There are two identical uniform spherical planets of radius R. The first has its center at the origin of the xyz coordinate system. The second has its center at (2R, 0, 0). The planets are touching.
A projectile is launched from the "North Pole" of the first planet at (0, 0, R) with its initial velocity pointed in the direction of the vector (1, 0, 1).
Let the escape speed relative to the planet's surface be ve. Note that here, the escape escape is for a single planet in isolation (following the typical convention).
With the given launch vector, let v0 be the minimum launch speed for the projectile to reach the "North Pole" of the second planet at (2R, 0, R).
How are the two speeds ve and v0 related?
I have now started launching the projectile in the correct direction:
vector (1, 0, 1), which is an elevation angle of 45 deg w.r.t. the 2nd N. pole viewed on the horizon. (I previously shot straight up, which also, incredibly, worked well).
At about 0.8146 * v_esc, I find that the projectile hits the N. Pole after traveling for 1888 sec (37.5 min).
The revised program is
here and the table is
here.
I am preparing plots, but before that, here are some ideas:
This is a
"restricted 3-body problem" and it is simplified further by the two massive objects being stationary - and thus imparting no centripetal acceleration on the projectile (and so angular momentum Omega = 0: see the link). This helps the math - and may even allow an analytic solution via solving a 2nd order differential equation with boundary conditions (a trajectory constrained to start at N. Pole 1 and end at N. Pole 2).
But there are some cautions: Finding a trajectory requires knowing the interior mass. While I don't yet have an understanding of the origin of broll's radius for the "notational" sphere as being R*2^(1/3), I do know that the N. Pole position is _within_ the mass distribution formed by the system and so the full 2M barycentric mass is an over estimation of an internal mass felt by the projectile - which is also in a field that is also not spherically symmetric. Overestimating the mass internal to the projectile starting at (0, 0, R) likely leads to broll's making of an "upper limit".
TBC
Edited on June 29, 2019, 12:40 pm
More Results (and ideas)
