Revision as of 13:06, 6 January 2026

Main Sequence to Red Giant to Planetary Nebula (Part 3)

In an accompanying discussion, we derived the so-called,

Polytropic LAWE

	$0 = \frac{d^{2} x}{d ξ^{2}} + [4 - (n + 1) Q] \frac{1}{ξ} \cdot \frac{d x}{d ξ} + (n + 1) [(\frac{σ_{c}^{2}}{6 γ_{g}}) \frac{ξ^{2}}{θ} - α Q] \frac{x}{ξ^{2}}$
where: $Q (ξ) \equiv - \frac{d \ln θ}{d \ln ξ},$ $σ_{c}^{2} \equiv \frac{3 ω^{2}}{2 π G ρ_{c}},$ and, $α \equiv (3 - \frac{4}{γ_{g}})$

Let's now insert an additive constant, $C_{0}$ , to this 2^nd-order ODE as follows:

	$C_{0} = \frac{d^{2} x}{d ξ^{2}} + [4 - (n + 1) Q] \frac{1}{ξ} \cdot \frac{d x}{d ξ} + (n + 1) [(\frac{σ_{c}^{2}}{6 γ_{g}}) \frac{ξ^{2}}{θ} - α Q] \frac{x}{ξ^{2}}$
where: $Q (ξ) \equiv - \frac{d \ln θ}{d \ln ξ},$ $σ_{c}^{2} \equiv \frac{3 ω^{2}}{2 π G ρ_{c}},$ and, $α \equiv (3 - \frac{4}{γ_{g}})$

@@ Line 23: / Line 23: @@
 <div align="center">
-<font color="maroon"><b>Yabushita68-Motivated Polytropic LAWE</b></font><br />
+<font color="maroon"><b>Polytropic LAWE</b></font><br />
 {{ Math/EQ_RadialPulsation02 }}
 </div>
+Let's now insert an additive constant, <math>C_0</math>, to this 2<sup>nd</sup>-order ODE as follows:
+<table border=0 cellpadding=2 align="center">
+<tr>
+<td align="right">
+[[Image:LSUkey.png|25px|link=Appendix/EquationTemplates#Stability:_Radial_Pulsation]]
+</td>
+<td align="center">
+<math>C_0 = \frac{d^2x}{d\xi^2} + \biggl[ 4 - (n+1) Q \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi}
++ (n+1) \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_g } \biggr)  \frac{\xi^2}{\theta}
+- \alpha Q\biggr]  \frac{x}{\xi^2} </math>
+</td>
+</tr>
+<tr>
+  <td align="center" colspan="2">
+where:&nbsp; &nbsp; <math>Q(\xi) \equiv - \frac{d\ln\theta}{d\ln\xi} \, ,</math> &nbsp;&nbsp; <math>\sigma_c^2 \equiv \frac{3\omega^2}{2\pi G\rho_c} \, ,</math> &nbsp; &nbsp; and, &nbsp; &nbsp; <math>\alpha \equiv \biggl(3 - \frac{4}{\gamma_\mathrm{g}}\biggr)</math>
+  </td>
+</tr>
+</table>
 =Related Discussions=
 {{ SGFfooter }}