The following Web Site features a calculation of the rotation curves of stars in galaxies, that matches the observed motion exactly, uses Newton's law and baryonic mass only, without the need for dark matter.
www.galaxymethods.net
The site also reviews the generally accepted calculation in Binney and Tremaine which concludes that spherical shells of dark matter are needed, and purports to show where they went wrong.
I have seen or brought attention to this site in at least 5 BAUT forums, and no one has ventured forth to challenge the calculation or raise objection to it in any way. In a phrase, "the silence is deafening". I am bringing it forward as a question to all, because I cannot find fault with the calculation, and would like to hear any scholarly discussion of it.
Thanks, TomT

Well, you can no longer say this (which I strongly doubt is even true), because:
I've looked at the "calculation", but unfortunately no details are given that would actually make it possible to check it. (Unless you want to buy the guy's book-- it's basically a shameless con game). However, assuming that the guy isn't a complete charlatan (which again, is impossible to check given the information he gives), I think the following quote is telling: (the "SMD" is the surface mass distribution, so is density integrated over disk height)
What I think this quote means is, he did not feel constrained to follow the normal mass-to-light ratio when constraining his baryonic mass! Well I have news for you, if you can use any mass-to-light ratio you want, obviously you can fit a galaxy rotation curve without "dark matter"! The dark matter is simply absorbed into a larger mass-to-light ratio. C'mon folks, this is pretty seriously brainless stuff.

Also, the link author makes great hay out of the fact that he doesn't use a spherical mass distribution, that he can get his fit using disklike mass distributions. But one-dimensional rotation curve calculations imply no constraints about whether or not the dark matter distribution is spherical (because in the plane of the disk, the gravity from a spherical shell is obviously the same, by symmetry, as the gravity from a ringlike shell that is in the plane of the disk).
You always have to make some assumption about the dark matter distribution if you are only analyzing a 1D rotation curve, a fact that the author appears to blissfully unaware of. Despite his (unsubstantiated) claims to the contrary, I'm sure that Binney and Tremaine (the conventional solution) are in fact aware of this, and chose a spherical distribution for the dark matter either for simplicity, or because it recovers the vertical disk density structure (a Gaussian distribution vertically). I don't know which, but I do note that the "solution" advocated in this link is a purely 1D rotation curve, so also does not analyze the vertical disk density structure, which is the only way to constrain non-spherical aspects of the mass distribution, as I stated.

Bottom line: the guy's an engineer, but not a very smart one, despite any credentials he might have. He may (or may not) have correctly solved the relevant equations, but it matters little because the conclusions he draws from them are completely useless, for the reasons I've stated. Given this, I wouldn't recommend driving on a bridge built by him. Sorry TomT, the link is a waste of time, and I suspect you'll be scolded for linking to a site that is merely trying to sell a book.

This is just a little comment from a complete layperson, but I was a bit turned off my his line that "astrophysics is simple compared to engineering problems." My reply would basically be, there are no trivial problems. The trivial problems get solved simply. The problems in any field are difficult, because it is the difficult problems that remain unsolved.

__________________
As above, so below

Thanks, Ken, for making any additional perusal of the linked site unnecessary.

__________________
Everyone is entitled to his own opinion, but not his own facts.

Good point. That's kind of a jerk statement. What is he, a transportation engineer? "Engineering" is a very, very wide field. Mechanical? Environmental? Hydro? Chemical? Bio? Structural?

__________________
Everyone is entitled to his own opinion, but not his own facts.

Besides, this raises the issue of what astrophysicists are really good at. The author of the link may be good at solving equations (and it is certainly possible that he solved them correctly), but he is not good at seeing the ramifications of his own solutions. And he has misrepresented the ramifications of the conventional solution. This is the key element that makes a good astrophysicist-- the ability to understand a larger context.

Hi Ken G,
Thanks for your response. I have a few comments on your reply, but first I see that Nicholson's web site does not give a direct link to his 3 papers which should answer your questions about the details of his calculations, and his criticism of the Binney and Tremaine calculation. I forgot that the links to the papers were given a Baut forum, so without these, I can understand that you can't say much about the calculation and assumptions. For starters, Nicholson does include variable thickness effects, and if I am not mistaken, it is Binney and Tremaine that use only a thin disk assumption to account for thickness. Correct me if I am wrong about B and T, with a reference please.
The papers can be obtained in PDF form by clicking on the following.

Errors in equations for galaxy rotation speeds

http://xxx.lanl.gov/ftp/astro-ph/pap...09/0309823.pdf

Galaxy statics without dark matter

http://xxx.lanl.gov/ftp/astro-ph/pap...09/0309762.pdf

Galaxy Mass Distributions from Rotation Speeds by Closed-Loop Convergence

http://xxx.lanl.gov/ftp/astro-ph/pap...03/0303135.pdf

One further comment for now, Nicholson will make available a CD where you can perform the calculation on you home computer and play with it to your hearts content. Notes are included. This is provided essentially at his cost.
You labelled this as a grab for money. Did you expect him to send these out for free?
Again, Thanks for your response, TomT

I don't know if he is motivated by greed, perhaps his products are fairly cheap. But there is certainly a better way to get the word out in the standard literature if one's only goal is mutual enlightenment! Anyway, I'll talk a look at those links, but note that the vertical structure business is a detail for constraining the nonspherical structure aspects. The real issue is whether or not you need dark matter, and I guarantee you that he has either made an error, or he has allowed the mass-to-light ratio to be a variable, or both. Even if there are no errors, it is perfectly obvious that increasing the mass-to-light ratio will allow solutions with no dark matter, and holding to the observed value will require dark matter. It's freshman physics.

Hi Ken G.,
I have further comments/questions on your reply.
1. Your mention of "normal mass to light ratio". I would like to dig into this one. Can you give me a reference that gives the accepted explanation and justification for this? I would like to know details, not just that it seems to be a good intuitive assumption.
2. Can you explain further what you mean by Nicholson "using a one dimensional rotation curve". I want to make sure I understand your point correctly before commenting.
3. Unfortunately, Nicholson, like so many others on either side of these debates, sometimes uses sarcastic comments in his writing. It immediately can cause hard feelings that get in the way for some. I have concluded that this is a self defeating practice, and scrupulously try to avoid doing it. I can only say, please ignore it and only pay attention to what is pertinent to the discussion.
Thanks, TomT

I've looked at the first link, and didn't get very far before I was quickly convinced this guy doesn't know much physics, even if he's good at programming equation solvers. What bothered me right off the bat was:

I learned why that statement is wrong in freshman physics, because that's where I learned how to use symmetry arguments. The geometry in question is in the plane of a thin disk at radius R from the center. If the mass distribution depends only on radius in the disk, then the gravitational force it will produce on a test mass at R in the plane of the disk will obviously be the same as if each circular shell in the mass disk was rotated around the axis from the center to the test mass, such that the circular shell becomes spread over a spherical shell. The only component of the force that can matter here is the radial component, and it is invariant under the rotation in question, proving that the circular shell yields the same gravity as the spherical shell. And we all know (I hope) that a spherical shell induces a gravity that is identical to if all the mass was concentrated at the center. Ergo, the force of gravity never diverges on a test mass in this calculation, no matter what the author may claim in the above quote.
In short, B + T never "assumed" anything, they applied the above geometric proof.

He also appears to be confused about what an integral is, because he refers to infinities that occur when the test mass R is at the same place as the integrated location r, which would only be true if the mass was actually concentrated into infinitely narrow rings rather then spread out over the plane of the disk. An integral postulates an arbitrarily narrow ring, and has no difficulty dealing with infinities that do not occur when the mass in each ring is formally zero. This is true about half of the integrals physicists deal with, but he is apparently not aware. Or is he claiming that half the integrals in physics are being done wrong? Please. I stopped reading at that point, this guy has no idea what he is talking about.

The entire idea behind dark matter is that you have matter that doesn't make light. Thus, the entire idea has presumed a mass-to-light ratio! Anything that would increase the mass-to-light ratio above what you observe from stars would therefore have to be construed as "dark matter". For a summary of the observed mass-to-light ratio, just google "mass to light ratio".
Simple-- the link you gave me only refers to a one dimensional form for the rotation curve. That is, v(r).
Personally, I'd have no problem with his sarcasm if he had been right. If he'd been right, he would have had good reason to be sarcastic. As he is not, it is others who have that right. Though I agree it shouldn't ever go "over the top", that's not productive. Believe me, I'm exercising restraint there.

[added on edit: I am looking for a better resource about the mass-to-light ratio than "googling it",which doesn't work very well in this case. Also I should be clear that the issue is a discrepancy between the M/L ratio you expect from luminous matter (stars) and what you get in actual observations of galaxy clusters (dominated by dark matter). Finally, let me note that I have no criticism of TomT here, only of the author of the badly reasoned arguments in the links. TomT's curiosity is well founded, given the extraordinary claims that abound even in mainstream astronomy!]

Hi again,
1. OK. I googled "mass to light ratio". I read through the first page of references and concluded it is not nearly as rosy a picture as you portrayed. But first could you give me your reaction to p1 and p2 of this reference from google.
"Photometric Mass-to-Light Ratio"
http://www.astro.psu.edu/users/rbc/a480/lec16n.pdf

In essence it says to me that the ML ratio can't be calculated accurately because small stars make up most of the galaxy population. What am I missing?

2. I still need clarification of what you mean by one dimensional velocity rotation curves.
Are you saying that:
(a) the rotation velocity of stars that we observe in galaxies is known in 3 dimensions (the radial, circumferential and z components). You obviously can't mean each individual star.
(b) Can you point out where Binney and Tremaine use something different than Nicholson for this. For starters, Nicholson points out the pertinent equations BandT use, B&T(2-146) through (2-174). What are they using for velocity that differs from N.

3. For further clarification of terms -
B and T find a galaxy mass distribution that simulates the observed rotation velocities. Their solution has a disc (with mass calculated from the ML ratio) that gives the Keplerian motion, plus a series of concentric spheres of dark matter. The outer sphere radius extends beyond the edge of the visible disc in the galactic plane, and the same distance above and below along the galaxy axis of rotation. The matter in the disc and spheres overlap in the galactic plane, i.e. dark matter permeates the disc also. The gravity effects from the spheres, when added to the disc effects, give the observed rotation curves. When you refer to galactic halo, are you referring to the dark matter spheres? I put the above down to make sure we are talking about the same thing. I have seen statements in the various forums that violate the B and T model, e.g. no dark matter (or bands of dark matter) in the Milky Way disc. As soon as you tinker with the model, you tinker with the velocity curves and they wont match what we observe.
Thanks, TomT

Wait a minute! B and T use a mass distribution based on the M/L ratio and it doesn't work. So they then add dark matter spheres to explain the observed rotation velocities. This has become the accepted solution.
Now you seem to be saying that there is a discrepancy between the M/L you expect from luminous matter and what we observe in actuality. This looks like the M/L ratio doesn't work after all. I thought you criticized Nicholson for not using it.
So use of the M/L ratio resulted in needing dark matter, but m/L is incorrect because dark matter effects the luminousity. Looks like pretty circular reasoning to me.
Maybe M/L is incorrect for the reason stated in the reference I quoted above, namely, most of the stars are small so the relationship doesn't hold for galaxies.
TomT

I have seen much written about this and in almost every discussion the difference between Baryonic Dark Matter (MACHOS) an Non Baryonic dark Matter is always involved.

Doesn't that need to be part of this and isn't the question really...are his calculations correct without using Non Baryonic Dark matter???

RussT

There are two totally independent issues here. The first is, how much mass do you need in your galaxy, and how must it be distributed, to get the rotation curve. That is a matter of straightforward gravity, although Nicholson seems to think the calculation has been done wrong (and in the process he makes incorrect statements that strongly suggest it is he who does not understand how to do the calculation). But none of this has anything to do with whether the matter is "dark" or not.

That brings up the second point, which is, once you know the mass, and you know the light from it, what is the "mass to light ratio"? This is where dark matter comes in. If you can't explain why you have so much mass, relative to the light, you have to say that some of the matter is "dark". So how can Nicholson say anything about dark matter without quoting his value of the M/L ration? That's what I criticized.

As a case in point, if you are right that the mass-to-light ratio is large because there are a lot of stars with very low masses that are being "missed", then that is your version of dark matter! It doesn't remove the need for dark matter, it just chooses a different explanation. It is certainly one possibility, but it is not favored for other reasons.

As for RussT's nonbaryonic dark matter, galaxy rotation curves have nothing to say about that. For that, you have to look at cosmological models, which currently strongly suggest that the dark matter must be nonbaryonic. I hope this clarifies the situation.

Hi Ken G,
I have taken the liberty to number 3 points of your reply and will give my thoughts to each.
1. I think you stopped reading Nicholson's paper based on a misunderstanding, but I will come back to that later in a future post.
2. You state "once you have so much mass". How do you know you have so much mass? Do you have an independent measure of galaxy mass, or are you saying this because the B&T calculation found dark matter necessary? I am saying that the Nicholson calculation requires only that mass be in the galactic disc, and if you read completely through the papers, you will find that his result for the total galaxy mass is about 1/3 of the value from the B&T result with dark matter. This is somewhat more than the B&T calculation for the disc contribution to the total galactic mass. This could easily be explained be the small star explanation. See point 3.
3. To clarify, I found the reference, actually a mathematical demonstration, that the M/L ratio doesn't work for galaxies when I took your suggestion to google for it. The reference I asked you to comment on was a few posts back (astro.psu.edu,etc). It isn't a question about me being right or my version of dark matter. I merely quoted a reference and what they had to say makes sense to me, and they demonstrated mathematically why they think it is correct. It also fits very nicely with Nicholson's result which requires little additional mass for the galactic disc to explain things without dark matter.
TomT

Typically, the dark matter in a galaxy is about 10 times the matter with a "normal" M/L ratio (and far more is needed for the whole galaxy cluster, but that's another matter altogether, no pun intended). Thus, if Nicholson gets 1/3 of that, he still needs dark matter! He has not changed the basic issue, just the degree. But it's still wrong, because simply putting 1/3 the mass into the disk, not spread over spherical shells, would be inadequate from the geometric proof I explained above. And whether it's 1/3 or the B+T mass, your claim that it may be explained by small stars is certainly a resolution that everyone would be very happy with, had it held up. It would have been the first resolution tried, not the last!
That reference is a nice lecture from a course that sounds pretty interesting, thanks for the link. It makes the (well-known) point that the stellar mass in a galaxy comes from low-mass stars, while the luminosity comes from high-mass stars. This indeed makes the determination of the stellar M/L ratio quite tricky. Nevertheless, efforts to determine this ratio using reasonable models of stellar populations find that the stellar M/L ratio is about a factor of 10 too low to explain the galactic M/L you need gravitationally. Had Nicholson been right, that would bring this down to a factor of 3-- you'd still have to monkey with the stellar M/L. How much leeway you have to do that is something I can't say much about, it's a very complex story involving galactic population synthesis. If you wanted to start a whole new thread on that, it would be much more interesting than the faulty Nicholson calculation, which doesn't even remove the need to consider changes in the stellar M/L!
If you think a factor of roughly 3 constitutes "little additional mass", perhaps!

Hi Ken G,
You are misusing my "factor of three" statement here, so I will clarify it further. The B and T solution finds a total galaxy mass which consists of the galaxy disc mass added to the dark matter sphere mass.
Let's call this Mb&t= Mdisc+Msphere

Nicholson finds a solution with the galaxy total mass in the disc only.
Let's call this Mnic.

Nicholson's solution for the total galactic mass is approximately
Mnic =1/3Mb&t.
I am saying that we need to compare Mnic with Mdisc to determine what the additional mass in the disc is needed from the small stars. This will be equal to Mnic -Mdisc, and it is certainly less than 1/3 of Mb&t.
Hope this clears that point up. I am digging to find the B&T number for Mdisc and will get back on this when I do.
In the meantiome, I will be getting back to your objection to Nicholson's calculation.
TomT

No, your explanation is exactly what I thought you meant, and I stand by my statement that Mnic is still about a factor of 3 higher than you can explain by the "normal" expectation for stellar M/L. Do the math. M/L has nothing to do with whether the mass is in the disk or in a halo, it is the total M divided by the total L. You could easily put your many small stars into a halo or a disk, it matters not, except for very detailed considerations about the vertical structure of the disk gravity, which is not being considered here because it is an unnecessary detail in the overall argument about M/L. Forget the Nicholson calculation, it is incorrect and has nothing to add. The only question of interest is, can you get a factor of 10 (or even a factor of 3, fine) increase in M/L by using many small stars. Would that make sense with everything else we know about stellar populations? I can't answer that part, it's a much more difficult issue and far more worthy of a thread than the Nicholson "calculation".

Hi Ken,
I have been pondering your many points about Nicholson's calculation and rather than ask for many clarifications, I want to boil all this down to one hypothetical question first.

Assume we have a galaxy disc cross section like that of NGC 4013. Let the disc be circular and symmetrical when viewed from above. Let's put aside the light to mass issue, and let all the mass of the galaxy be only in the galactic disc. This matter could be normal matter and dark matter. The dark matter could be small stars. Only Newton's gravitational law applies.


Are you saying that it makes no difference how much matter we put into the disc and no matter where we put it, there can't be a valid solution to the rotation velocity problem with matter only in the disc? For a valid solution we must place additional matter outside the disc.
TomT

No, what I'm saying is rather the opposite-- it makes no difference to the rotation curve whether the matter is in a disk or spread over spherical shells that have the same mass as the intersecting annulus of the disk. The only difference it would make is in the vertical structure in the disk, which is very much a higher order issue when compared to the all-important rotation curve. In contrast, Nicholson seems to think that the mass is in "rings", where the gravity becomes divergent close to each ring. This shows a basic lack of understanding of the idealizations involved in carrying out an integral. But I'll let that go, the point is, you can't get the same rotation curve using 1/3 of the mass simply by putting the mass in the disk. It's bogus, by the symmetry argument I gave.

Hi Ken,
I think I get your argument. Let me state it my way, to see if I do. B&T solved the rotation curve problem by adding mass in spheres around the galactic disk, and ended up with a solution that is about 10 times the mass in the disk. By symmetry, you could place all this additional mass in the disk, and get the same solution, but you must put all of it in. You can't get a solution with less mass.
Let me know if this is a correct representation of your argument. I will ponder it further, but if you are right, Nicholson owes you $100,000 in case you don't realize it. And if you are right , can I have a finders fee?(joke)

TomT

You understand the argument precisely. But I wouldn't count on Nicholson ever shelling out that $100,000-- you'd have to prove it to him. I'm sure many have tried! (If I get it, you can have half!)

Hi Ken,
Now that I understand your point, I can whittle my question/comment list down some. Here is how I see it.
I provided links to 3 papers by Nicholson. You started reading the first one, got half way through p.2 of the first paper, found something you didn't like in his critique of B&T, concluded he didn't know what he was talking about, and stopped reading his papers at that point.
Here is what you missed:
1.The B&T solution uses a thin, flat plate to simulate the galactic disc and adds concentric spheres of mass around the disc, extending even beyond the disk edge. They use a mass distribution for the disk based on the accepted mass to light ratio for galaxies. They solve this system by an elliptic integral method. Any more complicated geometry would probably be beyond such a solution method. This geometry yields a total mass about 10 times the disk mass.
2. Nicholson simulates the galaxy shape by using a much more accurate geometric description. He has an ellipsoidal looking object for the central bulge, and has a variable thickness plate for the outer disk. The surface of the plate is curved top and bottom, not flat. The plate tapers and has a finite edge thickness. The mass density varies in the r and z direction, but is constant in the circumferential direction at a given r. He has a method built in that fares the slope of the variable thickness plate to the central disk shape.
He then uses a numerical iteration method to solve for the mass and mass distribution required for this shape to yield the correct rotation curve. That solution has a total mass about 1/3 the B and T solution.

My conclusions:
1. The Nicholson solution is correct unless someone can dig through his calculation and find a error in theory or programming. There is nothing magic about B&T's mass figure. It is just the value they got from a cruder simulation of the disk shape.
2. If for some reason you want to move some of the mass from Nicholson's disk shape to spheres of mass around his disk by the argument of symmetry, I suppose you could, but I don't know what that would accomplish.
3. Nicholson's solution can be expanded to add the feature of variable mass density in the circumferentail (theta) direction. This would be necessary to more exactly simulate a spiral galaxy for example. His numerical mathod would allow adding this feature as a symmetrical sinusoidal function, for example, or even more accdurately a table of mass vs r and theta estimated by an actual spiral galaxy shape. My bet is with this last feature added, the solution will have even less mass, probably a value that is believable with the small star explanation.
TomT

I don't think that a "theta" component is necessary, since the density of the disk is fairly uniform. The spiral arms are just areas where stars are forming. Therefore, there is an overabundance of young, hot, bright type "O" and "B" stars.

__________________
Any day you wake up on "the right side of the dirt" is a good day.

T. Anderson

This is why I stopped reading at page 2. I was not born yesterday. I have seen countless examples of people who think the work of hundreds of accomplished astronomers is bogus because they screwed up a simple calculation that almost any grad student could do given enough time. Then they present a very complicated alternate solution that you can't possibly check simply because it would never be worth your time to do so. Then they support their answer based on arguments that have obvious factual and/or logical flaws (such as the claims about the B+T solution requiring an infinite mass, or the divergence of gravity near the "rings" in the integral. Hooey).
At the end of the day, they come up with a factor of 3 correction using a different model that in point in fact relies on extremely unimportant details (flaring disks, ellipsoidal cores, etc.-- all very minor deviations given the basic symmetry that a rotation curve is only sensitive to the mass in spherical shells, no matter how the mass is distributed in those shells, as long as the whole business has the azimuthal symmetry of the disk. And I agree with Kaptain K that bringing in spiral arms to try and wiggle out of that last bit would really be grasping at straws, not that I am implying this was the reason that you mentioned theta structure.) But I will sum up with a couple issues for you to keep in mind, even within the contexts of what you currently believe:
1) a factor of 3 reduction in the mass won't remove the need for dark matter in a galaxy, it just means you need less of it. Not an Earth-shattering result (and wholly implausible, for the reasons I gave above.) Yes, it might mean small stars can be more important, but you'd want to know if they are anyway-- you don't need additional motivation.
2) If you want to increase the baryonic matter by a factor of 3, you will need an entire new cosmology to go along with it. So this is what you are suggesting-- because one questionable integral solver thinks that every previous gravity calculation is wrong and he is right, you have to throw out a vast array of theoretical work on understanding general relativity and Big Bang nucleosynthesis, and reinterpret the WMAP and SN Ia data.
3) And here's the real kicker-- maybe all this would seem remotely worthwhile if it would relax the need to invent some unknown "dark matter", but it doesn't even save you from that, because most of the dark matter you need is not in the galaxies, it is in the galaxy clusters. You need even more dark matter to understand the orbits of galaxies in clusters than you need to understand rotation curves! I can't wait for Nicholson's arguments as to why the galaxy cluster calculations have been done wrong (they aren't disks!)

Hi Ken,
I stick to my contention that Nicholson's is correct until someone disproves his math, theory, or numerical calculation method. He has offered to provide a computer disc with the method on it that a person could check out on their home computer. I will volunteer to obtain a copy and give it to anyone willing to tackle it, provided they will let me know their findings. He has also offered to present his solution in person to a group, and probably will debate the issue. Not sure of this last one. So it is not impossible to check him out.
Secondly, we can invoke some dark matter to spare any changes in theory. Lets do this:
1. Assume that the light to mass ratio holds, and treat Nicholson's solution the same way I believe you said we have to look at the B&T solution. Take some matter out of Nicholson's disk solution and spread it around the galaxy in spherical dark matter shells. Account for the dimming effects of the dark matter. Adjust the luminosity and mass in m/l. Make the indicated correction to the spherical and disk masses. Repeat until the solution converges (hopefully) to the accepted m/l and viola we know how much mass is in the disk and shell, m/l works, and the total mass is 1/3 of the B&T result.
Another point, the use of dark matter within a galaxy to get the rotation
curves right is completely independent of using dark matter to get the intragalactic motion right. What difference does it make if we get a more correct value for the galaxy part?
As for improving the geometry accuracy by allowing variation with theta, the jury is still out. Kaptain K may be right and I don't know.
TomT

You obviously don't have much experience with "woo woos". But I agree that Nicholson has credentials that would seem to make his claims worthy of checking out. Problem is, so do a lot of woo woos. Who wants to waste that much time, when we all know they will never admit their error? (They never do.)
Because there would be value to astronomy if dark matter could be done away with, it is very bothersome. But this won't accomplish that, so again, it's an issue of cost/benefit of investing one's time. If I were a betting man, I'd give 99 to 1 odds that checking the calculation is a waste of time, given what I know. So how would anyone want to do it, unless there was a huge potential payoff? Indeed, why would Nicholson have bothered if he didn't feel that there was a big payoff? Note that he has not published in the conventional literature. Why not?

Ken,
To the above statement by me, you replied "you understand the argument precisely". This was in your post of 11-10 @ 4:28PM.
So from this response by you, I have to conclude that you can get a valid solution with all the mass in the disk.
Now in your post of 11-10 @ 7:58PM you say the Nicholson's more sophisticated geometry amounts to only minor deviations "given the basic symmetry that a rotation curve is only sensitive to the mass in spherical shells no matter how the mass is distributed in those shells, as long as the whole business has the azimuthal symmetry of the disk".
So if you really mean both statements you made above, you can move matter into andout of the disk and shells by symmetry, and a solution can be valid with all the mass in the disk, but for the other reasons you gave in the 7:58PM post, with the total mass in the disk, some must be baryonic, and some non baryonic. Correct or not?
Bear with me on all this, I only want to understand. My motive is only to find the truth as best we understand it. I consider everything subject to question, always. I have no personal motives, and am always ready to admit when I am wrong (eat crow if I have to). You might say I am an old dog trying to learn some new tricks.
TomT