Neutron star crustal properties from relativistic mean-field models and bulk parameters effects

Abstract

We calculate crustal properties of neutron stars, namely, mass ($M_\mathrm{crust}$), radius ($R_\mathrm{crust}$) and fraction of moment of inertia ($\Delta I/I$) from parametrizations of hadronic relativistic mean-field (RMF) model consistent with symmetric and asymmetric nuclear matter constraints, as well as some stellar boundaries. We verify which one are also in agreement with restrictions of $\Delta I/I \geqslant 1.4\%$ and $\Delta I/I \geqslant 7\%$ related to the glitching mechanism observed in pulsars, such as the Vela one. The latter constraint explains the glitches phenomenon when entrainment effects are taken into account. Our findings indicate that these parametrizations pass in the glitching limit for a neutron star mass range of $M\leqslant 1.82M_\odot$ ($\Delta I/I \geqslant 1.4\%$), and $M\leqslant 1.16M_\odot$ ($\Delta I/I \geqslant 7\%$). We also investigate the influence of nuclear matter bulk parameters on crustal properties and find that symmetry energy is the quantity that produces the higher variations on $M_\mathrm{crust}$, $R_\mathrm{crust}$, and $\Delta I/I$. Based on the results, we construct a particular RMF parametrization able to satisfy $\Delta I/I \geqslant 7\%$ even at $M=1.4M_\odot$, the mass value used to fit data from the softer component of the Vela pulsar X-ray spectrum. The model also presents compatibility with observational data from PSR J1614−2230, PSR J0348 + 0432, and MSP J0740 + 6620 pulsars, as well as, with data from the Neutron Star Interior Composition Explorer (NICER) mission.

Appendix

Here we derive the mainly equations of the approach used for the crustal star properties, based on the core equation of state and macroscopic properties of the core-crust interface [66]. This formalism is based on the assumption that the mass of the crust, $M_\mathrm{crust}$, is much smaller than the total mass of the star, M. We start by constructing the Tolman–Oppenheimer–Volkoff equations.

In order to describe a spherically symmetric object, we use the metric given by

$$\begin{aligned} ds^2 = e^{\varphi (r)} \, c^2 \, dt^2 - e^{\xi (r)} dr^2 - r^2 \left( d\theta ^2 + \sin ^2\theta d\phi ^2\right) . \end{aligned}$$

(18)

We consider the stress-energy tensor for a perfect fluid as $T_{\mu \nu } = \left( P/c^2+\rho \right) u_{\mu }\, u_{\nu } - g_{\mu \nu } \, P/c^2$, where P is the pressure and $\mathcal {E}$ is energy density. The quadrivelocity $u_{\mu }$ fulfills $u_{\mu } \, u^{\mu }=1$ and $\, u^{\mu } \nabla _{\nu }u_{\mu } =0$. By defining

$$\begin{aligned} e^{-\xi } = \left[ 1 - \frac{2 G m(r)}{r c^2}\right] , \end{aligned}$$

(19)

where $m=m(r)$ represents the mass inside a sphere of radius r, we obtain the system of equations that governs the hydrostatic equilibrium of the compact star, namely,

$$\begin{aligned} m'(r)&= 4 \pi \mathcal {E}(r) \, r^2, \end{aligned}$$

(20)

$$\begin{aligned} P'(r)&= \nonumber \\&- \frac{\left[ P(r)/c^2+ \mathcal {E}(r) \right] \left[ \frac{4 \pi G}{c^2}\, r \, P(r) + \frac{G \, m(r)}{r^2}\right] }{\left[ 1 - \frac{2 \, G\, m(r)}{r c^2}\right] } \, . \end{aligned}$$

(21)

In the limit $M_\mathrm{crust} \ll M$, we have $4\pi r^3 P(r)/[m(r)c^2] \ll 1$. Therefore, for the crust we can write Eq. (21) as

$$\begin{aligned} \frac{P'(r)}{P(r) + \mathcal {E}(r)\, c^2} = - \frac{G \, m(r)}{c^2 \, r^2} \left[ 1 - \frac{2 \, G\, m(r)}{r c^2}\right] ^{-1} \, . \end{aligned}$$

(22)

We can estimate the contribution of the crust mass by considering an approximation where $P(r)/c^2$ is much smaller than $\mathcal {E}(r)$. In the crust region, for $M_\mathrm{crust} \ll M$ and using $dm = \mathcal {E}(r) dv = \mathcal {E}(r) 4\pi r^2 dr$, we obtain

$$\begin{aligned} \frac{dP(r)}{dm} = - \frac{G \, M}{4\pi \, r^4} \left[ 1 - \frac{2 \, G\, M}{r c^2}\right] ^{-1} \, . \end{aligned}$$

(23)

Integrating Eq. (23) from the outer crust up to the core-crust interface, we have

$$\begin{aligned} M_\mathrm{crust} = \frac{4\pi P_t R^4_\mathrm{core}}{GM_\mathrm{core}}\left( 1-\frac{2GM_\mathrm{core}}{R_\mathrm{core}c^2}\right) \, . \end{aligned}$$

(24)

On the other hand, one can use the thermodynamic relation $\mu =d\mathcal {E}/d\rho$ and $\mu = \left( P(r) + \mathcal {E}(r)\, c^2\right) /\rho$ to write

$$\begin{aligned} \frac{d\mu }{\mu } = \frac{dP}{P(r) + \mathcal {E}(r)\, c^2} \, , \end{aligned}$$

(25)

which is precisely the left hand side of Eq. (22). Combining those expressions and using $M_\mathrm{core} \ll M$, one obtains, in the crust region, the following:

$$\begin{aligned} \frac{d\mu }{\mu } = - dr \, \frac{G \, M}{c^2 \, r^2} \left[ 1 - \frac{2 \, G\, M}{r c^2}\right] ^{-1} \, . \end{aligned}$$

(26)

Integrating from the outer crust up to the core-crust transition, we have

$$\begin{aligned} \int _{\mu _0}^{\mu _t} \frac{d\mu }{\mu }= & {} - \int _{R_\mathrm{core}}^{R} \frac{G \, M}{c^2 \, r^2} \left[ 1 - \frac{2 \, G\, M}{r c^2}\right] ^{-1} \, dr \, , \end{aligned}$$

(27)

$$\begin{aligned} \left( \frac{\mu _t}{\mu _0}\right) ^2= & {} \dfrac{1-\dfrac{2 \, G\, M}{R\, c^2}}{1- \dfrac{2 \, G\, M}{R_\mathrm{core}\, c^2}} \, , \end{aligned}$$

(28)

where $\mu _t$ is the baryon chemical potential at the core-crust transition and $\mu _0$ is the chemical potential at the surface of the neutron star.

Using Eq.(28), one can obtain

$$\begin{aligned} R = \frac{2 \, G \, M \, R_\mathrm{core}}{c^2 \, R_\mathrm{core} + 2 \, G \, M \, \left( \dfrac{\mu _t}{\mu _0}\right) ^2 - c^2 \, R_\mathrm{core} \left( \dfrac{\mu _t}{\mu _0}\right) ^2}\, . \end{aligned}$$

(29)

Finally, the crustal radius is [66]

$$\begin{aligned}&R_\mathrm{crust} = R - R_\mathrm{core} \nonumber \\&= \frac{2 \, G \, M \, R_\mathrm{core}}{c^2 \, R_\mathrm{core} + 2 \, G \, M \, \left( \dfrac{\mu _t}{\mu _0}\right) ^2 - c^2 \, R_{core} \left( \dfrac{\mu _t}{\mu _0}\right) ^2} - R_{core} \nonumber \\&= \phi R_\mathrm{core} \left[ \frac{1-R_\mathrm{s}/R_\mathrm{core}}{1-\phi \left( 1-R_\mathrm{s}/R_\mathrm{core}\right) }\right] , \end{aligned}$$

(30)

with $R_\mathrm{s} = 2GM/c^2$ and $\phi = [(\mu _t/\mu _0)^2 - 1]R_\mathrm{core}/R_\mathrm{s}$.