Photometry and models of selected main belt asteroids. VI. 160 Una, 747 Winchester, and 849 Ara

A. Marciniak; T. Michałowski; R. Hirsch; M. Polińska; K. Kamiński; T. Kwiatkowski; A. Kryszczyńska; R. Behrend; L. Bernasconi; J. Michałowski; S. Starczewski; M. Fagas; K. Sobkowiak

doi:10.1051/0004-6361/200811078

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Volume 498 / No 1 (April IV 2009)

A&A, 498 1 (2009) 313-320

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Issue		A&A Volume 498, Number 1, April IV 2009


Page(s)		313 - 320
Section		Planets and planetary systems
DOI		https://doi.org/10.1051/0004-6361/200811078
Published online		11 March 2009

Photometry and models of selected main belt asteroids.
VI. 160 Una, 747 Winchester, and 849 Ara

A. Marciniak¹ - T. Micha $\l$ owski¹ - R. Hirsch¹ - M. Polinska¹ - K. Kaminski¹ - T. Kwiatkowski¹ - A. Kryszczynska¹ - R. Behrend² - L. Bernasconi³ - J. Micha $\l$ owski⁴ - S. Starczewski⁵ - M. Fagas¹ - K. Sobkowiak¹

1 - Astronomical Observatory, Adam Mickiewicz University, S $\l$ oneczna 36, 60-286 Poznan, Poland
2 - Geneva Observatory, 1290 Sauverny, Switzerland
3 - Les Engarouines Observatory, 84570 Mallemort-du-Comtat, France
4 - UNIQA Software Service, Gdanska 132, 90-520 $\L$ ódz, Poland
5 - N. Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka 18, 00-716 Warsaw, Poland

Received 3 October 2008 / Accepted 29 January 2009

Abstract
We present a set of new photometric observations of three main belt asteroids: 160 Una, 747 Winchester, and 849 Ara. This, combined with the available data, allowed us to construct their physical models. The lightcurve inversion method was used to obtain their spins and shapes. We have resolved problems with the rotation period of 160 Una, and found it to be $11.033176 \pm 0.000011$ h, almost twice the value given in the literature.

Key words: techniques: photometric - minor planets, asteroids

1 Introduction

Construction of an increasing number of spin and shape models of asteroids allows for statistical studies that reveal many unknown facts about the Solar System's history, including the forces involved. Precise determinations of the spin periods and axis orientations are important in light of recent direct YORP-effect detections (Kaasalainen et al. 2007; Lowry et al. 2007; Durech et al. 2008) and the apparent proofs of the consequences this effect (Slivan 2002).

Traditional dense lightcurve observations are still an abundant source of asteroids' surface properties, as well as of the overall shapes with their various asymmetries, sharp edges, and other global shape features. Various geometries obtained during a few apparitions of a given body allow a full picture of its properties to be obtained. In particular, observations that cover the wide spans of phase angles carry most of the information on its shape. They allow construction of models based on fewer apparitions than the usual near-opposition lightcurves, which means a shorter data gathering period.

This is the sixth paper in a series that gives three new asteroid models constructed with the lightcurve inversion method. We combined data available in the literature, taken from the Asteroid Photometric Catalogue (Lagerkvist et al. 2001), with our own observations gathered over a campaign of a few years conducted at the Borowiec Observatory, and with amateur observations.

The models presented in this work will be added to two databases. An up-to-date database of all published asteroids spin parameters is available at our website: http://www.astro.amu.edu.pl/Science/Asteroids/ (Kryszczynska et al. 2007), and the models from lightcurve inversion are collected at: http://astro.troja.mff.cuni.cz/projects/asteroids3D.

2 Photometry of three main belt asteroids

Three main belt asteroids, 160 Una, 747 Winchester, and 849 Ara, were observed photometrically over an eight-year span, for a total of 60 nights. Most of these observations were conducted at the Borowiec Station of the Poznan Astronomical Observatory in Poland. The rest of the lightcurve data come from SAAO in South Africa and from amateur observers from France. It is very beneficial to combine the data from various observatories, even during the same apparition.

The equipment and the description of the reduction procedure can be found in the first paper of the present series: Micha $\l$ owski et al. (2004). The subsequent papers with different asteroid models are Micha $\l$ owski et al. (2005, 2006) and Marciniak et al. (2007, 2008). Almost all of the data were collected using small telescopes, like our 40-cm Newtonian one at the Borowiec Station.

Table 1: Aspect data.

In the asteroid observations from mid-northern latitudes, there is always a problem with the lack of data from the Southern Hemisphere. Asteroids are unobservable over long parts of their orbital paths which results in poor longitude coverage. Since the time when we started observations at SAAO in South Africa three years ago, these gaps have been filled successfully (Descamps et al. 2007; Marciniak et al. 2008). Thanks to these observations, we managed to determine unique rotation period of asteroid 160 Una, which turned out to be almost two times longer than previously thought (see Sect. 2.1).

Table 2: Asteroid parameters.

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fig1.eps} \end{figure}$	Figure 1: Composite lightcurve of 160 Una in 2000.
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$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fig2.eps} \end{figure}$	Figure 2: Composite lightcurve of 160 Una in 2001/2002.
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$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fig3.eps} \end{figure}$	Figure 3: Composite lightcurve of 160 Una in 2005.
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$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fig4.eps}\vspace*{0.6mm} \end{figure}$	Figure 4: Composite lightcurve of 160 Una in 2007.
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$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fig5.eps} \end{figure}$	Figure 5: Composite lightcurve of 160 Una in 2008.
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$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fig6.eps} \end{figure}$	Figure 6: Composite lightcurve of 747 Winchester in 2002.
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$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fig7.eps} \end{figure}$	Figure 7: Composite lightcurve of 747 Winchester in 2003.
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$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fig8.eps} \end{figure}$	Figure 8: Composite lightcurve of 747 Winchester in 2006.
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$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fig9.eps} \end{figure}$	Figure 9: Composite lightcurve of 747 Winchester in 2007.
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$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fi10.eps} \end{figure}$	Figure 10: Composite lightcurve of 747 Winchester in 2008.
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Aspect data for all of the observing runs are presented in Table 1. After the date of observation, given as the mid-time of the lightcurve, there follow the distances from the object to the Sun and to the Earth in AU. In the fourth column there is the Sun-asteroid-Earth phase angle, and the other two columns show the J2000 ecliptic coordinates of a given object at any given moment. The last column contains the observatory code.

Table 2 shows the physical properties of three asteroids. After the object's name there is its IRAS diameter, albedo, and the taxonomic type. The data for this table come from The Small Bodies Node of the NASA Planetary Data System (http://pdssbn.astro.umd.edu/), where the diameters and albedos we cite come from the IRAS Minor Planet Survey (Tedesco et al. 2004), and the taxonomic classifications are given after Tholen (1989).

Our observations are presented in the form of composite lightcurves, one for each apparition (Figs. 1-14). The composites were created by means of vertical shift, to obtain the smallest possible scatter, with a fixed synodic period, that is shown in the graphs. Different symbols denote different observing runs. The vertical scale, which is the same for all the lightcurves of one object, is the relative magnitude, while the horizontal scale is the phase of rotation. Such compositing allows one to find the first approximation of the rotation period for the inversion procedure.

2.1 160 Una

Una was first observed photometrically by Harris et al. (1999). On three consecutive nights in October 1982, the authors obtained three sparse lightcurves, later composited with the $5.55 \pm 0.02$ h period, which they considered the most probable, although they did not rule out periods that were two or three times longer. The 0.10 mag amplitude lightcurve was asymmetric, with two ``maxima'' very close to each other and with shallow minimum between them. The authors also gave its H-G values as $H=9.050 \pm 0.004$ and $G = 0.05 \pm 0.04$ . Next observations of this object were made by DiMartino et al. (1994) on two consecutive nights in October 1991, almost at the same place on this asteroid's orbit, due to its 9-year commensurability with the Earth's year. Their clumpy lightcurves were composited with a similar $5.61 \pm 0.01$ h period, had an amplitude of $0.14 \pm 0.01$ mag, and looked more symmetric than the previous one. Finally, Warner (2007b) estimated Una's H-G parameters as $H=8.95 \pm 0.02$ and $G=0.05 \pm 0.05$ .

We observed Una at five more apparitions: 2000, 2001/2002, 2005, 2007, and 2008. We usually restricted our observing runs according to its period from the literature: 5.61 h. But after the 2002 apparition, it was clear that the period must be longer than 6 h. Finally, we found that only a much longer period of $11.033 \pm 0.001$ h fits all the apparitions, so that is the period for all the composite lightcurves shown in Figs. 1 to 5. In most of the composites, the individual parts of the lightcurve were too short or not overlapping, so they were arbitrarily shifted in the vertical axis. Asteroid Una showed unusual, flat lightcurves with one pair of extrema in some apparitions, and bimodal curves with wide extrema in the other. The amplitudes ranged from 0.08 to 0.25 mag.

2.2 747 Winchester

The first observations of Winchester come from the years 1970 and 1973. Vesely & Taylor (1985) gave those two lightcurves, the first obtained on one night in March 1970 and the other in August 1973. Both of them are rather noisy and partial, with an amplitude not exceeding 0.13 mag. On two close nights in July and August 1978 Harris & Young (1980) obtained two, rather sparse lightcurves of this asteroid, with an amplitude of 0.13 mag. The lightcurves did not cover the whole rotational cycle, so the period of rotation was suggested to be of over 8 h. Next, Winchester was observed by Zappala et al. (1983). On six nights in February 1980 this object produced an asymmetric, 0.13 mag amplitude lightcurve. The extrema were at slightly different heights. Unfortunately, the data came as a composite lightcurve with a period of $9.40 \pm 0.01$ h, which was later corrected, so only one lightcurve of this set fits our model closely. On five nights in March-April 2007, Warner (2007a) observed Winchester obtaining a smooth, asymmetric composite lighcurve with $0.16 \pm 0.02$ mag amplitude and $9.4146 \pm 0.0002$ h period. All these data were included in our modelling.

We added five apparitions to the existing set, from the years: 2002, 2003, 2006, 2007, and 2008. All the lightcurves from the separate apparitions (Figs. 6-10) were composited with a synodic period of $9.414\pm 0.001$ h. Winchester always showed regular bi-modal lightcurves with similar amplitudes, ranging from 0.16 to 0.21 mag.

2.3 849 Ara

The only available lightcurves of this object can be found in Harris et al. (1992). There is one dense and four sparse lightcurves, coming from May-June 1981. The period determined by the authors was $4.11643 \pm 0.00005$ h, this precision followed from its shortness and a month-long observing span. The lightcurve itself had 0.34 mag amplitude and one of the maxima was asymmetrically shaped. The authors provided also the H-G values as $8.33 \pm 0.01$ and $0.21 \pm 0.01$ , respectively.

Our observations of this asteroid spanned four apparitions: 2004, 2006, 2007, and 2008. The period $4.116 \pm 0.001$ h has been confirmed and it fits all the composite lightcurves shown in Figs. 11-14. The look of the lightcurves changed profoundly from one apparition to the next, from a slightly irregular bi-modal lightcurve to the ones where one of the minima is almost flat. The amplitudes were in the range of 0.20 to 0.57 mag.

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fi11.eps} \end{figure}$	Figure 11: Composite lightcurve of 849 Ara in 2004.
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$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fi12.eps} \end{figure}$	Figure 12: Composite lightcurve of 849 Ara in 2006.
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$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fi13.eps} \end{figure}$	Figure 13: Composite lightcurve of 849 Ara in 2007.
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$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fi14.eps} \end{figure}$	Figure 14: Composite lightcurve of 849 Ara in 2008.
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3 Pole and shape results

To construct the spin and shape models, we used the lightcurve inversion method, whose details can be found in Kaasalainen & Torppa (2001) and Kaasalainen et al. (2001, 2002a). The procedure uses all the available brightness measurements directly. In the iteration process, the shape model and its spin axis are changed so as to reproduce the observed lightcurves as closely as possible. The resulting fits are usually very good, as the model can even reproduce the small features of the lightcurves. The key requirement for the data is the possibly widest span of the observing geometries. This may not always be the case, as the example of 747 Winchester shows (see below).

Table 3: Spin models.

Table 3 presents the resulting model parameters. In the first column there is the asteroid's sidereal period of rotation, with an uncertainty restricted to the last digit. Then both pole solutions follow with their J2000 ecliptic longitude $\lambda_{\rm p}$ and latitude $\beta_{\rm p}$ . The negative value of $\beta_{\rm p}$ indicates a retrograde rotation, which was confirmed by calculating obliquities $\beta_{\rm p}$ with respect to the orbits of individual objects. The next three columns of Table 3 give the observing span in years, the number of apparitions, and the number of lightcurves that were used to construct the model. The last two columns show the method used to construct the model and the reference (``L (com)'' denotes the lightcurve inversion applied to a combined dataset, with sparse datapoints and a few dense lightcurves). For 747 Winchester and 849 Ara, the pole results have been previously reported and are listed in Table 3 before the present solution.

We give the error values next to the pole coordinates, as they turned out to be unusual. Usually, the uncertainty of the pole solution in the lightcurve inversion is around $\pm$ $5\hbox{$^\circ$ }$ on the celestial sphere. In this method, the solution errors can only be estimated by starting the procedure with various parameters and determining the scatter of the proper solutions. (In Torppa et al. 2003 and Kaasalainen & Durech 2007 one can find more details on errors in the lightcurve inversion.) All three cases are described below with their errors.

To demonstrate how the model lightcurves fit the observed ones, we present lightcurves coming from three apparitions superimposed on the model lightcurves covering the whole rotational phase (Figs. 15, 17 and 19). The angles $\phi$ , $\phi_0$ , and $\alpha$ denote the aspect angles of the Earth (as the angle between asteroid's north pole and the direction to the Earth), the Sun, and the phase angle at the epoch of observation, respectively.

$\begin{figure} \par\includegraphics[width=18cm,clip]{1078fi15.eps} \end{figure}$	Figure 15: Observed lightcurves (points) superimposed on the lightcurves created by a model (curves) at the same epochs for 160 Una.
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3.1 160 Una

The model of 160 Una was successfully constructed with the newly found period of 11.033165 h. Twenty-four lightcurves taken over 7 apparitions were used (from 1982, 1991, 2000, 2001/2002, 2005, 2007, and 2008). Three of the apparitions happened at almost the same place on its orbit, so the actual number of different apparitions was five. This asteroid shows interesting changes in lightcurves, switching from a bimodal look with ditches in maxima to flat lightcurves with a wavy pair of extrema in them. The shape model that reproduces such data has a triangular polar cross-section (Fig. 16) and rather low pole inclination (Table 3), which allows large variations in the visible parts of the surface with the changing aspect. The interesting feature of this asteroid is that its amplitudes sometimes get smaller in aspects close to equatorial, while growing bigger in aspect closer to polar ones (Fig. 15). We usually expect the opposite. For most asteroids the amplitudes of the lightcurves increase at equatorial aspects. But this object's shape is so far from ellipsoidal that we observe smaller amplitudes and the disappearance of one pair of extrema in the equatorial aspects, not close to polar ones. Una proved to be a retrograde rotator. This is the first complete model of this asteroid.

$\begin{figure} \par\includegraphics[width=18cm,clip]{1078fi16.eps} \end{figure}$	Figure 16: Shape model of 160 Una, shown at equatorial viewing and illumination geometry, with rotational phases $90\hbox {$^\circ $ }$ apart (two pictures on the left) and the pole-on view on the right.
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$\begin{figure} \par\includegraphics[width=18cm,clip]{1078fi17.eps} \end{figure}$	Figure 17: Observed versus modelled lightcurves for 747 Winchester.
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$\begin{figure} \par\includegraphics[width=18cm,clip]{1078fi18.eps} \end{figure}$	Figure 18: Shape model of 747 Winchester.
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$\begin{figure} \par\includegraphics[width=18cm,clip]{1078fi19.eps} \end{figure}$	Figure 19: Observed versus modelled lightcurves for 849 Ara.
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$\begin{figure} \par\includegraphics[width=18cm,clip]{1078fi20.eps} \end{figure}$	Figure 20: Shape model of 849 Ara.
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3.2 747 Winchester

The 4-cycle semi-commensurability with the Earth's year makes 747 Winchester a difficult case. It was not until we had 44 lightcurves coming from 9 apparitions of this object (1970, 1973, 1978, 1980, 2002, 2003, 2006, 2007, and 2008) that we managed to obtain its unique model. Although the rotation period is precisely determined, the pole solution (Table 3) and the shape model's vertical dimension (Fig. 18) are not fully constrained. The error in the longitude of the pole solutions reaches $15\hbox{$^\circ$ }$ , while it is $10\hbox{$^\circ$ }$ in latitude, probably due to the restricted viewing geometry of this asteroid. New observations are not expected to improve the model until about ten years from now, when the repeating pattern shifts considerably along the orbit.

Our pole solution differs a lot from the previous pole determinations made by Micha $\l$ owski (1993) or by De Angelis (1995). However, the latter author wrote that the sense of rotation has not been determined, but he did not give an alternative pole solution (around $\lambda_{\rm p} = 173\hbox{$^\circ$ }$ , $\beta_{\rm p} = -39\hbox{$^\circ$ }$ ), which would then be close to our solution, because Winchester seems to be a retrograde rotator.

3.3 849 Ara

For a change 894 Ara was a graceful object to model and to observe. With its lightcurves of unusual shape and evenly distributed apparitions, a unique model already appeared after 5 of them (from 1981, 2004, 2006, 2007, and 2008). Here, the scatter of the pole solutions was unprecedently small, as low as $1\hbox{$^\circ$ }$ for the first pole and $3\hbox{$^\circ$ }$ for the other (see Table 3). This probably originates in the high orbital inclination of this asteroid, resulting in large changes in the ecliptic latitudes of the object in the sky (Table 1). Still, it did not allow rejection of any one of the solutions. The rotation proved to be retrograde, as for the two previous asteroids. Our poles are close to the poles found by Durech et al. (2009), who used the lightcurve inversion with sparse photometric measurements combined with a few traditional lightcurves, and the sidereal periods are identical. Our shape model of Ara (Fig. 20) contains large planar sections and has a cone-like appearance, possible indicators of a double-lobed or even a binary shape (Kaasalainen et al. 2002b).

Acknowledgements

Borowiec observations were reduced with the CCLRS STARLINK package. This work was partially supported by grants N N203 3959 33 and N N203 302535 from the Polish Ministry of Science and Higher Education. This paper uses observations made at the South African Astronomical Observatory (SAAO). The lightcurve inversion code was created by Mikko Kaasalainen and modified by Josef Durech, and is available at http://astro.troja.mff.cuni.cz/projects/asteroids3D.

References

De Angelis, G. 1995, Planet. Space Sci. 43, 649 (In the text)
Descamps, P., Marchis, F., Michaowski, T., et al. 2007, Icarus, 187, 482 [NASA ADS] [CrossRef] (In the text)
Di Martino, M., Blanco, C., Riccioli, D., & De Sanctis, G. 1994, Icarus, 107, 269 [NASA ADS] [CrossRef] (In the text)
Durech, J., Vokrouhlicky, D., Kaasalainen, M., et al. 2008, A&A, 489, L25 [NASA ADS] [CrossRef] [EDP Sciences] (In the text)
Durech, J., Kaasalainen, M., Warner, B., et al. 2009, A&A, 493, 291 [NASA ADS] [CrossRef] [EDP Sciences] (In the text)
Harris, A. W., & Young, J. W. 1980, Icarus, 43, 20 [NASA ADS] [CrossRef] (In the text)
Harris, A. W., Young, J. W., Dockweiler, T., et al. 1992, Icarus, 95, 115 [NASA ADS] [CrossRef] (In the text)
Harris, A. W., Young, J. W., Bowell, E., & Tholen, D. J. 1999, Icarus, 142, 173 [NASA ADS] [CrossRef] (In the text)
Kaasalainen, M., & Torppa, J. 2001, Icarus, 153, 24 [NASA ADS] [CrossRef] (In the text)
Kaasalainen, M., & Durech, J. 2007, in Near Earth Objects, our Celestial Neighbors: Opportunity and Risk, ed. A. Milani, G. B. Valsecchi, & D. Vokrouhlicky (Cambridge: Cambridge University Press), 151
Kaasalainen, M., Torrpa, J., & Muinonen, K. 2001, Icarus, 153, 37 [NASA ADS] [CrossRef] (In the text)
Kaasalainen, M., Mottola, S., & Fulchignoni, M. 2002a, In Asteroids III, ed. W. F. Bottke, A. Cellino, P. Paolicchi, & R. P. Binzel (Univ. Arizona Press), 139 (In the text)
Kaasalainen, M., Torrpa, J., & Piironen, J. 2002b, A&A, 383, L19 [NASA ADS] [CrossRef] [EDP Sciences] (In the text)
Kaasalainen, M., Durech, J., Warner, B. D., et al. 2007, Nature, 446, 420 [NASA ADS] [CrossRef] (In the text)
Kryszczynska, A., La Spina, A., Paolicchi, P., et al. 2007, Icarus, 192, 223 [NASA ADS] [CrossRef] (In the text)
Lowry, S. C., Fitzsimmons, A., Pravec, P., et al. 2007, Science, 316, 272 [NASA ADS] [CrossRef] (In the text)
Lagerkvist, C.-I., Magnusson, P., & Erikson, E. 2001, Asteroid Photometric Catalogue, fifth update, Uppsala Astronomical Observatory (In the text)
Marciniak, A., Michaowski, T., Kaasalainen, M., et al. 2007, A&A, 473, 633 [NASA ADS] [CrossRef] [EDP Sciences] (In the text)
Marciniak, A., Michaowski, T., Kaasalainen, M., et al. 2008, A&A, 478, 559 [NASA ADS] [CrossRef] [EDP Sciences] (In the text)
Michaowski, T. 1993, Icarus, 106, 563 [NASA ADS] [CrossRef] (In the text)
Michaowski, T., Kwiatkowski, T., Kaasalainen, M., et al. 2004, A&A, 416, 353 [NASA ADS] [CrossRef] [EDP Sciences] (In the text)
Michaowski, T., Kaasalainen, M., Marciniak, A., et al. 2005, A&A, 443, 329 [NASA ADS] [CrossRef] [EDP Sciences] (In the text)
Michaowski, T., Kaasalainen, M., Polinska, M., et al. 2006, A&A, 459, 663 [NASA ADS] [CrossRef] [EDP Sciences] (In the text)
Slivan, S. M. 2002, Nature, 419, 49 [NASA ADS] [CrossRef] (In the text)
Tedesco, E. F., Noah, P. V., Noah, M., & Price, S. D. 2004, IRAS Minor Planet Survey. IRAS-A-FPA-3-RDR-IMPS-V6.0. NASA Planetary Data System (In the text)
Tholen, D. J. 1989, In Asteroids II, ed. R. P. Binzel, T. Gehrels, & M. S. Matthews (Univ. Arizona Press), 1139 (In the text)
Torppa, J., Kaasalainen, M., Michalowski, T., et al. 2003, Icarus, 164, 346 [NASA ADS] [CrossRef] (In the text)
Vesely, C. D., & Taylor, R. C. 1985, Icarus, 64, 37 [NASA ADS] [CrossRef] (In the text)
Warner, B. D. 2007a, Minor Planet Bull., 34, 104 [NASA ADS] (In the text)
Warner, B. D. 2007b, Minor Planet Bull., 34, 113 [NASA ADS] (In the text)
Zappala, V., Scaltriti, F., Di Martino, M. 1983, Icarus, 56, 325 [NASA ADS] [CrossRef] (In the text)

Footnotes

... Ara: Photometric data are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/498/313

All Tables

Table 1: Aspect data.

Table 2: Asteroid parameters.

Table 3: Spin models.

All Figures

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fig1.eps} \end{figure}$	Figure 1: Composite lightcurve of 160 Una in 2000.
Open with DEXTER
In the text

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fig2.eps} \end{figure}$	Figure 2: Composite lightcurve of 160 Una in 2001/2002.
Open with DEXTER
In the text

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fig3.eps} \end{figure}$	Figure 3: Composite lightcurve of 160 Una in 2005.
Open with DEXTER
In the text

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fig4.eps}\vspace*{0.6mm} \end{figure}$	Figure 4: Composite lightcurve of 160 Una in 2007.
Open with DEXTER
In the text

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fig5.eps} \end{figure}$	Figure 5: Composite lightcurve of 160 Una in 2008.
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In the text

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fig6.eps} \end{figure}$	Figure 6: Composite lightcurve of 747 Winchester in 2002.
Open with DEXTER
In the text

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fig7.eps} \end{figure}$	Figure 7: Composite lightcurve of 747 Winchester in 2003.
Open with DEXTER
In the text

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fig8.eps} \end{figure}$	Figure 8: Composite lightcurve of 747 Winchester in 2006.
Open with DEXTER
In the text

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fig9.eps} \end{figure}$	Figure 9: Composite lightcurve of 747 Winchester in 2007.
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In the text

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fi10.eps} \end{figure}$	Figure 10: Composite lightcurve of 747 Winchester in 2008.
Open with DEXTER
In the text

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fi11.eps} \end{figure}$	Figure 11: Composite lightcurve of 849 Ara in 2004.
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In the text

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fi12.eps} \end{figure}$	Figure 12: Composite lightcurve of 849 Ara in 2006.
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In the text

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fi13.eps} \end{figure}$	Figure 13: Composite lightcurve of 849 Ara in 2007.
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In the text

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1078fi14.eps} \end{figure}$	Figure 14: Composite lightcurve of 849 Ara in 2008.
Open with DEXTER
In the text

$\begin{figure} \par\includegraphics[width=18cm,clip]{1078fi15.eps} \end{figure}$	Figure 15: Observed lightcurves (points) superimposed on the lightcurves created by a model (curves) at the same epochs for 160 Una.
Open with DEXTER
In the text

$\begin{figure} \par\includegraphics[width=18cm,clip]{1078fi16.eps} \end{figure}$	Figure 16: Shape model of 160 Una, shown at equatorial viewing and illumination geometry, with rotational phases $90\hbox {$^\circ $ }$ apart (two pictures on the left) and the pole-on view on the right.
Open with DEXTER
In the text

$\begin{figure} \par\includegraphics[width=18cm,clip]{1078fi17.eps} \end{figure}$	Figure 17: Observed versus modelled lightcurves for 747 Winchester.
Open with DEXTER
In the text

$\begin{figure} \par\includegraphics[width=18cm,clip]{1078fi18.eps} \end{figure}$	Figure 18: Shape model of 747 Winchester.
Open with DEXTER
In the text

$\begin{figure} \par\includegraphics[width=18cm,clip]{1078fi19.eps} \end{figure}$	Figure 19: Observed versus modelled lightcurves for 849 Ara.
Open with DEXTER
In the text

$\begin{figure} \par\includegraphics[width=18cm,clip]{1078fi20.eps} \end{figure}$	Figure 20: Shape model of 849 Ara.
Open with DEXTER
In the text

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