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Please bear with me, as I'm not in the field of physics, this question may seem a bit simple.

This question is concerning stable circular orbits around celestial bodies.

I know the equation relevant to my question is given by the equality of gravitational to centripetal force:

$$G\frac{mM}{r^2}=\frac{mv^2}{r}$$

where $m$ is the mass of the orbiting body, $M$ the mass of the celestial body, $G$ the gravitational constant, $r$ the distance between the centers of mass of both bodies and $v$ the tangential velocity.

  1. Assuming a celestial body without an atmosphere;

    When expecting the above formula, it seems that one, should be able to orbit at any
    given radius, however is this true in reality?

  2. Assuming a celestial body with an atmosphere;

    Will drag be the main reason why one cannot orbit at heights inside the atmosphere?

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2 Answers 2

up vote 6 down vote accepted

In Newtonian mechanics, with nothing to slow the orbiting object (or otherwise obstruct it), energy is conserved, so the orbit is maintained indefinitely, and can continue so at any altitude.

As soon as you introduce an atmosphere, there is a mechanism to remove energy from the orbiting object, so the orbit decays until it makes contact with the body it was orbiting.

In a real system, it can get a bit more complicated. Consider the Earth and Moon... there are tidal forces which are actually accelerating the Moon in its orbit (causing it to move further away) while slowing the Earth's rotation (albeit VERY gradually). Strictly speaking, the same tidal forces would act on small satellites too, and if they are in "below synchronous" orbits, they would be gradually slowed down and fall inward, but again, a very small and therefore gradual effect.

There are also non-uniformities in the density of the Earth, which produce "gravitational anomalies". They "perturb" the orbits of satellites. While there may be a cumulative effect to change a satellite's orbit, there would be no cumulative change to the energy of the orbit, so it should continue more or less indefinitely (ignoring the effect of atmospheric drag). However, it's conceivable that the cumulative effect of gravitational anomalies could make an orbit more eccentric over time, eventually leading to a significant encounter with the atmosphere.

Of course, in a real system, there would be other sources of perturbations - like the mass of the Sun and the Moon, which are known to alter the orbits of terrestrial satellites.

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Well, it is certainly possible. Very generally speaking, if a body has the correct acceleration to counteract the force of gravity, it can orbit at any altitude. This is of course as long as no collisions or unexpected events occur and the body is capable of reaching and maintaining this acceleration, along with being able to maintain a certain orbit.

I don't have any numbers for an example, but it is certainly able to be calculated.

For the second question, I would assume that the lowest possible orbit depends on the atmosphere and the body; that body's terminal velocity in the given atmosphere would determine its minimum orbit. But then again, I do believe it is possible as long as the body can achieve the needed acceleration and barring any collisions or unforeseen events.

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