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We have calculated a grid of empirically well tested evolutionary tracks with masses M between 0.5 and 50 M⊙, spaced by approximately 0.1 in log M, and with metallicities Z = 0.0001, 0.0003, 0.001, 0.004, 0.01, 0.02 and 0.03. We use a robust and fast evolution code with a self‐adaptive non‐Lagrangian mesh, which employs the mixing‐length theory but treats convective mixing as a diffusion process, solving simultaneously for the structure and the chemical composition. The hydrogen and helium abundances are chosen as functions of the metallicity: X = 0.76 − 3.0ZY = 0.24 + 2.0Z.  Two sets of models were computed, one without and one with a certain amount of enhanced mixing or ‘overshooting’. This amount has been empirically chosen by means of various sensitive tests for overshooting: (1) the luminosity of core helium burning (blue loop) giants of well‐known mass, (2) the width of the main sequence as defined by double‐lined eclipsing binaries with well‐measured masses and radii, and (3) the shape and implied stellar distribution of isochrones of various open clusters. The first two tests have been the subject of previous papers, the third test is discussed in this paper. On the basis of these tests, we recommend the use of the overshooting models for masses above about 1.5M ⊙.  We describe here the characteristics of the models, the procedure for constructing isochrones for arbitrary age and metallicity from the models, and the performance of these isochrones for several intermediate‐age and old open clusters. All original models are available in electronic form and we describe the means by which they may be obtained.

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We present a simple and e cient, yet reasonably accurate, equation of state, which at the moderately low temperatures and high densities found in the interiors of stars less massive than the Sun is substantially more accurate than its predecessor by Eggleton, Faulkner & Flannery. Along with the most recently available values in tabular form of opacities, neutrino loss rates, and nuclear reaction rates for a selection of the most important reactions, this provides a convenient package of input physics for stellar modelling. We brie y discuss a few results obtained with the updated stellar evolution code.

We derive from first principles equations governing (a) the quadrupole tensor of a star distorted by both rotation and the presence of a companion in a possibly eccentric orbit, (b) a functional form for the dissipative force of tidal friction, based on the concept that the rate of energy loss from a time-dependent tide should be a positive-definite function of the rate of change of the quadrupole tensor as seen in the frame which rotates with the star, and (c) the equations governing the rates of change of the magnitude and direction of the stellar rotation, and the orbital period and eccentricity, based on the concept of the Laplace-Runge-Lenz vector. Our analysis leads relatively simply to a closed set of equations, valid for arbitrary inclination of the stellar spin to the orbit. The results are equivalent to classical results based on the rather less clear principle that the tidal bulge lags behind the line of centres by some time determined by the rate of dissipation; our analysis gives the effective lag time as a function of the dissipation rate and the quadrupole moment.We discuss briefly some possible applications of the formulation.Subject headings: stars: binary, triple IntroductionWe present a partly novel derivation of some basic results, and some new results, for the 'equilibrium tide' model of tidal friction (Hut 1981). We derive the force and couple on a binary orbit that arises from dissipation of the equilibrium tide, starting only from the principles that (a) the rate of dissipation of energy should be a positive definite function of the rate of change of the tide, as viewed in a frame which rotates with the star, and (b) the total angular momentum is conserved. This model leads to a force and couple which are similar to those derived classically on the assumption that the tidal bulge lags the line of centres by some small time that is determined by the timescale of dissipation. However in our model the lag time is determined to be proportional to the quadrupole moment, rather than assumed to be independent of it. There is a clear relation between the rate of dissipation and the lag time. We also determine the relation between the hypothesised dissipation constant and the turbulent viscosity within the star. However there remains the more difficult task of determining the turbulent velocity field.There are two principle formalisms for discussing the effect of tidal dissipation on an orbit: the 'equilibrium tide' model

We present equations governing the way in which both the orbit and the intrinsic spins of stars in a close binary should evolve subject to a number of perturbing forces, including the effect of a third body in a possibly inclined wider orbit. We illustrate the solutions in some binary-star and triple-star situations: tidal friction in a wide but eccentric orbit of a radio pulsar about a B star (0045-7319), the Darwin and eccentricity instabilities in a more massive but shorter-period massive X-ray binary, and the interaction of tidal friction with Kozai cycles in a triple such as β Per, at an early stage in that star's life when all 3 components were ZAMS stars. We also attempt to model in some detail the interesting triple system SS Lac, which stopped eclipsing in about 1950. We find that our model of SS Lac is quite constrained by the relatively good observational data of this system, and leads to a specific inclination (29 • ) of the outer orbit relative to the inner orbit at epoch zero (1912). Although the intrinsic spins of the stars have little effect on the orbit, the converse is not true: the spin axes can vary their orientation relative to the close binary by up to 120 • on a timescale of about a century.

Binary systems of stars are as common as single stars. Stars evolve primarily by nuclear reactions in their interiors, but a star with a binary companion can also have its evolution influenced by the companion. Multiple star systems can exist stably for millions of years, but can ultimately become unstable as one star grows in radius until it engulfs another. This volume, first published in 2006, discusses the statistics of binary stars; the evolution of single stars; and several of the most important kinds of interaction between two (and even three or more) stars. Some of the interactions discussed are Roche-lobe overflow, tidal friction, gravitational radiation, magnetic activity driven by rapid rotation, stellar winds, magnetic braking and the influence of a distant third body on a close binary orbit. A series of mathematical appendices gives a concise but full account of the mathematics of these processes.

Tidal friction in close binaries, with periods of a few days, is expected to circularize the orbit on a time‐scale long compared with human observation but shorter than, or comparable to, the lifetimes of main‐sequence stars. In a hierarchical triple star, however, the perturbing effect of the distant third star may decircularize the inner orbit significantly on a time‐scale of the order of days (as in λ Tau) or centuries (as in β Per). If the inner pair is observed to be semidetached, however, it is plausible to assume that the eccentricity is small. This may be because tidal friction is operating on a comparably short time‐scale, and so it is in principle amenable to observation. We attempt to determine a lower limit to the strength of tidal friction in λ Tau and β Per, on the basis of this consideration. Tidal friction will also lead to a secular transfer of angular momentum from the inner orbit to the outer orbit. Too rapid a transfer may lead to orbital shrinkage that is fast compared with the nuclear time‐scales of the inner systems, and this can also be ruled out on observational grounds. Thus we may be able to set an upper as well as a lower limit to the strength of tidal friction, on the basis of observations. In a young hierarchical triple, provided that the orbits are fairly nearly orthogonal, tidal friction can serve to reduce the inner orbital period from months to days within a fairly short period of time, of order P2out/Pin. This may be a significant mechanism for producing young short‐period binaries.