3.5: The Virial Theorem and Stability

In the last section, I alluded to the effects that the surface terms have on the stability of the system being considered. This concept deserves some amplification as it represents one of the most productive applications of the virial theorem. However, before embarking on a detailed development of the virial theorem for this purpose, it is appropriate to review the use of the word stability itself.

When inquiring into the meaning of the word, it is customary to consult a dictionary. This approach provides the following definition:

Stability: "That property of a body which causes it, when disturbed from a condition of equilibrium or steady motion, to develop forces or moments which tend to restore the body to its original condition."

This definition is subject to several interpretations and serves to illustrate the danger of consulting an English dictionary to learn the meaning of a technical term. The word stability is usually associated with the word equilibrium. This is primarily because the concept of stability normally is first encountered during the study of statistics. However, there are many dynamical situations, which are not equilibrium situations that even the most skeptical person would call stable. One of the most obvious examples to an astronomer, are the stars themselves. Not all stars would be regarded as stable, but certainly most of the main sequence stars are. Since stars are not really equilibrium configurations, but rather steady state configurations, we see that we must extend our conceptualization of stability to include some dynamical systems. The normal definition of equilibrium requires that the sum of all forces acting on the system is zero. This concept may be broadened to dynamical systems if one requires that the generalized forces (Q_i) acting on the systems are zero. Here the concept of the generalized force may be most simply stated as

\[\mathrm{Q}_{\mathrm{i}}=\sum_{\mathrm{j}} \mathbf{F}_{\mathrm{j}} \cdot \frac{\partial \mathbf{r}}{\partial \mathrm{q}_{\mathrm{i}}},\label{3.5.1}\]

where \(\mathbf{F}_{\mathrm{j}}\) represents the physical forces of the system acting on the jth particle and the q_i's represent any set of linearly independent 'coordinates' adequate to describe the system. In a conservative system all the forces are derivable from a potential Φ .Thus, the generalized forces may be written as

\[\mathrm{Q}_{\mathrm{i}}=-\sum_{\mathrm{j}} \nabla_{\mathrm{j}} \Phi \cdot \frac{\mathrm{d}\mathbf{r}_{\mathrm{j}}}{\mathrm{dq}_{\mathrm{i}}}=\sum_{\mathrm{j}}\left(\frac{\partial \Phi}{\partial \mathrm{r}_{\mathrm{j}}}\right) \widehat{\mathbf{r}}_{\mathrm{j}} \cdot \left(\frac{\partial \mathbf{r}_{\mathrm{j}}}{\partial \mathrm{q}_{\mathrm{i}}}\right)=-\frac{\partial \Phi}{\partial \mathrm{q}_{\mathrm{i}}}.\label{3.5.2}\]

Thus, saying the generalized forces must vanish is equivalent to saying the potential energy must be in extremum.

\[\mathrm{Q}_{\mathrm{i}}=-\left.\left(\frac{\partial \Phi}{\partial \mathrm{q}_{\mathrm{i}}}\right)\right|_{\mathrm{q}_{\mathrm{i}}=\mathrm{q}_{\mathrm{i}}(0)}=0.\label{3.5.3}\]

Now, in terms of this definition of equilibrium we may proceed to a definition of stable equilibrium. If the potential extremum implied by Equation \ref{3.5.3} is a minimum, then the equilibrium is said to be stable. The conditions thus imposed on the potential are

\[\mathrm{Q}_{\mathrm{i}}=-\left.\left(\frac{\partial^2 \Phi}{\partial \mathrm{q}_{\mathrm{i}}^2}\right)\right|_{\mathrm{q}_{\mathrm{i}}=\mathrm{q}_{\mathrm{i}}(0)}>0.\label{3.5.4}\]

In order to see that this definition of stability is consistent with our dictionary definition, consider the following argument. Suppose a system is disturbed from equilibrium by an increase in the total energy dE above the total energy at equilibrium. If Φ is a minimum, any disturbances from equilibrium will produce an increase in the potential energy. Since the conservation of energy will apply to the system after the incremental energy dE has been applied, the kinetic energy must decrease. This implies that the velocities will decrease for all particles and eventually become zero. Thus, the motion of the system will be bounded (Note: the bound may be arbitrarily large). If, however, the departure from equilibrium brings about a decrease in the potential energy, then the velocities may increase without bound. We would certainly call such motion unstable motion.

However, simple and clear-cut as this definition of stability may seem, it is still inadequate to serve the needs of mathematical physicists in describing the behavior of systems of particles. Thus it is not uncommon to find modifying adjectives or compound forms of the word "stable" appearing in the literature. A few common examples are: Secular stability, global stability, quasi-stable, bi-stable, and over-stable. These terms are usually used without definition in the hope that the reader will be able to discern the correct meaning from the context. The introduction of these modifiers as often as not arises from the mode of analysis used to describe the system. It is a common practice to examine the response of the system to a continuous spectrum of perturbations. If any of these perturbations grow without bound the system is said to be unstable. This would seem in full accord with our dictionary definition and thereby wholly satisfying. Unfortunately, one is rarely able to calculate the response in general. It is usually necessary to linearize the Equations describing the system in order to solve them. Analysis of this type is called linear stability theory, and is actually the basis for most stability criterion. Thus, when analyzing a system not only must one correctly carry out the stability analysis, he must also decide on the applicability of the analysis to the system.

Recently, it has been quite fashionable to use the virial theorem as the vehicle to carry out linearized normal mode analysis of systems in order to determine their state of stability. However, the determination of a system state of stability seems to have inspired Jacobi to develop the n-body representation of Lagrange's identity from which it is a short step to the virial theorem. To see now how closely tied the virial theorem is connected to stability; let us summarize some of Jacobi's arguments. In Chapter I, [i.e. Equation \ref{1.4.12}], we arrived at simple statements of Lagrange's identity for self-gravitating systems as:

\[\frac{1}{2} \frac{\mathrm{d}^2 \mathrm{I}}{\mathrm{dt}^2}=2 \mathrm{T}+\Omega.\label{3.5.5}\]

One could say with some confidence that if \(\mathrm{d}^2 \mathrm{I} / \mathrm{dt}^2>0\) for all t the system would have to have at least one particle whose position coordinates increased without bound. That is to say, the system would be unstable. However, since both T and Ω vary with time it would be difficult to say something a priori about \(\mathrm{d}^2 \mathrm{I} / \mathrm{dt}^2\) from Lagrange's identity alone. Thus, Jacobi employed the constancy of the total energy (i.e., E = T + Ω = const.), and the fact that for self-gravitating systems Ω > 0, to modify Equation \ref{3.5.5} to give:

\[\mathrm{\frac{1}{2} \frac{d^2 I}{d t^2}=2 \mathrm{E}-\Omega>2 \mathrm{E}.}\label{3.5.6}\]

So, if \(\mathrm{E>0, d^2 I / d t^2>0}\) and the system is unstable. This is known as Jacobi's stability criterion and provides a sufficient (but not necessary) condition for a system to be called unstable.

It is the constancy of E with time that makes this a valuable criterion for stability and is the reason Jacobi used it. There is another approach to the problem of temporal variation which was not available to Jacobi. In Chapter 2 we discussed the extent to which time averages of quantities may be identified within their phase averages, through use of the Ergodic theorem. Integrating Equation \ref{3.5.5} over some time t , we get

\[\left.\frac{1}{\mathrm{t}_0}\left(\frac{\mathrm{dI}_0}{\mathrm{dt}}\right)\right|_0 ^{\mathrm{t}_0}=2 \overline{\mathrm{T}}+\overline{\Omega}\label{3.5.7}\]

If the system is to remain bounded, dI/dt must always be finite. If the system is to be always stable, then the limit of the left-hand side must tend to zero. Furthermore, the time averages on the right-hand side of Equation \ref{3.5.7} will tend to phase averages if the system is Ergodic. Thus

\[\mathrm{2<T>+<Q>=0},\label{3.5.8}\]

constitutes a stability criterion for Ergodic systems. That is, Equation \ref{3.5.8} must be satisfied in a stable Ergodic system and failure of Equation \ref{3.5.8} is sufficient for instability. It is not uncommon to find the statement in the literature that (2T + Q > 0) insures the instability of a system citing the virial theorem as the justification. Actually, it is Lagrange's identity that is the relevant expression and it only guarantees that at the moment the system is acceleratively expanding. What is really meant is that if (T + Q > 0) the system is indeed unstable as this is just a statement of Jacobi's stability criterion concerning the total energy of the system.

Now, let us turn the applications of the variational form of the virial theorem to stability, keeping in mind that the variational approach is essentially a first order or linearized analysis. The majority of Chapter III has been devoted to obtaining expressions for the frequency of a pulsating system. We obtained a value for the square of the frequency in terms of the equilibrium energies of the configuration. However, these expressions could be neither positive nor negative. In the earlier sections, we discussed only the meaning of the positive values, as negative squares of frequencies had no apparent physical meaning. Let us look again at the nature of the assumed pulsation in order to further investigate the meaning of these pulsation frequencies. In section 2 [i.e. Equations Equation \ref{3.2.12}], we assumed that the pulsation would be periodic and of the form

\[\frac{\delta \mathrm{r}}{\mathrm{r}}=\xi_0 \mathrm{e}^{\mathrm{i} \sigma t}.\label{3.5.9}\]

where σ was the frequency of pulsation and \(\mathrm{\xi_o}\) did not depend on time. Now, if we make the formal identification between σ and the period and take σ to be purely imaginary, we may write

\[\sigma= \pm 2 \pi \mathrm{i} / \mathrm{t}_0,\label{3.5.10}\]

where \(\mathrm{t_o}\) is a real number. Combining Equation \ref{3.5.10} and Equation \ref{3.5.9}, we have

\[\frac{\delta \mathrm{r}}{\mathrm{r}}=\xi_0 \mathrm{e}^{ \pm 2 \pi \mathrm{t} / \mathrm{t}_0}.\label{3.5.11}\]

Thus, the pulsation becomes exponential in nature. If the sign of σ is negative, then the sign of the exponential in Equation \ref{3.5.11} will be positive and the pulsation will grow without bound with a rate of growth determined by \(\mathrm{t_o}\).

One might be tempted to choose the negative sign of Equation \ref{3.5.11} saying that the system is stable as the pulsation will die out, even though \(\sigma^2<0\). This would be wrong. All classical Equations of dynamical symmetry exhibit full-time symmetry, thus those solutions which damp out in the future were unstable in the past and vice-versa. A specific solution would be fully determined by the boundary conditions at t = 0. Further, we assumed that a full continuum of perturbations are present, resulting from small but inevitable, departures from perfection produced by statistical fluctuations. Thus, if there exists even one mode with \(\sigma^2<0\), the instability associated with that mode will grow without bound. Therefore, this becomes a sufficient condition for the system to be unstable in the strictest sense of the word

\[\sigma^2<0.\label{3.5.12}\]

It is also worth noting that this criterion applies to the entire system, and thus is a "global" stability condition. However, it can be made into a local condition by taking an infinitesimal volume and including the surface terms discussed in section 4.

Now, let us see what implications this analysis has for the stability of stars. In section 2, we established an expression for the pulsation frequency of a gravitating gas sphere [Equation \ref{3.2.19}]. Now, applying the instability criterion Equation \ref{3.5.12}, we see that the sphere will become unstable, when

\[-\frac{(3 \gamma-4) \Omega_0}{\mathrm{I}_0}<0.\label{3.5.13}\]

Since the moment of inertia (I₀) is intrinsically positive, while the gravitational potential energy is intrinsically negative, Equation \ref{3.5.13} becomes

\[\left.\begin{array}{l}
(3 \gamma-4)<0 \\[4pt]
\gamma<4 / 3
\end{array}\right\}.\label{3.5.14}\]

Thus, a star will become unstable when \(\gamma\) is less than 4/3. This is the familiar instability criterion demonstrated by Chandrasekhar in Stellar Structure.⁷ He further demonstrates that a gas with \(\gamma\) equal to 4/3 corresponds to a gas where the total pressure is entirely due to radiation. If we consider the other stability criterion [Equation \ref{3.5.14}], we can see that a necessary condition for stability of a homogeneous non-rotating gas sphere is

\[\gamma>4/3.\label{3.5.15}\]

Thus, having extracted as much information as possible from the pulsation expression developed in section 2, let us turn to the more general formulae resulting from our analysis in section 3. Remember the final expression for the pulsational frequency was

\[\sigma^2=\frac{-(3 \gamma-4)\left(\Omega_0+\mathscr{M}_0\right)+(5-3 \gamma) \omega_0 \mathscr{L}_0}{\mathrm{I}_0}.\label{3.5.16}\]

Consider first a gas sphere which is not rotating, but which has a magnetic field. Equation \ref{3.5.16} becomes then

\[\sigma^2=\frac{-(3 \gamma-4)\left(\Omega_0+\mathscr{M}_0\right)}{\mathrm{I}_0}.\label{3.5.17}\]

Substituting this into the instability criterion Equation \ref{3.5.12}, we have

\[(3 \gamma-4)\left(\Omega_0+\mathscr{M}_0\right)<0.\label{3.5.18}\]

Now, if we assume \(\gamma>4/3\), we have a sufficient condition for instability due to the presence of magnetic energy as follows:

\[\mathscr{M}_0>\left|\Omega_0\right|.\label{3.5.19}\]

In the following manner, we may obtain a crude estimate of the magnitude of the magnetic fields necessary to disrupt a star. The gravitational potential energy for a sphere of uniform density is

\[\Omega=-\frac{3}{5} \frac{\mathrm{GM}^2}{\mathrm{R}},\label{3.5.20}\]

while the magnetic energy is

\[\mathscr{M}=\frac{1}{8} \iiint|\mathbf{H}|^2 \mathrm{dx}_1 \mathrm{dx}_2 \mathrm{dx}_3=\frac{\mathrm{R}^3 \overline{|\mathbf{H}|^2}}{6}.\label{3.5.21}\]

Combining Equation \ref{3.5.20} and Equation \ref{3.5.21}, we see that the root mean square value of the magnetic field sufficient to disrupt a uniformly dense sphere is

\[\sqrt{\overline{|\mathbf{H}|^2}}>2 \times 10^8 \frac{\mathrm{M}}{\mathrm{R}^2} \text { gauss },\label{3.5.22}\]

where M and R are given in solar units. Thus, for a main sequence A star with \(\mathrm{M}=4 \mathrm{M}_{\odot}\) and \(\mathrm{R}=5 \mathrm{R}_{\odot}\), we have

\[\mathrm{H}_{\mathrm{rms}}>3 \times 10^7 \text { gauss }.\label{3.5.23}\]

However, for a star like VV Cephei with \(\mathrm{M}=100 \mathrm{M}_{\odot}\), and \(\mathrm{R}=2600 \mathrm{R}_{\odot}\), we have

\[\mathrm{H}_{\mathrm{rms}}>3000 \text { gauss }.\label{3.5.24}\]

We may conclude from these arguments that for a main sequence star, an extremely large magnetic field would be sufficient to cause the star to become unstable. However, for an unusually large star, the required field becomes much smaller. In the case of VV Cephei, Babcock has measured a field ranging from +2000 to -1200 gauss. Thus, it would appear that VV Cephei is on the verge of being magnetically unstable. One might argue that our crude estimates of Ω are so crude as to be meaningless due to the large central concentration of the mass in giant stars. However, it should be pointed out that the magnetic field one can observe is, of necessity, a surface field and, therefore, provides us with a lower limit on the magnetic energy. Thus, we may have some hope that our limiting field values are not too far from being realistic.

It is interesting to note that the instability criterion Equation \ref{3.5.18} permits the existence of a gas with \(\gamma<4/3\) providing the magnetic energy exceeds the gravitational energy. Indeed, the stability criterion Equation \ref{3.5.18} would require a necessary condition for the stability of any configuration where \(\mathscr{M}>|\Omega|\) that \(\gamma\) be less than 4/3. However, it is also true that the physical meaning of a gas having a \(\gamma<4/3\) is a little obscure.

If we now consider a rotating configuration with no magnetic field, Equation \ref{3.5.14} combined with the stability criterion Equation \ref{3.5.12} becomes

\[(5-3 \gamma) \omega_0 \mathscr{L}_0>(3 \gamma-4) \Omega_0.\label{3.5.25}\]

If we restrict \(\gamma\) to be less than 5/3 we have

\[\omega_0 \mathscr{L}_0>\frac{(3 \gamma-4) \Omega_0}{(5-3 \gamma)}.\label{3.5.26}\]

Since \(\Omega_\mathrm{o}\) is intrinsically negative, we see that the stability condition will always be satisfied with any \(\omega_\mathrm{o}\). Thus, for all known stars the stability criterion for rotation is not particularly useful.

However, all this is not meant to imply that the rotational terms are unimportant. Indeed, Ledoux¹ has has shown that rotational velocities encountered in stars may lead to a variation in the pulsational period by as much as 20%. Let us briefly consider the instability criterion when both magnetic and rotational energy are present. This may be obtained by combining Equation \ref{3.5.15} and

\[(3 \gamma-4)\left(\Omega_0+\mathscr{M}_0\right)-(5-3 \gamma) \omega_0 \mathscr{L}_0>0.\label{3.5.27}\]

As before, this condition may never be satisfied unless \(\left|\Omega_0\right|>\mathscr{M}_0\). However, even in the event that \(\left|\Omega_0\right|>\mathscr{M}_0\), the condition may still not be satisfied because of the presence of the rotational term. Thus, it is evident that if \(4/3<\gamma<5/3\), then the presence of rotation will help stabilize stars. This result is certainly not intuitive. A physical explanation of the result might be supplied by the following argument.

Consider a pulsating configuration containing both rotational and magnetic energy. As the system expands or contracts, a certain amount of energy will be required to slow down or speed up the rotation in order to keep the angular momentum constant. This energy must be supplied by the kinetic energy of the gas itself, and, since this is supplied by the potential energies present, ultimately must come from the gravitational and magnetic energies. Therefore, the amount of this energy transferred from the magnetic and gravitational energies will depend on \(\gamma\). Also since the gravitational and magnetic energies must supply this energy to the rotation, the energy is no longer available to "feed" the pulsation and disrupt the star.

We may now ask what sort of increase in the maximum magnetic field can this additional rotational "stability" supply. From our previous investigation with the rotational stability criteria we might expect the result to be small. That is, since the rotational stability criteria did not supply us with as important a result as did the magnetic instability, we would expect the effects of rotation to be small compared with the magnetic energy. If one considers a uniform model with \(\gamma=3/2\), rotating at critical velocity, he will find the magnetic field may only be increased by about 0.3% before instability will again set in. Thus, even though stability is increased by the presence of rotation, it is not increased a great deal.

It is appropriate at this point to make some comments regarding all of the stability criteria relating to the stability of radial pulsations. It would have been more correct to employ the integral form of the expressions for the frequency of pulsation. However, the result one would obtain by using the integral expression and a specific model would only differ in degree from those derived here. It is hoped that the degree of differences would not be large. There is one respect in which the differences between the derived criteria and the 'correct' ones may result in a difference in kind. It must be remembered that the expressions developed for the pulsational frequencies were based on a first order theory as are the stability criteria developed in this section. However, the conditions at which one wishes to apply an instability criteria are generally such that the second, and higher, order terms are not small and should not be neglected. Chandrasekhar and Fermi³ have shown that a sphere under the influence of a strong dipole field will tend to be "flattened" in much the same way as it will be by rotation. Once the spherical symmetry has been destroyed, either by the presence of a strong, magnetic field or rapid rotation, the concept of radial pulsation becomes inconsistent. As mentioned before, analysis of such systems would require the use of the tensor virial theorem and considerable insight into the types of perturbations to employ.

I would be remiss if I left the subject of the virial theorem and stability without some discussion of the recent questions raised with regard to the appropriateness of the approach. To me, these questions appear to be partly substantive and partly semantic and revolve largely around one of these modifiers mentioned earlier, namely, secular stability. So far, our discussion has been restricted to problems involving dynamical stability about which there seems to be little argument. A dynamically unstable system will disintegrate exponentially, usually on a time scale related to the hydrodynamical time scale for the system. Such destruction is usually so unambiguous that no complications arise in the use of the word unstable. Such is not the case for the term secular stability.

The notion of secular stability involves the response of the system to small dissipative forces, such as viscosity and thus must depend to some extent, on the nature of those forces. Time scales for development of instabilities will be governed by the forces and hence may be very long. Perhaps one of the clearest contemporary discussions of the term is given by Hunter⁸ who notes that there is less than universal agreement on this meaning of the term. He points out that difficulties arise in rotating systems resulting from the presence of the coriolis forces, which lead to a clear distinction between dynamically and secularly stable systems. As we saw in Chapter II, section 5, the terms associated with the coriolis forces can be made to vanish by the proper choice of a coordinate frame and they would appear to play no role in the energy balance of the system. However, their variation does not vanish and hence they will affect the pulsational analysis. Since globally the forces are conservative the first result is not surprising and since radially moving mass in a rotating frame must respond to the conservation of angular momentum, neither is the second. Now if dissipative forces are present such as viscosity then it may be possible to redistribute local angular momentum while conserving it globally so that no equilibrium configuration is ever reached. In addition, except for the global constraint on the total angular momentum, no constraints are placed on the transfer of energy from the rotational field to the thermal field. Indeed the presence of dissipative forces guarantees that this must happen. Thus, instabilities associated with these forces might exist which would otherwise go undetected. This line of reasoning demonstrates a qualitative difference between the cases of uniform rotation and differential rotation in that in the former dissipative forces will be inactive and the analysis will be appropriate while in latter cases they must be explicitly included. This point is central to a lengthy series of papers^{9, lO, 11, 12, 13} by Ostriker and others, which discuss the stability of a variety of differentially rotating systems. However, the majority of these papers clearly state that the authors are dealing with systems with zero viscosity and so the problem is not one of the accuracy of the analysis but rather of the applicability of the analysis to physical systems. In practice, the viscosity of the gas in most stars is so extremely low that the time scales for the development of instability arising from viscosity driven instabilities will be very long.

One cannot hope to untangle in a few paragraphs a controversy which has taken more than a decade to develop and at a formal mathematical level is quite subtle. However, it is worth noting that recent¹⁴ statements which essentially say that the tensor virial approach to stability is wrong do nothing to clarify the situation. The presence of dissipative forces can be included in the Equations of motion and thus in the resulting tensor representation of Lagrange's identity. The resulting stability analysis would then correctly reflect the presence of these forces and thus be dependent on their specific nature. Insofar as the time averaged form of Lagrange's identity, which is technically the virial theorem, is used the arguments of Milne¹⁵ as presented in Chapter I still apply. The presence of velocity dependent forces does not affect the virial theorem unless those forces stop or destroy the system during the time over which the average is taken. At this level assailing the virial theorem is as useful an enterprise as denying the validity of a conservation law.