Photometric analysis for asteroid (346) Hermentaria: comparison of inverse methods
Authors: Wang Xiaobin , Muinonen Karri , Cellino Alberto 
Published: October 2020

Abstract&#160; We apply the statistical lightcurve inversion method developed&#160; by Muinonen et al. (2020, convex inversion in magnitude space CIM) to re-analyze the photometric data of&#160; (346) Hermentaria. As a main goal, we compare the results derived by the CIM and original convex inversion method in flux space (CIF) provided by Kaasalainen &amp; Torppa (2001) and Kaasalainen et al. (2001). The comparison concerns the solutions of spin and shape parameters.1. Introduction&#160;&#160; The idea to invert 3-D shapes of asteroids from their brightness variations can be traced to 1906 (Russell 1906). Now, several methods have been established to invert the shape of asteroids, the shape models range from triaxial ellipsoids to convex, and even non-convex shapes. Kaasalainen et al. (1992a,b) and Lamberg (1993) suggested a way to invert the convex shape of an asteroid by using the Gaussian surface density. Kaasalainen &amp; Torppa (2001) and Kaasalainen et al. (2001) implemented the idea by the Levenberg-Marquardt algorithm.&#160;&#160; Now, statistical techniques have been introduced into the photometry inversions of asteroids, e. g., a genetic algorithm (Cellino et al. 2015) and a Markov-chain Monte Carlo approach (MCMC, Muinonen et al. 2015).&#160; Wang et al. (2015b,a) presented a virtual observation method to figure out the uncertainties of parameters due to the observational uncertainties. Muinonen et al. (2020) developed&#160; a full statistical MCMC algorithm in the convex inversion of asteroids in the magnitude space (CIM), and gives a way to assess the uncertainties of spin parameters and convex shape parameters.&#160;&#160; We re-analyse the photometric data of (346) Hermentaria with CIM and CIF algorithms, and show the&#160; main differences of the results in the derivation of the convex shapes.2. Convex inversion with Bayesian inference&#160; The brightness model used in the CIM (Muninonen et a l. 2020)) involves a convex shape represented with the Gaussian surface density and the Lommel-Seeliger scattering law. In the CIM, the free parameters are the spin parameters, shape parameters, geometric albedo, and two parameters of the&#160;H,G1,G2 phase function. As for constructing the convex shape from its Gaussian surface density, Muinonen et al. (2020) developed a computationally easier, stochastic optimization metod.3. Results and discussion&#160; In total, 23 lightcurves of (346) Hermentaria on 4 apparitions are involved. 10 light curves are from the APC database (Lagerkvist et al. 1993) and rest of the lightcurves are ours observations. Based on these data, we performed the shape inversion with the CIM and CIF methods. As only relative intensities are involved, the parameters of the H G1 G2&#160;phase function in the CIM and the scattering parameters a, k,d and c in the CIF are not fitted. Table 1 list the derived spin parameters. Figure 1 shows the lightcurves fits for the case of the pole 2 solution.Table 1. Spin parameters of HermentariaMethod

Pole 1
(Ecliptic frame of J2000.0 )

Pole 2
(Ecliptic frame of J2000.0 )

Peroid (h)
&#160; Pole 1,&#160;&#160;&#160;&#160;&#160; Pole 2 &#160;&#160;&#160;&#160;&#160;&#160;

CIF

(133o.0,+17o.2)

(319o.5,+16o.8)

17.790043, 17.790039
CIM

(133o.8,+09o.6)

(321o.9,+17o.2)

17.790016, 17.790097
&#160;Figure 1. Lightcurves of Hermentaria (black circles) with the CIM and CIF model intensities ( red crosses and blue pluses respectively).&#160;&#160; The CIM gives a consistent result on spin parameters. From Figure 1, the modeled lightcurves with the CIF fit well to the features. We think this is partly due to the fact that we use a lower degree of spherical harmonics in the CIM and partly due to the differing model for the observational uncertainties.&#160;&#160;&#160; The theory by Minkowski (1903) provides a way to construct the three-dimensional shape of an object from its Gaussian surface density G.&#160; According to Minkowski, Muinonen et al.(2020) developed a stochastic optimization method to construct convex shape. Figure 2 shows the convex shapes of Hermentaria derived by the two methods.Figure 2. Convex shapes of Hermentaria corresponding to Pole 2 solution: the CIM (left) and CIF solutions (right)&#160;&#160; We have also compared the 3-D shapes constructed by the CIM and the CIF using the same Gaussian surface density originally derived by the CIM. Figure 3 shows the CIM and CIF results and by visual inspection, the results are very close to one another.Figure 3 &#160;Convex shapes of Hermentaria corresponding to the same Gaussian surface density:the CIF (left) and CIM reconstructions (right).4. Summary&#160; We applied the statistical convex inversion method (Muninonen et al. 2020) to analyze the photometric data of&#160; Hermentaria. The best-fit solution of the spin parameters and shape as well as the uncertainties of the estimated parameters can be derived.&#160; We compared the results to that of the CIF and found that the spin parameters are consistent with one another. The Gaussian surface density derived by the&#160; two methods is slightly different which leads to slightly different convex shapes.&#160; For validating the new shape reconstruction method in Muinonen et al. (2020), we compared the shape derived by the methods of Kaasalainen tel al.(2001) and Muinonen et al. (2020) from the same Gaussian surface density, and found them to be similar.ReferencesCellino, A., Hestroffer, D., Tanga, P., Mottola, S., &amp; Dell&#8217;Oro, A. 2009, A&amp;A, 506, 935Cellino, A., Muinonen, K., Hestroffer, D., &amp; Carbognani, A. 2015, PSS, 118, 221Kaasalainen, M., Lamberg, L., &amp; Lumme, K. 1992a, A&amp;A, 259, 333Kaasalainen, M., Lamberg, L., Lumme, K., &amp; Bowell, E. 1992b, A&amp;A, 259, 318Kaasalainen, M. &amp; Torppa, J. 2001a, Icarus, 153, 24Kaasalainen, M., Torppa, J., &amp; Muinonen, K. 2001b, Icarus, 153, 37Lagerkvist, C.-I., Erikson, A., Debehogne, H., et al. 1995, A&amp;A Supplement&#160;Series, 113, 115Lamberg, L. 1993, Ph.D. thesis, University of Helsinki, Finland, vol. 315LLumme, K. &amp; Bowell, E. 1981, Astron J, 86, 1694&#160;Minkowski, H. 1903, Mathematische Annalen, 57, 447Muinonen, K. &amp; Lumme, K. 2015a, A&amp;A, 584, A23Muinonen, K., Wilkman, O., Cellino, A., Wang, X., &amp; Wang, Y. 2015b, PSS, 118,&#160;227Muinonen ,K., Torppa, J., Wang, X.-B., Cellino,A., Penttil&#228; , A., 2020, submited to A&amp;A, in revisedRussell, H. N. 1906, Astrophys J, 24, 1Wang, X., Muinonen, K., &amp; Wang, Y. 2015a, PSS, 118, 242Wang, X., Muinonen, K., Wang, Y., et al. 2015b, A&amp;A, 581, A55