massive bodies, as well as the gravitational constant, are both equal to . Then, the motion of the planetoid is given by where . In this form the equations of motion carry an explicit time dependence through the coordinates . However, this time dependence can be removed through a transformation to a rotating reference frame, which simplifies any subsequent analysis. Solutions General solution There is no general closed-form solution to the three-body problem, meaning there is no general solution that can be expressed in terms of a finite number of standard mathematical operations. Moreover, the motion of three bodies is generally non-repeating, except in special cases. However, in 1912 the Finnish mathematician Karl Fritiof Sundman proved that there exists an analytic solution to the three-body problem in the form of a power series in terms of powers of . This series converges for all real , except for initial conditions corresponding to zero angular momentum. In practice, the latter restriction is insignificant since initial conditions with zero angular momentum are rare, having Lebesgue measure zero. An important issue in proving this result is the fact that the radius of convergence for this series is determined by the distance to the nearest singularity. Therefore, it is necessary to study the possible singularities of the three-body problems. As will be briefly discussed below, the only singularities in the three-body problem are binary collisions (collisions between two particles at an instant) and triple collisions (collisions between three particles at an instant). Collisions, whether binary or triple (in fact, any number), are somewhat improbable, since it has been shown that they correspond to a set of initial conditions of measure zero. However, there is no criterion known to be put on the initial state in order to avoid collisions for the corresponding solution. So Sundman's strategy consisted of the following steps: Using an appropriate change of variables to continue analyzing the solution beyond the binary collision, in a process known as regularization. Proving that triple collisions only occur when the angular momentum vanishes. By restricting the initial data to , he removed all real singularities from the transformed equations for the three-body problem. Showing that if , then not only can there be no triple collision, but the system is strictly bounded away from a triple collision. This implies, by using Cauchy's existence theorem for differential equations, that there are no complex singularities in