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__FORCETOC__ =Better Interface for 51BiPolytrope Stability Study= ==Content Pointing to Previous Work== ===Tilded Menu Pointers=== <ol type="I"> <li>Murphy & Fiedler (1985b): [[SSC/Stability/MurphyFiedler85|SSC/Stability/MurphyFiedler85]] <ol type="A"><li>[[SSC/Stability/MurphyFiedler85#Interface_Conditions|Interface Conditions]] as promoted by Ledoux & Walraven (1958)</li> <li>[[SSC/Stability/MurphyFiedler85#Numerical_Integration|Numerical Integration]] <ol type="1"><li>General Approach</li> <li>Special Handling at the Center</li> <li>Special Handling at the Interface</li></ol> </li> <li>[[SSC/Stability/MurphyFiedler85#Reconcile_Approaches|Reconcile Approaches]]</li> </ol> </li> <li>Excellent Foundation (no pointer from Tiled Menu): [[SSC/Stability/BiPolytropes#Radial_Oscillations_of_(nc,_ne)_=_(5,_1)_Models|SSC/Stability/Biipolytropes]]</li> <li>Our Broader Analysis: [[SSC/Stability/BiPolytropes/HeadScratching|SSC/Stability/BiPolytropes/HeadScratching]]</li> <li>Succinct Discussion: [[SSC/Stability/BiPolytropes/SuccinctDiscussion|SSC/Stability/BiPolytropes/SuccinctDiscussion]]</li> </ol> ===Ramblings: Analyzing Five-One Bipolytropes=== <ol> <li>[[Appendix/Ramblings/BiPolytropeStability|Assessing the Stability of Spherical, BiPolytropic Configurations]]</li> <li>[[Appendix/Ramblings/BiPolytrope51AnalyticStability|Searching for Analytic EigenVector for (5,1) Bipolytropes]]</li> <li>See (below) Discussing Patrick Motl's 2019 BiPolytrope Simulations</li> <li>[[Appendix/Ramblings/BiPolytrope51ContinueSearch|Continue Search]]</li> <li>[[SSC/Structure/BiPolytropes/Analytic51Renormalize|Renormalize Structure]]</li> <li>[[SSC/Structure/BiPolytropes/51RenormaizePart2|Renormalize Structure (Part 2)]]</li> <li>[[SSC/Structure/BiPolytropes/AnalyzeStepFunction|More Carefully Exam Step Function Behavior]]</li> <li>[[Appendix/Ramblings/51AnalyticStabilitySynopsis|More Focused Search]] for Analytic EigenVector if (5,1) Bipolytropes</li> <li>[[Appendix/Ramblings/51BiPolytropeStability/NoAnalytic|Do Not Confine Search to Analytic Eigenvector]]</li> <li>[[Appendix/Ramblings/51BiPolytropeStability/CleanMethodicalExaminatioin|Clean, Methodical Examination]]</li> <li>[[Appendix/Ramblings/51BiPolytropeStability/RethinkEnvelope|Rethink Handling of n = 1 Envelope]]</li> <li>[[Appendix/Ramblings/51BiPolytropeStability/BetterInterface|Improved Treatment of Core-Envelope Interface]]</li> </ol> ==Solid Foundation== Here we pull primarily from the chapters labeled II and III, above. ===Entire Configuration=== Beginning with the familiar, <div align="center" id="2ndOrderODE"> <font color="#770000">'''Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br /> {{Math/EQ_RadialPulsation01}} </div> where, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~g_0</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{G M_r^*}{(r^*)^2} \biggl[ \rho_c^{3 / 5} \biggl( \frac{K_c}{G}\biggr)^{1 / 2} \biggr] \, , </math> </td> </tr> </table> if we adopt the variable normalizations, <div align="center"> <table border="0" cellpadding="3"> <tr> <td align="right"> <math>~\rho^*</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~\frac{\rho_0}{\rho_c}</math> </td> <td align="center">; </td> <td align="right"> <math>~r^*</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~\frac{r_0}{[K_c^{1/2}/(G^{1/2}\rho_c^{2/5})]}</math> </td> </tr> <tr> <td align="right"> <math>~P^*</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~\frac{P_0}{K_c\rho_c^{6/5}}</math> </td> <td align="center">; </td> <td align="right"> <math>~M_r^*</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_c^{1/5})]}</math> </td> </tr> </table> </div> the LAWE takes the form, <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>~ \frac{d^2x}{dr*^2} + \frac{\mathcal{H}}{r^*} \frac{dx}{dr*} + \biggl[\biggl(\frac{\sigma_c^2}{\gamma_g}\biggr) \mathcal{K}_1 - \alpha_g \mathcal{K}_2\biggr] x \, , </math> </td> </tr> </table> where, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\mathcal{H}</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~ \biggl\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\} </math> </td> <td align="center"> , </td> <td align="right"> <math>~\mathcal{K}_1</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~ \frac{2\pi }{3}\biggl(\frac{\rho^*}{ P^* } \biggr) </math> </td> <td align="center"> and </td> <td align="right"> <math>~\mathcal{K}_2</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~ \biggl(\frac{\rho^*}{ P^* } \biggr)\frac{M_r^*}{(r^*)^3} \, , </math> </td> </tr> <tr> <td align="right"> <math>\sigma_c^2</math> </td> <td align="center"> <math>\equiv</math> </td> <td align="left"> <math> \frac{3\omega^2}{2\pi G\rho_c} </math> </td> <td align="center"> , </td> <td align="right"> <math>\alpha_g</math> </td> <td align="center"> <math>\equiv</math> </td> <td align="left"> <math> \biggl(3 - \frac{4}{\gamma_g}\biggr) \, . </math> </td> <td align="center" colspan="4"> </td> </tr> </table> ===Core=== Given that, in the core, <math>\gamma_g = 6/5</math> and, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>r^*</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \xi \, , </math> </td> </tr> </table> we can rewrite the LAWE to read, <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> \frac{d^2x}{d\xi^2} + \frac{\mathcal{H}}{\xi} \frac{dx}{d\xi} + \biggl(\frac{1}{4\pi}\biggr)\biggl[5\sigma_c^2 \mathcal{K}_1 + 2 \mathcal{K}_2\biggr] x \, , </math> </td> </tr> </table> where, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\mathcal{K}_1</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{2\pi }{3}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2} \, , </math> </td> </tr> <tr> <td align="right"> <math>\mathcal{H}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 4 - 2 \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \, , </math> </td> </tr> <tr> <td align="right"> <math>\mathcal{K}_2</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl(\frac{4\pi}{3}\biggr) \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \, . </math> </td> </tr> </table> ===Structure at the Interface=== Once <math>\mu_e/\mu_c</math> and <math>\xi_i</math> have been specified, other [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|parameter values at the interface]] are: <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\theta_i</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl( 1 + \frac{1}{3}\xi^2_i \biggr)^{-1 / 2} \, , </math> </td> </tr> <tr> <td align="right"> <math>\eta_i</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl(\frac{\mu_e}{\mu_c}\biggr) \sqrt{3}~\theta_i^2 \xi_i \, , </math> </td> </tr> <tr> <td align="right"> <math>\Lambda_i</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}} \, , </math> </td> </tr> <tr> <td align="right"> <math>A</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \eta_i(1+\Lambda_i^2)^{1 / 2} \, , </math> </td> </tr> <tr> <td align="right"> <math>B</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \eta_i - \frac{\pi}{2} + \tan^{-1}(\Lambda_i) \, , </math> </td> </tr> <tr> <td align="right"> <math>\eta_s</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> B + \pi \, . </math> </td> </tr> </table> ===Linearized Perturbation at the Interface=== At all radial locations throughout the equilibrium configuration, the three spatially dependent quantities — <math>p \equiv \delta p/P^*, d \equiv \delta\rho/\rho^*,</math> and <math>x \equiv \delta r/r^*</math> — are related to one another via the [[SSC/Perturbations#Summary_Set_of_Linearized_Equations|set of linearized governing relations]], namely, <div align="center"> <table border="1" cellpadding="10"> <tr><td align="center"> <font color="#770000">'''Linearized'''</font><br /> <span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br /> <math> r_0 \frac{dx}{dr_0} = - 3 x - d , </math><br /> <font color="#770000">'''Linearized'''</font><br /> <span id="PGE:Euler"><font color="#770000">'''Euler + Poisson Equations'''</font></span><br /> <math> \frac{P_0}{\rho_0} \frac{dp}{dr_0} = (4x + p)g_0 + \omega^2 r_0 x , </math><br /> <font color="#770000">'''Linearized'''</font><br /> <span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br /> <font color="#770000">'''First Law of Thermodynamics'''</font></span><br /> <math> p = \gamma_\mathrm{g} d \, . </math> </td></tr> </table> </div> Combining the 2<sup>nd</sup> and 3<sup>rd</sup> equations, we find, <table border="0" align="center" cellpadding="8"> <tr> <td align="right"><math>(4x + \gamma_g d)g_0 + \omega^2 r_0 x</math></td> <td align="center"><math>=</math></td> <td align="left"><math>\frac{P_0}{\rho_0} \frac{d (\gamma_g d)}{dr_0}</math></td> </tr> </table> ---- At the interface, presumably the ''dimensional'' structural variables, <math>P^*</math> and <math>r^*</math> have the same values, whether viewed from the perspective of the core or from the perspective of the envelope. But <math>\rho^*</math> has a different value, depending on the point of view. Specifically, <table border="0" align="center" cellpadding="8"> <tr> <td align="right"><math>\rho^*\biggr|_\mathrm{env}</math></td> <td align="center"><math>=</math></td> <td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)\rho^*\biggr|_\mathrm{core}\, .</math></td> </tr> </table> Hence, from the perspective of the core, the linearized equation of continuity may be written as, <table border="0" align="center" cellpadding="8"> <tr> <td align="right"><math>\biggl[ r_0 \cdot \frac{dx}{dr_0} + 3x \biggr]_\mathrm{core}</math></td> <td align="center"><math>=</math></td> <td align="left"><math>- \frac{\delta\rho}{[\rho^*]_\mathrm{core}} \, ;</math></td> </tr> </table> while, from the perspective of the envelope, the linearized equation of continuity may be written as, <table border="0" align="center" cellpadding="8"> <tr> <td align="right"><math>\biggl[ r_0 \cdot \frac{dx}{dr_0} + 3x \biggr]_\mathrm{env}</math></td> <td align="center"><math>=</math></td> <td align="left"><math>- \frac{\delta\rho}{[\rho^*]_\mathrm{core}} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}</math></td> </tr> </table> ---- <font color="red">Try again</font> From [[SSC/Perturbations#Entropy_Conservation|here]], we know … <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math> \frac{P}{\rho^{\gamma_g}} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \exp\biggl[ \frac{s(\gamma_g - 1)}{\mathfrak{R}/\bar\mu}\biggr]\, . </math> </td> </tr> </table> And, from my [[Appendix/Ramblings/PatrickMotl#Understanding_the_Step_Function_at_the_Core-Envelope_Interface|discussions with Patrick Motl]], we find … <font color="red"><b>CORE:</b></font> Throughout the core, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~P^*</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl(1+ \frac{\xi^2}{3}\biggr)^{-3}</math> </td> <td align="center"> and </td> <td align="right"> <math>~\rho^*</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl(1+ \frac{\xi^2}{3}\biggr)^{-5/2}</math> </td> <td align="center"> and </td> <td align="right"> <math>~\gamma_g</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{6}{5} \, .</math> </td> </tr> </table> Hence, independent of the radial location, <math>~\xi</math>, throughout the core, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{s}{\Re/\bar{\mu}}\biggr|_\mathrm{core}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 5\ln (5) \, . </math> </td> </tr> </table> <font color="red"><b>ENVELOPE:</b></font> Throughout the envelope, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~P^*</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\theta_i^6 [\phi(\eta)]^2</math> </td> <td align="center"> and </td> <td align="right"> <math>~\rho^*</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl(\frac{\mu_e}{\mu_c}\biggr)\theta_i^5 [\phi(\eta)]</math> </td> <td align="center"> and </td> <td align="right"> <math>~\gamma_g</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~2 \, .</math> </td> </tr> </table> Hence, independent of the radial location, <math>~\eta</math>, throughout the envelope, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{s}{\Re/\bar{\mu}}\biggr|_\mathrm{env}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \ln \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \theta_i^{-4} \biggr] </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \ln \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl( 1 + \frac{\xi_i^2}{3}\biggr)^2 \biggr] \, . </math> </td> </tr> </table> ===Envelope=== Given that, throughout the envelope <math>\gamma_g = 2</math> and, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>r^*</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2}\eta \, ,</math> </td> </tr> </table> we can rewrite the LAWE to read, <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> \frac{d^2x}{d\eta^2} + \frac{\mathcal{H}}{\eta} \frac{dx}{d\eta} + \frac{1}{2\pi \theta_i^4}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl[\biggl(\frac{\sigma_c^2}{2}\biggr) \mathcal{K}_1 - \mathcal{K}_2\biggr] x\, , </math> </td> </tr> </table> where, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\mathcal{K}_1</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>~ \frac{2\pi }{3}\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr] \, , </math> </td> </tr> <tr> <td align="right"> <math>\mathcal{H}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>~ 2\biggl[1 + \frac{\eta}{\tan(\eta-B)}\biggr] \, , </math> </td> </tr> <tr> <td align="right"> <math>\mathcal{K}_2</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 4\pi \biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \theta_i^4 \biggl[1 - \frac{\eta}{\tan(\eta-B)}\biggr]\frac{1}{\eta^2} \, . </math> </td> </tr> </table> =Entropy as a Step Function= Useful Chapters: <ul> <li>[[Appendix/Mathematics/StepFunction|Appendix/Mathematics/Stepfunction]]</li> <li>[[SSC/Structure/BiPolytropes/Analytic51Renormalize#From_Step-Function_Behavior_of_Specific_Entropy|Step-Function Behavior of Specific Entropy]]</li> </ul> ==Review== The unit — or, [https://en.wikipedia.org/wiki/Heaviside_step_function Heaviside] — step function, <math>H(x)</math>, is defined such that, <table border="0" align="center" cellpadding="8"> <tr> <td align="center"> <math> H(x) = \begin{cases} 0; & ~~ x < 0 \\ 1; & ~~ x > 0 \end{cases} </math> <p><br /> [<b>[[Appendix/References#MF53|<font color="red">MF53</font>]]</b>], Part I, §2.1 (p. 123), Eq. (2.1.6) </td> <td align="center" rowspan="2"> [[File:Heaviside01.png|300px|Heaviside Function]] </td> </tr> </table> In evaluating this function at <math>x=0</math>, we will adopt the ''half-maximum convention'' and set <math>H(0) = \tfrac{1}{2}</math>. As has been pointed out in, for example, [https://en.wikipedia.org/wiki/Heaviside_step_function a relevant Wikipedia discussion], the derivative of the unit step function is, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{dH(x)}{dx}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>\delta(x) \, ,</math> </td> </tr> </table> where, <math>\delta(x)</math> is the Dirac Delta function. ==Perturbed Density== Let, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\zeta</math> </td> <td align="center"> <math>\equiv</math> </td> <td align="left"> <math>\frac{r}{r_i} - 1 </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ [\rho^*]_\mathrm{eq}</math> </td> <td align="center"> <math>\equiv</math> </td> <td align="left"> <math>H(\zeta)\cdot \rho^*_\mathrm{env} + H(-\zeta) \cdot \rho^*_\mathrm{core} \, ;</math> </td> </tr> </table> and, more generally after a perturbation, <math>\delta\rho(\zeta)</math>, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\rho^* = [\rho^*]_\mathrm{eq} + \delta\rho</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>[\rho^*]_\mathrm{eq} \biggl\{1 + \frac{\delta\rho}{[\rho^*]_\mathrm{eq}}\biggr\} = [\rho^*]_\mathrm{eq} \biggl\{1 + de^{i\omega t}\biggr\} \, .</math> </td> </tr> </table> Hence, in the [[SSC/Perturbations#Continuity_Equation|linearized version of the continuity equation]], we recognize that, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>d e^{i \omega t}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>\frac{\delta\rho}{[\rho^*]_\mathrm{eq}} \, .</math> </td> </tr> </table> <font color="red"><b>CORE:</b></font> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\rho^*_\mathrm{core}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl(1+ \frac{\xi^2}{3}\biggr)^{-5/2}</math> </td> <td align="center"> and </td> <td align="right"> <math>~\gamma_g</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{6}{5} </math> </td> <td align="center"> and </td> <td align="right"> <math>~\mathcal{S}_\mathrm{core} \equiv \frac{s(\gamma_g-1)}{\Re/\bar{\mu}}\biggr|_\mathrm{core}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \ln (5) \, . </math> </td> </tr> </table> <font color="red"><b>ENVELOPE:</b></font> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\rho^*_\mathrm{env}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl(\frac{\mu_e}{\mu_c}\biggr)\theta_i^5 [\phi(\eta)]</math> </td> <td align="center"> and </td> <td align="right"> <math>~\gamma_g</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~2 </math> </td> <td align="center"> and </td> <td align="right"> <math>~\mathcal{S}_\mathrm{env} \equiv \frac{s(\gamma_g-1)}{\Re/\bar{\mu}}\biggr|_\mathrm{env}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \ln \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl( 1 + \frac{\xi_i^2}{3}\biggr)^2 \biggr] \, . </math> </td> </tr> </table> ==Perturbed Pressure== <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>P^*</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>(\rho^*)^{\gamma_g} \cdot \exp\biggl[ \frac{s(\gamma_g-1)}{\Re/\bar{\mu}} \biggr]</math> </td> </tr> </table> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>[P^*]_\mathrm{eq}</math> </td> <td align="center"> <math>\equiv</math> </td> <td align="left"> <math>H(\zeta)\cdot P^*_\mathrm{env} + H(-\zeta) \cdot P^*_\mathrm{core} \, ;</math> </td> </tr> </table> and, more generally after a perturbation, <math>\delta P(\zeta)</math>, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>P^* = [P^*]_\mathrm{eq} + \delta P</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>[P^*]_\mathrm{eq} \biggl\{1 + \frac{\delta P}{[P^*]_\mathrm{eq}}\biggr\} = [P^*]_\mathrm{eq} \biggl\{1 + pe^{i\omega t}\biggr\} \, .</math> </td> </tr> </table> Hence, in the [[SSC/Perturbations#Adiabatic_form_of_the_First_Law_of_Thermodynamics|linearized version of the first law of thermodynamics]], we recognize that, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>p e^{i \omega t}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>\frac{\delta P}{[P^*]_\mathrm{eq}} \, .</math> </td> </tr> </table> ==Obtaining Perturbed Density from Perturbed Pressure== Given that, quite generally, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math> \frac{P}{\rho^{\gamma_g}} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \exp\biggl[ \frac{s(\gamma_g - 1)}{\mathfrak{R}/\bar\mu}\biggr]\, , </math> </td> </tr> </table> let's define the density-like quantity, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\mathcal{Q}^*</math> </td> <td align="center"> <math>\equiv</math> </td> <td align="left"> <math> e^{-\mathcal{S/\gamma_g}}\biggl[P^*\biggr]^{1/\gamma_g} \, ,</math> </td> </tr> </table> in which case, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\rho^*_\mathrm{eq}</math> </td> <td align="center"> <math>\equiv</math> </td> <td align="left"> <math> H(\zeta)\cdot [ \mathcal{Q}^*_\mathrm{eq}]_\mathrm{env} + H(-\zeta) \cdot [ \mathcal{Q}^*_\mathrm{eq}]_\mathrm{core} \, .</math> </td> </tr> </table> What happens if we perturb the pressure? In either region (core or envelope), <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\mathcal{Q}^*</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> e^{-\mathcal{S/\gamma_g}} [P^*_\mathrm{eq}]^{1/\gamma_g}\biggl[ 1 + \frac{\delta P}{P^*_\mathrm{eq}}\biggr]^{1/\gamma_g} \approx e^{-\mathcal{S/\gamma_g}} [P^*_\mathrm{eq}]^{1/\gamma_g}\biggl[ 1 + \frac{1}{\gamma_g} \cdot \frac{\delta P}{P^*_\mathrm{eq}}\biggr] = [Q^*_\mathrm{eq}]\biggl[ 1 + \frac{1}{\gamma_g} \cdot \frac{\delta P}{P^*_\mathrm{eq}}\biggr] \, .</math> </td> </tr> </table> As a result, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\rho^* = \rho^*_\mathrm{eq} + \delta \rho</math> </td> <td align="center"> <math>\equiv</math> </td> <td align="left"> <math> H(\zeta)\cdot [ \mathcal{Q}^*]_\mathrm{env} + H(-\zeta) \cdot [ \mathcal{Q}^*]_\mathrm{core} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> H(\zeta)\cdot \biggl\{[Q^*_\mathrm{eq}]\biggl[ 1 + \frac{1}{\gamma_g} \cdot \frac{\delta P}{P^*_\mathrm{eq}}\biggr]\biggr\}_\mathrm{env} + H(-\zeta) \cdot \biggl\{[Q^*_\mathrm{eq}]\biggl[ 1 + \frac{1}{\gamma_g} \cdot \frac{\delta P}{P^*_\mathrm{eq}}\biggr]\biggr\}_\mathrm{core} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> H(\zeta)\cdot [Q^*_\mathrm{eq}]_\mathrm{env} + H(-\zeta) \cdot [Q^*_\mathrm{eq}]_\mathrm{core} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> +~ H(\zeta)\cdot \biggl\{[Q^*_\mathrm{eq}]\biggl[ \frac{1}{\gamma_g} \cdot \frac{\delta P}{P^*_\mathrm{eq}}\biggr]\biggr\}_\mathrm{env} + H(-\zeta) \cdot \biggl\{[Q^*_\mathrm{eq}]\biggl[ \frac{1}{\gamma_g} \cdot \frac{\delta P}{P^*_\mathrm{eq}}\biggr]\biggr\}_\mathrm{core} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \rho^*_\mathrm{eq} +~ H(\zeta)\cdot \biggl\{[Q^*_\mathrm{eq}]\biggl[ \frac{1}{\gamma_g} \cdot \frac{\delta P}{P^*_\mathrm{eq}}\biggr]\biggr\}_\mathrm{env} + H(-\zeta) \cdot \biggl\{[Q^*_\mathrm{eq}]\biggl[ \frac{1}{\gamma_g} \cdot \frac{\delta P}{P^*_\mathrm{eq}}\biggr]\biggr\}_\mathrm{core} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \frac{\delta\rho}{\rho^*_\mathrm{eq}}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{H(\zeta)}{\rho^*_\mathrm{eq}} \cdot \biggl\{[Q^*_\mathrm{eq}]\biggl[ \frac{1}{\gamma_g} \cdot \frac{\delta P}{P^*_\mathrm{eq}}\biggr]\biggr\}_\mathrm{env} + \frac{H(-\zeta)}{\rho^*_\mathrm{eq}} \cdot \biggl\{[Q^*_\mathrm{eq}]\biggl[ \frac{1}{\gamma_g} \cdot \frac{\delta P}{P^*_\mathrm{eq}}\biggr]\biggr\}_\mathrm{core} </math> </td> </tr> </table> ==Set of Linearized Equations== Borrowing from [[SSC/Perturbations#Summary_Set_of_Linearized_Equations|an accompanying discussion]], we have … <div align="center"> <table border="1" cellpadding="10"> <tr><td align="center"> <font color="#770000">'''Linearized'''</font><br /> <span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br /> <math> r_0 \frac{dx}{dr_0} = - 3 x - d , </math><br /> <font color="#770000">'''Linearized'''</font><br /> <span id="PGE:Euler"><font color="#770000">'''Euler + Poisson Equations'''</font></span><br /> <math> \frac{P_0}{\rho_0} \frac{dp}{dr_0} = (4x + p)g_0 + \omega^2 r_0 x , </math><br /> <font color="#770000">'''Linearized'''</font><br /> <span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br /> <font color="#770000">'''First Law of Thermodynamics'''</font></span><br /> <math> p = \gamma_\mathrm{g} d \, . </math> </td></tr> </table> </div> Rearranging terms in the "Linearized Euler + Poisson Equations" as follows … <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{1}{\rho_0} \cdot \frac{dp}{dr_0}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{P_0}\biggl[ (4x + p)g_0 + \omega^2 r_0 x \biggr] </math> </td> </tr> </table> we realize that the expression on the RHS has the same value at the interface, whether you're viewing the equation from the point of view of the core or the envelope; and we recognize as well that <math>\rho_0</math> is a simple step function at the interface. Hence, letting a prime indicate differentiation with respect to <math>r_0</math>, we can write, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math> H(\zeta)\cdot (p')_\mathrm{env} + H(-\zeta)\cdot (p')_\mathrm{core} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{[\rho_0]_\mathrm{core}}{P_0}\biggl[ (4x + p)g_0 + \omega^2 r_0 x \biggr] \biggl\{ H(\zeta)\cdot \biggl[ \frac{\mu_e}{\mu_c} \biggr] + H(-\zeta) \biggr\} \, . </math> </td> </tr> </table> Analogously, the "Linearized Equation of Continuity" can be rewritten as, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math> H(\zeta)\cdot (x')_\mathrm{env} + H(-\zeta)\cdot (x')_\mathrm{core} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - \frac{3x}{r_0} - \frac{1}{r_0}\biggl\{ H(\zeta)\cdot \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env} + H(-\zeta)\biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} \biggr\} \, . </math> </td> </tr> </table> Now, given that <math>\zeta = (r_0/r_i - 1)</math>, we see that, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math> \frac{dH(\zeta)}{dr_0} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{\partial H(\zeta)}{\partial \zeta} \cdot \frac{d\zeta}{dr_0} = r_o^{-1} \delta(\zeta) \, ; </math> </td> </tr> </table> and, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math> \frac{dH(-\zeta)}{dr_0} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> -\frac{\partial H(\zeta)}{\partial \zeta} \cdot \frac{d\zeta}{dr_0} = -r_o^{-1} \delta(\zeta) \, . </math> </td> </tr> </table> Hence, differentiation of the "Linearized Equation of Continuity" gives, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math> \frac{d}{dr_0}\biggl[H(\zeta)\cdot (x')_\mathrm{env}\biggr] + \frac{d}{dr_0}\biggl[H(-\zeta)\cdot (x')_\mathrm{core}\biggr] </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - \biggl[ \frac{3}{r_0} \cdot x' - \frac{3x}{r_0^2}\biggr] - \frac{1}{r_0} \frac{d}{dr_0}\biggl\{ H(\zeta)\cdot \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env} + H(-\zeta)\biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} \biggr\} + \frac{1}{r_0^2}\biggl\{ H(\zeta)\cdot \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env} + H(-\zeta)\biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} \biggr\} \, . </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ (x')_\mathrm{env}\frac{d}{dr_0}\biggl[H(\zeta)\biggr] + H(\zeta)\cdot (x'')_\mathrm{env} + (x')_\mathrm{core}\frac{d}{dr_0}\biggl[H(-\zeta)\biggr] + H(-\zeta)\cdot (x'')_\mathrm{core} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - \biggl[ \frac{3}{r_0} \cdot x' - \frac{3x}{r_0^2}\biggr] + \frac{1}{r_0^2}\biggl\{ H(\zeta)\cdot \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env} + H(-\zeta)\biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} \biggr\} \, . </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - \frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env}\frac{d}{dr_0}\biggl\{H(\zeta) \biggr\} - \frac{1}{r_0} \biggl\{ H(\zeta)\cdot \biggl[ \frac{p'}{\gamma_g} \biggr]_\mathrm{env} \biggr\} - \frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} \frac{d}{dr_0}\biggl\{H(-\zeta) \biggr\} - \frac{1}{r_0} \biggl\{H(-\zeta)\biggl[ \frac{p'}{\gamma_g} \biggr]_\mathrm{core} \biggr\} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ (x')_\mathrm{env} \biggl[\frac{\delta(\zeta)}{r_0}\biggr] + H(\zeta)\cdot (x'')_\mathrm{env} - (x')_\mathrm{core}\biggl[\frac{\delta(\zeta)}{r_0}\biggr] + H(-\zeta)\cdot (x'')_\mathrm{core} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - \biggl[ \frac{3}{r_0} \cdot x' - \frac{3x}{r_0^2}\biggr] + \frac{1}{r_0^2}\biggl\{ H(\zeta)\cdot \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env} + H(-\zeta)\biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} \biggr\} \, . </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - \frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env} \biggl\{\frac{\delta(\zeta)}{r_0} \biggr\} - \frac{1}{r_0} \biggl\{ H(\zeta)\cdot \biggl[ \frac{p'}{\gamma_g} \biggr]_\mathrm{env} \biggr\} + \frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} \biggl\{\frac{\delta(\zeta)}{r_0} \biggr\} - \frac{1}{r_0} \biggl\{H(-\zeta)\biggl[ \frac{p'}{\gamma_g} \biggr]_\mathrm{core} \biggr\} </math> </td> </tr> </table> ===From the Perspective of the Core=== When <math>r_0/r_i \le 1</math> — that is, from the perspective of the core while including the interface, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math> - (x')_\mathrm{core}\biggl[\frac{\delta(\zeta)}{r_0}\biggr] + H(-\zeta)\cdot (x'')_\mathrm{core} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - \biggl[ \frac{3}{r_0} \cdot x' - \frac{3x}{r_0^2}\biggr] + \frac{1}{r_0^2}\biggl\{ H(-\zeta)\biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} \biggr\} \, . </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - \frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env} \biggl\{\frac{\delta(\zeta)}{r_0} \biggr\} + \frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} \biggl\{\frac{\delta(\zeta)}{r_0} \biggr\} - \frac{1}{r_0} \biggl\{H(-\zeta)\biggl[ \frac{p'}{\gamma_g} \biggr]_\mathrm{core} \biggr\} </math> </td> </tr> </table> And examining only the interface, where <math>\delta(\zeta) = 1</math> while <math>H(-\zeta) = 0</math>, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math> - (x')_\mathrm{core}\biggl[\frac{\delta(\zeta)}{r_0}\biggr] </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - \biggl[ \frac{3}{r_0} \cdot x' - \frac{3x}{r_0^2}\biggr] - \frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env} \biggl\{\frac{\delta(\zeta)}{r_0} \biggr\} + \frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} \biggl\{\frac{\delta(\zeta)}{r_0} \biggr\} </math> </td> </tr> <tr> <td align="right"> <math> \Rightarrow ~~~ - (x')_\mathrm{core} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - \biggl[ 3x' - \frac{3x}{r_0}\biggr] - \frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env} + \frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} </math> </td> </tr> </table> ===From the Perspective of the Envelope=== When <math>r_0/r_i \ge 1</math> — that is, from the perspective of the envelope while including the interface, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math> (x')_\mathrm{env} \biggl[\frac{\delta(\zeta)}{r_0}\biggr] + H(\zeta)\cdot (x'')_\mathrm{env} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - \biggl[ \frac{3}{r_0} \cdot x' - \frac{3x}{r_0^2}\biggr] + \frac{1}{r_0^2}\biggl\{ H(\zeta)\cdot \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env} \biggr\} \, . </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - \frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env} \biggl\{\frac{\delta(\zeta)}{r_0} \biggr\} - \frac{1}{r_0} \biggl\{ H(\zeta)\cdot \biggl[ \frac{p'}{\gamma_g} \biggr]_\mathrm{env} \biggr\} + \frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} \biggl\{\frac{\delta(\zeta)}{r_0} \biggr\} </math> </td> </tr> </table> ==Focus on Nonlinear Continuity Equation== A spherical shell of ''core'' density, <math>\rho_0</math>, where the inner radius of the shell is <math>(r_0 - \Delta r)</math> and its outer radius is <math>(r_0+\Delta r)</math> has a ''shell'' mass given by the expression, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>(\Delta m)_\mathrm{core}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{4\pi}{3} \cdot \rho_0 \biggl[ (r_0+\Delta r)^3 - (r_0 - \Delta r)^3 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{4\pi}{3} \cdot \rho_0 r_0^3 \biggl[ \biggl(1 + \frac{\Delta r}{r_0} \biggr)^3 - \biggl(1 - \frac{\Delta r}{r_0} \biggr)^3 \biggr] </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \biggl[ \frac{3}{4\pi \rho_0 r_0^3} \biggr](\Delta m)_\mathrm{core}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[1 + \frac{2\Delta r}{r_0} + \biggl(\frac{\Delta r}{r_0}\biggr)^2\biggr] \biggl(1 + \frac{\Delta r}{r_0} \biggr) - \biggl[1 - \frac{2\Delta r}{r_0} + \biggl(\frac{\Delta r}{r_0}\biggr)^2\biggr] \biggl(1 - \frac{\Delta r}{r_0} \biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[1 + \frac{3\Delta r}{r_0} + 3\biggl(\frac{\Delta r}{r_0}\biggr)^2 + \biggl(\frac{\Delta r}{r_0}\biggr)^3\biggr] - \biggl[1 - \frac{3\Delta r}{r_0} + 3\biggl(\frac{\Delta r}{r_0}\biggr)^2 - \biggl(\frac{\Delta r}{r_0}\biggr)^3\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[ \frac{6\Delta r}{r_0} + 2\biggl(\frac{\Delta r}{r_0}\biggr)^3\biggr] </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ (\Delta m)_\mathrm{core}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[ \frac{4\pi \rho_0 r_0^3}{3} \biggr] \biggl[ \frac{6\Delta r}{r_0} + 2\biggl(\frac{\Delta r}{r_0}\biggr)^3\biggr] \, . </math> </td> </tr> </table> Similarly, as viewed from the perspective of the envelope, it has a shell mass of, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>(\Delta m)_\mathrm{env}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{4\pi}{3} \cdot \biggl(\frac{\mu_e}{\mu_c}\biggr) \rho_0 \biggl[ (r_0+\Delta r)^3 - (r_0 - \Delta r)^3 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl(\frac{\mu_e}{\mu_c}\biggr) \biggl[ \frac{4\pi \rho_0 r_0^3}{3} \biggr] \biggl[ \frac{6\Delta r}{r_0} + 2\biggl(\frac{\Delta r}{r_0}\biggr)^3\biggr] \, . </math> </td> </tr> </table> Better yet, pick the two edges of the shell, <math>r_+</math> and <math>r_-</math>, and let <math>r_0 \equiv (r_+ + r_-)/2</math> and <math>\Delta r = (r_+ - r_-)/2</math>. Given the value of <math>\rho_0</math>, the unperturbed mass in the shell is given by the above expression, that is, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>(\Delta m)_0</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[ \frac{4\pi \rho_0 r_0^3}{3} \biggr] \biggl[ \frac{6\Delta r}{r_0} + 2\biggl(\frac{\Delta r}{r_0}\biggr)^3\biggr] \, . </math> </td> </tr> </table> Now, let <math>r_+ \rightarrow (r_+)_0 + (\delta r_+), ~r_- \rightarrow (r_-)_0 + (\delta r_-), </math> and, <math>\rho_0 \rightarrow (\rho_0)_0 + (\delta \rho)</math>. We then have, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>2r_0</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> [(r_+)_0 + (\delta r_+)] + [(r_-)_0 + (\delta r_-)] = 2(r_0)_0 + [\delta r_+ + \delta r_-] \, ; </math> </td> </tr> <tr> <td align="right"> <math>2\Delta r</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> [(r_+)_0 + (\delta r_+)] - [(r_-)_0 + (\delta r_-)] = 2(\Delta r)_0 + [\delta r_+ - \delta r_-] \, ; </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \frac{\Delta r}{r_0}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{2(\Delta r)_0 + [\delta r_+ - \delta r_-]}{ 2(r_0)_0 + [\delta r_+ + \delta r_-] } = \frac{(\Delta r)_0}{(r_0)_0} \times \biggl\{1 + \biggl[ \frac{\delta r_+ - \delta r_-}{2(\Delta r)_0} \biggr] \biggr\} \biggl\{1 + \biggl[ \frac{\delta r_+ + \delta r_-}{2(r_0)_0} \biggr] \biggr\}^{-1} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>\approx</math> </td> <td align="left"> <math> \frac{(\Delta r)_0}{(r_0)_0} \times \biggl\{1 + \biggl[ \frac{\delta r_+ - \delta r_-}{2(\Delta r)_0} \biggr] \biggr\} \biggl\{1 - \biggl[ \frac{\delta r_+ + \delta r_-}{2(r_0)_0} \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>\approx</math> </td> <td align="left"> <math> \frac{(\Delta r)_0}{(r_0)_0} \times \biggl\{1 + \biggl[ \frac{\delta r_+ - \delta r_-}{2(\Delta r)_0} \biggr] - \biggl[ \frac{\delta r_+ + \delta r_-}{2(r_0)_0} \biggr] \biggr\} </math> </td> </tr> </table> In order for the shell to have the same <math>\Delta m</math> in both the unperturbed and perturbed cases, the following relation must hold: <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math> \biggl[ \frac{8\pi (\rho_0)_0 (r_0)_0^3}{3} \biggr] \biggl\{ \frac{3(\Delta r)_0}{(r_0)_0} + \biggl[ \frac{(\Delta r)_0}{(r_0)_0}\biggr ]^3 \biggr\} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl\{ \frac{\pi [(\rho_0)_0 + \delta\rho] [2r_0]^3}{2\cdot 3} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> \times 2\biggl\{ \frac{3\Delta r}{r_0} + \biggl(\frac{\Delta r}{r_0}\biggr)^3\biggr\} </math> </td> </tr> </table> =Work With Pair of First-Order Linearized Equations= ==Equilibrium Structures Using Preferred Normalizations== Working from our [[SSC/Structure/BiPolytropes/Analytic51Renormalize#BiPolytrope_with_(nc,_ne)_=_(5,_1)|earlier "new" normalization]] — which was done in the context of our [[SSC/Structure/BiPolytropes/Analytic51Renormalize#Model_Pairings|examination of the B-KB74 conjecture]] — that is, by setting, <table border="1" align="center" width="80%" cellpadding="5"><tr><td align="left"> <div align="center"><b>New Normalization</b></div> <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\tilde\rho</math></td> <td align="center"><math>\equiv</math></td> <td align="left"><math>\rho \biggl[\biggl( \frac{K_c}{G} \biggr)^{3 / 2} \frac{1}{M_\mathrm{tot}} \biggr]^{-5} \, ;</math></td> </tr> <tr> <td align="right"><math>\tilde{P}</math></td> <td align="center"><math>\equiv</math></td> <td align="left"><math>P \biggl[K_c^{-10} G^{9} M_\mathrm{tot}^{6} \biggr] \, ;</math></td> </tr> <tr> <td align="right"><math>\tilde{r}</math></td> <td align="center"><math>\equiv</math></td> <td align="left"><math>r \biggl[\biggl( \frac{K_c}{G} \biggr)^{5 / 2} M_\mathrm{tot}^{-2} \biggr]\, ,</math></td> </tr> <tr> <td align="right"><math>\tilde{M}_r</math></td> <td align="center"><math>\equiv</math></td> <td align="left"><math>\frac{M_r}{M_\mathrm{tot}} \, ;</math></td> </tr> <tr> <td align="right"><math>\tilde{H}</math></td> <td align="center"><math>\equiv</math></td> <td align="left"><math>H \biggl[K_c^{-5 / 2} G^{3 / 2} M_\mathrm{tot} \biggr] \, .</math></td> </tr> </table> </td></tr></table> and after adopting the notation, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\mathcal{m}_\mathrm{surf}</math></td> <td align="center"><math>\equiv</math></td> <td align="left"><math>\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \theta_i^{-1}\biggl(-\eta^2 \frac{d\phi}{d\eta}\biggr)_s = \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i} \, ,</math></td> </tr> </table> (see [[#interfaceValues|definitions of <math>A</math>, <math>\eta_s</math>, and <math>\theta_i</math> given below]]) we have throughout the core, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\tilde{\rho}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math>\mathcal{m}_\mathrm{surf}^5 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-10} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \, ;</math> </td> </tr> <tr> <td align="right"><math>\tilde{P}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math>\mathcal{m}_\mathrm{surf}^6 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3} \, ;</math> </td> </tr> <tr> <td align="right"><math>\tilde{r}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math>\mathcal{m}_\mathrm{surf}^{-2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi \, ;</math> </td> </tr> <tr> <td align="right"><math>\tilde{M}_r</math></td> <td align="center"><math>=</math></td> <td align="left"> <math>\mathcal{m}_\mathrm{surf}^{-1} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \, .</math> </td> </tr> </table> <table border="1" width="80%" align="center" cellpadding="8"><tr><td align="left"> [[#FromEarlier|For later use]], note that, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\frac{\tilde{\rho}}{\tilde{P}} \cdot \frac{\tilde{M}_r}{\tilde{r}^2}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ \mathcal{m}_\mathrm{surf}^5 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-10} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \biggr] \biggl[ \mathcal{m}_\mathrm{surf}^{-6} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{12} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{3} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>\times \biggl[ \mathcal{m}_\mathrm{surf}^{-1} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr] \biggl[ \mathcal{m}_\mathrm{surf}^{4} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-8} \biggl(\frac{3}{2\pi}\biggr)^{-1} \xi^{-2}\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \mathcal{m}_\mathrm{surf}^2 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-4} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \xi~ \biggl(\frac{2^3\pi}{3}\biggr)^{1 / 2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2\biggl[\frac{\xi}{\tilde{r}} \biggr] \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \xi </math> </td> </tr> </table> Note as well that, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\frac{\tilde{r}^3}{\tilde{M}_r}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ \mathcal{m}_\mathrm{surf}^{-1} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr]^{-1} \biggl[ \mathcal{m}_\mathrm{surf}^{-2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi \biggr]^{3} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ \mathcal{m}_\mathrm{surf} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl( \frac{4\cdot 3}{2\pi } \biggr)^{-1/2} \xi^{-3} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{3/2} \biggr] \biggl[ \mathcal{m}_\mathrm{surf}^{-6} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{12} \biggl(\frac{3}{2\pi}\biggr)^{3/2} \xi^3 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ \mathcal{m}_\mathrm{surf}^{-5} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{10} \biggl( \frac{3}{4\pi } \biggr) \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{3/2} \biggr] </math> </td> </tr> </table> </td></tr></table> Similarly, throughout the envelope, we find, <div align="center"> <table border="0" cellpadding="3"> <tr> <td align="right"> <math>\tilde\rho</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \mathcal{m}_\mathrm{surf}^5 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-9} \theta^{5}_i \phi \, ;</math> </td> </tr> <tr> <td align="right"> <math>\tilde{P}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \mathcal{m}_\mathrm{surf}^6 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \theta^{6}_i \phi^{2} \, ;</math> </td> </tr> <tr> <td align="right"> <math>\tilde{r}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \mathcal{m}_\mathrm{surf}^{-2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3} \theta^{-2}_i (2\pi)^{-1/2}\eta \, ;</math> </td> </tr> <tr> <td align="right"> <math>\tilde{M}_r</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \mathcal{m}_\mathrm{surf}^{-1}~ \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr) \, ,</math> </td> </tr> </table> </div> where, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\phi</math></td> <td align="center"><math>=</math></td> <td align="left"><math>A\biggl[ \frac{\sin(\eta - B)}{\eta}\biggr] \, ,</math></td> </tr> <tr> <td align="right"><math>\frac{d\phi}{d\eta}</math></td> <td align="center"><math>=</math></td> <td align="left"><math>\frac{A}{\eta^2}\biggl[ \eta \cos(\eta-B) - \sin(\eta-B)\biggr] \, ,</math></td> </tr> </table> <span id="interfaceValues">and,</span> <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\theta_i</math></td> <td align="center"><math>=</math></td> <td align="left"><math>\biggl(1+\frac{1}{3}\xi_i^2\biggr)^{-1 / 2} \, ,</math></td> </tr> <tr> <td align="right"><math>\eta_i</math></td> <td align="center"><math>=</math></td> <td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)~ \sqrt{3} \theta_i^2 \xi_i \, ,</math></td> </tr> <tr> <td align="right"><math>\Lambda_i</math></td> <td align="center"><math>=</math></td> <td align="left"><math>\frac{\xi_i}{\sqrt{3}} \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}\frac{1}{\theta_i^2 \xi_i^2} - 1\biggr] \, ,</math></td> </tr> <tr> <td align="right"><math>A</math></td> <td align="center"><math>=</math></td> <td align="left"><math> \eta_i\biggl(1 + \Lambda_i^2\biggr)^{1 / 2} \, ,</math></td> </tr> <tr> <td align="right"><math>B</math></td> <td align="center"><math>=</math></td> <td align="left"><math> \eta_i - \frac{\pi}{2} + \tan^{-1}(\Lambda_i) \, ,</math></td> </tr> <tr> <td align="right"><math>\eta_s = \pi + B</math></td> <td align="center"><math>=</math></td> <td align="left"><math> \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \, .</math></td> </tr> </table> Keep in mind, as well, that, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td> <td align="center"><math>=</math></td> <td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \sqrt{3} ~ \biggl[ \frac{\xi_i^3 \theta_i^4}{A\eta_s}\biggr] \, ,</math></td> </tr> <tr> <td align="right"><math>q \equiv \frac{r_\mathrm{core}}{R}</math></td> <td align="center"><math>=</math></td> <td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr) \sqrt{3} ~ \biggl[ \frac{\xi_i \theta_i^2}{\eta_s}\biggr] \, .</math></td> </tr> </table> ==Linearized Equations With Preferred Normalizations== ===Review and Elaborate=== <div align="center"> <table border="1" cellpadding="10"> <tr><td align="center"> <font color="#770000">'''Linearized'''</font><br /> <span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br /> <math> r_0 \frac{dx}{dr_0} = - 3 x - d , </math><br /> <font color="#770000">'''Linearized'''</font><br /> <span id="PGE:Euler"><font color="#770000">'''Euler + Poisson Equations'''</font></span><br /> <math> \frac{P_0}{\rho_0} \frac{dp}{dr_0} = (4x + p)g_0 + \omega^2 r_0 x , </math><br /> <font color="#770000">'''Linearized'''</font><br /> <span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br /> <font color="#770000">'''First Law of Thermodynamics'''</font></span><br /> <math> p = \gamma_\mathrm{g} d \, . </math> </td></tr> </table> </div> The LHS of the "linearized Euler + Poisson" equation is rewritten as, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\mathrm{LHS} = \frac{P_0}{\rho_0} \frac{dp}{dr_0}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[K_c^{-10} G^{9} M_\mathrm{tot}^{6} \biggr]^{-1}\tilde{P} \biggl[\biggl( \frac{K_c}{G} \biggr)^{3 / 2} \frac{1}{M_\mathrm{tot}} \biggr]^{-5} \tilde{\rho}^{-1} \biggl[\biggl( \frac{K_c}{G} \biggr)^{5 / 2} M_\mathrm{tot}^{-2} \biggr] \frac{dp}{d\tilde{r}} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\tilde{P}}{\tilde{\rho}} \frac{dp}{d\tilde{r}} \biggl[K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr] \biggl[ G^{15/2} K_c^{-15/2} M_\mathrm{tot}^5 \biggr] \biggl[K_c^{5/2} G^{-5/2} M_\mathrm{tot}^{-2} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\tilde{P}}{\tilde{\rho}} \frac{dp}{d\tilde{r}} \biggl[K_c^{5} G^{-4} M_\mathrm{tot}^{-3} \biggr] \, ;</math> </td> </tr> </table> and, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>g_0 = \frac{GM_r}{r_0^2}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> GM_\mathrm{tot} \tilde{M}_r \biggl[\biggl( \frac{K_c}{G} \biggr)^{5 / 2} M_\mathrm{tot}^{-2} \biggr]^2 \tilde{r}^{-2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\tilde{M}_r}{\tilde{r}^2} \biggl[\biggl( \frac{K_c}{G} \biggr)^{5 / 2} M_\mathrm{tot}^{-2} \biggr]^2 GM_\mathrm{tot} </math> </td> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\tilde{M}_r}{\tilde{r}^2} \biggl[K_c^5 G^{-4}M_\mathrm{tot}^{-3} \biggr] \, . </math> </td> </tr> </table> Therefore, multiplying the full equation through by <math>[K_c^{-5} G^4 M_\mathrm{tot}^3]</math> gives, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\frac{\tilde{P}}{\tilde{\rho}} \frac{dp}{d \tilde{r}} </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> (4x + p)\frac{\tilde{M}_r}{\tilde{r}^2} + \biggl[K_c^{-5} G^4 M_\mathrm{tot}^3 \biggr] \biggl[\biggl( \frac{K_c}{G} \biggr)^{5 / 2} M_\mathrm{tot}^{-2} \biggr]^{-1} \omega^2 \tilde{r} x </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> (4x + p)\frac{\tilde{M}_r}{\tilde{r}^2} + \tau_c^2 \omega^2 \tilde{r} x \, , </math> </td> </tr> </table> where the square of the characteristic timescale, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math> \tau_c^2</math></td> <td align="center"><math>\equiv</math></td> <td align="left"> <math> \biggl[ K_c^{-15/2} G^{13/2} M_\mathrm{tot}^{5} \biggr] \, . </math> </td> </tr> </table> <table border="1" align="center" cellpadding="8" width="75%"><tr><td align="left"> <font color="red">ASIDE:</font> Building on [[SSC/Stability/Polytropes#Numerical_Integration_from_the_Center,_Outward|an associated discussion]], the square of the dimensionless frequency also can be represented by the expression, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\sigma_c^2 \equiv \frac{3\omega^2}{2\pi G\rho_c}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{3\omega^2}{2\pi \tilde{\rho}_c} \biggl[\biggl( \frac{G}{K_c} \biggr)^{15 / 2} M_\mathrm{tot}^5 \biggr]G^{-1} = \frac{3\tau_c^2 \omega^2}{2\pi \tilde{\rho}_c} \, , </math> </td> </tr> </table> where, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\tilde{\rho}_c</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> m_\mathrm{surf}^5 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-10} \, . </math> </td> </tr> </table> </td></tr></table> <span id="FromEarlier">Hence we can write,</span> <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\frac{dp}{d \tilde{r}} </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\tilde{\rho}}{\tilde{P}} \cdot \frac{\tilde{M}_r}{\tilde{r}^2} \biggl[(4x + p) + \tau_c^2 \omega^2 \biggl(\frac{\tilde{r}^3 }{\tilde{M}_r}\biggr) x \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\tilde{\rho}}{\tilde{P}} \cdot \frac{\tilde{M}_r}{\tilde{r}^2} \biggl[ (4x + p) + \sigma_c^2 \biggl(\frac{2\pi}{3} \cdot \frac{\tilde{\rho}_c \tilde{r}^3 }{\tilde{M}_r}\biggr) x \biggr] \, . </math> </td> </tr> </table> ---- Focusing on the core … As demonstrated earlier, the leading term on the RHS of this expression can be rewritten to give, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\frac{dp}{d \tilde{r}} </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> 2\biggl[\frac{\xi}{\tilde{r}} \biggr] \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \xi \biggl[ (4x + p) + \sigma_c^2 \biggl(\frac{2\pi}{3} \cdot \frac{\tilde{\rho}_c \tilde{r}^3 }{\tilde{M}_r}\biggr) x \biggr] </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ \frac{dp}{d \xi} </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> 2\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \xi \biggl[ (4x + p) + \sigma_c^2 \biggl(\frac{2\pi}{3} \cdot \frac{\tilde{\rho}_c \tilde{r}^3 }{\tilde{M}_r}\biggr) x \biggr] \, . </math> </td> </tr> </table> Noting as well that, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\tilde{\rho}_c \biggl( \frac{\tilde{r}^3}{\tilde{M}_r}\biggr)</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> m_\mathrm{surf}^5 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-10} \biggl[ \mathcal{m}_\mathrm{surf}^{-5} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{10} \biggl( \frac{3}{4\pi } \biggr) \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{3/2} \biggr] = \frac{3}{4\pi } \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{3/2} \, , </math> </td> </tr> </table> we have, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\frac{dp}{d \xi} </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> 2\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \xi \biggl[ (4x + p) + \frac{\sigma_c^2}{2} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{3/2} x \biggr] \, . </math> </td> </tr> </table> ===At the Center=== ====All σ<sup>2</sup>==== According to our [[Appendix/Ramblings/PowerSeriesExpressions#Displacement_Function_for_Polytropic_LAWE|discussion in an appendix chapter]], starting from the center of the equilibrium configuration, the displacement function can be represented by a power-series expression of the form, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>x</math> </td> <td align="center"> <math>\approx</math> </td> <td align="left"> <math> 1 - \biggl[\frac{(n+1)\mathfrak{F}}{60}\biggr]\xi^2 \, ,</math> </td> </tr> </table> where, <math>\mathfrak{F} \equiv (\sigma_c^2/\gamma - 2\alpha)</math>, and (see, for example, [[SSC/Structure/BiPolytropes/Analytic51Renormalize#Core|here]]), <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\xi</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> m_\mathrm{surf}^2 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-4} \biggl(\frac{2\pi}{3}\biggr)^{1 / 2} \tilde{r} \, .</math> </td> </tr> </table> Note that, at the center of our <math>(n_c, n_e) = (5, 1)</math> bipolytrope, <math>\gamma_g = 6/5</math>, so <math>\alpha = -1/3</math>. Hence, for this particular investigation, the central boundary condition is, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>x</math> </td> <td align="center"> <math>\approx</math> </td> <td align="left"> <math> 1 - \biggl[ \frac{\sigma_c^2}{12} + \frac{1}{15} \biggr]\xi^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>\approx</math> </td> <td align="left"> <math> 1 - \biggl[ \frac{\sigma_c^2}{12} + \frac{1}{15} \biggr] \biggl[ m_\mathrm{surf}^4 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-8} \biggl(\frac{2\pi}{3}\biggr) \biggr] \tilde{r}^2 \, . </math> </td> </tr> </table> Also, the derivative of this displacement function is, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{dx}{d\xi}</math> </td> <td align="center"> <math>\approx</math> </td> <td align="left"> <math> - 2\biggl[ \frac{\sigma_c^2}{12} + \frac{1}{15} \biggr]\xi \, . </math> </td> </tr> </table> Hence, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>p = -\gamma_c\biggl[3x + \xi \cdot \frac{dx}{d\xi}\biggr]</math> </td> <td align="center"> <math>\approx</math> </td> <td align="left"> <math>-\frac{6}{5} \biggl\{ 3 \biggl[ 1 - \biggl( \frac{\sigma_c^2}{12} + \frac{1}{15} \biggr)\xi^2 \biggr] - 2\biggl( \frac{\sigma_c^2}{12} + \frac{1}{15} \biggr)\xi^2 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>-\frac{18}{5} \biggl[ 1 - \frac{5}{3}\biggl( \frac{\sigma_c^2}{12} + \frac{1}{15} \biggr)\xi^2 \biggr] \, . </math> </td> </tr> </table> Furthermore, we find, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>(4x + p)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 4\biggl[ 1 - \biggl( \frac{\sigma_c^2}{12} + \frac{1}{15} \biggr)\xi^2 \biggr] -\frac{18}{5} \biggl[ 1 - \frac{5}{3}\biggl( \frac{\sigma_c^2}{12} + \frac{1}{15} \biggr)\xi^2 \biggr] \, . </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[ \frac{20}{5} - 4\biggl( \frac{\sigma_c^2}{12} + \frac{1}{15} \biggr)\xi^2 \biggr] + \biggl[ - \frac{18}{5} + 6\biggl( \frac{\sigma_c^2}{12} + \frac{1}{15} \biggr)\xi^2 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{2}{5} + 2\biggl( \frac{\sigma_c^2}{12} + \frac{1}{15} \biggr)\xi^2 \, . </math> </td> </tr> </table> Hence we have, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\frac{dp}{d \xi} </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> 2\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \xi \biggl\{ \frac{2}{5} + 2\biggl( \frac{\sigma_c^2}{12} + \frac{1}{15} \biggr)\xi^2 + \frac{\sigma_c^2}{2} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{3/2} \biggl[ 1 - \biggl( \frac{\sigma_c^2}{12} + \frac{1}{15} \biggr)\xi^2 \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 2\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \xi \biggl\{ 24 + 2\biggl( 5\sigma_c^2 + 4 \biggr)\xi^2 + \frac{\sigma_c^2}{2} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{3/2} \biggl[ 60 - \biggl( 5\sigma_c^2 + 4 \biggr)\xi^2 \biggr] \biggr\}\frac{1}{60} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{30}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \xi \biggl\{ 24 + 8\xi^2 + 10\sigma_c^2 \xi^2 + 30\sigma_c^2\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{3/2} - \frac{\sigma_c^2}{2} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{3/2} \biggl( 4 + 5\sigma_c^2 \biggr)\xi^2 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{30}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \xi \biggl\{ \biggl[ 24 + 8\xi^2 + 10\sigma_c^2 \xi^2 \biggr] + \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{3/2}\biggl[ 30\sigma_c^2 - 2\sigma_c^2 \xi^2 - \frac{5}{2} \sigma_c^4 \xi^2 \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{15}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \xi \biggl[ 12 + 4\xi^2 + 5\sigma_c^2 \xi^2 \biggr] + \frac{\sigma_c^2 \xi}{30} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1/2}\biggl[ 30 - 2 \xi^2 - \frac{5}{2} \sigma_c^2 \xi^2 \biggr] </math> </td> </tr> </table> ====Just σ<sup>2</sup> = 0==== If we set <math>\sigma_c^2 = 0</math>, we have, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>p</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>-\frac{18}{5} + \frac{2}{5} \xi^2 </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \frac{dp}{d\xi}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>\frac{4}{5} \xi \, . </math> </td> </tr> </table> Alternatively, from immediately above, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\frac{dp}{d \xi}\biggr|_{\sigma_c^2=0} </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{15}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \xi \biggl[ 12 + 4\xi^2 + \cancelto{0}{5\sigma_c^2 \xi^2} \biggr] + \cancelto{0}{\frac{\sigma_c^2 \xi}{30} } \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1/2}\biggl[ 30 - 2 \xi^2 - \frac{5}{2} \sigma_c^2 \xi^2 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{12}{15}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \xi \biggl[ 1 + \frac{1}{3}\xi^2 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{4}{5}\xi \, . </math> </td> </tr> </table> Yes! It matches! ====Summary==== Moving from the center, outward thorough the core — that is, interior to the interface — we can assign values of <math>x(\xi)</math> and <math>p(\xi)</math> using the following approximate (''exact'' if <math>\sigma_c^2 = 0</math>) relations: <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>x</math> </td> <td align="center"> <math>\approx</math> </td> <td align="left"> <math> 1 - \biggl[ \frac{\sigma_c^2}{12} + \frac{1}{15} \biggr]\xi^2 \, ; </math> </td> </tr> <tr> <td align="right"> <math>p </math> </td> <td align="center"> <math>\approx</math> </td> <td align="left"> <math>-\frac{18}{5} \biggl[ 1 - \frac{5}{3}\biggl( \frac{\sigma_c^2}{12} + \frac{1}{15} \biggr)\xi^2 \biggr] \, . </math> </td> </tr> </table> For all radial shells throughout the entire bipolytropic configuration, the pair of first derivatives can be evaluated using the following relations: <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{dx}{d\tilde{r}}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> -\frac{1}{\tilde{r}}\biggl[3x + \frac{p}{\gamma_g}\biggr] \, ; </math> </td> </tr> <tr> <td align="right"><math>\frac{dp}{d \tilde{r}} </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\tilde{\rho}}{\tilde{P}} \cdot \frac{\tilde{M}_r}{\tilde{r}^2} \biggl[ (4x + p) + \sigma_c^2 \biggl(\frac{2\pi}{3} \cdot \frac{\tilde{\rho}_c \tilde{r}^3 }{\tilde{M}_r}\biggr) x \biggr] \, . </math> </td> </tr> </table> Near the center, this pressure-derivative expression can be checked against the relation, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\frac{dp}{d \xi} </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{15}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \xi \biggl[ 12 + 4\xi^2 + 5\sigma_c^2 \xi^2 \biggr] + \frac{\sigma_c^2 \xi}{30} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1/2}\biggl[ 30 - 2 \xi^2 - \frac{5}{2} \sigma_c^2 \xi^2 \biggr] \, ; </math> </td> </tr> </table> notice that, in order to make this comparison, you need to multiply this last expression through by the ratio, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\frac{\xi}{\tilde{r}} </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \mathcal{m}_\mathrm{surf}^{2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-4} \biggl(\frac{2\pi}{3}\biggr)^{1 / 2} \, .</math> </td> </tr> </table> The comparison should be especially accurate in the case of <math>\sigma_c^2 = 0</math>. ===At the Interface=== See [[#Interface|below]]. ===At the Surface=== Drawing from a [[SSC/Stability/Polytropes#Boundary_Conditions|separate discussion]], the surface boundary condition is, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~r_0 \frac{dx}{dr_0}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl( 4 - 3\gamma_g + \frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) \frac{x}{\gamma_g}</math> at <math>~r_0 = R \, ,</math> </td> </tr> </table> that is, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{d\ln x}{d\ln \tilde{r}}\biggr|_s</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>- \alpha + \frac{\omega^2 R^3}{\gamma_g GM_\mathrm{tot}} \, ,</math> </td> </tr> </table> where (see also, [[SSC/Stability/BiPolytropes#Review_of_the_Analysis_by_Murphy_&_Fiedler_(1985b)|here]]), <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>3 - \frac{4}{\gamma_g} \, .</math> </td> </tr> </table> Note that since, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>R^3</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \tilde{r}_s^3 \biggl[ G^{15/2} K_c^{-15/2} M_\mathrm{tot}^6\biggr] \, ,</math> </td> </tr> </table> in terms of our adopted normalizations, the frequency-squared term should be rewritten as, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{\omega^2 R^3}{\gamma_g G M_\mathrm{tot}}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \tilde{r}_s^3 \biggl[ \frac{\omega^2 \tau_c^2}{\gamma_g}\biggr] \, .</math> </td> </tr> </table> Note as well that, at the surface of our <math>(n_c, n_e) = (5, 1)</math> bipolytrope, <math>\gamma_g = 2</math>, so <math>\alpha = +1</math>. Hence, for this particular investigation, the surface boundary condition is, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{d\ln x}{d\ln \tilde{r}}\biggr|_s</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{2}\biggl[ \tilde{r}_s^3\omega^2 \tau_c^2\biggr] -1 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{\tilde{r}_s^3}{6}\biggl[ 2\pi \tilde{\rho}_c\sigma_c^2\biggr] -1 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{\tilde{r}_s^3}{6}\biggl[ 2\pi \tilde{\rho}_c\sigma_c^2\biggr]\frac{ 3 }{4\pi \tilde{r}_s^3 \tilde{\rho}_\mathrm{mean}} -1 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{4}\biggl( \frac{ \tilde{\rho}_c }{\tilde{\rho}_\mathrm{mean}}\biggr)\sigma_c^2 -1 \, . </math> </td> </tr> </table> This result should be compared with our [[SSC/Stability/BiPolytropes#Eigenfunction_Details|separate discussion of ''eigenfunction details'']]. ==Discretize for Numerical Integration== ===General Discretization=== ====First Approximation==== Now, let's set up a grid associated with a uniformly spaced spherical radius, where the subscript <math>J</math> denotes the grid zone at which all terms in the finite-difference representation of the governing relations will be evaluated. More specifically, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\tilde{r}_{J-1}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \tilde{r}_J - \Delta\tilde{r} </math> </td> <td align="center"> and </td> <td align="right"><math>\tilde{r}_{J+1}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \tilde{r}_J + \Delta\tilde{r} \, ; </math> </td> </tr> </table> also, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\biggl(\frac{dx}{d\tilde{r}}\biggr)_{J}</math></td> <td align="center"><math>\approx</math></td> <td align="left"> <math> \frac{(x_{J+1} - x_{J-1})}{2\Delta\tilde{r}} </math> </td> <td align="center"> and </td> <td align="right"><math>\biggl(\frac{dp}{d\tilde{r}}\biggr)_{J}</math></td> <td align="center"><math>\approx</math></td> <td align="left"> <math> \frac{(p_{J+1} - p_{J-1})}{2\Delta\tilde{r}} \, . </math> </td> </tr> </table> And at each grid location, the governing relations establish the local evaluation of the derivatives, that is, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\biggl(\frac{dx}{d \tilde{r}}\biggr)_J </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> - \frac{1}{\tilde{r}_J}\biggl[ 3x + \frac{p}{\gamma_g}\biggr]_J \, , </math> </td> <td align="center"> and </td> <td align="right"><math>\biggl(\frac{dp}{d \tilde{r}}\biggr)_J </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{\tilde{\rho}_J}{\tilde{P}_J}\biggl[ (4x + p)\frac{\tilde{M}_r}{\tilde{r}^2} + \tau_c^2 \omega^2 \tilde{r} x \biggr]_J \, . </math> </td> </tr> </table> <span id="1stapprox">So, integrating</span> step-by-step from the center of the configuration, outward, once all the variable values are known at grid locations <math>J</math> and <math>(J-1)</math>, the values of <math>x</math> and <math>p</math> at <math>(J+1)</math> are given by the expressions, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>x_{J+1}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> x_{J-1} - 2\Delta\tilde{r} \biggl\{ \frac{1}{\tilde{r}}\biggl[ 3x + \frac{p}{\gamma_g}\biggr] \biggr\}_J \, , </math> </td> <td align="center"> and </td> <td align="right"><math>p_{J+1}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> p_{J-1} + 2\Delta\tilde{r} \biggl\{ \frac{\tilde{\rho}}{\tilde{P}} \cdot \frac{\tilde{M}_r}{\tilde{r}^2} \biggl[ (4x + p) + \sigma_c^2 \biggl(\frac{2\pi}{3} \cdot \frac{\tilde{\rho}_c \tilde{r}^3 }{\tilde{M}_r}\biggr) x \biggr] \biggr\}_J\, . </math> </td> </tr> </table> Then we will obtain the "<math>x_J</math>" and "<math>p_J</math>" values via the ''average'' expressions, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>x_{J}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{2}(x_{J-1} + x_{J+1}) \, , </math> </td> <td align="center"> and </td> <td align="right"><math>p_{J}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{2}(p_{J-1} + p_{J+1}) \, . </math> </td> </tr> </table> <table border="1" align="center" cellpadding="8" width="80%"><tr><td align="left"> Consider implementing a more ''implicit'' finite-difference analysis. Wherever a "<math>J</math>" index appears in the source term, replace it with the ''average expressions.'' The general form of the source term expressions is, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>x_{J+1}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> x_{J-1} + 2\Delta\tilde{r} \biggl\{ \mathcal{A}x_J + \mathcal{B}p_J \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> x_{J-1} + \Delta\tilde{r} \biggl\{ \mathcal{A}\biggl[x_{J+1}+x_{J-1}\biggr] + \mathcal{B}\biggl[p_{J+1}+p_{J-1}\biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ x_{J+1}\biggl[1 - \Delta\tilde{r} \mathcal{A} \biggr]</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> x_{J-1}\biggl[1 + \Delta\tilde{r}\mathcal{A}\biggr] + \Delta\tilde{r} \mathcal{B}\biggl[p_{J+1}+p_{J-1}\biggr] \, , </math> </td> </tr> </table> where, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\mathcal{A}</math></td> <td align="center"><math>\equiv</math></td> <td align="left"> <math> - \frac{3}{\tilde{r}} \, , </math> </td> <td align="center"> and </td> <td align="right"><math>\mathcal{B}</math></td> <td align="center"><math>\equiv</math></td> <td align="left"> <math> - \frac{1}{\gamma_g \tilde{r}} \, ; </math> </td> </tr> </table> and, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>p_{J+1}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> p_{J-1} + 2\Delta\tilde{r} \biggl\{ \mathcal{C}x_J + \mathcal{D}p_J \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> p_{J-1} + \Delta\tilde{r} \biggl\{ \mathcal{C}\biggl[x_{J+1}+x_{J-1}\biggr] + \mathcal{D}\biggl[p_{J+1}+p_{J-1}\biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ p_{J+1}\biggl[1 - \Delta\tilde{r} \mathcal{D} \biggr]</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> p_{J-1}\biggl[1 + \Delta\tilde{r}\mathcal{D}\biggr] + \Delta\tilde{r} \mathcal{C}\biggl[x_{J+1}+x_{J-1}\biggr] \, , </math> </td> </tr> </table> where, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>\mathcal{C}</math></td> <td align="center"><math>\equiv</math></td> <td align="left"> <math> \biggl\{ \mathcal{D} \biggl[ 4 + \sigma_c^2 \biggl(\frac{2\pi}{3} \cdot \frac{\tilde{\rho}_c \tilde{r}^3 }{\tilde{M}_r}\biggr) \biggr] \biggr\}_J\, , </math> </td> <td align="center"> and </td> <td align="right"><math>\mathcal{D}</math></td> <td align="center"><math>\equiv</math></td> <td align="left"> <math> \biggl\{ \frac{\tilde{\rho}}{\tilde{P}} \cdot \frac{\tilde{M}_r}{\tilde{r}^2} \biggr\}_J\, . </math> </td> </tr> </table> In both cases, the two unknowns are <math>x_{J+1}</math> and <math>p_{J+1}</math>. Combining this pair of equations gives, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>x_{J+1}\biggl[1 - \Delta\tilde{r} \mathcal{A} \biggr]</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> x_{J-1}\biggl[1 + \Delta\tilde{r}\mathcal{A}\biggr] + \Delta\tilde{r} \mathcal{B}\biggl[p_{J-1}\biggr] + \Delta\tilde{r} \mathcal{B}\biggl[1 - \Delta\tilde{r} \mathcal{D} \biggr]^{-1}\biggl\{ p_{J-1}\biggl[1 + \Delta\tilde{r}\mathcal{D}\biggr] + \Delta\tilde{r} \mathcal{C}\biggl[x_{J+1}+x_{J-1}\biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ x_{J+1}\biggl\{ 1 - \Delta\tilde{r} \mathcal{A} - \Delta\tilde{r} \mathcal{B}\biggl[1 - \Delta\tilde{r} \mathcal{D} \biggr]^{-1}\biggl[ \Delta\tilde{r} \mathcal{C}\biggr] \biggr\}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> x_{J-1}\biggl[1 + \Delta\tilde{r}\mathcal{A}\biggr] + \Delta\tilde{r} \mathcal{B}\biggl[p_{J-1}\biggr] + \Delta\tilde{r} \mathcal{B}\biggl[1 - \Delta\tilde{r} \mathcal{D} \biggr]^{-1}\biggl\{ p_{J-1}\biggl[1 + \Delta\tilde{r}\mathcal{D}\biggr] + \Delta\tilde{r} \mathcal{C}\biggl[x_{J-1}\biggr] \biggr\} \, , </math> </td> </tr> </table> which determines <math>x_{J+1}</math>, which then allows the straightforward determination of <math>p_{J+1}</math>. Via the ''average'' expressions, we can also then determine — and record — the self-consistent values of <math>x_J</math> and <math>p_J</math>. ---- Dropping terms <math>\mathcal{O}[(\Delta\tilde{r})^2]</math> and higher gives, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>x_{J+1}\biggl[ 1 - \Delta\tilde{r} \mathcal{A} \biggr]_J</math></td> <td align="center"><math>\approx</math></td> <td align="left"> <math> x_{J-1}\biggl[1 + \Delta\tilde{r}\mathcal{A}\biggr]_J + \Delta\tilde{r} \mathcal{B}_J\biggl[p_{J-1}\biggr] \, , </math> </td> </tr> </table> and, in turn, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>p_{J+1}\biggl[1 - \Delta\tilde{r} \mathcal{D} \biggr]_J</math></td> <td align="center"><math>\approx</math></td> <td align="left"> <math> p_{J-1}\biggl[1 + \Delta\tilde{r}\mathcal{D}\biggr]_J + \Delta\tilde{r} \mathcal{C}_J\biggl[x_{J-1}\biggr] + \Delta\tilde{r} \mathcal{C}_J\biggl[ x_{J+1} \biggr] \, . </math> </td> </tr> </table> </td></tr></table> ====Second Approximation==== Let's assume that we know the three quantities, <math>x_{J-1}, x_J</math>, and <math>(x_J)^' \equiv (dx/d\tilde{r})_J</math> and want to project forward to determine, <math>x_{J+1}</math>. We should assume that, locally, the displacement function <math>x</math> is quadratic in <math>\tilde{r}</math>, that is, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>x</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> a + b\tilde{r} + c\tilde{r}^2 </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ \frac{dx}{d\tilde{r}}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> b + 2c\tilde{r} \, , </math> </td> </tr> </table> where we have three unknowns, <math>a, b, c</math>. These can be determined by appropriately combining the three relations, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>(x_J)^'</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> b + 2c\tilde{r}_J \, , </math> </td> </tr> <tr> <td align="right"><math>x_J</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> a + b\tilde{r}_J + c\tilde{r}_J^2 \, , </math> </td> </tr> <tr> <td align="right"><math>x_{J-1}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> a + b(\tilde{r}_{J}-\Delta\tilde{r}) + c(\tilde{r}_{J}-\Delta\tilde{r})^2 \, . </math> </td> </tr> </table> We have, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>b</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> (x_J)^' - 2c\tilde{r}_J \, , </math> </td> </tr> <tr> <td align="right"><math>- a</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> -x_J + [(x_J)^' - 2c\tilde{r}_J]\tilde{r}_J + c\tilde{r}_J^2 \, , </math> </td> </tr> <tr> <td align="right"><math>- a</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> -x_{J-1} + [(x_J)^' - 2c\tilde{r}_J](\tilde{r}_{J}-\Delta\tilde{r}) + c(\tilde{r}_{J}-\Delta\tilde{r})^2 \, . </math> </td> </tr> </table> Combining the last two expressions gives, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>-x_J + [(x_J)^' - 2c\tilde{r}_J]\tilde{r}_J + c\tilde{r}_J^2</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> -x_{J-1} + [(x_J)^' - 2c\tilde{r}_J](\tilde{r}_{J}-\Delta\tilde{r}) + c(\tilde{r}_{J}-\Delta\tilde{r})^2 </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ -x_J + (x_J)^'\tilde{r}_J - c\tilde{r}_J^2 </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> -x_{J-1} + (x_J)^'(\tilde{r}_{J}-\Delta\tilde{r}) - 2c\tilde{r}_J(\tilde{r}_{J}-\Delta\tilde{r}) + c[ \tilde{r}_{J}^2 - 2r_J \Delta\tilde{r} + (\Delta\tilde{r})^2 ] </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ -x_J - c\tilde{r}_J^2 </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> -x_{J-1} - (x_J)^'\Delta\tilde{r} + c[ \tilde{r}_{J}^2 - 2r_J \Delta\tilde{r} + (\Delta\tilde{r})^2 - 2\tilde{r}_J^2 + 2\tilde{r}_J(\Delta\tilde{r})] </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ c(\Delta\tilde{r})^2 </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> x_J -x_{J-1} - (x_J)^'\Delta\tilde{r} \, . </math> </td> </tr> </table> Therefore, also, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>b</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> (x_J)^' - \biggl[ x_J -x_{J-1} - (x_J)^'\Delta\tilde{r} \biggr]\frac{2\tilde{r}_J}{(\Delta\tilde{r})^2} \, , </math> </td> </tr> <tr> <td align="right"><math>a</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> x_J - (x_J)^' \tilde{r}_J + \biggl[ x_J -x_{J-1} - (x_J)^'\Delta\tilde{r} \biggr]\frac{\tilde{r}_J^2}{(\Delta\tilde{r})^2} </math> </td> </tr> </table> Hence, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>x_{J+1}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> a + b(\tilde{r}_J + \Delta\tilde{r}) + c(\tilde{r}_J + \Delta\tilde{r})^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> x_J - (x_J)^' \tilde{r}_J + \biggl[ x_J -x_{J-1} - (x_J)^'\Delta\tilde{r} \biggr]\frac{\tilde{r}_J^2}{(\Delta\tilde{r})^2} + (\tilde{r}_J + \Delta\tilde{r})\biggl\{ (x_J)^' - \biggl[ x_J -x_{J-1} - (x_J)^'\Delta\tilde{r} \biggr]\frac{2\tilde{r}_J}{(\Delta\tilde{r})^2} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + (\tilde{r}_J + \Delta\tilde{r})^2 \biggl[ x_J -x_{J-1} - (x_J)^'\Delta\tilde{r} \biggr]\frac{1}{(\Delta\tilde{r})^2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> x_J - (x_J)^' \tilde{r}_J + \biggl\{ \tilde{r}_J^2 + 2\tilde{r}_J(\tilde{r}_J + \Delta\tilde{r}) + (\tilde{r}_J + \Delta\tilde{r})^2 \biggr\} \biggl[ x_J -x_{J-1} - (x_J)^'\Delta\tilde{r} \biggr]\frac{1}{(\Delta\tilde{r})^2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> x_J - (x_J)^' \tilde{r}_J + \biggl[ 2\tilde{r}_J + \Delta\tilde{r}\biggr]^2 \biggl[ x_J -x_{J-1} - (x_J)^'\Delta\tilde{r} \biggr]\frac{1}{(\Delta\tilde{r})^2} \, . </math> </td> </tr> </table> <font color="red"><b>WRONG!!</b></font> Try again … ====Third Approximation==== Let's assume that we know the three quantities, <math>x_{J-1}, x_J</math>, and <math>(x_J)^' \equiv (dx/d\tilde{r})_J</math> and want to project forward to determine, <math>x_{J+1}</math>. We should assume that, locally, the displacement function <math>x</math> is quadratic in <math>\tilde{r}</math>, that is, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>x</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> a + b\tilde{r} + c\tilde{r}^2 </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ \frac{dx}{d\tilde{r}}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> b + 2c\tilde{r} \, , </math> </td> </tr> </table> where we have three unknowns, <math>a, b, c</math>. These can be determined by appropriately combining the three relations, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>(x_J)^'</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> b + 2c\tilde{r}_J \, , </math> </td> </tr> <tr> <td align="right"><math>x_J</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> a + b\tilde{r}_J + c\tilde{r}_J^2 \, , </math> </td> </tr> <tr> <td align="right"><math>x_{J-1}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> a + b(\tilde{r}_{J}-\Delta\tilde{r}) + c(\tilde{r}_{J}-\Delta\tilde{r})^2 \, . </math> </td> </tr> </table> The difference between the last two expressions gives, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>x_J - x_{J-1}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ b\tilde{r}_J + c\tilde{r}_J^2 \biggr] - \biggl[ b(\tilde{r}_{J}-\Delta\tilde{r}) + c(\tilde{r}_{J}-\Delta\tilde{r})^2 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> b\Delta\tilde{r} + c(2\tilde{r}_J \Delta\tilde{r} - \Delta\tilde{r}^2) \, . </math> </td> </tr> </table> Combining this with the first of the three expressions then gives, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"> <math> \Delta\tilde{r} \biggl[(x_J)^' - 2c\tilde{r}_J \biggr] + c(2\tilde{r}_J \Delta\tilde{r} - \Delta\tilde{r}^2) </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> x_J - x_{J-1} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ c(2\tilde{r}_J \Delta\tilde{r} - \Delta\tilde{r}^2) -c\biggl[ 2\tilde{r}_J \Delta\tilde{r} \biggr] </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> x_J - x_{J-1} - (x_J)^'\Delta\tilde{r} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ -c \Delta\tilde{r}^2 </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> x_J - x_{J-1} - (x_J)^'\Delta\tilde{r} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ c </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{\Delta\tilde{r}^2}\biggl[ -x_J + x_{J-1} + (x_J)^'\Delta\tilde{r} \biggr] \, . </math> </td> </tr> </table> Hence, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"> <math> b </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> (x_J)^' - 2c\tilde{r}_J </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> (x_J)^' + \frac{2\tilde{r}_J}{\Delta \tilde{r}^2}\biggl[ x_J - x_{J-1} - (x_J)^'\Delta\tilde{r} \biggr] \, ; </math> </td> </tr> <tr> <td align="right"> <math> a </math> </td> <td align="center"><math>=</math></td> <td align="left"> <math> x_J - b\tilde{r}_J -c\tilde{r}_J^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> x_J -\biggl\{ (x_J)^' + \frac{2\tilde{r}_J}{\Delta \tilde{r}^2}\biggl[ x_J - x_{J-1} - (x_J)^'\Delta\tilde{r} \biggr] \biggr\}\tilde{r}_J - \biggl\{ \frac{1}{\Delta\tilde{r}^2}\biggl[ -x_J + x_{J-1} + (x_J)^'\Delta\tilde{r} \biggr] \biggr\}\tilde{r}_J^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> x_J - (x_J)^'\tilde{r}_J -\biggl\{ \frac{2\tilde{r}_J^2}{\Delta \tilde{r}^2}\biggl[ x_J - x_{J-1} - (x_J)^'\Delta\tilde{r} \biggr] \biggr\} + \biggl\{ \frac{\tilde{r}_J^2}{\Delta\tilde{r}^2}\biggl[ x_J - x_{J-1} - (x_J)^'\Delta\tilde{r} \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> x_J - (x_J)^'\tilde{r}_J -\frac{\tilde{r}_J^2}{\Delta \tilde{r}^2}\biggl[ x_J - x_{J-1} - (x_J)^'\Delta\tilde{r} \biggr] \, . </math> </td> </tr> </table> As a result, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>x_{J+1}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl\{ a \biggr\} + (\tilde{r}_J +\Delta\tilde{r}) \biggl\{ b \biggr\} + (\tilde{r}_J+\Delta\tilde{r})^2 \biggl\{ c \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl\{ x_J - (x_J)^'\tilde{r}_J -\frac{\tilde{r}_J^2}{\Delta \tilde{r}^2}\biggl[ x_J - x_{J-1} - (x_J)^'\Delta\tilde{r} \biggr] \biggr\} + (\tilde{r}_J +\Delta\tilde{r}) \biggl\{ (x_J)^' + \frac{2\tilde{r}_J}{\Delta \tilde{r}^2}\biggl[ x_J - x_{J-1} - (x_J)^'\Delta\tilde{r} \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + (\tilde{r}_J^2 + 2\tilde{r}_J \Delta\tilde{r} +\Delta\tilde{r}^2) \biggl\{ \frac{1}{\Delta\tilde{r}^2}\biggl[ -x_J + x_{J-1} + (x_J)^'\Delta\tilde{r} \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> x_J - (x_J)^'\tilde{r}_J + \tilde{r}_J^2\biggl\{\frac{1}{\Delta \tilde{r}^2}\biggl[ -x_J + x_{J-1} + (x_J)^'\Delta\tilde{r} \biggr] \biggr\} + (x_J)^'(\tilde{r}_J +\Delta\tilde{r}) - 2\tilde{r}_J(\tilde{r}_J +\Delta\tilde{r}) \biggl\{ \frac{1}{\Delta \tilde{r}^2}\biggl[ -x_J + x_{J-1} + (x_J)^'\Delta\tilde{r} \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + (\tilde{r}_J^2 + 2\tilde{r}_J \Delta\tilde{r} +\Delta\tilde{r}^2) \biggl\{ \frac{1}{\Delta\tilde{r}^2}\biggl[ -x_J + x_{J-1} + (x_J)^'\Delta\tilde{r} \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> x_J + (x_J)^'\Delta\tilde{r} + \biggl[ \tilde{r}_J^2 - 2\tilde{r}_J(\tilde{r}_J +\Delta\tilde{r})+ (\tilde{r}_J^2 + 2\tilde{r}_J \Delta\tilde{r} +\Delta\tilde{r}^2)\biggr] \biggl\{\frac{1}{\Delta \tilde{r}^2}\biggl[ -x_J + x_{J-1} + (x_J)^'\Delta\tilde{r} \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> x_J + (x_J)^'\Delta\tilde{r} + \biggl[ -x_J + x_{J-1} + (x_J)^'\Delta\tilde{r} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> x_{J-1} + 2(x_J)^'\Delta\tilde{r} \, . </math> </td> </tr> </table> <font color="red"><b>GOOD!</b></font> This is the same as our [[#1stapprox|first approximation expression]] stated above. <table border=1 align="center" cellpadding="10" width="80%"><tr><td bgcolor="lightgreen" align="left"> This is test ... <table border="1" align="center" cellpadding="5"> <tr> <td align="center" bgcolor="white"><math>\Delta\tilde{r}</math></td> <td align="center" bgcolor="white"><math>x_J</math></td> <td align="center" bgcolor="white"><math>x_{J-1}</math></td> <td align="center" bgcolor="white"><math>(x_J)^'</math></td> <td align="center" bgcolor="white"><math>(x_{J-1})^'</math></td> </tr> <tr> <td align="center" bgcolor="white">0.001936393</td> <td align="center" bgcolor="white">-4.695376</td> <td align="center" bgcolor="white">-4.547832</td> <td align="center" bgcolor="white">-116.0119</td> <td align="center" bgcolor="white">-76.19513</td> </tr> </table> <tr><td bgcolor="white" align="center"> <table border="0" align="center" cellpadding="5"> <tr> <td align="right"><math>x_{J+1}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> x_{J-1} + 2(x_J)^'\Delta\tilde{r} = -4.997121 \, . </math> </td> </tr> </table> </td></tr> </td></tr></table> =Part 2= For a continuation (part 2) of this discussion, go [[Appendix/Ramblings/51BiPolytropeStability/BetterInterfacePt2|here]]. =See Also= {{ SGFfooter }}
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