E. Myra & D. Swesty 11/9/2004Supercomputing 2004The Storyline of the Project:How to turn this… …into this!Figure 1: Before and after pictures of SN1987a.Chapter 5Core-collapse Supernovae5.1 Infall phaseWe begin with a massive star, in excess of 10 solar masses, burning the hydrogen in its coreunder the conditions of hydrostatic equilibrium. When the hydrogen is exhausted, the corecontracts until the density and temperature are reached where 3α →12C can take place. TheHe is then burned to exhaustion. The pattern, fuel exhaustion, contraction, and ignition ofthe ashes of the previous burning cycle repeats several time, leading finally to the explosiveburning of28Si to Fe. For a heavy star, the evolution is rapid: the star has to work harderto maintain the hotter electron gas necessary to sustain itself against its own gravity, andtherefore consumes its fuel faster. Likewise, as the star contracts to higher density after eachburning stage, and because the energy liberated in late-stage burning cycles is modest (seebelow), the evolution accelerates as the star progresses to later stages. A 25 solar mass starwould go through all of these cycles in about 7 My, with the final explosion Si burning stagestaking a few days. The resulting ”onion skin” structure of the precollapse star is shown inFigure 2. Note that one can read off the nuclear history of the star by looking from thesurface inward.1E. Myra & D. Swesty 11/9/2004Supercomputing 2004He C,OC Ne,MgH HeO S,SiFeS,Si FeStellar Core Collapse-Massive stars evolve by burning lighter Elements, by thermonuclear fusion, into heavier elements-High mass stars will produce elementsUp to Silicon & Sulfur , which then burn into Iron-Star has an Onionskin-like structureWith layers of successively heavierelements-Each burning stage progressesMore rapidly -When the Iron core mass becomesabout 1.2-1.4 solar masses the corecan no longer sustain itself against the pull of gravity & it collapsesFigure 2: Qualitative depiction of the onionskin structure to which a massive star, aboveabout 8 solar masses, evolves. Core collapse is initiated when the inert iron core, augmentedby continuing silicon burning, reaches the Chandrasekar mass.2The source of energy for this evolution is nuclear binding energy. A plot of the nuclearbinding energy as a function of nuclear mass shows that the minimum is achieved at Fe. Ina scale where the12C mass is picked as zero:12C δ/nucleon = 0.000 MeV16O δ/nucleon = -0.296 MeV28Si δ/nucleon = -0.768 MeV40Ca δ/nucleon = -0.871 MeV56Fe δ/nucleon = -1.082 MeV72Ge δ/nucleon = -1.008 MeV98Mo δ/nucleon = -0.899 Mevwhere δ is the nuclear binding energy relative to C. This defines the ene rgy available fromburning carbon through iron, about 1 MeV. (Recall the energy liberated in burning protonsto He was about 6.5 MeV per nucleon.) Once the Si burns to produce Fe, there is no furthersource of nuclear energy adequate to support the star. So as the last remnants of nuclearburning take place, the core is largely supported by degeneracy pressure, with the energygeneration rate in the core being less than the stellar luminosity. The core density is about2 ×109g/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 burningstage, but continues. As gravity does work on the matter, the collapse leads to a rapidheating and compression of the matter. As the nucleons in Fe are bound by about 8 MeV,sufficient heating can release αs and a few nucleons. At the same time, the elec tron chemicalpotential is increasing. This makes electron capture on nuclei and any free protons favorablee−+ p → νe+ nNote that the chemical equilibrium condition isµe+ µp= µn+ hEνiThus the fact that neutrinos are not trapped plus the rise in the electron Fermi surface asthe density increases, lead to increased neutronization of the matter. The escaping neutrinocarry off energy and lepton number. Both the electron capture and the nuclear excitationand disassociation takes energy out of the electron gas, which is the star’s only source ofsupport. This means that the collapse is very rapid. Numerical simulations find that theiron 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 νes readily e scape. But neutrinos are trapped when adensity of ∼ 1012g/cm3is reached. At this point the neutrinos begin to scatter off the matterthrough both charged current and coherent neutral current processes. The neutral currentneutrino scattering off nuclei is particularly important, as the scattering cross section is offthe total nuclear weak charge, which is approximately N2, where N is the neutron number.3This pro c ess transfers very little energy because the mass energy of the nucleus is so muchgreater than the typical energy of the neutrinos. But momentum is exchanged. Thus theneutrino “random walks” out of the star. When the neutrino mean free path becomes suf-ficiently short, the “trapping time” of the neutrino begins to exceed the time scale for thecollapse to be completed. This occurs at a density of about 1012g/cm3, or somewhat lessthan 1% of nuclear density. After this point, the energy released by further gravitationalcollapse 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 itsbinding energy isGM22R∼ 2.5 × 1053ergsThus this is roughly the trapped energy that will later be radiated in neutrinos.The trapped lepton fraction YLis a crucial parameter in the explosion physics: a highertrapped YLleads to a larger homologous core, a stronger shock wave, and easier passage ofthe shock wave through the outer core, as will be discussed below. Most of the lepton numberloss of an infalling mass element occurs as it passes through a narrow range of densities justbefore trapping. The reasons for this are relatively simple: as we have seen in other plasmas,electron capture (and other weak interactions) goes as T5. Thus the electron capture rapidlyturns on as matter falls toward the trapping radius. So the lepton loss is maximal just priorto trapping. Inelastic neutrino reactions have an important effect on these losses (to bedescribed in detail in class).5.2 The shock waveThe velocity of sound in matter rises with increasing density. The inner homologous