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 $\nu$ and radiative flux density $S_\nu$, 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
:$\alpha \! \left( \nu \right) = \frac.$

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
:$\alpha \! \left( \lambda \right) =\frac.$
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
:$S=C_1(\nu_2^-\nu_1^)=C_2(\lambda_2^-\lambda_1^)=c^C_2(\nu_2^-\nu_1^)$
which implies that
:$\alpha_\lambda=-\alpha_\nu-2.$
The opposite sign convention is sometimes employed, in which the spectral index is given by
:$S_\nu\propto\nu^.$

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
:$B_\nu(T) \simeq \frac.$
Taking the logarithm of each side and taking the partial derivative with respect to $\log \, \nu$ yields
:$\frac \simeq 2.$
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
:$S \propto \nu^ T.$

References

{{Reflist

Radio astronomy
Equations of astronomy