Supernova Fallback onto Magnetars and Propeller-Powered Supernovae
aa r X i v : . [ a s t r o - ph . H E ] A p r Submitted for publication in The Astrophysical Journal.
Preprint typeset using L A TEX style emulateapj v. 11/10/09
SUPERNOVA FALLBACK ONTO MAGNETARS AND PROPELLER-POWERED SUPERNOVAE
Anthony L. Piro and Christian D. Ott
Theoretical Astrophysics, California Institute of Technology, 1200 E California Blvd., M/C 350-17, Pasadena, CA 91125;[email protected]; [email protected]
Submitted for publication in The Astrophysical Journal.
ABSTRACTWe explore fallback accretion onto newly born magnetars during the supernova of massive stars.Strong magnetic fields ( ∼ G) and short spin periods ( ∼ −
10 ms) have an important influenceon how the magnetar interacts with the infalling material. At long spin periods, weak magnetic fields,and high accretion rates, sufficient material is accreted to form a black hole, as is commonly found formassive progenitor stars. When B . × G, accretion causes the magnetar to spin sufficientlyrapidly to deform triaxially and produce gravitational waves, but only for ≈ −
200 s until it collapsesto a black hole. Conversely, at short spin periods, strong magnetic fields, and low accretion rates,the magnetar is in the “propeller regime” and avoids becoming a black hole by expelling incomingmaterial. This process spins down the magnetar, so that gravitational waves are only expected if theinitial protoneutron star is spinning rapidly. Even when the magnetar survives, it accretes at least ≈ . M ⊙ , so we expect magnetars born within these types of environments to be more massive thanthe 1 . M ⊙ typically associated with neutron stars. The propeller mechanism converts the ∼ ergsof spin energy in the magnetar into the kinetic energy of an outflow, which shock heats the outgoingsupernova ejecta during the first ∼ −
30 s. For a small ∼ M ⊙ hydrogen-poor envelope, this energycreates a brighter, faster evolving supernova with high ejecta velocities ∼ (1 − × km s − and mayappear as a broad-lined Type Ib/c supernova. For a large & M ⊙ hydrogen-rich envelope, the resultis a bright Type IIP supernova with a plateau luminosity of & ergs s − lasting for a timescale of ∼ −
80 days.
Subject headings: gravitational waves — stars: magnetic fields — stars: neutron — supernovae: general INTRODUCTION “Magnetars” are a subset of neutron stars with dipolemagnetic fields as strong as B ∼ − G (Dun-can & Thompson 1992; Thompson & Duncan 1993). Al-though at an age of 1 , − ,
000 years they have spinperiods of P = 5 −
12 s, as measured from soft gamma-ray repeaters and anomalous X-ray pulsars, it is an out-standing question of how rapidly they rotate when firstborn. Short initial spin periods ( P ∼ −
10 ms) havebeen favored theoretically so that the dynamo processthat creates these strong magnetic fields may operateefficiently (Duncan & Thompson 1992; Akiyama et al.2003; Thompson et al. 2005). Motivated by this, manygroups have investigated the possible impact of the spin-down of this newly formed magnetar in powering an ex-plosion (see, for example, Bodenheimer & Ostriker 1974;Wheeler et al. 2000; Thompson et al. 2004; Burrowset al. 2007; Dessart et al. 2008). Such short spin pe-riods may also be a source of ultra-high energy cosmic-rays (Arons 2003), create a collimated relativistic flow asneeded for gamma-ray bursts (Uzdensky & MacFadyen2007; Metzger et al. 2010, and references therein), orproduce a luminous supernova (Kasen & Bildsten 2010;Woosley 2010).An assumption of all of these studies is that the su-pernova which gave birth to the magnetar was successfulin ejecting the majority of the progenitor star’s enve-lope. This is clearly correct in many cases, since weknow that neutron stars with more modest magneticfields ( ∼ G) are created in supernovae. But it ispossible that some subset of supernovae which produce neutron stars have small injected explosion energies. Asis expected for massive stars that give rise to black holes,these would not be successful in ejecting the majority ofthe envelope and a sizable amount of fallback would oc-cur (as found for ≈ − M ⊙ stars by Heger et al. 2003).In addition, even in cases where the majority of the en-velope is ejected, asymmetries in the explosion may stillresult in significant fallback. For these reasons, it is plau-sible that there exists a population of massive stars thatgive birth to magnetars that are subsequently subject toaccretion of the envelope material.Another motivation for studying fallback accretiononto magnetars is the presence of magnetars near clus-ters of massive stars. SGR 1806 −
20 and CXOUJ164710 . − −
20 and Westerlund 1, respectively, and are in-ferred to have had progenitor masses of ≈ M ⊙ (Figeret al. 2005; Bibby et al. 2008; Muno et al. 2006). Fur-thermore, the expanding H I shell around the magnetar1E 1048 . − ≈ − M ⊙ pro-genitor (Gaensler et al. 2005). Such massive stars aretypically assumed to give rise to black holes (Fryer 1999;Heger et al. 2003), although we note that this will de-pend sensitively on the details of mass loss during stel-lar evolution (Smith et al 2010; O’Connor & Ott 2011)and on whether these magnetars have binary progenitors(Belczynski & Taam 2008). It is therefore worth explor-ing whether the presence of a highly-magnetized neutronstar qualitatively changes the outcome of the collapse ofmassive stars.In the following study we explore the interaction ofnewly born magnetars with supernova fallback. We be- PIRO & OTTgin in § § § § § FALLBACK VERSUS OUTFLOW
Before we investigate the effects of fallback accretion,it is pertinent to discuss when fallback is expected. Al-though these arguments are strictly applicable for onlyone-dimension, and we expect a multi-dimensional flowto provide more opportunities for fallback, this givessome intuition about how fallback depends on the ac-cretion rate, spin, and magnetic field strength.As the rapidly rotating, newly born magnetar spinsdown, it goes through stages in which it emits en-ergy in dipole spindown radiation and a neutrino-driven,magnetically dominated wind (Thompson, Chang, &Quataert 2004), both of which may hinder accretion. Fora magnetar with a dipole magnetic moment µ and spinΩ, the spindown luminosity is L dip = µ Ω c = 9 . × µ P − ergs s − , (1)where µ = µ/ G cm , as is appropriate fora neutron star with a 10 G magnetic field, and P = 2 π/ Ω = 1 P ms. Assuming this luminosity is car-ried by a relativistic wind, the associated pressure at aradius r is p dip = L dip / πcr . Fallback accretion exertsan inward ram pressure, and for the case of sphericallysymmetric accretion at a rate ˙ M onto a mass M , this isgiven by p ram = ˙ M π (cid:18) GMr (cid:19) / . (2)Since p dip ∝ r − and p ram ∝ r − / , the spindown lumi-nosity always wins at sufficiently large radii. If the fall-back accretion is already proceeding and then the spin-down luminosity is to disrupt this accretion flow, we canask what is the critical accretion rate above which thefallback ram pressure dominates at the magnetar radius R . This gives˙ M dip , crit = µ Ω c (cid:18) R GM (cid:19) / = 1 . × − µ P − M − / . R / M ⊙ s − , (3)where M . = M/ . M ⊙ and R = R/
12 km. Thisaccretion rate is well-exceeded in all cases we consider.During the Kelvin-Helmholtz cooling epoch for thenewly born magnetar, deleptonization and thermal neu-trino losses create a neutrino-driven wind that is magnet-
Fig. 1.—
The critical accretion rate, above which fallback dom-inates, as a function of the spin period. We consider two physicalprocesses for inhibiting the fallback: dipole spindown radiation(denoted by M dip , crit and given by eq. [3]), and a neutrino-drivenwind (denoted by M ν, crit and given by eq. [5]). In each case wevary the radius by a factor of 100 (as shown by the shaded regions)to represent uncertainty in the radius at which the accretion flowfirst comes into contact with this outgoing energy. ically flung by the magnetar’s dipole field. For a massloss rate ˙ M ν , the luminosity that goes into this processis (Thompson et al. 2004) L ν = (cid:18) µ Ω ˙ M ν (cid:19) / ˙ M ν = 4 . × µ / P − / ˙ M / ν, − ergs s − , (4)where ˙ M ν, − = ˙ M ν / − M ⊙ s − . Repeating the aboveanalysis of assuming this is a relativistic wind and com-paring to the ram pressure at the magnetar surface, wederive a critical accretion rate˙ M ν, crit = 2 ˙ M ν c (cid:18) R GM (cid:19) / (cid:18) µ Ω ˙ M ν (cid:19) / = 8 . × − µ / P − / ˙ M / ν, − M − / . R / M ⊙ s − . (5)This limit is a little more stringent than the one derivedfor dipole spindown (eq. [3]). Indeed some of the lowerfallback rates we consider are exceeded by this. WhenThompson et al. (2004) follow the spindown from aneutrino-driven wind, they find modest amounts of spin-down (an increase in the spin period of ∼ ∼ − − − M ⊙ s − , depending on the process thatis inhibiting the fallback. Comparing with the fallbackfound in numerical studies by MacFadyen et al. (2001)or Zhang et al. (2008), this implies a massive progeni-tor (in the range of ∼ − M ⊙ for solar metallicityand the progenitor models of Woosley et al. 2002) anda low explosion energy (higher explosion energies leadto weaker fallback; Dessart, Livne, & Waldman 2010).Although it is not well-known how progenitor mass andexplosion energy correlate with magnetar creation, evenwith these limitations, there is a wide parameter spacewhere fallback onto a magnetar seems inevitable.Even in cases where these scalings appear to argue thatfallback is inhibited, it is still worthwhile to investigatefallback on account that (1) the neutrino-driven windonly lasts ∼
10 s while the fallback occurs on a & SPIN EVOLUTION DUE TO FALLBACK ACCRETION
Accretion Versus Expulsion
The initial spin period of newly-born neutron stars de-pends on both the spin profile of the progenitor star andsubsequent processes that add, subtract, and redistributeangular momentum. Fryer & Heger (2000) performedsmoothed particle hydrodynamics simulations using a ro-tating progenitor model from Heger et al. (2000) andestimated an initial protoneutron star (PNS) spin periodon the order of 100 ms. It is, however, not clear how theydefined the extent of the PNS (see discussion in Ott etal. 2006). The subsequent cooling and contraction to aradius of ∼
12 km resulted in P ∼ M ⊙ is conserved as the PNS cools and contracts to aneutron star, finding periods of ∼ −
17 ms depending onthe progenitor model. Thompson et al. (2005) studiedthe action of viscous processes in dissipating the strongrotational shear profile produced by core collapse in arange of progenitors and for different initial iron core pe-riods. They showed that for rapidly rotating cores withpostbounce periods of . ∼ − P ∼ . −
10 ms and solid body ro-tation in the PNS core for progenitors with precollapseperiods .
50 s. We therefore consider initial magnetarspin periods in this range for our present study.Subsequent to the initial spin period being set as de-scribed above, the neutron star may be subject to fall-back accretion. Accretion comes under the strong in-fluence of the star’s dipole field at the nominal Alfv´enradius r m = µ / ( GM ) − / ˙ M − / , where µ is the dipolemagnetic moment of the magnetar. For typical magnetar parameters r m = 14 µ / M − / . ˙ M − / − km , (6)where ˙ M − = ˙ M / − M ⊙ s − , and the prefactor to r m can vary depending on the details of the interac-tion between the flow and magnetic field (Ghosh & Lamb1979; Arons 1986, 1993). The other critical radius,set by the magnetar’s spin Ω, is the corotation radius r c = ( GM/ Ω ) / , r c = 17 M / . P / km . (7)Roughly speaking, one expects that for r m < r c , materialis funneled by the magnetar’s dipole field before accretingonto the magnetar’s surface, while when r m > r c , ma-terial must spin at a super-Keplerian rate to come intocorotation with the magnetar and is thus expelled (the“propeller regime,” Illarionov & Sunyaev 1975). Setting r m > r c gives a critical accretion rate˙ M < . × − µ M − / . P − / M ⊙ s − . (8)Comparing to the 25 M ⊙ collapsar models of MacFadyenet al. (2001), they find early-time accretion rates of10 − − − M ⊙ s − by just varying the injected ex-plosion energy by (0 . − . × ergs. Whether amagnetar is in the propeller regime or not is thereforevery sensitive to how energetic the supernova is.This simplistic picture is not the complete story, ashas been detailed by a great many theoretical studies ofaccretion onto magnetic stars (see for example, Pringle &Rees 1972; Lynden-Bell & Pringle 1974; Ghosh & Lamb1979; Aly 1980; Wang 1987; Shu et al. 1994; Lovelace etal. 1995, 1999; Ikhsanov 2002; Rappaport et al. 2004;Ek¸si et al. 2005; Kluzniak & Rappaport 2007; D’Angelo& Spruit 2010). More recently, numerical simulationshave also been used to investigate this problem (Hayashiet al. 1996; Goodson et al. 1997; Miller & Stone 1997;Fendt & Elstner 2000; Matt et al. 2002; Romanova et al.2003, 2004, 2009). For our present work, we implementa simple model largely based on that used by Ek¸si etal. (2005), as described below. Their prescription hasthe advantage of being applicable and continuous overa wide range of parameters, while capturing the mainexpected features of the propeller regime.In cases where r c > r m > R , the inflowing materialis channeled onto the magnetar poles where it shocksand neutrino cools. We save a more detailed treatmentof the physics of this process for the Appendix, sinceit does not have a direct bearing on our results for thetime-dependent spin, which we consider next. Time-Dependent Spin From Fallback Accretion
Given this picture of accretion and expulsion describedabove, we solve for spin evolution under the influence offallback accretion by integrating the differential equation
I d Ω dt = N dip + N acc , (9)where I = 0 . M R is the moment of inertia (Lattimer& Prakash 2001), and N dip and N acc are the torques fromdipole emission and accretion, respectively. As discussedin §
2, we ignore spindown from neutrino-driven winds in PIRO & OTTequation (9). The dipole spindown torque is given by N dip = − µ Ω c = − . × µ P − ergs . (10)We assume that the magnetar is rotating as a solid body,as is likely the case within ∼ T / | W | in-stabilities (Watts et al. 2005; Ott et al. 2005) will limitdifferential rotation. When r m > R , material leaves thedisk with the specific angular momentum at a radius r m .Depending on the relative positions of the Alfv´en andcorotation radii, this can either spin up or spin down themagnetar, so we write the torque as N acc = n ( ω )( GM r m ) / ˙ M if r m > R, (11)where n ( ω ) is the dimensionless torque which dependson the fastness parameter ω = Ω / ( GM/r m ) / =( r m /r c ) / . Ek¸si et al. (2005) discuss different waysin which n ( ω ) can be set, but for simplicity we take n = 1 − ω . This has the advantage that the torque goesto zero at the corotation radius, is continuous for all ω ,and goes negative when r m > r c , corresponding to thespin down which occurs during the propeller regime. As ω gets larger, this prescription gives increasingly strongspindown, consistent with the more detailed simulationsof Romanova et al. (2004). When r m < R we set thetorque to N acc = (1 − Ω / Ω K ) ( GM R ) / ˙ M if r m < R, (12)where Ω K = ( GM/R ) / . The prefactor is includedto ensure that torque is continuous for all values of r m .The disadvantage is that since the prefactor is .
1, itwill underpredict the amount of torque, but this doesnot change our main conclusions, as we discuss in § β ≡ T / | W | , where T = I Ω /
2. We use the prescription given in Lattimer& Prakash (2001) for | W | , | W | ≈ . M c GM/Rc − . GM/Rc ) . (13)We keep R fixed even as M changes, which is roughlyconsistent with most equations of state, except when M gets near its maximum value (Lattimer & Prakash 2001).When β = 0 .
5, the neutron star is at breakup and can-not accept further angular momentum. Even prior tothis, dynamical bar-mode instabilities occur for β > . β & .
14, driven by gravitational radiation reaction or viscos-ity (Lai & Shapiro 1995). Since the dynamical bar-modeinstability is guaranteed to radiate and/or hydrodynami-cally re-adjust angular momentum, we set N acc = 0 when β > .
27. We ignore changes in spin due to the secularinstabilities since growth timescales are uncertain andmay be suppressed by competition between viscosity orgravitational radiation reaction (Lai & Shapiro 1995).We parameterize the fallback accretion rate to mimicthe results of MacFadyen et al. (2001) and Zhang et al.(2008). This can roughly be broken into two parts. Atearly times it scales as˙ M early = η − t / M ⊙ s − , (14) where η ≈ . −
10 is a factor that accounts for differentexplosion energies (a smaller η corresponds to a largerexplosion energy), and t is measured in seconds. The latetime accretion is roughly independent of the explosionenergy and is set to be˙ M late = 50 t − / M ⊙ s − . (15)The accretion rate at any given time is found from com-bining these two expressions˙ M = (cid:16) ˙ M − + ˙ M − (cid:17) − . (16)The mass of the neutron star increases at a rate ˙ M when r m < r c and is set fixed when r m > r c . For comparison,we also integrate ˙ M for all values of r m to follow howmuch matter the magnetar would have accreted if notfor the propeller mechanism.Equation (16) reflects fallback of the envelope, butmost likely this material must pass through a disk be-fore finally accreting onto the magnetar. To test thishypothesis and explore whether this leads to a quanti-tative change of the accretion rate, we built one-zone, α -disk models (similar to Metzger et al. 2008) usingthe angular momentum profiles of the massive, rotatingprogenitors of Woosley & Heger (2006) simulated withGR1D (O’Connor & Ott 2010). Our general finding wasthat (1) there is sufficient angular momentum to form adisk, and (2) the disk is nearly steady-state, where theaccretion rate onto the star differs from the infall rate byno more than a factor of ∼ α -viscosity, with a larger α resulting in higher accretionrates), and (3) the radius of the disk is typically welloutside of the Alfv´en radius. We therefore consider themediation of the disk to be degenerate with η and usethe direct infall rates as described above.In Figure 2, we compare integrations of equation (9)for values of η = 0 . , , and 10. The top panel shows theaccretion rate given by equation (16). The middle panelplots the time-dependent spin period. The bottom panelplots the fastness parameter, which reflects whether ornot the magnetar is in the propeller regime. For η = 0 . . M ⊙ is accreted out of a potential amount ofaccretion of 1 . M ⊙ , and for η = 1 only 1 . M ⊙ is ac-creted out of a potential amount of 3 . M ⊙ . Thereforeboth these cases are able to avoid becoming a black holevia the propeller mechanism (assuming a maximum neu-tron star mass of 2 . M ⊙ ). In contrast, the η = 10 case(which corresponds to a lower-energy explosion) accretes3 . M ⊙ out of 6 . M ⊙ , which means it likely becomesa black hole. Since the accretion rate is highest at earlytimes, black hole formation happens rather quickly dur-ing the runs, at ≈
34 s and ≈
46 s for maximum neutronstar masses of 2 . M ⊙ and 3 M ⊙ , respectively.In each of these cases, the spin eventually reaches anequilibrium value that simply tracks ˙ M with ω ≈
1. Set-ting r m = r c , we calculate an equilibrium spin period, P eq = 2 πµ / ( GM ) − / ˙ M − / = 5 . µ / M − / . ˙ M − / − ms , (17)where ˙ M − = ˙ M / − M ⊙ s − . MAGNETAR VERSUS BLACK HOLE FORMATION
UPERNOVA FALLBACK ONTO MAGNETARS 5
Fig. 2.—
The spin evolution of a magnetar with B = 10 G andan initial spin period of P = 1 ms. We compare values of η =0 . ,
1, and 10, demonstrating the strong effect early-time accretioncan have. The top panel shows the time-dependent accretion rate,the center panel shows the spin period, and the bottom panel showsthe fastness parameter ω , where ω > ω ≤ The Amount of Mass Accreted
The example models in the previous section demon-strate that the amount of mass accreted by the magne-tar depends strongly on whether the propeller regime isreached. Therefore, whether or not a magnetar even-tually becomes a black hole depends on its initial spinperiod and magnetic field. This is in stark contrast toneutron stars with dynamically unimportant magneticfields whose fates simply depend on the properties ofthe supernova and the compactness of the stellar core(Zhang et al. 2008; O’Connor & Ott 2011). To explorethese correlations, we plot contours for the amount ofmass accreted as a function of the initial spin period andmagnetic field in Figures 3 and 4 for values of η = 1 and0 .
1, respectively. In the η = 1 case, a magnetar remainsfor only a small fraction of the initial conditions (this ofcourse depends on the value of the maximum neutronstar mass). For η = 0 .
1, a magnetar is expected for themajority of the parameter space. Since these two valuesof η correspond to a factor of ∼ B ≈ G and P ≈ . increases for stronger magnetic fields, contrary to ourintuition for when the propeller mechanism should be Fig. 3.—
Contours show the amount of mass accreted (in so-lar masses) for different initial spin periods and magnetic fieldstrengths (all for η = 1). For every case we assume an initialmagnetar mass of 1 . M ⊙ with a radius of 12 km. If the propellerregime did not expel material, then 3 . M ⊙ would have been ac-creted. Fig. 4.—
The same as Fig. 3, but with η = 0 .
1. For this case,if the propeller regime did not expel material, then 1 . M ⊙ wouldhave been accreted. strongest. To explore what is happening here, we plotthe spin evolution for a collection of different magneticfields and initial spin parameters in Figure 5. We can seethat at sufficiently strong magnetic fields, the propelleris so strong that the star quickly spins down during thefirst ≈
20 s ( dotted line ). At this point the accretion rate PIRO & OTT
Fig. 5.—
Time-evolution of the spin period and fastness param-eter ω for a diverse selection of models. The accretion rate cor-responds to η = 1 for all cases (the solid line from the top panelin Fig. 2). The low magnetic field ( dot-dashed lines ) and slowlyspinning ( dashed line ) cases exceed a mass of 2 . M ⊙ at ≈
180 s,at which point they most likely become black holes. See the textfor further discussion of the features exhibited here. has increased dramatically, and the star now accretes andspins up until about ≈
200 s. It is due to this stage thatthe magnetar accretes more than was expected.One takeaway message of this parameter survey is thatfor all these models at least ≈ . M ⊙ is accreted. There-fore magnetars that are subject to the conditions of be-ing born within a massive star should on average be moremassive than the 1 . M ⊙ typically associated with neu-tron stars. Measuring the masses of magnetars wouldtherefore be useful for constraining whether some are in-deed born in massive progenitors. Prospects for Gravitational Wave Production
These results also have bearing on whether a youngmagnetar should be expected to be an important gravi-tational wave source (as discussed in Corsi & M´esz´aros2009, and references therein). For this to occur, it mustbe spinning sufficiently quickly that dynamical bar-modeinstabilities or secular instabilities are excited. To ex-plore this, we plot the spin parameter β for a selection ofmodels in Figure 6. The majority of the parameter spacewe probe experiences some time in the propeller regime,spinning down the magnetar, and making gravitationalwave emission unlikely. For magnetic fields . × G,the magnetar is spun up by accretion sufficiently to pro-duce gravitational waves, but the accretion then quicklyleads to collapse to a black hole. This is seen in thebottom panel of Figure 6, where β > .
14 for a time,but then exceeds a mass of 2 . M ⊙ at ≈
180 s. Thismodel never exceeds β = 0 .
27, but this is an artificialeffect of the 1 − Ω / Ω K factor for the torque prescrip-tion (see eq. [10]). If we instead assume the magnetaraccreted with the specific angular momentum at its sur- Fig. 6.—
The spin parameter β for a selection of models, all with η = 1. The dotted lines denote β = 0 .
14 (the critical value forsecular instabilities) and β = 0 .
26 (the critical value for dynamicalinstabilities). Gravitational wave emission is only expected abovethe dotted lines, and is generally seen for extremely short initialspin periods ( P . . B . × G). face of (
GM R ) / , β = 0 .
27 would be easily reached. Weestimate the timescale for gravitational wave emission byintegrating the early time accretion law, t gw = 140 η − / (cid:18) M max − M . M ⊙ (cid:19) / s , (18)where M max is the maximum neutron star mass beforeblack hole formation and M is the initial neutron starmass. The accretion peaks on a timescale t p = 150 η − / s , (19)which is found by equating equations (14) and (15). Soour two conditions for equation (18) to be valid are that B . × G and t gw < t p . If B & × G wedon’t expect appreciable spinup and gravitational waveemission, and if t gw > t p then the gravitational waveemission timescale is merely ≈ t p . PROPELLER-POWERED SUPERNOVAE
In cases that do not collapse into black holes, thematerial expelled by the propeller mechanism collideswith the supernova ejecta. This shock-heats the enve-lope and increases the energy budget of the supernova.We next estimate the observable signature of such pow-ering. The process we describe here is decidedly differentfrom what was explored by Kasen & Bildsten (2010) andWoosley (2010), who used dipole spindown luminosityto heat and power a more luminous supernova. In theircase the dipole spindown takes place on sufficiently longtimescales that it can directly power an extremely lumi-nous supernova. As we discuss below, the majority of theenergy from the propeller-mechanism is injected duringUPERNOVA FALLBACK ONTO MAGNETARS 7the first ∼ −
30 s, so it can be treated as a suddenimpulse of energy at early times. The majority of thisenergy is therefore lost to adiabatic expansion and notseen directly in the peak luminosity. Nevertheless, theenergy can accelerate the supernova ejecta to high veloc-ities of up to ∼ (1 − × km s − , which are observablein the spectra and alter the lightcurve shape. Propeller Energy Budget
The expelled material carries a kinetic energy equal tothe spindown energy of the magnetar. To help power thesupernova, this material must climb out of the magne-tar’s gravitational well, so we estimate the kinetic lumi-nosity of the propeller material as L prop = − N acc Ω − GM ˙ M /r m , (20)where the negative sign in the first term is because wehave defined N acc to be negative when the magnetar isspinning down (eq. [11]). With this equation we haveassumed that the majority of the outflow originates fromthe inner edge of the disk. While this is a reasonableassumption, it also means that the material has to travelthe furthest out of the potential well. If material canleave the disk at larger radii, it will require less energyto do so, thus this represents a lower limit. The totalenergy that can possibly be put into expelled material islimited by the magnetar rotation, E rot = 12 I Ω = 2 . × M . R P − . (21)In some cases the early-time accretion may even spin themagnetar up to sub-millisecond spin periods before thepropeller mechanism begins. In these cases the magnetarstores the accretion energy in its spin, which is tapped viathe propeller mechanism to help power the supernova.In Figures 7 and 8 we quantify the luminosity of thepropeller mechanism as well as what fraction of E rot isable to be tapped by this process. The top panels ofeach figure show L prop (eq. [20]), L dip (eq. [1]), and theradioactive decay of 0 . M ⊙ of Ni as a function of time.The propeller-powering only lasts ∼ −
30 s until r m ∼ r c . At this point the magnetar is not spinning sufficientlyrapidly to expel material to infinity and the luminosityquickly shuts off. The bottom panels of Figures 7 and 8show the integrated energy as a function of time, E i ( t ) = Z t L i ( t ) dt, (22)where i stands for either the propeller luminosity ordipole luminosity. In Figure 7, less than ∼
20% of therotational energy goes into expelling material. For astronger magnetic field the propeller regime is more ex-treme, and nearly all of the rotational energy is convertedinto energy of outflowing material, as shown in Figure 8.In either case, this additional energy may be greater thanthe typical supernova energy E sn of ∼ ergs. In thefollowing sections we explore how this additional energyalters the properties of the supernova depending on themass of the envelope material, representative of TypeIb/c and Type IIP supernovae. Low-Mass Envelopes
Fig. 7.—
The luminosity emitted by the magnetar, either fromkinetic energy of magnetically expelled material L prop ( solid lines ),dipole radiation L dip ( dashed lines ), and, for comparison, decay of0 . M ⊙ of Ni ( dotted lines ). The initial parameters are P = 1 mswith η = 1 with a magnetic field of 10 G. The bottom panelshows the integrated energy as a function of time for each case. Thetotal energy from radioactive heating is not sufficient to appear onthe bottom panel.
Fig. 8.—
The same as Fig. 7, but with a magnetic field of5 × G. This example is more strongly in the propeller-regimeat early times, so that a larger fraction of the spin energy goes intothe kinetic energy of the outflow.
PIRO & OTTWe consider a supernova with initial energy E sn , ejectamass M ej , subject to a sudden impulse of energy E prop .As we will show, the observational impact of E prop de-pends strongly on the ejecta mass and composition. Inthis section we focus on the properties of an event with ahydrogen-deficient envelope and a mass M ej . M ⊙ , asis expected for the progenitors of Type Ib/c supernovaethat have lost a large fraction of their envelope (includ-ing all of their hydrogen) to stellar winds, binary masstransfer, and/or outbursts (Smith et al. 2011).The collision of the propeller material with the super-nova ejecta shock heats and accelerates the ejecta. For atotal energy E tot = E sn + E prop , the final velocity, withwhich it coasts for the remainder of the expansion, is v f ≈ (2 E tot /M ej ) / = 2500 E / . M − / km s − , (23)where E . = E tot / × ergs and M = M ej / M ⊙ .The diffusion timescale of photons from this hot, expand-ing material is given by t d = (cid:18) M ej κ . v f c (cid:19) / = 11 κ / . M / E − / . days , (24)where κ is the opacity, which we scale to κ . = κ/ . g − (the typical opacity used for a gray calcu-lation, Pinto & Eastman 2000), and the factor of 13.78comes from detailed analytic studies of Type I super-novae (Arnett 1982; Pinto & Eastman 2000). The shellbecomes optically thin on a timescale t τ ≈ π M ej κv f ! / = 226 κ / . M E − / . days . (25)The diffusion approximation we will use is not applica-ble after this time. The increased velocity creates a fasterand more luminous supernova, but this higher luminosityis not directly from energy input from the propeller mech-anism. Instead, since the explosion velocity is higher, thediffusion time (eq. [24]) is shorter, and the Ni decay isbeing probed at earlier times.To understand the corresponding lightcurve createdby propeller energy being injected into the explosion,we construct a simple, one-zone model of the expan-sion, cooling and emission, following the mathematicalframework of Li & Paczy´nski (1998; also see Kulkarni2005, Kasen & Bildsten 2010). For an expanding shell,the internal energy E int satisfies the differential equation(rewritten from eq. [9] of Li & Paczy´nski 1998)1 t ddt [ E int ( t ) t ] = L prop ( t ) + L nuc e − t/t τ − L ( t ) , (26)where L ( t ) = E int ( t ) t/t d (27)is the emitted luminosity. The nuclear luminosityincludes contributions from Ni decay and subse-quent Co decay. We use the analytic expression(Pinto & Eastman 2000; Bersten et al. 2011) L nuc ( t ) = ǫ Ni M Ni e − t/t Ni + ǫ co M Ni h e − t/t Co − e − t/t Ni i , (28) Fig. 9.—
Bolometric luminosities and effective temperatures cal-culated using the one-zone model described by eq. (26). Eachcurve is labeled with a M ej , M Ni , and the total energy input. Seetext for further details. The general trend is that the energy injec-tion of the propeller mechanism creates a brighter, more quicklyevolving supernova. where M Ni is the mass of Ni synthesized, ǫ Ni = 3 . × ergs g − s − , t Ni = 7 . × s, ǫ Co = 6 . × ergs g − s − , and t Co = 9 . × s. The factorof e − t/t τ in equation (26) takes into account that thematerial eventually becomes optically thin to gamma-rays (although this factor only leads to small changesat late times). We have tested this simplified modelagainst a wide range of nuclear-powered explosion calcu-lations (Ensman & Woosley 1988; Iwamoto et al. 1998;Darbha et al. 2010), and found qualitatively good fits tothe timescales and magnitudes of the peak luminosity.For Figure 9, we integrate equation (26) numerically,setting E int = E sn = 10 ergs at t = 0. Each curveis labeled by different values for M ej , M Ni , and the to-tal energy input (initial supernova energy plus the pro-peller mechanism). The initial radius is 5 R ⊙ as ap-propriate for a compact Wolf-Rayet progenitor. We set T eff = ( L/ πr σ ) / , where r = v f t and σ is the Stefan-Boltzmann constant. This temperature is only accurateup to a time t ≈ t τ . The first two models ( solid and long-dashed lines ) explore the effect of a high input energy.The curve labeled with M ej = 10 M ⊙ and M Ni = 0 . M ⊙ ( dotted line ) is representative of a “hypernova” model(Nomoto et al. 2001). For this model we use an ini-tial radius of 10 R ⊙ , consistent with massive helium stars(Woosley et al. 1995). The model with merely 10 ergs( short-dashed line ) is meant to be representative of a nor-mal Type Ib/c supernova.From these calculations, we find the general trend thatadditional energy injection from the propeller mecha-nism creates a brighter, more quickly evolving supernova(it will also cool faster and show optically thin featuressooner). Within the framework we have described, it isnot necessary that these events produce more Ni thanUPERNOVA FALLBACK ONTO MAGNETARS 9average. We therefore expect propeller-powered TypeIb/c supernovae to be associated with a wide range ofpeak luminosities, but to generically exhibit high veloci-ties of ∼ (1 − × km s − . High-Mass Envelopes
If the envelope is more massive and has a hydrogen-rich composition, the lightcurve evolution can be sig-nificantly different, as is seen for Type IIP super-novae. The analytic features of these lightcurveswere well summarized by Popov (1993), whosework was confirmed and expanded upon by thenumerical simulations of Eastman et al. (1994) andKasen & Woosley (2009), and also studied with the firstnon-LTE time-dependent radiative-transfer simulationsby Dessart & Hillier (2011). The general picture is thatthe backward progression of a hydrogen recombinationwave through the expanding ejecta causes the supernovato radiate at a fixed effective temperature set by the ion-ization temperature T eff = 2 / T ion . This continues untilthe entire envelope has become neutral, which truncatesthe luminosity, revealing Co decay if it is sufficientlyavailable. Popov (1993) demonstrated that a hydrogen-rich envelope will exhibit a plateau phase when a cer-tain dimensionless parameter is greater than unity. Werewrite this condition in terms of a critical mass, finding M ej & E / . R / κ − / . T − / M ⊙ , (29)where κ . = κ/ .
34 cm g − is the electron-scatteringopacity for a solar composition, R is the initial stellarradius with R = R / R ⊙ , and T = T ion / L plat = 2 . × M − / E / . R / κ / . T / ergs s − , (30)and a plateau timescale t plat = 56 M / E − / . R / κ / . T − / days , (31)where we note that Kasen & Woosley (2009) find aslightly stronger scaling of t plat ∝ E − / in their nu-merical results. The large energy input would also resultin higher velocities of v f ∼ km s − (scaling eq. [23]to a mass of ∼ − M ⊙ ), which although not as highas in broad-lined supernovae, would be anomalously highfor a Type IIP supernova. The high luminosities we findare similar to what is seen for many Type IIn super-novae (see Fig. 3 of Smith et al. 2008), but our eventswould not have nebular features from the interaction withwinds and thus would appear distinct from Type IIn su-pernovae.To better demonstrate the impact of this energy injec-tion on the plateau phase, we plot example lightcurvesusing the analytic results of Popov (1993) in Figure 10.Beyond the plateau stage, the lightcurve may reveal apower-law decline from Co decay, which we include fora range of Ni masses, using equation (28). We do notplot the effective temperature since it is nearly constantat ∼ Fig. 10.—
Bolometric luminosity calculated using the analyticmodel of Popov (1993). The thick lines are models with an energyinjection of 3 × ergs, and the thin lines use 10 ergs (repre-sentative of a normal Type IIP supernova). The additional energyinjection results in a larger luminosity, but shorter timescale, con-sistent with eqs. (30) and (31). Beyond the plateau stage, thelightcurve is dominated by Co decay, which we plot for a rangeof Ni masses using eq. (28) as indicated ( dotted lines ). CONCLUSIONS AND DISCUSSION
We presented a study of the effect of supernova fallbackaccretion onto newly born magnetars. The combinationof spin, magnetic field, and fallback rate was used to cal-culate the time evolution of the magnetar spin, estimatehow much material was accreted, and determine whetherthe magnetar can expel enough material via the propellermechanism to prevent collapse to a black hole. Strongmagnetic fields and short spin periods are generally moreadvantageous for hindering black hole formation (but asFigures 3 and 4 show, there are subtle changes to thispicture depending on details of the time-dependent ac-cretion rate). Even in cases that avoid becoming blackholes, ∼ . M ⊙ or more of supernova fallback materialis accreted, so we expect magnetars formed in collaps-ing massive stars to be more massive than the canonical ∼ . M ⊙ neutron star mass. As discussed in §
1, thereare at least three observed cases in our Galaxy of mag-netars associated with ∼ − M ⊙ progenitors. Thepropeller mechanism suggests a natural connection forwhy neutron stars associated with massive progenitorsshould have magnetar-strength fields.Quickly spinning magnetars have been discussed aspromising candidate systems for gravitational wave pro-duction via the time-changing quadrupole moment cre-ated by dynamical or secular instabilities (see Corsi &M´esz´aros 2009, Ott 2009, and references therein). Weconclude that there are two main cases that may lead tothe emission of gravitational waves when fallback accre-tion is important. In the first case, if the propeller mech-anism is active (typically B & × G), the magnetarmust begin with a sufficiently short spin period by the0 PIRO & OTTprocess of cooling and contraction, as is found for somemodels explored by Ott et al. (2006, in particular, seetheir summary of β -values in Table 4). It will then emitgravitational waves until it is spun down by accretion ona timescale of ∼ −
100 s. Since low accretion rateswould extend the timescale for gravitational wave emis-sion, and these correspond to more energetic explosions,our model predicts that gravitational waves (if present)are most likely in the most energetic events that do notcollapse to black holes. We note that in such cases agamma-ray burst may be created directly by the magne-tar, as explored by Metzger et al. (2010, and referencestherein). In the second case, when accretion occurs di-rectly onto the magnetar surface, the magnetar is spunup sufficiently to emit gravitational waves. But as wediscussed at the end of §
5, this only proceeds until themagnetar collapses to a black hole after . −
200 s(see eq. [18]). The formation of a black hole and its sub-sequent accretion may then power a gamma-ray burst,again predicting a possible correlation between gravi-tational wave emission and a powerful electromagneticevent. Note however that in this case the gravitationalwaves would precede any sort of launching of a relativisticjet (in contrast to the model of Piro & Pfahl 2007, whichpredicts gravitational waves coincident with the promptgamma-ray emission, although both processes can occurin the same event).When a magnetar is in the propeller regime, the ex-pelled material collides with supernova ejecta, shock-heating it and energizing the supernova. Maeda et al.(2007) proposed that some ultraluminous supernovaemay be explained by dipole emission from a rapidly spin-ning magnetar, which was worked out in detail by Kasen& Bildsten (2010) and Woosley (2010). We emphasizethat our model is very different from theirs. In their casethe magnetar directly powers the observed supernova lu- minosity. In our case the spin energy is injected earlier,creating a faster evolving supernova. We explored tworegimes where this energy input may have a direct ob-servational consequence: (1) in the case of a low-mass( . M ⊙ ), hydrogen-deficient envelope, the additionalenergy gives rise to a broadlined Type Ib/c supernova,or potentially, a hypernova, depending on the amountof Ni synthesized, and (2) in the case of a massive( & M ⊙ ), hydrogen-rich envelope, we predict an eventsimilar to a Type IIP supernova, although brighter andwith higher velocities.Our predictions for the lightcurves of energetic super-novae are independent of the actual mechanism for inject-ing the energy, requiring only deposition at early times( ≪ t d ). Therefore, independent of our specific model forhow the energy is produced, Type IIP supernovae thathave similar lightcurves as we demonstrate in § ≈ × ergs(Botticella et al. 2010). The databases of high-cadencetransient surveys, such as the Palomar Transient Factory(Law et al. 2009) or Pan-STARRS (Kaiser et al. 2002),may reveal a larger population of Type IIP supernovaethat, although not as extreme as SN 2009kj, may stillrequire energy input beyond what is typically availablefor a supernova.We thank Lars Bildsten, Luc Dessart, Brian Metzger,Robert Quimby, Uli Sperhake, and Todd Thompson fortheir helpful suggestions. We also thank Evan O’Connorfor providing core collapse models for initial fallback cal-culations and estimates of the importance of an accretiondisk. This work was supported through NSF grants AST-0855535, PHY-0960291, and OCI-0905046, and by theSherman Fairchild Foundation. A.L.P. was supported inpart by NASA ATP grant NNX07AH06G. APPENDIX
NEUTRINO-COOLED ACCRETION COLUMNS
In cases where r c > r m > R , material is magnetically channeled before reaching the magnetar’s surface. Thisis traditionally called an accretion column in the study of accreting, magnetized white dwarfs and neutron stars(Frank et al. 1992). For a dipole field, sin θ/r is constant, so that the path of the flow is described by the equation1 /r m = sin θ/r (for simplicity we assume an aligned rotator). At a radius r from the magnetar, the material issqueezed into an area A ( r ) ≈ πr sin θ ≈ πr ( r/r m ) . (A1)Assuming that the flow comes in at approximately free-fall, the velocity and density are v in = (cid:18) GMr (cid:19) / , ρ = ˙ M Av in . (A2)where the factor of two is because there are two poles. From this we estimate v in = 1 . × M / . r − / cm s − , (A3)where r = r/
12 km. Combining equations (6), (A1), and (A2), the density is ρ = 1 . × µ / M / . r − / ˙ M / − g cm − . (A4)The flow will go through a shock before reaching the stellar surface. This is checked by estimating the Mach number M of the flow, M = v c s = 2 · / γ (cid:18) p B p (cid:19) / (cid:18) rr m (cid:19) / , (A5)UPERNOVA FALLBACK ONTO MAGNETARS 11where c s is the speed of sound, γ is the adiabatic coefficient, and p B /p is the ratio of the magnetic pressure to thepressure of the gas (including ideal gas, radiation, and degeneracy contributions). For typical parameters, M = 1 . µ / M / . ˙ M / − r / (cid:18) p B p (cid:19) / , (A6)where we set γ = 4 / M ≪ ∼ g cm − , but from continuity this implies an infall velocity much less thanfreefall ( v in ∼ cm s − ) contrary to our expectations. In addition, we estimate the mean free path for proton-protoncollisions to be ∼ − cm (using eq. [3.20] from Frank et al. 1992), much less than the width of the accretion column,so we expect the shock to be collisional.The flow can be broken into two main regions. The first is a supersonic flow starting from the edge of the magneto-sphere and then moving toward the magnetar pole. Then there is a shock, below which the subsonic flow settles ontothe star. The jump conditions at the shock interface are ρ sh = 7 ρ , p sh = 67 ρ sh v , v sh = 17 v in , (A7)for a strong shock with γ = 4 /
3. Therefore the post-shock density is ρ sh = 1 . × µ / M / . r − / ˙ M / − g cm − , (A8)and T sh = (cid:18) p sh a (cid:19) / = 1 . × µ / M / . r − / ˙ M / − K , (A9)is the post-shock temperature.The radiative diffusion timescale is approximately t ∼ κρr/c ∼ s, so the flow cannot cool via photons. Insteadwe consider neutrino cooling via electron-positron pair annihilation, which is given by (Popham et al. 1999)˙ q pairs = 5 × T ergs cm − s − , (A10)where T = T / K. The timescale for this cooling is t pairs = aT ˙ q pairs = 5 . × − µ − / M − / . r / ˙ M − / − s . (A11)It is also possible that Urca cooling is important, given by˙ q Urca = 9 × ρ T ergs cm − s − , (A12)for a composition of protons and neutrons (at these high temperature helium is photodisintegrated). The timescalefor this cooling is t Urca = aT ˙ q Urca = 6 . × − µ − / M − / . r / ˙ M − / − s . (A13)Urca cooling dominates when t Urca < t pair , implying an accretion rate˙
M > . × − µ − / M / . r / M ⊙ s − . (A14)This is a rather high accretion rate in comparison to what we consider, so it is sufficient to focus on pair cooling. Theheight of the shock above the magnetar surface is H sh ≈ v sh t pair = 1 . µ − / M − / . r / ˙ M − / − km . (A15)The shock therefore occurs at a radius of r sh = R + H sh . Note that the r on the righthand side of equation (A15)corresponds to r sh , so this equation is only accurate as long as H sh . r sh ≈ R .When H sh & R , we need to take into account adiabatic compression of material as it moves toward the magnetarpole. This gives a higher temperature at the magnetar surface in comparison to the shock radius, by an amount2 PIRO & OTT Fig. 11.—
Critical radii as a function of accretion rate for the problem of a neutrino-cooled accretion column. A given fluid elementmoves inward in radius (from top to bottom on the plot) at fixed accretion rate. In this way, one can read off what processes the fluidelement experiences. When it reaches r m ( dashed line ), its motion is determined by the magnetic field. If r m > r c , then it will be expelled(we plot r c for 1 and 10 ms spin periods as examples; dotted lines ). If r m < r c , then the flow will be channeled toward the magnetar poleand undergo a shock at r sh ( solid line ). It finally reaches the magnetar surface at R ( dot-dashed line ). As a comparison we also plot theapproximation give by eq. (A15) as the line labeled R + H sh . T = T sh ( R/r sh ) − , which revises the pair cooling timescale by a factor of ( R/r sh ) . We again write an equation forthe shock height, H sh = r sh − R ≈ v sh t pair ( r sh ) . (A16)This expression can be solved numerically for r sh , which we plot in Figure 11 in comparison to other critical radii. Thisshows that for all accretion rates of interest, r sh < r m , so the location of the shock is always within where funneledflow begins, as needed for consistency. REFERENCESAkiyama, S., Wheeler, J. C., Meier, D. L., & Lichtenstadt, I.2003, ApJ, 584, 954Aly, J. J. 1980, A&A, 86, 192Arons, J. 1986, in Plasma Penetration into Magnetospheres, ed.N. Kylafis, J. Papamastorakis, & J. Ventura (Iraklion: CreteUniv. Press), 115Arons, J. 1993, ApJ, 408, 160Arons, J. 2003, ApJ, 589, 871Arnett, W. D. 1982, ApJ, 253, 785Belczynski, K., & Taam, R. E. 2008, ApJ, 685, 400Bersten, M. C., & Hamuy, M. 2009, ApJ, 701, 200Bersten, M. C., Benvenuto, O., & Hamuy, M. 2011, ApJ, 729, 61Bibby, J. L., Crowther, P. A., Furness, J. P., & Clark, J. S. 2008,MNRAS, 386, L23Bodenheimer, P., & Ostriker, J. P. 1974, ApJ, 191, 465Botticella, M. T., et al. 2010, ApJ, 717, L52Burrows, A., Dessart, L., Livne, E., Ott, C. D., & Murphy, J.2007, ApJ, 664, 416Chandrasekhar, S. 1969, The Silliman Foundation Lectures, NewHaven: Yale University Press, 1969,Corsi, A., & M´esz´aros, P. 2009, ApJ, 702, 1171D’Angelo, C. R., & Spruit, H. C. 2010, MNRAS, 406, 1208Darbha, S., Metzger, B. D., Quataert, E., Kasen, D., Nugent, P.,& Thomas, R. 2010, MNRAS, 409, 846Dessart, L., Burrows, A., Livne, E., & Ott, C. D. 2008, ApJ, 673,L43Dessart, L., & Hillier, D. J. 2011, MNRAS, 410, 1739Dessart, L., Livne, E., & Waldman, R. 2010, MNRAS, 408, 827 Duncan, R. C., & Thompson, C. 1992, ApJ, 392, L9Eastman, R. G., Woosley, S. E., Weaver, T. A., & Pinto, P. A.1994, ApJ, 430, 300Ek¸si, K. Y., Hernquist, L., & Narayan, R. 2005, ApJ, 623, L41Ensman, L. M., & Woosley, S. E. 1988, ApJ, 333, 754Fendt, C., & Elstner, D. 2000, A&A, 363, 208Figer, D. F., Najarro, F., Geballe, T. R., Blum, R. D., &Kudritzki, R. P. 2005, ApJ, 622, L49Frank, J., King, A., & Raine, D. 1992, Accretion Power inAstrophysics (Cambridge: Cambridge Univ. Press)Fryer, C. L. 1999, ApJ, 522, 413Fryer, C. L., & Heger, A. 2000, ApJ, 541, 1033Fryer, C. L., & Warren, M. S. 2004, ApJ, 601, 391Gaensler, B. M., McClure-Griffiths, N. M., Oey, M. S., Haverkorn,M., Dickey, J. M., & Green, A. J. 2005, ApJ, 620, L95Ghosh, P., & Lamb, F. K. 1979, ApJ, 234, 296Goodson, A. P., Winglee, R. M., & B¨ohm, K.-H. 1997, ApJ, 489,199Hayashi, M. R., Shibata, K., & Matsumoto, R. 1996, ApJ, 468,L37Heger, A., Langer, N., & Woosley, S. E. 2000, ApJ, 528, 368Heger, A., Fryer, C. L., Woosley, S. E., Langer, N., & Hartmann,D. H. 2003, ApJ, 591, 288Ikhsanov, N. R. 2002, A&A, 381, L61Illarionov, A. F., & Sunyaev, R. A. 1975, A&A, 39, 185Iwamoto, K., et al. 1998, Nature, 395, 672Kaiser, N., et al. 2002, Proc. SPIE, 4836, 154Kasen, D., & Bildsten, L. 2010, ApJ, 717, 245