Why do Earth satellites stay up?

Understand

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.