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=Parabolic Density Distribution= <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]]<br /> [[ParabolicDensity/Axisymmetric/Structure/Try8thru10|Old: 8<sup>th</sup> thru 10<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== ===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) ===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> ===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> ===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> ===8<sup>th</sup> Try=== ====Foundation==== <table border="0" cellpadding="5" align="center"> <tr> <td align="left"><font color="orange"><b>Density:</b></font></td> <td align="right"> <math>\rho^* \equiv \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> </table> ====Complete the Square==== Again, 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> </table> in such a way that we effectively "complete the square." Assuming that the desired expression takes the form, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\biggl[ ~~ \biggr]_\mathrm{RHS}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[(A_{s s} a_\ell^2)^{1 / 2} \zeta^2 + B\chi^2 \biggr] \biggl[(A_{s s} a_\ell^2)^{1 / 2} \zeta^2 + C\chi^2 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> (A_{s s} a_\ell^2) \zeta^4 + (A_{s s} a_\ell^2)^{1 / 2} (B+C) \zeta^2\chi^2 + BC\chi^4 \, , </math> </td> </tr> </table> we see that we must have, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>(A_{s s} a_\ell^2)^{1 / 2} (B+C) </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 2(A_{\ell s}a_\ell^2 ) </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ B </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{2(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2)^{1 / 2} } - C \, ; </math> </td> </tr> </table> and we must also have, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>BC </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> (A_{\ell \ell} a_\ell^2) </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ B </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{(A_{\ell \ell} a_\ell^2)}{C} \, . </math> </td> </tr> </table> Hence, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{(A_{\ell \ell} a_\ell^2)}{C} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{2(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2)^{1 / 2} } - C </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ 0</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> C^2 - 2\biggl[ \frac{(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2)^{1 / 2} }\biggr]C + (A_{\ell \ell} a_\ell^2) \, . </math> </td> </tr> </table> The pair of roots of this quadratic expression are, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>C_\pm</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[ \frac{(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2)^{1 / 2} }\biggr] \pm \frac{1}{2}\biggl\{ 4\biggl[ \frac{(A_{\ell s}a_\ell^2 )^2}{(A_{s s} a_\ell^2) }\biggr] - 4(A_{\ell \ell} a_\ell^2) \biggr\}^{1 / 2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2)^{1 / 2} }\biggl\{1 \pm \biggl[ 1 - \frac{(A_{s s} a_\ell^2)(A_{\ell \ell} a_\ell^2) }{(A_{\ell s}a_\ell^2 )^2} \biggr]^{1 / 2} \biggr\} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \frac{C_\pm}{(A_{s s} a_\ell^2)^{1 / 2}}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2) }\biggl\{1 \pm \biggl[ 1 - \frac{(A_{s s} a_\ell^2)(A_{\ell \ell} a_\ell^2) }{(A_{\ell s}a_\ell^2 )^2} \biggr]^{1 / 2} \biggr\} \, . </math> </td> </tr> </table> Also, then, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{B_\pm}{(A_{s s} a_\ell^2)^{1 / 2}}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{2(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2) } - \frac{C_\pm}{(A_{s s} a_\ell^2)^{1 / 2}} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{2(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2) } - \frac{(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2) }\biggl\{1 \pm \biggl[ 1 - \frac{(A_{s s} a_\ell^2)(A_{\ell \ell} a_\ell^2) }{(A_{\ell s}a_\ell^2 )^2} \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 s}a_\ell^2 )}{(A_{s s} a_\ell^2) }\biggl\{1 \mp \biggl[ 1 - \frac{(A_{s s} a_\ell^2)(A_{\ell \ell} a_\ell^2) }{(A_{\ell s}a_\ell^2 )^2} \biggr]^{1 / 2} \biggr\} \, . </math> </td> </tr> </table> <table border="1" width="80%" cellpadding="8" align="center"><tr><td align="left"> NOTE: [[#Index_Symbol_Expressions|Given that]], <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>(A_{s s} a_\ell^2)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{2}{3(1-e^2)} - \frac{2}{3}(A_{\ell s} a_\ell^2) </math> </td> <td align="center"> and, </td> <td align="right"> <math>(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> </table> we can write, [[File:LambdaVsEccentricity.png|250px|right|Lambda vs Eccentricity]]<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\Lambda \equiv \frac{(A_{s s} a_\ell^2)(A_{\ell \ell} a_\ell^2) }{(A_{\ell s}a_\ell^2 )^2}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1 }{(A_{\ell s}a_\ell^2 )^2} \biggl\{ \biggl[\frac{2}{3(1-e^2)} - \frac{2}{3}(A_{\ell s} a_\ell^2)\biggr] \biggl[ \frac{1}{2} - \frac{1}{4}(A_{\ell s} a_\ell^2) \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1 }{6(A_{\ell s}a_\ell^2 )^2} \biggl\{ \frac{1}{(1-e^2)} \biggl[ 2 - (A_{\ell s} a_\ell^2) \biggr] - (A_{\ell s} a_\ell^2) \biggl[ 2 - (A_{\ell s} a_\ell^2) \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1 }{6(A_{\ell s}a_\ell^2 )^2} \biggl\{ \biggl[\frac{1}{(1-e^2)} - (A_{\ell s} a_\ell^2)\biggr] \biggl[ 2 - (A_{\ell s} a_\ell^2) \biggr] \biggr\} </math> </td> </tr> </table> </td></tr></table> In summary, then, we can write, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{B_\pm}{(A_{s s} a_\ell^2)^{1 / 2}}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2) }\biggl[ 1 \mp ( 1 - \Lambda )^{1 / 2} \biggr] </math> </td> <td align="center"> and, </td> <td align="right"> <math>\frac{C_\pm}{(A_{s s} a_\ell^2)^{1 / 2}}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2) }\biggl[ 1 \pm (1 - \Lambda )^{1 / 2} \biggr] \, , </math> </td> </tr> </table> where, as illustrated by the inset "Lambda vs Eccentricity" plot, for all values of the eccentricity <math>(0 < e \leq 1)</math>, the quantity, <math>\Lambda</math>, is greater than unity. It is clear, then, that both roots of the relevant quadratic equation are complex — i.e., they have imaginary components. But that's okay because the coefficients that appear in the right-hand-side, bracketed quartic expression appear in the combinations, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>(BC)_\pm</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{(A_{\ell s}a_\ell^2 )^2}{(A_{s s} a_\ell^2) } \biggl[ 1 - ( 1 - \Lambda )^{1 / 2} \biggr] \biggl[ 1 + ( 1 - \Lambda )^{1 / 2} \biggr] = \frac{(A_{\ell s}a_\ell^2 )^2}{(A_{s s} a_\ell^2) } \biggl[ \Lambda\biggr] = (A_{\ell \ell}a_\ell^2 ) \, , </math> </td> </tr> <tr> <td align="right"> <math>(B + C)_\pm</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2)^{1 / 2} }\biggl[ 1 \mp ( 1 - \Lambda )^{1 / 2} \biggr] + \frac{(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2)^{1 / 2} }\biggl[ 1 \pm (1 - \Lambda )^{1 / 2} \biggr] = \frac{2(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2)^{1 / 2} } \, , </math> </td> </tr> </table> both of which are real. ===9<sup>th</sup> Try=== ====Starting Key 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>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> </table> ====Play With Vertical Pressure Gradient==== <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> \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 - 2A_s )\zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr] - \chi^2 \biggl[ (2A_{\ell s}a_\ell^2 \chi^2 - 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 - 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> (2A_{\ell s}a_\ell^2 \chi^2 - 2A_s )\zeta + 2A_{ss} a_\ell^2 \zeta^3 - (2A_{\ell s}a_\ell^2 \chi^4 - 2A_s \chi^2)\zeta - 2A_{ss} a_\ell^2 \chi^2 \zeta^3 - (1-e^2)^{-1}\biggl[ (2A_{\ell s}a_\ell^2 \chi^2 - 2A_s )\zeta^3 + 2A_{ss} a_\ell^2 \zeta^5 \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 - 2A_{ss} a_\ell^2 \chi^2 - (1-e^2)^{-1}(2A_{\ell s}a_\ell^2 \chi^2 - 2A_s )\biggr]\zeta^3 + \biggl[ - (1-e^2)^{-1}2A_{ss} a_\ell^2 \biggr] \zeta^5 \, . </math> </td> </tr> </table> Integrate over <math>\zeta</math> gives … <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \int \biggl[\frac{\partial P}{\partial \zeta}\biggr] d\zeta </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \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 + \frac{1}{2}\biggl[ A_{ss} a_\ell^2 - A_{ss} a_\ell^2 \chi^2 - (1-e^2)^{-1}(A_{\ell s}a_\ell^2 \chi^2 - A_s )\biggr]\zeta^4 + \frac{1}{3}\biggl[ - (1-e^2)^{-1}A_{ss} a_\ell^2 \biggr] \zeta^6 + ~\mathrm{const} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[-A_s \zeta^2 + \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 + \frac{1}{2}(1-e^2)^{-1}A_s\zeta^4 - \frac{1}{3}(1-e^2)^{-1}A_{ss} a_\ell^2 \zeta^6 \biggr]\chi^0 + \biggl[ A_{\ell s}a_\ell^2 \zeta^2 + A_s\zeta^2 - \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 - \frac{1}{2}(1-e^2)^{-1}(A_{\ell s}a_\ell^2 \zeta^4 ) \biggr]\chi^2 + \biggl[- A_{\ell s}a_\ell^2 \zeta^2 \biggr]\chi^4 + ~\mathrm{const.} </math> </td> </tr> </table> ====Now Play With Radial Pressure Gradient==== <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\biggl[\frac{1}{(-\pi G\rho_c a_\ell^2)} \biggr] \frac{\partial \Phi}{\partial \chi}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\rho}{\rho_c} \cdot \biggl\{ - 2A_\ell \chi + \frac{1}{2}\biggl[ 4(A_{\ell s} a_\ell^2)\zeta^2\chi + 4(A_{\ell\ell} a_\ell^2)\chi^3 \biggl] \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_{\ell s} a_\ell^2 \zeta^2 - A_\ell )\chi + 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> 2\biggl[ (A_{\ell s} a_\ell^2 \zeta^2 - A_\ell )\chi + A_{\ell\ell} a_\ell^2 \chi^3\biggr] - 2\chi^2 \biggl[ (A_{\ell s} a_\ell^2 \zeta^2 - A_\ell )\chi + A_{\ell\ell} a_\ell^2 \chi^3\biggr] - 2\zeta^2(1-e^2)^{-1} \biggl[(A_{\ell s} a_\ell^2 \zeta^2 - A_\ell )\chi + 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> 2(A_{\ell s} a_\ell^2 \zeta^2 - A_\ell )\chi + 2\biggl[ A_{\ell\ell} a_\ell^2 + (A_\ell - A_{\ell s} a_\ell^2 \zeta^2 ) \biggr]\chi^3 - 2A_{\ell\ell} a_\ell^2 \chi^5 + 2(1-e^2)^{-1} \biggl[(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 )\chi - A_{\ell\ell} a_\ell^2 \zeta^2\chi^3\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2\biggl[ (A_{\ell s} a_\ell^2 \zeta^2 - A_\ell ) + (1-e^2)^{-1}(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 )\biggr]\chi + 2\biggl[ A_{\ell\ell} a_\ell^2 + (A_\ell - A_{\ell s} a_\ell^2 \zeta^2 ) - (1-e^2)^{-1}A_{\ell\ell} a_\ell^2 \zeta^2\biggr]\chi^3 - 2A_{\ell\ell} a_\ell^2 \chi^5 </math> </td> </tr> </table> Add a term <math>j^2 \sim (j_4^2\chi^4 + j_6^2\chi^6)</math> to account for centrifugal acceleration … <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 \chi} = \biggl[\frac{1}{(-\pi G\rho_c a_\ell^2)} \biggr] \frac{\partial \Phi}{\partial \chi} + \frac{j^2}{\chi^3}\biggl[\frac{\rho}{\rho_c}\biggr]</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> 2\biggl[ (A_{\ell s} a_\ell^2 \zeta^2 - A_\ell ) + (1-e^2)^{-1}(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 )\biggr]\chi + 2\biggl[ A_{\ell\ell} a_\ell^2 + (A_\ell - A_{\ell s} a_\ell^2 \zeta^2 ) - (1-e^2)^{-1}A_{\ell\ell} a_\ell^2 \zeta^2\biggr]\chi^3 - 2A_{\ell\ell} a_\ell^2 \chi^5 + \frac{j^2}{\chi^3}\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> 2\biggl[ (A_{\ell s} a_\ell^2 \zeta^2 - A_\ell ) + (1-e^2)^{-1}(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 )\biggr]\chi + 2\biggl[ A_{\ell\ell} a_\ell^2 + (A_\ell - A_{\ell s} a_\ell^2 \zeta^2 ) - (1-e^2)^{-1}A_{\ell\ell} a_\ell^2 \zeta^2\biggr]\chi^3 - 2A_{\ell\ell} a_\ell^2 \chi^5 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \frac{(j_4^2\chi^4 + j_6^2\chi^6)}{\chi^3} - \frac{(j_4^2\chi^4 + j_6^2\chi^6)}{\chi^3}\biggl[\chi^2 \biggr] - \frac{(j_4^2\chi^4 + j_6^2\chi^6)}{\chi^3}\biggl[\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> 2\biggl[ (A_{\ell s} a_\ell^2 \zeta^2 - A_\ell ) + (1-e^2)^{-1}(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 )\biggr]\chi + 2\biggl[ A_{\ell\ell} a_\ell^2 + (A_\ell - A_{\ell s} a_\ell^2 \zeta^2 ) - (1-e^2)^{-1}A_{\ell\ell} a_\ell^2 \zeta^2\biggr]\chi^3 - 2A_{\ell\ell} a_\ell^2 \chi^5 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + (j_4^2\chi + j_6^2\chi^3) - (j_4^2\chi + j_6^2\chi^3)\biggl[\zeta^2(1-e^2)^{-1} \biggr] - (j_4^2\chi^3 + j_6^2\chi^5) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2\biggl[ (A_{\ell s} a_\ell^2 \zeta^2 - A_\ell ) + (1-e^2)^{-1}(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 )\biggr]\chi + 2\biggl[ A_{\ell\ell} a_\ell^2 + (A_\ell - A_{\ell s} a_\ell^2 \zeta^2 ) - (1-e^2)^{-1}A_{\ell\ell} a_\ell^2 \zeta^2\biggr]\chi^3 - 2A_{\ell\ell} a_\ell^2 \chi^5 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - \biggl[j_4^2\zeta^2(1-e^2)^{-1} - j_4^2\biggr]\chi - \biggl[j_4^2 + j_6^2\zeta^2(1-e^2)^{-1} - j_6^2 \biggr]\chi^3 - \biggl[j_6^2\biggr]\chi^5 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ 2(A_{\ell s} a_\ell^2 \zeta^2 - A_\ell ) + 2(1-e^2)^{-1}(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 ) - j_4^2\zeta^2(1-e^2)^{-1} + j_4^2\biggr]\chi </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \biggl[ 2A_{\ell\ell} a_\ell^2 + 2(A_\ell - A_{\ell s} a_\ell^2 \zeta^2 ) - 2(1-e^2)^{-1}A_{\ell\ell} a_\ell^2 \zeta^2 - j_4^2 - j_6^2\zeta^2(1-e^2)^{-1} + j_6^2 \biggr]\chi^3 + \biggl[-j_6^2 - 2A_{\ell\ell} a_\ell^2 \biggr]\chi^5 </math> </td> </tr> </table> Integrate over <math>\chi</math> gives … <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \int \biggl[\frac{\partial P}{\partial \chi}\biggr] d\chi </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ (A_{\ell s} a_\ell^2 \zeta^2 - A_\ell ) + (1-e^2)^{-1}(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 ) - \frac{1}{2}j_4^2\zeta^2(1-e^2)^{-1} + \frac{1}{2}j_4^2\biggr]\chi^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \biggl[ \frac{1}{2}A_{\ell\ell} a_\ell^2 + \frac{1}{2}(A_\ell - A_{\ell s} a_\ell^2 \zeta^2 ) - \frac{1}{2}(1-e^2)^{-1}A_{\ell\ell} a_\ell^2 \zeta^2 - \frac{1}{4}j_4^2 - \frac{1}{4}j_6^2\zeta^2(1-e^2)^{-1} + \frac{1}{4}j_6^2 \biggr]\chi^4 - \biggl[\frac{1}{6}j_6^2 + \frac{1}{3}A_{\ell\ell} a_\ell^2 \biggr]\chi^6 </math> </td> </tr> </table> ====Compare Pair of Integrations==== <table border="1" align="center" cellpadding="8"> <tr> <td align="center" width="6%"> </td> <td align="center" width="47%">Integration over <math>\zeta</math></td> <td align="center">Integration over <math>\chi</math></td> </tr> <tr> <td align="center"><math>\chi^0</math></td> <td align="right"><math>-A_s \zeta^2 + \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 + \frac{1}{2}(1-e^2)^{-1}A_s\zeta^4 - \frac{1}{3}(1-e^2)^{-1}A_{ss} a_\ell^2 \zeta^6 </math></td> <td align="left">none</td> </tr> <tr> <td align="center"><math>\chi^2</math></td> <td align="right"> <math>A_{\ell s}a_\ell^2 \zeta^2 + A_s\zeta^2 - \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 - \frac{1}{2}(1-e^2)^{-1}(A_{\ell s}a_\ell^2 \zeta^4 )</math> </td> <td align="left"> <math>(A_{\ell s} a_\ell^2 \zeta^2 - A_\ell ) + (1-e^2)^{-1}(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 ) - \frac{1}{2}j_4^2\zeta^2(1-e^2)^{-1} + \frac{1}{2}j_4^2</math> </td> </tr> <tr> <td align="center"><math>\chi^4</math></td> <td align="right"> <math>- A_{\ell s}a_\ell^2 \zeta^2 </math> </td> <td align="left"> <math>\frac{1}{2}A_{\ell\ell} a_\ell^2 + \frac{1}{2}(A_\ell - A_{\ell s} a_\ell^2 \zeta^2 ) - \frac{1}{2}(1-e^2)^{-1}A_{\ell\ell} a_\ell^2 \zeta^2 - \frac{1}{4}j_4^2 - \frac{1}{4}j_6^2\zeta^2(1-e^2)^{-1} + \frac{1}{4}j_6^2 </math> </td> </tr> <tr> <td align="center"><math>\chi^6</math></td> <td align="right"> none </td> <td align="left"> <math> - \frac{1}{6}j_6^2 - \frac{1}{3}A_{\ell\ell} a_\ell^2 </math> </td> </tr> </table> Try, <math>j_6^2 = [-2A_{\ell\ell}a_\ell^2]</math> and <math>\frac{1}{2}j_4^2 = [A_\ell + (A_{\ell s} a_\ell^2) \zeta^2 ]</math>. <table border="1" align="center" cellpadding="8"> <tr> <td align="center" width="6%"> </td> <td align="center" width="47%">Integration over <math>\zeta</math></td> <td align="center">Integration over <math>\chi</math></td> </tr> <tr> <td align="center"><math>\chi^0</math></td> <td align="right"><math>-A_s \zeta^2 + \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 + \frac{1}{2}(1-e^2)^{-1}A_s\zeta^4 - \frac{1}{3}(1-e^2)^{-1}A_{ss} a_\ell^2 \zeta^6 </math></td> <td align="left">none</td> </tr> <tr> <td align="center"><math>\chi^2</math></td> <td align="right"><math>A_{\ell s}a_\ell^2 \zeta^2 + A_s\zeta^2 - \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 - \frac{1}{2}(1-e^2)^{-1}(A_{\ell s}a_\ell^2 \zeta^4 )</math></td> <td align="left"> <math> (A_{\ell s} a_\ell^2 \zeta^2 - A_\ell ) + (1-e^2)^{-1}(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 ) - \frac{1}{2}j_4^2\zeta^2(1-e^2)^{-1} + \frac{1}{2}j_4^2 </math> <br /><math>=</math><br /> <math> (A_{\ell s} a_\ell^2 \zeta^2 - A_\ell ) + (1-e^2)^{-1}(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 ) - [A_\ell + (A_{\ell s} a_\ell^2) \zeta^2 ]\zeta^2(1-e^2)^{-1} + [A_\ell + (A_{\ell s} a_\ell^2) \zeta^2 ] </math> <br /><math>=</math><br /> <math> 2(A_{\ell s} a_\ell^2) \zeta^2\biggl[1 - \zeta^2 (1-e^2)^{-1} \biggr] </math> </td> </tr> <tr> <td align="center"><math>\chi^4</math></td> <td align="right"> <math>- A_{\ell s}a_\ell^2 \zeta^2 </math> </td> <td align="left"> <math> \frac{1}{2}A_{\ell\ell} a_\ell^2 + \frac{1}{2}(A_\ell - A_{\ell s} a_\ell^2 \zeta^2 ) - \frac{1}{2}(1-e^2)^{-1}A_{\ell\ell} a_\ell^2 \zeta^2 - \frac{1}{4}j_4^2 - \frac{1}{4}[-2A_{\ell\ell}a_\ell^2]\zeta^2(1-e^2)^{-1} + \frac{1}{4}[-2A_{\ell\ell}a_\ell^2] </math> <br /><math>=</math><br /> <math> \frac{1}{4}\biggl[2(A_\ell - A_{\ell s} a_\ell^2 \zeta^2 ) - 2[A_\ell + (A_{\ell s} a_\ell^2) \zeta^2 ] \biggr] = - A_{\ell s}a_\ell^2 \zeta^2 </math> </td> </tr> <tr> <td align="center"><math>\chi^6</math></td> <td align="right"> none </td> <td align="left"> <math> 0 </math> </td> </tr> </table> What expression for <math>j_4^2</math> is required in order to ensure that the <math>\chi^2</math> term is the same in both columns? <table border="0" align="center" cellpadding="8"> <tr> <td align="right"><math> \frac{1}{2}j_4^2 \biggl[ 1 - \zeta^2(1-e^2)^{-1}\biggr]</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ A_{\ell s}a_\ell^2 \zeta^2 + A_s\zeta^2 - \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 - \frac{1}{2}(1-e^2)^{-1}(A_{\ell s}a_\ell^2 \zeta^4 )\biggr] - \biggl[(A_{\ell s} a_\ell^2 \zeta^2 - A_\ell ) + (1-e^2)^{-1}(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 ) \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ A_s\zeta^2 - \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 - \frac{1}{2}(1-e^2)^{-1}(A_{\ell s}a_\ell^2 \zeta^4 )\biggr] + \biggl[( A_\ell ) - (1-e^2)^{-1}(A_\ell\zeta^2 ) + (1-e^2)^{-1}( A_{\ell s} a_\ell^2 \zeta^4 ) \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ A_s\zeta^2 - \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 + \frac{1}{2}(1-e^2)^{-1}(A_{\ell s}a_\ell^2) \zeta^4\biggr] + A_\ell\biggl[1 - (1-e^2)^{-1}\zeta^2 \biggr] </math> </td> </tr> <tr> <td align="right"><math> \Rightarrow ~~~ \frac{1}{2}j_4^2 \biggl[ 1 - \zeta^2(1-e^2)^{-1}\biggr] - A_\ell\biggl[1 - \zeta^2(1-e^2)^{-1} \biggr] </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{2}(1-e^2)^{-1}(A_{\ell s}a_\ell^2) \zeta^4 + \biggl[ A_s \biggr]\zeta^2 - \frac{1}{2}\biggl[ A_{ss}a_\ell^2 \biggr] \zeta^4 </math> </td> </tr> </table> Now, considering the following three relations … <table border="0" align="center" cellpadding="8"> <tr> <td align="right"> <math> \frac{3}{2}(A_{ss}a_\ell^2) </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> (1-e^2)^{-1} - (A_{\ell s}a_\ell^2) \, ; </math> </td> </tr> <tr> <td align="right"> <math> A_s </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> A_\ell + e^2(A_{\ell s}a_\ell^2) \, ; </math> </td> </tr> <tr> <td align="right"> <math> e^2(A_{\ell s}a_\ell^2) </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2 - 3 A_\ell \, ; </math> </td> </tr> </table> we can write, <table border="0" align="center" cellpadding="8"> <tr> <td align="right"><math> \frac{1}{2}j_4^2 \biggl[ 1 - \zeta^2(1-e^2)^{-1}\biggr] - A_\ell\biggl[1 - \zeta^2(1-e^2)^{-1} \biggr] </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{2}(1-e^2)^{-1}(A_{\ell s}a_\ell^2) \zeta^4 + \biggl[ A_\ell + e^2(A_{\ell s}a_\ell^2) \biggr]\zeta^2 - \frac{1}{3}\biggl[ (1-e^2)^{-1} - (A_{\ell s}a_\ell^2)\biggr] \zeta^4 </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ 3j_4^2 \biggl[ 1 - \zeta^2(1-e^2)^{-1}\biggr] - 3A_\ell\biggl[2 - 2\zeta^2(1-e^2)^{-1} \biggr] </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> 3(1-e^2)^{-1}(A_{\ell s}a_\ell^2) \zeta^4 + 6\biggl[ A_\ell + e^2(A_{\ell s}a_\ell^2) \biggr]\zeta^2 - 2\biggl[ (1-e^2)^{-1} - (A_{\ell s}a_\ell^2)\biggr] \zeta^4 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> (A_{\ell s}a_\ell^2)\biggl\{2\zeta^4 + 3\zeta^4(1-e^2)^{-1} + 6 e^2\zeta^2 \biggr\} - 2\zeta^4 (1-e^2)^{-1} + 6A_\ell \zeta^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> - 2\zeta^4 (1-e^2)^{-1} + 6A_\ell \zeta^2 + \biggl[2 - 3A_\ell \biggr]\biggl\{2\zeta^4 + 3\zeta^4(1-e^2)^{-1} + 6 e^2\zeta^2 \biggr\}\frac{1}{e^2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> - 2\zeta^4 (1-e^2)^{-1} - 3A_\ell\biggl\{2\zeta^4 + 3\zeta^4(1-e^2)^{-1} + 4 e^2\zeta^2 \biggr\}\frac{1}{e^2} + \biggl\{4\zeta^4 + 6\zeta^4(1-e^2)^{-1} + 12 e^2\zeta^2 \biggr\}\frac{1}{e^2} </math> </td> </tr> </table> <table border="0" align="center" cellpadding="8"> <tr> <td align="right"><math>\Rightarrow ~~~ 3j_4^2 \biggl[ 1 - \zeta^2(1-e^2)^{-1}\biggr] </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> - 2\zeta^4 (1-e^2)^{-1} + \frac{3A_\ell(1-e^2)^{-1}}{e^2}\biggl\{ \biggl[2e^2(1-e^2) - 2e^2\zeta^2 \biggr] - \biggl[2\zeta^4(1-e^2) + 3\zeta^4 + 4 e^2(1-e^2)\zeta^2 \biggr] \biggr\} + \biggl\{4\zeta^4 + 6\zeta^4(1-e^2)^{-1} + 12 e^2\zeta^2 \biggr\}\frac{1}{e^2} </math> </td> </tr> </table> ===10<sup>th</sup> Try=== ====Repeating Key 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> <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> </table> From the [[#Starting_Key_Relations|above (9<sup>th</sup> Try) examination]] of the vertical pressure gradient, we determined that a reasonably good approximation for the normalized pressure throughout the configuration is given by the expression, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \int \biggl[\frac{\partial P}{\partial \zeta}\biggr] d\zeta </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[-A_s \zeta^2 + \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 + \frac{1}{2}(1-e^2)^{-1}A_s\zeta^4 - \frac{1}{3}(1-e^2)^{-1}A_{ss} a_\ell^2 \zeta^6 \biggr]\chi^0 + \biggl[ A_{\ell s}a_\ell^2 \zeta^2 + A_s\zeta^2 - \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 - \frac{1}{2}(1-e^2)^{-1}(A_{\ell s}a_\ell^2 \zeta^4 ) \biggr]\chi^2 + \biggl[- A_{\ell s}a_\ell^2 \zeta^2 \biggr]\chi^4 + ~\mathrm{const.} </math> </td> </tr> </table> If we set <math>\chi = 0</math> — that is, if we look along the vertical axis — this approximation should be particularly good, resulting in the expression, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>P_z \equiv \biggl\{ \biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \int \biggl[\frac{\partial P}{\partial \zeta}\biggr] d\zeta \biggr\}_{\chi=0}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math>P_c^* - A_s \zeta^2 + \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 + \frac{1}{2}(1-e^2)^{-1}A_s\zeta^4 - \frac{1}{3}(1-e^2)^{-1}A_{ss} a_\ell^2 \zeta^6 \, . </math> </td> </tr> </table> <table border="1" align="center" cellpadding="8" width="80%"><tr><td align="left"> Note that in the limit that <math>z \rightarrow a_s</math> — that is, at the pole along the vertical (symmetry) axis where the <math>P_z</math> should drop to zero — we should set <math>\zeta \rightarrow (1 - e^2)^{1 / 2}</math>. This allows us to determine the central pressure. <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>P_c^* </math></td> <td align="center"><math>=</math></td> <td align="left"> <math>A_s (1-e^2) - \frac{1}{2}A_{ss}a_\ell^2 (1-e^2)^2 - \frac{1}{2}(1-e^2)^{-1}A_s(1-e^2)^2 + \frac{1}{3}(1-e^2)^{-1}A_{ss} a_\ell^2 (1-e^2)^3 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math>A_s (1-e^2) - \frac{1}{2}A_s(1-e^2) + \frac{1}{3}A_{ss} a_\ell^2 (1-e^2)^2 - \frac{1}{2}A_{ss}a_\ell^2 (1-e^2)^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math>\frac{1}{2}A_s(1-e^2) - \frac{1}{6}A_{ss} a_\ell^2 (1-e^2)^2 \, . </math> </td> </tr> </table> </td></tr></table> This means that, along the vertical axis, the pressure gradient is, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>P_z \equiv \biggl\{ \biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \int \biggl[\frac{\partial P}{\partial \zeta}\biggr] d\zeta \biggr\}_{\chi=0}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math>P_c^* - A_s \zeta^2 + \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 + \frac{1}{2}(1-e^2)^{-1}A_s\zeta^4 - \frac{1}{3}(1-e^2)^{-1}A_{ss} a_\ell^2 \zeta^6 \, . </math> </td> </tr> </table> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\frac{\partial P_z}{\partial\zeta}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math>- 2A_s \zeta + 2A_{ss}a_\ell^2 \zeta^3 + 2(1-e^2)^{-1}A_s\zeta^3 - 2(1-e^2)^{-1}A_{ss} a_\ell^2 \zeta^5 \, . </math> </td> </tr> </table> This should match the more general "<font color="orange">vertical pressure gradient</font>" expression when we set, <math>\chi=0</math>, that is, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\biggl\{ \biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta} \biggr\}_{\chi=0}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ 1 - \cancelto{0}{\chi^2} - \zeta^2(1-e^2)^{-1}\biggr]\cdot \biggl[ 2A_{\ell s}a_\ell^2 \zeta \cancelto{0}{\chi^2} - 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_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr] + \zeta^2(1-e^2)^{-1} \biggl[2A_s \zeta - 2A_{ss} a_\ell^2 \zeta^3 \biggr] </math> </td> </tr> </table> <b><font color="red">Yes! The expressions match!</font></b> ====Shift to ξ<sub>1</sub> Coordinate==== In an [[ParabolicDensity/Axisymmetric/Structure/Try1thru7#Setup|accompanying chapter]], we defined the coordinate, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\biggl(\frac{\xi_1}{a_s}\biggr)^2</math></td> <td align="center"><math>\equiv</math></td> <td align="left"> <math> \biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_s}\biggr)^2 = \chi^2 + \zeta^2(1-e^2)^{-1} \, . </math> </td> </tr> </table> Given that we want the pressure to be constant on <math>\xi_1</math> surfaces, it seems plausible that <math>\zeta^2</math> should be replaced by <math>(1-e^2)(\xi_1/a_s)^2 = [(1-e^2)\chi^2 + \zeta^2]</math> in the expression for <math>P_z</math>. That is, we might expect the expression for the pressure at any point in the meridional plane to be, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>P_\mathrm{test01}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math>P_c^* - A_s \biggl[ (1-e^2)\chi^2 + \zeta^2 \biggr]^1 + \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]\biggl[ (1-e^2)\chi^2 + \zeta^2 \biggr]^2 - \frac{1}{3}(1-e^2)^{-1}A_{ss} a_\ell^2 \biggl[ (1-e^2)\chi^2 + \zeta^2 \biggr]^3 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math>P_c^* - A_s \biggl[ (1-e^2)\chi^2 + \zeta^2 \biggr]^1 + \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]\biggl[ (1-e^2)^2\chi^4 + 2(1-e^2)\chi^2\zeta^2 + \zeta^4 \biggr] - \frac{1}{3}(1-e^2)^{-1}A_{ss} a_\ell^2 \biggl[ (1-e^2)\chi^2 + \zeta^2 \biggr]\biggl[ (1-e^2)^2\chi^4 + 2(1-e^2)\chi^2\zeta^2 + \zeta^4 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> P_c^* - A_s \biggl[ (1-e^2)\chi^2 + \zeta^2 \biggr] + \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]\biggl[ (1-e^2)^2\chi^4 + 2(1-e^2)\chi^2\zeta^2 + \zeta^4 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - \frac{1}{3} A_{ss} a_\ell^2 \biggl[ (1-e^2)^2\chi^6 + 2(1-e^2)\chi^4\zeta^2 + \chi^2\zeta^4 \biggr] - \frac{1}{3}A_{ss} a_\ell^2 \biggl[ (1-e^2)\chi^4\zeta^2 + 2\chi^2\zeta^4 + (1-e^2)^{-1}\zeta^6 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \chi^0 \biggl\{ P_c^* -A_s\zeta^2 + \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]\zeta^4 - \frac{1}{3}A_{ss} a_\ell^2(1-e^2)^{-1}\zeta^6 \biggr\} + \chi^2 \biggl\{ -A_s(1-e^2) + \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]2(1-e^2)\zeta^2 -\frac{1}{3}A_{ss}a_\ell^2\zeta^4 - \frac{2}{3}A_{ss} a_\ell^2\zeta^4 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \chi^4 \biggl\{ \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr](1-e^2)^2 - \frac{2}{3}A_{ss} a_\ell^2(1-e^2)\zeta^2 - \frac{1}{3}A_{ss} a_\ell^2(1-e^2)\zeta^2 \biggr\} + \chi^6 \biggl\{ - \frac{1}{3} A_{ss} a_\ell^2 (1-e^2)^2 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \chi^0 \biggl\{ P_c^* -A_s\zeta^2 + \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]\zeta^4 - \frac{1}{3}A_{ss} a_\ell^2(1-e^2)^{-1}\zeta^6 \biggr\} + \chi^2 \biggl\{ -A_s(1-e^2) + \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]2(1-e^2)\zeta^2 - A_{ss}a_\ell^2\zeta^4 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \chi^4 \biggl\{ \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr](1-e^2)^2 - A_{ss} a_\ell^2(1-e^2)\zeta^2 \biggr\} + \chi^6 \biggl\{ - \frac{1}{3} A_{ss} a_\ell^2 (1-e^2)^2 \biggr\} </math> </td> </tr> </table> <table border="1" align="center" cellpadding="8"> <tr> <td align="center" width="6%"> </td> <td align="center" width="47%">Integration over <math>\zeta</math></td> <td align="center">Pressure Guess</td> </tr> <tr> <td align="center"><math>\chi^0</math></td> <td align="right"><math>-A_s \zeta^2 + \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 + \frac{1}{2}(1-e^2)^{-1}A_s\zeta^4 - \frac{1}{3}(1-e^2)^{-1}A_{ss} a_\ell^2 \zeta^6 </math></td> <td align="left"> <math> P_c^* -A_s\zeta^2 + \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]\zeta^4 - \frac{1}{3}A_{ss} a_\ell^2(1-e^2)^{-1}\zeta^6 </math> </td> </tr> <tr> <td align="center"><math>\chi^2</math></td> <td align="right"> <math>A_{\ell s}a_\ell^2 \zeta^2 + A_s\zeta^2 - \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 - \frac{1}{2}(1-e^2)^{-1}(A_{\ell s}a_\ell^2 \zeta^4 )</math> </td> <td align="left"> <math> -A_s(1-e^2) + \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]2(1-e^2)\zeta^2 - A_{ss}a_\ell^2\zeta^4 </math> </td> </tr> <tr> <td align="center"><math>\chi^4</math></td> <td align="right"> <math>- A_{\ell s}a_\ell^2 \zeta^2 </math> </td> <td align="left"> <math> \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr](1-e^2)^2 - A_{ss} a_\ell^2(1-e^2)\zeta^2 </math> </td> </tr> <tr> <td align="center"><math>\chi^6</math></td> <td align="right"> none </td> <td align="left"> <math> - \frac{1}{3} A_{ss} a_\ell^2 (1-e^2)^2 </math> </td> </tr> </table> ====Compare Vertical Pressure Gradient Expressions==== From our [[#Starting_Key_Relations|above (9<sup>th</sup> try) derivation]] we know that the vertical pressure gradient is given by the expression, <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> \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 - 2A_s ) - (2A_{\ell s}a_\ell^2 \chi^4 - 2A_s \chi^2)\biggr]\zeta + \biggl[ 2A_{ss} a_\ell^2 - 2A_{ss} a_\ell^2 \chi^2 - (1-e^2)^{-1}(2A_{\ell s}a_\ell^2 \chi^2 - 2A_s )\biggr]\zeta^3 + \biggl[ - (1-e^2)^{-1}2A_{ss} a_\ell^2 \biggr] \zeta^5 \, . </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ 2A_s (\chi^2-1) + 2A_{\ell s}a_\ell^2 (1 - \chi^2)\chi^2 \biggr]\zeta + \biggl[ 2A_{ss} a_\ell^2(1 - \chi^2 ) - 2A_{\ell s}a_\ell^2 (1-e^2)^{-1}\chi^2 + 2(1-e^2)^{-1}A_s \biggr]\zeta^3 + \biggl[ - 2A_{ss} a_\ell^2 (1-e^2)^{-1}\biggr] \zeta^5 \, . </math> </td> </tr> </table> By comparison, the vertical derivative of our "test01" pressure expression gives, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>P_\mathrm{test01}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \chi^0 \biggl\{ P_c^* -A_s\zeta^2 + \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]\zeta^4 - \frac{1}{3}A_{ss} a_\ell^2(1-e^2)^{-1}\zeta^6 \biggr\} + \chi^2 \biggl\{ -A_s(1-e^2) + \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]2(1-e^2)\zeta^2 - A_{ss}a_\ell^2\zeta^4 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \chi^4 \biggl\{ \frac{1}{2}\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr](1-e^2)^2 - A_{ss} a_\ell^2(1-e^2)\zeta^2 \biggr\} + \chi^6 \biggl\{ - \frac{1}{3} A_{ss} a_\ell^2 (1-e^2)^2 \biggr\} </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ \frac{\partial P_\mathrm{test01}}{\partial \zeta}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \chi^0 \biggl\{ -2A_s\zeta + 2\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr]\zeta^3 - 2A_{ss} a_\ell^2(1-e^2)^{-1}\zeta^5 \biggr\} + \chi^2 \biggl\{ 2\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr](1-e^2)\zeta - 4A_{ss}a_\ell^2\zeta^3 \biggr\} + \chi^4 \biggl\{ - 2A_{ss} a_\ell^2(1-e^2)\zeta \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \zeta^1\biggl\{ - 2A_s + 2\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr](1-e^2)\chi^2 - 2A_{ss} a_\ell^2(1-e^2)\chi^4 \biggr\} + \zeta^3\biggl\{ 2\biggl[ A_{ss}a_\ell^2 + (1-e^2)^{-1}A_s\biggr] - 4A_{ss}a_\ell^2\chi^2 \biggr\} + \zeta^5\biggl\{ - 2A_{ss} a_\ell^2(1-e^2)^{-1} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \zeta^1\biggl\{ 2A_s (\chi^2- 1) + 2A_{ss}a_\ell^2(1-e^2)\chi^2 (1-\chi^2) \biggr\} + \zeta^3\biggl\{ 2A_{ss}a_\ell^2(1-2\chi^2) + 2(1-e^2)^{-1}A_s \biggr\} + \zeta^5\biggl\{ - 2A_{ss} a_\ell^2(1-e^2)^{-1} \biggr\} </math> </td> </tr> </table> Instead, try … <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\frac{P_\mathrm{test02}}{P_c}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> p_2 \biggl(\frac{\rho}{\rho_c}\biggr)^2 + p_3\biggl(\frac{\rho}{\rho_c}\biggr)^3 </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ \frac{\partial}{\partial \zeta}\biggl[\frac{P_\mathrm{test02}}{P_c}\biggr]</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> 2p_2\biggl(\frac{\rho}{\rho_c}\biggr)\frac{\partial}{\partial\zeta}\biggl[ \frac{\rho}{\rho_c} \biggr] + 3p_3\biggl(\frac{\rho}{\rho_c}\biggr)^2 \frac{\partial}{\partial\zeta}\biggl[ \frac{\rho}{\rho_c} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl(\frac{\rho}{\rho_c}\biggr)\biggl\{2p_2 + 3p_3\biggl(\frac{\rho}{\rho_c}\biggr) \biggr\} \frac{\partial}{\partial\zeta}\biggl[ \frac{\rho}{\rho_c} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl(\frac{\rho}{\rho_c}\biggr)\biggl\{2p_2 + 3p_3\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] \biggr\} \frac{\partial}{\partial\zeta}\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> \biggl(\frac{\rho}{\rho_c}\biggr)\biggl\{(2p_2 + 3p_3) - 3p_3\chi^2 - 3p_3\zeta^2(1-e^2)^{-1} \biggr\} \biggl[ - 2\zeta(1-e^2)^{-1} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl(\frac{\rho}{\rho_c}\biggr)(1-e^2)^{-2}\biggl\{ 6p_3\chi^2\zeta(1-e^2) - 2(2p_2 + 3p_3)(1-e^2)\zeta + 6p_3\zeta^3 \biggr\} </math> </td> </tr> </table> Compare the term inside the curly braces with the term, from the beginning of this subsection, inside the square brackets, namely, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math> 2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{2}{e^4} \biggl[(3-e^2) - \Upsilon \biggr]\chi^2\zeta - \biggl[\frac{4}{e^2}\biggl(1-\frac{1}{3}\Upsilon\biggr)\biggr] \zeta + \frac{4}{3e^4}\biggl[\frac{4e^2-3}{(1-e^2)} + \Upsilon \biggr] \zeta^3 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{3e^4(1-e^2)}\biggl\{ 6 \biggl[(3-e^2) - \Upsilon \biggr](1-e^2)\chi^2\zeta - \biggl[12e^2\biggl(1-\frac{1}{3}\Upsilon\biggr)\biggr](1-e^2) \zeta + 4\biggl[(4e^2-3) + \Upsilon \biggr] \zeta^3 \biggr\} \, . </math> </td> </tr> </table> <font color="red"><b>Pretty Close!!</b></font> <table border="1" align="center" width="80%" cellpadding="5"><tr><td align="left"> Alternatively: according to the third term, we need to set, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math> 6p_3 </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> 4\biggl[(4e^2-3) + \Upsilon \biggr] </math> </td> </tr> <tr> <td align="right"><math> \Rightarrow ~~~ \Upsilon </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{3}{2}p_3 + (3 - 4e^2) </math> </td> </tr> </table> in which case, the first coefficient must be given by the expression, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math> \biggl[(3-e^2) - \Upsilon \biggr] </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> (3-e^2) - \frac{3}{2}p_3 + (4e^2 - 3 ) \biggr] = \biggl[ 3e^2 - \frac{3}{2}p_3 \biggr] \, . </math> </td> </tr> </table> And, from the second coefficient, we find, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math> 2(2p_2 + 3p_3) </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[12e^2\biggl(1-\frac{1}{3}\Upsilon\biggr)\biggr]</math> </td> </tr> <tr> <td align="right"><math> \Rightarrow ~~~ 2p_2 </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> 2e^2\biggl(3-\Upsilon\biggr) - 3p_3 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> - 3p_3 + 6e^2 - 2e^2\biggl[ \frac{3}{2}p_3 + (3 - 4e^2) \biggr]</math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> - 3p_3 + 6e^2 - \biggl[ 3e^2 p_3 + 6e^2 - 8e^4 \biggr]</math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 8e^4 - 3p_3(1+e^2) \, ;</math> </td> </tr> </table> or, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>p_2</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 4e^4 - (1+e^2)\biggl[(4e^2-3) + \Upsilon \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 4e^4 - (1+e^2)(4e^2-3) - (1+e^2)\Upsilon </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 4e^4 - [4e^2-3 + 4e^4-3e^2 ] - (1+e^2)\Upsilon </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 3 - e^2 - (1+e^2)\Upsilon </math> </td> </tr> </table> ---- SUMMARY: <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\frac{P_\mathrm{test02}}{P_c}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> p_2 \biggl(\frac{\rho}{\rho_c}\biggr)^2 + p_3\biggl(\frac{\rho}{\rho_c}\biggr)^3 \, , </math> </td> </tr> <tr> <td align="right"> <math>p_2</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 3 - e^2 - (1+e^2)\Upsilon = e^4(A_{\ell s}a_\ell^2) - e^2\Upsilon \, ,</math> </td> </tr> <tr> <td align="right"><math> p_3 </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{2}{3}\biggl[(4e^2-3) + \Upsilon \biggr] = e^4(A_{ss}a_\ell^2) + \frac{2}{3}e^2\Upsilon \, . </math> </td> </tr> </table> </td></tr></table> <table border="1" align="center" width="80%" cellpadding="5"><tr><td align="left"> Note: according to the first term, we need to set, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math> p_3 </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[(3-e^2) - \Upsilon \biggr] </math> </td> </tr> <tr> <td align="right"><math> \Rightarrow ~~~ \Upsilon </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[(3-e^2) - p_3 \biggr] \, , </math> </td> </tr> </table> in which case, the third coefficient must be given by the expression, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math> 4\biggl[(4e^2-3) + \Upsilon \biggr] </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> 4\biggl[(4e^2-3) + (3-e^2) - p_3 \biggr] = 4\biggl[3e^2- p_3 \biggr] \, . </math> </td> </tr> </table> And, from the second coefficient, we find, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math> 2(2p_2 + 3p_3) </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[12e^2\biggl(1-\frac{1}{3}\Upsilon\biggr)\biggr]</math> </td> </tr> <tr> <td align="right"><math> \Rightarrow ~~~ 2p_2 </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> 2e^2\biggl(3-\Upsilon\biggr) - 3p_3 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2e^2\biggl[3-[(3-e^2) - p_3]\biggr] - 3p_3</math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2e^2\biggl[e^2 + p_3\biggr] - 3p_3</math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2e^4 + (2e^2 - 3)p_3 \, ; </math> </td> </tr> </table> or, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math> 2p_2 </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2e^4 + (2e^2 - 3)\biggl[(3-e^2) - \Upsilon \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2e^4 + (2e^2 - 3)(3-e^2) - (2e^2 - 3)\Upsilon </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2e^4 + (6e^2 - 2e^4 -9 +3e^2) - (2e^2 - 3)\Upsilon </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 9(e^2 -1 ) - (2e^2 - 3)\Upsilon </math> </td> </tr> </table> </td></tr></table> Better yet, try … <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\frac{P_\mathrm{test03}}{P_c}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> p_2 \biggl(\frac{\rho}{\rho_c}\biggr)^2 \biggl[ 1 - \beta\biggl(1 - \frac{\rho}{\rho_c} \biggr)\biggr] = p_2 \biggl(\frac{\rho}{\rho_c}\biggr)^2 \biggl[ (1 - \beta) + \beta\biggl(\frac{\rho}{\rho_c} \biggr)\biggr] </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ \frac{\partial}{\partial \zeta}\biggl[\frac{P_\mathrm{test03}}{P_c}\biggr]</math></td> <td align="center"><math>=</math></td> <td align="left"> <math>\cdots</math> </td> </tr> </table> where, in the case of a [[SSC/Structure/OtherAnalyticModels#Pressure|spherically symmetric parabolic-density configuration]], <math>\beta = 1 / 2</math>. Well … this wasn't a bad idea, but as it turns out, this "test03" expression is no different from the "test02" guess. Specifically, the "test03" expression can be rewritten as, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\frac{P_\mathrm{test03}}{P_c}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> p_2 (1 - \beta)\biggl(\frac{\rho}{\rho_c}\biggr)^2 + p_2\beta \biggl(\frac{\rho}{\rho_c}\biggr)^3 \, , </math> </td> </tr> </table> which has the same form as the "test02" expression. ====Test04==== From above, we understand that, analytically, <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> \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 - 2A_s ) - (2A_{\ell s}a_\ell^2 \chi^4 - 2A_s \chi^2)\biggr]\zeta + \biggl[ 2A_{ss} a_\ell^2 - 2A_{ss} a_\ell^2 \chi^2 - (1-e^2)^{-1}(2A_{\ell s}a_\ell^2 \chi^2 - 2A_s )\biggr]\zeta^3 + \biggl[ - (1-e^2)^{-1}2A_{ss} a_\ell^2 \biggr] \zeta^5 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ 2A_s (\chi^2-1) + 2A_{\ell s}a_\ell^2 (1 - \chi^2)\chi^2 \biggr]\zeta + \biggl[ 2A_{ss} a_\ell^2(1 - \chi^2 ) - 2A_{\ell s}a_\ell^2 (1-e^2)^{-1}\chi^2 + 2(1-e^2)^{-1}A_s \biggr]\zeta^3 + \biggl[ - 2A_{ss} a_\ell^2 (1-e^2)^{-1}\biggr] \zeta^5 \, . </math> </td> </tr> </table> Also from above, we have shown that if, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\frac{P_\mathrm{test02}}{P_c}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> p_2 \biggl(\frac{\rho}{\rho_c}\biggr)^2 + p_3\biggl(\frac{\rho}{\rho_c}\biggr)^3 </math> </td> </tr> </table> <table border="1" width="60%" align="center" cellpadding="5"><tr><td align="left"> SUMMARY from test02: <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>p_2</math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 3 - e^2 - (1+e^2)\Upsilon = e^4(A_{\ell s}a_\ell^2) - e^2\Upsilon \, ,</math> </td> </tr> <tr> <td align="right"><math> p_3 </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{2}{3}\biggl[(4e^2-3) + \Upsilon \biggr] = e^4(A_{ss}a_\ell^2) + \frac{2}{3}e^2\Upsilon \, . </math> </td> </tr> </table> </td></tr></table> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\Rightarrow ~~~ \frac{\partial}{\partial \zeta}\biggl[\frac{P_\mathrm{test02}}{P_c}\biggr]</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl(\frac{\rho}{\rho_c}\biggr)(1-e^2)^{-2}\biggl\{ 6p_3\chi^2\zeta(1-e^2) - 2(2p_2 + 3p_3)(1-e^2)\zeta + 6p_3\zeta^3 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl(\frac{\rho}{\rho_c}\biggr)(1-e^2)^{-2}\biggl\{ 6\biggl[ e^4(A_{ss}a_\ell^2) + \frac{2}{3}e^2\Upsilon \biggr]\chi^2\zeta(1-e^2) - 2\biggl[2e^4(A_{\ell s}a_\ell^2) + 3e^4(A_{ss}a_\ell^2) \biggr](1-e^2)\zeta + 6\biggl[ e^4(A_{ss}a_\ell^2) + \frac{2}{3}e^2\Upsilon \biggr]\zeta^3 \biggr\} </math> </td> </tr> </table> ---- Here (test04), we add a term that is linear in the normalized density, which means, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\frac{P_\mathrm{test04}}{P_c}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{P_\mathrm{test02}}{P_c} + p_1 \biggl(\frac{\rho}{\rho_c}\biggr) </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ \frac{\partial}{\partial \zeta}\biggl[\frac{P_\mathrm{test04}}{P_c}\biggr]</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\partial}{\partial \zeta}\biggl[\frac{P_\mathrm{test02}}{P_c}\biggr] + \frac{\partial}{\partial \zeta}\biggl[p_1 \biggl(\frac{\rho}{\rho_c}\biggr)\biggr] = \frac{\partial}{\partial \zeta}\biggl[\frac{P_\mathrm{test02}}{P_c}\biggr] + p_1 \frac{\partial}{\partial \zeta}\biggl[ 1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr] </math> </td> </tr> </table>
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