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We present Chapman--Enskog and Hilbert expansions applied to the $\BigO(v/c)$Boltzmann equation for the radiative transfer of neutrinos in core-collapsesupernovae. Based on the Legendre expansion of the scattering kernel for thecollision integral truncated after the second term, we derive the diffusionlimit for the Boltzmann equation by truncation of Chapman--Enskog or Hilbertexpansions with reaction and collision scaling. We also give asymptoticallysharp results obtained by the use of an additional time scaling. The diffusionlimit determines the diffusion source in the \emph{Isotropic Diffusion SourceApproximation (IDSA)} of Boltzmann's equation for which the free streaminglimit and the reaction limit serve as limiters. Here, we derive the reactionlimit as well as the free streaming limit by truncation of Chapman--Enskog orHilbert expansions using reaction and collision scaling as well as timescaling, respectively. Finally, we motivate why limiters are a good choice forthe definition of the source term in the IDSA.

Derivation of the Isotropic Diffusion Source Approximation (IDSA) for Supernova Neutrino Transport by Asymptotic Expansions