to be considered. To leading order in , the Hill radius above also represents the distance of the Lagrangian point L1 from the secondary. A quick way of estimating the radius of the Hill sphere comes from replacing mass with density in the above equation: where and are the average densities of the primary and secondary bodies, and and are their radii. The second approximation is justified by the fact that, for most cases in the Solar System, happens to be close to one. (The Earth–Moon system is the largest exception, and this approximation is within 20% for most of Saturn's satellites.) This is also convenient, because many planetary astronomers work in and remember distances in units of planetary radii. True region of stability The Hill sphere is only an approximation, and other forces (such as radiation pressure or the Yarkovsky effect) can eventually perturb an object out of the sphere. This third object should also be of small enough mass that it introduces no additional complications through its own gravity. Detailed numerical calculations show that orbits at or just within the Hill sphere are not stable in the long term; it appears that stable satellite orbits exist only inside 1/2 to 1/3 of the Hill radius. The region of stability for retrograde orbits at a large distance from the primary is larger than the region for prograde orbits at a large distance from the primary. This was thought to explain the preponderance of retrograde moons around Jupiter; however, Saturn has a more even mix of retrograde/prograde moons so the reasons are more complicated. Further examples It is possible for a Hill sphere to be so small that it is impossible to maintain an orbit around a body. For example, an astronaut could not have orbited the 104 ton Space Shuttle at an orbit 300 km above the Earth, because a 104-ton object at that altitude has a Hill sphere of only 120 cm in radius, much smaller than a Space Shuttle. A sphere of this size and mass would be denser than lead, and indeed, in low Earth orbit, a spherical body must be more dense than lead in order to fit inside its own Hill sphere, or else it will be incapable of supporting an orbit. Satellites further out in geostationary orbit, however, would only need to be more than 6% of the density of water to fit