Photometry and astrometry

One of the basic astronomical pursuits throughout history has been to determine the amount and temporal nature of the flux emitted by an object as a function of wavelength. This process, termed photometry, forms one of the fundamental branches of astronomy. Photometry is important for all types of objects from planets to stars to galaxies, each with their own intricacies, procedures, and problems. At times, we may be interested in only a single measurement of the flux of some object, while at other times we could want to obtain temporal measurements on time scales from seconds or less to years or longer. Some photometric output products, such as differential photometry, require fewer additional steps, whereas to obtain the absolute flux for an object, additional CCD frames of photometric standards are needed. These standard star frames are used to correct for the Earth's atmosphere, color terms, and other possible sources of extinction that may be peculiar to a given observing site or a certain time of year (Pecker, 1970).

We start this chapter with a brief discussion of the basic methods of performing photometry when using digital data from 2-D arrays. It will be assumed here that the CCD images being operated on have already been reduced and calibrated as described in detail in the previous chapter. We will see that photometric measurements require that we accomplish only a few steps to provide output flux values. Additional steps are then required to produce light curves or absolute fluxes. Remember, for a photometrist, every photon counts but the trick is to count every photon.

As an introduction to the level of atmospheric extinction one might expect as a function of observational elevation and wavelength, Table 5.1 lists values of the extinction in magnitudes resulting from the Earth's atmosphere for an observing site at 2200 m elevation. Note that for observations made at reasonable airmass and redward of 4000 A, the effect of the Earth's atmosphere is, at worst, a few tenths of a magnitude. The details of photometric corrections

Table 5.1. Example Atmospheric Extinction Values (Magnitudes)

Altitude

Airmass

3000 Â

3500 Â

4000 Â

4500 Â

5000 Â

5500 Â

6000 Â

6500 Â

7000 Â

8000 Â

9000 Â

10 000 Â

90

1.00

1.2

0.65

0.4

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0.1

0.1

75

1.04

1.2

0.65

0.4

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0.1

0.1

60

1.15

1.3

0.75

0.5

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0.1

45

1.41

1.6

0.9

0.6

0.4

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

30

1.99

2.3

1.3

0.8

0.6

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

20

2.90

3.3

1.55

1.2

0.8

0.6

0.5

0.5

0.4

0.3

0.2

0.2

0.2

15

3.82

4.4

2.5

1.6

1.1

0.8

0.7

0.6

0.5

0.4

0.3

0.3

0.2

10

5.60

6.4

3.65

2.3

1.6

1.2

1.1

1.0

0.7

0.6

0.4

0.4

0.3

5

10.21

11.8

6.7

4.2

2.9

2.2

1.9

1.7

1.4

1.1

0.8

0.7

0.6

for the Earth's atmosphere and extinction effects are not germane to the topic of this book and their discussion here would be beyond the allowed space limitations. The interested reader is referred to the excellent presentations in Young (1974), Hendon & Kaitchuck (1982), Dacosta (1992), and Romanishin (2004). Further good discussions of photometric data handling are presented in Hiltner (1962), Howell & Jacoby (1986), Stetson (1987), Walker (1990), Howell (1992), Merline & Howell (1995), and Howell, Everett, & Ousley (1999).

5.1 Stellar photometry from digital images

Prior to the time when CCDs became generally available to the astronomical community, digital images of astronomical objects were being produced by detectors such as silicon intensified targets (SITs), video tube-type cameras, image tubes, and electronographic cameras. In addition, scanning of photographic plates with a microdensitometer resulted in large amounts of digital output. These mechanisms produced digital data in quantity and at rates far in excess of the ability of workers to individually examine each object of interest within each image. Today, the amount of CCD data greatly exceeds this limit. Thus, in the early 1980s, work began in earnest to develop methods by which photometry could be obtained from digital images in a robust, mostly automated manner.

One of the first such software packages to deal with digital images was written by Adams, Christian, Mould, Stryker, and Tody (Adams et al., 1980) in 1980. Numerous other papers and guides have been produced over the years containing methods, ideas, entire software packages that perform photometry, and specific detailed information for certain types of objects. I have tried to collect a fairly complete list of these in Appendix A. While details vary, the basic photometric toolbox must contain methods that perform at least the following primary tasks: (i) image centering, (ii) estimation of the background (sky) level, and (iii) calculation of the flux contained within the object of interest. We will assume below, for simplicity, that we are working with stellar images that to a good approximation are well represented by a point-spread function of more-or-less Gaussian shape. Deviations from this idealistic assumption and nonpoint source photometry will be discussed as they arise.

5.1.1 Image centering

Probably the simplest and most widely used centering approximation for a point-spread function (PSF) is that of marginal sums or first moment distributions. Starting with a rough pointer to the position of the center of the star (e.g., the cursor position, reading off the x, y coordinates, or even a good guess), the intensity values of each pixel within a small box centered on the image and of size 2L + 1 x 2L + 1 (where L is comparable to the size of the PSF) are summed in both x and y directions (see Figure 5.1). The x, y center is computed as follows: the marginal distributions of the PSF are found from

and i=L

where /y is the intensity (in ADU) at each x, y pixel; the mean intensities are determined from i=L

2L + 1 j and finally the intensity weighted centroid is determined using

For well-sampled (see Section 5.9), relatively good S/N (see Section 4.4) images, simple x, y centroiding provides a very good determination of the center position of a PSF, possibly as good as one fifth of a pixel. More sophisticated schemes to provide better estimations of image centers or applications appropriate to various types of non-Gaussian PSFs are given in Chiu (1977), Penny & Dickens (1986), Stone (1989), Lasker et al. (1990b), Massey & Davis (1992), Davis (1994), and Howell et al. (1996).

5.1.2 Estimation of background

The importance of properly estimating the background level on a CCD resides in the fact that the same pixels that collect photons of interest from an astronomical source also collect photons from the "sky" or background, which

0 ————————————————————

Fig. 5.1. An example of x, y centroiding. The idealized star image in the top box sits on a pixel grid with a center of (x, y) = (10.3, 10.6). The two other plots represent the x, y centroids for the star image normalized to a maximum height of one. In this case, the star center is well approximated by the peaks in the x, y centroids.

VCentroid

Fig. 5.1. An example of x, y centroiding. The idealized star image in the top box sits on a pixel grid with a center of (x, y) = (10.3, 10.6). The two other plots represent the x, y centroids for the star image normalized to a maximum height of one. In this case, the star center is well approximated by the peaks in the x, y centroids.

J_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_L

XCentroid

J_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_L

XCentroid

are of no interest. Remember that the background or sky value in a CCD image contains not only actual photons from the sky but also photons from unresolved astronomical objects, read noise, thermally generated dark current electrons, and other sources. All of these unwanted additional photons must be accounted for in some manner, estimated, and removed from the image before a final determination of the source flux is made. In order to determine this background level, a common technique is to place a software annulus around the source of interest and then use statistical analysis to estimate its mean level on a per pixel basis.

The background or sky annulus is usually defined by an inner and outer radius or by an inner radius and a width (see Figure 5.2). One simple, yet powerful, manner by which an estimation of the background level can be made is simply to extract the values of all the pixels within the annulus, sum them, and divide by the total number of pixels within the annulus. This provides an average value per pixel for the background level of the CCD image. For a good statistical determination of the background level, the total number of pixels contained within this annulus should be relatively large, about three times the number contained within that of the source aperture (Merline & Howell, 1995).1 A more robust estimator, requiring very little

1 Partial pixels arising from placing a circular annulus on a rectangular grid are usually not of concern here, as the number of annulus pixels is large. However, partial pixels cannot be so easily dismissed when we are determining the intensity within the much smaller source aperture.

Row Pixels

Fig. 5.2. Schematic drawing of a stellar image on a CCD pixel grid. The figure shows the location of the star, the "star" aperture (solid line), and the inner and outer "sky" annuli (dashed circles).

Row Pixels

Fig. 5.2. Schematic drawing of a stellar image on a CCD pixel grid. The figure shows the location of the star, the "star" aperture (solid line), and the inner and outer "sky" annuli (dashed circles).

additional work, is to collect all the pixel values from inside the sky annulus, order them in increasing value, and find the median intensity, BM .1 A nice touch here is to reexamine the list of annulus pixel values and toss out all those with values greater than from BM. This last step will eliminate cosmic ray hits, bad pixels, and contamination from close-by astronomical neighbors if they exist.

When applying a median filter and the cutoff technique to the list of background pixels, one can use the remaining annulus pixel values to construct a background histogram computed with a bin width resolution of say 0.5 ADU (Figure 5.3). The background histogram will be centered on the median background value with all contained pixel values within ±3o-. Since the detailed resolution of 0.5 ADU binning will likely produce a ragged histogram (since only a finite number of background pixels is used), some smoothing of the histogram may be useful, such as that done by Lucy (1975).

1 Note that the statistical values determined from a CCD image for the median or the mode must pick their answer from the list of the actual CCD pixel ADU values, that is, values that are integers containing only digitized levels and thus digitization noise. The statistical mean, however, allows for noninteger answers. This seemingly subtle comment is of great importance when dealing with partial pixels, undersampled data, or high CCD gain values.

Fig. 5.3. Histogram of the "sky" annulus around a star in the CCD image shown in Figure 4.5. Notice the roughly Gaussian shape to the sky distribution but with an extended tail toward larger values. This tail is due to pixels that were not completely calibrated in the reduction process, pixels with possible contamination due to dark current or cosmic rays, pixels with increased counts due to unresolved PSF wings from nearby stars, and contamination of sky annulus pixels by faint unresolved background objects. The need for some histogram smoothing, such as that described in the text, is apparent, especially near the peak of the distribution.

Fig. 5.3. Histogram of the "sky" annulus around a star in the CCD image shown in Figure 4.5. Notice the roughly Gaussian shape to the sky distribution but with an extended tail toward larger values. This tail is due to pixels that were not completely calibrated in the reduction process, pixels with possible contamination due to dark current or cosmic rays, pixels with increased counts due to unresolved PSF wings from nearby stars, and contamination of sky annulus pixels by faint unresolved background objects. The need for some histogram smoothing, such as that described in the text, is apparent, especially near the peak of the distribution.

Lucy smoothing will broaden the histogram distribution slightly after one iteration but application of a second iteration will restore the correct shape and provide a smooth histogram from which to proceed. This final step produces a statistically valid and robust estimator from which we can now compute the mean value of the background, B. A determination of the centroid of the final smoothed histogram, using equational forms for the centroid in one dimension such as those discussed above, can now be applied. Centroiding of this smoothed histogram is not influenced by asymmetries that may have been present in the wings of the initially unsmoothed values.

Correct estimation of the level of the CCD background on a per pixel basis is of increasing importance as the S/N of the data decreases and/or the CCD pixel sampling becomes poor. A background level estimation for each pixel that is off by as little as a few ADU can have large effects on the final result (Howell, 1989). Determination of the CCD background level has an associated error term a (1 + npix/nB)-1/2, which should be included in the S/N calculation of the final result. The "sky" is the limit.

5.1.3 Estimation of point source intensity

We now come to the pixel values of interest, that is, those that contain photons from the source itself. Using a software aperture of radius r centered on the x, y position of the centroid of the source PSF, we can extract the values from all pixels within the area A(= wr2) and sum them to form the quantity S, the total integrated photometric source signal. The sum S contains contributions from the source but also from the underlying background sources within A. To remove the estimated contribution to S from the background, we can make use of the value BB, discussed above. We can calculate an estimate of the collected source intensity, I, as I = S - npixBB, where npix is the total number of pixels contained within the area A. There are some additional minor considerations concerning this procedure but these will not be discussed here (Merline & Howell, 1995).

A final step usually performed on the quantity I, which we will discuss further below, is to determine a source magnitude. The value of a magnitude is defined by the following standard equation:

Magnitude = -2.5 log10(I) + C, where I is the source intensity per unit time, that is, the flux (universally given as per second), and C is an appropriate constant (usually ~23.5-26 for most earthly observing sites) and determined in such a manner so that the calculated source magnitude is placed on a standard magnitude scale such as that of the Johnson system or the Stromgren system.

As we mentioned above, when using circular apertures on rectangular pixel grids, partial pixels are inevitable. While we could toss them away for the large background area, we cannot follow a similar sloppy procedure for the smaller source aperture. Thus the question becomes, how do we handle partial pixels? This is not a simple question to answer and each photometric software package has its own methodology and approach. The three choices a software package can use are:

1. Do not use partial pixels at all. Any source intensity that falls into the source aperture but within a partially inscribed pixel is simply not used in the calculation of S.

2. Sum the values for every pixel within the source aperture regardless of how much or how little of the pixel area actually lies within A.

3. Make use of some computational weighting scheme that decides, in a predefined manner, how to deal with the counts contained within each partial pixel in the source aperture.

This last choice often uses the ratio of the pixel area inside the source aperture to that outside the aperture as a simple weighting factor. A computational scheme to handle partial pixels in a software package designed to perform digital photometry is the hardest of the above choices to implement, but it will provide the best overall final results. To know exactly how a certain software package handles partial pixels, the user is referred to the details presented within the documentation provided with the software. Many PC-type packages that perform photometry on CCD images do not detail their partial pixel and magnitude calculation methods and are therefore "black boxes" to be avoided.

There are two basic methods by which most observers estimate the total integrated signal within their source aperture: point-spread function fitting and digital aperture photometry. The first method relies on fitting a 2-D function to the observed PSF and using the integrated value underneath this fitted function as an estimate of S. The second method, digital aperture photometry, attempts to place a software aperture about the source profile (as shown in Figure 5.2), centered in some manner (e.g., x, y centroids), and then simply sums the pixel values within the source aperture to provide the estimation of S. We will discuss each of these methods in turn below and note here that it is unlikely that a single method of estimation will be the best to use in all possible situations. For example, for severely undersampled data the method of pixel mask fitting (Howell et al., 1996) provides the best solution.

The profiles of astronomical point sources that are imaged on two-dimensional arrays are commonly referred to as point-spread functions or PSFs. In order to perform measurements on such images, one method of attack is profile fitting. PSFs can be modeled by a number of mathematical functions, the most common include Gaussian,

5.2 Two-dimensional profile fitting modified Lorentzian, and Moffat,

W (1 + r 2/a2)b representations, where r is the distance from the center of the point source and a and b are fitting parameters (Stetson, Davis, & Crabtree, 1990). These types of functional forms can be used to define the PSF for each star within an image by the assumption that they provide a good representation of the data themselves. For example, adjustment of the values of a and b within one of these functions may allow an imaged PSF to be matched well in radius and profile shape (height and width), allowing a simple integration to be performed to measure the underlying flux.

Generally, the above functions are only a fair match to actual PSFs and so a second method of profile fitting can be applied. This method consists of using an empirical PSF fit to the actual digital data themselves, producing modified versions of the above functions. Depending on the application, PSFs may be evaluated at the center of a pixel or integrated over the area of each pixel. Even more general methods of allowing the data to produce completely analytic forms for the PSF functions have been attempted. The techniques and use of empirical PSFs could fill an entire chapter; we refer the reader to King (1971), Diego (1985), and Stetson (1987) for more details.

Both techniques, the use of completely mathematical forms for a PSF approximation and the more empirical method, have their advantages and disadvantages. Model PSF fitting allows the necessary integrations and pixel interpolations to be carried out easily as the functions are well known, while the empirical method, which makes hardly any assumptions about the actual shape of the PSF, is only defined on the CCD pixel grid and not in any general mathematical way. This latter complication can cause difficulties when trying to interpolate the empirical shape of one PSF somewhere on the CCD (say a bright reference star) to a PSF located somewhere else on the same image but that is likely to have a different pixel registration. For this reason, some implementations of empirical PSF fitting actually make use of the sum of an analytic function (such as one of those given as above) and a look-up table of residuals between the actual PSF and the fitting function. Figure 5.4 shows examples of some PSF models and some actual PSFs obtained with CCDs.

Procedurally, profile fitting techniques work by matching the implied PSF to the actual digital data in a 2-D fashion and within some radius, r, called the fitting radius. An attempt is then made to maximize some goodness-of-fit criteria between the assumed PSF and the observed one. PSF fitting can be further optimized by fitting N point sources within the image simultaneously

Fig. 5.4. Stellar PSFs are shown for various cases. The figure above shows two model PSFs, one for a bright star (S/N ~125) and one for a faint star (S/N ~ 20). The remaining two panels show similar brightness stars but are actual CCD data. Note that the models are shown as 3-D pixel histograms whereas the real data are represented as spline profile fits to the actual PSFs. The disadvantage of the latter type of plotting is that the true pixel nature of the image is lost.

Fig. 5.4. Stellar PSFs are shown for various cases. The figure above shows two model PSFs, one for a bright star (S/N ~125) and one for a faint star (S/N ~ 20). The remaining two panels show similar brightness stars but are actual CCD data. Note that the models are shown as 3-D pixel histograms whereas the real data are represented as spline profile fits to the actual PSFs. The disadvantage of the latter type of plotting is that the true pixel nature of the image is lost.

X-AXIS

Y-AXIS

X-AXIS

Y-AXIS

(usually one uses the brightest stars within the image) and using some combination of statistically weighted mean values for the final fitting parameters. PSF fitting can be very computationally demanding, much more so than the method of aperture photometry discussed below. However, for some types of imagery, for example crowded fields such as within star clusters for which some PSFs may overlap, PSF fitting may be the only method capable of producing scientifically valid results (Stetson, 1987, 1992, 1995).

5.3 Difference image photometry

One method used today for studies of photometric variability is the technique of difference image photometry (DIA), also called difference image analysis or image subtraction (Tomaney & Crotts, 1996). DIA is useful in (very) crowded field photometry or when searching for variable sources that may be blended with other possibly nonpoint sources. Most modern photometric searches for supernovae in external galaxies or gravitational lenses use DIA

as these studies involve imaging in very crowded stellar fields and searching for highly blended sources.

The basic idea of DIA is to take a reference image and subtract from it images of the same field of view but taken at different times. An example would be to take a CCD image of a star cluster at an airmass of one and use this as your reference image. Additional images taken of this same field over time are then each subtracted from the reference image and variable sources show up in the difference image. Figure 5.5 shows a nice example of DIA from the supermacho project being carried out at the CTIO 4-m telescope in Chile. The reference image may actually be a sum of some number of the best images obtained (say during the best seeing) or an image observed at the lowest airmass. In practice, DIA is not so simple and involves setting up a CPU intensive, fairly complex software pipeline.

Before the simple process of subtraction from the reference image can occur, each successive image must be positionally registered, photometric normalized, and adjusted for other offending effects such as differential refraction, seeing and telescope focus changes, and possibly sky conditions. The matching of the point-spread functions between frames can be accomplished by Fourier divison (Alcock et al., 1999) or a linear kernal decomposition in real space (Alard, 2000). Right away one can see that this is not a simple process. It involves setting up various transformation processes in software and a diligent eye to make sure they all work correctly in an automated

Fig. 5.5. An example of difference image analysis. The image was produced by C. Stubbs as a part of the high-z supernovae team using the supermacho data set. The image on the left is the reference image taken at epoch 1, the middle image is from epoch 2, 3 weeks later, and the right image is their difference. A supernova blended with its host galaxy image (middle frame) is clearly detected in the difference image.

Fig. 5.5. An example of difference image analysis. The image was produced by C. Stubbs as a part of the high-z supernovae team using the supermacho data set. The image on the left is the reference image taken at epoch 1, the middle image is from epoch 2, 3 weeks later, and the right image is their difference. A supernova blended with its host galaxy image (middle frame) is clearly detected in the difference image.

fashion. Errors in these flux manipulation steps can be large and are often unaccounted for in the final result.

DIA is somewhat akin to profile fitting but is done on a frame-by-frame basis. The two-dimensional profile of each object in one frame is transformed to match those of the reference frame. One can model the profiles in a given frame using some small fraction of the brightest uncrowded sources per frame and applying the same model mapping of these sources to the reference frame to all objects in the given frame (Alcock et al., 1999). This procedure saves CPU time but may introduce some uncertainty as it relies on a few of the brightest sources to be an exact match to the remaining sources. Spatial dependence, color dependence, pixel sampling, and seeing can all vary across an image and are hard to correct for perfectly. DIA has been used very effectively in a number of projects and continues to be the method of choice in certain regimes where crowding of some fashion is prevalent. The accuracy of the photometry delivered depends on how well the software processes of registration, convolution, and normalization are performed and what assumptions are used. For example, some DIA analysis assumes that stellar colors are well approximated by blackbody functions and in other cases that the bright stars well represent the remaining (fainter) stars. Both of these assumptions are valid to a point but are highly efficient in terms of processing the data. To date, photometric differences of 0.5 magnitude or better are easy to detect in a single difference frame. Thus DIA is a good technique for finding fairly large amplitude changes (i.e., newly brightened sources) but its ability to produce highly accurate light curves is yet to be fully explored.

The process of DIA is a mixture of photometry and astrometry plus profile fitting and using software to remap images to match the reference image. If observations are obtained of fields of interest that are not crowded or blended, than DIA is overkill and profile fitting or differential photometry (see below) work well and are easier to implement. These latter two techniques are also not subject to the addition of photon noise via the difference processing or any additional systematic effects as exist in DIA. But, as we note, every photometric technique has pros and cons and for highly blended or crowded fields DIA is a useful tool (see Tomaney & Crotts, 1996, Zebrun et al., 2001, and references therein).

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