in terms of the length . Circular orbits are possible when the effective force is zero i.e., when the two attractive forces — Newtonian gravity (first term) and the attraction unique to general relativity (third term) — are exactly balanced by the repulsive centrifugal force (second term). There are two radii at which this balancing can occur, denoted here as rinner and router which are obtained using the quadratic formula. The inner radius rinner is unstable, because the attractive third force strengthens much faster than the other two forces when r becomes small; if the particle slips slightly inwards from rinner (where all three forces are in balance), the third force dominates the other two and draws the particle inexorably inwards to r = 0. At the outer radius, however, the circular orbits are stable; the third term is less important and the system behaves more like the non-relativistic Kepler problem. When is much greater than (the classical case), these formulae become approximately Substituting the definitions of and rs into router yields the classical formula for a particle of mass orbiting a body of mass . where ωφ is the orbital angular speed of the particle. This formula is obtained in non-relativistic mechanics by setting the centrifugal force equal to the Newtonian gravitational force: Where is the reduced mass. In our notation, the classical orbital angular speed equals At the other extreme, when a2 approaches 3rs2 from above, the two radii converge to a single value The quadratic solutions above ensure that router is always greater than 3rs, whereas rinner lies between rs and 3rs. Circular orbits smaller than rs are not possible. For massless particles, a goes to infinity, implying that there is a circular orbit for photons at rinner = rs. The sphere of this radius is sometimes known as the photon sphere. Precession of elliptical orbits The orbital precession rate may be derived using this radial effective potential V. A small radial deviation from a circular orbit of radius router will oscillate stably with an angular frequency which equals Taking the square root of both sides and performing a Taylor series expansion yields Multiplying by the period T of one revolution gives the precession of the orbit per revolution where we have used ωφT = 2п and the definition of the length-scale a. Substituting the definition of the Schwarzschild radius rs gives This may be simplified using