Spectral index
Encyclopedia
In astronomy
, the spectral index of a source is a measure of the dependence of radiative flux density on frequency
. Given frequency
and radiative flux
, the spectral index 
is given implicitly by
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
Spectral index is also sometimes defined in terms of wavelength
. In this case, the spectral index 
is given implicitly by
and at a given frequency, spectral index may be calculated by taking the derivative
The opposite sign convention is sometimes employed, in which the spectral index is given by
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.
, intensity is given by
Taking the logarithm of each side and taking the partial derivative with respect to
yields
Using the positive sign convention, the spectral index of thermal radiation is thus
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
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
, the spectral index of a source is a measure of the dependence of radiative flux density on frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
. Given frequency
and radiative fluxRadiative flux
Radiative flux, or radiative flux density, is the amount of power radiated through a given area, in the form of photons or other elementary particles, typically measured in W/m2. It is used in astronomy to determine the magnitude and spectral class of a star...
, the spectral index is given implicitly byNote that if flux does not follow a power law
Power law
A power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...
in frequency, the spectral index itself is a function of frequency. Rearranging the above, we see that the spectral index is given by
Spectral index is also sometimes defined in terms of wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
. In this case, the spectral index is given implicitly byand at a given frequency, spectral index may be calculated by taking the derivative
The opposite sign convention is sometimes employed, in which the spectral index is given by
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 radiationThermal radiation
Thermal radiation is electromagnetic radiation generated by the thermal motion of charged particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation....
, intensity is given by
Taking the logarithm of each side and taking the partial derivative with respect to
yieldsUsing the positive sign convention, the spectral index of thermal radiation is thus
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