Time-Like Geodesic Motion in Schwarzschild Spacetime with Weak-Field Limit

We analyze the geodesic motion in Schwarzschild spacetime with the weak-field limit. We investigate all geodesic types of the test particle by solving the geodesic equation and analyzing the behavior of effective potential. At the same time, all kinds of orbits, which are allowed according to the energy level corresponding to the effective potential, are numerically simulated in detail. Then we discuss the effect of different parameters on the effective potential energy. We also find that the test particle falls rapidly along the fall-into orbit and the radius of stable (unstable) circular orbits become larger in the Schwarzschild spacetime with the weak-field limit than those in the Schwarzschild case.