Temperature and
Pressure in Stars

Stars are composed of hot gases in which the atoms and molecules are almost completely ionized in the interior (the state of matter called a plasma). The question of whether fusion reactions can occur in this plasma is primarily one of the density and the temperature of the gas, because the density controls the number of collisions and the temperature controls their average energy. Since for a fixed volume and temperature the density and pressure are directly related, we may also use the pressure instead of the density as a variable.

Kinetic Theory of Gases

According to the kinetic theory of gases, the temperature of a gas is just a measure of the average velocity of particles in the gas. For normal gases, this distribution is called a Maxwellian distribution, and is illustrated in the following figure for two different gas temperatures.

Maxwellian Velocity Distributions for Two Temperatures

In this plot the horizontal axis is the velocity in cm/sec and the vertical axis is proportional to the probability that a particle in the gas has that velocity.

Thus, we see that each temperature corresponds to a range of velocities, but the average velocity (indicated by the vertical lines) increases with temperature, and likewise the probability of having very high velocities for some gas particles increases with temperature.

Ideal Gas Law

The behavior of gases in most (but not all) cases in astronomy is well approximated by a simple set of rules and equations called the Ideal Gas Law. This law embodies much of our experience with gases under everyday conditions. For example, that if the volume is held constant, the pressure increases if the temperature increases. The ideal gas laws ensure us that the temperture in the center of stars will be quite high, and the kinetic theory of gases implies that these high temperatures mean that high velocity collisions are more likely between ions in the plasma.

The Energy Window for Nuclear Reactions

The Coulomb barrier for charged particle reactions and the distribution of velocities implied by the kinetic theory of gases imply that there is a narrow range of energies where nuclear reactions involving charged nuclei occur in stars. This window is called the Gamow window, and is illustrated schematically in the following image.

The Gamow window for charged-particle reactions

The peak is the product of the two curves decreasing in opposite directions: The probablility for penetrating the Coulomb barrier goes down rapidly with decreasing energy (the curve marked "barrier penetration"), but at a given temperature the possibility of having a particle of high energy (and therefore high velocity) decreases rapidly with increasing energy (the red curve).

The sum of these opposing effects produces an energy window for the nuclear reaction: only if the particles have energies approximately in this window can the reaction take place. This places very strong constraints on the charged-particle reactions responsible for producing fusion energy in stars.