Complete spin and orbital evolution of close-in bodies using a Maxwell viscoelastic rheology

In this paper, we present a formalism designed to model tidal interaction with a viscoelastic body made of Maxwell material. Our approach remains regular for any spin rate and orientation, and for any orbital configuration including high eccentricities and close encounters. The method is to integrate simultaneously the rotation and the position of the planet as well as its deformation. We provide the equations of motion both in the body frame and in the inertial frame. With this study, we generalize preexisting models to the spatial case and to arbitrary multipole orders using a formalism taken from quantum theory. We also provide the vectorial expression of the secular tidal torque expanded in Fourier series. Applying this model to close-in exoplanets, we observe that if the relaxation time is longer than the revolution period, the phase space of the system is characterized by the presence of several spin-orbit resonances, even in the circular case. As the system evolves, the planet spin can visit different spin-orbit configurations. The obliquity is decreasing along most of these resonances, but we observe a case where the planet tilt is instead growing. These conclusions derived from the secular torque are successfully tested with numerical integrations of the instantaneous equations of motion on HD 80606 b. Our formalism is also well adapted to close-in super-Earths in multiplanet systems which are known to have non-zero mutual inclinations.     

The orbital evolution of comet Bowell (1980b)

Planetary perturbations and orbital evolution of the elements of the comet Bowell (1980b) are calculated. The sudden change of all the elements of the orbit on February 1981 is caused by Jupiter's perturbation.     

Angular momentum and kinetic energy of rotating deformable bodies

The general equations of angular momentum and kinetic energy of a rotating deformable (or not rigid) body are discussed for a fixed and a rotating coordinate system. A new system of equations is developed for a deformable body of arbitrary form using the Lagrangian (vector) cisplacement up to the first order terms. The equations are, then, illustrated for a self-gravitating ceformable body perturbed by tides.     

The effect of orbital evolution on the Haumea (2003 EL61) collisional family

The Haumea family is currently the only identified collisional family in the Kuiper belt. We numerically simulate the long-term dynamical evolution of the family to estimate a lower limit of the family’s age and to assess how the population of the family and its dynamical clustering are preserved over Gyr timescales. We find that the family is not younger than 100 Myr, and its age is at least 1 Gyr with 95% confidence. We find that for initial velocity dispersions of 50–400 m s^?1, approximately 20–45% of the family members are lost to close encounters with Neptune after 3.5 Gyr of orbital evolution. We apply these loss rates to two proposed models for the formation of the Haumea family, a graze-and-merge type collision between two similarly sized, differentiated KBOs or the collisional disruption of a satellite orbiting Haumea. For the graze-and-merge collision model, we calculate that >85% of the expected mass in surviving family members within 150 m s^?1 of the collision has been identified, but that one to two times the mass of the known family members remains to be identified at larger velocities. For the satellite-break-up model, we estimate that the currently identified family members account for ～50% of the expected mass of the family. Taking observational incompleteness into account, the observed number of Haumea family members is consistent with either formation scenario at the 1σ level, however both models predict more objects at larger relative velocities (>150 m s^?1) than have been identified.     

The use of the distribution of the orbital energy to investigate the evolution of the Perseid meteoroid stream

Numerical integration of a meteor stream over a long period is a very time consuming process, especially if a number of different models have to be investigated. In this paper we demonstrate that using the distribution of orbital energy as a vector in a Markov process can be a useful tool and that meaningful results can be obtained with far less computation than is normally required.     

The orbital evolution of NEAs and the resonances with Jupiter

two near-earth-asteroids associated with resonances with Jupiter are studied over a time span of 10⁵ yrs. We found that asteroid (887) is temporary trapped in the 3:1 resonance; thus indicating that this resonance could be a source of short-lived NEAs. We also found that asteroid (3552) with a large eccentricity and a high inclination is wandering about the 1:1 resonant region.     

Properties of the moment of inertia in the problem of three bodies

Sundman's and Birkhoff's results are combined with a recently developed inequality and new qualitative results are given for the problem of three bodies.     

A study of dynamical effects in the observed second time-derivative of the spin or orbital frequencies of pulsars

The observed values of the time-derivatives of the spin or orbital frequency of pulsars are affected by their dynamical properties. We derive thorough analytical expressions for such dynamical contributions in terms of the Galactic coordinates, the proper motion, the pulsar distance, and the radial velocity. We find that the effects of the dynamical terms in the second-derivative of frequencies or parameters based on such second derivatives, e.g., braking index, are usually negligible. However, unique pulsars for which the effects of the dynamical terms are significant can exist. In particular, dynamical effects can make the magnitude of the observed value of the braking index to be in the order of thousand while the true value of it is close to the theoretically expected value three, especially if the pulsars lie close to the Galactic centre. Dynamics can also affect the value of the second derivative of the orbital frequency of a binary pulsar at the first decimal place. We also emphasize the fact that our expressions provide more accurate results than pre-existing approximate ones that exclude some of the terms. Comparison with a set of pulsars showed that the median value of the difference between the results obtained by our method and a pre-existing method is about 50 percent.     

Effect of the rotation and tidal dissipation history of stars on the evolution of close-in planets

Since 20 years, a large population of close-in planets orbiting various classes of low-mass stars (from M-type to A-type stars) has been discovered. In such systems, the dissipation of the kinetic energy of tidal flows in the host star may modify its rotational evolution and shape the orbital architecture of the surrounding planetary system. In this context, recent observational and theoretical works demonstrated that the amplitude of this dissipation can vary over several orders of magnitude as a function of stellar mass, age and rotation. In addition, stellar spin-up occurring during the Pre-Main-Sequence (PMS) phase because of the contraction of stars and their spin-down because of the torque applied by magnetized stellar winds strongly impact angular momentum exchanges within star–planet systems. Therefore, it is now necessary to take into account the structural and rotational evolution of stars when studying the orbital evolution of close-in planets. At the same time, the presence of planets may modify the rotational dynamics of the host stars and as a consequence their evolution, magnetic activity and mixing. In this work, we present the first study of the dynamics of close-in planets of various masses orbiting low-mass stars (from \(0.6~M_\odot \) to \(1.2~M_\odot \)) where we compute the simultaneous evolution of the star’s structure, rotation and tidal dissipation in its external convective envelope. We demonstrate that tidal friction due to the stellar dynamical tide, i.e. tidal inertial waves excited in the convection zone, can be larger by several orders of magnitude than the one of the equilibrium tide currently used in Celestial Mechanics, especially during the PMS phase. Moreover, because of this stronger tidal friction in the star, the orbital migration of the planet is now more pronounced and depends more on the stellar mass, rotation and age. This would very weakly affect the planets in the habitable zone because they are located at orbital distances such that stellar tide-induced migration happens on very long timescales. We also demonstrate that the rotational evolution of host stars is only weakly affected by the presence of planets except for massive companions.     

Cosmological evolution for dark energy models in f(T) gravity

In this paper, we investigate the behavior of equation of state parameter and energy density for dark energy in the framework of f(T) gravity. For this purpose, we use anisotropic LRS Bianchi type I universe model. The behavior of accelerating universe is discussed for some well-known f(T) models. It is found that the universe takes a transition between phantom and non-phantom phases for f(T) models except exponential and logarithmic models. We conclude that our results are relativity analogous to the results of FRW universe.     

On the effects of higher convection modes on the thermal evolution of small planetary bodies

The effects of higher modes of convection on the thermal evolution of a small planetary body is investigated. Three sets of models are designed to specify an initially cold and differentiated, an initially hot and differentiated, and an initially cold and undifferentiated Moon-type body. The strong temperature dependence of viscosity enhances the thickening of lithosphere so that a lithosphere of about 400 km thickness is developed within the first billion years of the evolution of a Moon-type body. The thermally isolating effect of such a lithosphere hampers the heat flux out of the body and increases the temperature of the interior, causing the solid-state convection to occur with high velocity so that even the lower modes of convection can maintain an adiabatic temperature gradient there. It is demonstrated that the effect of solid-state convection on the thermal evolution of the models may be adequately determined by a combination of convection modes up to the third or the fourth order harmonic. The inclusion of higher modes does not affect the results significantly.     

Late-stage impacts and the orbital and thermal evolution of Tethys

An inferred ancient episode of heating and deformation on Tethys has been attributed to its passage through a 3:2 resonance with Dione (Chen, E.M.A., Nimmo, F. [2008]. Geophys. Res. Lett. 35, 19203). The satellites encounter, and are trapped into, the e-Dione resonance before reaching the e-Tethys resonance, limiting the degree to which Tethys is tidally heated. However, for an initial Dione eccentricity >0.016, Tethys’ eccentricity becomes large enough to generate the inferred heat flow via tidal dissipation. While capture into the e-Dione resonance is easy, breaking the resonance (to allow Tethys to evolve to its current state) is very difficult. The resonance is stable even for large initial Dione eccentricities, and is not broken by perturbations from nearby resonances (e.g. the Rhea–Dione 5:3 resonance). Our preferred explanation is that the Tethyan impactor which formed the younger Odysseus impact basin also broke the 3:2 resonance. Simultaneously satisfying the observed basin size and the requirement to break the resonance requires a large (≈250 km diameter) and slow (≈0.5 km/s) impactor, possibly a saturnian satellite in a nearby crossing orbit with Tethys. Late-stage final impacts of this kind are a common feature of satellite formation models (Canup, R.M., Ward, W.R. [2006]. Nature 441, 834–839).     

The effect of the radiation pressure on the orbital evolution of geosynchronous objects

The effect of the radiation pressure and Poynting-Robertson effect on the evolution of the orbits of geosynchronous satellites is studied, depending on their area to mass ratio. The qualitative changes of the orbital evolution caused by these disturbances are considered. The reflection coefficient of the satellite’s surface was assumed to be 1.44. In the vicinity of the stable point with the longitude of 75° the exit from the libration resonance mode was registered when the area to mass ratio value changed from 5.9 to 6.0 m²/kg; in the vicinity of the unstable point at 345° with the area to mass ratio of 1.4 it occurred at 1.5 m²/kg. Re-entry to Earth occurs at values of the area to mass ratio above 32.2 m²/kg, and hyperbolic exit from the low-Earth orbit occurs at values of the area to mass ratio over 5267 m²/kg. At high values of the area to mass ratio, slopes of initially equatorial orbits can reach 49°. It is shown that due to the Poynting-Robertson effect the secular decrease in the semimajor axis of orbit in libration resonance region is 3–4 orders of magnitude less than outside of it.     

The evolution of bodies orbiting in the trans-Neptunian region on to intermediate-type orbits

We investigate the dynamical evolution of 210 hypothetical massless bodies initially situated between 10 and 30 au from the Sun in order to determine the general characteristics of the evolved system. This is of particular relevance to the understanding of the origin of Edgeworth–Kuiper belt objects on scattered intermediate orbits, such as 1996 TL ₆₆, which have high eccentricity and semimajor axis but nevertheless have perihelion in the region between 30 and 50 au from the Sun.     

The formation and habitability of terrestrial planets in the presence of close-in giant planets

‘Hot jupiters,’ giant planets with orbits very close to their parent stars, are thought to form farther away and migrate inward via interactions with a massive gas disk. If a giant planet forms and migrates quickly, the planetesimal population has time to re-generate in the lifetime of the disk and terrestrial planets may form [P.J. Armitage, A reduced efficiency of terrestrial planet formation following giant planet migration, Astrophys. J. 582 (2003) L47-L50]. We present results of simulations of terrestrial planet formation in the presence of hot/warm jupiters, broadly defined as having orbital radii ?0.5 AU. We show that terrestrial planets similar to those in the Solar System can form around stars with hot/warm jupiters, and can have water contents equal to or higher than the Earth's. For small orbital radii of hot jupiters (e.g., 0.15, 0.25 AU) potentially habitable planets can form, but for semi-major axes of 0.5 AU or greater their formation is suppressed. We show that the presence of an outer giant planet such as Jupiter does not enhance the water content of the terrestrial planets, but rather decreases their formation and water delivery timescales. We speculate that asteroid belts may exist interior to the terrestrial planets in systems with close-in giant planets.     

The simulation of the magnetic field and spin period evolution of accreting neutron stars

By means of the Monte Carlo method, we simulate the evolutionary distribution of accreting neutron stars (NSs) in the magnetic field versus spin period (B‐P) diagram where the accretion induced magnetic‐field decay model is exploited. The simulated results show that by mass accretion the B‐P distribution of the accreting NS would evolve along the equilibrium period line to a region with low field and short period. The B‐P distributions of the simulated accreting NSs are consistent with those of the observed millisecond pulsars (MSPs) after accretion of ∼ 0.1–0.2 M⊙. We also test the effects of the initial magnetic field and the spin period on the evolved B‐P distribution of the accreting NSs. It is shown that the evolved distributions of the simulated samples are independent of the selection of the initial condition when the NS magnetic field decays to a value less than ∼10¹⁰ G. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Recent orbital evolution and the internal structures of Enceladus and Dione

We study the orbital behavior of Saturn’s satellites Enceladus and Dione during their passage through the 2:1 mean-motion resonances to constrain their interior structures, parameterized by the quantity k₂/Q (assumed constant). Enceladus’ evolution after escape from the second-order e-Enceladus e-Dione resonance requires that (k₂/Q)_Enceladus<8×10^-4, for that Q_Saturn>18,000. This result is in agreement with [Meyer, J., Wisdom, J., 2008b. Icarus 193, 213-223]. The present-day libration amplitude of Enceladus requires that (k₂/Q)_Enceladus>1.2×10^-4, assuming that Q_Saturn<10⁵. Dione’s present-day eccentricity indicates that (k₂/Q)_Dione?3×10^-4 for Q_Saturn>18,000. Assuming Maxwellian viscoelastic behavior, we find that for Enceladus a convective ice shell overlying an ocean is too dissipative to match the orbital constraints. We conclude that a conductive shell overlying an ocean is more likely, and discuss the implications of this result. Dione’s ice shell is also likely to be conductive, but our results are less constraining.     

Orbit and spin evolution of synchronous binary stars on the main sequence(a theoretical improvement to the analytical method)

This paper provides a method to study the solution of equations for synchronous binary stars with large eccentricity on the main sequence.The theoretical results show that the evolution of the eccentricity is linear with time or follows an exponential form,and the semi-major axis and spin vary with time in an exponential form that are different from the results given in a previous paper.The improved method is applicable in both cases of large eccentricity and small eccentricity.In addition,the number of terms in the expansion of a series with small eccentricity is very long due to the series converging slowly.The advantage of this method is that it is applicable to cases with large eccentricity due to the series converging quickly.This paper chooses the synchronous binary star V1143 Cyg that is on the main sequence and has a large eccentricity(e = 0.54) as an example calculation and gives the numerical results.Lastly,the evolutionary tendency including the evolution of orbit and spin,the time for the speed up of spin,the circularization time,the orbital collapse time and the life time are given in the discussion and conclusion.The results shown in this paper are an improvement on those from the previous paper.