Research in Astron. Astrophys. 202 Vol. X No. XX, 000 000 http://www.raa-journal.org http://www.iop.org/journals/raa Research in Astronomy and Astrophysics Lightcurve inversion of asteroid (585) Bilkis with Lommel-Seeliger ellipsoid method Ao Wang,2,3, Xiao-Bin Wang,2, Karri Muinonen 4,5, Xianming L. Han 6, Yi-Bo Wang,2,3 Yunnan Observatories, Chinese Academy of Sciences, Kunming 6500, China; wangao@ynao.ac.cn 2 Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Scineces, Kunming 6500, China 3 Graduate University of Chinese Academy of Sciences, Beijing 00049, China 4 Department of Physics, University of Helsinki, Gustaf Hällströmin katu 2a, P.O. Box 64, FI-0004 U. Helsinki, Finland 5 Finnish Geospatial Research Institute, Geodeetinrinne 2, FI-02430 Masala, Finland 6 Department of Physics and Astronomy, Butler University, Indianapolis, IN 46208, USA Received 206 March 29; accepted 206 July 8 Abstract The basic physical parameters of asteroids, such as spin parameters, shape and scattering parameters, can provide us information on formation and evolution of both asteroids themselves and the entire solar system. In a majority of asteroids, the diskintegrated photometry measurement constitutes the primary source of the above knowledge. In the present paper, newly observed photometric data and existed data of (585) Bilkis are analysed based on a Lommel-Seeliger ellipsoid model. With a Markov Chain Monte Carlo(MCMC) method, we have determined its spin parameters (period, pole orientation) and shape ( b/a, c/a) of (585) Bilkis and their uncertainties. As a result, we obtained a rotational period of 8.5738209 h with an uncertainty of 0.0000009 h, and derived a pole of (36.46, 29.0 ) in ecliptic frame of J2000.0 with uncertainties of 0.67 and in longitude and latitude respectively. We also derived triaxial ratiosb/a andc/a of (585)Bilkis as 36 and 0 with uncertainties of 0.003 and 0.03. Key words: asteroids photometric observation spin parameter shape MCMC method INTRODUCTION Asteroids are thought to be remnants of planetesimals in the early stage of our solar system. It is very meaningful to study their basic physical properties/parameters, such as spin parameters, shape and scattering parameters, to find a clue to formation and evolution of both asteroids themselves and the entire solar system. The brightness of asteroid is due to the reflection of the solar light. A disk-resolved image of an asteroid can tell us information on its size, shape and ability of reflection. In practice, the disk-integrated intensities are obtained for most asteroids due to small size of asteroids and far distance from observers. The brightness of asteroid at a certain time is relate to the distances of asteroid from the source of light and observer, its size, shape and reflection ability of its surface. It is also relate to the Supported by the National Natural Science Foundation of China.
2 A. Wang et al. pose of asteroid in space if the asteroid is not spherical shape. In theory, above physical parameters can be inferred from lightcurves of an asteroid distributed at different viewing/illumination geometry. That is the reason why disk-integrated photometry data is and will remain a major source for most asteroids to understand their physical properties. The C-type main-belt asteroids are thought to be primitive small bodies of solar system, keep more information of the early stage of formation and evolution of the solar system (Wang et al. 205). We therefore choose asteroid (585) Bilkis as our target. Asteroid (585), a C-type main-belt asteroid of about 58 km size, was discovered by A. Kopff in 906 February. For this asteroid, several group have made photometric observations (Robinson & Warner 2002; Behrend 2005; Brincat 202; Behrend 202). Robinson & Warner (2002) obtained photometric data of Bilkis on 6 nights from May to August in 200 and gave a rotation-period of 6.442h. Behrend (202) suggested a period of 8.58h in their website. Brincat (202) also suggested a period of 8.583h. Warner (20) re-analysed previous lightcurves of (585) Bilkis and derived a spin period of 8.574 h. Obviously, there are slight differences among the spin periods derived from previous studies. Further more, no information on its pole-orientation and shape have been derived yet till now. To determine spin parameters and shape of (585) Bilkis accurately, new photometric observations are made in 7 nights in 202 and 204 with the.0 m telescope at Yunnan observatories of China and SARA s m telescope at Cerro Tololo Inter-American Observatory in Chile. In the present paper, based on the new observed and previously existing photometric data, we analyzed the spin parameters and shape of Bilkis considering a three-axial ellipsoid shape with Lommel- Seeliger scattering law (Muinonen et al. 205). The best values of parameters and their uncertainties are derived with the Markov chain Monte Carlo method developed by Muinonen et al. (202). In Section 2, the observations and data reduction for the target are briefly introduced. The lightcurve inversion method is described in Section 3. The analysis results for (585) Bilkis are presented in Section 4. In the last Section, a summary and future prospects close the present article. 2 OBSERVATION AND DATA REDUCTION For (585) Bilkis, 7 new lightcurves are obtained in 202 and 204. Among the new derived lightcurves, five were obtained by the.0m telescope at Yunnan Observatories in 202 and 204 and the rest two are obtained by the m SARA s telescope at Cerro Tololo Inter-American Observatory, Chile in 204. Additionally, 0 existing lightcurves (from the Light Curve Database of the Minor Planet Center web page ) are involved in our analysis. The detail information on data involved are listed in Table. The first column is the date of the observations in UT. The second and third columns present right ascension and declination of asteriod in J2000.0. and r are the geocentric and heliocentric distances of asteroid. ph means the phase angle of the object and V is the predicted visual magnitude. The new photometric data obtained at the.0m telescope at Yunnan Observatories is by 2k 2k CCD with a clear filter in 202 and 204 while the data from m SARA s telescope is by 2 2 bin mode of a4k 4k CCD with a R filter. The new images are reduced according to the standard procedure with IRAF 2 software. For all images, the effects of bias, flat and dark are corrected. The cosmic rays hit images are removed properly. The magnitude of objects are measured by apphot task of IRAF with an optimal aperture. Then, some systematic effects in the photometric data are simulated with reference stars in the field of asteroid and then removed from the target s photometric data (Wang et al. 203). The time stamps of involved photometric data were corrected by the light time and converted into JD in TDB system. The lightcurves of (585) Bilkis involved here are shown in Figure. 3 INVERSION METHOD As we know, the disk-integrated brightness of an asteroid at any time is the sum of reflected solar light by the visible surface of asteroid. The observed brightness of an asteroid is related to the ability http://www.minorplanetcenter.net 2 IRAF means Image Reduction and Analysis Facility package
Lightcurve Inversion using MCMC Method 3 02-02-202 0 0.02 0.04 0.06 0.08 0. 0.2 MJD(55959.80+) 03-02-202 0 0.05 0. 0.5 0.2 0.25 MJD(55960.67+) 09-03-202 0 0.05 0. 0.5 0.2 0.25 MJD(55995.49+) 0-03-202 0 0.05 0. 0.5 0.2 0.25 0.3 MJD(55996.49+) 04--204 0 0.02 0.04 0.06 0.08 0. 0.2 0.4 0.6 MJD(56965.6+) 08--204(SARA) 0 0.05 0. 0.5 0.2 0.25 0.3 0.35 MJD(56969.00+) 08--204 0 0.05 0. 0.5 0.2 0.25 0.3 0.35 MJD(56969.50+) 4-05-200 0 0.0 0.02 0.03 0.04 0.05 0.06 MJD(52043.3+) 5-05-200 0 0.05 0. 0.5 0.2 0.25 MJD(52044.3+) 23-06-200 0 0.0 0.02 0.03 0.04 0.05 0.06 0.07 0.08 MJD(52083.2+) 25-06-200 0 0.0 0.02 0.03 0.04 0.05 0.06 0.07 0.08 MJD(52085.2+) 09-07-200 0 0.0 0.02 0.03 0.04 0.05 0.06 0.07 MJD(52099.+) 5-07-200 0 0.0 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 MJD(5205.2+) 06.2-0-204 0 0.02 0.04 0.06 0.08 0. 0.2 0.4 0.6 MJD(56936.9+) 06.4-0-204 0 0.02 0.04 0.06 0.08 0. 0.2 0.4 0.6 MJD(56936.36+) 0 0.02 0.04 0.06 0.08 0. 0.2 0.4 0.6 MJD(56938.35+) 08.3-0-204 0 0.0 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0. MJD(56938.26+) 08.4-0-204 Fig. Lightcurves of (585)Bilkis. Scatter dots are observed data, lines are modeled ones using Lommel-Seeliger ellipsoid model.
4 A. Wang et al. Table The observation data of asteroid (585)Bilkis Date R.A. Decl. r Ph.( ) V Telescope Filter (UT) (J2000.0) (J2000.0) (AU) (AU) 200/05/4.4 3 49 2.3-02 49 53.347 2.288 2.0 3.6 0.25m, MPC C 200/05/5.4 3 48 49.0-02 45 44.353 2.290 2.4 3.6 0.25m, MPC C 200/06/23.4 3 47 26.7-02 49 52.739 2.344 23.3 4.6 0.25m, MPC C 200/06/25.4 3 48 24. -02 57 36.763 2.346 23.6 4.6 0.25m, MPC C 200/07/09.2 3 57 24.6-04 05 27.94 2.366 24.9 4.9 0.25m, MPC C 200/07/5.4 4 02 25.8-04 40 5 2.020 2.375 25. 5.0 0.25m, MPC C 202/02/02.8 09 52 +03 4 50 62 2.22 8.2 2.9.0m, YNAO C 202/02/03.67 09 5 25.9 +03 48 2 59 2.22 7.8 2.9.0m, YNAO C 202/03/09.50 09 25 3.7 +08 49 7 96 2.20 3. 3.2.0m, YNAO C 202/03/0.50 09 24 46.9 +08 57 36 0 2.20 3.6 3.2.0m, YNAO C 204/0/06.9 02 03 30.3 +07 7 40.65 2.58 7.3 4.0 0.35m, MPC C 204/0/06.36 02 03 28.4 +07 7 20.65 2.58 7.3 4.0 0.35m, MPC C 204/0/08.27 02 0 59.5 +07 02.606 2.578 6.4 4.0 0.35m, MPC C 204/0/08.36 02 0 55.5 +07 0 32.605 2.578 6.4 4.0 0.35m, MPC C 204//04.6 0 38 58.4 +03 37 03.584 2.546 7.0 3.9 m, SARA R 204//08.5 0 35 28.4 +03 09 04.600 2.540 8.9 4.0.0m, YNAO C 204//08.00 0 35 5.8 +03 2 4.598 2.54 8.7 4.0 m, SARA R Notes: We select the middle time of each lightcurves to give the position of asteroid of reflection of surface and the size of surface area both illuminated and visible. Theoretically, we can extract information on spin state, shape and surface property of asteroid from the observed light curves. Such a procedure is usually called the lightcurve inversion. The lightcurve inversion method used here is the Lommel-Seeliger ellipsoid method developed by Muinonen et al. (205). With this method, the rotation period, longitude and latitude of pole, semi major axis of ellipsoid and scattering parameters can be extracted from photometric data. In the present work, only spin parameters, axial ratios of ellipsoid are estimated because only relative intensities of (585) Bilkis are involved. The relative intensities of Bilkis are calculated from the magnitude differences/or reduced magnitudes. Then, they are normalized according the mean value of each lightcurve. Photometric data in different filter is processed with following equation. I rel = 0 ( (dm dm)/2.5) () where dm means magnitude differences and dm is the mean of lightcurves in each night. The modeled disk-integrated brightness of an asteroid at any time in the Lommel-Seeliger ellipsoid method(muinonen et al. 205), is calculated with Equation 2 L(α) = πpf 0 Φ HGG 2 (α) Φ LS (α) abc S S {cos(λ α )+cosλ + S sinλ sin(λ α )ln[cot 2 λ cot 2 (α λ )]}, (2) Φ HGG 2 (α) = G ( 6α π )+G 2( 9α π )+( G G 2 )exp( 4πtan 2 3 α), (3) 2 Φ LS (α) = sin 2 αtan 2 αln(cot 4 α). (4) In this formula, F 0 is incident intensity by the Sun, α is the solar phase angle of asteroid, p is the geometric albedo of an asteroid.φ HGG 2 (α) andφ LS (α) are the new adopt phase function and the Lommel-Seeliger phase function respectively.s,s are radii of sub-sun and sub-earth on the asteroid surface, α, λ and S are auxiliary quantities computed from vectors of light source and observer in the asteroid fixed frame (refer to equation and 2 of Muinonen et al. 205). a,b,c are three semi
Lightcurve Inversion using MCMC Method 5 major axis of an ellipsoid. H is absolute magnitude of asteroid and G, G 2 are the parameters of phase function. More information about the disk-integrated brightness of the Lommel-Seeliger ellipsoid can refer to papers ( Muinonen et al. 205; Muinonen & Lumme 205). In detail, the unknown vector( represented by P = (per,λ p,β p,φ 0,a,b,c,p,G,G 2,D) T (T is transpose)) compose of elements: the rotation period per, ecliptic longitude of pole λ p, ecliptic latitude of pole β p, rotational phase φ 0 at t 0, size of three semi major axis of ellipsoid a,b,c(a b c), geometric albedo p, parameters of phase function H,G,G 2, and equivalent diameter of asteroidd. In principle, we can infer these unknown parameters by comparing the observed intensities with the modeled ones. Because only relative intensities of (585) Bilkis are involved in the present work, only spin parameters (period and orientation of pole) and ellipsoid shape of (585) Bilkis (taken a=) are analysed in the present work. The whole analysis procedure contains two parts. Firstly, in order to find the global minimum RMS of observations, the rotation period and pole of (585) Bilkis are searched by scanning wide ranges. During the scanning for period and pole, a flexible Nelder-Mead downhill method is applied to test the different initial value of period and pole. Secondary, a MCMC method is applied to find the best solution for the multiple unknown parameters and their uncertainties. During the simulation of MCMC, the a posteriori probability density function(p.d.f.) p p corresponding to the unknown parameter P is derived according to the Bayesian inference. p p p pr (P)p ǫ ( L(P)) L(P) = L obs L(P) wherep pr, the priori probability density function, is assumed to be constant.p ǫ, observational errors p.d.f., is calculated by the residuals (O C). The final posteriori p.d.f. is (5) p p (P) exp[ 2 χ2 (P)] χ 2 (P) = L(P) T L(P) where means the covariance matrix of the errors. The RMS is (6) RMS = σ χ 2 (P)/N obs (7) where σ is observation error and N obs is the number of observations. The sampling of MCMC simulation is carried out using the Metropolis-Hastings algorithm. To derived a regular Gaussian shape of distribution of parameters efficiently, a proposal p.d.f of parameters is generated by the virtual observation Monte Carlo method. In detail, the virtual observations are generated by adding Gaussian noise into the original photometric data. Then, the corresponding virtual least-square solutions of parameters are derived with the optimization method Nelder-Mead downhill simplex method. 4 RESULTS 4. Downhill simplex solution To find the most possible value of period, we scan the period from 4 h to 20 h with a resolution of per 2 /2 T ( T denotes the time span of photometric data used) and find that the most possible period is near 8.574 h. (see Figure 2) Then, a range between 8.55 h and 8.6 h is scanned with a high resolution of per 2 /0 T. Finally the most significant value of period is found around 8.5738 h (see Figure 3). This value is very close to Warner s results (Warner 20). Every period scanning step is carried out roughly for hundreds of different pole orientations distributed as uniformly as possible on the unit sphere. For each rotation period, the pole orientation, rotational phase and axial ratios are optimised with the Nelder-Mead downhill simplex method. Optionally, the scattering parameters can also be optimized if the photometric data have been converted to absolute magnitude system. With the optimal period obtained, the most possible pole(λ p,β p ) is scanned systematically over the full solid angle with a high resolution about a few degrees on the unit sphere. Figure 4
6 A. Wang et al. 0.07 0.065 0.06 0.055 RMS 0.05 0.045 0.04 0.035 P=8.574 h 0.03 4 6 8 0 2 4 6 8 20 Period (hour) Fig. 2 Period distribution(4h to 20h) of (585) Bilkis vs. RMS of the fits. RMS 0.065 0.06 0.055 0.05 0.045 0.04 0.035 0.03 0.025 P = 8.5738 h 0.02 8.55 8.56 8.57 8.58 8.59 8.6 Period (hour) Fig. 3 Period distribution(8.55h to 8.60h) of (585) Bilkis vs. RMS of the fits. shows the tested poles. It is worth noting that the tested poles with the same pole latitude and 80 degree differences in pole longitude will have similar effect on disk-integrated brightness. By comparison, we find a pole of (35, 36 ) (white snowflake in Figure 4) gives a significant minimum RMS of observation among those tested poles. Taking the scanned spin parameters as initial values, unknown parameters are re-estimated with the Nelder-Mead downhill simplex method, and a pole of (35, 36 ) with a spin period of 8.5738 h is derived and an ellipsoid shape of axial ratios b/a=5 and c/a=0.64 are obtained as well.
Lightcurve Inversion using MCMC Method 7 90 0.065 Pole Latitude (degree) 60 30 0 30 60 * 0.06 0.055 0.05 0.045 0.04 0.035 0.03 rms 90 0 50 00 50 200 250 300 350 Pole Longitude (degree) 0.025 Fig. 4 Tested poles (λ p,β p ), the white cross is corresponding to the significant minimum of RMS. 4.2 Solution derived from the MCMC method. In order to understand the uncertainties of estimated parameters, a MCMC simulation is carried out for photometric data of (585)Bilkis. In detail, proposal distributions of parameters are derived with the virtual observation Monte Carlo method, then a random-walk Metropolis-Hastings algorithm is running based on those proposal distributions. The proposal distributions of parameters are composed of the least-square solutions of virtual observations generated by adding Gaussian noise into the original observations. The standard deviation of Gaussian added in one night s observation is the scatter of lightcurve compared to certain a modeled. Such a proposal distribution actually suggests us high probability regions in the parameters space corresponding to the observations, therefore it speeds the MCMC sampling procedure. Usually, 0000 virtual least-square solutions are contained in the proposal distributions. The detail information on the MCMC sampling can refer to Muinonen s work (Muinonen et al. 202) in asteroid orbital inversion. Finally, a joint distribution of parameters is derived with the MCMC simulation, in which more than 20000 samplers are contained. The best values and uncertainties of model parameters are estimated from the statistical feature of the marginal posterior distribution of individual parameter (see Figure 5). we ve obtained the peak value of Gaussian fitting for the distribution as the best value of corresponding parameter, and taken σ limit as the uncertainty of estimated parameter. With the method mentioned above, the best period of 8.5738209 h and its uncertainty of 0.0000009 h are derived. In the same way, the best values and corresponding uncertainties of pole orientation and three-axes ellipsoid parameters (b/a and c/a) are derived. We derive the most likely pole of (36.46, 29.0 ), the uncertainties of longitude and latitude are 0.67 and, respectively. For the shape of (585) Bilkis, axial ratiosb/a and c/a are 36 and 0 with uncertainties of 0.003 and 0.03. Easy to see clearly, the uncertainty of pole latitude is larger than that of pole longitude and the uncertainty of axial ratio c/a is significantly larger than that of b/a. It is because the latitude and the shape parameter c/a are coupled together to some extent in our model. 5 SUMMARY Using the Lommel-Seeliger ellipsoid model and the MCMC method, we derived the spin parameters (rotation period, pole orientation) and the shape of (585) Bilkis from new photometric data and existed data. Actually, such an analysis procedure will be applied in near earth asteroids (NEAs), especial potentially Hazardous Asteroids(PHAs) for fewer parameters involved compared to the convex inversion method. It is useful to analyse physical properties of those new discovered NEAs with the Lommel- Seeliger ellipsoid method, because of limit photometric data for the new discovered NEAs.
8 A. Wang et al. ' & &'(&& &'(& &'(& &'(& & '()* & ' ( ) &'() *()+)), & & & ' ( ) & &' (&')*''+ '( '( '( '( '( '( &' &( &' &' &' &'' &'( &) &) &) Fig. 5 Histogram distributions of spin period, pole orientation and axial ratios. The derived period of 8.5738209 h is consistent with previous results. For the first time, its pole and ellipsoid shape are determined. The ecliptic longitude of pole is 36.46 with a standard deviation of 0.67 and the ecliptic latitude is 29.0 with a standard deviation of. In the case of shape, the relative triaxial dimensions are b/a = 36, c/a = 0 with standard deviations of 0.003 and 0.03. At present, it is still a problem to decouple well the effects of the ecliptic latitude and the shape parameter c/a on the disk-integrated brightness. Anyway, we can estimate the spin parameters and shape in reasonable errors with relative small photometric data. In the near future, we ll have better
Lightcurve Inversion using MCMC Method 9 restrictions on the spin and shape parameters by the convex inversion method when more photometric observations and/or occultation data of (585)Bilkis are obtained. Acknowledgements This work was funded by the National Natural Science Foundation of China (contacts Nos. 07305 and 473066), and supported,in part, by the Academy of Finland (Project257966). We d like to thank assistants of the.0m telescope group of Yunnan Observatories. We appreciate help from Stephens,R.D. and Warner, B.D. for providing their photometric data of (585) Bilkis and help us to reduce existing data. References Behrend, R. 2005, Observatoire de Geneve web site, http://obswww.unige.ch/ behrend/ page_cou.html 2. 202, Observatoire de Geneve web site, http://obswww.unige.ch/ behrend/page_ cou.html 2 Brincat, S. 202, Posting on CALL site,http://www.minorplanet.info/call.html 2 Muinonen, K., Granvik, M., Oszkiewicz, D., Pieniluoma, T., & Pentikäinen, H. 202, Planet. Space Sci., 73, 5 2, 7 Muinonen, K., & Lumme, K. 205, A&A, 584, A23 5 Muinonen, K., Wilkman, O., Cellino, A., Wang, X., & Wang, Y. 205, Planet. Space Sci., 8, 227 2, 4, 5 Robinson, L. E., & Warner, B. D. 2002, Minor Planet Bulletin, 29, 6 2 Wang, X.-B., Gu, S.-H., Collier Cameron, A., et al. 203, Research in Astronomy and Astrophysics, 3, 593 2 Wang, X., Muinonen, K., Wang, Y., et al. 205, A&A, 58, A55 2 Warner, B. D. 20, Minor Planet Bulletin, 38, 52 2, 5