11institutetext: Xinjiang Astronomical Observatory, Chinese Academy of
Sciences, Urumqi 830011, Xinjiang, People’s Republic of China
11email<EMAIL_ADDRESS>(AK<EMAIL_ADDRESS>(NW<EMAIL_ADDRESS>(XZ) 22institutetext: Key Laboratory of Radio Astronomy, Chinese Academy of
Sciences, Urumqi 830011, Xinjiang, People’s Republic of China 33institutetext:
Xinjiang Key Laboratory of Radio Astrophysics, Urumqi 830011, Xinjiang,
People’s Republic of China 44institutetext: School of Astronomy and Space
Science, Nanjing University, Nanjing 210023, People’s Republic of China
55institutetext: Key Laboratory of Modern Astronomy and Astrophysics (Nanjing
University), Ministry of Education, Nanjing 210023, People’s Republic of China
66institutetext: School of Physics and Electronic Engineering, Sichuan
University of Science & Engineering, Zigong 643000, People’s Republic of China

# Repeating X-ray bursts: Interaction between a neutron star and clumps
partially disrupted from a planet

Abdusattar Kurban 112233 Xia Zhou 112233 Na Wang 112233 Yong-Feng Huang 445511
Yu-Bin Wang 66 Nurimangul Nurmamat 44

(Received Month xx, yyyy; accepted Month xx, yyyy)

Repeating X-ray bursts from the Galactic magnetar SGR 1806-20 have been
observed with a period of 398 days. Similarly, periodic X-ray bursts from SGR
1935+2154 with a period of 238 days have also been observed. Here we argue
that these X-ray bursts could be produced by the interaction of a neutron star
(NS) with its planet in a highly elliptical orbit. The periastron of the
planet is very close to the NS, so it would be partially disrupted by the
tidal force every time it passes through the periastron. Major fragments
generated in the process will fall onto the NS under the influence of
gravitational perturbation. The collision of the in-falling fragments with the
NS produces repeating X-ray bursts. The main features of the observed X-ray
bursts, such as their energy, duration, periodicity, and activity window, can
all be explained in our framework.

###### Key Words.:

Stars: neutron – Planet-star interactions – X-rays: bursts

##  1 Introduction

Magnetars are highly magnetized neutron stars (NSs; Thompson & Duncan, 1995)
that are characterized by recurrent emission of short-duration bursts (a few
milliseconds to seconds) in soft $\gamma$-rays and/or hard X-rays, with burst
energies in the range $10^{35}$ to $10^{46}$ erg (see the review articles
Turolla et al. 2015 and Kaspi & Beloborodov 2017 and reference therein). More
than 30 magnetars have been observed to
date,111http://www.physics.mcgill.ca/~pulsar/magnetar/main.html and they show
diverse burst properties (Olausen & Kaspi, 2014).

Many models have been proposed to account for magnetar burst activities.
According to their energy sources, these models can be divided into two
categories: internal and external mechanisms. In the internal mechanism
models, the burst energy mainly comes from the magnetic field, whose decay
leads to a rupture of the crust and rapid magnetic reconnection, which
releases a large amount of energy and causes particle acceleration and
radiation in the magnetosphere (Thompson & Duncan, 1995; Lyutikov, 2003; Gill
& Heyl, 2010; Turolla et al., 2015).

On the other hand, in the external mechanism models, the energy comes from the
interaction between the compact stars – for example NSs or strange stars (SSs)
– and external material. The bursts could also be due to the impact of massive
comet-like objects with a NS (Katz, 1996; Chatterjee et al., 2000; Marsden et
al., 2001) or a SS (Zhang et al., 2000; Usov, 2001; Ouyed et al., 2011). We
note that in the models the origin and location of these small bodies can be
different. They can come from a fossil disk formed after the birth of the NS
and/or SS (Chatterjee et al., 2000; Marsden et al., 2001) or from the comet
clouds around the central star (like the Oort cloud in the Solar System; Zhang
et al. 2000). They can even be formed during a quark nova explosion, an
explosive phenomenon that happens when a NS is converted into a SS due to
spin-down or accretion processes (Ouyed et al., 2011). We note that both NSs
and SSs could be involved in the bursts and super-Eddington activities (e.g.,
Katz, 1996; Geng et al., 2021).

At present, it is not clear which of the various mechanisms is in operation.
The richness of the phenomenology indicates that a combination of mechanisms
are responsible (Turolla et al., 2015).

Recent studies show that two soft gamma-ray repeaters (SGRs), SGR 1806-20 and
SGR 1935+2154, may have periodic burst activities. SGR 1806-20 was initially
known as GB790107 and was identified as a gamma-ray repeater by Laros et al.
(1986). It has a surface magnetic field of $B_{\star}=2\times 10^{15}$ G
(Olausen & Kaspi, 2014) and a spin period of $P_{\rm spin}=7.55$ s (Woods et
al., 2007). Zhang et al. (2021a) analyzed the arrival time of more than 3000
short bursts from SGR 1806-20 detected by different telescopes (Göǧüş et al.,
2000; Prieskorn & Kaaret, 2012; Kırmızıbayrak et al., 2017) and found a
possible period of about 398 days for the source.

SGR 1935+2154 was discovered by Stamatikos et al. (2014). The surface magnetic
field of this source is $B_{\star}=2.2\times 10^{14}$ G, and the spin period
is $P_{\rm spin}=3.245$ s (Israel et al., 2016). After analyzing the arrival
time of more than 300 X-ray bursts from SGR 1935+2154, Zou et al. (2021)
argued that these bursts exhibit a period of $\sim 238$ days, with a 150 day
active window (63.3% of the period). However, Xie et al. (2022) revisited the
periodicity of SGR 1935+2154 with an updated sample and found the period to be
126.88 days. Further investigation into the periodical activities of SGR
1935+2154 is still urgently needed.

The periodic activities of SGRs may be caused by the precession of a NS (Zhang
et al., 2021a; Zou et al., 2021), when the emission region periodically sweeps
our line of sight. In this framework, a burst occurs when the crust fractures
or a magnetic reconnection happens. The underlying cause of these activities
is likely the evolution of the magnetic field, which itself depends
dramatically on a large number of degrees of freedom for the large-scale
magnetospheric configuration (Turolla et al., 2015). Details concerning the
triggering mechanism, the efficient conversion of magnetic energy into thermal
energy, and the transportation of the thermal energy to the surface are still
largely uncertain (Kaspi & Beloborodov, 2017; Gourgouliatos & Esposito, 2018).

On the other hand, a wide variety of transients can be produced during the
tidal disruption processes. A tidal disruption happens when an object gets too
close to its host star, triggering so-called tidal disruption events (TDEs).
This could occur in any system if the tidal disruption condition is satisfied
(Hills, 1975; Rees, 1988). Several classes of TDEs with emissions in the
optical/UV, soft X-rays, hard X-rays, and gamma-rays have been reported (see
Gezari 2021 and references therein). It was recently reported by Cendes et al.
(2023) that many TDEs exhibit late-time radio emission, often months to years
after the initial optical/UV flare. It has also been argued that repeating
transient events, for example AT 2018fyk (Wevers et al., 2023), AT 2020vdq
(Somalwar et al., 2023), ASASSN-14ko (Payne et al., 2021; Cufari et al., 2022;
Huang et al., 2023), RX J133157.6-324319.7 (Malyali et al., 2023), eRASSt
J045650.3-203750 (Liu et al., 2023), and Swift J023017.0+283603 (Evans et al.,
2023), may be due to a repeating partial disruption.

Various emission mechanisms have been proposed to explain these observations.
It is widely believed that the interaction between an outflow and the dense
circumnuclear medium produces the radio emission in TDEs (Cendes et al., 2022,
2023). For the UV/optical emission, however, we still lack a clear picture.
Some models include the radiation from shocks generated in stream–stream
collisions (Piran et al., 2015; Shiokawa et al., 2015; Coughlin & Nixon,
2022b) or the reprocessing of X-rays in either outflowing (e.g., Strubbe &
Quataert, 2009; Metzger & Stone, 2016; Roth & Kasen, 2018) or static materials
(Loeb & Ulmer, 1997; Guillochon et al., 2014; Roth et al., 2016). The soft
X-ray emission is thought to come from a compact accretion disk (e.g., Komossa
& Bade, 1999; Auchettl et al., 2017). The hard X-rays in some sources may be
related to Comptonization in a relativistic outflow and amplification due to
the beaming effect (e.g., Bloom et al., 2011; Cenko et al., 2012; Pasham et
al., 2023), while in others it is likely due to sub-Eddington accretion and
the formation of a corona (e.g., Wevers et al., 2019; Lucchini et al., 2022).

It is interesting to note that the tidal disruption of a planet by its compact
host star under various circumstances has been investigated by many authors.
It may be related, for example, to the pollution of the white dwarf atmosphere
by heavy elements (Malamud & Perets, 2020b, a). It may be connected with the
close-in strange quark planetary systems (Geng & Huang, 2015; Huang & Yu,
2017; Kuerban et al., 2020) or be a mechanism that produces some kinds of fast
radio bursts (Kurban et al., 2022) or gamma-ray bursts (Colgate & Petschek,
1981; Campana et al., 2011). It may also lead to glitch and/or anti-glitches
(Huang & Geng, 2014; Yu & Huang, 2016) and X-ray bursts (Huang & Geng, 2014;
Geng et al., 2020; Dai et al., 2016; Dai, 2020). Recently, the dynamics of the
clumps generated during the partial disruption of a planet by its compact host
was studied in detail (Kurban et al., 2023). It is found that the clumps could
lose their angular momentum quickly and fall toward the central star on a
short timescale.

Repeating partial tidal disruption generally requires the planet to be in a
highly elliptic orbit with a small pericenter distance of $\sim 10^{11}$ cm.
How such a planet can be formed needs to be clarified. It is well known that
NSs are born during the death of massive stars, at which time close-in planets
would be engulfed by the expanding envelope of the star and destroyed. To
date, however, more than 20 pulsars accompanied by the candidate planets have
been observed (see the Extrasolar Planets
Encyclopaedia222https://exoplanet.eu/catalog/ and Schneider et al. 2011).
Several scenarios have been proposed to explain the existence of pulsar
planets. Firstly, a planet may survive the red-giant stage of a massive star
(Bailes et al., 1991) and acquire an eccentric orbit thanks to the kick of the
newborn NS produced in an asymmetric supernova explosion (Bhattacharya & van
den Heuvel, 1991; Thorsett et al., 1993). Secondly, the planet could be formed
in the fallback disk of the supernova (Currie & Hansen, 2007; Hansen et al.,
2009), or due to the destruction of a binary companion (Martin et al., 2016).
Thirdly, a NS could capture a passing-by planet most likely moving in a highly
eccentric orbit (Podsiadlowski et al., 1991). For example, the circumbinary
planet around PSR B1620-26 in a globular cluster is thought to form in this
way (Sigurdsson et al., 2003). Although the orbits of currently detected
pulsar planets generally have a small ellipticity, high-eccentricity pulsar-
planet systems can be formed from dynamical processes such as capturing (e.g.,
Goulinski & Ribak, 2018; Kremer et al., 2019; Cufari et al., 2022), scattering
(e.g., Hong et al., 2018; Carrera et al., 2019), or the Kozai-Lidov mechanism
(e.g., Kozai, 1962; Lidov, 1962; Naoz, 2016). They are more likely to be found
in globular clusters333https://www3.mpifr-
bonn.mpg.de/staff/pfreire/GCpsr.html. The current non-detection of highly
eccentric planets around pulsars may be due to their large distances from us
as well as various other observational biases.

Motivated by these studies, we propose a new model to explain the periodic
X-ray bursts of SGRs. In our scenario, a planet is orbiting around a magnetar
in a highly eccentric close-in orbit. Every time the planet passes through the
periastron, it is partially disrupted by the tidal force from its host. The
clumps generated in this way fall onto the magnetar and produce X-ray bursts.
The observed periodicity of SGRs can be explained by our model.

The structure of this paper is as follows. In Section 2 the basic picture of
our model is described, and key parameters of the planet and clumps disrupted
from the planet are introduced. In Section 3 the main properties of the X-ray
bursts are predicted, such as the energetics, durations, periodicity, and
activity window. In Section 4 we compare our theoretical results with
observations. Finally, Section 5 presents our conclusion and some brief
discussions.

##  2 Model and parameters

###  2.1 Model

Figure 1: Schematic illustration (not to scale) of the partial disruption of a
planet around a NS. The dashed ellipse represents the orbit of the planet. The
gray areas represent possible ranges of the clumps’ orbit in the inner and
outer bound stream. The dotted ellipses at the fringe of these gray areas
represent the orbits of the clumps generated near $\theta=\theta_{\rm c}$. The
solid ellipse in the inner stream represents the clumps produced near
$\theta=0$, while the solid ellipse in the outer stream represents the clumps
produced at somewhere between $\theta=0$ and $\theta=\theta_{\rm c}$,
depending on the structure of the planet. The curved arrow represents unbound
clumps.

We introduce a NS–planet interaction scenario to explain the periodic X-ray
bursts of SGRs. We considered a planetary system composed of a NS with a mass
of $M_{\star}$ (we take $M_{\star}=1.4M_{\sun}$ and $R_{\star}=10$ km in the
following calculations) and a rocky planet of mass $m_{\rm pl}$ and radius
$R_{\rm pl}$. The planet orbits around the host NS in a highly eccentric orbit
described by $r=a(1-e^{2})/(1+e\cos\theta)$, where $a$ is the semimajor axis
of the orbit, $e$ is the eccentricity, and $\theta$ is the true anomaly. The
periastron of the orbit is $r_{\rm p}=a(1-e)$. According to Kepler’s third
law, the orbital period is related to $a$ as $P_{\rm
orb}=\sqrt{4\pi^{2}a^{3}/[G(M_{\star}+m_{\rm pl})]}$. A planet will be tidally
disrupted if the tidal force exceeds the self-gravity during the orbiting
process. A planet whose periastron is only slightly larger than the tidal
disruption radius would be partially disrupted by the tidal force every time
it passes through the periastron. Figure 1 presents a schematic illustration
of the partial disruption process.

When the planet passes through the pericenter, a number of clumps will be
generated during the partial disruption process. The host NS, the surviving
main part of the planet, and the clumps form a many-body system. Ignoring the
interactions between the clumps, it can be simplified as a triple system
composed of the NS, the remnant planet, and a particular clump. In this case,
the gravitational perturbation from the planet plays an important role in the
evolution of the clump’s orbit, which could lose its angular momentum due to
the perturbation and fall toward the NS (Kurban et al., 2023). The Alfvén wave
drag caused by the magnetic field of the NS can further facilitate the in-
falling process (e.g., Geng et al., 2020). The interaction between the major
clumps and the NS will finally produce a series of X-ray bursts. We estimate
the burst energy, duration, periodicity, and activity window in Section 3.

###  2.2 Key parameters

In our framework, we considered a rocky planet that is a pure Fe object, a
pure MgSiO3 object, or a two-layer planet with an Fe core and an MgSiO3
mantle. The internal structure of such planets can be determined by using the
equations of state (EOSs) that are widely used in exoplanet modeling (see
Smith et al. 2018 for the EOS of Fe materials and Seager et al. 2007 for that
of MgSiO3 materials). Table 1 lists some typical parameters of the three kinds
of planets (see Kurban et al. 2023 for more details).

When the size of a planet is larger than 1000 km, self-gravity will dominate
over the internal tensile stresses (Brown et al., 2017). The tidal disruption
radius of such a self-gravity-dominated planet is $r_{\rm td}=R_{\rm
pl}(2M_{\star}/m_{\rm pl})^{1/3}$. However, the degree of disruption depends
on the impact parameter ($\beta=r_{\rm td}/r_{\rm p}$) as well as the planet’s
structure. A slight decrease in the pericenter may result in a complete
disruption, while a small increase will lead to no mass loss at all (e.g.,
Guillochon & Ramirez-Ruiz, 2013). To be more specific, a planet would be
completely destroyed only when the tidal force from the NS exceeds the maximum
self-gravitational force inside the planet but not the surface gravitational
field. This means that the dense planet core is most difficult to destroy and
could survive for slightly larger values of $\beta$ (e.g., Coughlin & Nixon,
2022a; Bandopadhyay et al., 2024). In short, a higher central density of the
planet makes a complete disruption less likely. As a result, partial
disruption could occur in a considerable range of $\beta$ for a planet. In
fact, numerical simulations (Guillochon et al., 2011; Guillochon & Ramirez-
Ruiz, 2013; Liu et al., 2013; Nixon et al., 2021) and analytic derivations
(Coughlin & Nixon, 2022a) show that a partial disruption occurs when $r_{\rm
p}\approx 2r_{\rm td}$ ($\beta\approx 0.5$). In this study, we took this
distance as a typical condition that the planet would be partially disrupted
near the periastron (Kurban et al., 2022, 2023).

Table 1: Typical parameters of the planets considered in our study and the
timing parameters of the clumps originated from the inner side of the planet.

|  $P_{\rm orb}=238$ days |  |  $P_{\rm orb}=398$ days
---|---|---|---
$m_{\rm pl}$ | $R_{\rm pl}$ | $\frac{m_{\rm pl}}{R_{\rm pl}}$ | $P_{\rm orb}^{\rm cl}$ | $t_{\rm trav}$ | $P_{\rm orb,c}^{\rm cl}$ | $t_{\rm trav,c}$ | $\Delta P_{\rm orb,up}^{\rm cl}$ | $\Delta t_{\rm trav,up}$ |  | $P_{\rm orb}^{\rm cl}$ | $t_{\rm trav}$ | $P_{\rm orb,c}^{\rm cl}$ | $t_{\rm trav,c}$ | $\Delta P_{\rm orb,up}^{\rm cl}$ | $\Delta t_{\rm trav,up}$
($M_{\oplus}$) |  ($R_{\oplus}$) |  | (days) | (days) | (days) | (days) | (days) | (days) |  | (days) | (days) | (days) | (days) | (days) | (days)
Fe planet
4.6 | 1.2 | 3.8 | 36.3 | 73.1 | 115.1 | 230.3 | 78.8 | 157.2 |  | 41.3 | 83.1 | 154.7 | 309.4 | 113.3 | 226.3
10.1 | 1.4 | 7.2 | 24.6 | 49.6 | 92.4 | 184.9 | 67.8 | 135.3 |  | 27.2 | 54.7 | 118.9 | 237.8 | 91.6 | 183.1
18.1 | 1.6 | 11.1 | 17.9 | 35.9 | 75.6 | 151.3 | 57.8 | 115.4 |  | 19.3 | 38.8 | 94.0 | 188.0 | 74.7 | 149.2
MgSiO3 planet
4.4 | 1.7 | 2.6 | 54.9 | 111.1 | 141.7 | 283.8 | 86.8 | 172.8 |  | 65.3 | 131.6 | 200.1 | 400.5 | 134.8 | 268.8
9.9 | 2.1 | 4.7 | 40.1 | 80.8 | 121.4 | 242.9 | 81.3 | 162.0 |  | 46.0 | 92.7 | 165.0 | 330.1 | 119.0 | 237.4
18.2 | 2.5 | 7.3 | 30.8 | 62.1 | 105.3 | 210.8 | 74.5 | 148.7 |  | 34.6 | 69.6 | 138.9 | 277.9 | 104.3 | 208.3
Two-layer planet
3.3 | 1.4 | 2.4 | 53.8 | 108.7 | 140.4 | 281.0 | 86.5 | 172.3 |  | 63.8 | 128.6 | 197.7 | 395.5 | 133.8 | 267.0
9.8 | 1.8 | 5.4 | 34.4 | 69.4 | 112.0 | 224.0 | 77.5 | 154.6 |  | 39.0 | 78.5 | 149.6 | 299.2 | 110.5 | 220.7
12.8 | 1.6 | 8.0 | 24.0 | 48.3 | 91.0 | 181.9 | 67.0 | 133.7 |  | 26.4 | 53.1 | 116.6 | 233.3 | 90.2 | 180.2
17.1 | 2.4 | 7.1 | 31.1 | 62.6 | 105.8 | 211.7 | 74.7 | 149.1 |  | 34.9 | 70.2 | 139.6 | 279.3 | 104.8 | 209.2

Note: The timing parameters of the clumps are calculated for two cases of the
planet’s orbital period ($P_{\rm orb}=238,398$ days) according to the methods
given in the main text.

Taking $r_{\rm p}\sim 10^{11}$ cm, the eccentricity is $e\sim 0.992$ for a
planet with an orbital period of $P_{\rm orb}=240$ day ($a=0.85$ au), while it
is $e\sim 0.994$ for $P_{\rm orb}=400$ day ($a=1.19$ au). One may obtain a
slightly wider parameter range when the diversity of the planet structure is
considered (Kurban et al., 2022, 2023). Such a highly eccentric orbit
satisfying $r_{\rm p}\approx 2r_{\rm td}$ can be formed in the dynamical
processes mentioned earlier. A simulation by Goulinski & Ribak (2018) showed
that more than 99.1% of the captured planets form orbits with $0.85<e\lesssim
1$ and $a\sim 1$ — $10^{4}$ au. The planet-planet scattering (e.g., Carrera et
al., 2019) and the capture of a planet from a nearby planetary system via the
Hills mechanism (e.g., Cufari et al., 2022) are the other two efficient ways
to form highly eccentric orbits. In addition, an extremely high eccentricity
can also be reached due to the Kozai-Lidov mechanism for a wide range of
semimajor axis (e.g., Naoz, 2016).

The clumps generated during the disruption will have slightly different orbits
(see Figure 1), which are determined by the planet’s binding energy and
pericenter distance (Norman et al., 2021). For example, the clumps originating
from the inner side of the planet will have a relatively small orbit. The
semimajor axis of their orbits can be expressed as (Malamud & Perets, 2020b;
Brouwers et al., 2022)

$a_{\rm cl}=a\left(1+a\frac{2R}{d(d-R)}\right)^{-1}\leavevmode\nobreak\
\leavevmode\nobreak\ (\rm for\leavevmode\nobreak\ inner\leavevmode\nobreak\
orbit\leavevmode\nobreak\ clumps),$ (1)

where $a$ is the planet’s original semimajor axis, $d$ is the distance between
the NS and the planet at the moment of the breakup (here $d=r_{\rm p}$), $R$
is the displacement of the clump relative to the planet’s mass center at the
moment of breakup ($R=0$ corresponds to the center of the planet). Because the
planet is in a bound orbit, the clumps in the inner stream are generally bound
to the NS so that they can be relevant to the X-ray bursts studied in the
work.

For clumps in the outer orbit stream, their semimajor axes can be calculated
as $a_{\rm cl}=a\left(1-a\frac{2R}{d(d+R)}\right)^{-1}$. From this relation, a
critical displacement can be derived as $R_{\rm crit}=d^{2}/(2a-d)$. A clump
is still bound to the NS if its displacement is $R<R_{\rm crit}$, while it is
unbound for $R>R_{\rm crit}$ (Malamud & Perets, 2020b). $R_{\rm crit}$ depends
on the pericenter and binding energy of the planet. Combining the partial
disruption condition of $d=r_{\rm p}=2r_{\rm td}$, we can assess the boundness
of the clumps in the outer stream. It is found that unbound (free) clumps
could be generated in the following cases considered here: an Fe planet with
$m_{\rm pl}=18.1\,M_{\oplus}$ in the case of $P_{\rm orb}=238$ day; an Fe
planet with $m_{\rm pl}=10.1,18.1\,M_{\oplus}$, an MgSO3 planet with $m_{\rm
pl}={18.2}\,M_{\oplus}$, and a two-layer planet with $m_{\rm
pl}={12.8}\,M_{\oplus}$ for the case $P_{\rm orb}=398$ day. The outer stream
clumps generated from other planets considered in our study are all bound to
the NS. The orbits of these bound clumps will also be affected by the remnant
planet’s gravity and can be altered significantly due to the scattering
effect. They may even be ejected from the system finally. As a result, the
clumps in the outer stream are less likely to collide with the NS (especially
on short timescales), and their involvement in the generation of X-ray bursts
is not expected.

In the case of a full disruption, the material in a bound orbit returns to the
periastron on a timescale of $T_{\rm ret}=(r_{\rm
td}^{2}/2R)^{3/2}(2\pi/\sqrt{GM_{\star}})$ (Lacy et al., 1982; Norman et al.,
2021). We note that this expression underestimates the return time if $r_{\rm
td}$ is replaced by $r_{\rm p}$ for deep encounters (Rossi et al., 2021;
Coughlin & Nixon, 2022a). For the partial disruption in our framework, the
clump’s return time approximately equals its orbital period, $t_{\rm
ret}\approx P_{\rm orb}^{\rm cl}=\sqrt{4\pi^{2}a_{\rm
cl}^{3}/[G(M_{\star}+m_{\rm cl})]}$. The clump’s orbit should satisfy $r_{\rm
p}=a_{\rm cl}(1-e_{\rm cl})\pm R$ (here “+” and “-” for the clumps in the
inner and outer stream, respectively), from which one can derive the
eccentricity as $e_{\rm cl}=1-(r_{\rm p}\mp R)/a_{\rm cl}$. Since the clumps
are stripped off from the surface of the planet, we took $R\sim R_{\rm pl}$
when calculating their orbital parameters.

After the partial disruption, clumps of different sizes may be generated.
Larger homogeneous monolithic clumps generally have larger cohesive strength
(e.g., Malamud & Perets, 2020b). Smaller ones may merge to form larger clumps
due to gravitational instabilities in the hydrodynamical evolution of the
debris disk (Coughlin & Nixon, 2015), which itself depends on the EOS and is
preferred for EOSs stiffer than $\gamma=5/3$ (Coughlin et al., 2016b, a, 2020;
Coughlin, 2023; Fancher et al., 2023). In our cases, the characteristics of
the massive clumps formed through the accumulation of debris due to
gravitational instabilities are similar to that of the rubble pile asteroids,
which have a material strength in the range of $\sim$ 1 Pa — 1000 Pa (Walsh,
2018).

Veras et al. (2014) have investigated the tidal disruption of rubble pile
asteroids by assuming that the asteroid is frictionless, which means they
essentially ignored the cohesive strength. The classical Roche approximation
based solely on self-gravity could be applied in that case. However, they have
also pointed out that cohesive strength could play a role in the process.
Especially, the breakup distance of small bodies ($<1000$ km) is mainly
determined by the material strength (Brown et al., 2017). The larger the
cohesive strength is, the closer it gets to the central star (Brouwers et al.,
2022). The homogeneous monolithic clumps and the rubble pile clumps formed in
our cases have different material strengths so that they break up at different
distances that are smaller than the partial disruption radius of the planet.
In short, the clumps can remain intact even when they are much closer to the
NS due to their cohesive strength. For them, the breakup distance is (Zhang et
al., 2021b)

$r_{\rm str}=\left(\frac{\sqrt{3}GM_{\star}r_{\rm cl}^{2}\rho_{\rm
cl}}{5k}\right)^{1/3},$ (2)

where $r_{\rm cl}$ and $\rho_{\rm cl}$ are the radius (here we assumed that
the clump is spherical for the sake of simplicity) and density of the clump,
respectively; $k$ is a function of the internal friction angle, $\phi$, and
the cohesive strength, $C$: $k=6C\cos\phi/\sqrt{3}(3-\sin\phi)$. The friction
angle of geological materials commonly ranges from $25^{\circ}$ to
$50^{\circ}$ (e.g., Holsapple & Michel, 2008; Bareither et al., 2008; Jiang et
al., 2018; Villeneuve & Heap, 2021). The typical strength is $C\sim 1$ Pa for
comets (Gundlach & Blum, 2016) and $C\sim 1$ Pa — 1000 Pa for rubble pile
asteroids (Walsh, 2018), while it is in the range 0.1 Mpa — 50 MPa for
monolithic asteroids/meteorites (Schöpfer et al., 2009; Ostrowski & Bryson,
2019; Pohl & Britt, 2020; Veras & Scheeres, 2020; Villeneuve & Heap, 2021). As
a result, the breakup distance for the clumps considered here is $\sim 10^{9}$
— $10^{10}$ cm.

##  3 X-ray bursts

###  3.1 Periodicity and activity window

Figure 2: Travel time ($t_{\rm trav}$) of the clumps in the innermost orbit as
a function of the planet’s orbital period ($P_{\rm orb}$). Three kinds of
planets are considered: (a) Fe planets, (b) MgSiO3 planets, and (c) two-layer
planets. The mass of the planet ($m_{\rm pl}$) is marked in each panel.

The periodicity and active window of X-ray bursts are determined by the
orbital evolution of the clumps generated in the NS planet system. It can be
simplified as a triple system composed of the NS, the remnant planet, and a
particular clump. The clump’s orbit evolves under the influence of strong
gravitational perturbation from the remnant planet.

In a triple system where a test particle revolves around its host in a close
inner orbit while a third object moves around in an outer orbit, the
eccentricity of the test particle can be significantly altered by the outer
object. This is known as the Kozai-Lidov mechanism, which can explain the
long-term orbital evolution of triple systems in the hierarchical limit
(Kozai, 1962; Lidov, 1962). In recent decades, it has been further developed
and applied to various mildly hierarchical or non-hierarchical systems. The
hierarchy of a system is mainly measured by a parameter defined as
$\epsilon=a_{1}e_{2}/[a_{2}(1-e_{2}^{2})]$, which actually is the coefficient
of the octupole-order interaction term (Lithwick & Naoz, 2011; Li et al.,
2014). Here the eccentricity ($e$) and semimajor axis ($a$) of the inner and
outer orbits are denoted by subscripts 1 and 2, respectively. $\epsilon$
parameterizes the size of the external orbit versus that of the internal
orbit. The initial orbital parameters of a triple system have an obvious
influence on its final fate after a long-term dynamic evolution. A system with
$\epsilon\leq 0.1$ is hierarchical and stable, a system with $\epsilon>0.3$ is
nonhierarchical and unstable, while $0.1<\epsilon\leq 0.3$ corresponds to the
mildly hierarchical condition (Naoz, 2016). We note that the Kozai-Lidov
mechanism (secular interaction) is broken down for the nonhierarchical and
unstable systems so that it cannot provide a meaningful description for the
orbital evolution at all (Perets & Kratter, 2012; Katz & Dong, 2012). Strong
perturbations from the outer orbit object (Toonen et al., 2022) lead the test
particle to effectively lose its angular momentum on a timescale on the order
of the inner orbit period, causing it to collide with the central object
(e.g., Antonini et al., 2014, 2016; He & Petrovich, 2018; Hamers et al.,
2022).

In our framework, the parameters of the clumps in the inner stream satisfy
$\epsilon>0.3$, which means that they are in a nonhierarchical and unstable
regime. The secular approximation (Kozai-Lidov mechanism) is broken down for
them. The surviving major portion of the planet plays the role of the outer
object, which can significantly alter the clump’s orbit and cause it to fall
onto the NS on a timescale comparable to a few orbital periods (Kurban et al.,
2023). In fact, in a recent study, it was shown that energy exchange between
the clump and the remnant planet could occur due to strong gravitational
interactions, which leads the clump’s periastron distance ($r_{\rm p}$) to
abruptly change to a very small value (e.g., Zhang et al., 2023). Below, we
present a rough estimate for the travel time that a clump experiences from its
birth to the final collision with the NS. The total duration for most clumps
to collide with the NS (i.e., the active window) can also be estimated.

Figure 3: Difference in the travel time ($\Delta t_{\rm trav}$) as a function
of the clump’s orbital period difference ($\Delta P_{\rm orb}^{\rm cl}$) for
planets in various conditions. The upper panels show the cases of Fe planets
for $P_{\rm orb}=238$ days (Panel a) and $P_{\rm orb}=398$ days (Panel b). The
middle panels show the cases of MgSiO3 planets for $P_{\rm orb}=238$ days
(Panel c) and $P_{\rm orb}=398$ days (Panel d). The lower panels show the
cases of two-layer planets for $P_{\rm orb}=238$ days (Panel e) and $P_{\rm
orb}=398$ days (Panel f). Note that the endpoint of each line corresponds to
$\Delta P_{\rm orb,up}^{\rm cl}$ and $\Delta t_{\rm trav,up}$.

After the partial disruption, a clump originating from the inner side of the
planet will move in an inner orbit. The travel time for it to fall onto the NS
depends on its return time ($t_{\rm ret}$) to the periastron as well as the
evolution timescale of its angular momentum ($t_{\rm evo}$). The return time
approximately equals its orbital period, $t_{\rm ret}\approx P_{\rm orb}^{\rm
cl}$. After the formation of the inner orbit, the clump loses its angular
momentum ($j_{\rm cl}=\sqrt{1-e_{\rm cl}^{2}}\rightarrow j_{\rm cl}\sim 0$) on
a timescale of $t_{\rm evo}$ due to gravitational perturbations. This is
expected to occur when the clump passes through the periastron at the end of
the second or third orbit because $t_{\rm evo}$ is much less than $P_{\rm
orb}^{\rm cl}$ under the condition of $m_{\rm pl}>2M_{\oplus}$ and $P_{\rm
orb}>100$ day (Kurban et al., 2023). Therefore, the clump’s travel time from
its birth at the planet to its collision with the NS can be approximated as
(Kurban et al., 2023)

$t_{\rm trav}\approx 2P_{\rm orb}^{\rm cl}+t_{\rm evo},$ (3)

where $t_{\rm evo}$ is expressed as (e.g., Antonini et al., 2014)

$t_{\rm evo}=\left(\frac{1}{j_{\rm cl}}\frac{dj_{\rm
cl}}{dt}\right)^{-1}\approx P_{\rm orb}^{\rm
cl}\frac{1}{5\pi}\frac{M_{\star}}{m_{\rm pl}}\left[\frac{a(1-e)}{a_{\rm
cl}}\right]^{3}\sqrt{1-e_{\rm cl}}.$ (4)

We note that different clumps with slightly different orbital periods will
arrive at the NS at different times.

Figure 2 shows the travel time of a clump in the innermost orbit for pure Fe
planets, MgSiO3 planets, and two-layer planets. From this figure, we see that
the clumps will fall onto the NS on a short timescale on the order of its own
orbital period, which is affected by the orbit and composition of the planet.

The activity widow of the bursts can be estimated by considering the
dispersion of the arrival time of various clumps with slightly different
orbital periods. We can consider the case of two clumps: one has an orbital
period of $P_{\rm orb}^{\rm cl}$, the other has a different orbital period of
$P_{\rm orb}^{\rm cl}+\Delta P_{\rm orb}^{\rm cl}$, where $\Delta P_{\rm
orb}^{\rm cl}$ is their difference in orbit. Their difference in arrival time
can be calculated from the derivative of travel time as $\Delta t_{\rm
trav}\approx 2\Delta P_{\rm orb}^{\rm cl}-4t_{\rm evo}(\Delta P_{\rm orb}^{\rm
cl}/P_{\rm orb}^{\rm cl})/3$, which is just the time interval between two
successive collisions.

The simulations by Malamud & Perets (2020b, a) show that the distribution of
the orbital period of inner bound clumps can range from $P_{\rm orb}^{\rm cl}$
to $P_{\rm orb}$ in a full disruption. However, the competition between the
tidal force and the self-gravity continuously evolves over the encounter. The
largest partial disruption distance for a planet is $\sim 2.7r_{\rm td}$
(Guillochon et al., 2011; Liu et al., 2013). During a strong encounter, the
mass loss from the planet continues till a separation of several times $r_{\rm
td}$ (e.g., Ryu et al., 2020). For the partial disruption cases discussed
here, the mass loss will continue until the planet’s separation from the NS is
$r_{\rm c}\sim 4r_{\rm td}$. In other words, the mass loss occurs until the
phase $\theta\sim\theta_{\rm c}$, where $\theta_{\rm c}$ is the true anomaly
at $r=r_{\rm c}$. When $\theta>\theta_{\rm c}$ (or $r>r_{\rm c}$), new clumps
could not be generated till the next periastron passage. For the inner and
outer bound streams, the clumps marginally bound to the remnant planet will be
re-accreted by the surviving core of the planet after a few dynamical
timescales defined by $t_{\rm dyn}=\sqrt{R_{\rm pl}^{3}/Gm_{\rm pl}}$ (e.g.,
Guillochon et al., 2011; Coughlin & Nixon, 2019). We note that the dynamical
timescale is typically $t_{\rm dyn}\sim$ 6 — 14 minutes, which depends on the
structure of the planets. On the other hand, the clumps that are not bound to
the remnant planet will return to the periastron and continue to orbit around
the NS (e.g., Guillochon et al., 2011). One can obtain the critical semimajor
axis $a_{\rm cl,c}$ (or the critical orbital period $P_{\rm orb,c}^{\rm cl}$)
using the critical separation of $d=r_{\rm c}\sim 4r_{\rm td}$. Thus, for the
clumps in the inner orbits, we expect their orbital periods to be between
$P_{\rm orb}^{\rm cl}$ and $P_{\rm orb,c}^{\rm cl}$. The upper limit of
$\Delta P_{\rm orb}^{\rm cl}$ would then be $\Delta P_{\rm orb,up}^{\rm
cl}=P_{\rm orb,c}^{\rm cl}-P_{\rm orb}^{\rm cl}$. We used this parameter to
estimate $\Delta t_{\rm trav,up}$, which is just the activity window of X-ray
bursts in our framework.

Figure 3 illustrates $\Delta t_{\rm trav}$ as a function of $\Delta P_{\rm
orb}^{\rm cl}$ for planets with different orbital period and composition. We
note that $\Delta P_{\rm orb}^{\rm cl}$ satisfies $0<\Delta P_{\rm orb}^{\rm
cl}\lesssim\Delta P_{\rm orb,up}^{\rm cl}$, which determines the range of the
active window for the X-ray bursts. $\Delta t_{\rm trav,up}$ is sensitive to
both $r_{\rm p}$ and $r_{\rm c}$. A too-small $r_{\rm p}$ may cause the period
to completely disappear.

Table 1 lists the timing parameters of the clumps disrupted from various
planets with different compositions, masses, and orbital periods. $P_{\rm
orb}^{\rm cl}$ and $P_{\rm orb,c}^{\rm cl}$ are the orbital periods of the
clumps in the innermost orbit and the clumps stripped off from the planet at
$\theta_{\rm c}$ (or $r=r_{\rm c}$), respectively. $t_{\rm trav}$ and $t_{\rm
trav,c}$ are their travel times, and $\Delta P_{\rm orb,up}^{\rm cl}$ and
$\Delta t_{\rm trav,up}$ are their orbital period difference and travel time
difference, respectively. It can be seen from this table that these parameters
are affected by the structure and orbital period of the planet. Generally, a
larger $P_{\rm orb}$ leads to a larger $\Delta t_{\rm trav,up}$. For planets
with a particular $P_{\rm orb}$, the structure is also an important factor:
$\Delta t_{\rm trav,up}$ decreases with the increase in compactness ($m_{\rm
pl}/R_{\rm pl}$).

###  3.2 Energy and duration

In the last section we investigated the properties of clumps and show that the
clumps fall onto the NS on a short timescale. The collision of a small body
(asteroid) with a NS can produce X-ray bursts (Huang & Geng, 2014; Geng et
al., 2020; Dai et al., 2016; Dai, 2020). The gravitational potential energy of
the small body will transform to X-ray burst energy as (e.g., Geng et al.,
2020; Dai, 2020)

$E_{X}=\eta\frac{GM_{\star}m_{\rm cl}}{R_{\star}},$ (5)

where $\eta$ ($\eta<1$) is the energy transforming efficiency.

In our case, the range of the mass of the clumps can be relatively wide. For
example, simulations show that the size of clumps generated during the partial
disruption ranges from a few kilometers to $\sim 100$ km (Malamud & Perets,
2020b, a). It depends on the distance to the NS as well as its intrinsic
material strength. On the other hand, if a clump is too large, it cannot
resist the tidal force and will further break up during its falling toward the
NS. The clumps will interact with the magnetosphere and produce Alfvén wings
at the light cylinder radius, $R_{\rm LC}=cP_{\rm spin}/2\pi$ (Cordes &
Shannon, 2008; Mottez & Heyvaerts, 2011; Mottez et al., 2013a, b; Chen & Hu,
2022), which further helps the NS capture the clumps (Geng et al., 2020) at
the Alfvén radius. The Alfvén radius is determined by assuming that the
kinetic energy equals the magnetic energy, namely $\rho_{\rm cl}\upsilon_{\rm
R_{A}}^{2}/2=B_{\rm R_{\rm A}}^{2}/8\pi$, where $\upsilon_{\rm
R_{A}}=\sqrt{2GM_{\star}/R_{\rm A}}$ and $B_{\rm R_{\rm
A}}=B_{\star}(R_{\star}/R_{\rm A})^{3}$ are the free-fall velocity and
magnetic field strength at the distance $R_{\rm A}$ from the NS, respectively.
The Alfvén radius $R_{\rm A}$ then can be calculated as $R_{\rm
A}=(B_{\star}^{2}R_{\star}^{6}/8\pi GM_{\star}\rho_{\rm cl})^{1/5}$. Taking
$R_{\star}$ = 10 km and $\rho_{\rm cl}=$ 5 — 8 g cm-3 (the surface density of
the planet with different compositions), we get $R_{\rm A}=$ (4.43 — 4.04)
$\times 10^{7}$ cm for SGR 1935+2154 and $R_{\rm A}=$ (1.83 — 1.67) $\times
10^{7}$ cm for SGR 1806-20.

Depending on the size and mass of the clumps near $R_{\rm LC}$, X-ray bursts
with different energies can be produced. Following Equation (2), we can
calculate the upper limit of the clump size at the light cylinder. Figure 4
shows the breakup distance as a function of the clump mass. The dotted,
dashed, and dash-dotted lines correspond to MgSO3 clumps ($\rho_{\rm
cl}=5\rm\,g\,cm^{-3}$) with cohesive strengths of $C=1$ kPa, 1 MPa, and 10
MPa, respectively. These $C$ values are typical for rocky materials (Gundlach
& Blum, 2016; Pohl & Britt, 2020; Veras & Scheeres, 2020; Villeneuve & Heap,
2021). Here, an internal friction angle of $\phi=45^{\circ}$ is used for rocky
materials (e.g., Holsapple & Michel, 2008; Bareither et al., 2008; Villeneuve
& Heap, 2021). The solid line represents iron clumps ($\rho_{\rm
cl}=8\rm\,g\,cm^{-3}$), for which $C=50$ MPa (Ostrowski & Bryson, 2019; Pohl &
Britt, 2020) and $\phi=37^{\circ}$ (Jiang et al., 2018) are adopted. The
horizontal lines represent the light cylinder radius ($R_{\rm LC}$) of NS with
a spin period of 11.79 s (1E 1841-045, Dib & Kaspi (2014)), 7.55 s (SGR
1806-20), and 3.245 s (SGR 1935+2154), respectively. From Fig. 4, we see that
the mass range of the clumps that safely enter $R_{\rm LC}$ is up to $\sim
10^{22}$ g, which satisfies the energy budget of X-ray bursts. We notice that
$R_{\rm LC}\sim 10^{10}$ cm is much larger than $R_{\rm A}$. But this is not a
problem in our case, since the clump can effectively lose its angular momentum
due to perturbation when it is still far from the light cylinder.

The energies of the X-ray bursts produced through the interactions between the
clumps and the NS can be calculated from Equation (5). The results are plotted
in Panel (a) of Fig. 5. It shows that the energetics can be up to $\sim
10^{43}$ erg, which is large enough to account for the energies of the short
X-ray bursts observed from SGR 1935+2154 and SGR 1806-20.

Figure 4: Breakup distance as a function of the clump mass. The clump is a
rocky (MgSO3) object with a density of $\rho_{\rm cl}=5\rm\,g\,cm^{-3}$ and an
internal friction angle of $\phi=45^{\circ}$. The dotted, dashed, and dash-
dotted line corresponds to the cohesive strength of $C=$ 0.1, 1, and 10 MPa,
respectively. The solid line represents an iron clump with $\rho_{\rm
cl}=8\rm\,g\,cm^{-3}$, $C=50$ MPa, and $\phi=37^{\circ}$. The horizontal lines
represent the light cylinder radii for NSs with different spin periods.
Figure 5: Energy (Panel a) and duration (Panel b) of X-ray bursts as a
function of the clump mass. In Panel (a) the dashed and solid lines correspond
to an energy transforming efficiency of $\eta=0.5$ and $\eta=10^{-4}$,
respectively. In Panel (b) the dotted and dashed lines represent the MgSO3
clumps with a cohesive strength of $C=1$ kPa and $C=1$ MPa, respectively. The
solid line corresponds to iron clumps with $C=50$ MPa.

The burst duration can be calculated as (e.g., Dai, 2020)

$\Delta t=\frac{12r_{\rm cl}}{5}\left(\frac{r_{\rm
str}}{GM_{\star}}\right)^{1/2},$ (6)

where $r_{\rm str}$ is the breakup distance of the clumps given by Equation
(2). Here, a relation of $r_{\rm cl}=(3m_{\rm cl}/4\pi\rho_{\rm cl})^{1/3}$ is
adopted. According to Equation (6), the burst duration can range from a few
milliseconds to seconds, as shown in Panel (b) of Fig. 5. We see that the
burst duration is also compatible with the observations.

##  4 Comparison with observations

The model described in Section 3 can explain the periodical X-ray bursts
observed from SGRs. Here we confront the theoretical burst energy, duration,
period, and activity window with the observations.

###  4.1 SGR 1935+2154

Figure 6 illustrates the burst energy as a function of duration for SGR
1935+2154. The observational data points of SGR 1935+2154 are shown by
assuming a distance of 9 kpc for the source (Lin et al., 2020b, a; Cai et al.,
2022b, a). The solid lines are calculated for different $C$ and $\eta$ values.
We see that the energy budgets can be easily satisfied when the two parameters
are evaluated in reasonable ranges.

The periodicity and activity window of the X-ray bursts from SGR 1935+2154 can
also be explained. From Fig. 3, we see that when we take the orbit period as
$P_{\rm orb}=238$ days and the upper limit of the period difference as $\Delta
P_{\rm orb,up}^{\rm cl}=\Delta P_{\rm orb}^{\rm cl}\approx 75$ days, then the
arrival time difference is $\Delta t_{\rm trav}=150$ days. This agrees well
with the observed $\sim$150 day activity window of the SGR.

Figure 6: X-ray bursts from SGR 1935+2154 plotted on the energy-duration
plane. The orange (Lin et al., 2020a), blue (Lin et al., 2020b), and green
(Cai et al., 2022b, a) dots represent the bursts observed by various groups.
The lines of different styles are the theoretical results calculated by taking
different values for the cohesive strength ($C$) and energy transforming
efficiency ($\eta$).

###  4.2 SGR 1806-20

More than 3000 X-ray bursts from SGR 1806-20 have been detected (see Göǧüş et
al. (2000); Prieskorn & Kaaret (2012); Kırmızıbayrak et al. (2017) and
references therein). Detailed spectroscopic analyses were performed on some
bursts (Atteia et al., 1987; Kouveliotou et al., 1987; Fenimore et al., 1994;
Kırmızıbayrak et al., 2017). The observation data of the Rossi X-ray Timing
Explorer (RXTE) show that the observed fluence ranges between $\sim 10^{-10}$
and $10^{-7}$ erg cm-2(Kırmızıbayrak et al., 2017). Similarly, the fluence of
the X-ray bursts detected by the International Cometary Explorer (ICE) is in
the range $\sim 10^{-8}$ — $10^{-5}$ erg cm-2 (Atteia et al., 1987;
Kouveliotou et al., 1987; Fenimore et al., 1994; Göǧüş et al., 2000). We
notice that the fluence of the ICE bursts is higher than that of the RXTE
bursts by about two orders of magnitude. This may be due to the fact that ICE
has a much higher sensitivity, meaning that weaker bursts can be recorded.

Figure 7 plots the burst energy as a function of the duration. The isotropic
energy of each burst is calculated by taking the distance as 8.7 kpc (Bibby et
al., 2008). The data points correspond to the bursts observed by RXTE, whose
energy ranges between $\sim 10^{37}$ and $10^{39}$ erg. The shaded area
illustrates the energy range of the X-ray bursts detected by ICE. We see that
the energetics of the observed X-ray bursts can be explained well by our
model.

Most of the X-ray bursts from SGR 1806-20 are concentrated near the phase of
$\sim$ 0.58 (Zhang et al., 2021a). However, the bursts could also appear at
all the period phases. In other words, the periodicity of SGR 1806-20 is not
as strict as that of SGR 1935+2154. The reason for such a relatively poor
periodicity may be that the system has a small $r_{\rm p}$, which can
effectively smear the periodicity to some extent (see Section 3).

Figure 7: X-ray bursts from SGR 1806-20 plotted on the energy-duration plane.
The black dots denote the observational data (Kırmızıbayrak et al., 2017). The
lines are the theoretical results calculated by taking different values for
the cohesive strength ($C$) and energy transforming efficiency ($\eta$). The
shaded area represents the energy range of the bursts detected by the ICE
detector (Göǧüş et al., 2000).

##  5 Conclusions and discussion

In this study, the periodicity and other properties of repeating X-ray bursts
from two magnetars are explained by the interaction between a NS and its
close-in planet. In our model, a rocky planet moves around a NS in a highly
eccentric orbit. The planet is partially disrupted every time it passes
through the periastron of the orbit due to the tidal force of its compact
host. During the process, clumps ranging from a few kilometers to $\sim$ 100
km are produced. The clumps can lose their angular momentum under the
influence of the remnant planet and fall toward the NS. They are further
disrupted and captured by the strong magnetic field when they enter the
magnetosphere of the magnetar and/or NS. The interaction between the NS and
the clumps produces a series of X-ray bursts. We show that the mass of the
clumps that can enter the light cylinder radius is up to $\sim 10^{22}$ g,
which is mainly determined by the clump’s shearing strength. The energy of the
X-ray bursts produced in this way can be as high as $\sim 10^{43}$ erg, and
their durations typically range from a few milliseconds to seconds. All these
features are consistent with observations. The bursts will show a clear
periodicity due to the orbital motion of the planet. The active window of the
two SGRs can also be explained (Zhang et al., 2021a; Zou et al., 2021; Xie et
al., 2022).

It was reported that there were no bursts during some of the periods (Zhang et
al., 2021a; Zou et al., 2021). In our framework, this can be caused by two
factors. First, the asymmetry of mass loss can alter the orbit. In the
planet’s subsequent passage through the periastron, the mass loss would not
occur if $r_{\rm p}$ were larger than the maximum partial disruption distance
($r_{\rm td,max}$). As a result, one would not be able to observe any bursts
from the system. Second, the Kozai-Lidov effect may be very complicated if the
system contains more than one planet. Multi-planet systems have been
identified in a few cases, such as PSR 1257+12 (Wolszczan & Frail, 1992), PSR
B0943+10 (Suleymanova & Rodin, 2014), and PSR J1807-2459 (Ransom et al., 2001;
Ray & Loeb, 2017). If the parameters of such systems satisfy some special
conditions, rapid eccentricity oscillation may occur due to the Kozai-Lidov
effect, causing complicated variations in the periastron distance ($r_{\rm
p}$). There would be no mass loss if $r_{\rm p}>r_{\rm td,max}$ during the
oscillation process, and thus no bursts would occur.

There is no clear correlation between the burst energy and waiting time for
the X-ray bursts from SGR 1806-20 (Göǧüş et al., 2000). This is consistent
with the expectation of self-organized criticality (Katz, 1986; Bak et al.,
1987). In our model, the waiting time is simply the interval time between two
successive collisions, which depends on the spatial distribution of in-falling
clumps. The breakup distance of the clumps is generally determined by their
composition, but the breakup procedure itself is a random process that leads
to a stochastic distribution of the size and in-fall time for the sub-clumps.
As a result, there is no correlation between the burst energy and the waiting
time in our framework, which agrees well with the observations of SGR 1806-20.

For SGR 1806-20, a giant flare, whose total energy release was $4.01\times
10^{46}$ erg, was observed on 27 December 2004 (Hurley et al., 2005). Such an
energetic flare is a rare event and one that had only been observed from this
source once before. If it were due to a collision event, then the clump mass
would be several times $10^{26}$ g. In the partial tidal disruption process
considered here, the generation of such a massive clump is possible (e.g.,
Malamud & Perets, 2020b, a). Considering the collisions between clumps, the
possibility of some large objects being scattered toward the host star exists,
but the probability is low and it should be a rare event (e.g., Cordes &
Shannon, 2008). If the clump is composed of materials with a high cohesive
strength, it can retain a high mass before arriving at the light cylinder and
will eventually collide with the NS and/or SS to produce a giant flare (e.g.,
Zhang et al., 2000; Usov, 2001). Therefore, giant flares are rare events in
our framework, but they could reoccur in the future.

It is interesting to note that periodic X-ray flares are observed from Jupiter
(Yao et al., 2021), which may be evidence that X-ray bursts can be generated
via the interaction of external materials with a planet in the Solar System.
It was argued that the charged ions that originate from the gas spewed into
space due to giant volcano events on Jupiter’s moon (Io) could flow onto
Jupiter along magnetic field lines, leading to an energy release in the form
of X-rays444https://phys.org/news/2021-07-scientists-year-mystery-jupiter-x-
ray.html (Cowley & Bunce, 2001; Yao et al., 2021). Such phenomena could be
universal and present across many different environments in space (Dunn et
al., 2017; Yao et al., 2021; Mori et al., 2022). The X-ray bursts generated
through the interaction of a NS with clumps disrupted from a planet are to
some extent similar phenomena.

Finally, we would like to mention that the dynamics of the clumps and the
evolution of their angular momentum is highly complicated. Even in the
asteroid-disruption explanation for the white dwarf pollution, it is still
unclear how disrupted asteroids ultimately lose enough angular momentum to
fall onto the white dwarf. There is a similar issue for the partial TDEs.
Self-intersection shocks arising from stream–stream collisions, which
themselves arise from relativistic apsidal precession, might play a role in
the dissipation. But this could be inefficient in some circumstances. In our
cases of partial disruption, the remnant planet may act as a third object
responsible for dissipation. Nevertheless, it is still possible that in some
cases the clumps may simply orbit around the NS many times, instead of
producing prompt accretion and X-ray bursts.

###### Acknowledgements.

We would like to thank the anonymous referee for helpful suggestions that led
to significant improvement of our study. This work was Sponsored by the
Natural Science Foundation of Xinjiang Uygur Autonomous Region (No.
2022D01A363), the National Natural Science Foundation of China (Grant Nos.
12033001, 12288102, 12273028 12041304, 12233002, 12041306), the Natural
Science Foundation of Xinjiang Uygur Autonomous Region (No. 2023D01E20), the
Major Science and Technology Program of Xinjiang Uygur Autonomous Region (Nos.
2022A03013-1, 2022A03013-3), National Key R&D Program of China
(2021YFA0718500), National SKA Program of China No. 2020SKA0120300, the
Tianshan Talents Training Program (Scientific and Technological Innovation
Team). YFH acknowledges the support from the Xinjiang Tianchi Program. AK
acknowledges the support from the Tianchi Talents Project of Xinjiang Uygur
Autonomous Region and the special research assistance project of the Chinese
Academy of Sciences (CAS). This work was also partially supported by the
Operation, Maintenance and Upgrading Fund for Astronomical Telescopes and
Facility Instruments, budgeted from the Ministry of Finance of China (MOF) and
administrated by the CAS, the Urumqi Nanshan Astronomy and Deep Space
Exploration Observation and Research Station of Xinjiang (XJYWZ2303).

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