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{\bf SIMPLIFIED DYNAMICS OF ASTEROID\\ACCESSING OUR SOLAR SYSTEM\\}
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{\it Lud\v{e}k Hyn\v{c}\'{\i}k $^1$, V\'aclav Uhl\'{\i}\v{r} $^2$\\}
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$^1$University of West Bohemia, $^2$Franti\v{s}ek Pe\v{s}ta's Observatory
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\noindent{\bf Abstract}

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Mankind explores the space from time out of mind. There are many bodies in the space influencing more or less the life on our Earth. To describe the motion of various spatial bodies may be useful even for future. This paper develops a very simple model based on multi-body dynamics that describes spatial motion of a asteroid flying inside our Solar system. Considering the dimension of modelled space, the whole Solar system is simulated as a system of mass points. There is the Sun that is fixed and all the other planets circulating around along the given trajectories. These trajectories are fixed by boundary conditions based on astronomical literature.\\

\noindent The asteroid is given by initial position and velocity. Each planet including Sun acts by gravitational force to the asteroid. No planet deflection is considered. Hence, one have defined three ordinary differential equations of motion of the second order with initial conditions specifying initial position vector and initial velocity vector. There is only a gravitational force on the right-hand side that depends on relative position of asteroid and particular planet.\\

\noindent The whole situation can be very shortly implemented into MATLAB. The basic geometrical and dynamical values are saved in a matrix with twelve rows for the Sun, the Earth's Moon, particular planets and the asteroid. The system of three ordinary differential equation of the second order is splitted into the system of six ordinary differential equation of the first order and solved. Furthermore, the the collision detection between asteroid and some other body is checked. Applying suitable initial conditions especially for position and velocity of the asteroid, one can simulate several cases such as the situation where the asteroids is caught by the Sun and it starts to revolute with decreasing semimajor axis or describing the situation where the asteroids crashes into the Sun. The average time of simulation describing two thousand years on regular PC is about ten till twenty seconds.\\

\noindent Using MATLAB and its inbuilt functions, we have simply numerically implemented the equation of motion of a mass point in the force field. The force filed is the gravitational field generated by planets of our Solar system with prescribed motion. Changing the initial conditions such as position in space and velocity, one can simulate the trajectory including collision detection for asteroid of a given mass.\\

\noindent\it\textbf{Contacts:} Ing. Lud\v{e}k Hyn\v{c}{\'\i}k, Ph.D., University of West Bohemia in Pilsen, Departments of Mechanics, Univerzitn{\'\i} 8, 30614 Plze\v{n}, Czech Republic, tel.:~\mbox{+420377634709}, fax:~\mbox{+420377634702}, e-mail: \href{mailto:hyncik3@ntc.zcu.cz}{hyncik3@ntc.zcu.cz}, Mgr. V\'aclav Uhl\'{\i}\v{r}, Hv\v{e}zd\'arna Franti\v{s}ka Pe\v{s}ty, P.O. Box 48, 391 02 Sezimovo �st�, tel.:~\mbox{+420605557731}, fax: ~\mbox{+420381261037}, e-mail: \href{mailto:nosuh@seznam.cz}{nosuh@seznam.cz}\\

\noindent\it\textbf{Acknowledgement:} The work is supported by the Ministry of Education of the Czech Republic, project MSM 4977751303.\\

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