Free Access
Issue
A&A
Volume 529, May 2011
Article Number A107
Number of page(s) 14
Section Planets and planetary systems
DOI https://doi.org/10.1051/0004-6361/201015365
Published online 12 April 2011

© ESO, 2011

1. Introduction

The traditional, long photometric observing runs of asteroids are no very time-efficient method, but remain the most abundant source of information we can infer from disk-integrated brightness measurements of these small bodies. Because thermal recoil forces (first described theoretically by Yarkovsky, and developed by Radzievskii 1954; Paddack 1969; and Rubincam 2000) were found to play a substantial role in asteroid dynamics and evolution (Kaasalainen et al. 2007; Lowry et al. 2007; Ďurech et al. 2008), there appeared more need for well determined asteroid pole and shape models. Therefore, to further increase the DAMIT,1 and “Poznań” database2, we present four new asteroid models obtained with the lightcurve inversion method.

This sample is different from the previous ones (included in Michałowski et al. 2004, 2005, 2006; Marciniak et al. 2007, 2008, 2009a,b) in the way that it contains only objects with poles of low inclination with respect to the ecliptic. These asteroids were long thought to be nonexistent, because modelling with the amplitude, magnitude, and epoch methods (Zappala & Di Martino 1986, and references therein) did not result in asteroid models with poles lying in the proximity of the ecliptic plane (having low |βp| values). However, Marciniak & Michałowski (2010) showed that these objects apparently do exist, only these methods were not capable of reproducing their spin axes position well. Since the lightcurve inversion method (Kaasalainen & Torppa 2001; and Kaasalainen et al. 2001) does not make any assumption on the shape of an asteroid, complex shapes are possible, and they can exhibit brightness variations even when viewed pole-on.

2. Photometry of four main-belt asteroids

Since 1997 we have been conducting photometric observations at the Borowiec station of the Poznań Astronomical Observatory in Poland. The equipment is a 0.4-m Newtonian telescope with a ST-7 CCD camera (see Michałowski et al. 2004, for the description of our observing routine and the reduction procedures). This resulted in an extensive dataset of lightcurves coming from many apparitions. More efficient asteroid modelling became possible after we included data obtained at the Southern Hemisphere and those coming from amateur astronomers. An asteroid needs to be observed at a large span of viewing geometries, including well-spread ecliptic longitudes, and a big span of phase angles, to present a unique spin and shape model.

The asteroids (94) Aurora, (174) Phaedra, (679) Pax and (714) Ulula were observed during a total of 134 runs. Unfortunately, these are mainly short pieces of their lightcurves, owing to the weather or other limitations. That is why creating composite lightcurves (shown in Figs. 126) became necessary. This step saves time in applying the lightcurve inversion routine, because it narrows the spin period range that needs to be scanned. Changes in the lightcurve amplitude and shape owing to the changing phase angle can be noticed; they bear valuable information on the asteroid’s shape and its shadowing properties, so we applied no corrections to this effect.

Table 1 contains the aspect data for all the observing runs. The mid-time of observation and the distances to the Earth and Sun in Astronomical Units are indicated in the first two columns. The table also shows the corresponding Sun-object-Earth phase angle and the J2000 ecliptic latitude and longitude of an asteroid in the sky. The next two columns provide data on the length and quality of each lightcurve, in the form of the number of points and the standard deviation in the relative brightnesses of one comparison star relative to another, usually the brightest ones in the image. The last column provides the observatory code.

Table 2 contains the physical parameters of these asteroids, including their diameters, albedos, taxonomic types and orbital parameters34.

Table 2

Asteroid parameters.

2.1. (94) Aurora

Asteroid (94) Aurora’s lightcurve was first presented by Harris & Young (1983). The data came from three nights in late August 1979 and were composited with 7.22 h period, showing regular 0.12 mag light variations. In 1984 Di Martino et al. (1987) observed over one night one full revolution of (94) Aurora, assuming the period of 7.22 h. This time the amplitude was also 0.12 mag, but the lightcurve was less regular, with one minimum visibly lower than the other. These were the only observations of this object from the southern Earth hemisphere. Schevchenko et al. (2006) and Marchis et al. (2006) refined the albedo and diameter values to 169 km and 0.0446; and to 168.88 km respectively.

Our observations of (94) Aurora spanned six apparitions: 1998, 1999, 2004, 2005, 2008, and 2009/2010. In many cases almost flat lightcurves were observed, so it was difficult to confirm its rotation period, not to mention any modelling attempts (see Figs. 1 to 6). The last apparition changed the situation, because (94) Aurora then displayed relatively large amplitudes (0.18 mag), and good quality lightcurves were obtained over a long period of time (see the composite lightcurve in Fig. 6). All composits were created using the period of 7.226 h, thus confirming previous determinations.

Table 3

Spin models parameters with their error values. See Sect. 3 for a description of the columns of this table.

2.2. (174) Phaedra

Magnusson & Lagerkvist (1991) observed (174) Phaedra on two consecutive nights in March 1987 at ESO, obtaining 5.75    ±    0.05 h period and a 0.53 mag amplitude. Sada & Cooney (2001) and Wang & Shi (2002) observed this asteroid in the year 2001. The first group obtained 5 runs, spanning one month at the beginning of 2001, and successfully composited them with a period of 5.744 ± 0.001 h. The lightcurve was highly asymmetric, with one minimum 0.2 mag deeper than the other. The other group observed (174) Phaedra later, on two nights close in time in March 2001. They obtained a similar period, 5.74 ± 0.01 h and slightly larger, overall amplitude of 0.40 mag. In 2003 Ivarsen et al. (2004) obtained data for (174) Phaedra with large amplitude of 0.52 mag, suggesting again a 5.75 ± 0.01 h period. On five nights in November 2008 more observations of this object were made by Ruthroff (2009), resulting in confirmation of the 5.75 h period.

We included lightcurve data in our modelling from the first three mentioned works since the other ones were unavailable.

We gathered lightcurves of (174) Phaedra on eight apparitions: 1998, 2000, 2001, 2005, 2006, 2007, 2008, and 2010 (see Table 1). This asteroid exhibited profound changes in its lightcurves: from simple, bimodal ones through the lightcurves of deep and shallow minima to monomodal, low-amplitude ones (Figs. 7 to 14). The amplitudes ranged from 0.18 to 0.58 mag, and the period of 5.750 h was the only one that could fit the entire set of composite lightcurves.

2.3. (679) Pax

The first lightcurve of (679) Pax was presented in Schober (1981). Two-night data came from October 1978 and were made at ESO in Chile using a photometer. The small-amplitude (0.07 mag) lightcurve was composited with a period 7.625 ± 0.005 h, but another period of 8.472 h could not be completely ruled out yet. No variations in colour during rotation were noticed. Another set of data on (679) Pax also comes from ESO and was published by Schober et al. (1994). In September 1982, it was observed during three nearby nights. This time the lightcurve was symmetric, with a much larger amplitude of 0.32 mag, and the period determination yielded a value of 8.452 ± 0.003. Next, Chiorny et al. (2003) confirmed this period. Marchis et al. (2006), using the W. M. Keck adaptive optics, estimated the general dimensions of this asteroid to be 78 × 47 km. Shevchenko et al. (2009) obtained a 0.15 mag light variation on three nights in May 1998, and composited these data with a 8.452 h period. Using data previously published, they estimated a pole position (see Table 3) with the nominal error of  ± 2° and with the ellipsoid axes values of a/b = 1.18 and b/c = 1.30.

We used photometric data extracted from the first two works and our recently acquired data for the modelling of the shape by lightcurve inversion.

We gathered data from six apparitions: 2002, 2004, 2005, 2006, 2009, and 2010. (679) Pax could only be observed in four limited places in its orbit, displaying only two states: large (around 0.30 mag); or very small (around 0.05 mag) amplitude lightcurves, without the intermediate stages. The composite lightcurves shown in Figs. 15 to 20 were created using the period of 8.4570 h, similar to the ones found by other authors.

thumbnail Fig. 27

Shape models of the Pole 1 and Pole 2 solutions for (94) Aurora (left and right image, respectively) compared to the Marchis’ et al. (2006) adaptive optics image (middle) at the same epoch and viewing geometry.

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thumbnail Fig. 28

Shape models of the Pole 1 and Pole 2 solutions for (679) Pax (left and right image, respectively) compared to the Marchis’ et al. (2006) adaptive optics image (middle) at the same epoch and viewing geometry.

Open with DEXTER

2.4. (714) Ulula

Schober and Stadler (1990) observed (714) Ulula photoelectrically during four nights in January 1983 at ESO, obtaining a 0.55 mag lightcurve with sharp minima. The data were composited with a 7.000    ±    0.005 h period. Photometric measurements for this target from the year 2005 were published by Lichelli (2006). Again, an amplitude of 0.55 mag was measured, and the synodic period was found to be 6.998 ± 0.001 h. Ďurech et al. (2009) published a model of this asteroid using the modified lightcurve inversion method, based mostly on sparse absolute brightness measurements. Their result is shown in Table 3.

We also extensively observed (714) Ulula in 2005 apparition, so we merged our data with the lightcurves from the first paper only.

Our database contains six apparitions for (174) Ulula: 2001, 2004, 2005, 2006, 2008, and 2009. Here also a commensurability with the Earth motion was present, so the apparitions of this asteroid took place in three places in its orbit only. In one of them practically flat lightcurves were observed, while in two others amplitudes of some 0.60 and 0.40 mag were noticed (see Figs. 21 to 26). The period found to fit all the composits, 6.9982 h, agrees with previous estimates.

3. Pole and shape results

All the studied objects exhibit clear signs of a low pole orientation (“low” in a sense of the ecliptic latitude, the |βp| value). In some apparitions lightcurves of substantial amplitude were observed, while in other small or almost no brightness variations were exhibited by the same object. This implies that these asteroids must have been sometimes viewed nearly pole-on. These asteroids are more problematic for modelling, because the rotation period cannot be determined well on flat lightcurves. But if other apparitions are well covered, this problem can be overcome.

We perform our asteroid modelling with the lightcurve inversion method described in Kaasalainen & Torppa (2001) and Kaasalainen et al. (2001). It allows the determination of both the pole and period solutions and the recovery of an approximated convex shape model. To mimic the surface reflectivity of the asteroids, we used a standard combination of Lommel-Seeliger and Lambert scattering laws, and to construct a convex reperesentatin of the shape model we used a spherical harmonics expansion, as described in the above-mentioned papers.

In Table 3 we present the parameters of the four models compared to previous results, if there are any. The first column contains the sidereal period with its uncertainty. The next four columns present the pole solutions (J2000 ecliptic longitude λp and latitude βp), with a “mirror” pole solution, because the uncertainty on two symmetrical pole solutions cannot be removed for asteroids that orbit close to the ecliptic plane. The pole solutions were also recalculated with respect to each asteroid’s orbital plane and placed in the next four columns (λo and βo). Usually, an asteroid ecliptic pole latitude is similar to its orbital pole latitude, because the orbit inclinations are not high for main-belt asteroids. However, with this sample of low-lying poles, the difference becomes important, thus we give both the ecliptic and orbital poles for future investigations.

For each of the ecliptic pole coordinates, a separate error analysis was performed, because errors in this method can only be estimated from the resulting parameter space (see Torppa et al. 2003; and Kaasalainen & Ďurech 2007, for more details). The error values are indicated in the second row of each asteroid solution. The next three columns of Table 3 contain the observing span in years, the number of all apparitions (Napp) and the number of lightcurves (Nlc) used for the modelling. In the two last columns, the method used (“L” for lightcurve inversion) is indicated and its reference.

To present how the model lightcurves reproduce the observed ones, we display in Figs. 29, 31, 33, and 35 three example observing runs, each from a different apparition (black dots), compared to the fits produced by the model (solid lines). The aspect angles of the Earth and the Sun (φ and φ0), and phase angle α are also given. The corresponding shape models are displayed in Figs. 30, 32, 34, and 36.

3.1. (94) Aurora

The model of (94) Aurora was constructed from 21 lightcurves from 8 apparitions (1979, 1984, 1998, 1999, 2004, 2005, 2008, and 2009/2010). The latest apparition, with a large span of the observing dates (from December until May!) and the phase angles (7° to 18°) proved to be crucial for obtaining a unique model. (94) Aurora then displayed large amplitudes in a lightcurve of a small scatter. All these factors influenced the quality of the solution shown in Table 3.

To obtain good fits to the observations, the degree and order of the spherical harmonics expansion needed to be increased, and many stray points had to be removed, because noisy lightcurves can spoil the solution or make it unretrievable. A compact shape model and the lightcurve fits are shown in Figs. 30 and 29, respectively.

In Fig. 27 we compare both our models with the adaptive optics image from Marchis et al. (2006). Unfortunately, because of the rather regular look of (94) Aurora in this viewing geometry, none of the solutions can be excluded on this basis.

thumbnail Fig. 29

Observed lightcurves (points) superimposed on the lightcurves created by a model (curves) at the same epochs for (94) Aurora. The overall rms residual of the fit was 0.0105 mag.

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thumbnail Fig. 30

Approximated convex shape model of (94) Aurora, shown at equatorial viewing and illumination geometry, with rotational phases 90° apart (two pictures on the left), and the pole-on view on the right. The reflectivity law is artificial, to reveal the shape details.

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thumbnail Fig. 31

Observed versus modelled lightcurves for (174) Phaedra. The rms residual was 0.0172 mag.

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thumbnail Fig. 32

Shape model of (174) Phaedra.

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thumbnail Fig. 33

Observed versus modelled lightcurves for (679) Pax. The rms residual was 0.0117 mag.

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thumbnail Fig. 34

Shape model of (679) Pax.

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thumbnail Fig. 35

Observed versus modelled lightcurves for (714) Ulula. The rms residual was 0.0210 mag.

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thumbnail Fig. 36

Shape model of (714) Ulula.

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3.2. (174) Phaedra

(174) Phaedra was a difficult case. Although we were able to find one unique sidereal period and a preliminary pole solution on a relatively small dataset, the model fit to the observed lightcurves and shape model appearance were unsatisfactory. The biggest problem was the shape model’s dimensions, where the z-axis was not coincident with the principal axis of the inertia tensor. The only solution to that problem was to add more photometric data. We created a 3D-shape convex model based on 36 lightcurves from 9 apparitions (1987, 1998, 2000, 2001, 2005, 2006, 2007, 2008, and 2010). Such a large dataset permitted the rejection of the mirror pole solution, since the fit of some of the lightcurves was poor and the shape models did not look “physical”. As another consequence we obtained an unusually small error of the final pole position on the celestial sphere (see Table 3).

Some lightcurves had to be averaged, to influence the solution with similar weight as the data coming from other apparitions. As a result, all model lighcurves fit the observations well (Fig. 31) and the shape model (Fig. 32) is irregular, as expected from the changes in the pattern of the lightcurve.

3.3. (679) Pax

The orbital motion of (679) Pax is commensurable with the Earth’s in terms of a mean motion, so the observing geometries were very limited. Still, it was possible to construct its model based on 42 lightcurves from 8 apparitions: 1978, 1982, 2002, 2004, 2005, 2006, 2009, and 2010. The pole coordinates are presented in Table 3.

Two more periods and some other pole orientations could provide a good numerical solution, but the visual assessment clearly proved these solutions wrong, as they gave time shifts or artificial lightcurve features compared to the observing points, especially in the last apparition in 2010 (unlike the adopted one shown in Fig. 33). The resulting elongated shape model is presented in Fig. 34.

The model obtained here confirms one of the solutions previously found by Shevchenko et al. (2009), while disagreeing with the other. However, our solutions are based on a richer dataset, and are consequently probably more reliable (see Table 3).

We also compared this model to the image of (679) Pax from AO observations made by Marchis et al. (2006), Fig. 28. Owing to the small aspect angle, the orientation of the model mostly depends on the phase of rotation, which is tied to the sidereal period. Assuming the period obtained here is correct, the comparison to the AO image suggests that the second solution (λp = 220°, βp =  + 32°) is preferred.

3.4. (714) Ulula

We constructed the model of asteroid (714) Ulula from a set of 40 partially averaged lightcurves from 7 apparitions: 1983, 2001, 2004, 2005, 2006, 2008, and 2009. A strongly preferred solution for the spin period and spin axis orientation was found in the solution space (see Table 3), but owing to the restricted viewing geometry, the shape model may have a slightly wrong inertia tensor in the c-axis direction (up to  ± 5%, see Fig. 36). The model fit to the observed lightcurves is shown in Fig. 35.

This model closely agrees with that of Ďurech et al. (2009), thus firming the reliability of a modified lightcurve inversion method, which combines a small number of dense lighcturves with many sparse data points.

4. Discussion and conclusion

Modelling of these asteroids with the classical EAM methods would most probably give higher |βp| values, because a smooth ellipsoid cannot exhibit brightness variations when viewed pole-on. It is possible that the observed ecliptic gap, with βp values from +8 to  − 8, mentioned by Pravec et al. (2002) and Kryszczyńska et al. (2007) is in fact due to the use of inaccurate EAM methods. Indeed, if the asteroid spin axes were randomly oriented, the histograms included in these works should be flat. The lack of asteroids with a low inclination with respect to the ecliptic remained an unexplained observational fact.

Marciniak & Michałowski (2010) have shown that this gap could be artificial, caused by the aforementioned EAM methods’ bias and by averaging various authors’ results in the spin axis database. These four convex 3D shape models of main-belt asteroids presented here further confirm that lowly-inclined asteroids exist in the solar system. This work complements the database of asteroids with known spin parameters that we are currently building. Today this set contains slightly more than one hundred asteroids. It is still not enough to draw conclusions on the spin axis of various sub-groups of asteroids by statistical analysis.


1

Database of Asteroid Models from Inversion Techniques, available at: http://astro.troja.mff.cuni.cz/projects/asteroids3D, described in Ďurech et al. (2010).

2

A regularly updated database of all published asteroid spin parameters obtained with various techniques available at: http://www.astro.amu.edu.pl/Science/Asteroids (Kryszczyńska et al. 2007).

3

Data for these properties come from The Small Bodies Node of the NASA Planetary Data System http://pdssbn.astro.umd.edu/), where the diameters and albedos are from the IRAS Minor Planet Survey (Tedesco et al. 2004), and the taxonomic classifications are given after Bus & Binzel (2002).

4

The data on orbital parameters come from the Minor Planet Center database available at: ftp://cfa-ftp.harvard.edu/pub/MPCORB/MPCORB.DAT

Acknowledgments

Borowiec observations were reduced with the CCLRSSTARLINK package. This work was partially supported by grants No. N N203 302535 and N N203 404139 from the Polish Ministry of Science and Higher Education. This work is partially based on observations made at the South African Astronomical Observatory (SAAO). The lightcurve inversion code was designed by Mikko Kaasalainen and modified by Josef Ďurech. It is available at http://astro.troja.mff.cuni.cz/projects/asteroids3D.

References

Online material

Table 1

Aspect data.

thumbnail Fig. 1

Composite lightcurve of (94) Aurora in 1998.

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thumbnail Fig. 2

Composite lightcurve of (94) Aurora in 1999.

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thumbnail Fig. 3

Composite lightcurve of (94) Aurora in 2004.

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thumbnail Fig. 4

Composite lightcurve of (94) Aurora in 2005.

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thumbnail Fig. 5

Composite lightcurve of (94) Aurora in 2008.

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thumbnail Fig. 6

Composite lightcurve of (94) Aurora in 2009 − 2010.

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thumbnail Fig. 7

Composite lightcurve of (174) Phaedra in 1998.

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thumbnail Fig. 8

Composite lightcurve of (174) Phaedra in 2000.

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thumbnail Fig. 9

Composite lightcurve of (174) Phaedra in 2001.

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thumbnail Fig. 10

Composite lightcurve of (174) Phaedra in 2005.

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thumbnail Fig. 11

Composite lightcurve of (174) Phaedra in 2006.

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thumbnail Fig. 12

Composite lightcurve of (174) Phaedra in 2007.

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thumbnail Fig. 13

Composite lightcurve of (174) Phaedra in 2008.

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thumbnail Fig. 14

Composite lightcurve of (174) Phaedra in 2010.

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thumbnail Fig. 15

Composite lightcurve of (679) Pax in 2002.

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thumbnail Fig. 16

Lightcurve of (679) Pax in 2004.

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thumbnail Fig. 17

Composite lightcurve of (679) Pax in 2005.

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thumbnail Fig. 18

Composite lightcurve of (679) Pax in 2006.

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thumbnail Fig. 19

Composite lightcurve of (679) Pax in 2009.

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thumbnail Fig. 20

Composite lightcurve of (679) Pax in 2010.

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thumbnail Fig. 21

Composite lightcurve of (714) Ulula in 2001.

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thumbnail Fig. 22

Lightcurve of (714) Ulula in 2004.

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thumbnail Fig. 23

Composite lightcurve of (714) Ulula in 2005.

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thumbnail Fig. 24

Composite lightcurve of (714) Ulula in 2006.

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thumbnail Fig. 25

Lightcurve of (714) Ulula in 2008.

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thumbnail Fig. 26

Composite lightcurve of (714) Ulula in 2009.

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All Tables

Table 2

Asteroid parameters.

Table 3

Spin models parameters with their error values. See Sect. 3 for a description of the columns of this table.

Table 1

Aspect data.

All Figures

thumbnail Fig. 27

Shape models of the Pole 1 and Pole 2 solutions for (94) Aurora (left and right image, respectively) compared to the Marchis’ et al. (2006) adaptive optics image (middle) at the same epoch and viewing geometry.

Open with DEXTER
In the text
thumbnail Fig. 28

Shape models of the Pole 1 and Pole 2 solutions for (679) Pax (left and right image, respectively) compared to the Marchis’ et al. (2006) adaptive optics image (middle) at the same epoch and viewing geometry.

Open with DEXTER
In the text
thumbnail Fig. 29

Observed lightcurves (points) superimposed on the lightcurves created by a model (curves) at the same epochs for (94) Aurora. The overall rms residual of the fit was 0.0105 mag.

Open with DEXTER
In the text
thumbnail Fig. 30

Approximated convex shape model of (94) Aurora, shown at equatorial viewing and illumination geometry, with rotational phases 90° apart (two pictures on the left), and the pole-on view on the right. The reflectivity law is artificial, to reveal the shape details.

Open with DEXTER
In the text
thumbnail Fig. 31

Observed versus modelled lightcurves for (174) Phaedra. The rms residual was 0.0172 mag.

Open with DEXTER
In the text
thumbnail Fig. 32

Shape model of (174) Phaedra.

Open with DEXTER
In the text
thumbnail Fig. 33

Observed versus modelled lightcurves for (679) Pax. The rms residual was 0.0117 mag.

Open with DEXTER
In the text
thumbnail Fig. 34

Shape model of (679) Pax.

Open with DEXTER
In the text
thumbnail Fig. 35

Observed versus modelled lightcurves for (714) Ulula. The rms residual was 0.0210 mag.

Open with DEXTER
In the text
thumbnail Fig. 36

Shape model of (714) Ulula.

Open with DEXTER
In the text
thumbnail Fig. 1

Composite lightcurve of (94) Aurora in 1998.

Open with DEXTER
In the text
thumbnail Fig. 2

Composite lightcurve of (94) Aurora in 1999.

Open with DEXTER
In the text
thumbnail Fig. 3

Composite lightcurve of (94) Aurora in 2004.

Open with DEXTER
In the text
thumbnail Fig. 4

Composite lightcurve of (94) Aurora in 2005.

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In the text
thumbnail Fig. 5

Composite lightcurve of (94) Aurora in 2008.

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In the text
thumbnail Fig. 6

Composite lightcurve of (94) Aurora in 2009 − 2010.

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In the text
thumbnail Fig. 7

Composite lightcurve of (174) Phaedra in 1998.

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In the text
thumbnail Fig. 8

Composite lightcurve of (174) Phaedra in 2000.

Open with DEXTER
In the text
thumbnail Fig. 9

Composite lightcurve of (174) Phaedra in 2001.

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In the text
thumbnail Fig. 10

Composite lightcurve of (174) Phaedra in 2005.

Open with DEXTER
In the text
thumbnail Fig. 11

Composite lightcurve of (174) Phaedra in 2006.

Open with DEXTER
In the text
thumbnail Fig. 12

Composite lightcurve of (174) Phaedra in 2007.

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In the text
thumbnail Fig. 13

Composite lightcurve of (174) Phaedra in 2008.

Open with DEXTER
In the text
thumbnail Fig. 14

Composite lightcurve of (174) Phaedra in 2010.

Open with DEXTER
In the text
thumbnail Fig. 15

Composite lightcurve of (679) Pax in 2002.

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In the text
thumbnail Fig. 16

Lightcurve of (679) Pax in 2004.

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In the text
thumbnail Fig. 17

Composite lightcurve of (679) Pax in 2005.

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In the text
thumbnail Fig. 18

Composite lightcurve of (679) Pax in 2006.

Open with DEXTER
In the text
thumbnail Fig. 19

Composite lightcurve of (679) Pax in 2009.

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In the text
thumbnail Fig. 20

Composite lightcurve of (679) Pax in 2010.

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In the text
thumbnail Fig. 21

Composite lightcurve of (714) Ulula in 2001.

Open with DEXTER
In the text
thumbnail Fig. 22

Lightcurve of (714) Ulula in 2004.

Open with DEXTER
In the text
thumbnail Fig. 23

Composite lightcurve of (714) Ulula in 2005.

Open with DEXTER
In the text
thumbnail Fig. 24

Composite lightcurve of (714) Ulula in 2006.

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In the text
thumbnail Fig. 25

Lightcurve of (714) Ulula in 2008.

Open with DEXTER
In the text
thumbnail Fig. 26

Composite lightcurve of (714) Ulula in 2009.

Open with DEXTER
In the text

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