Deflecting an Asteroid with a Low-Thrust Tangential Engine to the Orbit


In order to solve the problem of deflecting a dangerous asteroid from a collision orbit with the Earth, using a low-thrust engine directed tangentially to the trajectory is considered. The engine can be mounted on the asteroid or on a “gravity tractor.” The purpose of this study is to establish the fundamental possibility of steering away an asteroid to a safe distance over times of approximately a month and a year. This is acceptable since an asteroid with about a 100-m diameter is unlikely to strike immediately after its discovery. We limited ourselves to a model statement of the problem: the engine provides constant tangential acceleration. Previously, we transformed the respective Euler equations using the averaging method. Here, we solve them by the method of series in powers of “slow time” and demonstrate the adequacy of the solution on the time intervals of decades. It turns out that asteroids up to 55 m in diameter can be deflected in a year with an engine thrust of 1 N. With a thrust of 20 N, asteroids up to 50 m in diameter can be deflected in a month, and asteroids with a diameter of up to 150 m, in a year. Diverting larger asteroids requires more time or more powerful engines. The results are compared with the previously obtained similar data for the case of the transversal perturbing acceleration. The tangential traction leads to better results in all cases; however, both variants nearly coincide for orbits with eccentricities up to 0.4. The difference becomes significant at \(e > 0.5\).

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Fig. 1.


  1. 1.

    This is not the case for meter- and decameter-sized bodies that are discovered just in the immediate vicinity of the Earth. However, there is no need to deflect such small bodies; giving a public warning is sufficient.

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    All the values are given in the SI system, unless otherwise indicated.


  1. 1

    T. N. Sannikova and K. V. Kholshevnikov, Vestn. SPbGU, Ser. 1: Mat. Mekh. Astron., No. 4, 134 (2013).

  2. 2

    T. N. Sannikova and K. V. Kholshevnikov, Astron. Rep. 58, 945 (2014).

    ADS  Google Scholar 

  3. 3

    N. Batmunkh, T. N. Sannikova, and K. V. Kholshevnikov, Astron. Rep. 62, 288 (2018).

    ADS  Google Scholar 

  4. 4

    E. T. Lu and S. G. Love, Nature (London, U.K.) 438 (11), 177 (2005).

    ADS  Google Scholar 

  5. 5

    Yu. D. Medvedev, M. L. Sveshnikov, A. G. Sokol’skii, E. I. Timoshkova, Yu. A. Chernetenko, N. S. Chernykh, and V. A. Shor, Asteroid-Comet Hazard, Ed. by A. G. Sokol’skii (IPA RAN, St. Petersburg, 1996) [in Russian].

    Google Scholar 

  6. 6

    A. M. Mikisha and M. A. Smirnov, in Threat from the Sky: Rock or Chance?, Ed. by A. A. Boyarchuk (Kosmoinform, Moscow, 1999) [in Russian].

    Google Scholar 

  7. 7

    N. N. Gor’kavyi and A. E. Dudorov, Chelyabinsk Superbolid (Chelyab. Gos. Univ., Chelyabinsk, 2016) [in Russian].

    Google Scholar 

  8. 8

    A. V. El’kin and L. L. Sokolov, in Proceedings of the Conference on Asteroid Hazard-95, St. Petersburg, May 23–25,1995 (MIPAO and ITA RAN, St. Petersburg, 1995), Vol. 2, p. 41.

  9. 9

    G. L. Grodzovskii, Yu. N. Ivanov, and V. V. Tokarev, Space Flight Mechanics (Optimization Problems) (Nauka, Moscow, 1975) [in Russian].

    Google Scholar 

  10. 10

    V. N. Lebedev, Calculation of the Motion of a Spacecraft with Low Thrust (Vychisl. Tsentr AN SSSR, Moscow, 1968) [in Russian].

    Google Scholar 

  11. 11

    N. Batmunkh, K. I. Os’kina, T. N. Sannikova, V. B. Titov, and K. V. Kholshevnikov, Astron. Rep. 63, 954 (2019).

    ADS  Google Scholar 

  12. 12

    N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymptotic Methods in the Theory of Nonlinear Oscillations (Fizmatlit, Moscow, 1963; Gordon and Breach, 1961).

  13. 13

    D. Brower and G. Clemence, Methods of Celestial Mechanics (Academic, New York, 1961).

    Google Scholar 

  14. 14

    H. Poincare, Lectures on Celestial Mechanics (Springer, Berlin, 1971).

    Google Scholar 

  15. 15

    M. F. Subbotin, Introduction to Theoretical Astronomy (Nauka, Moscow, 1968) [in Russian].

    Google Scholar 

  16. 16

    W. Gröbner, Die Lie-Reihen und ihre Anwendungen (VEB Deutscher Verlag der Wissenschaften, Berlin, 1967).

    Google Scholar 

  17. 17

    G. E. O. Giacaglia, Perturbation Methods in Non-Linear Systems (Springer, New York, 1972).

    Google Scholar 

  18. 18

    A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973).

    Google Scholar 

  19. 19

    K. V. Kholshevnikov, Asymptotic Methods of Celestial Mechanics (Leningr. Gos. Univ., Leningrad, 1985) [in Russian].

    Google Scholar 

  20. 20

    K. V. Kholshevnikov, N. Batmunkh, K. I. Os’kina, and V. B. Titov, Astron. Rep. 64, 369 (2020).

    ADS  Google Scholar 

  21. 21

    N. Batmunkh, T. N. Sannikova, K. V. Kholshevnikov, and V. Sh. Shaidulin, Astron. Rep. 60, 336 (2016).

    ADS  Google Scholar 

Download references


The study was funded by the Russian Science Foundation (grant no. 18-12-00050).

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Correspondence to K. V. Kholshevnikov or D. V. Milanov or K. I. Os’kina or V. B. Titov.

Additional information

Translated by M. Chubarova



Let \(0 < {{e}_{0}} < 1\), \(0 \leqslant \alpha \leqslant 1\), \(0 \leqslant \psi < 2\pi \), \(z \in \mathbb{C}\), \({\text{|}}z{\text{|}} \leqslant 1\),

$$\begin{gathered} e(\psi ) = {{e}_{0}} + (1 - {{e}_{0}})\text{Exp} \psi , \\ {{{\text{g}}}_{1}}(\alpha ,{\text{z}}) = \frac{{1 - {\text{z}}}}{{1 - \alpha {\text{z}}}}, \\ {{{\text{g}}}_{2}}({\text{e}}) = {\text{e}}(1 - {{{\text{e}}}^{2}}){\mathbf{D}}({\text{e}}). \\ \end{gathered} $$

Let us estimate the absolute values of functions (37). Obviously, \({\text{|}}e(\psi ){\text{|}} \leqslant 1\),

$$\begin{gathered} 1 - {{e}^{2}}(\psi ) = (1 - {{e}_{0}})\{ 1 + {{e}_{0}} - 2{{e}_{0}}\cos\psi \\ - \;(1 - {{e}_{0}})\cos2\psi - \mathfrak{i}{\kern 1pt} \text{[}2{{e}_{0}}\sin\psi + (1 - {{e}_{0}})\sin2\psi ]\} , \\ {\text{|}}1 - {{e}^{2}}{{{\text{|}}}^{2}} = 2{{(1 - {{e}_{0}})}^{2}}{{g}_{3}}({{e}_{0}},\psi ), \\ {{g}_{3}} = 1 + 3e_{0}^{2} - 4e_{0}^{2}\cos\psi - (1 - e_{0}^{2})\cos2\psi . \\ \end{gathered} $$

The derivative

$$\frac{{\partial {{g}_{3}}}}{{\partial \psi }} = 4\sin\psi [e_{0}^{2} + (1 - e_{0}^{2})\cos\psi ]$$

vanishes at

$$\psi = 0,\quad \psi = \pi ,$$

and when \({{e}_{0}} < 1{\text{/}}\sqrt 2 \) also at

$$\cos\psi = - \frac{{e_{0}^{2}}}{{1 - e_{0}^{2}}},\quad \cos2\psi = - \frac{{1 - 2e_{0}^{2} - e_{0}^{4}}}{{{{{(1 - e_{0}^{2})}}^{2}}}}.$$

From here, we easily derive

$$\mathop {\max}\limits_\psi {\text{|}}1 - {{e}^{2}}(\psi ){\text{|}} = {{g}_{4}}({{e}_{0}}),$$


$${{g}_{4}}({{e}_{0}}) = \left\{ \begin{gathered} 2\sqrt {\frac{{1 - {{e}_{0}}}}{{1 + {{e}_{0}}}}} ,\quad {\text{if}}\quad {{e}_{0}} \leqslant 1{\text{/}}\sqrt 2 {\kern 1pt} , \hfill \\ 4{{e}_{0}}(1 - {{e}_{0}}),\quad {\text{if}}\quad {{e}_{0}} \leqslant 1{\text{/}}\sqrt 2 {\kern 1pt} {\kern 1pt} . \hfill \\ \end{gathered} \right.$$

The triangle axiom for points \(1\), \(z\), \(\alpha z\) of the complex plane gives

$${\text{|}}1 - z{\text{|}} \leqslant \left| {1 - \alpha z} \right| + \left| {z - \alpha z} \right|.$$


$${\text{|}}z - \alpha z{{{\text{|}}}^{2}} \leqslant {\text{|}}1 - \alpha z{{{\text{|}}}^{2}}.$$

Indeed, (40) follows from the obvious inequality

$$1 - 2\alpha \leqslant 1 - \alpha (z + \bar {z}).$$

Relations (39) and (40) imply the inequality

$${\text{|}}{{g}_{1}}(\alpha ,z){\text{|}} \leqslant 2.$$

Hence, we obtain the estimate of \({{g}_{2}}(e)\):

$$\begin{gathered} {\text{|}}{\kern 1pt} {{g}_{2}}(e){\kern 1pt} {\text{|}} \leqslant \left| {e\sqrt {1 - {{e}^{2}}} } \right|\int\limits_0^{\pi /2} \,\left| {\sqrt {{{g}_{1}}({{\sin}^{2}}x,{{e}^{2}})} } \right|{{\sin}^{2}}xdx \\ \, \leqslant \sqrt {2{{g}_{4}}({{e}_{0}})} \int\limits_0^{\pi /2} \,{{\sin}^{2}}xdx = \frac{\pi }{{\sqrt 8 }}\sqrt {{{g}_{4}}({{e}_{0}})} . \\ \end{gathered} $$

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Kholshevnikov, K.V., Milanov, D.V., Os’kina, K.I. et al. Deflecting an Asteroid with a Low-Thrust Tangential Engine to the Orbit. Astron. Rep. 64, 785–794 (2020).

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