4.2: The Influence of Rotation and Magnetic Fields on White Dwarf Gravitational Instability

At this point the reader is likely to complain that the derivation indicating the presence of an instability resulting from general relativity has been anything but brief. The length results largely from a somewhat different approach to the general relativistic term than used earlier. That the approach succeeds at all is largely a result of presumed spherical symmetry. However, to further demonstrate the efficacy of this approach let us consider what impact rotation and magnetic fields may have on the results of the last section. Fowler found that a very small amount of rotation would stabilize larger supermassive stars against the gravitational instability so one might wonder what would be the effect in white dwarfs. However, the situation for white dwarfs is quite different. Here the gravitational field is proportionally much stronger with γ being driven to 4/3 by the Equation of state and not the radiation field. Thus we may expect that a much larger rotational energy field is required to bring about stability than is the case for supermassive stars. In spite of this expectation, we shall assume that the effects of rotation and magnetic fields are not so extreme as to significantly alter the spherical symmetry.

Under these conditions, the Newtonian approach of Chapter III will suffice to calculate the terms to be added to the Equations of motion and to perform the required variational analysis. In Chapter III, we defined the rotational kinetic energy \(\mathcal{J}_3\) and magnetic energy \(\mathscr{M}_\mathrm{o}\) as

\[\left.\begin{array}{l}
\mathcal{J}_3=\int_0^{\mathscr{L}} \frac{1}{2} \omega \mathrm{d} \mathscr{L} \\[4pt]
\mathscr{M}=\int_{\mathrm{v}} \frac{\mathrm{H}^2}{8 \pi} \mathrm{dV}
\end{array}\right\},\label{4.2.1}\]

which have variations

\[\left.\begin{array}{l}
\delta \mathcal{J}_3=-2 \xi \mathcal{J}_3(0) \\[4pt]
\delta \mathscr{M}=-\xi \mathscr{M}
\end{array}\right\}.\label{4.2.2}\]

Adding this to the variational form of Lagrange's identity in section 1 [Equation \ref{4.1.21}], we get

\[\frac{1}{2} \frac{\mathrm{d}^2}{\mathrm{dt}^2}\left(\delta \mathrm{I}_{\mathrm{r}}\right)=3<\Gamma_1-1>\delta \mathcal{U}-\delta \Omega+2 \delta \mathcal{J}_3+\delta \mathscr{M}+<\Gamma_1-1>\delta \mathcal{U}_1-2 \delta \Omega_1.\label{4.2.3}\]

Now the condition on the variation of the total energy becomes

\[\delta \mathrm{E}=0=\delta \mathcal{U}-\delta \Omega+\delta \mathcal{J}_3+\delta \mathscr{M}+\delta \mathcal{U}_1-\delta \Omega_1,\label{4.2.4}\]

which enables us to re-write 4.2.3 as

\[\frac{1}{2} \frac{\mathrm{d}^2}{\mathrm{dt}^2}\left(\delta \mathrm{I}_{\mathrm{r}}\right)=<3 \Gamma_1-4>(\delta \mathcal{U}-\delta \mathscr{M})-2<\Gamma_1-1>\delta \mathcal{U}_1+<5-3 \Gamma_1>\left(\delta \mathcal{J}_3-\delta \Omega_1\right).\label{4.2.5}\]

Substituting in the values for the variations we get an expression analogous to Equation \ref{4.1.24}

\[\sigma^2 \mathrm{I}_{\mathrm{r}} \bar{\mathrm{y}}=<3 \Gamma_1-4>\left(\Omega_0-\mathscr{M}_0\right)-4<\Gamma_1-1>\mathcal{U}_1+2<3 \Gamma_1-5>\left[\Omega_1-\mathcal{J}_3(0)\right].\label{4.2.6}\]

These expressions differ from those in Chapter III only because the gravitational potential energy is taken here to be positive. In order for us to proceed further it will be necessary to normalize both the rotational energy and magnetic field by something. Let us consider the case for ridged rotation so that

\[\mathcal{J}_3=\frac{1}{2} \omega^2 \mathrm{I}_{\mathrm{z}}=\frac{1}{3} \omega^2 \mathrm{I}.\label{4.2.7}\]

Here we are ignoring the relativistic corrections to I and take \(\mathrm{I}=\alpha \mathrm{MR}^2\). In addition, let us normalize the angular velocity ω by the critical value for a Roche model. Then

\[\omega^2=\mathrm{w}^2\left(\frac{8 \mathrm{GM}}{27 \mathrm{R}_0^3}\right)=\frac{4 \mathrm{w}^2 \mathrm{c}^2}{27 \mathrm{R}_0}\left(\frac{2 \mathrm{GM}}{\mathrm{R}_0 \mathrm{c}^2}\right).\label{4.2.8}\]

This certainly does not imply that we are in any way assuming that white dwarfs are represented by a Roche model but rather that it merely provides us with a convenient scale factor. Thus

\[\mathcal{J}_3=\frac{4 \alpha}{81} \mathrm{w}^2 \mathrm{Mc}^2\left(\frac{\mathrm{R_s}}{\mathrm{R_0}}\right).\label{4.2.9}\]

In a similar manner let us normalize the magnetic energy \(\mathscr{M}_\mathrm{o}\) by the energy sufficient to bring about disruption of the star. In Chapter III we showed that if other effects were absent then \(\mathscr{M}_\mathrm{o}>|\Omega|\) would disrupt the star. Using this as the normalization constant we have

\[\mathscr{M}_0=\left[\frac{3}{2(5-\mathrm{n})}\right] \mathscr{H}^2 \mathrm{Mc}^2\left(\frac{\mathrm{R}_{\mathrm{S}}}{\mathrm{R}_0}\right).\label{4.2.10}\]

Under these conditions we can expect the maximum values

for w and \(\mathscr{H}\) to be

\[\left.\begin{array}{l}
\mathrm{w}=1 \\[4pt]
\mathscr{H}=1
\end{array}\right\},\label{4.2.11}\]

and in any event the assumption of sphericity will probably break down for w > 0.8 and \(\mathscr{H}>0.3\). Putting these values for \(\mathcal{J}_3\) and \(\mathscr{M}_\mathrm{o}\) along with the previously determined values for \(\mathrm{\Omega}\), \(\mathcal{U}_1\), and \(\Omega_1\), into Equation \ref{4.2.6} we can arrive at stability conditions analogous to Equation \ref{4.1.27}. Namely

\[\left(\frac{\mathrm{R}_{\mathrm{s}}}{\mathrm{R}}\right)\left(4 \varsigma_1<\Gamma_1-1>+2 \varsigma_2<5-3 \Gamma_1>\right)<\left(\frac{3<3 \Gamma_1-4>\left(1-\mathscr{H}^2\right)}{2(5-\mathrm{n})}\right)+\frac{8 \alpha \mathrm{w}^2}{81}.\label{4.2.12}\]

As before, let us pass to the case where n = 3, so that

\[\left(\frac{\mathrm{R}_{\mathrm{S}}}{\mathrm{R}}\right)\left(\frac{4}{3} \varsigma_1+2 \varsigma_2\right)<\frac{9}{2} \varepsilon\left(1-\mathscr{H}^2\right)+\frac{8 \alpha \mathrm{w}^2}{81}.\label{4.2.13}\]

With \(\alpha=0.113\) for polytropes of n = 3 and again using Fowler's⁴ values for \(\zeta_1\) and \(\zeta_2\) this becomes

\[\left(\frac{\mathrm{R}_{\mathrm{S}}}{\mathrm{R}_0}\right)<0.89 \varepsilon\left(1-\mathscr{H}^2\right)+4.4 \times 10^{-3} \mathrm{w}^2 .\label{4.2.14}\]

Using the same analysis for ε as before

\[\left(\frac{\mathrm{R}_{\mathrm{S}}}{\mathrm{R}_0}\right)<8.4 \times 10^{-8}\left(1-\mathscr{H}^2\right)\left(\mathrm{M} / \mathrm{M}_{\odot}\right)^{4 / 3}\left(\frac{\mathrm{R}_{\mathrm{S}}}{\mathrm{R}_0}\right)+4.4 \times 10^{-3} \mathrm{w}^2,\label{4.2.15}\]

and taking M to be near the Chandrasekhar limit, we have

\[\left(\frac{\mathrm{R}_0}{\mathrm{R}_{\mathrm{S}}}\right)^3+\left(\frac{\mathrm{R}_0}{\mathrm{R}_{\mathrm{S}}}\right) \frac{4.1 \times 10^4 \mathrm{w}^2}{\left(1-\mathscr{H}^2\right)}-\frac{9.3 \times 10^{-6}}{\left(1-\mathscr{H}^2\right)}>0 .\label{4.2.16}\]

For a point of reference it is worth mentioning the size of the normalization quantities so that various values of w and \(\mathscr{H}\) may be held in perspective. If we assume that we are dealing with objects on the order of 10³ km then the disruption field is of the order of 3 x 10¹⁵ gauss while the critical equatorial velocity would be about 10⁴ km/sec. The largest observed fields in white dwarfs reach 10⁸ gauss and although it can be argued that larger fields may be encountered in more massive white dwarfs, fields in excess of 10¹² gauss would not seem to be supported by observations. Thus, a plausible upper value of \(\mathscr{H}\) would be on the order of 10^-3. If, for the moment we neglect rotation, Equation \ref{4.2.14} can be written as

\[\left(\frac{\mathrm{R}_0}{\mathrm{R}_{\mathrm{s}}}\right)>210\left(1-\mathscr{H}^2\right)^{-1/3}.\label{4.2.17}\]

So it is clear that the only effect the field has is to act with general relativity to further destabilize the star. However, for the field to make any appreciable difference it will have to be truly large. For our plausible upper limit of H ≈ 10^-3 the effect is to increase the radius at which the instability sets in by about 1%.

The situation regarding rotation is slightly more difficult to deal with as the resulting inequality is a cubic. With \(\mathscr{H}=0\), Equation \ref{4.2.14} becomes

\[\left(\frac{\mathrm{R}_0}{\mathrm{R}_{\mathrm{S}}}\right)^3+\left(\frac{\mathrm{R}_0}{\mathrm{R}_{\mathrm{S}}}\right) 4.1 \times 10^4 \mathrm{w}^2-9.3 \times 10^{-6}>0.\label{4.2.18}\]

An additionally useful expression for the equatorial velocity which corresponds to a given w is

\[\mathrm{v}_{\mathrm{eq}}=\mathrm{R}_{\mathrm{eq}} \omega=\frac{2 \mathrm{cw}}{3 \sqrt{3}}\left(\frac{\mathrm{R}_{\mathrm{s}}}{\mathrm{R}_0}\right)^{\frac{1}{2}}=1.15 \times 10^5 \mathrm{w}\left(\frac{\mathrm{R}_{\mathrm{s}}}{\mathrm{R}_0}\right)^{\frac{1}{2}}(\mathrm{km} / \mathrm{sec}).\label{4.2.19}\]

If we pick a few representative values of w and solve Equation \ref{4.2.17} by means of the general cubic we get the results below for the associated values of \(\left(\mathrm{R}_{\mathrm{o}} / \mathrm{R}_{\mathrm{S}}\right)\) and \(\mathrm{V_{e q}}\).

Rotational Effects on the White Dwarf Instability Limit

w	0	0.1	0.5	1.0
\(\mathrm{V_{eq}(km/sec)}\)	0	794	4128	9453
\(\mathrm{R_o/R_s}\)	210	209	194	148

Even these few values are sufficient to indicate that although rotation helps to stabilize the star in the sense of allowing it to attain a smaller radius before collapse, nevertheless, like magnetic fields, the effect is small. One may choose to object to the assumption of rigid rotation as being too conservative. However, it is clear from the development that for either rotation or magnetic fields to really play an important role the total energy stored by either mechanism must approach that in the gravitational field. In order to do this with differential rotation, the differential velocity field would have to be alarmingly high. It seems likely that the resulting shear would produce significant dynamical instabilities.

Thus, we have seen that neither magnetic fields nor rotation can significantly alter the fact that a white dwarf will become unstable at or about 1000 km. Classically, the star reaches this point when it is within, but less than, a few percent of the Chandrasekhar limiting mass. So it is not the limiting mass resulting from the change in the Equation of state that keeps us from observing more massive white dwarfs. Rather it is the presence of general relativistic instability that destroys any more massive objects.

It is quite simple to dismiss this argument as 'nit-picking' as the mass at which the instability occurs is nearly identical to the Chandrasekhar limiting mass. However, when one tries to generalize the results of one problem to another, it is conceptual errors such as this that may lead to much more serious errors in the generalization. As we shall see in the next section, this is indeed the case with neutron stars.