In the 1930's considerable progress was made in understanding why most stars cluster about the main sequence line in an H-R diagram. While nuclear energy was not yet demonstrated on earth, it was evident that some such origin had to be assumed for the tremendous energy which the Sun continues to pour out. Even today the fusion process whereby in the Sun hydrogen is fused into helium has not been duplicated in a controlled manner on earth. Nevertheless, the hydrogen bomb is an example of uncontrolled fusion on earth. On the Sun this process is referred to as "hydrogen burning" although it is an abuse of the word "burning" for no chemical oxidation is implied.
Among the Dover paperbacks are several which describe models of stellar structure. One early book was Eddington's "The Internal Constitution of the Stars." More up-to-date are the books on stellar structure by S. Chandrasekhar.
At present all realistic stellar models are based upon numerical solution of differential equations which characterize hydrostatic equilibrium, conservation of energy, radiative transport, and convective transport. The study of such detailed models would be appropriate in the senior course in astrophysics.
It is now generally accepted that stars are formed out of interstellar gas and dust, and that multiple star formation occurs frequently, this giving rise to temporary associations of stars. While at this time it is still unclear how the individual stars begin to form, one has a fairly definite idea concerning how the evolution proceeds once a protostar has formed within a gas cloud. We can adequately describe the evolution from the point where the protostar has a radius about 104 RO, the very low density 10-12 g/cm3 and the temperature 103 oK.
Because of gravitational attraction the material of the protostar gradually falls inward. In several hundred years the radius of the protostar will decrease by a factor ten. Gravitational potential energy is converted into thermal energy, with a resulting rise of temperature of the gas. Hydrogen molecules are dissociated; the hydrogen atoms are ionized into their constituent protons and electrons. This happens especially at the center of the protostar where the temperature is higher. All the hydrogen in the core will be ionized when the temperature reaches 105 oK there. Meanwhile the surface of the protostar will only reach several thousand degrees Kelvin, so just below the surface there will be not only neutral hydrogen atoms but even H- ions. The latter ions play an important role in the thermodynamics of the protostar, because they contribute greatly to the opacity of the outer layers of the protostar.
The fact that the surface of the protostar has only a moderate temperature increase during the collapse means that the energy radiated per unit surface area goes up only by an order of magnitude (s wiggle T4), while the surface area decreases by several orders of magnitude. For this reason the luminosity of the protostar decreases as it contracts. The protostar, therefore, moves almost straight downward on the H-R diagram along a path at the far right. For a star of solar mass this process continues for about 107 years. At this point the temperature at the center of the star is so high that nuclear fusion of protons into Helium is initiated. The surface temperature of the star as well as its luminosity increases. The star moves upward to the left on the H-R diagram; it thus joins the main sequence.
If a similar analysis is carried out for stars of different total mass, one gets modified trajectories on the H-R diagram. Notice the spectacular difference in the time scales involved. A star of 50 MO forms in about 200 years. Once Helium begins to be formed it takes about 2.8 x 104 years for the star to assume its place on the main sequence. The blue giant stars are very young! The Sun took almost 1000 times longer to reach the main sequence, 27 x 106 years. It will remain on the main sequence some 9 billion years in all; it has already been there about 4 1/2 billion years.
Because stars spend a relatively short time reaching the main sequence, it is hard to find stars which are now in the process of reaching the main sequence. A few decades ago, however, direct evidence was found for this picture of the early life of a star.
On the main sequence two types of nuclear reaction compete. One is the process of fusion discussed earlier. The other involves Carbon, Nitrogen and Oxygen. The latter process dominates in the case of massive stars, while for less massive stars the fusion process is dominant. It is a question of the temperature dependence of the reactions. The C-N-O process gives rise to a convective core for massive stars, while the cores of stars like the Sun are nonconvective. In the latter type of star energy is transported outward through the core by radiation. However, the outer shell of the Sun is convective.
During the 9 billion years that the Sun spends on the main sequence, hydrogen is converted into helium. Right now the proportion of helium at the hot central core is about 70% by weight, while in the outer regions of the Sun the percentage is about 30%, which is not very different from the abundance in interstellar gas.
During this hydrogen burning phase the Sun will get increasingly dense in the center. The radius of the Sun will increase very slowly in order to compensate for the increased rate of buring of its nuclear fuel. As a result the temperature remains fairly constant, but the increasing radius implies an increasing luminosity too. The Sun is growing gradually brighter. During its 9 billion year life-time the Sun will double in size from its original size 4 1/2 billion years ago.
Once the central core becomes 100% helium, dramatic changes occur in the star. The helium core shrinks just like the protostar did before the hydrogen burning commenced early in the life of the star. This shrinking supplies thermal energy by converting gravitational potential energy, but we have already seen that this process cannot sustain a star for very long. However, one of the first effects of the core contraction will be the rapid burning of hydrogen at the fringes of the shrinking helium core. The luminosity of the star does not increase immediately, however. Instead, the hydrogen in the outlying areas of the star expands, and the surface temperature of the star decreases markedly. It reddens. This corresponds to horizontal motion to the right in the H-R diagram. Finally, however, the luminosity does increase as the hydrogen burns more furiously. The star enters the red giant phase of its life. For a star the mass of the Sun this takes about 1 billion years.
The core of the red giant has a density about 1 ton per cubic inch, and an overall size a couple times larger than the earth. 25% of the star's mass is located within this dense core. The hydrogen which burns is in a shell several thousand miles thick, surrounded by a very tenuous envelope with an average density of 10-7 ounces per cubic inch. That density is equivalent to a very good vacuum on earth! This envelope is so huge that the earth's entire orbit would fit inside.
Eventually the temperature at the center of the helium core of the red giant reaches 108 oK, at which point three helium nuclei combine to form a carbon nucleus. This is called "helium burning."
In the case of the hydrogen-burning star, thermal runaway is avoided by the expansion of the star. However, the dense helium core of a helium-burning star expands very little as it heats up. The reason for this is associated with the Pauli exclusion principle of quantum mechanics. In any event there is no safety valve for the helium-burning star. The core of a star of solar mass will simply explode like a bomb at this point. This is called the "helium flash." With its core greatly expanded, the star's temperature is no longer such as to burn hydrogen, so its luminosity drops precipitously. It moves downward on the H-R diagram. In addition, the star as a whole contracts under the influence of gravity. This slide continues for about 104 years until Helium burning ultimately succeeds in establishing itself. This point is the analog of the time early in the life of the star when hydrogen burning commenced. For a while the star moves horizontally to the left on the H-R diagram coming close to the main sequence line. When the helium in the core is exhausted, replaced by carbon, the star again moves up to the red giant region of the H-R diagram. Typically 108 years are spent going from point 7 to point 9 in the diagram, much less than the 1010 years spent on the main sequence as a hydrogen-burning star.
Eventually this shuttling back and forth between the main sequence line and the red giant region of the H-R diagram ceases. Next, we shall discuss the ultimate fate of stars after they are no longer welcome in the red giant region.