3.3: Time Scales

One of the most useful notions in stellar astrophysics for establishing an intuitive feel for the significance of various physical processes is the time required for those processes to make a significant change in the structure of the star. To enable us to estimate the relative importance of these processes, we shall estimate the time scales for several of them. In Chapter 2 we used the free-fall time of the sun to establish the fact that the sun can be considered to be in hydrostatic equilibrium. The statement was made that this time scale was essentially the same as the dynamical time scale. So let us now turn to estimating the time required for dynamical forces to change a star.

a. Dynamical Time Scale

The Virial theorem of Chapter 1 [equation \ref{1.2.34}] provides us with a ready way of estimating the dynamical time scale, for in the form given, it must hold for all \(\mathrm{1/r^2}\) forces. Consider a star which is not in equilibrium because the internal energy is too low. As it enters the non-equilibrium condition, the star's kinetic energy will also be small. Thus, the Virial theorem would require \[\frac{d^2 I}{d t^2} \approx \Omega\label{3.2.1}\]

implying a rapid collapse. If we take as an average value for the accelerative change in the moment of inertia \[\frac{d^2 I}{d t^2} \approx-\frac{I}{\tau_d^2}\label{3.2.2}\]

where t_d is the dynamical time by definition, then we get \[\tau_d^2=\frac{-I}{\Omega}=\frac{\frac{2}{5} M R^2}{\frac{3}{5} G M^2 / R}\label{3.2.3}\]

or \[\tau_d=\left(\frac{\frac{2}{3} R^3}{G M}\right)^{1 / 2}\label{3.2.4}\]

Now we compare this to the free-fall time obtained by direct integration of \[\frac{d^2 r}{d t^2}=-\frac{G M(r)}{r^2}\label{3.2.5}\]

remembering that, since the star is "free-falling", M(r) will always be the mass interior to r. Thus, a surface point will always be affected by the total mass M. With some attention to the boundary conditions [see equations \ref{5.2.12} through \ref{5.2.17}], direct integration yields a free-fall time of \[\tau_f=\frac{\pi}{2}\left(\frac{R^3}{2 G M}\right)^{1 / 2}=\left(\frac{3 \pi}{32 G\left\langle\rho_i\right\rangle}\right)^{1 / 2}\label{3.2.6}\]

which is essentially the same (within about a factor of 1.4) as the dynamical time.

Although we considered a star having zero pressure in order to derive both those time scales, the situation would not be significantly different if some pressure did exist. While a collapse will cause an increase in the pressure, the Virial theorem assures us that the gravitational energy will always exceed the internal energy of the gas unless there is a change in the equation of state resulting in a sudden increase in the internal energy. However, for the interior of the star to adjust to the collapse, it is necessary for information regarding the collapse to be communicated throughout the star. This will be accomplished by pressure waves which travel at the speed of sound. The sound crossing time is \[\tau_s=\frac{R}{\left\langle c_s\right\rangle}=R\left\langle\frac{\gamma P}{\rho}\right\rangle^{-1 / 2}=R\left\langle\left(\frac{\gamma k T}{\mu m_h}\right)^{-1 / 2}\right\rangle\label{3.2.7}\]

For a monatomic gas \(\gamma=5/3\). Hence \[\left\langle c_s\right\rangle=\left(\frac{5 k}{3 \mu m_h}\right)^{1 / 2}\left\langle T^{1 / 2}\right\rangle\label{3.2.8}\]

We may estimate the mean temperature for a uniform density sphere from the integral theorems [equations \ref{2.2.4} and \ref{2.2.7}] and obtain \[\tau_s=\left(\frac{3 R^3}{G M}\right)^{1 / 2}\label{3.2.9}\]

Although the sound crossing time is somewhat larger than the free-fall and dynamical time scales, they are all of the same order of magnitude, \(\sqrt{\left(\mathrm{R}^3 / \mathrm{GM}\right)}\). This is about 27 min for the sun. The similar magnitude for these times is to be expected since they have a common origin in dynamical phenomena. So we have finally justified our statement in Chapter 2 that any departure from hydrostatic equilibrium will be resolved in about 20 min. This short time scale is characteristic of the dynamical time scale; it is generally the shortest of all the time scales of importance in stars.

b. Kelvin-Helmholtz (Thermal) Time Scale

Now we turn to some of considerations that led Lord Kelvin to reject the Darwinian theory of evolution. These involve the gravitational heating of the sun. If you imagine the early phases of a star's existence, when the internal temperature is insufficient to ignite nuclear fusion, then you will have the physical picture of a cloud of gas which is slowly contracting and is thereby being heated. Ultimately some of the energy generated by this contraction will be released from the stellar surface in the form of photons. As long as the process is slow compared to the dynamical time scale for the object, the Virial theorem in the form of equation \ref{1.2.35} will hold and \(\langle\mathbf{T}\rangle \approx 0\). Thus \[1 / 2\langle\boldsymbol{\Omega}\rangle=-\langle\mathbf{U}\rangle\label{3.2.10}\]

which implies that one-half of the change in the gravitational energy will go into raising the internal kinetic energy of the gas. The other half is available to be radiated away. This was the mechanism that Lord Kelvin proposed was responsible for providing the solar luminosity and he suggested a lifetime for such a mechanism to be simply the time required for the luminosity to result in a loss of energy equal to the present gravitational energy. If we estimate the latter by assuming that the star of interest is of uniform density, then \[\tau_{\mathrm{K}-\mathrm{H}}=-\frac{\Omega}{L}=\frac{\frac{3}{5} G M^2}{R L}\label{3.2.11}\]

This is known as the Kelvin-Helmholtz gravitational contraction time, and it is the same as the lifetime obtained from the gravitational energy given in the previous section. Since the star is simply cooling off and having its internal energy re-supplied by gravitational contraction, some authors refer to this time scale as the thermal time scale. More properly, one could define the thermal time scale t_th as the time required for the luminosity to result in an energy loss equal to the internal heat energy, and then one could relate that to the Kelvin-Helmholtz time by means of the Virial theorem. That is, \[\tau_{\mathrm{th}}=\frac{\langle U\rangle}{L}=-\frac{\langle\Omega\rangle}{\langle 3(\gamma-1)\rangle L} \approx \frac{\tau_{\mathrm{K}-\mathrm{H}}}{\langle 3(\gamma-1)\rangle}\label{3.2.12}\]

Thus, we see that the two time scales are of the same order of magnitude differing only by a factor of 2 for a monatomic gas. For the sun, both time scales are of the order of 10¹¹ times longer than the dynamical time. In general the thermal time scale is very much longer than the dynamical time scale. The thermal time scale is the time over which thermal instabilities will be resolved, and so they are always less important than dynamical instabilities.

c. Nuclear (Evolutionary) Time Scale

In the beginning of this section we estimated the lifetime of the sun which could result from the dissipation of various sources of stored energy. By far the most successful at providing a long life was nuclear energy. The conversion of hydrogen to iron provided for a lifetime of some 140 billion years. However, in practice, when about 10 percent of the hydrogen is converted to helium in stars like the sun, major structural changes will begin to occur and the star will begin to evolve. We can define a time scale for these events in a manner analogous to our other time scales as \[\tau_n=\frac{K_n M c^2}{L}\label{3.2.13}\]

where \(K_n\) is just the fraction of the rest mass available to a particular nuclear process. While evolutionary changes often occur in one-tenth of the nuclear time scale, some stars show no significant change in less than 0.99_tn. While in the terminal phases of some stars' lives the nuclear time scale becomes rather shorter than the thermal time scale and conceivably shorter than the dynamical time scale, for the type of stars we will be considering the nuclear time scale is usually very much longer than the other two. Certainly for main sequence stars we may observe that \[\tau_d \approx \tau_f \approx \tau_s \ll \tau_{\mathrm{KH}} \ll \tau_n\label{3.2.14}\]

It is important to understand that the time scales themselves may change with time. The nuclear time scale will depend on the nature of the available nuclear fuel. However, the time scales do indicate the time interval over which you may regard their respective processes as approximately constant. They are useful, for they are easy to estimate, and they indicate which processes within the star will be important in determining its structure at any given time.