A way to compute the gravitational potential for near-spherical geometries

Grinfeld, Pavel; Wisdom, Jack

doi:10.1090/s0033-569x-06-01001-2

Cited by 12 publications

(6 citation statements)

References 6 publications

(7 reference statements)

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“…We restrict our view to those objects and functions required for our model problems. The CMS has been described in great detail in earlier publications [12,8]. The CMS, deeply rooted in tensor calculus [21], [27], [31], was originated by Jacques Hadamard.…”

Section: Calculus Of Moving Surfacesmentioning

confidence: 99%

A term rewriting system for the calculus of moving surfaces

Boady

Grinfeld

Johnson

2013

Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation

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The calculus of moving surfaces (CMS) is an analytic framework that extends the tensor calculus to deforming manifolds. We have applied the CMS to a number of boundary variation problems using a Term Rewrite System (TRS). The TRS is used to convert the initial CMS expression into a form that can be evaluated. The CMS produces expressions that are true for all coordinate spaces. This makes it very powerful but applications remain limited by a rapid growth in the size of expressions. We have extended results on existing problems to orders that had been previously intractable. In this paper, we describe our TRS and our method for evaluating CMS expressions on a specific coordinate system. Our work has already provided new insight into problems of current interest to researchers in the CMS.

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Section: Calculus Of Moving Surfacesmentioning

confidence: 99%

A term rewriting system for the calculus of moving surfaces

Boady

Grinfeld

Johnson

2013

Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation

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“…They studied a model similar to that of Busse (Busse 1974), but of a non-rotating body, and taking into account the translational motion of the mantle, that is to say, removing the motionless condition considered for this layer in Busse's investigation. The problem was solved using a perturbative approach of a linearized version of the Euler equations and a boundary technique based on the calculus of moving surfaces (Grinfeld & Wisdom 2006), from which the Newtonian dynamical equations of the mantle and the inner core were constructed under the assumption of small amplitude and velocity motion. The resulting expression of the Slichter mode turned out to be equivalent to that of Busse (1974) in the absence of rotation and when the mass of the inner core is negligible in comparison with that of the whole body.…”

Section: Earth Internal Translational Motionsmentioning

confidence: 99%

Free Translational Oscillations of Icy Bodies With a Subsurface Ocean Using a Variational Approach

Escapa

Fukushima

2011

The Astronomical Journal

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We analyze the influence of the interior structure of an icy body with an internal ocean on the relative translational motions of its solid constituents. We consider an isolated body differentiated into three homogeneous layers with spherical symmetry: an external ice-I layer, a subsurface ammonia-water ocean, and a rocky inner core. This composition represents icy bodies such as Europa, Titania, Oberon, and Triton, as well as Pluto, Eris, Sedna, and 2004 DW. We construct the equations of motion by assuming that the solid constituents are rigid and that the ocean is an ideal fluid, the internal motion being characterized by the relative translations of the solids and the induced flow in the fluid. Then we determine the dynamics of the icy body using the methods of analytical mechanics, that is, we compute the kinetic energy and the gravitational potential energy, and obtain the Lagrangian function. The resulting solution of the Lagrange equations shows that the solid layers perform translational oscillations of different amplitudes with respect to the barycenter of the body. We derive the dependence of the frequency of the free oscillations of the system on the characteristics of each layer, expressing the period of the oscillations as a function of the densities and masses of the ocean and the rocky inner core, and the mass of the icy body. We apply these results to previously developed subsurface models and obtain numerical values for the period and the ratio between the amplitudes of the translational oscillations of the solid components. The features obtained are quite different from the cases of Earth and Mercury. Our analytical formulas satisfactorily explain the source of these differences. When models of the same icy body, compatible with the existence of an internal ocean, differ in the thickness of the ice-I layer, their associated periods experience a relative variation of at least 10%. In particular, the different models for Titania and Oberon exhibit a larger variation of about 37% and 30%. This indicates an absolute difference of the order of three and two hours, respectively. This suggests that the free period of the internal oscillations might provide a new procedure to constrain the internal structure of icy bodies with a subsurface ocean.

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“…To a reader not familiar with the technique, our approach may seem complicated. Even so, we believe that our conclusions, as well as many of our intermediate expressions, are easily interpretable without a thorough understanding of the calculus of moving surfaces, as long as the reader has an intuitive understanding of the interface velocity C. We do not include a description of the calculus of moving surfaces here, but it can be found in [2], including the definitions of all the quantities that we use and a summary of the key identities.…”

Section: A Note About the Employed Techniquementioning

confidence: 99%

“…G, again, is the gravitational constant. The rate of change ∂ψ/∂τ in the potential field ψ at time τ = 0 can be expressed in terms of C lm n as follows [2]: …”

Section: Spherical Configurationsmentioning

confidence: 99%

Total gravitational energy of a slightly ellipsoidal trilayer planet

Grinfeld

Wisdom²

2006

Quart. Appl. Math.

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Abstract. We use a perturbational technique to compute the total gravitational energy of a slightly ellipsoidal trilayer planet as a function of the two sets of Euler angles. A second order computation is required since the torque is proportional to the product of the ellipticities of the inner core and the mantle. Although we focus on ellipsoidal perturbations, the intermediate analytical expressions are valid for arbitrary small deformations of the spherical configuration.The primary application of the expression for the total gravitational energy is in the Lagrangian formulation of dynamics. As a by-product, we determine the gravitationally stable equilibrium orientation of the rigid inner core. Due to symmetry, the six coaxial configurations are equilibrium. We show how to identify the stable configuration and to prove its uniqueness. and, quite likely, Mercury [3] are the solar system representatives of the class of planets that have a solid inner core, a fluid outer core and a solid mantle. For large planets these components are essentially spherical due to gravity, but not quite, and it is the deviations from the spherically symmetric configurations that are responsible for a number of interesting effects. A detailed description of a real planet's geometry is, of course, quite complicated -there are chemical inhomogeneities in the interior, uneven heating by the planet's star, and irregularities on the surface due to the elastic resistance of matter. We ignore these effects and employ an idealization that is designed to capture a specific small perturbation -the slight ellipticity of the planet. Ellipticity of planets is interesting because it causes precessions of the spin axis and librations which are deviations from uniform rotation. Ellipticity can be caused by the planet's rotation (the Earth) or heat convection (Mercury). While the Earth's ellipsoids are essentially axisymmetric, Mercury's famous resonant rotation [5] indicates that Mercury must be modeled as a triaxial ellipsoid. Introduction. The Earth

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A way to compute the gravitational potential for near-spherical geometries

Cited by 12 publications

References 6 publications

A term rewriting system for the calculus of moving surfaces

A term rewriting system for the calculus of moving surfaces

Free Translational Oscillations of Icy Bodies With a Subsurface Ocean Using a Variational Approach

Total gravitational energy of a slightly ellipsoidal trilayer planet

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