Unstable Orbits Research Articles

Probing correlation between photon orbits and phase structure of charged AdS black hole in massive gravity background

The phase structure of charged anti-de Sitter black hole in massive gravity is investigated using the unstable circular photon orbits formalism, concretely we establish a direct link between the null geodesics and the critical behavior thermodynamic of such black hole solution. Our analysis reveals that the radius and the impact parameter corresponding to the unstable circular orbits can be used to probe the thermodynamic phase structure. We also show that the latter are key quantities to characterize the order of Van der Waals-like phase transition. Namely, we found a critical exponent around [Formula: see text]. All these results support further that the photon trajectories can be used as a useful and crucial tool to probe the thermodynamic black holes criticality.

Can close-in giant exoplanets preserve detectable moons?

ABSTRACT Exoplanet discoveries have motivated numerous efforts to find unseen populations of exomoons, yet they have been unsuccessful. A plausible explanation is that most discovered planets are located on close-in orbits, which would make their moons prone to tidal evolution and orbital detachment. In recent models of tidally driven migration of exomoons, evolving planets might prevent what was considered their most plausible fate (i.e. colliding against their host planet), favouring scenarios where moons are pushed away and reach what we define as the satellite tidal orbital parking distance ($a_{\rm \mathrm{stop}}$), which is often within the critical limit for unstable orbits and depends mainly on the system’s initial conditions: mass ratio, semimajor axes, and rotational rates. Using semi-analytical calculations and numerical simulations, we calculate $a_{\rm \mathrm{stop}}$ for different initial system parameters and constrain the transit detectability of exomoons around close-in planets. We found that systems with Mm/Mp ≥ 10−4, which are less likely to form, are also stable and detectable with present facilities (e.g. Kepler and TESS) through their direct and secondary effects in planet + moon transit, as they are massive, oversized, and migrate slowly. In contrast, systems with lower moon-to-planet mass ratios are ephemeral and hardly detectable. Moreover, any detection, confirmation, and full characterization would require both the short cadence capabilities of TESS and high photometric sensitivity of ground-based observatories. Finally, despite the shortage of discovered long-period planets in currently available data bases, the tidal migration model adopted in this work supports the idea that they are more likely to host the first detectable exomoon.

Self-force effects on the marginally bound zoom-whirl orbit in Schwarzschild spacetime

For a Schwarzchild black hole of mass $M$, we consider a test particle falling from rest at infinity and becoming trapped, at late time, on the unstable circular orbit of radius $r=4GM/c^2$. When the particle is endowed with a small mass, $\mu\ll M$, it experiences an effective gravitational self-force, whose conservative piece shifts the critical value of the angular momentum and the frequency of the asymptotic circular orbit away from their geodesic values. By directly integrating the self-force along the orbit (ignoring radiative dissipation), we numerically calculate these shifts to $O(\mu/M)$. Our numerical values are found to be in agreement with estimates first made within the Effective One Body formalism, and with predictions of the first law of black-hole-binary mechanics (as applied to the asymptotic circular orbit). Our calculation is based on a time-domain integration of the Lorenz-gauge perturbation equations, and it is a first such calculation for an unbound orbit. We tackle several technical difficulties specific to unbound orbits, illustrating how these may be handled in more general cases of unbound motion. Our method paves the way to calculations of the self-force along hyperbolic-type scattering orbits. Such orbits can probe the two-body potential down to the "light ring", and could thus supply strong-field calibration data for eccentricity-dependent terms in the Effective One Body model of merging binaries.

Charged particle and strong cosmic censorship in Reissner–Nordström–de Sitter black holes

We investigate the instability of the unstable circular orbit of a charged null particle to test the strong cosmic censorship conjecture in Nariai-type near-extremal Reissner--Nordstr\"{o}m--de Sitter black holes. The instability is estimated as the Lyapunov exponent and found to depend on the mass and charge of the black hole. Then, we explicitly show that charged null particles in unstable circular orbits correspond to the charged massless scalar field in the eikonal limit. This provides a compact relationship representing the quasinormal frequency in terms of the characteristics of unstable circular orbits in near Nariai-type extremal conditions. According to this relationship, the strong cosmic censorship conjecture is valid.

Improvement of Stability in a PCM-Controlled Boost Converter with the Target Period Orbit-Tracking Method

Thee peak-current-mode (PCM) control strategy is widely adopted in pulse width-modulated (PWM) DC-DC converters. However, the converters always involve a sub-harmonic oscillating state or chaotic state if the active duty ratio is beyond a certain range. Hence, an extra slope signal in the inductor-current loop is used to stabilize the operation of the converter. This paper presents a new technique for enlarging the stable range of PCM-controlled DC-DC converters, in which the concept of utilizing unstable period-1 orbit (UPO-1) of DC-DC converters is proposed and an implementation scenario based on the parameter-perturbation method is presented. With the proposed technique, perturbations are introduced to the reference current of the control loop, and the converters operating in a chaotic state can be tracked, and thus be stabilized to the target UPO-1. Therefore, the stable operating range of the converters is extended. Based on an example of a PCM-controlled boost converter, simulations are presented as a guide to a detailed implementation process of the proposed technique, and comparisons between the proposed technique and techniques in terms of ramp compensation are provided to show the differentiation in the performance of the converter. Experimental results in the work confirm the effectiveness of the proposed technique.

Casimir effect around an Ellis wormhole

We discuss the Casimir effect in a small cavity, moving in a circular orbit around an Ellis wormhole. We show that the interplay between the spacetime geometry and the cavity orbital motion gives rise to a distortion in the Casimir energy density, causing a reduction of its absolute value. Quite interestingly, such effect can be observed also when the cavity moves on a circular geodesic (albeit unstable) orbit at the wormhole throat, where a comoving observer becomes locally inertial (namely, the observer’s reference frame reduces to a geodesic, Fermi-Walker transported one). In that respect, the discussed effect appears as a nonlocal quantum effect by means of which some properties of the underlying spacetime geometry, hidden to local classical measurements, can be captured and unveiled.

In situ scattering of warm Jupiters and implications for dynamical histories

ABSTRACT Many warm Jupiters (WJs) have substantial eccentricities, which are linked to their formation and migration histories. This paper explores eccentricity excitation of WJs due to planet–planet scattering, beginning with three to four planets in unstable orbits, with the innermost planet placed in the range (0.1−1) au. Such a setup is consistent with either in situ formation or arrival at sub-au orbits due to disc migration. Most previous N-body experiments have focused on ‘cold’ Jupiters at several au, where scattering results in planet ejections, efficiently exciting the eccentricities of surviving planets. In contrast, scattering at sub-au distances results in a mixture of collisions and ejections, and the final eccentricities of surviving planets are unclear. We conduct scattering experiments for a range of planet masses and initial spacings, including the effect of general relativistic apsidal precession, and systematically catalogue the scattering outcomes and properties of surviving planets. A comparable number of one-planet and two-planet systems are produced. Two-planet systems arise exclusively through planet–planet collisions, and tend to have low eccentricities/mutual inclinations and compact configurations. One-planet systems arise through a combination of ejections and collisions, resulting in higher eccentricities. The observed eccentricity distribution of solitary WJs (lacking detection of a giant planet companion) is consistent with roughly $60 {{\ \rm per\ cent}}$ of the systems having undergone in situ scattering, and the remaining experiencing a quiescent history.

Geodesic Structure of a Non-linear Magnetic Charged Black Hole Surrounded by Quintessence**Supported partly by the National Natural Science Foundation of China under Grant No. 11565016, the Special Training Program for Distinguished Young Teachers of the Higher Education Institutions of Yunnan Province under Grant No. 1096837802, and the Applied Basic Research Programs of Yunnan Provincial Science and Technology Department under Grant

This paper aims to investigate the geodesic motion in the spacetime of a non-linear magnetic charged black hole surrounded by quintessence. By varying the Lagrangian corresponding to the metric, the orbital motion equation has been obtained. The effects of the magnetic charge Q, positive normalization factor C, angular momentum b, and energy E on time-like and null geodesic motion are discussed from three aspects: orbital stability, orbital types, and circular orbits. By comparing the effects of the above parameters C, b on the effective potential, it is found that quintessence has an impact on the types and stability of orbits. In addition, for time-like orbital motion, when 3.443113 ≤ b ≤ 6.392 578 (for fixed C = 0.0002, M = 1, Q = 0.7), there are bound orbits, and within this range, the stable circular orbits exist, and the radii of the innermost and outermost stable circular orbit are r = 5.912 654 and r = 56.745 933, respectively. For null orbital motion, the orbital types have only unstable circular orbit which occur at r = 2.951 072 (), absorb orbits and escape orbits, but no stable circular orbits, and bound orbits.

Particle motion around generic black holes coupled to non-linear electrodynamics

We study spherically symmetric magnetically charged generic black hole solutions of general relativity coupled to non-linear electrodynamics (NED). For characteristic values of the generic spacetime parameters we give the position of horizons in dependence on the charge parameter, demonstrating separation of the black hole and no-horizon solutions, and possibility of existence of solutions containing three horizons. We show that null, weak and strong energy conditions are violated when the outer horizon is approaching the center. We study effective potentials for photons and massive test particles and location of circular photon orbits (CPO) and innermost stable circular orbit (ISCO). We show that the unstable photon orbit can become stable, leading to the possibility of photon capture which affects on silhouette of the central object. The position of ISCO approaches the horizon with increasing charge parameter q and the energy at ISCO decreases with increasing charge parameter. We investigate this phenomenon and summarize for a variety of the generic spacetime parameters the upper estimate on the spin parameter of the Kerr black which can be mimicked by the generic charged black hole solutions.

Geodesic structure of the Clifton–Barrow black hole space–time

In this paper, we investigate the geodesic structure of Clifton–Barrow black hole space–time. Through the numerical analysis of the effective potential and the motion equation, the orbital types of test particles and photons and the corresponding orbital motion diagrams of each orbital types under certain conditions are obtained. We find that angular momentum [Formula: see text] and [Formula: see text] determine the existence of bound orbits and circular orbits. And we also find that the radius of unstable circular orbit decreases with increases in [Formula: see text] while the radius of stable circular orbit increases. Furthermore, as [Formula: see text] increases, the radius of unstable circular orbit increases, while the radius of stable circular orbit decreases. For null geodesic, parameters [Formula: see text] and [Formula: see text] do not affect the types of null orbits. The radius of the unstable circular orbits increases with the increase of [Formula: see text]. However, the radius of the unstable circular orbits remains unchanged as [Formula: see text] increases. Also, we show that the precession direction of the bound orbits of the test particles is counterclockwise for [Formula: see text], but clockwise with [Formula: see text]. Moreover, different energy values have an effect on the curvature of escape and absorb orbits curve.

Trajectories of photons in modified Hayward black hole spacetime

We study the trajectories of photons in modified Hayward black hole spacetime. Three kinds of horizons are distinguished by analyzing the lapse function of modified Hayward black hole spacetime. Through plotting the effective potential with different values of parameters, it is found that the parameter [Formula: see text] has more conspicuous effects than parameters [Formula: see text] ([Formula: see text] is associated with the time-delay) and [Formula: see text] ([Formula: see text] is related to the 1-loop quantum corrections) for the effective potential. The change of the parameters ([Formula: see text], [Formula: see text] and [Formula: see text]) has an obvious influence on unstable circular orbits. The structure of null geodesic is simpler than that of time-like geodesic. The stable circular orbits and the bound orbits are not included for the photon trajectories. By analyzing the corresponding effective potential, all possible orbits of null geodesic are found. The radius of unstable circular orbit equals 2.8790 for fixed [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text].

The orbital Lense–Thirring precession in a strong field

We study the exact evolution of the orbital angular momentum of a massive particle in the gravitational field of a Kerr black hole. We show analytically that, for a wide class of orbits, the angular momentum's hodograph is always close to a circle. This applies to both bounded and unbounded orbits that do not end up in the black hole. Deviations from the circular shape do not exceed $\approx10\%$ and $\approx7\%$ for bounded and unbounded orbits, respectively. We also find that nutation provides an accurate approximation for those deviations, which fits the exact curve within $\sim 0.01\%$ for the orbits of maximal deviation. Remarkably, the more the deviation, the better the nutation approximates it. Thus, we demonstrate that the orbital Lense-Thirring precession, originally obtained in the weak-field limit, is also a valid description in the general case of (almost) arbitrary exact orbits. As a by-product, we also derive the parameters of unstable spherical timelike orbits as a function of their radii and arbitrary rotation parameter $a$ and Carter's constant $Q$. We verify our results numerically for all the kinds of orbits studied.

Hamiltonian Structure-Based Robust Station-Keeping for Unstable Libration Point Orbits

In this work, a controller for spacecraft station-keeping in an unstable libration point orbit was designed based on the Hamiltonian structure of the system. The proposed controller exploits the inherent Hamiltonian structure of the system, and therefore the performance of the controller can be easily adjusted while ensuring the stability of the closed-loop system. A filtered extended high-gain observer was also designed to compensate for the performance degradation of the controller due to external disturbances and model uncertainty. The filtered extended high-gain observer includes an integral state feedback to alleviate navigation error amplification due to the high gain of the observer. The global convergence of the proposed observer was demonstrated, and the tracking error was ultimately bounded to the reference libration point orbit by applying the Hamiltonian structure-based controller. The performance of the proposed controller and observer was demonstrated by numerical simulation of the Earth–Moon system.

Exploring the formation by core accretion and the luminosity evolution of directly imaged planets

Context. A low-mass companion to the two-solar mass star HIP 65426 has recently been detected by SPHERE at around 100 au from its host. Explaining the presence of super-Jovian planets at large separations, as revealed by direct imaging, is currently an open question. Aims. We want to derive statistical constraints on the mass and initial entropy of HIP 65426 b and to explore possible formation pathways of directly imaged objects within the core-accretion paradigm, focusing on HIP 65426 b. Methods. Constraints on the planet’s mass and post-formation entropy are derived from its age and luminosity combined with cooling models. For the first time, the results of population synthesis are also used to inform the results. Then a formation model that includes N-body dynamics with several embryos per disc is used to study possible formation histories and the properties of possible additional companions. Finally, the outcomes of two- and three-planet scattering in the post-disc phase are analysed, taking tides into account for small-pericentre orbits. Results. The mass of HIP 65426 b is found to be mp = 9.9−1.8+1.1 MJ using the hot population and mp = 10.9−2.0+1.4 MJ with the cold-nominal population. We find that core formation at small separations from the star followed by outward scattering and runaway accretion at a few hundred astronomical units succeeds in reproducing the mass and separation of HIP 65426 b. Alternatively, systems having two or more giant planets close enough to be on an unstable orbit at disc dispersal are likely to end up with one planet on a wide HIP 65426 b-like orbit with a relatively high eccentricity (≳ 0.5). Conclusions. If this scattering scenario explains its formation, HIP 65426 b is predicted to have a high eccentricity and to be accompanied by one or several roughly Jovian-mass planets at smaller semi-major axes, which also could have a high eccentricity. This could be tested by further direct-imaging as well as radial-velocity observations.

Analysis of lifetime extension capabilities for CubeSats equipped with a low-thrust propulsion system for Moon missions

In this paper, we analyze the mission lifetime extension capability for a CubeSat smaller than 3U in a circular lunar orbit at a 100-km altitude, assuming the utilization of a state-of-the-art low-thrust electric propulsion system such as pulsed plasma thrusters with an impulse bit (Ibit) and velocity change (ΔV) below 60 μNs and 120 m/s, respectively. Because of the non-spherical gravity field of the Moon and its strong influence on low-altitude lunar orbits, a long orbital lifetime is achievable only within a set of stable orbits which mainly depends on initial inclination and right ascension of the ascending node (RAAN); moreover, as a piggyback on a main mission, the deployment of CubeSats in those stable orbits is not guaranteed. For this reason, we propose an orbit correction strategy whose performance is constrained to the initial orbital parameters of the CubeSat (i.e., inclination and RAAN), its solar power generation capacity, its attitude control strategy, and its propulsion subsystem features, such as the thruster Ibit and budgeted ΔV. By analyzing the required time to perform the orbit correction maneuvers to extend the orbital lifetime and the minimum altitude achieved throughout the mission lifetime of the spacecraft, we demonstrated that a one-year mission can be achieved within initial orbital inclination values greater than 65°. For unstable orbits bounded by initial orbital inclination values smaller than 65°, the orbit lifetime can also be extended from a few days to up to one year. Better performance with our proposed orbit correction strategy can be achieved by using an electric propulsion system featuring Ibit and ΔV values greater than 40 μNs and 80 m/s, respectively. Our results show the feasibility of performing any orbit correction maneuver for the enhancement of the mission lifetime of a CubeSat, expanding the performance capabilities of CubeSats to any mission in a lunar orbit by reducing the limitation of deploying them in unstable orbits.

Probing the relationship between the null geodesics and thermodynamic phase transition for rotating Kerr-AdS black holes

In this paper, we aim to examine the relationship between the unstable circular photon orbit and the thermodynamic phase transition for a rotating Kerr-AdS black hole. On one side, we give a brief review of the phase transition for the Kerr-AdS black hole. The coexistence curve and the metastable curve corresponding to the phase transition are clearly shown. On the other side, we calculate the radius and the angular momentum of the unstable circular orbits for a photon by analyzing the effective potential. Then combining these two sides, we find the following results. i) The radius and the angular momentum of the unstable circular photon orbits demonstrate the non-monotonic behaviors when the thermodynamic phase transition takes place. So from the behavior of the circular orbit, one can determine whether there exists a thermodynamic phase transition. ii) The difference of the radius or the angular momentum for the coexistence small and large black holes can be treated as an order parameter to describe the phase transition. And near the critical point, it has a critical exponent of $\frac{1}{2}$. iii) The temperature and pressure corresponding to the extremal points of the radius or the angular momentum of the unstable circular photon orbit completely agree with that of the metastable curves from the thermodynamic side. Thus, these results confirm the relationship between the geodesics and thermodynamic phase transition for the Kerr-AdS black hole. Therefore, on one hand, we are allowed to probe the thermodynamic phase transition from the gravity side. On the other hand, the signature of the strong gravitational effect can also be revealed from the black hole thermodynamics.

Dynamics of a large population of red blood cells under shear flow

An exhaustive description of the dynamics under shear flow of a large number of red blood cells in a dilute regime is proposed, which highlights and takes into account the dispersion in cell properties within a given blood sample. Physiological suspending fluid viscosity is considered, a configuration surprisingly seldom considered in experimental studies, as well as a more viscous fluid that is a reference in the literature. Stable and unstable flipping motions well described by Jeffery orbits or modified Jeffery orbits are identified, as well as transitions to and from tank-treading motion in the more viscous suspending fluid case. Hysteresis loops upon shear rate increase or decrease are highlighted for the transitions between unstable and stable orbits as well as for the transition between flipping and tank-treading. We identify which of the characteristic parameters of motion and of the transition thresholds depend on flow stress only or also on suspending fluid viscosity.

Astrophysically relevant bound trajectories around a Kerr black hole

We derive alternate and new closed-form analytic solutions for the non-equatorial eccentric bound trajectories, , around a Kerr black hole by using the transformation . The application of the solutions is straightforward and numerically fast. We obtain and implement translation relations between the energy and angular momentum of the particle, (E, L), and eccentricity and inverse-latus rectum, (e, ), for a given spin, a, and Carter’s constant, Q, to write the trajectory completely in the (e, , a, Q) parameter space. The bound orbit conditions are obtained and implemented to select the allowed combination of parameters (e, , a, Q). We also derive specialized formulae for equatorial, spherical and separatrix orbits. A study of the non-equatorial analog of the previously studied equatorial separatrix orbits is carried out where a homoclinic orbit asymptotes to an energetically bound spherical orbit. Such orbits simultaneously represent an eccentric orbit and an unstable spherical orbit, both of which share the same E and L values. We present exact expressions for e and as functions of the radius of the corresponding unstable spherical orbit, rs, a, and Q, and their trajectories, for () separatrix orbits; they are shown to reduce to the equatorial case. These formulae have applications to study the gravitational waveforms from extreme-mass ratio inspirals (EMRIs) using adiabatic progression of a sequence of Kerr geodesics, besides relativistic precession and phase space explorations. We obtain closed-form expressions of the fundamental frequencies of non-equatorial eccentric trajectories that are equivalent to the previously obtained quadrature forms and also numerically match with the equivalent formulae previously derived. We sketch non-equatorial eccentric, separatrix, zoom-whirl, and spherical orbits, and discuss their astrophysical applications.

Parallelly generating halo orbit and its transfer trajectory in the full ephemeris model

This paper employs a parallelized genetic algorithm in generating long-term bounded halo orbit and optimizing transfer trajectory from earth to it in the full ephemeris model. While the conventional gradient methods are lacking in precision as well as time-consuming, the parallel genetic algorithm (PGA) offers an access to large-scale computing based on the advanced hardware capability. A bounded halo orbit without any stationkeeping maneuver in five periods is obtained inside a defined pipe region in the ephemeris model. Transfer trajectory from earth to the obtained halo orbit is also investigated in this paper. By optimizing insertion impulses in the backward propagation and midcourse in the forward propagation, a transfer trajectory from earth to halo with least fuel consumption is generated. Furthermore, launch window is investigated based on the periodic variation about inclination of the injection point on the parking orbit. A mapping method from spatial space to $i-\varOmega $ subspace is introduced for solving the time delay and position deviation problem. The numerical method employed in this paper is of high accuracy based on large-scale computing capability as well as efficient without any analytical initial guess, which makes it extensive in designing these unstable libration point orbits.

Probing Palatini-type gravity theories through gravitational wave detections via quasinormal modes

The possibility of testing gravity theories with the help of gravitational wave detections has become an interesting arena of recent research. In this paper, we follow this direction by investigating the quasinormal modes (QNMs) of the axial perturbations for charged black holes in the Palatini-type theories of gravity, specifically (1) the Palatini f(R) gravity coupled with Born-Infeld nonlinear electrodynamics and (2) the Eddington-inspired-Born-Infeld gravity (EiBI) coupled with Maxwell electromagnetic fields. The coupled master equations describing perturbations of charged black holes in these theories are obtained with the tetrad formalism. By using the Wentzel-Kramers-Brillouin (WKB) method up to 6th order, we calculate the QNM frequencies of the EiBI charged black holes, the Einstein-Born-Infeld black holes, and the Born-Infeld charged black holes within the Palatini R+alpha R^2 gravity. The QNM spectra of these black holes would deviate from those of the Reissner-Nordström black hole. In addition, we study the QNMs in the eikonal limit and find that for the axial perturbations of the EiBI charged black holes, the link between the eikonal QNMs and the unstable null circular orbit around the black hole is violated.