Quote:
Originally Posted by knicholson
To Grav
Set your disk data as follows:
|
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.