Interstellar: Why is Endurance still in "orbit"?
According to the flight plan in Interstellar the Endurance never entered orbit around <PERSON>'s planet, which is further evidenced by the time dilation not experienced by Romilly. So how is it that after a significant delay of 23 years, 4 months, 8 days the main characters still managed to fly the Ranger back up to space to find the Endurance? Does there exist a "synchronous" orbit at different radii from the same center of gravity? Otherwise an insane amount of fuel would be needed to actively adjust the orbit of the Endurance around Gargantua synchronized with <PERSON>'s planet and yet far enough to avoid significant time dilation for 23 years!
Original line around 1:04:29:
Cooper: "Here's Gargantua. Here's <PERSON>'s planet. Instead of taking the Endurance into orbit around <PERSON>'s planet, which would conserve fuel, but we would lose a lot of time. What if we take a wider orbit around Gargantua, parallel with Miller's planet outside of this time shift, to here? Then we take the Ranger down, we get Miller, we get her samples. We come back, we analyze, we debrief. 275
We're in, we're out. We lose a little fuel, but we save a whole lot of time."
Edit: Assuming a circular orbit, for each angular velocity, there exist a unique radius for that orbit. (Actually two, but they are in opposite directions.) The calculation is rudimentary in Newtonian mechanics equating centripetal force with gravitational force, giving r^3 = GM / w^2 where w is angular velocity with linear velocity v = rw. If you alter forward velocity to match angular velocity, you enter a completely different orbit at a different angular velocity, unless you actively expend fuel to compensate. Therefore the Endurance would not be in-sync with <PERSON>'s planet around Gargantua without spending fuel. And even though ordinary satellites do have to actively manage their orbit, they are merely compensating for friction to maintain the same orbit. Hence to clarify my question: Is this a plot hole, or does there exist an elliptical orbit that admits an absolutely wider distance from another planet's orbit which also has roughly identical angular velocities about the C.G.? Perhaps after relativistic effects are factored in? (By "absolutely" I mean no crossover at all, otherwise time dilation.) 319