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The spectral index of an observed electromagnetic-radiation source is a scalar measure describing an aspect of its spectrum, treating the observed spectrum as a power law. It is specifically the exponent of the frequency that produces a term proportional to the observed radiant flux at each frequency, i.e.,
S ∝ να
(Some use the term spectral index for an exponent of the wavelength rather than of the frequency, which is the negative of the above exponent. Some papers deliberately quote the formula they use to avoid confusion. Comments below use the above definition/formula.)
Such a spectral index always applies to a finite range/regime: otherwise it would imply infinite energy emission and a frequency range limited only by zero, with no upper limit. In actual practice, a spectral index is used to describe the spectrum viewed over a portion of the spectrum's full range, and describes an approximation of the slope characteristics of radiant-flux-per-unit-frequency over that particular portion. Black-body radiation, which clearly is not a power law over the whole spectrum, given its peak strength in the middle, does approximate a power law of α = 2 over the lower frequencies, as per the Rayleigh-Jeans law. If an observation is made in the regime where this holds true (called spectrum's the Rayleigh-Jeans regime), then black-body radiation is a possibility and parameters of the spectrum indicate the temperature it suggests.
Other spectral indexes suggest other types of EMR production, e.g., synchrotron radiation or bremsstrahlung.
The term emissivity index (β) refers to the spectral index minus two, basically comparing it to the low-frequency end of the black-body spectrum.
Radio astronomy makes much use of the spectral index, and the term radio spectral index is sometimes used to specify the spectral index in the radio regime. (Radio is generally within the Rayleigh-Jeans regime for sources bright enough to be observed.)