There are precise mathematical methods to piece together lightcurve segments into a complete lightcurve. One popular algorithm is based on a special application of
Fourier analysis. The two readily-available software packages that I'm aware of that implement this Fourier algorithm are Peranso by Tonny Vanmunster, and MPO Canopus by Brian Warner. What this algorithm does is simplify the process by automatically searching through all possible periods to find the best time-alignment. The result of doing that on 755 Quintilla is shown in Figure 4.26. The best-fit period is 4.55 hours (pretty close to our original very-approximate estimate of 4.6 hours!)
The advantage of programs such as Peranso or MPO Canopus is that they eliminate the need to re-format the spreadsheet for each new data set, they do all of the necessary calculations for you, they provide good estimates of the accuracy of the fit, and they each provide a wealth of other special features that are beyond the ability of a spreadsheet to offer. So, if you find yourself doing more than a few asteroid lightcurve projects, you will almost certainly want to invest in one or both of these fine programs.
Fourier analysis for determining an asteroid's rotation period
If you're familiar with Fourier analysis, you will recognize the principal that can be used to determine the "best-fit" lightcurve period. It is based on an algorithm developed by Dr. Alan Harris . Assume that the lightcurve is described by an equation of the form n
where t is time, P is the period, and n is the "order" of the Fourier fit to the data.
Take that equation, and your measured differential photometry (i.e., your series of data points of magnitude vs. time), pick a starting value of P, and perform a least-squares analysis to determine the values of at and bi that give the best fit to the data. Calculate the resulting mean-square error between your data and the best-fit equation M(t). Then increment P by a small amount (to P + AP), and repeat the least-squares analysis, and the calculation of the mean-square error for this new period estimate. Do that a zillion times, until you find the period estimate that minimizes the mean-square error. That's your best-estimate period P*.
126.96.36.199 "Unfiltered" photometry
Note that in this particular project, I didn't use a spectral filter, and I didn't do anything extraordinary to link the comp stars from one night to the next. With this example as background, you can see why asteroid lightcurves differ in these regards from variable star observations. The concept of an arbitrarily-selected "delta-comp" to bring different nights (with different comp stars) into alignment along the magnitude axis is usually successful because each night contains a large portion of the total light curve, and because the data is dense enough that you can visually recognize the portions that "overlap" between two or more nights.
Hence, aside from the challenge of placing your photometric measuring aperture over an object that is in a slightly different position in each image, asteroid photometry is in some ways less complex, and less demanding on your equipment, than is variable star photometry.
The type of result shown in Figure 4.26—the rotation period and lightcurve for a single asteroid—is a valuable (and publishable) contribution to Solar System science. If it is the first lightcurve for this object, then astronomers can add it to their statistical studies of asteroid rotation rates. If lightcurves for this object have already been determined at previous apparitions, then they may be able to combine the lightcurves to calculate the three-dimensional shape of the asteroid and determine the direction of its rotation and its pole orientation.
Asteroid lightcurve results are shared with the planetary science community by reporting them in the Minor Planet Bulletin, published quarterly by the Minor Planets Section of the Association of Lunar and Planetary Observers (ALPO). This is a peer-reviewed journal, so publication of your results not only serves the planetary science community, but may also serve to add a prestigious item to your curriculum vitae. The typical article in the Minor Planet Bulletin ranges from less than one page to a few pages long, and covers one to a half-dozen asteroid lightcurves. You can download recent copies of the Minor Planet Bulletin at no charge from http://www.minorplanet observer.com/mpb.default.htm This will enable you to see the type of reports that other astronomers (many of them amateurs) are making, and hopefully will encourage you to try your hand at asteroid lightcurve photometry. The same website can direct you to the Instructions for Authors when you're ready to submit your own observations.
How important are amateur astronomer's asteroid lightcurve observations? There are more than 100,000 known asteroids. As of February 2005, lightcurves have been reported for only about 2,400 of them. Of those, only about 1,100 are considered to be "secure results'' (i.e., full lightcurve and no ambiguity in the period determination). Pole orientations have been determined for fewer than 120 asteroids. So, your data is desperately needed to better understand the population of asteroids! You should definitely try at least one or two asteroid lightcurve projects, if only for the "chops". Who knows, you may find that you enjoy showing people the graphs made from your own data as much as you enjoy showing off the fruits of your astro-imaging. In that case, you will have joined the small community of active asteroid photometrists.
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