Spin-down of the magnetar in RCW 103

# Ejector and propeller spin-down: How might a superluminous supernova millisecond magnetar become the 6.67 hr pulsar in RCW 103

Wynn C. G. Ho, Nils Andersson
Mathematical Sciences and STAG Research Centre, University of Southampton, Southampton, SO17 1BJ, UK
Physics and Astronomy and STAG Research Centre, University of Southampton, Southampton, SO17 1BJ, UK
Email: wynnho@slac.stanford.edu
Accepted 2016 September 9. Received 2016 September 9; in original form 2016 August 10
###### Abstract

The X-ray source 1E 1613485055 in the supernova remnant RCW 103 recently exhibited X-ray activity typical of magnetars, i.e. neutron stars with magnetic fields . However, 1E 1613485055 has an observed period of 6.67 hr, in contrast to magnetars which have a spin period of seconds. Here we describe a simple model which can explain the spin evolution of 1E 1613485055, as well as other magnetars, from an initial period of milliseconds that would be required for dynamo generation of magnetar-strength magnetic fields. We propose that the key difference between 1E 1613485055 and other magnetars is the persistence of a remnant disk of small total mass. This disk caused 1E 1613485055 to undergo ejector and propeller phases in its life, during which strong torques caused a rapid increase of its spin period. By matching its observed spin period and  kyr age, we find that 1E 1613485055 has the (slightly) highest magnetic field of all known magnetars, with , and that its disk had a mass of , comparable to that of the asteroid Ceres.

###### keywords:
accretion, accretion discs – stars: magnetars – stars: magnetic field – stars: neutron – stars: individual (1E 1613485055; RCW 103) – supernovae: general.
pubyear: 2016pagerange: Ejector and propeller spin-down: How might a superluminous supernova millisecond magnetar become the 6.67 hr pulsar in RCW 103Ejector and propeller spin-down: How might a superluminous supernova millisecond magnetar become the 6.67 hr pulsar in RCW 103

## 1 Introduction

The recent detection by D’Aì et al. (2016); Rea et al. (2016) of high energy activity from the neutron star (NS) 1E 1613485055 in supernova remnant RCW 103 (also known as SNR G332.40.4) finally provides insights into its perplexing nature. Tuohy & Garmire (1980) discovered the X-ray point source 1E 1613485055 using Einstein, but no corresponding optical/IR or radio counterpart has been found to date (Tuohy et al., 1983; De Luca et al., 2008), which argues in part against the source being in a binary system. The age of RCW 103 is (Clark & Caswell, 1976) or within the range (Nugent et al., 1984; Carter et al., 1997). Garmire et al. (2000) find that 1E 1613485055 pulsates with a period of , with a more definitive and refined detection of 6.67 hr determined by De Luca et al. (2006). Continued monitoring of 1E 1613485055 yields a constraint on the time derivative of this period of (Esposito et al., 2011), which is higher than the of all known isolated pulsars. The 6.67 hr period makes 1E 1613485055 a particularly interesting object.

Some characteristics of 1E 1613485055 match those of the central compact object (CCO) class of NSs, which are found near the center of SNR and are only seen in X-rays, and thus 1E 1613485055 had been associated with this class (see De Luca 2008; Halpern & Gotthelf 2010; Gotthelf et al. 2013, for review; see also Ho 2013). CCOs have an inferred magnetic field , and three CCOs have a measured spin period : two have and one has . If the 6.67 hr period of 1E 1613485055 is ascribed to its spin, then this value is in sharp contrast to the spin period of CCOs.

However the recent high energy activity is very similar to activity seen in another class of NSs, that of the magnetars. Magnetars traditionally include two types of NSs observed at high energies, anomalous X-ray pulsars (AXPs) and soft gamma-ray repeaters (SGRs), almost all of which have an inferred magnetic field (see Mereghetti et al. 2015; Turolla et al. 2015, for review). Thus it is likely that 1E 1613485055 also possesses a magnetic field in this range, and hereafter we will assume this is the case. While may be similar to other magnetars, the 6.67 hr period of 1E 1613485055 (which we will assume is its spin period; see D’Aì et al. 2016; Rea et al. 2016) is drastically longer than the spin period of other magnetars, which are all in the range . Furthermore, this long period of 1E 1613485055 could be used to argue against the generation of magnetar-strength magnetic fields via a dynamo mechanism, since this mechanism requires an initial rapid (possibly millisecond) spin period (Thompson & Duncan, 1993; Bonanno et al., 2005; Spruit, 2009; Ferrario et al., 2015).

In Section 2, we present calculations of a simple scenario that can describe the spin evolution of the magnetar 1E 1613485055, starting from its birth with a spin period of a millisecond to its current 6.67 hr, and how 1E 1613485055 is different from other known magnetars. The scenario is as follows: A supernova gives birth to a rapidly rotating NS. The rapid spin rate allows us to retain the dynamo mechanism as a viable means to explain the magnetic fields of 1E 1613485055 and other magnetars (Thompson & Duncan, 1993; Bonanno et al., 2005; Spruit, 2009). We also note that our theoretically conceived millisecond magnetar connects observed magnetars (that are old) to those thought to power superluminous supernovae (see, e.g. Chatzopoulos et al. 2013; Inserra et al. 2013; Nicholl et al. 2014). Next, after an initial epoch during which the immediate environs surrounding the newborn NS settles into a relatively homogeneous low density plasma and the magnetic field organizes itself into an ordered dipole field, we begin at time with a millisecond magnetar (with and ). This millisecond magnetar evolves as a standard pulsar, i.e. it loses rotational energy and slows down as a result of the NS emitting electromagnetic dipole radiation. For most known magnetars, this spin-down continues for thousands of years until their present age and produces NSs that have a spin period of a few seconds (see after equation 3), just as observed. In contrast, we propose that for 1E 1613485055, there remained some material that was not ejected by the supernova (we estimate a total mass of about that of the asteroid Ceres; see Sect. 3), and it forms a remnant disk around the NS (see, e.g. Michel 1988; Lin et al. 1991; Perna et al. 2014). The rapid rotation of the NS causes it to be in an ejector state/phase and prevents the remnant disk from interacting with the NS (Illarionov & Sunyaev, 1975). The duration of the ejector phase can be hundreds to thousands of years (see equation 7), and all the while the NS emits dipole radiation and its spin period increases. Eventually its rotation becomes slow enough for disk material to couple to the NS magnetosphere, and the NS transitions to a propeller state/phase. In this state, matter is expelled by the (still) rapidly rotating NS, and the resulting spin-down torque on the NS is much stronger than that due to dipole radiation. The NS spin period increases at an exponential rate (see equation 12) for a short time (see equation 11), before reaching spin equilibrium, when torques on the NS balance. The result is a slowly spinning, strongly magnetized NS, like 1E 1613485055.

Here we briefly mention previous works which sought to explain the 6.67 hr period of 1E 1613485055 as its spin period. De Luca et al. (2006) (see also Esposito et al. 2011) ignore the ejector phase and begin their calculation of propeller phase spin-down at , finding that 1E 1613485055 has and a remnant disk mass of . Li (2007) describe an ejector and propeller evolution scenario and perform Monte Carlo simulations to obtain the magnetar spin period distribution. Pizzolato et al. (2008) consider the torque exerted by a binary companion star and find that 1E 1613485055 has and is in spin equilibrium. Ikhsanov et al. (2013) consider 1E 1613485055 to have and is accreting from a magnetized remnant disk. We also note the earlier studies of AXPs and SGRs as normal magnetic field () NSs that are accreting in the propeller phase, but near spin equilibrium, from a fossil disk (with constant mass; Alpar 2001; or with decreasing mass; Chatterjee et al. 2000; Ertan et al. 2009). While in the final stages of preparing our work, we became aware of the work of Tong et al. (2016), who consider a similar scenario as described here but obtain a much larger disk mass of (see Sec. 3).

## 2 Spin period evolution in ejector and propeller phases

The scenario for the evolution of the 1E 1613485055 spin period described in Section 1 requires a model for ejector and propeller phases (defined below). At early times in the ejector phase, a NS spins down in a similar fashion to an isolated pulsar, i.e. the NS emits electromagnetic dipole radiation and loses rotational energy. This energy loss produces a torque on the NS

 Nem = −2μ2Ω3sin2θ3c3=−B2R6Ω3sin2θ6c3=−βIΩ3 (1) = −1.5×1045 erg B215(P/1 ms)−3,

where () is spin frequency, is the angle between stellar rotation and magnetic axes, , , and we assume the magnetic dipole moment and an orthogonal rotator, i.e. . We take NS mass, radius, and moment of inertia to be , , and , respectively. For simplicity we use the traditional vacuum dipole formula of Pacini (1968); Gunn & Ostriker (1969). Corrections due to a plasma-filled magnetosphere and in the -dependence only introduce changes of order unity (see, e.g. Spitkovsky 2006; Contopoulos et al. 2014). Torque on the star is defined by , and, without additional sources of torque on the pulsar, the resulting evolution equation for spin frequency is , with solution

 Ω=Ω0(1+2βΩ20t)−1/2=Ω0(1+t/tem)−1/2for t0

where () is initial spin frequency and spin-down occurs on the timescale

 tem=1/2βΩ20=2.0×103 s B−215(P0/1 ms)2. (3)

From eq. (2) we see that, in isolation, 1E 1613485055 would spin down to , which coincides with the spin period range of other observed magnetars (Mereghetti et al., 2015; Turolla et al., 2015) but is much shorter than its current spin period of . This demonstrates that all magnetars except 1E 1613485055 could simply have spun down to their current spin period via the torque due to electromagnetic dipole radiation (equation 1). For 1E 1613485055, dipole radiation torque is too weak, and a stronger, additional or alternative torque, such as that due to mass accretion, is required to increase its spin period by its current age of a few thousand years.

Therefore let us suppose that when 1E 1613485055 was first born, it was surrounded by a disk of material from, e.g. supernova ejecta that did not escape the system (Chevalier, 1989). This material cannot interact with the pulsar as long as the pulsar light cylinder, defined by radius

 rlc=c/Ω=47.7 km (P/1 ms), (4)

is smaller than the magnetosphere, whose radial extent is approximately

 rm=ξrA=ξ(μ48GM˙M2)1/7=7.3×105 km ξB4/715˙M−2/7−12, (5)

where (see, e.g. Ghosh & Lamb 1979; Wang 1996), the Alfvén radius is derived from balancing ram pressure of the accreting material with pressure of the pulsar magnetic field (Lamb et al., 1973; Lipunov, 1992), is mass accretion rate, and . Thus this ejector phase takes place when . The transition between ejector and propeller phases occurs at spin period

 Pej=2πΩej=2πrmc=15 s ξB4/715˙M−2/7−12, (6)

and the duration of the ejector phase can be estimated from eqs. (2) and (6) and is

 tej=tem[(Ω0rmc)2−1]≈r2m2βc2=1.5×104 yr ξ2B−6/715˙M−4/7−12. (7)

Once , the propeller phase begins, and the total torque on the star is approximately

 N=Nacc+Nprop≡˙Mr2mΩK(rm)−˙Mr2mΩ=Nacc(1−^ωs), (8)

where is accretion (spin-up) torque and is propeller (spin-down) torque (see Ho et al. 2014, for derivation; see also, e.g. Alpar 2001; Esposito et al. 2011; Piro & Ott 2011; alternative prescriptions for total torque can be found in, e.g. Menou et al. 1999; Ertan et al. 2009; Parfrey et al. 2016). The Kepler orbital frequency at the magnetosphere radius has the corresponding period

 PK(rm)=2πΩK(rm)=(4π2r3mGM)1/2=9.0×103 s ξ3/2B6/715˙M−3/7−12. (9)

The fastness parameter [; Elsner & Lamb 1977] determines whether centrifugal force due to stellar rotation ejects matter and spins down the star (propeller phase with ) or matter accretes and spins up the star (accretor phase with ) (see, e.g. Wang 1995). The transition between these two phases () is where the total torque is approximately zero and the NS is in spin equilibrium (Davidson & Ostriker, 1973; Alpar et al., 1982) and occurs at spin period .

The evolution equation for spin frequency is obtained by equating eq. (8) to stellar torque , so that (see also Alpar 2001)

 dΩdt=−˙Mr2mI[Ω−ΩK(rm)]=−Ωtprop+ΩK(rm)tprop, (10)

where

 tprop≡I˙Mr2m=96 yr ξ−2B−8/715˙M−3/7−12. (11)

We can obtain a simple solution of the evolution equation by assuming and are constant (more sophisticated models with can be found in, e.g. Chatterjee et al. 2000; Ertan et al. 2009; Tong et al. 2016). Then the spin frequency as a function of time during the propeller phase, which starts from the end of the ejector phase at time with spin frequency , is

 Ω=[Ωej−ΩK(rm)]e−(t−tej)/tprop+ΩK(rm)for t>tej. (12)

Equations (2) and (12) thus describe the complete evolution of NS spin frequency (or spin period) through the ejector and propeller phases, respectively. Figures 1 and 2 plot this evolution, assuming , an initial spin period , and different combinations of magnetic field and average accretion rate . We note that, as long as , the evolution of spin period is unchanged for any , except at very early times. During the early evolution (at , depending on and ; see equation 7), the NS is in the ejector phase, and (see equation 2). At time when , the NS magnetosphere can interact with the remnant disk, and the NS enters the propeller phase. The spin period increases rapidly in this phase (; see equation 12) for a time (see equation 11). Finally, when approaches (see equation 9), propeller and accretion torques balance, so that the total torque on the star is zero and is approximately constant, and the NS is in spin equilibrium. Figure 1 shows that, for a given accretion rate, more strongly magnetized NSs reach longer periods, while Fig. 2 shows that, for a given magnetic field, lower accretion rates produce longer spin period NSs (see also equation 9).

Since we know the spin period and approximate age of 1E 1613485055, only particular combinations of magnetic field and average mass accretion rate will satisfy eq. (12), i.e. for

 lnΩejΩ−ΩK=∣∣age−tej∣∣tprop, (13)

where we take , the left-hand side must equal the right-hand side and spin frequency and age are set by and age = 1200–3200 yr, respectively. The values of and which satisfy the above are indicated by the shaded region in Fig. 3, along with the magnetic field of several magnetars, inferred from their and (values taken from the ATNF Pulsar Catalogue; Manchester et al. 2005; see also McGill Online Magnetar Catalog222http://www.physics.mcgill.ca/ pulsar/magnetar/main.html; Olausen & Kaspi 2014), the highest of which is for SGR 180620. If 1E 1613485055 has a slightly higher field of than SGR 180620 and is old, then it only requires an average accretion rate of to spin it down to a period of 6.67 hr (see also Fig. 2). If the accretion rate is much lower or higher (at this ), then 1E 1613485055 would be in the ejector phase or in spin equilibrium, respectively, with a spin period much shorter than 6.67 hr in both cases.

## 3 Discussion

Recent observations by D’Aì et al. (2016); Rea et al. (2016) of the X-ray source 1E 1613485055 in SNR RCW 103 strongly suggest it is a magnetar (NS with ) with an extremely long spin period , in contrast to all other known magnetars which have 2–12 s spin periods. The long spin period of 1E 1613485055 might argue against dynamo generation of magnetic fields because of a requirement for fast (millisecond) initial spin periods, and there is insufficient time for 1E 1613485055 to lose enough rotational energy via conventional electromagnetic dipole radiation.

Here we demonstrate, using a simple model with simple assumptions, that the spin period of 1E 1613485055 can increase from milliseconds to 6.67 hr over its 1.2–3.2 kyr lifetime by evolving through ejector and propeller phases while undergoing accretion from a disk. Our calculations show that a young NS, such as 1E 1613485055, can spend quite a long time in the ejector phase, and thus this phase should not be neglected. The requisite disk might have remained bound to the NS during its formation in a (superluminous) supernova and may have significant impact even when it has very low total mass, which we estimate to be . This disk may still be present or has been completely accreted/dissipated. In the case of the former, since 1E 1613485055 has evolved to be near spin equilibrium (when the total torque on the star is approximately zero), the long-term is very low and could satisfy the observed constraint of obtained by Esposito et al. (2011). 1E 1613485055 may on occasion accrete more or less material, which could explain the variability seen in X-rays (Gotthelf et al., 1999; De Luca et al., 2006). In fact, our derived corresponds to a luminosity , which is on the order of that observed (De Luca et al., 2006).

If the disk is no longer present (with the observed X-ray variability due to typical magnetar variability), then the dipole radiation torque yields , well below the observed constraint. We also note that, while the exact nature of the mechanism that causes radio emission is uncertain, it is thought that there exists a “death line” which demarcates when observable radio emission ceases (Ruderman & Sutherland, 1975; Bhattacharya et al., 1992). This death line is shown by dotted lines in Figs 1 and 2. It is clear that, while the NS is in the ejector phase, its spin period is below the death line, and as such, it could emit as a radio pulsar. However after transition to propeller phase, the spin period rapidly increases above the death line. Thus once the accretion disk material is exhausted and the propeller phase ceases, the NS will not emit as a radio pulsar.

Finally, our results suggest a possible unified formation scenario for various classes of observed NSs. This scenario is schematically described in Table 1 and depends on total accreted mass and time spent accreting following a chaotic and turbulent supernova (a scenario that includes more classes but is a function of accretion rate from a fallback disk is proposed in Alpar 2001). For short duration accretion (hours to possibly days) of a large amount of mass (), accreting matter can build up on the NS surface so fast that the magnetic field is buried temporarily. Once accretion slows or stops, the magnetic field re-emerges on a timescale of , depending on burial depth, and this increasing surface field could explain properties of CCOs (Ho, 2011, 2013). For small to no accretion, we transition from CCO formation to pulsars that have possible signatures of magnetic field growth, e.g. their braking index, to the majority of isolated radio pulsars (Pons et al., 2012; Ho, 2015). In the case of accretion for long durations, the magnetic field will not be buried if the total mass is small. For large total mass (e.g. , like that of the disk seen around magnetar 4U 0142+61; Wang et al. 2006), NSs with could end up with in . However, Perna et al. (2014) show that it is extremely difficult to retain such amounts for long durations during a supernova. For 1E 1613485055, only a small amount of mass () needs to be retained following its supernova. Thus 1E 1613485055 is possibly a very special system, and the interaction of its magnetic field with this small mass over a thousand years is what leads to its long spin period of 6.67 hr.

## Acknowledgements

The authors thank the anonymous referee for helpful comments. WCGH and NA acknowledge support from the Science and Technology Facilities Council (STFC) in the United Kingdom.

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