Insert the <photon temperature multipole> expansion into the Fourier-space transport equation. The <Legendre polynomial recurrence relation> turns multiplication by $\mu$ into nearest-neighbour multipole couplings, while <Orthogonality of Legendre polynomials> projects onto a fixed $\ell$. The monopole projection has no collision term because Thomson scattering conserves photon number:
$$
\boxed{\Theta_0'+k\Theta_1=\Phi'}.
$$
The dipole projection receives the electron-velocity source,
$$
\boxed{3\Theta_1'+k(\Theta_2-\Theta_0)
=k\Psi-\Gamma(3\Theta_1+v_e)}.
$$
For every $\ell\geq2$, the collision term damps the anisotropic multipole and the <photon Boltzmann hierarchy> is
$$
\boxed{
\Theta_\ell'+\frac{k}{2\ell+1}
\left[(\ell+1)\Theta_{\ell+1}-\ell\Theta_{\ell-1}\right]
=-\Gamma\Theta_\ell}.
$$
Thus the <Free-streaming photon Boltzmann equation> moves angular structure between neighbouring multipoles, whereas <Thomson scattering> suppresses all multipoles above the dipole in the <tight-coupling approximation>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 312 / 3 / b / ii / Solution

{c}

The Legendre polynomials are orthogonal on $[-1,1]$ with unit weight:
$$
\int_{-1}^1P_\ell(x)P_m(x)\,dx
=\frac{2}{2\ell+1}\delta_{\ell m}.
$$


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 Incoming links: Orthogonality of Legendre polynomials