Ballistic Motion in Space with One Gravitating Body.

Question

I had an interesting math problem presented to me some time ago by a friend (he stated it in non-mathematical terms). At what angle would you launch a projectile from a spaceship/satellite such that it left that object and went on to hit another orbiting object? Then as a supplemental question he asked at what angle would you launch that projectile to hit the other orbiting object in the least amount of time?

I assumed that the objects were only acted upon by a single spherically symmetric mass distribution such that I could treat it as a one-body problem for each object. Further, I assumed it all took place in the plane with a polar coordinate system so that I ended up with this simple system of nonlinear autonomous ODE's,

$$ \begin{bmatrix} \frac{d \theta}{dt} \\ \frac{dv}{dt}\\ \frac{dr}{dt} \end{bmatrix} = \begin{bmatrix} \frac{h}{r^{2}} \\ \frac{h^{2}}{r^{3}} -\frac{\mu}{r^{2}} \\ v \end{bmatrix}.$$

Where the inital conditions for the projectile would be $\{ \theta_{i} , r_{i}, v_{\beta}\cos(\phi - \theta_{i}) + v_{i}\}$ with an $h_{\phi} = r_{i} (v_{\beta}\sin(\phi - \theta_{i}) + v_{\theta i})$ and the target objects' initial conditions are $\{ \theta_{i}^{'} , r_{i}^{'}, v_{i}^{'}\}$ with $h'= r^{'}_{i} v_{\theta i}^{'}$. Where $\phi$ is the launch angle from the polar axis while $v_{\beta}$ is the magnitude of the projectiles velocity (which I assume to not change, only its launch direction) and $\mu$ is a constant relating to the gravitational field strength of the attracting object. Below is a picture depicting the general initial and final conditions.

After all these preliminaries, i'm basically asking if there is a variational calculus or other simpler solution to solving this problem as perhaps a boundary problem of sorts mixed with an initial ODE problem. That is, aside from computationally pouring on through thousands of trajectories with minutely differing $\phi$'s then numerically guessing at the appropriate approximate launch angle or angles that solve my question(s).

Which. . . isn't what I exactly want to do and I desire to know if there is an equation, a single or system of ODE's, that I could solve themselves for this launch angle which gives a least time of travel or gives launch angles that would lead to a hit (irrespective of the time of travel). If you can help in anyway, this would be most appreciated. I'm a sophomore College student with little knowledge of solving robustly ODE's or even programming solvers for them.

Answer 1

So, $v_\beta$ is fixed? There's just two parameters, $\phi$ and the launch time $t_0$?

For any given $\phi$, the set of $t_0$ which results in a collision will be a discrete set. The reverse is not quite true, but is true if you insist the collision occur within some time limit, or some finite number of orbits - for any given $t_0$, the set of $\phi$ that results in a collision within [time limit] is discrete.

Let's say you fix $t_0$ and solve for $\phi$. Once you have one solution, the travel time is determined, and you can (in principle) find the derivative of the travel time with respect to $t_0$. This allows you to use efficient optimisation methods (such as conjugate gradient), even if you can't solve it analytically.

Finding a neat representation of orbits (such as polar coordinates) really helps. However, the polar coordinate representation isn't the neatest. There's another way (and I can't find a link to it, alas) where (by modifying the time variable) all orbits become great circles on a 4-dimensional sphere $x^2+y^2+z^2+T^2=r^2$. If you can find that, it might make the problem even easier to solve, but alas, I can't help you find it.

Answer 2

Another approach is to assume all trajectories lie the same plane, and use Kepler's laws to turn this into a geometry problem. While it may be exactly solvable in some specific cases, after an initial dive it looks like this approach leads to a system of equations that can only be solved numerically. Still, that might be far preferable to numerically solving some ODEs.

Without getting into the messy algebra, the main tools of this approach would be that

• All the orbits are conic sections. In particular, the target is traveling in an ellipse (as is the launcher, less importantly), and the projectile has discrete possibilities of traveling in an ellipse or hyperbola (or, as a limiting case, a parabola). It seems like assuming the target is traveling in a circle rather than a general ellipse is a valid simplifying assumption, but doesn't make much of a difference for numerical solutions.
• Kepler's second law states that equal areas are swept by orbits in equal times. With not too much work, you can relate the area between the lines OA from the origin to the initial point A, OB, and a conic section C from A to B (call that TC). This could be generalized if the objects are allowed to complete one or more orbits before colliding, but that's quite likely to not be the least-time solution.
• Thus, the spatial coordinates of the collision will be at either an (Ellipse, Ellipse) intersection or (Hyperbola, Ellipse) intersection (or (Parabola, Ellipse), but that's a special case). Take the case where the projectile is launched into an elliptical orbit at fixed initial point A, and must hit a target traveling on a circular orbit from initial point B. The ellipse has three parameters in general (semi-major and semi-minor axes, and an angle of rotation). The total set of equations to solve is thus, schematically,

Ellipse intersects A
Ellipse intersects C
Circle intersects C
T[Ellipse](A,C) = T[Circle](B,C) = DeltaT

These are essentially four equations (as the first has two components) with five unknowns: the two coordinates of the intersection point C, the three parameters of the ellipse. So, one parameter - e.g., one of the ellipse parameters - can be scanned over, and a solution (if any) and corresponding DeltaT found for each value. This would have to be repeated for the other cases (where the projectile travels in a hyperbola or a parabola).