SSC/Stability/BiPolytropes/SuccinctDiscussion

BiPolytrope with n_c = 5 and n_e = 1

Here we construct and analyze the relative stability of a bipolytrope in which the core has an $n_c=5$ polytropic index and the envelope has an $n_e=1$ polytropic index.

Structure

1. Individual model profiles, taken from:
   • SSC/Structure/BiPolytropes/Analytic51#Examples
2. $(q,\nu )$ sequences of fixed ${\mu }_e/{\mu }_c$, taken from:
   • SSC/Structure/BiPolytropes/Analytic51#Model_Sequences
3. ${\nu }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ model, taken from:
   • SSC/Structure/BiPolytropes/Analytic51#Limiting_Mass
      
     

     Maximum Fractional Core Mass, $\nu =M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}/M_{\mathrm{t}\mathrm{o}\mathrm{t}}$ (solid green circular markers) for Equilibrium Sequences having Various Values of ${\mu }_e/{\mu }_c$
     $\frac{{\mu }_e}{{\mu }_c}$	${\xi }_i$	${\theta }_i$	${\eta }_i$	${\mathrm{\Lambda }}_i$	$A$	${\eta }_s$	LHS	RHS	$q\equiv \frac{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{R}$	$\nu \equiv \frac{M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}$
     $\frac{1}{3}$	$\mathrm{\infty }$	---	---	---	---	---	---	---	0.0	$\frac{2}{\pi }$
     0.33	24.00496	0.0719668	0.0710624	0.2128753	0.0726547	1.8516032	-223.8157	-223.8159	0.038378833	0.52024552
     0.316943	10.744571	0.1591479	0.1493938	0.4903393	0.1663869	2.1760793	-31.55254	-31.55254	0.068652714	0.382383875
     0.31	9.014959766	---	---	0.59835053	---	---	---	---	0.0755022550	0.3372170064
     0.3090	8.8301772	0.1924833	0.1750954	0.6130669	0.2053811	2.2958639	-18.47809	-18.47808	0.076265588	0.331475715
     $\frac{1}{4}$	4.9379256	0.3309933	0.2342522	1.4179907	0.4064595	2.761622	-2.601255	-2.601257	0.084824137	0.139370157
     Recall that, ${\ell }_i\equiv \frac{{\xi }_i}{\sqrt{3}};$       and       $m_3\equiv 3\left(\frac{{\mu }_e}{{\mu }_c}\right).$

   
    
   
   • SSC/Structure/BiPolytropes/Analytic51Renormalize#Model_Pairings
      
     

     file = Dropbox/WorkFolder/Wiki edits/Bipolytrope/Stability/qAndNuMax.xlsx --- worksheet = B-KB74 thru MinuPreparation Bipolytrope with $(n_c,n_e)=(5,1)$ Selected Pairings along the ${\mu }_e/{\mu }_c=0.31$ Sequence
     Pairing	${\xi }_i$	${\mathrm{\Lambda }}_i$	$\nu$	$q$
     A	$9.014959766$	$0.59835053$	$0.3372170064$	$0.0755022550$
     B1	$9.12744$	$0.60069262$	$0.3372001445$	$0.0746451491$
     B2	$8.90394$	$0.59610192$	$0.33720014467$	$0.0763642133$

     
     

     Bipolytropic (5, 1) Equilibrium Sequences

     Bipolytropic (5, 1) Equilibrium Sequences

Stability

1. LAWE taken from:   SSC/Stability/BiPolytropes/HeadScratching#Throughout_the_Configuration

   • $0$

     $=$

     $\frac{d^{2}x}{dr{*}^2}+\frac{{\mathscr{H}}}{r^{*}}\frac{dx}{dr*}+\left[\right(\frac{{\sigma }_c^2}{{\gamma }_g}){{𝒦}}_1-{\alpha }_g{{𝒦}}_2]x,$

     where,

     ${\mathscr{H}}$	$\equiv$	$\{4-(\frac{{\rho }^{*}}{P^{*}}\left)\frac{M_r^{*}}{(r^{*})}\right\}$	,	${{𝒦}}_1$	$\equiv$	$\frac{2\pi }{3}\left(\frac{{\rho }^{*}}{P^{*}}\right)$	and	${{𝒦}}_2$	$\equiv$	$\left(\frac{{\rho }^{*}}{P^{*}}\right)\frac{M_r^{*}}{(r^{*}{)}^3},$
     ${\sigma }_c^2$	$\equiv$	$\frac{3{\omega }^2}{2\pi G{\rho }_c}$	,	${\alpha }_g$	$\equiv$	$(3-\frac{4}{{\gamma }_g}).$

     Core:   ${\gamma }_g=6/5\Rightarrow {\alpha }_g=-1/3$
     Envelope:   ${\gamma }_g=2\Rightarrow {\alpha }_g=+1$

2. Central Boundary condition drawn from:   SSC/Stability/BiPolytropes/HeadScratching#Central_Boundary_Condition

   • $x_2$

     $=$

     $x_1[1-\frac{(n+1)\mathfrak{𝔉}{\mathrm{\Delta }}_{\xi }^2}{60}]=x_1[1-\frac{\mathfrak{𝔉}{\mathrm{\Delta }}_{\xi }^2}{10}],$

     where,

     $\mathfrak{𝔉}$

     $\equiv$

     $[\frac{{\sigma }_c^2}{{\gamma }_g}-2{\alpha }_g{]}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}=\frac{(8+{\sigma }_c^2)}{({\gamma }_g{)}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}-6=\frac{1}{6}[5{\sigma }_c^2+4].$

3. Interface conditions taken from:  
   • SSC/Stability/BiPolytropes#Core: As viewed from the perspective of the core, the slope at the interface is

     $[\frac{d{x}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{d\xi }{]}_i$

     $=$

     $\frac{1}{2\delta \xi }\left\{\right[2-(\delta \xi {)}^2(\frac{3{𝒦}}{2\pi }\left)\right]x_i-2x_{i-1}\left\}\right[1+\left(\frac{\delta \xi }{2\xi }\right){\mathscr{H}}{]}^{-1}.$

   • SSC/Stability/BiPolytropes/HeadScratching#Interface

     $0$	$=$	$\left[{\gamma }_c{x}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}\right(3+\frac{d\ln x_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{d\ln r^{*}})-{\gamma }_e{x}_{\mathrm{e}\mathrm{n}\mathrm{v}}(3+\frac{d\ln x_{\mathrm{e}\mathrm{n}\mathrm{v}}}{d\ln r^{*}}){]}_i$
     $\Rightarrow \left[x_{\mathrm{e}\mathrm{n}\mathrm{v}}\right(3+\frac{d\ln x_{\mathrm{e}\mathrm{n}\mathrm{v}}}{d\ln r^{*}}){]}_i$	$=$	$\frac{3}{5}\left[x_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}\right(3+\frac{d\ln x_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{d\ln r^{*}}){]}_i$
     $\Rightarrow [\frac{d\ln x_{\mathrm{e}\mathrm{n}\mathrm{v}}}{d\ln r^{*}}{]}_i$	$=$	$\frac{3}{5}[\frac{d\ln x_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{d\ln r^{*}}-2{]}_i.$

4. Surface boundary condition taken from:   SSC/Stability/BiPolytropes/HeadScratching#Surface_Boundary_Condition

   • $r_0\frac{d\ln x}{d{r}_0}$	$=$	$\frac{1}{{\gamma }_g}(4-3{\gamma }_g+\frac{{\omega }^2{R}^3}{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}})$         at         $r_0=R,$
     $\Rightarrow r^{*}\frac{d\ln x}{d{r}^{*}}$	$=$	$-{\alpha }_g+\frac{R^3}{{\gamma }_{g}G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}}\left[\frac{2\pi G{\rho }_c{\sigma }_c^2}{3}\right]$        at         $r^{*}=R^{*},$
     $\Rightarrow [\frac{d\ln x}{d\ln r^{*}}{]}_s$	$=$	$-({\alpha }_g{)}_{\mathrm{e}\mathrm{n}\mathrm{v}}+\frac{1}{2({\gamma }_g{)}_{\mathrm{e}\mathrm{n}\mathrm{v}}}[\frac{{\sigma }_c^2{\rho }_c}{\overline{\rho }}]$
     	$=$	$-1+\frac{1}{4}\left[\frac{{\sigma }_c^2}{(\overline{\rho }{)}^{*}}\right].$

Example Eigenvectors Along 0.31 Sequence

Sigma-Squared Scaled to Mean Density, and Better Error Analysis

From the K-BK74 Analysis

Keep in mind that, from the accompanying K-BK74 analysis, we obtained the following eigenvector for the ${\nu }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ model along the ${\mu }_e/{\mu }_c=0.31$ equilibrium model sequence:

File:K-BK74eigenfunctionVsRadius.png

ξ_i = 2.804179398

Excel file:Models031 & pages …			xi02804gMode0 ${\sigma }_c^2\cdot ({\rho }_c/\overline{\rho })=0.00000000$		xi02804gMode1 ${\sigma }_c^2\cdot ({\rho }_c/\overline{\rho })=11.60356417$		xi02804gMode2 ${\sigma }_c^2\cdot ({\rho }_c/\overline{\rho })=39.16484857$
	${\xi }_i$	2.804179398	Linear	Log10	Linear	Log10	Linear	Log10
	$q$	0.156742251
	$\nu$	0.199510368
	${\rho }_c/\overline{\rho }$	3.57005807 × 102
	Mode	${\sigma }_c^2\cdot ({\rho }_c/\overline{\rho })$
	0	0.0
	1	11.60356417
	2	39.16484857

ξ_i = 4

Excel file:Models031 & pages …			xi04gMode0 ${\sigma }_c^2\cdot ({\rho }_c/\overline{\rho })=-192.1449647$ CAUTION:  Surface B.C. not satisfied!		xi04gMode1 ${\sigma }_c^2\cdot ({\rho }_c/\overline{\rho })=10.58972472$		xi04gMode2 ${\sigma }_c^2\cdot ({\rho }_c/\overline{\rho })=38.57111996$
	${\xi }_i$	4	Linear	Log10	Linear	Log10	Linear	Log10
	$q$	0.136805779
	$\nu$	0.266175482
	${\rho }_c/\overline{\rho }$	1.656926473 × 103
	Mode	${\sigma }_c^2\cdot ({\rho }_c/\overline{\rho })$
	0	-192.1449647
	1	+10.58972472
	2	+38.57111996

ξ_i = 9

Excel file:Models031 & pages …			xi09gMode0 ${\sigma }_c^2\cdot ({\rho }_c/\overline{\rho })=-15553.34096$ CAUTION:  Surface B.C. not satisfied!		xi09gMode1 ${\sigma }_c^2\cdot ({\rho }_c/\overline{\rho })=11.16595582$		xi09gMode2 ${\sigma }_c^2\cdot ({\rho }_c/\overline{\rho })=39.57399551$
	${\xi }_i$	9	Linear	Log10	Linear	Log10	Linear	Log10
	$q$	0.075617469
	$\nu$	0.337216704
	${\rho }_c/\overline{\rho }$	1.155525994 × 105
	Mode	${\sigma }_c^2\cdot ({\rho }_c/\overline{\rho })$
	0	-15553.34096
	1	+11.16595582
	2	+39.57399551

ξ_i = 14

Excel file:Models031 & pages …			xi09gMode0		xi09gMode1 ${\sigma }_c^2\cdot ({\rho }_c/\overline{\rho })=11.01793184$		xi09gMode2 ${\sigma }_c^2\cdot ({\rho }_c/\overline{\rho })=39.30533842$
	${\xi }_i$	14	N/A xi09gMode0	N/A xi09gMode0Log	Linear	Log10	Linear	Log10
	$q$	0.04880185
	$\nu$	0.316868628
	${\rho }_c/\overline{\rho }$	1.4728825 × 106
	Mode	${\sigma }_c^2\cdot ({\rho }_c/\overline{\rho })$
	0	---
	1	+11.01793184
	2	+39.30533842

See Also