# ESA GNC Conference Papers Repository

**Title:**

##### Semi-analytical estimation of the probability of capture into 1:1 ground-track resonance of a low-thrust spacecraft around an asteroid.

**Authors:**

**Presented at:**

**Full paper:**

**Abstract:**

The DAWN mission demonstrated the feasibility of utilizing low-thrust propulsion for extended periods of time. The mission involves approaching the asteroid Vesta from a high-altitude mission orbit (HAMO) to a low-altitude mission orbit (LAMO). As the spacecraft descends towards the asteroid, there is a likelihood that it might be captured by the 1:1 ground-track resonance. This occurs when the spacecraft encounters the same gravitational configuration at each revolution, which leads to significant changes in the orbit's eccentricity and inclination. With the increasing trend of using low-thrust propulsion in space exploration, it is essential to investigate the probability of capture into resonance for low-thrust spacecrafts. Resonant orbits are defined as trajectories where the spacecrafts revolution period is commensurable with the asteroids rotation period, such as a 1:1 resonance orbit, where the spacecraft completes one orbit in the same period as the asteroid's rotation. This research focuses on developing a semi-analytical tool to efficiently estimate the probability of capture into 1:1 ground-track resonance of a low-thrust spacecraft around an asteroid. Firstly, the Hamiltonian model associated with the 1:1 ground-track resonance in the equatorial case is developed, taking into account perturbations from the irregular gravitational field up to the second order and continuous low thrust that is constant in magnitude and always in the opposite direction of the spacecrafts velocity. The equilibrium points, libration, and circulation regions are identified. Special attention is paid to the systems energy balances as the trajectory crosses the separatrix. As the trajectory approaches the separatrix, its evolution is not deterministic, making it appropriate to consider the probability of ending up in a libration region and being captured in resonance, or ending up in the lower circulation region and escaping the resonance. The Hamiltonian model has two degrees of freedom, and the effect of low-thrust on these degrees of freedom is characterized. The separatrices are numerically approximated using a fourth-order polynomial, and the system's energy balance is evaluated as it crosses the separatrix using a global adaptive quadrature method. Finally, the probability of capture into resonance is estimated as a function of the energy balance. The results are validated through comparison with numerical simulations using the equations of motion derived from the Hamiltonian. This research makes a significant contribution to the field of astrodynamics by systematically and efficiently analyzing the probability of low-thrust spacecraft capture into resonance around asteroids.