2022

Barbara Ercolano

These authors contributed equally to this work.

Universitäts-Sternwarte, Ludwig-Maximilians-Universität München, Scheinerstr.
1, München, 81679, Bayern, Germany

# Modelling photo-evaporation in planet forming discs

<EMAIL_ADDRESS>Giovanni Picogna<EMAIL_ADDRESS>*

###### Abstract

Planets are born from the gas and dust discs surrounding young stars.
Energetic radiation from the central star can drive thermal outflows from the
discs atmospheres, strongly affecting the evolution of the discs and the
nascent planetary system. In this context several numerical models of varying
complexity have been developed to study the process of disc photoevaporation
from their central stars. We describe the numerical techniques, the results
and the predictivity of current models and identify observational tests to
constrain them.

###### keywords:

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##  1 Introduction

The formation and evolution of planetary systems is strongly coupled to the
evolution and final dispersal of the protoplanetary discs in which they form.
This is driven to a large extent by irradiation from the central star, which
provides heating, ionization (thus magnetic field coupling) and can trigger
thermal outflows that disperse the disc material.

The dispersal of the gas disc imposes a final timescale within which giant
planets must form and is of crucial importance for the final architecture of
planetary systems. Photoevaporation operates by first opening a gap in the
disc, which allows the inner disc to drain onto the central star while the
outer disc is dispersed from the inside out. This provides a natural parking
mechanism for migrating planets (Alexander2012, ; ErcolanoRosotti2015, ;
Jennings2018, ; Monsch2019, ) as well as influencing the post-disc dynamical
evolution of the orbits (Moeckel2012, ; Liu2022, ). The final distribution of
the semi-major axis of giant planets in a population is extremely sensitive to
the mass-loss profile of the disc (ErcolanoRosotti2015, ; Jennings2018, ).

The role of photoevaporation in the early phases of planet formation, e.g. the
formation of planetesimals by the streaming instability is instead debated
(Ercolano2017, ; Carrera2017, ). For the streaming instability to occur a high
solid to gas ratio is needed. Photoevaporation is usually invoked as a
mechanism that preferentially removes gas, thus helping to locally increase
the dust-to-gas ratio. While (Ercolano2017, ) test this hypothesis and find
that the effect of photoevaporation on the total planetesimal mass is
negligible, the opposite conclusion is obtained by (Carrera2017, ), using a
similar method. The reason for this discrepancy lies in the different
photoevaporation mass-loss profiles assumed in these two different works.

Realistic mass-loss profiles are also a key ingredient in planet and disc
population synthesis models (e.g. Emsenhuber2021a, ; Emsenhuber2021b, ;
Manara2018, ), while a reliable disc surface density evolution consistent with
the thermodynamical wind structure is crucial for the interpretation of
observations of transition discs (Rosotti2015, ; Owen2016, ; Ercolano2017b, ;
Schaefer2022, ) and gas emission lines and dust observation (ErcolanoOwen2010,
; ErcolanoOwen2016, ; Weber2020, ; Franz2022a, ; Franz2022b, ).

In this context several theoretical and numerical models of the
photoevaporation process in planet-forming discs have been developed over the
years, having as the main objective the determination of accurate spatially-
resolved mass-loss rates.

Recent theoretical and observational advancements (see reviews
ErcolanoPascucci2017, ; Pascucci2022, ) which point to disc winds of magnetic
nature (MHD winds) in addition to thermal (photoevaporative) winds have added
to the urgency of developing quantitatively predictive thermal wind models in
order to distinguish them in the observations (Weber2020, ; Ricci2021, ). This
is of paramount importance to assess the relevance of MHD winds to angular
momentum extraction from discs.

In this paper, we will review the current stand of numerical photoevaporation
models. Special attention will be given to the applicability of the different
models, as well as to the divergence of the results. Present and future
observational tests of the models are also reviewed.

##  2 Numerical modelling

Disc photoevaporation is a coupled problem in radiative transfer,
thermochemistry, hydrodynamics and dust dynamics. Until recently, it was
deemed impossible to combine all these effects into comprehensive numerical
simulations, but in recent years it has been shown that the codes and
numerical capabilities are available (WangGoodman2017, ; Nakatani2018a, ),
though still with significant limitations in the radiative transfer.
Theoretical photoevaporation models have been developed for one or a
combination of three different portion of the stellar spectra that are capable
of driving a disc thermal wind: the far-UV (FUV)
$6\mathrm{eV}<h\nu<13.6\mathrm{eV}$; extreme-UV (EUV),
$13.6\mathrm{eV}<h\nu<100\mathrm{eV}$; and X-rays,
$100\mathrm{eV}<h\nu<10\mathrm{keV}$ (in particular its soft component
$100\mathrm{eV}<h\nu<1\mathrm{keV}$). Each of them presents its characteristic
features and numerical challenges, so it is important to distinguish between
them.

#### EUV radiation

EUV photons are defined as energetic enough to ionize atomic hydrogen from its
atomic state ($h\nu>13.6\,\mathrm{eV}$). Since the ionization cross-section
for neutral hydrogen is $10^{-17}\mathrm{cm}^{2}\mathrm{atom}^{-1}$, an EUV
photon will be absorbed after passing a neutral column density of
$N_{H}=10^{17}\,\mathrm{cm}^{-2}$. The interaction between a EUV photon and a
hydrogen atom is straightforward. The photon is absorbed, producing an ionized
hydrogen atom and thermal energy. Typical temperatures in ionized regions are
$\sim 10^{4}\,\mathrm{K}$. The production of the electron-ion pair is offset
by recombinations. Those to the electronic ground state produce a photon with
energy larger than 13.6 eV which can further ionize the medium, producing a
diffuse field. The recombinations to an excited state instead lead to the
destruction of the ionizing photon. In the case of irradiation by a central
star of a primordial disc with typical scale heights, photons from the central
star generate a hot hydrostatic atmosphere of $\sim 10^{4}\,\mathrm{K}$ above
the inner disc. It is then the diffuse field of recombination photons from
this atmosphere that irradiates and ionizes the outer disc. This problem has
to be solved as a 2D radiative transfer, which takes into account the direct
and diffuse ionizing fields while solving the photoionization problem.

#### FUV radiation

FUV photons have energies in the range of $6-13.6$ eV, they cannot ionize
hydrogen, but they effectively photodissociate molecular hydrogen, CO, and
other molecules, and ionize Carbon. Photoelectric heating from dust grains and
polycyclic aromatic hydrocarbons (PAHs) is the dominant gas heating mechanism
for FUV photons. Heating is proportional to the grain surface, thus in the
presence of abundant small grains or PAHs in the disc atmosphere, FUVs can
heat the gas and potentially contribute to driving a thermal wind (Gorti2009,
). The gas radiates only less than $1\%$ of the energy processed by the dust,
and it is heated mainly collisionally by photoelectrons from small dust grains
and PAHs, or by vibrationally excited H2. Thus its temperature is impacted by
both dust evolution and chemistry (Gorti2015, ; WangGoodman2017, ). FUV
photons can heat a primarily neutral layer of H and H2 to temperatures of
order $10-5000$ K, depending on the magnitude of the flux, the density of the
gas, and the chemistry. In order to correctly model this problem one has to
combine a 2D radiative transfer, with a sufficiently large chemical network to
account for the strong variation in density and temperature, and a proper dust
evolution model. Another complication arises from the fact that the component
of the stellar FUV field due to accretion onto the central star is can
dominate over the chromospheric component. This means that the FUV flux
reaching the disc, and thus the wind mass-loss rates, strongly depend on the
accretion rate which decreases with time and can stop completely once an inner
cavity is formed.

#### X-ray radiation

X-rays can be divided into two main components: the soft X-rays
($100\,\mathrm{eV}<h\nu<1\,\mathrm{keV}$), which are absorbed at a column
density of $\sim 10^{22}$ pp/cm2, heating up the disc’s upper layers where
they drive a thermal wind; the hard X-rays
($1\,\mathrm{keV}<h\nu<10\,\mathrm{keV}$), which reach much deeper into the
disc where dust and gas are thermally coupled, but they can increase the level
of ionization (together with cosmic rays) close to the disc mid-plane thus
affecting the coupling of the gas with the disc magnetic field. The main
interaction of X-ray photons is via photoionization of (the inner shells of)
atoms and molecules, which can lead to secondary ionisations (including of
Hydrogen), followed by the thermalisation of the kinetic energies of primary
and secondary electrons. X-ray heating produces a range of temperatures going
from a few hundred K in the dense X-ray PDRs up to $10^{4}$ K in the disc
upper layers. X-ray luminosities do not depend on stellar accretion rates and
in fact, they remain nearly constant with time during the first Myrs
(Getman2022, ), yielding a constant wind mass-loss rate regardless of changes
in the surface density of the disc.

###  2.1 Methods

Since the realisation of the importance of disc photoevaporation, more than 30
years ago (BallyScoville1982, ), the problem has been tackled placing more
emphasis either on radiative transfer modelling, or on the full time-dependent
radiation-hydrodynamics.

The first studies of disc photoevaporation focussed on EUV photoevaporation of
discs by OB stars, which are strong EUV emitters. Two lines of models started
contemporarily. The first adopting a simplified radiative transfer calculation
in the Eddington approximation (reducing the three-dimensional radiative
transfer problem to a “three stream” approximation coming from the star,
vertically upwards, and downwards at each point in the disc) applied to a
vertically hydrostatic disc (Hollenbach1993, ; Hollenbach1994, ). The second
one, more focused on the dynamics (YorkeWelz1993, ; YorkeKaisig1995, ),
started by developing a 2D hydrodynamic photoionization code, later improved
including the effects of UV dust scattering (RichlingYorke1997, ). Both lines
of studies found that the most important contribution to the wind comes from
the diffuse field from a puffed-up bound inner disc and that the thermal wind
is launched from the gravitational radius $R_{g}$111the gravitational radius
is defined as the location where a gas parcel becomes unbound from the central
star (Hollenbach1994, ; Liffman2003, ) $R_{g}=GM_{\star}/c_{s}^{2}$. For gas
at roughly 10000K in the atmosphere of a disc around a solar mass star this is
roughly 5 au. The wind however is launched at the so-called critical radius
$R_{c}\sim 0.2R_{g}\sim 1~{}au$ (e.g. Dullemond2007, ).  outwards. By adding
the analytical prescriptions for mass-loss rates as a function of disc radius
due to EUV photoevaporation as a sink term in the 1D disc viscous evolution
equation (Clarke2001, ; Matsuyama2003, ), the so-called ’EUV-switch’ was
demonstrated (see Section 3).

The first hydrodynamical models of EUV disc photoevaporation for T Tauri stars
(Font2004, ) were performed assuming an isothermal equation of state, and
imposing the density at the base of the flow from previous results
Hollenbach1994 . They found that the photoevaporative flow was launched
subsonically ($\sim 0.3\,c_{s}$) from a much smaller radius than the
gravitational radius ($R_{c}\simeq R_{g}/5$) (see also Dullemond2007, ), which
was not expected from the analytical prescription.

These studies were then followed by the first hydrostatic models of X-ray
irradiation from T Tauri stars, first with a simple heating model
(Alexander2004, ), and later with a more realistic two-dimensional Monte Carlo
photoionization and dust radiative transfer code (Ercolano2008b, ;
Ercolano2009, ). The latter calculations used a more realistic input spectrum
which included both the EUV and X-ray components, showing that the latter
completely dominates, with mass-loss rates some 2 orders of magnitude higher
than previous pure EUV models.

At the same time, a 1+1D hydrostatic equilibrium model was also developed
including EUV, X-rays and FUVs, and a comprehensive chemical network
(GortiHollenbach2009, ; Gorti2009, ). These models showed that FUV radiation
might be efficient at removing material from the outer disc, but only if PAHs
are abundant in discs. The input spectrum used by these models was however
very idealised, with a hard X-ray input spectrum, which was inefficient at
heating the gas and driving the wind.

Without proper hydrodynamical calculations, all these models estimated local
mass-loss rates by assuming that the local mass loss rate is roughly equal to
the product of the density at the sound speed at the base of the wind, the so-
called $\rho\dot{c}_{s}$ method. Without hydrodynamics, identifying the base
of the wind is rather arbitrary and different authors used different
approaches, which further exacerbated tension between the models.

The first (radiation-)hydrodynamical calculation including both X-rays and EUV
radiation (Owen2010, ; Owen2012, ) was performed using a parameterisation of
gas temperature as a function of the ionisation parameter obtained from
detailed thermal and ionisation calculations (Ercolano2009, ) performed with
the mocassin code (Ercolano2003, ; Ercolano2005, ; Ercolano2008a, ). The
ionisation parameter is defined as $\xi=\frac{L_{X}}{nr^{2}}$, where $L_{X}$
is the X-ray luminosity, $r$ and $n$ are the local disc radius and the volume
density of the gas. This method is generally known as the $\xi-T_{e}$
approach. These studies were later extended adopting the modern hydrodynamical
code pluto (Mignone2007, ), and better accounting for attenuation effects in
the disc using a column density dependant $\xi-T_{e}$ parameterisation
(Picogna2019, ; Woelfer2019, ; Picogna2021, ; Ercolano2021, ). With this
improved prescription it was shown that the X-ray irradiation could reach
several hundred au, and that the mass-loss rates for high X-ray luminosities
reaches a plateau as more energetic photons cannot heat up lower and denser
cold regions of the disc.

The advantage of the $\xi-T_{e}$ method is that, while being based on detailed
multi-frequency thermal calculations, introduces very little computational
overheads on the hydrodynamics. A drawback of this method is that the thermal
calculations are performed in radiative equilibrium, and thus the contribution
of adiabatic cooling cannot be accounted for. All models performed with the
$\xi-T_{e}$ approach should therefore perform a-posteriori check to ensure
that radiative equilibrium is justified throughout the simulation domain. This
has indeed been demonstrated for all the models obtained with this approach so
far (Owen2010, ; Picogna2019, ; Woelfer2019, ; Picogna2021, ; Ercolano2021, ).

Recently some numerical experiments have been carried out to perform (a very
streamlined) radiative transfer and thermochemical calculation on the fly in
hydrodynamical models (WangGoodman2017, ; Nakatani2018a, ). Due to the high
computational costs of this approach, only a handful of models exist with
limited spectral and spatial resolution. The results obtained by these models
diverge strongly from those obtained with the $\xi-T_{e}$ method, as discussed
in Section 3.1.

##  3 Results

The main results from all photoevaporation models developed to date can be
roughly summarised as follows: radiation from the central star heats the upper
layers of protoplanetary discs, which become unbound and are centrifugally
accelerated in a thermal outflow. Wind mass-loss rates peak around a specific
radius of the disc known as the gravitational radius, with profiles that can
be more or less extended to larger radii. No mass-loss is expected to occur
from regions that are close to the star, and thus more gravitationally bound,
which would require the gas to achieve very high temperatures to escape. The
combination of steady mass-loss rates due to photoevaporation and a steadily
decreasing accretion rate with time in a viscously accreting disc produces the
so-called photoevaporation switch: Discs evolve viscously for a few million
years, or until the viscous accretion rate has reached values comparable to
the wind mass-loss rate, at which point photoevaporation takes over and the
disc is quickly dispersed from the inside out, via the formation of a gap
first and then a clear-out cavity.

Modern hydrodynamical models of photoevaporation are performed (at least) in
2D (but see Weber2022, , for 3D simulations including planets) and thus allow
only to model the disc for a limited amount of orbits, over which a steady-
state solution for the outflow is (ideally) achieved. Modelling the evolution
of the surface density of the disc as the gap opens and the disc is dispersed
is generally well beyond feasibility for 2D hydrodynamical calculations. What
is done instead is to use the 2D steady-state solution to determine a one-
dimensional mass-loss profile, $\dot{\Sigma}_{\mathrm{w}}(R)$ (Clarke2001, ;
Matsuyama2003, ; AlexanderArmitage2009, ; Monsch2021, ), which is then
included as a sink term to the one-dimensional disc viscous evolution equation
(LyndenBellPringle1974, ; Pringle1981, ) to study the surface density
evolution of the disc as a function of time:

$\frac{\partial\Sigma}{\partial t}=\frac{1}{R}\frac{\partial}{\partial
R}\left[3R^{1/2}\frac{\partial}{\partial R}\left(\nu\Sigma
R^{1/2}\right)\right]-\dot{\Sigma}_{\mathrm{w}}(R,t).$ (1)

Figure 1: On the left panel, the 1D surface density evolution as a function of
disc radius is shown for a disc of $M_{d}=0.1M_{\odot}$, orbiting a Solar mass
star, with $L_{X}=2.04\cdot 10^{30}$ erg s-1 (Picogna2021, ). The different
lines are drawn at [0, 25, 50, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 90,
99] % of the corresponding total disc lifetime. The dotted line shows the
approximate location of the inner gap opening due to photoevaporation. On the
right panel, the correspoding Surface density mass-loss rate due to
photoevaporation is shown as a function of disc radii.

Figure 1, shows the mass-loss profile for a 1 M⊙ (right panel (Picogna2021, ))
and the corresponding surface density evolution (left panel). The surface
density evolution (time and location of gap opening, dispersal timescale etc.)
is extremely sensitive to $\dot{\Sigma}(R)$, which in turn affects also the
final architecture of planets formed in the disc (Alexander2012, ;
ErcolanoRosotti2015, ; Jennings2018, ), as well as the potential of
photoevaporation of triggering the formation of planetesimals by the streaming
instability (Carrera2017, ; Ercolano2017, ).

###  3.1 Divergence of the results

While the broad-brush picture described above is shared by most authors, the
$\dot{\Sigma}_{\mathrm{w}}(R)$, and the physical properties of the wind
derived by different projects (e.g. see Table 1) diverge significantly. A
recent comparison of the Surface density mass-loss between the different
models of disc photoevaporation (Pascucci2022, , see their Figure 7) has
highlighted an order of magnitude difference in the cumulative mass-loss rates
which depends not only on the location of the surface mass-loss rate maximum
but also on the extent of the profiles. These differences can be broadly
understood looking at the choices made for the frequency range adopted for the
stellar irradiation, and the different methods used. However, most of them
present a peak inside 10 au and a slow decline of the surface density mass-
loss rate in the outer regions.

As photoevaporation is driven by stellar irradiation, a large part of the
divergence can be understood by considering the input spectrum assumed by
different authors (see also discussion in (Lesur2022, ), section 4.3.1). A
pure EUV model (e.g. Alexander2006a, ; Alexander2006b, ) yields an almost
isothermal gas with a temperature around $10^{4}\,\mathrm{K}$. In this case
the mass-loss peaks around the gravitational radius which is about $9$ au for
a 1 $M_{\odot}$ star, the total mass-loss scales as the square root of the EUV
flux, and it is roughly 10-10 M⊙/yr assuming a EUV flux of 1041 phot/sec
(Alexander2006a, ; Alexander2006b, ).

Soft X-ray radiation penetrates deeper in the disc than EUV radiation and
yields mass-loss rates that are one or two orders of magnitude higher than the
classical EUV model, depending almost linearly on the X-ray luminosity of the
irradiating star (Owen2012, ; Picogna2019, ; Picogna2021, ). X-ray
photoevaporation models, all include a EUV component, but find it is
irrelevant to driving the wind (e.g. Ercolano2009, ). Models for 1 M⊙ star
yield total mass-loss rates of order 10${}^{-8}M_{\odot}/yr$ assuming a soft
spectrum and X-ray luminosities of 1030 erg/sec (Owen2010, ; Picogna2019, ).
The X-ray-driven mass-loss profile is much more extended than in the EUV case,
peaking around the gravitational radius, but extending out to $\sim 200$ au.
Carbon is one of the major contributors in the X-ray opacity, since X-ray
photons are mainly absorbed by the inner shells of the more abundant heavy
elements in the gas and dust (ErcolanoClarke2010, ). Carbon depletion is
expected as a natural consequence of disk evolution both chemical as the
Carbon turns from CO into more complex species, and physical due to grain
growth that locks up large fractions of Carbon in ice bodies. For Carbon
depleted disks, the magnitude and extent of the mass-loss rates are expected
to increase by a factor $\sim 2$ (Woelfer2019, ). Internal FUV radiation is
expected to drive photoevaporation from further out in the disc, and initial
estimates, based on the hydrostatic equilibrium models of Gorti2009 , found it
to be very efficient under the assumption that Polycyclic Aromatic
Hydrocarbons (PAHs) are abundant in the atmosphere of discs. However, the
atmospheric abundance of PAHs, which are rarely observed in T-Tauri discs
(Seok2017, ), is likely to be very small, furthermore stellar FUV flux depends
on the accretion rate onto the central star and decreases with time. For these
reasons, the role of FUV-driven winds on the final disc dispersal is
uncertain.

More recent calculations Nakatani2018a , while considering EUV, FUV and X-ray
simultaneously, come to the conclusion that _direct_ EUV flux dominates the
driving of the wind, but obtain two orders of magnitude higher mass-loss rates
than previous EUV-only models. These calculations are in direct contrast with
the classical picture that stellar (direct) EUV photons are absorbed in the
bound inner regions of the disc and the diffuse field coming from the puffed-
up inner disc drives photoevaporation (at low rates) at the gravitational
radius. The assumption that direct EUV are able to reach the disc at and
beyond the gravitational radius relies on previous calculations (Tanaka2013,
). The source of the discrepancy between these more recent calculations and
previous works might lie in the specific disc geometries assumed, where the
inner disc does not expand enough to screen the outer regions or where there
is significant flaring, which might not apply to the majority of discs
(Hollenbach2017, ). We finally note that these results are also in tension
with previous work (Ercolano2009, ) which by means of detailed 2D Monte Carlo
RT and thermal calculation in a hydrostatic disc, explicitly test the role of
EUV in the presence of an observationally derived X-ray spectrum and find EUV
effects to be negligible on the final mass-loss rates. A detailed comparison
in a dedicated work must however be carried out to finally determine the
nature of this discrepancy.

Strong divergence is also observed in the physical properties of the wind
obtained by recent calculations (WangGoodman2017, ) which use the Athena++
code (White2016, ; Stone2020, ) to perform self-consistent radiation
hydrodynamics and thermochemical calculation of photoionised disc compared to
models that use the $\xi-T_{e}$ approach Owen2010 ; Owen2012 ; Picogna2019 ;
Picogna2021 ; Ercolano2021 . While total mass-loss rates obtained are
comparable, the Athena++ models come to the conclusion that X-ray radiation is
inefficient at driving the wind and that EUV dominates. Also the temperature
structure of the wind is significantly different in the two cases. The
Athena++ models predict extremely high temperatures ($10^{5}$ K) at the base
of the wind which decreases higher up, in contrast with the Parker-like wind
obtained by the $\xi-T_{e}$ models, which has a few $10^{3}$ K launching
temperature and reaches top temperatures of $10^{4}$ K but only for the very
limited region of the wind that is heated (not launched!) by EUV radiation.
The divergence of these models has been often attributed to methodological
differences on how to approach to radiative transfer and thermal balance, the
choice of irradiating spectrum employed, and the processes available to cool
the gas. A recent detailed comparison (Sellek2022, ) has been able to shed
light on this question, demonstrating that the divergence between these models
is driven predominantly by the choice of the irradiating spectrum and the very
limited number of frequency points used to describe the radiation field by the
Athena++ model (7) compared to the $\xi-T_{e}$ models ($>$1000). The $\xi-
T_{e}$ models use an X-ray spectrum derived from deep Chandra observations of
T-Tauri stars (Getman2005, ; Ercolano2008b, ; Ercolano2021, ), while the
Athena++ model assumes an analytical distribution.

##  4 Observational tests

For complex theoretical models to become a realistic description of natural
phenomena rather than remaining numerical experiments, it is very important
that they provide predictions that can be tested against observations. In this
Section, we list a number of direct and indirect tests that have been used to
constrain theoretical photoevaporation models.

###  4.1 Direct Tests

Winds are tenuous and direct imaging of its gas component is not possible.
However, the gaseous component of the wind can still be ”observed” directly
via high-resolution spectroscopy of emission lines that are emitted in the
outflow and show a blue-shifted component (ErcolanoPascucci2017, ;
Pascucci2022, ). Line profiles of collisionally ionised lines of neutral and
low ionisation species ([OI], [NeII], [SII], [N2], [FeII]), as well as some
molecular tracers (CO and H2), are available for a statistically significant
number of T-Tauri stars (e.g. Simon2016, ; Banzatti2019, ; Gangi2020, ). The
majority of the observed profiles can be well-fitted by X-ray-driven
photoevaporative models (ErcolanoOwen2010, ; ErcolanoOwen2016, ; Rab2022, ),
in combination with a magnetically driven component for sources showing
composite profiles (Weber2020, ) (see Figure 2). Lines in the sub-mm region
(e.g. CO and CI) have also been used to infer the presence of disc winds in
highly inclined sources (Louvet2018, ) and some predictions from numerical MHD
wind models already exist (Gressel2020, ).

Figure 2: Left Panel: Full width at half maximum vs peak velocity of the
narrow low-velocity components (NLVC) which result from the multi-Gaussian
decomposition of [OI] 6300 Å line profiles. Grey plusses mark the observed
NLVCs (Banzatti2019, ). Blue circles and red triangles show NVLCs from the
photoevaporation and MHD wind models (Weber2020, ), respectively. Right Panel:
Same as Fig. 2 but for the NLVC of the
$\mathrm{o\\!-\\!H_{2}}\,1\\!-\\!0\,\mathrm{S}(1)$ at $2.12\;\mathrm{\mu m}$.
Grey pluses show the observed NLVCs [70]. The blue circles show the results
for the same physical photoevaporative wind models as shown in Fig. 2, but
post-processed with a thermo-chemical code to properly model the molecular
hydrogen chemistry and excitation [Rab et al., subm.].

Ionised gas in the wind and disc atmosphere region could also be detected and
spatially constrained via its free-free emission at cm-wavelengths using
observations from upcoming high spatial resolution facilities like ngVLA (e.g.
Ricci2021, ). This is a promising avenue to distinguish magnetic winds which
can be driven from regions much closer to the star than photoevaporative
winds.

Small ($\lessapprox$ 10$\mu$m) dust grains can be entrained in the wind from
the launching region (Owen2011, ; Ercolano2011, ; Hutchison2016, ;
Hutchison2017, ; Franz2020, ; Hutchison2021, ; Booth2021, ; Rodenkirch2022, ),
and it may be possible to detect their signature in scattered light
observations of highly inclined sources, particularly from discs with inner
dust cavities (Franz2022a, ; Franz2022b, ).

###  4.2 Indirect Tests

Indirect constraints that photoevaporation models should match might include
(i) observed disc dispersal timescales which can be traced by the evolution of
the accretion properties and surface density of observed disc populations,
including during the transition disc phase (e.g. Luhman2010, ; Ercolano2011, ;
Ercolano2015, ; Koepferl2013, ; Rosotti2013, ; Ercolano2018, ; Garate2021, );
(ii) metallicity dependence of disc lifetimes (Yasui2009, ; Yasui2010, ;
Yasui2014, ; ErcolanoClarke2010, ; Takagi2015, ; Nakatani2018a, ); (iii)
observed correlations in disc populations like the $\dot{M}-M_{\star}$
relation (e.g. Ercolano2014, ; Sellek2020, ; Somigliana2020, ).

It is important to note that in order to test photoevaporation models against
observed disc populations, wind properties (mass-loss rates as a function of
disc radius) must be calculated for a wide enough parameter space, spanning
the observed stellar masses and X-ray properties. Assumptions are then
necessary as to how angular momentum is transported in the disc. Most studies
to date have assumed a viscous model in combination with photoevaporation. It
would be interesting to repeat this body of work assuming angular momentum
transport by magnetised disc winds (e.g. Tabone2022, ; Trapman2022, ).

##  5 Outlook

As mentioned at the end of the last section, magnetised disc winds might be
present in combination with thermal winds and the former might provide the
dominant mechanism for angular momentum transport instead of viscosity. Direct
hints of their presence are seen in emission line profiles observed for some
sources which show multiple outflow components (e.g. Pascucci2022, , for a
recent review). Furthermore non-ideal magnetohydrodynamical models routinely
observe magnetised winds while they struggle to maintain magnetorotational
instability active in most regions of the disc (e.g. Lesur2022, , for a recent
review). Clearly the frontier of modelling disc winds is to account for both
processes simultaneously and at high enough spatial resolution to produce
models that have enough predictive power to be confronted with the
observations. Initial attempts are promising (e.g. Wang2019, ; Rodenkirch2020,
), but they still lack resolution in the inner disc regions (where most
observed diagnostics come from), have a large lower density threshold
(comparable to the wind density, making post-processing of the winds
problematic), oversimplify radiative transfer and rely on a number of
unconstrained parameters to describe the magnetic flux and its evolution. If
these problems can be overcome these models could acquire enough predictive
power for a meaningful comparison with the observations, as detailed in the
previous section, and thus provide us with powerful tools to understand how
protoplanetary discs evolve and finally disperse.

##  6 Acknowledgements

We thank the anonymous referee for a thorough and constructive report that
helped improve this review. We acknoweledge the support of hte Excellence
Cluster ORIGINS which is funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Germany´s Excellence Strategy –
EXC-2094-390783311. BE, GP and CHR acknowledge the support of the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) - 325594231

Table 1: Model comparison

Project | Hydro | Dust | Radiation | Chemistry | Spectrum
---|---|---|---|---|---
Wang et al.1 | yes | $r_{\mathrm{dust}}:5\AA$ | yes | 24 species of H,He,C,O,S, | 4 energy bins: 7 eV (FUV),
|  |  d/g: $7\cdot 10^{-5}$ | ray-tracing |  Si,Fe,dust grains,e-, | 12 eV (L–W), 25 eV (EUV),
|  | no dynamics/growth | (no diffuse field) |  $\sim 100$ chemical reactions | and 1 keV (X-ray band)
Nakatani et al.2 | yes |  MRN ($3.1\AA\to 0.01\mu m$) | yes | 8 species of H,C,O, grains |  $81$ freq. bins for FUV/EUV
|  |  d/g: $10^{-6}\to 10^{-1}$ | ray-tracing | 13 chemical reactions | X-ray SED derived from
|  | no dynamics/growth | (no diffuse field) |  | TW-Hya
$\xi-T_{e}$ | yes |  MRN ($50\AA\to 0.25\mu m$) | yes | no |  $>1000$ freq. bins
approach3 |  |  d/g: $2.5,4\cdot 10^{-4}$ | Monte Carlo RT |  | obs. derived input spectra
|  | no dynamics/growth | (radiative equilibrium) |  | different spec. hardness
Gorti et al.4 | no |  MRN ($50\AA\to 20\mu m$) | yes | 84 species of H, He, C, O, |
| vertical hydrostatic |  d/g: $0.01$ | 1+1D gas and dust | Ne,S, Mg, Fe, Si, Ar, S, Mg, |  LX(E)$\propto$ E [$0.1-2$] keV
| equilibrium | no dynamics/growth | radiative transfer | Fe, Si, Ar, |  $L_{X}(E)\propto E^{-1.75}$ [$2-10$] keV
|  |  |  |  $\sim 600$ chemical reactions |
Ercolano et al.5 | no |  MRN ($50\AA\to 0.25\mu m$) | yes | atomic and photoionised |  $>1000$ freq. bins,
| vertical hydrostatic |  d/g: $2.5,4\cdot 10^{-4}$ | full Monte Carlo RT | species of H, He, C, O, Ne, | obs. derived input spectrum
| equilibrium | no dynamics/growth | with dust and gas | S, Mg, Fe, Si, Ar |
Alexander et al.6 | yes |  | no | no | –
|  |  |  |  |

11footnotetext: 1(WangGoodman2017, ; Wang2019, ),2(Nakatani2018a, ;
Nakatani2018b, ; Komaki2021, ),3(Owen2010, ; Owen2012, ; Picogna2019, ;
Woelfer2019, ; Ercolano2021, ; Picogna2021, ),4(Gorti2009, ),5(Ercolano2009,
),6(Alexander2006a, ; Alexander2006b, )

Data Availability Statement: No Data associated in the manuscript

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