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The Crusts of Accreting Neutron Stars 0.01 Edward Brown, Michigan State University 1

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The Crusts of Accreting Neutron Stars 0.01 Edward Brown, Michigan State University 1 depth (m) envelope The Crusts of Accreting Neutron Stars 0.01 Edward Brown, Michigan State University 1 0.1 Neutron stars should have a km-thick crust composed of nuclei, electrons, and 1 free neutrons. unstable H/He 5 10 burning/rp-process Accretion pushes matter 10 through this crust and unstable 12C 8 induces nuclear reactions. 10 burning electron Observing the response of 100 captures 1011 the star to these reactions neutron drip allows us to infer the nuclear pasta properties of matter in the 1000 deep crust and core. ≈1014 -3 core A. Piro, Carnegie Obs. ρ (g cm ) SYMMETRY ENERGY AT SUBNUCLEAR DENSITIES AND ... PHYSICAL REVIEW C 75,015801(2007) 2 2 the nucleon part nw mnc nn mpc np [see Eq. (1)] and 0.20 + + the electron part (15). By comparing the resultant six energy inner crust densities, we can determine the equilibrium phase. 0.15 average proton fraction G H 0.10 I D III. EQUILIBRIUM SIZE AND SHAPE OF NUCLEI E proton fraction B A We proceed to show the results for the equilibrium nuclear 0.05 matter configuration obtained for various sets of the EOS F C 0.00 parameters L and K0 as shown in Fig. 1.Theseparametersare 5 6 2 3 4 5 6 2 3 4 5 6 still uncertain because they are little constrained from the mass 0.001 0.01 0.1 n (fm-3) and radius data for stable nuclei [2]. As we shall see, the charge b number of spherical nuclei and the density region containing FIG. 4. (Color online) The average proton fraction as a function bubbles and nonspherical nuclei have a strong correlation of nb,calculatedfortheEOSmodelsA–I. with L. We first focus on spherical nuclei, which constitute an obtained in earlier investigations [1,7,10], will be discussed equilibrium state in the low-density region. We calculate the below in terms of fission instability. charge number of the equilibrium nuclide as a function of n for b The average proton fraction, which is the charge number the EOS models A–I as depicted in Fig. 2. Note that the recent divided by the total nucleon number in the cell, is plotted in GFMC calculations of the energy of neutron matter based on Fig. 4.Weobservethatthedependenceoftheaverageproton the Argonne v8’ potential [15]areclosetothebehaviorof fraction on the EOS models is similar to that of Z.Wealso the model E. Hereafter we will thus call the model E as a find that the average proton fraction basically decreases with typical one. The result is shown in Fig. 3.Fordensitiesbelow baryon density. This is a feature coming from the fact that as 0.01 fm 3,thecalculateddensitydependenceofthecharge − the baryon density increases, the electron chemical potential number∼ Z is almost flat, a feature consistent with the results increases under charge neutrality and then the nuclei become in earlier investigations [1]. More important, the calculated more neutron-rich under weak equilibrium. charge number is larger for the EOS models having smaller L, We next consider the density region where bubbles and and this difference in Z is more remarkable at higher densities. nonspherical nuclei appear in equilibrium, i.e., the density As we shall see later in this section, this property of region of the “pasta” phases. We start with such a density Z is related to the tendency that with increasing L,the region calculated for the EOS models A–I. The results are nuclear density decreases while the density of the neutron plotted in Fig. 5.ExceptforthemodelC,weobtainthe gas increases. Note that Z is, within a liquid-drop model [1], successive first order transitions with increasing density: determined by the size equilibrium condition relating the sphere cylinder slab cylindrical hole spherical Coulomb and surface energies in such a way that Z increases hole →uniform matter.→ A marked→ correlation of the→ upper end with increasing surface tension. Because the Thomas-FermiPasta and crust-core transition of the→ density region with the parameter L can be observed by model adopted here can be mapped onto a compressible liquid- drop model [2], the present results may well be interpreted in Oyamatsu & Iida ’07 terms of the liquid-drop model. In fact we shall estimate the 0.14 G surface tension from the Thomas-Fermi model as a function of H L and discuss how the surface tension depends on the nuclear I ) pasta nuclei density and the neutron sea density. -3 0.12 fission instability We also note that the density at which the phase with proton clustering spherical nuclei ceases to be in the ground state is between 0.10 3 D 0.05 and 0.07 fm− . This result, consistent with the results E A 0.08 F 100 B C G 80 inner crust H 0.06 D nucleon number density (fm proton number Z I 60 E 0.04 40 0 20 40 60 80 100 120 140 160 A proton number 20 B L (MeV) C F FIG. 5. (Color online) The density region containing bubbles and 05 6 2 3 4 5 6 2 3 4 5 6 0.001 0.01 0.1 nonspherical nuclei as a function of L,calculatedfortheEOSmodels -3 nb (fm ) A–I. For comparison, the density corresponding to u 1/8 in the phase with spherical nuclei and the onset density, n(Q=), of proton FIG. 3. (Color online) The charge number of spherical nuclei as clustering in uniform nuclear matter, which will be discussed in afunctionofnb,calculatedfortheEOSmodelsA–I. Sec. IV, are also plotted by circles and crosses, respectively. 015801-5 schematic of Z, A in (outer) crust / (/ ;) ; #(;, /)=B7" B4" B" − B$ start with a simple liquid drop − − " − "/ µ = µ µ enforce beta-equilibrium, F O − Q / charge neutrality: Ye decreases µF B$" − :F + linearly with rising electron ≈ − B B ! " " ! " " chemical potential compute neutron separation ∂# / 4O = = B7 B" : B$: " energy Sn and set it zero to fnd ∂/ − − F − F ! "; density of neutron drip: # $ 4 = µ B .F7 ndrip ≈ 0.001 n0 O → ⇒ F ≈ 7 ≈ ionic lattice. Horowitz et al. (2007) nuclear pasta. Horowitz et al. (2004) µ kT, m c2 e e Z2e2 kT a ionic lattice. Horowitz et al. (2007) nuclear pasta. Horowitz et al. (2004) http://news.nationalgeographic.com µ kT, m c2 e e Z2e2 kT a structure of crust Rcore = 11 km, Mcore = 1.6 Msun, accreted composition outer inner 924 BROWN & BILDSTEN Vol. 496 5. OHMIC DIFFUSION IN THE DEEP OCEAN AND CRUST reactions occurring deep in the crust. As in that section, we assume that temperature is a function of only depth y, as at Present uncertainties in the composition of matter after these depths the accreted matter will have spread around hydrogen/helium burning prohibit a calculation of the sub- the star. We plot in Figure 9 (Q 1.0) the local Ohmic sequent chemical evolution of the ocean for hydrogen-rich \ di†usion time (solid lines) for accretion rates of 0.5, 1.0, and accretion. However, even though the composition is not 5.0 times Eddington. We also show the Ñow time over a well known, we can still use the thermal pro⇡les as estimates scale height, t y/m5 (dashed lines). A few conclusions are of the crust temperatures. This is important to the evolution 4 immediate. First,fl where the ions vibrate classically (T Z ), of the magnetic ⇡eld, as the accretion-induced heating of the # the ratio /t is nearly independent of depth until near crust reduces its conductivity and hastens the Ohmic di†u- q neutron dripdiff ; flmoreover, for accretion rates m5 /m5 [ sion of crust magnetic ⇡elds (Geppert & Urpin 1994). This 0.23(A/2Z)2(T /5 ] 108 K), the di†usion time is alwaysEdd heating also increases the mass of the ocean. These e†ects greater than the time for matter to Ñow through one scale have been considered (Romani 1990; Geppert & Urpin height. Second, impurity scattering is unimportant through- 1994; Pethick & Sahrling 1995; Urpin & Geppert 1995; out the crust for Q [ 1. Because we placed the heat sources Urpin & Geppert 1996; Konar & Bhattacharya 1997) for 904 G. Ushomirsky, C. Cutler and L. Bildsten at a ⇡xed depth y \ y , the thermal gradient changes sign stars accreting at M 10 M yr . w 0 [ ~9 _ ~1 there (see ° 4.3). Electron captures remove pressure support yields There are, however, a few neutron stars accreting globally crust | observables and therefore decrease the pressure scale height, causing the at or near the Eddington rate. There are two X-ray pulsars 3 2 2 abrupt decrease in the Ohmic timescale (solid line, Fig. 9). c Vd ha Ngw ; 8 (LMC X-4 and SMC X-1) and the six bright ““ Z ÏÏ sources Once neutron pressure dominates the equation of state, the G (Sco X-1, GX 5[1, GX 349]2, GX 17]2, GX 340]0, Cyg scale height again increases with depth. In this region, the which, when we write it in terms of Q quiescent(using equation 3)thermal gives emission from transients X-2). The accreted material will have spread over the sur- Ñow timescale is always longer than the di†usion timescale 22 faces of these star for column densities Z1014 g cm~2 (see The Radius of Neutron Stars 9 for . 1=2 2 (from Guillot et al. ’13) m5 [ m5 16 p GQ22V ° 6), so that a spherically symmetrical approach is war- Edd ha : 9 ranted for this calculation. We thus use our solutions for the magnetic field evolution 5 3 dc4 thermal pro⇡le of the deep crust at accretion rates m5 D m5 (from Brown & Bildsten 98) When the angular momentum1 loss by gravitational radiation Edd − 0.01 (see ° 4.3) to estimate the Ohmic di†usion timescales in the balances angular momentum gain by accretion, deep crust of these neutron stars.
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