scholarly journals Existence of a Smooth Hamiltonian Circle Action near Parabolic Orbits and Cuspidal Tori

2021 ◽  
Vol 26 (6) ◽  
pp. 732-741
Author(s):  
Elena A. Kudryavtseva ◽  
Nikolay N. Martynchuk
Keyword(s):  
Circle Action ◽  
Parabolic Orbits

scholarly journals DESTINY: Database for the Effects of STellar encounters on dIsks and plaNetary sYstems

2019 ◽  
Vol 6 (1) ◽  
Author(s):  
Asmita Bhandare ◽  
Susanne Pfalzner
Keyword(s):  
Planetary System ◽  
Planetary Systems ◽  
Gravitational Effect ◽  
Parameter Study ◽  
Orbital Inclination ◽  
Particle Properties ◽  
Disk Size ◽  
Stellar Group ◽  
Cluster Environment ◽  
Parabolic Orbits

Abstract Most stars form as part of a stellar group. These young stars are mostly surrounded by a disk from which potentially a planetary system might form. Both, the disk and later on the planetary system, may be affected by the cluster environment due to close fly-bys. The here presented database can be used to determine the gravitational effect of such fly-bys on non-viscous disks and planetary systems. The database contains data for fly-by scenarios spanning mass ratios between the perturber and host star from 0.3 to 50.0, periastron distances from 30 au to 1000 au, orbital inclination from 0∘ to 180∘ and angle of periastron of 0∘, 45∘ and 90∘. Thus covering a wide parameter space relevant for fly-bys in stellar clusters. The data can either be downloaded to perform one’s own diagnostics like for e.g. determining disk size, disk mass, etc. after specific encounters, obtain parameter dependencies or the different particle properties can be visualized interactively. Currently the database is restricted to fly-bys on parabolic orbits, but it will be extended to hyperbolic orbits in the future. All of the data from this extensive parameter study is now publicly available as DESTINY.

Circle Actions

2020 ◽  
pp. 161-166
Author(s):  
Loring W. Tu
Keyword(s):  
Differential Form ◽  
Circle Action ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Circle Actions ◽  
Algebra Isomorphism ◽  
Weil Algebra ◽  
Differential Graded Algebras ◽  
Cartan Model

This chapter focuses on circle actions. Specifically, it specializes the Weil algebra and the Weil model to a circle action. In this case, all the formulas simplify. The chapter derives a simpler complex, called the Cartan model, which is isomorphic to the Weil model as differential graded algebras. It considers the theorem that for a circle action, there is a graded-algebra isomorphism. Under the isomorphism F, the Weil differential δ‎ corresponds to a differential called the Cartan differential. An element of the Cartan model is called an equivariant differential form or equivariant form for a circle action on the manifold M.

scholarly journals Characteristics of Single Encounters of Long-Periodic Comets With Jupiter

1979 ◽  
Vol 81 ◽  
pp. 299-301
Author(s):  
Tsuko Nakamura
Keyword(s):  
Random Walk ◽  
Probability Distributions ◽  
Numerical Integrations ◽  
Elliptic Orbits ◽  
Parabolic Orbits ◽  
Statistical Results

Original nearly parabolic orbits of comets are known to be evolved toward short-periodic elliptic orbits as statistical results of hundreds of encounters with Jupiter. There seems to be two methods to handle the process, namely, the method by exact numerical integrations for each orbit (Everhart, 1972) and random walk approach by using probability distributions of perturbations after single encounters (Lyttleton and Hammersley, 1963; Shteins, 1972). Since both methods need a great number of input parabolic comets to have only a few tens of short-periodic ones, the second method may save time compared with the first one, which is in turn more accurate. The purpose of this paper is to clarify the characteristics of single-encounter effects, in order to develope the second method more elaborately and extensively.

scholarly journals On the Differential Correction of Nearly Parabolic Orbits

1972 ◽  
Vol 45 ◽  
pp. 123-123
Author(s):  
P. Herget
Keyword(s):  
Crucial Point ◽  
Partial Differential ◽  
Differential Correction ◽  
Parabolic Orbits

The differential correction of nearly parabolic orbits was discussed by the author (Herget, 1939) in the era of lead pencil computing. The Gauss-Marth method is the best one to use whenever the appropriate conditions exist, i.e., |E| < 64° and e nearly unity. The crucial point in the above-cited discussion is the use of the first differences from the Gauss-Marth tables in order to simplify the computation of the partial differential coefficients, namely dB/dA, dC/dA, and dD/dA.

scholarly journals On the Chern numbers and the Hilbert polynomial of an almost complex manifold with a circle action

2017 ◽  
Vol 19 (04) ◽  
pp. 1750043 ◽  
Author(s):  
Silvia Sabatini
Keyword(s):  
Fixed Points ◽  
Complex Manifold ◽  
Index Formula ◽  
Circle Action ◽  
Hilbert Polynomial ◽  
Dimensional Manifold ◽  
Hamiltonian Action ◽  
Almost Complex Manifold ◽  
Almost Complex ◽  
Chern Numbers

Let [Formula: see text] be a compact, connected, almost complex manifold of dimension [Formula: see text] endowed with a [Formula: see text]-preserving circle action with isolated fixed points. In this paper, we analyze the “geography problem” for such manifolds, deriving equations relating the Chern numbers to the index [Formula: see text] of [Formula: see text]. We study the symmetries and zeros of the Hilbert polynomial, which imply many rigidity results for the Chern numbers when [Formula: see text]. We apply these results to the category of compact, connected symplectic manifolds. A long-standing question posed by McDuff and Salamon asked about the existence of non-Hamiltonian actions with isolated fixed points. This question was answered recently by Tolman, with an explicit construction of a 6-dimensional manifold with such an action. One issue that this raises is whether one can find topological criteria that ensure the manifold can only support a Hamiltonian or only a non-Hamiltonian action. In this vein, we are able to deduce such criteria from our rigidity theorems in terms of relatively few Chern numbers, depending on the index. Another consequence is that, if the action is Hamiltonian, the minimal Chern number coincides with the index and is at most [Formula: see text].

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