arXiv subjects

Rudolf Dvorak

Publications and source records attributed to Rudolf Dvorak.

33 records · Page 2Linked to original sources

arXiv2013

SPH-based simulation of multi-material asteroid collisions

We give a brief introduction to smoothed particle hydrodynamics methods for continuum mechanics. Specifically, we present our 3D SPH code to simulate and analyze collisions of asteroids consisting of two types of material: basaltic rock and ice. We consider effects like brittle failure, fragmentation, and merging in different impact scenarios. After validating our code against previously published results we present first collision results based on measured values for the Weibull flaw distribution parameters of basalt.

astro-ph.EP↗

arXiv2013

Fugitives from the Hungaria region: close encounters and impacts with terrestrial planets

Hungaria asteroids, whose orbits occupy the region in element space between $1.78< a< 2.03$ AU, $e<0.19$, $12^\circ<i<31^\circ$, are a possible source of Near-Earth Asteroids (NEAs). Named after (434) Hungaria these asteroids are relatively small, since the largest member of the group has a diameter of just about 11 km. They are mainly perturbed by Jupiter and Mars, possibly becoming Mars-crossers and, later, they may even cross the orbits of Earth and Venus. In this paper we analyze the close encounters and possible impacts of escaped Hungarias with the terrestrial planets. Out of about 8000 known Hungarias we selected 200 objects which are on the edge of the group. We integrated their orbits over 100 million years in a simplified model of the planetary system (Mars to Saturn) subject only to gravitational forces. We picked out a sample of 11 objects (each with 50 clones) with large variations in semi-major axis and restarted the numerical integration in a gravitational model including the planets from Venus to Saturn. Due to close encounters, some of them achieve high inclinations and eccentricities which, in turn, lead to relatively high velocity impacts on Venus, Earth, and Mars. We statistically analyze all close encounters and impacts with the terrestrial planets and determine the encounter and impact velocities of these fictitious Hungarias.

astro-ph.EP↗

arXiv2012

The orbit of 2010 TK7. Possible regions of stability for other Earth Trojan asteroids

Recently the first Earth Trojan has been observed (Mainzer et al., ApJ 731) and found to be on an interesting orbit close to the Lagrange point L4 (Connors et al., Nature 475). In the present study we therefore perform a detailed investigation on the stability of its orbit and moreover extend the study to give an idea of the probability to find additional Earth-Trojans. Our results are derived using different approaches: a) we derive an analytical mapping in the spatial elliptic restricted three-body problem to find the phase space structure of the dynamical problem. We explore the stability of the asteroid in the context of the phase space geometry, including the indirect influence of the additional planets of our Solar system. b) We use precise numerical methods to integrate the orbit forward and backward in time in different dynamical models. Based on a set of 400 clone orbits we derive the probability of capture and escape of the Earth Trojan asteroids 2010 TK7. c) To this end we perform an extensive numerical investigation of the stability region of the Earth's Lagrangian points. We present a detailed parameter study in the regime of possible stable tadpole and horseshoe orbits of additional Earth-Trojans, i.e. with respect to the semi-major axes and inclinations of thousands of fictitious Trojans. All three approaches underline that the Earth Trojan asteroid 2010 TK7 finds himself in an unstable region on the edge of a stable zone; additional Earth-Trojan asteroids may be found in this regime of stability.

astro-ph.EP↗

arXiv2011

On the Mass of CoRoT-7b

The mass of CoRoT-7b, the first transiting superearth exoplanet, is still a subject of debate. A wide range of masses have been reported in the literature ranging from as high as 8 M_Earth to as low as 2.3 M_Earth. Although most mass determinations give a density consistent with a rocky planet, the lower value permits a bulk composition that can be up to 50% water. We present an analysis of the CoRoT-7b radial velocity measurements that uses very few and simple assumptions in treating the activity signal. By only analyzing those radial velocity data for which multiple measurements were made in a given night we remove the activity related radial velocity contribution without any a priori model. We demonstrate that the contribution of activity to the final radial velocity curve is negligible and that the K-amplitude due to the planet is well constrained. This yields a mass of 7.42 +/- 1.21 M_Earth and a mean density of rho = 10.4 +/- 1.8 gm cm^-3. CoRoT-7b is similar in mass and radius to the second rocky planet to be discovered, Kepler-10b, and within the errors they have identical bulk densities - they are virtual twins. These bulk densities lie close to the density - radius relationship for terrestrial planets similar to what is seen for Mercury. CoRoT-7b and Kepler-10b may have an internal structure more like Mercury than the Earth.

astro-ph.EP↗

arXiv2010

Dynamics of Neptune's Trojans: II. Eccentric orbits and observed ones

In a previous paper, we have presented a global view of the stability of Neptune Trojan (NT hereafter) on inclined orbit. We discuss in this paper the dependence of stability of NT orbits on the eccentricity. High-resolution dynamical maps are constructed using the results of extensive numerical integrations of orbits initialized on the fine grids of initial semimajor axis (a0) versus eccentricity (e0). The extensions of regions of stable orbits on the (a0, e0) plane at different inclinations are shown. The maximum eccentricities of stable orbits in three most stable regions at low (0, 12deg.), medium (22,36deg.) and high (51, 59deg.) inclination, are found to be 0.10, 0.12 and 0.04, respectively. The fine structures in the dynamical maps are described. Via the frequency analysis method, the mechanisms that portray the dynamical maps are revealed. The secondary resonances, concerning the frequency of the librating resonant angle and the frequency of the quasi 2:1 mean motion resonance between Neptune and Uranus, are found deeply involved in the motion of NTs. Secular resonances are detected and they also contribute significantly to the triggering of chaos in the motion. Particularly, the effects of the secular resonance v8, v18 are clarified. We also investigate the orbital stabilities of six observed NTs by checking the orbits of hundreds clones of them generated within the observing error bars. We conclude that four of them, except 2001 QR322 and 2005 TO74, are deeply inside the stable region. The 2001 QR322 is in the close vicinity of the most significant secondary resonance. The 2005 TO74 locates close to the boundary separating stable orbits from unstable ones, and it may be influenced by a secular resonance.

astro-ph.EP↗

arXiv2010

The SARS algorithm: detrending CoRoT light curves with Sysrem using simultaneous external parameters

Surveys for exoplanetary transits are usually limited not by photon noise but rather by the amount of red noise in their data. In particular, although the CoRoT spacebased survey data are being carefully scrutinized, significant new sources of systematic noises are still being discovered. Recently, a magnitude-dependant systematic effect was discovered in the CoRoT data by Mazeh & Guterman et al. and a phenomenological correction was proposed. Here we tie the observed effect a particular type of effect, and in the process generalize the popular Sysrem algorithm to include external parameters in a simultaneous solution with the unknown effects. We show that a post-processing scheme based on this algorithm performs well and indeed allows for the detection of new transit-like signals that were not previously detected.

astro-ph.EP↗

arXiv2010

Where are the Uranus Trojans?

The area of stable motion for fictitious Trojan asteroids around Uranus' equilateral equilibrium points is investigated with respect to the inclination of the asteroid's orbit to determine the size of the regions and their shape. For this task we used the results of extensive numerical integrations of orbits for a grid of initial conditions around the points L4 and L5, and analyzed the stability of the individual orbits. Our basic dynamical model was the Outer Solar System (Jupiter, Saturn, Uranus and Neptune). We integrated the equations of motion of fictitious Trojans in the vicinity of the stable equilibrium points for selected orbits up to the age of the Solar system of 5 billion years. One experiment has been undertaken for cuts through the Lagrange points for fixed values of the inclinations, while the semimajor axes were varied. The extension of the stable region with respect to the initial semimajor axis lies between 19.05 < a < 19.3 AU but depends on the initial inclination. In another run the inclination of the asteroids' orbit was varied in the range 0 < i < 60 and the semimajor axes were fixed. It turned out that only four 'windows' of stable orbits survive: these are the orbits for the initial inclinations 0 < i < 7, 9 < i < 13, 31 < i < 36 and 38 < i < 50. We postulate the existence of at least some Trojans around the Uranus Lagrange points for the stability window at small and also high inclinations.

astro-ph.EP↗

arXiv2009

An Overview of the 13:8 Mean Motion Resonance between Venus and Earth

It is known since the seminal study of Laskar (1989) that the inner planetary system is chaotic with respect to its orbits and even escapes are not impossible, although in time scales of billions of years. The aim of this investigation is to locate the orbits of Venus and Earth in phase space, respectively to see how close their orbits are to chaotic motion which would lead to unstable orbits for the inner planets on much shorter time scales. Therefore we did numerical experiments in different dynamical models with different initial conditions -- on one hand the couple Venus-Earth was set close to different mean motion resonances (MMR), and on the other hand Venus' orbital eccentricity (or inclination) was set to values as large as e = 0.36 (i = 40deg). The couple Venus-Earth is almost exactly in the 13:8 mean motion resonance. The stronger acting 8:5 MMR inside, and the 5:3 MMR outside the 13:8 resonance are within a small shift in the Earth's semimajor axis (only 1.5 percent). Especially Mercury is strongly affected by relatively small changes in eccentricity and/or inclination of Venus in these resonances. Even escapes for the innermost planet are possible which may happen quite rapidly.

astro-ph.EP↗

arXiv2009

Planetary Dynamics and Habitable Planet Formation In Binary Star Systems

Whether binaries can harbor potentially habitable planets depends on several factors including the physical properties and the orbital characteristics of the binary system. While the former determines the location of the habitable zone (HZ), the latter affects the dynamics of the material from which terrestrial planets are formed (i.e., planetesimals and planetary embryos), and drives the final architecture of the planets assembly. In order for a habitable planet to form in a binary star system, these two factors have to work in harmony. That is, the orbital dynamics of the two stars and their interactions with the planet-forming material have to allow terrestrial planet formation in the habitable zone, and ensure that the orbit of a potentially habitable planet will be stable for long times. We have organized this chapter with the same order in mind. We begin by presenting a general discussion on the motion of planets in binary stars and their stability. We then discuss the stability of terrestrial planets, and the formation of potentially habitable planets in a binary-planetary system.

astro-ph.EP↗

arXiv2009

The Dynamics of Neptune Trojan: I. the Inclined Orbits

The stability of Trojan type orbits around Neptune is studied. As the first part of our investigation, we present in this paper a global view of the stability of Trojans on inclined orbits. Using the frequency analysis method based on the FFT technique, we construct high resolution dynamical maps on the plane of initial semimajor axis $a_0$ versus inclination $i_0$. These maps show three most stable regions, with $i_0$ in the range of $(0^\circ,12^\circ), (22^\circ,36^\circ)$ and $(51^\circ,59^\circ)$ respectively, where the Trojans are most probably expected to be found. The similarity between the maps for the leading and trailing triangular Lagrange points $L_4$ and $L_5$ confirms the dynamical symmetry between these two points. By computing the power spectrum and the proper frequencies of the Trojan motion, we figure out the mechanisms that trigger chaos in the motion. The Kozai resonance found at high inclination varies the eccentricity and inclination of orbits, while the $ν_8$ secular resonance around $i_0\sim44^\circ$ pumps up the eccentricity. Both mechanisms lead to eccentric orbits and encounters with Uranus that introduce strong perturbation and drive the objects away from the Trojan like orbits. This explains the clearance of Trojan at high inclination ($>60^\circ$) and an unstable gap around $44^\circ$ on the dynamical map. An empirical theory is derived from the numerical results, with which the main secular resonances are located on the initial plane of $(a_0,i_0)$. The fine structures in the dynamical maps can be explained by these secular resonances.

astro-ph.EP↗

arXiv2008

On the Dynamics of Inclined Neptune's Trojans

The dynamics of artificial asteroids on the Trojan-like orbits around Neptune is investigated in this paper. We describe the dependence of the orbital stability on the initial semimajor axis a and inclination i by constructing a dynamical map on the (a,i)-plane. Rich details are revealed in the dynamical map, especially a unstable gap at i=45 deg is determined and the mechanism triggering chaos in this region is figured out. Our investigation can be used to guide the observations.

astro-ph↗

arXiv2006

A Planetary System with an Escaping Mars

The chaotic behaviour of the motion of the planets in our Solar System is well established. In this work to model a hypothetical extrasolar planetary system our Solar System was modified in such a way that we replaced the Earth by a more massive planet and let the other planets and all the orbital elements unchanged. The major result of former numerical experiments with a modified Solar System was the appearance of a chaotic window at $κ_E \in (4,6)$, where the dynamical state of the system was highly chaotic and even the body with the smallest mass escaped in some cases. On the contrary for very large values of the mass of the Earth, even greater than that of Jupiter regular dynamical behaviour was observed. In this paper the investigations are extended to the complete Solar System and showed, that this chaotic window does still exist. Tests in different 'Solar Systems' clarified that including only Jupiter and Saturn with their actual masses together with a more 'massive' Earth ($4 < κ_E < 6$) perturbs the orbit of Mars so that it can even be ejected from the system. Using the results of the Laplace-Lagrange secular theory we found secular resonances acting between the motions of the nodes of Mars, Jupiter and Saturn. These secular resonances give rise to strong chaos, which is the cause of the appearance of the instability window.

physics.class-ph↗

arXiv2005

The stability of the terrestrial planets with a more massive "Earth"

Although the long-term numerical integrations of planetary orbits indicate that our planetary system is dynamically stable at least +/- Gyr, the dynamics of our Solar System includes both chaotic and stable motions: the large planets exhibit remarkable stability on gigayear timescales, while the subsystem of the terrestrial planets is weekly chaotic with a maximum Lyapunov exponent reaching the value of 1/5 Myr. In this paper the dynamics of the Sun--Venus--Earth--Mars-Jupiter--Saturn model is studied, where the mass of Earth was magnified via a mass factor $κ_E$. The resulting systems dominated by a massive Earth may serve also as models for exoplanetary systems that are similar to our one. This work is a continuation of our previous study, where the same model was used and the masses of the inner planets were uniformly magnified. That model was found to be substantially stable against the mass growth. Our simulations were undertaken for more then 100 different values of K for a time of 20, in some cases for 100 Myrs. A major result was the appearance of an instability window at K = 5, where Mars escaped. This new result has important implications for the theories of the planetary system formation process and mechanism. It is shown that with increasing K the system splits into two, well separated subsystems: one consists of the inner, the other one consists of the outer planets. According to the results the model became more stable as K increases and only when K >= 540 Mars escaped, on a Myr timescale. We found an interesting protection mechanism for Venus. These results give insights also to the stability of the habitable zone of exoplanetary systems, which harbour planets with relatively small eccentricities and inclinations.

astro-ph↗

arXiv2002

Global dynamics and stability limits for planetary systems around HD 12661, HD 38529, HD 37124 and HD 160691

In order to distinguish between regular and chaotic planetary orbits we apply a new technique called MEGNO in a wide neighbourhood of orbital parameters determined using standard two-body Keplerian fits for HD 12661, HD 38529, HD 37124 and HD 160691 planetary systems. We show that the currently announced orbital parameters place these systems in very different situations from the point of view of dynamical stability. While HD 38529 and HD 37124 are located within large stability zones in the phase space around their determined orbits, the preliminary orbits in HD 160691 are highly unstable. The orbital parameters of the HD 12661 planets are located in a border region between stable and unstable dynamical regimes, so while its currently determined orbital parameters produce stable regular orbits, a minor change within the margin of error of just one parameter may result in a chaotic dynamical system.

astro-ph↗

arXiv1999

Diffusion and scaling in escapes from two-degree-of-freedom Hamiltonian systems

This paper summarises an investigation of the statistical properties of orbits escaping from three different two-degree-of-freedom Hamiltonian systems which exhibit global stochasticity. Each H=H_{0}+eH', with H_{0} integrable and eH' a nonintegrable correction, not necessarily small. For e below a critical e_{0} escapes are impossible energetically. For somewhat higher values, escape is allowed energetically but many orbits never escape and the escape probability P for a generic orbit ensemble decays exponentially. At or near a critical e_{1}>e_{0}, there is an abrupt qualitative change in behaviour. Above e_{1}, P typically exhibits (1) a rapid evolution towards a nonzero P_{0}(e) followed by (2) a much slower subsequent decay towards zero which, in at least one case, is well fit by a power law P=const x t^{m}, with m=0.35-0.40. In all three cases, P_{0} and the time T required to converge towards P_{0} scales in e-e_{1}, i.e., P_{0}=const x (e-e_{1})^{a} and T=const x (e-e_{1})^{b}, and T also scales in the size r of the region sampled for initial conditions, i.e., T=const x r^{-d}. To within statistical uncertainties, the best fit values of the critical exponents are the same for all three potentials, namely: a=0.5, b=0.4, and d=0.1, and satisfy a-b-d=0. The transitional behaviour observed near e_{1} is attributed to the breakdown of some especially significant KAM tori or cantori. The power law behaviour at late times is interpreted as reflecting intrinsic diffusion of chaotic orbits through cantori surrounding islands of regular orbits.

astro-ph↗