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3. Analysis

Standard reduction consisting of bias subtraction and flat fielding was performed on the images using the ESO-Midas package and our own software called "Mana'' (Magnier 1996). The INT data required special care in the bias subtraction as there were significant variations in the bias across the chip. The images were corrected using a template of the bias plus an offset determined from the overscan for each image.

The INT images had substantial variations in the point spread function (PSF) as a function of position on the image, possibly due to a mis-alignment of the CCD in the optical plane. We split each of these images into 9 subimages to minimize this problem. This problem was less severe for the MDM data, so that the entire images could be kept intact, but variations in the PSF with position for both telescopes continued to plague the analysis of all images during relative photometry, and forced some variation in the standard analysis, as discussed below.

Photometry was performed with the program DoPHOT (Mateo & Schechter 1989). This is a PSF fitting program which is designed to be easy to use automatically with a large number of images; DoPHOT needs a relatively small amount of information to start running, and it runs without any interaction with the user. A complete discussion of DoPHOT can be found in Mateo & Schechter (1989). Freedman (1989) also discusses DoPHOT in comparison with other commonly used photometry routines.

DoPHOT models the objects on an image using an elliptical Gaussian with 7 free parameters: a zero level (sky), the central intensity (I0), the centroid (X and Y), the semi-major and semi-minor axes of the ellipse (tex2html_wrap_inline1071 and tex2html_wrap_inline1073), and the angle between the semi-major axis of the ellipse and the pixel coordinate system (tex2html_wrap_inline1075). For a particular image, DoPHOT uses a single set of values for the 3 "shape'' parameters (tex2html_wrap_inline1071, tex2html_wrap_inline1073, and tex2html_wrap_inline1075) to represent the PSF. DoPHOT distinguishes between stars (which are well fit with the PSF) and extended objects (which are well fit by a profile significantly more extended than the PSF). DoPHOT reports instrumental magnitudes and positions in pixel coordinates for all objects as determined from these Gaussian fits.

Astrometry was performed on each image. Two databases were used as a reference. For most images, the MIT/Amsterdam CCD survey of M 31 (Magnier et al. 1992; Haiman et al. 1993) was used as an astrometric reference. For images without sufficient overlap with this survey, the survey by Berkhuijsen et al. (1988) was used, after correction for the systematic error reported in the astrometry (Magnier et al.\ 1993a). Linear astrometric conversions were used; i.e.\ translations, scaling, and rotations were included in the conversion between pixel coordinates (X, Y) and sky coordinates (tex2html_wrap_inline1075, tex2html_wrap_inline1089). The astrometry for certain images was determined from other program observations of the same field, for which the above astrometric parameters had been succesfully determined. Typically between 30 and 100 stars from either of the reference catalogs were identified on each CCD frame. The residuals of the fit were also measured to provide an estimation of the astrometric error, which we determined to be typically tex2html_wrap_inline1015 1.0 arcseconds, dominated by the errors in the original catalogs.

3.1. Relative photometry

Relative photometry was performed on the images to convert the instrumental magnitudes to a common system. This allows us to make corrections for a variety of effects which may alter the throughput for a given image, in particular small amounts of clouds.

For every star on every image, there exists a relationship between the observed instrumental magnitude m,
and the apparent magnitude in a common system tex2html_wrap_inline1095:
The subscripts i and j refer to a particular image and a particular star, respectively. tex2html_wrap_inline1101 is a correction for the throughput for a given image, and may incorporate effects such as cloud level. The goal of relative photometry is to determine tex2html_wrap_inline1101 for each image, then use this tex2html_wrap_inline1101 to find tex2html_wrap_inline1095 for all stars from Eq. (2 (click here)). Once one has tex2html_wrap_inline1095 for each star, one can then convert them to a standard system, such as the Johnson system, using appropriate color corrections. This last step is in general more inaccurate for a variety of reasons, particularly because of the undersampling of existing photometry bands (see Young 1992). For the purpose of identifying variable stars, however, it is more important to have an accurate relative magnitude in an ill-defined system than well-calibrated magnitudes in a commonly used system.

We performed relative photometry according to the scheme outlined above, using an iterative method to minimize the tex2html_wrap_inline1111, defined as
where tex2html_wrap_inline1113 is the error in the measurement mi,j. For the present dataset, some modifications were necessary. First, we measured the tex2html_wrap_inline1111 for each star independently and removed those stars with unusually high tex2html_wrap_inline1111 values. This is needed to remove both the true variable stars from the calibration, as well as those stars which have a single or a few extreme outlying points, due to, e.g., cosmic ray hits or the star falling on a bad column. We also found that the residuals for a given image were a clear function of the position on the image. We traced this problem to the variation of the PSF across the images, for both telescopes. Since the model PSF is kept fixed for a given image (or portion of an image in the case of the INT images), stars which fit the model less well than other stars on the same image will have their flux poorly measured. Thus, a trend across the image in the size of the PSF is translated to a trend in the effective magnitude of a star measured at that position. To compensate for this effect, we modified Eq. (2 (click here)) to incorporate a trend across the image:
where Ai through Fi are kept fixed for each image and x and y are the position of a measured star on an image (subscripts dropped for clarity). We solved the system for Ai through Fi along with the tex2html_wrap_inline1095 and tex2html_wrap_inline1101 terms. In fact, this last correction was not crucial; the actual magnitude of the correction introduced by the terms Ai through Fi was not very large compared to the variability of interest: typically only about tex2html_wrap_inline1141. The fact that every frame had an arbitrary zero point allows a simple connection of the data from the INT 2.5 m and the data from the MDM 1.3 m. In practice, we made initial guesses at the relative zero points of the solutions by correcting for the relative areas of the two telescopes. All data could then be processed simultaneously. The remaining scatter observed for non-variable stars which were bright enough (tex2html_wrap_inline1143) that photon noise was not significant was tex2html_wrap_inline1145. The relative photometry was converted to the Johnson system by calibrating each field relative to the MIT/Amsterdam CCD survey of M 31 (Magnier et al. 1992; Haiman et al. 1993).

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