Density of dark matter along the galaxy

Question

I was reading these 2 interesting articles about dark matter inside the solar system:

But I can't figure if:

A) Dark matter doesn't affect the planetary motion because this can't radiate, and thus has large orbits and then is less dense in the inner region of the galaxies and more dense in the outside regions. Or,

B) Dark matter has equal low density across the galaxy (compared with the higher density of baryonic mass inside the solar system), and simply his effects are only important if we sum all the matter in the vast space between stars.

I'm confused because I tend to think that if the dark matter is in a kind of shell, mainly outside the galaxy, this would help to spread the stars far from the galaxy center instead of contain everything in his relative rotation place. I can't find a distribution graph by the way.

Excuse my simple english and thanks in advance for your help!

Edit: I had heard a wrong idea in some documentaries, or I misinterpreted it, about a greater DM density in the outside region of the galaxies. Thanks to @KyleOman for pointing out DM is denser in the center.

<Heresy_mode=1> Dark matter problems are caused by the fact that it dos not exist. What is seen as being caused by the existence of dark matter is in fact due to the behaviour of gravity being misunderstood. Gravity is not simply described as a function of mass and the square of the inverse of the separation of the masses involved but has "other factors" which are not apparent at normal 'scales'. Note that this heresy is considered liable to be correct in some form by some relatively eminent investigators. As planetary motion demands inverse square law such effects must reduce to zero .... — Russell McMahon, Jul 29 '15 at 02:34
Russell is discussing extended theories of gravity that may have some merit, but do not produce any difference in the observables to distinguish one of them from dark matter. — Kyle Kanos, Jul 29 '15 at 02:40
@RussellMcMahon the alternative/extended gravity theories currently have less convincing evidence than DM, in my opinion. Not to say they have no merit; there are some big open questions in both DM research and alternative gravity research, but I think we're a long way from claiming conclusively that either GR is wrong or explaining what DM is in full detail. So yes, your statement is heretical, because you use "are" instead of "may be" ;) — Kyle Oman, Jul 29 '15 at 03:12
possible duplicate of Why doesn't dark matter affect planetary motion? — ProfRob, Jul 29 '15 at 07:38
@KyleOman "are" was no mistake :-). Nor: " ... does not exist ... " , " ... is in fact ...", " ... is not simply ... ", " ... but has ... " and " ...are not apparent ..." :-). And, yes, I know I don't KNOW (of course), but Occam cries out from his grave for this to be the "preferred" choice that the others are tested against. 'Broken' gravity or 5th force is, IMHO as always, liable to 'prove true'. TBD :-). — Russell McMahon, Jul 29 '15 at 08:10
@RussellMcMahon: of course, modified gravity may still be right, but you're asking way more of Occam with most realistic modified gravity schemes (seriously, look up the TeVeS Lagrangian) than you are by just saying "let's extend the standard model to include a stable 20 TeV [or whatever] WIMP plus a one-parameter cosmological constant". YMMV, of course, but non-radiative matter isn't a wild assumption. — Zo the Relativist, Jul 29 '15 at 19:23
Cosmological constant. Right!. Bring it on! :-).//Joking. The more interesting things that pop out of the woodwork along the way, the better. — Russell McMahon, Jul 30 '15 at 09:27
@RussellMcMahon I think the main point to keep in mind is this: Please don't present controversial theories as established truth. It's fine that you think the theory is correct, but comments like your first are kind of misleading to beginners. — Danu, Jul 30 '15 at 11:50
@Danu Hopefully being prefixed with "<Heresy_mode=1> ... " would give most people some clue. Followed by the material proper sandwiched between that and " ... Note that this heresy is considered liable to be correct in some form by some relatively eminent investigators." Suggests that the intervening material is considered incorrect by others. I'd hope that even someone who was not totally conversant with the finer points of heresy would still get the general adumbration. — Russell McMahon, Jul 30 '15 at 12:20
@RussellMcMahon Heh, okay :) I guess we're all on the same page then ;) — Danu, Jul 30 '15 at 12:34
@KyleKanos Thanks for the link. I saw the Navarro et al. distribution before asking, but I haven't the needed physic background to give value to the variables and then be able to graph the curve. — Leopoldo Sanczyk, Aug 04 '15 at 04:32
@RobJeffries I'm sorry, I visited the linked question before asking, but I'm an engineer from the IT forum, and I have not the reputation here to expand that thread with my doubts. — Leopoldo Sanczyk, Aug 04 '15 at 04:36

score 2 · Accepted Answer · edited Apr 13 '17 at 12:39

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Dark matter has a small/negligible influence in the Solar System because there isn't all that much of it in the Solar System, compared to say the mass of the Sun.

The NFW profile is the current default density profile for DM "haloes" (spherical-ish self-gravitating structures, such as the one in which the Milky Way resides). This is a fit to the density as a function of radius for dark matter haloes in cosmological simulations, and in a very broad sense it seems to work pretty well in the real Universe, most of the time (though see below about "cores"). The density is high in the centre, and decreases are $r^{-2}$ for a while, then $r^{-3}$ further out. The formula will give you infinite density at zero radius - clearly this is unphysical, but the point is that the central density rises sharply toward the centre to some high value. Plugging in parameters to the NFW profile for a Milky Way sized galaxy and evaluating the density at $8\,{\rm kpc}$ (distance of the Sun from the centre of the galaxy), I get about $8\times10^{6}\,{\rm M}_\odot\,{\rm kpc^{-3}}$, or about $9\times10^{-19}\,{\rm M}_\odot\,{\rm AU}^{-3}$. The volume of the Solar System is say about $30000\,{\rm AU}^3$, so the DM is outgunned in mass by the Sun by a factor of $4\times10^{13}$. In perhaps more familiar units, my estimate gives $5\times10^{16}\,{\rm kg}$ of DM - compare that with some Solar System bodies and you'll find that it's something along the lines of a medium asteroid. And the DM is diffuse all over the Solar System, so it's even more insignificant than a medium asteroid, gravitationally speaking.

So why is it such a big deal? Because space is big, all that interstellar space has similarly puny densities of DM, but there's so much space that it adds up to a lot of mass - typical estimates say that there should be about $10-100$ times more DM mass in the Milky Way than star and gas mass.

There are other density profiles that are proposed, e.g. Einasto profile, Di Cintio+2014 profile, and a handful of others. Qualitatively they're all fairly similar (except for "cored" profiles, for which I'll point you to a wiki article and, shamelessly, to my own work).

Just to cover all the points in your question, the DM distribution is certainly not a shell outside the galaxy - more like a cloud (denser in the middle) inside which the galaxy lives. And it must be diffuse - it cannot collapse to form dense structures like a DM star or a DM planet (provided something like the standard $\Lambda$CDM theory applies).

Please let me know if you'd like anything clarified, or if you have any followup questions post them and poke me, I'd be happy to have a look :)

edited Apr 13 '17 at 12:39

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answered Jul 29 '15 at 03:45

Kyle Oman

18,441

within the solar system it is my understanding that wrt "GISL" = "gravitational inverse square law" behaviour (1) If we consider only "conventional" matter then the GISLcan be shown to operate EXACTLY to the limits of precision available and over any time span chosen, subject only to relativistic effects. (2) If any dark matter is present the the conventional matter would not be able to fully account for GISL behaviour so conventional matter would have GISL apparently violated. BUT (3) Ongoing Newtonial orbital planetary behaviour depends on law being EXACTLY 1/2.0000000000000.... – Russell McMahon Jul 30 '15 at 12:29
... and any departure from true exact inverse law would be detectable by orbital motion subject to measurement limits - and that "even an asteroids worth" of DM should over enough time perturb orbital behaviour measurably with current capabilities. So (4) ie time and necessity for exact GISL behaviour acts as "a very long lever" to finely test for in system DM well beyong what we otherwsie could : SO QUESTIONS: (i) Is my above "understanding correct? (ii) If not why not and what are the fallacies in the understandingg? [I assume as of right that the above will be considered wrong]. – Russell McMahon Jul 30 '15 at 12:35
@RussellMcMahon "Exactly" does not exist in science. The inverse square law is exact up to our measurement limits. We don't even know where all the asteroids are yet - if we measured some perturbation to orbits, what's to say it isn't because of some asteroids? Of course people do try to constrain the density of DM in the Solar System. So far, there are some upper limits, but not sufficiently constraining limits to be worrisome for e.g. LCDM. – Kyle Oman Jul 30 '15 at 17:35
I'm aware of the point you are making re "exactly" (as you might expect). As far as can be told inverse square law is exact. ie just because we cannot measure it does not mean it cannot be exact - we just can't tell. | The arguments about not knowing where all asteroids are fails to address my question. We know where much mass is with good precision and running the system over and over and .... for a few million years allows us to see trends, should they exist, than our ability to measure the raw variable allows. IF ISL was not 2.000000000 there would be various disturbing hints by now. – Russell McMahon Jul 31 '15 at 09:11
If this is true (as I assert) then you need to decide whether very small quantities of DM in-system would be undetectable by this trivially simple tool OR if it may not be present in-system for whatever reason. If in fact DM is "allowed" in-system without being measureable by ISL deviations then presumably there are papers affirming this. Q: Are my assertions re ISL providing a very fine tool for determining DM effects "in system" correct? Why not? Any known references? – Russell McMahon Jul 31 '15 at 09:15
Note: Exponent > 2 - planets spiral out. Exponent < 2 - planets spiral in. Exponent = 2.00000000000... - orbits stable. – Russell McMahon Jul 31 '15 at 09:17
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There is a small inaccuracy in your post, close to the center of the galaxy the NFW gives $\rho\propto r^{-1}$ not $\rho\propto r^{-2}$. – Virgo Aug 04 '15 at 05:08
@KyleOman Thank you very much! Certainly the idea of DM outside the galaxy counteracting the gravity of baryonic mass mistook my reasoning. Then is a mix: the "space is big" from the option B and "DM can't collapse" from A, with the exeption that DM is not equal nor denser outside. Instead, DM is more dense in the inner galaxy. I had read about it in the "Cuspy Halo Problem" but I don't wanted to complicate my question until I understand the subject a bit better. – Leopoldo Sanczyk Aug 04 '15 at 05:42
@KyleOman Now I suppose the counteracting effect is due to DM extends far from the visible outside region of the galaxy. Is it right? Thanks again for your great help! – Leopoldo Sanczyk Aug 04 '15 at 05:43
@RussellMcMahon I think you're confusing extra mass: $F = G(M+dM)m/r^2$ with a change to the inverse square law $F = GMm/r^{2+\epsilon}$. The latter would probably rear its ugly head and make itself known; the former is much more subtle. – Kyle Oman Aug 04 '15 at 15:41
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@LeopoldoSanczyk I think you've got more or less the right idea now. One thing to think on: the DM distribution is spherical-ish, and there is the shell theorem. So the DM at super-solar galactic radii has almost no effect on anything at smaller radii (almost because spherical symmetry is only approximate). – Kyle Oman Aug 04 '15 at 15:50
@Kyle No - just noting that changes to the first caused by matter which was present and not seen would APPEAR to be violations of the second if you did not account for the extra mass. They would not be consistent "violations" due to the DM 's distribution - just things that did not work quite as they ought. eg Mercury is in slightly the wrong place :-) (but that is of course for 'other reasons'). As you say - it would probably be subtle. BUT as we have good reason to believe that the inverse square law exponent is exactly 2 ANY variations that cannot be explained points to something unusual. – Russell McMahon Aug 04 '15 at 16:07
@Kyle ... conversely, if the inverse square law appears to work very well indeed using observed mass then it implies a lack of unobserved mass. As orbits repeat indefinitely even vvv small violations of apparent ISR would become evident with enough time. The limit is probably uncertainties due to medium induced 'losses' - I imagine that they are small for planet sized masses. – Russell McMahon Aug 04 '15 at 16:11
@RussellMcMahon How do we know the mass of the Sun? By watching things orbit it. How do we know all that measured mass is in the Sun, and not ~uniformly distributed within the smallest orbit measured? We don't. This might be worth a question, rather than a massive comment string? – Kyle Oman Aug 04 '15 at 16:18
@KylenOman thanks for your remark! I've never intuited the second part of the shell theorem! Had a lot of sense now, haha. I will open a new question to not bother you with more comments. I expect to gain some reputation points soon to upvote your answer. Thanks a lot for your help! – Leopoldo Sanczyk Aug 04 '15 at 20:15

Density of dark matter along the galaxy

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