# Fast-varying time lags in the Quasi-periodic Oscillation in GRS 1915+105

Tomaso M. Belloni1,†, Mariano Méndez2, Federico García3 and Dipankar
Bhattacharya,4,5
1INAF - Osservatorio Astronomico di Brera, via E. Bianchi 46, I-23807 Merate,
Italy
2Kapteyn Astronomical Institute, University of Groningen, PO BOX 800, NL-9700
AV Groningen, the Netherlands
3Instituto Argentino de Radioastronomía (CCT La Plata, CONICET; CICPBA; UNLP),
C.C.5, (1894) Villa Elisa, Buenos Aires, Argentina
4Inter University Center for Astronomy and Astrophysics, Ganeshkhind, Post Bag
4, Pune 411007, India
5Ashoka University, Dept. Of Physics, Sonipat, Haryana-131029, India
†Deceased E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The properties of sub-second time variability of the X-ray emission of the
black-hole binary GRS 1915+105 are very complex and strictly connected to its
patterns of variability observed on long time scales. A key aspect for
determining the geometry of the accretion flow is the study of time lags
between emission at different energies, as they are associated to key time
scales of the system. In particular, it is important to examine the lags
associated to the strong low-frequency Quasi-periodic Oscillations (QPOs), as
the QPOs provide unambiguous special frequencies to sample the variability. We
have analyzed data from an observation with the AstroSat satellite, in which
the frequency of the low-frequency QPO varies smoothly between 2.5 and 6.6 Hz
on a time scale of $\sim$10 hours. The derived phase lags show the same
properties and evolution of those observed on time scales of a few hundred
days, indicating that changes in the system geometry can take place on times
below one day. We fit selected energy spectra of the source and rms and phase-
lag spectra of the QPO with a time-variable Comptonization model, as done
previously to RossiXTE data of the same source, and find that indeed the
derived parameters match those obtained for variations on much longer time
scales.

###### keywords:

accretion, accretion disks – black hole physics – relativistic processes –
X-rays: binaries – individual: GRS 1915+105

††pubyear: 2023††pagerange: Fast-varying time lags in the Quasi-periodic
Oscillation in GRS 1915+105–References

##  1 Introduction

The sub-second variability in the X-ray emission of black-hole binaries (BHBs)
displays a large range of features, ranging from broad-band noise components
to several types of Quasi-periodic Oscillations (QPOs) that can be observed on
characteristic time scales from a few millihertz to hundreds of Hz (see
Belloni & Stella 2014, Belloni & Motta 2016, Ingram & Motta 2020). While high-
frequency QPOs, with centroid frequencies above 40 Hz, are very rarely
detected (Belloni, Sanna & Méndez 2012), their low-frequency counterparts,
spanning frequencies typically in the $0.1-30$ Hz range, have been observed
several times from almost all objects. These QPOs have been divided into three
separate classes, called A, B and C (Wijnands & van der Klis 1999, Remillard
et al. 2002, Casella et al. 2004, Ingram & Motta 2020), usually mutually
exclusive although some cases of two different QPOs appearing at the same time
have been observed (see e.g. Motta et al. 2012). The most commonly observed
are type-C QPOs, which in the Power Density Spectra (PDS) are associated to
the presence of a strong band-limited noise that can be decomposed into a few
components (Belloni & Stella 2014). The centroid frequency of these QPOs has
been interpreted as being associated to the Lense-Thirring frequency at a
special radius in the accretion flow (Stella & Vietri 1998). A more complex
model based on the same physical frequency was later proposed (Ingram et al.
2009, see also Ingram & Motta 2020). Additional models have been proposed (a
review can be found in Belloni & Stella 2014). Recently, a different model was
introduced by Mastichiadis, Petropoulou & Kylafis (2022), in which the type-C
QPO is generated by the interaction between the corona and the accretion disk.
In this model, the frequency of the oscillation is associated to accretion
time scales in the interaction between the two physical components. All these
models address the observed timing properties and are not directly linked to
the energy spectra.

Frequency-dependent lags between different X-ray energy bands associated to
aperiodic variability in X-ray binaries have been observed since the 80s (van
der Klis et al. 1987; Miyamoto et al. 1988; Miyamoto et al. 1993). Lags
associated to type-C QPOs in BHBs have been measured with RossiXTE (see e.g.
Reig et al. 2000; Remillard et al. 2002; Casella et al. 2004; Muñoz-Darias et
al. 2010). Van den Eijnden (2017) studied type-C QPOs from a sample of sources
observed with RossiXTE and discovered a dependence on source inclination. In
the past few years, new results were obtained with data from other X-ray
missions, namely AstroSat, NICER and HXMT. Belloni et al. (2020) used NICER
data to obtain the first time lags for type-B QPOs down to energies below 2
keV, an energy range unreachable by the RossiXTE instruments, and showed that
in that band variations lag those at higher energies, with a lag increasing
with decreasing energy. This result was interpreted by García et al. (2021) in
light of a time-dependent Comptonization model with feedback developed by
Karpouzas et al. (2020). The model has been applied to type-C QPOs (see below)
and has been very successful in fitting the spectral-timing characteristics
and evolution, in particular for data from the bright transient GRS 1915+105
(see Fender & Belloni 2004 for a review on the source). The model is based on
the effects of the feedback between a Comptonizing corona and the accretion
disk, which is shown to lead to an oscillation whose spectral and timing
features are successfully fitted to the data (see Bellavita et al. 2022). The
main limitation of the model (at least for now) is that it assumes that the
corona is spherical with constant optical depth and electron temperature. The
model does not address the origin of the oscillation (see Mastichiadis et al.
2022), and is able to explain the observed features giving information on the
geometry of the accreting flow. The results of the application of this model
can be found in Karpouzas et al. (2021), Méndez et al. (2022), Zhang et al.
(2022a,b), García et al. (2022), Rawat et al. (2023) and Peirano et al.
(2023).

The results obtained on GRS 1915+105 with RossiXTE were based on a few hundred
observations spanning the full 16-year lifetime of the mission and showed a
clear evolution of the properties of the type-C QPO and its lag spectrum
(Zhang et al., 2020; Méndez et al. 2022; García et al. 2022). While the
observed correlations are solid and the interpretation with the model points
toward an evolution of the corona and the jet in the system, the evolution on
time scales shorter than days has not been studied yet. In this paper, we
present the results of the analysis of an observation of GRS 1915+105 made
with AstroSat that contains an interval of state-C (see Belloni et al. 2000
for a description of the states of GRS 1915+105), with a type-C QPO that
changed frequency between $\sim$6.6 and $\sim$2.5 Hz in a relatively short
period of several hours. We analyze the properties of the QPO and find a
similar evolution, to which we applied the same model.

##  2 Observations and data analysis

AstroSat (Agrawal 2006; Singh et al. 2014) is an astronomical satellite
launched by the Indian Space Research Organization (ISRO) on 28 September 2015
into a 97.6 min low-Earth orbit with $6^{\circ}$ inclination. It includes five
scientific instruments: a large-area X-ray instrument (LAXPC, $3-80$ keV), a
soft X-ray telescope (SXT, $0.3-8$ keV), a coded-mask hard X-ray instrument
(CZTI, $25-150$ keV), an all-sly monitor (SSM, $2.5-10$ keV) and an
ultraviolet telescope (UVIT, 130–180 nm, 200–300 nm, 320-5-50 nm). The Large-
Area X-ray Proportional Counter (LAXPC) is an X-ray proportional counter array
covering the energy range $3-80$ keV. It consists of three detectors (referred
to as LX10, LX20 and LX30 respectively), with a combined effective area of
6000 cm2 (Yadav et al. 2016a; Antia et al. 2017). At the time of the
observation, all three detectors were active, although with different
responses. The timing resolution of the LAXPC is 10 $\mu$s with a dead time of
42 $\mu$s. For the full observation, all information about single photons is
available. The characteristics of the LAXPC make it very effective for fast-
timing measurements.

Table 1: Log of the eight data intervals analysed here. The columns are: start and end times (seconds from MJD 57451.47437), number of averaged PDS, QPO frequency and FWHM (both in Hz), and statistical significance (in $\sigma$). QPO phase lags in radians are given for the 15–20 keV band with respect to the 3–4 keV band.  Int. | Ts | Te | NPDS | $\nu_{QPO}$ | $\Delta_{QPO}$ | $n_{\sigma}$ | phase lags (rad)
---|---|---|---|---|---|---|---
A | 0 | 918 | 14 | $5.67\pm 0.03$ | $0.70\pm 0.10$ | 10.2 | $-0.69\pm 0.04$
B | 3235 | 4087 | 13 | $6.61\pm 0.03$ | $0.67\pm 0.09$ | 10.7 | $-0.98\pm 0.05$
C | 6258 | 6717 | 7 | $4.60\pm 0.02$ | $0.45\pm 0.06$ | 12.1 | $-0.50\pm 0.05$
D | 9080 | 10391 | 20 | $4.50\pm 0.01$ | $0.50\pm 0.03$ | 27.7 | $-0.48\pm 0.03$
E | 14925 | 16629 | 27 | $3.54\pm 0.01$ | $0.46\pm 0.02$ | 36.9 | $-0.29\pm 0.02$
F | 20770 | 22933 | 33 | $2.55\pm 0.01$ | $0.32\pm 0.01$ | 40.2 | $-0.11\pm 0.02$
G | 26615 | 29171 | 39 | $2.79\pm 0.01$ | $0.36\pm 0.01$ | 80.8 | $-0.20\pm 0.02$
H | 32460 | 35475 | 46 | $3.22\pm 0.01$ | $0.41\pm 0.01$ | 47.9 | $-0.19\pm 0.02$

We analyzed the AstroSat/LAXPC data of GRS 1915+105 taken on 4–6 March 2016
(see Yadav et al. 2016b, where the same observation is analysed) using the
GHATS analysis package111http://astrosat-ssc.iucaa.in/uploads/ghats_home.html.
The observations span the period from 2016 March 04 11:22 UT to 2016 March 6
08:43 UT for a total net exposure of 63222 s. We show the full $3-50$ keV
light curve for this observation in Fig. 1. Strong variability typical of GRS
1915+105 (Belloni et al. 2000) is visible, but there is a long period when
this structured variability is absent and the count rate follows a ‘U’-shaped
evolution. Such an interval has been repeatedly observed from GRS 1915+105 and
is one of its ‘trademarks’ (Belloni et al. 1997, 2000). A preliminary
examination of this part of the light curve confirms that this interval is
characterized by fast timing properties compatible with state C (associated to
variability class $\chi$, Belloni et al. 2000), as previously shown in Yadav
et al. (2016). In order to study the properties of the low-frequency QPO
observed in state C, we limited our analysis to this part of the observation.
The last two intervals (corresponding to AstroSat separate orbits) were
affected by data drop outs and were excluded. This resulted in the selection
of the region marked in light blue in Fig. 1, which corresponds to nine
intervals, separated by gaps due to the orbital constraints of the satellite.
Since the fifth interval was very short (96 seconds), eight intervals were
finally considered, which we named with capital letters from A to H. An
observation log can be seen in Tab. 1.

Figure 1: Light curve ($3-50$ keV) for the full LAXPC observation with a time
resolution of 1 second. The light blue area and the red data points mark the
region considered in this work. The blue points are the QPO centroid
frequencies measured for each of the eight intervals considered here (see
text).

For each of the eight intervals A–H, we produced a Fourier Transform by
extracting a light curve with 1 ms time binning and dividing it in segments of
duration 65.536 seconds (the number of segments for each interval, $N_{PDS}$,
can be seen in Tab. 1). The PDS produced from each segment were then averaged
to obtain a single PDS per interval, with a Nyquist frequency of 500 Hz and a
lowest frequency of 15.3 mHz. This procedure was applied to data for the full
energy range ($3-50$ keV) as well as for 14 non-overlapping energy bands,
shown in Tab. 2. From each Fourier Transform, we produced a Power Density
Spectrum (PDS), normalized according to Leahy et al. (1983). We fitted the
full-band PDS for each interval with a combination of Lorentzian components
(Belloni et al. 2002) plus a constant component to account for Poissonian
noise.

Figure 2: Top panel: PDS of interval F, corresponding to the QPO with the
lowest centroid frequency (2.55 Hz, marked by a vertical line). The dashed
lines indicate the individual Lorentzian components used to fit the PDS, and
the full line their combination. Middle panel: phase-lag spectrum between the
$3-4$ keV energy band and the full energy band. Bottom panel: phase-lag
spectrum between the $15-20$ keV energy band and the full energy band.  Figure
3: Same as Fig. 2, but for interval B, corresponding to the QPO with the
highest centroid frequency (6.61 Hz).

For each of the eight intervals we also used the Fourier Transforms to compute
average cross-spectra, using the full $3-50$ keV band as reference (Uttley et
al. 2014; Ingram 2019). From the cross-spectra, we computed the phase lags at
the QPO frequency by averaging the real and imaginary parts over a frequency
band centred on the centroid frequency and one FWHM wide. As the reference
band $3-50$ keV included the photons of all other bands, there is a zero-lag
correlation for all cross-spectra, which we removed by estimating its
amplitude from the high-frequency part of the cross-spectra, where no source
signal is observed.

Two examples of PDS, corresponding to the intervals with the QPO at the lowest
and highest frequency are shown in the top panels of Figs. 2 and 3. The
centroid frequency of the QPO for all intervals are reported in Tab. 1 and
plotted in Fig. 1. From the best-fitting model, for each PDS we computed the
integrated fractional rms amplitude (Belloni & Hasinger (1990) of the QPO
component, which we show in Fig. 4. The QPO was not significant in the $40-50$
keV band for all intervals, while for other intervals significant detections
went up to 30 keV. In Fig. 4 we only include the rms amplitude of the QPO in
the energy bands in which it was significant, as the upper limits in the
remaining bands are not stringent.

The eight phase-lag spectra for the separate intervals are shown in Fig. 5. As
in the case of the fractional rms, we do not include upper limits for the
bands in which the QPO was not detected significantly.

The main results of the analysis are shown in Fig. 4 and 5. The fractional rms
of the QPO grows with energy until $\sim$10 keV, then flattens and possibly
decreases above 30 keV. The QPO phase lags all decrease with energy, but with
a very different slope, which increases as the QPO frequency increases. The
lags all have the full band ($3-50$ keV) as reference band, which sets the
zero level. The fact that all lag spectra cross zero around 8 keV is most
likely due to a combination of the QPO rms spectra and LAXPC response. From
Fig. 2 one can see that there is a positive-lag peak corresponding to the
second harmonic of the QPO, but the analysis of harmonics goes beyond the
scope of this paper.

We extracted LAXPC energy spectra from all the eight intervals in order to
check the evolution of Comptonization parameters with QPO frequency.

To obtain the energy spectra, we started from level1 files from the AstroSat
archive and obtained level2 products by using format (A)
software222http://astrosat-ssc.iucaa.in/laxpcData. We then used the same
software to extract a spectrum for each of the two intervals, using only
unit20 as it is the best calibrated of the three units (all active at the time
of observation). We extracted background spectra using the same time
intervals. A 2% systematic error was added to the spectra.

Instead of fitting only the energy spectra, we applied the model by Bellavita
et al.(2022) to the energy and timing data. Specifically, for each interval,
following García et al. (2022), we fitted simultaneously the average energy
spectrum of the source (in the $3-40$ keV energy range), and the rms the phase
lag spectrum of the QPO.

In XSPEC parlance, the model used to fit the average energy spectrum is PHABS
(DISKBB + NTHCOMP), where PHABS is the interstellar absorption (fitted
parameter is $N_{H}$), DISKBB a thermal disk component (fitted parameters are
the inner disk temperature $kT_{in}$ and the normalization $N_{dbb}$
proportional to the square of the inner radius), and NTHCOMP a thermal
Comptonization model (parameters are the photon index $\Gamma$, the electron
temperature $kT_{e}$ and a normalization, Zdziarski, Johnson & Magdziarz 1996;
Zycki, Done & Smith 1996). The temperature of the seed photons in the NTHCOMP
model was linked to the accretion-disk inner temperature, $kT_{in}$. The value
for interstellar absorption was left free, but tied between the eight
intervals, under the assumption that any local absorption would not vary on
such a short time scale.

For the lag and rms spectra, we fitted the model VKOMPTHDK * DILUTION, where
VKOMPTHDK333https://github.com/candebellavita/vkompth is the variable-
Comptonization spectral-timing model developed by Bellavita et al. (2022). Its
parameters are the same $\Gamma$ and $kT_{e}$ of NTHCOMP, plus the cloud size
$L$, the feedback fraction $\eta$, the external heating $\delta\dot{H}_{\rm
ext}$ and the reference lag reflag. DILUTION is a multiplicative model to take
into account the fact that the fractional rms of the (variable) Comptonization
component would be diluted by the (constant) contribution from another non-
variable component, in this case, the thermal accretion disk. The dilution
factor was fixed to 1 for the lag spectrum (where there is no dilution) and to
the ratio of contribution of the corona component and the total one for the
rms spectrum. The spectral parameters $kT_{in}$, $\Gamma$ and $kT_{e}$ of the
DISKBB and NTHCOMP spectral components were tied to the same parameters of the
VKOMPTHDK component used for rms and lag spectra. In this way, it was possible
to fit simultaneously the average energy spectrum of the source and the
spectra of the timing parameters of the QPO. Notice that García et al. (2022)
used a blackbody model for the accretion disk emission, while we adopted an
improved version of the model that assumes a disk-blackbody as the disk-
emitting component.

Table 2: Energy boundaries (in keV) for the full range ($T$) and the fourteen energy bands (1 to 14) used in the analysis.  Band | Elow | Ehigh
---|---|---
T | 3 | 50
1 | 3 | 4
2 | 4 | 5
3 | 5 | 6
4 | 6 | 7
5 | 7 | 8
6 | 8 | 9
7 | 9 | 10
8 | 10 | 12
9 | 12 | 15
10 | 15 | 20
11 | 20 | 25
12 | 25 | 30
13 | 30 | 40
14 | 40 | 50
Figure 4: Fractional rms of the QPO as a function of energy for the eight
intervals described in the text.  Figure 5: Phase lags of the QPO (in radians)
as a function of energy for the eight intervals described in the text. The
reference band is the total one (T), $3-50$ keV.

The best-fitting parameters are shown in Tab. 3. The joint best fit value for
the interstellar absorption was $5.3(6)\times 10^{22}$ cm-2 The model
represents the data reasonably well as far as the main trend is concerned,
although the chi-square values indicated that statistically the fits are not
good, in particular as the highest-energy points in the rms and lag spectra
show deviations. An example of best fit for the case of interval B, with
residuals, is shown in Fig. 6. A full detailed analysis of the spectra is
beyond the scope of this work and would require moving to the dual model by
García et al. (2021): the current fits are sufficient to identify the changes
at different QPO frequencies. It is clear that the as the QPO frequency
increases from 2.55 Hz (interval F) to 6.61 Hz (interval B), the parameter
that changes the most is the size of the corona, which increases by a factor
$\sim$5 (see Fig. 9, middle panel). At the same time the power-law index
$\Gamma$ increases as usually observed for type-C QPOs (Motta et al. 2009, see
bottom panel in Fig. 9), and the accretion disk becomes hotter and its inner
radius increases.

Figure 6: Left column: the best fits to flux, rms and phase-lag spectra for
interval B. In the top panel the curves for the two spectral components (disk
and Comptonization) are also shown. Right column: corresponding residuals.

The best-fitting values for the feedback fraction $\eta$ are similar across
the intervals. The parameter $\eta$ represents the fraction of disk flux due
to the photons from the corona that return to the disc. Such high values imply
that for both intervals the disk flux is almost entirely due to feedback. From
the best-fitting values, we can also estimate the fraction $\eta_{int}$ of
photons emitted by the corona that return to the disk (see Karpouzas et al.
2020 for a more detailed description), which is shown in the top panel of Fig.
9. Ignoring relativistic effects, an increase in the solid angle subtended by
the disk as seen by the corona will lead to an increase in $\eta_{int}$, which
is qualitatively what is observed. A more quantitative comparison would
require a much more detailed model.

##  3 Discussion

We have found an AstroSat observation of GRS 1915+105 in which the source went
through one of the typical ‘U’-shaped events, corresponding to state C, when
the type-C QPO is detected in this source. However, while in most of the cases
these events are either short, below $\sim$1 hr (Belloni et al. 1997;
Markwardt, Swank & Taam 1999; Belloni et al. 2000), or very long, days to
months (Fender et al. 1999; Trudolyubov 2001; Méndez et al. 2022), here the
full ‘U’ takes about ten hours. Therefore, it was possible to follow at high
signal-to-noise (also due to the high flux of the source and large effective
area of the LAXPC) the evolution of the rms and lag spectra of the type-C QPO.
As a function of increasing energy, the QPO rms increases, then flattens
around $\sim$10 keV, with a possible decrease above $\sim$30 keV (see Fig. 4).
Again as a function of increasing energy, the QPO lag spectrum decreases with
a slope that is higher for QPOs with a higher centroid frequency (see Fig. 5).
Remarkably, as a function of QPO frequency the parameters of the corona change
much more significantly than those of the disc (Table 3). Together with the
significant changes of the QPO rms and lag spectra with QPO frequency, this
bolsters the idea that the properties of the QPO are driven, predominantly, by
the corona, as proposed in the time-dependent Comptonisation model (Karpouzas
et al. 2020; Bellavita et al. 2022).

Figure 7: Plot of hardness ratio versus QPO frequency for RossiXTE
observations of GRS 1915+105 (from Méndez et al. 2022), with radio flux (in
mJy) coded in the symbol colors. The black stars represent our eight AstroSat
intervals, for which we have no radio information. The RossiXTE hardness
ratios have been computed using our best-fit spectral models and the response
of the PCA instrument.  Figure 8: Same as Fig . 7, with spectral index
$\Gamma$ coded in the symbol colors. The stars represent our eight AstroSat
intervals, with the same color coding according to our best-fit $\Gamma$
values.

The observation reported here was analyzed by Yadav et al. (2016b). In that
work, nine time intervals are included, although no light curve is shown and
the times do not seem to match ours. Three time-lag and rms spectra are shown
in their Fig. 9. From the QPO centroid frequency they appear to correspond to
our intervals F, D and B, although this identification does not match the
light curves. Strangely, their lag spectra extend to 70 keV, while the rms
spectra only reach 35 keV. Comparing the lag spectra we can see that they are
compatible with ours, although the one for the 2.55 Hz QPO has a smaller
amplitude. The rms spectra show a drop above 25 keV not present in ours and
overall are lower by about 60%. The description of their analysis is not
sufficiently detailed to identify the reasons for the differences. Only three
cases are presented and no interpretation or model application is made in that
work.

The results we found both for the phase lags and the rms are consistent with
those reported by Zhang et al. (2020). However, Zhang et al. (2020) analyzed
620 RossiXTE/PCA observations when GRS 1915+105 was in its state C,
irregularly sampled and covering the full operational life of RossiXTE, 16
years. Based on those results, Karpouzas et al. (2021) and García et al.
(2022) applied the Comptonization model developed by Karpouzas et al. (2020)
to the RossiXTE/PCA data, and together with the radio measurements, Méndez et
al. (2022) proposed a scenario for the accretion/ejection geometry in this
system: At high QPO frequency, $\sim 6-8$ Hz, the QPO lags are soft and the
magnitude of the lags is the largest, the X-ray corona is large (size $\approx
2000$ km) and relatively hot ($kT_{e}\gtrsim 15-20$ keV), and the radio
emission from the jet is low ($\lesssim 5$ mJy; see Fig. 4 and Extended Data
Fig. 4 in Méndez et al. 2022). As the QPO frequency decreases, the magnitude
of the lags decreases (the lags continue to be soft), $kT_{e}$ and the size of
the X-ray corona decrease and the radio flux increases. When the QPO frequency
crosses below $\sim 2$ Hz the lags become hard and their magnitude increases
with decreasing QPO frequency, the corona is the coolest ($kT_{e}\lesssim 5-6$
keV) and the jet emission is the strongest ($\gtrsim 100$ mJy). Based on that,
Méndez et al. (2022) proposed that the X-ray corona morphs into the radio jet
and vice versa.

Figure 9: Plot of selected model parameters versus QPO frequency for the eight
intervals. From top to bottom: $\eta_{int}$, representing the fraction of
photons emitted by the corona that return to the disk (this is not a direct
fit parameter, but it is derived from the others), the size of the corona in
km and the spectral index $\Gamma$.

The evolution of the spectral/timing parameters in these AstroSat observations
is consistent with what was observed by Méndez et al. (2022), whose work was
based on the analysis of a few hundred observations of GRS 1915+105 with
type-C QPOs made with RossiXTE. Their observations spanned the full
operational life of the satellite, more than 15 years, and therefore sampled
more or less randomly many events in the time evolution of the source. Here
the evolution of the disk and the corona is observed in a period as short as
11 hours, showing that the same behaviour takes place during a single event.
Even the evolution of the size of the corona with QPO frequency is the same
(see their Extended Data Fig. 4), increasing by a factor of $\sim$5 over the
observed range in QPO frequencies (there is an offset in absolute values,
which is likely caused by the difference in energy range between the RossiXTE
and AstroSat instruments). In this case the QPO frequency does not reach down
to 2 Hz and the lags do not switch from soft to hard. Since there was no
simultaneous radio coverage during the AstroSat observations, we cannot
compare the properties of the jet in our data and those of RossiXTE.

Because the RossiXTE observations were random pointings over the long time
evolution of the source whereas our observations cover an event lasting
several hours, our results confirm that the size of the corona is linked to
the QPO frequency and not dependent on the time scale of the events. Intervals
of this type have been observed on shorter time scales, from one hour (Belloni
et al. 1997; Markwardt et al. 1999), down to $\sim$10 s (variability class
$\mu$, Belloni et al. 2000). It is natural to speculate that the same
qualitative variations of the corona and the disk take place also for these
short-duration events, but the analysis of those requires a much higher signal
and is something that will be explored with future missions such as eXTP.

Table 3: Best-fitting parameters for all eight intervals.  Parameter | Interval A | Interval B | Interval C | Interval D
---|---|---|---|---
QPO $\nu_{0}$ (Hz) | $5.67$ | $6.61$ | $4.60$ | $4.50$
$N_{\rm H}~{}(10^{22}~{}\mathrm{cm}^{-2})$ | 5.3 $\pm 0.6$
$kT_{\rm in}~{}\mathrm{(keV)}$ | 1.199 ${}_{-0.027}^{+0.033}$ | 1.364 ${}_{-0.027}^{+0.030}$ | 1.202 ${}_{-0.021}^{+0.007}$ | 1.197 ${}_{-0.021}^{+0.007}$
$N_{dbb}$ | 316.9 ${}_{-15.4}^{+25.8}$ | 273.6 ${}_{-10.2}^{+21.2}$ | 323.8 ${}_{-9.2}^{+67.4}$ | 300.9 ${}_{-8.2}^{+30.5}$
$\Gamma$ | 2.784 ${}_{-0.013}^{+0.008}$ | 2.835 ${}_{-0.012}^{+0.010}$ | 2.540 ${}_{-0.014}^{+0.031}$ | 2.573 ${}_{-0.037}^{+0.008}$
$kT_{\rm e}~{}\mathrm{(keV)}$ | $>372$ | $>654$ | 20.4 ${}_{-1.5}^{+2.8}$ | 28.1 ${}_{-4.2}^{+0.9}$
$N_{\rm nthComp}$ | 3.74 ${}_{-0.25}^{+0.15}$ | 3.87 ${}_{-0.21}^{+0.20}$ | 2.31 ${}_{-0.08}^{+0.11}$ | 2.30 ${}_{-0.06}^{+0.01}$
$\mathrm{Size~{}(km)}$ | 4510.9${}_{-124.3}^{+163.4}$ | 4785.7${}_{-145.9}^{+167.8}$ | 2508.1${}_{-103.2}^{+135.3}$ | 2642.9${}_{-43.2}^{+36.1}$
$\eta$ | $>0.996$ | $>0.997$ | $>0.987$ | $>0.991$
$\delta\dot{H}_{\rm ext}$ | 0.079 $\pm$ 0.002 | 0.072 $\pm$ 0.003 | 0.122 $\pm$ 0.003 | 0.125 $\pm$ 0.002
$\eta_{int}(\%)$ | 26.5 $\pm$ 0.3 | 27.3 $\pm$ 0.2 | 25.0 $\pm$ 0.5 | 25.1 $\pm$ 0.3
$\chi^{2}(d.o.f.)$ | 106.7 (50) | 129.7 (52) | 72.8 (52) | 106.0 (52)
Parameter | Interval E | Interval F | Interval G | Interval H
QPO $\nu_{0}$ (Hz) | $3.54$ | $2.55$ | $2.79$ | $3.22$
$N_{\rm H}~{}(10^{22}~{}\mathrm{cm}^{-2})$ | 5.3 $\pm 0.6$
$kT_{\rm in}~{}\mathrm{(keV)}$ | 1.232 ${}_{-0.014}^{+0.001}$ | 1.234 ${}_{-0.020}^{+0.013}$ | 1.178 ${}_{-0.009}^{+0.007}$ | 1.198 ${}_{-0.014}^{+0.002}$
$N_{dbb}$ | 250.7 ${}_{-17.3}^{+43.6}$ | 180.4 ${}_{-10.4}^{+4.7}$ | 202.3 ${}_{-32.1}^{+62.0}$ | 241.0 ${}_{-52.1}^{+48.3}$
$\Gamma$ | 2.455 ${}_{-0.039}^{+0.014}$ | 2.324 ${}_{-0.036}^{+0.007}$ | 2.407 ${}_{-0.015}^{+0.005}$ | 2.465 ${}_{-0.015}^{+0.005}$
$kT_{\rm e}~{}\mathrm{(keV)}$ | 18.7 ${}_{-0.7}^{+0.2}$ | 13.8 ${}_{-1.3}^{+0.9}$ | 16.7 ${}_{-0.7}^{+0.4}$ | 20.0 ${}_{-1.1}^{+0.5}$
$N_{\rm nthComp}$ | 1.70 ${}_{-0.02}^{+0.01}$ | 1.42 ${}_{-0.14}^{+0.01}$ | 1.67 ${}_{-0.06}^{+0.01}$ | 1.89 ${}_{-0.01}^{+0.01}$
$\mathrm{Size~{}(km)}$ | 1865.9${}_{-45.6}^{+52.8}$ | 856.2${}_{-69.3}^{+88.7}$ | 1421.6${}_{-46.7}^{+43.4}$ | 1643.1${}_{-36.4}^{+31.8}$
$\eta$ | 0.957 ${}_{-0.023}^{+0.002}$ | $>0.915$ | 0.969 ${}_{-0.009}^{+0.008}$ | 0.940 ${}_{-0.011}^{+0.008}$
$\delta\dot{H}_{\rm ext}$ | 0.146 $\pm$ 0.002 | 0.137 $\pm$ 0.003 | 0.136 $\pm$ 0.001 | 0.142 $\pm$ 0.001
$\eta_{int}(\%)$ | 22.8 $\pm$ 0.4 | 22.4 $\pm$ 0.3 | 22.9 $\pm$ 0.3 | 22.8 $\pm$ 0.4
$\chi^{2}(d.o.f.)$ | 178.5 (54) | 209.0 (54) | 203.4 (54) | 267.7 (54)

## Acknowledgements

Unfortunately, Tomaso Belloni passed away before this paper was accepted. We
will miss his insightful ideas about these and other topics. Given that this
will likely be his final first-author paper, we would like to let Tomaso
express his gratitude to his daughter Alice and his son Alessandro who, as he
told us many times, were his most cherished treasures and his utmost source of
inspiration. This work makes use of data from the AstroSat mission of the
Indian Space Research Organisation (ISRO), archived at Indian Space Science
Data Centre (ISSDC). TMB acknowledges financial contribution from grant PRIN
INAF 2019 n.15. MM acknowledges support from the research programme Athena
with project number 184.034.002, which is (partly) financed by the Dutch
Research Council (NWO). F.G. is a CONICET researcher and acknowledges support
from PIP 0113 (CONICET), PICT-2017-2865 (ANPCyT) and PIBAA 1275 (CONICET). We
benefited from discussions during Team Meetings of the International Space
Science Institute (Bern), whose support we acknowledge. We would normally
thank the referee for their comments on the manuscript. This time, however,
this is not appropriate.

##  4 Data availability

The data used in this article are publicly available at the website of the
ISRO Science Data Archive for AstroSat Mission,
https://astrobrowse.issdc.gov.in/astro_archive/archive/Home.jsp. The vKompth
model is publicly available in a Github repository
(https://github.com/candebellavita/vkompth).

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