# C3 calculation in interplanetary missions

I'm starting to study Astrodynamics and I'm trying to make a Porkchop plot for a Mars mission to compare the results with actual porkchop plots which I found on the internet, so I know if my calculations are correctly done or not.  The thing is, that I don't understand how the C3 changes with the arrival date, as I understand, C3 has something to do with the escape velocity, being $$C_3=v^2-v_{esc}^2$$ But the thing is, how do I calculate this at the arrival? as I do not want to escape but intercept a planet. And, how do I represent this in the porkchop plot? The sum of both values? I don't really understand this, so if you can help me it would be greatly appreciated.

For reasons of convenience, departure excess energy is plotted as $C_3$, and arrival excess energy is plotted as $v_\infty$, where $C_3=v_\infty^2$. The reason is that launch vehicle performance is always quoted in terms of $C_3$, whereas computing things like orbit insertion $\Delta V$ or entry velocity is more straightforward with the arrival $v_\infty$. In any case, they represent the same thing.

$C_3$ and $v_\infty$ are constants of motion for any given elliptical orbit (where $C_3$ is negative) or hyperbolic trajectory (where $C_3$ is positive), which is why they are so useful. The relation is:

$$C_3=v_\infty^2=v^2-{2\mu\over r}$$

which says that given any location in the orbit, if you compute that expression in $v$, the magnitude of the velocity relative to the central body at that moment, and $r$, the distance from the center of that body at that moment, you will always get the same result. ($\mu$ is the $GM$ of the central body.)

The departure $C_3$ and the arrival $v_\infty$ are both determined by the solution to the Lambert problem for the chosen departure and arrival dates, for which there is one orbit about the Sun that connects those bodies at those times, under some constraints. The $v_\infty$ is the magnitude of the velocity difference between the trajectory at the body intercept and the velocity of the body.

• +1 I really appreciate answers that take the time to explain the meaning behind some fundamental concept that is intuitive and second nature to "orbit mechanics" but are tough for the rest of us. – uhoh Apr 10 '18 at 2:01
• Thank you very much for your answer! then, in theory if I solve Lambert's problem and get the velocity v2 and v1 (arrival and launch, respectively), if I calculate C3=v2^2-2*mu/r2 must be equal to v1^2-2*mu/r1, right? Because when I calculate it, I don't know why the result is different, and the graphic is not like a porkchop plot. I wonder where I made the mistake. – Alberto De Celis Romero Apr 10 '18 at 19:41
• No. Velocities are always relative to something. Your transit trajectories are Solar elliptical orbits, whereas your departure and arrival velocities are on hyperbolic orbits relative to the departure and arrival bodies. You need to be careful about your "v"'s and what they mean. You will need to convert from solar-relative to body-relative by subtracting the body velocity vector. And of course you need to use the appropriate $\mu$ for the current body of interest. – Mark Adler Apr 10 '18 at 22:48
• @AlbertoDeCelisRomero I've noticed you haven't accepted any answers on this site. That's certainly your prerogative, but if it's because you're not familliar with the function, then if you look at the arrows next to each answer (for up and down voting) there is also a check mark that is gray, until it is clicked. i.stack.imgur.com/h10t1.png It helps future readers know an answer has been accepted (it's display in different ways on different lists) and gives the user who wrote the answer another +15 bump in reputation. It's an additional way to say "Thank you very much for your answer!" – uhoh Mar 31 '19 at 9:42