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Spectral Index (α) ○◂｜Definition｜1st｜20251119205401-00-⌔

Spectral index - Wikipedia

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,¹ 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. 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.

Printed 2026-06-28.

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Footnotes

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. ↩

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