Nuclear Astrophysics

Nuclear Astrophysics

Physics and Astronomy Department at the University of Washington

Core-collapse Supernovae

4.1 Infall phase

We begin with a massive star, in excess of 10 solar masses, burning the hydrogen in its core under the conditions of hydrostatic equilibrium. When the hydrogen is exhausted, the core contracts until the density and temperature are reached where 3C can take place. The He is then burned to exhaustion. The pattern, fuel exhaustion, contraction, and ignition of the ashes of the previous burning cycle repeats several time, leading finally to the explosive burning of Si to Fe. For a heavy star, the evolution is rapid: the star has to work harder to maintain itself against its own gravity, and therefore consumes its fuel faster. A 25 solar mass star would go through all of these cycles in about 7 My, with the final explosion Si burning stages taking a few days. The resulting "onion skin" structure of the precollapse star is shown in the figure. Note that one can read off the nuclear history of the star by looking from the surface inward.

The source of energy for this evolution is nuclear binding energy. A plot of the nuclear binding energy as a function of nuclear mass shows that the minimum is achieved at Fe. In a scale where the C mass is picked as zero: C     /nucleon = 0.000 MeV
O     /nucleon = -0.296 MeV
Si    /nucleon = -0.768 MeV
Ca    /nucleon = -0.871 MeV
Fe    /nucleon = -1.082 MeV
Ge    /nucleon = -1.008 MeV
Mo    /nucleon = -0.899 Mev where is the nuclear binding energy relative to C. Thus once the Si burns to produce Fe, there is no further source of nuclear energy adequate to support the star. So as the last remnants of nuclear burning take place, the core is largely supported by degeneracy pressure, with the energy generation rate in the core being less than the stellar luminosity. The core density is about 2 g/cc and the temperature is kT 0.5 MeV.

Thus the collapse that begins with the end of Si burning is not halted by a new burning stage, but continues. As gravity does work on the matter, the collapse leads to a rapid heating and compression of the matter. As the nucleons in Fe are bound by about 8 MeV, sufficient heating can release s and a few nucleons. At the same time, the electron chemical potential is increasing. This makes electron capture on nuclei and any free protons favorable

Note that the chemical equilibrium condition is

Thus the fact that neutrinos are not trapped plus the rise in the electron Fermi surface as the density increases, lead to increased neutronization of the matter. The escaping neutrino carry off energy and lepton number. Both the electron capture and the nuclear excitation and disassociation takes energy out of the electron gas, which is the star's only source of support. This means that the collapse is very rapid. Numerical simulations find that the iron core of the star ( 1.2-1.5 solar mases) collapses at about 0.6 of the free fall velocity.

In the early stages of the infall the s readily escape. But neutrinos are trapped when a density of 10g/cm is reached. At this point the neutrinos begin to scatter off the matter through both charged current and coherent neutral current processes. The neutral current neutrino scattering off nuclei is particularly important, as the scattering cross section is off the total nuclear weak charge, which is approximately N, where N is the neutron number. This process transfers very little energy because the mass energy of the nucleus is so much greater than the typical energy of the neutrinos. But momentum is exchanged. Thus the neutrino ``random walks" out of the star. When the neutrino mean free path becomes sufficiently short, the ``trapping time" of the neutrino begins to exceed the time scale for the collapse to be completed. This occurs at a density of about 10 g/cm, or somewhat less than 1% of nuclear density. After this point, the energy released by further gravitational collapse and the star's remaining lepton number are trapped within the star.

If we take a neutron star of 1.4 solar masses and a radius of 10 km, a rough estimate of its binding energy is

Thus this is roughly the trapped energy that will later be radiated in neutrinos.

The trapped lepton fraction is a crucial parameter in the explosion physics: a higher trapped leads to a larger homologous core, a stronger shock wave, and easier passage of the shock wave through the outer core, as will be discussed below. The dashed curves in the second figure show that most of the lepton number loss of an infalling mass element occurs as it passes through a narrow range of densities just before trapping. The reasons for this are relatively simple: as we have seen in other plasmas, electron capture (and other weak interactions) goes as . Thus the electron capture rapidly turns on as matter falls toward the trapping radius. So the lepton loss is maximal just prior to trapping. Inelastic neutrino reactions have an important effect on these losses (to be described in detail in class).

4.2 The shock wave

The velocity of sound in matter rises with increasing density. The inner homologous core, with a mass solar masses, is that part of the iron core where the sound velocity exceeds the infall velocity. This allows any pressure variations that may develop in the homologous core during infall to even out before the collapse is completed. As a result, the homologous core collapses as a unit, retaining its density profile. That is, if nothing were to happen to prevent it, the homologous core would collapse to a point.

The collapse of the homologous core continues until nuclear densities are reached. Nuclear matter is rather incompressible ( 200 MeV/f!) densities of 3-4 times nuclear density are reached, e.g., perhaps g/cm. The innermost shell of matter reaches this supernuclear density first, rebounds, sending a pressure wave out through the homologous core. This wave travels faster than the infalling matter, as the homologous core is characterized by a sound speed in excess of the infall speed. Subsequent shells follow. The resulting series of pressure waves collect near the sonic point (the edge of the homologous core). As this point reaches nuclear density and comes to rest, a shock wave breaks out and begins its traversal of the outer core.

Initially the shock wave may carry an order of magnitude more energy than is needed to eject the mantle of the star (less than 10 ergs). But as the shock wave travels through the outer iron core, it heats and melts the iron that crosses the shock front, at a loss of 8 MeV/nucleon. The enhanced electron capture that occurs off the free protons left in the wake of the shock, coupled with the sudden reduction of the neutrino opacity of the matter (recall ), greatly accelerates neutrino emission. This is another energy loss. [Many numerical models predict a strong ``breakout" burst of s in the few milliseconds required for the shock wave to travel from the edge of the homologous core to the neutrinosphere at g/cm and km. See the figure. The neutrinosphere is the term from the neutrino trapping radius, or surface of last scattering.] The summed losses from shock wave heating and neutrino emission are comparable to the initial energy carried by the shock wave. Thus most numerical models fail to produce a successful ``prompt" hydrodynamic explosion.

Two explosion mechanisms were seriously considered in the last decade. In the prompt mechanism described above, the shock wave is sufficiently strong to survive the passage of the outer iron core with enough energy to blow off the mantle of the star. The most favorable results were achieved with smaller stars (less than 15 solar masses) where there is less overlying iron, and with soft equations of state, which produce a more compact neutron star and thus lead to more energy release. In part because of the lepton number loss problems discussed earlier, now it is widely believed that this mechanism fails for all but unrealistically soft nuclear equations of state.

The delayed mechanism begins with a failed hydrodynamic explosion; after about 0.01 seconds the shock wave stalls at a radius of 200-300 km. It exists in sort of equilibrium, gaining energy from matter falling across the shock front, but loosing energy to the heating of that material. However, after perhaps 0.5 sec, the shock wave is revived due to neutrino heating of the nucleon ``soup" left in the wake of the shock. This heating comes primarily from charged current reactions off the nucleons in that nucleon gas; quasielastic scattering also increases the energy transfer. This heated gas may reach 2 MeV in temperature; it has a very high entropy. Thus the energy is in the radiation, not the matter. The pressure exerted by this gas helps to push the shock outward. It is important to note that there are limits to how effective this neutrino energy transfer can be: if matter is too far from the core, the coupling to neutrinos is too weak to deposite significant energy. If too close, the matter may be at a temperature (or soon reach a temperature) where neutrino emission cools the matter as fast or faster than neutrino absorption heats it. It the parlance of the field, one hears the work ``gain radius" to describe the region where useful heating is done.

This subject is still very controversial and unclear. The problem is extremely difficult numerical, challenging modelers to handle the difficult hydrodynamics of a shock wave; the complications of the nuclear equation of state at densities not yet accessible to experiment; modeling in two or three dimensions; handling the slow diffusion of neutrinos; etc. Not all of these aspects can be handled reasonably at the same time, even with existing supercomputers. Thus there is considerable disagreement about whether we have any supernova model that succeeds in ejecting the mantle.

I should mention the term ``mass cut". This describes the bifurcation point of the star. Below the mass cut, matter falls into the neutron star (or black hole). Outside the mass cut, matter is ejected. Just above the mass cut, the matter is a very high entropy nucleon gas that is blown off the star by neutrinos, or the ``neutrino wind." We will have a homework problem on this wind.

4.3 Neutrinos

However the explosion proceeds, there is agreement that 99% of the 3 ergs released in the collapse is radiated in neutrinos of all flavors. The time scale over which the trapped neutrinos leak out of the protoneutron star is about 3 seconds. (Fits to SN1987A give, assuming an exponential cooling , 4.5 sec.) Through most of their migration out of the protoneutron star, the neutrinos are in flavor equilibrium

As a result, there is an approximate equipartition of energy among the neutrino flavors. After weak decoupling, the s and s remain in equilibrium with the matter for a brief time due to scattering off electrons and, especially, the charged current reactions

As a result, the heavy flavor neutrinos decouple from the matter first, at a somewhat higher temperature. Typical calculations yield

The difference between the and temperatures is a result of the neutron richness of the matter, which enhances the reactions of the s, thereby keeping them coupled to the matter to a larger, cooler radius.

This temperature hierarchy is crucially important to nucleosynthesis (we will see this when we look at the r-process) and also to possible neutrino oscillation scenarios. The three-flavor MSW level-crossing diagram is shown in the figure. One very popular scenario attributes the solar neutrino problem to transmutation; this means that a second crossing with a could occur at higher density. It turns out plausible seasaw mass patterns suggest a mass on the order of a few eV, which would be interesting cosmologically. The second crossing would then occur outside the neutrino sphere, that is, after the neutrinos have decoupled and have fixed spectra with the temperatures given above. Thus a oscillation would produce a distinctive MeV spectrum of s. This has dramatic consequences for terrestrial detection and for nucleosynthesis in the supernova.

4.4 Convection

A current hot topic is the possibility that convection is essential for understanding supernova explosions. There is substantial evidence from SN1987A and other supernovae that convection occurs during the explosion: explosion asymmetry, early emission of x-rays and rays, and outward mixing of Ni.

There has been a lot of excitement that 2D supernova simulations generate convection that might be helpful in producing a successful explosion. But not all groups have been able to reproduce early results; in fact, there is some evidence that better treatments of neutrino diffusion tend to suppress the convection. I'll explain in class why convection might enhance the transfer of energy from the neutrinos into the mantle of the star.

4.5 The light curve

Well after the explosion, observers can measure the light emitted by the supernova. At late times mich of the observed power comes from radioactive species synthesized in the explosion. The figure shows isotopes that we believe contribute to the light curve. One exciting, very recent result is the identification of a supernova remnant, about 300 years old, in our galaxy due to observation of the gamma ray line from Ti. This line was previously seen in a known supernova remnant (Cass A). The second source has no optical counterpart in the historical record.