classical mechanics - Maths behind gravity assist

What kind of maths is behind gravity assists and in general the theory of orbits, and how deep does it go?

I am just wondering if I know enough prerequisites!

Answer

A basic understanding of what's going on gain be gained just using Kepler's laws and Newtonian mechanics.

A simple way of dealing with multiple gravitational sources is to selectively ignore all but one of them. This is the patched conic approximation. Which gravitating body is in play? That depends on whether the spacecraft is inside the gravitational sphere of influence of one of the planets. If that is the case, you ignore the Sun and all the other planets. If the spacecraft is outside all planetary spheres of influence, you ignore all of the planets. With this treatment, the spacecraft is always subject to one and only one gravitating body. The spacecraft's trajectory is a piecewise continuous set of Keplerian segments.

Suppose a spacecraft is on an elliptical orbit about the Sun that brings the spacecraft inside of a planet's sphere of influence. At the point where the planet crosses that sphere, the trick is to switch from looking at the trajectory as an elliptical orbit about the Sun to a hyperbolic orbit about the planet. This is a reference frame change. The spacecraft's position and velocity with respect to the planet are the vector differences between the spacecraft's and planet's heliocentric positions and velocities.

The hyperbolic trajectory that results will soon carry the spacecraft out of the planet's sphere of influence. This trajectory will preserve the magnitude of the spacecraft's planet-centered velocity, but not the direction. Another change of reference frames is performed as the spacecraft exits the sphere of influence, but this time back to heliocentric coordinates. While the planetary encounter doesn't change the magnitude of the planet-centered velocity, it does change the magnitude of the Sun-centered velocity.

(Update) Details

I'll first give a brief overview of the Keplerian orbit of a test mass about a central mass. The mass of the test mass is many, many orders of magnitude smaller than that of the central body. A spacecraft orbiting a planet, for example, qualifies as a test mass (mass ratio is 10^-20 or smaller). Key concepts:

• $\mu$ - The central body's gravitational parameter, conceptually $\mu = GM$ but generally $\mu$ is known to much greater precision than are $G$ and $M$.



• $\vec r$ - The position of the test mass relative to the central body.


• $\vec v$ - The velocity of the test mass relative to the central body.


• $\vec h = \vec r \times \vec v$ - The specific angular momentum of the test mass.


• $\nu$ - The true anomaly of the test mass, measured with respect to the periapsis point.


• $e$ - The eccentricity of the orbit of the test mass about the central body.


• $r$ - The magnitude of $\vec r$.


• $v$ - The magnitude of $\vec v$.


• $a$ - The semi-major axis length of the test mass's orbit about the central body.


• $r = \frac {a(1-e^2)}{1+e\cos\nu}$ - Kepler's first law.


• $r_p = a(1-e)$ - Periapsis distance; closest approach of the test mass and central body.



• $v_\infty = \sqrt{\frac \mu {-a}}$ - Hyperbolic orbit excess velocity; the speed as $r\to\infty$.


• $\frac {v^2}{\mu} = \frac 2 r - \frac 1 a$ - The vis viva equation, which provides a mechanism for computing $a$.


• $\vec e = \frac {\vec v \times \vec h}{\mu} - \frac {\vec r}{r}$ - The test mass's eccentricity vector relative to the central body.


• $\hat x_o = \frac {\vec e}{e}$ - The x-hat axis of the orbit, which points from the central body toward the periapsis point.


• $\hat z_o = \frac {\vec h}{h}$ - The z-hat axis of the orbit, which points away from the orbital plane, in the direction of positive angular momentum.


• $\hat y_o = \hat z_o \times \hat x_o$ - The y-hat axis of the orbit, defined to complete an xyz right-handed coordinate system.

In the limit $r\to \infty$, the test mass will suffer no change in speed but will be subject to a change in velocity given by $\Delta \vec v = 2v_\infty \cos \nu_m \, \hat x_o$, where $\nu_m$ is the maximum true anomaly given by $1+e\cos \nu_m = 0$. Thus $\cos \nu_m = -1/e = -1/(1-r_p/a) = -1/(1+v_\infty^2 r_p/\mu)$. The change in velocity is thus given $\Delta \vec v = -v_\infty/(1+v_\infty^2 r_p / \mu) \, \hat x_0$. Note that this says that too low or too high of a hyperbolic excess velocity both result in a small $\Delta v$. The largest $\Delta v$ for a given periapsis distance results when $v\infty = \sqrt{\mu/r_p}$ (in which case the deflection angle is 60°).

Suppose the spacecraft enters a planet's sphere of influence (a sphere of radius $r_{\text{soi}} = a_p (m_p/m_\odot)^{2/5}$ about the planet) at a position $\vec r_0$ with respect to the planet and with some velocity $\vec v_0$ with respect to the planet. The semi-major axis length, specific angular momentum vector, and eccentricity vector of the spacecraft's hyperbolic orbit about the planet can be calculated given this initial state and the planet's gravitational parameter. Per the patched conic approximation, each of these will be a constant of motion. (Note that the "length" in "semi-major axis length" is a bit of a misnomer; it will be negative in the case of a hyperbolic orbit.) The periapsis distance can then be calculated.

Some of the above calculations simplify with the additional assumption that the initial velocity is very close to the hyperbolic excess velocity. (What's one more simplifying assumption on top of the huge assumption of patched conics?) With one more simplifying assumption, that the time spent inside the planet's sphere of influence is small, the $\Delta v$ from the flyby can be approximated as impulsive.

These key simplifying assumptions give mission planning programs something that can be dealt with. This is a very large and complex search space, and some of the optimization parameters are difficult to express numerically. No matter how good a plan is in terms of low Earth departure velocity, a plan that involves a correction burn on July 4 and a planetary encounter on Christmas Day is not a good plan. No matter how good a plan is in terms of nominally low Earth departure velocity, if the plan is extremely sensitive to errors it is not a good plan.

People are still better than machines with regard to weeding out plans that get in the way of people doing what they are wont to do (e.g., taking the Fourth of July off, along with not working from a day or two before Christmas to a day or two after New Years) and people are still better than machines with regard to weeding out plans that are ultra-sensitive to errors. Mission planners still like their porkchop plots. Unfortunately, porkchop plots are computationally expensive to produce. Multiple porkchop plots that string together (i.e., a planetary gravitational assist) are extremely expensive to produce. Multiple planetary encounters (e.g., Cassini) means stringing together a lot of porkchop plots. Hence all the simplifying assumptions.

All those simplifying assumptions mean that the nominal plan is ultimately flawed. Not badly flawed, but flawed nonetheless. A solver that doesn't make all those simplifying assumptions is needed. Unfortunately, there are no practical, generic closed-form solutions to the N-body problem. The only way around this is numerical propagation. Now we can throw all kinds of kinks at the solver: multiple gravitational bodies, some of which have a non-spherical gravity field, relativistic effects, and so on. This is not something that can be done from the start. It is something that can be done to polish up the solutions from an overly-simplified mission planning perspective. Note: I am not disparaging those mission planning efforts. The mission planning search space is so large that simplifying assumptions are an absolute necessity lest we have to wait until the next millennium (985 years away) for a solution.