Fourth-order gravity gradient torque of spacecraft orbiting asteroids

The dynamical behavior of spacecraft around asteroids is a key element in design of such missions. An asteroid's irregular shape, non-spherical mass distribution and its rotational state make the dynamics of spacecraft quite complex. This paper focuses on the gravity gradient torque of spacecraft around nonspherical asteroids. The gravity field of the asteroid is approximated as a 2nd degree and order-gravity field with harmonic coefficients C₂₀ and C₂₂. By introducing the spacecraft's higher-order inertia integrals, a full fourth-order gravity gradient torque model of the spacecraft is established through the gravitational potential derivatives. Our full fourth-order model is more precise than previous fourth-order model due to the consideration of higher-order inertia integrals of the spacecraft. Some interesting conclusions about the gravity gradient torque model are reached. Then a numerical simulation is carried out to verify our model. In the numerical simulation, a special spacecraft consisted of 36 point masses connected by rigid massless rods is considered. We assume that the asteroid is in a uniform rotation around its maximum-moment principal axis, and the spacecraft is on the stationary orbit in the equatorial plane. Simulation results show that the motion of previous fourth-order model is quite different from the exact motion, while our full fourth-order model fits the exact motion very well. And our model is precise enough for practical applications.