Spectral index explained

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$$S\_\backslash nu\backslash propto\backslash nu^\backslash alpha.$$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$$\backslash alpha\; \backslash !\; \backslash left(\backslash nu\; \backslash right)\; =\; \backslash frac.$$

Clearly the power law can only apply over a certain range of frequency because otherwise the integral over all frequencies would be infinite.

λ . In this case, the spectral index \alpha is given implicitly by$$S\_\backslash lambda\backslash propto\backslash lambda^\backslash alpha,$$and at a given frequency, spectral index may be calculated by taking the derivative$$\backslash alpha\; \backslash !\; \backslash left(\backslash lambda\; \backslash right)\; =\backslash frac.$$The spectral index using the S _{\nu} , which we may call \alpha _{\nu,} differs from the index \alpha _{λ} defined using S _{λ.} The total flux between two frequencies or wavelengths is$$S\; =\; C\_1\backslash left(\backslash nu\_2^-\backslash nu\_1^\backslash right)\; =\; C\_2\backslash left(\backslash lambda\_2^\; -\; \backslash lambda\_1^\backslash right)\; =\; c^\; C\_2\backslash left(\backslash nu\_2^-\backslash nu\_1^\backslash right)$$which implies that$$\backslash alpha\_\backslash lambda=-\backslash alpha\_\backslash nu-2.$$The opposite sign convention is sometimes employed,^[1] in which the spectral index is given by$$S\_\backslash nu\backslash propto\backslash nu^.$$

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$$B\_\backslash nu(T)\; \backslash simeq\; \backslash frac.$$Taking the logarithm of each side and taking the partial derivative with respect to

log \nu yields$$\backslash frac\; \backslash simeq\; 2.$$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] $$S\; \backslash propto\; \backslash nu^\; T.$$

Notes and References

1. Burke, B.F., Graham-Smith, F. (2009). An Introduction to Radio Astronomy, 3rd Ed., Cambridge University Press, Cambridge, UK,, page 132.
2. Web site: Radio Spectral Index . . 2011-01-19.