dyad.modified_delaunay_elements_from_orbital_elements

dyad.modified_delaunay_elements_from_orbital_elements(a, e, Omega, i, omega, theta, m)[source]

Return the modified 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 modified Delaunay elements are

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

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:

restuple: Modified Delaunay elements, \((J_{\varpi}, J_{\Omega}, J_{\lambda}, \Theta_{\varpi}, \Theta_{\Omega}, \Theta_{\lambda})\).

Examples

Scalar parameters.

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