Projectile angle (straight line) between 2 planets

[thomasvt]

TL;DR

Some tips in solving a linear math problem to predict the shooting angle of a projectile from one planet in orbit to another in orbit around the same center.

I have a planetary system with planets orbitting a central star (circular orbits). I want to shoot a projectile P in a straight line from planet A to B and need to calculate the angle or vector to shoot the projectile P. I know the straight line of the projectile is physically not correct, but it will probably get a lot more complex if add realistic orbits for P.

I know the orbit of the source planet A doesn't matter, since it has no influence on the problem once P has departed. So, I need to find the point in time t1 where the position of P equals the position of B.

I think that 2D problems can be split in 2 1D problems, so let's focus on the X-coordinate:

* the projectile's position in time:
xp = xp0 + v.t (xp0 = initial X position of P at t0)
* the planet B's angular position:
α = α0 + ω .t (α0 initial angle around the sun at t0, ω is angular velocity)
* the planet B's position:
xb = cos(α) = cos(α0 + ω .t)

I also think, I have to solve for t where xp == xb:

xp0 + v.t = cos(α0 + ω .t)

But i have no idea how to get the t out of that cosine :)

Thanks for any suggestion!

 

[thomasvt]

I just thought of the fact that I also don't know what the speed of P (v) is.. only that |v| is constant, but the angle and therefore its (x, y) are also unknown. damn :)