N E X E Xi

Similar expressions hold for the color index coefficient, k'c and c0. The method assumes that the second-order coefficient is constant and is known. See Schlosser et al. (1991, p. 129) for a detailed discussion of this method. Further refinements have been discussed by Sterken & Manfroid (1992). One of these involves the combinations of data from different nights to find the extra-atmosphere magnitudes and colors (assuming that the local system is stable from night to night).

A second method is referred to as the Hardie extinction method (Hardie 1962). It involves observations of groups of stars with a range of color indices at both low and high airmass. The equations of condition are dm- - dm-0 - k"d(Xc) = k'dX (2.1.6)

where the differential quantities refer to difference between star pairs at high and low airmass. The pairs must be of similar color index. The least-squares result for the one-unknown problem is

with a similar expression for k'c. This method works well with a cluster of stars which can be observed at different times of night and therefore at different airmasses, or with stars in two or more Harvard Selected Areas which have been systematically observed and standardized by Landolt [cf. Landolt (1983)].

The techniques are applicable to both photoelectric and CCD photometry. With meticulous care, relative photometric precision of the order of 2-3 millimagnitudes is possible, but absolute photometry is not so easily accomplished to this degree of accuracy. Transformation to a standard system is an important procedure which astronomers ignore to their peril. The result can be disastrous, especially for the combination of data from different observers. Failure to transform adequately is akin to writing a research paper in one's private code. Hardie's (1962) treatment of transformations from the local system (magnitudes and color indices transformed from raw instrumental values to outside the atmosphere values) involves the assumption of a linear relationship between the local and the standard systems. The relationships are as follows:

The e coefficient is usually small with an absolute value less than 0.1 for well-matched systems. For we expect a number near 1, typically in the range between 0.9 and 1.1. Z in each case is the zero point. If the filters and detector are similar to those used to establish the standard system, and no other major transmission element or exotic reflecting surface has been introduced in the light path, equations (2.1.9) and (2.1.10) are not bad approximations. However, the linearity of any set of transformations is always open to question [see Young (1974)]. Even when the local system is designed to approximate the standard system closely, there are occasions when it may fail to do so. Milone et al. (1980) discuss a situation where the zero points were time dependent. Apparently a broken heater wire for the fused quartz window of the PMT chamber5 led to a slow build up of frost on the window which resulted in a linear decrease in sensitivity of the photometer over time.

The solutions for the two unknowns for each equation are readily found. For the magnitude equations of condition, (2.1.9), they are nJ2[(^m)Cstdl - ECstd n E Cs2td - E Cstd E Cstd

Cs2td - E[(^m)Cstd] E Cstd fiA = ^ ^ 2 ^ ^ ^ , (2.1.11)

n E cstd - E cstd E Cstd where the quantity (Am) = mstd - m0. For the color equations of condition, they are n E[coCstd] - E coE Cstd ,, , , _

5 The chamber in which the photomultiplier tube in a photoelectric photometer is located. It is usually made of ^-metal and acts like a Faraday cage against the ambient magnetic field of the Earth, so that the latter does not affect the streaming of electrons across the dynodes. The window through which light enters is usually of fused quartz to permit ultraviolet radiation to pass. It must be heated to avoid dew and/or frost accumulation.

With the extinction and transformation coefficients and zero points, the raw differential magnitudes and color indices may be transformed to the standard system. Similar techniques can be carried out in infrared astronomy, but standardization has been problematic [see Milone (1989)]. New infrared passbands to minimize the effects of water vapor extinction and improve transformability have been proposed by the IAU Working Group on Infrared Astronomy (Young et al. 1994). The near-IR portion of this suite, namely iZ, iJ, iH, and iK, has been fabricated and tested (Milone & Young 2005, 2007, 2008).

2.1.6 Significance of Cluster Photometry

CCD photometry has proven its value in cluster studies. Observations of faint stars at or fainter than the main sequence turn-off in many globular clusters are being made for the first time.

Two powerful techniques for determining ages of star clusters are the following:

1. The fitting of isochrones to a cluster's color magnitude diagram (CMD) (see Fig. 7.5). An important test for the accuracy of this fitting is the agreement of the isochrones with the CMD locations of stars of known masses. This method is model dependent and requires accurate isochrones to be successful. An example of the method is provided in Milone et al. (2004).

2. The comparison of the sizes of evolved components with model predictions. This technique is not entirely independent of the first because evolutionary predictions are needed for this technique also, and the radius is a function of the luminosity. Substantial numbers of variables and nonvariables may be imaged in both direct and multi-object-spectrograph imaging.

Variable stars in clusters can provide independent assessments of cluster distance and permit checks on luminosity calibrations. Figure 2.3 shows what can be achieved. A group at the University of Calgary is analyzing UBVI data obtained with the SDSU Mt. Laguna Observatory 1-m telescope equipped with a back-illuminated, thinned 800 x 800 chip. Averages of the brightest nonsaturated, nonvariable stars in the field define an artificial "comparison star" for each frame. Typical (m.s.e.) uncertainties in each observation are about 0m05 for the faintest (V = 19?15) and about 0^005 for the brightest (V = 14m) variables. The study of variable stars in clusters can provide fundamental information about both the ensemble and the individual stars. The techniques are described in numerous sources, e.g., in Milone (2003) and Milone et al. (2004). Consult Howell et al. (2005) for an example of the use of CCD techniques to study an open cluster.

Fig. 2.3 CCD image frame of the globular cluster NGC 5466. This I (infrared) passband CCD image frame was obtained at Mt. Laguna by E. F. Milone. NGC 5466 contains at least three eclipsing binary blue stragglers
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