dyad.orbital_elements_from_delaunay_elements

dyad.orbital_elements_from_delaunay_elements(J_1, J_2, J_3, Theta_1, Theta_2, Theta_3, m)

Return the orbital elements given the Delaunay elements

Consider a body moving on an elliptical orbit in a gravitational central potential generated by a central mass of \(m\). The orbital elements are

\[\begin{split}a &= \dfrac{J_{3}^{2}}{\mathrm{G}m}\\ e &= \sqrt{1 - J_{2}^{2}/J_{3}^{2}}\\ \theta &= \theta(\Theta_{3})\\ \Omega &= \Theta_1\\ i &= \cos^{-1}(J_{1}/J_{2})\\ \omega &= \Theta_{2}\end{split}\]

where \(J_{1}/(\mathrm{AU}^{2}~\mathrm{d}^{-1}) \in (-\infty, \infty)\) is the first Delaunay action, \(J_{2}/(\mathrm{AU}^{2}~\mathrm{d}^{-1}) \in (0, \infty)\) is the second Delaunay action, \(J_{3}/(\mathrm{AU}^{2}~\mathrm{d}^{-1}) \in (0, \infty)\) is the third Delaunay action, \(\Theta_{1} \in (-\infty, \infty)\), is the first Delaunay angle (longitude of the ascending node), \(\Theta_{2} \in (-\infty, \infty)\) is the second Delaunay angle (argument of pericentre), \(\theta(\Theta_{3}) \in (-\infty, \infty)\) is the true anomaly corresponding to \(\Theta_{3} \in (-\infty, \infty)\), the third Delaunay angle (mean anomaly) and where \(|J_{1}| < J_{2}\) and \(J_{2} < J_{3}\).

Parameters:

J_1array-like

First Delaunay action

J_2array-like

Second Delaunay action

J_3array-like

Third Delaunay action

Theta_1array-like

First Delaunay angle (longitude of the ascending node)

Theta_2array-like

Second Delaunay angle (argument of pericentre)

Theta_3array-like

Third Delaunay angle (mean anomaly)

marray-like

Central mass

Returns:

restuple

Orbital elements \((a, e, \Omega, i, \omega, \theta)\)

Examples

Scalar parameters.

>>> dyad.orbital_elements_from_delaunay_elements(1., 0., 0., 0., 0.,
...     0., 1.)

Array-like parameters defining multiple orbits.

>>> J_1, J_2, J_3, Theta_1, Theta_2, Theta_3, m  = [1., 1.],
...     [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [1., 1.]
>>> dyad.orbital_elements_from_delaunay_elements(a, e, Omega, i,
...     omega, theta, m)