Two‐dimension potentials which generate spatial families of orbits

We consider the following case of the 3D inverse problem of dynamics: Given a spatial two‐parametric family of curves f (x, y, z) = c₁, g (x, y, z) = c₂, find possibly existing two‐dimension potentials under whose action the curves of the family are trajectories for a unit mass particle. First we establish the conditions which must be fulfilled by the family so that potentials of the form w (y, z) give rise to the curves of the family, and we present some applications. Then we examine briefly the existence of potentials depending on (x, z), respectively (x, y), which are compatible with the given family (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Szebehely's problem for holonomic systems involving generalized potential functions

We consider the problem of finding the generalized potential function V = U _i(q ¹, q ²,..., q ⁿ)q ⁱ + U(q ¹, q ²,...;q ⁿ) compatible with prescribed dynamical trajectories of a holonomic system. We obtain conditions necessary for the existence of solutions to the problem: these can be cast into a system of n – 1 first order nonlinear partial differential equations in the unknown functions U ₁, U ₂,...;, U _n, U. In particular we study dynamical systems with two degrees of freedom. Using adapted coordinates on the configuration manifold M ₂ we obtain, for potential function U(q ¹, q ²), a classic first kind of Abel ordinary differential equation. Moreover, we show that, in special cases of dynamical interest, such an equation can be solved by quadrature. In particular we establish, for ordinary potential functions, a classical formula obtained in different way by Joukowsky for a particle moving on a surface.Work performed with the support of the Gruppo Nazionale di Fisica Matematica (G.N.F.M.) of the Italian National Research Council.     

When did cosmic acceleration start? How fast was the transition?

Cosmic acceleration is investigated through a kink-like expression for the deceleration parameter (q). The new parametrization depends on the initial (q_i) and final (q_f) values of q, on the redshift of the transition from deceleration to acceleration (z_t) and the width of such transition (τ). We show that although supernovae (SN) observations (Gold182 and SNLS data samples) indicate, at high confidence, that a transition occurred in the past (z_t > 0) they do not, by themselves, impose strong constraints on the maximum value of z_t. However, when we combine SN with the measurements of the ratio between the comoving distance to the last scattering surface and the SDSS + 2dfGRS BAO distance scale (S_k/D_v) we obtain, at 95.4% confidence level, and for (S_k/D_v+Gold182), and and for (S_k/D_v+SNLS), assuming q_i = 0.5 and q_f = −1. We also analyze the general case, q_f  (−∞, 0) finding the constraints that the combined tests (S_k/D_v+SNLS) impose on the present value of the deceleration parameter (q₀).     

Decomposition of product of certain functions relevant to the solution of transfer equations by Wiener-Hopf technique

Decomposition of the product of functions like exp (– ⁰/z)S ⁺(z) [S ⁺(z = ₀ ¹ P(x) dx/(x–z)], obtained by Das Gupta (1978) and relevant to the solution of equations of radiative transfer or of transfer problems in finite media by Wiener-Hopf technique, is reviewed and transformed to quite simple integral forms amenable to easy numerical evaluations. The same forms are then shown to be directly obtainable in one step under a slightly stronger condition consistent with practical cases.     

Compatible potentials and orbits in synodic systems

For a given family of orbits f(x,y) = c ^* which can be traced by a material point of unit in an inertial frame it is known that all potentials V(x,y) giving rise to this family satisfy a homogeneous, linear in V(x,y), second order partial differential equation (Bozis,1984). The present paper offers an analogous equation in a synodic system Oxy, rotating with angular velocity . The new equation, which relates the synodic potential function (x,y), = –V(x, y) + % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSqaaSqaai% aaigdaaeaacaaIYaaaaaaa!3780!\[\tfrac{1}{2}\]²(x ² + y ²) to the given family f(x,y) = c ^*, is again of the second order in (x,y) but nonlinear.As an application, some simple compatible pairs of functions (x,y) and f(x, y) are found, for appropriate values of , by adequately determining coefficients both in and f.     

The geometry of the Roche coordinates

The exact geometry of the Roche curvilinear coordinates (, , ) in which corresponds to the zero-velocity surfaces is investigated numerically in the plane, as well as in the spatial, case for various values of the mass-ratio between the two point-masses (m ₁,m ₂) constituting a binary system.The geometry of zero-velocity surfaces specified by -values at the Lagrangian points are first discussed by taking their intersections with various planes parallel to thexy-, xz- andyz-planes. The intersection of the zero-velocity surface specified by the -value at the Lagrangian equilateral-triangle pointsL _4,5 with the planex=1/2 discloses two invariable curves passing through the pointsL _4,5 and situated symmetrically with respect to thexy-plane whose form is independent of the mass-ratio.The geometry of the remaining two coordinates (, ) orthogonal to the zero-velocity surfaces is investigated in thexy- andxz-planes from extensive numerical integrations of differential equations generated from the orthogonality relations among the coordinates. The curves (x, y)=constant in thexy-plane are found to be separated into three families by definite envelopes acting as boundaries whose forms depend upon the mass-ratio only: the inner -constant curves associated with the masspointm ₁, the inner -constant curves associated with the mass-pointm ₂ and the outer -constant curves. All the -constant curves in thexy-plane coalesce at either of the Lagrangian equilateraltriangle pointsL _4,5, except for a limiting case coincident with thex-axis. The curves (x, z)=constant in thexz-plane are also separated by definite envelopes depending upon the mass-ratio into different families: the inner -constant curves associated with the mass-pointm ₁, the inner -constant curves associated with the mass-pointm ₂ and the outer -constant curves on both sides out of the envelopes. For larger values ofz, the curves =constant tend asymptotically to the line perpendicular to thex-axis and passing through the centre of mass of the system, except for a limiting case coincident with thex-axis. The geometrical aspects of the envelopes for the curves (x, y)=constant in thexy-plane and the curves (x, z)=constant in thexz-plane are also discussed independently.In the three-dimensional space, the Roche coordinates can be conveniently defined in such a way as to correspond to the polar coordinates in the immediate neighbourhood of the origin, and to the cylindrical coordinates at great distances. From numerical integrations of simultaneous differential equations generating spatial curves orthogonal to the zero-velocity surfaces, the surfaces (x, y, z)=constant and the surfaces (x, y, z)=constant are constructed as groups of such spatial curves with common values of some parameters specifying the respective surfaces.On leave of absence from the University of Tokyo as an Honorary Fellow of the Victoria University of Manchester.     

The critical and the saturation content of magnetic monopoles in rotating relativistic objects

Both the critical content _c ( N _m /N _B , whereN _m ,N _B are the total numbers of monopoles and nucleons, respectively, contained in the object), and the saturation content _s of monopoles in a rotating relativistic object are found in this paper. The results are:

A note on the correspondence between Mathieu's transformations and redundant variables in lagrangian mechanics

It is shown that, to any change of variables:q _i=q_i(r, t) (i=1,..., n; =1,...,n+m; mn) increasing the number of variables, it is possible to associate a Mathieu's transformation and conversely. The results are applied to the theory of the osculating plane of motion.

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Resumé On montre qu'à toute transformation:q _i=q_i(r, t)(i=1,..., n; =1,...,n+m; mn) augmentant le nombre de variables, on peut associer une transformation de Mathieu et réciproquement. Les résultats sont appliqués à la théorie du plan osculateur du mouvement.

     

OnH-functions of radiative transfer

Chandrasekhar'sH-functionH(z) corresponding to the dispersion functionT(z)=| _rs –frs(z)|, where [f _rs (z)] is of rank 1, is obtained in terms of a Cauchy integral whose density functionQ(x, ₁, ₂,...) can be approximated by approximating polynomials (uniformly converging toQ(x)) having their coefficients expressed as known functions of the parameters _r 's. A closed form approximation ofH(z) to a sufficiently high degree of accuracy is then readily available by term by term integration.     

A class of periodic solutions of the N-body problem

We prove existence and multiplicity of T-periodic solutions (for any given T) for the N-body problem in ^m (any m 2) where one of the bodies has mass equal to 1 and the others have masses ₂,..., _N , small. We find solutions such that the body of mass 1 moves close to x = 0 while the body of mass _i moves close to one of the circular solutions of the two body problem of period T/k _i, where ki is any odd number. No relation has to be satisfied by k ₂,...,k _N.     

The origin of gravity

It has been shown by Atkinson (1965) that there is a rigorously exact euclidean interpretation of the general relativity field equations if certain arbitrary definitions of mass (m0 and the velocity of light (c) are invoked. With a preferred (euclidean) frame postulatedab initio, a particularly simple explanation in terms of classical physics may be found for very similar definitions ofm andc. It is not unexpected that with this scheme, all the usual tests of general relativity (light deflexion, perihelion motion, gravitational redshift, and radar delay time) are immediately satisfield. The preferred frame is however identified with a real aether and this requires a return to the Lorentzian interpretation of the special relativistic transformations of space and time variables. It is shown that gravity may be attributed to the action of a temperature gradient in the aether and an explanation of its origin in terms of an ideal relativistic gas is proposed. The temperature gradients are thermodynamically stable and do not diffuse if the relativistic aether ( _A ) is effectively adiabatic and matter is fundamentally a species of aether with instantaneous motion at high (> _A ) relative to the aethereal reast frame. To be consistent with such a picture, it is necessary to assume aether particles are capable of forming temporary associations (not recognized as matter) which take on some of the properties of crystalline solids and thereby become the means of transmitting electromagnetic radiation through space. The aether is essentially treated as a virtually incompressible fluid in which the pressure at any point arises from both random (temperature) and bulk (high ) motions. A number of specific predictions arising from this theory of gravity are indicated and these may serve to discriminate it from general relativity.     

Evaluation of gas production rates in some comets

Emission fluxes of CN, C₂ and C₃ carbon-bearing molecular species observed in the coma of comets Bennett (1969i 1970II), West (1975n 1976VI), P Halley (1982i), Hartley-Good (19851) and Bradfield (1987s) are analysed in the framework of Haser model. CN, C₂ and C₃ production rates are determined using recently derived fluorescence efficiencies. The dependence of CN, C₂ and C₃ production rates on the heliocentric distance and the possible correlations among these radicals is studied and briefly discussed.     

Satellite attitude acquisition by momentum transfer-the controlled wheel speed method

An implementation of the momentum transfer method for spacecraft attitude acquisition of momentum wheel stabilized geostationary satellites is presented, in which the wheel speed is varied in a predeterminable manner to reduce the nutation usually associated with the method. The implementation is found to be capable of achieving the transfer to the desired zero nutation end point with 5° to 20° of residual nutation in practical situations without additional nutation damping. The transfer time is typically 10–30 min. The implementation is described in terms of the momentum sphere and energy ellipsoid. The detailed functional dynamics and parametric relationships are given in terms of phase planes and elliptic integral solutions. Feasibility of the implementation is shown to be dependent on the moment of inertia configuration and the degree to which tolerances on moments of inertia, satellite spin rate, and wheel speed are predeterminable.Nomenclature A, B, C Principal moments of inertia of the total spacecraft - b ₁,b ₂,b ₃ Unit vectors parallel to principal axes - E, E(t) Energy of the spacecraft body (excludes the spin energy of momentum wheel) - E ₀ Initial energy of the spacecraft (wheel not spinning) - E _c A constant arbitrary value ofE(t) - E _i Energy after an impulse which causes to equal zero, Equation (13) - E ₂ The constant value ofE during Stage 2 - E ₃ Energy at the end of Stage 3 - E _s The separatrix value ofE, i.e.E when =0 - H ₀,H ₀ Angular momentum vector of the total spacecraft - H _x ,H _y ,H _z Variable components ofH ₀ resolved with respect tob ₁,b ₂, andb ₃ - J Moment of inertia of the momentum wheel - L The slope of the wheel speed profile, Equation (31) - L _y Momentum wheel torque - m Direction cosine of the angle, , betweenH ₀ andb ₂ - m ₁,m ₂ Constant values ofm - O The centre of the momentum sphere - Oxyz Spacecraft coordinate axes - P Represents a state on the momentum sphere and energy ellipsoid - s Momentum wheel speed - s ₂ Constant value ofs during Stage 2 - s _i Wheel speed after an impulse which causes to be zero - t Time - t ₁, t ₂, t ₃ Elapsed time for Stages 1, 2, and 3 respectively - ₁ to ₄ Roots of Equation (19) (poles on the phase plane) - Parameter defined in Equation (9) - cos^–1 m - ₂ The value of at the termination of Stage 2 - _min The minimum of the periodic function (t) (during Stage 2) - _x , _y , _z Angular rates of spacecraft body - ₀ Initial value of _z - K _m Curvature of the momentum sphere - K _xy ,K _yz Curvatures of the principal lines of the energy ellipsoid at (O,H ₀,O), in theOH _x H _y ) and (OH _y H _z ) planes respectively     

On the number of central configurations in the N-body problem

Central configurations are critical points of the potential function of the n-body problem restricted to the topological sphere where the moment of inertia is equal to constant. For a given set of positive masses m ₁,..., m _n we denote by N(m ₁, ..., m _n, k) the number of central configurations' of the n-body problem in ^k modulus dilatations and rotations. If m _{n 1},..., m _n, k) is finite, then we give a bound of N(m ₁,..., m _n, k) which only depends of n and k.     

On the far-ultraviolet radiation of the T Tauri-type stars

It is shown that the far-ultraviolet radiation (shorter than 2000 Å) discovered by ANS observations in the few T Tauri-type stars does not have any relation to the two-photon emission of hydrogen, as suggested by some authors. This is obtained from the observational data of the numerical values of the ratioQ ^*(2q)/Q ^*() for these stars, whereQ ^*(2q) is the complete number of the observed 2q-photons andQ ^*() is the number of observedH-photons. The observational values ofQ ^*(2q)/Q ^*() for four T Tauri-type stars turned out to be in the region of 20–90, while the theoretical value of this relation is 6. Hence, the observed fluxes in the region <2000 Å are 3–15 times larger than the theoretically expected values.The emission discovered in the region <2000 Å is of non-thermal origin, and can be identified with high probability with thetransition radiation. The latter originates as a result of the electromagnetic interaction of so-called fast electrons (E1.5 MeV) with dust particles in the gas-dust clouds surrounding these stars. The theoretical spectral curves of the transition radiation, for a few values of the plasma frequency ₀ for the dust particles, are calculated taking into account also the self-absorption effect of the radiation in the cloud and the absorption in the interstellar medium. Qualitatively, these curves (Figures 2, 3 and 4) are in good accord with the observed spectral distribution curves for the T Tauri-type stars (Figure 1). In particular, in both cases a minimum of radiation flux occurs near to 2200 Å, and a maximum near 1800 Å.The starting point of our analysis has been the concept of the identity of the processes, non-thermal and non-stationary in character, taking place at the time of the flare phenomenon of UV Cet-type stars in one case, and at the generation of continuous emission and the excitation of the emission lines in T Tauri-type stars on the other. In the latter case, the T Tauri-type stars can be regarded aspermanently flaring stars, with a very high frequency of flare events.     

Unsteady free-convective flow and mass transfer in a rotating porous medium with Hall current,I

The Hall effect on the unsteady hydromagnetic free-convection resulting from the combined effects of thermal and mass diffusion of an electrical-conducting liquid through a porous medium past an infinite vertical porous plate in a rotating system have been analysed. The expressions for the mean velocity, mean skin friction, and mean rate of heat transfer on the plate are derived. The effects of magnetic parameterM, Hall parameterm, Ekman numberE, and permeability parameterK ^* on the flow field are discussed with the help of graphs and tables.Nomenclature C _p specific heat at constant pressure - C the species concentration inside the boundary layer - C _w the species concentration at porous plate - C the species concentration of the fluid at infinite - C dimensionless species concentration - D chemical molecular diffusivity - E Ekman number - Ec Eckert number - g acceleration due to gravity - Gr Grashof number - Gm modified Grashof number - H ₀ applied magnetic field - (J _x, J_y, J_z) components of current density - M magnetic parameter - m Hall parameter - P Prandtl number - q _m mean rate of heat transfer - Sc Schmidt number - t time - t dimensionless time - T temperature of fluid - T _w temperature of the plate - T temperature of fluid at infinite - T dimensionless temperature - (u, v, w) components of the velocityq - w ₀ suction velocity - (x, y, z) Cartesian coordinates - z dimensionless coordinate normal to the plate Greek symbols coefficient of volume expansion - * coefficient of thermal expansion with concentration - frequency - dimensionless frequency - k thermal conductivity - K ^* permeability parameter - dinematic viscosity - density of the fluid in the boundary layer - coefficient of viscosity - _e magnetic permeability - angular velocity - electrical conductivity of the fluid - _m mean skin friction - _mn mean skin friction in the direction ofx - _mv mean skin friction in the direction ofy     

A problem of libration with two degrees of freedom

Moore (1983) presented a theory of resonance with two degrees of freedom based on the Bohlin-von Zeipel procedure. This procedure is now applied to librational motion with all constants re-evaluated in terms of values of the momenta given either by the initial conditions, or, in the case of the momentumy ₁ conjugate to the critical argument x₁, by its value at the libration centre. Numerical results are presented for a resonant satellite in a near 12 hr orbit and for a geosynchronous satellite. The theory is further developed to include near-circular orbits by recasting the problem in terms of the Poincaré eccentric variables.     

Isotropic flat space cosmology in Jordan-Brans-Dicke theory

The process of re-escalation of the scalar field as R ³, the energy density as R ³, and the pressurep aspR ³P, lends itself to obtain a reduced equation that represents, for a wide variety of equations of state, the cosmological evolution of an homogeneous and isotropic, flat Universe. A particular solution to this equation is presented.     

Magnetic and thermal pressures in the solar wind

Explorer 34 solar wind data for the period June to December, 1967 show that(a) The magnetic pressure, P _BB ²/8, and thermal pressure,P _kn _p kTp+n kT+n _e kTe,are variable and positively correlated on a scale of 2 days, but (b) changes in P _b and P _k are anticorrelated on a scale 1 hr (0.01 AU). Thus, dynamical hydromagnetic processes (dv/dto) must occur on the mesoscale, but the solar wind tends to be in equilibrium(P _B+P _Kconstant) on a smaller scale, the microscale. The 3-hr averages show that the most probable value of P _{k/P B} is =1.0±0.1, which implies that the most probable state of the solar wind at 1 AU is not one of equipartition between the thermal energy and magnetic energy. The average total pressure for a given bulk speed(P(V)=P ^k+P _k+P _B) is essentially independent of V, implying that P is not determined by the heating or acceleration mechanisms of the solar wind; the average pressure is P=(2.9±1.5)×10^-10dyne/cm².