Why Stellar Sequences Turn Over: Fixed Points, Instability, and Equation-of-State Universality

Abstract

We reformulate the stellar structure equations in the language of dynamical systems and show that the maximum mass of stellar sequences arises from the existence of a fixed point in the relativistic regime. In an appropriate representation of the Tolman-Oppenheimer-Volkoff equations, this fixed point becomes manifest and is directly associated with the turnover of the mass-radius curve. The existence of a fixed point implies an effective reduction in dimensionality near the onset of instability, which provides a simple explanation for several equation-of-state-insensitive relations and predicts new ones. In the weakly relativistic limit, we identify a complementary universal structure shared by stellar sequences at their maximum mass, which we term the "compressible limit," and derive distinct universal relations governing the maximum mass in the Newtonian and post-Newtonian regimes. Combining these theoretical results with current astrophysical constraints, we show that the J0740+6620 pulsar is unlikely to lie near the Tolman-Oppenheimer-Volkoff maximum mass unless the equation of state exhibits a strong first-order phase transition at densities just above its central density.

Figures (10)

• Figure 1: Spiraling behavior of the $M$--$R$ curve. Every EoS at extremely high densities has similar behavior, where it approaches a particular fixed point in $M$--$R$ space. In this paper, we discuss how this picture arises from the treatment of the stellar structure equations as a dynamical system, and how such a treatment explains certain quasi-universality. The SFHo Steiner:2012rk and BSk22 Pearson:2018tkr EoSs are given a constant speed of sound $c_s^2=1$ above the densities where they become acausal. $\text{SFHo}:2:0.33$ transitions from SFHo to a $c_s^2 = .33$ constant speed of sound EoS at twice nuclear saturation density. Gray dashed lines mark configurations of constant compactness. Among other pathologies, each EoS displays regions of decreasing compactness with respect to increasing central density.
• Figure 2: Partial solutions to the TOV equation (orange lines), integrated up to a maximum $\ln h$ or radius (orange circle), together with vector flows that represent the evolution of the TOV solution (i.e. the right-hand side of Eqs. \ref{['eq:tov-reformulated-etilde']} and \ref{['eq:tov-reformulated-v']}). The inset on the top, right corner of each panel indicates the maximum radius to which each partial integration goes relative to the size of the star (shaded in gray). Two dashed lines demarcate the regions where $\tilde{e}$ and $v$ change behavior (they change from increasing to decreasing or vice-versa). Top, left: when $\ln h$ is very close to $\ln h_c$, the functions $\tilde{e}(\ln h)$ and $v(\ln h)$ are nearly universal, and they are determined by the near-core expansion of Eqs. \ref{['eq:bc-u']} and \ref{['eq:bc-v']}. Top, right: the trajectory begins to subtly deflect from the near-core ($v = \tilde{e}/3$) expansion. Bottom, left: the solution is near the maximum value of $\tilde{e}=4\pi r^2 \, e$. This is the region of the star where most of the mass is accumulated. The system is complicated, depending sensitively on $c_s^2$ and $w$, which determine the structure of the slope field when both $\tilde{e}$ and $v$ are not small compared to unity. Bottom, right: the solution is very near the surface of the star. The equation for $\tilde{e}$ is nearly homogeneous and independent of $v$. The variable $v$ changes very little relative to $\tilde{e}$, since the right-hand side of the $\tilde{e}$ equation contains $1/c_s^2$, which is very large near the surface.
• Figure 3: Partial solutions to the TOV equations and flow vectors (with a $\Gamma=2$ polytropic EoS for illustrative purposes). This figure is qualitatively similar to Fig. \ref{['fig:demo-near-core']}), but carried out for many different central densities (different colored lines) and for snapshots that are integrated to different enthalpies (different final energy densities). Observe that the earliest snapshot (top, left panel) presents near-core universal behavior, with all solutions overlapping each other. As one integrates to lower energy densities (top, right panel), solutions with different central densities fan out, because the system stops being formally autonomous, although all trajectories are still attracted by the (slowly-drifting) fixed point. As one integrates further to even lower energy densities (bottom, left panel), the system becomes purely non-autonomous, and the spiral behavior becomes evident. When one integrates all the way to the surface (bottom, right panel), the spiral structure is maintained because the $\tilde{e}$ evolution is nearly homogeneous. The dashed curve is, to a good approximation, simply scaled horizontally.
• Figure 4: Parametric dependence of the fixed-point estimate for $M_{\max}$ [Eq. \ref{['eq:Mmax-scaling-units']}] as a function of $c_{s,\mathrm{typ}}^{\,2}$ for several representative choices of $w_{\mathrm{typ}}$ and $e_0$. We consider "neutron-star like" configurations with $e_0$ near $\rho_{\rm nuc}c^2$, the mass density of atomic nuclei ($\approx 2.8\times 10^{14}\,\rm{g}/{cm^3}$). Solid lines vary $w_{\rm typ}$ while keeping $e_0=\rho_{\rm nuc} c^2$ fixed, while dash-dot and dotted lines vary $e_0$ while keeping $w_{\rm typ} = 0.7 c_{s, \rm typ}^2$ fixed. These results are qualitatively consistent with, e.g.Rhoades:1974fnKalogera:1996ci. This plot is intended to illustrate the structure of the scaling, not to define a unique prescription for $(w_{\mathrm{typ}},e_0)$.
• Figure 5: Comparison between the true maximum masses $M_{\max}$ (left panel) and the true radii of the maximum mass stars (right panel) of $\sim 500$ RMFT-informed Gaussian-process EoSs relative to the predictions of Eq. \ref{['eq:Mmax-final-units']} and \ref{['eq:Rmax-units']}, evaluated using a single operational prescription for $(c_{s,\mathrm{typ}}^{\,2},w_{\mathrm{typ}},e_0)$ (see text). The lower panels show fractional residuals. Observe the high accuracy of the predictions of the maximum mass, and the deterioration of this accuracy for the radii of those stars.