Methodology for determining the pitch angle program for launching spacecraft into circular orbits

Circular orbits are widely used by spacecraft (SC) to fulfill their intended purposes. In particular, circular orbits are used by all satellite communication systems, navigation satellites, and Earth remote sensing satellites, where the stability of the constellation over time is a crucial element. The stability of the constellation in space at this stage can only be maintained on circular orbits. The placement of SC into circular orbits is carried out by launch vehicles (LV). The LV is launched from a launch pad in a vertical position, and the final phase involves achieving a velocity vector along the local horizon, with a speed corresponding to the circular velocity at the given altitude. Meeting these requirements is ensured by selecting a pitch program during the flight. The aim of this work is to develop a methodological framework for the automated determination of the pitch angle change program for the LV when placing the SC into circular orbits. The proposed methodology involves calculating the LV pitch angle change program using an inertial guidance system to place the SC into circular orbits by utilizing an analytical relationship describing velocities in the horizontal and vertical directions. A parabolic model describing the velocity components over time is proposed. Approaches to calculating the coefficients of the parabolic functions that describe the vertical and horizontal components of velocity are presented. The relationship between the pitch angles in the inertial and orbital coordinate systems is determined. The parabolic dependencies of velocity changes based on the flight data of the Falcon 9 Full Trust LV have shown that the angular movement of the LV according to the proposed methodology largely coincides with the actual movement. In the future, it is planned to find optimal pitch program options depending on various laws of change in the vertical and horizontal components of velocity.