Editing SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt3 (section)

====Analysis Philosophy====
The mass-radius relationship that derives from detailed force-balanced models is a physically meaningful and reliable statement of how a configuration's equilibrium radius will vary if its mass is changed.  (It must be accepted that the configuration's structural form factors will change as it settles into each new equilibrium state, so such an "evolution" must occur on a ''secular'' time scale.)  From the outset, however, the mass-radius relationship derived via the virial theorem &#8212; which, itself, derives from an analysis of the free energy function &#8212; should not be relied upon for the same physical insight.  Consider, for example, that the scalar virial theorem is obtained from an analysis of the free-energy function by varying a system's size ''while holding constant all coefficients in the free-energy expression''; this means that the system's mass as well as its structural form factors is held fixed while searching for an extremum in the free energy.  The temptation, then, is to use the virial theorem to predict what the configuration's new equilibrium size will be if the system's mass is changed while holding the coefficients in the virial theorem constant.  This means holding the structural form factors constant but not simultaneously holding the mass constant, and this differs from the constraints put on the free-energy function analysis that led to the virial theorem expression in the first place!

But we can combine the two analyses &#8212; the detailed force-balance analysis and the free-energy analysis &#8212; in the following meaningful way.  Use the detailed force-balance analysis to identify the properties of an equilibrium state, specifically, for a given mass, determine the system's equilibrium radius and its accompanying structural form factors.  (The virial theorem will be satisfied by this same set of determined parameter values.)  Then, holding both the mass and the structural form factors constant, see how the free energy of the system varies as the configuration's size changed.  In this manner the system's ''dynamical'' stability can be ascertained.

In summary:  The mass-radius relationship determined from an analysis of detailed force-balanced models defines the physically correct ''secular'' evolutionary track for the system; while, an analysis of the free energy variations about an equilibrium state will answer the question of ''dynamical'' stability.