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Given the orbital elements of two satellites and the delta-V capabilities of one of the satellites, how can you calculate the maneuvers required to have the earliest possible interception?

For reference of what I'm trying to do: You have two spacecraft, and one of them is attempting to fire a missile at the other. The orbits are known, as are the properties of the missile; the angle and timing are what's needed.

EDIT: The satellites are in the same orbital plane (I know it would be very different if they weren't).

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    $\begingroup$ Does the missile have a simple ballistic flightpath after the initial powered flight phase, or is it capable of mid-course corrections? Especially if unpowered ballistic flight, does the target satellite have a delta-v budget that allows it to move out of the way? $\endgroup$ – a CVn Jul 21 '16 at 8:56
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    $\begingroup$ "the fuel is a negligible percentage of the missile's weight" Congratulations! $\endgroup$ – a CVn Jul 21 '16 at 15:16
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    $\begingroup$ The "missile" version is much easier than the rendezvous which is a traditional problem, as only positions must be matched, and not the velocities. $\endgroup$ – SF. Jul 21 '16 at 16:36
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    $\begingroup$ I mean: if we simplify the problem: one central mass (no-gravitational-mass satellites), we can write the equations of motion of all the objecs, through quite moderately complex Lagrangians or Hamiltonians. Then we can quite easily add launch point equations, (position, velocity of origin sat. equal to these of missile) impact point equations (position of missile and target identical), and solve the set of equations, and we get the equation of motion. Except it will be a 2nd degree differential equation of about... 23 variables, and make every mathematician cry. $\endgroup$ – SF. Jul 22 '16 at 0:29
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    $\begingroup$ I am voting to close this question as too-broad as it is far to complicated to answer. Even the much easier subset of this question: "Can I transfer between two planar orbits given x m/s $\Delta v$" is unknown given arbitrary orbits, and this question complicates this by 1. requiring a specific point in the orbit (interception), and 2. the fastest way to do it. $\endgroup$ – Hohmannfan Jul 22 '16 at 10:36
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Assuming a restricted two-body problem (neglecting the mass of the satellites and no other forces at work) the problem can be solved with search combined with a solver for the Gauss' problem.

The gauss problem is the following: Given two positions and the time between the positions, find the velocites at both positions, which in other words finds the orbit. We will denote this as $(v_1, v_2) = GaussProblem(r_1, r_2, t)$.

So how can we use the solver to intercept the second satellite? This is done by assuming an intercept time ($t$) (we will iterate to find the best one later). So what we do now is that we calculates the position of the second satellite at the intercept time, which I assume you already can do (using numerical integration or solver for the Kepler problem). I will denote this as $(r', v') = KeplerProblem(r, v, t)$.

Now that we have two positions (the current position of the first satellite, the position of the second satellite at the intercept time), we can use the solver for the Gauss problem to find what the velocity at the first position should have been, if the first satellite was on the intercept orbit.

So our maneuver is simply then: $\Delta v = v_1 - v_{satellite 1}$.

The algorithm is the following:

  1. For each $t \in [min, max]$ (take some range)
  2. $(r', \_) = KeplerProblem(r_{satellite 2}, v_{satellite 2}, t)$
  3. $(v_1, v_2) = GaussProblem(r_{satellite 1}, r', t)$
  4. $\Delta v = v_1 - v_{satellite 1}$
  5. Select the maneuver that best matches your parameters (minimum intercept time given delta velocity requirements).

Some notes:

  1. For a given problem instance, a solver for the gauss problem may fail.
  2. The times are relative, not absolute.
  3. The maneuvers are assumed to be executed at the current time. Calculating the intercept at a later time may yield a better intercept in terms of time and deltaV.

Gauss Problem

I'm not 100% sure which is the correct name for the problem, as in Fundamentals of astrodynamics by Bate, Roger R., Donald D. Mueller, and Jerry E. White denotes this as the Gauss problem, others as Lambert's problem.

For a method for solving the problem, you should either look in the mentioned book, which contains several methods, or look at the following links: http://aerospacengineering.net/?p=1614, http://www.dept.aoe.vt.edu/~cdhall/courses/aoe4134/Apiteration.pdf, https://en.m.wikipedia.org/wiki/Lambert%27s_problem.

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The only answer for this is to build a simulator, and test out a variety of things. Here's a few things to keep in mind:

  1. A lower orbit is a faster orbit, which will allow you to catch up to your target. A higher orbit is slower, so you will slow down to intercept the object.
  2. It seems you want a high speed intercept, with some given initial delta v, and nothing else. That seems highly unlikely to work, as you'd have to be spot on, which is very difficult to achieve.

Bottom line, I would take and fire a missile, see how far it is away from the target when the intercept occurs, and then move it in time until it intersects with the target. Then try firing at different angles to see what happens. Iterate until you get the best solution.

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