Transcript MSci Astrophysics 210PHY412 - Queen's University Belfast

The structure and evolution of
stars
Lecture 9: Computation of stellar
evolutionary models
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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
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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
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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 Teff
X,Y,C12…NE22 = surface abundance of H,He, 12C …
22Ne (these are mass fractions)
2nd line
QCC = fraction of stellar mass within convective core
MDOT = mass loss rate:
.

.
-1
log(M)
where M  mass loss rate in M
sol yr
RHOC=central density
LOGTC = log Tc
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X,Y,C12…NE22 = central abundances
The Hayashi forbidden zone
The Hayashi line gives a
lower limit for the Teff of stars
in hydrostatic equilibrium.
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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 BöhmVitense, Ch. 11.2
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Example evolution of a 5M star
H-burning in main-sequence, Xc=0 at NB=13, =100 Myr (and Yc=0.98)
Star cools and moves across HRD on thermal timescale (20 - 13= 4.6x105
yrs). From Lecture 5, the thermal timescale of the Sun is ~1015 sec or
~30Myrs
GM 2
tth ~
LR
2


Rsol Lsol 
M
~ 1015
    s
M sol   R  L 
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
Thermal expansion
He-burning
9.4 x 107 yrs
4.6 x 105 yrs
16 x 106 yrs
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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 HRdiagram as the upper end, or turn-off of the mainsequence.
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
Time
0.1
6  1012
0.5
7  1010
1.0
1  1010
1.25
4  109
1.5
2  109
3.0
2  108
5.0
7  107
9.0
2  107
15
1  107
25
6  106
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).
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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
80MJup). 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 1-2x1010 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.
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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(Teff/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.
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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.
1/ 2 2


GM
l (   ad ) 3 / 2
2
Lconv  r c P T 2 
 r 
H P3 / 2
where
P
HP 
 the pressure scale height
(dP dr)
P dT
 1
 ad 
T dP

and c p is the specific heat at constant pressure

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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 :

l
HP
The value of  chosen can make a considerable difference to stellar structure,
particularly in cool stars. The structure of the Sun and its Teff can be
reproduced with  =1.6, But nothing definite known about this value for other
stars. In Schaller et al. they estimate  from the average location of the red giant
branch of 75 clusters, and obtained best fit for  =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.
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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. LogTeff 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 evolutionary
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track).
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
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Theoretical isochrones from Geneva models
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Examples of young and old clusters
NGC6231 young cluster
Age~ 6Myrs
Pleiades young open cluster
Age~ 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 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.
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