Spectrum of shear modes in the neutron-star crust: Estimating the nuclear-physics uncertainties

I construct a model of the inner crust of neutron stars using interactions from chiral effective field theory (EFT) in order to calculate its equation of state (EOS), shear properties, and the spectrum of crustal shear modes. I systematically study uncertainties associated with the nuclear physics input, the crust composition, and neutron entrainment, and estimate their impact on crustal shear properties and the shear-mode spectrum. I find that the uncertainties originate mainly in two sources: The neutron-matter EOS and neutron entrainment. I compare the spectrum of crustal shear modes to observed frequencies of quasiperiodic oscillations in the afterglow of giant $\gamma$-ray bursts and find that all of these frequencies could be described within uncertainties, which are, however, at present too sizable to infer neutron-star properties from observations.

Figure 1

Proton chemical potential as a function of the neutron chemical potential. The different bands show the uncertainties due to different sources: the variation of the surface parameters (red band), of $Z$ (green band), and of the neutron-matter EOS (blue band). As one can see, the uncertainties are dominated by the blue band (which overlaps the red and green bands almost always completely). I also show the results obtained by using a simpler parametrization for the nuclear interaction (dashed bands); see Sec. 2b.

Figure 2

Electron chemical potential as a function of the neutron chemical potential, where the bands are obtained as in Fig. 1.

Figure 3

Radii of the nucleus $R_0$ (dotted, lower bands) and of the Wigner-Seitz cell $R_W$ (filled, upper bands) as a function of baryon density. The bands are determined as in Fig. 1. I also show results by Douchin and Haensel [62] (orange points) and $R_W$ from the calculation by Chamel and Haensel [7] (grey points).

Figure 4

Nucleon numbers of the nucleus, $A_0$, (dotted, lower bands) and of the Wigner-Seitz cell, $A_W$, (filled, upper bands) as a function of baryon density. The bands are determined as in Fig. 1. I also show results by Douchin and Haensel [62] (orange points) and $A_W$ from the calculation by Chamel and Haensel [7] (grey points).

Figure 5

Inner-crust EOS (pressure as a function of baryon density) for the inner-crust model of this work. The uncertainty band for $P\left(n_B\right)$ solely originates in the uncertainty of the neutron-matter EOS. I also show the BPS EOS [25] and the EOS by Douchin and Haensel [62].

Figure 6

Shear modulus as a function of mass density, where the uncertainty bands are obtained as in Fig. 1.

Figure 7

Shear velocities as a function of density. In addition to the uncertainty bands from Fig. 1, I show the uncertainty in the entrainment parameter $R_e$ (light-red band) as well as the combined total uncertainty (grey band). I compare my results with the velocity band from Ref. [24] (SW09), which used various Skyrme interactions as input.

Figure 8

Frequencies of the fundamental ($n=0,l=2$) shear mode for the shear velocities of Fig. 7. All bands include a variation of the total neutron-star mass from $M=1.00–1.97M_{\odot }$ as described in the text. In addition, I show the frequency range when varying the neutron-star radius according to the uncertainties from Ref. [70] (light-blue band) and the frequencies when correcting $R_e$ by the suggested factor by Kobyakov and Pethick [68] (KP13, brown band). I compare with the results of Ref. [24] (SW09) and with observed QPO frequencies from SGR 1806-20 (black lines) and SGR 1900 + 14 (purple lines).

Figure 9

Lower panel: Frequencies of the $n=0,l=2$ fundamental shear mode for $R_e$ from Ref. [20] and various neutron-star masses (blue band). I also show the uncertainties from radius variation (light-blue band), and compare to the lowest observed QPO frequencies from SGR 1806-20 (black lines) and SGR 1900 + 14 (purple line). Upper panel: The same for the $n=1$ first radial overtone compared to the observed 626 Hz QPO.

Figure 10

Frequencies of the $n=0$ modes for $2\le l\le 16$ for $R_e$ from Ref. [20]. For each $l$, I show the frequencies obtained for a $1.00M_{\odot }$ neutron star (left boundary) up to a $1.97M_{\odot }$ neutron star (right boundary). I compare with observed QPO frequencies from SGR 1806-20 (black lines) and SGR 1900 + 14 (purple lines).