[Physics] Changing Planetary Orbits with Light

[Vaniver]

I was having a conversation with a friend about their idea for a sci-fi story, but I don't know enough orbital mechanics to give more than a basic guess of what'll happen, so I figured I'd turn to you.

Consider two-body system: a spherical planet covered with a mirror (or possibly just something like snow, with an albedo very near 1) orbiting a star. The light reflecting off the planet will, on net, push it away from the star- and so it'll rotate normally, following Kepler's laws, but as if gravity were slightly weaker than it actually is (based on the luminosity / gravity relationship of the star). That's pretty easy to work through- the luminosity of incident light varies inversely with r², just like gravity, and everything is radial.

Now let's make things more complicated. Instead of covering the planet with mirrors, let's cover it with fiber optics- so we can pipe the light around to wherever we want. (Alternatively, coat the planet with solar cells, and move around electrons rather than photons.) Now, shoot all the collected light from a point on the terminator in the plane of the planet's orbit, to create a positive angular acceleration.

What happens to the orbit of the planet? All of the orbital mechanics that I know hinge on conservation of energy and angular momentum, which aren't relevant to this system (since you ignore the energy and angular momentum of departing photons). It seems obvious that the planet's angular momentum will increase. But it's not clear how much of that increase comes through increasing eccentricity, and how much comes through increasing the radius of the periapsis. (Both terms used in the context of the orbit you would have if you shut the light collection system down.)

As well, the rate at which the angular momentum increases varies inversely with r². That seems like it would favor eccentricity over periapsis.

Any insight?

[eSOANEM]

Intuitively, assuming the orbit starts circular, it will just become more and more eccentric (although the exact rate will not be trivial to calculate because, as you say, the rate of gain of angular momentum varies with r^-2) before eventually reaching a hyperbolic escape trajectory.

[Qaanol]

Well, if I kept track of everything right, here is the set of differential equations you want, if you’re okay using Newtonian gravity. Let the star have mass M and emit light at a rate with total momentum flux 4πΦ. Let the planet have mass m and cross-sectional area A. The planet’s position is then given by [r, θ], satisfying:

\begin{align}\ddot{r} &= r{\dot{\theta}}^2 + \frac{\Phi \frac{A}{m} - GM}{r^2} \\
\ddot{\theta} &= \frac{\Phi \frac{A}{m}}{r^2} - \frac{\dot{\theta}\dot{r}}{r}\end{align}

This comes from the following forces:

\begin{align}
f_{light \; in} &= \frac{\Phi A}{r^2} \hat{r} \\
f_{light \; out} &= \frac{\Phi A}{r^2} \hat{\theta} \\
f_{gravity} &= \frac{-GmM}{r^2} \hat{r} \\
f_{centrifugal} &= mr{\dot{\theta}}^2 \hat{r} \\
f_{escape} &= \frac{-m\dot{\theta}\dot{r}}{r} \hat{\theta}
\end{align}

Note that the last three of those forces are fictitious, arising only as artifacts of our frame of reference. The very last one might not be obvious, but it represents the apparent change in angular velocity as a result of changing radius. Also, I assumed the planet is far enough from the star that all the light striking it can be taken as parallel.

The initial conditions are:

\begin{align}
r(0) &= r_0 \\
\dot{r}(0) &= 0 \\
\theta(0) &= 0 \\
\dot{\theta}(0) &= \sqrt{\frac{GM}{r^3}}\end{align}

You could try solving or simulating those equations, then checking what happens when you revert back to “normal” orbital mechanics with the radius and velocity that have been achieved. I expect that as long as the contribution from light is small compared to gravity, the result will be a very-nearly-circular orbit that slowly grows in radius.

[gmalivuk]

I expect that as long as the contribution from light is small compared to gravity, the result will be a very-nearly-circular orbit that slowly grows in radius.

Yeah, this is what I was thinking as well. A large burst of prograde momentum at one point in a circular orbit will make that the periapsis of an elliptical orbit. This would make thrust on the terminator not quite prograde except for periapsis and apoapsis, so part of it would go to changing eccentricity instead of adding to the angular momentum of the planet around the star. However, the system described here doesn't involve any large bursts of momentum, so starting with a circular orbit will continue in a pretty nearly circular orbit.