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 ${\displaystyle \nu }$ in Hz and radiative flux density ${\displaystyle S_{\nu }}$ in Jy, the spectral index ${\displaystyle \alpha }$ is given implicitly by $${\displaystyle S_{\nu }\propto \nu ^{\alpha }.}$$ 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 $${\displaystyle \alpha \!\left(\nu \right)={\frac {\partial \log S_{\nu }\!\left(\nu \right)}{\partial \log \nu }}.}$$

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 ${\displaystyle \lambda }$. In this case, the spectral index ${\displaystyle \alpha }$ is given implicitly by $${\displaystyle S_{\lambda }\propto \lambda ^{\alpha },}$$ and at a given frequency, spectral index may be calculated by taking the derivative $${\displaystyle \alpha \!\left(\lambda \right)={\frac {\partial \log S_{\lambda }\!\left(\lambda \right)}{\partial \log \lambda }}.}$$ The spectral index using the ${\displaystyle S_{\nu }}$, which we may call ${\displaystyle \alpha _{\nu },}$ differs from the index ${\displaystyle \alpha _{\lambda }}$ defined using ${\displaystyle S_{\lambda }.}$ The total flux between two frequencies or wavelengths is $${\displaystyle S=C_{1}\left(\nu _{2}^{\alpha _{\nu }+1}-\nu _{1}^{\alpha _{\nu }+1}\right)=C_{2}\left(\lambda _{2}^{\alpha _{\lambda }+1}-\lambda _{1}^{\alpha _{\lambda }+1}\right)=c^{\alpha _{\lambda }+1}C_{2}\left(\nu _{2}^{-\alpha _{\lambda }-1}-\nu _{1}^{-\alpha _{\lambda }-1}\right)}$$ which implies that $${\displaystyle \alpha _{\lambda }=-\alpha _{\nu }-2.}$$ The opposite sign convention is sometimes employed,^[1] in which the spectral index is given by $${\displaystyle S_{\nu }\propto \nu ^{-\alpha }.}$$

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. 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.

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 $${\displaystyle B_{\nu }(T)\simeq {\frac {2\nu ^{2}kT}{c^{2}}}.}$$ Taking the logarithm of each side and taking the partial derivative with respect to ${\displaystyle \log \,\nu }$ yields $${\displaystyle {\frac {\partial \log B_{\nu }(T)}{\partial \log \nu }}\simeq 2.}$$ Using the positive sign convention, the spectral index of thermal radiation is thus ${\displaystyle \alpha \simeq 2}$ 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^[2] $${\displaystyle S\propto \nu ^{\alpha }T.}$$