Editing VE (section)

==Preface==
The astrophysics community relies heavily on the virial equations &#8212; most often in the context of the [[#Scalar_Virial_Theorem|scalar virial theorem]] &#8212; to ascertain the basic properties of equilibrium systems.  As is described below, fundamentally the virial equations are obtained by taking moments of the Euler equation.  By examining the balance among various relevant energy reservoirs, the mathematical expression that defines virial equilibrium provides a means by which, for example, the radius of a configuration can be estimated, given a total system mass and mean system temperature.  It can also be used to estimate a system's maximum allowed rotation frequency and whether or not the properties of the equilibrium configuration will be significantly modified if the system is embedded in a hot tenuous external medium.

As is also discussed, below, it can be even more informative to examine how a system's global, Gibbs-like free energy, <math>\mathfrak{G}</math>, varies under contraction or expansion.  Extrema in the free energy identify equilibrium configurations, for example.  For spherically symmetric systems, in particular, the [[#Scalar_Virial_Theorem|scalar virial theorem]] is "derived" by identifying under what conditions <math>d\mathfrak{G}/dR = 0</math>.  Furthermore, the sign of the second derivative, <math>d^2\mathfrak{G}/dR^2</math>, tells whether or not the equilibrium state is stable or unstable.  Here we define relevant energy reservoirs that contribute to a system's global free energy.  In separate chapters we use the free energy function to help identify the properties of equilibrium systems and to examine their relative stability.

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'''Why Bother?'''<br />

Excerpts drawn from the introductory chapter (p. 3) of<br />
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<font color="#770000">'''Question'''</font>:  &nbsp; Why bother introducing the virial theorem and its allied free-energy expression, given that the astrophysical systems we are interested in analyzing can be fully described by solutions of the set of [[PGE#Principal_Governing_Equations|Principal Governing Equations]]?

<font color="#770000">'''Answer'''</font>: &nbsp; The [[PGE#Principal_Governing_Equations|Principal Governing Equations]] are, in general, <font color="#008899">non-linear, second-order, vector differential equations which exhibit closed form solutions only in special cases.  Although additional cases may be solved numerically, insight into the behavior of systems in general is very difficult to obtain in this manner.  The virial theorem</font> and its associated free-energy expression <font color="#008899">generally deals in scalar quantities and usually is applied on a global scale.  This reduction in complexity  &#8212; from a vector description to a scalar one &#8212; frequently enables us to solve the resulting equations</font> in closed form and to ascertain more straightforwardly what physical processes are most responsible for defining properties of the solution.

<font color="#770000">'''Caution'''</font>: &nbsp; We should always keep in mind that <font color="#008899">this reduction in complexity results in a concomitant loss of information and we cannot expect to obtain as complete a description of a physical system as would be possible from a full solution of the</font> [[PGE#Principal_Governing_Equations|Principal Governing Equations]].
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