The Andromeda-Milky Way collision is going to happen in approximately 4 billion years. What trajectory would the Andromeda galaxy follow on its path to collision with the Milky Way? How could this be calculated using General Relativity?
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3$\begingroup$ AFAIK, $N$-body simulations are usually sufficient for such a problem and these are generally Newtonian potentials $\endgroup$– Kyle KanosCommented Nov 23, 2016 at 14:50
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That actually depends on various numbers we still are not sure about
The mass of the Milky Way:
This is a whole subject by itself, including the dark halo, it can range from $0.8\times 10^{12}M_{\odot}$ to $2.0\times 10^{12}M_{\odot}$, depending on the tracer you use to estimate it, e.g. stars in the stellar halo, hypervelocity stars, dwarf satellite galaxies, stellar streams, $\cdots$
The mass of M31:
This is an even harder question to answer. We believe it is heavier than the Milky Way itself, but how much more is not very clear. There's, however, a good handle on the sum of these two masses, that comes from something called 'timing argument'. The problem is that this argument is very degenerated and clearly depends on us knowing what is the mass of the MW
The proper motion of M31:
If measuring the mass of M31 is difficult, measuring its velocity is quite the challenge. Turns out that measuring the radial component of the velocity is not that difficult with good spectroscopic observations (just need to measure the shift of the lines). But measuring the other two components (the ones projected on the sky) are way harder. You basically need to take many pictures for many years and see how it moves ... not a trivial task
In summary, you may get a good handle on all these three numbers, but an accurate representation of the orbit is not knows until you have very precise estimations of the mass and velocity of each body, which we currently do not have. There are a few simulations, but you should be careful how you interpret them
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$\begingroup$ Thanks for the answer! I have another question, can't we based on the magnitude of curvature of light around Andromeda determine its gravitational well and hence its mass? $\endgroup$ Commented Nov 23, 2016 at 15:11
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$\begingroup$ @NaveenBalaji You definitely can: [microlensing][1] is an option, not an infallible one though, it also has some drawbacks and limitations. [1] arxiv.org/abs/0912.2667 $\endgroup$– caveracCommented Nov 23, 2016 at 15:21
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2$\begingroup$ Out of date. arxiv.org/abs/1205.6863 $\endgroup$– ProfRobCommented Nov 23, 2016 at 15:55
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$\begingroup$ @caverac thanks! I have gone through the papers, assuming we know the unknowns in the future, with as much accuracy as possible, how would we go about calculating the trajectory ? $\endgroup$ Commented Nov 23, 2016 at 16:31
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1$\begingroup$ @NaveenBalaji that depends on what you want to do. Usually the best bet is generating an $N$-body realization of both galaxies and evolving it under their own gravity, this is pretty straightforward and there are plenty of open source codes for doing this. The advantage of using this approach is that it naturally encapsulates non-linear phenomena such as tidal disruption and mass transfer. But if you want a quicker answer just ignore this, and think of each galaxy as a point-mass evolving under the influence of the gravitational field of the other galaxy: two body problem $\endgroup$– caveracCommented Nov 23, 2016 at 16:56