Sails and electrodynamic tethers have been proposed as passive devices to deorbit dead satellites. Their implementation in satellites would diminish very much their deorbiting time, typically down to a few years, as opposed to several decades. However, they would also increase very much the collision cross section of the said satellites, which would therefore increase the probability of collision per unit time. This situation calls for a computation of probabilities and rates related to collision between sails and tethers with debris or satellites, in order to check that endowing satellites with sails or tethers is indeed advantageous.
The work done on probability of collision between spherical objects in orbit by various authors is extended here to the case of one spherical object and one flat object of circular or rectangular shape. The former is a model for spacecraft or debris, while the latter is a model for a sail or a tether. The work presented here is almost always analytical, that is, formulae are given. Two kinds of calculations are presented.
The first is the computation of the collision rate when the flux of one object (typically debris) with respect to the other object is known. Formulae are given both for a sail with random attitude and for a sail with fixed attitude. In the case of the tether formulae are given both for a tape tether and a tether of round cross-section. This information is important when planning a mission.
The second kind is the computation of the collision probability for a particular pair of objects whose probability density functions of the positions are known. This information is necessary to decide if an evasive maneuver is going to be performed or not.
All the work presented here is analytical except for the case of sail-sphere collision probability, which requires the integration of a bivariate Gaussian over a parallelogram. In order to compute it we have developed a program in which the fastest available algorithm to integrate a bivariate Gaussian is embedded. We have made this program available to the public as an app.