Description of 4 Spacecraft, Moving on Elliptic Kepler Orbits

Abstract

The four-spacecraft formation is essential for measurements of various physical fields. The use of this formation on substantially elliptical heliocentric Kepler orbits allows measuring gradients of gravitation field in Solar system. The accuracy of the measurements will be sufficient to confirm or to refute modified theories of gravity. In this paper a new approach for the description of this formation is presented. The analytical solutions of the linearized motion equations are obtained. The distinctive feature of the solutions is that they use Cartesian coordinates of one of the spacecraft, termed the chief. These solutions have a clear physical meaning. It is shown, that the volume of a tetrahedron formed by spacecraft is a polynomial of 3-rd degree of Cartesian coordinates of the chief. The polynomial's coefficients are functions of initial spacecraft coordinates and velocities and linearly depend on time. If all spacecraft have the same periods of rotation around the Sun, the volume is a polynomial of 2-nd degree of the chief coordinates with time-independent coefficients. In this case the volume can be zeroed from 0 to 4 times per the period. Suggested approach can significantly simplify planning missions for measurements of various interplanetary fields.

Figures (9)

• Figure : Fig. 1. Coordinate system. $R_{p}$ and $V_{p}$ are distance to the Sun and velocity of the chief at perihelion.
• Figure : Fig. 2. Dependencies of the coefficients in formulas \ref{['GrindEQ__36_']}, \ref{['GrindEQ__37_']} on $X$ for $Y\ge 0$ (movement from perihelion to aphelion). $e=0.6$.
• Figure : Fig. 3. Dependence ${\rm V}(t)/{\rm V}_{*}$ (left), $Q(t)$(center), and ${\rm V}(X,Y)/{\rm V}_{*}$ (right) in the case of $e= 0.6$. At perihelion ($t=0$) the tetrahedron is regular and ${\bf u}_{z} =0$, ${\bf w}=0$.
• Figure : Fig. 4. The same as Fig. 3, except for the start is at aphelion ($t_0 =\pi$).
• Figure : Fig. 5. The same as Fig. 3, except for $e= 0.3$.