Goofy wrote:Hi mneme.
...
3- moreover as you see there is a conspicuous central condensation that I feel a bit unsuitable with the overall shape.
Please remember that my feeling is just due to "aesthetic" reasons (perhaps it's scientifically correct, but I haven't the needed background to judge this).
...
And could you please restore what we discussed about the very nice results about globular clusters, making them possible as before, but with a round shape and following Fridger's suggestions from here
viewtopic.php?f=6&t=17126in order to give a scientifically correct overall distribution?
Beg your pardon Fridger, but this way we can show plausible globular clusters and navigate within them with our student, while waiting for your Celestia-Sci incredible globulars.
Thank you once again, waiting for #5 release.
Bye
Goofy
It is obvious that the stars generated randomly by Mneme
correspond to a completely WRONG probability distribution.
Unfortunately, advanced insight into statistics is required to get this right.
Many computer languages have a function built in that can generate random numbers within a certain prescribed range. If Mneme used it straightforwardly, the result cannot be right.
These built-in random generators are just the computer analogs of
throwing the dice, i.e. the resulting
random numbers follow a uniform (i.e. constant!) probability distribution.
Every child knows that in case of throwing the dice (possible random numbers 1,2,3,4,5,6), the probability to throw any number out of the 6 possibilities is identical, namely the constant 1/6.
The distribution of surface brightness and thus of GC stars as function of the radial distance r (replacing now the discrete numbers 1...6 of the dice example) is NOT constant. It rather equals the King function f(r). And that's Mneme's severe statistics fault!
Let us transform both the (stochastic) variable of the dice example and the GC task to a single dimensionless variable ? = {1/6, 2/6, 3/6, 4/6, 5/6, 1} with
0 <= ? <= 1 for the dice example and ? = r / r_t for the GCs following King's notation. Since the tidal radius r_t is the largest possible (star) distance from the GC center, obviously also here
0 <= ? <= 1.
Now we can directly compare: after throwing dice 1000 times, say, and counting how often each number has been thrown, you
find a constant as function of ?, as stated before.
On the other hand, if you throw the "dice" (aka random stars
) according to the
peaked probability function of King, then counting the thrown stars in each bin of ? will give you this probability function instead of a constant:
[Click on image by all means!]king1.jpg
The thick red curve is the original King function versus ?. The blue histogram you get after counting the actually generated stars in bins of ?. So you see that this time
the number distribution of random stars in ? follows precisely the desired theoretical curve by Ivan King.
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The big math task for you is then how to generate random data that follow a certain given probability function. Well, let me know if you consider trying it, then I'll help you further...
The trick of the famous mathematician Von Neumann was to use initially the [b]uniform random generators in the computer, BUT we must throw away certain generated stars. The accepted stars then correspond to the right probability. That's why Von Neumann's method is also known among experts as the
Acceptance-rejection method. It is the basis for the many Monte Carlo generators e.g. in my field of research...
But to do this in practice is not all that easy...
++++++++++++++++++++++++++++
That's not all: In order to generate the full spherical distribution of stars, you must use 3D polar coordinates (to make sure that the result is spherically symmetric!). The r-dependence is given by the above King values, while you must still generate UNIFORMLY the 2 missing angles (polar and azimuthal). Also here there is a trap...But for now, I leave it as an exercise to find out or to Google
how a uniform distribution of points on a unit sphere needs to be stochastically generated.
The end result with the above ? dependence following King then must look like this in x,y,z, with -0.5<=x,y,z <= +0.5:
[Click on image by all means!]king3d.jpg
Good luck,
Fridger
PS: And once you have done all this, you must remember that the brightness of GC stars also follows a statistical pattern that is described by the so-called
luminosity function d nStars/ d M_V. It has been measured for many GCs and needs to be fitted by a smooth mathematical function. Here is mine for celestia.Sci:
lumi.jpg
This function then is responsible for a realistic mixture of dim and bright stars. M_V is the absolute visual magnitude of a star. Again one generates stars randomly with this luminosity function as probability distribution.
Finally you need to generate color....
