The Supernova Problem

Cosmin Deaconu
Oct 28, 2008

(Submitted as coursework for Physics 204, Stanford University, Fall 2008)

Fig. 1: Supernova classification.

Supernovae, ephemeral and remarkably luminous, are celestial objects believed to be the result of the complete destruction of a star. While modern astronomical tools have resulted in many observations of supernovae, the exact mechanisms that can result in such energetic events are still unknown.

Historical Development

About 8 events thought to be supernovae have been recorded as visible to the naked eye. One of the first recorded by multiple cultures occurred in 1054, when a bright new "star" appeared in the sky. The star was visible in daylight for 23 days before getting dimmer and eventually disappearing entirely. A short-lived extraordinarily bright object in 1572 captured the imagination of Danish astronomer-to-be Tycho Brahe. In vain, he attempted to measure its parallax. His inability to do so suggested that the object was far away, an apparent violation of the then-hold Aristotlean notion that everything outside the planets was static. A mere 32 years later, a similar event occurred, this one studied heavily by Kepler (he also tried, in vain, to measure its parallax). The supernova in 1604 is believed to be the last which occurred in the Milky Way. [1]

The advent of telescopes led to further observations of "new stars", or novae. In 1885, a nova of apparent magnitude 6 was detected by chance in the Andromeda galaxy. At the time, Andromeda was thought to be relatively nearby, so the luminosity did not seem unreasonable. This changed, however, after Edwin Hubble's successful detection of Andromedan Cepheid variables (Cepheids are believed to have a consistent relationship between period and luminosity) and his subsequent distance measurement. Wilhelm Baade and Fritz Zwicke, working at Mt. Wilson, realized that the amount of energy necessary to produce such a large luminosity at a distance of 2.5 million light years was immense, much higher than previously observed novae. Their calculations yielded an energy release comparable to the total internal energy of a star. They decided to call such extremely energetic events "supernovae." The name stuck. [2]

Classification of Supernovae

Soon after Baade and Zwicke's "discovery" of supernovae, Baade and Rudolph Minkowski (not to be confused with his more famous uncle, Hermann Minkowski) began cataloguing them. Minkowski came up with a classification scheme based on the spectra of supernovae. Supernovae that lacked hydrogen absorption lines are Type I Supernovae while those with visible hydrogen absorption lines are Type II. [3] Later on, further differentiation was made within the Type I category. Type Ia supernovae have strong silicon absorption lines at a wavelength of 615 nm. Type Ib Supernovae do not. Type Ic have no silicon and also no helium. Type Ia are found everywhere in the universe and are of seemingly uniform luminosity and light curves (to the extent that type Ia supernovas have been used as standard candles to measure distance). Types Ib, Ic and II are found only in star-forming regions, suggesting that they come from younger stars. Type II supernovas may be further subdivided based on the shape of their light curves. [4]

While supernovae are easy to observe (they often outshine their parent galaxies), coming up with an explanation for such an exorbitant release of energy has proven more difficult. Before we discuss theories for the origin of supernovae, it is first necessary to review some stellar physics.

Fig. 2: Nuclear fusion in stars occurs at lower temperatures than one would expect if they one knew only of classical physics. An incoming proton need not exceed the peak energy of Coulomb repulsion as it can tunnel to the nuclear radius, where the Strong force takes precedence.

Stellar Physics Primer

Stars are believed to form from gravitational collapse. If the matter density within a certain volume of space is high enough, the material will start clumping from gravitational attraction. The gas will continue to collapse until the pressure of the gas is able to counteract the gravitational force, i.e., the hydrostatic equilibrium condition is met:

If the star has a mass greater than 0.08 solar masses, it will get hot enough to start burning hydrogen. Classically, in order for fusion to occur, the Coloumb barrier must be completely overcome (see Fig. 2), which would require immense temperatures. However, quantum tunneling allows fusion to happen at lower temperatures than classical mechanics would require. The predominant process occurring in stars with low metallicity is the proton-proton chain

The first step of the proton-proton chain is the slowest and dictates the lifetime of the hydrogen-burning epoch of a star. The temperature dependence of the reaction rate is T4. Stars with high metallicity (to the astronomer, anything other than hydrogen and helium is a metal) can undergo other, even more temperature sensitive, processes that also convert hydrogen to helium (for example, the carbon-nitrogen cycle has a reaction rate that goes as T18.) This means that more massive stars are much more luminous and short-lived than less massive stars.

After a star's core runs out of hydrogen, the fusion stops and hydrostatic equilibrium is broken. The star starts collapsing on itself, causing the core to get hotter and hotter. If the star has a mass of at least 0.5 solar masses, the core will eventually get hot enough that it will be able to burn helium. Helium fusion follows the triple-α process:

Also the carbon can further burn into oxygen:

Eventually, the helium in the core will also expire. Again, the core will collapse. If the mass of the star is less than 8 solar masses, it will never get hot enough to burn the oxygen and carbon. It cannot, however, collapse indefinitely. Eventually pressure from electron degeneracy will counteract the gravitational force and form a compact white dwarf from the carbon-oxygen core. The outer portion of the star is expelled in a planetary nebula. [4]

Catastrophic Core Collapse Proposal

If the mass of the star is greater than 8 solar masses, fusion will start up again, creating Neon. After the Neon expires, another collapse ensues, until the Neon may be fused into oxygen. The cycle of alternating core collapses followed by the burning heavier and heavier elements continues for a while, until finally silicon fuses to create iron (actually, nickel which eventually decays into iron). As iron and nickel have the highest binding energy of any elements, fusion cannot proceed past this point. At this point, the star has an onion-like elemental structure (see Fig. 3). The iron-nickel core is held up by electron degeneracy pressure from collapse.

However, as the iron-nickel core expands due to fusion of the surrounding silicon, it may approach the Chandrasekhar mass (this is the maximal mass that an object held up by electron degeneracy pressure can have before the gravitational force becomes too strong; this limit may be calculated by using the equation of state of a relativistic electron gas). When this limit is exceeded, the ensuing collapse provides a possible mechanism for supernovae to occur.

The model for a core collapse supernova is as follows. The collapsing core reaches very high temperatures and velocities, generating very high energy photons which purportedly photodisintegrate the heavy atoms into their constituent nuclear matter. It becomes energetically favorable for protons to undergo reverse beta decay (p + e- → n + νe), thus releasing neutrinos during the collapse phase. Also, thermal neutrinos are supposed to be generated.

With electron degeneracy out of the way, one might expect the matter to continue to collapse indefinitely. However, once the matter becomes dense enough, it becomes subject to neutron degeneracy pressure. Hydrostatic equilibrium is abruptly restored in the nascent neutron core. As infalling material moving at extremely high velocities hits this super-dense region, it is forced to stop very quickly, creating an outwardly propagating shock wave. With some luck, the shock wave might have enough energy to counteract the gravitational collapse and tear apart the hapless star. [4]

Fig. 3: The structure of a massive star in its last throes of life. The timescales are calculated for a star of 25 solar masses. [4]

Lacking the means to generate a bonafide supernova, scientists have turned to computer simulation to test the model. After 30 years of ever-more-sophisticated simulations of the core collapse and shock wave, it is believed that the shockwave is not energetic enough and stalls in the (former) outer core of the star. Another energy source is needed to complete the catastrophe. Some have posited that this energy may come from the neutrinos. If one percent of the neutrinos were to couple with the infalling material, then the shock wave can be resuscitated and the star properly exploded. How to get so many neutrinos to interact, however, is unclear. Simulations with attempts at realistic neutrino transfer have not been able to produce a proper (simulated) supernova (for example, see Liebendorfer et al.) [5]

In 1987, type Ib supernova occurred in the (relatively) nearby Large Magellanic Cloud (168,000 ly). About 20 neutrinos were claimed to have been observed originating from the supernova at three different locations (Kamiokande in Japan, Bakson in the USSR and Mt. Blanc) The fact that neutrinos were detected prior to optical sighting of the supernova is often taking as evidence that the core collapse model is more or less correct (neutrinos start to get emitted during the collapse; no light is produced until after the elusive shock wave breaches the exterior of the star). [6]

What about Type Ia?

Even if the core collapse supernova could be made to work, it could not fully explain all supernovae by itself. Core collapse requires an extremely massive star, which, due to reaction rates, necessarily die young. However, type Ia supernovas in particular have been observed in parts of the universe far from star forming regions. A massive star could not reach those areas before expiring.

In the 70's, Whelan and Iben came up with a possible explanation for type Ia supernovas. Less massive stars that can survive for a long time can eventually form white dwarves. Now, not all stars are solitary like our sun; over half of stars belong to multi-star systems. If one star is more massive than another, it is reasonable that at some point, one of the stars becomes a white dwarf while the other continues to fuse. In that case, it is possible for the surviving star to begin accreting part of its mass onto the white dwarf. As mass is gained by the white dwarf, its internal temperature and pressure start growing. Upon approaching the Chandresekhar limit, the white dwarf core's constituent matter becomes relativistic and the core's equation of state changes such that the core can contract even more. [7]

It may get hot enough in the core so that the carbon-oxygen mixture can ignite, causing a very energetic nuclear explosion. Rapid fusion of the dwarf follows, culminating in silicon fusing to create 56Ni, which then decays into 56Co, and finally 56Fe. The successful matching of white dwarf light curves to the half-lifes of the nickel and cobalt isotopes involved lend credence to this model. Moreover, the model explains the strong silicon absorption lines (from unfused silicon) and the seemingly homogenous luminosity.

Unfortunately, computer models of white dwarfs are of mixed success. While naive one-dimensional models have successfully yielded explosions, more sophisticated multidimensional models such as the one employed by Gamezo et al. are only able to produce an explosion an order of magnitude smaller than those observed. The conflagration does not seem to spread quickly enough to ignite enough of the star to spark something comparable to what has been observed. [8]


The supernova problem refers to the inability of computer simulations to produce a supernova. The implication is that either there is something wrong with the simulations or that there is something wrong with our understanding of supernovae.

© 2008 Cosmin Deaconu. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.


[1] D. H. Clark and F. R. Stephenson, The Historical Supernovae (Pergamon, 1977).

[2] W. Baade and F. Zwicky, "On Supernovae," Proc. Natl. Acad. Sci. 20, 254 (1934).

[3] R. Minkowski, "Spectra of Supernovae," Publ. Astron. Soc. Pacific 53, 224 (1941).

[4] A. C. Phillips, The Physics of Stars, 2nd Edition (Wiley, 1999).

[5] M. Liebendoerfer et al., "Probing the Gravitational Well: No Supernova Explosion in Spherical Symmetry with General Relativistic Boltzmann Neutrino Transport", Phys. Rev. D 63, 103004 (2001).

[6] M. Aglietta et al., "Correlation Analysis of the Data Recorded by the Baksan, Kamioka, and Mont Blanc Detectors During SN 1987A," in Particle Astrophysics: Forefront Experimental Issues (World, 1989), p. 324.

[7] J. A. Whelan and I. Iben, Jr., "Binaries and Supernovae of Type I," Astrophys. J. 186, 1007 (1973).

[8] V. N. Gamezo et al., "Thermonuclear Supernovae: Simulations of the Deflagration Stage and Their Implications," Science 299, 77 (2003).