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Photometric Determination Of
The Rotational Period of Asteroids

John E. Hoot
SSC Observatory
IAU #676

This Earth based view of Vesta was made in infrared lightwith adaptive optics by the Keck Observatory. It is the most detailed earth based image ever made of a minor planet and a taste of things yet to come.

 

 

Contents

Why Do We Bother To Measure Asteroid Rotational Periods

Aside from the satisfaction of aquiring emperical information on our universe for its own sake, asteriods have become a hot topic in the media because we have recently become aware that asteroid impacts on Earth have played a very significant roll in the development of out planet and the evolution and extinction of species. In developing a better understanding of the composition and keneticdynamics of asteroids, we will be better prepared to deal with the theat to Earth, that these bodies present.

Unlike the planets, asteroids exist in large enough numbers that statistical methods can allow us to gain insights into the evolution and behavior of asteroids. The population of asteroids for which we have good orbital data, has grown to approximately 10000 . This has doubled in the past decade and may more than double in the next 10 years with the applications of robotic telecopic searches and dedicated amateur asteroid hunters.

With this large a sample, if the spectral, rotational, and orbital characteristsc of the population were well known, the applicatoin of statistical methods should yield new knowledge of the families, origins, composition and evolution of swarm of minor planets that move within our solar system.

In addition to earth based observation, spacecraft have just started to visit the minor planets. This space based observation of asteroids, should provide baseline observations against which we can calibrate our earth based observations. The linking of space based observations with earth based observations will allow us to tune our models and therfore multiply the value of earth based observations. So far the Galileo spacecraft has visited Gaspara and Ida. The NEAR spacecraft has imaged 253 Mathilde and 433 Eros and will hopefully go into orbit around Eros in February 2000. Additionally, Deep Space I spacecraft is scheduled to rendevouz with asteroid 1992KD. And future mission plans include landing on asteroid Neriers. This agressive investigation of minor planets makes it an exciting time to be studying the minor planets.

At present, we know the rotational characteristics of only about 700 out of the 10000 asteroids with known orbits. The number of asteroids for which obervation is needed provides amateurs an opportunity to make a genuine contribution to the advancement of the science. The necessary observations are well within the capability of CCD equiped amateur telescopes.

Asteroid Characteristics

In order to infer a rotational period from asteroids based on photmetric data, we must have model of asteroids that explains the variations in reflected light from the asteroid. In general we assume that:

  1. Asteroids are ellipsoidal solid bodies.
  2. They rotate about their shortest axis
  3. The axis of rotation is invariant with time
  4. The albiedo of asteroid is roughly constant

This seems a reasonable set of assumptions. Only a few of the largest asteroids have enough self gravitation to force the asteroid to assume a shperical shape. Furthermore, if the asteroid is at all elastic, over time the conservation of angular momentum will move the rotational axis to pass though the narrowest part of the body. Baring collisions or fragmentation during the observation period, the asteroid's axis and period should remain invariant. Finally, that the albiedo should remain constant, is perhaps the weakest assumption, but at least to the first order, observations seem to support this assumption.

Taken together, I call these assumptions, the spinning spud model, since such bodies remind me of a potatoe spinning through space.

Spinning Spud Model

If you meditate a bit on the spinning spud model, illuminated by a distant point source (the Sun) and think about the total reflected light a seen by an earth based observer you conclude that:

  • The brightness will vary most then the axis of rotation is perpendicular to you line of observation.
  • The light curve will have two maxima. One when each of of the longer axis of the ellipisod is perendicular to observer, assuming the object is viewed at opposition. It gets a bit more complex when the object is view at other phase angles, but the peaks are still releated to longest axis being face on to a line that divides the angle formed by the sun, asteroid and earth..
  • The light curve will display two minima, when each of the shorter axis are perpendicular to the observer.
  • If the axis of rotation in pointed directly observer, the light curve should show no variation.

These properties of the spinning spud model allow you to infer much about an asteroids rotation from observation. If observed over several oppositions, its period of rotation, the orientation of its pole of rotation, and the ratio of its major and minor axis and, with color classification, its rough size and shape can be determined.

Asteroid Eros Flyby


Click on this image to see an animation of the flyby

The recent flyby of asteroid Eros by the NEAR spacecraft provides confirmation of this model of asteroid behavior.

The Distribution of Periods

Courtesy of A.W. Harris

The graph above shows the rough distribution of rotational periods vs amplitude of light curves for asteroids with well determined rotational rates. This scatter plot shows that the range of rotational rates is clustered around a centroid of about eight hours duration and the median variation is about 0.2 magnitudes. This puts the observation of these parameters well withing the range of amateurs. Most amateur CCD cameras are capable, of making photometic observations accurate to within 1%. Additionally, for typical asteroids, photometric observation over a couple of evenings should be sufficient to determine the rotational period.

This is a area of research where the odds are stacked in favor of the amateur. There are such an abundance of targets requiring sustained observation, professional telescope allocation committees are reluctant to allocate large telescope time for such observation. Furthermore, under the pressure of shrinking budgets, many institutions are shutting down sub meter class instruments suited to this type of observations.

Target Selection Process

It is easiest, and most useful to observe asteroids that are hear opposition. At this time, they are usually near their brightest. You should strive to observe them within with 10 degrees of there opposition. Normal ephemides available from the IAU Minor Planet Center, Lowell Obervatory or other sources contain a column indicating the phase illumination angle of the asteroid. Another advantage of picking asteroids at oppostion is that you have the longest available observing time, since the will transit about local midnight, being visible most of the night. An excellent list of minor planets near opposition can be found on a web page maintained by Turbo Power Software. Another source of brighter asteroids near opposition is Ron Baalke's Space Calendar.

To make the largest contribution, you would like to pick an asteroid for which the period is not well determined. A list of asteroids for which the period is already known can be found from the IAU Minor Planet Center's webpage. In practice, if you pick an asteroid with a number above 1000, you are very likely to contributing new observations. As a second priority is the observation of asteroids near opposition for which the pole orientation is undetermined. You observations can help in making that final determinations.

Observational Techniques

Photometric observation of minor planets is similar to variable star photometry performed with CCD cameras. Once you have located your target, you take a series of images of the object. To see a nighrs worth of obervations of 124 Alkeste collected into a time lapse animation, click on the image at left. You images should be spaced out in time so that you are getting about 100 observastions per rotation. For an asteroid with a typical rotation rate, one image every 5 minutes is about right. It is best if you take a couple of observartion and blink them to get a sense of the rate and direction of your target's motion relative to background stars. Then reposition your field of view so that the asteroid will stay within the frame for the entire night's observing session. This will allow you to use the same background stars as photometric references during data reduction and will simplify the differential photmetry.

Be sure to synchronize your computer's clock with an accurate time standard such as WWV, or GPS time signals prior to beginning your observations. Since we are critically interested in both time and intensity, using the best standards for both will insure that the most accurate results.

Be sure that your exposure time does not let the asteroid saturate your detector. For optimal signal to noise ratios, you want your peak asteroid flux to be 2/3 to 3/4 of your CCD's maximum dynamic range. Additionally, you want your star and asteroid images to cover several pixels. The FWHM should be 4 or more pixels. This assures that minor variations in pixel sensitivity (even after flat fielding) will not contribute to large sigmas (errors). If you seeing or focal lengths are such that you cannot acheive this, go ahead and slightly defocus your image to obtain a wider point spread function.

If you have access to photometric filters, use at least one of them to take your measurement. On at least on observation use both B and V filters or R and V filters to determine color indexs. This will allow you to reduce your instrumental magnitudes to standard magnitudes. This will make your observations useful to other observers and is essential if you wish your results to contribute to asteroid pole determination. If you do nor have photometric filters, your observations are still very useful in determining the rotational period of the asteroid.

You will want to gather data over the course of several evenings. This will help in several ways. It will provide a longer time baseline, so that your period can be determined much more accurately. Additionally, it will allow you to capture different segments of the rotational light curve, so that a complete curve will be able to be contructed from your observations. Finally, on any given night, try to get as long a sequence of observations as cirumstances allow. With a well aligned telescope and many of the CCD data capture programs taking multiple sequences of images, you may be able to capture long sequences of data while you sleep!

Data Reduction

To reduce your observations to data, you will need to process all of your images identically. You will need to perform dark subtraction and flat field all of your images. The quality of your flat fields are critical to obtaining good results. The best flat fields are obtained by combining many individual flat field images in a single master flat. Time spent on this effort will reward you with low systematic errors attibuted to object position in your frame. Do not do any other prettying up of your image such as contrast stretching, unsharp masking or filtering. What we are interested in is the photon count from the asteroid and reference stars during the image. Any other processing will corrupt this information.

It is easiest to do differential pohtometry each of your evening's observations. Select for your reference star a star of appoximately the same magnitude and color as your asteroid. This will minimize the extinction effects differential extinction (the tendency of the atmosphere to attenuate blue more than red at lower elevations) on your observations. You find out the color and background of your reference stars easily from the Hipparcos Catalog data available on the web. Additional reference stars can be found on the LONEOUS home page. As a good check point, you should also compute the differential magnitude of other stars in the field. Tabulate your results to show, Filename, UTC Observation Time, and Differential Magnidue.

Reduction of your data to just this level is sufficient to determine periods. If you have the ability to reduce your differential photometry to UBVI standards, by all means do so.

While this data reduction may seem a daughting task, there are several PC based packages that make this task managable. I use MIRA to do my data reduction. It allows you to batch process your whole evening's imagery. It also has an excellent aperature photometry module that computes magnitudes, extinction coefficients and error signals and collects the results in a text file that can be quickly edited for input into a spread sheet or period analysis programs.

If MIRA is beyond your means, you might consider bringing up LINUX on your PC. This free UNIX clone runs on most PC's. With it running on your PC your are able to run IRAF. This is the same software used at Kitt Peak and many other observatories to analyze astromical data. The IRAF source and object code for LINUX can be downloaded free off the internet or puchased on CD ROM for a nomial fee. The DAOPHOT module of IRAF is among the most capable photometric data reduction programs around.

While you are reducing your data, do not forget to perform the astometry. While you may not be observing an asteroid on the Minor Planet Center's critical list. Your observations of position for each night are important in keeping your asteroid off the critical list in the future.

Log Time & Differential Magnitude

When you are done, you are going to get a file similar to the fragment shown above. The next step is to import this information into your favorite spread sheet and graph the data. If all has gone well you will be looking at a fragment of an asteroid light curve. If your observations have been stretched out to far in time, you may want to correct your observation times for light travel time. You can do this by picking your first night as the reference distance, and adjusting the obsevation time according to the following formula:

Where

D is expressed in A.U.

T is expressed in Seconds

Estimate Rough Rotational Period

Using your favorite spread sheet program, graph each nights observation onto a separate transparency. Use the same time scale for the time dimension and span the same number of relative magnitudes in the Y dimension. You can now overlay theses tranparencies looking for a fit, and make your first rough estimate of the rotation period. From the example above you can see that we have captured a minima and maxima. Typcially, the asteroid's perdios will be somewhere around four times this value. In the case we would guess a period somewhere roughly arround 10 hours.

Determine Exact Period

Alan Harris of JPL has developed a program called Fourier Analysis of Light Curves that will take multiple light curve fragments and utilizing frequency domain techniques, determine the most likely period fit to your collection of light curve data. An explanation of this process in included with the program available for download here. Once the data has been formated for input into this program. You run successive trials, looking for the best period fit.

Above is an example of the output of the first run of the data. It shows that the best fit in the range of 9 to 11 hours is somewhere between 9.8 and 10 hours, confirming our intuition.

This next run narrows the period more to a value of 9.84hours. A final solution analysis output by the program will show the estimated period and the associated uncertainties.

Publishing Your Data

You data is important. Once you have made your observations, share them with professionals. If you have reduced a significant number of periods, consider publishing your work in Icarrus, PASP or Astronomy & Astrophysics. Otherwise, submit your observations to A.W. Harris or the Upsalal Asteroid Database

Milking The Numbers

Detailed scientific observation, takes dedication an care. When you go to the effort of making prolonged observations, you owe it to yourseld to milk all the data from you work. Be sure to do you astrometry and submit it to the IAU Minor Planet Center. Additionally, blink all of your images. You cannot stare at one spot in the sky for night on end and not get lucky enough to find other asteroids, or variable stars in you images. Blink all of your images, note and record new variable stars and other asteroids. Get to the blinking quickly. If you find a near earth asteroid, it will be too late to follow up if you wait more than a day or two report it.

Resources

Asteroid Ephimerides

Asteroids Near Opposition

Table of Know Asteroid Periods

Asteroid Magnitude vs Diameters

Period Calculators

Journal Articles

  1. Buie & Bus, Icarus 100:288-,
  2. Harris, A.W. “Rotations Rates of Small Asteroids:..” Lunar and Planetary Sci XXVVII (1996)
  3. Magnusson, “Pole determinations of asteroids” Icarus (ISSN 0019-1035), vol. 103, no. 1, p. 62-66.
  4. Magnusson, “Analysis of Asteroid Light Curves. I. Databases and Basic Reduction” Astron & Astrophys Sup. 86 45-51 (1990)
  5. Stellingwerf, “Period Determination Using Phase Dispersion Minimization”, Astrophy. Joun. 224:953-960 (1978)

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