Classical particle scattering for power-law two-body potentials

We present a rigorous study of the classical scattering for any two-body inter-particle potential of the form $v(r)=g/r^\gamma$, with $\gamma>0$, for repulsive ($g>0$) and attractive ($g<0$) interactions. We give a derivation of the complete power series of the deflection angle in terms of the impact factor for the weak scattering regime (large impact factors) as well as the asymptotic expressions for the hard scattering regime (small impact factors). We see a very different qualitative and quantitative behavior depending whether the interaction is repulsive or attractive. In the latter case, the families of trajectories depend also strongly on the value of $\gamma$. We also study carefully the modifications of the results when a regularization is introduced in the potential at small scales. We check and illustrate all the results with the exact integration of the equations of motion.