The five moons of Pluto form a sort of miniature planetary system — one that is unique in the solar system.

The largest, Charon, 750 miles wide, was discovered in 1978, and it is so large, about one-ninth the mass of Pluto, that the center of mass of the two lies outside Pluto. That has led some planetary scientists to regard Pluto and Charon as a double planet.

By contrast, Earth is 81 times the mass of the moon, and the center of mass is within Earth.

Two much smaller moons of Pluto, now named Nix and Hydra, were discovered in 2005 in images taken by the Hubble Space Telescope. In 2011, another moon, Kerberos, was discovered between the orbits of Nix and Hydra, and a year later, astronomers announced the fifth moon, Styx.

In an article published Wednesday in the journal Nature, Mark R. Showalter of the SETI Institute in Mountain View, Calif., and Douglas P. Hamilton of the University of Maryland calculated more precisely the orbits of the four smaller moons and turned up some surprises.