Stellar pulsations and asteroseismology

Pressure modes are predominantly acoustic, with pressure gradients providing the main restoring force. Gravity modes are predominantly restored by buoyancy in stably stratified regions; the word gravity here does not mean a gravitational wave of spacetime. Some evolved stars support mixed modes that sample both an acoustic envelope cavity and a buoyancy-dominated core cavity. Mode labels specify radial order and angular structure, but their precise sign and counting conventions vary, so a numerical comparison must declare how labels are assigned.

For adiabatic perturbations the sound speed squared is the adiabatic pressure derivative at fixed composition. It is not generally the isothermal derivative. Radial dynamical stability also depends on the pressure response during compression. A simplified homologous-compression argument gives a four-thirds threshold for a constant adiabatic exponent in a Newtonian model. A small outer region below that value does not alone prove global collapse; actual stability depends on the entire structure and mode equations. Relativistic corrections and nonadiabatic effects add further conditions.

For high radial order and low angular degree in an appropriate acoustic cavity, the radial-order spacing is approximately the inverse of the round-trip sound travel time between center and surface. The asymptotic frequency formula includes a phase offset from turning regions and the surface. This pattern makes the travel-time integral diagnostically useful, but its assumptions can fail for low-order modes, mixed modes, sharp structural features, or rapidly varying surface phase. A constant phase offset is an explicit toy simplification.

The large separation scales roughly with the square root of mean density when comparing homologous structures with the same dimensionless sound-speed profile. It is not an exact relation across all stellar masses and stages. Deviations from a regular spacing, rotational splitting, and gravity-mode period patterns can provide additional information, but interpreting them requires mode identification and an appropriate structural model. In particular, an acoustic spacing alone does not uniquely reveal chemical composition or distinguish all possible core-burning histories.

Writing perturbations with a minus-i-omega time convention makes a positive imaginary part of omega correspond to growth. In a damped mode the imaginary part is negative. Adiabatic stable-mode calculations give real frequencies but do not predict observed amplitudes. Nonadiabatic work from opacity variations, nuclear heating, radiative leakage, or convective heat exchange can drive or damp oscillations. Turbulent stochastic forcing can maintain a spectrum of otherwise damped modes. Strong amplitudes eventually require nonlinear dynamics and can invalidate linear eigenfunctions.

Observation adds geometric visibility and finite measurement duration: different angular patterns can cancel when the unresolved stellar disk is integrated. A broadened spectral peak is therefore interpreted through a model of excitation, damping, observation window, and noise rather than by identifying its width with a structural frequency spacing. The equal-frequency wave instrument can test coherent addition and cancellation, but it has no eigenfrequency search, spherical harmonics, internal cavities, or mode excitation. Its cancellation is an exact algebraic analogy, not evidence that a real star contains no oscillation energy.