Straight-line Orbits Research Articles

The linear response of stellar systems does not diverge at marginal stability

ABSTRACT The linear response of a stellar system’s gravitational potential to a perturbing mass comprises two distinct contributions. Most famously, the system will respond by forming a polarization ‘wake’ around the perturber. At the same time, the perturber may also excite one or more ‘Landau modes’, i.e. coherent oscillations of the entire stellar system which are either stable or unstable depending on the system parameters. The amplitude of the first (wake) contribution is known to diverge as a system approaches marginal stability. In this paper, we consider the linear response of a homogeneous stellar system to a point mass moving on a straight line orbit. We prove analytically that the divergence of the wake response is in fact cancelled by a corresponding divergence in the Landau mode response, rendering the total response finite. We demonstrate this cancellation explicitly for a box of stars with Maxwellian velocity distribution. Our results imply that polarization wakes may be much less efficient drivers of secular evolution than previously thought. More generally, any prior calculation that accounted for wakes but ignored modes – such as those based on the Balescu-Lenard equation – may need to be revised.

Experimental investigation for the dynamic mechanical properties of a vertical anisotropically supported rotor

The dynamic mechanical properties of rotors with anisotropic supporting stiffness have been intensively studied in recent decades. However, the transient unbalance response and the whirl motion when passing through backward critical speeds have not been adequately performed. In addition, there is little published experimental data on the influence of the support stiffness anisotropy. This paper attempts to investigate the steady-state and transient unbalance response of the rotor with anisotropic supports. Response characteristics are summarized based on a modified Jeffcott model first. Next, A vertical Jeffcott rotor test rig with anisotropic elastic supports is established, and the stiffness property of different supports is measured. The steady-state unbalance response and the transient response when accelerating are measured accurately and investigated thoroughly. The influence of stiffness anisotropy and the sensor mounting inclination are investigated experimentally, and whirl modes are observed and explained. The results reveal that due to the stiffness anisotropy in the supports, the amplitude and the number of peaks of the steady-state unbalance response in the x and y directions are different, and backward whirl motions can be aroused both in the steady-state and transient unbalance response. The diversity of critical speeds and range of backward whirl regions reduce for the decreasing of stiffness anisotropy. Then we found that the number of peaks and the amplitude of transient unbalance response are relevant to the sensor mounting inclination. Moreover, the experimental results confirm the appearance of straight-line orbits and the alternations of the whirl modes when passing through critical speeds, and the mechanism are specified using the beat vibration theory as well.

APLIKASI KOORDINAT PARALEL DI DALAM RUANG DIMENSI 4 PADA DISPLAY LALU-LINTAS PESAWAT

This research aims to describe ann-dimensional line on a parallel coordinate and uses the representation as a display of aircraft motion which flies at a straight line and constant velocity. A parallel coordinate of n-dimensional space depicted in the form of nparallel vertical lines which represent axis.Every two adjacent axes have the same distance. A horizontal line that cutsall axes indicates the initial pointsof each axis. In a parallel coordinate,an 15n"> -dimensional point is represented as a polygonal chainwhere the vertices located on its axis. Based on the representations of some of collinear points, a line is described on a parallel coordinate. On the other hand, one can consider a graph of an aircraft motion as a graph of a 4-dimensional space. At a constant speed with a straight line orbit, the graph of an aircraft movementis a graph of 4-dimensional line. The result shows that, on a parallel coordinate, an n-dimensional line represented as 15n-1"> dots. As a consequence, the graph of an aircraft that moveat a constant speed with a straight line orbitrepresented as3 dots. By this representation, the coordinate and altitude of the aircraft can be observes at anytime. It also shows whether the movement of an aircraft disturb (strike or too close to) another plane or not. Keywords: parallel coordinate, n-dimensional space, aircraft movement display

Straight line orbits in Hamiltonian flows

We investigate periodic straight-line orbits (SLO) in Hamiltonian force fields using both direct and inverse methods. A general theorem is proven for natural Hamiltonians quadratic in the momenta in arbitrary dimension and specialized to two and three dimension. Next we specialize to homogeneous potentials and their superpositions, including the familiar H\'enon-Heiles problem. It is shown that SLO's can exist for arbitrary finite superpositions of $N$-forms. The results are applied to a family of generalized H\'enon-Heiles potentials having discrete rotational symmetry. SLO's are also found for superpositions of these potentials.

Level Density of the Hénon-Heiles System Above the Critical Barrier Energy

We discuss the coarse-grained level density of the Henon-Heiles system above the barrier energy, where the system is nearly chaotic. We use periodic orbit theory to approximate its oscillating part semiclassically via Gutzwiller's semiclassical trace formula (extended by uniform approximations for the contributions of bifurcating orbits). Including only a few stable and unstable orbits, we reproduce the quantum-mechanical density of states very accurately. We also present a perturbative calculation of the stabilities of two infinite series of orbits (Rn and Lm), emanating from the shortest librating straight-line orbit (A) in a bifurcation cascade just below the barrier, which at the barrier have two common asymptotic Lyapunov exponents χR and χL. The two-dimensional Henon-Heiles (HH) Hamiltonian

Close triple approaches and escape in the three-body problem

We have studied a total of 5000 close triple approaches resulting in escape, for equal-mass systems with zero initial velocities. Escape is shown to take place in the majority of the cases after a ‘fly-by’ close triple approach when the escaper passes near the centre of mass along an almost straight-line orbit. A number of configurational and kinematical parameters are introduced in order to characterize the triple approach. The distributions of these parameters are investigated. A comparison with 831 examples in the vicinity of the so-called Pythagorean problem is carried out. We find that the general features of close triple approaches which result in escape are the same for both types of systems.

Practical evaluation of several cone beam orbits for SPECT

Several alternative orbits have been suggested for cone beam single photon emission computed tomography (SPECT), since the standard circular orbit provides incomplete sampling. The authors' computer simulation evaluates the practicality of several orbits, using the following criteria: spatial resolution, sensitivity, elimination of truncation, and theoretical completeness of the acquired data. All orbits are evaluated for brain imaging with a realistic human model. The results indicate that the simple transverse circular orbit has high sensitivity and resolution. The acquired data are incomplete. Modest improvement in resolution is obtained with a contoured transverse plane orbit. A single straight line orbit was inadequate. Three orbits, each composed of two noncoplanar curves, satisfy the completeness condition, and thus should produce more accurate reconstructions. Their resolution is similar to the circle, but the sensitivity is reduced by 15-30%. None of the orbits studied are as good in both resolution and sensitivity as the transverse plane orbits (circle and contoured), which do not satisfy the completeness condition. >

Newtonian and relativistic periodic orbits around two fixed black holes

We compare families of simple periodic orbits of test particles in the Newtonian and relativistic problems of two fixed centers (black holes). The Newtonian problem is integrable, while the relativistic problem is highly non-integrable. The orbits are calculated on the meridian plane through the fixed centersM 1 (atz=+1) andM 2 (atz=−1) for energies smaller than the escape energyE=1. We use prolate spheroidal coordinates (ψ, ϑ, ϕ=const) and also the variables λ=cosh ψ and μ=−cos ϑ. The orbits are inside a curve of zero velocity (CZV). The Newtonian orbits are also limited by an ellipse and a hyperbola, or by two eillipses. There are 3 main types of periodic orbits (1) elliptic type (around both centers), (2) hyperbolic-type, and (3) resonant-type. The elliptic type orbits are stable in the Newtonian case and both stable and unstable in the relativistic case. From the stable orbits bifurcate double period orbits both symmetric and asymmetric with respect to thez-axis. There are also higher order bifurcations. The hyperbolic-type orbits are unstable. The Newtonian resonant orbits are defined by the ratiot µ/t λ=n/m of oscillations along λ and μ during one period, and they are all marginally unstable. The corresponding relativistic orbits are stable, or unstable. The main families are figure eight orbits aroundM 1, or aroundM 2 (3/1 orbits); gamma, or inverse gamma orbits (4/2); higher resonant families 5/1,7/1,...,8/2,12/2,...;, more complicated orbits, like 5/3, and bifurcations from the above orbits. Satellite orbits aroundM 1, orM 2, and their bifurcations (e.g. double period) exist in the relativistic case but not in the Newtonian case. The characteristics of the various families are quite different in the Newtonian and the relativistic cases. The sizes of the orbits and their stabilities are also quite different in general. In the Appendix we study the various types of straight line orbits and prove that some subcases introduced by Charlier (1902) are impossible.

Stability analysis of some periodic orbits in the hydrogen atom in parallel electric and magnetic fields

The stability of some prominent periodic orbits in the hydrogen atom in parallel electric and magnetic fields is studied. For small field strengths, the application of classical perturbation theory allows for analytic treatments and provides useful information on possible types of classical motion and corresponding quantum states. Non-perturbative numerical and analytic stability analysis and calculations of the Liapunov exponent are performed for the straight-line orbit along the direction of the fields in the region of the zero-field ionization threshold. Bifurcation phenomena are discussed and related to recent photoexcitation experiments.

Local WKB dispersion relation for the Vlasov–Maxwell equations

A formalism for analyzing systems of integral equations, based on the Wentzel–Kramers–Brillouin (WKB) approximation, is applied to the Vlasov–Maxwell integral equations in an arbitrary-β, spatially inhomogenous plasma model. It is shown that when treating frequencies comparable with and larger than the cyclotron frequency, relevant new terms must be accounted for to treat waves that depend upon local spatial gradients. For a specific model, the response for very short wavelength and high frequency is shown to reduce to the straight-line orbit approximation when the WKB rules are correctly followed.

Line shapes for laser-induced collisions

The two-state model of laser-induced collisions introduced by Gudzenko and Yakovlenko is shown to lead to cross sections with a universal behavior in terms of the variables $z=|{C}_{3}|{E}_{0}{v}^{\ensuremath{-}\frac{3}{5}}{|{C}_{6}|}^{\ensuremath{-}\frac{2}{5}}$ and $d=\ensuremath{\delta}{|{C}_{6}|}^{\frac{1}{5}}{v}^{\ensuremath{-}\frac{6}{5}}\mathrm{sgn}({C}_{6})$. In terms of a physical description the dimensionless frequency-detuning variable is $d\ensuremath{\propto}$ (detuning of the laser from the large-$R$ resonance) (time of collision at the Weisskopf radius). The dimensionless variable $z$ is independent of laser frequency and measures the power dependence of the cross section. It is in fact proportional to $\ensuremath{\int}{\ensuremath{-}\ensuremath{\infty}}^{t}\frac{{C}_{3}{E}_{0}\mathrm{dt}}{R{(t)}^{3}}$ evaluated at an impact parameter given by the ${b}_{\ensuremath{\nu}}={(\frac{{C}_{6}}{v})}^{\frac{1}{5}}\ensuremath{\propto}$ Weisskopf radius = impact parameter where the phase shift due to the Van der Waals potential becomes $\ensuremath{\pi}$. Above, $\frac{{C}_{3}{E}_{0}}{{R}^{3}}$ is the coupling parameter at intranuclear separation $R$ and ${E}_{0}$ is the laser field amplitude. The cross section is of the form $\ensuremath{\sigma}={(\frac{|{C}_{6}|}{v})}^{\frac{2}{5}}H(d,z)$, where $H(d,z)$ is tabulated in detail. For large laser fields (i.e., $zg2$), the line shape for collisions at a particular relative velocity $v$, laser field amplitude ${E}_{0}$, and detuning (from the large-$R$ resonance frequency), $\ensuremath{\sigma}$ becomes symmetric about $\ensuremath{\delta}=0$ with the width decreasing with increasing laser power. The physical reason for the symmetric $H(d,z)$ at large $z$ is shown to be the decreased importance of curve-crossing effects for large positive $d$ corresponding to the onset of adiabatic behavior and the increased importance of contributions to $\ensuremath{\sigma}$ from such large impact parameters that the Van der Waals shifts can be neglected. Correspondingly, at large $z$ the linewidth is due entirely to time-of-collision effects. When $z\ensuremath{\ge}2$, both the long-range version of the atom-atom interaction and the assumption of straight-line orbits are excellent because of the dominant contribution to $\ensuremath{\delta}$ from impact parameters g 15 \AA{}.

A study of semiclassical approximations for heavy-ion transfer reactions

Semiclassical concepts have been used to obtain a formula for determining angular distributions for heavy-ion transfer reactions starting from the expression of the transition matrix in DWBA. The distorted waves are expanded in partial waves and the radial wavefunctions approximated by the WKB approximation. Further approximations enable one of the partial-wave summations to be performed analytically. The transfer matrix elements occurring in the resulting formula has been simplified to a one-dimensional integral by assuming a straight-line orbit for the relative motion. As an illustration, angular distributions have been calculated for the reactions 26Mg(11B,10B)27Mg and 26Mg(11B,10Be)27Al at 114 MeV laboratory energy and compared with experiment. The agreement is good.

Ray Tracing in Axisymmetric Toroidal Coordinates

A method is presented for determining the trajectory of a particle moving along a straight line orbit in axisymmetric toroidal coordinate systems. These geometries occur in problems associated with neutral, neutron, and radiation transport studies in tokamak fusion devices. Numerical solutions of the equations describing the trajectory are performed in geometric configurations generated by the solution of the plasma equilibrium problem. An example problem of the deposition of a pencil beam of high-energy neutral particles in a tokamak plasma with a noncircular cross section due to the necessity of incorporating a divertor and the desirability of operating at a high plasma energy density is described.

Damping of plasma waves in the lower hybrid frequency range

Damping of quasi-longitudinal plasma oscillations in the lower hybrid frequency range has been investigated. The complete dispersion equation governing small-amplitude quasi-longitudinal oscillations of hot homogeneous anisotropic plasma has been solved numerically for a wide range of plasma parameters. Temporal damping rates of stable eigen oscillations and spatial damping rates of externally driven plasma waves are given. The lower hybrid dispersion curves are presented. The straight-line orbit approximation and the influence of distant collisions on the ion resonant interaction are discussed. The results obtained indicate that the wave energy is preferentially given to the ion perpendicular motion while an efficient energy transfer to the electrons requires a large electron velocity spread along the field lines.

Electromagnetic Wave Scattering from a Magnetically Confined High-Temperature Plasma

Electromagnetic wave scattering from a charged particle confined to a circular orbit is studied for cases where relativistic corrections must be included in the analysis to obtain a valid description for the spatial and spectral distribution of scattered radiation. The results are compared with the scattering calculated on the basis of a two-dimensional isotropic distribution of straight-line orbits. It is shown that the two calculations produce the same spectrum of scattered radiation provided the fact that the scattering region is finite in the direction of the observer is included in the analysis.

Application of the Vlasov Equation to the High-Frequency Stability of Finite Relativistic Beam-Plasma Systems

The response of a finite relativistic beam-plasma system to small disturbances is analysed using the coupled Vlasov-Maxwell equations. It is shown that when |ω−kV0|2≫ωβ2, where ωβ is the betatron frequency and V0 the drift velocity of the beam, straight-line orbits are sufficiently accurate to describe the beam-particle trajectories. Dispersion relations are derived for a finite beam and for both infinite and finite plasma configurations. The effect of a finite but large beam radius is shown to increase the growth of the electrostatic instability.

Transconductance properties of plasmas

Radio-frequency effects arising from motions of the electrons and ions in plasmas are reviewed. Starting with cases of simple straight line orbits, the relationship between theory and experiment is cited in regard to wave dispersion, damping, amplification, and radiation. Circular and spiral orbits as well as linear and drifting-linear vibration orbits are considered. These examples show the common properties that linear drifting can produce damping or amplification while periodicities or near periodicities produce radiative coupling. The insight thus gained leads to some concluding remarks concerning more general orbits.