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0"MIRZO ULUG'BEK - BUYUK MUTAFAKKIR OLIM VA DAVLAT ARBOBI" MAVZUSIDAGI
XALQARO ILMIY-AMALIY ANJUMAN 2024-YIL 14-15 MAY
ABOUT MODELING SB-GALAXIES USING A LOCAL INTEGRAL
OF MOTION
Shamshiev F.T.
Department of Astronomy and Astrophysics of National University of Uzbekistan https://doi.org/10.5281/zenodo.11173543 The problem of describing the motion of a material point in a gravitational field with a sufficiently complex structure is undoubtedly important for all branches of celestial mechanics and stellar dynamics. However, this problem has not been solved in its full form, and therefore it is of great interest stellar dynamics and celestial mechanics [1-9].
It is known that there is no quadratic integral for motion in the equatorial plane of a rotating non-axisymmetric system. However, in some cases it is possible to construct an additional integral by analogy with a quadratic integral. In the work [6] similar integrals have been found, but they can be built on some surface of the phase space.
In this paper, we describe studies of a class of potentials that admit a local integral quadratic in the velocity components, which is different from the Vandervoort case. It should be noted that, in contrast to the case of Linden-Bell [2], the "local integral" in the sense used here was introduced by Antonov [1]. In general, these potentials depend on several arbitrary functions of one variable. The degree of arbitrariness of the local integral is determined by a finite number of parameters.
Some of the potentials obtained here can be used in the modeling of SB-galaxies. The existence of a local integral with a given Jacobi constant limits the mixing process, since it plays the role of a barrier.
The existence of a local integral makes it possible to partially find the trajectory, but in some particular cases its analytical description is possible.
Similar problems arise in axisymmetric galaxies, but taking into account the motion of its satellite in a circular orbit.
This is not the most general, but rather simple scheme for constructing a local integral with the specified topological properties of the velocity field. As explained in [4,5], under the above conditions, for any value of constant energy h, the permissible values of the velocity vector are completely determined if the initial conditions correspond to some definite invariant manifold in the phase space. This does not make it possible to get an explicit dependence of coordinates on time, since there is still need to integrate a system of differential equations
dx _ dy _ dz _ ^^
u y z
"MIRZO ULUG'BEK - BUYUK MUTAFAKKIR OLIM VA DAVLAT ARBOBI" MAVZUSIDAGI
XALQARO ILMIY-AMALIY ANJUMAN 2024-YIL 14-15 MAY
But the order of this system can be lowered if it is possible to "extend" the mentioned variety from zero to at least an infinitesimal width. With this expansion, we do not change the potential, but auxiliary functions can get their infinitesimal increments.
FUNDING. The work was partially supported by grant FZ- 20200929344, allocated by the Ministry of Higher Education, Science and Innovation of the Republic of Uzbekistan.
References:
1. Antonov, V.A.: 1981, Vestnik Leningrad University, 19, 97-205, (in Russian).
2. Lynden-Bell, D.: 1962, MNRAS, 124, 95-123.
3. Lynden-Bell, D.: 2016, MNRAS 458, 726-732.
4. Antonov, V.A., Shamshiev F.T., 1993, Cel. Mech. and Stellar Dyn., 56, 451469.
5. Antonov, V.A., Shamshiev F.T., 1994, Cel. Mech. and Stellar Dyn., 59, 209219.
6. Evans N. W., Sanders J. L., Williams A. A., An J., Lynden-Bell D., Dehnen W., 2016, MNRAS, 456, 4506.
7. An J., Evans N. W.: 2016, The Astrophysical Journal, 816, 35.
8. Vandervoort, P.O.: 1979, The Astrophysical Journal, 232, 91-105.
9. Contopoulos, G., Vandervoort, P.O.: 1992, The Astrophysical Journal, 389, 118-128.
