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.     

Special Families of Orbits in the Direct Problem of Dynamics

The direct problem of dynamics in two dimensions is modeled by a nonlinear second-order partial differential equation, which is therefore difficult to be solved. The task may be made easier by adding some constraints on the unknown function = f _y /f _x , where f(x, y) = c is the monoparametric family of orbits traced in the xy Cartesian plane by a material point of unit mass, under the action of a given potential V(x, y). If the function is supposed to verify a linear first-order partial differential equation, for potentials V satisfying a differential condition, can be found as a common solution of certain polynomial equations.The various situations which can appear are discussed and are then illustrated by some examples, for which the energy on the members of the family, as well as the region where the motion takes place, are determined. One example is dedicated to a Hénon—Heiles type potential, while another one gives rise to families of isothermal curves (a special case of orthogonal families). The connection between the inverse/direct problem of dynamics and the possibility of detecting integrability of a given potential is briefly discussed.This revised version was published online in October 2005 with corrections to the Cover Date.     

On the adelphic potentials compatible with a set of planar orbits

Given a planar potentialB=B(x, y), compatible with a monoparametric family of planar orbitsf(x, y)=c, we face the problem of producing potentialsA=A(x, y), adelphic toB(x, y), i.e. nontrivial potentials which have in common withB(x, y) the given set of orbits. We establish a linear, second order partial differential equation for a functionP(x, y) and we prove that, to any definite positive solution of this equation, there corresponds a potentialA(x, y) adelphic toB(x, y).     

Family boundary curves for autonomous dynamical systems

The notion of the family boundary curves (FBC), introduced recently for two-dimensional conservative systems, is extended to account for, generally, nonconservative autonomous systems of two degrees of freedom. Formulae are found for the force componentsX (x, y),Y (x, y) which produce a preassigned family of orbitsf(x, y)=c lying inside a preassigned, open or closed, regionB(x, y)0 of the xy plane.     

Generalization of Szebehely's equation

Szebehely's partial differential equation for the force functionU=U(x,y) which gives rise to a given family of planar orbitsf(x,y)=Constant is generalized to account for velocity-dependent potentials V^*=V^*(x,y, ). The new partial differential equation is quasi-linear and of the first order. An example is given and a comparison is made of the two equations.     

Families of orbits in planar anisotropic fields

The aim of the planar inverse problem of dynamics is: given a monoparametric family of curves f(x, y) = c, find the potential V (x, y) under whose action a material point of unit mass can describe the curves of the family. In this study we look for V in the class of the anisotropic potentials V(x, y) = v(a²x² + y²), (a=constant). These potentials have been used lately in the search of connections between classical, quantum, and relativistic mechanics. We establish a general condition which must be satisfied by all the families produced by an anisotropic potential. We treat special cases regarding the families (e. g. families traced isoenergetically) and we present certain pertinent examples of compatible pairs of families of curves and anisotropic potentials. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

New exact solutions for power-law inflation Friedmann models

We consider the spatially flat Friedmann model For a ≈ t^p, especially, if p ≥ 1, this is called power-law inflation. For the Lagrangian L = R^m with p = − (m − 1) (2m − 1)/(m − 2) power-law inflation is an exact solution, as it is for Einstein gravity with a minimally coupled scalar field ϕ in an exponential potential V(ϕ) = exp (μϕ) and also for the higher-dimensional Einstein equation with a special Kaluza-Klein ansatz. The synchronized coordinates are not adapted to allow a closed-form solution, so we write The general solutions reads Q(a) = (a^b + C)^f/b with free integration constant C (C = 0 gives exact power-law inflation) and m-dependent values b and f: f = −2 + 1/p, b = (4m − 5)/(m − 1). Finally, special solutions for the closed and open Friedmann model are found.     

Isovortical orbits in uniformly rotating coordinates

For the conservative, two degree-of-freedom system with autonomous potential functionV(x,y) in rotating coordinates; $$\dot u - 2n\upsilon = V_x , \dot \upsilon + 2nu = V_y $$ , vorticity (v _x -u _y ) is constant along the orbit when the relative velocity field is divergence-free such that: $$u(x,y,t) = \psi _y , \upsilon (x,y,t) = - \psi _x $$ . Unlike isoenergetic reduction using the Jacobi, integral and eliminating the time,non-singular reduction from fourth to second-order occurs when (u,v) are determined explicitly as functions of their arguments by solving for ψ (x, y, t). The orbit function ψ satisfies a second-order, non-linear partial differential equation of the Monge Ampere type: $$2(\psi _{xx} \psi _{yy} - \psi _{xy}^2 ) - 2(\psi _{xx} + \psi _{yy} ) + V_{xx} + V_{yy} = 0$$ . Isovortical orbits in the rotating frame arenot level curves of ψ because it contains time explicitly due to coriolis effects. Rather, (x, y) coordinates along the orbit are obtained, from (u, v) either by numerical integration of the kinematic equations, or by partial differentiation of the Legendre transform ? of ψ. In the latter case, ? is shown to satisfy a non-linear, second-order partial differential equation in three independent variables, derived from the Monge-Ampere Equation. Complete reduction to quadrature is possible when space-time symmetries exist, as in the case of central force motion.     

Surface photometry of theHii regions S254–S257

The surface photometry of S254–S257 has been carried out by means of a wide range image processing technique in the reduction system. The photographic plates in the H+[NII] andV-bands are taken with the Schmidt telescope. Especially, we have obtained the calibrated map of theHii region, superposing two or more plates with different exposure times, and removing the star images. Three kinds of calibrated maps of theHii regions are drawn: (1)E-map in the (H+[NII]+continuum) (2)V-map in the continuum atV-band, (3)(E-V)-map in the (H+[NII]) line emission. The intensity profiles across the nebular centers were also obtained. Based on calibrated maps, the morphological structure and mass distribution of S255 and S257 are discussed. The location of observed nebulae on the (m _H–m _v) diagram, wherem _H andm _v denote the surface brightness, expressed in the magnitude per square arcmin, is shown together with that of some other nebulae. Some arguements on the age sequence of observedHii regions are presented.     

Numerical analysis of the symmetric methods

Aimed at the initial value problem of the particular second-order ordinary differential equations,y =f(x, y), the symmetric methods (Quinlan and Tremaine, 1990) and our methods (Xu and Zhang, 1994) have been compared in detail by integrating the artificial earth satellite orbits in this paper. In the end, we point out clearly that the integral accuracy of numerical integration of the satellite orbits by applying our methods is obviously higher than that by applying the same order formula of the symmetric methods when the integration time-interval is not greater than 12000 periods.     

Motion of rigid bodies in a set of redundant variables

In this article we study the conditions for obtaining canonical transformationsy=f(x) of the phase space, wherey(y ₁,y ₂,...,y _2n ) andx(x ₁,x ₂,...,x _2m ) in such a way that the number of variables is increased. In particular, this study is applied to the rotational motion in functions of the Eulerian parameters (q ₀,q ₁,q ₂,q ₃) and their conjugate momenta (Q ₀,Q ₁,Q ₂,Q ₃) or in functions of complex variables (z ₁,z ₂,z ₃,z ₄) and their conjugate momenta (Z ₁,Z ₂,Z ₃,Z ₄) defined by means of the previous variables. Finally, our article include some properties on the rotational motion of a rigid body moving about a fixed point.     

Inhomogeneous potentials producing homogeneous orbits

We prove that, in general, a given two-dimensional inhomogeneous potential V(x,y) does not allow for the creation of homogeneous families of orbits. Yet, depending on the case at hand, if the given potential satisfies certain conditions, this potential is compatible either with one (or two) monoparametric homogeneous families of orbits or at most with five such familes. The orbits are then found on the grounds of the given potential.     

Programmed motion in the presence of homogeneity

In the framework of the inverse problem of dynamics, we face the following question with reference to the motion of one material point: Given a region T_orb of the xy plane, described by the inequality g (x, y) ≤ c₀, are there potentials V = V (x, y) which can produce monoparametric families of orbits f (x, y) = c (also to be found) lying exclusively in the region T_orb? As the relevant PDEs are nonlinear, an answer to this question (generally affirmative, but not with assurance) can be given by the procedure of the determination of certain constants specifying the pertinent functions. In this paper we ease the mathematics involved by making certain simplifying assumptions referring to the homogeneity of both the function g (x, y) (describing the boundary of T_orb) and of the slope function γ(x, y) = f_y/f_x (representing the required family f (x, y) = c). We develop the method to treat the so formulated problem and we show that, even under these restrictive assumptions, an affirmative answer is guaranteed provided that two algebraic equations have in common at least one solution (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Evidence for enhanced star formation in iras-detected markarian galaxies

The IR emission of 640 Markarian galaxies (MrkG), included in the IRAS Survey, is considered as an evidence for enhanced star formation rate (SFR) in these objects. About 73% of the MrkG have high far-infrared luminosities (ca. 10E + 44 erg s^–1) in 1–500 mcm IR spectral band. The distribution of log(f ₆₀/f ₁₀₀), peaked at about 45 K, shows that IRAS MrkGs have a tendency to extend the relationf ₆₀/f ₁₀₀ vsL _ir/L _bifor normal S glaxies. They emit up to hundred times more IR energy in 40–120 mcm band than in optics. The mean ratio log L _ir/L _b for 621 IRAS MrkG with known redshifts is 2.2.It is suggested that there are two IR emitting components in the IRAS MrkG - a warm one connected with the UV-fluxes of the newborn massive stars, re-radiated by dust, and a cool one, originated from the dust in galactic disks and heated by the general interstellar radiation field. The warm IR luminosities and warm IR fractions are determined on the basis of IR colour-colour diagrams(25/12),(60/25), and(100/60). The mean warm IR fraction for all Mrk IRAS detected galaxies with well-defined IR fluxes is 0.83 when the grain mass absorption coefficient model withn = 0.0 is used. The dust mass responsible for the IR flux at 60 mcm is derived to be about 10E + 5M , assuming the dust clouds are optically thin, and using the dust temperatureT _d 46 K (deduced from thef(60)/f(100) ratio). There is a relation betweenL _irandL _blwhich points out that the most IRAS MrkG have rather enhanced SFR.     

B,V surface photometry of the UV continuum galaxy NGC 838

Surface photometry of the UV continuum galaxy NGC 838 has been carried out in theB, V system using photographic plates obtained with the 74 Kottamia telescope, Egypt. Isophotes, luminosity profiles, integrated photographic magnitudes, effective diameters and other photometric parameters are derived.The photoelectrically calibrated total apparent magnitudes areB _T =13.57 with maximum diameters 1.57×1.34 (at threshold _m =27.7 mag.//) andV _T =12.91 with maximum diameters 1.54×1.32 (at threshold _m =27.7 mag./). The integrated colour index(B–V) _T =0.66 and the effective surface brightness _e (B=19.0 mag./) and _e (V=19.7 mag./. The major axis is at position angle =85°±1°.The nucleus of NGC 838 is quite blue (integrated colour(B–V)=0.41 forr ^*<0.1) compared to normal galaxies while the colour becomes redder from the nucleus outwards. The UV excess, H emission and radio continuum emission previously observed from this galaxy by other investigators may be attributed to a recent burst of star formation in the nucleus of the galaxy of duration slightly greater than 2×10⁷ yr.     

Anisotropic solar cosmic ray propagation in an inhomogeneous medium,II

The author's model for anisotropic solar cosmic ray propagation gives 2 coupled, partial differential equations for the intensity and anisotropy of solar cosmic rays propagating with finite speed V in an inhomogeneous medium. The model is used to study the effect of the solar shell on solar cosmic ray propagation. It predicts an exponential decay, regardless of the observer's position. It predicts that when the observer is near the center of the shell, t _D/t ₀ 20 to 30, (t _D= decay time, t ₀ = onset time) and A _m(anisotropy) 15%, if t _m/t ₀ 3 to 5 (t _m= time of maximum), consistent with observations of relativistic particles on Feb. 23, 1956. When the observer is between the shell and the sun, the model predicts that oscillations might be observed near maximum intensity. When the observer moves away from the sun and the shell, the propagation is diffusive, but there is an increasingly large persistent anisotropy which serves as a measure of the width of the shell.     

Chemical evolution of the Galaxy at the initial rapid-collapse phase

Equations for the chemical evolution of the Galaxy are derived, accounting for (i) the dynamical evolution of the Galaxy (i.e. the collapse of the proto-galaxy), and (ii) either a variable mass-spectrum in the birth-rate stellar function of the type (m, t)=(t)(m, t), or a constant mass-spectrum with variable lower mass limit for star birth:m _mf=m_mf(Z). Simple equations are adopted for the collapse of the proto-galaxy, accounting for the experimental data (i.e. axial ratio and major semi-axis) relative to the halo and to the disk, and best fitted for a rapid collapse; gas density is assumed to be always uniform. Numerical computations of several cases show that there is qualitative agreement with the experimental data relative to theZ(t) function when: (i) the mass-spectrum is nearly constant in time: (m, t)(m)=m ^–2.35; (ii) the efficiency (t) is sufficiently high; moreover, the super metallic effect (SME) takes place for greater than a given value (1.5); (iii) the shorter the collapse timeT _c, the more rapid is the initial increase of metallicity, the asymptotic value being left nearly unaltered. The theoretical present-day values of gas density and metallicity so obtained differ from the experimental values by a factor of 2 or 3. Leaving aside other possible explanations, such a discrepancy is within the range of the uncertainties concerning the amount of gas returned back into space by the decay of the stars. Our theoretical results are not in complete agreement with the observed data bearing on theN _n(Z) function (N _n is the number of stars whose Main-Sequence lifetime is not less than the age of the Galaxy), while a hypothesis of star formation with different efficiencies in different zones of the Galaxy, and successive stellar mixing from zone to zone, is not inconsistent with such data.     

A direct numerical approach to the Chandrasekhar'sH-functions for arbitrary characteristic functions

The purpose of this paper is to present a numerical technique to directly compute the Chandrasekhar'sH ()-function for anisotropic scattering in terms of the roots of the characteristic equations as well as the quadrature points of a certain degreen employed to approximate the definite integral involved in the basic equation. The principal feature of the algorithm proposed here is a compact computer code to enumerate _n C _m combinations ofn distinct integers {1,...,n} takenm at a time. With these quantities available, the coefficients of the polynomial equation of the characteristics equation can be readily computed for any given characteristic function, so that a standard technique such as the Laguerre method can be applied to find all the roots.It is shown that the results obtained for some representativeH()-functions using the present technique with relatively low-order formula (e.g.,n=7) are sufficiently accurate for all practical purposes.     

A method of colour excess determination for classical cepheids

By use of the reddening free [m ₁], [c ₁], and indices data inuvby photometric system for three classical cepheids whose reddening values had been determined with the aid of photometry of field stars, three intrinsic relations of [m ₁]–(b–y), [c ₁]–(b–y), and –(b–y) have been established. It was shown that these three relations can be used to determine the colour excesses for other classical cepheids.     

Generation of a d.c. field by nonlinear electromagnetic waves in relativistic plasmas

A finite amplitude linearly polarized electromagnetic wave propagating in a relativistic plasma, is found to generate the longitudinal d.c. as well as the oscillating electric field at the second harmonic. In a plasma consisting of only electrons and positrons, these fields cannot be generated.The evolution of the electromagnetic waves is governed by the non-linear Schrödinger equation which shows that the electromagnetic solitons are always possible in ultra-relativistic plasmas (electron-ion or electron-positron) but in a plasma with relativistic electrons and nonrelativistic ions, these solitons exist only if 1(KT _e/m_eC²)<(2m _i/15m_e);m _e andm _i being the electron and ion mass andT _e the electron temperature. Both the d.c. electric field and the solitons provide a nonlinear mechanism for anomalous acceleration of the particles. This model has direct relevance to some plasma processes occurring in pulsars.