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Spacecraft and an object are rotating on orbit. The spacecraft is controlled via the velocity vector change, may change its velocity to intersect another object.
Needed to find the time and coordinates of meeting, as both of them are dynamic objects. The optimal solution- minimum velocity change.
Ideas?
Another question: What is the maximum value of the delta V? It depend on the fuel, according to Cialcovski equation. Obviously, its value should be restricted by something else. Isp? And how?
As far as I know, no analytical solution to the problem has been found, and it's normally obtained through numerical optimization - like the Simplex method.
First, let's take care of the Tsiolkovsky's rocket equation: delta-V depends on specific impulse, dry mass of the craft (without fuel), and amount of fuel (or wet mass of the craft.) Since the first two don't change during the flight, only the third is actually a variable, not a constant. Still, delta-V may convert to different change of trajectory, and different change of energy, depending on what maneuver and where is performed - in particular, look up Oberth Effect.
Now for your problem: you're asking for minimal burn that creates a fly-by encounter between two arbitrary initial orbits; encounter speed doesn't matter, the crafts don't need to dock to each other.
You will need orbital motion equations of both crafts - for one central body you can use the conic section formulas / orbit equation, for a more complex system you'll use hamiltonian mechanics to derive equations of motion in the non-uniform gravitational field. You'll also need a third orbit, a generalized parametric equation for one, which will be your transfer orbit.
Then you need to bind the three: write a set of equations where:
Add an equation of delta-V of the departure burn: difference of velocities at the moment of transfer burn between the initial orbit and the transfer orbit.
Your set of equations is under-constrained: your free variables are time of departure, and departure delta-V (3 variables, a vector). This set of equations then is the subject to numerical optimization: finding such parameters that the magnitude of the departure burn is least - and where TIME IS REASONABLY CONSTRAINED!
This last part will be a big headache. That departure burn must occur before encounter, this is obvious. But the encounter largely depends on relative position of the two craft, and if their orbits aren't resonant (orbital period ratio isn't a simple fraction) one can always find a more optimal departure burn - if it's to happen enough millions of years away.
Simply, the best, most optimal transfer exists, which requires the craft to be in certain alignment to each other. Least delta-V to reach nearest point of the target orbit, and the target just happens to be there at the point of time when you arrive. Usually it won't - it will be in an entirely different place of its orbit. But if you wait long enough, you will be aligned to encounter it pretty close to that optimal point, and you will waste very little fuel vs the optimal for that adjustment. If you wait longer though, it will pass even closer, so you can do this more optimally. And so on, to infinity.
