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Energetics of Thermonuclear Supernovae

Degenerate (white) dwarfs come in three flavors: helium, carbon-oxygen, and oxygen-neon-magnesium.  The helium white dwarfs are the evolutionary endpoint of low-mass main-sequence stars that evolve over tens of billions of years and of intermediate-mass main-sequence stars in binary systems that lose their outer envelopes of hydrogen after they have converted their core hydrogen into helium.  Helium white dwarfs are rare, since only the second evolutionary scenario has had time to run to completion over the 16 billion year age of the universe.  The carbon-oxygen white dwarfs are the most common white dwarfs, because they evolve from solar mass stars, which complete their evolution in only about 10 billion years; the prevalence of carbon-oxygen white dwarfs makes them the natural origin of the type Ia supernovae.  The oxygen-neon-magnesium white dwarfs descend from main-sequence stars of several solar masses.   Because the lives of these main-sequence stars is much shorter than the age of the universe, most of these stars born over the lifetime of our Galaxy have already evolved to white dwarfs.  Because relatively few high-mass stars are born relative to solar-mass stars, the oxygen-neon-magnesium white dwarfs are less common than the carbon-oxygen white dwarfs.  The oxygen-neon-magnesium white dwarfs may be associated with other, less common, types of supernovae.

The energetics of an exploding white dwarf is well matched to the characteristics of the type Ia supernovae.  The amount of thermonuclear energy available in the conversion of intermediate elements into iron is considerable.  The conversion of carbon-12 (12C) to iron-56 (56Fe) releases 0.12% of the rest mass energy of carbon.  Other intermediate elements release less energy.  The conversion of oxygen-16 (16O) to  56Fe releases 0.084% of the rest mass energy of oxygen, while neon-20 (20Ne) releases 0.078% of its rest mass energy, and magnesium-24 (24Mg) releases 0.054% of its rest mass energy.  In comparison, the conversion of hydrogen (1H) into helium (4He) in the sun converts 0.7% of the rest mass of hydrogen into thermal energy.

The amount of thermonuclear energy available from the intermediate elements is just enough to blow a white dwarf apart.  The gravitational binding energy of a white dwarf is 3GM2/5R, where G is the gravitational constant, M is the mass of the star, and R is the radius of the star.  To blow the star apart, the amount of rest mass energy converted into thermonuclear energy must be greater than the binding energy, so the fraction α of rest mass energy that must be released is given by this inequality:

α > 3GM/5Rc2 = 0.042% (M/Mch) (R/3x108 cm)−1,

where c is the speed of light and Mch = 1.4 solar masses is the Chandrasekhar limit.

For a white dwarf of pure  12C with a mass of 1.4 solar masses and a radius of 3x108 cm, only 0.35% of the star must be converted to iron to blow the star apart.  The requirements become more stringent as we move to higher-mass elements, with a 50% conversion rate required of  16O, a 53% conversion rate required of  20Ne, and a 78% conversion rate required of  24Mg.  The precise values, of course, depend on the structure of the star; if the equation of state of the degenerate material in the star permits a larger radius for a given mass, then the amount of material that must burn to blow the star apart is smaller.

All of the thermonuclear energy immediately released in the explosion is converted into thermal energy that is trapped, unable to immediately escape from the white dwarf.  It exerts a pressure that exceeds the gravitational pressure of the white dwarf.  The white dwarf therefore expands, and all of the thermal energy trapped within the star goes to accelerate this expansion.  Virtually all of the thermal energy released in the explosion is therefore converted into kinetic energy.

We can get an upper limit on the velocity of the debris from the explosion by assuming that all of the thermonuclear energy is released and converted into kinetic energy after subtracting out the binding energy of the star.  For a star composed of  12C, the maximum average velocity is about 4% of the speed of light, while for a star composed of  16O, the maximum average velocity is about 3% of the speed of light.  A pure  20Ne star gives the same maximum average velocity as  16O, and a pure  24Mg star gives a maximum average velocity of about 2% the speed of light.  The actual maximum velocity can be a little higher than these values, because when the star expands, most of the kinetic energy goes to the outer regions of the star, while the core of the star moves out more slowly.  For this reason, the outer layers have a higher than average velocity.  Velocities for the stellar debris are derived from the Doppler shifts measured for the spectral lines.  The maximum observed velocities are less than 10% of the speed of light, so the estimated average velocities are consistent with maximum observed velocities.

Because most of the thermonuclear energy released in the explosion goes into expanding the star, relatively little of this energy is emitted as light.  But a type Ia supernova is brilliant, with a peak luminosity two weeks after the explosion of nearly 10 billion times the Sun's luminosity.  Somehow energy is generated within the stellar debris long after the shell has been accelerated to high velocities.  This energy source is radioactive decay.[1]

While iron is the lowest energy state for nuclear matter, transforming carbon and oxygen into iron on a very short timescale is not really possible.  The problem is that the stable isotopes of iron have more neutrons than protons, while the most abundant isotopes of carbon, oxygen, neon, and magnesium have equal numbers of neutrons and protons.  To get stable iron from these intermediate elements requires converting some protons into neutrons, which happens through very slow beta decays.  In the hot environment of a white-dwarf thermonuclear explosion, the dominant intermediate mass elements combine to give heavier elements with equal numbers of neutrons and protons; in fact, they combine to give heavier elements that are multiples of helium atoms—multiples of 2 neutrons and 2 protons—because they themselves are multiples of helium atoms.  Fusion of the intermediate atomic nuclei therefore does not produce much iron, but instead it produces massive amounts of the isotope nickel-56 (56Ni), which is the lowest-energy atomic nucleus that is a multiple of a helium nucleus.  This isotope is unstable, undergoing beta decay in six days to cobalt-56 (56Co).  Cobalt-56 is also unstable, decaying in 77 days to  56Fe, which is stable.  What this means is that the energy we see two weeks after the supernova explosion is from the decay of  56Ni, and the energy we see in the following months is from the decay of  56Co.  The total amount of energy released in these two decays is about 11% of the total thermonuclear energy available in carbon.  This means that the light emitted by a type Ia supernova accounts for no more than 0.013% of the white dwarf's rest mass energy.  Theorists find that this energy is sufficient to explain the brightness of type Ia supernovae if about 1 solar mass of  56Ni is created in the thermonuclear explosion.  The progression from 56Ni to 56Fe also matches the observed spectra, where iron and cobalt lines are seen 2 weeks after the explosion, and the cobalt lines weaken over the following months.

The fraction of thermonuclear energy tied up in nickel become more severe for the other intermediate elements.  The energy released by  56Ni constitutes 15% and 16% of the thermonuclear energy in  16O and  20Ne, and it constitutes 24% of the energy in  24Mg.  The fraction of thermonuclear energy from  24Mg that is tied up in  56Ni is so great that the fusion of  24Mg does not immediately release enough energy to overcome the binding energy of the star.  The the amount of thermonuclear energy from  16O and  20Ne  tied up in  56Ni is half of the amount of kinetic energy carried by the debris from the explosion.  A white dwarf without carbon should therefore produce an explosion with a much smaller ratio of kinetic energy to radiated energy than is seen.

The energetics and ubiquity of the carbon-oxygen white dwarfs make them favored among theorists over the oxygen-neon-magnesium white dwarfs as the bomb behind the type Ia supernovae.  Whether nature comes to the same conclusion, of course, is another, unresolved issue.

[1]Colgate, Sterling A., and McKee, Chester.  “Early Supernova Luminosity.”  Astrophysical Journal 157 (August 1969): 623–643.

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