Non-Perturbative Rayleigh-Jeans Limit

Expand a blackbody spectrum  about zero frequency to order [Graphics:SoftComptongr478.gif]

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substitute this into the spectrum-dependent part of the collision integral (N.B. this is only good to order [Graphics:SoftComptongr481.gif]

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or collecting the coefficients of [Graphics:SoftComptongr484.gif], [Graphics:SoftComptongr485.gif], [Graphics:SoftComptongr486.gif], [Graphics:SoftComptongr487.gif], [Graphics:SoftComptongr488.gif]

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In the low &agr;=0 (soft photon) limit, the rest of &Dgr; in the integrand is in [Graphics:SoftComptongr491.gif] which has the symmetry property [Graphics:SoftComptongr492.gif]. (N.B. in terms of Mathematica functions [Graphics:SoftComptongr493.gif] ). If, for negative &Dgr;, we make the substitution &Dgr;->[Graphics:SoftComptongr494.gif]  we can combine the negative and positive parts of the integral, and only integrate over positive &Dgr;.  Note that this mapping is the same which maps the fractional change in energy for a scattering to the fractional change in energy for the inverse scatteing:

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The coefficients for the positive and negative parts combined are

[Graphics:SoftComptongr497.gif]

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Thus a Taylor expansion of the distorion contains only odd powers.only the odd powers.  Here we consider only the lowest order contribution ~[Graphics:SoftComptongr499.gif], which we call the Rayleigh-Jeans distortion

The required integral is (note we drop the - sign!)

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or simplifying

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The Taylor series for small &bgr; to 20th order is

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which does very well

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Now plotting the error

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As a function of &ggr; this function is

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We also define a Taylor series in (&ggr;-1) by re-expanding the Taylor series in &bgr; (N.B. Mathematica seems to have a hard time directly doing a series in (&ggr;-1))

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Now we verify that 20th order in &bgr; is the same as 10th order in (&ggr;-1)

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We see that the series fails miserably for only moderate values of &ggr;

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Direct numerical evaluation of RJfuncOfGamma is noisy for &ggr; close to 1, whereas the Taylor series is not.  Therefore we see that is useful to switch between the two at &ggr;=1.01

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Note that RJfuncOfGamma[&ggr;] also does not behave well for large arguments

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Define a numerically accurate version of the function

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This will get underflow errors for &THgr;>1 - but this is to be expected.

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To get a Taylor series expression for RJfactor[&THgr;], we may use the series expansion RJfuncOfGammaSeries10[&ggr;].  Although this series is less well-behaved than RJfuncOfSeries10[&bgr;] , when integrated over a thermal distribution it can be expressed as a sum of [Graphics:SoftComptongr551.gif] which is equivalent ot a sum of modified Bessel functions. These are simpler to deal with than the hypergeometric functions which arise from the integrals [Graphics:SoftComptongr552.gif]  which appear with the &bgr; expansion.  The fomer integrals are already "defined" in ThermalIntegral[n,U]=[Graphics:SoftComptongr553.gif] for n>=0. Note U=[Graphics:SoftComptongr554.gif]

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which is indeed a polynomial in &ggr;.  Substituting in the pre-computed values of the integrals for each power of &ggr; (N.B. the following method of substitution doesn't work if there is a constant ([Graphics:SoftComptongr564.gif]) term)

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We see that this function is a good approximation for &THgr;<0.1, but lousy at larger temperatures

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Substitute the asymptotic series expansion for the Bessel functions

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and Taylor expand in temperature (N.B. we have checked that using a 10th order asymptotic expansion for the Bessel functions gives the same 10th order Taylor expansion as if we used higher order asymptotic expansions)

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Now a plot of the fractional error for each level of approximation

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A program which computes a table of RJfactors and the Taylor series approximants to it at different temperatures.  The 1st column is gives the electron temperatures used (in keV), the 2nd column the true RJfactor (computed numerically), the 3rd column the 1st order Taylor series, the 4th column the 2nd order Taylor expansion, etc.  There are Npoints different temperatures in the table, between at TmnkeV and TmxkeV, logarithmically spaced.  There maximum order Taylor series is OrderMax    

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A program which computes a table of the log of the relative error at different orders of the Taylor expansion at different temperatures.  The 1st column is gives the electron temperatures used (in keV), the 2nd column the log relative errors of the 1st order Taylor series, the 3rd column the log relative errors of the 2nd order Taylor expansion, etc.  There are Npoints different temperatures in the table, between at TmnkeV and TmxkeV, logarithmically spaced.  There maximum order Taylor series is OrderMax    

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This doesn't work

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Compute the &THgr; where the order n series becomes less accurate than the order n-1 series, for n=2,10

[Graphics:SoftComptongr621.gif]

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or in keV

[Graphics:SoftComptongr623.gif]

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Here is the minumum relative error at these temperatures

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Define futher truncated series

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This sequence should approach 1

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Document converted by Mathematica of Wolfram Research