Apollo and orbital mechanics: orbital decay if the Trans Earth Injection (TEI) burn had failed I'm reading Jim Lovell (Apollo 8 and 13) and Jeffrey Kluger's book Apollo 13, which is a fantastic read about a long past era I only have kindergarten memories of. On page 54 there is a paragraph that reads (emphasis mine):

Also, unlike the LOI burn, during TEI there would be no free-return
  slingshot to send the ship home in the event that the engine failed to
  light. If the hydrazine, the dimethylhydrazine, and nitrogen tetroxide
  did not mix and burn and discharge just so, Frank Borman, Jim Lovell,
  and Bill Anders would become permanent satellites of Earth's lunar
  satellite, expiring from suffocation in about a week and then continue
  to circle the moon, once every two hours, for hundreds — no
  thousands; no, millions — of years.

Apollo 8 was in an elliptical orbit of 169.1 × 60.6 miles. I am skeptic about how long such an orbit would be stable. My skills in orbital mechanics are rudimentary at best and I know there is no atmosphere to cause orbital decay due to atmospheric drag. But I have heard about other effects causing orbits to decay over time, such as variations in the moon's gravitational field (density anomalies, mascons). This may cause lunar satellites to eventually crash into the moon. Can someone estimate how long this would be for the case of the Apollo 8 lunar orbit? 
 A: Your skepticism is well founded in this case. For a low orbit such as this, the leading perturbation will be from the mass concentrations (colloquially "mascons") in the Moon, which produce large variations in the gravitational field. The relevant timescale is probably weeks at best.
The orbit of the Apollo 16 subsatellite PFS-2 decayed so rapidly that the instrument impacted the Moon after just 34 days. The GRAIL mission (at the time of writing this answer in mid-May 2012, in the science phase) planned to take advantage of this behavior in order to smoothly vary the orbits and study the gravitational field of the Moon. The spacecrafts were planned for decommission on May 29, 2012 and were expected to impact within one month.
It is worth noting that at some inclinations, low orbits may be much more stable (7º, 50º, 76º, and 86º), but Apollo 8 was inserted at 12º. At this inclination the orbit was actually observed to be perturbated by several kilometers over the course of the 20 hour mission.
