Exercice 3 - A Deep Observation of a Globular Cluster¶

Globular star clusters (GC’s) usually contain about $10^{5}$ stars. The stars are spherically distributed, and the central densities are about ten times larger than in open clusters. The globular clusters are among the oldest stars in the Milky Way, and are therefore of great importance for studies of stellar evolution. Very small heavy element abundances, down to about $10^{-3}$ times the solar value, have been detected in some halo globular clusters. They therefore give important information about the production of elements in the early Universe and during the formation of the Milky Way. There are about 150-200 globular clusters in the Milky Way.

Hertzsprung-Russell (HR) diagrams, or color-magnitude diagrams, of star clusters can be constructed in a self-consistent way without knowledge of the exact distances to them. Since the dimensions of a typical cluster are small relative to its distance from Earth, little error is introduced by assuming that each member of the cluster has the same distance modulus. As a result, plotting the apparent magnitude, rather than the absolute magnitude only amounts to shifting the position of each star in the diagram vertically by the same amount. By matching the observational main sequence of the cluster to a main sequence calibrated in absolute magnitude, the distance modulus of the cluster can be determined, giving the cluster’s distance from the observer. This method of distance determination is known as main-sequence fitting.

Rather than attempting to determine the effective temperatures of every member of a cluster by undertaking a detailed spectral line analysis of each star, it is much faster to determine their color indices (B-V). With knowledge of the apparent magnitude and the color index of each star, a colour-magnitude diagram (CMD) can be constructed.

$B$ and $V$ CCD images of the metal-poor Galactic GC M15 (= NGC 7078) have been obtained from the ESO FORS2 imager attached to one of the four Very Large Telescope (VLT) units. The images have already been reduced (i.e. bias substracted and flat-fielded). Analysing an HR diagram includes the following aspects:

3.1 - Aperture photometry¶

Aperture photometry (measure the observed flux) is the most straightforward way to compute magnitudes in uncrowded or moderately crowded stellar fields. The instrumental magnitude m of a star is computed as follows:

$m = -2.5 \, \log \left( \frac{C_{tot} - a c_{sky}}{\Delta t_{exp}} \right) + m_0$

where $C_{tot}$ is the total count (star + sky) in the photometry aperture; $a$ is area of the aperture in pixels squared and is roughly equal to $\pi r^2$ where $r$ is the radius of the photometry aperture in pixels; $c_{sky}$ is the estimated sky value in counts per pixel squared computed in a sky annulus centered on the star; $\Delta t_{exp}$ is exposure time and $m_0$ is zero point offset for the magnitude scale.

In practice the aperture photometry is carried out using the SExtractor software. It has also been installed on lesta, so you can execute module spider sextractor/2.19.5 to see how to load it. This software is able to provide a catalogue of objects that it identifies in the case of a single input file or it cross-matches in the case of double input files, with information that we require it to keep. An introduction to this software will be given in class. Some useful documents are also provided on our online LASTRO documentation.

Determine the optimal aperture diameter in pixels for images located in the images directory with the single-file mode of sextractor. This aperture represents the radius of a circle centered on the object for measuring the photons encompassed within it. The optimal value should include most of the flux from the centre object but exclude other nearby objects.
Extract the photometric information for aligned pairs of B-band and V-band images.
- Use ds9 to find the aligned images, i.e., those have the glubular cluster at the same place. You will have a set of images with longer exposures and that with shorter exposures.
- Use sextractor to measure the photons of stars in one band, and cross-match the object in this band to another band. Remember that the unit of the photometric information is in counts.

3.2 - Calibration or transforming to the standard system¶

The measurement done in the previous section is the instrumental magnitude. Conversion from instrumental magnitudes $(b,v)$ to the calibrated magnitudes $(B,V)$ follows a simple linear model:

$b = B + b_{0} + b_{1} X + b_{2} \left( B - V \right)$

$v = V + v_{0} + v_{1} X + v_{2} \left( B - V \right)$

The constants to account for the instrumental effect $b_{0}$ and $v_{0}$ can be determined using “standard stars”, which are stars with determined magnitudes and colours, observed with the same instrument and filters and under identical conditions as the M15 cluster. Don’t forget to have a look to headers of fits files as well!

For the atmospheric extinction coefficients $b_{1}$ and $v_{1}$ , one can use the average atmospheric absorption coefficients measured at Paranal : $b_1 = 0.26$ [mag / airmass], and $v_1 = 0.17$ [mag / airmass]. $X$ is the airmass ( $\approx \sec z$ where $z$ is the altitude of the star measured from zenith).

For simplicity, in this exercise we will ignore second-order colour coefficients $b_{2}$ and $v_{2}$ , which are negligible compared to other terms. They can be determined using “standard stars” as well.

Interstellar reddening and extinction are due to the presence of dust grains along the line-of-sight. While the former affects the measured colour, the latter affects the observed luminosity. We consider a reddening correction of $E_{B-V} = 0.10$ . As interstellar exctinction has generally the same origin as the reddening, the exctinction correction can be approximated by the relation $A_V = 3.2E_{B-V}$ .

Measure $b_{0}$ and $v_{0}$ for standard stars with the calibrated magnitude provided in Ru152.pdf under the standard_star directory.
Obtain the apparent magnitude in B- and V-band with corrections in instrument, atmospheric extinction, interstellar reddening and extinction

Now you should have two CMD at hand, one with longer exposure and the other with shorter exposure.

3.3 - Fiducial sequence and isochrones¶

Compare the CMDs with the fiducial sequence, choose the one that contains more information. Describe the morphology of that CMD, i.e. link the different regions of the diagram with the phases of stellar evolution and locate the main-sequence turn-off point (TO), i.e. the point where the stars are currently leaving the main-sequence. The so-called “fiducial sequence” is a smoothed sequence of M15 obtained by e.g. P.R. Durrell and W.E. Harris (1993, AJ 105, 1420), which just represents some kind of best fit to the observed sequence (evolution_models/fiducial.txt).
In this exercise, you need isochrones corresponding to the metallicity of M15, namely $[\rm{Fe/H}]=-2.15$ , which corresponds to a fraction of “heavy” elements $Z=1.2\cdot 10^{-4}$ , and its Helium composition of 0.2. Isochrones refer to curves connecting stars in CMD with the same composition and age, but various masses. Such isochrones were computed by Vandenberg (1985, ApJS 58, 561) for similar metallicities. See the file ReadMe.txt for a detailed description of files evolution_models/*85iso.txt.
- Determine the distance of M15 by compare your CMD with a calibrated zero-age main-sequence (ZAMS).
- Estimate the age of M15 by fitting isochrones to the CMD fiducial sequence.
Estimate the half-light radius of M15.
Estimate the luminosity and mass of M15 assuming the cluster consists in solar-type stars only.

Exercice 3 - A Deep Observation of a Globular Cluster¶

3.1 - Aperture photometry¶

3.2 - Calibration or transforming to the standard system¶

3.3 - Fiducial sequence and isochrones¶

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