Model Output

Solving the four-equation convective one-zone model.

Luminosity vs Phase

last two cycles

Radial Velocity vs Phase

last two cycles

Time Evolution

state variables

Luminosity Split

total, radiative, convective

Physical Model

The one-zone model compresses a pulsating stellar envelope into one moving 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 free-fall/dynamical time. The nonadiabatic pressure factor \(\ozPressure{P}\) stores thermal energy, 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.

Historically, this app follows R. F. Stellingwerf's 1986 ApJ paper “A Simple Model for Coupled Convection and Pulsation”, which generalized Baker's one-zone pulsation model with a time-dependent convection equation. The local S_Tran files also mention the overtone work connected to Stellingwerf, Gautschy, and Dickens 1987.

Variables

SymbolNameMeaning
\(\ozTau{\tau}\)Independent timeDimensionless clock scaled by the model's free-fall/dynamical time; derivatives such as \(dR/d\tau\) are per dynamical time unit.
\(\ozRadius{R}\)RadiusDimensionless radius of the moving shell.
\(\ozVelocity{V}\)Radial velocityTime derivative of the shell radius.
\(\ozPressure{P}\)Pressure factorNonadiabatic pressure or thermal state variable.
\(\ozConvective{U_c}\)Convective velocityTime-dependent turbulent/convective velocity scale.
\(\ozRadiative{L_r}\)Radiative luminosityRadiative luminosity-like flux leaving the shell.
\(\ozConvLum{L_c}\)Convective luminosityConvective luminosity-like flux proportional to \(U_c^3\).
\(\ozLuminosity{L}\)Total luminosityWeighted sum \(L=\gamma_rL_r+\gamma_cL_c\).

Equations Solved

\[ \begin{aligned} \frac{d\ozRadius{R}}{d\ozTau{\tau}} &= \ozVelocity{V},\\[0.35em] \frac{d\ozVelocity{V}}{d\ozTau{\tau}} &= \frac{\ozPressure{P}}{\ozRadius{R}^{q(\ozRadius{R})}} - \frac{1}{\ozRadius{R}^{2}} - \ozDamping{C_q}\ozVelocity{V}^{3},\\[0.35em] \frac{d\ozPressure{P}}{d\ozTau{\tau}} &= \ozZeta{\zeta}\, \ozRadius{R}^{\ozMass{m}(\ozRadius{R})(\ozGamma{\Gamma_1}-1)} \left[ \ozRadius{R}^{\ozSource{U}} - \ozNeutral{\gamma_r}\ozRadiative{L_r} - \ozGammac{\gamma_c}\ozConvLum{L_c} \right],\\[0.35em] \frac{d\ozConvective{U_c}}{d\ozTau{\tau}} &= \ozZetac{\zeta_c} \left[ \ozRadius{R}^{-d(\ozRadius{R})}D - \ozConvective{U_c} \right]. \end{aligned} \]
\[ \begin{aligned} \ozRadiative{L_r} &= \ozRadius{R}^{b(\ozRadius{R})} \ozPressure{P}^{\ozPink{s}+4},\\[0.35em] \ozConvLum{L_c} &= \ozRadius{R}^{-c(\ozRadius{R})} \ozConvective{U_c}^{3},\\[0.35em] \ozLuminosity{L} &= \ozNeutral{\gamma_r}\ozRadiative{L_r} + \ozGammac{\gamma_c}\ozConvLum{L_c},\\[0.35em] D &= \sqrt{\ozPressure{P}} \quad\text{or}\quad \sqrt{|\ozVelocity{V}|}. \end{aligned} \]

With fixed shell mass, \(m(R)=m\). With the toggle enabled, \(m(R)=3/[1-(\eta/R)^3]\), where \(\eta=(1-3/m)^{1/3}\).

Parameter Guide

ControlMeaning