scholarly journals Explicit Semianalytical Theory of Asteroid Motion

1999 ◽  
Vol 172 ◽  
pp. 87-96
Author(s):  
I.V. Tupikova ◽  
A.A. Vakhidov ◽  
M. Soffel
Keyword(s):  
Closed Form ◽  
Explicit Form ◽  
Minor Planets ◽  
Mean Motion Resonances ◽  
Resonant Case ◽  
Proper Elements ◽  
The Mean ◽  
The Masses ◽  
Three Body ◽  
Asteroid Orbit

AbstractA new semianalytical theory of asteroid motion is presented. The theory is developed on the basis of Kaula’s expansion of the disturbing function including terms up to the second order with respect to the masses of disturbing bodies. The theory is constructed in explicit form that gives the possibility to study separately the influence of different perturbations in the dynamics of minor planets. The mean-motion resonances with major planets as well as mixed three-body resonances can also be taken into account. For the non-resonant case the formulas obtained can be used for deriving the second transformation to calculate the proper elements of an asteroid orbit in closed form with respect to inclinations and eccentricities.

scholarly journals Numerical mean elements for asteroid orbits

1996 ◽  
Vol 172 ◽  
pp. 165-170 ◽  
Author(s):  
Anne Lemaitre
Keyword(s):  
Body Problem ◽  
Minor Planets ◽  
Proper Elements ◽  
N Body Problem ◽  
Double Averaging ◽  
The Mean ◽  
Osculating Elements ◽  
Mean Elements ◽  
Asteroid Families

In the context of the search of asteroid families (i.e. identification of minor planets as potential fragments of an old bigger body), the calculation of proper elements plays an important role. They are quasi-invariants of the motion, obtained by a double averaging process of the restricted N-body problem; firstly the osculating elements are averaged over the short periodic terms (namely the longitudes of the asteroid and of the perturbing planets) so to get the mean elements, and secondly, the mean elements are averaged over the long periodic terms (longitudes of the pericenters and of the nodes of the asteroid and of the perturbing planets) to obtain the proper elements.

scholarly journals Resonance libration and width at arbitrary inclination

2020 ◽  
Vol 493 (2) ◽  
pp. 2854-2871
Author(s):  
F Namouni ◽  
M H M Morais
Keyword(s):  
Analytical Tool ◽  
Resonance Width ◽  
Disturbing Function ◽  
Three Body Problem ◽  
Mean Motion Resonances ◽  
Resonance Modes ◽  
Pendulum Model ◽  
Powerful Analytical Tool ◽  
The Mean ◽  
Three Body

ABSTRACT We apply the analytical disturbing function for arbitrary inclination derived in our previous work to characterize resonant width and libration of mean motion resonances at arbitrary inclination obtained from direct numerical simulations of the three-body problem. We examine the 2:1 and 3:1 inner Jupiter and 1:2 and 1:3 outer Neptune resonances and their possible asymmetric librations using a new analytical pendulum model of resonance that includes the simultaneous libration of multiple arguments and their second harmonics. The numerically derived resonance separatrices are obtained using the mean exponential growth factor of nearby orbits (megno chaos indicator). We find that the analytical and numerical estimates are in agreement and that resonance width is determined by the first few fundamental resonance modes that librate simultaneously on the resonant time-scale. Our results demonstrate that the new pendulum model may be used to ascertain resonance width analytically, and more generally, that the disturbing function for arbitrary inclination is a powerful analytical tool that describes resonance dynamics of low as well as high inclination asteroids in the Solar system.

scholarly journals Mean Motion Resonances in The Asteroid Belt

1999 ◽  
Vol 172 ◽  
pp. 381-382
Author(s):  
D. Nesvorný ◽  
A. Morbidelli
Keyword(s):  
Asteroid Belt ◽  
Mean Motion Resonances ◽  
Mean Motion ◽  
Lyapunov Time ◽  
Asteroidal Belt ◽  
Kirkwood Gaps ◽  
The Mean ◽  
Three Body

The Kirkwood gaps in the main asteroidal belt (2 – 3.5 AU) coincide with the mean motion resonances with Jupiter (4/1, 3/1, 5/2, 7/3, 2/1). Similarly, several narrower gaps are observed in the outer asteroid belt (3.5 – 4 AU) at places of 11/6, 9/5, 7/4 and 5/3 Jovian resonances (Holman and Murray 1996). As it is now generally accepted, the formation and preservation of these gaps is due to the chaos of the resonant space and efficient ejection of the primordial and collisionaly injected bodies towards high eccentricities and planet-crossing orbits.The Jovian mean motion resonances are not the most important in what concerns the chaos of the observed (i.e. remaining) asteroid population. It was estimated by Šidlichovský and Nesvorný (1998) that about 40% of known objects have the Lyapunov time less than 105 years. It was later found (Nesvorný and Morbidelli 1998, 1999; Morbidelli and Nesvorný 1999) that the resonances responsible for this chaos are, in decreasing order of importance: 1) three-body resonances with Jupiter and Saturn, 2) exterior resonances with Mars, 3) moderate order Jovian resonances, and 4) three-body resonances with Mars and Jupiter.

scholarly journals Asteroid Masses and Densities

1971 ◽  
Vol 12 ◽  
pp. 33-39 ◽  
Author(s):  
Joachim Schubart
Keyword(s):  
Total Mass ◽  
Direct Determination ◽  
Absolute Magnitude ◽  
Strong Increase ◽  
Minor Planets ◽  
Solar Mass ◽  
A Value ◽  
The Mean ◽  
The Masses

Before 1966, when Hertz (1966) published his first direct determination of the mass of Vesta, all our knowledge on asteroid masses was based on estimates. The masses of the first four minor planets resulted from the measured diameters by Barnard (1900) (see the paper by Dollfus in this volume) and from estimated mean densities. The diameters of the smaller objects were derived from their brightness and an estimate of their reflectivity (usually the reflectivity of the Moon was adopted). In 1901, Bauschinger and Neugebauer (1901) derived a value for the total mass of the first 458 asteroids. All the diameters were computed from the brightness with an assumed value for the reflectivity. The diameter of Ceres found in this way is very close to Barnard’s (1900) value. The mean density of the 458 asteroids was put equal to that of Earth, and their total mass resulted as 3 X 10-9 solar mass. Stracke (1942) used the same method with an increased material, but the addition of more than 1000 faint asteroids did not bring a significant change in the estimate of the total mass. The report on the McDonald asteroid survey (Kuiper et al., 1958) does not contain another estimate of the total mass of the asteroid ring, but it points to the possibility of a very rapid increase in the number of asteroids with decreasing absolute brightness. If this increase is strong enough, each interval of 1 mag in absolute magnitude can contribute the same amount to the total mass. In the range of magnitudes covered by the Palomar-Leiden survey (PLS) (van Houten et al., 1970), there are no indications for such a strong increase.

scholarly journals Long-term evolution of the Galilean satellites: the capture of Callisto into resonance

2020 ◽  
Vol 639 ◽  
pp. A40 ◽  
Author(s):  
Giacomo Lari ◽  
Melaine Saillenfest ◽  
Marco Fenucci
Keyword(s):  
Galilean Satellites ◽  
Tidal Dissipation ◽  
Mean Motion Resonances ◽  
Tidal Forces ◽  
Recent Estimate ◽  
The Mean ◽  
A Chain ◽  
The Stability ◽  
Three Body

Context. The Galilean satellites have very complex orbital dynamics due to the mean-motion resonances and the tidal forces acting in the system. The strong dissipation in the couple Jupiter–Io is spread to all the moons involved in the so-called Laplace resonance (Io, Europa, and Ganymede), leading to a migration of their orbits. Aims. We aim to characterize the future behavior of the Galilean satellites over the Solar System lifetime and to quantify the stability of the Laplace resonance. Tidal dissipation permits the satellites to exit from the current resonances or be captured into new ones, causing large variation in the moons’ orbital elements. In particular, we want to investigate the possible capture of Callisto into resonance. Methods. We performed hundreds of propagations using an improved version of a recent semi-analytical model. As Ganymede moves outwards, it approaches the 2:1 resonance with Callisto, inducing a temporary chaotic motion in the system. For this reason, we draw a statistical picture of the outcome of the resonant encounter. Results. The system can settle into two distinct outcomes: (A) a chain of three 2:1 two-body resonances (Io–Europa, Europa–Ganymede, and Ganymede–Callisto), or (B) a resonant chain involving the 2:1 two-body resonance Io–Europa plus at least one pure 4:2:1 three-body resonance, most frequently between Europa, Ganymede, and Callisto. In case A (56% of the simulations), the Laplace resonance is always preserved and the eccentricities remain confined to small values below 0.01. In case B (44% of the simulations), the Laplace resonance is generally disrupted and the eccentricities of Ganymede and Callisto can increase up to about 0.1, making this configuration unstable and driving the system into new resonances. In all cases, Callisto starts to migrate outward, pushed by the resonant action of the other moons. Conclusions. From our results, the capture of Callisto into resonance appears to be extremely likely (100% of our simulations). The exact timing of its entrance into resonance depends on the precise rate of energy dissipation in the system. Assuming the most recent estimate of the dissipation between Io and Jupiter, the resonant encounter happens at about 1.5 Gyr from now. Therefore, the stability of the Laplace resonance as we know it today is guaranteed at least up to about 1.5 Gyr.

On nearly-commensurable periods in the restricted problem of three bodies, with calculations of the long-period variations in the interior 2:1 case

1966 ◽  
Vol 25 ◽  
pp. 197-222 ◽  
Author(s):  
P. J. Message
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Large Amplitude ◽  
Restricted Problem ◽  
Small Amplitude ◽  
Minor Planets ◽  
Long Period ◽  
Numerical Integrations ◽  
Period Variations ◽  
The Mean

An analytical discussion of that case of motion in the restricted problem, in which the mean motions of the infinitesimal, and smaller-massed, bodies about the larger one are nearly in the ratio of two small integers displays the existence of a series of periodic solutions which, for commensurabilities of the typep+ 1:p, includes solutions of Poincaré'sdeuxième sortewhen the commensurability is very close, and of thepremière sortewhen it is less close. A linear treatment of the long-period variations of the elements, valid for motions in which the elements remain close to a particular periodic solution of this type, shows the continuity of near-commensurable motion with other motion, and some of the properties of long-period librations of small amplitude.To extend the investigation to other types of motion near commensurability, numerical integrations of the equations for the long-period variations of the elements were carried out for the 2:1 interior case (of which the planet 108 “Hecuba” is an example) to survey those motions in which the eccentricity takes values less than 0·1. An investigation of the effect of the large amplitude perturbations near commensurability on a distribution of minor planets, which is originally uniform over mean motion, shows a “draining off” effect from the vicinity of exact commensurability of a magnitude large enough to account for the observed gap in the distribution at the 2:1 commensurability.

scholarly journals End-to-End Data Rate Performance of Decode-and-Forward Relaying with Different Resource Allocation Schemes

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Inam Ullah ◽  
Alexis Dowhuszko ◽  
Zhong Zheng ◽  
David González González ◽  
Jyri Hämäläinen
Keyword(s):  
Resource Allocation ◽  
Closed Form ◽  
Data Rate ◽  
Allocation Scheme ◽  
Optimal Resource Allocation ◽  
Decode And Forward ◽  
Optimal Resource ◽  
The Mean ◽  
Resource Allocation Scheme ◽  
Resource Allocation Schemes

This paper studies the end-to-end (e2e) data rate of dual-hop Decode-and-Forward (DF) infrastructure relaying under different resource allocation schemes. In this context, we first provide a comparative analysis of the optimal resource allocation scheme with respect to several other approaches in order to provide insights into the system behavior and show the benefits of each alternative. Then, assuming the optimal resource allocation, a closed form expression for the distribution of the mean and outage data rates is derived. It turns out that the corresponding mean e2e data rate formula attains an expression in terms of an integral that does not admit a closed form solution. Therefore, a tight lower bound formula for the mean e2e data rate is presented. Results can be used to select the most convenient resource allocation scheme and perform link dimensioning in the network planning phase, showing the explicit relationships that exist between component link bandwidths, SNR values, and mean data rate.

Leakage Prediction in Mechanical Seals Under Hydrostatic Operating Condition

2007 ◽  
Author(s):  
S. Elhanafi ◽  
K. Farhang
Keyword(s):  
Closed Form ◽  
Mass Flow Rate ◽  
Mass Flow ◽  
Flow Rate ◽  
Rough Surfaces ◽  
Mechanical Seals ◽  
Operating Condition ◽  
Form Equation ◽  
Macro Scale ◽  
The Mean

This paper considers leakage in mechanical seals under hydrostatic operating condition. A contact model based on the Greenwood and Williamson contact of rough surfaces is developed for treating problems involving mechanical seals in which both the micron scale roughness of the seal face and its macro scale profile are used to obtain either a closed-form equation or a nonlinear equation relating mean plane separation to the mass flow rate. The equations involve the micron scale geometry of the rough surfaces and physical parameter of the seal and carriage. Under hydrostatic condition, it is shown that there is an approximate closed-form solution in which mass flow rate in terms of the mean plane separation, or alternatively, the mean plane separation in terms of the leakage mass flow rate is found. Equations pertaining to leakage in nominally flat seal macro profile is considered and closed form equation relating to leakage flow rate to pressure difference is obtained that contain macro and micron geometries of the seal.

PARTON RECOMBINATION IN CLASSICAL CASCADE

1990 ◽  
Vol 05 (28) ◽  
pp. 2377-2383 ◽  
Author(s):  
A. V. BATUNIN ◽  
O. P. YUSHCHENKO
Keyword(s):  
Explicit Form ◽  
Heavy Ion Collisions ◽  
Charged Particles ◽  
Heavy Ion ◽  
Considerable Decrease ◽  
High Energies ◽  
Ion Collisions ◽  
Parton Cascade ◽  
Energy Formula ◽  
The Mean

An equation for parton multiplicity in cascade with the recombination 1 → 2 ⊕ 2 → 1 is derived from a Kolmogorov-Chapman equation and solved. An evolution parameter τ of the cascade depends on the c.m. energy [Formula: see text]; an explicit form of the dependence is obtained from the condition that the mean multiplicity of charged particles in pp, [Formula: see text] collisions be reproduced. A considerable decrease in the mean multiplicity in heavy-ion collisions per pair of the colliding nucleons at high energies is predicted and compared to the parton cascade with no recombination.

Coupled dynamics of populations supported by discrete sites and their continuum limit

2010 ◽  
Vol 466 (2121) ◽  
pp. 2561-2586 ◽  
Author(s):  
Timothy R. Field ◽  
Robert J. A. Tough
Keyword(s):  
Evolution Equation ◽  
Closed Form ◽  
Generating Function ◽  
Rate Equation ◽  
Continuum Limit ◽  
Infinite Chain ◽  
Migration Process ◽  
The Mean ◽  
The Continuum ◽  
Birth Death

The illumination of single population behaviour subject to the processes of birth, death and immigration has provided a basis for the discussion of the non-Gaussian statistical and temporal correlation properties of scattered radiation. As a first step towards the modelling of its spatial correlations, we consider the populations supported by an infinite chain of discrete sites, each subject to birth, death and immigration and coupled by migration between adjacent sites. To provide some motivation, and illustrate the techniques we will use, the migration process for a single particle on an infinite chain of sites is introduced and its diffusion dynamics derived. A certain continuum limit is identified and its properties studied via asymptotic analysis. This forms the basis of the multi-particle model of a coupled population subject to single site birth, death and immigration processes, in addition to inter-site migration. A discrete rate equation is formulated and its generating function dynamics derived. This facilitates derivation of the equations of motion for the first- and second-order cumulants, thus generalizing the earlier results of Bailey through the incorporation of immigration at each site. We present a novel matrix formalism operating in the time domain that enables solution of these equations yielding the mean occupancy and inter-site variances in the closed form. The results for the first two moments at a single time are used to derive expressions for the asymptotic time-delayed correlation functions, which relates to Glauber’s analysis of an Ising model. The paper concludes with an analysis of the continuum limit of the birth–death–immigration–migration process in terms of a path integral formalism. The continuum rate equation and evolution equation for the generating function are developed, from which the evolution equation of the mean occupancy is derived, in this limit. Its solution is provided in closed form.

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