№2 2021


Elements of ballistic calculation for spacecraft gravity assist


I.P. Popov


Kurgan State University
Kurgan, Russian Federation


The purpose of the study is an analytical description of the section of the ballistic trajectory corresponding to the normal fall of the spacecraft on the surface of an atmosphereless planet. In this case, the motion of a normally falling body is 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 reduced to solving a second-order differential equation, which is solved by the standard method. A feature of the solution is the formal use of the tabular integral at an intermediate stage. It turned out, however, that his formula is unreliable, namely, the derivative of the right-hand side is not equal to the integrand. It follows from this that the possible existing solutions to this problem, based on the use of the indicated tabular integral, are incorrect. The article presents the correction of this tabular integral, which is an incidental result of the study. In this work, the time equation of motion of a body normally falling on the surface of the planet in the absence of an atmosphere, as well as the time equations of its speed and acceleration are obtained. The results obtained can be useful in calculating passive gravity assist during interplanetary flights and calculating the sheer fall of small celestial bodies and spent structural element s of spacecraft.


ballistic trajectory, passive gravity assist, spacecraft, interplanetary flight, celestial body


[1] Vygodskiy M. Ya. Spravochnik po vysshey matematike [Handbook of Higher Mathematics]. Moscow, Science, 1977, 872 p. (In Russian)

[2] Bermant A. F., Aramanovich I. G. Kratkiy kurs matematicheskogo analiza dlya vtuzov [A short course in mathematical analysis for technical colleges]. Moscow, Science, 1971, 736 p. (In Russian)

[3] Spravochnik mashinostroitelya [Mechanical Engineer Handbook]. Moscow, 1963, 592 p. (In Russian)

[4] Starinova O. L., Sergaeva E. A., Shornikov A. Yu. Design and ballistic analysis of the mission for long-term study of the asteroid Apophis by a nanosatellite with an electric rocket propulsion system // Spacecrafts & Technologies, 2020, vol. 4, no. 3, pp. 161–170. doi: 10.26732/

[5] Panko S. P., Tsimbal M. S. Measurement of the velocity of the spacecraft // The Research of the Science City, 2015, no. 4, pp. 25–29.

[6] Korolev V. S. Problem optimum spaceship trajectory to inspect or service system of body // The Research of the Science City, 2015, no. 2, pp. 18–23.

[7] Popov I. P. Raschetnyye sistemy otscheta pri otnositel'nom dvizhenii kosmicheskikh ob"yektov [Computational reference systems for the relative motion of space objects] // Engineering Physics, 2019, no. 3, pp. 40–43. (In Russian)

[8] Popov I. P. Sistemy otscheta v navigatsii dvizhushchikhsya ob"yektov [Reference systems in the navigation of moving objects] // Mechatronics, automation, control, 2019, vol. 20, no. 3, pp. 189–192. (In Russian)

[9] Chebotarev V. Ye., Borisov V. I. Razrabotka algoritma rascheta trayektorii perekhvata raketoy asteroida, opasnogo dlya planety Zemlya [Development of an algorithm for calculating the trajectory of a rocket intercepting an asteroid dangerous for planet Earth] // Spacecraft & Technologies, 2012, no. 2, pp. 30–34. (In Russian)

[10] Levkina P. A., Sergeyev A. V. Kharakteristiki novykh ob"yektov kosmicheskogo musora, obnaruzhennykh v terskol'skoy observatorii [Characteristics of new objects of space debris discovered at the Terskol observatory] // Proceedings of the Institute of Astronomy of the Russian Academy of Sciences, 2019, vol. 4, pp. 306–311. (In Russian)

For citing this article

Popov I.P. Elements of ballistic calculation for spacecraft gravity assist // Spacecrafts & Technologies, 2021, vol. 5, no. 2, pp. 77-81. doi: 10.26732/

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