Rotational Period of Asteroid 755 Quintilla

Tomomi Watanabe

2004 June 4



Abstract

755 Quintilla, an asteroid in the main belt, was observed in the V filter. From the observations of the asteroid during the span of one night, a light curve was produced plotting the magnitude as a function of time. The rotational period was attempted to be estimated by fitting the light curve to a sine wave of different amplitudes and frequencies.  A contour plot of the standard deviations was produced to show what the period and amplitude were most likely for the asteroid's rotation.

1. Introduction

   There are only approximately 700 asteroids with recorded rotational periods out of the 10000 asteroids with known orbits. Many asteroids with unknown rotational properties have either never been studied, or produce too poor of light curves to allow the derivation of the rotational period.  Increasing the number of asteroids with known periods and amplitudes could greatly contribute to a more accurate understanding of asteroids including their composition, origin, and evolution.  Having a better understanding of asteroid properties for a wide range of characteristics is also beneficial in helping people prepare to deal with their threat to Earth in the future.  Because asteroids are so abundant in our solar system, a large sample is enough to give significant statistical results.  With well-known characteristics of spectral, rotational, and orbital properties, the application of statistical methods should yield new knowledge of minor planets.
   
   Light curves of asteroids can be produced by plotting their change in magnitude during a span of time.  Because most asteroids have ellipsoidal solid bodies, as opposed to perfectly spherical shapes, their reflected light is observed in varying magnitudes as a function of time.  In general it is assumed that they rotate about their shortest axis, and the axis of rotation is invariant with time.  As the asteroid spins, its surface that reflects the light changes over a course of time.  The magnitude will vary most rapidly when  the asteroid's axis of rotation becomes perpendicular to our line of light.  After observing the asteroid's variation of magnitude, the obtained  light curve can be fitted to a sinusoidal wave to determine its most likely period and amplitude of rotation.   Previous studies have shown that most asteroid periods fall within the range of 2 to 20 hours, with amplitudes densely crowding the range of 0 to 3 magnitudes.  

    Asteroid 755 Quintilla was discovered on April 06, 1908 by J.H. Metcalf.  It has been classified as an M class asteroid for its metallic composition, and is located in zone III of the main belt at 3.17 AU (Belskaya and Lagerkvist).  Its eccentricity is 0.1469 and has an inclination of 3.239 degrees.  Its magnitude at its brightest is 9.54, and completes one orbit in 2062.52 days.  The chosen asteroid of study was 755 Quintilla because of two main reasons.  For the planned time of observation, its opposition was favorable compared to many other candidates of study.  Its opposition was reached on April 14, so the asteroid promised potential in good observations for the spring of 2004.  The other appealing factor of 755 Quintilla was its lack of observations and recorded data.  No published data existed on the rotational characteristics of this object, so any set of observations could contribute greatly and prove to be useful.

    The object was planned to be observed over a course of two nights.   The object remained within 2 air masses in the sky from 9 p.m to 3 a.m. so the total observation time was guessed to be 6 hours at the most.  Some studies have shown that 50 well placed data points will define a good curve, or shooting the asteroid as an interval of 1 to 2 minutes (Warner, 2004). The interval from one data point to the next was calculated to be about 5 minutes, so the intention was to get approximately  30 to 40 data points in one night.   The  exposure time for the object was planned to be 100 seconds for a good signal-to-noise ratio and to avoid over-exposure.  Using the ephemeris for Quintilla, its location was predicted for every hour of the night to follow its movement efficiently (Bowell, 2004).

2. Observations

  755 Quintilla was observed in the V filter using the 0.25 meter Great Ohio Telescope on 2004 April 29 UT.  The sky was clear when the first flat frames were taken, but clouds were apparent in the western horizon.  The clarity remained for most of the night, but occasional high clouds were observed.  The CCD temperature was set to -15° C.  Calibration frames included the flat frames, zero frames, and dark frames.  The number of frames taken for flats was 5 during astronomical twilight with exposure times ranging from 10 to 60 seconds, and 3 before sunrise ranging from 10 to 200 seconds.  9 zero frames were taken after sunset and 4 before sunrise, all of 0.11 seconds.  3 dark frames were taken after sunset and before sunrise, all with exposure times of 100 seconds.   The estimated interval time from one exposure to the next of the object was much shorter than the actual time.  This was due to frequent refocusing and the sensitivity of the telescope to wind.  A total of 7 exposures were taken of 755 Quintilla, spanning over a period of 161.96 minutes.  Each exposure of the object was 100 seconds and were taken in V.

3. Reductions

    IRAF was used to perform data reduction with several software packages.  The zero frames were first inspected for deviant images, then the frames that seemed inconsistent with the others were deleted.  The same procedure was carried out with the darks.  The examined zeros were then combined, followed by the examined darks.  The flat frames were then dark and zero-corrected.  Next, the object frames were dark and zero-corrected.  The corrected flat frames were then examined in the similar way that zeros and darks were inspected.  The flats were combined after examination, and then the flat-field division was run on the object frames.  The object frames were inspected one by one to verify the asteroid's position.  Photometry was possible utilizing the aperture photometry.  The FWHM of the asteroid was observed to be 10.08.  Taking  the FWHM into account, the aperture size was set to 10 pixels.  Two objects in all 7 of the exposures had similar size and magnitude with the asteroid, so they were chosen as the comparison stars for the aperture photometry.  XEphem was used to find the identities of these comparison stars.  They were a) GSC 4972-0674 and b) GSC 4972-0250, both with magnitude 14.0. 

4. Results

    The following is a frame of one of the exposures taken of the asteroid along with the two comparison stars GSC 4972-0674 and GSC 4972-0250.   Throughout the course of the night, Quintilla drifted slowly to the Northwest.   

Figure 1. This is the fully corrected and calibrated image of  observed asteroid 755 Quintilla with comparison stars GSC 4972-0674 and GSC 2972-0250.  North is down, and West is right.

A total of three light curves were produced.  The first two were created using the error calculated from two separate comparison stars--GSC 4972-0674 for one, and GSC 4972-0250 for the other.  The last light curve was produced after combining the errors of both comparison stars to minimize the total error.  The last light curve was what was concentrated on for the remaining error analysis and discussion due to its shorter error bars in the y direction.  The error bars in the x direction of all plots represent the exposure time of all frames, 100 seconds.

Figure 2. These are the observations made of 755 Quintilla (magenta), GSC 4972-0674 (blue), and GSC 4972-0250 (red).  For a general behavior of each object's magnitude, the asteroid's magnitude was shifted upwards by one magnitude.  It is apparent that the magnitude of the asteroid does not follow the pattern of the two other stars as a function of time, therefore verifying that it is not a star-like object.

Figure 3. This plot exhibits the lightcurves of 755 Quintilla obtained from using two different comparison stars, GSC 4972-0674 (black) and GSC 4972-0250 (red).  

  
Figure 4. This is the light curve produced with errors in magnitudes calculated in relation to both comparison stars GSC 4972-0674 and GSC 4972-0250.

A program was made to run for fitting the light curve to the best sine function.  The used equation for the sine function was

y = q + Asin[ωx + p]
where q is the offset in the y direction, A is the amplitude, ω is 2π/(period), x is the time in minutes, and p is the phase shift.  The best parameters were given the following values by the fitting program:

 q
 6.23977 magnitudes
 A
 -0.249802 magnitudes
 ω 0.0460358 minutes-1
 p
 -0.42977 minutes

A program in IDL was used to obtain the best-fitting sine wave.  The program would calculate chi-square by using different parameters, and return the combination of the four parameters that would result in the smallest chi-square. To examine the light curve in relation to the best fitting sine wave, the sine wave was plotted over the light curve.  The sine wave ran through most of the error bars, and as observable, the wave lies closely to the individual points.

Figure 5. This is the light curve with both comparison stars fitted to the best corresponding sine wave of amplitude -0.249802 magnitudes and period 136.485 minutes.

By replacing the values for the period and phase parameters and running the program to find the best-fitting values for the remaining parameters, the lowest chi-square was obtained for each combination of omega and phase.  The following equation expresses how the chi-square was calculated.

χ2 = Σ [(mag - y)/σ] 2

where mag is the observed magnitude, y is the model magnitude calculated from the model sine wave, and σ is the error in the observed magnitude.  From the various values of χ2, a contour plot was created to show which combinations of omega and phase were most likely for the asteroid's rotation.  The most likely combinations are where  χ2  has the lowest value; it can be observed as the two plateaus in the contour plot. 

Figure 6. This is the contour plot showing χ2 as a function of omega and phase.  The star symbol represents the point at which the parameters produce the best fitting sine wave according to the fitting program.   Every fourth contour line is labeled with the corresponding value of χ2 .   The pattern repeats itself after every pi (3.14)  because the amplitude is fitted  to give the best fit, which means it will be given a negative or positive sign.

5. Discussion

    The best-fitting sine wave of the observed light curve of 755 Quintilla was found to be with the amplitude -0.249802 magnitudes and period of 136.485.  By plotting the chi-square as a function of phase and frequency, the most likely combinations appeared clearly to be within two plateaus within each pi.  Although the factors of phase shift and offset were not considered as significantly as amplitude and period, and the amplitude and period were calculated only within a limited range, the contour plot is helpful in presenting which combinations of frequency and phase are less likely, and which combinations are more likely.  The chi-square value clearly increases as the points deviate from the plateaus.  The reliability of these results cannot be determined until more observations are made of 755 Quintilla to verify the results made here with the few data points obtained.

    From the results, Quintilla is possibly a rapid rotating asteroid.  Its period places the asteroid on the lower end of the rotational period range.   The fast period may be a factor in determining the shape of the asteroid.  If more observations of the asteroid are repeated as close as possible to its oppositions, the orientation of its pole of rotation can be determined, and with color classification, the rough size and shape can also be determined. 

    For more accurate and definite results, the project could be repeated with several changes.  The exposure times could be increased slightly to produce brighter images--the concern of over-exposure was not an issue in any of the frames.  The telescope could be utilized more efficiently, with more practice before observations and better planning of observations.  The most precise coordinates for the asteroid should be obtained from the most reliable source to minimize searching.   The contour plotting could be extended to plot a wider range of amplitudes and periods.  Other contour plots could be made including the parameters of phase shift and offset.  The different contour plots plotting different combinations of parameters could be combined altogether to give a better idea of which combinations can be eliminated, and which can be kept as likely.

Acknowledgements

I would like to thank every member of the Astr 410/510 group for their kindness and generous help.  Despite everyone's busy schedule, no one ever hesitated to sacrifice their time to help me and make sure my questions were being answered.  I would also like to express my sincere appreciation for Dr. Statler for his patience and guidance in every step.  

References

Belskaya, I. N. and Lagerkvist, C.-I.  Physical Properties of M Class Asteroids.  http://earn.dlr.de/abstract/classe/ps.
Bowell, T. and Koehn, B.  2004, Asteroid Ephemeris, http://asteroid.lowell.edu/cgi-bin/koehn/asteph.
Hoot, J. E.  Photometric Determination of the Rotational Period of Asteroids.  http://68.5.152.104:800/observatory/asteroid/rotation.html.
Warner, B.  2004,  Guide to Minor Planet Photometry, http://www.minorplanetobserver.com/astic/PhotometryGuide.htm