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Energy is transported by radiation and convection. While the solar envelope is convective, radiative transport dominates in the core region where thermonuclear reactions take place. The radiative opacity depends sensitively on the solar composition, particularly the abundances of heavier elements. |
Thermonuclear reaction chains generate solar energy. The standard model predicts that 99% of this energy is produced from the pp chain conversion of four protons into He (see Fig. |
[EQUATION] The Sun is a large but slow reactor: the core temperature, [MATH] K, results in typical center-of-mass energies for reacting particles of [MATH] 10 keV, much less than the Coulomb barriers inhibiting charged particle nuclear reactions. Thus reaction cross sections are small: in most cases, as laboratory meas... |
The model is constrained to produce today’s solar radius, mass, and luminosity. An important assumption of the SSM is that the Sun was highly convective, and therefore uniform in composition, when it first entered the main sequence. It is furthermore assumed that the surface abundances of metals (nuclei with A [MATH] 5... |
Three cycles with quite different temperature dependences, reflecting the relative ease or difficulty of Coulomb barrier penetration, comprise the pp chain of Fig. . The competition between the cycles is very sensitive to the solar core temperature [MATH] . The initial interest in solar neutrinos came from the observat... |
The neutrino-producing reactions of the pp chain and CN cycle (a second process for burning protons to He, dominant in massive main-sequence stars, but responsible for only [MATH] 1% of solar energy) are summarized in Table . The first six reactions are [MATH] decays that produce continuous neutrino spectra. The last t... |
and Brun, Turk-Chieze, and Morel (BTCM98) . An update of the BS05(AGS,OP) SSM is in progress. 2.2 SNO and Super-Kamiokande Solar neutrino detection requires the combination of a large detector volume (to provide the necessary rate of events), very low backgrounds (so that neutrino events can be distinguished from backg... |
There are several possible detection modes for solar neutrinos, interesting because of their different sensitivities to flavor. The early radiochemical experiments using 37 Cl and 71 Ga targets were based on the charged current weak reaction |
[EQUATION] where the signal for neutrino absorption is the growth over time of very small concentrations of the daughter nucleus [MATH] in the detector. As the spectrum of solar neutrinos extends only to about 15 MeV, well below the threshold for producing muons, this reaction is sensitive only to electron neutrinos. |
A second possible nuclear detection channel is neutral-current scattering [EQUATION] a process independent of the neutrino flavor. If this scattering leaves the nucleus in an excited state, the observable would be the de-excitation of the nucleus, such as a decay [MATH] ray or the breakup of the nucleus. (An example wi... |
A third possibility is the scattering of neutrinos off electrons, [EQUATION] with detection of the recoiling scattered electron. Both electron- and heavy-flavor ( [MATH] [MATH] ) solar neutrinos can scatter off electrons, the former by charge and neutral currents, and the latter by neutral currents only. Consequently t... |
These various detection channels were exploited in two large-volume water Cerenkov detectors that recorded events in real time and provided flavor sensitivity. |
Super-Kamiokande (Fig. ) is a detector consisting of 50 kilotons of ultra-pure water within a cylindrical stainless steel tank, 39m in diameter and 42m tall. Two meters inside the walls a scaffold supports a dense array of 50-cm-diameter hemispherical photomultiplier tubes (PMTs), which face inward and view the inner 3... |
The detector has produced three data sets. In the first, obtained from 1996-2001, the detector was instrumented with an array of 11,146 50-cm PMTs, corresponding to about 40% coverage. In November 2001 one of the 50-cm PMTs imploded, creating a powerful shock wave that propagated through the tank, destroying 60% of the... |
for the high-energy B neutrino flux are [MATH] /cm s and [MATH] /cm /s, respectively, well below the SSM predictions of Table The Sudbury Neutrino Observatory (SNO) (Fig. ) was constructed at quite extraordinary depth, two kilometers underground in Inco’s Creighton nickel mine in Ontario, Canada. The detector took data... |
The choice of a heavy-water target allowed SNO experimentalists to exploit all three of the reaction channels described above, with their varying flavor sensitivities |
[EQUATION] The elastic scattering (ES) reaction is the same as that employed by Super-Kamiokande, with its differing sensitivities to electron- and heavy-flavor neutrinos. The charged-current (CC) reaction on deuterium is sensitive only to electron-flavor neutrinos, producing electrons that carry off most of the incide... |
The NC reaction, which is observed through the produced neutron, provides no spectral information, but does measure the total solar neutrino flux, independent of flavor. The SNO experiment has used three techniques for measuring the neutrons. In the initial pure-D phase the neutrons captured on deuterium, producing 6.2... |
SNO was constructed at very great depth and under clean-room conditions because of the need to suppress backgrounds. In particular, a minute amount of dust in the detector could have introduced environmental radioactivities that would have obscured the NC signal, a single neutron. The great advantage of the SNO detecto... |
The first results from SNO are shown in Fig. The final SNO Phase I (neutron capture on deuterium) deduced fluxes are [EQUATION] Fig. shows that the total flux (NC) is in agreement with the SSM prediction, but the neutrino flavor has been changed: about two-thirds of the electron neutrinos produced in the Sun arrive on ... |
2.3 Neutrino Mass and Oscillations The phenomenon by which a massive neutrino of one flavor changes into one of a second flavor is called neutrino oscillations. Neutrino oscillations have been shown to be responsible not only for the missing solar neutrinos in Davis’s experiment, but also for the missing atmospheric ne... |
Neutrino oscillations originate from two distinct sets of labels carried by neutrinos. One is flavor, a property of the weak interaction: an electron neutrino is defined as the neutrino accompanying a positron in [MATH] decay. The other possible label is mass. If a neutrino has a mass [MATH] , it propagates through fre... |
[MATH] . Thus neutrino states can be labeled according to flavor, and also labeled according to their masses. However nothing requires the neutrinos of definite flavor to be coincident with the neutrinos of definite mass. (In fact, in the analogous case of the quarks, it has long been known that the flavor (or weak int... |
[MATH] and [MATH] ) are related to the weak interaction eigenstates by [EQUATION] where [MATH] , the (vacuum) mixing angle, is nonzero. (Here, for simplicity, we consider just two neutrinos – the generalization to three flavors adds little new.) |
In this case a state produced as a [MATH] or a [MATH] at some time [MATH] — for example, a neutrino produced in [MATH] decay in the Sun’s core — does not remain a pure flavor eigenstate as it propagates away from the source. The different mass eigenstates comprising the neutrino will accumulate different phases as the ... |
[EQUATION] the accumulate phases depend on the mass [EQUATION] If the neutrino mass is small compared to the neutrino momentum/energy, one finds |
[EQUATION] There is a common average phase (which has no physical consequence) as well as a beat phase that depends on [EQUATION] |
From this one can find the probability that the neutrino state remains a [MATH] at time t [EQUATION] The probability oscillates from 1 to [MATH] and back to 1 over an oscillation length scale |
[EQUATION] as depicted in Fig. In the case of solar neutrinos, if [MATH] were comparable to or shorter than one astronomical unit, a reduction in the solar [MATH] flux would be expected in terrestrial detectors. |
The suggestion that the solar neutrino problem could be explained by neutrino oscillations was first made by Pontecorvo in 1958, who pointed out the analogy with [MATH] |
oscillations. If the Earth-Sun separation is much larger than [MATH] , one expects an average flux reduction due to oscillations of |
[EQUATION] For a 1 MeV neutrino, this requires [MATH] eV But such a reduction [MATH] particularly given the initial theory prejudice that neutrino mixing angles might be small [MATH] did not seem sufficient to account for the factor-of-three discrepancy that emerged from Davis’s early measurements. |
The view of neutrino oscillations changed when Mikheyev and Smirnov showed in 1985 that neutrino oscillations occurring in matter – rather than in vacuum – could produce greatly enhanced oscillation probabilities. This enhancement comes about because neutrinos propagating through matter acquire an additional mass due t... |
(who first described the phenomenon of neutrino effective masses). To explain this enhancement, consider the case where the mixing angle in vacuum, [MATH] , is small and [MATH] Then [MATH] |
where [MATH] is the local electron density, that is, the [MATH] and the light vacuum eigenstate [MATH] are almost identical. (Correspondingly, the heavy eigenstate [MATH] in vacuum.) Now what happens in matter? As matter makes the [MATH] |
heavier in proportion to the electron density, if that density is sufficiently high, clearly the electron neutrino must become the (local) heavy mass eigenstate. That is, |
[MATH] (and consequently [MATH] ). That is, we conclude that there must be a local mixing angle [MATH] that rotates from [MATH] in vacuum to [MATH] as [MATH] |
MSW enhancement occurs when the density changes between neutrino production and detection. In particular, electron neutrinos produced in the high-density solar core are created as heavy mass eigenstates. If these neutrinos now propagate to the solar surface adiabatically – this means that changes in the solar density s... |
[MATH] are small over an oscillation length, at all points along the neutrino trajectory – they will remain on the heavy-mass trajectory, and thus exit the Sun as [MATH] . That is, there will be an almost complete conversion of the [MATH] s produced in the solar core to [MATH] s. The MSW mechanism is an example of an a... |
A schematic comparison of vacuum and matter-enhanced oscillations is shown in Fig. . The matter transition between electron and muon flavors is centered around a density where the vacuum mass difference is just compensated by the matter contributions. |
The results from SNO and Super-Kamiokande, from earlier solar neutrino experiments, and from the reactor experiment KamLAND, have determined the parameters governing solar neutrino oscillations quite precisely |
. Unlike the example given above, the relevant mixing angle is rather large, [MATH] . The vacuum mass difference, [MATH] eV , leads to important matter effects in the higher energy portion of the solar neutrino spectrum, thus influencing the rates found in the SNO, Super-Kamiokande, and chlorine experiments. These effe... |
2.4 Solar Neutrinos: Outlook Neutrino oscillations proved to be responsible for both the solar neutrino problem and the atmospheric neutrino problem discussed in the next chapter. This phenomenon requires both flavor mixing and neutrino mass, phenomena that can be accommodated in various extensions of the standard mode... |
There remain important, unresolved issues in solar neutrino physics. The first direct (real-time) measurements of the dominant, low-energy neutrino fluxes – the pp and Be neutrinos – are just beginning. Borexino |
, a scintillation detector operating in the Italy’s Gran Sasso laboratory (Fig. ), began taking data on the Be flux in May 2007. One goal is to test the prediction that the survival probability of Be [MATH] will be larger than that for the higher energy neutrinos, since matter effects are more significant for the latte... |
There are also interesting developments involving the SSM. One of the important validations of the SSM has come through helioseismology, the measurement of solar surface fluctuations, as deduced from the Doppler shifts of spectral lines. The observed patterns can be inverted to determine properties of interior acoustic... |
But recently improved three-dimensional models of photospheric absorption lines have led to a 30% downward revision in convective-zone metal abundances (that is, abundances of elements heavier than helium). As discussed earlier, the SSM fixes the Sun’s zero-age metallicity to today’s surface abundances. If the new abun... |
Atmospheric Neutrinos The atmospheric neutrino problem developed very much in parallel with the solar neutrino problem and also involved missing neutrinos. The first definitive claim that neutrinos are massive came the atmospheric neutrino group associated with Super-Kamiokande, in 1998. The oscillations seen in atmosp... |
3.1 The Neutrino Source When primary cosmic-ray protons and nuclei hit the upper atmosphere, the ensuing nuclear reactions with atmospheric oxygen and nitrogen nuclei produce secondaries such as pions, kaons, and muons. Atmospheric neutrinos arise from the decay of these secondaries. For energies less than [MATH] 1 GeV... |
[EQUATION] Consequently one expects the ratio [EQUATION] to be approximately 0.5 in this energy range. Detailed Monte Carlo calculations, including the effects of muon polarization, give [MATH] . This ratio should be rather insensitive to theoretical uncertainties. It does not depend on absolute fluxes, and as a ratio ... |
Atmospheric neutrinos are a very attractive astrophysical source for experimenters. Apart from relatively minor geomagnetic effects, atmospheric neutrino production is uniform over the Earth. Thus an experimenter, operating an underground detector at some location, can make use of a set of nearly equivalent neutrino so... |
3.2 Atmospheric Neutrinos and Proton Decay Detectors The atmospheric neutrino anomaly grew out of efforts to build large underground detectors for proton decay, one of the phenomena expected in the Grand Unified Theories that were formulated in the late 1970s and early 1980s. As atmospheric neutrinos and proton decay w... |
and Kamiokande proton decay detectors. IMB first noticed a possible deficit of neutrino-induced muon events in 1986, while Kamiokande established a deficit in excess of 4 [MATH] by 1988. By 1998 this anomaly was also apparent in data from the Soudan detector and from Super-Kamiokande. |
The quantity determined in such experiments is a ratio (observed to predicted) of ratios [EQUATION] where the numerator is determined experimentally, and the denominator calculated. Agreement between data and theory thus requires [MATH] Early experimenters faced a difficulty in evaluating this ratio due to limited stat... |
[EQUATION] for sub-GeV events which were fully contained in the detector and [EQUATION] for fully- and partially-contained multi-GeV events. In addition, the collaboration presented an analysis in 1998, based on 33 kton-years of data, showing a zenith angle dependence inconsistent with theoretical calculations of the a... |
. This indicated a distance dependence in the muon deficit, a signature of oscillations. Furthermore the parameters of the oscillation, especially [MATH] eV [MATH] eV , differed from those that would later be determined from solar neutrinos. The collaboration concluded that the data were consistent with the two-flavor ... |
SK-I collected approximately 15,000 atmospheric neutrino events in nearly five years of running. The collaboration’s zenith-angle analysis of the data found evidence of a first oscillation minimum at [MATH] km/GeV, so that |
[MATH] 1000 km for a 1 GeV muon neutrino. The full analysis of oscillation parameters (see Fig. ) gives a best-fit [MATH] of [MATH] , a value clearly distinct from the solar neutrino mass difference. Most intriguing, the mixing angle appears to be very close to 45 – equal mixtures of two mass eigenstates. |
3.3 Outlook While a great deal of new physics has been learned from experiments on atmospheric and solar neutrinos, several important questions remain |
Oscillation experiments are sensitive to differences in the squared masses. They are not sensitive to absolute neutrino masses. We do know, from the atmospheric [MATH] that at least one neutrino must have a mass [MATH] 0.04 eV. But the only laboratory bound, from tritium [MATH] decay experiments, would allow neutrino m... |
Matter effects (from passage through the Earth) have not been seen in atmospheric neutrino experiments. This leaves open two possible orderings of the mass eigenstates, as illustrated in Fig. |
The solar and atmospheric mixing angles [MATH] and [MATH] have been determined, but a third mixing angle, [MATH] , is so far only bounded by the results from reactor experiments, [MATH] |
There are three CP-violating phases in the matrix that describes the relationship between neutrino mass and flavor eigenstates, one of which could be detected by looking for differences between certain conjugate oscillation channels, such as |
[MATH] and [MATH] . Finding such a difference is important to theories that attribute the excess of matter over antimatter in our universe to leptonic CP violation. |
The neutrino, lacking an electric or any other charge that must flip sign under particle-antiparticle conjugation, is unique among standard model particles in that it may be its own antiparticle. So far no measurement has been made that can distinguish this possibility (a Majorana neutrino) from the case where the [MAT... |
[EQUATION] could settle this issue, however. This process requires lepton number violation and Majorana masses. The two remaining CP-violating phases are Majorana phases that can affect neutrinoless double [MATH] decay rates. |
No compelling argument has been given to account for the large mixing angles deduced from atmospheric and solar neutrino oscillations. These angles differ markedly from their measured counterparts among the quarks. The special value of the atmospheric angle [MATH] |
is particularly curious. While some of these questions may be answered in terrestrial experiments, neutrino astrophysics will continue to offer unique environments for probing fundamental neutrino properties. Several examples are given in the chapter on neutrino cooling. |
Supernovae Neutrinos and Nucleosynthesis The bursts associated with a core collapse supernova are among the most interesting sources of neutrinos in astrophysics |
. A massive star, in excess of 10 solar masses, begins its lifetime burning the hydrogen in its core under the conditions of hydrostatic equilibrium. When the hydrogen is exhausted, the core contracts until the density and temperature are reached where 3 [MATH] C can take place. The helium is then burned to exhaustion.... |
12 C, 16 O and 20 Ne, 28 Si, and 56 Fe at the center. 4.1 The Explosion Mechanism and Neutrino Burst The source of energy for this evolution is nuclear binding energy. A plot of the nuclear binding energy [MATH] as a function of nuclear mass shows that the minimum is achieved at Fe. In a scale where the 12 C mass is pi... |
12 [MATH] /nucleon = 0.000 MeV 16 [MATH] /nucleon = -0.296 MeV 28 Si [MATH] /nucleon = -0.768 MeV 40 Ca [MATH] /nucleon = -0.871 MeV |
56 Fe [MATH] /nucleon = -1.082 MeV 72 Ge [MATH] /nucleon = -1.008 MeV 98 Mo [MATH] /nucleon = -0.899 Mev Once the Si burns to produce Fe, there is no further source of nuclear energy adequate to support the star. So as the last remnants of nuclear burning take place, the core is largely supported by degeneracy pressure... |
Thus the collapse that begins with the end of Si burning is not halted by a new burning stage, but continues. As gravity does work on the matter, the collapse leads to a rapid heating and compression of the matter. Sufficient heating of the Fe can release [MATH] s and a few nucleons, which are bound by [MATH] 8 MeV. At... |
[EQUATION] As the chemical equilibrium condition is [EQUATION] the increase in the electron Fermi surface with density will lead to increased neutronization of the matter, as long as neutrinos freely escape the star. These escaping neutrinos carry off energy and lepton number. Both the electron capture and the nuclear ... |
While the [MATH] s readily escape in the early stages of infall, conditions change once the density reaches [MATH] 10 12 g/cm At this point the neutrino scattering off the matter through both charged current and coherent neutral current processes begins to alter the transport. The neutral current neutrino scattering of... |
For a neutron star of 1.4 solar masses and a radius of 10 km, an estimate of its binding energy is [EQUATION] Thus this is roughly the trapped energy that will later be radiated in neutrinos, after core bounce, as the proto-neutron star formed in the collapse cools. |
The collapse produces a shock wave that is critical to subsequent ejection of the star’s mantle. The velocity of sound in matter rises with increasing density. Late in the collapse the sound velocity in the inner portion of the iron core, with [MATH] solar masses, exceeds the infall velocity. Any pressure variations th... |
The collapse of the core continues until nuclear densities are reached. As nuclear matter is rather incompressible ( [MATH] 200 MeV/f ), the nuclear equation of state is effective in halting the collapse: maximum densities of 3-4 times nuclear are reached, e.g., perhaps [MATH] g/cm . The innermost shell of matter reach... |
Initially the shock wave may carry an order of magnitude more energy than is needed to eject the mantle of the star (less than 10 51 |
ergs). But as the shock wave travels through the outer iron core, it heats and melts the iron that crosses the shock front, at a loss of [MATH] 8 MeV/nucleon. Additional energy is lost by neutrino emission, which increases after the melting. These losses are comparable to the initial energy carried by the shock wave. M... |
Most of the theoretical attention in the past decade has focused on the role of neutrinos in reviving this shock wave, a process that becomes more effective in multi-dimensional models that account for convection. In this delayed mechanism, the shock wave stalls at a radius of 200-300 km, some tens of milliseconds afte... |
Regardless of explosion details, neutrinos dominate supernova energetics. The kinetic energy of the explosion and supernova’s optical display account for less than 1% of the available energy. The remaining 99% of the 3 [MATH] ergs released in the collapse is radiated in neutrinos of all flavors. The timescale over whic... |
The burst of neutrinos produced in a galactic core-collapse supernova is detectable with instruments like Super-Kamiokande and SNO. On February 23, 1987, a neutrino burst from a supernova in the Large Magellanic Cloud was observed in the proton-decay detectors Kamiokande and IMB |
. The optical counterpart reached an apparent magnitude of about 3, and could be observed easily in the night sky with the naked eye. This supernova originated 160,000 light years from Earth. Approximately 20 events were seen in the Kamiokande and IBM detectors, spread over approximately 10 seconds. Within the limited ... |
Temperature differences between neutrino flavors are interesting because of oscillations and nucleosynthesis. The discussion of matter effects in the solar neutrino problem was limited to two flavors. But the higher densities found in core-collapse supernovae make all three flavors relevant. The three-flavor MSW level-... |
important for solar neutrinos, a second crossing of the [MATH] with the [MATH] . The higher density characterizing this second crossing, |
[MATH] g/cm , is determined by atmospheric mass difference [MATH] and by the typical energy of supernova neutrinos, [MATH] MeV. This density is beyond that available in the Sun ( [MATH] g/cm ), but far less than that of a supernova’s neutrinosphere. Consequently this second level crossing alters neutrino flavor only af... |
In fact this description oversimplifies the neutrino physics of supernovae. The enormous neutrino densities encountered in a supernova lead to a new aspect of the MSW effect – oscillations altered not by neutrino-electron scattering, but by neutrino-neutrino scattering |
. While the precise consequences of this neutrino-neutrino MSW potential are still being debated, the effects reach much deeper into the star and alter the flavor physics in distinctive ways. Supernovae likely provide the only environment in nature where neutrino-neutrino interactions dominate the MSW potential. |
4.2 Supernova Neutrino Physics This novel neutrino-neutrino MSW potential is one of many reasons core-collapse supernovae play an important role in neutrino astrophysics. Others include: |
The second level crossing involves the third, as yet unmeasured mixing angle [MATH] . This level-crossing occurs in the star’s mantle and remains adiabatic [MATH] |
the condition for flavor inversion [MATH] for mixing angles [MATH] This kind of sensitivity to small mixing angles is not achievable in the next generation of terrestrial neutrino experiments, which have goals of [MATH] . Thus supernovae may provide our only near-term hope of constraining this mixing angle, should it p... |
Neutrinos from galactic supernovae will not be obscured by intervening matter or dust, unlike optical signals. Thus supernova neutrino bursts should, over the next few hundred years, provide our most reliable measure of the contemporary rate of galactic core collapse. |
There exists a so-far undetected diffuse background of supernova neutrinos, produced by all past supernovae occurring in the universe. Future detectors that may approach the megaton scale should be able to see a few events from this source. Detection of these neutrinos would place a important constraint on the inventor... |
Supernovae are one of the most important engines for nucleosynthesis, controlling much of the chemical enrichment of the galaxy. As described in the next section, neutrinos are directly and indirectly involved in this synthesis. |
While most of the energy released in core collapse is radiated as neutrinos over the first several seconds, neutrino emission at a lower level continues as the proto-neutron star cools and radiates away its lepton number. It is quite possible that phase changes in the dense nuclear matter could occur several tens of se... |
[MATH] years. The neutrino burst could include other sharp features in time, marking interesting astrophysics. The melting of iron to nucleons with the passage of the shock wave through the outer iron core is predicted to produce a spike in the neutrino luminosity, lasting for a few milliseconds. Continued accretion on... |
Supernova cooling times place constraints on new physics associated with particles that also couple weakly to matter. For example, a light scalar called the axion could, in principle, compete with neutrinos in cooling a supernovae. The requirement that axion emission not shorting the cooling time too much, which would ... |
Neutrinos and Nucleosynthesis Neutrinos and nucleosynthesis are both associated with explosive environments found in astrophysics. This section discusses three examples, the Big Bang, the neutrino process, and the r-process. |
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