# De-orbiting a satellite

The same authors mentioned earlier, Philip Blanco and Carl Mungan, have written a stimulating paper entitled “Satellite splat: an inelastic collision with a surface-launched projectile,” (Eur. J. Phys. 36 (2015) 045004).  Blanco and Mungan consider the case of a totally inelastic collision between a satellite in circular orbit about a planet and a projectile launched vertically from the surface of that planet.  What are the conditions such that the combined object “de-orbits,” i.e., falls to the planet’s surface?  The paper contains a detailed analysis, but since our goal is to provide homework and exam exercises appropriate for introductory physics students, we will consider only the simplest cases.

Denote the satellite mass and orbital radius by m and r, while the planetary mass (pictured as Earth) and radius are M and R.  Let m’ be the projectile mass, and for simplicity, set the projectile launch speed to be the minimum speed sufficient to intercept the satellite; i.e., when m’ reaches the orbital radius r, its speed is zero.  See the figure.

1. What is the satellite speed before the collision? What is its speed immediately after the collision?
2. After the collision, the body follows an elliptical trajectory. Is the collision point its apogee, perigee, or neither?  Explain.
3. Suppose m’ is just sufficient to bring down the satellite. Where would it land?  Describe the combined body’s trajectory from the collision in space to touchdown on the planet’s surface.

In PPE, Equations 10.13 and 12.5 relate the semi-major axis a and semi-minor axis b of an elliptical orbit to the orbiting body’s total energy E and angular momentum L.  But a and b are independent of mass, as can be seen by re-writing those equations in terms of the specific energy $\large {E}'=E/m$ and specific angular momentum $\large {L}'=L/m$ of the orbiting body m:

$\LARGE {E}'=-\frac{GM}{2a}$

$\LARGE b=\frac{{L}'}{\sqrt{-2{E}'}}$

1. OK, back to the problem. For simplicity, let m’ = m.  In terms of the planet radius R, what is the maximum orbital radius r such that the combined body impacts the planet’s surface?   (Ans: rmax = 7R)
2. Now let’s generalize a bit. To reduce costs, we want to use the minimum projectile mass m’ that will de-orbit the satellite.  Derive a formula for this minimum mass in terms of the orbital radius r = nR. (Ans:  $\large {m}'_{min}/m=\sqrt{(n+1)/2}-1)$