I'm asked to predict the effect of a resistive force acting on a sattelite of mass $m$ in orbit of radius $r$ around a planet of mass $M$. I have come up with the following equations:

  1. Kinetic energy = $\frac{GMm}{2r}$ (Derived by equating equation for centripetal force to gravitation force acting on sattelite)

  2. Potential energy = $\frac{-GMm}{r}$ (Multiplied the gravitational potential energy eqution by $m$)

  3. Total energy = $\frac{-GMm}{2r}$ (Added equations 1 and 2)

What I predict will change:

  1. The kinetic energy will decrease because the resistive force will decrease the angular velocity
  2. From equation 1, the radius should increase
  3. From equation 2, the gravitational potential energy will increase since radius increases
  4. From equation 3, the total energy will increase since the radius increases

They are all wrong and the opposites are true. Where did I go wrong? I wrote all that since I feel I must be missing something very fundamental.


The error is right here:

The kinetic energy will decrease because the resistive force will decrease the angular velocity

This would happen only if the friction force was localized on a short segment of the orbit, such as if suddenly the satellite got hit by a dense cloud of dust or meteorites. And even then, the decrease in velocity would be only temporary, because the orbit has been altered and the satellite will start falling towards the attraction center and will increase its speed on the other side of the orbit.

Usually, the friction on satellites is due to atmosphere and is steady and very weak.

Let us assume initially circular orbit.

If there is no friction at all, the circular orbit will remain stable and the satellite will keep the same distance from the attracting center.

If, however, a weak friction opposed to velocity is present, what will happen is that the friction will slowly steal mechanical energy of the system satellite-planet. It will slowly put the satellite down. Because it is a slow process, the shape of the orbit will not change much. It will remain almost circular, but the circle will shrink in time (because mechanical energy is decreasing in time).

So the satellite, while orbiting, is also falling down, towards the center. And since circular orbit speed is inversely proportional to radius, the more the satellite falls, the higher its speed and kinetic energy becomes.


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