  Curves in a family derived from powers of the polar coordinate formula for
ellipses are found to provide good fits to bound orbits in a range of power-law
potentials. This range includes the well-known $1/r$ (Keplerian) and
logarithmic potentials. These approximate orbits, called p-ellipses, retain
some of the basic geometric properties of ellipses. They satisfy and generalize
Newton's apsidal precession formula, which is one of the reasons for their
surprising accuracy. Because of their simplicity the p-ellipses make very
useful tools for studying trends among power-law potentials, and especially the
occurence of closed orbits. The occurence of closed or nearly closed orbits in
different potentials highlights the possibility of period resonances between
precessing, eccentric orbits and circular orbits, or between the precession
period of multi-lobed closed orbits and satellite periods. These orbits and
their resonances promise to help illuminate a number of problems in galaxy and
accretion disk dynamics.
