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How to do photometry with SIP

One of the most basic tasks of astronomy is measuring amount of light energy arriving at Earth from a specific object. Investigations of the brightness of distant objects are necessary for understanding their structure and behavior. Furthermore, detailed studies of the time variations in brightness of certain variable stars has lead astronomers to extend our distance measurments to the farthest reaches of the universe, letting us measure the extent of the observable universe, and the age of the universe. Your instructor may have specific instructions on what you should do to perform photometric measurements in your images using SIP. This page gives a basic outline of the process.

In addition to discussing how you can carry out basic photometry tasks, this page discusses an Example Image Set for Photometry.

Precise photometric measurements of an object have as a goal the assignment of a precise "magnitude" to that object for each image taken, or at least a magnitude relative to some comparison stars in the image. Assigning a magnitude to a star is quite involved, requiring the separate imaging many "photometric standard stars" throughout the course of an evening along with images of the target star, and requiring very very clear skies. Most observatories cannot do such absolute photometry more than a few nights per year. Nevertheless, there is much that can be done by comparing the brightness of a target star with other "comparison" stars in the same digital image. This process is called "differential photometry" and is the subject of this page. By sticking with differences between stars within the same (small) image, even moderately cloudy conditions don't adversely affect the results.

The Magnitude System

If you already know about magnitudes in astronomy, you might want to skip this section.

Expressing stellar brightnesses (or the brightness of any object) in terms of "magnitudes" is a very ancient system. The Greek astronomer Hipparchos first used magnitudes, assigning 1st magnitude to the brightest stars visible to him, and 6th magnitude to the dimmest stars he could see. The system was put on a quantitative basis during the 19th century. It turns out that Hipparchos's assignment of magnitudes is a logarithmic system of ranking brightnesses, with its origin in the logarithmic response of the eye to light of different intensity.

In the modern system stars that differ by 5 magnitudes actually differ in the amount of radiation energy we receive by a multiplicative factor of 100. For example, a 6th magnitude star is 100 times fainter than 1st magnitude star. An 11th magnitude star is 10,000 times fainter than a 1st magnitude star (100 x 100). Each step up in magnitude is equivalent to a drop down in brightness by a factor of about 2.512 (a star with m = 4 is 2.512 times brighter than a star with m = 5).

Doing Differential Photometry

Let's assume you already have a set of images of a variable star, each image having been taken at different times throughout an observing night. Let's also assume these images have already been dark-corrected and flat-field corrected if you took them yourself (see How to process images with SIP). Or perhaps you downloaded already corrected images from the web (e.g., like those images in the section titled Example Image Set for Photometry on this page).

The idea is to analyze each image the same way, ultimately producing a differential magnitude for the variable star (relative to a set of comparison stars in the image) for each image. A plot of this differential magnitude should show the behavior of the variable star (as long as the comparison stars are not variables as well!).

To produce a differential magnitude for an image use the following steps:
  1. Use SIP to determine the "instrumental magnitude" of the variable star and a set of comparison stars of roughly similar brightness in the image. I recommend using at least 3 comparison stars. (You will be using this same set of comparison stars for every image.)
  2. Average the instrumental magnitudes of the comparison stars for that image.
  3. Subtract the average comparison star instrumental magnitude, determined in step 2, from the variable star's instrumental magnitude, the result is the differential magnitude for the variable star for that image.
  4. It is also worthwhile to pick one comparison star out of the set and determine it's differential magnitude relative to the average of the remaining comparison stars.
With these results in hand for each image in your set, plot the differential magnitude of the variable star versus the time the image was taken (usually found in the header of the image). As long as your measurement errors are not too large, you should see some systematic variation of the differential magnitude of the variable star. If the star is a periodically varying star with period less than one observing night you should see its differential magnitude go through at least one period of its periodic variation.

You can estimate the magnitude of your measurement errors by plotting the differential magnitude, versus time, of the one comparison star whose data you produced in step 4. That plot should show only random variations from one image to the next. The rms ("root-mean-square deviation from the mean") of these differential magnitudes for the one comparison star are an estimate of your measurement errors ("error bars" you should assign to the plot of the differential magnitude of the variable star.

The Details of Step 1: Determining Instrumental Magnitudes

The only tricky step in doing differential photometry is step 1 --- using SIP to determine the instrumental magnitude of the variable star and some number of comparison stars in the image. This section describes the details of step 1. The technique described here is called aperture photometry, since the brightness of the pixels in a particular region, or aperture, in the image are summed to obtain the brightness of the star in that aperture.

To see what's going on open up your first image in SIP. Set appropriate display parameters so you can clearly see the stars in the image (e.g., use the "Automatic Contrast Adjustment" selection under the View menu item, but note that fields containing only stars are generally best viewed with a higher Display Max than is produced by the "Automatic Constrast Adjustment" --- use "Change Image Display Parameters..." to set that value). Select the "Determine Centroid or Instrumental Magnitude..." selection under the Analyze menu item. Adjusting the location of the green (object) box so it is centered on one of your stars will produce a value for the "instrumental magnitude" m of that star. Technically, this value is

m = -2.5 log (S-B)

where the log is a "log to the base 10" of S (the sum of the pixel values in the green box) minus B (the sum of an equal number of pixels containing only the background mean value found in the red box on the image). This logarithmic expression is consistent with the definition of magnitudes, but the zero of the magnitude scale is not correctly incorporated in this case definition. Nevertheless, differences of these instrumental magnitudes will be not be dependent on the zero of the scale. Also, note that the minus sign in front of the 2.5 means the magnitude scale is reversed: brighter stars have smaller magnitudes (as Hipparchos chose). Note that you can play with the size of the green box, as well as the size and location of the red (background) box obtaining different values of the star's instrumental magnitude. The "mean" (average) of the values in the background box is used in computing the instrumental magnitude of the star in the green box as described above. Essentially, the background box's mean value is a "sea level" value for the determination of the total "mass" of the "mountain peak" of intensity in the green box. Clearly, it is important to set the red box near the green box to get background values of relevance to the star. For that reason the Background Annulus approach may be useful (the background mean is determined in a red square "ring" centered on the green box). In the end, you should ideally see values for the instrumental magnitude of the star that are not very sensitive to changes in the size or location of the green (object) or red (background) box or annulus. However, in practice the instrumental magnitude value is dependent upon the details of the green and red box sizes and separation, or the size and width of the red annulus. So, how is one to get some sort of reasonable precision in this game? The idea is to make your measurement process as consistent from one star to the next, and from one image to the next. Start with the variable star. Use the annulus background method. Center the green box on the variable star. Make sure the annulus is bigger than the green box. Now, keeping the annulus parameters fixed, try increasing the green box size, watching the instrumental magnitude value. At first as the green box increases in size the instrumental magnitude should decrease as more light is collected in the green box, but ultimately the instrumental magnitude will level off or even star to increase. The green box width where the instrumental magnitude levels off will be the green box width you stick with for all your magnitude measurements (for each star, for each image).

With a fixed green box width, and fixed annulus inner width and thickness, proceed to determine instrumental magnitudes for each comparison star in the image and your variable star. Move to the next image, and repeat. Once you are done analyzing each image in this way you can proceed to the remaining steps in the differential photometry process, summarized by steps 2 through 4.

Example Image Set for Photometry

[BL Cam] Images http://www.phys.vt.edu/~jhs/SIP/images/blcam/blcam_01.fit through blcam_21.fit are images of the field around the "ultra-fast" cepheid variable star BL Cam. Shown here is one of these images, with BL Cam labeled (the star just above the center of the image). These images were taken with an SBIG ST-7 CCD camera on the 0.4m f/4 reflector at the Martin Observatory, Virginia Tech, using a Bessell R photometric filter. In each image North is up, East is to the left (approximately). The field of view is approximately 15 arcminutes (East-West) and 10 arcminutes (North-South). The images were taken by students Michael Cooley, Eric Lang, and Chris Logie at Virginia Tech. Each exposure is 30 seconds long. The date and time of the observation (in the FITS header) are the beginning of the exposure in Universal Time (UT). These images are supplied so users can try out the astrometry techniques described on this page. These images are dark and flat field corrected (but not bias corrected --- which will not matter to the performance of the photometry).
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