Periodic perturbations in close binary systems due to tidal distortion (non-lagging tides)

In a previous paper (Zafiropoulos and Kopal, 1982; hereafter referred to as Paper I) we have studies the effects of rotational distortion on the orbital elements. The aim of the present paper is to investigate the secular and periodic perturbations of the orbital elements due to tidal distortion. For tidal distortion when tides do not lag, the Gaussian form of Lagrange's planetary equations has been employed to yield the first- and second-order approximations. The results obtained include the effects produced by the second, third and fourth harmonic distortions. The first order approximation for non-lagging tides has been expressed by means of Hansen coefficients.     

Periodic perturbations in close binary systems due to tidal lag

In two previous papers (Zafiropoulos and Kopal, 1983a, b; hereafter referred to as Papers I and II) we have investigated the effects of rotational and tidal distortion (for non-lagging tides) on the orbital elements of a close binary system. The present paper deals with secular and periodic perturbations caused by dynamical tides. The componentsR, S, andW of disturbing accelerations for tidal lag have been substituted in the Gaussian form of Lagrange's planetary equations to give the first-order approximation. The results obtained have been expressed by means of Hansen coefficients and include the effects produced by the second, third and fourth harmonic dynamical tides.     

Perturbations in close binary systems produced by the tides lagging in latitude

The aim of this investigation is to present the periodic and secular perturbations of the orbital elements of close binary systems due to tidal lag in latitude. The variational equations of the problem of plane motion will be set up in terms of the rectengular componentsR, S, andW of the disturbing accelerations. These equations are highly nonlinear with respect to the orbital elements and we present analytic approximations to the effects produced by the perturbing acceleration due to dynamical tides lagging in latitude. The perturbed elements of the orbit have been expressed by means of Hansen coefficients in the compact form of summations.     

On the tidal effects in the motion of artificial satellites

The general perturbations in the elliptic and vectorial elements of a satellite as caused by the tidal deformations of the non-spherical Earth are developed into trigonometric series in the standard ecliptical arguments of Hill-Brown lunar theory and in the equatorinal elements ω and Ω of the satellite. The integration of the differential equations for variation of elements of the satellite in this theory is easy because all arguments are linear or nearly linear in time. The trigonometrical expansion permits a judgment about the relative significance of the amplitudes and periods of different tidal ‘waves’ over a long period of time. Graphs are presented of the tidal perturbations in the elliptic elements of the BE-C satellite which illustrate long term periodic behavior. The tidal effects are clearly noticeable in the observations and their comparison with the theory permits improvement of the ‘global’ Love numbers for the Earth.     

Some Recurrence Relations Between Hansen Coefficients

A new system of recurrence relations for Hansen coefficients is obtained. This system gives a connection between only those coefficients which are included in the disturbing function of planetary or satellite motion and allows to compute efficiently the Hansen coefficients for perturbations both from internal and external bodies. The recurrence process can be realized both from high to low and from low to high harmonical terms of the disturbing function. The corresponding algorithms of evaluation of Hansen coefficients are presented. The efficiency of the obtained system of recurrence relations is discussed.     

Orbital evolution of a planet with tidal dissipation in a restricted three-body system

The angle between planetary spin and the normal direction of an orbital plane is supposed to reveal a range of information about the associated planetary formation and evolution. Since the orbit's eccentricity and inclination oscillate periodically in a hierarchical triple body and tidal friction makes the spin parallel to the normal orientation of the orbital plane with a short timescale in an isolated binary system, we focus on the comprehensive effect of third body perturbation and tidal mechanism on the angle. Firstly, we extend the Hut tidal model(1981) to the general spatial case, adopting the equilibrium tide and weak friction hypothesis with constant delay time, which is suitable for arbitrary eccentricity and any angle ? between the planetary spin and normal orientation of the orbital plane. Furthermore, under the constraint of angular momentum conservation, the equations of orbital and ratational motion are given. Secondly, considering the coupled effects of tidal dissipation and third body perturbation, and adopting the quadrupole approximation as the third body perturbation effect, a comprehensive model is established by this work. Finally, we find that the ultimate evolution depends on the timescales of the third body and tidal friction. When the timescale of the third body is much shorter than that of tidal friction, the angle ? will oscillate for a long time,even over the whole evolution; when the timescale of the third body is observably larger than that of the tidal friction, the system may enter stable states, with the angle ? decaying to zero ultimately, and some cases may have a stable inclination beyond the critical value of Lidov-Kozai resonance. In addition, these dynamical evolutions depend on the initial values of the orbital elements and may aid in understanding the characteristics of the orbits of exoplanets.     

Oort Cloud Comet Perihelion Asymmetries: Galactic Tide, Shower or Observational Bias?

We investigate the distribution of Oort cloud comet perihelia. The data considered includes comets having orbital elements of the two highest quality classes with original energies designated as new or young. Perihelion directions are determined in galactic, ecliptic and geocentric equatorial coordinates. Asymmetries are detected in the scatter and are studied statistically for evidence of adiabatic galactic tidal dynamics, an impulse-induced shower and observational bias. The only bias detected is the well-known deficiency of observations with perihelion distances q > 2.5 AU. There is no significant evidence of a seasonal dependence. Nor is there a substantive hemispherical bias in either ecliptic or equatorial coordinates. There is evidence for a weak stellar shower previously detected by Biermann which accounts for ≈ 10% of the total observations. Both the q bias and the Biermann star track serve to weaken the evidence for a galactic tidal imprint. Nevertheless, statistically significant asymmetries in galactic latitude and longitude of perihelia remain. A latitude asymmetry is produced by a dominant tidal component perpendicular to the galactic disk. The longitude signal implies that ≈ 20% of new comets need an additional dynamical mechanism. Known disk non-uniformities and an hypothetical bound perturber are discussed as potential explanations. We conclude that the detected dynamical signature of the galactic tide is real and is not an artifact of observational bias, impulsive showers or poor data. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Correlations of cosmic tidal fields

We study correlations amongst tidal fields originated by the large-scale distribution of matter in the Universe. The two-point tidal correlation is described as a rank-4 tensor, the elements of which can be written in terms of four fundamental scalar functions ranging, with respect to the spatial separation, from purely transversal to purely longitudinal correlations. Tidal fields, on both galaxy and cluster scales, are revealed to be correlated over distances larger than the mass–density correlation length, although traceless tidal fields show anti-correlation between diagonal terms along orthogonal directions. The cross-correlation between mass and tidal field is also analysed. These results are relevant for galaxy formation and the interpretation of large-scale weak lensing phenomena.     

Tidal tails of star clusters

Based on recent findings of a formation mechanism of substructure in tidal tails by Küpper et al., we investigate a more comprehensive set of N -body models of star clusters on orbits about a Milky Way like potential. We find that the predicted epicyclic overdensities arise in any tidal tail no matter which orbit the cluster follows as long as the cluster lives long enough for the overdensities to build up.
The distance of the overdensities along the tidal tail from the cluster centre depends for circular orbits only on the mass of the cluster and the strength of the tidal field, and therefore decreases monotonically with time, while for eccentric orbits the orbital motion influences the distance, causing a periodic compression and stretching of the tails and making the distance oscillate with time. We provide an approximation for estimating the distance of the overdensities in this case.
We describe an additional type of overdensity which arises in extended tidal tails of clusters on eccentric orbits, when the acceleration of the tidal field on the stellar stream is no longer homogeneous. Moreover, we conclude that a pericentre passage or a disc shock is not the direct origin of an overdensity within a tidal tail. Escape due to such tidal perturbations does not take place immediately after the perturbation but is rather delayed and spread over the orbit of the cluster. All observable overdensities are therefore of the mentioned two types. In particular, we note that substructured tidal tails do not imply the existence of dark matter substructures in the haloes of galaxies.     

Electron capture in carbon dwarf supernovae

The rates of electron capture on heavier elements under the extreme conditions predicted for dwarf star supernovae have been computed, incorporating modifications that seem to be indicated by present experimental results. An estimate of the maximum possible value of such rates is also given. The distribution of nuclei in nuclear statistical equilibrium has been calculated for the range of expected supernovae conditions, including the effects of the temperature dependence of nuclear partition functions. These nuclide abundance distributions are then used to compute nuclear equilibrium thermodynamic properties. The effects of the electron capture on such equilibrium matter are discussed. The results of supernova numerical hydrodynamics incorporating the computed equilibrium properties and the influence of electron capture are presented. In the context of the ‘carbon detonation’ supernova model, the dwarf central density required to assure core collapse to a neutron star configuration is found to be slightly higher than that obtained by Bruenn (1972) with the electron capture rates of Hansen (1966).     

Recursive calculation of hansen coefficients

Hansen coefficients are used in expansions of the elliptic motion. Three methods for calculating the coefficients are studied: Tisserand's method, the Von Zeipel-Andoyer (VZA) method with explicit representation of the polynomials required to compute the Hansen coefficients, and the VZA method with the values of the polynomials calculated recursively. The VZA method with explicit polynomials is by far the most rapid, but the tabulation of the polynomials only extends to 12th order in powers of the eccentricity, and unless one has access to the polynomials in machine-readable form their entry is laborious and error-prone. The recursive calculation of the VZA polynomials, needed to compute the Hansen coefficients, while slower, is faster than the calculation of the Hansen coefficients by Tisserand's method, up to 10th order in the eccentricity and is still relatively efficient for higher orders. The main advantages of the recursive calculation are the simplicity of the program and one's being able to extend the expansions to any order of the eccentricity with ease. Because FORTRAN does not implement recursive procedures, this paper used C for all of the calculations. The most important conclusion is recursion's genuine usefulness in scientific computing.     

The weak friction approximation and tidal evolution in close binary systems

The aim of this paper is to study the dynamical problem of tidal friction in a binary system consisting of deformable components, with the restriction that the angle of lag or advance of the tidal distortion with respect to the direction of the disturbing companion is small. The fractional distortion of the bodies due to rotation and tidal interaction is also treated as a first-order small quantity, and terms up to the fourth harmonic in the tidal potential are retained. In this linear approximation, the time-dependent tidal potential can be Fourier decomposed into a spectrum of simple harmonic terms, each of which is responsible for raising a partial wave in the body; each such partial wave can then be treated independently of the others. This is the method first employed by Darwin.In Section 2, it is assumed that the phase lag in the response of the body (due to dissipation of kinetic energy of deformation) is proportional to the forcing frequency, which is justified for small amplitude oscillations of a viscous fluid or visco-elastic body. A simple expression is then obtained for the potential function for the distortion in terms of the disturbing potential and the structure of the body.In Section 3, the distortion potential function is employed in deriving the componentsR, S andW of the disturbing force which are then substituted in the Gaussian form of the equations for variation of the elements. In Section 4, the Eulerian equations for motion of deformable bodies are derived, using the so-called mean axes of the body as the rotating axes of reference. In Section 5, it is shown that the dynamical effects of rotational distortion occur on a much shorter time scale than those arising from tidal friction, which allows one to consider the two phenomena as acting independently of one another. The collected set of Gaussian (orbital) and Eulerian (body) equations is re-written in terms of dimensionless variables for the tidal friction case, and the stability of the system is examined on the basis of these equations.In Section 6, the tidal friction equations are integrated numerically for the close binary system AG Persei and for the Earth-Moon system. In the former, the integrations were started from a highly elliptical orbit and the system was found to relax into a circular orbit, with synchronous rotation perpendicular to the orbit. In the latter, the integrations were performed backwards in time from the present day, and it was found that the lunar orbit rapidly becomes highly elliptical at the time of closest approach, thus indicating a probable capture of the Moon by the Earth. This result is in agreement with that obtained by other investigators; however, it is shown that the detailed behaviour of the system at the time of capture, in particular the inclination of the lunar orbit to the ecliptic, depends critically on the chosen rate of dissipation in the Moon's interior. A simple argument is presented which allows an estimation for the mean viscosity of a fluid body from the known age of the system: for the components of AG per, the result is 2×10¹¹ g cm^–1 s^–1, indicating that the stars must have possessed turbulent convective outer regions during some part of their tidal evolution, while for the Earth, the result, is 1.4×10¹² g cm^–1 s^–1. It is shown that the angle of tidal lag in nonsynchronous close binary systems in general is expected to be extremely small, and not observationally detectable.     

The uniform,regular differential equations of the KS transformed perturbed two-body problem

The Newtonian differential equations of motion for the two-body problem can be transformed into four, linear, harmonic oscillator equations by simultaneously applying the regularizing time transformation dt/ds=r and the Kustaanheimo-Stiefel (KS) coordinate transformation. The time transformation changes the independent variable from time to a new variables, and the KS transformation transforms the position and velocity vectors from Cartesian space into a four-dimensional space. This paper presents the derivation of uniform, regular equations for the perturbed twobody problem in the four-dimensional space. The variation of parameters technique is used to develop expressions for the derivatives of ten elements (which are constants in the unperturbed motion) for the general case that includes both perturbations which can arise from a potential and perturbations which cannot be derived from a potential. These element differential equations are slightly modified by introducing two additional elements for the time to further improve long term stability of numerical integration.Originally presented at the AAS/AIAA Astrodynamics Specialists Conference, Vail, Colorado, July 1973     

The haloes of merger remnants

We perform collisionless N -body simulations of 1:1 galaxy mergers, using models which include a galaxy halo, disc and bulge, focusing on the behaviour of the halo component. The galaxy models are constructed without recourse to a Maxwellian approximation. We investigate the effect of varying the galaxies' orientation, their mutual orbit and the initial velocity anisotropy or cusp strength of the haloes upon the remnant halo density profiles and shape, as well as on the kinematics. We observe that the halo density profile (determined as a spherical average, an approximation we find appropriate) is exceptionally robust in mergers, and that the velocity anisotropy of our remnant haloes is nearly independent of the orbits or initial anisotropy of the haloes. The remnants follow the halo anisotropy – local density slope (β–γ) relation suggested by Hansen & Moore in the inner parts of the halo, but β is systematically lower than this relation predicts in the outer parts. Remnant halo axis ratios are strongly dependent on the initial parameters of the haloes and on their orbits. We also find that the remnant haloes are significantly less spherical than those described in studies of simulations which include gas cooling.     

A simplified model for a nonlinear tidal effect on accretion disks in CVs

This paper investigates the tidal effect on accretion disk in CVs and sets up a simplified model in which the secondary's gravitation is substituted by a mean tidal torque. We find that a linear tidal torque will not be able to maintain an equilibrium disk. By using the result of the radius of the equilibrium disk approximately equals to the tidal radius, which was obtained by using the two dimensional numerical simulation invoking nonlinear tidal effect, we give the modified tidal dissipation function for our simplified model which could be used to interpret the outburst of the dwarf nova with tidal effect. The paper also shows that the radius of an equilibrium disk with a torus is slightly small than the Lubow-Shu radius, and the tidal effect may also cause the cycle of quiescence-superoutburst in addition to the cycle of quiescence-outbursts-superoutburst.     

Dependence of tidal effects on the mass distribution in binary stellar systems

Analytic expressions for tidal disruption and tidal coalescence for binary stellar systems have been derived, using impulsive approximation, as a function of the mass ratio and separation of the components. These expressions can be used for any mass distribution of the components for non-interpenetrating cases.     

Radiative transfer in spherical shell atmospheres: I. Rayleigh scattering

A comparison is made between the plane-parallel approximation and the more realistic spherical shell approximation for the radiance reflected from a planetary atmosphere. In this paper we have considered a planet of radius 6371 km (the Earth) with a homogeneous, conservative, Rayleigh scattering atmosphere extending to a height of 100 km. We have found significant departures from the plane-parallel approximation. Radiance versus height distributions for both single and multiple scattering are presented. Results are presented for the fractional radiance from altitudes in the atmosphere which contribute to the total unidirectional reflected radiance at the top of the atmosphere. We have referred to this as the radiance versus height distribution in the sequel. These data will be very useful for both remote sensing applications and planetary spectroscopy. We have also found that gross violations of the principle of reciprocity do occur in the spherical shell approximation.     

Comment on “Neutrino oscillations in the early universe: how can large lepton asymmetry be generated?” [Astropart. Phys. 14 (2000) 79–90]

We comment on the recent paper by A.D. Dolgov, S.H. Hansen, S. Pastor and D.V. Semikoz (DHPS) [Astropart. Phys. 14 (2000) 79] on the generation of neutrino asymmetries from active–sterile neutrino oscillations. We demonstrate that the approximate asymmetry evolution equation obtained therein is an expansion, up to a minor discrepancy, of the well-established static approximation equation, valid only when the supposedly new higher order correction term is small. In the regime where this so-called “back-reaction” term is large and artificially terminates the asymmetry growth, their evolution equation ceases to be a faithful approximation to the quantum kinetic equations simply because pure Mikheyev–Smirnov–Wolfenstein (MSW) transitions have been neglected. At low temperatures the MSW effect is the dominant asymmetry amplifier. Neither the static nor the DHPS approach contains this important physics. Therefore we conclude that the DHPS results have sufficient veracity at the onset of explosive asymmetry generation, but are invalid in the ensuing low temperature epoch where MSW conversions are able to enhance the asymmetry to values of order 0.2–0.37. DHPS do claim to find a significant final asymmetry for very large δm² values. However, for this regime the effective potential they employed is not valid.     

The tidal stripping of satellites

We present an improved analytic calculation for the tidal radius of satellites and test our results against N -body simulations.
The tidal radius in general depends upon four factors: the potential of the host galaxy, the potential of the satellite, the orbit of the satellite and the orbit of the star within the satellite . We demonstrate that this last point is critical and suggest using three tidal radii to cover the range of orbits of stars within the satellite. In this way we show explicitly that prograde star orbits will be more easily stripped than radial orbits; while radial orbits are more easily stripped than retrograde ones. This result has previously been established by several authors numerically, but can now be understood analytically. For point mass, power-law (which includes the isothermal sphere), and a restricted class of split power-law potentials our solution is fully analytic. For more general potentials, we provide an equation which may be rapidly solved numerically.
Over short times (≲1–2 Gyr ∼1 satellite orbit), we find excellent agreement between our analytic and numerical models. Over longer times, star orbits within the satellite are transformed by the tidal field of the host galaxy. In a Hubble time, this causes a convergence of the three limiting tidal radii towards the prograde stripping radius. Beyond the prograde stripping radius, the velocity dispersion will be tangentially anisotropic.