Methodological support for determining the pitch angle program of a launch vehicle during the insertion of spacecraft into circular orbits

Most modern satellite systems use circular orbits to maintain the stability of constellations in outer space. The insertion of spacecraft into such orbits is carried out using launch vehicles, where the choice of the optimal pitch angle is particularly important. This choice enables either an increase in the payload mass delivered to orbit or access to higher orbits with a given payload. An analysis of telemetry data on the pitch angle variation of the Falcon 9 launch vehicle (LV) revealed patterns in the changes of the horizontal and vertical velocity components of the LV, which were not considered in previous models.The purpose of this study is to develop methodological support for creating an optimal pitch angle program based on the initial data of the LV during the insertion of spacecraft into circular orbits. The study addresses the problem of designing a pitch angle program for a launch vehicle with an inertial guidance system intended for placing a spacecraft into a circular orbit. A relationship between the pitch angles in the inertial and orbital coordinate systems is established. To select a model for the variation of the horizontal and vertical velocity components, the velocity change characteristics of the Falcon 9 launch vehicle were used. These dependencies are proposed to be described by parabolic functions.As a result of the conducted research, methodological support was developed to determine the pitch angle program of the LV through vertical and horizontal velocity components using parabolic functions. An analytical relationship was obtained for determining the parameters of the proposed model based on the final motion data of the LV during spacecraft insertion into a circular orbit. The analytical dependencies of the model parameters are consistent with numerical simulation results, confirming the reliability of the obtained findings. The developed methodological support enables the formation of a pitch angle motion program for the LV that is close to optimal, based on the initial LV characteristics and the final data for spacecraft insertion into circular orbits.