Why do Earth satellites stay up?

Satellites in low Earth orbits must accurately conserve their orbital eccentricity, since a decrease in perigee of only 5-10% would cause them to crash. However, these satellites are subject to gravitational perturbations from the Earth's multipole moments, the Moon, and the Sun that are not spherically symmetric and hence do not conserve angular momentum, especially over the tens of thousands of orbits made by a typical satellite. Why then do satellites not crash? We describe a vector-based analysis of the long-term behavior of satellite orbits and apply this to several toy systems containing a single non-Keplerian perturbing potential. If only the quadrupole potential from the Earth's equatorial bulge is present, all near-circular orbits are stable. If only the octupole potential is present, all such orbits are unstable. If only the lunar or solar potential is present, all near-circular orbits with inclinations to the ecliptic exceeding 39 degrees are unstable. We describe the behavior of satellites in the simultaneous presence of all of these perturbations and show that almost all low Earth orbits are stable because of an accidental property of the dominant quadrupole potential. We also relate these results to the phenomenon of Lidov-Kozai oscillations.