User:Tohline/SSC/Virial/Isothermal
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Virial Equilibrium of Isothermal Spheres
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Review
In an introductory discussion of the virial equilibrium structure of spherically symmetric configurations — see especially the section titled, Energy Extrema — we deduced that a system's equilibrium radius, , measured relative to a reference length scale, , i.e., the dimensionless equilibrium radius,
is given by the root(s) of the following equation:
where the definitions of the various coefficients are,















Once the pressure exerted by the external medium (), and the configuration's mass (), angular momentum (), and specific entropy (via ) — or, in the isothermal case, sound speed () — have been specified, the values of all of the coefficients are known and can be determined.
Isolated, Nonrotating Configuration
For a nonrotating configuration that is not influenced by the effects of a bounding external medium , the statement of virial equilibrium is,
Isothermal Evolutions
For isothermal configurations , one and only one equilibrium state arises where,
that is,
Nonrotating Configuration Embedded in an External Medium
For a nonrotating configuration (C = J = 0) that is embedded in, and is influenced by the pressure P_{e} of, an external medium, the statement of virial equilibrium is,
Bounded Isothermal
For isothermal configurations , we deduce that equilibrium states exist at radii given by the roots of the equation,
Bonnor's (1956) Equivalent Relation
Inserting the expressions for the coefficients , , and gives,
or, because the volume for a spherical configuration, we can write,
It is instructive to compare this expression for a selfgravitating, isothermal equilibrium sphere to the one that was presented in 1956 by Bonnor (1956, MNRAS, 116, 351) as equation (1.2) in a paper titled, "Boyle's Law and Gravitational Instability":
Once we realize that, for an isothermal configuration, twice the thermal energy content, , can be written as just as well as via the product, , we see that our expression is identical to Bonnor's if we set the prefactor on Bonnor's last term, . (Indeed, later on the first page of his paper, Bonnor points out that this is the appropriate value for when considering a uniform density sphere.)
PV Diagram
Returning to the dimensionless form of this expression and multiplying through by , we obtain,
Now, taking a cue from the solution presented above for an isolated isothermal configuration, we choose to set the previously unspecified scale factor, , to,
in which case , and the quartic equation governing the radii of equilibrium states becomes, simply,
where,
For a given choice of and , can represent a dimensionless mass, in which case,
Alternatively, for a given choice of configuration mass and sound speed, this parameter, , can be viewed as a dimensionless external pressure; or, for a given choice of and , can represent a dimensionless sound speed. In most of what follows we will view as a dimensionless external pressure.
The above quartic equation can be rearranged immediately to give the external pressure that is required to obtain a particular configuration radius, namely,
The resulting behavior is shown by the black curve in Figure 2.
Figure 2: Equilibrium Isothermal PV Diagram 

The black curve traces out the function,
and shows the dimensionless external pressure, , that is required to construct a nonrotating, selfgravitating, isothermal sphere with an equilibrium radius . The pressure becomes negative at radii , hence the solution in this regime is unphysical. Figure 1 displays the free energy surface that "lies above" the twodimensional parameter space ; that is identified here by the thin, red rectangle. 
In the absence of selfgravity (i.e., ), the product of the external pressure and the volume should be constant. The corresponding relation, , is shown by the blue dashed curve in the figure. As the figure illustrates, when gravity is included the PV relationship pulls away from the PV = constant curve at sufficiently small volumes. Indeed, the curve turns over at a finite pressure, , and for every value of a second, more compact equilibrium configuration appears. The location of along the curve is identified by setting , that is, it occurs where,
Hence,
therefore, from above,
Quartic Solution
In the above diagram discussion, we rearranged the quartic equation governing equilibrium configurations to give for any chosen value of . Alternatively, the four roots of the quartic equation — , , and in the presentation that follows — will identify the radii at which a spherical configuration will be in equilibrium for any choice of the external pressure, , assuming the roots are real.
Roots of the quartic equation: χ^{4} − χΠ^{ − 1} + Π^{ − 1} = 0 

where,
and,
is the real root of the cubic equation,

Because must be positive in physically realistic solutions, we conclude that the two roots involving — that is, and — are imaginary and, hence, unphysical. The other two roots — and — will be real only if the arguments inside the radicals in the expression for are positive. That is, and will be real only for values of the dimensionless external pressure,
This is the same upper limit on the external pressure that was derived above, via a different approach, and translates into a maximum mass for a pressurebounded isothermal configuration of,
When combined, a plot of versus and versus will reproduce the solid black curve shown in Figure 2, but with the axes flipped. The topright quadrant of Figure 3 presents such a plot, but in logarithmic units along both axes; also is normalized to and is normalized to the equilibrium radius at that pressure. This is the manner in which Whitworth (1981, MNRAS, 195, 967) chose to present this result for uniformdensity, spherical isothermal configurations. Our solid and dashed curve segments — identifying, respectively, the and solutions to the above quadratic equation — precisely match the solid and dashed curve segments labeled "1" in Whitworth's Figure 1a (replicated here in the bottomright quadrant of Figure 3).
Figure 3: Equilibrium RP Diagram 

Top: The solid curve traces the function and the dashed curve traces the function , where and are the two real roots of the quartic equation,
Logarithmic units are used along both axes; is normalized to ; and is normalized to the equilibrium radius at . Bottom: A reproduction of Figure 1a from Whitworth (1981, MNRAS, 195, 967). The solid and dashed segments of the curve labeled "1" identify the equilibrium radii, , that result from embedding a uniformdensity, isothermal gas cloud in an external medium of pressure . Comparison: The curve shown above that traces out and should be identical to the "Whitworth" curve labeled "1". 

See Also
© 2014  2019 by Joel E. Tohline 