To Ken G

1. That bet sounds fine. Don't expect me to cave for things like dark-matter reports in the galaxy collisions, or other phony stuff, and I will expect you to defend to the end. Should be interesting.

2. You know MOND is still alive and well in some circles. Can you prove it wrong? They simply said that Newton's law had to be modified to get speed matches for galaxies and thus avoid dark matter, but their mods had to be special for nearly every case. Does that prove it wrong? I think so.

3. I'm back to reading Binney and Tremaine to find out how surface light is measured so I can talk with some knowledge on it. Pages 8,9,16,17 have good basic info. They conclude (p16) that our solar neighborhood of the galaxy has an SMD of 75 msuns/pc^2 (made up of all components, stars and non-star ordinary matter) and a surface brightness of 15 Lsunvs (sun visible light intensities/ pc^2)

They assume all the light comes from stars alone. Dividing 75/15 they get 5 msuns/Lsunv as a mass/light ratio. (So that's where the 5 came from). They then use 5 to estimate the mass of the Galaxy, assuming 5 as constant over the entire radius. This is where I think the problem lies. I believe the average M/L ratio is closer to 10 for most galaxies, and that the M/L varies a lot for different galaxies and over radius in a given galaxy. Because of the possible situation where there is very low average light, I think it should be a L/M ratio to be more realistic.

More later.

Ken Nicholson

It isn't necessary to prove one theory wrong to prefer another, all we have to do is pass judgement over which of the current possibilities makes the most useful predictions. I have never yet seen MOND make a significant prediction that worked, it is all retrofit. Dark matter makes predictions, and some have worked and others have not-- yet. So it is still pretty clearly the better of the two theories, which is the only question science asks. That doesn't mean it always will be.

At least it proves it's not our best theory, and really not much of a theory at all so far.
One thing I will freely admit in all this is that I have no idea what confidence we can put in L/M determinations. There are two problems with it that are potentially severe, one is that the L comes more from high-mass stars while the M comes more from low-mass stars, so there's really two sources of potential error there not just one. The other is that as soon as you presume the dark matter has to be spherical, it takes even more of it to fit the rotation curves. So even a fairly small shortfall in M/L in the disk will lead to a relatively large need for dark matter. That has a way of magnifying the importance of errors anywhere in the analysis.

The bottom line for me is, if galactic rotation curves were the only thing that forces us to think about dark matter, I would see it as being on pretty shaky footing. But there is an even stronger need for it in galaxy clusters and in cosmology, so extending that to galaxy rotation curves does not seem like a stretch. Indeed, if galaxy rotation curves worked fine without dark matter, using what seemed like a "normal" M/L, it would be a rather puzzling state of affairs.

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If we understood everything going on in the head of a pin... we still wouldn't know not to step on the pointy end.

People think the problem with models is that they are limited by our minds, but the greater problem is that our minds are limited by our models.

Thanks, Ken. Yes, of course. Looks like I will be going with your rods and rings program after all anyway. That thing with the summations of elliptic integrals basically only works out the way I need it to with special cases, such as results that only give complete elliptic integrals and so forth. But many to most of the results of the integrations I have worked through so far do not provide this. So I would still have to run them over points for the incomplete integrals, and it would then be the same thing but much simpler just to take the last part of an integration without the elliptical integrals involved and run that over points instead. But if I am going to do that, then I might as well run it for two integrations as you are doing for the rods in the rings and the rings over the radius, so that I can vary the height and density for each ring from the center, and of course there's the added benefit that we can stand on common ground in the discussion while using the same program. So it appears I have once again come full circle with this thing. I am anxious to get the ball rolling a little faster than I have been to this point, and I will write the program as you mentioned.

I was wondering about this, though. If I use a constant 120 rods per ring, it would seem I would have to increase the mass of the rods in each ring in proportion to the radius of the ring, then, in order to gain a constant density for each ring and therefore the disk. Either that or space them around the ring with the number of rods in proportion to the circumference of the ring each time, unless you are finding for a result in terms of the density and then multiplying that by the entire volume of the disk in the end or something like that. Otherwise, you would actually be finding for a disk with a density that falls with 1/r.

From post #72, I found the acceleration of gravity at the rim of a 1% thick disk with constant density to be about 3.62 times greater than for that of a sphere with the same mass and radius. That would give a rotation speed that is 1.9 times greater.

I still don't quite understand how one is to go about reversing that, especially since each ring affects each one before and after it, and so will any alterations of the mass densities of a ring to the rotation speeds of the others and so forth. The only trick I can see for that offhand is to apply a constant mass density to the disk, and then add or subtract some mass to a spherical central region, according to the rotation speed at that distance from the center to roughly the sphere's edge. Then add or subtract mass density to the rings one by one from the center outward according to what the mass density of each one would have to be to give the observed rotation speeds for those distances, with the added gravity of the central region and all of the rings before and after it, and do this all the way to the rim of the galaxy. But then the gravity of all of the additional rings past the central region will have changed the gravity applied to the edge of the central region accordingly, as well as the rotation speed for it, so one must add or subtract some of the mass from the central region again to balance out the effect of the gravity of all of the other rings on it, in order to gain the same original rotation speed for it. But then one must go back through all of the rings again, finding for the mass densities of each in the same way as applied the first time. And then back to the central region again and start over, continuing an iteration that will hopefully eventually level off at the corresponding mass density for each of the rings and central region that gives the rotation speeds for those distances.

__________________
Let's put together the pieces of The Grand Puzzle . (website)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."

To Grav

Set your disk data as follows:

rmax=10000 pcs, h=100 pcs, rho = 1 msun/pc^3 (density), total mass =pi*rmax^2*rho*h
20 rings with rods at the mid radius rr(i) of each ring i, rr(i)= dr*i-dr/2
and read the rotation speeds at the outer radius rs(j) of each ring j, rs(j)=j*dr
radial width dr = rmax/20 for all rings

mass of each ring becomes mr(i)= 2*pi*rr(i)*dr* h*rho
mass of each rod in ring i becomes mrod(i)= mr(i)/120
length of all rods = h, diameter=0 (not important and not used)

position your rods around the half-ring mid radii, theta=1.5,4.5,.....,178.5 deg, then
find the accelereration caused by a single rod as a function of distance from the rod along a line
perpendicular to and bisecting the rod, (if in doubt see my paper), note the acceleraton blows up if the test mass
touches the rod, so that never happens, then
find the acceleration of a test mass at rs(j) on the theta =0 axis caused by TWO rods in ring i at positions + -
theta around the ring
then add up the effects of the rods by summing the effects of the 60 pairs to get
acceleration a(i,j) of a test mass at radius rs(j) caused by ring i INCLUDING THE CASE i=j, then
add up the effects of all the rings (i=1 to 20) on the test mass at locations rs(j), (j=1 to 20), to get A(j) the total
accelertion of the test mass at all locations rs(j)

That will get you through the forward problem. Notice you should now have the effects of each ring (at
density of 1) on a test mass at each speed radius j, and you are ready for the reverse problem. I believe all will
become clear after we get you through the reverse problem.

Good luck, Ken Nicholson

While I appreciate your generosity, I would personally prefer you leave it to your heirs, or other family members (nieces, nephews, etc.). I'm sure they would, too. Unless, of course, you're giving them $10 million, in which case $100 k is chump change by comparison.

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I am Mugs, of the Alien clan of Usa, Nordamerica, a Terran, of Sol.

Perception isn't reality. It's merely an abstraction thereof, and quite often not a very good one at that.

"Staying young requires the unceasing cultivation of the ability to unlearn old falsehoods." - Heinlein

"Freedom begins when you tell Ms. Grundy to go fly a kite." - Heinlein

Thanks again, Ken N. Sorry it's taken me so long to respond. I set up the program in the way you stated, but I was a little skeptical about running the program for the same number of points for each ring, since the rods would become more and more dispersed for the rings at a greater radius using the same number of rods per ring and all. I generally think of the rods as all being of equal mass and distributed equally throughout the disk, encompassing equal areas and volumes each, so I wrote programs for both ways. The second one uses the same 1200 rods in all, so has the same running time, but the number of rods in each ring is proportional to the circumference and area of the ring, so starts with 6 rods for the first ring instead of 120, 18 for the second, 30 for the third, etc. As it turns out, there's not too much difference between them, though, but yours is more precise toward the center of the disk, because of the greater number of points used there, and the other is more precise toward the rim. By the way, I noticed you compare your thickness to the radius, not the diameter as I was doing, so the ratio of diameter to thickness would be 200 for what I was using, and the rotation speed for that would be just a touch over 2 times that of a sphere at the rim, then, not 1.9 as I said earlier.

Also, I guess you know that 120 rods per ring for 20 rings is nowhere near precise enough for finding the rotation speeds. I was actually getting a negative acceleration for the inner rings, for example. Ten times as many rods for ten times as many rings begins to come somewhat close to the actual rotation speeds, though, from center to rim. I used the program for the last integration along the x axis thing to compare results. That one is still the quickest and most precise by far. It only takes a few minutes to work through the first two-thirds of an integration with some specified mass distribution and height variance for a disk, and a few more minutes to change a couple of lines in the program to accommodate it, then let it run, gaining a result with amazing preciseness. Of course, I've already got the one for uniform density and height. In terms of using rings for the reverse problem, I was thinking I could just find for a disk with a radius equal to the outer rim of a ring with that program and subtract it from a disk with a radius equal to the inner rim of the ring, using the height and density for each ring, to find for a series of rings, using the rotation speeds for the resulting mass distribution. But then, that would require two integrations per ring all the way from the center to the radius of the ring, for each ring, so it might actually turn out to be less precise when running over the same number of points as your program would for the reverse problem, I don't know yet. I guess I will write a program for that next and compare the preciseness of the results over the same number of points and run time. I will want the most precise program possible over the shortest time when I run the iteration for the reverse problem.

I will do that next, and then I suppose I will use that last integration across x program to draw up a few quick graphs for the rotation speeds from center to rim for a few thicknesses of uniform density, uniform height disks and post the images for those. I can't believe I haven't even done that yet. Then, once I have determined which of these few programs is the quickest and most precise overall for the rings, I will run the iteration for the reverse problem to see what mass distribution will give the rotation speed curve as it is observed. I will have to do that a few times over varying thicknesses in order to account for larger thicknesses for gas and dust as well. The way I figure it, there may be an infinite number of ways to apply mass distributions for a particular rotation speed at a particular point using different geometries and so forth, but there is only one mass distribution that will generally give the entire rotation speed curve for a particular disk with a specified thickness. I aim to find it. With any luck, the total mass of the rings will then also match that measured for the galaxy. If nothing else, it should at least provide general information about how the mass distribution must fall from center to rim.

__________________
Let's put together the pieces of The Grand Puzzle . (website)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."

Oh, wait. If I'm thinking about this correctly, I should just be able to find for the acceleration of gravity at the edge of a ring using that last integration over x program by finding for the disk to rr + dr/2 and subtracting that for rr - dr/2, and just record those for a particle placed at the edge of every ring in respect to every other ring in the disk, for say, a 20 by 20 grid for every position and for every ring in a 20 ring disk, but I would need to do that only once. After that, the acceleration should vary in direct proportion with the mass density of the rings, so I would just add up the accelerations already found for a constant density disk by whatever their individual densities are multiplied for each ring as the iteration is performed to gain the rotation speeds. I hope I said that in a clear way.

__________________
Let's put together the pieces of The Grand Puzzle . (website)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."

To Grav

If you do it the way I set it up, then I can check your progress. After you do it that way, even if it doesn't look optimum to you, you can improve it later. What you need to do first is to get a simple working program for the forward problem. You'll learn a lot by doing it this easy way first.

Ken Nicholson

To Mugaliens

I was not being generous. I couldn't lose that bet. I had already shown that all mass could be put in the disk. My purpose was to smoke out those that claimed the extra mass needed could NOT be put in the disk and thus dark matter was required. It's a bet I'd be happy to make anytime, but it's probably immoral now since it's more of a steal than a bet.

Ken Nicholson

Yes, I've done that actually. It is what I was referring to in the first part of post #96. For a disk with a radius that is 100 times as large as the thickness, using 120 rods per ring for 20 rings, I found a rotation speed at the rim that is 1.7087 times as great as that of a sphere with the same radius. At the edge of the first ring, however, or 1/20 of the radius, the acceleration was negative, giving a complex rotation speed. When I ran it for ten times as many rods for each of ten times as many rings, however, I got a rotation speed at the rim of 1.9842 and at the edge of the first disk of .9232 as great as that of within a sphere with the same radius as the radius of the disk. The actual speeds are about 2.015 at the rim and 1.0005 at 1/20 of the radius.

__________________
Let's put together the pieces of The Grand Puzzle . (website)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."

To Grav

You're right! I also got the negative acceleration at r=rmax/20. The problem is one I fixed about 10 years back and forgot. The center ring mass is a circle, and the rods must be placed at the centroids of triangles that make up the circle to properly represent the circle. So for the first speed location the rods radius is 2*dr/3 and the rod location is asymptotic to rs(i)-dr/2 at large radii. I use the following:

rr(i) = (rs(i)-dr) + (rs(i)-dr/3)/(rs(i)-dr/2) * dr/2

Try again using this and the original data, and accuracy can be improved later.

Good work, Ken Nicholson

Has anybody looked for a correlation between the size and/or location of supermassive black holes and the inferred distribution of dark matter?

There would be a tight correlation, as supermassive black holes are primarily (exclusively?) found in galaxies, and the dark matter distribution is thought to be responsible for the forming of galaxies (it is concentrated near galaxies and galaxy clusters).

__________________
If we understood everything going on in the head of a pin... we still wouldn't know not to step on the pointy end.

People think the problem with models is that they are limited by our minds, but the greater problem is that our minds are limited by our models.

The part in bold is simply not correct. Milgrom made predictions about galaxy dynamics that have been shown to be correct. See section VIII of the paper linked to. For example prediction #6 was shown to right on. Low Surface brightness galaxies show mass discrepancies of the form predicted by Milgrom. The mass discrepancy starts at a smaller radius than for High SB galaxies. Also as predicted by MOND, LSB galaxies follow the same Tully-Fisher Relation as high SB galaxies.

McGaugh's papers provide an excellent read for the MOND issue. See this paper and this paper and this paper .

In section 4 of this paper it is shown that MOND cannot not be made to fit a hybrid galaxy that combines the rotation curve data for NGC 2403 with the photometry of UGC 128. MOND is unable to produce a fit to this hybrid galaxy (as would be expected since the galaxy is not real) whereas dark matter is able to make an acceptable fit to this galaxy. This is actually a problem for dark matter that it can be fit to a fictitious galaxy. There are too many free parameters to be fiddled with.

And then there are the tidal dwarf galaxies that show mass discrepancies unexpected in the DM framework but are expected in MOND. And Globular star clusters too. Another paper on MOND and globular star clusters.

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"The scientist who asks the right question reconnoiters a new patch of the unknown, and may, with luck, bring it within the constricted but expanding boundaries of the known."

~Timothy Ferris (The Red Limit) 1982

Actually, the part in bold was perfectly correct when it was written (it was the "it is all retrofit" part that you are actually taking issue with). However, now that you have pointed to those articles, I would have to say that the bolded statement is no longer correct, thank you. I don't suppose this thread should delve into the issue of how convincing this evidence really is.

__________________
If we understood everything going on in the head of a pin... we still wouldn't know not to step on the pointy end.

People think the problem with models is that they are limited by our minds, but the greater problem is that our minds are limited by our models.

Sorry I took so long again. I spent a day or so trying to find a geometry similar to that of a disk again which would just integrate into a single formula without running it over points, but still no luck so far. I also started to draw up a graph for the rotation speeds for a uniform disk as compared to a sphere, but it wasn't very interesting. The rotation speeds for a sphere just started off at (0,0) and ran in a straight line to (1,1), while that for a disk starts off in the same way, following the same line, then begins to deviate as it approaches the rim, until it curves up faster toward the rim to about twice the rotation speed as that of a sphere for the thickness of a disk we've been finding for. I might still be able to identify the formula for that curve to find a formula for a disk, though, if it turns out to be a hyperbola or something, but for now, I just want to get the iteration for the reverse problem going.

I tried that last formula you posted, and this time I got about 4 times the rotation speed for the first ring, so it's getting closer, but I don't trust the formula because I don't know where it came from, except as a clever rig to fit the rotation curve, which it might come close to doing over all of the rings, but I can't be sure about the effects of the individual rings on particular points, which I'll need for the reverse problem. Besides that, I still wanted something just a little more precise, so I went ahead and broke up that "last integration over x" program for a disk I had into individual rings. I then made an array for the acceleration of gravity for each ring as they would act on a particle placed outside of every ring as (i,j), where i denotes the ring and j the position of the particle. It builds the array in about 5 seconds for ten rings, with about 99.99965% accuracy at the inner ring and 99.987% at the rim, so it is still very precise also.

It needed to build the array only once, and I could then find for the acceleration and rotation speeds for any of the ten positions by simply having the program add up the accelerations for each of the ten rings for any particular position, using the array, which only takes a fraction of a second. I can also have the program go through the array over and over for the different positions, while giving each of the rings different densities, also in a fraction of a second, using the same array in combination with another array for the densities of the rings, and find for the rotations speeds over various density configurations. That is what I will use for the reverse problem. I will still be crossing my fingers that it will eventually level off at the corresponding rotation speeds and densities for the rings the way I'm hoping it will, though. If not, I will need to try other methods for the reverse problem as well.

__________________
Let's put together the pieces of The Grand Puzzle . (website)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."