Spectral index

In astronomy, the spectral index of a source is a measure of the dependence of radiative flux density on frequency. Given frequency $\nu$ and radiative flux $S$, the spectral index $\alpha$ is given implicitly by

$S\propto\nu^\alpha.$

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

$\alpha \! \left( \nu \right) = \frac{\partial \log S \! \left( \nu \right)}{\partial \log \nu}.$

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

$S\propto\lambda^\alpha,$

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

$\alpha \! \left( \lambda \right) =\frac{\partial \log S \! \left( \lambda \right)}{\partial \log \lambda}.$

The opposite sign convention is sometimes employed,^[1] 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 (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^[2]

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

References

1. Burke, B.F., Graham-Smith, F. (2009). An Introduction to Radio Astronomy, 3rd Ed., Cambridge University Press, Cambridge, UK, ISBN 978-0-521-87808-1, page 132.
2. "Radio Spectral Index". Wolfram Research. Retrieved 2011-01-19.