Proc. VI Serbian-Bulgarian Astronomical Conference, Belgrade 7-11 May 2008, Eds. M. S. Dimitrijeviü, M. Tsvetkov, L. ý. Popoviü, V. Golev Publ. Astr. Soc. "Rudjer Boškoviü", No. 9, 2009, 67-78 SOME ASPECTS OF ASTEROID MASS DETERMINATION ANDJELKA KOVAýEVIû Faculty of mathematics, Studentski trg 16, 11000 Belgrade, Serbia e-mail: [email protected] Abstract. There is great variety of astronomical objects in the Universe. Each of these classes of objects follows a certain distribution function in size, luminosity or mass. Most individual mass distributions approximately follow a power law of the form f(M)vM-2. A notable exception are planets and small bodies which seem to obey a flatter distribution. In spite of the rapidly growing number of newly detected extrasolar planets, our knowledge of the mass function of planetary and small bodies relay entirely on the our Solar System. If is there a ’universal’ mass distribution for astronomical objects on all scales, it will be very important to know mass distribtuion of small solar system bodies. Having in mind mentioned reasons we will present methods for asteroid mass determination as well as some of most interesting results. 1. INTRODUCTION As it is well known, the architecture of the Universe is made up by a great variety of astronomical objects. Usually, they are classified by increasing average size or mass: from asteroids (as the smallest objects) to clusters of galaxies (as the largest entities). Each class follows, generaly not well known, distribution function in size, luminosity or mass of objects. It could be noticed that the most common feature of all distribution of objects is that smaller objects of given kind are more abundant than larger ones. For example, there are about dozen of asteroids with radius greater than 200 km, and about 2·106 of these bodies with radius greater than 1km. Also, it is obvious for different class of objects per unit volume of space that there is more asteroids than planets, more planets than stars, etc. Mass is the most fundamental property of an astronomical object (luminosity is inappropiate because of the existence of dark objects). Astronomical methods for its determination depend on objects dimensions, structure and distance. For example, the mean distance R of the line emitting clouds and their velocity dispersion v derived from the mean width of the rms emission line profiles (FWHM) are needed for computing a central black hole mass in active galaxies (Koratkar and Gaskell 1991), while for objects in our Solar system, could be used 67 ANDJELKA KOVAýEVIû the motion of their natural satellites, perturbations on neighboring bodies (direct methods), or values of their diameters and densities (indirect methods), etc. The astrometric technique of mass determination of small Solar system bodies, in which the deflection of a smaller bodie’s trajectory enables us to calculate the mass of a larger perturbing body, may be entering a particularly fruitful period, as near-Earth asteroid (NEA) surveys coincidentally produce a flood of high- precision main-belt asteroid observations. The mass of an asteroid, when combined with its volume, yields information on its composition and structure. Densities of minor planets are valuable from the point of view of their evolution. Very low density can be consequence of ‘rublle pile’ structure which body obtained after fragmentation and accretion. Also, another explanation could be referred to cometary origin of some minor planets. The evaluation of percent of such objects in minor planet population is an impor- tant task for understanding main asteroid belt as a complex entity in our solar sys- tem. These data are needed for precise modeling of motion of Solar system bodies as well as for accurate navigation of space missions and their successful landing, particularly on Mars. As it is well known, the inability to accurately model astero- id perturbations due to their unknown masses represents the single greatest source of error in planetary ephemerides (Standish 2000). While indirect methods of mass calculation, such as assuming a given density based on taxonomic class, have proven extremely useful in dynamical modeling, such assumptions must be cali- brated against direct observation. As is well known, the method of minor planet mass determination that consid- ers gravitational perturbations produced by an asteroid on other bodies (major pla- nets, minor planets, spacecrafts) during mutual close encounter was developed first. However, this method is affected by significant formal errors of mass deriva- tion. For example, adopted masses of only five bodies in main asteroid belt: Ceres, Pallas, Vesta, Parthenope and Mathilde were determined with formal errors small- er than 5%. This may be a consequence of inhomogeneous distribution of observa- tions of perturbed bodies, their insufficient number and accuracy or low gravita- tional effects. Nevertheless, the majority of asteroid mass determination was based on single asteroid close encounters. In order to avoid problems of reliability of asteroid mass determination, Si- tarski and Todorovic-Juchniewicz (1992) used the method of asteroid gravitation- al perturbations on the orbits of many other perturbed bodies. A simplified method was aplied to asteroid mass determination by Kuznetsov (1999) and Michalak (2000). The mentioned reasons imply that new asteroid mass determinations (espe- cially based on new recorded close encounters) are needed. In this paper we present a modified method of asteroid mass determination and dynamical masses of some largest bodies in main asteroid belt. 68 SOME ASPECTS OF ASTEROID MASS DETERMINATION 2. OVERVIEW OF MASS DETERMINATION METHODS The most used astrometric mass determination method is a modification of conventional least-squares orbit determination, in which the mass of the perturbing body is added as a seventh solve-for parameter. Ideally, the process is applied to relatively close encounters between a large target asteroid and a small test asteroid, where precise observations of the test asteroid exist before, during, and after the encounter. According to this method, the system of linear equations could be expressed in the matrix space as: L'E R (1) where the matrix L depends on the partial derivatives of the coordinates (right ascensions and declinations) of the perturbed asteroid with respect to seven parameters (six osculating elements of the perturbed body and the perturbing mass). Further, '( '('( '( is a 7u1 matrix belonging to the space of system solutions, which contains the corrections of six orbital elements of the perturbed body and correction of the mass of the perturbing body. Finally, R is the matrix depending on (O-C) residuals in coordinates of the perturbed body. Elements of matrices L, R were computed for each epoch of observation. As it is well known, the procedure of solving the system (1) is an iterative one. At the first iteration, elements of matrices L and R were calculated using previously selected observations of perturbed bodies (based on 3V criterion described later in this section). Obtained corrections, the matrix ǻE, produced a new solution which was used as the initial condition for the next iteration. Only two iterations were performed until convergence. This technique, applied on Keplerian orbital elements, produces a correlation matrix with a large correlation between the mass of the perturbing body and the mean motion (or the semimajor axis) of the perturbed one. On the other hand, if the calculation is performed using Cartesian coordinates (initial position and velocity) such a characteristic is not common. A metric which could parameterize the uncertainty in the mass of the perturbing asteroid (Bowell et al. 1994) depends on the RMS of orbital residuals, semimajor axis and eccentricity of perturbed asteroid orbit, mass of perturbing asteroid, number of pre and post encounter observations used, length of corresponding orbital arcs covered by them, as well as the impact parameter and relative velocity of the close encounter. On the other hand, in our work we tried to find out is it possible to determine correction of perturbing mass separately from corrections of six osculating elements of perturbed asteroid. As a consequence we introduce the modified method of asteroid mass determination. The idea of our modification is to separate preencounter and postencounter sets of observations (parts of orbit) of perturbed asteroid. During this process it is not necessary to know the mass of the perturbing asteroid, because its perturbing effects are negligible. These two orbits are separated by an impulsive change due to the close encounter and have to be 69 ANDJELKA KOVAýEVIû connected by properly accounted gravitational effects of the perturbing body. If the pre and post encounter orbits are accurately determined, the same mass of the perturbing body will give the best representation of the postencounter observations with the preencounter orbit and vice-versa. Similarly to classical least squares method, solution of system of linear equations of modified method can be expressed in the matrix space as: 1 'm A B (2) where the matrix A depends on the partial derivatives of the coordinates of postencounter observations (right ascensions and declinations) of the perturbed asteroid with respect to the perturbing mass. 'm is the correction of the perturbing mass and B is the matrix depending on (O-C) residuals in postencounter coordinates of the perturbed body. Elements of matrices A,