Trajectories Of Test Particles Research Articles

Particle and light motion in Lyra–Schwarzschild spacetime

Abstract In this paper we recall the static and spherically symmetric solution derived within LyST gravity. LyST stands for “Lyra Scalar-Tensor” and is a scalar-tensor proposal for the gravitational interaction modifying general relativity by adopting Lyra manifold as the spacetime substrate. Lyra manifold is characterized both by the metric tensor gµν(x) and the scale φ(x); the latter being an integral part of the definition of reference frame. In a previous work, we have launched the formal geometrical basis of LyST, we have proposed an associated action integral for the gravitational interaction, we have derived the field equations generalizing Einstein field equation, and we have solved these equation to obtain the generalized version of Schwarzschild line element in Lyra manifold. This lead us to Lyra-Schwarzschild spacetime, which is further explored in the present paper. Herein, the trajectories of test particles is thoroughly studied. This is done for both massive particles and massless particles. The geodesic equations are built and solved. The effective potential associated to the motion of these particles around the source is determined and characterized. The possible trajectories include gravitational capture, scattering, bounded orbits, stable and unstable circular orbits, near-horizon motion. The innermost circular orbit allowed for massive particles is determined and compared to the one predicted by GR. The last photon orbit is also calculated for Lyra-Schwarzschild metric. The equation for the periastron shift in the context of Lyra-Schwarzschild solution is constructed; its application to observational data could be useful to constrain the parameters typical of LyST, telling it apart from GR.

The Ricci Rotation Coefficients in the Description of Trajectories of Spinning Test Particles Off-equatorial Plane in the Gravitational Field of a Rotating Source

We describe the trajectories of circular orbits of spinless and spinning test particles around a rotating body in equatorial and non-equatorial planes via the Mathisson-Papapetrou-Dixon equations. In this paper, these equations include the Ricci rotation coefficients with the purpose of describing not only the curvature of space time, but also the rotation of the spinning test particles that orbit around a rotating massive body. We found a numerical difference between the trajectories of spinless test particles and spinning test particles in the order of 10-7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{10}}^{\\varvec{-7}}$$\\end{document}. We take as parameters: radius, energy, Carter’s constant and angular momentum.

The Energy-dependent Gamma-Ray Light Curves and Spectra of the Vela Pulsar in the Dissipative Magnetospheres

We study the pulsar energy-dependent γ-ray light curves and spectra from curvature radiation in the dissipative magnetospheres. The dissipative magnetospheres with the combined force-free (FFE) and Aristotelian are computed by a pseudo-spectral method with a high-resolution simulation in the rotating coordinate system, which produces a near-FFE field structure with the dissipative region only near the equatorial current sheet outside the light cylinder. We use the test-particle trajectory method to compute the energy-dependent γ-ray light curves and phase-average and phase-resolved spectra by including both the accelerating electric field and radiation reaction. The predicted energy-dependent γ-ray light curves and spectra are then compared with those of the Vela pulsar observed by Fermi. Our results can generally reproduce the observed trends of the energy-dependent γ-ray light curves and spectra for the Vela pulsar.

Imprints of rastall gravity on particle dynamics

In this paper, we investigate the trajectories of test particle around the Kerr-Newman anti de-Sitter black hole enclosed by scalar field matter in the context of Rastall gravity. We limit our research to the equatorial plane and analyze the dynamics of both null as well as time-like geodesics. We explore the stable regions with respect to the event horizon, radius of photon orbit and the so called static radius for both null as well as time-like particles. We found that the stable point is always less than the cosmological horizon as well as the static radius. We also discuss the energy extraction as well as negative energy state via Penrose process. The efficiency of the process from Kerr-Newman anti de-Sitter black hole is also examined. We observe that the efficiency of the Penrose process increased with the spin parameter as well the intensity of the scalar field around the BH.

Non-stationary free-space geodesic dynamics due to the interaction with weak gravitational waves

In this paper, the author investigates how non-stationary reference frames affect the observed trajectories of test particles, and how these observed effects might allow for the detection of gravitational waves. Most gravitational wave experiments rely on observing geometric oscillations of the spacetime between stationary test masses (with respect to each other). The results of this study indicate that an impinging gravitational wave, even of very weak amplitude, can cause much more noticeable effects between moving reference frames. The concepts herein are loosely analogous to how the properties of a magnetic field can be obtained from observing a charged particle’s cyclotron motion through it. To this end, the linearized geodesic differential equations are solved to obtain the coordinates of a particle along a path, and explicit trajectories are calculated for a wide range of gravitational wave parameters. It is found that the angular deviation in trajectory is dependent on the polarization state of the wave and amplitude, leading to the possibly of extracting these parameters from the deviation for real-world experiments. The approach here represents a first step and as such, there are many simplifying assumptions, which will be relaxed slowly over time in future work.

Energy nonconservation and relativistic trajectories: Unimodular gravity and beyond

Energy conservation has the status of a fundamental physical principle. However, measurements in quantum mechanics do not comply with energy conservation. Therefore, it is expected that a more fundamental theory of gravity -- one that is less incompatible with quantum mechanics -- should admit energy nonconservations. This paper begins by identifying the conditions for a theory to have an energy-momentum tensor that is not conserved. Then, the trajectory equation for pointlike particles that lose energy is derived, showing that energy nonconservation produces a particular acceleration. As an example, the unimodular theory of gravity is studied. Interestingly, in spherical symmetry, given that there is a generalized Birkhoff theorem and that the energy-momentum tensor divergence is a closed form, the trajectories of test particles that lose energy can be found using well known methods. Finally, limits on the energy nonconservation parameters are set using Solar system observations.

ASSIST: An Ephemeris-quality Test-particle Integrator

We introduce ASSIST, a software package for ephemeris-quality integrations of test particles. ASSIST is an extension of the REBOUND framework and makes use of its IAS15 integrator to integrate test-particle trajectories in the field of the Sun, Moon, planets, and 16 massive asteroids, with the positions of the masses coming from the Jet Propulsion Laboratory’s DE441 ephemeris and its associated asteroid perturber file. The package incorporates the most significant gravitational harmonics and general-relativistic corrections. ASSIST also accounts for position- and velocity-dependent nongravitational effects. The first-order variational equations are included for all terms to support orbit fitting and covariance mapping. This new framework is meant to provide an open-source package written in a modern language to enable high-precision orbital analysis and science by the small-body community. ASSIST is open source, freely distributed under the GNU General Public license v3.0.

Gravitational wave of the Bianchi VII universe: particle trajectories, geodesic deviation and tidal accelerations

For the gravitational wave model based on the type III Shapovalov wave space-time, test particle trajectories and the exact solution of geodesic deviation equations for the Bianchi type VII universe are obtained. Based on the found 4-vector of deviation, tidal accelerations in a gravitational wave are calculated. For the obtained solution in a privileged coordinate system, an explicit form of transformations into a synchronous reference system is found, which allows time synchronization at any points of space-time with separation of time and spatial coordinates. The synchronous reference system used is associated with a freely falling observer on the base geodesic. In a synchronous coordinate system, an explicit form of the gravitational wave metric, a 4-vector of geodesic deviation, and a 4-vector of tidal accelerations in a gravitational wave are obtained. The exact solution describes a variant of the primordial gravitational wave. The results of the work can be used to study the plasma radiation generated by tidal accelerations of a gravitational wave.

Застосування числових методів газодинамічних розрахунків в задачах обтікання перешкод розрідженим струминним потоком

The development of competitive space hardware calls for continuing improvements in the accuracy of simulation of gas-dynamic processes in the space vehicle vicinity. This may contribute to extending the active life of spacecraft, thus improving the economic efficiency of space activities. In particular, quite a topical problem is the simulation of the interaction of rarefied jets from the propulsion system of a spacecraft with its individual components. To solve this problem in the case of a rather high surrounding vacuum, use is made of the molecular-kinetic concept of the gas structure based on the Boltzmann equation. The aim of this paper is to overview existing methods of simulation of gas-dynamic processes near spacecraft in a rarefied gas flow with account for propulsion system jets and to choose the most promising approaches to the solution of this problem. Among the methods considered, several main lines are set off: approximate, analytical, and numerical methods. Approximate methods use physical models of jet flow, approximation of numerical results, or a combination of both approaches. Analytical methods are based on essentially simplified assumptions and are intended for a very narrow class of problems. Numerical methods are the most universal tool of theoretical study. At the same time, each numerical method has a range of application of its own. At present, the most used and promising methods are statistical simulation methods: the direct simulation Monte Carlo method (DSMCM) and the test particle method (TPM). The former splits the continuous process of molecule motion and collisions in a rarefied gas into two successive independent stages (free-molecular transfer and relaxation) at each small time step. The simulation is done by time steps and in fact reproduces a nonstationary process. The latter, the TPM, consist in a statistical successive simulation of the wandering of test particles (molecules) on the background of field ones about the cells of the computational grid. Test particles, which move within the cells of the computational area, periodically collide with the obstacle in the flow and field particles, and in doing so they gradually change both their velocity and the field characteristics. For both statistical approaches, the simulation accuracy, as can be expected, is inversely proportional to the square root of the number of tests: the number of time steps and modeling particles for the DSMCM and the number of successively simulated test particle trajectories for the TPM. This may greatly affect the possibility of attaining a desired accuracy.

Cosmic-ray Transport in Magnetohydrodynamic Turbulence

Abstract This paper studies cosmic-ray (CR) transport in magnetohydrodynamic (MHD) turbulence. CR transport is strongly dependent on the properties of the magnetic turbulence. We perform test particle simulations to study the interactions of CR with both total MHD turbulence and decomposed MHD modes. The spatial diffusion coefficients and the pitch angle scattering diffusion coefficients are calculated from the test particle trajectories in turbulence. Our results confirm that the fast modes dominate the CR propagation, whereas Alfvén and slow modes are much less efficient and have shown similar pitch-angle scattering rates. We investigate the cross field transport on large and small scales. On large/global scales, normal diffusion is observed and the diffusion coefficient is suppressed by M A ζ compared to the parallel diffusion coefficients, with ζ closer to 4 in Alfvén modes than that in total turbulence, as theoretically expected. For the CR transport on scales smaller than the turbulence injection scale, both the local and global magnetic reference frames are adopted. Superdiffusion is observed on such small scales in all the cases. Particularly, CR transport in Alfvén modes show clear Richardson diffusion in the local reference frame. The diffusion transitions smoothly from the Richardson’s one with index 1.5 to normal diffusion as the particle mean free path decreases from λ ∥ ≫ L to λ ∥ ≪ L, where L is the injection/coherence length of turbulence. Our results have broad applications to CRs in various astrophysical environments.

The Pulsar Gamma-Ray Emission from High-resolution Dissipative Magnetospheres

Abstract The pulsar light curves and energy spectra in dissipative pulsar magnetospheres are explored with Aristotelian electrodynamics (AE), where particle acceleration is fully balanced with the radiation reaction. AE magnetospheres with nonzero pair multiplicity are computed using a pseudo-spectral method in the co-moving frame. The dissipative region near the current sheet outside the light cylinder is accurately captured by a high-resolution simulation. The pulsar light curves and spectra are computed using the test particle trajectory method, including the influence of both the consistent accelerating electric field and radiation reaction. Our results can generally reproduce the double-peak light curves and the GeV cutoff energy spectra in agreement with the Fermi observations for the pair multiplicity κ ≳ 1.

The reversibility of magnetic reconnection

The reversibility of the transfer of energy from the magnetic field to the surrounding plasma during magnetic reconnection is examined. Trajectories of test particles in an analytic field model demonstrate that irreversibility is associated with separatrix crossings and passages through regions of weaker magnetic field. Inclusion of a guide field enhances the magnetization of particles and the extent to which forward and reverse trajectories overlap. Full kinetic simulations with a particle-in-cell code support these results and demonstrate that while time-reversed simulations at first “un-reconnect,” they eventually evolve into a reconnecting state.

Orbital and epicyclic motion of charged test particles around non-rotating Einstein-Æther black holes

The study of charged test particle dynamics in the combined black hole gravitational field and magnetic field around it could provide important theoretical insight into astrophysical processes around such compact object. We have explored the orbital and epicyclic motion of charged test particles in the background of non-rotating Einstein-Æther black holes in the presence of external uniform magnetic field. We numerically integrate the equations of motion and analyze the trajectories of the charged test particles. We examined the stability of circular orbits using effective potential technique and study the characteristics of innermost stable circular orbits. We analyze the key features of quasi-harmonic oscillations of charged test particles nearby the stable circular orbits in an equatorial plane of the black hole, and investigate the radial profiles of the frequencies of latitudinal as well as radial harmonic oscillations in dependence on the strength of magnetic field, mass of the black hole and dimensionless coupling constants of the theory. We demonstrate that the magnetic field and dimensionless parameters of the theory have strong influence on charged particle motion around Einstein-Æther black holes.

Crystals of gauged solitons, force-free plasma, and resurgence

We show that the (3+1)-dimensional gauged non-linear sigma model minimally coupled to a U(1) gauge field possesses analytic solutions representing gauged solitons at finite Baryon density whose electromagnetic field is a Force Free Plasma. These gauged solitons manifest a crystalline structure and generate in a very natural way persistent currents able to support Force Free Plasma electromagnetic fields. The trajectories of charged test particles moving within these configurations can be characterized. Quite surprisingly, despite the non-integrable nature of the theory, some of the perturbations of these gauged solitons allow to identify a proper resurgent parameter. In particular, the perturbations of the solitons profile are related to the Lam\'e operator. On the other hand, the electromagnetic perturbations on the configurations satisfy a two-dimensional effective Schrodinger equation, where the soliton background interacts with the electromagnetic perturbations through an effective two-dimensional periodic potential. We studied numerically the band energy spectrum for different values of the free parameters of the theory and we found that bands-gaps are modulated by the potential strength. Finally we compare our crystal solutions with those of the (1+1)- dimesional Gross-Neveu model.

The Reversibility of Magnetic Reconnection

The reversibility of the transfer of energy from the magnetic field to the surrounding plasma during magnetic reconnection is examined. Trajectories of test particles in an analytic model of the fields demonstrate that irreversibility is associated with separatrix crossings and regions of weaker magnetic field. Inclusion of a guide field increases the degree of reversibility. Full kinetic simulations with a particle-in-cell code support these results and demonstrate that while time-reversed simulations at first un-reconnect, they eventually evolve into a reconnecting state.

Dynamical motion of matter around a charged black hole

We investigate the dynamical motion of test particles and accretion processes around spherically symmetric charged black hole. We first show the derivation of the Paczyňski–Wiita (PW) potential for any spherically symmetric spacetime and present analytical form of the PW potential in the Reissner–Nordström (RN) spacetime and the RN-type spacetime. We study the dependence of the characteristic orbits of test (neutral and charged) particles around a charged black hole from two main parameters, namely, charge of RN black hole and the interaction of charged particle with the external electromagnetic field of the black hole and present graphical trajectories of test particles around the black hole. It is shown that with increasing the absolute value of interaction parameter, [Formula: see text], the innermost stable circular orbit (ISCO) radius for charged particle always increases. It is also shown that the energy efficiency for a charged particles can reach up to [Formula: see text] in the presence of an external charged black hole, while for neutral particles it can reach up to [Formula: see text] for neutral particles. The capture cross-section of the photon is by the charged black and the optical appearance of the thin accretion disk around the charged black. Finally, we have studied the flux energy and the thermal spectrum of the accretion disk consisting of the polytropic gas, around the charged black hole.

Geodesic structure of the Euler-Heisenberg static black hole

We derive an electrically charged static black hole spacetime of the Einstein-Euler-Heisenberg theory, in terms of the Pleba\ifmmode \acute{n}\else \'{n}\fi{}ski dual variables. This solution is a nonlinear electromagnetic generalization of the Reissner-Nordstr\om solution, and it is characterized by the mass $M$, the electric charge $Q$ of the black hole, and the Euler-Heisenberg nonlinear constant, which includes the fine structure constant $\ensuremath{\alpha}$. We study all possible equatorial trajectories of test particles. Moreover, the orbits of photons are analyzed by means of the effective Pleba\ifmmode \acute{n}\else \'{n}\fi{}ski pseudometric related to the geometrical metric and to the electromagnetic energy-momentum tensor. The shape of the shadow of the black hole is also presented and discussed.

Three-dimensional general relativistic Poynting-Robertson effect. II. Radiation field from a rigidly rotating spherical source

We investigate the three-dimensional, general relativistic Poynting-Robertson effect in the case of rigidly rotating spherical source which emits radiation radially in the local comoving frame. Such radiation field is meant to approximate the field produced by the surface of a rotating neutron star, or by the central radiating hot corona of accreting black holes; it extends the purely radial radiation field that we considered in a previous study. Its angular momentum is expressed in terms of the rotation frequency and radius of the emitting source. For the background we adopt a Kerr spacetime geometry. We derive the equations of motion for test particles influenced by such radiation field, recovering the classical and weak-field approximation for slow rotation. We concentrate on solutions consisting of particles orbiting along circular orbits off and parallel to the equatorial plane, which are stabilized by the balance between gravitational attraction, radiation force and PR drag. Such solutions are found to lie on a critical hypersurface, whose shape may morph from prolate to oblate depending on the Kerr spin parameter and the luminosity, rotation and radius of the radiating sphere. For selected parameter ranges, the critical hypersurface intersects the radiating sphere giving rise to a bulging equatorial region or, alternatively, two lobes above the poles. We calculate the trajectories of test particles in the close vicinity of the critical hypersurface for a selected set of initial parameters and analyze the spatial and angular velocity of test particles captured on the critical hypersurface.

Pitch-angle diffusion coefficients in test particle simulations and the estimation of the particle parallel mean free path

The transport of energetic charged particles in turbulent magnetic fields is a topic of interest in various astrophysical contexts. In order to estimate the mean free path of a particle in the direction parallel to the mean magnetic field, one can use theoretical expressions that employ pitch-angle diffusion coefficients. In this work we review some of the methods used in estimating pitch-angle diffusion coefficients from test particle computer simulations. We examine if these methods and theoretical approaches are able to provide consistent estimates of the parallel mean free path, that can also be obtained directly from computer simulations. We perform test particle simulations for synthetic turbulence models over a range of turbulence parameters and particle energies. From the trajectories of test particles, pitch-angle distribution functions and statistics of pitch-angle displacements are obtained, which are then used to estimate the pitch-angle diffusion coefficients. We find that a method using the pitch-angle flux and derivative of the pitch-angle distribution is able to provide accurate values for the parallel mean free path over the range of parameters considered. Other methods considered are accurate only for a limited range of the turbulent fluctuation strength, or must be evaluated at a specific time to provide a reasonable estimate.

Optically thin outbursts of rotating neutron stars cannot be spherical

Abstract We investigate 3D relativistic trajectories of test particles in the spacetime of a slowly rotating compact star, under the combined influence of gravity and a strong, near-Eddington radiation field. While in the static case a spherically symmetric shell of matter suspended above the stellar surface can be formed at the location of radial equilibrium of effective forces, the same is not true for a rotating star. In the latter case the symmetry is broken by the interplay between motion in the non-static spacetime and the influence of strong radiation drag forces, pushing particles towards the equatorial plane. As a result an expanding spherical shell of matter ejected from the neutron star surface collapses on a short time-scale into a single stable equatorial ring supported by radiation. These findings have implications for the geometry of optically thin outflows during luminous neutron star bursts.