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6.3 Crust in rotating neutron stars

We consider stationary neutron-star rotation and assume, for the time being, perfect axial symmetry. Consequently, the spacetime metric is axially symmetric too. Using coordinates t, r, θ, and φ, we can write the spacetime metric in the form (we use the notation of Haensel, Potekhin and Yakovlev [184Jump To The Next Citation Point])
ds2 = c2dt2e2Φ − e2λr2sin2θ (dφ − ωdt )2 − e2α(dr2 + r2dθ2 ), (106 )
where the metric functions Φ, λ, ω, and α depend on r and θ. The metric function ω(r,θ) has the asymptotic behavior ω(r −→ ∞ ) = 0. It is usually referred to as the “angular velocity of the local inertial frames”. Einstein’s equations for a stationary rigid rotation of perfect fluid stars can be reduced to a set of 2-D coupled partial differential equations [53389]. Their numerical solution can now be obtained using publicly available domain codes, for example, the code rotstar from the LORENE library [270] and the code RNS [388].
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Figure 40: Cross section in the plane passing through the rotation axis of a neutron star of M = 1.82 M ⊙, rotating at 1200 Hz. The SLy EoS for crust and core [125Jump To The Next Citation Point] is used. The coordinates are defined by: x = rsinθ cosφ and z = rcos θ, while r, θ, and φ are metric coordinates, Equation (106View Equation). The contours are lines of constant density. Inner contour: crust-core interface. Intermediate contour: outer-inner crust interface. Outer contour: stellar surface. Figure made by J.L. Zdunik.
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Figure 41: Nonmagnetized and magnetized pure 56Fe crust in a neutron star with M = 1.4 M ⊙ and R = 10 km and at T = 0. Matter density (left vertical scale) and the mass of the outer shell ΔM (right vertical axis) versus depth below the surface z, for B = 0, B = 1012 G, and B = 1013 G. Arrows indicate densities (kinks) at which the n = 1 Landau level starts to be populated with increasing depth. Figure 6.15 from [184Jump To The Next Citation Point].

As can be seen on Figure 40View Image, the outer crust is the most strongly deformed by the centrifugal force. The spin frequency of 1200 Hz, significantly larger than the highest detected up to now, 716 Hz, was chosen to make the effect more spectacular. Let us nevertheless point out that some evidence suggesting the existence of a rapidly-rotating neutron star has recently been found in XTE J1739–285 [227Jump To The Next Citation Point] with a spin frequency of about 1122 Hz. Let us consider, for example, the difference between the equatorial (maximal) and polar (minimal) thickness of the crust, ΔReq − pol = ΔReq − ΔRpol. For a Ω, which is not too close to the mass-shedding limit, ΔReq − pol ∝ Ω2. Therefore, ΔReq −pol|716 Hz ≈ 0.4ΔReq − pol|1200 Hz.

The crustal baryon mass (not to be confused with the gravitational mass) M (Ω ) b,cr of a neutron star rotating at angular frequency Ω, is larger than the crustal baryon mass Mb,cr(0 ) of the static star (with the same total baryon mass). The baryon mass (also called the rest mass) of a star is equal to the number A of baryons it contains times an assumed baryon mass mb. One may take mb = mn or mb = mu. We take Mb,cr = Amu. For Ω not too close to Ω ms, we have M (Ω ) − M (0) ∝ Ω2 b,cr b,cr. Due to the radiation of electromagnetic waves and particles, a pulsar spins down, so that ˙ Ω < 0. Consequently, the baryon mass of the pulsar crust decreases in time, M˙b,cr ∝ ˙Ω Ω < 0. Nucleons pass from the crust to the liquid core, releasing some heat. As shown in Figure 40View Image, the crust is decompressed near the equator and compressed near the pole. These deformations trigger various nuclear reactions involving electrons, neutrons, and nuclei. These reactions tend to drive the deformed crust towards its equilibrium shape and release heat, which influences the cooling of a spinning down pulsar [205]. Additional heating results from crust cracking when local shear strain exceeds the maximal one.


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