Meteoritics & Planetary Science 44, Nr 12, 1937–1946 (2009) Abstract available online at http://meteoritics.org Asteroid photometric and polarimetric phase curves: Joint linear-exponential modeling K. MUINONEN1, 2, A. PENTTILÄ1, A. CELLINO3, I. N. BELSKAYA4, M. DELBÒ5, A. C. LEVASSEUR-REGOURD6, and E. F. TEDESCO7 1University of Helsinki, Observatory, Kopernikuksentie 1, P.O. BOX 14, FI-00014 U. Helsinki, Finland 2Finnish Geodetic Institute, Geodeetinrinne 2, P.O. Box 15, FI-02431 Masala, Finland 3INAF-Osservatorio Astronomico di Torino, strada Osservatorio 20, 10025 Pino Torinese, Italy 4Astronomical Institute of Kharkiv National University, 35 Sumska Street, 61035 Kharkiv, Ukraine 5IUMR 6202 Laboratoire Cassiopée, Observatoire de la Côte d’Azur, BP 4229, 06304 Nice, Cedex 4, France 6UPMC Univ. Paris 06, UMR 7620, BP3, 91371 Verrières, France 7Planetary Science Institute, 1700 E. Ft. Lowell Road, Tucson, Arizona 85719, USA *Corresponding author. E-mail: [email protected] (Received 01 April, 2009; revision accepted 18 August 2009) Abstract–We present Markov-Chain Monte-Carlo methods (MCMC) for the derivation of empirical model parameters for photometric and polarimetric phase curves of asteroids. Here we model the two phase curves jointly at phase angles գ25° using a linear-exponential model, accounting for the opposition effect in disk-integrated brightness and the negative branch in the degree of linear polarization. We apply the MCMC methods to V-band phase curves of asteroids 419 Aurelia (taxonomic class F), 24 Themis (C), 1 Ceres (G), 20 Massalia (S), 55 Pandora (M), and 64 Angelina (E). We show that the photometric and polarimetric phase curves can be described using a common nonlinear parameter for the angular widths of the opposition effect and negative-polarization branch, thus supporting the hypothesis of common physical mechanisms being responsible for the phenomena. Furthermore, incorporating polarimetric observations removes the indeterminacy of the opposition effect for 1 Ceres. We unveil a trend in the interrelation between the enhancement factor of the opposition effect and the angular width: the enhancement factor decreases with decreasing angular width. The minimum polarization and the polarimetric slope at the inversion angle show systematic trends when plotted against the angular width and the normalized photometric slope parameter. Our new approach allows improved analyses of possible similarities and differences among asteroidal surfaces. INTRODUCTION (Harris et al. 1989b; Rosenbush 2009), whereas in some cases negative polarization, in particular, extends over a wide range Two ubiquitous phenomena are observed for asteroids in phase angle (Cellino et al. 2006). and other atmosphereless solar system objects as well as for Coherent backscattering, single scattering, and cometary and interplanetary dust near opposition: a negative shadowing have been considered as mechanisms responsible branch in the degree of linear polarization (“negative for the phenomena (see review in Muinonen et al. 2002a). polarization”) and a nonlinear enhancement of brightness Coherent backscattering is an interference mechanism that (opposition effect). Negative polarization refers to the case can contribute to both brightness and polarization (Hapke where the scattered intensity component parallel to the Sun- 1990; Muinonen 1989, 1990; Shkuratov 1985, 1988, 1989; object-observer plane (scattering plane) predominates over Mishchenko and Dlugach 1993), single-scattering the one perpendicular to the plane. The negative-polarization interference effects can contribute to both phenomena over a and opposition-effect phenomena are constrained to Sun- wider range of phase angles (e.g., Muinonen et al. 2007; object-observer angles (phase angles) of գ25° and գ10°, Tyynelä et al. 2007) whereas shadowing is thought to respectively. In some cases, the phenomena show up at contribute to the opposition effect only (see Muinonen et al. extremely small phase angles within a degree from opposition 2002a). 1937 © The Meteoritical Society, 2009. Printed in USA. 1938 K. Muinonen et al. We analyze the photometric and polarimetric phase photometric phase curves must be constrained to positive curves jointly by using an empirical model developed to fit values at all phase angles. Nevertheless, the linear- the observations within the observational errors given. exponential model is suitable for the present study at small Empirical models are useful in planning further observations phase angles. of the objects and in grouping the objects based on the In order to distinguish between the photometric and similarities and differences in their phase curves. Note the polarimetric parameters, subscripts I (for intensity) and P (for important role played by the H, G magnitude system (Bowell polarization) are attached to the symbols a, b, k, and d. For et al. 1989) for asteroid research at large, and the ongoing photometry, the empirical parameters are the amplitude aI and efforts to improve the system (Muinonen et al. 2008). angular width dI of the opposition effect, the background The present study emerges from the earlier ones for brightness bI, and the slope kI. By normalization at zero phase estimating the parameters of the brightness opposition effect angle, the number of parameters decreases to three. For (e.g., Bowell et al. 1989; Lumme et al. 1993; Belskaya and polarimetry, the empirical parameters are the amplitude aP Shevchenko 2000; Muinonen et al. 2002b; Rosenbush et al. and angular width dP of the negative-polarization branch, the 2002; Kaasalainen et al. 2003; Avramchuk et al. 2007) and of balancing amplitude bP, and the slope kP. Based on the physics = the negative polarization (e.g., Muinonen et al. 2002b; of light scattering, we assume bP aP so that the degree of Rosenbush et al. 2002; Kaasalainen et al. 2003; Lumme and linear polarization is zero at zero phase angle. For the joint = = Muinonen 2003; Levasseur-Regourd 2003, 2004). We seek linear-exponential modeling, we assume d dI dP so that d answers to the question whether it is possible to explain the is the single common nonlinear parameter in the photometric and polarimetric phase curves with a joint interpretation of the photometric and polarimetric phase empirical model involving common parameters for curves. photometry and polarimetry. Here we present the results of There are additional dependent parameters that are of studies at phase angles գ25 °. We provide practical Markov- interest in comparative studies. For the opposition effect, such Chain Monte-Carlo methods (MCMC) for obtaining reliable parameters are, e.g., the enhancement factor ζ and the angular error estimates for nonlinear model parameters. Note that half-width at half-maximum d1 , ---I Penttilä et al. (2005) made use of MCMC methods in their 2 a + b statistical analyses of asteroidal and cometary polarization ζ = ---------------I I , phase curves using the trigonometric (Lumme and Muinonen bI 2003) and polynomial models (Levasseur-Regourd 2004). = In the Theoretical and Numerical Methods section, we d1 dI ln 2. (2) ---I summarize the linear-exponential model for the photometric 2 and polarimetric applications and describe the MCMC For the negative polarization branch, the phase angle methods for sampling the parameters. Section 3 includes the αmin and value of minimum polarization Pmin are of special application of the methods to the V-band photometric and interest: polarimetric phase curves of asteroids 419 Aurelia k d (taxonomic class F), 24 Themis (C), 1 Ceres (G), 20 Massalia α ⎛⎞P P min = –dP ln ----------- , (S), 55 Pandora (M), and 64 Angelina (E). We close the paper ⎝⎠aP with conclusions and future prospects. ⎛⎞kPdP THEORETICAL AND NUMERICAL METHODS Pmin = kPdP 1 – ln ----------- + bP .(3) ⎝⎠aP Linear-Exponential Modeling The inversion angle of polarization α0 needs to be solved from an implicit equation (using, e.g., Newton’s method): We start with the four-parameter empirical linear- exponential model for the photometric and polarimetric phase α a exp⎛⎞–-----0- ++b k α = 0. (4) curves close to the opposition (Muinonen et al. 2002b; P ⎝⎠d P P 0 Kaasalainen et al. 2003): P ′ α The corresponding first derivative P0 gives the slope of f (α) = a exp ⎛⎞–--- + b + kα, (1) ⎝⎠d the polarization curve at the inversion angle which is known to correlate with the geometric albedo and is used to derive where α is the phase angle, a, b, and k are the three linear reliable diameter estimates (see Cellino et al. 1999 and, in parameters, and d is the single nonlinear parameter. Note that particular, Belskaya et al. 2009 for wavelength dependences): the linear-exponential model does not describe polarimetric α and photometric phase curves at phase angles >30°: the ′ bP ++kP 0 kPdP P0 = ------------------------------------------ .(5) polarimetric phase curves show maxima near 90° and the dP Asteroid photometric and polarimetric phase curves 1939 Markov-Chain Monte-Carlo Methods RESULTS AND DISCUSSION The photometric and polarimetric phase curves are The photometric and polarimetric phase curves of analyzed using techniques developed further from those in asteroids 419 Aurelia (class F), 24 Themis (C), 1 Ceres (G), Muinonen et al. (2002b) and Kaasalainen et al. (2003). The a 20 Massalia (S), 55 Pandora (M), and 64 Angelina (E) in the posteriori probability density function (p.d.f.) for the V band are shown in Figs. 1 and 2, respectively, with the parameters P = (a, d, b, k) (for models separate for corresponding references