Epicyclic Model Research Articles

Generation of arbitrarily structured optical vortex arrays based on the epicycle model.

Optical vortex arrays (OVAs) are complex light fields with versatile structures that have been widely studied in large-capacity optical communications, optical tweezers, and optical measurements. However, generating OVAs with arbitrary structures without explicit analytical expressions remains a challenge. To address this issue, we propose an alternative scheme for customizing OVAs with arbitrary structures using an epicycle model and vortex localization techniques. This method can accurately generate an OVA with an arbitrary structure by pre-designing the positions of each vortex. The influence of the number and coordinates of the locating points on customized OVAs is discussed. Finally, the structures of the OVA and each vortex are individually shaped into specifically formed fractal shapes by combining cross-phase techniques. This unique OVA will open up novel potential applications, such as the complex manipulation of multiparticle systems and optical communication based on optical angular momentum.

Epicycle-model-guided arbitrary shaped customization of structured light

Structured light has been exploited as an important tool for particle manipulation along a desired complex path. However, generating the required structured light illumination for the creation of an arbitrary shape without an analytic expression as a guide is challenging, specifically for designing a structured beam by mapping the shape of an arbitrary object. To address this issue, we propose an effective scheme to customize structured light freely and precisely by modifying the epicycle model in astrophysics. Predesigned structured beams can be identified with or without explicit analytic expressions of the desired shapes. Moreover, we study the roles and relationships between the number of epicycles and the number of key points. The local stretching and transformation of a specific structured beam are also analyzed. The advantages of the proposed method are demonstrated by conducting dynamic manipulation experiments using polystyrene particles. This method is simple and intuitive and provides an effective toolkit for the design of structured light for more complex tasks, thus facilitating advanced applications in optical manipulations.

Epicycles in the Hyperbolic Sky

Consider a swiveling arm on an oriented complete riemannian surface composed of three geodesic intervals, attached one to another in a chain. Each interval of the arm rotates with constant angular velocity around its extremity contributing to a common motion of the arm. Does the extremity of such a chain have an asymptotic velocity? This question for the motion in the euclidian plane, formulated by J.-L. Lagrange, was solved by P. Hartman, E. R. Van Kampen, A. Wintner. We generalize their result to motions on any complete orientable surface of non-zero (and even non-constant) curvature. In particular, we give the answer to Lagrange’s question for the movement of a swiveling arm on the hyperbolic plane. The question we study here can be seen as a dream about celestial mechanics on any riemannian surface: how many turns around the Sun a satellite of a planet in the geliocentric epicycle model would make in 1 billion years?

Eccentricity samples: Implications on the potential and the velocity distribution

Planar and vertical epicycle frequencies and local angular velocity are related to the derivatives up to the second order of the local potential and can be used to test the shape of the potential from stellar disc samples. These samples show a more complex velocity distribution than halo stars and should provide a more realistic test. We assume an axisymmetric potential allowing a mixture of independent ellipsoidal velocity distributions, of separable or Staeckel form in cylindrical or spherical coordinates. We prove that values of local constants are not consistent with a potential separable in addition in cylindrical coordinates and with a spherically symmetric potential. The simplest potential that fits the local constants is used to show that the harmonical and non-harmonical terms of the potential are equally important. The same analysis is used to estimate the local constants. Two families of nested subsamples selected for decreasing planar and vertical eccentricities are used to borne out the relation between the mean squared planar and vertical eccentricities and the velocity dispersions of the subsamples. According to the first-order epicycle model, the radial and vertical velocity components provide accurate information on the planar and vertical epicycle frequencies. However, it is impossible to account for the asymmetric drift which introduces a systematic bias in estimation of the third constant. Under a more general model, when the asymmetric drift is taken into account, the rotation velocity dispersions together with their asymmetric drift provide the correct fit for the local angular velocity. The consistency of the results shows that this new method based on the distribution of eccentricities is worth using for kinematic stellar samples.

Models of quasi-periodic oscillations related to mass and spin of the GRO J1655-40 black hole

Frequencies of the three quasi-periodic oscillation (QPO) modes observed simultaneously in the accreting black hole GRO J1655-40 are compared with the predictions of models. Models in which all three QPO signals are produced at the same radius are considered: these include different versions of relativistic precession, epicyclic resonance, tidal disruption, and warped disc models. Models that were originally proposed to interpret only the twin high-frequency QPOs are generalized here to interpret also the low-frequency QPO in terms of relativistic nodal precession. Inferred values of the black hole mass and spin from each QPO model are compared with the mass measured from optical observations and the spin inferred from X-ray spectroscopy techniques. We find that along with the relativistic precession model predicting $M=(5.3\pm0.1)~M_{\odot}, a=0.286\pm0.004$, the so-called total precession model ($M=(5.5\pm0.1)~M_{\odot}, a=0.276\pm0.003$), and the resonance epicyclic model with beat frequency ($M=(5.1\pm0.1)~M_{\odot}, a=0.274\pm0.003$) also satisfy the optical mass test. We compare our results with those inferred from X-ray spectral measurements.

Mass of intermediate black hole in the source M82 X-1 restricted by models of twin high-frequency quasi-periodic oscillations

We apply the relativistic precession (RP) model with its variants and the resonance epicyclic model with its variants, based on the frequencies of the geodesic epicyclic motion in the field of a Kerr black hole, to put limits on the mass of the black hole in the ultraluminous X-ray source M82 X-1 demonstrating twin high-frequency quasi-periodic oscillations (HF QPOs) with the frequency ratio near 3:2. The mass limits implied by the geodesic HF QPO models are compared to those obtained due to the model of string loop oscillations around a stable equilibrium position. Assuming the whole range of the black hole dimensionless spin, 0 < a < 1, the restrictions on the black hole mass related to the twin HF QPOs are widely extended and strongly model dependent; nevertheless, they give the lower limit M_{M82 X-1} > 130 M_{sun} confirming existence of an intermediate black hole in the M82 X-1 source. The upper limit given by one of the variants of the geodesic twin HF QPO models goes up to M_{M82 X-1}<1500 M_{sun}. The range 37-210 mHz of the low-frequency QPOs observed in the M82 X-1 source introduces additional restrictive limits on the black hole mass, if we model the low-frequency QPOs by nodal precession of the epicyclic motion. The nodal precession model restrictions combined with those implied by the geodesic models of the twin HF QPOs give allowed ranges of the M82 X-1 black hole parameters, namely 140 M_{sun}<M_{M82 X-1}<660 M_{sun} for the mass parameter and 0.05<a_{M82 X-1}<0.6 for the spin parameter.

Black hole spin inferred from 3:2 epicyclic resonance model of high-frequency quasi-periodic oscillations

Estimations of black hole spin in the three Galactic microquasars GRS 1915+105, GRO J1655-40, and XTE J1550-564 have been carried out based on spectral and timing X-ray measurements and various theoretical concepts. Among others, a non-linear resonance between axisymmetric epicyclic oscillation modes of an accretion disc around a Kerr black hole has been considered as a model for the observed high-frequency quasi-periodic oscillations (HF QPOs). Estimates of spin predicted by this model have been derived based on the geodesic approximation of the accreted fluid motion. Here we assume accretion flow described by the model of a pressure-supported torus and carry out related corrections to the mass-spin estimates. We find that for dimensionless black hole spin a<0.9, the resonant eigenfrequencies are very close to those calculated for the geodesic motion. Their values slightly grow with increasing torus thickness. These findings agree well with results of a previous study carried out in the pseudo-Newtonian approximation. The situation becomes different for a>0.9, in which case the resonant eigenfrequencies rapidly decrease as the torus thickness increases. We conclude that the assumed non-geodesic effects shift the lower limit of the spin, implied for the three microquasars by the epicyclic model and independently measured masses, from a~0.7 to a~0.6. Their consideration furthermore confirms compatibility of the model with the rapid spin of GRS 1915+105 and provides highly testable predictions of the QPO frequencies. Individual sources with a moderate spin (a<0.9) should exhibit a smaller spread of the measured 3:2 QPO frequencies than sources with a near-extreme spin (a~1). This should be further examined using the large amount of high-resolution data expected to become available with the next generation of X-ray instruments, such as the proposed Large Observatory for X-ray Timing (LOFT).

The epicycle model

The solutions of the equations of motion for a point mass particle under a conservative force field are generally constrained by a basic set of integrals of motion, which depend on the knowledge about the potential function and on time and space properties and symmetries satisfied by the dynamical system. The epicycle approximation is a particular case of integration of the equations of motion under a minimum set of hypotheses, such as symmetry plane and axial symmetry, allowing to obtain solutions for nearly circular orbits in the three dimensional space. Under this approach, any orbit projected onto the symmetry plane describes an ellipse with origin at a guiding centre or epicentre, which moves uniformly in a circular orbit around the centre of the system. The approach is easy to model and allows to visualise the contribution of several parameters to the shape of the orbits, being an excellent example of how education and learning of science can take profit of mathematical modelling.

Solar Anomaly and Planetary Displays in the Antikythera Mechanism

Recent scholarship has greatly improved our understanding of the structure and function of the Antikythera mechanism.1 One unresolved issue is whether the mechanism included a display of the variability (or anomaly) in the angular motion of the Sun. In Price’s original reconstruction, the motion of the Sun was treated only in the mean. A single pointer indicated the day of the year in the Egyptian calendar and the position of the uniformly-moving mean Sun in the zodiac. Of course, the actual position of the Sun can depart from that of the mean Sun by up to 2° at some times of year (up to 2°23′ in Hipparchos’s solar theory). Wright seems to have been the first to propose that the mechanism treated the solar anomaly.2 And since the recognition that the mechanism includes a representation of the Moon’s variable motion,3 probably based on an epicycle model like that of Hipparchos, a number of similar proposals for the solar anomaly have been made (although no gear work for the Sun survives except a simple set of gears that produced a uniform angular motion). Generally these proposals involve separate pointers for the mean and true Sun. We will show that the mechanism did, indeed, treat the solar anomaly, but in a more economical fashion. A single pointer sufficed to give the date in the Egyptian year (indicative of the mean Sun) and the place of the true Sun in the zodiac. A second unresolved issue is the nature of the mechanism’s planetary displays. The inscription on the front cover appears to include detailed references to the synodic behaviour of Aphrodite (Venus).4 Some scholars have therefore proposed that the mechanism gave a complete, kinematic display of the motions of all five known planets around the zodiac. Freeth, Jones, Steele and Bitsakis, for example, wrote, “It seems likely that the Antikythera Mechanism also displayed some or even all of the five planets known in ancient times, but there is considerable debate about this”.5 Michael Wright had made a similar proposal and, in a mechanical tour de force, built a working model of the Antikythera mechanism that included pointers for the zodiacal positions of the five planets, along with the Sun and Moon, all with co-axial movement.6 While this proves that such a design is technically possible, it does not, of course, demonstrate that this is what the builder of Antikythera mechanism actually did (as Wright himself freely grants). It is important to remember that no gear

Accuracy of epicyclic description of dust grain orbits about Saturn

We investigate the metamorphosis of charged dust grain orbits in a planetary magnetosphere, the so‐called gravito‐electrodynamic problem. Our primary aim is to determine practical limits to the validity of an epicyclic model. We assume an axisymmetric planet with a centered dipole magnetic field aligned with the planetary spin axis, a good approximation for Saturn and a fair approximation for Jupiter. Nonideal forces, such as planetary oblateness, solar radiation pressure, plasma drag, and the effects of time‐dependent charging are neglected. A standard epicyclic expansion is used to calculate the mean motion and the aspect ratio of the epicyclic ellipse. For gravitationally dominated grains the mean motion is accurately given by the standard expression, but for magnetically dominated grains it can differ significantly even for small excursions from the equilibrium orbit. In some cases this finite gyroradius effect can even cause the mean motion to change direction. Previous studies are extended by examining a wide range of charge‐to‐mass ratios, for positively and negatively charged grains in prograde or retrograde equatorial orbits about Saturn, inside and outside of the synchronous radius. The results show that the epicyclic model is reasonably accurate for the majority of physically interesting orbits about Saturn. For grains suffering large radial excursions, however, major topological changes were often seen which defy an epicyclic description.

On the kinematic deconvolution of the local neighbourhood luminosity function

ABSTRA C T A method for inverting the statistical star counts equation, including proper motions, is presented; in order to break the degeneracy in that equation, it uses the supplementary constraints required by dynamical consistency. The inversion gives access to both the kinematics and the luminosity function of each population in three regimes: the singular ellipsoid, the constant ratio Schwarzschild ellipsoid plane-parallel models and the epicyclic model. This more realistic model is tailored to account for the local neighbourhood density and velocity distribution. The first model is fully investigated, both analytically and by means of a non-parametric inversion technique, while the second model is shown to be formally its equivalent. The effect of noise and incompleteness in apparent magnitude is investigated. The third model is investigated by a 5D 1 2D non-parametric inversion technique where positivity of the underlying luminosity function is explicitly accounted for. It is argued that its future application to data such as the Tycho catalogue (and in the upcoming satellite GAIA ) could lead - provided that the vertical potential and or the asymmetric drift or w( are known - to a non-parametric determination of the local neighbourhood luminosity function without any reference to stellar evolution tracks. It should also yield the proportion of stars for each kinematic component and a kinematic diagnostic to split the thin disc from the thick disc or the halo.

On the Greek Origin of the Indian Planetary Model Employing a Double Epicycle

The development of geometric models to explain the inequalities of planetary motion in terms of combinations of circular motions is characteristic of Greek astronomy, and derives its initial bias from statements of the Pythagoreans, Plato, and Aristotle regarding the structure of the universe and, in the case of Aristotle, the difference between the mechanics of the sublunar centre of the cosmos and that of the celestial spheres. It is hardly necessary to point out here that among the devices invented to account for these inequalities are the eccentric circle and the epicycle, whose properties were investigated by Apollonius of Perge in about 200 b.c.1 The occurrence of these devices in Sanskrit astronomical texts of the fifth century a.d. immediately suggests some Greek influence. And the supposition of such an influence is greatly strengthened by the fact that Greek adaptations of Babylonian linear astronomy2 and Greek treatises on genethlialogy3 were translated into Sanskrit between the second and fourth centuries a.d.; it is virtually confirmed by the fact that the earliest Sanskrit siddhânias to employ epicyclic models, the older Romaka (for the luminaries only4) and Paulisa (for the Sun only5), are largely of Greek or GrecoBabylonian origin. It is my intention here to investigate the Greek background of the common Indian model for the star-planets which involves two concentric epicycles.In the Paitâmahasiddhânta of the Visnudharmottarapurâria,6 which is our earliest extant exponent of the Indian double-epicycle planetary model (it was probably composed in the first half of the fifth century a.d.), the pattern was set for all later texts except for those belonging to or aware of the auduyaka System of the Âryapakca. The orbits of the planets are concentric with the centre of the Earth. The single inequalities recognized in the cases of the two luminaries are explained by manda-cpicycles (corresponding functionally to the Ptolemaic eccentricity of the Sun and lunar epicycle respectively), the two inequalities recognized in the case of each of the five star-planets by a mandaepicycle (corresponding to the Ptolemaic eccentricity) and a .%Arn-epicycle (corresponding to the Ptolemaic epicycle). The further refinements of the Ptolemaic models are unknown to the Indian astronomers.If one tries to imagine the geometric model utilized by the Paitâmahasiddhânta, one immediately realizes that it cannot be cinematic ; the planet cannot ride simultaneously on the circumferences of two epicycles. These two epicycles must be regarded simply as devices for calculating the amounts of the equations by which the mean planet on its concentric orbit is displaced to its true position. This interpretation is confirmed by the explanation offered in early texts of the mechanics of the unequal motions of the planets : demons stationed at the manda and ftghra points on their respective epicycles pull at the planets with chords of wind.7 The computation of the total effect of these two independent forces upon the mean planet varies somewhat from one school (pak$a) of astronomers to another, or even from astronomer to astronomer within a paksa. But the fundamental concept remains clear: the planet is always situated on the ciicumference of a deferent circle concentric with' the centre of the Earth, while two epicycles (one each for the Sun and Moon) revolve about it. As the planet progresses with its mean velocity about this deferent circle, at each instant it is pulled by the two epicycles away from its mean to its true longitude. These instantaneous true longitudes are subject to computation, but a true course of the planet over a period of time can only be conceived of as a series of such instantaneous true longitudes.This is not to deny that the equivalence of an eccentric to an epicyclic model had also been transmitted from Greece to India, though it would seem that the transmission was effected by a different text from that from which the doubleepicycje theory is derived. …