# 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

## 1 Answer

You do not get any spirals. In Newtonian gravity two-body motion is always along conic sections.

The "trick" here is to look at the problem from a moving reference frame. For simplicity, assume A and B to have equal mass. Then if you move with the centre of mass for the whole system you will see A and B start out with equal and opposite velocities and perform the usual elliptic, parabolic or hyperbolic orbits. With a bit more algebra one can turn the 2-body problem with unequal masses into a 1-body problem, and again the orbits are the standard conic ones. When you transform back to the original view you can certainly see A and B whirling around each other while the pair is also moving in some direction. But you never get any inspirals.

Three-body interactions are way messier, and I think there are at least mathematical cases where bodies collide after an infinite number of orbits around each other. Things are also slightly different in general relativity, where you can actually get spiral plunges close to black holes. In any of these cases I don't think the spirals are neatly described by any of the standard spiral formulas.