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Spectral index

## Summary

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 ${\displaystyle \nu }$ and radiative flux density ${\displaystyle S_{\nu }}$, 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

${\displaystyle S=C_{1}(\nu _{2}^{\alpha _{\nu }+1}-\nu _{1}^{\alpha _{\nu }+1})=C_{2}(\lambda _{2}^{\alpha _{\lambda }+1}-\lambda _{1}^{\alpha _{\lambda }+1})=c^{\alpha _{\lambda }+1}C_{2}(\nu _{2}^{-\alpha _{\lambda }-1}-\nu _{1}^{-\alpha _{\lambda }-1})}$

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. 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

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

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.