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=Parabolic Density Distribution (Tries 1 thru 7)= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" bgcolor="lightblue" width="25%"><br />[[ParabolicDensity/GravPot|Part I: Gravitational Potential]] </td> <td align="center" bgcolor="lightblue" width="25%"><br />[[ParabolicDensity/Spheres/Structure|Part II: Spherical Structures]] </td> <td align="center" bgcolor="lightblue" width="25%"><br />[[ParabolicDensity/Axisymmetric/Structure|Part III: Axisymmetric Equilibrium Structures]] [[ParabolicDensity/Axisymmetric/Structure/Try1thru7|Old: 1<sup>st</sup> thru 7<sup>th</sup> tries]] </td> <td align="center" bgcolor="lightblue"><br />[[ParabolicDensity/Triaxial/Structure|Part IV: Triaxial Equilibrium Structures (Exploration)]] </td> </tr> </table> ==Axisymmetric (Oblate) Equilibrium Structures== ===Setup=== Here we specifically discuss the case of configurations that exhibit concentric ellipsoidal iso-density surfaces of the form, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\rho</math> </td> <td align="center"> = </td> <td align="left"> <math>\rho_c \biggl[ 1 - \biggl( \frac{x^2 + y^2}{a_\ell^2} + \frac{z^2}{a_s^2}\biggr) \biggr] \, ,</math> </td> </tr> </table> that is, axisymmetric (<math> a_m = a_\ell</math>, i.e., oblate) configurations with ''parabolic density distributions''. Much of our presentation, here, is drawn from our separate, detailed description of what we will refer to as [[ThreeDimensionalConfigurations/FerrersPotential|Ferrers potential]]. <table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left"> This can be rewritten in terms of [[Appendix/Ramblings/T1Coordinates#T1_Coordinates|T1 Coordinates]]. In particular, defining, <math>q \equiv a_\ell/a_s</math> and, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\xi_1</math> </td> <td align="center"> <math>\equiv</math> </td> <td align="left"> <math> \biggl[ z^2 + \biggl(\frac{\varpi}{q}\biggr)^2\biggr]^{1 / 2} = a_s\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]^{1 / 2} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \frac{\rho}{\rho_c}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>\biggl[ 1 - \biggl(\frac{\xi_1}{a_s}\biggr)^2 \biggr] \, .</math> </td> </tr> </table> Because we expect contours of constant enthalpy <math>(H)</math> to coincide with contours of constant density in equilibrium configurations, we should expect to find that, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{H}{H_c}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>h(\xi_1) \, .</math> </td> </tr> </table> If the "radial" enthalpy profile resembles our [[SSC/Structure/OtherAnalyticModels#SphericalEnthalpyProfile|derived spherical enthalpy profile]], we should expect to find that, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>h(\xi_1)</math> </td> <td align="center"> <math>\sim</math> </td> <td align="left"> <math>h_0 \biggl[1 - h_2 \xi_1^2 - h_4 \xi_1^4 \biggr]</math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ 1 - \frac{h(\xi_1)}{h_0}</math> </td> <td align="center"> <math>\sim</math> </td> <td align="left"> <math>h_2 \xi_1^2 + h_4 \xi_1^4</math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> h_2 a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr] + h_4 \biggl\{ a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr] \biggr\}^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> h_2 a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr] + h_4 a_s^4 \biggl[ \biggl(\frac{z}{a_s}\biggr)^4 + 2\biggl(\frac{z}{a_s}\biggr)^2\biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^4 \biggr] </math> </td> </tr> </table> </td></tr></table> ===Total Mass=== The surface of the configuration with eccentricity, <math>e</math>, is defined by the expression, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>1</math> </td> <td align="center"> = </td> <td align="left"> <math> \frac{\varpi^2}{a_\ell^2} + \frac{z^2}{a_s^2} = \chi^2 + (1-e^2)^{-1}\zeta^2 \, . </math> </td> </tr> </table> When integrating over the volume elements, at each "radial" location, <math>\chi</math>,the vertical limit will be given by the expression, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\zeta_\mathrm{limit}</math> </td> <td align="center"> = </td> <td align="left"> <math> (1-e^2)^{1 / 2} (1 - \chi^2)^{1 / 2} </math> </td> </tr> </table> Specifically, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\mathrm{volume}</math> </td> <td align="center"> = </td> <td align="left"> <math> a_\ell^3 ~ 2\pi \int_0^1 \chi d\chi \int_0^{\zeta_\mathrm{limit}} 2d\zeta </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> = </td> <td align="left"> <math> a_\ell^3 ~ 4\pi \int_0^1 \chi \biggl[\zeta\biggr]_0^{\zeta_\mathrm{limit}}d\chi </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> = </td> <td align="left"> <math> a_\ell^3 ~ 4\pi (1-e^2)^{1 / 2}\int_0^1 \chi (1 - \chi^2)^{1 / 2} d\chi \, . </math> </td> </tr> </table> Make the variable substitution, <math>\chi ~\rightarrow ~ \sin\theta ~~ \Rightarrow ~~ d\chi = \cos\theta d\theta</math> … <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\mathrm{volume}</math> </td> <td align="center"> = </td> <td align="left"> <math> a_\ell^3 ~ 4\pi (1-e^2)^{1 / 2}\int_0^{\pi/2} \sin\theta \cos^2\theta d\theta </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> = </td> <td align="left"> <math> a_\ell^3 ~ 4\pi (1-e^2)^{1 / 2}\biggl[ - \frac{\cos^3\theta}{3} \biggr]_0^{\pi/2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> = </td> <td align="left"> <math> \frac{4\pi a_\ell^3}{3} ~ (1-e^2)^{1 / 2} \, . </math> </td> </tr> </table> Likewise, the mass is given by the expression, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>M_\mathrm{tot}</math> </td> <td align="center"> = </td> <td align="left"> <math> \biggl[\rho_c a_\ell^3\biggr] 2\pi \int_0^1 \chi d\chi \int_0^{\zeta_\mathrm{limit}} 2\biggl(\frac{\rho}{\rho_c}\biggr)d\zeta </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> = </td> <td align="left"> <math> \biggl[\rho_c a_\ell^3\biggr] 4\pi \int_0^1 \chi d\chi \biggl\{ \int_0^{\zeta_\mathrm{limit}} \biggl[ 1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]d\zeta \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> = </td> <td align="left"> <math> \biggl[\rho_c a_\ell^3\biggr] 4\pi \int_0^1 \chi d\chi \biggl\{ \int_0^{\zeta_\mathrm{limit}} \biggl[ 1 - \chi^2 \biggr]d\zeta - \int_0^{\zeta_\mathrm{limit}} \biggl[\zeta^2(1-e^2)^{-1} \biggr]d\zeta \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> = </td> <td align="left"> <math> \biggl[\rho_c a_\ell^3\biggr] 4\pi \int_0^1 \chi d\chi \biggl\{ \biggl[ (1 - \chi^2)\zeta \biggr]_0^{\zeta_\mathrm{limit}} - \biggl[\frac{\zeta^3}{3}(1-e^2)^{-1} \biggr]_0^{\zeta_\mathrm{limit}} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> = </td> <td align="left"> <math> \biggl[\rho_c a_\ell^3\biggr] 4\pi \int_0^1 \chi d\chi \biggl\{ (1 - \chi^2)\biggl[ (1-e^2)^{1 / 2} (1 - \chi^2)^{1 / 2} \biggr] - \frac{1}{3(1-e^2)}\biggl[ (1-e^2)^{1 / 2} (1 - \chi^2)^{1 / 2} \biggr]^3 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> = </td> <td align="left"> <math> \biggl[\rho_c a_\ell^3\biggr] 4\pi \int_0^1 \chi d\chi \biggl\{ \biggl[ (1-e^2)^{1 / 2} (1 - \chi^2)^{3 / 2} \biggr] - \frac{1}{3}\biggl[ (1-e^2)^{1 / 2} (1 - \chi^2)^{3 / 2} \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> = </td> <td align="left"> <math> \biggl[\rho_c a_\ell^3\biggr] 4\pi \int_0^1 \chi d\chi \biggl\{ \frac{2}{3}\biggl[ (1-e^2)^{1 / 2} (1 - \chi^2)^{3 / 2} \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> = </td> <td align="left"> <math> \biggl[\rho_c a_\ell^3\biggr] \frac{8\pi}{3}(1-e^2)^{1 / 2} \int_0^1 \chi \biggl[ (1 - \chi^2)^{3 / 2} \biggr]d\chi \, . </math> </td> </tr> </table> Again making the variable substitution, <math>\chi ~\rightarrow ~ \sin\theta ~~ \Rightarrow ~~ d\chi = \cos\theta d\theta</math> … <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>M_\mathrm{tot}</math> </td> <td align="center"> = </td> <td align="left"> <math> \biggl[\rho_c a_\ell^3\biggr] \frac{8\pi}{3}(1-e^2)^{1 / 2} \int_0^{\pi/2} \sin\theta \cos^4\theta ~d\theta </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> = </td> <td align="left"> <math> \biggl[\rho_c a_\ell^3\biggr] \frac{8\pi}{3}(1-e^2)^{1 / 2} \biggl[- \frac{\cos^5\theta}{5} \biggr]_0^{\pi/2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> = </td> <td align="left"> <math> \biggl[\rho_c a_\ell^3\biggr] \frac{8\pi}{15}(1-e^2)^{1 / 2} \, . </math> </td> </tr> </table> When we set <math>e = 0</math>, this result matches [[SSC/Structure/OtherAnalyticModels#TotalMass|the expression for the total mass]] of a spherically symmetric configuration with a parabolic density distribution. ===Gravitational Potential=== As we have detailed in [[ThreeDimensionalConfigurations/FerrersPotential|an accompanying discussion]], for an oblate-spheroidal configuration — that is, when <math>a_s < a_m = a_\ell</math> — the gravitational potential may be obtained from the expression, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c)}</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{2} I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) + \biggl( A_{12} x^2y^2 + A_{13} x^2z^2 + A_{23} y^2z^2\biggr) + \frac{1}{6} \biggl(3A_{11}x^4 + 3A_{22}y^4 + 3A_{33}z^4 \biggr) \, , </math> </td> </tr> </table> where, in the present context, we can rewrite this expression as, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c)}</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{2} I_\mathrm{BT} a_\ell^2 - \biggl[A_\ell (x^2 + y^2) + A_s z^2 \biggr] + \biggl[ A_{\ell \ell} x^2y^2 + A_{\ell s} x^2z^2 + A_{\ell s} y^2z^2\biggr] + \frac{1}{6} \biggl[3A_{\ell \ell} x^4 + 3A_{\ell \ell}y^4 + 3A_{ss}z^4 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{2} I_\mathrm{BT} a_\ell^2 - \biggl[A_\ell \varpi^2 + A_s z^2 \biggr] + \biggl[ A_{\ell \ell} x^2y^2 + A_{\ell s} \varpi^2 z^2 \biggr] + \frac{1}{2} \biggl[A_{\ell \ell} (x^4 + y^4) + A_{ss}z^4 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{2} I_\mathrm{BT} a_\ell^2 - \biggl[A_\ell \varpi^2 + A_s z^2 \biggr] + \frac{A_{\ell \ell}}{2} \biggl[(x^2 + y^2)^2\biggr] + \frac{1}{2} \biggl[ A_{ss}z^4 \biggr] + \biggl[ A_{\ell s} \varpi^2 z^2 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{2} I_\mathrm{BT} a_\ell^2 - \biggl[A_\ell \varpi^2 + A_s z^2 \biggr] + \frac{A_{\ell \ell}}{2} \biggl[\varpi^4\biggr] + \frac{1}{2} \biggl[ A_{ss}z^4 \biggr] + \biggl[ A_{\ell s} \varpi^2 z^2 \biggr] </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c a_\ell^2)}</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{2} I_\mathrm{BT} - \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr] + \frac{1}{2} \biggl[ A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) + A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr) + 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr) \biggr] \, . </math> </td> </tr> </table> ====Index Symbol Expressions==== The expression for the zeroth-order normalization term <math>(I_{BT})</math>, and the relevant pair of 1<sup>st</sup>-order index symbol expressions are: <table align="center" border=0 cellpadding="3"> <tr> <td align="right"><math>I_\mathrm{BT}</math> </td> <td align="center"><math>=</math> </td> <td align="left"> <math> 2A_\ell + A_s (1-e^2) = 2 (1-e^2)^{1/2} \biggl[ \frac{\sin^{-1}e}{e} \biggr] \, ; </math> </td> </tr> <tr> <td align="right"> <math> A_\ell </math> </td> <td align="center"> <math> = </math> </td> <td align="left"> <math> \frac{1}{e^2} \biggl[ \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1/2} \, ; </math> </td> </tr> <tr> <td align="right"><math>A_s</math> </td> <td align="center"><math>=</math> </td> <td align="left"> <math> \frac{2}{e^2} \biggl[ (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1 / 2} \, , </math> </td> </tr> </table> <div align="center"> [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 3, Eq. (36)</font><br /> [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], <font color="#00CC00">§4.5, Eqs. (48) & (49)</font> </div> where the eccentricity, <div align="center"> <math> e \equiv \biggl[1 - \biggl(\frac{a_s}{a_\ell}\biggr)^2 \biggr]^{1 / 2} \, . </math> </div> The relevant [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Index_Symbols_of_the_2nd_Order|2<sup>nd</sup>-order index symbol]] expressions are: <table align="center" border=0 cellpadding="3"> <tr> <td align="right"> <math> a_\ell^2 A_{\ell \ell} </math> </td> <td align="center"> <math> = </math> </td> <td align="left"> <math> \frac{1}{4e^4}\biggl\{- (3 + 2e^2) (1-e^2)+3 (1 - e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr] \biggr\} \, ; </math> </td> </tr> <tr> <td align="right"> <math>\frac{3}{2} a_\ell^2 A_{ss} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{( 4e^2 - 3 )}{e^4(1-e^2)} + \frac{3 (1-e^2)^{1 / 2}}{e^4} \biggl[\frac{\sin^{-1}e}{e}\biggr] \, ; </math> </td> </tr> <tr> <td align="right"> <math> a_\ell^2 A_{\ell s} </math> </td> <td align="center"> <math> = </math> </td> <td align="left"> <math> \frac{1}{ e^4} \biggl\{ (3-e^2) - 3 (1-e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr] \biggr\} \, . </math> </td> </tr> </table> We can crosscheck this last expression by [[ParabolicDensity/GravPot#Parabolic_Density_Distribution_2|drawing on a shortcut expression]], <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>A_{\ell s}</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> - \frac{A_\ell - A_s}{(a_\ell^2 - a_s^2)} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ a_\ell^2 A_{\ell s}</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{e^2}\biggl\{ A_s - A_\ell \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{e^2}\biggl\{ \frac{2}{e^2} \biggl[ (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1 / 2} - \frac{1}{e^2} \biggl[ \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1/2} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{e^4}\biggl\{ \biggl[ 2 - 2(1-e^2)^{1 / 2} \frac{\sin^{-1}e}{e} \biggr] - \biggl[ (1-e^2)^{1/2} \frac{\sin^{-1}e}{e} - (1-e^2) \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{e^4}\biggl\{(3-e^2) - 3(1-e^2)^{1 / 2} \frac{\sin^{-1}e}{e} \biggr\} \, . </math> </td> </tr> </table> ====Meridional Plane Equi-Potential Contours==== Here, we follow closely our separate discussion of equipotential surfaces for [[Apps/MaclaurinSpheroids#norotation|Maclaurin Spheroids, assuming no rotation]]. =====Configuration Surface===== In the meridional <math>(\varpi, z)</math> plane, the surface of this oblate-spheroidal configuration — identified by the thick, solid-black curve below, in Figure 1 — is defined by the expression, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{\rho}{\rho_c} </math> </td> <td align="center"> <math>=</math> </td> <td align="left" colspan="2"> <math>1 - \biggl[\frac{\varpi^2}{a_\ell^2} + \frac{z^2}{a_s^2} \biggr] = 0</math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \frac{\varpi^2}{a_\ell^2} + \frac{z^2}{a_s^2}</math> </td> <td align="center"> <math>=</math> </td> <td align="left" colspan="2"> <math>1 </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ z^2</math> </td> <td align="center"> <math>=</math> </td> <td align="left" colspan="2"> <math>a_s^2\biggl[1 - \frac{\varpi^2}{a_\ell^2} \biggr] = a_\ell^2 (1-e^2) \biggl[1 - \frac{\varpi^2}{a_\ell^2} \biggr]</math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \frac{z}{a_\ell}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>\pm ~(1-e^2)^{1 / 2} \biggl[1 - \frac{\varpi^2}{a_\ell^2} \biggr]^{1 / 2} \, ,</math> </td> <td align="right"> for <math>~0 \le \frac{| \varpi |}{a_\ell} \le 1 \, .</math></td> </tr> </table> =====Expression for Gravitational Potential===== Throughout the interior of this configuration, each associated <math>~\Phi_\mathrm{eff}</math> = constant, equipotential surface is defined by the expression, <!-- <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\phi_\mathrm{choice} \equiv \frac{\Phi_\mathrm{eff}}{\pi G \rho} + I_\mathrm{BT}a_1^2 </math> </td> <td align="center"> <math>=</math> </td> <td align="left" colspan="1"> <math>\biggl( A_1 - \frac{\omega_0^2}{2\pi G \rho}\biggr) \varpi^2 + A_3 z^2 </math> </td> </tr> </table> --> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\phi_\mathrm{choice} \equiv \frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(\pi G\rho_c a_\ell^2)} + \frac{1}{2} I_\mathrm{BT} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr] - \frac{1}{2} \biggl[ A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) + A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr) + 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr) \biggr] \, . </math> </td> </tr> </table> Letting, <div align="center"><math>\zeta \equiv \frac{z^2}{a_\ell^2}</math>,</div> we can rewrite this expression for <math>\phi_\mathrm{choice}</math> as, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\phi_\mathrm{choice} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \zeta - \frac{1}{2} A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) - \frac{1}{2} A_{ss} a_\ell^2 \zeta^2 - A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 }{a_\ell^2}\biggr)\zeta </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> - \frac{1}{2} A_{ss} a_\ell^2 \zeta^2 + \biggl[ A_s - A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 }{a_\ell^2}\biggr)\biggr]\zeta + A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) - \frac{1}{2} A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) \, . </math> </td> </tr> </table> =====Potential at the Pole===== At the pole, <math>(\varpi, z) = (0, a_s)</math>. Hence, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\phi_\mathrm{choice}\biggr|_\mathrm{mid} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> - \frac{1}{2} A_{ss} a_\ell^2 \biggl(\frac{a_s^2}{a_\ell^2}\biggr)^2 + \biggl[ A_s - A_{\ell s}a_\ell^2 \cancelto{0}{\biggl( \frac{\varpi^2 }{a_\ell^2}\biggr)}\biggr]\biggl(\frac{a_s^2}{a_\ell^2}\biggr) + A_\ell \cancelto{0}{\biggl(\frac{\varpi^2}{a_\ell^2}\biggr)} - \frac{1}{2} A_{\ell \ell} a_\ell^2 \cancelto{0}{\biggl(\frac{\varpi^4}{a_\ell^4}\biggr)} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> A_s \biggl(\frac{a_s^2}{a_\ell^2}\biggr) - \frac{1}{2} A_{ss} a_\ell^2 \biggl(\frac{a_s^2}{a_\ell^2}\biggr)^2 \, . </math> </td> </tr> </table> =====General Determination of Vertical Coordinate (ζ)===== <table border="1" align="center" cellpadding="8" width="80%"><tr><td align="left"> Given values of the three parameters, <math>e</math>, <math>\varpi</math>, and <math>\phi_\mathrm{choice}</math>, this last expression can be viewed as a quadratic equation for <math>\zeta</math>. Specifically, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>0</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \alpha \zeta^2 + \beta\zeta + \gamma \, , </math> </td> </tr> </table> where, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\alpha</math> </td> <td align="center"><math>\equiv</math></td> <td align="left"> <math> \frac{1}{2} A_{ss} a_\ell^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{3}\biggl\{ \frac{( 4e^2 - 3 )}{e^4(1-e^2)} + \frac{3 (1-e^2)^{1 / 2}}{e^4} \biggl[\frac{\sin^{-1}e}{e}\biggr] \biggr\} \, , </math> </td> </tr> <tr> <td align="right"> <math>\beta</math> </td> <td align="center"><math>\equiv</math></td> <td align="left"> <math> A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 }{a_\ell^2}\biggr) - A_s </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{e^4}\biggl\{(3-e^2) - 3(1-e^2)^{1 / 2} \frac{\sin^{-1}e}{e} \biggr\} \biggl( \frac{\varpi^2 }{a_\ell^2}\biggr) - \frac{2}{e^2} \biggl[ (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1 / 2} \, , </math> </td> </tr> <tr> <td align="right"> <math>\gamma</math> </td> <td align="center"><math>\equiv</math></td> <td align="left"> <math> \phi_\mathrm{choice} + \frac{1}{2} A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) - A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \phi_\mathrm{choice} + \frac{1}{8e^4}\biggl\{- (3 + 2e^2) (1-e^2)+3 (1 - e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr] \biggr\}\biggl(\frac{\varpi^4}{a_\ell^4}\biggr) - \frac{1}{e^2} \biggl[ \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1 / 2} \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) \, . </math> </td> </tr> </table> The solution of this quadratic equation gives, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\zeta</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{2\alpha}\biggl\{ - \beta \pm \biggl[\beta^2 - 4\alpha\gamma \biggr]^{1 / 2}\biggr\} \, . </math> </td> </tr> </table> Should we adopt the ''superior'' (positive) sign, or is it more physically reasonable to adopt the ''inferior'' (negative) sign? As it turns out, <math>\beta</math> is intrinsically negative, so the quantity, <math>-\beta</math>, is positive. Furthermore, when <math>\gamma</math> goes to zero, we need <math>\zeta</math> to go to zero as well. This will only happen if we adopt the ''inferior'' (negative) sign. Hence, the physically sensible root of this quadratic relation is given by the expression, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\zeta</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{2\alpha}\biggl\{ - \beta - \biggl[\beta^2 - 4\alpha\gamma \biggr]^{1 / 2}\biggr\} \, . </math> </td> </tr> </table> <!-- Given that in this physical system, <math>\zeta = z^2/a_\ell^2</math> must be positive, we must choose the superior root. We conclude therefore that, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{z^2}{a_\ell^2}</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{2\alpha}\biggl\{ \biggl[\beta^2 - 4\alpha\gamma \biggr]^{1 / 2} - \beta \biggr\} \, . </math> </td> </tr> </table> <font color="red">But check this statement because it appears that <math>\beta</math> will sometimes be negative.</font> --> </td></tr></table> <span id="QuantitativeExample">Here we present a quantitatively accurate depiction</span> of the shape of the (Ferrers) gravitational potential that arises from oblate-spheroidal configurations having a parabolic density distribution. We closely follow the discussion of [[Apps/MaclaurinSpheroids#Example_Equi-gravitational-potential_Contours|equi-gravitational potential contours that arise in (uniform-density) Maclaurin spheroids]]. In order to facilitate comparison with Maclaurin spheroids, we will focus on a model with … <table border="0" align="center" width="80%"> <tr> <td align="center"><math>\frac{a_s}{a_\ell} = 0.582724 \, ,</math></td> <td align="center"><math>e = 0.81267 \, ,</math></td> <td align="center"> </td> </tr> <tr> <td align="center"><math>A_\ell = A_m = 0.51589042 \, ,</math></td> <td align="center"><math>A_s = 0.96821916 \, ,</math></td> <td align="center"><math>I_\mathrm{BT} = 1.360556 \, ,</math></td> </tr> <tr> <td align="center"><math>a_\ell^2 A_{\ell \ell} = 0.3287756 \, ,</math></td> <td align="center"><math>a_\ell^2 A_{s s} = 1.5066848 \, ,</math></td> <td align="center"><math>a_\ell^2 A_{\ell s} = 0.6848975 \, .</math></td> </tr> </table> [<font color="red">NOTE:</font> Along the Maclaurin spheroid sequence, this is the eccentricity that marks bifurcation to the Jacobi ellipsoid sequence — see the first model listed in Table IV (p. 103) of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] and/or see Tables 1 and 2 of [[ThreeDimensionalConfigurations/JacobiEllipsoids|our discussion of the Jacobi ellipsoid sequence]]. It is unlikely that this same eccentricity has a comparably special physical relevance along the sequence of spheroids having parabolic density distributions.] The largest value of the gravitational potential that will arise inside (actually, on the surface) of the configuration is at <math>(\varpi, z) = (1, 0)</math>. That is, when, <!-- <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\alpha</math> </td> <td align="center"><math>\equiv</math></td> <td align="left"> <math> \frac{1}{2} A_{ss} a_\ell^2 </math> </td> </tr> <tr> <td align="right"> <math>\beta</math> </td> <td align="center"><math>\equiv</math></td> <td align="left"> <math> A_{\ell s}a_\ell^2 - A_s </math> </td> </tr> <tr> <td align="right"> <math>\gamma</math> </td> <td align="center"><math>\equiv</math></td> <td align="left"> <math> \phi_\mathrm{choice} + \frac{1}{2} A_{\ell \ell} a_\ell^2 - A_\ell </math> </td> </tr> </table> --> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\phi_\mathrm{choice}\biggr|_\mathrm{max} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> A_\ell - \frac{1}{2} A_{\ell \ell} a_\ell^2 = 0.3515026 \, . </math> </td> </tr> </table> So we will plot various equipotential surfaces having, <math>0 < \phi_\mathrm{choice} < \phi_\mathrm{choice}|_\mathrm{max} </math>, recognizing that they will each cut through the equatorial plane <math>(z = 0)</math> at the radial coordinate given by, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\phi_\mathrm{choice} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> - \frac{1}{2} A_{ss} a_\ell^2 \cancelto{0}{\zeta^2} + \biggl[ A_s - A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 }{a_\ell^2}\biggr)\biggr]\cancelto{0}{\zeta} + A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) - \frac{1}{2} A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ 0</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{2} A_{\ell \ell} a_\ell^2 \chi^2 - A_\ell \chi + \phi_\mathrm{choice} \, , </math> </td> </tr> </table> where, <div align="center"><math>\chi \equiv \frac{\varpi^2}{a_\ell^2} \, .</math></div> The solution to this quadratic equation gives, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\chi_\mathrm{eqplane} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{A_{\ell \ell} a_\ell^2}\biggl\{ A_\ell \pm \biggl[A_\ell^2 - 2A_{\ell \ell} a_\ell^2 \phi_\mathrm{choice}\biggr]^{1 / 2} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{A_\ell}{A_{\ell \ell} a_\ell^2}\biggl\{ 1 - \biggl[1 - \frac{2A_{\ell \ell} a_\ell^2 \phi_\mathrm{choice}}{A_\ell^2}\biggr]^{1 / 2} \biggr\} \, . </math> </td> </tr> </table> Note that, again, the physically relevant root is obtained by adopting the ''inferior'' (negative) sign, as has been done in this last expression. =====Equipotential Contours that Lie Entirely Within Configuration===== For all <math>0 < \phi_\mathrm{choice} \le \phi_\mathrm{choice} |_\mathrm{mid}</math>, the equipotential contour will reside entirely within the configuration. In this case, for a given <math>\phi_\mathrm{choice}</math>, we can plot points along the contour by picking (equally spaced?) values of <math>\chi_\mathrm{eqplane} \ge \chi \ge 0</math>, then solve the above quadratic equation for the corresponding value of <math>\zeta</math>. In our example configuration, this means … (to be finished) ===Hydrostatic Balance (Algebraic Condition)=== Following our [[Apps/MaclaurinSpheroids#Equilibrium_Structure|separate discussion of the equilibrium structure]] of Maclaurin spheroids, and given that our solution of the Poisson equation fixes the expression for <math>\Phi_\mathrm{grav} </math>, the algebraic expression ensuring hydrostatic balance is, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>H(\varpi, z)</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> C_B - \biggl[ \Phi_\mathrm{grav}(\varpi, z) + \Psi(\varpi, z) \biggr] \, , </math> </td> </tr> </table> where, <math>\Psi</math> is the centrifugal potential. <font color="red">NOTE:</font> Generally when modeling axisymmetric astrophysical systems (see our [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|accompanying discussion of ''simple'' rotation profiles]]) it is assumed that <math>\Psi</math> does not functionally depend on <math>z</math>. Here, our other constraints — for example, demanding that the configuration have a parabolic density distribution — may force us to adopt a <math>z</math>-dependent rotation profile. Here, we know that the adopted parabolic density distribution gives rise to a gravitational potential of the form, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c a_\ell^2)}</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{2} I_\mathrm{BT} - \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr] + \frac{1}{2} \biggl[ A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) + A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr) + 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr) \biggr] \, . </math> </td> </tr> </table> Hence, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\Psi(\varpi, z)</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> C_B - \Phi_\mathrm{grav}(\varpi, z) - H(\varpi, z) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> C_B + \pi G \rho_c a_\ell^2\biggl\{ \frac{1}{2} I_\mathrm{BT} - \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr] + \frac{1}{2} \biggl[ A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) + A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr) + 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr) \biggr] \biggr\} - H_c h(\xi_1) \, . </math> </td> </tr> </table> <table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left"> We presume that the enthalpy profile, as well as the density profile, can be rewritten in terms of [[Appendix/Ramblings/T1Coordinates#T1_Coordinates|T1 Coordinates]]. In particular, defining, <math>q \equiv a_\ell/a_s</math> and, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\xi_1</math> </td> <td align="center"> <math>\equiv</math> </td> <td align="left"> <math> \biggl[ z^2 + \biggl(\frac{\varpi}{q}\biggr)^2\biggr]^{1 / 2} = a_s\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]^{1 / 2} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \frac{\rho}{\rho_c}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>\biggl[ 1 - \biggl(\frac{\xi_1}{a_s}\biggr)^2 \biggr] \, .</math> </td> </tr> </table> Because we expect contours of constant enthalpy <math>(H)</math> to coincide with contours of constant density in equilibrium configurations, we should expect to find that, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{H}{H_c}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>h(\xi_1) \, .</math> </td> </tr> </table> If the "radial" enthalpy profile resembles our [[SSC/Structure/OtherAnalyticModels#SphericalEnthalpyProfile|derived spherical enthalpy profile]], we should expect to find that, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>h(\xi_1)</math> </td> <td align="center"> <math>\sim</math> </td> <td align="left"> <math>h_0 \biggl[1 - h_2 \xi_1^2 - h_4 \xi_1^4 \biggr]</math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ 1 - \frac{h(\xi_1)}{h_0}</math> </td> <td align="center"> <math>\sim</math> </td> <td align="left"> <math>h_2 \xi_1^2 + h_4 \xi_1^4</math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> h_2 a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr] + h_4 \biggl\{ a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr] \biggr\}^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> h_2 a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr] + h_4 a_s^4 \biggl[ \biggl(\frac{z}{a_s}\biggr)^4 + 2\biggl(\frac{z}{a_s}\biggr)^2\biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^4 \biggr] </math> </td> </tr> </table> </td></tr></table> Adopting this last expression for the enthalpy, we have, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{h(\xi_1)}{h_0}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 1 - h_2 a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr] - h_4 a_s^4 \biggl[ \biggl(\frac{z}{a_s}\biggr)^4 + 2\biggl(\frac{z}{a_s}\biggr)^2\biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^4 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 1 - h_2 a_s^2\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_\ell}\biggr)^2(1-e^2)^{-1}\biggr] - h_4 a_s^4 \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4 + \biggl(\frac{z}{a_\ell}\biggr)^4 (1-e^2)^{-2} + 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)^{-1} \biggr] \, . </math> </td> </tr> </table> Hence, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\Psi(\varpi, z)</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> C_B + \pi G \rho_c a_\ell^2\biggl\{ \frac{1}{2} I_\mathrm{BT} - \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr] + \frac{1}{2} \biggl[ A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) + A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr) + 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr) \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - H_c h_0 \biggl\{ 1 - h_2 a_s^2\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_\ell}\biggr)^2(1-e^2)^{-1}\biggr] - h_4 a_s^4 \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4 + \biggl(\frac{z}{a_\ell}\biggr)^4 (1-e^2)^{-2} + 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)^{-1} \biggr] \biggr\} \, . </math> </td> </tr> </table> At the pole of the configuration — that is, when <math>(\varpi, z) = (0, a_s)</math> — this statement of hydrostatic balance becomes, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\Psi(\varpi, z)</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> C_B + \pi G \rho_c a_\ell^2\biggl\{ \frac{1}{2} I_\mathrm{BT} - \biggl[A_\ell \cancelto{0}{\biggl(\frac{\varpi^2}{a_\ell^2}\biggr)} + A_s \biggl( \frac{a_s^2}{a_\ell^2}\biggr) \biggr] + \frac{1}{2} \biggl[ A_{\ell \ell} a_\ell^2 \cancelto{0}{\biggl(\frac{\varpi^4}{a_\ell^4}\biggr)} + A_{ss} a_\ell^2 \biggl(\frac{a_s^4}{a_\ell^4}\biggr) + 2A_{\ell s}a_\ell^2 \biggl( \frac{\cancelto{0}{\varpi^2} a_s^2}{a_\ell^4}\biggr) \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - H_c h_0 \biggl\{ 1 - h_2 a_s^2\biggl[ \cancelto{0}{\biggl(\frac{\varpi}{a_\ell}\biggr)^2} + \biggl(\frac{a_s}{a_\ell}\biggr)^2(1-e^2)^{-1}\biggr] - h_4 a_s^4 \biggl[\cancelto{0}{\biggl(\frac{\varpi}{a_\ell}\biggr)^4} + \biggl(\frac{a_s}{a_\ell}\biggr)^4 (1-e^2)^{-2} + 2\biggl(\frac{\cancelto{0}{\varpi^2} a_s^2}{a_\ell^4}\biggr) (1-e^2)^{-1} \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> C_B + \pi G \rho_c a_\ell^2\biggl[ \frac{1}{2} I_\mathrm{BT} - A_s (1-e^2) + \frac{1}{2} A_{ss} a_\ell^2 (1-e^2)^2 \biggr] - H_c h_0 \biggl[ 1 - h_2 a_s^2 - h_4 a_s^4 \biggr] \, . </math> </td> </tr> </table> For centrally condensed configurations, it is astrophysically reasonable to assume that <math>\Psi(\varpi, z)</math> is of the form such that the centrifugal potential goes to zero when <math>\varpi \rightarrow 0</math>. Adopting that assumption here means that the Bernoulli constant has the value, <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"> <td align="left"> <math> H_c h_0 \biggl[ 1 - h_2 a_s^2 - h_4 a_s^4 \biggr] - \pi G \rho_c a_\ell^2\biggl[ \frac{1}{2} I_\mathrm{BT} - A_s (1-e^2) + \frac{1}{2} A_{ss} a_\ell^2 (1-e^2)^2 \biggr] \, . </math> </td> </tr> </table> Plugging this expression for <math>C_B</math> back into the general statement of hydrostatic balance gives, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\Psi(\varpi, z)</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \pi G \rho_c a_\ell^2\biggl\{ \frac{1}{2} I_\mathrm{BT} - \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr] + \frac{1}{2} \biggl[ A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) + A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr) + 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr) \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - \pi G \rho_c a_\ell^2\biggl[ \frac{1}{2} I_\mathrm{BT} - A_s (1-e^2) + \frac{1}{2} A_{ss} a_\ell^2 (1-e^2)^2 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + H_c h_0 \biggl[ 1 - h_2 a_s^2 - h_4 a_s^4 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - H_c h_0 \biggl\{ 1 - h_2 a_s^2\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_\ell}\biggr)^2(1-e^2)^{-1}\biggr] - h_4 a_s^4 \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4 + \biggl(\frac{z}{a_\ell}\biggr)^4 (1-e^2)^{-2} + 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)^{-1} \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \pi G \rho_c a_\ell^2\biggl\{ \biggl[A_s (1-e^2)-A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) - A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr] + \frac{1}{2} \biggl[ A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) + A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr) + 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr) - A_{ss} a_\ell^2 (1-e^2)^2 \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + H_c h_0 \biggl\{ h_2 a_s^2\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_\ell}\biggr)^2(1-e^2)^{-1} - 1\biggr] + h_4 a_s^4 \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4 + \biggl(\frac{z}{a_\ell}\biggr)^4 (1-e^2)^{-2} + 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)^{-1} -1 \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \pi G \rho_c a_\ell^2\biggl\{ A_s \biggl[- \frac{A_\ell}{A_s} \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) - \biggl( \frac{z^2}{a_\ell^2}\biggr) + (1-e^2)\biggr] + \frac{A_{ss}a_\ell^2}{2} \biggl[ \frac{A_{\ell \ell}}{A_{ss}} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) + \biggl(\frac{z^4}{a_\ell^4}\biggr) + \frac{2A_{\ell s}}{A_{ss}} \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr) - (1-e^2)^2 \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + H_c h_0 \biggl\{ h_2 a_s^2(1-e^2)^{-1}\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2(1-e^2) + \biggl(\frac{z}{a_\ell}\biggr)^2 - (1-e^2)\biggr] + h_4 a_s^4 (1-e^2)^{-2} \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4(1-e^2)^{2} + \biggl(\frac{z}{a_\ell}\biggr)^4 + 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2) - (1-e^2)^{2} \biggr] \biggr\} </math> </td> </tr> </table> Let's set … <div align="center"> <math>H_c h_0 = \pi G \rho_c a_\ell^2 \, ;</math> <math>h_2 = \frac{A_s(1-e^2)}{a_s^2} \, ;</math> <math>h_4 = - \frac{ A_{ss}a_\ell^2 (1-e^2)^2 }{ 2a_s^4 } \, .</math> </div> This gives, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{\Psi(\varpi, z)}{\pi G \rho_c a_\ell^2}</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl\{ A_s \biggl[- \frac{A_\ell}{A_s} \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) - \biggl( \frac{z^2}{a_\ell^2}\biggr) + (1-e^2)\biggr] + \frac{A_{ss}a_\ell^2}{2} \biggl[ \frac{A_{\ell \ell}}{A_{ss}} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) + \biggl(\frac{z^4}{a_\ell^4}\biggr) + \frac{2A_{\ell s}}{A_{ss}} \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr) - (1-e^2)^2 \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \biggl\{ A_s\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2(1-e^2) + \biggl(\frac{z}{a_\ell}\biggr)^2 - (1-e^2)\biggr] - \frac{A_{ss} a_\ell^2}{2} \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4(1-e^2)^{2} + \biggl(\frac{z}{a_\ell}\biggr)^4 + 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2) - (1-e^2)^{2} \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl\{ A_s \biggl[- \frac{A_\ell}{A_s} \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) \biggr] + \frac{A_{ss}a_\ell^2}{2} \biggl[ \frac{A_{\ell \ell}}{A_{ss}} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) + \frac{2A_{\ell s}}{A_{ss}} \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr) \biggr] \biggr\} + \biggl\{ A_s\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2(1-e^2) \biggr] - \frac{A_{ss} a_\ell^2}{2} \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4(1-e^2)^{2} + 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2) \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl\{ - A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + \biggl[ \frac{A_{\ell \ell} a_\ell^2}{2} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) + A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr) \biggr] \biggr\} + \biggl\{ A_s\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2(1-e^2) \biggr] - \frac{A_{ss} a_\ell^2}{2} \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4(1-e^2)^{2} + 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2) \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ A_s (1-e^2) - A_\ell \biggr] \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + \frac{A_{\ell \ell} a_\ell^2}{2} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) - \frac{A_{ss} a_\ell^2}{2} \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4(1-e^2)^{2} \biggr] + A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr) - A_{ss} a_\ell^2 \biggl[ \biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2) \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ A_s (1-e^2) - A_\ell \biggr] \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + \frac{1}{2}\biggl\{ A_{\ell \ell} a_\ell^2 - A_{ss} a_\ell^2 (1-e^2)^{2} \biggr\} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) + \biggl\{ A_{\ell s}a_\ell^2 - A_{ss} a_\ell^2 (1-e^2) \biggr\}\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) \, . </math> </td> </tr> </table> ===2<sup>nd</sup> Try=== <table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left"> <div align="center">Keep in Mind, from Above</div> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c a_\ell^2)}</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{2} I_\mathrm{BT} - \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr] + \frac{1}{2} \biggl[ A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) + A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr) + 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr) \biggr] \, . </math> </td> </tr> </table> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\rho</math> </td> <td align="center"> = </td> <td align="left"> <math> \rho_c \biggl[ 1 - \biggl(\frac{\varpi^2}{a_\ell^2} + \frac{z^2}{a_s^2}\biggr) \biggr] </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \frac{\rho}{\rho_c}</math> </td> <td align="center"> = </td> <td align="left"> <math> 1 - \chi^2 - \zeta^2(1-e^2)^{-1} \, , </math> </td> </tr> </table> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\xi_1</math> </td> <td align="center"> <math>\equiv</math> </td> <td align="left"> <math> \biggl[ z^2 + \biggl(\frac{\varpi}{q}\biggr)^2\biggr]^{1 / 2} = a_s\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]^{1 / 2} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \frac{\rho}{\rho_c}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>\biggl[ 1 - \biggl(\frac{\xi_1}{a_s}\biggr)^2 \biggr] \, .</math> </td> </tr> </table> </td></tr></table> From our presentation of [[AxisymmetricConfigurations/PGE#Eulerian_Formulation_(CYL.)|the Eulerian formulation of the Euler equation in cylindrical coordinates]], we see that in steady-state axisymmetric flows, the two relevant equilibrium conditions are, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>~{\hat{e}}_\varpi</math>: </td> <td align="right"> <math>~ 0 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ - \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] + \frac{j^2}{\varpi^3} </math> </td> </tr> <tr> <td align="right"><math>~{\hat{e}}_z</math>: </td> <td align="right"> <math>~ 0 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ - \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] </math> </td> </tr> </table> ====Vertical Component==== We will focus, first, on the vertical component. Specifically, since both <math>\rho</math> and <math>\Phi_\mathrm{grav}</math> are known, the vertical gradient of the (unknown) scalar pressure is <table border="0" align="center" cellpadding="8"> <tr> <td align="right"><math>\frac{\partial P}{\partial z}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> - \rho ~ \frac{\partial}{\partial z} \biggl\{ \Phi_\mathrm{grav} \biggr\} </math> </td> </tr> </table> Multiply thru by <math>1/(\pi G \rho_c^2 a_\ell)</math>: <table border="0" align="center" cellpadding="8"> <tr> <td align="right"><math>\Rightarrow ~~~ \frac{1}{(\pi G\rho_c^2 a_\ell)} \cdot \frac{\partial P}{\partial z}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial z} \biggl\{ \frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell)} \biggr\} </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P}{\partial \zeta}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial \zeta} \biggl\{ \frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ 1 - \biggl( \frac{\varpi^2}{a_\ell^2} + \frac{z^2}{a_s^2}\biggr) \biggr] \cdot \frac{\partial}{\partial \zeta} \biggl\{ \frac{1}{2} I_\mathrm{BT} - \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr] + \frac{1}{2} \biggl[ A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) + A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr) + 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr) \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \cdot \frac{\partial}{\partial \zeta} \biggl\{ \frac{1}{2} I_\mathrm{BT} - \biggl[A_\ell \chi^2 + A_s \zeta^2 \biggr] + \frac{1}{2} \biggl[ A_{\ell \ell} a_\ell^2 \chi^4 + A_{ss} a_\ell^2 \zeta^4 + 2A_{\ell s}a_\ell^2 \chi^2\zeta^2 \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \cdot \biggl\{ - 2A_s \zeta + \biggl[ 2A_{ss} a_\ell^2 \zeta^3 + 2A_{\ell s}a_\ell^2 \chi^2\zeta \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2\biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \cdot \biggl[ A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta + A_{ss} a_\ell^2 \zeta^3 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2\biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \cdot \biggl[ A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta + A_{ss} a_\ell^2 \zeta^3 \biggr] </math> </td> </tr> </table> where (unlike above) we are using the dimensionless lengths, <math>\chi \equiv \varpi/a_\ell</math> and <math>\zeta \equiv z/a_\ell</math>. Continuing to streamline this function, we have, <table border="0" align="center" cellpadding="8"> <tr> <td align="right"><math>\frac{1}{(2\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P}{\partial \zeta}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta + A_{ss} a_\ell^2 \zeta^3 \biggr] - \chi^2\biggl[ A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta + A_{ss} a_\ell^2 \zeta^3 \biggr] - \zeta^2\biggl[ A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta + A_{ss} a_\ell^2 \zeta^3 \biggr](1-e^2)^{-1} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta + A_{ss} a_\ell^2 \zeta^3 \biggr] - \biggl[ A_{\ell s}a_\ell^2 \chi^4 \zeta - A_s \chi^2 \zeta + A_{ss} a_\ell^2 \chi^2 \zeta^3 \biggr] - \biggl[ A_{\ell s}a_\ell^2 \chi^2\zeta^3 - A_s \zeta^3 + A_{ss} a_\ell^2 \zeta^5 \biggr](1-e^2)^{-1} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ A_{\ell s}a_\ell^2 \chi^2 - A_s \biggr]\zeta + A_{ss} a_\ell^2 \zeta^3 + \biggl[ A_s \chi^2 - A_{\ell s}a_\ell^2 \chi^4 \biggr]\zeta - A_{ss} a_\ell^2 \chi^2 \zeta^3 + \biggl[ A_s \zeta^3 - A_{\ell s}a_\ell^2 \chi^2\zeta^3 - A_{ss} a_\ell^2 \zeta^5 \biggr](1-e^2)^{-1} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ A_{\ell s}a_\ell^2 \chi^2 - A_s + A_s \chi^2 - A_{\ell s}a_\ell^2 \chi^4 \biggr]\zeta + \biggl[ A_{ss} a_\ell^2 - A_{ss} a_\ell^2 \chi^2 + A_s(1-e^2)^{-1} - A_{\ell s}a_\ell^2 \chi^2(1-e^2)^{-1} \biggr]\zeta^3 - A_{ss} a_\ell^2(1-e^2)^{-1} \zeta^5 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ - A_s + (A_{\ell s}a_\ell^2 + A_s )\chi^2 - A_{\ell s}a_\ell^2 \chi^4 \biggr]\zeta + \biggl\{ [A_s(1-e^2)^{-1} + A_{ss} a_\ell^2] - [A_{ss} a_\ell^2 + A_{\ell s}a_\ell^2 (1-e^2)^{-1}]\chi^2 \biggr\}\zeta^3 - A_{ss} a_\ell^2(1-e^2)^{-1} \zeta^5 \, . </math> </td> </tr> </table> So, let's see what happens if we assume that the pressure has the form, <table border="0" align="center" cellpadding="8"> <tr> <td align="right"><math>\frac{P_\mathrm{vert}}{(2\pi G\rho_c^2 a_\ell^2)} </math></td> <td align="center"><math>=</math></td> <td align="left"> <math>p_0 + p_2 \zeta^2 + p_4\zeta^4 + p_6\zeta^6 </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ \frac{1}{(2\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P_\mathrm{vert}}{\partial \zeta}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math>2p_2 \zeta + 4p_4\zeta^3 + 6p_6\zeta^5 \, ,</math> </td> </tr> </table> in which case, <table border="0" align="center" cellpadding="8"> <tr> <td align="right"><math>\frac{P_\mathrm{vert}}{(2\pi G\rho_c^2 a_\ell^2)} </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> p_0 + \frac{1}{2}\biggl[ - A_s + (A_{\ell s}a_\ell^2 + A_s )\chi^2 - A_{\ell s}a_\ell^2 \chi^4 \biggr] \zeta^2 + \frac{1}{4}\biggl\{[A_s (1-e^2)^{-1} + A_{ss} a_\ell^2] - [A_{ss} a_\ell^2 + A_{\ell s}a_\ell^2 (1-e^2)^{-1} ]\chi^2 \biggr\}\zeta^4 + \frac{1}{6}\biggl[ - A_{ss} a_\ell^2 (1-e^2)^{-1} \biggr]\zeta^6 \, . </math> </td> </tr> </table> <table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left"> <font color="red">REMINDER:</font> From [[#2nd_Try|above]] … <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{\rho}{\rho_c}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 1 - \chi^2 - \zeta^2(1-e^2)^{-1} \, . </math> </td> </tr> </table> And, in the case of the spherically symmetric equilibrium configuration, the [[SSC/Structure/OtherAnalyticModels#Pressure|pressure distribution]] derived by {{ Prasad49 }} has the form, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{P}{P_c}</math> </td> <td align="center"> <math>\sim</math> </td> <td align="left"> <math> \biggl(\frac{\rho}{\rho_c}\biggr)^2 \biggl[1 + \biggl(\frac{\rho}{\rho_c}\biggr)\biggr] \, . </math> </td> </tr> </table> In the context of rotationally flattened configurations, therefore, we might expect the (vertical) pressure distribution to be of the form, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{P}{P_c}</math> </td> <td align="center"> <math>\sim</math> </td> <td align="left"> <math> \biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr] \biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr] \biggl[2 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>\sim</math> </td> <td align="left"> <math> \biggl[2 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr] \biggl\{ \biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr] -\chi^2\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr] - \zeta^2(1-e^2)^{-1}\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>\sim</math> </td> <td align="left"> <math> \biggl[2 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr] \biggl\{ \biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr] + \biggl[-\chi^2 + \chi^4 + \chi^2\zeta^2(1-e^2)^{-1}\biggr] + \biggl[- \zeta^2(1-e^2)^{-1} + \chi^2\zeta^2(1-e^2)^{-1} + \zeta^4(1-e^2)^{-2}\biggr] \biggr\} </math> </td> </tr> </table> </td></tr></table> ====Radial Component==== Start with, <table border="0" align="center" cellpadding="8"> <tr> <td align="right"> <math>- \frac{j^2 \rho}{\varpi^3} + \frac{\partial P}{\partial \varpi} </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> - \rho ~ \frac{\partial}{\partial \varpi} \biggl\{ \Phi_\mathrm{grav} \biggr\} </math> </td> </tr> </table> Multiply thru by <math>1/(\pi G \rho_c^2 a_\ell)</math>: <table border="0" align="center" cellpadding="8"> <tr> <td align="right"> <math>- \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell)} \biggr] \frac{j^2 \rho}{\varpi^3} + \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell)} \biggr]\frac{\partial P}{\partial \varpi} </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> - \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell)} \biggr]\rho ~ \frac{\partial}{\partial \varpi} \biggl\{ \Phi_\mathrm{grav} \biggr\} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ - \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} + \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial \chi} \biggl\{ \frac{\Phi_\mathrm{grav}}{(-\pi G \rho_c a_\ell^2)} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \cdot \frac{\partial}{\partial \chi} \biggl\{ \frac{1}{2} I_\mathrm{BT} - \biggl[A_\ell \chi^2 + A_s \zeta^2 \biggr] + \frac{1}{2} \biggl[ A_{\ell \ell} a_\ell^2 \chi^4 + A_{ss} a_\ell^2 \zeta^4 + 2A_{\ell s}a_\ell^2 \chi^2\zeta^2 \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \biggl\{ \biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi + 2 A_{\ell \ell} a_\ell^2 \chi^3 \biggr\} \, . </math> </td> </tr> </table> <table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left"> <div align="center"><font color="red">EXACT!</font></div> <table border="0" align="center" cellpadding="8"> <tr> <td align="right"> <math> - \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} + \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\rho}{\rho_c} \cdot \biggl\{ \biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi + 2 A_{\ell \ell} a_\ell^2 \chi^3 \biggr\} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\rho}{\rho_c} \cdot \biggl\{ \biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi + 2 A_{\ell \ell} a_\ell^2 \chi^3 + \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} \biggr\} </math> </td> </tr> </table> </td></tr></table> Continuing to streamline this function, we have, <table border="0" align="center" cellpadding="8"> <tr> <td align="right"> <math> - \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} + \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi} </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl\{\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi+ 2 A_{\ell \ell} a_\ell^2 \chi^3 \biggr\} - \biggl\{\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi+ 2 A_{\ell \ell} a_\ell^2 \chi^3 \biggr\}\chi^2 - \biggl\{\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi+ 2 A_{\ell \ell} a_\ell^2 \chi^3 \biggr\}(1-e^2)^{-1}\zeta^2 </math> </td> </tr> <tr> <td align="right"> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi+ 2 A_{\ell \ell} a_\ell^2 \chi^3 - \biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi^3 - 2 A_{\ell \ell} a_\ell^2 \chi^5 + \biggl[2A_\ell - 2A_{\ell s}a_\ell^2 \zeta^2 \biggr] \chi(1-e^2)^{-1} \zeta^2 - 2 A_{\ell \ell} a_\ell^2 \chi^3 (1-e^2)^{-1}\zeta^2 </math> </td> </tr> <tr> <td align="right"> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi +\biggl[2A_\ell - 2A_{\ell s}a_\ell^2 \zeta^2 \biggr] \chi(1-e^2)^{-1} \zeta^2 + \biggl[2 A_{\ell \ell} a_\ell^2 -2A_{\ell s}a_\ell^2 \zeta^2 + 2A_\ell - 2 A_{\ell \ell} a_\ell^2 \zeta^2\biggr] \chi^3 - 2 A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5 </math> </td> </tr> <tr> <td align="right"> <td align="center"><math>=</math></td> <td align="left"> <math> 2 \biggl[ - A_\ell(1-e^2)^{-1} + A_{\ell s}a_\ell^2 \zeta^2 + A_\ell\zeta^2 - A_{\ell s}a_\ell^2 \zeta^4(1-e^2)^{-1} \biggr] \chi + 2\biggl[A_{\ell \ell} a_\ell^2 + A_\ell -A_{\ell s}a_\ell^2 \zeta^2 - A_{\ell \ell} a_\ell^2 \zeta^2\biggr] \chi^3 - 2 A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5 \, . </math> </td> </tr> </table> ====Determine Specific Angular Momentum Distribution==== Now, from our analysis of the vertical component, we determined that, <table border="0" align="center" cellpadding="8"> <tr> <td align="right"><math>\frac{12P_\mathrm{vert}}{(2\pi G\rho_c^2 a_\ell^2)} </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> 12p_0 + 6\biggl[ - A_s + (A_{\ell s}a_\ell^2 + A_s )\chi^2 - A_{\ell s}a_\ell^2 \chi^4 \biggr] \zeta^2 + 3\biggl\{[A_s (1-e^2)^{-1} + A_{ss} a_\ell^2] - [A_{ss} a_\ell^2 + A_{\ell s}a_\ell^2 (1-e^2)^{-1} ]\chi^2 \biggr\}\zeta^4 + 2\biggl[ - A_{ss} a_\ell^2 (1-e^2)^{-1} \biggr]\zeta^6 \, . </math> </td> </tr> </table> <span id="RadialDerivative">The radial derivative of this function is</span>, <table border="0" align="center" cellpadding="8"> <tr> <td align="right"><math>\biggl[ \frac{12}{(2\pi G\rho_c^2 a_\ell^2)} \biggr]\frac{\partial P_\mathrm{vert}}{\partial \chi} </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> 6\biggl[ 2(A_{\ell s}a_\ell^2 + A_s ) \zeta^2 \chi - 4A_{\ell s}a_\ell^2\zeta^2 \chi^3 \biggr] + 6\biggl\{ - [A_{ss} a_\ell^2 + A_{\ell s}a_\ell^2 (1-e^2)^{-1} ] \zeta^4 \chi \biggr\} </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ \biggl[ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr]\frac{\partial P_\mathrm{vert}}{\partial \chi} </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ (2A_{\ell s}a_\ell^2 + 2A_s )\zeta^2 - A_{ss} a_\ell^2 \zeta^4 - A_{\ell s}a_\ell^2 (1-e^2)^{-1}\zeta^4 \biggr] \chi - 4A_{\ell s}a_\ell^2 \zeta^2\chi^3 \, . </math> </td> </tr> </table> We hypothesize that, <table border="0" align="center" cellpadding="8"> <tr> <td align="right"> <math> - \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr]\frac{\partial P_\mathrm{vert}}{\partial \chi} - \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\biggl[\frac{\partial P}{\partial \chi}\biggr]_\mathrm{rad} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ (2A_{\ell s}a_\ell^2 + 2A_s )\zeta^2 - A_{ss} a_\ell^2 \zeta^4 - A_{\ell s}a_\ell^2 (1-e^2)^{-1}\zeta^4 \biggr] \chi - 4A_{\ell s}a_\ell^2 \zeta^2\chi^3 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - 2 \biggl[ - A_\ell(1-e^2)^{-1} + A_{\ell s}a_\ell^2 \zeta^2 + A_\ell\zeta^2 - A_{\ell s}a_\ell^2 \zeta^4(1-e^2)^{-1} \biggr] \chi - 2\biggl[A_{\ell \ell} a_\ell^2 + A_\ell -A_{\ell s}a_\ell^2 \zeta^2 - A_{\ell \ell} a_\ell^2 \zeta^2\biggr] \chi^3 + 2 A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math>2\biggl\{ \biggl[ (A_{\ell s}a_\ell^2 + A_s )\zeta^2 + \frac{1}{2}[- A_{ss} a_\ell^2 - A_{\ell s}a_\ell^2 (1-e^2)^{-1}]\zeta^4 \biggr] \chi - 2A_{\ell s}a_\ell^2 \zeta^2\chi^3 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \biggl[ A_\ell(1-e^2)^{-1} - A_{\ell s}a_\ell^2 \zeta^2 - A_\ell\zeta^2 + A_{\ell s}a_\ell^2 \zeta^4(1-e^2)^{-1} \biggr] \chi + \biggl[ -A_{\ell \ell} a_\ell^2 - A_\ell + A_{\ell s}a_\ell^2 \zeta^2 + A_{\ell \ell} a_\ell^2 \zeta^2\biggr] \chi^3 + A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5 \biggr\} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ - \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(2\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ (A_{\ell s}a_\ell^2 + A_s )\zeta^2 + \frac{1}{2}[- A_{ss} a_\ell^2 - A_{\ell s}a_\ell^2 (1-e^2)^{-1}]\zeta^4 + A_\ell(1-e^2)^{-1} - A_{\ell s}a_\ell^2 \zeta^2 - A_\ell\zeta^2 + A_{\ell s}a_\ell^2 \zeta^4(1-e^2)^{-1} \biggr] \chi </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \biggl[ -A_{\ell \ell} a_\ell^2 - A_\ell + A_{\ell s}a_\ell^2 \zeta^2 + A_{\ell \ell} a_\ell^2 \zeta^2- 2A_{\ell s}a_\ell^2 \zeta^2\biggr] \chi^3 + A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl\{ A_\ell(1-e^2)^{-1} +(A_{\ell s}a_\ell^2 + A_s - A_{\ell s}a_\ell^2 - A_\ell)\zeta^2 + \frac{1}{2}\biggl[- A_{ss} a_\ell^2 - A_{\ell s}a_\ell^2 (1-e^2)^{-1} + 2A_{\ell s}a_\ell^2 (1-e^2)^{-1} \biggr]\zeta^4 \biggr\} \chi </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \biggl\{ (-A_{\ell \ell} a_\ell^2 - A_\ell) + (A_{\ell s}a_\ell^2 + A_{\ell \ell} a_\ell^2 - 2A_{\ell s}a_\ell^2) \zeta^2 \biggr\} \chi^3 + A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl\{ A_\ell(1-e^2)^{-1} + (A_s - A_\ell)\zeta^2 + \frac{1}{2}\biggl[- A_{ss} a_\ell^2 + A_{\ell s}a_\ell^2 (1-e^2)^{-1} \biggr]\zeta^4 \biggr\} \chi + \biggl\{ (-A_{\ell \ell} a_\ell^2 - A_\ell) + (A_{\ell \ell} a_\ell^2 - A_{\ell s}a_\ell^2) \zeta^2 \biggr\} \chi^3 + A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5 </math> </td> </tr> </table> Now, from [[ParabolicDensity/GravPot#Parabolic_Density_Distribution_2|our layout of relevant index symbol expressions]], let's see if the coefficients of various ζ-dependent terms go to zero. <font color="red">FIRST:</font> <table border="0" align="center" cellpadding="8"> <tr> <td align="right"> <math> A_{s\ell} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> - \frac{A_s - A_\ell}{(a_s^2 - a_\ell^2)} = \frac{A_s - A_\ell}{a_\ell^2 e^2} </math> </td> </tr> <tr> <td align="right"> <math> \Rightarrow ~~~ A_{s \ell}a_\ell^2 e^2 </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> (A_s - A_\ell) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl\{ \frac{2}{e^2} \biggl[ (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1 / 2} \biggr\} - \biggl\{ \frac{1}{e^2} \biggl[ \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1/2} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{e^2}\biggl\{ 2\biggl[ 1 - \frac{\sin^{-1}e}{e} (1-e^2)^{1 / 2}\biggr] - \biggl[ \frac{\sin^{-1}e}{e}(1-e^2)^{1/2} - (1-e^2) \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{e^2}\biggl[ 3 - e^2 - 3(1-e^2)^{1 / 2}\frac{\sin^{-1}e}{e} \biggr] \, ; </math> </td> </tr> </table> <font color="red">SECOND:</font> <table border="0" align="center" cellpadding="8"> <tr> <td align="right"> <math> 3A_{s s} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{2}{a_s^2} - 2A_{s \ell} </math> </td> </tr> <tr> <td align="right"> <math> \Rightarrow ~~~ \frac{3}{2}A_{s s}a_\ell^2 </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{a_\ell^2}{a_s^2} - A_{s \ell}a_\ell^2 = (1 - e^2)^{-1} - A_{s\ell}a_\ell^2 </math> </td> </tr> <tr> <td align="right"> <math> \Rightarrow ~~~ - A_{s s}a_\ell^2 </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{2}{3}A_{s\ell}a_\ell^2 - \frac{2}{3}(1 - e^2)^{-1} </math> </td> </tr> <tr> <td align="right"> <math> \Rightarrow ~~~ \biggl[ - A_{ss} a_\ell^2 + A_{\ell s}a_\ell^2 (1-e^2)^{-1} \biggr] </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{2}{3}A_{s\ell}a_\ell^2 - \frac{2}{3}(1 - e^2)^{-1} + A_{\ell s}a_\ell^2 (1-e^2)^{-1} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math>\frac{1}{3(1-e^2)}\biggl[ 2A_{s\ell}a_\ell^2 (1-e^2) - 2 + 3A_{\ell s}a_\ell^2 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{3(1-e^2)}\biggl[A_{s\ell}a_\ell^2 (5-2e^2) - 2 \biggr]\, ; </math> </td> </tr> </table> <font color="red">THIRD:</font> <table border="0" align="center" cellpadding="8"> <tr> <td align="right"> <math> 3A_{\ell \ell}</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{2}{a_\ell^2} - A_{\ell \ell} - A_{s\ell} </math> </td> </tr> <tr> <td align="right"> <math> \Rightarrow ~~~ 4A_{\ell \ell}a_\ell^2</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2 - A_{s\ell}a_\ell^2 </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ (A_{\ell \ell} a_\ell^2 - A_{\ell s}a_\ell^2)</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{2} - \frac{5}{4}A_{s\ell}a_\ell^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{4}\biggl[2 - 5A_{s\ell}a_\ell^2\biggr] \, . </math> </td> </tr> </table> ===3<sup>rd</sup> Try=== From the [[#Radial_Component|above, "2<sup>nd</sup> Try" discussion of the radial component]], we can write the following "EXACT!" relation, <table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left"> <div align="center"><font color="red">EXACT!</font></div> <table border="0" align="center" cellpadding="8"> <tr> <td align="right"> <math> - \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} + \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\rho}{\rho_c} \cdot \biggl\{ \biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi + 2 A_{\ell \ell} a_\ell^2 \chi^3 \biggr\} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\rho}{\rho_c} \cdot \biggl\{ \biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi + 2 A_{\ell \ell} a_\ell^2 \chi^3 + \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} \biggr\} \, . </math> </td> </tr> </table> </td></tr></table> Now, our [[#RadialDerivative|earlier examination of the radial derivative of]] <math>P_\mathrm{vert}</math> suggests that the left-hand-side of this expression should be of the form, <table border="0" align="center" cellpadding="8"> <tr> <td align="right"> LHS <math> \equiv \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi} </math> </td> <td align="center"><math>\sim</math></td> <td align="left"> <math> c_2\zeta^2 + c_4\zeta^4 \, , </math> </td> </tr> </table> where it is understood that the coefficients, <math>c_2</math> and <math>c_4</math>, are both functions of <math>\chi</math>. This should be compared with the "EXACT!" expression for the RHS after multiplying through by the expression for the dimensionless density, that is, <table border="0" align="center" cellpadding="8"> <tr> <td align="right"> RHS </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ 1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] \cdot \biggl\{ \biggl[ 2 A_{\ell \ell} a_\ell^2 \chi^3 - 2A_\ell \chi + \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} \biggr] + 2A_{\ell s}a_\ell^2 \chi \zeta^2 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> (1 - \chi^2)\biggl[ 2 A_{\ell \ell} a_\ell^2 \chi^3 - 2A_\ell \chi + \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} \biggr] + 2A_{\ell s}a_\ell^2 \chi (1 - \chi^2) \zeta^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - \biggl[ 2 A_{\ell \ell} a_\ell^2 \chi^3 - 2A_\ell \chi + \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} \biggr](1-e^2)^{-1}\zeta^2 - 2A_{\ell s}a_\ell^2 (1-e^2)^{-1}\chi \zeta^4 \, . </math> </td> </tr> </table> Because we are not expecting to see a term that is independent of <math>\zeta</math>, this suggests that the term inside the large square brackets must be zero. This leads to an expression for the distribution of specific angular momentum of the form, <table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left"> <table border="0" align="center" cellpadding="8"> <tr><td align="center" colspan="3"><font color="red">EXCELLENT !!</font></td></tr> <tr> <td align="right"> <math>0</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ 2 A_{\ell \ell} a_\ell^2 \chi^3 - 2A_\ell \chi + \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} \biggr] </math> </td> </tr> <tr> <td align="right"> <math> \Rightarrow ~~~ \frac{j^2 }{(\pi G \rho_c a_\ell^4)} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2A_\ell \chi^4 - 2 A_{\ell \ell} a_\ell^2 \chi^6 \, . </math> </td> </tr> </table> According to our [[AxisymmetricConfigurations/SolutionStrategies#Specifying_Radial_Rotation_Profile_in_the_Equilibrium_Configuration|accompanying discussion of ''Simple'' rotation profiles]], the corresponding centrifugal potential is given by the expression, <table border="0" align="center" cellpadding="8"> <tr> <td align="right"> <math>\Psi</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> - \int \frac{j^2(\varpi)}{\varpi^3} d\varpi = - (\pi G \rho_c a_\ell^2) \int \frac{1}{\chi^3} \biggl[2A_\ell \chi^4 - 2 A_{\ell \ell} a_\ell^2 \chi^6\biggr]d\chi </math> </td> </tr> <tr> <td align="right"> <math> \Rightarrow ~~~ \frac{\Psi }{(\pi G \rho_c a_\ell^2)} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> - \int \biggl[2A_\ell \chi - 2 A_{\ell \ell} a_\ell^2 \chi^3\biggr]d\chi = \frac{1}{2}\biggl[ A_{\ell \ell}a_\ell^2 \chi^4 - 2A_\ell \chi^2 \biggr]\, . </math> </td> </tr> </table> (Here, we ignore the integration constant because it will be folded in with the Bernoulli constant.) </td></tr></table> It also means that the RHS expression simplifies to the form, <table border="0" align="center" cellpadding="8"> <tr> <td align="right"> RHS </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2A_{\ell s}a_\ell^2 \chi (1 - \chi^2) \zeta^2 - 2A_{\ell s}a_\ell^2 (1-e^2)^{-1}\chi \zeta^4 \, . </math> </td> </tr> </table> This should be compared to our [[#RadialDerivative|earlier examination of the radial derivative of]] <math>P_\mathrm{vert}</math>, namely, <table border="0" align="center" cellpadding="8"> <tr> <td align="right"><math>\biggl[ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr]\frac{\partial P_\mathrm{vert}}{\partial \chi} </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ (2A_{\ell s}a_\ell^2 + 2A_s )\zeta^2 - A_{ss} a_\ell^2 \zeta^4 - A_{\ell s}a_\ell^2 (1-e^2)^{-1}\zeta^4 \biggr] \chi - 4A_{\ell s}a_\ell^2 \zeta^2\chi^3 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> (2A_{\ell s}a_\ell^2 + 2A_s )\chi\zeta^2- 4A_{\ell s}a_\ell^2 \chi^3\zeta^2 - \biggl[A_{\ell s}a_\ell^2 (1-e^2)^{-1} + A_{ss} a_\ell^2\biggr]\chi\zeta^4 </math> </td> </tr> </table> ===4<sup>th</sup> Try=== In our [[ThreeDimensionalConfigurations/FerrersPotential#The_Case_Where_n_=_1|accompanying discussion of Ferrers Potential]], we have derived the expression for the gravitational potential inside (and on the surface of) a triaxial ellipsoid with a parabolic density distribution. Specifically, for <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\rho(\mathbf{x})</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \rho_c \biggl[1 - \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}\biggr) \biggr] \, ,</math> </td> </tr> </table> [[ThreeDimensionalConfigurations/FerrersPotential#GravFor1|we find]], <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c)} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{2} I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) ~+ \biggl( A_{12} x^2y^2 + A_{13} x^2z^2 + A_{23} y^2z^2\biggr) ~+ \frac{1}{2} \biggl(A_{11}x^4 + A_{22}y^4 + A_{33}z^4 \biggr) \, . </math> </td> </tr> </table> In this [[ThreeDimensionalConfigurations/FerrersPotential#The_Case_Where_n_=_1|same accompanying discussion]], we plugged this expression for the gravitational potential into the Poisson equation and demonstrated that it properly generates the expression for the parabolic density distribution. For the axisymmetric configuration being considered here — with the short axis aligned with <math>c = a_3 = a_s</math> — these two relations become, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{\rho(\varpi, z)}{\rho_c}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[1 - \biggl( \frac{\varpi^2}{a_\ell^2} + \frac{z^2}{a_s^2}\biggr) \biggr] = \biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] \, ,</math> </td> </tr> <tr> <td align="right"> <math>\frac{ \Phi_\mathrm{grav}(\varpi,z)}{(-\pi G\rho_c a_\ell^2)} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{2} I_\mathrm{BT} - A_\ell \frac{\varpi^2}{a_\ell^2} - A_s \frac{z^2}{a_\ell^2} + (A_{\ell s}a_\ell^2 )\frac{ \varpi^2z^2 }{a_\ell^4} + \frac{1}{2}(A_{s s} a_\ell^2) \frac{z^4}{a_\ell^4} + \frac{A_{\ell \ell}a_\ell^2}{2} \biggl[ \frac{(x^4 + 2 x^2y^2 + y^4 )}{a_\ell^4} \biggr] \, . </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{2} I_\mathrm{BT} - A_\ell \chi^2 - A_s \zeta^2 + \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4 + 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2 + (A_{\ell \ell} a_\ell^2) \chi^4 \biggr] \, . </math> </td> </tr> </table> where, <math>\chi \equiv \varpi/a_\ell</math> and <math>\zeta \equiv z/a_\ell</math>. (This matches the [[#Gravitational_Potential|expression derived above]].) ---- Discuss scalar relationship between the enthalpy <math>(H)</math> and the effective potential. As has been detailed in [[AxisymmetricConfigurations/SolutionStrategies#Technique|an accompanying discussion of solution techniques]], a configuration will be in dynamic equilibrium if, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\nabla\biggl[ H + \Phi_\mathrm{grav} + \Psi \biggr]</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 0 </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ H + \Phi_\mathrm{grav} + \Psi </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> constant <math> = C_B </math> </td> </tr> </table> Given that, in our particular case, we have analytic expressions for <math>\Phi_\mathrm{grav}(\chi,\zeta)</math> and for <math>\Psi(\chi,\zeta)</math>, we deduce that, to within a constant, the enthalpy distribution is given by the expression, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\biggl[ \frac{H(\chi, \zeta) - C_B}{(\pi G\rho_c a_\ell^2)} \biggr] </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - \frac{\Phi_\mathrm{grav}}{{(\pi G\rho_c a_\ell^2)}} - \frac{\Psi}{{(\pi G\rho_c a_\ell^2)}} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{2} I_\mathrm{BT} - A_\ell \chi^2 - A_s \zeta^2 + \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4 + 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2 + (A_{\ell \ell} a_\ell^2) \chi^4 \biggr] - \frac{1}{2}\biggl[ A_{\ell \ell}a_\ell^2 \chi^4 - 2A_\ell \chi^2 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{2} I_\mathrm{BT} - A_s \zeta^2 + \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4 + 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{2} I_\mathrm{BT} - A_s \zeta^2 + \frac{\zeta^2}{2} \biggl[(A_{s s} a_\ell^2) \zeta^2 + 2(A_{\ell s}a_\ell^2 )\chi^2 \biggr] </math> </td> </tr> </table> Now, according to our [[ParabolicDensity/GravPot#Parabolic_Density_Distribution_2|related discussion of index symbols]], <table border="0" align="center" cellpadding="8"> <tr> <td align="right"><math>3A_{s s}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{2}{a_s^2} - 2A_{\ell s} </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ 3A_{s s}a_\ell^2</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> 2(1-e^2)^{-1} - 2A_{\ell s}a_\ell^2 </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~2(A_{\ell s}a_\ell^2)\chi^2 </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> 2(1-e^2)^{-1}\chi^2 - 3(A_{s s}a_\ell^2) \chi^2 \, . </math> </td> </tr> </table> Hence, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\biggl[ \frac{H(\chi, \zeta) - C_B}{(\pi G\rho_c a_\ell^2)} \biggr] - \frac{1}{2} I_\mathrm{BT} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - A_s \zeta^2 + \frac{\zeta^2}{2} \biggl[(A_{s s} a_\ell^2) \zeta^2 + 2(1-e^2)^{-1}\chi^2 - 3(A_{s s}a_\ell^2) \chi^2 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - A_s \zeta^2 + \frac{\zeta^2}{2} \biggl[(A_{s s} a_\ell^2) (\zeta^2 - 3\chi^2) + 2(1-e^2)^{-1}\chi^2 \biggr] \, . </math> </td> </tr> </table> <table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left"> Examining the radial derivative … <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{1}{(\pi G\rho_c a_\ell^2)} \frac{\partial H}{\partial \chi} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{\partial}{\partial \chi} \biggl\{ - A_s \zeta^2 + \frac{\zeta^2}{2} \biggl[(A_{s s} a_\ell^2) (\zeta^2 - 3\chi^2) + 2(1-e^2)^{-1}\chi^2 \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[-3(A_{s s} a_\ell^2) + 2(1-e^2)^{-1} \biggr]\zeta^2\chi </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 2(A_{\ell s} a_\ell^2)\zeta^2\chi \, . </math> </td> </tr> </table> <font color="red">YES !!!</font> This matches the "radial" pressure-gradient, below. Now, examining the vertical derivative … <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{1}{(\pi G\rho_c a_\ell^2)} \frac{\partial H}{\partial \zeta} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{\partial}{\partial \zeta} \biggl\{ - A_s \zeta^2 + \frac{\zeta^2}{2} \biggl[(A_{s s} a_\ell^2) (\zeta^2 - 3\chi^2) + 2(1-e^2)^{-1}\chi^2 \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{\partial}{\partial \zeta} \biggl\{ - A_s \zeta^2 + \frac{1}{2} \biggl[(A_{s s} a_\ell^2) \zeta^4 + [2(1-e^2)^{-1} - 3 (A_{s s} a_\ell^2)] \chi^2\zeta^2 \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - 2A_s \zeta + \biggl[2(A_{s s} a_\ell^2) \zeta^3 + [2(1-e^2)^{-1} - 3 (A_{s s} a_\ell^2)] \chi^2\zeta \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - 2A_s \zeta + \biggl[2(A_{s s} a_\ell^2) \zeta^3 + 2(A_{\ell s} a_\ell^2) \chi^2\zeta \biggr] </math> </td> </tr> </table> <font color="red">HURRAY !!!</font> This matches the "vertical" pressure-gradient, below. </td></tr></table> ---- <table border="0" align="center" cellpadding="8"> <tr> <td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\rho}{\rho_c} \cdot \biggl[ 2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr] </math> </td> </tr> <tr> <td align="right"> <math> \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\rho}{\rho_c} \cdot \biggl\{ \biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi + 2 A_{\ell \ell} a_\ell^2 \chi^3 + \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} \biggr\} </math> </td> </tr> </table> Plug in … <table border="0" align="center" cellpadding="8"> <tr> <td align="right"> <math> \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2A_\ell \chi - 2 A_{\ell \ell} a_\ell^2 \chi^3 \, . </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\rho}{\rho_c} \cdot \biggl\{ \biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi + 2 A_{\ell \ell} a_\ell^2 \chi^3 + 2A_\ell \chi - 2 A_{\ell \ell} a_\ell^2 \chi^3\biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\rho}{\rho_c} \cdot \biggl\{ 2A_{\ell s}a_\ell^2 \zeta^2 \chi \biggr\} </math> </td> </tr> </table> <!-- TEMPORARY PRESSURE (BEGIN) The result appears to be something like … <table border="0" align="center" cellpadding="8"> <tr> <td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] P</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\rho}{\rho_c} \cdot \biggl[ A_{\ell s}a_\ell^2 \chi^2\zeta^2 - A_s \zeta^2 + \frac{A_{ss} a_\ell^2}{2} \cdot \zeta^4 \biggr] </math> </td> </tr> </table> TEMPORARY PRESSURE (END) --> Hence, examination of the radial component leads to the following suggested expression for the pressure: <table border="0" align="center" cellpadding="8"> <tr> <td align="right"> <math> \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] \biggl[ 2A_{\ell s}a_\ell^2 \zeta^2 \chi\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ 2A_{\ell s}a_\ell^2 \zeta^2 \chi\biggr] - \chi^2 \biggl[ 2A_{\ell s}a_\ell^2 \zeta^2 \chi\biggr] - \zeta^2(1-e^2)^{-1} \biggl[ 2A_{\ell s}a_\ell^2 \zeta^2 \chi\biggr] </math> </td> </tr> <tr> <td align="right"> <math> \Rightarrow ~~~ \frac{P}{(\pi G \rho_c^2 a_\ell^2)} </math> </td> <td align="center"><math>\sim</math></td> <td align="left"> <math> \biggl[ A_{\ell s}a_\ell^2 \zeta^2 \chi^2\biggr] - \frac{1}{2}\biggl[ A_{\ell s}a_\ell^2 \zeta^2 \chi^4\biggr] - \zeta^2(1-e^2)^{-1} \biggl[ A_{\ell s}a_\ell^2 \zeta^2 \chi^2\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ 1 - \frac{\chi^2}{2} - \zeta^2(1-e^2)^{-1} \biggr] \biggl[ A_{\ell s}a_\ell^2 \zeta^2 \chi^2\biggr] \, . </math> </td> </tr> </table> While examination of the vertical component leads to the following suggested expression for the pressure: <table border="0" align="center" cellpadding="8"> <tr> <td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] \biggl[2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[1 - \frac{\chi^2}{2} - \zeta^2(1-e^2)^{-1} \biggr] \biggl[2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr] - \frac{\chi^2}{2}\biggl[2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr] </math> </td> </tr> </table> ===Tentative Summary=== ====Known Relations==== <table border="0" cellpadding="5" align="center"> <tr> <td align="left"><font color="orange"><b>Density:</b></font></td> <td align="right"> <math>\frac{\rho(\varpi, z)}{\rho_c}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] \, ,</math> </td> </tr> <tr> <td align="left"><font color="orange"><b>Gravitational Potential:</b></font></td> <td align="right"> <math>\frac{ \Phi_\mathrm{grav}(\varpi,z)}{(-\pi G\rho_c a_\ell^2)} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{2} I_\mathrm{BT} - A_\ell \chi^2 - A_s \zeta^2 + \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4 + 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2 + (A_{\ell \ell} a_\ell^2) \chi^4 \biggr] \, . </math> </td> </tr> <tr> <td align="left"><font color="orange"><b>Specific Angular Momentum:</b></font></td> <td align="right"> <math> \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2A_\ell \chi - 2 A_{\ell \ell} a_\ell^2 \chi^3 \, . </math> </td> </tr> <tr> <td align="left"><font color="orange"><b>Centrifugal Potential:</b></font></td> <td align="right"> <math> \frac{\Psi }{(\pi G \rho_c a_\ell^2)} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{2}\biggl[ A_{\ell \ell}a_\ell^2 \chi^4 - 2A_\ell \chi^2 \biggr]\, . </math> </td> </tr> <tr> <td align="left"><font color="orange"><b>Enthalpy:</b></font></td> <td align="right"> <math>\biggl[ \frac{H(\chi, \zeta) - C_B}{(\pi G\rho_c a_\ell^2)} \biggr] - \frac{1}{2} I_\mathrm{BT} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - A_s \zeta^2 + \frac{\zeta^2}{2} \biggl[(A_{s s} a_\ell^2) (\zeta^2 - 3\chi^2) + 2(1-e^2)^{-1}\chi^2 \biggr] \, . </math> </td> </tr> <tr> <td align="left"><font color="orange"><b>Vertical Pressure Gradient:</b></font></td> <td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\rho}{\rho_c} \cdot \biggl[ 2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr] </math> </td> </tr> <tr> <td align="left"><font color="orange"><b>Radial Pressure Gradient:</b></font></td> <td align="right"> <math> \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\rho}{\rho_c} \cdot \biggl\{ 2A_{\ell s}a_\ell^2 \zeta^2 \chi \biggr\} </math> </td> </tr> </table> where, <math>\chi \equiv \varpi/a_\ell</math> and <math>\zeta \equiv z/a_\ell</math>, and the relevant index symbol expressions are: <table align="center" border=0 cellpadding="3"> <tr> <td align="right"><math>I_\mathrm{BT}</math> </td> <td align="center"><math>=</math> </td> <td align="left"> <math> 2A_\ell + A_s (1-e^2) = 2 (1-e^2)^{1/2} \biggl[ \frac{\sin^{-1}e}{e} \biggr] \, ; </math> </td> </tr> <tr> <td align="right"> <math> A_\ell </math> </td> <td align="center"> <math> = </math> </td> <td align="left"> <math> \frac{1}{e^2} \biggl[ \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1/2} \, ; </math> </td> </tr> <tr> <td align="right"><math>A_s</math> </td> <td align="center"><math>=</math> </td> <td align="left"> <math> \frac{2}{e^2} \biggl[ (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1 / 2} \, ; </math> </td> </tr> <tr> <td align="right"> <math> a_\ell^2 A_{\ell \ell} </math> </td> <td align="center"> <math> = </math> </td> <td align="left"> <math> \frac{1}{4e^4}\biggl\{- (3 + 2e^2) (1-e^2)+3 (1 - e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr] \biggr\} \, ; </math> </td> </tr> <tr> <td align="right"> <math>\frac{3}{2} a_\ell^2 A_{ss} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{( 4e^2 - 3 )}{e^4(1-e^2)} + \frac{3 (1-e^2)^{1 / 2}}{e^4} \biggl[\frac{\sin^{-1}e}{e}\biggr] \, ; </math> </td> </tr> <tr> <td align="right"> <math> a_\ell^2 A_{\ell s} </math> </td> <td align="center"> <math> = </math> </td> <td align="left"> <math> \frac{1}{ e^4} \biggl\{ (3-e^2) - 3 (1-e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr] \biggr\} \, , </math> </td> </tr> </table> where the eccentricity, <div align="center"> <math> e \equiv \biggl[1 - \biggl(\frac{a_s}{a_\ell}\biggr)^2 \biggr]^{1 / 2} \, . </math> </div> ====Examine Behavior of Enthalpy==== <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\xi_1</math> </td> <td align="center"> <math>\equiv</math> </td> <td align="left"> <math> \biggl[ z^2 + \biggl(\frac{\varpi}{q}\biggr)^2\biggr]^{1 / 2} = a_s\biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_s}\biggr)^2 \biggr]^{1 / 2} = a_s\biggl[\chi^2 + \zeta^2 (1-e^2)^{-1}\biggr]^{1 / 2} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \frac{\rho}{\rho_c}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>\biggl[ 1 - \biggl(\frac{\xi_1}{a_s}\biggr)^2 \biggr] \, .</math> </td> </tr> </table> ====Try to Construct Pressure Distribution==== Drawing from the expression for the vertical pressure gradient, namely, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{\rho}{\rho_c} \cdot \biggl[ 2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]\biggl[ 2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr] - \chi^2 \biggl[2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr] - \zeta^2(1-e^2)^{-1} \biggl[2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr] + \biggl[-2A_{\ell s}a_\ell^2 \chi^4\zeta + 2A_s \chi^2\zeta - 2A_{ss} a_\ell^2\chi^2 \zeta^3 \biggr] + \biggl[-2A_{\ell s}a_\ell^2 \chi^2\zeta^3(1-e^2)^{-1} + 2A_s \zeta^3(1-e^2)^{-1} - 2A_{ss} a_\ell^2 \zeta^5(1-e^2)^{-1} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[2A_{\ell s}a_\ell^2 \chi^2 - 2A_s -2A_{\ell s}a_\ell^2 \chi^4 + 2A_s \chi^2 \biggr]\zeta + \biggl[ - 2A_{ss} a_\ell^2\chi^2 + 2A_{ss} a_\ell^2 -2A_{\ell s}a_\ell^2 \chi^2(1-e^2)^{-1} + 2A_s (1-e^2)^{-1} \biggr]\zeta^3 + \biggl[ - 2A_{ss} a_\ell^2 (1-e^2)^{-1} \biggr]\zeta^5 \, . </math> </td> </tr> </table> try the following pressure expression: <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{P}{(\pi G\rho_c^2 a_\ell^2)} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> f_0 + f_2 \biggl(\frac{\xi_1}{a_s} \biggr)^2 + f_4 \biggl(\frac{\xi_1}{a_s} \biggr)^4 + f_6 \biggl(\frac{\xi_1}{a_s} \biggr)^6 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> f_0 + f_2 \biggl[\chi^2 + \zeta^2 (1-e^2)^{-1}\biggr] + f_4 \biggl[\chi^2 + \zeta^2 (1-e^2)^{-1}\biggr]^2 + f_6 \biggl[\chi^2 + \zeta^2 (1-e^2)^{-1}\biggr]^3 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> f_0 + f_2 \biggl[\chi^2 + \zeta^2 (1-e^2)^{-1}\biggr] + f_4 \biggl[\chi^4 + 2\chi^2\zeta^2 (1-e^2)^{-1} + \zeta^4(1-e^2)^{-2}\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + f_6 \biggl[\chi^4 + 2\chi^2\zeta^2 (1-e^2)^{-1} + \zeta^4(1-e^2)^{-2}\biggr] \biggl[\chi^2 + \zeta^2 (1-e^2)^{-1}\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> f_0 + f_2 \biggl[\chi^2 + \zeta^2 (1-e^2)^{-1}\biggr] + f_4 \biggl[\chi^4 + 2\chi^2\zeta^2 (1-e^2)^{-1} + \zeta^4(1-e^2)^{-2}\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + f_6 \biggl[\chi^6 + 3\chi^4\zeta^2 (1-e^2)^{-1} + 3\chi^2\zeta^4(1-e^2)^{-2} + \zeta^6(1-e^2)^{-3} \biggr] \, . </math> </td> </tr> </table> The vertical derivative of this expression is, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\biggl[ \frac{1}{(\pi G\rho_c^2 a_\ell^2)}\biggr] \frac{\partial P}{\partial \zeta} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{\partial }{\partial \zeta}\biggl\{ f_2 \biggl[\zeta^2 (1-e^2)^{-1}\biggr] + f_4 \biggl[2\chi^2\zeta^2 (1-e^2)^{-1} + \zeta^4(1-e^2)^{-2}\biggr] + f_6 \biggl[3\chi^4\zeta^2 (1-e^2)^{-1} + 3\chi^2\zeta^4(1-e^2)^{-2} + \zeta^6(1-e^2)^{-3} \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl\{ f_2 \biggl[2\zeta (1-e^2)^{-1}\biggr] + f_4 \biggl[4\chi^2\zeta (1-e^2)^{-1} + 4\zeta^3(1-e^2)^{-2}\biggr] + f_6 \biggl[6\chi^4\zeta (1-e^2)^{-1} + 12\chi^2\zeta^3(1-e^2)^{-2} + 6\zeta^5(1-e^2)^{-3} \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl\{ \biggl[2f_2 (1-e^2)^{-1} + 4f_4\chi^2 (1-e^2)^{-1} + 6f_6\chi^4 (1-e^2)^{-1} \biggr]\zeta + \biggl[ 4f_4 (1-e^2)^{-2} + 12f_6\chi^2(1-e^2)^{-2}\biggr]\zeta^3 + \biggl[6f_6 (1-e^2)^{-3} \biggr]\zeta^5 \biggr\} \, . </math> </td> </tr> </table> Matching <math>\zeta^5</math> terms gives, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>6f_6 (1-e^2)^{-3} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - 2A_{ss} a_\ell^2 (1-e^2)^{-1} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ f_6 </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - \frac{1}{3}A_{ss} a_\ell^2 (1-e^2)^{2} \, . </math> </td> </tr> </table> Matching <math>\zeta^3</math> terms gives, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>4f_4 (1-e^2)^{-2} + 12f_6\chi^2(1-e^2)^{-2} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - 2A_{ss} a_\ell^2\chi^2 + 2A_{ss} a_\ell^2 -2A_{\ell s}a_\ell^2 \chi^2(1-e^2)^{-1} + 2A_s (1-e^2)^{-1} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ 4f_4 (1-e^2)^{-2} + 12 \biggl[- \frac{1}{3}A_{ss} a_\ell^2 (1-e^2)^{2} \biggr] \chi^2(1-e^2)^{-2} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> [2A_{ss} a_\ell^2 + 2A_s (1-e^2)^{-1}] - 2A_{ss} a_\ell^2\chi^2 -2A_{\ell s}a_\ell^2 \chi^2(1-e^2)^{-1} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ 4f_4 (1-e^2)^{-2} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> [2A_{ss} a_\ell^2 + 2A_s (1-e^2)^{-1}] + \biggl[2A_{ss} a_\ell^2 -2A_{\ell s}a_\ell^2 (1-e^2)^{-1} \biggr] \chi^2 \, .</math> </td> </tr> </table> Matching <math>\zeta^1</math> terms gives, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>2f_2 (1-e^2)^{-1} + 4f_4\chi^2 (1-e^2)^{-1} + 6f_6\chi^4 (1-e^2)^{-1} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 2A_{\ell s}a_\ell^2 \chi^2 - 2A_s -2A_{\ell s}a_\ell^2 \chi^4 + 2A_s \chi^2 </math> </td> </tr> </table> ===5<sup>th</sup> Try=== We should leave untouched the ''form'' of the expression for the centrifugal potential, but let its coefficient values remain unspecified. The enthalpy function will therefore remain flexible, and, in tern, so will the components of the pressure gradient. We should adjust these new coefficients in such a way that the gradient of the pressure is everywhere perpendicular to the surface of a constant-density contour; this means that the P-constant contours will be identical to the density-constant contours. ====Modifiable Relations==== <table border="0" cellpadding="5" align="center"> <tr> <td align="left"><font color="orange"><b>Density:</b></font></td> <td align="right"> <math>\frac{\rho(\varpi, z)}{\rho_c}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] \, ,</math> </td> </tr> <tr> <td align="left"><font color="orange"><b>Gravitational Potential:</b></font></td> <td align="right"> <math>\frac{ \Phi_\mathrm{grav}(\varpi,z)}{(-\pi G\rho_c a_\ell^2)} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{2} I_\mathrm{BT} - A_\ell \chi^2 - A_s \zeta^2 + \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4 + 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2 + (A_{\ell \ell} a_\ell^2) \chi^4 \biggr] \, . </math> </td> </tr> <tr> <td align="left"><font color="purple"><b>Specific Angular Momentum:</b></font></td> <td align="right"> <math> \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2j_1 \chi - 2 j_3 \chi^3 \, . </math> </td> </tr> <tr> <td align="left"><font color="purple"><b>Centrifugal Potential:</b></font></td> <td align="right"> <math> \frac{\Psi }{(\pi G \rho_c a_\ell^2)} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{2}\biggl[j_3 \chi^4 -2j_1 \chi^2 \biggr]\, . </math> </td> </tr> <tr> <td align="left"><font color="purple"><b>Enthalpy:</b></font></td> <td align="right"> <math>\biggl[ \frac{H(\chi, \zeta) - C_B}{(\pi G\rho_c a_\ell^2)} \biggr] - \frac{1}{2} I_\mathrm{BT} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - A_\ell \chi^2 - A_s \zeta^2 + \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4 + 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2 + (A_{\ell \ell} a_\ell^2) \chi^4 \biggr] - \frac{1}{2}\biggl[j_3 \chi^4 -2j_1 \chi^2 \biggr] </math> </td> </tr> <tr> <td align="left"><font color="orange"><b>Vertical Pressure Gradient:</b></font></td> <td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\rho}{\rho_c} \cdot \biggl[ 2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr] </math> </td> </tr> <tr> <td align="left"><font color="purple"><b>Radial Pressure Gradient:</b></font></td> <td align="right"> <math> \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\rho}{\rho_c} \cdot \biggl\{ \biggl[ 2j_1 - 2A_\ell + 2A_{\ell s}a_\ell^2 \zeta^2 \biggr] \chi + \biggl[ 2A_{\ell \ell} a_\ell^2 - 2j_3 \biggr]\chi^3 \biggr\} </math> </td> </tr> </table> where, <math>\chi \equiv \varpi/a_\ell</math> and <math>\zeta \equiv z/a_\ell</math>, and the relevant index symbol expressions are: ====Desired Slopes of Normal Vectors==== A vector that is normal to the surface of a constant-density (oblate-spheroidal) contour has the following components: <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{\partial}{\partial \chi}\biggl[\frac{\rho(\varpi, z)}{\rho_c} \biggr]</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{\partial}{\partial \chi}\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] = -2\chi \, ;</math> </td> </tr> <tr> <td align="right"> <math>\frac{\partial}{\partial \zeta}\biggl[\frac{\rho(\varpi, z)}{\rho_c} \biggr]</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{\partial}{\partial \zeta}\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] = -2\zeta (1-e^2)^{-1} \, .</math> </td> </tr> </table> Hence, the slope, <math>m</math>, of this normal vector is, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>m = \biggl\{\frac{\partial}{\partial \zeta}\biggl[\frac{\rho(\varpi, z)}{\rho_c} \biggr]\biggr\} \biggl\{\frac{\partial}{\partial \chi}\biggl[\frac{\rho(\varpi, z)}{\rho_c} \biggr]\biggr\}^{-1} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{ -2\zeta (1-e^2)^{-1}}{-2\chi} = \frac{\zeta}{\chi(1-e^2)} \, . </math> </td> </tr> </table> Now, if the constant-pressure contours are to lie precisely on top of our constant-density contours, the normals have to have the same slopes. This means that, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{\partial P}{\partial \zeta} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{\zeta}{\chi(1-e^2)} \biggl[\frac{\partial P}{\partial \chi}\biggr] </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \chi(1-e^2)\biggl\{ 2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr\} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \zeta \biggl\{ \biggl[ 2j_1 - 2A_\ell + 2A_{\ell s}a_\ell^2 \zeta^2 \biggr] \chi + \biggl[ 2A_{\ell \ell} a_\ell^2 - 2j_3 \biggr]\chi^3 \biggr\} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ 2A_{\ell s}a_\ell^2 (1-e^2) \chi^3\zeta - 2A_s(1-e^2) \chi\zeta + 2A_{ss} a_\ell^2 (1-e^2)\chi \zeta^3 </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> ( 2j_1 - 2A_\ell ) \chi \zeta + 2A_{\ell s}a_\ell^2 \chi \zeta^3 + (2A_{\ell \ell} a_\ell^2 - 2j_3 )\chi^3 \zeta </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \biggl[ 2A_{\ell s}a_\ell^2 (1-e^2) - (2A_{\ell \ell} a_\ell^2 - 2j_3 ) \biggr] \chi^3 \zeta + \biggl[- 2A_s(1-e^2) - ( 2j_1 - 2A_\ell ) \biggr]\chi\zeta </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[ 2A_{\ell s}a_\ell^2 - 2A_{ss} a_\ell^2 (1-e^2) \biggr] \chi \zeta^3 </math> </td> </tr> </table> <table border="1" width="80%" align="center" cellpadding="8"><tr><td align="left"> Note … <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>3A_{ss} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{2}{a_s^2} - 2A_{\ell s} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{2}{a_\ell^2(1-e^2)} - 2A_{\ell s} </math> </td> </tr> <tr> <td align="right"> <math> \Rightarrow ~~~ 3(1-e^2) (A_{ss} a_\ell^2) </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 2 - 2(1-e^2) (A_{\ell s}a_\ell^2) </math> </td> </tr> <tr> <td align="right"> <math> \Rightarrow ~~~ \mathrm{RHS} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl\{ 2A_{\ell s}a_\ell^2 - 2A_{ss} a_\ell^2 (1-e^2) \biggr\} \chi \zeta^3 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl\{ 2A_{\ell s}a_\ell^2 - \frac{2}{3}\biggl[ 2 - 2(1-e^2) (A_{\ell s}a_\ell^2)\biggr] \biggr\} \chi \zeta^3 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl\{ \biggl[2 + \frac{4}{3}(1-e^2)\biggr] (A_{\ell s}a_\ell^2) -\frac{4}{3} \biggr\} \chi \zeta^3 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{2}{3}\biggl[ (5-2e^2) (A_{\ell s}a_\ell^2) - 2 \biggr] \chi \zeta^3 \, . </math> </td> </tr> </table> </td></tr></table> In order for the <math>\chi^3\zeta</math> term on the LHS to be zero, we should set … <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>0 </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[ 2A_{\ell s}a_\ell^2 (1-e^2) - 2A_{\ell \ell} a_\ell^2 + 2j_3 \biggr] </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ j_3 </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> A_{\ell \ell} a_\ell^2 - A_{\ell s}a_\ell^2 (1-e^2) \, ; </math> </td> </tr> </table> and in order for the <math>\chi\zeta</math> term on the LHS to be zero, we should set … <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>0 </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[- 2A_s(1-e^2) - ( 2j_1 - 2A_\ell ) \biggr] </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ j_1 </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> A_\ell - A_s(1-e^2) \, . </math> </td> </tr> </table> ====Desired Slopes of Tangent Vectors==== Alternatively, if the constant-pressure contours are to lie precisely on top of our constant-density contours, the tangent vectors have to have slopes given by <math>-1/m</math>. This means that, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{\partial P}{\partial \zeta} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - \frac{1}{m}\biggl[\frac{\partial P}{\partial \chi}\biggr] = -\frac{\chi(1-e^2)}{\zeta} \biggl[\frac{\partial P}{\partial \chi}\biggr] </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \zeta \biggl\{ 2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr\} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> -\chi(1-e^2) \biggl\{ \biggl[ 2j_1 - 2A_\ell + 2A_{\ell s}a_\ell^2 \zeta^2 \biggr] \chi + \biggl[ 2A_{\ell \ell} a_\ell^2 - 2j_3 \biggr]\chi^3 \biggr\} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \biggl\{ \biggl[ A_{\ell s}a_\ell^2 \chi^2 - A_s \biggr]\zeta^2 + A_{ss} a_\ell^2 \zeta^4 \biggr\} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - (1-e^2) \biggl\{ \biggl[ j_1 - A_\ell + A_{\ell s}a_\ell^2 \zeta^2 \biggr] \chi^2 + \biggl[ A_{\ell \ell} a_\ell^2 - j_3 \biggr]\chi^4 \biggr\} </math> </td> </tr> </table> ===6<sup>th</sup> Try=== ====Euler Equation==== From, for example, [[PGE/Euler#in_terms_of_velocity:_2|here]] we can write the, <div align="center"> <span id="ConservingMomentum:Eulerian"><font color="#770000">'''Eulerian Representation'''</font></span><br /> of the Euler Equation, {{Template:Math/EQ_Euler02}} </div> In steady-state, we should set <math>\partial\vec{v}/\partial t = 0</math>. There are various ways of expressing the nonlinear term on the LHS; from [[PGE/Euler#in_terms_of_the_vorticity:|here]], for example, we find, <div align="center"> <math> (\vec{v}\cdot\nabla)\vec{v} = \frac{1}{2}\nabla(\vec{v}\cdot\vec{v}) - \vec{v}\times(\nabla\times\vec{v}) = \frac{1}{2}\nabla(v^2) + \vec{\zeta}\times \vec{v} , </math> </div> where, <div align="center"> <math> \vec\zeta \equiv \nabla\times\vec{v} </math> </div> is commonly referred to as the [https://en.wikipedia.org/wiki/Vorticity vorticity]. ====Axisymmetric Configurations==== From, for example, [[AxisymmetricConfigurations/PGE#CYLconvectiveOperator|here]], we appreciate that, quite generally, for axisymmetric systems when written in cylindrical coordinates, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math> (\vec{v} \cdot \nabla )\vec{v} </math> </td> <td align="center"> = </td> <td align="left"> <math> \hat{e}_\varpi \biggl[ v_\varpi \frac{\partial v_\varpi}{\partial\varpi} + v_z \frac{\partial v_\varpi}{\partial z} - \frac{v_\varphi v_\varphi}{\varpi} \biggr] + \hat{e}_\varphi \biggl[ v_\varpi \frac{\partial v_\varphi}{\partial \varpi} + v_z \frac{\partial v_\varphi}{\partial z} + \frac{v_\varphi v_\varpi}{\varpi} \biggr] + \hat{e}_z \biggl[ v_\varpi \frac{\partial v_z}{\partial\varpi} + v_z \frac{\partial v_z}{\partial z} \biggr] \, . </math> </td> </tr> </table> We seek steady-state configurations for which <math>v_\varpi =0</math> and <math>v_z = 0</math>, in which case this expression simplifies considerably to, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math> (\vec{v} \cdot \nabla )\vec{v} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \hat{e}_\varpi \biggl[ - \frac{v_\varphi v_\varphi}{\varpi} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \hat{e}_\varpi \biggl[ - \frac{j^2}{\varpi^3} \biggr] \, , </math> </td> </tr> </table> where, in this last expression we have replaced <math>v_\varphi</math> with the specific angular momentum, <math>j \equiv \varpi v_\varphi = (\varpi^2 \dot\varphi)</math>, which is a [[AxisymmetricConfigurations/PGE#Conservation_of_Specific_Angular_Momentum_(CYL.)|conserved quantity in dynamically evolving systems]]. NOTE: Up to this point in our discussion, <math>j</math> can be a function of both coordinates, that is, <math>j = j(\varpi, z)</math>. As has been highlighted [[AxisymmetricConfigurations/PGE#RelevantCylindricalComponents|here]] for example — for the axisymmetric configurations under consideration — the <math>\hat{e}_\varpi</math> and <math>\hat{e}_z</math> components of the Euler equation become, respectively,</span> <table border="1" align="center" cellpadding="10"><tr><td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>{\hat{e}}_\varpi</math>: </td> <td align="right"> <math> - \frac{j^2}{\varpi^3} </math> </td> <td align="center"> = </td> <td align="left"> <math> - \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] </math> </td> </tr> <tr> <td align="right"><math>{\hat{e}}_z</math>: </td> <td align="right"> <math> 0 </math> </td> <td align="center"> = </td> <td align="left"> <math> - \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] </math> </td> </tr> </table> </td></tr></table> ====Strategy==== <font color="red">STEP 1:</font> For the problem being tackled here, we start by recognizing that when considering hydrostatic balance in the <math>\hat{e}_z</math> direction, we have analytically known expressions for both <math>\rho(\varpi, z)</math> and <math>\partial\Phi/\partial z</math>. This means, therefore, that we can construct an analytical expression for the vertical component of the pressure gradient, specifically, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math> \frac{\partial P}{\partial z} </math> </td> <td align="center"> = </td> <td align="left"> <math> - \rho \cdot \frac{\partial \Phi}{\partial z} \, . </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P}{\partial \zeta}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial \zeta} \biggl\{ \frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)} \biggr\} </math> </td> </tr> </table> <font color="red">STEP 2:</font> Because we want the meridional-plane, constant-pressure contours to align with the meridional-plane, constant density contours, we can determine the radial component of the pressure gradient by forcing the slope of the tangent vector to match the tangent vector of the density contour. <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{\partial P}{\partial \zeta} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - \frac{1}{m}\biggl[\frac{\partial P}{\partial \chi}\biggr] = -\frac{\chi(1-e^2)}{\zeta} \biggl[\frac{\partial P}{\partial \chi}\biggr] </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P}{\partial \chi} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> -\frac{\zeta}{\chi(1-e^2)} \biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P}{\partial \zeta}\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> -\frac{\zeta}{\chi(1-e^2)} \biggl\{ \frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial \zeta} \biggl[ \frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)} \biggr] \biggr\} \, . </math> </td> </tr> </table> <font color="red">STEP 3:</font> Via the radial component of the hydrostatic balance expression, we can determine analytically the distribution of specific angular momentum. <table border="0" align="center" cellpadding="8"> <tr> <td align="right"> <math> \frac{j^2}{\varpi^3} </math> </td> <td align="center"> = </td> <td align="left"> <math> \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl(\frac{\rho}{\rho_c}\biggr)^{-1} \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi} - \frac{\partial}{\partial \chi} \biggl\{ \frac{\Phi_\mathrm{grav}}{(-\pi G \rho_c a_\ell^2)} \biggr\} </math> </td> </tr> </table> <font color="red">STEP 4:</font> From knowledge of both components of <math>\nabla P</math>, see if the expression for the pressure can be ascertained. ====Implication==== Hence, <table border="0" align="center" cellpadding="8"> <tr> <td align="right"> <math> \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> -\frac{\zeta}{\chi(1-e^2)} \cdot \frac{\partial}{\partial \zeta} \biggl[ \frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)} \biggr] - \frac{\partial}{\partial \chi} \biggl\{ \frac{\Phi_\mathrm{grav}}{(-\pi G \rho_c a_\ell^2)} \biggr\} \, . </math> </td> </tr> </table> Now, given that, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{ \Phi_\mathrm{grav}(\varpi,z)}{(-\pi G\rho_c a_\ell^2)} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{2} I_\mathrm{BT} - A_\ell \chi^2 - A_s \zeta^2 + \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4 + 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2 + (A_{\ell \ell} a_\ell^2) \chi^4 \biggr] \, , </math> </td> </tr> </table> we see that the pair of partial derivative expressions are: <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{\partial}{\partial \zeta} \biggl[ \frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)} \biggr] </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 2\biggl[(A_{s s} a_\ell^2) \zeta^3 + (A_{\ell s}a_\ell^2 )\chi^2 \zeta - A_s \zeta \biggr] \, ; </math> </td> <tr> <td align="right"> <math>\frac{\partial}{\partial \chi} \biggl[ \frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)} \biggr] </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 2\biggl[ (A_{\ell \ell} a_\ell^2) \chi^3 + (A_{\ell s}a_\ell^2 )\chi \zeta^2 - A_\ell \chi\biggr] \, . </math> </td> </tr> </table> As a result we find, <table border="0" align="center" cellpadding="8"> <tr> <td align="right"> <math> \frac{j^2 (1-e^2)}{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^2} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> -2\zeta \biggl[(A_{s s} a_\ell^2) \zeta^3 + (A_{\ell s}a_\ell^2 )\chi^2 \zeta - A_s \zeta \biggr] - 2\chi(1-e^2) \biggl[ (A_{\ell \ell} a_\ell^2) \chi^3 + (A_{\ell s}a_\ell^2 )\chi \zeta^2 - A_\ell \chi\biggr] </math> </td> </tr> <tr> <td align="right"> <math> \Rightarrow ~~~ \frac{j^2 (1-e^2)}{(2\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^2} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[-(A_{s s} a_\ell^2) \zeta^4 - (A_{\ell s}a_\ell^2 )\chi^2 \zeta^2 + A_s \zeta^2 \biggr] + (1-e^2) \biggl[ -(A_{\ell \ell} a_\ell^2) \chi^4 - (A_{\ell s}a_\ell^2 )\chi^2 \zeta^2 + A_\ell \chi^2\biggr] </math> </td> </tr> </table> Next, regarding <font color="red">STEP 4</font>, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P}{\partial \zeta}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial \zeta} \biggl\{ \frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] \biggl[(A_{s s} a_\ell^2) \zeta^3 + (A_{\ell s}a_\ell^2 )\chi^2 \zeta - A_s \zeta \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2\biggl[(A_{s s} a_\ell^2) \zeta^3 + (A_{\ell s}a_\ell^2 )\chi^2 \zeta - A_s \zeta \biggr] - 2\biggl[(A_{s s} a_\ell^2) \chi^2 \zeta^3 + (A_{\ell s}a_\ell^2 )\chi^4 \zeta - A_s \chi^2\zeta \biggr] - 2\biggl[(A_{s s} a_\ell^2) \zeta^5(1-e^2)^{-1} + (A_{\ell s}a_\ell^2 )\chi^2 \zeta^3(1-e^2)^{-1} - A_s \zeta^3(1-e^2)^{-1} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2 \biggl\{ (A_{s s} a_\ell^2) \zeta^3 + (A_{\ell s}a_\ell^2 )\chi^2 \zeta - A_s \zeta -(A_{s s} a_\ell^2) \chi^2 \zeta^3 - (A_{\ell s}a_\ell^2 )\chi^4 \zeta + A_s \chi^2\zeta -(A_{s s} a_\ell^2) \zeta^5(1-e^2)^{-1} - (A_{\ell s}a_\ell^2 )\chi^2 \zeta^3(1-e^2)^{-1} + A_s \zeta^3(1-e^2)^{-1} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2 \biggl\{ \biggl[ (A_{\ell s}a_\ell^2 )\chi^2 - A_s - (A_{\ell s}a_\ell^2 )\chi^4 + A_s \chi^2\biggr]\zeta + \biggl[ (A_{s s} a_\ell^2) -(A_{s s} a_\ell^2) \chi^2 - (A_{\ell s}a_\ell^2 )\chi^2 (1-e^2)^{-1} + A_s (1-e^2)^{-1}\biggr] \zeta^3 + \biggl[-(A_{s s} a_\ell^2) (1-e^2)^{-1} \biggr]\zeta^5 \biggr\} </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ \frac{12 P}{(2\pi G\rho_c^2 a_\ell^2)} </math></td> <td align="center"><math>\sim</math></td> <td align="left"> <math> 6\biggl[ (A_{\ell s}a_\ell^2 )\chi^2 - A_s - (A_{\ell s}a_\ell^2 )\chi^4 + A_s \chi^2\biggr]\zeta^2 + 3\biggl[ (A_{s s} a_\ell^2) -(A_{s s} a_\ell^2) \chi^2 - (A_{\ell s}a_\ell^2 )\chi^2 (1-e^2)^{-1} + A_s (1-e^2)^{-1}\biggr] \zeta^4 + 2\biggl[-(A_{s s} a_\ell^2) (1-e^2)^{-1} \biggr]\zeta^6 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 6(A_{\ell s}a_\ell^2 )\zeta^2\chi^2 - 6A_s\zeta^2 - 6(A_{\ell s}a_\ell^2 )\zeta^2\chi^4 + 6A_s \zeta^2 \chi^2 + 3(A_{s s} a_\ell^2)\zeta^4 -3(A_{s s} a_\ell^2) \zeta^4\chi^2 - 3(A_{\ell s}a_\ell^2 )\zeta^4\chi^2 (1-e^2)^{-1} + 3A_s \zeta^4(1-e^2)^{-1} - 2(A_{s s} a_\ell^2) \zeta^6(1-e^2)^{-1} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> + \biggl[ 3(A_{s s} a_\ell^2)\zeta^4 + 3A_s \zeta^4(1-e^2)^{-1} - 6A_s\zeta^2 - 2(A_{s s} a_\ell^2) \zeta^6(1-e^2)^{-1}\biggr] + \biggl[ 6(A_{\ell s}a_\ell^2 )\zeta^2 + 6A_s \zeta^2 - 3(A_{s s} a_\ell^2) \zeta^4 - 3(A_{\ell s}a_\ell^2 )\zeta^4 (1-e^2)^{-1}\biggr]\chi^2 + \biggl[ - 6(A_{\ell s}a_\ell^2 )\zeta^2 \biggr] \chi^4 \, ; </math> </td> </tr> </table> and, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P}{\partial \chi} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> -\frac{\zeta}{\chi(1-e^2)} \biggl\{ \frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial \zeta} \biggl\{ \frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)} \biggr\} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> -\frac{\zeta}{\chi(1-e^2)} \biggl\{ 2\biggl[(A_{s s} a_\ell^2) \zeta^3 + (A_{\ell s}a_\ell^2 )\chi^2 \zeta - A_s \zeta \biggr] - 2\biggl[(A_{s s} a_\ell^2) \chi^2 \zeta^3 + (A_{\ell s}a_\ell^2 )\chi^4 \zeta - A_s \chi^2\zeta \biggr] - 2\biggl[(A_{s s} a_\ell^2) \zeta^5(1-e^2)^{-1} + (A_{\ell s}a_\ell^2 )\chi^2 \zeta^3(1-e^2)^{-1} - A_s \zeta^3(1-e^2)^{-1} \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> -\frac{1}{(1-e^2)} \biggl\{ 2\biggl[(A_{s s} a_\ell^2) \chi^{-1}\zeta^4 + (A_{\ell s}a_\ell^2 )\chi \zeta^2 - A_s \chi^{-1}\zeta^2 \biggr] - 2\biggl[(A_{s s} a_\ell^2) \chi \zeta^4 + (A_{\ell s}a_\ell^2 )\chi^3 \zeta^2 - A_s \chi\zeta^2 \biggr] - 2\biggl[(A_{s s} a_\ell^2) \chi^{-1}\zeta^6(1-e^2)^{-1} + (A_{\ell s}a_\ell^2 )\chi \zeta^4(1-e^2)^{-1} - A_s \chi^{-1}\zeta^4(1-e^2)^{-1} \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{2}{(1-e^2)} \biggl\{ -(A_{s s} a_\ell^2) \chi^{-1}\zeta^4 - (A_{\ell s}a_\ell^2 )\chi \zeta^2 + A_s \chi^{-1}\zeta^2 + (A_{s s} a_\ell^2) \chi \zeta^4 + (A_{\ell s}a_\ell^2 )\chi^3 \zeta^2 - A_s \chi\zeta^2 + (A_{s s} a_\ell^2) \chi^{-1}\zeta^6(1-e^2)^{-1} + (A_{\ell s}a_\ell^2 )\chi \zeta^4(1-e^2)^{-1} - A_s \chi^{-1}\zeta^4(1-e^2)^{-1} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{2}{(1-e^2)} \biggl\{ \biggl[A_s \zeta^2 -(A_{s s} a_\ell^2) \zeta^4 - A_s \zeta^4(1-e^2)^{-1} + (A_{s s} a_\ell^2) \zeta^6(1-e^2)^{-1} \biggr]\chi^{-1} + \biggl[ - (A_{\ell s}a_\ell^2 )\zeta^2 - A_s \zeta^2 +(A_{s s} a_\ell^2) \zeta^4 + (A_{\ell s}a_\ell^2 )\zeta^4(1-e^2)^{-1} \biggr]\chi + \biggl[(A_{\ell s}a_\ell^2 )\zeta^2 \biggr]\chi^3 \biggr\} </math> </td> </tr> <tr> <td align="right"> <math> \Rightarrow ~~~ \frac{(1-e^2)P}{(2\pi G\rho_c^2 a_\ell^2)} </math> </td> <td align="center"> <math>\sim</math> </td> <td align="left"> <math> \biggl[A_s \zeta^2 -(A_{s s} a_\ell^2) \zeta^4 - A_s \zeta^4(1-e^2)^{-1} + (A_{s s} a_\ell^2) \zeta^6(1-e^2)^{-1} \biggr]\ln(\chi) + \frac{1}{2}\biggl[- (A_{\ell s}a_\ell^2 )\zeta^2- A_s \zeta^2+(A_{s s} a_\ell^2) \zeta^4+ (A_{\ell s}a_\ell^2 )\zeta^4(1-e^2)^{-1}\biggr]\chi^2 + \frac{1}{4}\biggl[(A_{\ell s}a_\ell^2 )\zeta^2 \biggr]\chi^4 \, . </math> </td> </tr> </table> ===7<sup>th</sup> Try=== ====Introduction==== <table border="0" cellpadding="5" align="center"> <tr> <td align="left"><font color="orange"><b>Density:</b></font></td> <td align="right"> <math>\frac{\rho(\chi, \zeta)}{\rho_c}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] \, ,</math> </td> </tr> <tr> <td align="left"><font color="orange"><b>Gravitational Potential:</b></font></td> <td align="right"> <math>\frac{ \Phi_\mathrm{grav}(\chi,\zeta)}{(-\pi G\rho_c a_\ell^2)} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{2} I_\mathrm{BT} - A_\ell \chi^2 - A_s \zeta^2 + \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4 + 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2 + (A_{\ell \ell} a_\ell^2) \chi^4 \biggr] \, . </math> </td> </tr> <tr> <td align="left"><font color="purple"><b>Specific Angular Momentum:</b></font></td> <td align="right"> <math> \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2j_1 \chi - 2 j_3 \chi^3 \, . </math> </td> </tr> <tr> <td align="left"><font color="purple"><b>Centrifugal Potential:</b></font></td> <td align="right"> <math> \frac{\Psi }{(\pi G \rho_c a_\ell^2)} </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{2}\biggl[j_3 \chi^4 -2j_1 \chi^2 \biggr]\, . </math> </td> </tr> </table> <table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left"> [[#Index_Symbol_Expressions|From above]], we recall the following relations: <table align="center" border=0 cellpadding="3"> <tr> <td align="right"> <math> 4e^4(A_{\ell \ell}a_\ell^2 ) </math> </td> <td align="center"> <math> = </math> </td> <td align="left"> <math> - (3 + 2e^2) (1-e^2) + \Upsilon \, ; </math> </td> </tr> <tr> <td align="right"> <math>\frac{3}{2} e^4(A_{ss}a_\ell^2 ) </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{( 4e^2 - 3 )}{(1-e^2)} + \Upsilon \, ; </math> </td> </tr> <tr> <td align="right"> <math> e^4(A_{\ell s}a_\ell^2 ) </math> </td> <td align="center"> <math> = </math> </td> <td align="left"> <math> (3-e^2) - \Upsilon \, . </math> </td> </tr> </table> where, <table align="center" border=0 cellpadding="3"> <tr> <td align="right"> <math> \Upsilon </math> </td> <td align="center"> <math> \equiv </math> </td> <td align="left"> <math> 3 (1 - e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr] \, . </math> </td> </tr> </table> <font color="red">Crosscheck</font> … Given that, <table align="center" border=0 cellpadding="3"> <tr> <td align="right"> <math> \Upsilon </math> </td> <td align="center"> <math> = </math> </td> <td align="left"> <math> (3-e^2) - e^4(A_{\ell s}a_\ell^2 ) \, . </math> </td> </tr> </table> we obtain the pair of relations, <table align="center" border=0 cellpadding="3"> <tr> <td align="right"> <math> 4e^4(A_{\ell \ell}a_\ell^2 ) </math> </td> <td align="center"> <math> = </math> </td> <td align="left"> <math> - (3 + 2e^2) (1-e^2) + (3-e^2) - e^4(A_{\ell s}a_\ell^2 ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math> = </math> </td> <td align="left"> <math> - (3-3e^2 + 2e^2 - 2e^4) + (3-e^2) - e^4(A_{\ell s}a_\ell^2 ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math> = </math> </td> <td align="left"> <math> 2e^4 - e^4(A_{\ell s}a_\ell^2 ) </math> </td> </tr> <tr> <td align="right"> <math> \Rightarrow ~~~ (A_{\ell \ell}a_\ell^2 ) </math> </td> <td align="center"> <math> = </math> </td> <td align="left"> <math> \frac{1}{2} - \frac{1}{4}(A_{\ell s}a_\ell^2 ) \, ; </math> </td> </tr> <tr> <td align="right"> <math>\frac{3}{2} e^4(A_{ss}a_\ell^2 ) </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{( 4e^2 - 3 )}{(1-e^2)} + (3-e^2) - e^4(A_{\ell s}a_\ell^2 ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{( 4e^2 - 3 )+(3-e^2)(1-e^2)}{(1-e^2)} - e^4(A_{\ell s}a_\ell^2 ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{e^4}{(1-e^2)} - e^4(A_{\ell s}a_\ell^2 ) </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ (A_{ss}a_\ell^2 ) </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{2}{3}\biggl[ \frac{1}{(1-e^2)} - (A_{\ell s}a_\ell^2 )\biggr] \, . </math> </td> </tr> </table> </td></tr></table> ====RHS Square Brackets (TERM1)==== Let's rewrite the term inside square brackets on the RHS of the expression for the gravitational potential. <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\biggl[ ~~ \biggr]_\mathrm{RHS}</math> </td> <td align="center"> <math>\equiv</math> </td> <td align="left"> <math> \biggl[(A_{s s} a_\ell^2) \zeta^4 + 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2 + (A_{\ell \ell} a_\ell^2) \chi^4 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> e^{-4} \biggl\{ \frac{2}{3}\biggl[ \frac{( 4e^2 - 3 )}{(1-e^2)} + \Upsilon\biggr] \zeta^4 + 2\biggl[ (3-e^2) - \Upsilon \biggr]\chi^2 \zeta^2 + \frac{1}{4}\biggl[ - (3 + 2e^2) (1-e^2) + \Upsilon \biggr] \chi^4 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> e^{-4} \biggl\{ \frac{2}{3}\biggl[ \frac{( 4e^2 - 3 )}{(1-e^2)} \biggr] \zeta^4 + 2\biggl[ (3-e^2) \biggr]\chi^2 \zeta^2 + \frac{1}{4}\biggl[ - (3 + 2e^2) (1-e^2) \biggr] \chi^4 + \frac{2}{3}\biggl[ \zeta^4 -3\zeta^2\chi^2 + \frac{3}{8}\chi^4 \biggr]\Upsilon \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - ~e^{-4} \biggl\{ \frac{2}{3}\biggl[ \frac{( 3-4e^2 )}{(1-e^2)} \biggr] \zeta^4 - 2\biggl[ (3-e^2) \biggr]\chi^2 \zeta^2 + \frac{1}{4}\biggl[ (3 + 2e^2) (1-e^2) \biggr] \chi^4 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> +~ e^{-4}\biggl\{ \frac{2}{3}\biggl[ (\zeta^2 - \chi^2)(\zeta^2-2\chi^2) - \frac{13}{8}\chi^4 \biggr]\Upsilon \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - ~e^{-4} \frac{2}{3(1-e^2)}\biggl\{ \biggl[ ( 3-4e^2 ) \biggr] \zeta^4 - 3\biggl[ (3-e^2) \biggr](1-e^2)\chi^2 \zeta^2 + \frac{3}{8}\biggl[ (3 + 2e^2) \biggr] (1-e^2)^2 \chi^4 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> +~ e^{-4}\biggl\{ \frac{2}{3}\biggl[ (\zeta^2 - \chi^2)(\zeta^2-2\chi^2) - \frac{13}{8}\chi^4 \biggr]\Upsilon \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - ~ \frac{2e^{-4}}{(1-e^2)}\biggl\{ \zeta^4 - 3 (1-e^2)\chi^2 \zeta^2 + \frac{3}{8} (1-e^2)^2 \chi^4 \biggr\} + ~ \frac{8e^{-2}}{3(1-e^2)}\biggl\{ \zeta^4 - \frac{3}{4} (1-e^2)\chi^2 \zeta^2 - \frac{3}{16} (1-e^2)^2 \chi^4 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> +~ \frac{2e^{-4}}{3}\biggl[ (\zeta^2 - \chi^2)(\zeta^2-2\chi^2) - \frac{13}{8}\chi^4 \biggr]\Upsilon </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - ~ \frac{2e^{-4}}{(1-e^2)}\biggl\{ \underbrace{ \biggl[\zeta^2 - (1-e^2)\chi^2\biggr]\biggl[ \zeta^2 - 2(1-e^2)\chi^2\biggr] - \frac{13}{8}(1-e^2)^2\chi^4}_{-0.038855} \biggr\} + ~ \frac{8e^{-2}}{3(1-e^2)}\biggl\{ \overbrace{ \biggl[\zeta^2 - (1-e^2)\chi^2\biggr]\biggl[ \zeta^2 + \frac{1}{4}(1-e^2)\chi^2\biggr] + \frac{1}{16}(1-e^2)^2\chi^4}^{-0.010124} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> +~ \frac{2e^{-4}}{3}\biggl[\underbrace{ (\zeta^2 - \chi^2)(\zeta^2-2\chi^2) - \frac{13}{8}\chi^4 }_{-0.061608} \biggr]\Upsilon </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 0.212119014 </math> ([[#Example_Evaluation|example #1]], below) . </td> </tr> </table> Check #1: <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math> (\zeta^2 - \chi^2)(\zeta^2-2\chi^2) - \frac{13}{8}\chi^4 </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \zeta^4 -3\chi^2\zeta^2 +2\chi^4 - \frac{13}{8}\chi^4 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \zeta^4 -3\chi^2\zeta^2 + \frac{3}{8}\chi^4 \, . </math> </td> </tr> </table> Check #2: <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math> (\zeta^2 - \chi^2)(\zeta^2 + \frac{1}{4}\chi^2) + \frac{1}{16}\chi^4 </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \zeta^4 - \frac{3}{4}\chi^2\zeta^2 - \frac{1}{4}\chi^4 + \frac{1}{16}\chi^4 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \zeta^4 - \frac{3}{4}\chi^2\zeta^2 - \frac{3}{16}\chi^4 </math> </td> </tr> </table> ====RHS Quadratic Terms (TERM2)==== The quadratic terms on the RHS can be rewritten as, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>A_\ell \chi^2 + A_s \zeta^2</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl\{ \frac{1}{e^2} \biggl[ \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1/2} \biggl\}\chi^2 + \biggr\{ \frac{2}{e^2} \biggl[ (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1 / 2} \biggr\}\zeta^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl\{ \frac{1}{e^2} \biggl[ (1-e^2)^{1/2}\frac{\sin^{-1}e}{e} - (1-e^2) \biggr] \biggl\}\chi^2 + \biggr\{ \frac{2}{e^2} \biggl[ 1 - (1-e^2)^{1 / 2} \frac{\sin^{-1}e}{e} \biggr] \biggr\}\zeta^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl\{ \frac{1}{3e^2} \biggl[ \Upsilon - 3(1-e^2) \biggr] \biggl\}\chi^2 + \biggr\{ \frac{2}{3e^2} \biggl[ 3 - \Upsilon \biggr] \biggr\}\zeta^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{(\Upsilon - 3)}{3e^2} \biggl[ \chi^2 - 2\zeta^2 \biggr] + \chi^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{(\Upsilon - 3)}{3e^2} (\chi + \sqrt{2}\zeta)(\chi - \sqrt{2} \zeta) + \chi^2 </math> </td> </tr> <tr> <td align="right"><math>\mathrm{TERM2}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> 0.401150 ~~~ </math> ([[#Example_Evaluation|example #1]], below) . </td> </tr> </table> where, again, <table align="center" border=0 cellpadding="3"> <tr> <td align="right"> <math> \Upsilon </math> </td> <td align="center"> <math> \equiv </math> </td> <td align="left"> <math> 3 (1 - e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr] = 2.040835 \, . </math> </td> </tr> </table> ====Gravitational Potential Rewritten==== In summary, then, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{ \Phi_\mathrm{grav}(\chi,\zeta)}{(-\pi G\rho_c a_\ell^2)} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{2} I_\mathrm{BT} - A_\ell \chi^2 - A_s \zeta^2 + \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4 + 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2 + (A_{\ell \ell} a_\ell^2) \chi^4 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{3} \Upsilon - \frac{(\Upsilon - 3)}{3e^2} (\chi + \sqrt{2}\zeta)(\chi - \sqrt{2} \zeta) - \chi^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - ~ \frac{e^{-4}}{(1-e^2)}\biggl\{ \biggl[\zeta^2 - (1-e^2)\chi^2\biggr]\biggl[ \zeta^2 - 2(1-e^2)\chi^2\biggr] - \frac{13}{8}(1-e^2)^2\chi^4 \biggr\} + ~ \frac{4e^{-2}}{3(1-e^2)}\biggl\{ \biggl[\zeta^2 - (1-e^2)\chi^2\biggr]\biggl[ \zeta^2 + \frac{1}{4}(1-e^2)\chi^2\biggr] + \frac{1}{16}(1-e^2)^2\chi^4 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> +~ \frac{e^{-4}}{3}\biggl[ (\zeta^2 - \chi^2)(\zeta^2-2\chi^2) - \frac{13}{8}\chi^4 \biggr]\Upsilon </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{3} \Upsilon - \frac{(\Upsilon - 3)}{3e^2} (\chi + \sqrt{2}\zeta)(\chi - \sqrt{2} \zeta) - \chi^2 + ~ \frac{4}{3e^{2}(1-e^2)}\biggl\{ \biggl[\zeta^2 - (1-e^2)\chi^2\biggr]\biggl[ \zeta^2 + \frac{1}{4}(1-e^2)\chi^2\biggr] + \frac{1}{16}(1-e^2)^2\chi^4 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - ~ \frac{1}{e^4(1-e^2)}\biggl\{ \biggl[\zeta^2 - (1-e^2)\chi^2\biggr]\biggl[ \zeta^2 - 2(1-e^2)\chi^2\biggr] - \frac{13}{8}(1-e^2)^2\chi^4 \biggr\} +~ \frac{1}{3e^4}\biggl[ (\zeta^2 - \chi^2)(\zeta^2-2\chi^2) - \frac{13}{8}\chi^4 \biggr]\Upsilon </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{3} \Upsilon - \frac{(\Upsilon - 3)}{3e^2} (\chi + \sqrt{2}\zeta)(\chi - \sqrt{2} \zeta) + ~ \frac{4}{3e^{2}(1-e^2)}\biggl\{ \biggl[\zeta^2 - (1-e^2)\chi^2\biggr]\biggl[ \zeta^2 + \frac{1}{4}(1-e^2)\chi^2\biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - ~ \frac{1}{e^4(1-e^2)}\biggl\{ \biggl[\zeta^2 - (1-e^2)\chi^2\biggr]\biggl[ \zeta^2 - 2(1-e^2)\chi^2\biggr] \biggr\} +~ \frac{\Upsilon}{3e^4}\biggl[ (\zeta^2 - \chi^2)(\zeta^2-2\chi^2) \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - \chi^2 + ~ \frac{4}{3e^{2}(1-e^2)}\biggl\{ \frac{1}{16}(1-e^2)^2\chi^4 \biggr\} + \frac{1}{e^4(1-e^2)}\biggl\{ \frac{13}{8}(1-e^2)^2\chi^4 \biggr\} - \frac{\Upsilon}{3e^4}\biggl\{ \frac{13}{8}\chi^4 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{3} \Upsilon - \frac{(\Upsilon - 3)}{3e^2} (\chi + \sqrt{2}\zeta)(\chi - \sqrt{2} \zeta) + ~ \frac{4(1-e^2)}{3e^{2}}\biggl\{ \biggl[(1-e^2)^{-1}\zeta^2 - \chi^2\biggr]\biggl[(1-e^2)^{-1} \zeta^2 + \frac{1}{4}\chi^2\biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - ~ \frac{(1-e^2)}{e^4}\biggl\{ \biggl[(1-e^2)^{-1}\zeta^2 - \chi^2\biggr]\biggl[ (1-e^2)^{-1}\zeta^2 - 2\chi^2\biggr] \biggr\} +~ \frac{\Upsilon}{3e^4}\biggl[ (\zeta^2 - \chi^2)(\zeta^2-2\chi^2) \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - ~\chi^2 + ~ \biggl\{ \frac{(1-e^2)}{12e^{2}} + \frac{13(1-e^2)}{8e^4} - \frac{13\Upsilon}{24e^4} \biggr\}\chi^4 \, . </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> 0.767874 (row 1) + 0.5678833 (row 2) - 0.950574 (row 3) = 0.3851876 . </td> </tr> </table> ====Example Evaluation==== Let's evaluate these expressions, borrowing from the [[#QuantitativeExample|quantitative example specified above]]. Specifically, we choose, <table border="0" align="center" width="80%"> <tr> <td align="center"><math>\frac{a_s}{a_\ell} = 0.582724 \, ,</math></td> <td align="center"><math>e = 0.81267 \, ,</math></td> <td align="center"> </td> </tr> <tr> <td align="center"><math>A_\ell = A_m = 0.51589042 \, ,</math></td> <td align="center"><math>A_s = 0.96821916 \, ,</math></td> <td align="center"><math>I_\mathrm{BT} = \frac{2}{3}\Upsilon = 1.360556 \, ,</math></td> </tr> <tr> <td align="center"><math>a_\ell^2 A_{\ell \ell} = 0.3287756 \, ,</math></td> <td align="center"><math>a_\ell^2 A_{s s} = 1.5066848 \, ,</math></td> <td align="center"><math>a_\ell^2 A_{\ell s} = 0.6848975 \, .</math></td> </tr> </table> Also, let's set <math>\rho/\rho_c = 0.1</math> and <math>\chi = \chi_1 = 0.75 ~~\Rightarrow ~~ \chi_1^2 = 0.5625</math>. This means that, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math> \zeta_1^2 </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> (1-e^2)\biggl[1 - \chi^2 - \frac{\rho(\chi, \zeta)}{\rho_c} \biggr] = \biggl[1 - (0.81267)^2)\biggr]\biggl[1 - 0.5625 - 0.1\biggr] = 0.11460 </math> </td> </tr> <tr> <td align="right"> <math> \Rightarrow ~~~ \zeta_1 </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 0.33853 \, . </math> </td> </tr> </table> So, let's evaluate the gravitational potential … <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{ \Phi_\mathrm{grav}(\chi_1,\zeta_1)}{(-\pi G\rho_c a_\ell^2)} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{2} I_\mathrm{BT} - \biggl[\overbrace{A_\ell \chi^2 + A_s \zeta^2}^{\mathrm{TERM2}} \biggr] + \frac{1}{2}\biggl[ \underbrace{(A_{s s} a_\ell^2) \zeta^4 + 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2 + (A_{\ell \ell} a_\ell^2) \chi^4 }_{\mathrm{TERM1}} \biggr] = 0.385187372 </math> </td> </tr> <tr> <td align="right"> <math>\mathrm{TERM1} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 0.019788921 + 0.088303509 + 0.104026655 = 0.212119085 </math> </td> </tr> <tr> <td align="right"> <math>\mathrm{TERM2} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 0.290188361 + 0.110961809 = 0.401150171 \, . </math> </td> </tr> </table> ====Replace ζ With Normalized Density==== First, let's readjust the last, 3-row expression for the gravitational potential so that <math>\zeta^2</math> can be readily replaced with the normalized density. <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{ \Phi_\mathrm{grav}(\chi,\zeta)}{(-\pi G\rho_c a_\ell^2)} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{3} \Upsilon - \frac{(\Upsilon - 3)}{3e^2} (\chi^2 - 2\zeta^2) + ~ \frac{4(1-e^2)}{3e^{2}}\biggl\{ \biggl[(1-e^2)^{-1}\zeta^2 - \chi^2\biggr]\biggl[(1-e^2)^{-1} \zeta^2 + \frac{1}{4}\chi^2\biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - ~ \frac{(1-e^2)}{e^4}\biggl\{ \biggl[(1-e^2)^{-1}\zeta^2 - \chi^2\biggr]\biggl[ (1-e^2)^{-1}\zeta^2 - 2\chi^2\biggr] \biggr\} +~ \frac{\Upsilon}{3e^4}\biggl[ (\zeta^2 - \chi^2)(\zeta^2-2\chi^2) \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - ~\chi^2 + ~ \frac{1}{24e^4}\biggl\{ 2 e^2(1-e^2) + 39(1-e^2) - 13\Upsilon \biggr\}\chi^4 \, . </math> </td> </tr> </table> Now make the substitution, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\zeta^2</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> (1-e^2)\biggl[1 - \chi^2 - \rho^*\biggr] \, ,</math> </td> </tr> </table> where, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\rho^*</math> </td> <td align="center"> <math>\equiv</math> </td> <td align="left"> <math> \frac{\rho(\chi, \zeta)}{\rho_c} \, .</math> </td> </tr> </table> We have, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{ \Phi_\mathrm{grav}(\chi,\zeta)}{(-\pi G\rho_c a_\ell^2)} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{3} \Upsilon - \frac{(\Upsilon - 3)}{3e^2} \biggl\{ \chi^2 - 2(1-e^2)\biggl[1 - \chi^2 - \rho^*\biggr] \biggr\} + ~ \frac{4(1-e^2)}{3e^{2}} \biggl\{\biggl[1 - \chi^2 - \rho^*\biggr] - \chi^2\biggr\}\biggl\{\biggl[1 - \chi^2 - \rho^*\biggr] + \frac{1}{4}\chi^2\biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - ~ \frac{(1-e^2)}{e^4} \biggl\{\biggl[1 - \chi^2 - \rho^*\biggr] - \chi^2\biggr\}\biggl\{ \biggl[1 - \chi^2 - \rho^*\biggr] - 2\chi^2\biggr\} +~ \frac{\Upsilon}{3e^4}\biggl\{ (1-e^2)\biggl[1 - \chi^2 - \rho^*\biggr] - \chi^2\biggr\} \biggl\{(1-e^2)\biggl[1 - \chi^2 - \rho^*\biggr] - 2\chi^2 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - ~\chi^2 + ~ \frac{1}{24e^4}\biggl\{ 2 e^2(1-e^2) + 39(1-e^2) - 13\Upsilon \biggr\}\chi^4 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{3} \Upsilon - \frac{(\Upsilon - 3)}{3e^2} \biggl\{ -2+2e^2 + (3-2e^2)\chi^2 + (2-2e^2)\rho^* \biggr\} + ~ \frac{4(1-e^2)}{3e^{2}} \biggl\{1 - 2\chi^2 - \rho^*\biggr\}\biggl\{1 - \frac{3}{4}\chi^2 - \rho^*\biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - ~ \frac{(1-e^2)}{e^4} \biggl\{1 - 2\chi^2 - \rho^*\biggr\}\biggl\{ 1 - 3\chi^2 - \rho^* \biggr\} +~ \frac{\Upsilon}{3e^4}\biggl\{ (1-e^2) - (2-e^2)\chi^2 - (1-e^2)\rho^* \biggr\} \biggl\{(1-e^2) - (3-e^2)\chi^2 - (1-e^2)\rho^* \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - ~\chi^2 + ~ \frac{1}{24e^4}\biggl\{ 39 - 37e^2 - 2e^4 - 13\Upsilon \biggr\}\chi^4 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{3} \Upsilon - \frac{(\Upsilon - 3)}{3e^2} \biggl\{ -2 + 3\chi^2 + 2\rho^* + 2e^2\biggl[1 -\chi^2 -\rho^* \biggr] \biggr\} + ~ \frac{4(1-e^2)}{3e^{2}} \biggl\{1 - 2\chi^2 - \rho^*\biggr\}\biggl\{1 - \frac{3}{4}\chi^2 - \rho^*\biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - ~ \frac{(1-e^2)}{e^4} \biggl\{1 - 2\chi^2 - \rho^*\biggr\}\biggl\{ 1 - 3\chi^2 - \rho^* \biggr\} +~ \biggl\{ \frac{\Upsilon}{3e^4}\biggl[ 1 - 2\chi^2 - \rho^*\biggr] + \frac{\Upsilon}{3e^2}\biggl[ - 1 + \chi^2 + \rho^* \biggr] \biggr\} \biggl\{(1-e^2) - (3-e^2)\chi^2 - (1-e^2)\rho^* \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - ~\chi^2 + ~ \frac{1}{24e^4}\biggl\{ 39 - 37e^2 - 2e^4 - 13\Upsilon \biggr\}\chi^4 \, . </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> 0.767874 (row 1) + 0.5678833 (row 2) - 0.950574 (row 3) = 0.3851876 . </td> </tr> </table> Now, let's group together like terms and examine, in particular, whether the coefficient of the cross-product, <math>\chi^2 \rho^*)</math>, goes to zero. <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{ \Phi_\mathrm{grav}(\chi,\zeta)}{(-\pi G\rho_c a_\ell^2)} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{3} \Upsilon - \frac{(\Upsilon - 3)}{3e^2} \biggl\{2e^2 -2 + (2 - 2e^2)\rho^* \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + ~ \biggl[1 - 2\chi^2 - \rho^*\biggr] \biggl\{ \frac{4(1-e^2)}{3e^{2}}\biggl[1 - \frac{3}{4}\chi^2 - \rho^*\biggr] - ~ \frac{(1-e^2)}{e^4}\biggl[ 1 - 3\chi^2 - \rho^* \biggr] + \biggl[\frac{\Upsilon}{3e^4} - \frac{\Upsilon}{3e^2}\biggr]\biggl[(1-e^2) - (3-e^2)\chi^2 - (1-e^2)\rho^* \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> -~ \biggl\{ \frac{\Upsilon}{3e^2} \biggr\}\biggl[(1-e^2) - (3-e^2)\chi^2 - (1-e^2)\rho^* \biggr]\chi^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - ~\chi^2 + ~ \frac{1}{24e^4}\biggl\{ 39 - 37e^2 - 2e^4 - 13\Upsilon \biggr\}\chi^4 - \frac{(\Upsilon - 3)}{3e^2} \biggl\{ 3\chi^2 - 2e^2\chi^2 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{3} \Upsilon + \frac{(\Upsilon - 3)}{3e^2} \biggl\{2(1 - e^2)(1 - \rho^*) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + ~ \biggl[1 - 2\chi^2 - \rho^*\biggr] \frac{(1-e^2)}{3e^{4}}\biggl\{ 4e^2\biggl[1 - \frac{3}{4}\chi^2 - \rho^*\biggr] - 3\biggl[ 1 - 3\chi^2 - \rho^* \biggr] + \Upsilon \biggl[(1-e^2) - (3-e^2)\chi^2 - (1-e^2)\rho^* \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> +~ \biggl\{ \frac{\Upsilon}{3e^2} \biggr\}\biggl[(1-e^2)\rho^* \biggr]\chi^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - ~\chi^2 + ~ \frac{1}{24e^4}\biggl\{ 39 - 37e^2 - 2e^4 - 13\Upsilon \biggr\}\chi^4 - \frac{(\Upsilon - 3)}{3e^2} \biggl\{ 3\chi^2 - 2e^2\chi^2 \biggr\} -~ \biggl\{ \frac{\Upsilon}{3e^2} \biggr\}\biggl[(1-e^2) - (3-e^2)\chi^2 \biggr]\chi^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{3} \Upsilon + \frac{(\Upsilon - 3)}{3e^2} \biggl\{2(1 - e^2)(1 - \rho^*) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + ~ \biggl[(1 - \rho^*) \biggr] \frac{(1-e^2)}{3e^{4}}\biggl\{ \biggl[4e^2 - 3 + \Upsilon (1-e^2)\biggr] (1 - \rho^* ) \biggr\} + ~ \biggl[- 2\chi^2\biggr] \frac{(1-e^2)}{3e^{4}}\biggl\{ \biggl[4e^2 - 3 + \Upsilon (1-e^2)\biggr] (1 - \rho^* ) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + ~ \biggl[(1 - \rho^*) \biggr] \frac{(1-e^2)}{3e^{4}}\biggl\{ \biggl[- 3e^2 +9 - (3-e^2)\Upsilon \biggr]\chi^2 \biggr\} + ~ \biggl[- 2\chi^2\biggr] \frac{(1-e^2)}{3e^{4}}\biggl\{ \biggl[- 3e^2 +9 - (3-e^2)\Upsilon \biggr]\chi^2 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> +~ \biggl[ \frac{\Upsilon(1-e^2)}{3e^2} \biggr]\rho^*\chi^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - ~\chi^2 + ~ \frac{1}{24e^4}\biggl\{ 39 - 37e^2 - 2e^4 - 13\Upsilon \biggr\}\chi^4 - \frac{(\Upsilon - 3)}{3e^2} \biggl\{ 3\chi^2 - 2e^2\chi^2 \biggr\} -~ \biggl\{ \frac{\Upsilon}{3e^2} \biggr\}\biggl[(1-e^2) - (3-e^2)\chi^2 \biggr]\chi^2 </math> </td> </tr> </table> ---- From our examination of spherically symmetric parabolic configurations, we have deduced that the [[ParabolicDensity/Spheres/Structure#Effective_Barotropic_Relations|effective enthalpy-density (barotropic) relation]] is, <!-- ORIGINAL STAB AT DETERMINING ENTHALPY <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\frac{H(\rho)}{H_\mathrm{norm}}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> 7 - 10\biggl[ 1 - \biggl(\frac{\rho}{\rho_c} \biggr) \biggr] + 3\biggl[ 1 - \biggl(\frac{\rho}{\rho_c} \biggr) \biggr]^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 4\biggl(\frac{\rho}{\rho_c} \biggr) + 3\biggl(\frac{\rho}{\rho_c} \biggr)^2 \, , </math> </td> </tr> </table> where, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>H_\mathrm{norm}</math> </td> <td align="center"> <math>\equiv</math> </td> <td align="left"> <math>\frac{GM_\mathrm{tot}}{8 R} \, .</math> </td> </tr> </table> Plugging in the 2D, axisymmetric density distribution gives <math>(h_1 = 4; h_2=3)</math>, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\frac{H(\chi, \zeta)}{H_\mathrm{norm}}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> h_1\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] + h_2\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[h_1 - h_1\chi^2 - h_1\zeta^2(1-e^2)^{-1} \biggr] + h_2\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> h_1 - h_1\chi^2 - h_1\zeta^2(1-e^2)^{-1} + h_2\biggl\{ \biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] - \chi^2 \biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] - \zeta^2(1-e^2)^{-1} \biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> h_1 - h_1\chi^2 - h_1\zeta^2(1-e^2)^{-1} + h_2\biggl\{ \biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] + \biggl[- \chi^2 + \chi^4 + \chi^2 \zeta^2(1-e^2)^{-1} \biggr] + \biggl[-\zeta^2(1-e^2)^{-1} + \chi^2\zeta^2(1-e^2)^{-1} + \zeta^4(1-e^2)^{-2} \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> h_1 - h_1\chi^2 - h_1\zeta^2(1-e^2)^{-1} + h_2\biggl\{ 1 - 2\chi^2 - 2\zeta^2(1-e^2)^{-1} + \biggl[\chi^4 + 2\chi^2 \zeta^2(1-e^2)^{-1} \biggr] + \zeta^4(1-e^2)^{-2} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> (h_1 + h_2) - (h_1 + 2h_2)\chi^2 - (h_1 + 2h_2)\zeta^2(1-e^2)^{-1} + h_2\biggl\{ \biggl[\chi^4 + 2\chi^2 \zeta^2(1-e^2)^{-1} \biggr] + \zeta^4(1-e^2)^{-2} \biggr\} </math> </td> </tr> </table> ORIGINAL STAB AT DETERMINING ENTHALPY -->
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