dyad.delaunay_elements_from_orbital_elements

dyad.delaunay_elements_from_orbital_elements(a, e, Omega, i, omega, theta, m)

Return the Delaunay elements given the orbital elements

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

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

where \(a/\text{AU} \in (0, \infty)\) is the semimajor axis, \(e = (0, 1)\) is the eccentricity, \(\Omega \in [0, 2\pi)\) is the longitude of the ascending node, \(i \in (0, \pi)\) is the inclination, \(\omega \in [0, 2\pi)\) is the argument of pericentre, and \(\mu(\theta) \in [0, 2\pi)\) is the mean anomaly corresponding to the true anomaly, \(\theta \in [0, 2\pi)\).

Parameters:

aarray-like

Semimajor axis

earray-like

Eccentricity

Omegaarray-like

Longitude of the ascending node

iarray-like

Incination

omegaarray-like

Argument of pericentre

thetaarray-like

True anomaly

marray-like

Central mass

Returns:

reslist

Delaunay elements, \((J_{1}, J_{2}, J_{3}, \Theta_{1}, \Theta_{2}, \Theta_{3})\).

Warning

Note that \(e \neq 0\) and \(i \neq 0\).

Examples

Scalar parameters.

>>> dyad.delaunay_elements_from_orbital_elements(1., 0., 0., 0., 0.,
...     0., 1.)
array([0.017202098944262, 0.017202098944262, 0.017202098944262,
       0.               , 0.               , 0.               ])

Array-like parameters defining multiple orbits.

>>> a, e, Omega, i, omega, theta, m  = [1., 1.], [0., 0.], [0., 0.],
...     [0., 0.], [0., 0.], [0., 0.], [1., 1.]
>>> dyad.delaunay_elements_from_orbital_elements(a, e, Omega, i,
...     omega, theta, m)
array([[0.017202098944262, 0.017202098944262, 0.017202098944262,
        0.               , 0.               , 0.               ],
       [0.017202098944262, 0.017202098944262, 0.017202098944262,
        0.               , 0.               , 0.               ]])