Direction of spin of Koronis family member (1289) Kutaïssi

Direction of Spin of Koronis Family Member (1289) Kutaissi

by

Alison J. Klesman

SUBMITTED TO THE DEPARTMENT OF EARTH, ATMOSPHERIC, AND PLANETARY SCIENCES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

BACHELOR OF SCIENCE IN PHYSICS AT THE

MASSACHUSETTS INSTITUTE OF TECHNOLOGY JUNE 2003

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Direction of Spin of Koronis Family Member (1289) Kutaissi

by

Alison J. Klesman

Submitted to the Department of Earth, Atmospheric, and Planetary Sciences on May 9, 2003 in Partial Fulfillment of the

Requirements for the Degree of Bachelor of Science in Physics

ABSTRACT

Observations of lightcurve of Koronis family member asteroid (1289) Kutaissi were taken in January, 2003, at MIT's Wallace Astrophysical Observatory. The goal of these observations included obtaining a precise rotation period and direction of rotation for the asteroid extracted from data obtained in two different filters.

By combining the new data with data from previous epochs, a rotation rate and direction of spin were found; results give a period of 3.62424 0.00001 hrs and the asteroid was found to be spinning retrograde to its orbital motion. The results obtained from this project will be useful in the future for determining a shape and pole solution for (1289) KutaYssi.

Thesis Supervisor: Richard Binzel Title: Professor of Planetary Science Research Mentor: Stephen M. Slivan (MIT)

Acknowledgements

First and foremost, I would like to thank Rick Binzel and Steve Slivan for all of

their help and guidance on this project, as well as in other areas of my life. I would also

like to thank Richard Ditteon for providing his original 2001 digital lightcurve data, Jim

Elliot for allocating telescope time, and Chelsey Logan for providing much-appreciated

technical support at Wallace Astrophysical Observatory. Thanks also go to Miquela Vigil

and Erica McEvoy for serving as observing partners on short notice.

In addition, I would like to thank all of those people who worked so hard to keep

me sane during the semester. Stacia Swanson and Nigel Laracuente did more than I think

they know, as did Jennifer Watson and Jennie Klesman; I can't begin to thank them

enough. Lisa Silverman, Sameer Gandhi, Lisa Wu, Ashley Freeman, and Alex Calhoun

also did their best to keep my spirits up. Thanks also go to Kakashi-sensei, who went

everywhere with me for days and made me do my work.

Lastly, I appreciate the officers of the MIT Anime Club for putting up with a

delinquent treasurer, as well as the rowers of the MIT women's varsity lightweight 4,

who put up with a very sleep-deprived coxswain for the better part of a week. And thanks

also to my parents, whose support most certainly helped to keep me going.

Thanks to everyone who helped me pull this off - I hope you can be as proud of

this work as I am!

Contents

1. In tro d u ctio n ... 8

2. O b servations... 9

2.1 G oals of the O bservations...10

2.2 Selection of Program Object...11

2.3 Planning and Scheduling Procedures...11

2.3.1 Lightcurve Observations...12

2.3.2 Comparison and Standard Star Observations...13

2.4 Instrumentation and Observing Procedures...14

2.4.1 Telescopes and Instruments...14

2.4.2 Lightcurve Observations...15

2.4.3 Comparison and Standard Star Observations...16

3. D ata A nalysis Procedures... 17

3.1 Im age P rocessing ... 17

3.2 Image Photometry of Lightcurves...18

3.3 Image Photometry of R and V Calibration Data...20

3.4 Reduction for Rotation Rate...21

3.5 Reduction for Direction of Spin...23

3.5.1 The Phase Angle Bisector...23

3.5.2 Determination of the Direction of Spin...25

3.6 Calculation of Asteroid Reduced R and V Magnitudes...26

4. Observational Results and Discussion...27

4.1 D ifferential L ightcurves...27

4.2 P eriod of R otation ... 29

4.3 D irection of Spin... 30

4 .4 D iscu ssion ... 32

Appendices:

A

Observing Conditions...33

B

(1289) Kutalssi Lightcurves...34

List of Figures

3.1: Illustration of the Phase Angle Bisector...24 4.1: UT 1/22/2003 Differential Lightcurve...28

List of Tables

3.1: Lightcurve Coverage Information...18

3.2: Apertures Used for Photometry...

19

3.3: Value of the Phase Angle Bisector...25

3.4: Magnitude and Light Travel Time Corrections...27

1. Introduction

Within the main asteroid belt, there exist sub-groupings of bodies in similar orbits which are believed to share a common origin, and are thus called "families" of asteroids. One such family, the Koronis family of asteroids, is believed to be the result of the collision of a single parent body because of similarities in composition. Asteroid families are enticing for study because they can tell of the outcomes of discrete collisions within the belt. Through understanding these events within the asteroid belt, the evolution of these asteroids can be traced to their origins and give a picture of the distribution of asteroids in the early solar system.

Asteroids are not perfectly spherical bodies; when an asteroid is observed over its period of rotation (when a "lightcurve" is obtained), two minima and two maxima appear, corresponding to the scattering of light off its narrower and broader sides, respectively. These variations depend not only on the shape of the asteroid, however, but on its

viewing aspect as well. Richard Binzel [1] calculated that if the asteroids in a family such as the Koronis were to have axes of rotation that were isotropically distributed in space, this would produce an mean lightcurve amplitude of 0.20 magnitude. Subsequent observations have shown, however, that the observed mean amplitude is larger than this expected magnitude value.

One explanation for this larger-than-expected mean amplitude is that the

distribution of the Koronis family members' spin vectors is not random, but that they are correlated. Koronis family members have low-inclination orbits, and these high

amplitude findings could then be due to low-obliquity axes of rotation. Lightcurve

observations of Koronis family members therefore continue to test the hypothesis that the relation of the spin axes of family members. The relation of spin axes tells of the origin and evolution of the bodies under scrutiny. One possibility for this occurrence is that the asteroids retained the original rotation axis of the body they came from, and is young enough that the axes have not yet been randomized by other collisions. Although this is not the only explanation for such a phenomenon, the continued investigation of the axes

of rotation for Koronis family asteroids is therefore important in understanding the origin

of these bodies.

It has been shown [6] that among the larger Koronis family members that have

been observed, the distribution of spin vectors is not random, but falls within

"nonrandom spin clusters." This means that there are two distinct sets of spin vector

obliquities among those determined so far: retrograde spin vectors near

1600

(almost

perpendicular to the ecliptic plane) and prograde spin vectors near

45'.

For 7 of 10

objects the retrograde and prograde spin obliquities are also correlated with spin periods

near 14 hours and 8.5 hours, respectively. There are two hypotheses that present possible

explanation for such groupings: either smaller gravitational aggregates from an initial

collision have fragmented to produce small groups with similar spin obliquities, or some

other process (such as possible thermal effects) is aligning the vectors.

The work presented in this thesis is designed to add another data point

-

the

rotation axis of asteroid (1289) Kutaissi

-

to the set of determined rotation axes of the

Koronis family [6]. By adding this data to previous sets, it is hoped that the conditions

under which the family existed and currently exist can be further understood.

2. Observations

Observations of (1289) Kutalssi were taken on 8 nights during January and

February of 2003 at MIT's Wallace Astrophysical Observatory (WAO). In addition to the

observations of Kutaissi, local comparison stars and known Landolt standard stars were

also observed during the nights in order to calibrate the data.

The observing circumstances are listed in Appendix A. The lightcurve results are

included in Appendix B.

2.1 Goal and Scope of the Observations

The main goal of the observations was to add another data point to the set of Koronis family observations in order to test the nonrandom distribution of spin axes among family members. Several data sets for (1289) have been published [1] [2] and other unpublished data exist, but the number of existing data sets is not yet enough to determine an accurate pole solution. Although the direction of the pole is not explicitly addressed for this thesis, the data obtained for this work will complete the final data set needed for the eventual pole solution of (1289).

In designing the observing program, the amount of observational data necessary to accomplish this goal controlled the arrangement of telescope time. During the planning stages, the following factors were taken into account:

1. Observe as much lightcurve as possible (preferably over 1 full rotation to obtain all possible extrema).

2. Observe lightcurve frequently enough to establish a well-defined brightness for each extremum.

3. Observe the asteroid over a range of solar phase angles (Earth-asteroid-Sun angle).

4. Observe the asteroid at previously unrecorded solar phase angles when possible.

The resolution of the spin axis is facilitated by obtaining numerous observations of both the amplitudes and the magnitudes of the lightcurve extrema. This process is further aided by recording lightcurve at a variety of different solar phase angles (the angle from the Earth to the asteroid to the Sun). In order to obtain well-defined extrema, a high time resolution is needed. To obtain data at various solar phase angles, observations were generally taken with large intervals between observing nights. The time frame of the observations taken for this thesis also coincided with a viewing aspect that had not yet been observed, yielding more data useful for determining the spin axis of (1289). Aside

from all of these considerations, telescope availability and weather conditions also played

large roles in the determination of observing dates.

2.2 Selection of Program Object

The determination of an object that could be used for this project took into

account many factors. A list of all possible Koronis objects was compiled and Kutaissi

was determined to be the best object for this project, requiring only one more set of

observations in order to determine its rotation axis.

One limiting constraint was that the object had to be observable from the Wallace

Astrophysical Observatory, and had to be observable during the early months of 2003.

The period of the program object had to be adequately small in order to be able to obtain

an entire rotation in one night.

For other observing constraints pertaining to the Wallace Astrophysical

Observatory and the 24" telescope system, see Section 2.3.1 below.

2.3 Planning and Scheduling Procedures

The scheduling of telescope time was a fairly easy procedure during the first

month of observation (January), but became progressively harder during subsequent

months. In general, there are few all-sky photometric nights at Wallace Astrophysical

Observatory that can be used to observe standard stars. There are increasingly fewer of

such nights after January; this coupled with the busier semester schedule of classes from

February onward made January the ideal observing time.

The 24" telescope at Wallace Astrophysical Observatory was not being used for

any other projects during January and February, so the telescope was free at any time

during the night with no other scheduling problems. Thus, a list was made of all possible

nights on which (1289) could be observed, and telescope time and transportation for these dates were reserved. Observations were attempted if there was a chance of obtaining usable data, even if conditions were unstable (such as thin clouds moving through, etc.). Data were analyzed after the fact and excluded if found to be too badly contaminated by adverse weather conditions during the night.

2.3.1 Lightcurve Observations

An online software application provided by the Jet Propulsion Laboratory (JPL) at http://ssd.jpl.nasa.gov/cgi-bin/eph was used to calculate the values of the following parameters (at WAO) that determined nights on which (1289) could potentially be observed.

1. Right Ascension (RA) and Declination (DEC) of (1289) Kutaissi. 2. The Sun-Target-Observer (STO) angle.

3. The Target-Observer-Moon angle and the percentage of the Moon illuminated.

The constraints using Wallace Astrophysical Observatory are as follows:

1. The object must have a predicted mean V magnitude greater than 16.5. 2. The object must be above +200 in altitude.

3. The altitude of the Sun must be less than -12'.

4. The object must be at least 600 away from the Moon or

5. The object must be at least 25 0 away from the Moon and percentage of the

Moon illuminated must be less than 84%.

technical and experimental limits at Wallace Observatory. Nights on which any of these

criteria were not met were automatically discarded and not considered "observable"

nights. These constraints are due partly to the design of the WAO 24" telescope system

-the telescope was generally used on -the west side of -the pier, and -the program that

controls the scope's movements (the MOVE program) had not yet been modified to

handle observing from the east side of the pier. This limited the amount of time the

telescope could track (1289) across the sky, and thus limited the amount of lightcurve

that could be observed during the later months.

The accuracy of the telescope's tracking system also limited the exposure time of

images that could be taken. It was found that exposure times longer than 120 seconds

were not viable because for exposures longer than 120 seconds the telescope's ability to

track was not reliable and the stars would often appear not as point sources but as streaks

on the chip. Especially deformed star images such as streaks are not usable when

extracting photometric data; thus, exposure times were limited to 120 seconds or less.

Lightcurve images were generally taken in sets of 10 and two of these sets could be taken

before the dome had to be moved in order to prevent occultation of the images by the

dome as the telescope tracked across the sky.

Lightcurves were taken in both V and R filters.

2.3.2 Comparison and Standard Star Observations

Comparison stars were used in differential photometry to produce the differential

lightcurve of Kutaissi for a single night. These comparison stars were chosen for each

separate night using stars on the (1289) image frames that were of about 12th magnitude

so they would not be overexposed during any of the images. The brightness of the

comparison stars was pinned to the R magnitude system using the same photometric

procedure used to determine the R-magnitude of (1289) Kutaissi.

Observations of at least one standard star were also needed to carry out the

transformation of a given night to the R or V magnitude system. The standard star used

for this purpose was a Landolt Standard Star, L096-235 [4]. The published R and V magnitudes for L096-235 can be found in Appendix C. Standard star observations

required extremely photometric conditions. On nights during which the standard star was observed, images of the standard were interspersed with images of the object for the first few hours of the night until the two fields had risen to very different airmasses, at which point only the night's asteroid field was pursued.

To calibrate nights which were not photometric enough to use standard star observations, images of a previous field were interspersed with the asteroid field for that night in order to calibrate the two later. In the case of these observations, standard star images in R were obtained on UT 1/21/2003 but not on UT 1/13/2003; the January 13 field was observed in short triples on January 21 in order to use the extinction solution for January 21 to calibrate the January 13 comparison star.

2.4 Instrumentation and Observing Procedures

2.4.1 Telescopes and Instruments

The data directly acquired for this project were obtained at MIT's Wallace Astrophysical Observatory (WAO) in Westford, MA, using the 24" telescope and the AP8p CCD camera in conjunction with an Optec MAXfilter slide. The camera is a

1024x 1024 pixel array, and the filter slide employs Clear (C), "Visual" (V), and Red (R) filters. Only one filter was used per night. The 24" telescope system is an f/14.9

Cassegrain system with a plate scale of 23.3 arcsec/mm for 0.56 arcsec/pixel. The detector has a field of view of 0.22' diagonal. The camera was cooled to -40' C each night before taking images. The Apogee camera is controlled by the MaxIM DL program, which runs through Windows 98. Images were saved to disk as FIT files and transferred to MIT for subsequent image processing.

As mentioned in Section 2.3.1, there was an obvious tracking error associated

with exposure times longer than 120 seconds. This was most likely caused by the limits

of the gears responsible for telescope motion, which become less reliable in very cold

temperatures such as those experienced during these observations.

In addition, there was a drift in right ascension of the field over time, most likely

caused by the incorrect tracking rate. This was not a problem for taking images, but the

telescope would routinely have to be moved in right ascension in order to keep the

original field on the chip (for all the comparison stars to remain in the images).

A last quirk of this camera system is twofold -

images taken with the Apogee

camera exhibit two strange features: "rings" of brightness and black spots, both of which

appear on all the images. Although there are several speculations as to the causes of these

problems, no good explanation or solution has yet been found to address them. An initial

run-through of the data indicated that neither of these features would adversely affect the

photometry for this particular observing program, however, and so these two problems

were ignored during the reduction.

2.4.2 Lightcurve Observations

Finder charts for both the asteroid and the standard star were made in advance

using the JPL ephemerides and the HST Guide Star Catalog CD-ROM program

"Pickles." A transparency printed with the outline of the CCD field size on such a scale

was overlaid on the charts and the boundary of the field for the night was recorded on the

chart itself for easy reference. When choosing the exact field to image, the following

objectives were taken into account:

1. Adjust the field around the asteroid so that as many comparison stars as

2. Keep the asteroid and possible comparison stars away from known areas of "bad pixels" on the CCD chip, such as hot pixels and bad columns (which had been previously mapped for the Apogee AP8p camera).

At the beginning of the night twilight flats were taken immediately after

sunset if possible, before the telescope was centered using a known standard star (most often Alpha Orionis, Betelgeuse) and moved to the asteroid field for the night. If observing began after sunset, the telescope was immediately centered and moved to the night's field, and either "dithered" images were taken at the end of the night (by moving the telescope in small increments of RA and DEC and taking one image after each move) or a screen and lamp in the dome were used to take flat frames.

Once the telescope had been pointed to the asteroid field for the night, "stare frames" of 120 second exposure times were taken continuously for the rest of the night, unless photometric conditions allowed for standard star observations as well, which were then interspersed with asteroid observations until the two fields had reached very

different airmasses. Other calibration frames - dark and bias images - were taken at the end of the night, or during cloudy periods in the hope that the sky would clear by the time they were done.

2.4.3 Comparison and Standard Star Observations

For each night of observation, several comparison stars were chosen from each frame containing the asteroid. The comparison stars that were afterwards chosen as ideal were marked on the finder charts along with the asteroid, so that these charts could be used on later nights to find the fields easily. In order to calibrate comparison stars from different nights to each other, an extinction curve was established for each night in the small area of sky that was observed. It can be assumed that for short enough periods of time and small enough areas of sky, this extinction curve would be accurate during stable observing conditions.

In order to calibrate between nights using the same filter, three images ("triples")

were taken of a previous night's field, taking care to include the comparison stars for the

previous night in the frame. These triples were taken at intervals throughout the night; the

telescope was moved from the asteroid to a previous field for a triple, and then moved

back in order to minimize the time during which lightcurve was not observed, and thus

minimize any "holes" in the lightcurve because of this procedure. Because only

comparison stars (brighter than the asteroid) were of interest in these frames, their

exposure time was less than 120 seconds, the time needed when the asteroid had been in

the frame.

On nights during which photometric conditions were present, observations of a

standard star (Landolt Standard L096-235) were also taken in addition to observations of

the asteroid and a previous night's field. Standard star images had shorter exposure times

of 40 seconds and were also interspersed in triples between asteroid observations in the

manner outlined above. The standard star was also observed at multiple times during the

night (during photometric conditions) in order to record it at a range of different

airmasses and produce a more complete extinction curve for use during analysis.

3. Data Analysis Procedures

3.1 Image Processing

Raw images were saved in FITS format on the hard disk of the control computer

for the Apogee AP8p camera. They were later transferred to hard disk at MIT and

burned on to CD-R for backup purposes, while the working files were left on the hard

drive. These FITS images were processed using the IRAF program on Athena. When

twilight or dome flats were not available (or viable), a selection of images spanning the night were combined using the "imcombine" package and used to flatten the frames, along with bias and dark corrections. Once the images were bias-, dark-, and flat-corrected, photometry could then be performed on them to extract the lightcurve information.

3.2 Image Photometry of Lightcurves

A total of 8 nights of observations for (1289) were gathered at Wallace, and the three best nights UT 1/13/2002, UT 1/21/2003, and UT 1/22/2003 are used in the discussion here. Lightcurves were obtained in both V and R filter; only one night was recorded in V filter so it was calibrated to the R filter data for the rest of the nights.

Table 3.1

Lightcurve Information Coverage

UT Date Filter Hours of Lightcurve

1/13/2003 R 5.0

1/21/2003 R 1.9

1/22/2003 V 3.8

Photometry.was performed on the image frames to extract instrumental

magnitudes for the asteroid, comparison, and standard stars using the "phot" package in IRAF. Aperture size was determined by inspection of star images using the "radprof" package. Radprof draws a radial plot of a chosen image; apertures were set at the point where signal from a star could no longer be discerned from background noise. Due to tracking problems (mentioned above), elongated images of objects caused more background noise and resulted in the use of a larger aperture to get any reasonable

number of data points. The aperture used for each night of data reduced is listed below in Table 3.2.

Table 3.2

Apertures Used for Photometry

UT Date Aperture Used

1/13/2003 6 pixels

1/21/2003 7 pixels

1/22/2003 7 pixels

For each image of the asteroid, instrumental magnitudes were extracted for the asteroid itself, two comparison stars, and four additional check stars used to test the reliability of the comparison stars. Comparison stars were chosen to be the brightest stars that appeared on all frames that were not overexposed. These comparison stars were used to extract the relative lightcurve of the asteroid. The four check stars were chosen to be stars which spanned the range of magnitudes of the asteroid throughout the course of the night. They were used for two purposes: to check that the comparison stars were not variable, and to estimate the uncertainty in the asteroid's magnitude (with respect to the comparison stars).

Observation was stopped when the asteroid came too close to a background star (called an appulse). Frames in which the asteroid's radius cannot be discerned from a background star (by the IRAF program) are not usable and were not included in the analysis.

In order to compute the asteroid's lightcurve for a given night, the following information was also computed or extracted from the FITS headers: file name, mid-exposure time of the image (in UT hours), airmass of the field at the mid-mid-exposure time. Using these values, in addition to the instrumental magnitudes of the objects in each frame, a spreadsheet was written in Xess to compute the final differential lightcurve of the asteroid for a night of data.

3.3 Image Photometry of R and V Calibration Data

Images of the standard star (L096-235) were taken on UT 1/21/2003 in R and on UT 1/22/2003 in V. Due to clouds during attempts at observing the standard on UT

1/13/2003, images of the January 13 field (to image the comparison stars after the asteroid had passed out of the field) were also taken on January 21 for calibration. In the calibration images, instrumental magnitudes were only extracted for the objects of interest: the standard star or the comparison and check stars. An extinction curve was calculated for the night using the following procedure:

1. The UT decimal hour mid-exposure time was calculated from the start time and exposure time listed in the FITS header of each image.

2. The airmass of each image was calculated using the calc-altitude program in the 12s.23 locker of athena, using the UT date, the mid-exposure times, and the RA/DEC coordinates of the object or field.

3. All instrumental magnitudes were corrected to the value they would have for an exposure time of 1 second, to account for different exposure times for the standard and comparison star fields.

4. The airmass correction obtained in step 2 was added to the 1-second

magnitude, and the difference taken between the standard's published value and the airmass-corrected observed value to give a differential magnitude.

5. A plot was then made of the standard star's differential magnitude vs. airmass. The extinction curve is the best-fit line plotted through these points.

The coefficients of the extinction curve were then used to compute the magnitude of the standard star at the airmass of each of the comparison star fields. The difference between the two values was the difference in magnitude between the comparison star and the standard. Since the magnitude of the standard was well-known, the magnitude

difference was added to the instrumental magnitude of the comparison star to give the comparison star's standard magnitude.

In order to check the variability of the comparison stars, on-chip differential

photometry was performed using four check stars in order to look for a change in the

difference of the magnitudes of the comparison stars over time. The difference between

the comparison star and a check star was calculated and the differences checked for

variability. In cases where variability was found, multiple check stars were used in case

one of the check stars (and not the comparison star) was the variable. Through this

method of on-chip differential photometry, only instrumental magnitudes were required

and variable comparison stars could be found and discarded so as not to be used in the

final differential lightcurve calculation.

3.4 Reduction for Rotation Rate

Previous observations of KutaIssi lightcurve in 2001 and 2002 provided by

Stephen Slivan (unpublished) and Richard Ditteon [1] were combined with the three nights

of Wallace observations in order to determine a precise period of rotation. Plots of each

night's differential lightcurve including uncertainties were printed on identical scales to

line up features in the lightcurves of separate nights using a light table. Once a feature

was matched, the elapsed time between the same extremum on any two nights was

recorded and used to calculate the rotation rate of the object. Kutaissi is an ideal object in

that its lightcurve is asymmetrical

-

the pairs of maxima and minima have different

magnitudes and thus make it relatively simple to match up features between nights with

little chance for ambiguity.

A preliminary solution for the period of rotation was found by overlaying two

identical plots of the UT 1/13/2003 data, which contained nearly a full period and a half

of rotation. By matching up the same feature at different UT times on the same night, the

difference in time between them could be read from the UT time plotted on the x-axis.

The uncertainty in the measurement was determined by sliding the plots left and right

until the curves could no longer be matched within the uncertainty of the data itself. This

first pass gave a rotational period of 3.64 0.06 hours, consistent with both Binzel's solution of 3.60 0.04 [1] hours and Ditteon's solution of 3.624 0.006 hours [2].

This preliminary solution was then applied to the consecutive nights UT

1/21/2003 and UT 1/22/2003. A similar method of overlaying the differential lightcurves to match up features on the two different nights was used to measure the number of hours that elapsed between the same point on the curve on January 21 and 22. The uncertainty in this measurement was again obtained by shifting the curves until they could no longer be matched within their individual uncertainties. The elapsed time between the same point on the curve was divided by the total range of the period determined from UT

1/13/2003 (i.e. its upper limits 3.58 hr and 3.70 hr), yielding a range of the possible number "n" of rotations between the two nights. This number must be an integer or half-integer value, and to be thorough all possible values of n were considered to account for half-period errors in lining up maxima or minima. The elapsed time between the same feature's appearance on January 21 and 22 measured by this method was then divided by each value of n (if more than one value was possible) to give possible periods of rotation. Periods that did not agree with the published Binzel [1] and Ditteon [2] results within their uncertainties were disregarded.

This method was then applied to the interval between UT 1/13/2003 and UT 1/21/2003, as well as UT 1/13/2003 and UT 1/22/2003, each time calculating the period of rotation over a longer interval of time. Using only the 2003 Wallace data yielded two possibilities (both within the published ranges specified above, so neither value could be

discarded). The results can be found in Table 4.1 in Section 4.2.

In order to measure the period over a much larger number of rotations, thus driving the uncertainty down further, the January 13, 2003 lightcurve was compared to a December 2002 lightcurve provided by Stephen Slivan using the same method described above. The number of rotations between January 13, 2003, and December 8, November 7, and October 15, 2002, were counted to yield an even more accurate measurement of the rotational period of Kutaissi. All results were checked against the published values for the period of rotation (Binzel [1], Ditteon [2]), and any values that did not agree with these values were again discarded.

3.5 Reduction for Direction of Spin

Once an accurate period of rotation had been calculated, the October 2001

lightcurve provided by Richard Ditteon [2] was used to determine both a very accurate

result for the period and the direction of spin of the asteroid. The features on UT

1/13/2003 were matched with those of the October lightcurve in the same manner as

described above.

3.5.1 The Phase Angle Bisector

The apparent rotation period of an asteroid observed from Earth changes slowly

throughout its orbit due to two main geometric contributions: the changing angle from

which it is viewed from the Earth and the changing angle at which it is illuminated by the

sun. Although for most asteroids, this effect cannot be completely modeled, one can

derive a sufficient model for the time shifts of lightcurve features throughout the orbit by

looking at the change in the "phase angle bisector" of the asteroid's position.

The phase angle bisector is the vector that bisects the asteroid's heliocentric

ecliptic longitude and its geocentric ecliptic longitude on a single night. (See Figure 3.1

below.) For the 2002 and 2003 data, the change in the phase angle bisector between any

two consecutive nights was small enough that it was not an important factor when solving

for the rotation period. When dealing with the comparison between the October 2001 data

and the ak04 data, however, the difference in the phase angle bisector could no longer be

neglected. (See Table 3.3.)

12S9

21>

1

K12

/

Illustration of the Phase Angle Bisector

The longitude of the Phase Angle Bisector is the value of the longitude which bisects the solar phase angle (cc). It is approximately the mean of the heliocentric and geocentric longitudes of the asteroid.

Table 3.3

Value of the Phase Angle Bisector

UT Date Heliocentric Geocentric Phase Angle Phase Angle

Longitude Longitude Bisector Bisector

(degrees) (degrees) Longitude Difference with (degrees) UT 1/13/2003 1/13/2003 101.1077 91.8156 96.4617 0.0462 1/21/2003 100.9328 91.9715 96.4522 0.0367

1/22/2003

99.4515

93.3795

96.4155

0

12/8/2002

92.7961

100.8804

96.8383

0.4228

11/7/2002

86.7331

103.6484

95.1908

1.2247

10/15/2002

82.7042

102.0681

94.3862

2.0293

10/26/2001

11.5011

0.55186

6.0265

90.389

3.5.2 Determination of the Direction of Spin

The difference in the phase angle bisectors of approximately 900 in longitude between October 2001 and January 13, 2003 corresponded to a little over /4 of Kutaissi's orbit. This meant that between October 2001 and January 2003, Kutaissi had covered 90' of its orbit, 1/4 of its path around the sun. Thus, when looking for the possible number of rotations "n" between October 2001 and January 2003, values of n + x*0.25 (where x =

1, 3) would tell the direction of rotation.

If a number of rotations found spanned more than one value of x - for example, a range of n equal to n.20 - n.79, thus including both n.25 and n.75, while an accurate period could be found, the direction of spin would be indiscernible. A unique solution allowing only n.25 or n.75 would lead, however, to only one possible direction of spin.

3.6 Calculation of Asteroid Reduced R and V Magnitudes

To calculate the reduced R and V magnitudes of the asteroid, the reduced comparison star magnitudes outlined in Section 3.3 were required. To calculate the reduced magnitude of the asteroid, the following formula was used:

Comparison Star Standard Magnitude + (Comparison Star Instrumental Magnitude

-Asteroid Instrumental Magnitude) + Earth-Asteroid Distance Magnitude Correction Asteroid Reduced Magnitude

The Earth-Asteroid distance magnitude correction included in the equation above is calculated by the following formula:

Earth-Asteroid Distance Magnitude Correction = -5logio(Ar)

where A = Earth-Asteroid distance (in AU) and r = Sun-Asteroid distance (in AU).

To plot the reduced lightcurve, a correction to the mid-exposure time also had to be made to account for the time it took light to travel to the Earth from the surface of the asteroid. The 1-way light travel time (also calculated by the JPL ephemeris program) in UT decimal hours was subtracted from the mid-exposure time of each frame. The

reduced lightcurve, Reduced Asteroid Magnitude vs. UT Mid-Exposure Time, could then be plotted.

Table 3.4

Magnitude and Light Travel Time Corrections

Night r (AU)

A

Correction Time (min)

ak04

3.007

2.06

-3.96

17.133

ak06

3.009

2.108

-4.011

17.532

ak07

3.009

2.115

-4.109

17.592

The standard calibrations of the lightcurves were not necessary for the

determination of the period or direction of spin. The V-R color to be determined using the

calibrated data, however, will provide the essential information needed to include R

lightcurves of KutaYssi from 1999 (Slivan, unpublished) in the pole determination of

(1289).

4. Observational Results and Discussion

4.1 Differential Lightcurves

The extraction of the shape of the lightcurve from the instrumental magnitudes in

order to determine the period of rotation and direction of spin was a fairly straightforward

step. Once a comparison star was checked for variability, the differential lightcurve could

simply be plotted as the difference between the magnitude of the comparison star and the

asteroid vs. the mid-exposure time of each frame. This simple method of on-chip

very close together in the sky and thus are subject to the same airmasses, exposure times, and any other conditions that may affect the magnitude of the objects. As both the comparison star and the asteroid would be affected in the same way, differential photometry was sufficient to extract the shape of the lightcurve for analysis.

A sample lightcurve (UT 1/22/2003) is provided here in Figure 4.1; the remaining lightcurves are provided in Appendix B.

(1289) Kutaissi -10--2 0

2

The differential lightcurve of the asteroid on UT 1/22/2003. The differential instrumental V magnitude of the asteroid is plotted on the y-axis; magnitude increases upwards on the axis. The mid-exposure time of

4.2 Period of Rotation

The tabulated values calculated for the rotation period of (1289) can be found below in Table 4.1. Going as far back as October of 2001 to match features in the lightcurve, a final period of 3.62424 0.00001 hrs was determined. This final value is within the limits of previously published periods for Kutaissi, namely Richard Ditteon's solution of 3.624 0.006 hrs [2] and Richard Binzel's solution of 3.60 0.04 hrs [1].

Table 4.1

Calculation of (1289) Kutaissi Period of Rotation

Nights Elapsed Time Change in Phase Possible Values Possible Periods

Compared (hr) Angle Bisector of n (hr)

(UT Date) Longitude

(degrees) 1/13/2003- X X 1 3.64 0.06 1/13/2003 1/21/2003- 21.85 0.13 0.16 6 3.62+0.02 1/22/2003 1/13/2003- 188.2798+ 1.41 51.5 3.624 0.003 1/21/2003 0.15 52 3.62 + 0.003 1/13/2003- 213.9005 1.56 59 3.6254 0.001 1/22/2003 0.06 1/13/2003- 862.822 7.50 238 3.6253

12/8/2002

0.057

0.0002

1/13/2003-

2156.537

8.69

595

3.62443 t

4.3 Direction of Spin

Following the calculations for the period of rotation down through the October 2001 data, the range of n between the nights of October 2 6th, 2001, and January 13, 2003

was found to be between 2939.434 and 2939.764. This range of n yielded only one possible value: n = 2939.75. This unique result, coupled with the fact that the lightcurve

of Kutaissi is asymmetric, allowed for only one possible period of rotation, as well as only one possible direction of spin.

As illustrated in Figure 4.2 on the following page, after 2939.75 rotations between October 2001 and January 2003, the only way to match up the same feature on both nights is if Kutaissi has turned the extra 0.75 rotation in the retrograde direction. The asymmetry of the lightcurve comes into play most importantly here; if the lightcurve were symmetric, there would be no way to distinguish a "spot" on the asteroid to match up on both nights, and the direction of motion could thus not be determined using this method.

Ii 3/ 2003

PA= 96. 4155" 1289

4

1289

10/262001

SpOt" -- > PAB- 6.2648 spot

Earth 1289 n- 2939 Safer n '2939,75 Irotadions **Where we see * Figure 4.2

Illustration of Direction of Rotation

If there was a "spot" on Kutaissi (i.e. maxima or minima of the lightcurve) that was seen head-on in

October of 2001, then a certain integer number of revolutions plus some extra (the 0.75 rotation found above) has to have passed so that the same spot (maximum or minimum) is visible in January of 2003.

Because 2939.75 is a unique solution for the number of elapsed rotations between the two days, the direction of rotation to put the spot head-on in January of 2003 is obvious.

One thing of note in the comparison between October 2001 and January 2003 was

that the difference in the amplitudes of the lightcurves was small. This feature suggests (although it does not guarantee) that the pole of Kutaissi is very nearly vertical with respect to the ecliptic. Because of the large difference in the phase angle bisector between the locations of KutaIssi on the night of January 13, 2003 and October 26, 2001, if the pole were not nearly or exactly vertical, one would expect to see a noticeable difference in the amplitude viewed on the two nights. Because the amplitude was not substantially different between the two nights, it is very likely that Kutaissi's pole points very nearly perpendicular with respect to the angle from which it is seen from the Earth.

4.4 Discussion

The work done in this thesis not only provides an accurate period of rotation and the direction of spin of asteroid (1289) KutaIssi, but also completes the data set required to compute a pole solution. The V-R color of the asteroid provided by the calibrated lightcurves will allow previous lightcurves recorded in R (Slivan, unpublished) to be incorporated into the pole solution as well. In the context of Slivan's 2002 paper [6], Kutaissi falls within the set of Koronis family members with retrograde spin vector obliquities almost perpendicular to the ecliptic.

5. Conclusions

1. The rotational period of (1289) Kutaissi was measured through very large n (number of rotations) to be 3.62424 0.00001 hours. This result is

consistent with previously published results.

2. The direction of spin of (1289) Kutaissi is in the retrograde direction, a result found by comparing the differential lightcurves from two epochs, namely 2001 and 2003.

3. Although not directly verifiable here, the spin axis alignment of (1289) Kutaissi is very likely to be nearly perpendicular to the ecliptic, as indicated by the close values of the differential lightcurve amplitudes between 2001 and 2003,

between which dates there occurred a significant change in the asteroid's phase angle bisector.

Appendix A

Observing Conditions

UT Date RA DEC Weather Photometric? Notes

1/13/2003 06 14 17.25 21 03 32.6 Very clear No Extensive

and cold. clouds

Thin clouds preventing moving in ~ standard star

11:20 observations.

1/21/2003 06 08 15.61 21 08 05.4 Clear and Yes Standard stars

very cold. observed.

1/22/2003 06 07 35.62 21 08 39.2 Clear with a Yes Standard stars

few small observed.

clouds near horizon, very cold.

Appendix B

(1289) Kutaissi Lightcurves

(1289) KutaYssi

UT 1/ 13/2003

0.2-S

(1289) Kutalssi

I

(1289) KutaYssi

N

(1289) KutaYssi

2 4 8

UT Hours on December 8, 2002

(unpublished) December 2002 Data Provided by Stephen Slivan

37 11.0->0 -ie 0 12.Al 0

(1289) Kutaissi

4 6 10

UT Hours on November 7, 2002

(unpublished) November 2002 Data Provided by Stephen Slivan

11. C) 1) U C) 12.

~{r

6

(1289) Kutalssi

4

4 6 a 16

UT Hours on October 15, 2002

(unpublished) October 2002 Data Provided by Stephen Slivan

II1 116- 11,8-

12.0--o

Appendix C

Standard Star R and V Magnitudes

Standard Star R Magnitude V Magnitude

Bibliography & References

[1] Binzel, R.P. (1987). "A Photoelectric Survey of 130 Asteroids." Icarus 72, 135-208, 1987

[2] Ditteon, Richard. "Asteroid Photometry at Oakley Observatory." Minor Planet Bulletin, Vol. 29, No. 3 (2002)

[3] A.W. Harris, J.W. Young, F. Scaltriti, and V. Zappali. Lightcurves and phase relations of the asteroids 82 Alkmene and 444 Gyptis. Icarus, 57(2):251-258, February 1984

[4] Landolt, A.U. UBVRI photometric standard stars around the celestial equator. Astronomical Journal, 88:439-460, March 1983

[5] Slivan, Stephen M. Spin-Axis Alignment of Koronis Family Asteroids. PhD Thesis,

1995

[6] Slivan, Stephen M. Spin vector alignment of Koronis family asteroids. Nature, 419, September 2002