8.3: Linear Nonadiabatic Radial Oscillations

The nonadiabatic processes that give rise to stellar pulsations are basically thermodynamic, so we should expect the time scale for their development to be roughly the thermal or Kelvin time scale. Since this is generally much longer than dynamical time scale of the resonant oscillation period, we might anticipate the energies involved in the nonadiabatic processes to be significantly less than those of the pulsational motions themselves. Small as these effects are, they are responsible for the origin of the pulsations.

Throughout the book we have generally treated the gas in stars as an ideal gas. Its thermodynamic properties could thus be represented by the parameter \(\gamma\), which is just the ratio of the specific heats of the gas. Now that we will be dealing with nonadiabatic processes, we should provide a more complete description of how a gas behaves when it departs from being an ideal gas.

a. Adiabatic Exponents

We will define several quantities that describe the change of the state variables of the gas with respect to one another when the gas is subject to an adiabatic change. It may seem a little odd that an adiabatic change is used to characterize the nonadiabatic behavior of a gas, but for the defined parameters to properly describe the behavior of the gas alone, it is necessary to separate the external environment of the gas from the gas itself. Hence we consider externally imposed changes to the gas that do no net work on the gas. If the gas is an ideal gas then the single parameter γ is sufficient to describe all the changes of the state variables with respect to each other. If the gas is not an ideal gas, then it is necessary to describe the change of each state variable with respect to the others. Thus we define \[\Gamma_1 \equiv\left(\frac{d \ln P}{d \ln \rho}\right)_{\mathrm{ad}} \quad \Gamma_3-1 \equiv\left(\frac{d \ln T}{d \ln \rho}\right)_{\mathrm{ad}} \quad \frac{\Gamma_2-1}{\Gamma_2} \equiv\left(\frac{d \ln T}{d \ln P}\right)_{\mathrm{ad}}\label{8.2.1}\]

Since there are three state variables, there can be only two linearly independent changes of one with respect to another. That is clear from the definition of \(\Gamma_2\). Since the \(\Gamma\)'s are defined in terms of logarithmic derivatives, they appear as exponents in the actual relations between the state variables themselves. Since the \(\Gamma\)'s describe the response of the gas to an adiabatic change, they are known as the adiabatic exponents of the gas. Under the conditions where the thermodynamic pressure is entirely due to an ideal gas, \(\Gamma_1=\gamma\).

As one proceeds inward through the outer envelopes of stars there exist regions where the dominant elements that make up the star (i.e., hydrogen and helium) change from being largely neutral to being largely ionized. This change usually makes for a relatively small change in temperature and so is physically a relatively narrow zone in the envelope. A gas that undergoes a change in its ionization state in response to a small change in temperature finds itself with more degrees of freedom than an ideal gas. This can be described as a change in the values of the adiabatic exponents, and it will play an important role in understanding stellar pulsation.

b. Nonadiabatic Effects and Pulsational Stability

In Section 8.1 we investigated the dynamical stability of a star to small pulsation. However, even where stable adiabatic pulsations are possible, the presence of nonadiabatic terms may force oscillations to grow or damp out in time. Thus we must investigate how the presence of nonadiabatic effects will affect the stability of stellar oscillations. We know that the thermodynamic nature of the nonadiabatic terms will cause them to effect the oscillations on a thermal time scale, but it would be useful to have a method of estimating what that time scale might be and to know whether the oscillations will grow or die out. To do this, we estimate the rate at which energy is transferred from the kinetic and potential energies associated with the oscillatory motion to the stellar gas by the nonadiabatic processes.

Let us begin by defining the nature of the perturbation to have the form \[\xi=\xi_0 e^{i \omega t} \quad \omega=\sigma+i \eta\label{8.2.2}\]

The pulsational frequency is complex so as to represent the dissipational losses through the quantity η as well as the pure oscillatory motion having a frequency \(\sigma\). The linear analysis of a simple harmonic oscillator shows that \[\frac{W}{\Psi}=-\frac{4 \pi \eta}{\sigma}=-2 P_p \eta\label{8.2.3}\]

where \(W\) is the total work done on the oscillator by the restoring forces over a complete cycle while \(\Psi\) is the total oscillation energy (potential and kinetic). We may rewrite this for η and replace \(W / P_{\mathrm{p}}\) by the average rate of energy loss \(<\mathrm{d} W / \mathrm{dt}>\). We get \[\eta=\frac{-\frac{1}{2}\left(W / P_p\right)}{\Psi}=\frac{-\frac{1}{2}\langle d W / d t\rangle}{\Psi}\label{8.2.4}\]

The instantaneous work done on the star by the oscillation can be represented as the sum of the gravitational and pressure forces times a differential displacement divided by a differential time and integrated over the entire mass of the star, so \[\begin{aligned}
\frac{d W}{d t}= & \int_0^M\left(g-\frac{1}{\rho} \frac{\partial P}{\partial r}\right) \frac{d r}{d t} d M(r)=\frac{d}{d t}\left(\int_0^M \frac{G M(r) d M(r)}{r}\right) \\
& -\int_0^M 4 \pi r^2 \frac{\partial P}{\partial M(r)} \dot{r} d M(r)
\end{aligned}\label{8.2.5}\]

Integrating the second term on the right by parts and using the condition that the surface pressure is zero, we get \[\frac{d W}{d t}=-\frac{d}{d t} \int_0^M \Omega(r) d M(r)+\int_0^M P \frac{\partial}{\partial t}\left(\frac{1}{\rho}\right) d M(r)\label{8.2.6}\]

Now, by averaging over one full pulsational period, the gravitational forces vanish since they are fully conservative, so that \[\left\langle\frac{d W}{d t}\right\rangle=\frac{1}{P_p} \int_0^{P_p} \frac{d W}{d t} d t=\frac{1}{P_p} \int_0^{P_p} \int_0^M P \frac{\partial(1 / \rho)}{\partial t} d M(r) d t\label{8.2.7}\]

However, since the heat increase over a complete cycle of the pulsation can be expressed in terms of the pressure and density changes, \[\int_0^{P_p} \frac{d q}{d t} d t=\int_0^{P_p} p \frac{\partial(1 / \rho)}{\partial t} d t\label{8.2.8}\]

which means that the average energy transfer can be written as \[\left\langle\frac{d W}{d t}\right\rangle=\frac{1}{P_p} \int_0^{P_p} \int_0^M \frac{d q}{d t} d M(r) d t\label{8.2.9}\]

Since the pulsation represents a closed cycle the star, behaves as if it were a Carnot engine. If <dW/dt>> 0, then the pressure forces are doing positive work on the star and some source of that work must be found. Indeed, each mass shell may be treated separately by calculating the \(\oint \mathrm{PdV}\) forces for that shell. So the star may be viewed as a sum of Carnot engines, some of which feed energy into the pulsation and others which remove it. If the sum of all the Carnot engines produce <dW/dt>> 0, the star is unstable and the oscillations will grow on a thermal time scale \(\tau_{\text {th }}\) [see equation \ref{3.2.12}] which we can estimate from the inverse of η. Thus \[\frac{\tau_{\mathrm{th}}}{P_p} \equiv \frac{1}{\eta P_p}=\frac{-2 \Psi}{\int_0^{P_p} \int_0^M(d q / d t) d M(r) d t}\label{8.2.10}\]

Cox⁷ (see p. 117) finds that \(\tau_{\mathrm{th}} / \mathrm{P}_{\mathrm{p}}\) ranges from the order of 1 to about 1000 for common variables extending from the Mira variables to the RR Lyrae stars, respectively. This would imply that the pulsations in these stars ought to damp out in less than 1000 periods due to thermal losses. Since this is clearly not the case, we must find some sort of driving mechanism.

c. Constructing Pulsational Models

While it is possible to develop a nonadiabatic wave equation (see Cox⁷, pp. 72, 73) similar to equation \ref{8.1.24}, we forgo doing so here. Instead let us consider a few aspects of the actual construction of a pulsating model. In principle, one has a complete equilibrium model which provides the run of state variables throughout the stars at equilibrium. It is against this background that one formulates the problem of how those state variables will vary with position and time. By assuming a variation of the form given in equation \ref{8.2.2} for each of the state variables, one has the four dependent variables \(\mathrm{\delta r/r_0}\), \(\mathrm{\delta \rho/\rho_0}\), \(\mathrm{\rho T/T_0}\), and \(\mathrm{\delta L/L_0}\), with \(M(\mathrm{r})\) as the independent variable. The situation is similar to that encountered in Chapter 4 when we arranged the variables in the same manner to utilize the Henyey scheme for the construction of equilibrium models. Just as the construction of those models involved the solution of a two-point boundary-value problem for a particular eigenvalue, so this problem involves finding the complex eigenfrequency ω.

It is clear that unlike the linear adiabatic wave equation, the linear nonadiabatic wave equation will be complex. Hence the four dependent variables will be complex, with the real part describing the oscillatory motion and the imaginary part describing the damping of the solution. The central boundary conditions can make use of the fact that all variations must vanish at the center. Indeed, many early models^{8, 9} simply required the solution to go over to the adiabatic solution in the deep interior of the star. However, the surface boundary conditions are rather trickier. Perhaps the simplest condition that can be employed at the surface is the radiative condition \[\left.\frac{\delta L}{L_0}\right|_R=\left.2 \frac{\delta r}{r_0}\right|_R+\left.4 \frac{\delta T}{T_0}\right|_R\label{8.2.11}\]

In addition to the difficulties of posing the correct boundary conditions, significant numerical problems are encountered in the actual solution. These problems were overcome by Castor¹⁰ and Iben¹¹ so that reliable pulsational models can now be obtained.

c. Pulsational Behavior of Stars

We have already indicated that the helium and hydrogen ionization zones might be expected to play a significant role in determining the actual nature of pulsating stars, since they represent zones where the value of \(\Gamma_3\) varies with position. However, such zones exist in all stars, but not all stars exhibit radial oscillations. Indeed, further thought about the role of ionization zones would lead one to believe that the destabilizing effect on a radial oscillation entering the ionization zone from below brought about by the rapidly declining value of \(\Gamma_3\) would be offset by the damping effect of a rising \(\Gamma_3\) at the top of the zone. For the majority of stars this appears to be the case. Something else must be operative, and to understand its nature, we must consider the nature of nonadiabatic forces near the surface of the star.

Consider the part of the stellar envelope that is relatively near the surface. This region is what Cox⁷ (pp. 140, 141) has called the transition zone. As one moves up through the stellar envelope toward the surface, the thermal cooling time steadily decreases. The nearer one is to the surface, the less time is required for energy to diffuse to the surface and escape. Thus the nonadiabatic effects will be more pronounced since pulsational energy that appears as thermal energy will be quickly radiated away. Changes in the luminosity generated deeper in the star will become "frozen in" the star and not travel with the wave. When this transition zone coincides with an ionization zone, the potential exists for the nonadiabatic effects of ionization to also be "frozen in". Thus, if the ionization zone lies at the right depth, the driving effects caused by the decline in \(\Gamma_3\) upon entering the zone will not be reversed upon leaving, because the luminosity changes so induced become "frozen in".

Since the conditions for ionization are set by atomic physics, we should expect that stars with the appropriate effective temperatures and gravities to produce an ionization zone at the correct diffusion depth corresponding to the transition zone will exhibit radial pulsations. There is indeed a region on the H-R diagram known as the instability strip which encompasses the Cepheid variables, W Virginis stars, δ Scuti stars, RR-Lyrae stars, and Dwarf Cepheids where this condition appears to be met for the He II ionization zone. The He II ionization zone seems to be the dominant zone for most of the common variable stars. The H and He I ionization zones lie so close together that they may be virtually considered as one. They invariably lie above the He II ionization zone in a region where the luminosity changes have already been frozen in and so play little role in the hot stars. However, this zone is suspected of introducing a phase shift between the luminosity variations and the radial velocity variation from that which is expected. Normally one would expect the time of maximum compression to be the time of maximum luminosity. However, the luminosity maximum lags somewhat, and this may result from its being delayed in the hydrogen ionization zone. In late-type variables where the He II zone is so deep as to be below the transition zone, the hydrogen ionization zone may be the primary driver of the pulsations. However, for these stars the situation becomes complicated by the large geometric extent of the atmosphere and the complex nature of the opacity.

We have seen how the onset of He II ionization at the appropriate place in the star can cause conditions to exist that will drive pulsations simply by a rapid decline of \(\Gamma_3\). This cause of pulsation is called the \(\gamma-mechanism\). However, there is another contributor to the destabilizing effect of ionization. In most of the stellar interior, an increase in temperature is accompanied by a decline in opacity. Certainly Kramer’s law [equation \ref{4.1.19}] implies this. Thus any temperature increase accompanying the compression of a passing pulsational perturbation would cause a decline in the local opacity and a release of the radiation trapped there. This would tend to stabilize the region against pulsations by removing the pulsational energy in the form of radiation. However, at low temperatures a rise in temperature results in an increase in opacity. This effect was largely responsible for the nearly vertical tracks of collapsing convective protostars (see Section 5.2c). If the opacity increases with increasing temperature, the opacity will tend to trap the energy at that point and prevent the energy from diffusing away. This is a destabilizing effect that tends to feed the pulsations, and it has been called the \(\kappa\) mechanism. Of these two, the \(\gamma\) mechanism seems to be the more important for the pulsation of Cepheid variables.

When the oscillations become large, the linear theory we have been describing becomes invalid. However, the complications introduced by these nonlinearities are well beyond the scope of this book, and we leave them to others to discuss.