Phi value analysis

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, 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

$S\propto {\nu }^{\alpha }T.$

References

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