From Alain Fournier:
On Sept/9/2018 at 04:25, Stuf4 wrote :
Reposting with corrections ("centrifugal force" is what was meant):

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It might help to think of two-body orbit dynamics in a way that most people don't think of it:

A satellite going around a planet acts like a mass hanging on the end of a spring.

Basic diagram:
https://i.ytimg.com/vi/lZPtFDXYQRU/maxresdefault.jpg

Gravity pulls down on the mass, but the mass can move down and up in an oscillation. The spring is pulling up on the mass, and this is how the centrifugal force works, pulling the satellite up and away from the Earth. The centrifugal force is a manifestation of the inertial property of the satellite's mass.

When gravity and the centrifugal force are in equilibrium, the mass remains at a constant altitude from the Earth. Circular orbits are static in this respect, in a reference frame that rotates at the same rate as the satellite is orbiting. And this is why you can bolt your DirecTV dish pointing to one point in the sky and the geometry does not change. The satellite is as still as the mass hanging on the end of the spring.

Elliptical orbits are not still. They have a continual tradeoff of Potential Energy & Kinetic Energy.

This is the situation you have in the lab, with the mass bobbing up and down on the end of the spring.

Hopefully this makes it clear exactly what is causing the altitude changes with the satellite. When the mass moves down past what would be the static equilibrium point, it has plenty of kinetic energy. And that is getting packed into "the spring" of inertia. It bottoms out at perigee when the spring force finally overcomes the motion from the gravitational force, and the direction reverses.

So yes, it is the inertia of the velocity that has built up during this downward part of the cycle that causes the altitude reversal. You can think of it as a spring that has been pulling on this satellite. You stretch the spring all the way down to perigee, and then its force will finally reverse the direction that the force of gravity was pulling in.

But from every moment that the mass was below the point of equilibrium - the altitude of the circular orbit - that spring force was greater than the force of gravity. The satellite's velocity toward the Earth was decelerating the entire time since it had passed that equilibrium point.


It isn't quite at the moment where the mass passes the altitude of the
circular orbit that the satellite's velocity towards Earth starts to
decelerates. It starts decelerating when the centrifugal force becomes
stronger than the force of gravity. When it crosses the altitude of the
circular orbit corresponding to its energy level, it has the same
potential energy as a satellite in the circular orbit since it is at the
same height. It also has the same speed as that satellite since its
total energy (potential energy + kinetic energy) is the same. But that
speed isn't in the right direction and therefore doesn't give as much
centrifugal force and the satellite's vertical speed is therefore still
increasing when it is on the way down or decreasing when it is on the
way up.

Else than that minor detail, your explanation is in my opinion correct.

Your reasoning looks sound to me. Thank you for highlighting my apparent error.

I had never seen anyone explain a two-body orbit in terms of a spring-mass system. It's clear that I needed to put more thought into my reply before posting.

Maybe some day someone will write a paper on this and nail it down. Or maybe such a paper was published long ago and I'm just not aware of it. I googled around and this was the closest I could find:

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Reactive centrifugal force
https://www.revolvy.com/page/Reactive-centrifugal-force
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Gravitational two-body case
In a two-body rotation, such as a planet and moon rotating about their common center of mass or barycentre, the forces on both bodies are centripetal. In that case, the reaction to the centripetal force of the planet on the moon is the centripetal force of the moon on the planet.[6]
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https://d1k5w7mbrh6vq5.cloudfront.ne...67f1ccca57.PNG
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The title of that page focuses on centrifugal force, but then describes the orbit case in terms of centripetal force.

~ CT