The angular momentum loss of low-mass close binary systems

The detailed evolution of low-mass main-sequence stars (M < 1M ) with a compact companion is studied. For angular momentum loss associated with magnetic braking it is found that about 10^–11–10^–12 M yr^–1 in stellar wind loss would be required. This wind is 10²–10³ times stronger than the solar wind, so we believe here magnetic stellar wind is insufficient. It is well known that there is mass outflow in low-mass close binary systems. We believe here that these outflows are centrifugal driven winds from the outer parts of the accretion disks. The winds extract angular momentum from these systems and therefore drive secular evolution. Disk winds are preferred to winds from the secondary, because of the lower disk surface gravity.     

Loss of orbital angular momentum and period changes in Algols

The observational data of Algols have been examined in order to clarify the implication of the assumption concerning the orbital angular momentum loss on their orbital period change. It is found that the agreement between theory and observational data of Algols is much better when a nonconservative approach of evolution is adopted.     

Rotationally-perturbed orbital elements of close binary systems

The aim of this investigation is to present the secular and periodic perturbations of the six orbital elements of a close binary system due to rotational distortion. In our study we consider very small inclinationst of the orbital plane of the system, whereas the eccentricity of the orbit may assume any value between 0<e<1. The final formulae for the various elements have been expressed by means of the unperturbed true anomaly measured from the ascending node.     

Tidal evolution in close binary systems

The aim of the present paper will be to give a mathematical outline of the theory of tidal evolution in close binary systems of secularly constant total momentum — an evolution activated by viscous friction of dynamical tides raised by the two components on each other. The first section contains a general outline of the problem; and in Section 2 we shall establish the basic expressions for the energy and momenta of close binaries consisting of components of arbitrary internal structure. In Section 3 we shall investigate the maximum and minimum values of the energy (kinetic and potential) which such systems can attain for given amount of total momentum; while in Section 4 we shall compare these results with the actual facts encountered in binaries with components whose internal structure (and, therefore, rotational momenta) are known to us from evidence furnished by the observed rates of apsidal advance.The results show that all such systems — be these of detached or semi-detached type — disclose that more than 99% of their total momenta are stored in the orbital momentum. The sum of the rotational momenta of the constituent components amounts to less than a percent of the total — a situation characteristic of a state close to the minimum energy for given total momentum. This appears, moreover, to be true not only of the systems with both components on the Main Sequence, but also of those possessing evolved components in contact with their Roche limits.Under such conditions, a synchronism between rotation and revolution (characteristic of both extreme states of maximum and minimum energy) is not only possible, but appears to have been actually approached — if not attained — in the majority of cases. In other words, it would appear that — in at least a large majority of known cases — the existing close binaries have already attained orbits of maximum distension consistent with their momenta; and tidal evolution alone can no longer increase the present separations of the components to any appreciable extent.The virtual absence, in the sky, of binary systems intermediate between the stages of maximum and minimum energy for given momentum leads us to conjecture that the process of dynamical evolution activated by viscous tides may enroll on a time-scale which is relatively short in comparison with their total age — even for systems like Y Cygni or AG Persei, whose total age can scarcely exceed 10⁷ yr. A secular increase of the semi-major axes of relative orbits is dynamically coupled with a corresponding variation in the velocity of axial rotation of both components through the tidal lag arising from the viscosity of stellar material. The differential equations of so coupled a system are given in Section 5; but their solution still constitutes a task for the future.The Lunar Science Institute Contribution No. 90. The Lunar Science Institute is operated by the Universities Space Research Association under Contract No. NSR 09-051-001 with the National Aeronautics and Space Administration.     

CaseA evolution of massive close binary systems

The caseA evolution of close binary systems with total mass of 20.4M is investigated by following the evolution of both components simultaneously. The evolution is followed up to the stage at which a system overflows the outer critical surface or evolves into the phase of modeBr mass-transfer. It is found that the evolution of the systems can be classified into six types. The ranges of initial parameters which lead systems to each type of evolution are shown on the initial-parameter plane. The evolutionary features of each evolution type are described in detail.     

Tidal evolution of close binary asteroid systems

We provide a generalized discussion of tidal evolution to arbitrary order in the expansion of the gravitational potential between two spherical bodies of any mass ratio. To accurately reproduce the tidal evolution of a system at separations less than 5 times the radius of the larger primary component, the tidal potential due to the presence of a smaller secondary component is expanded in terms of Legendre polynomials to arbitrary order rather than truncated at leading order as is typically done in studies of well-separated system like the Earth and Moon. The equations of tidal evolution including tidal torques, the changes in spin rates of the components, and the change in semimajor axis (orbital separation) are then derived for binary asteroid systems with circular and equatorial mutual orbits. Accounting for higher-order terms in the tidal potential serves to speed up the tidal evolution of the system leading to underestimates in the time rates of change of the spin rates, semimajor axis, and mean motion in the mutual orbit if such corrections are ignored. Special attention is given to the effect of close orbits on the calculation of material properties of the components, in terms of the rigidity and tidal dissipation function, based on the tidal evolution of the system. It is found that accurate determinations of the physical parameters of the system, e.g., densities, sizes, and current separation, are typically more important than accounting for higher-order terms in the potential when calculating material properties. In the scope of the long-term tidal evolution of the semimajor axis and the component spin rates, correcting for close orbits is a small effect, but for an instantaneous rate of change in spin rate, semimajor axis, or mean motion, the close-orbit correction can be on the order of tens of percent. This work has possible implications for the determination of the Roche limit and for spin-state alteration during close flybys.     

Tidal evolution in close binary systems,II

The aim of the present investigation will be to determine the explicit forms of differential equations which govern secular perturbations of the orbital elements of close binary systems in the plane of the orbit (i.e., of the semi-major axisA, eccentricitye, and longitude of the periastron ), arising from the lag of dynamical tides due to viscosity of stellar material. The results obtained are exact for any value of orbital eccentricity comprised between 0e<1; and include the effects produced by the second, third and fourth-harmonic dynamical tides, as well as by axial rotation with arbitrary inclination of the equator to the orbital plane.In Section 2 following brief introductory remarks the variational equations of the problem of plane motion will be set up in terms of the rectangular componentsR, S, W of disturbing accelerations with respect to a revolving system of coordinates. The explicit form of these coefficients will be established in Section 3 to the degree of accuracy to which squares and higher powers of quantities of the order of superficial distortion can be ignored. Section 4 will be devoted to a derivation of the explicit form of the variational equations for the case of a perturbing function arising from axial rotation; and in Section 5 we shall derive variational equations which govern the perturbation of orbital elements caused by lagging dynamical tides.Numerical integrations of these equations, which govern the tidal evolution of close binary systems prompted by viscous friction at constant mass, are being postponed for subsequent investigations.Prepared at the Lunar Science Institute, Houston, Texas, under the joint support of the Universities Space Research Association, Charlottesville, Virginia, and the National Aeronautics and Space Administration Manned Spacecraft Center, Houston, Texas, under Contract No. NSR 09-051-001. This paper constitutes Lunar Science Institute Contribution no. 100.Normally at the Department of Astronomy, University of Manchester, England.     

Tidal evolution in close binary systems,III

In this paper we shall investigate the energy of close binary systems of constant momentum takng into consideration the first-order effects of rotation and tidal attraction of the components of finite size. The equations for the momentum and the energy of the system will be set up in Section 2, making use of terms including the effects of finite size of the components of finite degree of central condensation. In Section 3 perturbation theory is applied to these equations using the results of Kopal (1972b) as our initial values. In Section 4 we shall compare our results with the initial values and then discuss variations in our constants and the application to various real systems.     

Case a evolution of massive close binary systems

We present in this paper the evolutionary characteristics of the systems belonging to two evolution types, designated as 1C2COf and 1Nc2COf, among seven evolution types which were studied in the preceding paper of this series. These two types are most complicated and interesting in the evolutionary behaviour, which consists of repeated detached, semi-detached, and contact phases. We discuss observational aspects of the results. Evolutionary behaviour of the systems are also discussed, compared with the thermal relaxation oscillation model.     

Case a evolution of massive close binary systems

The evolution of three close binary systems of total mass 20.4M in and after the phase of mode Br mass-transfer in caseA of mass exchange is investigated. In every case a secondary component evolves to interfere with the progress of primary's evolution and the system overflows the outer critical surface before the primary completes its nuclear-burning evolution. This strongly indicates the importance of simultaneous calculation of both components. A summary of evolution of the systems considered in this series of papers up to the stage ofL ₂-overflow is given. The observational aspects of the numerical models are also discussed.     

Internal stellar rotation and orbital period modulation in close binary systems

The orbital period modulation, observed in close binary systems with late-type secondary stars, is considered in the framework of a general model that allows us to test the hypothesis proposed by Applegate. It relates the orbital period variation to the modulation of the gravitational quadrupole moment of their magnetically active secondary stars produced by angular momentum exchanges within their convective envelopes. By considering the case of RS CVn binary systems, it is found that the surface angular velocity variation of the secondary component required by Applegate's hypothesis is between 4 and 12 per cent, i.e. too large to be compatible with the observations and that the kinetic energy dissipated in its convection zone ranges from 4 to 43 times that supplied by the stellar luminosity along one cycle of the orbital period modulation. Similar results are obtained for other classes of close binary systems by applying a scaling relationship based on a simplified internal structure model. The effect of rapid rotation is briefly discussed finding that it is unlikely that the rotational quenching of the turbulent viscosity may solve the discrepancy. Therefore, the hypothesis proposed by Applegate is not adequate to explain the orbital period modulation of close binary systems with a late-type secondary.     

On the angular momentum transfer on to compact stars in binary systems

Results of three-dimensional numerical simulations of the gas transfer in close binary systems show that, in addition to the formation of a tidally induced spiral shock wave, it is also possible for accretion streams to be produced, having low specific angular momentum in a region close to the accreting star. These streams are mainly placed above the orbital disc but are also unevenly present in the equatorial plane. The relevance of such flows is related to formation of hot coronae or bulges in regions very close to the accretor centre. The eventual formation of such bulges and shock-heated flows is interesting in the context of advection-dominated solutions and for the explanation of spectral properties of the black hole candidates in binary systems.     

On synchronism between axial rotation and orbital motion in close binary systems

In this paper we examine the possible outcome of the tidal evolution of a close binary system using a method from which the outline has already been given by Counselman (1973). If the value of the total angular momentum of the system is sufficiently large, two equilibrium states corresponding to synchronism between stellar rotation and orbital motion are possible. In one of these states the total energy attains no extreme value. The considered evolution can be visualized geometrically by the motion of a point along a hyperbolic cylinder in three-dimensional space. A comparison with some observational data reveals that most of the synchronously rotating detached systems have attained a stable equilibrium state of minimum total energy for the given value of total angular momentum.     

Perturbed orbital elements of close binary systems due to tidal lag in longitude

This paper deals with the perturbations which tidal lag in longitude can produce to the orbital elements of a close binary system. The expressions obtained for the six elements of the orbit have been presented as functions of the unperturbed true anomaly, measured from the periastron. Our study includes the effects produced by the second, third, and fourth tidal harmonic distortions. In order to save space these extremely lengthy equations are given in the compact form of summations, by means of Hansen coefficients. Various recurrence relations, which hold good for Hansen coefficients, are also presented. Finally, this paper includes a second-order approximation only for the secular terms of first-order approximation.     

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.     

Accretion, ablation and propeller evolution in close millisecond pulsar binary systems

A model for the formation and evolution of binary millisecond radio pulsars in systems with low mass companions (<0.1 M_⊙) is investigated using a binary population synthesis technique. Taking into account the non conservative evolution of the system due to mass loss from an accretion disk as a result of propeller action and from the companion via ablation by the pulsar, the transition from the accretion powered to rotation powered phase is investigated. It is shown that the operation of the propeller and ablation mechanisms can be responsible for the formation and evolution of black widow millisecond pulsar systems from the low mass X-ray binary phase at an orbital period of ～0.1 day. For a range of population synthesis input parameters, the results reveal that a population of black widow millisecond pulsars characterized by orbital periods as long as ～0.4 days and companion masses as low as ～0.005 M_⊙ can be produced. The orbital periods and minimum companion mass of this radio millisecond pulsar population critically depend on the thermal bloating of the semi-degenerate hydrogen mass losing component, with longer orbital periods for a greater degree of bloating. Provided that the radius of the companion is increased by about a factor of 2 relative to a fully degenerate, zero temperature configuration, an approximate agreement between observed long orbital periods and theoretical modeling of hydrogen rich donors can be achieved. We find no discrepancy between the estimated birth rates for LMXBs and black widow systems, which on average are ${\sim}1.3\times10^{-5}~{\rm yr}^{-1}$ and $1.3\times10^{-7}~{\rm yr}^{-1}$ respectively.     

Re-analysis of close binary systems

Line-forming regions around close binaries with strong winds ( /4r _* v 10^–4 g cm^–2) are large in extent compared with the stars, large enough to screen them. Their orbitally-modulated Doppler shifts can overestimate the mass function, because of a larger rotational lever arm. In particular, most of the black-hole candidates need not involve companions more massive than a neutron star.The solar-wind problem is reconsidered. An extrapolation to Wolf-Rayet stars suggests that their winds are centrifugally driven. Their mass-loss rates tend to have been overestimated.Seemingly single (massive) stars can hide a (compact) companion.     

How galaxies lose their angular momentum

The processes are investigated by which gas loses its angular momentum during the protogalactic collapse phase, leading to disc galaxies that are too compact with respect to the observations. High-resolution N -body/SPH simulations in a cosmological context are presented including cold gas and dark matter (DM). A halo with quiet merging activity since redshift   z ∼ 3.8  and with a high-spin parameter is analysed that should be an ideal candidate for the formation of an extended galactic disc. We show that the gas and the DM have similar specific angular momenta until a merger event occurs at   z ∼ 2  with a mass ratio of 5:1. All the gas involved in the merger loses a substantial fraction of its specific angular momentum due to tidal torques and dynamical friction processes falls quickly into the centre. In contrast, gas infall through small subclumps or accretion does not lead to catastrophic angular momentum loss. In fact, a new extended disc begins to form from gas that was not involved in the 5:1 merger event and that falls in subsequently. We argue that the angular momentum problem of disc galaxy formation is a merger problem: in cold dark matter cosmology substantial mergers with mass ratios of 1:1 to 6:1 are expected to occur in almost all galaxies. We suggest that energetic feedback processes could in principle solve this problem, however only if the heating occurs at the time or shortly before the last substantial merger event. Good candidates for such a coordinated feedback would be a merger-triggered starburst or central black hole heating. If a large fraction of the low angular momentum gas would be ejected, late-type galaxies could form with a dominant extended disc component, resulting from late infall, a small bulge-to-disc ratio and a low baryon fraction, in agreement with observations.