Waves in an Impossible Sea: AudioBook

2 Responses

1. The one interaction that dark matter does engage in with normal matter is gravitational pull:
   how does a Halo of dark matter around a Galaxy manage to exert a pull inwards allowing a Galaxy to spin as rapidly as it does?

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   1. To give you a rough understanding, let’s take a simple (but inaccurate) model. We can fix it later. Let’s imagine that the dark matter is just a sphere of constant density, total mass M, and radius R, centered around the galactic center, and that it far outweighs any stars and dust in mass. Call the distance from the galactic center r. Then, as Newton himself showed, the gravitational force exerted on any object of mass m at a distance r (less than R) from the center is simply equal to the gravitational pull of the sphere of dark matter that extends out to radius r (with all the dark matter at larger radius, from r out to R, having a gravitational effect that cancels out exactly.) In math,

      F = G M m /r^2 times (r^3/R^3) = G M m r/R^3

      and the resulting orbital speed is given by the circular acceleration formula

      a=v^2/r = F/m = G M r / R^3

      so that

      v = Sqrt[GM/R^3] r

      meaning that the rotation velocity actually increases with distance, rather than falling as 1/sqrt(r) as it does for planets in the solar system (or for any other system where most of the mass is at the center.)

      The rotational time T is (2 pi r)/v is then independent of r. Again, if most of the galaxy’s mass were dead center, then T would fall like 1/r^(3/2) as it does in the solar system’s planets.

      The data (as in https://en.wikipedia.org/wiki/Galaxy_rotation_curve#/media/File:Rotation_curve_of_spiral_galaxy_Messier_33_(Triangulum).png and most other galaxies) shows that the velocity does not increase at a completely constant rate, so the density of dark matter isn’t constant — but it must still be quite high, because the velocity of stars and gas at the outer visible edge of galaxies is still increasing or near-constant, showing that there is still plenty of dark matter beyond the visible part of the galaxy.

      All this assuming that Newton’s approximation to Einstein’s theory of gravity is applicable at these distances, as it is believed to be.

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