UNIVERSITY OF LJUBLJANAFACULTY OF MATHEMATICS AND PHYSICS

Supernovae

Jure Strleadvisor: doc. dr. Simon Sirca

March 2006

Figure 1: The remnant of the Kepler’s supernova of 1604. The picture is a false-colourcomposite of x-ray, visible and infrared light images.[9]

Abstract

Supernovae are stellar explosions of such immense power that it stretches even theboundaries of human imagination. There are two types of supernovae that have verydifferent explosion mechanics. The purpose of this seminar is to briefly describe the historyof study of supernovae and to explain the mechanics of both types, concentrating on theType II supernovae, which have been studied more. Presented are also the achievementsand the problems of modeling supernovae explosions with computers.

Contents

1 Introduction 3

2 History 3

3 Types of Supernovae 53.1 Type I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.1.1 Death of a Dwarf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.1.2 Light and Candles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Type II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2.1 Pre-supernova Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.2 Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2.3 Prompt Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2.4 Shock Revival . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2.5 Death of a Giant and Light Spectra . . . . . . . . . . . . . . . . . . . 12

4 Modeling of the Core Collapse in Type II supernova 134.1 “Grey” or “Multineutrino” . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Neutrino Transport Primer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3 Situation now and Future Challenges . . . . . . . . . . . . . . . . . . . . . . . 16

5 Conclusion 17

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1 Introduction

Supernovae are spectacular events. One of the most researched supernovae - SN1987A - emit-ted light at a rate 100 million times that of the Sun and it was one of the fainter supernovae.They are stars in the later stages of stellar evolution that suddenly contract and then ex-plode, increasing their energy output by several orders of magnitude, which is from afar seenas a great increase in brightness of the progenitor star. After weeks or months these brightobjects slowly decline to invisibility. Their importance is significant; most heavy elementsare created by nuclear reactions in supernovae and then returned to space and all elementsheavier than iron are produced in this process. They are also the principal heat source forinterstellar matter and may be a source of cosmic rays. The shock waves from supernovaeexplosions play a big role in triggering star formation. Based on the observed abundances ofheavy elements in our solar system, we can conclude that it was formed when remnants ofa supernova (or perhapse of multiple supernovae) settled in a nebula. Supernovae are quiterare. In the Milky Way only five have been observed in the last 1000 years and none in thelast 300.[9, 8]

2 History

Throughout history, humans have been observing and recording “guest stars”, which suddenlyappeared in the sky and then faded away. The brightest of these were probably supernovae,in which case a remnant should be somewhere. The Chinese “guest star” from 185 AD wasa supernova and its remnant gives a strong x-ray image. In 1006 they observed another one,also seen in the Middle East and Europe, whose remnant can now be observed as a radioimage. Another one recorded by Chinese is from year 1054 with the Crab Nebula remnant,which differs from those of 185 and 1006, which show only radiant shells, representing theshock wave these supernovae sent out into space. The remnant in Crab has a whole volumeluminous, which is connected to the fact that at the center there is a neutron star which emitselectromagnetic radiation and electrons which irradiate material throughout remnant, whichin turn emits visible light. The remnants of 185 and 1006 have no pulsars in the center.

Figure 2: (left to right): imperial Chinese astronomer[16], Tycho Brahe[17] and JohannesKepler[18].

In 1572 Danish astronomer Tycho Brahe discovered a “new star” in Cassiopeia and ob-served it for several months. He found that its position did not change relatively to thefixed stars. This was an important evidence against the Aristotelian dogma that nothing everchanges beyond the moon. German astronomer Johannes Kepler saw another supernova in1604 just before the invention of the telescope. It was visible for a whole year and it was

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observed extensively, so its approximate light curve is known. Another supernova explodedbetween 1650 and 1680, known as Cas A (in Cassiopeia constellation). Its remnant is a verystrong radio source, but it was not reported by contemporary observers. Some supernovae inother galaxies were observed between 1885 and 1930.

In 1934, Zwicky and Baade wrote a paper on supernovae and concluded that it shouldbe possible to find many more of them by a systematic survey of galaxies, since supernovaewould easily stand out above the background of ordinary stars. Within five years they foundnearly 20 supernovae by comparison of pictures of galaxies at different times; if a bright spotwas found on the later picture where the earlier one had none, it was likely to be a supernova.Minkowsky measured the spectra of the discovered supernovae and together they found thatthere are at least two types of supernovae: Type II have strong lines of hydrogen, while TypeI have none. There are also some subclasses within each type.[4]

Figure 3: (left): SN1987A after explosion; (right): progenitor of SN1987A, Sanduleak−69◦202.[12]

A very important year in the history of study of supernovae is 1987. On 23 Februaryastronomers discovered a new bright spot in the Large Magellanic Cloud, only 160,000 lightyears away. The light came from the SN1987A supernova, which was the closest one in 300years. It was of Type II and gave to astrophysicists a lot of valuable data on the deathof massive stars. The first sign of it was a burst of neutrinos detected by three neutrinoobservatories (Kamiokande II, IMB, Baksan) a few hours before arrival of visible light. Apartfrom the Sun, the SN1987A is the only extraterrestrial source of neutrinos known so far.[2]

Figure 4: Number of discovered supernovae per year.[10]

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The number of discovered supernovae (by amateur and professional astronomers alike)has been increasing exponentially. Until 1970 about 380 supernovae have been discovered intotal, by year 1990 this number has grown to about 800, by year 2000 to about 1880 and inthe last five years to about 3500 (Figure 4)![10]

3 Types of Supernovae

The light curve - the time dependence of the optical luminosity, has been measured for manysupernovae. Typical light curves for Type I and Type II both start with a sharp rise inluminosity, extending over a week or two, which is due to the expansion of the luminoussurface. Type I has a fairly narrow peak, while Type II peak is broader, of the order of 100days, due to the recombination of electrons in the excited H atoms of the ejecta. The intensitythen declines by roughly two orders of magnitude over a period of about a year in both types,due to radioactive decay of 56Co[11]. Not all supernovae of each type have the characteristiclight curve, but in Type I about 80% do; they are designated Type Ia.

Figure 5: Total luminosities of Type I (white dwarfs) and Type II (massives stars) supernovaeas as function of time. [13]

Zwicky and Baade suggested in 1934 that supernovae derive their tremendous energy fromgravitational collapse, in particular that the core of the star collapses to a neutron star, theconcept of which has been proposed by Landau in 1932. It is now generally believed thatthis mechanism is predominantly responsivle for Type II supernovae. On the other hand,Type I supernovae are believed to derive their energy from thermonuclear reactions, whichwas suggested by Hoyle and Fowler in 1960 and 1964.[4]

Type II supernovae are observed only in the arms of spiral galaxies and not in the ellipticalgalaxies. This suggests that the progenitors of Type II supernovae are relatively massivePopulation I stars with lifetimes of less than about 107 years. In contrast, Type I supernovaeoccur in all galaxies, with the rate in spiral galaxies compared to that in elliptical galaxies.Their progenitors are assumed to be old, relatively low-mass Population II stars. PopulationI stars are rich in heavy elements, while Population II stars are metal poor, nearly purehydrogen stars.[1]

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3.1 Type I

The mechanism of Type I supernovae is less well understood than that of Type II. Whileseveral models were proposed in the past, there is now agreement that Type I (or at leastthe subtype Ia) is due to the thermonuclear disruption of white dwarfs. White dwarfs format the end of the evolution of stars whose original masses are less than 8 M� (M� = solarmass). A star can lose a large fraction of its material by ejecting outer layers into space atthe final stages of evolution. The mass of the remaining white dwarf is always less than theChandrasekhar limit, 1.4 M�. A white dwarf, consisting mainly of C and O and detacheddegenerate electrons (the term degenerate means that electrons occupy all possible quantumstates below a certain energy), is stable and almost inert because its temperature is nothigh enough to induce any substantial nuclear reactions. When isolated it can exist almostindefinitely, slowly cooling down as it radiates its energy into space. However, most of thewhite dwarfs are not isolated, but belong to groups of two or more stars. In a close binarysistem, a white dwarf can increase its own mass by accreting material from a companion starand thereby reaching the Chandrasekhar mass. Such systems are considered to be the mostprobable supernova Type Ia progenitors, even though the exact nature of the companion starand the details of the mass accretion are still unclear.

Figure 6: A white dwarf accreting matter from a companion.[14]

3.1.1 Death of a Dwarf

When the mass of a white dwarf approaches the Chandrasekhar limit, any small mass increaseresults in a substantial contraction of the star, and the material near its center is compressed.This increases the temperature and accelerates thermonuclear reactions near the center. Re-leased energy further increases the temperature and aids the thermonuclear reactions. Theprocess is slowed down by neutrino emmision as well as convective and conductive cooling.Nevertheless, the temperature in the white dwarf core rises and reaches the point where theenergy release overwhelms the energy outflow. In a non-degenerate star, the energy releasewould be stabilized by a thermal expansion accompanied by the work against gravity. In awhite dwarf, however, the initial temperature increase does not affect the degenerate-electronpressure and therefore does not lead to any substantial expansion that could slow down ther-

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monuclear reactions and prevent the runaway process. Eventually, the temperature increasesto the level where the thermal and degenerate-electron pressure components become compa-rable and the material begins to expand, but at that time, the expansion is unable to quenchthe fast thermonuclear burning of carbon that is ignited in the center of a white dwarf.

Ignition starts a supernova Ia explosion that lasts only a few seconds but releases about1051 erg 1, about as much as the Sun would radiate during 8 billion years. The energyis produced by a network of thermonuclear reactions that begins from 12C and 16O nucleiand ends in 56Ni and in smaller quantities in other iron-group elements. Large amounts ofintermediate-mass elements (Ne, Mg, Si, S, Ca,. . . ) are created as well. It is believed thatthe main energy-producing reactions occur in a thin layer, called a thermonuclear flame, thatpropagates outwards. At the beginning the flame is laminar and its propagation velocity iscontrolled by thermal conductivity, but as the flame moves away from the center, it becomes 2

turbulent and accelerating. Eventually, the burning can undergo a transition from a relativelyslow, subsonic regime, called deflagration, into a supersonic regime, called detonation, wherethe reaction front is preceded by a shock wave. Most of the energy released during theexplosion transforms into kinetic and thermal energies of the expanding material. When thesum of those energies exceeds the potential energy, the star becomes unbound, and expandingmaterial will continue to expand indefinitely. This deflagration-to-detonation model was mostsuccesful in reproducing observed characteristics of supernovae Ia.[5]

3.1.2 Light and Candles

Thermonuclear reactions that occur during the explosion provide energy for the expansion,but not for the luminosity of the expanding gas observed as a supernova Ia. The energy sourcefor this is the slow radioactive decay sequence from the initially formed 56Ni, which decays to56Co, which in turn decays to 56Fe with a half-life of 77 days. The total optical energy observedcan be calculated from the assumption that most of the light is generated by the decay of56Co and 56Ni and that essentially all the mass of the star burns to these end products, themass being that of a white dwarf at the Chandrasekhar limit, 1.4 M�. The luminosity reachesits maximum 15 to 20 days after the explosion and then decreases slowly until all the 56Codecays. The maximum brightness is comparable to the brightness of an entire galaxy andcan vary by an order of magnitude from one supernova to another. Observations also showthat the maximum luminosity of Type Ia supernovae in visible wavelengths correlates withthe rate at which the luminosity decreases after the maximum. In combination with otherapproximate correlations this makes it possible to use supernovae Ia as “standard candles”for determining absolute magnitudes of galaxies and hence their distance. The absence ofhydrogen lines is to be expected from white dwarfs, since the hydrogen that is accreted fromthe companion is quickly converted into helium before the supernova explosion. In short, thewhite dwarf nuclear burn fits the observations.[5]

3.2 Type II

Type II supernovae occur in massive stars with mass greater than 8 M�. The basic mechanismis believed to be fairly well understood, however, there is still a lot of computation that needsto be done to confirm this belief. Zwicky and Baade proposed that supernovae derive their

1erg is unit for energy and mechanical work, widely used in astronomy, 1 erg= 10−7 J2due to gravity-induced Rayleigh-Taylor instabilities

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energy from the gravitational collapse of the core of a star, of approximately the mass ofthe sun, to a neutron star and thereby liberates enormous amounts of gravitational energy -on the order of 1053 erg - much more than the energy of the emitted light, which is of order1049 erg.

The life of a future Type II supernova could be schematicaly divided into four parts, threeof which happen almost instantly, while the preparation for the explosion takes millions ofyears:

Pre-supernova Star : massive stars are relatively short-lived, but in that time acquire thecharacteristic feature of onion-like structure of chemical elements with iron core, outerenvelope of hydrogen and other elements in between

Collapse : due to great mass of the iron core and slowing of nuclear fusion, the innercore collapses releasing a lot of gravitational energy in the process of becoming almostincompressible proto-neutron star

Prompt Shock : the infalling mass rebounds from the proto-neutron star and surges throughthe rest of the infalling matter as a shock wave, only to lose the energy in the still denseiron outer core

Shock Revival : the released gravitational energy heats up the core and the only way tocool itself is by emiting a vast number of neutrinos, which diffuse out of the opaque coreand deposit approximately 1% of their energy in the stalled shock, thus reviving it andliberating the matter from the huge gravitational pull of the proto-neutron star in onehuge explosion - supernova

These four parts are more thoroughly described below. As we will see, it is not quite assimple as that, especially in the shock revival part as will be shown in the following sectionof modeling the collapse.

3.2.1 Pre-supernova Star

Supernova’s characteristics are shaped by the progenitor star. Stars more massive than 8 M�evolve to an onion-like configuration. A good example for how this happens is the life of onesuch star - Sanduleak −69◦202, the progenitor of the forementioned SN1987A. It all beganabout 11 million years ago in a gas-rich part of the Large Magellanic Cloud, where a star wasborn with about 18M�. For the next 10 million years, this star generated energy by fusinghydrogen into helium. Because of its mass, the star “had to” maintain high temperaturesand pressures in its core to avoid collapse; as a result it was about 40000 times as bright asthe Sun and a voracious eater of nuclear fuel. When the innermost 30% of the star ran out ofhydrogen, the central regions began to gradually contract. As the core was compressed overtens of thousands of years, from a density of 6 g/cm3 to 1000 g/cm3, it heated up from about40 million K to 190 million K. The higher core temperature and pressure ignited a new andheavier nuclear fuel, helium. At the same time the outer layers of the star (mostly unburnedhydrogen) responded to the additional radiation from the core by expanding to a radius ofabout 300 million kilometers - the star had become a red supergiant.

The core’s suply of helium was exhausted in less than a million years, having burned tocarbon and oxygen. During the next few thousand years, the star went through the samescenario - core contraction, heating and ignition of a new heavier fuel (the ash of a previous

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cycle) a couple of times more. After helium came carbon, then neon and oxygen, followed bysilicon and sulphur. The fusion chains stop at the final nuclear product, 56Fe3, which has thehighest binding energy per nucleon. The iron core at this stage is surrounded by successivelayer of silicon, oxygen, carbon, helium and finally hydrogen. In addition to iron group nuclei,the core is composed of electrons and positrons, photons and a small fraction of protons andneutrons.

The core of the star thus passed through consecutive stages of burning at an acceleratingpace. Helium lasted a million years, burning of carbon took only 12,000 years, of neon 12years, of oxygen 4 years and of sillicon just a week. Each stock of nuclear fuel after hydrogenreleased about the same total energy, but at core temperatures above 500 million degrees K,the star found a far more efficient way to spend its energy. Very energetic gamma-ray photonswere transformed into particle pairs - an electron and a positron, as they passed near atomicnuclei. These pairs promptly annihilated each other, recreating gamma rays, but sometimesgiving rise to neutrinos due to reactions with nucleons.

Figure 7: The structure of a highly evolved star of 20 solar masses.[15]

Since neutrinos hardly interact with matter (a light year thick layer of lead would stopabout half of them), they escaped from the star far more easily than the gamma rays couldhave. Even during carbon burning, neutrino energy loss exceeded energy loss by radiation.As the core temperature rose during later stages, neutrino luminosity rose exponentially tobecome a ruinous energy drain, hastening the star’s demise.[2]

3.2.2 Collapse

The pressure in the core, which supports it against the inward pull of gravity, is dominatedby the electrons. Just before the collapse this balance is only marginally stable. As a result ofelectron capture on free protons and nuclei in the core and as a result of nuclear dissociationunder extreme densities and temperatures where high energy γ photons are abundant,

el. capture: e− + AZX → A

Z−1 Y + νe (1)

dissociation: γ + 56Fe → 13α + 4n (2)3also some other elements from iron group are produced

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Fuel Main Product Temperature [106 K] Density [g/cm3] Duration [years]H He 40 6 10,000,000He C, O 190 1,100 1,000,000C Ne, Mg 740 240,000 12,000Ne O, Mg 1,600 7400000 12O Si, S 2,100 16,000,000 4Si Fe (iron group) 3,400 50,000,000 0.02

Table 1: The burning process of Sanduleak −69◦202. These figures vary from star to star,because of the differences in mass and in composition of gas from which they were formed.All the numbers are approximate.[2]

electron and thermal pressure support are reduced, the core becomes unstable and collapses,which happens when the iron core exceeds the Chandrasekhar mass 1.4 M�. Gravity ulti-mately wins the contest of tens of millions of years.

The velocity of infalling matter in the core increases linearly with radius, which is acharacteristic of a homologous collapse expected of a fluid whose pressure is dominated byrelativistic, degenerate electrons. On the other hand, the sound speed decreases with den-sity (or radius) and thus with increasing radius the infall velocity eventually exceeds thelocal sound speed; the infall becomes supersonic. The core splits into a homologously andsubsonically infalling inner core and supersonically infalling outer core.[6]

In the process, the nuclei capture some electrons and become more neutron rich, butthere is a limit to this: the neutrinos formed in the process are scattered by the nuclei andat a density of about 1012 g/cm3 this scattering is sufficient to trap the neutrinos. Thenthe inverse reaction sets in, with neutrinos being captured by nuclei, giving electrons back.An equilibrium is reached at a certain electron fraction, Ye, which determines the resultingChandrasekhar mass after collapse. The initial densities of pre-supernova iron core of about109−10 g/cm3 thus proceed during collapse to about 1−3·1014 g/cm3. The inner core undergoesa phase transition from a two-phase systems of nucleons and nuclei to a one-phase system ofbulk nuclear matter. At this point one may view the inner core as one enormous nucleus.The pressure in the inner core increases as the result of Fermi effects and the repulsivenature of the nucleon-nucleon interaction potential at short distances; the inner core becomesincompressible and rebounds.[3]

3.2.3 Prompt Shock

Any information about the rebounding inner core would be conveyed to the outer core viapressure waves that propagate radially outward at the speed of sound. When these wavesreach the point at which infall is supersonic they are swept in as fast as they attempt topropagate outward. This means that no information about the rebounding inner core reachesthe infalling outer core, which in turn sets up a density, pressure and velocity discontinuityin the flow - a shock wave.[6]

This first shock is the so called “prompt shock”. Does the prompt shock cause the ultimatesupernova explosion? All realistic models completed to date suggest that this does not occur.Because the shock loses energy in dissociating the iron nuclei that pass through it as it prop-agates outward, the shock is enervated. Additional energy losses occur in the form of electron

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neutrinos, which occur when the core electrons capture on the newly dissociation-liberatedprotons. Eventually the shock moves out beyond the neutrinosphere4 and the previouslytrapped neutrinos escape.

Figure 8: Collapse of a massive star: 1: inner sub- and outer super-sonically infalling ironcore, 3-5: rebound of the very compressed core, 5-7: the shock wave (orange circle) is launchedand energized.[6]

This gives rise to the electron neutrino burst, which is the first of three major phasesof three-flavor neutrino emission during these events. As a result of these two enervatingmechanisms, the shock stalls in the iron core. How the shock is reenergized is currently thecentral question in core collapse supernova theory.[6]

3.2.4 Shock Revival

So, the prompt shock is insufficient to make the star explode. However, it does move outto some distance - 300 to 500 km from the center and in this way, the prompt shock isessential in preparation of the next stage. When the shock stalls, the core is composed of acentral radiating object, the proto-neutron star, which will go on to form a neutron star ora black hole, depending on the initial mass of the star. The ultimate source of energy in acore collapse supernova is the ∼ 1053 erg of gravitational binding energy associated with theformation of the neutron star - it is equivalent to the gravitational energy of the iron corewith mass 1.4 M�. We can estimate it with

E ∼ GM2NS/RNS = 2.6 · 1053 erg (MNS/M�)2 (10 km/RNS), (3)

where G is the gravitational constant and MNS and RNS are the mass and the radius of theneutron star, respectively.

This energy heats the core of proto-neutron star to the order of 1011 K, but not homoge-neously. The proto-neutron star has a relatively cold inner part composed of unshocked bulknuclear matter, together with a hot mantle of nuclear matter that has been shocked, but notexpelled. The only way this mantle can cool down is by emitting neutrinos, since electro-magnetic radiation is trapped in it, and it does so in the form of about 10 s long three-flavorneutrino pulse. This marks the second phase of the neutrino emission. Electron neutrinosare produced during stellar core collapse by electron capture on protons and nuclei, but afterthe bounce there are all three flavors of neutrinos and antineutrinos produced in the hotproto-neutron star mantle and are emitted as the mantle cools and contracts.

4neutrinosphere - the radius beyond which neutrinos can escape freely, place of last scattering

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Figure 9: During the shock reheating phase, the stellar core is composed of a central radiatingproto-neutron star whose surface is defined by the neutrinospheres (represented here by asingle sphere) and a region above the neutrinosphere consisting of a net cooling region anda net heating region below the stalled shock, separated by the gain radius at which heatingand cooling balance. They are mediated by electron neutrino and antineutrino absorptionand emission.[6]

The neutrinos are emitted from their respective neutrinospheres (we should talk aboutneutrinospheres in plural in the first place, since different energy and different flavour neutri-nos have neutrinospheres with different radii) and their total luminosities during this phase aremaintained at their average values around 1052 erg/s by mass accretion on the proto-neutronstar (kinetic energy of the infalling material is transformed into thermal energy).

The stalled supernova shock is thought to be revived by the current absorption of neu-trinos. We must not forget that the densities around the core might not be high enough tostop neutrinos completely, but a fraction, about 1% of the total luminosity or about 1051 ergof energy of neutrino pulse is still absorbed by protons and neutrons behind the shock andit starts the explosion of the star. This process is known as the delayed shock or neutrino-heating mechanism, but deciphering the precise role of it is difficult and the center of currentresearch.[6]

3.2.5 Death of a Giant and Light Spectra

After the explosion is initiated, the accretion luminosity decreases dramatically, and the neu-trino pulse enters its third and final stage: the exponential decay of the neutrino luminosities,the characteristic of a neutron star formation and cooling.

Once the shock has reached 3000 km, the influence of gravity becomes minor and theshock progresses through the rest of the star. When it breaks out of the star, light appears

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first in the far ultraviolet, then (about a day later) shifting to visible. Initial temperatures,as deduced from spectra are well above 100,000 K, but soon decrease to 5500 K. After a fewmonths, most of the radiation is in the infrared.[3]

When the shock has progressed some distance (on the order of few 1000 km), nuclearreactions take place, in which the pre-existing elements, O, Si and S, are converted into Feand intermediate elements. Because these reactions are extremely fast, taking place in abouta second, 56Fe is not formed directly. Similarly to supernovae of Type I, initially formed is56Ni, the most tightly bound nucleus that consists solely of alpha particles (it is a doubly-magic nucleus where both the proton and neutron number is 28). This material subsequentlydecays by positron emission into 56Co and then, with a half-life of 77 days, into 56Fe. Thislatter decay supplied most of the light energy of SN1987A. However, the majority of Type IIsupernovae is at the maximum light emission and perhaps two months beyond powered bythe energy left by the shock in the hydrogen envelope, and only later by the radioactive decayof 56Co.

Neutrinos, once they are emitted from the proto-neutron star, travel unimpeded throughthe rest of the star, while electromagnetic radiation is closely coupled to the matter and canonly emerge when the shock breaks out of the surface. The speed of the shock wave is a few10,000 km/s, but it still takes hours for it to reach the surface of a red (or blue) supergiant,thus the electromagnetic light starts with a few hour delay in the thousands or millions ofyears (depending on the distance to the supernova) long race with neutrinos, which it takesthem to reach Earth.

4 Modeling of the Core Collapse in Type II supernova

In the previous chapter we discussed the theory of Type II supernovae, but while most of itis accepted scientifically, there is still one thing not yet achieved - most of the simulations(with few rare exceptions (Figure 10)) have failed to reproduce the revival of the shockfollowing core collapse. After more than four decades of computational effort, the detailedmechanism remains elusive, although significant progress has been made in understandingthese multiscale events. 1D, 2D and 3D simulations of core collapse supernovae have shownthat there are many important ingredients which apply to the explosion mechanism. Theseingredients are: a) neutrino transport, b) fluid instabilities, c) rotation, d) magnetic fields, e)sub- and super-nuclear density equation of state, f) neutrino interactions and g) gravity.

Current 2D and 3D simulations have yet to include a)-d) with sufficient realism. 1D spher-ically symmetric models have achieved a significant level of sophistication, but (by definition)can not incorporate b)-d). Fully general relativistic spherically symmetric simmulations withBoltzmann neutrino transport do not yield explosions, demonstrating that some combinationof b), c) and d) is also required.[6]

4.1 “Grey” or “Multineutrino”

Neutrino heating and neutrino cooling have different radial profiles (Figure 9). The regionbetween the gain radius and the shock is the so called “gain region”. The neutrino heatingin the gain region can be written as

ε =Xn

λa0

Lνe

4πr2〈E2

νe〉⟨

1F

⟩+

Xp

λa0

Lνe

4πr2〈E2

νe〉⟨

1F

⟩, (4)

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Figure 10: One of the rare exceptions: Wilson’s (1985) “successful” spherically symmetricmodel that proposed the doubly diffusive neutron finger instabilities in the proto-neutron starto boost the neutrino luminosities. Each line on the graph traces a radial position of a shellof constant mass. The upper dashed curve is the shock, the lower one the neutrinosphere.Mass 1.665 M� is the first mass point propelled out by the second shock.[4]

where λa0 is mean free path, X fraction of nucleons, L luminosity, F flux and E energy; the

first term corresponds to the absorption of electron neutrinos and the second to antineutrinos.It depends linearly on the neutrino luminosities and inverse flux factors (which are a measureof the isotropy of the neutrino distribution) and quadratically on the neutrino spectrum.

All three quantities in the neutrino heating rate must be computed accurately, whichrequires that we solve the Boltzmann neutrino transport equations. In the “grey” approxi-mation the neutrino angles and energies are integrated out and neutrino specific energy andflux are functions of only spatial coordinates (in this case, radius). On the other hand, withthe “multineutrino” approach one can also accurately compute the neutrino spectrum.

The dependance of (4) on the neutrino spectrum means, that it is imperative to accuratelycompute also the spectrum, which requires the use of multineutrino (a.k.a. multifrequencyor multigroup) energy, dependent also on the neutrino angle and energy. The fundamentalshortcoming in a gray approach can be seen, if we specialize the neutrino Boltzmann equation(10), so it only includes absorption and then integrate over neutrino direction cosines andenergies:

∂εR∂t

=∫

dµdEE3 ∂F

∂t= −

∫dµdEE3χF ≡ − ¯χεR, (5)

where¯χ ≡

∫dµdEE3χF∫dµdEE3F

(6)

andεR ≡

∫dµdEE3F (7)

Equation (5) gives the local rate of change of the neutrino specific energy, εR, due toelectron neutrino absorption. This can be expressed in terms of the energy mean absorptionopacity, ¯χ, which depends on the electron neutrino spectra. In a gray approach these spectraare not computed, but they must be imposed, which means the neutrino specific distributionfunction, F , is also imposed. And in equation (6) it is evident that the mean absorption

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opacity is dependent on neutrino distribution function. This means the explosions can beinduced artificially, if an overly hard spectrum is imposed.

However, gray approach is not to be discarded. Owing to the dimensional reduction,simulations in two and even three spatial dimensions that implement grey neutrino transportare possible and allow us to explore the impact of physics beyond the neutrino transport,such as convection of the stellar matter and rotation of the core, which at this point presentan unsurpassable obstacle for the multineutrino approach. Thus, gray and multifrequencytreatments are complementary.[6]

4.2 Neutrino Transport Primer

Neutrinos propagate through the proto-neutron star and interact with the nucleons and elec-trons. Because the cross sections for neutrino interactions are energy dependent (reducedenergies generally mean reduced cross sections), neutrinos of lower energies have longer meanfree paths. In the vicinity of the neutrinosphere the neutrino mean free paths become com-parable to the size of the proto-neutron star. Deep within the core the high energy neutrinosinteract many times before escaping. This process is well described by diffusion theory. Onthe other hand, neutrinos, with mean paths much larger than the size of proto-neutron star,stream out of the core unimpeded and their transport is well described by free streaming.Neutrino mean free path for elastic scattering, λν , can be approximated by

λν = 1012 ρ−10 ε−2

ν

(N2/6A)Xh + Xncm, (8)

where ρ0 is the density in g/cm3, εν is the neutrino energy in MeV, Xh and Xn are themass fractions of heavy nucleons and neutrons, while N and A are numbers of neutrons andnucleons in an average nucleus. During infall nearly all matter is in heavy nucleons and fordensity 1012 g/cm3, we have N ' 50, N/A ' 0.6 and if we also take 20 MeV for neutrinoenergy, we get for mean free path:

λν ' 2 km (10 MeV

εν)2 = 0.5 km. (9)

At the neutrinospheres, the neutrinos are not transported by diffusion nor are they radiallyfree streaming. Their transport is significantly more complex and is well described only bysolutions of the full Boltzmann neutrino kinetic equations for neutrino distribution functions.Additional complexity is due to energy and flavor dependence of the cross sections.

A solution to the Boltzmann equation describes the time evolution of the neutrino distri-bution function for every time and space coordinate and gives the distributions of neutrinosin terms of direction cosines and energies. Thus it is a phase-space equation in the mul-tidimensional space of all spatial coordinates, angles and energies. Therefore, even a 1Dsupernova simulation, in which spherical symmetry is assumed, is in a sense a 3D simulation.If we include emission, absorption, isoenergetic scattering of neutrinos by nucleons and nu-clei, neutrino-electron scattering and pair emission and absorption, the Boltzmann equation

15

in spherical symmetry is5:

1c

∂F

∂t+ 4πµ

∂(r2ρF )∂m

+1r

∂[(1− µ2)F ]∂µ

+1c

(∂ ln ρ

∂t+

3v

r

)∂[µ(1− µ2)F ]

∂µ+

+1c

[µ2

(∂ ln ρ

∂t+

3v

r

)− v

r

]1

E2

∂(E3F )∂E

=

=j

ρ− χF +

1c

1h3c3

E2

∫dµ′RISF − 1

c

1h3c3

E2F

∫dµ′RIS +

+1

h3c4

(1ρ− F

) ∫dE′E′2dµ′Rin

NESF − 1h3c4

F

∫dE′E′2dµ′Rout

NES

(1ρ− F

)+

+1

h3c4

(1ρ− F

) ∫dE′E′2dµ′Rem

PAIR

(1ρ− F

)− 1

h3c4F

∫dE′E′2dµ′Rabs

PAIRF , (10)

where F (m,µ,E) is the specific neutrino distribution function, f/ρ, and f is the neutrinodistribution function (F is similarly for antineutrinos), m is the enclosed mass, µ is theneutrino direction cosine and E is the neutrino energy.

Let’s take a glance the equation (10) and begin on the left-hand side. The mass-derivativeterm describes the propagation of neutrinos with respect to the Lagrangian mass coordinate,m. The first µ-derivative term describes the change of the neutrino propagation directionwith respect to the outward radial direction. The second µ-derivative term describes theaberration in the propagation direction measured by an observer, who is moving with thefluid. The energy-derivative term describes the shift in the neutrino energy measured byco-moving observers, which is the Doppler shift, resulting from the change in the velocity ofan accelerated fluid. These aberration and especially frequency shift terms play a critical rolein the development of the neutrino distributions during core collapse and are refered to as“observer corrections”. On the-right hand side of equation (10), the first two terms describethe change in the neutrino distribution due to emission and absorption of neutrinos by nucle-ons and nuclei. The next two terms describe the isoenergetic inscattering and outscatteringof the same particles. The fifth and sixth term describe non-isoenergetic neutrino-electronscattering, and the last two terms describe pair emission and absorption.

For each component of the stellar core (photons, nucleons, nuclei, elctrons and positrons,. . . ),one can write down a kinetic equation for the distribution function, which induces an infiniteseries of moment equations. Under certain conditions and assumptions, these series close andgive rise to the familiar hydrodynamics equations for the component fluids.[6]

4.3 Situation now and Future Challenges

Because solving the Boltzmann equation is computationally intensive even in 1D simulations,which asume spherical symmetry, historically a number of increasingly more sophisticatedphysical approximations have been implemented. Spherically simetric simulations with avery accurate treatment of neutrino interactions and equation of state and multiangle, multi-frequency Boltzmann neutrino transport in full general relativity have been performed. Theresult was a failure to produce the explosion of the stellar core (Figure 11). However, untilthese simulations were completed, failure to produce explosions in past models that used ap-proximate treatments for the neutrino transport, could have resulted from either transport

5equation (10) is displayed solely for the purpose of demonstrating the complexity of the problem

16

approximations or from neglect of essential physics. We now know that the transport approx-imations were not the cause of these failures; we have to expand the physics in the modelsto include the forementioned fluid instabilities, rotation and magnetism. Systematic addingof the dimensionality and the physics will be needed to achieve a complete understanding ofthe supernova mechanism and phenomenology.

Figure 11: Fruition of very precise simulations with full general relativity approach, resultshould be typical for all spherically simetric models. Plotted are shock trajectories as afunction of time. In this case, progenitor was a star with mass 13 M�.[6]

The past modeling efforts have illuminated that core collapse supernovae may be neutrinodriven, magnetohydrodynamically (MHD) driven, or both. If a supernova is neutrino driven,magnetic fields will likely have an impact on the dynamics of the explosion. Similarly, if asupernova is MHD driven, the neutrino transport will dictate the dynamics of core collapse,bounce and the postbounce evolution, which in turn will create the environment in which anMHD-driven explosion would occur. Although reduction will allow us to sort out the rolesof each of the major physical components (4), we will not obtain a quantitative or perhapseven qualitative understanding of core collapse supernovae, until all components and theircoupling are included in the models with sufficient realism.[6]

5 Conclusion

The secret of supernovae explosions is still not completely uncovered, but with every passingmoment, we are closer to it. Projects such as GenASiS (General Astrophysical SimulationSystem[7]) have already been launched to attempt to solve the problem by including allrelevant physics - including magnetohydrodynamics, gravity, and energy- and angle-dependentneutrino transport in two or three spatial dimensions, making the problem a five or even six-dimensional one. And during the next five years, multidimensional supernova models willundergo a dramatic change in realism. However, fully general relativistic simulations will berequired to acquire quantitavely accurate models, which will need at least a decade or moreto develop.[6]

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References

[1] R. L. Bowers, T. Deeming, Astrophysics I, Jones and Bartlett Publishers, Inc., 1984

[2] S. Woosley, T. Weaver, The Great Supernova of 1987, Scientific American, August 1989

[3] H. A. Bethe, Supernovae, Physics Today, September 1990

[4] H. A. Bethe, Supernova mechanisms, Rev. Mod. Phys., Vol. 62, No. 4, October 1990

[5] W. N. Gamezo et al., Thermonuclear Supernovae: Simulations of Deflagration Stage andtheir Implications, Vol. 299, No. 5603, January 2003

[6] A. Mezzacappa, Ascertaining the Core Collapse Supernova Mechanism: The state of theArt and the Road Ahead, Annu. Rev. Nucl. Part. Sci., Vol 55, P. 467-515, 2005

[7] C. Y. Cardall, A. O. Razoumov et al., Toward Five-Dimensional Core-Collapse Super-nova Simulations, Preprint:astro-ph/0510706, Vol. v1, 25 October 2005

[8] MPE/Garching, Pogostost supernov, Spika, No. 2, P. 60, February 2006

[9] Wikipedia, Supernova, http://en.wikipedia.org/wiki/Supernova/, 2006

[10] CBAT, List of Supernovae, http://cfa-www.harvard.edu/cfa/ps/lists/Supernovae.html, 2006

[11] www, Spectra, http://www.astro.rug.nl/∼onderwys/ACTUEELONDERZOEK/JAAR2001/rico/spectra.html

[12] AAVSO, SN 1987A, http://www.aavso.org/vstar/vsots/0301.shtml, 2001

[13] Pearson Education, whitedwarf, http://boojum.as.arizona.edu/∼jill/NS102 2004/Lectures/Lecture34/whitedwarf.html, 2004

[14] Pearson Prentice Hall, Stellar Explosions, http://physics.uoregon.edu/∼jimbrau/astr122/Notes/Chapter21.html, 2005

[15] TRW Inc., Burning of Elements Heavier than Helium, http://observe.arc.nasa.gov/nasa/space/stellardeath/stellardeath 1c.html, 1999

[16] H. Aslaksen, Heavenly Mathematics & Cultural Astronomy, http://www.math.nus.edu.sg/aslaksen/teaching/heavenly.html, 2006

[17] www, Elementary Physics, http://www.pcs.cnu.edu/∼brash/phys103/

[18] The Imagine Team, Universe! Dictionary, http://imagine.gsfc.nasa.gov/docs/dictionary.html, 2005

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