I haven't seen any explanation for Dark Matter that explains its distribution. It's always "hey, if there's some stuff with this distribution it will explain this observation". An example would be a halo that fixes a rotation curve. If it interacts with visible matter gravitationally then why doesn't it take on the same distribution? That is never explained.
My other issue is that in discussions of rotation curves, I keep seeing reference to Kepler, which simply shouldn't apply. Where can I see the math behind the "expected" curve - I suspect an error.
"My other issue is that in discussions of rotation curves, I keep seeing reference to Kepler, which simply shouldn't apply. Where can I see the math behind the "expected" curve"
"Keplerian" in this context is an approximate term. It refers to the fact that most of a galaxy's visible mass is centrally concentrated, and so as you get further and further away, with virtually all of the mass inside whatever distance you're at, the rotation curve should converge on a true Keplerian one, because the difference between the effect of the true mass distribution and one where all the galaxy's (visible) mass is concentrated in a point at the center gets smaller and smaller.
Actual published fits to galaxy rotation curves always use the measured visible-mass distribution for a given galaxy to compute the non-dark-matter curve. No one working in this field is confused about this.
As long as physicists disagree on the explanation of anomalous physics, it is quite accurate to say that the physics community is in a state of doubt or confusion about the anomaly.
Note that this person explicitly asks for the dataset, and the computation of the expected curve, but hardly ever does anyone help such a person in such a direction, it is always taken as a suspected insult on the mental state of one camp of interpretation.
I too would love some kind of central register or portal for the most widely accepted anomalies (anomalous.physics/dark-matter/...), where people can get and inspect observation datasets, and competing models to fit the datasets.
Imagine Brahe & Kepler's, data & interpretation to be widely popular, but whenever someone asked for data or computations to compare circular orbits with elliptical ones, nobody would point them where to find such data and computations?
(The fourth link, for example -- http://astroweb.cwru.edu/SPARC/ -- has both observed rotation-curve data and computations of expected rotation curves.)
There is zero validity to treating galactic mass as a point mass. That is exactly one of the mistakes I suspect keeps being made. At best it is a misapplication of the divergence theorem. Disks dont behave like spheres and rings dont behave like uniform shells. Proximity matters.
It generally works a bit better if you say something like, "Hmm... it seems like this approach would be wrong, for this reason that just occurred to me. Am I missing something? Or: How do people in the field actually do it, so as to avoid this error?"
If, on the other hand, you assume they must all be stupider than you are and say things like "That is exactly one of the mistakes I suspect keeps being made", then you're basically saying, "I'll bet none of the hundreds or thousands of people working in this field for decades have ever thought of this one point that just occurred to me!" The latter is, shall we say, rather unlikely.
(In point of fact, Newton's shell theorem generalizes to the case of axisymmetric, flattened spheroids with homeoidal density distributions, a result that was derived by Laplace and others in the 18th and 19th Centuries. Which means that disks do behave somewhat like spheres.)
To get a sense of what's really involved, you could look at something like Brandt's 1960 paper, and then some of the papers that cite it (including some of the classic early papers by Vera Rubin and collaborators), to get an idea of how much more sophisticated than an simple Keplerian rotation curve:
what you say is obviously true, but could astronomers really be making this elementary kind of mistake? one would have to completely lose touch with elementary physics to make that mistake...
For modeling individual galaxies, that's more or less correct. (Of course, the same applied to things like MOND: "hey, if Newtonian acceleration is tweaked with this interpolation function and that free parameter, it will explain the observation".)
In a large-scale, statistical sense, the answer is: initial quantum fluctuations (as seen in the Cosmic Background Radiation) + gravitational collapse and fragmentation in an expanding universe. Simulations done under these assumptions have done an increasingly good job of describing the general distribution of dark matter, and the typical statistical distribution within galaxies.
This is why it's good to do independent fits of dark matter distributions to galaxies: it provides something you test the simulations against. After all, if the individual-galaxy fits say dark matter generally has distribution X, but the simulation say that gravitational collapse and mixing should almost always produce something different, then you know you've got a problem.
"If it interacts with visible matter gravitationally then why doesn't it take on the same distribution?"
Because visible matter interacts with itself via radiation and hydrodynamics (e.g., gas pressure). So it gets pushed around in ways the dark matter can't. These forces can be much stronger than gravity in some circumstances (gravity, after all, is the weakest fundamental force), so the gas atoms, ions, electrons, etc. experience forces the dark-matter particles do not.
> If it interacts with visible matter gravitationally then why doesn't it take on the same distribution?
Because unlike visible matter, it does not interact in non-gravitational ways, even with itself.
Visible matter can clump to form planets and stars because when two particles of it are attracted enough to hit each other there is some interaction other than gravity, which helps eat some of their kinetic energy. In contrast, when that happens with two particles of dark matter, they just fly through each other, and if they were moving fast enough, never meet again.
There are non-gravitational interactions even at the galactic scale, even though they are very weak indeed compared to gravity. As our sun plows through space, the particles it's sphere of influence hits have a preferred range of velocities and directions, the "galactic rest". Over billions of years, this does influence where in the galaxy our sun is. Dark matter has no such influence on it.
Do we actually know that dark matter doesn't interact with itself or other matter, or do we just have an upper bound on how much it could potentially interact? And if so, roughly what is it?
My other issue is that in discussions of rotation curves, I keep seeing reference to Kepler, which simply shouldn't apply. Where can I see the math behind the "expected" curve - I suspect an error.