Repulsion of fallback matter due to central energy source in supernova
aa r X i v : . [ a s t r o - ph . S R ] S e p Repulsion of fallback matter due to centralenergy source in supernova
Toshikazu Shigeyama , Kazumi Kashiyama Research Center for the Early Universe, Graduate School of Science, University of Tokyo,Bunkyo-ku, Tokyo 113-0033 Department of Physics, Graduate School of Science, University of Tokyo, Bunkyo-ku, Tokyo113-0033, Japan
Received ; Accepted
Abstract
The flow of fallback matter being shocked and repelled back by an energy deposition from acentral object is discussed by using newly found self-similar solutions. We show that thereexists a maximum mass accretion rate if the adiabatic index of the flow is less than or equal to4/3. Otherwise we can find a solution with an arbitrarily large accretion rate by appropriatelyshrinking the energy deposition region. Applying the self-similar solution to supernova fallback,we discuss how the fate of newborn pulsars or magnetars depends on the fallback accretionand their spin-down power. Combining the condition for the fallback accretion to bury thesurface magnetic field into the crust, we argue that supernova fallback with a rate of ˙ M fb ∼ − (4 - M ⊙ s − could be the main origin of the diversity of Galactic young neutron stars, i.e.,rotation-powered pulsars, magnetars, and central compact objects. Key words: hydrodynamics — accretion — stars: neutron — pulsars: general — supernovae: general
Compact objects such as neutron stars and black holes are formed in collapsing stars some of whicheventually end up with supernova explosions. In general, a fraction of supernova ejecta falls back1oward a newborn compact object (Colgate 1971; Zel’dovich et al. 1972; Michel 1988). The accretionrate and its temporal evolution can be determined by e.g., the strength of the supernova shock andthe progenitor structure (e.g., Ertl et al. 2016). On the other hand, newborn compact objects cancontinuously deposit energy into the fallback matter via e.g., neutrino emission, pulsar activity and/oraccretion-disk wind (e.g., Piro & Ott 2011). Competition between the fallback accretion onto and theenergy deposition from compact objects may result in a variety of outcomes and lead to a diversity ofcompact objects.In particular, the dynamics of the fallback accretion may be related to the diversity of rela-tively young ( ∼ -
10 kyr ) neutron stars; there are three distinct populations, ordinary pulsars, mag-netars, and central compact objects (CCOs) . Based on the spectral and timing observations, theyare considered to have different energy sources, the rotation, magnetic-field, and thermal energies,respectively. The most distinctive difference between them is the magnetic-field strength; young pul-sars typically have dipole fields of B d < ∼ G . Magnetars are considered to be powered by decaysof stronger fields of B ∗ > a few × G (Olausen & Kaspi 2014) while CCOs have considerablyweaker dipole fields of B d ≪ G (Pavlov et al. 2001; Park et al. 2006, and so on). Although thediversity of the magnetic field strength can be attributed to conditions before and during the neutronstar formation, i.e., the magnetic field strength of the iron core of a progenitor (the fossil scenario;Ferrario & Wickramasinghe 2006) or the magnetic field amplification by a dynamo process (Duncan& Thompson 1992; M ¨osta et al. 2015), it may be also possible to attribute the diversity to conditionsafter the formation, i.e., various rates of the fallback accretion onto the neutron star.If the accretion rate is sufficiently high, the fallback matter can compress and bury the mag-netic field into the neutron star crust (e.g., Bernal, Lee, & Page 2010; Torres-Forn´e et al. 2016). Sucha case may correspond to a CCO formation (the hidden magnetic field scenario; Muslimov & Page1995; Young & Chanmugam 1995). For example, Torres-Forn´e et al. (2016) obtained the criticalaccretion rate for burying the magnetic field by considering the competition between the magneticpressure of the surface field and the ram pressure of the fallback matter. In these studies, however,the energy deposition from the neutron star has been neglected. If the energy deposition rate is suffi-ciently large, the fallback matter can be repelled before reaching the surface. The critical conditionscan be obtained by considering the competition between the fallback accretion onto and the energydeposition from the neutron star. This may define additional bifurcations of the neutron star popula-tion. They sometimes show links to another population. For example, some pulsars underwent soft-gamma repeater like bursts (van der Horst et al. 2010), and aCCO was found to exhibit magnetar-like activity (D’A`ı et al. 2016; Rea et al. 2016). egion I(uniform) Region II(self-similar) Region III(free fall)Point mass M c Energy source dQdt = L l t l r R c R s Fig. 1.
Schematic view of the model consisting of three regions. Region I is adjacent to the central object. The energy from the central object is supposed tobe uniformly deposited in this region. Region II contains the shocked fallback matter and is separated from Region I by a contact surface. In Region III, thematter freely falls due to the gravity of the central object. An expanding shock front separates Region II from Region III. The black arrows indicate the directionof the flows while the green and red arrows indicate the propagations of the contact surface and the shock front, respectively. Note that the direction of theflow immediately behind the shock front depends on model parameters. See section 3 for details.
To this end, we here investigate the dynamics of fallback matter being pushed back by anenergy deposition from the central object with newly constructed spherically symmetric self-similarsolutions. We consider fallback matter marginally bound by the gravitational field of the central objectwith a mass M c and a power-law energy deposition rate from the central object; ˙ Q = L l t l . (1)Here L l and l are constants and t is the time measured from the onset of the energy deposition.Although our solutions are described by a single dimensionless variable ξ = r ( GM c t ) / , (2)where r is the radial coordinate and G denotes the gravitational constant, a variety of density andvelocity structures can be realized depending on the strength of the deposition and so on, which, toour knowledge, have not been seen in previous studies.The structure of the paper is as follows. The next section describes our model. Section 3presents our self-similar solutions with various values of parameters. In Section 4, we discuss someapplications of our solutions to neutron star formation. Section 5 summarizes the results and discussesrelations to previous works. We consider the flow of fallback matter affected by the energy deposition from a central object.Following Masuyama et al. (2016), we divide the flow into three distinct regions (see Fig. 1): theinnermost region (Region I) adjacent to the central object where the energy is deposited uniformly,3egion II where the fallback matter is shocked, and the outermost region in which cold matter freelyfalls due to the gravity of the central object (Region III). Regions I and II are separated by the contactdiscontinuity specified by the dimensionless variable introduced above as ξ = ξ c . The fluid in RegionII has a constant adiabatic index γ and turns into outflow at a certain point depending on ξ c and γ .The shock front divides the fallback matter into Regions II and III at ξ = ξ s . Such a flow structureacross Regions I and II has been reproduced by numerical simulations (e.g., Masuyama et al. 2016). The central object is assumed to deposit energy in a uniform spherical region with a radius R c = ξ c ( GM c t ) / . The internal energy E c and pressure P c in this region evolve with time t according tothe first law of thermodynamics as dE c dt + P c ddt πR ! = L l t l . (3)Here L l and l are constant. The subscript l of L l indicates that the dimension of this quantity dependson the value of l . This region becomes very hot and the equation of state can be approximated by thatfor ultra-relativistic gas, that is, E c = 4 πP c R . Accreted matter can be divided into the shocked fallback matter (Region II) and the surroundingaccreted matter (Region III). The motion of the surrounding matter is not affected by the energydeposition but controlled solely by the gravity from the central object. We describe the flow in RegionII by using a self-similar solution. We note that the problems treated in this paper involve two constantquantities with physical dimensions GM c and L l .Here we assume that the spherically symmetric flow depends on time t and r only through t and ξ ( r, t ) . Thus the density ρ , the velocity v , and the pressure p take the forms of ρ ( ξ, t ) = L l t l − D ( ξ )( GM c ) / , (4) v ( ξ, t ) = (cid:18) GM c t (cid:19) / V ( ξ ) , (5) p ( ξ, t ) = L l t l − Pr( ξ ) GM c , (6)as functions of ξ and t . Here we have introduced dimensionless functions D ( ξ ) , V ( ξ ) , and Pr( ξ ) .The governing equations are described as ∂ρ ( ξ, t ) ∂t + ∂r ρ ( ξ, t ) v ( ξ, t ) r ∂r = 0 , (7)4 ( ξ, t ) ∂v ( ξ, t ) ∂t + v ( ξ, t ) ∂v ( ξ, t ) ∂r ! + GM c ρ ( ξ, t ) r + ∂p ( ξ, t ) ∂r = 0 , (8) ∂∂t + v ( ξ, t ) ∂∂r ! " p ( ξ, t )( γ − ρ ( ξ, t ) + p ( ξ, t ) ∂∂t + v ( ξ, t ) ∂∂r ! " ρ ( ξ, t ) = 0 . (9)After some manipulations we obtain ordinary differential equations for dimensionless functions D ( ξ ) , V ( ξ ) , and Pr( ξ ) as ξ (2 ξ − V ( ξ )) D ′ ( ξ ) − D ( ξ ) (3 lξ + 3 ξV ′ ( ξ ) + 6 V ( ξ ) − ξ ) = 0 , (10) D ( ξ ) (cid:16) ξ (2 ξ − V ( ξ )) V ′ ( ξ ) + ξ V ( ξ ) − (cid:17) = 3 ξ ′ Pr( ξ ) , (11) γ (2 ξ − V ( ξ )) D ′ ( ξ ) D ( ξ ) + (3(1 − γ ) l + γ −
3) + (3 V ( ξ ) − ξ ) Pr ′ ( ξ )Pr( ξ ) = 0 , (12)where ′ denotes the derivative with respect to ξ . Eqs. (10-12) can be rewritten as D ′ ( ξ ) = D ( ξ ) ( D ( ξ ) (9 − ξV ( ξ )((3 l − ξ + 6 V ( ξ )) + (6 l − ξ )) ξ ( D ( ξ )(2 ξ − V ( ξ )) − γ Pr( ξ ))+ 9 ξ ( − γl + γ + 3 l − D ( ξ ) Pr( ξ ) ξ (2 ξ − V ( ξ )) ( D ( ξ )(2 ξ − V ( ξ )) − γ Pr( ξ )) , (13) V ′ ( ξ ) = 9 ξ Pr( ξ )(2 γV ( ξ ) + ( l − ξ ) − D ( ξ )(2 ξ − V ( ξ )) ( ξ V ( ξ ) − ξ ( D ( ξ )(2 ξ − V ( ξ )) − γ Pr( ξ )) , (14) ′ Pr( ξ ) = 3 D ( ξ ) Pr( ξ ) (3 ξV ( ξ )( ξ ( γ − l + 1) − γV ( ξ )) + 3 γ + 2( l − ξ ) ξ ( D ( ξ )(2 ξ − V ( ξ )) − γ Pr( ξ )) . (15)From the second term of the right hand side of equation (13), we can find that the derivative of thedensity diverges at ξ c where V ( ξ c ) = 2 ξ c , (16)is satisfied while the other derivatives of the pressure and the velocity do not diverge. Thus this point ξ = ξ c defines the contact surface between Regions I and II. To obtain a solution, we numericallyintegrate Eqs.(13-15) from ξ = ξ s to ξ = ξ c .The solution for the flow in Region III ( ξ ≥ ξ s ) is obtained by ignoring pressure in equations(10) and (11) (or (13) and (14)) as V ( ξ ) = − s ξ , (17) D ( ξ ) = D fb exp "Z ξξ s dx l − x − √ x − / x (2 x + 3 √ x − / ) . (18)Here a constant D fb denotes D ( ξ s ) in the un-shocked flow (Region III). It follows from substitutionsof these expressions into equations (4) and (5) that the distributions of the density and velocity in thefallback matter evolve as ρ ( r, t ) = L l D fb t l − / ( GM c ) exp "Z r ( GM c t / ξ s dx l − x − √ x − / x (2 x + 3 √ x − / ) , (19)5 ( r, t ) = − s GM c r , (20)and that the density approaches a power-law function proportional to r (3 l − / independent of t inthe limit of r → ∞ . Thus the flow approaches a stationary state for a large r ( ξ ). The velocityexpressed by equation (20) implies that the fallback matter is initially at rest at large distances fromthe center. Thus the total energy of the flow in Region III is equal to zero while the energy of the flowin Region II becomes positive due to the energy supply from the central source. This indicates thatall the solutions presented here describe the flow eventually repelled by the energy supply. Becausethe density distribution depends on the parameter l that specifies the nature of the central energysource, this density distribution is required to lead to a self-similar solution in which the shock frontexpanding with the radius proportional to t / for a given energy source. If the central source is moreenergetic, which corresponds to larger L l , the fall back matter can be denser, while if the gravity ofthe central object is stronger, which corresponds to a larger M c , then the fallback matter needs to bemore sparse to have an expanding shock front.We ignore physical processes like neutrino heating and cooling, radiative diffusion, photo-disintegration of nuclei, and electron-positron pair production and annihilation, some of which mightplay crucial roles in the fate of the fallback matter. Instead, we have presented solutions with varyingthe adiabatic index and the exponent l of ˙ Q in equation (1), which mimic some of the effects of theseprocesses. We require a condition that the flow has a continuous pressure distribution through the contact surfacebetween Regions I and II where equation (16) holds. From the evolution of the pressure derived fromequation (3), this requirement leads to
Pr( ξ c ) = 34 π (5 + 3 l ) ξ . (21)This boundary condition is equivalent to the condition that the energy deposited by the central sourceis equal to the total energy of the flow inside the shock front. Note here that the accreted mat-ter entering into the shock front carries no energy (the sum of the energy densities of the matter is v ( r, t ) / − GM c /r = 0 from equation 20). We found that the self-similar flow around the contactsurface behaves as D ( ξ ) ∼ D c ( ξ − ξ c ) − γ − l +3 γl γ − l ) , (22)6 ( ξ ) ∼ ξ c − γ − l γ ( ξ − ξ c ) , (23)where D c is a constant. From equation (22), the density distribution near the contact surface dras-tically changes if γ > / or not. If γ < / , the density decreases to zero toward the contactsurface. If γ > / , solutions with l = ( − γ ) / [3( γ − give constant densities at the contactsurface between Regions I and II. Otherwise the density at the contact surface diverges to infinity ( l < ( − γ ) / [3( γ − ) or goes to 0 ( l > ( − γ ) / [3( γ − ). We assume that a strong shock propagates in the accreted matter. The location of the shock front isspecified by ξ = ξ s , where ξ s is a constant. The Rankine-Hugoniot jump conditions at the shock frontcan be written as D ( ξ s ) = γ + 1 γ − D fb , (24) V ( ξ s ) = 1 − γγ + 1 s ξ s + 4 ξ s γ + 1) , (25) Pr( ξ s ) = 4 D fb q ξ l − γ + 1 q ξ . (26)Here we have used the fact that the dimensionless velocity of the accreted matter in Region III at theshock front is − q /ξ s (see eq. 17) and that the dimensionless density at the shock front is denotedby D fb . The value of D fb is determined to satisfy the boundary condition (21) at the contact surfacefor each given ξ s .Though it might be possible to extend this shock condition to that with an arbitrary strengthas was done in the context of failed supernovae by Coughlin et al. (2018), we take the strong shocklimit to simplify the procedure to obtain solutions. We can find solutions of equations (10)-(12) satisfying the boundary conditions at the shock front andthe contact surface for l > − and γ > . No solution exists in the other range of these parameters;the total deposited energy becomes infinite for l ≤ − and the density becomes negative or infinite atthe shock front for γ ≤ (see eq. 24). We obtain a solution for any positive value of ξ c . Note that ξ s monotonically increases with ξ c and the solutions in the limit of ξ c → give the minimum ξ s (denotedby ξ s , ). It follows that there are solutions with the maximum possible accretion rates at the shockfront when γ ≤ / for each l . Since the central source is supposed to deposit energy at the rate of ˙ Q ,it is convenient to normalize the accretion rate ˙ M at the shock front with ˙ Q as7 -7 -6 -5 -4 -3 -2 -1 -9 -7 -5 -3 -1 -2 -1 ! π " ξ %&’( π D fb ξ s1/2 " ξ % π D f b ξ s / & π D f b ξ s / D f b & ξ s ξ ) -7 -6 -5 -4 -3 -2 -1 -9 -7 -5 -3 -1 -5 -4 -3 -2 -1 π D fb ξ s3/2 π D fb ξ s1/2 ! " ξ s π D f b ξ s / & π D f b ξ s / D f b & ξ s ξ $ -7 -5 -3 -1 -9 -7 -5 -3 -1 -4 -3 -2 -1 π D fb ξ s3/2 π D fb ξ s1/2 D fb ξ s π D f b ξ s / & π D f b ξ s / D f b & ξ s ξ ! -5 -3 -1 -7 -5 -3 -1 -4 -3 -2 -1 ! π " ξ %&’( ! π " ξ %)’( " ξ s π D f b ξ s / & π D f b ξ s / ξ s & D f b ξ c Fig. 2.
Relations between some characteristic quantities and ξ c for solutions with four different values of γ : / (top left panel), / (top right panel), / (bottom left panel), and / (bottom right panel). l = 0 for all the solutions. D & P r v ξ D & P r v ξ D & P r v ξ Fig. 3.
Distributions of the density D , the pressure Pr , and the velocity V as functions of ξ for solutions with 3 different values of l : 0 (left panel), − / (middle panel), and − . (right panel). γ = 5 / for all the solutions. The values of ξ s are chosen arbitrarily. -1 -3.5-3-2.5-2-1.5-1-0.500.50 0.05 0.1 0.15 0.2 0.25DPrV D & P r V ξ -2 -1 -3.5-3-2.5-2-1.5-1-0.500.50.05 0.1 0.15 0.2 0.25 0.3DPr ! D & P r v ξ Fig. 4.
Distributions of the density D , the pressure Pr , and the velocity V as functions of ξ for solutions with the maximum accretion rates at the shock front.The left panel shows the solution with γ = 4 / and the right panel γ = 6 / . l = 0 for both solutions. GM c ˙ Mr s ˙ Q = 4 πD fb q ξ s , (27)where r s denotes the radius of the shock front and ˙ M = 4 πr ρ ( ξ s , t ) q GM c /r s is the mass accretionrate at the shock front. Another normalization is possible by taking account of the dimension as (cid:18) GM c t (cid:19) / ˙ M ˙ Q = 4 πD fb q ξ . (28)We show these dimensionless accretion rates, ξ s , and D fb as functions of ξ c for solutions with l = 0 and γ = 6 / , / , / , and / in Figure 2. Solutions with γ < / have maxima of the dimensionlessaccretion rates given by equation (27) or (28) at a finite ξ c . For γ = 4 / , solutions in the limit of ξ c → give the maximum accretion rates. Solutions with γ > / can sustain any accretion rates. In otherwords, the accretion rate monotonically increases to infinity as ξ c → in these solutions. At the sametime, the value of D fb becomes infinite in this limit. These characteristic values are listed in Table 1for some γ and l .Table 1 shows results of solutions with various l . The energy deposition rate usually decreaseswith time if a single physical process is involved and thus described with negative l . Such a depositionrate might mimic the declining phase of a sudden energy release due to a glitch in the crust followedby the transport of energy toward the surface (Eichler & Cheng 1989; van Riper et al. 1991). Apositive l may be realized when the central object starts to deposit energy from a slight excess of theheating rate compared with the cooling rate. Here we present solutions for the energy depositions withlinear and quadratic evolutions for simple examples of positive l . On the other hand, the solutionswith l = 0 are of particular importance because the magnetic dipole radiation from a pulsar deposits9nergy at a nearly constant rate in the early phase.As mentioned in section 2.3.1, some solutions have distinct distributions of the density nearthe contact surface. For example, Figure 3 shows solutions with γ = 5 / and three different values of l . The solution with l = − / has a finite density at the contact surface, while the density becomes0 at the contact surface in the solution with l = 0 and diverges in the solution with l = − . . Thesolution with l = 0 may be subject to the Rayleigh-Taylor instability because the density gradient nearthe contact surface has a sign opposite to that of the pressure gradient (see the left panel of Fig. 3).This instability tends to bring dense matter toward the contact surface. Thus the central activity mayfail to repel the fallback matter even if γ > / .Solutions with the maximum accretion rates are shown in Figure 4. These solutions havenegative velocities in some part of Region II because of their small ξ s ’s. From equation (25), thevelocity immediately behind the shock front becomes negative when ξ s < [3( γ − / / . If γ = 4 / ( γ = 6 / ), this criterion becomes ξ s < / ( ξ s < (5 / / / ∼ . ). On the other hand, the solutionswith γ = 5 / presented in Figure 3 happen to have large ξ s ’s that do not satisfy this criterion ( ξ s < / / for γ = 5 / ), thus the shocked matter flows outward while solutions with γ > / and negative l have ξ s , greater than the upper limit of the criterion as seen in Table 1. Shocked matter in a solutionwith a positive l tends to flow inward due to the weak central engine in the early phase. Because thecontact surface always moves outward in all the solutions (see eq. (16)), the pressure increased by thecentral energy deposition repels the fallback matter somewhere in Region II even for these solutionswith great accretion rates. In solutions with small accretion rates, the energy deposition can repel thefallback at the shock front.The kinetic energy E k ( t ) of the repelled ejecta evolves with time t following the equation E k ( t ) = 2 πL l t l +1 Z ξ s ξ c D ( ξ ) V ( ξ ) dξ. (29)Since the value of the dimensionless integral is of the order of unity here, the factor in front of theintegral gives a rough amount of the kinetic energy. Although the self-similar solutions constructed in the previous sections are valid under limited con-ditions, they can be practically applicable to some astrophysical phenomena. Here we mainly focuson supernova fallback pushed back by the energy deposition from a newborn neutron star, in partic-ular, a rotation-powered one. We discuss the conditions for repelling the fallback accretion by thespin-down luminosity and their implications on properties of associated supernovae and fates of theneutron stars. 10fter a successful supernova shock propagates through the progenitor star, a bulk of the stellarmaterial is ejected while a minor fraction falls back. The accretion rate and its temporal evolution canbe determined by the strength of the supernova shock and the inner structure of the progenitor. Forexample, Ertl et al. (2016) estimated the fallback rate based on one-dimensional numerical simulationsof neutrino-driven explosions. The fallback typically starts when the neutrino luminosity significantlydecreases, i.e., t fb > ∼ s after the core bounce, and the accretion rate subsequently decreases as ∝ ( t/t fb ) − / for t > ∼ t fb . The total mass M fb of the fallback matter ranges from ∼ − M ⊙ to ∼ − M ⊙ . More matter falls back in a more massive star. Correspondingly, the peak mass accretionrate is estimated to be ˙ M fb < ∼ − (3 - M ⊙ s − .In general, the maximum fallback accretion rate that can be repelled by the central energysource is described with M c , r s , (4 πD fb √ ξ s ) , and ˙ Q from equation (27). Since we consider a fallbackaccretion onto a neutron star, we take M c = M ∗ ∼ . M ⊙ and set the radius of the neutron star as R ∗ ∼
12 km . When the fallback accretion starts, the position of the shock will be r s ≈ ξ s ( GM ∗ t fb2 ) / ,or r s ∼ . × cm ξ s (cid:18) t fb
20 s (cid:19) / . (30)In the critical situation corresponding to the maximum mass accretion rate, the radius of the innercontact surface should be set as the neutron star radius, i.e., ξ c , crit ≈ R ∗ / ( GM ∗ t fb2 ) / , or ξ c , crit ∼ . × − (cid:18) t fb
20 s (cid:19) − / . (31)The temperature in the shocked region is typically high and the radiation pressure dominates so thatthe adiabatic index will be slightly larger than / . Combining this fact with equation (31), weestimate the critical values of ξ s and (4 πD fb √ ξ s ) from Figure 2 as ξ s , crit ∼ . πD fb q ξ s ) crit ∼ . . (32)Hereafter we will assume l = 0 which is typically valid for t ∼ t fb > ∼ since the timescale t sd of the spin-down is estimated to be much longer than that of the fallback from the following formula: t sd = − (cid:18) Ω˙Ω (cid:19) ∼ Ic B ∗ R ∗ Ω ∼ × s (cid:18) B ∗ G (cid:19) − (cid:18) P
10 ms (cid:19) , (33)where I denotes the moment of inertia of the neutron star, Ω(= 2 π/P ) the angular frequency ( P thespin period), B ∗ the magnetic field at the magnetic pole on the surface, and c denotes the speed oflight. Finally, the maximum energy deposition rate from a rotating neutron star can be described as ˙ Q crit ≈ ( µ Ω /c ) × ( R lc /R ∗ ) , or ˙ Q crit ∼ . × erg s − (cid:18) B ∗ G (cid:19) (cid:18) P
10 ms (cid:19) − , (34)11here µ = B ∗ R ∗ is the magnetic dipole moment and R lc = c/ Ω is the light cylinder radius. Wenote that the above energy deposition rate is different from the classical dipole spin-down rate, ˙ Q dipole ≈ ( µ Ω /c ) . The additional factor ( R lc /R ∗ ) represents the enhancement of spin-down lu-minosity (Parfrey, Spitkovsky, & Beloborodov 2016); the magnetic fields are maximally open, like asplit monopole, due to the accretion. From equations (30-34), we obtain the critical accretion rate as ˙ M crit , repul ∼ × − M ⊙ s − ξ s , crit . πD fb √ ξ s ) crit . (cid:18) B ∗ G (cid:19) (cid:18) P
10 ms (cid:19) − (cid:18) t fb
20 s (cid:19) / . (35)If ˙ M fb < ∼ ˙ M crit , repul , a bulk of the fallback matter will be directly repelled by the spin-downpower. Otherwise, it is accreted on the neutron-star surface.If ˙ M fb > ∼ ˙ M crit , repul , the fallback matter reaches the neutron star surface. If the accretion rateis so large, then the surface magnetic field can be buried and the spin-down power is significantlyreduced (e.g., Bernal, Lee, & Page 2010; Torres-Forn´e et al. 2016). A necessary condition for buryingthe magnetic field is set by the balance between the magnetic pressure and the ram pressure at thesurface; B ∗ π < ∼ ρv ∼ ˙ M πR ∗ s GM c R ∗ , (36)where ρ and v are the density and velocity of the accreted matter near the surface. The second equalityassumes the steady accretion of freely falling matter. Numerically, this yields the critical accretionrate of ˙ M crit , bury ∼ × − M ⊙ s − (cid:18) B ∗ G (cid:19) . (37)Whether the magnetic fields are actually buried into the crust may be addressed by comparing theposition of the magnetopause and the crust radius. In this way, Torres-Forn´e et al. (2016) estimatedthe threshold value as ˙ M crit , bury ∼ − M ⊙ s − (cid:18) B ∗ G (cid:19) / . (38)In the case of ˙ M crit , repul < ∼ ˙ M fb < ∼ ˙ M crit , bury , the situation will be more complicated. At first,the fallback matter can be accreted on the neutron star; the magnetic fields and spin-down energy areconfined in a near surface region. As the fallback rate decreases with time, large-scale fields emergeand the spin-down power pushes back the fallback matter. In Figure 5, we show how the consequencesof a fallback accretion depend on B ∗ and P for ˙ M fb = 10 − M ⊙ s − (left), − M ⊙ s − (center), and − M ⊙ s − (right).Let us now discuss possible connection between the diversities of neutron-star formation withfallback accretion and the observed young neutron stars. From Figure 5, the condition ˙ M fb < ˙ M crit , repul There is a typo in their eq. (25). ig. 5. Consequences of fallback accretion onto a neutron star and their dependencies on the magnetic field and rotation. The left, center, and right panelsshow the cases of ˙ M fb = 10 − M ⊙ s − , − M ⊙ s − , and − M ⊙ s − , respectively. is always satisfied for fast-spinning strongly-magnetized neutron stars with B ∗ > ∼ G and P < ∼ afew ms. Such cases have been proposed as a plausible central engine of extragalactic transientslike gamma-ray bursts, superluminous supernovae, and fast radio bursts (see e.g., Metzger et al.2015; Kashiyama et al. 2016; Kashiyama & Murase 2017; Margalit et al. 2018 and referencestherein) . A similar range of B ∗ and P has been also considered in the context of Galactic mag-netar formation; the magnetic field amplification can be attributed to the proto-neutron-star convec-tion coupled with a differential rotation (Duncan & Thompson 1992; Thompson & Duncan 1993)or the magneto-rotational instability (e.g., M ¨osta et al. 2015). Note, however, that so far there is noobservational support to the dynamo scenario (e.g., Vink & Kuiper 2006).Neutron stars with relatively weak magnetic fields ( B ∗ < ∼ G ( ˙ M fb / − M ⊙ s − ) / ) andslow spins ( P > ∼ a few 10 ms) satisfy the condition ˙ M fb > ˙ M crit , bury . Both the magnetic field andspin-down power are strongly suppressed by the accreted matter. As long as the situation continues,the thermal energy stored in the neutron star will be the main source of the emission. Such neutronstars will be observed as CCOs as proposed by Torres-Forn´e et al. (2016).The boundary between the former and latter cases is set by equation (35), i.e., ( B ∗ / G) × ( P/
10 ms) − ∼ ( ˙ M fb / − M ⊙ s − ) / , which might correspond to the boundary between rotation-powered pulsars and CCOs. For example, the initial magnetic field and spin period of the Crab pulsarhave been estimated to be B ∗ ∼ G and P = a few 10 ms, respectively, and consistent with theabove point if ˙ M fb < ∼ − M ⊙ s − . Such a relatively small fallback accretion rate might be alsoconsistent with a relatively low-mass progenitor inferred for the Crab pulsar.In the intermediate cases where ( ˙ M fb / − M ⊙ s − ) / < ∼ (B ∗ / G) < ∼ (P /
10 ms) × ( ˙M fb / − M ⊙ s − ) / , the magnetic field eventually emerges and the energy source of the emissioncan be either spin-down power or decay of the magnetic-field. Interestingly, the parameter range of B ∗ ∼ B QED = 4 . × G and P > ∼ a few 10 ms, especially for ˙ M fb > ∼ − M ⊙ s − is somewhat Also see Metzger, Beniamini, & Giannios (2018) for the impact of fallback accretion on the time evolution of spin in the early stage. ( B ∗ , P, ˙ M fb ) at their birth. We speculate that ordinary pulsars, CCOs, and magnetars originate fromthe blue, red, and yellow shaded regions in Figure 5, respectively. The fact that the three classes havea comparable population (e.g., Keane & Kramer 2008) can be naturally explained since the typicalparameters at their birth, B ∗ ∼ G and P ∼ a few 10 ms, roughly coincide with the intersectionof the boundaries of three regions. We should note, however, that the self-similar solutions in Sec.3 and calculations by Torres-Forn´e et al. (2016) are one dimensional. Multi-dimensional effectsin fallback accretion have to be taken into account consistently to make the above criterions morequantitatively accurate. For instance, as mentioned in section 3, solutions with l = 0 might be subjectto the Rayleigh-Taylor instability. Nevertheless, multi-dimensional effects are not expected to changethe above criterions because the energy of the shocked flow in Region II is always positive. Of course,one needs to perform multi-dimensional calculations to know detailed outcomes of this instability. Weshould also note that, in order to connect their states at birth to observational properties at t age > ∼ a few100 yrs, the long-term evolution of spins and magnetic fields become important. We will investigatethese topics elsewhere. We have presented series of self-similar solutions for fallback matter being shocked and repelled bythe energy deposition from a central object. The behavior of the solutions changes depending onthe adiabatic index γ in the shocked region. For γ > / , we can find a self-similar solution for anarbitrarily high fallback accretion rate by taking the radius of the contact surface correspondinglysmall. On the other hand, for γ ≤ / , there are upper-bounds to the accretion rate at the shock front r = r s : ˙ M crit = r s ˙ QGM c × (4 πD fb q ξ s ) crit . (39)The critical values of πD fb √ ξ s , ranging from ∼ − , are shown in Table 1. Note that πD fb √ ξ s =1 corresponds to the cases where the accretion luminosity G ˙ M M c /r s is equal to the energy depositionrate from the central object ˙ Q . For ˙ M > ˙ M crit , the fallback matter will plunge into the central object.We have applied the self-similar solution to neutron star formation with fallback accretionwhen the neutron star deposits energy due to its spin-down power. The solution suggests that the14ritical mass accretion rate above which the fallback matter can be accreted on the surface is given asa function of the initial spin rate and surface magnetic filed. It is shown that this condition togetherwith the criterion that the accreted matter can bury the magnetic field and suppress the spin-downpower may determine the fate of the newborn neutron star to become a magnetar, a pulsar, or a CCO. We here discuss relations between the self-similar solution constructed in this paper and previousstudies on the dynamics of fallback accretion.The shock propagates outward with the radius proportional to t / in our models. A simi-lar behavior of the shock front was obtained from numerical computations for fallback presented inZel’dovich et al. (1972). Thus our model can capture some features of realistic models that treat theenergy source originating from gravitational energy of accreted matter. Zel’dovich et al. (1972) dis-cussed the accretion of up to 10 − M ⊙ assuming a static power-law density distribution proportionalto r − / ( r denotes the radial coordinate) as the initial condition, which results in a constant accretionrate. They computed hydrodynamics of the fallback matter including radiative diffusion of photonsand neutrino emission. Colgate (1971) emphasized the role of neutrino emission in driving intensefallback motion. These investigations focused on the early phase affected by the unknown explosionmechanism of a core collapse supernova.Chevalier (1989) discussed the fallback phenomenon after a reverse shock wave reaches theneutron star. He argued that the accretion rate of the bound debris decreases with time t following apower law as t − / provided that the mass fraction of the debris is uniformly distributed in bindingenergy. This power law evolution of the accretion rate had been already pointed out from a dimen-sional analysis (Michel 1988). Though Chevalier (1989) started his computations from the uniformmotion of uniform matter to obtain this accretion rate, he pointed out that this situation can be realizedif the density ρ d ( r ) of the debris initially has a static power-law distribution ρ d ( r ) ∝ r − (Sakashita& Yokosawa 1974). He also discussed the effects of the pulsar activity on the fallback and concludedthat the effects are negligible if only the magnetic pressure is considered.On the contrary, newborn spinning neutron stars should deposit energy in the surroundingmatter as a result of the activities originating from rotating strong magnetic fields. This interactionhas been discussed with self-similar solutions in 1D spherical symmetry (Chevalier 2005; Suzuki &Maeda 2017) and numerical simulations in 2D axi-symmetry (Chen et al. 2016; Suzuki & Maeda2017; Blondin & Chevalier 2017) or 3D (Blondin & Chevalier 2017). All of these studies have dealtwith interactions between pulsar winds and expanding supernova ejecta without fallback matter. In15eality, the innermost part of once ejected matter falls back and the energy deposited by the pulsarwind should affect the motion of the fallback matter, which is important for determining the nature ofyoung neutron stars as discussed in section 4. Our solutions cover some of these aspects though wesimplified the situation by introducing the power law energy deposition rate and assuming a specificmotion of the fallback matter above the shock front, which is different from the assumptions in theabove previous work. Thus our solutions have a different temporal evolution of the mass accretionrate. Acknowledgments
This work is partially supported by JSPS KAKENHI Grant Number JP17K14248, JP18H04573, 16H06341, 16K05287, 15H02082, MEXT, Japan.
References
Bernal C. G., Lee W. H., Page D., 2010, RMxAA, 46, 309Blondin, J. M., & Chevalier, R. A. 2017, ApJ, 845, 139Chen, K.-J., Woosley, S. E., & Sukhbold, T. 2016, ApJ, 832, 73Chevalier, R. A. 1989, ApJ, 346, 847Chevalier, R. A. 2005, ApJ, 619, 839Colgate, S. A. 1971, ApJ, 163, 221Coughlin, E. R., Quataert, E., & Ro, S. 2018, ApJ, 863, 158D’A`ı, A., Evans, P. A., Burrows, D. N., et al. 2016, MNRAS, 463, 2394Duncan R. C., Thompson C., 1992, ApJ, 392, L9Eichler, D., & Cheng, A. F. 1989, ApJ, 336, 360Ertl, T., Janka, H.-T., Woosley, S. E., Sukhbold, T., & Ugliano, M. 2016, ApJ, 818, 124Ertl, T., Ugliano, M., Janka, H.-T., Marek, A., & Arcones, A. 2016, ApJ, 821, 69Ferrario L., Wickramasinghe D., 2006, MNRAS, 367, 1323Figer D. F., Najarro F., Geballe T. R., Blum R. D., Kudritzki R. P., 2005, ApJ, 622, L49Gaensler B. M., McClure-Griffiths N. M., Oey M. S., Haverkorn M., Dickey J. M., Green A. J., 2005, ApJ,620, L95Kashiyama K., Murase K., 2017, ApJ, 839, L3Kashiyama K., Murase K., Bartos I., Kiuchi K., Margutti R., 2016, ApJ, 818, 94Keane, E. F., & Kramer, M. 2008, MNRAS, 391, 2009Margalit B., Metzger B. D., Thompson T. A., Nicholl M., Sukhbold T., 2018, MNRAS, 475, 2659 asuyama, M., Shigeyama, T., & Tsuboki, Y. 2016, PASJ, 68, 22Metzger B. D., Beniamini P., Giannios D., 2018, ApJ, 857, 95Metzger B. D., Margalit B., Kasen D., Quataert E., 2015, MNRAS, 454, 3311Michel, F. C. 1988, Nature, 333, 644M¨osta P., Ott C. D., Radice D., Roberts L. F., Schnetter E., Haas R., 2015, Natur, 528, 376Muno M. P., et al., 2006, ApJ, 636, L41Muslimov A., Page D., 1995, ApJ, 440, L77Olausen S. A., Kaspi V. M., 2014, ApJS, 212, 6Park, S., Mori, K., Kargaltsev, O., et al. 2006, ApJL, 653, L37Parfrey K., Spitkovsky A., Beloborodov A. M., 2016, ApJ, 822, 33Pavlov, G. G., Sanwal, D., Kızıltan, B., & Garmire, G. P. 2001, ApJL, 559, L131Piro, A. L., & Ott, C. D. 2011, ApJ, 736, 108Rea, N., Borghese, A., Esposito, P., et al. 2016, ApJL, 828, L13Sakashita, S., & Yokosawa, M. 1974, Ap&SS, 31, 251Suzuki, A., & Maeda, K. 2017, MNRAS, 466, 2633Thompson C., Duncan R. C., 1993, ApJ, 408, 194Torres-Forn´e, A., Cerd´a-Dur´an, P., Pons, J. A., & Font, J. A. 2016, MNRAS, 456, 3813van der Horst, A. J., Connaughton, V., Kouveliotou, C., et al. 2010, ApJL, 711, L1van Riper, K. A., Epstein, R. I., & Miller, G. S. 1991, ApJL, 381, L47Vink J., Kuiper L., 2006, MNRAS, 370, L14Young E. J., Chanmugam G., 1995, ApJ, 442, L53Zel’dovich, Y. B., Ivanova, L. N., & Nadezhin, D. K. 1972, Soviet Ast., 16, 209 able 1. Critical dimensionless quantities in solutions with some l and γ . For γ = 5 / and / , values of ξ s , of the solutions in thelimit of ξ c → are listed. For γ = 4 / , in addition to ξ s , , the corresponding eigenvalue of D fb (eq. (24)) as well as the correspondingmaximum values of the two accretion rates (eqs. (27) and (28)) are listed. For γ = 6 / , values of ξ s ( ξ s , and ξ s , ) that give themaximum values of the two accretion rates and the corresponding values of D fb . γ / / / / l ξ s , ξ s , ξ s , D fb , π (cid:16) D fb p ξ s (cid:17) π (cid:16) D fb p ξ (cid:17) ξ s , D fb , π (cid:16) D fb p ξ s (cid:17) ξ s , D fb , π (cid:16) D fb p ξ (cid:17) − . − . − / − .1.1 0.49 0.25 1.2 7.8 2.0 0.2129 0.2550 1.479 0.4766 0.1176 0.48620 0.80 0.36 0.18 1.3 6.9 1.2 0.1691 0.2332 1.205 0.4010 0.1003 0.32011 0.57 0.26 0.13 1.3 5.8 0.73 0.1191 0.2471 1.072 0.3170 0.09583 0.21492 0.46 0.20 0.10 1.4 6.1 0.24 0.1000 0.2588 1.029 0.2661 0.09859 0.1701