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The structure and evolution ofstars

Lecture 9: Computation of stellarevolutionary models

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Learning OutcomesThe student will learn• How to interpret the models of modern calculations -

(in this case the models from the Geneva theoreticalstellar 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

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Introduction and recapSecond 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 andsolutions complex

• We will concentrate on comparing the observable properties of stars(Lecture 1) and testing models by comparing to HR diagram and all itsaspects

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Example set of models - “the Geneva Group”

See handout of paper of Schaller et al.(1992): the “standard” set of stellarevolutionary models form the Genevagroup.1st line in tableNB = model number (51)AGE = age in yrsMASS = current massLOGL = log L/L

LOGTE = log Teff

X,Y,C12…NE22 = surface abundance of H,He, 12C …22Ne (these are mass fractions)

2nd lineQCC = fraction of stellar mass within convective coreMDOT = mass loss rate:

RHOC=central densityLOGTC = log Tc

X,Y,C12…NE22 = central abundances

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The Hayashi forbidden zone

The Hayashi line gives alower limit for the Teff ofstars in hydrostaticequilibrium.

First determined whenevolution of protostarsconsidered - collapsingmolecular cloud to form amain-sequence star.

We will not treat itmathematically in this course:Further reading in Böhm-Vitense, Ch. 11.2

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Example evolution of a 5M starH-burning in main-sequence, Xc=0 at NB=13, τ=100 Myr (and Yc=0.98)

Star cools and moves across HRD on thermal timescale (τ20 - τ13= 4.6x105yrs). From Lecture 5, the thermal timescale of the Sun is ~1015 sec or~30Myrs

For 5M tth~2x105 yrs - similar to rapid movement timescale on HRD.

He burning begins at NB=20, ends at NB=43. Comparison of lifetimes:

H-burning 9.4 x 107 yrsThermal expansion 4.6 x 105 yrsHe-burning 16 x 106 yrs

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Main-sequence lifetimesApproximate main-sequence lifetimes (fromPrialnik, P. 142 - note some differences withGeneva models )

Stellar clusters (from Lecture 1) large group ofstars born at same time, age of cluster will showon HR-diagram as the upper end, or turn-off of themain-sequence.

We can use this as a tool (clock) for measuringage of star clusters. Stars with lifetimes less thancluster age, have left main sequence. Stars withmain-sequence lifetimes longer than age, still dwellon main-sequence.

1 × 10715

6 × 10625

2 × 1079.0

2 × 1091.52 × 1083.0

4 × 1091.25

7 × 1075.0

TimeMass M

1 × 10101.0

7 × 10100.5

6 × 10120.1

Stars of all masses live on the main-sequence, but subsequentevolution differs enormously. We can divide the HRD into four sections,defined by mass ranges within which the evolution is similar (or related).

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The five sections of the HRDNote all masses approximate, boundaries overlap depending ondefinition.

Brown dwarfs (and planets): estimated lower stellar mass limit is 0.08 M (or80MJup). Lower mass objects have core T too low to ignite H.

Red dwarfs: stars whose main-sequence lifetime exceeds the present age of theUniverse (estimated as 1-2x1010 yr). Models yield an upper mass limit of stars thatmust 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 considerableamount of mass, they will end their lives as white dwarfs and possibly planetarynebulae. 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 pathsto low-mass stars, but always at higher luminosity. Give planetary nebula and highermass white dwarfs. Complex behaviour on the AGP branch.

High mass (or massive) stars: M >8-10 M. Distinctly different lifetimes andevolutionary paths huge variation, will study in Lecture 11.

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SCLOCK simulation of Geneva modelsVisual tool for interpolation and plotting of Geneva models, works onWindows PCs. Link from module page on QOL.

Animates the evolution of stars (0.8 to 25 solar masses, solar metallicity) in theHertzsprung-Russell (H-R) diagram, more exactly in the log(L/L)vs. log(Teff/K) plane. The evolution is followed from the initialmain sequence (also called zero age main sequence, ZAMS) up to the endof the core carbon burning phase for the most massive stars, to theearly asymptotic giant branch (E-AGB) for the intermediate mass stars,and to the core helium ignition for the solar-type stars.

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Convection processes and uncertaintiesIn 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 interiorscontains a free parameter called the mixing length (l). Assume that theconvective elements of a characteristic size l rise or fall through a distance that iscomparable with their size, before they exchange heat with their surroundings.If it assumed that elements move adiabatically and in pressure balance with theirsurroundings, and that they are accelerated freely by buoyancy force.

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This expression is only useful if a value can be chosen for l. Often assumedthat an appropriate value is of order a pressure scale height, and a value isdefined :

The value of α chosen can make a considerable difference to stellar structure,particularly in cool stars. The structure of the Sun and its Teff can bereproduced with α =1.6, But nothing definite known about this value for otherstars. In Schaller et al. they estimate α from the average location of the redgiant branch of 75 clusters, and obtained best fit for α =1.6 ±0.1. Note that thisis an empirical fit, a theory of convection is not yet developed that can predict l.

Convective OvershootingOne more important property of convection. What happens at the boundarybetween a convective region and non-convecitve region ? A risingconvective element will still have a finite velocity as it enters the regionwhere the convective criterion is not satisfied. This process is calledconvective overshooting.This is generally not important for energy transport, but means that mixingcan occur between the regions which can be significant for later evolution.

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Modelling star clustersAs discussed in Lecture 1, best way to check stellar evolutionary calculations if tocompare 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 theproperties 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 timewith the same composition e.g. the Schaller et al. models. Consider stars ofdifferent masses but with the same age . Lets make a plot of Log(L/L)vs. LogTeff for an age of 1Gyr. The result is an isochrone.

Important - think about what we are looking at when we observe a cluster. Weare seeing a “freeze-frame” picture at a particular age. We see how stars ofdifferent masses have evolved up to that fixed age (this is not equivalent to anevolutionary track).

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Modelling star clustersMeynet et al. 1993 (Astr. & Astr. Supp. Ser., 98,477)“New dating of Galactic Open Clusters”

Using the Genevamodels, they fitisochrones to realstellar clusters

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Theoretical isochrones from Geneva models

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Examples of young and old clusters

NGC6231 young clusterAge~ 6Myrs

Pleiades young open clusterAge~ 100Myrs

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47 Tuc : globular cluster.Age= 8-10Gyrs NGC188: old open cluster .

Age= 7Gyrs

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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 HR-

diagrams• 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.