Rotation Rates of Koronis Family Asteroid

(1029) La Plata

Alessondra Springmann

Advisor: Stephen M. Slivan

Wellesley College Department of Astronomy

May 18, 2007

For Astronomy 350, Spring 2007

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ABSTRACT

Members of the Koronis family of asteroids have been noted for their odd

spin behavior in that these objects have “marked alignments of their spin vec-

tors, which has interesting implications for the evolution of small asteroids and

the age of the family (Slivan et al. 2007).” To expand the sample of ten objects,

the rotation periods and spin vectors of more family members need to be mea-

sured. By combining previously collected data of Koronis family member (1029)

La Plata with new observations, we can better constrain its rotation period in

addition to determining its spin vector, increasing the number of Koronis family

objects for which spin vector solutions have been performed.

1. Introduction

Asteroids, or small solar system bodies, are leftovers from the formation of our solar

system. Once referred to as “vermin of the skies” due to their unpredictable nature and

for disrupting astronomical observations of galaxies and other objects outside of the Solar

System, these bodies are now known to be scientifically interesting instead of a nuisance.

As some asteroids are mostly unchanged since their formation, studying these bodies allows

us to better comprehend the early conditions of the Solar System and therefore the Earth,

and also predict future impacts with our planet.

Asteroid families, or groupings, were first remarked upon by Hirayama (1918), and are

now widely believed to be caused by a collision between a large parent body and a smaller

asteroid. Family members tend to retain similar orbital elements to their parent body after

the disrupting collision, such as orbital eccentricity, inclination, and semimajor axis. The

Koronis family of asteroids, located in the main belt, is a Hirayama family thought to be

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on the order of billions of years old (Bottke et al. 2001). Through laboratory and numerical

experiments, astronomers predict that the rotation rates of Koronis family members should

follow a Maxwellian distribution, in addition to having random spin orientations in space

(Bottke et al. 2006).

Koronis family members do not behave in this predicted manner: in contrast, at least

ten objects in this family demonstrate a bimodal distribution of spin vectors (Figure 1)

and appear to be trapped in spin-orbit resonances with the longitude of Saturn’s s6 orbit

node (Vokrouhlicky et al. 2003). Objects rotating in the prograde sense (i.e. rotating in the

same direction as their orbit) have periods ranging from 7.5 to 9.5 hours and obliquities

between 42◦ and 50◦, while objects rotating in a retrograde sense have rotation periods less

than 5 hours or greater than 13 hours, with obliquities greater than or equal to 154◦ (Slivan

2002; Slivan et al. 2003). These spin vectors are inconsistent with the predictions of the

laboratory model and to explain this behavior we must look at forces other than collisions.

It is assumed that this odd spin grouping is due to Yarkovsky or YORP (Yarkovsky-

O’Keefe-Radzievskii-Paddack) effects, which are thermal torques due to solar radiation

that can change the rotation rates and spin vector alignments of small bodies. Thermal

forces may cause Koronis objects to be trapped in spin-orbit resonances, where spin axes

are less likely to be perturbed by thermal radiation effects. Models by Vokrouhlicky

et al. (2003) show that the YORP effects are plausible explanations for the unexpected

bimodal alignment of these spin vectors and that YORP forces work most efficiently on

asteroids less than 40 kilometers in diameter. As the largest member of the Koronis family

is 42 kilometers across (Slivan et al. 2003), it would not be unreasonable to assume that

Yarkovsky or YORP forces are responsible for perturbing the spin axis alignment of Koronis

family objects into the present observed distribution.

Spin axis alignments of Koronis family objects are calculated by inversion of rotation

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lightcurves to determine shape, pole, sidereal period, and sense of rotation (prograde versus

retrograde) (Slivan et al. 2003). However, there only exist models for 10 Koronis family

objects, biased against low amplitude objects or ones with longer periods (Slivan et al.

2007). Thus, in order to reduce sampling bias, more Koronis family objects need to be

observed and their spin axes determined. In this paper, we present observations of Koronis

family object (1029) La Plata from 1975 to 2007 to better constrain the sidereal rotation

period and thus eventually expand the sample of objects for which exist shape models and

pole solutions.

2. Observations & Data Reduction

In this paper, we present rotation lightcurves of Koronis family member (1029) La

Plata obtained between 1975 and 2007. Observations from 1975 were recorded at the

Kvistaberg Observatory, Uppsala-Bro, Sweden by Lagerkvist (1978) using photographic

plates and measurements were made with an iris photometer. Binzel (1987) recorded

observations at McDonald Observatory, Ft. Davis, Texas using the 0.91-m Cassegrain

telescope there and took measurements with a photoelectric photometer. The majority

of observations were recorded at Whitin Observatory at Wellesley College, Wellesley,

Massachusetts. Observations were made in the V filter with four minute integrations, using

a 1024-pixel square CCD detector with an image scale of 0.9 arcseconds per pixel and a

field size of 16’ square at Cassegrain focus. The nightly observing circumstances for the six

previously unpublished individual lightcurves are given in Table 1.

Composite lightcurve plots are shown in Figures 2-3 with the horizontal (time) axis

marked in UT hours on the date of the composite curve, covering one cycle of the rotation

period. Data from different nights have been translated into the plotted time span modulo

the given rotation period.

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2.1. (1029) La Plata

La Plata had been previously observed by Lagerkvist (1978), Binzel (1987), and Slivan

et al. (2007). We observed La Plata during its 2007 apparition on 2007 Mar 12 and Mar 25.

The refined synodic period of 15.310 ± 0.003 h was used for folding together observations

from different nights. Daylight constraints prevented observing the entire rotation period

of (1029) in one night, so by using the known synodic period of (1029), it was possible to

determine over which time spans on a given night that the missing lightcurve portions could

be observed. Data images taken during 2007 were reduced using IRAF1 and calibrated to

standard stars using an extinction solution.

3. Results

From Slivan et al. (2007) the refined synodic period was determined to be 15.310±0.003

h, more precise than the period of 15.37 ± 0.10 h determined by Binzel (1987). Using

this synodic period, observations during one apparition were folded together to generate

a lightcurve over one rotation period. Observations taken between 2005 and 2007 are

presented in Figures 2-3. Reduced V magnitude is plotted versus UT hours on the first

night of observation during the 2005 apparition (Fig. 2) while relative V magnitude is

plotted versus hours for the 2007 apparition (Fig. 3).

Observations were corrected for the changing solar phase angle during apparitions by

using the Lumme-Bowell relation (Binzel et al. 1989). For (1029) La Plata, we assumed a

mean solar phase slope parameter of G = 0.179. For each night of observations, effects due

1IRAF is distributed by the National Optical Astronomy Observatories, which are oper-

ated by the Association of Universities for Research in Astronomy, Inc., under cooperative

agreement with the National Science Foundation.

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to the changing solar phase angle were removed from the reduced magnitudes.

To link apparitions, the amount of time elapsed between maxima must be known in

order to correctly determine the number of rotation cycles elapsed between apparitions.

For doubly periodic lightcurves of approximately symmetric asteroids, it suffices to observe

down to half of a period in order to determine the location of lightcurve extrema.

In order to best estimate the times of maxima we fit a Fourier series to the lightcurves,

then used only the constant term and the second-order harmonics to determine the times

corresponding to lightcurve maxima. In the cases where the lightcurve coverage was

insufficient to fit a doubly-period Fourier series at the synodic period, lightcurve data were

fit using half the synodic period with a singly periodic Fourier model (Slivan et al. 2003).

Both non-linear and linear least-squares fitting methods were used to plot Fourier series

on top of the observations of the 2004 apparition, yielding indistinguishable results, but as

the linear algorithm took less time computationally and did not require initial guesses as to

the fit parameters unlike the non-linear method, the linear one was used to fit data from the

rest of the apparitions (see Figures 4-10). As we were not fitting for the period, a non-linear

parameter, it was sufficient to use a linear fit to determine Fourier series coefficients.

To link apparition maxima, the time elapsed between apparitions was calculated then

divided by the lower and upper limits of the known synodic period to better constrain the

sidereal rotation period. From this, the minimum and maximum number of cycles elapsed

between two apparitions could be calculated. The difference in the ecliptic longitude of the

phase angle bisector, which determines the longitude of the point on the asteroid facing the

Earth, was also calculated.

When calculating the sidereal rotation period from the synodic period, we assume that

the same feature in the lightcurve occurs at the same part in the asteroid’s rotation, no

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matter where Earth is relative to the asteroid. Looking at the asteroid from Earth, it makes

most sense to measure the rotational phase relative light reflected normal to the surface of

the asteroid at a specific point (Figure 11).

Between t1 and t2, observers on Earth see a point (with a specific longitude of the phase

angle bisector) on the surface of the asteroid revolve less than 360◦ about the asteroid’s axis

if the asteroid is rotating prograde, but more than 360◦ if the asteroid is rotating in the

retrograde sense. In this figure if the asteroid is spinning prograde, it has to rotate some

distance more to complete its 360◦ rotation whereas for rotating in the retrograde sense,

it has already completed one full rotation and thus arrived “early” as seen by an observer

on Earth measuring the change in phase angle bisector. The angle k is the difference in

longitude of the phase angle bisector—if the asteroid is late, it will have to rotate through

this angle k for the same point on its surface to face Earth again whereas if the asteroid is

early, it will have already rotated through this angle k past 360◦.

By determining what fraction of an orbit the asteroid has already completed or has yet

to complete between apparitions based on its orbital position and change in the longitude of

the phase angle bisector, the number of cycles elapsed can be constrained based on whether

the asteroid is rotating in either a prograde or retrograde sense. When determining the

change in phase of a lightcurve maxima, we assume that the the asteroid is spinning with

its pole normal to the plane it makes with the Earth. The times of maxima and longitude

of the phase angle bisector are shown in Table 2.

The fraction of turns the asteroid would have to complete to face Earth again, combined

with the number of cycles elapsed between features, can be used to calculate a new sidereal

period for rotation in either the prograde or retrograde sense and eventually constrain these

values. The error in the sidereal period is equivalent to the inverse of the number of cycles

elapsed, so as the sidereal period is calculated between apparitions that are further and

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further away from each other are used, the error in the sidereal period decreases. Table 3

shows the fractional change in the phase angle bisectors between apparitions while Table 4

shows the calculated sidereal period between apparitions.

The refined sidereal period is sufficient to link the 1975 and the 1983 apparitions, which

are six apparitions apart. Even though the periods appear to be converging in both the

retrograde and prograde sense, it was impossible to determine whether (1029)’s rotation

sense was prograde or retrograde.

Given the possible range of cycles elapsed between the 1975 and 1983 apparitions

(between 4301.38 and 4301.53 cycles), the only period that is consistent with the fractional

change in the phase angle bisector (0.46 or 0.96 if retrograde; 0.04 or 0.54 if prograde)

is the sidereal period corresponding rotation in the retrograde sense, providing a value of

15.3110 ± 0.0002 hours for the sidereal period. If the asteroid was rotating in a prograde

sense, the fractional change in the longitude of the phase angle bisector would be 0.01

rotations larger than the upper bound of the number of cycles elapsed, whereas if the

asteroid is rotating in a retrograde sense, the fractional change of the phase angle bisector

falls comfortably within the range of possible cycles elapsed.

4. Conclusion

Calculating the sidereal period from the synodic rotation period of (1029) La Plata

using the time elapsed between apparitions yields a retrograde period of 15.3110± 0.0002 h,

slightly larger than the synodic period of 15.310± 0.003 h calculated by Slivan et al. (2007)

and consistent with the other calculated sidereal periods between apparitions (as shown in

Table 4).

For Koronis family objects with existing shape solutions and known polar alignments,

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objects that rotate in the retrograde sense have rotation periods less than 5 hours or greater

than 13 hours (Slivan 2002; Slivan et al. 2003; Bottke et al. 2006). This calculated sidereal

period seems consistent with other retrograde Koronis family objects, though the possibility

remains that this object is rotating in the prograde sense, as the prograde sidereal period

is 0.01 rotation cycles away from the range of elapsed cycles between apparitions for which

this sidereal period was calculated.

Regardless, the pole orientation and thus the spin vector direction of (1029) La Plata

is not yet known. The sidereal rotation period was calculated assuming that the asteroid’s

rotation vector is normal to the plane between the asteroid and the Earth; knowing the true

pole alignment would thus change the number of elapsed cycles and thus the calculated

period of (1029). There exist a number of methods in the literature (Slivan et al. 2003) that

can further refine the sidereal period in addition to calculating the spin vector orientation of

(1029). Only when the shape model and spin vector of (1029) La Plata are known can it be

determined whether this object fits into the known distribution of Koronis family objects.

More Koronis family members must be observed to expand upon the sample with

known spin vectors and shape solutions to determine whether the behavior of this small

sample is indicative of the spin alignments of the rest of the family. Understanding Koronis

family rotation dynamics not only gives us a better understanding of the formation of this

family, but a better comprehension of the implications of Yarkovsky and YORP effects

acting not only on Koronis family objects but on other bodies in the Solar System.

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Acknowledgments

I am grateful to Professor Slivan for his guidance, patience, encouragement, and

knowledge; for Tim Smith’s ability to convince Matlab to parse text; for Kate and Bekki’s

keeping the asteroids in check; for Dick French reminder about Julian dates; and for the

rest of the Astronomy Department’s support.

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REFERENCES

Binzel, R. P. 1987, Icarus, 72, 135

Binzel, R. P., Gehrels, T., & Matthews, M. S., eds. 1989, Asteroids II (The University of

Arizona Press)

Bottke, W. F., Vokrouhlicky, D., Broz, M., Nesvorny, D., & Morbidelli, A. 2001, Science,

294, 1693

Bottke, Jr., W. F., Vokrouhlicky, D., Rubincam, D. P., & Nesvorny, D. 2006, Annual

Review of Earth and Planetary Sciences, 34, 157

Hirayama, K. 1918, AJ, 31, 185

Lagerkvist, C.-I. 1978, A&AS, 31, 361

Slivan, S. M. 2002, Nature, 419, 49

Slivan, S. M., Binzel, R. P., Boroumand, S. C., Pan, M. W., Simpson, C. M., Tanabe, J. T.,

Villastrigo, R. M., Yen, L. L., Ditteon, R. P., Pray, D. P., & Stephens, R. D. 2007,

Icarus

Slivan, S. M., Binzel, R. P., Crespo da Silva, L. D., Kaasalainen, M., Lyndaker, M. M., &

Krco, M. 2003, Icarus, 162, 285

Vokrouhlicky, D., Nesvorny, D., & Bottke, W. F. 2003, Nature, 425, 147

This manuscript was prepared with the AAS LATEX macros v5.0.

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Tables

Table 1: Observing circumstances for (1029) La PlataUT date α◦ Filter Telescope Observer(s) Fig.

2005 12 04.1 2.8 V Whitin Observatory 0.61-m Springmann 212 08.1 1.3 V Whitin Observatory 0.61-m Maynard 212 09.0 1.1 V Whitin Observatory 0.61-m Slivan 212 10.1 0.9 V Whitin Observatory 0.61-m Slivan 2

2007 03 12 1.2 V Whitin Observatory 0.61-m Springmann 303 26 5.1 V Whitin Observatory 0.61-m Springmann 3

Table 2: Times of maximaYear Reference Day Time of First Maximum λPAB

(hours after 0 UT) (◦)

1975 Nov 07 0.16 033.21983 May 13 4.40 228.41987 Mar 31 4.89 141.51989 Sep 07 2.45 327.92004 Sep 11 6.37 349.82005 Dec 04 4.93 078.22007 Mar 12 5.40 172.9

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Table 3: Rotations elapsed between apparitionsYear range Apparitions elapsed Cycles elapsed Fractional difference of λPAB

2005 - 2004 1 703 0.251987 - 1989 2 1396 0.521983 - 1987 3 2222 0.761983 - 1989 5 3619 0.281975 - 1983 6 4301 0.54

Table 4: Calculation of sidereal periodsYear range Apparitions elapsed Calculated sidereal period

if retrograde (h) if prograde (h)

2005 - 2004 1 15.3103± 0.0014 15.3101± 0.00141987 - 1989 2 15.3106± 0.0007 15.3110± 0.00071983 - 1987 3 15.3109± 0.0004 15.3108± 0.00041983 - 1989 5 15.3110± 0.0002 15.3106± 0.00021975 - 1983 6 15.3110± 0.0002 15.3106± 0.0002

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Figures

Fig. 1.— Shape models and north pole directions of 10 Koronis family members with spinvector determinations from Slivan (2002) and Slivan et al. (2003).

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0 2 4 6 8 10 12 14 16

10.9

10.95

11

11.05

11.1

11.15

11.2

11.25

UT hours on 2005 Dec 04

diffe

rent

ial V

mag

nitu

de

(1029) La Plata (P = 15.310 h)

Fig. 2.— Observations from 2005 courtesy of S. M. Slivan.

0 2 4 6 8 10 12 14 16

8.9

9

9.1

9.2

9.3

9.4

9.5

UT hours on 2007 Mar 11

rela

tive

V m

agni

tude

(1029) La Plata (P = 15.310 h)

Fig. 3.— Observations from 2007.

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0 2 4 6 8 10 12 14 16

!4.2

!4.1

!4

!3.9

!3.8

!3.7

!3.6

!3.5

!3.4

UT hours on 1975 Nov 07

diffe

rent

ial V

mag

nitu

de

(1029) La Plata (P = 15.310 h, n = 1)

Fig. 4.— Fourier fit (n = 1) to 1975 data.

0 2 4 6 8 10 12 14 16

11.7

11.8

11.9

12

12.1

12.2

UT hours on 1983 May 13

diffe

rent

ial V

mag

nitu

de

(1029) La Plata (P = 15.310 h, n = 1)

Fig. 5.— Fourier fit (n = 1) to 1983 data.

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0 2 4 6 8 10 12 14 16

12.4

12.5

12.6

12.7

12.8

12.9

13

UT hours on 1987 Mar 31

diffe

rent

ial V

mag

nitu

de

(1029) La Plata (P = 15.310 h, n = 2)

Fig. 6.— Fourier fit (n = 2) to 1987 data.

0 2 4 6 8 10 12 14 16

12.2

12.25

12.3

12.35

12.4

12.45

12.5

12.55

12.6

12.65

12.7

UT hours on 1989 Sep 07

diffe

rent

ial V

mag

nitu

de

(1029) La Plata (P = 15.310 h, n = 2)

Fig. 7.— Fourier fit (n = 2) to 1989 data.

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0 2 4 6 8 10 12 14 16

10.8

10.9

11

11.1

11.2

11.3

11.4

11.5

11.6

11.7

11.8

UT hours on 2004 Sep 11

diffe

rent

ial V

mag

nitu

de

(1029) La Plata (P = 15.310 h, n = 7)

Fig. 8.— Fourier fit (n = 7) to 2004 data.

0 2 4 6 8 10 12 14 16

10.9

10.95

11

11.05

11.1

11.15

11.2

11.25

UT hours on 2005 Dec 04

diffe

rent

ial V

mag

nitu

de

(1029) La Plata (P = 15.310 h, n = 4)

Fig. 9.— Fourier fit (n = 4) to 2005 data.

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0 2 4 6 8 10 12 14 16

8.9

9

9.1

9.2

9.3

9.4

9.5

UT hours on 2007 Mar 11

rela

tive

V m

agni

tude

(1029) La Plata (P = 15.310 h, n = 2)

Fig. 10.— Fourier fit (n = 2) to 2007 data.

Fig. 11.— If the asteroid completes one sidereal rotation as it orbits the Sun, a feature onits surface will either be “early” or “late” in facing an observer on the Earth. The angle k isthe fractional difference of the longitude of the phase angle bisector. Courtesy S. M. Slivan.