Preface     

Rudolf Dvorak  Sylvio Ferraz-Mello 《Celestial Mechanics and Dynamical Astronomy》2009,104(1-2):1-2

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Numerical results to the Sitnikov-problem     

R. Dvorak 《Celestial Mechanics and Dynamical Astronomy》1993,56(1-2):71-80

We present numerical results of the so-called Sitnikov-problem, a special case of the three-dimensional elliptic restricted three-body problem. Here the two primaries have equal masses and the third body moves perpendicular to the plane of the primaries' orbit through their barycenter. The circular problem is integrable through elliptic integrals; the elliptic case offers a surprisingly great variety of motions which are until now not very well known. Very interesting work was done by J. Moser in connection with the original Sitnikov-paper itself, but the results are only valid for special types of orbits. As the perturbation approach needs to have small parameters in the system we took in our experiments as initial conditions for the work moderate eccentricities for the primaries' orbit (0.33e _primaries 0.66) and also a range of initial conditions for the distance of the 3 ^rd body (= the planet) from very close to the primaries orbital plane of motion up to distance 2 times the semi-major axes of their orbit. To visualize the complexity of motions we present some special orbits and show also the development of Poincaré surfaces of section with the eccentricity as a parameter. Finally a table shows the structure of phase space for these moderately chosen eccentricities.  相似文献   

Stability of fictitious Trojan planets in extrasolar systems     

R. Schwarz  R. Dvorak  . Süli  B. rdi 《Astronomische Nachrichten》2007,328(8):785-788

Our work deals with the dynamical possibility that in extrasolar planetary systems a terrestrial planet may have stable orbits in a 1:1 mean motion resonance with a Jovian like planet. We studied the motion of fictitious Trojans around the Lagrangian points L₄/L₅ and checked the stability and/or chaoticity of their motion with the aid of the Lyapunov Indicators and the maximum eccentricity. The computations were carried out using the dynamical model of the elliptic restricted three‐body problem that consists of a central star, a gas giant moving in the habitable zone, and a massless terrestrial planet. We found 3 new systems where the gas giant lies in the habitable zone, namely HD99109, HD101930, and HD33564. Additionally we investigated all known extrasolar planetary systems where the giant planet lies partly or fully in the habitable zone. The results show that the orbits around the Lagrangian points L₄/L₅ of all investigated systems are stable for long times (10⁷ revolutions). (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

On the global stability of single‐planet systems     

. Süli  R. Dvorak  B. rdi 《Astronomische Nachrichten》2007,328(8):781-784

In this paper, we estimate the global stability properties of single‐planet systems by using a catalogue of stability maps. The data of the catalogue were used to generate probability values on the mass parameter–eccentricity plane for the occurrence of stable orbits. We showed that the probability data can be well approximated by a second order surface. Using the resulted formula the likelihood of finding Earth‐like planets in single‐planet systems can be easily estimated. As an example, we derived estimations for four known exoplanetary systems. Our formula can be useful in selecting target stars for future space missions. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

A planetary system with an escaping Mars     

. Süli  R. Dvorak 《Astronomische Nachrichten》2007,328(1):4-9

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 ∈ (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. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

Integrating GRASS GIS and Jupyter Notebooks to facilitate advanced geospatial modeling education     

Caitlin Haedrich  Vaclav Petras  Anna Petrasova  Stefan Blumentrath  Helena Mitasova 《Transactions in GIS》2023,27(3):686-702

Open education materials are critical for the advancement of open science and the development of open-source software. These accessible and transparent materials provide an important pathway for sharing both standard geospatial analysis workflows and advanced research methods. Computational notebooks allow users to share live code with in-line visualizations and narrative text, making them a powerful interactive teaching tool for geospatial analytics. Specifically, Jupyter Notebooks are quickly becoming a standard format in open education. In this article, we introduce a new GRASS GIS package, grass.jupyter , that enhances the existing GRASS Python API to allow Jupyter Notebook users to easily manage and visualize GRASS data including spatiotemporal datasets. While there are many Python-based geospatial libraries available for use in Jupyter Notebooks, GRASS GIS has extensive geospatial functionality including support for multi-temporal analysis and dynamic simulations, making it a powerful teaching tool for advanced geospatial analytics. We discuss the development of grass.jupyter and demonstrate how the package facilitates teaching open-source geospatial modeling with a collection of Jupyter Notebooks designed for a graduate-level geospatial modeling course. The open education notebooks feature spatiotemporal data visualizations, hydrologic modeling, and spread simulations such as the spread of invasive species and urban growth.  相似文献   

Transition spectra of dynamical systems     

G. Contopoulos  N. Voglis  C. Efthymiopoulos  C. Froeschlé  R. Gonczi  E. Lega  R. Dvorak  E. Lohinger 《Celestial Mechanics and Dynamical Astronomy》1997,67(4):293-317

The spectra of ‘stretching numbers’ (or ‘local Lyapunov characteristic numbers’) are different in the ordered and in the chaotic domain. We follow the variation of the spectrum as we move from the centre of an island outwards until we reach the chaotic domain. As we move outwards the number of abrupt maxima in the spectrum increases. These maxima correspond to maxima or minima in the curve a(θ), where a is the stretching number, and θ the azimuthal angle. We explain the appearance of new maxima in the spectra of ordered orbits. The orbits just outside the last KAM curve are confined close to this curve for a long time (stickiness time) because of the existence of cantori surrounding the island, but eventually escape to the large chaotic domain further outside. The spectra of sticky orbits resemble those of the ordered orbits just inside the last KAM curve, but later these spectra tend to the invariant spectrum of the chaotic domain. The sticky spectra are invariant during the stickiness time. The stickiness time increases exponentially as we approach an island of stability, but very close to an island the increase is super exponential. The stickiness time varies substantially for nearby orbits; thus we define a probability of escape P_n(x) at time n for every point x. Only the average escape time in a not very small interval Δx around each x is reliable. Then we study the convergence of the spectra to the final, invariant spectrum. We define the number of iterations, N, needed to approach the final spectrum within a given accuracy. In the regular domain N is small, while in the chaotic domain it is large. In some ordered cases the convergence is anomalously slow. In these cases the maximum value of a_k in the continued fraction expansion of the rotation number a = [a₀,a₁,... a_k,...] is large. The ordered domain contains small higher order chaotic domains and higher order islands. These can be located by calculating orbits starting at various points along a line parallel to the q-axis. A monotonic variation of the sup {q}as a function of the initial condition q₀ indicates ordered motions, a jump indicates the crossing of a localized chaotic domain, and a V-shaped structure indicates the crossing of an island. But sometimes the V-shaped structure disappears if the orbit is calculated over longer times. This is due to a near resonance of the rotation number, that is not followed by stable islands. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

Exchange orbits: an interesting case of co-orbital motion     

Barbara Funk  Rudolf Dvorak  Richard Schwarz 《Celestial Mechanics and Dynamical Astronomy》2013,117(1):41-58

In this investigation we treat a special configuration of two celestial bodies in 1:1 mean motion resonance namely the so-called exchange orbits. There exist—at least—theoretically—two different types: the exchange-a orbits and the exchange-e orbits. The first one is the following: two celestial bodies are in orbit around a central body with almost the same semi-major axes on circular orbits. Because of the relatively small differences in semi-major axes they meet from time to time and exchange their semi-major axes. The inner one then moves outside the other planet and vice versa. The second configuration one is the following: two planets are moving on nearly the same orbit with respect to the semi-major axes, one on a circular orbit and the other one on an eccentric one. During their dynamical evolution they change the characteristics of the orbit, the circular one becomes an elliptic one whereas the elliptic one changes its shape to a circle. This ‘game’ repeats periodically. In this new study we extend the numerical computations for both of these exchange orbits to the three dimensional case and in another extension treat also the problem when these orbits are perturbed from a fourth body. Our results in form of graphs show quite well that for a large variety of initial conditions both configurations are stable and stay in these exchange orbits.  相似文献   

Long-term stability of planetary orbits between Jupiter and Saturn     

Rudolf Dvorak  Manfred Cuntz 《Astronomische Nachrichten》2024,345(1):e230147

We extend our two previous studies on the existence of stable orbits in the Solar System by examining the domain between Jupiter and Saturn. We place (1) a massless object, (2) a Moon-mass object, (3) a Mars-mass object, (4) an Earth-mass object, and (5) a Uranus-mass object in the said region. Note that these objects are considered separately in the framework of our simulations. Our goal is to explore the orbital stability of those objects. We employ the Lie-integration method, which is fast and well established, allowing us to solve the respective differential equations for the $N$-body system. Hence, we consider the celestial bodies spanning from Jupiter to Neptune, including the aforementioned test object, the main focus for our model simulations. The integrations indicate that in some models the test objects placed in the region between Jupiter and Saturn reside in that region for more than 600 Myr. Between 5 and 10 au, mean-motion resonances (MMRs) take place acting upon the test objects akin to simulations of Paper I and II. Our models indicate relatively small differences for the long-term stability of the five test objects notwithstanding their vastly different masses. Generally, it is found that between $a_{\text{ini}}=7.04$ and 7.13 au the orbits become unstable mostly within 5 million years and further out, that is, up to $a_{\text{ini}}=7.29$ au, the duration of stability lengthens to up to hundreds of millions of years.  相似文献   

Year‐round estimation of soil moisture content using temporally variable soil hydraulic parameters          下载免费PDF全文

Vaclav Sipek  Miroslav Tesar 《水文研究》2017,31(6):1438-1452

Soil moisture plays a key role in the hydrological cycle as it controls the flux of water between soil, vegetation, and atmosphere. This study is focused on a year‐round estimation of soil moisture in a forested mountain area using the bucket model approach. For this purpose, three different soil moisture models are utilised. The procedure is based on splitting the whole year into two complement periods (dormant and vegetation). Model parameters are allowed to vary between the two periods and also from year to year in the calibration procedure. Consequently, two sets of average model parameters corresponding to dormant and vegetation seasons are proposed. The process of splitting is strongly supported by the experimental data, and it enables us to variate saturated hydraulic conductivity and pore‐size characterisation. The use of the two different parameter sets significantly enhances the simulation of two (Teuling and Troch model and soil water balance model‐green–ampt [SWBM‐GA]) out of three models in the 6‐year period from 2009 to 2014. For these two models, the overall Nash‐Sutcliffe coefficient increased from 0.64 to 0.79 and from 0.55 to 0.80. The third model (the Laio approach) proved to be insensitive to parameter changes due to its insufficient drainage prediction. The variability of the warm and cold parameter sets between particular years is more pronounced in the warm periods. The cold periods exhibited approximately similar character during all 6 years.  相似文献