# A NICER look at the jet-like corona of MAXI J1535$-$571 through type-B
quasi-periodic oscillations

Yuexin Zhang,1 Mariano Méndez,1 Federico García,2 Diego Altamirano,3 Tomaso M.
Belloni,4 Kevin Alabarta,5,6 Liang Zhang,7 Candela Bellavita,8 Divya Rawat,9
and Ruican Ma1,7,10
1Kapteyn Astronomical Institute, University of Groningen, P.O. BOX 800, 9700
AV Groningen, The Netherlands
2Instituto Argentino de Radioastronomía (CCT La Plata, CONICET; CICPBA; UNLP),
C.C.5, (1894) Villa Elisa, Buenos Aires, Argentina
3School of Physics and Astronomy, University of Southampton, Southampton, SO17
1BJ, UK
4INAF-Osservatorio Astronomico di Brera, via E. Bianchi 46, I-23807 Merate,
Italy
5Center for Astro, Particle and Planetary Physics, New York University Abu
Dhabi, PO Box 129188, Abu Dhabi, United Arab Emirates
6New York University Abu Dhabi, PO Box 129188, Abu Dhabi, United Arab Emirates
7Key Laboratory of Particle Astrophysics, Institute of High Energy Physics,
Chinese Academy of Sciences, Beijing 100049, People’s Republic of China
8Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La
Plata, Paseo del Bosque, B1900FWA La Plata, Argentina
9Inter-University Centre for Astronomy and Astrophysics, Ganeshkhind, Pune
411007, India
10University of Chinese Academy of Sciences, Chinese Academy of Sciences,
Beijing 100049, People’s Republic of China E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

MAXI J1535$-$571 is a black-hole X-ray binary that in 2017 exhibited a very
bright outburst which reached a peak flux of up to 5 Crab in the 2–20 keV
band. Given the high flux, several X-ray space observatories obtained
unprecedented high signal-to-noise data of key parts of the outburst. In our
previous paper we studied the corona of MAXI J1535$-$571 in the hard-
intermediate state (HIMS) with Insight-HXMT. In this paper we focus on the
study of the corona in the soft-intermediate state (SIMS) through the
spectral-timing analysis of 26 NICER detections of the type-B quasi-periodic
oscillations (QPOs). From simultaneous fits of the energy, rms and lag spectra
of these QPOs with our time-dependent Comptonization model, we find that in
the SIMS the corona size is $\sim$ 6500 km and vertically extended. We detect
a narrow iron line in the energy spectra, which we interpret to be due to the
illumination of the outer part of the accretion disk by this large corona. We
follow the evolution of the corona and the radio jet during the HIMS-SIMS
transition, and find that the jet flux peaks after the time when the corona
extends to its maximum vertical size. The jet flux starts to decay after the
corona contracts vertically towards the black hole. This behavior points to a
connection between the X-ray corona and the radio jet similar to that seen in
other sources.

###### keywords:

accretion, accretion discs – stars: individual: MAXI J1535$-$571 – stars:
black holes – X-rays: binaries

††pagerange: A NICER look at the jet-like corona of MAXI J1535$-$571 through
type-B quasi-periodic oscillations–A

##  1 Introduction

Black-hole low-mass X-ray binaries (BH LMXBs) consist of a stellar-mass black
hole with a low-mass companion star overflowing its Roche lobe, forming an
accretion disk (see Belloni et al., 2011; Kalemci et al., 2022, for reviews).
During an outburst, a black-hole X-ray transient traces an anticlockwise ‘q’
path in the hardness-intensity diagram (HID; Homan & Belloni, 2005; Belloni et
al., 2005), during which the proportion of the thermal and Comptonized
components changes leading to several well-defined spectral states (Belloni et
al., 2005). The accretion disk emits thermal emission in the soft X-ray band
with the disk spectrum peaking at around 0.2–2.0 keV; a fraction of the soft
disk photons are Compton up-scattered in the corona, forming a power-law like
component in the energy spectrum in the hard X-ray band up to 100 keV (for
reviews, see Done et al., 2007; Gilfanov, 2010). When the outburst starts, the
X-ray luminosity of the source increases by several orders of magnitude
compared to the quiescent state and the source enters the low-hard state (LHS)
with a dominant hard Comptonized component and a relatively weak thermal disk
component in the X-ray spectrum (e.g. Sharma et al., 2018; Wang-Ji et al.,
2018). As the mass accretion rate increases, the source becomes brighter and
quickly goes into the hard-intermediate state (HIMS), the soft-intermediate
state (SIMS), the high-soft state (HSS), and sometimes an anomalous state at
the highest luminosity (e.g. Méndez & van der Klis, 1997; Belloni et al.,
2005; Belloni, 2010; Motta et al., 2012). From the LHS to the HSS, the
truncated disk moves closer to the innermost circular stable orbit (ISCO)
around the black hole (Esin et al. 1997; but see Reis et al. 2008; Wang-Ji et
al. 2018 and Miller et al. 2018 for a different interpretation) and the
thermal component becomes dominant, while the spectrum of the hard component
becomes steeper and weaker (e.g. Sharma et al., 2018; Dong et al., 2022). With
the gradual decay of the mass accretion rate, the source evolves back to the
LHS and finally returns to the quiescent state. In some so-called failed-
transition outbursts the source stays in the LHS and HIMS and never enters the
SIMS (Alabarta et al., 2021).

The Comptonized photons in the corona can impinge back onto the accretion
disk, resulting in the reprocessing and Compton back-scattering of the hard
photons, which leads to a relativistic reflection component (Fabian et al.,
1989; Bambi et al., 2021). The reflection spectrum includes characteristic
X-ray emission lines among which the most prominent is the iron $K_{\alpha}$
line around 6.4–7.0 keV and the Compton hump around 20 keV (García et al.,
2014). Recent studies have also modeled the relativistic reflection
considering the incident photons coming from a hot blackbody or a returning
reflection spectrum due to the strong gravitational bending (e.g. Connors et
al., 2020; García et al., 2022b; Dauser et al., 2022). A relativistic
reflection component can appear through the spectral evolution from the LHS to
the HSS (e.g. Wang-Ji et al., 2018; Connors et al., 2020; Dong et al., 2022).

The X-ray light curves of black-hole binaries (BHBs) show variability on time
scales from milliseconds to years (see Ingram & Motta, 2019, for a recent
review). In the Fourier domain, the power density spectrum (PDS) of the X-ray
light curve shows different kinds of variability, e.g. broadband noise and
narrow peaks called quasi-periodic oscillations (QPOs; Van der Klis, 1989;
Nowak, 2000; Belloni et al., 2002). The QPOs observed in BHBs are classified
as low-frequency QPOs (LFQPOs), with central frequencies ranging from mHz to
$\sim$ 30 Hz, and high-frequency QPOs (HFQPOs), with central frequencies
$\gtrsim$ 60 Hz (Belloni & Stella, 2014). Depending on the shape of the
broadband noise, the fractional root-mean-square amplitude (hereafter rms),
the phase lags of the QPOs and the spectral state of the source, LFQPOs are
divided into three classes, namely type A, B, and C (Remillard et al., 2002;
Casella et al., 2004; Casella et al., 2005). Type-C QPOs generally appear in
the hard states (both LHS and HIMS) with central frequency in the $\sim$
0.01–30 Hz range, high rms (up to 20%), strong band-limited noise, and usually
second and sub harmonics. Type-B QPOs only appear in the SIMS (Belloni &
Motta, 2016), with central frequency below $\sim$ 6 Hz, low rms ($\lesssim
5\%$), weak red noise, and sometimes the second and sub harmonics. Type-A
QPOs, which are rarely detected, appear in the SIMS and HSS, with central
frequencies in the $\sim$ 6–8 Hz range, very weak rms and no harmonics. In the
short-lived SIMS, fast transitions are sometimes observed between the type-B
and either other types of QPOs or the disappearance of QPO (e.g. Casella et
al., 2004; Zhang et al., 2021).

The Fourier cross spectrum of two simultaneous light curves in different
energy bands (subject and reference band) can be used to compute the phase
lags as a function of Fourier frequency. It is common to report lags of the
signals identified in the PDS extending over a given frequency range, e.g. the
broadband noise and the QPOs (Nowak et al., 1999). Hard (positive) lags
(Miyamoto et al., 1988) can be produced by propagation of fluctuations of the
mass accretion from the outer part towards the inner part of the disk and the
corona (e.g. Arévalo & Uttley, 2006; Ingram & van der Klis, 2013). Soft
(negative) lags can be due to reverberation or thermalization of hard photons
when the corona photons impinge back onto the accretion disk (e.g. Uttley et
al., 2014; Karpouzas et al., 2020).

Apart from the X-ray emission, BHBs radio emission from a jet is sometimes
prominent, and can be classified into two types depending on the radio
spectral index and the morphology of the jet: a small-scale, optically thick,
compact jet and an extended, optically thin, transient jet (for a review, see
Fender, 2006). The relation between the spectral states and the radio emission
indicates the existence of an accretion-ejection coupling. In the LHS and the
HIMS, a compact jet is observed, while during the state transition from the
HIMS to the SIMS, the compact jet can be quenched for a few days (Fender et
al., 2004). In the SIMS, there is no longer a compact jet but a bright
transient jet appears with observable discrete relativistic ejecta, while in
the HSS, the jet disappears (e.g. Mirabel & Rodríguez, 1994; Corbel et al.,
2004; Ingram, 2019). Time variability in the radio band sometimes appears, but
much less often than in the X-ray band (e.g. Tetarenko et al., 2019, 2021).
Type-B QPOs in the X-ray band are usually thought to be connected to the
relativistic transient jet but the exact mechanisms that explain the
connection are still unknown (e.g. Fender et al., 2009; Homan et al., 2020;
García et al., 2021).

There are still many open questions regarding the the disk-corona-jet
evolution, for instance, the disk truncation during the evolution of the black
hole transients (Esin et al., 1997), the nature of the corona (the inner hot
flow, disk sandwich, or the base of the jet; Galeev et al., 1979; Haardt &
Maraschi, 1991; Markoff et al., 2005), and the geometry of the corona and its
typical size (Martocchia & Matt, 1996; Lee & Miller, 1998). The geometry of
the corona and its connection with the disk and the jet are still to be
understood. A universal radio–X-ray correlation in the LHS provides evidence
that the corona can be related to the radio jet (e.g. Gallo et al., 2003;
Fender et al., 2004). Using data of GRS 1915+105, Méndez et al. (2022)
proposed that (part of) the spread in the radio–X-ray correlation (Gallo et
al., 2012) could be due to changes of the corona temperature. Studies of the
spectral energy distribution (SED) from the radio to the X-ray band shows that
the corona emission can originate from a shock acceleration region of tens of
gravitational radii ($R_{g}$) as the jet base (Markoff et al., 2001; Connors
et al., 2019; Cao et al., 2022). The spectral analysis of the reflection
component in MAXI J1820+070 suggests that the corona outflows with a higher
relativistic velocity as it is closer to the black hole (You et al., 2021).
Through X-ray variability studies, the size of the corona shows a continuous
evolution during the outburst that could be related to the change of the radio
jet emission, suggesting a disk-corona-jet connection (Kara et al., 2019; Wang
et al., 2021; Méndez et al., 2022; Zhang et al., 2022).

Many corona models of time variability have been proposed to explain the
corona geometry. Using the broadband noise, reltrans models the reverberation
lags and measures the corona height assuming a lamppost geometry of the corona
(Ingram et al., 2019). Assuming the corona is a wide, low-velocity, wind-like
structure, a corona outflow model is proposed to explain the observed
correlations of, for instance, the power-law photon index and time lags and
the photon index and radio flux (Kylafis et al., 2008; Kylafis & Reig, 2018).
The propagation of mass accretion rate fluctuations model assumes that the
corona lies in a truncated disk. This model follows the variability (both QPOs
and broadband noise) and also fits the time-averaged energy spectrum (e.g.
Ingram & van der Klis, 2013; Zdziarski et al., 2021; Kawamura et al., 2022).
The JED-SAD model assumes that the hard part of the spectrum comes from a jet-
emitting disk (JED) and the soft part from a standard accretion disk (SAD)
(Ferreira, 1997; Petrucci et al., 2008; Marcel et al., 2018). The JED-SAD
model not only explains the ‘q’ path in the HID in terms of transitions
between accretion modes, but also matches the observed variability like the
LFQPOs and the hard-soft lags (Marcel et al., 2019, 2020). The dynamical
origin of the LFQPOs can be explained by the Lense-Thirring (L-T) precession
of the corona which also restricts the corona region within a truncated disk
or a jet base (Stella & Vietri, 1998; Ingram et al., 2009; You et al., 2018;
Ma et al., 2021), or instabilities in the disk accretion flow (Tagger &
Pellat, 1999).

Recently Karpouzas et al. (2020) and Bellavita et al. (2022) developed a time-
dependent Comptonization model called vKompthdk that explains the radiative
properties (rms and phase lags) of QPOs and measures the corona geometry
around the black hole. The vKompthdk model assumes that the temperatures of
the disk and the corona and the rate at which the corona is heated up
oscillate coherently at the frequency of the QPO. The unspecified external
heating source is required to keep the temperature of the corona that,
otherwise, would cool down very quickly and become undetectable. In this
model, a hot spherical corona partially covers the soft disk and scatters the
seed photons from the disk into the Comptonized hard photons. Note that we
assume that the temperature of the corona is constant with radius (Sunyaev &
Titarchuk, 1980). While inverse-Comptonization scattering in the corona may
lead to a radial dependence of the corona temperature, this effect is likely
negligible, especially at a large radii (Meyer-Hofmeister et al., 2012). Part
of the out-going hard photons are observed while the others feedback onto the
soft disk, are reprocessed, and finally reach thermal equilibrium with the
disk. The steady state energy spectrum produced by vKompthdk is the same as
that of nthcomp (Zdziarski et al., 1996; Życki et al., 1999). The inverse-
Compton scattering of the soft photons in the corona results in hard lags
while the reprocessing of the hard photons in the disk leads to soft lags,
both of which can be reproduced by vKompthdk to predict the size of the
corona. We note that the model defines the flux of the feedback hard photons
divided by the flux of the total Comptonized photons as an intrinsic feedback
fraction, $\eta_{\text{int}}$. Apart from the $\eta_{\text{int}}$, in the
model an explicit feedback fraction parameter, $\eta$, is the flux of the
feedback hard photons divided by the flux of the observed soft disk, thus
$\eta$ is in the range of 0–1. The intrinsic feedback fraction indicates the
efficiency of hard photons that feedback onto the disk. Combining the
measurements of the corona size and the feedback fraction, we can understand
to what extent the corona covers the disk and whether the shape of the corona
is sphere-like or jet-like (Karpouzas et al., 2021; García et al., 2021;
Méndez et al., 2022; Zhang et al., 2022).

MAXI J1535$-$571 is an X-ray transient discovered by MAXI/GSC and Swift/BAT
independently when it went into outburst on 2017 September 2 (Kennea et al.,
2017; Negoro et al., 2017). The source reached a peak flux of up to 5 Crab in
the 2–20 keV band (Nakahira et al., 2017). It has been proposed that MAXI
J1535$-$571 has a rapidly spinning (> 0.84) black hole and subtends a high
inclination angle (> 60∘) through the modeling of the relativistic reflection
component (Miller et al., 2018; Xu et al., 2018; Dong et al., 2022). Dong et
al. (2022) also reported that the corona temperature increases from 18 keV to
> 300 keV as the source evolves from the LHS to the SIMS. In the soft state,
the luminosity of MAXI J1535$-$571 is near the Eddington luminosity and the
structure of the standard disk is likely to become slim (Tao et al., 2018).
Timing studies of MAXI J1535$-$571 show that through the outburst, different
types of LFQPOs appear and the LFQPOs evolve as the spectral state changes
(Huang et al., 2018; Stevens et al., 2018). Bhargava et al. (2019) showed a
correlation between the frequency of type-C QPOs and the hard photon index,
indicating a connection between the timing features and the spectral
parameters. From the radio observations of MAXI J1535$-$571, the jet is first
a compact jet in the HIMS, then quenches during the transition from the HIMS
to the SIMS, and finally a transient jet appears in the SIMS (Russell et al.,
2019, 2020; Chauhan et al., 2019).

In this paper, we continue our previous study of the corona geometry of MAXI
J1535$-$571 and the connection between the X-ray corona and the radio jet
through the type-C QPOs in the HIMS using Insight-HXMT observations (Zhang et
al., 2022). We further explore the corona properties in the SIMS through the
type-B QPOs, which are weak and much less frequently detected, using NICER
observations. We fit jointly the rms and phase-lag spectra of the type-B QPO
and the time-averaged energy spectra of the source using the latest version of
the time-dependent Comptonization model vKompthdk (Bellavita et al., 2022).
This paper is organized as follows: In section 2 we describe the data
reduction of the NICER observations of MAXI J1535$-$571, and explain how we
calculate the rms and phase lags of the type-B QPO in different energy bands.
We also explain the parameter settings of the model used to fit jointly the
rms and phase-lag spectra of the QPO and the time-averaged energy spectra of
the source. In section 3 we show the X-ray temporal evolution of MAXI
J1535$-$571, the rms and the phase-lag spectra of the identified type-B QPO,
and the spectral parameters measured from the joint spectral fitting. Finally,
in section 4 we discuss our results and compare them with previous studies of
type-B QPOs and the corona geometry. In that section, we combine the
properties of the corona measured in this work with our previous results and
propose a more complete picture of the corona-jet connection during the whole
evolution of MAXI J1535$-$571 from the HIMS to the SIMS.

##  2 Observations and data analysis

The Neutron Star Interior Composition Explorer (NICER; Gendreau et al., 2016)
observed MAXI J1535$-$571 from 2017 September 7 to 2019 May 11 for a total of
219 observations. We use the standard NICER reduction routine nicerl2 with
CALDB version xti20210707 to process the data. We remove the data of detectors
# 14 and # 34 which are affected by episodes of increased electronic noise. We
require the pointing direction of the instrument to be less than 0.015∘
offset, at least 40∘ above the bright Earth limb, and at least 30∘ above the
Earth limb. For a bright source like MAXI J1535$-$571, we apply an undershoot
count rate range 0–200 per module (underonly range). We set the column types
of the prefilter to be NICERV3 and 3C50 as the recommended background columns.
We require each GTI to be longer than 16 s and split the observations into
separate orbits.

###  2.1 Light curve and hardness-intensity diagram

We focus on the 36 observations in the period MJD 58008–58036, i.e. the
outburst before the four reflares (Cúneo et al., 2020). We use XSELECT to
extract light curves at 1-s resolution for each orbit in the 1–3 keV and the
3–10 keV bands. We exclude the data below 1 keV since we find that the
interstellar absorption towards the source is very high (see subsection 3.2),
and there is no significantly detected emission from the source below that
energy. In order to obtain the HID, for each orbit we use the average count
rate in the 1–10 keV band and calculate the hardness ratio using the ratio of
the average count rate in the 3–10 keV band and the 1–3 keV band.

###  2.2 Energy spectra

We extract the energy spectra of the source and the background in each orbit
using the nibackgen3C50 tool (Remillard et al., 2022), and use nicerarf and
nicerrmf to generate the ancillary files and the response files, respectively.
We group the spectrum such that we have at least 25 counts per bin and we
oversample the intrinsic resolution of the instrument by a factor of 3.

We fit the energy spectra using XSPEC v12.12.1 (Arnaud, 1996). For the time-
averaged energy spectrum of MAXI J1535$-$571, following Miller et al. (2018)
we use the 2.3–10 keV energy band in order to avoid the calibration in the Si
band (1.7–2.1 keV) and the Au band (2.2–2.3 keV), but we notice the energy
range between 2.1 keV and 2.2 keV. A systematic error of 0.5% is added in
quadrature to the model when performing the spectral fitting.

We apply the model TBfeo*(diskbb+vKompthdk+gaussian), hereafter Model 1, to
fit the energy spectra. The component TBfeo models the Galactic absorption
towards the source with variable oxygen and iron abundances. We set the cross
section and the solar abundance of the ISM using the tables of Verner et al.
(1996) and Wilms et al. (2000), respectively. The component diskbb (Mitsuda et
al., 1984) represents the emission of a multi-temperature optically thick and
geometrically thin disk with parameters being the disk temperature,
$kT_{\text{in}}$, and a normalization. The time-averaged or steady-state
version of the time-dependent Comptonization model vKompthdk (Karpouzas et
al., 2020; Bellavita et al., 2022) is the same as nthcomp (Zdziarski et al.,
1996; Życki et al., 1999). The parameters of this component are the seed
photon temperature, $kT_{\text{s}}$, the corona temperature, $kT_{\text{e}}$,
the photon index, $\Gamma$, and a normalization. The optical depth, $\tau$, is
a function of $kT_{\text{e}}$ and $\Gamma$ in vKompthdk:

$\tau=\sqrt{\frac{9}{4}+\frac{3}{\left(kT_{\text{e}}/m_{\text{e}}c^{2}\right)\left(\left(\Gamma+1/2\right)^{2}-9/4\right)}}-\frac{3}{2},$
(1)

where $m_{\text{e}}$ and $c$ are the mass of electron and the speed of light,
respectively, and the Compton y-parameter (Zel’dovich & Shakura, 1969; Shapiro
et al., 1976),
$y=\text{max}(\tau,\tau^{2})\,4kT_{\text{e}}/(m_{\text{e}}c^{2})$, drives the
shape of the spectrum. The seed photon temperature, $kT_{\text{s}}$, in
vKompthdk is linked to the inner disk temperature, $kT_{\text{in}}$, in
diskbb. In fact vKompthdk contains four extra parameters that only affect the
time-dependent spectrum produced by this model, and describe the corona size,
$L$, the feedback fraction, $\eta$, the amplitude of the variability of the
rate at which the corona is heated by an (unspecified) external source,
$\delta\dot{H}_{\text{ext}}$, and an additive parameter, reflag, that gives
the phase lag in the 2–3 keV band111Since in the data the reference band used
to compute the lags is arbitrary, the lags are defined up to an additive
constant.. Note that none of these four parameters changes the steady-state
spectrum produced by vKompthdk. These four parameters, plus $kT_{\text{s}}$,
$kT_{\text{e}}$, and $\Gamma$, describe the radiative properties of the QPOs,
i.e. the rms and the phase lags. We add a Gaussian component to fit the broad
iron $K_{\alpha}$ emission line in the spectrum. We do not use more
complicated reflection models like relxill, since in the energy band of NICER
we only detect the iron line feature and we do not have data above 10 keV
where the Compton hump appears.

###  2.3 Power spectrum

We generate the PDS for each orbit in the 1–10 keV energy band and in five
narrow bands, 1–2.5, 2.5–3.9, 3.9–5.5, 5.5–7.3, 7.3–10.0 keV bands. The length
of each PDS segment is 8.192 s and the Nyquist frequency is 125 Hz. In each
orbit we average all the PDS segments, subtract the Poisson noise using the
average power from 100 Hz to 125 Hz in the PDS and normalize the PDS to
fractional rms amplitude (Belloni & Hasinger, 1990):

$\text{rms}=\frac{\sqrt{P(S+B)}}{S},$ (2)

where $P$ is the power in Leahy units (Leahy et al., 1983), and $S$ and $B$
are, respectively, the source and background count rates. We apply a
logarithmic rebin in frequency to the PDS such that the bin size increases by
a factor $\exp(1/100)$ compared to the previous one.

We fit the averaged PDS of each orbit in the 1–10 keV band with Lorentzian
functions (Nowak, 2000; Belloni et al., 2002) in the frequency range of 0.1–30
Hz. The parameters of a Lorentzian function are the central frequency, the
full width at half maximum (FWHM), and the normalization. We fit the PDS with
four Lorentzians representing the QPO fundamental, the second harmonic, and
two broadband noise components. An extra Lorentzian is needed if there is a
QPO sub harmonic. All parameters in all the Lorentzians are free, but we
freeze the central frequency of one Lorentzian function which fits the
broadband noise at 0. If there is no QPO, we reduce the number of Lorentzian
functions to one or two to only fit the broadband noise. Using the models that
we apply to fit the PDS of each orbit in the 1–10 keV band as baselines, we
further fit the PDS of each orbit in the five narrow energy bands (see above).
We fix all central frequencies and the FWHMs of all the Lorentzians to the
values we obtained for the PDS in the 1–10 keV band and only let the
normalizations free to fit the PDS in those five narrow energy bands. We also
check that the central frequency and width of those Lorentzians do not change
significantly with energy. Finally we take the square root of the
normalizations of the Lorentzians that represent the variability components to
calculate the rms.

###  2.4 Phase lag spectrum

We generate and average the Fast Fourier Transformation (FFT) in each orbit
using the 1–10 keV energy band as reference band to compute the phase lags.
The length of each FFT segment is 8.192 s and the Nyquist frequency is 125 Hz.
We also compute the FFT of the data in the energy bands 1–2.5 keV, 2.5–3.9
keV, 3.9–5.5 keV, 5.5–7.3 keV, 7.3–10.0 keV that we use as the subject bands
to compute the cross spectrum for each subject band with respect to the
reference band.

Considering that the cross spectrum as a function of Fourier frequencies is
$G(f)=\text{Re}\,G(f)+i\,\text{Im}\,G(f)$, where the phase lag is
$\phi(f)=\tan^{-1}\left(\text{Im}\,G(f)/\text{Re}\,G(f)\right)$, we can
calculate the phase lags of the variability components that we measure in the
PDS (Vaughan & Nowak, 1997; Nowak et al., 1999; Ingram, 2019). As in Peirano &
Méndez (2022) and Alabarta et al. (2022), we perform a simultaneous fitting of
both the real and the imaginary parts of $G(f)$ using the same Lorentzian
components that we used to fit the PDS. We add a constant in the model to fit
the correlated part of the signal that arises because the subject bands are
always within the 1-10 keV reference band (Ingram, 2019). This constant is not
linked between the real and the imaginary parts of the cross spectra. We fix
the central frequency and the FWHM of all the Lorentzians at the same values
that we get from fitting the PDS. For the real part, we fit the normalizations
of the Lorentzians, while for the imaginary part, we define an extra free
parameter, $\phi$, for each Lorentzian so that the normalization of a
Lorentzian in the imaginary part is equal to $\tan{\phi}$ times the
normalization of the corresponding Lorentzian in the real part. We note that
this assumes that the phase lag of each component is constant with frequency
(Peirano & Méndez, 2022). Using this method, for the energy bands we defined,
we compute all the phase lags of all the variability components fitted by the
Lorentzian functions.

##  3 Results

###  3.1 Light curve and hardness-intensity diagram

Figure 1: Light curve (left) and hardness-intensity diagram (right) of MAXI
J1535$-$571 during the main outburst from MJD 58008 to 58037. Each point
corresponds to a NICER orbit. The grey, red, and blue points indicate
observations with, respectively, no QPO, type-B QPOs, and type-C QPOs. The
error bars indicating 68% confidence level are smaller than the size of the
points. The grey dashed lines connect the data points in time sequence.

Fig. 1 shows the light curve (left) and the HID (right) of MAXI J1535$-$571 in
the time period MJD 58008–58036. Each data point corresponds to one NICER
orbit. During the outburst, the source moves in an anticlockwise direction in
the HID. From MJD 58008 to 58011, MAXI J1535$-$571 stays in the LHS with a
count rate $\sim$ 8000 cts/s and a hardness ratio increasing from $\sim$ 0.6
to $\sim$ 0.64. From MJD 58011 to 58017 the count rate increases quickly from
$\sim$ 7000 cts/s to $\sim$ 18000 cts/s while the hardness ratio increases
slowly from $\sim$ 0.63 to $\sim$ 0.64 with some excursions up to a maximum of
0.66. At this point the source moves to the left in the HID and transits from
the LHS to the HIMS (e.g. Belloni et al., 2005). From MJD 58017 to 58024 the
count rate gradually decreases to $\sim$ 15000 cts/s, while the hardness ratio
decreases from $\sim$ 0.62 to $\sim$ 0.56, indicating a state transition from
the HIMS to a softer state. During the final part of the outburst, from MJD
58024 to 58036, the count rate continues decreasing gradually to 10000 cts/s,
while the hardness ratio decreases from 0.6 to 0.56.

We mark with blue in Fig. 1 the observations where we detect the type-C QPOs.
The data of the type-C QPOs are from Rawat et al. (2023) who performed a
systematic study of the type-C QPOs in the NICER observations of MAXI
J1535$-$571\. The observations mark the hard and the hard-intermediate states,
and do not include the observations of the soft-intermediate and the soft
states where weak QPOs may appear (e.g. Belloni & Stella, 2014, for a review).
In the top part of the HID (Fig. 1) we plot in grey observations with no
significant QPO in the PDS. The remainder of the orbits, 26 in total, show a
QPO with a rather constant frequency of $\sim$ 5–6 Hz. The information of the
26 orbits is listed in Tab. 3. We plot these observations in Fig. 1 with red.
Since these QPOs in the SIMS are weak and their central frequencies are within
a small range, to improve the signal-to-noise ratio we average the PDS of the
26 orbits. The left panel of Fig. 2 shows the averaged PDS. Based on the fact
that both the QPO fundamental and the second harmonic appear and given the
weak rms of the zero-centered broadband noise (0.7%; see Tab. 1), we
tentatively identify these QPOs as type-B QPOs.

Since the red points in the HID (the right panel of Fig. 1) extend to a very
soft region with hardness ratio less than 0.59, some of the QPOs we detected
may be type-A (e.g. Casella et al., 2004). To check whether all our detections
are type-B QPOs we divide the observations into two groups with hardness ratio
either greater or smaller than 0.59. We find that in both groups the averaged
PDS are consistent with the PDS shown in Fig. 2. We therefore conclude that
all the QPOs that appear in the observations marked with red in Fig. 1 are
type-B QPOs.

###  3.2 Initial fitting to the time-averaged energy spectra

We first fit all the energy spectra of the 26 orbits with the type-B QPOs
separately using the model TBfeo*(diskbb+vKompthdk+gaussian), as introduced in
subsection 2.2. The best-fitting values of $\Gamma$, $kT_{\text{in}}$ and the
iron line profiles in the separate orbits are consistent with being the same
within errors and only the normalizations of the Comptonized and the disk
components change. Given that $kT_{\text{e}}$ cannot be constrained using the
NICER spectra, we fix it to 250 keV.

Since the only parameters that change for the different orbits are the
normalizations of the disk and the corona components, to improve the signal-
to-noise ratio (SNR) we fit the energy spectra of the 26 orbits together,
using the same model TBfeo*(diskbb+vKompthdk+gaussian) linking all the
parameters in each of the 26 data groups, except for the normalizations of
diskbb and vKompthdk components. Note that this fitting can also make it more
convenient for our later joint fitting in subsection 3.4. This spectral
fitting shows that, at this stage, the electron temperature of the corona
$kT_{\text{e}}$ still cannot be constrained, therefore we fix it again to 250
keV. We find that the best-fitting hydrogen column density is $4.2\times
10^{22}$ $\text{cm}^{-2}$, consistent with previous measurements (Miller et
al., 2018). The disk temperature is 1.14 keV, while the corona photon index is
2.84, indicating that the source is in a relatively soft state (Dong et al.,
2022). The gaussian used to fit the iron line has a central energy of 6.8 keV
with a width of 0.98 keV. For the 26 orbits, the 0.5–10 keV averaged flux of
the disk and the Comptonized components is, respectively, $1.6\times 10^{-7}$
erg cm-2 s-1 and $0.4\times 10^{-7}$ erg cm-2 s-1.

###  3.3 Fractional rms and phase-lag spectra

Figure 2: Left panel: Averaged PDS of MAXI J1535$-$571 in the 1–10 keV band
for the NICER orbits listed in Tab. 3, with the best-fitting model and the
residuals. The black points are the data and the red line is the best-fitting
model. The dashed lines from the left to the right show two broadband noise,
the type-B QPO fundamental and the second harmonic components. The residuals
are the data minus the model divided by the error. Right panel: The QPO
central frequency vs. energy for the type-B QPO. The error bars indicate the
68% confidence level. Figure 3: Fractional rms amplitude spectrum (left panel)
and phase-lag spectrum (right panel) of the type-B QPO of MAXI J1535$-$571\.
The dashed grey line in the right panel marks the zero phase lag. In both
panels, the vertical error bars indicates the 68% confidence level, while the
horizontal error bars indicate the width of the energy channels.

As shown in the left panel of Fig. 2, we fit the PDS with four Lorentzians
representing two broadband noise components, the QPO fundamental, and the
second harmonic. Tab. 1 gives the best-fitting parameters of the Lorentzians.
The average type-B QPO has a central frequency of $5.64\pm 0.07$ Hz with a
FWHM of $2.9\pm 0.3$ Hz. The type-B QPO is weak, with an rms of $1.1\pm
0.05\%$. The second harmonic of the type-B QPO appears at a central frequency
of $10.5\pm 0.5$ Hz, consistent with being twice the central frequency of the
QPO fundamental.

We plot the rms spectrum of the QPO in the left panel of Fig. 3. The
horizontal error bars are the width of the energy channels. As the energy
increases from 1 keV to 10 keV, the rms increases monotonically from $\sim
1\%$ to $\sim 6\%$, indicating that the type-B QPO is mainly modulated by the
hot corona.

The right panel of Fig. 3 shows the phase-lag spectrum of the type-B QPO. The
horizontal error bars are the width of the energy channels. As the energy
increases from 1 keV to 7.3 keV, the phase lags decrease from 0.8 rad to -0.7
rad, while above 7.3 keV the phase lags increase slightly to around -0.6 rad,
resulting in a minimum in the phase-lag spectrum at around 6–7 keV. The energy
where the minimum phase lag appears is consistent with the results of Stevens
et al. (2018). The phase lag with respect to the the 1–10 keV reference band
crosses the zero line (grey dashed line in the right panel of Fig. 3) from the
positive to the negative at an energy of $\sim$ 3 keV.

Table 1: The parameters of the Lorentzians used to fit the averaged PDS of
MAXI J1535$-$571\. The error bar indicates the 1-$\sigma$ confidence level.

Component | Frequency (Hz) | FWHM (Hz) | rms (%)
---|---|---|---
QPO | $5.64\pm 0.07$ | $2.9\pm 0.3$ | $1.11\pm 0.05$
Harmonic | $10.5\pm 0.5$ | $3\pm 2$ | $0.43\pm 0.08$
BBN1∗ | $0$ | $0.5\pm 0.1$ | $0.73\pm 0.08$
BBN2∗ | $0.8\pm 0.1$ | $2.5\pm 0.2$ | $1.40\pm 0.07$

  * *

BBN: Broadband noise component.

###  3.4 Joint fitting of the rms and phase-lag spectra of the QPO and the
energy spectra of the source

Similar to what we did in Zhang et al. (2022), we fit jointly the rms and
phase-lag spectra of the QPO and the energy spectra of the 26 orbits of MAXI
J1535$-$571 (the latter are initially fitted in subsection 3.2) using the
model TBfeo*(diskbb+vKompthdk+gaussian).

Note that for the fitting of the rms spectrum, vKompthdk is multiplied by a
dilution component which is not explicitly written in the total model. The
introduction of the dilution model is justified since the time-dependent
Comptonization model computes the rms spectrum of a variable corona but the
observed rms is reduced by all the non-variable spectral components. Here, we
assume that the diskbb and the gaussian components are not variable, and
therefore the dilution component is
$\text{Flux}_{\text{Compt}}(E)/\text{Flux}_{\text{Total}}(E)$ such that
$\text{rms}_{\text{Obs}}=\text{rms}_{\text{Compt}}*\text{Flux}_{\text{Compt}}/\text{Flux}_{\text{Total}}$.
(Note that this dilution component does not introduce any new parameters to
the fits.)

Initially, when we fit simultaneously the energy spectra of the 26 orbits and
the rms and the phase-lag spectra of the QPO, $kT_{\text{s}}$ in vKompthdk is
linked to $kT_{\text{in}}$ in diskbb. The fit yields a relatively large
$\chi^{2}=104.0$ using 10 bins for the timing spectra and a corona size of
$\sim 10^{4}$ km. The large corona size indicates that the softer seed photons
from the outer part of the disk play an important role in producing the phase
lags. Therefore, we let $kT_{\text{s}}$ in vKompthdk linked with
$kT_{\text{in}}$ in diskbb to fit the energy spectra of the 26 orbits, but
allow $kT_{\text{s}}$ to vary freely when fitting the rms and phase-lag
spectra. We show a comparison of the best-fitting results of the cases with
$kT_{\text{s}}=kT_{\text{in}}$ and $kT_{\text{s}}$ free in Fig. 4. Letting
$kT_{\text{s}}$ free gives a significantly better fit, $\chi^{2}=62.3$ (Tab.
2), with $kT_{\text{s}}=0.59\pm 0.07$ keV, lower than $kT_{\text{in}}=1.139\pm
0.002$ keV; the corona is still relatively large with a size of $6500\pm 500$
km. In the time-averaged energy spectrum $kT_{\text{in}}$ and $\Gamma$ are the
same as when $kT_{\text{s}}=kT_{\text{in}}$. The feedback fraction pegs at the
upper boundary 1, which gives $\eta_{\text{int}}=0.33\pm 0.02$. (For an
explanation of the difference between the $\eta$ and $\eta_{\text{int}}$, see
section 1.) This means that $\sim 33\%$ of the corona photons return to the
accretion disk where they are thermalized and re-emitted, resulting in a soft
lag, while the other 67% of the corona photons are the observed Comptonized
photons. The temperature of the inner disk, $kT_{\text{in}}=1.139\pm 0.002$
keV, and the photon index, $\Gamma=2.91\pm 0.06$, indicate that the source is
in a relatively soft state. The spectral state matches well with the fact that
the type-B QPO appears in the SIMS. Since we now include the rms and phase-lag
spectra of the QPO and let $kT_{\text{s}}$ free in the fits, we try to let
$kT_{\text{e}}$ free. From the joint fitting, we measure a hot corona with
temperature $kT_{\text{e}}=330\pm 50$ keV, which is consistent with the corona
temperature measured by Dong et al. (2022) in the SIMS of MAXI J1535$-$571
using the broadband data of NuSTAR and Insight-HXMT.

From the values of $kT_{\text{e}}$ and $\Gamma$ from Model 2, the optical
depth of the corona is $\tau=0.16$ and the Compton y-parameter is $y=0.41$,
indicating that the system is in the unsaturated Comptonization regime. This
is consistent with the assumptions of the time-dependent Comptonization model
vKompthdk that we used.

We note that, although a fit with a dual corona (see García et al., 2021) may
reduce further the $\chi^{2}$, we do not explore it here because of the
limited number of energy bands in the rms and phase-lag spectra of the QPO and
the large number of free parameters that would be involved.

Since the corona is large, it should illuminate the outer part of the disk and
produce a narrow iron line due to reflection off cold material there (e.g.
Miller et al., 2018). Therefore, we add an extra Gaussian line to Model 1,
which then becomes Model 2: TBfeo*(diskbb+vKompthdk+gaussian1+gaussian2). We
find that a second Gaussian line centered at $6.62\pm 0.02$ keV with a width
of $0.12\pm 0.02$ keV fits the data well. The significance of the narrow
Gaussian line, measured as the ratio of the line normalization to its error,
is 5 $\sigma$ and an F-test yields a probability of $1.8\times 10^{-9}$, which
indicates that the narrow Gaussian line improves the fit significantly. The
final results of the fitting are plotted in Fig. 5 and the parameters are
given in Tab. 2.

To test a possible degeneracy of the parameters, we run an MCMC simulation for
Model 2, using the Goodman-Weare chain algorithm (Goodman & Weare, 2010).
After testing the convergence of the chain, we set the chain length to 240000
with 1200 walkers. We discard the first 120000 steps and record the last
120000 steps. The entire covariance matrix is divided by a factor of 10000 to
ensure that the walkers initially sample a large range of the parameter space.
Fig. 6 shows the results of the MCMC simulation. The inner disk temperature,
$kT_{\text{in}}$, is somewhat covariant with the Galactic absorption,
$N_{\text{H}}$, which is expected since the absorption influences mainly the
soft part of the spectrum, while the outer disk temperature, $kT_{\text{s}}$,
is covariant with the corona temperature, $kT_{\text{e}}$, since these two
temperatures play an important role when computing the phase lags (Bellavita
et al., 2022). For the corona geometry, the size of the corona may be
modulated by the seed photons from the outer disk and the hot corona
temperature. The feedback fraction reaches its upper limit and a 3-$\sigma$
error also constrain the feedback fraction to be larger than 0.9. Since the
external heating rate maintains the temperature of the system when describing
the timing spectra, $\delta\dot{H}_{\text{ext}}$ is correlated with the outer
disk temperature, $kT_{\text{s}}$, the corona temperature, $kT_{\text{e}}$,
and the corona size, $L$; the latter is always large, between 5600 km and 7200
km.

Figure 4: A comparison of the best-fitting model to the rms spectra (upper
panels) and phase-lag spectra (lower panels) in the case in which
$kT_{\text{s}}=kT_{\text{in}}=1.141$ keV (left panels) and $kT_{\text{s}}$ is
free (right panels). In the case in which $kT_{\text{s}}=kT_{\text{in}}$,
$\chi^{2}=104.0$ for 10 bins, while in the case in which $kT_{\text{s}}$ is
free, $\chi^{2}=62.3$ for 10 bins. The data and the best-fitting model are in
black and red, respectively. Figure 5: The joint fitting of MAXI J1535$-$571
using NICER data. From top to bottom, the left panels show, respectively, the
energy spectra of the 26 orbits, and the rms and phase-lag spectra of the
type-B QPO with the best-fitting models. The right panels show the residuals
with respect to the best-fitting model. Figure 6: MCMC simulation of the
spectral parameters of MAXI J1535$-$571\. The contours in each panel indicate
the 1-$\sigma$, 2-$\sigma$ and 3-$\sigma$ confidence ranges. The parameters
are the same as those in the Model 2 in Tab. 2. Table 2: Spectral parameters
of the joint fitting for MAXI J1535$-$571\. The error indicates the 1-$\sigma$
confidence level. See the text for more details about the parameters.

Component | Parameter | Model 1 | Model 2
---|---|---|---
TBfeo |  $N_{\text{H}}$ ($10^{22}$ cm-2) | $4.18\pm 0.08$ | $4.18\pm 0.08$
| $A_{\text{O}}$ | $0.44\pm 0.05$ | $0.52\pm 0.05$
| $A_{\text{Fe}}$ | $1.74\pm 0.08$ | $1.51\pm 0.09$
diskbb |  $kT_{\text{in}}$ (keV) | $1.138\pm 0.002$ | $1.139\pm 0.002$
vKompthdk |  $kT_{\text{s}}$ (keV) | $0.59\pm 0.06$ | $0.59\pm 0.07$
|  $kT_{\text{e}}$ (keV) | $330\pm 50$ | $330\pm 30$
| $\Gamma$ | $2.91\pm 0.06$ | $2.88\pm 0.04$
|  $L$ ($10^{3}$ km) | $6.5\pm 0.4$ | $6.5\pm 0.5$
| $\eta^{\text{a}}$ | $1_{-0.07}$ | $1_{-0.07}$
| $\delta\dot{H}$ | $0.15\pm 0.02$ | $0.16\pm 0.02$
gaussian1 | LineE (keV) | $6.80\pm 0.04$ | $6.75\pm 0.05$
|  $\sigma$ (keV) | $0.96\pm 0.04$ | $1.01\pm 0.05$
| Norm ($10^{-2}$) | $6.2\pm 0.5$ | $5.9\pm 0.4$
gaussian2 | LineE (keV) |  | $6.62\pm 0.02$
|  $\sigma$ (keV) |  | $0.14\pm 0.02$
| Norm ($10^{-3}$) |  | $3.0\pm 0.7$
$\chi^{2}/\text{bin}$ (SSS${}^{\text{b}}$) | 4137.84 / 4576 | 4098.42 / 4576
$\chi^{2}/\text{bin}$ (rms${}^{\text{c}}$) | 46.76 / 5 | 45.81 / 5
$\chi^{2}/\text{bin}$ (phase lag${}^{\text{d}}$) | 15.57 / 5 | 15.47 / 5
$\chi^{2}/\text{d.o.f.}$ | 4200.08 / 4520 | 4159.70 / 4517
Disk flux${}^{\text{e}}$ (erg cm-2 s-1) | $1.577\pm 0.004\times 10^{-7}$
Corona flux${}^{\text{e}}$ (erg cm-2 s-1) | $0.444\pm 0.003\times 10^{-7}$

  * a

We give only the negative error of the feedback fraction, $\eta$, because the
parameter is consistent with 1, the maximum possible value at the 1-$\sigma$
confidence level.

  * b

Steady-state spectrum.

  * c

Fractional rms spectrum.

  * d

Phase-lag spectrum.

  * e

Unabsorbed flux.

##  4 Discussion

We have analyzed 36 NICER observations of the black hole candidate MAXI
J1535$-$571 during the 2017/2018 outburst. We identify a type-B QPO in 26
NICER orbits where the source is in the SIMS, during which the inner disk is
relatively hot, $kT_{\text{in}}\sim 1.1$ keV, and the corona photon index is
relatively high, $\Gamma\sim 2.9$. From the simultaneous fit of the energy
spectra of the source and the rms and phase-lag spectra of the type-B QPO, we
find that the corona has a size $L=6500\pm 500$ km and a feedback fraction
$\eta=1_{-0.07}$222This value of the feedback fraction gives
$\eta_{\text{int}}=0.33\pm 0.02$ which means that $\sim 33\%$ of the total
Comptonized photons impinge back onto the disk, while the remaining 67% are
the observed Comptonized photons.. We detect an additional significant narrow
iron line with central energy of $\sim 6.6$ keV and FWHM of $\sim 0.1$ keV,
consistent with a large corona illuminating the outer and cooler parts of the
accretion disk.

###  4.1 The phase-lag spectrum of type-B QPOs

By systematically studying the type-B QPOs using RXTE data, the lags of type-B
QPOs are hard for the BHB systems of low inclination angle, while they are
either hard or soft for the BHB systems of high inclination angle (Van den
Eijnden et al., 2017; Gao et al., 2017). The hard lags are likely due to
inverse-Compton scattering of the soft disk photons in the corona (Kylafis et
al., 2008), while the soft lags are likely due to the down-scattering of the
hard photons in the disk (Uttley et al., 2014). Stevens & Uttley (2016)
studied the hard lags of type-B QPOs in GX 339$-$4 using RXTE data and
suggested that a corona with a large-scale height ($\sim 1.8\times 10^{4}$ km)
is the jet base. This scenario of a large corona derived from the phase-
resolved spectroscopy of the type-B QPO is later modeled by Kylafis et al.
(2020) who proposed that the type-B QPO in GX 339$-$4 originates from a
precessing jet. Note that RXTE is not sensitive to photons below 2–3 keV.
Using NICER observations of MAXI J1348$-$630, Belloni et al. (2020) found that
the phase lags are positive both in the 3–10 keV and the 0.7–2 keV with
respect to the reference band at 2–3 keV. The positive phase lags in the 0.7–2
keV band exclude the possibility that the type-B QPOs are due to propagation
of mass accretion rate fluctuations which, in this scenario, the phase lags
below 2 keV relative to the 2–3 keV band should be negative. Comptonization of
the soft photons in the corona naturally explains the positive lags at energy
above 3 keV, while Belloni et al. (2020) suggested that the positive lags of
the type-B QPO below 2 keV in MAXI J1348$-$630 could be due to Compton down-
scattering in the corona by using Monte Carlo simulations assuming a flat seed
spectrum between 2 and 3 keV. However, because their seed spectrum does not
emit below 2 keV, they neglected the effect of the direct emission of the seed
source on the phase-lag spectrum. Indeed, the direct emission of a more
realistic seed source (e.g. an accretion disc) leads to a flat phase-lag
spectrum below $\sim$ 3 keV (Kylafis, priv. comm.; see also Figure 1 in
Bellavita et al., 2022), contrary to the observations.

From the phase-lag spectrum (right panel of Fig. 3) in MAXI J1535$-$571, the
phase lags generally decrease as the energy increases, with the lags being a
minimum at around 6 keV. If we take the lowest energy band (1–2.5 keV) as the
reference band, in MAXI J1535$-$571 all the lags are soft. Stevens et al.
(2018) proposed that the soft lags of the type-B QPOs in MAXI J1535$-$571 are
due to the phase offset between the peaks in the corona emission and the
modulation of the disk spectrum. Based on the fitting results of our vKompthdk
model, we find that $33\%\pm 2\%$ of the corona photons return to and are
reprocessed in the accretion disk, producing the soft lags. Therefore, the
observed soft lags can be explained as the light-crossing time of a large
corona illuminating the disk. Note that the minimum in the phase-lag spectrum
at $\sim$ 6 keV could be related to the iron line feature, but since the
type-B QPO in this source is weak, we do not have good enough resolution to
perform a detailed line analysis.

###  4.2 Comparison of corona models of variability

X-ray variability in the accreting X-ray binaries is generally classified as
broadband noise and QPOs (Ingram & Motta, 2019, for a review). The broadband
noise at low frequencies usually shows large hard lags that are thought to be
due to Comptonization, while at relatively high frequencies it sometimes shows
soft lags, which have been proposed to be caused by X-ray reverberation of
corona photons reflected off the accretion disk (Uttley et al., 2014). In a
lamppost geometry of the corona above a black hole, the model reltrans
calculates the difference in the light-travel time of the photons reflected
off the accretion disk relative to the corona photons that travel directly to
the observer (Ingram et al., 2019). This model is able to fit the time-lag
spectrum of the broadband noise in the black hole X-ray transient MAXI
J1820+070, giving a corona height of up to $\sim$ 500 $R_{\text{g}}$ (Wang et
al., 2021), equivalent to $\sim$ 6400 km for an 8.5-$M_{\odot}$ black hole
(Torres et al., 2020). The lamppost geometry of the corona is a simplification
to allow the calculation of the ray tracing in the spacetime around the black
hole. On the other hand, this model cannot explain the hard lags observed
sometimes in these systems that are therefore assigned to either
Comptonization or fluctuation of accretion propagation (Arévalo & Uttley,
2006; Kylafis et al., 2008). Since the soft (reverberation) and the hard
(inverse-Compton scattering and mass accretion rate fluctuation) are treated
separately, the model is in essence two separate mechanisms that are connected
by the fitting procedure.

From the perspective of QPOs that dynamically originate from the L-T
precession, Ingram et al. (2016) and Nathan et al. (2022) developed a
tomographic model that fits the QPO phase-dependent energy spectrum,
explaining the energy shifts of the observed iron $K_{\alpha}$ emission line
in different QPO phases. Using NICER and NuSTAR data of GRS 1915+105 with a
QPO frequency at 2.2 Hz, Nathan et al. (2022) measured a thermalization time
delay of 70 ms, which is too long since in this long time delay the QPO signal
would be washed out. Nathan et al. (2022) attributed this problem to the
oversimplification of the precessing corona model. For instance, the authors
did not take into account the precessing corona/jet obscuring different disk
azimuths, which would result in a variation of the shape of the observed disk
spectrum, and they ignored the light-crossing delays which can be of the order
of milliseconds. Systematic error in the spectral modeling would potentially
affect the measured time lags as well. Nathan et al. (2022) also measured an
inner radius, $R_{\text{in}}=1.5R_{\text{g}}$, of the accretion disk which is
too small to produce the 2.2-Hz QPO predicted by the L-T precession of the
corona lying inside a truncated disk (Ingram et al., 2009). It could be that
the corona is not horizontally, but more likely vertically, extended in the
form of a jet-like structure, namely the outflow (Stevens & Uttley, 2016;
Kylafis et al., 2020).

Regardless of the dynamical origin of the QPOs, the time-dependent
Comptonization model vKompthdk developed by Karpouzas et al. (2020) and
Bellavita et al. (2022) describes both the hard and soft lags by considering
inverse-Compton scattering in the corona and thermal reprocessing in the disk.
Since the model solves the linearized Kompaneets equation, it provides the
distribution of photons in energy at any given time, regardless of the history
of these photons via a single mechanism333In that respect this model is
simpler than the reverberation model that requires two separate mechanisms to
explain both the soft and hard lags at different Fourier frequencies.. This
model can successfully fit the steady-state Comptonization spectrum and the
rms and phase-lag spectra of QPOs with energies in the 1–100 keV range
(Karpouzas et al., 2020; García et al., 2021; Méndez et al., 2022; García et
al., 2022a; Zhang et al., 2022). A limitation of the model is that even though
it considers the thermal reprocessing of hard photons in the accretion disk,
it ignores the relativistic reflection that produces the atom florescent
emission lines and Compton hump (García et al., 2014). The corona size
measured by vKompthdk is generally big compared with the corona size predicted
by L-T precession inside the truncated disk (see supplementary Fig. 4 in
Méndez et al. 2022 and Fig. 5 in García et al. 2022a). If the corona size is
large, regardless of the feedback fraction, a big corona is likely to be
vertically extended (Méndez et al., 2022; Zhang et al., 2022). A dual-
Comptonization model is also used to fit the NICER data of the type-B QPO of
MAXI J1348$-$630 (García et al., 2021), showing that the inner part of the
corona is small ($L\simeq 400$ km) and spherical or slab, with the hard
photons efficiently returning to the accretion disk, while the outer part of
the corona is large ($L\simeq 10^{4}$ km) and jet-like with nearly no hard
photon feedbacking onto the accretion disk. The corona geometry, e.g. the
size, measured from different models mentioned above has features in common
(see below).

Figure 7: Left panel: The evolution of the corona geometry and the jet flux
from the HIMS to the SIMS in MAXI J1535$-$571\. The black, red, and blue
points indicate the corona size, the intrinsic feedback fraction, and the jet
flux density, respectively. The blue dashed line is discontinuous since the
jet is quenched from MJD 58009 to 58017. The green dashed line indicates the
time of the transition from the HIMS to the SIMS. Right panel: A schematic
figure of the corona geometry of MAXI J1535$-$571 in the SIMS. The inner part
of the disk provides the seed photons to the Comptonized emission that
dominates the steady-state energy spectrum of the source, while the outer
parts of the disk provide the seed photons to the Comptonized emission that
dominates the rms and phase-lag spectra of the QPO. See text for more details.

###  4.3 Corona geometry

Fast transitions between type-B and other types (A and C) QPOs are often
observed (e.g. Casella et al., 2004; Zhang et al., 2021). The detailed
spectral-timing analysis of H 1743$-$322 indicates that the transition from
type-B to type-C QPOs could be explained by the presence of a jet or a
vertically extended optically thick Comptonization region (Harikrishna &
Sriram, 2022). From a spectral-timing analysis of the Insight-HXMT data of
MAXI J1348$-$630 in the SIMS, Liu et al. (2022) proposed a vertically extended
corona that is the base of the jet, and explained the disappearance and
appearance of the type-B QPO as the jet being parallel to the BH spin axis or
not, due to the Bardeen-Petterson effect. In our study of MAXI J1535$-$571 in
the SIMS, we notice that the type-B QPOs disappear in some NICER orbits as
shown in Fig. 1 which may be explained by the L-T precession of the corona or
the jet proposed by Liu et al. (2022).

The modeling of the corona through type-B QPOs in MAXI J1348$-$630 shows that
the size of the jet-like corona is $\sim 10^{4}$ km (García et al., 2021).
This size is comparable with the corona size of $6500\pm 500$ km that we find
in MAXI J1535$-$571\. The intrinsic feedback fraction in this work is $33\%\pm
2\%$, which indicates that the corona should be covering the accretion disk to
some extent, as shown in the right panel of Fig. 7. From the fitting results
in Tab. 2, the inner disk provides the seed photons and the Comptonized
photons contribute mainly to the steady-state energy spectrum, while the outer
disk provides relatively cold seed photons and the long light-crossing time of
the Comptonized photons contribute mainly to the phase-lag spectrum. In MAXI
J1535$-$571 we have already measured the corona geometry through type-C QPOs
in the HIMS from MJD 58008 to 58017 (Zhang et al., 2022). The left panel of
Fig. 7 shows the corona size, intrinsic feedback fraction, and the 9-GHz jet
flux density (Russell et al., 2019) using the results both in Zhang et al.
(2022) and this work444In this work, we improve the fitting model compared to
Zhang et al. (2022). For more information about the refined model, see
subsection 3.4.. Note that we have converted the feedback fraction, $\eta$,
into the intrinsic feedback fraction, $\eta_{\text{int}}$. As discussed in
Zhang et al. (2022), the corona on MJD 58017 is a vertically extended, jet-
like corona, which expands vertically to its maximal size, $L\sim 9300$ km,
with $\eta_{\text{int}}\sim 28\%$, two days before the transient jet reaches
the maximum flux density. After MJD 58017, MAXI J1535$-$571 transits into the
SIMS. In the SIMS from MJD 58018 to 58024, we measure a corona size $L\sim
6500$ km with $\eta_{\text{int}}\sim 33\%$. Compared to the corona geometry in
the end of the HIMS on MJD 58017, in the SIMS the corona size contracts and
the hard photons feedback onto the disk more efficiently, indicating that the
corona contracts vertically and expands horizontally. After MJD 58024 no QPO
appears in the intermediate and high-soft states and the transient jet flux
density gradually decays until it is no longer detected. The change of the
morphology of the corona geometry in the SIMS, together with that in the HIMS,
and the change of the jet flux density, suggests that the increasing size of
the jet-like corona may give rise to the large-scale transient jet ejecta
lagging behind the change of the corona size. After the ejection, the corona
in the jet base contracts and the transient jet loses its energy source so the
observed radio flux density drops. This suggests a connection between the
corona and the jet.

The corona-jet connection has been investigated using vKompthdk through type-C
QPOs in GRS 1915+105 with 12-year RXTE observations (Karpouzas et al., 2021;
Méndez et al., 2022; García et al., 2022a). The connection between the corona
and the jet in the persistent source GRS 1915+105 is similar to that in the BH
X-ray transient MAXI J1535$-$571\. In GRS 1915+105, when the radio emission is
weak the corona covers large parts of the accretion disk and the hard photons
efficiently feedback onto the disk, while when the radio emission is strong
the corona is vertically extended and jet-like, and less hard photons feedback
onto the disk (Méndez et al., 2022). The only difference is that, in the SIMS
of MAXI J1535$-$571, the hard photons feedback onto the disk more efficiently
than in the HIMS. The $\sim 33\%$ intrinsic feedback fraction in the SIMS of
MAXI J1535$-$571 is similar to the $\sim 35\%$ intrinsic feedback fraction in
the SIMS of MAXI J1348$-$630 (García et al., 2021; Bellavita et al., 2022). In
fact the feedback fraction of hard photons may either decrease or increase,
depending on the balance between the Compton cooling process in the disk and
the heating up of the corona (Merloni & Fabian, 2001; Karpouzas et al., 2020).

The evolution of the corona height using the light-crossing delays in
reverberation shows quite a similar trend to our measurements using vKompthdk
through the QPOs. Wang et al. (2022) studied archival data of 10 black hole
candidates with NICER and found that during the hard to intermediate state
transition the soft lags first decrease and then increase (see Fig. 6 in Wang
et al., 2022). Assuming a lamppost geometry, the corona height in the hard
state decreases from $\sim 1000$ km to $\sim 200$ km, in the HIMS it increases
monotonically up to 9000 km, and in the SIMS it decreases slightly to $\sim
6000$ km. Comparing these results with those in the left panel of Fig. 7, we
notice a general consistence of the measurements of the corona size. This
consistence suggests the possibility to combine reltrans and vKompthdk to
measure the corona geometry when reverberation lags and QPOs appear
simultaneously.

## Acknowledgements

We thank the referee for constructive comments that helped us improve the
quality of this paper. YZ acknowledges support from the China Scholarship
Council (CSC 201906100030). MM acknowledges support from the research
programme Athena with project number 184.034.002, which is (partly) financed
by the Dutch Research Council (NWO). FG is a CONICET researcher. FG
acknowledges support by PIP 0113 (CONICET) and PIBAA 1275 (CONICET). This work
received financial support from PICT-2017-2865 (ANPCyT). DA acknowledges
support from the Royal Society. TMB acknowledges financial contribution from
PRIN INAF 2019 n.15. RM acknowledges support from the China Scholarship
Council (CSC 202104910402). MM, FG and TMB have benefited from discussions
during Team Meetings of the International Space Science Institute (Bern),
whose support we acknowledge.

## Data Availability

The X-ray data used in this article are accessible at NASA’s High Energy
Astrophysics Science Archive Research Center https://heasarc.gsfc.nasa.gov/.
The software GHATS for Fourier timing analysis is available at
http://www.brera.inaf.it/utenti/belloni/GHATS_Package/Home.html. The time-
dependent Comptonization model and the generator of the MCMC corner plot are
available at the GitHub repositories https://github.com/candebellavita/vkompth
and https://github.com/garciafederico/pyXspecCorner.

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##  Appendix A Observations and orbits used for concatenation

Table 3: NICER observations and QPO frequency of MAXI J1535$-$571\. The error bars indicate the 1-$\sigma$ confidence level. Observation ID | Orbit | MJD start | QPO frequency (Hz)
---|---|---|---
1050360114 | 1 | 58018.88 | $6.1\pm 0.1$
| 2 | 58018.94 | $6.7\pm 0.4$
1050360115 | 1 | 58019.01 | $6.2\pm 0.4$
| 4 | 58019.22 | $5.6\pm 0.4$
| 5 | 58019.28 | $5.5\pm 0.5$
| 6 | 58019.47 | $5.7\pm 0.3$
| 7 | 58019.54 | $5.5\pm 0.3$
| 8 | 58019.60 | $5.3^{+0.6}_{-0.1}$
| 13 | 58019.92 | $5.2\pm 0.5$
| 14 | 58019.99 | $6.0\pm 0.2$
1050360116 | 1 | 58020.05 | $5.6\pm 0.5$
| 4 | 58020.25 | $5.1\pm 0.2$
| 5 | 58020.31 | $5.5\pm 0.4$
| 6 | 58020.37 | $5.8\pm 0.2$
| 8 | 58020.50 | $6.2\pm 0.2$
1050360117 | 2 | 58021.15 | $6.2\pm 0.4$
| 3 | 58021.40 | $5.3\pm 0.2$
| 4 | 58021.60 | $5.8\pm 0.3$
| 5 | 58021.77 | $5.5^{+0.1}_{-0.5}$
1050360118 | 1 | 58022.24 | $6.0\pm 0.2$
1050360119 | 1 | 58023.26 | $6.1\pm 0.1$
| 3 | 58023.77 | $5.1\pm 0.1$
1050360120 | 1 | 58020.57 | $6.6\pm 0.4$
| 2 | 58020.63 | $5.8\pm 0.2$
| 4 | 58020.76 | $5.6\pm 0.2$
| 7 | 58020.95 | $4.8\pm 0.1$