Reachable domain of spacecraft with a single tangent impulse considering trajectory safety

The reachable domain of the two-body transfer orbit with a single upper-bounded tangent impulse is studied. Three cases are analyzed for either the magnitude of the tangent impulse or the initial impulse point being free, or both being free. For a fixed impulse magnitude and a free initial impulse point, the initial orbit is proved to be one of the envelopes of the reachable domain. Moreover, the trajectory safety for the transfer orbit requires a lower bound on the perigee altitude and an upper bound on the apogee altitude. Then the ranges of the impulse magnitude and the initial true anomaly can be obtained by solving quadratic and cubic inequalities, respectively. If both constraints are required for an arbitrary impulse point, the range of the impulse magnitude is obtained with impulses at the perigee and the apogee. Several numerical examples with different eccentricities are provided to show the geometry of the reachable domain and to verify the proposed method.