celeries.perham3pla.PerHam3pla.angle

PerHam3pla.angle(k)

Returns the terms of the perturbative part of the Hamiltonian proportional to \(e^{i(k_1 \lambda_1 + k_2 \lambda_2 + k_3 \lambda_3)}\). A factor \(m_1 m_2 n_3 \Lambda_3 / m_0^2\), common to all terms, is left out.

The returned serie depends on \(X_i\), \(Y_i\), \(\bar{X}_i\), \(\bar{Y}_i\), for \(1 \leq i \leq 3\). These quantities are defined in Laskar&Robutel, 1995 (Eq. 13 and Appendix).

If ev is False, the output also depends on the mean motions \(n_j\) for \(1 \leq i \leq 3\), as well as on alpha12 \(=\alpha_{12} = a_1/a_2\), on alpha23 \(=\alpha_{23} = a_2/a_3\), on sqalp12 \(= \sqrt{\alpha_{12}}\) and on sqalp23 \(= \sqrt{\alpha_{23}}\).

If ev is False, the returned serie depends on the Laplace coefficients bs0_ij \(= b_{s/2}^{(0)}(\alpha_{ij})\) and bs1_ij \(= b_{s/2}^{(1)}(\alpha_{ij})\), for \(1 \leq i < j \leq 3\), where, at degree \(r\) in eccentricity and inclination, \(s\) is an odd integer verifying \(1 \leq s \leq 3 + r + (r \mod 2)\).

If disregardInd is True, the returned serie does not depend on \(\sqrt{\alpha_{23}}\) (but it may still depend on \(\sqrt{\alpha_{12}}\) due to the substitution \(\Lambda_1 m_2/(\Lambda_2 m_1) = \sqrt{\alpha_{12}}\)).

Author : Jeremy Couturier. https://jeremycouturier.com

Parameters:

k: Tuple of int

Coefficients \((k_1, k_2, k_3)\) of the mean longitudes for the 3-planet angle.

Returns:

RSeries

The contribution to the perturbative part of the Hamiltonian for the required 3-planet angle.