Orbital Angular Velocity Research Articles

Landau Tidal Damping and Major-Body Clustering in Solar and Extrasolar Subsystems

Major (exo)planetary and satellite bodies seem to concentrate at intermediate areas of the radial distributions of all the objects orbiting in each (sub)system. We show that angular-momentum transport during secular evolution of (exo)planets and satellites necessarily results in the observed intermediate accumulation of the massive objects. We quantify the ‘middle’ as the mean of mean motions (orbital angular velocities) when three or more massive objects are involved. Radial evolution of the orbits is expected to be halted when the survivors settle near mean-motion resonances and angular-momentum transfer between them ceases (gravitational Landau damping). This dynamical behavior is opposite in direction to what has been theorized for viscous and magnetized accretion disks, in which gas spreads out and away from either side of any conceivable intermediate area. We present angular momentum transfer calculations in few-body systems, and we also calculate the tidal dissipation timescales and the physical properties of the mean tidal field in planetary and satellite (sub)systems.

Long-term double synchronization in close-in gas giant planets

ABSTRACT Hot Jupiters, orbiting their host stars at extremely close distances, undergo tidal evolution, with some being engulfed by their stars due to angular momentum exchanges induced by tidal forces. However, achieving double synchronization can prolong their survival. Using the mesa stellar evolution code, combined with the magnetic braking model of Matt et al. (2015), we calculate 25 000 models with different metallicity and study how to attain the conditions that trigger the long-term double synchronization. Our results indicate that massive planets orbiting stars with lower convective turnover time are easier to achieve long-term double synchronization. The rotation angular velocity at the equilibrium point (Ωsta) is almost equal to orbital angular velocity of planet (n) for the majority of the main sequence lifetime if a system has undergone a long-term double synchronization, regardless of their state at this moment. We further compared our results with known parameters of giant planetary systems and found that those systems with larger planetary masses and lower convective turnover time seem to be less sensitive to changes in the tidal quality factor $Q^{\prime }_{_*}$. We suggest that for systems that fall on the state of Ωsta ≈ n, regardless of their current state, the synchronization will persist for a long time if orbital synchronization occurs at any stage of their evolution. Our results can be applied to estimate whether a system has experienced long-term double synchronization in the past or may experience it in the future.

Time characteristics of microsatellite attitude control at intervals between reac-tion wheels desaturations

The article is devoted to the processes study of changing the microsatellite reaction wheels angular velocities during its stabilization, turn and detumbling. Satellite control is considered under conditions of constant and harmonic external torques, as well as under gravitational, aerodynamic and magnetic disturbing torques. Expressions that describe the change in the reaction wheels angular velocities during satellite stabilization under the action of constant and harmonic torques at a frequency equal to the orbital angular velocity are given. The influence of the magnetic disturbing torque caused by the constant component of the satellite's dipole moment on the accumulation of the reaction wheels angular momentum has been studied. The formula for determining the magnitude of the reaction wheels angular velocities oscillations was obtained using a model of the Earth's magnetic field in the form of the straight dipole and taking into account a circular orbit. Moreover, its use for calculating the maximum values of angular velocities gives results that are very close to the results of modeling taking into account accurate models of the Earth’s magnetic field and the satellite’s center of mass trajectory. The maximum possible operating time intervals of the control system without forced desaturation of its controls are estimated.

General formulae for the periapsis shift of a quasi-circular orbit in static spherically symmetric spacetimes and the active gravitational mass density

We study the periapsis shift of a quasi-circular orbit in general static spherically symmetric spacetimes. We derive two formulae in full order with respect to the gravitational field, one in terms of the gravitational mass [Formula: see text] and the Einstein tensor and the other in terms of the orbital angular velocity and the Einstein tensor. These formulae reproduce the well-known ones for the forward shift in the Schwarzschild spacetime. In a general case, the shift deviates from that in the vacuum spacetime due to a particular combination of the components of the Einstein tensor at the radius [Formula: see text] of the orbit. The formulae give a backward shift due to the extended-mass effect in Newtonian gravity. In general relativity, in the weak-field and diffuse regime, the active gravitational mass density, [Formula: see text], plays an important role, where [Formula: see text], [Formula: see text] and [Formula: see text] are the energy density, the radial stress and the tangential stress of the matter field, respectively. We show that the shift is backward if [Formula: see text] is beyond a critical value [Formula: see text][Formula: see text]g/cm3[Formula: see text], while a forward shift greater than that in the vacuum spacetime instead implies [Formula: see text], i.e. the violation of the strong energy condition, and thereby provides evidence for dark energy. We obtain new observational constraints on [Formula: see text] in the Solar System and the Galactic Centre.

Back Cover: Predestined Steady‐State Entanglement under Pure Vacuum Noise (Adv. Quantum Technol. 3/2023)

For a pair of rotating qubits under pure vacuum noise with appropriate orbital radius and angular velocity, entanglement is not only resilient against vacuum noise for initially entangled states, but is also predestined to be created and be preserved for arbitrary initially separable states. More details can be found in article number 2200159 by Yuebing Zhou, Jiawei Hu, and Hongwei Yu.

Predestined Steady‐State Entanglement under Pure Vacuum Noise

AbstractEver since the seminal work by Yu and Eberly [Phys. Rev. Lett. 93, 140404 (2004)], it has become widely accepted that nonlocal entanglement is more fragile than local coherence, in the sense that even under the influence of pure vacuum noise, the two‐qubit entanglement may end within a finite time, while the single‐qubit coherence always decays exponentially. By contrast, in this paper, it is shown that, for a pair of rotating qubits under pure vacuum noise with appropriate orbital radius and angular velocity, nonlocal entanglement as compared to local coherence is not only more resilient against vacuum noise to be preserved in the steady‐state for initially entangled states, but more importantly is also predestined to be created and be preserved for initially separable states. The generation of predestined entanglement here is essentially different from the dissipative steady‐state entanglement generation in existing schemes insofar as it is inevitably generated in pure vacuum instead of an elaborately designed environment.

Steady-state entanglement for rotating Unruh-DeWitt detectors

The Unruh-DeWitt model of particle detector is widely used to probe quantum effects in noninertial frames and curved spacetimes. In this paper, we investigate the entanglement dynamics of a quantum system composed of two rotating Unruh-DeWitt detectors in interaction with a fluctuating massless scalar field in vacuum. We obtain the necessary and sufficient condition for the detectors to achieve steady-state entanglement, and systematically investigate the steady-state entanglement of two Unruh-DeWitt detectors rotating in coaxial orbits with the same orbital radius and angular velocity. When the separation between the detectors is vanishing, the detectors can obtain steady-state entanglement dependent on the initial state, as the detectors in uniform acceleration and static detectors in a thermal bath do. Remarkably, however, when the separation between the detectors is nonvanishing but small compared with the transition wavelength of detectors, the detectors can obtain steady-state entanglement irrespectively of what the initial state is. This is the unique phenomenon caused by the centripetal acceleration and is not present in the uniform acceleration case and the case of static detectors in a thermal bath.

Exploratory scenarios for the differential nuclear and tidal evolution of TZ Fornacis

Aims. TZ Fornacis is a double-lined eclipsing binary system with similar masses (2.057 ± 0.001 and 1.958 ± 0.001 M⊙) but characterized by very different radii (8.28 ± 0.22 and 3.94 ± 0.17 R⊙). This similarity in terms of mass makes it possible to study the system’s differential stellar evolution as well as some aspects of its tidal evolution. With regard to its orbital elements, it was recently confirmed that its orbit is circular with an orbital period of 75.7 days. The less massive component rotates about 17 times faster than the primary one, which is synchronized with the mean orbital angular velocity. Our main objective in this work is to study both the nuclear and the tidal evolution of the system. Methods. To model the TZ For system, we used the MESA package, computing the grids using the exact observed masses, radii, and effective temperatures as input, and then varying the metallicity, the core overshooting amount, and the mixing-length parameter. A χ2 statistic was used to infer the optimal values of the core overshooting and the mixing-length parameters. The same procedure was used to generate rotating models with the GRANADA code. The respective errors in the average age of TZ For were less than 5%. On the other hand, the differential equations that govern the tidal evolution were integrated using the fifth-order Runge–Kutta method, ith a tolerance of 1 × 10−7. Results. We explored two scenarios regarding the initial eccentricities: a high one (0.30) and a case of an initial circular orbit. A good agreement has been found between the observational values of the eccentricity, synchronism levels, and orbital period with the values predicted by the integration of the tidal evolution equations. The influence of the friction timescale on the evolution of the orbital elements of TZ For is also studied here. The orbital elements most affected by the uncertainties in the friction timescale are the synchronism levels of the two components. On the other hand, we used the properties of the rotating models generated by the GRANADA code as the initial angular velocities instead of using trial values. In this case, comparisons between the theoretical values of the orbital elements and their observed counterparts also lead to a good interagreement.

Tidal effect on the gyroscopic precession around a compact star

General relativistic effects around massive astrophysical objects can be captured using a test gyro orbiting the object in a circular geodesic. This paper discusses how the tidal field due to a companion object affects the spin precession frequency and orbital angular velocity of a spinning gyro orbiting around a compact astrophysical object. The precession frequency is studied in a region of space around the central object using a perturbative approach. The central object is either a neutron star or a white dwarf in this study. The test gyro is any planetary or asteroid-like object orbiting a neutron star or a white dwarf. Moreover, the companion object that causes the tidal field can be a neutron star, white dwarf or a stellar black hole. It is seen that the tidal effect significantly affects the spacetime around the central object, which affects the gyro precession frequency and the orbital angular velocity. Slow rotation approximation has been considered for the central object, which creates negligible deformation. The change in the gyro’s precession frequency and the orbital angular velocity due to the tidal field increases with an increase in the companion object’s mass and decreases as the separation between the central star and the companion star increases. The tidal effect also varies with the stiffness of the equation of state of matter describing the host star. The lower the compactness of the host star, the greater is the tidal response; thus the greater is the change in the gyro’s precession and angular velocity of the geodesic.

Providing Nanosatellite Triaxial Gravitational Orientation Using Magnetic Actuators

Three-axis gravity stabilization of 3U CubeSat is achieved due to selection of the nanosatellite moments of inertia at the design stage, as well as special modes included in the algorithm to provide stabilization of CubeSat relative to each motion channel separately. In this paper, we propose a modified algorithm based on the magnetic stabilization algorithm B-dot. The modified algorithm provides three modes intended to damp the initial angular velocity to the value of the orbital angular velocity, to keep the angular velocity at a value close to that of the orbital angular velocity, and to provide the nanosatellite gravitational triaxial stabilization by using one magnetic coil located on the axis with the transversal moment of inertia, which is possible due to the small angle between the magnetic field line and the satellite's trajectory. We propose two modifications for forming a control loop for orientation and stabilization of the 3U CubeSat: the first one uses measurements from magnetometers and angular rate sensors as feedback, and the second one, only magnetometers. The efficiency of the two modifications of modifications was studied by means of statistical modeling.

Reducing orbital eccentricity in initial data of black hole–neutron star binaries in the puncture framework

We develop a method to compute low-eccentricity initial data of black hole--neutron star binaries in the puncture framework extending previous work on other types of compact binaries. In addition to adjusting the orbital angular velocity of the binary, the approaching velocity of a neutron star is incorporated by modifying the helical Killing vector used to derive equations of the hydrostationary equilibrium. The approaching velocity of the black hole is then induced by requiring the vanishing of the total linear momentum of the system, differently from the case of binary black holes in the puncture framework where the linear momentum of each black hole is specified explicitly. We successfully reduce the orbital eccentricity to <~0.001 by modifying the parameters iteratively using simulations of ~3 orbits both for nonprecessing and precessing configurations. We find that empirical formulas for binary black holes derived in the excision framework do not reduce the orbital eccentricity to ~0.001 for black hole--neutron star binaries in the puncture framework, although they work for binary neutron stars.

The role of galactic dynamics in shaping the physical properties of giant molecular clouds in Milky Way-like galaxies

ABSTRACT We examine the role of the large-scale galactic-dynamical environment in setting the properties of giant molecular clouds in Milky Way-like galaxies. We perform three high-resolution simulations of Milky Way-like discs with the moving-mesh hydrodynamics code arepo, yielding a statistical sample of ${\sim}80\, 000$ giant molecular clouds and ${\sim}55\, 000$ H i clouds. We account for the self-gravity of the gas, momentum, and thermal energy injection from supernovae and H ii regions, mass injection from stellar winds, and the non-equilibrium chemistry of hydrogen, carbon, and oxygen. By varying the external gravitational potential, we probe galactic-dynamical environments spanning an order of magnitude in the orbital angular velocity, gravitational stability, mid-plane pressure, and the gradient of the galactic rotation curve. The simulated molecular clouds are highly overdense (∼100×) and overpressured (∼25×) relative to the ambient interstellar medium. Their gravoturbulent and star-forming properties are decoupled from the dynamics of the galactic mid-plane, so that the kpc-scale star formation rate surface density is related only to the number of molecular clouds per unit area of the galactic mid-plane. Despite this, the clouds display clear, statistically significant correlations of their rotational properties with the rates of galactic shearing and gravitational free-fall. We find that galactic rotation and gravitational instability can influence their elongation, angular momenta, and tangential velocity dispersions. The lower pressures and densities of the H i clouds allow for a greater range of significant dynamical correlations, mirroring the rotational properties of the molecular clouds, while also displaying a coupling of their gravitational and turbulent properties to the galactic-dynamical environment.

Influence of bias magnetic field direction on band structure of magnetic photonic crystal and its role in constructing topological edge states

The topological transitions in two-dimensional photonic crystals (PCs) originate from the opening-closing-reopening of the bandgap, accompanied with the band order inversion. The topological bandgap in magnetic PC can be created by applying a bias magnetic field or deforming the geometry structure of the PC. In this paper, we demonstrate that the direction of the bias magnetic field also plays a key role in modifying the band structure in a two-dimensional magnetic PC. The results show that by reversing the direction of the bias magnetic field, the eigenstates with the same parity may exchange their orders in the band structure. We investigate this type of band order exchange in the applications of constructing topological edge states and its influence on the properties of edge states. We find, for example, reversing the direction of the bias magnetic field can create two almost degenerated topological edge modes, which propagate in the same direction but have opposite orbital angular momenta. The edge modes and their characteristics can be determined by the schematics of the band orders for the photonic crystals on the two sides of the boundary. The relative relationship of the band orders determines the emergence of the topological edge states, the number of edge states, and edge modes’ properties such as the orbital angular momentum and group velocity. Also, it affects the transmission efficiency of the electromagnetic wave on the boundary. The direction effect of the bias magnetic field on band order exchange presented in this paper provides us with a new way to change the feature of topological edge states and helps us to better understand the influence of band order on topological phases of photonic crystals. It may have potential applications, such as in pseudo-spin splitter and reflection-free one-way optical switch.

Kepler´s Ellipse Observed from Newton´s Evolute (1687), Horrebow´s Circle (1717), Hamilton´s Pedal Curve (1847), and Two Contrapedal Curves (28.10.2018)

Johannes Kepler discovered the very elegant elliptical path of planets with the Sun in one focus of that ellipse in 1605. Kepler inspired generations of researchers to study properties hidden in those elliptical paths. The visible elliptical paths belong to the Aristotelian World. On the other side there are invisible mathematical objects in the Plato&amp;acute;s Realm that might describe the mechanism behind those elliptical paths. One such curve belonging to the Plato&amp;acute;s Realm discovered Isaac Newton in 1687 - the locus of radii of curvature of that ellipse (the evolute of the ellipse). Are there more curves in the Plato&amp;acute;s Realm that could reveal to us additional information about Kepler&amp;acute;s ellipse? W.R. Hamilton in 1847 discovered the hodograph of the Kepler&amp;acute;s ellipse using the pedal curve with pedal points in both foci (the auxiliary circle of that ellipse). This hodograph depicts the moment of the tangent momentum of orbiting planets. Inspired by the hodograph model we propose newly to use two contrapedal curves of the Kepler&amp;acute;s ellipse with contrapedal points in both the Kepler&amp;acute;s occupied and Ptolemy&amp;acute;s empty foci. Observers travelling along those contrapedal curves might bring new valuable experimental data about the orbital angular velocity of planets and a new version of the Kepler&amp;acute;s area law. Based on these contrapedal curves we have defined the moment of the normal momentum. The first derivation of the moment of the normal momentum reveals the torque of the ellipse. This torque of ellipse should contribute to the precession of the Kepler&amp;acute;s ellipse. In the Library of forgotten works of Old Masters we have re-discovered the Horrebow&amp;acute;s circle (1717) and the Colwell&amp;acute;s anomaly H (1993) that might serve as an intermediate step in the solving of the Kepler&amp;acute;s Equation (KE). Have we found the Arriadne&amp;acute;s Thread leading out of the Labyrinth or are we still lost in the Labyrinth?

Эволюция вращательного движения спутника с гибкими вязкоупругими стержнями на эллиптической орбите

The rotational motion of an artificial satellite in the central Newtonian force field is investigated. The satellite is modeled as a symmetric solid with rigid viscoelastic rods rigidly attached along the symmetry axis and with a spherical inertia tensor. A limited statement of the problem is considered: the center of mass of the satellite moves along a given Keplerian elliptic orbit. As is known, some artificial Earth satellites (AES) have whip radio antennas for a length of hundreds of meters. The problem considered in this paper is a model for studying the rotational motion of this type of satellites. To solve this problem, the method of separation of motions and averaging is used for mechanical systems with an infinite number of degrees of freedom. We obtain an averaged system of differential equations in the Andoyer variables, which describes the evolution of the satellite's rotational motion. Partial solutions of the evolutionary system of the equations are found. For each such class of solutions phase portraits are built. It is shown that in the process of dissipative evolution caused by energy dissipation in rods, the angle between the vector of the angular momentum of the satellite's rotational motion and the normal to the orbital plane decreases to zero. Besides, new classes of stationary motions are found: the axis of symmetry of the satellite makes an arbitrary angle with the normal to the orbit plane, or the module of the angular velocity of the satellite rotation does not depend on the orbital angular velocity of the radius vector of its center of mass.

Structural integrity of rubble asteroidal satellites

We investigate the structural integrity of asynchronous, rubble asteroidal satellites, which may or may not be in synchronous motion, by identifying necessary conditions for the structural equilibrium of such objects. Asynchronous motion may occur when the satellite’s rotation rate about its mass center does not match its orbital angular velocity about the asteroid. The asteroid’s shape is modeled as a triaxial ellipsoid to account for variability in asteroid shapes. The satellites are considered to be oblate or prolate objects, whose rubble interiors are modeled as a rigid-plastic, cohesive material obeying the Drucker–Prager yield criterion. We employ volume-averaging to obtain average stresses in the satellite, which are combined with the yield criterion to obtain necessary conditions for the satellite’s structural equilibrium. Structural equilibrium regions are obtained in terms of the satellite’s shape through the extent of its flattening/elongation, the satellite’s orientation and orbital location with respect to the central asteroid, as characterized by the lead angle of the satellite’s long axis, and an equivalent radius, which accounts for the asteroid’s irregular shape and possibly different density from the satellite. When required, the satellite’s orbital angular velocity is estimated from its distance from the asteroid, assuming that satellite’s orbit is circular. We demonstrate that necessary conditions for the satellite’s structural integrity can be discussed in terms of two particular configurations, viz. the long- and intermediate axis configurations in which the satellite’s long and intermediate axis point towards the asteroid. This reduces a complex dynamical analysis to a simpler quasi-static one. The effects of inter-grain cohesion are explicitly included, as cohesion can play an important role in the extremely low-gravity environments of asteroidal satellites. The final results are applied to recently discovered asteroidal satellites to support or refute the possibility them being rubble piles.

The complex kinematics of rotating star clusters in a tidal field

Abstract We broaden the investigation of the dynamical properties of tidally perturbed, rotating star clusters by relaxing the traditional assumptions of coplanarity, alignment, and synchronicity between the internal and orbital angular velocity vector of their initial conditions. We show that the interplay between the internal evolution of these systems and their interaction with the external tidal field naturally leads to the development of a number of evolutionary features in their three-dimensional velocity space, including a precession and nutation of the global rotation axis and a variation of its orientation with the distance from the cluster centre. In some cases, such a radial variation may manifest itself as a counter-rotation of the outermost regions relative to the inner ones. The projected morphology of these systems is characterized by a non-monotonic ellipticity profile and, depending on the initial inclination of the rotation axis, it may also show a twisting of the projected isodensity contours. These results provide guidance in the identification of non-trivial features which may emerge in upcoming investigations of star cluster kinematics and a dynamical framework to understand some of the complexities already hinted by recent observational studies.

Close binary evolution

Context. Massive stars with solar metallicity lose important amounts of rotational angular momentum through their winds. When a magnetic field is present at the surface of a star, efficient angular momentum losses can still be achieved even when the mass-loss rate is very modest, at lower metallicities, or for lower-initial-mass stars. In a close binary system, the effect of wind magnetic braking also interacts with the influence of tides, resulting in a complex evolution of rotation. Aims. We study the interactions between the process of wind magnetic braking and tides in close binary systems. Methods. We discuss the evolution of a 10 M⊙ star in a close binary system with a 7 M⊙ companion using the Geneva stellar evolution code. The initial orbital period is 1.2 days. The 10 M⊙ star has a surface magnetic field of 1 kG. Various initial rotations are considered. We use two different approaches for the internal angular momentum transport. In one of them, angular momentum is transported by shear and meridional currents. In the other, a strong internal magnetic field imposes nearly perfect solid-body rotation. The evolution of the primary is computed until the first mass-transfer episode occurs. The cases of different values for the magnetic fields and for various orbital periods and mass ratios are briefly discussed. Results. We show that, independently of the initial rotation rate of the primary and the efficiency of the internal angular momentum transport, the surface rotation of the primary will converge, in a time that is short with respect to the main-sequence lifetime, towards a slowly evolving velocity that is different from the synchronization velocity. This “equilibrium angular velocity” is always inferior to the angular orbital velocity. In a given close binary system at this equilibrium stage, the difference between the spin and the orbital angular velocities becomes larger when the mass losses and/or the surface magnetic field increase. The treatment of the internal angular momentum transport has a strong impact on the evolutionary tracks in the Hertzsprung-Russell Diagram as well as on the changes of the surface abundances resulting from rotational mixing. Our modelling suggests that the presence of an undetected close companion might explain rapidly rotating stars with strong surface magnetic fields, having ages well above the magnetic braking timescale. Our models predict that the rotation of most stars of this type increases as a function of time, except for a first initial phase in spin-down systems. The measure of their surface abundances, together, when possible, with their mass-luminosity ratio, provide interesting constraints on the transport efficiencies of angular momentum and chemical species. Conclusions. Close binaries, when studied at phases predating any mass transfer, are key objects to probe the physics of rotation and magnetic fields in stars.

BD-19 5044L: discovery of a short-period SB2 system with a magnetic Bp primary in the open cluster IC 4725

Until recently almost nothing was known about the evolution of magnetic fields found in upper main sequence Ap/Bp stars during their long main sequence lifetime. We are thus studying magnetic Ap/Bp stars in open clusters in order to obtain observational evidence of how the properties of Ap/Bp magnetic stars, such as field strength and structure, evolve with age during the main sequence. One important aspect of this study is to search for the very rare examples of hot magnetic stars in short-period binary systems among magnetic cluster members. In this paper we characterize the object BD-19~5044L, which is both a member of the open cluster IC 4725 = M~25, and a short-period SB2 system containing a magnetic primary star. We have obtained a series of intensity and circular polarisation spectra distributed through the orbital and rotation cycles of BD-19 5044L with the ESPaDOnS spectropolarimeter at CFHT. We find that the orbit of BD-19 5044L AB is quite eccentric (e = 0.477), with a period of 17.63 d. The primary is a magnetic Bp star with a variable longitudinal magnetic field, a polar field strength of ~1400 G and a low obliquity, while the secondary is probably a hot Am star and does not appear to be magnetic. The rotation period of the primary (5.04 d) is not synchronised with the orbit, but the rotation angular velocity is close to being synchronised with the orbital angular velocity of the secondary at periastron, perhaps as a result of tidal interactions. The periastron separation is small enough (about 12 times the radius of the primary star) that BD-19 5044L may be one of the very rare known cases of a tidally interacting SB2 binary system containing a magnetic Ap/Bp star.

Tidally driven mean flows in slowly and uniformly rotating massive main-sequence stars

We calculate tidally driven mean flows in a slowly and uniformly rotating massive main sequence star in a binary system. We treat the tidal potential due to the companion as a small perturbation to the primary star. We compute tidal responses of the primary as forced linear oscillations, as a function of the tidal forcing frequency $\omega_{\rm tide}=2(\Omega_{\rm orb}-\Omega)$, where $\Omega_{\rm orb}$ is the mean orbital angular velocity and $\Omega$ is the angular velocity of rotation of the primary star. The amplitude of the tidal responses is proportional to the parameter $f_0\propto (M_2/M)(a_{\rm orb}/R)^{-3}$, where $M$ and $M_2$ are the masses of the primary and companion stars, $R$ is the radius of the primary and $a_{\rm orb}$ is the mean orbital separation between the stars. For a given $f_0$, the amplitudes depend on $\omega_{\rm tide}$ and become large when $\omega_{\rm tide}$ is in resonance with natural frequencies of the star. Using the tidal responses, we calculate axisymmetric mean flows, assuming that the mean flows are non-oscillatory flows driven via non-linear effects of linear tidal responses. We find that the $\phi$-component of the mean flow velocity dominates. We also find that the amplitudes of the mean flows are large only in the surface layers where non-adiabatic effects are significant and that the amplitudes are confined to the equatorial regions of the star. Depending on $M_2/M$ and $a_{\rm orb}/R$, the amplitudes of mean flows at the surface become significant.