arXiv:0902.3111v1 [astro-ph.HE] 18 Feb 2009 e-uHu Ren-Yu progenitors magnetar as stars massive Magnetized
o.Nt .Ato.Soc. Astron. R. Not. Mon. cetd20 eray1.Rcie 09Fbur 6 nori in 06; February 2009 Received 16. February 2009 Accepted antcfil tegh osdrbysrne hnthe than surface stronger with of stars value considerably critical neutron strengths quantum be field to magnetic believed are Magnetars INTRODUCTION 1 oy@snhaeuc,luodo.ciaoeu(Y-QL) [email protected] [email protected], very as well ⋆ as review extensive an ten and list and Mereghetti latest SGRs (see a six observationally for now, 2008 identified to been have Up AXPs (AXPs). Anoma- Pulsars (ii) X-ray and magne- (SGRs) lous for Repeaters Gamma-ray manifestations Soft observational (i) tars: of types main two c 3 2 1 ainlAtooia bevtre,CieeAaeyo S of Academy o Chinese University Observatories, The Astronomical Astrophysics, (T National and Astrophysics Astronomy for of Centre Department Tsinghua and Department Physics -al [email protected] RH and (RYH) [email protected] E-mail: 08RAS 2008 1 ⋆ n uQn Lou Yu-Qing and 000 B QED , ?? – opc bet(..nurnsa)lf eida h etewt a with centre the at we behind consid envelope, left the and star) and interior neutron surface stellar (i.e. stellar the object progenitor through compact the breaking on shock temperatu rate density, bound loss mass as mass such symmet wind parameters quasi-spherical physical a of self-simila with progenitor fication a self-gravity of massive the framework of under theoretical dynamics the magnetofluid in collapse mystery fields core a magnetic is surface the magnetars model on We fields magnetic physics. ultra-intense of origin The ABSTRACT madams ag of range mass a and cm tr:nurn—sproarmat ht dwarfs white — remnants supernova — neutron stars: — h neligcue o aiu bevdhg-nryactivities words: high-energy Key pie observed broken various and of for energies causes rearrangement magnetic underlying accumulated as of the lik well releases equa most violent as and rotation, is ropes Sporadic Couple differential magnetar flux convection, extents. a magnetic various exposed interior of to of mag configuration pieces presence a magnetospheric into the inside star the force neutron spin, Lorentz a tar intense of magne The crust of magnetars. continuum the fi a to magnetic allows pulsars surface scenario of from our range stars, magn a progenitor strongly With massive the stars. from progenitor magnetars massive forming inside of scenario field’ ‘fossil olpeadrbudsok u hsclmcaim hc osno does which g mechanism, the physical st of Our massive evolution quire shock. magnetized rebound magnetohydrodynamic that and the collapse propose on We based progenitor. progenitors crit massive factor the enhancement of index field scaling (S magnetic self-similar repeaters The the gamma-ray (AXPs). soft pulsars include X-ray which magnetars for ferred fsc ido opc bet a be can objects compact of kind such of 4 = ?? 20)Pitd2 coe 08(NL (MN 2018 October 24 Printed (2008) . 4 × dhoc ad 10 13 .Teeare There G. yaoapicto ihnafs pnignurnsa,fvusth favours star, neutron spinning fast a within amplification dynamo 1 anthdoyais(H)—sokwvs—sas antcfie magnetic stars: — waves shock — (MHD) magnetohydrodynamics , 2 , 3 ⋆ ia om20 eebr16 December 2008 form ginal ine 2,DtnRa,Biig 002 China 100012, Beijing, Road, Datun A20, cience, C) snhaUiest,Biig108,China 100084, Beijing University, Tsinghua HCA), ∼ USA 60637, IL Chicago, Avenue, Ellis South 5640 Chicago, f 1 − n iha nerditnemgei edof J1846-0258 field magnetic PSR intense pulsar inferred an also rotation-powered with are a emissions from X-ray Magnetar-like detected 2005). al. et Palmer il o aiu iheeg ciiis o xml h g the example respon- for are activities, they ant high-energy and various Universe on for the fields sible in magnetic unique surface are ultra-intense magnetars The al stars neutron magnetars. 2008), isolated al. to dim ordinary et from (Kouveliotou Castro-Tirado continuum flares a 2008; luding optical al. fast et very Stefanescu mag- Galactic 2008; with new reported a recently, is Most s). netar short- 2.07 a of with period J1550-5418 spin est SGR of explosions powerful recent hc lodtrie h nta est distribution density initial the determines also which , M 3 γ ⊙ ryflr fSR10-0(..Hre ta.2005; al. et Hurley (e.g. 1806-20 SGR of flare -ray oevr efidta ufc antcfields magnetic surface that find we Moreover, . ∼ A 10 T E tl l v2.2) file style X 14 − 10 15 ,cnitn ihtoein- those with consistent G, oilble ussof bursts bulge, torial e antcfil and field magnetic re, R)adanomalous and GRs) fmagnetars. of eea polytropic general r tzdcr collapse core etized ihtemagne- the with d y ihtespeci- the With ry. rkncutare crust broken a l tegh over strengths eld clydpnson depends ically i edstrengths field tic ea a break may netar aiainlcore ravitational nmdr astro- modern in r smagnetar as ars e ftecrust. the of ces ∼ aisof radius eesrl re- necessarily t rto fare- a of eration tr ihhigh with stars n remnant a find l aibein variable ely 4 . 9 × 10 13 ∼ at G 10 lds i- e 6 - 2 R.-Y. Hu and Y.-Q. Lou the centre of supernova remnant Kes75 (e.g. Gavriil et al. couple of early B stars, for example the B0.5V star HD 37061 2008; Archibald et al. 2008). (∼ 650 G, e.g. Hubrig et al. 2006). Petit et al. (2008a,b) Recent observations have also provided clues connecting carried out systematic spectropolarimetric observations to magnetars with very massive progenitor stars, for example search for magnetic fields on all massive OB stars in the an infrared elliptical ring or shell was discovered surround- Orion Nebula Cluster star-forming region. Strong magnetic ing the magnetar SGR 1900+14 (e.g. Wachter et al. 2008). fields of the order of kG were inferred on 3 stars out of a However, the formation of magnetars, especially the origin of sample of 8. The existence of strong magnetic fields on OB the ultra-intense magnetic field, remains an important open stars even appears somewhat overwhelming in contrast to issue. There are two major contending physical scenarios, very few magnetars that have been discovered so far. viz. the dynamo scenario versus the fossil-field scenario. With the assumption that neutron stars form dur- Duncan & Thompson (1992) and Thompson & Duncan ing the collapse of massive progenitors in the Galac- < < (1993) explored the turbulent dynamo amplification, occur- tic disc with 8 ∼ M/M⊙ ∼ 45 (stellar masses in ring primarily in the convection zone of the progenitor, as the main-sequence phase), and ∼ 8 percent of massive well as in a differentially rotating nascent neutron star, and stars have surface magnetic fields higher than ∼ 1000 concluded that very strong magnetic field, in principle up G, Ferrario & Wickramasinghe (2006) estimated that these to ∼ 3 × 1017 G, may be created. The dynamo mechanism high-field massive progenitors gave birth to 24 neutron stars > 14 requires an extremely rapid rotation of a nascent neutron with magnetic field ∼ 10 G, consisting a major part of star with a spin period of a few milliseconds. However, the magnetars. While the fossil-field scenario appears plausible current population of magnetars appears to be slow rota- from the perspective of statistics, it is highly instructive to tors, having spin periods in the range of ∼ 2 − 12 s (e.g. have a more direct magnetohydrodynamic (MHD) model Mereghetti 2008). Therefore, neutron star dynamo scenario description for the core collapse of high-field massive pro- for magnetars faces a considerable challenge to account for genitor stars and to check whether compact remnants left the fact of slowly rotating magnetars as observed so far. behind MHD rebound shocks do possess ultra-intense mag- The fossil-field scenario for the mag- netic fields. netism of compact objects was first pro- In this paper, we attempt to model magnetized massive posed to explain magnetic white dwarfs (e.g. progenitor stars with a quasi-spherical general polytropic Braithwaite & Spruit 2004; Wickramasinghe & Ferrario magnetofluid under the self-gravity (Wang & Lou 2008; Lou 2005; Ferrario & Wickramasinghe 2005; Lou & Wang 2007). & Hu 2009). We examine semi-analytic and numerical so- It is conceivable that the magnetic field of white dwarfs may lutions to explore the self-similar MHD evolution emerging be of fossil origin from the main-sequence phase of their from dynamic processes of core collapse and rebound shock progenitors, and the attempt to link magnetic white dwarfs travelling in the stellar envelope with a wind mass loss. More with their main-sequence progenitors naturally makes the specifically, we adopt a general polytropic equation of state chemically peculiar Ap and Bp stars as plausible candi- (EoS) p = κ(r, t)ργ with p, ρ, γ, and κ respectively be- dates. Observations of Auri`ere et al. (2003) have shown ing the gas pressure, mass density, polytropic index and a that chemically peculiar Ap and Bp stars are generally proportional coefficient dependent on radius r and time t. magnetic indeed, with a surface magnetic field of ∼ 100 G Here, κ is closely related to the ‘specific entropy’ and is not by Zeeman splittings. In general, magnetic field strengths necessarily a global constant. By ‘specific entropy’ conser- fall in the range of ∼ 3 × 102 − 3 × 104 G (e.g. Braithwaite & vation along streamlines, another key parameter q arises in Spruit 2004 and references therein). Magnetic white dwarfs self-similar dynamics (see Wang & Lou 2008). For κ being may be created as a result of rebound shock explosion a global constant, or equivalently q = 0, the general poly- (Lou & Wang 2007) and may further give rise to novel tropic EoS reduces to a conventional polytropic EoS. By magnetic modes of global stellar oscillations (Lou 1995). By further setting γ = 1, a conventional polytropic gas reduces the magnetic flux conservation during the stellar evolution, to an isothermal gas (e.g. Lou & Shen 2004). We also require Ferrario & Wickramasinghe (2005) argued that stellar γ > 1 to ensure a positive specific enthalpy p/(γ − 1). magnetic fields (∼ 100 G) in their main-sequence phase Chiueh & Chou (1994) studied the isothermal MHD by can be enhanced up to the range of ∼ 106 − 109 G on the including the magnetic pressure gradient force in the radial surface of magnetic white dwarfs. This fossil-field scenario momentum equation. Yu & Lou (2005), Yu, Lou, Bian & Wu is supported by the statistics for the mass and magnetic (2006), Wang & Lou (2007) and Wang & Lou (2008) gener- field distributions of magnetic white dwarfs. alized the self-similar hydrodynamic framework by including Based on the same scenario and a similar physical argu- a completely random transverse magnetic field with the ap- ment, Ferrario & Wickramasinghe (2006) further suggested proximation of a ‘quasi-spherical’ symmetry (e.g. Zel’dovich that the ultra-intense magnetic field over the surface of mag- & Novikov 1971); the radial component of such magnetic netars may also come from fossil magnetic fields. The pro- field is much weaker than the transverse components. We genitors of magnetars are expected to be O stars and early conceive a simple ‘ball of thread’ scenario for random mag- B stars with high surface magnetic fields of ∼ 1000 G. netic fields in a massive progenitor star. In other words, a Ferrario & Wickramasinghe (2008) presented a population magnetic field line follows the ‘thread’ meandering within synthesis study of the observed properties of magnetars and a thin spherical ‘layer’ in space in a completely random found that magnetars arise from high mass progenitor stars manner. Strictly speaking, there is always a weak radial < < (20M⊙ ∼ M ∼ 45M⊙). magnetic field component such that field lines in adjacent Up to now magnetic fields are directly measured for two ‘layers’ remain physically connected throughout in space. O stars, namely θ1Ori C (∼ 1 kG, e.g. Donati et al. 2002) In particular, we emphasize that the nature of the ran- and HD 191612 (∼ 1.5 kG, e.g. Donati et al. 2006), and a dom magnetic fields inside a progenitor star may be ei-
c 2008 RAS, MNRAS 000, ??–?? Intense Magnetic Fields on Magnetars 3
2 ther those generated by dynamo mechanism probably linked where u is the bulk radial flow speed and < Bt > is the en- with convection and differential rotation (e.g. Spruit 2002; semble mean square of a random transverse magnetic field. Heger, Woosley & Spruit 2005), or those of ‘fossil fields’ en- The weak radial component of the magnetic field is deter- trained from molecular clouds in dynamic processes of star mined by equations (10) and (11) of Yu & Lou (2005). With formation (e.g. Wang & Lou 2007, 2008). the self-similar MHD transformation of Wang & Lou (2008), According to numerical simulations of differentially ro- the ideal MHD equations together with the magnetic flux tating magnetized stars by Heger et al. (2005), the dynamo- frozen-in condition and the general polytropic EoS can be powered radial magnetic fields in a progenitor star are about readily reduced to a set of nonlinear MHD ODEs in the 3 to 4 orders of magnitude weaker than the transverse mag- highly compact form of netic fields during the pre-supernova evolution. In line with ′ ′ X (x, α, v)α = A(x, α, v) , X (x, α, v)v = V(x, α, v) , (2) this hint, we simplify the treatment by only dealing with the dominant transverse field and focus on their dynamic effects where the prime ′ stands for the first derivative d/dx, and on the bulk motion of gas in the radial direction. By taking the three functionals X , A and V are defined by the ensemble average of magnetic fields in each thin spherical (3n − 2) ‘layer’, we smooth out small-scale magnetic field structures X (x, α, v) ≡ C 2 − n + q » 2 – and are left with ‘layers’ of large-scale transverse magnetic − fields. We also presume that small-scale non-spherical flows ×α1 n+3nq/2x2q (nx − v)q + hαx2 − (nx − v)2 , as a result of the magnetic tension force may be neglected as compared to large-scale mean radial bulk flow motions. x − v A(x, α, v) ≡ 2 α Cqα1−n+3nq/2x2q (nx − v)q−1 Therefore on large scales, a completely random magnetic x ˆ field contributes to the dynamics in the form of the average (nx − v) +(nx − v) − α (n − 1)v + α + 2hαx magnetic pressure gradient force and the average magnetic » (3n − 2) tension force in the radial direction. ˜ This theoretical model framework of self-similar MHD +Cqα1−n+3nq/2x2q−1(nx − v)q−1(3nx − 2v) , – has been applied to gravitational core collapse and rebound shock processes within progenitor stars for supernovae. (x − v) 3n V(x, α, v) ≡ 2 α C 2 − n + q Lou & Wang (2006, 2007) modelled the hydrodynamic and x » „ 2 « MHD rebound shocks of supernovae in the self-similar phase of evolution. Hu & Lou (2008b) presented preliminary re- ×α−n+3nq/2x2q(nx − v)q + hx2 sults for a shock breakout to reproduce the early X-ray light – curve of supernova SN 2008D (e.g. Soderberg et al. 2008; (nx − v) −(nx − v) (n − 1)v + α + 2hαx Mazzali et al. 2008). In this paper, we demonstrate that such » (3n − 2) a self-similar MHD process may give birth to a compact +Cqα1−n+3nq/2x2q−1(nx − v)q−1(3nx − 2v) . (3) remnant with a nuclear density and a range of ultra-intense – surface magnetic fields. We will see that massive progenitor stars whose collapsing cores have magnetic fluxes similar In this straightforward yet somewhat tedious derivation, we 2 to those of magnetars will eventually collapse into neutron have useful relations m = αx (nx − v) and q ≡ 2(n + γ − stars with a magnetar level of magnetic fluxes because of 2)/(3n − 2) and we have performed the following MHD self- the magnetic flux conservation. In our model scenario, the similar transformation for a general polytropic gas, viz. neutron star dynamo processes and the required initial rapid r = k1/2tnx, u = k1/2tn−1v , spins of nascent neutron stars may not be necessary.
α kt2n−4 ρ = , p = Cαγ mq, 4πGt2 4πG 2 GENERAL POLYTROPIC SELF-SIMILAR MAGNETOHYDRODYNAMICS k3/2t3n−2m kt2n−4 M = , < B2 >= hα2x2, (4) (3n − 2)G t G Under the approximation of quasi-spherical symmetry and based on the physical idea outlined in the introduction, where G = 6.67×10−8 g−1 cm3 s−2 is the gravitational con- the ideal MHD equations involve mass conservation, radial stant, M is the enclosed mass at time t within radius r, x momentum equation, specific entropy conservation along is the independent self-similar variable, v(x) is the reduced streamlines and the magnetic induction equation (see Wang flow speed, α(x) is the reduced mass density, m(x) is the re- & Lou 2008). We highlight the essential parts of this formu- duced enclosed mass, k, n and C are three parameters, and lation of nonlinear MHD equations below. the dimensionless coefficient h is referred to as the magnetic 2 2 2 2 parameter such that < Bt >= 16hπ Gρ r . It follows that q = 2/3 leads to γ = 4/3 for a relativistically hot gas. Only 2.1 Theoretical MHD Model Formulation for the case of q = 2/3, can the parameter C be indepen- dently chosen; otherwise for q =6 2/3, C can be set to 1 with- The ideal magnetic induction equation (without the resistiv- out loss of generality (Lou & Cao 2008; Lou & Hu 2009). The ity) implying the frozen-in condition for the magnetic flux magnetosonic critical curve (MCC) is determined by the si- can be cast into the following form multaneous vanishing of the numerator and denominator on ∂ ∂ ∂u the right-hand sides (RHS) of ODEs (2) and (3). Once so- + u (r2 < B2 >) + 2r2 < B2 > = 0 , (1) „ ∂t ∂r « t t ∂r lutions are obtained for v(x) and α(x), the mean magnetic
c 2008 RAS, MNRAS 000, ??–?? 4 R.-Y. Hu and Y.-Q. Lou
2 1/2 field strength < Bt > can be readily determined from q = 0 or n + γ = 2 is included here and corresponds to a self-similar MHD transformation (4). With proper asymp- conventional polytropic gas in magnetostatic equilibrium.A totic conditions as well as eigen-derivatives across the MCC, further special case of n = γ = 1 corresponds to a magneto- nonlinear MHD ODEs (2) and (3) can be numerically in- static singular isothermal sphere (SIS). Physically, expres- 2 tegrated by using the standard fourth-order Runge-Kutta sion (7) requires q =6 2/3 and h < hc ≡ n /[2(1 − n)(3n − 2)] scheme (e.g. Press et al. 1986). for the existence of the global magnetostatic SPS solution It is straightforward to treat MHD shock conditions in a general polytropic gas. For n = 4/5, hc reaches the for self-similar solutions to cross the magnetosonic singu- minimum value hc = 4. This places a constraint only when lar surface X (x, α, v) = 0. By the conservations of mass, n< 1; while for n > 1, parameter h> 0 is fairly arbitrary. radial momentum, and energy as well as magnetic induc- There exists an analytic asymptotic MHD solution ap- tion equation in the comoving framework of reference across proaching the magnetostatic SPS solution at small x (re- an MHD shock front, we obtain a set of jump conditions ferred to as the ‘quasi-magnetostatic’ asymptotic solution), K −2/n K−1−2/n for an MHD shock that can be cast into a self-similar form namely v = Lx and α = A0x + Nx where K (see Appendix A, Wang & Lou 2008 and Lou & Hu 2009). is the root of the following quadratic equation Note that the so-called ‘sound’ parameter k in self-similar n2 (3n − 2) (3n − 4) MHD transformation (4) is related to the polytropic sound + nh + Q K2 + K − speed and changes across a shock front, with the relations »2(3n 2) 2 –» n – 2 k2 = λ k1, h1 = h2 = h, x1 = λx2 where subscripts 1 and 2(2 − n)(1 − n) n2 + (3n − 2)2(1 − 4/n)Q 2 refer to the immediate upstream and downstream sides of + h+ = 0 , (8) a shock and λ is a dimensionless scaling parameter. Strictly n (3n − 2) speaking, magnetic fields have very weak radial components where Q ≡ q n2/[2(2 − n)(3n − 2)] − (1 − n)h/(2 − n) is normal to the shock front. Our treatment of magnetic field introduced for˘ notational clarity (Lou & Wang 2006, 2007).¯ coplanar with the shock front represents a very good ap- In a certain regime, at least one root of quadratic equation proximation for our purposes. (8) satisfies ℜ(K) > 1 and therefore quasi-magnetostatic solutions do exist. The two coefficients L and N are simply 2.2 Analytic Asymptotic MHD Solutions related by the following algebraic expression
The analytic asymptotic solutions at large x of nonlinear n(K − 1)N =(K + 2 − 2/n)A0L . (9) coupled MHD ODEs (2) and (3) are In this case, the magnetic Lorentz force (i.e. magnetic pres- α = Ax−2/n + · · · , sure and tension forces together) and the gas pressure force are in the same order of magnitude in the regime of small x. It can be proved that the parameter regimes where 1−1/n n 2h(n − 1) v = Bx + − + A quasi-magnetostatic solutions exist are γ > 1, h < hc, »(3n − 2) n – q < 2/3, n < 0.8 and γ > 1, h < hc, q > 2/3. With pa- +2(2 − n)nq−1A1−n+3nq/2 x1−2/n + · · · , (5) rameters outside these two regimes, the so-called strong-field ff asymptotic solutions at small x, for which the magnetic force where A and B are two integration constants, referred to as dominates over the thermal pressure force, have been shown the mass and velocity parameters (Wang & Lou 2008). To to exist (Yu et al. 2006; Wang & Lou 2007, 2008). ensure the validity of asymptotic MHD solution (5), we re- quire 2/3 < n 6 2; the inequality n> 2/3 is directly related to self-similar MHD transformation (4) where a positive M 3 FORMATION OF COMPACT MAGNETARS is mandatory on the ground of physics. For 2/3 < n 6 1, both mass and velocity parameters A > 0 and B are fairly 3.1 Model Progenitors and Compact Remnants arbitrary. In case of 1 < n 6 2, we should require B = 0 To be specific, the radial range of our model solutions is 6 to avoid a divergent v(x) at large x unless a flow system set within ri
c 2008 RAS, MNRAS 000, ??–?? Intense Magnetic Fields on Magnetars 5 stellar surface can be inferred from the observed range. From field solutions (Wang & Lou 2007, 2008). It is desirable that these quantities, one estimates the dimensionless magnetic after a long lapse (i.e. t → ∞), the enclosed mass within ri parameter h, which plays an important role in the MHD evolves towards a constant value and form a compact rem- evolution of quasi-magnetostatic solutions. By self-similar nant of nuclear mass density. Asymptotically, the enclosed MHD transformation (4) and the ideal gas law, we have mass takes the form of 1/n k T nk A0 − Ck1−3q/2 = B , (10) M = r3 2/n (14) µ(4πG)γ−1Gq (3n − 2)q ργ−1M q (3n − 2)G where T is the gas temperature, kB is Boltzmann’s con- for quasi-magnetostatic solutions and becomes independent stant, and µ is the mean molecular (atomic) weight for gas of t. This appears consistent with the scenario of form- particles. For a gas mainly of ionized hydrogen and a stellar ing a central compact object with a strong magnetic field mass of several solar masses, one can estimate parameter k (Lou & Wang 2007). In a companion paper, we will show according to expression (10) from γ and the local density ρ that the asymptotic enclosed mass for strong-field solutions and temperature T . For typical values of ρ ∼ 10−5 g cm−3 depends on t, and may be invoked to model a continuous and T ∼ 105 − 106 K on the stellar surface of a massive pro- accretion or outflow around a nascent neutron star (Hu & genitor in the late phase (just before the gravitational core Lou 2009, in preparation). As quasi-magnetostatic solutions collapse) and γ = 1.3 and n = 0.7, we estimate a range of require that h < hc, we arrive at an interesting situation, i.e. k ∼ 1016 − 1017 cgs unit. in order to give birth to a stable neutron star with an ultra- With t → 0+ and/or r → ∞ (i.e. for the regime of intense magnetic field, the massive progenitor star needs to large x), self-similar MHD solutions follow asymptotic form be magnetized but not too much. We shall see presently that (5). For such analytic asymptotic solutions, the mass density h < hc is generally satisfied for massive main-sequence stars. simply scales as We now introduce the outer initial mass Mo,ini and the inner ultimate mass M in the same manner as done Ak1/n i,ult ρ = r−2/n (11) in Lou & Wang (2006, 2007) and regard them as rough 4πG estimates for the the initial progenitor mass and the fi- and is independent of time t. The reason to refer A as the nal mass of the remnant compact object, respectively. The mass parameter is now apparent. With asymptotic mass ratio of these two masses is given by Mo,ini/Mi,ult = density scaling (11), one can estimate A from the surface ∗ (3−2/n) ∗ −2/n λ (ro/r ) with λ ≡ (A/A0)λ . The ratio of outer mass density of a progenitor star. We have the radial flow i initial magnetic field at the surface of the progenitor star to velocity the inner final magnetic field of the central compact rem- 1/(2n) 1−1/n 2 1/2 2 1/2 ∗ (1−2/n) u = Bk r , (12) nant is < Bo,ini > / < Bi,ult > = λ (ro/ri) . The factor λ∗ is insignificant as compared to the radial vari- also independent of t. The radial flow velocity near the stel- ation of magnetic field, i.e. the very radial dependence of lar surface relates to the mass accretion rate or mass loss r(1−2/n). As n → 2/3 and the polytropic index γ approaches rate by − 4/3, this scaling approaches r 2. With ro = 1012 cm and ˙ 2 6 M = −4πr ρu . (13) ri = 10 cm, the magnetic field strength can be rapidly en- hanced by a factor up to ∼ 1012. Thus, for a magnetar (i.e. With expressions (12) and (13), one can determine velocity neutron star) to have a surface magnetic field strength of parameter B from the stellar mass loss rate for B > 0 or < B2 >1/2∼ 1015G, we need a magnetic field of ∼ 103G mass accretion rate for B < 0. Observationally, mass loss i,ult − − over the progenitor stellar surface, which is attainable for rates of Galactic OB stars fall in the range of ∼ 10 5 −10 7 −1 magnetic massive OB stars. M⊙ yr (e.g. Lamers & Leitherer 1993). Mass loss rates of Wolf-Rayet stars are much higher than massive main- −4 −6 sequence stars, falling in the range of ∼ 10 − 10 M⊙ 3.2 Numerical Model Calculations yr−1 (e.g. Singh 1986). Very recently, Puls, Vink & Najarro (2008) provide an extensive review on mass loss rates of The initial time to apply our solutions is estimated by massive stars. the time when the rebound shock crosses ri, viz. t1 = 1/2 1/n In summary, with the mass density, temperature, mag- [ri/(k xs)] (Lou & Wang 2006). Here we suppose that netic field and mass loss rate specified at the surface of a roughly from t1 on the collapse and rebound shock inside massive progenitor star, one can determine all parameters of the progenitor star have already evolved into a self-similar asymptotic solution (5) and integrate nonlinear MHD ODEs phase (typically this process takes a few milliseconds). The (2) and (3) inwards to produce an MHD profile for the pro- rebound shock travels outwards through the stellar interior 4 6 genitor interior and envelope. We still have the freedom to into the envelope in ∼ 10 − 10 s (e.g. Lou & Wang 2007; choose the rebound shock position after the gravitational Lou & Hu 2009; Hu & Lou 2008a). We set t2 as the time core collapse. Physically, the speed of the rebound shock de- when the rebound shock reaches ro, roughly around the pends on the core collapse, for example the EoS and the neu- shock breakout. We will also show self-similar MHD solu- trino reheating, etc. Here we simply treat it as an adjustable tions at tm1 = 1 s as an intermediate time between t1 and parameter to search for downstream solutions within the t2, and at t = ∞. shock front as x → 0+. As an example of illustration, we choose n = 0.673, q = With t → ∞ and/or r → 0+ (i.e. for the regime of 0, γ = 1.327 for a conventional polytropic gas. From the small x), the final evolution may approach either a quasi- analysis in subsection 3.1, solutions with a smaller n (> 2/3) 2 1/2 magnetostatic solution (Lou & Wang 2007) or the strong- tend to give a larger Mi,ult and < Bi,ult > . The choice of
c 2008 RAS, MNRAS 000, ??–?? 6 R.-Y. Hu and Y.-Q. Lou
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4 2 t 10 3 5 log
2 0 6 7 8 9 10 11 12 6 7 8 9 10 11 12 log r [cm] log r [cm]
Figure 1. The radial profiles of mass density, radial flow velocity, enclosed mass and transverse magnetic field at four different epochs. For all panels, the dash-dotted, solid, dashed and dotted curves are at t1, tm1, t2, and t = ∞, respectively. Mass density, velocity and magnetic field profiles are multiplied by numerical factors marked along the curves for a compact presentation. parameters here is a compromise among multiple numerical the outgoing rebound shock emerges and travels within the tests. With a typical surface mass density of 2.5 × 10−5 g stellar envelope. Figure 1 clearly shows this discontinuity cm−3, a surface temperature 3×104 K, a mass loss rate 10−6 across the outgoing MHD shock front. The rebound shock −1 3 M⊙ yr , and a surface magnetic field strength 10 G for a evolves in a self-similar manner with an outgoing speed de- 16 massive progenitor star, we estimate k1 = 1.55 × 10 cgs creasing with time t for n < 1 (see Lou & Hu 2008 and unit, A = 8.4378, B = 1.27×10−7 and h = 1.52×10−4 . Such Hu & Lou 2008b for a comparison with numerical simula- parameter h ensures that a stellar core collapse evolves into tions). The immediate downstream side of the shock has a quasi-magnetostatic manner for small x. Here, inequality an outward velocity and the immediate upstream side has h < hc is readily satisfied for this set of adopted parameters. an inward velocity. The enclosed mass of the downstream We set a rebound shock reaching a radius r = 109 cm at side decreases rapidly towards the centre while that of the t = 1 s, which fixes the shock location and travel speed. upstream side remains nearly unchanged. Across the shock With these parameters, a global self-similar rebound shock front, both mass density and magnetic field strength are en- 2 1/2 solution can be constructed and the temporal evolution is hanced by a factor of 6.96. It can be derived that < Bt >1 × −5 2 1/2 2 shown in Figure 1 at sampled epochs t1 = 3.49 10 s, / < Bt >2 = ρ1/ρ2 = 2/[(γ+1)M1]+(γ−1)/(γ+1) where 4 tm1, t2 = 2.87 × 10 s and t = ∞. M1 is the upstream Mach number in the comoving shock The initial progenitor mass is around 5.59 M⊙, consis- framework of reference. The maximum enhancement across tent with observations of Soderberg et al. (2008). At epoch the shock is (γ + 1)/(γ − 1) = 7.12 for our adopted value of t1 (dash-dotted curve), the rebound shock has not yet polytropic index γ = 1.327. emerged. The radial velocities around the surface of the proto-neutron star point inwards, corresponding to a core This rebound shock breaks out from the stellar envelope 4 collapse process leading to a subsequent rebound shock trav- in t2 ∼ 3 × 10 s. We see at that moment the flow velocity elling outwards inside the progenitor envelope. Meanwhile, within the spherical volume previously occupied by the pro- the outer part of the progenitor envelope still flows slowly genitor star becomes very much reduced in the wake of the outwards for a stellar wind. Such kind of self-similar shock rebound shock, and the gas there gradually approaches the manifestation with a collapsing inner part and an expand- quasi-magnetostatic phase of evolution. Coupled with radi- ing outer part is made possible from the quasi-magnetostatic ation mechanisms, for instance the thermal bremsstrahlung > 7 asymptotic solutions (see Lou & Wang 2006) and is another of hot electrons with a temperature T ∼ 10 K, and using form of envelope expansion with core collapse (EECC) pro- the dynamic profiles shown in Figure 1, one may compute posed by Lou & Shen (2004). At epoch tm1 (solid curve), the radiation detected and reproduce the X-ray or γ-ray
c 2008 RAS, MNRAS 000, ??–?? Intense Magnetic Fields on Magnetars 7
11 x 10 0.8 8 0.75 0.7 2 1/2 2 1/2 12 7 / [10 ] 1/2
0.65 i,ult o,ini >
0.6 2 o,ini 6 0.55 / 0.45 2 i,ult 0.4 M /M
2 4 6 8 10 12 3 8 R [cm] at t=1 s x 10 0.4 0.5 0.6 0.7 0.8 s M /M i,ult o,ini Figure 2. The dependence of the ensemble averaged magnetic field strength enhancement factor < B2 >1/2 / < B2 >1/2 Figure 3. The ensemble averaged magnetic field strength en- i,ult o,ini hancement < B2 >1/2 / < B2 >1/2 versus the mass ra- and the mass ratio Mi,ult/Mo,ini on the MHD rebound shock i,ult o,ini radii for n = 0.673, q = 0, γ = 1.327, a surface mass density tio Mi,ult/Mo,ini after a long time of rebound shock evolution. 2.5 × 10−5 g cm−3, a surface temperature 3 × 104 K, and a mass The star points mark our numerical explorations with different −6 −1 loss rate 10 M⊙ yr . shock properties in the case of n = 0.673, q = 0, γ = 1.327, a surface mass density 2.5 × 10−5 g cm−3, a surface temperature 4 −6 −1 3 × 10 K and a mass loss rate 10 M⊙ yr . Our numerical data can be best fitted (solid curve) with a quadratic relation light curves observed (e.g. Hu & Lou 2008b; Soderberg et al. < B2 >1/2 / < B2 >1/2= −8.8 × 1011(M /M )2 + 2008; Mazzali et al. 2008). Eventually, flow velocities of the i,ult o,ini i,ult o,ini 12 10 entire system tend to zero and the enclosed masses at all 1.8 × 10 (Mi,ult/Mo,ini) − 7.1 × 10 . radii remain unchanged. From Figure 1, we see that for the initial and final stages of the MHD evolution, the mass den- sity and magnetic field distributions obey power laws, con- 0.6 sistent with the asymptotic analysis in subsection 3.1. Fi-
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Another major limit arises as the fact that the magnetic en- flare activities and coronal mass ejections. For compact stel- hancement factor correlates to the mass ratio. If this mass lar object like neutron stars, this type of dramatic magnetic ratio is too high, the compact remnant mass may exceed the energy releases might fuel ‘magnetic fireballs’ which pro- Tolman-Oppenheimer-Volkoff (TOV) limit (∼ 3 − 3.2 M⊙; duce short-hard gamma-ray bursts, such as those reported Rhoades & Ruffini 1974), and the core object would collapse giant flares (e.g. Hurley et al. 2005) as well as very recent further to form a black hole. The upper bound for the mass outbursts of SGR J1550-5418 and SGR 1627-41. After gi- of neutron stars then places a limit on their surface magnetic ant flares, a neutron star may still recover surface magnetic field strengths. fields of order of ∼ 1014 − 1015 G manifested as either an AXP or a SGR. For an intense magnetic field buried inside a magne- tar, there is yet another possible source of activities. The 4 CONCLUSIONS AND DISCUSSION magnetic Lorentz force may break the crust of a neutron We combine semi-analytic and numerical self-similar MHD star into pieces to various levels. Coupled with the mag- solutions to model the gravitational core collapse, the re- netar spin, a broken crust can give rise to various activi- bound shock explosion of a magnetized massive progenitor ties. For examples, chunks of crust may pile up around the star, and the formation of a central remnant compact mag- equatorial bulge and rearrange themselves to generate stel- netar. As natural extension and generalization to recent su- lar seismic activities in a random manner; interior magnetic pernova rebound shock models of Lou & Wang (2006, 2007), flux ropes may burst into the magnetar magnetosphere ran- we invoked quasi-magnetostatic asymptotic solutions and domly at weak points of a crust; if the crust is more or less asymptotic solutions far away from the centre for a general destroyed by the Lorentz force, then footpoints of magnetic polytropic magnetofluid. With the magnetic frozen-in condi- field lines can be moved around by convective motions and tion imposed, the surface magnetic field of a nascent neutron possible differential rotations of a magnetar; the manoeuvre star can be very much stronger than that of its progenitor of magnetic footpoints over the surface of a magnetar leads by a factor of ∼ 1011 − 1012 during processes of the grav- to variable configurations of the magnetosphere and thus itational core collapse and the self-similar rebound shock produces magnetic activities including analogues of ‘flares’ breakout. Therefore if the progenitor is a magnetic massive and ‘coronal mass ejections’ mentioned earlier. star with a surface magnetic field strength of ∼ 103 G, it Regarding observed magnetar-like X-ray emissions from would have a good chance to produce a magnetar at the a rotation-powered pulsar PSR J1846-0258 inside supernova centre of its supernova remnant. Here we propose that mag- remnant Kes75 (e.g. Gavriil et al. 2008; Archibald et al. netars may be produced through powerful supernova explo- 2008), our model can accommodate this pulsar resulting sions of magnetized massive progenitor stars. Such physical from a magnetized massive progenitor yet with a lower sur- origin is also supported by statistical inferences from ob- face magnetic field strength. Such magnetar-like activities servational surveys (e.g. Ferrario & Wickramasinghe 2006). are physically associated more with strong magnetospheric While the magnetic flux of the collapsing core inside the field strengths. With the current observational evidence, it progenitor star may come from either main-sequence stellar appears not necessary to postulate that magnetars evolve dynamo processes or ‘fossil fields’ of molecular clouds, it will from fast spinning radio pulsars. be dragged and squeezed into the newborn neutron star by In our ‘ball of thread’ scenario for a random magnetic the conservation of magnetic fluxes. At least for magnetic field within a massive progenitor star, the large-scale mean massive stars as magnetar progenitors, the post-supernova of such a random magnetic field is idealized as dominantly dynamo processes inside the remnant neutron star may not transverse with a fairly weak radial component. By the be necessary for producing the ultra-intense surface mag- approximation of quasi-spherical symmetry, low-amplitude netic field. Magnetic field strengths in the interior of such small-scale deviations, oscillations or fluctuations are ran- magnetars thus produced are expected to be even stronger domly distributed about the mean flow profile and are ex- and are gravitationally buried and confined by the nuclear- pected to co-evolve with large-scale MHD profiles (e.g. Lou density matter. & Bai 2009 in preparation; Cao & Lou 2009 in preparation). If the surface magnetic field strength of a massive pro- During the processes of the gravitational core collapse and genitor happens to be even higher, possibly up to ∼ 3 × 104 the MHD rebound shock breakout, a remnant neutron star G or more, a nascent compact magnetar may then possess forms with high nuclear density and ultra-intense magnetic an ultra-intense surface magnetic field up to ∼ 1015 −3×1016 field, while a major portion of the interior and envelope of G or higher. Such strong magnetic fields can give rise to var- the massive progenitor star is driven out into the interstellar ious stellar activities and magnetic reconnection can release space by the rebound shock with entrained strong magnetic stored magnetic energy sporadically and violently (e.g. Low field. Our semi-analytic model describes a large-scale self- & Lou 1990). For example, if one approximates the mag- similar MHD evolution for the supernova explosion of a mag- netospheres of SGRs, AXPs, and radio pulsars as in gross netized massive progenitor. Along with this large-scale evo- force-free equilibria (i.e. electric currents parallel to mag- lution, the central magnetar and the thrown-out stellar ma- netic fields), then the magnetic energies retained in such terials may certainly follow their own courses of adjustment magnetospheric systems are higher than the corresponding or rearrangement (e.g. Lou 1994). For example, considerable potential field configurations with the same footpoint mag- radial components of magnetic field should emerge in a ran- netic field at the stellar surface. A loss of equilibrium most dom turbulent manner. In fact, magnetars are expected to likely triggered by magnetic reconnections may then lead to possess magnetospheres with various possible configurations outbursts of available magnetic energies. In the solar and (see Low & Lou 1990 for constructing force-free stellar mag- stellar contexts, such processes correspond to solar/stellar netic field configurations). By numerical simulations with
c 2008 RAS, MNRAS 000, ??–?? Intense Magnetic Fields on Magnetars 9
‘fossil fields’, Braithwaite & Spruit (2004) illustrated exam- ing systems, including magnetars, pulsars, magnetic white ples of processes for magnetic field rearrangement to occur dwarfs, protostars, planets and so forth. within a few Alfv´en timescales. The emergent stable struc- In addition to the quasi-magnetostatic asymptotic so- ture of magnetic field for magnetic Ap stars appears always lutions adopted and exemplified in this paper, it may be of ‘offset dipole’ type (with complex and twisted magnetic possible that a magnetized massive progenitor star evolves field configurations inside), consistent with observations. By towards the strong-field asymptotic solutions ultimately this analogy, we presume that such magnetic field rearrange- (Yu et al. 2006; Wang & Lou 2007, 2008), involving a ma- ment processes could take place very rapidly during the for- terial fall-back process towards the central remnant neutron mation of intensely magnetized neutron stars, whose Alfv´en star. Under this situation, the enclosed mass within a certain timescale is of the order of ∼ 0.1 s. Eventually, a magnetar radius keeps increasing until the mass of the neutron star ex- can possess a variety of magnetic field configurations (e.g. ceeds the TOV limit. Such an MHD fall-back process offers Low & Lou 1990). a possible means to form stellar mass black holes as compact Magnetars observed so far appear to be slow rotators, remnants in supernova and hypernova explosions. We em- while massive stars are in general rapid rotators with typi- phasize that such a mechanism requires a strong magnetic cal equatorial speeds of ∼ 200 km s−1 (e.g. Fukuda 1982). field inside a progenitor and the magnetic force becomes Therefore, significant angular momentum transfer may have dominant during the fall-back process. taken place in the stellar evolution of magnetic massive stars. Spruit (1999, 2002) has shown that magnetic fields can be created in stably stratified layers inside a differen- ACKNOWLEDGMENTS tially rotating star. Heger et al. (2005) gave detailed rotating stellar evolution calculations for stars in the mass range of This research was supported in part by Tsinghua Centre ∼ 12 − 35 M⊙ incorporating the dynamo-powered magnetic for Astrophysics (THCA), by the National Natural Sci- field. In general, it is found that magnetic breaking decreases ence Foundation of China (NSFC) grants 10373009 and the final spin rate of the collapsing iron core by a factor of 10533020 and by the National Basic Science Talent Train- ∼ 30 − 50 when compared with the nonmagnetic case. The ing Foundation (NSFC J0630317) at Tsinghua University, ‘fossil’ (or primordial) magnetic fields may have similar dy- and by the SRFDP 20050003088 and 200800030071 and namic effects regarding the re-distribution of angular mo- the Yangtze Endowment from the Ministry of Education mentum inside a massive star. In particular, for magnetar at Tsinghua University. The hospitality of Institut f¨ur The- formation, high magnetic fields may lead to stronger core- oretische Physik und Astrophysik der Christian-Albrechts- envelope coupling during the hydrogen and helium burning Universit¨at Kiel Germany and of International Center for phase of the SN progenitor, and the collapsing iron core Relativistic Astrophysics Network (ICRANet) Pescara, Italy and the compact supernova remnant is expected to be even is gratefully acknowledged. slower rotators. This is in accordance with the population of slowly rotating magnetars observed. 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APPENDIX A: MHD SHOCK CONDITIONS
In the shock comoving framework of reference, the MHD shock jump conditions in terms of the reduced self-similar
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