If a satellite is in a sufficiently low orbit, it will encounter air drag from the earth’s atmosphere. Since air drag does negative work (the force of air drag is directed opposite the motion), the mechanical energy will decrease. According to Eq. (13.13), if E decreases (becomes more negative), the radius r of the orbit will decrease. If air drag is relatively small, the satellite can be considered to be in a circular orbit of continually decreasing radius.

(a) According to Eq. (13.10), if the radius of a satellite’s circular orbit decreases, the satellite’s orbital speed increases. How can you reconcile this with the statement that the mechanical energy decreases? 

(b) Due to air drag, the radius of a satellite’s circular orbit decreases from r to r – Δr, where the positive quantity Δr is much less than r. The mass of the satellite is m. Show that the increase in orbital speed is Δv = + (Δr/2) √Gm_E/r³; that the change in kinetic energy is ΔK = + (Gm_E m/2r²) Δr; that the change in gravitational potential energy is ΔU = –2 ΔK = – (Gm_Em/r²) Δr; and that the amount of work done by the force of air drag is W = – (Gm_Em/2r²) Δr. Interpret these results in light of your comments in part (a).

(c) Asatellite with mass 3000 kg is initially in a circular orbit 300 km above the earth’s surface. Due to air drag, the satellite’s altitude decreases to 250 km. Calculate the initial orbital speed; the increase in orbital speed; the initial mechanical energy; the change in kinetic energy; the change in gravitational potential energy; the change in mechanical energy; and the work done by the force of air drag.

(d) Eventually a satellite will descend to a low enough altitude in the atmosphere that the satellite burns up and the debris falls to the earth. What becomes of the initial mechanical energy?