Fast neutrino cooling of nuclear pasta in neutron stars: molecular dynamics simulations
Zidu Lin, Matthew E. Caplan, Charles J. Horowitz, Cecilia Lunardini
FFast neutrino cooling of nuclear pasta in neutron stars: molecular dynamicssimulations
Zidu Lin, Matthew E. Caplan, Charles J. Horowitz, and Cecilia Lunardini Department of Physics, Arizona State University,450 E. Tyler Mall, Tempe, AZ 85287-1504 USA Department of Physics, Illinois Sate University, Normal, IL 61790, USA Center for the Exploration of Energy and Matter and Department of Physics,Indiana University, Bloomington, IN 47405, USA (Dated: June 11, 2020)The direct URCA process of rapid neutrino emission can occur in nonuniform nuclear pastaphases that are expected in the inner crust of neutron stars. Here, the periodic potential for anucleon in nuclear pasta allows momentum conservation to be satisfied for direct URCA reactions.We improve on earlier work by modeling a rich variety of pasta phases (gnocchi, waffle, lasagna,and anti-spaghetti) with large-scale molecular dynamics simulations. We find that the neutrinoluminosity due to direct URCA reactions in nuclear pasta can be 3 to 4 orders of magnitude largerthan that from the modified URCA process in the NS core. Thus neutrino radiation from pastacould dominate radiation from the core and this could significantly impact the cooling of neutronstars.
I. INTRODUCTION
Neutron stars (NS) cool primarily by neutrino emis-sion from their dense interiors [1, 2]. Therefore, X-rayobservations of NS surface temperatures can provide in-sight into exotic high density phases that may be present.Many neutron stars are thought to cool relatively slowlyby the modified URCA process where two correlated nu-cleons undergo a cycle of beta decay followed by electroncapture that radiates neutrino anti-neutrino pairs [2].Two nucleons are needed in order to conserve both mo-mentum and energy during the weak interactions. How-ever, this restricts the available phase space and reducesthe neutrino emissivity Q mUrca of the modified URCAprocess.Alternatively, if the proton fraction in dense matteris very high, above a critical value of Y CP = 0 . − . Y CP is achieved. If direct URCA isallowed, it can serve as the most important cooling chan-nel and its presence can be tested by X-ray observationsof NS thermal radiations. For example, recently the neu-tron star in MXB 1659-29 was observed to have a verylow surface temperature, despite large accretion heating.This strongly suggests enhanced neutrino cooling from adirect URCA or similar process [3].The original direct URCA process occurs in the innercores of massive NSs. More recently a number of directURCA like processes that take place at lower densitiesare being explored. Schatz et al. discuss a possible Urca cycle where a nucleus with odd mass-number A in theouter crust of a NS, undergoes first beta decay and thenelectron capture [6]. Here nuclear recoil allows the con-servation of both momentum and energy and neutrinoradiation from this cycle could rapidly cool a layer of thecrust.Nonuniform phases of nuclear matter may allow an-other way to conserve both momentum and energy for theweak interactions. At just below nuclear density, compe-tition between coulomb repulsion and nuclear attractioncan rearrange nuclear matter into rod-like, slab-like, orother complex shapes that are known as nuclear pasta[7, 8]. Nuclear pasta is expected at the base of the NScrust, just before the transition to uniform nuclear mat-ter in the NS core [9, 10].In [11], Gusakov et al. showed that the direct Urcaprocess can possibly occur in nuclear pasta. Due to theperiodic potential created by the inhomogeneous densitydistributions of nuclear pasta, nucleons in the inner crustwould acquire large quasi-momenta, and in this way sat-isfy the momentum conservation required by the directUrca process. Gusakov et al . [11] use a liquid drop modelby K. Oyamatsu [12] to describe the pasta and focus ontwo high density pasta phases (inverted cylinder and in-verted sphere) when calculating the neutrino emissivity.Gusakov et al. found that the neutrino emissivity dueto the direct Urca process in a layer of nuclear pastacan be 2 orders of magnitude stronger than the modifiedUrca process (although still about 5 orders of magnitudeweaker than the URCA process in the neutron star core).Thus, in a neutron star where the central density is toolow to support the direct Urca reaction in the core, thisneutrino emission reaction in the NS crust can profoundlyaffect NS cooling.In addition, URCA emission from nuclear pasta couldmodify the cooling of neutron star crusts in similar waysto Ref. [6]. Here the surface layers could thermally de-couple from the deeper regions so that X-ray bursts and a r X i v : . [ nu c l - t h ] J un other surface phenomena might be independent of thestrength of deep crustal heating.In this paper, we present improved calculations of theneutrino emissivity of pasta based on large-scale molecu-lar dynamics (MD) simulations. These semiclassical sim-ulations allow us to freely explore more complex nuclearpasta shapes and to directly calculate the emissivity. Themethod we use in this work has been extensively usedin the past to study the thermal conductivity, electricalconductivity, shear viscosity and neutrino opacity of nu-clear pasta [13–15]. We investigate the effect of four mainpasta phases (gnocchi, waffle, lasagna, anti-spaghetti)observed in our MD nuclear pasta model on the directURCA process in the inner crust. The paper is organizedas follows: in Section II, we discuss our molecular dynam-ics simulations of the nuclear pasta, and the physics ofthe direct URCA process in the pasta layer. In SectionIII, we present the calculation of the neutrino emissivity,which is very sensitive to the nuclear pasta structure.We then calculate the neutrino luminosity due to directURCA process in the pasta and compare it with the lu-minosity due to the modified Urca process in the NS core.Finally, we conclude in Section IV. II. METHODA. Direct Urca emissivity and its reduction factorin neutron star crust
To calculate the neutrino emissivity of the directURCA process in neutron star crusts, we first determinethe wave functions of protons and neutrons in nuclearpasta layer, which is roughly approximated using pertur-bation theory and is expressed as a Bloch wave functionas in [11]. The nucleon wave function is written as:Φ j = χ s √ V ( e i p · r + (cid:88) q (cid:54) =0 C q e i p (cid:48) · r ) , (1)where V is the normalization volume, q is the inverselattice vector, p is the momentum of a nucleon, C q = V j ( q ) / ( E p − E p (cid:48) ) and p (cid:48) = p + q . Finally V j ( q ) isthe Fourier transformed nucleon potential in the nuclearpasta with j = N, P , where N stands for a neutron and P stands for a proton. Given the nucleon wave functions,the neutrino emissivity Q is calculated similaly as in [11],and we get: Q ( T, n ) = Q ( T, n ) R ( n ) , (2)where Q is the direct Urca emissivity in uniform mat-ter without the momentum conservation constraint, T istemperature and n is baryon number density. Specifi- cally, Q is written as: Q ( T, n ) = 457 π G F cos θ C ( f V + 3 g A ) m N m P m e T ≈ × (cid:18) n e n (cid:19) / T erg cm − s − , (3)with T = T / K. In the calculations we assume that m ∗ N,P = m N,P , where m ∗ N,P is the effective mass of aneutron or a proton at the Fermi surface. The effect ofthe nuclear pasta structure on Q is manifested in thefunction R ( n ), which is: R ( n ) = (cid:88) j = N,P (cid:88) q ( m j V j ( q )) α j P F j × [ F (2 α j D maxj + 2 α j ) − F (2 α j D minj + 2 α j )] × Θ , (4)where P F j is the Fermi momentum of a nucleon, α j = q/P F j , D N ± = [( P F P ± P F e ) − P F N − q ] / P F N q , D P ± =[( P F N ± P F e ) − P F P − q ] / P F P q , D maxj = min [1 , D j + ], D minj = max [ − , D j − ], and F ( x ) = 12 ln | ( √ x +1) / ( √ x − |−√ x/x . Following [11], a simplifiedThomas-Fermi approximation is used in our calculation,and the Fermi momentum of a neutron and a proton iscalculated as: P F N = (3 π n N ) / and P F P = P F e =(3 π n P ) / . Finally, Θ is a step function: Θ = 1 if themomentum conservation is satisfied in the direct URCAlike reactions, and Θ = 0 otherwise. The step functionconstrains the region of allowed momentum transfer q indirect Urca reactions in nuclear pasta layer: P F N − P F P − P F e ≤ q ≤ P F N + P F P + P F e . (5)To determine R , we need to specify the baryon density n , the electron fraction Y e as well as the Fourier trans-formed nucleon potential V j ( q ) in pasta phases. In thiswork our MD simulations are used to find the baryondensity n at which the nuclear pasta phases of gnocchi,lasagna, waffle, and anti-spaghetti form. A detailed de-scription of the MD simulation is presented in sectionII C. The electron fraction Y e in the pasta layer of neu-tron stars should be applied in eq. (2). Oyamatsu [12]studied the nuclear pasta at beta equilibrium, and foundthat the pasta forms at proton fraction Y P ≈ .
03. Cor-respondingly we calculated the function R ( n ) at around Y P = 0 .
03, to study the direct Urca process in realisticconditions of inner NS crust.The Fourier transformed nucleon potential V j ( q ) is ob-tained directly from our numerical simulations of differ-ent pasta phases and will be described in more details insection II B. Interestingly, we found high peaks of V j ( q )based on our large scale nuclear pasta simulations, whichcould potentially amplify the value of R in eq. (4), andcould give a much larger neutrino emissivity. More de-tails about the impact from peaks of V j ( q ) on the neu-trino emissivity will be discussed in section. III C. B. Nucleon Potential in Pasta
In section II A, we show that the effect of nuclear pastastructure on the direct URCA like process is manifestedin eq. (4), where R ∝ V j ( q ). In this section we fur-ther calculate the potential energy of a nucleon V j innuclear pasta numerically. The Indiana University semi-classical Molecular Dynamics simulation IUMD [14] isused to study the nuclear pasta structure, and the codeis described with more details in section II C. In MD sim-ulations the dynamical evolution of N tot nucleons is sim-ulated in a box with side length L , and the structure ofnuclear pasta is depicted by the time-dependent spatialdistributions of nucleons in the box, which are called tra-jectory configurations of the nuclear pasta. The potentialenergy of a nucleon V j ( j = N, P ) at r l is: V j ( r l ) = N tot (cid:88) m =1 V ( l, m ) , (6)where V ( l, m ) is a semi-classical potential for a two bodynucleon interaction with the spacing of nucleons being r lm = | r l − r m | (see eq. (8) for detailed definition of V ( l, m )), and r l = sd ˆ i + td ˆ j + ud ˆ k , with d being thespacing of the potential grids, s , t , u being integers andˆ i , ˆ j , ˆ k are orthogonal unit vectors. Consequently, wehave N grid = ( L/d ) grid points on which we calculatethe nucleon potential of nuclear pasta. The grid pointspacing d is chosen so that d (cid:28) L and is approximately 2fm, near the characteristic nucleon spacing in our model.Given V j ( r l ), we calculate the Fourier transformed nu-cleon potential V j ( q ) numerically, as: V j ( q ) = (cid:80) N grid j =1 V N ( r j ) × exp ( i q · r j ) d L , (7)where q = ( πL M )ˆ i + ( πL N )ˆ j + ( πL O )ˆ k , with M , N , O being integers.We use 100 trajectory configurations from the MD nu-clear pasta simulations spaced by 1000 MD timesteps.The nucleon potential V xj ( r j ) of each configuration x is calculated per eq. (6), and is averaged by V j ( r i ) = (cid:80) x =1 V xN ( r i ) / C. Molecular Dynamics of Nuclear Pasta
We use the Indiana University Molecular Dynamicscode (IUMD) to simulate nuclear pasta, as in past work[10, 14, 16–24]. For completeness, we include a brief re-view here. IUMD uses a semi-classical potential V ( l, m ) for a two body nucleon interaction, which is: V ( l, m ) = ae − r lm / Λ +[ b + cτ z ( l ) τ z ( m )] e − r lm / + V c ( l, m ) . (8)Here a = 110 MeV, b = −
26 MeV, c = 24 MeV, Λ = 1 . , and V c ( l, m ) = e r lm e − r lm /λ τ P ( l ) τ P ( m ) (9)is the Coulomb repulsion between protons. We set λ = 10fm as the Coulomb screening length, τ z = 1 for protonand τ z = − τ p = τ z = 1. Note that r ij = (cid:112) [ x i − x j ] + [ y i − y j ] + [ z i − z j ] where the pe-riodic distance (given by [ l ] = Min( | l | , L −| l | )) is used. Allsimulations described in this work use periodic boundaryconditions in a cubic box, with side length L . All sim-ulations are isothermal and at constant density with anMD timestep of 2 fm/c.This two-body interaction is simple and the nuclearattraction is short ranged, allowing us to efficiently sim-ulate hundreds of thousands of nucleons [10, 21, 21, 23].The geometric pasta phase can be specified by three ther-modynamic parameters: the nucleon number density n ,temperature T , and the proton (electron) fraction Y e ,though hysteresis effects and formation history can berelevant for determining the exact structure of large vol-umes of pasta [10, 14]. This model has now been usedextensively to study the phases and structure of nuclearpasta. It is known to form a variety of phases simi-lar to diblock copolymers including gnocchi (spheres),lasagna (planar or lam), waffles (perforated lam), andantispaghetti (uniform matter with cylindrical holes),which will be the subject of this work [10, 14, 20, 21]. Ourmodel, having finite temperature, also exhibits a large va-riety of additional phases and ‘defects’ as well, such ashelicoids that connect lasagna (structurally identical toTerasaki ramps), and may buckle over large lengths anddisrupt long range order [21–24]. This work is thereforenot confined to the unit cell; our MD model allows usto study both the simple idealized cases and phases withlong range disorder self consistently, which is not possi-ble with fully quantum mechanical simulations which arelimited to small numbers of particles [25, 26].We address the robustness of our semi-classical modelfor this problem. Past work with this model has focusedon the parameter space near T = 1 MeV, nucleon densi-ties between n =0.01 and 0.12 fm − , and electron (pro-ton) fractions between Y e = 0.3 and 0.5 because this isthe parameter range for which our model produces pasta[10]. At significantly higher temperature the nucleonsdissolves into a gas, while at temperatures near 0.5 MeVthe nucleons crystallize and become locked into a lattice.Similarly, at lower proton fractions our model forms agas of nucleons [23]. Therefore, we are confined to thisparameter range to study pasta when using IUMD sim-ulations, although the proton fraction range applied inthe simulations is higher than expected in inner crustof NSs. Nevertheless, our pasta model is still consistentwith mean field and fully quantum mechanical simula-tions which produce all the same pasta phases we ob-serve at similar densities [25, 27]. We note that at verylow proton fractions of Y e = 0 .
05 and Y e = 0 .
1, whichare close to the beta equilibrium conditions, a large scalequantum simulation of pasta phases [25] gives similar nu-clear pasta structures when comparing them with resultsfrom IUMD. The pasta phases may have many minimain their energy landscape separated by large tunnelingbarriers, and so configurations which are stable on MDtimescales may not be true ground states. Nevertheless,initial and final configurations are generally equivalentto each other in all simulations reported in this work sothese configurations are stable on MD timescales and fur-thermore we do not observe any trend in total simulationenergy.As one of our primary goals in this work is to calculatethe temperature independent function R in eq. (2), whichcontrols the magnitude of neutrino emissivity relating topasta structures, the exact thermodynamic parameters ofour pasta simulations do not heavily bias our results; weuse them to generate structural conditions which containsufficiently large numbers of nucleons to be in classicallimit. III. RESULTSA. Molecular Dynamics Simulations
We study 12 MD simulations of nuclear pasta in thiswork, two gnocchi, four lasagna, four waffles, and twoantispaghetti. We give each an identifier for readabil-ity, such as G1 and G2 for the gnocchi simulations, etc.Initial conditions for our MD simulations are assembledfrom or derived from our body of past work and archivaldata, though a few new configurations were generatedfor this work. The preparation of these simulations arebriefly described in the following, while a summary of themolecular dynamics conditions is included in Tab. I.The initial conditions for these simulations were allevolved for at least 10 MD timesteps prior to collectingdata to guarantee they were dynamically equilibrated.For consistency, all configurations used to calculate R were generated from equilibrium MD simulations specif-ically for this work.G1 and G2 were taken from past studies (ref. [28])which considered phases of nuclear pasta at different pro-ton fractions and are shown in Fig. 1. In that work,high density matter was expanded by incrementally in-creasing the box size after each timestep. This resultsin much more regularly distributed gnocchi than in sim-ulations equilibrated from random, as they fission fromlarge structures generally more symmetrically. A clearbody-centered-cubic (BCC) lattice is visible, with nuclearseparations comparable to nuclear radii.Simulations of lasagna can be seen in Fig. 2. L1 andL2 were likewise taken from past work (ref. [19, 22, 23] and allow us to study finite size effects and how the orien-tation of pasta within the simulation volume may affectour calculations. L1 was prepared by including a sinu-soidal external potential during a brief initial simulation,while L2 is generated from L1 by random switching neu-trons for protons, resulting in plates with spontaneoussplay at a higher Y e . L3, instead, is unique to this work,though prepared similarly to L1. L4, while having sim-ilar parameters to our other simulations of lasagna, wasprepared by simulating at the slightly lower temperatureof 0.8 MeV. At this temperature many defects are frozenin, including helicoids and buckles which present a sortof ‘fingerprint’ defect. This allows us to compare moreidealized plates to a structure without long ranged order.L4 is long lived in MD.Our waffle configurations, which are similar to lasagnabut with holes perforating the plates, are shown in Fig.3. W1 is a trivial variation of L1, obtained by reducing Y e . W2 is similarly produced from past work (ref. [22]),and allows us to study how the orientation of the plates inthe simulation volume may affect our calculations of thereduction factor. The plates in W2, however, do not showa regular lattice of holes like W1, the higher temperatureresults in many short-lived holes as thermal fluctuationsof the pasta surface. W3 is similar to L3 and is a variationon past work (ref. [19]), allowing us to resolve finite sizeeffects, while W4 is obtained from L4 by reducing theproton fraction from Y e = 0 . Y e = 0 . B. Nucleon potential in real space and momentumspace
First, we show the nucleon number density distribu-tions and nucleon potential energy distributions in nu-clear pasta, in Fig. 1-4. Interestingly, the potential en-ergy distributions of nuclear pasta exhibit similar non-uniform characteristics when they are compared to thenumber density distributions, due to the short-range na-ture of the nuclear force. In this way, one might expectthat the structural information of nuclear pasta will beimprinted on its Fourier transformed nucleon potential V ( q ) and on the magnitude of neutrino emissivity in di-rect URCA process (see Eq. (4)).We show Fourier transformed proton and neutron po-tentials as a function of momentum transfer q in Fig. 5and Fig. 6. These potentials for different pasta phasesare then compared in Fig. 7. As shown in Fig. 7, theFourier transformed nucleon potentials display a largepeak at | q | (cid:39) −
80 MeV due to the fact that the peri-
TABLE I: Summary of molecular dynamics configurations studied in this work. We include nucleon number density n , tem-perature T , number of nucleons N , proton (electron) fraction Y e , and their source in the literature.Identifier n (fm − ) T (MeV) N Y e SourceG1 0.015000 1.0 51200 0.3 Refs. [28, 29]G2 0.014951 1.0 51200 0.4 Refs. [28, 29]L1 0.050000 1 102400 0.4 Refs. [19, 22]L2 0.050000 1.2 102400 0.5 Present work, derivative of L1 and Ref. [23]L3 0.050000 1 204800 0.4 Present workL4 0.050007 0.8 204800 0.4 Present workW1 0.050000 1 102400 0.3 Present work, derivative of L1 and Ref. [29]W2 0.05 1.6 102400 0.4 Present work, derivative of Ref. [22]W3 0.05 1 204800 0.3 Present work, derivative of Ref. [19]W4 0.050007 0.8 204800 0.3 Present work, derivative of L4AS1 0.0882 0.8 51200 0.3 Present workAS2 0.0882 0.8 51200 0.4 Refs. [28, 29] (a) G1 (b) G2 (c) G1 (d) G2
FIG. 1: (Color online) Neutron density distributions and potential energy distributions for the gnocchi phase. Panel (a) and(b) represent neutron density distribution in gnocchi simulated with 51200 nucleons at Y e = 0 . Y e = 0 . odic spacing of nuclear pasta potential is comparable tothe wavelength of the nucleon momentum transfer q . Tohave a clearer understanding of the relationship betweenthe pasta structure and the properties of the peaks of V ( q ), in Appendix A we analytically evaluated the posi-tion and the height of the peak of a gnocchi phase assum-ing it is composed of perfectly spherical nuclei and has aclear BCC lattice structure. We further discuss the pos-sible relationship between the V ( q ) and the static struc-ture factor S ( q ) of the nuclear pasta, where the latterembodies the coherence effect on nuclear pasta electronscattering and on the nuclear pasta neutrino scattering inNSs [14, 15]. The static structure factor of nuclear pastadisplays large peaks in q domain when the wavelengthof q is comparable to the inter-particle spacing. Due tothe structural similarities between distributions of nu-cleon densities and distributions of nucleon potentials innuclear pasta (see Figs. 1–4), the peaks of V ( q ) and thepeaks of nuclear pasta static structure factor [13, 15] areapproximately in the same region of | q | .Let us now discuss the relationship between the nu-clear pasta potentials in real space and the correspond-ing Fourier transformed potentials. Firstly, in Fig. 1the gnocchi phases G1 and G2 are simulated at differentelectron fractions, namely Y e = 0 . Y e = 0 .
4. How- ever the distributions of nucleon potential in real spaceare very similar to each other , and correspondingly thepeaks of V ( q ) of these two simulations approximatelyoverlap with each other, as shown in the upper left panelof Fig. 5 and 6. Secondly, the lasagna phases L1-4 issimulated with different number of nucleons, different Y e and orientations of the lasagna plates, as shown in Fig.2. In the left lower panel of Fig. 5 and Fig. 6, the peakof V ( q ) based on simulation L1 looks very similar to thatbased on L3, which indicates that the number of nucle-ons involved in our simulations will not severely affectthe outcome, and that the finite-size effect of our MDsimulations is minor. However, although the location ofthe peaks based on these four simulations basically agreewith each other, the height of peaks based on L2 and L4is obviously smaller than those based on L1 and L3. Thisis due to the fact that lasagna simulations of L2 and L4exhibit more irregular local structures such as the connec-tion between two plates and the curvature of the plates,while keeping about the same spacing of plates as in L1and L3. Thirdly, the waffle phase are simulated withdifferent number of nucleons, electron fractions Y e , andorientations of the waffle plates. In the upper right panelof Fig. 5 and Fig. 6, the distribution of V ( q ) based onW1 and W3 are similar, which once again demonstrate (a) L1 (b) L2(c) L3 (d) L4 (e) L1 (f) L2(g) L3 (h) L4 FIG. 2: Results are shown for density and real space potential energy distribution of a neutron, due to lasagna structure.Panel (a), (b), (c) and (d) represent neutron density distribution in the lasagna simulated with 102400 nucleons at Ye=0.4,102400 nucleons at Ye=0.5, 204800 nucleons at Ye=0.4 and 204800 nucleons at Ye=0.4 but not aligned with the box surfacerespectively. Also, in panel (e), (f), (g) and (h) we show the potential energy distribution of a neutron from 0 (deep blue) to50 (red) MeV in the lasagna. (a) W1 (b) W2(c) W3 (d) W4 (e) W1 (f) W2(g) W3 (h) W4
FIG. 3: Results are shown for density and real space potential energy distribution of a neutron, due to waffle structure. Panel(a), (b), (c) and (d) represent neutron density distribution in the in ’waffle’ with 102400 nucleons at Ye=0.3, 102400 nucleonsat Ye=0.4, 204800 nucleons at Ye=0.3 and 204800 nucleons at Ye=0.3 but not aligned with the box surface respectively. Also,in panel (e), (f), (g) and (h) we show the potential energy distribution of a neutron from 0 (deep blue) to 50 (red) MeV in thewaffle. (a) AS1 (b) AS2 (c) AS1 (d) AS2
FIG. 4: Results are shown for neutron density distribution due to anti-spaghetti structure. Panel (a) and (b) represent neutrondensity distribution in anti-spaghetti simulated with 51200 nucleons at Ye=0.3 and 51200 nucleons at Ye=0.4 respectively. Also,Results are shown for potential energy distribution of a neutron, due to anti-spaghetti structure. In panel (c) and (d) we showthe potential energy distribution of a neutron from 0 (deep blue) to 50 (red) MeV in anti-spaghetti. the finite size effect on our evaluations is small. Butthe peak of W1 is obviously higher than the other threesimulations, which is possibly due to a more regular dis-tribution of these waffle plates and short-lived holes inthis specific simulation, as shown in Fig. 3. Finally, wepresent the distribution of V ( q ) corresponding to anti-spaghetti in the lower right panal in Fig. 5 and Fig. 6.Although the location of the main peaks based on AS1and AS2 are approximately the same, the height of peaksbased on these two simulations are very different. Thisis because AS1 is actually a disordered form of AS2, andthe latter shows clearly the long-range correlations thatAS1 lacks and exhibits a much clearer periodic structurethan AS2 does, as shown in Fig. 4. C. Neutrino emissivity
In this section we calculate the direct URCA neu-trino emissivity in nuclear pasta. The effect of nuclearpasta structure on the neutrino emissivity is illustratedin Fig. 8 and Fig. 9, where R varies as a functionof Y e , at fixed baryon densities. As Y e decreases, thelower bound of allowed q rises up, and the contribu-tion from the peaks of V ( q ) to the R will be excludedif the lower bound is higher than q peak , with q peak be-ing the corresponding momentum transfer at the peakof V ( q ). As a result, the function R decreases rapidlyat around Y e =0.01, 0.035, 0.035 and 0.045 for gnocchi,waffle, lasagna and anti-spaghetti phase respectively inFig. 8 and Fig. 9. We note that the electron fractionsfor nuclear pasta in inner crust at beta equilibrium is ap-proximately 0 . < Y e < . Y e at beta equilib-rium in inner NS crust demonstrate the close proximityto the enhancement of R and hence the enhancement ofneutrino emissivities via direct URCA reactions due tonuclear pasta structures. In Tab. II we summarize the R s of different pasta phases. To illustrate the contribu-tion from V ( q ) peaks, R is calculated in Table II at twodifferent electron fractions, Y e = 0 .
03 and Y e = 0 .
05. As shown in Fig. 5 and Fig. 6, at Y e = 0 .
03 the R ofmost nuclear pasta phases (except those corresponding toG1 and G2) do not include the contributions from V ( q )peaks because of momentum conservation. At Y e = 0 . R due to the contribution from peaks of their V ( q ). Corre-spondingly, the neutrino emissivity is greatly enhanced,and is only 1-2 orders of magnitude weaker than thatvia direct URCA reactions. However, when the peaksof V ( q ) are excluded due to momentum conservation re-quirement, the R decreases by 3-4 magnitude of orders. In the latter case our results about R are reasonablyclose to the calculations in [11], while the remaining de-viations between our results of R and that reported in[11] are possibly due to the differences between the pastamodels applied.Finally, the neutrino emissivity Q from the core viamodified URCA reactions are compared to Q from thenuclear pasta layer in the inner crust due to direct URCAreactions. The neutrino emissivities of the modifiedURCA process, in both the neutron and proton branches(which is denoted as Q MN and Q MP respectively), aresummarized below (see detailed description of modifiedURCA in [30]): Q MN ≈ . × (cid:18) n P n (cid:19) / T α n β n erg cm − s − , (10) Q MP ≈ Q MN ( P F e + 3 P F P − P F N ) P F e P F P Θ MP , (11)where α n = 1 . β n = 0 .
68, and Θ MP is the thresholdfor the proton branch, allowing the modified Urca processwhen P F N < P F e . We calculate Q MN and Q MP at coredensity n core = 2 n , where n is the saturation density0 . f m − . The neutron star is assumed to be isothermaland neutrino emissivity from the core and the crust areboth calculated at T = 3 × K.Given Q MN and Q MP , in Tab. II we list an order-of-magnitude estimate on the ratio of crust neutrino lu-minosity to core neutrino luminosity at Y e = 0 .
03 and
G1G2 ( MeV ) V P ( M e V ) W1W2W3W4 ( MeV ) V P ( M e V ) L1L2L3L4 ( MeV ) V P ( M e V ) AS1AS2 ( MeV ) V P ( M e V ) FIG. 5: Fourier transformed potential energy distribution of a proton in gnocchi (upper left), waffle (upper right), lasagna(lower left), and anti-spaghetti (lower right) respectively. The vertical dot dashed and dotted black lines are lower bounds ofallowed q when Y e = 0 .
03 and Y e = 0 .
05 respectively, above which V ( q ) is included in function R (see eq. 4). The upperbounds of q lie beyond the range of the plots and are not shown here.Properties of these pasta phases are summarized in Table.I Y e = 0 .
05. The neutrino luminosity of modified Urcafrom the core is: L core = 4 π R ( Q MN + Q MP ) , (12)where R = 10 km is approximately the radius of neutronstar cores. The neutrino luminosity of direct Urca in thecrust is: L crust = 4 πR hQ, (13)where h = 100 m is approximately the width of nuclearpasta layer [27]. In the calculations of L crust , we assumethat the inner crust is composed of nuclear pasta of aspecific phase, e.g. , only lasagna or only anti-spaghetti.A more accurate estimation of L crust might require con-sidering the co-existence of multi-nuclear pasta phases inthe inner crust, so that the total luminosity would bean appropriate average of the luminosities of the vari-ous pasta phases. In Tab. II we summarize the neutrinoluminosities of direct URCA process due to different nu-clear pasta phases in neutron star crusts, and see that they can be about 1-2 magnitude of orders stronger thanthat in neutron star cores from modified URCA process,if the contribution from peaks of V ( q ) to neutrino emis-sivity Q is excluded by momentum conservation. At suffi-ciently high Y e (for example at Y e = 0 . V ( q ) to the function R can greatly am-plify the neutrino luminosity in neutron star inner crusts,making it even stronger, which is about 3-4 magnitude ofoders higher than that due to the modified Urca reactionsin the cores of NSs. IV. CONCLUSION
In this paper we calculated the neutrino emissivity dueto a direct Urca process in nuclear pasta. This nonuni-form phase is expected near the base of the neutron starcrust. Different shaped pasta phases were explored usingmolecular dynamics simulations containing 51,200 and204,800 nucleons. In our semi-classical simulations, bothneutrons and protons are free to explore a variety of
G1G2 ( MeV ) V N ( M e V ) W1W2W3W4 ( MeV ) V N ( M e V ) L1L2L3L4 ( MeV ) V N ( M e V ) AS1AS2 ( MeV ) V N ( M e V ) FIG. 6: Fourier transformed potential energy distribution of a neutron in gnocchi (upper left), waffle (upper right), lasagna(lower left), and anti-spaghetti (lower right) respectively. The vertical dot dashed and dotted black lines are lower bounds ofallowed q when Y e = 0 .
03 and Y e = 0 .
05 respectively, above which V ( q ) is included in function R (see eq. 4). The upperbounds of q lie beyond the range of the plots and are not shown here. Properties of these pasta phases are summarized inTable. G1W1L1AS1 ( MeV ) V p ( M e V ) G1W1L1AS1 ( MeV ) V n ( M e V ) FIG. 7: Fourier transformed potential energy distribution of a neutron and a proton in gnocchi, waffle, lasagna and anti-spaghettiare compared. The properties of these pasta phases are summarized in Table. I TABLE II: Summary of Direct URCA emissivity in this work. We include function R , neutrino emissivity Q (in unit of10 erg cm − s − and the ratio of neutrino luminosity L via direct URCA from the crust at Y e = 0 .
03 and at Y e = 0 .
05 to thatvia modified URCA from the core. The temperature corresponding to the calculations here is T = 3 × K.Identifier R Y e =0 . R Y e =0 . Q crustY e =0 . (10 erg cm − s − ) Q crustY e =0 . (10 erg cm − s − ) L crustY e =0 . /L core L crustY e =0 . /L core G1 0.046 0.58 19 280 1 . × . × G2 0.11 0.95 44 470 4 . × . × L1 9 . × − . × −
590 5.8 6 . × L2 1 . × − . × −
200 7.5 2 . × L3 1 . × − . × −
620 1.0 6 . × L4 2 . × − .
17 260 17 2 . × W1 5 . × − .
35 120 36 1 . × W2 1 . × − .
11 190 12 2 . × W3 9 . × − .
56 100 58 1 . × W4 4 . × − . . × AS1 7 . × − .
57 41 58 4 . × AS2 5 . × − . . × G1G2 - - Y e R W1W2W3W4 - - Y e R L1L2L3L4 - - - Y e R AS1AS2 - - Y e R FIG. 8: The factor R as a function of Y e in gnocchi (upper left), waffle (upper right), lasagna (lower left) and anti-spaghetti(lower right). shapes. We approximated the wave functions of nucle-ons in our pasta simulations using perturbation theoryas in Ref. [11]. Given these nucleon wave functions, theneutrino emissivity of the direct Urca process was calcu-lated for various nuclear pasta phases, including gnocchi,waffle, lasagna, and anti-spaghetti. We found that theneutrino luminosity due to a direct Urca process in nu-clear pasta can be 3-4 orders of magnitude larger thanthat from the modified URCA process in neutron star cores. Thus neutrino radiation from pasta could domi-nate over radiation from the core. This enhanced emis-sion could have a pronounced effect on the cooling ofneutron stars and on X-ray observations of NS thermalradiations. Therefore, future work should explore fur-ther the neutrino emissivity of nuclear pasta includingthat from fully quantum mean field calculations. Thiswill allow calculations directly at low beta equilibriumvalues of Y e and should provide a better understanding1 G1W1L1AS1 - - Y e R G1W1L1AS1 Y e Q ( e r g s - c m - ) FIG. 9: The left panel shows R as a function of Y e in different nuclear pasta phases, which are G1, W1, L1 and AS1 respectively.The right panel shows neutrino emissivities in these phases. The corresponding densities of G1, L1, W1 and AS1 pasta phasesare listed in Table 1. The temperature at which Q is calculated is at T = 3 × K . on how neutrino emissivitiy depends on Y e .In the near future we expect more X-ray observationsof thermal radiation from NS. These cooling observationsmay depend on a variety of NS features such as the pres-ence or absence of a heavy element envelope, of a DirectURCA process in the core, and on a variety of superfluidand superconducting pairing gaps [2]. In addition therecould be a sizable contribution to cooling from nuclearpasta. It should be possible to use X-ray observations ofboth isolated and accreting NS to sort out some of thesefeatures and gain insight into the dense phases of matterpresent in NS. Acknowledgments
ZL thanks D. G. Yakovlev for helpful discussions. ZLand CL acknowledge funding from the National ScienceFoundation Grant No. PHY-1613708. ZL acknowl-edges funding from US Department of Energy grants de-sc0019470. CH is supported in part by US Department ofEnergy grants DE-FG02-87ER40365 and de-sc0018083.
Appendix A: Analytic model of nucleon potentialsfor the gnocchi phase
In the appendix we aim to gain an analytical under-standing of the large-scale molecular dynamics numericalsimulations. We choose the gnocchi phase as the test bed,which forms a body-centered-cubic (BCC) lattice whenthe simulation is equilibrated. The Fourier transformedpotential V j ( q ) is defined as: V j ( q ) = 1 V (cid:90) V dV V j ( r ) e i q · r . (A1)In a well equilibrated gnocchi phase a reciprocal latticestructure is formed and we assume that V j ( r ) = V j ( r + T ), where T is the lattice vector. V j ( r ) can be expressed in terms of a Fourier decomposition, given its periodicstructure V j ( r ) = (cid:88) G V G e − i G · r , (A2)where G is the reciprocal lattice vector. Given eq. A2,eq. A1 becomes V j ( q ) = (cid:88) G (cid:90) V dV V G e i ( G − q ) · r . (A3)We see that V ( q ) reaches its peak at the diffractionpoints where q = G , and find: V j ( G ) = 1 V (cid:88) T (cid:90) cell dV V j ( r + T ) e i G · ( r + T ) = NV (cid:90) cell dV V j ( r ) e i G · r , (A4)where N is the number of unit lattice cells in a MD simu-lations and V is the volume of the box in MD simulations.Assuming that we have s gnocchi in a unit cell located at r k , it is convenient to write potential energy as the su-perposition of potential energy V j associated with eachgnocchi k of the basis, so that V j ( r ) = (cid:80) Sk =1 V j ( r − r k )).We then have (cid:90) cell dV V j ( r ) e − i G · r = S (cid:88) k =1 (cid:90) cell dV V j ( r − r k ) e − i G · r = (cid:90) cell dV V j ( R ) e − i G · R × S (cid:88) k =1 e − i G · r k , (A5)where R = r − r k . Assuming that the nucleon potentialenergy V j in the gnocchi is distributed uniformly with asharp surface radius R , eq. 9 could be further simplified,since (cid:90) cell dV V j ( R ) e − i GR = 4 πV j − GRcos ( GR ) + sin ( GR ) G . (A6)2For a BCC lattice, we have (cid:80) Sk =1 e − i G · r j = 1 + e − i Gr ,where G = 2 πa ( m ˆ i + m ˆ j + m ˆ k ) and r = a i +ˆ j + ˆ k ), with a being the center-to-center distance ofthe BCC lattice. It turns out that (cid:80) Sk =1 e − i G · r j =1 + ( − m + m + m , and we have V N ( G ) = 4 πV j − GRcos ( GR ) + sin ( GR ) G × [1 + ( − m + m + m ] NV . (A7)Finally, we compare the analytical expression of V N ( G )with that from numerical simulations. In the gnocchi phase G1, the gnocchi center-to-center distance is ap-proximately 30 fm, the radius of the sphere is approxi-mately 7.5 fm, and the mean potential energy of neutronsin gnocchi is approximately 30 MeV. Plugging these num-bers into eq. A7, we found that the first peak locates at | q | = | G | ≈
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