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 ν {\displaystyle \nu } in Hz and radiative flux density S ν {\displaystyle S_{\nu }} in Jy, the spectral index α {\displaystyle \alpha } is given implicitly by

S ν ∝ ν α . {\displaystyle 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
α ( ν ) = ∂ log ⁡ S ν ( ν ) ∂ log ⁡ ν . {\displaystyle \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 λ {\displaystyle \lambda } . In this case, the spectral index α {\displaystyle \alpha } is given implicitly by

S λ ∝ λ α , {\displaystyle S_{\lambda }\propto \lambda ^{\alpha },}
and at a given frequency, spectral index may be calculated by taking the derivative
α ( λ ) = ∂ log ⁡ S λ ( λ ) ∂ log ⁡ λ . {\displaystyle \alpha \!\left(\lambda \right)={\frac {\partial \log S_{\lambda }\!\left(\lambda \right)}{\partial \log \lambda }}.}
The spectral index using the S ν {\displaystyle S_{\nu }} , which we may call α ν , {\displaystyle \alpha _{\nu },} differs from the index α λ {\displaystyle \alpha _{\lambda }} defined using S λ . {\displaystyle S_{\lambda }.} The total flux between two frequencies or wavelengths is
S = C 1 ( ν 2 α ν + 1 − ν 1 α ν + 1 ) = C 2 ( λ 2 α λ + 1 − λ 1 α λ + 1 ) = c α λ + 1 C 2 ( ν 2 − α λ − 1 − ν 1 − α λ − 1 ) {\displaystyle 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
α λ = − α ν − 2. {\displaystyle \alpha _{\lambda }=-\alpha _{\nu }-2.}
The opposite sign convention is sometimes employed,[1] in which the spectral index is given by
S ν ∝ ν − α . {\displaystyle 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. 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 ν ( T ) ≃ 2 ν 2 k T c 2 . {\displaystyle 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 ν {\displaystyle \log \,\nu } yields
∂ log ⁡ B ν ( T ) ∂ log ⁡ ν ≃ 2. {\displaystyle {\frac {\partial \log B_{\nu }(T)}{\partial \log \nu }}\simeq 2.}
Using the positive sign convention, the spectral index of thermal radiation is thus α ≃ 2 {\displaystyle \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 ∝ ν α T . {\displaystyle S\propto \nu ^{\alpha }T.}

References

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