# What happens if the gravitational force on a satellite is greater than the centripetal force needed to keep it in circular motion?

Will the satellite continue to orbit the Earth, albeit not in a circle? Why? I would greatly appreciate it if someone could help me address this query. :) Also, I think someone asked a similar question sometime ago and one of the answers said that the satellite will travel around the earth in an elliptical orbit, but I'm not sure why.

• In short it will start to slowly spiral towards the earth. With each pass it slows down a bit more and the pull gets a bit higher. Eventually it crashes. – Bob van de Voort Jun 19 '18 at 14:49
• @BobvandeVoort - What you wrote is rarely true, and is never true for objects orbiting well above the Earth's atmosphere. For most objects whose orbits are decaying due to atmospheric drag, the orbit is very close to circular and the gravitational force is very close $m v^2/r$. The reason the orbit decays is that the atmospheric drag exerts a force orthogonal to the gravitational force, directed against the velocity vector. – David Hammen Jun 19 '18 at 17:33
• There is no such thing as a centripetal force, just gravity. Centripetal acceleration is a name applied to the acceleration experienced when undergoing circular motion. – honeste_vivere Jul 25 '18 at 14:09

Assuming the satellite is far enough away from the Earth that the atmosphere doesn't matter:

The centripetal force required to keep something moving in a circle depends on two things: the radius of the circle and the speed of the moving object. The bigger the radius, the less force is needed; likewise, for faster speeds, more force is needed.

If the gravitational force is greater than the required centripetal force, then the satellite will begin to be pulled inward; if, on the other hand, the gravitational force is less than the centripetal force, then the satellite will begin to fly outward.

So if we start in a position where gravity is greater than the centripetal force, then the satellite will begin to move inward. As it does so, it picks up speed; therefore, the required centripetal force becomes greater and greater, until it eventually is equal to gravity. At this distance and speed, if the satellite was moving completely laterally (i.e. not moving inward or outward), a circular orbit could be sustained. But the satellite is not moving completely laterally here; due to the fact that it was pulled inward by gravity, it's still moving inward, so it overshoots this "equivalent circular orbit" and keeps on moving inward. But now gravity is less than the required centripetal force, and so the satellite's inward motion starts to decelerate, until its inward motion completely stops and it starts flying outward again. By the same logic, it overshoots its "equivalent circular orbit" again, flying outward past it. This process of oscillating about the "equivalent circular orbit" repeats indefinitely.

Now that we've demonstrated that the satellite's distance from Earth oscillates, it should make sense that it would travel in an ellipse (which is a shape which, as you travel around it, has a distance from its focus that also oscillates). It's actually somewhat nontrivial to prove that the shape of a gravitational orbit is an ellipse; that has to do with the particular character of the gravitational force, and the calculus is somewhat involved (you can find the relevant proof in most classical mechanics textbooks). But this should give you an intuitive idea of why an elliptical orbit would make sense.

I think you are confusing centripetal force with centrifugal force. Centripetal force is the gravity force projected along the instantaneous radius of the orbit of planet. I a circular orbit it is the same as force of gravity.

Centrifugal force is the force due to inertia of the satellite and will try to pull it out and fight gravity. Its magnitude is the same as centripetal but with reverse direction. $$F= \frac { m_{satellite} \times v^2}{r}$$ It is a fictitious force which must be added to correct the balance of forces in an inertial frame rotating with the satellite.

If this centrifugal force is less than the gravity at a given point the stellate will descend into lower orbit while accelerating, because of the additional pull of gravity which is gradually stronger at lower orbits. As it accelerates the centrifugal force will increase as well and the satellite will start to climb to higher orbit.

This will lead to a elliptical orbit, which is the typical orbit of solar system planets .

Usually in planetary mechanics they just use Newton's gravity force which is more direct and exact vector:$$F = -GMm/r^2 \times (unit\space vector\space of R)$$ negative sign indicates direction of the force is in to the earth.

In certain cases and depending on the angle and velocity of descent, the stellate in descending to lower orbits, may pick up so much velocity that it will go around the earth while having taken advantage of the great earth velocity and leave the other side and escape earth's gravity. They call this gravitational assist and NASA uses it as a way to send objects far away into space and save on fuel.

• Centripetal and centrifugal force are two sides of the same coin in this instance. The only difference is that you're working in a rotating reference frame and the OP is not. Also, a gravitational slingshot is only very loosely analogous to "bouncing off the other side" of the Earth, and is a misleading phrasing for someone who is not already well-versed in physics (and who therefore might interpret your words literally). – probably_someone Jun 19 '18 at 17:14
• @probably_someone, I agree and will modify my answer if you have better phrasing for gravitational slingshot. By bouncing off I meant an image you see if you were looking from the sun to the earth, in such distance even though the satellite never touches the earth and never gets closer than hundreds of miles, looks like a ball bouncing off a big globe. – kamran Jun 19 '18 at 17:25
• I would suggest something like "the satellite, in descending to lower orbits, can take advantage of the gigantic inertia of the Earth's forward motion to pull itself forward, gaining a lot of speed." – probably_someone Jun 19 '18 at 17:30
• Centrifugal force is a bad way of looking at circular orbits, and is very bad way of looking at orbits in general. – David Hammen Jun 19 '18 at 17:34
• The gravitational slingshot effect is only valid when an object which is not in closed orbit of a planet encounters the planet closely enough to gain kinetic energy but not be trapped by the planet's gravity. If the object is already in a closed orbit around a planet, it cannot be slung away by that same planet. – Bill N Jun 19 '18 at 19:48