A new non-convex model of the binary asteroid (809) Lundia obtained with the SAGE modelling technique
P. Bartczak, A. Kryszczyńska, G. Dudziński, M. Polińska, F. Colas, F. Vachier, A. Marciniak, J. Pollock, G. Apostolovska, T. Santana-Ros, R. Hirsch, W. Dimitrow, M. Murawiecka, P. Wietrzycka, J. Nadolny
MMNRAS , 1–6 (2016) Preprint 22 May 2019 Compiled using MNRAS L A TEX style file v3.0
A new non-convex model of the binary asteroid (809)Lundia obtained with the SAGE modelling technique
P. Bartczak (cid:63) , A. Kryszczy´nska , G. Dudzi´nski , M. Poli´nska , F. Colas ,F. Vachier , A. Marciniak , J. Pollock , G. Apostolovska , T. Santana-Ros ,R. Hirsch , W. Dimitrow , M. Murawiecka , P. Wietrzycka , and J. Nadolny , Astronomical Observatory Institute, Faculty of Physics, Adam Mickiewicz University, S(cid:32)loneczna 36, 60-286 Pozna´n, Poland IMCCE, Observatoire de Paris, Av. Denfert-Rochereau 77, 75014 Paris, France Physics and Astronomy Department, Appalachian State University, Boone, NC 28608, USA Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia Instituto de Astrof´ısica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain Departamento de Astrof´ısica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain NaXys, Department of Mathematics, University of Namur, 8 Rempart de la Vierge, 5000 Namur, Belgium.
Accepted XXX. Received YYY; in original from 2016 Dec 8
ABSTRACT
We present a new non-convex model of the binary asteroid (809) Lundia. A SAGE(Shaping Asteroids with Genetic Evolution) method using disc-integrated photome-try only was used for deriving physical parameters of this binary system. The modelof (809) Lundia improves former system’s pole solution and gives the ecliptic co-ordinates of the orbit pole – λ = 122 ◦ , β = 22 ◦ , σ = ± ◦ – and the orbital pe-riod of 15 . ± . D eff = 9 . ± . . ± . and macroporosity of 13-23%.The intermediate-scale features of the model may also offer new clues on the compo-nents’ origin and evolution. Key words: methods: numerical – techniques: photometric. (809) Lundia was classified as a V-type asteroid in the Floradynamical family (Florczak et al. 2002). The discovery ofits binary nature in September 2005 was based on pho-tometric observations carried out at Borowiec observatory(Kryszczynska et al. 2005). The first modelling of the Lun-dia synchronous binary system was based on 23 lightcurvesobtained at Borowiec and Pic du Midi Observatories dur-ing two oppositions in 2005/2006 and 2006/2007. The twomethods of modelling - modified Roche ellipsoids and kine-matic - gave similar parameters of the system (Kryszczy´nskaet al. 2009). The poles of the orbit in the ecliptic coordinatesfound were: longitude 118 ± ◦ , and latitude 28 ± ◦ in themodified Roche model and 120 ± ◦ , 18 ± ◦ respectively inthe kinematic model (Kryszczy´nska et al. 2009). The orbitalperiod obtained from the lightcurve analysis as well as frommodelling was 15 . ± . h . The obtained bulk density (cid:63) E-mail:[email protected] of both components was 1 .
64 or 1 . g/cm . The compari-son with HED meteorites gave very high macroporosity of42 − c (cid:13) a r X i v : . [ a s t r o - ph . E P ] M a y P. Bartczak et al. lifetime (Britt et al. 2002). Macroporosity can be found bycomparing meteorite analogs’ density and porosity with thedensity derived from independent method and data like pho-tometry based shape modelling.Density estimation relies on the estimations of the massand volume. As summarized by Carry (2012), mass canbe estimated using several methods: orbit deflection duringclose encounters, planetary ephemeris, spacecraft trackingor studying orbit of a satellite. Although spacecraft tracingtechnique is the most precise, it is limited to small numberof space missions’ targets. Very good results (of 10-15% ac-curacy) can be achieved with binary asteroids, where onecan derive total mass of the system from Kepler’s 3 rd lawonce the satellite’s orbit is known.To know the asteroid’s volume, detailed shape modeland its size are needed. Among the methods of size esti-mation, about 85% of asteroids’ sizes were obtained withthermal modeling. Majority of those estimates have relativeuncertainty of about 5%, however there are some indicationsof these values being underestimated (Carry 2012). Meth-ods capable of deriving convex shapes only (e.g. spheres,tri-axial ellipsoids or Roche ellipsoids) put lower constraintson the density, as introducing concavities will increase den-sity of the body with the same equivalent sphere diameterand period. In-situ observations revealed concave nature ofasteroids in general, which makes non-convex methods moreadequate and accurate.According to a list of binaries’ parameters describedfirst in (Pravec & Harris 2007), there are only 12 doublesynchronous asteroids discovered so far. Synchronous binaryasteroids with circular orbits are special cases of binary sys-tems where rotational periods of both components are equalto orbital one. Additionally, angular momenta of compo-nents, orbit and resulting system’s one are parallel. Thisreduces the amount of free parameters of the model and al-low for detailed shape modelling leading to more accuratevolume estimates.In this paper we present a new non-convex model of the(809) Lundia system using SAGE (Shaping Asteroids withGenetic Evolution) method using disc-integrated photome-try only, described by Bartczak et al. (2014). This methodwas successfully applied to model the binary asteroid (90)Antiope. We continued observations of (809) Lundia system in 2007,2008, 2009/2010, 2011, and 2012 oppositions at Borowiec,Pic du Midi, PROMPT, South African Astronomical Ob-servatory, and Bulgarian National Observatory Rozhen. Aspredicted the well visible eclipses/occultation events wereobserved only in 2011. Signs of partial eclipses/occultationare visible in the lightcurves from 2012 opposition. InFig. 1 we show positions of the Earth in the referenceframe of the asteroid. Blue dots represent observationswith eclipse/occultation events and green without events.Open circles represent future observing geometries and Figure 1.
Positions of the Earth in the reference frameof the asteroid. Blue dots represent positions with observedeclipse/occultation events and green without events. Open circlesrepresent future observing geometries. show that in 2018 only there will be a chance to observeeclipses/occultation events.Observations at Borowiec observatory were carried outwith 0 . m Newton telescope equipped with the KAF402MECCD camera and clear filter. The details of the Borowiecsystem were decsribed by Micha(cid:32)lowski et al. (2004). Ob-servations from SAAO were carried out at 0 . m reflectorequipped with the University of Cape Town (UCT) CCDcamera and R filter.All CCD frames from Borowiec and SAAO were reducedfor bias, dark current, and flatfield using CCLR STARLINKpackage. The aperture photometry was performed to mea-sure the instrumental brightness of the asteroid and thecomparison and check stars. Lightcurves observed at Pic duMidi in 2009 were obtained using 1 . m . / . m Schmidt telescope andKAF1602E CCD camera and R filter. For the data reductionand aperture photometry the IDL software was used. Threelightcurves were obtained by PROMPT 4, a 0 . m Ritchey-Chr´etien telescope located in Cerro Tololo Inter-AmericanObservatory in Chile, equipped with Alta U47+ camera. Theaspect data of Lundia are listed in Table 1.The obtained composite lightcurves are presented inFigs. 2-6. Lighcurves are composed with the synodic periodof 15 . ± . h . For good comparison between lightcurvesthe scale on each graph is the same. In Fig 2. we present alightcurve from NAO Rozhen in comparison with alreadypublished lightcurves from Pic du Midi. Despite the timespan between the lightcurves is as much as four months theinternal fit is very good and it confirms the derived syn-odic period. Currently, our dataset consists of 41 individ-ual lightcurves obtained during 6 oppositions and covering MNRAS000
Positions of the Earth in the reference frameof the asteroid. Blue dots represent positions with observedeclipse/occultation events and green without events. Open circlesrepresent future observing geometries. show that in 2018 only there will be a chance to observeeclipses/occultation events.Observations at Borowiec observatory were carried outwith 0 . m Newton telescope equipped with the KAF402MECCD camera and clear filter. The details of the Borowiecsystem were decsribed by Micha(cid:32)lowski et al. (2004). Ob-servations from SAAO were carried out at 0 . m reflectorequipped with the University of Cape Town (UCT) CCDcamera and R filter.All CCD frames from Borowiec and SAAO were reducedfor bias, dark current, and flatfield using CCLR STARLINKpackage. The aperture photometry was performed to mea-sure the instrumental brightness of the asteroid and thecomparison and check stars. Lightcurves observed at Pic duMidi in 2009 were obtained using 1 . m . / . m Schmidt telescope andKAF1602E CCD camera and R filter. For the data reductionand aperture photometry the IDL software was used. Threelightcurves were obtained by PROMPT 4, a 0 . m Ritchey-Chr´etien telescope located in Cerro Tololo Inter-AmericanObservatory in Chile, equipped with Alta U47+ camera. Theaspect data of Lundia are listed in Table 1.The obtained composite lightcurves are presented inFigs. 2-6. Lighcurves are composed with the synodic periodof 15 . ± . h . For good comparison between lightcurvesthe scale on each graph is the same. In Fig 2. we present alightcurve from NAO Rozhen in comparison with alreadypublished lightcurves from Pic du Midi. Despite the timespan between the lightcurves is as much as four months theinternal fit is very good and it confirms the derived syn-odic period. Currently, our dataset consists of 41 individ-ual lightcurves obtained during 6 oppositions and covering MNRAS000 , 1–6 (2016)
AGE non-convex model of 809 Lundia Table 1.
Aspect data. Columns give dates of observations with respect to the middle of the lightcurve, asteroid’s distances to the Sun( r ) and Earth (∆) in AU, phase angle ( α ), ecliptic longitude ( λ ) and latitude ( β ) for J2000.0 and the observatory. r ∆ Phase λ β ObservatoryDate (UT) angle (J2000)(AU) (AU) ( ◦ ) ( ◦ ) ( ◦ )2007 Apr 04.91 2.7186 1.9977 17.29 159.95 3.73 NAO Rozhen2008 May 07.32 2.1530 1.5361 25.38 292.88 9.27 PROMPT2008 May 08.29 2.1509 1.5242 25.25 293.05 9.31 PROMPT2008 May 09.30 2.1487 1.5118 25.11 293.22 9.36 PROMPT2008 Jun 10.92 2.0776 1.1682 16.68 294.94 10.59 SAAO2009 Nov 20.10 2.4282 1.9806 23.26 133.39 -6.17 Pic du Midi2009 Nov 25.14 2.4336 1.9276 22.57 133.93 -6.23 Pic du Midi2011 Apr 07.07 2.5479 1.6394 11.81 226.89 9.72 Pic du Midi2011 Apr 08.10 2.5462 1.6310 11.86 226.71 9.79 Pic du Midi2011 Apr 09.06 2.5446 1.6334 11.09 226.55 9.85 Pic du Midi2011 Apr 11.10 2.5414 1.6080 10.34 226.17 9.97 Pic du Midi2011 Apr 12.08 2.5399 1.6009 9.98 225.98 10.03 Pic du Midi2012 Oct 09.04 1.9847 1.2466 24.71 71.60 -10.87 Borowiec2012 Oct 18.05 2.0017 1.1901 21.59 71.60 -11.69 Borowiec2012 Oct 18.02 2.0035 1.1846 21.21 71.55 -11.77 Borowiec2012 Oct 20.07 2.0056 1.1787 20.80 71.49 -11.86 Borowiec2012 Nov 11.05 2.0495 1.1012 10.87 67.95 -13.31 Borowiec2012 Nov 26.03 2.0810 1.1086 6.36 63.95 -13.50 Borowiec different observing geometries. Details of the observing ge-ometries of the (809) Lundia system for each opposition aregiven in Table 2. We used genetic-algorithm-based modelling method SAGEthat given solely photometric observations recreates non-convex shape, spin axis orientation and rotational periodof synchronous binary asteroid (Bartczak et al. 2014).The asteroid system is assumed to be synchronous withcircular orbit about center of mass. Each of the bodies isdescribed by 62 vectors with fixed directions; the length ofeach vector is a free parameter during modelling process. Tocreate lightcurves of the system a refined and more detailedshapes are used, created using surface smoothening algo-rithm (Catmull & Clark 1978). To fully describe the systemthere are two additional free parameters: bodies size ratioand separation. During modelling process the best fit fororbital period is searched and than used to establish massratio using the separation, shapes and size ratio as well.In every step of modelling process synthetic lightcurvesare compared with observed ones. To calculate a lightcurvefor specific moment of time, a geometry of the observations(i.e. the positions of the Sun, asteroid and Earth) is recon-structed using position vectors in heliocentric, ecliptic refer-ence frame based on an ephemeris of the asteroid. When a3D scene is constructed a system is rotated 360 ◦ to createa full rotation lightcurve that is later used for comparison.The flux of an asteroid is obtained in rasterisation process:the image is created as if a telescope with an infinite res-olution observed an asteroid creating a CCD image. Thesum of the pixels gives a synthetic relative photometry mea-surement. A Lommel-Seeliger scattering law is used and noalbedo variations are assumed. The modelling process uses genetic algorithm to ar-rive at global minimum. The modelling starts with the twospheres, random spin axis orientation and approximate rota-tional period. In every step a random changes to the shapes,period, spin axis orientation, separation and size ratio areapplied creating a random population (generation) of sys-tems. Each body of the system is a physical model assuminghomogeneous distribution of mass; the axes of largest in-ertia of the bodies are aligned with spin axis of the wholesystem. Than, for every model of the system in the popu-lation a synthetic lightcurves are calculated and comparedwith observations using χ test. The model with smallest χ is chosen as the seed for the next population and thewhole process repeats until χ value no longer changes frompopulation to population. In order to assure the modellingprocess not to fall into local minimum a weighting processis applied in every step. The lightcurve with largest χ isgiven the largest weight to steer the process towards theglobal minimum; the weights change in every step.Additionally, the whole modelling process is run multi-ple times creating a family of solutions. This is a standardprocedure in genetic evolution algorithms to ensure the re-sult being in global, rather than local, minimum. The pathleading to a model in each run is different, but the modelsshould be alike in the end. If the models differ significantly,it indicates that not enough observational data has beensupplied for the modelling. The Lundia shape model projections can be seen in Fig. 4.Synthetic lightcurves generated by the model fit the ob-served ones very well (Fig. 4). Mutual eclipse events areperfectly timed and the lightcurves’ shape is reproduced ongeneral and detail levels. The uncertainty of photometry was
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Table 2.
Details of the lightcurves used for our modelling of (809) Lundia The table columns describe the observations time range,the number of observing nights, the phase angle range, the ecliptic longitudes and latitudes of the asteroid around the oppositiondates, the observed eclipsing amplitudes and the references. The value of ’0’ for the eclipsing amplitudes means that no eclipseswere observed for these apparitions (i.e. 2006/2007, 2008 and 2009).Time range N lc α ( ◦ ) λ ( ◦ ) β ( ◦ ) Eclipsing Referenceamplitude (mag)Sep 2005 – Jan 2006 19 7.3 – 25.9 45 -11 0.75 – 0.30 Kryszczynska et al. (2005)Dec 2006 – Apr 2007 4+1 17.3 – 21.4 168 2 0 Kryszczynska et al. (2005) and this paperMay 2008 – Jun 2008 4 16.7 – 25.3 294 10 0 this paperNov 2009 2 22.6 – 23.3 133 -6 0 this paperApr 2011 5 10.0 – 11.8 226 10 0.58 this paperOct 2012 – Nov 2012 6 6.4 – 24.7 68 -12 0 – 0.15 this paper Figure 2.
Composite lightcurves of 809 Lundia from 2006/2007opposition (top) with corresponding view of the system fromEarth (bottom).
Figure 3.
Composite lightcurves of 809 Lundia from 2008 op-position (top) with corresponding view of the system from Earth(bottom). MNRAS000
Composite lightcurves of 809 Lundia from 2008 op-position (top) with corresponding view of the system from Earth(bottom). MNRAS000 , 1–6 (2016)
AGE non-convex model of 809 Lundia Figure 4.
Composite lightcurves of 809 Lundia from 2009 op-position (top) with corresponding view of the system from Earth(bottom). not reported by the observers and therefore it is assumed tobe 0 . mag .Using Spitzer Space telescope Marchis et al. (2012) cal-culated effective diameter of Lundia D eff = 9 . km with3 σ uncertainty of 1 . km . Combining this result with in-homogeneous Roche ellipsoids model by Descamps (2010)they obtained equivalent sphere diameters for the primaryand secondary components D p = 7 . ± . km and D s =6 . ± . km , with system separation d = 14 . km and av-erage bulk density of the system ρ = 1 . g/cm .Applying said effective diameter we scaled the new non-convex Lundia model by assuming the same volume of thesystem. Mass was determined from the orbital period P andsystem separation d using Kepler’s third law.The parameters of non-convex model of Lundia systemare: Figure 5.
Composite lightcurves of 809 Lundia from 2011 op-position (top) with corresponding view of the system from Earth(bottom). • system’s spin axis ecliptic coordinates: λ = 122 . ± ◦ , β = 22 ± ◦ • sidereal period: P = 15 . ± − h • primary equivalent sphere diameter: D p = 8 . ± . km • secondary equivalent sphere diameter: D s = 7 . ± . km • D s /D p = 0 . • system separation: d = 18 . ± . km • total mass: (1 . ± . kg • bulk density: ρ = 2 . ± . g/cm • macroporosity: 13-23%Spectroscopic observations’ analysis (Birlan et al. 2014)indicates similar mineralogical composition as howardite-diogenite meteorites. The obtained density of 2 . g/cm MNRAS , 1–6 (2016)
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Figure 6.
Composite lightcurve of 809 Lundia from 2012 oppo-sition (top) with corresponding view of the system from Earth(bottom). is much higher than determined before, and this value incomparison with the density of HED meteorites of 2 .
86 to3 . g/cm reported by Britt & Consolmagno (2003) andMcCausland & Flemming (2006) infers the macroporosityof (809) Lundia of only 13-23%, rather than 40-50% as re-ported by Marchis et al. (2012). The obtained non-convex model of (809) Lundia perfectlyreproduces the obtained set of photometric lightcurves. Byscaling volume of the system components we found density50% higher than the one calculated in the previous stud-ies and macroporosity of about 60% smaller than beforeKryszczy´nska et al. (2009). However the internal structureof asteroids is presently unclear and newly obtained values may serve as an input to the theories of the Solar Systemevolution. Higher values of the bulk density are due to non-convex shape of the system. The more shape deviates froma sphere the larger density value we get.The shape and size of Lundia could be refined givendirect measurements, like stellar occultation events. Predic-tions for 2017 based on new GAIA DR1 catalog yield twoevents on 27 April and 28 May 2017. Unfortunately Lundia’ssmall size makes accurate prediction difficult, as uncertaintyof star position on the level of 7 mas translates into 15 kmuncertainty in the position on Earth which is about the sizeof the Lundia system. Nevertheless, such observations couldput better constrains on asteroid’s size and density.
ACKNOWLEDGEMENTS
The research leading to these results has received fundingfrom the European Union’s Horizon 2020 Research and In-novation Programme, under Grant Agreement no 687378.This work was partialy supported by grant no.2014/13/D/ST9/01818 from the National Science Centre,PolandThis paper uses observations made at the South AfricanAstronomical Observatory (SAAO). The reduction of CCDframes from Borowiec and SAAO were performed with theCCLR STARLINK package.G. Apostolovska gratefully acknowledge observing grantsupport from the Institute of Astronomy and Rozhen Na-tional Astronomical Observatory, Bulgarian Academy of Sci-ences
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Bartczak P., Micha(cid:32)lowski T., Santana-Ros T., Dudzi´nski G., 2014,MNRAS, 443, 1802Birlan M., Nedelcu D. A., Popescu M., Vernazza P., Colas F.,Kryszczy´nska A., 2014, MNRAS, 437, 176Britt D. T., Consolmagno G. J., 2003, Meteoritics and PlanetaryScience, 38, 1161Britt D. T., Yeomans D., Housen K., Consolmagno G., 2002, As-teroid Density, Porosity, and Structure. pp 485–500Carry B., 2012, Planet. Space Sci., 73, 98Catmull E., Clark J., 1978, Computer-Aided Design, 10, 350Descamps P., 2010, Icarus, 207, 758Florczak M., Lazzaro D., Duffard R., 2002, Icarus, 159, 178Kryszczynska A., Kwiatkowski T., Hirsch R., Polinska M., Kamin-ski K., Marciniak A., 2005, Central Bureau Electronic Tele-grams, 239Kryszczy´nska A., et al., 2009, A&A, 501, 769Marchis F., et al., 2012, Icarus, 221, 1130McCausland P. J. A., Flemming R. L., 2006, in Mackwell S.,Stansbery E., eds, Lunar and Planetary Science ConferenceVol. 37, 37th Annual Lunar and Planetary Science Confer-ence.Micha(cid:32)lowski T., et al., 2004, A&A, 416, 353Pravec P., Harris A. W., 2007, Icarus, 190, 250This paper has been typeset from a TEX/L A TEX file prepared bythe author. MNRAS000 , 1–6 (2016)
AGE non-convex model of 809 Lundia Figure 7. xz , yz , xy , − xz , − yz , − xy projections of the Lundia model.MNRAS , 1–6 (2016) P. Bartczak et al.
Figure 8.
Observations (black crosses) versus synthetic lightcurves of the model (solid red line).MNRAS000