Boundary Conditions

To form a closed system, the separated equations are augmented by algebraic relations at the inner and outer boundaries of the computational domain, and possibly at interior points as well.

Inner Boundary

When the inner boundary is placed at the stellar origin (\(r=0\)), the requirement that solutions remain finite leads to the set of regularity conditions

\[\begin{split}\begin{aligned} \txir - \ell \txih = 0, \\ \ell \tPhi' - r \deriv{\tPhi'}{r} = 0, \\ \delta \tS = 0. \end{aligned}\end{split}\]

Sometimes it’s desirable that the inner boundary is instead placed at \(r > 0\) — for instance, to excise the stellar core from the oscillation calculations. Then, there is much more flexibility in the choice of inner boundary condition. Possible options include setting \(\txir = 0\) or \(\txih=0\) instead of the first equation above.

Outer Boundary

The outer boundary typically corresponds to the stellar surface. Under the assumption that the density vanishes on and above this surface, the gravitational potential must match onto a solution to Laplace’s equation that remains finite at infinity, leading to the potential boundary condition

\[(\ell + 1) \tPhi' + r \deriv{\tPhi'}{r} = 0.\]

Likewise, the assumption that there is no external pressure acting on the star (consistent with the vanishing surface density) gives the momentum boundary condition

\[\delta \tP = 0.\]

Finally, the thermal boundary condition can be derived from the equation

\[T^{4}(\tau) = \frac{4\sigma}{3} \Fradr \left( \tau + \frac{2}{3} \right)\]

describing the thermal structure of an atmosphere under the combined plane-parallel, grey, Eddington, local thermodynamic equilibrium and radiative equilibrium approximations. Here, \(\tau\) is the vertical optical depth and \(\sigsb\) the Stefan-Boltzmann constant. Setting \(\tau=0\) (again, consistent with the vanishing surface density) and perturbing this equation yields the desired boundary condition

\[4 \frac{\delta \tT}{T} = \frac{\delta \tFradr}{\Fradr}.\]

Complications arise when realistic stellar models are considered, because these typically extend only out to an optical depth \(\tau=2/3\) (or some similar value) corresponding to the photosphere. In such cases the density does not vanish at the nominal stellar surface, and the outer boundary conditions must be modified to account for the effects of the atmosphere region lying above the surface. Many stellar oscillation codes, including GYRE, can adopt more sophisticated formulations for the momentum boundary condition — for instance, based on the assumption that the outer atmosphere has an isothermal stratification. However, the atmospheric effects on the potential and thermal boundary conditions are usually neglected.

Internal Boundaries

Internal boundaries arise when the density and other related quantities show a radial discontinuity within the star. Across such a discontinuity \(\txir\), \(\delta \tP\), \(\delta \tPhi\) and \(\delta \tT\) must remain continuous[1]. Internal boundary conditions on other perturbations are given by

\[\begin{split}\begin{aligned} \tP^{\prime +} - \tP^{\prime -} &= \deriv{\Phi}{r} \left( \rho^{+} - \rho^{-} \right) \txir, \\ \left. \frac{\delta \tS}{\cP} \right|^{+} - \left. \frac{\delta \tS}{\cP} \right|^{-} &= - \left( \nabad^{+} - \nabad^{-} \right) \frac{\delta \tP}{P}, \\ \left. \deriv{\tPhi'}{r} \right|^{+} - \left. \deriv{\tPhi'}{r} \right|^{-} &= - 4 \pi G \left( \rho^{+} - \rho^{-} \right) \txir. \\ \end{aligned}\end{split}\]

Here, + (-) superscripts indicate quantities evaluated on the inner (outer) side of the discontinuity. The first of these conditions arises from applying equation (7) to the pressure perturbations on either side of the discontinuity; the second from evaluating the linearized equation-of-state on either side of the discontinuity; and the third from integrating the separated continuity and Poisson equations (9 and 10, respectively) across the discontinuity.