# Spectral index

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

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

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

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

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

${\displaystyle S\propto \lambda ^{\alpha },}$

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

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

The opposite sign convention is sometimes employed,[1] in which the spectral index is given by

${\displaystyle S\propto \nu ^{-\alpha }.}$

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

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

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]

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

## 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.