scholarly journals Effect of Perturbed Potentials on the Stability of Libration Points in the Restricted Problem

1979 ◽  
Vol 81 ◽  
pp. 57-57
Author(s):  
K. B. Bhatnagar ◽  
P. P. Hallan
Keyword(s):  
Restricted Problem ◽  
Classical Problem ◽  
Oblate Spheroid ◽  
Libration Points ◽  
The Other ◽  
Triangular Points ◽  
Oblate Spheroids ◽  
The Stability

The location and the stability of the libration points in the restricted problem have been studied when there are perturbations in the potentials between the bodies. It is seen that if the perturbing functions involving the parameters α,α1,α2 satisfy certain conditions, there are five libration points, two triangular and three collinear. It is further observed that the collinear points are unstable and for the triangular points, the range of stability increases or decreases depending upon whether the perturbation point (α,α1,α2) lies on one or the other side of the plane Aα + Bα1 + Cα2 = 0, and it remains the same if the point lies on the plane, where A,B,C depend on the perturbations. The theory is verified in the following four cases: (1) there are no perturbations in the potentials (classical problem), (2) only the bigger primary is an oblate spheroid, (3) both the primaries are oblate spheroids, and (4) the primaries are spherical in shape and the bigger is a source of radiation.

Eulerian libration points of restricted problem of three oblate spheroids

1993 ◽  
Vol 63 (1) ◽  
pp. 23-28 ◽  
Author(s):  
S. M. El-Shaboury ◽  
M. A. El-Tantawy
Keyword(s):  
Restricted Problem ◽  
Libration Points ◽  
Oblate Spheroids

scholarly journals Unveiling Perturbing Effects of P-R Drag on Motion around Triangular Lagrangian Points of the Photogravitational Restricted Problem of Three Oblate Bodies

2021 ◽  
pp. 10-31
Author(s):  
Tajudeen Oluwafemi Amuda ◽  
Oni Leke ◽  
Abdulrazaq Abdulraheem
Keyword(s):  
Radiation Pressure ◽  
Equilibrium Points ◽  
Three Body Problem ◽  
Small Perturbations ◽  
Lagrangian Points ◽  
Triangular Points ◽  
Oblate Spheroids ◽  
The Stability ◽  
Zero Velocity Surfaces ◽  
Three Body

The perturbing effects of the Poynting-Robertson drag on motion of an infinitesimal mass around triangular Lagrangian points of the circular restricted three-body problem under small perturbations in the Coriolis and centrifugal forces when the three bodies are oblate spheroids and the primaries are emitters of radiation pressure, is the focus of this paper. The equations governing the dynamical system have been derived and locations of triangular Lagrangian points are determined. It is seen that the locations are influenced by the perturbing forces of centrifugal perturbation and the oblateness, radiation pressure and, P-R drag of the primaries. Using the software Mathematica, numerical analysis are carried out to demonstrate how the dynamical elements: mass ratio, oblateness, radiation pressure, P-R drag and centrifugal perturbation influence the positions of triangular equilibrium points, zero velocity surfaces and the stability. Our investigation reveals that, though the radiation pressure, oblateness and centrifugal perturbation decrease region of stability when motion is stable, however, they are not the influential forces of instability but the P-R drag. In the region when motion around the triangular points are stable an inclusion of the P-R drag of the bigger primary even by an almost negligible value of 1.04548*10-9 overrides other effect and changes stability to instability. Hence, we conclude that the P-R drag is a strong perturbing force which changes stability to instability and motion around triangular Lagrangian points remain unstable in the presence of the P-R drag.

scholarly journals The ultimate shape of pebbles, natural and artificial

1942 ◽  
Vol 181 (985) ◽  
pp. 107-118 ◽  
Keyword(s):  
Oblate Spheroid ◽  
The Other ◽  
Spherical Form ◽  
Oblate Spheroids ◽  
The One ◽  
Specific Curvature

Flint pebbles that have been worn down enough to be symmetrical may be spherical, or may approximate to prolate or oblate spheroids or to ellipsoids. When, however, the section is at all elongated, it is not as a rule accurately elliptical, but except at the axial points it lies entirely outside an ellipse adjusted to the same axes. Thus, if one of the axes is much smaller than the other, the pebble is much flatter than an ellipsoid. Considering the quasi-spheroidal pebbles, whether prolate or oblate, these are always flattened at the poles, and the very oblate ones become tabular or even concave at the poles. These statements hold for flint pebbles, but a large quartzite pebble is figured which is pretty accurately an oblate spheroid, the meridional sections being truly elliptical. Experiments are described with chalk pebbles, initially shaped as prolate or oblate spheroids, and the change of figure under abrasion is observed. This depends in some degree on what is the abrasive. Steel nuts, nails ('tin tacks’) and small shot were used. In general the axes tend to approach equality, but not rapidly enough for the spherical form to be attained before the pebble has disappeared. The form initially spheroidal becomes flattened at the poles just like the natural flint pebbles, and may become concave, as flints sometimes do. The circumstances determining the rate of abrasion at any point are considered, and it is shown that this abrasion cannot be merely a function of the local specific curvature. The figure at points other than the one under consideration comes into question, so that the problem in this form has no determinate answer. It is shown by simple mechanical considerations how the concave form arises.

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