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.

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    $\begingroup$ 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. $\endgroup$ – Kyle Kanos Jul 29 '15 at 2:40
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    $\begingroup$ @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" ;) $\endgroup$ – Kyle Oman Jul 29 '15 at 3:12
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    $\begingroup$ possible duplicate of Why doesn't dark matter affect planetary motion? $\endgroup$ – Rob Jeffries Jul 29 '15 at 7:38
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    $\begingroup$ See also physics.stackexchange.com/q/194107 $\endgroup$ – Rob Jeffries Jul 29 '15 at 7:40
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    $\begingroup$ @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. $\endgroup$ – Jerry Schirmer Jul 29 '15 at 19:23

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 :)

  • $\begingroup$ 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.... $\endgroup$ – Russell McMahon Jul 30 '15 at 12:29
  • $\begingroup$ ... 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]. $\endgroup$ – Russell McMahon Jul 30 '15 at 12:35
  • $\begingroup$ @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. $\endgroup$ – Kyle Oman Jul 30 '15 at 17:35
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    $\begingroup$ There is a small inaccuracy in your post, close to the center of the galaxy the NFW gives $\rho\propto r^{-1}$ not $\rho\propto r^{-2}$. $\endgroup$ – Virgo Aug 4 '15 at 5:08
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    $\begingroup$ @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). $\endgroup$ – Kyle Oman Aug 4 '15 at 15:50

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