# The variable radio jet of the accreting neutron star the Rapid Burster

J. van den Eijnden1, D. Robins122footnotemark: 2, R. Sharma122footnotemark: 2,
C. Sánchez-Fernández2, T. D. Russell3<EMAIL_ADDRESS>authors contributed equally to this work N. Degenaar4, J. C. A. Miller-Jones5,
and T. Maccarone6
1Department of Physics, University of Warwick, Coventry CV4 7AL, UK
2 Science Operations Department, European Space Astronomy Centre (ESA/ESAC),
Madrid, Spain
3 INAF, Istituto di Astrofisica Spaziale e Fisica Cosmica, Via U. La Malfa
153, I-90146 Palermo, Italy
4 Anton Pannekoek Institute for Astronomy, University of Amsterdam, Amsterdam,
The Netherlands
5 International Centre for Radio Astronomy Research, Curtin University, Perth,
Western Australia, Australia
6 Department of Physics and Astronomy, Texas Tech University, Lubbock, TX, USA

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The Rapid Burster is a unique neutron star low-mass X-ray binary system,
showing both thermonuclear Type-I and accretion-driven Type-II X-ray bursts.
Recent studies have demonstrated how coordinated observations of X-ray and
radio variability can constrain jet properties of accreting neutron stars –
particularly when the X-ray variability is dominated by discrete changes. We
present a simultaneous VLA, Swift, and INTEGRAL observing campaign of the
Rapid Burster to investigate whether its jet responds to Type-II bursts. We
observe the radio counterpart of the X-ray binary at its faintest-detected
radio luminosity, while the X-ray observations reveal prolific, fast X-ray
bursting. A time-resolved analysis reveals that the radio counterpart varies
significantly between observing scans, displaying a fractional variability of
$38\pm 5$%. The radio faintness of the system prevents the robust
identification of a causal relation between individual Type-II bursts and the
evolution of the radio jet. However, based on a comparison of its low radio
luminosity with archival Rapid Burster observations and other accreting
neutron stars, and on a qualitative assessment of the X-ray and radio light
curves, we explore the presence of a tentative connection between bursts and
jet: i.e., the Type-II bursts may weaken or strengthen the jet. The former of
those two scenarios would fit with magneto-rotational jet models; we discuss
three lines of future research to establish this potential relation between
Type-II bursts and jets more confidently.

###### keywords:

accretion: accretion discs – stars: individual (MXB 1730-335) – stars: neutron
– X-rays: binaries – radio continuum: transients – X-rays: bursts

††pubyear: 2023††pagerange: The variable radio jet of the accreting neutron
star the Rapid Burster–The variable radio jet of the accreting neutron star
the Rapid Burster

##  1 Introduction

Jets are a fundamental component of a wide range of accretion-driven and
cataclysmic transients in the universe, from, e.g., forming stars (see e.g.,
Pudritz et al., 2012; Frank et al., 2014, for reviews) and active galactic
nuclei (Blandford et al., 2019) to gamma-ray bursts (Kumar & Zhang, 2015) and
compact object mergers (Combi & Siegel, 2023). Different jet-launching objects
may provide complementary constraints on the physics of jet formation,
collimation, and impact, by measuring the dynamic, spectral, and geometric
properties of the jet across various time and size scales. Within the Milky
Way, relativistic jets may be studied in depth in X-ray binaries: systems
wherein a compact object accretes from a non-degenerate or white dwarf donor
star. Low-mass X-ray binaries (LMXBs), where the donor star is typically
defined to be $<1M_{\odot}$, have historically been the most common target for
such studies: their relative proximity and time scales of variability – from
sub-second variability in their accretion flow to weeks to months-lasting
outbursts – provide the opportunity to study the evolution of their jets
across accretion rate and accretion flow state.

The population of known Galactic LMXBs is divided between systems hosting a
black hole or a neutron star, as the compact object (Bahramian & Degenaar,
2023; Avakyan et al., 2023; Fortin et al., 2024), with the latter forming the
majority. Both types of LMXBs are known to launch jets, dominant at the low-
frequency end of the spectral energy distributions (radio and mm bands),
albeit with different typical luminosities (Fender & Hendry, 2000; Migliari &
Fender, 2006; van den Eijnden et al., 2021). Their ‘compact jets’ show a
coupling to the accretion flow, observed as a correlation between accretion-
driven X-ray and the jet-driven radio luminosities (Hannikainen et al., 1998;
Corbel et al., 2003; Gallo et al., 2003; Migliari & Fender, 2006; Tudor et
al., 2017). In black hole systems, this correlation is found to be $\sim 20$
times radio brighter than in neutron star LMXBs at the same X-ray
luminosity(Fender & Kuulkers, 2001; Gallo et al., 2018). Another essential
difference between both types of LMXBs is the presence of a solid surface and
potential rotating stellar magnetic field in neutron stars: both this surface
and magnetosphere interact with and affect the accretion flow, especially in
the inner regions where the radio jet is thought to originate (Blandford &
Payne, 1982; Parfrey et al., 2016).

Coordinated radio and X-ray monitoring throughout LMXB outbursts has revealed
a wide range of phenomenological properties of the coupling between accretion
flow and jet. These properties include the aforementioned X-ray – radio
luminosity correlation, radio flaring, and the jet quenching observed in soft-
state black hole LMXBs (Vadawale et al., 2003; Fender et al., 2004; Russell et
al., 2020). Additionally, jet ejecta launched during periods of radio flaring
have been revealed systematically (Mirabel & Rodríguez, 1994), particularly in
recent years (Russell et al., 2019; Carotenuto et al., 2021; Wood et al.,
2021). Such studies are typically performed by conducting repeated
observations at a cadence of days to weeks, which allows for tracking the
average radio and X-ray behaviour between the different observing epochs. In
recent years, however, these studies have been complemented by low-frequency
variability studies at higher time resolution (e.g., Tetarenko et al., 2021;
van den Eijnden et al., 2020), opening a complementary avenue to study the
geometry and dynamics of LMXB jets and moving beyond the small number of
radio-bright sources where such studies were performed earlier (e.g., Pooley &
Fender, 1997).

In particular, coordinated studies tracking both the accretion flow and jet at
high time resolution have yielded new insights into the propagation of
accretion variability into the outflow. Observations of a set of accreting
stellar-mass black hole systems, have revealed variability in compact jets as
they respond to mass-accretion-rate fluctuations observed in X-rays (Cyg X-1
and MAXI J1820+070; Tetarenko et al., 2019, 2021) and in the transient jets
launched by V404 Cyg (Tetarenko et al., 2017, 2021). In the neutron star LMXB
Swift J1858.6-0814, Vincentelli et al. (2023) presented the detection of a jet
response, measured in dual-band radio observations, to instabilities in the
accretion flow, as probed by fast IR observations. In particular, assuming the
launch of discrete ejecta to coincide with each transition between different
stages of the accretion flow instability, Vincentelli et al. (2023) were able
to reproduce the observed radio variability with a superposition of four
launch epochs of outflowing ejecta. Their same approach was also shown to
reproduce the jet variability observed in the black hole LMXB GRS 1915+105,
which undergoes the same accretion flow instabilities (Pooley & Fender, 1997).

The key to these multi-band variability studies lies in the combination of
spectral and timing information, tracking outflowing material as it moves down
the jet and shifts its emission to lower frequencies (Blandford & Königl,
1979). The high-quality, multi-wavelength data sets required for these studies
have restricted them predominantly to black hole systems. Apart from Swift
J1858.6-0814, no other neutron star LMXB has shown accretion instabilities
that could be associated with jet ejecta (Vincentelli et al., 2023). The
relative faintness of the compact jets from neutron star LMXBs has furthermore
prevented detections of compact jet variability resulting from continuous
accretion rate fluctuations.

However, the presence of a stellar surface and rotating magnetic field in
neutron star LMXBs offers a complementary approach to track the short-time-
scale jet responses to the dynamics of the inner accretion flow. Firstly,
accreted material on the neutron star surface may ignite in a runaway
thermonuclear explosion, lasting several seconds followed by a cooling tail
lasting tens of seconds, known as a Type-I burst (e.g., Galloway & Keek,
2021). Type-I bursts generate soft X-ray photons that cool down the accretion
flow via successive inverse-Compton scatterings (Kajava et al., 2017; Sánchez-
Fernández et al., 2020) and cause brief increases of the accretion rate
(Degenaar et al., 2018; Fragile et al., 2018, 2020). The brightness and short
duration of Type-I bursts make them powerful signposts of the accretion flow
changes they cause. Through an extensive and simultaneous X-ray and radio
monitoring campaign, Russell et al. (2024) recently found that the radio jets
in two neutron star LMXBs (4U 1728-34 and 4U 1636-536) show bright radio
flares in the minutes succeeding Type-I bursts. This jet response, delayed
increasingly towards lower radio frequencies, yielded the first measurement of
the compact jet speed in a neutron star LMXB: $v=0.38^{+0.11}_{-0.08}c$, which
is consistent with the expected escape speed of a neutron star.

The rotating neutron star magnetic field may also affect the inner accretion
flow. In LMXBs, where the neutron stars typically are weakly magnetized
($B\lesssim 10^{9}$ G), these interactions most often take the form of
magnetic truncation of the inner flow (Ghosh & Lamb, 1978; Ludlam, 2024). Gas
is subsequently channelled onto the neutron star’s magnetic poles, leading to
X-ray pulsations (see e.g., Di Salvo et al., 2023, for a recent review). In
rare cases, however, neutron star LMXBs may display Type-II X-ray bursts.
While the nature of these bursts, which typically last between seconds and a
minute (Bagnoli et al., 2015), remains not fully understood, one common
explanation lies in the disk-magnetosphere interaction: in the trapped disk
model (Spruit & Taam, 1993; D’Angelo & Spruit, 2010), the accretion flow
encounters a centrifugal magnetic barrier, when its inward pressure is
balanced by the magnetic field slightly outside the radius where the disk’s
rotation equals the neutron star spin (the ‘co-rotation’ radius). As material
is trapped and accumulates, its inward pressure increases, pushing the
magnetic barrier inwards until it crosses the co-rotation radius and overcomes
the centrifugal barrier. The resulting increase in accretion rate manifests as
a Type-II bursts. This trapped disk model naturally explains the observed
relation between Type-II burst fluence and wait time between bursts (Bagnoli
et al., 2015); it is further consistent with the measurement of large disk
truncations in the only two sources showing Type-II bursts : the Rapid Burster
(MXB 1730-335; van den Eijnden et al., 2017) and the Bursting Pulsar (GRO
J1744-28; Degenaar et al., 2014). In addition, it is consistent with the brief
dips in X-ray emission seen before and after some Type-II, which could signal
the depletion of the accretion flow (Bagnoli et al., 2015; Younes et al.,
2015; Court et al., 2018). On the other hand, a range of open questions
remains regarding the nature of Type-II bursts; for instance, both the trapped
disk and alternative models have difficulty to explain the rarity of the Type-
II bursts amongst the neutron star LMXB population (see e.g. Lyutikov, 2023,
for a recent discussion). We therefore refer the reader to Bagnoli et al.
(2015) for a more extensive introduction into Type-II bursts, including
alternative models.

During Type-II bursts, observational evidence suggests that the accretion rate
increases significantly (Bagnoli et al., 2015; Court et al., 2018), consistent
with the trapped disk model. This change in accretion rate may, subsequently,
lead to observable changes in the jets launched by the LMXB. The nature of
this response depends on the mechanism underlying the neutron star jet launch
itself: in the classic Blandford & Payne (1982) model, the rotating inner
accretion flow is responsible for tangling up the toroidal disk magnetic
field, leading to the launch of jets. In more recent models, introduced by
Parfrey et al. (2016) and explored numerically in Das et al. (2022), Parfrey &
Tchekhovskoy (2023), Murguia-Berthier et al. (2024), and Das & Porth (2024),
the spin and magnetic field of the neutron star instead provide the jet power,
as the accretion flow opens up field lines to launch an outflow. In the former
case, the significant truncation of the inner disk in between bursts may lead
to a weaker jet, that brightens during bursts as more material reaches the
jet-launch regions. In the latter type of model, the swift accretion of
material trapped by the centrifugal barrier in between bursts may reduce the
gas influx and weaken the jet; alternatively, it may not affect the jet
substantially, if the power and matter are provided over a range of radii
beyond the barrier. The detection of a potential jet response to Type-II
bursts can therefore constrain both properties of the jet, such as its speed,
and inform the underlying launch mechanism.

Both the Rapid Burster and Bursting Pulsar are transient LMXBs. The radio
counterpart of the Bursting Pulsar is not known, and due to its location close
to the Galactic Centre and relatively long recurrence time between outbursts
(five outbursts since 1995), limits on its radio emission are shallower than
for other neutron star LMXBs (Russell et al., 2017; van den Eijnden et al.,
2021). The Rapid Burster, on the other hand, is a prolific transient LMXB,
showing quasi-regular outbursts lasting weeks and recurring every $\sim$ 80 –
105 days (Guerriero et al., 1999). In the past $\sim 5$ decades, radio studies
of the Rapid Burster or its environment have been carried out repeatedly at
increasing sensitivity (Johnson et al., 1978; Rao & Venugopal, 1980; Grindlay
& Seaquist, 1986; Johnston et al., 1991; Tudor et al., 2022). Its radio
counterpart was first confidently identified by Moore et al. (2000). They
further assessed whether the radio counterpart responds to the Type-II bursts,
but their findings were limited by the signal-to-noise ratio in establishing
the presence of such a response. The Rapid Burster is located in the globular
cluster Liller 1, at a distance of 7.9 kpc (Valenti et al., 2010). Due to this
crowded stellar environment and high absorption, no optical or infrared
counterpart of the LMXB is known (Homer et al., 2001).

Making use of the quasi-predictable outburst occurance and evolution, we
designed a coordinated radio and X-ray campaign of the Rapid Burster to
investigate the effect of Type-II bursts on its radio jet at unprecedented
sensitivity. In this coordinated campaign on 19 March 2020, the Karl G. Jansky
Very Large Array (VLA), the Neil Gehrels Swift Observatory (Swift; Gehrels et
al., 2004), and the INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL;
Winkler et al., 2003) observed the LMXB simultaneously. In this paper, we
present the radio and X-ray results of this campaign and search for a jet
response to Type-II bursts, as a probe of both the neutron star jet launching
and origin of Type-II bursts.

##  2 Observations and data analysis

###  2.1 2020 Campaign: VLA, Swift, and INTEGRAL

To study the time-averaged and variable radio jet of the Rapid Burster, we
carried out a coordinated, simultaneous observing campaign with the VLA,
Swift, and INTEGRAL observatories. The observations were taken on 19 March
2020, where multi-wavelength data were collected simultaneously between
12:11:48 and 14:14:17 UTC, with the exception of a number of gaps for radio
calibration and X-ray Earth occultation (see below). This overlap between
observatories was limited by the duration of the radio observation.

Triggered by the potential and expected onset of an outburst in all-sky
monitoring by the Monitor of All-Sky X-ray Image (MAXI; Matsuoka et al., 2009)
aboard the International Space Station (ISS), Swift monitored the Rapid
Burster (target ID 31360) with six short ($\sim 0.4-0.5$ ks) observations
between 3 and 19 March 2020. These observations confirmed the outburst and
motivated the scheduling of the coordinated epoch, optimized to catch the
target in a phase with a large number of Type-II bursts. During this
coordinated epoch, Swift observed the Rapid Burster in two consecutive
exposures separated by an Earth occultation: ObsIDs 00031360171 and
00031360172, each lasting 1.6 ks and starting at 12:06:34 and 13:41:36 UTC,
respectively. The X-ray Telescope (XRT) was employed in WT mode.

We employed the online Swift XRT products
pipeline111https://www.swift.ac.uk/user_objects/ (Evans et al., 2007) to
extract a combined, time-averaged X-ray spectrum from ObsIDs 00031360171 and
00031360172. We also extracted a light curve at a 1-second time resolution
between 1 and 10 keV, in order to determine the presence of Type-II (or
Type-I) bursts and measure their start and end times. Type-II bursts are
mostly achromatic below 10 keV, as the increase in mass accretion rate
predominantly affects the spectral normalisation (Bagnoli et al., 2015);
therefore, we do not extract separate burst and non-burst spectra. When
analysing the X-ray spectrum, as described in Section 3.1, we used xspec
v12.13.0c (Arnaud, 1996). We assumed abundances and cross-sections from Wilms
et al. (2000) and Verner et al. (1996), respectively, when accounting for
interstellar absorption. We quote all measured X-ray fluxes and spectral
parameters in this paper at their $1\sigma$ uncertainties; all reported fluxes
are corrected for absorption.

We observed the Rapid Burster with the VLA between 12:00:00 and 14:15:37 UTC
on 19 March 2020 under program 20A-172. The first scan of the Rapid Burster
started at 12:11:48. In total, we obtained 11 scans of the target field, each
lasting 519 seconds, with the final scan finishing at 14:14:17. As primary and
secondary calibrators, we observed 3C286=J1331+305 and J1744-3116,
respectively. In order to optimize the simultaneous coverage in radio and
X-rays, we observed the primary calibrator after eight target scans, during
Swift’s Earth occultation, before returning to the target. The observation was
carried out at C band in 3-bit mode, yielding 4 GHz of bandwidth between 4 and
8 GHz. The data were collected with a dump time of 5 seconds; the array was in
its C configuration. We used the Common Astronomy Software Application
v6.5.5.21 (CASA; CASA Team et al., 2022) to further analyse the data. Using a
combination of automated routines and manual inspection, we removed RFI from
the uncalibrated measurement set, before applying standard routines to perform
the calibration222Detailed flagging and calibration logs are included in this
paper’s reproduction package; see the Data Availability Statement..

To analyse the time-averaged observation, we imaged the target field using
tclean, where we used Briggs weighting with a robust parameter of zero to
optimize the balance between increased sensitivity and minimising the effects
of side-lobes in the telescope beam pattern as well as diffuse emission within
the field. Due to the faintness of the target, we did not split the
observation into multiple frequency bands. As discussed in Section 3.1, we
removed the inner baselines, covering the inner $3.5\times 10^{3}\lambda$,
from the measurement set before both the time-averaged and time-resolved
analysis: this uv-plane cut optimizes the removal of diffuse emission present
in the full image, without introducing a significant loss in sensitivity. To
measure flux densities in the image plane, we used the casa tool imfit to fit
an elliptical Gaussian source profile with its FWHM and orientation fixed to
the size of the synthesized beam ($6.07\times 1.98$ arcsec, position angle
$2.39\degree$ East of North, in the full time-averaged observation; for images
of subsets of the data, as discussed below, the synthesized beam changes
shape). The RMS sensitivity of each analysed image was measured in a source-
free region close to the target position; the time-averaged observation
reaches a sensitivity of $5.5$ $\mu$Jy.

To study the time-resolved properties of the Rapid Burster, on a range of time
scales (introduced in Section 3.2), we measured the source’s flux density in
the uv-plane via the CASA tool uvmodelfit. For this purpose, we first
identified all other sources in the time-averaged image, after removing the
inner baselines. We measured their positions and flux densities using imfit
before using the ft tool to define a model containing these background
sources. Using uvsub, we created a measurement set where these sources are
subtracted, after which we used phaseshift to shift the pointing centre of
that measurement set at the position of the Rapid Burster. The resulting
measurement set was used for the uv-plane fitting at the different employed
time resolutions. In order to assess whether any observed variability is
intrinsic to the target, we repeated all steps above to create a second
measurement set, replacing the Rapid Burster by a check source of similar flux
density (no substantially brighter sources are present in the field) located
at $17\text{h }33\text{m }10.75\text{s}$, $-33^{\rm o}24\text{' }26\text{"}$.
As discussed in Section 3.2, we also checked the validity of these uv-plane
results with image-plane comparisons.

INTEGRAL observations of the Rapid Burster were executed between 10:22:03 and
18:04:29 UTC on 19 March 2020. The observations were performed using the
hexagonal (HEX) dithering mode, which consists of one pointing on the source,
and six pointings surrounding it, arranged an hexagonal shape. The angular
distance between consecutive pointings is $2.17\degree$, and the pointing
duration 1872 seconds. We used data from the Joint European X-ray Monitor
(JEM-X) (Lund et al., 2003). JEM-X consists of two identical units X1 and X2,
is sensitive in the 3–35 keV range where it provides an angular resolution of
3$\arcmin$. The data were reduced with the INTEGRAL Offline Science Analysis
(OSA) v11.0 (Courvoisier et al., 2003), provided by the INTEGRAL Science Data
Center (ISDC). The data were analysed using standard procedures. The light
curves of the Rapid Burster were generated in in the 3–25 keV band, with a
time resolution of 5 seconds.

###  2.2 Archival radio and X-ray observations

Date | $\nu_{\rm R}$ | $F_{\rm R}$ |  $L_{\rm R}$ [5 GHz] | $R_{\rm PCA}$ | $F_{\rm X}$ | $L_{\rm X}$ | Fluence in bursts
---|---|---|---|---|---|---|---
| [GHz] | [$\mu$Jy] | [erg/s] | [cts/s] | [erg/s/cm2] | [erg/s] | [%]
Archival data
1996 Nov 6 | 8.4 | $370\pm 45$ | $(1.4\pm 0.2)\times 10^{29}$ | $2830\pm 150$ | $(7.2\pm 1.7)\times 10^{-9}$ | $(5.3\pm 1.2)\times 10^{37}$ | $0$
1996 Nov 11 | 4.9 | $190\pm 45$ | $(7.1\pm 1.7)\times 10^{28}$ | $1660\pm 100$ | $(4.2\pm 1.0)\times 10^{-9}$ | $(3.1\pm 0.7)\times 10^{37}$ | $0$
1997 Jun 29 | 4.9 | $210\pm 45$ | $(7.8\pm 2.6)\times 10^{28}$ | $3680\pm 200$ | $(9.3\pm 2.2)\times 10^{-9}$ | $(7.0\pm 1.6)\times 10^{37}$ | $0$
1997 Jul 24 | 8.4 | $<90$ | $<3.4\times 10^{28}$ | $210\pm 100$ | $(5.3\pm 2.8)\times 10^{-10}$ | $(4.0\pm 2.1)\times 10^{36}$ | $>78$
1998 Feb 19 | 8.4 | $<90$ | $<3.4\times 10^{28}$ | $360\pm 120$ | $(9.1\pm 3.7)\times 10^{-10}$ | $(6.8\pm 2.7)\times 10^{36}$ | $100$
2015 Jan 06 | 7.25 | $<13.2$ | $<4.9\times 10^{27}$ | – | – | – | –
2020 campaign
2020 Mar 19 | 6.0 | $58.1\pm 5.5$ | $(2.2\pm 0.2)\times 10^{28}$ | – | $(8.0\pm 2.5)\times 10^{-9}$ | $(6.0\pm 1.9)\times 10^{37}$ | 85
Table 1: Overview of the observation-averaged X-ray and radio properties of
the Rapid Burster in archival observations (based on Moore et al. (2000) and
Tudor et al. (2022) and the 2020 campaign (this work). The columns list the
epoch date, radio observing frequency $\nu_{\rm R}$, radio flux density
$F_{\rm R}$, inferred radio luminosity $L_{\rm R}$ at 5 GHz, RXTE/PCA count
rate (summed over five PCUs) $R_{\rm PCA}$, unabsorbed X-ray flux $F_{\rm X}$,
inferred unabsorbed X-ray luminosity $L_{\rm X}$, and fraction of fluence in
the X-ray observation detected during Type-II bursts. The X-ray flux and
luminosity are bolometric (archival data) or in the 0.5-10 keV band (2020
campaign); the former include a systematic uncertainty capturing the effect of
converting PCA rates to fluxes. Upper limits are reported at $3\sigma$. All
listed epochs are based on simultaneous X-ray and radio observations.

Since its discovery, the Rapid Burster has been the subject of multiple
coordinated X-ray and radio observing campaigns. The lower sensitivity of
these observations compared to the 2020 campaign, particularly in the radio
band, yielded relatively unconstraining limits on the response of the radio
jet to the presence of X-ray bursts. However, the time-averaged properties of
these observations, as well as information on the prominence of Type-II bursts
during these campaigns, provide crucial context for the interpretation of the
results from 2020 campaign. In our analysis, we do not include the radio
observations333We also do not include the results from Calla et al. (1980),
which claims the detection of radio bursts with flux densities of $400$-$600$
Jy, as such remarkably large flux densities have not been confirmed or
observed since. presented in Johnson et al. (1978), Rao & Venugopal (1980),
Johnston et al. (1991), and Grindlay & Seaquist (1986): these studies do not
confidently detect a radio counterpart of the Rapid Burster, nor a radio
response to its Type-II bursts. More recent studies (e.g., Moore et al., 2000)
demonstrate that, indeed, the radio sensitivity in these studies was
insufficient to detect this radio counterpart. In addition, these studies do
not present X-ray flux measurements, but at most discuss the number of Type-II
bursts, which further limits a comparison with later studies.

Moore et al. (2000) presented the detection of the Rapid Burster counterpart
using VLA, the Australia Telescope Compact Array (ATCA), and James Clerk
Maxwell Telescope/Sub-millimeter Common User Bolometer Array observations. As
we discussed in more detail later, the Rapid Burster did not show Type-II
bursts during the radio-detected epochs. Their campaign includes five epochs
with coordinated Rossi X-ray Timing Explorer (RXTE) observations, providing a
count rate measurement from its Proportional Counter Array (PCA) instrument.
To convert these reported X-ray count rate measurements to X-ray fluxes, we
adopt the conversion of PCA count rate to bolometric X-ray flux derived by
Bagnoli et al. (2015): 1 count/s/PCU equals $(1.26\pm 0.07)\times 10^{-11}$
erg/s/cm2, where the error is statistical; in our further calculations, we
also include a $23$% systematic uncertainty in the conversion, as further
derived by Bagnoli et al. (2015), by quadratically combining both
uncertainties444Generally, PCA observations can consist of up to five
Proportional Counter Units, or PCUs; in all five observations discussed here,
five PCUs were active, as shown by RXTE’s mission-long data products for the
Rapid Burster. The PCA count rates reported by Moore et al. (2000) are the sum
of all PCUs and have therefore been divided by five in our analysis.. With
this systematic uncertainty, we also aim to capture the difference between
this bolometric X-ray flux and the 0.5-10 keV X-ray flux typically calculated
for other X-ray binaries. In the radio band, three of the five coordinated
observations in Moore et al. (2000) are reported at a single frequency of
$8.4$ GHz: one detection and two non-detections. For those three epochs, we
assume a flat spectrum to convert the reported flux density (limit) to a $5$
GHz luminosity. In the other two radio epochs, with detections at both $4.9$
and $8.4$ GHz, we adopt the former, with its uncertainty, for this
calculation. The measured values from Moore et al. (2000) and derived X-ray
fluxes and X-ray/radio luminosities are listed in Table 1.

Figure 1: The observation-averaged, full-band, complete-FoV radio image of the
Rapid Burster in Liller, generated from the 2020 VLA observation. The radio
counterpart of the Rapid Burster is detected as the point source at the centre
of the image. The synthesized beam is shown in the bottom left corner. The
half-light and core radius of Liller 1 are indicated by the dashed and solid
circle, respectively (Tudor et al., 2022).

More recently, Tudor et al. (2022) presented deep ATCA observations of the
globular cluster Liller 1, as part of the MAVERIC survey of such clusters. The
observations, taken in 2015, show the absence of compact radio sources at the
core of the cluster, i.e., consistent with the position of the Rapid Burster,
down to an RMS sensitivity of $4.4$ $\mu$Jy/beam at $7.25$ GHz (combining the
5.5 and 9 GHz ATCA basebands). No pointed X-ray observations where taken
simultaneous with or close to this ATCA epoch. However, inspection of the
long-term X-ray monitoring light curve from MAXI reveals no evidence for X-ray
activity or an outburst. During quiescence, the MAXI data of the Rapid Burster
are dominated by contaminating flux from the persistent accreting neutron star
4U 1728-34, separated by an angular distance of $\sim 0.5\degree$. Therefore,
the MAXI light curve and spectrum do not provide constraining upper limits on
the Rapid Burster’s X-ray flux in quiescence. However, the lack of evidence of
an outburst – which are routinely detected by MAXI above the persistent
emission from 4U 1728-34 – and the radio limit below our 2020 VLA detection
can be used to associate the 2020 radio emission to the Rapid Burster (see
Section 3.1).

##  3 Results

###  3.1 Time-averaged analysis

In Figure 1, we show the entire field of view of the time-averaged 2020 VLA
observation of the Rapid Burster and its surroundings. A faint point source
can be identified in the centre of the image, which we will discuss in detail
below. In addition to a number of other point sources across the field, the
image also reveals apparent structure on scales larger than the image
resolution: the residual structure of the background surrounding these point
sources appears to deviate from a pure noise profile with the beam size as the
characteristic size scale. To further demonstrate the presence of such
extended emission in the field, we show three zooms of the field in Figure 2.
The left and middle panels show the image, re-generated by selecting only
baselines smaller or larger than $2\times 10^{4}\lambda$, respectively. The
left panel, as expected with reduced spatial resolution, confirms the presence
of extended emission on scales beyond the half-light radius of Liller 1
(indicated by the dashed line; Tudor et al., 2022). The middle, high-
resolution panel, instead shows only the point sources seen in Figure 1. For
completeness and comparison, the right panel shows the same zoom in the image
without any baselines cuts applied. Due to the presence of diffuse emission
visible on the shortest baselines, we apply a $3.5\times 10^{3}\lambda$
baseline cut in all further analysis of the VLA observation: we converged to
this value by trialing a range of uv-plane cuts, optimizing the trade-off
between the detection of diffuse emission and loss of sensitivity.

Figure 2: A zoom of the central $4\times 4$ arcminutes around the position of
the Rapid Burster in the time-averaged VLA observation. The three panels show
the result of imaging the measurement set with different baseline selections:
only baselines within $2\times 10^{4}\lambda$ (left), only baselines outside
of $2\times 10^{4}\lambda$ (middle), and all baselines (right; e.g., the same
as Figure 1). All three panels are shown on the same color scale. The lowest
spatial resolution image on the left, as indicated by the size of the
synthesized beam in the bottom left corner of each panel, shows the presence
of extended emission on scales beyond the half-light radius of Liller 1; this
half-light radius is indicated by the dashed circle, while the solid circle
shows the core radius of the cluster (Tudor et al., 2022).

As shown in Figure 2, a clear but faint radio point source is present at the
centre of Liller 1. Using imfit, we measure this source’s flux density as
$58.1\pm 5.5$ $\mu$Jy. We measure a source position of:

$\displaystyle\text{RA (J2000)}=17\text{h }33\text{m }24.61\text{s }\pm
0.01\text{s}$ $\displaystyle\text{Dec (J2000)}=-33^{\rm o}23\text{'
}20.1\text{" }\pm 0.6\text{"}$

where the uncertainties reflect $10$% of the beam size when imaging the field
using all baselines. This position is consistent within $1\sigma$
uncertainties with the radio position of the Rapid Burster reported in Moore
et al. (2000): the offset between the two positions is $0.3$ arcseconds in
declination, and $0$ (at the quoted number of significant digits) in right
ascension. In Table 2, we list the coordinates and 6-GHz flux densities of the
other significantly-detected point sources in the image as well. In order to
constrain the spectral index, we also image the full observation in two sub-
bands, from $4-6$ and $6-8$ GHz. We measure flux densities of $63\pm 8$
$\mu$Jy and $55\pm 8$ $\mu$Jy, respectively, where the uncertainty again
reflects the image RMS sensitivity. These values imply a poorly constrained
spectral shape of $\alpha=-0.4\pm 1.3$, consistent with the full range of
expected indices for LMXB jets.

Right Ascension | Declination | 6 GHz flux density
---|---|---
17h 33m 24.61s $\pm$ 0.01s |  $-$33$\degree$ 23’ 20.1" $\pm$ 0.6" |  $58.1\pm 5.5$ $\mu$Jy
17h 33m 43.50s $\pm$ 0.01s |  $-$33$\degree$ 23’ 48.3" $\pm$ 0.6" |  $155.5\pm 5.5$ $\mu$Jy
17h 33m 30.44s $\pm$ 0.01s |  $-$33$\degree$ 22’ 41.4" $\pm$ 0.6" |  $42.5\pm 5.5$ $\mu$Jy
17h 33m 25.92s $\pm$ 0.01s |  $-$33$\degree$ 20’ 08.9" $\pm$ 0.6" |  $40.1\pm 5.5$ $\mu$Jy
17h 33m 25.51s $\pm$ 0.01s |  $-$33$\degree$ 24’ 49.0" $\pm$ 0.6" |  $31.5\pm 5.5$ $\mu$Jy
17h 33m 24.11s $\pm$ 0.01s |  $-$33$\degree$ 23’ 21.4" $\pm$ 0.6" |  $18.7\pm 5.5$ $\mu$Jy
17h 33m 11.66s $\pm$ 0.01s |  $-$33$\degree$ 25’ 38.4" $\pm$ 0.6" |  $77.6\pm 5.5$ $\mu$Jy
17h 33m 10.70s $\pm$ 0.01s |  $-$33$\degree$ 24’ 25.5" $\pm$ 0.6" |  $80.4\pm 5.5$ $\mu$Jy
17h 33m 08.59s $\pm$ 0.01s |  $-$33$\degree$ 20’ 13.6" $\pm$ 0.6" |  $127.3\pm 5.5$ $\mu$Jy
17h 33m 01.16s $\pm$ 0.01s |  $-$33$\degree$ 22’ 21.7" $\pm$ 0.6" |  $96.1\pm 5.5$ $\mu$Jy
Table 2: The significantly detected point sources in the 20A-172 VLA
observation of the Rapid Burster in Liller 1. The Rapid Burster is listed as
the first source. The flux densities are reported at 6 GHz. The positional
uncertainties reflect $10\%$ of the beam.

However, despite this positional agreement, the 2020 VLA observation alone
does not necessarily imply that the radio source corresponds to the Rapid
Burster: the centre of Liller 1 presents a dense environment, where lower
frequency and lower resolution observations have revealed emission that likely
originates from an unresolved population of steep-spectrum radio pulsars
(Fruchter & Goss, 1995). The source revealed by the VLA in 2020 is fainter
than all earlier radio detections of the Rapid Burster, and lies below the
upper limits of the earlier radio non-detections discussed in Moore et al.
(2000). Therefore, we need to consider the scenario that this faint source
corresponds to a persistent, unresolved population of radio sources in the
cluster. As discussed in Section 1, a deep ATCA observation, taken in between
outbursts in 2015, did not detect any radio emission from the core of Liller 1
down to a $3\sigma$ limit of $13.2$ $\mu$Jy. This limit, significantly below
the 2020 VLA detection and at similar frequency, rules out this scenario of an
unresolved source population; we therefore conclude that this VLA source is
the counterpart of the Rapid Burster, observed at its faintest detected radio
luminosity.

To measure the X-ray flux during the 2020 VLA observation, we fit the time-
averaged Swift/XRT spectrum observed simultaneously with the radio
observation. We find that out of single component models, such as an absorbed
thermal or power law spectrum, an absorbed accretion disc model (tbabs*diskbb)
provides the best fit with $\chi^{2}_{\nu}=1040/886=1.18$. Models with two
simple components, however, provide statistically superior fits, with the best
fit provided by a tbabs*(bbody+po) model at $\chi^{2}_{\nu}=957/884=1.08$. As
we aim to use the Swift/XRT spectrum only to obtain a measurement of the X-ray
flux, we refer the reader to earlier works for a more in-depth discussion on
X-ray spectral modelling in the Rapid Burster, making use of spectra with
longer exposures and broader energy coverage (see e.g. van den Eijnden et al.,
2017, and references therein). For this best-fit model, we find an absorbing
column density of $N_{H}=(6.8\pm 0.4)\times 10^{22}$ cm-2, black body
temperature and normalizations of $T_{\rm BB}=1.84\pm 0.04$ keV and $N_{\rm
BB}=(2.7\pm 0.1)\times 10^{-2}$, and a power law index and normalization of
$\Gamma=3.7\pm 0.3$ and $N_{\rm po}=2.1^{+0.8}_{-0.6}$. Using the convolution
model cflux, we measure an unabsorbed 0.5-10 keV X-ray flux of $(8.0\pm
2.5)\times 10^{-9}$ erg/s/cm2, corresponding to a luminosity of $(6.0\pm
1.9)\times 10^{37}$ erg/s.

Figure 3: The X-ray – radio luminosity plane for neutron star and black hole
X-ray binaries. The open squares and open stars show hard-state atoll neutron
stars and accreting millisecond X-ray pulsars, respectively; the open circles
show black holes. The Rapid Burster, based on archival and 2020 data, is shown
as the filled pentagons. The dashed and dash-dotted lines show the black hole
and neutron star correlations as determined by Gallo et al. (2006) and Gallo
et al. (2018) Strikingly, the 2020 Rapid Burster observation (during a
prolific bursting state), shows the source at its faintest detected radio
luminosity, while its X-ray luminosity is similar to the radio-brighter epochs
presented in Moore et al. (2000, when no bursting was observed). The
comparison samples of black holes and neutron stars were collated by Bahramian
& Rushton (2022).

In Figure 3, we show the X-ray – radio luminosity diagram for LMXBs, based on
the database by Bahramian & Rushton (2022). Capturing the Rapid Burster’s
observation-averaged behaviour, we plot the archival points, adapted from
Moore et al. (2000) and listed in Table 1, as the five purple pentagons (due
to the lack of X-ray information, we do not include the data from Tudor et
al., 2022). The 2020 campaign is plotted as the green pentagon. The three
archival radio detections are consistent with the X-ray brightest hard state
neutron star LMXBs; the two archival upper limits are only consistent with the
radio-faintest neutron star LMXBs in the range between $L_{X}=5\times 10^{36}$
and $10^{37}$ erg/s. Interestingly, the Rapid Burster was significantly radio
fainter, by a factor $\sim 5$, during the 2020 epoch than during the archival
radio monitoring, despite a similar average X-ray luminosity.

Figure 4: Radio and X-ray light curves of the Rapid Burster and the check
source from the coordinated 2020 campaign. The top panel shows the radio light
curve of the Rapid Burster, at three time resolutions: one and four
measurements per scan (filled squares and open circles, respectively), and
eight measurements per scan (the shaded band, indicating the region with the
$1\sigma$ flux density uncertainties). The middle panel shows the radio light
curve, at the former two time resolutions (following the same marker
convention), for the check source. All radio flux densities shown in both
panels are measured in the uv-plane. The bottom panel shows the X-ray light
curves measured by Swift/XRT in black (1 second time resolution) and
INTEGRAL/JEM-X (5 second time resolution), where the latter have been
multiplied by $1.5$ to better match the vertical extent of the curves. We
identify Type-II X-ray bursts in the bottom panel as instances where the X-ray
count rate exceeds 15 and 30 counts/second, for Swift and INTEGRAL,
respectively. Those times are indicated as the shaded vertical bands in all
panels. Above the top panel, the four qualitative phases of the Rapid
Burster’s radio light curve are indicated, as introduced in Section 3.2.
Figure 5: A logarithmically-binned histogram of the Swift/XRT count rates,
showing two clear peaks due to burst and non-burst emission. The red line
indicates the burst-rate-threshold, above which the bursts first start to
contribute and then dominate the count rates.

###  3.2 Time-resolved analysis

Figure 6: The inner $0.5\times 0.5$ arcminute field of view around the Rapid
Burster (top two rows) and check source (bottom two rows). The top left panel
for both sources shows the time averaged image; the remaining eleven panels
per source show the image for each of the scans of the target field (numbers
for the top two rows as well). Both sources and all panels are shown with the
same extent and scaling of their colormaps. The panels show that the Rapid
Burster varies between $\sim 100$ $\mu$Jy and non-detection (scans 5, top
right panel), as is similarly seen in the uv-plane measurements underlying the
light curve in Figure 4. Similarly, the bottom two rows show how the check
source is always detected on top of a varying background. In all panels, the
synthesized beam is shown in the bottom left: it can, as expected, be seen to
rotate throughout the observation. The position of the source is shown with
the point in each panel. Figure 7: The difference between the flux density of
the Rapid Burster measured during all burst times and all non-burst times,
plotted as function of delay. The delay is defined as the shift in burst times
used when measuring the combined burst and non-burst flux densities. The
measured flux density differences are consistent with statistical variations
around zero.

During the coordinated 2020 observing campaign, the Rapid Burster resided in a
state of prolific bursting. In this Section, we explore the time-resolved
properties of the X-ray and radio observations. In Figure 4, we show the radio
and X-ray light curves of the Rapid Burster and the radio check source. The
upper panel shows the Rapid Burster’s radio flux density as a function of time
at three time resolutions: one, four, and eight measurements per target scan,
plotted as the squares, and open circles, and shaded band (indicating the
range included within the $1\sigma$ uncertainties), respectively. The middle
panel shows the flux densities at for the same two resolutions, but for the
check source. Finally, the bottom panel shows the Swift/XRT (black) and
INTEGRAL/JEM-X (green) X-ray light curves. The small gaps in the two radio
light curves indicate secondary calibrator scans, while the larger gap
corresponds to the primary calibrator scan; the latter was coordinated to
overlap with the Earth occultation in the Swift observation, visible as the
gap in the bottom panel.

In the X-ray light curve, a large number of Type-II bursts are evident. The
Swift/XRT light curve shows that, when the burst recurrence time allows, the
intra-burst rate slowly increases after the end of the previous burst, before
a dip preceeds the next bursts. Such behaviour, for instance seen after the
Sun constraint window, is consistent with the pre- and post-burst dips known
in both the Rapid Burster and Bursting Pulsar (Bagnoli et al., 2015; Younes et
al., 2015; Court et al., 2018). No Type-I bursts, with a characteristic
exponential cooling tail, are observed. Due to their brightness, the Type-II
bursts are easily identified by eye. After trialling several values, we find
that a $15$ cts/s and $30$ cts/s threshold for Swift and INTEGRAL,
respectively, select all bursts without mistaking stochastic variability in
the quiescent emission for bursts (the difference in driven by the larger
noise in the JEM-X light curve). We therefore adopt those values to define the
burst start and end times, and indicate these times with light vertical bands
in all panels. Most Type-II bursts occur with a relatively short recurrence
time ($\sim 1$ minute) and duration (tens of seconds); the exceptions are the
first plotted Type-II burst and the burst occurring around MJD 58927.53, both
observed by INTEGRAL: both are of longer duration and are followed by a
comparatively long period without bursting. Using the Swift burst threshold,
we measure that $\sim 85$% of the X-ray fluence in the observation arises
during the times of Type-II bursts, which occur during $\sim 26$% of the time.
We plot a logarithmic histogram of Swift/XRT fluxes in Figure 5 to further
clarity.

The radio light curve of the Rapid Burster shows variability inconsistent with
statistical variations around a constant: the $\chi^{2}_{\nu}$ compared to a
constant mean flux density is $\chi^{2}_{\nu}=2.7$ for the lowest time
resolution (one measurement per 519-second scan). In comparison, the check
source, at the same time resolution, is consistent with a constant at
$\chi^{2}_{\nu}=0.8$. At shorter time resolutions, the uncertainties on the
Rapid Burster’s flux density increase such that any apparent variability
becomes consistent with a constant. The trends of these shorter duration
measurements, however, follow the behaviour indicated by the longer time
resolution light curve, further strengthening the argument that these
variations at the lowest time resolution are physical and do not arise from
statistical variations. Following Vaughan et al. (2003), we calculate the
fractional variability in the Rapid Burster’s lowest time resolution light
curve as $F_{\rm var}=38\pm 5$%. For the other time resolutions and the check
source, we find $\chi^{2}_{\nu}<1$, implying that the variance of the light
curve is smaller than the mean square errors on the data. Therefore, this
fractional variability is consistent with zero and cannot be calculated.

The radio light curves in Figure 4 show the flux densities calculated through
the uv-plane method described in Section 2.1. In particular, the Rapid Burster
displays variations between $\sim 100$ $\mu$Jy to non-detections,
characterised by a uv-plane flux density measurement consistent with zero. The
first segment of the VLA observations, prior to the primary calibrator scan,
in particular displays a striking evolution, first decreasing from this
maximum to minimum radio level before recovering on a comparable time scale.
To confirm that this evolution is not an artefact of the analysis method, we
generate images of each individual scan, plotted for both the Rapid Burster
(top half) and the check source (bottom half) in Figure 6. The top left panel
for both sources shows the time-averaged image. The check source is detected
in each image, on top of a varying background, consistent with the uv-plane
light curve in the middle panel of Figure 4. The Rapid Burster, on the other
hand, is indeed observed to change significantly throughout the observation,
with a non-detection during the fifth target scan (top right panel).

Linking the variability in the radio light curve to individual Type-II X-ray
bursts is challenging for a host of reasons: firstly, the relatively low
signal-to-noise ratio of the full observation, corresponding to a detection
significance of $\sim 10\sigma$, causes any variability plotted for time
scales shorter than individual scans, to be statistically insignificant.
Furthermore, the majority of bursts last for less than 30 seconds. This short
duration not only reduces the amount of radio data collected per burst, but
also implies that the following non-burst period is shorter. Bursts that
follow each other on time scales shorter than the characteristic time scale of
any potential jet response, may not individually affect the jet but instead
end up in a resonance, such that their effect may instead be collective. As a
test to overcome this limitation of individual bursts, we fit the flux density
of the Rapid Burster in the uv-plane using only the combined burst times or
the combined non-burst times, as identified by the grey shaded regions in
Figure 4. We do not find a statistically significant difference between the
collective burst and non-burst intervals in the radio flux density of the
Rapid Burster: $54\pm 15$ $\mu$Jy and $59\pm 6$ $\mu$Jy, respectively. As the
radio emission originates from far down the jet, a delay is expected between
any radio jet response to the X-ray behaviour. However, systematically
shifting the start and end times of the burst and non-burst intervals with an
increasing delay, does not reveal a systematic difference at any delay time
scale either: in Figure 7, we show the flux density difference as a function
of delay, finding a curve that is consistent with statistical scatter around
zero. Given the faintness of the Rapid Burster in comparison with the
sensitivity in the cumulative burst and non-burst intervals, this negative
result is not surprising.

Turning, therefore, to a more qualitative assessment, the scan-by-scan radio
variability in Figure 4 can be summarized, broadly, in four (slightly
overlapping/continuous) phases: (i) a constant, relatively bright phase around
the source’s maximum flux density in scans 1-3; (ii) a decay phase down to
non-detection in scans 3-5; (iii) a recovery phase to maximum flux density in
scans 6-7; and finally (iv) a constant phase consistent with the observation-
averaged flux density in scans 9-11. As argued in the previous paragraph,
linking the radio variability to individual X-ray bursts is challenging.
However, it is worth exploring how these four radio phases relates to the
overall bursting behaviour in X-rays.

One possible interpretation of the radio variability is as follows: the two
brightest radio phases – phase (i) and the endpoint of the recovery in phase
(iii) – follow after the longest burst-free periods in the observation: the
burst-free segments after the first INTEGRAL burst and the after the most
fluent detected burst, centred on MJD 58927.53. The decay in phase (ii), on
the other hand, appears to follow from the transition from rapidly repeating,
short bursts (MJD 58927.51 – 58927.52) to stronger, more fluent, and more
slowly recurring bursts (after MJD 58927.52); the minimum in the light curve,
i.e. the non-detection, occurs after the strongest Type-II burst, before
recovery sets in. Finally, phase (iv) coincides with a relatively stable
bursting period, where the bursts recur with slightly longer wait times and
duration than between MJD 58927.51 – 58927.52. This potential interpretation
could tentatively be summarized as follows: X-ray bursting tends to weaken the
jet, with more fluent and slower recurring bursts having a stronger effect,
while longer periods of non-bursting lead to the recovery of the jet to its
maximum level. Such potential, tentative links would affect the radio emission
with a delay, estimated visually from the light curves, of the order of $\sim
5\times 10^{-3}-10^{-2}$ MJD, or $\sim$ 7 to 14 minutes.

An alternative scenario should be considered as well, however: the response
time in the jet may be substantially shorter, as expected when driven by the
expected escape velocity from the neutron star (e.g., $\sim 3$ minutes, as
seen in response to Type-I bursts in 4U 1728-34 by Russell et al., 2024). For
such more immediate response, we observe that the radio brightest phases
mentioned above occur during prolific X-ray bursting (Phase (i)), or during a
period lacking X-ray coverage. The radio-faintest segment, during the fifth
target scan, instead coincides with a lack of X-ray bursting. This alternative
interpretation fits with a scenario where the jet is weakened in between the
bursts.

This ambiguity in interpretation is strengthened by the lack of radio
observations during the aforementioned first long burst-free period (i.e.
after the first INTEGRAL burst) and the lack of both radio and X-ray
information on the transition between phase (iii) and (iv). We also note that,
due to the non-significance of the variability on shorter time scales, we do
not attempt to extend these lines of qualitative arguments to this faster
variability (or lack thereof). Therefore, the above scenarios should be
assessed critically in the light of all (including archival) information on
the Rapid Burster, which we will do in Section 4.

##  4 Discussion

While the unique X-ray bursting properties of the Rapid Burster have been
known for decades, their effect on the binary’s jet launching properties has
remained elusive. The main results of our 2020 investigation of this question,
using simultaneous VLA, Swift, and INTEGRAL observations, can be summarized as
follows:

  1. 1.

The Rapid Burster is detected at a time averaged 0.5–10 keV X-ray flux of
$(8.0\pm 2.5)\times 10^{-9}$ erg/s/cm2 and radio flux density of $58.1\pm 5.5$
$\mu$Jy; the latter is the faintest radio detection of the Rapid Burster to
date.

  2. 2.

The 2020 campaign coincides with a phase of prolific Type-II X-ray bursting,
with Swift detecting $85$% of its total X-ray fluence during burst times.

  3. 3.

The radio luminosity of the Rapid Burster was significantly lower in 2020 than
in the observations presented in Moore et al. (2000) at a comparable time-
averaged X-ray luminosity; interestingly, no Type-II bursts occurred during
those archival, radio-brighter observations.

  4. 4.

The radio flux density of the Rapid Burster shows significant variability
between target scans, with a fractional variability of $F_{\rm var}=38\pm 5$%,
evolving between $96\pm 16$ $\mu$Jy to non-detection. On shorter time scales
(i.e., within scans), no statistically-significant variations are found.

In this Section, we will first compare these observed properties of the Rapid
Burster with other neutron star LMXBs, focusing particularly on its
comparative radio faintness and radio variability. Secondly, we will discuss
the implications of our results and their broader context on the origin of
Type-II bursts and on jet launching; discussing the latter topic both for the
Rapid Burster specifically and in neutron star LMXBs more generally.

###  4.1 The 2020 campaign in context: historical observations and other LMXBs

As shown in the X-ray – radio luminosity diagram, the 2020 campaign reveals an
unexpectedly faint radio counterpart of the Rapid Burster. To assess this
faintness more quantitatively (particularly comparing across X-ray
luminosities), we calculate the relative position of the Rapid Burster and
other neutron star LMXBs with respect to the neutron star the X-ray – radio
luminosity correlation found by Gallo et al. (2018):

$\log L_{\rm R}-\log L_{\rm R,0}=\alpha+\beta\left(\log L_{\rm X}-\log L_{\rm
X,0}\right)\text{ ,}$ (1)

where we do not include the scatter parameter, as we intend to quantify this
scatter. We adopt the parameters from Gallo et al. (2018), who find $\log
L_{\rm R,0}=28.57$, $\log L_{\rm X,0}=36.20$, $\alpha=-0.17$ and $\beta=0.44$.
Introducing renormalized luminosities $\overline{L_{\rm i}}=L_{\rm i}/L_{\rm
i,0}$, the above Equation is equivalent to

$\log\left(\frac{\overline{L_{\rm R}}}{\overline{L_{\rm
X}}^{\beta}}\right)=\alpha\text{.}$ (2)

Calculating the left hand side of this equation for all individual
observations of neutron star LMXBs, we can assess their deviation from the
best-fit correlation with respect to the overall correlation’s logarithmic
normalization $\alpha$.

In Figure 8, we plot this quantity for the Rapid Burster, as a function of the
X-ray fluence observed during times of Type-II bursts (left panel). We also
show, using a rotated histogram, all other neutron star LMXBs in the
comparison sample in Figure 3 (right panel). For this latter histogram, we
show two realisations: a histogram calculated directly from the observed
luminosities, and one calculated from $10^{4}$ re-constructions of the
comparison sample by varying each luminosity within its uncertainty (assuming
Gaussian luminosity errors and a 10% error for luminosities originally
reported without uncertainties). As expected, the peak of this distribution is
consistent with $\alpha=-0.17$ (dotted line), as found by Gallo et al. (2018).

For the 2020 Rapid Burster observation, we directly measured the fluence
during times of Type-II bursts from the Swift light curve; for the archival
data, we used the information reported in Moore et al. (2000) regarding the
RXTE/PCA light curves: the number of Type-II bursts, their average duration,
and the average count rates during the entire observation and during the
bursts. Combining this with the exposure of each observation and assuming that
the bursts are achromatic (implying the same rate to flux conversion during
and in between bursts), we calculated the fractional fluence during burst
times.

During the 2020 observation, the Rapid Burster showed a deviation of
$\log\left(\overline{L_{\rm R}}/\overline{L_{\rm
X}}^{\beta}\right)=-0.93^{+0.07}_{-0.08}$, amongst the radio faintest
deviations seen in the larger neutron star LMXB sample. The two archival upper
limits of the Rapid Burster, instead, lie closer to the mean $\alpha$, as do
the three measurements from the archival radio detections. We therefore
conclude that the 2020 observation of the Rapid Burster was radio faint
compared to (i) archival studies of the same source without Type-II bursts;
and (ii) to the full neutron star LMXB sample, when correcting for the
sample’s dependence on X-ray luminosity.

Before turning to the Rapid Burster’s variability, we should consider several
important caveats to the above analysis. Firstly, neutron star LMXB show
significant scatter around their best-fit relation, whose slope may be
affected by a bias against publication of radio upper limits, particularly at
low X-ray luminosities. Therefore, the correction of the X-ray luminosity
dependence may be similarly biased by assuming $\beta=0.44$. Furthermore, as
Gallo et al. (2018) explore, different sub-classes of neutron star LMXB may
show different accretion flow – jet couplings. Finally, the histograms plotted
in Figure 8 are based on radio-detected neutron stars LMXBs. The low-$\alpha$
tail may therefore be artificially over-pronounced as the number low-$\alpha$
detections can be be limited by radio sensitivity. A striking example of this
behaviour is given by Aql X-1, for which (Gusinskaia et al., 2020b) present a
rapid decrease of its radio luminosity during the decay of its 2016 outburst.
The three radio non-detections of Aql X-1 during this decay show a strong
suppression compared to the correlation we applied in our analysis, with upper
limits on $\alpha$ between $-0.9$ and $-1.0$. These values, similar to the
Rapid Burster 2020 campaign, highlight these sensitivity-limited biases in our
approach. While these effects do not counter the conclusion that the Rapid
Burster was radio faint during the 2020 campaign, they should be kept in mind
when interpreting the exact value of $\alpha$.

While radio faint, the Rapid Burster showed remarkable radio variability of
its radio jet on the time scale of individual target scans. While rarely
observed (or investigated; see Section 4.3 and below) amongst atoll sources,
radio variability is regularly seen in other (classes of) neutron stars in
LMXBs and similar systems. Z sources, for instance, have long been known to
vary strongly in radio properties, changing between different branches of
their Z track (Hjellming et al., 1990a, b; Migliari & Fender, 2006) and
potentially launching transient ultra-relativistic outflows (Fomalont et al.,
2001a, b; Motta & Fender, 2019). Similar levels of fractional radio
variability to the Rapid Burster were observed in Swift J1858.6-0814 (van den
Eijnden et al., 2020), which Vincentelli et al. (2023) modelled as arising
from jet ejecta following instabilities in the inner accretion flow. Recently,
Russell et al. (2024) reported the discovery of radio jet brightenings
directly succeeding thermonuclear (Type-I) bursts in 4U 1728-34. During X-ray-
dominated states of (candidate) transitional milli-second pulsars (tMSPs),
significant and even structured radio variability has been observed: for
instance, the prototypical tMSP PSR J1023+0038 shows both radio flaring and
radio moding, sharply transitioning between low and high states which are
strongly anti-correlated with similarly-sharp X-ray moding (Bogdanov et al.,
2018). The candidate tMSP CXOU J110926.4-650224 was more recently reported to
show radio brightening throughout a coordinated X-ray and radio observation,
potentially in response to the onset of X-ray flaring (Coti Zelati et al.,
2021).

The radio variability in the Rapid Burster therefore fits within a sub-class
of accreting neutron stars: those that display a jet/outflow response,
following sudden changes in the inner accretion flow that are triggered by
instabilities in the flow itself (Swift J1858.6-0814), by track changes of
Z-sources (Penninx et al., 1988; Hjellming et al., 1990a, b), by the influence
of the neutron star’s surface (thermonuclear bursts in 4U 1728-34), or by
magnetic fields (likely playing a central role in tMSPs; see e.g., Papitto &
de Martino, 2022, for a recent review). It should be noted that the literature
contains few detailed investigations of the radio variability on short time
scales for atoll neutron stars and AMXPs that show no peculiar X-ray
variability. In the atoll systems and AMXPs where such radio variability of
their compact jet was investigated, no evidence for significant variability
was detected within radio observation (Russell et al., 2018; Gusinskaia et
al., 2020a). However, the subset of sources where such searches are performed
and reported remains small, raising the question to what extent peculiar X-ray
behaviour introduces a strong bias motivating radio variability searches. Yet,
even within this context of other accretion neutron stars, the Rapid Burster
and its radio variability remain unique: it remains the only radio-detected
source with Type-II source, allowing for a further comparison with
expectations from Type-II burst and neutron star jet models.

Figure 8: Left panel: the relative radio brightness of the Rapid Burster as a
function the fractional X-ray fluence detected during the times of Type-II
bursts. The relative radio brightness is plotted on a logarithmic scale and
defined as the ratio between the normalized radio luminosity $\overline{L_{\rm
R}}$ and the normalized X-ray luminosity $\overline{L_{\rm X}}^{0.44}$, which
captures the difference between the radio luminosity and the best-fit $L_{\rm
X}$–$L_{\rm R}$ relation for neutron star LMXBs as measured by Gallo et al.
(2018) and shown in Figure 3 (see main text for details and normalization
luminosities). The six data points show the observation-averaged data from the
five archival observations from Moore et al. (2000) and the 2022 campaign; the
shaded horizontal band shows the maximum radio luminosity observed in the per-
scan light curve of the Rapid Burster in 2020. In the latter, the fractional
X-ray fluence in Type-II bursts is ill-defined, as we calculate this for full
observations. Right panel: a vertical histogram of the same relative radio
brightness, calculated using the X-ray and radio luminosities of the
comparison sample of neutron star LMXBs from Bahramian & Rushton (2022). The
filled histogram shows this comparison sample; the unfilled histogram shows
the same data after including their uncertainties on X-ray and radio
luminosity via a Monte-Carlo approach. In both panels, this dotted line shows
the normalization value measured by Gallo et al. (2018) for neutron star
LMXBs, unsurprisingly matching the histogram peak.

###  4.2 Implications for Type-II bursts and neutron star jets

The relative position of the Rapid Burster in the X-ray – radio luminosity
diagram may be understood as either relatively X-ray bright or relatively
radio faint, or a combination of both. The X-ray bright scenario may arise if
the radio jet is coupled to the accretion flow in between Type-II bursts, but
does not substantially respond to the increased mass accretion during the
bursts. In other words, the jet is de-coupled from the accretion rate during
bursts, and calculating a mean X-ray luminosity during the observation over-
estimates the accretion rate driving the jet. In the radio-faint scenario, on
the other hand, the presence of the Type-II bursts would weaken the jet either
during or in between bursts). As a result, while the a-chromatic nature of
Type-II bursts implies that the average X-ray luminosity traces the average
mass accretion rate, the resulting jet radio luminosity is weaker.

Our observational results argue that the first scenario is unlikely. Scaling
the mean non-burst Swift rate, the non-burst X-ray luminosity of the Rapid
Burster is $(1.2\pm 0.4)\times 10^{37}$ erg/s. Even at that level, the
observed radio luminosity is substantially fainter than the mean neutron star
X-ray – radio luminosity correlation at that non-bursting X-ray luminosity.
This remains in contrast with the archival, non-bursting observations, where
the observed radio luminosity is consistent with the mean neutron star LMXB
correlation (see also Figure 8). On the other hand, the strength of this
argument is limited by the range that individual neutron star LMXBs show in
the inferred slopes of their X-ray – radio luminosity correlations: a
correlation for the Rapid Burster steeper than the best-fitting slope of the
full sample would counter this argument. However, the X-ray bright scenario
does also not account for the observed radio variability: in this scenario,
the jet is effectively de-coupled from the bursts, and may potentially not be
expected to show the significant levels of variability observed in the 2020
campaign. As discussed earlier, a larger number of radio variability
measurements in atoll sources would provide a crucial further test of this
variability argument, by fully addressing the uniqueness of such variability
amongst neutron star LMXBs.

Alternatively, the radio-faint scenario, where the jet weakens as a result of
the presence of Type-II bursts, can account for both the faintness and
variability. This scenario can fit the two possible interpretations of the
four phases introduces in the radio light curve in Section 3.2: either the jet
is weakened by bursts, and the fading and re-brightening of the jet
respectively follow an intense phase of bursting, including the most powerful
burst observed in the campaign and the longest burst-free period;
alternatively, the jet is weakened in between bursts, as the radio-faintest
segment overlaps with a burst-less period. In principle, the radio faintness
and variability may be explained by both scenarios: these two observables
reveal that the jet is not constantly launched at a constant and high
luminosity, but do not necessarily constrain whether the jet is correlated or
anti-correlated with the Type-II bursts. Therefore, causality comes into play,
through the delay that is physically expected, and routinely observed in any
radio response to X-ray variability (Tetarenko et al., 2019; Vincentelli et
al., 2023; Russell et al., 2024). A longer delay implies that the causal
burst-jet relation is more likely to be be destructive (i.e., an anti-
correlation); instead, associating the phase of radio non-detection with the
coincident burst-free period, requires a shorter time for the jet response to
reach the radio-emitting regions. We finally note that a third, but minor
feature is present in the X-ray light curve: brief pre- and post-burst dips.
Due to their relative weakness compared to the fluence changes between burst
and non-burst phases, we deem it unlikely that these dips cause a significant
effect in the radio light curve; testing the association explicitly is beyond
our current data.

It is worth emphasizing that this potential effect of the presence of Type-II
bursts – weakening or even destroying the jet – is opposite to the effect seen
in most other LMXBs. Typically, increases in accretion rate and instabilities
in the accretion flow are observed and modelled to lead to a more powerful
compact jet or to the emergence of a second type of outflow (transient ejecta,
for instance, or expanding plasmoids suggested in tMSPs; Papitto & de Martino,
2022). As shown in Figure 8, the maximum flux density reached by the system in
the scan-by-scan radio light curve does not bridge the luminosity gap with the
archival observations. This remaining faintness can be explained by the short
recurrence time scale of the bursts, not allowing the jet to fully build up
after or during successive bursts, depending the causal relation. These two
properties of the observed radio response warrant a further discussion in the
light of more specific jet and Type-II bursts models. So far, we have only
associated the Type-II bursts with an increase in accretion rate, following
model-independent empirical results. We will now briefly discuss our results
specifically in the context of magnetic gating models for the bursts.

The magnetic gating model for Type-II bursts (Spruit & Taam, 1993; D’Angelo &
Spruit, 2010) poses that the bursts are the result of a centrifugal barrier
created by the neutron star’s magnetosphere: in between bursts, the
magnetospheric radius where the accretion flow is truncated, is located
slightly outside the co-rotation radius. The resulting centrifugal barrier
leads to a gradual build-up of transferred material trapped in the inner disk,
resulting in a low overall accretion rate and an increasing ram pressure from
the flow. The latter pushes the magnetospheric radius inwards, up until the
point where it crossed the co-rotation radius and the barrier vanishes,
resulting in a burst wherein the build-up reservoir depletes and the system
returns to its original configuration. Empirically, this scenario is supported
by the observed correlation between recurrence time and burst fluence (Bagnoli
et al., 2015); moreover, X-ray reflection spectroscopic studies have revealed
a significant truncation of the inner accretion flow in between the Type-II
bursts of both the Rapid Burster and the Bursting Pulsar (Degenaar et al.,
2014; van den Eijnden et al., 2017). On the other hand, the uniqueness of the
Rapid Burster and Bursting Pulsar within the much larger LMXB population
remains poorly understood and cannot be straightforwardly explained by the
gating model. Still, the gating model arguably remains the most well-developed
and empirically-tested Type-II burst model in the literature.

Combining this magnetic gating mechanism with the first potential scenario –
the tentative inference that the jet is fainter during bursts – we can
envision a scenario where the jet is launched from regions close to or beyond
the magnetospheric radius, as the mass reservoir builds up. Given the large
observed disk truncation in both the Rapid Burster ($41.8^{+6.7}_{-5.3}$
$R_{g}$; van den Eijnden et al., 2017) and the Bursting Pulsar ($85.0\pm 10.9$
$R_{g}$; Degenaar et al., 2014), that scenario implies the jet is not launched
from the inner tens of gravitational radii around the neutron star. Such a
launch radius appears at odds with its expected location in the classical
Blandford & Payne (1982) jet model, where toroidal magnetic field lines
threading the innermost accretion flow are spun up by the flow’s differential
rotation. Instead, the central role for the neutron star magnetic field in
controlling the accretion flow dynamics is consistent with models where the
jet is powered by the spin of the neutron star, as its magnetic field lines
are opened up by the inflowing material Parfrey et al. (2016). In such models,
as more recently also demonstrated in computational simulations (Parfrey &
Tchekhovskoy, 2023; Das et al., 2022; Das & Porth, 2024; Murguia-Berthier et
al., 2024), material is launched from outside the magnetospheric radius,
regardless of its exact location. Moreover, when discussing jet quenching,
Parfrey et al. (2016) discuss how quenching is expected once the accretion
rate becomes sufficiently high and the disk reaches the neutron star surface,
such that the entire magnetosphere is dominated by accreting matter and its
ability to launch an outflow is thereby suppressed. During Type-II bursts, the
accretion rate increase may then indeed cause the jet to quench. In other
words, such magneto-rotational neutron star jet models may explain both the
launching of the jet in between bursts, and their potentially detected decay
caused by bursts. We note that these models operate differently than
propeller-type scenarios, where the rotating magnetosphere expels mass from
the binary (Illarionov & Sunyaev, 1975): a mass reservoir should instead
gradually build up beyond the magnetospheric radius in between the bursts.

In the alternative scenario, where the jet responds faster and is weaker in
between Type-II bursts, the above reasoning leads to a different conclusion.
In that case, a stronger jet is observed when material is rapidly accreting
within the co-rotation radius, therefore arguing instead for (Blandford &
Payne, 1982) type jet models. The limited time scale of the bursts may explain
the relative faintness of the jet. In this scenario, a more accurate spectral
index measurement, constraining the jet type, is of particular interest: does
the swiftness of the accretion-driven burst lead to the launch of transient
ejecta with a steep spectrum? Additionally, this scenario may suggest a
different jet-launch scenario compared to more strongly-magnetized neutron
stars, where magneto-rotational models likely operate (van den Eijnden et al.,
2021).

If future observations confirm the potential jet response to Type-II bursts,
we cannot directly conclude that this same jet model underlies all neutron
star LMXB jet launching. Importantly, the relatively strong magnetic field,
truncating and gating the accretion flow, does not play the same central role
in all accreting neutron star systems. Without the strong disk truncation,
which is rarely observed to the extent of the Rapid Burster and Bursting
Pulsar (Ludlam, 2024), the above arguments concerning the Blandford & Payne
(1982) type models disappear. Jet launching in accreting neutron stars may
instead be caused by a mixture of these various mechanisms, depending on the
dynamical influence of the magnetic field; even in the Rapid Burster itself,
when no bursts occur (i.e., at high accretion rate), different jet launch
models may be dominant.

Understanding whether all neutron star jets are launched through similar
models requires a more detailed understanding of the jet properties
themselves: moving beyond their mere detection and luminosity evolution to
constrain outflow speeds, geometries, and powers. The recent study by Russell
et al. (2024) reports the response of a radio jet to Type-I bursts, which
opens up a new method to measure such properties systematically in a large
number of Type-I bursters. In our 2020 observations, the Rapid Burster does
not reach sufficient signal-to-noise to obtain similar measurements, nor to
allow us to break up the VLA observations in different frequency bands and
track changes as they propagate down the jet. While the order-of-magnitude
time scales of the two potential burst-jet associations discussed in Section
3.2 ($7$–$14$ minutes versus significantly shorter, for a jet weakened during
or in between bursts, respectively) are similar to the response time scale in
4U 1728-34 and Swift J1858.6-0814, we cannot perform the detailed modeling
that subsequently constrains the jet’s properties: such modelling requires an
accurate delay measurement and multi-band radio monitoring.

###  4.3 Future outlook

To further investigate these potential burst and jet model implications, we
envision three types of future observing campaigns. Firstly, the majority of
published measurements or mentions of radio variability within observations
are motivated by peculiar X-ray behaviour: bursting of different types,
flaring, or moding. Therefore, we currently lack a comparison sample
characterizing the radio variability in ordinary atoll neutron stars. Without
such a base-level comparison, we are limited in assessing whether and to what
extent the Rapid Burster’s radio variability is unique. Constructing such a
comparison sample does not require new observations, but instead a dedicated
and systematic re-analysis of published VLA observations of atolls at higher
time resolution.

Secondly, the low signal-to-noise during the 2020 campaign may be countered by
a longer coordinated observation during a phase dominated by longer and more
slowly recurring Type-II bursts. If a causal relation indeed exists between
the X-ray fluence during burst times and the suppression of the jet
luminosity, it may not be possible to catch the source at higher radio
luminosity while bursting – in that case, such a longer campaign would benefit
substantially from the improved sensitivity of future instruments such as the
Square Kilometre Array or the next-generation Very Large Array (Selina et al.,
2018). However, the longer recurrence time between powerful Type-II bursts,
typically seen more early in the outburst (Bagnoli et al., 2015), would ease
the detection of a causal association between individual bursts and the
response of the jet – distinguishing between a constructive or destructive
effect of the bursts. As a subsequent step, performing such a campaign at
multiple observing frequencies would allow for the first measurements of jet
speed and geometry in the Rapid Burster.

Finally, instead of searching for a causal relation between the jet and
individual bursts, a full outburst monitoring campaign could instead focus on
the evolution of other, indirect observables of the jet-burst connection: the
X-ray and radio luminosity, the fractional radio variability, and the fluence
during burst times, for instance. Tracking such properties in tandem across
the outburst could further test whether Type-II bursts suppress the jet. Such
observations would require less strict simultaneity between X-ray and radio,
where the maximum allowable time difference would be driven by the typical
time scales of significant variations in X-ray or radio luminosity, and in
bursting rates and properties, across the outburst.

##  5 Acknowledgements

For the purpose of open access, the authors have applied a Creative Commons
Attribution (CC-BY) licence to any Author Accepted Manuscript version arising
from this submission. The authors want to express their gratitude to the VLA,
Swift, and INTEGRAL schedulers in their work to set up and perform the
coordinated observations underlying this work, particularly during the
unprecedented circumstances of the then-emerging COVID-19 pandemic. We thank
the INTEGRAL PI for approving the DDT observations used in this work. JvdE
acknowledges a Warwick Astrophysics prize post-doctoral fellowship made
possible thanks to a generous philanthropic donation. This research has made
use of the MAXI data provided by RIKEN, JAXA, and the MAXI team. This research
has made use of NASA’s Astrophysics Data System Bibliographic Services. We
acknowledge the use of public data from the Swift data archive. This research
was partly based on observations with INTEGRAL, and ESA project with
instruments and science data centre funded by ESA member states (especially
the PI countries: Denmark, France, Germany, Italy, Switzerland, Spain), and
Poland, and with the participation of Russia and the USA. The National Radio
Astronomy Observatory is a facility of the National Science Foundation
operated under cooperative agreement by Associated Universities, Inc.

## Data Availability

A GitHub reproduction repository will be made public upon acceptance of this
paper and will contain CASA analysis scripts, X-ray spectra and model files,
X-ray and radio light curve files, as well as analysis and plotting scripts.
An archived and stable release of this reproduction repository at the time of
paper accceptance will also be made available on Zenodo via DOI. Due to size
constraints, the unreduced VLA observations can instead be accessed publicly
via the NRAO data archive (data.nrao.edu) under project code 20A-172.

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