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Astronomy & Astrophysics manuscript no. hallinh3 February 5, 2008 (DOI: will be inserted by hand later) The Occurrence of the Hall–Instability in Crusts of Isolated Neutron Stars 1 2 1 M. Rheinhardt , D. Konenkov and U. Geppert 1 Astrophysikalisches Institut Potsdam, Ander Sternwarte16, 14482 Potsdam, Germany; e-mail: [email protected] 2 Ioffe Physico-Technical Institute, 194021 Af. F., Politechnicheskaya 26, St.Petersburg, Russia 4 e-mail: [email protected] 0 0 Received date ; accepted date 2 n Abstract. Informerpapersweshowedthatduringthedecayofaneutronstar’smagneticfieldundertheinfluence a oftheHall–drift,anunstableriseofsmall–scalefieldstructuresattheexpenseofthelarge–scale backgroundfield J may happen. This linear stability analysis was based on the assumption of a uniform density throughout the 2 neutronstarcrust,whereasinrealitythedensityandalltransportcoefficientsvarybymanyordersofmagnitude. Here, we extend the investigation of the Hall–drift induced instability by considering realistic profiles of density 2 v andchemicalcomposition,aswellasbackgroundfieldswithmorejustifiedradialprofiles.Twoneutronstarmodels 7 areconsidereddifferingprimarilyintheassumptiononthecorematterequationofstate.Fortheircoolinghistory 1 and radial profiles of density and composition we use known results to infer the conductivity profiles. These 9 were fed into linear calculations of a dipolar field decay starting from various initial configurations. At different 0 stages of the decay, snapshots of the magnetic fields at the equator were taken to yield background field profiles 1 for the stability analysis. The main result is that the Hall instability may really occur in neutron star crusts. 3 Characteristic growth times are in the order of . 104...106 yrs depending on cooling age and background field 0 strength. The influenceof the equation of state and of theinitial field configuration is discussed. / h p Key words.stars: neutron – stars: magnetic fields - o r t 1. Introduction 2002) showed that for a (large–scale) magnetic back- s a ground field characterized by a sufficiently curved radial : v Neutronstars(NSs)arecarriersofthestrongestmagnetic profilebeyonda certainmarginalfield strength,the Hall– i fields which occur in nature. But, astonishingly enough, drift may cause an unstable growth of small–scale per- X most of the quantitative studies of magnetic field decay turbations.For a homogeneousmedium these are concen- r a in NS crusts consider only the linear induction equa- trated towards the boundary adjacent to vacuum. tion, i.e., a field decay caused solely by Ohmic dissipa- We performed this stability analysis based on the lin- tion(see,e.g.,Urpin et al.1994;Urpin & Konenkov1997; earized induction equation with Hall–drift for a homoge- Page et al. 2000). neous plane slab of finite thickness, infinitely extended in Yakovlev & Shalybkov (1991) showed that in a two- both horizontal dimensions and bound by vacuum and a componentplasmatheresistivitycomponentsparalleland medium of infinite conductivity at its upper and lower perpendicular to the magnetic field coincide, as a result sides,respectively.Whilethismodelisperhapsanaccept- of which in turn the ambipolar drift disappears. This re- able approximationofthe NS crustwith respectto geom- sultis applicabletothe electronsinthefully ionizedcrys- etry, the assumption of a spatially uniform density and tal matter of the NS crust, too (see Urpin & Shalybkov chemical composition, which result in a uniform scalar 1999). Further, convective motions will not exist since conductivity and Hall–drift coefficient, is surely a very the crust is almost completely crystallized after, say, 104 crude one. Actually, the density and thus the coefficients yrs. Therefore, the magnetic field evolution in the crust depending on it may vary throughout the crust by many is solely determined by Ohmic diffusion and the so–called ordersofmagnitude(see,e.g.,Page et al.2000).Lesspro- Hall–drift where the latter is the only non–linearity in nouncedly, the chemical composition (that is, mass num- this process (if the weak and therefore weakly nonlinear ber A and atomic number Z) varies, too. Additionally, back–reaction of the magnetic field on the conductivity the scalar conductivity and the Hall–drift coefficient are tensor via Joule heating is discounted). Recently, two of dependent on the temperature and the impurity concen- us (Rheinhardt & Geppert 2002; Geppert & Rheinhardt tration. 2 Rheinhardt,Konenkov& Geppert: Hall–Instability in Neutron stars The effect of the Hall–drift on the magnetic field Alsounansweredis,howfaritmaybehiddenbyotheref- evolution in NSs has been considered by a number of fects of the field evolutionas,e.g., the “normal”Hall cas- authors (Goldreich & Reisenegger 1992; Vainshtein et al. cade. The first of these questions will be addressed here. 2000; Urpin & Shalybkov 1995, 1999; Shalybkov & Urpin One of the most convincing indicators for the impor- 1995, 1997; Naito & Kojima 1994; Haensel et al. 1990; tance of the Hall–drift in the crust is the evolution of Hollerbach & Ru¨diger 2002). The only study, however, the magnetization parameter ω τ there. In Fig. 1 of B e which includes quantitatively the effect of the crustal Geppert & Rheinhardt (2002) this quantity is shown for density stratification in this context is the one by differenttemperatures,thatis,fordifferentagesasafunc- Vainshtein et al. (2000). They showed that due to a den- tion of density. These results are based on linear field de- sity gradient, the Hall–drift may create current sheets, cay calculations for a standard NS model with a medium wherethefieldcanbedissipatedveryefficiently.However, EOS as presented by Page et al. (2000). Clearly, as soon the occurrence of the mentioned Hall–instability could as the magnetic field strength exceeds 1012 G, ω τ > 1 B not be detected by them because in their linear analy- in some regions of the crust, and the Hall–drift begins to sis they considered a uniform background field only, i.e., dominate the Ohmic decay. a field which will not become unstable regardless which In order to answer the question whether at all and strength it has. In considering the non–linear evolution howintensivetheHall–instabilityoccursinrealNScrusts, of the toroidal field they neglected the coupling with the we will consider crustal density profiles resulting from poloidalone,thusmakinganinstabilityimpossibleeither, NS models based on stiff and medium equations of state see (9) below. (EOSs)ofthecorematter.Asbackgroundmagneticfields Observational evidence for a decay of the large– whichmust–incomparisonwiththeunstablefieldpertur- scale (dipolar) magnetic field of middle aged pul- bations – evolve very slowly, we use those, calculated by sars, being drastically accelerated in comparison with themethodsdescribedinPage et al.(2000).Theevolution the purely Ohmic decay, has been discussed in of those large–scale fields is affected only by the density Geppert & Rheinhardt (2002). Our reasoning there is profile ofthe NS crustandits coolinghistory,both deter- based on the detection of braking indices greater than mined essentially by the EOS. Furthermore, the crustal threebyJohnston & Galloway(1999),whousedmeasure- field evolution is affected by the chemical composition mentsoftherotationalperiodP,anditstemporalderiva- and impurity concentration within the crust, as well as tive P˙, dating from different observational epochs. As a by the initial strength and structure of the field which, in typicalresultwefoundthatthe decaytimesinferredfrom turn,reflecttheprocessesatbirthoftheNS.Theseresults P and P˙ may be smallerthan 10−4 times the (estimated) are insofar incomplete, as they just do not take into ac- Ohmicdecaytime.Apossibleexplanationforsucharapid countthe veryefficientdrainofmagnetic energyfromthe fielddecay maybe the Hall–instabilitywhichdrains mag- backgroundfieldduringtheperiodofthe Hall–instability. netic energy out of the dipolar field and uses it for the They should, however, provide hints under which condi- build up of small–scale magnetic field structures, which tions, more realistic than those considered up to now, an eventually decay by Ohmic dissipation. episode of strongly non–linear magnetic field decay may The occurrence of small–scale magnetic field struc- take place. turesinthecrustallayersofNSsmaycausefurthereffects Thispaperfollowsthelinesofthoughtaspresentedby which are potentially observable. Beyond the decrease of Rheinhardt & Geppert(2002),andisorganizedasfollows: the braking power of the magnetic field, the increased In the next section the basic equations of the model are small–scale Lorentz forces can trigger a cracking of the derived,andthe assumedproperties ofthe crustalmatter crystallized crust. Also, due to a rapid and spatially con- are described together with the background field profiles centratedOhmic decayofthe small–scalefieldstructures, we used. Section 3 briefly outlines the method of solving hot spots may appear in the surface layers.Recent obser- theeigenvalueproblemandprovidesanddiscussesthenu- vations,bothintheX–ray(Becker et al.2003),andinthe merical results. Conclusions are drawn in Sect. 4 with a radio range (Gil et al. 2002), support the idea of the ex- focus on possible observational consequences of the Hall– istence of strong small–scale magnetic field structures at instability. the NS surface. Of course, none of the mentioned phenomena can 2. Description of the model be unambiguously attributed to the effect of the Hall– instability.Somepossiblealternativeexplanationsaredis- 2.1. Basic equations, geometry and boundary cussed in Geppert & Rheinhardt (2002). conditions However,thesimultaneousexistenceofasufficientcur- In the absence of convective motions and of ambipolar vature of the backgroundfield profile and of a sufficiently diffusion, conditions found in the solid crust of NSs, the large magnetization parameter ω τ related to that field B e equations governing the magnetic field evolution read (where ω τ >1 is a necessary condition) will very likely B e c lead to the appearance of the Hall–instability. How vig- B˙ =−c curl curlB+ω τ (curlB×e ) B e B orously it develops for a realistically modelled crust and (cid:18)4πσ0 (cid:19) (1) at which strength it saturates, remain as open questions. divB =0, (cid:0) (cid:1) Rheinhardt,Konenkov& Geppert: Hall–Instability in Neutron stars 3 where τ is the relaxationtime of the electrons and ω = atthe surfaceofthe NS. Thebackgroundfieldis assumed e B e|B|/m∗c the electron Larmor frequency, with e being to be parallel to the surfaces of the slab pointing, say, in e the elementary charge, m∗ the effective mass of an elec- x–directionandto be dependent onthe depthz only,i.e., e tron, and c the speed of light. e is the unit vector in B B–direction. The scalar conductivity σ is given by B =B e =f(z)e , (5) 0 0 0 x x Z e2ρ(1−X ) where in comparison to the homogeneous density model n σ = τ . (2) 0 A m m∗ e the minimum demand on f(z) can now be relaxed from u e quadratic to linear in z, if only α is not a constant (see Here ρ denotes the (depth depending) density of the above).Note,that due to this assumptionαcurlB ×B 0 0 crustal matter, and mu the atomic mass unit. By virtue represents a gradient. Thus the unperturbed evolution of of their direct dependences on density, both m∗e and τe thebackgroundfieldisnotatallaffectedbytheHall–drift; depend on the crustal depth, too. The mass and atomic in the absence of any electromotive force it would decay numbers A and Z, respectively, also depend on depth purely ohmically. as well as the fraction of free neutrons Xn (see, e.g., XXX The choice(5)is notonly motivatedby the sake Negele & Vautherin 1973; Haensel & Zdunik 1990). Eq. of simplicity, but to some extent justified by the physical (1) lets one expect that the Hall–drift described by the conditions during the proto–NS stage (after the end of term ωBτe(curlB ×eB) may become important for the a possible field–generating phase): As long as the layers magnetic field decay only if ωBτe &1. which later on form the crust are liquid, the magnetic Linearizationof (1)withrespecttoabackgroundfield field adjusts to be close to a magnetostatic equilibrium. (= reference state) B0 yields: Then it approximately obeys the condition b˙ =−curl(η curlb) 1 curl curlB×B =0 (6) −curl α(curlB0×b+curlb×B0) (3) (cid:18)ne (cid:19) divb=0 (cid:0) (cid:1) with ne being the electron number density. (see Thompson & Duncan 1993, Sect. 14.1)1. As the Hall– describing the behavior of small perturbations b of the coefficientαcanbe writteninthe formα=c/4πn e(see, e reference state. Here, the magnetic diffusivity η, and the e.g., Goldreich & Reisenegger 1992) this is at the same Hall coefficient α are given by time the condition for the Hall e.m.f. in (1) to be a gra- c2 ω τ A m c dient and thus ineffective. However, as the modes of free B e u η = 4πσ , α=η |B| = Z 4πeρ(1−X ) . (4) decayinthecrystallizedcrustareingeneralviolating(6), 0 n the crustalfieldwill tend to deviate increasinglyfrom the Note,thattheHallparameterαistime–independentfora magnetostaticconfiguration.Butat leastfor earlystages, non–accretingNSwhileη depends stronglyonthe crustal one may suppose that although ω τ might already be B e temperature via τ and, therefore, on time. In general, biggerthan unity,the effect ofthe Halltermonthe back- e B is a decaying field, i.e., time–dependent, too. For the ground field is dominated by ohmic dissipation. XXX 0 instability analysis we nevertheless want to treat them as In Fig. 1, the model geometry is shown with three constants. Then the following necessary condition has to differentdepthprofilesofB ,depictingqualitativelysome 0 be satisfied to take the results for valid: The growth time of those we employed. They will be specified later. of an unstable mode has to be significantly shorter than We decompose a perturbation b into poloidal and all the characteristic times of changes of coefficients in toroidal constituents, b = b +b , which can be repre- p t (3) especially significantly shorter than the decay time of sentedbyscalarfunctionsS andT,respectively,byvirtue the backgroundfield. In the following this restriction will of the definitions bereferredtoasthe“backgrounddynamicspermissibility condition”. bp =−curl(ez ×∇S), bt =−ez ×∇T , (7) Employingtheargumentsfromdynamotheoryquoted ensuring divb=0 for arbitrary S,T. in Rheinhardt & Geppert (2002), one can conclude that For the sake of simplicity, we will confine ourselves to αcurlB mustnotbeahomogeneousfield.Moreprecisely, 0 the study of plane wave solutions with respect to the x– wheninterpretingthistermasavelocityfield,itmustnot and y–directions, thus making the ansatz bearigidbodymotion,inordertobecapableofdelivering energytob.Hence,incontrasttothehomogeneousdensity S s model,nowanyinhomogeneousB maypotentiallyenable (x,τ)= (z)exp(ik˜x˜+pτ), (8) 0 the instability. (T ) (t) Let us now specify the geometry of our model. where τ denotes the time, k˜ = (k ,k ), x˜ = (x,y) and p Idealizing the spherical shell of the NS crust, we consider x y is a complex time increment. a plane slab which is infinitely extended both into the x– and y–directions, but has a finite thickness d in z– 1 Thereexistevendipolarfieldsinsphericalshellssatisfying direction.We identify z with the crustaldepth being zero (6) with ne =ne(r) exactly. 4 Rheinhardt,Konenkov& Geppert: Hall–Instability in Neutron stars z =0 However,for k =0 growing solutions with their poloidal -------------B0 --------B0(cid:27)(cid:27)(cid:27)(cid:27)(cid:27)-----B---0 (cid:0)(cid:9)(cid:0)y ?0z -x aℑn{dkxtoRr0do(iαdfal/ηp)ayr(sts∗tm)duztu}a>lly0.coupled seem well possible if 2.2. Crustal matter properties z =d The main goal of the present paper is to enhance our Fig.1. Sketch of the model geometry (B0 - background knowledge on the conditions under which the Hall– fields). In z > d a perfect conductor, in z < 0 vacuum is instability occurs. As one step in this direction we con- assumed. sider a stratified crust characterizedby a realistic density profile instead of a homogeneous crust. Inserting (7) with (8) into (3), we finally obtain two coupled ordinary differential equations for the scalars s and t: FP PS pt−η(t′′−k˜2t)−η′t′ +iαk (f′′s+k˜2fs−fs′′) x EOS medium stiff +iα′(kyft+kxf′s)=0 (9) compactness GM/c2R 0.2 0.13 ps−η(s′′−k˜2s) +iα(k ft−k f′s)=0, central density (gcm−3) 1.17·1015 3.64·1014 x y NS radius (km) 10.64 16.4 where the dash denotes the derivative with respect to z, crust thickness(m) 730 3860 and k˜2 = k2 + k2. In comparison with the correspond- Friedman & Pandharipande x y ing Eqs. (6) in Rheinhardt & Geppert (2002), the terms reference Pandharipande, & Smith, ∝ η′ and α′ occur additionally, but they don’t affect the 1981 1975 energetic conclusions relevant for the existence of the instability. Table 1. Properties of the Pandharipande–Smith (PS) When completed with appropriate boundary condi- and Friedman–Pandharipande (FP) NS models; M = NS tions, these equations define an eigenvalue problem with 1.4M⊙ . respecttothetimeincrementp.Theboundaryconditions chosen here are transition to vacuum at z = 0, and to a The resistivity tensor is strongly dependent on den- perfect conductor at z = d, respectively, mimicking the sity, temperature, and chemical composition of the superconducting core below the bottom of the crust and crust (see Eqs. (2), (4); τ , as far as collisions with e an atmosphere with low conductivity above its surface. phonons are concerned depends on temperature, too, They read see Urpin & Yakovlev (1980)). In turn, these quantities s′+k˜s=t =0 for z =0 are in their temporal behavior mainly determined by (10) the EOS of the NS matter, especially that of the core. s=ηt′−ik αft =0 for z =d. y Therefore, one has first to make a choice among the mul- These conditions are equivalent to the requirements that titude of proposed core EOSs (see, e.g., van Riper 1991). allcomponentsofthemagneticfieldarecontinuousacross We decided to consider two of them, which are gener- the vacuum boundary, and that neither the normal com- ally accepted to be typical representatives of a medium ponent of the magnetic field nor the tangential compo- (Friedman & Pandharipande1981, henceforthFP)anda nents of the electric field penetrate the core. For details stiff(Pandharipande & Smith1975, henceforthPS)EOS, see Geppert & Rheinhardt (2002). respectively, thus probably covering an essential part of The signs of the wavenumbers are irrelevant for the the observed NSs. (The extreme soft EOSs, e.g., that de- eigenvalues of (9) with (10), since the transformations rived by Baym et al. (1971), are now considered unlikely to be realized in nature, see Page et al. 2000) Further on, kx →−kx, p→p, s→s, t→−t we specify the crust in the FP case to consist of cold and (11) catalyzed matter, and in the PS case to be dominated k →−k , p→p∗, s→s∗, t→−t∗ by matter accreted and processed in the past. The cor- y y responding EOSs of the crustal matter and its chemical hold (∗ is the complex conjugate). Thus it is sufficient compositions were derived in Negele & Vautherin (1973) to consider the quadrant kx,ky > 0 of the kx–ky–plane and Haensel & Zdunik (1990), respectively. Since these only.On the basis of (11),it canbe easily concludedthat two models are referred to and utilized in numerous pa- changing the sign of f is irrelevant, too, because pers,we thushope tofacilitate discussionandapplication f →−f, p→p∗, s→s∗, t→t∗ . (12) of our results. Some characteristic properties of the selected mod- As can be inferred from (9) using standard energy els resulting from their density structure specified for arguments all solutions for kx = 0 are damped oscil- MNS = 1.4M⊙ , are summarized in Table 1; Figure 2 lations, being either purely toroidal or purely poloidal. showsthedensityprofiles.Additionally,theprofilesofthe Rheinhardt,Konenkov& Geppert: Hall–Instability in Neutron stars 5 Fig.3. Cooling curves for the FP (squares) and PS (triangles)models.Thesymbolsdenotetheagesatwhichthe stability analysis is performed. Finally, Fig. 4 gives the profiles of the magnetic diffusivity η and the Hall parameter α for both models. For η different profiles are shown corresponding to the ages at which the stability analysis will be performed, but only one profile for α since it doesn’t depend on temperature and hence not on age.For η an impurity concentrationof 1% was assumed. Note, that in contrast to the PS model merelytheimplicitinfluenceofAandZ onηandαviathe densityisincludedfortheFPmodel,theexplicitonebeing Fig.2.Radialdensity(solid)profilesfortheFP(top)and neglectedas wellas the influence ofX . Since the cooling n PS(bottom)models.Additionally,forthelatterthechem- of a NS proceeds faster in the PS model, further cooling ical composition, A, Z, (dashed and dash–dotted, respec- will not affect the conductivity after about 5×105 yrs in tively) is given. this case. Then phonons are practically no longer excited in the crustal crystal and the conductivity is determined by electron–impurity collisions alone. Hence, the diffusiv- chemicalcomposition,that is,A and Z, for the PS model ity doesn’t change any more, while for the FP–model a as derived from the data in Haensel & Zdunik (1990) are significant number of phonons may be excited up to an given. There exist a number of discrete depths (i.e., den- age of approximately 5×106 yrs; therefore η depends on sities and pressures) where the preferred species of nuclei time until this age. change almost abruptly causing discontinuities in A and AccordingtothediscontinuousbehaviorofAandZ in Z. For the FP model, the corresponding data were given the PS model (see Fig. 2), a non–smooth behavior of its onlyimplicitlyintheformofasmoothfittingfunctionfor conductivityparametersappearsinFig.4.Wedecidednot ρ(z). In both cases the crust–coreboundary was assumed to smooth out these discontinuities because it had meant to be at a density of 2 ×1014gcm−3 whereas the crust introducing unnecessarily some further arbitrariness into surface was defined by the density 1010gcm−3. the model. The most important of the EOS’s direct consequences is the degree of compactness: The smaller compactness 2.3. Background field (i.e., larger radius and crust thickness) of the PS model compared with the FP model of equal mass, results from As the second major ingredient of a more realistic crust its stiffer EOS. Correspondingly, the latter model cools model we have to specify background field profiles more down significantly slower than the former for ages higher justified with a view to NS physics than the ad–hoc as- than 105 yrs, since a large compactness (strong gravity) sumptions employed in our former papers. That’s why inhibits the heat transfer (see, e.g., Misner et al. 1973). we turn at this point to a model simulating the Ohmic Fig. 3 shows the cooling curves for both models as cal- decay of the magnetic field in a cooling NS’s crust culated by van Riper (1991). Throughout this paper we (i.e., in a spherical shell), which is just the situation confine ourselves to NSs old enough (say, some 103 yrs) where we expect the most significant effects of the Hall– that the crustal temperature can be considered uniform, instability to occur. When thinking about initial con- and the density profile is no longer changing in time. ditions for such simulations, one has to cope with the 6 Rheinhardt,Konenkov& Geppert: Hall–Instability in Neutron stars (which is partly taken over by the gradient of α in the present case as discussed above). Atthis pointitisnecessaryto discussthe relationship betweentheonlyz–dependentB profileusedintheplane 0 slab geometry, and a both r– and θ–depending profile resulting from decay calculations in a spherical shell. Since B as assumed in (5) must be parallel to the boundaries, 0 the planeslabmodelisonlymeaningfulintheclosevicin- ity of the magnetic equator θ = π/2 (θ – polar angle). f(z) for a certain instant τ has therefore to be identi- fied with the θ–component of B(r,θ,τ) where r =R−z, θ =π/2. Accordingly, the initial backgroundfield is completely specified in geometry and strength after having prescribed f(z) for τ = 0, which we denote as f . (The in restrictiontobackgroundfieldsparalleltotheslabbound- ariescanberelaxedtofieldswithaconstantverticalcom- ponent. We will treat this case in a forthcoming paper.) In view of all these aspects, we generally chose initial profiles showing at least one maximum inside the crust andbeingzerobeneathacertaininitialpenetrationdepth z . We fixed the latter by prescribing the corresponding in densityρ tobeeither1013or1012gcm−3.Thesmallerρ in in resultsinlargerderivativesofthebackgroundfieldprofiles thus possibly favoring the instability at early instances. On the other hand the decay will be accelerated and we have to expect smaller growth rates at higher ages (see Sect.3).Thefollowinginitialprofilesofthe3rdandhigher degrees were chosen: 2 2 R z z Fig.4. Radial profiles of diffusivity η (solid) for different fin(z)=−2Bd (1− ) cubic (13) z R−z z ages of the NS, and Hall coefficient α (broken). Top: FP, (cid:18) in(cid:19) (cid:18) in(cid:19) bottom: PS model. All η profiles for ages ≥107 yrs (FP) R 4 z3 z 4 f (z)=−4B (1− ) heptic(14) and ≥106 yrs (PS) practically coincide. in d z R2(R−z) z (cid:18) in(cid:19) (cid:18) in(cid:19) R R z f (z)= πB sin 4π sinusoidal(15) in d z R−z z in (cid:18) in(cid:19) fact that, unfortunately, there is less certainty about the Here, B is just the initial polar surface magnetic field. d very origin of the NSs’ magnetic fields. Favorite mecha- Note, that the denominator R−z in (13)–(15) doesn’t in- nisms ofmagnetic fieldgenerationareproto–NSdynamos fluence the degree of f with respect to z significantly in (Thompson & Duncan 1993) and thermoelectric instabil- since inside the crust R−z ≈R. ities (see, e.g., Wiebicke & Geppert 1996), but inheriting Thedecaycalculationsarestraightforward(fordetails the magnetic field from the NS’s progenitor seems well see Page et al. 2000); we only mention that the back– possible, too. Since the way of field generation is surely reactionofthemagneticfieldonthetemperaturedistribu- significant with respect to the details of the field struc- tion and cooling history via Joule heating was neglected. ture, there seems to be no other resort when having to Hence, the results depend linearly on the initial condi- specify it than the principle of greatest possible simplic- tions. The same boundary conditions as referred to for ity. Therefore, we decided to consider dipolar fields only. the magnetic field perturbations were used. However, we Further on, it seems to be reasonable to assume with re- refrained from adjusting the initial profiles (13)–(15) to specttotheirradialprofilesthatnotallpartsofthe(inthe the vacuum boundary condition at the surface since the case of a proto–NS dynamo: later) crust equally partici- decaysimulationwilldothisjobautomatically.(Likewise, pate in the field generation. Another process to be taken diffusionsmoothesoutthe‘kink’atz =z .)Allrunswere in into account, is the fallback accretion burying the mag- performed with B =1013G and snapshots of the decay- d neticfieldagainwithinthecrustsoonafteritsemergence. ingfieldweretakenatnineinstantsfrom3×103to3×107 Ontheotherhandandsomewhatconflictingwithmax- yrs in order to define the profiles f(z). Additionally, we imum simplicity, we had to ensure that enough curvature scaled them (that is, B ) by factors 2 or 5 to study the d is ‘injected’ into the radial magnetic field profile initially. dependence of the instability on the initial field strength. See Rheinhardt & Geppert (2002) for the essential role Figures 5 to 7 show typical examples of the result- of the second derivative of the background field’s profile ing f(z) for both models. (See also Fig. 1 for schematic Rheinhardt,Konenkov& Geppert: Hall–Instability in Neutron stars 7 sketches of characteristic background fields.) Moreover, these figures present the magnetization parameter ω τ B0 e and — only for the FP model, in Fig. 5 — the quantity (αf′)′/η,furtheronreferredtoas“curvatureparameter”. Fig.5. Background field B0, magnetization parameter ωB0τe and curvature parameter (αB′)′/η resulting from the FP 0 model with sinusoidal initial profile (13) and initial penetration density ρin = 1013gcm−3, and Bd = 1013G. Red parts indicate adherence to thesign condition (16). 8 Rheinhardt,Konenkov& Geppert: Hall–Instability in Neutron stars Fig.7. As Fig. 6, but with sinusoidal initial profile (15). Fig.6.BackgroundfieldB andmagnetizationparameter 0 ω τ resulting from the PS model with cubic initial pro- B0 e file (13) and initial penetration density ρ =1013gcm−3, in and B =1013G. d On the other hand, we want to stress that the magnitude of the curvature parameter alone is again not sufficient to infer the existence of the instability, which relies It is an estimate for the ratio of the potentially energy– onacomplicatedinterplayofvectorquantities.Therefore, deliveringtermcurl(αcurlB ×b)tothe(anyway)energy– 0 an additional qualitative criterion must be satisfied in or- dissipating term curl(ηcurlb) in (3). (Note that the term der to have the potentially energy–delivering term in (3) curl(αcurlb×B0) is energetically neutral.) Therefore we reallyenergy-delivering.Fromsimplequalitativeconsider- suppose that the curvature parameter rather than ω τ B0 e ationswithaxisymmetricfieldsinahomogeneousmedium itselfisthe keyindicatorwithrespectto the vigourofthe it is possible to conclude that f′′(z)f(z) < 0 has to be instability. For the PS model, the curvature parameter is satisfied,atleastoveracertainz-interval.Orgeneralized: only piecewise continuous due to the non–smooth profiles Backgroundfield and curvature parametermust havedif- of η which resultin f′′ profiles only piecewise continuous, ferent signs: too. Nevertheless, the solutions of (9) are well defined, at least in the mathematically weak sense. (αf′)′(z)f(z)<0. (16) Rheinhardt,Konenkov& Geppert: Hall–Instability in Neutron stars 9 3. Results In Figs. 9 to 14 the growth times of the fastest growing modes 1/max(ℜ(p)) (simply referred to as “growth 3.1. Numerics time” τ of a specific model) for the most important growth cases considered, are presented as functions of age and The system of ordinary differential equations (9) was dis- initial field strength B . In addition, the tangential pe- cretized by help of symmetric difference formulae of sec- d riod lengths of these modes, λmax, are given; because of ond order; near boundaries unsymmetric formulae were kmax =0 this quantity is defined as employed, if necessary. In most cases an equidistant grid y withatypicalnodenumberof1200wasused,butinquite 2π a number of spot checks of convergence the node number λmax = . (17) kmax was repeatedly doubled up to 9600 maximum. x For the numerical solution of the resulting com- It should be regarded as one of the dominating scales of plex non-Hermitian standard matrix eigenvalue prob- the unstable modes since,incontrastto the homogeneous lem we took benefit of the package ARPACK (routines density model, in the majority of cases no prominently znaupd, zneupd).Someresultswerecheckedapplyingthe small radial scales were found (see Fig. 15). Instead, the LAPACK routine zgeev. The results from both packages radialscalesoftheunstablemodesseemtobedetermined agreein at leastsix digits (but comparisonswere possible simply by the radialscaleof the backgroundfield.Except for moderate node numbers only). for some earlier stages of the PS model, λmax depends in Unfortunately,theorderofthedifferenceformulaewas generalonlyslightlyonB .Since thewavenumberk was d x not always reflected by the convergence rate of the eigen- given only discrete equidistant values, the period lengths valueswithrespecttothenodenumber:ForthePSmodel for different values of B frequently even coincide (indi- d convergencewas usually significantly slower.Perhapsthis catedinthefiguresbyfilledsymbolsinsteadofopenones). behavior is connected with the discontinuities of the coefficients η′, α′ and f′′ described above. This supposition 3.3. Growth times is supported by the fact that the convergencerates of the FP model are close to (although not exactly) quadratic. We stress again that the growth times have to be considered as referring to snapshots only. That is, the value for 3.2. General aspects of the results a given NS age has been calculated assuming that the instability startsjustatthatage,with the backgroundfield As with the homogeneous density model, both oscillat- given at exactly that moment. Clearly, the results must ing and non-oscillating unstable modes exist, where the not be interpreted as a sort of temporal evolution of the fastest growing mode of any specific model turned out to growth times: Any occurrence of the Hall–instability will bealwaysamongstthelatter.Asanothercommonfeature affect the strength and structure of the background field, of the fastest growing modes, their wavenumber k has thereby changing the conditions for the occurrence of the y always been found to be zero. When deriving the back- instabilityitselfatlatermoments.Thisshortcomingofthe ground field from an axisymmetric poloidal one in the resultsinducesustoemphasizeoncemorethenecessityof vicinity of its magnetic equator (as we did), association full non–linear calculations. of the y–co–ordinate with longitude is surely the proper Asanoverallproperty,westatethatthedependenceof choice. Thus, with some care one may suppose that in a τ upon the initial polar magnetic field B is always growth d sphericalshell the most unstable modes are preferentially monotonically falling. Hence, we never entered the range axisymmetric. of B where higher values may yield larger growth times d Figure 8 shows growth rate ℜ(p) and oscillation fre- (see Rheinhardt & Geppert 2002). quency ℑ(p) as functions of k and k for a typical ex- x y ample (PS model, age = 3×105 yrs, cubic initial field, 3.3.1. FP model B =5×1013G,ρ =1013gcm−3).Therectangleenclos- d in ing the unstable region in the kx–ky–plane is defined by The growthtimes of the FP model (Figs.9 to 11, Tab.2) 0.4/d ≤ kx ≤ 18.4/d, 0 ≤ ky ≤ 34.4/d. But, when obey- have in common that they start with small values (most ingthebackgrounddynamicspermissibilitycondition(see in the order of magnitude 103 yrs and smaller) at the Sect. 2.1) only a considerablysmaller unstable regioncan youngest age, exhibit a nearly linear dependence on age be considered feasible. (a power law with an exponent between 0.8 and 1) until In contrast to the growth rates, the oscillation fre- 105... 3×105yrs,reachamaximumat106... 3×106yrs, quencies grow in general with k ; their highest values and fall progressively with age later on. At early stages y seem to occur always at the edge of the unstable re- the background dynamics permissibility condition is well gion in the k –k –plane, that is, for ℜ(p) → 0 (and fulfilled for B = 5× 1013G but must be put in ques- x y d k 6= 0). Note, that the oscillating unstable modes can tion in most cases with B = 1013G. Analogously, the y d be regarded as helicoidal waves with growing amplitudes. maximum growth times are reliable for most cases with (Cf. Vainshtein et al. (2000) who considered damped he- ρ = 1013gcm−3, but questionable for some of the cases in licoidal waves in a stratified crust.) with ρ = 1012gcm−3 and B = 1013G. At the latest in d 10 Rheinhardt,Konenkov& Geppert: Hall–Instability in Neutron stars Fig.8.Eigenvaluepasfunctionofthewavenumbersk ,k forthePSmodelwithcubicinitialbackgroundfieldprofile x y (13), initial field B = 5×1013G, initial penetration density ρ = 1013gcm−3, age 3×105 yrs. Left: growth rate d in ℜ(p), negative values were set to zero; right: frequency ℑ(p), not shown for ℜ(p) < 0. p in units of 4/(d/cm)2s−1 = 2.68×10−11s−1, wave numbers in units of 2/d=5.18×10−6cm−1. stage considered all the growth times satisfy the condition. We note that the dependence of the growth time on B is not far from being linear for ρ = 1013gcm−3, d in whereas for ρ =1012gcm−3 the dependence is of higher in order.Inthiscase,withthecubicprofileandB =1013G d even a “gap” in the growth time curve occurs: From the ageof104totheageof3×105yearstherewerenogrowing modes to be found at all. Fig.9.Growthtimesandtangentialperiodlengthsofthe fastest growing modes for the FP model with cubic initial background field profile (13), initial penetration density ρ =1013gcm−3. Solid, diamonds: B =5×1013G; Comparing the growth times for different values of in d dashed, triangles: B = 1013G. Thick lines, big symbols: ρ , one has to state that the smaller initial penetration d in depth (ρ =1012gcm−3), althoughbeing connected with growth times; thin lines, small symbols: period lengths. in Full smalldiamonds:coincidenceofthe periodlengths for strongergradients of the backgroundfield, is nevertheless different values of B . disfavoringtheinstability.Thistendencybecomesincreas- d ingly apparent with growing age. We explain it with the accelerated decay of these “shallower”profiles. Comparingthegrowthtimesfordifferentinitialprofile types, one can see that the values for the heptic and the sinusoidalprofilesareclosetogetheratthe earlierstages Fig.12. Growth times and tangential period lengths of thefastestgrowingmodesforthePSmodelwithcubicini- tial background field profile (13), initial penetration density =1013gcm−3. Solid, diamonds: =5×1013G;