# Stellar Structure

It is said that Fred Hoyle once described the evolution of a star as a continual war between nuclear physics and gravity. The structure of a star can be characterized as a polarized battle in that war. The gravity of the stellar material pulls on all the other stellar material striving to bring about a collapse. However, the gravity is opposed by the internal pressure of the stellar gas which normally results from heat produced by nuclear reactions. This balance between the forces of gravity and the pressure forces is called hydrostatic equilibrium, and the balance must be exact or the star will quickly respond by expanding or contracting in size. So powerful are the separate forces of gravity and pressure that should such an imbalance occur in the sun, it would be resolved within half an hour. That fact that the sun is about five billion years old emphasizes just how exactly and continuously that balance is maintained.

In addition to its reliance on balance between gravity and pressure, the internal structure depends on the behavior of the stellar material itself. Most stars are made primarily of hydrogen, since it is the dominant form of matter in the universe. However, the behavior of hydrogen will depend on the temperature, pressure, and density of the gas. Indeed, the quantities temperature, pressure, and density, are known as state variables, since they describe the state of the material. Any equation or expression that provides a relationship between these variables is called an equation of state. The relevant equation of state and hydrostatic equilibrium go a long way toward specifying the structure of the star. However, there is one more ingredient necessary in order to determine the structure of the star.

Most of the energy which flows from a star originates at its center. The way in which this energy travels to the surface will influence the internal structure of the star.

There are basically three ways by which energy flows outward through a star. They are conduction, convection, and radiation. The flow of energy up an iron poker, which has one end in a fire, is energy transfer by conduction. This mode of energy flow is too inefficient to be of any interest in most stars. Most people have watched the energy flow from the bottom of a heated pot of water to the top. One sees large motion of heated material rising which is balanced by cooler material falling to the bottom where it, in turn, becomes heated and rises. This organized churning is called convection. In the interior of stars, when convection takes place it dominates the transport of energy over radiation. When the conditions are such that no convective churning takes place, no energy is carried by convection and the only remaining alternative is radiative transport.

The heat produced by an infrared heat lamp or the warmth of the sun on a clear day are good examples of energy transport by radiation. Radiative transport simply means the direct flow of radiant energy from one place to another. If the material through which the energy flows is relatively transparent, the flow takes place at virtually the speed of light. However, the more opaque the material is, the slower the flow of energy will be. In the sun, where light flowing out from in the core will travel less than a centimeter before it is absorbed, it may take a million years for the light energy to make its way to the surface.

The mode of energy transport, equation of state, and hydrostatic equilibrium can be quantified and self-consistent solutions found numerically on a computer for a star of given mass, composition and age. Such solutions are called model stellar interiors, and supply the detailed internal structure of a particular star. For the vast majority of stars which derive their energy from the nuclear fusion of hydrogen into helium the internal structure is quite similar. We call such stars main sequence stars.

The sun is such a star and has a central region where the material is quiescent and the energy flows through by radiative diffusion. The radiative core is surrounded by a churning convective envelope which carries the energy to within a few thousand kilometers of the surface. This outer percent or so of the sun is called the atmosphere, and here the energy again flows primarily by radiation as it escapes into space. This structure is common to all main sequence stars with mass less than about one and a half time the mass of the sun. The structure of more massive stars is nearly reversed. They consist of a convective core surrounded by a radiative envelope, which in turn gives way to an atmosphere where the energy also leaves the star by radiation.

There is a class of very dense stars known as white dwarfs. These stars originate as main sequence stars, but have changed dramatically over time. As a result, their internal structure has changed drastically. The matter of a white dwarf no longer behaves like a normal gas, and it has a different equation of state. The structure of such a star is vastly different from main sequence star. For example, while a white dwarf has a comparable mass to that of the sun, it is not much larger than the earth. This means that their average density is huge. A cubic inch of white dwarf material might weigh forty tons and exert a tremendous pressure to balance the fierce gravity resulting from as much material as one finds in the sun in a sphere no larger than the earth. Material in such a state is said to be degenerate. In these weird stars, conduction by free moving electrons becomes extremely efficient so that any internal energy is transported almost instantly within the star. The entire interior is at the same temperature and only a thin outer blanket of more normal non-degenerate material keeps the energy from leaking quickly into interstellar space.

More than 90% of all stars fall into the categories described above. The remaining stars are called red giants or red supergiants. These names are derived from the external appearance, for they are relatively cool and therefore red, but are extremely distended. If one were to replace the sun with the typical red giant, it would occupy most of the inner solar system. Should the replacement involve a red supergiant, even the planet Mars would find itself within the supergiant's atmosphere and Jupiter itself would be heated to the point where much of its substance would evaporate away. However, so huge is the volume occupied by these stars that their average density would be considered a pretty good vacuum by most physicists. The situation is made even worse since the inner cores of these stars are about the size of the earth and contain about ninety percent of the mass of the giant. Most red giants have an object much like a white dwarf located at their center. Therefore, their structure would best be described as a white dwarf surrounded by an extensive fog. The energy for most of these stars is supplied by the nuclear fusion of helium into carbon, which is a much less energy efficient source than the fusion of hydrogen to helium. This is the basic reason why there are so many more main sequence stars than red giants and super giants. Another reason is that a star uses up its helium much more quickly and spends relatively little time in the red giant phase. By determining how the structure of a star changes with time, the evolutionary history and fate of that star can be constructed. This is called the theory of stellar evolution.

## Resources

### Books

Abell, G.O. Exploration of the Universe. Philadelphia, New York: Saunders College Publishing, 1982.

Arny, T.T. Explorations: An Introduction to Astronomy. St. Louis, MO: Mosby, 1994.

Collins, G.W. II. Fundamentals of Stellar Astrophysics. New York: W.H. Freeman, 1989.

Seeds, M.A. Horizons-Exploring the Universe. Belmont, CA: Wadsworth, 1995.

George W. Collins

## KEY TERMS

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Hydrostatic equilibrium

—Expresses a static balance between the forces of gravity and the forces due to the internal pressure.

Ideal gas

—A term used to describe the behavior of a gas composed of simple atoms under standard conditions. An ideal gas obeys the following relation for the state variables of temperature T, number density n, and pressure P. The constant k is known as the Boltzmann constant. P = nkT.

Main sequence

—A sequence of stars of differing mass, but all deriving their energy from the nuclear fusion of hydrogen into helium.

Nuclear fusion

—The combining of the nuclei of two elements so as to produce an new more massive element. The process is accompanied by the release of energy as long as the end product of the reaction is less massive than iron.