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Spectral index
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<p>In <a href="/facts/Astronomy/0ArJVM1p">astronomy</a>, 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 <a href="/facts/Frequency/QUSbPaW8">frequency</a>. Given frequency ν {\displaystyle \nu } in Hz and <a href="/facts/Radiative_flux/oXqzX3Ij">radiative flux</a> 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 <a href="/facts/Power_law/DhAjYwwz">power law</a> 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 }}.} </p><p>Clearly the power law can only apply over a certain range of frequency because otherwise the integral over all frequencies would be infinite. </p><p>Spectral index is also sometimes defined in terms of <a href="/facts/Wavelength/kcJU0ymz">wavelength</a> 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, in which the spectral index is given by S ν ∝ ν − α . {\displaystyle S_{\nu }\propto \nu ^{-\alpha }.} </p><p>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 <a href="/facts/Thermal_emission/1PD9jtCv">thermal emission</a>, while a steep negative spectral index typically indicates <a href="/facts/Synchrotron_emission/mTsSKmRz">synchrotron emission</a>. 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. </p>