Intercept probability of satellites in circular orbits

Abstract

An analytical and computational approach is provided to calculate the probability that a designated transmitter satellite in circular orbit will intercept a particular receiver satellite also in circular orbit. The term intercept implies that the transmitter satellite passes within some minimum distance of the receiver satellite for a minimum period of time. The probability calculated is the probability of interception within some maximum time interval. This paper discusses the dynamics and definitions leading to intercept probability calculations and compares the predicted intercept probabilities to estimates determined through Monte Carlo techniques. HE problem posed in this paper can be defined as a form of communication s problem. Assume that one satellite in a circular orbit must transmit information to another satellite in circular orbit within some specified period of time, 7}. Furthermore, assume that a transmission time of at least tt s must be allowed, and the transmitter must be within a range R of the receiver for the communication to take place. Then, an important parameter of this communications problem is the probability that the transmitter will be within R for at least tt within 7} starting from some random initial time. Throughout the remainder of this paper the transmitter will be called a platform and the receiver will be called the target. An intercept will be defined to be the event of the platform passing within the distance R of the target for at least tt s. Therefore, the problem can be restated to be the calculation of the probability of interception within the interval 7} assuming a random starting time. An obvious solution to this problem is to simply simulate the trajectories of the platform and target and, using random initial positions of the satellites, estimate the probability of intercept using a Monte Carlo technique. However, Monte Carlo techniques, although effective, provide little insight into the understanding of the problem. A similar problem involving the collision of satellites has been studied by several authors.13 A simplified collision problem treats essentially the same types of parameters: the probability that two orbiting objects will pass within some specified distance within some specified time. However, the distance is typically on the order of meters in the collision problem and kilometers in the intercept problem, and the time is on the order of months and years in the collision problem and hours in the intercept problem. Perhaps a more significant difference is that an intercept is an event that occurs over a time interval, whereas a collision is essentially a point event. Despite these differences both problems share many areas of common interest. The primary purpose of this paper is to provide an analytical model for understanding and calculating the probability of intercept. It will be assumed throughout this paper that the platform and target are both in circular orbits. Although this assumption appears rather restrictive, it does lead to a tractable problem and provides a close approximation for low eccentricity orbits. Furthermore, the solution also provides some interesting and potentially valuable insights into the variation of intercept probability. It should be noted at this point that the probability calculations also ignore the effects of precession. The error introduced by this omission will be