Reducing and analyzing your observations

Having made a night-long series of images, you're ready for image reduction and data analysis. Image reduction should follow the same procedure as I described in Section 4.3.3 for variable star photometry: save your raw images (including darks and flats) on a non-volatile medium; do the routine CCD image reduction of flat-fielding and dark-subtraction (and bias subtraction, if you're using "scaled" dark frames); and do not do any kind of image enhancement algorithm. With your reduced images, you're ready for data analysis. This has three steps: finding and following the asteroid, measuring its brightness, and plotting the results.

To start, you'll need to find the asteroid in each image so that you can place your photometric measuring aperture directly on it in each image. The standard way to do this is to "blink" several images in your sequence. The "blinking" algorithm that comes with most astronomical CCD image-processing software is the digital-age equivalent of the mechanical "blink comparator" that decades of 20th century astronomers used to search their images (on glass plates!) for moving objects. The idea is that two images taken at different times are aligned, and then the screen rapidly shifts to display first one, then the second, then back to the first, etc. If the images are finely aligned, what you'll see on your screen is the stars unchanging (or changing or shifting only very slightly) as the image "blinks". The asteroid, on the other hand, will bounce back and forth as the screen shifts from image to image. A little care in watching, and a few notes on scratch paper, should be sufficient for you to identify the asteroid on any of your images as it moves across the FOV. My habit is to select one of the first images of the night, an image about mid-way through the session, and one of the final images, and blink the three in sequence. That helps me determine the asteroid's path across the stars. It's also a chance to search the field for any other moving or changing objects (e.g., an undiscovered asteroid). It hasn't happened to me yet, and the odds are very much against such a discovery (see Chapter 6), but I figure that I'll never know if I don't look.

Now that you know the location and path of the asteroid, you are in a position to place your photometry aperture onto the comp stars and the asteroid, in frame after frame. If you're doing this manually, it can be something of a chore. A continuous series of 2-minute exposures for a 6-hour observing session, you'll have 180 images to reduce! For my first asteroid lightcurve, I used a rudimentary CCD imaging program that allowed me to put the measuring aperture over one comp star to display the integrated ADUs. I typed this value into an Excel spreadsheet, then moved the measuring aperture to the asteroid, and entered its ADU into the spreadsheet. Then I loaded the next image, and repeated the process ... until all 180 images were reduced. This laborious routine had several meritorious consequences: I learned a lot about the various flaws that you will find from time to time in your images, I avoided the cost of buying special-purpose software until after I had tried my hand at photometry, and (happily) I got a very nice lightcurve from an asteroid that fortuitously happened to have a large amplitude (over 0.5 magnitude peak-to-peak). This effort showed me that photometry was within my grasp, and made my sub sequent investments in specialized photometry software a lot more palatable! (It also was the initiating force that eventually led to the purchase of a better CCD, a filter wheel, a set of photometric filters, a new telescope, and a backyard observatory; but that's another story.)

Some widely-available and modestly-priced software will make the photometric reductions much easier. TheSky has an asteroid lightcurve routine that will automatically move the measuring aperture to follow the asteroid, and generate the lightcurve. MPO Canopus also does that, as well as providing several features that make it relatively easy to merge several nights' data and interpret the lightcurve after it's been measured. If your software doesn't offer to follow a moving object with the photometry aperture, than you'll have to do it by hand, manually placing the aperture over the object in each image. I've done it both ways, and they give equivalent results, but if a few projects convince you that asteroid photometry is "your thing'', then you'll definitely want to invest in a software package such as MPO Canopus that simplifies the lightcurve reduction process.

Plotting the differential lightcurve from a single night is pretty straightforward— any spreadsheet can do it as well as the more sophisticated photometry packages. With luck, your first night will show a noticeable variation in the asteroid's brightness relative to the comp stars, and may even give a tantalizing hint of a periodic cycle. Most likely, in order to observe the complete cycle of brightness variation, you'll need to gather two or three more nights of data. It is preferable to make your observations on consecutive nights, but that isn't a hard and fast requirement. If you miss a night or two between observing runs, chances are that you'll still be able to construct a complete lightcurve and determine the asteroid's rotation period.

Merging data from several nights and combining them together to determine the rotation period and complete lightcurve shape can be done with a spreadsheet, but it is a bit of a challenge. Here, special-purpose software such as MPO Canopus shines. The nature of the problem can be best explained by an example. Figure 4.14 shows the data from two nights' observation of asteroid 755 Quintilla. This is a plot of "target minus comp'' where "comp" in this case is the average instrumental magnitude of five comp stars.

There are a few items to note on this graph. First, there's a huge gap containing no data. That's the daytime between these two adjacent nights. Second, the x-axis reports time in terms of "Julian Days'', to simplify aligning observations across long periods of time. (See Appendix A for an explanation of JD.) Third, the time is "corrected for asteroid-Earth light time'' so that the time scale represents the time that the light left the asteroid, rather than the time that the light arrived at the Earth. (See Appendix A for a discussion of this topic, also.) Fourth, the two nights did not have any comp stars in common, because the asteroid moved through a distance greater than my FOV between the two nights.

The use of different comp stars for the two nights explains the vertical offset between the two nights' curves. The standard asteroid photometrist's method of dealing with this is simple: select one night as the reference curve, and adjust the other nights up or down by a "delta-comp" that brings their curves into line. "Delta-comp'' is a purely arbitrary vertical offset. Each night has its own "delta-comp", and

Figure 4.14. Raw differential photometry from two nights' observation of an asteroid. The apparent brightness difference between the two nights is an artifact of using different comp stars on the two nights (because the asteroid had moved through a distance larger than the imager's FOV).

you use a "try and check'' approach to get the curves lined up in the vertical axis. It sounds a little sloppy, but it almost always works.

Figure 4.15 shows the two nights' curves lined up in magnitude. Each data point in the second night was adjusted by:

and in this example, delta comp « 0.06 magnitude.

Careful study of the shape of the curve gives you some clues about how to time-align the two nights. The idea is that the asteroid is rotating, sort of like a poorly thrown American football (or a potato). When we view it "point on'', it is faint. When we view it "side on'', it is brightest. So we expect that the lightcurve will go up and down, in time with the asteroid's rotation. Since a football has two "points" and two "side on'' orientations, we expect the lightcurve to be "double-humped". That is, one complete rotation of the asteroid normally gives two "peaks" and two "bottoms" in the lightcurve. Usually, these "peaks" and "bottoms" are not exactly the same magnitude. The asteroid's deviation from a perfect triaxial ellipsoid shape will make one "peak" brighter than the other, and one "bottom" fainter than the other.

In order to get the two nights time-aligned in terms of the asteroid's rotation

Figure 4.15. The raw photometry from Figure 4.14, with the second night adjusted by "delta-comp'' = 0.06 mag so that it lines up with the first night.

period, let's take a close look at that "brightest maximum'' that appears on both nights. Reading off the graph, the times of the peak are approximately:

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