Vol.0 (20xx) No.0, 000–000

11institutetext: School of Physics and Materials Science, Guangzhou
University, Guangzhou 510006, China<EMAIL_ADDRESS>

# A note on the anti-glitch of magnetar SGR 1935+2154

H. Tong

(Received 2009 month day; accepted 2009 month day)

###### Abstract

The magnetar SGR 1935+2154 is reported to have an anti-glitch, accompanied by
fast radio bursts, and transient pulsed radio emission. In the wind braking
model, this triplet event tell people that (1) SGR 1935+2154 does not have a
strong particle wind and can be approximated by magnetic dipole braking in the
persistent state; (2) Its anti-glitch is due to an enhanced particle wind,
similar to the first anti-glitch in magnetars; (3) Its transient pulsed radio
emission may due to enlarged and decreasing emission beam during the outburst.
The enhanced particle acceleration potential and pulsar death line may not be
the dominate factor.

###### keywords:

stars: magnetar – pulsars: general – pulsars: individual (SGR 1935+2154)

##  1 Introduction

The magnetar SGR 1935+2154 is reported to have an anti-glitch (i.e., spin-down
event) (Younes et al. 2022). The anti-glitch is also followed by fast radio
bursts and pulsed radio emission (Younes et al. 2022; Good et al. 2020; Zhu et
al. 2020). This will make SGR 1935+2154 the sixth magnetar to show pulsed
radio emission (Camilo et al. 2006; Huang et al. 2021). At the same time, SGR
1935+2154 is also the first Galactic magnetar to show a fast radio burst
(CHIME/FRB collaboration et al. 2020; Bochenek et al. 2020). This triplet of
anti-glitch, fast radio bursts and pulsed radio emission may shield light on
the physics of magnetar magnetospheres (Younes et al. 2022).

Previously, the magnetar spin-down behaviors is proposed to be dominated by
wind braking (Harding et al. 1999; Tong et al. 2013). When the anti-glitch was
firstly discovered in the magnetar 1E 2259+586 (Archibald et al. 2013), it was
proposed that anti-glitch is due to wind braking of the magnetar (Tong 2014).
The magnetic fields of magnetars may be twisted compared with that of normal
pulsars (Thompson et al. 2002). This may account for the outburst activities
of magnetars, i.e. decreasing X-ray luminosity, shrinking hot spot, decreasing
torque etc (Beloborodov 2009; Pavan et al. 2009; Tong 2019; Tong & Huang
2020).

Based on previous experiences on magnetar anti-glitch and magnetospheres, the
following three questions of magnetar SGR 1935+2154 will be addressed: (1)
What’s its spin-down mechanism during the persistent state? (2) Can its anti-
glitch be understood similarly to the first anti-glitch? (3) Why its pulsed
radio emission only last about one month?

##  2 Spin-down mechanism of SGR 1935+2154 during the persistent state

The magnetar SGR 1935+2154 has a period of $P=3.245\ \rm s$, and period
derivative of $\dot{P}=1.43\times 10^{-11}\ \rm s\ s^{-1}$ (Israel et al.
2016). Assuming that the neutron star is spin-down by magnetic dipole
radiation in vacuum, the surface magnetic field at the equatorial region is
(i.e., the characteristic magnetic field):

$B_{\rm c}=3.2\times 10^{19}\sqrt{P\dot{P}}=2.2\times 10^{14}\ \rm G.$ (1)

Note that the magnetic field at the polar region is two times higher for a
dipole field (Lyne & Graham-Smith 2012). The polar region magnetic field is
more relevant to particle acceleration and radio emission (Ruderman &
Sutherland 1975). The rotational energy loss rate of SGR 1935+2154 is about:
$\dot{E}_{\rm rot}=1.7\times 10^{34}\ \rm erg\ s^{-1}$. The soft X-ray flux of
SGR 1935+2154 is about: $3\times 10^{12}\ \rm erg\ cm^{-2}\ s^{-1}$ (Younes et
al. 2022). Assuming a distance of $10\ \rm kpc$ (Zhou et al. 2020), the
isotropic soft X-ray luminosity of SGR 1935+2154 is: $L_{\rm x}=3.6\times
10^{34}\ \rm erg\ cm^{-2}\ s^{-1}$. Therefore, the X-ray luminosity is
slightly higher than the rotational energy loss rate.

The magnetic dipole braking assumption does not consider the effect of the
magnetosphere. According to the wind braking model of magnetars, the
rotational energy loss rate is enhanced due to an outflow of particle wind
(Harding et al. 1999; Tong et al. 2013):

$\dot{E}_{\rm w}=\dot{E}_{\rm d}\left(\frac{L_{\rm p}}{\dot{E}_{\rm
d}}\right)^{1/2},$ (2)

where $\dot{E}_{\rm w}$ is the rotational loss rate due to the particle wind,
$\dot{E}_{\rm d}$ is the rotational energy loss rate due to magnetic dipole
braking, and $L_{\rm p}$ is the particle wind luminosity. The rotational
energy loss rate is enhanced for a strong particle wind $L_{\rm
p}\gg\dot{E}_{\rm d}$. The magnetic field in the strong wind case will be much
lower than characteristic magnetic field (Tong et al. 2013). When the particle
wind luminosity is comparable with the dipole rotational energy loss rate
$L_{\rm p}\sim\dot{E}_{\rm d}$, the wind braking rotational energy loss rate
is similar to the magnetic dipole braking case. The magnetic field may be
approximated by the characteristic magnetic field.

Since both the particle wind and X-ray luminosity are powered by the
magnetar’s magnetic energy release, it is possible that: $L_{\rm p}\sim L_{\rm
x}$ (Tong et al. 2013). For most magnetars, their X-ray luminosity are much
higher than their rotational energy loss rate (i.e., the name “magnetar”),
therefore the true magnetic field of magnetars may be much lower than the
characteristic magnetic field. For SGR 1935+2154, its X-ray luminosity is
slightly higher than its rotational energy loss rate. Therefore, the existence
of a particle wind will only slightly enhance the rotational energy loss rate
compared with the magnetic dipole case. Therefore, the true magnetic field of
SGR 1935+2154 may be approximated by its characteristic magnetic field. The
spin-down mechanism of SGR 1935+2154 may be approximated by the magnetic
dipole braking in the persistent state.

##  3 Anti-glitch of SGR 1935+2154 in the wind braking scenario

After the discovery of the first anti-glitch (Archibald et al. 2013), it was
proposed that: there is no anti-glitches, and anti-glitch is just a period of
enhanced spin-down in the wind braking model. Due to the sparse of
observations, a discrete decrease of frequency is seen (i.e., anti-glitch111In
the following, the name “anti-glitch” will still be used to denote the
corresponding timing event in the magnetar., Tong 2014).

For a net decrease of frequency: $|\Delta\nu|=1.8\times 10^{-6}\ \rm Hz$, the
decrease of rotational energy is: $|\Delta E_{\rm rot}|=2\times 10^{40}\ \rm
erg$. In the wind braking model, the corresponding rotational energy is
carried away by the particle wind (eq. (3) in Tong 2014):

$|\Delta E_{\rm rot}|=\dot{E}_{\rm d}^{1/2}L_{\rm p}^{1/2}\Delta
t=\dot{E}_{\rm d}^{1/2}\Delta E_{\rm w}^{1/2}\Delta t^{1/2},$ (3)

where $\Delta t$ is the time duration for the particle wind, and $\Delta
E_{\rm w}$ is total energy of the particle wind. The anti-glitch (or timing
event) of SGR 1935+2154 occurred between October 1st and October 6th (Younes
et al. 2022), therefore the maximum possible time duration is about 5 days.
Similar to the calculation for 1E 2259+586 (section 3.2 in Tong 2014), the
required particle wind energy for the anti-glitch in SGR 1935+2154 is:

$\Delta E_{\rm w}=8\times
10^{40}I_{45}^{2}b_{0}^{-2}\left(\frac{|\Delta\nu|}{1.8\times 10^{-6}\,\rm
Hz}\right)^{2}\left(\frac{5\,\rm days}{\Delta t}\right)\,\rm erg,$ (4)

where $b_{0}$ is the star’s polar magnetic field in units of $4\times 10^{14}\
\rm G$ (the value by assuming magnetic dipole braking). The corresponding wind
luminosity is:

$L_{\rm p}=1.8\times
10^{35}I_{45}^{2}b_{0}^{-2}\left(\frac{|\Delta\nu|}{1.8\times 10^{-6}\,\rm
Hz}\right)^{2}\left(\frac{5\,\rm days}{\Delta t}\right)^{2}\,\rm erg\,s^{-1}.$
(5)

The particle wind luminosity is about five times higher than the magnetar’s
persistent X-ray luminosity. Therefore, it is may be easy to achieve. A
slightly higher particle luminosity is also consistent with the non-detection
of significant X-ray variations in SGR 1935+2154 (Younes et al. 2022).

In the discovery paper (Younes et al. 2022), the authors also discussed the
possibility of wind braking for the anti-glitch. However, (1) they assumed
mass loading of the particle wind. The wind from normal pulsars may come from
pair production in open field line regions. The wind of magnetars may also
come from pair production in the open or closed field line regions
(Beloborodov 2009; Tong 2019). There is no need for mass loading of the
particle wind. (2) They implicitly assume that anti-glitch is a short duration
event, like that of the glitches in pulsars and magnetars. Therefore, the
authors try to constrain the time for the anti-glitch event observationally.
However, in the wind braking model, anti-glitch is just a period of enhanced
spin-down. It’s an enhanced spin-down process, not an instantaneous event.

In the magnetospheric model for magnetar anti-glitches, a particle wind
lasting for some time interval (Tong 2014) or corona-mass-ejection-like event
(Lyutikov 2013) can both account for the anti-glitches. The anti-glitch in SGR
1935+2154 is accompanied by radio activities. This is consistent with the
magnetospheric model for anti-glitches (also noted in the discovery paper).
Furthermore, no magnetar burst is seen around the anti-glitch epoch (Younes et
al. 2022). The absence of burst may be understood more naturally in the wind
braking model.

##  4 Transient pulsed radio emission of SGR 1935+2154

Figure 1: Evolution of the magnetar polar cap area with time. The polar cap
area is normalized to the area at $t=\tau=30\ \rm days$. The typical duration
of transient pulsed radio emission in SGR 1935+2154 is about 30 days.

Transient pulsed radio emission is detected lasting for about 30 days after
the epoch of anti-glitch (Younes et al. 2022; Zhu et al. 2022). This makes SGR
1935+2154 the sixth magnetar to have pulsed radio emissions (Camilo et al.
2006; Huang et al. 2021). The coherent radio emission mechanism of pulsars,
magnetars and fast radio bursts is still uncertain at present (Lyne & Graham-
Smith 2012). It may be easier to answer why the radio emission of SGR
1935+2154 only turns on for about 30 days.

Magnetars may have twisted magnetospheres compared with that of normal pulsars
(Thompson et al. 2002). The untwisting of the magnetic field may be
responsible for the X-ray outburst of magnetars (Beloborodov 2009; Pavan et
al. 2009; Tong 2019; Tong & Huang 2020). Due to the twist of magnetic field,
magnetars may have larger polar cap than the pure dipole magnetic field case
(Tong 2019). During the untwisting process, the polar cap shrinks with time,
which may corresponds to the shrinking hot spot of magnetars X-ray
observations (Tong 2019; Tong & Huang 2020). At the same time, the beam of the
radio emission will also be proportional to the polar cap area at a given
emission height (Lyne & Manchester 1988; Rankin 1993; Wang et al. 2013). A
time varying emission beam may account for the nulling and intermittence of
pulsars (Timokhin 2010; Huang et al. 2016). Similarly, an enlarged and
decreasing emission beam during magnetar outburst may also explain the
transient behavior of magnetar pulsed radio emissions.

According to the analytical treatment of magnetar outburst (Tong & Huang 2020,
section 5 there), for a self-similar globally twisted dipole magnetic field
(Wolfson 1995; Thompson et al. 2002; Pavan et al. 2009; Tong 2019), the polar
cap opening angle is:

$\theta_{\rm pc}=\left(\frac{R}{R_{\rm lc}}\right)^{n/2},$ (6)

where $R$ the neutron star radius, $R_{\rm lc}$ is the light cylinder radius,
$n$ is the parameter characterising the twist of the magnetic field. For a
twisted magnetic field, the magnetic field varies with radius as: $B(r)\propto
r^{-(2+n)}$ (Wolfson 1995). It can be seen that $n=1$ corresponds to the
dipole case, $n=0$ cooresponds to the split monopole case, $0<n<1$ corresponds
to the twisted dipole case. From eq.(6), it can be seen that magnetars with
twisted magnetic field will have larger polar caps compared with the pure
dipole case (Tong 2019). By introducing a typical decaying timescale of the
magnetic energy, the twist of the magnetic field evolves with time as (Tong &
Huang 2020):

$n(t)=1-(1-n_{0})e^{-t/\tau},$ (7)

where $n_{0}$ is the initial value of $n$ (which may be related to the hot
spot radius), $\tau$ is the typical decaying timescale of the magnetic energy
(which may be determined by the particle acceleration process in the twisted
magnetosphere, Beloborodov 2009; Tong 2019). For an initial value of
$n_{0}=0.5$ (which corresponds to a polar cap about $1\ \rm km$), and
$\tau=30\ \rm days$ (which is about the duration of the transient radio
emission), the polar cap area evolution with time is shown in figure 1. It can
be seen that the polar cap area decreased by a factor of 20. At a given
emission height, the radio emission beam radius is proportional to the polar
cap area. Then the radio emission beam will also decrease by a factor of 20
during the magnetar outburst. The pulsed radio emission is turned on when the
enlarged emission beam encloses the line of sight. When the emission beam
decreased significantly, the line of sight is out of the emission beam. The
radio emission will be “turned off”.

Here the appearance and disappearance of pulsed radio emission is mainly
discussed. The fast radio burst from SGR 1935+2154 may be similar to the
magnetar giant flares (Lyutikov 2006; Yu 2012; Yu & Huang 2013 in the magnetar
context; Popov & Postnov 2007; Katz 2016; Beloborodov 2017; Margalit & Metzger
2018 in the fast radio burst context). It needs independent studies. Similar
things in the case of magnetars are: modeling bursts and giant flares is
different from that of outburst and persistent emissions.

##  5 Discussions: other possibilities that are less likely

Although magnetars generally have a rotational period about several seconds,
most of them still line above the radio pulsar death line (see figure 3). This
is because their dipole magnetic field can be as high as $10^{14}-10^{15}\ \rm
G$. If the particle wind with luminosity is as high as $10^{38}\ \rm erg\
s^{-1}$ (i.e., $10^{4}$ times higher than the rotational energy los rate),
then the required magnetic will be $100$ times lower (see eq.(2)).
Furthermore, if the magnetar’s spin-down torque is dominated by a fallback
disk (Chatterjee et al. 2000; Alpar 2001; Benli & Ertan 2016), the star’s true
magnetic field may be in the range $10^{12}-10^{13}\ \rm G$. With such a
normal magnetic field strength, for a period about $10\ \rm s$, the star may
already lies below the radio pulsar death line. During the outburst, a larger
polar cap is obtained due to the twist of the magnetic field (see eq.(6)).
This will result in a higher acceleration potential across the polar cap and
may explain the radio turn-on of SGR 1935+2154. During the outburst decaying
phase, a shrinking polar cap will result in an decreasing acceleration
potential. When the acceleration potential decreases below a threshold value,
the radio emission may cease to operate.

A fiducial definition of the radio pulsar death line is that the maximum
acceleration potential across the polar cap is about $10^{12}\ \rm V$
(Ruderman & Sutherland 1975; Zhou et al. 2017). The maximum acceleration
potential is (Ruderman & Sutherland 1975; Zhou et al. 2017):

$\Phi_{\rm max}=\frac{B_{\rm p}R^{2}\Omega}{2c}\sin^{2}\theta_{\rm pc},$ (8)

where $B_{\rm p}$ is the surface magnetic field at the magnetic pole region.
For a given magnetic field strength $4\times 10^{12}\ \rm G$ (one hundred
times smaller than the characteristic magnetic field of SGR 1935+2154),
employing a decreasing polar cap (eq.(6)), the evolution of acceleration
potential is shown in figure 2. In the figure, a higher threshold acceleration
potential of $10^{13}\ \rm V$ is shown, due to uncertainties in the
definitions of pulsar death line (Chen & Ruderman 1993; Zhang et al. 2000) or
it can be viewed as fine tuning of parameters. When the maximum acceleration
potential drops below the threshold value, the star can no longer sustain the
required potential of pair production for coherent radio emissions. The
position of the death line is also lower on the $P$-$\dot{P}$ diagram of
pulsars, shown in figure 3. As can be seen from the figure, most of the
magnetars lie already above the fiducial death line. A lower death line for a
twisted magnetic field only strengthens this point.

This possibility (decreasing acceleration potential) may be less likely
compared with the previous one (decreasing emission beam). The reason is that
additional assumptions are required to make the magnetar have much lower
dipole field and lie below the death line in the persistent state. In the case
of decreasing emission beam, the SGR 1935+2154 already lies above the pulsar
death line with a magnetar strength magnetic field. An even higher
acceleration potential during the outburst state may help to generate the
pulsed radio emissions. But the acceleration potential is not the dominate
factor in this case. A similar point is that some pulsars are only detected at
high energy band (other than radio) (Manchester et al. 2005). This may because
their radio emission beam does not point toward us, not because they are radio
quiet.

Figure 2: Evolution of the maximum acceleration potential with time, for a
twisted magnetosphere.
Figure 3: Radio pulsar death line on the $P$-$\dot{P}$ diagram. Both the
fiducial death line and the death line for a twisted dipole field (with
$n=0.8$) are shown. For $n=0.5$, the position of the death line will be even
lower. The blue star is the magnetar SGR 1935+2154. Adapted from calculations
in Tong (2022).

##  6 conclusions

The triplet of an anti-glitch, fast radio bursts and pulsed radio emission in
SGR 1935+2154 can unveil the physics of the magnetar magnetosphere:

  1. 1.

The magnetar does not have a strong particle wind during its persistent state.
Its spin-down mechanism can be approximated by the magnetic dipole braking in
the persistent state.

  2. 2.

The anti-glitch may due to an enhanced particle wind, similar to the first
anti-glitch. The anti-glitch in SGR 1935+2154 is consistent with the wind
braking model of anti-glitches (Tong 2014).

  3. 3.

The transient pulsed radio emission may due to an enlarged and decreasing
emission beam after outburst. An enhanced and decreasing acceleration
potential may not be the dominate factor.

The mechanism of fast radio burst needs independent studies.

## acknowledgments

This work is supported by National SKA Program of China (No. 2020SKA0120300)
and NSFC (12133004).

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