Solar Fusion & Neutrinos


An introduction to nuclear fusion in the sun, and the solar neutrino problem.

This page last updated October 11, 2005


Note added October 2005: In creating this page I have been helped much by the journal papers, and web resources of John Bachall. John passed away in August, 2005. At least for the time being, his webpage at the Institute for Advanced Study in Princeton, remains active, and I will keep those links here as long as I can. He still continues to teach.

By the late 19th century a conflict had arisen between geology and astronomy. Geologists had firmly determined that the Earth was extremely old, hundreds of millions, perhaps even billions of years old. But physicists studying the sun could think of no source for the sun's heat except for gravitational contraction, and that would not allow the sun to be so old (Thomson, 1862; Lane, 1870).

The mystery remained intact until Albert Einstein derived the surprising equivalence between mass and energy (E=mc2) as a curious side effect of special relativity theory. That, plus the discovery of radioactivity at about the same time, led to new ideas about the sun's heat. The well known American astronomer Henry Norris Russell speculated about the properties that the sun's mysterious heat source must have (Russell, 1919) and made several surprisingly accurate conclusions. But it appears to have been Sir Arthur Eddington, Einstein's English champion, who first noticed the mass deficit of helium as compared to hydrogen, and suggested the fusion of hydrogen into helium might be the ultimate source of the sun's power (Hale, 1920; Eddington, 1926).

It is Hans Bethe who usually gets credit for first quantifying the reality of nuclear fusion as the sun's energy source, beyond Eddington's level of speculation, in a one page letter to the editor of Physical Review (Bethe, 1939). This paper came on the heels of Chandrasekhar's book, which really established the full scale of mathematical physics in stellar structure and evolution (Chandrasekhar, 1939). But Bethe's work came too late, after the relevant chapters of Chandrsekhar's book were already written.

But while the basic energetics seemed to work, reaction rates and reaction channels remained unknown. The post WWII boom in nuclear physics, sparked by war research and the Manhatten Project, changed all of that. With the new knowledge firmly in hand, it could finally be determined, once and for all, if nuclear fusion would really generate the needed energy fast enough to serve as the energy source for stars. The answer was yes. Fred Hoyle first made the determination (Hoyle, 1954), but that paper was eclipsed in importance by a later paper in which he collaberated (Burbidge, Burbidge, Fowler & Hoyle, 1957). Martin Schwarzschild's book (Schwarzschild, 1958) was the first major update to the study extended by Chandrasekhar 20 years earlier, and the first book to include a complete treatment of nuclear fusion as the source of stellar energy.

All of this work established the theoretical reality of fusion powered stars, but only stellar surface could be observed, not stellar interiors. That has since changed, with the advent of both helioseismology and neutrino astronomy, both of which allow us to peer into the solar interior in real time. It is against this background of discovery that I will use this webpage to introduce the basics of stellar fusion, and the solar neutrino problem. This is not an exhaustive treatment, but should be accessible to non scientist readers, and the references will lead to treatments that should satisfy the most advanced readers.

Pre Main Sequence Collapse

The standard model of stellar birth has long maintained that stars are the end result of the gravitational collapse of large clouds of dust and gas. These clouds are well known to exist, and are indeed prevalent throughout this galaxy, and all other galaxies we can see in sufficient detail.

The exact mechanism of collapse and star formation remains the topic of considerable discussion amongst astrophysicists. However, that the overall model is correct was confirmed by a series of dramatic images returned by the Hubble Space Telescope. Images of objects called "proplyds" in the Orion Nebula (M42) show a bright protostar embedded in a disk of dust; the images show these objects both edge on and face on. In both cases the shape & form of the disk and protostar are as anticipated by theory. But even more dramatic images of a set of Herbig-Haro objects provide definitive evidence that stars form as expected. These images define the shape of the surrounding dust disk much more sharply. They also show the polar jets expected in the later stages of protostar formation. Protostars in the last stages before joining the main sequence are knwon as T-Tauri stars. I will not try to describe the details of how protostars form here. The fact that we have stars leaves little doubt that star formation occurs. The Hubble images emphasize that nature and theory conform as well as we can observe conformity, so we believe that the standard theory is valid.

The collapse of a great cloud into a star or a cluster of stars is in principle no mystery. The force that does it is gravity. The outer parts of the clouds fall inward as a result of their own weight. That weight is miniscule at first, but not zero. It happens, but it takes time.

As the protostar collapses, the interior of the collapsing cloud naturally heats up. As the temperature rises, the speed with which the particles that make up the cloud collide with each other also rises. As the collisions become more energetic, the atoms deep inside the cloud collide so strongly that the electrons are stripped away from all of the nuclei. The gas becomes a completely ionized plasma, a mix of free electrons and free atomic nuclei, the vast majority of which are hydrogen nuclei, which are protons. Eventually, the temperature gets high enough for the protons to stick together. That's when proton fusion begins, and that's when the collapsing cloud becomes a true star.

The Proton - Proton Cycle

A Note on the PP Tables

Tables 1-3 below outline the proton-proton fusion cycle as it occurs in all main sequence stars. Each table includes a time scale characteristic of the conditions found in our sun. Some time scales, like the decay of unstable boron and beryllium nuclei are independent of pressure and temperature, or only weakly dependent, and therefore essentially the same in all main sequence stars. But the initial proton-proton fusion reaction is very temperature dependent, and will vary strongly, being much shorter in hot high mass stars, and very much longer in cool low mass stars. The time scales represent how long an average particle will survive, in our sun, before experiencing the reaction. But of course many react much more quickly, many will last far longer, and many will never react at all.

For consistency, all time scales reported here are taken from Clayton (1968). The only reaction that is of uncertain time scale is the initial proton-proton fusion, which is too slow to measure in a laboratory. So the time scale is computed from basic theory. Hansen & Kawaler (1994) give the time scale 6,000,000,000 years, whereas Clayton gives 7,900,000,000. Bohme-Vitense (1992) gives 14,000,000,000 years, but for the lower temperature of 14,000,000 Kelvins. The one thing that is certain is that the reaction is slow.

Table 1
86% of 3He goes out this way
p + p --> d + e+ + nu 7.9 x 109 years
p + p + e- --> d + nu 1012 years
d + p --> 3He + g 1.4 sec
3He + 3He --> 4He + 2p 2.4 x 105 years


The first fusion reaction to occur in the collapsing cloud is deuteron-proton fusion, the third line in table 1. This happens because there is a small background abundance of deuterons in the cloud already. But the abundance is extremely small, and the energy released is insignificant. Once proton-proton (pp) fusion begins, enough heat is produced by the fusion to provide an outward pressure that stops the collapse. Since this is the first stable object produced, this is when a star is born.

Protons repel each other electrostatically because they both have a positive charge. But at extremely close range (like 10-15 meters) they will attract each other, because of the strong nuclear force (usually shortened to strong force). Because of this, the temperature has to get high enough to overcome that repulsion, and push the protons close enough together for the strong force to take over. Once that happens, the strong force overpowers the electrostatic repulsion, and pulls the two protons together, initiating proton-proton fusion. This requires temperatures in excess of 10,000,000 Kelvins. When that happens, one of the protons changes into a neutron, and the resultant nucleus of one proton and one neutron is called a deuteron ("d"). The excess positive charge is expelled in the form of a positron (e+). That positron rapidly runs into an electron (e-), and the two anhilliate each other into gamma rays (the positron is the anti-matter analog for an electron). The excess momentum is carried away by a neutrino ("nu").

The second line in table 1 illustrates the pep or proton-electron-proton chain. It has a characteristic time scale rather larger than the age of the universe at this time, and so is insignificant in the sun as far as energy generation is concerned. Nevertheless, the pep fusion accounts for about 0.25% of the deuterons created in the PPI chain. Enough pep fusions happen to produce a detectable number of neutrinos, so the reaction must be accounted for by those interested in the solar neutrino problem (discussed below).

Even at temperatures in the sun's core, 15,000,000 Kelvins or 27,000,000 Fahrenheit, the average lifetime of a proton against pp fusion is about 8,000,000,000 years. It is an extremely slow reaction, and it is this time scale that sets the stellar clock, so to speak, by determining how long the star will remain a stable main sequence object. In contrast, the deuteron created will only last about 1.3 seconds before it plows into another proton and fusion creates a helium-3 (3He) nucleus (2 protons and 1 neutron) and a gamma ray ("g") as the 3He nucleus relaxes to the ground state. That 3He nucleus will last about 250,000 years before it hits another 3He nucleus hard enough for the two to stick together. But the momentary conglomeration of 4 protons and 2 neutrons is a highly unstable 6Be nucleus that falls apart at once, tossing away two protons, and leaving behind a stable 4He nucleus. This process illustrated in table 1 is the main route for creating helium from hydrogen by nuclear fusion.

The fusion process generates a great deal more energy than does gravitational collapse. Once fusion starts, gravitational collapse stops. This happens because the outward flow of fusion energy creates an outward pressure that balances the inward pressure of gravitational collapse. The central temperature of the resulting stable star is a function of the star's mass. Heavier stars have a stronger gravity, and will compress more before they reach a stable equilbrium with the outward pressure of fusion energy. The rate of pp fusion is roughly proportional to T4, where T is the temperature. So double the temperature and the rate of fusion goes up 16 times. For this reason, our sun will sit stable on the main sequence for about 10,000,000,000 years, but a star only 5 times more massive might be expected to remain stable for less than 100,000,000 years. But the smallest mass main sequence stars, only about 0.08 solar masses, are barely able to reach fusion temperatures, and so can remain stable for as long as 100,000,000,000,000 (that's a hundred trillion) years.

The sun, and any other main sequence star, will remain relatively stable until the hydrogen in its core is used up. Stars, like anything else, eventually just run out of gas.

Table 2
14% of 3He goes out this way.
99.89% of the 7Be goes the 7Li route
3He + 4He --> 7Be + g 9.7 x 105 years
7Be + e- --> 7Li + nu + g 142 days
7Li + p --> 4He + 4He 9.5 minutes


But the 3He nucleus does not always have to hit another 3He nucleus. it could hit an 4He nucleus. This is illustrated above, in table 2. Fusion with 4He is less likely, because there is more 3He around deep inside the stellar core. But in heavier stars, where the temperatures exceed about 24,000,000 Kelvins, the PPII chain can rival the PPI chain for energy production inside the star. This is because at higher temperatures the 3He gets used up faster, driving down its abundance compared to 4He. In our sun, about 14% of the 3He avoids the PPI chain, and shows up here in PPII.

In this case, instead of forming unstable 6Be, the fusion with 4He forms stable 7Be. But 7Be has an affinity for electron capture, and can absorb free electrons, as in the 2nd line of table 2. The electron turns one of the Be protons into a neutron, changing the 7Be into 7Li, while tossing out a neutrino and a gamma ray photon. The lithium nucleus will then quickly fuse with a free proton, resulting in unstable 8Be which immediately falls apart into two stable 4He nuclei.

Table 3
0.11% of the 7Be goes this route
7Be + p --> 8B + g 66 years
8B --> 8Be + e+ + nu 0.9 sec
8Be --> 4He + 4He 9.7 x 10-17 sec


The last part of the proton-proton chain, PPIII, is illustrated in table 3. The 7Be has two ways to go - it can absorb an electron, as in PPII (99.89%), or it can absorb a proton, as in PPIII (0.11%). Absorbing a proton raises the nucleus from beryllium to boron, and the 7Be becomes 8B. But 8B is unstable and takes about 0.8 seconds, fairly independent of temperature, to spit out a positron (e+) and a neutrino to become beryllium again, only this time it's 8Be. But 8Be falls apart in a hurry into two 4He nuclei, and once again we have turned hydrogen into helium.

All three of the reaction chains are active in the sun, but PPI dominates. In low mass stars the internal temperature is not high enough to finish the cycle. They produce the first stage of pp fusion up to 3He, but are unable to force the last stage of 3He fusion, either with another 3He or an 4He. So they fuse hydrogen into 3He instead of 4He. This fact is confirmed by the observation that low mass stars are often anomalously rich in 3He compared to 4He.

The CNO Bi-Cycle

The material that formed the sun had already been cycled through one or more generations of stars before becoming our sun. We know that since we can see elements up to, and beyond 56Fe in the solar atmosphere, yet the pp fusion chain can't get past 4He. The presence of impurities in the solar core opens the door to another fusion cycle, which is responsible for as much as 2% of the sun's total output. The CNO cycle in illustrated in Table 4 below.

CNO Bi-Cycle I
Table 4
The Carbon-Nitrogen-Oxygen (CNO) main cycle
Time scales from Böhm-Vitense 1992
12C + p --> 13N + g 106 years
13N --> 13C + e+ + nu 14 minutes
13C + p --> 14N + g 3 x 105 years
14N + p --> 15O + g 3 x 108 years
15O --> 15N + e+ + nu 82 seconds
15N + p --> 12C + 4He 104 years

This is the main CNO cycle. It uses carbon, nitrogen and oxygen as catalysts to suck up four protons and build an 4He nucleus out of them. The relative abundances of C, N and O do not change. The cycle does not start until pp fusion has begun, and provides the energy necessary to allow a low level of proton fusions onto the heavier nuclei. The CNO cycle lacks significance at the low temperatures in the sun. But in the later stages of the solar lifetime, when the temperature exceed 108 Kelvins, and helium fusion into carbon begins (the triple alpha process), then the CNO cycle will also increase in importance as an energy generator.

There is a small chance (0.0004) that instead of 15N + p producing 12C + 4He, it will produce a single 16O. That will set off the second part of the CNO bi-cycle. Unlike the main cycle, this branch of the CNO bi-cycle does change the abundance ratios of 16O and 17O; they are both stable isotopes, and not every single nucleus is going to wind up fusing with a proton, so the net effect is to increase the abundances. Part II of the CNO bi-cycle is shown below in table 5.

CNO Bi-Cycle II
Table 5
The Carbon-Nitrogen-Oxygen (CNO) secondary cycle
No time scales are available
15N + p --> 16O + g ??
16O + p --> 17F + g ??
17F --> 17O + e+ + g ??
17O + p --> 14N + 4He ??

This completes the picture of nuclear fusion in the sun. As the sun ages it will gain a hotter core, and eventually begin fusing helium into carbon, once the temperature reaches about 108 Kelvins. At those higher temperatures, the CNO cycle will also become more important. The sun is not massive enough to go beyond the stage of fusing helium into carbon (and a little oxygen perhaps). It will eventually pass through the stage of Red Giant, followed by planetary nebula, and quietly fade away as a carbon white dwarf. More massive stars will, depending on their mass, fuse heavier nuclei, all the way to iron, which requires central temperatures as high as 3,000,000,000 Kelvins. The most massive stars, those with masses in excess of about 8 to 10 solar masses, will be unable to produce a stable core at all in the normal course of the fusion cycles. Those stars will end as supernova explosions, leaving behind a neutron star or black hole.

The Solar Neutrino Problem

Figure 1
Updated 4/2/2004
This figure comes from a collection of neutrino vuegraphs maintained by John Bahcall and illustrates the solar neutrino problem. See accompanying text for details.

What's the Problem?

It has been known for a long time that only nuclear fusion will allow for the sun as a stable star for as long as the earth has been around. But scientists are fussy about details, and have long sought a way to prove it. So off they went looking for the elusive neutrinos. If the sun is powered by fusion, it has to make neutrinos, which will streak right out of the solar core, at the speed of light (or close enough), to terrestrial neutrino detectors. Furthermore, if the sun does make neutrinos, fusion is the only way. So the very existence of neutrinos proves that fusion is happening. And, furthermore, the enrgies of the detected neutrinos matches the energy predicted by theory for solar nuclear reactions. However, the total number of neutrinos does not, and therein lies the solar neutrino problem.

The solar neutrino problem is usually cast into a statement like "the sun does not produce enough neutrinos", and is left at that. But reality is a little bit more complicated, and Bahcall (one of the world's leading neutrino astrophysicists) points out that figure 1 above illustrates not just one "neutrino problem", but three "neutrino problems".

The first experimental detection of neutrinos was announced in 1968 (See Haxton, 1995). Those results indicated an upper bound of about 3 SNU (Solar Neutrino Unit; 1 SNU = 10-36 captures per target atom per second). The same year it was determined that theory predicted 7.5 ± 3.3 SNU. This is illustrated by the first set of vertical bars (on the left) in figure 1 above, with numbers updated by Bahcall to reflect more accurate observations and calculations. The theoretically predicted count of about 7.7 SNU is about a factor of 3 greater than the observed value of roughly 2.6 SNU. This is the first neutrino problem, and is what most people are talking about (whether they know it or not) when they talk about the neutrino problem being that "the sun does not produce enough neutrinos".

The second neutrino problem is illustrated by the center set in figure 1, compared with the chlorine detector results already mentioned, on the left. We now know that, if the energy spectrum of the neutrinos is not altered from standard theory, then the experimental results are impossible. The Kamiokande water detector sees more neutrinos than the chlorine detector, but it sees too many more. Since the detectors have different threshold energies, this means that the distribution of neutrinos by number as a function of energy is not in fact as standard theory says it should be.

The third neutrino problem is illustrated by the third set in figure 1. The total number of neutrinos detected by the gallium experiments is accounted for by the pp and pep neutrinos, implying that there are no 7Be neutrinos at all. But the detectors are calibrated against a known reactor source that produces similar energy neutrinos, so it is not a case of the detector just missing them, it's a case of they really don't appear to be there.

Paragraph added 2/4/2004: This updaterd figure shows two mode vertical bar plots, illiustrating the SNO results discussed below. The 4th bar shows that if one counts only electron neutrinos, then the detector sees only about 1/3 of the expected total. But, as the 5th bar shows, if we add up all of the electron and non-electron neutrinos, the total equals the total of electron neutrinos that the sun should produce (it should produce only electron neutrinos). This result supports the hypothesis that neutrinos "oscillate" from one type top another.

So, first & foremost, we see that there is not just a neutrino problem, but rather more then one. And it's not just that the count is low, which is all the first experiments could tell, but that the energy spectrum is distorted, and that some kinds of neutrino (namely the 7Be kind) seem to be not there at all.

What's the Answer?

The existence of the solar neutrino problem raises three questions:

1) Is there something wrong with standard solar models?
2) Is there something wrong with standard neutrino physics?
3) Is there something wrong with standard nuclear fusion physics?

It's easy to answer the 3rd question in the negative. The physics of nuclear fusion was intensively studied by the military, leading to the development of hydrogen fusion weapons. It has been deeply studied in laboratories world wide. Indeed, the fusion reactions that typically occur in stars, both the PP and CNO cycles, and more, are all low energy reactions, and easily duplicated in a modest accelerator facility. Experience is already well beyond the level of quality required to show that there are no such large errors in the parameters of standard nuclear fusion physics.

So for many years the fight has been between the solar models (the astrophysical solution), and neutrino physics (the particle physics solution). There are a number of "tweakable" parameters in solar models, including relative abundances of helium and heavier elements, the effect of diffusion and convective mixing, electron screening of nuclear fusion, and more. They are all "tweakable" because, like any measured or derived quantity, they all carry intrinsic uncertainties along for the ride. As long as the parameters are "tweaked" within the limits set by the known uncertainties, the models are still unchanged. That process has been carried out extensively, with the result being that no allowed amount of "tweaking" can reproduce the observed neutrino results. The astrophysical solution does not work.

At this point it's worth noting that our models of the solar interior are not just wild guesses, but are based on pretty basic ideas in physics. There are a lot of particular complications, but the basic ideas aren't all that weird. If you have a big fat thing like the sun, it has to be really hot inside just because its own weight will compress and heat the interior. And if it gets hot enough there has to be nuclear fusion (a fact dramatically demonstrated by the existence of hydrogen bombs). So the basic picture is pretty obvious. But the physical conditions (mostly pressure, temperature and convective flow) at any specific place in the star have to be computed with attention to much detail. Still, wouldn't it be nice to know that we did it right? Alas, a spacecraft probe will not penetrate the sun, even if it could survive the approach. But there is another way.

That way is helioseismology. Light photons take about 1,000,000 years or more to make the journey from the center of the sun to the surface, but sound waves can make the trip in a matter of hours. Helioseismology is the study of the jumble of sound waves that play around on the surface of the sun. Each frequency of vibration is sensitive to a different layer of the sun. Combining the whole collection allows us to determine the physical state of the whole interior, and test by these observations whether or not our models are right. The result is that they were actually pretty good to start with, and now they are right on the money. We now know what the solar interior looks like, certainly well enough to know that fusion has got to happen, and well enough to calculate the rate at which it should happen.

Of course, we can imagine a more radical form of astrophysical solution, such as abandoning the idea of fusion in the solar core altogether. But that causes more problems than it solves. If not fusion, then what is the sun's internal power source? And if there is no fusion in the solar core, why does it make any neutrinos at all? If there were no fusion, there would be no neutrinos; if the solar neutrino problem had been "no neutrinos at all", then we might have gone that route. But "not enough neutrinos" is a worse problem than 'no neutrinos", because the proposed solutions are nowhere near as obvious.

So it would appear that the only candidate for a solution that remains is the particle physics solution, the physics of the neutrino. But what does the solution look like?

The MSW Effect

If you have done any reading on neutrino oscillations already, then you have probably run across the expression "the MSW Effect". MSW stands for "Mikheyev-Smirnov-Wolfenstein", and the MSW effect, also known as "matter induced oscillations", is based on an idea first voiced by Wolfenstein in 1979. The basic idea of non-MSW oscillations is that a neutrino has a quantum mechanical description (called an Eigenstate) that is actually a mixture of two different kinds. So, there is always a small probability that a neutrino might spontaneously change ("oscillate") from one kind to the other. In a vacuum the probability is too small to reasonably explain the solar neutrino deficit. But, it turns out that in the presence of high density of matter (which the solar interior has in abundance), then the probability of oscillation is strongly ehnanced, almost to a certainty. A detailed explanation beyond this is well beyond the scope of this webpage, but there are good descriptions of the MSW mechanism, and all of the attendent quantum mechanics, in Haxton (2000) and Haxton & Holstein (2000); the more you already know about quantum mechanics, the easier it will be to follow. But this explanation shows that the neutrino actually makes its change, from one kind to another, well inside the sun, and not while on its journey between the sun & Earth.

Problem Solved?

The search for solar neutrinos was motivated by a desire to detect neutrinos and thereby prove that fusion powered the sun as expected. But the results went beyond expectations. Not only was solar fusion confirmed, but the peculiar and unexpected shortfall in beryllium & boron neutrinos has led to an unanticipated new discovery in particle physics: neutrinos oscillate. That means they change from one kind of neutrino into another on their way here from the sun. If that conclusion survives further scrutiny, as seems likely, it leads to the corollary necessity that neutrinos have a rest mass. What started out as an innocent attempt to confirm solar suspicions has ended up radically altering our view of particle physics and cosmology, because a neutrino with mass adds to the mass density of the universe, and that plays a role in determining the long term fate of the expanding universe.

There is now enough experimental & theoretical evidence abroad, that the particle physics solution is no longer questioned. Evidence has come forth for the oscillation of atmospheric neutrinos (Fukuda et al., 1998b, Hatakeyama et al., 1998), and for neutrinos produced by a test nuclear reactor (Athanassopoulos et al., 1998). Subsequent investigations have only strengthened these early results (Maris & Petcov, 2000; Haxton, 2000; Haxton & Holstein, 2000). Both theory & experiment agree that neutrinos must not behave as theory had previously considered, and the new theory has already reached the level where reliable calculations support the observed asymmetries in neutrino distributions. At this poitn, it seems a fait accompli that the solar neutrino problem has been solved, by the realization of new principles of neutrino physics.

There was already quite a bit of convincing evidence that neutrino oscillations were the key to solving the solar neutrino problem, when in June 2001, news from the Sudbury Neutrino Observatory added the coup de grace. Such are the nuclear reactions inside the sun, that they generate only one kind of neutrino: electron neutrinos. At Sudbury, they were able to more accurately measure the electron neutrino flux than has yet been done. And as before, they produce an undercount of neutrinos with respect to expectations of theory. However, by combining those results with results from other neutrino observatories (notably Super Kamiokande), which are (potentially) "contaminated" by other types of neutrinos, they are able to determine the level of "contamination". By combining the observed flux of electron neutrinos, with the observed non-electron neutrinos, the total is the theoretically expected flux of electron neutrinos from the sun. This is the first model independent indication of neutrino oscillation. If the result stands up to further inquiry, as seems likely, then at this point we can say with some conviction that the solar neutrino problem has been solved. Solar electron neutrinos are "oscillating" into tau or muon neutrinos, on the trip from the sun to the Earth.

I will finish off with a paragraph lifted from the Sudbury press release, dated June 18, 2001.

"We now have high confidence that the discrepancy is not caused by problems with the models of the Sun but by changes in the neutrinos themselves as they travel from the core of the Sun to the earth," says Dr. Art McDonald, SNO Project Director and Professor of Physics at Queen's University in Kingston, Ontario. "Earlier measurements had been unable to provide definitive results showing that this transformation from solar electron neutrinos to other types occurs. The new results from SNO, combined with previous work, now reveal this transformation clearly, and show that the total number of electron neutrinos produced in the Sun are just as predicted by detailed solar models."

Note added July 2002

The Sudbury results referred to above, were published in August 2001 ("Measurement of the Rate of nu(e) + d -> p + p + e Interactions Produced by 8B Solar Neutrinos at the Sudbury Neutrino Observatory", Q.R. Ahmad et al. (SNO Collaboration), Physical Review Letters 87(7): 071301, August 13, 2001). The only missing piece was the neutral current (NC) experiment results.

Those results have been published in July 2002: "Direct Evidence for Neutrino Flavor Transformation from Neutral-Current Interactions in the Sudbury Neutrino Observatory", Q.R. Ahmad et al. (SNO Collaboration), Physical Review Letters 89(1): 011301, July 1, 2002; "Measurement of Day and Night Neutrino Energy Spectra at SNO and Constraints on Neutrino Mixing Parameters", Q.R. Ahmad et al. (SNO Collaboration), Physical Review Letters 89(1): 011302, July 1, 2002.

SNO is the first site to be able to do the NC experiment, which is equally sensitive to all 3 neutrino "flavors". Their NC results are 5.09 +0.44,-0.43(stat) +0.46,-0.43(syst), in units of 106 cm-2sec-1. The total flux determined theoretically from standard solar models is 5.05 +1.01,-0.8 in the same units. So the SNO experiment sees a total number of neutrinos that agrees with the expectations of standard models. The charged current (CC) experiment, which sees only electron neutrinos, sees a flux of 1.76 ± 0.05(stat) ± 0.09(syst) in the same units.

The obvious implication is that electron neutrinos, created inside the sun, are transforming, on their way to Earth, into neutrinos of another type. The total number of neutrinos does not change, and that total is seen directly by the SNO NC experiment, and the deficit shows up in the CC data.

So in July, 2002, it seems even more apparent than it was a years ago, that the solar neutrino problem has now been solved. Neutrinos have a small mass, and do "oscillate" from one type to another.

Note added April 2004

There are two mode encouraging bits of news to support the theory of neutrino oscillation to report.

First, there is now direct, earthbound experience, that supports the particle physics theory of neutrino oscillation. The KAMLAND experiment has detected a fall-off in electron neutrino counts, as a function of the distance from the reactor that produces them. That fall off is consistent with the "large mixing angle" solution in neutrino oscillation equations. So we now have both theoretical and laboratory results in hand, supporting the theory of solar neutrino oscillation.

Second, the evidence is now strong, that the neutrinos observed really are solar neutrinos, and not from some outside source, that fortuitously adds up to the expected number. The "day night asymmetry" correlates neutrino arrival direction with the angular position of the sun. There is also a seasonal signal in the neutrino counts, correlated with sun-earth distance (which changes because the earth has a slightly eccentric orbit around the sun). See Precise measurement of the solar neutrino day-night and seasonal variation in Super-Kamiokande-I, Physical Review D 69(1): 011104(R), January 2004.


By no means intended to be an exhaustive list. Includes the sources I used to construct this article, and more. I have tried to include enough for everybody, both general sources for the "lay reader", as well as technical sources for buffed-out physics majors! I have also tried too concentrate on web based resources where possible.

General references on the sun, solar structure & evolution

  • Our Sun and Stellar Structure.
    A chapter out of Astronomy Notes, an online astronomy text by Nick Strobel, professor of physics at Bakersfield College in California.

  • The Formation of the Elements
  • Neutrino Astrophysics
    Both hosted by the CMB Astrophysics Research Program at the Lawrence-Berkeley National Laboratory. Element Formation includes a page on the formation of elements in the Big Bang, and another on stellar nucleosynthesis. A good introduction to the nuclear activity inside stars. Neutrino Astrophysics discusses current projects in using neutrino detection as a general tool in astrophysics, not just solar neutrinos.

  • Yokoh Public Outreach Page.
    Courtesy of the Lockheed-Martin Solar and Astrophysics Lab. A host of information, images, and learning activities about the sun. Includes a detailed look inside the sun with the Structure of the Sun Tour.

  • Solar Data Analysis Center.
    NASA's all purpose sun page. Links to all of the active NASA solar missions, and a host of solar resources on the web.

  • Solar FAQ - with solar neutrinos and other solar oddities
    Primarily a solar neutrino FAQ. But other topics are adressed.

  • The Neutrino Oscillation Industry
    From the High Energy Physics Group at Argonne National Laboratory. A host of links to experiments the world over studying neutrino oscillations. Not all of these are strictly technical, so nontechnical types might benefit from poking around the various webpages.

  • ABC's of Nuclear Science, an introduction to nuclear physics from the Lawrence Berkeley National Laboratory.

  • Reaction Rates for Stellar Nucleosynthesis - For science & technical users. Includes data pages, and discussion of reaction rates as modeled by nuclear physics theory.

  • An Educational Tour of Nuclear Data - From the Nuclear Information Service group at Los Alamos National Laboratory. Also includes a webpage on Helium Burning in Stars.

  • Nuclear Data for Nuclear Astrophysics - From the Astrophysics Program at Oak Ridge National Laboratory.

    Scientific & technical references on the sun, solar models & neutrinos
  • Bhatnagar, Arvind & Livingston, William
    Fundamentals of Solar Astronomy
    World Scientific, Singapore, 2005. A less technical book, with only some basic algebra. Topics include ancient & modern observational methods & observatories. The quiet sun, active sun, eclipses, and helioseismology are also included.

  • Böhme-Vitense, Erika
    Introduction to Stellar Astrophysics
    Cambridge University Press, 1989, 1992
    Vol 1: Basic stellar observations and data, 1989
    Vol 2: Stellar Atmospheres, 1989
    Vol 3: Stellar structure and evolution, 1992
    Excellent and not outdated overview of the field of stellar astrophysics.

  • Chandrasekhar, S. An Introduction to the Study of Stellar Structure
    University of Chicago Press, 1939. This book is still available as a Dover reprint, and is quite characteristic of Chandrasekhar's detailed and exhaustive approach to all of the topics he chose to write about.

  • Clayton, Donald D.
    Principles of Stellar Evolution and Nucleosynthesis
    McGraw-Hill, 1968.
    Still recognized as the standard introduction to stellar nucleosynthesis.

  • Dwivedi, Bhola N. (editor)
    Dynamic Sun
    Cambridge University Press, 2003. Chapters written by different authors. Deals more with flares, prominences, CMEs, physics of the chromosphere & corona. Also includes helioseismology & solar oscillations.

  • Eddington, Sir Arthur
    The Internal Constitution of the Stars
    A technical book, first published in 1926 and revised in 1930. Mine is a Dover reprint of the 1959 edition edited by Loyd Motz. See chapter XI, The Source of Stellar Energy.

  • Foukal, Peter V.
    Solar Astrophysics
    Wiley-VCH, 2004 (2nd, revised edition). A complete study of solar physics. Topics include radiative transfer, spectroscopy, solar plasmas, internal structure, energy generation, prominences, flares, solar variability & etc.

  • Green, Simon F. & Jones, Mark H.
    An Introduction to the Sun and Stars
    Cambridge University Press, 2003 & 2004. Less technical, only some basic algebra. Lavishly illustrated. A good general introduction to the sun, and stellar evolution.

  • Hale, George Ellery
    The New Heavens
    A non-technical book for the general reader; first published in 1920, mine is a 1922 second edition. For reference to Eddington & fusion see pages 80-82.

  • Hansen, Carl J. & Steven D. Kawaler
    Stellar Interiors - Physical Principles, Structure and Evolution
    Springer, 1994.
    Advanced reading for those who can't get enough. All about what stars do and why they do it. Intended for university level astrophysics types.

  • Mestel, Leon
    Stellar Magnetism
    Oxford University Press, 2003. As the title implies, it's all about magnetism. Includes dynamos, magnetic fields & convection, magnetohydrodynamics, magnetic braking. Also includes pulsar electrodynamics, and star formation.

  • Schwarzschild, Martin
    Structure and Evolution of the Stars
    Princeton University Press, 1958. This book is still available as a Dover reprint, of which my copy is one. Chandrasekhar's 1939 book concentrates on stellar structure, but has little to say about stellar evolution. Schwarzschild covers nuclear reactions in chapter II, section 10, but goes deeply into stellar evolution as well. So far as I know, this is the first book length treatment of stellar evolution in the "modern" sense. This is not surprising, as stellar evolution could hardly be characterized without first knowing the reaction rates, temperature sensitivity, threshold energies, & etc.

  • Thompson, Michael J. & Christensen-Dalsgaard, Jorgen
    Stellar Astrophysical Fluid Dynamics
    Cambridge University Press, 2005. Highly mathematical, this book is primarily devoted to helioseismology and asteroseismology, and other fluid dynamical topics, such as magnetic field generation, rotation and convection.

  • Zirker, Jack B.
    Journey from the Center of the Sun
    Princeton University Press, 2002. Not exactly non-technical, but certainly non-mathematical. An ideal book for beginners, or more general readers, without the background in physics and mathematics that other books assume. Written in the style of a journey outwards, starting at the center of the sun.

  • Adelberger, Eric G. et al.
    Solar fusion cross sections
    Reviews of Modern Physics 70(4): 1265-1291 (October 1998)
    A review paper which outlines the current state of theory and experiment on solar nuclear fusion reaction rates. 207 References.

  • Athanassopoulos, C. et al.
    Evidence for nu_mu -> nu_e Neutrino Oscillations from LSND
    Physical Review Letters 81(9): 1774-1777 (31 August 1998)

  • Bahcall, John N.
    How the Sun Shines
    Published by the Nobel E-museum (where you can read the article online). An excellent review for general readers that covers the basic physics, as well as the history (a more extended discussion than my introduction).

  • Bahcall, John N., M.H. Pinsonneault & Sarbani Basu
    Solar Models: current epoch and time dependences, neutrinos, and helioseismological properties
    An excellent & thorough, though technical review of the current state of the art in solar interior modeling, including constraints from helioseismology. Published before the Sudbury neutrino results tended to confirm the reality of neutrino oscillation, which results do not adversely affect this study.

  • Bethe, Hans
    Energy Production in Stars
    Physical Review, v55 p103(L), January 1, 1939.
    A one page letter that outlines the basic nuclear physics, and suggests both the CNO cycles and the triple-alpha process, for synthesizing heavier elements.

  • Brun, A.S.; S. Turck-Chièze, P. Morel
    Standard Solar Models in Light of New Helioseismic Constraints I: The Solar Core
    Astrophysical Journal 506: 913-925 (1998 October 20)

  • Burbidge, Burbidge, Fowler & Hoyle
    Synthesis of the Elements in Stars
    Reviews of Modern Physics 29, 548 (1957)

  • Fukuda, Y. et al. (1998a)
    Measurements of the Solar Neutrino Flux from Super-Kamiokande's First 300 Days
    Physical Review Letters 81(6): 1158-1162 (10 August 1998)

  • Fukuda, Y. et al. (1998b)
    Evidence for Oscillation of Atmospheric Neutrinos
    Physical Review Letters 81(8): 1158-1162 (24 August 1998)

  • Gruzinov, Andrei V. & John N. Bahcall
    Screening in Thermonuclear Reaction Rates in the Sun
    Astrophysical Journal 504: 996-1001 (1998 September 10)

  • Haxton, W.C.
    The Solar Neutrino Problem
    Annual Review of Astronomy and Astrophysics 33: 459-503 (1995)

  • Haxton, W.C.
    Neutrino Oscillations and the Solar Neutrino Problem
    April 28, 2000; Los Alamos Preprint Archive.

  • Haxton, W.C. & Holstein, B.R.
    Neutrino Physics
    American Journal of Physics 68(1): 15-32 (January 2000)

  • Hatekayama, S. et al.
    Measurement of the Flux and Zenith-Angle Dependence of Upward Through-Going Muons in Kamiokande II + III
    Physical Review Letters 81(10): 2016-2019 (7 September 1998)

  • Hoyle, F.
    On nuclear reactions occuring in very hot stars. I. The synthesis of elements from carbon to nickel
    Astrophysical Journal Supplement, 1, 121 (1954)

  • Lane, J. Homer
    On the Theoretical Temperature of the Sun, under the Hypothesis of a Gaseous Mass maintaining its Volume by its Internal Heat, and depending on the Laws of Gases as known to Terrestrial Experiment
    American Journal of Sciences and Arts, v2 n4 p57

  • Maris, M. & S.T. Petcov
    The Super-Kamiokande Day-Night Effect Data and the MSW Solutions of the Solar Neutrino Problem
    20 May 2000; Los Alamos Preprint Archive.

  • Russell, H.N.
    On the Sources of Stellar Energy
    Publications of the Astronomical Society of the Pacific, August 1919

  • Schou, J. et al.
    Helioseismic Studies of Differential Rotation in the Solar Envelope by the Solar Oscillations Investigation Using the Michelson Doppler Imager
    Astrophysical Journal 505: 390-417 (1998 September 20)

  • Takata, Masao & Hiromoto Shibahashi
    Solar Models Based on Helioseismology and the Solar Neutrino Problem
    Astrophysical Journal 504: 1035-1050 (1998 September 10)

  • Thomson, W. (Lord Kelvin)
    On the Age of the Sun's Heat
    Macmillan Magazine, March 5, 1862, pp288-293

  • Turcotte, S. et al.
    Consistent Solar Evolution Model Including Diffusion and Radiative Acceleration Effects
    Astrophysical Journal 504: 539-558 (1998 September 1)

  • John Bahcall's Home Page
    Bahcall is one of the most prolific and authoritative researchers on the solar neutrino problem. You can download copies of his papers, and specially made vuegraph sets. A lot of technical discussion is digested here too.

  • The Ultimate Neutrino Page
    Extensive collection of papers, articles, and hyperlinks. Solar neutrinos, atmospheric neutrinos, neutrino oscillations, it's all here, and a lot more.

    The End

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