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I'm wondering whether there is a similar effect between Galaxies as we have between a planet and it's star. I know there are tidal waves between galaxies and that gravity seems to always be attractive therefore, there should be Lagrange like points between Galaxies.

However, the tidal waves seem to wrack havoc galaxies so I could imagine that no such stable points would exist, at least not for long (a few million years maybe?)

Is that at all possible?

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    $\begingroup$ Standard Lagrange point calculations treat the star & planet as points. A galaxy & its satellite galaxy aren't well-modeled as points, since the separation distance is fairly small relative to the sizes of the galaxies. $\endgroup$
    – PM 2Ring
    Apr 27 at 9:34
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    $\begingroup$ Taking that comment a bit farther: we can treat sun, earth, moon as points because they are compact (two are solid; one is gravitationally held to be a sphere). Galaxies are more analogous to a star surrounded with thousands of small asteroids. $\endgroup$ Apr 27 at 12:12
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    $\begingroup$ different but related: Are there places in the Universe without gravity? See also Density and dimensionality of zeros in inverse square force fields of randomly distributed sources in (at least) 1, 2 and 3 dimensions? But of course you've now in a rotational frame. $\endgroup$
    – uhoh
    Apr 27 at 16:21
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    $\begingroup$ Keep in mind that Lagrangian point stability is a long-term matter, the perturbations tend to cancel out over many orbits. The universe isn't old enough for galaxies to have had large numbers of orbits. On an intergalactic scale, the timescales can be measured in star lifetimes. The Magellanic Clouds, if they're actually gravitationally bound to the Milky Way, have an orbital period of at least 4 billion years. Even the sun takes ~240 million years to complete an orbit within the Milky Way. $\endgroup$ Apr 30 at 13:11

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Short answer: likely sometimes.

The gravitational potential of a galaxy is not going to be a spherically symmetric $1/r$ potential, since it is the sum of the potentials of all the stars, plus from a diffuse dark matter component. In particular it can be flattened.

That said, in practice one can decompose the potential using a multipole expansion where we in essence make a Taylor expansion in terms of powers of $1/r$. A flattened distribution will have a nonzero $1/r^2$ potential component, a more lumpy one quadrupole ($1/r^3$) and higher order moments. Since at large $r$ these terms vanish faster than the monopole $1/r$ term we well-separated, fairly spherical galaxies will in the large behave as mass-points. Hence there would be Lagrange points for the circular or elliptic restricted three body problem for them.

One can wonder if the corrections due to the multipole moments break the Lagrange points anyway. Without doing numerical simulations it is hard to tell, but a simple heuristic argument is that the L4 and L5 Lagrange points do exist for any $1/r^\alpha$ potential (I have still not found a neat way of showing this except reams of algebra). Hence extra multipole terms locally just perturb the potential as if it was not quite $1/r$, but there will be Lagrange points in the vicinity.

There is likely no hard criterion for the existence of Lagrange points in the general galaxy problem since even defining stability is subtle (real galactic mass distributions orbiting each other will change each other's shape, making the problem non-periodic). In the Local Group case Andromeda and the Milky Way are on their way to a multi-encounter merger that isn't possible in mass-point mechanics and would mean any Lagrange points would have to follow into the merger and then disappear. Plus M33 and other nearby galaxies are turning it into a n-body problem even if everything were just well-behaved mass points*.

[* In a n-body problem with chaos, generically there will be an infinity of unstable periodic orbits for the main bodies. Adding a test body to the resulting potential from one such periodic orbit generically also has an infinity of unstable orbits, so in a sense you can get "Lagrange trajectories" that return to the same place once a certain integer number of main body orbits have passed. However, I don't know if there is any guarantee that there is any trajectory that does this in one orbit, and I think it is super-unlikely (unless we choose the problem cleverly) that it would retain constant distance to two of the bodies.]

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