# The evolution of circumstellar discs in the Galactic Centre: an application
to the G-clouds

James E. Owen1 and Douglas N. C. Lin 2,3
1Astrophysics Group, Imperial College London, Blackett Laboratory, Prince
Consort Road, London SW7 2AZ, UK
2 Department of Astronomy and Astrophysics, University of California, Santa
Cruz, CA 95064, USA
3Institute for Advanced Studies, Tsinghua University, Beijing, 100086, China
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The Galactic Centre is known to have undergone a recent star formation episode
a few Myrs ago, which likely produced many T Tauri stars hosting circumstellar
discs. It has been suggested that these discs may be the compact and dusty
ionized sources identified as “G-clouds”. Given the Galactic Centre’s hostile
environment, we study the possible evolutionary pathways these discs
experience. We compute new external photoevaporation models applicable to
discs in the Galactic Centre that account for the sub-sonic launching of the
wind and absorption of UV photons by dust. Using evolutionary disc
calculations, we find that photoevaporation’s rapid truncation of the disc
causes them to accrete onto the central star rapidly. Ultimately, an accreting
circumstellar disc has a lifetime $\lesssim 1~{}$Myr, which would fail to live
long enough to explain the G-clouds. However, we identify a new evolutionary
pathway for circumstellar discs in the Galactic Centre. Removal of disc
material by photoevaporation prevents the young star from spinning down due to
magnetic braking, ultimately causing the rapidly spinning young star to torque
the disc into a “decretion disc” state which prevents accretion. At the same
time, any planetary companion in the disc will trap dust outside its orbit,
shutting down photoevaporation. The disc can survive for up to $\sim$10 Myr in
this state. Encounters with other stars are likely to remove the planet on Myr
timescales, causing photoevaporation to restart, giving rise to a G-cloud
signature. A giant planet fraction of $\sim 10\%$ can explain the number of
observed G-clouds.

###### keywords:

Galaxy: centre – stars: pre-main-sequence – protoplanetary discs – accretion,
accretion discs – black hole physics

††pubyear: 2015††pagerange: The evolution of circumstellar discs in the
Galactic Centre: an application to the G-clouds–B

##  1 Introduction

The centre of our galaxy, the Milky Way, contains a supermassive black hole
(Sgr A*). Through the analysis of the orbits of stars, either statistically or
individually, in the vicinity of Sgr A*, its mass has been determined to be
$\sim 4\times 10^{6}$ M⊙ (e.g. Eckart & Genzel, 1997; Ghez et al., 1998; Ghez
et al., 2003; Gillessen et al., 2009; Genzel et al., 2010; Gravity
Collaboration et al., 2018). As such, the Galactic Centre provides a unique
window into astrophysical processes that are prevalent in the majority of
galaxies. The Galactic Centre contains a nuclear star cluster (e.g. Genzel &
Townes, 1987; Rieke & Rieke, 1988; Ghez et al., 2005a; Genzel et al., 2010),
which contains a large number of massive and hot stars indicative of recent
star formation (e.g. Krabbe et al., 1995; Paumard et al., 2001; Ghez et al.,
2003; Eisenhauer et al., 2005; Paumard et al., 2006; Bartko et al., 2009; Do
et al., 2013). Follow-up analysis suggests that the most recent star formation
event occurred 3-5 Myr ago, giving birth to a large number of massive stars
with a top-heavy IMF (e.g. Paumard et al., 2006; Bartko et al., 2010; Lu et
al., 2013).

Detailed analysis of the Galactic Centre with near diffraction-limited
observations (e.g. Ghez et al., 2005b; Gillessen et al., 2009) has produced
some unexpected results. Gillessen et al. (2012) reported the discovery of a
compact, dusty ionized cloud, G2. Multi-epoch observations allowed its orbit
around Sgr A* to be determined, showing this object was on an extremely
eccentric orbit (e.g. Gillessen et al., 2012, 2013; Pfuhl et al., 2015; Plewa
et al., 2017) and connections were made to an earlier detected cloud, G1,
(e.g. Clénet et al., 2005). Recently, additional objects have been discovered
(Eckart et al., 2013; Ciurlo et al., 2020; Peißker et al., 2020) taking the
number of known “G-clouds111Throughout the literature, these objects have been
given different nomenclature (G-clouds, DSOs, D-clouds); in this work, we
simply refer to them as G-clouds.” to around 5-10. The more recently detected
objects do not possess the same extreme eccentricity as G1/G2 and do not
appear to be located in a common orbital plane (Ciurlo et al., 2020). The
objects show similar observational characteristics: NIR emission with a black-
body temperature of $\sim 600$ K and luminosities less than a few times Solar,
indicative of dust, and Br-$\gamma$/HeI emission indicating the presence of
ionized gas with a temperature of $\sim 10^{4}$ K, suggesting this gas is
photoionized by the EUV background in the Galactic Centre (e.g. Gillessen et
al., 2012). This recombination emission implies densities of $\sim
10^{5}-10^{6}$ cm-3 on size scales of 10-100 AU (e.g. Gillessen et al., 2012).

Despite a decade of study, the origin and astrophysical nature of the G-clouds
remain a mystery. Burkert et al. (2012) discusses two possible classes: a
diffuse cloud scenario, where the G-clouds are pressure-confined clouds, or a
compact source scenario, where the G-clouds represent the diffuse gas and dust
surrounding an undetected central mass. Simulations of the pressure-confined
diffuse gas cloud scenario showed agreement with the position-velocity
properties of the Br-$\gamma$ emission (e.g. Burkert et al., 2012; Schartmann
et al., 2012, 2015). However, these simulations demonstrated that, due to the
strong tidal forces from Sgr A*, these diffuse gas clouds could not survive
pericentre passage if they were on an extremely eccentric orbit like G2.
During its pericentre passage in 2014, observations of G2 (Plewa et al., 2017)
showed that it did survive pericentre passage, although becoming tidally
distorted on its closet approach. The fact that G2 (and G1) did survive
pericentre passage strengthens the case for the compact-source scenario.

Several different versions of the compact source scenario have been presented:
photoevaporating circumstellar discs, either around T Tauri stars (Murray-Clay
& Loeb, 2012), or created through tidal interactions (Miralda-Escudé, 2012); a
nova (Meyer & Meyer-Hofmeister, 2012); winds launched from young T Tauri stars
(e.g. Scoville & Burkert, 2013; Ballone et al., 2013, 2018; De Colle et al.,
2014; Valencia-S. et al., 2015; Zajaček et al., 2017) or merger products (e.g.
Witzel et al., 2014; Prodan et al., 2015).

There have been claims of a central source detected in the K-band for G2 (e.g.
Peißker et al., 2020, 2021) indicative of a young T Tauri star; however, these
claims have been disputed, with other authors finding a non-detection (e.g.
Gillessen et al., 2012; Pfuhl et al., 2015; Plewa et al., 2017). Nevertheless,
the constraints on the central luminosity of any T Tauri star are weak, with
luminosities $\lesssim 5$ L⊙ consistent with the observations (e.g. Gillessen
et al., 2012). A photoevaporating circumstellar disc is appealing as the
origin of the G-clouds. The Galactic Centre is known to be a site of star
formation (e.g. Genzel et al., 2010), which naturally includes the emergence
of T Tauri stars. The recent star formation episode could have produced, at
maximum, a few thousand T Tauri stars (e.g. Nayakshin & Sunyaev, 2005; Lu et
al., 2013), which would initially host circumstellar discs. The background UV
environment of the Galactic Centre produces photoevaporation rates and
densities consistent with the observational constraints. This agreement led
Murray-Clay & Loeb (2012) to propose that G2 is a protoplanetary disc of size
$\sim 10$AU that is being photoevaporated and tidally stripped as it
approached Sgr A*. With an alternative scenario, Miralda-Escudé (2012)
proposed that G2 was a photoevaporating circumstellar disc created by a close
encounter between a black hole and a star. While these early works were
suggestive, they did not precisely address the evolution of such
photoevaporating discs, nor did they use detailed photoevaporation models. In
particular, previous works have ignored the ongoing accretion process, which
channels gas from the discs to their central hosts. Typically, accretion plays
an important (if not the dominant) role in the removal of disc material, not
photoevaporation (e.g. Clarke, 2007). Therefore, simply comparing
photoevaporation rates to disc masses (as done in previous work) is likely to
overestimate their lifetimes. Additionally, photoevaporation causes disc radii
to shrink to the point where the photoionized gas is formally bound to the
host star. It is well-known photoevaporation can still occur, driven by sub-
sonic pressure gradients in the bound gas (e.g. Adams et al., 2004). However,
no photoevaporation model applicable in this bound limit has been developed
for the high-EUV fluxes prevalent in the Galactic Centre. Thus, in this work,
we study the properties and evolution of circumstellar discs in the Galactic
Centre by (i), developing a photoevaporation model applicable to circumstellar
discs in the Galactic Centre and (ii), considering the secular evolution of
circumstellar discs through numerical computation. In doing this work, we also
use our results to critically assess the photoevaporating disc scenario for
the G-clouds.

##  2 The radiation environment

The photoevaporation of objects is controlled by the amount of UV radiation
they receive. In the Galactic Centre, as we shall demonstrate explicitly
later, the UV radiation environment is so strong that ionizing EUV radiation
plays the controlling role. For example, the EUV luminosity emanating from the
central parsec is $\sim 10^{51}$ ph s-1 (e.g. Martins et al., 2007); however,
this arises from the combination of numerous sources and will vary with
position. Therefore, in order to calculate the evolution of a photoevaporating
object, we must estimate the EUV flux as a function of position in the
galactic centre.

There are two populations of young stars that are sufficiently massive to emit
large amounts of ionizing radiation. First, the S-star cluster, which is
dominated by main sequence B-stars (e.g. Schödel et al., 2003; Ghez et al.,
2003) and is approximately spherically symmetric (e.g. Genzel et al., 2010).
The second population is the O/WR stars, this stellar cluster is located in a
ring (e.g. Levin & Beloborodov, 2003) with an inner edge of $\sim 0.04~{}$pc
and an outer edge of $\sim 0.5$ pc (e.g. Paumard et al., 2006; Lu et al.,
2009; Bartko et al., 2009). It is currently debated whether there are one or
multiple rings, or the rings are warped (e.g. Genzel et al., 2003; Paumard et
al., 2006; Lu et al., 2009; Cuadra et al., 2008; Bartko et al., 2010; Naoz et
al., 2018); however, in our calculation, we simply assume a single ring and
calculate EUV fluxes within and perpendicular to the disc’s plane. This
approach then covers the range of expected EUV fluxes for multiple rings.

Since the number of ionizing sources is not very large ($N_{*}$ is a few
hundred), the EUV fluxes as a function of distance from Sgr A* can vary due to
Poisson noise in the exact positions of all the individual stars.
blackAdditionally, the use of smooth density distributions is sensitive to any
chosen smoothing length due to the small number of stars. Thus, we choose to
estimate the EUV fluxes through Monte Carlo modelling of the distribution of
the star’s positions, masses and ionizing luminosities. Each Monte Carlo
realisation consists of multiple steps. Firstly, based on observed stellar
surface density distribution of $\Sigma_{S}\propto R^{-1.5}$ and
$\Sigma_{O}\propto R^{-1}$ for the S-star cluster and O/WR disc respectively
(e.g. Bartko et al., 2010; Genzel et al., 2010; Do et al., 2013) we randomly
draw positions for individual stars. We assume the S-star cluster is
spherically symmetric and consider stars out to distances from Sgr A* of 1 pc.
For the O/WR disc, we assume it extends from 0.04 to 0.5 pc and has a constant
thickness of $H/R=0.2$222Provided $H/R\ll 1$ we find its choice does not
affect our results.. For the S-stars, we consider 110 stars (the value
obtained by integrating the observed surface density profile out to 1 pc, and
comparable to the 112 stars identified by Ali et al. 2020), and for the O/WR
disc, we use the number required black(typically $\sim$250 stars) to give an
initial stellar mass cluster of $10^{4}$ M⊙ (Lu et al., 2013),.

The next step is to find the star’s initial mass. For this, we draw from the
measured IMFs for the two clusters. For the S-stars, we use a Salpeter IMF
(e.g. Bartko et al., 2010). The O/WR disc is thought to have a top-heavy mass
function (e.g. Nayakshin & Sunyaev, 2005; Paumard et al., 2006), thus we use a
top-heavy IMF of ${\rm d}N_{*}/{\rm d}M_{*}\propto M_{*}^{-1.7}$ (with $M_{*}$
the stellar mass, e.g. Lu et al., 2013). With our random sampling, we exclude
stars less massive than 8 M⊙ as they are not observable (nor do they
contribute to the EUV flux), and we exclude stars more massive than 150 M⊙ as
they were not included in the young star properties estimated by Lu et al.
(2013). Based on the initial masses, we use a $\sim 3$ Myr isochrone to
extract the current stellar properties (approximately the age of the young
stars from Lu et al. 2013). This 3 Myr isochrone is obtained from the mist
stellar tracks (Dotter, 2016; Choi et al., 2016), created using mesa (Paxton
et al., 2011; Paxton et al., 2013, 2015) for solar metallicity stars333The Lu
et al. (2013) characterisation of the stellar population, which we follow,
used solar-metallicity tracks. with initial rotation rates of $v/v_{\rm
crit}=0.4$ (the default value). We then use the stellar properties to estimate
the star’s EUV Luminosity using the stellar atmospheres of Thompson (1984),
which were tabulated and displayed graphically in Figure 4 of Kuiper et al.
(2020), accounting for absorption of EUV photons in the stellar atmosphere.

Finally, we include attenuation of the EUV photons between stars due to dust
in the Galactic Centre. Throughout this work, we adopt an EUV dust
opacity444We specifically mean per unit mass of dust, rather than the standard
quoted value of per mass of a gas and dust mixture with a specified dust-to-
gas ratio. This is because our dust-to-gas mass ratio will vary later. of
$\kappa_{\rm d,EUV}=10^{5}~{}$cm2 g-1, appropriate for ionizing photons being
absorbed by small sub-micron sized silicate dust particles (Ali, 2021)555Note
Ali (2021) present their dust opacity in terms of the dust and gas mixture,
see footnote 4.. Since the nature of the dust-and-gas distribution in the
galactic centre remains uncertain we adopt a dust-to-gas mass ratio of 0.01
and consider a gas distribution of $n(r)=n_{0}(r/r_{0})^{-\gamma}$ where
$n_{0}=10^{4}$ cm-3 and $r_{0}=10^{15}$ cm. We take either $\gamma=-1$ as
proposed by Xu et al. (2006) or adopt a flatter distribution $\gamma=-0.5$ as
proposed by Gillessen et al. (2019).

Based on this dust distribution and the positions and luminosities of all the
stars for a given realisation, we then use ray tracing to compute the EUV flux
impinging on a specific point at a known distance from Sgr A*. This point is
taken to either lie in the plane of the O/WR stellar disc or perpendicular to
it. We repeat this analysis 500 times to extract the expected range of EUV
fluxes as a function of distance from Sgr A*. The result of this exercise is
shown in Figure 1.

Figure 1: The lines show the median EUV flux as a function of distance from
the Galactic Centre. The lines for the perpendicular models are for the EUV
flux perpendicular to the plane of the O/B young discs and the parallel models
are for the EUV flux in the plane of the disc. The different $\gamma$’s show
the effect of attenuation due to dust in the Galactic centre for two profiles
with $n\propto r^{-\gamma}$. The shaded region around the lines represents the
5–95% range from the realisations. The region of the young disc is shown
shaded in grey.

This analysis shows that the massive stars in the O/WR young disc dominate the
EUV flux throughout the Galactic Centre. This results in an essentially flat
EUV flux distribution inside the inner edge of the disc. These fluxes can be
compared to the extreme end of the EUV fluxes experienced by the “proplyds”
(known photoevaporating protoplanetary discs) in the Orion Nebula Cluster of
$10^{13}$ to a few $10^{14}$ photons s-1 cm-2 (Bally et al., 1998; Johnstone
et al., 1998). Our model produces an EUV luminosity emanating from the inner
parsec ranging from $5\times 10^{50}-3\times 10^{51}$ photons s-1 in agreement
with the observational estimate of $\sim 10^{51}$ photons s-1 (e.g. Martins et
al., 2007; Genzel et al., 2010). Our result shows that dust attenuation plays
a small role, only reducing the fluxes by a factor of two in the inner
regions. Finally, we do note that our Monte-Carlo sampling does indicate cases
where an individual star is nearby and completely dominates the EUV flux,
resulting in fluxes that can exceed $10^{17}$ photons s-1 cm-2. If this were
to happen, it was likely to be only short-lived. In the inner 0.1 pc, the
average distance between stars is $\sim 10^{16}$ cm. Thus, the travel time to
move out of the range of a single star and closer to the average is $t_{\rm
travel}\sim 10^{16}{\rm~{}cm}/(4\times 10^{7}~{}{\rm cm~{}s^{-1}})\sim 10$
years, short compared to the lifetime of any star/object in the galactic
centre. In general, we expect the EUV flux received at a given point in the
Galactic Centre to be dominated by the combination of a large fraction of the
massive stars in the O/WR disc. We will take our nominal flux to be $3\times
10^{15}$ photons s-1 cm-2 and consider a low value of $10^{14}$ photons s-1
cm-2.

##  3 The anatomy of the evaporating disc

Our goal is to explore the photoevaporating disc scenario as the origin of the
G-clouds in detail. To achieve this goal, we need to know the photoevaporative
mass-loss rates as a function of star and disc parameters, as well as how they
vary with the UV environment. However, the EUV fluxes we have derived for the
Galactic Centre in the previous section exceed those of typical star-forming
regions. As discussed in the Introduction, such modern external
photoevaporation calculations are not applicable to this extreme environment.
Thus, we calculate a new family of external photoevaporation models in this
section. blackAdditionally, to launch a photoevaporative outflow, the external
pressure needs to be sufficiently low to allow the wind to be accelerated to
super-sonic speeds. While this is generally true in the Galactic Centre, as we
discuss in Section 9 it is not true near Sgr A* (and hence for G2 near
pericentre).

We calculate the photoevaporation models using a modified version of the
Johnstone et al. (1998) framework. While there has been significant recent
work advancing external photoevaporation calculations (e.g. Adams et al.,
2004; Facchini et al., 2016; Haworth et al., 2018b; Haworth & Clarke, 2019;
Winter & Haworth, 2022) the focus has been to improve the calculations at
lower UV irradiation levels, more appropriate for typical galactic star-
forming environments (e.g. Fatuzzo & Adams, 2008; Adams, 2010). In these
cases, the EUV irradiation is unable to penetrate deep into the FUV heated
region and does not affect the dynamics (e.g. Adams et al., 2004; Haworth et
al., 2018b). blackHowever, at high EUV fluxes, the ionization front is able to
penetrate below where the sonic point in a pure FUV heated flow would be.
Thus, the flow structure is sub-sonic in the FUV heated region, and the EUV
flux controls the outflow properties (e.g. Johnstone et al., 1998). blackIn
the Appendix, we show that the EUV fluxes in the Galactic Centre are
significantly above the critical flux required for the outflow to be in the
EUV-controlled limit.

The original Johnstone et al. (1998) formalism did not include thermal
launching of the wind inside the “critical-radius”, $R_{g}\equiv
GM_{*}/c_{s}^{2}$, where $c_{s}$ is the isothermal sound speed (e.g. Adams et
al., 2004). This radius is approximately $10$ AU for a Solar mass star when
the EUV heated region reaches $10^{4}$ K. As described above, it is possible
the G-clouds can be smaller than this value, and as we show below, discs are
likely to be rapidly evaporated to sizes smaller than this radius. While EUV
heated gas in the outer disc is still formally bound inside $R_{g}$, provided
that there’s sufficient energy input (from an external radiation field),
thermally driven outflows can be launched from inside $R_{g}$ (Parker, 1965;
Lamers & Cassinelli, 1999). Previous work on disc evaporation in the Galactic
Centre (e.g. Murray-Clay & Loeb, 2012; Miralda-Escudé, 2012) applied the
Johnstone et al. (1998) model to discs with radii below $R_{g}$, even though
this approach does not give the correct flow structure. Nor did these
approaches include the FUV-heated sub-sonic layer.

Therefore, in this section, we update the theory of externally
photoevaporating discs in the EUV regime to include wind launching inside
$R_{g}$. In addition, we account for the attenuation of the UV radiation field
due to dust. At dust-to-gas ratios of order $\sim 0.01$, absorption of EUV
photons by neutral hydrogen typically dominates over dust in ionized gas.
However, it can become important at larger dust-to-gas ratios. In some of our
evolutionary simulations, we find the outer regions of our discs can reach
dust-to-gas ratios approaching unity, and therefore we cannot neglect the
absorption of EUV photons by dust. These complications increase the complexity
of the models; however, to quickly scan the parameter space, we still adopt
the Johnstone et al. (1998) formalism as the basis of our calculation.

The flow topology is taken as follows: from the disc, an isothermal FUV heated
outflow is launched that remains sub-sonic, the flow then transitions through
a thin ionization front and becomes an ionized isothermal outflow. We take the
sound speed of the gas to be 3 km s-1 in the FUV heated outflow, appropriate
for gas in strong FUV fields (e.g. Hollenbach & Tielens, 1999; Adams et al.,
2004) and 10 km s-1 in the EUV regions, appropriate for ionized gas at 104 K.
We assume the EUV outflow has spherical velocity divergence and the velocity
and density profile follow an isothermal Parker wind (e.g. Parker, 1958;
Cranmer, 2004). We modify the isothermal sound speed of the outflowing fluid
(of gas and dust) to be $c_{s,{\rm fluid}}=c_{s,{\rm gas}}/\sqrt{1+X_{d}}$
where $X_{d}$ is the dust-to-gas mass ratio in the wind, this accounts for the
fact the effective pressure felt by a dusty-fluid is lower than one without
dust. As it is the small dust grains that dominate the EUV opacity, we assume
that the dust particles are 0.1$\mu$m. Dust grains of this size are easily
entrained in photoevaporative outflows (Facchini et al., 2016; Owen & Altaf,
2021; Winter & Haworth, 2022), hence we take the dust-to-gas mass ratio to be
constant throughout the outflow.

The FUV heated region is strongly sub-sonic ($R_{g}$ for FUV heated gas is
$\sim 10$ times larger than for EUV heated gas), hence we model the density
structure in this region as being hydrostatic, including rotational support at
constant specific angular momentum. Thus, the density profile in the FUV
heated region, as a function of distance ($r$) becomes:

$\rho_{\rm
FUV}=\rho_{d}\exp\left[\frac{GM_{*}}{c_{s,FUV}^{2}}\left(\frac{1}{r}-\frac{1}{R_{d}}\right)-\frac{h_{d}^{2}}{2c_{s,FUV}^{2}}\left(\frac{1}{r^{2}}-\frac{1}{R_{d}^{2}}\right)\right]$
(1)

where $\rho_{d}$ is the density in the FUV heated region at the disc’s outer
edge and $h_{d}$ is the specific angular momentum in the outflow, given by
$h_{d}=\sqrt{GM_{*}R_{d}}$. The fluxes for the EUV heated region are
sufficiently high that the outflow is in ionization-recombination balance. As
we include dust attenuation, we cannot simply make a Strömgren volume argument
like Bertoldi & McKee (1990); Johnstone et al. (1998), and we must calculate
the ionization structure explicitly. Making the on-the-spot approximation
blackfor pure hydrogen we calculate the ionization fraction ($X_{i}$) as:

$X_{i}=\frac{1}{2}\left[\sqrt{\left(\frac{\Gamma}{\alpha_{b}n_{H}}\right)^{2}+4\frac{\Gamma}{\alpha_{b}n_{H}}}-\frac{\Gamma}{\alpha_{b}n_{H}}\right]$
(2)

where $n_{H}$ is the density of hydrogen, $\alpha_{b}$ is the case-B
recombination coefficient, and $\Gamma$ is the ionization rate blackper
neutral hydrogen given by:

$\Gamma=\phi\>\\!\sigma_{\rm EUV}\exp\left(-\tau\right)$ (3)

with $\phi$ the EUV photon flux impinging on the disc. The optical depth
($\tau$) is calculated including both gas and dust to the ionization front:

$\tau=\int_{\infty}^{R_{IF}}n_{H}\left[\left(1-X_{i}\right)\sigma_{\rm
EUV}+\kappa_{\rm d,EUV}X_{d}m_{h}\right]{\rm d}r$ (4)

where $R_{IF}$ is the radius of the ionization front. This optical depth is
then evaluated by integrating the density profile of the isothermal Parker
wind solution out to twenty times the ionization front distance. The velocity
structure in this region is given by:

$u=c_{s,EUV}\sqrt{-W_{i}\left\\{-\left(\frac{R_{g}}{2r}\right)^{4}\exp\left[f(r)\right]\right\\}}$
(5)

where $W_{i}$ is the $i^{\rm th}$ branch of the Lambert W function (with the 0
branch applicable for the sub-sonic regions and the -1 branch applicable for
the supersonic regions Cranmer 2004). The function $f(r)$ is given by:

$f(r)=3-\frac{2R_{g}}{r}$ (6)

The density profile can then be obtained through mass conservation:

$\rho(r)=\rho_{IF}\frac{u_{IF}R_{IF}^{2}}{ur^{2}}$ (7)

where $\rho_{IF}$ and $u_{IF}$ are the density and velocity in the EUV heated
region at the ionization front, respectively. We define the ionization front
to be the radius at which the ionization fraction drops below $10^{-4}$;
however, since the ionization front is very thin, this choice makes no
practical difference. For simplicity, we assume a monochromatic spectrum for
the EUV field of $\sim$20 eV photons, as such $\sigma_{\rm EUV}=2\times
10^{-18}$ cm2. Following Johnstone et al. (1998), the size of the FUV heated
portion is chosen such that its column density provides an optical depth of
unity to FUV photons. We then solve this coupled problem such that momentum-
flux is conserved through the ionization front. We do this by matching:

$\rho_{FUV}\left(R_{IF}\right)c_{s,FUV}^{2}=\rho_{IF}\left(c_{s,EUV}^{2}+u_{IF}^{2}\right)$
(8)

Since the FUV heated region is extremely sub-sonic ($R_{d}/R_{g,FUV}$ is
always $\ll 1$), we have dropped the $\rho u^{2}$ term in the momentum flux of
FUV heated gas into the ionization front. This simplification means we don’t
need to explicitly solve for mass conservation across the ionization front.
Matching the momentum flux across the ionization front is an iterative
procedure for which we use the brentq solver provided in scipy (Virtanen et
al., 2020). The resulting mass-loss rates as a function of disc size are shown
in Figure 2 for our nominal incident EUV flux of $\phi=3\times 10^{15}$
photons s-1 cm-2 and a stellar mass of 1 M⊙.

Unsurprisingly, we find larger mass-loss rates for larger discs and lower
dust-to-gas mass ratios. Below a dust-to-gas mass ratio of $X_{d}\sim 0.03$,
the effect of the dust is weak. This weak dependence arises because below
$X_{d}\sim 0.03$ dust has a limited effect on the attenuation of EUV photons;
however, it still dominates the attenuation of FUV photons. In EUV
photoevaporation, the EUV flux plays the dominant role in setting the mass-
loss rates, and the size of the FUV-dominated region plays a minor role. Hence
the slow increase in mass-loss rate as the dust-to-mass ratio decreases at low
values is due to the slowly increasing size of the FUV-heated region. As the
density profile in the FUV heated region is an exponential profile (Equation
1), the depth of the region for a given column density (of dust) changes
logarithmically. However, above a dust-to-gas ratio of 0.03, the dust absorbs
a significant fraction of the EUV flux, resulting in a much lower EUV flux
reaching the ionization front. As such, the mass-loss rates fall off rapidly
with an increasing dust-to-gas mass ratio. Additionally, we find that mass-
loss rates depend on the square root of the ionizing flux (as expected for
ionization-recombination equilibrium - Johnstone et al. 1998) and
exponentially with stellar mass as expected for a sub-sonically launched
thermal wind (e.g. Adams et al., 2004). While our mass-loss calculations are
1D and assume spherical divergence, this approach has been shown to give order
unity correct mass-loss rates when compared to multi-dimensional simulations
(Haworth & Clarke, 2019). Instead of tabulating our mass-loss rates, we choose
to fit a physically motivated functional form to the results. This allows us
to avoid computationally expensive multi-dimensional interpolation in
evolutionary calculations. These functional forms are detailed in the
Appendix.

Figure 2: The external photoevaporation rates as a function of disc size and
dust-to-gas mass ratio at the outer edge of the disc for an EUV flux of
$3\times 10^{15}$ photons s-1. The photoevaporating mass-loss rates required
to approximately explain the G-clouds are shown as the open-dotted region,
with mass-loss rates between $10^{-8}$ and $10^{-7}$ M⊙ yr-1. Curves of
constant disc survival time (3 Myr) are shown for disc mass of of $10^{-3}$
and $10^{-2}$ M⊙ and a viscous time of 1 Myr (for $\alpha=10^{-3}$) are shown.
The dot-dashed line (labeled $R_{\rm disc}$) shows the radius an initially 0.1
M⊙ disc would be truncated to in 3 Myr by photoevaporation alone. As discussed
in Section 4, this figure demonstrates the inherent problem of having a
standard protoplanetary disc explain the G clouds. A large disc requires an
extremely high dust-to-gas ratio to survive photoevaporation. A small disc (to
the left of the $t_{\nu}=1$ Myr) line will drain onto the star via accretion
rapidly. Hence, for nominal dust-to-gas ratios, it is impossible to satisfy
both constraints simultaneously, let alone explain the required
photoevaporation rates for the G-Clouds.

###  3.1 Constraints from the observed G-cloud fluxes

One of the clues pointing to the fact the G-clouds are photoionized gas is the
detection of the hydrogen and helium recombination lines (Gillessen et al.,
2012; Burkert et al., 2012) as well collisionally excited metal lines in the
more recently detected G-clouds (Peißker et al., 2020; Ciurlo et al., 2020).
Following other authors (e.g. Gillessen et al., 2012), the measured
Br-$\gamma$ line luminosities of order 2–10 $\times 10^{30}$ erg s-1 imply
electron densities of order $10^{5}$–$10^{6}$ cm-3 on scales of $\sim 100$ AU.
Indeed, as pointed out by Murray-Clay & Loeb (2012), if one assumes that these
electron densities are outflows at $\sim 10$ km s-1 then one obtains a mass-
loss rate of a few $10^{-8}$ M⊙ yr-1. This value is roughly the mass-loss rate
a photoevaporating disc with a size of a few AU would give you in the Galactic
Centre.

Since our new photoevaporation models explicitly calculate the density and
ionization structure of the photoevaporating outflows, we are able to directly
compute the expected Br-$\gamma$ luminosity from our models. Assuming case-B
recombination, the Br-$\gamma$ luminosity is:

$L_{\rm Br\gamma}=\int\varepsilon X^{2}n_{H}^{2}{\rm d}V$ (9)

where $\varepsilon$ is a constant, with a value of $3.44\times 10^{-27}$ erg
s-1 cm3 for $10^{4}$ K gas (e.g. Ballone et al., 2018). Since our
photoevaporating simulations are 1D, we need to turn our density solutions
into 3D density structures to compute the line luminosity. Assuming the flow
covers the full $4\pi$ solid angle is likely to overestimate luminosity as
outflows are launched from the disc’s outer edge over a finite solid angle
(Richling & Yorke, 1998, 2000; Adams et al., 2004; Haworth & Clarke, 2019).
Inspection of the integrand of Equation 9 indicates that the contributions
from different radii scale as $n^{2}R^{3}$. Outside $R_{g}$, the outflow is
likely to expand spherically, inside $R_{g}$, the outflow is likely to expand
super-spherically with an approximately exponential density profile (Owen &
Altaf, 2021). This indicates that the Br-$\gamma$ luminosity should be
dominated at small radii inside $R_{g}$. We draw special attention to G2,
which, as we discuss in Section 9 is likely to have been pressure confined
during its plunge towards pericentre. As such, the argument that the
Br-$\gamma$ is dominated at small radii does not hold for G2. For a pressure-
confined isothermal structure, the density is approximately constant outside
$R_{g}$ (inspection of Equation 1 in the limit that $r\gg R_{g}$, indicates
the density will tend to a constant for a hydrostatic profile), and as such
the Br-$\gamma$ flux will be dominated at large radii.

For outflowing density structures, inside $R_{g}$, the flow is sub-sonic, and
the density profile is close to hydrostatic equilibrium. Thus, the density
perpendicular to the disc’s mid-plane will approximately obey
$n_{H}=n_{0}\exp(-z^{2}/2H^{2})$ with $H=c_{s}/\Omega$. This allows us to
approximate Equation 9 as:

$L_{\rm Br\gamma}\approx 2\pi^{3/2}\varepsilon\int
X(R)^{2}n_{0}(R)^{2}H(R)R{\rm d}R$ (10)

where $n_{0}(R)$ is our density solution to the 1D outflow problem. Finally,
we restrict $H(R)$ to be no larger than $2/\sqrt{\pi}R$ which allows an
approach to the spherical case in the limit of $H\longrightarrow R$, although
this has little impact on the result as $H$ only approaches $R$ outside
$R_{g}$.

Figure 3: The mass-loss rates from the photoevaporating wind scenario as a
function of Br-$\gamma$ line luminosity. The black-dotted line shows the
$1\sigma$ error of $2.5\times 10^{-4}$ L⊙ from the G2 discovery paper
(Gillessen et al., 2012), which is representative of the flux errors reported
in other works (e.g. Gillessen et al., 2013; Pfuhl et al., 2015; Peißker et
al., 2020). The orange dotted line shows the most luminous G-cloud, “G3” from
the discoveries reported by Ciurlo et al. (2020). Thus, to explain the
G-clouds with a photoevaporating disc, one requires mass-loss rates in the
approximate range $10^{-8}$–$10^{-7}$ M⊙ yr-1.

The results of this calculation are shown in Figure 3, where we show the
region of parameter space in mass-loss rate and Br-$\gamma$ flux that the disc
photoevaporation models can explain. Comparison between observed Br-$\gamma$
luminosities and the mass-loss rates indicates that if the G-clouds are
photoevaporating discs, they are likely losing mass at a rate between
$10^{-8}$ and $10^{-7}$ M⊙ yr-1.

Thus, throughout this work, we use the criteria of a photoevaporative mass-
loss rate in between $10^{-8}$ and $10^{-7}$ M⊙ yr-1 as an indication of
matching the observed Br-$\gamma$ line fluxes. As we shall see the
evolutionary calculations in our scenarios indicate the discs spend very
limited time with a mass-loss rate exceeding $10^{-7}$ M⊙ yr-1 and evolve
quickly once the mass-loss rate drops below $10^{-8}$ M⊙ yr-1, indicating the
time our models spend appearing as G-clouds is insensitive to the exact values
of these boundaries.

We also note a difference between our calculations and previous work. In the
original consideration of the photoevaporating disc scenario, Murray-Clay &
Loeb (2012) assumed that in the central region, the EUV flux was dominated by
the S-stars, and as such in their calculation the EUV flux impinging on the
disc rose as G2 approached pericentre, resulting in a Br-$\gamma$ flux that
increased by roughly a factor of five. This increase is inconsistent with the
observations (e.g. Gillessen et al., 2013; Pfuhl et al., 2015). However, as we
showed in Section 2 (Figure 1 we find the EUV-flux is dominated by the O/WR
disc and it remains approximately constant as one approaches Sgr A*. This
means the predicted Br-$\gamma$ fluxes from the photoevaporating disc scenario
would not actually strongly rise as inferred by Murray-Clay & Loeb (2012), but
would rather remain roughly constant or slightly increase. Although, we note
as mentioned above (and discussed in 9), we suspect G2 is actually a pressure-
confined photoevaporating flow throughout its observational history. This fact
means it’s not trivial to map our photoevaporating wind Br-$\gamma$ fluxes
directly onto G2’s.

####  3.1.1 NIR luminosities

The observed NIR luminosities of the G-clouds are less constraining. Zajaček
et al. (2017) used radiative transfer modelling to show that the NIR
properties of the G-clouds were consistent with those of a young T Tauri star
surrounded by a disc. The emission demonstrates the presence of dust, but it’s
still currently unclear what source(s) are heating the dust. In the T Tauri
star scenario, it could either be the young star itself, the combined
background emission from all the nearby stars, or both. blackWe can
investigate this more explicitly using our analysis from Section 2. During our
computation of the typical EUV fluxes, we also compute the properties of the
bolometric radiation field, finding fluxes in the range $10^{5}-10^{6}$ erg
s-1 cm-2, with mean temperatures $\sim 2-4\times 10^{4}$ K. Using these values
we can compute the temperatures of optically thin dust particles, using the
optical constants for astronomical silicate grains (Weingartner & Draine,
2001). The results are shown in Figure 4, where small sub-micron particle
sizes can be heated well above the black-body temperature to temperatures
$\sim$600 K due to their efficient absorption in the UV and inefficient
emission in the IR. However, without knowledge of the dust size distribution
in the G-clouds, it’s not possible to conclude whether the dust is externally
or internally heated.

Figure 4: The temperature of 0.001 and 0.1 $\mu$m size silicate grains in the
Galactic Centre as a function of bolometric flux, in addition to the blackbody
temperature. The bands represent ranges arising from the plausible range in
the temperature of the radiation field, as well as the range of solid angles
over which absorption could take place (between $4\pi$ for an isotropic
radiation field and $\pi$ of a plane-parallel radiation field). This
demonstrates that it is possible that the dust in the G-clouds is heated by
the external radiation field to temperatures $\sim$600 K.

However, we can place a few constraints on the requirement that there is
sufficient dust to match the required luminosities. The most constrainable
scenario we present in this work is where we explain the G-clouds as a dust-
depleted photoevaporating disc, and therefore we should check we have
sufficient dust mass to explain the observed NIR luminosities. The K band
luminosity for G2 is debated. However, there is little debate at longer
wavelengths with G-clouds typically presenting a $\sim 600~{}$K black-body-
like spectrum with a luminosity of a few Solar luminosities, corresponding to
a black body emitting radius of $\sim 1$ AU (e.g. Gillessen et al., 2012;
Peißker et al., 2020). Such a temperature and size rule out black-body
emission from an optically thick disc, internally heated by a central star, as
it is not possible to reach such high temperatures at such large distances
without violating the total luminosity constraints. Optically thin, small dust
particles that are heated above their blackbody temperature due to their
inefficient emission of IR radiation are more likely (Figure 4). Taking the
power emitted by a single dust grain to be $4\pi a^{2}Q\sigma T^{4}$, where
$Q$ is the radiative efficiency, which we approximate to be $Q\sim 2\pi
aT/b_{w}$ (with $b_{w}$ Wein’s displacement constant), we find the total mass
in dust must be:

$M_{\rm dust}\sim 1\times 10^{-3}~{}{\rm
M}_{\oplus}~{}\left(\frac{L_{G}}{1~{}{\rm
L}_{\odot}}\right)\left(\frac{T}{600~{}{\rm
K}}\right)^{-5}\left(\frac{\rho_{\rm in}}{1~{}{\rm g~{}cm}^{-3}}\right)$ (11)

Where we have inserted representative values for the G-clouds NIR luminosities
($L_{G}$), temperatures, and the internal density of the dust particles
($\rho_{\rm in}$). This mass of dust would become optically thick to external
irradiation if it had a size smaller than:

$R_{\tau=1}\approx 8~{}AU\left(\frac{M_{\rm dust}}{10^{-3}~{}{\rm
M}_{\oplus}}\right)^{1/2}\left(\frac{\kappa_{d}}{10^{4}~{}{\rm
cm^{2}~{}g^{-1}}}\right)^{1/2}$ (12)

where we have assumed that the heating source is UV photons from the nearby
massive stars, adopting an opacity of $10^{4}$ cm2 g-1 (e.g. Ali, 2021); for
cooler irradiating sources (e.g. an internal low-mass star) the opacity will
be lower and size-scale limit correspondingly smaller. Comparisons to our
outflow solutions indicate that they contain sufficient mass, and have large
enough length scales that they can satisfy the above mass and optical depth
requirements unless the disc is small ($\lesssim 3$ AU) and are significantly
depleted in dust $X_{d}\lesssim 10^{-4}$, a region rarely traversed by our
evolutionary models.

##  4 Standard Evolution Scenarios

Our mass-loss rates, calculated in Section 3 provide a fundamental problem to
the protoplanetary disc origin of the G-clouds. Specifically, it is difficult
to make a protoplanetary disc, born at a similar time to the young cluster,
live sufficiently long while providing a large enough mass-loss rate at late
times to be observable as a G-cloud.

Figure 2 shows the rough range of photoevaporation rates consistent with the
observed Br$\gamma$ fluxes for the G-clouds. The solid lines show contours of
a constant mass-loss timescale of 3 Myr. As discussed in Murray-Clay & Loeb
(2012) a massive disc ($\sim 0.1$ M⊙) would have a mass-loss timescale of a
few Myr for a disc with the standard dust-to-gas ratio of 0.01 and sizes of
$\sim 10$ AU, while being compatible with the observed Br$\gamma$ flux of G2.
While the above argument is indicative that one might be able to explain the
G-clouds properties, it neglects that photoevaporation causes the disc to
evolve and accrete rapidly (e.g. Clarke, 2007; Rosotti et al., 2017; Winter et
al., 2020). As photoevaporation removes material from the outer disc, reducing
its radius, the viscous transport time of the disc decreases ($t_{\nu}$
approximately scales linearly with radius for a constant $\alpha$, passively
heated disc – Hartmann et al. 1998). Shorter viscous transport timescales
allow the disc to lose angular momentum (by providing it to the
photoevaporating wind) more efficiently, allowing the disc to accrete more
efficiently. In Figure 2, we show the disc radius (line labelled $R_{\rm
disc}$) photoevaporation would reach by removing material from a 0.1 M⊙ disc
with a $\Sigma\propto R^{-1}$ profile and initial radius of 20 AU in 3 Myr as
the dot-dashed line (significantly bigger discs would be tidally stripped by
the black hole’s tidal field). In addition, we show the radius where the
viscous transport time of the disc is equal to 1 Myr (for $\alpha=10^{-3}$).
These two lines allow us to sketch an evolutionary pathway: for an initially
large disc, photoevaporation will strip the disc, reducing its radius. Once
the disc is photoevaporated to the point where the viscous transport time is
of order its age, accretion takes over, and the disc rapidly accretes onto the
star. This pathway is confirmed in our numerical calculations shown later. The
final accretion stage takes place at approximately fixed radii, where the
outward transport of angular momentum in the disc is balanced by the removal
of angular momentum in the wind (e.g. Winter et al., 2020; Owen & Altaf,
2021). Thus, accretion acts to shorten the disc’s lifetime from what would be
obtained by estimating instantaneous mass-loss timescales (the disc mass
depletes as $\propto\exp(-t/t_{\nu})$, rather than a power-law fashion for
discs with a fixed outer radius Rosotti & Clarke 2018). Figure 2 indicates
that it is difficult to simultaneously have a protoplanetary disc survive for
at least a few Myr while providing sufficient mass-loss rates at late times to
explain the Br$\gamma$ fluxes observed for the G-clouds. Indeed, these issues
are similar to the “propyld lifetime” problem identified in more benign star-
forming regions (e.g. Henney & O’Dell, 1999; Haworth et al., 2022). However,
the proposed solution for the Orion Nebula Cluster of dynamical evolution,
coupled with an extended period of star-formation (e.g. Winter et al., 2019),
is unlikely to hold in the Galactic Centre. This is because the EUV fluxes are
still large at distances of $\gtrsim 1$ pc, so it is difficult to argue that
recently formed stars have just been scattered in.

In the following sub-sections, we demonstrate these insights explicitly
through direct numerical calculation of the disc evolution. In the following
sections, we modify the standard scenario to arrive at an evolutionary picture
that allows a protoplanetary disc to survive for a few Myr in the Galactic
Centre and provides sufficient mass-loss rates at late times to explain the
observed G-clouds. We also comment on the evolution of any circumstellar disc
formed recently, perhaps through a stellar merger.

###  4.1 Numerical Models

To model the evolution of a photoevaporating circumstellar disc in the
Galactic Centre, we numerically solve the viscous thin disc model for both the
gas and dust components. In this formalism, the surface density of the gas,
$\Sigma_{g}$, and dust $\Sigma_{d}$, evolve according to:

$\frac{\partial\Sigma_{g}}{\partial t}=\frac{1}{R}\frac{\partial}{\partial
R}\left[3R^{1/2}\frac{\partial}{\partial
R}\left(\nu\Sigma_{g}R^{1/2}\right)-\frac{2\Lambda\Sigma_{g}}{\Omega}\right]-\dot{\Sigma}_{w,g},$
(13)

and

$\frac{\partial\Sigma_{d}}{\partial t}=-\frac{1}{R}\frac{\partial}{\partial
R}\left[Rv_{d}\Sigma_{d}-\frac{\nu}{Sc}R\Sigma_{g}\frac{\partial}{\partial
R}\left(\frac{\Sigma_{d}}{\Sigma_{g}}\right)\right]-\dot{\Sigma}_{w,d},$ (14)

where $R$ is the cylindrical radius, $t$ is the time, $\nu=\alpha c_{s}H$ is
the kinematic viscosity, $\Omega$ is the Keplerian angular frequency, $v_{d}$
is the velocity of the dust, $Sc$ is the dust’s Schmidt number which we set to
unity, $\Lambda$ is any external torque on the disc (e.g. from a planet) and
$\dot{\Sigma}_{w}$ is the photoevaporation rate of gas (sub-script $g$) and
dust (sub-script $d$). This model is numerically integrated using the method
outlined in (Owen, 2014; Booth & Owen, 2020), which is first-order in time and
second-order in space and uses a van-Leer limiter for advection terms. We use
a grid extending from 0.025 AU to 40 AU with 500 non-uniformly spaced cells
(the spacing follows $R^{1/3}$, yielding higher resolution at smaller radii).
At the outer boundary, we assume that the tidal field removes material
instantly from the disc and use outflow boundary conditions (in all cases,
this doesn’t matter as photoevaporation truncates the disc inside the inner
boundary). The dust-size distribution is evolved using the formalism of
Birnstiel et al. (2012), using a fragmentation velocity of 10 m s-1. This
dust-size evolution accounts for growth due to sticking and fragmentation due
to destructive collisions, we assume that 10% of collisions below the
fragmentation velocity result in growth (Booth & Owen, 2020). We note that
these choices for the fragmentation velocity and fraction of collisions that
result in growth do not change the general picture of the evolutionary results
shown below. The external photoevaporation rates from our calculation
described in section 3 are implemented using a formalism similar to Clarke
(2007). Our procedure is as follows: we define the disc radius to be the point
in the disc where the gas surface density drops below $10^{-4}$ g cm-2. We
then measure the dust-to-gas ratio by considering the amount of dust and gas
that resides within 10 cells of the disc radius to estimate the dust-to-gas
mass ratio in the photoevaporative outflow. We then remove material from
cells, moving inwards until we reach the required total mass-loss. Cells that
have material completely removed are assigned a surface density of the floor
value of 10-20 g cm-2. The calculation of the dust-to-gas ratio over a radial
range of 10 cells at the outer boundary is necessary as the finite nature of
the grid can change the exact value of the pressure gradient (and hence the
dust-to-gas ratio) in the last cell to change by large amounts from time-step
to time-step. Using just the last cell to measure the dust-to-gas mass ratio
causes it to fluctuate, and hence the mass-loss rate to fluctuate strongly,
causing the disc to occasionally nonphysically expand and then contract. By
experimentation, we find that 10 cells provide a sensible balance between
smoothing the calculated dust-to-gas ratio and using a small radial range near
the outer edge. We note that in these 10 cells, the dust advection is directed
outwards. We find the exact choice of the number of cells to measure the dust-
to-gas mass ratio over or the threshold surface density to define the disc’s
outer edge does not change the general results of our calculations.

###  4.2 Results: protoplanetary disc lifetimes in the Galactic Centre

In order to directly verify our insights that a protoplanetary disc is
unlikely to survive the $\gtrsim 3$ Myr required to appear as G-cloud today,
we simulate the evolution of a disc receiving a photon flux of $3\times
10^{15}$ photons s-1 cm-2. We make a number of choices to maximise the disc
lifetimes. The luminosity constraints on G2 of $\lesssim 5$L⊙ (Gillessen et
al., 2012) implies that if the central source were a young star, it would have
to be less massive than $\sim 1.4$ M⊙. As more massive stars have deeper
gravitational potential wells, making them harder to photoevaporate, we chose
our central star to have a mass of 1.4 M⊙. We assume an initially compact disc
(such that initial tidal removal of material is minimal, maximising its mass),
where the initial profiles follow a Lynden-Bell & Pringle zero-time similarity
solution (Lynden-Bell & Pringle, 1974):

$\Sigma_{g}=\Sigma_{0}\left(\frac{R}{R_{0}}\right)^{-1}\exp\left(-\frac{R}{R_{0}}\right)$
(15)

The initial disc mass is set to be 0.15 of the stellar mass, $R_{0}$ is set to
5 AU, the dust-to-gas mass ratio is assumed to be uniform everywhere with a
value of 0.01, and the dust grains are taken to initially be 0.1$\mu$m in
size. The disc’s mid-plane temperature structure follows a $T\propto R^{-1/2}$
profile, appropriate for a disc passively heated by the central star (e.g.
Kenyon & Hartmann, 1987). The disc temperature is set to 250 K at 1 AU,
appropriate for a flaring, passively heated disc around a 1.4 M⊙ star. While
there is debate as to whether the external radiation field contributes to
heating the dust in G-clouds (Section 3.1.1, e.g. Gillessen et al., 2012;
Peißker et al., 2020), a lower disc temperature causes the disc to evolve
slower, extending its lifetime. Since we are applying no external processes to
the disc in this first set of models, there is no external torque
($\Lambda=0$). In these calculations, the standard zero-torque boundary
condition is applied at the inner boundary (e.g. Pringle, 1981; Pringle et
al., 1986).

Figure 5: The evolution of the disc mass and the total masses lost through
accretion and in the photoevaporative wind. The thick portion of the disc mass
lines shows when the photoevaporative rate exceeds $10^{-8}$ M⊙ yr-1 (and is
visible as a G-cloud in Br-$\gamma$). The first three panels show different
values of the turbulent viscosity for a standard EUV photon flux of $3\times
10^{15}$ photons s-1 cm-2, the final panel shows a low EUV scenario (i.e
residing in the outer regions of the Young disc) with an $\alpha=10^{-3}$. It
is extremely difficult to have a disc survive for $\gtrsim 2$ Myr with
photoevaporation rates sufficient to explain the G-clouds.

The resulting evolutionary tracks for a range of plausible parameters are
shown in Figure 5. All these show disc lifetimes considerably shorter than the
required value of $\gtrsim 3$ Myr, and in the more nominal cases have
sufficiently small photoevaporation rates that they would not be observed as
G-clouds beyond 1 Myr.

These results, appear at first glance to not match the calculations that
appear to indicate photoevaporation will only deplete discs of 5-10 AU in
scale on timescales of several Myr (Murray-Clay & Loeb, 2012). However, as
discussed above, these calculations only focused on the removal of disc
material by photoevaporation. Whereas, the numerical calculations show that
disc accretion onto the central star is an equally, if not more powerful disc
depletion mechanism. Thus, through accretion, photoevaporation brings about a
rapid end to the disc’s life. Even in the cases of extremely low viscosities
($\alpha=10^{-4}$) or low photon-fluxes ($10^{14}$ photons s-1 cm-2) with
favourable choices in the initial conditions we cannot make discs survive
around young stars sufficiently long to provide the source population of
G-clouds unless we somehow modify the disc’s evolutionary pathway.

##  5 Instantaneous formation of a disc

One trivial solution to the disc lifetime problem would be to suppose that the
disc does not surround a young star born in the star formation episode $\sim
3$ Myr ago, but rather the disc has recently formed. This disc could have
formed either as the by-product of a stellar merger (e.g. Prodan et al., 2015;
Naoz, 2016), or through interactions with compact objects (such as black
holes), as suggested by Miralda-Escudé (2012). In the following discussions,
we assume that it is the disc, that creates the G-cloud signature666It could
be that this collisional byproduct itself is responsible for the G-cloud
signature; however, since the properties of these byproducts are uncertain, we
focus our discussion on any disc created.. In this case, if the formed disc
had initial conditions similar to those chosen in the previous section, i.e.
similar to discs formed through the star formation processes then there is no
lifetime issue. However, the discs used in the previous calculations are too
extended and massive to have formed recently through such processes. Thus,
here we consider more compact, lower-mass discs.

If the G-clouds were to have formed recently, they would be in similar orbits
to where we see them today, with most spending some time in their orbit inside
0.01 pc. The tidal radius at separations $\lesssim 0.01$ pc is now only a few
AU. Thus, we choose to set our initial disc scale radius for these
calculations to be $R_{0}=1$ AU. The initial disc masses are more uncertain,
but Miralda-Escudé (2012) appeal to planetary collision simulations (e.g.
Canup, 2004) which suggest disc masses of $M_{d}/M_{*}\lesssim 10^{-2}$.

Figure 6: The mass-accretion (solid) and photoevaporation rates (dashed) for a
low-mass, compact disc that has been recently created. When the
photoevaporation rate sits in the shaded region the photoevaporative flow is
sufficiently dense to explain the observed Br-$\gamma$ luminosities of the
G-clouds. Each panel shows a different evolutionary calculation where the
initial disc mass and viscous $\alpha$ parameter has been varied.

The resultant evolutionary calculations, shown in Figure 6 show that these
models can explain the lower range of mass-loss rates $\lesssim 3\times
10^{-8}$ M⊕ yr-1 for short periods of time $\lesssim 0.1$ Myr. However,
reproducing such cases requires massive initial discs and some amount of fine-
tuning for the value of the viscosity. Even so, we do struggle to reproduce
any model with sufficiently high photoevaporation rates to explain the
Br-$\gamma$ fluxes of the more luminous G-clouds (e.g. G3 and G4 Ciurlo et al.
2020). Thus, while this scenario could work to explain the G-clouds, the short
disc-lifetime of $\lesssim 10^{5}$ years, would require a production rate of
$\gtrsim 1/10^{4}$ yr-1 to explain the fact we current observe $\sim 5-10$
objects. This exceeds the rate estimated by Miralda-Escudé (2012) for
interactions between stars and compact objects. Alternatively, in a merger
scenario, the rate would imply that $\sim 300$ mergers need to have occurred
since the last star-formation episode. Since these young stars which formed in
binaries are the most likely progenitors for such mergers (Ciurlo et al.,
2020), as binaries do not survive long in the dense environment of the
Galactic Center (e.g. Stephan et al., 2019). Stephan et al. (2016) estimated
that $\sim 10$% of binaries would merge in the Galactic Center within a few
Myr. Thus, one would require a population of $\gtrsim 3000$ young, low-mass
binaries to have formed in the last star formation episode. Additionally,
Stephan et al. (2016) estimate that only 10% of Solar-like stars can exist in
binaries in the Galatic Center (as opposed to the 50% in the field). Thus, one
would require a population of $\gtrsim 30,000$ young T-Tauri stars to have
formed in the star-formation episode. This estimate exceeds the $\sim$2000
T-Tauri stars estimated to have formed based on extrapolations of the observed
IMFs, and X-ray surface brightness constraints (e.g. Lu et al. 2013, see
Section 8). Even if all the young binaries could be made to merge within a few
Myr, one could still not explain the number of observed G-clouds through the
instantaneous formation of an accretion disc.

##  6 Non accreting Disc

The problem identified in the previous Sections is photoevaporation enhances
accretion onto the star. Therefore, if accretion could be prevented, this
would provide a promising avenue for extending a disc’s lifetime.

Young, low-mass stars ($M_{*}\lesssim 1.5$ M⊙) are known to accrete through
magnetospheric accretion (Bouvier et al., 2007; Hartmann et al., 2016), where
the strong magnetic field of the young star ($B_{*}\sim 1$kG) truncates the
disc, funnelling material via accretion streams onto the stellar poles. This
truncation radii occurs when the magnetic torque from the star’s magnetic
field is sufficiently strong to overcome pressure (thermal and ram) of
material in the disc, disrupting it (e.g. Ghosh & Lamb, 1979a; Koenigl, 1991).
In general, this occurs at a radius where the orbital frequency of the disc is
faster than the star’s rotation period. However, if the star is spinning
sufficiently rapidly, the system can enter a “propeller” regime (e.g. Lovelace
et al., 1999; Romanova et al., 2003). In this case, the truncation occurs
outside the co-rotation point, now, material at the inner edge of the disc has
a rotation velocity slower than the stellar magnetosphere. In this case, the
disc material is prevented from accreting, and the star provides a torque to
the disc causing it to expand, resulting in a “decretion” disc (Pringle,
1991), and in some cases, a powerful wind from the inner regions can be
launched (e.g. Romanova et al., 2004).

In regular T Tauri stars, this process happens toward the end of the disc’s
lifetime when the accretion rate and disc mass are low (e.g. Armitage &
Clarke, 1996). However, there is ample reason to think it might happen earlier
in the Galactic Centre at higher disc masses. In the standard picture, it is
the accretion of disc material that modulates the stellar spin, maintaining
the disc inner edge close to co-rotation, allowing accretion, a process known
as disc locking (e.g. Shu et al., 1994; Hartmann, 2002). However, it has been
recently hypothesised that the removal of disc material, through
photoevaporation, stops the star from accreting as much material as normal,
preventing the slowing of its rotation. In addition, in the Galactic Centre,
the disc’s mass is lower at an earlier age, so when the young T Tauri star
begins to contract and spin up, there is less disc material to break the
star’s rotation. Thus, strong external photoevaporation produces a population
of T Tauri stars that are rapidly spinning (periods $\sim$ 1 day, rather than
the more typical periods of $\sim$ 10 days, Lee & Chiang 2017) by the age of
$\sim 1$ Myr (Roquette et al., 2021). Indeed, there is observational support
for this concept, where T Tauri stars in more massive clusters (which produce
higher UV fields, and hence stronger external photoevaporation) are more
rapidly rotating (Roquette et al., 2021).

With this in mind, we can estimate the accretion rate (and hence approximate
disc mass) at which a rapidly spinning T Tauri star would prevent accretion
and transition from an accretion disc to an outflow-dominated decretion disc.
The stellar magnetic field truncates a disc at a radius of approximately (e.g.
Ghosh & Lamb, 1979b):

$R_{T}\approx
R_{*}\left(\frac{B_{*}^{4}R_{*}^{5}}{GM_{*}\dot{M}_{*}^{2}}\right)^{1/7}$ (16)

requiring that the truncation radii is at a larger radius than the radius of
co-rotation, yields:

$\dot{M}_{*}<\frac{B_{*}^{2}R_{*}^{6}\Omega_{*}^{7/3}}{\left(GM_{*}\right)^{5/3}}$
(17)

or, putting in typical values:

$\displaystyle\dot{M}_{*}$ $\displaystyle\lesssim$ $\displaystyle 5.9\times
10^{-9}~{}{\rm M}_{\odot}~{}{\rm yr}^{-1}\left(\frac{B_{*}}{1~{}{\rm
kG}}\right)^{2}\left(\frac{R_{*}}{1.5~{}{\rm R}_{\odot}}\right)^{6}$ (18)
$\displaystyle\times$ $\displaystyle\left(\frac{M_{*}}{1.4~{}{\rm
M}_{\odot}}\right)^{-5/3}\left(\frac{P}{3~{}{\rm days}}\right)^{-7/3}$

This mass accretion rate yields a disc mass of order $10^{-3}$ M⊙ inside 10 AU
for a viscous alpha value of $10^{-3}$, consistent with our requirements from
Figure 2. At this mass-accretion rate, the angular momentum transfer rate
between the star and disc $\dot{L}\sim\dot{M}_{*}\sqrt{GM_{*}R_{T}}$ can be
compared to the angular momentum in the star ($L=I_{*}\Omega_{*}$) to
determine the spin-down timescale of the star. Evaluating for the same
representative parameters above yields a spin-down timescale of $\sim 5$ Myr.
This spin-down timescale is longer than, or comparable to the spin-up
timescale due to the pre-main-sequence contraction of the star (a few Myr).
This means that even though the star is transferring angular momentum to the
disc, the star’s contraction still means it will spin up, allowing accretion
to be halted indefinitely. However, if the star were to have a slower spin
rate, the reverse would be true and stellar contraction would be unable to
overcome the spin-down torque on the disc, the star’s spin would slow, and
accretion would proceed. Indeed, in Section 7.1, we find a critical value of
the star’s initial spin to be around 4-5 days, with spin periods less than
this value suppressing accretion at disc masses of order $10^{-2}-10^{-3}$ M⊙
and slower spins allowing accretion to continue to very low disc masses.

Figure 7: The evolution of the disc mass and the total masses lost in the
photoevaporative wind, in the case that the disc is prevented from accreting.
The thick portion of the disc masses shows when the photoevaporative rate
exceeds $10^{-8}$ M⊙ yr-1. The first three panels show different values of the
turbulent viscosity for a standard EUV photon flux of $3\times 10^{15}$ s-1,
and the final panel shows a low EUV scenario (i.e residing in the outer
regions of the Young disc) with an $\alpha=10^{-3}$. It is extremely difficult
to have a disc survive for $\gtrsim 2$ Myr with photoevaporation rates
sufficient to explain the G-clouds.

In order to assess whether this effect can shut off accretion and provide
sufficiently long disc lifetimes, we model the evolution of our discs as in
Section 4.1, but this time we set the initial disc mass to $0.01$ M∗. This
choice of initial condition represents an optimistic upper limit for the
remaining disc material at the point that accretion shuts down. In order to
prevent accretion in our numerical scheme, the inner boundary is given a zero-
flux boundary condition, causing the disc to behave like a decretion disc. In
this case, the torque from the star causes the disc to expand more rapidly,
enhancing the photoevaporation rate as the material is driven outwards by the
disc, where it is easier to escape. The resulting disc lifetimes (after
accretion ceases) are shown in Figure 7. These show that preventing accretion
alone cannot be the answer, as the enhanced photoevaporation now destroys the
discs fairly rapidly $\lesssim 0.5$ Myr. Thus, it would require considerable
amounts of fine-tuning for the observed G-clouds to be in the phase where
accretion has just ceased, and we’re witnessing their final destruction.

However, should the disc have just formed via a merger (e.g. Prodan et al.,
2015) (see discussion in Section 5), then the prevention of accretion by a
strong magnetic field could allow the disc to survive sufficiently long to
explain the G-cloud population. We speculate here that such a merger would
have to create a rapidly spinning, highly magnetised central object. The
prevention of accretion in the merger scenario solves several of the issues
posited in the previous section. Namely, we can now reach the higher mass-loss
rates $\sim 10^{-7}~{}$M⊙ yr-1 required to explain the more luminous G-clouds.
In addition, the longer disc lifetimes allow the discs to be observed as
G-clouds for $\sim 4\times 10^{5}$ yrs. In this case, we would require roughly
75 mergers to have taken place in the last 3 Myr, at a formation rate of $\sim
2.5\times 10^{-5}$ yr-1, which is still higher than the rate from Miralda-
Escudé (2012). Again following Stephan et al. (2016), who demonstrated that
$10\%$ of stars could be in binaries in the Galactic Centre, then our 2000
Tauri stars could yield $\sim 200$ low-mass binaries. A 10% merger fraction
within a few Myr would again fail to explain the number of observed G-clouds.
However, unlike the case discussed in Section 5, there is at least a
sufficient number of progenitor binaries in this scenario. Thus, we speculate
it may be possible to increase the merger rate. For example, through the use
of evection resonances, where even co-planar binaries can be excited to high
eccentricity, this mechanism has been suggested as a process to enhance the
rate of binary black-hole mergers (Gautham Bhaskar et al., 2022).

This merger scenario should be considered in the future to determine whether
the binary origin for G-clouds can match the requirements in disc mass and
create a rapidly spinning magnetised central object with sufficient frequency.
We shall not do such calculations here and instead suggest an alternative
scenario that naturally arises out of the formation and evolution of young
T-Tauri stars in the following section.

##  7 Shutdown of accretion and photoevaporation

To make a circumstellar disc around a T Tauri star survive for several Myr, we
both need to shut down accretion and photoevaporative mass-loss. While this
might seem like a difficult proposition, there is a natural solution. As
already discussed in Section 3 the photoevaporation rate can be significantly
reduced if the gas becomes enhanced in dust. Furthermore, as discussed in the
previous section, T Tauri stars in the Galactic Centre are likely to be rapid
rotators, preventing accretion at disc masses of order $\sim 10^{-3}$ M∗,
forcing the disc into the propeller regime at higher disc masses that standard
T Tauri stars.

There is a well-known sub-population of protoplanetary discs that appear to
have high dust-to-gas mass ratios in their outer regions: “transition discs”
(Andrews et al., 2011; van der Marel et al., 2015; van der Marel et al.,
2016). In these discs, the gas contains a pressure maximum that traps dust
preventing it from drifting towards the star, resulting in high, localised
dust-to-gas ratios that can approach unity (Bruderer et al., 2014; Ubeira
Gabellini et al., 2019). This trapping of dust has been confirmed
observationally, where the material actively accreting onto the star is
observed to be depleted in refractory elements (Kama et al., 2015). While it
is still not entirely clear what causes the pressure maximum (Owen, 2016); a
favoured explanation is a planet which is sufficiently massive to open a gap
in the gas disc, resulting in a pressure maximum exterior to its orbit (Rice
et al., 2006; Zhu et al., 2012; Pinilla et al., 2012). This picture matches
the disc morphology and architecture in PDS 70 system (Keppler et al., 2019).

Therefore, we consider a final scenario where the disc forms a giant planet at
a few AU and follow the combined evolution of the star, disc, and planet. As
indicated above, the disc switches from an accretion disc to a decretion disc
once the disc becomes sufficiently depleted, and the star is spinning rapidly.
In order to capture this process, we model the combined interaction between
the magnetic star and the disc. To do this, we follow the formalism laid out
in Livio & Pringle (1992) and Armitage & Clarke (1996) for the interaction
between the stellar magnetic field and disc. This allows us to insert a non-
zero torque $\Lambda$ in the inner disc, allowing the disc to exist in either
the accretion or propeller regime and switch between the two based on the
disc’s properties. This star-disc coupling also spins up or spins down the
star, and we calculate the evolution of the star’s spin in an identical way to
Armitage & Clarke (1996). We take the star’s radius to evolve in an identical
manner to Armitage & Clarke (1996), which itself is a fit to stellar evolution
models from Pols et al. (1995). Since we are focused on the early evolution of
the star on timescales less than ($\lesssim 10$ Myr), shorter than the
timescales for typical Sun-like stars to form radiative cores, we fix the
star’s moment of inertia to $0.2M_{*}R_{*}^{2}$. The planet’s effect on the
disc is included as another torque, arising from the impulse approximation
(Lin & Papaloizou, 1986) and is implemented, including migration, using the
approach of (Alexander & Pascucci, 2012), as done by Booth & Owen (2020). One
critical issue in 1D simulations of planet-disc interactions is how much
material flows across the planetary gap. This flow must be parameterised and
is uncertain even in the case of regular disc evolution. We use the approach
from Alexander & Armitage (2009) where the steady-state viscous mass-flux
(i.e. $3\pi\nu\Sigma$) is measured outside the planet’s orbit. A fraction of
this mass-flux $f_{\dot{M}}$ is then manually transferred across the planetary
gap, here we nominally set that value to 0.5, which is consistent with the
range of values measured from multi-dimensional simulations (D’Angelo et al.,
2002; Lubow & D’Angelo, 2006). We note the position at which the measurement
of the steady-state viscous flux is determined is typically three times the
planetary separation; however, sometimes, the disc becomes truncated by
photoevaporation inside this radius. Hence here we choose one and a half times
the planet’s separation. While we could include a parameterised model for how
the planet additionally accretes and grows in mass (Li et al., 2021), this
process is again uncertain. Thus, in order to isolate physical effects in our
parameter study, we choose to fix the planet’s mass for the duration of the
simulations. In our initial calculation, the planet is inserted at 4 AU, with
a mass of 1 MJ at the beginning of the simulation.

The evolution of the dust and gas surface density are shown in Figure 8 as
snapshots of their profiles at relevant points in the evolution. In addition,
we show the evolution of the disc mass and the mass lost to accretion and the
wind in Figure 9. Finally, the accretion rate onto the star and the
photoevaporation rate is shown as a function of time in Figure 10.

Figure 8: Dust and gas surface density shown at different snapshots in time
throughout the disc’s evolution. The planet’s location is indicated by the
position of the vertical dotted line. The dust-to-gas ratio is significantly
enhanced in the outer disc by the giant planet, suppressing the
photoevaporative mass-loss. Between the snapshot at 1.3 Myr and 2.45 Myr the
disc switches from an accretion to a decretion disc (signified by the change
in surface density profile in the inner regions). Once the outer disc is
depleted of gas (at $\sim 5$ Myr), the dust is assumed to be blown away, and
the disc starts rapidly photoevaporating from outside in, lasting a few
$10^{5}$ years before it eventually disperses.  Figure 9:  Disc mass and mass
lost to accretion and photoevaporation as a function of time for the evolution
calculation shown in Figure 8. The combination of the planetary and magnetic
torque holds the disc at a constant mass from $\sim 2-5$ Myr. When the disc
mass line is thick, the photoevaporative mass-loss rates are sufficient to
match the required Br-$\gamma$ fluxes of the G-clouds.  Figure 10: The mass
accretion rate (solid) and photoevaporative mass-loss rates (dashed) for the
disc evolutionary calculation shown in Figure 8. The variable accretion period
occurs during a disc-locking phase, where the mass accretion rate and stellar
spin are modulated to fix the disc’s inner edge close to co-rotation.

These Figures show that dust is trapped outside the planet’s orbit in this
scenario, locally enhancing the dust-to-gas ratio at about twice the planet’s
semi-major axis (Pinilla et al., 2012). Photoevaporation will then remove disc
material until the outer disc edge approaches the region where the dust-to-gas
ratio is enhanced, suppressing the mass-loss rate. While this is occurring,
the disc will continue to accrete onto the star, evolving to the point where
the stellar magnetic field is strong enough to prevent accretion. As the star
is preventing accretion, angular momentum will be transferred to the disc,
causing it to expand, which will ultimately cause the planet to migrate
outwards. In this way, the evolution of the system will be governed by the
viscous time of the disc, which as the planet’s orbit expands, will increase.
Ultimately, we expected the disc could survive for several Myr in this
scenario. However, since we have suppressed or shut off photoevaporation, it
will not appear as a G-cloud at this stage. What we find in our simulations is
that our manually implemented leakage slowly drains the outer disc of gas.
Once the outer disc is fully drained of gas, the dust particles are assumed to
be blown out, and the EUV radiation can penetrate to the disc inside the
planet’s orbit. This disc is now dust-depleted, and photoevaporation can
proceed again at an enhanced rate, which can exceed a rate of $10^{-7}$ M⊙
yr-1. Photoevaporation now destroys the disc on a timescale of a $\sim$105
years with the evolution looking similar to the case of a non-accreting disc
discussed in Section 6 (Figure 7). This agreement is not surprising as the
initial conditions for our non-accreting discs are similar to the disc
properties once the disc material outside the planet has been depleted. We do
wish to highlight that this switch from a disc which is not accreting or
photoevaporating to one which appears like a G-cloud is slightly pathological
in our modelling in that is arising due to the implementation of a leaky
planetary gap. If we did not include any leakage across the gap in our
simulation, the simulated disc could essentially live for a very long time. We
wish to highlight that in our simulation, the leakage prescription moves disc
material from the outer disc to the inner disc even in the decreation disc
phase, where the net viscous flow is outwards; however, the gas disc profile
in the vicinity of the planet promotes inward viscous flow. Thus, it may be
that, in this case, disc material does not flow inwards. Nevertheless, this
pathological transition from a non-accreting, non-photoevaporating disc to a
rapidly photoevaporating G-cloud phase in our simulations is not a
particularly concerning problem. In section 8 we discuss several other
processes that are prevalent in the Galactic Centre that will naturally cause
the disc to transition to a G-cloud whether the leakage works as we
implemented or not. In this discussion, we actually suggest the more likely
scenario to transition into the G-cloud phase is to remove the planet through
dynamical encounters.

###  7.1 Sensitivity to parameters

Figure 11: This figure shows the evolution of the disc’s mass and the
cumulative masses accreted onto the star and lost to the photoevaporative
wind. Each panel varies a single one of the listed parameters away from or
“nominal” case shown in Figure 9. The thick portions of the disc mass line
indicate the time at which the disc would be photoevaporating sufficiently
strongly to appear as a G-cloud in Br-$\gamma$.

Since we have identified a successful pathway to explain the G-clouds
originating from circumstellar discs around T-Tauri stars formed in the most
recent star formation episode a few Myr ago, it is worth checking how
sensitive to the parameters the scenario is. Specifically, we wish to
determine whether the scenario is fine-tuned or not. Here we vary the viscous
$\alpha$, initial planet mass and separation, the leakage parameter across the
gap, the planet’s disc mass and the initial spin period of the star. The
resulting evolution of the disc mass and the times the disc will appear as a
G-cloud are shown in Figure 11. We find that, in general, the scenario is not
fine-tuned and while the overall disc lifetime and the length of the G-cloud
phase (ranging from $\sim 10^{5}$ to $10^{6}$ years) are sensitive to the
parameters, the resulting evolutionary pathway is often the same. We find, as
one would expect, lower viscosities, higher planet masses, and smaller
leakages result in longer disc lifetimes. Additionally, the star’s initial
spin period and viscosity play a strong role in determining the disc mass at
which accretion is shut off, with faster spin rates and lower viscosities
shutting off accretion at higher masses.

Our parameter study does identify a couple of “failure modes” of the scenario,
which arise from the inability to shut off either accretion and/or
photoevaporation. The planet with an initial starting radius of 3 AU shown in
Figure 11 indicates a case where the planet is able to migrate inwards towards
the star, becoming a hot Jupiter. In this example, photoevaporation can never
shut off and ultimately destroys the disc. Another case is shown for a viscous
$\alpha$ value of 3$\times 10^{-3}$, here the viscosity is high enough that
accretion can continue onto the star until later times. Finally, although not
shown, if we make the initial spin period longer than $\sim 4-5$ days,
accretion can only be prevented at extremely low disc masses, and the
remaining disc reservoir is insufficient to explain the mass-loss rates of
observed G-clouds. blackThese failure modes highlight the need for the disc to
become a decretion disc at sufficiently high mass. Otherwise, the planet
continues its migration towards the star, and the dust trap migrates inward,
allowing accretion and photoevaporation to proceed. Ultimately, we expect star
formation in the Galactic Centre to give us a range of different initial
star/disc/planet parameters, many of which evolve into G-clouds at different
times, giving a continual supply of G-clouds that last $10^{5}$ to a few
$10^{5}$ years each over a period of Myrs.

Additionally, it is worth investigating this evolutionary picture for discs in
more typical star-forming environments, as it may play a role in resolving the
propyld lifetime problem.

##  8 Transition to a G-cloud

We have shown in the previous section that the presence of a giant planet
allows the disc to survive a few Myr in the Galactic Centre. However, by using
the giant planet to suppress photoevaporation, allowing the disc to survive,
the disc must transition to a G-cloud phase at late times. As a basis for the
following discussion, we consider that there are currently around 5-10
observed G-clouds ($N_{G}$) in the Galactic Centre (e.g. Peißker et al., 2020;
Ciurlo et al., 2020).

Additionally, extrapolating the observed IMF from the young cluster in the
Galactic Centre (Lu et al., 2013), we estimate roughly 2000 T Tauri stars
($N_{TT}$) with masses between 0.3 and 1.5 M⊙ were born in the young disc.
These young stars roughly represent the progenitor population of the G-clouds.
This number is in agreement with the value estimated by Nayakshin & Sunyaev
(2005), assuming a fraction of the diffuse X-ray emission from the Galactic
Centre is produced by T Tauri stars, as demonstrated by Lu et al. (2013).
Certainly, more than a few thousand T Tauri stars are ruled out by the X-ray
emission constraints of Nayakshin & Sunyaev (2005).

In the scenario where we use an embedded planet to suppress photoevaporation,
adopting a G-cloud lifetime of $t_{G}$, we may write the number of G-clouds at
any instant as:

$N_{\rm clouds}\sim f_{p}N_{TT}\Gamma_{G,*}t_{G}$ (19)

where $f_{p}$ is the fraction of the stars that form planets that allow the
disc to survive photoevaporation and $\Gamma_{G,*}$ is the rate at which an
individual planet-hosting star becomes a G-cloud. Assuming the production of
G-clouds has roughly been uniform over the age of the young cluster ($t_{\rm
age}$), we have $\Gamma_{G,*}\lesssim 1/t_{\rm age}$. This implies that the
G-cloud lifetime must be $\gtrsim f_{p}^{-1}\,1.5\times 10^{4}$ yr for $t_{\rm
age}=3$ Myr and $N_{\rm clouds}=10$. Taking $f_{p}\sim 0.1$, the value
obtained from exoplanet surveys, for a fraction of stars that host giant
planets on AU scale orbits (e.g. Cumming et al., 2008; Winn & Fabrycky, 2015;
Fernandes et al., 2019) we arrive at a G-cloud lifetime of order $\sim 10^{5}$
yr. This value is exactly what we find in our calculations. Thus, provided T
Tauri stars in the Galactic Centre can form giant planets at a similar rate to
those studied in the Solar Neighbourhood then our presented scenario can
produce G-clouds at the required rate.

As discussed previously, in order to appear as a G-cloud the disc must exit
the phase where photoevaporation has been shut off by the giant planet’s dust
trap. Below, we consider three possible scenarios and discuss their relative
likelihood.

###  8.1 Draining of the outer disc

The first scenario is the one identified in our simulations, that the gas in
the outer gas disc is destroyed both by draining into the inner disc via
accretion and photoevaporation. This scenario is the most difficult to
calculate as the gas transfer rate across the gap is poorly known, even in the
more standard case of a planet embedded in an extended disc. As discussed in
section 4.1, in our simulations, we parameterised this transfer rate in terms
of the fraction of the mass accretion rate a steady-state disc would provide
with a given surface density. In the majority of our simulations, we found
that accretion draining dominates over photoevaporation (as photoevaporation
is significantly suppressed at high dust-to-gas ratios blacksince the dust
absorbs the EUV photons); hence in our estimates below, we neglect
photoevaporation.

The disc mass in the outer regions trapped outside the planet is given by
approximately:

$M_{d}\sim 2\pi RH_{R}\Sigma$ (20)

where $H_{R}$ is the radial size of the gas disc exterior to the planet. The
accretion draining rate is given by:

$\dot{M}=f_{\dot{M}}3\pi\nu\Sigma$ (21)

hence the depletion timescale is:

$t_{\rm dep}\sim\frac{1}{\alpha
f_{\dot{M}}}\left(\frac{R}{H}\right)\left(\frac{H_{R}}{H}\right)\Omega^{-1}$
(22)

Our simulations indicate that $H_{R}/H\sim 3-5$ (Figure 8), thus for
$\alpha=10^{-3}$ if $f_{\dot{M}}$ is of order a few tenths then the depletion
timescale is a few Myr at around 10 AU.

Thus, if $f_{\dot{M}}$ is of order a few-tenths then it will deplete the outer
discs on the required timescale of a few Myr. If $f_{\dot{M}}$ is $\sim 1$ or
larger, then the discs will deplete too rapidly and if $f_{\dot{M}}\ll 1$ then
discs will not deplete quickly enough. It is unclear if this is a fine-tuning
problem, since $f_{\dot{M}}$ of a few tenths is roughly in agreement with
values provided from simulations of planet-disc interactions (albeit in more
extended discs, Lubow & D’Angelo 2006).

###  8.2 Tidal stripping of the planet by Sgr A*

Since our T Tauri scenario results in an outwardly migrating planet, if the
planet were to migrate to sufficiently large distances it would be tidally
stripped by the black hole, allowing the transition to a G-cloud phase.

The disc interior to the planet is not allowed to accrete, and hence will
behave like a steadily expanding decretion disc (e.g. Pringle, 1991; Nixon &
Pringle, 2021) with a constant outward angular momentum flux of
$F_{J}=3\pi\nu\Sigma h$. This angular momentum flux will be absorbed by the
planet causing it to migrate outwards at a rate:

$\frac{a_{p}}{\dot{a}_{p}}\sim\frac{M_{p}}{M_{\rm d,in}}t_{\nu}$ (23)

We find the tidal stripping scenario to be the least likely scenario for
creating the G-clouds. At a distance of $0.1$ pc from Sgr A* the Hill radius
for a solar-mass star is approximately 80 AU, meaning the planet will be
tidally stripped at a distance of roughly 40 AU. The viscous time-scale at
$\sim 40$ AU is long ($t_{\nu}\sim 1/\alpha(R/H)^{2}\Omega^{-1}$),
necessitating a low-mass planet. However, the planet’s mass cannot be
arbitrarily low. At too low planetary masses the perturbed disc structure is
unable to trap dust exterior to its orbit. This happens at the “pebble-
isolation mass” (Lambrechts et al., 2014), which is closely related to the
planetary thermal mass. For a typical disc around a solar mass T Tauri star,
the disc’s aspect ratio is approximately $H/R\sim 0.07$ at 40 AU. For
$\alpha=10^{-3}$, Bitsch et al. (2018) find the pebble-isolation mass to be
$\sim 0.2$ MJ. Our simulations indicate that accretion is shut off and the
planet begins its outward migration for disc masses of few Jupiter masses,
indicating a minimum planetary migration timescale of $\sim 1~{}$Myr. This
timescale would be just about compatible with the observations. However, it is
unlikely the planet could remain at this low mass for the entire evolution.
sub-Jovian mass planets are subject to run-away accretion in a gas-rich
environment (e.g. Pollack et al., 1996) and can accrete rapidly reaching
Jovian masses, slowing the outward migration. In addition, the
photoevaporation rates for a 40 AU disc are extremely large ($\sim 10^{-6}$ M⊙
yr-1, resulting in a lower disc mass once it is rapidly photoevaporated down
to radii of a few AU and shorter disc lifetimes. Thus, while roughly
compatible with the required observational rates, tidal stripping of the
planet by the black hole might be fine-tuned, unless the progenitor stars have
pericentre distances less than 0.1 pc. Finally, we note that by the time
planet has migrated out to $\sim 40$ AU, the disc/planet system might have
extracted so much angular momentum from the star that it is no longer spinning
rapidly enough to prevent accretion. Ultimately, this may cause the disc to
begin accreting and potentially allow the planet to migrate inwards again.

###  8.3 Removal of the planet via encounters

The Galactic Centre is a reasonably dense stellar environment, with around a
stellar mass of $10^{6}$ M⊙ inside 1 pc (e.g. Genzel et al., 2010). Therefore,
the encounter rate with another object is high. The majority of these objects
are old main sequence stars and compact remnants that have migrated towards
the Galactic Centre (e.g. Miralda-Escudé & Gould, 2000; Alexander & Hopman,
2009; Linial & Sari, 2022). An encounter with a closest approach distance
similar to the orbital separation of the planet is likely to be highly
disruptive. Such close encounters often unbind the planetary companion from
the host star and tidally truncate the disc (e.g. Clarke & Pringle, 1993;
Pfalzner et al., 2005; Malmberg et al., 2007; Davies et al., 2014). Due to the
sheer number of old stars and remnants, they are likely to dominate the close
encounters, rather than other young stars777$H/R$ of the young disc would need
to be $\sim 10^{-3}$ for the local density of young stars to be similar to old
stars.. This population of old stars and remnants is relaxed, and thus has a
velocity dispersion ($\sigma_{d}$) of order the Keplerian speed around the
black hole, significantly higher than the escape velocity of the planet. As
such, the collision cross-section on AU scales or greater is simply the
geometric cross-section (the Safronov number $\ll 1$). In this case, the
encounter rate is $\Gamma_{\rm en}\sim n_{\rm old}\pi b^{2}\sigma_{d}$, where
$n_{\rm old}$ is the density of old stars/remnants and $b$ is the impact
parameter of the encounter. Miralda-Escudé (2012) provide estimates of the
density of old stars and stellar mass black holes in the Galactic Centre,
since their velocity dispersion are similar and the density of old stars
exceeds that of the remnants, close encounters on AU scales are dominated by
those with the old stars. Miralda-Escudé (2012) argued that the relaxed old
stars will approach a $r^{-3/2}$ density profile (in agreement with the
calculations of Vasiliev 2017), thus the encounter rate for an individual T
Tauri star is:

$\Gamma_{\rm en}\sim 1~{}{\rm Myr}^{-1}\left(\frac{r}{0.1~{}{\rm
pc}}\right)^{-2}\left(\frac{b}{5~{}{\rm AU}}\right)^{2}$ (24)

where we have followed Miralda-Escudé (2012) in normalising the old star
density profile assuming that the population consists entirely of 1 M⊙ stars.
Now since an order unity fraction of close encounters with impact parameter
$b$ can strip a planet with an orbital semi-major axis of order $b$, then the
derived encounter rate is sufficient to remove planets on 1-10 AU orbits on
Myr timescales, transitioning the planet-disc system into a G-cloud. The
$b^{2}$ scaling necessitates that encounters that completely destroy the disc,
resulting in too little mass to form a G-cloud are rare. Alternatively,
instead of a single disruptive encounter, repeated longer-distance
interactions with other objects could “evaporate” the star-planet system.
Frequent interactions cause the planet’s semi-major axis to grow by giving
energy to the planet, eventually unbinding it. Rose et al. (2020), estimate
this evaporation timescale to be $\sim 10^{6}$ years for objects with distance
from Sgr A* to be of order 0.1pc. Hence, based on the discussion above, we
suspect that either a single disruptive, close encounter with a nearby old
star or frequent interactions with other stars in the Galactic Centre are the
most likely mechanisms for transitioning the disc into a G-cloud.

##  9 Pressure confinement: the special case of G2

The G2 cloud deserves special attention. G2 is the most extreme of the
identified G-clouds with an eccentricity of $\sim 0.98$ (e.g. Gillessen et
al., 2012; Plewa et al., 2017) resulting in a pericentre distance that takes
it to within 150 AU of the black hole. This extremely close passage has two
important consequences for our scenario. In our scenario, the G-clouds have
lifetimes longer than their Keplerian orbital periods, which range between 100
and 1000’s years (e.g. Gillessen et al., 2012; Ciurlo et al., 2020), thus, it
is likely G2 has performed many pericentre passages already. The Hill radius
for a Solar mass star at a distance of $\sim 150$ AU around a $4\times
10^{6}~{}$M⊙ black hole is only around 1 AU. Repeated pericentre passages are
likely to result in tidal truncation of the disc down to a similar size scale,
with the exact tidal stripping radius depending on the orientation of the
disc’s angular momentum axis relative to its orbital angular momentum axis
around Sgr A*. Figure 2 indicates that a disc with a dust-to-gas ratio of
order $\sim 10^{-4}$ can explain the Br-$\gamma$ flux of G2 with a disc radius
of $\sim 1.5$ AU, slightly larger than the nominal 1 AU truncation radii. We
speculate that this mild tension is resolved by the fact the pericentre
passage of G2 occurs in a fairly high-pressure environment, and as discussed
in Section 3.1, our Br-$\gamma$ luminosities are underestimates for pressure-
confined outflows. The vicinity of Sgr A* contains dense gas, with densities
of $\gtrsim 10^{4}$ cm-3 expected in the vicinity of pericentre passage (e.g.
Xu et al., 2006; Murchikova et al., 2019). While these densities are too low
to result in significant ram-pressure stripping over the G-cloud’s lifetime,
they may provide additional pressure support during pericentre passage
preventing the disc from being tidally truncated to small sizes. High-
resolution simulations of the compact-source scenario for G2, have a minimum
resolution of $\sim 5~{}$AU, thus new, higher resolution simulations are
required to fully explore the truncation of a disc at pericentre passage.

This realisation that the ambient pressure environment in the Galactic Centre
matters raises an important question: can you actually drive a
photoevaporative outflow? To obtain a photoevaporative outflow, the external
pressure at large radii needs to be sufficiently low that the outflow can pass
through a sonic point (e.g. Parker, 1958; Lamers & Cassinelli, 1999). If the
external pressure is too large, the photoionized gas will be held in
hydrostatic equilibrium by the external environment. The external pressure
will be too weak to prevent a hydrodynamic outflow if the hydrostatic pressure
of the photoionized gas from the disc exceeds the external pressure at large
radii, or:

$P_{\rm ext}<P_{w,\infty}\approx P_{b}\exp\left(-\frac{GM_{*}}{c_{s}^{2}R_{\rm
disc}}\right)$ (25)

where $P_{b}$ is the pressure at the base of the photoionized region at the
edge of the disc. In Figure 12,

Figure 12: The solid and dashed lines show the ram pressure (assuming a static
medium) and thermal pressure acting on G2 as a function of time before
pericentre passage. The dot-dashed line shows the thermal pressure at the
photoevaporative wind base and the dotted line shows the ambient pressure to
prevent the wind from escaping for disc sizes of 2 and 5 AU. The shaded
regions show those arising from the 5–95% EUV fluxes shown in Figure 1. We use
fluxes for objects perpendicular to the O/WR disc; choosing the parallel model
does not change the times at which the wind becomes pressure confined. G2’s
spends most of its time able to drive a hydrodynamic photoevaporative wind;
however, on approach to pericentre the wind region becomes pressure confined,
and finally, at pericentre passage, the disc becomes pressure confined.

we show the external thermal and ram pressures (assuming that the external
medium’s relative velocity with G2 is just the Keplerian value of G2 orbit),
adopting the orbital parameters of G2 from Plewa et al. (2017). Additionally,
we use our photoevaporation models and EUV flux estimates to show both the
hydrostatic pressure of the photoionized region at large distances
($P_{w,\infty}$) and the pressure of the base of the photoionized wind. What
this comparison demonstrates is that for most of its life, the external
pressure is too weak to prevent a hydrodynamic outflow. However, once G2
begins its plunge towards Sgr A*, the photoevaporative outflow is pressure
confined a few to 10s years before pericentre passage, with smaller discs
being confined earlier. At pericentre passage, the entire photoionized region
has a pressure lower than the external environment. Hence, at pericentre
passage, it is likely that the photoionized gas around the disc is ram
pressure stripped. It would then take $\sim R_{g}/c_{s}\approx 4~{}$ years for
the photoionized region to be reestablished around the disc. Thus, we
hypothesize the following orbital evolution for G2 in the photoevaporative
disc scenario. At apocentre, the disc behaves like a standard photoevaporating
disc; however, as it falls towards Sgr A* the outflow would be shutoff and
would result in a pressure confined gas cloud of size $\sim 100$ AU around the
disc/star, around 10 years before pericentre passage for a disc of a couple of
AU in size. As G2 approaches pericentre, this gas cloud would be further
pressure confined and tidally stripped. Since the change in external pressure
is fast compared to the sound crossing time of the photoionized gas cloud it
is likely to highly dynamic. At pericentre passage, the photoionized region is
stripped, allowing a new photoionized region to be created on a timescale of a
few years; once G2 is sufficiently far from Sgr A* (after $\sim 10$ years),
the photoevaporative outflow will no longer be pressure confined and wind will
be re-established. Within $\sim 100$ years before the next pericentre passage,
the wind will output a few Earth-masses of ionized gas, sufficient to explain
the Br-$\gamma$ fluxes every orbit (as if the photoionized gas is stripped
every pericentre passage it must be replenished every orbit). We additionally
add the pressure confinement helps ease the tension between the disc’s size
required to match the Br-$\gamma$ flux and truncation radius. Being pressure
confined region results in a higher density at large radii than an expanding
outflow, and as such will give higher Br-$\gamma$ fluxes for the same disc
radius.

This evolutionary sketch perhaps explains a couple of the mysteries of G2. The
pressure-confined gas cloud scenario (without a central compact source) nicely
explains the position-velocity structure of the Br-$\gamma$ emission as G2
approached pericentre (e.g. Schartmann et al., 2015). However, since the
object contains a central source that can reform a pressure-confined
photoionized cloud after pericentre passage, one can explain how a G2 survived
pericentre passage (e.g. Plewa et al., 2017). Thus a photoevaporating disc
behaves like the pressure-confined diffuse gas cloud scenario in the decade
preceding pericentre passage, but like the compact source scenario during
pericentre passage. As mentioned above, the sound crossing time for the
photoionized region is longer than the timescale on which these transitions
are likely to happen. Thus, a 3D simulation must be performed to explore the
dynamical evolution of a disc and its photoionized region as it orbits Sgr A*.

Further, it has been suggested that G2 is potentially slowing in its orbit as
if it is experiencing a drag from the surroundings (e.g. Gillessen et al.,
2019). This is problematic for the compact source scenario as it is difficult
to understand how an object with a mass of $\sim 1$ M⊙ would experience
sufficient drag to slow it down enough for a deviation from the Keplerian
orbit to be observed. However, as described above, we may expect the
photoevaporative wind to be stripped from the star during pericentre passage,
and it takes a few tens of years for the disc to relaunch a new wind. The
photoionized region itself has a mass of only a few Earth masses, sufficiently
low that it could experience a strong deceleration from drag. Thus, we
speculate that a solution to this potential paradox is that the G2’s pressure-
confined photoevaporative region experiences sufficient drag during pericentre
passage to either cause it to trail behind the star or even detach entirely,
making it appear as if G2 has slowed down.

Finally, taking the orbital parameters from Ciurlo et al. (2020), the majority
of the remaining G-clouds do not appear to be orbits that take them
sufficiently close to Sgr A* for them to become pressure confined. Thus, we
suspect they will behave like photoevaporating discs for the entirety of their
lifetimes.

##  10 Discussion

In this work, we have studied the photoevaporating circumstellar disc
hypothesis for the G-clouds in the Galactic Centre. Through the use of
explicit calculation of these discs’ secular evolution, we have shown that
accretion discs around T Tauri stars that formed in the same star formation
epoch that created the young, massive stars cannot survive sufficiently long
enough to be observed as G clouds. This brief lifespan is primarily driven by
the fact that photoevaporation truncates the discs to small radii (several
AU), causing them to accrete rapidly onto their host stars. This process is
well understood in even more benign star-forming environments, where
photoevaporation brings about more rapid disc evolution, not due to the
explicit removal of mass, but the shrinking of the disc’s size causing it to
evolve faster (e.g. Clarke, 2007). Hence, our evolutionary calculations
demonstrate that the G clouds are unlikely to be circumstellar discs around
young T Tauri stars that follow the typical evolutionary pathway identified in
nearby star-forming environments. Since the photoevaporating disc scenario for
the G-clouds has many desirable characteristics when compared to observations
(e.g. Murray-Clay & Loeb, 2012; Miralda-Escudé, 2012), we instead propose that
the G-clouds are not photoevaporating accretion discs, but rather
photoevaporating decretion discs, where accretion is prevented by a large
magnetic torque from the central star.

###  10.1 The G-clouds as photoevaporating decretion discs around young stars

Young T Tauri stars are well known to host strong magnetic fields. In our
G-cloud scenario where the decretion disc is created by a young star, we have
speculated that the removal of disc mass by photoevaporation results in the
young star spinning rapidly at an age of a few Myr. This is because as the
star contracts towards the main sequence, it accretes less material than in a
typical star-forming environment, allowing it to approach breakup velocity.
This means that the star is able to magnetically disrupt the disc outside co-
rotation while the disc’s mass is still significant ($\gtrsim 10^{-3}$M∗). The
magnetic torque prevents accretion and results in the disc expanding as a
decretion disc (e.g. Livio & Pringle, 1992), where both angular momentum and
mass are transported outwards. Without any additional processes, this outward
transport of mass causes the disc to undergo more vigorous photoevaporation
resulting in disc lifetimes that are too short to explain the G-clouds.
However, a forming giant planet, which can trap dust exterior to its orbit,
significantly suppresses photoevaporation once the dust-to-gas mass ratios
exceed $\sim 0.1$ exterior to the planet. The suppression of photoevaporation
occurs because dust particles dominate the absorption of EUV photons,
resulting in low densities of photoionized gas and weak mass-loss rates. At
this stage, these non-accreting, non-photoevaporating discs can survive for
$\gtrsim 5-10$ Myr as the planet migrates outwards slowly.

Ultimately, we suspect encounters with old stars in the Galactic Centre will
eject or unbind the planet, removing the dust trap, and allowing the disc to
begin photoevaporating again. It is this photoevaporating remnant we
hypothesise that creates the G-cloud signature, and they have lifetimes of
$\sim 10^{5}$ yrs.

####  10.1.1 Planet formation in Galactic Centre

In order for the T Tauri star scenario to work, we need a planet to form in
the disc that is sufficiently massive to trap dust exterior to its orbit
($\gtrsim 0.2$ MJ). While the required fraction of discs that form such
planets is consistent with both the giant planet occurrence rate on AU scales
and the fraction of protoplanetary discs that appear as “transition” discs in
nearby star-forming regions, it’s unclear whether planet formation in the
Galactic Centre will proceed in a similar manner. The tidal effects of the
black hole or the impulses injected from more frequent close encounters may
increase the velocity dispersion in any planetesimals, reducing the efficiency
of planetary core formation. Alternatively, the smaller disc radii would
result in more rapid dust growth at the disc’s outer edge, giving rise to a
higher pebble flux (provided significant dust mass is not lost to
photoevaporation early – Haworth et al. 2018a), resulting in more efficient
planetary core formation.

Additionally, in the early stages of the disc’s lifetime, while it’s still
self-gravitating, planet formation via direct collapse could occur. A disc
that is stable to fragmentation and planet formation, can be made to undergo
fragmentation and planet formation by a tidal perturbation (e.g. Pfalzner &
Olczak, 2007; Meru, 2015; Cadman et al., 2022). This mechanism is particularly
relevant for our scenario, as it causes fragmentation in the inner regions of
the disc (Meru, 2015), allowing giant planet formation to occur close-in,
early in the disc’s lifetime. We speculate that both, the tidal forcing from
the black hole and stellar flybys on 50 AU scales which occur frequently
(Equation 24), could result in enhanced giant planet formation through
gravitational fragmentation. Ultimately, planet formation simulations that
account for the unique properties of the environment need to be performed
before one can assess whether giant planet formation is more or less likely in
the Galactic Centre.

###  10.2 The merger scenario

The stellar merger scenario has been suggested as a possible origin for the
G-clouds (Witzel et al., 2014; Prodan et al., 2015; Ciurlo et al., 2020),
where the dense Galactic Centre environment causes binaries to merge
frequently. Most of these works focus on calculating the properties and
frequency of such mergers, what is not clear is how exactly such a merger
product creates a G-cloud signature. It could either be the merger product
itself or, as we explore here, a photoevaporating decretion disc. Thus, our
discussion and rate calculations are focused on exclusively on disc origin.
Provided the merger remnant created is rapidly spinning and magnetised, then
any bound high-angular momentum material left over in the collision is likely
to form a disc. Like the T Tauri scenario, the strong magnetic torque prevents
the disc from accreting, causing it to expand and photoevaporate. We show that
such a disc could live for a few $10^{5}$ years, potentially explaining the
G-cloud populations. If the progenitor binaries formed in the most recent
star-forming episode, we estimate roughly $\sim 100$ potential binaries
formed, a sufficient population to explain the G-clouds if they can all be
made to merge in a few Myrs. While current calculations do not indicate such
high merger rates (e.g. Stephan et al., 2016), there are other dynamical
processes that could be included to enhance the merger rate.

###  10.3 Limitations and further work

The goal of this work has been to explore the photoevaporating circumstellar
disc scenario as the origin of the G-clouds. In order to span such a large
range of possible different individual pathways, we have necessarily
simplified our approach. We have focused on simple 1D models of both the
photoevaporative flow structure and the secular evolution of the discs.
blackWe did not consider the role of radiation pressure on dust in the
photoevaporative outflows. Using the bolometric fluxes expected in the
Galactic Centre (Section 3.1.1) the typical radiation acceleration (for $Q\sim
1$) on a 0.1$\mu$m size dust grain is 0.1 cm s-2. This is comparable to the
gravitational acceleration from a Solar-mass star on AU scales. However, in
the case of an approximately isotropic radiation field (as applicable in the
Galactic Centre) this radiative acceleration will be significantly reduced. In
addition, the stopping time of the dust is $\sim 10^{4}-10^{5}$ s at the sonic
point, significantly shorter than the flow timescale ($\sim 10^{7}-10^{8}$ s)
meaning the dust will be tightly coupled to the gas, and the net radiative
acceleration reduced by the dust-to-gas ratio. Thus, we do not expect
radiation pressure to be able to affect photoevaporation at low dust-to-gas
ratios, although it may affect the dust size distribution. At high-dust-to-gas
ratios it may help shut down photoevaporation; however, since photoevaporation
drops off at high dust-to-gas ratios due to the absorption of EUV photons,
neglecting radiation pressure is unlikely to affect our evolutionary
calculations. No photoevaporation models applicable for compact discs in the
high EUV environment of the Galactic Centre had been calculated; since we
added the complication of self-consistently including entrained dust into
these photoevaporation calculations we took a simplified approach. We
considered the photoevaporative flow structure to consist of two isothermal
winds, an FUV-heated region that transitions through an ionization front into
a EUV-heated outflow. While the isothermal approximation for the EUV heated
outflow is probably acceptable, the FUV heated region can have a complicated
thermal structure (Adams et al., 2004; Haworth et al., 2018b) with feedback
between the entrained dust and outflow structure (Facchini et al., 2016).
While the size of the FUV heated region only provides an order unity
correction to the mass-loss rates (Johnstone et al., 1998); future, work
should consider more realistic temperature and flow structures. This would
allow both more accurate mass-loss rates to be obtained and more detailed
observational signatures to be computed (such as the metal lines).
Additionally, we have assumed the outflows are exclusively photoevaporative.
Simulations of discs in the propeller regime (e.g. Romanova et al., 2004;
Ustyugova et al., 2006) have additionally shown that material could be ejected
from the inner disc in the form of a wind. Several authors (e.g. Scoville &
Burkert, 2013; Ballone et al., 2018) have considered the G-cloud signature to
arise from winds from a young star. Future modelling should include both this
potential wind component in addition to the photoevaporative wind.

One weakness inherent in any secular evolutionary calculation of externally
photoevaporating discs, such as those performed in this work, is how to match
the outflow solution onto the standard thin-disc evolutionary model, and how
to calculate dust entrainment (see discussion in, Sellek et al., 2020). Our
approach to dust entrainment was to estimate the dust-to-gas ratio at the
outer edge by measuring it in a few cells close to the point where
photoevaporation truncates the disc providing a reasonable first approach.
Recently, Owen & Altaf (2021) used a “slim-disc” model to show that you can
self-consistently compute the flow structure from the disc into the outflow
and any size-dependent entrainment of dust grains. However, these calculations
were performed in steady-state, and no evolutionary method for the slim-disc
approach is available. Since the entrainment of dust grains is critical to the
evolution of the disc calculations presented here, this should provide such
motivation to develop such an evolutionary model. Additionally, we neglected
the dynamical feedback of the dust on the gas which can become important when
the dust-to-gas ratio exceeds unity (provided the dust’s stopping time is
sufficiently short). This leads to localised regions in our disc calculations
where the dust-to-gas ratio reaches large values in the centre of the pressure
trap. While this is probably not physical for the extremes found in the
calculation it is unlikely to strongly affect our results. At high dust-to-gas
ratios, the advection of dust into the centre of the dust trap will expel gas
(by momentum conservation) weakening the strength of the trap and resulting in
a more uniform dust-to-gas mixture in the disc exterior to the planet. This
process will reduce the dust-to-gas ratio in the centre of the dust-trap, but
actually, enhance the dust-to-gas ratio at the edges of the pressure trap.
This is the region where the photevaporative wind is entraining dust
particles; thus we suspect the inclusion of dust feedback on the gas would
actually decrease the photoevaporation rates from the outer edge when the dust
trap is present.

Finally, the T Tauri star and planet scenario may have reconciled the
difference between the pressure-confined diffuse cloud scenario and compact
source scenario for G2, by demonstrating that even though a photoevaporating
disc is a compact source scenario, while G2 has been observed over the last
decade its likely behaved in a manner similar to the pressure confined diffuse
cloud scenario. This is because the external pressure within the central 0.01
pc is sufficiently high to pressure confine a photoevaporative wind. However,
we have still presented no scenario as to how the progenitor disc of G2 ended
up on such a highly eccentric orbit. As argued by Miralda-Escudé (2012) single
scattering events are likely to be highly distributive and could destroy the
disc, and it’s unlikely we’ve just happened to catch G2 on its first plunge
toward Sgr A*. One way to increase the eccentricity of G2, without a
disruptive single event would be through a secular process; for example
through secular resonance sweeping (e.g. Zheng et al., 2020). This possibility
should be explored in further work.

##  11 Summary

We have studied the hypothesis that the Galactic Centre G-clouds are
photoevaporating circumstellar discs. To calculate the photoevaporation rates,
and subsequent emission properties, such as the Br-$\gamma$ flux, we have
computed a new suite of external photoevaporation models in the EUV-driven
regime. These new models account for the sub-sonic launching of the outflow
and account for the entrainment of small dust particles. These models indicate
shut-down in photoevaporation at dust-to-gas mass ratios that exceed $\sim
0.1$. This drop in the photoevaporation rates occurs because the dust
particles themselves absorb the incident EUV photons, reducing the density of
photoionized gas. These photoevaporation flow structures indicate that mass-
loss rates of order $10^{-8}-10^{-7}$ M⊙ yr-1 are required to explain the
observational characteristics of the G-clouds.

By coupling these new photoevaporation models to numerical evolutionary
calculations of both the gas and dust components of circumstellar discs, we
have explored various hypothesises for the origin of G-clouds. An accreting
circumstellar disc around a young low-mass star formed a few Myr’s ago in the
most recent star formation episode in the Galactic Centre is ruled out. This
arises, not because photoevaporation is able to entirely strip the disc, but
rather because photoevaporation truncates the edge causing it to accrete
rapidly onto the central star on a timescale of $\lesssim 1~{}$Myr.

We demonstrate that is unlikely that the G-clouds are any form of accreting,
photoevaporating circumstellar disc, including those formed recently through a
merger. This is because of photoevaporative truncation of the disc, which
promotes rapid accretion and destruction of the disc. In both the young star
and merger hypothesis there is unlikely to be a large enough source population
of progenitors to explain the fact we currently observe 5-10 G-clouds.
blackInstead, if G-clouds are circumstellar discs a large magnetic torque from
the central star is required to prevent accretion, causing the disc to evolve
as a decretion disc transporting material outwards into the photoevaporating
wind.

If the G-cloud progenitors are young T Tauri stars, to overcome the lifetime
issue, photoevaporation must be suppressed. If the disc around a T Tauri star
formed a giant planet, which would trap dust exterior to orbit,
photoevaporation and accretion could both be suppressed for Myr timescales
while the planet slowly migrates outwards. Once the planet is ejected by
encounters, which occur on Myr timescales per disc, then the star/disc system
would enter a photoevaporating decretion disc that would last up to a few
$10^{5}$ years, while having observational signatures reminiscent of the
G-clouds. If T Tauri stars form giant planets at a similar rate to the stars
in the Solar Neighbourhood, then this scenario can match the occurrence rate
of G-clouds we observe in the Galactic Centre today. Finally, we demonstrate
that if G2 can be explained as a photoevaporating disc, it is likely that
during the last few decades of observations, on its approach to Sgr A* its
photoevaporating outflow was pressure confined. In this scenario, G2 survival
of pericentre passage is explained by the compact-source scenario, while its
observational characteristics would match the pressure-confined diffuse cloud
scenario.

## Acknowledgements

We thank the anonymous referee for comments which improved the manuscript. We
are grateful to Andreas Burkert and Smadar Naoz for interesting discussions.
JEO is supported by a Royal Society University Research Fellowship. This work
benefited from the 2022 Exoplanet Summer Program in the Other Worlds
Laboratory (OWL) at the University of California, Santa Cruz, a program funded
by the Heising-Simons Foundation. For the purpose of open access, the authors
have applied a Creative Commons Attribution (CC-BY) licence to any Author
Accepted Manuscript version arising.

## Data Availability

The disc evolution code used in this work is freely available at
https://bitbucket.org/jo276/viscous_panda/. The simulation results themselves
will be shared on reasonable request to the corresponding author.

## References

  * Adams (2010) Adams F. C., 2010, ARA&A, 48, 47
  * Adams et al. (2004) Adams F. C., Hollenbach D., Laughlin G., Gorti U., 2004, ApJ, 611, 360
  * Alexander & Armitage (2009) Alexander R. D., Armitage P. J., 2009, ApJ, 704, 989
  * Alexander & Hopman (2009) Alexander T., Hopman C., 2009, ApJ, 697, 1861
  * Alexander & Pascucci (2012) Alexander R. D., Pascucci I., 2012, MNRAS, 422, L82
  * Ali (2021) Ali A. A., 2021, MNRAS, 501, 4136
  * Ali et al. (2020) Ali B., et al., 2020, ApJ, 896, 100
  * Andrews et al. (2011) Andrews S. M., Wilner D. J., Espaillat C., Hughes A. M., Dullemond C. P., McClure M. K., Qi C., Brown J. M., 2011, ApJ, 732, 42
  * Armitage & Clarke (1996) Armitage P. J., Clarke C. J., 1996, MNRAS, 280, 458
  * Ballone et al. (2013) Ballone A., et al., 2013, ApJ, 776, 13
  * Ballone et al. (2018) Ballone A., et al., 2018, MNRAS, 479, 5288
  * Bally et al. (1998) Bally J., Sutherland R. S., Devine D., Johnstone D., 1998, AJ, 116, 293
  * Bartko et al. (2009) Bartko H., et al., 2009, ApJ, 697, 1741
  * Bartko et al. (2010) Bartko H., et al., 2010, ApJ, 708, 834
  * Bertoldi & McKee (1990) Bertoldi F., McKee C. F., 1990, ApJ, 354, 529
  * Birnstiel et al. (2012) Birnstiel T., Klahr H., Ercolano B., 2012, A&A, 539, A148
  * Bitsch et al. (2018) Bitsch B., Morbidelli A., Johansen A., Lega E., Lambrechts M., Crida A., 2018, A&A, 612, A30
  * Booth & Owen (2020) Booth R. A., Owen J. E., 2020, MNRAS, 493, 5079
  * Bouvier et al. (2007) Bouvier J., Alencar S. H. P., Harries T. J., Johns-Krull C. M., Romanova M. M., 2007, in Reipurth B., Jewitt D., Keil K., eds, Protostars and Planets V. p. 479 (arXiv:astro-ph/0603498)
  * Bruderer et al. (2014) Bruderer S., van der Marel N., van Dishoeck E. F., van Kempen T. A., 2014, A&A, 562, A26
  * Burkert et al. (2012) Burkert A., Schartmann M., Alig C., Gillessen S., Genzel R., Fritz T. K., Eisenhauer F., 2012, ApJ, 750, 58
  * Cadman et al. (2022) Cadman J., Hall C., Fontanive C., Rice K., 2022, MNRAS, 511, 457
  * Canup (2004) Canup R. M., 2004, ARA&A, 42, 441
  * Choi et al. (2016) Choi J., Dotter A., Conroy C., Cantiello M., Paxton B., Johnson B. D., 2016, ApJ, 823, 102
  * Ciurlo et al. (2020) Ciurlo A., et al., 2020, Nature, 577, 337
  * Clarke (2007) Clarke C. J., 2007, MNRAS, 376, 1350
  * Clarke & Pringle (1993) Clarke C. J., Pringle J. E., 1993, MNRAS, 261, 190
  * Clénet et al. (2005) Clénet Y., Rouan D., Gratadour D., Marco O., Léna P., Ageorges N., Gendron E., 2005, A&A, 439, L9
  * Cranmer (2004) Cranmer S. R., 2004, American Journal of Physics, 72, 1397
  * Cuadra et al. (2008) Cuadra J., Armitage P. J., Alexander R. D., 2008, MNRAS, 388, L64
  * Cumming et al. (2008) Cumming A., Butler R. P., Marcy G. W., Vogt S. S., Wright J. T., Fischer D. A., 2008, PASP, 120, 531
  * D’Angelo et al. (2002) D’Angelo G., Henning T., Kley W., 2002, A&A, 385, 647
  * Davies et al. (2014) Davies M. B., Adams F. C., Armitage P., Chambers J., Ford E., Morbidelli A., Raymond S. N., Veras D., 2014, in Beuther H., Klessen R. S., Dullemond C. P., Henning T., eds, Protostars and Planets VI. p. 787 (arXiv:1311.6816), doi:10.2458/azu_uapress_9780816531240-ch034
  * De Colle et al. (2014) De Colle F., Raga A. C., Contreras-Torres F. F., Toledo-Roy J. C., 2014, ApJ, 789, L33
  * Do et al. (2013) Do T., Lu J. R., Ghez A. M., Morris M. R., Yelda S., Martinez G. D., Wright S. A., Matthews K., 2013, ApJ, 764, 154
  * Dotter (2016) Dotter A., 2016, ApJS, 222, 8
  * Eckart & Genzel (1997) Eckart A., Genzel R., 1997, MNRAS, 284, 576
  * Eckart et al. (2013) Eckart A., et al., 2013, A&A, 551, A18
  * Eisenhauer et al. (2005) Eisenhauer F., et al., 2005, ApJ, 628, 246
  * Facchini et al. (2016) Facchini S., Clarke C. J., Bisbas T. G., 2016, MNRAS, 457, 3593
  * Fatuzzo & Adams (2008) Fatuzzo M., Adams F. C., 2008, ApJ, 675, 1361
  * Fernandes et al. (2019) Fernandes R. B., Mulders G. D., Pascucci I., Mordasini C., Emsenhuber A., 2019, ApJ, 874, 81
  * Gautham Bhaskar et al. (2022) Gautham Bhaskar H., Li G., Lin D. N. C., 2022, arXiv e-prints, p. arXiv:2204.07282
  * Genzel & Townes (1987) Genzel R., Townes C. H., 1987, ARA&A, 25, 377
  * Genzel et al. (2003) Genzel R., et al., 2003, ApJ, 594, 812
  * Genzel et al. (2010) Genzel R., Eisenhauer F., Gillessen S., 2010, Reviews of Modern Physics, 82, 3121
  * Ghez et al. (1998) Ghez A. M., Klein B. L., Morris M., Becklin E. E., 1998, ApJ, 509, 678
  * Ghez et al. (2003) Ghez A. M., et al., 2003, ApJ, 586, L127
  * Ghez et al. (2005a) Ghez A. M., Salim S., Hornstein S. D., Tanner A., Lu J. R., Morris M., Becklin E. E., Duchêne G., 2005a, ApJ, 620, 744
  * Ghez et al. (2005b) Ghez A. M., et al., 2005b, ApJ, 635, 1087
  * Ghosh & Lamb (1979a) Ghosh P., Lamb F. K., 1979a, ApJ, 232, 259
  * Ghosh & Lamb (1979b) Ghosh P., Lamb F. K., 1979b, ApJ, 234, 296
  * Gillessen et al. (2009) Gillessen S., Eisenhauer F., Trippe S., Alexander T., Genzel R., Martins F., Ott T., 2009, ApJ, 692, 1075
  * Gillessen et al. (2012) Gillessen S., et al., 2012, Nature, 481, 51
  * Gillessen et al. (2013) Gillessen S., et al., 2013, ApJ, 763, 78
  * Gillessen et al. (2019) Gillessen S., et al., 2019, ApJ, 871, 126
  * Gravity Collaboration et al. (2018) Gravity Collaboration et al., 2018, A&A, 615, L15
  * Hartmann (2002) Hartmann L., 2002, ApJ, 566, L29
  * Hartmann et al. (1998) Hartmann L., Calvet N., Gullbring E., D’Alessio P., 1998, ApJ, 495, 385
  * Hartmann et al. (2016) Hartmann L., Herczeg G., Calvet N., 2016, ARA&A, 54, 135
  * Haworth & Clarke (2019) Haworth T. J., Clarke C. J., 2019, MNRAS, 485, 3895
  * Haworth et al. (2018a) Haworth T. J., Facchini S., Clarke C. J., Mohanty S., 2018a, MNRAS, 475, 5460
  * Haworth et al. (2018b) Haworth T. J., Clarke C. J., Rahman W., Winter A. J., Facchini S., 2018b, MNRAS, 481, 452
  * Haworth et al. (2022) Haworth T. J., et al., 2022, MNRAS, 512, 2594
  * Henney & O’Dell (1999) Henney W. J., O’Dell C. R., 1999, AJ, 118, 2350
  * Hollenbach & Tielens (1999) Hollenbach D. J., Tielens A. G. G. M., 1999, Reviews of Modern Physics, 71, 173
  * Johnstone et al. (1998) Johnstone D., Hollenbach D., Bally J., 1998, ApJ, 499, 758
  * Kama et al. (2015) Kama M., Folsom C. P., Pinilla P., 2015, A&A, 582, L10
  * Kenyon & Hartmann (1987) Kenyon S. J., Hartmann L., 1987, ApJ, 323, 714
  * Keppler et al. (2019) Keppler M., et al., 2019, A&A, 625, A118
  * Koenigl (1991) Koenigl A., 1991, ApJ, 370, L39
  * Krabbe et al. (1995) Krabbe A., et al., 1995, ApJ, 447, L95
  * Kuiper et al. (2020) Kuiper R., Yorke H. W., Mignone A., 2020, ApJS, 250, 13
  * Lambrechts et al. (2014) Lambrechts M., Johansen A., Morbidelli A., 2014, A&A, 572, A35
  * Lamers & Cassinelli (1999) Lamers H. J. G. L. M., Cassinelli J. P., 1999, Introduction to Stellar Winds. Cambridge University Press
  * Lee & Chiang (2017) Lee E. J., Chiang E., 2017, ApJ, 842, 40
  * Levin & Beloborodov (2003) Levin Y., Beloborodov A. M., 2003, ApJ, 590, L33
  * Li et al. (2021) Li Y.-P., Chen Y.-X., Lin D. N. C., Zhang X., 2021, ApJ, 906, 52
  * Lin & Papaloizou (1986) Lin D. N. C., Papaloizou J., 1986, ApJ, 309, 846
  * Linial & Sari (2022) Linial I., Sari R., 2022, arXiv e-prints, p. arXiv:2206.14817
  * Livio & Pringle (1992) Livio M., Pringle J. E., 1992, MNRAS, 259, 23P
  * Lovelace et al. (1999) Lovelace R. V. E., Romanova M. M., Bisnovatyi-Kogan G. S., 1999, ApJ, 514, 368
  * Lu et al. (2009) Lu J. R., Ghez A. M., Hornstein S. D., Morris M. R., Becklin E. E., Matthews K., 2009, ApJ, 690, 1463
  * Lu et al. (2013) Lu J. R., Do T., Ghez A. M., Morris M. R., Yelda S., Matthews K., 2013, ApJ, 764, 155
  * Lubow & D’Angelo (2006) Lubow S. H., D’Angelo G., 2006, ApJ, 641, 526
  * Lynden-Bell & Pringle (1974) Lynden-Bell D., Pringle J. E., 1974, MNRAS, 168, 603
  * Malmberg et al. (2007) Malmberg D., de Angeli F., Davies M. B., Church R. P., Mackey D., Wilkinson M. I., 2007, MNRAS, 378, 1207
  * Martins et al. (2007) Martins F., Genzel R., Hillier D. J., Eisenhauer F., Paumard T., Gillessen S., Ott T., Trippe S., 2007, A&A, 468, 233
  * Meru (2015) Meru F., 2015, MNRAS, 454, 2529
  * Meyer & Meyer-Hofmeister (2012) Meyer F., Meyer-Hofmeister E., 2012, A&A, 546, L2
  * Miralda-Escudé (2012) Miralda-Escudé J., 2012, ApJ, 756, 86
  * Miralda-Escudé & Gould (2000) Miralda-Escudé J., Gould A., 2000, ApJ, 545, 847
  * Murchikova et al. (2019) Murchikova E. M., Phinney E. S., Pancoast A., Blandford R. D., 2019, Nature, 570, 83
  * Murray-Clay & Loeb (2012) Murray-Clay R. A., Loeb A., 2012, Nature Communications, 3, 1049
  * Naoz (2016) Naoz S., 2016, ARA&A, 54, 441
  * Naoz et al. (2018) Naoz S., Ghez A. M., Hees A., Do T., Witzel G., Lu J. R., 2018, ApJ, 853, L24
  * Nayakshin & Sunyaev (2005) Nayakshin S., Sunyaev R., 2005, MNRAS, 364, L23
  * Nixon & Pringle (2021) Nixon C. J., Pringle J. E., 2021, New Astron., 85, 101493
  * Owen (2014) Owen J. E., 2014, ApJ, 789, 59
  * Owen (2016) Owen J. E., 2016, Publ. Astron. Soc. Australia, 33, e005
  * Owen & Altaf (2021) Owen J. E., Altaf N., 2021, MNRAS, 508, 2493
  * Parker (1958) Parker E. N., 1958, ApJ, 128, 664
  * Parker (1965) Parker E. N., 1965, Space Sci. Rev., 4, 666
  * Paumard et al. (2001) Paumard T., Maillard J. P., Morris M., Rigaut F., 2001, A&A, 366, 466
  * Paumard et al. (2006) Paumard T., et al., 2006, ApJ, 643, 1011
  * Paxton et al. (2011) Paxton B., Bildsten L., Dotter A., Herwig F., Lesaffre P., Timmes F., 2011, ApJS, 192, 3
  * Paxton et al. (2013) Paxton B., et al., 2013, ApJS, 208, 4
  * Paxton et al. (2015) Paxton B., et al., 2015, ApJS, 220, 15
  * Peißker et al. (2020) Peißker F., Hosseini S. E., Zajaček M., Eckart A., Saalfeld R., Valencia-S. M., Parsa M., Karas V., 2020, A&A, 634, A35
  * Peißker et al. (2021) Peißker F., et al., 2021, ApJ, 923, 69
  * Pfalzner & Olczak (2007) Pfalzner S., Olczak C., 2007, A&A, 462, 193
  * Pfalzner et al. (2005) Pfalzner S., Vogel P., Scharwächter J., Olczak C., 2005, A&A, 437, 967
  * Pfuhl et al. (2015) Pfuhl O., et al., 2015, ApJ, 798, 111
  * Pinilla et al. (2012) Pinilla P., Benisty M., Birnstiel T., 2012, A&A, 545, A81
  * Plewa et al. (2017) Plewa P. M., et al., 2017, ApJ, 840, 50
  * Pollack et al. (1996) Pollack J. B., Hubickyj O., Bodenheimer P., Lissauer J. J., Podolak M., Greenzweig Y., 1996, Icarus, 124, 62
  * Pols et al. (1995) Pols O. R., Tout C. A., Eggleton P. P., Han Z., 1995, MNRAS, 274, 964
  * Pringle (1981) Pringle J. E., 1981, ARA&A, 19, 137
  * Pringle (1991) Pringle J. E., 1991, MNRAS, 248, 754
  * Pringle et al. (1986) Pringle J. E., Verbunt F., Wade R. A., 1986, MNRAS, 221, 169
  * Prodan et al. (2015) Prodan S., Antonini F., Perets H. B., 2015, ApJ, 799, 118