Time Development Of Differential Charging On A spacecraft during electron beam emission

The time development of differential charging on a spacecraft during electron beam emission is studied in a capacitance coupling model. The spacecraft is assumed to be spherical and partially covered with a thin layer of dielectric. The incoming space plasma currents are modeled using Langmuir's orbit limiting plasma probe equations. The current flow between the space plasma, beam, and spacecraft surfaces is modeled as a circuit. The differential equations governing the potentials of the circuit are formulated. The response of the circuit system to a step function representing a sudden onset or a turn off of beam emission or the energetic space plasma environment is obtained by solving the Laplace transform of a coupled differential equation. It is found that with a sudden onset, the potential of the dielectric follows that of the ground transiently but differential charging gradually develops. With a sudden turn off, the transient behavior appears again, but with an initial condition of differential charging. As a result, overshoot of potential may occur. Numerical results of a model of a worst case scenario is presented. It is concluded that using an electron beam to mitigate spacecraft potential may end up in a worse condition: the dielectric potential may overshoot its original value transiently.