Can satellites stay in orbit when the Earth is being accelerated?

In a TV show I'm watching, the people of Earth have decided to leave the solar system, and take Earth with them, so they've built a large number of powerful rocket engines on one side of the Earth (as one does) and fire them to accelerate Earth out to a larger orbit (and eventually, to the stars).

Meanwhile, some of the characters are living in a space station that is orbiting the Earth.

My question is, in a scenario like this (ignoring for now the plausibility of the rocket engines), would a space station find it difficult to maintain a stable orbit, given that the planet it is orbiting around is accelerating to a larger orbit itself?

• The Earth orbits the Sun, undergoing a constant centripetal acceleration. Satellites continue orbiting the Earth just fine despite Earth’s acceleration. Commented May 15, 2019 at 3:36
• @G.Smith, the device in the question accelerates only the earth. I wouldn't equate that to the sun, which is accelerating the satellites also. Commented May 15, 2019 at 5:52
• Yeah, i'm also not convinced, I'm pretty sure the satellites orbits relative to the earth would be affected. Commented May 15, 2019 at 5:53

For simplicity, ignore the effect of the sun. Think of a planet (like the earth) being pushed by rockets with uniform acceleration through otherwise empty space — empty except for the hopeful satellite (space station).

Answering this question in quantitative detail would probably be difficult. It would amount to studying solutions of the differential equation $$\frac{d^2}{dt^2}\vec x(t) = -\frac{GM \hat x(t)}{|\vec x(t) - \vec X(t)|^2} \tag{1}$$ where $$\vec x(t)$$ is the time-dependent position of the satellite (the thing we're solving for), $$\hat x(t)$$ is its unit-vector version, and $$\vec X(t)$$ is the prescribed time-dependent position of the planet, such as $$\vec X(t) = \vec A\frac{t^2}{2} \tag{2}$$ for a planet with acceleration $$\vec A$$. As usual, $$G$$ is the gravitational constant and $$M$$ is the mass of the planet.

I don't have any explicit solutions, but here are some guesses based on intuition:

• If the planet's acceleration is slow enough, so that $$|\vec A|$$ is much less than the magnitude of the right-hand side of (1), then the satellite will remain bound to the planet. (The adiabatic approximation probably has something to say about this, combined with the fact that orbits are stable when $$\vec A=0$$.) The orbits will probably be complicated, though. I'll bet there are no orbits that are both circular and planar, not even in a frame accelerating with the planet.

• If the planet's acceleration is large enough, so that $$|\vec A|$$ is much greater than the magnitude of the right-hand side of (1), then the satellite will either be left behind or will collide with the planet, depending on where in its would-be orbit the satellite happens to be when the planet's rocket engines are ignited.

I don't know exactly what value of the acceleration represents the boundary between these two extremes, but I'm guessing it depends on the satellite's initial conditions. Solving equations (1)-(2) numerically could be a fun exercise, and I'd be interested in seeing the results if anybody decides to try this. I'd be especially interested in knowing if my guesses are wrong.

• A simple orbit is to imagine the satellite is simply dragged behind the Earth. If the Earth is being accelerated at 1 $g$, placing the satellite near the surface would have it also pulled at 1 $g$ and keep up with the Earth, but perhaps this is unstable. I can't prove but can hand-wave that an orbit that goes around the Earth is not possible when it's accelerating; "errors" build up that eventually leave the satellite behind or sling it away at high speed. Commented May 20, 2019 at 22:40
• @HiddenBabel The dragged-behind example is a good one. It's certainly unstable, and of course the meaning of "1 g" needs to be adjusted for the satellite's altitude. In fact, the accelerating-earth scenario is equivalent (via a time-dependent coordinate transform) to a stationary-earth scenario with a constant-and-uniform force-field that affects only the satellite. Maybe that perspective makes the problem easier to analyze. Commented May 21, 2019 at 0:59