We compare Planck’s distribution for spectral radiation with Callendar’s. We cover the two regimes of wavelengths and frequencies. In each of the four cases we evaluate the dependence of the spectral radiance on the temperature for the maximum, the standard deviation and the distance between the two inflection points. The astrophysical comparison of Planck’s and Callendar’s distributions covers the temperature of the sun, the temperature of the cosmic microwave background, the (B-V) color and the bolometric correction.
References
[1]
Planck, M. (1900) On the Theory of the Energy Distribution Law of the Normal Spectrum. Verhandlungen der Deutschen Physikalischen Gesellschaft, 2, 237-245.
Tran, M. (2020) Planck’s and Callendar’s Blackbody Radiation Formulas and Their Fitness to Experimental Data. EuropeanJournalofPhysics, 41, Article 025102. https://doi.org/10.1088/1361-6404/ab513b
[5]
Hernandez, H. (2025) Modelling Thermal Radiation 3. Spectral Radiation, Fors Chem Research Reports, 1-17
[6]
Mohr, P.J., Newell, D.B., Taylor, B.N. and Tiesinga, E. (2025) CODATA Recommended Values of the Fundamental Physical Constants: 2022. ReviewsofModernPhysics, 97, Article No. 025002. https://doi.org/10.1103/revmodphys.97.025002
[7]
Phillips, K.J.H. (1995) Guide to the Sun. Cambridge University Press.
[8]
Fixsen, D.J. (2009) The Temperature of the Cosmic Microwave Background. TheAstrophysicalJournal, 707, 916-920. https://doi.org/10.1088/0004-637x/707/2/916
[9]
Planck, M. (1959) The Theory of Heat Radiation. Dover Publications.
[10]
Kraus, J.D. (1986) Radio Astronomy. Cygnus-Quasar Books.
[11]
Rybicki, G. and Lightman, A. (1991) Radiative Processes in Astrophysics. Wiley-Interscience.
[12]
Condon, J.J. and Ransom, S.M. (2016) Essential Radio Astronomy. Princeton University Press.
[13]
Stefan, J. (1879) über die beziehung der w?rmestrahlungund der temperature. Sitzungs-berichte der KaiserlichenAkademie der Wissenschaften, Mathe-matische-Naturwissenschaftliche Classe Abteilung II, Vol. 79, 391-428.
[14]
Boltzmann, L. (1884) Ableitung des stefan’schen gesetzes, betreffend die abh?ngigkeit der w?rmestrahlung von der temperatur aus der electromagnetischen lichttheorie. AnnalenderPhysik, 258, 291-294. https://doi.org/10.1002/andp.18842580616
Olver, F.W.J., Lozier, D.W., Boisvert, R.F. and Clark, C.W. (2010) NIST Handbook of Mathematical Functions. Cambridge University Press.
[17]
Reed, B.C. (2025) Characterizing the Breadth of the Planck Function. AmericanJournalofPhysics, 93, 1000-1004. https://doi.org/10.1119/5.0281796
[18]
Xu, G., Ke, Z., Zhuang, C., Li, Y., Cai, R., Yang, Y., et al. (2023) Measurements and Analysis of Solar Spectrum in near Space. EnergyReports, 9, 1764-1773. https://doi.org/10.1016/j.egyr.2023.04.229
[19]
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.
[20]
Fixsen, D.J. and Mather, J.C. (2002) The Spectral Results of the Far‐Infrared Absolute Spectrophotometer Instrument on Cobe. TheAstrophysicalJournal, 581, 817-822. https://doi.org/10.1086/344402
[21]
Mohr, P.J. and Taylor, B.N. (2005) CODATA Recommended Values of the Fundamental Physical Constants: 2002. ReviewsofModernPhysics, 77, 1-107. https://doi.org/10.1103/revmodphys.77.1
Zaninetti, L. (2008) Semi-Analytical Formulas for the Hertzsprung-Russell Diagram. SerbianAstronomicalJournal, 177, 73-85. https://doi.org/10.2298/saj0877073z
[24]
Hernandez, H. (2026) Analysis of the am0 Spectrum of Solar Radiation. ForsChem Research Reports, 11, 1-13.