# 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, et al.

► To cite this version:

P. Bartczak, A. Kryszczyńska, G. Dudziński, M. Polińska, F. Colas, et al. A new non-convex model of the binary asteroid (809) Lundia obtained with the SAGE modelling technique. Monthly Notices of the Royal Astronomical Society, 2017, 471 (1), pp.941-947. 10.1093/mnras/stx1603. hal-02262293

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# A new non-convex model of the binary asteroid (809) Lundia obtained with the SAGE modelling technique

P. Bartczak,<fnref n="1" />★ A. Kryszczyńska,<fnref n="1" /> G. Dudziński,<fnref n="1" /> M. Polińska,<fnref n="1" /> F. Colas,<fnref n="2" /> F. Vachier,<fnref n="2" />

A. Marciniak,<fnref n="1" /> J. Pollock,<fnref n="3" /> G. Apostolovska,<fnref n="4" /> T. Santana-Ros,<fnref n="1" /> R. Hirsch,<fnref n="1" />

W. Dimitrow,<fnref n="1" /> M. Murawiecka,<fnref n="5" /> P. Wietrzycka<fnref n="1" /> and J. Nadolny<fnref n="6" />,<fnref n="7" />

Accepted 2017 June 21. Received 2017 June 21; in original form 2017 March 17

## 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 photometry only was used for deriving physical parameters of this binary system. The model of (809) Lundia improves former system's pole solution and gives the ecliptic coordinates of the orbit pole – $\lambda = 122^\circ$, $\beta = 22^\circ$, $\sigma = \pm 5^\circ$ – and the orbital period of $15.415\,74 \pm 0.000\,01\,h$. For scaling our results, we used an effective diameter ($D_{\text{eff}}$) of $9.6 \pm 1.1\,\text{km}$ obtained from *Spitzer* observations. The non-convex shape description of the components permitted a refined calculation of the components' volumes, leading to a density estimation of $2.5 \pm 0.2\,\text{g cm}^{-3}$ and a macroporosity of 13–23 per cent. The intermediate-scale features of the model may also offer new clues on the components' origin and evolution.

**Key words:** methods: numerical – techniques: photometric.

## 1 INTRODUCTION

(809) Lundia was classified as a V-type asteroid in the dynamical Flora family (Florczak, Lazzaro & Duffard 2002). The discovery of its binary nature in September 2005 was based on photometric observations carried out at the Borowiec Observatory (Kryszczynska et al. 2005). The first modelling of the Lundia synchronous binary system was based on 23 lightcurves obtained at Borowiec and Pic du Midi Observatories during two oppositions in 2005/2006 and 2006/2007. The two methods of modelling – modified Roche ellipsoids and kinematic – gave similar parameters of the system (Kryszczyńska et al. 2009). The poles of the orbit in the ecliptic coordinates found were as follows: longitude $118 \pm 2^\circ$ and latitude $28 \pm 2^\circ$ in the modified Roche model and longitude $120 \pm 2^\circ$ and latitude $18 \pm 2^\circ$ in the kinematic model (Kryszczyńska et al. 2009). The orbital period obtained from the light-curve analysis as well as from the modelling was $15.418 \pm 0.001\,h$. The obtained bulk density of both components was $1.64$ or $1.71\,\text{g cm}^{-3}$. The comparison with HED meteorites gave a very high macroporosity of 42–49 per cent. Spectroscopic observations of the (809)

Lundia system were performed using NASA Infrared Telescope Facility and SpeX spectrograph in 2005, 2007 and 2010. One of these spectra, observed during the total eclipse of the components, allowed to investigate the homogeneity/heterogeneity of both bodies. A detailed analysis of all spectra confirmed similar mineralogy of both components. By applying different mineralogical models, a composition similar to one of the howardite–diogenite meteorites was found (Birlan et al. 2014).

Finding bulk density is the main goal of modelling asteroids' physical properties, as it gives insights into their internal structure and composition. Determining the latter based on spectroscopy or albedo probing only the surface of the body leads to strongly biased bulk densities, but one can use these data to find meteorite analogs for further comparison. Besides density, macroporosity is another valuable information putting constraints on e.g. evolution or collisional lifetime (Britt et al. 2002). Macroporosity can be found by comparing meteorite analogs' density and porosity with the density derived from an independent method and data like photometry-based shape modelling.

Density estimation relies on the estimations of the mass and volume. As summarized by Carry (2012), the mass can be estimated using several methods: orbit deflection during close encounters, planetary ephemeris, spacecraft tracking or studying the orbit of

<footnote n="1">¹Astronomical Observatory Institute, Faculty of Physics, Adam Mickiewicz University, Sloneczna 36, PL-60-286 Poznań, Poland</footnote>

<footnote n="2">²IMCCE, Observatoire de Paris, Av. Denfert-Rochereau 77, F-75014 Paris, France</footnote>

<footnote n="3">³Physics and Astronomy Department, Appalachian State University, Boone, NC 28608, USA</footnote>

<footnote n="4">⁴Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia</footnote>

<footnote n="5">⁵NaXys, Department of Mathematics, University of Namur, 8 Rempart de la Vierge, B-5000 Namur, Belgium</footnote>

<footnote n="6">⁶Instituto de Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain</footnote>

<footnote n="7">⁷Department Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain</footnote>

<footnote>* E-mail: przebar@amu.edu.pl</footnote>
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Positions of the Earth in the reference frame of the asteroid. Blue dots represent positions with observed eclipse/occultation events and green without events. Open circles represent future observing geometries.</figcaption></figure>

a satellite. Although the spacecraft tracing technique is the most precise, it is limited to a small number of space missions' targets. Very good results (of 10–15 per cent accuracy) can be achieved with binary asteroids, where one can derive the total mass of the system from Kepler's third law once the satellite's orbit is known.

To know the asteroid's volume, a detailed shape model and its size are needed. Amongst the methods of size estimation, about 85 per cent of asteroids' sizes were obtained with thermal modelling. The majority of those estimates have a relative uncertainty of about 5 per cent; however, there are some indications of these values being

underestimated (Carry 2012). Methods capable of deriving convex shapes only (e.g. spheres, tri-axial ellipsoids or Roche ellipsoids) put lower constraints on the density, as introducing concavities will increase the density of the body with the same equivalent sphere diameter and period. *In situ* observations revealed concave nature of asteroids in general, which makes non-convex methods more adequate and accurate.

According to a list of binaries' parameters<fnref n="1" /> described first by Pravec & Harris (2007), there are only 12 double synchronous asteroids discovered so far. Synchronous binary asteroids with circular orbits are special cases of binary systems where rotational periods of both components are equal to orbital one. Additionally, angular momenta of components, orbit and resulting system's one, are parallel. This reduces the amount of free parameters of the model and allows for detailed shape modelling, leading to more accurate volume estimates.

In this paper, we present a new non-convex model of the (809) Lundia system applying a SAGE (Shaping Asteroids with Genetic Evolution) method using disc-integrated photometry only, described by Bartczak et al. (2014). This method was successfully applied to model the binary asteroid (90) Antiope.

## 2 OBSERVATIONS

We continued observations of the (809) Lundia system in 2007, 2008, 2009/2010, 2011 and 2012 oppositions at Borowiec, Pic du Midi, PROMPT, South African Astronomical Observatory and Bulgarian National Observatory Rozhen. As predicted, the well visible eclipses/occultation events were observed only in 2011. Signs of partial eclipses/occultation are visible in the lightcurves from 2012 opposition. In Fig. 1, we show positions of the Earth in the reference frame of the asteroid. Blue dots represent observations

<table><caption>**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 (*Δ*) in au, phase angle (*α*), ecliptic longitude (*λ*) and latitude (*β*) for J2000.0 and the names of the observatory.</caption><thead><tr><th>Date (UT)</th><th>r<br>(au)</th><th>Δ<br>(au)</th><th>Phase<br>angle<br>(°)</th><th>λ<br>(°)<br>(J2000)</th><th>β<br>(°)</th><th>Observatory</th></tr></thead><tbody><tr><td>2007 Apr 04.91</td><td>2.7186</td><td>1.9977</td><td>17.29</td><td>159.95</td><td>3.73</td><td>NAO Rozhen</td></tr><tr><td>2008 May 07.32</td><td>2.1530</td><td>1.5361</td><td>25.38</td><td>292.88</td><td>9.27</td><td>PROMPT</td></tr><tr><td>2008 May 08.29</td><td>2.1509</td><td>1.5242</td><td>25.25</td><td>293.05</td><td>9.31</td><td>PROMPT</td></tr><tr><td>2008 May 09.30</td><td>2.1487</td><td>1.5118</td><td>25.11</td><td>293.22</td><td>9.36</td><td>PROMPT</td></tr><tr><td>2008 Jun 10.92</td><td>2.0776</td><td>1.1682</td><td>16.68</td><td>294.94</td><td>10.59</td><td>SAAO</td></tr><tr><td>2009 Nov 20.10</td><td>2.4282</td><td>1.9806</td><td>23.26</td><td>133.39</td><td>-6.17</td><td>Pic du Midi</td></tr><tr><td>2009 Nov 25.14</td><td>2.4336</td><td>1.9276</td><td>22.57</td><td>133.93</td><td>-6.23</td><td>Pic du Midi</td></tr><tr><td>2011 Apr 07.07</td><td>2.5479</td><td>1.6394</td><td>11.81</td><td>226.89</td><td>9.72</td><td>Pic du Midi</td></tr><tr><td>2011 Apr 08.10</td><td>2.5462</td><td>1.6310</td><td>11.86</td><td>226.71</td><td>9.79</td><td>Pic du Midi</td></tr><tr><td>2011 Apr 09.06</td><td>2.5446</td><td>1.6334</td><td>11.09</td><td>226.55</td><td>9.85</td><td>Pic du Midi</td></tr><tr><td>2011 Apr 11.10</td><td>2.5414</td><td>1.6080</td><td>10.34</td><td>226.17</td><td>9.97</td><td>Pic du Midi</td></tr><tr><td>2011 Apr 12.08</td><td>2.5399</td><td>1.6009</td><td>9.98</td><td>225.98</td><td>10.03</td><td>Pic du Midi</td></tr><tr><td>2012 Oct 09.04</td><td>1.9847</td><td>1.2466</td><td>24.71</td><td>71.60</td><td>-10.87</td><td>Borowiec</td></tr><tr><td>2012 Oct 18.05</td><td>2.0017</td><td>1.1901</td><td>21.59</td><td>71.60</td><td>-11.69</td><td>Borowiec</td></tr><tr><td>2012 Oct 18.02</td><td>2.0035</td><td>1.1846</td><td>21.21</td><td>71.55</td><td>-11.77</td><td>Borowiec</td></tr><tr><td>2012 Oct 20.07</td><td>2.0056</td><td>1.1787</td><td>20.80</td><td>71.49</td><td>-11.86</td><td>Borowiec</td></tr><tr><td>2012 Nov 11.05</td><td>2.0495</td><td>1.1012</td><td>10.87</td><td>67.95</td><td>-13.31</td><td>Borowiec</td></tr><tr><td>2012 Nov 26.03</td><td>2.0810</td><td>1.1086</td><td>6.36</td><td>63.95</td><td>-13.50</td><td>Borowiec</td></tr></tbody></table>

<footnote n="1">¹ http://www.asu.cas.cz/asteroid/binastdata.htm</footnote>
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 2.** Composite lightcurves of (809) Lundia from 2006/2007 opposition (top) with a corresponding view of the system from Earth (bottom). The line with a circle represents the orbital spin vector with spin direction.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 3.** Composite lightcurves of (809) Lundia from 2008 opposition (top) with a corresponding view of the system from the Earth (bottom). The line with a circle represents the orbital spin vector with spin direction.</figcaption></figure>

with eclipse/occultation events and green without events. Open circles represent future observing geometries and show that in 2018 only there will be a chance to observe eclipses/occultation events.

Observations at the Borowiec Observatory were carried out using the 0.4 m Newton telescope equipped with the KAF402ME CCD camera and a clear filter. The details of the Borowiec system were described by Michałowski et al. (2004). Observations from SAAO (South African Astronomical Observatory) were carried out using the 0.76 m reflector equipped with the University of Cape Town (UCT) CCD camera and an *R* filter.

All CCD frames from Borowiec and SAAO were reduced for bias, dark current and flat-field using the CCLR Starlink package. The aperture photometry was performed to measure the instrumental brightness of the asteroid and comparison stars. Lightcurves observed at Pic du Midi in 2009 were obtained using the 1.05 m Cassegrain telescope equipped with the THX 7863 CCD camera and an *L* filter. Lightcurves from 2011 were taken using the Andor iKon-L CCD camera and an *L* filter. A standard reduction was performed using the Audela software (http://www.audela.org) whose photometry analysis was developed at IMCCE in Paris.

Observations at the Bulgarian National Observatory at Rozhen were carried out using the 0.5/0.7 m Schmidt telescope with the KAF1602E CCD camera and an *R* filter. For the data reduction and aperture photometry, the IDL software was used. Three lightcurves were obtained by PROMPT 4, a 0.41 m Ritchey-Chrétien telescope located at the Cerro Tololo Inter-American Observatory in Chile, equipped with the Alta U47+ camera. The aspect data of Lundia are listed in Table 1.

The obtained composite lightcurves are presented in Figs 2–6. Lightcurves are composed of the synodic period of 15.418 ± 0.001 h. For good comparison between lightcurves, the scale on each graph is the same. In Fig. 2, we present a lightcurve from NAO Rozhen in comparison with already published lightcurves from Pic du Midi. Despite the time span between the lightcurves as much as of four months, the internal fit is very good and it confirms the derived synodic period. Currently, our data set consists of 41 individual lightcurves obtained during 6 oppositions and covering different observing geometries. Details of the observing geometries of the (809) Lundia system for each opposition are given in Table 2.
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 4.** Composite lightcurves of (809) Lundia from 2009 opposition (top) with a corresponding view of the system from the Earth (bottom). The line with a circle represents the orbital spin vector with spin direction.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 5.** Composite lightcurves of (809) Lundia from 2011 opposition (top) with a corresponding view of the system from Earth (bottom). The line with a circle represents the orbital spin vector with spin direction.</figcaption></figure>

## 3 METHOD

We used the genetic-algorithm-based modelling method SAGE that, given solely photometric observations, recreates non-convex shape, spin axis orientation and rotational period of synchronous binary asteroid (Bartczak et al. 2014).

The asteroid system is assumed to be synchronous with a circular orbit about the centre of mass. Each of the bodies is described by 62 vectors with fixed directions; the length of each vector is a free parameter during the modelling process. To create lightcurves of the system, refined and more detailed shapes are used, created using a surface smoothening algorithm (Catmull & Clark 1978). To fully describe the system, there are two additional free parameters: bodies size ratio and separation. During the modelling process, the best fit for the orbital period is searched and then used to establish the mass ratio using the separation, shapes and size ratio as well.

In every step of the modelling process, synthetic lightcurves are compared with the observed ones. To calculate a lightcurve for the specific moment of time, a geometry of the observations (i.e. the positions of the Sun, asteroid and Earth) is reconstructed using position vectors in the heliocentric, ecliptic reference frame based

on an ephemeris of the asteroid. When a 3D scene is constructed, a system is rotated 360° to create a full rotation lightcurve that is later used for comparison. The flux of an asteroid is obtained in a rasterization process: the image is created as if a telescope with an infinite resolution observed an asteroid creating a CCD image. The sum of the pixels gives a synthetic relative photometry measurement. A Lommel-Seeliger scattering law is used and no albedo variations are assumed.

The modelling process uses a genetic algorithm to arrive at the global minimum. The modelling starts with the two spheres, random spin axis orientation and approximate rotational period. In every step, a random changes to the shapes, period, spin axis orientation, separation and size ratio are applied, creating a random population (generation) of systems. Each body of the system is a physical model assuming a homogeneous distribution of mass; the axes of largest inertia of the bodies are aligned with the spin axis of the whole system. Then, for every model of the system in the population, synthetic lightcurves are calculated and compared with observations using the $\chi^2$ test. The model with smallest $\chi^2$ is chosen as the seed for the next population and the whole process repeats until the $\chi^2$ value no longer changes from population to population. In order to
<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /><figcaption>**Figure 6.** Composite lightcurve of (809) Lundia from 2012 opposition (top) with a corresponding view of the system from Earth (bottom). The line with a circle represents the orbital spin vector with spin direction.</figcaption></figure>

ensure the modelling process not to fall into the local minimum, a weighting process is applied in every step. The lightcurve with largest $\chi^2$ is given the largest weight to steer the process towards the global minimum; the weights change in every step.

Additionally, the whole modelling process is run multiple times creating a family of solutions. This is a standard procedure in genetic

evolution algorithms to ensure the result being in the global, rather than the local, minimum. The path leading to a model in each run is different, but the models should be alike in the end. If the models differ significantly, it indicates that not enough observational data have been supplied for the modelling.

## 4 MODEL OF (809) LUNDIA

The Lundia shape model projections can be seen in Fig. 7. Synthetic lightcurves generated by the model fit the observed ones very well (Fig. 8). Mutual eclipse events are perfectly timed and the shape of lightcurves is reproduced on general and detail levels. The uncertainty of photometry was not reported by the observers and therefore it is assumed to be 0.02 mag.

Using the *Spitzer Space Telescope*, Marchis et al. (2012) calculated the effective diameter of Lundia $D_{\text{eff}} = 9.6$ km with $3\sigma$ uncertainty of 1.1 km. Combining this result with inhomogeneous Roche ellipsoids model by Descamps (2010), they obtained equivalent sphere diameters for the primary and secondary components $D_p = 7.2 \pm 1.4$ km and $D_s = 6.4 \pm 1.3$ km, with the system separation $d = 14.6$ km and the average bulk density of the system $\rho = 1.77 \text{ g cm}^{-3}$.

Applying the effective diameter of the whole system $D_{\text{eff}} = 9.6$ km, we scaled the new non-convex Lundia model by assuming the same cross-section of the system at the time of *Spitzer* observations. Mass was determined from the orbital period *P* and the system separation *d* using Kepler's third law.

The parameters of non-convex model of the Lundia system are as follows:

(i) system's spin axis ecliptic coordinates:

$$ \lambda = 122.5 \pm 5^{\circ}, \\ \beta = 22 \pm 5^{\circ}, $$

(ii) sidereal period: $P = 15.41574 \pm 10^{-5}$ h,

(iii) primary equivalent sphere diameter: $D_p = 7.3 \pm 0.8$ km,

(iv) secondary equivalent sphere diameter: $D_s = 6.2 \pm 0.7$ km,

(v) $D_s/D_p = 0.85$,

(vi) system separation: $d = 14.65 \pm 0.8$ km,

(vii) total mass: $(6 \pm 0.4)10^{14}$ kg,

(viii) bulk density: $\rho = 2.5 \pm 0.2 \text{ g cm}^{-3}$ and

(ix) macroporosity: 13–23 per cent.

Spectroscopic observations' analysis (Birlan et al. 2014) indicates similar mineralogical composition as howardite–diogenite meteorites. The obtained density of $2.5 \text{ g cm}^{-3}$ is much higher than that determined before, and this value in comparison with the

<table><caption>**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 opposition dates, the observed eclipsing amplitudes and the references. The value of '0' for the eclipsing amplitudes means that no eclipses were observed for these apparitions (i.e. 2006/2007, 2008 and 2009).</caption><thead><tr><th>Time range</th><th>N<sub>lc</sub></th><th>α (°)</th><th>λ (°)</th><th>β (°)</th><th>Eclipsing amplitude (mag)</th><th>Reference</th></tr></thead><tbody><tr><td>2005 Sep to 2006 Jan</td><td>19</td><td>7.3–25.9</td><td>45</td><td>-11</td><td>0.75–0.30</td><td>Kryszczynska et al. (2005)</td></tr><tr><td>2006 Dec to 2007 Apr</td><td>4 + 1</td><td>17.3–21.4</td><td>168</td><td>2</td><td>0</td><td>Kryszczynska et al. (2005) and this paper</td></tr><tr><td>2008 May to 2008 Jun</td><td>4</td><td>16.7–25.3</td><td>294</td><td>10</td><td>0</td><td>This paper</td></tr><tr><td>Nov 2009</td><td>2</td><td>22.6–23.3</td><td>133</td><td>-6</td><td>0</td><td>This paper</td></tr><tr><td>Apr 2011</td><td>5</td><td>10.0–11.8</td><td>226</td><td>10</td><td>0.58</td><td>This paper</td></tr><tr><td>2012 Oct to 2012 Nov</td><td>6</td><td>6.4–24.7</td><td>68</td><td>-12</td><td>0–0.15</td><td>This paper</td></tr></tbody></table>
<figure><img src="image_14.png" /><figcaption>Figure 7. xz, yz, xy, -xz, -yz and -xy projections of the Lundia model.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Figure 8. Set of observations (black crosses) versus synthetic lightcurves (solid red line) showing the model's fit to the data.</figcaption></figure>
density of HED meteorites of $2.86–3.26 \text{ g cm}^{-3}$ reported by Britt & Consolmagno (2003) and McCausland & Flemming (2006) infers the macroporosity of (809) Lundia of only 13–23 per cent, rather than 40–50 per cent as reported by Marchis et al. (2012).

## 5 CONCLUSIONS

The obtained non-convex model of (809) Lundia perfectly reproduces the obtained set of photometric lightcurves. By scaling volume of the system components, we found density 50 per cent higher than the one calculated in the previous studies and macroporosity of about 60 per cent smaller than before (Kryszczyńska et al. 2009). However, the internal structure of asteroids is presently unclear and newly obtained values may serve as an input to the theories of the Solar system evolution. Higher values of the bulk density are due to the non-convex shape of the system. The more shape deviates from a sphere the larger density value we get.

The shape and size of Lundia could be refined given direct measurements, like stellar occultation events. Predictions for 2017, based on new GAIA DR1 catalogue, yield two events on 2017 April 27 and May 28. Unfortunately, Lundia's small size makes accurate prediction difficult, as uncertainty of the star position on the level of 7 mas translates into 15 km uncertainty in the position on the Earth, which is about the size of the Lundia system. Nevertheless, such observations could put better constraints on asteroid's size and density.

## ACKNOWLEDGEMENTS

The research leading to these results has received funding from the European Union's Horizon 2020 Research and Innovation Programme under grant agreement no. 687378.

This work was partially supported by grant no. 2014/13/D/ST9/01818 from the National Science Centre, Poland.

This paper uses observations made at the South African Astronomical Observatory (SAAO). The reduction of CCD frames

from Borowiec and SAAO were performed with the CCLR Starlink package.

GA gratefully acknowledges observing grant support from the Institute of Astronomy and Rozhen National Astronomical Observatory, Bulgarian Academy of Sciences.

This work is based in part on observations made with the *Spitzer Space Telescope*, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA.

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This paper has been typeset from a *TEX/LATEX* file prepared by the author.