4.4: Additional Topics and Final Thoughts

It would be possible and perhaps even tempting to continue demonstrating the efficacy of the virial theorem in stellar astrophysics. However, attempting to exhaust the possible applications of the virial theorem is like trying to exhaust the applicability of the conservation of momentum. I would be remiss if I did not indicate at least some other possible areas in which the virial theorem can lend insight.

In Chapter III, section 4 we discussed the variational effect of the surface terms resulting from the application of the divergence theorem. These terms are generally neglected and for good reason. In most cases the bounding surface can be chosen so as to include the entire configuration. In instances where this is not the case, such as with magnetic fields, the term is still generally negligible. If one considers the form of the surface terms given by Equation \ref{2.5.19} (Chapter II), compared to the volume contribution [i.e. Equation \ref{2.5.18}], the ratio of the scalar value is

\[\aleph=\frac{\int_{\mathrm{s}} \mathrm{P}_0 \mathbf{r} \cdot \mathrm{d} \mathbf{S}}{\int_{\mathrm{v}} \mathrm{PdV}}=\frac{\mathrm{P}_0}{\langle\mathrm{P}\rangle},\label{4.4.1}\]

where <P> is the average value of the internal pressure. Since, in any equilibrium configuration the pressure must be a monotone increasing function as one moves into the configuration, \(\aleph<1\). In general it is very much less than one. However, in the case where \(\gamma \rightarrow 4 / 3\) the variational contribution of the surface pressure approaches \(3 \mathrm{P}_0 \int \mathbf{r} \cdot \mathbf{d} \mathbf{S}\), while the internal pressure and Newtonian gravity contributions vanish. Such terms are then available to combine with the effects of general relativity. Since they are of the same sign as the general relativistic terms the surface terms will only serve to increase the instability of the entire configuration. This results in an increase of the radius at which white dwarfs become unstable. Thus, the accretion of matter onto the surface of such objects may cause them to collapse sooner than one might otherwise expect.

Since magnetic fields also are usually assumed to increase inward, the influence of magnetic surface terms will be similar to that of a non-zero surface pressure. Like the surface pressure terms, they will in general be small compared to the internal contributions. Only in case of a system with an effective \(\gamma\) approaching 4/3 could these small terms be expected to exert a trigger effect on the resulting configuration.

There is one instance in which the use of the surface terms can be significant. One of the most attractive aspects of this entire approach is that it can deal with the properties of an entire system. Indeed the spatial moments are taken in order to achieve that result. However it is interesting to consider the effects of applying the virial theorem to a sub-volume of a larger configuration. Clearly, as one shrinks the volume to zero he recovers the Equations of motion themselves multiplied by the local positional coordinate. If one considers a case intermediate to these limits and investigates the stability of a sub-volume which could include the surface of the star, it would be possible to analyze the outer layer for instabilities which might not be globally apparent. It is true that a local stability criterion would be sufficient to locate such instabilities but one could not be sure how the instabilities would propagate without carrying out a large structural analysis. This latter effect can be avoided by utilizing a sub-global form of the virial theorem. Under these conditions one might expect the surface terms to be the dominant terms of the resulting expression.

In discussing some of the more bizarre and contemporary aspects of stellar structure it is easy to overlook the role played by the virial theorem in the development of the classical theory of stellar structure. It is the virial theorem which provides the theoretical basis for the definition of the Kelvin-Helmholtz contraction time. This is just the time required for a star to radiate away the available gravitational potential energy at its present luminosity. It is the virial theorem which essentially tells how much of the gravitational energy is available. Thus if the contraction liberating the potential energy is uniform and \(\mathrm{d}^2 \mathrm{I} / \mathrm{dt}^2\) is zero then the total kinetic energy always must be

\[\mathrm{T}=-1 / 2 \Omega .\label{4.4.2}\]

This makes the other half of the gravitational energy available to be radiated away. The Kelvin-Helmholtz contraction time for poly tropes is thus

\[\mathbf{T}=\frac{3 \mathrm{GM}^2}{2(5-\mathrm{n}) \mathrm{RL}} \cong \frac{4.5 \times 10^7}{5-\mathrm{n}}\left(\mathrm{M} / \mathrm{M}_{\odot}\right)^2\left(\mathrm{R}_{\odot} \mathrm{L}_{\odot} / \mathrm{RL}\right) \text { (years). }\label{4.4.3}\]

Reasoning that this provided an upper limit to the age of the sun Lord Kelvin challenged the Darwinian theory of evolution on the sound ground that 23 million years (i.e. KHT for a polytrope with an internal density distribution of n=3)⁸ was not long enough to allow for the evolutionary development of the great diversity of life on the planet. The reasoning was flawless, only the initial assumption that the sun derived its energy from gravitational contraction which was plausible at the time, failed to withstand the development of stellar astrophysics.

Another aspect of classical stellar evolution theory is clarified by application of the virial theorem. All basic courses in astronomy describe post-main sequence evolution by pointing out that the contraction of the core is accompanied by an expansion of the outer envelope. Most students find it baffling as to why this should happen and are usually supplied with unsatisfactory answers such as "it's obvious" or "it's the result of detailed model calculations" which freely translated means "the computer tells me it is so." However, if the virial theorem is invoked, then once again any internal re-arrangement of material that fails to produce sizable accelerative changes in the moment of inertia will require that

\[2 \mathrm{T}+\Omega=2 \mathrm{E}-\Omega=0.\label{4.4.4}\]

Since the only way that the star can change its total energy E without outside intervention is by radiating it away to space, any internal changes in the mass distribution which take place on a time scale less than the Kelvin Helmholtz contraction time will have to keep the total energy and hence the gravitational potential energy constant. Now

\[\Omega=-\alpha \mathrm{M}^2 / \mathrm{R}.\label{4.4.5}\]

where \(\alpha\) is a measure of the central condensation of the object, so, as the core contracts and \(\alpha\) increases, R will have to increase in order to keep Ω constant. In general the evolutionary changes in a star do take place on a time scale rather less than the contraction time and thus we would expect a general expansion of the outer layer to accompany the contraction of the core. The microphysics which couples the core contraction to the envelope expansion is indeed difficult and requires a great deal of computation to describe it in detail. However the mass distribution of the star places constraints on the overall shape it may take on during rapid evolution processes. It is in the understanding of such global problems that the virial theorem is particularly useful.

I have attempted throughout this book to emphasize that global properties are the very essence of the virial theorem. The centrality of taking spatial moments of the Equations of motion to the entire development of the theorem demonstrates this with more clarity than any other aspect. Although this global structure provides certain problems when the development is applied to continuum mechanics nothing is encountered within the framework of Newtonian mechanics which is insurmountable. Only within the context of general relativity may there lie fundamental problems with the definition of spatial moments. Even here the first order theory approximation to general relativity yields an unambiguous form of the virial theorem for spherical objects. In addition, certain specific time independent or at least slowly varying cases of the non-approximated Equations also yield unique results. Thus one can realistically hope that a general formulation of the virial theorem can be made although one must expect that the interpretation of the resultant space-time moments will not be intuitively obvious.

The rather recent development of the virial theorem provides us with a dramatic example of the fact that theories do not develop in an intellectual vacuum. Rather they are pushed and shoved into shape by the passage of the time. Thus we have seen the virial theorem born in an effort to clarify thermodynamics and arising in parallel form in classical dynamics. However the similarity did not become apparent until the implications of the ergodic theorem inspired by statistical mechanics were understood. Although sparsely used by the early investigators of stellar structure, the virial theorem did not really attract attention until 1945 when the global analysis aspect provided a simple way to begin to understand stellar pulsation. The attendant stability analysis implied by this approach became the main motivation for further development of the tensor and relativistic forms and provides the primary area of activity today. Only recently has the similarity of virial theorem development to that of other conservation laws been clearly expounded.

Recent criticism of some work utilizing the virial theorem, incorrectly attacks the theorem itself as opposed to analyzing the application of the theorem and the attendant assumptions. This is equivalent to attacking a conservation law and serves no useful purpose. Indeed it may, by rhetorical intimidation, turn some less sophisticated investigators aside from consideration of the theorem in their own problems. This would be a most unfortunate result as by now even the most skeptical reader must be impressed by the power of the virial theorem to provide insight into problems of great complexity. Although there is a trade-off in that a complete dynamical description of the system is not obtainable, certain general aspects of the system are analyzable. Even though some might claim a little knowledge to be a dangerous thing, I prefer to believe that a little knowledge is better than none at all. Thus, the perceptive student of science will utilize the virial theorem to provide a 'first look' at problems to see which are of interest. Used well this first look will not be the last.

Through the course of this book we have examined the origin of the virial theorem, noted its development and applicability to a wide range of astrophysical problems, and it is irresistible to contemplate briefly its future growth. In my youth the course of future events always seemed depressingly clear but turned out to be generally wrong. Now, in spite of a better time base on which to peer forward, the new future seems at best "seen through a glass darkly", and I am mindful that astronomers have not had an exemplary record as predictors of future events^†. Nevertheless, there may be one or two areas of growth for the virial theorem on which we can count with some certainty.

Immediate problems which seem ideally suited to the application of the virial theorem certainly include exploration into the nature of the energy source in QSO's. Perhaps one will finally observe that the gravitational energy of assembly of a galaxy or its components is of the same order as the estimated energy liberated by a QSO during its lifetime. The virial theorem implies that half of this energy may be radiated away. Thus, it would appear that one need not look for the source of such energy but rather be concerned with the details of the "generator".

†

It is said that the great American astronomer, Simon Newcomb "proved" that heavier than air flight was impossible and that after the Wright Brothers flew, it was rumored that he maintained it would never be practical as no more than two people could be carried by such means.

Perhaps future development will consider applications of the virial theorem as represented by Lagrange's identity. To date the virial theorem has been applied to systems in or near equilibrium. It is worth remembering that perhaps the most important aspect of the theorem is that it is a global theorem. Thus systems in a state of rapid dynamic change are still subject to its time dependent form.

In the mid twentieth century, as a consequence of discovering that the universe is not a quiet place, theoreticians became greatly excited about the properties of objects undergoing unrestrained gravitational collapse. It is logical to suppose that sooner or later they will become interested in the effects of such a collapse upon fields other than gravitation (i.e. magnetic or rotational), that may be present. The virial theorem provides a clear statement on how the energy in such a system will be shifted from one form to another as soon as one has determined \(\mathrm{d}^2 \mathrm{I} / \mathrm{dt}^2\). Future investigation in this area may be relevant to phenomena ranging from novae to quasars. Perhaps the most exciting and at the same time least clear and speculative development in which the virial theorem may play a role involves its relationship to general relativity. This is a time of great activity and anticipatory excitement in fundamental physics and general relativity in particular.

Perhaps through the efforts of Stephan Hawking and others, and as Denis Sciama has noted, we are on the brink of the unification of general relativity, quantum mechanics, and thermodynamics. Thermodynamics is the handmaiden of statistical mechanics and it is here through the application of the ergodic theorem that the virial theorem may play its most important future role.

You may remember that in Chapter II, difficulty in the interpretation of moments taken over space-time frustrated a general development of the virial theorem in general relativity and it was necessary to invoke first order approximations to the relativistic field Equations. In addition the ergodic theorem seems inexorably tied to the nature of reversible and irreversible processes. The advances in relating general relativity to thermodynamics bring these areas and theorems into direct conceptual confrontation and may perhaps provide the foundations for the proper understanding of time itself.