Optimising interplanetary travel times (for a game)

[Kael42]

Calculating the optimum angle (resulting in the minimum time) for a linear-pathed spacecraft (unaffected by gravities) to leave a planet performing uniform circular motion to reach another planet undergoing uniform circular motion.

I am really struggling to work my head around this problem, I was hoping someone on here might have a crack at it, and see what they end up with.

So in this game I am working on, planets orbit a central star, performing uniform circular motion around it. The planets have various different periods (a randomised variable) measured in "steps", and different sizes (also randomised) measured in units unspecified (please identify the unit symbol used). For the purposes of the 2D game, mass is directly proportional to circumference.
Distance is also in unspecified units, so again please identify the unit symbol used.

So, call the initial planet "A", with initial position (x_A, y_A) on a circular path.
The destination planet "B", with final position (x_B, y_B) on a circular path.
The circular paths will require 2 parametric equations each, all containing time.

From this, the direction from A to B will be found, and a parametric linear equation fitting this. Assume the craft has a constant velocity, v.

The final values I require are travel time, travel distance, and the parametric linear formula for the path taken by the craft.

Did I miss anything?

Thanks, Kael.

 

[D H]

What you missed is that gravity also affects spacecraft . Since your problem is non-physical, nobody has studied it.

If you want gravity to also affect your spacecraft you might want to look into Lambert targeting. Also google the term "porkchop plot".

 

[HallsofIvy]

As D H says, a spacecraft cannot fly from one planet to another along a straight line. You might also want to look at "Hohmann transfer orbit".