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I'm thinking of a system like an object in orbit around a planet. Say a 100kg mass orbiting the earth. If I were to impart 1 m/s to the object down toward the center of the mass it is orbiting, what would happen?

What I'm assuming is that the object would move closer to the earth which would cause an increasing imbalance of the objects orbital speed and its altitude. This would cause it to feel an increasing acceleration away from the planet, come to a stop, and eventually begin accelerating towards its proper altitude.

Inertia being what it is, it seems that this would set up an oscillation centered on the altitude/speed balance. I'm assuming in most cases this would decay into a stable altitude.

How can the period of the oscillations be calculated? Are there stable resonances possible in this modality and how can they be found?

If we were to plot the y value of a circular orbit over time we would get a nice sin wave. What I'm effectively wondering is if an oscillation can be added to that, a la:

orbit waves

Is it possible to create such an oscillation in the object's orbital altitude with a downward impulse?

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  • $\begingroup$ Resonance is the amplification of an oscillation by a driving force applied at the same frequency. Are you applying a periodical force to the orbiting object, or are you imparting a one-time impulse? Your graph does not show a growth in amplitude so this is not resonance. What it shows results from a one-time impulse. $\endgroup$ – sammy gerbil Jan 11 at 7:46
  • $\begingroup$ Hello @sammygerbil! Good read, yes, I am trying to figure out how to set up an impulse resonance. However, if there's no oscillation it means no possibility for resonance. $\endgroup$ – joshperry Jan 13 at 2:27
  • $\begingroup$ So you want to apply an impulse to the satellite at regular intervals? If the impulses are synchronised with the orbital period then the orbit grows more and more eccentric, eventually crashing into the planet's surface. Is that what you are intending? I don't understand what you mean by stable resonances - a continual growth in amplitude is not a stable outcome. $\endgroup$ – sammy gerbil Jan 13 at 3:41
  • $\begingroup$ Imagine a continuous river encircling the planet and a ship sailing along this river, it sits on the water at a balance between its buoyancy and weight; this represents a circular orbit. If we were to apply a momentary downward impulse on the ship, the balance of its buoyancy would be upset and it would bob back upwards beyond the balance point (because of the inertia in the system), then back down, quickly dampening to balance. However, if a repeating impulse was applied at the correct point, a resonance could be realized around the balance, bobbing many times per "orbit". $\endgroup$ – joshperry Jan 31 at 21:26
  • $\begingroup$ There is damping in a river because of friction and because the waves carry away energy. But there are no equivalent damping mechanisms in space. A single impulse would be enough to make the satellite bob up and down indefinitely. $\endgroup$ – sammy gerbil Jan 31 at 21:45
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Orbits are elliptical.

An orbit that’s just a little bit elliptical, like the one perturbed by 1 m/s here, looks like bobbing up and down around the original circular orbit. And the period of that bobbing oscillation is equal to the orbital period: the motion goes down once and up once before returning to the same point and repeating.

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  • $\begingroup$ Isn't the period of the bobbing oscillation twice the orbital period? In an ellipse there are two points which are furthest from the centre and two which are nearest. $\endgroup$ – sammy gerbil Jan 11 at 7:38
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    $\begingroup$ The earth is at a focus of the ellipse, not its center. So the periods are the same, once “up” per “around”. In really elliptical orbits you can see this: the satellite goes really high just once. (As the orbit gets more elliptical, the satellite sampled gravity that’s different on the very high and very low sides, so the oscillation isn’t symmetric: it can go arbitrarily high, but not arbitrarily low, past the earth) $\endgroup$ – Bob Jacobsen Jan 11 at 17:19
  • $\begingroup$ Ah yes I didn't think of that. $\endgroup$ – sammy gerbil Jan 11 at 17:22
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1 m/s is a small change to modify too much an orbit. In general, the elliptic orbit would change. If it was in a perfectly circular orbit, it would become slightly elliptic.

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  • $\begingroup$ I added a graph to perhaps clarify what I'm getting at. $\endgroup$ – joshperry Jan 9 at 1:37

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