Spectral index

In astronomy, the spectral index of a source is a measure of the dependence of intensity on frequency. Given frequency $\nu$ and Specific radiative intensity $S_{\nu }$, the spectral index $\alpha$ is given implicitly by

$S\propto \nu ^{\alpha }.$

Note that if intensity does not follow a power law in frequency, the spectral index itself is a function of frequency. Rearranging the above, we see that the spectral index is given by

$\alpha ={\frac {\partial \log S}{\partial \log \nu }}.$

Spectral index is also sometimes defined in terms of wavelength $\lambda$. In this case, the spectral index $\alpha$ is given implicitly by

$S\propto \lambda ^{n},$

and at a given frequency, spectral index may be calculated by taking the derivative

$\alpha ={\frac {\partial \log S}{\partial \log \lambda }}.$

The opposite sign convention is sometimes employed, in which the spectral index is given by

$S\propto \nu ^{-\alpha }.$

The spectral index of a source can hint at its properties. For example, using the positive sign convention, a spectral index of 0 to 2 at radio frequencies indicates thermal emission, while a steep negative spectral index typically indicates synchrotron emission.

Spectral Index of Thermal emission

At radio frequencies, where the Rayleigh–Jeans law is a good approximation to the spectrum of thermal radiation, intensity is given by

$S_{\nu }(T)={\frac {2\nu ^{2}kT}{c^{2}}}.$

Taking the logarithm of each side and taking the partial derivative with respect to $\log \,\nu$ yields

${\frac {\partial \log S_{\nu }(T)}{\partial \log \nu }}=2.$

Using the positive sign convention, the spectral index of thermal radiation is thus $\alpha \approx 2$ in the Raleigh-Jeans regime. The spectral index departs from this value at shorter wavelengths, for which the Raleigh-Jeans law becomes an increasingly inaccurate approximation, tending towards zero as intensity reaches a peak at a frequency given by the frequency-dependent formulation of Wien's displacement law.