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Modeling of double asteroids with PIKAIA algorithm Przemysław Bartczak Astronomical Observatory of A. Mickiewicz University

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Idea of modelling Observation data Model of binary system simulation

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Model of system Cayley-Klein parameters:Euler angles: Rotation angle α Nutation angle β Precession angle γ Body frame: The axes are directed along the principal moments of interia of the primary. Fixed frame: the axes are aligned with some suitably chosen astronomical coordinate system. Both system of axes are Cartesian, right-handed and share the same origin 0, located at the center of mass of the primary Drawback: undetermined for β=0 or β=π

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Model of system When the primary rotates, the Cayley-Klein parameters change according to the differential equations where is the angular rate vector in body frame.

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Model of system Dynamics equations describe the orbital motion of the satelite with respect to the primary and rotation of primary. Ω - Angular rate vector R - Satelites radius vector P - Momentum vector Γ - Angular momentum vector J 1,J 2,J 3 – principal moments

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Model of system Constans of motion: Hamiltonian: Total angular momentum vector: Cayley-Klein parameters: Integrating the equations of motion by means of the Raudau-Everhart RA-15 procedure, we have obtained highly accurate results within a fairly short computation time.

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Model of shape The dynamical part of the model (free or forced precession) Primary: Three-axial ellipsoid Satellite: Spherical

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Model of shape The synchronous double asteroids Primary and satellite: Three-axial elipsoids Primary and satellite: Three-axial elipsoids plus two craters.

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Model of shape YORP Only one body: Triangular faces

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Input parameters Date of observation Position of asteroid (Orbital elements ) Orientation of binary system Model of shape and binary system Modelling of lightcurve Position of Sun and Earth

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Model of lightcurve Ray tracing is a technique for generating an image by tracing the path of light through pixels in an image plane and simulating the effects of its encounters with virtual objects. Scattering : Lommel-Seeliger law

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Model of lightcurve Ray tracing

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Modelling of lightcurve Z-buffering is the management of image depth coordinates in three-dimensional (3-D) graphics. The depth of a generated pixel (z coordinate) is stored in a buffer (the z-buffer or depth buffer)

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Modelling of lightcurve Z-buffering

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PIKAIA – genetic algorithm Genetic algorithms are a class of search techniques inspired from the biological process of evolution by means of natural selection.

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PIKAIA – genetic algorithm Determined parameters of model (blue): System:Shape: Period, primary: a, b/a, c/a density, secoundary: a, b/a, c/a Rotation angle α, Nutation angle βDeformation: Precession angle γ2 craters: (8 parameters)

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Parallel computing SQL database PC System: Debian Compilator: gcc,c++ SQL database: MySql, oracleXe Librares: CORBA, POSIX Threads

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