Analytic perturbation solutions to the Venusian orbiter due to the nonspherical gravitational potential

The method of the geopotential parameters determination with the use of the gradiometry data is considered. The second derivative of the gravitational potential in the correction equation on the rectangular coordinates x, y, z is used as a measured variable. For the calculated value of the measured quantity required for the formation of a free member of the correction equation, the the Cunningham polynomials were used. We give algorithms for computing the second derivatives of the Cunningham polynomials on rectangular coordinates x, y, z, which allow to calculate the second derivatives of the geopotential at the rectangular coordinates x, y, z.Then we convert derivatives obtained from the Cartesian coordinate system in the coordinate system of the gradiometer, which allow to calculate the free term of the correction equation. Afterwards the correction equation coefficients are calculated by differentiating the formula for calculating the second derivative of the gravitational potential on the rectangular coordinates x, y, z. The result is a coefficient matrix of the correction equations and corrections vector of the free members of equations for each component of the tensor of the geopotential. As the number of conditional equations is much more than the number of the specified parameters, we go to the drawing up of the system of normal equations, from which solutions we determine the required corrections to the harmonic coefficients.

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Evaluation of the Second, Fourth and Sixth Harmonics in the Earth's Gravitational Potential

Nature ◽

10.1038/187490b0 ◽

1960 ◽

Vol 187 (4736) ◽

pp. 490-491 ◽

Cited By ~ 5

Author(s):

D. G. KING-HELE

Keyword(s):

Gravitational Potential

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Gravitational Interaction in the Chimney Lattice Universe

Universe ◽

10.3390/universe7040101 ◽

2021 ◽

Vol 7 (4) ◽

pp. 101

Author(s):

Maxim Eingorn ◽

Andrew McLaughlin ◽

Ezgi Canay ◽

Maksym Brilenkov ◽

Alexander Zhuk

Keyword(s):

Helmholtz Equation ◽

Gravitational Potential ◽

Fourier Expansion ◽

Gravitational Interaction ◽

Alternative Solution ◽

The Third ◽

Present Universe ◽

Delta Functions ◽

Yukawa Potentials ◽

The Universe

We investigate the influence of the chimney topology T×T×R of the Universe on the gravitational potential and force that are generated by point-like massive bodies. We obtain three distinct expressions for the solutions. One follows from Fourier expansion of delta functions into series using periodicity in two toroidal dimensions. The second one is the summation of solutions of the Helmholtz equation, for a source mass and its infinitely many images, which are in the form of Yukawa potentials. The third alternative solution for the potential is formulated via the Ewald sums method applied to Yukawa-type potentials. We show that, for the present Universe, the formulas involving plain summation of Yukawa potentials are preferable for computational purposes, as they require a smaller number of terms in the series to reach adequate precision.

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Gravitational potential: a thought experiment

Physics Education ◽

10.1088/0031-9120/50/4/397 ◽

2015 ◽

Vol 50 (4) ◽

pp. 397-398 ◽

Cited By ~ 1

Author(s):

William H Baird

Keyword(s):

Gravitational Potential ◽

Thought Experiment

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The Ballistic Particle Model

Symposium - International Astronomical Union ◽

10.1017/s0074180900032629 ◽

1983 ◽

Vol 100 ◽

pp. 133-134

Author(s):

Frank N. Bash

Keyword(s):

Radial Velocity ◽

Gravitational Potential ◽

Molecular Clouds ◽

Milky Way ◽

Terminal Velocity ◽

Particle Model ◽

The Sun ◽

Giant Molecular Clouds ◽

Spiral Pattern ◽

Armed Spiral

Bash and Peters (1976) suggested that giant molecular clouds (GMC's) can be viewed as ballistic particles launched from the two-armed spiral-shock (TASS) wave with orbits influenced only by the overall galactic gravitational potential perturbed by the spiral gravitational potential in the arms. For GMC's in the Milky Way, the model predicts that the radial velocity observed from the Sun increases with age (time since launch). We showed that the terminal velocity of CO observed from l ≃ 30° to l ≃ 60° can be understood if all GMC's are born in the spiral pattern given by Yuan (1969) and live 30 × 106 yrs. Older GMC's were predicted to have radial velocities which exceed observed terminal velocities.

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Perturbation Solution for One-Dimensional Flow to a Constant-Pressure Boundary in a Stress-Sensitive Reservoir

Transport in Porous Media ◽

10.1007/s11242-021-01570-w ◽

2021 ◽

Author(s):

Amarjot Singh Bhullar ◽

Gospel Ezekiel Stewart ◽

Robert W. Zimmerman

Keyword(s):

Approximate Solution ◽

Pore Pressure ◽

Constant Pressure ◽

Initial Permeability ◽

Order Perturbation ◽

Zeroth Order ◽

One Dimensional ◽

First Order ◽

Outflow Boundary ◽

Perturbation Solutions

Abstract Most analyses of fluid flow in porous media are conducted under the assumption that the permeability is constant. In some “stress-sensitive” rock formations, however, the variation of permeability with pore fluid pressure is sufficiently large that it needs to be accounted for in the analysis. Accounting for the variation of permeability with pore pressure renders the pressure diffusion equation nonlinear and not amenable to exact analytical solutions. In this paper, the regular perturbation approach is used to develop an approximate solution to the problem of flow to a linear constant-pressure boundary, in a formation whose permeability varies exponentially with pore pressure. The perturbation parameter αD is defined to be the natural logarithm of the ratio of the initial permeability to the permeability at the outflow boundary. The zeroth-order and first-order perturbation solutions are computed, from which the flux at the outflow boundary is found. An effective permeability is then determined such that, when inserted into the analytical solution for the mathematically linear problem, it yields a flux that is exact to at least first order in αD. When compared to numerical solutions of the problem, the result has 5% accuracy out to values of αD of about 2—a much larger range of accuracy than is usually achieved in similar problems. Finally, an explanation is given of why the change of variables proposed by Kikani and Pedrosa, which leads to highly accurate zeroth-order perturbation solutions in radial flow problems, does not yield an accurate result for one-dimensional flow. Article Highlights Approximate solution for flow to a constant-pressure boundary in a porous medium whose permeability varies exponentially with pressure. The predicted flowrate is accurate to within 5% for a wide range of permeability variations. If permeability at boundary is 30% less than initial permeability, flowrate will be 10% less than predicted by constant-permeability model.

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