Spectral index - Alchetron, The Free Social Encyclopedia

In astronomy, the spectral index of a source is a measure of the dependence of radiative flux density on frequency. Given frequency ν and radiative flux S , the spectral index α is given implicitly by

S ∝ ν α .

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 ) ∂ log ⁡ ν .

Spectral index is also sometimes defined in terms of wavelength λ . In this case, the spectral index α is given implicitly by

S ∝ λ α ,

and at a given frequency, spectral index may be calculated by taking the derivative

α ( λ ) = ∂ log ⁡ S ( λ ) ∂ log ⁡ λ .

The opposite sign convention is sometimes employed, in which the spectral index is given by

S ∝ ν − α .

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 ν ( T ) ≃ 2 ν 2 k T c 2 .

Taking the logarithm of each side and taking the partial derivative with respect to log ν yields

∂ log ⁡ B ν ( T ) ∂ log ⁡ ν ≃ 2.

Using the positive sign convention, the spectral index of thermal radiation is thus α ≃ 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 ∝ ν α T .

References

Spectral index Wikipedia

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