On parameters calculation of passive gravity-assisted maneuvers of interplanetary spacecraft

Theoretical mechanics


DOI: 10.34759/trd-2021-118-01

Аuthors

Popov I. P.

Kurgan State University, 63/4, Sovetskaya str., Kurgan, 640020, Russia

e-mail: ip.popow@yandex.ru

Abstract

The purpose of the study consists in analytical description of the ballistic trajectory section corresponding to the normal fall of the spacecraft on the surface of an atmosphere-free planet. The motion of a normally falling body herewith is being characterized by an increasing acceleration of gravity. The problem of the speed, time and acceleration of the normal fall of a body on the planet’s surface in the absence of an atmosphere is being reduced to solving a second-order differential equation, which is solved by the standard method. The solution specificity is the formal use of the table integral at an intermediate stage. It turned out, however, that his formula is inauthentic, namely, the derivative of the right-hand side is not equal to the integrand. From this, It necessary follows that possible existing solutions to this problem based on employing the above said table integral are incorrect. The article presents correction of this table integral, which is an incidental result of the study. Temporal equation of a normal body falling on the planet surface in the absence of the atmosphere, as well as temporal equations of its speed and acceleration were obtained in this work. Expressions for the distance, speed and acceleration were obtained as implicit functions of time. The article presents a numerical example with regard to a planet with parameters of the Earth, according to which the period of normal fall of a body on the planet’s surface from the altitude of 629 km is 387.275 s = 6.455 minutes, while the body’s velocity at the planet’s surface is 3353.297 m/s. The results obtained may be handy for calculating the normal incidence of small celestial bodies and spent spacecraft structural elements.

Keywords:

planet, body, equation of motion, speed, acceleration, mass, distance

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