Lecture 12: The Interior of the Sun

Sometime too hot the eye of heaven shines,
And often is his gold complexion dimm'd:
And every fair from fair sometime declines,
By chance, or nature's changing course, untrimm'd.

-- William Shakespeare, Eighteenth Sonnet

12.1 Basic Characteristics of the Sun

(Discovering the Universe, 5th ed., §9.0)

Photo Information

• The Sun is just a star like all of the others, with one very important difference: it is very close to us, and is therefore our primary source of heat and light.
  
  A small change in the Sun's behavior could melt our icecaps or start another Ice Age.
  
  Click here for a real-time image of the Sun from the National Solar Observatory/Sacramento Peak (available when the telescope is operating).
  
   
  
• The Sun is an average star, in any of the different ways we might describe it:
  
  
  • Size: R_Sun = 7.0 x 10⁵ Km = 1/200 AU = 110 x R_Earth
    
    
  • Mass: M_Sun = 2.0 x 10³⁰ Kg = 3.3 x 10⁵ x M_Earth
    
    
  • Density: d_Sun = 1.4 g/cm³
    
    
  • Composition: 74% H, 25% He, 1% other elements
    
    
  • Temperature: T_surface = 5800 K
    
    
  • Luminosity: L_Sun = 3.9 x 10²⁶ W
    
    
  • Age: 5 Gy
    

12.2 The Sun's Structure

(Discovering the Universe, 5th ed., §9.0, 9.8)

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• The Sun can be divided into two basic parts.
  
  The interior of the Sun is a spherical region that is opaque to visible light and is relatively high density.
  
  The atmosphere of the Sun surrounds the interior, and is transparent to visible light and is relatively low density.
  
  The surface of the Sun is the boundary between the interior and the atmosphere, and is more or less what we "see" when we look at the Sun.
  
  The "size" of the Sun given above, R_Sun, is the radius of the interior and the surface.
  
  Although the interior is relatively high density and the atmosphere is relatively low density, there is no sharp drop-off in density such as occurs at the Earth's surface.
  
   
  
• The Sun is so hot that molecules are torn apart into atoms, which are, in turn, mostly torn apart into a plasma of charged particles: positive ionized atoms (including "bare" nuclei) and negative electrons.
  
  The Sun is therefore a large ball of fluid plasma, held together by its own gravity.
  
   
  
• The size of the Sun does not change, so it must be in mechanical equilibrium.
  
  Because the Sun consists of a fluid, this is also called hydrostatic equilibrium.
  
  For this equilibrium to exist, there must be a balance between the inward force of gravity, trying to compress the Sun, and the outward fluid pressure, which resists compression.
  
  The deeper you go inside the Sun, the greater the weight of the mass above you.
  
  Therefore, the opposing pressure must also increase with depth.
  
  As the pressure increases, the density and temperature will also increase:
  
  P ~ dT
  
   
  
• So, at the center of the Sun it must be very hot and very dense.
  
  It is calculated that:
  
  P_center = 340 Gbar
  T_center = 15.5 MK
  d_center = 160 g/cm³.
  
  Question: if the average density of the Sun is only 1.4 g/cm³, what does that say about the density of the atmosphere of the Sun?
  
   
  
• Because of the Sun's fluid nature, it is possible for its density to increase with little or no increase in pressure; this would require its temperature to decrease:
  
  P (no change) ~ d (up)T (down)
  
  So, it's possible for gravity to compress the Sun even further than it currently is.
  
  To maintain its mechanical equilibrium, then, the Sun must prevent its temperature from decreasing.
  
  The Sun maintains its temperature by producing energy in its core.
  
  Question: if the Sun didn't maintain its temperature, what would prevent it from completely collapsing?
  

12.3 The Sun's Energy Source

(Discovering the Universe, 5th ed., §9.7)

• The source of the Sun's energy was not understood until this century.
  
  It was known that familiar sources of energy could not be responsible.
  
  For example, if the Sun was made of oil, to produce the observed luminosity its mass would be burned up in a few thousand years, and the Earth was known to be much, much older than that.
  
   
  
• Einstein's Theory of Special Relativity provided the key when it showed that energy and mass are convertible:
  
  E = mc²
  
   
  
• Because c is so large, a small amount of mass can provide a huge amount of energy.
  
  The only question then was how this conversion from mass to energy might occur.
  
   
  
• The subsequent development of nuclear physics provided the necessary understanding: thermonuclear fusion of hydrogen into helium.
  
  Basically, this is nuclear fusion at a very high temperature, which is necessary to overcome the nuclei's electrostatic repulsion.
  
   
  
• Recall that the positive charge on hydrogen nuclei (protons) ordinarily keeps them far apart.
  
  The proton's electric potential energy increases as they get closer together, so they will slow down, stop, and begin to separate again.
  
  This is like trying to roll a ball up a hill; with too little energy, it will turn around and roll back down.
  
   
  
• If the protons are hot enough, however, they will have enough energy that that they can get quite close together.
  
   
  
• At short (nuclear) distances, the strong nuclear force will dominate the electric force, so that the protons are now attracted to each other, and they can fuse together into a new nucleus.
  
  This would be like a ball with enough energy to roll up and over the top of a hill, and down the other side.
  
   
  
• The energy required for protons to overcome their electric repulsion corresponds to a temperature of 8.5 MK.
  
  Since the Sun's core has a temperature of 15.5 MK, it can easily sustain hydrogen fusion, and in fact burns hydrogen faster than it would at a lower temperature.
  
   
  
• The actual process by which hydrogen fusion occurs is somewhat involved and is called the proton-proton cycle, summarized by the equation:
  
  
  
  This equation says that four hydrogen atoms (with one nucleon each) are combined to produce a helium atom (with four nucleons).
  
  In addition, the fusion produces four photons (represented by the Greek letter gamma, because they are gamma radiation), two neutrinos (represented by the Greek letter nu), and energy.
  
   
  
• According to Einstein, the proton-proton cycle must conserve mass + energy.
  
  In particular, because
  
  
  
  the overall reaction begins with more mass than it ends up with, and the difference is converted into other forms of energy.
  
  Question: what other forms of energy might result from this fusion?
  
   
  
• Note that in the above reaction, the hydrogen on the left has four protons, while the helium on the right has two protons and two neutrons.
  
  The weak nuclear force governs the reaction which turns protons into neutrons, resulting in the neutrinos:
  
  
  
  Question: where did we previously see a similar process? how did it differ?
  
  Question: why does this reaction require energy to occur? (Hint: consider the masses involved.)
  
   
  
• In addition to a neutrino, this reaction produces a positron, which is a positive electron.
  
  The positron ensures conservation of electric charge by carrying away the proton's positive charge.
  
  Positrons are a form of antimatter, particles which have the same mass as regular matter but the opposite electric charge.
  
  Antimatter does not normally exist in nature, because it will usually quickly collide with matter and be destroyed in a process called pair annihilation:
  
  
  
  Note that here mass is now converted completely back into energy in the form electromagnetic radiation.
  
   
  
• The proton-neutron conversion occurs when two protons ("ordinary" hydrogen) fuse into a nucleus of deuterium ("heavy" hydrogen).
  
  Deuterium is an isotope of hydrogen, ²H, with one proton and one neutron.
  
  
  
  The energy to convert the proton into a neutron comes from a net mass loss in the conversion from two protons to deuterium.
  
   
  
• The deuterium quickly fuses with another proton to form an isotope of helium, ³He, which has only one neutron:
  
  
  
  Another photon of gamma radiation is released in this process.
  
   
  
• Finally, two "light" helium nuclei fuse together, producing "ordinary" helium, and two protons:
  
  
  
   
  
• The fusions (1) and (2) occur twice, and together with fusion (3) the result is:
  
  
  
  Note that two protons are "recycled" for use in later fusions, which is how the proton-proton cycle gets its name.
  
  The entire series of reactions is displayed at the right.
  
   
  
• The energy resulting from thermonuclear fusion is distributed in several ways:
  
  
  • kinetic energy of ⁴He and the two "recycled" protons: 91%
    
    
  • electromagnetic energy of the photons: 8%
    
    
  • kinetic energy of the neutrinos: 1%
    
     
    
• To produce the Sun's luminosity, 600 million tons of H must be burned by this process every second.
  
  Because the Sun is so massive, though, it will last for another 5 billion years!

12.4 The Flow of Energy

(Discovering the Universe, 5th ed., §9.8)

• The energy produced at the center of the Sun must eventually be radiated away at its surface.
  
  Otherwise, the Sun would get hotter over time, which is not observed.
  
  This balance is called thermal equilibrium.
  
   
  
• There are three means by which energy might travel outward to the surface:
  
  
  • Conduction: this process occurs in solids or dense fluids.
    
    It is a "domino" effect, with one atom bumping into another atom and passing on its energy, which then bumps into another atom, etc.
    
    Question: where have you seen conduction take place in everyday life?
    
    
  • Convection: this process occurs in fluids.
    
    It involves the flow of matter itself, with hotter, less-dense material rising upward while cooler, more-dense material sinks downward to replace it.
    
    A stable structure of convection rolls will typically form, as shown.
    
    Question: where have you seen convection take place in everyday life?
    
    
    
  • Radiative diffusion: this process occurs in all materials to a varying degree.
    
    As we have seen before, they act like blackbodies, so their heat is converted to electromagnetic energy and radiated away as a stream of photons.
    
    Depending on the materials through which they pass, the photons may be absorbed and reemitted multiple times, so that they may travel at something less than the speed of light.
    
    Question: where have you experienced radiative diffusion in everyday life?
    
     
    
• Theoretical models of the Sun's interior tell us quite a bit about where its energy is produced and how it is transported to the surface.
  
   
  
• The Sun's core, the region where thermonuclear fusion occurs and energy is produced, extends out to about a quarter of the Sun's radius.
  
   
  
• About 99% of the Sun's mass lies within 60% of the Sun's radius.
  
   
  
• In the radiative zone, extending out to about 80% of the Sun's radius, radiative diffusion is the primary means by which the energy is transported to the surface.
  
   
  
• As fusion-produced photons travel outward, they interact with the nuclei and electrons which make up the Sun's plasma, and they are absorbed and reemitted time and time again.
  
  Travelling extremely slowly, photons can take hundreds of thousands of years to reach the surface.
  
   
  
• As the photons move outward from hotter to cooler regions, they will lose energy and their wavelength will decrease.
  
  Initially they are gamma photons; by the time they reach the surface, they have become a blackbody distribution, with about 46% visible, 43% infrared, and 11% ultraviolet.
  
   
  
• In the convective zone, the outermost 20% of the Sun's radius, the density is very low and convection becomes the most important means of energy transport.
  
   
  
• Although conduction certainly occurs to some degree in the denser parts of the Sun, it is not an important effect compared to radiative diffusion and convection.

12.5 The Solar Neutrino Problem

(Discovering the Universe, 5th ed., §9.9)

• Another type of radiative diffusion involves the neutrinos produced in the Sun's core, about 2 x 10³⁸ every second.
  
  Becaue neutrinos don't interact easily with matter, they will mostly zip right through the Sun, carrying away their small amount of energy at close to the speed of light.
  
   
  

Picture Information

• A small fraction of these neutrinos reach the Earth, and scientists have set up neutrino detectors to observe them.
  
  To avoid interference from other forms of radiation in the Earth's atmosphere, these detectors are built deep underground (typically in old mines), where only neutrinos can penetrate.
  
  For example, the new Sudbury Neutrino Observatory in Ontario is a 12-m sphere buried 2.1 Km below the ground.
  
  It is filled with heavy water, which consists of oxygen and deuterium instead of "ordinary" hydrogen.
  
   
  
• On rare occasions, about 1 in every 10¹⁵ neutrinos, an interaction will occur with one of the neutrons in the deuterium to produce a proton and an electron:
  
  
  
  Neutrinos can also simply collide with existing electrons.

Picture Information

In either case, the electrons are ejected at close to c, the speed of light in vacuum, which is much faster than the speed of light in the surrounding water (0.75c).

These high-speed electrons therefore produce a shock wave of light called Cherenkov radiation, which can be observed by the 9600 detectors mounted around the surface of the sphere.

Information about the neutrino's energy and direction can then be determined.

Question: what other physical phenomenon have we seen which is due to the reduced speed of light in a material?

• The first neutrino detectors demonstrated that the Sun does, in fact, emit neutrinos, confirming that nuclear fusion occurs at its core.
  
  Extra: The developers of the first neutrino detectors recently won the 2002 Nobel Prize in Physics.
   
  The early neutrino experiments also resulted in one problematic observation: they only counted between 1/3 to 2/3 of the calculated number of neutrinos, depending on their energy.
  
  This unexpected deficit is known as the solar neutrino problem.
  
   
  

• Experimental results from the Super Kamiokande neutrino detector, reported in 1998, have provided a solution to the solar neutrino problem by demonstrating that neutrinos have mass (although we still don't know the actual values).
  
  There are actually three types (or "flavors") of neutrinos, distinguished by their mass; the Sun produces the lightest neutrino.
  
  Because they have mass, neutrinos can easily "oscillate" from one flavor of neutrino to another.
  
  
  
  Since older neutrino detectors couldn't detect the heavier neutrinos after an oscillation occurred, there appeared to be a deficit of the Sun's lighter neutrinos.

12.6 Helioseismology

(Discovering the Universe, 5th ed., §9.8)

Picture Information

• In the 1960s astronomers discovered that the Sun's surface vibrates slightly, with a period of five minutes.
  
  Other vibrations were subsequently discovered, with periods of 30 minutes and longer.
  
  The study of these vibrations is known as helioseismology, from the Greek for "Sun" and "to shake".
  
  The image to the right of the Sun's surface reveals a mixture of these vibrations over a one-hour period.
  
   
  

Picture Information

• The Sun's vibrations are resonant sound waves, like those that occur inside the chambers of musical instruments.
  
  At any given time different portions of the Sun's surface are moving outward while other portions are moving inward.
  
  Question: which color (red or blue) corresponds to which motion?
  
  These vibrations can extend all the way into the interior of the Sun.
  
  If these vibrations were in air, they would sound like this. (AU format; Sound Information.)
  
  
  
• Analysis of these vibrations provides information about the interior of the Sun, in particular how its movements, density, temperature, and chemical composition vary with depth.
  
  As a result, observations of these vibrations provide important tests for theories of stellar structure and evolution.
  
   
  

Picture Information

• One important result of these observations are descriptions of the motion of the Sun's interior.
  
  The Sun experiences a differential rotation, i.e. different parts rotate at different rates, as shown in the picture at the right.
  
  This is not surprising, given that the Sun is fluid.
  
  There are two different regions: an outer shell, which correspond to the convective zone, and an inner sphere, which correspond to the radiative zone.
  
  In the outer shell, rotation varies with latitude.
  
  Outer equatorial regions rotate relatively rapidly, with a period of about 25 days (red).
  
  Outer polar regions rotate relatively slowly, with a period of about 35 days (blue).
  
  Intermediate latitudes (orange/yellow/green) rotate with a period in between these extremes.
  
  In the inner sphere, there is little variation in rotation, with a relatively short period of 25 days (red).
  
  Question: why might the inner regions of the Sun have much less variation in rotation speed?
  

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