Chapter 14 Supernovae - University of...

Chapter 14

Supernovae

Supernovae represent the catastrophic death of certain stars. They areamong the most violent events in the Universe,

• They typically produce about1053 erg, with a large fraction ofthis energy released in the first second of the explosion.

• The total luminosity of the Sun is only about1033 erg s−1.

• Even a nova outburst releases only of order1047 ergover a char-acteristic period of a few hundred seconds.

There is more than one type of supernova, with two general method-ologies for classification:

1. According to the spectral and lightcurve properties,

2. According to the fundamental mechanism responsible for the en-ergy release.

629

630 CHAPTER 14. SUPERNOVAE

In addition to their intrinsic interest, supernovae of vari-ous types are of fundamental importance for a variety ofastrophysical phenomena, including

• element production and galactic chemical evolution,

• potential relationship to some types of gamma-raybursts,

• energizing and compressing the interstellar medium(implying a connection with star formation),

• gravitational wave emission, and

• applications in cosmology associated with standard-izable candle properties.

We begin our discussion by considering the taxonomy ofthese events.

14.1. CLASSIFICATION OF SUPERNOVAE 631

14.1 Classification of Supernovae

The traditional classification of supernovae is based on ob-servational evidence, primarily

• Spectra

• Lightcurves.

Some representative supernova spectra are displayed inFig. 14.1 on the following page.


(Early)

(Late)

A. V. Filippenko, Annu. Rev. Nucl. Part. Sci. 35, 309 (1997)

Figure 14.1:Early time and late time spectra for several classes of supernovae.


Days after maximum light

Blu

e m

agnitude

0 100 200 300 400

II-P

II-L

Ia

Ib

Figure 14.2:Schematic lightcurves for different classes of supernovae.

Some typical lightcurves are illustrated in Fig. 14.2.


In most cases now we have at least a schematic model thatcan be associated with each class and that can account forthe observational characteristics of that class. Those mod-els suggest that all supernova events derive their enormousenergy from either

• gravitational collapse of a massive stellar core or

• a thermonuclear runaway in dense, degenerate mat-ter.


The observational characteristics of supernovae deriveboth from

• the internal mechanism causing the energy release(for example, collapse of a stellar core) and

• the interaction of the initial energy release with thesurrounding outer layers or extended atmosphere ofthe star.

Therefore

• some observational characteristics are direct diag-nostics of the explosion mechanism itself,

• while others are only indirectly related to the explo-sion mechanism and instead are diagnostics for thestate of the star and its surrounding medium at thetime of the outburst.

The standard classes of supernovae and some of their char-acteristics are illustrated in Fig. 14.3.

63

6C

HA

PT

ER

14.S

UP

ER

NO

VAE

H

He

H

He

(H)

HeMantle

WDAccretion

He

Hydrogen in

Early Spectra?YesNo

SN I

Silicon Lines?

SN II

Detailed Lightcurve Shape?

YesNo

SN Ia

Helium

Rich?

YesNo

SN Ic SN Ib SN II-b SN II-L SN II-P

Core Collapse

Type II lightcurve:

linear after

maximum

Core Collapse

Type II lightcurve:

plateau after

maximum

Core Collapse

Most H removed

before collapse

(bridge II to Ib, Ic)

Thermonuclear

Thermonuclear

runaway, accreting

C/O white dwarf

Core Collapse

H removed before

collapse

Core Collapse

H and much of He

removed before

collapse

Fig

ure

14

.3:C

lassificationofsupernova

events.N

otethat

TypeII-n

isno

tshown

herebutdiscussed

inthe

text.


The primary initial distinction concerns whether hydro-gen lines are present in the spectrum, which divides su-pernovae into

• Type I (no hydrogen lines) and

• Type II (significant hydrogen lines).

The standard subclassifications then correspond to the fol-lowing characteristics:


14.1.1 Type Ia

A Type Ia supernova is thought to be associated with athermonuclear runaway under degenerate conditions in awhite dwarf that is accreting matter from a companion.

• This class of supernovae is sometimes termed ather-monuclear supernova,to distinguish it from all otherclasses that derive their power from gravitational col-lapse and not from thermonuclear reactions.

• No hydrogen is observed but calcium, oxygen, andsilicon appear in the spectrum near peak brightness.

• Type Ia supernovae are found in all types of galax-ies and their standardizable candle properties makethem a valuable distance-measuring tool (see follow-ing Box).


Box 14.9 Standard and Standardizable Candles

We may distinguish

• A standard candle, which is a light source that always has the same intrinsicbrightness under some specified conditions.

• A standardizable candle, which is a light source that may vary in brightnessbut that can be standardized (normalized to a common brightness) by somereliable method.

Standard candles, or standardizable candles, then permit distance measurement bycomparing observed brightness with the standard brightness.

• Different Type Ia supernovae have similar but not identical lightcurves.Hence they are not standard candles.

• However, there are empirical methods that allow the lightcurves of differentType Ia supernovae to be collapsed to a single curve. Thus, they are stan-dardizable candles (see Fig. 14.4 for an example).


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a

b

As measured

Light-curve timescale

"stretch factor" corrected

Calan-Tololo SNe Ia

-20 0 20 40 60-15

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Days

Figure 14.4:Empirical rescaling of Type Ia supernova lightcurves to make themstandardizable candles. B-band lightcurves for low-redshift Type Ia supernovae(Calan-Tololo survey; Hamuy, et al, 1996). As measured, theintrinsic scatter is0.3 mag in peak luminosity. After 1-parameter correction the dispersion is 0.15mag. See the directory ../astrofigs/standardizableCandleSNIa for information andreferences. From Ann. Rev. Astronomy and Astrophysics, 46,385 (2008)


Type Ia standardizable candles are particularly valuablebecause their extreme brightness makes them visible atvery large distances.

• The standardizable candle and brightness propertiesof Type Ia supernovae have made them a central toolin modern cosmology.

• For example, they are the most direct indicator thatthe expansion of the Universe is currently accelerat-ing.

• This implies that the Universe is permeated by a mys-teriousdark energythat can effectively turn gravityinto antigravity.


14.1.2 Type Ib and Type Ic

Type Ib and Ic supernovae are thought to represent thecore collapse of a massive star that has lost much of itsouter envelope because of strong stellar winds or inter-actions with a binary companion (these are calledWolf–Rayet stars).

For both Type Ia and Ib supernovae hydrogen and siliconspectral lines are absent, but helium lines are present forType Ib supernovae. The distinction between Types Ia andIb is thought to lie in whether

• only the hydrogen envelope has been lost before corecollapse (Type Ib), or

• whether most of the helium layer has also been ex-pelled (Type Ic).


14.1.3 Type II

Type II supernovae are characterized by prominent hydrogenlines.

• They are thought to be associated with the core collapse of amassive star.

• They are found only in regions of active star formation (they arealmost never found in elliptical galaxies, for example).

Type II supernovae are further subdivided according to detailed spec-tral and light-curve properties:

1. Type II-P: In the designation Type II-P, the P refers to a plateauin the light curve.

2. Type II-L: In the designation Type II-L, the L refers to a lin-ear decrease of the light curve in the region where a Type II-Plightcurve has a plateau.

3. Type II-b: In a Type II-b event the spectrum contains promi-nent hydrogen lines initially, but the spectrum then transitionsinto one similar to that of a Type Ia,b supernova. The suspectedmechanism is core collapse in a red giant that has lost most butnot all of its hydrogen envelope through strong stellar winds, orthrough interaction with a binary companion.

4. Type II-n: In this class of core-collapse supernova, narrow emis-sion lines and a strong hydrogen spectrum are present. Type II-nsupernovae are thought to originate in the core collapse of amas-sive star embedded in dense shells of material ejected by thestarshortly before the explosion.


Thus, all of the Type-II subcategories, and the Type Iband Type Ic subcategories, correspond to a similar core-collapse mechanism. The observational differences deriveprimarily from differences in the outer envelope and theirinfluence on the spectrum and lightcurve, not in the pri-mary energy-release mechanism.

14.2. TYPE IA SUPERNOVAE 645

14.2 Type Ia Supernovae

A Type Ia supernova is thought to correspond to a ther-monuclear explosion of an electron-degenerate, carbon–oxygen white dwarf that is triggered by accretion from acompanion star in a binary system. Thus it differs fun-damentally from all of the other classes of supernovae inFig. 14.3.


AccretionRing

White

Dwarf

Primary

White

Dwarf

Accretion

(a)

(b)

Ignition of thermonuclear

runaway in C-O under

degenerate conditions

Hydrogen and helium accreted

onto white dwarf through

accretion ring and burned to C

and O without triggering nova

explosion

Chandrasekhar-mass

white dwarf consumed in

thermonuclear runaway

that burns the C-O white

dwarf to NSE

White dwarf grows to

Chandrasekhar mass

Companion

Star

Figure 14.5:Mechanism for a Type Ia supernova.

14.2.1 The Favored Type Ia Mechanism

The Type Ia scenario is illustrated in Fig. 14.5. It is related to the novascenario. The difference is that

• In a nova a thermonuclear runaway is initiated in a thin surfacelayer after a certain amount of accretion and the white dwarfremains largely intact after the layer is ejected in the explosion.

• In the Type Ia situation the matter accumulates on the surface ofthe white dwarf over a long period without triggering a runawayin the accumulated surface layers.


AccretionRing

White

Dwarf

Primary

White

Dwarf

Accretion

(a)

(b)

Ignition of thermonuclear

runaway in C-O under

degenerate conditions

Hydrogen and helium accreted

onto white dwarf through

accretion ring and burned to C

and O without triggering nova

explosion

Chandrasekhar-mass

white dwarf consumed in

thermonuclear runaway

that burns the C-O white

dwarf to NSE

White dwarf grows to

Chandrasekhar mass

Companion

Star

• As the unburned matter accumulates, the mass of the white dwarfgrows and can eventually approach the Chandrasekhar limit.

• Near the Chandrasekhar limit the very high density can triggera thermonuclear runaway in the interior of the star that initiallyignites carbon and oxygen.

• Quickly (in a matter of a second or less) the runaway burns alarge percentage of the mass of the white dwarf to iron-groupnuclei, with an enormous release of energy

• Most of the iron in the Universe probably originates in TypeIa,and to a somewhat lesser degree, core-collapse, supernovae.


Thus, unlike a nova, or a core-collapse supernova thatwe discuss further below, a Type Ia supernova does notleave behind a significant remnant of the obliterated whitedwarf. An explanation of how this mechanism can leadto the standardizable candle properties of Type Ia super-novae, and a possible alternative explanation for the mech-anism, are discussed in Box 15.2.


Box 14.10 Single-Degenerate and Double-Degenerate Scenarios

The Type Ia mechanism proposed here is sometimes termed the “singly-degeneratemodel”, since it involves only one degenerate object (the white dwarf).

• Alternative mechanism: triggering of a thermonuclear burn by merger of twowhite dwarfs.

• This is called the “double-degenerate” model, because it involves two degen-erate objects.

• At present the singly-degenerate model appears to be in better accord withdata, but neither model can yet describe all aspects of a TypeIa explosionwithout assumptions.

• It is possible that more than one progenitor scenario can lead to Type Iasupernovae.

One side benefit of the singly-degenerate model is that it canprovide a plausibleexplanation for the observation that all Type Ia explosionsare similar in intrinsicbrightness (standardizable candle property).

• The Chandrasekhar mass is almost the same for all white dwarfs.

• Thus, if the white dwarf that explodes is always near the Chandrasekhar massit makes sense that the total energy produced by different Type Ia events issimilar.

• In contrast, for the doubly-degenerate model there is no obvious reason forthe sum of the masses of the two white dwarfs that merge to be similar indifferent events.


However, the preceding may be an oversimplified analy-sis.

• The later-time Type Ia lightcurve is largely deter-mined by how much56Ni is produced in the explo-sion.

• Thus the standardizable candle property could resultfrom any mechanism that causes a similar amount of56Ni to be made in all Type Ia explosions.


14.2.2 Thermonuclear Burning under Extreme Conditions

Because of the gigantic energy release in a small regionover a very short period of time, the conditions in a TypeIa explosion are extreme. Simulations indicate that

• Temperatures in the hottest parts can approach1010 K, with densities as large as 109 g cm−3.

• In the thermonuclear burn front, temperature changesas large of 1017 K/s are seen in simulations.

The physics of the Type Ia explosion presents a numberof issues that are difficult to deal with in the large numer-ical simulations that are required to model such events, asdiscussed in Box 15.3.


Box 14.11 Thermonuclear Burns: Different Scales

In the Type Ia explosion there is a thermonuclear burn corresponding to conversionof carbon and oxygen fuel into heavier elements by nuclear reactions that releaselarge amounts of energy. This burn is extremely violent and involves energy andtemperature scales far beyond our everyday experience, butit shares many qualita-tive properties with ordinary chemical burning.

• There is aburn front that proceeds through the white dwarf, with “cooler”(that being a relative term!) unburned fuel in front and hot burned products(ash) behind.

• This burn front can be remarkably narrow—as small as millimeters.

Thus there are two extremely different distance scales characterizing the explosion:

• the size of the white dwarf, which is of order 104 km, and

• the width of the burn front that consumes it, which can be billions of timessmaller.

This presents severe difficulties in accurately modelingType Ia explosions, since standard numerical approaches tosolving the equations governing the explosion cannot han-dle such disparate scales without drastic approximation.


Box 14.12 DeFlagration and Detonation Waves

In thermonuclear and ordinary chemical burning there is an important distinctionassociated with the speed of the burn front.

• If the burn front advances through the fuel at a speed less than the speed ofsound in the medium (subsonic), it is termed adeflagration wave.

• If the burn front advances at greater than the speed of soundin the medium(supersonic) it is called adetonation wave.

Deflagration and detonation waves have different characteristics:

• In a deflagration, fuel in front of the advancing burn is heated to the ignitiontemperature by conduction of heat across the burn front (recall that matterdescribed by a degenerate equation of state is a very good thermal conductor,much like a metal).

• In a detonation a shock wave forms and the fuel in advance of the burn frontis heated to ignition temperature by shock heating.

• Generally detonation is much more violent than deflagration.


Box 14.13 DeFlagration, Detonation and Isotopic Abundances

Deflagrations and detonations produce different isotopic abundance signatures inthe ash that is left behind.

• Detailed observational characteristics of Type Ia supernovae (in particular,the elemental abundances detected in the expanding debris)could be ac-counted for most naturally if we assume that part of the burn is a deflagrationand part of it is a detonation.

• This is a difficulty for the theory because general considerations suggest thatthe explosion starts off as a deflagration and it is not easy toget the burn incomputer simulations to transition to a detonation withoutmaking significantuntested assumptions.


Thus, we believe that the proposed Type Ia mechanism isplausible in outline, but there are bothersome details thatleave some doubt about whether we understand fully themechanism of these gigantic explosions.


14.2.3 Element and Energy Production

The energy released in a Type Ia supernova explosion de-rives primarily from the thermonuclear burning of carbonand oxygen to heavier nuclei.

• If the explosion lasts long enough to achieve nuclearstatistical equilibrium (NSE), the primary final prod-ucts of this burning will be iron-group nuclei (unlessthe temperature becomes so high that all nuclei pho-todisintegrate into alpha particles).

• An example of network evolution under conditionstypical of the Type Ia explosion in the deep interiorof the white dwarf is illustrated in Fig. 14.6.


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Z=12 N=11

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N e u t r o n s

Protons

ρ = 1 x 108 g/cm3

-6 -5 -4

log time

1

3

5

7

T9

Figure 14.6:Element production in a Type Ia explosion. Upper: mass fractionsXfor 468 isotopes and integrated energy production∑E in MeV per nucleon. Lower:distribution of abundancesY for all isotopes at end of calculation in the upperfigure. Inset on left shows the variation of temperature withtime (density remainsalmost constant over this time period).


• In this calculation the initial temperature wasT9 = 2, the initialdensity wasρ = 1× 10−8 g cm−3, and the initial compositionwas assumed to be equal mass fractions of12C and16O.

• The explosion is initiated by carbon burning, which quickly raisesthe temperature (see the inset to the figure) and initiates burningof oxygen and all the reaction products that are produced by car-bon and oxygen burning.

• The rapid temperature rise is associated with the couplingof thelarge energy release from the thermonuclear burning describedby the reaction network to the fluid of the white dwarf, which isdescribed by hydrodynamics.

• This energy release (through the equation of state) causesa rapidrise in temperature in the fluid representing the white dwarf, andthis in turn increases rapidly the rate of nuclear reactionsin thenetwork.

• The net result is the almost vertical rise in temperature fromT9 ∼

2 to T9 ∼ 6.6 in a period of less than 10−5 s.

• During this time the isotopic species in the network have in-creased from two to about five hundred.

• Significant population of the iron group of nuclei already evi-dent.


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Z=9 N=9

Z=29 N=30

Z=8 N=7

Z=13 N=13

Z=0 N=1

Z=22 N=27

Z=19 N=22

Z=28 N=33

Z=20 N=24

Z=6 N=7

Figure 14.7:Calculation of Fig. 14.6 with the timescale greatly expanded in theregion where much of the energy is released and there is a rapid increase in tem-perature.

• The very narrow range in time in Fig. 14.6 over which the net-work releases much of its energy is illustrated in greatly ex-panded scale in Fig. 14.7.

• Notice the rapid increase in population for hundreds of newele-ments.


Under these conditions, as the thermonuclear flame burnsthrough the white dwarf the carbon and oxygen fuel ineach region is burned in a tiny fraction of a second, andthe entire white dwarf is consumed on a timescale of lessthan a second.

14.3. CORE-COLLAPSE SUPERNOVAE 661

14.3 Core-Collapse Supernovae

A core-collapse supernova is one of the most spectacularevents in nature, and is almost certainly the source of theheavy elements that are produced in the rapid neutron cap-ture or r-process.

• Considerable progress has been made over the pasttwo decades in understanding the mechanisms re-sponsible for such events.

• This understanding was tested both qualitativelyand quantitatively by the observation of Supernova1987A in the nearby Magellanic Cloud—the bright-est supernova observed from earth since the time ofKepler.


14.3.1 The “Supernova Problem”

The observations of Supernova 1987A, and the continuing studies ofits aftermath,

• Provide compelling evidence that a core-collapse supernova rep-resents the death of a massive star in which a degenerate ironcore of approximately 1.4 solar masses collapses catastrophi-cally on timescales of tens of milliseconds.

• This gravitational collapse is reversed as the inner core exceedsnuclear densities because of the properties associated with thestiff nuclear equation of state.

• A pressure wave reflects from the center of the star and prop-agates outward, steepening into a shock wave as it passes intoincreasingly less dense material of the outer core.

• This shock blows off the outer layers of the star, producingthespectacular explosion seen in observations


However, the most realistic simulations of this event indicate that

• The shock wave loses energy rapidly as it propagates throughthe outer core.

• If the core has a mass of more than about 1.1M⊙, the shock stallsinto an accretion shock within several hundred milliseconds ofthe bounce at a distance of several hundred kilometers from thecenter.

• Thus the “prompt shock” does not blow off the outer layers ofthe star and fails to produce a supernova.

This is the“supernova problem”: there is good evidencethat we understand the basics, but the details fail to workrobustly. In this chapter we summarize the present statusof this important problem.


Stellar Burning Shells

Hydrogen

Oxygen

Carbon

Silicon

Iron

Helium Center of 25 Solar

Mass Star

25 M

T = 4 ×109 K

ρ = 107 gcm

-3

T = 2 ×107 K

ρ = 102 gcm

-3

Figure 14.8:Center of a 25 solar mass star late in its life. See also Table 5.5.

14.3.2 The Death of Massive Stars

Because of the sequential advanced burning stages described in Ch. 5,massive stars near the ends of their lives build up the layered structuredepicted in Fig. 14.8.

• The iron core that is produced in the central region of the starcannot produce energy by fusion (the curve of binding energypeaks in the iron region).

• Thus, it must ultimately be supported by electron degeneracypressure.

• As the silicon layer undergoes reactions the central iron corebecomes more massive.



Hydrogen

Oxygen

Carbon

Silicon

Iron


Mass Star

25 M

T = 4 ×109 K

ρ = 107 gcm

-3

T = 2 ×107 K

ρ = 102 gcm

-3

• Electron degeneracy can support the iron core against gravita-tional collapse only if its mass remains below the Chandrasekharlimit, which depends on the electron fraction but is approxi-mately 1.1–1.4 solar masses for typical iron cores.

• When the iron core exceeds this critical mass it begins to col-lapse because the electron degeneracy can no longer balancethegravitational forces.

• At the point where the collapse begins, the iron core of a 25M⊙ star has a mass of about 1.4 solar masses and a diameter ofseveral thousand kilometers.

This is but a tiny fraction of the total diameter. The mas-sive star is typically a supergiant by this time, with its ten-uous outer layers spread over a volume that would encom-pass much of the inner Solar System if it were placed atthe position of the Sun.



Hydrogen

Oxygen

Carbon

Silicon

Iron


Mass Star

25 M

T = 4 ×109 K

ρ = 107 gcm

-3

T = 2 ×107 K

ρ = 102 gcm

-3

• The core density is about 6×109 g cm−3,

• the core temperature is approximately 6×109 K, and

• the entropy per baryon per Boltzmann constant of the core isabout 1, in dimensionless units.

• This is a very small entropy, as discussed on the following page.


Box 14.14 Entropy of the Iron Core

This entropy of the iron core in a pre-supernova star is remarkably low: the entropyof the original main-sequence star that produced this iron core is about 15 in unitsof entropy per baryon per Boltzmann constant.

• At first glance, it may seem contradictory for the entropy todecrease as thestar burns its fuel.

• However, the star is not a closed system: as nuclear fuel is consumed, energyleaves the star in the form of photons and neutrinos.

• In the process, the nucleons in the original main-sequencestar are convertedto iron nuclei.

• In 56Fe, the 26 protons and 30 neutrons are highly ordered compared with56 free nucleons in the original star, because they are constrained to movetogether as part of the iron nucleus.

• Thus, the core of the star becomesmore orderedcompared with the originalstar as the nuclear fuel is consumed.

• The entire universe tends togreater disorder,as required by the second lawof thermodynamics, because the star radiates energy in the form of photonsand neutrinos as it builds its ordered core.


14.3.3 Sequence of Events in Core Collapse

The collapse of the core as the Chandrasekhar limit is exceeded trig-gers a rapid sequence of events on a timescale of less than a second.

1. When the mass of the Fe core exceeds∼ 1.4M⊙, it collapsesgravitationally. The collapse is accelerated by two factors:

(a) As the core heats up, high-energyγ-rays are produced.

• These photodisintegrate some of the iron-peak nucleiinto α-particles.

• The corresponding process is highly endothermic; forexample,56Fe→ 13α +4nhas aQ = −124.4 MeV.

• This decreases the kinetic energy of the electrons, whichlowers the pressure and hastens the collapse.

(b) As the density and temperature increase, the rate forp++

e− → n+ν, is greatly enhanced.

• This reaction is calledneutronization.

• It converts protons and electrons into neutrons and neu-trinos, decreasing the electron fractionYe of the core,lowering the electronic contribution to the pressure.

• The neutrinos produced in this reaction easily escapethe core during the initial phases of the collapse becausetheir mean free path is much larger than the initial radiusof the core.

• These neutrinos carry energy with them, decreasing thecore pressure and accelerating the collapse even further.

• They also deplete the lepton fraction (which is definedanalogous to the electron fraction, but for all leptons,which in this context means mostly electrons, positrons,neutrinos, and antineutrinos).


2. The accelerated core collapse proceeds on a timescale of mil-liseconds, with velocities that are significant fractions of thefree-fall velocities. The core separates into

• An inner corethat collapses subsonically and homologously(Homologous means that the collapse is “self-similar”, inthat it can be described by changing a scale factor.)

• An outer corethat collapses largely in free-fall, with a ve-locity exceeding the local velocity of sound in the medium(it is supersonic).

3. This collapse is

• Rapid on timescales characteristic of most stellar evolution,

• Slow compared with the reaction rates and the core is ap-proximately in equilibrium during all phases of the collapse.

Thusentropy is constant, and the highly-ordered iron core beforecollapse (S≃ 1) remains ordered during the collapse.


Mass (solar units)

λ (

km

)

0 0.5 1.0 1.5

Radius

Conditions

Just After

Bounce

λe

λµ

10-6

10-2

102

Figure 14.9:Mean free path for neutrinos shortly after core bounce (Burrows andLattimer).

4. As the collapse proceeds and the temperature and density rise,a point is reached where the neutrino interactions become suffi-ciently strong that the mean free path of the neutrinos becomesless than the radius of the core.

• Shortly thereafter, the time for neutrinos to diffuse outwardbecomes longer than the characteristic time of the collapseand the neutrinos are effectively trapped in the collapsingcore (see Fig. 14.9, where we see that neutrino mean freepaths in the collapsed core can become as short as a fractionof a meter).

• The radius at which this occurs is termed thetrapping ra-dius, and it is closely related to the neutrinosphere to bedefined below.


5. Because the collapse proceeds with low entropy,

• There is little excitation of the nuclei

• the nucleons remain in the nuclei until densities are reachedwhere nuclei begin to touch.

At this point, the collapsing core begins to resemble a gigantic“macroscopic nucleus”.

6. This core of extended nuclear matter

• is a nearly degenerate fermi gas of nucleons, and

• has a very stiff equation of state because nuclear matter ishighly incompressible.

At this point, the pressure of the nucleons begins to dominatethat from the electrons and neutrinos.


Sonic Point

SoundSpeed

Infall

Velocity

|v|

Matter at NuclearDensity

Shock WaveBegins to Form

SoundSpeed

InfallVelocity

|v|

Radius Radius

Figure 14.10:The sonic point during collapse (left) and the beginning of shock-wave formation following the bounce (right).

7. Somewhat beyond nuclear density,

• the incompressible core of nearly-degenerate nuclear mat-ter rebounds violently as a pressure wave reflects from thecenter of the star and proceeds outward.

• This wave steepens into a shock wave as it moves outwardthrough material of decreasing density (and thus decreasingsound speed),

• The shock wave forms near the boundary between the sub-sonic inner core and supersonic outer core (this point iscalled thesonic point;see Fig. 14.10).

In the simplest picture this shock wave would eject the outerlayers of the star, resulting in a supernova explosion. Thisiscalled theprompt shock mechanism.


The gravitational binding energy of the core at reboundis about1053 erg, and the typical observed energy of asupernova (the expanding remnant plus photons) is about1051 erg. Thus, only about 1% of the gravitational energyneed be released in the form of light and kinetic energy toaccount for the observed properties of supernovae.

1051 erg defines a unit of energy commonlyused in supernova discussions that is termedthe foe: 1 foe≡ 1051 ergs, with the name foederiving from the first letters offifty-one ergs.In more modern usage, this unit is termed thebethe, in honor of Hans Bethe.


NeutrinosDiffusive

NeutrinosRadiative

NeutrinosRadiative

Neutrinosphere

Neutrinosphere = the radius after

which a neutrino will, on the

average, suffer one more scattering

before exiting from the star.

Figure 14.11:The neutrinosphere.

8. Unfortunately, a simple prompt-shock mechanism appearsto beenergetically prohibited:

(a) The shock dissociates Fe nuclei as it passes through theouter core; this highly endothermic reaction saps it of a largeamount of energy.

(b) As the shock wave passes into increasingly less dense ma-terial,

• the mean free path for the trapped neutrinos increasesuntil the neutrinos can once again be freely radiated fromthe core.

• The radius at which the neutrinos change from diffusiveto radiative behavior is termed theneutrinosphere.

• When the shock wave penetrates the neutrinosphere, aburst of neutrinos is emitted from the core, carrying withit large amounts of energy (about half the energy re-leased in the gravitational collapse resides in the neu-trinos).

This further deprives the shock of its vitality by loweringthe pressure behind it.


Shock

Gain Radius

Matter Flow

ν-Sphere

Cooling

Heating

Neutrino

Radiation

Figure 14.12:Conditions at time of shock stagnation in a core-collapse supernova.The neutrinosphere and the gain radius (defined in the text) are indicated.

9. The most realistic calculations indicate that the shock wave stallsinto an accretion shock(a standing shock wave at a constantradius) before it can exit the core, unless the original ironcorecontains less than about 1.1 solar masses.

• In a typical calculation, the accretion shock forms at about200–300 km from the center of the star within about 10 msof core bounce.

• Since there is considerable agreement that SN1987A re-sulted from the collapse of a core having 1.3–1.4 solar masses,the prompt shock mechanism is unlikely to be a generic ex-planation of Type-II supernovae.

Figure 14.12 illustrates the conditions characteristic ofa stalledaccretion shock in a modern calculation of shock propagation ina core-collapse supernova.


14.3.4 Neutrino Reheating

The idea that neutrinos might play a significant role in supplying en-ergy to eject the outer layers of a star in a supernova event isan oldone.

• Failure of the prompt mechanism to yield supernova explosionsconsistently led to a revival of interest in such mechanisms.

• This evolved into what is generally termed thedelayed shockmechanismor neutrino reheating mechanism.

• In this picture, the stalled accretion shock is re-energized throughheating of matter behind the shock by interactions with neutrinosproduced in the region interior to the shock.

• This raises the pressure sufficently to impart an outward velocityto the stalled shock on a timescale of approximately one second,and the reborn shock then proceeds through the outer envelopeof the star.


Envelope Outer Core

Inner Core

Precollapse Infall Bounce

Shock Formation Shock Propagation

through Outer Core

and Stagnation

Neutrino Heating and

Shock Revival

Electron-NeutrinoBurst

Shock

Shock

3 km

Figure 14.13:Illustration of the neutrino reheating mechanism for a supernovaexplosion. Figures are approximately to scale, except thatthe surface of the starwould lie some 3 km from the center, if represented on this scale.

The schematic mechanism for the supernova event thus becomes thetwo-stage process depicted in Fig. 14.13.


Log t (ms)

Log L

n (

erg

/s)

Accretion on

Protoneutron Star

Cooling of

Neutron Star

Co

llap

se

Sh

ock

Breakout Burst

νe

νµ ,ν µ

ντ ,ν τ

ν e

0 1 2 3 4 5

50

51

52

53

54

Figure 14.14:Neutrino luminosities in a supernova explosion. The zero ofthetime scale is at core bounce.

Reheating of Shocked Matter

Neutrinos and antineutrinos are produced copiously in the hot, densecenter of the collapsed core. Luminosities are shown in Fig.14.14. Inthe post-bounce phase, the neutrinos can impart energy to the shockedmatter behind the stalled shock in three ways:

1. Electronν andν̄ can undergo charged current interactions withnuclei and free nucleons,e+

+ p↔ n+νe ande++n↔ p+ ν̄e.

2. All flavors undergo the charged and neutral current interactionsν +e→ ν +e.

3. All neutrino flavors can participate in the pair interaction ν +

ν̄ ↔ e++e−.

For the conditions found in supernova cores, this latter interaction istypically less important.


By these types of interactions, the neutrinos that are produced in thecore can interact with the matter behind the shock wave. In discussingthe details of that interaction, it is useful to introduce two character-istic radii.

• The first we have already met in Fig. 14.11: Theneutrinosphereis a sphere defined by a radius beyond which an average neutrinowill suffer one more scattering before it leaves the star. Itmarksa boundary between diffusion and free streaming for neutrinopropagation.

• The second is associated with the observation that the neutrinointeractions with the shocked matter could either cool or heat thematter.

General considerations suggest that there is always a ra-dius outside of which the net effect of the neutrino in-teractions is to heat the matter. This break-even radius,beyond which the neutrino interactions become effectivein increasing the pressure behind the shock is termed thegain radius.


Shock

Gain Radius

Matter Flow

ν-Sphere

Cooling

Heating

Neutrino

Radiation

The neutrinosphere and gain radius are indicated schemat-ically in the figure above. Revival of the shock is favoredby deposition of neutrino energy in the region between thegain radius and the shock.


Calculations with Neutrino Reheating

The result of a large number of calculations is that

• Neutrino reheating helps, but generally does not pro-duce successful explosions without artificial boostsof the neutrino luminosities.

• This suggests that there still are missing ingredientsin the supernova mechanism that must be included toobtain a quantitative description.

• One possibility is that convection in the region inte-rior to the shock wave alters the neutrino spectrumand luminosity in a non-negligible fashion.

Let us now turn to a general discussion of what role con-vection might play in supernova explosions.


14.3.5 Convection and Neutrino Reheating

In Ch. 6 we discussed general conditions under which stars can be-come unstable toward convection overturn.

• We may conjecture that substantial convection inside the stalledshock could have significant influence on the possibility of neu-trinos reenergizing the shock, and on the quantitative character-istic of a reenergized shock.

• In particular, we note that to boost the stalled shock it is neces-sary for the neutrinos to deposit energy behind the shock front(outside the gain radius, but inside the shock).

• Convective motion inside the shock front could, by overturninghot and cooler matter, cause more neutrino production and alterthe neutrino spectrum.

• The convection could also move neutrino-producing matterbe-yond the neutrinosphere, so that the neutrinos that are producedwould have a better chance to propagate into the region closelybehind the shock where deposition of energy would have themost favorable influence in increasing the pressure and reener-gizing the shock.

• This would provide a possible method to produce a supernovaexplosion with the requiredO

(

1051)

ergsof energy.


0

2

4

6

8

10

10 100 1000

0.1

0.2

0.3

0.5

0.4

0.0

Radius (km)

S(k

/NB) Y

e ,Y

LEntropy

ShockNeutrinosphere

Electron

Fraction

Lepton

Fraction

Figure 14.15:Lepton fractions and entropy 6.3 ms after bounce in a supernovacalculation. The progenitor had a mass of 15M⊙.

14.3.6 Convectively Unstable Regions in Supernovae.

Armed with our pedagogical results from § 6 for predicting whichregions may be convectively unstable, let us now examine theleptonfraction and entropy gradients produced during the period of shockstagnation in typical supernova calculations.

Fig. 14.15 shows results for a 15M⊙ star 6 ms after bounce. Thereare two potentially unstable regions:

• In the supernova 6 ms after the bounce, there is a Schwarzschildunstable region lying inside the shock front at about 100 kmwhere there is a large negative entropy gradient.

• There is a region near the neutrinospheres where both the en-tropy and the lepton fraction exhibit strong negative gradientsand thus favor an instability against Ledoux convection.


Figure 14.16:Simulation of convection in the core of a core-collapse supernova.The color scale indicates entropy, with maximum values in red and minimum valuesin blue. The boundary between the blue and purple regions indicates the locationof the shock wave.

The preceding arguments only identify regions that are unstable.

• Whether such regions develop convection, the timescale for thatconvection, and the quantitative implications for supernova ex-plosions can only be settled by detailed calculations.

• Many calculations have demonstrated that convection is signifi-cant in the core-collapse supernova problem.

• An example is shown in Fig. 14.16, where we see the onset ofspectacular convection below the stalled shock.

• Such violent and large-scale convection breaks sphericalsym-metry and cannot be treated with approximations like the mixing-length theory introduced in Ch. 6.

• It can only be modeled adequately using numerical multidimen-sional hydrodynamics simulations.


The present feeling is that

• A successful model of core-collapse supernovae willrequire both 3-dimensional hydrodynamics and acorrect treatment of neutrino transport.

• It has not been possible to include both in currentcodes because of inadequate computing power.

• It is thought that the generation of supercomputersjust beginning to become available may provide suf-ficient computational power to see whether these in-gredients lead to a succesful model of core-collapsesupernovae, or whether they point to new physics notyet incorporated into the models.


n

Rapid Neutron Capture

n n

Beta Decay

n n

. . .

. . .

e

n n n n n

-

n

Neutron NumberPro

ton

Nu

mb

er

Figure 14.17: A schematic representation of the rapid neutron capture or r-process.

14.4 Producing Heavy Elements: the r-Process

A very important question concerns the origin of the heaviest nuclei.

• They cannot be made by normal charged-particle reactions inequilibrium in stars because the peak of the binding energy curveoccurs for the iron-group nuclei and because of Coulomb barriereffects.

• We have noted that neutron capture reactions could circumventthe Coulomb barrier problem (since neutrons carry no charge).

• It is thought that many of the heavier elements are made in therapid neutron capture or r-process illustrated in Fig. 14.17.

14.4. PRODUCING HEAVY ELEMENTS: THE R-PROCESS 687

0 20 40 60 80 100 120 140 160

0

20

40

60

80

100

Neutron Number

Pro

ton N

um

ber

Neutron Drip Line

Proto

n Drip

Lin

e

Stability Valley

r-Process Path

Figure 14.18:The path for the r-process.

This is similar to the s-process occurring in red giants, except that

• Now we assume that the conditions are such that there is a veryhigh flux of neutrons and they can be captured very rapidly com-pared with the time forβ decay. As illustrated in Fig. 14.18.

• This has a tendency to take the population very very far to theneutron-rich side of the chart of the nuclides before it begins toβ -decay back toward the stability valley.

• Thus the r-process can populate many neutron-rich isotopes outof the stability valley that cannot be reached by the s-process.

• Further (unlike the s-process), the path illustrated in Fig. 14.18can populate isotopes beyond the gap in the stability valleyfoundnear lead and bismuth, thus accounting for the actinide isotopes.


• The most likely astrophysics site for the r-process is a core-collapse supernova, which may be expected to produce a hugeflux of neutrons.

• It is also thought that some r-process nuclei may be produced inthe merger of two neutron stars (though it is somewhat uncer-tain how much r-processed material could escape the resultingcollapse to a rotating black hole).

• However, our understanding of both core-collapse supernovaeand neutron star mergers is insufficient so far to confirm eitherof these as actual sites of the r-process.

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