1. science
2. physics

STELLAR STRUCTURE AND EVOLUTION
FOR DAVES’IES
A guide made for passing the course of Astro4
The used book notes were Stellar Structure and Evolution by Jørgen
Christensen-Dalsgaard,6 t h Edition.
“This course has changed my life.”
-
Daves Hoszowski
Notes made by Daves Hoszowski, [email protected]. I hope these notes will help the reader to get
a better understanding of the course, Astro4 – run by, Anja A., and Jens V.C, at the University of
Copenhagen.
Made by Daves Hoszowski
third year on B.Sc
University of Copenhagen
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Contents
Different funny properties ................................................................................................................. 3
Different laws .................................................................................................................................... 3
Exam Questions................................................................................................................................. 4
1. ................................................................................................................................................... 4
Equation of state ....................................................................................................................... 4
Fundamental Equations for stellar structure .............................................................................. 4
Basic stellar models ................................................................................................................... 5
2. ................................................................................................................................................... 7
Introduction of the Sun .............................................................................................................. 7
Stellar observations and the results for stellar structure and evolution ...................................... 7
HR-diagrams .............................................................................................................................. 8
3. ................................................................................................................................................... 9
ZAMS (Zero Age Main Sequence) ............................................................................................... 9
Evolution of main Sequence ....................................................................................................... 9
Radiative transport .................................................................................................................. 10
Evolutionary timescales ........................................................................................................... 10
Mass-luminosity relation ......................................................................................................... 10
4. ................................................................................................................................................. 11
Evolution of a 1 Msun ................................................................................................................ 11
5. ................................................................................................................................................. 13
Evolution of a 5 Msun ................................................................................................................ 13
6. ................................................................................................................................................. 15
Evolution of a 20 Msun .............................................................................................................. 15
7. ................................................................................................................................................. 17
The stellar phases .................................................................................................................... 17
s-/r- and p-process ................................................................................................................... 18
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Different funny properties
To increase opacity on the surface, you have to increase the density of the H - - absorption.
Core stability is achieved when the mass is much under the Schönberg-Chandrasekhar limit.
As the surface radius decreases, the effective temperature increases.
Every star that has convective envelopes have core temperatures that make the CNO process more
effective than the pp-process.
Different laws
The shell-burning law:
When a region within an active shell contracts, then the region outside will expand. So
if there is a H-shell, and the core contracts  the star will expand. If there is a Heshell as well, then the region between He- and H-shell will expand, while the star itself
will contract.
3
Exam Questions
1.
Give the fundamental equations describing stellar structure, including the boundary
conditions, as well as a short description of the basic principles for doing stellar
evolution models. Give examples of simple evolution models, i.e. the polytropic
models.
Equation of state
To explain the stellar structure we need to assume thermodynamical equilibrium. It results in that
we do not need to think of reactions in the gas. This assumption is okay, as long as there is full
ionization, and therefore we can assume this almost in the whole star except the surface.
Another assumption we take the equation of state is that the gas is spherical and homogenous and
can be assumed to be under high density.
Fully ionized ideal gas; In the core, and means that T and the ionization degree are high.
Partially ionized ideal gas; In the colder regions, and we use the saha equation.
One of the differences is that under adiabatic compression the partial ionized gas will work on to
increase its degree of ionization instead of temperature and pressure, while in a fully ionized gas, the
temperature and pressure will raise more.
Both these equations make up the equation of state, since they both tell us about the state
variables; P,T, p and u (internal energy). Another equation of state which is very different is for
degeneracy. It is important when talking about very high density and low temperature.
Degeneracy: The pressure rises due to the Pauli exclusion principle, meaning that it prevents
particles to enter identical quantum-states  forcing higher energy level/state  resist pressure. If
a degenerate core gets pressured, under degeneracy, and obtains a high enough energy to burn the
next process then it can be active, since per definition a degenerate core is ‘passive’.
Fundamental Equations for stellar structure
The first equation: (the pressure gradient at a = 0  Hydrostatic equilibrium.)
𝐸𝑄 4.4
𝑑𝑃
𝐺𝑚𝑝
=− 2
𝑑𝑟
𝑟
It is important to note that the pressure, P, is the total pressure, meaning P(gas) and P(rad). The
equation comes from taking NII law and saying that the two forces affecting the area are the
gravitational (contracting) and the pressure (expanding). The equation above tells us what the
pressure gradient is in the shell.
This equation can be rewritten to relating the mass, and if we do so, we get:
𝑑𝑃
𝐺𝑚𝑝
𝑑𝑃 𝑑𝑟
𝑑𝑃
𝐺𝑚𝑝
1
𝐺𝑚
= − 2 → 𝑑𝑚 = 4𝜋𝑟 2 𝑝𝑑𝑟 →
=
=− 2 ∙
=−
↔ (𝐸𝑄. 9.3)
2
𝑑𝑟
𝑟
𝑑𝑟 𝑑𝑚 𝑑𝑚
𝑟
4𝜋𝑟 𝑝
4𝜋𝑟 4
EQ. 9.3 tells us how the pressure relates to the mass of the shell. This equation tells us that the
pressure will decrease with increasing mass.
4
The second equation: (the continuity equation)
𝑑𝑚
= 4𝜋𝑟 2 𝑝
𝑑𝑟
𝐸𝑄 4.5
This equation tells us that the mass increases with increasing radius, which is logical.
The third Equation: (The energy equation)
𝐸𝑄 5.27
𝑑𝐿
= 4𝜋𝑟 2 𝑝𝜀 + 2𝑡𝑒𝑟𝑚𝑠 ,
𝑑𝑟
2𝑡𝑒𝑟𝑚𝑠 = −𝑝
𝑑 𝑢
𝑃 𝑑𝑝
+
𝑑𝑡 𝑝
𝑝 𝑑𝑡
These two terms which are mentioned are if the start is undergoing gravitational contraction and if
the star has energy hidden in the gas. The 2 terms are only contributing before the nuclear burning.
After nuclear burning the temperature will raise  u/p increase, but that is compensated by dp/dt.
The 2 terms are negligible and can be seen since,
𝑡 𝑘ℎ
𝑡 𝑛𝑢𝑐
≪ 1. This equation is used to obtain the
temperature gradient for radiation. It also tells us that the Luminosity changes increase with
increased radius.
The fourth Equation: (radiative)
𝐸𝑄 5.8
𝑑𝑇
3𝑘𝑝𝐿 𝑟
=−
𝑑𝑟
16𝜋𝑎𝑐𝑟 2 𝑇 3
The equation tells us that the temperature decreases rapidly with increasing radius when the opacity
is high or the luminosity is high. These conditions lead to instability, therefore the radiative equation
is only applied on the low mass stars,  low temp (mostly, low radius).
The fifth Equation: (convection)
𝐸𝑄 6.30
𝑑𝑇
𝑑𝑟
=
𝑎𝑑
𝛤2 − 1 𝑇 𝐺𝑚𝑝
∙ ∙ 2
𝛤2
𝑃 𝑟
This equation is not legitimate in regions with nuclear reactions, because of our assumption;
constant chemical composition. This equation is used when there is an instable layer in the star,
which means that the radiation cannot travel through, since the radiative temperature gradient is
not steep enough.
All the above mentioned fundamental equations have some boundary conditions, and these
conditions are the following:
𝐶𝑒𝑛𝑡𝑒𝑟: 𝑟 = 0
→ 𝑚 = 0,
𝐿=0
𝐵𝑜𝑢𝑛𝑑𝑎𝑟𝑦: 𝑚 = 𝑀 → 𝐿 = 4𝜋𝑅2 𝜍𝑇 4 , 𝑃 =
𝑎 + 1 𝐺𝑀
𝑘𝑅2
Basic stellar models
To create a stellar model we have to determine p, since the first two fundamental equations cannot
be used as they stand, without knowing the density. We can determine that either p is a known
function of r or a known function of P.
The first assumption gives us the ability to integrate with new boundaries (r,R) with the
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boundary condition P(R)=0.  Linear model
The second assumption tells us a special case of the equation of state, the degenerate matter.
This model is used for W.D. and is called the  polytropic model
The reason for creating stellar models is that we would like to make qualified guess work on the
values of the stellar interiors. Using the above mentioned fundamental equations will increase our
chances of guesswork to calculate the values for the center values, due to numerical values.
Polytropic model
The polytropic model is great approximation for models where the pressure is dominated by
degenerate electrons. This is the case for White dwarfs, as earlier mentioned. To obtain the equation
which satisfies the polytrop model, we will use the first two fundamental equations.
6
2.
Make a comparison between stellar observations and the results of models for stellar
structures and evolutions, with special emphasis on the Sun, interpretation of HRdiagrams for stellar clusters, binary stars and variable/pulsating stars.
The stellar models, evolution of the sun – Compare with HR-model - Tests of solar models – Rest
Then compare it to the HR model, talk about the models describing it then go into how do we find
these models; observe > < numerical calculations. What can we do this on ? Binaries => mass,
Clusters => same properties, closed => young, open => old. Pulsating stars tell us about neutrino –
why and what?
Chapter 11.5, Chapter 13 – HR diagram, isochrones.
Introduction of the Sun
The Sun is our closest star with a mass of 1 solar mass, or ~2*1030kg. The mass of it, categorizes it
under the lower mass stars. It has a very long timescale, 1010years, and an convective envelope. All
our fundamental equations are all a result from the measurements to the sun; T, P, p, f(frequency),
ve. The sun consists of around 75% Hydrogen, 23% He, 2 % other.
Stellar observations and the results for stellar structure and evolution
Much of our observational data comes from clusters, since they have some features which make
them good testing-models, but let’s focus on the Star that can give us most information, the Sun.
There are two basic features about the sun which we can observe that will reveal some of the details
of the stellar interior; the oscillations and the solar neutrinos.
Solar oscillations, helioseismology
A normal string has many places where it stands still, the eigenmodes. It is a one dimensional figure,
and then easy to decide. However, the Sun is a three dimensional figure, so it has 3 different
eigenmodes; sound waves, gravity waves, and internal gravity waves. To make life a bit easier, we
will neglect the rotation of the star (1 frequency), so we are left with 2. One of those two is the
sound waves, which tells us about the temperature in the stellar interiors, with the needs of
additional constraints; chemical composition and hence mean molecular weight.
The results we got out of the models showed that two parameters were very sensitive; the coulomb
interaction in the EOS and the opacity. By adjusting these two parameters we have got the right fits,
see fig. 11.10. As seen on the figures on page 158, the sound waves computed by the model are
almost the exact as the ones observed, while the density only varies in the convective envelope,
since the opacity is more doubtable in these layers.
Solar neutrinos,
Since the neutrinos do not react with anything on their path outwards, then they are a direct
measure of the nuclear reactions rate. It can be seen from the pp-process having neutrino excitation.
The problem with the neutrino capture is the very small cross section and that the results were a lot
less the ones predicted. It is indicated in two problems, one, that the observed capture is different,
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depending on the solar activity (solar minimum  large rate). If you decreased the temperature by
5% in the core, the models would fit the observed, but then the oscillation-models would not fit. A
possible solution is that there are different types of neutrinos; muon and tau. This could also affect
the initial experiments.
RR Lyrae and Cepheids are also good to measure on, since they are very luminous and oscillate by
pulsation. The difference is that the oscillation of a Cepheid is due to ionization of He.
HR-diagrams
The reason for the use of HR-diagrams on clusters is that the only thing difference, the stars within
the cluster, has is their mass (neglecting: effects of rotation or a magnetic field). This makes the
clusters perfect laboratories for testing models, since it means that they all have the same chemical
composition and age ( all made from same Inter Stellar Medium).
Since all open clusters (we use open clusters, since they are old stars) only vary on mass, we can use
isochrones to predict their paths. As seen on the paths, the isochrones predict the same as the
observations do /theory  the higher mass, the shorter their main sequence gets, fig.13.1. By EQ
13.5 we see that an estimate for the isochrones is not even that bad. At fig.13.4, we see an
isochrones suddenly jumping from 1.2 – 1.25, and this is due to the convective core, and as you see
the isochrones will give you the mass of stars for an age of 3.5*109yr. And as can be seen on the
observed (left fig.) and the theoretical isochrones (right) is that they fit quite well. This also means
that by applying isochrones we can determine ages.
The mass is a big problem in astronomy to define, and can ‘only’ be defined by the use of binary
systems (exercise 2.1). If distance is know the Luminosity and temperature can be defined.
Maybe more on variable/pulsating stars
8
3.
Describe the structure and evolution of main sequence stars, with emphasis on energy
transport, energy formation (fusion processes), evolution time scales and mass-luminosity
relations.
+ Problem 2.1 (Alpha Centauri, without the uncertainty calculations).
Chapter 7 + 11
Start out by talking about the structure of the star and how it expands for a low mass star, (core –
slowly exceeding H-abundance) => temperature arises for He-core – then contract expand..
Degenerate core, and the death in a white dwarf (at same time show where we are on a diagram).
Tell meanwhile how the energy is transported and how the convection zones evolve and the
processes going around in the core PP => CNO => and no further since degeneracy can’t lift it up to
the amount needed. Then talk about the mass luminosity and we see 7.7 in effect.
It is important to remember that the hydrogen burning phase is the longest time scale for a star,
independent of the mass. It is due to hydrogen’s enormous efficiency in nuclear reactions.
ZAMS (Zero Age Main Sequence)
At figures 11.1, 11.2, 11.3; it is seen how the different relations depend on each other; most
measurements taken by binaries (due to mass on x.). ZAMS should not be directly interpreted as a
stabile sequence as the Main Sequence, but more as a point, where hydrogen burning establishes.
Evolution of main Sequence
In the start of the main sequence, we start having a nuclear reaction in the core, where hydrogen
gets transformed into Helium by the pp-process.In higher mass stars, we see another process doing
the same, the CNO-process; it requires convective core (to temp. sensitivity) and Z > 0. The general
tendency is that the luminosity increase along the main sequence path, point 1 to point 2 at fig.
11.5. This tendency can be understood by EQ 3.27, since µ increases, and to obtain the pressure, the
Temperature needs to rise as well, and therefore the core contracts a bit, leading to higher p.
EQ 3.24:
𝑃=
𝜌 𝑘𝑏 𝑇
µ𝑚 𝑢
;
tells us: µ inc.  pT inc.  stabile core
The mentioned contraction will increase the values by virial theorem; which results in increasing the
thermal energy of the gas  inc. Tcore. The inc. temperature and the contraction will then increase
the energy rate  increases the radiative flux of energy, through partly reduction in opacity  inc.
luminosity
𝒦~𝑇 −3.5
EQ 5.14:
The increases we see for luminosity and mean molecular weight were predicted by 7.7 and 7.8,
which will be mentioned later.
As it is known, the evolvement of a star results in increasing radius;
It is modest seen in low masses, as a result of 𝐿 = 4𝜋𝑅2 𝜍𝑇 4 .
In higher mass stars the expansion is more rapid therefore the temperature
decreases.
A theory for these reactions is that the high mass stars are dominated by CNO
9
process, which is very temperature sensitive, that’s why the cores temperature is not
allowed to raise so much as for low mass stars. Therefore the high mass stars need to
expand more than the lower mass stars to become stabile.
NOTE: Remember to have the fourth fundamental equation in mind when talking about contraction
increasing temperature, increasing energy production and density.
𝑑𝐿
= 4𝜋𝑟 2 𝑝𝜀 + 2𝑡𝑒𝑟𝑚𝑠
𝑑𝑟
The two terms aren’t used since there is no hidden energy in gas and energy from contraction. (the
contraction energy goes purely to heat and radiation).
Radiative transport
In a core with Radiative transport, it is assumed that no mixing occurs, so the products from nuclear
reactions stay in the same position, and since the center of the core is hottest, it losses hydrogen
abundance from the center and outwards towards the end of the core, see fig 11.6 (for convective
envelope). The changes in all values can be seen in, fig 11.8.
As for convective core, the chemical composition can be assumed uniform; resulting in a uniform
reduction of hydrogen abundance, see fig 11.7.
EQ 5.15 𝒦𝑒 = 0,2 1 + 𝑋 𝑐𝑚 2 𝑔−1
The reduction in hydrogen tells us that the dominant opacity by electron scattering will also be
reduced. The reduction in opacity will cause the convective cores to decrease in size as seen on fig.
12.3. The reduction will mean that the Radiative transport can take over.
Evolutionary timescales
Use table 11.1, p. 146.
Mass-luminosity relation
High mass stars have the following relation; 𝐿𝑠 ~𝑀3 . Higher MS, electron scattering.
Lower mass have the the following relation, 𝐿𝑠 ~𝑅−0.5 𝑀5.5 . Good on lower MS, Kramer
approximation.
Another thing to remember is that 7.7 and 7.8 essentially assumed same chemical composition, and
that’s not the case for inc. µ.
We have the following energy transportation, chapter 14, Radiation (photons), Convection (gas
cells), Conduction (atoms, electrons) and particle radiation (neutrinos).
10
4.
Describe the structure and evolution of low mass stars (typically one solar mass) from the
zero age main sequence to the end stages.
+ Problem 1.4 (Fundamental stellar parameters, questions a–e)
Chapter 11+12
Evolution of a 1 Msun
Before point 1, the star has been undergoing ZAMS, and here it is quite uncertain what happens, but
the most striking feature is the rapid increase in luminosity with stellar mass as predicted by the
mass-luminosity equation, see fig. 11.3. (M inc.  R inc.  Teff inc.  L inc.)
As a low mass star, we take a star with the same mass as our sun, 1 solar mass. A low mass star has a
convective envelope according to fig. 11.4, so this leads to that the PP-process in the core uses only
the Hydrogen in the core, since no ‘mixing’ is applied. This is a slow, non violent Hydrogen main
sequence phase, where the star just burns Hydrogen and slowly increases its Luminosity and
temperature, point 1 to point 2.
After point 3 in fig. 11.5, the core gets a convective thick shell, according to fig. 12.9. The figure
illustrates that the hydrogen abundance increases after around 0.05Rsun, and stays at a high level till
the surface. And the high production of H-burning can be seen by the amount of 3He in the shell. In
should also be noticed how the luminosity follows the hydrogen abundance. There is a small
contraction compared to the instability of the 5 Msun core, and as the core exceeds 0.1Msun.The
speed of the contraction increases and therefore the thickness of the H-shell decreases rapidly as
seen in fig. 12.10. Here we can first notice the shell-burning law; saying core contraction leads to
shell expansion, which can be seen by a dive in luminosity at 0.9Rsun. Secondly we can notice that the
shell gets thinner since the luminosity increases before the abundance of hydrogen.
At point 5, the core continues to contract therefore leading to two things; increased density 
degeneracy, and the Hayashi track. The core slowly gets degenerate which is seen on fig 12.11 by the
P(gas)/P(deg. e-). Another appearance is the deep convective envelope seen by the abundances of
14
N and 1H (steeply falls, and steep raise, respectively). This deep convective envelope results in a fall
in Xsurface (0.708  0.693). The core now stays degenerate, and because of the conductivity; the core
doesn’t get heated by contraction (deg. e- are independent of T). However, the H-shell still keeps
burning and increases the stars Luminosity. While the star moves up the Hayashi track, the core
temperature raises, and when set to 108K, we get helium burning.
The transition occurs as a thermal-run-away, point 6. Since the pressure of the degenerate
matter is independent of T, then the nuclear burning can set in, since the temperature can rise
without being effected by pressure, and therefore no reaction of the core itself. Since nothing stops
the temperature from rising, we get  Thermal run-away (high luminosity increase in hours). The
gradient of temperature gets stopped when the degeneracy is lifted due to the high temperature of
the core (EQ. 3.61, the radiation pressure exceeds the deg. pressure?). This violent phase IS NOT
seen at surface because of its dynamical timescale, and the radius of the star. The core will then end
in expanding and cooling to reach equilibrium, and enter a quite burning He-phase  He-main
sequence (horizontal branch, fig. 2.6).
When the star reaches the HB-phase, then the helium will start burning into carbon, and its
luminosity will increase due to the burning of hydrogen in the thin shell. When the helium core is
11
exhausted as it was with hydrogen, it will establish itself a He-shell and leave the Carbon-Oxygen
core, and slowly approach the AGB. The following result of an AGB is a thermal pulse-AGB; A thirddredge up (a convective zone from surface to the he-shell)  mixing material, to let the burning
processes go on. These deep convective envelopes give us data on the inner parts of the star and
observations have shown that we get stronger absorption of blue light by the carbon-rich stars (C/O
>1), and red for the other type, Oxygen-rich stars (C/O < 1).
Stars which are below 0.7 Msun will end up being He-rich stars, since their degeneracy will never be
lifted by the temperature.
It is important to notice that the lower mass stars may not transport material by supernova, but
when undergoing the last thermal instability (before W.D.) they have a massive loss of mass.
12
5.
Describe the structure and evolution of moderate mass stars (typically five solar masses)
from the zero age main sequence to the end stages.
+ Problem 7.1 (First stars).
Due to convective core this star will only have a thin H-shell, since its core gets supplied with
hydrogen from the star itself, opposite to a low mass star.
Note: table 11.1 – lifetime of all processes
Evolution of a 5 Msun
Before the main sequence, it can see the position of the star on fig. 11.1; its luminosity seems to fit
with the mass-luminosity relation. Another place to see how it evolves to a star is by the illustration
on fig 11.5, where it can be seen that by the help of gravitational energy it heats up the gas, and
therefore by increasing the thermal energy it can evolve into
Point 1 till 2, here we have the main sequence phase (TAMS, terminal age main sequence), where
the star will spend most of its time burning hydrogen into helium by the use of the CNO-process.
During this point it will exhaust the hydrogen throughout the star because of a convective core. Acc.
to fig. 12.3 the convective core gets smaller.
Point 2, here the star is near exhausting its hydrogen supplies, so it will try to increase its central
temperature by an overall contraction, which results in gravitational energy, where have is
converted into thermal energy and other half into heat. This is seen by turning into higher
temperature to point 3. Hence it can be compared to the process 2-3 on fig. 11.1, but with only a
moderate use of gravitational energy (seen by inc. in temperature  inc. luminosity). While moving
towards point 4, the star slowly establishes a hydrogen shell. The stellar structure inside the core
can be seen on fig. 12.3. Here it is seen that the convective core falls, and gets ‘filled’ with Helium.
Point 4, this point is symbolized because the star gets a new shell source, h-shell, which becomes
dominant, this means that by the shell-law, we will have an expansion of the star, since the core
contracts, therefore the temperature will fall, but luminosity will increase.
Point 5, due to the increased luminosity by the shell source, the core starts to exceed the
Schönberg-Chandrasekhar mass limit, and therefore the core contracts and the outer parts expand
to cool the star down, and moving it towards point 6.
Point 6, here we start seeing the convective envelope, fig. 12.3, and the source shell gets thinner
when moving towards the Hayashi track, since the luminosity increases rapidly. At this point the
helium ignition also starts, during the RGB-phase. Till point 7, where a triple alpha process comes in
(works in the He-burning core), and the star slowly decreases its luminosity due to its expansion, and
the nuclear burning tries to work against the gravity, which tries to expand the star, till point 8,
where the convective envelope disappears, and it happens due to the efficiency of the helium core
(not isothermal anymore), at point 9.
Point 9, the released energy (radiative) from the nuclear core processes lets the star expand, but
because of the shell law, the hydrogen layer will expand and the star contract, fig 12.3, while still
increasing luminosity. As the helium burning proceeds the convection core ‘somehow’ gets smaller.
13
Point 10, from 9 to 10, the helium core slowly establishes itself in a helium-main sequence, and
the contraction/expansion stops. Acc. to fig.12.3, the H-burning shell still contributes. As the helium
burning slowly increases the mean molecular weight; the core will contract and heat up.
Point 11, At this point the helium core will as said, contract and star making a new shell, a helium
burning shell. Point 12, the helium burning shell kicks in, as seen in a quick luminosity snap at fig.
12.2. Now the star has two shells and an exhausted shell.
Point 13 – 14, the core now needs to heat up to make carbon-reactions, and therefore leading to a
contraction, meaning that the region beyond the H-shell expands and the region beyond He-shell
contracts, outer parts. To see the interior of the sun during this phase, we can take a look at fig 12.5.
The inner luminosity is due to contraction while the two others are due to shells, while between the
two rising luminosities it is the expansion between the He-shell and H-shell. The expansion will be
the end of the h-shell because it gets cooled down, so the process cannot react, then entering the
AGB-phase. Before this phase, the stars here are called Cepheid, because of luminosity and slowly
upcoming instability.
Point 14, Here there will be an enormous convective envelope. This enormous convective
envelope helps us, observe what lies beneath the surface – and was the first sign of stars undergoing
s,r-processes (when seen Technetium.) in bigger stars, respectively.
After point 14, the luminosity greatly increases as well with the radius  thermal pulses, which
leads to rapid mass rate  planetary nebula. And the only material left from the star, is the
degenerate matter, which cools degenerately and ends as a WD. While the star is cooling, the outer
layers will dispers as a planetary nebula, as mentioned.
14
6.
Describe the structure and evolution of high mass stars (typically above ten solar masses)
from the zero age main sequence to the end stages, including a description of supernovae.
+ Problem 1.1 (Stellar timescales).
Firstly it should be mentioned that these stellar objects have the shortest nuclear timescale,
therefore they are the hardest to observe, since they are few. In these scenarios most of our theory
and predictions are built out of evolutionary tracks (moderate stars) and isochrones, fig 13.5.
Evolution of a 20 Msun
Before point 1, the star is believed to be very luminous compared to the other stars; around 104
orders of the suns luminosity, for a 20 Msun. These stars evolve like most other stars, just in a shorter
timescale. As seen on fig. 12.6, they will undergo the same ignitions, but never come into a point,
where they are in new reactions. It can also be seen for a 15 solar mass; the density of the matter in
the centre is lower, but the temperature for ignition to start, by contraction, is higher.
The main sequence (Hydrogen, Helium…), since the star is larger, the core will also be larger, and
the convective core will be enormous, according to fig. 11.4. It will expand to almost have of the
star, 0.7 mass is within the convective core. This means roughly that half of the hydrogen abundance
is within the convective core (20 Msun). The star will according to fig.11.5, have a very boring
evolvement during its many sequences, since its temperature after the H-burning shell just
decreases, and the luminosity increases as the shells get added on. All these shells which are added
on will end up in an Onion-like-structure, see fig.14.3.
The later stages,
0.5 GK the carbon burning sets in, resulting in different end products (12C + 12C  a lot
different).The high temperature and density effects the protons and alpha particles to react and
produce additional elements, with the free neutrons. After exhaustion, the core contracts and heats
leading to Ne-burning though photodisintegration.
1 GK, Oxygen burning sets in, here again we have the buildup of new elements, involving p, n, and
4
He. After the oxygen burning, we will start a new contraction and heating. Another dominant factor
at this energy level is the photo-dissociation in the nuclei (photons can react now.)
The photo-dissociation and then following photo-ejection will then make a complex network will
end up in having more tightly bound and hence heavier nuclei, and this set of quasi-equilibrium
processes is known as silicon burning. When the core ends up being in the iron group which is the
most tightly one, then no reactions can break the newly formed atoms, so the core gets no energy,
therefore it will end its phase.
It should be noted that within these large changes in the later stages, the stellar envelope has almost
the same composition since it started, apart from the first dredge up (RGB). These stages are very
fast, according to fig 1.3.
15
The end,
16
7.
Depict the formation of elements in stars (all masses) by fusion processes and by r and sprocesses, including a description of the corresponding evolutionary stellar phases.
+ Problem 6.2 (Solar convection, question 1).
All of the elements, fig. 14.1, we know and observe are remnants from most of the stars in the
universe. It is assumed that 50% of the elements in the lower category are formed by low/moderate
mass stars by the solar wind and during the Thermal pulses/dredge up (Uffe Jørgensen,
Astrobiologi). The other parts come from the supernova, where only type II can create the r-/pprocess elements. The other parts up to iron are made by all stars which can end up in a SI-burning
state.
Low mass stars
moderate stars
50 %
Stars
High mass stars
50 %
0.8 Msun He-rich
star
unproccessed
hydrogen
Carbon-rich
Oxygen-rich
Unprocessed
Hydrogen, Helium
and maybe C or O.
Supernovae
The rest of
elements, s-,r- and
p-process.
The stellar phases
The stellar phases are, hydrogen burning (CNO), helium burning (triple-alpha), Carbon burning (a lot
of end products), Ne-burning (photodisintegration), Oxygen burning (buildup of everything).
15 MK, here we have the pp-process, which fuels the younger stars. It simply says that 4 H  He +
2p.
20 MK, the CNO process, which is another hydrogen burning process, which fuels the massive
stars (> 1.5 Msun) through its lifetime. It tells us through a complex cycle that 12C + 4 H  12C + He.
0.1 – 0.2 GK, the triple alpha process can start making the carbon needed, to set the next phase in
action. It is noted as, 3α  12C.
0.5 GK the carbon burning sets in, resulting in different end products (12C + 12C  a lot
different).The high temperature and density affects the protons and alpha particles to react and
produce additional elements, with the free neutrons. After exhaustion, the core contracts and heats
leading to Ne-burning though photodisintegration (~1 GK).
> 1 GK, Oxygen burning sets in, here again we have the buildup of new elements, involving p, n,
17
and 4He. After the oxygen burning, we will start a new contraction and heating. Another dominant
factor at this energy level is the photo-dissociation in the nuclei (photons can react now.)
s-/r- and p-process
The processes are, while under AGB towards the Supernova (r-process, p-process), three processes;
s-process, r-process and p-process, fig. 15.1.
S-process, slow process; meaning that the neutron capture is slower than the beta decay. These
processes start appearing from the AGB-phase.
𝑍, 𝐴 + 𝑛 → 𝑍, 𝐴 + 1 + 𝛾
𝑍, 𝐴 + 1 → 𝑍 + 1, 𝐴 + 1 + 𝑒 − + 𝜈𝑒
The result of one neutron capture, is the decay, and therefore we get lifted up the stairs, leaving an
electron and a neutrino.
R-process, rapid process; meaning that the neutron capture is faster than the beta decay. These
nuclei are more neutron rich, since they become stabile by neutron capture.
𝑍, 𝐴 + 𝑛 → 𝑍, 𝐴 + 1 + 𝛾
𝑍, 𝐴 + 1 + 𝑛 → 𝑍, 𝐴 + 2 + 𝛾
The result, of one neutron capture of another, is that we stay on the same atomic level, but as a
stable isotope, and emit a photon.
To produce R-processes there is a large need of high neutron flux, which can only be fulfilled in a
type II supernovae / new born neutron star. Since the models are not consisted with the solar
abundance of r-process material, there has been thoughts on if r-process could come, as well, from
neutron-star mergers or supermassive stars in the early stage of our galaxy.
P-process; Atoms made under these processes are very rare, and are proton rich. And therefore
to the left of the steps, fig. 15.1. Their formation is uncertain, but it is believed to be from either
type I-SN or ISM.
It is important to notice the odd numbers have the largest cross-section, since they would be more
stable with filled neutron shells. This effect can be seen by fig. 15.2, and it is important to notice that
the most stable nuclei are achieved by the magic numbers, 50, 82 and 126.
The evidence for these processes was observed when a star should signs of technetium ( 43Tc) on
the surface due to the third dredge up.
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