Past exam of the physics 1 paper of the University of Oxford / 2016 resit / 10 / Solution

Use the test-particle approximation, neglecting the planet's recoil. The conserved specific angular momentum is $j=bv$ and the specific energy at infinity is $v^2/2$. At closest approach the radial velocity vanishes, so the effective potential givesThe positive root and the tangential speed areThe second form for $r_0$ avoids subtracting two nearly equal numbers for a small impact parameter. These formulas describe a nonradial orbit; $b=0$ gives a collision rather than a finite nonzero pericentre in the point-planet model.