Perhaps the best known mass profile is the Navarro-Frenk-White (NFW) profile (https://en.wikipedia.org/wiki/Navarro%E2%80%93Frenk%E2%80%93White_profile). The Wikipedia article also mentions an alternative (the Einasto profile).

Note that the Halo Dark Matter distribution clearly extends past the edge of the visible disk (i.e., the total galaxy is much bigger than the visible galaxy).

There is also the "Dark Disk," which is thought to have been feed by dark matter captured during mergers and collisions with other galaxies. The dark disk has a smaller surface density than the Halo dark matter, but roughly comparable volume density to the Halo material, if you happen to be nearly in the galactic plane, as is the Solar System. My single favorite reference for this is

Just, A., Jahreiß, H., 2010. Towards a fully consistent Milky Way disc model - I. The local model based on kinematic and photometric data. Mon. Not. R.A.S. 402, 461–478. 0910.3481.

rho(r) = rho_0 / (r/Rs) / (1+ r/Rs)^2 is what is given in wikipedia

for the NFW mass profile.

at r --> infinity it is vanishing

ar r--> 0 it is too big. Perhaps a singularity.

If there is a very high density dark matter core then,

even visible matter will fall to the galaxy core by the

gravitational attraction. This will be observable

by ordinary telescopes. How to solve this anomaly ?

Any suggestions ?

I must point out that the question presumes that proper gravitational evaluation of spiral galaxy rotation would inficate that rotational velocity should diminish as a function of radial distance - therefore requiring some explanation for the observed flat rotation curves.

This is certainly the case if simplistic two-body Keplerian equations are applied to disk objects, but if the distribution of mass and gravitational force imparted to objects within spiral galaxy disks are properly evaluated there is no two body inverse square diminishment of rotational velocity indicated. For example, see http://arxiv.org/abs/1101.3224.

When observed flat rotation curves can be reproduced by proper gravitational evaluations, there is no need for a vast, extended halo of compensatory mass - any compensatory mass may be found within the visible spiral galaxy structures.

Please see http://news.sciencemag.org/2013/01/has-milky-way-lost-weight and its subject research http://arxiv.org/abs/1205.6203v2. This research evaluated the Keplerian motion of stars within the Milky Way's stellar halo, at distances from 50-150 kpc, to determine the mass within their orbital radii. In section 6.3, "A low mass, high concentration halo?" It states:

"Given that the total mass within 50 kpc is ~ 4 ×10^11 [solar masses], our results suggest that there may be little mass between 50 and 150 kpc."

Of course, if 'missing' galactic mass was less than commonly expected (perhaps equivalent to that of luminous masses) and might all be contained within the visible galactic structure, it might all consist of ordinary, baryonic matter - dim objects such as black holes, cold neutron stars, brown dwarfs, planets, etc., plus even undetected cold plasma and luminous stars obscured by foreground dust, etc.

I think it is fair to say that the Cooperstock and Tieu work has not met with universal acceptance : http://arxiv.org/pdf/astro-ph/0701019v1.pdf http://arxiv.org/pdf/astro-ph/0601191.pdf http://arxiv.org/pdf/gr-qc/0511082.pdf http://arxiv.org/pdf/astro-ph/0510750.pdf http://arxiv.org/pdf/astro-ph/0508377.pdf

By contrast, the Deason et al work (1205.6203) seems fairly sound, but they are talking about a change of ~ 50% in the mass of the Milky Way. If we actually knew it to a factor of 2, I would be pleased, I don't think it changes much in what I said.

Note that there _is_ a very high density dark matter core to the Galaxy, thought to be a supermassive black hole plus "hangers on," with a total mass of ~ 10^6 Solar Masses

Marshall,

Yes, as had to be explained in the Carrick & Cooperstock preprint referenced above, or http://www.dx.doi.org/10.1007/s10509-011-0854-z, published five years after the Cooperstock & Tieu paper:

"Understandably, our work was problematic for the many researchers who had relied upon traditional Newtonian gravity theory for their research. They could ignore the alternative theories as being outside of their sphere of accepted science but general relativity was accepted by virtually all researchers as the very best theory of gravity. Thus our work was the subject of intense scrutiny and criticism. It is to be emphasized that we have responded to all the issues of criticism and to this point, as far as we are aware, there has not been any new rebuttal offered by any of these critics. As well, there has been some work by others lending support to our findings.

To round out a unified picture of our work, it is worth a brief recount of the key areas of criticism..."

If you have some specific issues with the later work, I'm sure you could take it up with the authors, rather than dismissing their work in such an off-hand way. If you have some specific issues please let me know.

Marshall,

Regarding the The Deason, et al., work, there are two significant findings: the reduced mass estimate and that there is little mass from 50 to 150 kpc.

This last conflicts with the existence of a massive dark matter halo extending into and beyond the observed galactic halo composed of ancient stars, star clusters and dwarf galaxies. Again, this finding supports models that indicate that flat rotation curves can be produced solely from matter within the boundaries of observed galaxy structures.

Marshall,

Regarding the expected very high density dark matter core, this has nothing to do with Sgr A*.

As I understand, models of expected dark matter behavior indicate that it should become denser near a halo's center - as a result of self-gravitation. Please see http://en.wikipedia.org/wiki/Cuspy_halo_problem.

I didn't see that in Carrick & Cooperstock, but, by resorting to innuendo they have lost a lot of credibility in my estimation. I started out doing GR, and am well aware of the tricky nature of doing calculations in GR. These tend to revolve around the choice of gauge, mixing gauges, and confusing coordinate effects for observable effects. So, frankly, when I see something like this I assume that there is a coordinate / gauge error somewhere until convinced otherwise, and it seems, from a read through of the counter articles, that that was likely the case.

Now, maybe they are correct, and everyone else is wrong, although my GR inspired physical intuition says no. Then they are simply going to have a convince enough people in the field that this is so, Until then, I, myself, will trust my physical intuition.

Note that the galactic rotation curve is only firm out to about 20 Kpc - see http://arxiv.org/pdf/0811.0859v2.pdf - and it is fairly flat all the way out to there. If there is little mass beyond 50 Kpc, that is fine by me.

Marshall,

I think that Fred Cooperstock has far more expertise in the field of GTR research than you might have gained in you brief experience. I have no idea what 'innuendo' you refer to - IMO it is you who are resorting to innuendo in your casual dismissals of the work referenced.

The passage I quoted is in section 5, "Perspectives", pages 16 - 19 in the preprint and pages 327 and 328 in the published article. That passage is but a prelude to a 'brief' discussion describing the criticisms they have responded to and references to the detailed responses. There is no need for you to rely on your intuition - just read the later article before passing judgement.

Marshall,

Note that the diameter of the Milky Way disk is considered to be between 31 an 38 kpc. As I understand, dark matter halos are generally considered to produce flat rotation curves by virtue of their generally flat radial densities. Of course, a dark matter halo could not be truncated at the largest radii of measured disk rotation - it must gradually disperse to form a smooth, elliptical halo. That requires that halos extend far beyond the largest radii of measured flat rotational velocities. For example, http://en.wikipedia.org/wiki/Milky_way#Size_and_mass states that a "dark matter halo is spread out relatively uniformly to a distance beyond one hundred kiloparsecs from the Galactic Center."

The Daeason, et al., finding - that there is little mass from 50 to 150 kpc - is problematic for any dark matter halo whose densities are flat throughout the measurable galaxy disk region, since it implies a very rapid peripheral extinction of dark matter.

FYI - a model of spiral galaxy rotation that employs Newtonian dynamics to obtain flat rotation curves from observed galactic structures can be found at http://arxiv.org/pdf/1104.3236v4.pdf.

Going out a little from the (likely) black hole in the center of the galaxy, there is the so called "core-cusp problem," where the numerical integration of Cold Dark Matter (which makes for cusps in the dark matter density near the galactic center) does not agree very well with observations (which indicate a fairly flat dark matter density there) - see http://arxiv.org/pdf/0910.3538v1.pdf

There was a fair amount of hope that Warm Dark Matter (WDM - dark matter with a small, but non negligible velocity at recombination) could fix problem (streaming of DM particles with velocity erases structure below a certain scale). These velocities are generally expresses as v_rms(0) - i.e., the rms velocity propagated to now - and are of the order of 100 m / sec (although this can be confusing as these velocities are also expressed in terms of keV limits of the WDM mass, assuming the WDM to be microphysical). Limits from the so-called Lyman alpha forest are now making this idea less viable, as the WDM is limited to being so cold (v_rms(0) < 56 m/sec) it has the same cusp problem as the standard Cold Dark Matter. So, while the measurements seem (to me) not to be in question, how you obtain the flat core remains an open question.

See http://arxiv.org/abs/1309.6971, http://arxiv.org/abs/0812.0010 and http://arxiv.org/abs/1309.5960 for more on this.

>> If there is a very high density dark matter core then,

>even visible matter will fall to the galaxy core by the

>gravitational attraction. This will be observable

>by ordinary telescopes. How to solve this anomaly ?

>Any suggestions?

In Atomic physics also one came across such problems.

Why the electron does not fall into the nucleus by

electrostatic attraction? At that time we postulated that

there are some allowed orbits only (OLD QUANTUM THEORY).

Here we are revisiting such a scenario. Why stars do not

fall into the galactic core which has almost infinite dark matter

density in NFW model.

Some repulsive force must be there. But part of repulsion is

the rotational centrifugal force. This will increase for

large angular velocities of course. That means to

balance very high gravitational attraction, the rotational

velocity also have to be very large to balance it.

Again very high angular velocity will not be allowed

by the flat rotation curves !! So one goes from one

problem to another problem.

Can the general theory of relativity help ?

Biswajoy,

"Some repulsive force must be there. But part of repulsion is the rotational centrifugal force. This will increase for large angular velocities of course. That means to balance very high gravitational attraction, the rotational velocity also have to be very large to balance it. Again very high angular velocity will not be allowed by the flat rotation curves !! So one goes from one problem to another problem."

It seems to me that you are referring to a conceptual model that presumes only a centralized mass, two-body (planetary) relation between galactic mass and centrifugal effects.

Please consider that a peripheral disk object observed to be rotating at high velocity will be subject to large centrifugal effects, but that it is not actually gravitationally bound only to a single mass at a single separation distance (r). In actuality, a peripheral disk object is gravitationally bound to many, perhaps hundreds of billions, of comparably massive neighboring objects - at a vast distribution of separation distances.

As a result, the centrifugal effects imparted to a peripheral object rotating at high velocity are not balanced against a single attraction to the distant galactic center (perhaps at a distance of dozens of kpc) but to a vast array of attractions to massive objects distributed throughout the galaxy, especially at relatively small separation distances (higher net G force) to neighboring objects within the disk.

This perspective might be more completely explained in my brief, informal profile essay. Sorry that I cannot better explain in physics and math terms.

Excellent points, James. Food for a different intuition!

Biswajoy,

"Can the general theory of relativity help ?"

In "General relativistic dynamics applied to the rotation curves of galaxies," Carrick & Cooperstock argue that it is inherent limitations in Newtonian physics that produce the improper representations of mass distribution and resulting gravitational interactions that produce spiral galaxy gravitational evaluation results that seem to require the presence of dark matter.

See http://arxiv.org/abs/1101.3224, page 4:

"In dismissing general relativity in favour of Newtonian gravitational theory for the study of galactic dynamics, insufficient attention has been paid to the fact that the stars that compose the galaxies are essentially in motion under gravity alone (\gravitationally bound"). It had been known for many years, in fact since the time of Eddington, that the gravitationally bound problem in general relativity is an intrinsically non-linear problem even when the conditions are such that the field is weak and the motions are non-relativistic, at least in the time-dependent case. Most significantly, we have found that under these conditions, the general relativistic analysis of the problem is also non-linear for the stationary (non-time-dependent) case at hand. Thus the intrinsically linear Newtonian-based approach used prior to our work has been inadequate for the description of the galactic dynamics and Einstein's general relativity must be brought into the analysis within the framework of established gravitational theory."

However, I think that disk mass distributions' gravitational interactions can be properly represented using Newtonian dynamics. The methods of GTR may or may not better facilitate proper representations - I can't evaluate.

Again, a model of spiral galaxy rotation that employs Newtonian dynamics to obtain flat rotation curves from observed galactic structures can be found at http://arxiv.org/pdf/1104.3236v4.pdf.

@James

Thank you for the references. I must look at them.

Perhaps one should separately consider hot component

and cold component of the dark matter while considering

GTR effects. I know that a linearization of GTR can give

normal Newtonian gravity. Perhaps one should keep

terms which are neglected while one is doing linear approximation.

This paper just came out today : http://arxiv.org/abs/1401.5799

"Where Do Galaxies End?" - the author says for the Milky Way "for half-mass halo

assembly at z = 1.06±0.03" its 153 +- ~ 20 kPc..

Marshall,

The paper's intent is to refine the definition of galactic components.

Its abstract also states:

"... Physical criteria are proposed to define bound structures, including a more realistic definition of Rvir(M*,Mh, za) for stellar mass M* and halo mass Mh, half of which formed at “assembly redshifts” ranging from za ≈ 0.7 − 1.3. We estimate the extent of bound gas and dark matter around L* galaxies [like the MW & M31] to be ~ 200 kpc..."

- it continues:

"... This formalism is intended to clarify semantic differences arising from observations of extended gas in galactic halos, circumgalactic medium (CGM), and filaments of the intergalactic medium (IGM). Astronomers should refer to bound gas in the galactic halo or CGM, and unbound gas at the CGM-IGM interface, on its way into the IGM..."

Section 4, "Discussion and Conclusions" states on page 17:

"Their mass sum, log M ≈ 12.55±0.14, is comparable to recent estimates of the timing mass of the Local Group, log MLG = 12.69+0.13 −0.19 (van der Marel 2012), although a recent study (Gonzalez et al. 2013) used cosmological simulations of halo pairs and a likelihood correction to find a lower value, log MLG = 12.38+0.09 −0.07. For either of these masses, the Milky Way and Andromeda are likely to have radial extents ~ 200 kpc."

There are widely varying estimates of Milky Way mass in recent studies, and new studies will soon be published...

https://arxiv.org/pdf/1609.05917v1.pdf

What now if the dark mass has the strong correlation with the mass of baryonic matter in galaxy? Do every wimp have a conventional partner partice (uh!) or must we throw the whole dark matter away?

My old side-understanding on GR that there is a need for serial signalling between all spacetime events for updating correct state of changed gravitational field seems to be tempting to study again, including expotential delay affection on longer distances and manages with cluster collisions too...

Esa, thank you for the link you provided. From a quick glance at the paper what I gather is that there is a link between observed acceleration in spiral arms to what is predicted by the presence of baryons in the region with radius less than the point of observation. At the end of the paper there is a short paragraph on the possible effects of this (experimental) result on models of dark matter. The authors think that if dark matter behaves like a fluid, it should explain the result. I guess that means that dark matter should have self interaction properties other than that provided by gravity.