2.6.35 | before revision #20203 | Savchenko Solutions

Statement

2.6.35^∗. Descending in a spiral from a circular orbit to the surface of the planet in the rarefied layers of the atmosphere, the satellite makes almost circular turns of decreasing radius. At the same time, its speed increases as if the force of atmospheric resistance pushes the satellite forward, in the direction of its flight! Explain qualitatively and quantitatively this paradoxical behavior of the satellite.

Solution

When a satellite with mass $$m$$ is in a circular orbit with radius $$r$$ around a planet with mass $M\gg m$ , the gravitational force serves as the centripetal force

$\frac{GMm}{r^2}=\frac{mv^2}{r}.$

In this orbit, the gravitational potential energy is

$-\frac{GMm}{r}=-2\cdot\frac{1}{2}mv^2;$

that is, the gravitational potential energy is negative but its magnitude is twice the kinetic energy. Consequently, if the total energy of the satellite changes by $-\Delta E$ , then the gravitational potential energy changes by $-2\cdot\Delta E$ and the kinetic energy changes by $\Delta E$ . Therefore, the speed of the satellite increases even though its total energy decreases.

In terms of forces, the atmospheric resistance does decrease the tangential velocity of the satellite. However, the decrese in orbital radius means that the gravitational force increases the inward radial velocity, resulting in an overall increase in the total speed of the satellite.

Answer

See the solution.