Spectral Index of Thermal Emission
At radio frequencies (i.e. in the low-frequency, long-wavelength limit), where the Rayleigh–Jeans law is a good approximation to the spectrum of thermal radiation, intensity is given by
Taking the logarithm of each side and taking the partial derivative with respect to yields
Using the positive sign convention, the spectral index of thermal radiation is thus in the Rayleigh-Jeans regime. The spectral index departs from this value at shorter wavelengths, for which the Rayleigh-Jeans law becomes an increasingly inaccurate approximation, tending towards zero as intensity reaches a peak at a frequency given by Wien's displacement law. Because of the simple temperature-dependence of radiative flux in the Rayleigh-Jeans regime, the radio spectral index is defined implicitly by
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