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I am designing an Earth-Mars transfer. The launch window is 03-Jul-2026 (departure from Earth) & 22-Jun-2028 (Mars arrival).
The declination and right ascension for the departure asymptote from earth were given in NASA reports but I am not aware of any method to calculate these for arrival into Mars.
Any help would be appreciated.
To compute the arrival asymptotes, convert the heliocentric position and velocity into the planetocentric frame (the Mars J2000 frame in your case). Then, compute the B-Plane parameters as derived here , and a decent visualization is here.
Once you are in that frame, you may compute the declination $\delta$ and right ascension $\alpha$ at arrival as follows:
$$ \delta = \sin^{-1} \frac {z} {||\mathbb{r}||}$$
$$ \alpha = \tan^{-1} \frac y x $$
If you have the arrival hyperbolic excess velocity vector $\vec{v_{\infty}}$ and its magnitude $\big|\vec{v_{\infty}}\big|$, then you can calculate the unit vector $\hat{s}$ parallel to it by doing the following:
$$\hat{s} = \frac{\vec{v_{\infty}}}{\big|\vec{v_{\infty}}\big|}$$
You can then use the algorithm to calculate the right-ascension and the declination of a certain point on the geocentric equatorial frame. An explanation of the algorithm (ALGORITHM 4.1) can be found here: https://www.sciencedirect.com/topics/engineering/geocentric-equatorial-frame
The same method is also applicable for outbound velocity vectors.
