Approximate distribution of nearly circular orbits

The theory of eirors is employed to assess the precision of a guidance system designed to place a pay load in a prescribed circular orbit Subject to certain hypotheses about the parameters of the vehicle's propulsion and guidance systems at burnout, the form of the distribution of the square of the orbital eccentricity is obtained, and several approximations to it are presented It is indicated how knowledge of this distribution may be used to estimate the probability of mission success, suitably denned, and how this quantity may be used to study various questions that ai e of intei est to the system planner OR ceitain missions it is required to place a pay load in circular or nearly circular orbit at a prescribed altitude No orbit will be perfect, and it is desirable to have available a satisfactory method for predicting the eccentricity of the actual orbit from knowledge of the statistical characteristics of the numerous fundamental sources of error, such as motor characteristics, specific impulse, propellant weight, case weight, guidance errors, etc Such a methodology would be of value in determining to what extent existing equipment and techniques, for which performance data are either available or obtainable through controlled experimentation, are capable of carrying out intricate space missions, and to assess the performance capabilities of proposed systems The characteristics of the payload's free orbit depend upon the values, at the time of burnout or fuel shutoff, characterizing the preceding group of error sources We will call this set of variables the "fundamental guidance parameters " The guidance and propulsion systems of the vehicle are designed to achieve burnout with the fundamental guidance parameters at certain nominal values so chosen that a free circular orbit at a prescribed altitude is attained The fact that any orbit will fail to be circular, to a greater or lesser extent, is due to the failure of the fundamental guidance parameters to be exactly at their nominal, or planned, values at the instant of burnout The procedure given in this paper is based on the statistical theory of errors It provides, subject to certain hypotheses about the values of the vehicle's fundamental guidance parameters at the time of burnout, the form of the distribution of the square of the orbital eccentricity, given in terms of the tolerances associated with the (production or guidance) mechanisms influencing the fundamental guidance parameters The exact distribution is difficult to obtain in explicit form, and several approximations to it are given Illustrations indicate how knowledge of this distribution may be used to provide a rational basis for estimating the probability of mission success, suitably defined, the number of launchings that must be planned in order to achieve, with high probability, at least one successful mission, and related quantities of interest to the system planner The basic methodology is not limited to the study of "single-shot" launchings; it can be extended to treat synchronous orbits and other more complicated space missions A reasonable application of the procedure would be to assist in the planning of a mission in which a trial orbit is established by a main stage guidance system, subsequent corrections being commanded following analysis of the trial orbit The