Physical Processes in the Sun
The Sun is the astronomer's basic model for stars. We feel confident that we know the physical laws operating inside the Sun and therefore presumably inside other main-sequence stars. For example, if the Sun emitted radiant energy simply because it is a hot body without any internal source of energy, it should cool at a rate fast enough for us to measure the decrease. But the evidence suggests that the Sun is not cooling significantly; consequently the Sun and other main-sequence stars must have a source of energy deep in the interior that replaces energy radiated away from their surfaces, thereby keeping the deep interior hot.
Hydrostatic Equilibrium and Ideal Gases
For a star to resist the compression of gravity, or for that matter to neither expand nor contract, the upward forces on each layer from its center to its surface must equal the downward force. Such a condition is called hydrostatic equilibrium, and is described by the statement that:
Hydrostatic Equilibrium: At any distance from the center of a star, if the weight of overlying layers is balanced by the upward pressure of hot gas in layers closer to the center, then the star is said to be in a state of hydrostatic equilibrium.
Through most of a star's existence, the gases inside the star conform to what is called the perfect-gas law, which says that:
Perfect Gas Law: In an ideal or perfect gas, the particles composing the gas occupy only a negligible portion of the volume, and when the gas particles collide, they do so quickly bouncing off each other like tiny billiard balls. For a perfect gas, the pressure of the gases (which results from the forces imparted by collisions) is directly proportional to the density and temperature of the gases.
At the high temperatures inside stars collisions between gas particles will strip electrons from atoms, leaving a plasma of bare nuclei and free electrons. Because most of an atom is just empty space, free electrons and bare nuclei can crowd much closer together in a plasma than they can in a gas composed of atoms, allowing the material inside stars to remain gaseous even at very high density. In the high-temperature and high-density regions near a star's center, gas particles increase their speed and so collide more often and more violently. Consequently, the pressure exerted by the gas is greater in the center and declines outward. If the gas pressure exceeds the force of gravity, a star must expand; if gravity is greater, a star should contract. Thus each layer inside a star will undergo adjustments if a net force exists until hydrostatic equilibrium is restored.
For the hottest stars on the main sequence, the high-energy photons of very intense radiation fields inside these stars can add an additional pressure, a phenomenon known as radiation pressure, which along with the gas pressure helps balance the weight of overlying layers.
Thermal Equilibrium
During most of a star's existence, it maintains itself in a state that we call thermal equilibrium. By this we mean that:
Thermal Equilibrium: In thermal equilibrium, the amount of energy generated inside a star equals the amount radiated away into space as luminosity at its surface. As a result, the temperature and pressure at all points inside such a star remain reasonable constant during such a period.
Thermal equilibrium is self-regulating in that if more energy is released in a star's center than is radiated away at its surface, the temperature inside the star rises. Because gas pressure depends directly on temperature, the pressure increases, and such a star should expand, with heat or thermal energy being transformed into gravitational potential energy. This change in turn cools the gas, decreasing the gas pressure, and hydrostatic equilibrium is achieved at a larger radius. If, however, too little energy is produced, a star will contract and heat up, which increases the gas pressure and stops further contraction. Hydrostatic equilibrium is now found at a smaller radius.
What would happen if gas pressure did not depend on temperature? Heating or cooling from changes in the amount of energy generated would not be checked by increasing or decreasing the gas pressure followed by subsequent changes in a star's size. Such a situation does happen and is called a degenerate gas. It may apply to all constituents of a gas or to only one of them, such as to free electrons, in which case it would then be called a degenerate electron gas. In Section 18.2, we shall see the significance of the development of degeneracy in stars.
Degenerate Gases
In Section 12.2 it was pointed out that there is a limit to the number of electrons that may occupy a particular orbit, or energy level, in an atom. This exclusion principle, originally proposed by the theoretical physicist Wolfgang Pauli (1900-1958), also applies to electrons that are not bound to a nucleus but merely confined to a fixed volume, such as the deep interior of a star. In such a volume there are only certain discrete energy states available to electrons. When the density of a gas is quite low, as in the air in a room, there are always enough unoccupied energy states available that electrons may readily gain or lose energy in order to move from one energy state to another. However, when the density of a gas is quite high, as in the interior of a star (Table 17.1), the lowest energy states may all be filled, leaving only the highest still unoccupied. In this case, electrons cannot readily gain or lose energy, and they will become highly incompressible. Even though thermal energy may be extracted from the gas, it still may not cool down, since electrons cannot give up energy by moving to a lower energy state, and thus they exert a dominant influence on the behavior of the overall gas.
In a degenerate gas, the average pressure is high enough to keep material from being compressed by gravity. In addition, because the kinetic energies of electrons are quite high and the rate of collision between electrons and other particles is quite low, degenerate electrons can travel great distances at velocities that approach the speed of light. And unlike pressure in a perfect gas, the pressure that degenerate electrons exert has little to do with their temperature. When electron degeneracy occurs within stellar matter, major changes in the thermal balance can make a star unstable.
Transporting Energy
Energy may be transported in one or more of three distinct ways, although usually one means is more efficient than the others under the prevailing conditions. One of these means is conduction, which is the way heat is transferred along a metal rod placed in a fire. In the interior of main-sequence stars obeying the perfect-gas law, this is a very ineffective means of energy transfer. In the cores of stars where the material has become degenerate, however, conduction becomes the primary means of transporting energy.
A more frequently occurring means for transporting energy in stars is convection, in which gas circulates between hot and cool regions, transferring thermal energy to the cool region (Figure 17.1 and Section 16.1). Once a pattern of circulation is established, convection can be a very efficient mechanism of energy transport. For layers in which temperature changes quite rapidly with depth, convection develops as the principal means of carrying energy. These layers are referred to as a convective zone, and in them the atomic constituents are well mixed by the continual stirring.
Radiation is the third way of moving energy. Inside a star, photons resulting from energy-generation processes diffuse outward through overlying material. After a photon is emitted, it very soon is either absorbed by an atom (or ion) or scattered by free electrons after traveling a characteristic distance that ranges from very small fractions of a centimeter deep in a star's interior to several kilometers in its photosphere. Although the direction in which a reemitted or scattered photon can move is generally arbitrary, there will be a net drift of photons outward from the center to the surface. Where energy is transported by radiation, such layers are known as a radiative zone. In a radiative zone, chemical elements go through very little mixing; any chemical inhomogeneity developing here should persist.
Opacity
Radiation and matter continually interact by absorption, reemission, and scattering of photons, processes that impede the outward flow of radiant energy (Figure 17.1). Such processes constitute what is known as the matter's opacity.
Opacity: Matter's resistance to the flow of radiation through it is measured by its opacity. If the resistance is almost complete, then we say that the matter is opaque to radiation of that particular wavelength or interval in wavelength. If the resistence is minimal, then we say that the matter is transparent to radiation of that particular wavelength. For example, concrete block walls are opaque to visible light but transparent to radio-wavelength radiation.
For stars, in regions of large opacity, the temperature drops rapidly outward, and convection will take over as the primary way of transporting energy. When the density is low, radiation travels more freely through a star.
Energy is liberated in the deep interior of stars as a comparatively small number of high-energy gamma-ray photons. As these photons work their way outward, absorption and reemission by overlying matter degrades the gamma-ray photons into millions of lower-energy ones by the time they reach the photosphere. Radiant energy takes hundreds of thousands of years to diffuse through the Sun from its energy-generating core.
Thermonuclear Fusion
Over a century ago astronomers understood that the energy already radiated by the Sun could never have been supplied by ordinary combustion (for example, by the burning of wood or coal). Another way of producing energy known to astronomers was by the conversion of gravitational potential energy into heat during contraction. In fact, in the nineteenth century, this was thought to be the only source for the Sun's energy. We now know that contraction is a vital source of energy on which a star can draw at various stages in its life. But at its current luminosity, our Sun could not have survived on gravitational potential energy alone for more than about 15 million years.
For stars like the Sun, a source of energy must keep the luminosity approximately constant for billions (not just millions) of years. The question plaguing astronomers in the early part of this century was what that source is. The answer is the fusion of small-mass nuclei to form more massive nuclei. Sir Arthur Eddington (1882-1944) suggested in 1920 that fusion of hydrogen could form helium and that this could be the long-sought fuel. After it was found that stars have vast quantities of hydrogen, the physicist Hans Bethe (b. 1906) proposed in 1938 a way in which four hydrogen nuclei (four protons) could be converted into a helium nucleus, releasing energy. If many hydrogen nuclei are converted, they will release sufficient energy through this process (known as thermonuclear fusion) to keep stars shining for billions of years.
What determines whether hydrogen can be fused to form helium? The answer is the temperature and density of a gas; the higher the temperature and density, the more readily the process will proceed. Thermonuclear reactions will therefore be most numerous in a star's central region, where the temperature and density are highest. The reactions will gradually decline to zero somewhere out from the center, where temperature and density are too low to sustain them. This distance from the center, then, defines the energy-generating core of a star.
The p-p Chain and CNO Cycle
Hydrogen burning proceeds by two principal schemes (Figure 17.2): the proton-proton chain (p-p chain) and the carbon-nitrogen-oxygen cycle (CNO cycle). In each process, four protons are fused into one helium nucleus with a slight loss in mass, which is converted into energy. Which thermonuclear process produces more energy depends on the temperature. Up to about 16 million K the p-p chain dominates. Beyond that temperature, however, the CNO cycle takes over as the most important thermonuclear process. The average rate of energy generation for the entire Sun, which depends primarily on the p-p chain, is about 2 erg/g.s. For a star of 10M., the average rate of energy generation, supplied principally by the CNO cycle, is about 1000 times greater than that of the Sun.
The mass of the end product of hydrogen burning, He4, is 0.71 percent less than the combined masses of four reacting protons (H41). What has happened to the rest of the mass? Early in this century Einstein pointed out that there is an equivalence between mass and energy. Mass is just one more manifestation of energy, and what is conserved in any type of interaction between particles of matter is the total energy, including the energy equivalent of the mass. The equivalence is symbolized in Einstein's equation E = mc2, where E is the energy, m is the mass, and c is the velocity of light.
In hydrogen burning 1 g of hydrogen is converted into 0.9929 g of helium plus 6.4 x 1018 erg of energy--exactly 0.71 percent of the original 1 g of hydrogen times c2. In what form does the energy appear? In the various steps, several gamma-ray photons are created and degraded by absorption and reemission into many photons having the same total energy. Also, some of the material particles created have large kinetic energies, which they will soon redistribute to other particles by collisions. Thus both radiant energy and heat energy come from the mass that is lost in these thermonuclear fusion processes.
Translated into practical units, every second the Sun converts 600 million metric tons of hydrogen into 596 million metric tons of helium and 4 million metric tons of mass into energy. This energy will diffuse to the surface, where it will supply the 3.83 x 1033 erg of energy radiated away into space each second. In its core the Sun has enough hydrogen to keep it shining for about 10 billion years. In 4.6 billion years the Sun has existed so far, it has used up about half its core's hydrogen supply and lost about 0.043 percent of its mass.
Steps in Hydrogen Burning
Step 1 in the p-p chain (Table 17.2) is fusion of two colliding protons (H1) to form a deuteron (H2), which is the nucleus of the hydrogen isotope deuterium, resulting in the emission of a positron (e+) and a neutrino ( ). This reaction happens, on average, once every 14 billion years for each isolated pair of protons. The time for the entire thermonuclear process is determined by this first step, and it is only the enormous quantity of hydrogen in the cores of stars that makes this process a significant source of energy.
The positron is a positively charged particle with the mass and other characteristics of an electron; it is the antiparticle for the electron. The collision of a positron with an electron destroys them as matter and creates two gamma-ray photons. The neutrino, however, is a massless, chargeless particle that moves at the speed of light. Since the neutrino has a low probability of interacting with matter, it immediately escapes from the star, carrying away about 2 percent of the energy released in the p-p chain of reactions.
Step 2 in the p-p chain is the collision within a few seconds of another proton with the deuteron to fuse and form the light isotope of helium (He3), resulting in the emission of a gamma-ray photon. Finally, in step 3, two He3 nuclei collide every few million years and fuse to form the heavy isotope of helium (He4), accompanied by the return of two protons. (We should point out that there are other branches of these reactions leading to the same end.)
All told, six protons have taken part in producing two He3 nuclei, from which one He4 nucleus is produced and two protons are returned to the reservoir of fusionable matter.
The other hydrogen-burning reaction, the CNO cycle, has six steps occurring at rates between 80 seconds and 300 million years but leading to the same result as the p-p chain; that is, the conversion of four protons to produce one helium nucleus and to liberate energy. The cycle begins with C12 and closes with the return of C12, so that carbon is thus only a catalyst that makes the reaction go.
Constructing Stellar Models
Each of the physical processes described in the preceding section depends on several physical quantities, among them temperature, density, pressure, and the mass and luminosity of a star. We can use a symbol to represent the numerical value of each quantity and combine these symbols into mathematical equations embodying their relationships. These equations, called equations of stellar structure, describe how mass, pressure, temperature, and luminosity vary outward from the center of a star. Within these equations are additional quantities, such as density, chemical composition, opacity, and the rate at which energy is generated. To construct models of stars, we take their observed properties--such as mass, radius, luminosity, surface temperature, and estimated chemical composition--as constraints in solving the equations of stellar structure at a discrete number of points along the radius. The solution is a mathematical model, or a stellar model. The computer makes it possible to calculate these models in reasonable lengths of time for several hundred points. Without the high-speed digital computer, much of the progress we have made in the last 20 years in this field would not have been possible.
An example of such a mathematical model for the Sun at present is given in Table 17.3 and is shown schematically in Figure 17.3. As a star evolves, it alters the structure of its layers from center to surface. The chemical composition is also changing in the layers responsible for nuclear burning.
We can calculate a sequence of mathematical models simulating the restructuring that real stars presumably undergo as they age during their life spans of millions of human generations. For each model in a sequence, the surface temperature and luminosity are a point in the H-R diagram for a particular stellar age. How do we test the validity of our models? We see how well this time sequence of points, or evolutionary track, for one star helps us predict the distribution of real stars in the H-R diagram. Both open and globular clusters are important observational keys in checking our results. The color-magnitude diagrams of clusters, such as Figures 14.16 and 14.17, give the best evidence on stellar aging because a cluster is a group of stars with a definite range of masses that began their existence at about the same time and in the same place. They also formed from the same material and so at first were reasonably similar chemically. From the distribution of cluster stars in the H-R diagram, we can then deduce details of how individual stars age.
Oscillations of the Sun
The Sun radiates as much energy away from its surface as is liberated in its interior. If it did not, the photosphere would cool down or heat up, which does not appear to be happening. The Sun is in a state of thermal equilibrium, with the temperature declining from some 15 million K at the center to about 6000 K in the photosphere. The density also declines from a central value of about 160 g/cm3 to a mere 10-7 g/cm3 in the photosphere. The drop-off in density is so rapid that most of the Sun's mass is in its deep interior, close to the center; in fact, some 90 percent of its mass lies within 50 percent of the radius.
Sparked by the discovery of 5-minute solar oscillations in the early 1960s a number of astronomers were able to show that the Sun is also vibrating with periods of tens of minutes up to 160 minutes. Thus the Sun vibrates much as a ringing bell does (Figure 17.4). Just as oscillations of the Earth in the form of seismic waves after earthquakes can be used to probe the Earth's interior, these solar oscillations are indicative of the physical conditions inside our Sun. As such, they can be used to verify the theoretical interior structure that we described earlier. This relatively new field of solar research is called solar seismology, and such efforts as setting up telescopes at the South Pole to observe solar oscillations around the clock will eventually provided a first-hand understanding of the solar interior.
Solar Neutrino Experiment
How certain are astronomers about the thermonuclear processes going on inside stars? So far, the only experiment designed to test the theory directly is one that tries to detect the neutrinos created in thermonuclear processes. Neutrinos pass freely out of the Sun into space, and so carried away with them, at the speed of light, is a small fraction of the energy generated in the Sun. By detecting solar neutrinos, we can have first-hand information on the average temperature in the hydrogen-burning core.
One scheme for detecting solar neutrinos uses a huge tank filled with 400,000 liters of dry-cleaning fluid (Figure 17.5). It is located deep in a South Dakota gold mine to shield the chlorine atoms in the solvent from cosmic-ray particles; nothing is allowed to reach them but neutrinos. When the nucleus of a chlorine atom (Cl37) captures a neutrino, which is not often, it is transformed into radioactive argon (Ar37), which has a half-life of about 35 days. The argon nucleus is recovered, and its decay is monitored. We find that the observed flux of solar neutrinos is about four times smaller than the rate predicted from standard mathematical models of the Sun. By manipulating the solar model, which is subject to some uncertainties anyway, the discrepancy can be reduced but not eliminated. Other explanations for the discrepancy have been proposed, but none has received widespread endorsement. Plans are being made for a new experiments that will measure the lower-energy neutrinos that are not counted in the experiment just described.
Recently, another explanation has arisen as a result of two studies of the historical measurements of the Sun's diameter. Both studies seem to find that the Sun has been shrinking for the last 100 or so years. The question is by how much. The contraction, if real, amounts to about 0.01 to 0.1 percent per century. Further speculation prompted by this finding suggests that the Sun may undergo a long-term cycle of very slow expansion and contraction. During contraction, the Sun derives heat energy from gravitational potential energy and lowers its rate of hydrogen burning. This would account for the emission of fewer neutrinos. These results are a long way from being thoroughly verified.
The solar neutrino problem is serious, for it casts doubt on our knowledge of the details of structure and/or energy generation in main-sequence stars. Thus we are forced to look more carefully at the details of these processes in the Sun if the experiment is completely correct. But it is unlikely that the solar neutrino problem forecasts the failure of the present theory of stellar evolution. The details of the theory will change over time as a result of the solar neutrino problem and others, but the general outline, which is covered in the remainder of this chapter and the next two, seems likely to endure.