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ASTR 5610, Majewski [SPRING 2012]. Lecture Notes

ASTR 5610 (Majewski) Lecture Notes


STARCOUNTS AND THE EQUATION OF STELLAR STATISTICS

References: Binney & Merrifield, Sections 3.6, 10.4.3.

Also helpful are Reid & Majewski (1993, ApJ, 409, 635).

Closeup view of the Galaxy's disk and halo structure perpendicular to the Galactic plane near the solar neighborhood (shaded circle centered around dot). Relative space densities, ρ, of the stars belonging to the different components are shown schematically (left) and analytically (right) as exponential functions of vertical distance, z: ρ(z) = ρ(0)ez/hz. Horizontal dashed lines indicate the exponential scale heights of the thin disk (hz,D = 300 pc) and the thick disk (hz,T = 1000 pc), where the densities have decreased to a fraction 1/e ~ 0.37 of their maximum values at z = 0. In the solar neighborhood, for 1000 thin-disk stars there are about 20 thick-disk stars and only about 1 halo star, while at a height |z| = 1000 pc, the numbers of thin-disk and thick-disk stars per unit volume are practically the same. At even larger distances, the thick-disk stars dominate far into the halo. From Buser (2000, Science, 287, 5450, 69).

Spatial distributions (Density Laws) of Galactic Stellar Populations:

One way this is done is with the method of starcounting:


A HISTORY OF STARCOUNTING

1600's: Galileo uses telescope to resolve Milky Way into a sea of stars.

In his book "Sidereus Nuncius" Galileo demonstrated that there were many stars that were previously unknown because they were fainter than the unaided eye could see. To make his point forcefully, he drew figures (as on the right above) of familiar constellations, like Orion's belt here, with the new stars he discovered shown. Compare to the figure on the left of naked eye stars in Orion.
1700's: Thomas Wright, Immanuel Kant speculate that the Milky Way (the "sidereal universe") is disk-shaped.

Late 1700s-1800s: Herschel began the science of statistical astronomy (carried on by his son John).

Eventually, some problems began to become evident in the 1800s:

  1. Herschel noted that with a larger telescope, he saw more stars in each field of view.

    Herschel's "40 foot telescope".
  2. Herschel also discovered that stars are not all of equal brightness (how?).

  3. Herschel then concludes that the number of stars observed is related both to the extent of the Galaxy and the density of stars.

  4. In an (incorrect!) attempt to solve Olber's Paradox, Friedrich Struve invoked the idea of interstellar absorption (he was right, but for the wrong reasons!).
Despite these shortcomings, and even with the development of extensive catalogues of hundreds of thousands of stars (e.g., as in the Bonner Durchmusterung and the even more systematic, photographically-based surveys like the Carte du Ciel and the Southern Cape Photographic Durchmusterung by David Gill) the "Herschel Method" was used for a "direct" analysis of starcount data for more than another century!

1847: Struve performed the first statistical analysis of density laws.

1870s: A Swede named Gylden introduced the idea of the luminosity function:

The Gylden invention is an advance, but introduces a new problem: How do we determine Φ(M )?? By starcounting !

This brought things to what has been described as "the vicious circle of starcounting":


A Brief Aside on the Luminosity Function

We do know that there are stars of different intrinsic brightness.

We also know that the density of stars varies from point to point.

Let dN be the number of stars with absolute magnitudes between M and M+dM in the volume d 3x around point x.

Now it is useful to imagine that the relative mix of stars of different luminosities does not vary by position in the Galaxy (this is untrue, but...).


1900: von Seeliger, after about 15 years using the same, frustrating, Herschel-like techniques, introduces a powerful new statistical, integral approach that fixes the problems plaguing the earlier work.

1900-1930: Jacobus Kapteyn understood the power of the von Seeliger approach and set out to make the "Selected Areas" program to undertake a systematic starcounts survey (expanded to include astrometry, spectroscopy) to understand the Galaxy (i.e. derive D[r]).

Jacobus Cornelius Kapteyn (my great, great, great thesis grandfather!) on the left and his Milky Way model from 1922 (in Plan of Selected Areas), on the right. The model shows ellipsoids of constant star density that he derived using starcounts in his Selected Areas.
A modern version of the same kind of density model from a starcounts project by UVa graduate student Michael Siegel (Siegel et al. 2002, ApJ, 578, 151) -- see discussion below.
A few comments on the classical von Seeliger/Kapteyn approach to getting density laws of the Galaxy:

  1. The von Seeliger integral is manageable if we know:

    • A(m) - obtained by observation

    • Φ(M) - somehow (assumed, iteratively arrived at)

    • a(r) - reddening variation with distance known

  2. Although the integral is from 0 to infinity, practically we truncate at some distance that is physically limited by the maximum distance we can see with the brightest stars:

    log rmax ~ 0.2 (m-Mmin ) + 1 - 0.2a(rmax )

    where Mmin ~ -6 for normal stars and a(r) is about 1 mag/kpc on average in the disk.

  3. This "starcount" methodology applies to counting anything (stars, galaxies, QSOs, etc.).

  4. As written so far, the methodology ignores known variations in Φ(M ).

    For example,

    • we know that bright supergiants are Pop I, disk objects not seen in the halo.

    • abundance variations change Φ(M ) -- or galaxy types by environment if counting galaxies.

  5. The von Seeliger integral is a convolution.

    • The larger the range in Φ(M ), the larger the range of r that contribute to A(m) .

    • The larger the range of r accessed, the more severely information about D(r) is smeared out and hard to recover.

    That is, a fluctuation in D(r) at r' can be washed out by an opposing fluctuation at r".

    Combined with observational errors in A(m) and Φ(M ), the great width in Φ(M ) means that D(r) is very poorly determined with this methodology.

  6. Clearly, we gain resolution in D(r) if we confine ourselves to a counting of star with a narrow Φ(M ) range.

    • Thus we prefer to work with a luminosity function for a single spectral type, S, having a narrow range of Φ(M ).

    • Thus, starcount analyses are most successful when limited to so-called "tracers" of a population (e.g., RR Lyrae stars, giant stars, star clusters, G dwarf stars).

    • In this case, we are actually approaching something like Gylden's Gaussian-like luminosity function, which may be appropriate for a single tracer star species.


Modern Usage of the von Seeliger Equation (The Photometric Parallax and Computer Simulation Methods)

An early plot demonstrating the existence of the thick disk using starcounts, from data by Gilmore & Reid (1983). Figure from Binney & Merrifield.

Recall that there has long been recognition of a need for a stellar population "intermediate" between the halo and the disk.

Note that one of the shortcomings of the "photometric parallax method" is that information on metallicities is typically missing and needs to be assumed.

A lot of work has been done towards the Galactic poles, since this simplifies the problem to only understanding the vertical structure of Galactic populations (except, of course, the bulge!).

But of course, one can (and should!) explore other directions to probe radial variations.

Some luminosity functions adopted in the Reid & Majewski (1993) modeling.

Examples of Hess diagrams for the thin disk, IPII/thick disk and halo as used in an earlier version of the Besancon starcounts model.
Need to assume functional forms for the density laws of the "non-disk" components:

After defining some guesses at density laws, luminosity functions (Hess diagrams) and reddening laws (and in the case of the next figure, a star formation rate ψ), one starts to integrate along the line of sight to find a prediction of starcounts as a function of magnitude (and spectral type or color).

The figure below shows how the von Seeliger integral is then calculated in a computer model.

Example of a flow chart for the Besancon Galactic structure model.

A similar flow chart for the Trilegal Galaxy model by Girardi et al. (2005, A&A) -- see below.

One then compares the results (e.g., distribution by color and magnitude) of the artificially generated starcounts to actual data, and then iterates to improve fits by varying assumed input parameters (such as the luminosity function, the color-absolute magnitude relations for stars of different types, the dust extinction law and, of course, the much sought after density law).

A number of computer models have been created throughout the years, and some are still in common use:

When various groups use such models on Milky Way data, they commonly find certain preferred models that fit best:

The SDSS database provides one of the largest databases applied to the starcounts problem. Here are the results from Juric et al. (2008, ApJ, 673, 864).

Figures from Juric et al. (2008) showing in left column the derived density distribution for stars in four different r- color bins, the middle column is the derived Milky Way model fit, and the right column is the ratio of data to model.

Illustration of degeneracies present in fitting of Galactic models. The two panels in the left column are showing the vertical (Z) distributions of SDSS stellar counts at the position of the Sun and for different color bins, as marked. The lines are exponential models fitted to the points. The dashed lines in the top panel correspond to a fit with a single, exponential disk having a 270 pc scale height. The vertical dot-dashed line marks the position of the density maximum and implies a solar offset from the Galactic plane of ~20 pc. The dashed line in the middle panel corresponds to a sum of two disks with scale heights of 270 and 1200 pc and a relative normalization of 0.04 (the "thin" and the "thick" disks). The dot-dashed line is the contribution of the 1200 pc disk. Note that its contribution becomes important for |Z| > 1000 pc. The right panels show a different decomposition of the counts with a 250 pc scaleheight thin disk and an ~700 pc scaleeheight think disk with a relative normalization of 0.12 (and including a halo as well for one color bin). In spite of substantially different best-fit values, the two models are virtually indistinguishable when fitting the solar position, +/-Z-direction of the data. (Captions and figures modified from Juric et al. 2008).

Why does knowing the IPII parameterization matter for understanding stellar populations?


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All material copyright © 2003,2006,2008,2010,2012 Steven R. Majewski. All rights reserved. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 551 and Astronomy 5610 at the University of Virginia.