The structure and evolution of stars Lecture 9: Computation of stellar evolutionary models 1 Learning Outcomes The student will learn How to interpret the models of modern calculations - (in this case the models from the Geneva theoretical stellar evolution group) How a realistic theoretical HRD is constructed Understand how stars of different masses schematically evolve To appreciate how stellar lifetime varies with mass How clusters are used to test models of stellar evolution 2 1
Introduction and recap Second part of course: Previous lectures analytical - now we will be more descriptive. Account of results for full-scale numerical calculations of the set of equations Numerical studies date back to 1960s (Icko Iben - momentous efforts over 30 years, often illustrated in text books) Results of these computations are not always anticipated or intuitively expected from fundamental principles - equations are non-linear and solutions complex We will concentrate on comparing the observable properties of stars (Lecture 1) and testing models by comparing to HR diagram and all its aspects 3 Example set of models - the Geneva Group See handout of paper of Schaller et al. (1992): the standard set of stellar evolutionary models form the Geneva group. 1st line in table NB = model number (51) AGE = age in yrs MASS = current mass LOGL = log L/L LOGTE = log T eff X,Y,C12 NE22 = surface abundance of H,He, 12 C 22 Ne (these are mass fractions) 2nd line QCC = fraction of stellar mass within convective core MDOT = mass loss rate: RHOC=central density LOGTC = log T c X,Y,C12 NE22 = central abundances 4 2
The Hayashi forbidden zone The Hayashi line gives a lower limit for the Teff of stars in hydrostatic equilibrium. First determined when evolution of protostars considered - collapsing molecular cloud to form a main-sequence star. We will not treat it mathematically in this course: Further reading in Böhm- Vitense, Ch. 11.2 5 Example evolution of a 5M star H-burning in main-sequence, X c =0 at NB=13, τ=100 Myr (and Y c =0.98) Star cools and moves across HRD on thermal timescale (τ 20 - τ 13 = 4.6x105 yrs). From Lecture 5, the thermal timescale of the Sun is ~10 15 sec or ~30Myrs For 5M t th ~2x10 5 yrs - similar to rapid movement timescale on HRD. He burning begins at NB=20, ends at NB=43. Comparison of lifetimes: H-burning Thermal expansion He-burning 9.4 x 10 7 yrs 4.6 x 10 5 yrs 16 x 10 6 yrs 6 3
Main-sequence lifetimes Approximate main-sequence lifetimes (from Prialnik, P. 142 - note some differences with Geneva models ) Stellar clusters (from Lecture 1) large group of stars born at same time, age of cluster will show on HR-diagram as the upper end, or turn-off of the main-sequence. We can use this as a tool (clock) for measuring age of star clusters. Stars with lifetimes less than cluster age, have left main sequence. Stars with main-sequence lifetimes longer than age, still dwell on main-sequence. Mass M 0.1 0.5 1.0 1.25 1.5 3.0 5.0 9.0 15 25 Time 6 10 12 7 10 10 1 10 10 4 10 9 2 10 9 2 10 8 7 10 7 2 10 7 1 10 7 6 10 6 Stars of all masses live on the main-sequence, but subsequent evolution differs enormously. We can divide the HRD into four sections, defined by mass ranges within which the evolution is similar (or related). 7 The five sections of the HRD Note all masses approximate, boundaries overlap depending on definition. Brown dwarfs (and planets): estimated lower stellar mass limit is 0.08 M (or 80M Jup ). Lower mass objects have core T too low to ignite H. Red dwarfs: stars whose main-sequence lifetime exceeds the present age of the Universe (estimated as 1-2x10 10 yr). Models yield an upper mass limit of stars that must still be on main-sequence, even if they are as old as the Universe of 0.7M Low-mass stars: stars in the region 0.7 M 2 M. After shedding considerable amount of mass, they will end their lives as white dwarfs and possibly planetary nebulae. In Lecture 10 we will follow the evolution of a 1M star in detail. Intermediate mass stars: stars of mass 2 M 8-10 M. Similar evolutionary paths to low-mass stars, but always at higher luminosity. Give planetary nebula and higher mass white dwarfs. Complex behaviour on the AGP branch. High mass (or massive) stars: M >8-10 M. Distinctly different lifetimes and evolutionary paths huge variation, will study in Lecture 11. 8 4
SCLOCK simulation of Geneva models Visual tool for interpolation and plotting of Geneva models, works on Windows PCs. Link from module page on QOL. Animates the evolution of stars (0.8 to 25 solar masses, solar metallicity) in the Hertzsprung-Russell (H-R) diagram, more exactly in the log(l/l ) vs. log(t eff /K) plane. The evolution is followed from the initial main sequence (also called zero age main sequence, ZAMS) up to the end of the core carbon burning phase for the most massive stars, to the early asymptotic giant branch (E-AGB) for the intermediate mass stars, and to the core helium ignition for the solar-type stars. 9 Convection processes and uncertainties In Schaller et al. there is some discussion on Convection Parameters ( 2.5). Mixing length theory of convection: The description of convection which is commonly used in stellar interiors contains a free parameter called the mixing length (l). Assume that the convective elements of a characteristic size l rise or fall through a distance that is comparable with their size, before they exchange heat with their surroundings. If it assumed that elements move adiabatically and in pressure balance with their surroundings, and that they are accelerated freely by buoyancy force. 10 5
This expression is only useful if a value can be chosen for l. Often assumed that an appropriate value is of order a pressure scale height, and a value is defined : The value of α chosen can make a considerable difference to stellar structure, particularly in cool stars. The structure of the Sun and its T eff can be reproduced with α =1.6, But nothing definite known about this value for other stars. In Schaller et al. they estimate α from the average location of the red giant branch of 75 clusters, and obtained best fit for α =1.6 ±0.1. Note that this is an empirical fit, a theory of convection is not yet developed that can predict l. Convective Overshooting One more important property of convection. What happens at the boundary between a convective region and non-convecitve region? A rising convective element will still have a finite velocity as it enters the region where the convective criterion is not satisfied. This process is called convective overshooting. This is generally not important for energy transport, but means that mixing can occur between the regions which can be significant for later evolution. 11 Modelling star clusters As discussed in Lecture 1, best way to check stellar evolutionary calculations if to compare calculated and observed tracks. But can t observe stars as they evolve - need to use star clusters. Isochrones: A curve which traces the properties of stars as a function of mass for a given age. Be clear about the difference with an evolutionary track - which shows the properties of a star as a function of age for a fixed mass. Isochrones are particularly useful for star clusters - all stars born at the same time with the same composition e.g. the Schaller et al. models. Consider stars of different masses but with the same age. Lets make a plot of Log(L/L ) vs. LogT eff for an age of 1Gyr. The result is an isochrone. Important - think about what we are looking at when we observe a cluster. We are seeing a freeze-frame picture at a particular age. We see how stars of different masses have evolved up to that fixed age (this is not equivalent to an 12 evolutionary track). 6
Modelling star clusters Meynet et al. 1993 (Astr. & Astr. Supp. Ser., 98,477) New dating of Galactic Open Clusters Using the Geneva models, they fit isochrones to real stellar clusters 13 Theoretical isochrones from Geneva models 14 7
Examples of young and old clusters NGC6231 young cluster Age~ 6Myrs Pleiades young open cluster Age~ 100Myrs 15 47 Tuc : globular cluster. Age= 8-10Gyrs NGC188: old open cluster. Age= 7Gyrs 16 8
Summary We have seen examples of modern stellar evolutionary calculations (the Geneva Group) The main-sequence lifetimes are very dependent on initial stellar mass Isochrones rather than tracks for each mass. They are equivalent, but give a snapshot of the cluster at a particular age Excellent agreement between models, and the observed HRdiagrams Can be confident that we are predicting the real behaviour of these stars. Next two lectures will look in detail at a low-mass star, and a high mass star as case studies. 17 9