Since I wrote the Perl code long time ago, I should know what it all means.
A proper understanding of the underlying theory would require reading and digesting a number of astrophysical research papers as well as knowing about advanced methods in Stochastic math. This is probably not what you guys might have in mind. I actually have collected a long long
clickable list of scientific references that went into my entirely new coding of globular clusters in celestia.Sci.
Here is the link, just in case
http://www.celestialmatters.org/users/t ... apers.htmlBut I suppose there is little interest for that here...
As to the
King concentration c = log10(r_t/r_c), this is easy to explain. As the famous astrophysicist Prof. Ivan King found out a long long time ago, one can fit the skyplane-projected radial surface luminosity profiles of ALL known GCs very well with a so-called King formula
king.jpg
that involves besides the radial distance
r from the GC center , TWO radius parameters
r_c and
r_t. These characterize the shape of the luminosity distribution of stars in a given GC.
r_c = core radius
r_t = tidal radius and usually, r_t >> r_c.
Instead of these 2 radii, it has become customary to use equivalently ONE radius and the so-called dimensionless King concentration c = log10(r_t/r_c). Just replace (r_t/r_c)^2 in the second term of the King formula by 100 ^ c.
The tidal radius is defined as follows:
For core distances r > r_t strictly NO stars can be gravitationally bound by the GC core such that any stars beyond r_t do not belong to the GC! In other words the above King formula satisfies the constraint that it vanishes for r = r_t (as well as for r > r_t) as you may easily check.
The core radius r_c characterizes the size of the bright core of the GC and mathematically
r_c is implicitly defined as distance r =r_c where f(r_c)/f(0) ~ 1/2. Thus c = log10(r_t/r_c) has indeed the meaning of a concentration parameter.
For the regular Celestia distribution I just used the parameters r_c and c by Harris. They may be found in my globulars.dsc file.
For
celestia.Sci the next step was to take the King luminosity formula and perform a least-square fit by computer to all corresponding measured data for the 156 galactic GCs. In the paper of Harris, the resulting values for r_c and c actually refer to a more complicated variant of the King formula, the use of which would be too slow for Celestia. Since empirically it does not describe the data better, I did myself a re-fit of all data and extracted the best r_c and c values based on the above formula.
With a complex Maple program I did all these many fits automatically with excellent results. Here are two examples illustrating how well the King formula (
blue curve) fits the
measured surface luminosity data (
red dots):
1) famous NGC 104 = 47 Tuc (=> r_c = 0.51, c=1.94)
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n104_king.jpg
This one is MUCH dimmer and far less concentrated (r_c = 2.29, c= 0.52), whence you see that a large amount of different profiles can be fit by the King formula!
2) Palomar 4 (=> r_c = 2.29, c= 0.52)
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Palomar_4_king.jpg
The computer determined the resulting r_c and r_t radii for all these fits. They are to be found in the globulars.dsc file of celestia.Sci. Note that in the plots the radius scale is logarithmic and the units are arcmin! The quantity on the y-axis is the visual surface brightness ?_V in [magnitudes/arcsec^2]. The black lines indicate the physical so-called ?_V = 25m/arcsec^2 radius where the surface brightness isophote has reached the very dim standard value of 25m/arcsec^2. This is the physical GC radius where Celestia(.Sci) puts its four red marking triangles.
With these determined profiles (yet a very different color-magnitude distribution) the two representative GCs above look very different in celestia.Sci: NGC 104 on top, Palomar 4 below,
both being displayed with the SAME field of view (FoV) to allow for a meaningful comparison:
The next step is to interpret the King formula as a probablity distribution for finding stars at distance r from the GC center! By means of the standard
Von Neumann method I then randomly generate
~100 000 stars, whose distribution in distance from the GC center is according to the King formula. Note that we are not talking here about
uniform distributions of random numbers (like when you throw dice), but rather a specific probability function (<=> King) that is strongly peaked in the center.
Von Neumann tought us long ago, how to generate random numbers according to an
arbitrary probability distribution in Stochastics. It's NOT simple.
The latter part is done within the Celestia(.Sci) code for efficiency reasons. Of course in celestia.Sci there is much more involved stuff, like the GC stars must also have the correct luminosity, mass and color distributions etc...But that would lead too far here.
Such GC stars could be also generated with Perl scripting but then reading in the data for 100 000 stars, say, would take a long long time. And one wants at least 156 galctic GCs hence 156 * 100 000 stars to generate and to read in... Finally, also Perl requires advanced programming knowledge.
. However, learning Perl is substantially easier than learning good C++
All these steps represent familiar "handicraft" for professionals like myself. For lay persons I am afraid this might be far too involved...
Fridger
