The libration points for the motion of a star inside an elliptical galaxy

Lukyanov, L. G.

doi:10.1134/1.1515096

Cited by 10 publications

(9 citation statements)

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“…Then, any isolated singular point L exhibits Lyapunov stability in a first approximation, since all the roots of (39) are purely imaginary (in our case, this concerns the central libration point L 1 , for which A 1 < 0, A 2 < 0, and A 3 < 0). Such a point will also be stable in a non-linear formulation [9]. Any conical singular point L with the cone axis OZ for which A 1 > 0, A 2 > 0, and A 3 < 0 exhibits Lyapunov stability in a first approximation.…”

Section: Type and Stability Of The Libration Pointsmentioning

confidence: 96%

“…If the units of time, mass, and distance are chosen so that G = 1, M * = 1, and Ω = 1, and we neglect the gravitation of the homeoid, setting B = 0, we obtain the same expressions for X 0 and Y 0 as those in [5][6][7]9]: X 0 = ±(1 + αε) and Y 0 ± (1 + βε), or equivalently a 1 = 1 and a 2 = 1.…”

Section: Stationary Solutions (Libration Points)mentioning

confidence: 99%

“…A non-linear analysis showed that L 4 and L 5 are stable for most initial conditions according to the Lebeg measure, apart from some resonance cases when these points become unstable [7]. Moreover, the infinitely distant libration points L 6,7 , which exist only when B = 0 [9] and in the restricted three-body problem [10], are isolated singular points.…”

Section: Examplesmentioning

confidence: 99%

“…In place of the exact formula (5) for U * , we must use the expression 0). If the luminous part of the EG is close to a sphere, we can use the potential presented in [9] to determine the central libration point. In addition to the central point L 1 and the points L n (n = 2, 3, 4, 5), there also exist the infinitely distant libration points L 6,7 , for X 0 = 0, Y 0 = 0, Z 0 = ±∞, but only when B = 0 [9], and in the restricted three-body problem [10].…”

Section: Stationary Solutions (Libration Points)mentioning

confidence: 99%

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Steady-state solutions for the motion of a globular cluster in an inhomogeneous, rotating elliptical galaxy

Gasanov

2012

Astron. Rep.

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Steady-state solutions for the motion of a passively gravitating globular cluster (GC) inside an inhomogeneous, rotating, ellipsoidal elliptical galaxy (EG) are considered. It is assumed that an EG with a halo is comprised of a triaxial ellipsoid consisting of two layers. The first is formed by an inner, uniform ellipsoid representing the luminous part of the galaxy, while the second corresponds to the space between an inner and outer ellipsoid, which is uniformly filled with dark matter. The triaxial ellipsoids are taken to be homothetic and to have a common center; the space between them is called a homeoid. The outer boundary of the homeoid is the boundary of the galaxy halo. The densities of the luminous part of the EG and the homeoid are different. This picture of an EG is in agreement with our current understanding of galactic structure. The motion of the GC occurs outside the luminous part of the EG, but inside the homeoid, which is treated like a perturbing body. Steady-state solutions (libration points) are found for the GC, and its Lyapunov stability determined. The elliptical galaxies NGC 4472 (M49), NGC 4636, and NGC 4374, which contain a large number of GCs, are used as examples. Analysis of these galaxies shows that the exact expression for the potential of the luminous part of the EG must be used to find the libration points and study their stability, rather than an approximate expression for this potential.

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Section: Type and Stability Of The Libration Pointsmentioning

confidence: 96%

Section: Stationary Solutions (Libration Points)mentioning

confidence: 99%

Section: Examplesmentioning

confidence: 99%

Section: Stationary Solutions (Libration Points)mentioning

confidence: 99%

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Steady-state solutions for the motion of a globular cluster in an inhomogeneous, rotating elliptical galaxy

Gasanov

2012

Astron. Rep.

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“…Here, b 1 ≥ 0 is an arbitrary constant, G is the gravitational constant, and c is the polar semiaxis of the elliptical galaxy E. In addition, the relation between the second eccentricities λ and µ (µ ≤ λ ≤ 1) and the semiaxes a, b, and c of this galaxy and the positive coefficients R k (k = 1, 2, · · · , 6) are given in Gasanov (2006) and Gasanov and Luk'yanov (2002). The system of equations (2) admits of an analogue of the Jacobi integral (Gasanov 2007),…”

Section: Autonomized Equationsmentioning

confidence: 99%

10.1007/s11443-008-3005-2

2010

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We have found libration points and investigated their Lyapunov stability in the problem of the motion of a star inside a layered inhomogeneous rotating elliptical galaxy with a variable mass. We have constructed the surfaces of zero velocity and obtained stability conditions for unsteady motion in the first approximation. We analyze general case where the densities of the galactic nucleus and layers vary with time according to different laws.

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