Download Supernovae

UNIVERSITY OF LJUBLJANA
FACULTY OF MATHEMATICS AND PHYSICS
Supernovae
Jure Strle
advisor: doc. dr. Simon Širca
March 2006
Figure 1: The remnant of the Kepler’s supernova of 1604. The picture is a false-colour
composite of x-ray, visible and infrared light images.[9]
Abstract
Supernovae are stellar explosions of such immense power that it stretches even the
boundaries of human imagination. There are two types of supernovae that have very
different explosion mechanics. The purpose of this seminar is to briefly describe the history
of study of supernovae and to explain the mechanics of both types, concentrating on the
Type II supernovae, which have been studied more. Presented are also the achievements
and the problems of modeling supernovae explosions with computers.
Contents
1 Introduction
3
2 History
3
3 Types of Supernovae
3.1 Type I . . . . . . . . . . . . . . . . . . . . .
3.1.1 Death of a Dwarf . . . . . . . . . . .
3.1.2 Light and Candles . . . . . . . . . .
3.2 Type II . . . . . . . . . . . . . . . . . . . .
3.2.1 Pre-supernova Star . . . . . . . . . .
3.2.2 Collapse . . . . . . . . . . . . . . . .
3.2.3 Prompt Shock . . . . . . . . . . . .
3.2.4 Shock Revival . . . . . . . . . . . . .
3.2.5 Death of a Giant and Light Spectra
4 Modeling of the Core Collapse in Type II
4.1 “Grey” or “Multineutrino” . . . . . . . .
4.2 Neutrino Transport Primer . . . . . . . .
4.3 Situation now and Future Challenges . . .
5 Conclusion
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17
2
1
Introduction
Supernovae are spectacular events. One of the most researched supernovae - SN1987A - emitted 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 explode, increasing their energy output by several orders of magnitude, which is from afar seen
as a great increase in brightness of the progenitor star. After weeks or months these bright
objects slowly decline to invisibility. Their importance is significant; most heavy elements
are created by nuclear reactions in supernovae and then returned to space and all elements
heavier than iron are produced in this process. They are also the principal heat source for
interstellar matter and may be a source of cosmic rays. The shock waves from supernovae
explosions play a big role in triggering star formation. Based on the observed abundances of
heavy elements in our solar system, we can conclude that it was formed when remnants of
a supernova (or perhapse of multiple supernovae) settled in a nebula. Supernovae are quite
rare. In the Milky Way only five have been observed in the last 1000 years and none in the
last 300.[9, 8]
2
History
Throughout history, humans have been observing and recording “guest stars”, which suddenly
appeared 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 was
a 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 radio
image. 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 the
shock wave these supernovae sent out into space. The remnant in Crab has a whole volume
luminous, which is connected to the fact that at the center there is a neutron star which emits
electromagnetic radiation and electrons which irradiate material throughout remnant, which
in 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 Johannes
Kepler[18].
In 1572 Danish astronomer Tycho Brahe discovered a “new star” in Cassiopeia and observed it for several months. He found that its position did not change relatively to the
fixed stars. This was an important evidence against the Aristotelian dogma that nothing ever
changes beyond the moon. German astronomer Johannes Kepler saw another supernova in
1604 just before the invention of the telescope. It was visible for a whole year and it was
3
observed extensively, so its approximate light curve is known. Another supernova exploded
between 1650 and 1680, known as Cas A (in Cassiopeia constellation). Its remnant is a very
strong radio source, but it was not reported by contemporary observers. Some supernovae in
other galaxies were observed between 1885 and 1930.
In 1934, Zwicky and Baade wrote a paper on supernovae and concluded that it should
be possible to find many more of them by a systematic survey of galaxies, since supernovae
would easily stand out above the background of ordinary stars. Within five years they found
nearly 20 supernovae by comparison of pictures of galaxies at different times; if a bright spot
was 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 that
there are at least two types of supernovae: Type II have strong lines of hydrogen, while Type
I 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 February
astronomers discovered a new bright spot in the Large Magellanic Cloud, only 160,000 light
years away. The light came from the SN1987A supernova, which was the closest one in 300
years. It was of Type II and gave to astrophysicists a lot of valuable data on the death
of massive stars. The first sign of it was a burst of neutrinos detected by three neutrino
observatories (Kamiokande II, IMB, Baksan) a few hours before arrival of visible light. Apart
from the Sun, the SN1987A is the only extraterrestrial source of neutrinos known so far.[2]
Figure 4: Number of discovered supernovae per year.[10]
4
The number of discovered supernovae (by amateur and professional astronomers alike)
has been increasing exponentially. Until 1970 about 380 supernovae have been discovered in
total, by year 1990 this number has grown to about 800, by year 2000 to about 1880 and in
the 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 many
supernovae. Typical light curves for Type I and Type II both start with a sharp rise in
luminosity, extending over a week or two, which is due to the expansion of the luminous
surface. Type I has a fairly narrow peak, while Type II peak is broader, of the order of 100
days, due to the recombination of electrons in the excited H atoms of the ejecta. The intensity
then declines by roughly two orders of magnitude over a period of about a year in both types,
due to radioactive decay of 56 Co[11]. Not all supernovae of each type have the characteristic
light 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) supernovae
as as function of time. [13]
Zwicky and Baade suggested in 1934 that supernovae derive their tremendous energy from
gravitational collapse, in particular that the core of the star collapses to a neutron star, the
concept of which has been proposed by Landau in 1932. It is now generally believed that
this mechanism is predominantly responsivle for Type II supernovae. On the other hand,
Type I supernovae are believed to derive their energy from thermonuclear reactions, which
was 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 elliptical
galaxies. This suggests that the progenitors of Type II supernovae are relatively massive
Population I stars with lifetimes of less than about 107 years. In contrast, Type I supernovae
occur 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. Population
I stars are rich in heavy elements, while Population II stars are metal poor, nearly pure
hydrogen stars.[1]
5
3.1
Type I
The mechanism of Type I supernovae is less well understood than that of Type II. While
several models were proposed in the past, there is now agreement that Type I (or at least
the subtype Ia) is due to the thermonuclear disruption of white dwarfs. White dwarfs form
at the end of the evolution of stars whose original masses are less than 8 M (M = solar
mass). A star can lose a large fraction of its material by ejecting outer layers into space at
the final stages of evolution. The mass of the remaining white dwarf is always less than the
Chandrasekhar limit, 1.4 M . A white dwarf, consisting mainly of C and O and detached
degenerate electrons (the term degenerate means that electrons occupy all possible quantum
states below a certain energy), is stable and almost inert because its temperature is not
high enough to induce any substantial nuclear reactions. When isolated it can exist almost
indefinitely, slowly cooling down as it radiates its energy into space. However, most of the
white dwarfs are not isolated, but belong to groups of two or more stars. In a close binary
sistem, a white dwarf can increase its own mass by accreting material from a companion star
and thereby reaching the Chandrasekhar mass. Such systems are considered to be the most
probable supernova Type Ia progenitors, even though the exact nature of the companion star
and 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 increase
results 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. Released energy further increases the temperature and aids the thermonuclear reactions. The
process 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 the
energy release overwhelms the energy outflow. In a non-degenerate star, the energy release
would be stabilized by a thermal expansion accompanied by the work against gravity. In a
white dwarf, however, the initial temperature increase does not affect the degenerate-electron
pressure and therefore does not lead to any substantial expansion that could slow down ther-
6
monuclear reactions and prevent the runaway process. Eventually, the temperature increases
to the level where the thermal and degenerate-electron pressure components become comparable and the material begins to expand, but at that time, the expansion is unable to quench
the 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 about
1051 erg 1 , about as much as the Sun would radiate during 8 billion years. The energy
is produced by a network of thermonuclear reactions that begins from 12 C and 16 O nuclei
and ends in 56 Ni and in smaller quantities in other iron-group elements. Large amounts of
intermediate-mass elements (Ne, Mg, Si, S, Ca,. . . ) are created as well. It is believed that
the main energy-producing reactions occur in a thin layer, called a thermonuclear flame, that
propagates outwards. At the beginning the flame is laminar and its propagation velocity is
controlled 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 relatively
slow, subsonic regime, called deflagration, into a supersonic regime, called detonation, where
the reaction front is preceded by a shock wave. Most of the energy released during the
explosion transforms into kinetic and thermal energies of the expanding material. When the
sum of those energies exceeds the potential energy, the star becomes unbound, and expanding
material will continue to expand indefinitely. This deflagration-to-detonation model was most
succesful 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 source
for this is the slow radioactive decay sequence from the initially formed 56 Ni, which decays to
56 Co, which in turn decays to 56 Fe with a half-life of 77 days. The total optical energy observed
can be calculated from the assumption that most of the light is generated by the decay of
56 Co and 56 Ni and that essentially all the mass of the star burns to these end products, the
mass being that of a white dwarf at the Chandrasekhar limit, 1.4 M . The luminosity reaches
its maximum 15 to 20 days after the explosion and then decreases slowly until all the 56 Co
decays. The maximum brightness is comparable to the brightness of an entire galaxy and
can vary by an order of magnitude from one supernova to another. Observations also show
that the maximum luminosity of Type Ia supernovae in visible wavelengths correlates with
the rate at which the luminosity decreases after the maximum. In combination with other
approximate correlations this makes it possible to use supernovae Ia as “standard candles”
for determining absolute magnitudes of galaxies and hence their distance. The absence of
hydrogen lines is to be expected from white dwarfs, since the hydrogen that is accreted from
the companion is quickly converted into helium before the supernova explosion. In short, the
white 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 mechanism
is believed to be fairly well understood, however, there is still a lot of computation that needs
to be done to confirm this belief. Zwicky and Baade proposed that supernovae derive their
1
2
erg is unit for energy and mechanical work, widely used in astronomy, 1 erg= 10−7 J
due to gravity-induced Rayleigh-Taylor instabilities
7
energy from the gravitational collapse of the core of a star, of approximately the mass of
the 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 order
1049 erg.
The life of a future Type II supernova could be schematicaly divided into four parts, three
of which happen almost instantly, while the preparation for the explosion takes millions of
years:
Pre-supernova Star : massive stars are relatively short-lived, but in that time acquire the
characteristic feature of onion-like structure of chemical elements with iron core, outer
envelope of hydrogen and other elements in between
Collapse : due to great mass of the iron core and slowing of nuclear fusion, the inner
core collapses releasing a lot of gravitational energy in the process of becoming almost
incompressible proto-neutron star
Prompt Shock : the infalling mass rebounds from the proto-neutron star and surges through
the rest of the infalling matter as a shock wave, only to lose the energy in the still dense
iron outer core
Shock Revival : the released gravitational energy heats up the core and the only way to
cool itself is by emiting a vast number of neutrinos, which diffuse out of the opaque core
and deposit approximately 1% of their energy in the stalled shock, thus reviving it and
liberating the matter from the huge gravitational pull of the proto-neutron star in one
huge explosion - supernova
These four parts are more thoroughly described below. As we will see, it is not quite as
simple as that, especially in the shock revival part as will be shown in the following section
of 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 one
such star - Sanduleak −69◦ 202, the progenitor of the forementioned SN1987A. It all began
about 11 million years ago in a gas-rich part of the Large Magellanic Cloud, where a star was
born with about 18 M . For the next 10 million years, this star generated energy by fusing
hydrogen into helium. Because of its mass, the star “had to” maintain high temperatures
and pressures in its core to avoid collapse; as a result it was about 40000 times as bright as
the Sun and a voracious eater of nuclear fuel. When the innermost 30% of the star ran out of
hydrogen, the central regions began to gradually contract. As the core was compressed over
tens of thousands of years, from a density of 6 g/cm3 to 1000 g/cm3 , it heated up from about
40 million K to 190 million K. The higher core temperature and pressure ignited a new and
heavier nuclear fuel, helium. At the same time the outer layers of the star (mostly unburned
hydrogen) responded to the additional radiation from the core by expanding to a radius of
about 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 to
carbon and oxygen. During the next few thousand years, the star went through the same
scenario - core contraction, heating and ignition of a new heavier fuel (the ash of a previous
8
cycle) a couple of times more. After helium came carbon, then neon and oxygen, followed by
silicon and sulphur. The fusion chains stop at the final nuclear product, 56 Fe3 , which has the
highest binding energy per nucleon. The iron core at this stage is surrounded by successive
layer 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 and
neutrons.
The core of the star thus passed through consecutive stages of burning at an accelerating
pace. Helium lasted a million years, burning of carbon took only 12,000 years, of neon 12
years, of oxygen 4 years and of sillicon just a week. Each stock of nuclear fuel after hydrogen
released 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 photons
were transformed into particle pairs - an electron and a positron, as they passed near atomic
nuclei. These pairs promptly annihilated each other, recreating gamma rays, but sometimes
giving 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 stop
about half of them), they escaped from the star far more easily than the gamma rays could
have. Even during carbon burning, neutrino energy loss exceeded energy loss by radiation.
As the core temperature rose during later stages, neutrino luminosity rose exponentially to
become 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 dominated
by the electrons. Just before the collapse this balance is only marginally stable. As a result of
electron capture on free protons and nuclei in the core and as a result of nuclear dissociation
under extreme densities and temperatures where high energy γ photons are abundant,
A
el. capture: e− + A
Z X → Z−1 Y + νe
(1)
γ + 56 F e → 13α + 4n
(2)
dissociation:
3
also some other elements from iron group are produced
9
Fuel
H
He
C
Ne
O
Si
Main Product
He
C, O
Ne, Mg
O, Mg
Si, S
Fe (iron group)
Temperature [106 K]
40
190
740
1,600
2,100
3,400
Density [g/cm3 ]
6
1,100
240,000
7400000
16,000,000
50,000,000
Duration [years]
10,000,000
1,000,000
12,000
12
4
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 ultimately wins the contest of tens of millions of years.
The velocity of infalling matter in the core increases linearly with radius, which is a
characteristic of a homologous collapse expected of a fluid whose pressure is dominated by
relativistic, degenerate electrons. On the other hand, the sound speed decreases with density (or radius) and thus with increasing radius the infall velocity eventually exceeds the
local sound speed; the infall becomes supersonic. The core splits into a homologously and
subsonically infalling inner core and supersonically infalling outer core.[6]
In the process, the nuclei capture some electrons and become more neutron rich, but
there is a limit to this: the neutrinos formed in the process are scattered by the nuclei and
at a density of about 1012 g/cm3 this scattering is sufficient to trap the neutrinos. Then
the 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 resulting
Chandrasekhar mass after collapse. The initial densities of pre-supernova iron core of about
109−10 g/cm3 thus proceed during collapse to about 1−3·1014 g/cm3 . The inner core undergoes
a phase transition from a two-phase systems of nucleons and nuclei to a one-phase system of
bulk 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 repulsive
nature of the nucleon-nucleon interaction potential at short distances; the inner core becomes
incompressible and rebounds.[3]
3.2.3
Prompt Shock
Any information about the rebounding inner core would be conveyed to the outer core via
pressure waves that propagate radially outward at the speed of sound. When these waves
reach the point at which infall is supersonic they are swept in as fast as they attempt to
propagate outward. This means that no information about the rebounding inner core reaches
the infalling outer core, which in turn sets up a density, pressure and velocity discontinuity
in the flow - a shock wave.[6]
This first shock is the so called “prompt shock”. Does the prompt shock cause the ultimate
supernova 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 propagates outward, the shock is enervated. Additional energy losses occur in the form of electron
10
neutrinos, which occur when the core electrons capture on the newly dissociation-liberated
protons. Eventually the shock moves out beyond the neutrinosphere4 and the previously
trapped neutrinos escape.
Figure 8: Collapse of a massive star: 1: inner sub- and outer super-sonically infalling iron
core, 3-5: rebound of the very compressed core, 5-7: the shock wave (orange circle) is launched
and energized.[6]
This gives rise to the electron neutrino burst, which is the first of three major phases
of three-flavor neutrino emission during these events. As a result of these two enervating
mechanisms, the shock stalls in the iron core. How the shock is reenergized is currently the
central 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 out
to some distance - 300 to 500 km from the center and in this way, the prompt shock is
essential in preparation of the next stage. When the shock stalls, the core is composed of a
central radiating object, the proto-neutron star, which will go on to form a neutron star or
a black hole, depending on the initial mass of the star. The ultimate source of energy in a
core collapse supernova is the ∼ 1053 erg of gravitational binding energy associated with the
formation of the neutron star - it is equivalent to the gravitational energy of the iron core
with mass 1.4 M . We can estimate it with
2
53
2
E ∼ GMN
S /RN S = 2.6 · 10 erg (MN S /M ) (10 km/RN S ),
(3)
where G is the gravitational constant and MN S and RN S are the mass and the radius of the
neutron star, respectively.
This energy heats the core of proto-neutron star to the order of 1011 K, but not homogeneously. The proto-neutron star has a relatively cold inner part composed of unshocked bulk
nuclear matter, together with a hot mantle of nuclear matter that has been shocked, but not
expelled. The only way this mantle can cool down is by emitting neutrinos, since electromagnetic radiation is trapped in it, and it does so in the form of about 10 s long three-flavor
neutrino pulse. This marks the second phase of the neutrino emission. Electron neutrinos
are produced during stellar core collapse by electron capture on protons and nuclei, but after
the bounce there are all three flavors of neutrinos and antineutrinos produced in the hot
proto-neutron star mantle and are emitted as the mantle cools and contracts.
4
neutrinosphere - the radius beyond which neutrinos can escape freely, place of last scattering
11
Figure 9: During the shock reheating phase, the stellar core is composed of a central radiating
proto-neutron star whose surface is defined by the neutrinospheres (represented here by a
single sphere) and a region above the neutrinosphere consisting of a net cooling region and
a net heating region below the stalled shock, separated by the gain radius at which heating
and cooling balance. They are mediated by electron neutrino and antineutrino absorption
and emission.[6]
The neutrinos are emitted from their respective neutrinospheres (we should talk about
neutrinospheres in plural in the first place, since different energy and different flavour neutrinos have neutrinospheres with different radii) and their total luminosities during this phase are
maintained at their average values around 1052 erg/s by mass accretion on the proto-neutron
star (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 neutrinos. We must not forget that the densities around the core might not be high enough to
stop neutrinos completely, but a fraction, about 1% of the total luminosity or about 1051 erg
of energy of neutrino pulse is still absorbed by protons and neutrons behind the shock and
it starts the explosion of the star. This process is known as the delayed shock or neutrinoheating mechanism, but deciphering the precise role of it is difficult and the center of current
research.[6]
3.2.5
Death of a Giant and Light Spectra
After the explosion is initiated, the accretion luminosity decreases dramatically, and the neutrino 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 the
shock progresses through the rest of the star. When it breaks out of the star, light appears
12
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 few
months, most of the radiation is in the infrared.[3]
When the shock has progressed some distance (on the order of few 1000 km), nuclear
reactions take place, in which the pre-existing elements, O, Si and S, are converted into Fe
and intermediate elements. Because these reactions are extremely fast, taking place in about
a second, 56 Fe is not formed directly. Similarly to supernovae of Type I, initially formed is
56 Ni, the most tightly bound nucleus that consists solely of alpha particles (it is a doublymagic nucleus where both the proton and neutron number is 28). This material subsequently
decays by positron emission into 56 Co and then, with a half-life of 77 days, into 56 Fe. This
latter decay supplied most of the light energy of SN1987A. However, the majority of Type II
supernovae is at the maximum light emission and perhaps two months beyond powered by
the energy left by the shock in the hydrogen envelope, and only later by the radioactive decay
of 56 Co.
Neutrinos, once they are emitted from the proto-neutron star, travel unimpeded through
the rest of the star, while electromagnetic radiation is closely coupled to the matter and can
only emerge when the shock breaks out of the surface. The speed of the shock wave is a few
10,000 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 of
years (depending on the distance to the supernova) long race with neutrinos, which it takes
them 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 it
is 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 shock
following core collapse. After more than four decades of computational effort, the detailed
mechanism remains elusive, although significant progress has been made in understanding
these multiscale events. 1D, 2D and 3D simulations of core collapse supernovae have shown
that there are many important ingredients which apply to the explosion mechanism. These
ingredients 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 spherically symmetric models have achieved a significant level of sophistication, but (by definition)
can not incorporate b)-d). Fully general relativistic spherically symmetric simmulations with
Boltzmann neutrino transport do not yield explosions, demonstrating that some combination
of 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 region
between the gain radius and the shock is the so called “gain region”. The neutrino heating
in the gain region can be written as
Xp Lν̄e
Xn Lνe
1
1
2
2
˙ = a
hE i
+ a
hEν̄e i
,
(4)
2
λ0 4πr2 νe
F
4πr
F̄
λ̄0
13
Figure 10: One of the rare exceptions: Wilson’s (1985) “successful” spherically symmetric
model that proposed the doubly diffusive neutron finger instabilities in the proto-neutron star
to boost the neutrino luminosities. Each line on the graph traces a radial position of a shell
of 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 measure
of the isotropy of the neutrino distribution) and quadratically on the neutrino spectrum.
All three quantities in the neutrino heating rate must be computed accurately, which
requires that we solve the Boltzmann neutrino transport equations. In the “grey” approximation the neutrino angles and energies are integrated out and neutrino specific energy and
flux are functions of only spatial coordinates (in this case, radius). On the other hand, with
the “multineutrino” approach one can also accurately compute the neutrino spectrum.
The dependance of (4) on the neutrino spectrum means, that it is imperative to accurately
compute also the spectrum, which requires the use of multineutrino (a.k.a. multifrequency
or multigroup) energy, dependent also on the neutrino angle and energy. The fundamental
shortcoming 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 and
energies:
Z
Z
∂R
3 ∂F
¯ R,
= dµdEE
= − dµdEE 3 χ̃F ≡ −χ̃
(5)
∂t
∂t
where
R
dµdEE 3 χ̃F
¯
χ̃ ≡ R
dµdEE 3 F
and
Z
R ≡
dµdEE 3 F
(6)
(7)
Equation (5) gives the local rate of change of the neutrino specific energy, R , due to
electron neutrino absorption. This can be expressed in terms of the energy mean absorption
¯ which depends on the electron neutrino spectra. In a gray approach these spectra
opacity, χ̃,
are not computed, but they must be imposed, which means the neutrino specific distribution
function, F , is also imposed. And in equation (6) it is evident that the mean absorption
14
opacity is dependent on neutrino distribution function. This means the explosions can be
induced 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 transport
are 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 present
an unsurpassable obstacle for the multineutrino approach. Thus, gray and multifrequency
treatments are complementary.[6]
4.2
Neutrino Transport Primer
Neutrinos propagate through the proto-neutron star and interact with the nucleons and electrons. Because the cross sections for neutrino interactions are energy dependent (reduced
energies generally mean reduced cross sections), neutrinos of lower energies have longer mean
free paths. In the vicinity of the neutrinosphere the neutrino mean free paths become comparable to the size of the proto-neutron star. Deep within the core the high energy neutrinos
interact many times before escaping. This process is well described by diffusion theory. On
the 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
−2
ρ−1
0 ν
cm,
(N 2 /6A)Xh + Xn
(8)
where ρ0 is the density in g/cm3 , ν is the neutrino energy in MeV, Xh and Xn are the
mass fractions of heavy nucleons and neutrons, while N and A are numbers of neutrons and
nucleons in an average nucleus. During infall nearly all matter is in heavy nucleons and for
density 1012 g/cm3 , we have N ' 50, N/A ' 0.6 and if we also take 20 MeV for neutrino
energy, we get for mean free path:
λν ' 2 km (
10 MeV 2
) = 0.5 km.
ν
(9)
At the neutrinospheres, the neutrinos are not transported by diffusion nor are they radially
free streaming. Their transport is significantly more complex and is well described only by
solutions 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 distribution function for every time and space coordinate and gives the distributions of neutrinos
in terms of direction cosines and energies. Thus it is a phase-space equation in the multidimensional space of all spatial coordinates, angles and energies. Therefore, even a 1D
supernova 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 nuclei, neutrino-electron scattering and pair emission and absorption, the Boltzmann equation
15
in spherical symmetry is5 :
∂(r2 ρF ) 1 ∂[(1 − µ2 )F ] 1 ∂ ln ρ 3v ∂[µ(1 − µ2 )F ]
1 ∂F
+ 4πµ
+
+
+
+
c ∂t
∂m
r
∂µ
c
∂t
r
∂µ
v 1 ∂(E 3 F )
1 2 ∂ ln ρ 3v
−
+ µ
+
=
c
∂t
r
r E 2 ∂E
Z
Z
j
1 1
1 1
= − χ̃F +
E 2 dµ0 RIS F −
E 2 F dµ0 RIS +
3
3
ρ
ch c
c h3 c3
Z
Z
1
1
1
1
out
in
+ 3 4
−F
−F +
dE 0 E 02 dµ0 R̃NES
F − 3 4 F dE 0 E 02 dµ0 R̃NES
h c ρ
h c
ρ
Z
Z
1
1
1
1
abs
em
F̄ ,
+ 3 4
−F
− F̄ − 3 4 F dE 0 E 02 dµ0 R̃PAIR
dE 0 E 02 dµ0 R̃PAIR
h c ρ
ρ
h c
(10)
where F (m, µ, E) is the specific neutrino distribution function, f /ρ, and f is the neutrino
distribution function (F̄ is similarly for antineutrinos), m is the enclosed mass, µ is the
neutrino 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-derivative
term 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 direction
with respect to the outward radial direction. The second µ-derivative term describes the
aberration in the propagation direction measured by an observer, who is moving with the
fluid. The energy-derivative term describes the shift in the neutrino energy measured by
co-moving observers, which is the Doppler shift, resulting from the change in the velocity of
an accelerated fluid. These aberration and especially frequency shift terms play a critical role
in 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 describe
the change in the neutrino distribution due to emission and absorption of neutrinos by nucleons and nuclei. The next two terms describe the isoenergetic inscattering and outscattering
of the same particles. The fifth and sixth term describe non-isoenergetic neutrino-electron
scattering, 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 infinite
series of moment equations. Under certain conditions and assumptions, these series close and
give 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 sophisticated
physical approximations have been implemented. Spherically simetric simulations with a
very accurate treatment of neutrino interactions and equation of state and multiangle, multifrequency Boltzmann neutrino transport in full general relativity have been performed. The
result was a failure to produce the explosion of the stellar core (Figure 11). However, until
these simulations were completed, failure to produce explosions in past models that used approximate treatments for the neutrino transport, could have resulted from either transport
5
equation (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 approximations were not the cause of these failures; we have to expand the physics in the models
to include the forementioned fluid instabilities, rotation and magnetism. Systematic adding
of the dimensionality and the physics will be needed to achieve a complete understanding of
the supernova mechanism and phenomenology.
Figure 11: Fruition of very precise simulations with full general relativity approach, result
should be typical for all spherically simetric models. Plotted are shock trajectories as a
function 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 neutrino
driven, 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 a
supernova 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 an
MHD-driven explosion would occur. Although reduction will allow us to sort out the roles
of each of the major physical components (4), we will not obtain a quantitative or perhaps
even qualitative understanding of core collapse supernovae, until all components and their
coupling are included in the models with sufficient realism.[6]
5
Conclusion
The secret of supernovae explosions is still not completely uncovered, but with every passing
moment, we are closer to it. Projects such as GenASiS (General Astrophysical Simulation
System[7]) have already been launched to attempt to solve the problem by including all
relevant physics - including magnetohydrodynamics, gravity, and energy- and angle-dependent
neutrino transport in two or three spatial dimensions, making the problem a five or even sixdimensional one. And during the next five years, multidimensional supernova models will
undergo a dramatic change in realism. However, fully general relativistic simulations will be
required to acquire quantitavely accurate models, which will need at least a decade or more
to develop.[6]
17
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 and
their Implications, Vol. 299, No. 5603, January 2003
[6] A. Mezzacappa, Ascertaining the Core Collapse Supernova Mechanism: The state of the
Art 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 Supernova 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,
Supernovae.html, 2006
http://cfa-www.harvard.edu/cfa/ps/lists/
[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.html, 2005
Dictionary, http://imagine.gsfc.nasa.gov/docs/
18