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__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Virial Equilibrium of Adiabatic Spheres= Highlights of the rather detailed discussion presented below have been [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Virial_Equilibrium_of_Adiabatic_Spheres_.28Summary.29|summarized in an accompanying chapter]] of this H_Book. ==Review== ===Adopted Normalizations=== In an [[SSCpt1/Virial#Virial_Equilibrium_of_Spherically_Symmetric_Configurations|introductory discussion]] of the virial equilibrium structure of spherically symmetric configurations, we adopted the following physical parameter normalizations for adiabatic systems. <div align="center"> <table border="1" align="center" cellpadding="5"> <tr><th align="center" colspan="2"> Adopted Normalizations for Adiabatic Systems </th></tr> <tr><th align="center"> In Terms of <math>~\gamma_g</math> </th> <th align="center"> In Terms of <math>~n</math> </th></tr> <tr> <td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~R_\mathrm{norm}</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~\biggl[ \biggl( \frac{G}{K} \biggr) M_\mathrm{tot}^{2-\gamma_g} \biggr]^{1/(4-3\gamma_g)} </math> </td> </tr> <tr> <td align="right"> <math>~P_\mathrm{norm}</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~\biggl[ \frac{K^4}{G^{3\gamma_g} M_\mathrm{tot}^{2\gamma_g}} \biggr]^{1/(4-3\gamma_g)} </math> </td> </tr> <tr> <td align="center" colspan="3"> ---- </td> </tr> <tr> <td align="right"> <math>~E_\mathrm{norm}</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~ P_\mathrm{norm} R_\mathrm{norm}^3 = \biggl[ KG^{3(1-\gamma_g)}M_\mathrm{tot}^{6-5\gamma_g} \biggr]^{1/(4-3\gamma_g)} </math> </td> </tr> <tr> <td align="right"> <math>~\rho_\mathrm{norm}</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~\frac{3M_\mathrm{tot}}{4\pi R_\mathrm{norm}^3} = \frac{3}{4\pi} \biggl[ \frac{K^3}{G^3 M_\mathrm{tot}^2} \biggr]^{1/(4-3\gamma_g )} </math> </td> </tr> <tr> <td align="right"> <math>~c^2_\mathrm{norm}</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~\frac{P_\mathrm{norm}}{\rho_\mathrm{norm}} = \frac{4\pi}{3} \biggl[ \frac{K}{(G^3 M_\mathrm{tot}^2)^{\gamma_g-1}} \biggr]^{1/(4-3\gamma_g )} </math> </td> </tr> </table> </td> <td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~R_\mathrm{norm}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)} </math> </td> </tr> <tr> <td align="right"> <math>~P_\mathrm{norm}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)} </math> </td> </tr> <tr> <td align="center" colspan="3"> ---- </td> </tr> <tr> <td align="right"> <math>~E_\mathrm{norm}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[ K^n G^{-3}M_\mathrm{tot}^{n-5} \biggr]^{1/(n-3)} </math> </td> </tr> <tr> <td align="right"> <math>~\rho_\mathrm{norm}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{3}{4\pi} \biggl[ \frac{K^3}{G^3 M_\mathrm{tot}^2} \biggr]^{n/(n-3 )} </math> </td> </tr> <tr> <td align="right"> <math>~c^2_\mathrm{norm}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> ~\frac{4\pi}{3} \biggl[ \frac{K^n}{G^3 M_\mathrm{tot}^2} \biggr]^{1/(n-3 )} </math> </td> </tr> </table> </td> </tr> </table> </div> ===Virial Equilibrium=== Also in our introductory discussion — see especially the section titled, [[SSCpt1/Virial#Energy_Extrema|''Energy Extrema'']] — we deduced that an adiabatic system's dimensionless equilibrium radius, <div align="center"> <math>~\chi_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \, ,</math> </div> is given by the root(s) of the following equation: <div align="center"> <math> 2\mathcal{C} \chi_\mathrm{eq}^{-2} + ~3 \mathcal{B}\chi_\mathrm{eq}^{3 -3\gamma_g} -~3\mathcal{A}\chi_\mathrm{eq}^{-1} -~ 3\mathcal{D} \chi_\mathrm{eq}^3 = 0 \, , </math> </div> where the definitions of the various coefficients are, <div align="center"> <table border="0" cellpadding="5"> <tr> <td align="right"> <math>~\mathcal{A}</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>\frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W \, ,</math> </td> </tr> <tr> <td align="right"> <math>~\mathcal{B}</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math> \frac{4\pi}{3} \biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]_\mathrm{eq}^{\gamma} \cdot \mathfrak{f}_A </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> \frac{4\pi}{3} \biggl[ \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr)\chi^{3\gamma} \biggr]_\mathrm{eq} \cdot \mathfrak{f}_A \, , </math> </td> </tr> <tr> <td align="right"> <math>~\mathcal{C}</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math> \frac{3\cdot 5}{2^4 \pi} \biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr] \cdot \frac{\mathfrak{f}_T}{\mathfrak{f}_M} \, , </math> </td> </tr> <tr> <td align="right"> <math>~\mathcal{D}</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math> \biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \, . </math> </td> </tr> </table> </div> (The [[SSCpt1/Virial#Structural_Form_Factors|dimensionless structural form factors]], <math>\mathfrak{f}_i,</math> that appear in these expressions are defined ''for isolated polytropes'' in our accompanying introductory discussion and are [[SSC/Virial/Polytropes#Role_of_Structural_Form_Factors|discussed further, below]].) Once the pressure exerted by the external medium (<math>P_e</math>), and the configuration's mass (<math>M_\mathrm{tot}</math>), angular momentum (<math>J</math>), and specific entropy (via <math>K</math>) have been specified, the values of all of the coefficients are known and <math>\chi_\mathrm{eq}</math> can be determined. <div align="center" id="ABratios"> <table border="1" cellpadding="10" align="center"> <tr><td align="left"> For later use, we note that after making the substitution, <math>~\gamma_g \rightarrow (n+1)/n</math>, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\biggl( \frac{\mathcal{A}}{\mathcal{B}} \biggr)^n</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl\{ \frac{3}{20\pi} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W \biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]^{-(n+1)/n} \cdot \mathfrak{f}_A^{-1} \biggr\}^n</math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~5^{-n} \biggl( \frac{3}{4\pi} \biggr)^n \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^{2n} \biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]^{-(n+1)} \cdot \biggl( \frac{\mathfrak{f}_W}{\mathfrak{f}_A} \biggr)^{n} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{1}{5^n} \biggl( \frac{4\pi}{3} \biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \cdot \frac{\mathfrak{f}_W^n}{\mathfrak{f}_A^n \mathfrak{f}_M^{n-1}} \, ; </math> </td> </tr> </table> </div> and, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl\{ \frac{1}{5} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W \biggr\}^{-3(n+1)} \biggl\{ \frac{4\pi}{3} \biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]^{(n+1)/n} \cdot \mathfrak{f}_A \biggr\}^{4n}</math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 5^{3(n+1)} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^{-6(n+1)} \biggl( \frac{4\pi}{3} \biggr)^{4n-4(n+1)} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]^{4(n+1)} \cdot\frac{\mathfrak{f}_A^{4n}}{\mathfrak{f}_W^{3(n+1)}} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 5^{3(n+1)} \biggl( \frac{3}{4\pi} \biggr)^{4} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2(n+1)} \cdot\frac{\mathfrak{f}_A^{4n} \mathfrak{f}_M^{2(n+1)} }{\mathfrak{f}_W^{3(n+1)}} </math> </td> </tr> </table> </div> </td></tr> </table> </div> ==Isolated Nonrotating Adiabatic Configuration== For a nonrotating configuration <math>~(C=J=0)</math> that is not influenced by the effects of a bounding external medium <math>~(D=P_e = 0)</math>, the statement of virial equilibrium is, <div align="center"> <math> 3 B\chi_\mathrm{eq}^{3 -3\gamma_g} -~3A\chi_\mathrm{eq}^{-1} = 0 \, . </math> </div> Hence, one equilibrium state exists for each value of <math>~\gamma_g</math> and it occurs where, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\chi_\mathrm{eq}^{4-3\gamma_g} = \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{4-3\gamma_g} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> \frac{A}{B} \, . </math> </td> </tr> </table> </div> <div align="center" id="TwoPointsOfView"> <table border="1" width="100%" cellpadding="8"> <tr> <th align="center" colspan="2"> Two Points of View </th> </tr> <tr> <td align="center" colspan="1"> In terms of <math>~K</math> and <math>~M_\mathrm{limit} ~(= M_\mathrm{tot})</math> </td> <td align="center" colspan="1"> In terms of <math>~P_c</math> and <math>~M_\mathrm{limit} ~(= M_\mathrm{tot})</math> </td> </tr> <tr> <td align="center" colspan="1"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\chi_\mathrm{eq}^{4-3\gamma_g}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> \biggl\{ \frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> \times \biggl\{ \frac{3}{4\pi} \biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]_\mathrm{eq}^{-{\gamma_g}} \cdot \frac{1}{\mathfrak{f}_A} \biggr\} </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow~~~R_\mathrm{eq}^{4-3\gamma_g}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> \frac{4\pi}{3\cdot 5} \biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]_\mathrm{eq}^{2-{\gamma_g}} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A} \biggl[ \frac{GM_\mathrm{tot}^{2-\gamma_g}}{K} \biggr] </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~\frac{KR_\mathrm{eq}^{4-3\gamma_g}}{GM_\mathrm{limit}^{2-\gamma_g}}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> \frac{1}{5} \biggl( \frac{4\pi}{3} \biggr)^{\gamma_g-1} \frac{\mathfrak{f}_W}{\mathfrak{f}_A \mathfrak{f}_M^{2-\gamma_g}} </math> </td> </tr> <tr> <td align="center" colspan="3"> — — — — — — or, inverted and setting <math>~\gamma_g = 1 + 1/n</math> — — — — — — </td> </tr> <tr> <td align="center" colspan="3"> <math>~ 4\pi \biggl( \frac{G}{K} \biggr)^n M_\mathrm{limit}^{n-1} R_\mathrm{eq}^{3-n} = \biggl( \frac{5 \mathfrak{f}_A \mathfrak{f}_M}{\mathfrak{f}_W} \biggr)^n \frac{3}{\mathfrak{f}_M} </math> </td> </tr> </table> </td> <td align="center" colspan="1"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\chi_\mathrm{eq}^{4-3\gamma_g}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> \biggl\{ \frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> \times \biggl\{ \frac{3} {4\pi} \biggl[ \biggl( \frac{P_\mathrm{norm}}{P_c} \biggr)\chi^{-3\gamma} \biggr]_\mathrm{eq} \cdot \frac{1}{\mathfrak{f}_A} \biggr\} </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow~~~\chi_\mathrm{eq}^{4}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> \frac{3}{20\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \biggl( \frac{P_\mathrm{norm}}{P_c} \biggr) \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A \cdot \mathfrak{f}_M^2} </math> </td> </tr> </table> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Rightarrow~~~\frac{P_c R_\mathrm{eq}^{4}}{P_\mathrm{norm} R_\mathrm{norm}^4} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{limit}} \biggr)^{2}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> \frac{3}{20\pi} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A \cdot \mathfrak{f}_M^2} </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow~~~\frac{P_c R_\mathrm{eq}^{4}}{G M_\mathrm{limit}^2} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> \frac{3}{20\pi} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A \cdot \mathfrak{f}_M^2} </math> </td> </tr> </table> </td> </tr> </table> </div> According to the solution shown in the left-hand column, for fluid with a given specify entropy content, the equilibrium mass-radius relationship for adiabatic configurations is, <div align="center"> <math> M_\mathrm{tot}^{(\gamma_g - 2)} \propto R_\mathrm{eq}^{(3\gamma_g -4)} \, . </math> </div> We see that, for <math>~\gamma_g=2</math>, the equilibrium radius depends only on the specific entropy of the gas and is independent of the configuration's mass. Conversely, for <math>~\gamma_g = 4/3</math>, the mass of the configuration is independent of the radius. For <math>~\gamma_g > 2</math> or <math>~\gamma_g < 4/3</math>, configurations with larger mass (but the same specific entropy) have larger equilibrium radii. However, for <math>~\gamma_g</math> in the range, <math>~2 > \gamma_g > 4/3</math>, configurations with larger mass have smaller equilibrium radii. (Note that the related result for [[SSC/Virial/Isothermal#Virial_Equilibrium_of_Isothermal_Spheres|isothermal configurations]] can be obtained by setting <math>~\gamma_g = 1</math> in this adiabatic solution, because <math>~K = c_s^2</math> when <math>~\gamma_g = 1</math>.) ==Role of Structural Form Factors== When employing a virial analysis to determine the radius of an equilibrium configuration, it is customary to set the structural form factors, <math>~\mathfrak{f}_M</math>, <math>~\mathfrak{f}_W</math> and <math>~\mathfrak{f}_A</math>, to unity and accept that the expression derived for <math>~R_\mathrm{eq}</math> is an estimate of the configuration's radius that is good to within a factor of order unity. As has been demonstrated in our [[SSC/VirialEquilibrium/UniformDensity#Comparison_with_Detailed_Force-Balance_Model|related discussion of the equilibrium of uniform-density spheres]], these form factors can be evaluated if/when the internal structural profile of an equilibrium configuration is known from a complementary detailed force-balance analysis. In the case being discussed here of isolated, spherical polytropes, [[SSC/Structure/Polytropes#Lane-Emden_Equation|solutions to the]], <div align="center"> <span id="LaneEmdenEquation"><font color="#770000">'''Lane-Emden Equation'''</font></span> <br /> {{Math/EQ_SSLaneEmden01}} </div> can provide the desired internal structural information. Here we draw on [[Appendix/References|Chandrasekhar's [C67]]] discussion of the structure of spherical polytropes to show precisely how our structural form factors can be expressed in terms of the Lane-Emden function, <math>~\Theta_H</math>, dimensionless radial coordinate, <math>~\xi</math>, and the function derivative, <math>~\Theta^' = d\Theta_H/d\xi</math>. ===Mass=== We note, first, that [[Appendix/References|Chandrasekhar [C67]]] — see his Equation (78) on p. 99 — presents the following expression for the mean-to-central density ratio: <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{\bar\rho}{\rho_c}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\xi_1} \, ,</math> </td> </tr> </table> </div> where the notation at the bottom of the closing square bracket means that everything inside the square brackets should be, "evaluated at the surface of the configuration," that is, at the radial location, <math>~\xi_1</math>, where the Lane-Emden function, <math>~\Theta_H(\xi)</math>, first goes to zero. But, as we pointed out when [[SSCpt1/Virial#Structural_Form_Factors|defining the structural form factors]], the form factor associated with the configuration mass, <math>~\mathfrak{f}_M</math>, is equivalent to the mean-to-central density ratio. We conclude, therefore, that, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\mathfrak{f}_M</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\xi_1} \, .</math> </td> </tr> </table> </div> ===Gravitational Potential Energy=== Second, we note that [[Appendix/References|Chandrasekhar's [C67]]] expression for the gravitational potential energy — see his Equation (90), p. 101 — is, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~-W</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{3}{5-n} \biggl( \frac{GM^2}{R} \biggr) \, ,</math> </td> </tr> </table> </div> whereas our analogous expression is, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~-W_\mathrm{grav}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{3}{5} \biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}} \biggr) \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} \, .</math> </td> </tr> </table> </div> We conclude, therefore, that, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{5}{5-n} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~~\mathfrak{f}_W </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\xi_1} \, .</math> </td> </tr> </table> </div> ===Mass-Radius Relationship=== Third, [[Appendix/References|Chandrasekhar [C67]]] shows — see his Equation (72), p. 98 — that the general mass-radius relationship for isolated spherical polytropes is, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~GM^{(n-1)/n} R^{(3-n)/n}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> ~\frac{(n+1)K}{(4\pi)^{1/n}} \biggl[ - \xi^{(n+1)/(n-1)} \frac{d\Theta_H}{d\xi} \biggr]^{(n-1)/n}_{\xi=\xi_1} \, , </math> </td> </tr> </table> </div> which we choose to rewrite as, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~4\pi \biggl( \frac{G}{K}\biggr)^n M^{(n-1)} R^{(3-n)}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> ~(n+1)^n\biggl[ \xi^{(n+1)} (-\Theta^')^{(n-1)}\biggr]_{\xi=\xi_1} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> - \biggl( \frac{\xi}{\Theta^'} \biggr)_{\xi=\xi_1} \biggl[ (n+1) \xi ~(-\Theta^') \biggr]^n_{\xi=\xi_1} \, . </math> </td> </tr> </table> </div> By comparison, the expression for the equilibrium radius that has been derived, above, from an analysis of extrema in the free energy function — specifically, see the last expression in the left-hand column of the [[SSC/Virial/Polytropes#TwoPointsOfView|table titled "Two Points of View"]] — we obtain, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~4\pi \biggl( \frac{G}{K}\biggr)^n M^{(n-1)} R_\mathrm{eq}^{(3-n)}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> ~\frac{3}{\mathfrak{f}_M} \biggl( \frac{5\mathfrak{f}_A \mathfrak{f}_M}{\mathfrak{f}_W} \biggr)^n \, . </math> </td> </tr> </table> </div> Hence, it appears as though, quite generally, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~ \frac{1}{\mathfrak{f}_M} \biggl( \frac{5\mathfrak{f}_A \mathfrak{f}_M}{\mathfrak{f}_W} \biggr)^n </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> - \biggl( \frac{\xi}{3\Theta^'} \biggr)_{\xi=\xi_1} \biggl[ (n+1) \xi ~(-\Theta^') \biggr]^n_{\xi=\xi_1} \, . </math> </td> </tr> </table> </div> Or, taking into account the expressions for <math>~\mathfrak{f}_M</math> and <math>~\mathfrak{f}_W</math> that have just been uncovered, we conclude that, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~ \frac{5\mathfrak{f}_A \mathfrak{f}_M}{\mathfrak{f}_W} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> \biggl[ (n+1) \xi ~(-\Theta^') \biggr]_{\xi=\xi_1} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~~ \frac{\mathfrak{f}_A}{\mathfrak{f}_W} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> \frac{(n+1) }{3\cdot 5} ~\xi_1^2 \, . </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~~ \mathfrak{f}_A </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> \frac{(n+1) }{3\cdot 5} ~\xi_1^2 \biggl\{ \frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\xi_1} \biggr\}\, . </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> \frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\xi_1} \, . </math> </td> </tr> </table> </div> ===Central and Mean Pressure=== It is also worth pointing out that [[Appendix/References|Chandrasekhar [C67]]] — see his Equations (80) & (81), p. 99 — introduces a dimensionless structural form factor, <math>~W_n</math>, for the central pressure via the expression, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~P_c</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~W_n \biggl( \frac{GM^2}{R^4} \biggr) \, ,</math> </td> </tr> </table> </div> and demonstrates that, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{1}{W_n}</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~4\pi (n+1) \biggl[ \Theta^' \biggr]^2_{\xi_1} \, .</math> </td> </tr> </table> </div> It is therefore clear that a spherical polytrope's central pressure is expressible in terms of our structural form factor, <math>~\mathfrak{f}_A</math>, as, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~P_c</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{3}{4\pi (5-n)} \biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr) \frac{1}{\mathfrak{f}_A} \, .</math> </td> </tr> </table> </div> Looking back at our [[SSCpt1/Virial#FormFactors|original definition of the structural form factors]], we note that, <div align="center"> <math>\mathfrak{f}_A = \biggl( \frac{\bar{P}}{P_c} \biggr)_\mathrm{eq} \, .</math> </div> Hence, this last equilibrium relation can be rewritten as, <div align="center"> <math> \frac{\bar{P} R_\mathrm{eq}^4}{GM_\mathrm{tot}^2} = \frac{3}{4\pi (5-n)} \, .</math> </div> ===Alternate Derivation of Gravitational Potential Energy=== <!-- BEGINNING OF DELETION: The following subsubsection is redundant, now that the "Two Points of View" table has been included. ====Central Pressure==== As is pointed out, above, in our [[SSC/Virial/Polytropes#Mass-Radius_Relationship|discussion of the mass-radius relationship]], the expression for the equilibrium radius that has been derived from our analysis of extrema in the free energy function can be written as, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~4\pi \biggl( \frac{G}{K}\biggr)^n M^{(n-1)} R_\mathrm{eq}^{(3-n)}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> ~\frac{3}{\mathfrak{f}_M} \biggl( \frac{5\mathfrak{f}_A \mathfrak{f}_M}{\mathfrak{f}_W} \biggr)^n \, . </math> </td> </tr> </table> </div> Via the polytropic equation of state, we can relate <math>~K</math> to the central pressure as follows: <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~P_c</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> ~K \rho_c^{1+1/n} = K \bar\rho^{1+1/n} \biggl( \frac{\rho_c}{\bar\rho} \biggr)^{1+1/n} = K \biggl[ \frac{3M_\mathrm{tot}}{4\pi R_\mathrm{eq}^3} \biggr]^{(n+1)/n} \biggl( \frac{1}{\mathfrak{f}_M} \biggr)^{(n+1)/n} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~~ K^{n}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> P_c^{n} \biggl[ \frac{4\pi R_\mathrm{eq}^3}{3M_\mathrm{tot}} \biggr]^{(n+1)} \mathfrak{f}_M^{(n+1)} \, . </math> </td> </tr> </table> </div> Hence, in the mass-radius relationship we can replace <math>~K</math> with this expression to obtain, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~4\pi \biggl( \frac{G}{P_c}\biggr)^n M^{(n-1)} R_\mathrm{eq}^{(3-n)} \biggl[ \frac{3M_\mathrm{tot}}{4\pi R_\mathrm{eq}^3} \biggr]^{(n+1)}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> ~3 \biggl( \frac{5\mathfrak{f}_A \mathfrak{f}_M^2}{\mathfrak{f}_W} \biggr)^n </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~~ \biggl( \frac{G M_\mathrm{tot}^2}{P_c R_\mathrm{eq}^4}\biggr)^n \biggl( \frac{3}{4\pi } \biggr)^{n}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> ~\biggl( \frac{5\mathfrak{f}_A \mathfrak{f}_M^2}{\mathfrak{f}_W} \biggr)^n </math> </td> </tr> <tr> <td align="right" colspan="3"> <math> \Rightarrow ~~~~ P_c = \frac{3 }{20\pi }\biggl( \frac{G M_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr) \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A \mathfrak{f}_M^2} \, . </math> </td> </tr> </table> </div> Now, from our above [[SSC/Virial/Polytropes#Gravitational_Potential_Energy|examination of the expression for the gravitational potential energy]], we know that, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{\mathfrak{f}_W}{\mathfrak{f}_M^2}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{5}{5-n} \, .</math> </td> </tr> </table> </div> Hence, our expression for the central pressure becomes, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right" colspan="3"> <math> ~P_c = \frac{3 }{4\pi (5-n)}\biggl( \frac{G M_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr) \cdot \frac{1}{\mathfrak{f}_A } \, . </math> </td> </tr> </table> </div> As it should, this precisely matches [[SSC/Virial/Polytropes#Central_Pressure|the expression that we derived, above]], starting from [[Appendix/References|Chandrasekhar's [C67]]] presentation. ====Gravitational Potential Energy==== END OF DELETION --> As has been [[SSCpt1/Virial#AlternateGravPotEnergy|discussed elsewhere]], we have learned from [[Appendix/References|Chandrasekhar's discussion of polytropic spheres [C67]]] — see his Equation (16), p. 64 — that if a spherically symmetric system is in hydrostatic balance, the total gravitational potential energy can be obtained from the following integral: <div align="center"> <table border="0" cellpadding="5"> <tr> <td align="right"> <math>~W_\mathrm{grav}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ + \frac{1}{2} \int_0^R \Phi(r) dm \, .</math> </td> </tr> </table> </div> Using "[[SSCpt2/SolutionStrategies#Technique_3|technique #3]]" to solve the differential equation that governs the statement of hydrostatic balance, we know that in any polytropic sphere, <math>~\Phi(r)</math> is related to the configuration's radial enthalpy profile, <math>~H(r)</math>, via the algebraic expression, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Phi(r) + H(r)</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~C_B \, ,</math> </td> </tr> </table> </div> where, <math>~C_B</math>, is an integration constant. At the surface of the equilibrium configuration, <math>~H = 0</math> and <math>~\Phi = - GM_\mathrm{tot}/R_\mathrm{eq}</math>, so the integration constant is, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~C_B</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~- \frac{GM_\mathrm{tot}}{R_\mathrm{eq}} \, ,</math> </td> </tr> </table> </div> which implies, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Phi(r) </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ - H(r) - \frac{GM_\mathrm{tot}}{R_\mathrm{eq}} \, .</math> </td> </tr> </table> </div> Now, from [[SR#Barotropic_Structure|our general discussion of barotropic relations]], we can write, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~H(r)</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~(n+1) \frac{P(r)}{\rho(r)} \, .</math> </td> </tr> </table> </div> Hence, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~-\Phi(r)</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~(n+1) \frac{P(r)}{\rho(r)} + \frac{GM_\mathrm{tot}}{R_\mathrm{eq}} \, ,</math> </td> </tr> </table> </div> and, <div align="center"> <table border="0" cellpadding="5"> <tr> <td align="right"> <math>~W_\mathrm{grav}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ - \frac{1}{2} \int_0^R \biggl[ (n+1) \frac{P(r)}{\rho(r)} + \frac{GM_\mathrm{tot}}{R_\mathrm{eq}} \biggr] 4\pi \rho(r) r^2 dr </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ - 2\pi \biggl\{ (n+1) \int_0^R P(r) r^2 dr + \frac{GM_\mathrm{tot}}{R_\mathrm{eq}} \int_0^R \rho(r) r^2 dr \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ - 2\pi \biggl\{ \frac{1}{3} (n+1)P_c R_\mathrm{eq}^3 \int_0^1 3\biggl[ \frac{P(x)}{P_c} \biggr] x^2 dx + \frac{GM_\mathrm{tot}}{3R_\mathrm{eq}} \biggl( \rho_c R_\mathrm{eq}^3 \biggr) \int_0^1 3 \biggl[ \frac{\rho(x)}{\rho_c} \biggr] x^2 dx \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ - 2\pi \biggl\{ \frac{1}{3} (n+1)P_c R_\mathrm{eq}^3 \mathfrak{f}_A + \frac{GM_\mathrm{tot}}{3R_\mathrm{eq}} \biggl( \rho_c R_\mathrm{eq}^3 \biggr) \mathfrak{f}_M \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ - \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}} \biggl\{ \frac{2\pi}{3} (n+1) \biggl[ \frac{P_c R_\mathrm{eq}^4}{GM_\mathrm{tot}^2} \biggr] \mathfrak{f}_A + \frac{1}{2} \biggl[ \frac{4\pi \rho_c R_\mathrm{eq}^3}{3M_\mathrm{tot}} \biggr] \mathfrak{f}_M \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ - \frac{1}{2} \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}} \biggl\{ \frac{4\pi}{3} (n+1) \biggl[ \frac{P_c R_\mathrm{eq}^4}{GM_\mathrm{tot}^2} \biggr] \mathfrak{f}_A + 1 \biggr\} \, .</math> </td> </tr> </table> </div> We now recall two earlier expressions that show the role that our structural form factors play in the evaluation of <math>~W</math> and <math>~P_c</math>, namely, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~W_\mathrm{grav}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ - \frac{3}{5} \biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}} \biggr) \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2}</math> </td> </tr> </table> </div> and, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~P_c</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{3 }{20\pi }\biggl( \frac{G M_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr) \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A \mathfrak{f}_M^2} \, .</math> </td> </tr> </table> </div> Plugging these into our newly derived expression for the gravitational potential energy gives, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~- \frac{3}{5} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ - \frac{1}{2} \biggl\{ \frac{4\pi}{3} (n+1) \biggl[ \frac{3 }{20\pi }\cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A \mathfrak{f}_M^2} \biggr] \mathfrak{f}_A + 1 \biggr\} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~~ (2\cdot 3) \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ (n+1) \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} + 5 </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~~ (5 - n) \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 5 </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~~ \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{5}{5-n} \, . </math> </td> </tr> </table> </div> As it should, this agrees with the expression for the ratio, <math>\mathfrak{f}_W/\mathfrak{f}_M^2</math>, that was [[SSC/Virial/Polytropes#Gravitational_Potential_Energy|derived in our above discussion of the gravitational potential energy]]. ===Summary=== In summary, expressions for the three structural form factors associated with isolated, spherically symmetric polytropes are as follows: <div align="center"> <table border="1" align="center" cellpadding="5"> <tr><th align="center" colspan="1"> Structural Form Factors for <font color="red">Isolated</font> Polytropes </th></tr> <tr><td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\mathfrak{f}_M</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\xi_1} </math> </td> </tr> <tr> <td align="right"> <math>\mathfrak{f}_W </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\xi_1} </math> </td> </tr> <tr> <td align="right"> <math>\mathfrak{f}_A </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> \frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\xi_1} </math> </td> </tr> </table> </td> </tr> </table> </div> ==Nonrotating Adiabatic Configuration Embedded in an External Medium== For a nonrotating configuration <math>~(C=J=0)</math> that is embedded in, and is influenced by the pressure <math>~P_e</math> of, an external medium, the statement of virial equilibrium is, <div align="center"> <math> 3 B\chi_\mathrm{eq}^{3 -3\gamma_g} -~3A\chi_\mathrm{eq}^{-1} -~ 3D\chi_\mathrm{eq}^3 = 0 \, . </math> </div> ====Solution Expressed in Terms of K and M (Whitworth's 1981 Relation)==== This is precisely the same condition that derives from setting equation (3) to zero in [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth's] (1981, MNRAS, 195, 967) discussion of the ''Global Gravitational Stability for One-dimensional Polytropes''. The overlap with Whitworth's narative is clearer after introducing the algebraic expressions for the coefficients <math>~A</math>, <math>~B</math>, and <math>~D</math>, to obtain, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~4\pi \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \chi_\mathrm{eq}^3 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~3 \biggl( \frac{4\pi}{3} \biggr)^{1-\gamma_g} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{\gamma_g} \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{\gamma_g}} \cdot \chi_\mathrm{eq}^{3 -3\gamma_g} ~-~\frac{3}{5} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \cdot \chi_\mathrm{eq}^{-1} \, ;</math> </td> </tr> </table> </div> dividing the equation through by <math>~(4\pi \chi_\mathrm{eq}^3/P_\mathrm{norm})</math>, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~P_e </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~P_\mathrm{norm} \biggl[ \biggl( \frac{3}{4\pi} \biggr)^{\gamma_g} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{\gamma_g} \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{\gamma_g}} \cdot \chi_\mathrm{eq}^{-3\gamma_g} ~- \biggl(\frac{3}{20\pi} \biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \cdot \chi_\mathrm{eq}^{-4} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~P_\mathrm{norm} R_\mathrm{norm}^4\biggl[ \biggl( \frac{3}{4\pi R_\mathrm{eq}^3} \biggr)^{\gamma_g} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{\gamma_g} \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{\gamma_g}} \cdot R_\mathrm{norm}^{3\gamma_g-4} - \biggl(\frac{3}{20\pi R_\mathrm{eq}^4} \biggr)\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \biggr] \, ;</math> </td> </tr> </table> </div> and inserting expressions for the [[User:Tohline/SphericallySymmetricConfigurations/Virial#Normalizations|parameter normalizations]] as defined in our accompanying [[User:Tohline/SphericallySymmetricConfigurations/Virial#Virial_Equilibrium_of_Spherically_Symmetric_Configurations|introductory discussion]] to obtain, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~P_e </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~GM_\mathrm{tot}^2\biggl[ \biggl( \frac{3}{4\pi R_\mathrm{eq}^3} \biggr)^{\gamma_g} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{\gamma_g} \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{\gamma_g}} \cdot \frac{K M_\mathrm{tot}^{\gamma_g-2}}{G} - \biggl(\frac{3}{20\pi R_\mathrm{eq}^4} \biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~K \biggl( \frac{3M_\mathrm{limit}}{4\pi R_\mathrm{eq}^3} \biggr)^{\gamma_g} \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{\gamma_g}} - \biggl(\frac{3GM_\mathrm{limit}^2}{20\pi R_\mathrm{eq}^4} \biggr) \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \, .</math> </td> </tr> </table> </div> If the structural form factors are set equal to unity, this exactly matches equation (5) of [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth], which reads: <div align="center"> <table border="2"> <tr><td> [[File:Whitworth1981Eq5.jpg|500px|center|Whitworth (1981, MNRAS, 195, 967)]] </td></tr> </table> </div> Notice that, when <math>~P_e \rightarrow 0</math>, this expression reduces to the solution we obtained for an isolated polytrope, expressed in terms of <math>~K</math> and <math>~M_\mathrm{limit}</math> (see the left-hand column of our [[User:Tohline/SSC/Virial/Polytropes#TwoPointsOfView|table titled "Two Points of View"]]). ====Solution Expressed in Terms of M and Central Pressure==== Beginning again with the [[User:Tohline/SSC/Virial/Polytropes#Nonrotating_Adiabatic_Configuration_Embedded_in_an_External_Medium|relevant statement of virial equilibirum]], namely, <div align="center"> <math> A = B\chi_\mathrm{eq}^{4 -3\gamma_g} -~ D\chi_\mathrm{eq}^4 \, , </math> </div> but adopting the alternate expression for the coefficient, <math>~B</math>, [[User:Tohline/SSC/Virial/Polytropes#Virial_Equilibrium|given above]], that is, <div align="center"> <math> B = \frac{4\pi}{3} \biggl[ \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr)\chi^{3\gamma} \biggr]_\mathrm{eq} \cdot \mathfrak{f}_A \, , </math> </div> we can write, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{4\pi}{3} \biggl[ \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr)\chi^{3\gamma} \biggr]_\mathrm{eq} \mathfrak{f}_A \cdot \chi_\mathrm{eq}^{4 -3\gamma_g} -~ \biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \chi_\mathrm{eq}^4 </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow~~~\frac{3}{20\pi} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[ \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr) \mathfrak{f}_A -~ \frac{P_e}{P_\mathrm{norm}} \biggr] \chi_\mathrm{eq}^4 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[ \mathfrak{f}_A P_c -~ P_e\biggr] \frac{R_\mathrm{eq}^4}{G M_\mathrm{tot}^2} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \frac{3}{20\pi} \biggl( \frac{G M_\mathrm{limit}^2}{R_\mathrm{eq}^4}\biggr) \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \mathfrak{f}_A P_c -~ P_e \, . </math> </td> </tr> </table> </div> Again notice that, when <math>~P_e \rightarrow 0</math>, this expression reduces to the solution we obtained for an isolated polytrope, but this time expressed in terms of <math>~P_c</math> and <math>~M_\mathrm{limit}</math> (see the right-hand column of our [[User:Tohline/SSC/Virial/Polytropes#TwoPointsOfView|table titled "Two Points of View"]]). ====Contrast with Detailed Force-Balanced Solution==== As has just been demonstrated, the virial theorem provides a mathematical expression that allows us to ''relate'' the equilibrium radius of a configuration to the applied external pressure, once the configuration's mass and either its specific entropy or central pressure are specified. In contrast to this, as has been [[User:Tohline/SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|discussed in detail in another chapter]], [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)], [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)] and [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)] have each derived separate analytic expressions for <math>~R_\mathrm{eq}</math> and <math>~P_e</math> — given in terms of the Lane-Emden function, <math>~\Theta</math>, and its radial derivative — without demonstrating how the equilibrium radius and external pressure directly relate to one another. That is to say, solution of the detailed force-balanced equations provides a pair of equilibrium expressions that are parametrically related to one another through the Lane-Emden function. For example — see our [[User:Tohline/SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|related discussion for more details]] — Horedt derives the following set of parametric equations relating the configuration's dimensionless radius, <math>~r_a</math>, to a specified dimensionless bounding pressure, <math>~p_a</math>: <div align="center"> <table border="0" cellpadding="3"> <tr> <td align="right"> <math> ~r_a \equiv \frac{R_\mathrm{eq}}{R_\mathrm{Horedt}} </math> </td> <td align="center"> <math>~=~</math> </td> <td align="left"> <math> \tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} \, , </math> </td> </tr> <tr> <td align="right"> <math> ~p_a \equiv \frac{P_\mathrm{e}}{P_\mathrm{Horedt}} </math> </td> <td align="center"> <math>~=~</math> </td> <td align="left"> <math> \tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} \, , </math> </td> </tr> </table> </div> where, <div align="center"> <table border="0" cellpadding="3"> <tr> <td align="right"> <math> ~R_\mathrm{Horedt} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> \biggl[ \frac{4\pi}{(n+1)^n}\biggl( \frac{G}{K} \biggr)^n M_\mathrm{limit}^{n-1} \biggr]^{1/(n-3)} \, , </math> </td> </tr> <tr> <td align="right"> <math> ~P_\mathrm{Horedt} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> K^{4n/(n-3)}\biggl[ \frac{(n+1)^3}{4\pi G^3 M_\mathrm{limit}^2} \biggr]^{(n+1)/(n-3)} \, . </math> </td> </tr> </table> </div> It is important to appreciate that, in the expressions for <math>~r_a</math> and <math>~p_a</math>, the tilde indicates that the Lane-Emden function and its derivative are to be evaluated, not at the radial coordinate, <math>~\xi_1</math>, that is traditionally associated with the "first zero" of the Lane-Emden function and therefore with the surface of the ''isolated polytrope,'' but at the radial coordinate, <math>~\tilde\xi</math>, where the internal pressure of the isolated polytrope equals <math>~P_e</math> and at which the ''embedded'' polytrope is to be truncated. The coordinate, <math>~\tilde\xi</math>, therefore identifies the surface of the embedded — or, pressure-truncated — polytrope. We also have taken the liberty of attaching the subscript "limit" to <math>~M</math> in both defining relations because it is clear that Hoerdt intended for the normalization mass to be the mass of the pressure-truncated object, not the mass of the associated ''isolated'' (and untruncated) polytrope. In anticipation of further derivations, below, we note here the ratio of Hoerdt's normalization parameters to ours, assuming <math>~\gamma = (n+1)/n</math>: <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\biggl( \frac{R_\mathrm{Hoerdt}}{R_\mathrm{norm}} \biggr)^{n-3}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[ \frac{4\pi}{(n+1)^n}\biggl( \frac{G}{K} \biggr)^n M_\mathrm{limit}^{n-1} \biggr] \biggl[ \biggl( \frac{K}{G}\biggr)^n M_\mathrm{tot}^{1-n}\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{4\pi}{(n+1)^n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \, , </math> </td> </tr> <tr> <td align="right"> <math>~\biggl( \frac{P_\mathrm{Hoerdt}}{P_\mathrm{norm}} \biggr)^{n-3}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ K^{4n}\biggl[ \frac{(n+1)^3}{4\pi G^3 M_\mathrm{limit}^2} \biggr]^{n+1} \biggl[ \frac{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}}{K^{4n}} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[ \frac{(n+1)^3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2}\biggr]^{n+1} \, . </math> </td> </tr> </table> </div> Next, we demonstrate that this pair of parametric relations satisfies the virial theorem and, in so doing, demonstrate how <math>~r_a</math> and <math>~p_a</math> may be ''directly'' related to each other. Given that Hoerdt's chosen normalization radius and normalization pressure are defined in terms of <math>~K</math> and <math>~M_\mathrm{limit}</math>, we begin with the [[User:Tohline/SSC/Virial/Polytropes#Solution_Expressed_in_Terms_of_K_and_M_.28Whitworth.27s_1981_Relation.2|virial theorem derived above]] in terms of <math>~K</math> and <math>~M_\mathrm{limit}</math>, setting <math>~\gamma_g = (n+1)/n</math>. <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~P_e </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~K \biggl( \frac{3M_\mathrm{limit}}{4\pi R_\mathrm{eq}^3} \biggr)^{(n+1)/n} \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{(n+1)/n}} - \biggl(\frac{3GM_\mathrm{limit}^2}{20\pi R_\mathrm{eq}^4} \biggr) \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \, .</math> </td> </tr> </table> </div> After setting <math>~R_\mathrm{eq} = r_a R_\mathrm{Horedt} </math>, a bit of algebraic manipulation shows that the first term on the right-hand side of the virial equilibrium expression becomes, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~ K \biggl( \frac{3M_\mathrm{limit}}{4\pi R_\mathrm{eq}^3} \biggr)^{(n+1)/n} \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{(n+1)/n}} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ r_a^{-3(n+1)/n} \mathfrak{f}_A \biggl[ \biggl( \frac{3}{\mathfrak{f}_M} \biggr)^{n-3} \frac{(n+1)^{3n}}{(4\pi)^n}\biggr]^{(n+1)/[n(n-3)]} [K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)} ]^{1/(n-3)} \, , </math> </td> </tr> </table> </div> while the second term on the right-hand side becomes, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~ \biggl(\frac{3GM_\mathrm{limit}^2}{20\pi R_\mathrm{eq}^4} \biggr) \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{3}{5} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} \cdot r_a^{-4} (4\pi)^{-(n+1)/(n-3)} (n+1)^{4n/(n-3)}~ [K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)} ]^{1/(n-3)} \, . </math> </td> </tr> </table> </div> But, using Horedt's expression for <math>~P_e</math>, the left-hand side of the virial equilibrium equation becomes, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~P_e = p_a P_\mathrm{Hoerdt}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ p_a~(4\pi)^{-(n+1)/(n-3)} ~(n+1)^{3(n+1)/(n-3)} [K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)} ]^{1/(n-3)} \, . </math> </td> </tr> </table> </div> Hence, the statement of virial equilibrium is, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~ p_a~ </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggr\{ r_a^{-3(n+1)/n} \mathfrak{f}_A \biggl[ \biggl( \frac{3}{\mathfrak{f}_M} \biggr)^{n-3} \frac{(n+1)^{3n}}{(4\pi)^n}\biggr]^{(n+1)/[n(n-3)]} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~~ - \frac{3}{5} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} \cdot r_a^{-4} (4\pi)^{-(n+1)/(n-3)} (n+1)^{4n/(n-3)} \biggr\}(4\pi)^{(n+1)/(n-3)} ~(n+1)^{-3(n+1)/(n-3)} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \mathfrak{f}_A \biggl( \frac{3}{\mathfrak{f}_M \cdot r_a^3} \biggr)^{(n+1)/n} - \frac{3(n+1)}{5} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} \cdot r_a^{-4} \, ; </math> </td> </tr> </table> </div> or, multiplying through by <math>~r_a^4</math> and rearranging terms, <div align="center"> <table border="1" align="center" cellpadding="8"> <tr><td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~ \mathfrak{f}_A \biggl( \frac{3}{\mathfrak{f}_M} \biggr)^{(n+1)/n} r_a^{(n-3)/n} - p_a r_a^4 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{3(n+1)}{5} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} \, . </math> </td> </tr> </table> </td></tr> </table> </div> Now, Hoerdt has given analytic expressions for <math>~r_a</math> and <math>~p_a</math> in terms of the Lane-Emden function and its first derivative. The question is, what should the expressions for our structural form factors be in order for this virial expression to hold true for all pressure-truncated polytropic structures? As has been [[User:Tohline/SSC/Virial/Polytropes#Summary|summarized above]], in the case of an ''isolated'' polytrope, whose surface is located at <math>~\xi_1</math> and whose global properties are defined by evaluation of the Lane-Emden function at <math>~\xi_1</math>, we know that (see the [[User:Tohline/SSC/Virial/Polytropes#Summary|above summary]]), <div align="center"> <table border="0" align="center" cellpadding="5"> <tr><th align="center" colspan="1"> Structural Form Factors for <font color="red">Isolated</font> Polytropes </th></tr> <tr><td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\mathfrak{f}_M</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\xi_1} </math> </td> </tr> <tr> <td align="right"> <math>\mathfrak{f}_W </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\xi_1} </math> </td> </tr> <tr> <td align="right"> <math>\mathfrak{f}_A </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> \frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\xi_1} </math> </td> </tr> </table> </td> </tr> </table> </div> These same expressions may or may not work for pressure-truncated polytropes, even if the evaluation radius is shifted from <math>~\xi_1</math> to <math>~\tilde\xi</math>. Let's see … Inserting Hoerdt's expressions for <math>~r_a</math> and <math>~p_a</math> into the viral equilibrium expression, we have, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~ \frac{3(n+1)}{5} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \mathfrak{f}_A \biggl( \frac{3}{\mathfrak{f}_M} \biggr)^{(n+1)/n} [ \tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} ]^{(n-3)/n} ~-~ \tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} [ \tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} ]^{4} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \mathfrak{f}_A \biggl( \frac{3}{\mathfrak{f}_M} \biggr)^{(n+1)/n} [ \tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} ]^{(n-3)/n} ~-~ \tilde\theta_n^{n+1} \tilde\xi^4 [( -\tilde\xi^2 \tilde\theta' )^{2(n+1)+4(1-n)} ]^{1/(n-3)} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \mathfrak{f}_A \biggl( \frac{3}{\mathfrak{f}_M} \biggr)^{(n+1)/n} [ \tilde\xi^{(n-3)} ( -\tilde\xi^2 \tilde\theta' )^{(1-n)} ]^{1/n} ~-~ \tilde\theta_n^{n+1} \tilde\xi^4 [( -\tilde\xi^2 \tilde\theta' )^{-2} ] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \mathfrak{f}_A \biggl( \frac{3}{\mathfrak{f}_M} \biggr)^{(n+1)/n} \tilde\xi^{-(n+1)/n} ( -\tilde\theta' )^{(1-n)/n} ~-~ \frac{\tilde\theta_n^{n+1} }{( -\tilde\theta' )^{2} } \, . </math> </td> </tr> </table> </div> Assuming that the structural form factor, <math>~\mathfrak{f}_M</math>, has the same functional expression as in the case of isolated polytropes (but evaluated at <math>~\tilde\xi</math> instead of at <math>~\xi_1</math>), the virial relation further reduces to the form, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~ \frac{\tilde\theta_n^{n+1} }{( -\tilde\theta' )^{2} } </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \mathfrak{f}_A \biggl( \frac{\tilde\xi}{- \tilde\theta'} \biggr)^{(n+1)/n} \tilde\xi^{-(n+1)/n} ( -\tilde\theta' )^{(1-n)/n} - \biggl[ \frac{(n+1)}{3\cdot 5} \biggr] \frac{\tilde\xi^2}{(-\tilde\theta^')^2} \cdot \mathfrak{f}_W </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{\mathfrak{f}_A}{(- \tilde\theta' )^2} - \biggl[ \frac{(n+1)}{3\cdot 5} \biggr] \frac{\tilde\xi^2}{(-\tilde\theta^')^2} \cdot \mathfrak{f}_W </math> </td> </tr> <tr> <td align="right"> <math>~ \Rightarrow~~~\tilde\theta^{n+1} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \mathfrak{f}_A - \biggl[ \frac{(n+1)}{3\cdot 5} \biggr] \tilde\xi^2 \cdot \mathfrak{f}_W </math> </td> </tr> <tr> <td align="right"> <math>~ \Rightarrow~~~\frac{\mathfrak{f}_A - \tilde\theta^{n+1} }{\mathfrak{f}_W} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[ \frac{(n+1)}{3\cdot 5} \biggr] \tilde\xi^2 \, . </math> </td> </tr> </table> </div> While this does not give us individual expressions for the form factors, <math>~\mathfrak{f}_W</math> and <math>~\mathfrak{f}_W</math>, the expression derived for the ratio of the form factors makes sense because the term that has been subtracted from <math>~\mathfrak{f}_A</math> in the numerator on the lefthand side, that is, <math>\tilde\theta^{n+1}</math>, naturally goes to zero in the limit of <math>~\tilde\xi \rightarrow \xi_1</math>, producing the correct expression for the ratio, <math>\mathfrak{f}_A/\mathfrak{f}_W</math>, in ''isolated polytropes''. In summary, then, we have, <div align="center" id="PTtable"> <table border="1" align="center" cellpadding="5"> <tr><th align="center" colspan="1"> Structural Form Factors for <font color="red">Pressure-Truncated</font> Polytropes </th></tr> <tr><td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\tilde\mathfrak{f}_M</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\tilde\xi} </math> </td> </tr> <tr> <td align="right"> <math>~ ~\frac{\tilde\mathfrak{f}_A - \tilde\theta^{n+1} }{\tilde\mathfrak{f}_W} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[ \frac{(n+1)}{3\cdot 5} \biggr] \tilde\xi^2 </math> </td> </tr> </table> </td> </tr> </table> </div> Notice that, in an effort to differentiate them from their counterparts developed earlier for "isolated" polytropes, we have affixed a tilde to each of these three form-factors, <math>~\mathfrak{f}_i</math>. ====Renormalization==== =====Grunt Work===== Returning to the dimensionless form of the virial expression and multiplying through by <math>~[-\chi_\mathrm{eq}/(3D)]</math>, we obtain, <div align="center"> <math> \chi_\mathrm{eq}^4 = \frac{B}{D} \chi_\mathrm{eq}^{4-3\gamma_g} - \frac{A}{D} \, , </math> </div> or, after plugging in [[User:Tohline/SphericallySymmetricConfigurations/Virial#Gathering_it_all_Together|definitions of the coefficients]], <math>~A</math>, <math>~B</math>, and <math>~D</math>, and rewriting <math>~\chi_\mathrm{eq}</math> explicitly as <math>~R_\mathrm{eq}/R_\mathrm{norm}</math>, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> \biggl(\frac{3}{4\pi}\biggr)^{\gamma_g} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{\gamma_g} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr)^{-1} \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{\gamma_g}} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{4-3\gamma_g} - \frac{3}{20\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr)^{-1} \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \, .</math> </td> </tr> </table> </div> This relation can be written in a more physically concise form, as follows. First, normalize <math>~P_e</math> to a new pressure scale — call it <math>~P_\mathrm{ad}</math> — and multiply through by <math>~(R_\mathrm{norm}/R_\mathrm{ad})^4</math> in order to normalizing <math>~R_\mathrm{eq}</math> to a new length scale,<math>~R_\mathrm{ad}</math>: <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{ad}} \biggr)^4 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> \biggl(\frac{3}{4\pi}\biggr)^{\gamma_g} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{\gamma_g} \biggl( \frac{P_\mathrm{norm}}{P_\mathrm{ad}} \biggr) \biggl( \frac{P_e}{P_\mathrm{ad}} \biggr)^{-1} \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{\gamma_g}} \biggl( \frac{R_\mathrm{norm}}{R_\mathrm{ad}} \biggr)^{3\gamma_g}\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{ad}} \biggr)^{4-3\gamma_g} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> ~- \frac{3}{20\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2} \biggl( \frac{P_\mathrm{norm}}{P_\mathrm{ad}} \biggr) \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \biggl( \frac{P_e}{P_\mathrm{ad}} \biggr)^{-1} \biggl( \frac{R_\mathrm{norm}}{R_\mathrm{ad}} \biggr)^{4} \, , </math> </td> </tr> </table> </div> or, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Chi_\mathrm{ad}^4 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> \biggl(\frac{3}{4\pi}\biggr)^{\gamma_g} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{\gamma_g} \biggl( \frac{P_\mathrm{norm}}{P_\mathrm{ad}} \biggr) \biggl( \frac{R_\mathrm{norm}}{R_\mathrm{ad}} \biggr)^{3\gamma_g} \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{\gamma_g}}\cdot \frac{\Chi_\mathrm{ad}^{4-3\gamma_g} }{\Pi_\mathrm{ad}} - \frac{3}{20\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2} \biggl( \frac{P_\mathrm{norm}}{P_\mathrm{ad}} \biggr) \biggl( \frac{R_\mathrm{norm}}{R_\mathrm{ad}} \biggr)^{4} \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \cdot \frac{1}{\Pi_\mathrm{ad}} \, ,</math> </td> </tr> </table> </div> where, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Chi_\mathrm{ad}</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~\frac{R_\mathrm{eq}}{R_\mathrm{ad}} \, ,</math> </td> </tr> <tr> <td align="right"> <math>~\Pi_\mathrm{ad}</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~\frac{P_\mathrm{e}}{P_\mathrm{ad}} \, .</math> </td> </tr> </table> </div> By demanding that the leading coefficients of both terms on the right-hand-side of the expression are simultaneously unity — that is, by demanding that, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\biggl(\frac{3}{4\pi}\biggr)^{\gamma_g} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{\gamma_g} \biggl( \frac{P_\mathrm{norm}}{P_\mathrm{ad}} \biggr) \biggl( \frac{R_\mathrm{norm}}{R_\mathrm{ad}} \biggr)^{3\gamma_g} \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{\gamma_g}}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~1 \, ,</math> </td> </tr> </table> </div> and, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{3}{20\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2} \biggl( \frac{P_\mathrm{norm}}{P_\mathrm{ad}} \biggr) \biggl( \frac{R_\mathrm{norm}}{R_\mathrm{ad}} \biggr)^{4} \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~1 \, ,</math> </td> </tr> </table> </div> we obtain the expressions for <math>~R_\mathrm{ad}/R_\mathrm{norm}</math> and <math>~P_\mathrm{ad}/P_\mathrm{norm}</math> as shown in the following table. <div align="center"> <table border="1" align="center" cellpadding="5"> <tr><th align="center" colspan="1"> Renormalization for Adiabatic (''ad'') Systems </th></tr> <tr><td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{R_\mathrm{ad}}{R_\mathrm{norm}}</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~\biggl[ \frac{1}{5} \biggl( \frac{4\pi}{3} \biggr)^{\gamma_g-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2-\gamma_g} \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_A \cdot \tilde\mathfrak{f}_M^{2-\gamma_g}} \biggr]^{1/(4-3\gamma_g)} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[ \frac{1}{5^n} \biggl( \frac{4\pi}{3}\biggr)\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{n-1} \frac{\tilde\mathfrak{f}_W^n}{\tilde\mathfrak{f}_A^n \cdot \tilde\mathfrak{f}_M^{n-1}} \biggr]^{1/(n-3)} </math> </td> </tr> <tr> <td align="right"> <math>~\frac{P_\mathrm{ad}}{P_\mathrm{norm}}</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~ \biggl[ \tilde\mathfrak{f}_A^4 \cdot \biggl( \frac{3\cdot 5^3}{4\pi} \cdot \frac{\tilde\mathfrak{f}_M^2}{\tilde\mathfrak{f}_W^3} \biggr)^{\gamma_g} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2\gamma_g} \biggr]^{1/(4-3\gamma)} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[ \tilde\mathfrak{f}_A^{4n} \biggl(\frac{3\cdot 5^3}{4\pi} \biggr)^{n+1} \biggl( \frac{\tilde\mathfrak{f}_M^2}{\tilde\mathfrak{f}_w^3} \biggr)^{n+1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2(n+1)} \biggr]^{1/(n-3)} </math> </td> </tr> </table> </td> </tr> </table> </div> Using these new normalizations, we arrive at the desired, concise virial equilibrium relation, namely, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Pi_\mathrm{ad} = \Chi_\mathrm{ad}^{-3\gamma_g} - \Chi_\mathrm{ad}^{-4}</math> </td> <td align="center"> <math>\Leftrightarrow</math> </td> <td align="left"> <math>~ \Pi_\mathrm{ad} = \Chi_\mathrm{ad}^{-3(n+1)/n} - \Chi_\mathrm{ad}^{-4}</math> </td> </tr> </table> </div> or, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~~\Chi_\mathrm{ad}^{4-3\gamma_g} - \Pi_\mathrm{ad} \Chi_\mathrm{ad}^4 = 1</math> </td> <td align="center"> <math>\Leftrightarrow</math> </td> <td align="left"> <math>~\Chi_\mathrm{ad}^{(n-3)/n} - \Pi_\mathrm{ad} \Chi_\mathrm{ad}^4 = 1 \, .</math> </td> </tr> </table> </div> =====In Terms of Free-Energy Coefficients===== Referring back to relations between our free-energy coefficients, as [[User:Tohline/SSC/Virial/PolytropesEmbedded/FirstEffortAgain#ABratios|presented earlier]], we note that, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{R_\mathrm{ad}}{R_\mathrm{norm}}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl( \frac{\mathcal{A}}{\mathcal{B}}\biggr)^{n/(n-3)} \, ,</math> </td> </tr> <tr> <td align="right"> <math>~\frac{P_\mathrm{ad}}{P_\mathrm{norm}}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{3}{4\pi} \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]^{1/(n-3)} \, .</math> </td> </tr> </table> </div> Hence, we can write, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Chi_\mathrm{ad}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\chi_\mathrm{eq} ~\biggl( \frac{\mathcal{B}}{\mathcal{A}}\biggr)^{n/(n-3)} \, ,</math> </td> </tr> <tr> <td align="right"> <math>~\Pi_\mathrm{ad}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\mathcal{D} ~\biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr]^{1/(n-3)} \, .</math> </td> </tr> </table> </div> =====In Terms of Horedt's Equilibrium Parameters===== For later use it is also worth developing expressions for both <math>~\Chi_\mathrm{ad}</math> and <math>~\Pi_\mathrm{ad}</math> that are in terms of our structural form factors and Horedt's two dimensionless functions, <math>~r_a</math> and <math>~p_a</math>. (Adopting a unified notation, we will set <math>~\gamma_g \rightarrow (n+1)/n</math>.) <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Chi_\mathrm{ad} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{ad}}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{R_\mathrm{eq}}{R_\mathrm{Hoerdt}} \cdot \frac{R_\mathrm{Hoerdt}}{R_\mathrm{norm}} \cdot \frac{R_\mathrm{norm}}{R_\mathrm{ad}}</math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ r_a \cdot \biggl[ \frac{4\pi}{(n+1)^n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \biggr]^{1/(n-3)} \cdot \biggl[ 5^n \biggl( \frac{3}{4\pi} \biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{1-n} \frac{\tilde\mathfrak{f}_A^n}{\tilde\mathfrak{f}_W^n \cdot \tilde\mathfrak{f}_M^{1-n}} \biggr]^{1/(n-3)} </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow~~~\Chi_\mathrm{ad}^{n-3} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 3 r_a^{n-3} \cdot \biggl[ \frac{5}{(n+1)} \cdot \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_W} \biggr]^n \tilde\mathfrak{f}_M^{n-1} \, . </math> </td> </tr> </table> </div> And, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Pi_\mathrm{ad} \equiv \frac{P_e}{P_\mathrm{ad}}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{P_e}{P_\mathrm{Hoerdt}} \cdot \frac{P_\mathrm{Hoerdt}}{P_\mathrm{norm}} \cdot \frac{P_\mathrm{norm}}{P_\mathrm{ad}}</math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ p_a \cdot \biggl[ \frac{(n+1)^3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2}\biggr]^{(n+1)/(n-3)} \cdot \biggl[ \tilde\mathfrak{f}_A^{-4n} \cdot \biggl( \frac{4\pi}{3\cdot 5^3} \cdot \frac{\tilde\mathfrak{f}_W^3}{\tilde\mathfrak{f}_M^2} \biggr)^{(n+1)} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2(n+1)} \biggr]^{1/(n-3)} </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow~~~\Pi_\mathrm{ad}^{n-3} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ p_a^{n-3}~ \tilde\mathfrak{f}_A^{-4n} \cdot \biggl[ \frac{(n+1)^3}{3\cdot 5^3} \biggl( \frac{\tilde\mathfrak{f}_W^3}{\tilde\mathfrak{f}_M^2} \biggr) \biggr]^{n+1} \, . </math> </td> </tr> </table> </div> Plugging in the expressions for <math>~r_a</math> and <math>~p_a</math>, as [[User:Tohline/SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Contrast_with_Detailed_Force-Balanced_Solution|reprinted, for example, above]], along with our [[User:Tohline/SSC/Virial/PolytropesEmbedded/FirstEffortAgain#PTtable|deduced expressions for <math>~\mathfrak{f}_M</math> and <math>~\mathfrak{f}_A</math> (in terms of <math>~\mathfrak{f}_W</math>)]], these two relations become: <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Chi_\mathrm{ad}^{n-3} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 3 \biggl[ \tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} \biggr]^{n-3} \cdot \biggl[ \frac{5}{(n+1)} \cdot \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_W} \biggr]^n \biggl[- \frac{3\tilde\theta^'}{\tilde\xi} \biggr]^{n-1} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \tilde\xi^{[(n-3)+2(1-n)-(n-1)]} \cdot \biggl[ \frac{3\cdot 5}{(n+1)} \cdot \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_W} \biggr]^n </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[ \frac{3\cdot 5}{(n+1)\tilde\xi^2} \cdot \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_W} \biggr]^n \, ; </math> </td> </tr> </table> </div> and, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Pi_\mathrm{ad}^{n-3} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[ \tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} \biggr]^{n-3}~ \tilde\mathfrak{f}_A^{-4n} \cdot \biggl[ \frac{(n+1)}{3\cdot 5} \cdot \tilde\mathfrak{f}_W \biggr]^{3(n+1)} \biggl( \frac{3}{\tilde\mathfrak{f}_M}\biggr)^{2(n+1)} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \tilde\theta_n^{(n+1)(n-3)}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)} ~ \tilde\mathfrak{f}_A^{-4n} \cdot \tilde\xi^{-6(n+1)} \biggl[ \frac{(n+1)\tilde\xi^2}{3\cdot 5} \cdot \tilde\mathfrak{f}_W \biggr]^{3(n+1)} \tilde\xi^{2(n+1)} (-\tilde\theta^')^{-2(n+1)} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \tilde\theta_n^{(n+1)(n-3)}~ \tilde\mathfrak{f}_A^{-4n} \cdot \biggl[ \frac{(n+1)\tilde\xi^2}{3\cdot 5} \cdot \tilde\mathfrak{f}_W \biggr]^{3(n+1)} \, . </math> </td> </tr> </table> </div> =====Summary===== If we define, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~a_\mathrm{ad}</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~\frac{(n+1)\tilde\xi^2 }{3\cdot 5} \cdot \mathfrak{f}_W \, ,</math> and, </td> </tr> <tr> <td align="right"> <math>~b_\mathrm{ad}</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~\tilde\theta^{n+1} \, ,</math> </td> </tr> </table> </div> in which case the [[User:Tohline/SSC/Virial/PolytropesEmbedded/FirstEffortAgain#PTtable|relationship between <math>~\mathfrak{f}_A</math> and <math>~\mathfrak{f}_A</math> for pressure-truncated polytropes]] can be rewritten as, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\mathfrak{f}_A </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~a_\mathrm{ad} + b_\mathrm{ad} \, .</math> </td> </tr> </table> </div> In addition, the expressions for the dimensionless equilibrium radius and the dimensionless external pressure, as just derived, may be written as, respectively, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Chi_\mathrm{ad}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[ 1 + \frac{b_\mathrm{ad}}{a_\mathrm{ad}} \biggr]^{n/(n-3)} \, , </math> and, </td> </tr> <tr> <td align="right"> <math>~\Pi_\mathrm{ad} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ b_\mathrm{ad} \biggl[ \frac{a_\mathrm{ad}^{3(n+1)} }{( a_\mathrm{ad} + b_\mathrm{ad} )^{4n} } \biggr]^{1/(n-3)} = \frac{b_\mathrm{ad}}{a_\mathrm{ad}} \biggl[ 1 + \frac{b_\mathrm{ad}}{a_\mathrm{ad}} \biggr]^{-4n/(n-3)} \, . </math> </td> </tr> </table> </div> Using these expressions, it is easy to demonstrate that the virial equilibrium relation is satisfied, namely, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Pi_\mathrm{ad} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\Chi_\mathrm{ad}^{-3(n+1)/n} - \Chi_\mathrm{ad}^{-4} \, .</math> </td> </tr> </table> </div> ====P-V Diagram==== For an arbitrary value of the adiabatic exponent, <math>~\gamma_g</math>, it isn't possible to invert this virial relation to obtain an analytic expression for <math>~\chi_\mathrm{ad}</math> as a function of <math>~\Pi_\mathrm{ad}</math>. But, as written, the virial relation dictates the behavior of <math>~\Pi_\mathrm{ad}</math> as a function of <math>~\Chi_\mathrm{ad}</math>. Figure 4 displays this <math>~\Pi_\mathrm{ad}(\chi_\mathrm{ad})</math> behavior for a number of different values of <math>~\gamma_g</math>. <div align="center"> <table border="2" cellpadding="8"> <tr> <td align="center" colspan="2"> '''Figure 4:''' <font color="darkblue">Equilibrium Adiabatic P-V Diagram </font> </td> </tr> <tr> <td valign="top" width=450 rowspan="1"> The curves shown here, on the right, trace out the function, <div align="center"> <math> ~\Pi_\mathrm{ad} = (\Chi_\mathrm{ad}^{4-3\gamma_g} - 1)/\Chi_\mathrm{ad}^4 \, , </math> </div> for six different values of <math>~\gamma_g</math> — specifically, for <math>2, ~5/3, ~7/5, ~6/5, ~1, ~2/3</math>, as labeled — and show the dimensionless external pressure, <math>~\Pi_\mathrm{ad}</math>, that is required to construct a nonrotating, self-gravitating, pressure-truncated adiabatic sphere with an equilibrium radius <math>~\Chi_\mathrm{ad}</math>. The solid red curve identifies the behavior of an isothermal <math>~(\gamma_g=1)</math> system. The mathematical solution becomes unphysical wherever the pressure becomes negative. </td> <td align="center" bgcolor="white"> [[File:AdabaticBoundedSpheres_Virial.jpg|450px|center|Equilibrium Adiabatic P-R Diagram]] </td> </tr> </table> </div> For physically relevant solutions, both <math>~\Chi_\mathrm{ad}</math> and <math>~\Pi_\mathrm{ad}</math> must be nonnegative. Hence, as is illustrated by the curves in Figure 4, the physically allowable range of equilibrium radii is, <div align="center"> <math> 1 \le \Chi_\mathrm{ad} \le \infty \, ~~~~~\mathrm{for}~ \gamma_g < 4/3 \, ; </math> <math> 0 < \Chi_\mathrm{ad} \le 1 \, ~~~~~~\mathrm{for}~ \gamma_g > 4/3 \, . </math> </div> Each of the <math>~\Pi_\mathrm{ad}(\Chi_\mathrm{ad})</math> curves drawn in Figure 4 exhibits an extremum. In each case this extremum occurs at a configuration radius, <math>~\Chi_\mathrm{extreme}</math>, given by, <div align="center"> <math> \frac{\partial\Pi_\mathrm{ad}}{\partial\Chi_\mathrm{ad}} = 0 \, , </math> </div> that is, where, <div align="center"> <math> 4 - 3\gamma_g \Chi_\mathrm{ad}^{4-3\gamma_g} = 0 ~~~~\Rightarrow ~~~~~ \Chi_\mathrm{extreme} = \biggl[ \frac{4}{3\gamma_g} \biggr]^{1/(4-3\gamma_g)} \, . </math> </div> For each value of <math>~\gamma_g</math>, the corresponding dimensionless pressure is, <div align="center"> <math> ~\Pi_\mathrm{extreme} = \biggl(\frac{4}{3\gamma} - 1 \biggr) \biggl[ \frac{3\gamma_g}{4} \biggr]^{4/(4-3\gamma_g)} \, . </math> </div> In terms of the polytropic index, the equivalent limiting expressions are, <div align="center"> <table border="0" cellpadding="5"> <tr> <td align="center"> <math>~\Chi_\mathrm{extreme} = \biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)}</math> </td> <td align="center"> and </td> <td align="center"> <math>~\Pi_\mathrm{extreme}^{(n-3)} = (4n)^{-4n} (n-3)^{n-3} [3(n+1)]^{3(n+1)} \, .</math> </td> </tr> </table> </div> (In a [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Stability|separate, related discussion of the free-energy function]], we demonstrate that this "extremum" also serves as a dividing line between dynamically stable and unstable models along a given curve.) In examining the group of plotted curves, notice that, for <math>~\gamma_g > 4/3</math>, an equilibrium configuration with a positive radius can be constructed for all physically realistic — that is, for all positive — values of <math>~\Pi_\mathrm{ad}</math>. Also, consistent with the behavior of the curves shown in Figure 4, the extremum arises in the regime of physically relevant — ''i.e.,'' positive — pressures only for values of <math>~\gamma_g < 4/3</math>; and in each case it represents a ''maximum'' limiting pressure. {{ SGFworkInProgress }} ====Maximum Mass==== =====n = 5 Polytropic===== When <math>~\gamma_a = 6/5</math> — which corresponds to an <math>~n=5</math> polytropic configuration — we obtain, <div align="center"> <math> ~\Pi_\mathrm{max} = \Pi_\mathrm{ad}\biggr|_\mathrm{extreme}^{(\gamma_g = 6/5)} = \biggl( \frac{3^{18}}{2^{10}\cdot 5^{10}} \biggr) \, , </math> </div> which corresponds to a maximum mass for pressure-bounded <math>~n=5</math> polytropic configurations of, <div align="center"> <math>~M_\mathrm{max} = \Pi_\mathrm{max}^{1/2} \biggl(\frac{3\cdot 5^3}{2^2\pi} \biggr)^{1/2} \biggl( \frac{\bar{c_s}^8}{G^3 P_e} \biggr)^{1/2} = \biggl(\frac{3^{19}}{2^{12}\cdot 5^{7}\pi} \biggr)^{1/2} \biggl( \frac{\bar{c_s}^8}{G^3 P_e} \biggr)^{1/2} \, .</math> </div> This result can be compared to [[SSC/Structure/LimitingMasses#Bounded_Isothermal_Sphere_.26_Bonnor-Ebert_Mass|other determinations of the Bonnor-Ebert mass limit]]. {{ SGFfooter }}
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