Supernovas: Making Astronomical History

Fusion and Building Chemical Elements

Binding Energies

Stars, including our own Sun, produce heavy elements by smashing together the nuclei of lighter ones. Here, we will look at how this process operates, and what conditions constrain the production of new elements.

To understand nuclear reactions in more detail, we use Einstein's equation E = mc2. When we combine protons and neutrons to make a nucleus, the final nucleus is lighter than the sum total of the individual particles' masses. The difference, the so-called "mass defect", goes into an energy, the energy of the strong-force bonds holding the nucleus together.

Graph of Energy vs. time in a reaction  (Blake Stacey)

A fusion reaction requires some amount of energy to go, which depends upon the specific nuclei being fused. Since all positive charges repel, we must put energy into the reaction, pushing the nuclei together against their electric repulsion. The more strongly charged the nuclei, the more energy this requires; therefore, we expect heavier elements with more protons to be harder to fuse. In chemists' language, we have to overcome the "activation energy", a sort of "hill" which we must climb over to reach the final state (a fused nucleus). The high activation energy for nuclear fusion reactions implies that fusion is a high-temperature process. What we call "heat" is, on the atomic scale, the motion of atoms: the kinetic energy of their motion appears to us as the "thermal energy" of the substance. The higher the temperature, the faster the atoms are moving; to achieve fusion, the temperature must be sufficiently high that atoms impact strongly enough to overcome their electrical repulsion. In other words, the thermal energy must be large enough to overcome the activation energy. (At these temperatures, the electrons are stripped off the atoms, leaving a "soup" of bare nuclei and free electrons zinging about, a state of matter we call a plasma.)

A fusion reaction can also release energy: suppose we combine some small nuclei, which each have some mass defect, to make a larger nucleus, which will have its own characteristic mass defect. The conservation of mass-energy states that the total mass-energy cannot change--mass may convert to energy and vice versa, but the total remains fixed. Therefore, since the mass defects are not likely to be exactly the same, we expect to find some "excess mass", which must exist to balance the equation.

Mprotons + Mneutrons - (mass defect of small nuclei) = Mprotons + Mneutrons - (mass defect of large nucleus) + (excess)

We can subtract the mass of the protons and neutrons from each side. The excess mass is given by the difference of mass defects:

(excess) = (mass defect of large nucleus) - (mass defect of small nuclei)

By Einstein's equation, this excess mass shows up as an amount of energy. Scientists typically write a change in a quantity using the Greek letter Δ, "delta", so a change of energy can be written ΔE.

ΔE = c2 * ((mass defect of large nucleus - (mass defect of small nuclei)).

If ΔE is larger than the activation energy (the energy required to make the reaction go), then the fusion reaction can be self-sustaining, with the output from one fusion event fueling the next. The excess above and beyond the activation energy can show up as heat, light or other radiation. Therefore, the fusion reactions in stars are governed by the mass defects of the elements.

By convention, scientists speak of "binding energies", which are really just mass defects times c2. What, then, determines the binding energies of the different elements? This depends upon the interplay between the electromagnetic and the strong nuclear forces. As mentioned earlier, the positively charged protons all repel one another. The electromagnetic force is what we call an "inverse-square" interaction: its strength drops off with the square of the distance between charges. If the distance between protons is doubled, the repulsive force weakens by a factor of 22 = 4. If the distance is increased by a factor of 10, the force drops to 1/100 of its previous value. The sizes of atomic nuclei, however, are small enough that all the protons can essentially "feel" each other. The "electrostatic" energy due to all these repulsive forces therefore grows without limit as the nuclei get larger and larger.

On the other hand, the strong nuclear force is a "short-range" interaction. Protons and neutrons only "feel" a strong-force effect due to their immediate neighbors. Interestingly, the strong force attraction between a proton and a neutron is the same as that between two protons or between two neutrons. (A physicist would say, "The strong force is symmetric under exchanges of protons and neutrons.") We can therefore lump both kinds of particles under one term, "nucleons". Inside the nucleus, nucleons have partners on all sides, but those nucleons on the surface have partners on only one side. A larger nucleus, with more "room" on the inside compared to its surface area, has comparatively more attractive bonds, so the attractive strong-force binding energy increases with nuclear size (up to a point).

These two contributions have opposing effects, one attractive and the other repulsive. The overall result is that the binding energy per nucleon increases as the nucleus size goes up, until we reach the element iron, at which point it goes down again.

This implies that elements heavier than iron cannot fuse spontaneously. It "costs more" to squeeze two heavy nuclei together than the reaction can produce. It is physically possible to do this, but the reaction requires more energy than it can supply, and in nature it only happens when another reaction produces a large surplus of energy.

Producing the Elements

As we mentioned earlier, the activation energy is larger for reactions involving bigger nuclei, since the more protons we have, the more electric repulsion we will encounter as we bring the nuclei close together. This means that reactions with light nuclei, particuarly hydrogen and helium, are the easiest to achieve. Fusion can occur with heavier elements, all the way up to iron, but as we indicated earlier, fusing iron soaks up more energy than it gives out.

a brown dwarf

Artist's rendering of a brown dwarf with an accretion disk

The mass of a star is the single most important criterion for determining what elements it will fuse, how long it will fuse them, and consequently, how long the star will "live". A star's life begins when a cloud of gas (a nebula) contracts under its own gravity, pulling together and warming up until the pressure and temperature at its center is great enough to initiate nuclear fusion. A brown dwarf, with a mass seventy or so times that of the planet Jupiter, can manage to fuse deuterium (heavy hydrogen), but not the ordinary hydrogen isotope. This requires more mass.

A star with about the mass of the Sun (i.e., with about "1 solar mass" of material) spends most of its existence fusing hydrogen to make helium. Three helium nuclei can, in turn, be fused into a carbon nucleus. (Other reactions, involving two helium nuclei or a hydrogen-plus-helium combinations, lead to unstable nuclei which break down again.) When a star in this mass range nears the end of its life, it has accumulated enough helium to begin fusing that helium into carbon. The extra energy produced by helium fusion causes the star to expand, radically: when the Sun enters this phase, four billion years or so from now, it will swell to engulf the orbits of Mercury, Venus and perhaps Earth. Stars in this phase are known as red giants, thanks to their color. Their surfaces are relatively cool (and therefore radiate red light, by Planck's blackbody law), but because of their large total surface area, they can be very bright.

A star with one solar mass of material cannot sustain fusion past this point. Eventually, it will exhaust its helium supply (in much less time than it spent fusing hydrogen to make helium). The red giant will then collapse, leaving a shell of gas and a very small, tightly packed central remnant, a "white dwarf" mostly made of carbon and oxygen. The white dwarf glows with the leftover energy of the star, but it does not sustain fusion of its own. In a white dwarf, atoms are packed extremely closely together, so much so that they "degenerate": nuclei are so small compared to atoms that atoms are mostly empty space, but in white-dwarf material this empty space is filled.

see caption

Spitzer Telescope image of infant stars in the Triffid Nebula

Consider a star with several times the mass of the Sun. Such a star can achieve fusion of heavier elements, including carbon and beyond. However, it exhausts its fuel supply in less time: heavier stars are shorter lived. A one-solar-mass star can "survive" for ten billion years, while heaver stars "die" in a thousandth of that time. Their death throes are also more impressive.

This table shows the "burning" process for a star of twenty solar masses. (We hasten to clarify that nuclear fusion is not the same thing as burning ordinary fuel, which is a chemical reaction involving electrons in the outer regions of atoms, and which delivers much less energy. However, astronomers and astrophysicists feel free to speak about stars "burning hydrogen" or "burning helium", and as long as we keep the basic physics in mind, there really isn't much chance of confusion.) Note that the initial "burning" stage, in which hydrogen is fused into helium, is by far the longest, enduring about eight million years.

Source: "Massive Star Evolution Through the Ages"

Star of 20 Solar Masses
Fuel Main Product T (109 K) Duration (yr)
H He 0.037 8.1 x 106
He O, C 0.19 1.2 x 106
C Ne, Mg 0.87 9.8 x 102
Ne O, Mg 1.6 0.60
O Si, S 2.0 1.3
Si Fe 3.3 0.031 (11 days)

As time progresses, a massive star comes to resemble a nuclear-fusion onion, with reactions progressing in layers. The very center of the star will be "burning" the heaviest elements, with lighter nuclei "burning" in successive layers, out from the center. We might find, for example, a core in which oxygen fuses to silicon and sulfur, surrounded by layers in which neon fuses to oxygen and magnesium, carbon fuses to neon and magnesium, and so forth.

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Cut-away schematic of a massive star in its last days (Blake Stacey, based upon this SOHO (ESA & NASA) image(Replaced).)

The process ends, as advertised, with iron (Fe). What happens when the star builds up a core of iron? In brief, a catastrophe: the iron (which exists in a gaseous plasma state, like all elements within the hot star) soaks up more energy than fusing it can produce. Stifled at its heart, the star cannot hold itself up against its own gravity. (It is the thermal energy of these large stars which keeps them from collapsing under their own weight, and this thermal energy requires constant fusion to maintain.) When the iron chokes off its energy source, the star collapses.

We have never had the chance to see what happens next up close, or even from a safe distance of several tens of light-years. The only such events which have been witnessed using modern observing technology have occured in other galaxies. Scientists are, therefore, still guessing about some of the details. What we do know is the following:

The collapse of the star triggers an explosion as bright as an entire galaxy, which can glow for weeks. About 1046 joules of energy are released as neutrinos, and about one one-hundredth of this (1044 joules) becomes the visible explosion. This number is so large that illustrative analogies are hard to find: the first atomic bombs (those used at Alamogordo, Hiroshima and Nagasaki) released about 1014 joules. The fusion "devices" produced in later years can be a thousand times more powerful than the original fission-powered bombs, but Nature still does better, by about twenty-nine zeroes before the decimal point.

Such a blast is called a Type II supernova. Astronomers have observed other types of large stellar explosions, which they sort into other categories. The primary division, between type I and type II, is made using spectroscopy: type II supernova show hydrogen lines in their spectra, and type I do not. Supernovae are classified into subcategories (Ia, Ib and so forth) based upon finer details of their spectra and the shapes of their light curves, that is, how they grow bright and dim again over time.

Type I supernovae lack hydrogen lines in their spectra, which presumably means the original star was poor in hydrogen. They are often due, directly or indirectly, to binary star systems, in which two stars are in orbit around their common center of gravity. (This situation resembles the Earth-Moon system, except that the distance between objects is larger.) If both partners of a binary system form at the same time, the more massive one will speed through its nuclear fuel more rapidly, reaching the end of its life cycle sooner. This means that one star may have collapsed to a white dwarf "ember", slowly radiating away its leftover energy, while the other is still burning hydrogen and has not yet begun burning helium--the phase of stellar development astronomers call the "Main Sequence".

Gravity plays a major role in what happens to binary star systems. First of all, it is the gravitational attraction between the stars which binds them into orbit and keeps them from flying apart into space. Second, it may happen that one star's gravity draws gases off the other star, draining away the second star's outer layers and adding mass to the first star. Recalling the "onion" model of massive stars, we expect these outer regions to be mostly hydrogen. If a partner star strips away the hydrogen from a heavy giant star (one which is at least fusing helium to make carbon), we will be left with a "stub" of the heavy star, a body called a Wolf-Rayet star.

Losing its outer, hydrogen-rich layers does not affect what happens inside a Wolf-Rayet star. When the WR core begins to "burn" silicon, it will create an iron buildup, which will choke off the stellar fusion, just as we discussed above. The WR star will then go supernova, but the blast will be missing hydrogen, since the WR star's hydrogen was all drained away! (It is also possible for a star to lose its outer layers due to a strong "stellar wind": something about the star's interior makes it "blow off" gases from its surface, thinning the outer regions.)

Supernovae which do not show hydrogen lines and which also lack a strong silicon signature are thought to be exploding Wolf-Rayet stars. This category is termed type Ib; there is also a type Ic supernova, in which the spectrum has no helium lines (or very weak ones). We might expect that type Ic explosions are due to WR stars which have lost their hydrogen and helium layers, so that the outermost part of the "onion" contains carbon plasma, burning to make neon and magnesium.

Deploying our knowledge of the Latin alphabet, we expect that having type Ib and Ic supernovae means that astronomers also have a type Ia supernova category, and so they do. As with all type I events, Ia bursts lack hydrogen. Explosions are classified as type Ia if they lack helium (as with Ic) but have a silicon absorption line in their spectrum, at about the time when they are the brightest (the "peak" of the light curve). In spectroscopy, we speak of "absorption lines" when cooler gas, between our telescope and the source of the light, absorbs frequencies from the light, making dark lines across the spectrum. (If the absorbing gas were heated, we would see bright lines in the same positions as the dark ones we find from the cool gas.) In a type Ia supernova, therefore, we see cooler silicon atoms, being illuminated by fusion reactions in material which lacks both hydrogen and helium. The commonly accepted theory behind this phenomenon involves a little more subatomic physics, which is why we treat type Ia supernovas last here.

All subatomic particles fall into two broad categories: bosons and fermions. The former group includes particles like the photons of light, while the latter group includes electrons, neutrinos and the quarks which are the inner constituents of protons and neutrons. In general, matter is made of fermions, while the forces between matter particles involve bosons. If we gather a large pile of particles together, bosons and fermions behave in different ways. An indefinitely large number of bosons can gather together in the same energy state, all occupying more or less the same location. However, no two fermions can occupy the same vicinity. In the language of quantum mechanics we discussed earlier, the wavefunctions of two fermions, say two electrons, do not "like to" overlap. (Technically, we can "squeeze" two electrons into the same small region, if we have them spinning in opposite directions, but this is a minor point for our purposes here. In astrophysics, this detail typically means that we just have a 2 in some equations instead of a 1.)

This effect is not related to the electric repulsion between electrons, which occurs because they have the same charge. If we had a new kind of neutral electrons, their fermionic nature would still tend to keep them apart. The rule which says that fermions do not occupy the same location (with the same spin) is called the Pauli exclusion principle. It requires several years of physics to be able to derive the exclusion principle from any other fact: up until graduate school, physics students typically treat it as given, and try to understand its effects.

In a white dwarf, as we mentioned earlier, atoms are squeezed so hard that they lack the empty space present in normal matter. This "degenerate matter" consists of atomic nuclei, immersed in a "sea" of electrons, all packed very closely together. Gravity would tend to make this material collapse still further, under its own weight, but the odd behavior of quantum particles keeps gravity at bay: the exclusion principle prevents white dwarfs from collapsing! In principle, this happens as follows.

When we have a large "sea" of electrons, the exclusion principle means that their wavefunctions tend to avoid overlapping. Stated more precisely, we do not find two electrons with the same spin and the same velocity.) We have to "squeeze" this electron aggregate to make it any smaller, doing work against a "force" which is really an effect of Pauli exclusion. We say that the fermions' distaste for overlapping creates an "exclusion pressure", which acts to keep the electron sea from shrinking. The white dwarf is in equilibrium, balanced between gravity trying to pull it tighter and exclusion pressure keeping it large.

What happens when gravity is too intense for exclusion pressure to overcome? Calculating the details of this phenomenon, the Indian physicist Subrahmanyan Chandrasekhar (1910-1995) deduced that any white dwarf with more than 1.44 solar masses will collapse under its own gravitational pull. As with the case of a type II supernova's iron core, this collapse can trigger an explosion.

Suppose we have again a binary star system. Imagine that one of the stars has collapsed into a white dwarf, composed mostly of carbon (with some oxygen present as well), while the other star continues to burn its nuclear fuel. If the white dwarf is close enough, it will draw gases from the larger star, adding mass to itself. This extra mass may push it pass the 1.44 solar-mass "Chandrasekhar limit". The collapse increases the pressure inside the dwarf, and (through details not yet fully understood) a "flame" of fusing carbon and oxygen erupts. The energy of this flame causes the dwarf to explode, releasing about 1044 joules and ejecting matter at speeds of around 10,000 km/s.

This explains the features of type Ia spectra. The explosion lacks hydrogen and helium lines, since the white dwarf is composed of carbon and oxygen. The silicon is produced by fusion of the elements in the exploding dwarf.

All supernovae spread matter into the space around them. For type Ia bursts, this matter would contain mostly carbon, oxygen and silicon.

Hubble Heritage composite Composite image of supernova remnant N132D in the Large Magellanic Cloud, the ashes of a 10-15 solar mass star which exploded about 160,000 years ago.

For a type II blast, we find all the elements produced in the "onion" reactions, all the way up to iron. In addition, the sheer flood of energy this blast releases allows energetically unfavorable reactions (the kind which cannot sustain themselves) to occur: the force of the shock wave can slam together medium-mass nuclei to make elements heavier than iron. This is the process by which heavy elements are produced and spread throughout the universe.

Here, we note an interesting fact: the original star for a type II supernova can have a mass anywhere within a broad range. Even without knowing the details of the explosion mechanism, we can expect a fairly broad variability in type II explosions. However, if we look at type Ia blasts, they are all due to white dwarfs having about 1.44 solar masses, the Chandrasekhar mass. We might expect that such dwarfs, which all have roughly the same composition and the same mass, will go "boom" in roughly the same way. Astronomers exploit this fact by using type Ia supernovas as "standard candles": because their parent stars are so similar, type Ia supernovae all have about the same intrinsic brightness. If we identify a supernova as type Ia (using its spectral features), we can gauge how far away it is by measuring how bright it appears.

References and Further Reading

For information on the topics this page treats, we recommend the following books:

Researched, written and maintained by Blake Stacey.