004-6361/201527875
[21]
Panwar, N., Pandey, A.K., Samal, M.R., Battinelli, P., Ogura, K., Ojha, D.K., et al. (2018) Young Cluster Berkeley 59: Properties, Evolution, and Star Formation. TheAstronomicalJournal, 155, 44. https://doi.org/10.3847/1538-3881/aa9f1b
[22]
Brandner, W., Calissendorff, P. and Kopytova, T. (2023) Astrophysical Properties of 600 Bona Fide Single Stars in the Hyades Open Cluster. TheAstronomicalJournal, 165, 108. https://doi.org/10.3847/1538-3881/acb208
[23]
Gunn, J.E., Carr, M., Rockosi, C., Sekiguchi, M., Berry, K., Elms, B., et al. (1998) The Sloan Digital Sky Survey Photometric Camera. TheAstronomicalJournal, 116, 3040-3081. https://doi.org/10.1086/300645
[24]
Blanton, M.R., Hogg, D.W., Bahcall, N.A., Brinkmann, J., Britton, M., Connolly, A.J., et al. (2003) The Galaxy Luminosity Function and Luminosity Density at Redshift z = 0.1. TheAstrophysicalJournal, 592, 819-838. https://doi.org/10.1086/375776
[25]
Zaninetti, L. (2025) New Probability Distributions in Astrophysics: XIV. Truncation of the Modified Lognormal Distribution. InternationalJournalofAstronomyandAstrophysics, 15, 19-42. https://doi.org/10.4236/ijaa.2025.151003
[26]
Zaninetti, L. (2024) New Probability Distributions in Astrophysics: XIII. Truncation for the Benini Distribution. InternationalJournalofAstronomyandAstrophysics, 14, 203-219. https://doi.org/10.4236/ijaa.2024.143013
[27]
Zaninetti, L. (2024) New Probability Distributions in Astrophysics: XII. Truncation for the Gompertz Distribution. InternationalJournalofAstronomyandAstrophysics, 14, 101-119. https://doi.org/10.4236/ijaa.2024.142007
[28]
Zaninetti, L. (2023) New Probability Distributions in Astrophysics: XI. Left Truncation for the Topp-Leone Distribution. InternationalJournalofAstronomyandAstrophysics, 13, 154-165. https://doi.org/10.4236/ijaa.2023.133009
[29]
Zaninetti, L. (2022) New Probability Distributions in Astrophysics: X. Truncation and Mass-Luminosity Relationship for the Frèchet Distribution. InternationalJournalofAstronomyandAstrophysics, 12, 347-362. https://doi.org/10.4236/ijaa.2022.124020
[30]
Zaninetti, L. (2021) New Probability Distributions in Astrophysics: V. The Truncated Weibull Distribution. InternationalJournalofAstronomyandAstrophysics, 11, 133-149. https://doi.org/10.4236/ijaa.2021.111008
[31]
Zaninetti, L. (2021) New Probability Distributions in Astrophysics: VI. The Truncated Sujatha Distribution. InternationalJournalofAstronomyandAstrophysics, 11, 517-529. https://doi.org/10.4236/ijaa.2021.114028
[32]
Zaninetti, L. (2020) New Probability Distributions in Astrophysics: II. The Generalized and Double Truncated Lindley. InternationalJournalofAstronomyandAstrophysics, 10, 39-55. https://doi.org/10.4236/ijaa.2020.101004
[33]
Zaninetti, L. (2019) New Probability Distributions in Astrophysics: I. The Truncated Generalized Gamma. InternationalJournalofAstronomyandAstrophysics, 9, 393-410. https://doi.org/10.4236/ijaa.2019.94027
[34]
Zaninetti, L. (2017) A Left and Right Truncated Lognormal Distribution for the Stars. Advances in Astrophysics, 2, 197.
[35]
Zaninetti, L. (2013) The Initial Mass Function Modeled by a Left Truncated Beta Distribution. TheAstrophysicalJournal, 765, 128. https://doi.org/10.1088/0004-637x/765/2/128