A case of commensurability induced by oblateness

In the three-dimensional restricted three-body problem, it is known that there exists a near one-to-one commensurability ratio between the planar angular frequencies (s _{1, 2, 3}) and the corresponding angular frequency (S ₂) in thez-direction at the three collinear equilibria (L _{1, 2, 3}), which is significant for small and practically important values of the mass parameter (). When the more massive primary is treated as an oblate spheroid with its equatorial plane coincident with the plane of motion of the primaries, it is established that oblateness induces a one-to-one commensurability at the exterior pointL ₃ (to the right of the more massive primary) and at the interior pointL ₂ for 01/2 and that atL ₁ no such commensurability exists. However, the values of the oblateness coefficient (A ₁) involved atL ₂ are too high to have any practical significance, while those atL ₃ being small for small values of may be useful for generating periodic orbits of the third kind.     

The linear stability of libration points of the photogravitational restricted three-body problem when the smaller primary is an oblate spheroid

This paper deals with the stationary solutions of the planar restricted three-body problem when the more massive primary is a source of radiation and the smaller primary is an oblate spheroid with its equatorial plane coincident with the plane of motion. The collinear equilibria have conditional retrograde elliptical periodic orbits around them in the linear sense, while the triangular points have long- or short-periodic retrograde elliptical orbits for the mass parameter 0 < _crit, the critical mass parameter, which decreases with the increase in oblateness and radiation force. Through special choice of initial conditions, retrograde elliptical periodic orbits exist for the case = _crit, whose eccentricity increases with oblateness and decreases with radiation force for non-zero oblateness.     

Stability of the triangular libration points of the circular restricted problem in the presence of resonances

The stability of triangular libration points, when the bigger primary is a source of radiation and the smaller primary is an oblate spheroid. has been investigated in the resonance cases ₁ = 2₂ and ₁ = 3₂. The motion is unstable for all the values of parameters q and A when ₁ = 2₂ and the motion is unstable and stable depending upon the values of the parameters q and A when ₁ = 3₂. Here q is the radiation parameter and A is the oblateness parameter.     

Bifurcations of Plane to 3d Periodic Orbits in the Restricted Three-Body Problem with Oblateness

We consider the bifurcation of 3D periodic orbits from the plane of motion of the primaries in the restricted three-body problem with oblateness. The simplest 3D periodic orbits branch-off at the plane periodic orbits of indifferent vertical stability. We describe briefly suitable numerical techniques and apply them to produce the first few such vertical-critical orbits of the basic families of periodic orbits of the problem, for varying mass parameter and fixed oblateness coefficent A₁ = 0.005, as well as for varying A₁ and fixed = 1/2. The horizontal stability of these orbits is also determined leading to predictions about the stability of the branching 3D orbits.     

The effect of oblateness of the bigger primary on collinear libration points in the restricted problem of three bodies

In the restricted problem of three bodies, the effect of oblateness of the bigger primary appears as an additional term in the potential. As a result, the location of libration points and the roots of the characteristic equation at these points depend not only upon the mass parameter but also on the oblateness termI of the bigger primary. Series solutions are developed in terms of andI which are used for locating the collinear libration points and for determining the mean motions and characteristic exponents at these points.The work is supported by a fellowship awarded to the second author by University Grant Commission, India.     

Parameter values for stable low-inclination periodic motion in the restricted three-body problem with oblateness

The known intervals of possible stability, on the mgr-axis, of basicfamilies of 3D periodic orbits in the restricted three-body problem areextended into -A₁ regions for oblate larger primary, A ₁ beingthe oblateness coefficient. Eight regions, corresponding to the basicstable bifurcation orbits l1v, l1v, l2v, l3v, m1v, m1v,m2v, i1v are determined and related branching 3D periodic orbits arecomputed systematically and tested for stability. The regions for l1v,m1v and m2v survive the test emerging as the regions allowing thesimplest types of stable low inclination 3D motion. For l1v, l2v,l3v, m1v and m2v oblateness seems to have a stabilising effect,while stability of i1v survives only for a very small range of A ₁values.     

An analysis of 1983 observations of facular contrast with an extreme limb photometer

Analysis of facular contrast I/I _qs from Extreme Limb Photometer (ELP) data of the summer of 1983 yield a mean contrast of 0.91 ± 0.19% and 1.57 ± 0.16% for apertures 1 and 2 located at = 0.198 and at = 0.111, respectively. The ratio of the mean contrast in the outer aperture (closer to the limb) to that of the inner one is 1.71 ± 0.40, indicating an increase in the mean facular contrast toward the limb. This result is in agreement with observations made in 1975, 1979, and 1982. The errors are dominated by the random presence of solar active regions. The combined results from all seasons follow an approximately ^–1 curve. Facular excess solar oblateness signals for 1983 are 33.8 ± 6.6 arc ms and 16.5 ± 2.1 arc ms for ELP apertures 1 and 2, respectively, in reasonable agreement with the 1983 excess solar oblateness results of Dicke, Kuhn, and Libbrecht (1985).     

Satellite theory

In this paper dynamical characteristics of satellites are outlined by classifying the satellites into three categories according to the values of the solar tidal factor (n/n)² which is the disturbing factor due to the sun and the oblateness factor of the primary planetJ ₂/a ². For inner satellites (n/n)² is much smaller thanJ ₂/a ² and there are several pairs among them, for which the mean motions are commensurable to each other, and for some of them secular accelerations in the mean longitudes have been detected. For outer satellites (n/n)² is much larger and the solar perturbations are dominant. For intermediary satellites the motion of the pole of the orbital plane is not so simple as those of the satellites of the other categories.     

Position and velocity sensitivities at the triangular libration points in the restricted problem of three bodies when the bigger primary is an oblate body

In this paper we have examined the stability of triangular libration points in the restricted problem of three bodies when the bigger primary is an oblate spheroid. Here we followed the time limit and computational process of Tuckness (Celest. Mech. Dyn. Mech. 61, 1–19, 1995) on the stability criteria given by McKenzie and Szebehely (Celest. Mech. 23, 223–229, 1981). In this study it was found that in comparison to other studies the value of the critical mass μ _c has been reduced due to oblateness of the bigger primary, i.e. the range of stability of the equilateral triangular libration points reduced with the increase of the oblateness parameter I and hence the order of commensurability was increased.     

A new outlook on the `Differential Theory' of the solar Quadrupole Moment and Oblateness

The modeling of the quadrupole moment J ₂ and of the oblateness , two key solar parameters, derives from the development in successive spherical harmonics of the gravitational potential. These harmonics are representative of the shape of the Sun, by studying the local distortion of the internal layers, under their distribution of mass and velocity. The first aim of this paper is to study, over the radius r and the colatitude , the structure of the internal layers of the Sun through a geometrical approach, considering J ₂ and under a differential form. The second aim is to determine their theoretical values, after integration over r and , taking the best available models of density and rotation into account constrained by helioseismic data. The novelty of our approach lies in the analysis of the profiles of the two above-mentioned solar parameters, under differential form, from the core to the surface. This analysis allows us to comply with the physical processes located in the transition regions, namely the tachocline and maybe a new subsurface layer which could be called the leptocline. The profiles of tildeJ ₂ show two sharp decreases, which are directly connected to the shear layers located at 0.7 R and beneath the surface. The profiles of tilde show five changes of curvature, which seem to be connected to solar processes, such as the matter circulation flows, seismic events or the storage of the magnetic field, phenomena taking place in the transition regions. These sets of profiles allow us to propose a configuration scenario composed of a double layer. Moreover, as a result of the theoretical determination of tildeJ ₂ and tilde, the values at the surface of the quadrupole moment and of the oblateness can be deduced, which are 1.60×10^–7 and 8.77×10^–6, respectively. As a result of an analysis of available data, we may admit J ₂=(2.0±0.4)×10^–7. The theoretical computations of J ₂ and at the surface will be confronted in the near future with the values measured in space by means of the Picard microsatellite. This mission should permit one to measure at the same time both the total solar irradiance and the latitudinal diameters in any position angle (after removing the passing spots or faculae at the limb).     

Existence and stability of triangular points in the restricted three-body problem with numerical applications

In this paper, we prove that the locations of the triangular points and their linear stability are affected by the oblateness of the more massive primary in the planar circular restricted three-body problem, considering the effect of oblateness for J ₂ and J ₄. After that, we show that the triangular points are stable for 0<μ<μ _c and unstable when , where μ _c is the critical mass parameter which depends on the coefficients of oblateness. On the other hand, we produce some numerical values for the positions of the triangular points, μ and μ _c using planets systems in our solar system which emphasis that the range of stability will decrease; however this range sometimes is not affected by the existence of J ₄ for some planets systems as in Earth–Moon, Saturn–Phoebe and Uranus–Caliban systems.     

Wall temperature oscillation on convection flow in a rotating fluid with hall and ion-slip currents

An analysis of Hall and ion-slip current effects on the MHD free-convection flow of a partiallyionised gas past an infinite vertical porous plate in a rotating frame of reference is carried out. A strong magnetic field is applied perpendicular to the plate and the plate temperature oscillates in time about a constant non-zero mean. The problem has been solved for the velocity and temperature fields and the effects of _e (the Hall parameter), _i (the ion-slip parameter),E _r (rotation parameter), and have been discussed and shown graphically.     

Stability of the Triangular Points Under Combined Effects of Radiation and Oblateness in the Restricted Three-Body Problem

This paper deals with the existence of triangular points and their linear stability when the primaries are oblate spheroid and sources of radiation considering the effect of oblateness up to 10^?6 of main terms in the restricted three-body problem; we see that the locations of the triangular points are affected by the oblateness of the primaries and solar radiation pressure. It is further seen that these points are stable for 0 ≤ μ ≤μ _c ; and unstable for μ _c  ≤ μ ≤1/2; where μ _c is the critical mass value depending on terms which involve parameters that characterize the oblateness and radiation repulsive forces such that $ \mu_{c} \in (0,1/2) $ ; in addition to this an algorithm has been constructed to calculate the critical mass value.     

Finite-difference analysis of the MHD stokes problem for a vertical plate with Hall and ion-slip currents

An explicit finite-difference method is employed to study the MHD free convection heat generating fluid past an impulsively started vertical infinite plate when a strong magnetic field is applied perpendicular to the plate. The velocity and temperature profiles are shown on graphs and the results are discussed in terms of the non-dimensional parameters _e (Hall parameter), _i (ionslip parameter), (heat source parameter), and Gr (Grashof number).     

Stability of equilibrium points in the generalized perturbed restricted three-body problem

This paper studies the existence and stability of equilibrium points under the influence of small perturbations in the Coriolis and the centrifugal forces, together with the non-sphericity of the primaries. The problem is generalized in the sense that the bigger and smaller primaries are respectively triaxial and oblate spheroidal bodies. It is found that the locations of equilibrium points are affected by the non-sphericity of the bodies and the change in the centrifugal force. It is also seen that the triangular points are stable for 0<μ<μ _c and unstable for \(\mu_{c}\le\mu <\frac{1}{2}\), where μ _c is the critical mass parameter depending on the above perturbations, triaxiality and oblateness. It is further observed that collinear points remain unstable.     

Is there an oblique magnetic rotator inside the Sun?

The size of the rotational splitting recently observed (Claverie et al., 1981) is correlated with the 12.2^d variation in the measurements of solar oblateness observed by Dicke (1976) and implies a convection zone of depth of 0.1 R . The near equality of amplitudes of global velocity oscillations (Claverie et al., 1981) of the various m components of the l = 1 and l = 2 modes as seen from the Earth viewing the Sun nearly along the equator is unexpected for pure rotational splitting. It is suggested that a magnetic perturbation is present and an oblique asymmetric magnetic rotator with magnetic fields of a few million gauss is responsible. A more detailed account was submitted to Nature.Proceedings of the 66th IAU Colloquium: Problems in Solar and Stellar Oscillations, held at the Crimean Astrophysical Observatory, U.S.S.R., 1–5 September, 1981.     

Spin parameter and scale-free density perturbations in hierarchical clustering

The present attempt aims to predict the dependence of the spin parameter, , the angular momentum,J, and the typical radius,a _vir, on the mass,M, which have been found inN-body simulations of expanding density perturbations in hierarchical clustering, when virialization is attained. We show that M ⁰ for systems with same adimensional density distribution and velocity distribution, and in particular for scale-free density perturbations in hierarchical clustering. In the special case of ellipsoidal perturbations, it is also found:J M ^7/4,a _vir M ^1/2. All these results turn out to be in close agreement withN-body simulations, despite the simple model adopted. Expanding and virialized perturbations are modelled, respectively, by homogeneous and heterogeneous, similar ellipsoids which allow flat rotation curves far from the centre. Both energy and angular momentum maintain from a given time on, lying between the beginning of strong decoupling from the Hubble flow and the occurrence of maximum volume. Then the whole set of virialized ellipsoidal configurations with same energy and angular momentum are derived, and the dependence of the spin parameter on the anisotropy parameter, _pec is investigated. Turning our attention to the formation of galaxies, we derive the total mass as a function of the collapse factor, using the empirical anticorrelation between dark to visible mass ratio within the optical radius of disk galaxies and their luminous masses. Observational data related to a sample of elliptical galaxies provide evidence that the contraction in these bodies occurred in proportion to the square root of the ratio of total to luminous mass. On the contrary, it is deduced that dissipation of angular momentum in elliptical galaxies occurred more or less at the same rate. If both shape and anisotropy parameter are preserved during the collapse, typical axis rations ₂₁ = 0.98, ₃₁ = 0.69, are found to correspond to a moderate anisotropy, _pec 0.27, with a small dependence on the spin parameter in the range allowed.     

A study of the degradation of daytime astronomical images due to turbulence in the lower atmosphere by measurement of the standard deviation of the angle of arrival

Fluctuations in the angle-of-arrival of a luminous wavefront are investigated by placing a narrow slit over the image of the solar limb, and observing the entrance pupil through this diaphragm. Measurement of the standard deviation _AA of these fluctuations, in a direction perpendicular to the solar limb, provides sufficient information to estimate an absolute parameter characterizing the seeing, i.e. Fried's parameter r ₀ or the equivalent parameter _L C _N ²(h) dh which gives the integral, over an atmosphere thickness L which depends on the spatial and angular filterings performed by the slit, of the structure constant C _N ² characterizing fluctuations in the air's refractive index. Simultaneous estimations of _AA ² and _L C _N ² (h) dh permit a linear relationship to be established between these two quantities. The result is interpreted theoretically by integration of the two-dimensional power spectrum for the angle-of-arrival fluctuations. The influence of filtering effects is discussed.Such a calibration can be performed for a given telescope and slit, enabling the system to be used as a seeing monitor.     

Unsteady MHD flow past a porous plate with step function change in suction at slip flow regime

Unsteady two-dimensional hydromagnetic flow of an electrically conducting viscous incompressible fluid past a semi-infinite porous flat plate with step function change in suction velocity is studied allowing a first order velocity slip at the boundary condition. The solution of the problem is obtained in closed form and the results are discussed with the aid of graphs for various parameters entering in the problem.Notations B intensity of magnetic field - H magnetic field parameter,H=(M+1/4)^1/2–1/2 - h rarefaction parameter - L ₁ slip coefficient; ;I, mean free path of gas molecules;f, Maxwell's reflection coefficient - M magnetic field parameter - r suction parameter - t time - t dimensionless time - u velocity of the fluid - u dimensionless velocity of the fluid - U velocity of the fluid at infinity - v suction velocity - v ₁ suction velocity att<=0 - v ₂ suction velocity att>0 - x distance parallel to the plate - y distance normal to the plate - y nondimensional distance normal to the plate - v kinematic viscosity - electric conductivity of the fluid - density of the fluid - shear stress at the wall - nondimensional shear stress at the wall - erf error function - erfc complementary error function     

Third-order secular Solution of the variational equations of motion of a satellite in orbit around a non-spherical planet

We constructed an analytical theory of satellite motion up to the third order relative to the oblateness parameter of the Earth (J ₂). Equations of secular variations was developed for the first three orbital elements (a, e, i) of an artificial satellite. The secular variations are solved in a closed form.