- #1

kimbyd

Science Advisor

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- Summary:
- In the two-body problem, circular orbits can be stable with a cosmological constant. But can elliptical orbits?

For background, consider this paper, which describes circular orbits for the two-body problem in the presence of a cosmological constant:

https://arxiv.org/abs/1906.05861

What they describe is a system with three regimes of behavior: stable circular orbits below a certain radius, unstable circular orbits above that radius but below another, and no orbits beyond that second radius.

What I'm wondering is: do elliptical orbits screw this up? My searches so far haven't come up with an answer. My reasoning for why elliptical orbits might always exhibit instability in the presence of a cosmological constant starts with the circular case. In particular, it's possible to have a stable circular orbit for any spherically-symmetric potential with the right derivatives.

But an elliptical orbit spends some of its time closer to the center of mass, some of its time further away. The elliptical orbit always spends more of its time further away from the center of mass than closer. Might this not lead to the objects in elliptical orbit gaining some small impulse away from one another?

This wouldn't immediately cause the orbit to get larger in average radius, but I believe it would cause the orbit to become more eccentric over time. Eventually the eccentricity would grow large enough that that it enters the unstable regime and the objects actually do go apart (unless the objects crash due to their finite radii).

Anyway, if elliptical orbits are unstable in the presence of a cosmological constant, it would have to be a very, very slight instability for something like our solar system, so much so that I'd bet it's unmeasurable at present.

https://arxiv.org/abs/1906.05861

What they describe is a system with three regimes of behavior: stable circular orbits below a certain radius, unstable circular orbits above that radius but below another, and no orbits beyond that second radius.

What I'm wondering is: do elliptical orbits screw this up? My searches so far haven't come up with an answer. My reasoning for why elliptical orbits might always exhibit instability in the presence of a cosmological constant starts with the circular case. In particular, it's possible to have a stable circular orbit for any spherically-symmetric potential with the right derivatives.

But an elliptical orbit spends some of its time closer to the center of mass, some of its time further away. The elliptical orbit always spends more of its time further away from the center of mass than closer. Might this not lead to the objects in elliptical orbit gaining some small impulse away from one another?

This wouldn't immediately cause the orbit to get larger in average radius, but I believe it would cause the orbit to become more eccentric over time. Eventually the eccentricity would grow large enough that that it enters the unstable regime and the objects actually do go apart (unless the objects crash due to their finite radii).

Anyway, if elliptical orbits are unstable in the presence of a cosmological constant, it would have to be a very, very slight instability for something like our solar system, so much so that I'd bet it's unmeasurable at present.