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Stars

The Gravitational Collapse of Stars

We are accustomed to longevity in astronomy. The Sun has burned for 4.5 billion years, orbited by planets of equal age. Many of the stars in our galaxy are over 10 billion years old. These stars will eventually burn out and grow cold, but they will change only slowly. But this is not that fate of all stars, for some stars are vulnerable to a catastrophic collapse. Gravitational collapse produce supernovae from blue giants and degenerate dwarfs. Gravitational collapse sets a maximum mass for both the degenerate dwarf and the neutron star. Gravitational collapse is the creator of black holes. And gravitational collapse, whether of a blue giant, degenerate dwarf, or a neutron star, in all cases has the same principal cause—special relativity.

A small body such as Saturn is stable against gravitational collapse because of the way the pressure exerted by the material at its core changes with density. If we gave Saturn a slight squeeze, both the gravitational force and the pressure within its core would increase. The gravitational force would rise simply as the inverse-square of the radius, but the force of the pressure would rise faster than the inverse-square of the radius. This imbalance of forces would cause Saturn to expand back to its equilibrium radius. This is true regardless of how cold Saturn grows.

But what if the material at the core of an object behaved differently than the material in Saturn. If the force of the pressure rose less rapidly than the inverse-square of the radius, this force would be less than the force exerted by gravity, and the object would collapse.

We can put a firm number on the condition for gravitational collapse if we let the pressure change with density as p ∝ ργ, where p is the pressure, ρ is the density, and γ is a parameter called the adiabatic index. A material obeying this equations is stable to gravitational collapse if the pressure times the surface area increases more rapidly than R-2. Because the density is proportional to R-3, the force exerted by the pressure is proportional to R2-3γ. This force increases more rapidly than the gravitational force when γ > 4/3.

The adiabatic index hides a tremendous amount of physics. Issues such as ionization can cause dramatic changes in the value of the index. In general, however, the values encountered in astronomical sources in the absence of thermonuclear fusion hover around either 5/3 or 4/3. Which of these values occurs depends on the temperature and pressure of the material. If the temperature and pressure are low, the adiabatic index has a value of 5/3, and an object composed of this material is stable against gravitational collapse, but if either the temperature or the pressure is high, the adiabatic index has a value of 4/3, and an object composed of this material is unstable to gravitational collapse. Why do the temperature and pressure play such an important role? If both the temperature and density are low, the average speed of the particles in the material is much less than the speed of light, and the material has a 5/3 adiabatic index. But if the temperature is high, the average speed of the particles is close to the speed of light, and the material has a 4/3 adiabatic index. If the density is high, an effect of quantum mechanics, the Pauli exclusion principle, causes the average speed of the particles to be close to the speed of light, giving the material a 4/3 adiabatic index. The specific property of special relativity that causes change in adiabatic index is the relationship between energy and momentum. For a velocity much less than the speed of light, a particle's kinetic energy is proportional to the square of its momentum, but for a velocity close to the speed of light, the kinetic energy is directly proportional to the momentum. This simple change in the relationship between energy and momentum is sufficient to destabilize a star.

A star's mass is the property that drives a star into instability. At a low mass, gravity can be counteracted by a low pressure, which means low pressure and density. As the mass increases, the pressure increases, and eventually either the temperature or the density is sufficiently high to cause gravitational collapse. For this reason, large stars collapse and supernovae, but small stars do not.

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