= Solution
Take a static, nonrotating, nonmagnetic spherical equilibrium with $\nabla P=-\rho\nabla\Phi$ and $\nabla^2\Phi=4\pi G\rho$. Use a <fluid displacement> $\boldsymbol\xi(\mathbf r)e^{-i\omega t}$. Write $p_1,\rho_1,\phi_1$ for <Eulerian fluid perturbations>. The <Lagrangian pressure perturbation> and corresponding <mass density> change obey $\Delta P=p_1+\boldsymbol\xi\cdot\nabla P$, $\Delta\rho=\rho_1+\boldsymbol\xi\cdot\nabla\rho$. Linearizing <mass conservation>, the <Euler equations for an inviscid fluid> and <Poisson equation for Newtonian gravity> gives
$$
\boxed{\begin{aligned}
\rho_1&=-\nabla\cdot(\rho\boldsymbol\xi),\\
-\omega^2\rho\boldsymbol\xi&=-\nabla p_1-\rho_1\nabla\Phi-\rho\nabla\phi_1,\\
\nabla^2\phi_1&=4\pi G\rho_1.
\end{aligned}}
$$
There is no equilibrium acceleration to multiply a perturbed <mass density>. The <adiabatic process> condition, with composition carried by the parcel, closes the system:
$$
\boxed{\frac{\Delta P}{P}=\Gamma_1\frac{\Delta\rho}{\rho},\qquad\Delta\rho=-\rho\nabla\cdot\boldsymbol\xi,\qquad p_1=-\Gamma_1P\nabla\cdot\boldsymbol\xi-\boldsymbol\xi\cdot\nabla P.}
$$
These are the complete <linear adiabatic stellar oscillation equations>, including the perturbation of self-gravity. Neglect of heat exchange is appropriate when oscillation periods are short compared with relevant thermal relaxation times; it does not determine nonadiabatic excitation or damping.
For the pressure and buoyancy modes, separate angular dependence using <spherical harmonics>:
$$
\boldsymbol\xi=\xi_r(r)Y_\ell^m\hat{\mathbf r}+\xi_h(r)\nabla_\Omega Y_\ell^m,\qquad (p_1,\rho_1,\phi_1)=(p_\ell,\rho_\ell,\phi_\ell)Y_\ell^m.
$$
The horizontal amplitude $\xi_h$ has dimensions of length. Angular differentiation gives $\nabla\cdot\boldsymbol\xi=[r^{-2}(r^2\xi_r)'-\ell(\ell+1)\xi_h/r]Y_\ell^m$, and tangential momentum gives $\omega^2r\xi_h=p_\ell/\rho+\phi_\ell$. Define the <adiabatic sound speed>, <stellar buoyancy frequency> and <Lamb frequency> by
$$
c_s^2=\frac{\Gamma_1P}\rho,\qquad N^2=g\left(\frac1{\Gamma_1}\frac{d\log P}{dr}-\frac{d\log\rho}{dr}\right),\qquad S_\ell^2=\frac{\ell(\ell+1)c_s^2}{r^2}.
$$
The adiabatic <mass density> relation becomes $\rho_\ell/\rho=p_\ell/(\rho c_s^2)+(N^2/g)\xi_r$. Substitution produces a radial form of the full oscillation equations for nonzero $\omega$:
$$
\begin{aligned}
\frac{d\xi_r}{dr}&=\left(\frac g{c_s^2}-\frac2r\right)\xi_r+\frac1{\rho c_s^2}\left(\frac{S_\ell^2}{\omega^2}-1\right)p_\ell+\frac{\ell(\ell+1)}{r^2\omega^2}\phi_\ell,\\
\frac{dp_\ell}{dr}&=\rho(\omega^2-N^2)\xi_r-\frac g{c_s^2}p_\ell-\rho\frac{d\phi_\ell}{dr},\\
\frac1{r^2}\frac d{dr}\left(r^2\frac{d\phi_\ell}{dr}\right)-\frac{\ell(\ell+1)}{r^2}\phi_\ell&=4\pi G\left(\frac{p_\ell}{c_s^2}+\rho\frac{N^2}{g}\xi_r\right).
\end{aligned}
$$
The apparent $N^2/g$ factor is evaluated through its defining gradient at the centre rather than by dividing two zeros.
Regularity at the centre excludes singular solutions. At a free surface the <Lagrangian pressure perturbation> vanishes, $\Delta P=0$; outside the star the gravitational perturbation decays as $r^{-\ell-1}$. For a model with <mass density> tending to zero at its surface, continuity of $\phi_\ell$ and its radial derivative gives $\phi_\ell'(R)=-(\ell+1)\phi_\ell(R)/R$. If the equilibrium <mass density> jumps to vacuum, include the displaced-surface mass sheet: the outward-minus-inward derivative jump is $4\pi G\rho(R)\xi_r(R)$, so the interior condition is $\phi_\ell'(R)=-(\ell+1)\phi_\ell(R)/R-4\pi G\rho(R)\xi_r(R)$. An atmospheric <boundary condition> can replace the ideal free surface.
These conditions make $\omega^2$ an <eigenvalue>, not an arbitrary local sound frequency. With conservative <boundary conditions> the adiabatic operator is <self-adjoint>, giving real $\omega^2$; negative values describe instability. The frequencies depend on $P(r),\rho(r),\Gamma_1(r)$, self-gravity, stratification and boundaries, as well as angular degree $\ell$ and radial order. In a spherical nonrotating star they are degenerate in $m$. For homologous equilibrium structures, their scale is
$$
\boxed{\omega\sim\left(\frac{GM}{R^3}\right)^{1/2};}
$$
thus the typical oscillation time measures inverse square root of mean <mass density>, while individual frequencies probe the interior <adiabatic sound speed> and <stellar buoyancy frequency> profiles.
For radial modes, write $\xi_r=r\eta$. Eliminating the <pressure> and gravitational perturbations gives the <radial stellar pulsation equation>
$$
\boxed{\frac d{dr}\left(\Gamma_1Pr^4\frac{d\eta}{dr}\right)+r^3\frac d{dr}[(3\Gamma_1-4)P]\eta+\rho r^4\omega^2\eta=0.}
$$
This is a <Sturm-Liouville problem>. Multiplication by $\eta$ and integration, with vanishing boundary terms, gives its <Rayleigh quotient>
$$
\omega^2=\frac{\int_0^R\Gamma_1Pr^4(\eta')^2\,dr-\int_0^Rr^3[(3\Gamma_1-4)P]'\eta^2\,dr}{\int_0^R\rho r^4\eta^2\,dr}.
$$
For constant $\Gamma_1>4/3$, $P'<0$ makes both numerator contributions nonnegative. At $\Gamma_1=4/3$, a homologous displacement is neutral; for constant $\Gamma_1<4/3$ the same trial displacement makes the quotient negative. For the <uniform-density stellar model>, $\eta=$ constant is an exact mode and $\omega^2=(3\Gamma_1-4)GM/R^3$. With varying $\Gamma_1$, the integral criterion, rather than a universal pointwise threshold, controls radial stability.
The <Cowling approximation> neglects $\phi_\ell$ while retaining the equilibrium gravitational field. It is useful for short-wavelength modes, but is not needed for the full derivation above. In a locally slowly varying region, take both remaining amplitudes proportional to $e^{i\int k_rdr}$ and retain the leading derivative terms. Then
$$
ik_r\xi_r\simeq\frac{S_\ell^2/\omega^2-1}{\rho c_s^2}p_\ell,\qquad ik_rp_\ell\simeq\rho(\omega^2-N^2)\xi_r.
$$
Eliminating either amplitude gives the <acoustic-gravity propagation relation>
$$
\boxed{k_r^2\simeq\frac{(\omega^2-S_\ell^2)(\omega^2-N^2)}{c_s^2\omega^2}.}
$$
Positive $k_r^2$ is oscillatory propagation; negative $k_r^2$ means an <evanescent wave>. The high-frequency branch, $\omega^2>S_\ell^2,N^2$, describes <stellar acoustic modes>, restored chiefly by compressibility and <pressure>. At frequencies well above $N$, this gives $\omega^2\simeq c_s^2[k_r^2+\ell(\ell+1)/r^2]$. The low-frequency propagating branch in stable stratification, $\omega^2<N^2,S_\ell^2$, instead describes <stellar gravity modes>, restored by <buoyancy>. At $\ell=0$ there is no such nonradial gravity-wave cavity.
A <stellar acoustic mode> is trapped between an inner turning point near $\omega=S_\ell$ and an outer reflecting region. Low-$\ell$ modes penetrate deeply; radial modes reach the centre. Higher-degree modes turn farther out. Standing waves require the <WKB quantization condition>
$$
\int_{r_1}^{r_2}k_r\,dr\simeq\pi(n+\alpha),
$$
where the phase $\alpha$ depends on the central or turning-point behaviour and surface reflection. For high radial order and small degree, the leading acoustic travel-time result is the <large frequency separation>
$$
\boxed{\Delta\nu\simeq\left(2\int_0^R\frac{dr}{c_s}\right)^{-1},\qquad\nu_{n\ell}\simeq\Delta\nu\left(n+\frac\ell2+\varepsilon\right),\qquad\nu=\frac\omega{2\pi}.}
$$
The $\ell/2$ term is the leading central angular phase shift; smaller frequency separations depend on detailed interior gradients. The near-surface <mass density> stratification sets the <acoustic cutoff frequency>. In a plane-parallel isothermal atmosphere with <mass density> scale height $H_\rho$, $\omega_{\rm ac}\simeq c_s/(2H_\rho)$. Modes below this cutoff can reflect and form a cavity; waves sufficiently above it escape and need an outgoing-wave <boundary condition>. Nonadiabatic damping, driving and rotation alter real-star mode properties, but the adiabatic frequency problem isolates their dependence on the equilibrium structure.
Back to article page
