Why is the density of matter greater near the galactic center?
What did you discover so far ?
Given the evidence that a supermassive black hole exists at the center, the density there is not really knowable.
A singularity represents a situation which cannot be described by relativity or other mathematical models.
(or rather the math leads to a nonsense conclusion that the density would be infinite.)
Probably what really is happening inside a black hole is physics that we don't know about, so there is no way to model it.
What do you mean by near? Inside the black hole (which likely isn't quite at the MW's center of mass)? In gas clouds within 100 LY of the center of mass? In stars+gas within such a distance? ...?
As for why, the simplest answer is: because that's where gravity concentrates matter.
The simple answer?
Because matter tends to gravitate to the common center of mass/gravity that is shared by all bodies in that area of collective matter.
I was using 'near' the center in a relative sense - I had in mind the distributiion of regular matter and dark matter in spiral galaxies, where visible matter tends to concentrate 'near' the centre, while dark matter halo extends maybe upto five times the radius of the optical disk.
Ah! Well, in that case, the reason seems to be gas: For spirals, most of the bright stars are more concentrated than the halo and DM are, because most of the bright stars are new, having formed within, say, the last 1e9 years. They formed out of gas that had collapsed to a "pancake" due to dissipation of energy through collisions. And that's the key difference from stars and DM: They don't collide, they just move past each other. So they don't dissipate their orbital energy, they maintain widely distributed orbits.
For intermediate-age stars like the Sun, they were formed out of that same disk of gas. Although they aren't especially bright, so they don't dominate our perception of the galaxy's size, they are still part of the collapsed disk. And although they don't collide and dissipate their orbital motion (very much), they were already in the disk, so they don't change the story about the apparent (visible) size of the galaxy.
Very old stars, the galaxy's halo, are widely distributed, like DM: They apparently formed from gas that had not yet collapsed to form a disk. But they aren't bright, so for spirals, they don't dominate our perception of the visible size of the galaxy. (I do not know if the halo is as widely distributed as DM, however. It may be that this has not yet been determined.)
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