Angular momentum and tidal evolution

Five satellites of Neptune orbit under the synchronous zone. In this sense the Neptune's system is similar to that of Uranus (nine satellites) and differs from Jupiter (two) and Saturn (zero). The basic parameters describing the angular momentum within the Neptune's system and of its tidal evolution are estimated. The main character of the tidal dynamics is due to the retrograde Triton. The total tidal decrease in the spin angular momentum of Neptune is compared with those of Uranus, Jupiter and Saturn.     

Precession of the orbital nodes of Jupiter and Saturn triggered by the mutual perturbation: A model of two rings

The problem of the precession of the orbital planes of Jupiter and Saturn under the influence of mutual gravitational perturbations was formulated and solved using a simple dynamical model. Using the Gauss method, the planetary orbits are modeled by material circular rings, intersecting along the diameter at a small angle α. The planet masses, semimajor axes and inclination angles of orbits correspond to the rings. What is new is that each ring has an angular momentum equal to the orbital angular momentum of the planet. Contrary to popular belief, it was proved that the orbital resonance 5: 2 does not preclude the use of the ring model. Moreover, the period of averaging of the disturbing force (T ≈ 1332 yr) proves to be appreciably greater than a conventionally used period (≈900 yr). The mutual potential energy of rings and the torque of gravitational forces between the rings were calculated. We compiled and solved the system of differential equations for the spatial motion of rings. It was established that a perturbing torque causes the precession and simultaneous rotation of the orbital planes of Jupiter and Saturn. Moreover, the opposite orbit nodes on the Laplace plane coincide and perform a secular movement in retrograde direction with the same velocity of 25.6″/yr and the period T _J = T _S ≈ 50687 yr. These results are close to those obtained in the general theory (25.93″/yr), which confirms the adequacy of the developed model. It was found that the vectors of the angular velocity of orbital rings move counterclockwise over circular cones and describe circles on the celestial sphere with radii β₁ ≈ 0.8403504° (Saturn) and β₂ ≈ 0.3409296° (Jupiter) around the point which is located at an angular distance of 1.647607° from the ecliptic pole.     

Exchange of global mean angular momentum between an atmosphere and its underlying planet

This paper investigates the exchange of global mean angular momentum between an atmosphere and its underlying planet by a simple model. The model parameterizes four processes that are responsible for zonal mean momentum budget in the atmospheric boundary layer for a rotating planet: (i) meridional circulation that redistributes the relative angular momentum, (ii) horizontal diffusion that smoothes the prograde and retrograde winds, (iii) frictional drag that exchanges atmospheric angular momentum with the underlying planet, and (iv) internal redistribution of the zonal mean momentum by wave drag. It is shown that under a steady-state or a long-term average condition, the global relative angular momentum in the boundary layer vanishes unless there exists a preferred frictional drag for either the prograde or the retrograde zonal wind. We further show quantitatively that one cannot have either a predominant steady prograde or retrograde wind in the boundary layer of a planetary atmosphere. The parameter dependencies of the global relative angular momentum and the strength of the atmospheric circulation in the boundary layer are derived explicitly and used to explain the observational differences between the atmospheres of Earth and Venus.     

Diurnal thermal tides in a non-synchronized hot Jupiter

We perform a linear analysis to investigate the dynamical response of a non-synchronized hot Jupiter to stellar irradiation. In this work, we consider the diurnal Fourier harmonic of the stellar irradiation acting at the top of a radiative layer of a hot Jupiter with no clouds and winds. In the absence of the Coriolis force, the diurnal thermal forcing can excite internal waves propagating into the planet's interior when the thermal forcing period is longer than the sound crossing time of the planet's surface. When the Coriolis effect is taken into consideration, the latitude-dependent stellar heating can excite weak internal waves (g modes) and/or strong baroclinic Rossby waves (buoyant r modes) depending on the asynchrony of the planet. When the planet spins faster than its orbital motion (i.e. retrograde thermal forcing), these waves carry negative angular momentum and are damped by radiative loss as they propagate downwards from the upper layer of the radiative zone. As a result, angular momentum is transferred from the lower layer of the radiative zone to the upper layer and generates a vertical shear. We estimate the resulting internal torques for different rotation periods based on the parameters of HD 209458b.     

Effects of differential rotation on the gravitational figures of Jupiter and Saturn

It is assumed that observed zonal currents in the atmospheres of Jupiter and Saturn correspond to a state of permanent rotation, and that the angular velocity is constant on cylindrical surfaces parallel to the rotation axis. The equation of hydrostatic equilibrium for a rotating planet is solved under these restrictive assumptions, and the effect of the hypothesized rotation state on the planet's gravity harmonics and external shape is investigated. Spacecraft data on zonal currents are used to derive nearly model-independent corrections to the first four zonal gravity harmonic coefficients, which can be used to correct observed gravity harmonics to values appropriate for solid-body rotation. If the assumed rotation state is applicable, then zonal currents lead to measurable topography of isopycnic surfaces with respect to the reference fihure defined by the magnetospheric rotation period and the gravity harmonics. The amplitude of the topography is on the order of 5 km for Jupiter and 60 km for Saturn.     

On the dynamical derivation of the Titius-Bode law

The Principle of Least Action Interaction, developed by the dynamical astronomer Michael W. Ovenden, is tested using a new algorithm based on the ergodic hypothesis that the time mean of the disturbing function is equal to the space mean. This algorithm is an improvement over the one that Ovenden (1972) used in testing his principle, i.e. it can be applied to systems having more than three satellites without violating the conservation law of angular momentum and these satellites may have significant inclinations. This algorithm treats the problem of finding the configuration of least action interaction as a Lagrange multiplier problem. Renormalization group techniques and existing non-gradient optimization algorithms are incorporated into this new algorithm to reduce some of the numerical complexities.This algorithm is tested on the planets and asteroids in our solar system and on the satellite systems of Jupiter, Saturn, and Uranus. In most cases the results show that the current distances of the satellites from their primary is very close to the minimum interaction-action configuration for that system. The possibility of a planet lying beyond Pluto is investigated using this algorithm.Finally, some of my results are compared with those of Ovenden (1972) for our solar system. The results indicate that the interaction-action potential is lower using this new algorithm than the potential obtained from Ovenden's. Also, greater skepticism is raised concerning the one-time existence of a planet of 90 earth masses lying between Mars and Jupiter.     

Relativistic Precession of Elliptical Orbits of a Circumbinary Planet can be Cancelled

In publications presenting analytical results on the non-coplanar motion of a circumbinary planet it was shown that the unperturbed elliptical orbit of the planet undergoes simultaneously two kinds of the precession: the precession of the orbital plane and the precession of the orbit in its own plane. It is also well-known that there is also the relativistic precession of the planetary orbit in its own plane. In the present paper we study a combined effect of the all of the above precessions. For the general case, where the planetary orbit is not coplanar with the stars orbits, we analyzed the dependence of the critical inclination angle i_c, at which the precession of the planetary orbit in its own plane vanishes, on the angular momentum L of the planet. We showed that the larger the angular momentum, the smaller the critical inclination angle becomes. We presented the analytical result for i_c(L) and calculated the value of L, for which the critical inclination value becomes zero. For the particular case, where the planetary orbit is not coplanar with the stars orbits, we demonstrated analytically that at a certain value of the angular momentum of the planet, the elliptical orbit of the planet would become stationary: no precession. In other words, at this value of the angular momentum, the relativistic precession of the planetary orbit and its precession, caused by the fact that the planet revolves around a binary (rather than single) star, cancel each other out. This is a counterintuitive result.     

Some dynamical aspects of the accretion of Uranus and Neptune: The exchange of orbital angular momentum with planetesimals

The final stage of the accretion of Uranus and Neptune is numerically investigated. The four Jovian planets are considered with Jupiter and Saturn assumed to have reached their present sizes, whereas Uranus and Neptune are taken with initial masses 0.2 of their present ones. Allowance is made for the orbital variation of the Jovian planets due to the exchange of angular momentum with interacting bodies (“planetesimals”). Two possible effects that may have contributed to the accretion of Uranus and Neptune are incorporated in our model: (1) an enlarged cross section for accretion of incoming planetesimals due to the presence of extended gaseous envelopes and/or circumplanetary swarms of bodies; and (2) intermediate protoplanets in mid-range orbits between the Jovian planets. Significant radial displacements are found for Uranus and Neptune during their accretion and scattering of planetesimals. The orbital angular momentum budgets of Neptune, Uranus, and Saturn turn out to be positive; i.e., they on average gain orbital angular momentum in their interactions with planetesimals and hence they are displaced outwardly. Instead, Jupiter as the main ejector of bodies loses orbital angular momentum so it moves sunward. The gravitational stirring of planetesimals caused by the introduction of intermediate protoplanets has the effect that additional solid matter is injected into the accretion zones of Uranus and Neptune. For moderate enlargements of the radius of the accretion cross section (2–4 times), the accretion time scale of Uranus and Neptune are found to be a few 10⁸ years and the initial amount of solid material required to form them of a few times their present masses. Given the crucial role played by the size of the accretion cross section, questions as to when Uranus and Neptune acquired their gaseous envelopes, when the envelopes collapsed onto the solid cores, and how massive they were are essential in defining the efficiency and time scale of accretion of the two outer Jovian planets.     

Measurement of the angular momentum of Jupiter and the Sun by use of the Lense-Thirring effect

We investigate the feasibility of using the Lense-Thirring effect to measure the rotational angular momentum of Jupiter and the Sun. This experiment uses gyroscopes in close Jovian and solar orbits. It is important because it provides direct, unique information. The angular momentum is not derivable from the gravitational moments when non-uniform rotation is present. Analysis shows that this experiment could be done around Jupiter with current technology, but could not be done around the Sun for some years.Supported in part, by a Dissertation Research Assistantship of the Graduate College, Iowa State University.     

Angular momentum exchange during secular migration of two-planet systems

We investigate the secular dynamics of two-planet coplanar systems evolving under mutual gravitational interactions and dissipative forces. We consider two mechanisms responsible for the planetary migration: star-planet (or planet-satellite) tidal interactions and interactions of a planet with a gaseous disc. We show that each migration mechanism is characterized by a specific law of orbital angular momentum exchange. Calculating stationary solutions of the conservative secular problem and taking into account the orbital angular momentum leakage, we trace the evolutionary routes followed by the planet pairs during the migration process. This procedure allows us to recover the dynamical history of two-planet systems and constrain parameters of the involved physical processes.     

Calculations of the early evolution of Jupiter

The evolution of the protoplanet Jupiter is followed, using a hydrodynamic computer code with radiative energy transport. Jupiter is assumed to have formed as a subcondensation in the primitive solar nebula at a density just high enough for gravitational collapse to occur. The initial state has a density of 1.5 × 10^?11 g cm^?3 and a temperature of 43 K; the calculations are carried to an equilibrium state where the central density reaches 0.5 g cm^?3 and the central temperature reaches 2.5 × 10⁴ K. During the early part of the evolution the object contracts in quasi-hydrostatic equilibrium; later on hydrodynamic collapse occurs, induced by the dissociation of hydrogen molecules. After dissociation is complete, the planet regains hydrostatic equilibrium with a radius of a few times the present value. Further evolution beyond this point is not treated here; however the results are consistent with the existence of a high-luminosity phase shortly after the planet settles into its final quasistatic contraction.     

Dynamics of the 3/1 planetary mean-motion resonance: an application to the HD60532 b-c planetary system

In this paper, we use a semi-analytical approach to analyze the global structure of the phase space of the planar planetary 3/1 mean-motion resonance. The case where the outer planet is more massive than its inner companion is considered. We show that the resonant dynamics can be described using two fundamental parameters, the total angular momentum and the spacing parameter. The topology of the Hamiltonian function describing the resonant behaviour is investigated on a large domain of the phase space without time-expensive numerical integrations of the equations of motion, and without any restriction on the magnitude of the planetary eccentricities. The families of the Apsidal Corotation Resonances (ACR) parameterized by the planetary mass ratio are obtained and their stability is analyzed. The main dynamical features in the domains around the ACR are also investigated in detail by means of spectral analysis techniques, which allow us to detect the regions of different regimes of motion of resonant systems. The construction of dynamical maps for various values of the total angular momentum shows the evolution of domains of stable motion with the eccentricities, identifying possible configurations suitable for exoplanetary systems.     

Outcomes of tidal evolution for orbits with arbitrary inclination

Tides raised by a satellite on a rotating planet dissipate energy and result in an exchange of angular momentum between the orbit and the spin. A set of diagrams is constructed which shows the evolution of the angular momentum vectors. The results are applied to possible histories of the Uranus system.     

Bulges versus discs: the evolution of angular momentum in cosmological simulations of galaxy formation

We investigate the evolution of angular momentum in simulations of galaxy formation in a cold dark matter universe. We analyse two model galaxies generated in the N -body/hydrodynamic simulations of Okamoto et al. Starting from identical initial conditions, but using different assumptions for the baryonic physics, one of the simulations produced a bulge-dominated galaxy and the other one a disc-dominated galaxy. The main difference is the treatment of star formation and feedback, both of which were designed to be more efficient in the disc-dominated object. We find that the specific angular momentum of the disc-dominated galaxy tracks the evolution of the angular momentum of the dark matter halo very closely: the angular momentum grows as predicted by linear theory until the epoch of maximum expansion and remains constant thereafter. By contrast, the evolution of the angular momentum of the bulge-dominated galaxy resembles that of the central, most bound halo material: it also grows at first according to linear theory, but 90 per cent of it is rapidly lost as pre-galactic fragments, into which gas had cooled efficiently, merge, transferring their orbital angular momentum to the outer halo by tidal effects. The disc-dominated galaxy avoids this fate because the strong feedback reheats the gas, which accumulates in an extended hot reservoir and only begins to cool once the merging activity has subsided. Our analysis lends strong support to the classical theory of disc formation whereby tidally torqued gas is accreted into the centre of the halo conserving its angular momentum.     

On the early history of the asteroids

Among the major features of the asteroids as a group is the fact that they are small and numerous rather than being a single planet, and that they have unusually high eccentricities and inclinations. Regarding the first, this paper presents two lines of argument concerning the concept that the asteroids formed with a mass-distribution similar to what we observe today. Considerations of planet accretion are used to clarify the role of Jupiter in preventing coalescence of asteroids into a single planet. Regarding the eccentricities and inclinations, it is proposed that they were excited by secular resonances associated with the presence of Jupiter within a dissipating solar nebula.     

Orbital migrations in planetesimal discs: N-body simulations and the resonant dynamical friction

We have performed N -body numerical simulations of the exchange of angular momentum between a massive planet and a 3D Keplerian disc of planetesimals. Our interest is directed at the study of the classical analytical expressions of the lineal theory of density waves, as representative of the dynamical friction in discs 'dominated by the planet' and the orbital migration of the planets with regard to this effect. By means of a numerical integration of the equations of motion, we have carried out a set of numerical experiments with a large number of particles  ( N ≥10 000)  , and planets with the mass of Jupiter, Saturn and one core mass of the giant planets in the Solar system  ( M _c=10 M_⊕)  . The torque, measured in a phase in which a 'steady forcing' is clearly measurable, yields inward migration in a minimum-mass solar disc  (Σ∼10 g cm^-2  ), with a characteristic drift time of ∼ a few 10⁶ yr. The planets predate the disc, but the orbital decay rate is not sufficient to allow accretion in a time-scale relevant to the formation of giant planets. We found reductions of the measured torque on the planet, with respect to the linear theory, by a factor of 0.38 for M _c, 0.04 for Saturn and 0.01 for Jupiter, due to the increase in the perturbation on the disc. The behaviour of planets whose mass is larger than M _c is similar to the one of type II migrators in gaseous discs. Our results suggest that, in a minimum mass, solar planetesimals disc, type I migrations occur for masses smaller than M _c, whereas for this mass value it could be a transition zone between the two types of migration.     

Rotational evolution of exoplanets under the action of gravitational and magnetic perturbations

We investigate the evolution of the rotational axes of exoplanets under the action of gravitational and magnetic perturbations. The planet is assumed to be dynamically symmetrical and to be magnetised along its dynamical-symmetry axis. By qualitative methods of the bifurcation theory of multiparametric PDEs, we have derived a gallery of 69 phase portraits. The portraits illustrate evolutionary trajectories of the angular momentum of a planet for a variety of the initial conditions, for different values of the ratio between parameters describing gravitational and magnetic perturbations, and for different rates of the orbital evolution. We provide examples of the phase portraits, that reveal the differences in topology and the evolutionary track of in the vicinity of an equilibrium state. We determine the bifurcation properties, i.e., the way of reorganisation of phase trajectories in the vicinities of equilibria; and we point out the combinations of parameters’ values that permit ip-overs from a prograde to a retrograde spin mode.     

The formation and stability of the configuration of the planetary system HD 126611

Recent Doppler velocity measurements have revealed the existence of two planets orbiting the star HD 12661 on medium-eccentricity orbits. The inner planet has a period of 263.6 d and a mass of 2.3 Jupiter masses, and the outer planet has a period of 1444.5d and a mass of 1.57 Jupiter masses. The stability of this system requires the two planets to be in a state of mean-motion orbit resonances. By numerical method we have studied the orbit migration and stability of the system in its early ages under the action of the proto-stellar disk, and calculated the probabilities of the planets being captured into the mean -motion resonances during their migrations. It is found that at present the two planets are probably situated at the edge of the 11:2 mean-motion resonance and are in chaotic motions. This result may be helpful to clarify the arguments on the present configuration. Besides, it is indicated that very probably, after the formation of the system, the gaseous disk has almost disappeared before the planets migrated to the present configuration.     

Regions of stability in rotational dynamics

We investigate the rotational dynamics of a triaxial planet moving on a Keplerian orbit around its star. The dynamics is ruled by several parameters, like the eccentricity, the obliquity, the non-principal rotation, the angular momentum, etc. We consider two specific cases in which the planet is symmetric or asymmetric, according to whether two moments of inertia coincide or differs from each other. We study the dynamics by constructing maps of dynamical stability based on the computation of the maximum Lyapunov characteristic number versus some typical parameters. The results show that only specific resonances appear in the symmetric case, while the asymmetric case shows a much richer phenomenology.     

On the migration of a system of protoplanets

The evolution of a system consisting of a protoplanetary disc with two embedded Jupiter-sized planets is studied numerically. The disc is assumed to be flat and non-self-gravitating; this is modelled by the planar (two-dimensional) Navier–Stokes equations. The mutual gravitational interaction of the planets and the star, and the gravitational torques of the disc acting on the planets and the central star are included. The planets have an initial mass of one Jupiter mass M _Jup each, and the radial distances from the star are one and two semimajor axes of Jupiter, respectively.
During the evolution a joint wide annular gap is created by the planets. Both planets increase their mass owing to accretion of gas from the disc: after about 2500 orbital periods of the inner planet it has reached a mass of 2.3  M _Jup, while the outer planet has reached a mass of 3.2  M _Jup. The net gravitational torques exerted by the disc on the planets result in an inward migration of the outer planet on time-scales comparable to the viscous evolution time of the disc. The semimajor axis of the inner planet remains constant as there is very little gas left in its vicinity to induce any migration. When the distance of close approach eventually becomes smaller than the mutual Hill radius, the eccentricities increase strongly and the system may become unstable.
If disc depletion occurs rapidly enough before the planets come too close to each other, a stable system similar to our own Solar system may remain. Otherwise the orbits may become unstable and produce systems like υ And.