# QPE or QPO? – Quasiperiodic Activity in Low–Mass Galaxy Nuclei

Andrew King1,2,3
1 School of Physics & Astronomy, University of Leicester, Leicester LE1 7RH UK
2 Astronomical Institute Anton Pannekoek, University of Amsterdam, Science
Park 904, NL-1098 XH Amsterdam, The Netherlands
3 Leiden Observatory, Leiden University, Niels Bohrweg 2, NL-2333 CA Leiden,
The Netherlands

###### Abstract

Quasiperiodic eruptions (QPEs) from low–mass galaxy centres may result from
accretion from a white dwarf in a very eccentric orbit about the central
massive black hole. Evolution under gravitational radiation losses reduces the
separation and eccentricity. I note that below a critical eccentricity $e_{\rm
crit}\simeq 0.97$, the accretion disc’s viscous timescale at pericentre
passage is probably longer than the orbital period $P$, and periodic eruptive
behaviour is no longer possible. These QPE descendant systems (QPEDs) are then
likely to produce quasiperiodic oscillations (QPOs) rather than eruptions,
varying more smoothly over the orbital cycle, with duty cycles $\sim 1$. I
identify 2XMM J123103.2+110648 ($P\simeq 3.9$ hr) and (more tentatively) RE
J1034+396 ($P\simeq 1$ hr) as candidate systems of this type, and find
agreement with their deduced eccentricities $e<e_{\rm crit}$. The absence of
eruptions and the lower accretion luminosities resulting from the smaller
gravitational radiation losses may make QPED systems harder to discover.
Ultimately they must evolve to have viscous times much longer than the orbital
period, and either remain steady, or possibly have infrequent but large
outbursts. The latter systems would be massive analogues of the soft X–ray
transients produced by low stellar–mass X–ray binaries.

###### keywords:

galaxies: active: supermassive black holes: black hole physics: X–rays:
galaxies

11footnotetext: E-mail<EMAIL_ADDRESS>
##  1 Introduction

Accreting systems frequently produce quasiperiodic luminosity variations.
Recent observations show that several low–mass galaxy nuclei produce
quasiperiodic eruptions (QPEs) in ultrasoft X–rays. Here much of the total
accretion luminosity is emitted in short but very bright bursts, which recur
at slightly varying intervals. The number of identified QPE systems is likely
to grow rapidly, both from new surveys, and from searches through existing
databases.

A possible model for these QPE systems (King, 2020; 2022; 2023) invokes a
white dwarf (WD) in a very eccentric orbit around the central moderately
massive black hole (MBH). This binary system is assumed to originate in a
tidal capture (possibly of a stellar–mass binary containing the WD – see
Cufari et al., 2022) which narrowly avoids completely disrupting the WD.
Gravitational radiation (GR) losses shrink the MBH – WD binary on a timescale
of a few thousand years and drive mass transfer. The eruptions occur as the WD
fills its tidal lobe at pericentre on each orbit, with some variation,
possibly because of angular momentum redistribution within the disc through
interaction with resonant disc orbits (King, 2023).

The restriction to WD orbiters comes from observational selection: these stars
fill their lobes at separations small enough that GR drives the orbital
evolution and mass transfer at rates providing significant accretion
luminosity. The highly visible nature of the eruptions makes them easily
observable. Main sequence (MS) orbiters must presumably be much more common,
as in mature stellar populations the mass in MS stars is typically a factor
$\sim 100$ times that in WDs, but do not give bright eruptions because they
fill their tidal lobes in wider orbits and have consequently lower GR losses.
They nevertheless must collectively contribute to – and presumably dominate –
the total black hole mass gain in the long term. This possibility gives QPE
systems wider importance in perhaps revealing a new channel for central black
hole growth in galaxies.

This picture appears to be quite successful in describing the observed
properties of all known QPE systems (including the predicted CNO enhancement
in at least one case (Sheng, 2021)), and extending the sample to include other
objects not previously recognised as QPE systems, such as HLX–1, with a period
$\sim 1$ yr (see King, 2022). The current QPE sample now has periods ranging
from $\sim 1$ yr down to $\sim 2.4$ hr (King 2022; 2023). For given
observational input (period, black hole mass, and luminosity) the model
specifies the companion WD mass and its orbital eccentricity $e$ (see Table
1). In all cases $e$ is very close to unity, and there is a strong tendency
for the eccentricity to be less extreme for QPE systems with shorter orbital
periods. This is consistent with a population undergoing orbital evolution
through GR losses.

The main point of this paper is to extend this picture. I show that continued
GR losses are likely to evolve the binary into a state that ultimately
suppresses the eruptions and turns them into rather less noticeable
quasiperiodic oscillations (QPOs). I propose two known systems (one rather
tentatively) as candidates for these QPE ‘descendant’ systems.

##  2 Light Curves

The quasi–periodic eruptions themselves generally have fast rises and
quasi–exponential decays, and duty cycles $<1$. This is similar in form to
soft X–ray transient (SXT) outbursts from stellar–mass X–ray binary systems
with low–mass companions, although in SXTs the observed timescale is much
longer (weeks to months) than in QPEs. It is now well established that SXT
events result from thermal–viscous accretion disc instabilities in which
irradiation by the central X–ray source traps the disc in the high
accretion–rate state until a large fraction of the disc mass is accreted (King
& Ritter, 1998; see Lasota 2001 for a review). The similar light curves and
strong ultrasoft emission seen in QPEs suggests that a similar origin is
likely for them too. Provided that the system erupts on all or most orbital
cycles this means that the disc mass probably does not grow monotonically,
explaining why these systems do not seem eventually to have more prolonged
outbursts separated by much longer intervals.

This analogy with SXTs raises questions. SXTs are stellar–mass binary systems
in which the viscous timescale $t_{\rm visc}$ characterizing the outbursts is
far longer than the orbital period. But in QPE systems the reverse holds: the
eruptions occupy only a small fraction of the orbital period, which is itself
often short compared with those of the SXTs. Evidently this stark difference
must be related to the extreme eccentricities $e$ of the QPE sources (see
Table 1), which make their accretion discs fundamentally distinct from the
circular discs of SXT systems.

We will see that orbital evolution under GR is likely to lead to a reversal of
the inequality $t_{\rm visc}<P$ characterizing QPE systems. So to get a full
picture of non–disruptive tidal captures and their possible role in black hole
growth, we need to consider these systems, which I denote as QPE descendants
(QPEDs).

Since they have $t_{\rm visc}>P$, QPEDs cannot have eruptive light curves, as
the accretion events last longer than the times separating them. The
thermal–viscous disc instabilities underlying QPEs must in these systems
instead produce quasiperiodic (QPO) luminosity variations occupying most or
all of the orbital cycle (i.e. with duty cycle $\sim 1$). But the discussion
is inevitably complicated because there are a number of other plausible
mechanisms which may produce QPOs. In particular I note that there is at least
one observed case – GSN 069 (see Miniutti et al. (2023) and references
therein) which shows both QPEs and QPOs.

In the limit that $t_{\rm visc}>>P$ in a QPED system we would expect the
variations induced by the orbital motion to become negligible, so that the
sources might be observed as steady, or possibly with bright outbursts
separated by long intervals (i.e. much longer than any plausible orbital
period). In this last case they would be high–mass versions of SXTs.

In the rest of this paper I investigate the condition defining the border
between QPE and QPED systems, and identify two candidates (one rather
tentative) as possible examples of the latter.

##  3 The viscous timescale in QPE/QPED systems

The accretion discs in QPE sources are evidently able to cool rapidly compared
with dynamical timescales, and so are likely to be thin, i.e. have $H<<R$,
where $H$ is the disc scaleheight at disc radius $R$. In a circular thin
accretion disc the diffusion of surface density $\Sigma$ is well described in
term of the viscous timescale

$t_{\rm
visc}\simeq\frac{1}{\alpha}\left(\frac{R}{H}\right)^{2}\left(\frac{R^{3}}{GM}\right)^{1/2},$
(1)

where $M$ is the controlling central black hole mass, and $\alpha$ is the
standard Shakura–Sunyaev viscosity parameter (see e.g. Frank et al., 2002).

In judging the effects on the X–ray light curve in a highly eccentric QPE
accretion disc we are particularly interested in the viscous timescale at
radii close to the pericentre separation $p=a(1-e)$, where $a$ is the
semi–major axis of the QPE binary. Here the periodic passage of the WD
companion transfers mass to the accretion disc. In addition it probably also
triggers the infall of gas on to the MBH by destabilizing the disc through
extraction of angular momentum from resonant disc orbits (King, 2023)111In
contrast, the long–term formation history of the disc is controlled by the
longer viscous timescale at large disc radii, when the WD is far away from
pericentre – see King, 2022)..

The dynamical timescale at the disc edge closest to the pericentre passage of
the WD is given by setting $R=p=a(1-e)$ in the last factor of (1). Then the
viscous timescale there is

$t_{\rm visc,\
peri}\simeq\frac{1}{2\pi\alpha}\left(\frac{R}{H}\right)^{2}(1-e)^{3/2}P$ (2)

where $P$ is the orbital period, and I have used Kepler’s 2nd law to write
$a^{3}=GMP^{2}/4\pi^{2}$.

Miniutti et al (2023) evaluated this expression numerically for the case of
GSN 069, assuming $H/R=\alpha=0.1$, and found that $t_{\rm visc,\ peri}$
(there denoted as $t_{\rm visc}$) is equal to the orbital period $P$ for a
critical eccentricity

$e=0.97$ (3)

(see the discussion below their Fig. 19).

This is in fact the universal borderline between QPE and QPED systems, as

$t_{\rm visc,\ peri}\simeq P$ (4)

defines the critical maximum eccentricity $e_{\rm crit}$ for QPED systems,
which gives $e_{\rm crit}=0.97$ if we assume $H/R\sim\alpha\sim 0.1$ (cf King
et al., 2007) in (2).

Clearly this criterion is not precise, not least because of the uncertainties
in estimating $H/R$ and $\alpha$. There is the related question of using
$(R^{3}/GM)^{1/2}$ as the dynamical timescale in regions of the disc where
non–circular motions may occur. Since dissipation tends to circularize orbits,
we would expect this problem to be less serious in cases where dissipation is
strongest – for example if, as is likely, the disc plane set by the WD orbit
is significantly inclined to the black hole spin. Despite the uncertainties,
we will see that the rough criterion (3) appears to be satisfied by the two
candidate QPED systems.

An important possible consequence of the inequality $t_{\rm visc,\ peri}<P$
satisfied by QPE systems is that viscous evolution of parts of the disc away
from pericentre may produce an eccentric disc. This may make it possible (cf
Lu & Quataert, 2022) that gas streams from lobe–filling MS stars could
intersect the accretion disc near pericentre and give another way of
generating QPEs.

##  4 QPED systems

Two systems have appeared in the literature which are candidates for QPED
systems: 2XMM J123103.2+110648 (Lin et al., 2020; Webbe & Young, 2023) and RE
J1034+396 (Gierliński et al., 2008). 2XMM J123103.2+110648 has significant
repetitive variation throughout its observed cycle, and RE J1034+396 has a
near–sinusoidal light curve with a rather coherent period $P\simeq 1$ hr.

One can use the same methods to derive WD masses and orbital eccentricities
for these two systems as for the QPE systems (i.e. Chen et al., 2022; King
2022). Webbe & Young (2023) do this for 2XMM J123103.2+110648, but remark
explicitly that it is difficult to reconcile its observed duty cycle $\sim 1$
with the much shorter duty cycles of the QPE sources.

The interpretation of RE J1034+396 as a QPED system is significantly more
problematic than for 2XMM J123103.2+110648. Its fairly low–mass black hole
makes tidal interaction possible, and the derived parameters for an orbiting
WD mass donor are encouraging. But unlike any other QPE–related system it has
a well–developed broad–line region, which may indicate that its accretion disc
is more extended, perhaps allowing for direct impacts with the orbiting donor
star. In addition, and possibly even more significantly, the QPO in RE
J1034+396 dominates the hard power–law X–ray emission and is not detected in
the soft X–rays which presumably originate in the accretion disc. Further, the
rms variability is observed to drop (Middleton et al., 2009) as the soft
excess starts to dominate the emission, diluting the QPO variability.
Evidently more work is needed to see if these differences are compatible with
QPED status.

Table 1 below gives the observed and derived properties of these two candidate
QPED sources, together with those of the known QPE systems. The Table shows
that both candidate QPED systems formally satisfy the critical eccentricity
condition (3), and indeed RE J1034+396 has the smallest eccentricity of any of
the sources in Table 1.

But we also see that three QPE systems out of the seven known also lie above
the line in Table 1, whereas I argued above that systems with $t_{\rm visc,\
peri}>P$ (which is equivalent to being above the line) cannot have eruptions.

One likely possibility here is that this may simply reflect the difficulty in
estimating the fairly variable quantity $(L\Delta t)_{45}$ characterizing the
total emitted radiation energy per orbit. But other factors such as the black
hole spin orientation may affect the true physical criterion, as suggested in
Section 3 above.

The fact that GSN 069 shows both QPEs and QPOs suggests another possibility (I
am grateful to the anonymous referee for emphasizing this). It is unlikely
that the accretion disc is only present when the WD is close to pericentre,
and so it presumably evolves viscously during the remainder of the orbital
cycle. Since the outer parts of the disc probably lie in the WD’s orbital
plane this star may interact gravitationally with it.

One result is that the WD extracts angular momentum from the accretion disc,
particularly from resonant gas orbits within it, so limiting its outward
spread. This is probably required for mass transfer to be stable (King, 2023).
But in addition, this interaction may induce precession of the disc, as it is
closely analogous to the superhump interaction found in short–period
cataclysmic variables – see King (2023) for a discussion. Superhumps
themselves are a photometric modulation with a period slightly longer than the
(circular) orbit, resulting from tidal flexing of the now precessing disc.

It seems possible that for example the presence of a QPO in the QPE system GSN
069 results from phenomena analogous to superhumps, but now involving a very
eccentric binary orbit. One needs to show that it would appear in X–rays,
rather than in the optical emission that characterizes superhumps.

As mentioned in the Introduction, a further possible complication is that in
some cases there may be accretion from one or more orbiting main–sequence
stars in addition to the WD. Another potential source of QPO variability comes
from the very likely possibility that the WD orbit – and therefore the
accretion disc – is inclined wrt the MBH spin. Then the disc may tear (cf
Nixon et al., 2012) into rings which then precess at different frequencies.
Raj & Nixon (2021) investigate this idea. Finally, we should note that the
viscous spreading of the disc at locations far from pericentre may allow
physical impact of the orbiter with the disc – Xian et al. (2021) have
suggested this as a possible cause of QPEs.

The discussion here and in the previous Section illustrates that both QPEs and
QPOs may potentially arise in a number of different ways, particularly
depending on the competition between viscous spreading of the disc and the
tendency of resonant gravitational interactions with the orbiting WD to
inhibit it. Investigating this in general requires a full fluid–dynamics
treatment of the disc. Despite this, one reasonably secure inference is that
the impact of the mass transfer stream on the accretion disc at pericentre
passage does not produce a luminosity which is competitive withe the total
accretion luminosity. These are respectively given by $\sim GM\dot{M}/a(1-e)$
and $\sim GM\dot{M}/R_{g}$, and physical consistency requires that the disc
radius at pericentre is much larger than the gravitational radius $R_{g}$. In
fact we see from Table 1 that for all except possibly one of the QPE/QPED
systems we have $R_{g}\ll a(1-e)$.

##  5 Discussion

We have seen that as GR evolution decreases both $e$ and $P$, QPE systems are
likely to evolve into QPED systems, where the eccentricity is below a critical
value $e_{\rm crit}\simeq 0.97$ and the accretion disc’s viscous timescale is
longer than the orbital period. Then the accretion disc instabilities which
made QPEs probably cause quasiperiodic oscillations (QPOs) instead. In
agreement with this, the two objects identified here as candidate QPED systems
both have $e<e_{\rm crit}$ and fairly short orbital periods.

The data of Table 1 do not yet give a clear test of these ideas. Although
there is a general tendency for $e$ and $P$ to decrease together, the source
properties are defined by three input parameters subject to random and
possibly systematic errors, so the sample is currently too small to test this
formally. These uncertainties alone may account for the fact that three of the
seven QPE sources appear to have eccentricities formally slightly below
$e_{\rm crit}$. A larger observational sample would allow one to test for the
presence of angular momentum transfer between the orbiting WD and the
accretion disc gas, and some of the other physical effects mentioned at the
end of the previous Section.

QPED sources may become almost steady once the viscous timescale is much
longer than the orbital period, provided that the disc can smoothly transport
all the mass gained from the WD to the MBH.

If this last condition fails, these systems may have large outbursts recurring
at long intervals. A requirement for outbursts is that the effective
temperature $T_{\rm irr}$ at the edge of the irradiated accretion disc should
in a steady state be less than the hydrogen ionization temperature
$T_{H}\simeq 6500$ K (cf King, Kolb & Burderi (1996), who investigate this for
the circular discs of stellar–mass SXTs). We use equation (15) of King (2022)
to give the accretion rate, and find $T_{\rm irr}$ from

$T_{\rm irr}^{4}\simeq\frac{\eta c^{2}(1-\beta)(-\dot{M}_{2})}{4\pi\sigma
R^{2}}\left(\frac{{\rm d}H}{{\rm d}R}-\frac{H}{R}\right).$ (5)

Here $\eta=0.1$ is the accretion efficiency, $\beta$ the X–ray albedo,
$\sigma$ the Stefan–Boltzmann constant, and $R=a(1-e)$ is the (virtually
constant) disc edge size opposite the WD at pericentre, and the term in large
brackets is $\sim H/R\sim 0.1$, (cf eqn (3) of King, Kolb & Burderi, 1996).
Since $R=a(1-e)$ is virtually constant until $e$ is $\ll 1$ (the GR losses are
effectively a point interaction) we find that $T_{\rm irr}\gtrsim T_{H}$ for
all likely QPED systems. It appears then that SXT–analogue outbursts of QPED
systems are probably rare, but remain a possibility to bear in mind.

Aside from these possibilities, QPE systems seem likely to become less
observable as they evolve into QPED objects. This is not a serious restriction
on observable lifetimes, as the eccentricity decreases on the GR timescale
$-M_{2}/\dot{M}_{2}$, which is of order a few thousand years for all QPE/QPED
systems (cf King, 2022).

## DATA AVAILABILITY

No new data were generated or analysed in support of this research.

## ACKNOWLEDGMENTS

It is a pleasure to thank the anonymous referee for very thoughtful and
stimulating comments, which led to significant improvements in the paper.

Table 1: Parameters of the Current QPE/QPED Sample

Source | $P_{4}$ | $m_{5}$ | $(L\Delta t)_{45}$ | $m_{2}$ | $1-e$ | duty cycle | light curve
---|---|---|---|---|---|---|---
RE J1034+396a[?] | 0.37 | 25 | 7.5 | 0.19 | $0.139$ | 1 | O
eRO–QPE2b | 0.86 | 2.5 | 0.8 | 0.16 | $9.0\times 10^{-2}$ | 0.2 | E
XMMSL1 J024916.6–04124c | 0.90 | 0.85 | 0.34 | 0.15 | $8.9\times 10^{-2}$ | 0.2 | E
RXJ1301.9+2747d | 1.65 | 18 | 1.7 | 0.14 | $6.5\times 10^{-2}$ | 0.15 | E
2XMM J123103.2+110648e | 1.37 | 1 | 30 | 0.51 | $3.0\times 10^{-2}$ | 1 | O
GSN 069f | 3.16 | 4.0 | 10 | 0.29 | $2.5\times 10^{-2}$ | 0.06 | E + O
eRO–QPE1g | 6.66 | 9.1 | 300 | 0.62 | $9.0\times 10^{-3}$ | 0.4 | E
ASASSN–14koh | 937 | 700 | $3388$ | 0.56 | $3.7\times 10^{-4}$ | 0.1 | E
HLX–1i | 2000 | [10] | 1000 | 0.81 | $1.5\times 10^{-4}$ | 0.3 | E
. . . |  . . . | [0.5] |  . . . | 1.43 | $1.2\times 10^{-4}$ | 0.3 | E

  * This table is adapted from Table 1 of King (2023), but now ordered by eccentricity $e$, which decreases upwards (in the direction of GR evolution, which also shortens the orbital periods). The systems are characterized by their periods $P_{4}=P/10^{4}~{}{\rm s}$, black hole masses $m_{5}=M/10^{5}\rm M_{\odot}$, and the total energy $(L\Delta t)_{45}=L\Delta t/10^{45}\,{\rm erg}$ emitted per orbit. Assuming these quantities are related by the orbital evolution given by GR emission and the resulting mass accretion, the method of Chen et al. (2022) then gives the quantities $m_{2}$ (the WD mass in $\rm M_{\odot}$) and $1-e$, with $e$ the eccentricity of the WD orbit. The values of some uncertain parameters have been adjusted slightly, with consequent minor changes in the results for $m_{2},1-e$, compared with Table 1 of King (2023). The last column on the right specifies the type of light curve, as E or O for QPE/QPO. The uncertain status of the QPO system RE J13034+396 as a candidate QPED system is indicated by the [?] sign. The horizontal line dividing the Table is the rough maximum eccentricity criterion (3): the two candidate QPED systems defined by duty cycles $\sim 1$ lie above this line, with $e<e_{\rm crit}$, in agreement. The position of three QPE systems above the line may reflect uncertainties in the input parameters $P_{4},m_{5},(L\Delta t)_{45}$, or possibly further relevant physics, such as disc breaking, excitation of resonances within the disc by the WD, or possibly the effects of other MS companion stars.

All systems satisfy the criterion $m_{5}(L\Delta t)_{45}^{1/3}\lesssim 10^{4}$
(King, 2023) required to avoid the model formally predicting that the WD
pericentre distance $a(1-e)$ is smaller than the innermost stable orbital
radius, which is itself $\simeq R_{g}=GM/c^{2}$, i.e. the black hole’s
gravitational radius, although ASASSN–14ko is very close to this limit. King
(2023) shows this explicitly for all the sources in the present Table other
than the two ‘O’ systems, and it is simple to verify for these directly.
References to observations: $a=$ Gierliński et al. (2008), $b=$ Arcodia et al.
(2021), $c=$ Chakraborty et al. (2021), $d=$ Giustini et al. (2020), $e=$ Lin
et al. (2020) and Webbe & Young (2023), $f=$ Miniutti et al (2019), $g=$
Arcodia et al. (2021), $h=$ Payne et al. (2021), $i=$ Farrell et al. (2009).
Note that there is no secure mass for the black hole in HLX–1, and $m_{5}=0.5$
is the minimum allowing a WD mass respecting the Chandraskhar limit. Assuming
a larger $m_{5}$ gives smaller $m_{2}$ and $e$.

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