dyad.orbital_elements_from_modified_delaunay_elements

dyad.orbital_elements_from_modified_delaunay_elements(J_pi, J_Omega, J_lambda, Theta_pi, Theta_Omega, Theta_lambda, m)

Return the orbital elements given the modified 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

\[\begin{split}a &= \dfrac{J_{\lambda}^{2}}{\mathrm{G}m}\\ e &= \sqrt{1 - \left(1 - \dfrac{J_{\varpi}}{J_{\lambda}}\right)^{2}}\\ \theta &= \theta(\Theta_{\lambda} + \Theta_{\varpi})\\ \Omega &= -\Theta_{\Omega}\\ i &= \cos^{-1}\left( 1 - \dfrac{J_{\Omega}}{J_{\lambda} - J_{\varpi}} \right)\\ \omega &= -\Theta_{\varpi} + \Theta_{\Omega}\end{split}\]

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

Parameters:

J_piarray-like

First modified Delaunay action

J_Omegaarray-like

Second modified Delaunay action

J_lambdaarray-like

Third modified Delaunay action

Theta_piarray-like

First modified Delaunay angle (longitude of pericentre)

Theta_Omegaarray-like

Second modified Delaunay angle (longitude of the ascending node)

Theta_lambdaarray-like

Third modified Delaunay angle (mean longitude)

marray-like

Central mass

Returns:

restuple

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

Examples

Scalar parameters.

>>> dyad.orbital_elements_from_modified_elements(1., 0., 0., 0., 0.,
...     0., 1.)
array([3379.38068342,    0.        ,    0.        ,    0.        ,
          0.        ,    0.        ])

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_modified_delaunay_elements(a, e,
...     Omega, i, omega, theta, m)