Possible physical realizations of the Tolman VII solution

The Tolman VII solution for a static perfect fluid sphere to the Einstein equations is reexamined, and a closed form class of equations of state (EOSs) is deduced for the first time. These EOSs allow further analysis to be carried out, leading to a viable model for compact stars with arbitrary boundary mass density to be obtained. Explicit application of causality conditions places further constraints on the model, and recent observations of masses and radii of neutron stars prove to be within the predictions of the model. The adiabatic index predicted is $\gamma \ge 2$, but self-bound crust solutions are not excluded if we allow for higher polytropic indices in the crustal regions of the star. The solution is also shown to obey known stability criteria often used in modeling such stars. It is argued that this solution provides realistic limits on models of compact stars, maybe even independently of the type of EOS, since most of the EOSs usually considered do show a quadratic density falloff to first order, and this solution is the unique exact solution that has this property.

Figure 1

Variation of density with the radial coordinate inside the star. The parameter values are ${\rho }_c=1\times {10}^{15}\mathrm{g}{\mathrm{cm}}^{-3}$, $r_b=1\times {10}^6\mathrm{cm}$ and $0.6\le \mu \le 1$, taking the various values shown in the legend.

Figure 2

Variation of pressure with the radial coordinate inside the star. The parameter values are ${\rho }_c=1\times {10}^{15}\mathrm{g}{\mathrm{cm}}^{-3}$, $r_b=1\times {10}^6\mathrm{cm}$ and $0.6\le \mu \le 1$, taking the various values shown in the legend.

Figure 3

Variation of speed of sound with the radial coordinate inside the star. The parameter values are ${\rho }_c=1\times {10}^{15}\mathrm{g}{\mathrm{cm}}^{-3}$, $r_b=1\times {10}^6\mathrm{cm}$ and $0.6\le \mu \le 1$, taking the various values shown in the legend.

Figure 4

Variation of bulk modulus with the radial coordinate inside the star. The parameter values are ${\rho }_c=1\times {10}^{15}\mathrm{g}{\mathrm{cm}}^{-3}$, $r_b=1\times {10}^6\mathrm{cm}$ and $0.6\le \mu \le 1$ taking, the various values shown in the legend.

Figure 5

Variation of $Y$ metric variable with the radial coordinate inside the star. The parameter values are ${\rho }_c=1\times {10}^{15}\mathrm{g}{\mathrm{cm}}^{-3}$, $r_b=1\times {10}^6\mathrm{cm}$ and $0.6\le \mu \le 1$, taking the various values shown in the legend.

Figure 6

Variation of $Z$ metric variable with the radial coordinate inside the star. The parameter values are ${\rho }_c=1\times {10}^{15}\mathrm{g}{\mathrm{cm}}^{-3}$, $r_b=1\times {10}^6\mathrm{m}$ and $0.6\le \mu \le 1$, taking the various values shown in the legend.

Figure 7

Variation of $\lambda$ metric variable with the radial coordinate inside the star. The parameter values are ${\rho }_c=1\times {10}^{15}\mathrm{g}{\mathrm{m}}^{-3}$, $r_b=1\times {10}^6\mathrm{cm}$ and $0.6\le \mu \le 1$, taking the various values shown in the legend.

Figure 8

Variation of $\nu$ metric variable with the radial coordinate inside the star. The parameter values are ${\rho }_c=1\times {10}^{15}\mathrm{g}{\mathrm{cm}}^{-3}$, $r_b=1\times {10}^6\mathrm{m}$ and $0.6\le \mu \le 1$, taking the various values shown in the legend.

Figure 9

The redshift $z_s$ at the surface of the sphere for different values of $\mu$.

Figure 10

Log-log plot of pressure versus density for neutron star models determined by different $\mu$, but the same ${\rho }_c$ and $r_b$. The densities and pressures are in cgs units, and the $\mathbf{\Pi }$ is fixed by the following: $r_b={10}^6\mathrm{cm}$, ${\rho }_c={10}^{15}\mathrm{g}{\mathrm{cm}}^{-3}$. Since pressure is a decreasing function of distance from the center, large densities indicate points closer to the center of the star.

Figure 11

The adiabatic index variation from the center to the boundary of the star for different values of the parameter $\mu$. The other parameter values are the same as those in Fig. 10.

Figure 12

The effect of changing the central density ${\rho }_c$ through 1 order of magnitude on the EOS with a boundary radius kept fixed at 10 km, for the natural EOS with $\mu =1$. We notice that the slopes of the lines change by very little. The legend also provides the corresponding masses associated with each parameter choice in solar mass units.

Figure 13

The effect of changing the boundary radius $r_b$ through 1 order of magnitude, while the central density is kept fixed at $1.5\times {10}^{15}\mathrm{g}{\mathrm{cm}}^{-3}$. We notice that the lines remain mostly parallel, suggesting that only the $k$ value in the polytrope is changing, while the adiabatic index is remaining about the same. The masses of the corresponding stars are also provided in solar units in the legend.

Figure 14

The mass of possible stars just obeying causality. The grey surface obeys the equation $v_s(r=0)=\mathrm{c}$. Every point below the surface is a possible realization of a star, and we can potentially read off the mass, central density, and $\mu$ value of that star. The numbered lines represent stars with the same mass that are causal, i.e. they are projections of the causal surface onto the ${\rho }_c-\mu$ plane. Glendenning’s [22] curve is shown in orange and represents a limit in the natural case only, and according to our results is acausal, being above our surface. The $\mu =1$ plane’s intersection with our graph is the graph given in Ref. [20], and here too our prediction is more restrictive.

Figure 15

The compactness as a function of the self-boundedness parameter $\mu$. This plot was generated by varying $r_b$ from 4 km to 20 km for fixed $\mu$ and finding ${\rho }_c$ and subsequently the compactness each time, such that the sound speed was causal at the center of the star. The curve shown is a polynomial fit, and the box-and-whisker plots (very small in green) show the variation of $\beta$ for fixed $\mu$, but different $r_b$. The very small whiskers justify the pertinence of $\beta$ as a useful measure in the analysis of the behavior of the model.

Figure 16

The mass $M$ in solar units versus radius $r_b$ in kilometers of a few stars for which these values have been measured. We use error bars to denote observational uncertainties, and colored bands in the case where only the mass is known. The lines we show and the ones that are at the causality limit in the Tolman VII model, and the whole area below those lines can be causal parameter choices for stars.