In that limit, the metric returns to that defined by special relativity. In practice, this ratio is almost always extremely small. For example, the Schwarzschild radius of the Earth is roughly 9 mm ( inch); at the surface of the Earth, the corrections to Newtonian gravity are only one part in a billion. The Schwarzschild radius of the Sun is much larger, roughly 2953 meters, but at its surface, the ratio is roughly 4 parts in a million. A white dwarf star is much denser, but even here the ratio at its surface is roughly 250 parts in a million. The ratio only becomes large close to ultra-dense objects such as neutron stars (where the ratio is roughly 50%) and black holes. Orbits of test particles We may simplify the problem by using symmetry to eliminate one variable from consideration. Since the Schwarzschild metric is symmetrical about , any geodesic that begins moving in that plane will remain in that plane indefinitely (the plane is totally geodesic). Therefore, we orient the coordinate system so that the orbit of the particle lies in that plane, and fix the coordinate to be so that the metric (of this plane) simplifies to Two constants of motion (values that do not change over proper time ) can be identified (cf. the derivation given below). One is the total energy : and the other is the specific angular momentum: where is the total angular momentum of the two bodies, and is the reduced mass. When , the reduced mass is approximately equal to . Sometimes it is assumed that . In the case of the planet Mercury this simplification introduces an error more than twice as large as the relativistic effect. When discussing geodesics, can be considered fictitious, and what matters are the constants and . In order to cover all possible geodesics, we need to consider cases in which is infinite (giving trajectories of photons) or imaginary (for tachyonic geodesics). For the photonic case, we also need to specify a number corresponding to the ratio of the two constants, namely , which may be zero or a non-zero real number. Substituting these constants into the definition of the Schwarzschild metric yields an equation of motion for the radius as a function of the proper time : The formal solution to this is Note that the square root will be imaginary for tachyonic geodesics. Using