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

Idea of modelling

Observation data Model of binary system simulation

Model of system

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 Cayley-Klein parameters: Euler angles: Rotation angle Nutation angle α β Precession angle γ Drawback: undetermined for β=0 or β=π

Model of system

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

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 - Satelite’s radius vector P - Momentum vector Γ - Angular momentum vector J 1 ,J 2 ,J 3 – principal moments

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.

Model of shape

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

Model of shape

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

Primary and satellite: Three-axial elipsoids

Model of shape

YORP Only one body: Triangular faces

Input parameters

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

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

Model of lightcurve

• Ray tracing

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)

Modelling of lightcurve

• Z-buffering

PIKAIA – genetic algorithm

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

System:

PIKAIA – genetic algorithm

Determined parameters of model (blue): Shape: Period , density , Rotation angle Nutation angle α, β Precession angle γ primary: a, b/a, c/a secoundary: a, b/a, c/a Deformation: 2 craters: (8 parameters)

Parallel computing

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