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[117.02] A New Approach to Blackbody Radiation

M.E. Cahill (U. Wisconsin, Washington County)

The current model of blackbody radiation, due to Planck, Bose & Einstein (PBE theory), has been remarkably accurate and valuable in Astronomy & Physics. It is possible to make several improvements in the treatment of the radiation which agree well with PBE theory for most of the energy range but which differ at the low and high energy ends of the distribution.

First it is possible to obtain an explicit microcanonical distribution for the radiation. The distribution has the form

dn = kN(1-x)^3N-4x²dx, k^-1= {\displaystyle\int\limits₀¹} (1-x) ^3N-4x²dx, x=\varepsilon/E

where E is the total energy, \varepsilon is the energy of interest, x is a unitless energy, N is the total number of photons and dn is the number of photons with energy x in range dx. The parameter N is not deternimed by the microcanonical method.

It is also possible to determine an improved most probable distribution based on BE statistics using photon populations of cells in 6 dimensional phase space. This requires not using the Stirling approximation in the derivation since a finite total number of photons implies that most of the cells in an infinite phase space will be empty. The cell size, phase space number density, phase space volume element, the thermodynamic permutability and the number of cells in an element are given by

C = \Omegah³, \rho^\prime = Cdn/dw, dw=V4\pip²dp, ln(P) = \int ln(\Omega-1+\rho^{\prime})!/(\Omega-1)!(\rho^{\prime})! dC, dC = dw/C

The distribution follows from making the permutability a maximum for arbitrary variations in photon number density with fixed total energy

\delta (ln(P)-\beta E) = 0 \Longrightarrow D(\Omega-1+\rho^\prime) - D(\rho^\prime) = \epsilon, D(x) = dln(x!)/dx, \epsilon = \beta \varepsilon

The distributions differ from PBE for they have energy cutoffs (indicated by a c subscript) and for large N

\varepsilon_c ~ lnN, \varepsilon_c^\prime = \gamma +D(\Omega-1) ~ ln\Omega

showing that the cells have N Planck bins. The distributions differ from each other because there is no infinite photon density at 0 energy in the more fundamental microcanonical distribution. If the distributions are compared where they are in good agreement, N is the standard N=8\piV((T/(hc)))³. Wavelength cutoffs for the sun and for the cosmic background radiation are roughly \lambda _\sun \approx 10nm, \lambda_CBR\approx 10\mum The cosmic background radiation cutoff is beyond current observations.

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Bulletin of the American Astronomical Society, 35 #5
© 2003. The American Astronomical Soceity.