Spectral index

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 $\nu$ in Hz and radiative flux density $S_{\nu }$ in Jy, 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 $$S=C_{1}\left(\nu _{2}^{\alpha _{\nu }+1}-\nu _{1}^{\alpha _{\nu }+1}\right)=C_{2}\left(\lambda _{2}^{\alpha _{\lambda }+1}-\lambda _{1}^{\alpha _{\lambda }+1}\right)=c^{\alpha _{\lambda }+1}C_{2}\left(\nu _{2}^{-\alpha _{\lambda }-1}-\nu _{1}^{-\alpha _{\lambda }-1}\right)$$ 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 $$S\propto \nu ^{\alpha }T.$$