There are several types of trajectories that a satellite can take around the primary object. Hyperbolic, parabolic, elliptical and circular. The first two are “open” in that the satellite makes a single pass, but then never comes back. The last two are “closed” orbits. In the absence of any atmosphere, particles, other forms of friction or other disturbing gravitation, these closed orbits are stable.
The circular orbit is just a special case of the elliptical one. It is where both the major and minor axis of the ellipse is the same.
So, that means that an elliptical orbit covers any closed orbit. However in space science, we generally use the term to mean orbits with a reasonably difference between the major and minor axis.
An ellipse has two “foci” (like the centre of a circle, but there are two of them). The centre of mass of BOTH the planet and the satellite will be located at one of the foci. Because the planet is often very much greater than that of the satellite, the focus will often be very close to the centre of mass of the planet itself.
Often small moons have elliptical orbits. Comets have elliptical orbits around the sun. For artificial satellites, geostationary satellites tend to be on circular orbits, where as polar satellites need elliptical ones.
Most of the planets that orbit the Sun have an elliptic orbit (oval-shaped) — this is because of the interaction of all of the other bodies in the solar system (and perfect spherical orbits are very rare).
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