This chapter was copied with permission from Nick Strobel’s Astronomy Notes. Go to his site at www.astronomynotes.com for the updated and corrected version. This chapter has been edited for content. Paragraphs that have been altered from the original text are denoted in green font.

The Sun and Stellar Structure

This chapter covers: The Sun, interiors of stars, and nuclear fusion, neutrinos, the solar neutrino problem, and helioseismology. The concept of hydrostatic equilibrium is used to explain the mass-luminosity relation and the reason for the mass cut-off at the high and low ends. Brown dwarf discussion expanded.

12.1 Introduction

Much of what is known about the stars comes from studying the star closest to us, the Sun. At a distance of almost 150 million kilometers, the Sun is a few hundred thousand times closer to us than the next nearest star. Because of its proximity, astronomers are able to study our star in much, much greater detail than they can the other stars.

The Sun is a G2-type main sequence star that has been shining for almost 5 billion years. It is known from radioactive dating of the Earth, Moon, and meteorites, that these objects have been around for about that length of time and temperatures on the surface of the Earth have been pleasant since it formed. The Sun’s energy has made this possible. What could power something as big as the Sun for so long? The process called nuclear fusion is now known to be the source of the Sun’s enormous energy, as well as that of other stars. This is a relatively recent discovery. However, using simple physical principles of gas physics, astronomers knew about the density and temperature structure of the interior of the stars long before they unlocked the secret to what could power them for so long. This chapter will cover these topics. I will first give a brief description of the Sun to give you an idea of what a star is like and then go into the basic principles of what the interiors of stars are like and what powers them. The vocabulary terms are in boldface.

12.2 The Sun–The Closest Star

The Sun and planets are shown to the same scale. The small terrestrial planets and tiny Pluto are in the box—the Earth is the blue dot near the center of the box (montage created by Nick Strobel using NASA images).

12.2.1 Size

The Sun is by far the biggest thing in the solar system. From its angular size of about 0.5° and its distance of almost 150 million kilometers, its diameter is determined to be 1,392,000 kilometers. This is equal to 109 Earth diameters and almost 10 times the size of the largest planet, Jupiter. All of the planets orbit the Sun because of its enormous gravity. It has about 333,000 times the Earth’s mass and is over 1,000 times as massive as Jupiter. It has so much mass that it is able to produce its own light. This feature is what distinguishes stars from planets.

12.2.2 Composition

What is the Sun made of? Spectroscopy shows that hydrogen makes up about 94% of the solar material, helium makes up about 6% of the Sun, and all the other elements make up just 0.13% (with oxygen, carbon, and nitrogen the three most abundant “metals”—they make up 0.11%). In astronomy, any atom heavier than helium is called a “metal” atom. The Sun also has traces of neon, sodium, magnesium, aluminum, silicon, phosphorus, sulfur, potassium, and iron. The percentages quoted here are by the relative number of atoms. If you use the percentage by mass, you find that hydrogen makes up 78.5% of the Sun’s mass, helium 19.7%, oxygen 0.86%, carbon 0.4%, iron 0.14%, and the other elements are 0.54%.

12.2.3 The Sun’s Interior

Here are the parts of the Sun starting from the center and moving outward.

Core

The core is the innermost 10% of the Sun’s mass. It is where the energy from nuclear fusion is generated. Because of the enormous amount of gravity compression from all of the layers above it, the core is very hot and dense. Nuclear fusion requires extremely high temperatures and densities. The Sun’s core is about 16 million K and has a density around 160 times the density of water. This is over 20 times denser than the dense metal iron which has a density of “only” 7 times that of water. However, the Sun’s interior is still gaseous all the way to the very center because of the extreme temperatures. There is no molten rock like that found in the interior of the Earth.

Radiative Zone

The radiative zone is where the energy is transported from the superhot interior to the colder outer layers by photons. Technically, this also includes the core. The radiative zone includes the inner approximately 85% of the Sun’s radius.

Convective Zone

Energy in the outer 15% of the Sun’s radius is transported by the bulk motions of gas in a process called convection. At cooler temperatures, more ions are able to block the outward flow of photon radiation more effectively, so nature kicks in convection to help the transport of energy from the very hot interior to the cold space. This part of the Sun just below the surface is called the convection zone.

12.2.4 The Sun’s Surface

The deepest layer of the Sun you can see is the photosphere. The word “photosphere” means “light sphere.” It is called the “surface” of the Sun because at the top of it, the photons are finally able to escape to space. The photosphere is about 500 kilometers thick. Remember that the Sun is totally gaseous, so the surface is not something you could land or float on. It is a dense enough gas that you cannot see through it. It emits a continuous spectrum. Several methods of measuring the temperature all determine that the Sun’s photosphere has a temperature of about 5840 K.

Measuring the Sun’s Temperature

One method, called Wien’s law, uses the wavelength of the peak emission, λ_peak, in the Sun’s continuous spectrum. The temperature in Kelvin = 2.9 × 10⁶ nanometers/λ_peak.

Another method uses the flux of energy reaching the Earth and the inverse square law. Recall from the Stellar Properties chapter that the flux is the amount of energy passing through a unit area (e.g., 1 meter²) every second. From the inverse square law of light brightness, you find that the solar flux at the Earth’s distance = the Sun’s surface flux × (Sun’s radius/Earth’s distance)² = 1380 Watts/meter². Since the Sun’s photosphere is approximately a thermal radiator, the flux of energy at its surface = σ × (the Sun’s surface temperature)⁴, where σ is the Stefan-Boltzmann constant. Rearranging the equation, the photosphere’s temperature = [(solar flux at Earth)/σ) × (Earth distance/Sun’s radius)²]^1/4.

These two methods give a rough temperature for the Sun of about 5800 K. The upper layers of the photosphere are cooler and less dense than the deeper layers, so you see absorption lines in the solar spectrum. Which element absorption lines are present and their strength depends sensitively on the temperature. You can use the absorption line strengths as an accurate temperature probe to measure a temperature of about 5840 K.

Features on the Photosphere

Sunspots + granulation in the photosphere (courtesy of Peter N. Brandt)

Galileo discovered that the Sun’s surface is sprinkled with small dark regions called sunspots. Sunspots are cooler regions on the photosphere. Since they are 1000–1500 K cooler than the rest of the photosphere, they do not emit as much light and appear darker. They can last a few days to a few months. Galileo used the longer-lasting sunspots to map the rotation patterns of the Sun. Because the Sun is gaseous, not all parts of it rotate at the the same rate. The solar equator rotates once every 25 days, while regions at 30° above and below the equator take 26.5 days to rotate and regions at 60° from the equator take up to 30 days to rotate.

Animation begins with sunspots at different latitudes lined up. The sequence ends after one rotation of the equator—the sunspots near the poles have not appeared yet—and the animation starts over.

Hundreds of years of observing the sunspots on the Sun shows that the number of sunspots varies in a cycle with an average period of 11 years. At the start of a sunspot cycle the number of sunspots is at a minimum and most of them are at around 35° from the solar equator. At solar maximum when the sunspot number peaks about 5.5 years later, most of the sunspots are within just 5° of the solar equator.

Sunspots are regions of strong magnetic fields. This affects the spectral lines in the sunspot spectra. Each absorption lines will split up into multiple components. The amount of separation between the components measures the strength of the magnetic field. The magnetic field is somehow responsible for the sunspot cycle. In one 11-year cycle the leading sunspot in a sunspot group will have a north magnetic pole while the trailing sunspot in the group will have a south magnetic pole. In the next 11-year cycle the poles will switch so the total cycle is 22 years long. Sunspots form where twisted magnetic field lines rise out of the photosphere and then loop back down into the photosphere and deeper layers. The magnetic field lines suppress the convection at those points on the photosphere so energy has a harder time leaking out at those points on the photosphere—they are cooler than the rest of the photosphere.

How many sunspots are there today on the Sun? See today’s image of the photosphere at the SOHO website or the National Solar Observatory’s GONG website or the Solar Dynamics Observatory website.

At solar maximum there are more prominences and solar flares. Prominences are bright clouds of gas forming above the sunspots in the chromosphere that follow the magnetic field line loops. So-called “quiet” ones form in the corona (the Sun’s atmosphere) about 40,000 kilometers above the surface. Sometimes they form loops of hydrogen gas as the gas follows the loops in the magnetic field. Quiet prominences last several days to several weeks. “Surge” prominences lasting up to a few hours shoot gas up to 300,000 kilometers above the photosphere.

The smallest of the loop prominences shown here is over three times bigger than the Earth (courtesy of National Solar Observatory/Sacramento Peak Observatory).

Solar flares are eruptions more powerful than surge prominences (a flare is shown in the Sun + planets montage above). They will last only a few minutes to a few hours. They probably form when the magnetic field lines get so twisted, that they snap violently, releasing the trapped material. A lot of ionized material is ejected in a flare. Unlike the material in prominences, the solar flare material moves with enough energy to escape the Sun’s gravity. When this burst of ions reaches the Earth, it interferes with radio communication. Sometimes a solar flare will cause voltage pulses or surges in power and telephone lines. Brownouts or blackouts may result. Humans traveling outside the protection of the Earth’s magnetic field will need to have shielding from the powerful ions in a flare.

High resolution observations of the solar surface show a honeycomb pattern called granulation made of bright spots of convection 700 to 1000 kilometers across (see the picture above). Hot gas rises in the middle of each granule bringing energy from the interior to the surface and sinks back down on the border of a granule. The hot gas rising in the center is brighter than the cooler gas sinking at the borders. Each granule will last for about 8 minutes. High resolution images and movies of the Sun’s surface around a sunspot are available on The Institute for Solar Physics webpage for their 2002 Nature article and Picture of the month section of the Kiepenheuer-Institut fur Sonnenphysik (check the archive).

12.2.5 Solar Atmosphere

Moving outward from the core to the surface of the Sun, the temperature and density of the gas decreases. This trend in the density continues outward in the Sun’s atmosphere. However, the temperature increases above the photosphere. The cause of the temperature increase is not well known but it involves some combination of sonic waves and magnetic waves from shaking magnetic loops above sunspots, numerous nanoflares, and wiggling jets in the chromosphere known as spicules to heat the atmosphere.

Chromosphere

During solar eclipses a thin pink layer can be seen at the edge of the dark Moon. This colorful layer is called the chromosphere (it means “color sphere”). The chromosphere is only 2000 to 3000 kilometers thick. Its temperature rises outward away from the photosphere. Because it has a low density, you see emission lines of hydrogen (mostly at the red wavelength of 656.3 nanometers)

The thin chromosphere is visible in this solar eclipse picture.

Images of today’s chromosphere are available on the National Solar Observatory’s GONG website.

Corona

When the new Moon covers up the photosphere during a total solar eclipse, you can see the pearly-white corona around the dark Moon. This is the rarefied upper atmosphere of the Sun. It has a very high temperature of one to two million Kelvin. Despite its high temperature, it has a low amount of heat because it is so tenuous.

Total solar eclipse in 1973 showing the corona (courtesy of Fred Espenak).

The corona is known to be very hot because it has ions with many electrons removed from the atoms. At high enough temperatures the atoms collide with each other with such energy to eject electrons. This process is called ionization. At very high temperatures, atoms like iron can have 9 to 13 electrons ejected. Nine-times ionized iron is only produced at temperatures of 1.3 million K and 13-times ionized iron means the temperature gets up to 2.3 million K! During strong solar activity, the temperature can reach 3.6 million K and lines from 14-times ionized calcium are seen.

Most of the corona is trapped close to Sun by loops of magnetic field lines. In X-rays, those regions appear bright. Some magnetic field lines do not loop back to the Sun and will appear dark in X-rays. These places are called “coronal holes”.

More details in the corona are seen when looking in the higher energy regions of the electromagnetic spectrum than visible light (extreme ultraviolet image from the SOHO spacecraft, courtesy of NASA and ESA).

Images of today’s corona and the transition zone between the chromosphere and corona are available on the SOHO website and the Solar Dynamics Observatory (also has movies)

Other pictures in stereo (use 3D cyan-red glasses), movies, and other data of the Sun’s corona are available from NASA’s STEREO mission (and other site)—two nearly identical space observatories, one ahead of the Earth in its orbit and one trailing behind, will trace the flow of energy and matter from the Sun to the Earth.

Fast-moving ions can escape the Sun’s gravitational attraction. Moving outward at hundreds of kilometers/second, these positive and negative charges travel to the farthest reaches of the solar system. They are called the solar wind. The solar wind particles passing close to a planet with a magnetic field are deflected around the planet. Fluctuations in the solar wind can give energy to the trapped charged particles in the planet’s radiation belts. Particles with enough energy can leave the belts and spiral down to the atmosphere to collide with molecules and atoms in the thermosphere of the planet. When the charged particles hit the planet’s atmosphere, they make the gas particles in the atmosphere produce emission spectra—the aurorae (see the aurorae section in the planets chapter for more details). During solar maximum the increased number and energy of the solar wind particles produce more extensive auroral displays in the Earth’s atmosphere—the aurorae can even be seen by those at latitudes near 30° north or south! Usually, aurorae are seen by only those above 50° N latitude (or 50° S latitude for the aurora australis). The effects of the solar wind on the Earth is described more fully in the Space Weather site at Rice University (Houston, TX; will display in another window) and NASA’s Solar Dynamics Observatory.

Vocabulary

• aurorae
• chromosphere
• convection zone
• core (stellar)
• corona
• granulation
• ionization
• photosphere
• radiative zone
• solar wind
• sunspots

Formulae

• Temperature from Wien’s Law: T (in K) = 2.9 × 10⁶ nm / λ_peak, where λ_peak is the wavelength of peak emission in a star’s spectrum given in nanometers.
• Temperature from solar flux: T (in K) = [(solar flux at Earth/σ) × (Earth distance/Sun’s radius)²]^1/4, where σ is the Stefan-Boltzmann constant and the solar flux at Earth = 1380 Watts/meter².

Review Questions 1

1. What are the two main gases in the Sun? How does the Sun’s mass and size compare with Jupiter?
2. What goes on in the core, radiative zone, and convection zone of the Sun?
3. Describe the three ways astronomers use to find that the photosphere is about 5800 K.
4. What are some of the characteristics of sunspots? What is the sunspot cycle?
5. Do all surface layers of the Sun rotate at the same rate? How can you tell?
6. What produces the granulation on the surface of the Sun?
7. What are prominences and flares? How are they associated with solar activity? How is their number correlated with the number of sunspots?
8. How can we tell that the chromosphere and corona are over 6000 K (some parts reaching a few million degrees). What are “coronal holes”?
9. What is the association of the magnetic field with the sunspots and solar atmosphere?
10. How is the solar wind associated with aurorae?

12.3 The Sun’s Power Source

The Sun produces a lot of light every second and it has been doing that for billions of years. How does it or any other star produce so much energy for so long? This section will cover how stars produce their energy. Astronomers have known for a long time that the Sun produces a tremendous amount of energy. The first part of this section will try to give you an idea of how much energy it produces. Do not feel bad if you have trouble grasping the amount. It is mind-boggling! There are several ways to generate the amount of energy coming from the Sun. What distinguishes the correct explanation from the other models is how long it can power the Sun.

12.3.1 Solar Luminosity—huge energy output!

The first basic question about the Sun is how bright is it? It puts out A LOT of energy every second. How much? The answer from our measurements is 4 × 10²⁶ watts. Such a large number is beyond most of our comprehension, so let’s put the Sun’s total energy output (i.e., its luminosity) in more familiar units. It is equal to 8 × 10¹⁶ of the largest power plants (nuclear or hydroelectric) on the Earth. Our largest power plants now can produce around 5,000 Megawatts of power. Another way to look at this is that the sun puts out every second the same amount of energy as 2.5 × 10⁹ of those large power plants would put out every year—that’s over two billion!

12.3.2 Possible Sources of Energy

What could produce that much energy every second? Let’s first rule out other likely candidates. How about chemical reactions? The most efficient chemical reaction is combining two hydrogen atoms and one oxygen atom to make a water molecule plus some energy. Such a reaction has a very small “efficiency” (something like 1/66,000,000 of one percent). The efficiency = amount of energy released/(mass× c²), where “mass” is the total mass of all of the atoms involved and c is the speed of light. The amount of energy the Sun has stored = the efficiency × (the mass of the fuel source) × c².

To find out how long the Sun would last, you need to find out how much energy the Sun has stored in its account and know how fast it makes withdrawals on its account. The amount of time it would last is the amount of energy stored divided by the rate of withdrawal: lifetime = energy stored/consumption rate = E stored/Luminosity. Makes sense, yes? If the Sun could use all of its hydrogen to make water, the chemical reactions would only power the Sun for about 18,000 years. However, the amount of oxygen is much less than the hydrogen, so the chemical reactions can power the Sun for only 30 years.

We need a reaction with a higher efficiency. How about the ultimate in efficiency—a complete matter to energy conversion with 100% efficiency. Such a reaction could power the Sun for 10¹³ years. Unfortunately, there are problems with this because the number of heavy particles (protons + neutrons) in the Sun must stay the same and protons are extremely stable—they do not spontaneously change into energy (photons).

12.3.3 Gravitational Contraction Doesn’t Power the Sun Long Enough

How about gravitational settling? This is a fancy way of referring to the converting of the potential energy of the falling layers to kinetic energy. When you hold a rock above the ground it has stored energy (“potential energy”—it has the potential to do some work). The stored energy is released as you let it fall. The rock gets kinetic energy because it is moving. Kinetic energy can heat things up. This is what would happen to the layers of the Sun if they were to fall inward toward the center of the Sun. The gas would be compressed and, therefore, would heat up. In addition to the expected heating, the gas would also radiate light.

Until the beginning of this century, this was the idea physicists strongly argued for. This gravitational energy (with an efficiency of 1/10,000 of one percent) could power the Sun for 30 million years—a nice long time, but eventually, physicists had to change their minds about the age of the Sun (and Earth) as radioactive dating indicated a 4.6 billion year age for the solar system and, therefore, the Sun. It was the fact that the Sun could not last long enough being powered by gravitational contraction that motivated the search for nuclear power sources.

12.3.4 Nuclear Fusion Needs Extreme Temperatures and Densities

Nuclear power is the only thing left to power the Sun for as long as it has been shining. There are two types possible: fusion and fission. They both transform the nucleus of an atom into another type of nucleus. Fission produces energy by breaking up massive nuclei like uranium into less massive nuclei like helium and lead. Fusion produces energy by fusing together light nuclei like hydrogen to make more massive nuclei like helium. Atomic power plants and the Atom Bomb use fission to get the energy. Stars and hydrogen bombs use fusion.

To get the positively-charged nuclei to fuse together, their electrical repulsion must be overcome (remember that like charges repel and opposite charges attract—something that rarely happens in human interactions). Once the positively-charged nuclei are close enough together (within several 10^-13 centimeters of each other), another fundamental force of nature called the strong nuclear force takes over. It is much more powerful than the electric force and makes the nuclei stick together.

To get those nuclei close enough together requires high temperatures and high densities. At high temperatures the nuclei move fast enough to be driven close enough together for them to fuse. The high densities ensure that there are enough nuclei within in a small volume for the collisions to take place at all. The only place these extreme conditions occur naturally is in the cores of stars.

The temperatures in the cores of stars are above the approximately 8 million K needed to fuse hydrogen nuclei together. The amount of repulsion is larger for nuclei with more positive charge so the fusion of elements with greater positive charge requires greater temperatures and densities than that needed for fusing elements with small positive charge. This is why stars fuse hydrogen nuclei before they fuse other nuclei. For example, the fusion of helium nuclei requires temperatures above 100 million K and heavier nuclei require even higher temperatures. You will see in the next chapter that these ultra-extreme conditions occur in the final stages of a star’s life cycle after the main sequence stage.

12.3.5 Some Mass is Converted to Energy in Fusion Reactions

Fusion involves low-mass nuclei whose combined mass is more than the resulting fused massive nucleus. The mass that was given up to form the massive nucleus was converted to energy. Remember E=mc²? That tells you how much energy (E) can be made from matter with mass m. Remember that c is the speed of light and it’s squared (!) so a little bit of mass can make a lot of energy.

In the cores of main sequence stars, four hydrogen nuclei, each with the mass of one proton, are fused together to form a single helium nucleus (two protons and two neutrons) that has a mass of 3.97 times the mass of one proton. An amount of mass equal to 0.03 times the mass of one proton was given up and converted to energy equal to 0.03 × (mass one proton) × c². The efficiency of this reaction is about 4/5 of one percent. The Sun could last for about 10 billion years on hydrogen fusion in its core. This is plenty long enough to satisfy the modern geologists.

12.3.6 Why Stars Use a Complicated Chain Reaction

The fusion process in stars is a little more complicated than what was described above. Rather than creating the helium nucleus in a single reaction, nature uses a series of reactions to build up the helium nucleus step-by-step. In most stars a three-step chain reaction is used called the “proton-proton chain.” It is described in the animation below. Massive stars also use a reaction that uses carbon, nitrogen, and oxygen nuclei in a chain process, called the Carbon-Nitrogen-Oxygen chain, with several more steps. Regardless of the chain process used, the net result of the fusion process is to fuse four hydrogen nuclei (protons) to create one helium nucleus (2 protons + 2 neutrons) plus some energy.

Three step nuclear reaction chain. Selecting the image will bring up a single frame that summarizes the chain reaction in another window.

Why does nature use a long complicated chain reaction process to fuse four protons into one helium nucleus? Would it not be much simpler if four protons would collide simultaneously to make one helium nucleus? Simpler, but not very likely is the answer. Getting four objects to collide simultaneously each with high enough energy is very hard to do—the chances of this happening are very, very small (as one from a family of 8 boys I can attest to the difficulty of getting just half of us together for a mini-family reunion!). The chances of this type of collision are too small to power the Sun, so nature has found a cleverer scheme. The chances of two particles colliding and fusing is much higher, so nature slowly builds up the helium nucleus.

Nuclear fusion is something of a holy grail for utility companies because it produces no nasty waste products and has the potential of getting more energy out of it than you put in—free energy! Unfortunately, the conditions to get fusion to happen are very extreme by our standards. A major problem is containing the very hot gas for extended periods of time to provide a sustained energy source. We have been only able to tap the fusion process with the hydrogen bomb, but that is a one shot deal. The hydrogen bomb still needs an atomic bomb trigger to create the extreme temperatures needed for the fusion process. At least you can get the waste product of the Sun’s fusion process for free with solar power collectors. The Sun can have a controlled fusion process and not blow up all at once because of the hydrostatic equilibrium “thermostat.”

12.3.7 Hydrostatic Equilibrium Controls the Reaction Rates

Hydrostatic equilibrium is the balance between the thermal pressures from the heat source pushing outwards and gravity trying to make the star collapse to the very center. I will discuss hydrostatic equilibrium in more depth (no pun intended) in a later section. The nuclear fusion rate is very sensitive to temperature. It increases as roughly temperature⁴ for the proton-proton chain and even more sharply (temperature¹⁵) for the Carbon-Nitrogen-Oxygen chain. So a slight increase in the temperature causes the fusion rate to increase by a large amount and a slight decrease in the temperature causes a large decrease in the fusion rate.

Now suppose the nuclear fusion rate speeds up for some reason. Then the following sequence of events would happen: 1) the thermal pressure would increase causing the star to expand; 2) the star would expand to a new point where gravity would balance the thermal pressure; 3) but the expansion would lower the temperature in the core—the nuclear fusion rate would slow down; 4) the thermal pressure would then drop and the star would shrink; 5) the temperature would rise again and the nuclear fusion rate would increase. Stability would be re-established between the nuclear reaction rates and the gravity compression.

A similar type of scheme would occur if the nuclear fusion rate were to slow down for some reason. The fusion rate stays approximately constant for stars that are fusing hydrogen to make helium + energy in the core. Once the hydrogen fuel in the core has been used up, hydrostatic equilibrium can no longer stabilize the star. What happens next will have to wait until I talk about stellar development.

Summary

A. Need an energy source that lasts a long time: lifetime = energy stored/luminosity. Nuclear fusion is the only process that can do this. With nuclear fusion, lower-mass nuclei fuse together to form a single more massive nucleus + energy. The sum of the low-mass nuclei masses = the massive nucleus mass + energy/c² (remember E = mc²). The “c” is the symbol for the speed of light. Nuclear fusion can power the Sun for about 10 billion years.

B. To overcome the mutual electrical repulsion of positively-charged nuclei, a star needs extremely high temperatures and densities. These conditions are found only in the core of a star. Under these extreme conditions, particles move fast enough to get close enough for the strong nuclear force to overcome electrical repulsion. Repulsive force increases with more positive charges. Hydrogen is fused first because it requires less extreme conditions, than the fusion of more massive nuclei.

C. Stars use a chain process to fuse four hydrogen nuclei to create one helium nucleus. A chain process is much more probable than a process that fuses four hydrogen nuclei simultaneously. Most stars use a proton-proton chain that is described in the animation above. Stars with enough mass will also use the “Carbon-Nitrogen-Oxygen chain” process. The net process is the fusion of four hydrogen nuclei to make one helium nucleus plus some energy.

D. The balance between gravity compression and outward thermal pressure controls the rate of the nuclear fusion reactions. The star does not blow up like a bomb.

12.4 Neutrinos

Helium is produced in the fusion of hydrogen. As shown in the proton-proton fusion chain diagram above, there are two other particles produced. One is the “positron” and the other is a “neutrino”. A positron is the antimatter counterpart of the electron. It has the same mass as an electron but the opposite charge. When it collides with an electron, they annihilate each other converting all of their mass into energy.

The photons produced in nuclear reactions take about a million years to move from the core to the surface. The photons scatter off the dense gas particles in the interior and move about a centimeter between collisions. In each collision they transfer some of their energy to the gas particles. By the time photons reach the photosphere, the gamma rays have become photons of much lower energy—visible light photons. Because the photons now reaching the surface were produced about a million years ago, they tell us about the conditions in the core as it was a million years ago. The other particle produced in nuclear reactions has a less tortuous path out of the core.

A neutrino is a massless (or very nearly massless) particle that rarely interacts with ordinary matter. Neutrinos travel extremely fast—the speed of light if they have zero mass or very close to the speed of light if they have a small mass. Because they travel so fast and interact so rarely with matter, neutrinos pass from the core of the Sun to the surface in only two seconds. They take less than 8.5 minutes to travel the distance from the Sun to the Earth. If you could detect them, the neutrinos would tell you about the conditions in the Sun’s core as it was only 8.5 minutes ago (much more current information than the photons!).

The problem with neutrinos is that they have a very low probability of interacting with matter. A neutrino could pass through a light year of lead and not be stopped by any of the lead atoms! However, there are A LOT of neutrinos produced by the Sun. Take a look at your pinky finger. In one second several trillion neutrinos passed through your pinky (did you feel them?). Do not worry, the neutrinos did not damage anything. The great majority of neutrinos pass right through the materials around you.

Homestake Gold Mine Neutrino Experiment (courtesy of R. Davis, Brookhaven National Laboratory).

A few of them will interact with some matter on the Earth. You can increase the odds of detecting a few of them by using a LARGE amount of a material that reacts with neutrinos in a certain way. A chlorine isotope will change to a radioactive isotope of argon when a neutrino interacts with it. In the same way a gallium isotope will change to a radioactive isotope of germanium. Water molecules will give off a flash of light when struck by a neutrino. Neutrino detectors use hundreds of thousands of liters of of these materials in a container buried under many tens of meters of rock to shield the detectors from other energetic particles from space called cosmic rays. Even the largest detectors detect only a few dozen neutrinos in a year.

Sudbury Neutrino Observatory

Super-Kamiokande Neutrino Detector water tank showing the thousands of photon detectors each about the size of a beach ball.

12.4.1 Solar Neutrino Problem

As shown in the animation describing the proton-proton chain above, the number of neutrinos produced in the Sun is directly proportional to the number of nuclear reactions that are taking place in the Sun’s core. The same can also be said of the number of neutrinos produced via the Carbon-Nitrogen-Oxygen chain. The more reactions there are, the more neutrinos are produced and the more that should be detected here on the Earth. The number of neutrinos detected coming from the Sun was smaller than expected. Early experiments detected only 1/3 to 1/2 of the expected number of neutrinos. These experiments used hundreds of thousands of liters of cleaning fluid (composed of chlorine compounds) or very pure water. They were sensitive to the high-energy neutrinos produced in less than one percent of the nuclear fusion reactions. Later experiments using many tons of gallium were able to detect the more abundant low-energy neutrinos. However, those experiments also found the same problem—too few neutrinos (the gallium experiments found about 2/3 the expected number). The puzzling lack of neutrinos from the Sun was called the solar neutrino problem. There were several possible reasons for this discrepancy between the observations and our predictions:

1. Nuclear fusion is not the Sun’s power source. This reason was not supported by other observations, so it is not likely to be the correct reason.
2. The experiments were not calibrated correctly. It was unlikely that all of the carefully-tuned experiments were tuned in the same wrong way. The experiments used three very different ways to detect neutrinos and produced the same lack of neutrinos. The experiments were independently verified by many other scientists, so astronomers think that the results are correct, even if they are disappointing.
3. The nuclear reaction rate in the Sun is lower than what our calculations say. This was possible but many people checked and re-checked the physics of the reaction rates. There are some strong constraints in how much you can lower the temperature in the core of the Sun to slow down the reactions. Astronomers know how much total energy is emitted by the Sun, so they know very accurately how many nuclear reactions are needed to produce all of those photons seen coming from the Sun. Those reactions also produce the neutrinos. Astronomers think they have a good idea of how stars produce their energy. That left another alternative.
4. Neutrinos produced in the core of the Sun change into other types of neutrinos during their flight from the Sun to the Earth. Our neutrino detectors can detect a certain kind of neutrino, called the “electron neutrino”, that are produced from nuclear fusion. Some of these electron neutrinos may change into another of two types of neutrino (the “muon neutrino” or the “tau neutrino”) that do not interact with the detection material as well as the electron neutrino. Some experiments in high-energy particle accelerators and the water tank Super-Kamiokande neutrino detector (Japan) suggested that the neutrinos could change into other types. The Sudbury Neutrino Observatory (Canada), using almost 900,000 liters of heavy water, was built to detect all three types of neutrinos. It firmly established that some of the electron neutrinos produced by the nuclear fusion in the Sun do change into the other neutrino types and that the solar nuclear fusion models do predict the correct number of neutrinos. It looks like the 30-year mystery has been solved! A neutrino can change into another type of neutrino only if the neutrino has some mass. If the neutrino has mass, then it cannot travel at the speed of light, but can get darn close. Recent experiments have shown that the neutrino does have a tiny amount of mass (several million times less than an electron). A neutrino with even as small a mass as this has important consequences for the development of the universe (more about that later) and our understanding of the structure of matter. It is amazing that in their effort to check their nuclear fusion theory, astronomers have learned totally unexpected things about fundamental physics and this has changed what is known about the structure and behavior of the entire universe itself. Wow!

12.4.2 Explorations of Neutrino Detectors

Here are some links to the homepages of neutrino “observatories” that are in operation around the world. All of the sites will be displayed in another window.

1. The Borexino detector uses 300 tons of liquid scintillator (an organic liquid much like mineral oil that gives off light when charged particles interact in it) to detect low-energy neutrinos produced in the Beryllium-7 electron-capture process in nuclear fusion in the Sun. It is buried far below a mountain in Italy.
2. The Sudbury Neutrino Observatory Plus (SNO+) will use 1000 tons of liquid scintillator (specifically, linear alkyl benzene, LAB) with dissolved neodymium buried far below ground outside of Sudbury, Ontario (Canada) to search for a particular neutrino reaction as well as being able to detect low-energy neutrinos. SNO+ is the followup to the Sudbury Neutrino Observatory that used 1000 tons of “heavy water” in the same container as SNO+. Heavy water uses the deuterium isotope of hydrogen instead of the ordinary isotope of hydrogen in the water molecule (H₂O). Deuterium has 1 proton+1 neutron in its nucleus instead of just the 1 proton of ordinary hydrogen. The extra neutron makes deuterium twice as massive as ordinary hydrogen, so the “heavy water” molecule is about 10% heavier than ordinary water. The original experiment is now finished and is being replaced by the SNO+.
3. The Super-Kamiokande experiment is a joint-project of the United States and Japan. The detector uses 50,000 tons of water buried deep underground in Japan. The link takes you to the United States homepage at the University of Washington. The University of California at Irvine is also involved in the project.
4. KamLAND is a liquid scintillator detector in Japan in the Kamioka Mine 1 kilometer below the surface.

Two other sites worth exploring about neutrinos follow:

• John Bahcall probably contributed the most to our understanding of the properties of solar neutrinos and what they tell us about the Sun’s core and fundamental physics before his death in 2005. Select his Solar Neutrino viewgraphs to see his presentations on the solar neutrino problem. Links to articles and talks for the general public are also available there.
• The University of Pennsylvania Physics Department distributes an excellent set of articles about neutrino physics and detectors.

12.5 Helioseismology

Another probe of the Sun’s interior uses the pulsating motions of the Sun. The pulsations are too small to be seen just by looking at the Sun. But the pulsations can be seen if the Doppler shifts are measured across the face of the Sun. Some parts of the Sun expand towards the Earth and adjacent regions contract away from the Earth. These regions are several thousands of kilometers across and the pulsation periods are just a few minutes long. Different types of oscillating waves combine to produce the complicated patterns of pulsation seen.

One type of pulsation is shown here. The blue regions are approaching and the red regions are receding from you. The pulsations are thought to extend far into the Sun’s interior (courtesy of the National Solar Observatory).

If you disentangle the different oscillation modes from each other, you can use these waves to probe the solar interior. How those waves propagate through the Sun and interact with each other depends on the temperature, density, and composition of the material they pass through. By observing the effects of these waves on the photosphere of the Sun, you can determine the temperature, density, and composition of the different layers inside the Sun. Geologists on the Earth use similar techniques to study the interior of our planet from earthquake waves in the research field called seismology. Modifying the name for solar studies, the study of the Sun’s interior using the solar oscillations is called helioseismology.

Solar astronomers have set up a global network of stations to continuously monitor the Sun’s pulsations. This network is called the Global Oscillations Network Group (GONG). Links to web sites describing GONG and other helioseismology sites are given below. Instruments to detect solar oscillations have also been placed on satellites. Check the links below for more information about them.

Links to Centers Probing the Sun’s Interior

All of the sites will be displayed in another window.

1. The GONG homepage at the National Optical Astronomy Observatories is a must see. A concise fact sheet for GONG is available, as well as, information about helioseismology in general.
2. The Solar Oscillations Investigation at Stanford is another major center for helioseismology research.
3. The Stanford group have also constructed an excellent resource site for K-12 students called The Solar Center. Many educational activities are available, along with excellent images, movies, and audio (yes, you can hear the Sun pulsate!—the Doppler observations have been converted into sound).
4. The Marshall Space Flight Center’s Solar Physics web site is an excellent starting point for all the research about the Sun. Links to the space missions and the science background about the Sun are given here.

Sections Review

Vocabulary

• helioseismology
• luminosity
• neutrino
• nuclear fusion
• proton-proton chain
• solar neutrino problem

Review Questions 2

1. How does nuclear fusion produce energy?
2. Why does nuclear fusion need high temperatures and densities?
3. Why is it so hard to develop nuclear fusion as a dependable power source on Earth?
4. Why will chemical reactions or gravitational contraction not work for powering the Sun?
5. What is the net result of the nuclear fusion chain process? Why does nature use the complicated chain process instead of a one-step fusion procedure?
6. Where are neutrinos produced? What information can they tell you about interior conditions in the Sun?
7. What was the solar neutrino problem? How was the problem solved and what are the implications of that solution?
8. How can you use pulsations of the Sun to find out about the structure and composition of its interior?

12.6 Interior Structure of Stars

Observations of the stars in all regions of the electromagnetic spectrum and careful observations of the Sun’s pulsation modes and neutrinos provide the data needed to construct models of the interiors of stars. This section is about how to find out what the interior of a star is like without physically taking one apart (a rather difficult thing to do).

12.6.1 Mathematical Models

Astronomers construct mathematical models of the interior of a star using the information pouring from the surfaces of stars (especially the Sun) and their knowledge of how gases behave under different conditions. The mathematical models are a set of equations that describe how things work layer by layer in a star. Fortunately, the interior of stars is completely gaseous all the way to the center, so the equations are relatively simple (whew!). The physics of gases can be described with just three parameters:

1. Temperature—a measure of the random motion energy (the average kinetic energy) of the gas particles. The higher the temperature, the more random kinetic energy is present.
2. Pressure—the amount of force/area. Hot gas expands to create pressure on its surroundings. For example, the gas inside a hot air balloon pushes out on the material of the balloon enclosing the gas.
3. Mass Density—the amount of mass/volume. Gaseous material can be compressed to smaller volumes and higher densities.

12.6.2 Equation of State

How the three parameters work together to describe the material you are studying is determined by the equation of state of the material. This is an equation that relates density, pressure, and temperature. The equation of state for solids and liquids is very complex and uncertain. The equation of state for the gas is simple: the pressure = (a constant × the mass density × the temperature) / (the molecular weight of the gas). The molecular weight of a particular type of gas is the combined mass of all of the isotopes of that type of gas in the proportions found in nature. For hydrogen, the molecular weight is very close to 1; for helium, the molecular weight is very close to 4. For a gas made of different types of atoms (such as that found in stars), the molecular weight is the weighted mean of the different atomic types, taking into account the relative proportions of the different types of atoms. This equation of state for simple gases is also called the ideal gas law.

12.6.3 Gravity Holds a Star Together

Stars are held together by gravity. Gravity tries to compress everything to the center. What holds an ordinary star up and prevents total collapse is thermal and radiation pressure. The thermal and radiation pressure tries to expand the star layers outward to infinity.

Hydrostatic equilibrium: gravity compression is balanced by pressure outward.

Greater gravity compresses the gas, making it denser and hotter, so the outward pressure increases.

In any given layer of a star, there is a balance between the thermal pressure (outward) and the weight of the material above pressing downward (inward). This balance is called hydrostatic equilibrium. A star is like a balloon. In a balloon the gas inside the balloon pushes outward and the elastic material supplies just enough inward compression to balance the gas pressure. In a star the star’s internal gravity supplies the inward compression. Gravity compresses the star into the most compact shape possible: a sphere. Stars are round because gravity attracts everything in an object to the center. Hydrostatic equilibrium also explains why the Earth’s atmosphere does not collapse to a very thin layer on the ground and how the tires on your car or bicycle are able to support the weight of your vehicle.

Long before astronomers knew about nuclear fusion, they had a good idea of how the density and temperature of stars increased toward their cores. Deeper layers have more gravity compression from the overlying layers. The greater gravity compression raises the density of the gas. In order to balance the greater gravity compression, the outward pressure of the gas and radiation is increased by raising the temperature. Calculating the change in density and temperature layer by layer toward the center of a star, you find the temperature at the core of a star = 8 to 28 million K and the densities = 10 to 130 times the density of water. As stars age, these numbers increase! You have already seen in the previous section that hydrostatic equilibrium also provides a “thermostatic control” on the energy generation inside a star and keeps the star stable.

12.6.4 Other Pieces

Other basic physical principles are put into the mathematical models:

1. Continuity of Mass: the total stellar mass = sum of all of the shell layer masses. Mass is distributed smoothly throughout star’s interior (there are no gaps or pockets of “negative” mass). Also, the law of the conservation of mass says that the total amount of mass does not change with time.
2. Continuity of Energy: the amount of energy flowing out the top of each shell layer in a star = the amount of energy flowing in at bottom of the shell layer. No energy is magically destroyed or created from nothing. A star’s luminosity = sum of all of the shell layer energies. Also, the law of the conservation of energy says that the total amount of energy does not change with time. Energy can change from one form to another form of energy, but the total amount is a constant.
3. Energy Transport: Recall from the discussion about how energy flows in planetary atmospheres that energy moves from hot to cold via conduction, radiation, or convection. Nature will first try to use radiation (photons) to move energy from the very hot interior to the very cold space. If radiation cannot transport all of the energy over the distance from the center to the surface of the star, then nature will also use convection. Convection is the bulk motion of gases used to transport energy. Hot gases rise to the upper levels and radiate their extra energy at the upper levels while cooler gases sink to pick up more energy from the hot interior. Conduction transports energy by having each atom transfer its energy to the atom next to it. Conduction is not an efficient process in a gas so it transports a very small amount of energy in stars and is usually ignored.
4. Opacity: It takes a LONG time for photons produced by nuclear reactions in the core to reach the surface. In the opaque interior a photon travels only about 1 centimeter before it runs into an atom or ion and is absorbed. A measure of the gas’ ability to absorb the photons is called its opacity. You cannot see into the interior of a star because the gas has a high opacity. The photon is later re-emitted but in a random direction. It may be re-emitted in the direction it came from! So the photon travels a very zig-zag sort of path outward. It takes about a million years for a photon to travel from where it was created in the core to the surface where it is finally released into space. Along the way the photon has transferred some its energy to the gas particles, so the photon has changed from very high energy gamma rays to the lower energy visible light photons. Some of the radiation is also in the form of neutrinos. The gas has almost zero opacity with the neutrinos so they pass right on through the star’s gas in just a few seconds.
5. The equation of state, hydrostatic equilibrium and the other physical principles are put together for each layer in a star. The equations are solved for each layer starting from the layer there is direct information of, the surface. That result gives the conditions for the next layer’s equations. Solving the layer’s equations gives the conditions for the layer below it and this process continues on down toward the center layer by layer. In order to get sufficient detail for accurate results, the star’s interior is divided into hundreds of layers. To save on time, the equations are solved using a computer.

Mass-Luminosity Relation Explained

The mass-luminosity relation for 192 stars in double-lined spectroscopic binary systems.

Observations of thousands of main sequence stars show that there is definite relationship between their mass and their luminosity. The more massive main sequence stars are hotter and more luminous than the low-mass main sequence stars. Furthermore, the luminosity depends on the mass raised to a power that is between three and four (Luminosity ~ Mass^p, where p is between 3 & 4). This means that even a slight difference in the mass among stars produces a large difference in their luminosities. For example, an O-type star can be only 20 times more massive than the Sun, but have a luminosity about 10,000 times as much as the Sun. Putting together the principle of hydrostatic equilibrium and the sensitivity of nuclear reaction rates to temperature, you can easily explain why.

Massive stars have greater gravitational compression in their cores because of the larger weight of the overlying layers than that found in low-mass stars. The massive stars need greater thermal and radiation pressure pushing outward to balance the greater gravitational compression. The greater thermal pressure is provided by the higher temperatures in the massive star’s core than those found in low-mass stars. Massive stars need higher core temperatures to be stable!

The nuclear reaction rate is very sensitive to temperature so that even a slight increase in temperature makes the nuclear reactions occur at a MUCH higher rate. This means that a star’s luminosity increases a lot if the temperature is higher. This also means that a slight increase in the mass of the star produces a large increase in the star’s luminosity.

Mass Cutoff Explained

The principle of hydrostatic equilibrium and nuclear fusion theory also explain why stars have a certain range of masses. The stars have masses between 0.08 and about 100 solar masses.

Stars with too little mass do not have enough gravitational compression in their cores to produce the required high temperatures and densities needed for fusion of ordinary hydrogen. The lowest mass is about 0.08 solar masses or about 80 Jupiter masses. A star less massive than this does not undergo fusion of ordinary hydrogen but if it is more massive than about 13 Jupiters it can fuse the heavier isotope of hydrogen, deuterium, in the first part of its life. Stars in this boundary zone between ordinary stars and gas planets are called brown dwarfs. After whatever deuterium fusion it does while it is young, a brown dwarf then just slowly radiates away the heat from that fusion and that left over from its formation. Among the first brown dwarfs discovered is the companion orbiting the star Gliese 229. Selecting the picture below of Gliese 229 and its companion, Gliese 229B, will take you to the caption for the picture at the Space Telescope Institute.

With the discovery of several hundred brown dwarfs in recent infrared surveys, astronomers have now extended the spectral type sequence to include these non-planets. Just beyond the M-stars are the L dwarfs with surface temperatures of about 1400 K to 2200 K with strong absorption lines of metal hydrides and alkali metals. Cooler than the L dwarfs are the T dwarfs. At their cooler temperatures, methane lines become prominent.

Stars with too much mass have so much radiation pressure inside pushing outward on the upper layers, that the star is unstable. It blows off the excess mass. The limit is roughly about 100 to perhaps 150 solar masses. Stars like Eta Carinae and the “Pistol star” are examples of these supermassive stars. The picture of Eta Carinae below shows two dumbbell-shaped lobes of ejected material from the star in an earlier episode of mass ejection. Selecting the image will take you to more information about the image at the Space Telescope Institute (will display in another window).

The picture below from the Hubble Space Telescope shows the violet Pistol Star surrounded by hydrogen gas fluorescing from the copious ultraviolet light coming from the star. Selecting the image will bring up the press release from the Space Telescope Institute in another window.

Sections Review

Vocabulary

• brown dwarfs
• equation of state
• hydrostatic equilibrium
• ideal gas law
• mass density
• mathematical models
• opacity
• pressure
• temperature

Review Questions 3

1. How can you determine what the interiors of stars are like?
2. What three quantities does an equation of state relate?
3. What is the equation of state for gases? (Almost any gas has this equation of state, even the air in your automobile tires or air-filled ball.)
4. Use the equation of state of a gas to explain in what way the temperature of the gas changes as the pressure exerted on the gas is increased. Explain why the pressure in your automobile tires is slightly less when they are cold than right after a long drive.
5. What is being equilibrated in hydrostatic equilibrium? How does hydrostatic equilibrium explain why the temperature and density increases inward toward the core of a star?
6. How does hydrostatic equilibrium control the fusion rate in the Sun?
7. What would happen to the size of a star if its core steadily produced more energy than it did at some earlier time (e.g., when a main sequence star becomes a red giant)?
8. What would happen to the size of a star if its core steadily produced less energy than it did at some earlier time (e.g., when a star stops fusing nuclei in its core)?
9. Do photons produced in the core zip right out from the Sun or does it take longer? Explain why.
10. Why do brown dwarfs not undergo fusion?
11. What are some basic differences between stars and planets?

Is this page a copy of Strobel’s Astronomy Notes?