Expanding the <disturbing function> in <orbital elements> gives a Fourier series of the form
$$
\mathcal R_1=
\sum_{\mathbf j}C_{\mathbf j}(a_1,a_2),
e_1^{m_1}e_2^{m_2}
\sin^{n_1}\!\frac{I_1}{2}
\sin^{n_2}\!\frac{I_2}{2}
\cos\Phi_{\mathbf j},
$$
where
$$
\Phi_{\mathbf j}=j_1\lambda_1+j_2\lambda_2
+j_3\varpi_1+j_4\varpi_2+j_5\Omega_1+j_6\Omega_2.
$$
The radial coefficients are combinations of <Laplace coefficients>. Rotational and reflection symmetry impose the <D'Alembert characteristic>: the integer coefficients sum to zero, nodal coefficients obey a parity rule, and each harmonic begins at the corresponding order in eccentricities and inclinations.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 316 / 4 / ii / Solution

{c}
{title2=$b_s^{(j)}(\alpha)$}

A Laplace coefficient is the Fourier coefficient
$$
b_s^{(j)}(\alpha)=\frac1\pi\int_0^{2\pi}
\frac{\cos(j\psi)\,d\psi}{(1-2\alpha\cos\psi+\alpha^2)^s}.
$$
It packages the radial dependence of a disturbing-function expansion.


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