STUDIO FOTOMETRICO A 3 DIMENSIONI PER ASTEROIDI 

Di seguito l'articolo presentato alla rivista Astronomy & Astrophysics  a cui abbiamo collaborato con l'Osservatorio SAS del Prof , Federico Manzini, dall'Osservatorio personale a Tradate di Roberto Crippa e dalla Fondazione Osservatorio Astronomico di Tradate FOAM13

In questo articolo si presenta la possibilità di poter determinare la forma di un asteroide con immagini distanziate nel tempo che possono dare parametri fotometrici anche molto diversi.

Con opportuni software sviluppati dall'ESO si possono trarre ipotesi a tre dimensioni dello sviluppo volumetrico di un asteroide.

Il Software è stato sviluppato dal dipartimento di matematica applicata dell'Università di Helsinki.

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Astronomy & Astrophysicsmanuscript no. ten models  ESO 2006

July 25, 2006

Physical models of ten asteroids from observers collaboration network

J. ¡ Durech1;2, M. Kaasalainen2, A. Marciniak3, B. Allen23, R. Behrend22, C. Bembrick4, L. Bernasconi5, J.

Berthier25, G. Bolt6, R. Crippa21, M. Crow7, R. Durkee8, R. Dymock9, M. Fagas3, M. Fauerbach11, S. Fauvaud10 , M.

Frey12, R. Goncalves20, R. Hirsch3, D. Jardine15, K. Kami´nski3, R. Ko_13, T. Kwiatkowski3, A. L´opez14,

F. Manzini21, T. Micha³owski3, R. Pacheco14, F. Pilcher15, R. Poncy19, D. Pray16, W. Pych24, R. Roy17, G. Santacana10,

S. Slivan12 , R. Stephens18, B. Warner26, and M. Wolf1

ABSTRACT

Aims.We present physical models of ten asteroids obtained by means of lightcurve inversion. A substantial part of the photometric data was

observed by amateur astronomers. We emphasize the importance of a coordinated network of observers that will be of extreme importance for

future all-sky asteroid photometric surveys.

Methods. The lightcurve inversion method was used to derive spin states and shape models of the asteroids.

Results.We derived spin states and shape model for ten new asteroids. This increases the number of asteroid models up to nearly one hundred.

Key words. asteroids – photometry – models

1. Introduction

The lightcurve inversion method has become a standard tool for asteroid shape and spin state determination (Kaasalainen & Torppa 2001; Kaasalainen et al. 2001, 2002a). Convex modelsare a good representation of real shapes of asteroids, as has been proven by ground truths from, e.g., Kaasalainen et al.(2001, 2005); Marchis et al. (2006). Slightly less than one hundredasteroid models have been derived so far (Kaasalainenet al. 2002c, 2004; Torppa et al. 2003). However, the number of asteroid models increases slowly, mainly due to the fact that at least three well covered apparitions are necessary for a main-belt asteroid to be modelled. The Uppsala Asteroid Photometric Catalogue (UAPC, Lagerkvist et al. (2001)) – the main source of asteroid photometric data – has been already exploited and all well-observed asteroids were modelled. The UAPC still contains valuable photometric data of many asteroids but the amount of the data is not su_cient for a unique physical model. For many such targets, observations from only one more apparition are su_cient for a model, and many of those targets are within the reach of amateur astronomers. We present new observations and physical models of asteroids:

110 Lydia, 125 Liberatrix, 130 Elektra, 165 Loreley,196 Philomela, 218 Bianca, 306 Unitas, 423 Diotima, 776 Berbericia, and 944 Hidalgo.

In the last section, we discuss Send o_print requests to: J. ¡ Durech the possibility of combining the ordinary lightcurves with the sparse photometric data that will be available from all-sky photometric

surveys in the near future.

2. Observations

In order to derive unique spin state solutions and shape models we combined photometric data published in the UAPC with the new observations that were carried out by a large number of both amateur and pro_esional observers. Some lightcurves from the UAPC that were too noisy, consisted of only a few points, or were clearly wrong, were not included in the analysis.

All the new observations are listed in Table 3. For each lightcurve, there is the date of observation, aspect data, asteroid’s ecliptic coordinates and the code number of the observatory. This number refers to Table 1 where information about observers and telescopes is listed. All photometric observations were treated as relative and we used a combination of the Lommel-Seeliger and Lambert light-scattering laws (Kaasalainen et al. 2002a) as our scattering model.

We do not present all the lightcurves in a graphical form but select only tree representative lightcurves for each asteroid. Most of the lightcurves can be found at the Collaborative Asteroid Lightcurve Link (http://www. MinorPlanetObserver.com/astlc/default.htm) or at http://obswww.unige.ch/~behrend/page cou.html.

There is not enough room for full report of observed data.

2 J. ¡ Durech et al.: Physical models of ten asteroids from observers collaboration network

Table 1. The list of observatories and telescopes, D is the telescope aperture diameter.

code observing site D [cm] observers

1 Blauvac Observatory , France 31 R. Roy

2 Borowiec Station, Poznan Observatory , Poland 40 A. Marciniak, R. Hirsch, K. Kaminski,

M. Fagas, T. Micha³owski, T. Kwiatkowski

3 Carbuncle Hill Observatory , Rhode Island , USA 35 D. Pray

4 Egan Observatory, Florida , USA 40 M. Fauerbach

5 Observatori Astronomic de Consell, Mallorca A. Lopez, R. Pacheco

7 Ond¡rejov Observatory , Czech Republic 65 M. Wolf, J. ¡ Durech

8 Ostrowik, Poland 60 W. Pych

9 Pic du Midi , France 105 T. Micha³owski, J. Berthier

11Santana Observatory , CA , USA 35 R. Stephens

12 Shed of Science Observatory, Minneapolis , USA 25 R. Durkee

13 Pic de Chateau-Renard Observatory, France 62 S. Fauvaud, G. Santacana

14 Waterlooville, Hampshire , England 25 R. Dymock

15 Whitin Observatory , Massachusetts , USA 61 S. Slivan , M. Frey

16 Mt Tarana Observatory, Bathurst , Australia 40 C. Bembrick

17 Craigie, Australia 25 G. Bolt

18 Pleasant Plains, Illinois, USA 35 F. Pilcher, D. Jardine

19 R. Poncy

20 Linhaceira Observatory , Portugal R. Goncalves

21Stazione Astronomica di Sozzago, Italy R. Crippa, F. Manzini

22 Vintage Lane Observatory, Blenheim, New Zeland 30 B. Allen

23 UK ? M. Crow

24 Antelope Hills Observatory 25 R. Ko_

25 Les Engarouines Observatory , France L. Bernasconi

26 Goat Mountain Astronomical Research Station , USA 35 R. Stephens

Internet databases will be the only possibility of data presentation

after Pan-STARRS and other surveys start.

3. Models

The spin solutions and shape models were derived using the lightcurve inversion method developed by Kaasalainen & Torppa (2001); Kaasalainen et al. (2001). The spin axis direction in J2000 ecliptic coordinates _p; _p and the rotation Period P for each asteroid are listed in Table 2. In the case of the lightcurve inversion the systematic errors in lightcurves and model errors dominate over the  observational noise. Thus it is pointless to report error estimations derived from statistical tools (for example confidence limits based on _2 distribution).

A good estimation of a typical error in the pole direction is about 5_ of arc. The accuracy of the period determination is of the order of the last unit digit of the period value given in Table 2. For more detailed discussion of error estimation see Torppa et al. (2003). In Figs. 1 to 20 we plot the the shape model of each asteroid viewed from the plane of its equator (two views 90_ apart) and pole-on and the corresponding lightcurve fit. In some cases, the are two possible pole solutions with the ecliptic longitudes _ about 180_ apart and with similar values of the pole ecliptic latitude _. This ambiguity is inevitable for disk-integrated measurements of objects orbiting near the plane of ecliptic (see Kaasalainen & Lamberg 2006). Due to the fact that we used only relative photometry, the dimensions along the rotation axis are not well constrained. The pole-on silhouettes are very good approximations of asteroids real shapes whereas the silhouettes viewed from the equatorial plane can be significantly stretched or squeezed along the rotation axis. The principal axis of the inertia tensor (assuming uniform density) corresponding to the maximum moment of inertia is very close to the rotation axis for every model. The models together with the spin vector solutions are available at http://astro.troja.mff.cuni.cz/ projects/asteroids3D. 110 Lydia Lightcurve amplitudes do not exceed 0.2 mag. The shape is flat with a regular pole-on silhouette. There are two pole solutions. 125 Liberatrix The rotation axis is almost perpendicular to the plane of ecliptic and the orbit is close to the ecliptic (for most lightcurves j_j _ 5_). Liberatrix has been seen equator-on all the time and the lightcurves hardly change from one opposition to another, they have the same amplitude 0.4 mag. Relative lightcurves and the restricted geometry do not allow us to constrain the dimension along the rotation axis accurately – the shape model can be more or less stretched along this axis and the lightcurve fits remain almost the same. 130 Elektra Lightcurves of Elektra are typical double sinusoidal, the shape is regular and elongated. Although the rotation axis is perpendicular to the plain of ecliptic, the viewing/illumination geometry is not restricted to equatorial view/illumination (contrary to the previous case of Liberatrix) due to the high ecliptic longitudes Elektra reached (_35_ < _  <J. ¡ Durech et al.: Physical models of ten asteroids from observers collaboration network 3

Table 2. The table lists the ecliptic coordinates of asteroid’s spin axis direction (_p; _p), its sidereal rotation period P, the span of observations

in years, the umber of oppositions Nopp, the number of lightcurves Nlc, and the rms residual of the fit.

Asteroid _p _p P years of obs. Nopp Nlc rms

[deg] [deg] [hr] [mag]

110 Lydia 331 _61 10.92580 1958–2003 4 26 0.011 149 _55

125 Liberatrix 280 +74 3.968199 1981–2005 7 34 0.024 95 +68

130 Elektra 65 _88 5.224664 1980–2003 10 52 0.013

165 Loreley 346 +29 7.22667 1981–2006 6 29 0.015

196 Philomela 276 _49 8.332827 1964–2005 8 27 0.013 111 _41

218 Bianca 305 +17 6.33717 1979–2005 10 50 0.015 121 _10

306 Unitas 79 _35 8.73875 1979–2003 5 15 0.015 254 _18

423 Diotima 351 +4 4.775377 1981–2006 10 36 0.019

776 Berbericia 347 +12 7.66701 1977–2006 8 36 0.012

944 Hidalgo 281 +5 10.058634 1976–2004 4 14 0.013

25_). Although the pole direction reported in Table 2 gives the best fit to the data, there is a second pole solution (241_; _33_) that gives only a slightly worse fit and cannot be completely

ruled out. 165 Loreley The shape model has many flat areas, the lightcurves have small amplitude of 0.2 mag at most and a complicated structure. The pole direction solution (346_; +29_) is clearly the best one but there is the second solution (165_; +15_) giving only a slightly worse fit. 196 Philomela The shape model is asymmetric and smooth, the geometry varies a lot, lightcurves vary from almost flat to those with amplitudes up to 0.4 mag. 218 Bianca There are two pole solutions corresponding to almost the same spin axis with prograde and retrograde sense of rotation. The shape is asymmetric. 306 Unitas The shape is regular, lightcurves exhibit typical two extrema per rotation. 423 Diotima The lightcurves vary a lot – some are almost flat and others exhibit 0.2 mag deep minima. From the photometric data we obtained two solutions for the pole direction: (349_; +3_) and (173_; +34_). But only the first one is consistent with the adaptive optics image obtained by Marchis et al. (2006). 776 Berbericia Lightcurves are very di_erent for di_erent apparitions – sometimes there is only one maximum per period.

The corresponding shape model is asymmetric with sharp edges. 944 Hidalgo Although our model is based on only 14 lightcurves from three oppositions, the pole and period solution is unique. The shape model has very large flat areas and a ‘rectangular’ pole-on silhouette, which are strong indications of a highly nonconvex shape (Kaasalainen et al. 2002b; ¡ Durech & Kaasalainen 2003). Also the sharp minima of some lightcurves support the idea of a two-lobed shape.

4. Future work

The number of asteroid models available so far is very small when compared with the whole asteroid population. The classical approach of observing selected target (or a few targets) during the night in order to densely cover the lightcurve in the rotation phase is time consuming. The number of asteroids with enough observations to derive a model increases only slowly. The situation is going to change in the near future with the asteroid photometric surveys (for example Pan-STARRS). It has been shown (see Kaasalainen 2004; ¡ Durech et al. 2006) that asteroid models can be derived from calibrated photometric measurements sparse in time. This kind of data will be provided by future photometric surveys – instead of tens of lightcurves covering several apparitions we will have typically a hundred or more individual brightness measurements spread over several years. A di_culty that appears when analyzing sparse data is that the rotation period of an asteroid is not ‘visible’ from the data as it is in the case of an ordinary well-covered lightcurve. Thus a very wide interval of all possible periods must be densely scanned for the correct value. The time consuming process of period search can be sped up dramatically by adding just one ordinary lightcurve that constrains the search to a narrow interval of periods. more...

Acknowledgements. This work was supported, in part, by CIMO and

the Academy of Finland . The observations carried out at the Borowiec

Station were supported by Polish Grant 1 P03D 020 27.

4 J. ¡ Durech et al.: Physical models of ten asteroids from observers collaboration network