Building the model

**The goals**

Student are stimulated to think about and realize a model, considering
all the parameters studied in the previous lessons. The model, derived
by simple mathematical tools, permits an easy visualization of an
abstract idea, but also a deeper insight about the physical concepts
which stay behind it.

**The main items**

During the first minutes of Universe's life, the rapid expansion didn't
permit the synthesis of all the elements of the periodic table: only
Hydrogen, Helium and a very little quantity of light elements can have a
cosmological origin.

The other elements that have been formed during stellar evolution. A
star can be imagined as a big 'cauldron' in which the lighter elements
are 'cooked' to get the heavier ones: the stars usually spend a long
time (e.g. ten billion years) in which the thermonuclear reactions
transform Hydrogen into Helium; then, according to the stellar mass,
other nuclear exothermic reactions are able to the synthesize elements
until Iron, beyond which all the fusion processes become endothermic.

Sometimes the end of a star's life is signaled by a massive explosion,
resulting in an extremely bright and short-lived object that emits vast
amounts of energy, typically as much as an entire galaxy in several
years: this giant outburst is called *supernova*. The great energy
involved during a supernova event allows the formation of elements
heavier than iron through a process named *capture reactions*. The
remains of the matter that is exploded away from the star during the
supernova is known as a *supernova remnant*.

The classification of supernovae is quite complicated. Astronomers use
observational criteria, not theoretical criteria, to classify
supernovae. Type I supernovae do not have hydrogen lines in their
spectra, but Type II do.

A type II supernova is caused when a star with at 10 times the mass of
the sun becomes a supergiant. The process whereby the star starts to die
and grow in size and temperature follows the same path as a [red
giant](../Star%20Death.htm). The more massive the star, the quicker the
star will burn fuel.

As the star burns fuel, it will go through a series of phases whereby
after one element has been burned (fused) a heavier element that has
been produced starts to fuse itself i.e. Hydrogen fuses into Helium
(10\'s of millions of years). As the core gets hotter (100 million
degrees K) and pressures increase, the Helium will fuse into Carbon (1
million years). The Star now contains five fusing rings. As the core has
ceased in producing helium it no longer contains either Hydrogen or
Helium. The outer ring of the Star now fuses what Hydrogen there is into
helium. The next ring inward where the temperature and pressure are
slightly higher, fuse helium into Carbon and Oxygen (a few hundred
years). The next ring inward fuses that Carbon into Neon and Magnesium.
The next ring inward fuses Oxygen into Silicon and Sulphur (a few
months). The last ring before the core fuses the Silicon and Sulphur
into Iron (a few days). At this stage the core temperature is almost 3
trillion grades Kelvin. At the core no more fusing takes place because
this requires an input rather than an output of energy. The pressures
and temperature needed for this enormous input of energy are only
available during the Supernova explosion.

Regardless of the enormous pressures and temperatures created now that
the core is all Iron, it cannot fuse any more. The star is now in a
sense on a life support machine, and is not producing it\'s own energy.
All the energy is being fed by the star's incredible gravity. The entire
star now is supported by the pressure of the degenerate electrons in the
core (a white dwarf). Due to the enormous sizes of supergiant stars, the
weight of the star is too much for this small white dwarf to cope with.
As the matter in the core gets compressed passed the Chandrasekhar limit
(named after Dr S. Chandrasekhar who discovered it in 1930) the
degenerate electrons from the outer atom become un-degenerate and are
squeezed into the nuclei of the atom. Here they react with protons to
produce neutrons and neutrinos. With the loss of the electrons, the
atoms at the core are further weakened helping gravity compress it ever
more. The iron nuclei begin to break up, using energy from the heat in
the core. This slows down the increase in pressure, which again helps
gravity. The core of the star now sits at around 1.4 times the mass of
the sun yet is only about 2/3\'s the size of Earth. Now that the
electrons are no longer degenerate, the neutrons in the core of the
atoms become degenerate (neutron degeneracy or neutronization).

The iron is now compressed to 10 trillion times the density of water.
Although gravity has had the upper hand all the way through the process,
pressure has put up a very good struggle. But as the electrons and
protons disappear there are fewer and fewer to resist the gravitational
pressure, all of a sudden Gravity rapidly gains the upper hand and the
entire core collapses in less than 1 second, from around 5,000 miles to
just 10-15 miles. It happens so fast that the shells outside of the core
have no time to follow suit. As the core (neutron star) reaches it\'s
minimum size during it\'s collapse it violently rebounds and pushes
against the rest of the core and inward rushing shells. The shockwave
produced travels at about 1/6th the speed of light. But, due to the
enormous density of the star, the shockwaves normally die out long
before they reach the surface of the star.

Outward rushing neutrino\'s that were produced by the loss of protons
and electrons give the shockwave the energy required to reach the
surface much quicker. At the peak of the explosion about 90% of the
energy produced is by these neutrinos. The collision of the outward
rushing shells is explosive on an unimaginable scale. The force of the
explosion will throw the shells outward at over 6,000 miles per second.
The explosion in general is so powerful that this one star can easily
outshine many billion others for several weeks. The energy released
during core collapse can equal all of the energy produced by 100 stars
in 10 billion years. The visible light given out from a supernova
amounts to less than 1% of the total energy of the explosion.

Type Ia supernovae are fundamentally different from the Type II. The
Type Ia supernovae cannot be caused by the deaths of massive stars.
First, no compact neutron stars are found in the remnants of type Ia
supernovae. Second, no physical process, no matter how violent, can
produce the luminosity observed from Type Ia by core collapse of even a
supermassive star. The outer layers of a giant star would absorb too
much of the energy. Rather, it is necessary that a mass of something
more than the Sun\'s mass be completely destroyed while unshielded by
overlaying material for anything as bright as a Type Ia supernova to be
possible.

A model that permit to explain all these features is based on a binary
system of small stars ($M_{\text{Star}} \leq 2M_{\text{Sun}}$), that
transfer some mass between them during their evolution. As we have seen
in the last lesson the gravitational potential play a fundamental part
in the binary system's evolution. As long as the separate stars are in
the main sequence (Hydrogen burning), their size is smaller than Roche
lobe's size and stellar evolution develops normally. When the Hydrogen
burning is finished, the star becomes a red giant, which sometime
increases its size a factor of one hundred. If the system is
sufficiently close, the stellar size is comparable to size of the Roche
lobe: the primary star (more massive), evolving earlier, can fill its
Roche lobe and produce a fall of material on its companion star. This
process is a kind of "decorticate" of the external shells, made
principally of Hydrogen.

After this passage the original secondary star become the primary thanks
to material received by the more massive star of the system: the
original primary star degenerate as a white dwarf of Helium with a mass
less of 1.4 solar mass (*the Chandrasekhar limit*). When the secondary
star ends its Hydrogen, it evolves analogously as a red giant: if its
size become larger than Roche lobe's size, we can see a second exchange
of material, from the secondary star to the first one, that finds itself
to have a degenerate structure.

As soon as the mass of the white dwarf exceeds 1.4 solar masses, with a
complex and still not completely understood mechanism, the star collapse
provoking a massive explosion that produces an extremely massive
outburst of energy, some $\text{10}^{\text{44}}\text{Joule}$, with an
absolute magnitude of about -19.5. The star literally blows itself to
bits, leaving nothing behind except a rapidly expanding remnant.

Recently, some astronomers[^1] have suggested that the binary star KPD
1930 + 2752 is a future candidate for a future Type Ia supernova event.
Now the primary star in this system is a subdwarf-B star, and it has an
unseen companion star that is almost certainly a white dwarf. This means
that the primary star is burning Hydrogen in the main sequence, while
the secondary star (which probably long ago was the primary) is a white
dwarf. Thought the white dwarf is more massive than the subdwarf-B star,
its luminosity is higher because the white dwarf has finished the
Hydrogen burning.

Since the orbital period is only 2 hours 17 minutes and the total mass
of the system is 1.47 Solar Masses (above the Chandrasekhar limit) many
scientist proposed that the binary will merge within approximately 200
million years due to orbital shrinkage and the evolutionary expansion of
the primary. When this happens, helium and other elements heavier than
hydrogen 'fall' onto the white dwarf from the subdwarf-B star, that will
evolve as red giant. When the mass of the white dwarf of Helium exceed
the 1.4 solar mass, probably a Type Ia supernova would occur.

The development in classroom

Recent statistical data estimate that about 30% of all supernovae are
Supernovae Type Ia: this means that, if many elements that we can see
everyday on the Earth are 'cooked' in a supernova process, it is
possible that this process has involved a binary system of stars.

As you can see in this model a strategic role is played by the
gravitational potential: the Roche lobe and the equipotential surface
are fundamental in the binary system's evolution and for the
comprehension of Supernova Type Ia. The challenge of this lesson is to
make the student understand the tight relationship between the
gravitational potential and the evolution of a binary system of stars.
In order to reach this target, we suggest to build a simple model, that
permit the visualization of the gravitational potential and the stellar
evolution.

According to us is important that the idealization and the construction
of the model are developed by the students in classroom. Perhaps this
can be seen as a loss of time by the teacher, but it represents a moment
in which the students try to visualize an abstract idea through a
concrete model. Initially the mathematical approach can appear too
abstract for a young student: a deep meditation on the physical meaning,
a concrete visualization and the construction of a model with his hands
lead the student to a deep comprehension of the phenomenon.

See
<http://imagine.gsfc.nasa.gov/docs/teachers/Elements2002/StellarEvol/images.html>
for a set of pictures which illustrate the building of model of a binary
star system using polystyrene foam.

[^1]: P.F. Maxted, T.R. Marsh & R.C. North, \"KPD1930 + 2732 - a
    candidate Type Ia Supernova Progenitor\", Monthly Notices of the
    Royal Astronomical Society **317**, September 2000.