scholarly journals Survey of secular resonances in the asteroid belt

2021 ◽  
pp. 4-4
Author(s):  
Z. Knezevic
Keyword(s):  
Asteroid Belt ◽  
Secular Resonance ◽  
Secular Resonances ◽  
Mean Motion Resonances ◽  
Fundamental Frequencies ◽  
Cycle Slips ◽  
Polynomial Fit ◽  
Special Cases ◽  
Order Four ◽  
Future Work

Using a recently introduced synthetic method to compute the asteroid secular frequencies (Knezevic and Milani 2019), in this paper we survey the locations of secular resonances in the 9 dynamically distinct zones of the asteroid belt. Positions of all resonances up to order four, of a significant fraction of the order six resonances, and of a several order eight ones were determined, plotted in the space of proper elements, and discussed in relation to the local dynamics and to the structure and shape of the nearby asteroid collisional families. Only the resonant combinations with fundamental frequencies of Jupiter and Saturn were considered, with a few special cases involving other planets and largest asteroids. Accuracy of the polynomial fit to determine the frequencies was found to be satisfactory for the purpose of determination of secular resonance positions. This enabled a precise identification of dynamical mechanisms affecting the computation of frequencies (close vicinity of the mean motion resonances and libration in secular resonances), and of the \cycle slips" as a primary technical drawback causing deterioration of the results. For each zone we also presented and discussed a fairly complete sample of recent works dealing with interaction of the secular resonances with asteroid families present in that zone. Finally, a few words were devoted to possibilities for future work.

scholarly journals The Secular Resonances in the Solar System

1994 ◽  
Vol 160 ◽  
pp. 189-204
Author(s):  
Christiane Froeschle ◽  
Alessandro Morbidelli
Keyword(s):  
Solar System ◽  
Secular Resonance ◽  
Small Bodies ◽  
Fourth Degree ◽  
Secular Resonances ◽  
Mean Motion Resonances ◽  
Non Linear ◽  
Resonant Dynamics ◽  
Dynamical Nature ◽  
Element Algorithm

In the last three years new studies on secular resonances have been done. The second–order and fourth–degree secular perturbation theory of Milani and Knežević allowed to point out the effect of mean motion resonances on the location of the linear and non linear secular resonances. Moreover this theory improved the knowledge of the exact location of the g = g6 (i.e. ν6) resonance at low inclination. Morbidelli and Henrard revisited the semi–numerical method of Williams, taking into account the quadratic terms in the perturbing masses. They computed not only the location of secular resonances, but also provided a global description of the resonant dynamics in the main secular resonances namely g = g5 (i.e. ν5), g = g6 (i.e. ν6) and s = s6 (i.e. ν16). The resonant proper element algorithm developed by Morbidelli allows to identify the dynamical nature of resonant objects, and is a powerful tool to study the mechanisms of meteorite transport to the inner Solar System. Purely numerical experiments have been done, which show: (i) the complexity of the dynamics when two resonances overlap; (ii) the efficiency of successive crossings of non linear resonances in pumping up the inclination of small bodies; (iii) the efficiency of the secular resonance ν6 as a source of meteorites up to 2.4 AU.

On the accuracy of position of the linear secular resonance g − g5 in the proper elements’ space

2020 ◽  
Vol 497 (4) ◽  
pp. 4921-4936
Author(s):  
Zoran Knežević
Keyword(s):  
Resonant State ◽  
Resonance Position ◽  
Secular Resonance ◽  
New Approach ◽  
Cycle Slips ◽  
Proper Elements ◽  
Polynomial Fit ◽  
Depth Analysis ◽  
Proper Values

ABSTRACT An in-depth analysis is presented of the accuracy of position of the linear secular resonance g − g5 in the phase space of proper elements, as determined by the recently introduced polynomial fit method. Different attempts to pinpoint the exact location of this resonance are described, leading to improvement in the accuracy of resonance position achieved via local adjustments of the new method and measured in comparison with the corresponding positions of selected asteroids. The resonant state and proper frequencies of the longitude of perihelion of these asteroids are determined, and compared to the catalogue values computed in the course of determination of their synthetic proper elements. The problem of cycle slips, affecting the computation of frequencies, is thoroughly examined and successfully explained, and the procedure of double filtering of the time series of proper values to remove the cycle slips proposed. The results of tests of the new approach have shown that the accuracy of newly determined frequencies is significantly improved with respect to the previously available values.

scholarly journals Asteroid Proper Elements: The Big Picture

1994 ◽  
Vol 160 ◽  
pp. 143-158 ◽  
Author(s):  
Zoran Knežević ◽  
Andrea Milani
Keyword(s):  
Point Of View ◽  
Dynamical Structure ◽  
Main Belt ◽  
Secular Resonances ◽  
Mean Motion Resonances ◽  
Mean Motion ◽  
Proper Elements ◽  
Big Picture ◽  
Linear And Nonlinear ◽  
Asteroid Families

Four perturbation theories presently used to compute asteroid proper elements are reviewed, and their results are briefly discussed (Milani and Knežević, 1990, 1992, 1994, for low to moderate eccentricity/inclination main belt objects; Lemaitre and Morbidelli, 1994, for high e, I objects; Milani, 1993, for Trojans; Schubart, 1982, 1991 for Hildas). The most important recent improvements are described, in particular those pertaining to the upgrades of the previous analytic and semianalytic solutions. The dynamical structure of the asteroid main belt, as defined by the low order mean motion resonances and by linear and nonlinear secular resonances, is considered from the point of view of the effects of these resonances on the accuracy and/or reliability of the computation of proper elements and on the reliability of the identification of asteroid families.

scholarly journals A Hirota bilinear equation for Painlevé transcendents PIV, PII and PI

2018 ◽  
Vol 07 (04) ◽  
pp. 1840001
Author(s):  
A. N. W. Hone ◽  
F. Zullo
Keyword(s):  
Taylor Series ◽  
Taylor Expansion ◽  
Tau Function ◽  
Tau Functions ◽  
Sigma Function ◽  
Special Cases ◽  
Hirota Bilinear Equation ◽  
Order Four ◽  
Hirota Bilinear ◽  
Bilinear Equation

We present some observations on the tau-function for the fourth Painlevé equation. By considering a Hirota bilinear equation of order four for this tau-function, we describe the general form of the Taylor expansion around an arbitrary movable zero. The corresponding Taylor series for the tau-functions of the first and second Painlevé equations, as well as that for the Weierstrass sigma function, arise naturally as special cases, by setting certain parameters to zero.

scholarly journals The Distribution of Asteroidal Dust in the Inner Solar System

2004 ◽  
Vol 202 ◽  
pp. 184-186
Author(s):  
Keith Grogan ◽  
S.F. Dermott ◽  
T.J.J. Kehoe
Keyword(s):  
Solar System ◽  
Parent Body ◽  
Asteroid Belt ◽  
Dust Particles ◽  
Secular Resonances ◽  
Level Of Confidence ◽  
Asteroidal Dust

In this paper we demonstrate how the action of secular resonances near the inner edge of the asteroid belt strongly effects the inclinations and eccentricities of asteroidal dust particles, such that they lose the orbital characteristics of their parent body and are dispersed into the zodiacal background. As a consequence, it may not be possible to relate the distribution of interplanetary material at 1 AU to given asteroidal or cometary sources with the level of confidence previously imagined.

The Solar Nebula, Secular Resonances, Gas Drag, and the Asteroid Belt

Icarus ◽  
1997 ◽  
Vol 129 (1) ◽  
pp. 134-146 ◽  
Author(s):  
Myron Lecar ◽  
Fred Franklin
Keyword(s):  
Solar Nebula ◽  
Asteroid Belt ◽  
Secular Resonances ◽  
Gas Drag

scholarly journals The Chaotic Motion of the Solar System

1993 ◽  
Vol 132 ◽  
pp. 21-21
Author(s):  
J. Laskar
Keyword(s):  
Lyapunov Exponent ◽  
Solar System ◽  
Initial Conditions ◽  
Maximum Lyapunov Exponent ◽  
Secular Resonance ◽  
Critical Argument ◽  
Outer Planets ◽  
Fundamental Frequencies ◽  
Chaotic Nature ◽  
Evolution With Time

AbstractIn a previous paper (Laskar, Nature, 338, 237-238), the chaotic nature of the solar system excluding Pluto was established by the numerical computation of the maximum Lyapunov exponent of its secular system over 200 Myr. In the present an explanation is given for the exponential divergence of the orbits: it is due to the transition from libration to circulation of the critical argument of the secular resonance 2(g4−g3)−(s4−s3) related to the motions of perihelions and nodes of the Birth and Mars. An other important secular resonance is identified: (g1−g5)−(s1−s2). Its critical argument stays in libration over 200 Myr with a period of about 10 Myr and amplitude from 85° to 135°. The main features of the solutions of the inner planets are now identified when taking these resonances into account. Estimates of the size of the chaotic regions are determined by a new numerical method using the evolution with time of the fundamental frequencies. The size of the chaotic regions in the inner solar system are large and correspond to variations of about 0.2 arcsec/year in the fundamental frequencies. The chaotic nature of the inner solar system can thus be considered as robust against small variations of the initial conditions or of the model. The chaotic regions related to the outer planets frequencies are very thin except for g6 which present variations sufficiently large to be significant over the age of the solar system.

scholarly journals The Locations of Secular Resonances and the Evolution of Small Solar System Bodies

1992 ◽  
Vol 152 ◽  
pp. 123-132
Author(s):  
Ch Froeschle ◽  
P. Farinella ◽  
C. Froeschle ◽  
Z. Knežević ◽  
A. Milani
Keyword(s):  
Perturbation Theory ◽  
Solar System ◽  
Small Body ◽  
Kuiper Belt ◽  
Dynamical Evolution ◽  
Asteroid Belt ◽  
Secular Resonances ◽  
Outer Planets ◽  
Proper Elements ◽  
Solar System Bodies

Generalizing the secular perturbation theory of Milani and Knežević (1990), we have determined in the a — e — I proper elements space the locations of the secular resonances between the precession rates of the longitudes of perihelion and node of a small body and the corresponding eigenfrequencies of the secular perturbations of the four outer planets. We discuss some implications of the results for the dynamical evolution of small solar system bodies. In particular, our findings include: (i) the fact that the g = g6 resonance in the inner asteroid belt lies closer than previously assumed to the Flora region, providing a plausible dynamical route to inject asteroid fragments into planet-crossing orbits; (ii) the possible presence of some low-inclination “stable islands” between the orbits of the outer planets; (iii) the fact that none of the secular resonances considered in this work exists for semimajor axes > 50 AU, so that these resonances do not provide a mechanism for transporting inwards possible Kuiper–belt comets.

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