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Edit: I moved this topic here, since it concerns a recent development feature in celestia.Sci. In irregular time intervals, members of the celestia.Sci dev team will report here about topical research underlying some important new code development.

This material will often be addressed to people with some background in Astrophysics/ Mathematics, who want to know in more concise terms about the theoretical concepts and astrophysical/astronomical data on which new celestia.Sci code will be based.
Throughout it is encouraged to present links to used scientific papers..
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I needed a smooth function fit of the best available Luminosity Function (LF) data for nearby MilkyWay stars (< 25 pc). After proper normalization, its purpose is to serve as probability distribution for statistically generating realistic populations of faint stars in the ~ spherical halos around galaxies in celestia.Sci. Perhaps my analysis results are of some interest to others.

The best LF data (including Hipparcos stars!) are tabulated (with statistical errors) in this paper
http://adsabs.harvard.edu/abs/1997ESASP.402..675J
by H. Jahreiß and R. Wielen (1997).

Here I have done a plot of these data with Maple 17:
[Click on image for a bigger display]


On the x-axis you have the absolute visual magnitude M_V while on the y-axis you see the corresponding star counts per unit volume [pc^3].

Note the conspicuous peak around M_V~ 12 -14 and the shoulder around M_V~ 5.6. In this region left of the main peak, the errors are pretty small and the least-square fit should be very good here.

It should be quite obvious that a very good ansatz is in terms of a sum of two Gaussians, one multiplied with another factor of M_V^2. Each Gaussian brings 3 parameters along like so

G1(x) = a[1] *x^2*exp(-(x-x0)^2/a[2]^2);
G2(x) = a[3] * exp(-(x-x0)^2/a[4]^2);

LF = G1(M_V) + G2(M_V)

with parameters a[1]..a[4] and x0, x1 to be determined by a least square optimization algorithm in Maple. In the paper also the errors were given, so these can be accounted for as usual in the fit.

Note: Gaussian fit functions have the big advantage over polynomial-based fits that outside the proper fitting range they quickly tend towards zero, while (high-order) polynomials VERY quickly run out of control by adopting non-sensical values!

Here is the beautiful result, with all parameters determined:
[Click on image for a bigger display]


Here is the link to my well-commented Maple17 worksheet from where all determined parameters etc may be copied.

http://www.celestialmatters.org/users/t00fri/files/luminosityfunction_HIP_PMSU4_fit.pdf

Interested readers may also get a concise idea about the typical working steps in such type of scientific analysis...

For your convenience, here is a summary of the used function and its determined parameters
[Click on image for a bigger display]


Fridger

Perhaps some of you are interested in the important role the Luminosity Function plays in the context of the generation of globular cluster stars in celestia.Sci. Here are the data taken for some prominent globular clusters NGC 7078 (M 15) and NGC 6341 (M 92) along with a respective least square fit done with Maple some time ago (blue curve).

[Click for a bigger image]


Upon comparison with the above Luminosity Function for nearby MW stars, the similarity of the function shape is apparent. Yet there are differences in detail. After normalization of the area to 1 and plenty of statistical math, it was used in the celestia.Sci code as a probability distribution for generating cluster stars.

After these many thousand stars are randomly generated with this probability function, one can do a crucial cross-check. Here it is:

Place the stars of a globular with absolute magnitudes between (M_V - w) and (M_V + w) in bins of suitable width w (e.g. 0.25), and plot the corresponding star counts you find in each bin versus M_V. The shape of the resulting distribution should be identical to the normalized Luminosity Function!

Here is the amazing result from celestia.Sci:

[Click for a bigger image]


As you see, the agreement of the actual bin distribution (dark blue) with the above fit to the measured Luminosity Function (bright green curve) is just perfect as it should be...This check warrants that in each of the rendered globulars the brightness distribution of their thousands of gc stars follows precisely the measured Luminosity Function!

Fridger

PS: For people interested in generating realistic globular clusters, it is crucial to learn how one randomly generates stars whose brightness is NOT uniformly distributed but follows an arbitrary given probability distribution (<- luminosity function!)! If just a built-in random generator like Mathf::frand() in C++ is used to create the stars, the results get GROSSLY wrong of course. The corresponding stars would all have about the same brightness, rather than being strongly peaked around M_V ~ 10-11.

from "LF_data_nearbystars_fit.jpg" the plot has a rather large statistical difference in the 10 to 17 range

I take it you have looked at including a +- noise function so that some of he stars are brighter/ dimmer than the ones around it .

John,

a noise function is not needed here, since we are doing statistics with the LF as a probability. This implies automatically that the generated stars will properly fluctuate in brightness also in the region between MV=10 to 17. In the fit the error bars have been included as usual in least-square optimizations; chi^2 ~ sum( (data-fit)/errors )^2

Fridger

Now that this analysis is finished, let me show you an instructive comparison between the Luminosity Function of nearby MW stars (< 25pc) and that of two prominent globular clusters:

[Click on image for bigger display]


One clearly observes a considerable similarity in the shapes, yet also marked differences. In particular, compared to the blue MW stars, the absolute magnitude (MV) scale for GC stars (green) seems compressed. Correspondingly the peak positions differ by about 3 magnitudes and the peak for GC stars is narrower! Hence the most abundant stars outside of GCs are dimmer with a peak absolute magnitude of MV~ 13.2 compared to the peak in GCs around MV~10. Furthermore for the nearby MW stars, the shoulder left of the main peak is more pronounced than for the GC stars.

When the LF is used as a probability distribution for a random generation of faint galactic stars, the shoulder corresponds to an enhanced probability for somewhat brighter stars around MV ~ 5-6 that stick out among the many much fainter ones fluctuating around MV ~ 13. Very bright stars and very dim ones are highly improbable.

Here is another comparison of two different data sets for the LF of nearby MW stars.

[Click on image for a bigger display]


The blue points denote my preferred data set (Jahreiss & Wielen 1997) already displayed above, while the red circles now show LF data from

THE PALOMAR/MSU NEARBY STAR SPECTROSCOPIC SURVEY
I. Neill Reid, John E. Gizis, Suzanne L. Hawley (2002)

in addition. Note the latter don't include error bars. Yet after some renormalization, the two sets are pretty well compatible.


Fridger

Last not least:

Meanwhile, I have implemented the new Luminosity Function fit to the measured nearby MW star data (cf above) into the celestia.Sci code. As stated, it serves as probability function for the generation of realistic populations of faint stars within the spherical halos of galaxies.

Now here comes the crucial code check:

As described for GCs further above, I ask the celestia.Sci code to first print out the absolute magnitudes (MV) of 16384 generated stars into a file which is read in subsequently by Maple.

Using Maple's 'Histogram(...)' command, on this vector of MV values, neighboring values are placed in the same bins and their star contents are counted and plotted versus MV. The area (= integral) of the distribution is normalized to 1, as required for a probability.

Here is the PERFECT result:

[Click on image for bigger display]


Again, in green I plotted for comparison the best fit of the corresponding LF as determined above. Its agreement with the values from my C++ code --after actual star generation-- is excellent. This shows that in celestia.Sci the randomly generated brightness values of faint halo stars match closely the best existing corresponding measurements!

Fridger

=================================================================
Please note: I moved this thread here, since it concerns a recent development feature in celestia.Sci. In irregular time intervals, members of the celestia.Sci dev team will report here about topical research underlying some important new code development.

This material will often be addressed to people with some background in Astrophysics/ Mathematics, who want to know in more concise terms about the theoretical concepts and astrophysical/astronomical data on which new celestia.Sci code will be based.

Throughout it is encouraged to present links to used scientific papers..
=================================================================