Linear adiabatic stellar oscillation equations ($\Delta P/P={\Gamma }_1\Delta \rho /\rho$)

For a static spherical star, a fluid displacement $\mathbf{\xi }{e}^{-i\omega t}$ gives ${\rho }_1=-\nabla \cdot (\rho \mathbf{\xi })$, $-{\omega }^2\rho \mathbf{\xi }=-\nabla p_1-{\rho }_1\nabla \Phi -\rho \nabla {\varphi }_1$ and ${\nabla }^2{\varphi }_1=4\pi G{\rho }_1$. The adiabatic relation is $\Delta P=-{\Gamma }_{1}P\nabla \cdot \mathbf{\xi }$. Centre regularity and surface mechanical/gravitational boundary conditions select discrete eigenvalues in a conservative model.

For a nonrotating self-gravitating star, let $p_1,{\rho }_1,{\varphi }_1$ be Eulerian perturbations and $\Delta p=-\gamma p\nabla \cdot \mathbf{\xi }$ the adiabatic Lagrangian perturbation. Integration of the linear momentum equation against ${\mathbf{\xi }}^{\ast }$, using hydrostatic balance and the perturbed Poisson equation, yields the displayed Rayleigh quotient. The gravitational integral is over all space and the other integrals over the star. The compression denominator is $\gamma p$, not $\gamma \rho$. Surface terms vanish for compactly supported trials or the physical free-surface and exterior potential conditions.

Choose a smooth radial bump $f$ supported inside a shell where the stellar buoyancy frequency has $N^2<0$. The displayed fluid displacement has $\nabla \cdot \mathbf{\xi }=(\rho g/\gamma p){\xi }_r$, hence its Eulerian pressure perturbation vanishes identically. Its buoyancy contribution to the stellar displacement energy identity is strictly negative and self-gravity is nonpositive. The variational principle therefore proves convective instability with no unsupported assumption that incompressibility alone removes pressure work. Large spherical-harmonic degree also makes perturbed self-gravity small.

In the Cowling approximation and local short-wavelength limit, the radial wave number satisfies $k_r^2\simeq ({\omega }^2-S_{\ell }^2)({\omega }^2-N^2)/(c_s^2{\omega }^2)$. The Lamb frequency $S_{\ell }$ and stellar buoyancy frequency $N$ delimit acoustic and gravity-wave cavities. This leading interior relation does not replace the near-surface acoustic cutoff or the full gravitational boundary problem.

The Cowling approximation neglects the perturbation of the gravitational potential while retaining the equilibrium gravitational force in a stellar oscillation. It simplifies short-wavelength mode propagation, but can be inaccurate for low-order modes whose perturbed self-gravity matters.

Take depth $z$ positive downward, vertical fluid displacement $\xi$ positive upward, and Lagrangian pressure perturbation $\Pi$. In a plane-parallel atmosphere with $p^{\prime }=\rho g$, the Cowling approximation and adiabatic relation giveIndeed the Eulerian perturbations are $p_1=\Pi +\rho g\xi$ and ${\rho }_1=\Pi /c^2+{\rho }^{\prime }\xi$. Horizontal momentum gives ${\mathbf{\xi }}_h=i\mathbf{k}{p}_1/(\rho {\omega }^2)$; combine this with mass conservation and vertical momentum to eliminate the horizontal displacement. Reversing the sign convention for vertical displacement reverses both displacement-pressure coupling signs.

The plane-parallel Cowling displacement-pressure equations admit this stellar oscillation for every compatible equilibrium density and adiabatic sound speed profile. In the upward-displacement convention its horizontal amplitude is $i\mathbf{k}\xi /k$, its Eulerian pressure amplitude is $\rho g\xi$, and its Lagrangian density and pressure perturbations vanish. Its displacement decays into the star but grows upward without bound, so a pure linear continuation to infinite height is not physical.

For a high-sound-speed, low-density atmosphere over a denser stellar envelope, set ${\omega }^2=gk(1-\epsilon )$ in the plane-parallel Cowling displacement-pressure equations. Define $I_a={\int }_{-\infty }^{z_0}{e}^{2k(z-z_0)}{\rho }^{-1}dz$ and $I_b={\int }_{z_0}^{\infty }{e}^{-2k(z-z_0)}\rho dz$. The lower leading solution has $\Pi /\xi =-2\epsilon gk{I}_b$ at the match. Above, write $\rho =\epsilon \hat{\rho }$, $\Pi =\epsilon \psi$, $\xi =\mu \psi$. The upward-decaying solution satisfies ${\mu }^{\prime }+2k\mu =-k/(g\hat{\rho })$, so $\mu (z_0)=-(k/g)\epsilon I_a$. Writing $f=\Pi /(\epsilon \xi )$ below, continuity of displacement and Lagrangian pressure requires $\mu (z_0)f(z_0)=1$, giving the displayed condition. The integrals must converge and the upper displacement must actually decay. For two constant-density incompressible layers it gives $\epsilon \simeq 2{\rho }_a/{\rho }_b$, the expansion of the exact interfacial gravity-wave dispersion relation.

For a normalized spherical harmonic of degree $\ell \ge 1$, write ${\rho }_1=s(r)Y$ and ${\varphi }_1=F(r)Y$. Integration of the radial Poisson equation gives $A=-4\pi G\int r^2{F}^{\ast }sdr$, where $A=\int [r^2|F^{\prime }{|}^2+\ell (\ell +1)|F{|}^2]dr$. Cauchy-Schwarz inequality and $\int |F{|}^2\le A/[\ell (\ell +1)]$ give the displayed bound on $|E_g|=A/(4\pi G)$. For bounded density amplitudes with fixed radial support this decays as ${\ell }^{-2}$, giving a quantitative angular short-scale justification of the Cowling approximation.

For a radial stellar oscillation, put ${\xi }_r=r\eta$. The self-gravitating adiabatic equation is $({\Gamma }_{1}P{r}^4{\eta }^{\prime }{)}^{\prime }+r^3[(3{\Gamma }_1-4)P{]}^{\prime }\eta +\rho r^4{\omega }^2\eta =0$. It is a Sturm-Liouville problem whose Rayleigh quotient tests radial stability. The familiar constant-exponent threshold is ${\Gamma }_1=4/3$; a variable exponent requires the full integral test.

For constant adiabatic exponent $\gamma >4/3$ and a star whose central mass density is its maximum, the homologous radial fluid displacement ${\xi }_r=r$ makes the derivative term in the weighted stellar pulsation Rayleigh quotient vanish. The remaining quotient is $(3\gamma -4)$ times a positive weighted average of $g/r$. Since $m(r)\le 4\pi {\rho }_c{r}^3/3$, $g/r\le 4\pi G{\rho }_c/3$, proving the bound. For $\gamma <4/3$, the same trial quotient is negative and proves radial instability; the inequality with the central density must not be extended with its sign unchanged.

For radial adiabatic stellar oscillations, the quotient is $\mathcal{R}[\xi ]=\int [\gamma p{r}^4|\xi \prime {|}^2+r^3((4-3\gamma )p)\prime |\xi {|}^2]dr/\int \rho r^4|\xi {|}^{2}dr$. Regular-centre and free-surface endpoint conditions remove the boundary term. The infimum is the fundamental squared frequency; any negative trial value proves radial instability. A variable stellar adiabatic exponent must remain inside the derivative.

Integrate the pressure-gradient term in the weighted stellar pulsation Rayleigh quotient by parts, with regular centre and zero surface pressure. Writing $\chi =r{\xi }^{\prime }+3\xi$ uses $3{\xi }^2+2r\xi {\xi }^{\prime }=({\chi }^2-r^2({\xi }^{\prime }{)}^2)/3$ and gives the displayed expression. Thus radial instability requires $\gamma <4/3$ somewhere in the positive-pressure interior. A constant trial function gives $K=3{\xi }^2{\int }_0^R(3\gamma -4)p{r}^{2}dr$, so strict radial stability requires this weighted integral to be positive, in particular $\gamma >4/3$ somewhere. When $\gamma =4/3$ throughout, constant $\xi$ is an exact zero-frequency homologous mode: the star is marginal, rather than strictly stable. Local threshold crossings alone do not determine stability for a mixed profile; the full Rayleigh-Ritz variational principle does.

The homologous trial function $\xi =1$ in the weighted stellar pulsation Rayleigh quotient has numerator $-3{\int }_0^{R_s}{r}^{2}p(4-3\gamma )dr$. Positivity of that integral therefore proves radial instability. It is a sufficient test: a nonpositive integral does not guarantee stability against every radial shape. For constant $\gamma$, the homologous threshold is $4/3$.