Phi value analysis: Difference between revisions
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In [[astronomy]], the '''spectral index''' of a source is a measure of the dependence of radiative flux density on [[frequency]]. Given frequency <math>\nu</math> and [[radiative flux]] <math>S</math>, the spectral index <math>\alpha</math> is given implicitly by | |||
:<math>S\propto\nu^\alpha.</math> | |||
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 | |||
:<math>\alpha \! \left( \nu \right) = \frac{\partial \log S \! \left( \nu \right)}{\partial \log \nu}.</math> | |||
Spectral index is also sometimes defined in terms of [[wavelength]] <math>\lambda</math>. In this case, the spectral index <math>\alpha</math> is given implicitly by | |||
:<math>S\propto\lambda^\alpha,</math> | |||
and at a given frequency, spectral index may be calculated by taking the derivative | |||
:<math>\alpha \! \left( \lambda \right) =\frac{\partial \log S \! \left( \lambda \right)}{\partial \log \lambda}.</math> | |||
The opposite sign convention is sometimes employed,<ref>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.</ref> in which the spectral index is given by | |||
:<math>S\propto\nu^{-\alpha}.</math> | |||
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 | |||
:<math>B_\nu(T) \simeq \frac{2 \nu^2 k T}{c^2}.</math> | |||
Taking the logarithm of each side and taking the partial derivative with respect to <math>\log \, \nu</math> yields | |||
:<math>\frac{\partial \log B_\nu(T)}{\partial \log \nu} \simeq 2.</math> | |||
Using the positive sign convention, the spectral index of thermal radiation is thus <math>\alpha \simeq 2</math> 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#Frequency-dependent_formulation|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<ref>{{cite web|title=Radio Spectral Index |publisher=[[Wolfram Research]] |url=http://scienceworld.wolfram.com/astronomy/RadioSpectralIndex.html |accessdate=2011-01-19}}</ref> | |||
:<math>S \propto \nu^{\alpha} T.</math> | |||
==References== | |||
{{Reflist}} | |||
[[Category:Radio astronomy]] | |||
{{astronomy-stub}} | |||
Latest revision as of 19:16, 23 August 2013
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,[1] 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 radiation, 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[2]
References
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- ↑ 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.
- ↑ Template:Cite web