then the orbit is an ellipse where and represent the distances of furthest and closest approach, respectively. If is zero or a negative real number, the orbit is a parabola or a hyperbola, respectively. In these latter two cases, represents the distance of closest approach; since the orbit goes to infinity (), there is no distance of furthest approach. Roots and overview of possible orbits A root represents a point of the orbit where the derivative vanishes, i.e., where . At such a turning point, reaches a maximum, a minimum, or an inflection point, depending on the value of the second derivative, which is given by the formula If all three roots are distinct real numbers, the second derivative is positive, negative, and positive at u1, u2, and u3, respectively. It follows that a graph of u versus φ may either oscillate between u1 and u2, or it may move away from u3 towards infinity (which corresponds to r going to zero). If u1 is negative, only part of an "oscillation" will actually occur. This corresponds to the particle coming from infinity, getting near the central mass, and then moving away again toward infinity, like the hyperbolic trajectory in the classical solution. If the particle has just the right amount of energy for its angular momentum, u2 and u3 will merge. There are three solutions in this case. The orbit may spiral in to , approaching that radius as (asymptotically) a decreasing exponential in φ, , or . Or one can have a circular orbit at that radius. Or one can have an orbit that spirals down from that radius to the central point. The radius in question is called the inner radius and is between and 3 times rs. A circular orbit also results when is equal to , and this is called the outer radius. These different types of orbits are discussed below. If the particle comes at the central mass with sufficient energy and sufficiently low angular momentum then only will be real. This corresponds to the particle falling into a black hole. The orbit spirals in with a finite change in φ. Precession of orbits The function sn and its square sn2 have periods of 4K and 2K, respectively, where K is defined by the equation Therefore, the change in φ over one oscillation of (or, equivalently, one oscillation of ) equals In the classical limit,