THE OUTBURST DECAY OF THE LOW MAGNETIC FIELD MAGNETAR SGR 0418+5729

The X-Ray Telescope (XRT; Burrows et al. 2005) on board Swift uses a front-illuminated CCD detector sensitive to photons between 0.2 and 10 keV. Two main readout modes are available: photon counting (PC) and windowed timing (WT). PC mode provides two-dimensional imaging information and a 2.5073 s time resolution; in WT mode only one-dimensional imaging is preserved, achieving a time resolution of 1.766 ms. The XRT data were uniformly processed with xrtpipeline (version 12, in the heasoft software package version 6.11), and filtered and screened with standard criteria, correcting for effective area, dead columns, etc. The source counts were extracted within a 20 pixel radius (one XRT pixel corresponds to about 236). For the spectroscopy, we used the spectral redistribution matrices in caldb (20091130; matrices version v013 and v014 for the PC and WT data, respectively), while the ancillary response files were generated with xrtmkarf, and they account for different extraction regions, vignetting, and point-spread function (PSF) corrections.

The Proportional Counter Array (PCA; Jahoda et al. 1996) on board RXTE consists of five collimated xenon/methane multi-anode Proportional Counter Units (PCUs) operating in the 2–60 keV energy range. Raw data were reduced using the ftools package (version 6.11). To study the timing properties of SGR 0418+5729, we restricted our analysis to the data in Good Xenon mode, with a time resolution of 1 μs and 256 energy bins. The event-mode data were extracted in the 2–10 keV energy range from all active detectors (in a given observation) and all layers, and binned into light curves of 0.1 s resolution. We use here 46 RXTE/PCA observations of SGR 0418+5729, spanning the first six months of the outburst, until the source flux decayed below the instrument detection level. The total 194.2 ks exposure time is divided in observations of 0.6 to 13.6 ks exposure each. See Esposito et al. (2010) for further details on the Swift and RXTE observations.

The Chandra X-Ray Observatory monitored SGR 0418+5729 five times during the past three years. The first observation was with the High Resolution Imaging Camera (HRC-I; Zombeck et al. 1995) and the following four observations were with the Advanced CCD Imaging Spectrometer (ACIS-S; Garmire et al. 2003). Data were analyzed using standard cleaning procedures¹⁷ and CIAO version 4.4. The HRC-I camera does not have a sufficient spectral resolution, and it was used only for the timing analysis; it has a timing resolution of ∼16 μs. All ACIS-S observations were performed in VERY FAINT mode, with only the S7 CCD on, resulting in a timing resolution of 0.44 s. Photons were extracted from a circular region with a radius of 3'' around the source position, including more than 90% of the source photons, and background was extracted from a similar region far from the source position.

SGR 0418+5729 was observed six times with XMM-Newton (Jansen et al. 2001). Data have been processed using SAS version 12, and we have employed the most up-to-date calibration files available at the time the reduction was performed (2012 August). Standard data screening criteria are applied in the extraction of scientific products. For our spectral analysis we used only the EPIC-pn camera (Turner et al. 2001) which provides the spectra with the best statistics, while the MOS cameras (Strüder et al. 2001) were added in the timing analysis. The EPIC-pn camera was set in Small Window (timing resolution of 6 ms) and Full Frame (73 ms) modes in the first two observations, respectively, and in Large Window mode for all the following ones (48 ms), with the source at the aim-point of the camera, and the MOS cameras in Small Window mode (0.3 s). We extracted the source photons from a circular region of 30'' radius, and a similar region was chosen for the background in the same CCD. We restricted our spectral analysis to photons having PATTERN ⩽ 4 and FLAG = 0 for the EPIC-pn data.

For the spectral analysis we used source and background photons from the Swift, Chandra, and XMM-Newton observations extracted as described in the previous section (we also checked our results using a larger extraction region for the background spectra). The response matrices were built using ad hoc bad-pixel files built for each observation. We used the XSPEC package (version 12.4) for all fittings, and the phabs absorption model with the Anders & Grevesse (1989) abundances, and the Balucinska-Church & McCammon (1992) photoelectric cross-sections. We restricted our spectral modeling to the 0.7–10 keV energy band, excluding bad channels when needed. The Swift spectra were binned in order to have at least 30 counts per spectral bin. XMM-Newton spectra were grouped such to have at least 100, 50, and 40 photons per bin in the first three observations, respectively, and a minimum of 30 counts in the subsequent observations. On the other hand, all Chandra spectra have at least 25 counts per bin.

We started the spectral analysis by fitting all the spectra together (see Table 1) with a single component model: an absorbed blackbody or a power-law model. While the former gave a good fit, a single power law could not reproduce all the spectra at the same time. Fixing the absorption value to be the same for all spectra, for a single blackbody model (phabs*bbodyrad) we find an acceptable fit with N_H = (1.15 ± 0.06) × 10²¹ cm⁻² and $\chi _{\nu }^2 = 1.19$ (940 dof; errors on the spectral parameters are all reported at 90% confidence level). However, not unexpectedly, the best collected spectrum (the first XMM-Newton observation on 2009-08-12; see Table 1) gave bad residuals at lower and higher energies (see Figure 2, left panel). We tried to model this observation alone, and indeed a single absorbed blackbody or power-law components were not reproducing this spectrum properly ($\chi _{\nu }^2 > 2$). We then used a composite model. Good fits were found using both an absorbed blackbody plus a power law (phabs*(bbodyrad + power); N_H = (6.32 ± 0.04) × 10²¹ cm⁻², kT = 0.91 ± 0.07 keV, Γ = 2.82 ± 0.16, and $\chi _{\nu }^2 = 0.97$ for 392 dof) and an absorbed resonant cyclotron scattering model (RCS: Rea et al. 2007, 2008, or NTZ: Zane et al. 2009). Two blackbodies were also producing acceptable reduced chi-square values ($\chi _{\nu }^2 = 1.01$ for 392 dof) but with worse residuals at higher energies (this is compatible with what was found in Esposito et al. 2011 and Turolla et al. 2011). The parameters we found for the resonant cyclotron scattering models are N_H = (1.9 ± 0.3) × 10²¹ cm⁻², τ = 8.8 ± 1.2, β = 0.21 ± 0.08, and kT = 0.63 ± 0.11 keV ($\chi _{\nu }^2 = 1.08$ for 392 dof) for the RCS model; and N_H = (1.8  ±  0.2) × 10²¹ cm⁻², Δϕ = 1.9  ±  1.0, β_bulk = 0.13  ±  0.05, and kT = 0.88  ±  0.07 keV ($\chi _{\nu }^2 = 1.11$ for 392 dof) for the NTZ model.

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We then continued our spectral modeling of all spectra together by adding a further component only for this observation (adding a further component for all spectra did not significantly change the goodness of the fit; $\chi _{\nu }^2 = 1.13$ (938 dof); see Figure 2, middle and right panels). In Table 1 we report the values of the single absorbed blackbody model (see also Figure 2, left panel, and Figures 3 and 4), since when fitting all the spectra by using a composite model only for the first XMM-Newton observation, we find no change in the parameters of the other spectra with respect to the single blackbody fit. However, although a blackbody plus power-law model gives a good fit when fitting the first XMM-Newton observation alone, it is not so when fitting all data together. This is because the power-law component produces an unrealistic N_H increase, which does not match the value required by all the other observations modeled by a single blackbody. We then use one of the resonant cyclotron scattering models, the RCS model, for the joint fit as an empirical model for the first XMM-Newton observation.¹⁸

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In addition to the joint spectral modeling, we also fitted all the spectra individually. Beside the first XMM-Newton observation discussed above, the last XMM-Newton observation, when fitted alone with a single blackbody model, did not give a good chi-square ($\chi _{\nu }^2 = 2.2$ for 16 dof). Given the low number of counts in the spectrum of this observation (∼400 background-subtracted counts), this deviation from the blackbody model had only a marginal effect on the joint fit. A better fit was found adding a second blackbody ($\chi _{\nu }^2 = 1.2$ for 14 dof) or (with a slightly worse chi-square) a power-law component ($\chi _{\nu }^2 = 1.5$ for 14 dof). However, given the reduced number of counts collected in this observation, a detailed modeling of the quiescent spectrum of SGR 0418+5729 will be possible only when more data is accumulated.

We also tried to (1) model all the spectra with two blackbodies leaving one of the blackbodies with a fixed area mimicking the whole surface emission, and (2) fix one blackbody to the value observed in the last observation (see Table 1) and leave the second blackbody free to vary. In neither of those two cases could we find any improvement in the modeling of the data. Note that although a joint two blackbody model can fit the first few observations (Turolla et al. 2011), this is no longer the case when modeling together all the data collected in the entire 1200 day time span.

All the Chandra and XMM-Newton event files collected between 2010 November and 2012 August were used in order to extend the coherent timing solution we derived in Rea et al. (2010): P = 9.07838827(4) s 90% c.l., and 3σ first period derivative upper limit of $|\dot{P}|< 6.0\times 10^{-15}$ s s⁻¹ at epoch 54993.0 MJD. Photon arrival times were corrected to the barycenter of the solar system.¹⁹ Timing analysis was carried out by means of a phase-fitting technique (details on this technique are given in Dall'Osso et al. 2003; see also Esposito et al. 2010 for further details on this source). Given the intrinsic variability of the pulse shape as a function of time (see Figure 6), we inferred the phase of the modulation by fitting the average pulse shape of each observation with a number of harmonics, the exact number of which is variable and determined by requesting that the inclusion of any higher harmonic is statistically significant (by means of an F test). All data reported in Table 1 were folded using a reference period 9.07838880562798 s at epoch 54993 MJD, and fitted with one or more harmonics. In Figure 1 we plot the phases at which the fundamental sine function is equal to zero in its ascending part (positive derivative).

The fit of the resulting pulse phases with a linear component gives a reduced $\chi ^2_r \sim 3.2$ for 26 degrees of freedom (dof hereafter). The inclusion of a quadratic term, corresponding to a first period derivative component, was found to be significant at a confidence level of 3.5σ (by means of an F test). The resulting best-fit solution corresponds to P = 9.07838822(5) s (1σ c.l., two parameters of interest; epoch 54993.0 MJD) and $\dot{P}=4(1)\times 10^{-15}$ s s⁻¹ with a reduced $\chi ^2_r \sim 2.1$ (for 25 dof; see also Figure 1). The new timing solution implies an rms variability of only 0.2 s. As depicted above, the time evolution of the phase can be described by a relation of the form $\phi =\phi _0+2\pi (t-t_0)/P-\pi (t-t_0)2\dot{P}/P^2$.

To further assess the significance of the quadratic component, reflecting the period derivative, we performed detailed Monte Carlo (MC) simulations assuming as the null model a simple linear relation (see Protassov et al. 2002 for further details). By running 10⁵ MC simulations we verified that the quadratic component is significant at >99.96% confidence level, which is in very good agreement with what was estimated by means of the F test. We notice that examining the simulated data with different sampling distributions does not significantly change the results. Furthermore, we performed the same MC simulations using the whole set of observations but without considering the last XMM-Newton observation. We find a chance probability for the addition of a quadratic component of 0.65% (<3σ). Given the strong influence of the last of our observations in the determination of the period derivative, we will perform further X-ray observations in the next few years in order to increase the significance of the current $\dot{P}$ measurement.

Using this $\dot{P}$ measurement, we infer a surface dipolar magnetic field strength of B_dip = (6 ± 2) × 10¹² G, calculated at the neutron star equator. This value is fully consistent with the 3σ upper limit reported in Rea et al. (2010). We also estimate a characteristic age of $\tau _{\rm c} \simeq P/2\dot{P} \sim 35$ Myr, and a rotational power of $\dot{E} \simeq 3.9\times 10^{46} \dot{P}/P^3 \sim 2\times 10^{29}$ erg s⁻¹.

Based on the above phase coherent timing solution, we also studied the pulse shape and pulsed fraction evolution. Figure 1 shows the pulsed fraction evolution as a function of time. There is an evident increase starting soon after the burst detection with a recovery toward an asymptotic quiescent value which appears to be at about the 70%–80% level.

In Figures 1, 5, and 6 we study in detail the shape of the pulse profile as it evolves in time and in energy. By looking at the profile shapes of all X-ray observations performed so far, the source appears to be switching among a triple/double/single peak shape during the early outburst phases, with no clear trend in time (Figure 1). The pulse profile stabilizes to a single peak about three months after the outburst onset. However, studying in detail the first and the last XMM-Newton observations, a few key pieces of information can be extracted: (1) at lower energies (<1 keV), the pulse profile is mainly single peaked, while a second, and possibly also a third peak, appears at higher energies (see Figure 5); (2) the main component of the pulse profile continues to be at the same phase over the whole outburst decay (see Figure 7).

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We performed a pulse phase spectroscopy of the first XMM-Newton observation. A clear pulse phase dependence of the spectrum is already observed by simply looking at the pulse profile changes as a function of the energy (see Figure 5). In order to quantify the spectral variability as a function of the rotational phase, we performed a pulse phase spectroscopy extracting the spectra from phases 0–0.4, 0.4–0.6, and 0.6–1. These phase intervals were chosen by looking at Figure 5 in order to isolate the dip in the 1–4 keV pulse profiles at phase ∼0.55. In Table 2 we report the results of our modeling. The phase-averaged spectrum is not well fit by either a single blackbody nor a power law. We then used an absorbed blackbody plus power-law modeling (note that the RCS and NTZ models are not suited for phase resolved analysis since they are intrinsically phase-average), using for the photoelectric absorption model the same cross-section and abundances as for the phase-average spectrum (see Section 3.1). The blackbody temperature and radius were consistent in all three spectra, hence we fixed them to be the same for all spectra (kT_BB = 0.91 ± 0.01 keV, BB norm = 0.82 ± 0.05), while a variability >3σ has been observed in the photon index (it changed from about 2.9 to 4.1 between the spectra of the first peak and the dip).

However, from Figure 7 it is clear that the main difference in the spectra is at lower energies. In particular, above 5 keV the three spectra are very similar, while the 0.4–0.6 phase-resolved spectrum seems to have less counts than the other below such energy.

In Turolla et al. (2011) it was shown that the rotational properties of SGR 0418+5729 can be reproduced if the source is an aged magnetar, which experienced substantial field decay but still retains a strong enough internal toroidal field. The most updated magneto-thermal evolutionary models discussed in Viganò et al. (2013; but see also Pons et al. 2009 and Aguilera et al. 2008), confirm this scenario. The evolution of an initial dipolar magnetic field of $B_{\rm dip}^0\sim 1.5\times 10^{14}$ G (surface value at the pole) correctly provides the observed P and $\dot{P}$ at an age of ∼550 kyr, which is probably the real age of this source.

Although different combinations of the initial components of the magnetic field are possible, in all the models the magnetic field must have been large in the past (≳ 10¹⁴ G) to explain at the same time the long spin period, the bright X-ray emission at this old age, and the flaring activity of the source. The characteristic age overestimates the real one by almost two orders of magnitude. In Figure 9, we show the evolution of period, period derivative, the source track in the $P\hbox{--}\dot{P}$ diagram, and the bolometric thermal luminosity. In this scenario we estimate that SGR 0418+5729's mean surface temperature should now be ∼0.05 keV, unfortunately undetectable by current X-ray observations (which are observing only a hot tiny region on the star surface).

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For the evolution of the timing properties, we assume the magneto-dipole braking formula given by Spitkovsky (2006): $I\Omega \dot{\Omega } \approx ({B_d^2R^6\Omega ^4}/{4c^3}) (1+\sin {\chi }^2)$, where R is the NS radius, χ is the angle between the rotational and the magnetic axis, c is the speed of light, Ω = 2π/P is the angular velocity, and I is the moment of inertia of the star.

An alternative possibility is that the neutron star was born with an external magnetic field close to the present one, but its large core poloidal field slowly diffuses out, i.e., by ambipolar diffusion (Soni 2012). Although from the timing properties alone it is hard to discriminate between a hidden strong crustal magnetic field, a hidden strong core field or an intrinsically low-B neutron star, the magnitude of the X-ray luminosity, and the spectral properties and light curves may be used to distinguish the different scenarios.

In particular, the low magnetic field scenario cannot explain the high luminosity, large pulsed fraction, and the flaring activity of the source. As a matter of fact, if there is little field decay and the real age corresponds to the characteristic age (which in this scenario would be needed to reach the present period of 9 s), no existing non-magnetic cooling model can account for an X-ray luminosity of ≈10³¹ erg s⁻¹ at a characteristic age of ≈35 Myr.

The scenario in which a large core field diffuses out has the same problem: if the real age of the star is similar to its old characteristic age, no cooling model in the literature predicts such high quiescent luminosity. In addition, while the timescales used in Soni (2012) are correct for normal, non-superfluid nuclear matter, recent work (Glampedakis et al. 2011) shows that in the presence of superfluidity in the neutron star core, the timescales for ambipolar diffusion are many orders of magnitude longer, and therefore ambipolar diffusion does not play any role during the active age of the star.

An important question is whether a relatively low dipolar field is consistent with the star-quake model in which the primary cause of the outburst is an internal deposition of energy following a crust fracture. It is often overlooked that the magnetic stress needed to break the crust is strongly dependent on density (it is much easier to break the outer crust than the inner crust) and that the crust thickness grows as the temperature drops with age. In Figure 10 we show an estimate of the minimum magnetic field variation required to induce a fracture. As assumed in Perna & Pons (2011), this estimate is obtained assuming that the crust moves through a series of equilibrium states in which its elastic stress balances the (time-dependent) magnetic stress. The deviation of the magnetic field with respect to the last unstressed configuration (δB_c) may be large enough to break the crust when

$$\begin{equation} \delta B_c \approx \left(4 \pi \sigma _b^{\rm max} \right)^{1/2} \end{equation} \tag{ 1 }$$

where $\sigma _b^{\rm max}$ is the maximum stress that a neutron star crust can sustain. For young, relatively hot magnetars (crustal temperatures of 5 × 10⁸ K), only the inner crust is solid, and strong field variations δB_c ≳ 10¹⁴ G are required to fracture the crust. However, for old, cold neutron stars (crustal temperatures of 10⁸ K), the solid crust extends down to 10⁸ g cm⁻³, and is much easier to break, even with variations of the magnetic field of the order of δB_c ≳ 10¹² G. Note also that fractures close to the inner crust are much more energetic (because of both the higher available elastic energy and larger volume involved) than fractures in the low density region. In the first case, one can reach up to 10⁴⁴ erg, while in the second case, events of ≈10⁴¹ erg are expected. A rough prediction of the expected outburst rate (Perna & Pons 2011; Pons & Perna 2011) for the solution model mentioned above gives ≲ 10⁻³ star-quakes yr⁻¹ for an object as SGR 0418+5729. Assuming that there are about 10⁴ neutron stars in the Galaxy with similar age, and that (approximatively) 10% of them are born as magnetars, a naive extrapolation of this event rate to the whole neutron star population leads to an expected low-B magnetar outburst rate of ≲ 1 per year. Therefore, we expect that more and more objects of this class will be discovered in the upcoming years (as, e.g., Swift 1822.3–1606; Rea et al. 2012; Scholz et al. 2012).

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By modeling the flux evolution in time we tested if the crustal cooling model presented in Pons & Rea (2012) can fit the flux decay of SGR 0418+5729, on the wave of what done for Swift J1822.3–1606 (Rea et al. 2012). We assume a dipolar field of 6 × 10¹² G (equatorial), an internal toroidal field of 10¹⁴ G (at maximum) as inferred in Section 8.1, and an average surface temperature of 0.05 keV, which is the temperature we expect for the surface of such an old magnetar (note that in the 0.5–10 keV band we are only seeing a tiny hot spot). The best modeling was found by injecting 2.5 × 10²⁶ erg cm⁻³ in a thin layer in the outer crust between 4.5 × 10⁹ and 10¹⁰ g cm⁻³, and in the region contained within a cone with axis in the direction of the magnetic pole, and aperture a ≈ 0.4 rad, for a total energy deposition of 2.5 × 10⁴¹ erg (compatible with typical magnetar outbursts; Pons & Rea 2012). The evolution of the bolometric, and of the 0.5–10 keV flux is shown in Figure 10 (solid and dashed lines, respectively). We have shown with red squares how the observed flux decay would appear when adding the contribution of a blackbody component at 0.05 keV, mimicking the entire neutron star surface. It is clear that crustal cooling can easily explain the decay only if this further component is taken into account. In particular, the solid line is fitting the red points because the entire neutron star surface is taken into account, while it is not in the observed black data, which in fact cannot be reproduced by the dashed line. This is indicative of the difficulty of comparing theoretical cooling curves with data obtained in a certain energy band. We also note that no theoretical model predicts a surface temperature as high as 0.3 keV on a timescale of years, unless a continuous energy release is assumed (e.g., by long-lived internal currents). We finally mention that all the previous considerations are based on the (implicit) assumption that the blackbody temperature is a measure of the physical temperature of the emitting region. If this turned out not to be the case, e.g., because the spectrum is thermal but not Planckian so that a color correction is required, the physical surface temperature may be smaller than the measured blackbody temperature.

An alternative model to the crustal cooling scenario consists in the presence of currents flowing into the magnetosphere through a gradually shrinking magnetic bundle heating the neutron star surface from the top. In particular, the deepest available XMM-Newton observation of SGR 0418+5729, performed two months after the outburst onset, revealed that the 0.5–10 keV spectrum of SGR 0418+5729 is best reproduced by a blackbody component plus an additional non-thermal component, or by a resonant cyclotron scattering model. This suggests the presence of twisted magnetic field lines, at least in the first outburst stages. Furthermore, the limited spatial extent of the heated region (<1 km) is suggestive of a scenario in which the twist is confined within a small part of the magnetosphere, a thin current-carrying bundle, or j-bundle (Beloborodov 2009). As the j-bundle untwists during the outburst decay, the spectrum becomes more and more blackbody-like, as indeed observed.

Resonant cyclotron scattering from a thin, decaying j-bundle appears also capable of explaining (qualitatively) the spectrum and its evolution during the first outburst stages, but whether it can explain the double-peaked pulse profiles is unclear. However, this scenario has some further difficulties: (1) the total luminosity produced by currents in the bundle is, for such a low-B and a small thermal spot, well below the one observed at early times, at least if the spot is at the polar cap (see again Beloborodov 2009 and also Turolla et al. 2011), (2) the timescale for the twist decay is much shorter (<1 yr) than that implied by the long outburst of SGR 0418+5729, and (3) an approximate relation between the emitting area and the luminosity exists (A ∼ L²; Beloborodov 2009) if most of the luminosity is produced by current dissipation, but SGR 0418+5729 data show a somewhat flatter dependence when the first stages of the outburst are fitted (see Figure 11).

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In summary, the crustal cooling model, when including also a possible hidden contribution from the entire neutron star surface, appears favorable in explaining the outburst decay of SGR 0418+5729. However, it is likely that a combination of crustal cooling and magnetospheric untwisting bundle can be operating at the first stages of the outburst.

A fallback disk around SGR 0418+5729 was suggested by Alpar et al. (2011) as a way to aid the spin-down of the pulsar and explain the 9 s periodicity of this source. As an alternative, our results show (see Section 8.1) that both the thermal and the timing properties of this source can be reproduced for a pulsar age of ∼550 kyr by properly accounting for magnetic field evolution and dissipation, which also imply that the neutron star was born with a much higher dipolar field than the one measured today (see Section 8.1). Hence, in principle, the timing properties of this source would not necessarily require an additional spin down torque by a disk. However, given the suggestion that fallback disks around isolated neutron stars might be common (Michel 1988; Chevalier 1989; Lin et al. 1991), it is worthwhile to use the current multi-band upper limits to set constraints on the presence of a fallback disk around SGR 0418+5729.

Since the pulsar is currently spinning down, any disk-magnetosphere interaction must probably occur in the propeller regime, with the pulsar transferring angular momentum to the disk. For this condition to be satisfied, the inner boundary of the disk, located at about the magnetospheric radius $R_m=2.5\times 10^8[\dot{M}/(10^{16}\;{\rm g}\,{\rm s}^{-1})]^{-2/7} (M_{\rm NS}/M_\odot)^{-1/7}[B/(10^{12}\,{\rm G})]^{4/7}$, must be equal to or larger than the corotation radius R_co = (GM_NS)^1/3Ω^2/3. The strongest constraint on the disk emission is obtained when the inner radius of the disk obtains its minimum value, i.e., R_in = R_m = R_co. For the outer radius, we assume R_out = 10¹⁴ cm. We found that larger values do not result in appreciably larger emission at the frequency of interest, and hence this value allows to set the tightest constraint on the disk emission.

With the inner and outer disk radii fixed as discussed above, the emission spectra from the fallback disk is computed using the model of Perna et al. (2000). The disk is assumed to be optically thick and geometrically thin, and the anisotropy in the X-ray luminosity from the source (which irradiates the disk) is neglected, since it is found to be of second order (Perna & Hernquist 2000). The disk is assumed to be still "active," i.e., viscously accreting (see Menou et al. 2001). The disk emission is the result of both viscous dissipation and re-radiation of the pulsar X-ray luminosity. In order for the magnetospheric radius not to exceed the corotation radius, the accretion rate must be limited to $\dot{M}\lower.5ex\hbox{$\; \buildrel< \over \sim \;$}10^{15}$ g s⁻¹. With this value, the disk luminosity in the millimeter band is dominated by reprocessing of the pulsar X-ray luminosity. At 166 GHz, the predicted flux is about 0.01 mJy for a face-on disk, below the measured limit of 0.24 mJy. Hence the presence of a fossil disk cannot be ruled out by the current millimeter measurements. Even adding the contribution from the whole surface of the star by a putative thermal component at 0.05 keV, would bring the predicted millimeter flux just around the measured flux limit for a face-on disk.

The field around SGR 0418+5729 was also observed with the Grantecan and Hubble telescopes (Esposito et al. 2010; Durant et al. 2011). In particular, the latter observations were performed in two wide filters, the optical, with a pivot wavelength of 5921 Å, and in the NIR, with pivot wavelength of 11534 Å. The source was not detected down to the flux of f_O < 2.3 × 10⁻³¹ erg s⁻¹ cm⁻² Hz⁻¹ and f_NIR < 4.4 × 10⁻³¹ erg s⁻¹ cm⁻² Hz⁻¹, respectively. We found that this optical limit (nor the Grantecan or WHT limit) is not sufficiently constraining for a disk with the properties described above (the predicted emission for a face-on disk is about a factor of four below the limit). On the other hand, in the NIR, the observational limit is already able to rule out a face-on disk, which would yield an emission about twice larger than the measured flux limit. However, a disk inclined with respect to the observer by an angle $\cos \theta \lower.5ex\hbox{$\; \buildrel< \over \sim \;$}0.5$ would still be allowed by the observations (although falling short in explaining the X-ray bursts of this object).

At the time of this writing, in the ATNF pulsar catalogue (Manchester et al. 2005) 138 isolated radio pulsars have a dipolar magnetic field larger than that inferred for SGR 0418+5729. Our results imply that some of these objects might hide a strong toroidal component of the internal field, not measurable via the pulsar timing properties. A hint for such strong fields might be a high surface temperature, hotter than what would be predicted by standard cooling models at the pulsar age. However, only a few of those pulsars have had dedicated X-ray observations, and the shallow surveys do not suffice to detect such emission (expected to be as luminous as L_X ∼ 10³¹ erg s⁻¹). Furthermore, our calculation of the outburst rate of a low magnetic field magnetar also suggests that roughly once a year a quiet neutron star might turn on with magnetar-like activity.

On the other hand, if indeed a large number of neutron stars is hiding a strong magnetic field component, there would be important consequences also for other branches of astrophysics. In particular, it would imply that supernova explosions should generally produce strong magnetic fields, and that most massive stars are either producing fast rotating cores during the explosion to activate a dynamo, or are strongly magnetized themselves. Furthermore, in this scenario a non-negligible fraction of gamma-ray bursts might be due to the formation of magnetars, and the gravitational wave background produced by magnetar births should then be larger than predicted so far (important for future instruments as the Advanced-LIGO).