11institutetext: MIT Kavli Institute for Astrophysics and Space Research, 70
Vassar Street, Cambridge, MA 02139, USA
11email<EMAIL_ADDRESS>22institutetext: Max-Planck-Institut für
extraterrestrische Physik (MPE), Gießenbachstraße 1, 85748 Garching bei
München, Germany 33institutetext: International Centre for Radio Astronomy
Research – Curtin University, GPO Box U1987, Perth, WA 6845, Australia
44institutetext: South African Astronomical Observatory, P.O. Box 9,
Observatory 7935, Cape Town, South Africa 55institutetext: Astronomical
Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland
66institutetext: Astrophysics Science Division, NASA Goddard Space Flight
Center, 8800 Greenbelt Road, Greenbelt, MD 20771, USA 77institutetext: INAF-
Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807 Merate (LC),
Italy 88institutetext: Leibniz-Institut für Astrophysik Potsdam (AIP), An der
Sternwarte 16, 14482 Potsdam, Germany

# The more the merrier: _SRG_ /eROSITA discovers two further galaxies showing
X-ray quasi-periodic eruptions

R. Arcodia NASA Einstein fellow1122 Z. Liu 22 A. Merloni 22 A. Malyali 22 A.
Rau 22 J. Chakraborty 11 A. Goodwin 33 D. Buckley 44 J. Brink 44 M. Gromadzki
55 Z. Arzoumanian 66 J. Buchner 22 E. Kara 11 K. Nandra 22 G. Ponti 7722 M.
Salvato 22 G. Anderson 33 P. Baldini 22 I. Grotova 22 M. Krumpe 88 C. Maitra
22 J. C. A. Miller-Jones 33 M. E. Ramos-Ceja 22

(Received ; accepted )

X-ray quasi-periodic eruptions (QPEs) are a novel addition to the group of
extragalactic transients. With only a select number of known sources, and many
more models published trying to explain them, we are so far limited in our
understanding by small number statistics. In this work, we report the
discovery of two further galaxies showing QPEs, hereafter named eRO-QPE3 and
eRO-QPE4, with the eROSITA X-ray telescope on board the Spectrum Roentgen
Gamma observatory, followed by _XMM-Newton_ , _NICER_ , _Swift-XRT_ , _SALT_
($z=0.024$ and $z=0.044$, respectively), and _ATCA_ observations. Among the
properties in common with those of known QPEs are: the thermal-like spectral
shape in eruption (up to $kT\sim 110-120$ eV) and quiescence ($kT\sim 50-90$
eV) and its evolution during the eruptions (with a harder rise than decay);
the lack of strong canonical signatures of active nuclei (from current
optical, UV, infrared and radio data); and the low-mass nature of the host
galaxies ($\log M_{*}\approx 9-10$) and their massive central black holes
($\log M_{\rm BH}\approx 5-7$). These discoveries also bring several new
insights into the QPE population: i) eRO-QPE3 shows eruptions on top of a
decaying quiescence flux, providing further evidence for a connection between
QPEs and a preceding tidal disruption event; ii) eRO-QPE3 exhibits the longest
recurrence times and faintest peak luminosity of QPEs, compared to the known
QPE population, excluding a correlation between the two; iii) we find
evidence, for the first time, of a transient component that is harder, albeit
much fainter, than the thermal QPE spectrum in eRO-QPE4; and iv) eRO-QPE4
displays the appearance (or significant brightening) of the quiescence disk
component after the detection of QPEs, supporting its short-lived nature
against a preexisting active galactic nucleus. These new properties further
highlight the need to find additional QPE sources to increase the sample size
and draw meaningful conclusions about the intrinsic population. Overall, the
newly discovered properties (e.g., recent origin and/or transient nature of
the quiescent accretion disk; lack of correlation between eruption recurrence
timescales and luminosity) are qualitatively consistent with recent models
that identify QPEs as extreme mass-ratio inspirals.

##  1 Introduction

The advent of wide-field cadenced surveys across the electromagnetic spectrum
during the last decade opened a new window to the realm of extra-galactic
transients, with the discovery of a few classes of repeaters. Some galactic
nuclei have been shown to undergo outbursts recurring on timescales of months
to decades in optical and/or X-rays (Payne et al., 2021; Wevers et al., 2023;
Liu et al., 2023; Malyali et al., 2023), others over tens of days (Evans et
al., 2023; Guolo et al., 2024) and even down to surprisingly fast recurrences
of a few hours, such as the X-ray quasi-periodic eruptions (QPEs; Miniutti et
al., 2019; Giustini et al., 2020; Arcodia et al., 2021, hereafter ; ; ).
Whether all these flavors are connected within a single recipe remains to be
seen, although the leading models of each subclass appear to relate somehow to
the dynamics of stellar objects in galactic nuclei (e.g., Linial & Quataert,
2024).

In particular, QPEs are sharp X-ray bursts that last a few hours and repeat in
a quasi-periodic manner every several hours to about a day. So far, these
peculiar bursts have been observed from massive black holes (MBHs) of $M_{\rm
BH}\approx 10^{5-6.7}\,M_{\astrosun}$ (Shu et al., 2017; ; ; A21;Wevers et
al., 2022) residing in the nuclei of nearby low-mass galaxies (e.g., stellar
masses of $M_{*}\approx 1-3\times 10^{9}\,M_{\astrosun}$; A21, ). To date, no
associated optical, UV, IR or radio flares have been observed, although most
of the current multiwavelength photometry is likely contaminated or dominated
by either the host galaxy emission or that of the accretion disk. In fact,
when in quiescence in between QPEs, most sources are still detected in X-rays,
with a soft spectrum reminiscent of the Wien tail of a radiatively efficient
accretion disk (with a peak temperature $kT\sim 40-70\,$eV; ; ; ). Eruptions
reach a soft X-ray luminosity of $\sim 10^{42}-10^{43}$ erg s-1 at their peak,
$\sim 10-100$ times brighter than the quiescence level, with a spectrum that
is hotter when brighter and remains soft and thermal in shape (with a peak
temperature $kT\sim 100-200\,$eV; ; ; ). While the timing properties show
significant diversity in how regular the eruption arrival times are (; ; ),
all eruptions from all the different QPE sources seem to follow the same
spectral evolution: bursts progress with a harder rise than decay at the same
X-ray count rate or, modeling the emission as a black body, with a hotter rise
than decay at the same luminosity (Arcodia et al., 2022; Miniutti et al.,
2023b; Giustini et al., in prep.).

Here, we consider secure QPE sources GSN 069 (M19), RX J1301.9+2747 (Sun et
al., 2013; G20), eRO-QPE1 and eRO-QPE2 (A21). In addition, the QPE candidate
XMMSL1 J024916.6-041244 (Chakraborty et al., 2021) showed remarkably similar
spectral properties but only 1.5 eruptions, impeding its secure classification
to date. Recently, the start of a flare consistent, in terms of spectral and
timing properties, with those of eRO-QPE1 was seen in an optical tidal
disruption event (TDE) “Tormund” (Quintin et al., 2023); however, the lack of
repetition to date makes its classification still ambiguous. Further, the
source Swift J023017.0+283603 shows repeating soft X-ray outbursts every $\sim
21\,$d (Evans et al., 2023; Guolo et al., 2024), although its opposite
asymmetry in the burst shape and opposite spectral evolution during the bursts
(with a harder decay than rise) does not allow a secure association with QPEs
for the time being.

The origin of QPEs is still actively debated, with some models proposing some
kind of accretion disk instability (Raj & Nixon, 2021; Śniegowska et al.,
2023; Kaur et al., 2023; Pan et al., 2023), whilst most suggest that QPEs are
triggered by a binary system including a central MBH and a much smaller body
orbiting it (King, 2020; Suková et al., 2021; Xian et al., 2021; Zhao et al.,
2022; Wang et al., 2022; Metzger et al., 2022; King, 2022; Krolik & Linial,
2022; Linial & Sari, 2023; Lu & Quataert, 2023; Franchini et al., 2023; Tagawa
& Haiman, 2023; Linial & Metzger, 2023). For instance, most of the
observational properties can be qualitatively reproduced by shocks caused by
the interaction between the smaller orbiter and the accretion flow around the
MBH (Xian et al., 2021; Franchini et al., 2023; Tagawa & Haiman, 2023; Linial
& Metzger, 2023). In this framework, the disk is pierced by the orbiter once
or twice per orbit, and shocks are caused by an initially optically thick
cloud of gas ejected by the collisions (Linial & Metzger, 2023). Orbital
precession and Lense-Thirring precession of the disk would provide the
required departure from exact periodicity (Franchini et al., 2023). A
requirement, if not a prediction, for the collisions model is that the
accretion flow around these MBHs is not an extended flow typical of active
galactic nuclei (AGN), but rather a more compact density distribution.
Otherwise the orbiter would sink into the disk plane and/or be ablated by the
collision in a more extended disk. This led the latest models to suggest that
the disk originates from a TDE, whose role in the QPE emission is solely to
throw gas at the preexisting extreme mass ratio inspiral (EMRI) orbiters
(Franchini et al., 2023; Linial & Metzger, 2023). Interestingly, this
theoretical connection between QPEs and a preexisting behavior akin to that of
TDEs was previously suggested based on observational properties alone: both
GSN 069 (Shu et al., 2018; Sheng et al., 2021; Miniutti et al., 2023b) and the
two candidate QPEs (Chakraborty et al., 2021; Quintin et al., 2023) show
evidence of past TDEs prior to the onset of QPE (or candidate QPE) behavior.

Clearly, at this stage it is fundamental to find more QPEs to draw any
significant conclusions on the population and its physical origin. After the
serendipitous discovery of the first QPE emitting source (M19), QPEs were
either found through dedicated searches in the X-ray archives (G20;
Chakraborty et al., 2021; Quintin et al., 2023) or with a blind search within
the live data stream of the eROSITA X-ray telescope (A21). As much as the
former method is important to make sure no interesting source was overlooked,
wide-area surveys in X-rays provide the only way to systematically detect new
QPE sources as they happen in the sky. In A21, we reported results from the
first two eROSITA all-sky surveys (Merloni et al., 2023). Here, we report on
two new discoveries based on the subsequent eROSITA surveys.

##  2 Data processing and analysis

###  2.1 X-ray spectral analysis

Spectral analysis is performed with the Bayesian X-ray Analysis software (BXA)
version 4.0.7 (Buchner et al., 2014), which connects the nested sampling
algorithm UltraNest (Buchner, 2019, 2021) with the fitting environment XSPEC
version 12.13.0c (Arnaud, 1996), in its Python version PyXspec111Link to
PyXspec. The continuum model adopted is absorbed by Galactic column density
from HI4PI (HI4PI Collaboration et al., 2016) and redshifted to rest-frame
using spectroscopic redshifts. These are $z=0.024$ for eRO-QPE3 and $z=0.044$
for eRO-QPE4 (more details in Sect. 2.3). We quote median and 16th and 84th
percentiles ($\sim 1\sigma$) uncertainties from fit posteriors, unless
otherwise stated, for fit parameters, flux and luminosity. The bolometric
luminosity (labeled with “bol”, e.g., $L_{disk,bol}$) is obtained integrating
the adopted source model between 0.001 and 100 keV. For non-detections, we
quote $\sim 1\sigma$ ($\sim 3\sigma$) upper limits using the 84th (99th)
percentiles of the fit posteriors, unless otherwise stated. eROSITA source
plus background spectra (Sect. 2.2) were fit including a model component for
the background, which was determined via a principal component analysis (e.g.,
Simmonds et al., 2018) from a large sample of eROSITA background spectra
(e.g., Liu et al., 2022). _XMM-Newton_ spectra (Sect. 2.4) were instead fit
using wstat, namely XSPEC implementation of the Cash statistic (Cash, 1979),
given the good counts statistics in both source and background spectra. In
plots (e.g., Appendix A.1), data are rebinned, with uncertainty on the summed
counts ($CTS_{\rm tot}$) computed as $1+\sqrt{CTS_{\rm tot}+0.75}$, only for
visualization purposes.

As discussed in Miniutti et al. (2023a, b), we interpret the QPE emission as
thermal-like and we use a simple model black body (zbbody in XSPEC). This is a
rather model-independent interpretation given the observed spectral shape,
with consequences only on the QPE’s bolometric emission. The soft X-ray
emission in quiescence, namely in between the eruptions, is often detected in
QPE-sources and interpreted as the inner regions of a radiatively efficient
accretion disk. Therefore, we model the quiescence, if detected, with a diskbb
in XSPEC222We refer to this model (zashift(diskbb) in XSPEC) as disk or diskbb
throughout the paper.. This component is held fixed whilst fitting of the
eruption, therefore QPE fluxes are integrated only under the QPE model. For
spectra with good counts statistics (e.g., _XMM-Newton_ spectra) we allow the
quiescence parameters free to vary within the 10th and 90th percentiles of the
posterior distributions of the quiescence-only spectral fits. In this way, the
spectral fit parameters and fluxes of the eruptions are marginalized over the
uncertainties of the quiescence model. For spectra with lower counts
statistics (e.g., eRASS spectra) we freeze the quiescence model in the QPE
fits. Additional spectral components are added, if needed to model residuals,
and discussed individually in the following Sections. The best-fit model among
the ones adopted is selected by inspection of the residuals and by comparing
the logarithmic Bayesian evidence (_Z_), adopting the model with the highest
value. In general, these hard residuals, if present, are here interpreted as
thermal Comptonization of the disk emission. We use either a simple
phenomenological powerlaw (zpowerlw in XSPEC) or a more physically motivated
model (e.g., nthComp in XSPEC; Zdziarski et al., 1996; Życki et al., 1999),
depending on the counts statistics in the spectra. We note that alternative
models may also be used in future work, although we make the simple assumption
that disk residuals are due to Comptonization of the disk photons. This is for
the sake of simplicity and it is motivated by its nearly ubiquitous presence
in accretion flows around black holes.

Figure 1: eRASS1-5 light curves of eRO-QPE3 in the $0.2-2.0\,$keV band, each
scaled by a reference starting time (MJD $\sim$58874.111, $\sim$59057.576,
$\sim$59233.903, $\sim$59422.701 and $\sim$59604.611 from $t_{\rm eRO,1}$ to
$t_{\rm eRO,5}$, respectively). Orange data points highlight the putative
detection of eruptions, whilst the gray line connects the background level
across the eROdays in each eRASS.

###  2.2 eROSITA

eROSITA (Predehl et al., 2021) is the soft X-ray instrument aboard the
Spectrum-Roentgen-Gamma (_SRG_) mission (Sunyaev et al., 2021). On 13 December
2019 it started observing in survey mode and to this date it has completed
four (and started the fifth) of the foreseen eight all-sky surveys (eRASS1-8),
each of which is completed in six months. In each survey, every source in the
sky is observed for $\sim 40\,$s every $\sim 4\,$h (i.e., a so-called eROday)
for a total number of times within a single eRASS that depends on its location
in the sky: around six times on the ecliptic plane and increasing toward
higher (ecliptic) latitudes. We have developed an algorithm to look for
significant and repeated high-amplitude variability in the eROSITA sources.
Light curves are systematically extracted by the eROSITA team with the srctool
task of the eROSITA Science Analysis Software System (eSASS; Brunner et al.,
2022) from event files version 020. In a nutshell, we select sources showing
high-amplitude variability within eROdays of the single eRASS, or across
multiple eRASS. After excluding secure Galactic objects using Gaia DR3 (Gaia
Collaboration et al., 2016, 2021), we visually inspect their X-ray and
multiwavelength properties, although only the former (truly alternating
variability pattern and a soft spectrum) are used to consider a source as QPE
candidate. A first version of this method was described in A21 and we present
more details in a companion paper (Arcodia et al., 2023) presenting the
calculation of intrinsic volumetric rates based on the eROSITA QPE search
method.

####  2.2.1 eRO-QPE3

The source eRASSt J140053-284557 (hereafter eRO-QPE3) is located at the
astrometrically corrected X-ray position of (RAJ2000, DECJ2000)=(210.2222,
-28.7665), with a total 1$\sigma$ positional uncertainty (including a
systematic error of $\sim 1.5$”) of 1.6”, based on the latest-available
internal catalog from the first four surveys (eRASS:4). The source is detected
in eRASS1 with the identifier 1eRASS J140053.3-284558 (Merloni et al., 2023).
It was also independently found in a search for TDEs, as a new, relative to
the archival upper limit, soft eRASS1 source (Grotova et al, in prep). During
eRASS1 the source was detected in each eROday with a compatible flux level,
with the exception of one eROday showing count rate higher by a factor $\sim
2-3$ (top left of Fig. 1). However, this intra-eROday variability was not
deemed significant by the flare searching code. eRO-QPE3 then triggered an
alert from the QPE-candidates searcher in eRASS2, eRASS3 and eRASS4. In
eRASS2, the source was fainter compared to eRASS1 (in terms of average count
rate), while still detected at all eROdays with an overall constant flux with
the exception of one (top middle panel in Fig. 1). In eRASS3-4-5, nearly all
the counts were detected in a single eROday, with most of (or all, depending
on the eRASS) the remaining eROdays consistent with background (see Fig. 1).

Figure 2: Same as Fig. 1, but for eRASS4 data of eRO-QPE4. The reference
starting time is $MJD\sim 59457.807$. eRO-QPE4 was undetected in the previous
eRASS1-3 survey.

Figure 3: _Left panels:_ $45$”$\times\,45$” cutout of the DESI Legacy Imaging
Surveys Data Release 10 [Legacy Surveys / D. Lang (Perimeter Institute)] with
the X-ray 1$\sigma$ position circles in red (eROSITA) and green (XMM-Newton),
for eRO-QPE3 (_top_) and eRO-QPE4 (_bottom_). _Right panels:_ SALT optical
spectra of eRO-QPE3 (_top_) and eRO-QPE4 (_bottom_), with inferred spec-z of
$\sim 0.024$ and $\sim 0.044$, respectively. The sky spectrum of the red
spectral exposure is shown in cyan in arbitrary units in both subpanels.

Details on the spectral fits performed on eROSITA data of eRO-QPE3 are
reported in Appendix A.1 and Table 2. The eRASS1 spectrum in quiescence was
modeled with a diskbb plus zpowerlw (top panel of Fig. 16). The median (and
16th, 84th percentiles) peak temperature of the disk model is $kT_{\rm
in}=kT_{\rm disk}=100_{-20}^{+18}\,$eV, with a rest-frame flux of $F_{\rm
0.2-2.0\,keV}=1.4_{-0.6}^{+0.4}\times 10^{-12}\,$erg s-1 cm-2. Here, we
consider the orange point in the top-left panel of Fig. 1 as a bright state
part of an eruption. However, we do not exclude the possibility that in eRASS1
no QPEs were present, as the total eRASS1 spectrum can be also modeled with
the quiescence model alone (see Appendix. A.1 and Fig. 15). The quiescence
model was held fixed during the putative eruption and a black body component
was added (bottom panel of Fig. 16). The median (and 16th, 84th percentiles)
peak temperature of the QPE component is $kT=kT_{\rm QPE}=122_{-15}^{+19}\,$eV
and its flux is $F_{\rm 0.2-2.0\,keV}=(3.0\pm 0.7)\times 10^{-12}\,$erg s-1
cm-2. The median temperature is harder than the quiescence disk’s peak
temperature, but compatible within uncertainties. This confirms that the
contrast between the putative quiescence and the QPE, if present, is low
during eRASS1. The total flux during the bright eROday, quiescence included,
in the observed X-ray band is $F_{\rm 0.2-8.0\,keV}=4.9_{-0.6}^{+0.7}\times
10^{-12}\,$erg s-1 cm-2. The eRASS2 spectrum in quiescence is fainter than in
eRASS1 (see top panel of Fig. 18 and Fig. 1). The median (and related 16th,
84th percentiles) temperature of the disk is $kT_{\rm
disk}=89_{-60}^{+42}\,$eV, with a flux $F_{\rm
0.2-2.0\,keV}=3.12_{-3.11}^{+1.51}\times 10^{-13}\,$erg s-1 cm-2. The total
flux in the observed X-ray band is $F_{\rm
0.2-8.0\,keV}=5.8_{-2.9}^{+2.7}\times 10^{-13}\,$erg s-1 cm-2. As done for
eRASS1, this quiescence model is held fixed during the brighter eROday (the
orange point in the top-medium panel of Fig. 1) and a black body component is
added (bottom panel of Fig. 18). The median (and 16th, 84th percentiles) peak
temperature of the QPE component is $kT_{\rm QPE}=82_{-9}^{+10}\,$eV and its
flux is $F_{\rm 0.2-2.0\,keV}=3.2_{-0.7}^{+1.0}\times 10^{-12}\,$erg s-1 cm-2.
The median temperature is colder compared to the bright eROday in eRASS1,
despite the flux being compatible. However, given the short $\sim 40$s
exposure of an eROday compared to the typical QPE duration ($0.5-7\,$h, A21),
eRASS data catch the eruption at different phases and they must be considered
as lower limits of the true QPE peaks. The total flux during the bright
eROday, quiescence included, in the observed X-ray band is $F_{\rm
0.2-8.0\,keV}=3.9_{-0.8}^{+1.0}\times 10^{-12}\,$erg s-1 cm-2. Therefore,
despite the decreasing quiescence level, the flux in the bright eROday is
compatible between eRASS1 and eRASS2. In eRASS3, eRASS4 and eRASS5, most of
the signal is observed during a single eROday, therefore no quiescent state is
detected. We adopt a disk model to compute upper limits (Table 2) and we fit
the bright state with a black body QPE component only. In eRASS5, the fit is
compatible with background (within $3\sigma$, Fig. 21) even in the putative
bright state. We report the fit values in Table 2 for completeness.

####  2.2.2 eRO-QPE4

The source eRASSt J044534-101201 (hereafter eRO-QPE4) is located at the
astrometrically corrected X-ray position of (RAJ2000, DECJ2000)=(71.3921,
-10.2005), with a total 1$\sigma$ positional uncertainty (including a
systematic error of $\sim 1.5$”) of 4.9”, based on the latest-available
internal catalog from the first four surveys (eRASS:4). eRO-QPE4 was
undetected in the first three eRASS, with a $1\sigma$ ($3\sigma$) upper limit
of $F_{\rm 0.2-2.0\,keV}<3.8\times 10^{-15}$ erg s-1 cm-2 ($<2.8\times
10^{-14}$ erg s-1 cm-2). In eRASS4, most of the X-ray signal was observed in a
single eROday (see Fig. 2). Adopting a black body model for the QPE emission
(a behavior in the source confirmed with follow-up observations), we obtain
$kT_{\rm QPE}=93_{-14}^{+18}\,$eV and a flux $F_{\rm
0.2-2.0\,keV}=3.4^{+1.7}_{-1.1}\times 10^{-12}$ erg s-1 cm-2. Excluding this
eROday, the source is undetected with a $1\sigma$ upper limit of $F_{\rm
0.2-2.0\,keV}<6.9\times 10^{-14}$ erg s-1 cm-2, using a diskbb model. The
position of eRO-QPE4 was not observed in the partial eRASS5 survey.

###  2.3 Optical identification

####  2.3.1 eRO-QPE3

eRO-QPE3 is associated with the galaxy 2MASS 14005331-2846012 at (RA, Dec) =
(14:00:53.315, -28:46:01.26). We show the Legacy Survey DR10 optical image in
the top left panel of Fig. 3, centered at the X-ray coordinates. The eROSITA
position and 1$\sigma$ accuracy (based on the cumulative eRASS:4 image) are
shown with a red cross and circle, respectively, whilst in green we show the
analog from the deeper, hence more accurate, _XMM-Newton_ observation.
Similarly to other QPE sources, eRO-QPE3 is consistent with the nucleus
position, given the available positional accuracy (; ; ). 2MASS
14005331-2846012 was observed in the grizJHK bands with the seven-channel
imager GROND (Gamma-Ray Burst Optical/Near-Infrared Detector; Greiner et al.,
2008) at the MPG 2.2 m telescope at La Silla Observatory (Chile) on 2020
February 27th. The data were reduced with the standard GROND pipeline (Krühler
et al., 2008), which performs the bias and flat-field corrections, image
stacking and astrometric calibration. The photometric calibration was achieved
against PanSTARRS (griz) and 2MASS (JHK; Skrutskie et al., 2006). The observed
Kron magnitudes in the AB system are $18.37\pm 0.06$, $17.60\pm 0.05$,
$17.29\pm 0.07$, $17.05\pm 0.05$, $16.82\pm 0.11$, $16.60\pm 0.13$ and
$16.95\pm 0.14$ mag, respectively.

We instigated optical spectroscopic follow-up with the Robert Stobie
Spectrograph (RSS, Burgh et al., 2003) on the Southern African Large Telescope
(SALT, Buckley et al., 2006) on the night of April 24 2022 (top right panel of
Fig. 3). Data were reduced following the procedure outlined in A21. The host
galaxy shows a seemingly inactive spectrum, with $z=0.024$ based on Calcium H
and K absorption lines (top right panel of Fig. 3). We fit the SALT optical
spectrum with Firefly, which is a fitting code for deriving the stellar
population properties of galaxy spectra (Wilkinson et al., 2017; Maraston et
al., 2020). SALT spectra are not calibrated to absolute values (Buckley et
al., 2018), therefore we first renormalized the spectrum flux using optical
photometry from the GROND data using the Specutil Python package (Earl et al.,
2023). Firefly was run twice using two different stellar population models
(Maraston & Strömbäck, 2011). The run with the population model ELODIE
(Maraston & Strömbäck, 2011) was adopted to estimate the mean values for
stellar-mass ($M_{*}$) and star formation rate ($SFR$). We added in quadrature
to the measured statistical errors a systematic error to account for the
choice between the two stellar population models, which was computed from the
difference between the two mean values. The stellar mass inferred with Firefly
for eRO-QPE3 is $M_{*}=2.56^{+0.24}_{-1.40}\times 10^{9}\,M_{\astrosun}$,
whist SFR is $0.20^{+0.02}_{-0.14}\,M_{\astrosun}\,$yr-1. The low $M_{*}$
value is in line with the very compact nature of the host galaxy (Fig. 3,
top), and is in line with those inferred for eRO-QPE1 and eRO-QPE2 (A21),
which is remarkable given the blind nature of our search. Using scaling
relations between black hole and total stellar mass of the galaxy (Reines &
Volonteri, 2015), we obtain $M_{\rm BH}=5.3^{+0.7}_{-3.5}\times
10^{6}\,M_{\astrosun}$. Although these scaling relations are not necessarily
well-calibrated at low masses, the relation with bulge stellar mass has been
shown to hold sufficiently well at lower masses too (Schutte et al., 2019). We
performed SED photometry fitting to have an independent estimate of $M_{*}$.
We collated Legacy Survey DR10 photometry with
RainbowLasso333https://github.com/JohannesBuchner/RainbowLasso and fit the SED
with Genuine Retrieval of AGN Host Stellar Population (GRAHSP; Buchner et al.,
in prep.). The model used in GRAHSP includes AGN components (continuum,
emission lines, torus) and galaxy components, both attenuated by dust and
redshifted. We obtain $M_{*}=0.74^{+0.41}_{-0.29}\times
10^{9}\,M_{\astrosun}$, which is slightly lower than the Firefly estimate.
This would correspond (Reines & Volonteri, 2015) to a black hole mass of
$M_{\rm BH}=0.93^{+0.79}_{-0.47}\times 10^{6}\,M_{\astrosun}$.

No significant emission lines are apparent in the spectrum (top right panel of
Fig. 3), although subtracting the continuum model from Firefly the presence of
faint narrow H$\alpha$, [O iii] $\lambda$5007 and O i $\lambda$6302 lines, and
very faint $H\beta$ can be inferred. Since we are mainly interested in
computing line flux ratios for narrow-lines diagnostics, we computed them from
the continuum-subtracted spectrum. We used the lmfit package (Newville et al.,
2014) and adopted a polynomial model plus Gaussians for emission lines. The
algorithm fits a narrow line to H$\beta$, both [O iii] lines, O i
$\lambda$6302 and H$\alpha$, whilst no narrow [N ii] lines are visible. We
obtain from this $\log([{O\textsc{iii}}]/{\rm H}\beta)\sim 0.42$ and
$\log([{N\textsc{ii}}]/{\rm H}\alpha)\lesssim-0.87$, after tentatively
computing an upper limit on the [N ii] $\lambda$6549 line by imposing the
presence of a narrow line. These values would place eRO-QPE3 in the star-
forming region of narrow lines classifications, whilst using the
$\log({O\textsc{i}}/{\rm H}\alpha)=-0.45$ the galaxy is instead classified as
a LINER (Kewley et al., 2006). Given the limited resolution and tentative
detection of some narrow lines, we defer a conclusive classification to future
work. However, this result confirms that for these nuclear transients narrow
lines classifications are ambiguous and have to be interpreted with care,
particularly so since they trace past nuclear activity potentially unrelated
to the current transient activity. We obtained an independent estimate of the
black hole mass by fitting the SALT spectrum with the Penalized PiXel-Fitting
method (pPXF; Cappellari & Emsellem, 2004; Cappellari, 2017, 2023). The fit
velocity dispersion is $\sim 204\,$km s-1. Estimating an instrumental
broadening of $6\AA$ from the narrowest arc lines for the grating angle
settings adopted, the instrumental velocity dispersion computed around the
median wavelength $\sim 150\,$km s-1. Subtracting this dispersion in
quadrature and summing in quadrature the template dispersion of $\sim 83\,$km
s-1, we obtain a corrected velocity dispersion of $\sim 152\,$km s-1. Using
the scaling relation from Gültekin et al. (2009), we infer $M_{\rm BH}\sim
4\times 10^{7}\,M_{\astrosun}$, which is significantly larger than the
estimate obtained from the stellar mass. Given the good agreement of the $\log
M_{*}\sim 9$ estimate from both spectroscopy and photometry fitting, we adopt
the $M_{\rm BH}$ estimate from stellar mass as reference for eRO-QPE3 at this
stage (thus in the range $M_{\rm BH}\sim(0.9-5.3)\times
10^{6}\,M_{\astrosun}$). Further analysis with spectra obtained at higher
resolution is needed for a more robust narrow-line classification and black
hole mass estimate from velocity dispersion.

####  2.3.2 eRO-QPE4

eRO-QPE4 was associated with the galaxy 2MASS 04453380-1012047 at (RA, Dec) =
(04:45:33.80, -10:12:04.74). We show the 1$\sigma$ eROSITA positional accuracy
(based on the cumulative eRASS:4 image) in the bottom left panel of Fig. 3
with a red circle, whilst in green the more accurate _XMM-Newton_ circle. We
took an optical spectrum with RSS on SALT on February 23 2023 (bottom right
panel of Fig. 3), data were reduced as reported in A21. Similarly to the other
eRO-QPEs, the galaxy does not show prominent emission lines and a redshift of
0.0437 is estimated from Calcium absorption lines (bottom right panel of Fig.
3). We fit the salt spectrum with Firefly, after renormalizing it using
optical photometry from VEXAS DR2 catalogs (Khramtsov et al., 2021), which are
also consistent with SkyMapper petrosian magnitudes (Wolf et al., 2018). We
refer to the eRO-QPE3 fit for more details on the stellar population models
and the related systematic uncertainties. The stellar mass inferred with
Firefly for eRO-QPE4 is $M_{*}=1.6^{+0.7}_{-0.6}\times
10^{10}\,M_{\astrosun}$, whist SFR is
$2.26^{+1.06}_{-2.21}\,M_{\astrosun}\,$yr-1. Using Reines & Volonteri (2015),
we obtain $M_{\rm BH}=6.8^{+4.8}_{-3.2}\times 10^{7}\,M_{\astrosun}$. This is
the highest $M_{*}$ estimate inferred from any of the eROSITA QPEs (A21) and
the highest black hole mass ever inferred for a QPE source (Wevers et al.,
2022). With the GRAHSP SED fitting we obtain $M_{*}=0.60^{+0.29}_{-0.12}\times
10^{10}\,M_{\astrosun}$, which is slightly lower than the Firefly estimate,
even including uncertainties, but still much higher than what is estimated for
eRO-QPE3 with the same method. This corresponds (Reines & Volonteri, 2015) to
$M_{\rm BH}=1.74^{+1.28}_{-0.47}\times 10^{7}\,M_{\astrosun}$. From the SALT
spectrum we can only identify faint narrow H$\alpha$, [O ii] and [O iii]
$\lambda$5007 in emission, while no H$\beta$ nor [N ii] lines are visible,
even from the residual spectrum provided by Firefly after subtracting the
continuum. Hence, we are unable to securely classify the galaxy at this stage,
although in similarity with other eROSITA QPEs a strong preexisting AGN is
disfavored. We obtained a corrected velocity dispersion of $\sim 133\,$km s-1
with pPXF. Using the scaling relation from Gültekin et al. (2009), we infer
$M_{\rm BH}\sim 2.4\times 10^{7}\,M_{\astrosun}$, which is reasonably
consistent with that obtained via the $M_{*}$ estimate. Hence, for eRO-QPE4
all diagnostics used point to $M_{\rm BH}$ in the range $\sim(1.7-6.8)\times
10^{7}\,M_{\astrosun}$.

Figure 4: _XMM-Newton_ $0.2-10.0\,$keV EPICpn light curve of eRO-QPE3, of the
first (eRO3-XMM1, _left_) and second observation (eRO3-XMM2, _right_). The two
eruptions in eRO3-XMM1 are separated by $\sim 20.4$ h. Here, $t_{\rm
eRO3-XMM1,0}$ corresponds to MJD $\sim 59779.799$ and $t_{\rm eRO3-XMM2,0}$ to
$\sim 59799.745$. Figure 5: Source plus background spectrum of eRO-QPE3 in
quiescence (dark gray, background dominated) and at the peak of the first
burst in the observation eRO3-XMM1 (orange, following the color-coding of Fig.
22). The light-blue line and contour represent the background spectrum and its
uncertainties. Source plus background data points are shown to put the source-
only models (lines and contours with the same colors) into context. Darker
model contours are $1\sigma$ percentiles, lighter ones are $3\sigma$.

###  2.4 XMM-Newton

Two _XMM-Newton_ observations were performed on eRO-QPE3, namely ObsID
0883770101 starting on 19 July 2022 (hereafter eRO3-XMM1) and 0883770701
starting on 08 August 2022 (hereafter eRO3-XMM2). A single _XMM-Newton_
observation was taken on eRO-QPE4, namely ObsID 0883770401 starting on 10
March 2023 (hereafter eRO4-XMM). We reduced _XMM-Newton_ data of EPIC-MOS1 and
2 (Turner et al., 2001) and EPIC-PN (Strüder et al., 2001) cameras and the
Optical Monitor (OM; Mason et al., 2001) using standard tools and
prescriptions (SAS v. 20.0.0 and HEAsoft v. 6.29). Event files from EPIC
cameras were screened for flaring particle background. For eRO-QPE3, source
(background) regions were extracted within a circle of 30” centered on the
source (in a nearby source-free region). For eRO-QPE4, apertures of 40” were
used. The X-ray position obtained from eROSITA during the survey was refined
using _XMM-Newton_ data and the task eposcorr. We cross-correlated the X-ray
sources in the EPIC-PN image with external optical and infrared catalogs
(available through the Processing Pipeline Subsystem). The counterparts of
eRO-QPE3 and eRO-QPE4 were excluded to obtain a more unbiased estimate of the
possible offset from the nucleus. The resulting 1$\sigma$ positional circle
from _XMM-Newton_ is shown in green in Fig. 3 for both eRO-QPE3 (accuracy of
$1.5$”) and eRO-QPE4 (accuracy of $4.9$”). For both sources, the refined X-ray
position is consistent with being nuclear within the current uncertainties.

####  2.4.1 eRO-QPE3

In eRO3-XMM1 two eruptions are observed, with faster rise than decay,
separated by $\sim 20\,$h, while in eRO3-XMM2 only one burst was detected
(Fig. 4). We use the burst model described in Arcodia et al. (2022), since it
has shown itself to be successful in parametrizing asymmetric QPE light
curves:

$\cases{}{eq:1}A\,\lambda\,e^{\tau_{1}/(t_{\rm peak}-t_{\rm as}-t)}&\text{if
$t<t_{\rm peak}$}\\\ A\,e^{-(t-t_{\rm peak})/\tau_{2}}\text{if $t>=t_{\rm
peak}$}\\\ {}$ (1)

which is evaluated at zero for times smaller than the asymptote at $t_{\rm
peak}-t_{\rm as}$, where $t_{\rm as}=\sqrt{\tau_{1}\,\tau_{2}}$. $\tau_{1}$
and $\tau_{2}$ are the characteristic timescales, although only the latter is
directly related to the decay timescale. $A$ is the amplitude at the peak and
$\lambda=e^{t_{\lambda}}$ a normalization, where
$t_{\lambda}=\sqrt{\tau_{1}/\tau_{2}}$. Rise and decay times can be defined as
a function of $1/e^{n}$ factors with respect to the peak flux, with $n$ being
an integer. Rise and decay are obtained from the width and asymmetry factors
($w$ and $k$), where $w(n)=\tau_{2}\,n-\tau_{1}/(t_{\lambda}\,n)+t_{\rm as}$
and $k(n)=[\tau_{2}\,n+\tau_{1}/(t_{\lambda}\,n)-t_{as}]/w$. We define rise
and decay timescales as $\tau_{\rm rise}=w\,(1-k)/2$ and $\tau_{\rm
decay}=w\,(1+k)/2$ using $n=1$. We quote median values and related 16th and
84th percentiles. Referring to the three eruptions observed by _XMM-Newton_
with time, the rise is $2908_{-435}^{+95}$ s, $2522_{-192}^{+290}$ s,
$2903_{-175}^{+219}$ s, respectively, and the decay is $5896_{-308}^{+681}$ s,
$5194_{-357}^{+672}$ s, $5015_{-512}^{+395}$ s, respectively. The two
eruptions in eRO3-XMM1 are separated by a median recurrence time (and related
16th, 84th percentiles) $20.40_{-0.11}^{+0.25}$ h. We show an example of this
model and its application to QPEs in Fig. 22, for the first burst of
observation eRO3-XMM1. Given the relatively low signal-to-noise in eRO-QPE3,
we test the QPEs’ energy dependence dividing the light curve in two energy
bins only, namely $0.2-0.6\,$keV and $0.8-2.0\,$keV. Adopting the first
eruption of eRO3-XMM1 as representative, we obtain a rise (decay) of
$3661_{-244}^{+202}$ s ($6020_{-342}^{+445}$ s) in the low-energy bin and of
$2237_{-398}^{+895}$ s ($4946_{-1234}^{+1288}$ s) in the high-energy bin.
Furthermore, we obtain that the peak time of the low-energy light curve is
$(13.3\pm 0.3)$ ks after the start of eRO3-XMM1, while it is
$12.0_{-0.5}^{+0.6}$ ks for the high-energy light curve. Therefore, the
eruptions of eRO-QPE3 follow the known energy-dependence (; ; ; Chakraborty et
al., 2021; Arcodia et al., 2022), in that they are wider and start later at
lower energies.

Figure 6: _XMM-Newton_ $0.2-10.0\,$keV EPICpn light curve of eRO-QPE4. The
three eruptions are separated by $\sim 9.82$ h and $\sim 14.70\,$h. Here,
$t_{\rm eRO4-XMM,0}$ corresponds to MJD $\sim 60\,013.127$. Figure 7: Spectrum
of eRO-QPE4 in quiescence (dark gray) and at the peak of the second burst in
the observation eRO4-XMM (light green, following the color-coding of Fig. 22).
The dark-red line and contour represent the background spectrum and its
uncertainties. Source plus background data points are shown to put the source-
only models (lines and contours with the same colors) into context. Darker
model contours are $1\sigma$ percentiles, lighter ones are $3\sigma$.
Individual model components are labeled.

The quiescent state of eRO-QPE3 is too faint to be securely detected above
background. We adopt a disk spectrum to provide a flux upper limit of $F_{\rm
0.2-2.0\,keV}<1.1\times 10^{-14}$ erg s-1 cm-2 (Fig. 5). The fit posterior on
the flux, or disk normalization, is unconstrained, while the disk temperature
is loosely constrained to $kT_{\rm disk}=46_{-18}^{+21}$ eV. Despite the weak
constraint on the temperature, we conservatively consider the quiescence of
eRO-QPE3 to be undetected at the _XMM-Newton_ epoch with a soft X-ray
luminosity upper limit of $L_{0.2-2.0\,\rm keV}<1.6\times 10^{40}$ erg s-1. We
divided the first eruption of eRO3-XMM1, as example, in five epochs, namely
rise1, rise2, peak, decay1 and decay2 (see Fig. 22). We fit each epoch with a
black body (see more details in Appendix A.2) and report best-fit parameters
in Table 2. We present more details on the spectral evolution during the
eruptions in Sect. 3. Here, we quote results of the peak spectrum (see Fig.
5): we obtained $kT_{\rm QPE}=111_{-5}^{+6}$ eV and $F_{\rm
0.2-2.0\,keV}=3.1_{-0.2}^{+0.3}\times 10^{-13}$ erg s-1 cm-2 ($L_{\rm
0.2-2.0\,keV}=(4.2\pm 0.4)\times 10^{41}$ erg s-1), thus $\gtrsim 25-30$ times
brighter than the quiescence flux in the soft X-ray band. The bolometric
luminosity of the black body component is $L_{\rm QPE,bol}=(4.9\pm 0.5)\times
10^{41}$ erg s-1.

The OM UVW1 (U) filter was used throughout eRO3-XMM1 (eRO3-XMM2) with a series
of $\sim 4400$ s-long exposures. The subpanels of Fig. 4 show the OM light
curves in the respective filters. Similarly to other QPE sources (; ; ), no
simultaneous variability is observed, with the caveat that these data are
likely contaminated by the host galaxy and possibly nuclear star cluster. The
mean flux is $\lambda F_{\lambda,344\,nm}=(2.2\pm 0.5)\times 10^{-13}$ erg s-1
cm-2 and $\lambda F_{\lambda,291\,nm}=(1.0\pm 0.5)\times 10^{-13}$ erg s-1
cm-2 in the U (344 nm) and UVW1 (291 nm) filter, respectively, with magnitudes
of $19.44\pm 0.09$ and $19.87\pm 0.11$, respectively.

####  2.4.2 eRO-QPE4

In eRO4-XMM, three eruptions are observed (Fig. 6). Their duration appears
shorter compared to eRO-QPE3 and they appear more symmetric. We fit each
eruption in eRO4-XMM with both a Gaussian model and the asymmetric model of
Eq. LABEL:eq:1. The latter model performs significantly worse (in terms of
residuals and goodness of fit) and we adopt the Gaussian model for the
eruptions in eRO-QPE4. We show an example in Fig. 22, for the second burst of
observation eRO4-XMM. The median (with related 16th, 84th percentile values)
of the Gaussian width is $1678_{-90}^{+83}$ s, $1765_{-23}^{+24}$ s and
$2081_{-29}^{+30}$ s, for the three eruptions, respectively. The peak-to-peak
separation between the QPEs is $9.82_{-0.02}^{+0.03}$ h and
$14.70_{-0.01}^{+0.02}\,$h. We divided the light curve in energy bins of
$0.2-0.4\,$keV, $0.4-0.6\,$keV, $0.6-0.8\,$keV, $0.8-1.0\,$keV and
$1.0-2.0\,$keV (hereafter $E_{1}$ to $E_{5}$, respectively). Adopting the
second eruption in eRO4-XMM as example, given its higher signal-to-noise, we
obtain that the median (with related 16th, 84th percentile values) of the
Gaussian width is $(1759\pm 22)$ s, $1630_{-27}^{+25}$ s, $1417_{-32}^{+33}$
s, $1292_{-52}^{+51}$ s and $1299_{-116}^{+120}$ s, respectively, hence
decreasing from $E_{1}$ to $E_{5}$. Adopting the $E_{5}$ Gaussian peak as
reference (which occurs $10.28_{-0.03}^{+0.04}$ h after the start of
eRO4-XMM), we obtain delays of $603_{-30}^{+31}$ s, $464_{-30}^{+35}$ s,
$165_{-40}^{+43}$ s and $-30_{-70}^{+67}$ s, for $E_{1}$, $E_{2}$, $E_{3}$ and
$E_{4}$, respectively. Apart from $E_{4}$ which is consistent with the
properties of $E_{5}$ within uncertainties, the eruptions of eRO-QPE4 follow
the known energy-dependence (; ; ; Chakraborty et al., 2021; Arcodia et al.,
2022), in that they are wider and start later at lower energies.

The quiescent state of eRO-QPE4 is detected with a soft spectrum (Fig. 7)
similar to that of other QPE sources in quiescence (; ; , Chakraborty et al.,
2021). We fit the quiescence with a disk model and subsequently add a
Comptonization component (Fig. 7), which yields a disk temperature of $kT_{\rm
disk}=(43\pm 2)\,$eV with a flux $F_{\rm 0.2-2.0\,keV}=(3.9\pm 0.4)\times
10^{-13}$ erg s-1 cm-2. This corresponds to $L_{\rm 0.2-2.0\,keV}=(1.7\pm
0.3)\times 10^{42}$ erg s-1 for the disk component alone, or to a bolometric
luminosity of $L_{\rm disk,bol}=2.0_{-0.3}^{+0.6}\times 10^{43}$ erg s-1. The
Comptonization component is much fainter, with $F_{\rm
0.2-2.0\,keV}=1.4_{-0.4}^{+0.8}\times 10^{-14}$ erg s-1 cm-2, and its slope is
unconstrained. More details are presented in Appendix A.2 and Table 3. As done
for eRO-QPE3, the QPE flares of eRO-QPE4 were separated in five epochs, namely
rise1, rise2, peak, decay1 and decay2 (bottom panel of Fig. 22 for the second
burst of the observation eRO4-XMM). The quiescence model was held fixed by
imposing its parameters to vary only within the 10th-90th percentile interval
of the posteriors obtained during the quiescence fit alone. The fit results
for the different burst phases are reported in Table 3. For the epochs rise1,
rise2, decay1 and decay2 the best-fit model adopted for the burst is that of
the quiescence plus a black body component. For the peak spectrum we add a
Comptonization component (see Fig. 7). More details are presented in Appendix
A.2 and Table 3. This model yields $kT_{\rm QPE}=(123\pm 2)$ eV and $F_{\rm
0.2-2.0\,keV}=(2.9\pm 0.1)\times 10^{-12}$ erg s-1 cm-2 ($L_{\rm
0.2-2.0\,keV}=(1.27\pm 0.03)\times 10^{43}$ erg s-1). The bolometric
luminosity of the black body component is $L_{\rm QPE,bol}=(1.44\pm
0.04)\times 10^{43}$ erg s-1. Similarly to eRO-QPE3, the OM UVW1 filter was
used throughout and no simultaneous variability is observed. The mean flux is
$\lambda F_{\lambda,291\,nm}=(2.1\pm 0.5)\times 10^{-13}$ erg s-1 cm-2 and the
mean magnitude $18.96\pm 0.15$.

###  2.5 NICER

_NICER_ data were processed using HEAsoft v6.32.1, using NICERDAS v11a. We
grouped the data into 200-second Good-Time Intervals (GTIs) and adopted custom
filtering choices of unrestricted undershoot (underonly_range=*-*) and
overshoot rates (overonly_range=*-*), with per-FPM and per-MPU autoscreening
disabled to prevent aggressive event filtering. We manually discarded focal
plane modules (FPMs) with $0-0.2$ keV rates or $5-18$ keV count rates
$>3\sigma$ higher than average, or above an absolute threshold of 5 counts
sec-1. As both eRO-QPE3 and eRO-QPE4 are super-soft and faint, the emission
above 5 keV is entirely background-dominated, whereas the $0-0.2$ keV band is
undershoot-dominated, making these bands effective proxies for assessing the
background conditions.

Following the methodology of Chakraborty et al. (2023) we create the light
curves in Figs. 8 and 9 with time-resolved spectroscopy across the GTIs. After
screening the event lists, we used the
SCORPEON444https://heasarc.gsfc.nasa.gov/lheasoft/ftools/
headas/niscorpeon.html background model to estimate the contribution from the
diffuse X-ray background and non X-ray background. We fit the entire broadband
($0.2-18$ keV) array counts with
PyXspec555https://heasarc.gsfc.nasa.gov/docs/xanadu/xspec/
python/html/index.html. Along with the SCORPEON background we fit each GTI
with a source model represented by tbabs$\times$zbbody. We consider a source
detection as any GTI in which the blackbody normalization is $>1\sigma$
inconsistent with zero, namely a non-background component is required by the
fit at the $1\sigma$ level. The source count rates are the summed counts
contained only in the blackbody component. We consider all other cases as non
detections and we adopt as upper limit the $3\sigma$ upper error uncertainty
on the flux of the blackbody model that did not make the significance cut.

Figure 8: _NICER_ background-subtracted $0.2-2.0$ keV light curve of eRO-QPE3,
in the epoch eRO3-NICER1 (_top_ , starting at MJD$\sim 59697.644$) and
eRO3-NICER2 (_bottom_ , starting at MJD$\sim 59822.782$). Black points
represent secure detections of a source component on top of background
emission, contrary to the epochs shown in red which are consistent with
background-only. Vertical dashed lines represent evenly spaced recurrences of
$20\,$h, as a guide for the eye.

###  2.6 eRO-QPE3

eRO-QPE3 was observed for a total of $\sim 26.2$ ks between 28 April 2022 and
3 May 2022, and again for a total of $\sim 40.3$ ks between 31 August 2022 and
11 September 2022, namely a few months before and about a month after
eRO3-XMM1 (Fig. 8). These observation sets are hereafter named eRO3-NICER1 and
eRO3-NICER2, respectively. In both sets, the source is significantly detected
in a transient fashion, although in most cases the full eruption is not
resolved, at times due to nonoptimal background conditions preventing a secure
detection of a source component.

Only two clear bright eruptions are seen in eRO3-NICER1 (Fig. 8, top),
separated by roughly $\sim 17$ h, which is significantly shorter than the
$\sim 20.4\,$h recurrence seen in eRO3-XMM1 (Fig. 4). eRO-QPE3 is then
significantly detected above background for a few short exposures in
eRO3-NICER1. As these transient detections are roughly separated by $\sim
20\,$h (see vertical dashed lines in Fig. 8, as guide for the eye), we are
confident that they all correspond to eruption phases. Although current data
do not allow for a more precise constraint on the exact recurrence times, we
can conclude that in eRO-QPE3 QPEs have a typical recurrence in the range
$\sim 17-20\,$h, with significant scatter as seen in other sources (e.g., eRO-
QPE1; A21). The brightest eruptions in eRO3-NICER1 reach a maximum flux of
$F_{\rm 0.2-2.0\,keV}=9.8_{-1.3}^{+1.2}\times 10^{-13}$ erg s-1 cm-2 and a
temperature $kT=133_{-8}^{+10}\,$eV, whilst the average of all bursts
including the other transient detections is $F_{\rm
0.2-2.0\,keV}=6.5_{-3.1}^{+2.3}\times 10^{-13}$ erg s-1 cm-2 and
$kT=107_{-20}^{+24}\,$eV. The maximum values are larger than those observed in
eRO3-XMM1, perhaps indicating a long-term decay of peak flux and temperature.
In eRO3-NICER2, only part of a bright eruption is observed (Fig. 8, bottom),
with a peak flux and temperature roughly consistent with those of the maximum
values observed in eRO3-NICER1. At several other epochs the source is found
with a flux of $F_{\rm 0.2-2.0\,keV}\sim 5\times 10^{-13}$ erg s-1 cm-2,
similar to the peak flux observed in eRO3-XMM1. This would suggest that most
QPEs peak at a level consistent with eRO3-XMM1, but occasional brighter
eruptions reach a brighter flux, as that seen at the maximum of eRO3-NICER1
and eRO3-NICER2. The repetition is unclear and not as close to $\sim 20\,$h as
in eRO3-NICER1, although in most snapshots we were not able to securely assess
the presence of a source component. However, these lower-flux detections close
to the background count rate are much harder to be securely disentangled from
the latter, compared to brighter eruptions. Hence, we refrain from further
interpretations of eRO3-NICER2 at this stage and defer more thorough analysis
to future work.

###  2.7 eRO-QPE4

Figure 9: As in Fig. 8 but for eRO-QPE4, starting at MJD$\sim 59971.637$.
Vertical dotted lines represent approximate locations for the eruptions’ peak
(thus they are not evenly spaced), as a guide for the eye.

eRO-QPE4 was observed for a total of $\sim 66$ ks between 27 January 2023 and
2 February 2023 (Fig. 9), hereafter named eRO4-NICER. The source is
significantly detected in a transient fashion, although in most cases the full
eruption is not resolved due to the short duration of eruptions ($\approx 30$
min of FWHM, based on _XMM-Newton_ , Fig. 6) compared to the typical gap in
_NICER_ ’s monitoring. In Fig. 9, vertical dotted lines represent approximate
locations for the eruptions, as a guide for the eye. Inferred separations are
in the range $\sim(11-15.5)$ h and appear somewhat regular (considering that
an eruption was likely lost in the gap around MJDeRO4-NICER \+ 1.5 in Fig. 9).
However, they do not come in a clear alternating pattern such as in GSN 069
and eRO-QPE2 (; ; Arcodia et al., 2022), nor do they show very large scatter
such as the eruptions in eRO-QPE1 (A21; Chakraborty et al., 2023) or
RXJ1301.9+2747 (G20; Giustini et al., in prep.), but rather show an
intermediate behavior. The peak flux and temperature range from $F_{\rm
0.2-2.0\,keV}=1.6_{-0.1}^{+0.2}\times 10^{-12}$ erg s-1 cm-2 and
$kT=121_{-7}^{+6}\,$eV for the lowest peaks, to $F_{\rm 0.2-2.0\,keV}=(2.4\pm
0.2)\times 10^{-12}$ erg s-1 cm-2 and $kT=(118\pm 5)\,$eV. This is reasonably
consistent with _XMM-Newton_ (Fig. 6 and Table 3), given that not all NICER
eruptions are resolved and some peaks have been missed.

###  2.8 _Swift_

Two serendipitous _Swift-XRT_ observations were taken with eRO-QPE3 in the
field of view, on 2020-08-25 and 2020-12-26, for an exposure of $\sim 584$ s
and $\sim 767$ s, respectively. Aperture photometry was performed at the
location of eRO-QPE3 with the Python package photutils. The source aperture
adopted was $40$”, while background counts were extracted in an annulus with
inner and outer radius of $120$” and $360$”, respectively. Count rates were
converted to fluxes using WebPIMMS and adopting the best-fit models of the
closest eROSITA observation. eRO-QPE3 was detected during the first
observation, close to eRASS2, at about the same flux level of the eRASS2
bright state, perhaps indicative of a serendipitous QPE observation (see Sec.
4.1). The second observation did not detect the source, at the same depth of
the eRASS3 upper limit of the faint phase. eRO-QPE3 was in the UVOT field of
view only in the second observations, for a total exposure of $\sim 390\,$s
with the UVW1 filter at 268 nm. Performing aperture photometry with the task
uvotsource, eRO-QPE3 is only detected at 2.6$\sigma$, hence it is formally
undetected. The corresponding flux upper limit is $<1.3\times 10^{-13}$ erg
s-1 cm-2, so consistent with the OM-UVW1 detection of $\sim 1.0\times
10^{-13}$ erg s-1 cm-2 at 291 nm.

We triggered _Swift_ ToO follow up observations of eRO-QPE4 after the eRASS4
detection. eRO-QPE4 was also serendipitously observed with _Swift_ before the
eRASS4 discovery. We used the XRT online data analysis
tool666http://www.swift.ac.uk/user_objects (Evans et al., 2009) to check
whether eRO-QPE4 was detected for each observation. The tool was also used to
extract the X-ray spectra for observations in which it was detected, and to
estimate the $3\sigma$ count rate upper limits for non-detections. The X-ray
spectra were rebinned to have at least one count in each bin. An absorbed
accretion disk model (diskbb) model was used to calculate the flux and upper
limits. Most observations are non detections, shallower than the eROSITA and
_XMM-Newton_ fluxes, hence uninformative. In between the eRASS4 detection and
eRO4-XMM, there are a few _Swift-XRT_ detections in the range $F_{\rm
0.2-2.0\,keV}\sim(1-4)\times 10^{-12}$ erg s-1 cm-2, which is compatible with
the eruptions fluxes observed in eRASS4 and with _XMM-Newton_ , thus
supporting a serendipitous detection during an eruption. eRO-QPE4 was not in
the UVOT FoV of the archival observations. We thus only analyzed the UVOT data
for the ToO observations taken after eRASS4. We used the UVOT analysis
pipeline provided in HEASoft software (version 6.31) with UVOT calibration
version 20201215 to analyze the UVOT data. Source counts were extracted from a
circular region with a radius of $5^{\prime\prime}$ centered at the source
position. A $20^{\prime\prime}$ radius circle from a source-free region close
to the position of eRO-QPE4 was chosen as the background region. The task
uvotsource was used extract the photometry. The UVOT/UVW1 (UVOT/UVW2) flux is
fairly consistent among the several snapshots, with an average flux at $\sim
2.8\times 10^{-13}$ erg s-1 cm-2 ($\sim 2.4\times 10^{-13}$ erg s-1 cm-2).

###  2.9 Radio observations

We observed the coordinates of eRO-QPE3 and eRO-QPE4 on two occasions each
with the Australia Telescope Compact Array (ATCA) between October 2022 and
August 2023 (proposal ID C3513/C3527). All radio data were reduced using
standard procedures in the Common Astronomy Software Application (CASA v5.6.3,
CASA Team et al., 2022) including flux and bandpass calibration with PKS
1934-638 and phase calibration with PKS 1406-267 (eRO-QPE3), PKS 0458-020
(eRO-QPE4 4 cm), and PKS 0420-014 (eRO-QPE4 16 cm). For eRO-QPE3 we observed
the target on two occasions separated by approximately 8 months with the ATCA
dual 4 cm receiver and the CABB correlator producing 2x2 GHz of bandwidth
split into 2048x1 MHz channels centered on 5.5 GHz and 9 GHz. For eRO-QPE4 we
initially observed at 5.5 and 9 GHz with the dual 4 cm receiver and then
followed up 3 months later with an ATCA observation with the 16 cm receiver
and the CABB correlator producing 2 GHz of bandwidth split into 2048x1 MHz
channels centered on 2.1 GHz. Images of the target field were made with the
CASA task tclean. No radio emission was detected at the coordinates of either
QPE source in any of the observations. A summary of the ATCA observations
including the 3$\sigma$ flux density upper limits is given in Table 1. The
radio non-detections of eRO-QPE3 and eRO-QPE4 indicate it is unlikely the host
galaxies have strong AGN activity in the nuclei, in agreement with the optical
proxies.

Table 1: Summary of ATCA radio observations of eRO-QPE3 and eRO-QPE4. Date | Array config. | Frequency | Flux Density
---|---|---|---
|  | (GHz) | ($\rm{\mu}$Jy)
eRO-QPE3 |  |  |
2022-Oct-01 | 6D | 5.5 | $<54$
2022-Oct-01 | 6D | 9 | $<49$
2023-Jun-13 | 6D | 5.5 | $<51$
2023-Jun-13 | 6D | 9 | $<49$
eRO-QPE4 |  |  |
2023-May-05 | 1.5A | 5.5 | $<48$
2023-May-05 | 1.5A | 9 | $<39$
2023-Aug-20 | 6D | 2.1 | $<298$

##  3 Spectral evolution of QPEs

Figure 10: Spectral evolution of QPEs in eRO-QPE3 (_left_) and eRO-QPE4
(_right_). The top-right (bottom) subpanels show the luminosity (temperature)
evolution with time of the QPE component. Darker to lighter colors represent
the evolution from start to end of the eruptions. The first eruptions of the
eRO3-XMM1 observation and the second of the eRO4-XMM observation are taken as
reference for the QPEs in eRO-QPE3 and eRO-QPE4, respectively. The top-left
subpanels show the luminosity-temperature coevolution, analogous to the rate-
hardness evolution in Arcodia et al. (2022) which first reported the
hysteresis cycle.

Given the growing number of repeating X-ray transients, it is useful to
identify common properties that indicate a common emission process, or suggest
otherwise. For instance, Arcodia et al. (2022) first reported the specific
spectral evolution of QPEs in eRO-QPE1 as a soft X-ray spectrum which, for the
same count rate or flux, is harder during the rise than during the decay.
Moreover, the peak hardness of the spectrum is reached before the peak flux of
the eruption. The same behavior was identified in GSN 069 (Miniutti et al.,
2023b) and RX J1301.9+2747 (Giustini et al., in prep.). Identifying this
hysteresis cycle in a repeating X-ray transient is the smoking gun of the same
emission process. This is particularly important since the timing properties
of the known QPEs are quite diverse (; ; ; Arcodia et al., 2022), and can even
significantly change in the same source (Miniutti et al., 2023a, b).

We show the spectral evolution of eRO-QPE3 and eRO-QPE4 in Fig. 10. We choose
model-dependent quantities such as luminosity and temperature, compared to
pure data-driven quantities such as hardness-ratio and count rate. This is
because, at least in eRO-QPE4, the quiescence component needs to be decomposed
(e.g., as done for GSN 069, Miniutti et al., 2023b). This was not required for
eRO-QPE1 (Arcodia et al., 2022) as the source is not detected between the
eruptions. Therefore, Fig. 10 assumes that QPEs can be modeled by a black
body. Results would be identical with any model with an exponential decay due
to a characteristic temperature, for instance Bremsstrahlung or an accretion
disk. In Fig. 10, the first eruption of eRO3-XMM1 and the second of eRO4-XMM
are used as representative, although other eruptions are shown in Fig. 24. The
eruption evolves from darker to lighter colors of the respective color maps,
as shown in Fig. 22. The eruptions were divided in 5 epochs with the goal to
compare rise and decay at similar count rate or luminosity. As there is not a
one-to-one relation between count rate and luminosity, this is not necessarily
achieved. As a matter of fact, as much as the $L_{\rm bol}$ and $kT$ values
would change slightly depending on the definition of the epochs, the
hysteresis behavior in the $L_{\rm bol}-kT$ plane (or count rate versus
hardness ratio) would remain evident. This is confirmed by the same pattern in
other eruptions of both eRO-QPE3 and eRO-QPE4 (Fig. 24) and other QPE sources
(Arcodia et al., 2022; Miniutti et al., 2023b). We also note that, similarly
to eRO-QPE1 (Arcodia et al., 2022) and GSN 069 (Miniutti et al., 2023b), the
peak temperature is reached before the peak luminosity.

As done in Miniutti et al. (2023b), we estimate the size of this emitting
region, under the assumption that it is indeed black body emission. We assume
that the observer sees half of a radiating spherical surface. For eRO-QPE3
(Fig. 10, left) we obtain radii of $\sim 1.2\times 10^{10}\,$cm2, $\sim
1.2\times 10^{10}\,$cm2, $\sim 1.7\times 10^{10}\,$cm2, $\sim 1.6\times
10^{10}\,$cm2 and $\sim 1.8\times 10^{10}\,$cm2 for rise1, rise2, peak, decay1
and decay2, respectively. Therefore, we infer an increase in size of a factor
$\sim 1.5$ from start to end of the eruptions, lower than what was inferred
for GSN 069 (Miniutti et al., 2023b). For eRO-QPE4 (Fig. 10, right), we obtain
radii of $\sim 2.4\times 10^{10}\,$cm2, $\sim 7.0\times 10^{10}\,$cm2, $\sim
9.5\times 10^{10}\,$cm2, $\sim 17.3\times 10^{10}\,$cm2 and $\sim 11.6\times
10^{10}\,$cm2 for rise1, rise2, peak, decay1 and decay2, respectively.
Therefore, we infer an increase in size of a factor $\sim 7$ from rise1 to
decay1 and of a factor $\sim 5$ from rise1 to decay2, which is larger than
what was inferred for GSN 069 (Miniutti et al., 2023b). Regardless of the fine
details which are deferred to future homogeneous work extended to all QPE
sources, the common thread is that during the $L_{\rm bol}-kT$ hysteresis the
size of the emitting region increases, which is qualitatively consistent with
the latest models of disk-orbiter collisions (Franchini et al., 2023; Linial &
Metzger, 2023; Tagawa & Haiman, 2023). The absolute numbers are, of course,
model-dependent and any departure from this spectral model would likely change
the absolute values of the size, but not the qualitative evolution during the
eruptions, provided the X-ray spectrum is indeed the exponential tail of a
thermal spectrum.

##  4 Long-term evolution

###  4.1 eRO-QPE3: Quiescent flux decays until disappearance

Figure 11: Long-term evolution of eRO-QPE3, separating flux states between
quiescence and eruption (which includes the former). eROSITA points are shown
as circles, squares for _XMM-Netwon_ , diamonds for _NICER_ (the brightest
eruptions in the light curves of Fig. 8) and stars for _Swift-XRT_. All
uncertainties are $3\sigma$. Red contours on orange points highlight
observations in which the identification of an eruption is less secure. White
symbols for eRASS1 and eRASS2 correspond to the flux of the full observation
without separating different states. Horizontal lines highlight _ROSAT_ and
_XMM-Newton_ archival upper limits, as stated in the legend.

In Fig. 11, we show the long-term evolution of eRO-QPE3 inferred with eROSITA,
_XMM-Newton_ , _NICER_ and _Swift-XRT_. We separate flux states between
quiescence (dark red data points) and eruption (orange). This separation is
straight-forward in _XMM-Newton_ and _NICER_ data and at least in eRASS3 and
eRASS4 data (see Fig. 1). In eRASS1 and eRASS2, since the quiescence is
detected and given eROSITA’s sampling, the identification of the eruption
state is informed by the presence of QPEs at later times. However, formally
one cannot exclude high-amplitude variability of the quiescent state with no
eruptions. This potential no-QPE flux state is shown in white with dark red
contours and the related possible eruption states are shown with red contours
on orange points. For eRASS5, we consider the eruption state as ambiguous
since the variability is not as significant and the flux chain has a tail
extending to faint fluxes, therefore it is formally not well constrained at
$3\sigma$ (hence the large uncertainty in Fig. 11). The _Swift-XRT_ detection
is also ambiguous: its flux is as bright as the eruption state in eRASS2, but
no variability can be inferred from the few hundred seconds of exposure to
unambiguously identify it as an eruption. Despite these caveats, throughout
the rest of the discussion we assume that all the orange points caught eRO-
QPE3 in eruption, and the dark red points represent the flux of the intra-
QPEs’ quiescence.

Quite interestingly, eRO-QPE3 showed a disappearing quiescence component (Fig.
1 and Table 2), being detected in eRASS1 and eRASS2 and never since, including
_Swift-XRT_ and eRASS3 observations taken within 6 months. This is consistent
with what has been observed for GSN 069 (Shu et al., 2018; Miniutti et al.,
2023b), albeit over a decade and with a much better data coverage before the
QPEs’ discovery. Furthermore, the two QPE candidates presented in Chakraborty
et al. (2021) and Quintin et al. (2023) also showed evidence of eruptions on
top of a decaying baseline flux. In general, a possible interpretation of the
decaying quiescence in GSN 069 and the two candidates is that the accretion
flow was induced by a TDE. The presence of a precursor TDE is secure in
AT2019vcb (“Tormund”; Quintin et al., 2023), but since only the possible start
of an X-ray eruption was observed, the association between QPEs and TDEs is
still open. Were this connection to be unambiguously proven, one can interpret
the long-term evolution of eRO-QPE3 in a similar way. However, the available
data (only two epochs are detected in eRO-QPE3) do not allow us to achieve a
precise constraint or prediction on this long-term evolution. We can test
whether the spectral evolution between the disk component in eRASS1 and eRASS2
is consistent with a cooling thin disk. The disk temperature decreased from
$\sim 100\,$eV to $\sim 89\,$eV from eRASS1 to eRASS2 (Table 2). Approximating
the disk emission as a black body with constant area with the temperature of
the inner radius, the eRASS2 bolometric emission would be
$\sim(89/100)^{4}\sim 0.6$ times that of eRASS1. The disk bolometric emission
of eRASS1 and eRASS2, estimated from the spectral fits, is $L_{\rm
bol}=4.1_{-2.3}^{+1.6}\times 10^{42}\,$erg s-1 and $L_{\rm bol}\sim
8.4_{-8.3}^{+4.7}\times 10^{41}\,$erg s-1, respectively. This corresponds to a
decrease to $\approx 20\%$ of the eRASS1 luminosity, although given the large
$1\sigma$ uncertainties of the eRASS2 value, a 60$\%$ decrease is still
compatible. Regardless of the exact type and nature of the observed decay,
eRO-QPE3 is the first QPE source in which the intra-QPE quiescence is detected
at some earlier phases and never again. Other secure QPE sources have either
always shown it, or never (; ; ). Hence, eRO-QPE3 would bridge the gap between
QPE sources that always showed a (possibly time-evolving) quiescence spectrum
in between QPEs (e.g., GSN 069, Miniutti et al., 2023b) and QPE sources which
never showed a quiescence spectrum (e.g., eRO-QPE1, A21). This suggests that
the absence of intra-QPE quiescence in some QPE sources is merely
observational, as it might have faded compared to an earlier epoch.

Furthermore, we also show in Fig. 11 $3\sigma$ archival upper limits from
ROSAT (from 1991) and the _XMM-Newton_ Slew Survey (from 2000 and 2013; Saxton
et al., 2008). Comparing these with the eRASS1 flux (both if the quiescence
includes or not the putative eruption), we notice that ROSAT would have
detected the source if it were as bright as eRASS1. Therefore, the accretion
flow of eRO-QPE3 likely became brighter (e.g., perhaps radiatively efficient)
in the soft X-rays some time between 1991 and 2020.

Figure 12: Long-term evolution of eRO-QPE4, separating flux states between
quiescence and eruption. eROSITA points are shown as circles, squares for
_XMM-Netwon_ , diamonds for _NICER_ and stars for _Swift-XRT_. _Swift-XRT_ and
_NICER_ eruptions points are considered such based on their flux level similar
to the range shown by _XMM-Netwon_. All uncertainties are $3\sigma$. The
lower-panel shows _Swift-UVOT_ data with the UVW1 (light blue) and UVW2 filter
(azure).

Despite _XMM-Newton_ data and most _NICER_ data (unless there are large gaps
in the monitoring) allow the identification of the peak, the eROSITA and
_Swift-XRT_ data points would catch the source in an unknown part of the
eruptions. Therefore, all data points except _XMM-Newton_ and _NICER_ are
lower limits to the peak. Without _NICER_ monitoring covering several burst
cycles, one might have concluded that QPEs must have significantly faded over
only a few months. However, _NICER_ data taken both before and after eRO3-XMM1
and eRO3-XMM2 (which showed $F_{\rm 0.2-2.0\,keV}\sim 3.1\times 10^{-13}$ erg
s-1 cm-2) unveil the presence of some eruptions significantly brighter
($F_{\rm 0.2-2.0\,keV}\sim 1.0\times 10^{-12}$ erg s-1 cm-2) than the others,
which are instead compatible with the eRO3-XMM1 peak flux. This indicates that
whilst on average the peak flux might still be overall decreasing, current
data also unveil the presence of significant diversity in amplitudes, as seen
in other QPE sources (; ). We show the flux of these brigthest eruptions in
Fig. 8. Finally, we note that no significant optical variability is apparent
within the available public datasets from optical sky monitors (e.g., ASAS-SN,
Shappee et al., 2014) covering the epochs shown in Fig. 11.

###  4.2 eRO-QPE4: quiescent flux appears or brightens after the eruptions

Fig. 12 shows the long-term evolution of the X-ray and UV flux of eRO-QPE4.
The UV flux is overall constant within the available exposuress. On the
contrary, the X-ray quiescence flux (dark gray points) is only constrained by
_XMM-Newton_ with a detection. However, it is interesting to note that both
the archival ROSAT (dotted line) and eROSITA 3$\sigma$ upper limits suggest
that the quiescent accretion disk must have been much fainter or absent at the
ROSAT/eROSITA epochs. Current data surely rule out the presence of a
radiatively efficient accretion flow, as bright as that seen by _XMM-Newton_ ,
prior to the eRASS4 QPE discovery. This would be in agreement with many other
QPE sources with no evidence of bright X-ray sources much earlier than the
onset of QPEs. However, we are not able to constrain the start of the QPE
behavior. Regarding the peaks of the eruptions, the eROSITA, _Swift-XRT_ and
_NICER_ fluxes are compatible with the range spanned by the faintest and
brightest eruption in eRO4-XMM (Fig. 6). Finally, we note that similarly to
eRO-QPE3 no significant optical variability is observed (e.g., ASAS-SN and
ZTF; Shappee et al., 2014; Bellm et al., 2019) during the epochs shown in Fig.
12.

##  5 Discussion

###  5.1 QPEs in preexisting AGN?

The X-ray properties of all QPE sources suggest that the nucleus is currently
active, as the soft X-ray spectrum observed in quiescence is indicative of
radiatively efficient accretion (; ; , Chakraborty et al., 2021). In eRO-QPE4
the quiescence emission was a few times fainter, or nonexistent, prior to the
QPE discovery (Fig. 12). This would support a transient nature for the
quiescent accretion disk and disfavor a long-lived radiatively efficient
accretion flow. However, in other QPE sources it is much harder to say whether
or not this was the case. In fact, most QPE sources do not have constraining
archival X-ray observations prior to the current activity phase, and results
from the radio and optical bands are currently ambiguous. For instance, eRO-
QPE2 is detected as a faint radio source, although its radio emission is
consistent with that predicted from star-formation alone (Arcodia et al., in
prep.). All other eROSITA QPE sources are undetected in radio (e.g., this
work), which also rules out a preexisting powerful AGN in these nuclei and
agrees with the lack of a canonical hard and bright X-ray power-law component
in QPE sources. From the optical band, given the lack of broad lines detected,
we do not have information on the recent nuclear activity. Narrow-line
diagnostics place the select number of QPE sources in different places of the
typical diagnostic diagrams including regions indicating the need of a nuclear
ionizing source (e.g., an AGN, Wevers et al., 2022). However, these
diagnostics cannot securely determine whether these lines were excited by the
current activity epoch, while spatially resolved observations will help
solving this ambiguity. As a matter of fact, GSN 069 was found to be evidently
powered by an AGN using narrow-line diagnostics (Wevers et al., 2022).
However, recent work by Patra et al. (2023) making use of archival HST data of
GSN 069 found evidence of a compact nuclear [OIII] region, which indicates
that the accretion activity is only $\sim 10-100\,$y old. More radially
extended [OIII] regions, which are not indicative of the current accretion
activity, would instead contaminate typical spectra which populate the narrow-
line diagnostic diagrams. In absence of a comparable systematic study with
spatially resolved observations on all QPE sources, simplistic narrow-line
diagnostics have to be interpreted with caution. Significant progress can be
made in the near future using integral-field unit data, and high angular
resolution spectroscopic and photometric observations, to understand the
spatial distribution of the gas in these galaxies and how it relates to the
current QPE activity.

###  5.2 What is new on QPEs based on these discoveries

Figure 13: Relation between recurrence and peak luminosity (_top_) and
recurrence and duration timescales (_bottom_) in secure QPE sources, shown as
labeled. The extent of shaded areas represents the standard deviation, when
available, or the span within observed values in a given source. eRO-QPE3 is a
clear outlier of the top panel, ruling out the presence of a predictive
correlation between timescales and peak luminosity.

The discovery of QPEs on top of a decaying X-ray continuum in eRO-QPE3 (Fig.
11) offers a new piece of evidence that strengthens the connection between
QPEs and previous TDEs in the same nucleus (Shu et al., 2018; Chakraborty et
al., 2021; Sheng et al., 2021; Miniutti et al., 2023b; Quintin et al., 2023).
Moreover, eRO-QPE3 showed the faintest peak luminosity (average of $L_{\rm
0.5-2.0\,keV}^{\rm peak}\sim 9.7\times 10^{41}\,$erg s-1 in the eRASS1-4
epoch, down to $\sim 4.2\times 10^{41}\,$erg s-1 in the eRO3-XMM1 epoch) and
the highest recurrence times ($\approx 20\,h$) among secure QPEs, therefore
invalidating any strict relation between peak luminosity and recurrence time
across the QPE population. In fact, eRO-QPE1 shows similar recurrence times
but a luminosity up to 10-100 times higher (A21). We show this in the top
panel of Fig. 13, where the extent of shaded areas represents the standard
deviation, when available, or the span within observed values in a given
source (; ; ; Arcodia et al., 2022; Miniutti et al., 2023a, b; Chakraborty et
al., 2023; Giustini et al., in prep.). This lack of correlation disfavor
accretion-based QPE emission models, in which both peak luminosity and
accretion timescales would depend on the black hole mass. Instead, for EMRI
models, the somewhat more diverse orbital parameters (semi-major axis,
eccentricity) and the orbiter’s mass and nature (i.e., stars of different
kind, degenerate stars and/or black holes) would more easily accommodate
scenarios in which the peak luminosity and average recurrence are decoupled.
Conversely, as shown in previous work (Chakraborty et al., 2021; Guolo et al.,
2024) recurrence and duration timescales do appear to be somewhat correlated
(bottom panel of Fig. 13), although we refrain from attempting to fit any
scaling due to the low number of sources and the lesson learned from the top
panel of Fig. 13. Nonetheless, for the sake of designing a somewhat informed
follow-up monitoring based on a single observed eruption (e.g., Quintin et
al., 2023), this tentative relation may be used based on the current QPE
sources.

eRO-QPE4 shows evidence of a harder component appearing close to the QPE peak
(Fig. 7), although at this stage it is unclear what its origin might be
(Appendix A.2). Modeling it as Comptonization of the QPE thermal continuum, we
infer a flux ratio of $\sim 1:94$ in the $0.2-2.0\,$keV band compared to the
latter. However, this is merely a possible interpretation, as after all the
inferred slope is rather unconstrained and its signal very weak. Nonetheless,
we note that until now, a harder component was only seen in quiescence (and
not ubiquitously, ; ; ; Chakraborty et al., 2021), but never during the
eruptions, let alone in a transient fashion.

In eRO-QPE4 we see clearly, for the first time, that the quiescence emission
as detected during the QPE epoch must have been either absent or much fainter
previous to the QPE behavior (Fig. 12). What is particularly intriguing is
that in eRASS4, when the first QPE was observed, the quiescence emission was
also much fainter than the eRO4-XMM epoch. Given the lower sensitivity with
_Swift-XRT_ and _NICER_ , their upper limits are shallower than the eRO4-XMM
level therefore we are not able to constrain how fast the brightening was.

The long-term emission of both eRO-QPE3 and eRO-QPE4 can only be constrained
qualitatively. In the former case the fainter peak flux, close to _NICER_ ’s
background level, prevents us from securely resolving the fainter eruptions,
which appear to be more frequent than the bright ones around the _NICER_ and
_XMM-Newton_ epochs (Fig. 4 and 8). In eRO-QPE4 eruptions have a FWHM of
$\approx 30\,$min, which is on the order of the typical separation in a
_NICER_ monitoring given its $\sim 90\,$-min orbit. Nonetheless, based on
_NICER_ data we are able to identify the timing properties of eRO-QPE4 as
somewhat intermediate compared to the other known QPEs. The eruptions are not
strictly regular or in alternation, as from Fig. 9 we roughly constrain the
last five uninterrupted recurrences to be $\sim 12.1$ h, $\sim 14.2$ h, $\sim
14.0$ h, $\sim 12.4$ h and $\sim 15.5$ h, combined with the $\sim 9.8\,$h and
$\sim 14.7\,$h separations seen by _XMM-Newton_ (Fig. 6). They are somewhat
similar to the irregular recurrences in eRO-QPE1 (A21; Chakraborty et al.,
2023) or RXJ1301.9+2747 (G20; Giustini et al., in prep.), although in eRO-QPE4
they do not show burst superposition such as in eRO-QPE1 (Arcodia et al.,
2022), nor closely spaced eruptions such as RXJ1301.9+2747 (Giustini et al.,
in prep.). Hence, eRO-QPE4 shows intermediate properties (between regular and
irregular arrival times) confirming what the latest orbital models suggest
(e.g., Linial & Metzger, 2023; Franchini et al., 2023), namely that precession
would induce different timing behavior in a continuum from regular (no disk
precession, lower eccentricity) and irregular (disk and orbital precession,
higher eccentricity).

We also note that in eRO-QPE3, in which the bursts recur every $\approx 20$ h,
the asymmetry (with a fast rise and slow decay) appears evident (Fig. 22,
top), whereas in eRO-QPE4 bursts are consistent with being symmetric within
the current statistics (Fig. 22, bottom). Interestingly, the QPE sources with
the clearest asymmetry (eRO-QPE1 and eRO-QPE3) are also the ones with
undetected quiescence.

The two discoveries in this work confirm the trend of finding QPEs from low-
mass galaxies ($\log M_{*}\approx 9-10$) and low-mass black holes ($\log
M_{\rm BH}\approx 5-7$), where the high-mass end has been now pushed to larger
values by eRO-QPE4. We remind readers that the eROSITA search applied to
discover the first four eROSITA QPEs is blind in terms of the host galaxy
nature, as we only select extragalactic sources with Gaia without selecting
specific galaxy counterparts. This might be a selection effect due to
observational and model-dependent reasons. An observational bias could be that
for more massive galaxies and black holes, the Wien tail of the accretion flow
would be shifted out of band. In this case, if the temperature of the QPE
component is somewhat dependent on the temperature of the accretion flow, the
QPE emission would also be softer and currently unobservable. So far, the
observed disk and QPE temperatures have spanned a fairly consistent range
within all the known QPE sources, although the absence of harder eruptions is
remarkable, despite being easily observable. Larger sample statistics on the
QPE population is required for more conclusive statements. In addition, for
partial TDEs and stellar EMRI models the detectability of QPEs is limited to
smaller galaxies and MBHs for which the tidal radius is not within the
innermost stable circular orbit (e.g., Hills, 1975; van Velzen, 2018; King,
2023; Linial & Metzger, 2023).

##  6 Summary and prospects

We report on the discovery of two more galaxies showing QPEs in the eROSITA
all-sky survey data, eRO-QPE3 and eRO-QPE4. eRO-QPE3 was found to flare once
in multiple eROSITA surveys (Fig. 1) and its nature as a QPE emitter was later
confirmed by _XMM-Newton_ (Fig. 4) and _NICER_ (Fig. 8). Eruptions are
observed to last $\sim 2.1-2.4\,$h and to recur every $\sim 17-20\,$h,
although the exact recurrence pattern is currently poorly constrained. eRO-
QPE4 was instead detected only once in a transient fashion across all eROSITA
surveys (Fig. 2) and both _XMM-Newton_ (Fig. 6) and _NICER_ (Fig. 9) confirmed
the repeating nature of the source. Eruptions in eRO-QPE4 are observed to last
$\approx 0.5\,$h and recur every $\sim 9.8-15.5\,$h.

Many of their properties are in agreement with those of known secure QPEs (; ;
), for instance:

  * •

Their X-ray spectra during the eruptions are soft and thermal in shape, with
typical peak temperatures of $kT=111_{-5}^{+6}$ eV and $kT=(123\pm 4)$ eV for
eRO-QPE3 and eRO-QPE4, respectively, as observed by _XMM-Newton_

  * •

The emission in quiescence, when detected, is consistent with the Wien tail of
a radiatively efficient accretion disk, with inner temperatures $kT_{\rm
in}\sim(90-100)$ eV and $kT_{\rm in}\sim 50$ eV for eRO-QPE3 and eRO-QPE4,
respectively (all spectral parameters are reported in Table 2 and 3)

  * •

Both sources show a clear energy dependence during the eruptions, with a
harder rise than decay at the same flux level (Fig. 10 and 24), which was
first reported for eRO-QPE1 in Arcodia et al. (2022) and then recovered in
both GSN 069 (Miniutti et al., 2023b) and RXJ1301.9+2747 (Giustini et al., in
prep.). At this stage, we consider it the only consistent property across all
secure QPE sources, and therefore it is a key requirement to name an X-ray
repeater as such.

  * •

Their optical counterparts are local galaxies with a seemingly inactive
spectrum (Fig. 3) and even if a secure classification is compromised by the
current data quality (e.g., see Wevers et al., 2022), a preexisting powerful
AGN is ruled out. This is in agreement with the lack of a bright hard power-
law spectrum indicative of a corona, the lack of broad lines, infrared
photometry suggestive of an AGN-like torus obscurer and with the deep radio
non-detections (Table 1).

  * •

Both eRO-QPE3 and eRO-QPE4 are found in low-mass galaxies. In eRO-QPE3 $M_{*}$
is in the range $(0.7-2.6)\times 10^{9}\,M_{\astrosun}$, which in turn implies
$M_{\rm BH}$ in the range $(0.9-5.3)\times 10^{6}\,M_{\astrosun}$ (Reines &
Volonteri, 2015). Using the stellar velocity dispersion, the inferred $M_{\rm
BH}$ (Gültekin et al., 2009) is however $\sim 4\times 10^{7}\,M_{\astrosun}$,
which only confirms how these scaling relations are poorly calibrated at low
masses. In eRO-QPE4, all estimates are more consistent and point toward a
slightly more massive galaxy, with $M_{*}$ being in the range $(0.6-1.6)\times
10^{10}\,M_{\astrosun}$ and black hole, with $M_{\rm BH}$ in the range
$(1.7-6.8)\times 10^{7}\,M_{\astrosun}$, with the velocity dispersion also
yielding $\sim 2.7\times 10^{7}\,M_{\astrosun}$. eRO-QPE4 therefore extends
the known QPE population to slightly higher masses.

Furthermore, there are many novelties brought by the new discoveries, which
only further highlight the importance to find more QPE sources to build
meaningful statistics for a population study. For instance, we report the
following:

  * •

eRO-QPE3 showed QPEs on top of a decaying quiescence emission (Fig. 11), in
support of other evidence that connects QPEs to a previous TDE (Chakraborty et
al., 2021; Sheng et al., 2021; Miniutti et al., 2023b; Quintin et al., 2023).

  * •

eRO-QPE3 shows the longest recurrence time ($\approx 20$ h) and the lowest
peak luminosity ($L_{\rm 0.5-2.0\,keV}^{\rm peak}\sim\rm{few}\times
10^{41}\,$erg s-1) among known QPEs (Fig. 13, top), thus further strengthening
that there is no correlation between these two quantities. Such timescale-
luminosity correlations are instead predicted in many accretion-instability
models.

  * •

eRO-QPE4 shows evidence, for the first time, of a harder component appearing
close to the QPE peak (Fig. 7), although it is very faint and of an unclear
origin (Appendix A.2).

  * •

In eRO-QPE4 we see clearly, for the first time, that the quiescence emission
detected during the QPE epoch must have been either absent or much fainter
previous to the QPE behavior (Fig. 12), supporting a short-lived nature of the
radiatively efficient accretion flow seen in-between QPEs.

  * •

The long-term recurrence pattern of both eRO-QPE3 and eRO-QPE4 cannot be
accurately constrained based on current data, although in eRO-QPE4 (Fig. 9) it
appears somewhat intermediate between regular alternating and irregular,
perhaps bridging the gap between the apparent dichotomy that would have been
inferred from the only four sources available (e.g., Arcodia et al., 2022); a
continuum of variability patterns would be expected in the case of orbital
phenomena with orbital and/or disk precession in play.

The detection of eRO-QPE4 as a non-repeating bright and soft extragalactic
flare, compared to the repeated nature of eRO-QPE1, eRO-QPE2 (A21) and eRO-
QPE3 (this work), opens up a new way to select more candidates within the
available eROSITA surveys. Furthermore, the wealth of eROSITA data can still
be used for searches for QPEs performed in a more informed way, by targeting
galaxies strikingly similar to those of the known QPE sources (e.g., Wevers et
al., 2022). Looking at the future, wide-area X-ray telescopes that are
sensitive in the soft X-rays ($<2\,$keV) are needed to efficiently detect
QPEs. In the meantime, thorough archival searches are still very useful (Webbe
& Young, 2023) and have already proved successful in providing QPE candidates
(Chakraborty et al., 2021; Quintin et al., 2023). Finally, we note that the
homogeneous nature of our search with the eROSITA telescope allows us to
estimate the intrinsic abundance rate of QPEs, which will be presented in a
companion paper (Arcodia et al., 2023).

###### Acknowledgements.

We thank the teams of both _XMM-Newton_ and _NICER_ for their prompt efforts
in the scheduling of our observations. R.A. thanks G. Miniutti and M. Giustini
for the many lively discussions about QPEs. R.A. received support for this
work by NASA through the NASA Einstein Fellowship grant No HF2-51499 awarded
by the Space Telescope Science Institute, which is operated by the Association
of Universities for Research in Astronomy, Inc., for NASA, under contract
NAS5-26555. R.A. thanks T. Nowa and B. Banks and their sets for inspiring the
writing. GP acknowledges funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No 865637), support from Bando per il Finanziamento della
Ricerca Fondamentale 2022 dell’Istituto Nazionale di Astrofisica (INAF): GO
Large program and from the Framework per l’Attrazione e il Rafforzamento delle
Eccellenze (FARE) per la ricerca in Italia (R20L5S39T9). This work was
supported by the Australian government through the Australian Research
Council’s Discovery Projects funding scheme (DP200102471). This work is based
on data from eROSITA, the soft X-ray instrument aboard SRG, a joint Russian-
German science mission supported by the Russian Space Agency (Roskosmos), in
the interests of the Russian Academy of Sciences represented by its Space
Research Institute (IKI), and the Deutsches Zentrum für Luft- und Raumfahrt
(DLR). The SRG spacecraft was built by Lavochkin Association (NPOL) and its
subcontractors, and is operated by NPOL with support from the Max Planck
Institute for Extraterrestrial Physics (MPE). The development and construction
of the eROSITA X-ray instrument was led by MPE, with contributions from the
Dr. Karl Remeis Observatory Bamberg & ECAP (FAU Erlangen-Nuernberg), the
University of Hamburg Observatory, the Leibniz Institute for Astrophysics
Potsdam (AIP), and the Institute for Astronomy and Astrophysics of the
University of Tuebingen, with the support of DLR and the Max Planck Society.
The Argelander Institute for Astronomy of the University of Bonn and the
Ludwig Maximilians Universitaet Munich also participated in the science
preparation for eROSITA. The eROSITA data shown here were processed using the
eSASS software system developed by the German eROSITA consortium. This work
has made use of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This research has made use
of the VizieR catalogue access tool, CDS, Strasbourg, France (DOI :
10.26093/cds/vizier). The original description of the VizieR service was
published in 2000, A&AS 143, 23. Part of the funding for GROND (both hardware
as well as personnel) was generously granted from the Leibniz-Prize to Prof.
G. Hasinger (DFG grant HA 1850/28-1). MK acknowledges support from DLR grant
FKZ 50 OR 2307. We acknowledge the use of the matplotlib package (Hunter,
2007).

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##  Appendix A X-ray spectral analysis

###  A.1 eROSITA data of eRO-QPE3

Figure 14: Spectral fit for the eRASS1 (_top_) and eRASS2 (_bottom_) spectra
in quiescence modeled with a diskbb. Black points are source plus background
data, empty gray points show the background alone. The green line and related
light green (dark green) shaded regions are the source plus background model
median and 1st-99th (16th-84th) percentiles, respectively. The orange dashed
lines shows the background model alone. In the lower panel, the data-model
ratio is shown. The individual absorption-corrected source model component
(here a diskbb) is shown with a red line, as indicated by the legend. Figure
15: As in Fig. 14, but for the quiescence model (diskbb and zpowerlw) applied
to the eRASS1 spectrum of the full observations, without separating faint
eROdays from the bright (top-left panel of Fig. 1).

The eRASS1 spectrum in quiescence, if modeled with a simple accretion disk,
shows marginal residual at high-energies (top panel of Fig. 14) and a median
peak temperature of $kT_{disk}\sim 133\,$eV. This temperature would be
unusually high compared to other QPE-sources in quiescence, in which the
presence of a fainter harder component is often observed (; ; ; Chakraborty et
al. 2021). Therefore, we add a power-law component to account for the
residuals, mimicking thermal Comptonization of the accretion disk photons (top
panel of Fig. 16). Compared to a disk model alone, $\log Z$ improves by 2.4,
however we note that spectral simulations are needed to calibrate the
$\delta\log Z$ for this model comparison. This more complex model is mostly
chosen as best-fit model due to the comparison with other quiescence spectra
of QPE sources, obtained with higher signal-to-noise spectra (; ; ;
Chakraborty et al. 2021). With the addition of the power-law component at
higher energies, the median (and 16th, 84th percentiles) peak temperature of
the disk model becomes $kT_{\rm disk}=100_{-20}^{+18}\,$eV. The slope of the
power-law component $\Gamma_{X}$ is unconstrained, with a slight preference
toward softer values777A maximum value of $\Gamma_{X}=3.5$ is allowed for the
fit.. The median (and related 16th, 84th percentiles) flux of the disk
component is $F_{\rm 0.2-2.0\,keV}=1.4_{-0.6}^{+0.4}\times 10^{-12}\,$erg s-1
cm-2, while that of the power-law component is $F_{\rm
0.2-2.0\,keV}=2.7_{-1.7}^{+8.4}\times 10^{-13}\,$erg s-1 cm-2. The total flux
in the observed X-ray band is $F_{\rm 0.2-8.0\,keV}=1.9_{-0.6}^{+0.7}\times
10^{-12}\,$erg s-1 cm-2. This model is held fixed during the eruptions and a
black body component is added (bottom panel of Fig. 16). Since the eRASS1
light curve (top-left panel of Fig. 1) does not show variability of the same
amplitude compared to eRASS2-3-4, we explore the possibility that no QPEs were
present during eRASS1 and that the quiescence continuum was simply highly
variable. For instance, in GSN 069, the only other secure QPE source with a
clearly decaying quiescence emission over months-years as in eRO-QPE3, QPEs
were not detected when the quiescence emission was the brigthest (Miniutti et
al. 2023b). Therefore we also fit the quiescence model (diskbb and zpowerlw)
to the full eRASS1 observation, without separating faint eROdays from the
bright one (top-left panel of Fig. 1). In this work, we do not aim to reach a
conclusive statement on the presence of QPEs in eRASS1 or not. Model
comparison cannot be performed due to the use of different good-time intervals
(GTIs) for the spectra in the two scenarios. We show this fit in Fig. 15 and
we report the fit parameters in Table 2 together with other eRASS1 fit
results.

Table 2: Spectral fit results for eRO-QPE3.

Epoch | Spectrum | Model | $kT_{\rm disk}$ | $F^{\rm disk}_{\rm 0.2-2.0\,keV}$ | $F^{\rm pl}_{\rm 0.2-2.0\,keV}$ | $kT_{\rm QPE}$ | $F^{\rm QPE}_{\rm 0.2-2.0\,keV}$ | $\Delta\log Z$
---|---|---|---|---|---|---|---|---
|  |  | [eV] |  [erg s-1 cm-2] |  [erg s-1 cm-2] | [eV] |  [erg s-1 cm-2] |
eRASS1 | Quiescence | disk | $133_{-12}^{+13}$ | $(1.4\pm 0.2)\times 10^{-12}$ | – | – | – | 2
|  | disk+pl | $100_{-20}^{+18}$ | $1.4_{-0.6}^{+0.3}\times 10^{-12}$ | $2.9_{-1.9}^{+6.7}\times 10^{-13}$ | – | – | 0
| QPE | disk+pl+bb | $100$ | $1.4\times 10^{-12}$ | $2.9\times 10^{-13}$ | $122_{-15}^{+19}$ | $(3.0\pm 0.7)\times 10^{-12}$ | –
| Full | disk+pl | $110_{-27}^{+15}$ | $1.7_{-1.3}^{+0.4}\times 10^{-12}$ | $5.3_{-3.8}^{+14.3}\times 10^{-13}$ | – | – | –
eRASS2 | Quiescence | disk | $151_{-24}^{+29}$ | $3.9_{-0.8}^{+1.1}\times 10^{-13}$ | – | – | – | 1.5
|  |  disk+pl (free) | $80_{-59}^{+85}$ | $<4.2\times 10^{-13}$ | $4.2_{-4.0}^{+1.7}\times 10^{-13}$ | – | – | 0.5
|  | disk+pl | $89_{-60}^{+42}$ | $3.12_{-3.11}^{+1.51}\times 10^{-13}$ | $2.4_{-1.1}^{+0.9}\times 10^{-13}$ | – | – | 0
| QPE | disk+pl+bb | $89$ | $3.1\times 10^{-13}$ | $2.4\times 10^{-13}$ | $82_{-9}^{+10}$ | $3.2_{-0.7}^{+1.0}\times 10^{-12}$ | –
eRASS3 | Quiescence | disk | – | $<2.7\times 10^{-14}$ | – | – | – | –
| QPE | bb | – | – | – | $72_{-9}^{+12}$ | $3.2_{-0.7}^{+1.0}\times 10^{-12}$ | –
eRASS4 | Quiescence | disk | – | $<3.5\times 10^{-13}$ | – | – | – | –
| QPE | bb | – | – | – | $112_{-16}^{+18}$ | $1.4_{-0.3}^{+0.5}\times 10^{-12}$ | –
eRASS5 | Quiescence | disk | – | $<7.2\times 10^{-15}$ | – | – | – | –
| QPE | bb | – | – | – | $\approx 294_{-103}^{+240}$ | $3.2_{-1.6}^{+2.4}\times 10^{-13}$ | –
XMM1 burst1 | Quiescence | disk | – | $<1.2\times 10^{-14}$ | – | – | – | –
| QPE rise1 | bb | – | – | – | $108_{-13}^{+16}$ | $5.3_{-1.1}^{+1.3}\times 10^{-14}$ | –
| QPE rise2 | bb | – | – | – | $(121\pm 6)$ | $1.7_{-0.1}^{+0.2}\times 10^{-13}$ | –
| QPE peak | bb | – | – | – | $111_{-5}^{+6}$ | $3.1_{-0.2}^{+0.3}\times 10^{-13}$ | –
| QPE decay1 | bb | – | – | – | $97_{-8}^{+9}$ | $1.7_{-0.2}^{+0.3}\times 10^{-13}$ | –
| QPE decay2 | bb | – | – | – | $(86\pm 7)$ | $1.1_{-0.1}^{+0.3}\times 10^{-13}$ | –

  * •

Fit values show the median and related 16th-84th percentiles of the fit
posteriors. We mark in boldface the adopted best-fit model, if more than one
is reported, which has the highest Bayesian evidence. The difference between
the logarithmic Bayesian evidence values is shown in the last column for these
cases. For eRASS2, the disk+pl(free) model has no constraint on the power-law
normalization, whilst in the disk+pl model the power-law is tied to that of
the disk with the same ratio observed in the eRASS1 spectrum. During the QPE
fit, the quiescence model is held fixed, if detected. For eRASS5, the QPE fit
is compatible with background within $3\sigma$ and the fit parameters should
be interpreted with caution. For _XMM-Newton_ , the different phases are shown
in Fig. 10 and 22. Given the spectroscopic redshift and the cosmology adopted
(Hinshaw et al. 2013), the conversion for related luminosity values for eRO-
QPE3 is $1.34\times 10^{54}$ cm2 in this paper.

Figure 16: As in Fig. 14, but for the chosen best-fit models for both eRASS1
in quiescence (_top_ , diskbb and zpowerlw) and in eruption (_bottom_ , adding
a zbbody).

The eRASS2 spectrum in quiescence appears fainter than in eRASS1 (see Fig. 14
and Fig. 1) and can be adequately fit by either a diskbb or a power-law
continuum. Models provide equally good fits due to the low counts statistics
(32 counts within $0.2-8.0\,$keV versus, e.g., the 95 in eRASS1). Adopting the
disk model, based on results of other QPE sources, we obtain $kT_{\rm
disk}\sim 151\,$eV, which is higher than the disk temperature in eRASS1 (Table
2). We consider unlikely that the quiescence flux is lower and the temperature
higher (e.g., Miniutti et al. 2023a, b, for the long-term evolution in GSN
069). Therefore, we attribute this result on the eRASS2 quiescence spectrum to
the use of an incomplete source model. Based on eRASS1 and results of other
QPEs in quiescence, we adopt the diskbb plus zpowerlw model as reference,
despite not being statistically required compared to a single component. If
left free to vary, the power-law component dominates the fit and the disk
component is fit at background level (Fig. 17), with a highly-uncertain
temperature ($kT_{\rm disk}=80_{-59}^{+85}\,$eV). The median (and related
16th, 84th percentiles) flux of the power-law component is $F_{\rm
0.2-2.0\,keV}=4.2_{-4.0}^{+1.7}\times 10^{-13}\,$erg s-1 cm-2 while the disk
component is an upper limit at $<4.2\times 10^{-13}\,$erg s-1 cm-2, with a
median value of the flux posterior at $\sim 1\times 10^{-14}\,$erg s-1 cm-2.
Based on eRASS1 data of eRO-QPE3 and results obtained for other QPE sources
with higher signal-to-noise spectra (; ; ; Chakraborty et al. 2021), we assume
that this fit result is likely due to the greater flexibility of the power-law
model and model degeneracies. A more realistic result with a diskbb plus
zpowerlw would find the former brighter than the latter. We therefore impose
that the normalization of the power-law is tied to that of the disk with the
same ratio observed in the eRASS1 spectrum. The assumption is that both
components have faded. We note that, for the scope of this work, the main aim
is to obtain a reliable quiescence model to be held fixed in the QPE spectrum,
to isolate this component. We verified a posteriori that the use of either
free or tied power-law components does not affect the fit of the QPE spectrum
in the bright state. We therefore adopt this tied diskbb plus zpowerlw model
as reference for the quiescence (top panel of Fig. 18). The median (and
related 16th, 84th percentiles) temperature of the disk becomes $kT_{\rm
disk}=89_{-60}^{+42}\,$eV, thus slightly more constrained and still compatible
with the former value. Its flux becomes $F_{\rm
0.2-2.0\,keV}=3.12_{-3.11}^{+1.52}\times 10^{-13}\,$erg s-1 cm-2, therefore
still compatible with background. This confirms that we did not artificially
imposed the presence of a disk, but merely allowed the posterior of its flux
to be more reasonably brighter. The flux of the power-law component is $F_{\rm
0.2-2.0\,keV}=2.4_{-1.1}^{+0.9}\times 10^{-13}\,$erg s-1 cm-2. The total flux
in the observed X-ray band is $F_{\rm 0.2-8.0\,keV}=5.8_{-2.9}^{+1.7}\times
10^{-13}\,$erg s-1 cm-2. As done for eRASS1, this quiescence model is held
fixed during the brighter eROday (the orange point in the top-medium panel of
Fig. 1) and a black body component is added (see bottom panel of Fig. 18). The
median (and 16th, 84th percentiles) peak temperature of the QPE component is
$kT_{\rm QPE}=82_{-9}^{+10}\,$eV and its flux is $F_{\rm
0.2-2.0\,keV}=3.2_{-0.7}^{+1.0}\times 10^{-12}\,$erg s-1 cm-2. The median
temperature is colder compared to the bright eROday in eRASS1, despite the
flux being compatible. However, given the short $\sim 40$s exposure of an
eROday compared to the typical QPE duration ($0.5-7\,$h, A21), eRASS data
catch the eruption at different phases and they are lower limits of the QPE
peaks. The total flux during the bright eROday, quiescence included, in the
observed X-ray band is $F_{\rm 0.2-8.0\,keV}=3.9_{-0.8}^{+1.0}\times
10^{-12}\,$erg s-1 cm-2. Therefore, despite the decreasing quiescence level,
the flux in the bright eROday is compatible between eRASS1 and eRASS2.

Figure 17: As in Fig. 14, but for the quiescence spectrum of eRASS2, where
both disk and power-law components are free to vary.

Figure 18: As in Fig. 16, but showing the best-fit models for eRASS2, in
quiescence (_top_) and eruption (_bottom_). Figure 19: As in Fig. 14, but for
the QPE spectrum of eRASS3. Figure 20: As in Fig. 14, but for the QPE spectrum
of eRASS4. Figure 21: As in Fig. 14, but for the QPE spectrum of eRASS5. As
the fit model is compatible with background within the $3\sigma$ contours,
eRASS5 fit results are to be interpreted with caution.

###  A.2 _XMM-Newton_ spectral analysis

####  A.2.1 eRO-QPE3

Figure 22: _Top:_ a chunk of the eRO3-XMM1 observation (see Fig. 4). The best-
fitting model (Eq. LABEL:eq:1) is superimposed as a green line (median) and
related percentile contours (equivalent to $1\sigma$ and $3\sigma$). The
vertical shaded areas represent the phases of the burst (namely rise1, rise2,
peak, decay1, decay2, with the same color-coding as Fig. 10 for eRO-QPE3).
_Bottom:_ same as the top panel, but for the eRO4-XMM observation. The best-
fitting model is here shown in red, and the phases of the burst follow the
color-coding of eRO-QPE4 in Fig. 10.

The quiescent state of eRO-QPE3 was selected from GTIs excluding the bursts
from both eRO3-XMM1/2 observations: namely taking GTIs between times
$774645000-774651000$ s and $774675000-774724000$ s of eRO3-XMM1 and GTIs
$<776410000$ s and $>776440000\,$s of eRO3-XMM2. The resulting quiescence
spectrum is too faint to be securely detected above background (dark-gray
points in Fig. 5). We adopt a disk spectrum to provide a flux upper limit of
$F_{\rm 0.2-2.0\,keV}<1.1\times 10^{-14}$ erg s-1 cm-2. The fit posterior on
the flux, or disk normalization, is unconstrained (dark-gray line and contours
in Fig. 5), while the disk temperature is loosely constrained to $kT_{\rm
disk}=46_{-18}^{+21}$ eV. We conservatively consider the quiescence of eRO-
QPE3 to be undetected at the _XMM-Newton_ epoch, although the weak constraints
of temperature and luminosity ($L_{\rm disk,bol}<2.3\times 10^{41}$ erg s-1,
with a median value of the unconstrained posterior at $L_{\rm disk,bol}\sim
3.5\times 10^{40}$ erg s-1) are consistent with quiescent states of other QPEs
(; ; ). Therefore, we suggest that the quiescent state of eRO-QPE3 is
compatible with background (hence not formally detected), but not
significantly fainter.

The QPE flares are separated in two rise states, one peak and two decays. We
show an example of this in Fig. 22 for the first burst of the observation
eRO3-XMM1. The color-coding, darker to lighter going from rise1 to decay2,
follows that of Fig. 10. Separations are defined in terms of total count rate,
so that, for instance, the separation between rise1 and rise2 occurs roughly
at the same count rate as that between decay1 and decay2. Naturally, they are
somewhat arbitrary and, as much as temperature or flux values would slightly
change with different definitions, no major conclusion of this work would, not
the hysteresis shown in Fig. 10. For the example in Fig. 22, the separating
times are 774651000, 774654500, 774657500, 774661000, 774664000 and 774668500
s, respectively. Since the quiescent state is formally undetected, we fit all
phases of all bursts with a black body model, representative of the QPE. The
individual temperature and luminosities are shown in Fig. 10 and Table 2. For
the peak spectrum (orange in Fig. 5), we obtain $kT_{\rm QPE}=111_{-5}^{+6}$
eV and $F_{\rm 0.2-2.0\,keV}=3.1_{-0.2}^{+0.3}\times 10^{-13}$ erg s-1 cm-2
($L_{\rm 0.2-2.0\,keV}=(4.2\pm 0.4)\times 10^{41}$ erg s-1), thus $\gtrsim
25-30$ times brighter than the quiescence flux upper limit in the soft X-ray
band. The bolometric luminosity of the black body component is $L_{\rm
QPE,bol}=(4.9\pm 0.5)\times 10^{41}$ erg s-1. Based on the eRASS1 and eRASS2
results (and _XMM-Newton_ analysis of eRO-QPE4; see Sect. 2.4.2 and see
below), we test the addition a Comptonization component to the QPE phases.
However, during all phases the flux of this component is compatible with
background (thus, with zero flux) and the $logZ$ of the fit is orders of
magnitude worse. Therefore, this component is not statistically required given
the current data quality. We adopt the simple black body as reference model
for the eruptions of eRO-QPE3 observed by _XMM-Newton_.

####  A.2.2 eRO-QPE4

Table 3: Spectral fit results for eRO-QPE4.

Epoch | Spectrum | Model | $kT_{\rm disk}$ | $F^{\rm disk}_{\rm 0.2-2.0\,keV}$ | $F^{\rm disk,comp}_{\rm 0.2-2.0\,keV}$ | $kT_{\rm QPE}$ | $F^{\rm QPE}_{\rm 0.2-2.0\,keV}$ | $F^{\rm QPE,comp}_{\rm 0.2-2.0\,keV}$ | $\Delta\log Z$
---|---|---|---|---|---|---|---|---|---
|  |  | [eV] |  [erg s-1 cm-2] |  [erg s-1 cm-2] | [eV] |  [erg s-1 cm-2] |  [erg s-1 cm-2] |
eRASS:3 | Full | disk | – | $<3.8\times 10^{-15}$ | – | – | – | – | –
eRASS4 | Quiescence | disk | – | $<6.9\times 10^{-14}$ | – | – | – | – | –
| QPE | bb | – | – | – | $93_{-14}^{+18}$ | $3.4_{-1.1}^{+1.7}\times 10^{-12}$ | – | –
XMM burst2 | Quiescence | disk | $51_{-2}^{+3}$ | $(2.8\pm 0.2)\times 10^{-13}$ | – | – | – | – | 91
|  | disk+comp | $43\pm 2$ | $(3.9\pm 0.4)\times 10^{-13}$ | $1.4_{-0.4}^{+0.8}\times 10^{-14}$ | – | – | – | 0
| QPE rise1 | Q+bb | $\sim 43$ | $\sim 3.9\times 10^{-13}$ | $\sim 1.4\times 10^{-14}$ | $116_{-10}^{+14}$ | $1.3_{-0.2}^{+0.3}\times 10^{-13}$ | – | –
| QPE rise2 | Q+bb | $\sim 43$ | $\sim 3.9\times 10^{-13}$ | $\sim 1.4\times 10^{-14}$ | $123_{-3}^{+4}$ | $(1.5\pm 0.1)\times 10^{-12}$ | – | 0
|  | Q+bb+comp | $\sim 43$ | $\sim 3.9\times 10^{-13}$ | $\sim 1.4\times 10^{-14}$ | $120_{-6}^{+5}$ | $(1.5\pm 0.1)\times 10^{-12}$ | $<1.3\times 10^{-13}$ | 2
| QPE peak | Q+bb | $\sim 43$ | $\sim 3.9\times 10^{-13}$ | $\sim 1.4\times 10^{-14}$ | $125_{-1}^{+2}$ | $(2.9\pm 0.1)\times 10^{-12}$ | – | 0
|  | Q+bb+comp | $\sim 43$ | $\sim 3.9\times 10^{-13}$ | $\sim 1.4\times 10^{-14}$ | $123\pm 2$ | $(2.9\pm 0.1)\times 10^{-12}$ | $3.1_{-2.1}^{+4.6}\times 10^{-14}$ | 0.5
| QPE decay1 | Q+bb | $\sim 43$ | $\sim 3.9\times 10^{-13}$ | $\sim 1.4\times 10^{-14}$ | $89\pm 2$ | $(2.1\pm 0.1)\times 10^{-12}$ | – | –
| QPE decay2 | Q+bb | $\sim 43$ | $\sim 3.9\times 10^{-13}$ | $\sim 1.4\times 10^{-14}$ | $71\pm 5$ | $3.2_{-0.6}^{+0.7}\times 10^{-13}$ | – | –

  * •

See notes in Table 2. For _XMM-Newton_ , the different phases are shown in
Fig. 10 and the bottom panel of 22. Here, the quiescence spectrum (”Q”) is
held fixed during the QPE epochs by letting its parameters free to vary only
within the 10th-90th percentiles of the posteriors of the quiescence fit alone
(hence the ”$\sim$”). Given the spectroscopic redshift and the cosmology
adopted (Hinshaw et al. 2013), the conversion for related luminosity values
for eRO-QPE4 is $4.42\times 10^{54}$ cm2 in this paper.

The quiescent state of eRO-QPE4 was selected from GTIs excluding the bursts
from the eRO4-XMM observation: namely taking GTIs between times
$794814000-794837000$ s and $794850500-794889800$ s. The source is detected,
with a soft spectrum (Fig. 7) reminiscent of that of other QPE sources in
quiescence. We initially fit the quiescence with a disk model (Fig. 23) and
subsequently add a Comptonization component (Fig. 7). As it can be noted in
Fig. 23, a thermal disk model alone fails to account residual signal between
$\sim 0.6-1.5\,$keV. This signal is significantly above background (Fig. 7),
as it is comparable to it after background subtraction (Fig. 23). Adding a
Comptonization component vastly improves the $\log Z$ of the spectral fit by
92. Therefore, we adopt the disk plus Comptonization model as reference for
the quiescent state of eRO-QPE4, which yields a disk temperature of $kT_{\rm
disk}=(43\pm 2)$ eV with a flux $F_{\rm 0.2-2.0\,keV}=(3.9\pm 0.4)\times
10^{-13}$ erg s-1 cm-2. This corresponds to $L_{\rm
0.2-2.0\,keV}=1.7_{-0.1}^{+0.2}\times 10^{42}$ erg s-1 for the disk component
alone, or to a bolometric luminosity of $L_{disk,bol}=2.0_{-0.3}^{+0.6}\times
10^{43}$ erg s-1. The Comptonization component is much fainter, with $F_{\rm
0.2-2.0\,keV}=1.4_{-0.4}^{+0.8}\times 10^{-14}$ erg s-1 cm-2, and its slope is
unconstrained with preference for the softest values allowed (i.e.,
$\Gamma_{X}=3.5$). As discussed for eRASS data of eRO-QPE3, Comptonization is
merely one possible interpretation for the excess signal observed. In this
work, the scope of the quiescence fit is to obtain a good-enough model in
order to better isolate the QPE component. We report both models in Table 3.
We have also tested a Comptonization component alone to fit the observed X-ray
photons, with the assumption that a colder disk is present. Residuals are much
poorer and $\Delta\log Z\sim 156$ compared to the disk plus Comptonization
quiescence model, hence the latter is preferred.

Figure 23: Same as Fig. 7, but showing background-subtracted spectra. The
models shown are a simple disk spectrum for quiescence and a black body in
addition for the peak, both showing significant residuals at higher energies.
These residuals are comparable with background or even brighter. The
background spectrum (subtracted to yield the data shown) is represented in in
red for visualization.

The QPE flares of eRO-QPE4 were separated in five epochs, namely rise1, rise2,
peak, decay1 and decay2 (bottom panel of Fig. 22 for the second burst of the
observation eRO4-XMM). The color-coding, darker to lighter going from rise1 to
decay2, follows that of Fig. 10. For the example in Fig. 22, the separating
times are 794837000, 794839000, 794840500, 794843500, 794845500 and 794850500
s, respectively. The quiescence model was held fixed by imposing its
parameters to vary only within the 10th-90th percentile interval of the
posteriors obtained during the quiescence fit alone. Therefore, the bursts
parameters are marginalized over the uncertainties of the quiescence model.
The fit results for the different burst phases are reported in Table 3. We fit
each phase with a black body alone and also with a Comptonization component of
the QPE spectrum in addition. For all phases excluding the peak, the
additional Comptonization component is not required by the fit, as established
by comparing the $\log Z$ and visualizing the normalization or flux posteriors
of the component. Therefore, for these epochs (rise1, rise2, decay1 and
decay2) the best-fit model adopted for the burst is that of the quiescence
plus a black body component. For the peak spectrum the $\log Z$ of the more
complex (black body plus Comptonization) is compatible with that of the
simpler model (black body) within typical uncertainties of 0.3 on $\log Z$,
thus the former is not formally required. However, the flux and normalization
posteriors of the Comptonization component are constrained and the marginal
higher-energy residuals accounted for (comparing Fig. 7 and 23), all at the
expense of a marginal change in the black body properties: as it can be noted
in Table 3, both temperature and flux of the much brighter black body QPE
component are unaffected by the much fainter harder component. Therefore, we
adopt the more complex model as reference for the QPE peak of eRO-QPE4 at the
_XMM-Newton_ epoch (see Fig. 7). This model yields $kT_{\rm QPE}=(123\pm 2)$
eV and $F_{\rm 0.2-2.0\,keV}=(2.9\pm 0.1)\times 10^{-12}$ erg s-1 cm-2
($L_{\rm 0.2-2.0\,keV}=(1.27\pm 0.03)\times 10^{43}$ erg s-1), thus $\sim 7-8$
times brighter than the quiescence luminosity in the same soft X-ray band. The
bolometric luminosity is $L_{\rm QPE,bol}=(1.44\pm 0.04)\times 10^{43}$ erg
s-1. In this case, the additional Comptonizaton component requires further
discussion, since it is not ubiquitous to QPE sources, not throughout the
burst of eRO-QPE4 itself. Again, for the scope of this paper this is
irrelevant since the QPE black body properties do not change. Future work
should establish whether this signal is significant and physical, or whether,
for instance, it could be pile-up effect due to the bright and soft QPE
spectrum, since it is only observed at the peak.

Figure 24: Same as Fig. 10, but for the eruption of the eRO3-XMM2 observation
(_top_) and the third of the eRO4-XMM observation (_bottom_).