Physical Model

The one-zone model compresses a pulsating stellar envelope into a single shell. The shell radius \(\ozRadius{R}\) and radial velocity \(\ozVelocity{V}\) describe the mechanical motion. The independent clock \(\ozTau{\tau}\) is scaled by the model's dynamical time. The nonadiabatic pressure factor \(\ozPressure{H}\) carries the thermal/pressure response, while the convective velocity \(\ozConvective{U_c}\) relaxes toward a mixing-length-like value on the convective time scale.

The star brightens through a combined luminosity \(\ozLuminosity{L}\), built from radiative \(\ozRadiative{L_r}\) and convective \(\ozConvLum{L_c}\) channels. Varying the time scales and convective fraction changes whether the pulse damps, settles into a limit cycle, or runs into a dynamic instability.

Mechanically, the shell behaves like a nonlinear oscillator: pressure support pushes outward, gravity pulls inward, and turbulent damping removes kinetic energy. The thermal equation updates \(\ozPressure{H}\) by comparing the inner luminosity source with the radiative and convective luminosity leaking out of the shell. If energy is trapped and released at the right phase of the motion, the pulse grows; if leakage and damping win, it decays.

Convection adds a delayed cooling channel. The convective velocity \(\ozConvective{U_c}\) relaxes toward a local value set mainly by radius and \(\ozPressure{H}\), while the convective luminosity scales like \(\ozConvective{U_c}^{3}\). That delay is why changing \(\ozZeta{\zeta}\), \(\ozZetac{\zeta_c}\), and \(\ozGammac{\gamma_c}\) can move the model between damping, a stable limit cycle, and dynamic instability. Phase-folded plots are most meaningful once the model has reached a repeatable cycle; a still-growing pulse should be read as transient evolution, not a final waveform.

For the one-zone convection framework, see Baker (1966), Stellingwerf (1986), and Stellingwerf, Gautschy, and Dickens (1987).