I am doing an Introductory course in astrophysics UG level. I have studied that mass of the gases ($M_{gas}$) decreases with time as new stars are formed but mass of stars ($M_{star}$) doesn't increase because stars die out too. What about the dark matter halo ($M_{halo}$)? Does its mass increase or decrease over time?. Also how do $M_{gas}/M_{halo}$ and $M_{star}/M_{halo}$ change as the galaxy ages?
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There is recycling of mass going on. As stars form they deplete the gas mass, but as they age they replenish the gas mass with a fraction through mass loss and supernovas.
Current thinking is that most halo formation has already occured, so the mass stays constant. There is perhaps a trickle of loss due to capture in massive objects (if there is some cross section between the DM particles and dense matter) including black holes, but this is vanishingly small over the stelliferous era (the next $\sim 10^{14}$ years or so). Over the degenerate era beyond this may matter more: if dark matter consists of particles and antiparticles they would slowly annihilate in the halo, and more rapidly inside the dense objects. Even if it doesn't annihilate, it will eventually very slowly fall into the central black hole in the black hole era due to gravitational radiation.
In the large, $M_{star}/M_{halo}$ begins low, as stars form it increases towards the current value and will keep on increasing across the stelliferous era (but still, it is $\ll 1$). Then around $10^{20}$ years in the future it will start declining as stars are randomly ejected from the galaxy or absorbed by the central black hole due to rare encounters, eventually becoming zero (or halving, if you still count the ejected stars). $M_{gas}$ will keep declining across the stelliferous era as it is used up, but infalling interstellar gas may slow the decline and make more stars for some time; in the degenerate era the small amount of remaining gas will keep on condensing on bodies or get ejected with them.
(See the excellent Adams & Laughlin paper for an overview of the long-term issues.)
Anders Sandberg
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1I don't think the far future is what the OP had in mind (that's rarely something you'd concern yourself with in an intro course). Also, your statement about Mstar/Mhalo increasing isn't necessarily true for all masses, since you also have baryons being blown out of small halos, as well as accretion of pristine material. I think the ratio is a much stronger function of halo mass than it is on age. – pela Nov 02 '20 at 13:10
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You can sometimes have $M_{\rm star}/M_{\rm halo}$ get larger if the halo is tidally stripped, as in this paper on a possible formation mechanism for dark-matter-deficient dwarf galaxies – Peter Erwin Nov 02 '20 at 23:22
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@Anders Sandberg would Mg/Mh would be <<1 in the stelliferous era too? – Nitin Kumar Chauhan Nov 15 '20 at 14:08
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@space_nkc - Yes, the gas content of the galaxy has been smaller than the star content for a long time now. – Anders Sandberg Nov 15 '20 at 22:22