Download Chapter 14. Stellar Structure and Evolution

Chapter 14. Stellar Structure and Evolution
The four equations of stellar structure discussed in Chapter 12, along with the principles
of nuclear energy generation (and opacity, which we did not discuss in detail and postpone
to a more advanced course) are the elements of stellar structure theory. The equations need
to be solved by a computer...they have no analytical solution. There are also free choices or
“parameter” that the modeler must make to generate a specific model. These include the
mass of the star that it is desired to model as well as the chemical composition. Fortunately,
we have found that the chemical composition of stars does not vary too much from one to
another. Basically, stars are (initially) composed of about 73% H and 25% He. The other 2%
is all the other elements, which astronomers lump together under the term “metals”. The
“metal content” or “metallitcity” of stars can range from nearly (although not precisely)
zero to about 2%. Most stars in the vicinity of the Sun have the 2% value and are referred
to as Population I stars. Stars that are deficient in metals are referred to as “metal poor”
and are called Population II stars. We shall see later that they are also older stars that
were formed earlier in the history of the Galaxy before a lot of metals had been created by
supernovae.
1.
Main Sequence Stars: Hydrogen Core Burning
When a star first forms it is chemically homogeneous and composed mostly of Hydrogen.
The temperature and density must be highest at its core to provide the pressure needed to
support the full weight of the star. Naturally, it is deep in the core, then, that conditions
first become suitable for thermonuclear fusion. Also, naturally, it is Hydrogen that begins
to fuse first, since it is most abundant and has the smallest Coulomb barrier. The particular
reactions that occur were described in the last chapter and are either the p-p chain or CNO
cycle. It turns out that the p-p chain produces more energy in the lower mass stars (below
about 2 solar masses) while the CNO cycle produces of the star’s energy for the higher
mass stars. Once the nuclear fusion turns on, the star can stabilize and is no longer forced
to contract to re-supply the energy it is constantly radiating from its surface. The star
stabilizes at a constant radius and enters its longest-lived phase of stellar evolution, the
so-called “main sequence”. During this phase, it gains its energy through conversion of H to
He in its core. That energy, in the form of photons, takes about a million years to random
walk its way to the surface, where it is radiated to space as the star’s luminosity. The set
of stars in this phase of evolution define the main sequence on the H-R diagram. What sets
one apart from the other is primarily the mass of the star. Chemical composition plays only
a minor role because there is not much variation in initial chemical composition among stars
–2–
– they are all mostly H and He in the same basic proportions.
The range of stellar masses is set by two conditions. The upper mass limit is about
50-100 solar masses and is set by the luminosity of the star. Very massive stars are so
luminous that the radiation pressure from their luminosity is sufficient to stop further mass
from falling on them. They therefore are self-limiting in terms of the mass to which they can
grow. Low mass stars, on the other hand, can be so small in mass that they do not get their
central temperatures high enough to start nuclear fusion of Hydrogen. The lower mass limit
is about 0.07 solar masses. Lower than that limit, objects are called “brown dwarfs”. They
can be seen for billions of years, shining with very low luminosity that is generated entirely
by their collapse and release of gravitational potential energy. Eventually their cores become
dense enough to support the star and they become like a massive planet (e.g. Jupiter is an
example).
Main sequence stars “burn” the Hydrogen in their cores (fuse it actually) at a rate that
depends on their mass. Since the luminosity of a star depends to a high power of its mass on
the main sequence (basically, L ∝ M 3.3 ), the lifetime of a star on the main sequence depends
strongly on its mass. Higher mass stars (e.g. O stars with masses above 20 solar masses) will
burn out the H in their cores in a period of 10-100 million years or less. They evolve rapidly
by astronomical standards. A star like the Sun (G-type of 1 solar mass) will take around 10
billion years to exhaust the H in its core, since its rate of energy production and, therefore,
luminosity is so much smaller. The very lowest mass stars (M-type of, say, 0.1 solar mass)
can last for trillions of years, much longer than the age of the Universe (13.7 billion years).
Hence, O and B stars are all relatively recently formed and do not last long – there are also
not that many of them. – whereas every M star that ever formed in the Universe is still with
us as a main sequence star. By far, the most common stars in the Universe are M-type stars
with masses of only a few tenths of a solar mass. There are roughly 100 million M stars for
every O star. On the other hand, the stars that we see at night and that light up a galaxy
are predominantly the O and B stars, as well as the supergiants and giants (see below). It
is the rare, massive stars that provide most of the light of a galaxy and the common, low
mass stars, that provide most of its mass.
We turn now to a discussion of the evolution of low mass stars (i.e. those with initial
masses of around 3-5 solar masses or less). The evolution of higher mass stars is similar
up to the point where a planetary nebula forms. Higher mass stars do not form planetary
nebulae, but explode as supernovae. This difference in their evolution will be discussed in
the next chapter. Here we first describe how most stars (i.e. low mass stars – the common
form of star) evolve.
–3–
2.
Hydrogen Exhaustion and the Red Giant Branch (RGB)
Inevitably a star will exhaust the H in its core, having converted it to He. The Sun
is about half way through that process. In the core of the Sun, we believe the present
composition is about 50% He. As the He is created, the core of the star must move to
slightly higher temperatures and pressures to keep up its pressure (see next chapter on ideal
gas law). When the Hydrogen is completely exhausted in the core, it becomes difficult for the
core to support the star and the compression is rather dramatic. Somewhat paradoxically,
the compression of the core, actually leads to a great expansion (and, therefore, cooling)
of the outer parts of the star. This is because the compressing core allows H-rich material
in the vicinity of the core to come closer in and reach higher temperatures and densities,
therefore increasing the rate at which nuclear fusion reactions occur. In other words, the
star transitions from producing most of its energy in its core to producing most of its energy
in a H-rich shell around the He-rich core. It enters a phase of “H shell burning”.
The increase luminosity of this phase, accompanied by the greatly extended outer envelope of the star (and its cooler surface temperature) moves the star on the H-R diagram
to the regime of red giants. The star becomes an RGB (Red Giant Branch) star. Its core
begins to be supported by electron degeneracy pressure (see next chapter). Helium cannot
yet be used as a fuel because the Coulomb barrier is so high and because the 4 He atom is
such a stable structure that it will not fuse with another 4 He atom to produce a more stable
atom. The 8 Be atom actually has less binding energy per nucleon (proton or neutron) than
the 4 He atom, so a star cannot gain any energy by fusing two 4 He atoms. Its core must
continue to contract and heat until it reaches temperatures near 100 million K and extreme
densities. Finally the star can undergo a set of nuclear reactions that gains it some energy!
This is called the Triple-alpha process.
3.
Horizontal Branch (HB) Stars: Core Helium plus Shell Hydrogen Burning
Physicists studying radioactive elements noted that there were two types of particles
that were emitted by radioactive isotopes. One is called the β particle and is now known to
be an electron. Electrons are, therefore, still often known as Beta-particles. The other type
of particle emitted commonly in radioactive decays is a 4 He atom. These were called Alpha
particles and, therefore, their fusion in a 3-body reaction is known as the Triple-α process.
The reaction can be written as follows:
4
He + 4 He + 4 He → 12 C + γ
–4–
When this reaction finally turns on, at the high luminosity “tip” of the red giant branch
(RGB), it has a major and rapid effect on the structure of the star. The star moves rapidly
on the H-R diagram to a position characterized by lower luminosity and higher surface
temperature. That is because, again somewhat paradoxically, the new energy source within
the core, expands the inner regions of the star a bit and lowers the amount of energy that the
star gets through H-shell burning. This more than offsets the gain from He-core burning,
so the star’s luminosity drops. Also, since there is less energy input going into heating
the envelope of the star, it contracts inward and, therefore, heats. This leads to a smaller
radius but higher surface temperature. Hence the star moves to the location of the so-called
“Horizontal Branch (HB)” on the H-R diagram. It is getting its energy at this state from
core He-burning plus shell H-burning. In addition to the triple alpha process, that is creating
Carbon from Helium, some of the C goes through an additional reaction with He to form
Oxygen, ie.
4
He + 12 C → 16 O + γ.
4.
Asymptotic Giant Branch (AGB) Stars: Shell Helium plus Shell Hydrogen
Burning
During the HB phase, the star begins to deplete its He in the core and gradually transitions to a core that is composed almost entirely of Carbon and Oxygen. As was the case
with Helium, the temperature and density are not high enough to burn C and O into higher
elements, because the Coulomb barrier is to steep. So the inner core continues to compress
as it cannot generate enough energy to support the weight of the star above it. This likewise brings the He-burning shell and H-burning shell to higher densities and temperatures,
pushing up the luminosity of the star. The outer envelope begins to go through the same
kind of transition as during the Red Giant phase. It expands outward and cools. The star
follows a path on the H-R diagram that takes it close to the RGB, but a bit above it. This
is referred to as the Asymptotic Giant Branch (AGB) phase of evolution. AGB stars have
Carbon/Oxygen cores in which nuclear fusion is not taking place, as well as He-burning and
H-burning shells that produce the star’s luminosity. Eventually, the luminosity from these
shells becomes so great that radiation pressure lifts the envelope right off of the star and
expands it into space, forming the beautiful phenomenon known as a “planetary nebula”
(see slides on that lecture or Google it!) The envelope of the star, containing a good fraction
of its mass, is gradually diffused back into the interstellar medium (gas and dust between
the stars). The He and H burning shells are gradually quenched and the Carbon-Oxygen
core settles down into the final phase of a (low mass) star’s evolution – a white dwarf.
–5–
5.
Testing Models of Stellar Evolution: Cluster H-R Diagrams
What we have described above is our understanding of stellar evolution that arises from
the computer models of stars created on the basis of the physical laws discussed. All theories
such as this need to be tested empirically against the data that we can observe in nature.
Since the evolution of any one star takes millions to billions (to trillions, in some case!) of
years, we obviously cannot test the models by watching any one star evolve. In practice, we
make use of the fact that there are whole clusters of stars, containing up to 1 million stars,
that can be found around the sky. A cluster represents a group of stars of varying mass
that have all formed together and, hence, have the same age. Since the rate at which stars
evolve depends strongly on their initial mass, clusters have stars at all different evolutionary
stages, depending on their initial mass. A cluster H-R diagram, therefore, looks almost like
the evolutionary track of a single star (but not exactly). What we actually compute to
compare with cluster H-R diagrams are so-called isochrones. This is a set of models of stars
of different mass but the same age. Comparing the isochrones generated by the models with
the cluster H-R diagrams we can test the models and get an accurate idea of the age of the
cluster.
6.
Testing Models of Stellar Evolution: Solar Neutrinos
Another, more direct, way of testing our ideas of stellar evolution is to try to look
directly into the core of the Sun by detecting the neutrinos that are predicted to be created
by the nuclear reactions there. In the slides accompanying the lectures there are pictures
of some of the neutrino detectors that we have on Earth. As noted in class, neutrinos have
very little interaction with normal matter and, therefore, unlike the photons do not have to
random walk their way out of the Sun. They come directly out and if they can be captured
by the neutrino detectors they can be useful. (Only a very tiny fraction can be captured, but
it is enough to detect them and test the models.) The first neutrino detector was installed
in a gold mine in South Dakota and comprised of a huge tank of “cleaning fluid”. It made
use of the following reaction to capture neutrinos:
ν + 37 Cl → 37 Ar + e−
Initially not enough neutrinos were detected to match the models. But this turned out to
be because of the physics of neutrinos, not because of a failure of the solar model. It turns
out that neutrinos can actually oscillate between three different forms – electron, muon and
tauon – and this experiment is only sensitive to electron neutrions. That accounts for the
paucity of neutrinos detected.
–6–
7.
Testing Models of Stellar Evolution: The Main Sequence
Finally, we note that the models of stellar evolution discussed do an excellent job of
explaining the location of the main sequence on the H-R diagram. As noted above, this is
the locus of stars that are in their core Hydrogen-burning phase of evolution. Since this is
where they spend most of their lives, as stars, this is where the main sequence of stars is
on the H-R diagram. Recall that higher mass stars, also known as “early type” (of O, B, A
and F type) use primarily the CNO cycle to burn their H, while lower mass stars, so-called
“late-type” stars (of G, K and M type) use the p-p chain. The nice match that occurs
between the details of the shape of the observed main sequence and the predictions based
on the models gives us confidence that the models are reasonably good. Of course, when
looked at in detail, there are always lots of little things that can be changed and improved
and many factors need to be ignored (such as rotation and magnetic fields). But, basically,
the models of stars at least low mass stars through most of their evolutionary lives are in
good shape.