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. keV 1

3=4 − 1=2 1=2 R 300 Hz −3 F h 3 5 10227 6 10 x a : 1=4 28 22 21 5.1. T he Microphysics in the Crust Â ns 10 erg cm s M1:4 The conductivity in the crust is set by electron-phonon 10 and electron-impurity scattering. In the relaxation-time 10−4 approximation, the conductivity is (Yakovlev & Urpin (Wagoner 1984; Bildsten 1998a). Here we have replaced MÇ and d 1980) with the observed X-ray flux, F GMM_ =4pRd2: The gravi- x−5 tational wave strength for an NS10 accreting at the Eddington limit 1 e2n normalized counts s 227 p \ e , (28) at the Galactic Centre is then ha < 5 10 : Prior accurate l Â me J1 ] x2 knowledge of the position on the10 sky−6 and orbital periods of many of these X-ray binaries will allow for deep searches with the suite Figure 1. The schematic structure of the neutron star, displaying the inner where l is the sum of the electron-phonon (l ) and ph of laser-interferometric gravitational2 wave detectors currently and outer crust lying on the liquid core. The ocean on top of the crust ends electron-impurity (l ) collision frequencies, for which we under construction (LIGO, VIRGO, GEO-600 and TAMA-300; when the material is dense enough to crystallize. The outer crust consists use the expressions fromimp Urpin & Yakovlev (1980), ignition of long X-ray bursts of nuclei in a lattice and a background of degenerate, relativistic electrons, see Brady et al. 1998 and Brady & Creighton 2000). The nearby whereas the inner crust [at densities * 4±6 1011 g cm23 has free (fromsource ScorpiusKuulkers X-1 is ‘01) the obvious0 first target. Its X-ray flux is Â 1 +2c # 2 1@2 χ neutronscrust in addition mountains to the nuclear lattice. B 1 ] , (29a) F < 2 1027 erg cm22 s21 and the spin period is still unknown, l 13e2k T 3.5T x Â ph B C A B D but probably in the range of 300 Hz (van der Klis 1998), giving (from Ushomirsky et al. ‘00) 226 −2 1 3 + Z 1 ha < 1:6 10 : Bradshaw, Fomalont & Geldzahler (1999) n 3 Â B 1 . (29b) recently determined that Sco X-1 is at a distance d 2:8 ^ l 8me e4 Q x imp 0:3 kpc; giving a luminosity close to the Eddington0.5 value for 1 2 5 10 38 21 _ Here Q 4 nI~1 ;j (Zj [ Z1 )2nj parameterizes the impurities, cosmic abundances, 2 10 erg s ; and implying M < Energy (keV) described by charge and number density . We assume 28 21 Â Zj nj 2 10 M( yr and a required quadrupole Q22 < 1:6 that x 1 and neglect anisotropies in the relaxation time Â38 2 5/2 Â ? 10 g cm (300 Hz/Fig.n 3.—s) Thisfor equilibrium figure shows GW the spectra emission. resulting from Run #7, obtained with the model wabs*(nsatmos+pow),forwhichthebest-fit caused by the magnetic ⇡eld. From these conductivities, we 2 statistic is χν /dof (prob.) = 0.98/628 (0.64). then calculate the local Ohmic di†usion time over a scale height, 1.2 Origin ofwill the better crustal explore quadrupole the elongated moment parts of the parameter space, i.e., one of the parameters has the same value space, but it decreases the likelihood that the next step is within all the chains causing all following steps to evolve H2 The crust of anaccepted. NS is a thin A ( smaller<1 km) value layer of ofa crystallineonly produces òrdinary' small excur- in the same plane. However, increasing the number of q \ 4np . (30) (albeit neutron-rich)sions from matter the that previous overlies value the liquid but increases core composed the likelihood walkers also increases the convergence time. diff c2 of neutrons, protonsthat the and next electrons step(see is accepted. Fig. 1). The Effi compositionciency is optimized of at The resulting posterior distributions are then an intermediate value of a. The Stretch-Move algorithm marginalized over nuisance parameters. While necessary We are using the pressure scale height H (eq. [16]) the crust, i.e., the mass number A and the charge Z of the nuclei, Figure 2. A cartoon description of how a transverse temperature gradient can be fine-tuned with only two parameters: a and thein the crustfor the will spectral lead to a fitting, varying these altitude parameters for the electron do not captures. provide The as the characteristic lengthscale. At neutron drip, H/R FIG. 9.ÈOhmic di†usion in the crust as a function of column depth for varies with depth. As an accreted nucleus gets buried under an 2 B number of simultaneous chain. By comparison, the MHdashedphysical line denotes information the unperturbed (e.g., locationα, the of the pile-up e capture parameter). boundary accretion rates m5 /m5 \ 0.5, 1.0, and 5.0. The conductivity includes both increasingly thick layer of more recently accreted material, it 0.01(2Z/A)4@3(o/1011 g cm~3)1@3, so that a plane-parallel Edd algorithm requires N(N +1)/2 tuning variables, wherebetweenThe the layer results of material are presented composed in of Section nuclei (A 4,, whereZ ), a mass the fractionvalues approach is valid throughout the crust. electron-phonon scattering and electron-impurity (Q \ 1.0) scattering. We undergoes aN series is the of number nuclear of reactions,free parameters. including electron quoted correspond to the the median1 value1 (i.e., 50% show the timescale for Ohmic di†usion (solid line) over a scale height and Xn1 of free neutrons, and free electrons, and a layer with nuclei (A2, Z2), captures, neutronThe emissions, validity and of thispycnonuclear MCMC reactionsalgorithm (Sato is assessed quantile) of each parameter. For highly skewed param- the timescale for the crust to be pushed through a scale height, y/m5 (dashed Xn2 fraction of free neutrons, and free electrons. When a lateral 1979; Blaes etby al. performed 1990; Haensel a test & run Zdunik with 1990b). a single The source crust (U24 in eter posterior distributions, the median value can be 5.2. Ohmic Di†usion T imes in the Crust line). The two timescales become comparable, above neutron drip, when NGC 6397, with fixed distance) and comparing the re-temperaturevery gradient different7T fromis introduced, the most the probable capture boundary value. The shifts error to a M0 B 3 ] 10~9 M yr~1. Below neutron drip, the Ñow timescale is always therefore consists of layers of different nuclear composition, as new location shown by the solid line. Note that this cartoon presumes that For the temperatures in the crust, we used the estimated _ sulting MNS–RNS contours with those obtained from a regions (90%-confidence) of each quoted parameters larger than the di†usion timescale for sub-Eddington accretion rates. The indicated schematicallysimple grid-search in Fig. 1. method (steppar in XSPEC ). Specif-the crustare is infinitely also calculated rigid and with therefore quantiles. does not adjust In some elastically cases, to the the pro⇡les from ° 4.3 for the case of nonequilibrium nuclear density as a function of column depth is approximately given by eq. (17). Essentially allically, the pressure the obtained in the outerMNS– crust,RNS andcontours an appreciable as well as othershift Dznormalizedd of the capture probability boundary. of a parameter posterior distri- fraction of theposterior pressure distributions in the inner match crust, those is supplied obtained by from a bution does not converge to zero within the parameter’s XSPEC XSPEC degenerate relativisticsteppar electrons.grid-search The in pressure must. The be additioncontinuous of Gaus-largerhard radius) limits than in the colder. This regions. is indicated This effect by is a ’ illustratedp’ in the sian Bayesian priors on the distance is also tested with tables listing the parameters. across the boundaries of the compositional layers. However, schematically in Fig. 2, which shows a patch of the crust near a U24, which results in MNS–RNS contours broadened in electron-capturethe reactionsR direction. reduce This the number is because of theelectrons normalization per ofcapture layer3.7. whereDistances nucleus to the(A1, GlobularZ1) is transformed Clusters and into their (A2, Z2). nucleon, and hencethe thermal require∞ spectrum density jumps is approximately between the∝ composi-(R /d)2. If there is no lateral temperatureUncertainties gradient, then the capture layer is ∞ tional layers depictedFor the in Fig. Stretch-Move 1. In the outer algorithm, crust, thesethe minimum density num-sphericallyWhile symmetric, most GCs as indicated have distances by the dashed estimated line. from The lateral pho- jumps are as largeber of as simultaneous<10 per cent, chains while is in equal the inner to N crust + 1, where the N istemperaturetometry gradient – using (shown RR Lyrae by the variable arrow) stars induces (Marconi some et extra al. the number of free parameters. However, increasing the density contrast is smaller, &1 per cent. captures2003; on the Bono left (hotter)et al. 2007), side, and horizontal reverses some branch captures stars on number of simultaneous chains ensures a more complete Bildsten & Cumming (1998) showed that at the typical crustal the right(Valenti (cooler) et al.side. 2007; If the Gratton lateral et temperature al. 2010), or contrast the carbon- is dT, coverage of the parameter space, when comparing the re- oxygen white-dwarf (CO-WD) sequence (Hansen et al. temperatures ofsults accreting of the neutron Stretch-Move stars, * algorithm2 108 K; tothe contours electron- obtainedthen e2 captures on the hot side of the star happen at the Fermi Â 2007) – these methods suffer from systematic uncertain- capture rates arewith sensitivesteppar to. the In local addition, temperature it reduces (see the discussion chances of hav-energyties that that is roughly are diYffikcultBdT tolower, quantify. and captures In fact, on many the cold recent side ing the N + 1 walkers collapsing to a N – 1 dimensional in Section 2.1). In this case, regions of the crust that are hotter proceedphotometric at the Fermi studies energy of GCs about doY notkBd quoteT higher the than amount in the of undergo electron-capture transitions at a lower density (and thus unperturbed case (Bildsten 1998a). Typical values of Y are 10±20

q 2000 RAS, MNRAS 319, 902±932 transients

fg. from Cackett et al. ‘06 2001: RXTE discovers quasi-persistent transients

2002: Rutledge et al. suggest looking for crust thermal relaxation

2002–: cooling detected! (many: Wijnands, Cackett, Degenaar, Fridriksson) 10 Homan et al.

TABLE 7 FITS TO COOLING CURVES WITH AN EXPONENTIAL DECAY TO A CONSTANTa

Source τ (days) A (eV) B (eV) Data References MAXI J0556–332 161±5151±2184.5±1.5 this work (model I) 197±10 137±2174±2thiswork(modelII) IGR J17480–2446 157±62 21.6±484.3±1.4 Degenaar et al. (2013) EXO 0748–676 172±52 18±3114.4±1.2 Degenaar et al. (2014) XTE J1701–462 230±46 35.8±1.4 121.9±1.5 Fridriksson et al. (2011) KS 1731–260 418±70 39.8±2.3 67.7±1.3 Cackett et al. (2010a) MXB 1659–29 465±25 73±254±2Cackettetal.(2008) a ∞ −t/τ kTeff(t)=A × e + B,wheret is time since the end of the outburst in days. below our estimated distance range (e.g., 20 kpc) we find the neutron star in MAXI J0556–332 did not allow us to ex- temperatures (134–218 eV for model I and 131–195 eV for plore values of the neutron-star parameters other than Mns = model II) that are substantially higher than those observed in 1.4M⊙ and Rns = 10 km, as these parameters are advised to XTE J1701–462during its first ∼500 days (125–163 eV). The remain fixed at those values (Zavlin et al. 1996). While other short cooling timescale observedin MAXI J0556–332implies neutron-star atmosphere models allow for changes in Mns and ahighthermalconductivityofthecrust,similartotheother Rns,noneoftheavailablemodelsareabletohandlethehigh cooling neutron stars that have been studied. temperatures observed during the first ∼200 days of quies- Given the similarities between the outbursts of MAXI cence. It is, of course, possible that the properties of the J0556–332 and XTE J1701–462, it is interesting to compare neutron star in MAXI J0556–332 are significantly different these two systems in more detail, as it may help us under- from those in the other cooling neutron-star transients that stand what causes the neutron-star crust in MAXI J0556–332 have been studied. Lower temperatures would be measured to be so hot. MAXI J0556–332 was in outburst for ∼480 if one assumed a lower Mns and/or a larger Rns.Toestimate 38 2 days with a time-averaged luminosity of ∼1.7×10 (d45) the effects of changes in neutron-star parameters we used the erg s−1,whileXTEJ1701–462wasinoutburstfor∼585 days nsatmos model to fit the spectrum of observation 11, initially 38 2 −1 with a time-averaged luminosity ∼2.0×10 (d8.8) erg s assuming Mns = 1.4M⊙ and Rns =10 km. While keeping the (Fridriksson et al. 2010). The total radiated energies of the distance from this fit fixed, and changing Mns to 1.2 M⊙ and MAXI J0556–332 and XTE J1701–462 outbursts are there- Rns to 13 km (values that are still reasonable), the measured 45 2 46 2 fore 7.1×10 (d45) erg and 1.0×10 (d8.8) erg, respec- temperature was reduced by only ∼10%. Such changes are tively. Despite the fact that the radiated energies and time- not large enough to reconcile the temperatures measured in averaged luminosities of the two systems are comparable,observed the MAXI cooling J0556–332 with those of oftransients the other sources. post- initial luminosity of the thermal component (which reflects An alternative explanation for the high inferred tempera- the temperature at shallow depths in the crust at the endoutburst of the tures could be that part of the quiescent thermal emission is outburst) is an order of magnitude higher in MAXI J0556– caused by low-level accretion. Indications for low-level ac- 332 than in XTE J1701–462. This suggests the presence of additional shallow heat sources in the crust of MAXI J0556– 332 and/or that the shallow heat sources in MAXI J0556–332 were more efficient per accreted nucleon. The high observed temperatures are difficult to explain with current crustal heating models. Bringing the initial temperatures down to those seen in XTE J1701–462 requires a distance of ∼10–15 kpc (depending on the assumed model). Such distances are problematic for several reasons. First it implies that Z source behavior in MAXI J0556–332 is observed at much lower luminosities (by factors of 9 or more) than in other Z sources. Second, fits to the quiescent spectra with such a small distance are of poor quality. Finally, a smaller distance does not solve the fact that crustal heatingap- pears to have been much more efficient per accreted nucleon than in other sources. A reduction in distance by a factor of 3 results in a reduction in luminosity and presumably then, by extension, the total mass accreted onto the neutron star and total heat injected into the crust by a factor of 9. Given that we inferred ∼30% less mass accreted onto the neutron star in MAXI J0556–332 during its outburst than in XTE J1701–462 for our preferred distance of ∼45 kpc, this would mean ∼13 FIG.5.—Evolutionoftheeffectivetemperatureofthequiescentneutron star in MAXI J0556–332, based on fits with model II (purpleHoman stars). et Temper- al. ‘14 times less mass accreted onto MAXI J0556–332 than XTE ature data for five other sources are shown as well. The solid lines represent J1701–462 yet similar initial temperatures. the best fits to the data with an exponential decay to a constant. See Table 7 The nsa model that we used to fit the thermal emission from for fit parameters and data references. Infer crust properties from cooling Ushomirsky & Rutledge, Shternin et al., Brown & Cumming, Page & Reddy, Turlione et al., Deibel et al. data from Cackett et al. 2008 fits from Brown & Cumming 2009

● ● 120 MXB 1659-29 obs. ● KS 1731-260 obs.

110 MXB 1659-29 model

● KS 1731-260 model 100 ) eV ( 90

↵ ● 1 e ● kT 80 ● ● ● ● ●● 70 ●

60 ● ● ● 50 102 103 t (d) cooling timescale

3 lg(/g cm ) 9.0 10.0 11.0 12.0 13.0 14.0 1021 )

1 from Brown & Cumming ‘09

K 20

1 10

Thermal diﬀusion cm 1 19 2 = (; ; ) 10 ⟨ −⟨ ⟩ ⟩ total 5 phonon ergs s 18 $ = (, 5) ( 10 U · K impurity

/ 1 $ )

= E[ /A 1 , A 10 N total

B 2 k

( 10 (Z, A) ⇥ 3 e P 10 C n 4 10

1012 1013 1014 1015 1016 1017 1018 1019 2 P/g (g cm ) VOLUME 86, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 16 APRIL 2001

(γ,a) helium production in the SnSbTe cycle broadens this distribution further. 105 106 107 108 Te Te Te Te To summarize, we have shown that the synthesis of heavy nuclei via the rp process is limited to nuclei with 104Sb 105Sb 106Sb 107Sb Z # 54 due to our newly discovered SnSbTe cycle. The existence of a SnSbTe cycle under all rp process condi- (p,γ ) tions is a consequence of the low, experimentally known 103Sn 104Sn 105Sn 106Sn 106,107,108,109 β+ [24] a separation energies of the Te isotopes and is therefore not subject to nuclear physics uncertainties. However, because of the uncertainties in the proton 102In 103In 104In 105In separation energies of the Sb isotopes there is some uncertainty in the relative strength of the SnSbTe subcycles FIG. 2. The reactions in the SnSbTe cycles during an x-ray 106 107 burst. In the case of proton captures the arrows indicate the closed by (g, a) photodisintegration on Te, Te, and direction of the net flow, the difference of the flow via proton 108Te. This will be discussed in a forthcoming paper. capture, and the reverse flow via (g, p) photodisintegration. The A likely consequence of the SnSbTe cycle for accret- line styles are the same as in Fig. 1. ing neutron stars is that the matter entering the crust is composed of nuclei lighter than A ഠ 107. The only way obtain a broad distribution of nuclei in the A ෇ 64 107 to bypass the SnSbTe cycle would be a pulsed rp pro- mass range. This is a result of the long-lived waiting cess, where between pulses matter could decay back to point nuclei along the rp process reaction path which stable nuclei. This could happen during so-called dwarf store some material until the burning is over. The late bursts, which have been suggested to be secondary bursts produced by reignition of the ashes [25]. However, this would require some unburned hydrogen in the burst ashes 6x1038 Luminosity (see discussion below) or extensive vertical mixing [14]. NuclearEnergy 4x1038 Our calculations give a strong indication that the synthesis of nuclei beyond 56Ni and especially into the A ෇ 100 2x1038 mass regionbury in hydrogen the rich bursts ashes leads to extended of en- H, He burning ergy production. This might explain the long duration Luminosity(er g/s) 0 0 100 200 300 (100 sec) bursts seen from, for example, GS 1826-24 [26].

39 10 Schatz et al. ‘01 10-2 38 10 X-ray burst This makes a really -3 37 10 10 dirty crust Nucl.E nergy(er g/s) 0100200300 -1 10 -4 64 10 -2 56 Ge 68 2 = (; ; ) 10 Ni 104 Se Sn ⟨ −⟨ ⟩ ⟩∼

-3 Abundance 10 10-5 72 -4 Kr Abundance 10 -6 -5 10 10 0 20 40 60 80 100 120 0 100 200 300 -2 10 0 10 1 Steady state burning -1 H 10 -3 10 4 -2 He 10 Abundance 10-4 -3 10 0 100 200 300 -5

Time (s) Abundance 10 FIG. 3. Luminosity, nuclear energy generation rate, and the 10-6 abundances of hydrogen, helium, and the important waiting 0 20 40 60 80 100 120 point nuclei as functions of time during an x-ray burst. For comparison, the nuclear energy generation rate is also shown as Mass number a dashed line together with the luminosity, though it is out of FIG. 4. Final abundance distribution as functions of mass num- scale during the peak of the burst. The mass of the accreted ber for an x-ray burst, and for steady-state burning at an accretion 21 layer is 4.9 3 10 g. rate of 40mᠨ Edd.

3473 12C ignition requires high crust temperatures Brown 2004, Cooper & Narayan 2005, Cumming et al. 2006 8 Plot from Cumming et al. 2006 superbursts ignite TABLE 2 at this depth CORE NEUTRINO EMISSION

Label Typea Prefactorb Comment Thermal Instability (erg cm−3 s−1) a fast 1026 fast cooling b slow 3 × 1021 enhanced c slow 1020 mURCA factor of ≈30 in d slow 1019 nn Bremsstrahlung e slow 1017 suppressed ignition mass a 9 6 Fast and slow cooling laws are of the form Qν = Q f (Tc/10 K) and Qν = 9 8 Qs(Tc/10 K) respectively. b Either Qs or Q f for slow or fast cooling, respectively.

FIG. 12.— The effect of crust composition and conductivity on superburst ignition conditions. Temperature proﬁles for superburst ignition models at m˙ = 0.3 m˙ Edd. We show two cases of core neutrino emissivity: slow cooling 19 −3 −1 26 −3 −1 with Qs = 10 erg cm s and fast cooling with Q f = 10 erg cm s . Solid lines are for a composition of 56Fe and a disordered lattice. Short- dashed lines have a heavier composition (A = 106,Z = 46), and dot-dashed lines are for a larger thermal conductivity (Q = 100). The long-dashed line shows the carbon ignition curve for XC = 0.2, and the vertical dotted line indicates a column depth of 1012 g cm−2.

that the neutron and/or proton Tc in the core are of the order of 109 K there is intense neutrino emission from the Cooper pair formation, resulting in an enhanced slow cooling rate 21 −3 −1 which we model by considering Qs ∼ 3 × 10 erg cm s FIG. 11.— The effect of core neutrino emissivity on superburst ignition (see, e.g., Figures 20 and 21 in Page et al. 2004). Finally, we ˙ ˙ 26 −3 −1 conditions at m = 0.3 mEdd. We assume a disordered lattice in the crust, also consider a fast cooling rate with Q f ∼ 10 erg cm s and do not include Cooper pairing. The accreted composition is 20% 12C 56 corresponding, e.g., to the direct Urca process. These mod- (XC = 0.2) and 80% Fe by mass. From top to bottom, the temperature profiles are for increasing core neutrino emissivity; the letters refer to Table els are summarized in Table 2. The core temperature Tc 2. The long-dashed line shows the carbon ignition curve for XC = 0.2, and the can be estimated in each case. For slow cooling, we find vertical dotted line indicates a column depth of 1012 g cm−2. 8 1/8 1/8 ˙ ˙ 1/8 Tc ≈ 4.9 × 10 K ( fin /Qs,20) m/mEdd and fast cooling 7 1/6 1/6 ˙ ˙ 1/6 Tc ≈ 5.0 × 10 K ( fin /Q f,26)!m/mEdd" where fin is the fraction of heat released in the crust that is conducted into the and a larger maximum temperature, but the results are simi- core. ! " lar and so we do not show them here. Cooper pair emission For the composition of the crust, we use the composition was not considered by Brown (2004) and Cooper & Narayan calculated by either Haensel & Zdunik (1990) or Haensel & (2005); however we show here that it has a dramatic effect on Zdunik (2003). The difference between these two calcula- the crust temperature profile. tions is the nucleus assumed to be present at low densities, ei- For the core neutrino emissivity, we consider the “fast” ther 56Fe (Haensel & Zdunik 1990), or a heavy nucleus 106Pd 9 6 and “slow” cooling laws Qν = Q f (Tc/10 K) and Qν = (Z = 46) (Haensel & Zdunik 2003), as would be appropriate 9 8 Qs(Tc/10 K) (e.g. Yakovlev & Haensel 2003; Yakovlev & if rp-process hydrogen burning is able to run to its endpoint Pethick 2004, Page et al. 2005). The “standard” slow cool- (Schatz et al. 2001). We calculate results for these two cases 20 −3 −1 ing by modified URCA processes has Qs ∼ 10 erg cm s . to illustrate the variation expected from changes in composi- However, if either the core protons or neutrons are super- tion. For the conductivity, we consider two cases. The first 9 fluid, with very high values of Tc (≫ 10 K), then this pro- is a “disordered” crust, for which we take the conductivity cess is totally suppressed, leading to cooling by nucleon- to be that of a liquid phase, in the second case, we calculate nucleon Bremsstrahlung (involving the non-superfluid com- the contributions from phonons (Baiko & Yakovlev 1996) and ponent). This process is roughly a factor of ten slower than electron-impurity scattering (Itoh & Kohyama 1993), taking modified URCA, and so we take Q ∼ 1019 erg cm−3 s−1 in the impurity parameter Q = 100 (see Itoh & Kohyama 1993 for s 2 this case. If both protons and neutrons are strongly super- a definition of the impurity parameter, written as ⟨(∆Z) ⟩ in fluid in the core, the neutrino emission will be supressed their notation). Note that a crust with Q = 100 is very impure. further. To model this case, we assume that the core neu- However, we do not consider smaller values of the impurity trino emission is suppressed by a further factor of 100, giving parameter because as we will show they would not agree with 17 −3 −1 Qs ∼ 10 erg cm s . However, in the more reasonable case observed X-ray burst properties. crust cooling | surface temperatures after a 12 yr accretion outburst

The following 8 slides were made using the open- source code dStar (https://github.com/nworbde/dStar). code to generate plots is posted at https://github.com/nworbde/dStar In this case, crust takes decades to cool heating from pycnonuclear Ushomirsky & Rutledge ‘01 reactions in the inner crust

/ $ = E[ , set Q = 4 Crust cools in a few years; temperature rise is less pronounced after outburst Very little evolution of surface temperature until cooling front reaches inner crust. Add a heat source, L = 1.7 MeV • dM/dt

Fit to MXB 1659 Brown & Cumming ‘09

140 ● Obs.

numerical ● 120 model

eq. (12) ) eV

( 100 ↵ 1 e kT ●

80 ●

●

60 ● ● ●

10 102 103 t (d) How impure is the crust? Q < 10 Shternin et al. 2007; Brown & Cumming 2009

● Observations ● Observations

Q = 0, Tb,8 = 3.8 Q = 0, Tb,8 = 4.5 120 ● ● 120 Q = 1, Tb,8 = 3.8 Q = 1, Tb,8 = 4.2

Q = 4, Tb,8 = 3.8 Q = 4, Tb,8 = 3.8

Q = 10, Tb,8 = 3.8 Q = 10, Tb,8 = 3.5 ) 100 100 ) eV eV ( ( ↵ ↵ 1 1 e e ● ● kT kT

80 ● ● 80

● ●

60 ● ● 60 ● ● ● ●

102 103 102 103 time since outburst end (d) what have we learned?

Neutron stars have crusts (including the inner part)

These crusts are good conductors of heat

The neutrons must be superfuid cooling timescale

3 lg(/g cm ) 9.0 10.0 11.0 12.0 13.0 14.0 1021 )

1 from Brown & Cumming ‘09

K 20

1 10

Thermal diﬀusion cm 1 19 2 = (; ; ) 10 ⟨ −⟨ ⟩ ⟩ total 5 phonon ergs s 18 $ = (, 5) ( 10 U · K impurity

/ 1 $ )

= E[ /A 1 , A 10 N total

B 2 k

( 10 (Z, A) ⇥ 3 e P 10 C n 4 10

1012 1013 1014 1015 1016 1017 1018 1019 2 P/g (g cm ) what have we learned?

Neutron stars have crusts (including the inner part)

These crusts are good conductors of heat

The neutrons must be superfuid

There appears to be additional heating 10 Homan et al.

TABLE 7 FITS TO COOLING CURVES WITH AN EXPONENTIAL DECAY TO A CONSTANTa

Source τ (days) A (eV) B (eV) Data References MAXI J0556–332 161±5151±2184.5±1.5 this work (model I) 197±10 137±2174±2thiswork(modelII) IGR J17480–2446 157±62 21.6±484.3±1.4 Degenaar et al. (2013) EXO 0748–676 172±52 18±3114.4±1.2 Degenaar et al. (2014) XTE J1701–462 230±46 35.8±1.4 121.9±1.5 Fridriksson et al. (2011) KS 1731–260 418±70 39.8±2.3 67.7±1.3 Cackett et al. (2010a) MXB 1659–29 465±25 73±254±2Cackettetal.(2008) a ∞ −t/τ kTeff(t)=A × e + B,wheret is time since the end of the outburst in days. below our estimated distance range (e.g., 20 kpc) we find the neutron star in MAXI J0556–332 did not allow us to ex- temperatures (134–218 eV for model I and 131–195 eV for plore values of the neutron-star parameters other than Mns = model II) that are substantially higher than those observed in 1.4M⊙ and Rns = 10 km, as these parameters are advised to XTE J1701–462during its first ∼500 days (125–163 eV). The remain fixed at those values (Zavlin et al. 1996). While other short cooling timescale observedin MAXI J0556–332implies neutron-star atmosphere models allow for changes in Mns and ahighthermalconductivityofthecrust,similartotheother Rns,noneoftheavailablemodelsareabletohandlethehigh cooling neutron stars that have been studied. temperatures observed during the first ∼200 days of quies- Given the similarities between the outbursts of MAXI cence. It is, of course, possible that the properties of the J0556–332 and XTE J1701–462, it is interesting to compare neutron star in MAXI J0556–332 are significantly different these two systems in more detail, as it may help us under- from those in the other cooling neutron-star transients that stand what causes the neutron-star crust in MAXI J0556–332 have been studied. Lower temperatures would be measured to be so hot. MAXI J0556–332 was in outburst for ∼480 if one assumed a lower Mns and/or a larger Rns.Toestimate 38 2 days with a time-averaged luminosity of ∼1.7×10 (d45) the effects of changes in neutron-star parameters we used the erg s−1,whileXTEJ1701–462wasinoutburstfor∼585 days nsatmos model to fit the spectrum of observation 11, initially 38 2 −1 with a time-averaged luminosity ∼2.0×10 (d8.8) erg s assuming Mns = 1.4M⊙ and Rns =10 km. While keeping the (Fridriksson et al. 2010). The total radiated energies of the distance from this fit fixed, and changing Mns to 1.2 M⊙ and MAXI J0556–332 and XTE J1701–462 outbursts are there- Rns to 13 km (values that are still reasonable), the measured 45 2 46 2 fore 7.1×10 (d45) erg and 1.0×10 (d8.8) erg, respec- temperature was reduced by only ∼10%. Such changes are tively. Despite the fact that the radiated energies and time- not large enough to reconcile the temperatures measured in averaged luminosities of the two systems are comparable,observed the MAXI cooling J0556–332 with those of oftransients the other sources. post- initial luminosity of the thermal component (which reflects An alternative explanation for the high inferred tempera- the temperature at shallow depths in the crust at the endoutburst of the tures could be that part of the quiescent thermal emission is outburst) is an order of magnitude higher in MAXI J0556– caused by low-level accretion. Indications for low-level ac- 332 than in XTE J1701–462. This suggests the presence of additional shallow heat sources in the crust of MAXI J0556– 332 and/or that the shallow heat sources in MAXI J0556–332 were more efficient per accreted nucleon. The high observed temperatures are difficult to explain with current crustal heating models. Bringing the initial temperatures down to those seen in XTE J1701–462 requires a distance of ∼10–15 kpc (depending on the assumed model). Such distances are problematic for several reasons. First it implies that Z source behavior in MAXI J0556–332 is observed at much lower luminosities (by factors of 9 or more) than in other Z sources. Second, fits to the quiescent spectra with such a small distance are of poor quality. Finally, a smaller distance does not solve the fact that crustal heatingap- pears to have been much more efficient per accreted nucleon than in other sources. A reduction in distance by a factor of 3 results in a reduction in luminosity and presumably then, by extension, the total mass accreted onto the neutron star and total heat injected into the crust by a factor of 9. Given that we inferred ∼30% less mass accreted onto the neutron star in MAXI J0556–332 during its outburst than in XTE J1701–462 for our preferred distance of ∼45 kpc, this would mean ∼13 FIG.5.—Evolutionoftheeffectivetemperatureofthequiescentneutron star in MAXI J0556–332, based on fits with model II (purpleHoman stars). et Temper- al. ‘14 times less mass accreted onto MAXI J0556–332 than XTE ature data for five other sources are shown as well. The solid lines represent J1701–462 yet similar initial temperatures. the best fits to the data with an exponential decay to a constant. See Table 7 The nsa model that we used to fit the thermal emission from for fit parameters and data references. The Astrophysical Journal Letters, 809:L31 (6pp), 2015 August 20 Deibel et al. the distance to the source, accretion rate variations during outburst, and the choice of gravity. In Section 4, we use the MAXI light curve to constrain the presence of Urca cooling pairs. We discuss our results in Section 5, in particular the implications of such a strong heat source for models of the shallow heating mechanism.

2. CRUST THERMAL RELAXATION MODELS OF THE MAXI J0556-332 LIGHT CURVE We solve the thermal evolution of the neutron star crust numerically by evolving the thermal diffusion equation. To provide a check on our results,Shallow we do this with Heating two different in MAXI J0556-332 numerical implementations.Deibel The fietrst al. is ‘15 the open-source code 6 which solves the fully general relativistic heat The Astrophysical Journal Letters, 809:L31 (6pp), 2015dStar August 20 Deibel et al. diffusion equation for the crust using the method of lines, Figure 1. Model fit to the quiescent light curve of MAXI J0556-332 . The solid black curve corresponds to a model with M = 1.5 M, R = 11 km, the distance to the source, accretion rate variationsimplemented during using stiff ODE solvers in the MESA numerical break in light curve gives8 depth of shallow Qshallow = 6.0 MeV, and Tcore = 10 K; the dashed black curve is for the same outburst, and the choice of gravity. In Section 4,library we use the(Paxton et al. 2011, 2013). The second code 7 7 modelheating with Tcore =´ (4–10310K MeV. The per black accreted dotted curves nucleon!) are light curves with a MAXI light curve to constrain the presence of Urcacrustcool cooling , assumes a constant Newtonian gravity and reheating event »170 days into quiescence for Q = 6.0 MeV (upper pairs. We discuss our results in Section 5, in particular the shallow applies a global redshift to the observer frame. This is a good curve) and Qshallow = 3.0 MeV (lower curve). The blue dashed curve is for a fi implications of such a strong heat source for modelsapproximation of the because the crust is thin and is more efficient M = 2.1 M, R = 12 km neutron star t to the observations by changing the shallow heating mechanism. fi shallow heating depth and strength. the data above the light curve are computationally for tting purposes. To perform Markov chain ¥ 14 fi contamination from residual accretion. Note that Tgeff µ+(1 z), which Monte Carlo ts, we have coupled the crustcool calcula- leads to different observed core temperatures for different gravities. 2. CRUST THERMAL RELAXATION MODELStions OF THE to the emcee code (Foreman-Mackey et al. 2013). The MAXI J0556-332 LIGHT CURVE microphysics input in both codes (equation of state, thermal We solve the thermal evolution of the neutronconductivity, star crust superfluid critical temperatures, and neutrino numerically by evolving the thermal diffusion equation.emissivities To ) is similar and follows Brown & Cumming (2009). provide a check on our results, we do this with twoThe different temperature Tb at the top of the computational grid numerical implementations. The first is the open-source(typically code taken at a column depth y = 1010 g cm- 2) is mapped 6 dStar which solves the fully general relativisticto the photosphere heat temperature Teff using a separately computed diffusion equation for the crust using the methodset of of lines, envelopeFigure models 1. Model withfit to a the helium quiescent light top-layer curve of MAXI and J0556-332 iron . The solid black curve corresponds to a model with M = 1.5 M, R = 11 km, implemented using stiff ODE solvers in the MESA numerical 8 bottom-layer (followingQshallow = 6.0 Brown MeV, and etT al.core =200210 K).; the At dashed the tempera- black curve is for the same library (Paxton et al. 2011, 2013). The second code 7 tures observed formodel MAXI, with Tcore the=´310KT –T .relation The black is dotted insensitive curves are light to curves with a crustcool,7 assumes a constant Newtonian gravity and eff b / the helium massreheating in the envelope. event »170 Whereasdays into quiescence Brown$ & for CummingQshallow = 6.0 MeV (upper applies a global redshift to the observer frame. This is a good curve) and Qshallow = 3.0 MeV (lower curve).E The[ blue dashed curve is for a (2009) held T fixed during = accretion to simulatefi the effect of a approximation because the crust is thin and is more efficient b M = 2.1 M, R = 12 km neutron, star t to the observations by changing the computationally for fitting purposes. To perform Markovshallow chain heat source,shallow we heating instead depth include and strength. the heat the source data above directly the light curve are contamination from residual accretion. Note that Tg¥ µ+14 1 z, which fi and allow Tb to evolve as accretion proceeds. The shalloweff heat ( ) Monte Carlo ts, we have coupled the crustcool calcula- leads to different observed core temperatures for different gravities. tions to the emcee code (Foreman-Mackey et al. source2013). The is uniformly distributed in log y centered on a value 13- 2 10- 3 microphysics input in both codes (equation of state,yh =´ thermal6.5 10 g cm (r »´1.2 10 g cm ) and ranging conductivity, superfluid critical temperatures, andfrom neutrinoy 3 to y ´ 3. The strength of shallow heating is h h Figure 2. Solid black curves indicate the evolution of the crust temperature emissivities) is similar and follows Brown & Cummingassumed(2009 to). vary proportionally with the accretion rate. during quiescence for the M = 1.5 M and R = 11 km model, shown in The temperature Tb at the top of the computationalTo fi gridt the cooling light curve, we assume that a MM= 1.5 10- 2 Figure 1. The red dotted curve is the melting line of the crust (G = 175) for the (typically taken at a column depth y = 10 g cm and) is mappedR = 11 km neutron star accreted at the local Eddington crust composition in Haensel & Zdunik (1990), the black dotted curve is the to the photosphere temperature Teff using a separatelyrate computedmm=º´8.8 10421 g cm-- s for 16 months, match- transition from an electron-dominated heat capacity to an ion-dominated heat set of envelope models with a helium top-layer and˙˙ iron Edd e ion ing the duration of the MAXI outburst (Homan et al. 2014), capacity (CV = CV ), and the blue dotted curve is where the local neutrino bottom-layer (following Brown et al. 2002). At the tempera- cooling time is equal to the thermal diffusion time (tt= ). The gray before cooling began. We find that the subsequent cooling of n therm tures observed for MAXI, the Teff–Tb relation is insensitive to dashed curve shows the lattice Debye temperature QD; when T QD electron- the helium mass in the envelope. Whereas Brown &the Cumming crust naturally reproduces the shape of the light curve if we impurity scattering influences the thermal conductivity. include a strong shallow heat source. The solid curve in (2009) held Tb fixed during accretion to simulate the effect of a shallow heat source, we instead include the heat sourceFigure directly1 shows a model with a Qshallow = 6.0 MeV heat source determines the electron-impurity scattering contribution to the and allow Tb to evolve as accretion proceeds. The shallow(the other heat curves will be discussed in Section 3). The 9 thermal conductivity, to be Q = 1. source is uniformly distributed in log y centeredtemperature on a value at the top of the crust reaches Tb 2´ 10 K imp 13- 2 10- 3 A few analytic estimates help outline the location and yh =´6.5 10 g cm (r »´1.2 10 g cm ) andby the ranging end of outburst, as shown in Figure 2. At the high from y 3 to y ´ 3. The strength of shallowtemperatures heating is found in the crust of MAXI the electron thermal strength of shallow heating needed to explain the light curve. h h Figure 2. Solid black curves indicate the evolution of the crust temperature – assumed to vary proportionally with the accretionconductivity rate. is controlledduring quiescence by electron-ion for the M = 1.5 andM electron-phononand R = 11 km model, shownThe in break in the light curve at »10 20 days into quiescence fi To t the cooling light curve, we assume that a MMscattering= 1.5 in theFigure liquid1. andThe red solid dotted phase, curve is respectively. the melting line of It the is crust only(G = 175)occursfor the when the inward propagating cooling front reaches the and R = 11 km neutron star accreted at the localwhen Eddington Q , wherecrust compositionQ is the in Haensellattice & Debye Zdunik temperature,(1990), the black that dotted curveshallow is the heating depth (i.e., the peak of the crust temperature 421-- T D transition fromD an electron-dominated heat capacity to an ion-dominated heat rate mm˙˙=ºÉdd 8.8 10 g cm s for 16 months, match- fl profile). The time for the region of the crust with mass density ρ electron-impuritycapacity scattering(C e = C inion)uences, and the the blue thermaldotted curve conductiv- is where the local neutrino ing the duration of the MAXI outburst (Homan et al. 2014), V V 12 2 ity. In MAXI, thecooling crust time temperature is equal to the is thermal always diffusion well above time (ttn Q= therm). Theto gray cool is its thermal time, trtherm =¢14 CV K dz before cooling began. We find that the subsequent cooling of D ()[(òz ) ] and electron-impuritydashed curve scattering shows the plays lattice Debye no role temperature in theQD thermal; when T QD electron-(Henyey & L’Ecuyer 1969), where C is the specific heat the crust naturally reproduces the shape of the light curve if we impurity scattering influences the thermal conductivity. V include a strong shallow heat source. The solidconductivity. curve in Therefore, we set the impurity parameter, which and K is the thermal conductivity given by 222 Figure 1 shows a model with a Q = 6.0 MeV heat source KnckTE= ()(pn3 e B F )with ne the electron number shallow 6 (the other curves will be discussed in Section https:3).// Thegithub.comdetermines/nworbde/dStar the electron-impurity scattering contributiondensity, to the EF the electron fermi energy, and ν the electron 7 9 thermal conductivity, to be Q = 1. temperature at the top of the crust reaches Tb 2https:´ 10//github.com K /andrewcumming/crustcool imp collision frequency. Brown & Cumming (2009) showed that by the end of outburst, as shown in Figure 2. At the high A few analytic estimates help outline the location and temperatures found in the crust of MAXI the electron thermal strength of shallow heating needed to explain the light2 curve. conductivity is controlled by electron-ion and electron-phonon The break in the light curve at »10–20 days into quiescence scattering in the liquid and solid phase, respectively. It is only occurs when the inward propagating cooling front reaches the shallow heating depth (i.e., the peak of the crust temperature when T QD, where QD is the lattice Debye temperature, that electron-impurity scattering influences the thermal conductiv- profile). The time for the region of the crust with mass density ρ 12 2 to cool is its thermal time, trtherm =¢14 CV K dz ity. In MAXI, the crust temperature is always well above QD ()[(òz ) ] and electron-impurity scattering plays no role in the thermal (Henyey & L’Ecuyer 1969), where CV is the specific heat conductivity. Therefore, we set the impurity parameter, which and K is the thermal conductivity given by 222 KnckTE= ()(pn3 e B F )with ne the electron number 6 https://github.com/nworbde/dStar density, EF the electron fermi energy, and ν the electron 7 https://github.com/andrewcumming/crustcool collision frequency. Brown & Cumming (2009) showed that

2 Questions

Is there residual accretion during quiescence?

What causes the shallow heating?

Why is the inner crust so nearly pure?

How quickly does pasta cool?

What do the inferred core temperatures imply? 12C ignition requires high crust temperatures Brown 2004, Cooper & Narayan 2005, Cumming et al. 2006 8 Plot from Cumming et al. 2006 superbursts ignite TABLE 2 at this depth CORE NEUTRINO EMISSION

that the neutron and/or proton Tc in the core are of the order of 109 K there is intense neutrino emission from the Cooper pair formation, resulting in an enhanced slow cooling rate 21 −3 −1 which we model by considering Qs ∼ 3 × 10 erg cm s FIG. 11.— The effect of core neutrino emissivity on superburst ignition (see, e.g., Figures 20 and 21 in Page et al. 2004). Finally, we ˙ ˙ 26 −3 −1 conditions at m = 0.3 mEdd. We assume a disordered lattice in the crust, also consider a fast cooling rate with Q f ∼ 10 erg cm s and do not include Cooper pairing. The accreted composition is 20% 12C 56 corresponding, e.g., to the direct Urca process. These mod- (XC = 0.2) and 80% Fe by mass. From top to bottom, the temperature profiles are for increasing core neutrino emissivity; the letters refer to Table els are summarized in Table 2. The core temperature Tc 2. The long-dashed line shows the carbon ignition curve for XC = 0.2, and the can be estimated in each case. For slow cooling, we find vertical dotted line indicates a column depth of 1012 g cm−2. 8 1/8 1/8 ˙ ˙ 1/8 Tc ≈ 4.9 × 10 K ( fin /Qs,20) m/mEdd and fast cooling 7 1/6 1/6 ˙ ˙ 1/6 Tc ≈ 5.0 × 10 K ( fin /Q f,26)!m/mEdd" where fin is the fraction of heat released in the crust that is conducted into the and a larger maximum temperature, but the results are simi- core. ! " lar and so we do not show them here. Cooper pair emission For the composition of the crust, we use the composition was not considered by Brown (2004) and Cooper & Narayan calculated by either Haensel & Zdunik (1990) or Haensel & (2005); however we show here that it has a dramatic effect on Zdunik (2003). The difference between these two calcula- the crust temperature profile. tions is the nucleus assumed to be present at low densities, ei- For the core neutrino emissivity, we consider the “fast” ther 56Fe (Haensel & Zdunik 1990), or a heavy nucleus 106Pd 9 6 and “slow” cooling laws Qν = Q f (Tc/10 K) and Qν = (Z = 46) (Haensel & Zdunik 2003), as would be appropriate 9 8 Qs(Tc/10 K) (e.g. Yakovlev & Haensel 2003; Yakovlev & if rp-process hydrogen burning is able to run to its endpoint Pethick 2004, Page et al. 2005). The “standard” slow cool- (Schatz et al. 2001). We calculate results for these two cases 20 −3 −1 ing by modified URCA processes has Qs ∼ 10 erg cm s . to illustrate the variation expected from changes in composi- However, if either the core protons or neutrons are super- tion. For the conductivity, we consider two cases. The first 9 fluid, with very high values of Tc (≫ 10 K), then this pro- is a “disordered” crust, for which we take the conductivity cess is totally suppressed, leading to cooling by nucleon- to be that of a liquid phase, in the second case, we calculate nucleon Bremsstrahlung (involving the non-superfluid com- the contributions from phonons (Baiko & Yakovlev 1996) and ponent). This process is roughly a factor of ten slower than electron-impurity scattering (Itoh & Kohyama 1993), taking modified URCA, and so we take Q ∼ 1019 erg cm−3 s−1 in the impurity parameter Q = 100 (see Itoh & Kohyama 1993 for s 2 this case. If both protons and neutrons are strongly super- a definition of the impurity parameter, written as ⟨(∆Z) ⟩ in fluid in the core, the neutrino emission will be supressed their notation). Note that a crust with Q = 100 is very impure. further. To model this case, we assume that the core neu- However, we do not consider smaller values of the impurity trino emission is suppressed by a further factor of 100, giving parameter because as we will show they would not agree with 17 −3 −1 Qs ∼ 10 erg cm s . However, in the more reasonable case observed X-ray burst properties. Heating by acoustic modes Philippov et al. ’16; Inogamov & Sunyaev ‘10 The Astrophysical Journal, 817:62 (20pp), 2016 January 20 Philippov, Rafikov, & Stone

The Astrophysical Journal, 817:62 (20pp), 2016 January 20 vorticity Philippov, Raﬁkov, & Stone ¶E ++=-F-L·((EP )vv )rr ·.3 ( ) ¶t

Here v, ρ, andP are the two-dimensional velocity, density, and pressure, respectively, Φ is the gravitational potential, E =-+Pv()gr1 2 2 is the full energy of the flow (we use adiabatic index g = 53throughout this work, except for isothermal runs), and Λ is the specific cooling rate (see Section 2.3). In addition, mixing between the freshly accreted and stellar material is explored by following the evolution of passive scalar S advected with the flow. We consider the gravitational acceleration g to be constant throughout the simulation domain, which corresponds to a potential

F=gz.4()

In the absence of stellar rotation gGMR= 2, but we leave g * * as a free parameter in general, to allow the possibility of including the centrifugal acceleration due to a rapidly spinning central object. In the case of the NS, radiation pressure plays a major role in Figure 1. Description of the geometry and coordinate system used in this work. the flow dynamics (Inogamov & Sunyaev 1999). In this paper Our 2D simulation domain is aligned perpendicular to the meridional plane of we do not take into considerationthe radiation pressure, the spherical coordinate system, with ez and ex axes aligned with the local er leaving study of its role for future work. and ef, respectively. A more detailed view of the box in the orthogonal plane with its main constituent parts is shown in the bottom of the figure. The main goal of this work is to identify the mechanism of figures from Philippov et al. ‘16 the angular momentum transport in the SL geometry. For this reason, we do not include any explicit viscosity in our that these effects do not change the physics of the sonic simulations. It is also important for our work that Athena instability and can be neglected in the local approximation. Figure 20. Snapshot of r vz illustrating the structure of acoustic modes in a We note that because of these simplifications the SL does notsimulationexhibits with very cooling low levels time t of numerical» 0.5cg viscosity.at t » This160c makesg. At it this time the Figure 21. Same as Figure 19, but for a simulation with cooling time very well suited for studyingcool nonlinears evolution of thes sonic achieve a steady state in our model. This is because dissipationinstability hasreached saturation and one can clearly see that its morphology tcool » 0.5cgs at t = 400cs g. Note the very well pronounced shock structure in our simulation box should in practice drive the meridionalcorrespondsmodes, to as shown the middle in Dong branch et al. (2011 of) acousticandBelyaev modes et al.(Belyaev & in the SL, shock penetration into the upper atmosphere, and reflection off the motion of accreted gas toward the poles. In a 3D SL this gas fi (2012, 2013a). High levels of numerical diffusion may be a interface between the SL and the atmosphere. Ra kov 2012). fi gets replaced by the lower-latitude fluid carrying higher angular reason why sonic instability has not been identi ed in earlier momentum, which a enforces (quasi-)steady state in the low-resolution studies of the BLs (Armitage 2002; Steinacker system. By explicitly neglecting the meridional flow for the fl& Papaloizou 2002). fl reasons outlined above and not imposing external forcing inthe ow behavior very regular, which is re ected in the small vorticity is generated only at the shocks and at the boundary our simulation box, we lose the ability to properly describe theshock amplitude in Figure 8(c). We call this regime between the hot atmosphere and the SL. Vorticity production is steady-state SL at a given latutude. intermediate and2.3. discuss Treatment it of in Thermodynamics more detail in Section 6.1. fi Nevertheless, since the evolution of the SL is slow less ef cient in the bulk compared to the adiabatic simulation. Finally,In this for papertcool welonger pay special than attention several to scale the effects height of crossing (compared to the dynamical timescale), we are still able to thermodynamics on the wave excitation, propagation, and There is essentially no mixing in the intermediate cooling understand the development and operation of the sonictimesback-reactionH cs the on system the global evolution properties is of similar the flow. to Just the to case with regime(see Figure 11(c)). This is because the shocks into instability in this work (Section 3). Also, in Section 8.3 we tcool =¥reiterate,. As here described we consider inP Sectionto be the gas5, pressure, in this regime neglecting the SL gets provide a qualitative interpretation of the SL evolution seen in which acoustic modes evolve are relatively weak, and contribution from the radiation pressure, which can be our simulations in terms of the global (meridionally dependentheated) by dissipation of acoustic modes in shocks and puffs up. momentum transport is carried mainly via sonic modes, which important in the NS accretion. properties of the SL. We leave the detailed exploration of theThe flow becomes subsonic and rather turbulent owing to the Previous studies of sonic instability (Belyaev et al. 2012) are not good at mixing. Also, no bursts of activity are seen 3D structure of the SL to future work. excitationfocused of primarily the KH on purely instability. dynamical effects This and results employed in a small-scale during the simulation. This is related to the fact that the variationssimple of isothermal temperature EOS. In and this density work we(see go Figures beyond that8(d) and (e) fi 2.2. Numerical Scheme temperature in the layer does not increase suf ciently because for theapproach shock and structure consider) three. different thermodynamical setups of efficient cooling. As a result, the Mach number of the flow Our simulations employ the grid-based Godunov code to get an idea of how thermodynamics affects the existence and Athena (Stone et al. 2008) to solve the equations of fluid operation of acoustic modes. does not go down substantially (see Figure 6), and the dynamics in Cartesian geometry, including the energy transport To provide a baseline, our first setup employs the pure 6.1. Intermediate Cooling Regime 2 secondary KH instability is not excited during our simulation equation. These equations are isothermal EOS used in Belyaev et al. (2012), so that Pc= r s , with c being the sound speed, which is constant across the run, unlike the isothermal and adiabatic cases. For this reason Figure 20s shows a snapshot of r vz in the intermediate ¶r simulation domain. In this case the energy Equation (3) is no turbulence is generated in the intermediate cooling case and += ·(rv )0, ( 1cooling ) regime for tcool » 0.5Hcs at time tc= 160 s g. One ¶t not used. material mixing between the SL and the layer of previously can clearlySecond, see we consider acoustic a no-cooling modes propagatingsetup, in which the both full in the SL accreted matter is very weak. (predominantlyenergy in the simulation below the box ò atmosphere()E+Fr dV )isand conserved. in the In layer of ¶()rv +++F=·(rrvv )P 0, ( 2previously ) this case accreted the thermal energymatter, of the at flzuid< is0 no. longer The constant, structure of the ¶t but increases owing to shock dissipation. velocity perturbation is easily identified as corresponding to the 7. NS BOUNDARY CONDITIONS middle3 branch of the dispersion relation of the acoustic mode (Belyaev & Rafikov 2012), with kz being the same both below We also perform simulations with the NS boundary and above the shear layer. condition mentioned in Section 2.6. In this setup we impose In the intermediate regime the shocks are rather weak a reflective BC at the lower boundary of the simulation box (at (weaker than in the isothermal case) but clearly visible(see z =-1 H), very close to the shear layer, which now extends Figure 8(c)), and the shock pattern is stable. Dissipation at from z=0 to z = 0.6 H. This setup is supposed to mainly shocks is easily balanced by cooling, resulting in only a weak illustrate the effect of a different BC on the operation of the change of the density distribution in the simulation box. acoustic modes and is not intended to closely emulate the Figure 21 shows the snapshot of the baroclynic vorticity conditions near the condensed surface of the NS (as envisaged production taken at the same time as the r vz snapshot in in Inogamov & Sunyaev 1999), which should be located at Figure 20. One can see that in the intermediate cooling regime much greater depth. With this new BC the simulations are run

14 Questions

Is there residual accretion during quiescence?

What causes the shallow heating?

Why is the inner crust so nearly pure?

How quickly does pasta cool?

What do the inferred core temperatures imply? stable nuclei at 6×1012 g cm-3

courtesy A. Deibel 8 Ni (28) Co (27) 6 Fe (26) [MeV] Mn (25) 78 80 A

/ Cr (24) 76 74 4 BE V (23) Ti (22) 72 Sc (21) 70 Ca (20) 68 K (19) 66 Z Ar (18) 64 " Cl (17) 62 S (16) 58 60 P (15) 56 Si (14) 54 Al (13) 52 Mg (12) 50 Na (11) 40 42 44 46 48 Ne (10) 38 F (9) 34 36 O (8) 32 14 16 18 20 22 24 26 28 30 N !

➤ Facility for Rare Isotope Beams at Michigan State University Facility for Rare Isotope Beams at Michigan State University JINA/JINA-CEE experiments across the Cd (48) Ag (47) Pd (46) crust nuclear landscape Rh (45) Ru (44) Tc (43) Mo (42) search for gs-gs pairs Nb (41) Zr (40) Y (39) charge exchange for transition Sr (38) Rb (37) strength Kr (36) s Br (35) se Se (34) as 61 – As (33) l m proposed NSCL V β decay study Ge (32) ta Ga (31) en Zn (30) rim es Cu (29) pe ass Ni (28) ex m Co (27) f or 68 70 Fe (26) r o f Mn (25) e ch rd ea 62 64 66 Cr (24) bo B r 56 60 V (23) RI 58 Ti (22) F 54 Sc (21) Ca (20) 52 K (19) Ar (18) 48 Cl (17) 46 drip50 S (16) 42 44 on mass measurements P (15) utr Si (14) 40 ne completed NSCL TOF Al (13) 38 for Mg (12) ach Na (11) 30 32 34 re36 Ne (10) RIB F (9) 24 26 F O (8) 20 N (7) C (6) 22 fusion28 reactions towards n-rich nuclei B (5) 16 18 Be (4) ANL fusion measurements 6 8 10 12 14 Questions

Is there residual accretion during quiescence?

What causes the shallow heating?

Why is the inner crust so nearly pure?

How quickly does pasta cool?

What do the inferred core temperatures imply? depth (m) envelope Pasta delays late-time cooling 0.01 1 Horowitz et al. ‘15

140 MXB 1659-29 0.1 120

100 (eV) eff

T 1 80 screw defect unstable H/He 5 60 10 burning/rp-process “lasagne” plates 140 40 MXB 1659-29 10 10 100 1000 10000 Time (days) unstable 12C 8 120 10 burning Figure 1: (left) MD simulation of nuclear pasta showing ﬂat plates (lasagne) of nuclear matter connected electron by defects that reduce the conductivity.100 The simulation used 409,600 nucleons in a volume 200 fm on a 100 captures side. (right) Surface temperature of MXB 1659-29 versus the time since accretion stopped. Dashed curve1011 (eV)

assumes a high thermal conductivityeff for nuclear pasta while the solid curve has a reduced conductivity neutron drip T (Horowitz et al. 2015). 80 nuclear pasta how long it takes for a cooling wave60 to di↵use from the surface to the base of the crust, where the nuclear1000 pasta resides (Horowitz et al. 2015). This is illustrated in the right panel of Fig. 1, in which the solid curve≈1014 shows the cooling when the thermal conductivity has been reduced. The presence of a layer with low ther- core 40 -3 mal conductivity can extend the cooling time of the crust by an order of magnitude, from 103 d toρ (g10 cm4 d.) 10 100 1000 ⇠ 10000 ⇠ Note that in this figure, the actual value of the thermal conductivityTime (days) was not calculated over the expected density range. Figure 1: (left) MD simulationThis of nuclear proposal pasta details showing the first e flat↵ort plates to calculate (lasagne) the thermal of nuclear conductivity matter of connected the base of the by defects that reduce the conductivity.neutron star The crust simulation over a realistic used range 409,600 of densities nucleons and generate in a realistic volume lightcurves 200 fm on for testing a side. (right) Surface temperatureagainst of MXB observed 1659-29 quiescent versus neutron the star time transients. since accretion stopped. Dashed curve assumes a high thermal conductivity for nuclear pasta while the solid curve has a reduced conductivity (Horowitz et al. 2015). 2Plan of Work We propose for the PI (C. Horowitz) and a graduate student at Indiana University to simulate nuclear pasta how long it takes for a coolingand determine wave to its di thermal↵use from conductivity the surface using our to existing the base molecular of the crust, dynamics where code the IUMD. nuclear This GPU based pasta resides (Horowitz etcode al. 2015). runs e Thisciently is on illustrated very large computers.in the right The panel Co-I of (E. Fig. Brown) 1, in and which a second the graduate solid curve student at Michi- shows the cooling when thegan thermal State University conductivity will incorporate has been this reduced. information The into presence our existing of a neutron layer with star thermal low ther- evolution code mal conductivity can extenddStar the cooling(Brown 2015) time to of predict the crust light by curves an order for a variety of magnitude, of systems. from 103 d to 104 d. We anticipate performing this work as follows. ⇠ ⇠ Note that in this figure, the actual value of the thermal conductivity was not calculated over the expected density range. 1. The PI and IU student will perform pasta simulations for a range of densities and particle numbers based on extensive previous work (Schneider et al. 2014; Horowitz et al. 2004b,a, 2005; Horowitz & This proposal details theBerry first 2008; e↵ort Horowitz to calculate et al. 2011). the thermal Simulations conductivity with up to 409,600 of the base nucleons of the will be run locally on neutron star crust over a realisticIndiana University’s range of densities Big Red II. and This generate is a Cray realistic XE6/XK7 lightcurves with a peak for performance testing of 1 petaFLOPS. against observed quiescentTo neutron study finite star size transients. e↵ects, a few larger simulations will be run on the supercomputer TITAN at the Oak Ridge Leadership Computing Facility (OLCF) using an existing INCITE allocation under project NPH008.

2Plan of Work 2. From simulation trajectories, the PI and IU student will calculate the static structure factor Sq that describes electron-pasta scattering, and from Sq the thermal conductivity (Horowitz & Berry 2008; We propose for the PI (C. Horowitz) and a graduate student at Indiana University to simulate nuclear pasta and determine its thermal conductivity using our existing molecular dynamics2 code IUMD. This GPU based code runs eciently on very large computers. The Co-I (E. Brown) and a second graduate student at Michi- gan State University will incorporate this information into our existing neutron star thermal evolution code dStar (Brown 2015) to predict light curves for a variety of systems. We anticipate performing this work as follows. 1. The PI and IU student will perform pasta simulations for a range of densities and particle numbers based on extensive previous work (Schneider et al. 2014; Horowitz et al. 2004b,a, 2005; Horowitz & Berry 2008; Horowitz et al. 2011). Simulations with up to 409,600 nucleons will be run locally on Indiana University’s Big Red II. This is a Cray XE6/XK7 with a peak performance of 1 petaFLOPS. To study ﬁnite size e↵ects, a few larger simulations will be run on the supercomputer TITAN at the Oak Ridge Leadership Computing Facility (OLCF) using an existing INCITE allocation under project NPH008.

2. From simulation trajectories, the PI and IU student will calculate the static structure factor Sq that describes electron-pasta scattering, and from Sq the thermal conductivity (Horowitz & Berry 2008;

2 Questions

Is there residual accretion during quiescence?

What causes the shallow heating?

Why is the inner crust so nearly pure?

How quickly does pasta cool?

What do the inferred core temperatures imply? 1426 HEINKE ET AL. Vol. 660

TABLE 2 Luminosities and Mass Transfer Rates

NH kT D M˙ LNS 22 2 1 1 Source (10 cmÀ ) (eV) (kpc) Outbursts Years (M yrÀ ) (ergs sÀ ) References

21 10 33 Aql X-1 ...... 4:2 ; 10 94 5 8 10.7 4 ; 10À 5:3 ; 10 1, 2, 3, 4 20 11 32 Cen X-4 ...... 5:5 ; 10 76 1.2 ...... <3:3 ; 10À 4:8 ; 10 5, 3 21 10 33 4U 1608 522 ...... 8 ; 10 170 3.6 4 10.7 3:6 ; 10À 5:3 ; 10 6, 3, 4 À 22 9 32 KS 1731 260...... 1:3 ; 10 70 7 1 30 <1:5 ; 10À 5 ; 10 7, 4 À 21 10 32 MXB 1659 29...... 2:0 ; 10 55 10? 2 10.7 1:7 ; 10À 2:0 ; 10 7, 4 À 21 11 31 EXO 1747 214...... 4 ; 10 <63 <11 ...... <3 ; 10À <7 ; 10 8 À 22 10 33 Terzan 5 ...... 1:2 ; 10 <131 8.7 2 10.7 3 ; 10À <2:1 ; 10 9, 10, 4 21 10 32 NGC 6440...... 7 ; 10 87 8.5 3 35 1:8 ; 10À 3:4 ; 10 11, 4 22 10 33 Terzan 1 ...... 1:4 ; 10 74 5.2 ...... <1:5 ; 10À <1:1 ; 10 12 20 11 32 XTE 2123 058 ...... 6 ; 10 <66 8.5 1 10.7 <2:3 ; 10À <1:4 ; 10 3, 4 À 21 11 32 SAX J1810.8 2609...... 3:3 ; 10 <72 4.9 1 10.7 <1:5 ; 10À <2:0 ; 10 13, 3, 4 À 21 10 33 RX J1709 2639...... 4:4 ; 10 122 8.8 2 10.7 1:8 ; 10À 2:2 ; 10 14, 15, 4 À 21 10 31 1H 1905+000...... 1:9 ; 10 <50 10 ...... <1:1 ; 10À <4:8 ; 10 16, 15 21 11 31 SAX J1808.4 3658...... 1:3 ; 10 <34 3.5 5 10.7 1:0 ; 10À <1:1 ; 10 17, 4, 15 À Notes.—Estimates of quiescent thermal luminosities from neutron star transients, and mass transfer rates (inferred from RXTE ASM observations for systems with RXTE-era outbursts). Quiescent thermal luminosities are computed for the unabsorbed NS component in the 0.01Y10 keVrange. Outbursts and years columns give the number of outbursts and the time baseline used to compute M˙ , if this calculation was performed in this work (indicated by referring to reference 4). References.—(1) Rutledge et al. 2001b; (2) Campana & Stella 2003; (3) Tomsick et al. 2004; (4) Mass transfer rate computed in this work; (5) Rutledge et al. 2001a; (6) Rutledge et al. 1999; (7) Cackett et al. 2006a; (8) Tomsick et al. 2005; (9) Wijnands et al. 2005; (10) Heinke et al. 2006b; (11) Cackett et al. 2005; (12) Cackett et al. 2006b; (13) Jonker et al. 2004b; (14) Jonker et al. 2004a; (15) Quiescent bolometric luminosity computed in this work; (16) Jonker et al. 2006; (17) Galloway & Cumming 2006.

31 1 NS luminosity LNS < 1:1 ; 10 ergs sÀ . Choosing a NS radius entire 12.5 yr outburst. For KS 1731 260 and the transient in of 12 km, or a mass of 2.0 M , varies this constraint by only 3%. Terzan 1 (for which we take a 12 yr outburst),À we take a mini- The rather tight distance limits of Galloway & Cumming (2006; mum recurrence time of 30 yr. 3:5 0:1 kpc) produce only a 6% uncertainty. Allowing the NH For 1808 we derive a time-averaged mass transfer rate of 1:0 ; Æ 11 1 to ﬂoat freely permits a thermal 0.01Y10 keV NS luminosity 10À M yrÀ , an excellent match to the prediction of general 32 1 21 2 11 1 LNS < 1:0 ; 10 ergs sÀ (for NH 1:7 ; 10 cmÀ ). relativity of 0:95 ; 10À (M2/0:05 M ) M yrÀ (Bildsten & ¼ Chakrabarty 2001). We note that the true mass transfer rate cannot 4. RAMIFICATIONS Heinke et al. 2007, following Yakovlev et al. 2004 We have estimated the time-averaged mass transfer rates for

1808 and several other transient LMXBs (Aql X-1, Cen X-4, 4U / ⟨M˙ ⟩ 1 8 1608 52, KS 1731 260, RX 1709 2639, MXB 1659 29, T ≈ 2 × 108 K core −10 M −1 XTEÀ 2123 058, SAXÀ 1810.8 2609,À and those in Terzan 5À and !10 ⊙ yr "

core

NGC 6440)À from the RXTE ÀAll-Sky Monitor (ASM) record T

∼ (1996 to November 2006), under the assumption that the time- averaged mass accretion rate over the last 10 yr reflects the time- averaged mass transfer rate (Table 2). We use PIMMS and a power law of photon index 2 to convert the ASM count rates dur- 9 ing outbursts into 0.1Y20 keV fluxes. This is, of course, a rough ˙ ⟩ ⟨M approximation, as the spectral shapes of LMXBs in outburst Q = vary substantially. Additional sources of potential error include L q poor ASM time coverage of some outbursts, uncertainty in the NS mass and radius (affecting the energy released per accreted gram and thus the conversion from LX to mass accretion rate), ˙ 1/6 7 ⟨M⟩ Tcore ≈ 2 × 10 K −10 −1 variability in the mass transfer rate, and uncertain distances (which 10 M⊙ yr will equally affect the quiescent luminosity). We plot an arbitrary ! " uncertainty of 50% in both mass transfer rate and quiescent luminosity for each point in Figure 2. For Cen X-4 we use the lowest measured quiescent luminosity and the mass transfer rate limit inferred if Cen X-4 undergoes outbursts every 40 yr with a fluence similar to its 1969 outburst (Chen et al. 1997). The NS component ∼crustal heating flux for Aquila X-1 is somewhat uncertain and possibly variable Fig. 2.—Cooling curves for various NS interior neutrino emission scenarios, (Rutledge et al. 2002; Campana & Stella 2003). We assume compared with measurements (or 95% confidence upper limits) of the quiescent that all outbursts from NGC 6440 since 1971 have been detected. 0.01Y10 keV NS luminosity and time-averaged mass transfer rate for several NS For KS 1731 260, we assume that the average flux seen with transients (see Table 2). The cooling curves are taken from Yakovlev & Pethick À (2004); the dotted curve represents a low-mass NS, while the lower curves rep- RXTE/ASM during outburst was the average flux during the resent high-mass NSs with kaon or pion condensates or direct Urca (Durca) processes mediated by nucleons or hyperons. Limits on the quiescent NS luminosity of 9 We have verified that this conversion is correct to within 50% for outbursts SAX J1808.4 3658 are given for the 2001 and 2006 observations. The effect of a of the transients EXO 1745 245 and Aquila X-1. distance errorÀ as large as a factor of 1.5 is also indicated (upper left). À Questions

Is there residual accretion during quiescence?

What causes the shallow heating?

Why is the inner crust so nearly pure?

How quickly does pasta cool?

What do the inferred core temperatures imply?