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__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =LAWE= ==Most General Form== In an [[SSC/Perturbations#2ndOrderODE|accompanying discussion]], we derived the so-called, <div align="center" id="2ndOrderODE"> <font color="#770000">'''Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br /> {{Math/EQ_RadialPulsation01}} </div> <!-- <div align="center" id="2ndOrderODE"> <font color="#770000">'''Adiabatic Wave Equation'''</font><br /> <math> \frac{d^2x}{dr_0^2} + \biggl[\frac{4}{r_0} - \biggl(\frac{g_0 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{dr_0} + \biggl(\frac{\rho_0}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{r_0} \biggr] x = 0 \, , </math> </div> --> where the [[SSC/Perturbations#g0|gravitational acceleration]], <table border="0" align="center" cellpadding="8"> <tr> <td align="right"><math>g_0</math></td> <td align="center"><math>\equiv</math></td> <td align="left"> <math> \frac{GM_r}{r_0^2} = - \frac{1}{\rho_0} \frac{dP_0}{dr_0} ~~~\Rightarrow ~~~ \frac{g_0\rho_0 r_0}{P_0} = - \frac{d\ln P_0}{d\ln r_0} \, . </math> </td> </tr> </table> The solution to this equation gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations. The boundary condition conventionally used in connection with the adiabatic wave equation is, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>r_0 \frac{d\ln x}{dr_0}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>\frac{1}{\gamma_g} \biggl( 4 - 3\gamma_g + \frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) </math> at <math>r_0 = R \, .</math> </td> </tr> </table> ==Polytropic Configurations== ===Part 1=== If the initial, unperturbed equilibrium configuration is a [[SSC/Structure/Polytropes#Polytropic_Spheres|polytropic sphere]] whose internal structure is defined by the function, <math>\theta(\xi)</math>, that provides a solution to the, <div align="center"> <span id="LaneEmdenEquation"><font color="#770000">'''Lane-Emden Equation'''</font></span> <br /> {{Math/EQ_SSLaneEmden01}} </div> then, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>r_0</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>a_n \xi \, ,</math> </td> </tr> <tr> <td align="right"> <math>\rho_0</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>\rho_c \theta^{n} \, ,</math> </td> </tr> <tr> <td align="right"> <math>P_0</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>K\rho_0^{(n+1)/n} = K\rho_c^{(n+1)/n} \theta^{n+1} \, ,</math> </td> </tr> <tr> <td align="right"> <math>g_0</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>\frac{GM(r_0)}{r_0^2} = \frac{G}{r_0^2} \biggl[ 4\pi a_n^3 \rho_c \biggl(-\xi^2 \frac{d\theta}{d\xi}\biggr) \biggr] \, ,</math> </td> </tr> </table> </div> where, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>a_n</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>\biggl[\frac{(n+1)K}{4\pi G} \cdot \rho_c^{(1-n)/n} \biggr]^{1/2} \, .</math> </td> </tr> </table> </div> Hence, after multiplying through by <math>~a_n^2</math>, the above adiabatic wave equation can be rewritten in the form, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{d^2x}{d\xi^2} + \biggl[\frac{4}{\xi} - \frac{g_0}{a_n}\biggl(\frac{a_n^2 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{d\xi} + \biggl(\frac{a_n^2\rho_0}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{a_n\xi} \biggr] x </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~0 \, .</math> </td> </tr> </table> </div> In addition, given that, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{g_0}{a_n}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~4\pi G \rho_c \biggl(-\frac{d \theta}{d\xi} \biggr) \, ,</math> </td> </tr> </table> </div> and, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{a_n^2 \rho_0}{P_0}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{(n+1)}{(4\pi G\rho_c)\theta} = \frac{a_n^2 \rho_c}{P_c} \cdot \frac{\theta_c}{\theta}\, ,</math> </td> </tr> </table> </div> we can write, <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} + \biggl[\frac{4 - (n+1)V(\xi)}{\xi} \biggr] \frac{dx}{d\xi} + \biggl[\omega^2 \biggl(\frac{a_n^2 \rho_c }{\gamma_g P_c} \biggr) \frac{\theta_c}{\theta} - \biggl(3-\frac{4}{\gamma_g}\biggr) \cdot \frac{(n+1)V(x)}{\xi^2} \biggr] x </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>\frac{d^2x}{d\xi^2} + \biggl[\frac{4 - (n+1)V(\xi)}{\xi} \biggr] \frac{dx}{d\xi} + (n+1)\biggl[\omega^2 \biggl(\frac{a_n^2 \rho_c }{\gamma_g P_c} \biggr) \frac{\xi^2 \theta_c}{(n+1)\theta} - \biggl(3-\frac{4}{\gamma_g}\biggr) \cdot V(x) \biggr] \frac{x}{\xi^2} </math> </td> </tr> </table> where we have adopted the function notation, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>V(\xi)</math> </td> <td align="center"> <math>\equiv</math> </td> <td align="left"> <math>- \frac{\xi}{\theta} \frac{d \theta}{d\xi} \, .</math> </td> </tr> </table> </div> ===Part 2=== Drawing from an [[SSC/Stability/InstabilityOnsetOverview#Polytropic_Stability|accompanying discussion]], we have the following: <div align="center"> <font color="maroon"><b>Polytropic LAWE (linear adiabatic wave equation)</b></font><br /> {{ Math/EQ_RadialPulsation02 }} </div> <table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left"> In order to reconcile with the "Part 1" expression, we note first that <math>V(\xi) \leftrightarrow Q(\xi)</math>. We note as well that since, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\biggl(\frac{a_n^2 \rho_c }{P_c} \biggr)\theta_c</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>\frac{(n+1)}{4\pi G\rho_c}\, , </math> </td> </tr> </table> we have, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\omega^2 \biggl(\frac{a_n^2 \rho_c }{\gamma_g P_c} \biggr) \frac{\xi^2 \theta_c}{(n+1)\theta}</math> </td> <td align="center"> <math>\leftrightarrow</math> </td> <td align="left"> <math> \frac{\omega^2}{\gamma_g} \biggl[\frac{(n+1)}{4\pi G\rho_c} \biggr] \frac{\xi^2 }{(n+1)\theta} = \frac{1}{6\gamma_g} \biggl[\frac{3\omega^2}{2\pi G\rho_c} \biggr] \frac{\xi^2 }{\theta} = \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \frac{\xi^2 }{\theta} \, . </math> </td> </tr> </table> </td></tr></table> All physically reasonable solutions are subject to the inner boundary condition, <div align="center"> <math>\frac{dx}{d\xi} = 0</math> at <math>\xi = 0 \, ,</math> </div> but the relevant outer boundary condition depends on whether the underlying equilibrium configuration is isolated (surface pressure is zero), or whether it is a "pressure-truncated" configuration. As is the case with the pressure-truncated isothermal spheres, discussed above, if the polytropic configuration is truncated by the pressure, <math>P_e</math>, of a hot, tenuous external medium, then the solution to the LAWE is subject to the outer boundary condition, <div align="center"> <math>-\frac{d\ln x}{d\ln\xi} = 3</math> at <math>\xi = \tilde\xi \, .</math> </div> But, for ''isolated'' polytropes, the sought-after solution is subject to the more conventional boundary condition, <div align="center"> <math>- \frac{d\ln x}{d\ln \xi} = \biggl(\frac{3-n}{n+1}\biggr) + \frac{n\sigma_c^2}{6(n+1)} \biggl[\frac{\xi}{\theta^'}\biggr] </math> at <math>\xi = \xi_\mathrm{surf} \, .</math><br /> </div> =Radial Pulsation Neutral Mode= ==Background== The integro-differential version of the statement of hydrostatic balance is <div align="center"> {{Math/EQ_SShydrostaticBalance01}} </div> [[SSC/Stability/InstabilityOnsetOverview#Analyses_of_Radial_Oscillations|From our separate discussion]], we have found that, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="center" colspan="3"><font color="maroon"><b>Exact Solution to the <math>~(3 \le n < \infty)</math> Polytropic LAWE</b></font></td> </tr> <tr> <td align="right"> <math>~\sigma_c^2 = 0</math> </td> <td align="center"> and </td> <td align="left"> <math>~x_P \equiv \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\xi \theta^{n}}\biggr) \frac{d\theta}{d\xi}\biggr] \, .</math> </td> </tr> </table> </div> Let's rewrite the significant functional term in this expressions in terms of basic variables. That is, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\biggl( \frac{1}{\xi \theta^n}\biggr)\frac{d\theta}{d\xi}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> -\biggl(\frac{a_n \rho_c}{r_0 \rho_0}\biggr)\frac{g_0}{4\pi G \rho_c a_n} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> -\frac{M(r_0)}{4\pi r_0^3 \rho_0 } \, . </math> </td> </tr> </table> ==Trial Eigenfunction & Its Derivatives== Let's adopt the following ''trial'' solution: <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>x_t</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> a -\frac{bM_r}{4\pi r_0^3 \rho_0 } = a - \frac{bg_0}{4\pi G r_0 \rho_0 } \, . </math> </td> </tr> </table> Then we have, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>-\biggl(\frac{1}{b}\biggr)\frac{dx_t}{dr_0}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{d}{dr_0} \biggl[\frac{M_r}{4\pi r_0^3 \rho_0 }\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[\frac{1}{4\pi r_0^3 \rho_0 }\biggr]\frac{dM_r}{dr_0} - \biggl[\frac{M_r}{4\pi r_0^3 \rho_0^2 }\biggr] \frac{d\rho_0 }{dr_0} - \biggl[\frac{3M_r}{4\pi r_0^4 \rho_0 }\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{r_0 } - \biggl[\frac{3M_r}{4\pi r_0^4 \rho_0 }\biggr] - \biggl[\frac{M_r}{4\pi r_0^3 \rho_0^2 }\biggr] \frac{d\rho_0 }{dr_0} </math> </td> </tr> <tr> <td align="right"> <math>-\biggl(\frac{1}{b}\biggr)\frac{d^2 x_t}{dr_0^2}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{d}{dr_0}\biggl\{ \frac{1}{r_0 } - \biggl[\frac{3M_r}{4\pi r_0^4 \rho_0 }\biggr] - \biggl[\frac{M_r}{4\pi r_0^3 \rho_0^2 }\biggr] \frac{d\rho_0 }{dr_0} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> -\frac{1}{r_0^2 } - \biggl[\frac{3}{4\pi r_0^4 \rho_0 }\biggr]\frac{dM_r}{dr_0} + \biggl[\frac{3M_r}{4\pi r_0^4 \rho_0^2 }\biggr]\frac{d\rho_0}{dr_0} +4 \biggl[\frac{3M_r}{4\pi r_0^5 \rho_0 }\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - \biggl[\frac{M_r}{4\pi r_0^3 \rho_0^2 }\biggr]\frac{d^2\rho_0 }{dr_0^2} - \biggl[\frac{1}{4\pi r_0^3 \rho_0^2 }\biggr]\frac{d\rho_0 }{dr_0} \cdot \frac{dM_r}{dr_0} + \biggl[\frac{3M_r}{4\pi r_0^4 \rho_0^2 }\biggr]\frac{d\rho_0 }{dr_0} + \biggl[\frac{2M_r}{4\pi r_0^3 \rho_0^3 }\biggr]\biggl(\frac{d\rho_0 }{dr_0}\biggr)^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> -\frac{4}{r_0^2 } + \frac{3M_r}{4\pi r_0^5 \rho_0 }\biggl[4 + \frac{d\ln \rho_0}{d \ln r_0}\biggr] + \frac{1}{r_0^2 }\biggl[ \frac{3M_r}{4\pi r_0^3 \rho_0 } - 1 \biggr] \frac{d \ln \rho_0 }{d\ln r_0} - \biggl[\frac{M_r}{4\pi r_0^3 \rho_0^2 }\biggr]\frac{d^2\rho_0 }{dr_0^2} + \biggl[\frac{2M_r}{4\pi r_0^5 \rho_0 }\biggr]\biggl(\frac{d\ln \rho_0 }{d \ln r_0}\biggr)^2 \, . </math> </td> </tr> </table> Given that, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\Delta \equiv \frac{M_r}{4\pi r_0^3\rho_0}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{4\pi G} \biggl( \frac{g_0}{r_0\rho_0}\biggr) = - \biggl[\frac{P_0}{4\pi G r_0^2 \rho_0^2} \cdot \frac{d\ln P_0}{d \ln r_0} \biggr] \, , </math> </td> </tr> </table> these expression can be rewritten as, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>-\biggl(\frac{r_0^2}{b}\biggr)\frac{dx_t}{dr_0}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>r_0 \biggl\{ 1 - 3\Delta - \Delta \cdot \frac{d\ln \rho_0 }{d\ln r_0} \biggr\} \, , </math> </td> </tr> </table> and, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>-\biggl(\frac{r_0^2}{b}\biggr)\frac{d^2 x_t}{dr_0^2}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - 4 + 3\Delta \biggl[4 + \frac{d\ln \rho_0}{d \ln r_0}\biggr] + \biggl[ 3\Delta - 1 \biggr] \frac{d \ln \rho_0 }{d\ln r_0} - \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr)\frac{d^2\rho_0 }{dr_0^2} + 2\Delta \biggl(\frac{d\ln \rho_0 }{d \ln r_0}\biggr)^2 \, . </math> </td> </tr> </table> ==Plug Trial Eigenfunction Into LAWE== <br /> <table border="1" width="60%" align="center" cellpadding="8"><tr><td align="center"> <div align="center">'''LAWE'''</div> {{Math/EQ_RadialPulsation01}} </td></tr></table> Plugging our ''trial'' radial displacement function, <math>x_t</math>, into the LAWE gives, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> LAWE </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> -\biggl(\frac{r_0^2}{b}\biggr)\frac{d^2 x_t}{dr_0^2} -\biggl(\frac{r_0}{b}\biggr)\biggl[ 4 + \frac{d\ln P_0}{d\ln r_0}\biggr]\frac{dx_t}{dr_0} - \biggl(\frac{r_0^2}{b}\biggr) \biggl( \frac{\rho_0}{\gamma_g P_0}\biggr) \biggl[ (4-3\gamma_g)\frac{g_0}{r_0} + \sigma_c^2\biggl(\frac{2\pi G\rho_c}{3}\biggr)\biggr]x_t </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> -\biggl(\frac{r_0^2}{b}\biggr)\frac{d^2 x_t}{dr_0^2} -\biggl(\frac{r_0}{b}\biggr)\biggl[ 4 + \frac{d\ln P_0}{d\ln r_0}\biggr]\frac{dx_t}{dr_0} + \biggl(\frac{1}{b}\biggr) \frac{1}{\gamma_g} \biggl( \frac{d\ln P_0}{d \ln r_0}\biggr) (4-3\gamma_g)x_t - \biggl(\frac{r_0^2}{b}\biggr) \biggl( \frac{\rho_0}{\gamma_g P_0}\biggr) \biggl[ \sigma_c^2\biggl(\frac{2\pi G\rho_c}{3}\biggr)\biggr]x_t </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - 4 + 3\Delta \biggl[4 + \frac{d\ln \rho_0}{d \ln r_0}\biggr] + \biggl[ 3\Delta - 1 \biggr] \frac{d \ln \rho_0 }{d\ln r_0} - \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr)\frac{d^2\rho_0 }{dr_0^2} + 2\Delta \biggl(\frac{d\ln \rho_0 }{d \ln r_0}\biggr)^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> +\biggl[ 4 + \frac{d\ln P_0}{d\ln r_0}\biggr]\biggl\{ 1 - 3\Delta - \Delta \cdot \frac{d\ln \rho_0 }{d\ln r_0} \biggr\} + \biggl(\frac{1}{b}\biggr) \frac{1}{\gamma_g} \biggl( \frac{d\ln P_0}{d \ln r_0}\biggr) (4-3\gamma_g)(a - b\Delta) - \biggl(\frac{1}{b}\biggr) \biggl( \frac{\rho_0r_0^2}{\gamma_g P_0}\biggr) \biggl[ \sigma_c^2\biggl(\frac{2\pi G\rho_c}{3}\biggr)\biggr](a - b\Delta) \, . </math> </td> </tr> </table> Now, if we set <math>\sigma_c^2 = 0</math> and <math> d\ln P_0/d\ln r_0 = \gamma_g(d\ln \rho_0/d\ln r_0)</math>, this expression becomes, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> LAWE </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - 4 + 3\Delta \biggl[4 + \frac{d\ln \rho_0}{d \ln r_0}\biggr] + \biggl[ 3\Delta - 1 \biggr] \frac{d \ln \rho_0 }{d\ln r_0} - \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr)\frac{d^2\rho_0 }{dr_0^2} + 2\Delta \biggl(\frac{d\ln \rho_0 }{d \ln r_0}\biggr)^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> +\biggl[ 4 + \frac{d\ln P_0}{d\ln r_0}\biggr]\biggl\{ 1 - 3\Delta - \Delta \cdot \frac{d\ln \rho_0 }{d\ln r_0} \biggr\} + \biggl(\frac{1}{b}\biggr) \frac{1}{\gamma_g} \biggl( \frac{d\ln P_0}{d \ln r_0}\biggr) (4-3\gamma_g)(a - b\Delta) - \biggl(\frac{1}{b}\biggr) \biggl( \frac{\rho_0r_0^2}{\gamma_g P_0}\biggr) \biggl[ \sigma_c^2\biggl(\frac{2\pi G\rho_c}{3}\biggr)\biggr](a - b\Delta) \, . </math> </td> </tr> </table> Notice that the key components of this last term may be rewritten as, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math> \biggl( \frac{\rho_0r_0^2}{\gamma_g P_0}\biggr) \biggl[ \sigma_c^2\biggl(\frac{2\pi G\rho_c}{3}\biggr)\biggr] </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl( \frac{4\pi G \rho_0^2r_0^2}{P_0}\biggr) \biggl[ \frac{\sigma_c^2}{6\gamma_g} \biggl(\frac{\rho_c}{\rho_0}\biggr)\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - \biggl( \frac{1}{\Delta}\biggr) \frac{d\ln P_0}{d\ln r_0}\biggl[ \frac{\sigma_c^2}{6\gamma_g} \biggl(\frac{\rho_c}{\rho_0}\biggr)\biggr] \, . </math> </td> </tr> </table> So, for our ''trial'' eigenfunction, we have … <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> LAWE </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[ 2\Delta - 1 \biggr] \frac{d \ln \rho_0 }{d\ln r_0} - \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr)\frac{d^2\rho_0 }{dr_0^2} + 2\Delta \biggl(\frac{d\ln \rho_0 }{d \ln r_0}\biggr)^2 + \frac{d\ln P_0}{d\ln r_0} \cdot \biggl\{\biggl[ 1 - 3\Delta - \Delta \cdot \frac{d\ln \rho_0 }{d\ln r_0} \biggr] + \frac{(4-3\gamma_g)}{\gamma_g} \biggl[ \frac{a}{b} - \Delta \biggr] + \frac{\sigma_c^2}{6\gamma_g} \biggl(\frac{\rho_c}{\rho_0}\biggr) \biggl[ \frac{a}{b\Delta} - 1\biggr] \biggr\} \, . </math> </td> </tr> </table> ==Consider Polytropic Structures== Referring back to, for example, [[SSC/Stability/Polytropes#Groundwork|a separate review of polytropic structures]], we recognize that, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\Delta = \frac{1}{4\pi G} \biggl( \frac{g_0}{r_0\rho_0}\biggr)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{1}{\xi^3} \biggl[ \biggl(-\xi^2 \frac{d\theta}{d\xi}\biggr) \biggr]\theta^{-n} = \frac{1}{\xi} \biggl(- \frac{d\theta}{d\xi}\biggr) \theta^{-n} = -\frac{\theta^'}{\xi \theta^n} \, , </math> </td> </tr> <tr> <td align="right"> <math>\frac{d\ln \rho_0}{d\ln r_0}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> n\, , </math> </td> </tr> <tr> <td align="right"> <math>\frac{d\ln P_0}{d\ln r_0}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> (n+1) \, . </math> </td> </tr> </table> Also, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\biggl(\frac{r_0^2}{\rho_0}\biggr) \frac{d^2\rho_0}{dr_0^2}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl(\frac{\xi^2}{\rho_c \theta^n}\biggr) \frac{d}{d\xi}\biggl[n\rho_c \theta^{n-1} \theta^'\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl(\frac{n\xi^2}{\theta^n}\biggr) \biggl[ (n-1)\theta^{n-2} (\theta^')^2 + \theta^{n-1} \theta^{''} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl( \frac{n\xi^2}{\theta^2}\biggr) \biggl[ (n-1)(\theta^')^2 + \theta \cdot \theta^{''} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl( \frac{n\xi^2}{\theta^2}\biggr) \biggl[ (n-1)(\theta^')^2 - \biggl( \theta^{n+1} + \frac{2\theta ~\theta^'}{\xi} \biggr) \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl( \frac{n\xi^2}{\theta^2}\biggr) \biggl[ (n-1)(\xi \theta^n \Delta)^2 + \theta^{n+1} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> n(n-1)(\xi^{n+1} \theta^{n-1} \Delta)^2 + n\xi^2 \theta^{n-1}\, . </math> </td> </tr> </table> Hence, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> LAWE </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> n( 2\Delta - 1 ) - \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr)\frac{d^2\rho_0 }{dr_0^2} + 2n^2 \Delta + (n+1) \biggl[ 1 - 3\Delta - n\Delta \biggr] + (n+1) \biggl\{\frac{(4-3\gamma_g)}{\gamma_g} \biggl[ \frac{a}{b} - \Delta \biggr] + \frac{\sigma_c^2}{6\gamma_g} \biggl(\frac{\rho_c}{\rho_0}\biggr) \biggl[ \frac{a}{b\Delta} - 1\biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 2n\Delta - n + 2n^2 \Delta + n - 3n\Delta - n^2\Delta + 1 - 3\Delta - n\Delta - \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr)\frac{d^2\rho_0 }{dr_0^2} + (n+1) \biggl\{\frac{(4-3\gamma_g)}{\gamma_g} \biggl[ \frac{a}{b} - \Delta \biggr] + \frac{\sigma_c^2}{6\gamma_g} \biggl(\frac{\rho_c}{\rho_0}\biggr) \biggl[ \frac{a}{b\Delta} - 1\biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 1 + n^2 \Delta - (2n+3)\Delta - \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr)\frac{d^2\rho_0 }{dr_0^2} + (n+1) \biggl[ \frac{4}{\gamma_g} -3\biggr] \biggl[ \frac{a}{b} - \Delta \biggr] + (n+1) \biggl\{\frac{\sigma_c^2}{6\gamma_g} \biggl(\frac{\rho_c}{\rho_0}\biggr) \biggl[ \frac{a}{b\Delta} - 1\biggr] \biggr\} \, . </math> </td> </tr> </table> If, <math>\gamma_g = (n+1)/n</math>, we can further simplify and obtain, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> LAWE </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 1 + n^2 \Delta - (2n+3)\Delta - \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr)\frac{d^2\rho_0 }{dr_0^2} + \biggl[ n-3\biggr] \biggl[ \frac{a}{b} - \Delta \biggr] + (n+1) \biggl\{\frac{\sigma_c^2}{6\gamma_g} \biggl(\frac{\rho_c}{\rho_0}\biggr) \biggl[ \frac{a}{b\Delta} - 1\biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 1 + (n-3)\frac{a}{b} + n^2 \Delta + (3-n)\Delta - (2n+3)\Delta - \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr)\frac{d^2\rho_0 }{dr_0^2} + (n+1) \biggl\{\frac{\sigma_c^2}{6\gamma_g} \biggl(\frac{\rho_c}{\rho_0}\biggr) \biggl[ \frac{a}{b\Delta} - 1\biggr] \biggr\} </math> </td> </tr> </table> =Try Again= ==General Form of Wave Equation== <br /> <table border="1" width="60%" align="center" cellpadding="8"><tr><td align="center"> <div align="center">'''LAWE'''</div> {{Math/EQ_RadialPulsation01}} </td></tr></table> Employing the substitutions, <table border="0" align="center" cellpadding="5"> <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> </tr> <tr> <td align="right"><math>\alpha</math></td> <td align="center"><math>\equiv</math></td> <td align="left"> <math> 3 - \frac{4}{\gamma_g} = \frac{3-n}{n+1} \, , </math> </td> </tr> <tr> <td align="right"><math>g_0</math></td> <td align="center"><math>\equiv</math></td> <td align="left"> <math> \frac{GM_r}{r_0^2} = - \frac{1}{\rho_0} \frac{dP_0}{dr_0} ~~~\Rightarrow ~~~ \frac{g_0\rho_0 r_0}{P_0} = - \frac{d\ln P_0}{d\ln r_0} \, , </math> </td> </tr> <tr> <td align="right"> <math>\Delta </math> </td> <td align="center"> <math>\equiv</math> </td> <td align="left"> <math> \frac{M_r}{4\pi r_0^3\rho_0} = \frac{1}{4\pi G} \biggl( \frac{g_0}{r_0\rho_0}\biggr) = - \biggl[\frac{P_0}{4\pi G r_0^2 \rho_0^2} \cdot \frac{d\ln P_0}{d \ln r_0} \biggr] \, , </math> </td> </tr> </table> we have, <table border="0" cellpadding="5" align="center"> <tr> <td align="right">LAWE</td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{d^2x}{dr_0^2} + \frac{1}{r_0}\biggl[4 - \frac{g_0 \rho_0 r_0}{P_0} \biggr] \frac{dx}{dr_0} + \biggl[ \biggl(\frac{4}{\gamma_g} - 3 \biggr)\frac{g_0 \rho_0 r_0}{ P_0} \biggr] \frac{x}{r_0^2} + \biggl(\frac{\rho_0}{ P_0} \biggr)\biggl[ 4\pi G \rho_c \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggr] x </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{d^2x}{dr_0^2} + \frac{1}{r_0}\biggl[4 + \frac{d\ln P_0}{d\ln r_0} \biggr] \frac{dx}{dr_0} + \biggl[ \alpha \cdot \frac{d\ln P_0}{d\ln r_0} \biggr] \frac{x}{r_0^2} - \frac{1}{\Delta} \biggl[\frac{d\ln P_0}{d\ln r_0}\cdot \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr)\biggr] \frac{x}{r_0^2} \, . </math> </td> </tr> </table> <table border="1" width="80%" align="center" cellpadding="8"><tr><td align="left"> In the context of polytropic configurations (see more [[#Assume_Polytropic_Relations|below]]), we appreciate that, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\frac{\rho_0}{\rho_c} </math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \theta^n \, , </math> </td> </tr> <tr> <td align="right"><math>\frac{d \ln P_0}{d\ln r_0}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> (n+1) \frac{d\ln \theta}{d\ln\xi} = - (n+1)Q \, , </math> and, </td> </tr> <tr> <td align="right"><math>\frac{1}{\Delta} \cdot \frac{d \ln P_0}{d\ln r_0}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> - (n+1) \xi^2 \theta^{n-1} \, . </math> </td> </tr> </table> Inserting these into the LAWE expression and multiplying through by the square of the polytropic length scale, <math>a_n^2</math>, we obtain, <table border="0" cellpadding="5" align="center"> <tr> <td align="right">LAWE</td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{d^2x}{d\xi^2} + \frac{1}{\xi}\biggl[4 -(n+1)Q \biggr] \frac{dx}{d\xi} - \biggl[ \alpha (n+1)Q \biggr] \frac{x}{\xi^2} + \biggl[(n+1)\frac{\xi^2}{\theta} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr)\biggr] \frac{x}{\xi^2} \, . </math> </td> </tr> </table> This is identical to what has been referred to in [[SSC/Stability/InstabilityOnsetOverview#Polytropic_Stability|a separate discussion]], as the <div align="center">'''Polytropic LAWE'''<br /> {{Math/EQ_RadialPulsation02}} </div> </td></tr></table> ==Derivatives of Δ== Here we evaluate the first derivative of <math>\Delta</math> with respect to <math>r_0</math>, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\frac{d\Delta}{dr_0}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{d}{dr_0} \biggl\{ \frac{M_r}{4\pi r_0^3\rho_0} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{4\pi r_0^3 \rho_0} \cdot \frac{dM_r}{dr_0} - \frac{3M_r}{4\pi r_0^4 \rho_0} - \frac{M_r}{4\pi r_0^3 \rho_0^2}\cdot \frac{d\rho_0}{dr_0} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{r_0} - \frac{3M_r}{4\pi r_0^4 \rho_0} - \frac{M_r}{4\pi r_0^3 \rho_0^2}\cdot \frac{d\rho_0}{dr_0} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{r_0} \biggl\{ 1 - \Delta\biggl[3 + \frac{d\ln \rho_0}{d\ln r_0} \biggr]\biggr\} </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ r_0 \cdot \frac{d\Delta}{dr_0}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ 1 - 3\Delta - \Delta \cdot \frac{d\ln \rho_0}{d\ln r_0} \biggr] \, ; </math> </td> </tr> </table> and the second derivative of <math>\Delta</math> with respect to <math>r_0</math>, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\frac{d^2\Delta}{dr_0^2}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{d}{dr_0} \biggl\{ \frac{1}{r_0} \biggr\} - \frac{d}{dr_0} \biggl\{ \frac{3M_r}{4\pi r_0^4 \rho_0} \biggr\} - \frac{d}{dr_0} \biggl\{ \frac{M_r}{4\pi r_0^3 \rho_0^2}\cdot \frac{d\rho_0}{dr_0} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> -\frac{1}{r_0^2} - \frac{3}{4\pi} \biggl\{ \frac{1}{r_0^4 \rho_0}\cdot \frac{dM_r}{dr_0} - \frac{4M_r}{ r_0^5 \rho_0} - \frac{M_r}{ r_0^4 \rho_0^2} \cdot \frac{d\rho_0}{dr_0} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - \frac{1}{4\pi} \biggl\{ \frac{1}{r_0^3 \rho_0^2}\cdot \frac{d\rho_0}{dr_0} \cdot \frac{dM_r}{dr_0} - \frac{3M_r}{r_0^4 \rho_0^2}\cdot \frac{d\rho_0}{dr_0} - \frac{2M_r}{r_0^3 \rho_0^3}\cdot \biggl[\frac{d\rho_0}{dr_0}\biggr]^2 + \frac{M_r}{r_0^3 \rho_0^2}\cdot \frac{d^2\rho_0}{dr_0^2} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> -\frac{1}{r_0^2} - \frac{3}{r_0^2} \biggl\{ 1 - \Delta \biggl[4 + \frac{d\ln \rho_0}{d\ln r_0}\biggr] \biggr\} - \frac{1}{r_0^2} \biggl\{ \biggl[ 1 - 3\Delta \biggr] \frac{d\ln \rho_0}{d\ln r_0} - 2\Delta\cdot \biggl[\frac{d\ln \rho_0}{d\ln r_0}\biggr]^2 + \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr) \frac{d^2\rho_0}{dr_0^2} \biggr\} </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ - r_0^2 \cdot \frac{d^2\Delta}{dr_0^2}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> 1 + 3 - 3\Delta \biggl[4 + \frac{d\ln \rho_0}{d\ln r_0}\biggr] + \biggl[ 1 - 3\Delta \biggr] \frac{d\ln \rho_0}{d\ln r_0} - 2\Delta\cdot \biggl[\frac{d\ln \rho_0}{d\ln r_0}\biggr]^2 + \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr) \frac{d^2\rho_0}{dr_0^2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 4 - 12\Delta + \biggl[ 1 - 6\Delta \biggr] \frac{d\ln \rho_0}{d\ln r_0} - 2\Delta\cdot \biggl[\frac{d\ln \rho_0}{d\ln r_0}\biggr]^2 + \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr) \frac{d^2\rho_0}{dr_0^2} \, . </math> </td> </tr> </table> ==Trial Eigenfunction== As [[#Trial_Eigenfunction_.26_Its_Derivatives|above]], let's adopt a ''trial'' eigenfunction of the form, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>x_t</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> a -\frac{bM_r}{4\pi r_0^3 \rho_0 } = a - b\Delta \, . </math> </td> </tr> </table> Then we have, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\frac{1}{b} \biggl[ r_0^2 ~\times~ \mathrm{LAWE}~\biggr]_\mathrm{trial}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> - r_0^2 \cdot \frac{d^2\Delta}{dr_0^2} - \biggl[4 + \frac{d\ln P_0}{d\ln r_0} \biggr]r_0\cdot \frac{d\Delta}{dr_0} + \biggl[ \alpha \cdot \frac{d\ln P_0}{d\ln r_0} \biggr] \biggl(\frac{a}{b} - \Delta \biggr) - \frac{1}{\Delta} \biggl[\frac{d\ln P_0}{d\ln r_0}\cdot \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr)\biggr] \biggl(\frac{a}{b} - \Delta \biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl\{ 4 - 12\Delta + \biggl[ 1 - 6\Delta \biggr] \frac{d\ln \rho_0}{d\ln r_0} - 2\Delta\cdot \biggl[\frac{d\ln \rho_0}{d\ln r_0}\biggr]^2 + \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr) \frac{d^2\rho_0}{dr_0^2} \biggr\} - ~ \biggl[4 + \frac{d\ln P_0}{d\ln r_0} \biggr] \biggl[ 1 - 3\Delta - \Delta \cdot \frac{d\ln \rho_0}{d\ln r_0} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + ~\biggl[ \alpha \cdot \frac{d\ln P_0}{d\ln r_0} \biggr] \biggl(\frac{a}{b} - \Delta \biggr) - \frac{1}{\Delta} \biggl[\frac{d\ln P_0}{d\ln r_0}\cdot \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr)\biggr] \biggl(\frac{a}{b} - \Delta \biggr) \, . </math> </td> </tr> </table> ===Assume Polytropic Relations=== If we assume that the equilibrium models are polytropes, then we know that, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\rho_0 \propto \theta^n</math> </td> <td align="center"> <math>\Rightarrow</math> </td> <td align="left"> <math> \frac{d\ln \rho_0}{d\ln r_0} = n \cdot \frac{d\ln\theta}{d\ln\xi} \, ; </math> </td> </tr> <tr> <td align="right"> <math>P_0 \propto \theta^{n+1}</math> </td> <td align="center"> <math>\Rightarrow</math> </td> <td align="left"> <math> \frac{d\ln P_0}{d\ln r_0} = (n+1)\cdot \frac{d\ln\theta}{d\ln\xi} \, . </math> </td> </tr> </table> We also deduce that, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\biggl( \frac{r_0^2}{\rho_0}\biggr) \frac{d^2\rho_0}{dr_0^2}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl( \frac{\xi^2}{\theta^n}\biggr) \frac{d}{d\xi}\biggl[ \frac{d\theta^n}{d\xi}\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl( \frac{\xi^2}{\theta^n}\biggr) \frac{d}{d\xi}\biggl[ n\theta^{n-1} \theta^'\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl( \frac{n\xi^2}{\theta^n}\biggr) \biggl[ (n-1)\theta^{n-2} (\theta^' )^2 + \theta^{n-1} \theta^{''} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[ \frac{n(n-1)\xi^2}{\theta^2}\biggr] (\theta^' )^2 - \biggl( \frac{n\xi^2}{\theta}\biggr) \biggl[ \theta^n + \frac{2}{\xi} \theta^' \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> n(n-1) \cdot \biggl[ \frac{d\ln\theta}{d\ln\xi}\biggr]^2 - n \xi^2\theta^{n-1} - 2n \cdot \frac{d\ln \theta}{d\ln \xi} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> n(n-1) \cdot \Xi^2- n \xi^2\theta^{n-1} - 2n \cdot \Xi \, , </math> </td> </tr> </table> where we have introduced the shorthand notation, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\Xi</math> </td> <td align="center"> <math>\equiv</math> </td> <td align="left"> <math> \frac{d\ln\theta}{d\ln \xi}\, . </math> </td> </tr> </table> Drawing from our [[SSC/Stability/Polytropes#Groundwork|accompanying discussion]], for example, we note as well that, <table border="0" align="center" cellpadding="5"> <tr> <td align="right"> <math>\Delta </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - \biggl[\frac{P_0}{4\pi G r_0^2 \rho_0^2} \cdot \frac{d\ln P_0}{d \ln r_0} \biggr] </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ - \frac{1}{\Delta} \cdot \frac{d\ln P_0}{d \ln r_0} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{4\pi G r_0^2 \rho_0^2}{P_0} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 4\pi G (a_n^2 \xi^2) (\rho_c \theta^n)^2 [K^{-1} \rho_c^{-(n+1)/n} \theta^{-(n+1)}] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 4\pi G \biggl[ \frac{(n+1)K}{4\pi G} \cdot \rho_c^{(1-n)/n} \biggr] (\rho_c )^2 \biggl[K^{-1} \rho_c^{-(n+1)/n} \biggr] \xi^2 \theta^{ (n-1)} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> (n+1)\xi^2 \theta^{ (n-1)} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \Delta \cdot \xi^2 \theta^{n-1} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - \frac{1}{(n+1)} \cdot \frac{d\ln P_0}{d \ln r_0} = - \Xi \, . </math> </td> </tr> </table> Hence, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\frac{1}{b} \biggl[ r_0^2 ~\times~ \mathrm{LAWE}~\biggr]_\mathrm{trial}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> 4 - 12\Delta + \biggl[ 1 - 6\Delta \biggr] n\Xi - 2\Delta\cdot \biggl[n\Xi\biggr]^2 + \Delta \biggl[ n(n-1) \cdot \Xi^2- n \xi^2\theta^{n-1} - 2n \cdot \Xi \biggr] - ~ \biggl[4 + (n+1)\Xi \biggr] \biggl[ 1 - 3\Delta - \Delta \cdot n\Xi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + ~\biggl[ \alpha \cdot (n+1)\Xi \biggr] \biggl(\frac{a}{b} - \Delta \biggr) - \frac{1}{\Delta} \biggl[(n+1)\Xi \cdot \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr)\biggr] \biggl(\frac{a}{b} - \Delta \biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> n\Xi - 6\Delta n\Xi - 2\Delta\cdot \biggl[n\Xi\biggr]^2 + \Delta \biggl[ n(n-1) \cdot \Xi^2- n \xi^2\theta^{n-1} + 2n \cdot \Xi \biggr] + ~ 2(n+1)\Xi + ~ n(n+1)\Xi^2 \cdot \Delta </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + ~\biggl[ \alpha \cdot (n+1)\Xi \biggr] \biggl(\frac{a}{b} \biggr) - ~\biggl[ \alpha \cdot (n+1)\Xi \biggr] \Delta - \frac{1}{\Delta} \biggl[(n+1)\Xi \cdot \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr)\biggr] \biggl(\frac{a}{b} - \Delta \biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math>\Xi \biggl[ 1 + n + ~ 2(n+1) + \alpha \cdot (n+1) \biggl(\frac{a}{b} \biggr) \biggr] + \Delta \biggl[ - [4n + \alpha \cdot (n+1)]\Xi + n(n-1) \cdot \Xi^2 + n(n+1)\Xi^2 - 2n^2 \Xi^2 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - \frac{1}{\Delta} \biggl[(n+1)\Xi \cdot \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr)\biggr] \biggl(\frac{a}{b} - \Delta \biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math>\Xi \biggl[ 3(n+1) + \alpha \cdot (n+1) \biggl(\frac{a}{b} \biggr) \biggr] - \Delta [4n + \alpha \cdot (n+1)]\Xi - \frac{1}{\Delta} \biggl[(n+1)\Xi \cdot \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr)\biggr] \biggl(\frac{a}{b} - \Delta \biggr) </math> </td> </tr> </table> =Third Time= ==General Relations== Various ''general'' relations taken from above derivations: <table border="0" cellpadding="5" align="center"> <tr> <td align="right">LAWE</td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{d^2x}{dr_0^2} + \frac{1}{r_0}\biggl[4 + \frac{d\ln P_0}{d\ln r_0} \biggr] \frac{dx}{dr_0} + \biggl[ \alpha \cdot \frac{d\ln P_0}{d\ln r_0} \biggr] \frac{x}{r_0^2} - \frac{1}{\Delta} \biggl[\frac{d\ln P_0}{d\ln r_0}\cdot \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr)\biggr] \frac{x}{r_0^2} \, , </math> </td> </tr> </table> where, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\Delta</math></td> <td align="center"><math>\equiv</math></td> <td align="left"> <math> \frac{M_r}{4\pi r_0^3 \rho_0} </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ \frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> -~\frac{4\pi G r_0^2 \rho_0^2}{P_0} \, ; </math> </td> </tr> <tr> <td align="right"><math>r_0 \cdot \frac{d\Delta}{dr_0}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ 1 - 3\Delta - \Delta \cdot \frac{d\ln \rho_0}{d\ln r_0} \biggr] \, ; </math> </td> </tr> <tr> <td align="right"><math>- r_0^2 \cdot \frac{d^2\Delta}{dr_0^2}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> 4 - 12\Delta + \biggl[ 1 - 6\Delta \biggr] \frac{d\ln \rho_0}{d\ln r_0} - 2\Delta\cdot \biggl[\frac{d\ln \rho_0}{d\ln r_0}\biggr]^2 + \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr) \frac{d^2\rho_0}{dr_0^2} \, . </math> </td> </tr> </table> ==Polytropes== If polytropic relations are adopted: <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\Delta</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{Q}{\xi^2 \theta^{n-1}} \, ; </math> </td> </tr> <tr> <td align="right"><math>\frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> -(n+1) \xi^2 \theta^{n-1} \, ; </math> </td> </tr> <tr> <td align="right"><math>r_0^2 \times~ \mathrm{LAWE}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> r_0^2 \frac{d^2x}{dr_0^2} + \biggl[4 -(n+1)Q \biggr] r_0 \cdot \frac{dx}{dr_0} + \biggl[ (n-3)Q \biggr] x + \frac{1}{\Delta} \biggl[(n+1) Q \cdot \frac{1}{\theta^n} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr)\biggr] x \, ; </math> </td> </tr> <tr> <td align="right"><math>r_0 \cdot \frac{d\Delta}{dr_0}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[ 1 - 3\Delta + n \Delta Q \biggr] =1 + (nQ - 3)\Delta \, ; </math> </td> </tr> <tr> <td align="right"><math>- r_0^2 \cdot \frac{d^2\Delta}{dr_0^2}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> 4 - 12\Delta - n\biggl[ 1 - 6\Delta \biggr] Q - 2n^2 \Delta Q^2 + \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr) \frac{d^2\rho_0}{dr_0^2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 4 - 12\Delta - n\biggl[ 1 - 6\Delta \biggr] Q - 2n^2 \Delta Q^2 + \Delta \biggl[ n(n-1)Q^2 +2nQ-\frac{nQ}{\Delta} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 4 -nQ - 12\Delta + 6nQ\Delta - 2n^2 \Delta Q^2 + \Delta \biggl[ n(n-1)Q^2 +2nQ \biggr] - nQ </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 4 - 2nQ + \Delta \biggl[6nQ -2n^2Q^2 + n(n-1)Q^2 +2nQ - 12 \biggr] \, . </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 4 - 2nQ + \Delta \biggl[8nQ - n^2Q^2 - nQ^2 - 12 \biggr] \, . </math> </td> </tr> </table> ==Eigenfunction Choice== Again, let's try the ''trial'' eigenfunction, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>x_t</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> a - b\Delta \, , </math> </td> </tr> </table> in which case, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\frac{1}{b}\biggl[ r_0^2 \times~ \mathrm{LAWE} \biggr]</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> - r_0^2 \frac{d^2\Delta}{dr_0^2} - \biggl[4 -(n+1)Q \biggr] r_0 \cdot \frac{d\Delta}{dr_0} + \biggl[ (n-3)Q \biggr] \biggl( \frac{a}{b} - \Delta\biggr) + \frac{1}{\Delta} \biggl[(n+1) Q \cdot \frac{1}{\theta^n} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr)\biggr] \biggl( \frac{a}{b} - \Delta\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 4 - 2nQ + \Delta \biggl[8nQ - n^2Q^2 - nQ^2 - 12\biggr] - \biggl[4 -(n+1)Q \biggr] \biggl[1 + (nQ - 3)\Delta \biggr] + \biggl[ (n-3)Q \biggr] \biggl( \frac{a}{b} - \Delta\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \frac{1}{\Delta} \biggl[(n+1) Q \cdot \frac{1}{\theta^n} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr)\biggr] \biggl( \frac{a}{b} - \Delta\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 4 - 2nQ + \Delta \biggl[8nQ - n^2Q^2 - nQ^2 - 12\biggr] - 4 - (4nQ - 12)\Delta + (n+1)Q + (n+1) (nQ^2 - 3Q)\Delta + (n-3)Q \biggl( \frac{a}{b} \biggr) - (n-3)Q \Delta </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \frac{1}{\Delta} \biggl[(n+1) Q \cdot \frac{1}{\theta^n} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr)\biggr] \biggl( \frac{a}{b} - \Delta\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[(n+1) - 2n + (n-3) \biggl( \frac{a}{b} \biggr)\biggr]Q + \Delta \biggl[8nQ - n^2Q^2 - nQ^2 - 12 - 4nQ + 12 + (n+1) (nQ^2 ) -3Q (n+1) + (3-n)Q \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \frac{1}{\Delta} \biggl[(n+1) Q \cdot \frac{1}{\theta^n} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr)\biggr] \biggl( \frac{a}{b} - \Delta\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl[(1-n) + (n-3) \biggl( \frac{a}{b} \biggr)\biggr]Q + \Delta \biggl[ 0 \biggr] + \frac{1}{\Delta} \biggl[(n+1) Q \cdot \frac{1}{\theta^n} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr)\biggr] \biggl( \frac{a}{b} - \Delta\biggr) \, . </math> </td> </tr> </table> Hence, we are left with only the <math>\sigma_c^2</math> term if we set, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>0</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> (1-n) + (n-3) \biggl( \frac{a}{b} \biggr)</math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ \biggl( \frac{a}{b} \biggr)</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{n-1}{n-3} \, . </math> </td> </tr> </table> We conclude, therefore, that the radial displacement function (''i.e.,'' the eigenfunction) for the neutral <math>(\sigma_c^2 = 0)</math> mode of all polytropic configurations is, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>x_\mathrm{neutral}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{b} \biggl[ \frac{n-1}{n-3} - \Delta \biggr] = \frac{1}{b} \biggl[ \frac{n-1}{n-3} - \frac{Q}{\xi^2 \theta^{n-1}} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{b} \biggl( \frac{n-1}{n-3} \biggr) \biggl[1 + \biggl(\frac{n-3}{n-1}\biggr)\frac{1}{\xi^2 \theta^{n-1}} \cdot \frac{d\ln \theta}{d\ln \xi} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{b} \biggl( \frac{n-1}{n-3} \biggr) \biggl[1 + \biggl(\frac{n-3}{n-1}\biggr)\frac{1}{\xi \theta^{n}} \cdot \frac{d \theta}{d \xi} \biggr] \, . </math> </td> </tr> </table> This last expression exactly matches [[SSC/Stability/InstabilityOnsetOverview#NeutralMode|our earlier result found for polytropic configurations]] if we choose an overall amplitude coefficient of the form, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>b</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{2n}{3(n-3)} \, . </math> </td> </tr> </table> '''<font color="red">Hooray!</font>''' =Summary= ==Setup== We begin with the traditional, <div align="center" id="2ndOrderODE"> <font color="#770000">'''Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br /> {{Math/EQ_RadialPulsation01}} </div> This linear, adiabatic wave equation (LAWE) can straightforwardly be rewritten in the form we will refer to as the, <div align="center" id="Delta_Highlighted"> <font color="#770000">'''Δ-Highlighted LAWE'''</font><br /> {{Math/EQ_RadialPulsation04}} </div> Multiplying this ''Δ-Highlighted LAWE'' through by <math>a_n^2 = (r_0/\xi)^2</math> and recognizing that, for polytropic configurations, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\Delta</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> -~\frac{1}{\xi^2 \theta^{n-1}} \frac{d\ln \theta}{d\ln \xi} \, , </math> </td> <td align="center"> </td> <td align="right"><math>\frac{d\ln P_0}{d\ln r_0}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> ~(n+1)\frac{d\ln \theta}{d\ln \xi} \, , </math> </td> <td align="center"> <math>\Rightarrow</math> </td> <td align="right"><math>\frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> -(n+1) \xi^2 \theta^{n-1} \, , </math> </td> </tr> </table> we immediately obtain what we have frequently referred to as the, <div align="center" id="PolytropicLAWE"> <font color="#770000">'''Polytropic LAWE'''</font><br /> {{Math/EQ_RadialPulsation02}} </div> ==Neutral-Mode Eigenfunction== In the preceding subsections of this chapter, we have demonstrated that if <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\gamma_g = \frac{n+1}{n}</math></td> <td align="center"><math>~~~\Rightarrow ~~~</math></td> <td align="left"> <math> \alpha = \frac{3-n}{n+1} \, , </math> </td> </tr> </table> the radial displacement function (''i.e.,'' the eigenfunction) for the neutral <math>(\sigma_c^2 = 0)</math> mode of all polytropic configurations is, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>x_\mathrm{neutral}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{3(n-3)}{2n} \biggl[ \frac{n-1}{n-3} - \Delta \biggr] \, , </math> </td> </tr> </table> to within an arbitrarily chosen leading scaling coefficient. More completely, if we let "LAWE" stand for the RHS of our Δ-Highlighted LAWE, then setting <math>x = x_\mathrm{neutral}</math> results in the expression, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\frac{1}{b}\biggl[ r_0^2 \times~ \mathrm{LAWE} \biggr]</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> (n+1) Q \cdot \frac{1}{\theta^n} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl[ \frac{1}{\Delta} \biggl( \frac{n-1}{n-3}\biggr) - 1 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> -(n+1) \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl[ \frac{\xi^2}{ \theta} \biggl( \frac{n-1}{n-3}\biggr) - \frac{1}{\theta^n} \cdot \frac{d\ln\theta}{d\ln\xi} \biggr] \, , </math> </td> </tr> </table> which goes to zero if <math>\sigma_c^2 = 0</math>. =Parabolic Density Distribution= Here, we build upon our [[SSC/Structure/OtherAnalyticModels#Parabolic_Density_Distribution|separate discussion]] of equilibrium configurations with a parabolic density distribution. ==Equilibrium Structure== In an article titled, "Radial Oscillations of a Stellar Model," [http://adsabs.harvard.edu/abs/1949MNRAS.109..103P C. Prasad (1949, MNRAS, 109, 103)] investigated the properties of an equilibrium configuration with a prescribed density distribution given by the expression, <div align="center"> <math>\rho_0 = \rho_c\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr] \, ,</math> </div> where, <math>\rho_c</math> is the central density and, <math>R</math> is the radius of the star. Both the mass distribution and the pressure distribution can be obtained analytically from this specified density distribution. Specifically, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>M_r</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>\int_0^{r_0} 4\pi r_0^2 \rho_0 dr_0</math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>\frac{4\pi\rho_c r_0^3}{3} \biggl[1 - \frac{3}{5} \biggl( \frac{r_0}{R} \biggr)^2 \biggr] \, ,</math> </td> </tr> </table> </div> in which case we can write, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~g_0 \equiv \frac{G M_r }{r_0^2} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{4\pi G \rho_c r_0}{3} \biggl[1 - \frac{3}{5} \biggl( \frac{r_0}{R} \biggr)^2\biggr] \, ,</math> </td> </tr> </table> </div> and, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\Delta \equiv \frac{M_r }{4\pi r_0^3\rho_0} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> \frac{1 }{3} \biggl[1 - \frac{3}{5} \biggl( \frac{r_0}{R} \biggr)^2 \biggr]\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-1} \, .</math> </td> </tr> </table> Hence, proceeding via what we have labeled as [[SSCpt2/SolutionStrategies#Technique_1|"Technique 1"]], and enforcing the surface boundary condition, <math>~P(R) = 0</math>, [http://adsabs.harvard.edu/abs/1949MNRAS.109..103P Prasad (1949)] determines that, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>P_0</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{4\pi G\rho_c^2 R^2}{15} \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^2 \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \, ,</math> </td> </tr> </table> </div> where, it can readily be deduced, as well, that the central pressure is, <div align="center"> <math>P_c = \frac{4\pi}{15} G\rho_c^2 R^2 \, .</math> </div> <table border="1" width="90%" cellpadding="8" align="center"><tr><td align="left"> <div align="center">'''Specific Entropy Distribution'''</div> For purposes of later discussion, we find from [[Appendix/Ramblings/PatrickMotl#Tying_Expressions_into_H_Book_Context|a separate examination of specific entropy distributions]], <math>s_0(r_0)</math>, that, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{s_0}{\Re/\bar{\mu}}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{1}{(\gamma_g-1)}\ln \biggl(\frac{\tau_0}{\rho_0}\biggr)^{\gamma_g} = \frac{1}{(\gamma_g-1)}\ln \biggl[ \frac{P_0}{(\gamma_g-1)\rho_0^{\gamma_g}} \biggr] </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \biggl[ \frac{\gamma_g - 1}{\Re/\bar{\mu}} \biggr]s_0</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \ln \biggl[ \frac{P_c}{(\gamma_g-1)\rho_c^{\gamma_g}} \biggr] + \ln \biggl\{ \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^2 \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggr\} + \ln \biggl\{ \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-\gamma_g}\biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \ln \biggl[ \frac{P_c}{(\gamma_g-1)\rho_c^{\gamma_g}} \biggr] + \ln \biggl\{ \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggr\} + (2- \gamma_g) \ln \biggl\{ \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]\biggr\} \, . </math> </td> </tr> </table> Notice that, independent of the value of <math>\gamma_g</math>, the specific entropy varies with <math>r_0</math> throughout the structure. According to the [[2DStructure/AxisymmetricInstabilities#Schwarzschild_Criterion|Schwarzschild criterion]], spherically symmetric equilibrium configurations will be stable against convection if the specific entropy increases outward, and unstable toward convection if the specific entropy decreases outward. Let's examine the slope, <math>ds_0/dr_0</math>, throughout configurations that have a parabolic density distribution. <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\biggl[ \frac{\gamma_g - 1}{\Re/\bar{\mu}} \biggr]\frac{ds_0}{dr_0}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ -\frac{r_0}{R^2} \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} - \frac{2(2- \gamma_g)r_0}{R^2} \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-1} </math> </td> </tr> <tr> <td align="right"> <math> \Rightarrow ~~~ \frac{R^2}{r_0} \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr] \biggl[ \frac{\gamma_g - 1}{\Re/\bar{\mu}} \biggr]\frac{ds_0}{dr_0} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> -~\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr] - 2(2- \gamma_g) \biggl[ 1 - \frac{1}{2}\biggl(\frac{r_0}{R} \biggr)^2 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~(2\gamma_g- 5) + (3 - \gamma_g) \biggl[\biggl(\frac{r_0}{R} \biggr)^2 \biggr] </math> </td> </tr> </table> [[File:EntropyDistribution245.png|right|400px]] The slope is zero when, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math> \biggl(\frac{r_0}{R} \biggr)^2 </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{5 - 2\gamma_g}{3 - \gamma_g} \, . </math> </td> </tr> </table> Moving from the center of the configuration to its surface, <math>0 < (r_0/R)^2 < 1</math>, the slope will go to zero — hence, the slope of the entropy will change sign </td></tr></table> ==Some Relevant Structural Derivatives== We note for later use that, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{1}{P_c} \cdot \frac{dP_0}{dr_0}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^2 \frac{d}{dr_0}\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr] + \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \frac{d}{dr_0}\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> -\frac{1}{2R^2} \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^2 \frac{d}{dr_0}\biggl[r_0^2\biggr] + 2\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \frac{d}{dr_0}\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>\biggl\{ -\frac{1}{2R^2} \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^2 - \frac{2}{R^2}\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggr\}\frac{d}{dr_0}\biggl[r_0^2\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>-\frac{r_0}{R^2}\biggl\{ \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^2 + 4\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>-\frac{r_0}{R^2}\biggl\{ \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr] + 4 \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggr\} \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>-\frac{5r_0}{R^2} \biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr] \, . </math> </td> </tr> </table> <table border="1" align="center" width="80%" cellpadding="10"><tr><td align="left"> Checking for detailed force-balance, we note that, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>- ~\frac{1}{\rho_0} \cdot \frac{dP_0}{dr_0}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>\frac{5r_0}{R^2}\biggl[ \frac{4\pi G \rho_c^2 R^2}{15} \biggr] \biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr] \cdot \frac{1}{\rho_c} \biggl[1 - \biggl(\frac{r_0}{R} \biggr)^2\biggr]^{-1} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>\frac{4\pi G \rho_c r_0}{3} \biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \, , </math> </td> </tr> </table> which is exactly the expression that we have just derived for <math>g_0 = GM_r/r_0^2</math>. </td></tr></table> Hence, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{d\ln P_0}{d \ln r_0} = \frac{r_0}{P_0/P_c} \biggl[ \frac{1}{P_c}\cdot \frac{dP_0}{dr_0} \biggr]</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>-\frac{5r_0^2}{R^2} \biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggl\{\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^2 \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]\biggr\}^{-1} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> -5 \biggl(\frac{r_0}{R}\biggr)^2 \biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} \, ; </math> </td> </tr> </table> and, given that, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\Delta^{-1} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math> 3 \biggl[1 - \frac{3}{5} \biggl( \frac{r_0}{R} \biggr)^2 \biggr]^{-1} \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr] \, ,</math> </td> </tr> </table> we can write, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{1}{\Delta}\cdot \frac{d\ln P_0}{d \ln r_0} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>- 15 \biggl(\frac{r_0}{R}\biggr)^2 \biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} \biggl[1 - \frac{3}{5} \biggl( \frac{r_0}{R} \biggr)^2 \biggr]^{-1} \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>- 15 \biggl(\frac{r_0}{R}\biggr)^2 \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} \, . </math> </td> </tr> </table> Also, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>3 \cdot \frac{d\Delta}{dr_0} </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{d}{dr_0}\biggl\{ \biggl[1 - \frac{3}{5} \biggl( \frac{r_0}{R} \biggr)^2 \biggr] \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-1} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[1 - \frac{3}{5} \biggl( \frac{r_0}{R} \biggr)^2 \biggr]\frac{d}{dr_0} \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-1} + \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-1}\frac{d}{dr_0} \biggl[1 - \frac{3}{5} \biggl( \frac{r_0}{R} \biggr)^2 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[1 - \frac{3}{5} \biggl( \frac{r_0}{R} \biggr)^2 \biggr] \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-2} \frac{2r_0}{R^2} + \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-1} \biggl[- \frac{6}{5} \cdot \frac{r_0}{R^2} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{r_0}{R^2} \biggl\{ 2\biggl[1 - \frac{3}{5} \biggl( \frac{r_0}{R} \biggr)^2 \biggr] \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-2} - \frac{6}{5} \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-1} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{r_0}{5R^2} \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-2}\biggl\{ \biggl[10 - 6 \biggl( \frac{r_0}{R} \biggr)^2 \biggr] - \biggl[ 6 - 6\biggl(\frac{r_0}{R} \biggr)^2 \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{4r_0}{5R^2} \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-2} </math> </td> </tr> <tr> <td align="right"> <math>\Rightarrow ~~~ \frac{d\Delta}{dr_0}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{4r_0}{15R^2} \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-2} \, ; </math> </td> </tr> </table> and, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{15R^2}{4} \cdot \frac{d^2\Delta}{dr_0^2}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-2} -2r_0 \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3}\biggl[ -\frac{2r_0}{R^2} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3} \biggl\{ \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr] +\frac{4r_0^2}{R^2} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3} \biggl[1 + 3 \biggl(\frac{r_0}{R}\biggr)^2 \biggr] \, . </math> </td> </tr> </table> ==Neutral Mode== Again, adopting the ''trial'' eigenfunction, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>x_t</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> a - b\Delta \, , </math> </td> </tr> </table> from the, <div align="center" id="Delta_Highlighted"> <font color="#770000">'''Δ-Highlighted LAWE'''</font><br /> {{Math/EQ_RadialPulsation04}} </div> we can write, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\mathrm{LAWE}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> - b \cdot \frac{d^2\Delta}{dr_0^2} - \frac{b}{r_0}\biggl[4 + \frac{d\ln P_0}{d\ln r_0} \biggr] \frac{d\Delta}{dr_0} + \alpha\biggl[ \frac{d\ln P_0}{d\ln r_0} \biggr] \frac{(a - b\Delta)}{r_0^2} - \frac{1}{\Delta} \biggl[\frac{d\ln P_0}{d\ln r_0}\cdot \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr)\biggr] \frac{(a - b\Delta)}{r_0^2} </math> </td> </tr> </table> ===First Attempt=== <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\Rightarrow ~~~ \frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> - r_0^2 \cdot \frac{d^2\Delta}{dr_0^2} - r_0\biggl[4 + \frac{d\ln P_0}{d\ln r_0} \biggr] \frac{d\Delta}{dr_0} + \alpha\biggl[ \frac{d\ln P_0}{d\ln r_0} \biggr] \biggl( \frac{a}{b} - \Delta \biggr) - \frac{1}{\Delta} \biggl[\frac{d\ln P_0}{d\ln r_0}\cdot \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr)\biggr] \biggl(\frac{a}{b} - \Delta \biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> - r_0^2 \cdot \frac{d^2\Delta}{dr_0^2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - r_0\biggl[4 + \frac{d\ln P_0}{d\ln r_0} \biggr] \frac{d\Delta}{dr_0} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \alpha\biggl[ \frac{d\ln P_0}{d\ln r_0} \biggr] \biggl( \frac{a}{b} - \Delta \biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{ \biggl[\frac{d\ln P_0}{d\ln r_0}\biggr] - \frac{a}{b}\biggl[\frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}\biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> - \frac{4}{15} \biggl(\frac{r_0}{R}\biggr)^2 \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3} \biggl[1 + 3 \biggl(\frac{r_0}{R}\biggr)^2 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> - \frac{16}{15}\biggl(\frac{r_0}{R}\biggr)^2 \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-2} - \frac{4}{15}\biggl(\frac{r_0}{R}\biggr)^2 \biggl\{ -5 \biggl(\frac{r_0}{R}\biggr)^2 \biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} \biggr\} \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \alpha \biggl\{ -5 \biggl(\frac{r_0}{R}\biggr)^2 \biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} \biggr\} \biggl\{ \frac{a}{b} - \Delta \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{ \biggl[\frac{d\ln P_0}{d\ln r_0}\biggr] - \frac{a}{b}\biggl[\frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}\biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> - \frac{4}{3}\biggl(\frac{r_0}{R}\biggr)^2\biggl[ 1 - \frac{1}{5} \biggl(\frac{r_0}{R}\biggr)^2 \biggr] \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> +~ \frac{4}{3}\biggl(\frac{r_0}{R}\biggr)^4 \biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + ~ \frac{5\alpha}{3} \biggl(\frac{r_0}{R}\biggr)^2 \biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-2} \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} \biggl\{ 1 - \frac{3a}{b} -\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{ \biggl[\frac{d\ln P_0}{d\ln r_0}\biggr] - \frac{a}{b}\biggl[\frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}\biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> - \frac{4}{3}\biggl(\frac{r_0}{R}\biggr)^2\biggl[ 1 - \frac{1}{5} \biggl(\frac{r_0}{R}\biggr)^2 \biggr] \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3} + ~ \frac{5\alpha}{3} \biggl(\frac{r_0}{R}\biggr)^2 \biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-2} \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} \biggl(1 - \frac{3a}{b} \biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> +~ \frac{4}{3}\biggl(\frac{r_0}{R}\biggr)^4 \biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3} - ~ \alpha \biggl(\frac{r_0}{R}\biggr)^4 \biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-2} \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{ \biggl[\frac{d\ln P_0}{d\ln r_0}\biggr] - \frac{a}{b}\biggl[\frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}\biggr] \biggr\} </math> </td> </tr> </table> Continuing … <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl(\frac{r_0}{R}\biggr)^2 \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3} \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} \biggl\{ \frac{5\alpha}{3}\biggl(1 - \frac{3a}{b} \biggr) \biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2 \biggr] \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr] - \frac{4}{3}\biggl[ 1 - \frac{1}{5} \biggl(\frac{r_0}{R}\biggr)^2 \biggr]\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> +~ \biggl(\frac{4}{3} - \alpha \biggr)\biggl(\frac{r_0}{R}\biggr)^4\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1}\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> -~ \alpha \biggl(\frac{r_0}{R}\biggr)^6\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1}\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{ \biggl[\frac{d\ln P_0}{d\ln r_0}\biggr] - \frac{a}{b}\biggl[\frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}\biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl(\frac{r_0}{R}\biggr)^2 \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3} \biggl\{ \frac{5\alpha}{3}\biggl(1 - \frac{3a}{b} \biggr) \biggl[1-\frac{8}{5}\biggl(\frac{r_0}{R}\biggr)^2 + \frac{3}{5} \biggl( \frac{r_0}{R}\biggr)^4\biggr] - ~\frac{4}{3}\biggl[ 1 - \frac{7}{5} \biggl(\frac{r_0}{R}\biggr)^2 + \frac{1}{10}\biggl(\frac{r_0}{R}\biggr)^4\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> +~ \biggl(\frac{4}{3} - \alpha \biggr)\biggl(\frac{r_0}{R}\biggr)^2 \biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] -~ \alpha \biggl(\frac{r_0}{R}\biggr)^4\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{ \biggl[\frac{d\ln P_0}{d\ln r_0}\biggr] - \frac{a}{b}\biggl[\frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}\biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl(\frac{r_0}{R}\biggr)^2 \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3} \biggl\{ \biggl[\frac{5\alpha}{3}\biggl(1 - \frac{3a}{b} \biggr) - \frac{4}{3} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \biggl(\frac{r_0}{R}\biggr)^2 \biggl[ -\frac{8\alpha}{3}\biggl(1 - \frac{3a}{b} \biggr) + \frac{28}{15} \biggr] + \biggl( \frac{r_0}{R}\biggr)^4 \biggl[\alpha\biggl(1 - \frac{3a}{b} \biggr) - ~\frac{2}{15} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> +~ \biggl(\frac{4}{3} - \alpha \biggr)\biggl(\frac{r_0}{R}\biggr)^2 \biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] -~ \alpha \biggl(\frac{r_0}{R}\biggr)^4\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{ \biggl[\frac{d\ln P_0}{d\ln r_0}\biggr] - \frac{a}{b}\biggl[\frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}\biggr] \biggr\} </math> </td> </tr> </table> ===Second Attempt=== <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> - r_0^2 \cdot \frac{d^2\Delta}{dr_0^2} - r_0\biggl[4 \biggr] \frac{d\Delta}{dr_0} + \biggl[\frac{d\ln P_0}{d\ln r_0} \biggr]\biggl[ - r_0\cdot \frac{d\Delta}{dr_0} + \alpha \biggl( \frac{a}{b} - \Delta \biggr)\biggr] + \biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]\biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{1 - \frac{a}{b}\biggl[\frac{1}{\Delta} \biggr] \biggr\} \frac{\rho_c}{\rho_0} \, , </math> </td> </tr> </table> where, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\frac{\rho_0}{\rho_c}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \biggl(1 - x^2\biggr) \, , </math> </td> </tr> <tr> <td align="right"><math>\Delta</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{1}{3} \biggl(1 - \frac{3}{5} x^2\biggr)\biggl(1 - x^2\biggr)^{-1} \, , </math> </td> </tr> <tr> <td align="right"><math>\frac{d\ln P_0}{d\ln r_0}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> -5x^2 \biggl(1 - \frac{3}{5} x^2\biggr)\biggl(1 - x^2\biggr)^{-1} \biggl(1 - \frac{1}{2}x^2\biggr)^{-1} = -15 x^2 \biggl(1 - \frac{1}{2}x^2\biggr)^{-1} \Delta \, , </math> </td> </tr> <tr> <td align="right"><math>r_0 \cdot \frac{d\Delta}{dr_0}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{4}{15} x^2 \biggl(1 - x^2\biggr)^{-2} \, , </math> </td> </tr> <tr> <td align="right"><math>r_0^2 \cdot \frac{d^2\Delta}{dr_0^2}</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{4}{15} x^2 \biggl(1 - x^2\biggr)^{-3}\biggl( 1 + 3x^2\biggr) \, . </math> </td> </tr> </table> Hence, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> - r_0^2 \cdot \frac{d^2\Delta}{dr_0^2} - r_0\biggl[4 \biggr] \frac{d\Delta}{dr_0} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \biggl[\frac{d\ln P_0}{d\ln r_0} \biggr]\biggl[ - r_0\cdot \frac{d\Delta}{dr_0} + \alpha \biggl( \frac{a}{b} - \Delta \biggr)\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]\biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{ 1 - \frac{a}{b}\biggl[\frac{1}{\Delta} \biggr] \biggr\} \frac{\rho_c}{\rho_0} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> - \frac{4}{15} x^2 \biggl(1 - x^2\biggr)^{-3}\biggl( 1 + 3x^2\biggr) - \frac{16}{15} x^2 \biggl(1 - x^2\biggr)^{-2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \biggl[\frac{d\ln P_0}{d\ln r_0} \biggr] \biggl[\frac{a\alpha}{b} - \frac{4}{15} x^2 \biggl(1 - x^2\biggr)^{-2} - \frac{\alpha}{3} \biggl(1 - \frac{3}{5} x^2\biggr)\biggl(1 - x^2\biggr)^{-1}\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> + \biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]\biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{ 1 - \frac{3a}{b}\biggl(1 - \frac{3}{5}x^2\biggr)^{-1} \biggl(1-x^2\biggr)\biggr\} \biggl(1 - x^2\biggr)^{-1} \, . </math> </td> </tr> </table> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>\Rightarrow ~~~ \frac{1}{x^2}\biggl(1 - x^2\biggr) \frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> - \frac{4}{15} \biggl(1 - x^2\biggr)^{-2}\biggl( 1 + 3x^2\biggr) - \frac{16}{15} \biggl(1 - x^2\biggr)^{-1} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> -5 \biggl(1 - \frac{3}{5} x^2\biggr) \biggl(1 - \frac{1}{2}x^2\biggr)^{-1} \biggl[\frac{a\alpha}{b} - \frac{4}{15} x^2 \biggl(1 - x^2\biggr)^{-2} - \frac{\alpha}{3} \biggl(1 - \frac{3}{5} x^2\biggr)\biggl(1 - x^2\biggr)^{-1}\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> -5 \biggl(1 - \frac{3}{5} x^2\biggr) \biggl(1 - \frac{1}{2}x^2\biggr)^{-1}\biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{ 1 - \frac{3a}{b}\biggl(1 - \frac{3}{5}x^2\biggr)^{-1} \biggl(1-x^2\biggr)\biggr\} \biggl(1 - x^2\biggr)^{-1} </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ \frac{1}{x^2}\biggl(1 - x^2\biggr)^3 \frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> - \frac{4}{15} \biggl( 1 + 3x^2\biggr) - \frac{16}{15} \biggl(1 - x^2\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> -5 \biggl(1 - \frac{3}{5} x^2\biggr) \biggl(1 - \frac{1}{2}x^2\biggr)^{-1} \biggl[\frac{a\alpha}{b}\biggl(1 - x^2\biggr)^2 - \frac{4}{15} x^2 - \frac{\alpha}{3} \biggl(1 - \frac{3}{5} x^2\biggr)\biggl(1 - x^2\biggr)\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> -5 \biggl(1 - \frac{1}{2}x^2\biggr)^{-1}\biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{ \biggl(1 - \frac{3}{5} x^2\biggr) - \frac{3a}{b} \biggl(1-x^2\biggr)\biggr\} \biggl(1 - x^2\biggr) </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ \frac{15}{x^2}\biggl(1 - x^2\biggr)^3 \biggl(1 - \frac{1}{2}x^2\biggr) \frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td> <td align="center"><math>=</math></td> <td align="left"> <math>- 2 \biggl(2 - x^2\biggr)\biggl(5 - x^2\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> -~ \biggl(5 - 3 x^2\biggr) \biggl[\frac{15a\alpha}{b}\biggl(1 - 2x^2 + x^4\biggr) - 4 x^2 - \alpha \biggl(5 - 8 x^2 + 3x^4 \biggr)\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> -~ \biggl( \frac{5\sigma_c^2}{2\gamma_g}\biggr) \biggl\{ \biggl(5 - 3 x^2\biggr) - \frac{15a}{b} \biggl(1-x^2\biggr)\biggr\} \biggl(1 - x^2\biggr) </math> </td> </tr> <tr> <td align="right"><math>\Rightarrow ~~~ -~\frac{15}{x^2}\biggl(1 - x^2\biggr)^3 \biggl(1 - \frac{1}{2}x^2\biggr) \frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> 20 - 14x^2 + 2x^4 +~ \biggl(5 - 3 x^2\biggr) \biggl[ \biggl(\frac{15a\alpha}{b}-5\alpha \biggr) + x^2\biggl( \frac{30 a\alpha}{b} -4 + 8\alpha \biggr) + \alpha x^4 \biggl(\frac{15a}{b} - 3 \biggr) \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> +~ \biggl( \frac{5\sigma_c^2}{2\gamma_g}\biggr) \biggl[ \biggl(5 - \frac{15a}{b}\biggr) + x^2 \biggl(-3 + \frac{15a}{b} \biggr) \biggr] \biggl(1 - x^2\biggr) \, . </math> </td> </tr> </table> Now, if we set <math>(15a/b) = 3</math>, this last expression reduces to, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"><math>-~\frac{15}{x^2}\biggl(1 - x^2\biggr)^3 \biggl(1 - \frac{1}{2}x^2\biggr) \frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> 20 - 14x^2 + 2x^4 -~ 2\biggl(5 - 3 x^2\biggr) \biggl[ \alpha + x^2 (2 - 7\alpha ) \biggr] +~ \biggl( \frac{5\sigma_c^2}{\gamma_g}\biggr) \biggl(1 - x^2\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math> 20 - 14x^2 + 2x^4 -~ 2 \biggl[ 5\alpha + x^2 (10-38\alpha ) + x^4(21\alpha - 6) \biggr] +~ \biggl( \frac{5\sigma_c^2}{\gamma_g}\biggr) \biggl(1 - x^2\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"><math>=</math></td> <td align="left"> <math>x^0 \biggl[20 - 10\alpha + \biggl( \frac{5\sigma_c^2}{\gamma_g}\biggr) \biggr] + x^2\biggl[76\alpha -34 - \biggl( \frac{5\sigma_c^2}{\gamma_g}\biggr) \biggr] + 14 x^4 \biggl[1 - 3\alpha \biggr] </math> </td> </tr> </table> =See Also= {{ SGFfooter }}
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