Comets and Dyson spheres

Factual Questions
1

So the Overlords come to our solar system and, out of the goodness of their hearts, install a Dyson sphere for us overnight. They don’t mention exactly how massive it is, but it’s as big around as Earth’s orbit and it’s probably pretty massive. They mention in passing as they leave that they removed all of the planets out past Earth as part of the construction, but they did the job in a hurry and didn’t bother to clear out all the Earth-crossing comets or asteroids.

Still, not to worry: the sphere is made up of panels that can be shifted around to allow such bodies to pass through on the way in our out without hitting anything, so no Fists-of-God to mess up the nice landscaping.

But what happens to the comets and asteroids over the ‘long haul’ now? The Dyson sphere is massive, but when a comet is inside it, the sphere exerts no net gravitational force on the comet, so the comet’s motion is controlled by the Sun’s gravity. Outside the sphere, though, the comet feels the gravity of Sun + Sphere, so it’s orbit must get warped, right?

Is it possible to have a ‘stable’ cometary orbit under such conditions? It’s not a two-body problem anymore, so it wouldn’t be an ellipse, but maybe something like two ellipses bolted together? My gut says that the orbits would become quite chaotic and that no comet or asteroid would make more than one pass through the sphere before being ejected from the system or falling into the Sun.

What does the math say?

2

If we’re neglecting all other solar system objects besides the Sun and the sphere, and we’re neglecting interactions between the comets, then it shouldn’t go chaotic. Each comet will have the same aphelion distance and the same perihelion distance on every orbit. The orbits will not be closed, but will spirograph around, but that’s not really a problem for stability.

3

I have learned something new today – before this I only knew about the Dyson ball.

4

Thanks, Chronos. I like the description of the orbit as ‘spirographing’ - a great way to describe precession of the perihelion. I guess I can see how the orbit would sort of ‘refract’ as it passed through the shell, with the comet returning along a different heading w/r/t the fixed stars on each subsequent pass. I suppose it remains a two-body problem even though the effective mass of one body changes over time.

5

In a way, the biggest change would be the lack of disruption. If Jupiter et al are now components of the Dyson sphere, then net gravity would be very similar even when the comets are far from the sun. You’d have eliminated all those little localized high gravity zones that would be experienced by a comet passing near to a planet like Jupiter, and those are would be most likely to cause a disruption in orbit.