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If a satellite is in a stable circular orbit and goes about 41% faster (escape velocity) then it leaves its host forever. I get that. However, what if it speeds up by less than 41%?

Intuitively, it would seem to make the satellite move farther away from the host, and thus enter a higher (more distant) orbit.

However, according to my understanding, a stable orbit requires the satellite to move more slowly the farther away it is from the host. For example, the earth moves more slowly around than the sun than Venus because it is farther away from the sun than Venus.

So, if a satellite speeds up then the stable orbit would be closer to the host, not farther away. What am I missing here?

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    $\begingroup$ @JohnForkosh That's an answer $\endgroup$ – FGSUZ Apr 20 '19 at 10:56
  • $\begingroup$ @JohnForkosh that is indeed a good answer. $\endgroup$ – Ján Lalinský Apr 20 '19 at 22:24
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The parameters of stationary orbits depend on the energy and orbital momentum. For circular orbits, we have a simple relationship for orbital velocity and orbit radius $v^2 =\frac {GM}{r}$. It follows that Venus moves faster than Earth, and Mercury moves faster than Venus. However, for elliptical orbits, the speed does not depend only on the radius. Maneuvering with increasing speed at a given point of the orbit leads to a change in the shape of the orbit. For example, a circular orbit becomes an ellipse, parabola or hyperbola. Figure 1 shows examples of maneuvers with a transition to an elliptical orbit (the speed increases by 30%) and a parabolic orbit (the speed increases by 41%).

fig1

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Since the stable velocity v for a given orbital radius $R$ is given by $v = \sqrt\frac{GM}{R}$, I would assume that the satellite spirals outwards at an accelerating rate. Since its velocity is greater than the stable velocity for that orbit, it will start to increase the radius of that orbit. As the radius increases, the magnitude of the gravitational force decreases and thus the rate of spiralling increases (therefore, an accelerating spiral). I'm not sure myself either because this would also imply that the satellite will eventually leave the host, but this is my best guess.

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    $\begingroup$ "best guesses" should be comments, not answers. $\endgroup$ – garyp Apr 20 '19 at 18:22

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