Secular dynamics of multiplanetary circumbinary systems: stationary solutions and binary-planet secular resonance

2018 ◽  
Vol 130 (1) ◽  
Author(s):  
Eduardo Andrade-Ines ◽  
Philippe Robutel
Keyword(s):  
Stationary Solutions ◽  
Secular Resonance ◽  
Secular Dynamics

scholarly journals A simplified model for the secular dynamics of eccentric discs and applications to planet–disc interactions

2019 ◽  
Vol 490 (3) ◽  
pp. 4353-4365 ◽  
Author(s):  
Jean Teyssandier ◽  
Dong Lai
Keyword(s):  
Evolution Equations ◽  
Giant Planet ◽  
Giant Planets ◽  
Simplified Model ◽  
Secular Resonance ◽  
Secular Dynamics ◽  
Long Time ◽  
Hydrodynamical Simulations ◽  
Self Gravity

ABSTRACT We develop a simplified model for studying the long-term evolution of giant planets in protoplanetary discs. The model accounts for the eccentricity evolution of the planets and the dynamics of eccentric discs under the influences of secular planet–disc interactions and internal disc pressure, self-gravity, and viscosity. Adopting the ansatz that the disc precesses coherently with aligned apsides, the eccentricity evolution equations of the planet–disc system reduce to a set of linearized ordinary differential equations, which allows for fast computation of the evolution of planet–disc eccentricities over long time-scales. Applying our model to ‘giant planet + external disc’ systems, we are able to reproduce and explain the secular behaviours found in previously published hydrodynamical simulations. We re-examine the possibility of eccentricity excitation (due to secular resonance) of multiple planets embedded in a dispersing disc, and find that taking into account the dynamics of eccentric discs can significantly affect the evolution of the planets’ eccentricities.

scholarly journals Secular evolution of asteroid families: the role of Ceres

2015 ◽  
Vol 10 (S318) ◽  
pp. 46-54 ◽  
Author(s):  
Bojan Novaković ◽  
Georgios Tsirvoulis ◽  
Stefano Marò ◽  
Vladimir Đošović ◽  
Clara Maurel
Keyword(s):  
Numerical Simulations ◽  
Main Belt ◽  
Secular Resonance ◽  
Secular Evolution ◽  
Secular Dynamics ◽  
Proper Elements ◽  
Post Impact ◽  
Actual Shape ◽  
Asteroid Families

AbstractWe consider the role of the dwarf planet Ceres on the secular dynamics of the asteroid main belt. Specifically, we examine the post impact evolution of asteroid families due to the interaction of their members with the linear nodal secular resonance with Ceres. First, we find the location of this resonance and identify which asteroid families are crossed by its path. Next, we summarize our results for three asteroid families, namely (1726) Hoffmeister, (1128) Astrid and (1521) Seinajoki which have irregular distributions of their members in the proper elements space, indicative of the effect of the resonance. We confirm this by performing a set of numerical simulations, showcasing that the perturbing action of Ceres through its linear nodal secular resonance is essential to reproduce the actual shape of the families.

scholarly journals Stationary Non-equilibrium Solutions for Coagulation Systems

2021 ◽  
Vol 240 (2) ◽  
pp. 809-875
Author(s):  
Marina A. Ferreira ◽  
Jani Lukkarinen ◽  
Alessia Nota ◽  
Juan J. L. Velázquez
Keyword(s):  
Power Law ◽  
Source Term ◽  
Continuous Model ◽  
Stationary Solutions ◽  
Weight Functions ◽  
Equilibrium Solutions ◽  
Coagulation Rate ◽  
Diffusion Limited Aggregation ◽  
Coagulation Equations ◽  
Non Equilibrium

AbstractWe study coagulation equations under non-equilibrium conditions which are induced by the addition of a source term for small cluster sizes. We consider both discrete and continuous coagulation equations, and allow for a large class of coagulation rate kernels, with the main restriction being boundedness from above and below by certain weight functions. The weight functions depend on two power law parameters, and the assumptions cover, in particular, the commonly used free molecular and diffusion limited aggregation coagulation kernels. Our main result shows that the two weight function parameters already determine whether there exists a stationary solution under the presence of a source term. In particular, we find that the diffusive kernel allows for the existence of stationary solutions while there cannot be any such solutions for the free molecular kernel. The argument to prove the non-existence of solutions relies on a novel power law lower bound, valid in the appropriate parameter regime, for the decay of stationary solutions with a constant flux. We obtain optimal lower and upper estimates of the solutions for large cluster sizes, and prove that the solutions of the discrete model behave asymptotically as solutions of the continuous model.

Periodic defect wall structures in irradiated solids: The spectrum of stationary solutions

1989 ◽  
Vol 40 (18) ◽  
pp. 12531-12534 ◽  
Author(s):  
H. Trinkaus ◽  
C. Abromeit ◽  
J. Villain
Keyword(s):  
Stationary Solutions ◽  
Wall Structures

Any large solution of a non-linear heat conduction equation becomes monotonic in time

1991 ◽  
Vol 118 (1-2) ◽  
pp. 13-20 ◽  
Author(s):  
Victor A. Galaktionov ◽  
Sergey A. Posashkov
Keyword(s):  
Parabolic Equation ◽  
Initial Function ◽  
Heat Conduction Equation ◽  
Stationary Solutions ◽  
Classical Solutions ◽  
Smooth Functions ◽  
Large Solution ◽  
Additional Hypothesis ◽  
The Cauchy Problem ◽  
Degenerate Parabolic

SynopsisIn this paper we prove a certain monotonicity in time of non-negative classical solutions of the Cauchy problem for the quasilinear uniformly parabolic equation u1 = (ϕ(u))xx + Q(u) in wT = (0, T] × R1 with bounded sufficiently smooth initial function u(0, x) = uo(x)≧0 in Rl. We assume that ϕ(u) and Q(u) are smooth functions in [0, +∞) and ϕ′(u) >0, Q(u) > 0 for u > 0. Under some additional hypothesis on the growth of Q(u)ϕ′(u) at infinity, it is proved that if u(to, xo) becomes sufficiently large at some point (to, xo) ∈ wT, then ut(t, x0) ≧0 for all t ∈ [t0, T]. The proof is based on the method of intersection comparison of the solution with the set of the stationary solutions of the same equation. Some generalisations of this property for a quasilinear degenerate parabolic equation are discussed.

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