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## Summary

In astronomy, the spectral index of a source is a measure of the dependence of radiative flux density (that is, radiative flux per unit of frequency) on frequency. Given frequency $\nu$ and radiative flux density $S_{\nu }$ , the spectral index $\alpha$ is given implicitly by

$S_{\nu }\propto \nu ^{\alpha }.$ Note that if flux 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

$\alpha \!\left(\nu \right)={\frac {\partial \log S_{\nu }\!\left(\nu \right)}{\partial \log \nu }}.$ Clearly the power law can only apply over a certain range of frequency because otherwise the integral over all frequencies would be infinite.

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

$S_{\lambda }\propto \lambda ^{\alpha },$ and at a given frequency, spectral index may be calculated by taking the derivative

$\alpha \!\left(\lambda \right)={\frac {\partial \log S_{\lambda }\!\left(\lambda \right)}{\partial \log \lambda }}.$ The spectral index using the $S_{\nu }$ , which we may call $\alpha _{\nu },$ differs from the index $\alpha _{\lambda }$ defined using $S_{\lambda }.$ The total flux between two frequencies or wavelengths is

$S=C_{1}(\nu _{2}^{\alpha _{\nu }+1}-\nu _{1}^{\alpha _{\nu }+1})=C_{2}(\lambda _{2}^{\alpha _{\lambda }+1}-\lambda _{1}^{\alpha _{\lambda }+1})=c^{\alpha _{\lambda }+1}C_{2}(\nu _{2}^{-\alpha _{\lambda }-1}-\nu _{1}^{-\alpha _{\lambda }-1})$ which implies that

$\alpha _{\lambda }=-\alpha _{\nu }-2.$ The opposite sign convention is sometimes employed, in which the spectral index is given by

$S_{\nu }\propto \nu ^{-\alpha }.$ The spectral index of a source can hint at its properties. For example, using the positive sign convention, the spectral index of the emission from an optically thin thermal plasma is -0.1, whereas for an optically thick plasma it is 2. Therefore, a spectral index of -0.1 to 2 at radio frequencies often indicates thermal emission, while a steep negative spectral index typically indicates synchrotron emission. It is worth noting that the observed emission can be affected by several absorption processes that affect the low-frequency emission the most; the reduction in the observed emission at low frequencies might result in a positive spectral index even if the intrinsic emission has a negative index. Therefore, it is not straightforward to associate positive spectral indices with thermal emission.

## 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

$B_{\nu }(T)\simeq {\frac {2\nu ^{2}kT}{c^{2}}}.$

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

${\frac {\partial \log B_{\nu }(T)}{\partial \log \nu }}\simeq 2.$

Using the positive sign convention, the spectral index of thermal radiation is thus $\alpha \simeq 2$  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

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