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The Rotational Motion of Vesta A&A 556, A151 (2013) Astronomy DOI: 10.1051/0004-6361/201220732 & c ESO 2013 Astrophysics The rotational motion of Vesta N. Rambaux1,2 1 Université Pierre et Marie Curie, UPMC – Paris 06, 4 place Jussieu, 75005 Paris, France e-mail: [email protected] 2 IMCCE, Observatoire de Paris, CNRS UMR 8028, 77 Avenue Denfert-Rochereau, 75014 Paris, France Received 13 November 2012 / Accepted 4 March 2013 ABSTRACT Context. Vesta is the second largest body of the main asteroid belt, and it has been studied recently in great detail by the Dawn mission. It was the first time that this asteroid, or protoplanet, has been explored by a space mission, and it revealed a differentiated body. The knowledge of the rotational motion and, especially, its precession-nutation and length-of-day variations may add precious information on its interior. Aims. The objective of this paper is to present the first rotational model of Vesta based on the data acquired by the Dawn mission. The Dawn mission determined the orientation of Vesta with 0.01 degree accuracy, as well as the second degree of the gravity coefficients with a few percent accuracy. Methods. We built a semi-analytical model of the rotational motion of Vesta based on the orbital perturbations and the large triaxial shape of Vesta. The rotational motion is then described through the polar motion, precession, nutation, and length-of-day variations. The sensitivity of the precession-nutation is given as a function of the polar moment of inertia, which has not yet been determined. Results. We find that the amplitude of the nutation is about 2000 milli-arcsec at the semi-annual period, whereas the amplitude of the length-of-day variation is about 3 mas/days, in the semi-diurnal period. Finally, we show that the signature of the polar moment of inertia, which is crucial for constraining geophysical model, is on the order of 200 milli-arcsec for the semi-annual nutations for two extrema of polar moment with inertia values of 0.38–0.42. Key words. minor planets, asteroids: individual: Vesta – celestial mechanics 1. Introduction models used in this paper. Then, we explain the rotational the- ory, and in Sect. 5, we discuss the range of solutions. Finally, we The protoplanet Vesta is the second largest asteroid of the main conclude. asteroid belt with a mean radius of 262.7 ± 0.1 km (Russell et al. 2012), and since July 2011, it has belonged to the short list of as- teroids visited by a spacecraft. The Dawn mission orbited Vesta 2. Input data for almost one year and investigated the main characteristics of 2.1. Orientation data this body in great detail. Notably, the observational data allow assessing its origin as a remnant protoplanet from the forma- The Dawn mission has determined the orientation of the spin tion of the early solar system (Russell et al. 2012). Therefore, axis of Vesta during its mission from July 2011 to August 2012. it is crucial to determine its interior properties. One way to in- The spin axis orientation is described in terms of the right ascen- vestigate the interior structure is to accurately determine the sion α and declination δ. That means that the spin axis is given rotational motion, which is the objective of this paper. in a reference frame defined by the ICRF (International Celestial The Dawn mission has mapped the surface of Vesta with an Reference Frame). Table 1 presents the values of the right ascen- unprecedented resolution of 0.1 km that leads to determining sion and declination for Vesta obtained by Russell et al. (2012), the orientation of the spin axis at 0.01 degree accuracy (Russell Li et al. (2011), and Thomas et al. (1997). The data from Thomas et al. 2012). This accuracy improves the precision of the spin et al. (1997) comes from images obtained during a campaign position upon the Hubble Space Telescope (HST) measurements with the HST in December 1996, whereas Li et al. (2011)used by Thomas et al. (1997) and from the reanalysis of the com- combined ground data and HST data from 1986 to 2003. bined HST/ground data by Li et al. (2011) collected from 1983 to Figure 1 shows the projection of Vesta’s spin axis on the 2006 by a factor of 500. In parallel to Vesta’s mapping, the radio- ICRF XY plane reported by the different authors. We plot the science experiment provides the gravity field of Vesta (Russell ellipses of uncertainties related to each set of observations, in- et al. 2012) and thereby leading to constraints on the mass distri- cluding the five degrees uncertainties obtained by Thomas et al. bution inside Vesta and proving that the interior is differentiated. (1997) and Li et al. (2011), whereas the uncertainty ellipse ob- The objective of this paper is to present the first rotational tained by Dawn is 0.01 degree. The ellipse for the Dawn data model of Vesta based on the recent data obtained by Dawn and to is therefore too small to be visible on the scale of the figure. predict the expected precession-nutation and length-of-day vari- The figure also shows the projection of the orbital axis in the ations from various interior models. In return, the knowledge of XY plane located at the coordinates 0.124, –0.375. The determi- the rotational motion can be used to infer the interior properties nation of the orbital axis comes from the Horizons ephemerides of this protoplanet. The article is divided into four parts. First, we (Acton et al. 1996). In this figure, the position of the spin axis present the input data coming from Dawn and from geophysical with respect to the orbital axis can be described by two spherical Article published by EDP Sciences A151, page 1 of 8 A&A 556, A151 (2013) Table 1. Estimation of the orientation of the spin axis of Vesta. α (deg) β (deg) Ω0 (deg/day) Russell et al. (2012) 309.03 ± 0.01 42.23 ± 0.01 1617.333119 ± 0.000003 Li et al. (2011) 305.8 ± 3.1 41.4 ± 1.5 – Thomas et al. (1997) 301 ± 541± 5 1617.332776 -0.35 where M and R are the mass and radius of Vesta, respec- / 2 -0.4 tively. Because Vesta has a non zero C22, the values of A MR 2 ρ and B/MR are different. In addition, C22 is large, around 0.4 -0.45 of the C20, meaning that Vesta has a large triaxial shape. To ob- λ -0.5 tain the set of the three moments of inertia, it is necessary to add an external constraint on the value of the polar moment of / 2 ICRF -0.55 inertia C MR . According to Russell et al. (2012), Vesta is dif- Y ferentiated into two layers, a mantle and a core. In addition, they -0.6 suggest that the core could still be in hydrostatic equilibrium. As -0.65 a consequene, we compute the expected core’s flattening by us- ing Clairaut equation for a two-layer Vesta. We obtain the core’s -0.7 flattening equal to (ac − cc)/ac = 0.122, where ac and cc are the -0.75 equatorial and polar axis of the oblate core. Then, the total polar 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 moment of inertia of Vesta is expressed as X ICRF π π 4 2 2 8 4 Fig. 1. XY C = abc a + b ρm + a cc(ρc − ρm), (2) Projection in the ICRF plane of the spin axis of Vesta from 15 15 c Thomas et al. (1997) (green curve), Li et al. (2011) (blue curve) and Russell et al. (2012) (red curve). The ellipses represent the measured where a, b,andc are the major, intermediate, and minor axes uncertainties and in the case of Dawn the ellipse is not visible on the of the ellipsoidal shape of Vesta measured by the Dawn mis- scale of the figure. The black point represents the orientation of the ρ ρ λ ρ sion, and c and m are the density of the core and of the man- orbital axis of Vesta. The angles and represent the longitude and tle. The mass of the ellipsoid is equal to M = 4π/3abcρ projection of the obliquity of Vesta into the ICRF from the orbital axis. mean with ρmean the mean density of the body. We compute the polar moment of inertia of Vesta for two sets of interior models: ρc = 3 3 7100 kg/m , rc = 113 km and ρc = 7800 kg/m , rc = 107 km angles, the projected obliquity ρ and longitude angle λ repre- (Russell et al. 2012). By conserving the mass, we obtain a den- / 3 / 3 sented in the figure. The values of ρ for each set of observa- sity for the mantle of 3148.2 kg m and 3148.4 kg m that is in tion are in very good agreement, whereas the longitude angles the range of mantle density determined in Zuber et al. (2011). / 2 = . present a larger error than the ellipse of uncertainties. The longi- The resulting polar moment of inertia is C MR 0 4168 / 2 = . tude angle uncertainty may be related to the angle of illumination and C MR 0 4161, which are very close. For the present at the surface of Vesta by the Sun. Thus, the uncertainty in the study, we fix the value of the polar moment of inertia equal / 2 = .
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