### Question --- Make a dimensional analysis of the equation of hydrostatic equilibrium using a polytropic equation of state to find a general mass-radius relation for spherically-symmetric, self-gravitating bodies. For which two polytropic indices is the configuration unstable? ### Answer --- We begin by using [[Stellar Structure#Mass Continuity|Mass Continuity]], [[Stellar Structure#Hydrostatic Equilibrium|Hydrostatic Equilibrium]], and the [[Polytropes|Polytrope Equation of State]]... $ \underbrace{\frac{\mathrm{d} M}{\mathrm{d} r} = 4 \pi r^{2} \rho}_{\require{enclose}\enclose{circle}{1}} \hspace{2cm} \underbrace{\frac{\mathrm{d} P}{\mathrm{d} r} = - \frac{G M_{r} \rho}{r^{2}}}_{\require{enclose}\enclose{circle}{2}} \hspace{2cm} \underbrace{P = K \rho^{\gamma}}_{\require{enclose}\enclose{circle}{3}} $ ...and doing some loose dimension-matching. $\require{enclose}\enclose{circle}{1} \hspace{1cm} \frac{\mathrm{d} M}{\mathrm{d} r} = 4 \pi r^{2} \rho \hspace{1cm} \Rightarrow \hspace{1cm} M_{\rm tot} \; \propto \; R^{3} \, \bar{\rho}$ $\require{enclose}\enclose{circle}{2} \hspace{1cm} \frac{\mathrm{d} P}{\mathrm{d} r} = - \frac{G M_{r} \rho}{r^{2}} \hspace{1cm} \Rightarrow \hspace{1cm} \frac{P_{\rm c}}{R} \; \propto \; \frac{G M_{\rm tot} \bar{\rho}}{R^{2}}$ $\require{enclose}\enclose{circle}{3} \hspace{1cm} P = K \rho^{\gamma} \hspace{1cm} \Rightarrow \hspace{1cm} P_{\rm c} \sim K \bar{\rho}^{\ \gamma} \hspace{1cm} \Rightarrow \hspace{1cm} \frac{P_{\rm c}}{R} \; \propto \; \frac{K \bar{\rho}^{\ \gamma}}{R}$ Combining $\require{enclose}\enclose{circle}{1}$, $\require{enclose}\enclose{circle}{2}$, and $\require{enclose}\enclose{circle}{3}$... $ \frac{P_{\rm c}}{R} \; \propto \; \frac{G M_{\rm tot} \bar{\rho}}{R^{2}} \; \propto \; \frac{K \bar{\rho}^{\ \gamma}}{R} \hspace{1cm} \Rightarrow \hspace{1cm} \begin{aligned}[t] M_{\rm tot} \; &\propto \; R \, \bar{\rho}^{\ (\gamma-1)} \\ M_{\rm tot} \; &\propto \; R \, \left(\frac{M_{\rm tot}}{R^{3}}\right)^{\ (\gamma-1)} \\ M_{\rm tot}^{\ 1 - (\gamma-1)} \; &\propto \; R^{\ -3(\gamma-1)+1} \\ M_{\rm tot} \; &\propto \; R^{\ (4-3\gamma)/(2-\gamma)} \end{aligned} $ $\boxed{ \; M_{\rm tot} \; \propto \; R^{\ (4-3\gamma)/(2-\gamma)} \; }$ The configuration is **unstable** when - At $\gamma = 2$ the mass scales "infinitely quickly" with $R$ - At $\gamma = \frac{4}{3}$, the mass is independent of $R$ - At $\gamma < \frac{4}{3}$, the mass is directly proportional to radius (should be inversely proportional) > [!note]- More on stability... > If we were to "squeeze" the star to perturb the radius ($r' \to r (1 - \varepsilon)$), then the density should be perturbed ($\rho' \to \rho (1-\varepsilon)^{-3} \approx \rho' (1+3\varepsilon)$). The star will remain *stable* in hydrostatic equilibrium if the new outward gas pressure is greater than the new inward gravitational pressure. *(See 8.901 Notes, Lecture 10)* > $P'_{\rm gas} \; \propto \; P_{\rm gas} (1 + 3 \varepsilon)^{\gamma}$ > $P'_{\rm grav} = \int_{M_{r}}^{M} \frac{\mathrm{d} P}{\mathrm{d} M_{r}} \; \mathrm{d} M_{r} = \int_{M_{r}}^{M} \frac{\mathrm{d} P}{\mathrm{d} r} \frac{\mathrm{d} r}{\mathrm{d} M_{r}} \; \mathrm{d} M_{r} = -\int_{M_{r}}^{M} \frac{G M_{r}}{4 \pi r'^{4}} \; \mathrm{d} M_{r} = P_{\rm grav} (1+4\varepsilon)$ > $\Downarrow$ > $P'_{\rm grav} = \quad \boxed{P_{\rm grav} (1+4\varepsilon) < P_{\rm gas} (1 + 3 \varepsilon)^{\gamma}} \quad = P'_{\rm gas}$ > > In a similar argument, we can take a more dimensional analysis approach. > $P_{\rm gas} \propto \rho^{\gamma} \propto \left(\frac{M}{R^{3}}\right)^{\gamma} \hspace{2cm} P_{\rm HSE} \propto \frac{M \rho}{R} \propto \frac{M^{2}}{R^{4}}$ > $\mathrm{d} P_{\rm gas} \propto - R^{(-3\gamma-1)} \; \mathrm{d} R \hspace{2cm} \mathrm{d} P_{\rm HSE} \propto - R^{-5} \; \mathrm{d} R$ > $ > \begin{align} > \mathrm{d} P_{\rm gas} &> \mathrm{d} P_{\rm HSE} \\ > - R^{(-3\gamma-1)} \; \mathrm{d} R &> - R^{-5} \; \mathrm{d} R \\ > R^{-3\gamma} &< R^{-4} \\ > R^{\gamma} &> R^{4/3} \\ > \gamma \ln R &> \frac{4}{3} \ln R \\ > \gamma &> \frac{4}{3} > \end{align} > $ > This is the criteria for the gas pressure to be able to bounce back to equilibrium. Additionally, this is the criteria for the total mass to have a inverse proportionality relationship to radius ($M_{\rm tot} \propto R^{-n}$), which is the desired state. (think of density for justification) > [!note] > All of the assumptions made by the question are hidden within the polytropic assumptions or the stellar structure equations.