The Astrophysical Journal, 740:105 (7pp), 2011 October 20 doi:10.1088/0004-637X/740/2/105
C
2011. The American Astronomical Society. All rights reserved. Printed in the U.S.A.
IS SGR 0418+5729 INDEED A WANING MAGNETAR?
R. Turolla
1,2
,S.Zane
2
,J.A.Pons
3
,P.Esposito
4
, and N. Rea
5
1
Department of Physics, University of Padova, Via Marzolo 8, I-35131 Padova, Italy
2
Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey, RH5 6NT, UK
3
Department de Fisica Aplicada, Universitat d’Alacant, Ap. Correus 99, 03080 Alacant, Spain
4
INAF-Astronomical Observatory of Cagliari, Localit
`
a Poggio dei Pini, Strada 54, I-09012 Capoterra, Italy
5
Institut de Ci
`
encies de l’Espai (CSIC-IEEC), Campus UAB, Facultat de Ci
`
encies, Torre C5-parell, E-08193 Barcelona, Spain
Received 2011 June 1; accepted 2011 July 27; published 2011 October 4
ABSTRACT
SGR 0418+5729 is a transient soft gamma-ray repeater which underwent a major outburst in 2009 June, during which
the emission of short bursts was observed. Its properties appeared quite typical of other sources of the same class
until long-term X-ray monitoring failed to detect any period derivative. The present upper limit on
˙
P implies that the
surface dipole field is B
p
7.5 × 10
12
G, well below those measured in other soft gamma-ray repeaters (SGRs) and
in the Anomalous X-ray Pulsars (AXPs), a group of similar sources. Both SGRs and AXPs are currently believed
to be powered by ultra-magnetized neutron stars (magnetars, B
p
≈ 10
14
–10
15
G). SGR 0418+5729 hardly seems to
fit in such a picture. We show that the magneto-rotational properties of SGR 0418+5729 can be reproduced if this
is an aged magnetar, ≈1 Myr old, which experienced substantial field decay. The large initial toroidal component
of the internal field required to match the observed properties of SGR 0418+5729 ensures that crustal fractures,
and hence bursting activity, can still occur at the present time. The thermal spectrum observed during the outburst
decay is compatible with the predictions of a resonant Compton scattering model (as in other SGRs/AXPs) if the
field is low and the magnetospheric twist is moderate.
Key words: pulsars: individual (SGR 0418+5729) – stars: magnetic field – stars: neutron
Online-only material: color figure
1. INTRODUCTION
Soft gamma repeaters (SGRs) and Anomalous X-ray Pulsars
(AXPs) form a small, but rapidly expanding, class of isolated
neutron star (NS) sources characterized by the emission of
short (≈0.1 s), energetic (≈10
40
erg s
−1
) bursts of X-rays.
Their persistent luminosity (L
X
≈ 10
32
–10
36
erg s
−1
in the
∼0.5–10 keV range) is often variable, with flux enhancements
up to several hundreds during the outburst phases of transient
sources (e.g., Rea & Esposito 2011). SGRs and AXPs share
similar timing properties, with periods in a narrow range (P ∼
2–12 s), and large period derivatives (
˙
P ≈ 10
−13
to 10
−10
ss
−1
).
In most of these sources the X-ray luminosity exceeds the
rate of rotational energy losses (
˙
E ≈ 10
32
–10
35
erg s
−1
), and
highly variable radio activity has been detected in three objects.
X-ray spectra are often characterized by a blackbody (BB;
kT ≈ 0.5 keV) plus a high-energy power-law (PL; photon index
Γ ≈ 1.5–3) component (e.g., Rea et al. 2008), or by two thermal
components (e.g., Gotthelf & Halpern 2007; Tiengo et al.
2008). Both the spectral parameters and pulse profiles appear
to vary with the source flux (see, e.g., Woods & Thompson
2006; Mereghetti 2008; Rea & Esposito 2011, for reviews on
SGR/AXP properties).
6
The large values of the magnetic field derived from the
standard dipole formula (5 × 10
13
G B
p
10
15
G), the lack
of detected stellar companions, and their large X-ray output as
compared to
˙
E led to the suggestion that SGRs and AXPs are
powered by a young (age ≈ 10
3
–10
4
yr), ultra-magnetized NS,
or magnetar (Duncan & Thompson 1992; Thompson & Duncan
1993). Indeed, the magnetar scenario has been largely successful
in explaining many of the observed properties of SGRs/AXPs,
6
See also http://www.physics.mcgill.ca/∼pulsar/magnetar/main.html for an
updated catalog of SGRs/AXPs.
including those of the bursts (Thompson & Duncan 1995) and
of the persistent emission (the twisted magnetosphere model,
Thompson et al. 2002; Zane et al. 2009; Albano et al. 2010, and
references therein).
Despite that SGRs and AXPs are far from being a homoge-
neous class, in particular the inferred surface dipolar field spans
nearly two orders of magnitude, their observational behavior is
now commonly associated with that of (active) magnetars, to
the point that often the terms SGR/AXP and magnetar are used
as synonyms. This, actually, reflects the original definition of
a magnetar as an NS which is powered by its (large) magnetic
field (Thompson & Duncan 1993). In this respect, a super-strong
magnetic field is not per se a sufficient condition for trigger-
ing SGR/AXP-like activity, as testified by the existence of NS
sources, for instance most of the so-called High-B radio pulsars
(HBPSRs; e.g., Kaspi 2010), and possibly some of the ther-
mally emitting isolated NSs (XDINSs; e.g., Turolla 2009), with
surface magnetic fields comparable to those of SGRs/AXPs
but having substantially different properties and not showing
any bursting/outbursting activity over the ∼10–20 yr time span
during which they were observed.
Over the last few years, the family of the magnetar candidates
grew with the addition of several new objects, most of them
transient. X-ray bursting activity or peculiar radio emission
similar to that of SGRs/AXPs was detected in allegedly rotation-
powered pulsars, such as PSR J1846−0258 (Gavriil et al.
2008; Kumar & Safi-Harb 2008) and PSR 1622−4950 (Levin
et al. 2010). This suggests that magnetic energy substantially
contributes to their emission at certain stages. The magnetic field
inferred from their rotation properties (B
p
∼ (0.5–3) × 10
14
G)
is, in fact, close to that of SGRs/AXPs, contributing to the
widespread belief that magnetar-like activity has to be associated
with super-strong magnetic fields, typically higher than the
quantum field B
Q
4.4 × 10
13
G.
1
The Astrophysical Journal, 740:105 (7pp), 2011 October 20 Turolla et al.
In this respect, the recent discovery of a low-field SGR,
SGR 0418+5729 (Rea et al. 2010), came as a surprise.
SGR 0418+5729 was observed for the first time in 2009 June
when it entered an outburst state during which X-ray bursts
were detected (van der Horst et al. 2010). The enhanced flux
level allowed for a measure of the source periodicity, P ∼ 9.1s,
but, despite that the source was then monitored in X-rays for
∼500 days, no significant evidence for a period derivative was
found (Esposito et al. 2010; Rea et al. 2010, and references
therein). The published upper limit is
˙
P 6 × 10
−15
ss
−1
,
leading to an inferred dipole field B
p
7.5 × 10
12
G.
The very low magnetic field of SGR 0418+5729 (when
compared with other SGRs/AXPs) raises a number of questions
as to if, and how, the observed phenomenology of this source can
be accommodated within the magnetar picture. A crucial point is
whether an NS with a surface field well below 10
13
G can harbor
an internal toroidal magnetic field strong enough to produce
crustal displacements, which are believed to be responsible for
the bursting/outbursting episodes in SGRs/AXPs (Thompson
& Duncan 1995; Thompson et al. 2002; Beloborodov 2009). In
this paper, we address this issue and discuss both the spectral
and timing properties of SGR 0418+5729 in the framework of
an aging magnetar.
2. SGR 0418+5729 AS AN OLD MAGNETAR
The suggestion that SGR 0418+5729 may be an aged mag-
netar was already put forward by Rea et al. (2010; see also
Esposito et al. 2010) on the basis of the low persistent luminos-
ity (most likely well below that observed during the outburst
decay, L
X
∼ 6 × 10
31
(D/2 kpc) erg s
−1
), weak bursting activity
(only two faint bursts were recorded; van der Horst et al. 2010),
and large spin-down age (t
c
24 Myr). Rea et al. (2010) esti-
mated that an internal toroidal field B
tor
≈ 5×10
14
G is required
to power the persistent emission over a source lifetime ∼t
c
, and
concluded that B
tor
can be still large enough to overcome the
crustal yield.
The main appeal of the “old magnetar” scenario is that
it can offer an interpretation of the observed properties of
SGR 0418+5729 within an already established framework,
validating the magnetar model also for (surface) field strengths
quite far away from those of canonical SGRs/AXPs. However
intriguing, the considerations presented by Rea et al. (2010)
necessary rely on quite crude estimates. A more thorough
investigation on the behavior of evolved magnetars, in which
a substantial decay of the magnetic field occurred, is definitely
in order before claiming that SGR 0418+5729 is indeed powered
by the last hiccups of a once-ultra-magnetized NS.
In the following we focus on three points which we deem
central in order to test the “old magnetar” hypothesis. First,
assuming that SGR 0418+5729 was born with a magnetar-
like surface field, B
p
must have decayed by a factor ≈100 to
match the current upper limit. Roughly the same reduction is
expected in the internal field. Although the latter can initially
be ≈10–100 times higher than B
p
(at least locally), one may
wonder if at late times internal magnetic stresses are still strong
enough to crack the crust. A second and related question is if
realistic models of field decay in magnetars can account for the
observed rotational properties (period and period derivative) of
SGR 0418+5729. This also has direct bearing on the true age
of the source, which is most probably much younger than the
characteristic age, estimated assuming a non-decaying field. A
final point concerns the persistent emission of SGR 0418+5729.
XMM-Newton observations show evidence for a two-component
thermal spectrum (similar to those observed in other transient
magnetars), although the presence of a non-thermal tail cannot
be excluded (Esposito et al. 2010, 2011; see Section 2.4). In
the magnetar model, spectra are expected to exhibit a PL tail
that originates from resonant scattering in a twisted magneto-
sphere (Thompson et al. 2002). However, no calculations have
been performed yet in the range of surface fields implied by
SGR 0418+5729.
2.1. Magneto-rotational Evolution
A major issue in establishing the magnetic evolution of NSs
(and of magnetars in particular) is that observations place very
little, if any, constraint on the structure and strength of the
internal magnetic field. While there are several indications that
the large-scale, external field can be reasonably assumed to
be dipolar, with a moderate amount of twist in magnetars, the
different mechanisms proposed for the generation of the internal
field in the earlier phases of the NS life (differential rotation,
dynamo, magneto-rotational instability) most likely give rise to
both toroidal and poloidal components (e.g., Geppert et al. 2004,
2006 and references therein). The presence of a toroidal field,
roughly in equipartition with the poloidal one, is also required
by general stability arguments (e.g., Braithwaite & Spruit 2006
and references therein). A further complication comes from the
present poor knowledge of where the internal field resides. The
field can either permeate the entire star (“core” fields), or be
mostly confined in the crust (“crustal” fields), depending on
where its supporting (super) currents are located. The highly
anisotropic surface temperature distribution required in some
XDINSs has been taken as observational evidence in favor of a
complex field geometry in the external layers (crust, envelope,
atmosphere) of NSs, either in the form of strong crustal toroidal
fields, multipolar poloidal components, or both (Geppert et al.
2006;P
´
erez-Azor
´
ın et al. 2006; Zane & Turolla 2006).
The more general configuration for the internal field in a
NS will then be that produced by the superposition of current
systems in the core and the crust. As stressed by Pons & Geppert
(2007), the relative contribution of the core/crustal fields is
likely different in different types of NSs. In old radio pulsars,
where no field decay is observed, the long-lived core component
may dominate, while a sizable, more volatile crustal field is
probably present in magnetars, for which substantial field decay
over a timescale ≈10
3
–10
5
yr is expected (e.g., Goldreich &
Reisenegger 1992).
A particularly important result (Glampedakis et al. 2011)
is the lesser role that ambipolar diffusion plays in magnetar
cores (after crystallization, the absence of convective motions
already quenched ambipolar diffusion in the crust) on their
active lifetimes, contradicting an assumption often made in
the modeling of the flaring activity. Therefore, if the decay/
evolution of the magnetic field is indeed the cause of magnetar
activity, it is likely to take place outside the core and will be
governed by Hall/Ohmic diffusion in the stellar crust. The
relative importance of these two mechanism is strongly density
and temperature dependent. Thus, any self-consistent study of
the magnetic field evolution must be coupled to a detailed
modeling of the NS thermal evolution, and vice versa. Other
mechanisms, e.g., flux expulsion from the superconducting core,
due to the interaction between neutron vortices and magnetic
flux tubes, are highly uncertain and very difficult to model.
For these reasons, recent investigations of the magnetic field
evolution in magnetars focused only on the crustal component
of the field.
2
The Astrophysical Journal, 740:105 (7pp), 2011 October 20 Turolla et al.
Figure 1. From top left to bottom right, the evolution of the luminosity, surface dipole field, period, and period derivative according to the model discussed in the text.
The three cases refer to B
tor
(t = 0) = 0 (solid lines), B
tor
(t = 0) = 4 × 10
14
G (dotted lines), and B
tor
(t = 0) = 4 × 10
16
G (dashed lines).
The first attempts in this direction used a split approach.
Pons & Geppert (2007) studied the evolution of the field by
solving the complete induction equation in an isothermal crust,
but assuming a prescribed time dependence for the temperature.
They found that crustal magnetic fields in NSs suffer significant
decay during the first ≈10
6
yr and that the Hall drift, although
inherently conservative (i.e., alone it cannot dissipate magnetic
energy), plays an important role since it may reorganize the
field from the larger to the smaller (spatial) scales where Ohmic
dissipation proceeds faster.
The cooling of magnetized NSs with field decay was investi-
gated by Aguilera et al. (2008) by adopting a simple, analytical
law for the time variation of the field which incorporates the
main features of the Ohmic and Hall terms in the induction
equation. The fully coupled magneto-thermal evolution of an
NS was finally addressed by Pons et al. (2009), including all
realistic microphysics. However, owing to numerical difficul-
ties in treating the Hall term, their models include only Ohmic
diffusion. This can be a limitation because, as they note, the
Hall drift likely drastically affects the very early evolution of
ultra-magnetized NSs with surface field values of B
p
10
15
G,
and also that of “normal” NSs at late times (10
6
yr), when
the temperature in the crust has dropped. On the other hand, for
initial values of B
p
10
15
G, still well within the magnetar
range, the effect of the Hall drift is expected to introduce at
most quantitative changes (a somewhat faster dissipation) with
respect to the purely Ohmic picture.
2.2. The Case of SGR 0418+5729
To explore if, and to which extent, the magneto-thermal
evolution of (initially) highly magnetic NSs can lead to objects
with properties compatible with those of SGR 0418+5729, we
performed some runs using the code of Pons et al. (2009). We
refer to Section 2 in Pons et al. (2009) and Section 4 of Aguilera
et al. (2008) for all details about the code and the microphysical
input. We evolved a 1.4 M
NS assuming the minimal cooling
scenario (Page et al. 2004), with no exotic phases or fast
neutrino cooling processes, but including enhanced neutrino
emission from the breaking and formation of neutron Cooper
pairs in the NS core, as recent observations of the Cassiopeia A
supernova remnant seem to require (Page et al. 2011; Shternin
et al. 2011). The initial period was fixed at 10 ms and the
initial dipole field to B
p
= 2.5 × 10
14
G. Note that the
internal poloidal field is actually higher, with a maximum value
B
pol
(t = 0) 2.5 × 10
15
G in the inner crust.
We considered three models with different values of the
(maximum) internal toroidal field, B
tor
(t = 0) = 0, 4 × 10
14
and 4 × 10
16
G, which turns out to be the crucial parameter, as
shown in Figure 1. The four panels illustrate the evolution of
luminosity, dipole field B
p
, period P, and period derivative
˙
P .
Indeed the properties of SGR 0418+5729 are recovered in the
case of B
tor
(t = 0) = 4 × 10
16
G and age ∼1.5 × 10
6
yr. The
main conclusions can be summarized as follows.
1. The low quiescent luminosity is easily explained consider-
ing that the object is relatively old: even an NS born as a
bright, hot magnetar becomes cool and dim at this age.
2. On the other hand, the observed period constrains the
dipolar field: the initial dipole field cannot be much higher
than that the one considered here in order to prevent the
star from spinning down too fast and reaching periods
longer than that observed at present. Obviously there are
other large uncertainties, such as the angle between rotation
and magnetic axis, that may reduce the period (we assumed
an orthogonal rotator here).
3. Although the components of the initial internal field
B
tor
(t = 0) can be varied to some extent, a quite large
value is required. A large toroidal field, in fact, implies
strong currents, which, in turn, produce more heating and
higher temperatures. This drives a faster global field decay,
3
The Astrophysical Journal, 740:105 (7pp), 2011 October 20 Turolla et al.
which makes it possible to match the observed upper limit
on
˙
P and B
p
.
We stress that, while there are no stringent arguments against
such large internal fields in the NS crust, their real occurrence in
magnetars is an open issue. A possibility is that if the Hall drift
becomes very important and it results in much faster dissipation,
then one can obtain the same results starting with lower initial
toroidal fields. Finally, we note that the Hall term is bound to
become important again for objects like SGR 0418+5729 at late
stages (1 Myr) as the star cools down and the conductivity
increases by several orders of magnitude. No calculations are
available in this regime but the possible occurrence of a second
“Hall-active” phase could lead to enhanced bursting activity and
rapid field decay. This may be an indication that the estimate of
the bursting rate in Perna & Pons (2011) is a lower limit.
2.3. Occurrence of Bursts
Very recently Perna & Pons (2011) used the magnetic
evolution code of Pons & Geppert (2007) together with the
cooling models by Pons et al. (2009) to compute the magnetic
stress acting on the NS crust at different times. Their baseline
model has B
p
(t = 0) = 8 × 10
14
G and B
tor
(t = 0) = 10
15
G.
They found that the occurrence of crustal fractures (and hence
of bursts) is not restricted to the early NS life, during which
the surface field is ultra-strong, but can extend to late phases
(age ≈10
5
–10
6
yr; see their Figure 2). Both the energetics and
the recurrence time of the events evolve as the star ages. For
“old” magnetars about 50% crustal fractures release ≈10
41
erg
and the waiting time between two successive events is ≈1–10 yr.
They also made a longer run with a model with B
p
(t = 0) =
2 × 10
14
G and B
tor
(t = 0) = 10
15
G, for which the event rate
is about a factor of 10 smaller.
The model we considered in Section 2.1 as representative of
SGR 0418+5729 has B
p
(t = 0), very close to this latter con-
figuration, while B
tor
(t = 0) is larger. The present (maximum)
value of the internal toroidal field is ∼9 × 10
14
G and, although
we did not perform any detailed simulations, we argue that the
bursting rate of our model, at its present age, is similar to the
second model of Perna & Pons (2011), because the internal con-
figuration of the magnetic field is similar. It is important to note
that, despite the much larger initial toroidal fields of the model
presented in this paper, this leads to faster decay and therefore
similar values are reached when the NS is a million years old.
Comparing both models at the estimated age of 1.5 Myr, the in-
ternal toroidal field of the model presented in this paper is only
two times larger than that discussed in Perna & Pons (2011),
and we estimate that the typical lapse time between events for
an object like SGR 0418+5729 is ∼20–50 years.
2.4. Persistent Emission
In order to investigate the spectral properties of the persistent
emission from SGR 0418+5729 and its time evolution, we
analyzed eight Swift X-Ray Telescope (XRT)
7
and one XMM-
Newton/EPIC spectra (see Table S1 of Rea et al. 2010 for more
details). Preliminary results for some of these data sets were
already reported in Esposito et al. (2010, 2011). Spectra for
all the epochs were fitted simultaneously using XSPEC v.12.6,
with the value of the column density N
H
tied within the different
observations. Several one- and two-component models were
7
Each Swift data set contains several individual observations in order to
obtain good enough statistics.
tried, including a single blackbody, a one-dimensional (RCS;
Rea et al. 2008) and a three-dimensional (NTZ; Zane et al. 2009)
resonant scattering model, a double blackbody and a blackbody
plus a PL. All single component models give rather poor fits.
While for the RCS and NTZ models the residuals and the χ
2
values were not acceptable, a single BB decomposition properly
reproduces all the data but the XMM-Newton spectrum, which
is the only one responsible for the relatively large χ
2
of the
simultaneous fit. Since the highest-quality available spectrum
argues against the source having (at least at the early stage of
the outburst) a single thermal spectrum, we decided to add a
second component to the multi-instrument fit.
A BB+BB and a BB+PL model with all parameters free
(except N
H
, see above) provide acceptable fits of comparable
quality (χ
2
red
= 1.15 for 601 degrees of freedom (dof) and
χ
2
red
= 1.12 for 601 dof, respectively). However, we stress
again that both these spectral representations contain a large
number (32 in total) of free parameters, and those associated
with the second component are not required by the seven Swift
observations but only by the XMM-Newton one. Moreover,
despite that on a statistical ground there is no reason to prefer
the BB+BB over the BB+PL model, we note that in the latter:
(1) the spectral index Γ changes dramatically and in a totally
erratic way from one observation to another and (2) Γ can be
as large as ∼6, arguing against a PL as a physically motivated
representation of the second spectral component. On the other
hand, the values of the spectral parameters in the BB+BB model
appear to be reasonable and their time evolution is monotonic
(see below).
Since in the BB+BB best-fit model the temperature and nor-
malization of the colder BB component appear not to vary sensi-
bly in time (again, possibly because they are poorly constrained
by the Swift observations), we performed a fit with these two
parameters tied across the various data sets. This resulted in a
similarly good fit (χ
2
red
= 1.18 for 617 dof) and has the advan-
tage of containing 16 degrees of freedom less. In the case of
the BB+PL model the goodness of the fit worsens considerably
(χ
2
red
= 1.42 for 617 dof) by requiring that the parameters of
the (single) BB are the same at the different epochs.
For these reasons, in the following we take the BB+BB
model (with the colder BB constant in time) as the most likely
representation of the data, and discuss the ensuing implications
in the framework of an evolved magnetar. It is worth mentioning
that the failure of the resonant scattering models to fit the data
may be due to the fact that both RCS and NTZ were originally
developed for much higher fields than that likely present in
SGR 0418+5729 (the NTZ version used here assumes B
p
=
10
14
G). A more detailed spectral analysis will be the subject of
a forthcoming paper (N. Rea et al. 2011, in preparation).
The picture which emerges from the spectral analysis is that
of thermal emission from two regions on the star surface, a
cold one, with more or less constant size and temperature
(R
c
∼ 0.75 km for a fiducial distance D = 2 kpc and
T
c
∼ 0.31 keV), and a hot one, which shrinks during the
outburst decay. The evolution of the temperature and size of
the two components is shown in Figure 2. The temperature
of the hot region is more or less constant at kT
h
∼ 0.93 and
its area changes from ∼0.2to∼0.03 times that of the cold
region (R
h
∼ 0.15–0.30 km, again for D = 2 kpc). The
overall behavior is quite reminiscent of those seen in other
transient magnetar sources, notably the AXPs XTE J1810−197
and CXOU J164710.2−455216 (e.g., Albano et al. 2010 and
references therein).
4
The Astrophysical Journal, 740:105 (7pp), 2011 October 20 Turolla et al.
Figure 2. Time evolution of the temperature and emitting radius of the two
BB components in the spectrum of SGR 0418+5729; a source distance of
2 kpc is assumed. Diamonds refer to the hot and filled circles to the cold
component. The solid line shows the ratio of the emitting areas, A
hot
/A
cold
.Time
is counted in days from the outburst onset, on 2009 June 5 20:38:24.000 UTC
(MJD 54987.862).
Within the magnetar model this is interpreted as being due to
the sudden development of a twist in the external magnetic field
which then progressively decays. The twist likely affects only
a limited bundle of (closed) field lines and the charges flowing
along the current-carrying bundle heat the surface layers as
they impact upon the star. When the magnetosphere untwists,
the size of the heated region decreases (Beloborodov 2009).
This picture is compatible with the results we obtained for
SGR 0418+5729 with the BB+BB model assuming that the
heated region corresponds to the area emitting the hotter BB.
This is superimposed (or close) to a cooler, larger cap which is
responsible for the emission of the softer BB. It is interesting to
note that the analysis of the pulse profiles during the first stages
of the outburst supports this view. The double-peaked pulse
profile of SGR 0418+5729 suggests, in fact, that the surface
thermal map of the star comprises two warm caps, only one of
which was involved in the heating process (Esposito et al. 2010).
The predicted characteristic time for the outburst evolution is
≈5(Φ/10
9
V)
−1
(B/10
14
G)Δφ(A/10
12
cm
2
) yr, where Φ is the
discharge voltage and A is the area of the surface region involved
by the twist (Beloborodov 2009). Taking B ∼ 5×10
12
G, Δφ ∼
0.4 rad, and A ∼ 10
11
cm
2
, we get for the characteristic time
≈0.02(Φ/10
9
V)
−1
yr. A low discharge voltage, Φ ≈ 10
8
V,
is then required to obtain a decay time ≈ a few months. We
warn that, as already noted by Esposito et al. (2010), the
luminosity produced by Ohmic dissipation appears to be too
low to reproduce that observed at the beginning of the outburst,
∼10
34
(D/2 kpc)
2
erg s
−1
. However, if the twist affected a region
different from a polar cap (e.g., a ring confined between
two values θ
1
and θ
2
of the magnetic colatitude) the value
of the luminosity can be higher.
8
Alternatively, other heating
mechanisms may be operating, e.g., the release of magnetic
energy in the star outer layers (Lyubarsky et al. 2002).
As discussed in Section 2.3, the internal field of
SGR 0418+5729 can be still large enough to produce crustal
displacements so that magnetic helicity is transferred to the
external field, twisting up the magnetosphere. The appearance
8
We thank A. Beloborodov for bringing this point to our attention.
of a twist is usually accompanied by the formation of a high-
energy spectral tail, due to resonant cyclotron up-scattering of
thermal surface photons, which is, however, not unambiguously
detected in SGR 0418+5729. In order to investigate the prop-
erties of resonant cyclotron scattering spectra in the low-field
regime (B
p
5 × 10
13
G), we run a series of three-dimensional
Monte Carlo simulations, using the relativistic transport code
of Nobili et al. (2008a, 2008b), to which we refer for all de-
tails. We considered four values of the (polar) surface field
(B
p
= 10
12
, 5× 10
12
, 10
13
, 5× 10
13
G), two values of the seed
photon temperature (kT = 0.3, 0.9 keV), and several values of
the twist angle
9
in the range 0.1rad < Δφ<1.2 rad. The elec-
tron temperature and bulk velocity were fixed to kT
el
= 10 keV
and v/c = 0.5 in all cases. A different choice of these parame-
ters produce similar results provided that the scattering particles
are mildly relativistic, as indeed required to reproduce the ob-
served 1–10 keV spectra of SGRs/AXPs (see, e.g., Nobili et al.
2008a, 2008b; Zane et al. 2009, and references therein). Results
are summarized in Figure 3, which shows the photon index of the
non-thermal tail (computed in the 6–8 keV range) as a function
of the twist angle for the different values of B
p
. The (average)
index (in the same energy range) of the blackbody spectrum is
marked by a dashed horizontal line: when the photon index ap-
proaches the line the spectrum becomes indistinguishable from
a blackbody and no tail is present. As can be seen, while for
kT = 0.3 keV a non-thermal tail below 10 keV appears for all
the values of the twist, unless B
p
= 10
12
G and Δφ 0.3 rad,
the upscattering of seed photons associated with the hotter com-
ponent only produces a tail if Δφ 0.5 rad and B
p
> 10
12
G.
We stress that here we are considering photon energies below
10 keV, so the lack of a non-thermal spectral component for
kT = 0.9 keV only reflects the fact that now resonant Comp-
tonization tends to move photons at energies higher than 10 keV.
A tail, in fact, may be present above 10 keV also if it does not
show up below 10 keV.
Although the resonant scattering spectrum produced by the
reprocessing of soft photons coming from two NS surface
regions at different temperatures is not exactly given by the
superposition of the two individual models (see the discussion
in Albano et al. 2010), we adopt this approach to get some
insight into the spectral properties of SGR 0418+5729. If the
observed spectrum of the source is best modeled in terms of
the superposition of two blackbodies with kT ∼ 0.3, 0.9keV
the twist angle must be 0.5radforB
p
> 10
12
G to not
produce a PL tail in the hot component (see Figure 3). This,
however, is only a necessary condition because a PL tail may
still appear in the cold component. The emergence of such a
PL is related to the relative magnitude of the hot and cold
components. The total spectrum resulting from the superposition
of the models with kT ∼ 0.3 and 0.9 keV is shown in Figure 4
for B
p
= 5 × 10
12
G and two values of the ratio of the emitting
areas, A
cold
/A
hot
= 15, 30, typical of those measured during
the evolution of SGR 0418+5729. We note that the largest area
ratio (= 30) corresponds to the most unfavorable case: if the
tail does not appear now it is not present for smaller values of
the area ratio, when the cold component contributes less. As can
be seen, the total spectrum is very close to the superposition
of two blackbodies, with no high-energy tail. The same result
holds for different values of the magnetic field, provided that
Δφ 0.5 rad, and for even larger values of the twist if the
field is as low as 10
12
G. We conclude that the strong evidence
9
Here the magnetosphere is assumed to be globally twisted.
5
