Download Chapter 15. The Chandrasekhar Limit, Iron-56 and Core

Chapter 15. The Chandrasekhar Limit, Iron-56 and Core Collapse
Supernovae
1.
The Equation of State: Pressure of an Ideal Gas
Before discussing results of stellar structure and stellar evolution models further, we
describe one additional component of the models. That is, the relationship that exists
between pressure and other variables, normally density and temperature. This is often
called an equation of “state”. In most parts of most stars it is sufficient to treat them as
an ideal gas and the appropriate equation of state is the “Ideal Gas Law” that most of
you have probably studied in Chemistry or Physics classes before. Normally, it is described
as the pressure (P) that exists on the walls of a container of volume, V, if the gas is at a
temperature, T. The equation is often written as:
P V = NRT
where N is the number of “moles” of a gas (remember that a “mole” is the amount of gas
that contains Avogadro’s number worth of molecules) and R is the “gas constant”.
This form of the ideal gas law is not so useful to astronomers because there are no
containers inside stars that specify fixed volumes. Therefore, we prefer to divide both sides
of the equation above by V to get the number of particles per unit volume (i.e. the “number
density”). We also prefer to express this as the number of free particles per unit volume
(n) rather than the number of moles. Therefore, the constant in the equation changes and
becomes, in fact, the Boltzmann constant, k. The ideal gas law, as astronomers write it,
therefore, is:
P = nkT.
Note that a free particle here means that the particle is contributing to the pressure. For
example, an electron, if free of its atom, needs to be counted, but the electrons that are
bound to the atom are not. That is because an electron free of its atom will be moving
at high speed (owing to its light weight), while an electron attached to an atom will be
moving at much lower speed. This comes about because of the principle of equipartition of
energy. Each free particle gets 23 kT worth of energy and that dictates its mean speed, or
√
“root-mean-square speed (vrms = <v 2 >) according to:
3
1
m < v 2 >= kT
2
2
To complete this section, we note that in the equations of stellar structure discussed
previously, we used a slightly different expression of density, namely the “mass density”, ρ.
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This has the units of gm per cm3 (in cgs untis, or equivalently kg per m3 in mks units). The
relationship between mass density and number density is:
n=
ρ
<m>
where < m > is the average mass of a free particle (i.e. one contributing to the pressure).
Written in these terms, the ideal gas law for the pressure becomes:
P =
2.
ρ
kT
<m>
Electron Degeneracy Pressure and the Chandrasekhar Mass Limit
While the ideal gas law covers most situations in stellar astrophysics it does break down
at the very high densities that exist in the cores of some stars, especially as they evolve
towards the red giant branch and beyond. Physically, what happens is that the free electrons
that have been ionized from their atoms by the high temperatures in the core begin being
packed so closely together that the Pauli exclusion principle begins to be a factor. This is
what limits the number of electrons in any one shell of an atom. Normally it does not apply
to free electrons because normally free electrons are so far apart from one another that their
wave character and uncertain location (due to quantum effects) is not at issue. However, at
very high densities, in stellar cores, it can be an issue. Under these conditions, electrons resist
further compression and develop a pressure, called electron degeneracy pressure, that can
help to support a star. This pressure does not depend at all on the temperature of the gas,
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but only on its density. It has the form P = const ρ 3 or P = const ρ 3 . The first form applies
when the density is high, but not exceedingly high and conditions are non-relativistic. The
second form is called “relativistic degeneracy”. A full description of how these relationship
are derive and their meaning must await a more advanced course. The important thing here
is that there is such a thing as electron degeneracy pressure and it is key in supporting a
star against gravitational collapse without requiring an increase in temperature.
The other key thing about electron degeneracy pressure is that there is a maximum
amount of mass that can be supported by it. This is known as the “Chandrasekhar mass
limit” after a famous Indian/British/American astronomer. The value of the mass limit is
something that can be derived (and will be a problem set!) in an advanced class. It is 1.44
solar masses. Stars more massive than that cannot be supported by electron degeneracy
pressure alone. They will be forced to collapse further to a neutron star configuration. Note
that there is an analogous physical phenomenon called neutron degeneracy that can support
a neutron star against further collapse. The maximum mass of a neutron star that can
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be supported is about 3 solar masses. If a degenerate stellar core ever exceeds that, there
is no further known degeneracy gas law to support it and it will collapse to a black hole.
These physical facts about pressure give rise to the observed phenomena of core collapse and
thermonuclear supernovae and o X-ray binaries, as we will discuss further below.
3.
Final Phase of Evolution of a Low Mass Star: White Dwarfs
Returning now to the evolution of low mass (3-5 solar mass stars) we consider the
“burnt out core” that is left behind after the planetary nebula stage. Recall that this will
be composed of Carbon and Oxygen. Its dimensions will be roughly that of the Earth, but
its mass will be typically about 0.5 solar masses or more. Its maximum mass is 1.4 solar
masses, namely the Chandrasekhar limit. White dwarfs are supported by electron degeneracy
pressure and therefore do not have to keep their core temperatures high via nuclear reactions
to keep from collapsing. Their pressure depends only on their density, which is already high.
They do not have any form of nuclear energy generation and they radiate only their thermal
energy, cooling off slowly over time. They have enough thermal energy and parse it out
slowly enough (given their low luminosities) that they can continue to shine for billions of
years. Basically, any white dwarf created during the Universe can still be seen (if close
enough to the Earth – we cannot, of course, see very distant ones). These are the stellar
corpses of the galaxy. They represent the final evolutionary stage of low mass stars. Unless
they are in a binary system, where they can get a new lease on life by accumulating mass
dumped on them from their companion star, they have nowhere to go in their evolution.
4.
The Evolution of High Mass Stars: Core Collapse Supernovae
The initial evolution of high mass stars, as they deplete the Hydrogen in their cores
follows a similar path to the low mass stars. Because of their higher luminosity, they populate
the region of the supergiants, rather than the giants. Supergiants are rare because high mass
stars are much rarer than low mass ones and because the evolutionary phases represented by
such stars are relatively short. It is hard to catch many such stars in such phases. However,
because of their huge luminosity, when a star is in such a rare phase it is easy to detect it.
So, we do see quite a few supergiants and they are an important contributor to the light of
many galaxies.
The main difference in the evolution of high mass stars occurs after the development of
a C/O core (Carbon/Oxygen). Unlike the low mass stars, these objects have high enough
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temperature and density in their cores to initiate nuclear fusion among the C/O atoms.
That is, they can overcome the repulsive force of the Coulomb barrier. Hence, a higher mass
star proceeds to additional stages of the core - shell phenomenon, eventually developing an
Iron core. It cannot go beyond iron, because 56 F e is the most stable atom, in terms of
binding energy per nucleon (see slide accompanying the lecture). It represents the dividing
point between fission and fusion as sources of nuclear energy. Lighter atoms can be fused
together to form more massive atoms, gaining energy in the process. Heavier atoms can be
split (nuclear fission) gaining energy in that process. 56 F e is the most stable atom and one
cannot get energy from it by either fusion or fission. It is, therefore, useless to the star as a
source of energy and once the iron core forms the star has no choice but to continue shrinking
its core, releasing gravitational potential energy, and then relying on electron degeneracy to
support the iron core.
So, a high mass star moves inexorably to a situation where it has the growing iron core,
surrounded by many shells of nuclear burning. It is sometimes referred to as an “onion model
of a star” because of all these layers of nuclear burning. It can go on with its life until its iron
core reaches the Chandrasekhar limit of 1.44 solar masses. When that happens, the core can
no longer support itself through electron degenerate pressure and the electrons are forced into
the atoms, creating neutrons. On a time scale of seconds, the Earth-sized electron degenerate
core, which has now lost its electrons and, therefore, its pressure support, collapses under
gravity to a point where the neutrons become degnerate. This is at a radius of about 10
km. The gravitational potential energy released in taking 1.4 solar masses of matter from
the size of the Earth to a sphere of about 10 km, is enormous. This energy is deposited into
the inner part of the star within seconds and results in a catastrophic explosion of the outer
parts of the star. In the process, the star emits more energy than the Sun will produce over
its entire lifetime. Most of the energy goes into neutrinos and into the kinetic energy of the
explosion. The outer parts of the star are propelled into space at speeds approaching the
speed of light. The light from the supernova, although a mere 1% or so of its total energy,
nonetheless may be enough to outshine its entire galaxy for a period of days or weeks. This
is a core-collapse supernova. It represents the final evolutionary phase of a high mass star.
5.
Neutron Stars and Pulsars
While the outer envelope of the supernova is ejected into space, it does leave a remnant,
namely the neutron star. With a radius of only about 10 km, neutron stars are too small to
observe directly by their light, even when they first form. However, we can detect them as
“pulsars” in some cases. What happens during the collapse is that conservation of angular
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momentum and magnetic flux, greatly intensifies the magnetic field of the neutron star over
what existed in the star prior to the core collapse. Also the star spins much faster (about
30 times per second when first formed). If the magnetic and rotation axes are even slightly
misaligned (as they are on Earth and in most objects), then the rotation of the pulsar may
cause the magnetic pole to sweep across our line of sight in periodic fashion, much as a light
house does for ships. The magnetic poles are intense sources of optical and/or radio emission
caused by non-thermal processes, in particular “synchrotron radiation”. This is the emission
of high energy electrons spiraling in a magnetic field. The bursts of radiation that occur
as the magnetic poles of rotating neutron stars sweep across our line of sight are detected,
usually by radio telescopes, but in some cases (e.g. the Crab nebula) optically. These are
called “pulsars” because of the pulses of radiation detected (not because they are pulsing
stars....they are not!). Not every supernova leaves a pulsar, because not every supernova is
a core collapse supernova (the other type – thermonuclear supernovae – is discussed in the
next chapter) and also not every rotating neutron star will have its magnetic axis oriented
so that we can see it.
Neutron stars also represented a cosmic corpse – a final resting place for matter, as long
as the corpse is not part of a binary system. If it is part of a binary then further evolution
of the neutron star may occur as matter from its binary companion accretes onto it during
the advanced evolutionary phases of its companion. Pulsars gradually slow down their spin
rates with time due to magnetic dragging with their local interstellar medium. The age of
a pulsar can be estimated by its spin rate and other properties determined by watching its
spin. Neutron stars are supported by neutron degeneracy and are essentially in the solid
state. Sometimes their crusts can undergo “quakes” analogous to Earth quakes and these
will result in readjustments of their spin rates. They can be detected by monitoring the
radio pulses for the pulsars. It is also possible to test predictions of general relativity by
monitoring pulsars in binary systems, since one has essentially orbiting “clocks”, where the
spin rate acts as the clock and the detected pulses as the time beats. A link to the site
describing Nobel prize-winning research along these lines is given on the links page.