# MeerKAT radio observations of the neutron star low-mass X-ray binary Cen X–4
at low accretion rates

J. van den Eijnden,1 R. Fender,1,2 J. C. A. Miller-Jones,3 T. D. Russell,4 P.
Saikia,5 G. R. Sivakoff,6 and F. Carotenuto,1
1Department of Physics, Astrophysics, University of Oxford, Denys Wilkinson
Building, Keble Road, Oxford OX1 3RH, UK
2Department of Astronomy, University of Cape Town, Private Bag X3, Rondebosch
7701, South Africa
3International Centre for Radio Astronomy Research, Curtin University, GPO Box
U1987, Perth, WA 6845, Australia
4INAF, Istituto di Astrofisica Spaziale e Fisica Cosmica, Via U. La Malfa 153,
I-90146 Palermo, Italy
5Center for Astro, Particle and Planetary Physics (CAP3), New York University
Abu Dhabi, PO Box 129188, Abu Dhabi, UAE
6Department of Physics, CCIS 4-181, University of Alberta, Edmonton, AB, T6G
2E1, Canada

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

###### Abstract

Centaurus X–4 (Cen X–4) is a relatively nearby neutron star low-mass X-ray
binary that showed outbursts in 1969 and 1979, but has not shown a full
outburst since. Due to its proximity and sustained period of quiescence, it is
a prime target to study the coupling between accretion and jet ejection in
quiescent neutron star low-mass X-ray binaries. Here, we present four MeerKAT
radio observations at $1.3$ GHz of Cen X–4, combined with NICER and Swift
X-ray monitoring. During the first and most sensitive observation, Cen X–4 was
in a fully quiescent X-ray state. The three later and shorter observations
targeted a brief period of faint X-ray activity in January 2021, which has
been referred to as a ‘mis-fired’ outburst. Cen X–4 is not detected in any of
the four MeerKAT observations. We place these radio non-detections on the
X-ray – radio luminosity diagram, improving the constraints on the correlation
between the two luminosities from earlier quiescent radio studies. We confirm
that Cen X–4 is radio fainter than the transitional milli-second pulsar PSR
J1023+0038 at the same X-ray luminosity. We discuss the radio behaviour of
accreting neutron stars at low X-ray luminosity more generally and finally
comment on future observing campaigns.

###### keywords:

accretion, accretion discs – stars: neutron – X-rays: binaries – radio
continuum: transients

††pubyear: 2021††pagerange: MeerKAT radio observations of the neutron star
low-mass X-ray binary Cen X–4 at low accretion rates–MeerKAT radio
observations of the neutron star low-mass X-ray binary Cen X–4 at low
accretion rates

##  1 Introduction

Low-mass X-ray binaries (LMXBs), wherein a black hole (BH) or neutron star
(NS) accretes from a binary companion star, are routinely used to study the
connection between accretion and the launch of outflows. Such outflows can
take two forms, namely highly-collimated and often relativistic outflows from
the inner accretion flow called jets, and slower, more massive disk winds with
a wider opening angle and a range of launch radii. A large fraction of LMXBs
are transient sources, spending most of their lifetimes in quiescent states
where little to no accretion takes place. Intermittently, these transient
LMXBs show outbursts, where the accretion disk transitions into a hot, ionized
state and the accretion rate increases by orders of magnitude. Alternatively,
persistently-accreting LMXBs do not show outburst-quiescence cycles, and their
accretion disk instead remains in the ionized outburst state (see e.g. Done et
al., 2007; Gilfanov, 2010, for reviews).

LMXBs in outburst can reside in different spectral-timing states (Fender et
al., 2004): the system rises in accretion rate in the hard state, where its
X-ray emission is highly variable and dominated by the Comptonized emission
from an optically-thin population of hot electrons often called the corona.
Via intermediate spectral-timing states, it then often transitions into the
soft state, characterized by weak X-ray variability and emission from an
optically-thick, geometrically-thin disk. While it may transition to a high
Eddington accretion state or make multiple transitions between the hard and
soft state, during the outburst decay, it transitions from the soft into the
hard state at lower X-ray luminosities than the hard-to-soft transition
(Maccarone & Coppi, 2003). Finally the outburst sinks back into quiescence.
The spectral-timing states of NSs are historically named differently, with the
main classification into atoll and Z-sources based on the shape of the tracks
they display in their X-ray color-color diagrams. Within both the atoll and
Z-classifications, several sub-states have been identified (see e.g. Hasinger
& van der Klis, 1989; Van der Klis, 2006, for details). The discovery of a NS
LMXBs transitioning between the atoll and Z-source classifications showed that
mass-accretion rate fundamentally underlies the difference between the
classes, with the latter at the highest accretion rates (Homan et al., 2007).
Despite these different historical spectral-timing classifications between NS
and BH systems, several authors have identified equivalencies between the
states across both source types (e.g. Migliari & Fender, 2006; Muñoz-Darias et
al., 2014).

These LMXB states are intricately linked to their outflow properties. In BH
systems, the hard state is associated with the continuous launch of a compact
jet, followed by the launch of discrete ejecta during the hard-to-soft
transitional states, while the compact jet in quenched in the soft state (e.g.
Fender et al., 2004; Russell et al., 2019b). In atolls, which can (but do not
always) change between Comptonization-dominated (‘hard’, island state) and
thermal-dominated (‘soft’ banana state) states, compact jets are observed
routinely in the former (Migliari et al., 2003; Migliari & Fender, 2006;
Gusinskaia et al., 2020). However, whether jet quenching in the latter state
always occurs remains unclear (Rutledge et al., 1998; Migliari et al., 2004;
Miller-Jones et al., 2010; Fender, 2016; Gusinskaia et al., 2017; van den
Eijnden et al., 2021a). The Z-sources show compact jets and discrete ejecta,
depending on the exact branch of their Z-track (Migliari & Fender, 2006). Disk
winds were initially observed in the X-ray band, where they show up in BH soft
states and are easiest to observe in high inclination systems (Ponti et al.,
2012). However, disk winds are also observed in the optical (e.g. Muñoz-Darias
et al., 2016) and IR band (e.g. Sánchez-Sierras & Muñoz-Darias, 2020): optical
winds are observed in the hard state, while IR winds are seen across the
outburst. Recently, the first UV wind detection was reported in a NS LMXB
(Castro Segura et al., 2022). Finally, outburst light curve modelling also
suggests strong outflows (Tetarenko et al., 2018).

The connection between the accretion flow and compact jets is routinely
studied in the X-ray – radio luminosity ($L_{X}$–$L_{R}$) diagram. Here, the
former traces the accretion luminosity and is a proxy for accretion rate,
while the latter traces the jet and its brightness. Compact BH jets show an
$L_{X}$–$L_{R}$ correlation across $\sim$eight orders of magnitude in X-ray
luminosity (Hannikainen et al., 1998; Corbel et al., 2000, 2003; Gallo et al.,
2006); a subset of sources follows a radio-bright correlation with a power law
slope of $\beta\approx 0.6$ (where $L_{R}\propto L_{X}^{\beta}$, while others
follow a steeper correlation with $\beta\gtrsim 1$ at high X-ray luminosities,
before re-joining the other track around $L_{X}\approx 10^{35}$ erg/s (e.g.
Soleri & Fender, 2011; Coriat et al., 2011; Carotenuto et al., 2021, although
discussion exists regarding the statistical robustness of this separation;
e.g. Gallo et al. 2018). The situation for NS LMXBs is even more complex.
While, as a sample, NS LMXBs are $\sim 22$ times radio-fainter than BH
systems, the individual NS systems do not appear to follow a single
correlation; significant scatter exists both between sources and between
outbursts of the same source. Furthermore, due to their radio faintness, few
sources have been monitored over a large range of X-ray luminosity —
particularly below $L_{X}\approx 10^{35}$ erg/s, few NS LMXBs have been
detected in radio (e.g. Gusinskaia et al., 2020; Tudor et al., 2017). For the
sample of NS LMXBs, however, a power law slope of $\beta\approx 0.4--0.5$ has
been measured (Gallo et al., 2018), which is similar to BHs. Different
radiative efficiencies may be expected for the two types of LMXBs — BHs can
advect a fraction of the liberated gravitational energy across the event
horizon, while the presence of a NS surface implies that all this liberated
energy should, eventually, be radiated. Therefore, a similarity in the
$L_{X}$–$L_{R}$ coupling of the two source classes, which depends on this
radiative efficiency, is surprising.

While BH systems have been radio-detected in quiescent states down to
$L_{X}<10^{31}$ erg/s (Gallo et al., 2006; Dinçer et al., 2018), no NS LMXB
radio detections have been obtained below $L_{X}=10^{34}$ erg/s.111Here, we
ignore the transitional millisecond pulsars, which we will discuss in Section
4. Moreover, few radio upper limits exist in this X-ray regime. Therefore, the
correlation between the accretion flow and compact jets is poorly known for
weakly-accreting and quiescent NS LMXBs, and either radio detections or deeper
upper limits in this regime are necessary to advance our understanding. In
this work, we present a dedicated radio and X-ray campaign of the NS LMXB
Centaurus X–4 (Cen X–4) to constrain the low-$L_{X}$ inflow-outflow coupling.

Cen X–4 is a close-by transient NS LMXB, located at a relatively close-by
distance: Chevalier et al. (1989) determine $d=1.2\pm 0.3$ kpc, while recent
Gaia measurements imply a slightly larger distance (see Section 4). It was
discovered originally in outburst in 1969 (Conner et al., 1969) and showed a
second outburst in 1979 (Kaluzienski et al., 1980). Observations with the Very
Large Array detected its radio counterpart and monitored it during the 1979
outburst (Hjellming, 1979; Hjellming et al., 1988). Searches for radio
emission during quiescence yielded no detections, despite its small distance
(Kulkarni et al., 1992; Tudor et al., 2017). However, the limit found by Tudor
et al. (2017) provides the deepest radio constraint on a quiescent NS LMXB and
therefore provides the best low-$L_{X}$ anchor to the NS $L_{X}$–$L_{R}$
relation to date. In January 2021, long-term optical monitoring of Cen X–4 in
the XB-NEWS program (Waterval et al., 2020) revealed the onset of activity in
all optical bands (Saikia et al., 2021). X-ray follow up observations
confirmed the activity (van den Eijnden et al., 2021b). However, the source
did not enter a full outburst phase, but instead returned to quiescent levels
at all wavelengths two weeks later (van den Eijnden et al., 2021c). This
behaviour has been referred to as a ‘mis-fired’ outburst, caused either by an
inside-out heating front stalled by low levels of irradiation in the outer
accretion disk, or by a local thermal-viscous instability in the disk (Baglio
et al., 2022). In this paper, we present MeerKAT radio observations of Cen X–4
taken in its quiescent state in September 2020 and during the brief period of
activity in January 2021.

##  2 Observations and data reduction

Table 1: Summary of the MeerKAT radio observations of Cen X-4. The Start MJD column refers to the first on-target scan, and the On-source time does not include setup or calibration scans. The radio luminosity $L_{R}$ is calculated assuming a distance of $1.2$ kpc, and limits are quoted at $3$-$\sigma$ significance. | Start MJD | On-source time | RMS sensitivity |  $5$-GHz $L_{R}$
---|---|---|---|---
|  |  | [$\mu$Jy/bm] | [erg/s]
1 | 59118.54 |  $4$ hour | $4.3$ | $<1.1\times 10^{26}$
2 | 59221.43 |  $15$ min | $23$ | $<5.9\times 10^{26}$
3 | 59223.36 |  $15$ min | $18$ | $<4.7\times 10^{26}$
4 | 59230.32 |  $15$ min | $16$ | $<4.1\times 10^{26}$

###  2.1 Radio: MeerKAT

We performed a total of four observations of the field surrounding Cen X–4
(J2000 14h58m21.935s, $-$31∘40${}^{{}^{\prime}}$07.52${}^{{}^{\prime\prime}}$)
with MeerKAT. The first observation was taken on 26 September 2020 (capture
block 1601122564), while Cen X–4 was in its usual quiescent state, with a
longer, $4.5$-hour observation time including overheads ($\sim 4$ hour on
target). The three remaining observations were taken during the brief period
of activity in January 2021, specifically on 7, 9, and 16 January 2021
(capture blocks 1610014143, 1610180483, 1610782322, respectively). Those
latter three observations were all shorter, providing $15$ minutes on-target
observing time. All observations were taken as part of the ThunderKAT Large
Survey Program (Fender et al., 2016), which monitors active transient X-ray
binaries on a weekly basis and performs additional dedicated observations of
individual X-ray binaries, such as the quiescent observation of Cen X–4.
Observational details are also summarized in Table 1.

All four observations were performed using the L-band receivers, providing an
observing band between $856$ and $1712$ MHz, for a central frequency of $1.3$
GHz. The data were collected in 32k-mode (i.e. 32768 frequency channels) and
standard integration time of 8 seconds. The (standard) primary and nearby
secondary calibrator sources were J1939-6342 and J1501-3918, respectively. The
former was observed at the start of each observation for $5$ minutes.
$2$-minute scans of the latter source bookended the target scan during the
three short observations, while such scans were performed every $30$ minutes
during the longer September 2020 observation.

We used the OxKAT suite of analysis scripts (Heywood,
2020)222https://github.com/IanHeywood/oxkat to perform flagging, calibration,
and imaging of the observations. Through OxKAT we used the common astronomy
software application (casa; McMullin et al., 2007) to perform initial data
averaging, flagging, and calibration. Afterwards, we applied the bandpass,
flux-scale, complex gain, and delay corrections to the target scans, and split
of the target data, before flagging and imaging these data using the tricolour
and wsclean (Offringa et al., 2014) packages, respectively. Next, we applied
self-calibration and re-imaged the field. Finally, (as no source was detected
at the position of Cen X-4), we placed upper limits on the flux density at
three times the RMS sensitivity over a region covering the source position
that was also devoid of detected point sources in the longer observation.

###  2.2 X-rays: Swift and NICER

During the time frame of the MeerKAT observations, the Neil Gehrels Swift
Observatory (hereafter Swift) X-ray Telescope (XRT) observed Cen X–4 seventeen
times. On 26 September 2020 (ObsID 00088937006), a coordinated observation was
performed simultaneously with the deep MeerKAT observation, for a total
exposure of $\sim 3$ ks in Photon-Counting (PC) mode. Later, during the brief
period of activity around January 2021, Swift/XRT observed a further sixteen
times in PC mode between 28 December 2020 and 4 February 2021 with shorter
exposures ranging from $\sim 0.2$ to $\sim 1.0$ ks. To monitor the profile of
the X-ray activity, we extracted count rates for all observations using the
Online Data Products Pipeline (Evans et al.,
2007)333https://www.swift.ac.uk/user_objects/.

To measure the X-ray flux evolution during the observations, we also extracted
the X-ray spectrum for each observation. We downloaded the raw data from the
heasarc and used the run_xrtpipe.pl script to perform standard data reduction.
We then created source and background regions by defining a circular source
region with a $59$ arcsec diameter (25 pixels; Evans et al., 2007) centered on
the brightest target pixel and an annular background region with inner and
outer diameters of 59 and 400 arcsecond, respectively. Due to the low flux of
Cen X–4 and the short Swift exposures, five observations did not detect
sufficient counts for a spectral analysis (i.e. between 2 and 17 counts in the
source region). Therefore, we only extracted source and background spectra
from the remaining twelve observations, using xselect. We then created the
observation-specific arf-response files using the xrtmkarf tool and downloaded
the appropriate rmf-response file, swxpc0to12s6_20130101v014.rmf, from the
caldb. Finally, we rebinned all twelve spectra to at least $1$ count per bin
using grppha. In all steps we used HEASoft v6.29. We also used Swift’s Online
Data Products Pipeline to extract the X-ray spectra for the same observations,
to re-perform our spectral analysis and confirm that the results are
consistent.

Swift/XRT did not observe within 24 hours from the 9 January 2021 MeerKAT
observation. However, the Neutron Star Interior ExploreR (NICER) observed with
a $\sim 0.53$ ks cumulative444NICER ObsIDs typically combine data from
multiple exposures on the same day, whose individual durations are short due
to the complex visibility of targets as seen from the International Space
Station. exposure on that day (ObsID 3652010901). NICER is a single-pixel
instrument with a lower angular resolution than Swift/XRT (particularly in PC
mode), a softer response, and higher background count rate. Therefore, NICER
spectra should be examined carefully when observing faint sources such as Cen
X–4, for instance to determine the energy range where the source dominates
over the background. Hence, we only extract the NICER spectrum when no Swift
data is available. To extract the NICER spectrum, we downloaded the heasarc
datafile and ran the nicerl2 tool to re-apply the level-2 calibration using
the latest version of the caldb. We then used xselect to extract the target
spectrum, without specific energy cuts, as we checked the energy range where
the source dominates explicitly before spectral fitting. We then created
response files using the nicerrmf and nicerarf tools, before calculating the
background spectrum using the nicer_bkg_estimator tool555See
https://heasarc.gsfc.nasa.gov/docs/nicer/tools/nicer_bkg_est_tools.html. We
finally rebinned the spectrum to 20 counts per bin.

All X-ray spectral fits, for both Swift and NICER observations, were performed
in xspec v12.12.0 (Arnaud, 1996), assuming interstellar abundances from Wilms
et al. (2000) and cross-sections from Verner et al. (1996). We included the
interstellar absorption via the tbabs model and used the convolution model
cflux whenever we calculated fluxes from the fitted models. We calculated and
report all errors at the $1$-$\sigma$ level. Due to the low X-ray count rates,
we applied C-statistics (Cash, 1979) for all X-ray spectral fitting.

##  3 Results

###  3.1 Radio observations

No significant radio emission is detected from the position of Cen X–4 in any
of the four MeerKAT observations. In Figure 1, we show the large-scale field
surrounding Cen X–4 ($1.4\times 1.4$ degree), as observed in the deep,
$4.5$-hour observation. The black cross indicates the position of the X-ray
binary. While a large number of unresolved point sources and extended
(background) sources are visible in the image, the zoom inset displayed in
Figure 2 ($4\times 4$ arcminutes) confirms that no radio emission associated
with Cen X–4 is detected. In the deep, $4.5$ hour observation, this non-
detection implies a $3$-$\sigma$ radio flux density upper limit of $13$
$\mu$Jy at $1.3$ GHz, as determined from the image RMS calculated over the
target position. In the three shorter observations on 7, 9, and 16 January
2021, the non-detection of Cen X–4 implies $3$-$\sigma$ upper limits of $69$,
$54$, and $48$ $\mu$Jy, respectively. These upper limits are plotted in the
bottom panels of the light curves in Figure 3. For a distance of $1.2$ kpc and
assuming a flat radio spectrum, these limits correspond to $5$-GHz radio
luminosity limits of $L_{R}<1.1\times 10^{26}$ erg/s in the deep observation,
and $5.9\times 10^{26}$ erg/s, $4.7\times 10^{26}$ erg/s, and $4.1\times
10^{26}$ erg/s in the three shorter observations, respectively.

To test whether short-time-scale flaring is present in the observations, we
split the three short observations into two equal halves and re-imaged the
field. Again, no radio emission is detected in any image at the position of
Cen X-4, at RMS sensitivities a factor $\sqrt{2}$ higher than in the full
observations. Similarly, we break up the long quiescent observation into four
equal segments lasting two observing scans ($\sim 1$ hour), optimizing the
trade-off between time resolution and sensitivity at these low X-ray
luminosities. None of the four radio images show significant emission at the
target position, at a sensitivity of $9$ $\mu$Jy/bm.

Figure 1: The large-scale $1.3$ GHz MeerKAT view of the field surrounding the
position of Cen X–4, shown by the black cross. The RMS noise of the image is
$4.4$ $\mu$Jy/beam, obtained in the $4.5$-hour observation on 26 September
2020. The beam is shown in the lower left corner. A large number of unresolved
point sources and extended objects, are visible throughout the $1.4\times 1.4$
degree field. A zoom around the position of Cen X–4 is shown in Figure 2,
confirming the absence of a MeerKAT radio detection. Figure 2: Zoom of the
$4\times 4$-arcminute field around the position of Cen X–4 in the MeerKAT
field shown in Figure 1. The beam is shown in the bottom left corner. No point
source or extended emission is visible at the position of Cen X–4, shown by
the black cross, at an RMS sensitivity of $4.4$ $\mu$Jy/beam.

###  3.2 X-ray light curves and spectra

Figure 3: X-ray and radio light curves of Cen X–4 in September 2020 (left) and
January 2021 (right). The top panel shows the Swift/XRT count rates, all
measured in PC mode. The middle panel shows the $1$–$10$ keV flux measured
from the spectral fits to either Swift (black circles) or NICER (blue square)
spectra. The bottom panel shows the radio flux density upper limits for the
four MeerKAT observations at $1.3$ GHz.

In the top panel of Figure 3, we plot the Swift/XRT light curve around the
deep September 2020 radio and X-ray observations (left) and during the period
of faint activity in January 2021 (right). The red dashed lines indicate the
times of the four MeerKAT observations. The latter activity shows a rise in
X-ray rates over approximately one week, peaking around $0.65$ cts/s in PC
mode (see also Saikia et al., 2021; van den Eijnden et al., 2021b). Cen X–4
then decayed over a comparable time scale (e.g., van den Eijnden et al.,
2021c), returning to its typical quiescent count rate levels between $\sim
10^{-2}$ and $\sim 10^{-1}$ cts/s (Bernardini et al., 2013).

To calculate the X-ray flux from the Swift and NICER spectra, we followed the
approach of Bernardini et al. (2013), who studied the quiescent Swift/XRT PC-
mode spectra of Cen X–4 taken daily over a $60$-day period. Bernardini et al.
(2013) decomposed the spectrum into a soft component emitted by the NS
atmosphere and a harder component attributed to the low-level accretion flow.
To perform X-ray modelling at the low cumulative number of counts in the Swift
spectra, they combined observations in three $0.5$–$10$ keV count-rate ranges
to derive typical spectral parameters for the atmosphere and accretion-driven
emission: low rates below $7\times 10^{-2}$ ct/s, medium between $7\times
10^{-2}$–$1.1\times 10^{-1}$ ct/s, and high above $1.1\times 10^{-1}$ ct/s. As
shown by Figure 3, our observations probe a similar range in X-ray activity of
Cen X–4. As we aim to measure the time-dependent flux, we cannot similarly
combine observations. However, we can use the template spectra determined in
the above three count rate ranges to infer fluxes, when necessary.

Practically speaking, this means we define three ranges for our Swift spectra,
each with their own approach: (i) for observations exceeding $100$ source
counts, we directly fit the spectral models to the data; (ii) for observations
with $<100$ counts and a count rate below $7\times 10^{-2}$ ct/s, we fit the
overall normalization of the low-rate template spectral model from Bernardini
et al. (2013); (iii) for observations with $<100$ counts and a count rate
between $7\times 10^{-2}$-$1.1\times 10^{-1}$ ct/s, we fit the normalization
of the medium-rate template spectral model instead. We explain the fits for
these three ranges below in more detail. In Table 2, we list the spectral
analysis details for each ObsID.

For the three Swift spectra with sufficient counts for an actual fit – case
(i) – we attempted model fits with three models based on Bernardini et al.
(2013): a phenomenological, absorbed power law model (tbabs*powerlaw); a
physical, absorbed NS atmosphere model (tbabs*nsatmos); and their combination,
representing the addition of NS atmospheric and accretion-driven emission
(tbabs*(nsatmos + powerlaw)). In all X-ray spectral fits, we fixed the
absorption column to $N_{H}=8\times 10^{20}$ cm-2. In all three cases of the
high-rate regime, when comparing the single-component models, we find that the
power law model provides a statistically better fit than the NS atmosphere
model (i.e. a lower C statistic for the same number of free parameters). The
composite model does not provide a statistically better fit in any of the
spectra, given the C-statistic improvement for the two extra free parameters.
To mirror the approach in cases (ii) and (iii) (see below), we use the
$1$–$10$ keV unabsorbed flux from this composite model. However, this flux is
consistent with the power-law-only flux in all three cases. All fitted
parameters, fit statistics, and fluxes are listed in Table 2. We also
confirmed that the fitted parameters and measured fluxes were consistent
between the run_xrtpipe.pl and online-pipeline data reduction.

In cases (ii) and (iii), the template model is defined as the third model
above, i.e. tbabs*(nsatmos + power law). The difference between the low and
medium rate case lies in the parameters of the NS atmosphere and the relative
contributions of the two spectral components. In practice, we took the
parameters from Table 4 in Bernardini et al. (2013), converted the
$kT^{\infty}$ values to units of Kelvin and the thermal fraction to an nsatmos
normalization, and defined a constant*tbabs*(nsatmos + powerlaw) model in
xspec. We then fitted the constant for each considered spectrum, keeping all
other parameters frozen to their template values. Finally, we calculated the
X-ray flux by multiplying the $1$–$10$ keV unabsorbed model fluxes of
$1.31\times 10^{-12}$ erg/cm2/s (low rates) or $1.60\times 10^{-12}$ erg/cm2/s
(medium rates) with the fitted constant. We finally calculated the flux errors
by propagating the $1$-$\sigma$ error on the constant to the flux. Our
approach implies that we use the full X-ray flux; in the Discussion we will
explicitly discuss the contributions of the different spectral components to
the flux.

Finally, we turned to the single NICER spectrum. Cen X–4 is only detected
above the background between $0.5$ and $1.5$ keV. We assume the medium-rate
template spectrum from Bernardini et al. (2013) and again fit the constant
scaling factor. This spectral shape gives a significantly better fit than the
low-rate template. Moreover, it fits with the rates in the surrounding Swift
observations, which exceed the maximum rate for the low-rate template.

The light curve of the $1$–$10$ keV unabsorbed X-ray flux is plotted in the
middle panels of Figure 3. The evolution clearly follows the count rate
profile, although minor differences are visible. For instance, due to
differences in spectral shape, the X-ray flux during the September 2020
observation is significantly lower than that after the return to quiescence in
January 2021, despite both sharing similar count rates. Also, five count-rate
data points do not have an associated X-ray flux due to the limited number of
source counts. Finally, the NICER flux fits well with the general decay trend
during that phase of the light curve, despite the small energy range where Cen
X–4 is detected.

Table 2: Details of the X-ray flux determination. For the first three ObsIDs, we list the fitted spectral parameters for the three attempted models: tbabs*powerlaw, tbabs*nsatmos, and tbabs*(nsatmos + powerlaw), per respective row. For the other ten observations, we instead list the measured Swift/XRT PC-mode count rate, the adopted template model shape, and the measured model normalization. For all observations, we list the MJD, C-statistic, number of degrees of freedom (dof), and unabsorbed 1–10 keV X-ray flux. In all fits, we fixed interstellar absorption to $N_{H}=8\times 10^{20}$ cm-2. Due to a low total number of source counts, we did not perform a spectral analysis for Swift ObsIDs 00035324073, 00035324077, 00035324078, 00035324079, and 00035324083. All Swift/XRT count rates in the top panels of Figure 3 are available via the link provided in the Data Availability statement. *Parameter pegged at lower limit. Obs | ObsID | MJD | $\log T_{\rm eff}$ | $N_{\rm nsa}$ | $\Gamma$ | $N_{\rm po}$ | C (dof) |  $1$–$10$ keV flux
---|---|---|---|---|---|---|---|---
|  |  | [K] |  |  | [keV-1cm-2s-1] |  | [erg cm-2s-1]
Swift | 00088937006 | 59118.52 | – | – | $3.0\pm 0.3$ | $(2.3\pm 0.3)\times 10^{-4}$ | 40.2 (93) | $(3.3\pm 0.9)\times 10^{-13}$
|  |  | $6.3\pm 0.1$ | $\left(2.1^{+3.4}_{-1.1}\right)\times 10^{-2}$ | – | – | 49.8 (93) | $(2.8\pm 0.6)\times 10^{-13}$
|  |  | 5.0* |  $\geq 0$* | $2.8\pm 0.4$ | $\left(2.1^{+0.4}_{-0.6}\right)\times 10^{-4}$ | 39.5 (91) | $(3.5\pm 1.0)\times 10^{-13}$
Swift | 00035324070 | 59218.96 | – | – | $2.4\pm 0.1$ | $(4.4\pm 0.2)\times 10^{-3}$ | 195.9 (189) | $(1.0\pm 0.1)\times 10^{-11}$
|  |  | $>6.47$ | $\left(5.8^{+3.6}_{-0.4}\right)\times 10^{-2}$ | – | – | 268.2 (189) | $(8.9\pm 0.6)\times 10^{-12}$
|  |  | $6.25^{+0.11}_{-0.07}$ | $\left(3.2^{+3.6}_{-2.6}\right)\times 10^{-1}$ | $1.9^{+0.4}_{-0.6}$ | $(2.0\pm 0.1)\times 10^{-3}$ | 190.6 (187) | $(1.1\pm 0.1)\times 10^{-11}$
Swift | 00035324071 | 59221.15 | – | – | $2.7\pm 0.2$ | $(3.4\pm 0.3)\times 10^{-3}$ | 86.0 (89) | $(6.4\pm 0.9)\times 10^{-12}$
|  |  | $6.45\pm 0.04$ | $\left(6.2^{+3.0}_{-2.0}\right)\times 10^{-2}$ | – | – | 132.3 (89) | $(6.0\pm 0.7)\times 10^{-12}$
|  |  | $6.0\pm 0.1$ | $3.6^{+7.2}_{-2.2}$ | $1.6\pm 0.6$ | $\left(9.1^{+1.1}_{-0.6}\right)\times 10^{-4}$ | 82.3 (87) | $(7.4\pm 1.5)\times 10^{-12}$
Obs | ObsID | MJD | PC Rate | Template | Template | C (dof) |  $1$–$10$ keV flux
|  |  | [cts s-1] | spectrum | normalization |  | [erg s-1cm-2]
Swift | 00035324068 | 59211.46 | $1.1\times 10^{-1}$ | Medium | $1.84\pm 0.22$ |  | 61.4 (64) | $(2.9\pm 0.6)\times 10^{-12}$
Swift | 00035324069 | 59214.25 | $1.1\times 10^{-1}$ | Medium | $2.79\pm 0.41$ |  | 42.9 (41) | $(4.5\pm 1.8)\times 10^{-12}$
NICER | 3652010901 | 59223.46 | N/A | Medium | $2.07\pm 0.06$ |  | 102.1 (63) | $(3.2\pm 0.2)\times 10^{-12}$
Swift | 00035324072 | 59225.93 | $7.3\times 10^{-2}$ | Medium | $0.74\pm 0.12$ |  | 31.0 (35) | $(1.2\pm 0.1)\times 10^{-12}$
Swift | 00035324074 | 59227.06 | $4.2\times 10^{-2}$ | Low | $0.91\pm 0.17$ |  | 29.4 (30) | $(1.2\pm 0.1)\times 10^{-12}$
Swift | 00035324075 | 59228.13 | $5.1\times 10^{-2}$ | Low | $1.03\pm 0.18$ |  | 22.4 (35) | $(1.3\pm 0.2)\times 10^{-12}$
Swift | 00035324076 | 59229.98 | $6.0\times 10^{-2}$ | Low | $1.52\pm 0.25$ |  | 38.5 (34) | $(2.0\pm 0.5)\times 10^{-12}$
Swift | 00035324081 | 59233.76 | $3.6\times 10^{-2}$ | Low | $0.68\pm 0.17$ |  | 21.0 (22) | $(8.9\pm 0.2)\times 10^{-13}$
Swift | 00035324082 | 59234.96 | $4.8\times 10^{-2}$ | Low | $1.15\pm 0.21$ |  | 27.2 (34) | $(1.5\pm 0.3)\times 10^{-12}$
Swift | 00035324084 | 59237.61 | $3.8\times 10^{-2}$ | Low | $1.06\pm 0.23$ |  | 26.0 (23) | $(1.4\pm 0.3)\times 10^{-12}$

###  3.3 The X-ray – radio luminosity plane

To further investigate the low-luminosity inflow/outflow coupling in Cen X–4,
we combine X-ray and radio observations and place them on the X-ray–radio
luminosity diagram. We match up each radio observation with the X-ray
observation taken within $24$ hours (ObsIDs 00088937006, 00035324071,
3652010901, and 00035324076, chronologically). To convert the X-ray fluxes to
$1$–$10$ keV luminosities, we assume a distance of $1.2$ kpc (Chevalier et
al., 1989, although see Section 4 for a discussion on the distance). To
calculate the radio luminosity upper limits from the flux densities, we
calculate $L_{R}=\nu L_{\nu}$ at a frequency of 5 GHz. This X-ray energy band
and radio frequency are chosen to be the same as in the X-ray binary
$L_{X}$–$L_{R}$ catalogue by Bahramian & Rushton (2022).

In Figure 4, we plot the X-ray–radio luminosity diagram for low-mass X-ray
binary systems, based largely on Bahramian & Rushton (2022). The green circles
show hard-state BH systems, while NSs in equivalent states666More
specifically, atolls in the island state and the Accreting Millisecond X-ray
Pulsars. are shown in blue squares. The transitional millisecond pulsars
(tMSPs) in their X-ray bright state are shown as the light-green upwards
triangles. A number of NS systems are specifically highlighted. Cen X–4 in
shown by the stars, where red stars show archival data from Tudor et al.
(2017), Hjellming et al. (1988), and Kaluzienski et al. (1980), and turquoise
stars show the four observations presented in this work. Aql X-1 is shown with
yellow circles, taken from the detailed study by Gusinskaia et al. (2020),
which is the most commonly observed system of the NSs plotted here (note that
we only plot points where Gusinskaia et al. 2020 calculate a hardness ratio of
at least 0.75). Three observations of GRS 1747-312, that we will explicitly
discuss in Section 4, are shown by the magenta squares (Panurach et al.,
2021). Using distances from Tremou et al. (2018), we also include the quasi-
simultaneous data from Panurach et al. (2021) for the globular cluster sources
4U 1746-37, XB 1832-330, X 1850-087, and M15 X-3, which were all observed when
actively accreting, although they were relatively X-ray faint.

The deepest MeerKAT radio upper limit of Cen X–4 is comparable to the deep
Very Large Array limit obtained by Tudor et al. (2017), but at a $\sim$four
times higher X-ray luminosity. As a result, the new non-detection reported
here is more constraining to the low-luminosity end of the X-ray–radio
luminosity correlation for the source. The radio upper limits from the further
three MeerKAT observations are at higher luminosities, but as they are taken
during the period of faint X-ray activity, they may similarly constrain the
slope of this correlation. More generally, radio detections of confirmed NS
below $L_{X}=10^{34}$ erg/s are rare. Our observations add four more points to
this poorly-explored regime, offering a particularly interesting comparison to
the tMSPs: this source class has been radio-detected at similar X-ray
luminosities, with radio luminosities above or close to the detection
threshold of our Cen X–4 observations (see Discussion).

To assess the effect of our new observations on the inferred $L_{X}$–$L_{R}$
coupling, we follow the linmix-approach, originally developed by (Kelly,
2007). This MCMC-method fits a linear model to data with errors in both the
dependent and independent variable, fully accounting for upper limits in the
former. It was first applied to the study of the X-ray binary $L_{X}$–$L_{R}$
plane by Gallo et al. (2014), and further adapted by Gusinskaia et al. (2020)
and Van den Eijnden et al. (submitted) to estimate the uncertainties and
account for distance errors. To apply this method, we linearize the
correlation model $L_{R}/L_{R,0}=\xi(L_{X}/L_{X,0})^{\beta}$ to

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

We then apply the python-implementation of the method777Available via
https://github.com/jmeyers314/linmix to fit the offset and slope of this
linear equation, as well as the data’s intrinsic scatter, parameterized by a
Gaussian standard deviation $\sigma$. In particular, we determine for each
parameter the median and $16^{\rm th}$/$84^{\rm th}$ percentile from $10^{4}$
draws of the posterior distribution. We then follow Gusinskaia et al. (2020)
and Van den Eijnden et al. (submitted) and repeat this approach $500$ times,
each time re-drawing the assumed distance (i.e. scaling the fitted
luminosities by a factor $(D_{i}/D)^{2}$ for iteration i) from a Gaussian
distribution centred at $1.2$ kpc with a $0.3$ kpc standard deviation. We then
determine the fitted parameters and their $1$-sigma uncertainties as the mean
value of the $500$ saved medians and $16^{\rm th}$/$84^{\rm th}$ percentiles,
respectively.

In Figure 4, the black line shows the best fit to all Cen X–4 observations,
with $\log\xi=-0.14^{+0.22}_{-0.27}$ and $\beta=0.98^{+0.60}_{-0.28}$. The
lightly-shaded area indicates the $1$-$\sigma$ confidence interval of the
fit888We note that combining the two X-ray brightest radio epochs, which also
returns a non-detection, does not lead to significantly different results of
the fit.. To compare these values to the archival data from Tudor et al.
(2017), we exactly repeat these fits without the four MeerKAT data points. In
that case, we find $\log\xi_{\rm archival}=-0.12^{+0.23}_{-0.27}$ and
$\beta_{\rm archival}=0.84^{+0.64}_{-0.28}$, which set the $1$-$\sigma$ region
enclosed by the red dotted lines in Figure 4. Expectedly, the fitted
normalisations are consistent, as these are most strongly constrained by the
radio detections during outburst. Similarly, the upper limits on the slope
$\beta$ are almost identical, as these are not affected by the addition of
radio upper limits at low $L_{X}$. The minimum slope is, on the other hand,
more strongly constrained. This can be deduced from comparing the $1$-$\sigma$
regions in Figure 4 at low $L_{X}$. Alternatively, we can compare the $90$%
lower limit on $\beta$ from our fit: this limit increases from $\beta\geq
0.51$999This value is consistent with the lower limit reported by Tudor et al.
(2017), i.e. $\beta>0.5$. to $\beta\geq 0.66$ with the addition of the MeerKAT
observations.

Figure 4: The $1$–$10$ keV X-ray – 5-GHz radio luminosity diagram for low-mass
X-ray binaries. The grey circles represent hard-state black hole systems.
Light blue squares show neutron stars, particularly hard-state atoll sources
and AMXPs. tMSPs are shown as the light green upwards pointing triangles,
while four tMSPs are shown as the upwards dark green triangles. We also
highlight several sources in particular: Cen X–4 as the stars (red for
archival data, turquoise for data from this work), Aql X-1 as the yellow
circles, and GRS 1747-312 as the purple squares. For Aql X-1, the compilation
of data is taken from Gusinskaia et al. (2020), where we only plot points with
an associated hardness ratio exceeding 0.75. The black line indicates the best
linmix fit to all Cen X–4 data, with the shaded area showing the $1$-$\sigma$
confidence region. The red dotted lines indicate the $1$-$\sigma$ confidence
region without the new MeerKAT observations. Plot based on the catalogue by
Bahramian & Rushton (2022) with more recent data added (see text).

##  4 Discussion

In this paper, we have presented four MeerKAT radio observations of the NS
LMXB Cen X–4 during quiescence and a period of faint X-ray activity. Cen X–4
is not detected in any of the radio observations, providing deep limits on the
correlation between X-ray and radio luminosity in the low-luminosity regime
(e.g. $L_{X}\lesssim 10^{33}$ erg/s). Here, we will discuss the implication of
these measurements for our understanding of (different types of) NSs in the
$L_{X}$–$L_{R}$ plane, the origin of the measured X-ray luminosity, and the
prospects for low-luminosity NS LMXB radio studies using more sensitive future
arrays.

###  4.1 A note on the distance to Cen X-4

In our analysis, we assumed the $1.2\pm 0.3$ kpc distance measured by
Chevalier et al. (1989) to convert fluxes and flux densities to luminosities.
Recent Gaia parallax measurements (Gaia Collaboration et al., 2021), however,
imply a slightly larger distance: at $68$% confidence, converting the parallax
to a distance using the prior for Galactic X-ray binaries from Atri et al.
(2019) and applying zero-point corrections, we find $d=1.87^{+0.75}_{-0.42}$
kpc. To assess the effect of this larger distance on our inferences, we repeat
our fits in the $L_{X}$–$L_{R}$ plane after correcting all luminosities by a
factor $(1.87/1.2)^{2}$. This calculation returns consistent parameters: we
find $\beta_{\rm gaia}=0.98^{+0.58}_{-0.28}$ and $\log\xi_{\rm
gaia}=-0.13^{+0.23}_{-0.38}$. Finding the same slope is unsurprising, given
that both $L_{X}$ and $L_{R}$ scale in the same manner with distance. Finding
a consistent normalization at $L_{X,0}$ occurs because the best-fit slope is
consistent with linear. Therefore, the main effect of the larger Gaia distance
is in the comparison with other sources. We will therefore note, in the below
discussions, where this may play a role.

###  4.2 The origin of the X-ray emission

In this work, we have followed the common approach to the X-ray spectra, where
the unabsorbed X-ray fluxes from the entire spectral model are calculated to
obtain the $L_{X}$ measurements. However, at the low X-ray luminosities of Cen
X–4, the soft X-rays are fitted by a NS atmosphere model, while only the
harder X-rays are dominated by the fitted power law component. When our Swift
spectra contain enough counts for a model fit, we find that this two-component
model does not fit better than a power-law-only model. However, this could
likely be caused by the low signal-to-noise, given that analyses by Bernardini
et al. (2013) and Chakrabarty et al. (2014) find significant soft thermal
emission at similar X-ray luminosity in higher quality spectra. For this
reason, and for consistency with the lower quality spectra analysed using two-
component template spectral models, we used the two-component X-ray fluxes for
all observations.

Since $L_{X}$–$L_{R}$ correlations are interpreted as the observational
signature of a coupling between inflow, i.e. mass accretion rate, and outflow,
i.e. jet power, we can wonder whether the two-component X-ray flux correctly
traces the mass accretion rate. The detailed modeling of the high-energy
component by Chakrabarty et al. (2014) is consistent with this component
originating in a radiatively-inefficient accretion flow, while Bernardini et
al. (2013) similarly infer that the power-law-emission seen in Swift spectra
likely originates from the accretion flow at low mass accretion rate.
Moreover, Bernardini et al. (2013) find that the thermal X-rays from the NS
atmosphere and the harder X-ray component are variable in a coupled fashion,
leading to their conclusion that the full X-ray spectrum is powered by the
accretion flow: either directly in the hard component or indirectly, after
accreting onto the neutron star surface, in the soft, thermal component. In
that interpretation, the combined X-ray luminosity can indeed be used in our
$L_{X}$–$L_{R}$ modelling.

For a scenario where all components of the X-ray spectrum are directly or
indirectly powered by accretion (instead of e.g. the neutron star surface
cooling from previous activity but not actively heated by the instantaneous
accretion), the correlation between accretion rate and X-ray luminosity is
expected to be linear: as advection of energy across an event horizon is not
possible, the process is ultimately expected to be radiatively efficient. In
that scenario, although the relative contributions of different spectral
components may differ as a function of accretion rate, the correlation between
$L_{X}$ and $L_{R}$ would remain the same from quiescence to the outburst
peak. The lack of radio detections of NS LMXBs below $L_{X}=10^{34}$ erg/s has
made it challenging to test this hypothesis; a recent analysis at higher X-ray
luminosities for the NS LMXB Aql X-1 by Fijma et al. (2022) did not reveal
evidence for any significant effects of the relative contributions of spectral
components on the coupling. However, that study also showed how signal-to-
noise with current facilities, at distances of several kpc, makes it hard to
disentangle such X-ray components, and how the presence of accretion state
changes dominates the changes in radio behaviour. Therefore, observations
covering quiescence and the rise of the outburst may be most suitable for such
a study. However, our MeerKAT results show that current sensitivities in
monitoring observations are insufficient for low-$L_{X}$ radio detections,
even at $1.2$ kpc.

###  4.3 NS LMXB jets at low X-ray luminosity

####  4.3.1 Phenomenological source comparisons

With our radio limits at low X-ray luminosity, we constrain the slope of the
$L_{X}$–$L_{R}$ correlation for Cen X–4 to $\beta=0.98^{+0.59}_{-0.28}$. Using
the same fitting approach, Gallo et al. (2018) found a correlation slope of
$\beta=0.44^{+0.05}_{-0.04}$ for the entire sample of NS LMXBs, significantly
shallower than our Cen X–4 result. As Cen X–4 is likely an atoll source, a
more apt comparison, however, may be with the atoll-only fit performed by
Gallo et al. (2018). This atoll-only correlation slope,
$\beta=0.71^{+0.11}_{-0.09}$ is consistent within its $1$-$\sigma$ errors with
Cen X–4. As discussed in Gallo et al. (2018), the measured slopes may be
affected by the X-ray luminosity distribution of the fitted observations: for
the total sample of atolls, which are predominantly observed (and especially
radio detected) above $10^{36}$ erg/s, this may lead to shallower slopes. With
the largest range in X-ray luminosity of any (likely) atoll source, we expect
that such issues do not affect our Cen X–4 measurement to the same extent.
Recently, Gusinskaia et al. (2020) presented a detailed look at the behaviour
of Aql X-1 in the $L_{X}$–$L_{R}$ plane, across various outbursts. When, for
the first time, including deep radio limits (i.e. $L_{R}\lesssim 10^{27}$
erg/s) between $L_{X}=6\times 10^{34}$ and $3\times 10^{35}$ erg/s, they find
a correlation slope of $\beta=1.17^{+0.30}_{-0.21}$ – again consistent with
our Cen X–4 measurement.

Moving beyond comparisons with atoll source, we can turn to other sub-classes
of accreting NSs. For instance, our Cen X–4 radio non-detections during its
period of faint activity provide stricter confirmation that tMSPs are radio-
brighter than other NSs at these low X-ray luminosities (note that the same
does not necessarily hold above, e.g., $L_{X}=10^{35}$ erg/s; Russell et al.
2018; van den Eijnden et al. 2018). In particular, the observations of the
first-discovered tMSP, PSR J1023+0038, by Deller et al. (2015) lie above our
radio upper limits and are excluded from the $1$-$\sigma$ region for Cen X–4’s
$L_{X}$–$L_{R}$ correlation, for the distance of $1.2\pm 0.3$ kpc. This
finding confirms the suspicion from Tudor et al. (2017) that observations at
higher $L_{X}$ than theirs, would either detect or disprove the formation of
jets in Cen X–4 as radio-bright as tMSPs (although we note that a jet origin
for the radio emission of tMSPs is not confirmed; e.g. Bogdanov et al., 2018).

In Figure 4, we also show four candidate but unconfirmed tMSPs that were
recently studied in the $L_{X}$–$L_{R}$ plane as the dark green upward
triangles: 3FGL J0427.9-6704 (Li et al., 2020), 3FGL J1544.6-1125 (Jaodand et
al., 2021), NGC 6652B (Paduano et al., 2021), and CXOU J110926.4-650224 (Coti
Zelati et al., 2021, assuming a $4$ kpc distance). We plot one point per
source: the radio brightest of four observations for 3FGL J1544.6-1125, and
the average X-ray and radio luminosity of the other three. All four are also
detected as radio-bright systems, sometimes even consistent with the BH
population in the $L_{X}$–$L_{R}$ diagram. Those observations, if the systems
are indeed confirmed to be tMSPs, further indicate a clear difference with Cen
X–4 at low $L_{X}$. If the relative radio brightness of tMSPs is related to
interactions between the low-level accretion flow and the pulsar
magnetosphere, those interactions should be markedly different than in non-
transitioning LMXBs.

Finally, we note that the recent work by Panurach et al. (2021), studying NS
LMXBs in globular clusters with the MAVERIC survey, has revealed that low-
level-accreting neutron stars can be radio variable. In particular, the
transient NS LMXB GRS 1747-312, shown in Figure 4 as the magenta squares,
shows a mixture of radio detections and non-detections, implying radio
variability by more than an order of magnitude despite a relatively small
range in X-ray luminosity ($5\times 10^{33}$–$5\times 10^{34}$ erg/s). At a
slightly higher $L_{X}\approx 10^{36}$ erg/s, Panurach et al. (2021) find that
the persistent NS LMXB X 1850-087 is similarly radio variable. In the tMSP
population, PSR J1023+0038 also famously shows strong radio variability during
its X-ray-bright state: highly-structured moding behaviour, anti-correlated
with the X-ray band, as well as erratic radio flaring, both on time scales of
minutes (e.g. Bogdanov et al., 2018). During the faint activity period of Cen
X-4, we do not find evidence for variability during our observations, when
splitting the observing block in two. Longer observations during faint periods
of accretion may, in the future, provide better constraints on such
variability. Similarly, the longer MeerKAT observation did not reveal evidence
for radio variability, during the source’s quiescent state.

####  4.3.2 Correlation breaks and radio-dark propeller outflows

The phenomenological measurements of the $L_{X}$–$L_{R}$ relation discussed in
the previous section (i.e. for Aql X-1, all atolls, and ours for Cen X–4),
fundamentally assume a single correlation to hold between quiescence and the
peak of the outburst (or the state transition where the source may show radio
quenching). Gusinskaia et al. (2020) instead also discuss an alternative
scenario, where the $L_{X}$–$L_{R}$ correlation of Aql X-1 shows a sharp
cutoff a X-ray luminosities below $\sim 5\times 10^{35}$ erg/s101010We note
that (Plotkin et al., 2017) rule out a similar cutoff in the X-ray – radio
luminosity relation for the black hole LMXB V404 Cyg. In that work, such a
cutoff would be associated with the jet dominating the X-ray emission a low
accretion rate.. Generally, NS LMXBs appear to change from regular radio
detections above $L_{X}\approx 10^{36}$ erg/s to mostly radio non-detections
at lower X-ray luminosities. While this change is most simplistically
explained via a relatively steep, single $L_{X}$–$L_{R}$ correlation, one can
explore an explanation for a break in the correlation as well. For Cen X–4 (or
any other source, for that matter), current data do not allow to distinguish
between these scenarios: this is particularly difficult due to the $\sim 3$
order of magnitude in X-ray luminosity that has never been covered with radio
observations. Similarly, if such a break in the correlation would exist,
current data do not constrain whether it occurs at the same X-ray luminosity
in different sources.

The X-ray luminosity range where radio non-detections become dominant, is of
similar order of magnitude as the range where the onset of the propeller
regime is expected. In this regime, the magnetospheric radius, where the NS
magnetic field and accretion disk pressure are in balance, moves outside the
co-rotation radius due to a decrease in the accretion rate. At the co-rotation
radius, the NS spin equals the Keplerian frequency of the disk. Therefore,
when the magnetospheric radius moves beyond the co-rotation radius, a
centrifugal barrier may be created, halting the accretion flow. The accretion
luminosity where this is expected to occur can be written as

$L_{\rm prop}=1.6\times 10^{33}\text{ }B_{8}^{2}\text{ }\nu_{100}^{7/3}\text{
erg/s,}$ (2)

where we assume disk accretion (instead of spherical/wind accretion), $B_{8}$
is the magnetic field divided by $10^{8}$ G, and $\nu_{100}$ is the spin
frequency divided by $100$ Hz (Tsygankov et al., 2017). For e.g, Aql X-1, with
its $550$ Hz spin, $L_{\rm prop}$ can easily match $\sim 5\times 10^{35}$
erg/s for a realistic magnetic field of a few times $10^{8}$ G; more
generally, for AMXPs with spins at hundreds of Hz, $L_{\rm prop}$ can
reasonably be expected to lie between $10^{35}$ and $10^{36}$ erg/s. In the
propeller scenario, decaying below $L_{\rm prop}$ would expel material as
angular momentum is transferred to the accreting material (Illarionov &
Sunyaev, 1975). Alternatively, in other models and circumstances, the disk may
instead be trapped around the magnetospheric radius, preventing efficient
accretion (Spruit & Taam, 1993; D’Angelo & Spruit, 2010).

The onset of the propeller regime has regularly been suggested to be
associated with enhanced radio emission from the outflow. For instance, to
explain the difference between the radio luminosity during the decay of
outbursts of the AMXP SAX J1808.4-3658, Tudor et al. (2017) invoke a
propeller-driven outflow to explain radio detections below $L_{X}=10^{36}$
erg/s. However, one may wonder if additional radio emission, or any radio
emission, is always expected in this regime. For instance, if the disk is
trapped (mostly expected when $L_{X}$ drops slightly below $L_{\rm prop}$), it
may prevent effective jet launching altogether. Moreover, simulations of the
propeller regime by Ustyugova et al. (2006), Romanova et al. (2009), and Lii
et al. (2014) show the onset of a two-component outflow: a fast and collimated
outflow combined with a more massive, wider-angled and slower wind-like
outflow. While radio emission may be expected from the former, it is unclear
where the energetic balance between the outflows lies. If most of the energy
is carried away by the wind-like slow outflow, which is not expected to emit
brightly at radio frequencies, the radio luminosity of NS LMXBs may be
suppressed when decreasing below $L_{\rm prop}$.

We reiterate that the onset of such a ‘radio-dark’ propeller outflow is not
necessary to explain current observations; as stated earlier, a relatively
steep, single $L_{X}$–$L_{R}$ correlation (per source or the full sample) can
similarly account for the currently available data. As discussed below, more
sensitive and regular radio monitoring of this low-luminosity regime could
provide the data to observationally test the single-correlation hypothesis.
However, two brief points are worth stating: firstly, if a steepening/breaking
of the $L_{X}$–$L_{R}$ correlation would be observed in future campaigns, the
explanation will likely be related to the NS properties, given the lack of
such steepening in black hole systems. Secondly, the idea of the propeller
regime suppressing the radio luminosity in an X-ray binary was speculated on
earlier by van den Eijnden et al. (2019) for strongly-magnetic but slowly-
spinning accreting NSs (i.e. $B\gtrsim 10^{12}$ G, $\nu<1$ Hz). In particular,
in Swift J0243.6+6124, the archetypal example of a transient, radio-detected
X-ray binary in that class, radio emission is observed to turn on and off
rapidly across a narrow range in X-ray luminosity. This luminosity could be
associated with the propeller transition for reasonable magnetic field
strengths for its NS, which are similar to the recent cyclotron line estimates
of its magnetic field (Kong et al., 2022).

####  4.3.3 Observational tests with future observatories

With the planned sensitivity of future arrays, such as the next-generation VLA
(ngVLA) or the Square Kilometer Array (SKA, specifically SKA-Mid), we will be
able to probe at least an order of magnitude deeper to search for quiescent
radio emission from Cen X–4. For instance, the ngVLA’s intended $3$-$\sigma$
detection limit at a distance of $1.2$ kpc in a short, $15$-minute observation
is $1.2\times 10^{25}$ erg/s (Selina et al., 2018). At these sensitivities, it
is possible to directly test for the presence of a break in the
$L_{X}$–$L_{R}$ correlation at low X-ray luminosities, and therefore test the
idea of dark propeller outflows. If such a correlation break is not observed,
it can instead probe the radiative efficiency of the accretion flow down to
low accretion rate by measuring or constraining the correlation slope further:
a single radio non-detection at $L_{X}=10^{32}$ erg/s, with the $15$-minute
ngVLA sensitivity, would constrain the slope to $\beta\geq 0.8$ ($1$-$\sigma$
limit). Taking this further, not detecting Cen X–4 during an exact repeat of
our MeerKAT campaign — i.e. at the same X-ray luminosities but with the
$15$-minute ngVLA sensitivity — will imply that $\beta\geq 1.1-1.2$: a limit
that starts to constrain the accretion flow to be radiatively efficient
(Migliari & Fender, 2006).

On a more systematic level, these future radio sensitivity levels could allow
for systematic monitoring of quiescent (NS) LMXBs to detect any low-level
activity at larger distances than that of Cen X–4. Such systematic programs
could (i) detect more examples of low-level activity (i.e. $L_{X}<10^{34}$
erg/s) associated with mis-fired outbursts, and therefore reveal how common
such mis-fired outbursts are; (ii) detect the early onset of outbursts to
allow for detailed $L_{X}$–$L_{R}$ monitoring; or (iii) provide deep combined
images of the quiescent LMXB if no activity is detected. The low-level
activity in Cen X–4 was identified by Waterval et al. (2020) due to its small
distance in combination with long-term optical monitoring by the XB-NEWS
project (Russell et al., 2019a). Currently, no complementary program to XB-
NEWS exists at other wavelengths, and the low-level activity falls below the
sensitivity of current and planned all-sky X-ray monitors. The ngVLA and SKA-
Mid would be suitable for such complementary low-level activity programs,
aimed at the jets instead of the inflow observed in optical: in short,
$5$-minute observations with the ngVLA, one reaches $3$-$\sigma$ detection
thresholds of $L_{R}\approx 2\times 10^{26}$ erg/s at 4 kpc. A similar program
for BH LMXBs may require even shorter observing times or be possible with sub-
arrays to increase observing efficiency: the better established
$L_{X}$–$L_{R}$ relation towards quiescence for BHs also removes uncertainty
caused by the uncertain nature of this relation at low $L_{X}$ for NSs.

##  5 Acknowledgments

The authors thank the anonymous referee for a constructive report. JvdE thanks
Joe Bright for discussions on MeerKAT data reduction. JvdE is supported by a
Lee Hysan Junior Research Fellowship awarded by St. Hilda’s College. We thank
the staff at the South African Radio Astronomy Observatory (SARAO) for
schedul- ing these observations. The MeerKAT telescope is operated by the
South African Radio Astronomy Observatory, which is a facility of the National
Research Foundation, an agency of the Department of Science and Innovation.
This work was carried out in part using facilities and data processing
pipelines developed at the Inter-University Institute for Data Intensive
Astronomy (IDIA). IDIA is a partnership of the Universities of Cape Town, of
the Western Cape and of Pretoria. The authors acknowledge the use of public
data from the Swift data archive. This research has made use of data and
software provided by the High Energy Astrophysics Science Archive Research
Center (HEASARC) and NASA’s Astrophysics Data System Bibliographic Services.
This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. TDR acknowledges financial
contribution from the agreement ASI-INAF n.2017-14-H.0. GRS is supported by
NSERC Discovery Grant RGPIN-2021-04001. This work makes use of several python
packages, namely numpy (Oliphant, 2006), astropy (Astropy Collaboration et
al., 2013, 2018), matplotlib (Hunter, 2007), and aplpy (Robitaille & Bressert,
2012).

## Data Availability

A GitHub repository with a jupyter notebook and all underlying data to
reproduce the figures and analysis in this work will be made public upon
acceptance and publication via
https://github.com/jvandeneijnden/MeerKATCampaignOfCenXm4.

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# Criteria for the Application of Double Exponential Transformation

Arezoo Khatibi  University of Kashan,Kashan,Iran  Omid Khatibi††correspond
email: md_khn@yahoo.com,a1449933@unet.univie.ac.at,Faculty of Mathematics,
University of Vienna, Oskar morgernstern platz 1, 1090, Vienna, Austria.
University of Vienna,Vienna,Austria

###### Abstract

The double exponential formula was introduced for calculating definite
integrals with singular point oscillation functions and Fourier-integrals. The
double exponential transformation is not only useful for numerical
computations but it is also used in different methods of Sinc theory. In this
paper we use double exponential transformation for calculating particular
improper integrals. By improving integral estimates having singular final
points. By comparison between double exponential transformations and single
exponential transformations it is proved that the error margin of double
exponential transformations is smaller. Finally Fourier-integral and double
exponential transformations are discussed.
MATHEMATICS SUBJECT CLASSIFICATION: 65D30, 65D32.
Key words: numerical integral, double exponential transformation, single
exponential, Fourier integral

††

## Introduction:

The double exponential transformations (DE) is used for evaluation of
integrals of an analytic function has end point singularity. This
transformation improved frequently transformation like Fourier transformation.
The fact that double exponential transformations error is smaller than single
exponential transformation before this was seen as intuitive, but here we have
proved it through theoretical analysis. Our numerical results are achieved in
the maple software with different numerical results compared with Mori [1].
This article defines the criteria for using double exponential
transformations. This technique can not be applied to sharp points. Sinc
calculations are very accurate and complex. Regarding the above we have
reached a numerical conclusion. In the following sections, we have expanded on
the application of double exponential transformations worked in the Fourier
integrals and also boundary value conditions. In this regard by incorporating
new innovations in the lemma and theorems, we have improved on the above
results. In [4] Ooura expended animt-type quadrature formula with the same
asymptotic performance as the de-formula. Muhammad, Nurmuhammad and Mori the
process by which a numerical solution of integral equations by means of the
Sinc collocation method based on the double exponential transformation has
attracted considerable attention recently [5]. In a mathematically more
rigorous manner, optimality of the double exponential formula-functional
analysis approach is established by Sugihara [6]. Koshihara and Sugihara made
a full study of A Numerical Solution for the Sturm–Liouville Type eigenvalue
Problems employing the double exponential Transformation [8]. Ooura expended a
double exponential formula for the Fourier transformation [10]. Stenger used
DE Formula in Sinc approximations [7]. This article is organized as follows:
In section 2 we state and prove a comparison between errors in DE and SE
transformation. In section 3 we use DE transformation to compute several
integrals and compare the result with that of Mori [1]. In section 4 we
demonstrate an application of DE in solving a boundary value problem in
section 5 we apply the DE transformation to Fourier integral operators and
give a novel proof for the auxiliary theorem 2.

##  1 Conception of Double Exponential Transformation

Let the following integral be

$\int_{a}^{b}\\!f(x)\,\mathrm{d}x,$ (1.1)

such that the interval $(a,b)$ is infinite or half infinite and the function
under integral is analytic on $(a,b)$ and perhaps has singular point in $x=a$
or $x=b$ or both, now consider the change of variables below [9]:

$x=\phi(t),a=\phi(-\infty),b=\phi(+\infty),$ (1.2)

where $\phi$ is analytic on $(-\infty,+\infty)$ and

$I=\int_{a}^{b}\\!f(\phi(t))\phi\prime(t)\,\mathrm{d}t.$ (1.3)

Hence, after the change of variable the integrand decays double exponential:

$\mid f(\phi(t))\phi\prime(t)\mid\approx e^{-ce^{\mid t\mid}},\mid
t\mid\rightarrow\infty,c\textgreater 0.$ (1.4)

By using the trapezoidal formula with mesh size h on (1.3) we have

$I_{h}=h\sum_{k=-\infty}^{+\infty}f(\phi(kh))\phi\prime(kh).$ (1.5)

The above infinite summation is truncated from $k=〖-N〗^{-}$ to $k=N^{+}$ in
computing (1.5) we get

$I_{h}^{(N)}=h\sum_{k=-N^{-}}^{N^{+}}f(\phi(kh))\phi\prime(kh),N=N^{+}+N^{-}+1$
(1.6)

The N+ 1 is due to zero point, for example, $N=30$ we have $N^{-}=14,N^{+}=15$
finally 30=15+14+1. What gives us the authority to truncate n numbers from the
infinite summation? In fact $e^{(-ce^{\mid t\mid})}$ quickly approaches zero.

For the integral over (-1, 1).

$\int_{-1}^{1}\\!f(x)\,\mathrm{d}x,$ (1.7)

consider,

$\phi(t)=\tanh(\frac{\pi}{2}\sinh(t)),$ (1.8)

by substituting the transformation $x=\phi(t)=\tanh(\pi/2\sinh(t))$ into (1.5)
we obtain

$\phi(t)\prime=\tanh\prime(\pi/2\sinh(t))\pi/2\cosh(t)=\left(\frac{1}{\cosh^{2}(\pi/2\sinh(t))}\right)\pi/2\cosh(t).$
(1.9)

Consequently, by using the double exponential formula we will have:

$I_{h}^{(N)}=h\sum_{k=-N^{-}}^{N^{+}}f(\tanh(\pi/2\sinh(kh))\frac{\pi/2\cosh(kh)}{\cosh^{2}(\frac{\pi\sinh(kh)}{2})}.$
(1.10)

##  2 Numerical Experience and Comparison between D.E Approximation Error and
S.E Transformation Approximation Error in Sinc Theory

In this section we prove the main results of our paper. We compare two error
estimates for the second-orders two point boundary problem exponential due to
Horiuchi and Sugihara [2], which combine the double exponential transformation
with the Sinc-Galerkin method, and prove the superiority of one estimate over
the other.

As we laid out in [1], one considers

$\tilde{y}^{{}^{\prime\prime}}(x)+\tilde{\mu}(x)\tilde{y}^{\prime}(x)+\tilde{\nu}(x)\tilde{y}(x)=\tilde{\sigma}(x)\quad
a<x<b,$ (2.1)

$\tilde{y}(a)=\tilde{y}(b)=0$,
by using the variable transformation

$x=\varphi(t)\qquad a=\varphi(-\infty)\qquad b=\varphi(+\infty),$ (2.2)

since, together with the change of notation we have

$y(t)=\tilde{y}(\varphi(t)),$ (2.3)

transforms to

${y}^{{}^{\prime\prime}}(x)+{\mu}(x){y}^{\prime}(x)+{\nu}(x){y}(x)={\sigma}(x)\quad-\infty<t<+\infty,y(-\infty)=y(+\infty)=0,$
(2.4)

the Sinc-Galerkin method have ability to approximate the solution of (2.4) by
combination of Sinc function:

$y_{N}(t)=\sum_{k=-n}^{n}w_{k}S(k,h)(t),N=2n+1.$ (2.5)

Where

$S(k,h)(t)=\frac{\sin(\pi)/h(x-kh)}{\pi/h(x-kh)},k=0,\pm 1,\pm 2,\dotsc$ (2.6)

based on numerical result and theoretical proof the approximation error
estimate by

$|y(t)-y_{N}(t)|\leq c^{\prime}N^{\frac{5}{2}}exp(-c\sqrt{N}),$ (2.7)

when the true solution y(t) of the transform problem decays single
exponentially leads to

$|y(t)|\leq\alpha exp(-\beta|t|).$ (2.8)

Furthermore, the approximation error estimate by

$|y(t)-y_{N}(t)|\leq c^{\prime}N^{2}\exp(\frac{-cN}{\log N}),$ (2.9)

when the true solution y(t) of the transform problem decays double
exponentially leads to

$|y(t)|\leq\alpha\exp(-\beta\exp(\gamma|t|)).$ (2.10)

######  Theorem 2.1.

In The above situation we prove that the approximation error arising from
double exponential decay is less than the approximation error arising from
single exponential decay.

###### Proof.

The single exponential approximation error can be estimated by:

$|y(t)-y_{N}(t)|\leq c^{\prime}N^{\frac{5}{2}}exp(-c\sqrt{N}).$ (2.11)
$y_{N}(t)=\sum_{k=-n}^{n}w_{k}S(k,h)(t),N=2n+1.$ (2.12)

$S(k,h)(t)$ is Sinc function.

The downside is established in accordance with the following decay single
exponential $y(t)$,

$|y(t)|\leq\alpha exp(-\beta|t|).$ (2.13)

The double exponential approximation error can be estimate by

$|y(t)-y_{N}(t)|\leq c^{\prime}N^{2}\exp(\frac{-cN}{\log N}),$ (2.14)

if the true solution of $y(t)$ the transformed problem decay double
exponentially like[1]

$|y(t)|\leq\alpha\exp(-\beta\exp(\gamma|t|)),$ (2.15)

we will prove that the error in the D.E transformation is much less than the
error in the S.E transformation.

For this purpose we need lemma 2.2.

######  Lemma 2.2.

There exists $N_{0}$ such that for all $N>N_{0}$, the inequality
$c^{\prime}N^{2}e^{\frac{-cN}{\log
N}}<c^{\prime\prime}N^{\frac{5}{2}}e^{-c\sqrt{N}}$ hold true.

###### Proof.

$\frac{c^{\prime}}{c^{\prime\prime}}<N^{\frac{1}{2}}e^{\frac{cN}{\log
N}{-c\sqrt{N}}}.$ (2.16)

It is necessary to be prove:

$1<e^{\frac{cN}{\log
N}-c\sqrt{N}},\frac{c^{\prime}}{c^{\prime\prime}}<N^{\frac{1}{2}},$ (2.17)
$1<e^{\frac{cN}{\log N}-c\sqrt{N}}\Longleftrightarrow 0<\frac{cN}{\log
N}-c\sqrt{N},$ (2.18)
$\frac{c^{\prime}}{c^{\prime\prime}}<N^{\frac{1}{2}}\Longleftrightarrow(\frac{c^{\prime}}{c^{\prime\prime}})^{2}<N,$
(2.19) $0<\frac{cN}{\log N}-c\sqrt{N}\Longrightarrow cN>c\sqrt{N}\log
N\Longleftrightarrow\sqrt{N}>\log N\overset{N=\log
N}{\Longleftrightarrow}e^{\frac{\log N}{2}}>\log N,$ (2.20)

according to lemma 2.3 there exists $x_{0}$ such that for,

$x>x_{0},e^{\frac{x}{2}}>x,$ (2.21)

is now

$N>e^{x_{0}}\Longrightarrow\log N>x_{0}\overset{N=\log
N}{\Longleftrightarrow}e^{\frac{\log N}{2}}>\log N,$ (2.22)

then put $N_{0}=max\\{(\frac{c^{\prime}}{c^{\prime\prime}})^{2},e^{x_{0}}\\}$
and the theorem satisfied. ∎

For complete the proof we need other lemma.

######  Lemma 2.3.

For each $a>0$ such that $t>2a$ we have $e^{t}>at$.

###### Proof.

$e^{t}=1+\frac{t}{1!}+\frac{t^{2}}{2!}+\frac{t^{3}}{3!}+\dots>1+t+\frac{t^{2}}{2},\quad
t>0,$ (2.23)

for this we prove that,

$1+t+\frac{t^{2}}{2}>at,$ (2.24)

this yields,

$1+(1-a)t+\frac{t^{2}}{2}>0,$ (2.25) $t>\frac{a-1+\sqrt{(a-1)^{2}-2}}{1},$
(2.26)

and

$t>\frac{a-1-\sqrt{(a-1)^{2}-2}}{1},$ (2.27)

the discriminant of the quadratic equation

$1+(1-a)t+\frac{t^{2}}{2}=0,$ (2.28)

is

$\Delta=(1-a)^{2}-2=a^{2}-2a-1=0,$ (2.29)

if $\Delta\geq 0$ we put $t_{0}$ equal max root in quadratic equation
$t_{0}=a-1+\sqrt{(a-1)^{2}-2}$. Then the statement is satisfied. And note that

$t_{0}<a+\sqrt{a^{2}}=2a.$ (2.30)

If $\Delta<0$ that is the roots are complex and the coefficient of $t^{2}$ is
positive then for all of t inequality is true. Hence we can set $t_{0}=0$ and
this completes the proof.

∎

∎

##  3 NUMERICAL EXPERIENCE

For the following four integral we get:

DEFINT(Table 1) |  |
---|---|---
INTEGRAL | N | abs. error
$I_{1}$ | 25 | 0.370582258
$I_{2}$ | 387 | 0.154551243
$I_{3}$ | 387 | 0.021640875
$I_{4}$ | 259 | 0.001118879

In the above table, the absolute error tolerance is $10^{-8}$ where $N$ is the
number of function evaluations and abs. Error is the actual absolute error of
the result and it gives an approximate value which is correct up to 16
significant digits.

$\displaystyle I_{1}=\int_{0}^{1}X^{\frac{-1}{4}}log\frac{1}{X}dx.\quad
I_{2}=\int_{0}^{1}\frac{1}{16(X-\frac{\pi}{4})^{2}+\frac{1}{16}}dx.\qquad
I_{3}=\int_{0}^{\pi}cos(64sinX)dx.$ $\displaystyle
I_{4}=\int_{0}^{1}exp(20(x-1))sin(256x)dx.$

Our results are very different from [1] which was Previously defined. Compare
quadrature or Horner method, give at least $10^{-5}$ errors so these method
are better than the double exponential transformation method.

##  4 •Application of the Double-Exponential Transformation for the Sinc-
Galerkin Method and the Second-Order Two-Point Boundary Problem

Let

$\tilde{y}^{{}^{\prime\prime}}(x)+\tilde{\mu}(x)\tilde{y}^{\prime}(x)+\tilde{\nu}(x)\tilde{y}(x)=\tilde{\sigma}(x)\quad
a<x<b,$ (4.1)

$\tilde{y}(a)=\tilde{y}(b)=0$,
by using the variable transformation

$x=\varphi(t)\qquad a=\varphi(-\infty)\qquad b=\varphi(+\infty),$ (4.2)

since, together with the change of notation we have

$y(t)=\tilde{y}(\varphi(t)),$ (4.3)

and

$y^{\prime}(t)=\varphi^{\prime}(t)\tilde{y}^{\prime}(\varphi(t)),$ (4.4)

hence

$y^{\prime\prime}(t)=\varphi^{\prime\prime}(t)\tilde{y}^{\prime}(\varphi(t))+\varphi^{\prime}(t)^{2}\tilde{y}^{\prime\prime}(\varphi(t)),$
(4.5)

in virtual point that 4.4 we obtain:

$\tilde{y}^{\prime}(\varphi(t))=\frac{1}{\varphi^{\prime}(t)}y^{\prime}(t).$
(4.6)

In note that 4.5 we have:

$\tilde{y}^{\prime\prime}(x)=-\tilde{\mu}(x)\tilde{y}^{\prime}(t)-\tilde{\nu}(x)\tilde{y}(x)+\tilde{\sigma}(x),$
(4.7)
$y^{\prime\prime}(t)=\varphi^{\prime\prime}(t)\tilde{y}^{\prime}(\varphi(t))+\varphi^{\prime}(t)^{2}(-\tilde{\mu}(\varphi(t))\tilde{y}^{\prime}(\varphi(t))-\tilde{\nu}(\varphi(t))\tilde{y}(\varphi(t))+\tilde{\sigma}(\varphi(t))).$
(4.8)

Our aim is change $\tilde{y}$ to y by using 4.3, 4.8 we get follow equation:

$y^{\prime\prime}(t)=\varphi^{\prime\prime}(t)\frac{1}{\varphi^{\prime}(t)}y^{\prime}(t)+\varphi^{\prime}(t)^{2}(-\tilde{\mu}(\varphi(t))\frac{1}{\varphi^{\prime}(t)}y^{\prime}(t)-\tilde{\nu}(\varphi(t))y(t)+\tilde{\sigma}(\varphi(t))),$
(4.9)

this however, leads to

$y^{\prime\prime}(t)+(\varphi^{\prime}(t)\tilde{\mu}(\varphi(t))-\frac{\varphi^{\prime\prime}(t)}{\varphi^{\prime}(t)})y^{\prime}(t)+\varphi^{\prime}(t)^{2}\tilde{\nu}(\varphi(t))y(t)=\varphi^{\prime}(t)^{2}\tilde{\sigma}(\varphi(t)),$
(4.10)
$\mu(t)=(\varphi^{\prime}(t)\tilde{\mu}(\varphi(t))-\frac{\varphi^{\prime\prime}(t)}{\varphi^{\prime}(t)},$
(4.11) $\nu(t)=\varphi^{\prime}(t)^{2}\tilde{\nu}(\varphi(t)),$ (4.12)
$\sigma(t)=\varphi^{\prime}(t)^{2}\tilde{\sigma}(\varphi(t)),$ (4.13)

and observ that

$y^{\prime\prime}(t)+\mu(t)y^{\prime}(t)+\nu(t)y(t)=\sigma(t).$ (4.14)

By using the boundary conditions and 4.4 we have:

$\begin{cases}\tilde{y}(a)=\tilde{y}(\varphi(-\infty))=y(-\infty)=0\\\
\tilde{y}(b)=\tilde{y}(\varphi(\infty))=y(\infty)=0\end{cases},$ (4.15)

The Sink Galerkin approximation method provides solutions to transform such as
the one in 4.14 by combining it linearly with the Sinc function we have:

$y_{N}(t)=\sum_{k=-n}^{n}W_{k}S(k,h)(t),\quad N=2n+1.$ (4.16)

Every method for trapezoidal numerical integration suggest one method in Sinc
approximation. Horiuchi and Sugihara combine the Sinc-Galerkin method with the
double exponential transformation for the second-order two-point boundary
problem [2]. Mori and Nurmuhammad and Muhammad studied the performance of de-
sinc method when used in the second order singularly perturbed boundary value
problems [3].

##  5  Evaluation of Fourier-type integrals

The double exponential is useful for several integrals for example Fourier
integral but it does not suitable for Fourier-type integrals slowly
oscillatory function. Fourier integral can compute from follow formula:

$\hat{f}(w)=\int_{-\infty}^{+\infty}f(x)e^{ixw}dx.$ (5.1) $e^{ixw}=coswx-
isinwx.$ (5.2)

Therefore we can show sin section with $I_{s}$ sign and cos section with
$I_{c}$ sign then:

$\begin{cases}Is=\int_{0}^{\infty}f_{1}(x)sinwx\\\
Ic=\int_{0}^{\infty}f_{1}(x)coswx\end{cases}.$ (5.3)

we choose $\varphi(t)$ function applies to the following conditions.

$\varphi(-\infty)=0,\varphi(-\infty)=\infty,$ (5.4)

if $t\longrightarrow-\infty$ double exponentially

$\varphi^{\prime}(t)\longrightarrow 0,$ (5.5)

if $t\longrightarrow\infty$ double exponentially

$\varphi(t)\longrightarrow t,$ (5.6)

if $t\longrightarrow 0$ then

$\exists D|\varphi^{\prime}(t)|\leq
Dexp(-cexp|t|)|t|>0\Longleftrightarrow\varphi^{\prime}(t)\approx
exp(-cexp|t|)\longrightarrow 0$ (5.7)

by using 5.6 we have:

$|\frac{\varphi(t)}{t}|\approx exp(-cexp|t|),$ (5.8)

this means having the same exponentially growth ,on the other hand,

$\varphi(t)-t\approx exp(-cexp|t|).$ (5.9)

Now by using change follow variable in transformation $I_{S}$, $I_{C}$ we
have:

$\begin{cases}I_{S}:X=M\varphi(t)/w\\\
I_{C}:X=M\varphi(t-\frac{\pi}{2M})/w\end{cases}(M=constant),$ (5.10)
$X=M\varphi(t)/w\Rightarrow dx=M\varphi^{\prime}(t)/wdt.$ (5.11)

By using 5.3 we obtain:

$I_{S}=\int_{0}^{\infty}f_{1}(x)sinwx=\int_{0}^{\infty}f_{1}(M\frac{\varphi(t)}{w})sin(M\varphi(t))\frac{M\varphi^{\prime}(t)}{w}dt.$
(5.12)

######  Theorem 5.1.

Consider

$\varphi(t)=\frac{t}{1-exp(-ksinht)},$ (5.13)

if $t\longrightarrow-\infty$ double exponentially
$\varphi^{\prime}(t)\longrightarrow 0$.

###### Proof.

$\varphi^{\prime}(t)=\frac{(1-e^{-ksinht})-kcoshte^{-ksinht}t}{(1-e^{-ksinht})^{2}},$
(5.14)

if $t\longrightarrow-\infty$ then

$sinht=\frac{e^{+t}-e^{-t}}{2}\longrightarrow-\infty,$ (5.15)

and observe that

$\varphi^{\prime}(t)=\frac{1}{1-e^{-ksinht}}-\frac{ktcoshte^{-ksinht}}{(1-e^{-ksinht})^{2}},$
(5.16)

if $t\longrightarrow-\infty$ then

$\frac{1}{1-e^{-ksinht}}\longrightarrow 0,$ (5.17)

since by 5.16 have

$=-\frac{ktcoshte^{-ksinht}}{(1-e^{-ksinht})^{2}}.$ (5.18)

However, if the second fraction of 5.16 is zero. Condition 5.5 is satisfied.

######  Lemma 5.2.

If $t\longrightarrow-\infty$ term
$\frac{ktcoshte^{-ksinht}}{(1-e^{-ksinht})^{2}}$ double exponentially approach
to zero.

###### Proof.

Consider

$A=\frac{ktcosht}{1-e^{-ksinht}}\times\frac{e^{-ksinht}}{1-e^{-ksinht}},$
(5.19) $|A|=|A_{1}|\times|A_{2}|,$ (5.20)

such that

$A_{1}=\frac{ktcosht}{1-e^{-ksinht}},$ (5.21)
$A_{2}=\frac{e^{-ksinht}}{1-e^{-ksinht}},$ (5.22)

and

$\lim_{t\longrightarrow-\infty}A_{2}=\lim_{x\longrightarrow\infty}\frac{x}{1-x}=\frac{\infty}{-\infty}=-1,$
(5.23)

so that

$\lim_{t\longrightarrow\infty}|A_{2}|=1.$ (5.24)

For $t_{0}$ large enough, $|A_{2}|<2$,
numerator $A_{1}$ term $ktcosht$ exponentially approach to $\infty$ but
denominator double exponentially approach to $\infty$ then antecedent disable
for neutralize denominator therefore term obtain such that double
exponentially approach to zero with $c$ less than $k$.∎

### Claim:

For $c=\frac{k}{4}$ we must prove

$|A_{1}|<De^{Ce^{|t|}}.$ (5.25)

$sinht$ has constant 2 in its denominator hence everything less than the value
$\frac{k}{2}$ will work well but for the value $k$ itself, the term equal to
$\infty$.

######  Lemma 5.3.

There is $t_{0}$ such that for
$t>t_{0},|1-e^{-ksinht}|>\frac{1}{2}e^{-ksinht}.$

###### Proof.

$\lim_{t\longrightarrow-\infty}\frac{|1-e^{-ksinht}|}{e^{-ksinht}}=\lim_{x\longrightarrow\infty}\frac{|1-x|}{x}=1,$
(5.26)

since $1>\frac{1}{2}$ therefore for $t>t_{0}$ we have:

$\frac{|1-e^{-ksinht|}}{e^{-ksinht}}>\frac{1}{2},$ (5.27)

and

$|A_{1}|<\frac{ktcosht}{\frac{1}{2}e^{-ksinht}}.$ (5.28)

It is trivial instead denominator $A_{1}$ in 5.3 from lemma 5.2 We must prove

$|t|>t_{0},|A_{1}|<De^{-\frac{k}{4}e^{|t|}},$ (5.29)

for this aim we must prove

$|2ktcoshte^{ksinht}e^{\frac{k}{4}e^{|t|}}|<D,$ (5.30)

as a matter of fact we change denominator $A_{1}$ 5.28 to numerator then
implies 5.29 we get:

$t\longrightarrow-\infty,|t|=-t,\text{left
side}=|2ktcoshte^{\frac{k}{2}}e^{t}e^{\frac{-k}{2}}e^{-t}e^{\frac{k}{4}}e^{|t|}|<D,$
(5.31)

this yields

$=|2ktcoshte^{\frac{k}{2}e^{t}}e^{\frac{k}{4}e^{-t}}|<D,$ (5.32)

in view of 5.32 we infer that

$=|2ktcoshte^{\frac{k}{2}e^{t}}e^{\frac{-k}{4}e^{-t}}|<D,$ (5.33)

such that

$=\lim_{t\longrightarrow-\infty}2ktcoshte^{\frac{k}{2}e^{t}}e^{\frac{-k}{4}e^{-t}},$
(5.34)

indeed,we have

$\lim_{t\longrightarrow-\infty}e^{\frac{k}{2}e^{t}}=1,$ (5.35)

and notice that we can write the equation

$=2k\lim_{t\longrightarrow-\infty}coshte^{\frac{-k}{4}e^{-t}},$ (5.36)

instead $cosht=\frac{e^{t}+e^{-t}}{2}$ in 5.36:

$=2k\lim_{t\longrightarrow-\infty}\frac{1}{2}(e^{t}e^{\frac{-k}{4}e^{-t}}+e^{-t\frac{k}{4}e^{-t}}),$
(5.37)

we infer that

$=k\lim_{t\longrightarrow-\infty}(e^{t\frac{k}{4}e^{-t}}+e^{-t\frac{k}{4}e^{-t}})=0.$
(5.38)

∎

######  Lemma 5.4.

$\lim_{t\longrightarrow-\infty}t-\frac{k}{4}e^{-t}=-\infty,$ (5.39)
$\lim_{t\longrightarrow-\infty}-t-\frac{k}{4}e^{-t}=-\infty.$ (5.40)

###### Proof.

$\lim_{t\longrightarrow-\infty}t\times\frac{t-\frac{k}{4}e^{-t}}{t}=-\infty,$
(5.41)

for using L’Hôpital’s rule we multiply $t$ in denominator and numerator in
fact we create $\frac{\infty}{\infty}$.

$\lim_{t\longrightarrow-\infty}t=-\infty,$ (5.42)

and observe that

$\lim_{t\longrightarrow-\infty}\frac{t-\frac{k}{4}e^{-t}}{t}=\lim_{t\longrightarrow-\infty}\frac{1+\frac{k}{4}e^{-t}}{1}=\infty,$
(5.43)

we obtain

$\lim_{t\longrightarrow-\infty}t\times\frac{-t-\frac{k}{4}e^{-t}}{t}=-\infty.$
(5.44)

Hence 5.38 approach to zero therefore 5.31, 5.32 for $t>t_{0}$ is true.
We proved 5.18 approach to zero then we will prove 5.17 approach to zero.
According the 5.2 we have:

$|\frac{1}{1-e^{-ksinht}}|<\frac{1}{\frac{1}{2}e^{-ksinht}},$ (5.45)

and

$=\frac{2}{e^{-k(\frac{e^{t}-e^{-t}}{2})}},$ (5.46)

so that

$=\frac{2}{e^{-k\frac{e^{t}}{2}}e^{k\frac{e^{-t}}{2}}}=\frac{2e^{k\frac{e^{t}}{2}}}{e^{k\frac{e^{-t}}{2}}},$
(5.47)

this , however,leads to

$\lim_{t\longrightarrow-\infty}2e^{k\frac{e^{t}}{2}}=1\Longrightarrow|t|>t_{0},2e^{k\frac{e^{t}}{2}}<1,$
(5.48)

since $t<0$, absolute $t$ equal $-t$ gives :

$|\frac{1}{1-e^{-ksinht}}|<\frac{2}{e^{k\frac{e^{-t}}{2}}}=2e^{-\frac{k}{2}e^{-t}}<De^{-Ce^{|t|}}.$
(5.49)

And the proof is finished, hence the statement is satisfied.∎

∎

### Conclusion.

In this paper the concept of the double exponential transformation was used to
establish this method in spite of usefulness and being attractive it is not
generally, we cannot use this method in all conditions especially in sharp
point case and convergence rate slowly. The result in the oscillatory case is
shown much better in $I_{4}$ than in the other cases $I_{1},I_{2},I_{3}$
(Table 1).

### Appendix A:GALERKIN METHOD

The Galerkin method, which approximates a solution of an equation of the form

$(1-\lambda k)f=g.$ (5.50)

defined on a Banach space $X$, is simple to describe. We start with a set of
basis functions $\\{\psi_{k}\\}_{k=1}^{n}$, and we assume that for each $f\in
X$, we can determine a unique set of numbers $c_{1},\dots,c_{n}$ in
$\mathbb{C}$, such that the projection $P_{n:X\longrightarrow X}$ is defined,
with

$P_{n}f=\sum_{k=1}^{n}c_{k}\psi_{k}.$ (5.51)

It means $p_{n}^{2}=p_{n}$. The Galerkin method then enables us to replace
equation 5.50 by the approximate one, namely 5.52,

$(1-\lambda p_{n}k)f_{n}=p_{n}g,$ (5.52)

why did we instead? Because system 5.50 is infinite system but new system is
system of n equations in n indeterminate.

$(1-\lambda p_{n}k)\sum_{k=1}^{n}c_{k}\psi_{k}=p_{n}g,$ (5.53)

and observe that

$p_{n}g=\sum_{k=1}^{n}d_{k}\psi_{k},$ (5.54)

such that

$(1-\lambda p_{n}k)\sum_{k=1}^{n}c_{k}\psi_{k}=\sum_{k=1}^{n}d_{k}\psi_{k},$
(5.55)

this, however, leads to

$\sum_{k=1}^{n}c_{k}\psi_{k}-\lambda\sum_{k=1}^{n}p_{n}k\psi_{k}=\sum_{k=1}^{n}d_{k}\psi_{k},$
(5.56)

we write $p_{n}k\psi_{k}$ in terms of $\psi_{j}$,

$p_{n}k\psi_{k}=\sum_{k=1}^{n}c_{kj}\psi_{j},$ (5.57)

we infer that

$\sum_{k=1}^{n}c_{k}\psi_{k}-\lambda\sum_{k=1}^{n}\sum_{j=1}^{n}c_{kj}\psi_{j}=\sum_{k=1}^{n}d_{k}\psi_{k},$
(5.58)

we will be equal the coefficients $\psi_{i}$ on both sides,

$c_{i}-\lambda\sum_{k=1}^{n}c_{ki}=d_{i},\qquad i=1,\dots,n$ (5.59)

this process produces n equations, $n$ indeterminate.
In practice, we try to select projection as $p_{n}$, to be comfortable working
with them and $\|k-p_{n}k\|$ being so small [7].

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