A&A 493, 291–297 (2009) Astronomy DOI: 10.1051/0004-6361:200810393 & c ESO 2008 Astrophysics Asteroid models from combined sparse and dense photometric data J. Durechˇ 1, M. Kaasalainen2,B.D.Warner3, M. Fauerbach4,S.A.Marks4,S.Fauvaud5,6,M.Fauvaud5,6, J.-M. Vugnon6, F. Pilcher7, L. Bernasconi8, and R. Behrend9 1 Astronomical Institute, Charles University in Prague, V Holešovickáchˇ 2, 18000 Prague, Czech Republic e-mail: [email protected] 2 Department of Mathematics and Statistics, Rolf Nevanlinna Institute, PO Box 68, 00014 University of Helsinki, Finland 3 Palmer Divide Observatory, 17995 Bakers Farm Rd., Colorado Springs, CO 80908, USA 4 Florida Gulf Coast University, 10501 FGCU Boulevard South, Fort Myers, FL 33965, USA 5 Observatoire du Bois de Bardon, 16110 Taponnat, France 6 Association T60, 14 avenue Edouard Belin, 31400 Toulouse, France 7 4438 Organ Mesa Loop, Las Cruces, NM 88011, USA 8 Observatoire des Engarouines, 84570 Mallemort-du-Comtat, France 9 Geneva Observatory, 1290 Sauverny, Switzerland Received 16 June 2008 / Accepted 15 October 2008 ABSTRACT Aims. Shape and spin state are basic physical characteristics of an asteroid. They can be derived from disc-integrated photometry by the lightcurve inversion method. Increasing the number of asteroids with known basic physical properties is necessary to better understand the nature of individual objects as well as for studies of the whole asteroid population. Methods. We use the lightcurve inversion method to obtain rotation parameters and coarse shape models of selected asteroids. We combine sparse photometric data from the US Naval Observatory with ordinary lightcurves from the Uppsala Asteroid Photometric Catalogue and the Palmer Divide Observatory archive, and show that such combined data sets are in many cases sufficient to derive a model even if neither sparse photometry nor lightcurves can be used alone. Our approach is tested on multiple-apparition lightcurve inversion models and we show that the method produces consistent results. Results. We present new shape models and spin parameters for 24 asteroids. The shape models are only coarse but describe the global shape characteristics well. The typical error in the pole direction is ∼10–20◦. For a further 18 asteroids, inversion led to a unique determination of the rotation period but the pole direction was not well constrained. In these cases we give only an estimate of the ecliptic latitude of the pole. Key words. minor planets, asteroids – techniques: photometric – methods: data analysis 1. Introduction sparse in time is fully sufficient for modelling as long as the photometric accuracy of the data is better than ∼5% (Durechˇ The lightcurve inversion method is a powerful tool to de- et al. 2005, 2007). Sparse data that will be available from all- sky surveys such as Pan-STARRS, LSST or Gaia are extremely rive basic physical properties of asteroids (rotation periods, ffi spin axis orientations, and shapes) from their disc-integrated time-e cient in obtaining new asteroid models compared to the lightcurves (Kaasalainen & Torppa 2001; Kaasalainen et al. standard approach of observing individual objects. However, ac- 2001, 2002a). As has been proven by space probes, laboratory curate sparse data are not yet available, and all asteroid mod- models, and adaptive optics images (Kaasalainen et al. 2001, els based on photometry were derived by inversion of standard 2005; Marchis et al. 2006), models derived by this method are dense lightcurves (e.g., Kaasalainen et al. 2002b, 2004; Torppa good approximations of the real shapes of asteroids. In the stan- et al. 2003). dard approach, the lightcurve inversion method is applied to The main challenge to be solved when inverting sparse data a set of typically tens of lightcurves observed during at least is the correct determination of the rotation period. Because there three or four apparitions – only then can the spin state and the is no direct information about the rotation period in the data, corresponding shape be derived uniquely. Kaasalainen (2004) the correct value has to be found by scanning the whole in- showed that to derive a unique physical model it is not neces- terval of possible values and looking for the best fit. This is sary to have dense lightcurves – i.e., brightness measurements a time consuming process that yields a unique period solution that densely sample brightness variations during one revolu- only if there are enough data points (more than about a hun- tion. Lightcurve inversion can also be used on so-called “sparse dred) and if the data photometric accuracy is better than about data” that only sparsely sample brightness variations. Typically, ∼5%. If these conditions are not fulfilled, there is usually more sparse data consist of at most a few measurements per night. than one period with the same fit to the data (for more de- A set of more than about one hundred calibrated measurements tails see Durechˇ et al. 2005, 2007). On the other hand, just one Article published by EDP Sciences 292 J. Durechˇ et al.: Asteroid models from combined sparse and dense photometric data ordinary dense lightcurve usually defines the possible range of about five years and typically consist of 50–200 individual mea- the rotation period well. As has been shown by Kaasalainen & surements. Durechˇ (2007), noisy sparse data can be combined with a few ordinary lightcurves and such combined data sets give in many cases a unique solution. When combining sparse data with dense 3. Tests lightcurves, inversion leads to a unique solution for the period even if sparse data are noisy and the number of dense lightcurves Because our aim was to derive asteroid models based on noisy is small. sparse data and a small number of dense lightcurves, we carried In this paper we present new asteroid models derived from out various tests to determine how accurate and reliable the ob- combined data sets. We used sparse data from the US Naval tained results were. Observatory, Flagstaff (USNO) and dense lightcurves from the For our tests, we used the Database of Asteroid Models Uppsala Asteroid Photometric Catalogue (UAPC, Lagerkvist from Inversion Techniques (DAMIT)3 now containing about et al. 2001) and from the archive of the Palmer Divide 80 models derived from dense lightcurves. From DAMIT, we Observatory (PDO). selected 60 asteroids that had enough sparse USNO data (more We also present the results of various tests that estimate the than 30 points). From the full set of dense photometry for each errors in pole directions and show that our results are reliable. asteroid, we randomly selected individual lightcurves and com- bined them with the sparse data to create test data sets. For each asteroid, we created twelve different test data sets: one set = 2. Combined data with six lightcurves (Nlc 6) from three apparitions, two sets with four lightcurves (Nlc = 4) from two apparitions, three sets with two lightcurves (N = 2) from one apparition, and six indi- As stated above, combining sparse and dense photometry can lc vidual lightcurves (N = 1). Each data set also contained sparse yield a unique model even in cases when neither sparse data nor lc USNO data. We then processed the test data sets the same way as lightcurves would be sufficient for modelling alone. Sparse data the real data using the lightcurve inversion method (Kaasalainen typically cover more apparitions carrying thus information about ff 2004; Durechˇ et al. 2005, 2007). The inversion algorithm derives brightness variations for di erent geometries. Dense lightcurves, / ff on the other hand, well define the rotation period and enable us aspinshape model that minimizes the di erence between ob- to narrow down the search interval. served and computed brightness. The minimization process can use dense and sparse data and we decided to weight the sparse data one third of the dense data. We used information about ro- 2.1. Standard lightcurves tation periods and reliability codes listed in the Minor Planet Lightcurve Parameters database4. All asteroids from DAMIT Most of the photometric lightcurves we used were archived in have reliability codes 3 or 4. For each asteroid, we scanned an UAPC. Apart from this source, we also used the database of interval of periods centered at the reported period P with a width 5 lightcurves of more than 200 asteroids observed at the Palmer of ±1% and ±3% of P for reliability codes 4 and 3, respectively . Divide Observatory1. We also included new observations of As expected, in most cases the limited amount of data was (34) Circe and (416) Vaticana. Circe was observed by F. Pilcher not sufficient to provide us with a unique solution – out of the (Las Cruces, New Mexico, 35-cm telescope) in 2007 and by 60 asteroids and 12 tests for each asteroids there were only 22 M. Fauerbach and S. A. Marks (Egan Observatory, Florida, asteroids that had at least one unique solution. A unique solution 40-cm telescope) in 2007/2008. Vaticana was observed in was defined as follows: the best period had at least 10% lower 2006/2007 by M. Fauerbach (Egan Observatory, Florida, 40-cm χ2 than all other periods from the scanned interval and, for this telescope) and by S. Fauvaud, J.-M. Vugnon, and M. Fauvaud period, there was only one pole solution with at least 10% lower 2 ◦ (Pic du Midi Observatory, France, 60-cm telescope). χ than the others (with a possible mirror solution for λ ± 180 ). We compared the obtained results with the models from DAMIT based on full lightcurve sets – the DAMIT models were 2.2. Sparse data taken as established. Results listed in Table 1 report the maxi- mum difference Δmax (in degrees) between the pole solution for Sparse photometric data accurate enough to provide a unique so- the corresponding test data set with Nlc lightcurves and the orig- lution of the inverse problem are not yet available.