Symbolic-Numeric Computation in Modeling the Dynamics of the Many-Body System TRAPPIST

Modeling the dynamics of the exoplanetary system TRAPPIST with seven bodies of variable mass moving around a central parent star along quasi-elliptic orbits is discussed. The bodies are assumed to be spherically symmetric and attract each other according to Newton’s law of gravitation. In this case, the leading factor of dynamic evolution of the system is the variability of the masses of all bodies. The problem is analyzed in the framework of Hamiltonian’s formalism and the differential equations of motion of the bodies are derived in terms of the osculating elements of aperiodic motion on quasi-conic sections. These equations can be solved numerically but their right-hand sides contain many oscillating terms and so it is very difficult to obtain their solutions over long time intervals with necessary precision. To simplify calculations and to analyze the behavior of orbital parameters over long time intervals we replace the perturbing functions by their secular parts and obtain a system of the evolutionary equations composed by 28 non-autonomous linear differential equations of the first order. Choosing some realistic laws of mass variations and physics parameters corresponding to the exoplanetary system TRAPPIST, we found numerical solutions of the evolutionary equations. All the relevant symbolic and numeric calculations are performed with the aid of the computer algebra system Wolfram Mathematica.