Fitting a triaxial ellipsoid to a geoid model

The aim of this work is the determination of the parameters of Earth’s triaxiality through a geometric fitting of a triaxial ellipsoid to a set of given points in space, as they are derived from a geoid model. Starting from a Cartesian equation of an ellipsoid in an arbitrary reference system, we develop a transformation of its coefficients into the coordinates of the ellipsoid center, of the three rotation angles and the three ellipsoid semi-axes. Furthermore, we present different mathematical models for some special and degenerate cases of the triaxial ellipsoid. We also present the required mathematical background of the theory of least-squares, under the condition of minimization of the sum of squares of geoid heights. Also, we describe a method for the determination of the foot points of the set of given space points. Then, we prepare suitable data sets and we derive results for various geoid models, which were proposed in the last fifty years. Among the results, we report the semi-axes of the triaxial ellipsoid of geometric fitting with four unknowns to be 6378171.92 m, 6378102.06 m and 6356752.17 m and the equatorial longitude of the major semi-axis –14.9367 degrees. Also, the parameters of Earth’s triaxiality are directly estimated from the spherical harmonic coefficients of degree and order two. Finally, the results indicate that the geoid heights with reference to the triaxial ellipsoid are smaller than those with reference to the oblate spheroid and the improvement in the corresponding rms value is about 20 per cent.

Introducing a New 3D Dynamical Model for Barred Galaxies

The regular or chaotic dynamics of an analytical realistic three dimensional model composed of a spherically symmetric central nucleus, a bar and a flat disk is investigated. For describing the properties of the bar, we introduce a new simple dynamical model and we explore the influence on the character of orbits of all the involved parameters of it, such as the mass and the scale length of the bar, the major semi-axis and the angular velocity of the bar, as well as the energy. Regions of phase space with ordered and chaotic motion are identified in dependence on these parameters and for breaking the rotational symmetry. First, we study in detail the dynamics in the invariant planez = pz= 0 using the Poincaré map as a basic tool and then study the full three-dimensional case using the Smaller Alignment index method as principal tool for distinguishing between order and chaos. We also present strong evidence obtained through the numerical simulations that our new bar model can realistically describe the formation and the evolution of the observed twin spiral structure in barred galaxies.

A New Design Method for the Rotor Profile Curve of the Cam Pump

The cam pumps are especially suitable for the transmission of the liquids with different impurities or pure solid materials. A new design method for the rotor profile curve of the cam pump was put forward to improve the wear-proof of the addendum of the rotors. A rotor with a simple profile curve was designed and then the profile curve of another rotor would be generated by a tool whose cutting profile curve was the same as the designed rotor profile curve. This method was stated and a mathematic model was built based on gear meshing theory by choosing an ellipse as the designed rotor profile curve. The influence of the design parameters on the generated rotor profile curve, the area utilization coefficient of the pump and the relative normal curvature of the two profile curves at the contact points were analyzed. The results demonstrated as: the designed profile curve can be some kind of simple continuous curve and the generated profile curve is continuous curve as well; the area utilization coefficient, C, increases by increasing the major semi-axis of the ellipse, a, or decreasing the minor semi-axis, b, if the designed profile surface is an elliptic one; the maximum of the area utilization coefficient, Cmax, will decrease when the blade number of the rotor 2, z2, increases because of the undercutting; the maximum of the area utilization coefficient is 0.48 when z2 is 2; the blade number of the rotor 2 is no more than 4.

Calculation of the Stress Intensity Factors for Elliptical Cracks Embedded in a Weld under Tensile Loading II：Double Cracks

In this paper, double embedded elliptical cracks in a weld of pressure vessels under tension was taken into consideration, and stress intensity factors at the crack tip were calculated with the emphasis on the interaction between cracks. It is found that when the distance between the double embedded elliptical cracks is larger than the major semi-axis of the ellipse, influence between the cracks can be neglected. Unlike the single embedded crack, owing to the crack interactions, the point with maximum stress intensity factor is not always at the end of the minor axis of the ellipse, it may swift to the end of major axis, especially when the ellipse is close to a circle.

On the Fluctuations of the Total Solar Irradiance

The fundamental quantity for the total solar irradiance is the solar constant J which is determined by the mean Sun-Earth distance and by the energy budget in the interior of the sun. The mean distance is the major semi-axis of the earth orbit and therefore a constant of celestial mechanics. The energy production and transport in the interior of the sun must be constant at least during a Helmholtz-Kelvin period. Actually, the heat budget of the sun is constant during some billion years.

On Asymmetric Periodic Solutions of the Plane Restricted Problem of Three Bodies, and Bifurcations of Families

Previous work on the plane circular restricted problem of three bodies (Message 1953, 1959, 1970, and Fragakis 1973) has shown the existence, in association with each of the commensurabilities 2:1 and 3:1 of the orbital periods, of a pair of families of asymmetric periodic solutions, branching from the stable series of symmetric periodic solutions of Poincaré’s second sort associated with that commensurability. (Each solution of either family is the mirror image, in the line of the two finite bodies, of a member of the other family of solutions associated with the commensurability.) The stability is transferred at the bifurcation to the two series of asymmetric orbits, each of which is therefore stable. Recent numerical integrations carried out by one of us (P.J.M.) have found such asymmetric periodic orbits associated also with the 4:1 commensurability, and quantities describing orbits of one of the two series are given in Table 1, showing the run of such orbits up to a second bifurcation with the same series of symmetric periodic orbits from which it sprang. Quantities describing some members of this series of symmetric orbits are given in Table 2. It is seen that stability is transferred back to the symmetric series at the second bifurcation. (The unit of distance is the distance between the two finite bodies, the unit of speed is the speed, of their relative motion, and the initial conditions given (x°, ẋ°, ẏ°) are for a crossing of the line of the two finite bodies, this line being taken as axis of “x” in a rotating Cartesian frame in the usual way. The mean values of the major semi-axis and eccentricity are denoted by ā and ē, respectively, C is Jacobi’s constant, and ȳ2 is the mean value of the critical argument ȳ2 = 4λ – λ′ – 3ω. The mass ratio used is 0.000954927, T is the period of the solution in units of the period of the motion of the two finite bodies, and 2π c/T is the non-zero characteristic exponent.)

Influence of the Dynamical Figure of the Moon on Its Rotational-Translational Motion

The influence of the dynamical figure of the Moon on its rotation with respect to its mass centre (the physical libration) is determined by means of the theorem on the angular moment of a rigid body. In the expansion of the Moon's force function in spherical harmonics all the second and the third order harmonics are taken into consideration. For the determination of the Moon's physical libration components a linear system of differential equations of the second order with constant coefficients is constructed.The integration displays the essential influence of the new terms in the force function expansion. For evaluation of the disturbed elements of the lunar orbit due to the nonsphericity of the Moon's dynamical figure the Lagrange's equations are solved. The disturbing function is taken in an expansion form in powers of the eccentricity of the lunar orbit and of the inclinations of the Moon's equator and its orbit with respect to the ecliptic. The commensurability of the Moon's mean motion and its angular velocity of rotation produces in the major semi-axis of the lunar orbit secular perturbations of the first order.

Note on the central differential equation in the relativity theory of gravitation

The critical equation in Prof. Einstein’s relativity theory of gravitation is given in the form ( du / dϕ ) 2 + u 2 = c 2 — 1/ h 2 + 2 mu / h 2 + 2 mu 3 , where m is the mass of the sun, u is the reciprocal of the distance of the moving body from the sun, h 2 = ma (1— e 2 ), c 2 — 1 = — m / a , a is the “major semi-axis of the orbit,” the units being astronomical, and e is the “excentricity” of the orbit. The foregoing equation can be integrated exactly, the functions involved being elliptic and not circular; and the approximations for the solar system are easily obtained, because the modulus of the elliptic functions is small.