because the torque exerted by the force is zero. As a consequence, the body moves on the plane perpendicular to the angular momentum vector and containing the origin, and obeys Kepler's second law. (If the angular momentum is zero, the body moves along the line joining it with the origin.)
As a consequence of being conservative, a central force field is irrotational, that is, its curl is zero, except at the origin:
The force field of a spatial harmonic oscillator is central with F(r) proportional to r and negative.
By Bertrand's theorem, these two, F(r) = −k/r2 and F(r) = −kr, are the only possible central force fields where all orbits are stable closed orbits. However there exist other force fields, which have some closed orbits.