Amplifying magnetic fields of a newly born neutron star by stochastic angular momentum accretion in core collapse supernovae
DDraft version August 23, 2019
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Amplifying magnetic fields of a newly born neutron star by stochastic angular momentum accretion in core collapsesupernovae
Noam Soker
1, 2 Departmet of Physics, Technion, Haifa, 3200003, Israel Guangdong Technion Israel Institute of Technology, Shantou 515069, Guangdong Province, China
ABSTRACTI present a novel mechanism to boost magnetic field amplification of newly born neutron stars incore collapse supernovae. In this mechanism, that operates in the jittering jets explosion mechanismand comes on top of the regular magnetic field amplification by turbulence, the accretion of stochasticangular momentum in core collapse supernovae forms a neutron star with strong initial magneticfields but with a slow rotation. The varying angular momentum of the accreted gas, which is uniqueto the jittering jets explosion mechanism, exerts a varying azimuthal shear on the magnetic fieldsof the accreted mass near the surface of the neutron star. This, I argue, can form an amplifyingeffect which I term the stochastic omega (S ω ) effect. In the common αω dynamo the rotation hasconstant direction and value, and hence supplies a constant azimuthal shear, while the convection hasa stochastic behavior. In the S ω dynamo the stochastic angular momentum is different from turbulencein that it operates on a large scale, and it is different from a regular rotational shear in being stochastic.The basic assumption is that because of the varying direction of the angular momentum axis from oneaccretion episode to the next, the rotational flow of an accretion episode stretches the magnetic fieldsthat were amplified in the previous episode. I estimate the amplification factor of the S ω dynamo aloneto be ≈
10. I speculate that the S ω effect accounts for a recent finding that many neutron stars areborn with strong magnetic fields. Keywords: stars: neutron — stars: magnetic field — (stars:) supernovae: general INTRODUCTIONCore collapse supernovae (CCSNe) occur when thecore of massive stars collapses to form a neutron star(NS) or a black hole (e.g., Woosley, & Weaver 1986). Afraction of the gravitational energy that the collapsinggas releases powers the explosion by ejecting the restof the core and the envelope (e.g., Janka 2012). Thereis no consensus yet on the processes that channel thegravitational energy to explosion, as I discuss next.There are two contesting theoretical models to channelthe gravitational energy to explosion, the delayed neu-trino mechanism (Bethe & Wilson 1985) and the jitter-ing jets explosion mechanism (Soker 2010). There are,what I see as, some difficulties in the delayed neutrinomechanism (e.g., Papish et al. 2015; Kushnir 2015), e.g.,in the classical one-dimensional delayed neutrino mech-
Corresponding author: Noam [email protected] anism the heating by neutrinos has no time to acceleratethe ejecta to high energies (Papish et al. 2015). Three-dimensional effects seem to partially solve this problem(e.g., M¨uller et al. 2019). There are also seeminglycontradicting results in obtaining explosions with the de-sired explosion energy as some obtain explosions (e.g.,M¨uller et al. 2017; Vartanyan et al. 2019) and some donot (e.g. O’Connor & Couch 2018). For these, I con-sider the jittering jets explosion mechanism as a moresuccessful explosion model, or the two mechanisms ofheating by neutrinos and jittering jets should act to-gether (Soker 2019b).The jittering jets explosion mechanism includes a neg-ative feedback component (e.g., Gilkis et al. 2016; Soker2017), the jet feedback mechanism (for a review seeSoker 2016), in the sense that when the jets expel therest of the core and the envelope, they shut themselvesdown. This implies that the explosion energy is of theorder of, or several times, the binding energy of the corefor an efficient feedback process, or many tens times thebinding energy of the core in rare cases when the feed- a r X i v : . [ a s t r o - ph . H E ] A ug Soker, N. back process is not efficient (Gilkis et al. 2016). Thisexpected behavior of the jittering jets explosion mecha-nism is compatible with observations.As well, neutrino heating does play a role in the jitter-ing jets explosion mechanism (Soker 2018, 2019a,b), butnot the dominant role. When the pre-collapse core isslowly rotating the angular momentum of the accretionflow onto the newly born NS will be highly stochasticdue to fluctuations in the convective regions of the pre-collapse core or envelope (Gilkis & Soker 2014, 2015;Quataert et al. 2019), that the spiral standing accretionshock instability (SASI) modes (for studies of the spiralSASI see, e.g., Blondin & Mezzacappa 2007; Rantsiouet al. 2011; Iwakami et al. 2014; Kuroda et al. 2014;Fern´andez 2015; Kazeroni et al. 2017) further amplify.When the pre-collapse core is rapidly rotating, (i.e., thespecific angular momentum of the gas allows it to forman accretion disk around the newly born NS), the jit-tering will have relative small amplitudes around a fixedangular momentum axis (Soker 2017).Many studies have found indications, like polarizationin some CCSNe and the morphology of some supernovaremnants, for some roles that jets play in possibly mostCCSNe (e.g., Wang et al. 2001; Maund et al. 2007; Smithet al. 2012; Lopez et al. 2011; Milisavljevic et al. 2013;Gonz´alez-Casanova et al. 2014; Margutti et al. 2014; In-serra et al. 2016; Mauerhan et al. 2017; Grichener &Soker 2017; Bear et al. 2017; Garc´ıa et al. 2017; Lopez& Fesen 2018). As well, there are many studies of jet-driven CCSNe that do not consider jittering, and henceare aiming at rare cases, e.g., of progenitors having avery rapidly rotating pre-collapse core (e.g., Khokhlovet al. 1999; Aloy et al. 2000; H¨oflich et al. 2001; Mac-Fadyen et al. 2001; Obergaulinger et al. 2006; Burrowset al. 2007; Nagakura et al 2011; Takiwaki & Kotake2011; Lazzati et al. 2012; Maeda et al. 2012; L´opez-C´amara et al. 2013; M¨osta et al. 2014; Ito et al. 2015;Bromberg & Tchekhovskoy 2016; L´opez-C´amara et al.2016; Nishimura et al. 2017; Feng et al. 2018; Gilkis2018; Obergaulinger et al. 2018).The jittering-jets explosion mechanism differs fromthe processes that these studies of jets consider in havingsome unique properties. (1) The jittering jets explosionmechanism supposes to explode all CCSNe with kineticenergies of (cid:38) × erg, and many (or even all) of CC-SNe below that energy, rather than only a small percent-age of all CCSNe (e.g., Soker 2016). (2) The pre-collapsecore can have any value of rotation, from non-rotatingto rapidly rotating, rather than having rapid rotationonly (e.g., Gilkis & Soker 2014). (3) The jets can beintermittent, and for slowly rotating pre-collapse coresthey are also strongly jittering (i.e., having large vari- able directions; e.g., Soker 2017). (4) The jets operate ina negative feedback mechanism. Namely, the jets reducethe accretion rate and hence their power while removingmass from the core and envelope (e.g., Soker 2016). (5)In cases of a strong jittering, each jet-launching episodeis active for a short time and the direction of the jetschanges rapidly. Therefore, the jets do not break outfrom the ejecta of the explosion (e.g., Papish & Soker2011). In some cases of strong jittering the jets fromthe last jet-launching episode or two might break out ofthe ejecta and inflate two opposite small lobes (calledears) on the outskirts of the supernova remnant (e.g.,Bear et al. 2017; Grichener & Soker 2017).In the present study I continue my exploration of thejittering jets explosion mechanism and I raise the pos-sibility that in the jittering jets explosion mechanismthere is a process that contributes to magnetic field am-plification in the material that the newly born neutronstar accretes. I term this the stochastic- ω (S ω ) effect. Idescribe this effect in section 2, and discuss some plau-sible typical quantitative parameters in section 3. TheS ω that I propose differs from the αω dynamo, as I ex-plain in the following sections. In the αω dynamo the ω refers to the stretching of poloidal magnetic field linesto azimuthal lines by an ordered differential rotation,while the α effect refers to stochastic motion, like tur-bulence, that entangles the azimuthal magnetic fields toform poloidal magnetic field lines to close the dynamocycle.I mention that simulations that do not consider the jit-tering jets explosion mechanism also find the accretion ofstochastic angular momentum onto the newly born NS(e.g., Kazeroni et al. 2016; M¨uller et al. 2017). Hence,the S ω effect might take place also in the neutrino drivenexplosion mechanism. However in the jittering jets ex-plosion mechanism the S ω effect is a generic outcome.As well, the amplification of magnetic fields by the S ω effect might help the launching of jets. In the presentstudy I scale quantities according to the expactation ofthe jittering jets explosion mechanism.In section 4 I summarise the main results and discussthe general picture of forming NSs with strong magneticfields and the broader relation to the jittering jets ex-plosion mechanism. THE STOCHASTIC-OMEGA (S ω ) EFFECT2.1. General description
I consider the following general flow of a CCSN wherethe pre-collapse core rotational velocity is low, and sothe collapsing core gas that feeds the newly born NS hasa stochastic angular momentum. The total mass thatflows on to the NS during this phase is (cid:39) . − . M (cid:12) agnetic fields of new NS αω dynamo (e.g., Soker 2018, 2019a). The two ingre-dients of the αω dynamo are the turbulent motion thatentangles the azimuthal magnetic fields to form poloidalmagnetic field lines (the α effect), and the differentialrotation of the toroidal (azimuthal) flow that stretchespoloidal magnetic field lines to azimuthal lines. In theregular αω dynamo the direction of the angular momen-tum of the toroidal flow does not change. As well, theazimuthal velocity depends only the poloidal location( (cid:36), z ), where (cid:36) is the distance from the symmetry axisand z is the distance from the equatorial plane.Here I consider the case where the angular momentumaxis changes in a stochastic manner, and I study the ef-fect that this might have on magnetic field amplification.I term this the Stochastic- ω (S ω ) effect. Both the α ef-fect and the ω effect still exist, and I argue below thatthe S ω effect adds to the magnetic field amplificationduring the periods when the angular momentum axischanges its direction. According to the jittering jetsexplosion mechanism there are about ten to few tensof accretion episodes, and the variations in the angu-lar momentum axis take place in short periods betweenconsecutive accretion episodes (see relevant references insection 1).Specifically, in the present study I focus on the ampli-fication near the surface of the newly born NS just asthese magnetic fields are dragged onto the NS. In Fig1 I present a schematic description of the flow, showingthe NS and two consecutive accretion episodes, number n − n . The upper panel presents only the earlyaccretion episode, where the differential rotation ampli-fies an azimuthal magnetic field that I present as thinmagnetic field lines. This is the regular ω effect of the αω dynamo. In the lower panel I present one stream line(thick red line) of the next accretion flow, which has itsangular momentum axis inclined by an angle β n to thatof the flow in the previous accretion episode. The newflow drags and stretches the magnetic field lines on theouter part of the flow of the previous accretion episode,which I represent by one thin red line. By that stretch-ing the flow further amplifies the magnetic field. This isthe S ω effect. The stochastic accretion of angular mo-mentum of the S ω effect implies that the NS is born with slow rotation, but that nonetheless might have astrong magnetic field. Namely, it might be born as aslowly rotating magnetar. Neutron star B field lines Flow: n-1 accretion episode
Flow : n accretion episode Stretched B field lines Neutron star
Figure 1.
A schematic drawing of the flow interaction be-tween accretion episodes n and n −
1. The upper panel showsthe flow in the accretion episode n − n . The angular momen-tum axes of the two episodes are inclined to each other by anangle β n . The field lines from episode n − n episode. In bothepisodes the toroidal flow is much thicker than what is drawnhere, both in the radial direction away from the symmetryaxis and perpendicular to the rotational plane. Relevant equations
Consider the ω effect in the induction equation (e.g.,Priest 1987) ∂ B ∂t = curl( v × B ) + η ∇ B , (1) Soker, N. where v is the plasma velocity and η is its magneticdiffusivity. For a case where the main flow during the n accretion episode is azimuthal v φ , i.e., toroidal flowalong coordinate φ , and neglecting magnetic dissipation,i.e., a very small value of η , the induction equation forthe toroidal magnetic field component reads (e.g., Priest1987) ∂B φ ∂t = R B p · ∇ (cid:16) v φ R (cid:17) , (2)where B p is the poloidal component of the magneticfield, and R is the distance from the rotation axis. Whatmatters here is only the magnitude of equation (2). Inthe above equation I assume that within each accretionepisode the flow is steady and axisymmetric. However,in the jittering jets explosion mechanism the flow is notsteady and the axisymmetry axis changes direction be-tween consecutive accretion episodes.I assume that in accretion episode n − B n − (cid:39) B φ ,n − , where here φ n −
1. In the jitteringjets explosion mechanism there are about ten to few tensof accretion episodes (e.g., Papish & Soker 2011). I takethe azimuthal field of accretion episode n − n , where n = 2 , ... few × β n be the angle between the angular momentum axisof episode n and episode n −
1, such that the seed mag-netic field at the beginning of accretion episode n in theregion where the two azimuthal consecutive flows crosseach other is B n, = B φ ,n − cos β n ˆ φ + B φ ,n − sin β n ˆ z (3)where here φ is the direction of the toroidal flow inepisode n , and ˆ z is a unit vector perpendicular to thetoroidal plane.For the gradient of the velocity in the relevant direc-tion ˆ z I take ∇ (cid:16) v φ,n R (cid:17) z = v φ,n χ n R ˆ z, (4)where χ n ≈
1. Namely, the distance along the ˆ z direc-tion over which v φ,n changes is χ n R .I take the poloidal magnetic field component fromequation (3), and for the gradient of the flow from equa-tion (4), and substitute both in the absolute value ofequation (2). This gives the magnitude of the az-imuthal magnetic field component at the end of accre-tion episode nB φ, n (cid:39) B φ ,n − sin β n v φ,n χ n R ∆ t n , (5)where ∆ t n is the duration of accretion episode n . I emphasis here that equation (5) represents the am-plification due only to the change of the angular mo-mentum axis. There are two other effects, which are theusual ω effect, resulting from the velocity gradient dur-ing the considered accretion episode, and the α effectdue to turbulence within the accretion flow.There are two considerations that reduce the effectivevolume in which the stochastic accretion flow amplifiesthe magnetic field. (1) The flow in episode n stretchesthe seed magnetic fields of episode n − β n >
0. (2) The interaction betweentwo consecutive accretion episodes is in the interface be-tween them. During each accretion episode the regular αω dynamo takes place, as mentioned above.I take the effective volume in which the S ω effect oper-ates to be a fraction δ (cid:28) F S ω due to the S ω ef-fect during the entire accretion process, from an initialmagnetic field B , to a final one of B f, S ω , by substitut-ing for N ae accretion episodes in equation (5), by multi-plying by the effective amplification volume fraction δ ,and by averaging over the relevant quantities F S ω ≡ B f, S ω B , ≈ v φ χR N ae ∆ tδ sin β. (6)In equation (6) the quantities, v φ , sin β , χ , δ , ∆ t aretheir respective values averaged over the N ae accretionepisodes QUANTITATIVE ESTIMATES3.1.
Plausible numerical values
I now very crudely estimate the values of the differentparameters that appear in equation (6) for the amplifi-cation factor of the S ω effect in the jittering jets explo-sion mechanism of CCSNe. I scale equation (6) near thesurface of the newly born NS at R (cid:39)
20 km as follows F S ω ≈ χ − (cid:16) v φ v Kep (cid:17) (cid:0) R
20 km (cid:1) − / (cid:16) M NS . M (cid:12) (cid:17) / (7) × (cid:0) N ae (cid:1) (cid:0) ∆ t . (cid:1) (cid:0) δ . (cid:1) (cid:16) sin β . (cid:17) . The final radius of the NS is about 12 km, but duringthe accretion process the NS is still hot and its radiusis somewhat larger than its final radius, hence I scalewith R = 20 km. I elaborate on the scaling of theother different quantities below (see Table 1), and thenI compare to some non-dimensional ratios in the solar αω dynamo. agnetic fields of new NS Quantity Symbol Crude valueVelocity variation distance χR R (cid:39)
20 km ( χ ≈ v φ v φ (cid:46) v Kep
Number of accretion episodes N ae − t . − . β ◦ Fraction of effective volume δ . F S ω Table 1.
The typical parameters of equation (7), which arethe typical values averaged over N ae accretion episodes, andthe typical amplification factor in the last line (see also Fig.1). The typical Keplerian velocity on the surface of the NSis v Kep (cid:39) km s − . The relation N ae ∆ t ≈ − The distance scale of velocity variation χR . I simplyassume that the velocity varies along the direction per-pendicular to the toroidal flow over a distance of (cid:39) R ,i.e., χ = 1. This can be smaller even, but then the effec-tive volume fraction δ might be smaller (see below). I donote that the velocity gradient between the accreted gasand the surface of the NS might be much larger becauseover a short radial distance the velocity changes from aslowly rotating NS to v φ . This, however, is related tothe αω dynamo as it does not directly need the stochas-tic angular momentum accretion. It indirectly requiresthe stochastic angular momentum accretion to ensurethat the newly born NS is a slow rotator and its angularmomentum is not aligned with that of the accreted gas. The toroidal velocity v φ . I scale it with the Keplerianvelocity at a radius of R around a newly born NS ofmass M NS . The velocity in the jittering jets explosionmechanism might be lower than the Keplerian velocity(e.g., Schreier & Soker 2016; Soker 2019a), even by afactor of a few. In that case though, the accretion flowis thicker and the effective volume fraction of the S ω effect δ will be larger (see below). The number of accretion episodes N ae and their av-erage duration ∆ t . The total duration of the explosionprocess is about a second to few seconds. The number ofepisodes might be somewhat larger. In that case the av-erage duration ∆ t is smaller, such that N ae ∆ t (cid:39) − The angle between consecutive accretion episodes β . The angle is not completely random (Papish & Soker2014) but tends to be smaller than the average valueof completely random angular momentum directions. Itcan be smaller than sin β = 0 .
5, but then the overlapbetween the toroidal flow regions of consecutive episodesis larger, and hence δ will be larger. The fraction of effective volume δ . The S ω effect oper-ates when the symmetry axis of the toroidal flow changes direction. We can think of a torus-like region throughwhich there is a toroidal flow of accretion episode n − n . They each have a volume of Vol n . The two torus-likevolumes are incline to each other, and hence overlap ina small fraction of the volume δ i Vol n . In addition thestretching of the magnetic field lines of the torus-like re-gion of episode n − n occurs in the interface between them. This is a smallfraction δ w of the width of the torus. Overall, the S ω effect operates in a volume that is a fraction of δ = δ i δ w of the inflow volume. This value is highly uncertain, andI simply take δ ≈ . δ cannot be much smaller, as this requiresa thin accretion disk at each episode. This can be thecase only if the accreted gas has a specific angular mo-mentum that allows it to form an accretion disk. Thisin turn can be the case only if the angular momentum ofthe pre-collapse core was high. This brings the situationto another regime of the jet feedback mechanism wherethere is a more or less constant angular momentum axis.I do not consider this case here (this case will make jetlaunching easier even). The effective volume fraction δ can be made larger by considering regions with lowertoroidal velocity, e.g., an accretion belt rather than anaccretion disk. This will reduce v φ . The value of δ is larger if on average two consecutive accretion episodeare almost aligned with each other. But this will makesin β smaller.3.2. Some hints from the solar αω dynamo In the S ω effect the time scale of magnetic fieldstretching is t B , str (cid:39) πR/v φ which is about equalto the Keplerian time t Kep (cid:39) .
001 s, or somewhatlonger. For the parameters I use here this time scale is t B , str ≈ .
002 s or somewhat longer. I crudely estimatethe stretching of the filed during the time t B , str to beby a distance of 2 πR sin β . In the jittering jets explo-sion mechanism there are several to few tens of accretionepisodes over a total time of about a second to severalseconds (e.g., Papish & Soker 2011). Each episode lastsfor a time of ∆ t (cid:39) few × .
01 s to ∆ t (cid:39) few × . ≈ . − . ≈ −
30 times the stretching time t B , str at a radiusof R (cid:39)
20 km. Taking ten to several tens of accretionepisodes, I find that the activity of the S ω effect lastsfor t S ω ≈ (30 − t B , str . (8)Let us consider the stretching and entangling timescalesin the Sun. In main sequence stars the strength of the Soker, N. magnetic activity is related to the Rossby number Ro(or to the dynamo number N D = Ro − ; e.g., for theSun, Kim & Demarque 1996; Landin et al. 2010). TheRossby number is defined as Ro ≡ P rot /τ c , where P rot is the rotation period and τ c = α ml H P /v c (cid:39) H P /v c is the local convective turnover time. Here α ml H P isthe mixing length, H P is the pressure scale height, and v c is the convective velocity. The magnetic activity ofmain sequence stars increases as the Rossby number de-creases, until a saturation for Ro (cid:46) . P rot (cid:39)
25 day ≈ P Kep , where P Kep is the Keplerian orbital period onthe surface of the star, and τ c (cid:39)
20 days (e.g., Landinet al. 2010). In the Sun itself the magnetic cycle periodis about 22 years (e.g., Howard, & Labonte 1980). Mostof the rise in the intensity of magnetic activity withineach half a cycle occurs within several years, t rise , (cid:12) ≈ P rot , (cid:12) (cid:39) τ c, (cid:12) .Overall in the Sun, the surface magnetic field intensityrises on a timescale of several tens times the stretchingtime scale of the field lines. Considering equation (8)in relation to this ratio, hints that the total time of op-eration of the S ω effect in the jittering jets explosionmechanism allows this mechanism to contribute to theamplification of the magnetic fields in the material thatthe newly born NS accretes. This can increase the ini-tial magnetic field of newly born NSs by an order ofmagnitude. The final field intensity depends on otherfactors beside the operation of the S ω dynamo. DISCUSSION AND SUMMARYThe evolution of magnetic fields from core collapse toNS formation involves four phases of magnetic field am-plification. (1) In the pre-collapse core where a dynamoin the convective zones amplifies magnetic fields (e.g.,Wheeler et al. 2015) and radiative zones store magneticfields till collapse (Peres et al. 2019). (2) During the col-lapse itself where the converging inward flow amplifiesthe radial component of the magnetic fields, as mag-netic flux conservation implies. (3) In the unstableregion behind the stalled shock, where in particular thespiral-SASI can amplify the magnetic fields (e.g., Endeveet al. 2010, 2012; Rembiasz et al. 2016a,b; Obergaulingeret al. 2018). (4) Near and on the surface of the newlyborn NS, e.g., Obergaulinger & Aloy (2017) who con-sider only axisymmetrical effects. In the present paperI addressed the last magnetic field amplification phase.I considered the contribution of the stochastic angu-lar momentum of the accreted mass to the magneticfield amplification as the mass reaches the surface of the NS. Figure 1 presents the basic process, that I term theStochastic- ω (S ω ) effect. The toroidal (azimuthal) flowof two consecutive accretion episodes are inclined to eachother. As a result of that the toroidal flow of the laterepisode stretches the magnetic field lines that the earlytoroidal flow amplified. Within each accretion toroidalflow the regular αω dynamo might operate.Simulations (that do not consider the jittering jetsexplosion mechanism) find stochastic angular momen-tum accretion onto newly born NSs (e.g., Kazeroni etal. 2016; M¨uller et al. 2017). As the jittering jets ex-plosion mechanism must include accretion of stochasticangular momentum with large amplitudes, the S ω effectis expected to take place in the jittering jets explosionmechanism. I derived an approximate expression forthe extra magnetic field amplification of the S ω effect inequation (6), and substitute typical values (with largeuncertainties) in equation (7). This equation suggeststhat in many cases, for which the typical values of thedifferent parameters are crudely listed in Table 1, thejittering jets explosion mechanism comes along with theformation of a NS with strong magnetic fields.The stochastic angular momentum of the accreted gasimplies that in many cases the newly born NS will havea slow rotation (relative to breakup rotation velocity).Overall, according to the jittering jets explosion mech-anism many NSs are born with strong magnetic fields,being even magnetars, but with a slow rotation. In a re-cent study Beniamini et al. (2019) conclude that a frac-tion of 0 . +0 . − . of NSs are born as magnetars with mag-netic fields at birth of B (cid:38) × G. They claim thatthis high fraction challenges existing theories for form-ing magnetars, as these theories require extreme andrare conditions, i.e., pre-collapse rapid rotation and/orstrong magnetic fields. The challenge is stronger evenif we take into account that the initial rotation periodof most NSs are two orders of magnitudes longer thantheir maximum possible period (breakup period; e.g.,Popov & Turolla 2012; Igoshev & Popov 2013; Gull´onet al. 2015). I here propose that the S ω effect that oper-ates in the jittering jets explosion mechanism, and evenin cases where jets are not launched, might account forthe finding of Beniamini et al. (2019) that many NSs areborn as magnetars.I thank Avishai Gilkis for helpful comments, and ananonymous referee for many useful and detailed com-ments. 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