Planar Hill Research Articles

Regular Quaternion Equations of the Spatial Hill Problem in Kustaanheimo–Stiefel Variables and Quaternion Osculating Elements

Regular quaternion equations of the spatial Hill problem (a variant of the limited three-body problem (Sun, Earth, Moon (or another low-mass moving cosmic body under study)) are obtained, when the distance between two bodies with finite masses is considered very large, in four-dimensional Kustaanheimo-Stiefel variables (KS-variables) within the framework of the elliptical and circular spatial bounded three-body problem, as well as the regular quaternion equations of the planar Hill problem in two-dimensional Levi-Civita variables. In these equations, the variables are KS-variables or Levi-Civita variables and the energy of relative motion of the body under study, or a variable that converts for the circular Hill problem into a constant of motion of this body (the Jacobi integration constant), as well as the planetocentric distance of the Sun and real time associated with a new independent variable by the Sundman differential transformation of time or other more complex differential ratio. These equations are supplemented by the equation of the Earth’s orbit in polar coordinates and the equation for the true anomaly characterizing the Earth’s position in the orbit. The first integral of the obtained equations in KS-variables in the case of a circular problem is established. Another first partial integral in the general case is a bilinear relation connecting KS-variables and their first derivatives. Three new forms of regular equations of the spatial Hill problem in quaternion osculating elements (slowly changing quaternion variables) are proposed. The proposed regular quaternion equations have an oscillatory form or the form of equations with slowly changing variables, which makes it possible to effectively use analytical and numerical methods of oscillation theory and methods of nonlinear mechanics in the study of the Hill problem.

Orbital dynamics in the Hill problem with oblateness

In this paper, we investigate the impact of the oblateness of the secondary body on the motion of a test particle in the near vicinity of the secondary body, specifically in the context of the extended version of the planar Hill problem. To achieve this objective, we conduct a comprehensive survey by systematically classifying the initial conditions of trajectories and scanning the phase space using two-dimensional (2D) maps on various planes. We then numerically integrate the starting conditions on these maps and classify the final states of the test particles as either bounded or unbounded. Bounded orbits are further subclassified as collision orbits and regular (or chaotic), while unbounded orbits are sub-classified according to the sector of the (x,y) plane through which the particle escapes the potential well. Our results reveal that increasing values of the oblateness coefficient reduce the number of escaping orbits and limit the extension of islands of regular motion (in the Liouville–Arnold sense). Regular bounded motion occurs only at larger distances from the secondary body.

Integrability of the generalised Hill problem

We consider a certain two-parameter generalisation of the planar Hill lunar problem. We prove that for nonzero values of these parameters the system is not integrable in the Liouville sense. For special choices of parameters the system coincides with the classical Hill system, the integrable synodical Kepler problem or the integrable parametric Hénon system. We prove that the synodical Kepler problem is not super-integrable, and that the parametric Hénon problem is super-integrable for infinitely many values of the parameter.

Numerical investigation on the Hill’s type lunar problem with homogeneous potential

We consider the planar Hill's lunar problem with a homogeneous gravitational potential. The investigation of the system is twofold. First, the starting conditions of the trajectories are classified into three classes, that is bounded, escaping, and collisional. Second, we study the no-return property of the Lagrange point $L_2$ and we observe that the escaping trajectories are scattered exponentially. Moreover, it is seen that in the supercritical case, with $\alpha \geq 2$, the basin boundaries are smooth. On the other hand, in the subcritical case, with $\alpha < 2$ the boundaries between the different types of basins exhibit fractal properties.

Laboratory experiments on three-dimensional deformable granular landslides on planar and conical slopes

Landslides of subaerial and submarine origin may generate tsunamis with locally extreme amplitudes and runup. While the landslides themselves are dangerous, the hazards are compounded by the generation of tsunamis along coastlines, in enclosed water bodies, and off continental shelves and islands. Tsunamis generated by three-dimensional deformable granular landslides were studied on planar and conical hill slopes in the three-dimensional NEES tsunami wave basin at Oregon State University based on the generalized Froude similarity. A unique pneumatic landslide tsunami generator (LTG) was deployed to control the kinematics and acceleration of the naturally rounded river gravel and cobble landslides to simulate broad ranges of landslide shapes and velocities along the slope. Lateral and overhead cameras are used to measure the landslide shapes and kinematics, while acoustic transducers provide the shape of the subaqueous deposits. The subaerial landslide shape is extracted from the camera images as the landslide propagates under gravity down the hill slope, and surface reconstruction of the landslide is conducted using the stereo particle image velocimetry (PIV) system on the conical hill slope. Subaerial landslide surface velocities are measured with a planar PIV system on the planar hill slope and stereo PIV system on the conical hill slope. The submarine deposits are characterized by the runout distances and the deposit thickness distributions. Larger cobbles are observed producing hummock type features near the maximum runout length. These unique laboratory landslide experiments serve to validate deformable landslide models as well as provide the source characteristics for tsunami generation.

Runup of granular landslide‐generated tsunamis on planar coasts and conical islands

AbstractLarge‐scale physical model experiments of tsunamis generated by granular landslides and volcanic flank collapses are conducted to study the wave runup on both the hill slope laterally adjacent to the landslide and an opposing hill slope. A pneumatic landslide tsunami generator was deployed on planar and convex conical hill slopes to simulate deformable landslides with various geometries and kinematics. On the landslide hill slope, maximum runup and rundown were observed in the landslide impact region followed by adjacent second maxima after the lateral waves were fully formed. The runup and rundown decayed asymptotically from the second maxima. In the conical island scenario, a localized runup amplification was measured on the lee side of the island. Outside the landslide impact region, the effects of the landslide granulometry on the lateral wave runup are minimal. The lateral wave runup on the planar hill slope was generally larger than on the convex conical hill slope outside the landslide impact region. The convex conical hill slope traps less lateral wave energy. The zeroth mode of the edge wave dispersion relation matched the first and second lateral waves on the planar hill slope and the first wave on the convex conical hill slope. Predictive equations for the laterally propagating wave characteristics are derived and a method to predict the runup on an opposing hill slope is presented. The predictive wave and runup equations are benchmarked against the 2007 landslide‐generated tsunami in Chehalis Lake, British Columbia, Canada.

Families of periodic orbits in the planar Hill’s four-body problem

In this work we perform a numerical exploration of the families of planar periodic orbits in the Hill's approximation in the restricted four body problem, that is, after a symplectic scaling, two massive bodies are sent to infinity, by mean of expanding the potential as a power series in m3^1/3, (the mass of the third small primary) and taking the limit case when m3 tends to zero. The limiting Hamiltonian depends on a parameter mu (the mass of the second primary) and possesses some dynamical features from both the classical restricted three- body problem and the restricted four-body problem. We explore the families of periodic orbits of the infinitesimal particle for some values of the mass pa- rameter, these explorations show interesting properties regarding the periodic orbits for this problem, in particular for the Sun-Jupiter-asteroid case. We also offer details on the horizontal and vertical stability of each family.

Physical modelling of tsunamis generated by three-dimensional deformable granular landslides on planar and conical island slopes.

Tsunamis generated by landslides and volcanic island collapses account for some of the most catastrophic events recorded, yet critically important field data related to the landslide motion and tsunami evolution remain lacking. Landslide-generated tsunami source and propagation scenarios are physically modelled in a three-dimensional tsunami wave basin. A unique pneumatic landslide tsunami generator was deployed to simulate landslides with varying geometry and kinematics. The landslides were generated on a planar hill slope and divergent convex conical hill slope to study lateral hill slope effects on the wave characteristics. The leading wave crest amplitude generated on a planar hill slope is larger on average than the leading wave crest generated on a convex conical hill slope, whereas the leading wave trough and second wave crest amplitudes are smaller. Between 1% and 24% of the landslide kinetic energy is transferred into the wave train. Cobble landslides transfer on average 43% more kinetic energy into the wave train than corresponding gravel landslides. Predictive equations for the offshore propagating wave amplitudes, periods, celerities and lengths generated by landslides on planar and divergent convex conical hill slopes are derived, which allow an initial rapid tsunami hazard assessment.

On the periodic orbits and the integrability of the regularized Hill lunar problem

The classical Hill's problem is a simplified version of the restricted three-body problem where the distance of the two massive bodies (say, primary for the largest one and secondary for the smallest one) is made infinity through the use of Hill's variables. The Levi-Civita regularization takes the Hamiltonian of the Hill lunar problem into the form of two uncoupled harmonic oscillators perturbed by the Coriolis force and the Sun action, polynomials of degree 4 and 6, respectively. In this paper, we study periodic orbits of the planar Hill problem using the averaging theory. Moreover, we provide information about the C1 integrability or non-integrability of the regularized Hill lunar problem.

The planar Hill problem with oblate primary

The regularized equations of motion of the planar Hill problem which includes the effect of the oblateness of the larger primary body, is presented. Using the Levi-Civita coordinate transformation as well as the corresponding time transformation, we obtain a simple regularized polynomial Hamiltonian of the dynamical system that corresponds to that of two uncoupled harmonic oscillators perturbed by polynomial terms. The relations between the synodic and regularized variables are also given. The convenient numerical computations of the regularized equations of motion, allow derivation of a map of the group of families of simple-periodic orbits, free of collision cases, of both the classical and the Hill problem with oblateness. The horizontal stability of the families is calculated and we determine series of horizontally critical symmetric periodic orbits of the basic families g and g'.

Central stable/unstable manifolds and the destruction of KAM tori in the planar Hill problem

The classical Hill’s problem is a simplified version of the restricted three body problem (RTBP) where the distance of the two massive bodies (say, primary for the largest one and secondary for the smallest) is made infinity through the use of Hill’s variables and a limiting procedure so that a neighborhood of the secondary can be studied in detail. In this way it is the zeroth-order approximation in powers of μ 1/3. The Levi-Civita regularization takes the Hamiltonian into the form of two uncoupled harmonic oscillators perturbed by the Coriolis force and the Sun action, polynomials of degree 4 and 6, respectively. The goal of this paper is multiple. It presents a detailed description of the main features, including a global description of the dynamics, when the zero velocity curve (zvc) confines the motion. Then it focuses on the collinear equilibrium points and its nearby periodic orbits. Several homoclinic and heteroclinic connections are displayed. Persistence of confined motion when the zvc opens is one of the major concerns. The geometrical behavior of the center-stable/unstable manifolds of the libration points L 1 and L 2 is studied. Suitable Poincaré sections make apparent the relation between these manifolds and the destruction of the invariant KAM tori surrounding the secondary. These results extend immediately to the RTBP. Some practical applications to astronomy and space missions are mentioned. The methodology presented here can be useful on a more general framework for many readers in other areas and not only in celestial mechanics.

Groundwater drainage flow in a soil layer resting on an inclined leaky bed

We study subsurface storm flow from a planar hill slope, a problem that is similar hydraulically to lateral flow toward drains in landfills. Our analysis is based on the linearized one‐dimensional Boussinesq equation (Dupuit‐Forchheimer approximation), which is extended to allow for leakage through the underlying barrier. This linear advection‐diffusion equation has a greater range of validity than the kinematic wave equation. Stating it in terms of the discharge, the variable of primary hydrologic interest, we integrate it numerically, using an adaptation of the Muskingum‐Cunge routing scheme. A single‐step computation of the outflow hydrograph, which combines the convenience of an analytical solution formula with acceptable accuracy, is proposed as a design tool and as a means of parameterization of drainage from hill slopes. Depth profiles are determined afterwards by a simple integration of Darcy's law. Examples of the buildup and recession phases, with and without leakage, demonstrate the application of the computational method.

Generalized Hill's problem: Some cases of complete integrability

In this paper, we are investigating cases of integrability in the planar Hill's problem. The external potential Uextis supposed to be time independent in a given uniformly rotating frame. Cases of integrability of the relative motion of two interacting particles in the vicinity of an equilibrium solution of Uextare found. In all these cases, the form of the second integral is explicitly given, the first being the Jacobian one. Cases in which the interacting potential U between the two particles is of newtonian type are particularized.

Myothermal economy of rat myocardium, chronic adaptation versus acute inotropism.

By means of rapid planar Hill type antimony-bismuth thermophiles the initial heat liberated by papillary muscles was measured synchronously with developed tension for control (C), pressure-overload (GOP), and hypothyrotic (PTU) rat myocardium (chronic experiments) and after application of 10(-6) M isoproterenol or 200 10(-6) M UDCG-115. Economy of force production was analyzed by the ratio of initial heat versus developed tension-time integral. This ratio was found to be reduced by 34% in GOP and by 43% in PTU myocardium (P less than 0.01, respectively) indicating increased economy of force production. In contrast, isoproterenol increased initial heat versus tension-time integral by 70% (P less than 0.01) indicating reduced economy of force production. No change in this ratio was found for UDCG-115. The presented data indicates that long and short term modulation of myocardial energetic costs of force generation is possible. The basic mechanisms for these myocardial alterations are discussed.

Hill's problem: Families of three-dimensional periodic orbits (part I)

Starting from the vertical critical periodic orbits of the planar Hill's problem we computed five families of three-dimensional periodic orbits. It is found that their behaviour is qualitatively similar to that of the corresponding families of the restricted problem.