Nonradial f-, p-, and g-Modes

Nonradial stellar oscillations are classified most reliably by their restoring force and eigenfunction, not by frequency alone. Pressure produces acoustic p-modes, buoyancy produces g-modes, and the fundamental f-mode lies between them without a radial node in its dominant fluid displacement. Relativity, composition gradients, avoided crossings, and the Cowling approximation shift the spectrum, but they do not erase this physical taxonomy when its assumptions are stated.

Helpful background. Stress–Energy and Conservation supplies the perturbed fluid equations; the Einstein equation supplies the stellar geometry; Relativistic Matter and Compact Objects orients the material closure; Black-Hole Perturbation Theory supplies mode boundary conditions; Gravitational Radiation and Waveforms supplies radiative multipoles; and Computational Relativity supplies eigenvalue and evolution checks.

Harmonics and restoring forces

Begin with a spherical equilibrium and expand a displacement in scalar and vector spherical harmonics of degree $\ell$. Polar perturbations contain radial and tangential fluid motion and couple to density and metric perturbations; the fluid part of an axial perturbation is toroidal in a nonrotating perfect fluid and has zero frequency until rotation, elasticity, or magnetic stresses supply a restoring force. Relativistic axial metric perturbations can still support nonzero-frequency w-modes without an oscillatory fluid motion. With $e^{-i\omega t}$, a damped outgoing mode has $\operatorname{Im}\omega<0$.

In a local short-wavelength limit, acoustic disturbances satisfy

$$\omega_{\rm p}^2\simeq c_s^2k^2, \qquad c_s^2=\left(\frac{\partial p}{\partial\varepsilon}\right)_{s,Y_i},$$

and pressure is the restoring force. Their frequencies generally increase with radial overtone. Buoyancy modes instead depend on the difference between the displaced element’s frozen-composition response and the background stratification. Schematically,

$$\omega_{\rm g}^2\lesssim N^2,$$

where $N$ is the relativistic Brunt–Väisälä frequency. In a cold barotrope with instantaneous composition equilibration, $N=0$ and ordinary composition g-modes disappear. That limit is a powerful check on any claimed g-mode spectrum.

The Newtonian limit makes buoyancy concrete. With outward coordinate $r$, inward gravitational acceleration of magnitude $g=GM(r)/r^2$, and mass density $\rho$, the squared buoyancy frequency is

$$N^2=g\left(\frac{1}{\Gamma_1}\frac{d\ln p}{dr} -\frac{d\ln\rho}{dr}\right).$$

A displaced parcel rapidly matches the ambient pressure but retains its entropy and slowly reacting composition. Its density therefore follows $d\ln\rho_{\rm parcel}=d\ln p/\Gamma_1$, while its surroundings follow the second gradient. If the lifted parcel is denser than its new surroundings, gravity restores it: $N^2\gt0$. Negative $N^2$ gives convective growth, not a stable g-mode. In a locally uniform, stably stratified patch the low-frequency dispersion relation is $\omega^2\simeq N^2k_\perp^2/(k_r^2+k_\perp^2)$, explaining why g-mode frequencies lie below $N$ and decrease as radial wavelength becomes shorter.

The fundamental branch

The f-mode is the nodeless fluid branch for each $\ell\ge2$. Its natural scale is the dynamical frequency,

$$\omega_f\sim\sqrt{\frac{GM}{R^3}},$$

with a dimensionless coefficient set by compactness and the density profile. It is neither a zero-pressure p-mode nor a high-order g-mode. In an incompressible Newtonian star it is a surface-gravity oscillation; in a relativistic neutron star it involves the whole fluid and radiates through its time-dependent mass multipole.

Kokkotas and Schmidt organize fluid and spacetime stellar modes, including their boundary conditions and damping, in Kokkotas and Schmidt 1999, §§4.1–4.2.1, especially Eq. (57). Their discussion also shows why the frequency alone is insufficient: f- and g-branches can approach and exchange character when stratification or stellar parameters vary.

For the incompressible Newtonian benchmark the coefficient is known exactly:

$$\omega_f^2=\frac{2\ell(\ell-1)}{2\ell+1}\frac{GM}{R^3}, \qquad \ell\ge2.$$

The quadrupole thus has $\omega_f^2=4GM/(5R^3)$. This is useful for checking units, harmonic labels, and a weak-field calculation. It is not a precision formula for a compressible neutron star, and infinite sound speed is only a Newtonian idealization.

Full relativity, Cowling, and avoided crossings

A complete polar calculation perturbs both matter and metric, enforces regular center data, imposes vanishing Lagrangian pressure at the surface, matches to the vacuum exterior, and selects outgoing waves at infinity. The relativistic Cowling approximation freezes the metric perturbation. It reduces the system and can be useful for high-order fluid modes, but it shifts the f-mode and cannot describe curvature-led w-modes or gravitational-wave damping self-consistently.

Mode identity should therefore be followed by eigenfunction overlap or energy partition as parameters change. At an avoided crossing, two eigenfrequencies repel while the modes exchange localization or restoring-force character. Labeling a continuous numerical branch only by its ordering can silently swap “p” and “g.” A robust calculation tracks nodes, fluid compression, buoyancy work, metric amplitude, and overlap with neighboring parameter points.

Rotation splits azimuthal components and couples different $\ell$ values; magnetic fields introduce Alfvén continua or mixed magnetoelastic modes; a solid crust adds shear traction; and superfluidity adds counter-moving degrees of freedom. Those effects generate additional branches rather than licensing the reuse of a nonrotating barotropic label without qualification.

Discriminating checks

Three limits expose common mistakes. First, taking $N\to0$ must remove composition g-modes. Second, taking compactness $GM/R\to0$ must recover the Newtonian fluid spectrum and the scale $\sqrt{GM/R^3}$. Third, restoring the metric after a Cowling calculation should change the f-mode and provide a radiative damping time, not merely add an unrelated exterior solution.

All frequencies must be quoted in a declared frame and normalization. A dimensionless value $M\omega$ cannot be compared with a frequency in hertz until the physical mass and the convention for $G$ are restored.

Check the mode label

A computed low-frequency branch persists when the composition gradient is artificially set to zero, but its eigenfunction becomes concentrated at a density discontinuity. Is it still an ordinary composition g-mode?

Solution

No. Removing the buoyancy frequency eliminates the restoring force of an ordinary composition g-mode. Persistence with displacement localized at a density jump instead identifies an interface or discontinuity mode. The correct check is to vary the jump strength: its frequency and localization should vanish or merge into another family when the discontinuity is removed.

References

• Kokkotas, Kostas D., and Bernd G. Schmidt. “Quasi-Normal Modes of Stars and Black Holes.” Living Reviews in Relativity 2 (1999): 2. DOI.