On this work we prove the presence of chaotic dynamics for the classical two-body Kepler problem with a time-periodic variable gravitational coefficient using two different methods, the stretching along the path (SAP) technique and the Melnikov method. For the SAP method, a piecewise periodic constant function is taken as the time-periodic coefficient, and the chaotic dynamics are obtained for small values of angular momentum or large periods. For the Melnikov method, a sinusoidal perturbation function is taken as the time-periodic coefficient. To guarantee chaotic dynamics big values of angular momentum or small periods of the function are necessary. The classical Melnikov method could not be applied, as the critical point of the system was parabolic, not hyperbolic. Hence, some modifications to the method have been necessary.