# Spirals in newtonian celestial mechanics?

I know Kepler's laws, Newton's laws, and that conic sections are the trajectories of noncolliding two point masses. But I wonder about a point mass A eventually colliding with point mass B.

In particular, suppose B begins at rest while A has an initial velocity, in 2D Newtonian gravity. What do the trajectories look like for A, and how do we derive them?

I think we would get a spiral path towards B. Do we get logarithmic spirals? What is the parametric description of the path? What are the analogues of Kepler's laws?

• What do you mean by 2D Newtonian gravity? Do you mean that the force of attraction is $F\propto 1/r$ instead of $F \propto 1/r^2$? You find the trajectories in the same way as for 3D with the $1/r^2$ force law. IE you write down an equation for $F=ma$ in polar co-ordinates and you solve it. – sammy gerbil Apr 7 '18 at 13:20