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.     

Secular ring instability in the protoplanetary accretion disk

The basic parameters describing the angular momentum distribution within the Uranus system and of its tidal evolution have been estimated. The nine satellites orbiting under the synchronous zone of Uranus is the maximum number in the solar system and it makes the Uranus system different compared with any other in the Solar system, however the satellites in question are relatively small and their contribution of the tidal dynamics of the system is small compared with that due to UI and UV. The time for existence of the nine satellites as integrated bodies can be estimated as 1.4 × 10⁹ y (UVI) and more. The total tidal decrease in the Uranus angular velocity of rotation is estimated as 7 × 10^–9s^–1.     

Zonal winds near Venus' cloud top level: A model study of the interaction between the zonal mean circulation and the semidiurnal tide

A primitive equation wave-mean flow interaction model, designed by J. R. Holton and used originally to study Earth's middle atmosphere, has been adapted to Venus in order to clarify our understanding of the interaction between the semidiurnal tide and the thermally driven mean meridional circulation near the cloud top level. With or without the tide the model produces midlatitude jets whose structure is insensitive to vertical shear of the background angular velocity above and below the cloud top level, but it is sensitive to background angular velocity at the cloud top level. When this background angular velocity is close to that of Venus, the latitudes and speeds of these jets are similar to the latitudes and speeds of jets at the Venus cloud top level as inferred from observed temperatures and the cyclostrophic balance condition. In agreement with the hypothesis of Fels and Lindzen, the model tide accelerates the equatorial zonal wind near the cloud top level and decelerates it at higher levels. The tidal vertical wavelength, maximum amplitude, dissipative decay length, and zonal flow accelerations are sensitive functions of background angular velocity, in agreement with elementary gravity wave theory. In the equatorial cloud top region, tidal acceleration is comparable in magnitude to the decelerative effects of vertical advection and the model's Rayleigh friction damping. For sufficiently rapid initial zonal flow near the cloud top level, the area-weighted global mean cloud top level zonal wind increases with time over a 50-day model run as a result of tidal acceleration. Agreement between the model tide and the observed tide, or the tide determined in the more detailed calculations of Pechmann and Ingersoll, is best when the background angular velocity at the jet level is about 30% larger than that observed.     

Orientation of Galaxies in Clusters

A detailed analysis of the geometry of galaxies in clusters has been undertaken in both two and three dimensional space. The procedure was applied to the three Abell clusters A1644, A548 E and A548 W. No significant alignment trend of galaxies in clusters confirmed in three dimensional space was found. This result is consistent with the mixed dark matter model MDM of galaxy formation. The result is also consistent with Peebles suggestion that protogalaxies acquired angular momentum from tidal torques exerted by their neighbours in the early universe. The amount of angular momentum predicted by this mechanism could be described by a single dimensionless papameter λ≈ 0.8. N-body experiments have shown that λ has approximately normal distribution with mean value about 0.07 and with a standard deviation of about 0.03. From the analyic fit to flat rotation curves it is found that tidal torque can provide the observed angular momentum if the mean collapse factor is about 20 and the mean halo-to-disk mass ratio is of order 10. The MDM model does not predict any systematic relation between the tidal torques among the halos of galaxies and the large scale structures such as the clusters, superclusters of galaxies and giant voids in between. Further work in this area is still required. This revised version was published online in July 2006 with corrections to the Cover Date.     

A note on the tidal expansion of the orbit of Mercury

Under the assumption that the secular decrease in the sidereal mean motion of Mercury is due to the Sun-Mercury tidal torque, the expansion of the orbit of Mercury and the tidal phase-lag angle of the tidal bulge are estimated.     

An Estimate of the Long-term Tendency of Tidal Evolution of Earth-lunar System

According to the conservation principle of angular momentum, we calculate in this paper the revolution period and the distance between the Earth and the Moon in the equilibrium state of the tidal evolution in the Earth-Moon system. The difference of energy between the current state and the equilibrium state is used to compute the time needed to fulfil the equilibrium state. Then the long-term variations of the Earth-Moon distance and of the Earth rotation rate are further estimated.     

Protostars and the origin of the angular momentum of the solar system

Protostars in a group exert gravitational tidal torques on an aspherical nebula located in the group. The net torque transfers angular momentum from the orbital motions of the stars to rotation of the nebula. A relation can be derived between the parameters describing the protostars and the final angular momentum of the nebula. While the parameters concerned are uncertain, a conservative choice results in a value for the angular momentum equal to about 1/3 of that of the present solar system. This suggests that if the Sun formed in a group, tidal interactions with other protostars may account for a significant part of the angular momentum of the solar system.     

Time Scales and Tidal Effects in Minor Mergers

We use controlled N-body simulation to investigate the dynamical processes (dynamical friction, tidal truncation, etc.) involved in the merging of small satellites into bigger halos. We confirm the validity of some analytic formulae proposed earlier based on simple arguments. For rigid satellites represented by softened point masses, the merging time scale depends on both the orbital shape and concentration of the satellite. The dependence on orbital ellipticity is roughly a power law, as suggested by Lacey & Cole, and the dependence on satellite concentration is similar to that proposed by White. When merging satellites are represented by non-rigid objects, Tidal effects must be considered. We found that material beyond the tidal radius are stripped off. The decrease in the satellite mass might mean an increase in the merging time scale, but in fact, the merging time is decreased, because the stripped-off material carries away a proportionately larger amount of of orbital energy and angular momentum.     

潮汐能量耗散与地月系统演化

对地月系统而言, 在很大程度上角动量守恒是正确的. 地月距离的变化主要是受到月球引起的潮汐能量耗散的影响. 根据月球的平均运动和它的长期加速度, 就可以计算出月潮能量耗散的数值. 海洋是潮汐能量耗散的主要区域. 由于潮汐的高度正比于月球对潮汐隆起的万有引力, 由此可导出总的月球潮汐摩擦力正比于月球平均运动的平方. 如果采用月球平均加速度数值-20.72$''\cdot$cy^-2, 就可以推算出35亿年来地月之间的距离以及回归年日数和朔望月日数的演化. 此理论结果与古生物钟的数据进行比对, 两者符合较好.     

Tidal Angular Momentum in Close Binary Systems in presence of Wind Driven Non-conservative Mass Transfer with Uniform Mass Accretion Rate

A very well-known property of close binary stars is that they usually rotate slowly than a similar type single star. Massive stars in close binary systems are supposed to experience an exchange of mass and angular momentum via mass transfer and tidal interaction, and thus the evolution of binary stars becomes more complex than that of individual stars. In recent times, it has become clear that a large number of massive stars interact with binary companions before they die. The observation also reveals that in close pairs the rotation tends to be synchronized with the orbital motion and the companions are naturally tempted to invoke tidal friction. We here introduce the effect of tidal angular momentum in the model of wind driven non-conservative mass transfer taking mass accretion rate as uniform with respect to time. To model the angular momentum evolution of a low mass main sequence companion star can be a challenging task. So, to make the present study more interesting, we have considered initial masses of the donor and gainer stars at the proximity of bottom-line main sequence stars and they are taken with lower angular momentum. We have produced a graphical profile of the rate of change of tidal angular momentum and the variation of tidal angular momentum with respect to time under the present consideration.     

On the characteristics of tidal structures of interacting galaxies

We present the results of our analysis of the geometrical tidal tail characteristics for nearby and distant interacting galaxies. The sample includes more than two hundred nearby galaxies and about seven hundred distant ones. The distant galaxies have been selected in several deep fields of the Hubble Space Telescope (HDF-N, HDF-S, HUDF, GOODS, GEMS) and they are at mean redshift 〈z〉 = 0.65. We analyze the distributions of lengths and thicknesses for the tidal structures and show that the tails in distant galaxies appear shorter than those in nearby ones. This effect can be partly attributed to observational selection, but, on the other hand, it may result from the general evolution of the sizes of spiral galaxies with z. The positions of interacting galaxies on the galaxy luminosity (L)-tidal tail length (l) plane are shown to be explained by a simple geometrical model, with the upper envelope of the observed distribution being \(l \propto \sqrt L\). We have solved the problem on the relationship between the observed distribution of tail flatting and the tail length in angular measure by assuming the tidal tails to be arcs of circumferences visible at arbitrary angles to the line of sight. We conclude that the angular length of the tidal tails visually distinguished in nearby and distant galaxies, on average, exceeds 180°.     

On the history of the lunar orbit

A frequency-dependent model of tidal friction is used in the determination of the time rate of change of the lunar orbital elements and the angular velocity of the Earth. The variational equations consider eccentricity, the solar tide on the Earth, Earth oblateness, and higher-order terms in the Earth's tidal potential. A linearized solution of the equations governing the precission of the Earth's rotational angular momentum and the lunar ascending node is found. This allows the analytical averaging of the variational equations over the period of relative precession which, though large, is necessarily small in comparison to the time step of the numerical integrator that yields the system history over geological time. Results for this history are presented and are identified as consistent with origin of the Moon by capture. This model may be applied to any planet-satellite system where evolution under tidal friction is of interest.     

Effects of external tidal field on the evolution of the outer regions of multi-mass star clusters

We present N -body simulations (including an initial mass function) of globular clusters in the Galaxy in order to study effects of the tidal field systematically on the properties of the outer parts of globular clusters. Using nbody6 , which correctly takes into account the two-body relaxation, we investigate the development of tidal tails of globular clusters in the Galactic tidal field. For simplicity, we have employed only the spherical components (bulge and halo) of the Galaxy, and ignored the effects of stellar evolution which could have been important in the very early phase of the cluster evolution. The total number of stars in our simulations is about 20 000, which is much smaller than the realistic number of stars. All simulations had been done for several orbital periods in order to understand the development of the tidal tails. In our scaled-down models, the relaxation time is sufficiently short to show the mass segregation effect, but we did not go far enough to see the core collapse, and the fraction of stars lost from the cluster at the end of the simulations is only ∼10 per cent. The radial distribution of extra-tidal stars can be described by a power law with a slope around −3 in surface density. The directions of tidal tails are determined by the orbits and locations of the clusters. We find that the length of tidal tails increases towards the apogalacticon and decreases towards the perigalacticon. This is an anti-correlation with the strength of the tidal field, caused by the fact that the time-scale for the stars to respond to the potential is similar to the orbital time-scale of the cluster. The escape of stars in the tidal tails towards the pericentre could be another reason for the decrease of the length of tidal tails. We find that the rotational angular velocity of tidally induced clusters shows quite different behaviour from that of initially rotating clusters.     

Quantitative analysis of clumps in the tidal tails of star clusters

Tidal tails of star clusters are not homogeneous but show well-defined clumps in observations as well as in numerical simulations. Recently, an epicyclic theory for the formation of these clumps was presented. A quantitative analysis was still missing. We present a quantitative derivation of the angular momentum and energy distribution of escaping stars from a star cluster in the tidal field of the Milky Way and derive the connection to the position and width of the clumps. For the numerical realization we use star-by-star N -body simulations. We find a very good agreement of theory and models. We show that the radial offset of the tidal arms scales with the tidal radius, which is a function of cluster mass and the rotation curve at the cluster orbit. The mean radial offset is 2.77 times the tidal radius in the outer disc. Near the Galactic Centre the circumstances are more complicated, but to lowest order the theory still applies. We have also measured the Jacobi energy distribution of bound stars and showed that there is a large fraction of stars (about 35 per cent) above the critical Jacobi energy at all times, which can potentially leave the cluster. This is a hint that the mass loss is dominated by a self-regulating process of increasing Jacobi energy due to the weakening of the potential well of the star cluster, which is induced by the mass loss itself.     

Empirical evidence for tidal evolution in transiting planetary systems

Most transiting planets orbit very close to their parent star, causing strong tidal forces between the two bodies. Tidal interaction can modify the dynamics of the system through orbital alignment, circularization, synchronization and orbital decay by exchange of angular moment. Evidence for tidal circularization in close-in giant planet is well known. Here, we review the evidence for excess rotation of the parent stars due to the pull of tidal forces towards spin-orbit synchronization. We find suggestive empirical evidence for such a process in the present sample of transiting planetary systems. The corresponding angular momentum exchange would imply that some planets have spiralled towards their star by substantial amounts since the dissipation of the protoplanetary disc. We suggest that this could quantitatively account for the observed mass–period relation of close-in gas giants. We discuss how this scenario can be further tested and point out some consequences for theoretical studies of tidal interactions and for the detection and confirmation of transiting planets from radial velocity and photometric surveys.     

The role of tidal interactions in star formation

Nearly all of the initial angular momentum of the matter that goes into each forming star must somehow be removed or redistributed during the formation process. The possible transport mechanisms and the possible fates of the excess angular momentum are discussed, and it is argued that transport processes in discs are probably not sufficient by themselves to solve the angular momentum problem, while tidal interactions with other stars in forming binary or multiple systems are likely to be of very general importance in redistributing angular momentum during the star formation process. Most, if not all, stars probably form in binary or multiple systems, and tidal torques in these systems can transfer much of the angular momentum from the gas around each forming star to the orbital motions of the companion stars. Tidally generated waves in circumstellar discs may contribute to the overall redistribution of angular momentum. Stars may gain much of their mass by tidally triggered bursts of rapid accretion, and these bursts could account for some of the most energetic phenomena of the earliest stages of stellar evolution, such as jet-like outflows. If tidal interactions are indeed of general importance, planet-forming discs may often have a more chaotic and violent early evolution than in standard models, and shock heating events may be common. Interactions in a hierarchy of subgroups may play a role in building up massive stars in clusters and in determining the form of the upper initial mass function (IMF) . Many of the processes discussed here have analogues on galactic scales, and there may be similarities between the formation of massive stars by interaction-driven accretion processes in clusters and the buildup of massive black holes in galactic nuclei.     

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.     

Corrections stemming from the non-osculating character of the Andoyer variables used in the description of rotation of the elastic Earth

We explore the evolution of the angular velocity of an elastic Earth model, within the Hamiltonian formalism. The evolution of the rotation state of the Earth is caused by the tidal deformation exerted by the Moon and the Sun. It can be demonstrated that the tidal perturbation to spin depends not only upon the instantaneous orientation of the Earth, but also upon its instantaneous angular velocity. Parameterizing the orientation of the Earth figure axis with the three Euler angles, and introducing the canonical momenta conjugated to these, one can then show that the tidal perturbation depends both upon the angles and the momenta. This circumstance complicates the integration of the rotational motion. Specifically, when the integration is carried out in terms of the canonical Andoyer variables (which are the rotational analogues to the orbital Delaunay variables), one should keep in mind the following subtlety: under the said kind of perturbations, the functional dependence of the angular velocity upon the Andoyer elements differs from the unperturbed dependence (Efroimsky in Proceedings of Journées 2004: Systèmes de référence spatio-temporels. l’Observatoire de Paris, pp 74–81, 2005; Efroimsky and Escapa in Celest. Mech. Dyn. Astron. 98:251–283, 2007). This happens because, under angular velocity dependent perturbations, the requirement for the Andoyer elements to be canonical comes into a contradiction with the requirement for these elements to be osculating, a situation that parallels a similar antinomy in orbital dynamics. Under the said perturbations, the expression for the angular velocity acquires an additional contribution, the so called convective term. Hence, the time variation induced on the angular velocity by the tidal deformation contains two parts. The first one comes from the direct terms, caused by the action of the elastic perturbation on the torque-free expressions of the angular velocity. The second one arises from the convective terms. We compute the variations of the angular velocity through the approach developed in Getino and Ferrándiz (Celest. Mech. Dyn. Astron. 61:117–180, 1995), but considering the contribution of the convective terms. Specifically, we derive analytical formulas that determine the elastic perturbations of the directional angles of the angular velocity with respect to a non-rotating reference system, and also of its Cartesian components relative to the Tisserand reference system of the Earth. The perturbation of the directional angles of the angular velocity turns out to be different from the evolution law found in Kubo (Celest. Mech. Dyn. Astron. 105:261–274, 2009), where it was stated that the evolution of the angular velocity vector mimics that of the figure axis. We investigate comprehensively the source of this discrepancy, concluding that the difference between our results and those obtained in Ibid. stems from an oversimplification made by Kubo when computing the direct terms. Namely, in his computations Kubo disregarded the motion of the tide raising bodies with respect to a non-rotating reference system when compared with the Earth rotational motion. We demonstrate that, from a numerical perspective, the convective part provides the principal contribution to the variation of the directional angles and of length of day. In the case of the x and y components in the Tisserand system, the convective contribution is of the same order of magnitude as the direct one. Finally, we show that the approximation employed in Kubo (Ibid.) leads to significant numerical differences at the level of a hundred micro-arcsecond.     

The circumnuclear material in the Galactic Centre: a clue to the accretion process

On the basis of 'sticky particle' calculations, it is argued that the gas features observed within 10 pc of the Galactic Centre — the circumnuclear disc (CND) and the ionized gas filaments — as well as the newly formed stars in the inner 1 pc can be understood in terms of tidal capture and disruption of gas clouds on low angular momentum orbits in a potential containing a point mass. The calculations demonstrate that a dissipative component forms a 'dispersion ring', an asymmetric elliptical torus precessing counter to the direction of rotation, and that this shape can be maintained for many orbital periods. For a range of plausible initial conditions, such a structure can explain the morphology and kinematics of the CND and of the most conspicuous ionized filament. While forming the dispersion ring, a small cloud with low specific angular momentum is drawn into a long filament which repeatedly collides with itself at high velocity. The compression in strong shocks is likely to lead to star formation even in the near tidal field of the point mass. This process may have general relevance to accretion on to massive black holes in normal and active galactic nuclei.     

An observational test for the origin of the Titan-Hyperion orbital resonance

If Hyperion's radius is near the upper limit of recent estimates, and tidal dissipation in Hyperion is reasonably well represented by a frequency-independent Q ? 2–300, finding Hyperion rotating in the 3:2 spin-orbit resonance like Mercury would imply a primordial origin for the Titan-Hyperion 4:3 orbital resonance. Independent of this test, observation of Hyperion's spin rate will place an upper bound on the average tidal effective Q for the satellite as a function of its assumed initial angular velocity.