On the normalization of perturbed Keplerian systems

Normal Forms for Perturbed Keplerian Systems

Journal of Differential Equations ◽

10.1006/jdeq.2001.4068 ◽

2002 ◽

Vol 180 (2) ◽

pp. 471-519 ◽

Cited By ~ 16

Author(s):

Jesús Palacián

Keyword(s):

Normal Forms ◽

Keplerian Systems

Reduced 4D oscillators and orbital elements in Keplerian systems: Cushman–Deprit coordinates

Celestial Mechanics and Dynamical Astronomy ◽

10.1007/s10569-020-09995-z ◽

2020 ◽

Vol 132 (11-12) ◽

Author(s):

S. Ferrer ◽

F. Crespo ◽

J. L. Zapata

Keyword(s):

Orbital Elements ◽

Keplerian Systems

Mission design through averaging of perturbed Keplerian systems: the paradigm of an Enceladus orbiter

Celestial Mechanics and Dynamical Astronomy ◽

10.1007/s10569-010-9286-2 ◽

2010 ◽

Vol 108 (1) ◽

pp. 1-22 ◽

Cited By ~ 19

Author(s):

Martín Lara ◽

Jesús F. Palacián ◽

Ryan P. Russell

Keyword(s):

Mission Design ◽

Keplerian Systems

Coordinates for perturbed keplerian systems with axial symmetry

Celestial Mechanics and Dynamical Astronomy ◽

10.1007/bf00049358 ◽

1992 ◽

Vol 53 (1) ◽

Cited By ~ 1

Author(s):

Sebasti�n Ferrer ◽

BruceR. Miller

Keyword(s):

Axial Symmetry ◽

Keplerian Systems

The secular evolution of discrete quasi-Keplerian systems

Astronomy and Astrophysics ◽

10.1051/0004-6361/201731088 ◽

2018 ◽

Vol 609 ◽

pp. A38 ◽

Cited By ~ 2

Author(s):

J.-B. Fouvry ◽

C. Pichon ◽

P.-H. Chavanis

Keyword(s):

Black Hole ◽

Landau Equation ◽

Collective Effects ◽

Galactic Centre ◽

Central Black Hole ◽

Massive Black Hole ◽

Secular Evolution ◽

Gauss Method ◽

And Diffusion ◽

Keplerian Systems

A discrete self-gravitating quasi-Keplerian razor-thin axisymmetric stellar disc orbiting a massive black hole sees its orbital structure diffuse on secular timescales as a result of a self-induced resonant relaxation. In the absence of collective effects, such a process is described by the recently derived inhomogeneous multi-mass degenerate Landau equation. Relying on Gauss’ method, we computed the associated drift and diffusion coefficients to characterise the properties of the resonant relaxation of razor-thin discs. For a disc-like configuration in our Galactic centre, we showed how this secular diffusion induces an adiabatic distortion of orbits and estimate the typical timescale of resonant relaxation. When considering a disc composed of multiple masses similarly distributed, we have illustrated how the population of lighter stars will gain eccentricity, driving it closer to the central black hole, provided the distribution function increases with angular momentum. The kinetic equation recovers as well the quenching of the resonant diffusion of a test star in the vicinity of the black hole (the “Schwarzschild barrier”) as a result of the divergence of the relativistic precessions. The dual stochastic Langevin formulation yields consistent results and offers a versatile framework in which to incorporate other stochastic processes.

Alternative Reduction by Stages of Keplerian Systems. Positive, Negative, and Zero Energy

SIAM Journal on Applied Dynamical Systems ◽

10.1137/19m1264060 ◽

2020 ◽

Vol 19 (2) ◽

pp. 1525-1539 ◽

Cited By ~ 1

Author(s):

Francisco Crespo ◽

Sebastián Ferrer

Keyword(s):

Zero Energy ◽

Keplerian Systems

Fractal geometry of angular momentum evolution in near-Keplerian systems

Monthly Notices of the Royal Astronomical Society Letters ◽

10.1111/j.1745-3933.2010.00994.x ◽

2010 ◽

Vol 411 (1) ◽

pp. L56-L60 ◽

Cited By ~ 1

Author(s):

M. Atakan Gürkan

Keyword(s):

Angular Momentum ◽

Fractal Geometry ◽

Keplerian Systems

Simplifications toward Integrability of Perturbed Keplerian Systems

Annals of the New York Academy of Sciences ◽

10.1111/j.1749-6632.1988.tb51569.x ◽

1988 ◽

Vol 536 (1 Integrability) ◽

pp. 127-139 ◽

Cited By ~ 2

Author(s):

SEBASTIAN FERRER ◽

CAROL A. WILLIAMS

Keyword(s):

Keplerian Systems

An improved and generalized theory for the collisional evolution of Keplerian systems

Astrophysics and Space Science ◽

10.1007/bf00644532 ◽

1978 ◽

Vol 58 (2) ◽

pp. 477-519 ◽

Cited By ~ 36

Author(s):

K. A. H�meen-Anttila

Keyword(s):

Keplerian Systems ◽

Collisional Evolution

A new method of calculating the mean value of 1/ r 3 for keplerian systems in quantum mechanics

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character ◽

10.1098/rspa.1934.0027 ◽

1934 ◽

Vol 143 (850) ◽

pp. 679-681 ◽

Cited By ~ 28

Keyword(s):

Quantum Mechanics ◽

Diagonal Element ◽

Mean Value ◽

Usual Method ◽

Matrix Elements ◽

Generating Series ◽

The Mean ◽

Schrödinger Form ◽

Interesting Alternative ◽

Keplerian Systems

The usual method of calculating the diagonal matrix elements of an integral power of the radius r in an inverse square quantum system is that due to Waller. His procedure is based on the Schrödinger form of the theory, and utilizes in particular the generating series for the Langueree polynomials. For negative powers, Dirac's very elegant theory of "q-numbers," developed in these 'proceedings' during the early days of quantum mechanics, furnishes an interesting alternative method which appears to have been overlooked, and which we believe is easier. From his theory the following rule can be derived: Suppose that we desire the mean value (diagonal element) of 1/r s , where s is an integer greater than unity. We write down the experssion with A(α l + β l e - i x + γ l e i x ) s-2 A = 16π 4 m 2 z 2 e 4 /( l + ½) n 3 h 4 , α l = 4π 2 m z e 2 / h 2 l ( l +1), β l = 2π 2 m Z e 2 / h 2 ( l +½)( l +1)[1-( l +1) 2 / n 2 ]½, γ l = 2π 2 m z e 2 / h 2 l ( l +½)[1- l 2 / n 2 ]½.