arXiv subjects

Christoph Lhotka

Publications and source records attributed to Christoph Lhotka.

At least 19 recordsLinked to original sources

arXiv2025

The dynamics of the spin-spin problem in Celestial Mechanics

This work investigates different models of rotational dynamics of two rigid bodies with the shape of an ellipsoid, moving under their gravitational influence. The focus of this study is on their behavior, their linear stability, and numerical investigation of the main resonances. We assume that the spin axes of the two bodies are perpendicular to the orbital plane and coinciding with the direction of their shortest physical axis. In the basic approximation, we assume that the orbits of the centers of mass are Keplerian and we retain the lowest order of the potential, according to which the rotational motions of the two bodies are decoupled, the so-called spin-orbit problem. When considering highest order approximation of the potential, the rotational motions become coupled giving rise to the so-called spin-spin problem. Finally, we release the assumption that the orbit is Keplerian, which implies that the rotational dynamics is coupled to the variation of the orbital elements. The resulting system is called the full spin-spin problem. We also consider the above models under the assumption that one or both bodies are non rigid; the dissipative effect is modeled by a linear function of the rotational velocity, depending on some dissipative and drift coefficients. We consider three main resonances, namely the (1:1,1:1), (3:2,3:2), (1:1,3:2) resonances and we start by analyzing the linear stability of the equilibria in the conservative and dissipative settings (after averaging and keeping only the resonant angle), showing that the stability depends on the value of the orbital eccentricity. By a numerical integration of the equations of motion, we compare the spin-orbit and spin-spin problems to highlight the influence of the coupling term of the potential in the conservative and dissipative case...

math.DS↗

arXiv2025

The dynamics around the collinear points of the elliptic three-body problem: A normal form approach

We study the dynamics of the collinear points in the planar, restricted three-body problem, assuming that the primaries move on an elliptic orbit around a common barycenter. The equations of motion can be conveniently written in a rotating pulsating barycentric frame, taking the true anomaly as independent variable. We consider the Hamiltonian modeling this problem in the extended phase space and we imple ment a normal form to make a center manifold reduction. The normal form provides an approximate solution for the Cartesian coordinates, which allows us to construct several kinds of orbits, most notably planar and vertical Lyapunov orbits, and halo orbits. We compare the analytical results with a numerical simulation, which requires special care in the selection of the initial conditions.

math.DS↗

arXiv2023

Breakdown of rotational tori in 2D and 4D conservative and dissipative standard maps

We study the breakdown of rotational invariant tori in 2D and 4D standard maps by implementing three different methods. First, we analyze the domains of analyticity of a torus with given frequency through the computation of the Lindstedt series expansions of the embedding of the torus and the drift term. The Padé approximants provide the shape of the analyticity domains by plotting the poles of the polynomial at the denominator of the approximants. Secondly, we implement a Newton method to construct the embedding of the torus; the breakdown threshold is then estimated by looking at the blow-up of the Sobolev norms of the embedding. Finally, we implement an extension of Greene method to get information on the breakdown threshold of an invariant torus with irrational frequency by looking at the stability of the periodic orbits with periods approximating the frequency of the torus. We apply these methods to 2D and 4D standard maps. The 2D maps can either be conservative (symplectic) or dissipative ( more precisely, conformally symplectic, namely a dissipative map with the geometric property to transform the symplectic form into a multiple of itself). The 4D maps are obtained coupling $(i)$ two symplectic standard maps, or $(ii)$ two conformally symplectic standard maps, or $(iii)$ a symplectic and a conformally symplectic standard map. Concerning the results, Padé and Newton methods perform well and provide reliable and consistent results (although we implemented Newton method only for symplectic and conformally symplectic maps). Our implementation of the extension of Greene method is inconclusive, since it is computationally expensive and delicate, especially in 4D non-symplectic maps, also due to the existence of Arnold tongues.

math.DS↗

arXiv2022

Trojan asteroids and co-orbital dust ring of Venus

The long-standing co-orbital asteroids have been thought to be the possible source of zodiacal dust ring around the orbit of Venus, but inconsistent conclusions on the orbital stability thus existence of Venus Trojans are found in literature. We present in this paper a systematic survey of the orbital stability of Venus Trojans, taking into account the dynamical influences from the general relativity and the Yarkovsky effect. The orbits of thousands of fictitious Venus Trojans are numerically simulated. Using the method of frequency analysis, their orbital stabilities and the dynamical mechanisms behind are described in detail. The influences of general relativity and of Yarkovsky effect, which were either neglected or oversimplified previously, are investigated by long-term numerical simulations. The stability maps on the $(a_0,i_0)$ plane and $(a_0,e_0)$ plane are depicted, and the most stable Venus Trojans are found to occupy the low-inclination horseshoe orbits with low eccentricities. The resonances that carve the fine structures in the stability map are determined. The general relativity decreases the stability of orbits by little but the Yarkovsky effect may drive nearly all Venus Trojans out of the Trojan region in a relatively short time. The Venus Trojans have poor orbital stability and cannot survive the age of the Solar system. The zodiacal dust ring found around the orbit of Venus is more likely a sporadic phenomenon, as the result of temporarily capture into the 1:1 mean motion resonance of dust particles produced probably from passing comets or asteroids, but not Venus Trojans.

astro-ph.EP↗

arXiv2021

Tadpole type motion of charged dust in the Lagrange problem with planet Jupiter

We investigate the dynamics of charged dust interacting with the interplanetary magnetic field in a Parker spiral type model and subject to the solar wind and Poynting-Robertson effect in the vicinity of the 1:1 mean motion resonance with planet Jupiter. We estimate the shifts of the location of the minimum libration amplitude solutions close to the location of the L4 and L5 points of the classical - gravitational - problem and provide the extension of the 'librational regimes of motion' and the width of the resonance in dependency of the nongravitational parameters related to the dust grain size and surface potential of the particles. Our study is based on numerical simulations in the framework of the spatial, elliptic restricted three-body problem and semi-analytical estimates obtained by averaging of Gauss' planetary equations of motion.

astro-ph.EP↗

arXiv2020

Dynamics of charged dust in the orbit of Venus

We study the dynamics of co-orbital dust in the inner solar system, i.e. the role of the solar radiation pressure, Poynting-Robertson effect, solar wind, and the interplanetary magnetic field on the location, width and stability of resonant motion of charged, and micron sized dust grains situated in the 1:1 mean motion resonance with planet Venus. We find deviations and asymmetry between $L_4$ and $L_5$ in locations of libration centers and libration width under the influence of non-gravitational effects via both analytical and numerical methods. The triangular Lagrangian points become unstable once we take into consideration solar radiation pressure, the Poynting-Robertson effect and solar wind drag. The Lorentz force could further destabilize the orbits, especially for small dust particles. We also make a comparison between the circular, elliptic restricted three-body model and a more complete model including all planets.

astro-ph.EP↗

arXiv2020

Sub-ion scale Compressive Turbulence in the Solar wind: MMS spacecraft potential observations

Compressive plasma turbulence is investigated at sub-ion scales in the solar wind using both the Fast Plasma Investigation (FPI) instrument on the Magnetospheric MultiScale mission (MMS), as well as using calibrated spacecraft potential data from the Spin Plane Double Probe (SDP) instrument. The data from FPI allow a measurement down to the sub-ion scale region ($f_{sc}\gtrsim 1$ Hz) to be investigated before the instrumental noise becomes significant at a spacecraft frame frequency of $f_{sc}\approx 3$Hz, whereas calibrated spacecraft potential allows a measurement up to $f_{sc}\approx 40$Hz. In this work, we give a detailed description of density estimation in the solar wind using the spacecraft potential measurement from the SDP instrument on MMS. Several intervals of solar wind plasma have been processed using the methodology described which are made available. One of the intervals is investigated in more detail and the power spectral density of the compressive fluctuations is measured from the inertial range to the sub-ion range. The morphology of the density spectra can be explained by either a cascade of Alfvén waves and slow waves at large scales and kinetic Alfvén waves at sub-ion scales, or more generally by the Hall effect. Using electric field measurements the two hypotheses are discussed.

physics.space-ph↗

arXiv2020

Asteroid migration due to the Yarkovsky effect and the distribution of the Eos family

Based on a linearized model of the Yarkovsky effect, we investigate in this paper the dependence of the semimajor axis drift $Δa$ of a celestial body on its size, spinning obliquity, initial orbit and thermal parameters on its surface. With appropriate simplification and approximation, we obtain the analytical solutions to the perturbation equations for the motion of asteroids influenced by the Yarkovsky effect, and they are then verified by numerical simulations of the full equations of motion. These solutions present explicitly the dependencies of $Δa$ on the thermal and dynamical parameters of the asteroid. With these analytical formulae for $Δa$, we investigate the combined seasonal and diurnal Yarkovsky effects. The critical points where the migration direction reverses are calculated and the consequent selective effects according to the size and rotation state of asteroids are discussed. %Solely the Yarkovsky effect is found to be able to produce some ring structure in the aged circumstellar debris disk. Finally, we apply the analytical formulae to calculate the migration of Eos family members. The space distribution of asteroids is well reproduced. Our calculations suggest that statistically the orientations of spin axes of family members satisfy a random-obliquity distribution, and the rotation rate $ω_{\rm rot}$ of asteroid depends on its size $R$ by $ω_{\rm rot}\propto R^{-1}$.

astro-ph.EP↗

arXiv2019

Resonances in the Earth's Space Environment

We study the presence of resonances in the region of space around the Earth. We consider a massless body (e.g, a dust particle or a small space debris) subject to different forces: the gravitational attraction of the geopotential, the effects of Sun and Moon. We distinguish different types of resonances: tesseral resonances are due to a commensurability involving the revolution of the particle and the rotation of the Earth, semi-secular resonances include the rates of variation of the mean anomalies of Moon and Sun, while secular resonances just depend on the rates of variation of the arguments of perigee and the longitudes of the ascending nodes of the perturbing bodies. We characterize such resonances, giving precise statements on the regions where the resonances can be found and provide examples of some specific commensurability relations.

astro-ph.EP↗

arXiv2019

Charged dust close to outer mean-motion resonances in the heliosphere

We investigate the dynamics of charged dust close to outer mean-motion resonances with planet Jupiter. The importance of the interplanetary magnetic field on the orbital evolution of dust is clearly demonstrated. New dynamical phenomena are found that do not exist in the classical problem of uncharged dust. We find changes in the orientation of the orbital planes of dust particles, an increased amount of chaotic orbital motions, sudden 'jumps' in the resonant argument, and a decrease in time of temporary capture due to the Lorentz force. Variations in the orbital planes of dust grain orbits are found to be related to the angle between the orbital angular momentum and magnetic axes of the heliospheric field and the rotation rate of the Sun. These variations are bound using a simplified model derived from the full dynamical problem using first order averaging theory. It is found that the interplanetary magnetic field does not affect the capture process, that is still dominated by the other non-gravitational forces. Our study is based on a dynamical model in the framework of the inclined circular restricted three-body problem. Additional forces include solar radiation pressure, solar wind drag, the Poynting-Robertson effect, and the influence of a Parker spiral type interplanetary magnetic field model. The analytical estimates are derived on the basis of Gauss' form of planetary equations of motion. Numerical results are obtained by simulations of dust grain orbits together with the system of variational equations. Chaotic regions in phase space are revealed by means of Fast Lyapunov Chaos Indicators.

astro-ph.EP↗

arXiv2019

Orbital stability of ensembles of particles in regions of magnetic reconnection in Earth's magneto-tail

We investigate the collective behaviour of particle orbits in the vicinity of magnetic reconnection in Earth's magneto-tail. Various regions of different kinds of orbital stability of particle motions are found. We locate regimes of temporary capture of particle orbits in configuration space as well as locations, where strong particle accelerations take place. With this study we are able to provide a detailed map, i.e. the topology, of high and low acceleration centers close to the reconnection site. Quasi-regular and chaotic kinds of motions of elementary particles can be determined as well. The orbital stability of particle orbits is obtained by a statistical analysis of the outcome of the system of variational equations of particle orbits within the framework of particle-in-cell simulations. Using the concept of Lyapunov Characteristic Numbers to ensembles of particle orbits we introduce Lyapunov Ensemble Averages to describe the response of particle orbits to local perturbations induced by the electro-magnetic field.

physics.plasm-ph↗

arXiv2016

Gravity field and solar component of the precession rate and nutation coefficients of Comet 67P Churyumov-Gerasimenko

The aim of this study is first to determine the gravity field of the comet 67P Churyumov-Gerasimenko and second to derive the solar component of the precession rate and nutation coefficients of the spin axis of the comet nucleus, i.e. without the direct, usually larger, effect of outgassing. The gravity field, and related moments of inertia, are obtained from two polyhedra, that are provided by the OSIRIS and NAV-CAM experiments on Rosetta, and are based on the assumption of uniform density for the comet nucleus. We also calculate the forced precession rate as well as the nutation coefficients on the basis of Kinoshita's theory of rotation of the rigid Earth and adapted it to be able to indirectly include the effect of outgassing on the rotational parameters. The 2nd degree denormalized Stokes coefficients of comet 67P C-G turn out to be (bracketed numbers refer to second shape model) C20 = -6.74 [-7.93] * 10^-2, C22 = 2.60 [2.71] * 10^-2 consistent with normalized principal moments of inertia A/MR^2 = 0.13 [0.11], B/MR^2 = 0.23 [0.22], with polar moment c = C/MR^2 = 0.25, depending on the choice of the polyhedron model. The obliquity between the rotation axis and the mean orbit normal is 52 degree, and the precession rate only due to solar torques becomes 20-30 "/y. Oscillations in longitude caused by the gravitational pull of the Sun turn out to be of the order of 1', oscillations in obliquity can be estimated to be of the order of 0.5'.

astro-ph.EP↗

arXiv2016

Charged dust grain dynamics subject to solar wind, Poynting-Robertson drag, and the interplanetary magnetic field

We investigate the combined effect of solar wind, Poynting-Robertson drag, and the frozen-in interplanetary magnetic field on the motion of charged dust grains in our solar system. For this reason we derive a secular theory of motion by the means of averaging method and validate it with numerical simulations of the un-averaged equations of motions. The theory predicts that the secular motion of charged particles is mainly affected by the z-component of the solar magnetic axis, or the normal component of the interplanetary magnetic field. The normal component of the interplanetary magnetic field leads to an increase or decrease of semi-major axis depending on its functional form and sign of charge of the dust grain. It is generally accepted that the combined effects of solar wind and photon absorption and re-emmision (Poynting-Robertson drag) lead to a decrease in semi-major axis on secular time scales. On the contrary, we demonstrate that the interplanetary magnetic field may counteract these drag forces under certain circumstances. We derive a simple relation between the parameters of the magnetic field, the physical properties of the dust grain as well as the shape and orientation of the orbital ellipse of the particle, which is a necessary conditions for the stabilization in semi-major axis.

astro-ph.EP↗

arXiv2016

Poynting-Robertson drag and solar wind in the space debris problem

We analyze the combined effect of Poynting-Robertson and solar wind drag on space debris. We derive a model within Cartesian, Gaussian and Hamiltonian frameworks. We focus on the geosynchronous resonance, although the results can be easily generalized to any resonance. By numerical and analytical techniques, we compute the drift in semi-major axis due to Poynting-Robertson and solar wind drag. After a linear stability analysis of the equilibria, we combine a careful investigation of the regular, resonant, chaotic behavior of the phase space with a long-term propagation of a sample of initial conditions. The results strongly depend on the value of the area-to-mass ratio of the debris, which might show different dynamical behaviors: temporary capture or escape from the geosynchronous resonance, as well as temporary capture or escape from secondary resonances involving the rate of variation of the longitude of the Sun. Such analysis shows that Poynting-Robertson and solar wind drag must be taken into account, when looking at the long-term behavior of space debris. Trapping or escape from the resonance can be used to place the debris in convenient regions of the phase space.

astro-ph.EP↗

arXiv2015

Effective stability around the Cassini state in the spin-orbit problem

We investigate the long-time stability in the neighborhood of the Cassini state in the conservative spin-orbit problem. Starting with an expansion of the Hamiltonian in the canonical Andoyer-Delaunay variables, we construct a high-order Birkhoff normal form and give an estimate of the effective stability time in the Nekhoroshev sense. By extensively using algebraic manipulations on a computer, we explicitly apply our method to the rotation of Titan. We obtain physical bounds of Titan's latitudinal and longitudinal librations, finding a stability time greatly exceeding the estimated age of the Universe. In addition, we study the dependence of the effective stability time on three relevant physical parameters: the orbital inclination, $i$, the mean precession of the ascending node of Titan orbit, $\dotΩ$, and the polar moment of inertia, $C$.

math.DS↗

arXiv2014

The effect of Poynting-Robertson drag on the triangular Lagrangian points

We investigate the stability of motion close to the Lagrangian equilibrium points L4 and L5 in the framework of the spatial, elliptic, restricted three- body problem, subject to the radial component of Poynting-Robertson drag. For this reason we develop a simplified resonant model, that is based on averaging theory, i.e. averaged over the mean anomaly of the perturbing planet. We find temporary stability of particles displaying a tadpole motion in the 1:1 resonance. From the linear stability study of the averaged simplified resonant model, we find that the time of temporary stability is proportional to beta a1 n1 , where beta is the ratio of the solar radiation over the gravitational force, and a1, n1 are the semi-major axis and the mean motion of the perturbing planet, respectively. We extend previous results (Murray (1994)) on the asymmetry of the stability indices of L4 and L5 to a more realistic force model. Our analytical results are supported by means of numerical simulations. We implement our study to Jupiter-like perturbing planets, that are also found in extra-solar planetary systems.

astro-ph.EP↗

arXiv2013

The influence of orbital dynamics, shape and tides on the obliquity of Mercury

Earth-based radar observations of the rotational dynamics of Mercury (Margot et al. 2012) combined with the determination of its gravity field by MESSENGER (Smith et al. 2012) give clues on the internal structure of Mercury, in particular its polar moment of inertia C, deduced from the obliquity (2.04 +/- 0.08) arcmin. The dynamics of the obliquity of Mercury is a very-long term motion (a few hundreds of kyrs), based on the regressional motion of Mercury's orbital ascending node. This paper, following the study of Noyelles & D'Hoedt (2012), aims at first giving initial conditions at any time and for any values of the internal structure parameters for numerical simulations, and at using them to estimate the influence of usually neglected parameters on the obliquity, like J3, the Love number k2 and the secular variations of the orbital elements. We use for that averaged representations of the orbital and rotational motions of Mercury, suitable for long-term studies. We find that J3 should alter the obliquity by 250 milli-arcsec, the tides by 100 milli-arcsec, and the secular variations of the orbital elements by 10 milli-arcsec over 20 years. The resulting value of C could be at the most changed from 0.346mR^2 to 0.345mR^2.

astro-ph.EP↗

arXiv2012

Stability of nearly-integrable systems with dissipation

We study the stability of a vector field associated to a nearly-integrable Hamiltonian dynamical system to which a dissipation is added. Such a system is governed by two parameters, named the perturbing and dissipative parameters, and it depends on a drift function. Assuming that the frequency of motion satisfies some resonance assumption, we investigate the stability of the dynamics, and precisely the variation of the action variables associated to the conservative model. According to the structure of the vector field, one can find linear and exponential stability times, which are established under smallness con- ditions on the parameters. We also provide some applications to concrete examples, which exhibit a linear or exponential stability behavior.

math.DS↗