7.2: Classical Distortion- The Structure Equations

The loss of spherical symmetry will change the familiar equations of stellar structure to vector form. Before developing the specific equations for axial distortion, let us consider the general form of these equations. In Chapter 6 we compared the relativistic equations of stellar structure to the classical spherical equations. In a similar manner, let us begin our discussion of distortion with a comparison of the classical spherical equations with their counterparts for distorted stars.

a. A Comparison of Structure Equations

Below is a summary of the equations of stellar structure for spherically symmetric stars and stars which suffer a general distortion.

\[\label{7.1.1}\]

The variable M(r) that is so useful for spherical structure is replaced by the potential, given here as the gravitational potential. In principle, the potential could contain a contribution from other physical phenomena such as magnetism or rotation. Poisson's equation is a second-order partial differential equation and replaces the first-order total differential equation for spherical structure. So the price we pay for the loss of spherical symmetry is immediately obvious. While the conservation of energy equation remains a scalar equation, as it should, it now involves a vector quantity, the radiative flux, and an additional term that anticipates some results from later in the chapter. The quantity S is the entropy of the gas, and in Chapter 4 [see equation \ref{4.6.9}] we saw that this term had to be included when the models were changing rapidly in time. In this case, the term is required to describe the flow of energy due to mass motions resulting from the distortion itself. Both radiative and hydrostatic equilibrium become vector equations where we have explicitly indicated the presence of a perturbing force by the vector \(\vec{\mathrm{D}}\) which, should it be derivable from a potential, could be included directly in the potential. This perturbing force is assumed to be known. The quantities such as \(\kappa\) and \(\varepsilon\), which depend on the local microphysics, presumably will not be directly affected by the presence of a macroscopic perturbing force. A possible exception could be the case of distortion by a magnetic field where the local field would contribute to the total pressure and in extreme cases, could affect the opacity.

b. Structure Equations for Cylindrical Symmetry

To minimize the complexity, we consider those cases resulting in the loss of only one symmetry coordinate, and we deal with those systems exhibiting axial symmetry. This is clearly appropriate for rapidly rotating stars as well as stars distorted by the presence of a companion. In addition, we shall see that it also is appropriate for the distortion introduced by an ordered magnetic field that itself exhibits axial symmetry.

To specifically see the effects that result from a distortion force, we have to express that force in some appropriate coordinate system. The distortion force was represented in the structure equations, (7.1.1), by the vector \(\vec{\mathrm{D}}\) in the equation of hydrostatic equilibrium. For axial symmetry, cylindrical and spherical polar coordinates both form suitable coordinate systems for this description (see Figure 7.1). We express the components of the perturbing force in terms of Legendre polynomials of the polar angle θ . Once the perturbing force has been characterized, we shall indicate, in the next section, how the solution of the structure equations proceeds.

The Legendre polynomials form an orthogonal set of polynomials over a finite, defined range. Specifically, let \[\mu=\operatorname{Cos} \theta\label{7.1.2}\]

Figure 7.1 shows suitable coordinate systems to describe an axially symmetric perturbing force \(\vec{\mathrm{D}}\) with components \(\mathrm{D_r}\) and \(\mathrm{D_\theta}\). We have chosen to illustrate rotational distortion so that Z represents the spin axis. However, we could have illustrated gravitational distortion by an external object in which case Z would lie along the line of centers of the system and \(\vec{\mathrm{D}}\) would point to the center of the other object.

Then the Legendre polynomials form an orthonormal set in the interval −1 ≤ µ ≤ +1 subject to the normalization condition \[\int_{-1}^{+1} P_n(\mu) P_m(\mu) d \mu=\delta_{m, n}\left(\frac{2}{2 n+1}\right)\label{7.1.3}\]

Here \(\delta_{\mathrm{m}, \mathrm{n}}\) is the Kronecker delta which is 1 if m = n and 0 otherwise. Various members of the set of Legendre polynomials can be generated from the recursion relation \[P_{m+1}(\mu)=\frac{2 m+1}{m+1} \mu P_m(\mu)-\frac{m}{m+1} P_{m-1}(\mu)\label{7.1.4}\]

where the first three members of the set are \[\begin{aligned}
& P_0(\mu)=1 \\
& P_1(\mu)=\mu=\operatorname{Cos} \theta \\
& P_2(\mu)=\frac{3}{2} \mu^2-\frac{1}{2}=1-\frac{3}{2} \operatorname{Sin}^2 \theta
\end{aligned}\label{7.1.5}\]

Before we can specify the effects of the perturbing force in detail, we must indicate its nature. So let us turn to some simple examples of distorting forces and their effects on the structure equations.

Rigid Rotation For our first example, we consider the case where the star is rotating as a rigid body. This yields a simple expression for the magnitude of the distorting force produced by the local centripetal acceleration, which is \[|D|=\omega^2 r \operatorname{Sin} \theta\label{7.1.6}\]

where ω is the angular velocity of the star and is assumed to be constant. The components of the acceleration are then \[D_r=\omega^2 r \operatorname{Sin}^2 \theta \quad D_\theta=\omega^2 r \operatorname{Sin} \theta \operatorname{Cos} \theta\label{7.1.7}\]

which can be expressed in terms of Legendre polynomials and their derivatives as \[D_r=\frac{2}{3} \omega^2 r\left[1-P_2(\mu)\right] \quad D_\theta=-\frac{1}{3} \omega^2 r \frac{\partial P_2(\mu)}{\partial \theta}\label{7.1.8}\]

Due to the axial symmetry, \(\mathrm{D_\phi=0}\) and it is a simple matter to show that the curl of D, \(\nabla \times \overrightarrow{\mathrm{D}}\), is 0 so that \(\vec{\mathrm{D}}\) is derivable from a scalar potential by \[\overrightarrow{\mathrm{D}}=-\nabla \Lambda\label{7.1.9}\]

where \[\Lambda=-\frac{1}{2} \omega^2 r^2 \operatorname{Sin}^2 \theta\label{7.1.10}\]

Thus, the components of the perturbing force and the rotational potential can be expressed in terms of the Legendre polynomials as \[\begin{aligned}
D_r & =A(r)+B(r) P_2(\mu) \\
D_\theta & =C(r) \frac{\partial P_2(\mu)}{\partial \theta} \\
\Lambda & =Q_0(r)+Q_1(r) P_2(\mu)
\end{aligned}\label{7.1.11}\]

Although the above relations are correct for ω = constant, it is worth considering the functional dependence of ω for which it is true in general. Consider the nature of centripetal acceleration in a cylindrical coordinate system where the radial coordinate is denoted by s. The components of \(\vec{\mathrm{D}}\) are \[\mathrm{D}_{\mathrm{z}}=\mathrm{D}_\phi=0, \quad \mathrm{D}_{\mathrm{s}}=\omega^2 \mathrm{~s}\label{7.1.12}\]

In order for the rotational force to be derivable from a scalar potential, its curl must be zero. The cylindrical components of the curl are \[\begin{aligned}
& (\nabla \times \vec{D})_s=\frac{1}{s} \frac{\partial D_z}{\partial \phi}-\frac{\partial D_\phi}{\partial s}=0 \\
& (\nabla \times \vec{D})_z=\frac{\partial D_s}{\partial \phi}=0 \\
& (\nabla \times \vec{D})_\phi=\frac{\partial D_s}{\partial z}-\frac{\partial D_z}{\partial s}=\frac{\partial\left(\omega^2 s\right)}{\partial z}
\end{aligned}\label{7.1.13}\]

The radial component is identically zero, so we may suspect that if the object exhibits axial symmetry, ω cannot be a function of \(\phi\). In this case, the z component of the curl would also be zero. Thus, the condition that the rotational force be derivable from a scalar potential boils down to the \(\phi\) component of the curl being zero, so that \[\frac{\partial\left(s \omega^2\right)}{\partial z}=0\label{7.1.14}\]

Thus, \[\omega \neq \omega(z)\label{7.1.15}\]

so the angular velocity must be constant on cylinders. It can be shown, that if the perturbing force is not derivable from a potential, then no equilibrium solution of the structure equations exists. This is sometimes called the Taylor-Proudman theorem¹ and it basically guarantees that if the star has reached an equilibrium angular momentum distribution, the angular velocity will be constant on cylinders.

Gravitational Distortion by an External Point Mass Now let us return to the spherical polar coordinates that we used to obtain the components of the rotational force. The force will be directed toward an external point mass located along the z axis at a distance d from the center of the star (see Figure 7.2).

Figure 7.2 shows the type of distortion to be expected from the presence of a companion such as would be found in a close binary system. For simplification, the rotational distortion is considered to be negligible. The distance from any point in the star to the perturbing mass is denoted by \(\rho\)

Now the perturbing potential of the point mass \(M_2\) is \[\Omega_2=\frac{G M_2}{\rho}\label{7.1.16}\]

where \[\rho^2=d^2+r^2-2 r d \operatorname{Cos} \theta=d^2\left[1+\left(\frac{r}{d}\right)^2-2 \frac{r}{d} \operatorname{Cos} \theta\right]\label{7.1.17}\]

So the potential can be written in terms of our coordinates and the stellar separation as \[\Omega_2=\frac{G M_2}{d}\left[1-2 \frac{r}{d} \operatorname{Cos} \theta+\left(\frac{r}{d}\right)^2\right]^{-1 / 2}\label{7.1.18}\]

Equation \ref{7.1.18} is rather non-linear in the θ coordinate so, in order to express the potential in term of Legendre polynomials, we can make use of the "generating function" (see Arfken² for the development of this generating function) for the Legendre polynomials, \[\left(1-2 \alpha \mu+\alpha^2\right)^{-1 / 2}=\sum_{i=0}^{\infty} P_i(\mu) \alpha^i\label{7.1.19}\]

so the perturbing potential becomes \[\Omega_2=\frac{G M_2}{d} \sum_{i=0}^{\infty}\left(\frac{r}{d}\right)^i P_i(\operatorname{Cos} \theta)\label{7.1.20}\]

Since the perturbing force is conservative, we may obtain it from \[\vec{D}=+\nabla \Omega_2\label{7.1.21}\]

which has components \[\begin{aligned}
& D_r=\frac{G M_2}{d} \sum_{i=1}^{\infty}\left(\frac{i}{d}\right)\left(\frac{r}{d}\right)^{i-1} P_i(\operatorname{Cos} \theta) \\
& D_\theta=\frac{G M_2}{d} \sum_{i=0}^{\infty}\left(\frac{1}{r}\right)\left(\frac{r}{d}\right)^i \frac{\partial P_i(\operatorname{Cos} \theta)}{\partial \theta} \\
& D_\phi=0
\end{aligned}\label{7.1.22}\]

So far, the only approximation that we have made is that the perturbing potential is that of a point mass. To simplify the remaining discussion, we assume that the point mass is distant compared to the size of the object so that \[\begin{aligned}
& D_r=\frac{G M_2}{d}\left[\frac{\operatorname{Cos} \theta}{d}+\frac{2 r}{d^2} P_2(\operatorname{Cos} \theta)+\cdots+\right] \\
& D_\theta=\frac{G M_2}{d}\left[\frac{-\operatorname{Sin} \theta}{d}+\frac{r}{d^2} \frac{\partial P_2(\operatorname{Cos} \theta)}{\partial \theta}+\cdots+\right]
\end{aligned}\label{7.1.23}\]

Note that the zeroth order terms of the components can be added vectorially to give \[\vec{D}_0=\frac{G M_2}{d^2} \hat{d}\label{7.1.24}\]

This is just the gravitational force that is balanced by the acceleration resulting from the orbital motion of the system, and so this force can be made to vanish by going to a rotating coordinate system. In such a system, the components of the perturbing force will just be the first-order terms, so that \[\tilde{D}_r=\frac{2 G M_2 r}{d^3} P_2(\operatorname{Cos} \theta) \quad \tilde{D}_\theta=\frac{G M_2 r}{d^3} \frac{\partial P_2(\operatorname{Cos} \theta)}{\partial \theta}\label{7.1.25}\]

These components of the perturbing force have the same form as those of rotation [see equation \ref{7.1.11}], and any method which is applicable to the solution of the structure equations for rotational distortion will also be applicable to the problem of gravitational distortion.

Distortion Resulting from a Toroidal Magnetic Field Consider the Lorentz force of an internal magnetic field on the material of the star: \[\vec{f}=\frac{\vec{j} \times \vec{B}}{c}=-(4 \pi c)^{-1}[\vec{B} \times(\nabla \times \vec{B})]\label{7.1.26}\]

The perturbing acceleration due to this force will be \[\vec{D}=\frac{\vec{f}}{\rho}=(4 \pi \rho c)^{-1}[\vec{B} \times(\nabla \times \vec{B})]\label{7.1.27}\]

Now assume a special, but not implausible, geometry for the internal stellar magnetic field. Specifically, let us choose a toroidal field that exhibits pure axial symmetry so that \[\vec{B}=\psi(r)(\operatorname{Sin} \theta)(\hat{\phi})\label{7.1.28}\]

where \(\psi(\mathrm{r})\) contains the arbitrary, but presumed known, variation of the field with the radial coordinate r. Since the field only has a \(\phi\) component, the vector part of equation \ref{7.1.27} is \[\vec{B} \times(\nabla \times \vec{B})=-\hat{r}\left[B_\phi(\nabla \times \vec{B})_\theta\right]+\hat{\theta}\left[B_\phi(\nabla \times \vec{B})_r\right]\label{7.1.29}\]

The θ and r components of the curl of \(\vec{\mathrm{B}}\) are \[\begin{aligned}
& (\nabla \times \vec{B})_\theta=-r^{-1} \frac{\partial\left(r B_\phi\right)}{\partial r}=-\frac{\left[B_\phi+r \operatorname{Sin} \theta\left(\frac{\partial \psi}{\partial r}\right)\right]}{r} \\
& (\nabla \times \vec{B})_r=(r \operatorname{Sin} \theta)^{-1}\left[\psi(r) \operatorname{Sin} \theta \operatorname{Cos} \theta+B_\phi \operatorname{Cos} \theta\right]=\frac{2 \psi(r) \operatorname{Cos} \theta}{r}
\end{aligned}\label{7.1.30}\]

which yields for the vector components of the perturbing field \[\begin{aligned}
& \mathrm{D_r=(4 \pi \rho c)^{-1}\left[\frac{\psi^2(r)}{r}+\psi(r) \frac{\partial \psi}{\partial r}\right] \operatorname{Sin}^2 \theta=\widetilde{A}(r)+\widetilde{B}(r) P_2(\operatorname{Cos} \theta)} \\
& \mathrm{D_\theta=((4 \pi \rho c)^{-1} \frac{\psi^2(r)}{r} \frac{\partial P_2(\operatorname{Cos} \theta)}{\partial \theta}=\widetilde{C}(r) \frac{\partial P_2(\operatorname{Cos} \theta)}{\partial \theta}}
\end{aligned}\label{7.1.31}\]

Again, these components have the same form as those of rotational distortion.

Thus we can expect to be able to solve a wide variety of distortion problems by considering the single case of an axis-symmetric perturbing force of the form given in equations \ref{7.1.11}, \ref{7.1.25}, and \ref{7.1.31}. We now consider some aspects of the solution of such problems.