My specific example for the question is the future collision of the galaxies Milky Way (our own galaxy) and Andromeda in a couple billion years. The star in question is obviously the sun in this case. I want to know the chances of a collision with another star and if it's significant or not.
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1$\begingroup$ The sun might be dead by the time that happens so the specific part of that question might not be relevant. But the density of stars in a galaxy is not high in the arms where the the sun is but higher in the galactic core. The stars like the sun will very likely not collide with anything because the volume that the star actually occupy is much smaller than the space between stars in both galaxies. $\endgroup$– A. C. A. C.Commented Sep 6, 2017 at 23:23
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3$\begingroup$ @A.C.A.C. Why do you say might? Which bit is sufficiently uncertain to attract a qualifying word? I say the Sun will be alive when the collision takes place. $\endgroup$– ProfRobCommented Sep 6, 2017 at 23:31
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3$\begingroup$ There is a whole subsection on this question on the relevant Wikipedia page. en.m.wikipedia.org/wiki/Andromeda–Milky_Way_collision $\endgroup$– ProfRobCommented Sep 6, 2017 at 23:32
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2$\begingroup$ astronomy.stackexchange.com/questions/1911/… contains several answers which address this question. $\endgroup$– ProfRobCommented Sep 7, 2017 at 9:10
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2$\begingroup$ 5.5 billion years would be to reach the end of the main sequence. More like 7.7 billion to become a white dwarf. e.g arxiv.org/abs/0801.4031 @A.C.A.C. $\endgroup$– ProfRobCommented Sep 7, 2017 at 19:01
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1 Answer
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Let's see what we get from some back-of-the-envelope estimates.
Imagine throwing one star (e.g., the Sun) at the other galaxy. How likely is it we'll hit a star in the other galaxy? Well, it's basically proportional to how big a target each star in the other galaxy is (its cross-sectional area) compared to the size of the whole galaxy, multiplied by the total number of stars in the target galaxy.
Let's assume it's the Milky Way-Andromeda scenario, so each galaxy has about 100 billion stars, and each star is roughly the same size as the Sun (some are much larger, most are smaller). The actual target area for an individual star is a circle with twice the star's radius (we're counting one star just grazing the other as a collision). Let's also assume the stars are more or less evenly distributed in a circular disk. Since "100,000 light years" is a common (and not completely crazy) estimate of the Milky Way's size, that's a circle of radius = 50,000 light years (about $10^{16}$ meters).
So: 100 billion stars in the target galaxy, each with target radius $\sim 2 R_{\odot}$, gives us a total target area of $10^{11} \times \pi (2 R_{\odot})^{2} \approx 10^{30}$ m$^{2}$.
The area of the target galaxy is $\pi R_{gal}^{2} \approx 10^{42}$ m$^{2}$. So the chance of our Sun hitting a star in the other galaxy is $\approx 10^{30} / 10^{42} = 10^{-12}$ -- or about one in a trillion.
The odds of any star from our galaxy not hitting a star in the other galaxy would be $(1 - 10^{-12})^{10^{11}} \approx 0.90$.
So there's only about a 10% chance of one (or more) of the galaxy's 100 billion stars hitting a star in the other galaxy. And the chances of any one particular star (like our Sun) hitting a star in the other galaxy is about one in a trillion.
answered Sep 11, 2017 at 20:47
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2$\begingroup$ Factoring in Roche limit breakups (leading to partial collisions?) and the denser center parts of the two galaxies colliding, which is expected to happen I would think the odds of collision goes up, but running the math on that gets a bit problematic for me. Still, props on doing the math. $\endgroup$– userLTKCommented Sep 11, 2017 at 21:22
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2$\begingroup$ Well, the original question was about "collisions", so I went with that. I also neglected gravitational focussing, which increases the effective impact parameter and thus the odds of collisions. But that's still not going to get you more than one or two orders of magnitude, which means the chances go from 1 in a trillion to, say, one in 10 or 100 billion. Still utterly insignificant. $\endgroup$ Commented Sep 12, 2017 at 9:37
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2$\begingroup$ Increased central density doesn't change the base chance for a random individual star (like the Sun) to collide: sure, more stars per square meter in one part of the target galaxy, but they're in a smaller area now, so it's harder to hit the region with those stars. (The math cancels out.) $\endgroup$ Commented Sep 12, 2017 at 9:40
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3$\begingroup$ It is true that the dense centers of the two galaxies will, if you let the whole merger play out (beyond the first collision) spiral into the common center via dynamical friction and merge, so the odds that some stars will collide will be higher. $\endgroup$ Commented Sep 12, 2017 at 9:43