What is the likely closest encounter for a typical star during the Milky Way Andromeda Merger?

Question

This is closely related, but distinct question to What are the chances of a star colliding with another during a galactic collision?.

The accepted answer to that question places the odds of the Sun colliding with another star at about $10^{-12}$, what I want to ask is how near a near-miss is expected? In other words, what is the expected closest encounter between the Sun and another star during the collision? A more sophisticated answer would give the expected distribution of close encounters by distance.

A more general version of the question is -- what disruptive events would a typical star such as the Sun expect to experience during the merger?

Answer 1

Suppose the sun moves through a column of Andromeda stars $L$ parsecs long with density $\rho$ and random positions. The probability of encountering at least one star within $r$ of the sun along this column would be $$F(r)=1-e^{-\pi r^2 L \rho}$$ (basically, what is the probability of no "success" when doing a Poisson sample of the column volume times density).

Setting $F(r_{1/2})=1/2$ to find the distance where we should expect a 50% chance of a star more nearby we get $r_{1/2}=\sqrt{\ln(2)/\pi L \rho}$. If we plug in $\rho=0.14 $ per cubic parsec and $L=1000$ parsec (assuming a flat disk-disk collision) I get $r_{1/2}=0.02$ parsec. That is 4125.296 AU, reassuringly far.

To get the mean distance we can use the not very well known but useful formula $E[r]=\int_0^\infty (1-F(r)) dr =\int_0^\infty e^{-\pi r^2 L \rho} dr = 1/2\sqrt{L \rho}\approx 0.042$ parsec.

Depending on your views of disk thickness and angle on merger $L$ and $\rho$ may vary a fair bit. A sideways collision might have $L=67$ kpc reducing the distances by a factor of 8.1. Direct collisions still remain very unlikely, but messed up Oort clouds seem very likely.