Stellar oscillation

A stellar oscillation is a normal mode of a star or stellar model. Pressure, buoyancy, rotation, self-gravity, or an imposed gravitational potential supplies its restoring force.

Helioseismology infers the Sun's structure and dynamics from its stellar oscillations. Spatial projection of resolved surface observations onto spherical harmonics identifies angular degree and azimuthal order; a temporal Fourier transform measures mode frequencies. Stellar acoustic modes constrain the adiabatic sound speed, while rotational splitting constrains stellar rotation. Their finite wavelength, surface reflection phases and finite observational coverage limit inversion resolution.

Slow stellar rotation breaks the azimuthal degeneracy of a stellar oscillation. First-order rotational frequency shifts are weighted integrals of the angular velocity with kernels computed from the unperturbed mode. Multiple radial orders and angular degrees provide overlapping constraints on internal rotation. Surface rotation alone cannot replace these internal measurements, and a finite kernel set does not give pointwise uniqueness.

For a high-order stellar acoustic mode, neglecting buoyancy and representing surface reflection by a phase, WKB quantization gives $F(w)={\int }_{r_t}^R\sqrt{1-c^2/(w^2{r}^2)}dr/c$, where $w=\omega /L$ and $L=\sqrt{\ell (\ell +1)}$. Thus the scaled frequencies depend mainly on $\omega /L$, rather than independently on degree and radial order. A frequency-dependent surface phase can be fitted when a constant $\alpha$ is inadequate.

Assume $a=c/r$ decreases outward, tends to zero at $R$, and define $h(a)=-d\ln r/da$. Differentiation of the Duvall law gives $F^{\prime }(w)={\int }_0^{w}h(b)b/[w^2\sqrt{w^2-b^2}]db$. Interchanging integrals and using ${\int }_b^{a}dw/[w\sqrt{w^2-b^2}\sqrt{a^2-w^2}]=\pi /(2ab)$ proves the displayed Abel transform inversion. For a finite outer reference value $w_s$, both lower limits become $w_s$, after removing the known outer-layer contribution to the travel time. Inversion requires a monotone branch and amplifies errors in differentiated frequency data.

In the Cowling approximation, an axisymmetric meridional fluid displacement in a cylindrically rotating barotropic star obeys $-{\omega }^2\xi =\mathcal{F}\xi$, with $\delta \rho =-\nabla \cdot (\rho \xi )$ and $\delta p=-\xi \cdot \nabla p-\gamma p\nabla \cdot \xi$. Conservation of specific angular momentum supplies the epicyclic term, with ${\kappa }^2=R^{-3}d(R^4{\Omega }^2)/dR$. The Cowling energy principle for a rotating barotropic star gives its symmetric quadratic form.

With vanishing surface pressure/mass density and regular admissible displacements, integration by parts makes the axisymmetric adiabatic displacement operator symmetric in the mass density-weighted inner product. Completing the pressure square gives the displayed energy form. Nonnegative form for every admissible displacement excludes exponentially growing axisymmetric adiabatic modes in the Cowling approximation. A negative trial Rayleigh quotient proves negative spectrum by the Rayleigh-Ritz variational principle. Nonnegative stratification and epicyclic coefficients give a simple sufficient stability condition.

For barotropic equilibrium $dp/d\Psi =-\rho$, this coefficient equals $-{\rho }^{-1}(dp/d\Psi )[{\gamma }^{-1}d\ln p/d\Psi -d\ln \rho /d\Psi ]$. It weights displacement across effective-potential surfaces in the Cowling energy principle for a rotating barotropic star. Its sign distinguishes stabilizing from destabilizing buoyancy there. It is not itself a squared frequency: the physical local buoyancy-frequency square contains an additional factor $|\nabla \Psi {|}^2$.

A stellar gravity mode is a nonradial stellar oscillation restored by buoyancy in stable stratification. In the local interior approximation it propagates where ${\omega }^2<N^2,S_{\ell }^2$. Such modes are distinct from stellar acoustic modes and do not have a radial $\ell =0$ gravity-wave branch.

A stellar acoustic mode is a pressure-restored stellar oscillation. Its local high-frequency propagation branch lies above the stellar buoyancy frequency and Lamb frequency. Inner refraction and near-surface reflection form an acoustic cavity; its standing-wave frequencies probe the adiabatic sound speed profile.

The leading separation between successive high-order low-degree stellar acoustic mode frequencies is the inverse round-trip acoustic travel time, $\Delta \nu \simeq [2{\int }_0^{R}dr/c_s{]}^{-1}$. For homologous stars it scales approximately as $(M/R^3{)}^{1/2}$ and therefore measures mean density.

The Lamb frequency is the horizontal acoustic angular frequency of a spherical-harmonic component of degree $\ell$. With adiabatic sound speed $c_s$, $S_{\ell }^2=\ell (\ell +1)c_s^2/r^2$. It helps determine inner acoustic turning points and vanishes for radial modes.

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 $4/3$.

The fluid displacement $\mathbf{\xi }$ determines Eulerian density through $\delta \rho =-\nabla \cdot (\rho \mathbf{\xi })$. The pressure law is $\delta p+\mathbf{\xi }\cdot \nabla p=-\gamma p\nabla \cdot \mathbf{\xi }$. The linear momentum equation must include both $-\delta \rho \nabla \Phi$ and $-\rho \nabla \delta \Phi$, with ${\nabla }^2\delta \Phi =4\pi G\delta \rho$. Retaining the last term incorporates perturbed self-gravity.

A constant-density gas sphere in hydrostatic equilibrium has $g={\omega }_d^{2}r$ and $p=\rho {\omega }_d^2(R^2-r^2)/2$. Its stellar oscillations can be expanded using regular solid harmonics and radial displacement amplitudes. Uniform density does not imply uniform specific entropy, so this equilibrium can be radially stable and still exhibit convective instability.

For the even polynomial family of degree $n\ge 2$, the physical radial squared frequency is $[\gamma n(n+1)/2-4]{\omega }_d^2$. The fundamental $n=2$ mode is stable when $\gamma >4/3$ and marginal at $\gamma =4/3$. At angular degree zero the gradient of the regular solid harmonic vanishes, so the auxiliary displacement amplitude multiplying it is redundant.

When the amplitudes of uniform-density stellar oscillation are finite even polynomials, their highest coefficients close algebraically. With $L=n(2l+n+1)$ and $x={\omega }^2/{\omega }_d^2$, the nonradial coefficient determinant is $x^2+(4-\gamma L/2)x-l(l+1)$. A degree balance gives a necessary frequency relation without solving every lower coefficient or imposing surface conditions.

The frequency ${\omega }_d=(GM/R^3{)}^{1/2}=(4\pi G\rho /3{)}^{1/2}$ measures the inverse gravitational timescale of a uniform-density sphere. It is the natural normalization for uniform-density stellar oscillation and the linear restoring or destabilizing accelerations.

An incompressible stellar surface mode has vanishing Lagrangian volume perturbation. In a harmonic fixed potential, a displacement generated by a solid spherical harmonic has a frequency fixed by its angular degree.

The Kelvin stellar modes are the potential incompressible stellar surface modes of a nonrotating self-gravitating uniform-density sphere with incompressible flow. For a regular displacement potential $U=A{r}^l{Y}_l^m$, the free-surface Lagrangian pressure perturbation condition and the surface gravity perturbation of a uniform-density sphere giveIn detail $\delta p/\rho ={\omega }^{2}U-\delta \Phi$ and $\delta p/\rho =(GM/R^2){\xi }_r$ at $r=R$, so ${\omega }_l^2=lGM/R^3-4\pi G\rho l/(2l+1)$. The $l=1$ branch is rigid translation with zero restoring force. This potential family does not exhaust vortical zero-frequency displacements: $\mathbf{\xi }=\mathbf{\Omega }\times \mathbf{r}$ is tangent to the sphere and has zero restoring force but nonzero curl.

For a uniform-density sphere with incompressible flow of radius $R$ with one surface spherical harmonic displacement ${\xi }_r(R)=lA{R}^{l-1}{Y}_l^m$, the bulk density perturbation vanishes but the surface contributes $\delta \rho =\rho {\xi }_r(R)\delta (r-R)$. Continuity of the Newtonian gravitational potential and the Poisson equation give $[{\partial }_r\delta \Phi {]}_{\mathrm{out}-\mathrm{in}}=4\pi G\rho {\xi }_r(R)$. The regular interior and decaying exterior solutions areThe coefficient follows because the derivative jump is $-(2l+1)C{R}^{l-1}{Y}_l^m$. This surface source is why neglecting the bulk density perturbation does not justify neglecting the perturbed gravity.

Periodic tidal forcing excites a stellar normal mode with response proportional to $({\omega }^2-{\omega }_{\ell }^2{)}^{-1}$ in an undamped linear model. Dissipation regularizes the divergence and introduces a phase shift near resonance.