Development of the numerical approach in the theory of physical libration within the framework of the 'main problem'

A.A. Zagidullin*, N.K. Petrova**, V.S. Usanin***, Yu.A. Nefediev****, M.V. Glushkov****

Kazan Federal University, Kazan, 420008 Russia

E-mail: *arhtur.zagidullin@ya.ru, **nk_petrova@mail.ru,  ***vusanin@yandex.ru, ****star1955@mail.ru, *****sh345sqrt@gmail.com

Received March 27, 2017

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Abstract

A numerical theory of lunar rotation has been developed. The mathematical model of the rotation of the Moon has been considered within the “main problem”. The equations of the rotation have been constructed on the basis of the Hamiltonian approach. The resulting differential equations have been solved using the 10th order Runge–Kutta method. The analysis of the obtained data has been carried out on the basis of residual differences (between the numerical and analytical solutions). As a result, we have found that the range of residual differences does not exceed in modulus 1.8 and 0.9 arcsec. by longitude and latitude, respectively. This relatively high divergence occurs due to the fact that the frequencies are close to the natural frequencies of the system.

Keywords: theory of physical libration of Moon, main problem, Hamilton's equations, Runge–Kutta method, residual differences, resonant frequencies

Figure Captions

Fig. 1. Selenocentric coordinate system: XYZ – ecliptic coordinate system, X axis directed to the vernal point (γ), Z – to the ecliptic pole; (X, Y, Z) – ecliptic coordinate system rotating continuously at the speed of n = dL/dt; L – the mean longitude of the Moon. Trihedron (x, y, z) – dynamic coordinate system: x axis is oriented in the direction of the smallest moment of inertia A; z – in the direction of the largest moment C; the direction of y axis was selected so that it forms the right Cartesian coordinate system; μ, ν, π – libration angles.

Fig. 2. Physical libration of the Moon by longitude μ.

Fig. 3. Physical libration of the Moon by latitude: for ν – on the left, for π – on the right.

Fig. 4. Residual differences in the physical libration angles of the Moon q₁ = μ, q₂ =  ν, q₂ =  π.

Fig. 5. Residual differences between the solutions of G.I. Eroshkin and D.H. Eckhardt by longitude.

Fig. 6. Residual differences between the solutions of G.I. Eroshkin and M. Moons by longitude.

Fig. 7. Frequency dependence for the residual differences.

Fig. 8. The accuracy of the 10th order Runge–Kutta method for different steps. on the left – difference between the steps of 0.05 and 0.01 days, on the right – difference between the steps of 0.05 and 0.1 days.

Fig. 9. Residual differences in the directional cosines of the ecliptic pole.

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For citation: Zagidullin A.A., Petrova N.K., Usanin V.S., Nefediev Yu.A.,Glushkov M.V. Development  of the numerical approach in the theory of physical libration within the framework of the “main problem”. Uchenye Zapiski Kazanskogo Universiteta.Seriya Fiziko-Matematicheskie Nauki, 2017, vol. 159, no. 4, pp. 529–546. (In Russian)

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