Motor integrals of a generalized Kepler motion

The ordinary spinor differential Equation (20) of the unperturbed Kepler motion is obtained from the classical equation of motion (19) if one uses the spinor regularization (9) and postulates an essential subsidiary condition (10). A natural generalization for the Kepler motion follows by dropping this subsidiary conditions; it is the 8-parameter set of solutions of the spinor equation of motion (20). The sixteen natural extensive integrals (30)–(35) for this generalized Kepler motion are here deduced by means of the relativistic motors (2), (7) of the Spinor Ring Algebra. These integrals form, with respect to the Poisson bracket operation, a 15-dimensional Lie algebra (40)–(44), closely related to the Lie algebras in quantum mechanics.Dedicated to Professor G. Järnefelt on his 70th anniversary.     

具有旋量场的量子宇宙学

本文利用Hartle和Hawking的方法,讨论了具有旋量场的量子宇宙学,得到了相应的Wheeler-De Witt方程。求出了具有旋量场的宇宙波函数。从波函数可以看出,当标度因子α很小时,旋量场的影响很强,具体的形式与初始条件有关,而当标度因子α很大时,旋量场的行为和标量场一样。     

On Spinor Equations of Motion and their “Possible Integrals”

The motion of one point mass of the classical mechanics is treated by means of the relativistic spinor regularization (KUSTAANHEIMO 1975). Most general spinor equations of motion (2.9)‒(2.10)and the differential equations(3.16)‒(3.25)for the “possible integrals” (3.2)‒(3.11)of these quations of motion are deduced. If the force is a superposition of a conservative central force and of another force perpendicular to radius vector and velocity (Chapter 4, Case D), then the theory yields scalar and spinor integrals (4.7), (4.10)‒(4.12), (4.14)‒(4.15), (4.17), (4.28) that enable a parametric representation of the orbit by quadratures, as soon as one solution of a RICCATI differential equation (4.33) has been found.     

Nonlinear spinor field in Bianchi type-I Universe filled with viscous fluid: numerical solutions

We consider a system of nonlinear spinor and a Bianchi type I gravitational fields in presence of viscous fluid. The nonlinear term in the spinor field Lagrangian is chosen to be λ F, with λ being a self-coupling constant and F being a function of the invariants I an J constructed from bilinear spinor forms S and P. Self-consistent solutions to the spinor and BI gravitational field equations are obtained in terms of τ, where τ is the volume scale of BI universe. System of equations for τ and ε, where ε is the energy of the viscous fluid, is deduced. This system is solved numerically for some special cases.      

Interacting Scalar And Spinor Fields In Bianchi Type I Universe Filled With Magneto-Fluid

Self-consistent system of spinor, scalar and BI gravitational fields in presence of magneto-fluid and Λ term is considered. Assuming that the expansion of the BI universe is proportional to the σ₁ component of the shear tensor, exact solutions for the metric functions, as well as for scalar and spinor fields are obtained. For a non-positive Λ, the initially anisotropic space–time becomes isotropic one in the process of expansion; whereas for Λ > 0, an oscillatory mode of expansion of the BI model occurs. PACS numbers: 03.65.Pm and 04.20.Ha     

FRW and Bianchi type I cosmology of f-essence

F-essence is a generalization of the usual Dirac model with the nonstandard kinetic term. In this paper, we introduce a new model of spinor cosmology containing both Ricci scalar and the non minimally coupled spinor fields in its action. We have investigated the cosmology with both isotropy and anisotropy, where the equations of motion of FRW and Bianchi type-I spacetimes have been derived and solved numerically. Finally the quantization of these models through Wheeler-De Witt (WD) wave function has been discussed.     

Celestial mechanics with geometric algebra

Geometric algebra is introduced as a general tool for Celestial Mechanics. A general method for handling finite rotations and rotational kinematics is presented. The constants of Kepler motion are derived and manipulated in a new way. A new spinor formulation of perturbation theory is developed.This work was partially supported by JPL under contract with the National Aeronautics and Space Administration.     

Reflection of Alfvén waves by non-uniform ionospheres

Magnetospheric Alfvén waves are reflected by the ionosphere. We investigate the effect of horizontally varying ionospheric conductivity on the process of Alfvén wave reflection. Four idealised ionospheric models are considered in detail. We find that the reflection process is strongly dependent on the orientation of the wave electric field vector with respect to the boundary between high and low conductivities, and under certain conditions subsidiary Alfvén waves are generated. The field-aligned currents in the subsidiary Alfvén waves serve to close divergent horizontal currents resulting from the non-uniform ionospheric conductivity. The implications for ground-based pulsation studies are discussed.     

Rotational dynamics with geometric algebra

Celestial Mechanics and Dynamical Astronomy - A new spinor formulation of rotational dynamics is developed. A general theorem is established reducing the theory of the symmetric top to that of the...     

Semi-classical Gravitational Effects Around Global Monopoles in Lyra Geometry

W.A. Hiscock (1990, Class. Quantum Gravitation 7, L235) obtained the vacuum expectation value of the stress-energy tensor of an arbitrary collection of conformal massless free quantum fields (scalar, spinor and vectors) in the space-time of a static global monopole. With this stress-energy tensor, the semi-classical Einstein equations are solved retaining terms up to first order in ħ in Lyra geometry. This revised version was published online in July 2006 with corrections to the Cover Date.     

Lie groups of motor integrals of generalized Kepler motion

The singularity of the Kepler motion can be eliminated by means of the spinor regularization. The extensive integrals of the Kepler motion form a Lie algebra with respect to the Poisson bracket operation. Mayer-Gürr has shown that in the caseH>0 the corresponding Lie group is the multiplicative group of all real 4×4 unimodular matrices SL(4,R). Kustaanheimo has posed the problem of the identification of the corresponding Lie groups in the elliptic and parabolic cases. We solve this problem here and use the opportunity to introduce the concept of the Clifford algebra which is needed in our solution.     

The optical–infrared colour distribution of a statistically complete sample of faint field spheroidal galaxies

In hierarchical models, where spheroidal galaxies are primarily produced via a continuous merging of disc galaxies, the number of intrinsically red systems at faint limits will be substantially lower than in 'traditional' models where the bulk of star formation was completed at high redshifts. In this paper we analyse the optical–near-infrared colour distribution of a large flux-limited sample of field spheroidal galaxies identified morphologically from archival Hubble Space Telescope data. The I ₈₁₄− HK ' colour distribution for a sample jointly limited at I ₈₁₄<23 mag and HK '<19.5 mag is used to constrain their star formation history. We compare visual and automated methods for selecting spheroidals from our deep HST images and, in both cases, detect a significant deficit of intrinsically red spheroidals relative to the predictions of high-redshift monolithic-collapse models. However, the overall space density of spheroidals (irrespective of colour) is not substantially different from that seen locally. Spectral synthesis modelling of our results suggests that high-redshift spheroidals are dominated by evolved stellar populations polluted by some amount of subsidiary star formation. Despite its effect on the optical–infrared colour, this star formation probably makes only a modest contribution to the overall stellar mass. We briefly discuss the implications of our results in the context of earlier predictions based on models where spheroidals assemble hierarchically.     

Variations in air density from January 1972 to April 1975 at heights near 200 km

Variations in air density have been determined using the orbit of the satellite Cosmos 462, 1971-106A, which entered orbit on 3 December 1971 with an initial perigee near 230 km and inclination 65.75°, and decayed on 4 April 1975. Accurate orbits determined at 85 epochs give perigee height correct to about 200 m throughout the satellite's lifetime. Using these values of perigee height and orbital decay rates from NORAD elements, 604 values of air density at half a scale height above perigee have been evaluated. These densities have been compared with values from the COSPAR International Reference Atmosphere 1972, taking account of variations due to solar activity and geomagnetic disturbances, and day-to-night variations, to reveal the residual variations in density at a series of standard heights, 245, 240, 232 and 213 km.The main residual variation is semi-annual, with maxima usually in April and October, and minima usually in January and July; but it is irregular in phase and shape. The amplitude of the semi-annual variation is remarkably constant from year to year between 1972 and 1975, and considerably greater than that given by CIRA 1972: the April/July density ratio is 1.68, not 1.32 as in CIRA; the October–November maxima are all lower than the April maxima, whereas CIRA gives the opposite; the July minima are 18% lower than the January minima, as opposed to 10% in CIRA.A standardized semi-annual density variation for the early 1970s is presented, with January minimum of 0.94, April maximum of 1.28, July minimum of 0.77 and October–November maximum of 1.22. In addition, three other recurrent variations are recognizable: in each year the density has a subsidiary minimum in May and maximum in June; there are low values in mid November and high values in late December.     

Parametric excitation induced by solar pressure torque on the roll-yaw attitude motion of a gravity-gradient stabilized spacecraft

The parametric excitation of a gravity gradient stabilized spacecraft induced by the periodic solar pressure torque is discussed. The solar pressure torque in the linearized equations of motion appears as linear terms with periodic coefficients. The attitude stability is analyzed numerically through the calculation of the Floquet multiplier. The perturbation method is also applied to identify the instability condition analytically. It is made clear that the periodic solar pressure torque can destabilize the coupled roll and yaw attitude motion of the spacecraft. It is also shown that the conditions of parametric resonance are included in the gravity gradient stability condition. Nonlinear simulations are also carried out to verify the effect of the parametric resonance. The numerical simulation using actual parameters shows that the spacecraft inevitably experiences a large amplitude attitude motion due to the periodic solar pressure torque even if the gravity gradient stability condition is satisfied.     

The semi-annual variation in air density for 1974–1978 from the orbit of 1972-05B

Values of air density at 712 epochs between August 1973 and September 1978 have been determined using orbital elements of 1972-05B with orbital decay rates from NORAD bulletins. Normalised to a series of fixed heights and cleared of the effects of solar activity, geomagnetic activity and the diurnal variation, the residual air density was analysed for the semi-annual variation. This variation exhibited maxima usually in April and October and minima usually in January and July.

For 1974–1978, this study revealed near-identical values of the April and October maxima and a July minimum 12% stronger than the January minimum. Further, the shape and phase of the variation exhibited an irregular pattern from year-to-year. Overall the amplitude of the variation was considerably greater than that given in the atmospheric models (CIRA, 1972; Jacchia, 1977). Other observations included the presence of subsidiary minima and maxima in late May and June respectively during 1977 and 1978 and a general increase in air density from mid 1977 onwards, relative to the atmospheric models.     

Near distance approximation in astrodynamical applications of Lambert’s theorem

The smallness parameter of the approximation method is defined in terms of the non-dimensional initial distance between target and chaser satellite. In the case of a circular target orbit, compact analytical expressions are obtained for the interception travel time up to third order. For eccentric target orbits, an explicit result is worked out to first order, and the tools are prepared for numerical evaluation of higher order contributions. The possible transfer orbits are examined within Lambert’s theorem. For an eventual rendezvous it is assumed that the directions of the angular momenta of the two orbits enclose an acute angle. This assumption, together with the property that the travel time should vanish with vanishing initial distance, leads to a condition on the admissible initial positions of the chaser satellite. The condition is worked out explicitly in the general case of an eccentric target orbit and a non-coplanar transfer orbit. The condition is local. However, since during a rendezvous maneuver, the chaser eventually passes through the local space, the condition propagates to non-local initial distances. As to quantitative accuracy, the third order approximation reproduces the elements of Mars, in the historical problem treated by Gauss, to seven decimals accuracy, and in the case of the International Space Station, the method predicts an encounter error of about 12 m for an initial distance of 70 km.     

Error estimates with L1 solutions

A method is given, based on the pseudoinverse of the equations of condition, to obtain error estimates for the solution in the normL ₁ of an over-determined linear system. The computational labor to obtain the errors, while not trivial, is less than that for various competing methods, particularly if there are many more equations of condition than unknowns. The error estimates for anL ₁ solution are substantially larger than those for a least squares solution of the some data. It is suggested that a complete discussion of a linear system include at least bothL ₁ and least squares solutions with their respective errors and the condition number of the linear system.     

An iterative method for obtaining a nonlinear solution for the temperature distribution of a spinning spherical body irradiated by a central star

We developed an iterative method for determining the three-dimensional temperature distribution in a spherical spinning body that is irradiated by a central star. The seasonal temperature change due to the orbital motion is ignored. It is assumed that material parameters such as the thermal conductivity and the thermometric conductivity are constant throughout the spherical body. A general solution for the temperature distribution inside a body is obtained using spherical harmonics and spherical Bessel functions. The surface boundary condition contains a term obtained using the Stefan–Boltzmann law and is nonlinear with respect to temperature because it is dependent on the fourth power of temperature. The coefficients of the general solution are fitted to satisfy the surface boundary condition by using the iterative method. We obtained solutions that satisfy the nonlinear boundary condition within 0.1% accuracy. We calculated the rate of change in the semimajor axis due to the diurnal Yarkovsky effect using the linear and nonlinear solutions. The maximum difference between the rates calculated using the two sets of solutions is 13%. Therefore current understanding of the diurnal Yarkovsky effect based on linear solutions is fairly good.