**1. Introduction**

Even in ancient times, scientists, studying the relative movements of various bodies, have always tried to determine the reasons, principles and patterns that determine these movements. The combination of modern methods of mathematics, physics and mechanics makes it possible to form mathematical models of the interaction of material objects and predict possible new states based on the observed initial values of parameters and quality characteristics for the selected system of bodies.

The gravitational field for determining the forces of mutual attraction was originally considered as the field of gravity on the surface of the Earth. The famous experiments of Galileo Galilei (1564–1642) made it possible to determine the magnitude of acceleration and the direction of free fall, which is considered the same for any falling body. This defines a uniform gravitational field, which gives the simplest version of a mathematical model of gravity. The appearance of the basic laws of Johannes Kepler (1571–1630) after processing the results of many years of astronomical observations by the Danish astronomer Tycho Brahe (1546–1601) makes it possible to describe the motions of the planets of the solar system and other bodies. As a result of the publication of "Mathematical Foundations of Natural Philosophy" by Isaac Newton (1642–1727), a substantiation of the law of universal gravitation appeared, which led to the creation of new models of gravitational forces and new possible solutions to the problems of celestial mechanics [1].

Theoretical astronomy studies various variants of the motion of a selected system of bodies: stars and planets, asteroids or comets. A special class of problems appears when the forces of attraction are supplemented by the forces of light pressure [1–7], acting on the surface of bodies. The history of the emergence of new directions of science in the form of photogravitational celestial mechanics (PhCM) is determined by remarkable scientists: Kepler, Bessel, Maxwell, Bredikhin, Lebedev and some authors.

The principle of movement in space under a solar sail is based on the effect of light pressure, which Johannes Kepler guessed about when observing the movement of comets. Kepler was the first to suggest that cometary tails are a stream of particles thrown by the action of light away from the Sun as the comet approaches the Sun (as was noticed by ancient Chinese astronomers, who did not try to explain this, however). Back in 1619, he wrote: "*Dirty matter clumps together, forming the head of a comet. The sun's rays, falling on it and penetrating through its thickness, again transform it into the thinnest substance of the ether and, leaving it, form a strip of light on the other side, which we call a comet's tail. Thus, a comet, throwing out a tail from itself, thereby destroys itself and is destroyed* "[2]. Kepler was the first to formulate the basic laws of planetary motion, and also realized and pointed out the essential role of solar radiation in the evolution of bodies in the solar system, in particular for comets.

In 1836, Bessel published a work on the motion of cometary particles under the influence of the Sun's gravitational force and its repulsive force, which (as assumed) changes like the gravity force inversely proportional to the square of the distance from the center of the Sun to the point of observation. Later, it was noted that the magnitude of the force of light pressure depends on the shape and surface area of the body, as well as on the reflection coefficient and location relative to the flow of sunlight, all other things being equal.

The mechanical theory of cometary forms was further developed in the works of the Moscow astronomer academician F.A. Bredikhin. Improving Bessel's theory, he corrected a number of inaccuracies in it and found the law of motion of a particle under various physical conditions.

New possibilities in explaining the nature of these forces appeared after it was predicted and then experimentally confirmed the effect of light (as a particular manifestation of an electromagnetic field) on material bodies. D.K. Maxwell in the middle of the nineteenth century, developing the theory and equations of the action of the electromagnetic field of forces, showed that light must produce pressure on the surface placed in the path of the light flux. Maxwell in 1873 succeeded in theoretically predicting the magnitude of the light repulsion force and substantiating the dynamic essence of light pressure as a physical effect. Experiments that confirmed Maxwell's prediction were carried out by the Russian physicist Pyotr Nikolaevich Lebedev (1866–1912).

Prominent Russian experimental physicist P.N. Lebedev, the founder of the first Russian scientific physics school, who made a significant contribution to the development of astrophysics, primarily by his virtuoso experimental proof and measurement of light pressure on solids and gases (1899; 1910). He worked in close contact with the largest Russian astrophysicist F.A. Bredikhin in the study of comets. It has been proven that comet tails are a real formation from matter flowing from the comet's nucleus. He began to experimentally prove and measure the pressure of light on solids in 1897. In 1899, for the first time in the history of science, he experimentally discovered and measured the pressure of light on solids and
