# Chapter 15. The Chandrasekhar Limit, Iron-56 and Core Collapse Supernovae

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1 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: PV = 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 3 kt worth of energy and that dictates its mean speed, or 2 root-mean-square speed (v rms = <v 2 >) according to: 1 2 m < v2 >= 3 2 kt 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.

2 2 This has the units of gm per cm 3 (in cgs untis, or equivalently kg per m 3 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 = ρ < m > kt 2. 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, but only on its density. It has the form P = const ρ 5 3 or P = const ρ 4 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

3 3 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

4 4 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 Fe 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 Fe 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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