spectral index について

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.

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.

The opposite sign convention is sometimes employed,〔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.〕 in which the spectral index is given by
:

S\propto\nu^.

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.

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〔(【引用サイトリンク】publisher=Wolfram Research )〕
:

S \propto \nu^ T.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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