the relation and . The time dilation factor between the bookkeeper and the moving test-particle can also be put into the form where the numerator is the gravitational, and the denominator is the kinematic component of the time dilation. For a particle falling in from infinity the left factor equals the right factor, since the in-falling velocity matches the escape velocity in this case. The two constants angular momentum and total energy of a test-particle with mass are in terms of and where and For massive testparticles is the Lorentz factor and is the proper time, while for massless particles like photons is set to and takes the role of an affine parameter. If the particle is massless is replaced with and with , where is the Planck constant and the locally observed frequency. Exact solution using elliptic functions The fundamental equation of the orbit is easier to solve if it is expressed in terms of the inverse radius The right-hand side of this equation is a cubic polynomial, which has three roots, denoted here as , , and The sum of the three roots equals the coefficient of the term A cubic polynomial with real coefficients can either have three real roots, or one real root and two complex conjugate roots. If all three roots are real numbers, the roots are labeled so that . If instead there is only one real root, then that is denoted as ; the complex conjugate roots are labeled and . Using Descartes' rule of signs, there can be at most one negative root; is negative if and only if . As discussed below, the roots are useful in determining the types of possible orbits. Given this labeling of the roots, the solution of the fundamental orbital equation is where represents the function (one of the Jacobi elliptic functions) and is a constant of integration reflecting the initial position. The elliptic modulus of this elliptic function is given by the formula Newtonian limit To recover the Newtonian solution for the planetary orbits, one takes the limit as the Schwarzschild radius goes to zero. In this case, the third root becomes roughly , and much larger than or . Therefore, the modulus tends to zero; in that limit, becomes the trigonometric sine function Consistent with Newton's solutions for planetary motions, this formula describes a focal conic of eccentricity If is a positive real number,