4.3: Stability of Neutron Stars

A second class of objects whose existence became well established during the 1960's is the neutron stars. It is a commonly held misconception that a neutron star is nothing more than a somewhat collapsed white dwarf, since the masses are thought to be similar. In reality the ratios of typical white dwarf radii to neutron stars is suspected to be nearly 1000 which is just about the ratio of the sun's radius to that of a typical white dwarf. A similar misconception relates to the notion of the neutron star’s limiting mass. It is popular to suggest that since the mass limit in white dwarfs arises as a result of the change in the Equation of state so a similar change in the Equation of state for a neutron star yields a limiting mass for these objects. It is true that a limiting mass exists for neutron stars but this limit does not primarily arise from a change in the Equation of state.

Let us consider a very simple argument to dramatize this point. The Equation of the state change that results in the Chandrasekhar limit occurs because the electrons achieve relativistic velocities. If this were to happen in neutron stars, the configuration would still have to satisfy the virial theorem.

For the moment let us ignore the effects of general relativity and just consider the special relativistic virial theorem as we derived in Chapter II [Equation Equation \ref{2.3.9}].

\[\mathrm{\frac{1}{2} \frac{d^2 I_r}{d t^2}=\Omega+T+\int_v(\tau / \gamma) d V},\label{4.3.1}\]

where now \(\gamma=\left(1-\mathrm{v}^2 / \mathrm{c}^2\right)^{-1 / 2}\).

As the neutrons become relativistic \(\gamma^{-1} \rightarrow 0\) and \(\mathrm{T}=\alpha \mathrm{Mc}^2\) where \(\alpha \gg1\). We may write the gravitational potential energy as

\[\Omega=-\eta \frac{\mathrm{GM}^2}{\mathrm{R}_0}=-\frac{\eta \mathrm{Mc}^2}{2}\left(\frac{\mathrm{R}_{\mathrm{s}}}{\mathrm{R}_0}\right) .\label{4.3.2}\]

The variational form of the virial theorem will require that

\[\mathrm{T}+\Omega<0,\label{4.3.3}\]

so that

\[\begin{gathered}
\mathrm{Mc}^2\left[\alpha-(\eta / 2)\left(\mathrm{R}_{\mathrm{S}} / \mathrm{R}_0\right)\right]<0, \\[4pt]
\text{or }\left(\mathrm{R}_0 / \mathrm{R}_{\mathrm{S}}\right)>\eta / 2 \alpha .
\end{gathered}\label{4.3.4}\]

Since \(\eta\) is of the order of unity and \(\alpha \gg 1\), this would require that the object have a radius less than the Schwarzschild radius in order to be stable against radial pulsations. This is really equivalent to invoking Jacobi's stability condition on the total energy.

This simplistic argument can be criticized on the grounds that it ignores general relativity which can be viewed as increasing the efficiency of gravity. Perhaps the "increased gravity" would help stabilize the star against the rapidly increasing internal energy. This is indeed the case for awhile. However, based on the analysis in section 1, as the value of \(Γ\) approaches 4/3 the same type of instability which brought about the collapse of the white dwarfs will occur in the neutron stars. The exact value of \(\mathrm{R_o / R_S}\) for which this happens will depend on the exact nature of the Equation of state as well as details of model construction. However, since the general relativistic correction terms will be much larger than in the case of white dwarfs, we should expect the value of \(\Gamma\) to depart farther from the relativistic limit of 4/3 than before. That this is indeed the case is clearly shown by Tooper⁷ in considering the general properties of relativistic adiabatic fluid spheres. He concludes that the instability always occurs before the gas has become relativistic at high pressures. Unfortunately we cannot quantitatively apply the results of Section 1 since the way in which \(\Gamma\) approaches 4/3 (more properly the way in which \(\Gamma\) departs from 5/3) depends in detail on the Equation of state. However, we may derive some feeling for the way in which the instability sets in by assuming the compression has driven the value of \(\Gamma\) down from 5/3 to 3/2 (i.e., just half way to its relativistic value. Substitution of \(\Gamma=3 / 2\) in Equation \ref{4.1.27} and using Fowler’s⁴ values of \(\zeta_1\) and \(\zeta_2\) for a polytrope of index 2 gives a stability limit of

\[\left(\frac{\mathrm{R}_0}{\mathrm{R}_{\mathrm{s}}}\right)>4.3.\label{4.3.5}\]

Thus \(\mathrm{R_o}\) for a neutron star would have to be greater than about 12 km. Since typical model radii are of the order of 10 km, 3/2 is probably a representative value of \(\Gamma\), yet it is still far from the relativistic value of 4/3. This argument further emphasizes the fact that it is the general relativistic instability which places an upper limit on the size of the configuration, not the Equation of state becoming relativistic.

The discussion in section 2 would lead us to believe that neither rotation nor magnetic fields can seriously modify the onset of the general relativistic instability. This can be made somewhat quantitative by evaluating Equation \ref{4.2.10} for a polytrope of index \(\mathrm{n=2}\). However, in order to do this, we must re-evaluate the moment of inertia weighting factor \(a\). A crude estimate here will suffice since we are neglecting an increase of perhaps a factor of 2 due to general relativistic terms. One can show by integrating \(4 \pi \int_0^{\mathrm{R}} \mathrm{r}^2 \rho \mathrm{dr}\) by parts in Emden polytropic variables that

\[\alpha=\left[1+\frac{\frac{6}{\xi_1^2} \int_0^{\xi_1} \xi \theta \mathrm{d} \xi}{\left.\xi_1^2 \frac{\mathrm{d} \theta}{\mathrm{d} \xi}\right|_{\xi_1}}\right],\label{4.3.6}\]

which for \(n = 2\) very approximately gives \(α = 0.345\). Substitution into Equation \ref{4.2.13} then yields

\[\left(\frac{\mathrm{R}_{\mathrm{S}}}{\mathrm{R}_0}\right)<0.234\left(1-\mathscr{H}^2\right)+3.2 \times 10^{-2} \mathrm{w}^2.\label{4.3.7}\]

The effects of magnetic fields and rotation are qualitatively the same for neutron stars as for white dwarfs. However, as \(\mathrm{R_o}\) is only a few times the Schwarzschild radius, the normalizing fields and rotational velocities are truly immense. For a neutron star with a 10 km radius the magnetic field corresponding to \(\mathscr{H}=1\) in of the order of 10¹⁸ gauss while the rotational velocity corresponding to \(\mathrm{w=1}\) would be about 10% the velocity of light. These values vastly exceed those for the most extreme pulsar. Thus, barring modification to the Equation of state resulting from these effects, they can largely be ignored in investigating neutron star stability. This result is exactly in accord with what one might have expected on the basis of the white dwarf analysis.

We began this discussion by indicating that the analysis would be very simplistic and yet we have attained some very useful qualitative results. Since the mass-radius law for any degenerate Equation of state (excepting small technical wiggles) will provide for stars whose radius decreases with increasing mass, we can guarantee that the resulting decrease in \(\Gamma\) will give rise to an unstable configuration at a few Schwarzschild radii. Thus, there will exist an upper limit to the mass allowable for a neutron star. The origin of this limit is conceptually identical to that for white dwarfs. Furthermore, as for white dwarfs, this limit can be modified by the presence of magnetic fields and rotation only for the most extreme values of each. One may argue that in discussing effects of general relativity we have included terms of \(O (1/c^2)\) and that higher order effects may be important. While this is true regarding such items as gravitational radiation, none of these terms should be important unless the configuration becomes smaller than several Schwarzschild radii.

Even then, they are unlikely to affect the qualitative behavior of the results. Very little has been said about the large volume of work relating to the Equation of state for neutron degenerate matter. This is most certainly not to deny its existence, just its relevance. One of the strong points of this approach is that insight can be gained into the global behavior of the object without undue concern regarding the microphysics. This type of analysis is a probe intended to ascertain what effects are important in the construction of a detailed model and what may be safely ignored. It cannot hope to provide the information of a detailed structural model but only point the way toward successful model construction.