Equatorial Orbit Research Articles

Charged particles in dipole magnetosphere of neutron stars: epicyclic oscillations in and off-equatorial plane

Abstract We study the motion of charged test particles around a magnetized neutron star described by the Schwarzschild gravitational field of mass M and a dipole magnetic field characterized by parameter b, representing the ratio of electromagnetic to gravitational forces. The neutron star’s radius is assumed at $$R=3M$$ R = 3 M . Circular orbits “in” and “off” the equatorial plane, determined by the dipole field’s symmetry plane, are analyzed based on background and particle parameters; their stability against radial and latitudinal perturbations is established. The existence of chaotic charged particle motion in belts around off-equatorial orbits, influenced by sufficiently strong repulsive magnetic force, is demonstrated. The belts’ distance from the neutron star varies with parameter b for different charged particles. Frequencies of epicyclic oscillatory motion related to “in” and “off” equatorial circular orbits, alongside the orbital frequency, are determined. The magnetically modified geodesic model is applied to fit observational data from twin-peak, high-frequency quasi-periodic oscillations in binary systems containing neutron stars. Rough fitting to the data for circular orbits for particles under magnetic attraction (repulsion) is shown. The possible fittings imply strong limits on the magnetic parameter, $$b \sim 0.01$$ b ∼ 0.01 ( $$b \sim -10$$ b ∼ - 10 ) for magnetic attraction (repulsion), suggesting minor influence of electromagnetic forces and test particles with small specific charges like dust or plasmoids in strong magnetic fields around neutron stars.

Topological arrangements in the equatorial timelike circular orbits of regular black holes

The study of topology opens a new frontier in understanding the characteristics of light rings in rotating black holes, as well as the equatorial timelike circular orbits found in non-rotating black holes. This work is devoted to exploring the configurations of circular orbits within the temporal dimension, characterized by the different topology in regular black holes, and determining the corresponding zero points. Using effective potential, we built the vector in the r-θ plane where the zeros of ϕ define location of the timelike circular orbits. This special feature allows one to relate timelike circular orbits with the topology. To each zero point of the vector it is possible to assign a winding number, which defines a topological characteristic for timelike circular orbits. Winding numbers of -1 and +1 are assigned for unstable and stable timelike circular orbits respectively.

Orbital motion and QPOs testing around rotating Hairy black holes in Horndeski gravity

We investigate the orbital and oscillatory motion of test particles around a rotating hairy black hole in the context of Horndeski gravity. This black hole consists of three parameters, namely, the mass M of the black hole, spin parameter a of the black hole, and the hairy parameter q that results from the Horndeski theory of gravity. Analytical solutions are obtained for the radial profiles of specific energy and angular momentum corresponding to equatorial stable circular orbits, in terms of the model parameters. Using the effective potential method, we also examine the stability of these orbits. Additionally, we derive the analytical expressions for frequencies of radial and latitudinal harmonic oscillations, as a function of the black hole’s mass, charge, and spin of the black hole, for both local and distant observers. The key characteristics of quasi-periodic oscillations near stable circular orbits in the equatorial plane are explored. Furthermore, we analyze the precession of periastron and the Lense-Thirring effect. Our findings reveal that the particle motion around the black hole is significantly shaped by the parameters of the model.

Radiative back-reaction on charged particle motion in the dipole magnetosphere of neutron stars

The motion of charged particles under the Lorentz force in the magnetosphere of neutron stars, represented by a dipole field in the Schwarzschild spacetime, can be determined by an effective potential, whose local extrema govern circular orbits both in and off the equatorial plane, which coincides with the symmetry plane of the dipole field. In this work, we provide a detailed description of the properties of these “conservative” circular orbits and, using the approximation represented by the Landau-Lifshitz equation, examine the role of the radiative back-reaction force that influences the motion of charged particles following both the in and off equatorial circular orbits, as well as the chaotic orbits confined to belts centered around the circular orbits. To provide clear insight into these dynamics, we compare particle motion with and without the back-reaction force. We demonstrate that, in the case of an attractive Lorentz force, the back-reaction leads to the charged particles falling onto the neutron star's surface in all scenarios considered. For the repulsive Lorentz force, in combination with the back-reaction force, we observe a widening of stable equatorial circular orbits; the off-equatorial orbits shift toward the equatorial plane and subsequently widen if they are sufficiently close to the plane. Otherwise, the off-equatorial orbits evolve toward the neutron star surface. The critical latitude, which separates orbital widening from falling onto the surface, is determined numerically as a function of the electromagnetic interaction's intensity.

Joule-Thomson expansion, motion of particles and QPOs around Bardeen-AdS black hole immersed in a fluid of strings

In this work, we investigate the dynamics of particles around a Bardeen AdS black hole immersed in a fluid of strings, focusing on how the black hole parameters affect particle motion. We observe the black hole's Joule-Thomson expansion and the impact of physical parameters on the cooling and heating zones. Using Joule-Thomson coefficients, we also discuss the stable and unstable configuration of the considered black hole for both cases. The stability of circular equatorial orbits is analyzed using the effective potential approach. We derive analytical expressions for the energy and angular momentum of these circular orbits as functions of the black hole parameters. We also explore the impact of these parameters on the innermost stable circular orbits and discuss the effective forces acting on the particles. In addition, we examine the epicyclic oscillations of particles near a stable equatorial orbits and calculate the corresponding oscillation frequencies as function of black hole parameters. The periastron frequency is also analyzed. Furthermore, we study particle collisions and the resulting center of mass-energy in the vicinity of the black hole. We show that the parameters of the model significantly influence particle motion. Lastly, we compare the particle dynamics around the Bardeen AdS black hole immersed in a fluid of strings with those around the Bardeen black hole and the Bardeen Reissner-Nordström black hole.

Particle dynamics with trajectories and epicyclic oscillations around a piece-wise black hole immersed in dark matter

We investigate the dynamics of neutral test particles revolving around a non-rotating black hole immersed in dark matter DM. We derive exact solutions for the radial profiles of specific angular momentum and energy for equatorial stable circular orbits as a function of the parameters of the black hole. Using the effective potential approach, we explore the stability of these circular orbits and the effective force acting on particles in the presence of dark matter. Additionally, we examine the impact of dark matter on the innermost stable circular orbits. By numerically integrating the equations of motion, we plot the trajectories of particles around the black hole and analyze how dark matter affects these trajectories. We also calculate the analytical frequencies of radial and latitudinal harmonic oscillations as functions of the dark matter parameter for both local and distant observers. We compare these results with scenarios where dark matter is absent. Furthermore, we study the effects of dark matter on periastron precession. We present an analysis of the constraints on the dark matter and black hole mass parameters using Markov Chain Monte Carlo (MCMC) methods. Our study focuses on QPOs observed in X-ray binaries, in particular, the microquasars GRO J1655-40 and XTE J1550-564, and the centers of the galaxies M82 X-1 and Sgr A⁎. We also analyze the particle collisions, discussing the center-of-mass energy of the colliding particles. Our observations reveal that the motion of particles revolving around the black hole is significantly affected by varying the model parameters.

Magnetic Perturbations on Cable-Connected Satellites in Equatorial Orbit: A Review of the Current State of Research

Cable-connected satellites in equatorial orbit offer enhanced capabilities for various space applications, but their stability is susceptible to magnetic perturbations from the Earth's magnetic field. This literature review examines the current state of research on the effects of geomagnetic field on the dynamics of cable-connected satellites in equatorial orbit. We synthesize findings from previous studies on the impact of magnetic field on orbital dynamics, attitude control, and communication. Our review highlights the significance of considering magnetic perturbations in the design and operation of cable-connected satellites, and identifies knowledge gaps for future research. The results of this review can inform the development of strategies to mitigate the effects of magnetic perturbations and ensure the stability of cable-connected satellites in equatorial orbit.

Nonlinear Stability of Cable-Connected Satellites: A Review of the Combined Effects of Solar Radiation Pressure and Earth’s Magnetic Field

The nonlinear stability of cable-connected satellites in equatorial orbit is crucial for maintaining their operational efficiency and extending their mission duration. This literature review aims to investigate the combined effects of solar radiation pressure and Earth’s magnetic field on the nonlinear stability of cable-connected satellites. We examine the current state of research on the dynamics of cable-connected satellites under the influence of solar radiation pressure and Earth’s magnetic field, focusing on nonlinear stability analysis. The review highlights the key findings, methodologies, and challenges in this research area. Our analysis reveals that the combined effects of solar radiation pressure and Earth’s magnetic field can significantly impact the nonlinear stability of cable-connected satellites, leading to complex dynamics and potential instability. The review identifies areas for future research, emphasizing the need for advanced modeling and simulation techniques to accurately predict and mitigate the effects of these environmental factors on cable-connected satellite systems. This comprehensive review provides a valuable resource for researchers and engineers working on the design and operation of cable-connected satellites in equatorial orbit.

Extreme mass-ratio inspiral around the horizonless massive object

In this paper, we calculated the extreme mass-ratio inspiral (EMRI) waveform radiated from a binary composed of a massive horizonless object (MHO) and a compact object (CO), where CO is spiraling on a circular equatorial orbit around the MHO. Due to the absent of horizon, there exist the ingoing and outgoing waves near the reflective surface of MHO, which have significantly influence on the evolution of orbital parameters. We observe that there indeed exist the differences of EMRI trajectories between the massive Kerr black hole and MHO cases. By calculating the mismatch of gravitational wave (GW) waveforms from massive black hole (MBH) and MHO, our result indicates that the space-borne gravitational wave detector could distinguish the modified effect of reflectivities from the BH case, which allows to put an upper constraint on the reflectivity R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {R}}$$\\end{document} of MHO, at the level of R≳10-4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {R}} > rsim 10^{-4}$$\\end{document}.

Photon emissions from Kerr equatorial geodesic orbits

We consider the light emitters moving freely along the geodesics on the equatorial plane near a Kerr black hole and study the observability of these emitters. To do so, we assume these emitters emit photons isotropically and monochromatically, and we compute the photon escaping probability (PEP) and the maximum observable blueshift (MOB) of the photons that reach infinity. We obtain numerical results of PEP and MOB for the emitters along various geodesic orbits, which exhibit distinct features for the trajectories of different classes. In particular, we find that the plunging emitters could have considerable observability even in the near-horizon region. This interesting observational feature becomes more significant for the high-energy emitters near a high-spin black hole. As the radiatively-inefficient accretion flow may consist of plunging emitters, the present work could be of great relevance to the astrophysical observations.

Orbital motion and epicyclic oscillations around Bardeen black hole surrounded by perfect fluid dark matter

We study the orbital and oscillatory motion of test particles moving around a rotating Bardeen black hole immersed in perfect fluid dark matter. We obtain the analytical solutions for the radial profiles of specific energy as well as the specific angular momentum of the equatorial stable circular orbits. Using the effective potential approach, we also discuss the stability of circular orbits. We compute the frequencies of radial and latitudinal harmonic oscillations as a function of mass, charge, and angular momentum of the black hole as well as the parameter describing the perfect fluid dark matter. The main characteristics of test particle quasi-periodic oscillations near stable circular orbits are examined in the equatorial plane. Furthermore, we study precessions of Periastron and Lense-Thirring. It is observed that the particle's motion around the black hole is strongly influenced by the model parameters.

Metric perturbations of Kerr spacetime in Lorenz gauge: circular equatorial orbits

We construct the metric perturbation in Lorenz gauge for a compact body on a circular equatorial orbit of a rotating black hole (Kerr) spacetime, using a newly-developed method of separation of variables. The metric perturbation is formed from a linear sum of differential operators acting on Teukolsky mode functions, and certain auxiliary scalars, which are solutions to ordinary differential equations in the frequency domain. For radiative modes, the solution is uniquely determined by the s=±2 Weyl scalars, the s = 0 trace, and s=0,1 gauge scalars whose amplitudes are determined by imposing continuity conditions on the metric perturbation at the orbital radius. The static (zero-frequency) part of the metric perturbation, which is handled separately, also includes mass and angular momentum completion pieces. The metric perturbation is validated against the independent results of a 2+1D time domain code, and we demonstrate agreement at the expected level in all components, and the absence of gauge discontinuities. In principle, the new method can be used to determine the Lorenz-gauge metric perturbation at a sufficiently high precision to enable accurate second-order self-force calculations on Kerr spacetime in future. We conclude with a discussion of extensions of the method to eccentric and non-equatorial orbits.

A Multistep Method for Integration of Perturbed and Damped Second-Order ODE Systems

Based on the Ψ-functions series method, a new numerical integration method for perturbed and damped second-order systems of differential equations is presented. This multistep method is defined for variable step and variable order (VSVO) and maintains the good properties of the Ψ-functions series method. In addition, it incorporates a recurring algebraic procedure to calculate the algorithm’s coefficients, which facilitates its implementation on the computer. The construction of Ψ-functions and the Ψ-functions series method are presented to address the construction of both explicit and implicit multistep methods and a predictor–corrector method. Three problems analogous to those solved by the Ψ-functions series method are analyzed, contrasting the results obtained with the exact solution of the problem or with its first integral. The first example is the integration of a quasi-periodic orbit. The second example is a Structural Dynamics problem associated with an earthquake, and the third example studies an equatorial satellite with perturbation J2. This allows us to compare the good behavior of the new code with other prestige codes.

Revisiting the gravitomagnetic clock effect

To the first post-Newtonian order, if two test particles revolve in opposite directions about a massive, spinning body along two circular and equatorial orbits with the same radius, they take different times to return to the reference direction relative to which their motion is measured: it is the so-called gravitomagnetic clock effect. The satellite moving in the same sense of the rotation of the primary is slower, and experiences a retardation with respect to the case when the latter does not spin, while the one circling in the opposite sense of the rotation of the source is faster, and its orbital period is shorter than it would be in the static case. The resulting time difference due to the stationary gravitomagnetic field of the central spinning body is proportional to the angular momentum per unit mass of the latter through a numerical factor which so far has been found to be 4π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4\\pi $$\\end{document}. A numerical integration of the equations of motion of a fictitious test particle moving along a circular path lying in the equatorial plane of a hypothetical rotating object by including the gravitomagnetic acceleration to the first post-Newtonian order shows that, actually, the gravitomagnetic corrections to the orbital periods are larger by a factor of 4 in both the prograde and retrograde cases. Such an outcome, which makes the proportionality coefficient of the gravitomagnetic difference in the orbital periods of the two counter-revolving orbiters equal to 16π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$16\\pi $$\\end{document}, confirms an analytical calculation recently published in the literature by the present author. It is an important result in view of the astrophysical implications of the gravitomagnetic clock effect around Kerr black holes.

Extreme mass-ratio inspiral as a probe of extra dimensions: The case of spinning massive object

Extreme mass-ratio inspiral (EMRI), as one of most important and promising gravitational wave (GW) sources, which is the ideal tool to test fundamental theories and nature of massive black hole (MBH). In this paper, we consider the EMRI system consisting of a rotating barneworld MBH with tidal charge Q and a stellar-mass compact object (CO), where a non-spinning CO inspirals on the equatorial and circular orbits around the MBH. Under the adiabatic approximate condition, we show the evolution of circular orbital radius by computing the EMRI fluxes and orbital phase modified by tidal charge. We find that the positive and negative values of tidal charges have the difference influences on the EMRI orbital dynamics. In particular, the inspiral motion of CO in the rotating barneworld MBH with a negative tidal charge would be end more earlier than the positive tidal charge case. Accordingly, the EMRI waveforms from the positive tidal charge cases can put more rigorous constraint results comparing with the negative tidal charge cases. Finally, we analyze the phase differences of two EMRI waveforms from Kerr BH and rotating braneworld BH with tidal charge by computing dephasing and mismatch. Our results indicate that LISA can distinguish the EMRI waveforms from rotating braneworld BH with the negative tidal charge as small as Qmin∼−10−6 under the suitable scenarios, which can sever as the unique discriminator of detecting the physics of extra dimensions.

Examining the reliability of selected materials for the Indonesian small satellite by electron beam machine

The interaction of charged particles and satellites can result in partial or total loss, leading to the satellite’s inoperable. Charging on the satellite is the dominant cause of malfunctions and failures of satellite components. Damage mitigation can be done through a series of experiments in which materials and satellite components are bombarded with high- energy electrons. In this study, electrons are generated and accelerated using an electron beam machine (EBM). The test material and electronic components are placed beneath the EBM’s beam window, where a 300 keV electron beam bombards them uniformly. The experimental results show that the electron beam can penetrate all the test materials used as shields within 20 s, and the absorbance levels range between 67% and 98%. The penetration depths span between 0.16 and 0.46 mm and agree with the Kanaya-Okayama calculation. The values of stopping power for most materials show the ineffectiveness of materials for shielding. However, increasing the thickness, such as in the MLI case, can reduce the level of electron radiation absorption. This experiment provides recommendations on the selection of materials used for equatorial satellite shielding, particularly the next Indonesian LEO equatorial satellite, both in terms of material and size.

Relativistic Electron Precipitation Driven by Mesoscale Transients, Inferred From Ground and Multi‐Spacecraft Platforms

AbstractPrecipitation of relativistic electrons into the Earth's atmosphere regulates the outer radiation belt fluxes and contributes to magnetosphere‐atmosphere coupling. One of the main drivers of such precipitation is electron scattering by whistler‐mode waves. Such waves typically originate at the equator, where they can resonate with and scatter sub‐relativistic (tens to a few hundred keV) electrons. However, they can occasionally propagate far away from the equator along field lines, reaching middle latitudes, where they can resonate with and scatter relativistic (>500 keV) electrons. Such a propagation is typical for the dayside, but statistically has not been found on the nightside where the waves are quickly damped along their propagation due to Landau damping. Here we explore two events of relativistic electron precipitation from low‐altitude observations on the nightside. Combining measurements of whistler‐mode waves from ground observatories, relativistic electron precipitation from low‐altitude satellites, total electron content maps from GPS receivers, and magnetic field and electron flux from equatorial satellites, we show wave ducting by plasma density gradients is the possible channel that allows the waves to reach middle latitudes and scatter relativistic electrons. We suggest that both whistler‐mode wave generation and ducting can be driven by equatorial mesoscale (with spatial scales of about one Earth radius) transient structures during nightside injections. We also compare these nightside events with observations of ducted waves and relativistic electron precipitation at the dayside, where wave generation and ducting are driven by ultra‐low‐frequency waves. This study demonstrates the potential importance of mesoscale transients in relativistic electron precipitation, but does not however unequivocally establish that ducted whistler‐mode waves are the primary cause of the observed electron precipitation.

Post-Newtonian Orbital Effects Induced by the Mass Quadrupole and Spin Octupole Moments of an Axisymmetric Body

The post-Newtonian orbital effects induced by the mass quadrupole and spin octupole moments of an isolated, oblate spheroid of constant density that is rigidly and uniformly rotating on the motion of a test particle are analytically worked out for an arbitrary orbital configuration and without any preferred orientation of the body’s spin axis. The resulting expressions are specialized to the cases of (a) equatorial and (b) polar orbits. The opportunity offered by a hypothetical new spacecraft moving around Jupiter along a Juno-like highly elliptical, polar orbit to measure them is preliminarily studied. Although more difficult to be practically implemented, also the case of a less elliptical orbit is considered since it yields much larger figures for the relativistic effects of interest. The possibility of using the S-stars orbiting the supermassive black hole in Sgr A* at the Galactic Center as probes to potentially constrain some parameters of the predicted extended mass distribution surrounding the hole by means of the aforementioned orbital effects is briefly examined.

Black holes shielded by magnetic fields

Black holes (BHs) formed by collapsing and/or merging of magnetized progenitors, have magnetic fields penetrating the event horizon, and there are several possible scenarios. Thus, the no-hair theorem that assumes the outside medium is a vacuum, is not applicable in this case. Bearing this in mind and considering a Schwarzschild BH of mass M immersed in a uniform magnetic field B, we show that all three frequencies related to the equatorial circular orbit of a test particle become imaginary for the orbits of radii rB>2B−1. It signifies that if a BH is surrounded by a magnetic field of order B∼Rg−1 (where Rg is the gravitational radius of the BH), a test particle could unable to continue its regular geodesic motion from/at r>rB, hence the accretion disk could not be formed, and the motion of other stellar objects around the BH could be absent. As the BHs are generally detected by watching for their effects on nearby stars and gas, a magnetic field of order B∼Rg−1 could be able to shield a BH in such a way that it could remain undetectable. Motivated with this theoretical investigation and considering the sphere (of radius rf) of magnetic influence around an astrophysical BH, we constrain B, above which a magnetized BH could remain undetectable. For example, M=109M⊙ BH surrounded by B>106 G and M=10M⊙ BH surrounded by B>1014 G could remain undetectable for rf∼105Rg. In other words, our result also explains why a detected SMBH has surprisingly weak magnetic field.

Simulation of proton-induced primary displacement damage in GaAs under different ambient temperatures

The performance of on-orbit GaAs-based solar cells is susceptible to the displacement damage effect. The proton-induced primary displacement damage in GaAs on a geosynchronous equatorial orbit (GEO) was simulated and analyzed by combining the Monte Carlo (MC) and molecular dynamics (MD) methods. The MC simulation provided the distribution of primary knock-on atoms (PKAs) in GaAs induced by GEO-related protons to the MD simulation. In MD simulations, the effects of radiation fluence and ambient temperature on the displacement damage were considered. The simulation results showed that GEO-related protons generated a significant number of PKAs within an energy range of below 10 keV in GaAs. The high-fluence radiation emulated by the binary PKA could generate more point defects and cluster defects in GaAs than the low-fluence radiation emulated by single PKAs. As compared to room temperature (300 K), both a low (100 K) and high (500 K) ambient temperature could deteriorate the displacement damage. In addition, a high ambient temperature of 500 K could induce widespread thermal spikes in GaAs as compared to 100 and 300 K. This work can provide useful insight into the proton-induced displacement damage in GaAs and the radiation hardening of GaAs-based photoelectric devices in space.