The STAROX stellar evolution code

This paper describes the STAROX stellar evolution code for the calculation of the evolution of a model of a spherical star. The code calculates a model at time t _k , that is the run of pressure, density, temperature, radius, energy flux and related variables on a mesh in mass M _i , given the distribution of chemical elements X _j (i) at t _k and the model at the previous time step t _k?1. It then advances the chemical composition to the next time step t _k+1 and calculates a new model at time t _k+1. This process is iterated to convergence. The model equations are solved by Newton–Raphson relaxation; the chemical equations are solved by an iterative procedure, each element being advanced in turn, and the process repeated to convergence. Convection is modelled by a mixing length model and convective mixing is treated as a diffusive process; chemical overshooting can be incorporated in parametric form. The equation of state is taken from OPAL tables and the opacity from a blend of OPAL and Alexander tables. Nuclear reaction rates are from NACRE but only cover the p–p chain and CNO cycle. The atmospheric layers are incorporated in the model by applying the surface boundary condition at small optical depth (τ≈0.001). The mesh in mass M _i is usually taken as fixed except that there is a moveable mesh point at the boundary of a convective core. Results are given for models of mass 0.9 and 5.0M _⊙ with initial composition X=0.7,Z=0.02 evolved to a state where the central hydrogen abundance is X _c =0.35, and for a model of mass 2.0M _⊙ with initial X=0.72,Z=0.02, evolved to X _c =0.01 and with core overshooting. In this latter case we compute two models one with and one without a moveable mesh point at the boundary of the convective core to illustrate the importance of having such a moveable mesh point for the determination of the Brunt–Väisälä frequency in the layers outside the core.     

The YNEV stellar evolution and oscillation code

We have developed a new stellar evolution and oscillation code YNEV,which calculates the structures and evolutions of stars,taking into account hydrogen and helium burning.A nonlocal turbulent convection theory and an updated overshoot mixing model are optional in this code.The YNEV code can evolve low-and intermediate-mass stars from the pre-main sequence to a thermally pulsing asymptotic branch giant or white dwarf.The YNEV oscillation code calculates the eigenfrequencies and eigenfunctions of the adiabatic oscillations for a given stellar structure.The input physics and numerical scheme adopted in the code are introduced.Examples of solar models,stellar evolutionary tracks of low-and intermediate-mass stars with different convection theories(i.e.mixing-length theory and nonlocal turbulent convection theory),and stellar oscillations are shown.     

A new, efficient stellar evolution code for calculating complete evolutionary tracks

We present a new stellar evolution code and a set of results, demonstrating its capability at calculating full evolutionary tracks for a wide range of masses and metallicities. The code is fast and efficient, and is capable of following through all evolutionary phases, without interruption or human intervention. It is meant to be used also in the context of modelling the evolution of dense stellar systems, for performing live calculations for both normal star models and merger products.
The code is based on a fully implicit, adaptive-grid numerical scheme that solves simultaneously for structure, mesh and chemical composition. Full details are given for the treatment of convection, equation of state, opacity, nuclear reactions and mass loss.
Results of evolutionary calculations are shown for a solar model that matches the characteristics of the present sun to an accuracy of better than 1 per cent; a  1 M_⊙  model for a wide range of metallicities; a series of models of stellar Populations I and II, for the mass range 0.25 to  64 M_⊙  , followed from pre-main-sequence to a cool white dwarf or core collapse. An initial–final mass relationship is derived and compared with previous studies. Finally, we briefly address the evolution of non-canonical configurations, merger products of low-mass main-sequence parents.     

MODEST-1: Integrating stellar evolution and stellar dynamics

We summarize the main results from MODEST-1, the first workshop on MOdeling DEnse STellar systems. Our goal is to go beyond traditional population synthesis models, by introducing dynamical interactions between single stars, binaries, and multiple systems. The challenge is to define and develop a software framework to enable us to combine in one simulation existing computer codes in stellar evolution, stellar dynamics, and stellar hydrodynamics. With this objective, the workshop brought together experts in these three fields, as well as other interested astrophysicists and computer scientists. We report here our main conclusions, questions and suggestions for further steps toward integrating stellar evolution and stellar (hydro)dynamics.     

The OSCROX stellar oscillaton code

This paper describes the OSCROX stellar oscillation code for the calculation of the adiabatic oscillations of low degree ? of a spherical star. There are two principal versions: one in Lagrangian variables (oscroxL), the second in Eulerian variables (oscroxE). The Lagrangian code does not require values of the Brunt Väisälä frequency or equivalently the density gradient. For ?=1 the oscillation equations have both an exact integral and an exact partial wave solution, and codes oscroxL1 and oscroxE1 incorporate these exact solutions. The difference in the frequencies obtained with the various codes gives some estimate of the uncertainty in the results due both to limited accuracy of hydrostatic support of the stellar model, and the limited accuracy of the integration of the oscillation equations. We compare the results of the different methods by calculating the frequencies in the range 20–2500 μHz of a model of a 1.5 M_⊙ main-sequence star (ModelJC) kindly provided by J. Christensen-Dalsgaard for the purposes of cross comparison of codes, a modified version of this model (ModelJCA) with improved hydrostatic support, and of a highly accurate n=3 polytropic model of a star with the same mass and radius. For the polytropic model the frequencies as calculated by all codes agree to within 0.001 μHz, whereas for the 1.5 M_⊙ main sequence model the frequency differences reach a maximum of 0.04 μHz, due primarily to the limited accuracy of hydrostatic support in the model; this is reduced to 0.01 μHz for ModelJCA.     

The FRANEC stellar evolutionary code

We summarize the main physical assumptions and numerical procedures adopted by the FRANEC code to compute stellar models in all the evolutionary phases at hydrostatic and thermal equilibrium. An application to the Standard Solar Model is also briefly presented.     

Trends of stellar entropy along stellar evolution

This paper is devoted to discussing the difference in the thermodynamic entropy budget per baryon in each type of stellar object found in the Universe. We track and discuss the actual decrease of the stored baryonic thermodynamic entropy from the most primitive molecular cloud up to the final fate of matter in black holes, passing through evolved states of matter as found in white dwarfs and neutron stars.We then discuss the case of actual stars with different masses throughout their evolution, clarifying the role of the virial equilibrium condition for the decrease in entropy and related issues. Finally, we discuss the role of gravity in driving the composition and the structural changes of stars with different Main Sequence masses during their evolution up to the final product. Particularly, we discuss the entropy of a black hole in this context arguing that the dramatic increase in its entropy, differently from the other cases, is due to the gravitational field itself.     

Relating stellar cycle periods to dynamo calculations

Stellar magnetic activity in slowly rotating stars is often cyclic, with the period of the magnetic cycle depending critically on the rotation rate and the convective turnover time of the star. Here we show that the interpretation of this law from dynamo models is not a simple task. It is demonstrated that the period is (unsurprisingly) sensitive to the precise type of non-linearity employed. Moreover the calculation of the wave-speed of plane-wave solutions does not (as was previously supposed) give an indication of the magnetic period in a more realistic dynamo model, as the changes in length-scale of solutions are not easily captured by this approach. Progress can be made, however, by considering a realistic two-dimensional model, in which the radial length-scale of waves is included. We show that it is possible in this case to derive a more robust relation between cycle period and dynamo number. For all the non-linearities considered in the most realistic model, the magnetic cycle period is a decreasing function of | D | (the amplitude of the dynamo number). However, discriminating between different non-linearities is difficult in this case and care must therefore be taken before advancing explanations for the magnetic periods of stars.     

Isentropic models for final stages of stellar evolution

Using isentropic and homogeneous stellar models the final stages of stellar evolution are discussed. These primitive models are a natural generalization of polytropic models to cases of complex equations of state. The equations of state used in the present work include the effects of radiation, quantum degeneracy, relativistic electrons and pairs, and of a Fe-He transition.A limiting mass-entropy relation is discussed. This relation is a direct generalization of Chandrasekhar's limiting mass for white dwarfs, and is useful for obtaining a general picture of the evolutionary tracks at late stages.Tracks of central conditions on thep, T plane are calculated for isentropic models and the occurrence of various instabilities is discussed.Supported in part by the National Science Foundation [GP-7976] formerly [GP-5391] and the Office of Naval Research [Nonr-220 (47)].     

Does simultaneous solution matter for stellar evolution codes?

A version of the stars stellar evolution code has been developed that uses a non-simultaneous solution of the equations of stellar structure and evolution. In all other respects it is identical to the normal, fully simultaneous version. It is therefore possible to test the dependence of the solution on how the equations are solved. Two cases are investigated: a 5- and a  3-M_⊙  star, both of metallicity   Z = 0.02  . Prior to the asymptotic giant branch, the models are almost identical. However, once thermal pulses start, the two methods of solution yield diverging results with the non-simultaneous technique predicting longer interpulse periods. This is traced to difficulties associated with hydrogen burning caused by the use of a moving mesh. It is shown that, with careful control of the temporal resolution, the results of the simultaneous technique can be recovered.     

The dynamical evolution of stellar superclusters

Recent images taken with the Hubble Space Telescope ( HST ) of the interacting disc galaxies NGC 4038/4039 (the Antennae) reveal clusters of many dozens and possibly hundreds of young compact massive star clusters within projected regions spanning about 100 to 500 pc. It is shown here that a large fraction of the individual star clusters merge within a few tens to a hundred Myr. Bound stellar systems with radii of a few hundred parsecs, masses ≲ 10⁹ M⊙ and relaxation times of 10¹¹ − 10¹² yr may form from these. These spheroidal dwarf galaxies contain old stars from the pre-merger galaxy and much younger stars formed in the massive star clusters, and possibly from later gas accretion events. The possibility that star formation in the outer regions of gas-rich tidal tails may also lead to superclusters is raised. The mass-to-light ratio of these objects is small, because they contain an insignificant amount of dark matter. After many hundred Myr such systems may resemble dwarf spheroidal satellite galaxies with large apparent mass-to-light ratios, if tidal shaping is important.     

McScatter: A simple three-body scattering package with stellar evolution

We describe a simple computer package which illustrates a method of combining stellar dynamics with stellar evolution. Though the method is intended for elaborate applications (especially the dynamical evolution of rich star clusters) it is illustrated here in the context of three-body scattering, i.e. interactions between a binary star and a field of single stars. We describe the interface between the dynamics and the two independent packages which describe the internal evolution of single stars and binaries. We also give an example application, and introduce a stand alone utility for the visual presentation of simulation results.     

Pre-main sequence stellar evolution with mass loss

A semi-empirical formulation is given of the rate of stellar mass loss by stellar winds. Evolutionary studies of stars in the pre-main sequence (T Tauri) stage are presented for a variety of rates of mass loss. It has been found that different mass loss rates produce only small changes in the positions of equal evolutionary time lines in HR diagrams. Thus it is concluded that the spread of points in HR diagrams of young clusters results from a spread in their times of formation. This is consistent with the initiation of star formation by violent hydrodynamic compression of a typical interstellar cloud.