7.3: Solution of Structure Equations for a Perturbing Force

The equations given by equations \ref{7.1.1}, and which arise from the perturbations discussed in Section 7.1, are partial differential equations and must be solved numerically. The numerical solution of partial differential equations constitutes a major area of study in its own right and is beyond the scope of this book. So we leave the numerical methods required for the actual solution to others and another time. Instead, we concentrate on the conditions required for the equations to have a solution and some of the implications of those solutions.

Since the perturbing forces derived in Section 7.1 are all conservative forces (that is, \(\nabla \times \overrightarrow{\mathrm{D}}=0\)), they are all derivable from some scalar potential which we can call Λ. This can be added to the gravitational potential so that we have a generalized potential to enter into the structure equations which we can call \[\Phi(r, \theta, \phi)=\Omega+\Lambda\label{7.2.1}\]

Since all the forces exhibited axial symmetry, there will be no explicit dependence of the generalized potential on \(\phi\). There will be sets of values of θ and r, for which Φ is constant. For the unperturbed gravitational potential alone these would be spheres of a given radius. For the generalized potential, they will be surfaces that exhibit axial symmetry. Such surfaces are known as level surfaces since a particle placed on one would feel no forces that would move it along the surface. Thus, if \(\hat{n}\) represents a normal to such a surface, the gradient of the potential can be expressed as \[\nabla \Phi=\hat{n} \frac{d \Phi}{d n}\label{7.2.2}\]

As long as the chemical composition is constant, the state variables will be constant on level surfaces. This is sometimes known as Poincare's theorem which we prove for rotation in the next section. However, the result is entirely reasonable. The values of the state variables change in response to forces acting on the gas. Since the potential is constant on a level surface and its gradient is always normal to the surface, there are no forces along the surface to produce such differences.

If we take the state variables to be constant along level surfaces of constant potential, we can expect the variables to have the same functional dependence on the coordinates as the potential itself. Thus, from the form of Λ given by equation \ref{7.1.11}, the state variables, and those parameters that depend directly on them, can be written as \[\begin{aligned}
\mathrm{P}(r, \theta)&=\mathrm{P}_0(r)+\mathrm{P}_2(r) P_2(\operatorname{Cos} \theta) \\
\rho(r, \theta)&=\rho_0(r)+\rho_2(r) P_2(\operatorname{Cos} \theta) \\
\epsilon(r, \theta)&=\epsilon_0(r)+\epsilon_2(r) P_2(\operatorname{Cos} \theta) \\
\kappa(r, \theta)&=\kappa_0(r)+\kappa_2(r) P_2(\operatorname{Cos} \theta) \\
\Omega(r, \theta)&=\Omega_0(r)+\Omega_2(r) P_2(\operatorname{Cos} \theta)
\end{aligned}\label{7.2.3}\]

The gravitational potential must also be written with a θ dependence, because the perturbing force will rearrange the matter density so that the potential is no longer spherically symmetric.

We now regard equations \ref{7.2.3} as perturbative equations in the traditional sense in that the terms with subscript 2 will be considered to be small compared to the terms with subscript ₀.

a. Perturbed Equation of Hydrostatic Equilibrium

Substituting the perturbed form of the structure variables given by equation \ref{7.2.3}, into the equation of hydrostatic equilibrium [equation \ref{7.1.1 d}], we get \[\begin{aligned}
\nabla P= & \nabla\left[\mathrm{P}_0(r)+\mathrm{P}_2(r) P_2(\operatorname{Cos} \theta)\right]=-\rho \nabla \Phi=-\rho \nabla \Omega+\rho \vec{D} \\
= & -\left\{\rho_0(r) \nabla \Omega_0(r)+\rho_0(r) \nabla\left[\Omega_2(r) P_2(\operatorname{Cos} \theta)\right]\right. \\
& \left.+\rho_2(r) P_2(\operatorname{Cos} \theta) \nabla \Omega_0(r)\right\}+\rho_0(r) \vec{D} \\
& +\rho_2(r) P_2^2(\operatorname{Cos} \theta) \nabla \Omega_2(r)+\rho_2(r) P_2(\operatorname{Cos} \theta) \vec{D}
\end{aligned}\label{7.2.4}\]

The terms on the last line of equation \ref{7.2.4} are small "second-order" terms by comparison to the other terms, so, in the tradition of perturbative analysis, we will ignore them. Since the equations must hold for all values of θ , the r component of the gradient yields two distinct equations and the θ component yields one equation. These are basically the zeroth and second-order equations from the two components of the gradient. However, in general, there will be no zeroth-order θ equation, since the unperturbed state is spherically symmetric. Remembering the form for \(\vec{\mathrm{D}}\) from equation \ref{7.1.11}, we see that the partial differential equations for hydrostatic equilibrium are \[\begin{aligned}
\mathrm{\frac{\partial P_0(r)}{\partial r}}&=\mathrm{-\rho_0 \frac{\partial \Omega_0(r)}{\partial r}+\rho_0(r) A(r)} \\
\mathrm{\frac{\partial P_2(r)}{\partial r}}&=\mathrm{-\rho_0(r) \frac{\partial \Omega_2(r)}{\partial r}+\rho_2(r) \frac{\partial \Omega_0(r)}{\partial r}+\rho_0(r) B(r)} \\
\mathrm{P_2(r)}&=\mathrm{-\rho_0(r) \Omega_2(r)+\rho_0(r) C(r) / r}
\end{aligned}\label{7.2.5}\]

b. Number of Perturbative Equations versus Number of Unknowns

The number of independent partial differential equations generated by the vector equation of hydrostatic equilibrium is 3. In general, the vector equations of stellar structure will yield three such independent equations while the scalar equations will produce only two, since there is no θ component. In Table 7.1 we summarize the number of equations we can expect from each of the structure equations.

Each of the perturbed variables P, T, and Ω will produce a first- and second-order unknown function of r for a total of six unknowns. The perturbations in the density \(\rho\) are not linearly independent since they are related to those of P and T by the equation of state. A similar situation exists for the opacity κ and energy generation ε. However the radiative flux is a vector quantity and will yield two unknown perturbed quantities, \(\mathrm{F_{0r}}\) and \(\mathrm{F_{2r}}\), from the r-component and one, \(\mathrm{F_{1\theta}}\), from the θ component. Thus the total number of unknowns in the problem is 9 and the problem is over determined and has no solution. This implies that we have left some physics out of the problem.

In counting the unknowns resulting from perturbing equations \ref{7.1.1}, we implicitly assumed that there were no mass motions present in the star, with the result that \(\partial \boldsymbol{S} / \partial \mathrm{t}\) in equation \ref{7.1.1b} was taken to be zero. If we assume that a stationary state exists, then we can represent the local time rate of change of entropy by a velocity times an entropy gradient, so equation \ref{7.1.1b} becomes \[\nabla \cdot \vec{F}=\rho \epsilon-\vec{v} \cdot T \nabla S\label{7.2.6}\]

Thus, we have added a velocity with three components each of which will have two perturbed parameters. However, in general \(\vec{\mathrm{v}}_0\) will be zero, since we assume no circulation currents in the unperturbed model. In addition, \(\nabla \boldsymbol{S}\) will exhibit axial symmetry and have no \(\phi\) component. Thus the \(\mathrm{v_{2\phi}}\) perturbed parameter will be orthogonal to \(\nabla \boldsymbol{S}\) and not appear in the final equations. This leaves us with 11 unknowns and 10 equations. However, we have not included the fact that mass conservation must be involved with any transport of matter, and modifying the conservation of mass equation to include mass motions will provide one more equation, completing the specification of the problem.

Table 7.1 The number of Independent Scalar Structure Equations

Equation	Zeroth-order r	Second-order r	Second-order \(\theta\)	Total
(7.1.1a) (7.1.1b) (7.1.1c) (7.1.1d)	1 1 1 1	1 1 1 1	1 1	2 2 3 3
Total	4	4	2	10