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 $$B_\nu(T) \simeq \frac{2 \nu^2 k T}{c^2}.$$ 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 $$S \propto \nu^{\alpha} T.$$