Comprehensive Mathematical Model of the Ballistic Trajectory of an Operational-Tactical Missile

Flat trajectories that do not exceed the Karman line and maneuvering in the terminal (descending) phase of the flight have become the main requirements for the characteristics of modern operational-tactical missiles. Fulfilling these requirements makes quasi-ballistic trajectories of operational-tactical missiles less predictable, reducing their detection range and the effectiveness of tracking and interception by modern air defense/ballistic missile defense systems. Such flight characteristics are currently possessed by Russian-made operational-tactical missiles such as the X-47M2 “Kinzhal”, 9M723 “Iskander-M,” and the North Korean KN-23 “Hwasong-11A”. A ballistic trajectory model with a controlled active phase has been developed to study the parameters of the quasi-ballistic trajectory of modern operational-tactical missiles. Its integration with the ballistic trajectory model based on Kepler's differential equations is carried out, which made it possible to calculate the operational-tactical missile parameters with the corrected active phase from the start to the endpoint. Based on the tactical and technical characteristics of the 9M723 “Iskander-M”, variants of its flight trajectory in the active phase were calculated using different pitch angle control programs: cubic (n = 3), quadratic (n = 2), and linear (n = 1) pitch angle versus time. The parameters of the pitch angle control programs for three variants of the trajectory apogee of the operational-tactical missile 9M723 “Iskander-M” are obtained: H1 = 40 km, Н2 = 50 km, and H3 = 60 km, and the corresponding graphs of parameters change (height-distance, speed-distance, speed-time) are presented. The developed model of the ballistic trajectory with a controlled active phase will further enhance adaptive algorithms for tracking quasi-ballistic missiles using defense acquisition radars and missile seekers.  The purpose of this article is to develop and study a comprehensive mathematical model of the quasi-ballistic trajectory of modern operational-tactical missiles, which will combine a classic ballistic trajectory and a section with a corrected active flight. The main requirements for modeling the flight of an operational-tactical missile were that the missile's flight does not cross the Kármán line, taking into account various pitch angle control programs in the active section, as well as determining the influence of these programs on the main parameters of the trajectory (apogee, range, speed, and time characteristics). The results obtained will make it possible to further improve the accuracy of modeling and create the conditions for improving algorithms for radar detection, tracking, and interception of quasi-ballistic missiles by modern air defense/missile defense systems. The materials of the article can be used by scientists, engineers, and specialists in the field of radar, as well as by specialists in the field of air and missile defense. The results obtained during modeling can be useful for the following purposes: analysis of hypersonic aircraft as targets for air defense/missile defense systems; development of a mathematical model of the trajectory of a single hypersonic quasi-ballistic missile for air defense/missile defense systems trajectory processing algorithms; calculation of the trajectory parameters of modern operational-tactical ballistic missiles; мethodology development for determining a set of characteristics of operational-tactical ballistic missiles for calculating the parameters of hypersonic quasi-ballistic flight trajectories.