Does Reducing a Satellite's Speed Guarantee Its Descent to Earth?

[goh]

Suppose we have a satellite orbiting in circular motion around the Earth (steady velocity) in a radius R.Everything is ok now.
If suddenly we reduce the speed of the satellite is it bound to fall to the Earth ? Or it can continue moving in circular orbit with smaller R ? If i want it to fall , how can i calculate where on Earth does it fall ? And what should be the reduced speed so it will fall there ?
Ok I am not expecting from you to tell me all the exact equations to solve this but some general eplain would be useful. Thanks

 

[jtbell]

If suddenly we reduce the speed of the satellite is it bound to fall to the Earth ? Or it can continue moving in circular orbit with smaller R ?

It will start to move in an elliptical orbit whose apogee (point of maximum R) is at the point where the speed was suddenly reduced. If the perigee (point of minimum R) of the new orbit lies below the Earth's surface (or better for practical purposes, within the Earth's atmosphere), then the satellite will hit the earth.

 

[charng]

I would like to first clarify that there are a few assumptions and simplifications being made in this scenario. In reality, the motion of a satellite around the Earth is not perfectly circular and there are other forces at play such as gravitational pull from other celestial bodies. However, for the sake of this problem, let's assume that the satellite is in a perfect circular orbit around the Earth and that there are no other significant forces acting on it.

Now, to answer the questions raised - if the speed of the satellite is suddenly reduced, it will not necessarily fall to the Earth. It can continue to move in a circular orbit with a smaller radius, as long as the gravitational force from the Earth is still strong enough to keep it in orbit. This is similar to how planets in our solar system have different orbital radii but still maintain their orbits around the sun.

If you want the satellite to fall to the Earth, you would need to reduce its speed enough so that the gravitational force from the Earth is greater than the centripetal force keeping it in orbit. This can be calculated using the equation F = ma, where F is the gravitational force, m is the mass of the satellite, and a is the acceleration towards the Earth. The reduced speed needed would depend on the initial speed and the mass of the satellite.

To calculate where on Earth the satellite would fall, you would need to consider the Earth's rotation and curvature, as well as the initial position and velocity of the satellite. This would involve using equations of motion and taking into account the Earth's shape and gravity. Again, the exact calculations would depend on the specifics of the scenario.

In conclusion, while some general explanation can be provided, the exact equations and calculations would depend on the specific details of the scenario. It is important to note that celestial mechanics is a complex and constantly evolving field of study, and it would require thorough analysis and consideration of various factors to accurately predict the motion of a satellite in a given scenario.