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Comprehensive Analytic Formulae for Stellar Evolution As a Function Of Mon. Not. R. Astron. Soc. 000, 1–29 (1999) Printed 28 May 2018 (MN LATEX style file v1.4) Comprehensive analytic formulae for stellar evolution as a function of mass and metallicity Jarrod R. Hurley1, Onno R. Pols2 and Christopher A. Tout1 1Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK 2Instituto de Astrof´ısica de Canarias, c/ Via L´actea s/n, E-38200 La Laguna, Tenerife, Spain E-mail: [email protected], [email protected], [email protected] 28 May 2018 ABSTRACT We present analytic formulae that approximate the evolution of stars for a wide range of mass M and metallicity Z. Stellar luminosity, radius and core mass are given as a function of age, M and Z, for all phases from the zero-age main-sequence up to, and including, the remnant stages. For the most part we find continuous formulae accurate to within 5% of detailed models. These formulae are useful for purposes such as population synthesis that require very rapid but accurate evaluation of stellar properties, and in particular for use in combination with N-body codes. We describe a mass loss prescription that can be used with these formulae and investigate the resulting stellar remnant distribution. Key words: methods: analytic – stars: evolution – stars: fundamental parameters – stars: Population II – galaxies: stellar content 1 INTRODUCTION evolve a model of just one star. Thus it is desirable to gener- ate a large set of detailed models and present them in some The results of detailed stellar evolution calculations are re- convenient form in which it is relatively simple to utilise the quired for applications in many areas of astrophysics. Ex- results at a later stage. amples include modelling the chemical evolution of galaxies, There are two alternative approaches to the problem determining the ages of star clusters and simulating the out- of using the output of a series of stellar-evolution runs as comes of stellar collisions. As stellar evolution theory, and data for projects that require them. One approach is to con- our ability to model it, is continually being improved (the arXiv:astro-ph/0001295v1 17 Jan 2000 struct tables (necessarily rather large, especially if a range treatment of convective overshooting and thermal pulses, of metallicities and/or overshoot parameter is to be incor- for example) there is an ongoing need to update the results porated) and interpolate within these tables. The other is to of these calculations. For a recent overview of problems in approximate the data by a number of interpolation formulae stellar evolution see Noels et al. (1995). as functions of age, mass and metallicity. Both procedures As with all theories our understanding of stellar evo- have advantages and disadvantages (Eggleton 1996), so we lution must be tested against observations. One way to do have worked on both simultaneously. Stellar models have this is to attempt to reproduce the findings of large-scale been available in tabular form for many years (Schaller et star surveys, such as the Bright Star Catalogue (Hoffleit al. 1992; Charbonnel et al. 1993; Mowlavi et al. 1998). Stel- 1983) and the Hipparcos Catalogue (Perryman et al. 1997), lar populations cover a wide range of metallicity so the ideal using population synthesis. The Hipparcos Catalogue is an is to have a set of models that cover the full range of pos- excellent example of how improved observing techniques can sible compositions and stellar masses. In a previous paper initiate a re-evaluation of many aspects of stellar evolution (Pols et al. 1998) we presented the results of stellar evolu- theory (Baglin 1997; de Boer et al. 1997; Van Eck et al. tion calculations for a wide range of mass and metallicity 1998). In order to make population synthesis statistically in tabular form. In the present paper we report on the re- meaningful it is necessary to evolve a large sample of stars sults of the second approach, construction of a set of single so as to overcome Poisson noise. If we synthesize n exam- star evolution (SSE) formulae, thus expanding the work of ples of a particular type of star we have an error of √n Eggleton, Fitchett & Tout (1989) along the lines of Tout et ± which means that for rarer stars often millions of possible al. (1996). It is more difficult in practice to find analytic ap- progenitors are required to get a sufficently accurate sample. proximations of a conveniently simple nature for the highly However, detailed evolution codes can take several hours to non-uniform movement of a star in the Hertzsprung-Russell c 1999 RAS 2 J. R. Hurley, O. R. Pols and C. A. Tout diagram (HRD) than it is to interpolate in tables, but the ing as they evolve off the MS but for higher mass stars with resulting code is very much more compact and adaptable to convective cores the transition is not so smooth. Owing to the requirements of, for example, an N-body code (Aarseth mixing in the core there is a sudden depletion of fuel over 1996) or variable mass loss. This is reinforced in the circum- a large region which leads to a rapid contraction over the stance where one wishes to include binary-star interactions, inner region at core hydrogen-exhaustion. This causes the such as Roche-lobe overflow, common-envelope evolution, hydrogen-exhausted phase gap, or MS hook, which occurs and magnetic braking with tidal friction, for example (Tout on a thermal timescale. The different features of MS evo- et al. 1997). lution are illustrated by comparing the evolution tracks for In Section 2 we provide a brief overview of how stars the 1.0 M⊙ and 1.6 M⊙ stars in Figure 1. behave as they evolve in time which introduces some of the The immediate post-MS evolution towards the right in terminology that we use and will hopefully facilitate the un- the HRD occurs at nearly constant luminosity and is very derstanding of this paper. Section 3 describes the detailed rapid. For this reason very few stars are seen in this phase, models from which the formulae are derived and justifies and this region of the HRD is called the Hertzsprung gap the inclusion of enhanced mixing processes. In Section 4 we (HG), or the sub-giant branch. During this HG phase the ra- outline the procedure to be used for generating the SSE dius of the star increases greatly causing a decrease in Teff . package. The evolution formulae are presented in Section 5 For cool envelope temperatures the opacity increases causing for all nuclear burning phases from the main-sequence to the a convective envelope to develop. As the convective envelope asymptotic giant branch. Our formulae are a vast improve- grows in extent the star will reach the giant branch (GB) ment on the work of Eggleton, Fitchett & Tout (1989) not which is the nearly vertical line corresponding to a fully only due to the inclusion of metallicity as a free parameter convective star, also known as the Hayashi track. All stars but also because we have taken a great deal of effort to pro- ascend the GB with the hydrogen-exhausted core contract- vide a more detailed and accurate treatment of all phases of ing slowly in radius and heating while the hydrogen-burning the evolution. Features such as main-sequence formulae that shell is eating its way outwards in mass and leaving behind are continuous over the entire mass range and the modelling helium to add to the growing core. As the stars move up of second dredge-up and thermal pulses will be discussed. the GB convection extends over an increasing portion of the Section 6 discusses the behaviour of a star as the stellar en- star. The convective envelope may even reach into the previ- velope becomes small in mass and outlines what happens ously burnt (or processed) regions so that burning products when the nuclear evolution is terminated. We also provide are mixed to the surface in a process called dredge-up. formulae which model the subsequent remnant phases of Eventually a point is reached on the GB where the core evolution. In Section 7 we describe a comprehensive mass temperature is high enough for stars to ignite their central loss algorithm which can be used in conjunction with the helium supply. For massive stars, M > 2.0 M⊙, this takes evolution formulae, as well as a method for modelling stellar place gently. When core helium burning∼ (CHeB) begins the rotation. Various uses for the formulae and future improve- star descends along the GB until contraction moves the star ments are discussed in Section 8 along with details of how away from the fully convective region of the HRD and back to obtain the formulae in convenient subroutine form. towards the MS in what is called a blue loop. During CHeB, carbon and oxygen are produced in the core. Eventually core helium is exhausted and the star moves back to the right in the HRD. The size of the blue loop generally increase with 2 STELLAR EVOLUTION OVERVIEW mass, as can be seen by comparing the 4.0 M⊙ and 10.0 M⊙ A fundamental tool in understanding stellar evolution is the tracks in Figure 1. Lower mass stars have degenerate helium Hertzsprung-Russell diagram (HRD) which provides a cor- cores on the GB leading to an abrupt core-helium flash at relation between the observable stellar properties of lumi- helium ignition (HeI). The star then moves down to the nosity, L, and effective surface temperature, Teff . Figure 1 zero-age horizontal branch (ZAHB) very quickly.
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