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}kT}{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

↑ 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.
↑ "Radio Spectral Index". Wolfram Research. Retrieved 2011-01-19.

This article is issued from Wikipedia - version of the 11/26/2013. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.