Free Fall

The free fall timescale can be estimated using the law of gravitation. If we have a uniform sphere of mass, mutual gravitation will accelerate each point of the sphere toward the center. We can then use Newton's laws to calculate how long it would take to collapse the sphere if no pressure opposes it. The answer is

tfree fall = constant x (1/GD)1/2

where D is the average density and G is the universal gravitational constant. Thus, the free fall timescale is proportional to the inverse square root of the density and higher densities mean shorter timescales.

Dynamical Timescales

An important concept in the structure of stars is what we term a dynamical timescale. This is a fancy name for a simple idea: a dynamical timescale is a characteristic time for a particular change to take place. Dynamical timescales are important because they play a major role in determining stability. The stability of stars depends primarily on a balance between gravity and pressure, so the dynamical timescales for the star to respond to changes in these quantities are particulary important.
Free-Fall and Expansion
We may define one useful dynamical timescale by asking the following question: if the pressure keeping a star from collapsing gravitationally were suddenly taken away, how long would it take the star to collapse by a significant amount? This is called the free fall timescale, because we are imagining the matter to be falling freely toward the center of the star with nothing to slow it. Another dynamical timescale can be obtained by considering the opposite extreme: if gravity were taken away suddenly so that there was nothing to counteract the pressure, how long would it take the star to expand by a significant amount? We may call this the expansion timescale.

Hydrodynamical Timescale
We may combine these two timescales into a single one by the following reasoning. To maintain hydrostatic equilibrium, with its delicate balance between pressure and gravity, the free fall and expansion timescales must be comparable, since stability requires that the star be able to respond with equal rapidity to expansions or contractions away from the equilibrium point. This combined dynamical timescale is termed the hydrodynamical timescale for the star. It can be estimated using the gravitational law (see the box above) and one finds that different kinds of stars have very different hydrodynamical timescales. Some examples are summarized in the table given below.

Characteristic Hydrodynamical Timescales
Object ~M / MSun ~ R / RSun Hydro Timescale
Red Giant 1 100 36 days
Sun 1 1 55 minutes
White Dwarf 1 1/50 9 seconds

where the mass M and radius R are measured in solar units, and the symbol ~ means approximately.

The Role of Density
As described in the box above, the critical quantity that dictates this large difference in timescales is the average density. This is, in turn, determined by the mass and the radius. A white dwarf is much more dense than the Sun, which is in turn much more dense than a red giant. This causes the response time of a white dwarf to be much faster than that of a main sequence star, which is in turn much faster than that for a giant star. As noted in the right panel, the response time of a main sequence star is central to the stability of such stars. We shall find in later chapters that the hydrodyamical timescales for giants and white dwarfs can be important for what happens to stars after they leave the main sequence.