11institutetext: Institut für Astronomie und Astrophysik, Universität
Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany
11email<EMAIL_ADDRESS>22institutetext: Physikalisches
Institut, Universität Bern, Gesellschaftsstr. 6, CH-3012 Bern, Switzerland
33institutetext: Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117
Heidelberg, Germany 44institutetext: Institute for Advanced Study, Tsinghua
University, Beijing 100084, People’s Republic of China 55institutetext:
Department of Astronomy, Tsinghua University, Beijing 100084, People’s
Republic of China66institutetext: Institut für Theoretische Astrophysik (ITA),
Universität Heidelberg, Albert-Ueberle-Str. 2, D-69120 Heidelberg, Germany
77institutetext: Physics and Astronomy Department, Amherst College, 25 East
Drive, Amherst, MA 01002, USA 88institutetext: Jet Propulsion Laboratory,
California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109,
USA 99institutetext: European Southern Observatory, Karl-Schwarzschild-Straße
2, D-85748 Garching bei München, Germany 1010institutetext: Institute for
Particle Physics and Astrophysics, ETH Zürich, Wolfgang-Pauli-Strasse 27,
CH-8093 Zürich, Switzerland 1111institutetext: University of Arizona, 1200 E
University Blvd., Tucson, AZ 85721, USA 1212institutetext: Centre for Space
and Habitability, Universität Bern, Gesellschaftsstr. 6, CH-3012 Bern,
Switzerland 1313institutetext: Institutionen för astronomi, AlbaNova
universitetscentrum, Stockholms universitet, SE-106 91 Stockholm, Sweden
1414institutetext: Department of Astronomy and Steward Observatory, University
of Arizona, 933 N Cherry Ave, Tucson, AZ 85719, USA 1515institutetext:
Department of Physics and Astronomy, University of California, 4129 Frederick
Reines Hall, Irvine, CA 92697-4575, USA 1616institutetext: Hamburger
Sternwarte, Gojenbergsweg 112, D-21029 Hamburg, Germany 1717institutetext:
NASA Ames Research Center, Moffett Blvd, Mountain View, CA 94035, USA
1818institutetext: SETI Institute, 189 Bernardo Avenue, Suite 200, Mountain
View, CA 94043, USA 1919institutetext: Observatoire astronomique de
l’Université de Genève, 51 ch. des Maillettes, CH-1290 Versoix, Switzerland

# Accreting protoplanets: Spectral signatures and magnitude
of gas and dust extinction at H $\alpha$

G.-D. Marleau Accreting protoplanets: Spectral signatures and magnitude of gas
and dust extinction at H $\alpha$Accreting protoplanets: Spectral signatures
and magnitude of gas and dust extinction at H $\alpha$Accreting protoplanets:
Spectral signatures and magnitude of gas and dust extinction at H
$\alpha$Accreting protoplanets: Spectral signatures and magnitude of gas and
dust extinction at H $\alpha$Accreting protoplanets: Spectral signatures and
magnitude of gas and dust extinction at H $\alpha$Accreting protoplanets:
Spectral signatures and magnitude of gas and dust extinction at H $\alpha$ Y.
Aoyama Accreting protoplanets: Spectral signatures and magnitude of gas and
dust extinction at H $\alpha$Accreting protoplanets: Spectral signatures and
magnitude of gas and dust extinction at H $\alpha$Accreting protoplanets:
Spectral signatures and magnitude of gas and dust extinction at H
$\alpha$Accreting protoplanets: Spectral signatures and magnitude of gas and
dust extinction at H $\alpha$ R. Kuiper Accreting protoplanets: Spectral
signatures and magnitude of gas and dust extinction at H $\alpha$Accreting
protoplanets: Spectral signatures and magnitude of gas and dust extinction at
H $\alpha$Accreting protoplanets: Spectral signatures and magnitude of gas and
dust extinction at H $\alpha$Accreting protoplanets: Spectral signatures and
magnitude of gas and dust extinction at H $\alpha$Accreting protoplanets:
Spectral signatures and magnitude of gas and dust extinction at H
$\alpha$Accreting protoplanets: Spectral signatures and magnitude of gas and
dust extinction at H $\alpha$ K. Follette Accreting protoplanets: Spectral
signatures and magnitude of gas and dust extinction at H $\alpha$Accreting
protoplanets: Spectral signatures and magnitude of gas and dust extinction at
H $\alpha$ N. J. Turner Accreting protoplanets: Spectral signatures and
magnitude of gas and dust extinction at H $\alpha$Accreting protoplanets:
Spectral signatures and magnitude of gas and dust extinction at H $\alpha$ G.
Cugno Accreting protoplanets: Spectral signatures and magnitude of gas and
dust extinction at H $\alpha$Accreting protoplanets: Spectral signatures and
magnitude of gas and dust extinction at H $\alpha$ C. F. Manara Accreting
protoplanets: Spectral signatures and magnitude of gas and dust extinction at
H $\alpha$Accreting protoplanets: Spectral signatures and magnitude of gas and
dust extinction at H $\alpha$ S. Y. Haffert NASA Hubble Fellowship Program
Sagan FellowAccreting protoplanets: Spectral signatures and magnitude of gas
and dust extinction at H $\alpha$Accreting protoplanets: Spectral signatures
and magnitude of gas and dust extinction at H $\alpha$ D. Kitzmann Accreting
protoplanets: Spectral signatures and magnitude of gas and dust extinction at
H $\alpha$Accreting protoplanets: Spectral signatures and magnitude of gas and
dust extinction at H $\alpha$ S. C. Ringqvist Né S. C. ErikssonAccreting
protoplanets: Spectral signatures and magnitude of gas and dust extinction at
H $\alpha$Accreting protoplanets: Spectral signatures and magnitude of gas and
dust extinction at H $\alpha$ K. R. Wagner11footnotemark: 1, NASA NExSS Alien
Earths TeamAccreting protoplanets: Spectral signatures and magnitude of gas
and dust extinction at H $\alpha$Accreting protoplanets: Spectral signatures
and magnitude of gas and dust extinction at H $\alpha$ R. van Boekel Accreting
protoplanets: Spectral signatures and magnitude of gas and dust extinction at
H $\alpha$Accreting protoplanets: Spectral signatures and magnitude of gas and
dust extinction at H $\alpha$ S. Sallum Accreting protoplanets: Spectral
signatures and magnitude of gas and dust extinction at H $\alpha$Accreting
protoplanets: Spectral signatures and magnitude of gas and dust extinction at
H $\alpha$ M. Janson Accreting protoplanets: Spectral signatures and magnitude
of gas and dust extinction at H $\alpha$Accreting protoplanets: Spectral
signatures and magnitude of gas and dust extinction at H $\alpha$ T. O. B.
Schmidt Accreting protoplanets: Spectral signatures and magnitude of gas and
dust extinction at H $\alpha$Accreting protoplanets: Spectral signatures and
magnitude of gas and dust extinction at H $\alpha$ L. Venuti Accreting
protoplanets: Spectral signatures and magnitude of gas and dust extinction at
H $\alpha$Accreting protoplanets: Spectral signatures and magnitude of gas and
dust extinction at H $\alpha$Accreting protoplanets: Spectral signatures and
magnitude of gas and dust extinction at H $\alpha$Accreting protoplanets:
Spectral signatures and magnitude of gas and dust extinction at H $\alpha$ Ch.
Lovis Accreting protoplanets: Spectral signatures and magnitude of gas and
dust extinction at H $\alpha$Accreting protoplanets: Spectral signatures and
magnitude of gas and dust extinction at H $\alpha$ C. Mordasini Accreting
protoplanets: Spectral signatures and magnitude of gas and dust extinction at
H $\alpha$Accreting protoplanets: Spectral signatures and magnitude of gas and
dust extinction at H $\alpha$

( – / 17 November 2021)

###### Abstract

Context. Accreting planetary-mass objects have been detected at H $\alpha$,
but targeted searches have mainly resulted in non-detections. Accretion
tracers in the planetary-mass regime should originate from the shock itself,
making them particularly susceptible to extinction by the accreting material.
High-resolution ($R>50,000$) spectrographs operating at H $\alpha$ should soon
enable one to study how the incoming material shapes the line profile.

Aims. We calculate how much the gas and dust accreting onto a planet reduce
the H $\alpha$ flux from the shock at the planetary surface and how they
affect the line shape. We also study the absorption-modified relationship
between the H $\alpha$ luminosity and accretion rate.

Methods. We computed the high-resolution radiative transfer of the H $\alpha$
line using a one-dimensional velocity–density–temperature structure for the
inflowing matter in three representative accretion geometries: spherical
symmetry, polar inflow, and magnetospheric accretion. For each, we explored
the wide relevant ranges of the accretion rate and planet mass. We used
detailed gas opacities and carefully estimated possible dust opacities.

Results. At accretion rates of $\dot{M}\lesssim 3\times
10^{-6}\leavevmode\nobreak\ M_{\mathrm{J}}\,\mathrm{yr}^{-1}$, gas extinction
is negligible for spherical or polar inflow and at most
$A_{\mathrm{H}\,\alpha}\lesssim 0.5$ mag for magnetospheric accretion. Up to
$\dot{M}\approx 3\times 10^{-4}\leavevmode\nobreak\
M_{\mathrm{J}}\,\mathrm{yr}^{-1}$, the gas contributes
$A_{\mathrm{H}\,\alpha}\lesssim 4$ mag. This contribution decreases with mass.
We estimate realistic dust opacities at H $\alpha$ to be $\kappa\sim 0.01$–10
cm2 g-1, which is 10–$10^{4}$ times lower than in the interstellar medium.
Extinction flattens the ${L_{\textnormal{H}\,\alpha}}$–$\dot{M}$ relationship,
which becomes non-monotonic with a maximum luminosity
${L_{\textnormal{H}\,\alpha}}\sim 10^{-4}\leavevmode\nobreak\ {L_{\odot}}$
towards $\dot{M}\approx 10^{-4}\leavevmode\nobreak\
M_{\mathrm{J}}\,\mathrm{yr}^{-1}$ for a planet mass $\sim
10\leavevmode\nobreak\ M_{\mathrm{J}}$. In magnetospheric accretion, the gas
can introduce features in the line profile, while the velocity gradient smears
them out in other geometries.

Conclusions. For a wide part of parameter space, extinction by the accreting
matter should be negligible, simplifying the interpretation of observations,
especially for planets in gaps. At high $\dot{M}$, strong absorption reduces
the H $\alpha$ flux, and some measurements can be interpreted as two $\dot{M}$
values. Highly resolved line profiles ($R\sim 10^{5}$) can provide (complex)
constraints on the thermal and dynamical structure of the accretion flow.

###### Key Words.:

accretion — planets and satellites: gaseous planets — planets and satellites:
detection — planets and satellites: formation — radiative transfer — line:
profiles

##  1 Introduction

In order to test and constrain planet formation models, it is crucial to
detect planets not only shortly after their formation, but also during. The
timeline of the mass assembly of gas giants—their accretion rate history—is an
important and barely constrained aspect of planet formation with consequences
for the migration history and thus chemical composition as well as dynamics of
forming planetary systems, and thus of the assembly of possibly life-bearing
planets. The detection of accretion tracers such as H $\alpha$ provides a
unique window into this key phase.

So far, a few low-mass accreting objects have been detected through their H
$\alpha$ emission. Overall, dedicated surveys with different strategies have
not yet been successful in revealing accreting planets. However, upcoming
instrumental upgrades offer the hope of soon uncovering a large population of
forming planets. We review this in Section 2.

The interpretation of both detections and non-detections requires models of
the emission and radiation transfer of the accretion tracers. Despite decades
of work by a few groups (e.g. Muzerolle et al., 2001; Romanova et al., 2004;
Kurosawa & Romanova, 2012), the equivalent problem in low-mass star formation
is not entirely solved. The planetary case is even less well understood but
there have been recent theoretical developments. Indeed, Aoyama et al. (2018)
and Aoyama et al. (2020) presented detailed models of the emission of tracers
from an accretion shock on the circumplanetary disc (CPD) or the planet
surface, respectively. Thanathibodee et al. (2019) presented the first
predictions of a model of magnetospheric accretion for low-mass stars
(Classical T Tauri Stars; CTTSs) applied to planets.

What these studies have not modelled in detail is absorption by the material,
both gas and dust, accreting onto the planet. This is the subject of this
study for the case of the planet-surface accretion shock. Recently, in Sanchis
et al. (2020) and Szulágyi & Ercolano (2020), this was undertaken using global
disc simulations with a very different density and temperature structure near
the planet. We discuss this in Section 7.6.

We study H $\alpha$ since it is the most commonly used accretion tracer and
indeed usually exhibits the strongest signal. Estimating the strength of the
absorption will inform studies of accretion tracers both in a static and in a
(more realistic) time-varying picture.

The paper is laid out as follows: Section 2 highlights some aspects of the
current observational and instrumental landscape. In Section 3 we present
three possible accretion geometries and the details of the macroscopic and
microscopic quantities for the accretion flow. The effect of the absorption by
the gas is studied in Section 4, which presents integrated fluxes and line
profiles at extremely high spectral resolution across the parameter space.
Then, we deal with absorption by dust in Section 5. In Section 6, we discuss
the resulting H $\alpha$-luminosity–accretion rate relationship and apply our
results to known accreting low-mass objects. Section 7 presents a discussion
of our model and in Section 8 we summarise and conclude. Appendix A discusses
the contribution of the emission from the accreting material in the radiative
transfer, and Appendix B presents additional line profiles.

##  2 Current and future instrumentation

Thanks to instrumentation advances (VLT/SPHERE, VLT/MUSE, LBT/LBTI,
Magellan/MagAO; Bacon et al., 2010; Close et al., 2014a, b; Schmid et al.,
2017, 2018), a handful of low-mass companions to young stars have been
detected and explored that (possibly) show signs of ongoing accretion: LkCa 15
b (Sallum et al., 2015; but see also Mendigutía et al. 2018; Currie et al.
2019), PDS 70 b and c (Keppler et al., 2018; Müller et al., 2018; Wagner et
al., 2018; Haffert et al., 2019; Zhou et al., 2021), and Delorme 1 (AB)b
(Eriksson et al., 2020). On the other hand, recent surveys searching for
further accreting planets through their H $\alpha$ signal have returned null
results, from five (Cugno et al., 2019) and eleven (Zurlo et al., 2020)
different sources111Zurlo et al. (2020) do not include PDS 70 in order to
analyse the known planets in that system separately. (see also Xie et al.
2020).

However, more companions might be revealed by ongoing and future searches at H
$\alpha$ with various instruments. One is Subaru/SCExAO+VAMPIRES (Lozi et al.,
2018; Uyama et al., 2020), with $R\gtrsim 1000$. Another is VLT/MUSE, which
provides at H $\alpha$ in narrow-field mode (NFM) the currently highest
available spectral resolution of $R=2516$, corresponding to an instrumental
spectral full width at half maximum (FWHM) of $\approx 0.26$ nm (see for
example Figure B.3 of Eriksson et al. 2020). A similar resolution will be
afforded by HARMONI, the first-light spectrograph on the Extremely Large
Telescope (ELT)222See http://harmoni-elt.physics.ox.ac.uk. (Thatte et al.,
2016; Rodrigues et al., 2018), with $R\approx 3300$ blueward of 0.8 $\upmu$m.

Also, there are hopes from upgrades to existing instruments or upcoming or
proposed instruments. Indeed, several efforts focussed on pushing visible
direct imaging (photometry) or spectroscopy to the limit are being developed
with H $\alpha$ as their main science case. Imaging instruments include:
Magellan/MagAO-X (Males et al., 2018; Close et al., 2018) and GMagAO-X on the
Giant Magellan Telescope (GMT) (Males et al., 2019), the Keck Planet Imager
and Characterizer (KPIC) at the Keck II telescope (Jovanovic et al., 2019),
and also possibly the near-infrared (NIR) spectrograph NIRSpec333Only the
$R\sim 100$ prism can access H $\alpha$ (see https://jwst-docs.stsci.edu),
making it challenging but still possible. on the James Webb Space Telescope
(JWST) or the Coronograph
Instrument444https://wfirst.ipac.caltech.edu/sims/Param_db.html (CGI) aboard
the Nancy Grace Roman Space Telescope (formerly known as WFIRST). A few
spectrographs covering H $\alpha$ are also being planned and developed: an
integral-field spectrograph of $R=15,000$ named Visible Integral-field
Spectrograph eXtreme (VIS-X) at MagAO-X (Haffert et al., 2021); a proposed
Extreme-Adaptive-Optics- (XAO)-assisted high-resolution spectrograph for the
VLT, dubbed RISTRETTO555See https://zenodo.org/record/3356296. (PI: Ch. Lovis;
Chazelas et al., 2020), with $R\geqslant 130,000$–$150,000$; as well as the
Replicable High-resolution Exoplanet and Asteroseismology (RHEA) spectrograph
for Subaru/SCExAO (Rains et al., 2016, 2018; Anagnos et al., 2020), which
should provide $R\approx 60,000$. For both RISTRETTO and RHEA, a low
throughput could, however, make observations challenging.

We note that VLT/UVES (Dekker et al., 2000) has a resolution of approximately
$R=100,000$ at H $\alpha$ (and $R=80,000$ at H $\beta$), with some variations
depending on the exact setting. One of the main limitations for using UVES for
exoplanet purposes is that it is seeing limited, so that unlike AO-assisted
instruments such as MUSE, UVES generally cannot spatially resolve the
planetary flux. In that case, the only way to study the H $\alpha$ line would
be if it were strong enough to be comparable to the flux of the primary star.
However, in cases where a companion is a few arcseconds (i.e. a few hundreds
of astronomical units at 150 pc) away and its H $\alpha$ line is reasonably
strong, it could still be possible to get a spatially resolved signature with
UVES, Delorme 1 (AB)b (Eriksson et al., 2020) being one potential example.

Finally, one should also mention the Extremely Large Telescope (ELT), expected
to come online in the next decade, with its second-generation High Resolution
Spectrograph (HIRES). While it does not cover H $\alpha$, its Integral Field
Unit (IFU) is sensitive to $1.0$–1.8 $\upmu$m at $R\approx 100,000$–$150,000$
(Marconi et al., 2016, 2018; Tozzi et al., 2018), which includes shock
emission lines such as Pa $\beta$ (Aoyama et al., 2018) and He i $\lambda$
10830\. Thanks to this high resolution, it could also be a powerful help in
characterising accreting planets.

##  3 Physical model: Macro- and microphysics

Here we detail the geometries we consider for the accretion (Section 3.1), the
parameter space of accretion rate, planet mass, and planet radius (Section
3.2), the structure of the flow (Section 3.3), the calculation of the input
line profiles (Section 3.4), the radiative transfer in the accretion flow
(Section 3.5), and the gas opacities used (Section 3.6).

###  3.1 Accretion geometry

The geometry of accretion onto forming gas giants is an open question. In the
classical, simplified picture of accreting gas giants, matter begins at a
finite starting radius ${R_{\textnormal{acc}}}$ and falls freely onto the
planet in a spherically symmetric fashion (e.g. Pollack et al., 1996;
Bodenheimer et al., 2000; Mordasini et al., 2012b; Helled et al., 2014;
Marleau et al., 2017, 2019). Based on the arguments in, for exmaple, Ginzburg
& Chiang (2019a), may be a reasonable model, at least on the scale of the
Bondi radius ${R_{\textnormal{Bondi}}}$ for sub-thermal planets, that is, for
planets whose Hill radius
${R_{\textnormal{Hill}}}=a(M_{\mathrm{p}}/3{M_{\star}})^{1/3}$ is smaller than
the pressure scale height of the circumstellar disc (CSD), with $a$ the semi-
major axis and ${M_{\star}}$ the stellar mass.

More realistically, or at least for higher-mass planets, radiation-
hydrodynamical simulations have suggested the presence of meridional
circulation around an accreting planet: motion from the upper layers of the
protoplanetary disc towards the planet at its location, where it has opened at
least a partial gap (e.g. Kley et al., 2001; Machida et al., 2008; Tanigawa et
al., 2012; Gressel et al., 2013; Morbidelli et al., 2014; Fung et al., 2015;
Fung & Chiang, 2016; Szulágyi et al., 2016, 2019; Dong et al., 2019; Schulik
et al., 2019, 2020; Bailey et al., 2021; Rabago & Zhu, 2021; see also Béthune,
2019). Thanks to the exquisite sensitivity of ALMA, Teague et al. (2019)
provided the first observational evidence for such a pattern over a scale of a
few Hill radii in a young disc thought to contain accreting planets, HD
163296, and recently obtained a similar result for HD 169142 (Yu et al.,
2021). For such a geometry to hold, however, the gap opened by the planet
needs to be not too wide, and once higher masses have been reached, the large
width of the gap ($\propto{M_{\mathrm{p}}}^{0.5}$; Kanagawa et al., 2017, but
see also Bergez-Casalou et al., 2020, who explore how time-dependent disc
models lead to a different opening criterion than in equilibrium discs) should
lead to accretion from the CSD onto a circumplanetary disc (CPD), and from
this onto the planet. This will hold especially in multiple-planet systems in
which forming gas giants have opened a common gap (Close, 2020). As Ginzburg &
Chiang (2019b) point out, this might be a long phase of the accretion process,
over which a significant fraction of the planet mass could be assembled.

How the gas then ultimately reaches the forming planet, with a size of order
$R_{\mathrm{p}}\sim(0.01\textrm{--}0.001){R_{\textnormal{Hill}}}$, is an open
question. From angular momentum conservation, and by analogy with objects
across a large range of mass scales, a part of the matter likely goes through
a CPD (which in its outer regions may be a decretion disc, i.e. exhibit an
outflow; Tanigawa et al., 2012; the general theory of decretion discs is
presented in Lynden-Bell & Pringle, 1974; Pringle, 1991; Nixon & Pringle,
2021) while the rest could fall directly onto the proto-gas-giant with a polar
shock (see Béthune & Rafikov, 2019; Béthune, 2019). However, the fraction of
the gas processed through a CPD is unknown, as is how the gas leaves the CPD
to be incorporated into the planet. Obvious possibilities are processes
invoked for CTTSs (see review in Hartmann et al., 2016), including boundary-
layer accretion (e.g. Lynden-Bell & Pringle, 1974; Kley & Lin, 1996; Piro &
Bildsten, 2004; see brief review in Geroux et al., 2016), or magnetospheric
accretion (Lovelace et al., 2011; Batygin, 2018; Cridland, 2018; see also
Fendt, 2003). As in the stellar context (see reviews in Romanova & Owocki,
2015; Mendigutía, 2020), which of these two mechanisms dominates will depend
on the magnetic field strength of the forming planet and the coupling of the
gas to the magnetic field. In the boundary-layer accretion (BLA) scenario
there is no accretion shock on the planet surface. Therefore, we do not treat
BLA here; in this case the observed H $\alpha$ would need to come from a shock
on the CPD (Aoyama et al., 2018).

There are suggestions that both the magnetic field and the coupling of the gas
to the magnetic field could be strong enough for magnetospheric accretion to
occur (Lovelace et al., 2011; Batygin, 2018). Applying the Christensen et al.
(2009) scaling to the luminosities of young planets, Katarzyński et al. (2016)
found that they should have a surface dipole field $B\sim 0.5$–1 kG. Using
typical accretion rates (Mordasini et al., 2012b), the resulting estimate of
the magnetospheric (or Alfvén) radius is usually larger than the planet
radius, which holds even more for forming planets, which might have an even
higher luminosity (Mordasini et al., 2017). Taken together with the estimate
of a relatively low magnetic diffusivity, this suggests that planets could
indeed accrete magnetospherically (Batygin, 2018; Hasegawa et al., 2021).
Nevertheless, one should keep in mind that the validity of the Christensen et
al. (2009) scaling for forming planets has yet to be shown, especially since
they are not necessarily fully convective (Berardo et al., 2017; Berardo &
Cumming, 2017). Estimating whether planets can accrete magnetospherically
might also depend on the field topology, which for CTTSs has significant non-
dipole components (Hartmann et al., 2016).

If magnetospheric accretion onto planets does take place, it could be in
analogy to the stellar case, in which the material is lifted from the disc,
following the magnetic field lines connecting it to the protostar.
Alternatively, or simultaneously, mass newly supplied could be coming from
above, in the downward part of a meridional flow (of size
$\gtrsim{R_{\textnormal{Hill}}}$), and thus falling onto the apex of the
magnetic fields lines (Batygin, 2018). However, for the shock this detail
should not matter much since in both cases the velocity of the infalling
matter will be essentially the free-fall velocity, even though possibly
starting effectively from different accretion radii.

Since in all, the accretion geometry is very uncertain, we consider three
basic geometries for the accretion shock on the planet surface, as illustrated
in Figure 1:

  1. 1.

SpherAcc: spherically symmetric accretion,

  2. 2.

Polar: accretion concentrated towards the planet’s magnetic poles, with no
accretion outside,

  3. 3.

MagAcc: magnetospheric accretion, with accretion only along one or several
column(s), with overall a small filling factor $f_{\mathrm{fill}}$, which is
the fraction of the planet’s surface covered by the (global) accretion rate.

The MagAcc case differs from the Polar case by the filling factor (only
quantitatively) but also by the length of the accretion flow over which the
gas can affect the radiation (see the end of this subsection).

We treat the hydrodynamics and the radiative transfer approximately with one-
dimensional models by following in each geometry the radiative transfer along
the flow. For SpherAcc this is natural, for Polar this is effectively averaged
over the cone angle with a non-zero accretion rate, and for MagAcc it is as
for Polar, but for an even smaller region (a thin accretion stream or
several). Table 1 describes these different scenarios in terms of the filling
factor $f_{\mathrm{fill}}$, the size of the “accretion radius”
${R_{\textnormal{acc}}}$ from which the gas is starting at rest (Bodenheimer
et al., 2000; Mordasini et al., 2012b), and the integration outer limit
${r_{\rm max}}$ for the optical depth calculation (see Equation (14c)).

Figure 1: Schematic view of the accretion scenarios considered in this work.
In spherical symmetry (SpherAcc), the gas (straight thick lines) is assumed to
be distributed uniformly on the surface of the planet
($f_{\mathrm{fill}}=100\leavevmode\nobreak\ \%$), whereas in the case of polar
infall (Polar) the accreting gas is concentrated near the pole regions
(assumed to be uniform within $f_{\mathrm{fill}}\approx 30\leavevmode\nobreak\
\%$, and zero outside of this). In the magnetospheric accretion case (MagAcc),
the magnetic field of the protoplanet leads the gas onto a small fraction of
the surface (here shown as polar-ring hot spots; $f_{\mathrm{fill}}\sim
10\leavevmode\nobreak\ \%$). See Table 1. At slightly larger scales (the
insets show out to ${R_{\textnormal{acc}}}$) the gas could be coming from
above, falling onto the apex of the magnetic field lines (Batygin, 2018), or
from a circumplanetary disc (purposefully not shown). In all cases, the
radiation (H $\alpha$-coloured squiggly lines) reaching the observer passes
through the matter (gas and dust) accreting onto the planet and bears the
spectral imprint of this material, which is the subject of this paper.
Illustration by Th. Müller (MPIA/Haus der Astronomie). Table 1: Accretion
geometries considered in this work.

Model name |  $f_{\mathrm{fill}}$aa$a$Fraction of the planet’s surface covered by the accretion flow. |  $\theta_{\mathrm{min}}$bb$b$Opening angles of the accretion cone covering each pole (Equation (1)); for illustration. |  $\theta_{\mathrm{max}}$bb$b$Opening angles of the accretion cone covering each pole (Equation (1)); for illustration. |  ${R_{\textnormal{acc}}}$cc$c$Starting position of the infalling gas with a radial velocity $v_{r}=0$ there. |  ${r_{\rm max}}$dd$d$Outer limit of the optical radiative transfer calculation, limited to ${r_{\rm max}}\approx 100\leavevmode\nobreak\ R_{\mathrm{p}}$ due to uncertainty in the structure of the far layers. The exact value is however inconsequential for SpherAcc and Polar.
---|---|---|---|---|---
Spherical accretion
SpherAcc | 1.0 | $0{\mbox{\textdegree}}$ | $90{\mbox{\textdegree}}$ | $\infty$ | $100\leavevmode\nobreak\ R_{\mathrm{p}}$
Polar inflow
Polar | 0.3 | $0{\mbox{\textdegree}}$ | $46{\mbox{\textdegree}}$ | $\infty$ | $100\leavevmode\nobreak\ R_{\mathrm{p}}$
Magnetospheric accretion
MagAcc | 0.1 | $20{\mbox{\textdegree}}$ | $50{\mbox{\textdegree}}$ | $5\leavevmode\nobreak\ R_{\mathrm{p}}$ | $\approx\frac{1}{2}{R_{\textnormal{acc}}}$
666 The different geometries are illustrated in Figure 1. Model names in the
paper are suffixed with Warm or Cold depending on whether the radius fit
$R_{\mathrm{p}}(\dot{M},M_{\mathrm{p}})$ to cold-accreting or warm-accreting
planets (see Section 3.2) is used.

In the polar case, the accretion is assumed to be uniform within an
axisymmetric cone or ring and zero outside, with the observer looking down
into this region. The accreting region is defined by

$f_{\mathrm{fill}}=\cos\theta_{\mathrm{min}}-\cos\theta_{\mathrm{max}},$ (1)

where $\theta_{\mathrm{min}}=0$ for the pole ($\theta_{\mathrm{min}}\neq 0$
for a ring; see e.g. Kulkarni & Romanova 2013) and
$\theta_{\mathrm{max}}\leqslant 90{\mbox{\textdegree}}$ is the equivalent
opening angle of the accreting region at each pole. For
$f_{\mathrm{fill}}=0.3$ (e.g. 15 % of the total area at the north pole and 15
% at the south pole), the equivalent opening angle from
$\theta_{\mathrm{min}}=0{\mbox{\textdegree}}$ is
$\theta_{\mathrm{max}}=46{\mbox{\textdegree}}$. For a ring with
$f_{\mathrm{fill}}=0.1$ starting and at
$\theta_{\mathrm{min}}=20{\mbox{\textdegree}}$ (for example),
$\theta_{\mathrm{max}}=50{\mbox{\textdegree}}$, while $f_{\mathrm{fill}}=0.01$
would require $\theta_{\mathrm{max}}=21.6{\mbox{\textdegree}}$, or
$\theta_{\mathrm{max}}=12.9{\mbox{\textdegree}}$ if instead
$\theta_{\mathrm{min}}=10{\mbox{\textdegree}}$.

In reality, there is evidence for multiple accretion components in
observations, in agreement with simulations (e.g. Ingleby et al., 2013;
Robinson & Espaillat, 2019; Robinson et al., 2021), but our assumption of a
constant local accretion rate and an infinitely sharp transition between the
accreting and the non-accreting region is a minor one. Also, given that very
edge-on systems are less likely to be detected or observed, there is a
relatively large probability of viewing the planet indeed within
$45{\mbox{\textdegree}}$ of pole-on. Therefore, we assume that the radiation
is travelling only radially (i.e. away from the planet) through the accretion
region towards the observer, and neglect the possibility of scattering out of
the accretion cone into the line of sight towards an observer not looking down
into the accretion cone.

For the magnetospheric case, in analogy to CTTSs, we consider
${R_{\textnormal{acc}}}=5\leavevmode\nobreak\ R_{\mathrm{p}}$ (Calvet &
Gullbring, 1998), but also ${R_{\textnormal{acc}}}=2\leavevmode\nobreak\
R_{\mathrm{p}}$ for comparison, as the magnetic field may be weaker, whether
in total or in its dipole component. We assume that the flux coming from the
postshock region (the accretion “hot spots”) passes tangentially through the
accretion arc to the observer and thus travels a distance ${r_{\rm
max}}\approx
1/3\times\pi{R_{\textnormal{acc}}}/2\approx{R_{\textnormal{acc}}}/2$, as
illustrated in Figure 1. Therefore, curvature can be ignored at the level of
our approximation, and the radiative transfer can be performed also here
purely radially.

This assumption of observing along the accretion column leads to a strong
estimate of the amount of absorption by the accreting gas. If the optical
depth is high along the accretion column, it might be more realistic for the
photons to escape preferentially at an angle from the base of the accretion
footpoints (see the pole-side photons in Figure 1). In this case, they would
travel relatively unimpeded towards the observer. Scattering out of the
accretion arc close to the planet could also contribute to the flux, depending
on the system’s inclination. Thus our approach of integrating along a segment
of the flow is a simple one that should maximise potential signatures.

###  3.2 Parameter space

For a geometry given by $(f_{\mathrm{fill}},{R_{\textnormal{acc}}},{r_{\rm
max}})$, the main quantities defining the parameter space are the mass
accretion rate $\dot{M}$, the planet mass $M_{\mathrm{p}}$, and the planet
radius $R_{\mathrm{p}}$. We consider a range of accretion rates
$\dot{M}\approx 3\times 10^{-8}$–$3\times 10^{-4}\leavevmode\nobreak\
M_{\mathrm{J}}\,\mathrm{yr}^{-1}$. The high end of the $\dot{M}$ range is
higher than usually found in planet formation models (Mordasini et al., 2012b;
Tanigawa & Tanaka, 2016) but could be relevant to an accretion outburst akin
to the situation of FU Orionis stars (Audard et al., 2014). With these high
accretion rates, we can address whether we would indeed observe a signal in
the (perhaps unlikely) event a planet were caught outbursting. This range also
covers the accretion rate inferred for the PDS 70 planets (Haffert et al.,
2019; Manara et al., 2019).

Concerning the mass, we focus on the gas-giant and low-mass-brown-dwarf regime
with $M_{\mathrm{p}}\approx 1$–$20\leavevmode\nobreak\ M_{\mathrm{J}}$. At
much higher masses, the velocity of the gas at the shock is too high for H
$\alpha$ to be generated in the postshock region, and the hydrogen lines are
thought to be emitted by the accreting gas (Hartmann et al., 2016; Aoyama et
al., 2021).

Both to reduce the dimensionality and to provide guidance as to a possible
trend, we do not keep the planet radius as a free parameter but instead adopt
for definiteness the fits777For convenience, they can be found in a few
popular languages in the “Suite of Tools to Model Observations of accRetIng
planeTZ” at https://github.com/gabrielastro/St-Moritz. Radii of up to
$R_{\mathrm{p}}\approx 9$–12 $R_{\mathrm{J}}$ are reached for $\dot{M}=3\times
10^{-4}\leavevmode\nobreak\ M_{\mathrm{J}}\,\mathrm{yr}^{-1}$. This might seem
high but is in line with the high-entropy models of Spiegel & Burrows (2012)
and Marleau & Cumming (2014). by Aoyama et al. (2020) of the radius as a
function of accretion rate and planet mass in the population synthesis
calculations888Since the planet structure model is the same for gas giants,
the newest-generation population synthesis of Emsenhuber et al. (2020a, b)
yields the same results, only with fewer synthetic gas giants and therefore
less statistically robust fits. of Mordasini et al. (2012b, a), for their
scenarios of “cold accretion” and of “hot-accretion” (see also Mordasini et
al., 2017). We call these fits respectively “Cold-population fit” and “Warm-
population fit”. Towards high accretion rates and masses, the radius increases
with either quantity, and the warm-population fit is larger than the cold-
population fit. Typical values are $R_{\mathrm{p}}=1.5$–3 $R_{\mathrm{J}}$ and
2–4 $R_{\mathrm{J}}$, with a stronger dependence of $R_{\mathrm{p}}$ on
$\dot{M}$ in the “hot” case. The “cold-” and “hot accretion” scenarios
represent extreme outcomes of the accretion process in which the entire
accretion energy is respectively radiated away at the shock or on the contrary
brought into the planet (Marley et al., 2007; Mordasini et al., 2012b; Marleau
et al., 2017, 2019). We assume that the radius depends only on the total
accretion rate and the mass but not on the filling factor.

A complication is that in reality, the accretion history, not only the
instantaneous values, likely sets the radius. This is suggested by the results
of Berardo et al. (2017), who find that during formation, planets can have a
radiative (and not convective) structure for a large fraction of their outer
mass layers. Thus the results of classical convective-planet structures that
let us write $R_{\mathrm{p}}=R_{\mathrm{p}}(\dot{M},M_{\mathrm{p}})$, with no
time dependence, might be a simplification. However, we effectively mitigate
this by considering the two different fits (“cold” and “hot”), and will find
that the radius does not have a major influence in any case.

Finally, there is another, minor parameter: the interior flux from the planet,
characterised by an effective temperature ${T_{\textnormal{int}}}$. We assumed
somewhat arbitrarily that ${T_{\textnormal{int}}}\approx 1100$ K for all
models. However, the interior flux is relevant only at the lowest accretion
rates, for which there will be essentially no absorption, so that in practice
it is not important.

###  3.3 Structure of the accretion flow

Figure 2: Overview of the structure of the accretion flow and of the shock
region (see labels). Top panel: temperature (double logarithmic scale),
showing the shock temperature $T_{0}$, the Zel’dovich spike up to $T\approx
T_{1}$, and the dust destruction front at radius ${R_{\textnormal{dest}}}$.
The shock defines the planet’s radius. The temperature flattening at
$T\approx{T_{\textnormal{dest}}}$ (e.g. Marleau et al., 2019) is not shown.
The next two panels focus on the dashed box. Middle panel: temperature near
the shock on a linear scale. Below the hydrodynamic jump, in the NLTE cooling
and settling zone, the “population temperature” of the electrons
${T_{\textnormal{pop}}}$ and the kinetic temperature ${T_{\textnormal{gas}}}$
differ at first. The temperature structure is simplified; Figure 8 of Vaytet
et al. (2013b) gives a more accurate schematic. Bottom panel: upward-moving
flux at one restframe wavelength within the H $\alpha$ line (red line), from
the microphysical models of Aoyama et al. (2018). We illustrate the line-
formation region (red) and the self-absorption (dark grey) in the settling
zone, and the absorption by the accretion flow, calculated in this work. The
features in the opacity (dotted grey curve; arbitrary linear scale) come from
the Doppler-shift gradient (Equation (15); Figure 4).

For all accretion geometries, the supersonic infalling matter makes up the
accretion flow999In Marleau et al. (2017, 2019), this whole region (out to
${R_{\textnormal{acc}}}$) is often called the “preshock region”, meant as a
synonym. However, for clarity, we keep here the term “preshock” for the layers
immediately before (i.e. upstream of) the shock, as is more common in the
literature. before hitting the planet’s surface, defined as the location of
the radiative hydrodynamical shock (Marleau et al., 2017, 2019). This is
illustrated in Figure 2. Immediately below the hydrodynamical shock is the
postshock region proper, a spatially thin “settling zone” and cooling region.
There, the gas is subsonic and gradually slows down, reaching hydrostatic
equilibrium at depths, where the structure is that of a non-accreting planet.
The postshock region consists of the layers close to but below the shock,
especially the thin cooling region. Note that an accreting planet does not
have an atmosphere in the classical sense (for an isolated object), or does so
at most only on the parts of its surface that are not accreting.

We now describe the mechanical and thermal structure of the accreting matter.
We assume that the supersonic accretion flow is essentially spherically
symmetric with purely radial motion. In general, the accretion is non-zero
within a region whose boundaries in angle are set by $f_{\mathrm{fill}}$. For
$f_{\mathrm{fill}}=1$ we have true spherical symmetry. Therefore, the velocity
and density in the accretion flow are given by

$\displaystyle v(r)=$ $\displaystyle\leavevmode\nobreak\
\sqrt{2GM_{\mathrm{p}}\left(\frac{1}{r}-\frac{1}{{R_{\textnormal{acc}}}}\right)}$
(2a) $\displaystyle=$ $\displaystyle\leavevmode\nobreak\
73\leavevmode\nobreak\
\mathrm{km}\,\mathrm{s^{-1}}\left(\frac{M_{\mathrm{p}}}{3\leavevmode\nobreak\
M_{\mathrm{J}}}\right)^{1/2}\left(\frac{r}{2\leavevmode\nobreak\
R_{\mathrm{J}}}\right)^{-1/2}\zeta^{1/2},$ (2b) $\displaystyle\rho(r)=$
$\displaystyle\leavevmode\nobreak\ \frac{\dot{M}}{4\pi
r^{2}f_{\mathrm{fill}}v(r)}$ (3a) $\displaystyle=$
$\displaystyle\leavevmode\nobreak\ 3\times 10^{-12}\leavevmode\nobreak\
\mathrm{g\,cm}^{-3}\frac{1}{f_{\mathrm{fill}}}\left(\frac{\dot{M}}{\upmu
M_{\mathrm{J}}\,\mathrm{yr}^{-1}}\right)$
$\displaystyle\left(\frac{M_{\mathrm{p}}}{3\leavevmode\nobreak\
M_{\mathrm{J}}}\right)^{1/2}\left(\frac{r}{2\leavevmode\nobreak\
R_{\mathrm{J}}}\right)^{-3/2}\zeta(r)^{-1/2},$ (3b)

where

$\zeta(r)\equiv\left(1-\frac{r}{{R_{\textnormal{acc}}}}\right),$ (4)

with $\zeta=0.8$ for ${R_{\textnormal{acc}}}=5R_{\mathrm{p}}$ and
$\zeta\rightarrow 1$ when ${R_{\textnormal{acc}}}\gg r$. For a given accretion
rate, the density depends on the filling factor but the velocity does not.
These are the classical formulae for an accretion flow (e.g. Calvet &
Gullbring, 1998; Zhu, 2015) and have been verified to apply to planets by
Marleau et al. (2017, 2019) through one-dimensional radiation-hydrodynamical
simulations. For reference, the shock Mach number is in the range
$\mathcal{M}\approx 10$–40 (Marleau et al., 2019), and the ram pressure of the
incoming gas is given by (e.g. Berardo et al., 2017)

$\displaystyle P_{\textnormal{ram}}=$ $\displaystyle\leavevmode\nobreak\
\rho(R_{\mathrm{p}})v(R_{\mathrm{p}})^{2}=\frac{\dot{M}v(R_{\mathrm{p}})}{4\pi
R_{\mathrm{p}}^{2}f_{\mathrm{fill}}}$ (5a) $\displaystyle\approx$
$\displaystyle\leavevmode\nobreak\ 171\leavevmode\nobreak\
\mathrm{erg\,cm}^{-3}\frac{1}{f_{\mathrm{fill}}}\left(\frac{\dot{M}}{\upmu
M_{\mathrm{J}}\,\mathrm{yr}^{-1}}\right)$
$\displaystyle\left(\frac{M_{\mathrm{p}}}{3\leavevmode\nobreak\
M_{\mathrm{J}}}\right)^{1/2}\left(\frac{R_{\mathrm{p}}}{2\leavevmode\nobreak\
R_{\mathrm{J}}}\right)^{-5/2}\zeta^{1/2}.$ (5b)

For the temperature structure of the flow, we use the results of Marleau et
al. (2017, 2019). They have found that the radiative precursor (Drake, 2006;
Vaytet et al., 2013b) of the shock extends to the Hill sphere (formally to
infinity). This means that the infalling gas is preheated by the radiation
escaping from the shock, which the work of Aoyama et al. (2018) suggests
occurs mainly through the absorption of Ly $\alpha$ photons since they carry
most of the shock emission flux. This contrasts to the stellar case, in which
the precursor region is thin and located close to the star (see e.g. Calvet &
Gullbring 1998; Colombo et al. 2019; de Sá et al. 2019) and is due to the
absorption of photoionising radiation. This infinite precursor implies that
the temperature profile is given by

$T(r)=T_{0}\left(\frac{r}{R_{\mathrm{p}}}\right)^{-1/2}.$ (6)

In turn, the equilibrium gas temperature at the shock101010A quick tool to
compute these temperature and density profiles is provided at
https://github.com/gabrielastro/St-Moritz, which also computes the time since
the beginning of free-fall and provides a more accurate temperature profile
than Equation (6) when the opacity is high (see Section 7.3). $T_{0}$ has the
value needed to radiate the mechanical energy converted to radiation as well
as the interior flux:

$\displaystyle acT_{0}^{4}$
$\displaystyle={F_{\textnormal{acc}}}+{\sigma}{T_{\textnormal{int}}}^{4}$ (7a)
$\displaystyle=\frac{GM_{\mathrm{p}}\dot{M}}{4\pi
R_{\mathrm{p}}^{3}f_{\mathrm{fill}}}\zeta+{\sigma}{T_{\textnormal{int}}}^{4},$
(7b)

where $v_{0}=v(R_{\mathrm{p}})$, $\rho_{0}=\rho(R_{\mathrm{p}})$, and $a$ and
${\sigma}$ are respectively the radiation and Stefan–Boltzmann constants.
Marleau et al. (2017) found that the entire mechanical energy goes into
radiation. Thus the radiation flux from accretion is
${F_{\textnormal{acc}}}=0.5\rho_{0}v_{0}^{3}$, leading to Equation (7b). When
$\dot{M}\approx 0$, the choice of ${T_{\textnormal{int}}}\approx 1100$ K sets
a minimum of $T_{0}\approx 800$ K (since $4{\sigma}=ac$;
${T_{\textnormal{int}}}$ is an effective temperature but $T_{0}$ a material
temperature).

Equation (7) uses the result from Marleau et al. (2017, 2019) that the
planetary accretion shock is supercritical and thus isothermal, in the sense
that upstream and downstream of the Zel’dovich spike the gas temperature is
set by the balance of the incoming flux of material energy and the outgoing
radiative flux. This single temperature $T_{0}$ is shown in Figure 2. The
shock is a “thick–thin” shock in the usual classification (e.g. Drake, 2006):
the radiation is diffusive below and free-streaming above the shock. (For
recent discussion of sub- and supercritical shocks, see Commerçon et al.
(2011) and Vaytet et al. (2013a, b), and importantly Drake (2007) for a brief
critical review of inadequate uses of these terms in the literature.) In the
limit that the kinetic energy dominates over the internal flux, the shock
temperature is (Marleau et al., 2019)

$\displaystyle T_{0}\approx$ $\displaystyle\leavevmode\nobreak\
1280\leavevmode\nobreak\ {\rm K}\leavevmode\nobreak\
\frac{1}{f_{\mathrm{fill}}^{1/4}}\left(\frac{R_{\mathrm{p}}}{2\leavevmode\nobreak\
R_{\mathrm{J}}}\right)^{-3/4}$ $\displaystyle\times\left(\frac{\dot{M}}{\upmu
M_{\mathrm{J}}\,\mathrm{yr}^{-1}}\right)^{1/4}\left(\frac{M_{\mathrm{p}}}{3\leavevmode\nobreak\
M_{\mathrm{J}}}\right)^{1/4}\zeta^{1/4}.$ (8)

Since we focus on a line, H $\alpha$, that carries a negligible fraction of
the total flux (Aoyama et al., 2018), we assume that the temperature in the
accretion flow is fixed and independent of the absorption or non-absorption of
the H $\alpha$ photons.

Note that Equation (6) assumes a radially constant luminosity in the accretion
flow given by

$\displaystyle L=$ $\displaystyle\leavevmode\nobreak\
{L_{\textnormal{acc}}}+L_{\mathrm{int}}=\frac{GM_{\mathrm{p}}\dot{M}}{R_{\mathrm{p}}}\zeta+L_{\mathrm{int}},$
(9a) $\displaystyle=$ $\displaystyle\leavevmode\nobreak\ 4\times
10^{-4}\leavevmode\nobreak\ {L_{\odot}}\leavevmode\nobreak\
\left(\frac{\dot{M}}{\upmu
M_{\mathrm{J}}\,\mathrm{yr}^{-1}}\right)\left(\frac{M_{\mathrm{p}}}{3\leavevmode\nobreak\
M_{\mathrm{J}}}\right)\left(\frac{R_{\mathrm{p}}}{2\leavevmode\nobreak\
R_{\mathrm{J}}}\right)^{-1}\zeta+L_{\mathrm{int}},$ (9b)

where $L_{\mathrm{int}}=4\pi r^{2}{\sigma}{T_{\textnormal{int}}}^{4}$ is the
interior luminosity from the planet. Marleau et al. (2019) showed that a
constant $L(r)$ is a good approximation. However, Equation (6) does not
reflect the region of a flatter, almost constant, $T(r)$ profile where the
dust is destroyed, near a destruction temperature
${T_{\textnormal{dest}}}=1220\times{\rho_{-11}}^{0.0195}$ K, where
$\rho_{-11}\equiv\rho\times 10^{11}\leavevmode\nobreak\
\mathrm{cm}^{3}\,\mathrm{g}^{-1}$ (Isella & Natta, 2005). In particular,
Equation (6) assumes that the frequency-averaged radiation is free-streaming
throughout. Thus, if the accretion flow is devoid of dust, Equation (6) will
hold, while in the presence of dust this will slightly underestimate the
temperature at a given radial position (and thus density). However, at these
low temperatures $T\lesssim 1200$ K, the gas opacity is small anyway and we
find (see Section 4) that, in the relevant part of the parameter space, the H
$\alpha$ is not extincted. This justifies approximately the simplification.

We have scaled Equations (2), (3), and (3.3) using reasonable values (with
$\dot{M}$ close to the minimum $\dot{M}\gtrsim 5\times
10^{-7}\leavevmode\nobreak\ M_{\mathrm{J}}\,\mathrm{yr}^{-1}$ derived by
Hashimoto et al., 2020 for PDS 70 b) but one should remember that the
parameter space is large, with especially $\dot{M}$, $M_{\mathrm{p}}$, and
$f_{\mathrm{fill}}$ varying by orders of magnitude. Profiles in $\rho$–$T$
space are shown for a range of parameters in Figure 3. Densities are $\rho\sim
10^{-16}$–$10^{-9}$ g cm-3 and temperatures $T\sim 100$–$10^{4}$ K, with
approximately $T\propto\rho^{1/3}$, as can be seen from Equations (3) and
(3.3) for $\zeta\approx 1$. An extensive discussion of the profiles, in
particular against radial distance from the planet, is given in Marleau et al.
(2019).

Figure 3:  Structure of the accretion flow (solid lines) and gas opacity
(background greyscale). The profiles are for a grid of accretion rates
(isochromatic curve groups; from $\dot{M}=3\times 10^{-4}$ (top right) to
$3\times 10^{-8}\leavevmode\nobreak\ M_{\mathrm{J}}\,\mathrm{yr}^{-1}$ (bottom
left)) and masses ($M_{\mathrm{p}}=1$, 3, 5, 10, 20 $M_{\mathrm{J}}$ from
bottom to top within each group) in the Polar-Cold case. Each profile begins
on the left at ${r_{\rm max}}\approx 100R_{\mathrm{p}}$ and ends at the shock
at $R_{\mathrm{p}}$ (dot), at $(\rho_{0},T_{0})$. Only the continuum opacity
at H $\alpha$ (no resonance; see text) is shown averaged over $\Delta v=\pm
50\leavevmode\nobreak\ \mathrm{km}\,\mathrm{s^{-1}}$. The dust destruction
line ${T_{\textnormal{dest}}}(\rho)$ is indicated (Isella & Natta, 2005; long-
dashed line), and line segments show pressures $P=10^{-3}$, 1, $10^{3}$ erg
cm-3. Black dotted lines show where half of the hydrogen is molecular, atomic,
or ionised. The opacity depends mostly on temperature, and most models
$(\dot{M},M_{\mathrm{p}})$ include a region of high opacity, often near the
shock.

###  3.4 Spectral profiles of shock line emission

The hydrogen line emission at the shock is taken from the models of Aoyama et
al. (2020). These apply the non-LTE radiation-hydrodynamical simulations of
Aoyama et al. (2018) to the shock at the surface of an accreting planet.
Through detailed calculations of chemical reactions and electron transitions
in hydrogen atoms, the Aoyama et al. (2018) models calculate the cooling in
the disequilibrium region immediately below the hydrodynamical shock,
corresponding roughly to the downstream part of the Zel’dovich spike (Vaytet
et al., 2013b). See Figure 2c. This provides high-resolution profiles and
line-integrated fluxes for 55 hydrogen lines in the series of Lyman, Balmer,
Paschen, Brackett, etc., which are emitted from the shock towards the
observer. These line profiles thus serve as the input for our calculation of
the radiative transfer (Equation (14c) below), and will be seen as the black
dashed lines in Figures 6–8. The line-integrated luminosity
${L_{\textnormal{H}\,\alpha}}$ as a function of $\dot{M}$ and $M_{\mathrm{p}}$
is shown in Figure 7 of Aoyama et al. (2020) and constitutes one of their main
results.

The Aoyama et al. (2018) microphysical shock models depend only on the number
density of hydrogen protons $n_{0}$ and the velocity $v_{0}$, both evaluated
immediately before the shock. Thus $v_{0}=v(R_{\mathrm{p}})$ from Equation (2)
and $n_{0}=X\rho(R_{\mathrm{p}})/m_{\mathrm{H}}$ from Equation (3), where $X$
is the hydrogen mass fraction and $m_{\mathrm{H}}$ is the hydrogen atomic
mass. The number ratios are given by
$\textrm{H}:\textrm{He}:\textrm{C}:\textrm{O}=1:10^{-1.07}:10^{-3.48}:10^{-3.18}$
(Cox, 2000). The present iteration of the models (Aoyama et al., 2018) covers
a range $n_{0}=10^{9}$–$10^{14}$ cm-3 and $v_{0}=20$–200
$\mathrm{km}\,\mathrm{s^{-1}}$.

With the fits of the planet radius
$R_{\mathrm{p}}=R_{\mathrm{p}}(\dot{M},M_{\mathrm{p}})$ that were mentioned in
Section 3.2, Equations (2) and (3) evaluated at $r=R_{\mathrm{p}}$ relate the
macrophysical parameters $(\dot{M},M_{\mathrm{p}},f_{\mathrm{fill}})$ to the
microphysical ones, $(n_{0},v_{0})$. For $f_{\mathrm{fill}}=1$, this leads to
ranges of $n_{0}\sim 10^{10}$–$10^{14}$ cm-3 and $v_{0}\sim 50$–200
$\mathrm{km}\,\mathrm{s^{-1}}$ for typical
$(\dot{M},M_{\mathrm{p}},R_{\mathrm{p}})$ values. For small filling factors
($f_{\mathrm{fill}}\lesssim 10\leavevmode\nobreak\ \%$) or the largest
accretion rates ($\dot{M}/f_{\mathrm{fill}}\gtrsim 1\times
10^{-4}\leavevmode\nobreak\ M_{\mathrm{J}}\,\mathrm{yr}^{-1}$ for the cold-
population radius fit and $\dot{M}/f_{\mathrm{fill}}\gtrsim 3\times
10^{-3}\leavevmode\nobreak\ M_{\mathrm{J}}\,\mathrm{yr}^{-1}$ for
$M_{\mathrm{p}}\gtrsim 10\leavevmode\nobreak\ M_{\mathrm{J}}$ in the warm-
population fit), the resulting $n_{0}$ was higher than available in the
previously mentioned grid and we extrapolated the spectra in preshock density.
Nevertheless, this should not introduce much inaccuracy on the spectral shape
or total flux.

The H $\alpha$ line is surrounded by a continuum, set by the flux from the
planet’s interior. However, it is usually insignificant compared to the strong
line emission. Therefore, we will neglect the continuum in the main part of
this work, but do discuss this approximation in Section 4.3.3 and Appendix A.

###  3.5 Radiative transfer

We calculate the radiative transfer only within the accreting region, assuming
spherical symmetry. For the Polar or MagAcc case, this ignores edge effects
for the radiation travelling close to the walls of the accretion cone or the
accretion columns, respectively; the matter and radiation properties are
assumed to be independent of the angle within that region. This simplification
should lead only to a modest overestimate of the amount of absorption, and is
in line with other approximations in our approach.

In spherical symmetry, the radial radiative flux $F_{\lambda}$ is given in
general by

$F_{\lambda}(r)=2\pi\int_{-1}^{1}I_{\lambda}(r,\mu)\mu\,d\mu,$ (10)

where $I_{\lambda}$ is the specific intensity in a given direction and
$\mu=\cos\xi$, with $\xi$ the angle between the given direction and the radial
direction. The intensity $I_{\lambda}$ is set by the radiation transfer
equation, which reads (Davis et al., 2012)

$\hat{n}\cdot\nabla
I_{\lambda}(\hat{n})=\frac{dI_{\lambda}}{ds}=\alpha_{\lambda}\left(S_{\lambda}-I_{\lambda}\right),$
(11)

where $\hat{n}$ is a unit vector defining a direction,
$S_{\lambda}=j_{\lambda}/\alpha_{\lambda}$ is the source function with
$j_{\lambda}$ the emissivity, $\alpha_{\lambda}$ is the coefficient of
extinction including scattering and true absorption, and $s$ is the position
along the ray defined by $\hat{n}$. The middle term in Equation (11) is
written for the specific intensity in the direction of the ray.

We assume that the accretion flow is in local thermodynamic equilibrium (LTE),
i.e. that collisions determine the electron populations in the accretion flow.
Therefore, $S_{\lambda}=B_{\lambda}$ by Kirchhoff’s law, with $B_{\lambda}$
the Planck function.

In general, Equation (11) must be integrated numerically. However, the peak
intensity of the H $\alpha$ line corresponds to that of a blackbody usually at
a much higher temperature than the gas anywhere in the accretion flow, so that
$I_{\lambda}\gg S_{\lambda}=B_{\lambda}$. Therefore, in Equation (11) the
absorption term dominates over the emission, leading to

$\frac{dI_{\lambda}}{ds}\approx-\alpha I_{\lambda}.$ (12)

Since we are concerned with the flux at the observer (at infinity), $r\gg
R_{\mathrm{p}}$ and $dI_{\lambda}/ds$ needs to be integrated only along a
radial ray. This effectively neglects limb darkening. As mentioned before, we
assume that the radiative quantities are independent of angle within the
accretion flow. Therefore, we only need to integrate Equation (12) radially
outwards from the shock at $R_{\mathrm{p}}$. The solution is
$I_{\lambda}(r)=I_{\lambda}(R_{\mathrm{p}})\exp\left(-\Delta\tau_{\lambda}(r)\right)$,
where

$\Delta\tau_{\lambda}(r)\equiv\int_{R_{\mathrm{p}}}^{r}\alpha_{\lambda}(r^{\prime})\,dr^{\prime}.$
(13)

With this, Equation (10) implies that the flux at large $r$ is

$\displaystyle F_{\lambda}$
$\displaystyle=2\pi\int_{-1}^{1}I_{\lambda}(R_{\mathrm{p}})e^{-\Delta\tau_{\lambda}}\mu\,d\mu$
(14a) $\displaystyle\approx 2\pi
I_{\lambda}(R_{\mathrm{p}})e^{-\Delta\tau_{\lambda}}\int_{1-0.5(R_{\mathrm{p}}/r)^{2}}^{1}\mu\,d\mu$
(14b)
$\displaystyle=F_{\lambda}(R_{\mathrm{p}})\frac{R_{\mathrm{p}}^{2}}{r^{2}}e^{-\Delta\tau_{\lambda}},$
(14c)

taking $I_{\lambda}(R_{\mathrm{p}})$ for Equation (14b) to be constant
(neglecting limb darkening) and non-zero only over the small angle
$\Delta\xi\approx R_{\mathrm{p}}/r$ (around $\hat{n}$) subtended by the
planet, so that also $F_{\lambda}(R_{\mathrm{p}})=\pi
I_{\lambda}(R_{\mathrm{p}})$. We use Equation (14c) to calculate the flux at
the observer throughout this work and do in Appendix A a comparison with the
full solution to Equation (11). For most cases the approximation is excellent.

###  3.6 Gas opacity

The coefficient of absorption $\alpha_{\lambda}=\kappa_{\lambda}\rho$ (with
dimensions of inverse length), where $\kappa_{\lambda}$ is the opacity, is
divided into two contributions for the gas: the H $\alpha$ resonant (i.e.
particularly strong) opacity due to the electrons in the $n=2$ quantum energy
level, as well as a (pseudo-)continuum, made up of a true continuum and the
superposition of many line wings. The resonant and continuum components are
described in the following subsections.

The Doppler shift due to the bulk motion of the infalling gas is taken into
account by evaluating for a given observer-frame (restframe) frequency $f$ the
opacity at frequency

$f^{\prime}(v)=f\left(1+\frac{v}{c}\right),$ (15)

where $v$ is the velocity at a given position. This explains the strong
variations of the monochromatic opacity shown as a grey dashed line in Figure
2c.

Given the massive uncertainties on the dust absorption, we treat it separately
in Section 5. Note that for simplicity we do not include scattering as this
would introduce a disproportionate level of complexity (especially for the
realistic case of anisotropic scattering) compared to the rest of our
approach.

####  3.6.1 Resonant opacity

We use standard formulae to calculate the resonant opacity (e.g. Carson, 1988;
Hilborn, 2002; Sharp & Burrows, 2007; Wiese & Fuhr, 2009; Hubeny & Mihalas,
2014). Since we deal with temperatures much lower than what corresponds to H
$\alpha$, we do not include stimulated emission, and approximate the ground
state to be dominantly populated, which implies that the partition function is
$Q(T)\approx 2$. With this approximation, we do not need to handle the well-
known divergence of $Q$, which, however, can be corrected easily by the
occupation probability formalism (Hubeny et al., 1994). We assume a Doppler
line profile, appropriate for our temperature regime.

The strength of the resonant opacity is proportional to the number of
absorbers, calculated from the Saha equation (e.g. D’Angelo & Bodenheimer,
2013). Only for the highest values of $\dot{M}/f_{\mathrm{fill}}$ and
$M_{\mathrm{p}}$ is the gas at least partially ionised when reaching the
shock; for most $(\dot{M},M_{\mathrm{p}})$ combinations, the hydrogen is
atomic (see black dotted lines in Figure 3). At low
$\dot{M}/f_{\mathrm{fill}}$ the hydrogen is molecular.

####  3.6.2 Continuum opacity

For the continuum gas opacity, we use very-high-resolution LTE absorption
coefficients with a constant step size in wavenumber of 0.01 cm-1,
corresponding to a spectral resolution $R=1.5\times 10^{6}$. This is
sufficient to resolve line cores over the relevant $(P,T)$ domain (Mollière et
al., 2015, 2019). We assume a solar-metallicity (Asplund et al., 2009)
mixture. The opacities of the individual species were calculated with HELIOS-K
(Grimm & Heng, 2015; Grimm et al., 2021). All atoms and ions up to a proton
number $Z=40$ are included, and so are the continua of H- and collision-
induced absorption (CIA) of H2–H2, He–H, and H2—He. Molecular absorption is
included for H2O , CO, TiO, SH, and VO. Abundances were determined by chemical
equilibrium calculations for the gas phase as provided by FastChem (Stock et
al., 2018). Where needed, we extrapolate the fit of the equilibrium constants
beyond their tabulated range of $T=100$–6000 K to calculate opacities from
$T=100$ to $10^{4}$ K. A detailed description of chemical and opacity data for
the atoms and ions can be found in Hoeijmakers et al. (2019). Molecular
absorption coefficients are based on the Exomol line lists where available
(Polyansky et al., 2018; McKemmish et al., 2016, 2019; Gorman et al., 2019)
and on HITEMP (Li et al., 2015) otherwise. The CIA data are taken from the
HITRAN database (Karman et al., 2019).

Assuming solar metallicity might not be accurate because (i) important opacity
sources could be locked up in large dust grains that remain in the midplane
despite meridional circulation and thus do not accrete onto the planet; (ii)
molecular abundances at a given position might not correspond to the chemical-
equilibrium values at the local density and temperature, if the dynamical
timescale is shorter than the chemical timescale (Booth & Ilee, 2019; Cridland
et al., 2020); and (iii) the accretion luminosity of the planet, in particular
in the UV via photochemical reactions, can also affect the chemical abundances
(Rab et al., 2019). Whether any of these effects will increase or decrease the
opacity cannot be said in general and is likely not robust against the details
of the modelling.

The opacity tables used in this work assume that all molecules and atoms are
in the gas phase even at low temperatures. Whether for equilibrium or non-
equilibrium abundances, this should not be of consequence for the absorption
by the gas because below $T\approx 1000$ K, the gas opacity is very low (see
Figure 3). Thus we keep this simplification, keeping in mind that the exact
molecular abundances are uncertain, as discussed previously.

A formal limitation is that the absorption coefficients of molecules are
tabulated only up to $T\approx 3000$ K, with the value at the highest
temperature used for higher temperatures. However, in practice this is not an
issue because molecules are usually not important anymore at such high
temperatures due to dissociation. For atoms and ions, the tables go up to
$T=6100$ K. We only consider thermal broadening and the natural line widths,
except for the case of the Na and K resonance line wings, where we use the
pressure-broadened line profiles provided by Allard et al. (2016) and Allard
et al. (2019). We note, however, that pressure broadening is not important for
this study given the range of relevant pressures ($P\sim 10^{-12}$–10-2 bar;
see Figure 3, discussed below).

We will compute the monochromatic radiative transfer (Equation (14c)) and then
integrate over the line width to obtain the total line extinction. However,
the (flux-)averaged opacity provides an estimate of the strength of the
absorption. The wavelength range is small (the width of the emerging line is
of the order of 1/2000th of the wavelength, which is
$\lambda_{\mathrm{H}\,\alpha}=656.464$ nm in vacuum; Wiese & Fuhr, 2009), so
that it is even sufficient to compute the direct mean opacity because the
Planck function does not vary much. Indeed, at $T\approx 10^{4}$ K, the
highest temperatures, the Planck function changes near H $\alpha$ over a scale
of $H_{B}\equiv B_{\lambda}/(dB_{\lambda}/d\lambda)\approx
260\leavevmode\nobreak\ \mathrm{nm}$, which is much larger than the line width
$\Delta v/c\times\lambda_{\mathrm{H}\,\alpha}\approx 0.1$ nm for $\Delta
v=50\leavevmode\nobreak\ \mathrm{km}\,\mathrm{s^{-1}}$. Even at $T\approx
10^{5}$ K, the scale would still be $H_{B}\approx 160$ nm.

Figure 3 displays the continuum gas opacity near
$\lambda_{\mathrm{H}\,\alpha}$ averaged directly111111I.e.,
$\langle\kappa_{\lambda}\rangle=\int\kappa_{\lambda}\,d\lambda/\int d\lambda$.
Our resolution is high enough for this to yield the correct result (Malygin et
al., 2014). over $\Delta v=\pm 50\leavevmode\nobreak\
\mathrm{km}\,\mathrm{s^{-1}}$. Shown is also the $\rho$–$T$ structure of the
accreting gas for SpherAcc-Cold as an example. This wavelength range covers
most of the flux emerging from the shock for all models, as Figures 6–8 will
show. The opacity is at most $\kappa\sim 0.1$ cm2 g-1 at H $\alpha$
(greyscale), but note that for $T\approx 800$–1200 K the wavelength dependence
(not shown) is large. In Figure 3, the resonant opacity is not included
because of its strong wavelength and temperature dependence; because of the
Doppler shift (Equation (15)), even an average would not provide a meaningful
estimate of the typical opacity at a $(\rho,T)$ position in the structure.

We can now combine these elements to calculate the line shape of accreting
planets, which we do in the following sections.

##  4 Absorption by the gas

We have calculated the absorption of H $\alpha$ (Equation (14c)) for a grid of
accretion rates $\dot{M}$ and masses $M_{\mathrm{p}}$ for the different
accretion geometries discussed in Section 3.1: a spherically symmetric inflow
(SpherAcc), accretion onto the polar regions (Polar), and magnetospheric
accretion (MagAcc). They are illustrated in Figure 1 and summarised in Table
1. In each case we take the emerging flux from the Aoyama et al. (2018) models
and integrate the extinction along a radial radiative path starting at the
shock and going out to ${r_{\rm max}}$. The density and temperature are as
shown in Figure 3 for one geometry.

Section 4.1 presents and discusses in detail one combination of $\dot{M}$,
$M_{\mathrm{p}}$, and $R_{\mathrm{p}}$ in the Polar-Cold geometry. Then,
Sections 4.2 and following present the results for the grid of models.

###  4.1 One example of gas absorption

Figure 4:  Top panel: H $\alpha$ line profile at spectral resolution
$R=1.5\times 10^{6}$ for one case in the Polar-Cold scenario with
$\dot{M}=3\times 10^{-5}\leavevmode\nobreak\
M_{\mathrm{J}}\,\mathrm{yr}^{-1}$, $M_{\mathrm{p}}=10\leavevmode\nobreak\
M_{\mathrm{J}}$, and $R_{\mathrm{p}}=2.04\leavevmode\nobreak\ R_{\mathrm{J}}$
(solid red line) when observing into the accreting area at 150 pc. The flux in
the absence of absorption by the incoming matter is also shown (dashed green
line). The background colour shows as a function of distance from the planet
(right vertical axis) and observer-frame wavelength the strength of the
absorption by the gas $\dot{\tau}(r,\lambda)$ (Equation (16)), capped at
$\dot{\tau}=5$. Few regions have a higher $\dot{\tau}$, the main one being the
H i resonance near the shock (reaching $\dot{\tau}\approx 5\times 10^{3}$).
Indicated are also the central peak of the H $\alpha$ line in the rest frame
$\lambda_{\mathrm{H}\,\alpha}$ (vertical dotted grey line) and, as a function
of radial distance from the planet, the wavelength that becomes blueshifted to
$\lambda_{\mathrm{H}\,\alpha}$ (see Equation (15); curved solid grey line). No
dust nor ISM extinction is included. Over the brighter region at $r=15$–30
$R_{\mathrm{J}}$, the temperature is $T\approx 2000$–1500 K and the pressure
$P\approx 1$–0.2 erg cm-3 (cf. Figure 3). Bottom panel: Extinction
$A_{\lambda}$ from the accretion flow against velocity offset relative to H
$\alpha$.

We show the spectrally resolved line flux in Figure 4 for one case in the
Polar accretion geometry, with $\dot{M}=3\times 10^{-5}\leavevmode\nobreak\
M_{\mathrm{J}}\,\mathrm{yr}^{-1}$, $M_{\mathrm{p}}=10\leavevmode\nobreak\
M_{\mathrm{J}}$, and $R_{\mathrm{p}}=2.04\leavevmode\nobreak\ R_{\mathrm{J}}$.
The flux is the one seen by an observer looking into the accretion cone
(within $\theta_{\mathrm{max}}=46{\mbox{\textdegree}}$ of the pole; see Figure
1) and at 150 pc, similar to the distance of well-known young star-forming
regions such as Taurus, Lupus, or $\rho$ Ophiucus–Upper Scorpius. As
throughout this work, no absorption by the interstellar medium (ISM) is
included, to separate the two effects121212For the PDS 70 planets ($d=113$
pc), Müller et al. (2018) derived a $V$-band extinction $A_{V}\approx 0.05$
mag, or $A_{R}=0.04$ mag (at the $R$ band, which is around H $\alpha$) using
Cardelli et al. (1989) with $R_{V}=5$.. Due to the absorption by the accreting
gas, the line-integrated flux drops by about 32 %, corresponding to an
extinction131313Note that $A\equiv 2.5\log_{10}(F_{0}/F)$ and
$\Delta\tau=\ln(F_{0}/F)=\ln(10)\times\log_{10}(F_{0}/F)$ are nearly equal
($\Delta\tau=0.92A$) since $\ln(10)=2.30\approx 2.5$.
$A_{R}/\left(\mathrm{mag}\right)\approx\Delta\tau=0.4$. We now discuss the
wavelength-dependent extinction.

In contrast with the line shape leaving the shock, the observable line shows
small-scale structures. What dominates the absorption can be easily seen by
looking at a measure of the strength of the local absorption in the flow
$\dot{\tau}$, the optical depth per decade in $r$. It is given by

$\dot{\tau}(r,\lambda)\equiv\frac{d\tau}{d\log_{10}r}=\ln(10)\,\kappa(r,\lambda)\rho(r)r.$
(16)

This is indicated as a colourscale in Figure 4. Thus, on a logarithmic radial
scale every unit $\dot{\tau}$ contributes equally to the total extinction at a
given wavelength. A radial cut at one restframe wavelength is shown in Figure
2c. The obvious curvature of the opacity features is due to the Doppler shift
gradient, described by Equation (15).

The quantity $\dot{\tau}$ reveals that most of the absorption occurs at
$r=15$–30 $R_{\mathrm{J}}$. There, the temperatures are $T\approx 2000$–1500 K
respectively (not shown), a factor $\approx 2$ lower than the shock
temperature. The extinction comes mainly from the continuum opacity with a
small contribution from the resonant hydrogen absorption. In general, because
of the high temperatures in the postshock region (Aoyama et al., 2018), the
emerging line is usually much broader than the thermal broadening of the
incoming gas. Equation (15)) shows that there are only redshifted absorbers;
layers moving towards the planets absorb photons blueshifted to H $\alpha$ in
the frame of the accreting gas (see the curved grey line in Figure 4).
Therefore, resonant absorption (by the incoming $n=2$ electrons) can occur
only redward of the rest central wavelength, approximately equal to the centre
of the line emerging from the postshock region. This is the yellow region in
Figure 4.

Because the velocity decreases away from the planet, there is the possibility
that the complete red wing be absorbed by the incoming $n=2$ electrons.
However, the temperature and thus the number fraction of absorbers decrease
outwards much faster than the Doppler shift. Therefore, only a small
wavelength range can be absorbed by the incoming $n=2$ electrons.Also, as we
calculate in Appendix A, the emission by the accretion gas in that wavelength
range would be important, so that in reality the increase in the extinction
would be smaller than what Figure 4b suggests (see Figure 12a).

Figure 4 shows that the Doppler shift gradient smears the strong wavelength
dependence of $\dot{\tau}(r,\lambda)$ for the integrated extinction and that
the resulting absorption (bottom panel) has less structure. However, in this
example, there happens to be a clear downward slope in the extinction from
656.45 to 656.55 nm ($\Delta v\approx 0$–$100\leavevmode\nobreak\
\mathrm{km}\,\mathrm{s^{-1}}$), which exacerbates the asymmetry in the input
line profile141414This is partly due to a clear resonance near
$\lambda_{\mathrm{H}\,\alpha}+0.05$ nm, present from large $r$ down to
$r\approx 4\leavevmode\nobreak\ R_{\mathrm{J}}$ ($T\approx 3000$–4000 K).
Comparing with Sharp & Burrows (2007), it likely comes from TiO or maybe VO..
The result is a crudely gaussian-looking blue wing but a linearly decreasing
red wing. This leads to an apparent offset in the line peak but only by 25
$\mathrm{km}\,\mathrm{s^{-1}}$, much less than the line width.

Our temperature structure is based on frequency-integrated equations and
supported by the grey flux-limited diffusion (FLD) simulations of Marleau et
al. (2017, 2019). FLD is an approximate method (e.g. Ensman, 1994; Turner &
Stone, 2001), but the temperature structure should be robust because it
reflects global energy conservation151515As a corroboration, in the stellar
context Vaytet et al. (2013a) found that non-grey collapse simulations, which
also feature a shock, yielded the same structures and evolution as a
frequency-averaged approach.. In particular, in our case the temperature must
drop from a value of order $\approx T_{0}$ (Equation (7)) to a much lower
value in the CSD and therefore cross at some point this range $T\approx
2000$–1500 K where the opacity is particularly strong. It depends on the
density only weakly (see Figure 3), and for a given temperature profile $T(r)$
and thus161616This ignores the density dependence of dust opacity transitions.
opacity profile $\kappa(r)$, $\dot{\tau}$ goes as $\rho r\propto r^{-1/2}$ (in
the limit $\zeta=1$; Equations (3) and (16)), which is not a strong
dependence. Thus if the temperature of maximum absorption were at another
distance from the planet, the integrated absorption should be similar. The
frequency dependence would be different (because of the radial dependence of
the velocity) but only quantitatively. Within the other assumptions, these
results are somewhat robust since the maximum absorption does not occur in the
outermost parts of the flow, where the accretion geometry is much more
uncertain.

We have not considered absorption by the dust here. Taking the approximate
expression for $\rho$ valid in the limit ${R_{\textnormal{acc}}}=\infty$,
Equation (3) with $\zeta=1$, the gas column density is

$\Sigma=\int_{R_{\mathrm{p}}}^{r_{\rm max}}\rho\,dr\approx\frac{\dot{M}}{2\pi
f_{\mathrm{fill}}\sqrt{2GM_{\mathrm{p}}R_{\mathrm{p}}}}$ (17)

since ${r_{\rm max}}\gg R_{\mathrm{p}}$, which yields $\Sigma=5$ g cm-2 for
this example. Thus, the dust will not be able to absorb much radiation if the
dust opacity (cross-section per gram of gas, not of dust)
$\kappa_{\mathrm{dust,\,H}\,\alpha}\equiv
f_{\textrm{d/g}}\kappa_{\bullet,\,\mathrm{H}\,\alpha}\ll 1/\Sigma\approx 0.2$
cm2 g-1, where $\kappa_{\bullet,\,\mathrm{H}\,\alpha}$ is the dust material
opacity (cross-section per gram of dust) at H $\alpha$. Comparing to the
opacity values of Woitke et al. (2016), who study the effect of the variation
of several relevant opacity parameters, this situation seems possible.
However, the parameter space is large and the opacity itself is very
uncertain. We explore the absorption by the dust in more detail and
systematically in Section 5.

In the following subsections, we discuss the results from the grid of models
covering the relevant planet-formation parameter space (see Section 3.2), as
discussed at the beginning of this section.

###  4.2 Line-integrated fluxes

Figure 5:  Percent reduction in the line-integrated flux for the SpherAcc,
Polar, and MagAcc accretion geometries (top to bottom rows) from the gas
opacity only. Shown are the results for the warm- (left column) and cold-
population radii (right column). The dotted lines highlight an extinction
$A_{R}=0.5$, 1, 2, and 4 mag (bottom to top). At a given accretion rate, high-
mass planets suffer less from absorption. For moderate $\dot{M}\lesssim
10^{-5}\leavevmode\nobreak\ M_{\mathrm{J}}\,\mathrm{yr}^{-1}$, the extinction
is at most $A_{R}\approx 0.5$ mag. For SpherAcc and Polar, both radius fits
yield similar results.

The first two rows of Figure 5 display the relative drop in the flux $\Delta
F$ across the accretion flow for the SpherAcc and Polar geometries. The drop
is mainly an increasing function of $\dot{M}/f_{\mathrm{fill}}$ and ranges
between $\Delta F=0\leavevmode\nobreak\ \%$ (no absorption) and nearly $\Delta
F=100$ % over the parameter space considered. Already for $\dot{M}\approx
3\times 10^{-5}\leavevmode\nobreak\ M_{\mathrm{J}}\,\mathrm{yr}^{-1}$, there
is noticeable absorption with $\Delta F\approx 20$–50 % depending on the mass,
with more absorption at lower planet masses. This corresponds to an extinction
$A_{R}\approx 2$–4 mag. The increase of extinction with decreasing planet mass
(and thus, at a given $\dot{M}$, with the gas column density) can be seen from
Equation (17). Also, for the three highest $\dot{M}$ values in Figure 3, the
opacity increases with decreasing temperature (due to the contribution from
water; Marleau et al., 2017) and increasing density, that is, decreasing mass.
For moderate $\dot{M}\lesssim 10^{-5}\leavevmode\nobreak\
M_{\mathrm{J}}\,\mathrm{yr}^{-1}$, the extinction is at most $A_{R}\approx
0.5$ mag.

Interestingly, the dependence of $\Delta F$ on the mass becomes larger towards
smaller filling factors. The choice of warm- or cold-population radii barely
changes the outcome but the trend is as expected: $\Delta F$ is larger for the
cold-population radii since they are smaller, leading to higher preshock
temperatures and densities ($\tau\propto\rho$), with the temperature peaking
at a few thousand kelvin (see Figure 3).

The MagAcc scenario, shown in the third row, leads to qualitatively similar
but quantitatively different results. The minimum accretion rate needed to
have significant absorption ($A_{R}\gtrsim 1$ mag) can be as low as
$\dot{M}\sim 3\times 10^{-6}\leavevmode\nobreak\
M_{\mathrm{J}}\,\mathrm{yr}^{-1}$, at low masses $M_{\mathrm{p}}\approx 1$–5
$M_{\mathrm{J}}$. The mass dependence of $\Delta F$ is stronger for the warm-
population radii, and for the cold-population radii $\Delta F$ shows clear
non-monotonic behaviour. In particular, there is a “window” near
$M_{\mathrm{p}}=3$–7 $M_{\mathrm{J}}$ in which the absorption corresponds only
to $A_{R}\lesssim 0.5$ mag even for a colossal accretion rate
$\dot{M}=10^{-4}\leavevmode\nobreak\ M_{\mathrm{J}}\,\mathrm{yr}^{-1}$; for
only slightly smaller masses of $M_{\mathrm{p}}\approx 1$–2 $M_{\mathrm{J}}$,
the extinction is $A_{R}\gg 4$ mag. Thus these $M_{\mathrm{p}}\approx
5\leavevmode\nobreak\ M_{\mathrm{J}}$ planets could be particularly
observable. However, this depends on the viewing geometry.

Overall, these results show that absorption can be significant (several
magnitudes of extinction) at higher accretion rates or for low-mass planets in
the MagAcc case, particularly for smaller (cold-population) radii. However,
the extinction from the gas is negligible if $\dot{M}\lesssim 3\times
10^{-6}\leavevmode\nobreak\ M_{\mathrm{J}}\,\mathrm{yr}^{-1}$.

The outer integration limit for the absorption calculation (i.e. in Equation
(14c)), ${r_{\rm max}}$, is a poorly known quantity. However, one can argue
that this is of little consequence: Figure 4 suggests that most of the
absorption, if there is any, occurs at tens, not hundreds of Jupiter radii.
This is corroborated heuristically by Figure 3, which shows that profiles with
$\dot{M}\gtrsim 3\times 10^{-6}\leavevmode\nobreak\
M_{\mathrm{J}}\,\mathrm{yr}^{-1}$ cross the high-opacity region near the shock
in density space, and thus also in radial distance from the planet since
$\rho$ is a monotonic function of radius for $r\ll{R_{\textnormal{acc}}}$
(Equation (3)).

###  4.3 Line profiles

Figure 6:  Effect of the extinction by the accreting gas on the H $\alpha$
profile. Shown is the flux for sources at 150 pc as a function of planet mass
and accretion rate (outer axes) for the SpherAcc-Warm case of Table 1. For
each subpanel, we plot the profile without extinction (black dashed line) and
the profile after passing through the accreting material (red solid line). No
ISM absorption is considered. The observable profiles are also shown convolved
with the resolution of MUSE ($R=2516$; dashed pale red line), and of VIS-X
($R=15,000$; dashed dark red line). The heated photosphere (BT-Settl model,
with ${T_{\textnormal{eff}}}$ from the fit of Aoyama et al., 2020; green
dotted line and label) is too weak to be seen in any panel. The horizontal
axes are the velocity offset from the line centre. The flux and velocity
ranges differ from panel to panel.  Figure 7:  As in Figure 6, but for the
Polar-Warm case. The observer is looking into the accreting regions, along the
accretion flow. In the top row, the flux without absorption $F_{0}$ (black
dashed line) has been reduced by a factor of ten to make the profiles with
absorption visible.

Figure 8:  As in Figure 6, but for the MagAcc cases (cold and warm; left and
right groups, respectively). The mass in $M_{\mathrm{J}}$ and accretion rate
in $M_{\mathrm{J}}\,\mathrm{yr}^{-1}$ are indicated in each subpanel. The
spectra are for an observer looking into the accretion flow towards the base
of the shock or more generally for radiation passing through only the layers
closest to the planet and escaping (see Figure 1). Possible rotation and
inclination (between the observer and the accretion column) are not taken into
account. The strong photospheric absorption (green) in some cases may not be
realistic; see text.

####  4.3.1 Spherical and polar cases

High-resolution ($R=1.5\times 10^{6}$) line profiles are shown in Figures 6
and 7 for several cases from Figure 5 for the SpherAcc and Polar cases. The
absolute spectral densities are for sources at 150 pc (relevant e.g. for
Taurus, Lupus, or $\rho$ Oph–U Sco) and are mostly of order $F_{\lambda}\sim
10^{-14}$–$10^{-13}$ erg s-1 cm-2 Å-1 for the range of $\dot{M}$ and
$M_{\mathrm{p}}$ shown. We also show an estimate of the photospheric emission
from the fit of Aoyama et al. (2020), to which we return in Section 4.3.3.

For a given $(\dot{M},M_{\mathrm{p}})$ and therefore, by assumption, radius,
the line-integrated flux at the observer decreases when going from SpherAcc to
Polar, that is, when reducing $f_{\mathrm{fill}}$. This is due to the
increased preshock density
($n_{0}\propto\rho\propto\dot{M}/f_{\mathrm{fill}}$), which leads to more
self-absorption in the postshock region (Aoyama et al., 2020; sketched in
Figure 2c), thereby reducing the luminosity in the H $\alpha$ line.

As discussed in Aoyama & Ikoma (2019), the line widths (at 10 and 50 % of the
maximum) of the profiles in Figures 6 and 7 reflect both the temperature of
the region at which most of the line is formed as well as the absorption by
the layers above this in the postshock region. We can now extend this
statement by noting that the accretion flow too can change the width of the
line, for $\dot{M}>3\times 10^{-5}$ ($3\times 10^{-3}$)
$M_{\mathrm{J}}\,\mathrm{yr}^{-1}$ in the SpherAcc (Polar) case.

The profiles as they emerge from the accretion flow reveal further
information. They exhibit, in part as expected from Figure 5, a range of
qualitative outcomes: no absorption (low $\dot{M}$), moderate wavelength-
independent (high $\dot{M}$ and intermediate $M_{\mathrm{p}}$), moderate
wavelength-dependent absorption (intermediate $\dot{M}$ and low
$M_{\mathrm{p}}$), and very strong absorption (high $\dot{M}$ and low
$M_{\mathrm{p}}$). Also, different line shapes are visible: roughly Gaussian
(towards the bottom right); flattened (middle right); or with self-absorption
in the peak taking place in the postshock region (top and middle left).

In Figures 6 and 7, the lines at (somewhat) high accretion rates
$\dot{M}\gtrsim 3\times 10^{-5}\leavevmode\nobreak\
M_{\mathrm{J}}\,\mathrm{yr}^{-1}$ display some asymmetry, as in the example
shown in Figure 4. In all examples considered, an asymmetry, if present, is
characterised by a peak shifted blueward by around 10 to
$20\leavevmode\nobreak\ \mathrm{km}\,\mathrm{s^{-1}}$, which results from
stronger absorption in the red wing. This asymmetry is present in the opacity
and comes at least in part from a slight slope in the continuum of the
opacity, as can be seen in Figure 4. In that example, there happen to be
several stronger features in the opacity between $\Delta v\approx-10$ and
$+150\leavevmode\nobreak\ \mathrm{km}\,\mathrm{s^{-1}}$ relative to the line
centre. These features are physically unrelated to the H $\alpha$ line and
might not be present for other elemental mixtures or atomic and molecular
abundances (for instance due to disequilibrium chemistry). The asymmetry
already present in the input profile grows towards lower masses. Only in a few
cases is there a small dip at the central position of H $\alpha$, but this is
not due to resonant absorption because the latter is redshifted as discussed
in Section 4.1. For the highest accretion rates, especially at
$M_{\mathrm{p}}\lesssim 5\leavevmode\nobreak\ M_{\mathrm{J}}$, the signal is
reduced by orders of magnitude. This implies that a planet accreting at such
high rates would be indistinguishable from the continuum, especially taking
the emission from the accreting gas into account (see Figure 12b).

For the cold-population radii, the results (shown in Figures 14 and 15) are
similar, with a moderate amount of extinction imprinting spectral structure at
high accretion rates $\dot{M}\gtrsim 3\times 10^{-5}\leavevmode\nobreak\
M_{\mathrm{J}}\,\mathrm{yr}^{-1}$. However, the line fluxes (at the observer’s
position) are either very similar or smaller, with very few exceptions. In the
Cold cases, the smaller radius leads to a higher accretion luminosity
(${L_{\textnormal{acc}}}\propto 1/R_{\mathrm{p}}$), but in the Warm cases the
conversion of ${L_{\textnormal{acc}}}$ to ${L_{\textnormal{H}\,\alpha}}$ turns
out to be more efficient by a factor greater than the ratio of the radii.
Thus, overall, the flux is usually smaller in the Cold cases.

####  4.3.2 Magnetospheric accretion

Spectra for the MagAcc case are shown in Figure 8. In this geometry, we have
assumed that the flux from the postshock region passes through the accretion
column out to a distance ${r_{\rm max}}\approx{R_{\textnormal{acc}}}/2$ (i.e.
tangentially through the accretion arc; see Figure 1). The high spectral
resolution $R\sim 10^{6}$ of our calculations reveals pronounced features for
the warm-population radii that are absent for the cold-population radii. This
is due to the different highest temperatures in the flow (Equations (6 and
(7)). Indeed, the shock temperature (where the maximum is reached) is
$T_{0}\approx 2000$–3000 K for the warm-population case as opposed to
$T_{0}\approx 4000$–7000 K for the cold case, due to the smaller radii in the
latter case. For these higher temperatures in the cold-population case, close
to the planet molecules are absent and only a few atomic lines and the
continuum leave a spectral imprint.

There are two consequences of the short integration length. One is that the
temperature does not drop into a region where molecules are important. The
second consequence is that the features are sharper because they are not
blurred by a gradient in the Doppler shift; in other words, over the region of
the strongest absorption, roughly $R_{\mathrm{p}}$ to ${r_{\rm
max}}\approx{R_{\textnormal{acc}}}/2=2.5R_{\mathrm{p}}$, the change in the
Doppler shift $\Delta(\Delta\lambda)\approx dv/dr\times({r_{\rm
max}}-R_{\mathrm{p}})\times\lambda_{0}/c\sim(3/4)(v_{0}/c)\lambda_{0}$ is
smaller than the typical spacing between the spectral features. Finally, in a
few of the examples selected (e.g. cold-population radii, $\dot{M}=3\times
10^{-5}\leavevmode\nobreak\ M_{\mathrm{J}}\,\mathrm{yr}^{-1}$,
$M_{\mathrm{p}}=10$ and $5\leavevmode\nobreak\ M_{\mathrm{J}}$,
$f_{\mathrm{fill}}=10\leavevmode\nobreak\ \%$), the spectral footprint of the
H $\alpha$ resonance is clearly visible at $\Delta
v\gtrsim+60\leavevmode\nobreak\ \mathrm{km}\,\mathrm{s^{-1}}$ as a strong
absorption.

More generally, how smooth or ragged the line profile is, is determined by how
large the Doppler shift gradient in the flow (not the Doppler shift itself) is
over the length over which there are stronger spectral signatures, which in
turn is set by the temperature structure in the flow. Very roughly speaking,
the opacity is low both for $T\lesssim 800$ K and at $T\approx 3000$–6000 K
(see Figure 3) and does not depend strongly on density. If the maximal
temperature in the flow (at the shock) is above or close to the upper edge of
the opacity maximum, a sufficiently large fraction of the flow will be in the
high-opacity regime. This matters especially because both the density (thus
the optical depth) and the velocity gradient are stronger closer to the shock.
A maximum temperature near the opacity maximum therefore leads to a blurring
of the features by the Doppler shift (i.e. velocity) gradient.

####  4.3.3 Importance of the continuum

So far, we have effectively considered continuum-subtracted emission lines by
calculating the radiative transfer only for the shock excess. However,
currently used high-resolution spectral differential imaging (HRSDI)
techniques involve the subtraction of the continuum and of low-frequency
spectral components (in H $\alpha$: Haffert et al., 2019; Xie et al., 2020; in
the NIR: Snellen et al., 2014; Hoeijmakers et al., 2018; Petrus et al., 2021;
Cugno et al., 2021). Therefore, we need to ensure that subtracting the
photosphere, especially if it is somewhat mismatched, will still leave the
line emission detectable. For spectrally resolved observations, what matters
here is, to first order, not the ratio of the line peak to the continuum, but
rather its ratio to the spread of the local (pseudo)continuum features, the
“photospheric noise”.

To assess whether the photospheric emission could hinder the line measurement,
we plot in Figures 6–8 the approximate photospheric emission of the planet.
This is to be compared to the line emission at the planet’s surface. The more
complete approach would require inputting the line together with the continuum
to the radiative transfer including emission from the accretion flow (Appendix
A), but for our purposes the current estimate should suffice.

We use the fit171717Available at https://github.com/gabrielastro/St-Moritz. of
Aoyama et al. (2020) of ${T_{\textnormal{eff}}}(\dot{M},M_{\mathrm{p}})$.
These fits consider the downward-travelling radiative fluxes from the detailed
shock models and ensure that the total emitted luminosity from the planet is
equal to the sum of the interior and the incoming kinetic energy. Thus, for
the photosphere, $\sigma{T_{\textnormal{eff}}}^{4}$ is not simply given by
${L_{\textnormal{acc}}}/(4\pi R_{\mathrm{p}}^{2}f_{\mathrm{fill}})$ because a
part of this accretion energy is already emitted in the hydrogen lines and
continua. Rather, the luminosity from the photosphere at
${T_{\textnormal{eff}}}$ is
$L_{\mathrm{phot}}={L_{\textnormal{acc}}}+L_{\mathrm{int}}-L_{\mathrm{shock}}$,
where $L_{\mathrm{shock}}$ is the total outward travelling luminosity from the
shock models. This holds for the accreting region of fractional area
$f_{\mathrm{fill}}$, while the rest of the planet surface has
${T_{\textnormal{eff}}}={T_{\textnormal{int}}}$.

For the spectrum, we use the high-resolution (at H $\alpha$:
$\Delta\lambda=0.01$–0.05 Å, i.e. $R\approx 130,000$–$660,000$) solar-
metallicity BT-Settl/AGSS2009 models (Allard et al., 2012) obtained from the
SVO Theory Server181818See http://svo2.cab.inta-csic.es/theory/newov2/. and
round $\log g$ to the nearest 0.5 dex and ${T_{\textnormal{eff}}}$ up (to be
conservative) to the nearest multiple of 100 K. To plot against the Doppler
distance $\Delta v$ from the line centre, we use as the central wavelength of
the photospheric models $\lambda_{\mathrm{H}\,\alpha}=656.283$ nm in air (for
the BT-Settl models on the SVO server; Wiese & Fuhr, 2009).

For the SpherAcc-Warm case (Figure 6), the continuum is never visible on the
linear scale of the line peak and would need to be $\sim 10$ stronger to be
even barely noticeable. Thus the photospheric emission is entirely negligible.
In the Polar-Warm case (Figure 7) for $\dot{M}\lesssim 3\times
10^{-4}\leavevmode\nobreak\ M_{\mathrm{J}}\,\mathrm{yr}^{-1}$, the continuum
noise is not important, but for the highest accretion rate it makes the H
$\alpha$ line undetectable, except at 15 $M_{\mathrm{J}}$. In the SpherAcc-
Cold and Polar-Cold cases (Figures 14 and 15), the photospheric noise is
mostly very low, with some exceptions at the highest accretion rate.

In the MagAcc-Warm or -Cold cases shown in Figure 8, the photospheric noise
(i.e. the amplitude of the features) is small to insignificant for most cases
shown. The exception is for the combination
($M_{\mathrm{p}}=5\leavevmode\nobreak\ M_{\mathrm{J}}$, $\dot{M}=3\times
10^{-5}\leavevmode\nobreak\ M_{\mathrm{J}}\,\mathrm{yr}^{-1}$), whether in the
Warm or in the Cold scenario, in which the H $\alpha$ line is in absorption in
the atmospheric spectrum, with the other features smaller. This is maybe not
realistic because the BT-Settl models were calculated with the
pressure–temperature structure of an isolated, not accreting, atmosphere. From
Equation (5a), the ram pressure for the cases in Figure 8 is
$P_{\textnormal{ram}}\approx 3\times 10^{-3}$–$3\times 10^{-2}$ bar. These
pressures are likely neither very far up in the atmosphere nor very deep, in
which cases the shock would, respectively, not or very much change the
structure. Therefore, without detailed calculations of the resulting
pressure–temperature structure of the atmosphere and the radiation transport,
it is difficult to estimate the effect of the heating by the shock on the line
emerging from the atmosphere below the shock.

####  4.3.4 Observability

Line-integrated luminosities of our models are presented in Aoyama et al.
(2020), and the line-integrated extinction in Section 4.2. Up to now, the few
known accreting low-mass objects have been observed in H $\alpha$ filters with
a width of order of 1–2 nm (Wagner et al., 2018; Zhou et al., 2021) or with
MUSE at a resolution $\Delta\lambda\approx 0.3$ nm (Haffert et al., 2019;
Eriksson et al., 2020). However, as seen in Section 2, much more powerful
instruments are underway or expected soon. For example, with $R\geqslant
130,000$, the resolution of RISTRETTO is comparable to the effective
resolution of the model curves given the native width of the spectral features
and the slight smear coming from the velocity gradient. Therefore, we discuss
in this section the observability at high resolution of accreting planets
whose line profiles are in part shaped by the accreting gas.

In Figures 6, 7, and 8, we show line profiles $F_{\lambda}(\lambda)$ for our
grid of models convolved to $R=2516$ (for MUSE and slightly broader than
HARMONI) and to $R=15,000$ (for VIS-X). The full-resolution curve (for
RISTRETTO) was discussed above, and Subaru+SCExAO/RHEA with $R=60,000$ is in
an intermediate range. The flux densities are for an observer on Earth for a
source $d=150$ pc away, typical of several star-forming regions.

The resolution of MUSE corresponds to $\Delta v=120\leavevmode\nobreak\
\mathrm{km}\,\mathrm{s^{-1}}$, which is much larger than the spectral
features. Therefore, they are not recognisable at this resolution. In some
cases, very slight asymmetries between the red and blue wings are visible but
they would be undetectable given any amount of noise and given the finite
number of pixels per resolution element. Unfortunately, the higher resolution
of HARMONI with $\Delta v=100\leavevmode\nobreak\
\mathrm{km}\,\mathrm{s^{-1}}$ will not yield a significant improvement. This
emphasises the need for very-high-resolution spectrographs.

Even at $R=15,000$ ($\Delta v=20\leavevmode\nobreak\
\mathrm{km}\,\mathrm{s^{-1}}$), the finest spectral features are not
distinguishable. Nevertheless, some asymmetries are preserved, for instance in
the Polar case at $(\dot{M}=3\times 10^{-5}\leavevmode\nobreak\
M_{\mathrm{J}}\,\mathrm{yr}^{-1},M_{\mathrm{p}}=2$–$5\leavevmode\nobreak\
M_{\mathrm{J}})$, or more clearly at the highest $\dot{M}$ and
$M_{\mathrm{p}}$. This leaves the line centre shifted by approximately
$-20\leavevmode\nobreak\ \mathrm{km}\,\mathrm{s^{-1}}$. Such large offsets are
not expected from other effects such as planetary spin or orbital motion (see
Section 7.4). Therefore, a line displaced from the theoretical value by a
large amount could be a tell-tale sign of extinction from the surrounding gas.

In summary, the spectral resolving power of planned instruments such as
RISTRETTO and VIS-X will make it possible in the next few years to start
studying in detail the line profiles of low-mass companions, revealing the
spectral imprint of the absorbing gas in the accretion flow.

##  5 Absorption by the dust

So far, we have neglected the contribution from the dust in our radiative
transfer calculation. Since dust exists only at low temperatures and these
temperatures tend to be reached at larger distances from the planet, the outer
parts of the accretion flow can be important for setting the total optical
depth. In the case of accretion directly onto the planet, it seems likely that
the last stages, that is, close to the planet surface, are radial and
supersonic. This leaves only the filling factor as the main free parameter of
the accretion geometry relevant for the gas. In the case of the dust, however,
the fact that parts further out can contribute makes the analysis more
tentative.

Given these uncertainties, we estimate what dust opacity is needed to have a
significant optical depth in the accretion flow (Section 5.1). We then derive
a range of plausible values for the dust opacity based on the relevant recent
literature (Section 5.2), and in Section 5.3 compare this to other recent
estimates. We derive the total dust optical depth in Section 5.4.

There is only a small range of temperatures in which (i) gas–magnetic field
coupling is possible thanks to sufficient ionisation ($T\gtrsim 1000$ K; Desch
& Turner, 2015) and (ii) at the same time the dust has not been evaporated
($T\lesssim{T_{\textnormal{dest}}}\approx 1500$ K, ${T_{\textnormal{dest}}}$
being the temperature at which the last component of the dust evaporates;
Pollack et al., 1994; Semenov et al., 2003). In other words, in the MagAcc
scenario the temperature is usually too high for dust to survive. Therefore,
we will put MagAcc aside for this section.

###  5.1 Calculating the optical depth

We first calculate the optical depth of the dust in the accretion flow. Given
the large uncertainties on the dust opacity (see Section 5.2), we take a crude
approach and let the dust opacity be independent of density and temperature in
the region where dust is present ($T<{T_{\textnormal{dest}}}$, or
$r>{R_{\textnormal{dest}}}$)191919A test with the weak powerlaw dependence
$\kappa\propto T^{1/2}$ appropriate for the “metal grains” regime of Bell &
Lin (1994) indeed did not lead to a significantly different optical depth..
The contribution of the dust to the optical depth through the accreting matter
is given by

$\tau_{\mathrm{dust,\,H}\,\alpha}=\int_{R_{\textnormal{dest}}}^{r_{\rm
max}}f_{\textrm{d/g}}\kappa_{\bullet,\,\mathrm{H}\,\alpha}\,\rho\,dr,$ (18)

where we remind that $\kappa_{\bullet,\,\mathrm{H}\,\alpha}$ is the opacity of
the dust at H $\alpha$ as a cross-section per unit mass of dust. We have
defined in Section 4.1 the dust opacity as a cross-section per gram of gas as
$\kappa_{\mathrm{dust,\,H}\,\alpha}\equiv
f_{\textrm{d/g}}\kappa_{\bullet,\,\mathrm{H}\,\alpha}$ and will use this
below. The dust-to-gas mass ratio is essentially zero at
$r<{R_{\textnormal{dest}}}$, which is why the lower limit of the integration
is the “dust destruction front” ${R_{\textnormal{dest}}}$ (Stahler et al.,
1980). From Equation (6), this is:

${R_{\textnormal{dest}}}=\left(\frac{T_{0}}{{T_{\textnormal{dest}}}}\right)^{2}R_{\mathrm{p}}$
(19)

if the shock temperature is larger than ${T_{\textnormal{dest}}}$. At low
$\dot{M}$ there is no destruction radius in the classical sense because the
dust survives down to the shock (see Figure 3 and Section 6.2 of Marleau et
al. 2019). In that case the lower limit of the integral in Equation (18) is
$R_{\mathrm{p}}$ instead of ${R_{\textnormal{dest}}}$. For simplicity, we do
not take the weak density dependence of ${T_{\textnormal{dest}}}$ (Isella &
Natta, 2005) into account. The dust destruction front is illustrated in Figure
2.

Since $\rho\propto r^{-3/2}$ (Equation (3b) when $\zeta=1$), the integral in
Equation (18) will converge, as ${r_{\rm max}}$ tends to infinity, if
$\kappa_{\mathrm{dust,\,H}\,\alpha}$ is constant or does not increase faster
than $r^{1/2}$. However, this also assumes that the accretion radius
${R_{\textnormal{acc}}}$ is sufficiently large; if not, the outer parts of the
flow contribute significantly because the density $\rho\propto v$ is higher
there due to the smaller velocity $v\propto(1/r-1/{R_{\textnormal{acc}}})$.

If ${R_{\textnormal{dest}}}\geqslant R_{\mathrm{p}}$ and in the limit ${r_{\rm
max}}\gg{R_{\textnormal{dest}}}$ and ${R_{\textnormal{acc}}}\gg
R_{\mathrm{p}}$, the optical depth (Equation (18)) becomes

$\displaystyle\tau_{\mathrm{dust,\,H}\,\alpha}=$
$\displaystyle\leavevmode\nobreak\
f_{\textrm{d/g}}\kappa_{\bullet,\,\mathrm{H}\,\alpha}{T_{\textnormal{dest}}}\left(\frac{{\sigma}\dot{M}^{3}R_{\mathrm{p}}}{4\pi^{3}f_{\mathrm{fill}}^{3}G^{3}M_{\mathrm{p}}^{3}}\right)^{1/4}$
(20a) $\displaystyle=$ $\displaystyle\leavevmode\nobreak\
0.2\,\left(\frac{f_{\textrm{d/g}}}{10^{-4}}\right)\left(\frac{\kappa_{\bullet,\,\mathrm{H}\,\alpha}}{10^{4}\leavevmode\nobreak\
\mathrm{cm}^{2}\,\mathrm{g}_{\mathrm{dust}}^{-1}}\right)\left(\frac{{T_{\textnormal{dest}}}}{1500\leavevmode\nobreak\
\mathrm{K}}\right)$
$\displaystyle\times\left(\frac{f_{\mathrm{fill}}}{0.30}\right)^{-3/4}\left(\frac{\dot{M}}{\upmu
M_{\mathrm{J}}\,\mathrm{yr}^{-1}}\right)^{3/4}\left(\frac{M_{\mathrm{p}}}{5\leavevmode\nobreak\
M_{\mathrm{J}}}\right)^{-3/4}$
$\displaystyle\times\left(\frac{R_{\mathrm{p}}}{2\leavevmode\nobreak\
R_{\mathrm{J}}}\right)^{1/4},$ (20b)

where the subscript “dust” in the units of
$\kappa_{\bullet,\,\mathrm{H}\,\alpha}$ indicates that this is the cross-
section per gram of dust, that is, the intrinsic (material) cross-section.
Equation (20b) was written with $R_{\mathrm{p}}$ as an independent quantity
while in our fits it depends on $\dot{M}$ and $M_{\mathrm{p}}$. However,
$\tau_{\mathrm{dust,\,H}\,\alpha}$ does not depend strongly on
$R_{\mathrm{p}}$ (only as the fourth root). The $f_{\mathrm{fill}}$ factor on
the denominator in these expressions comes from the fact that
$\tau_{\mathrm{dust,\,H}\,\alpha}$ is the optical depth through the accretion
flow, with $\rho\propto\dot{M}/f_{\mathrm{fill}}$ (see Equation (3)). Thus the
optical depth should increase with increasing accretion rate (all the more
since $R_{\mathrm{p}}$ also grows with $\dot{M}$), whereas it should decrease
somewhat with mass (in part counterbalanced by the growth of $R_{\mathrm{p}}$
with $M_{\mathrm{p}}$, but only weakly because
$\tau_{\mathrm{dust,\,H}\,\alpha}\propto R_{\mathrm{p}}^{1/4}$).
Unfortunately, $\tau_{\mathrm{dust,\,H}\,\alpha}$ has a non-negligible
dependence (linear) on the very uncertain quantities $f_{\textrm{d/g}}$ and
$\kappa_{\bullet,\,\mathrm{H}\,\alpha}$. In this light, the uncertainty on
${T_{\textnormal{dest}}}$ and its weak density dependence do not matter.

The prefactor in Equation (20b) implies that for the nominal values in that
equation, the dust is not able to absorb much H $\alpha$ emission, with a flux
decrease $\Delta F=1-\exp(-0.2)\approx 20$ %. The parameter space is large,
however, and we explore it more systematically in this and the following
sections.

One can ask what the minimum average dust opacity (as a cross-section per unit
mass of gas) is needed to have $\tau_{\mathrm{dust,\,H}\,\alpha}\sim 1$.
Equation (18) can be written as

$\tau_{\mathrm{dust,\,H}\,\alpha}=\tilde{\Sigma}_{\textrm{gas}}\langle\kappa_{\mathrm{dust,\,H}\,\alpha}\rangle,$
(21)

where

$\tilde{\Sigma}_{\textrm{gas}}\equiv\int_{R_{\textnormal{dest}}}^{r_{\rm
max}}\rho\,dr$ (22)

is the gas column density of the accretion flow where the dust is present (as
opposed to the full column density) and
$\langle\kappa_{\mathrm{dust,\,H}\,\alpha}\rangle=\langle
f_{\textrm{d/g}}\kappa_{\bullet,\,\mathrm{H}\,\alpha}\rangle$ is the dust
absorption cross-section per unit gas mass averaged over that part of the
accretion flow. Since we assumed the dust material opacity to be constant
where it is non-zero, and if we also take $f_{\textrm{d/g}}$ constant in the
accretion flow, it holds simply that
$\langle\kappa_{\mathrm{dust,\,H}\,\alpha}\rangle=\kappa_{\mathrm{dust,\,H}\,\alpha}$.
Equation (21) implies that $\tilde{\Sigma}_{\textrm{gas}}^{-1}$ is equal to
the minimum dust opacity (as a cross-section per unit mass of gas) needed to
have $\tau_{\mathrm{dust,\,H}\,\alpha}\approx 1$ and thus contribute
noticeably to the absorption. Over the narrow width of the line
($\Delta\lambda\lesssim 0.001\leavevmode\nobreak\ \upmu$m), the dust opacity
is constant, so that it can affect only the total flux but not the line shape.

Figure 9a displays the minimum average dust opacity required to have an
optical depth of unity. We focus on the SpherAcc and Polar cases. We find that
over the parameter range considered but restricting ourselves to
$\dot{M}\lesssim 3\times 10^{-5}\leavevmode\nobreak\
M_{\mathrm{J}}\,\mathrm{yr}^{-1}$, the minimum dust opacity required in order
for the dust to absorb a significant fraction of the H $\alpha$ signal is of
order $\tilde{\Sigma}_{\textrm{gas}}^{-1}\gtrsim 1$–100 cm2 g-1. The results
of Figure 9a are only moderately sensitive to the planet mass and scale
roughly as $\Sigma_{\mathrm{dust}}\sim\dot{M}^{1.5}$. The choice of the cold-
or warm-population radii is not crucial (not shown), and the filling factor
only has a moderate effect (solid vs. dashed lines). For most of parameter
space, the shock is hot enough that the dust destruction radius is at several
times the planet radius, which typically corresponds to
${R_{\textnormal{dest}}}\approx 10$–30 $R_{\mathrm{J}}$ (see grey dotted
lines). For increasing accretion rate, the size of the dust-free inner region
relative to the planet radius, ${R_{\textnormal{dest}}}/R_{\mathrm{p}}$,
grows. Nevertheless, the dust column density increases with $\dot{M}^{1.5}$,
so that the minimum opacity required becomes smaller.

We have assumed that the dust opacity has a constant value throughout the
accretion flow at $r>{R_{\textnormal{dest}}}$ and is zero within the dust
destruction front. If it is not too far out (i.e. if
${R_{\textnormal{dest}}}\ll{r_{\rm max}}$), and assuming that
${R_{\textnormal{acc}}}\gg R_{\mathrm{p}}$, the dust optical depth
$\tau_{\mathrm{dust,\,H}\,\alpha}$ can be expressed analytically (Equation
(20)). In this case, but also in general, $\tau_{\mathrm{dust,\,H}\,\alpha}$
depends linearly on the dust opacity. We now turn to the task of estimating
its value.

Figure 9:  Estimate of the dust absorption in the accretion flow. (a) Minimum
dust opacity at H $\alpha$,
$f_{\textrm{d/g}}\kappa_{\bullet,\,\mathrm{H}\,\alpha}$ (per unit gas mass;
contour labels), needed to have $\tau_{\mathrm{dust,\,H}\,\alpha}\approx 1$ in
the accretion flow onto a growing gas giant (Equation (21)) in the Polar
(solid lines) and SpherAcc (dashed lines) cases. We take the cold-population
radii and fix ${T_{\textnormal{dest}}}=1500$ K. Grey dotted lines show where
${R_{\textnormal{dest}}}/R_{\mathrm{p}}=1.5$ and 5 (Equation (19)). (b)
Material opacity $\kappa_{\bullet,\,\mathrm{H}\,\alpha}$. We show the ISM fits
of Cardelli et al. (1989) for $R_{V}=3.1$ (dotted black) and 5.5 (dotted
grey), with the absolute scale from Güver & Özel (2009); Wang & Chen (2019)
(solid black); and Chiar & Tielens (2006) (dotted grey, up to 8 $\upmu$m). Red
and blue curves are for size distributions set by the slope $q$ and minimum
size (see legend). Curves for each model are for 20 or 80 % carbon (bottom to
top), with silicate completing. At the bottom, hydrogen lines, two HST
filters, and NACO IR filters are shown. The grey area is a BT-Settl model with
${T_{\textnormal{eff}}}=1200$ K and $\log g=4$. (c) Estimate of the dust
opacity
$\kappa_{\mathrm{dust,\,H}\,\alpha}=f_{\textrm{d/g}}\kappa_{\bullet,\,\mathrm{H}\,\alpha}$
in the accretion flow. Pale dotted lines are for $a_{\rm max}=1$ mm instead of
$0.1$ mm. Red curves represent recent simulation results (see text), with a
low dust abundance $f_{\textrm{d/g}}\sim 10^{-5}$–$10^{-4}$, implying
$\kappa_{\mathrm{dust,\,H}\,\alpha}\sim 0.3$ cm2 g-1 with a half-spread
$\sigma=1.5$ dex (grey shaded region). The opacity of Flock et al. (2016,
“F+16”), Rab et al. (2019, “R+19”), and Sanchis et al. (2020, “ISM”) is shown
for $f_{\textrm{d/g}}=0.01$ (grey symbols, shifted left) and the pure-
graphite, “mixture”, and pure-silicate opacity of Szulágyi & Ercolano (2020,
“SzE20”) is shown (green circles; top to bottom). All but Flock et al. (2016)
assume $f_{\textrm{d/g}}=0.01$ in their work.

###  5.2 Realistic estimates of dust opacity

####  5.2.1 General considerations about dust parameters

The dust monochromatic opacity is set amongst others by the composition,
shape, space- and time-dependent size distribution, and dynamics of dust
particles in CSDs (Andrews, 2020) and specifically in the vicinity of a
growing and migrating gas giant (e.g. Pollack et al., 1994). In particular,
the dust-to-gas mass ratio $f_{\textrm{d/g}}$ and—to a lesser extent (Chachan
et al., 2021)—the minimum and maximum grain sizes $a_{\rm min}$ and $a_{\rm
max}$ impact the opacity (e.g. Cuzzi et al., 2014; Kataoka et al., 2014;
Woitke et al., 2016; Krapp et al., 2021). In turn, these properties are set by
many processes (e.g. Flock et al., 2016; Birnstiel et al., 2018; Vorobyov et
al., 2018; Drażkowska et al., 2019; see review in the latter work).

To obtain accurate dust opacities would require global-disc radiation-
hydrodynamic simulations following the dust growth, drift, and evaporation
with sufficiently high resolution both in space and in the dust size, while
covering the full range of grain sizes that set the opacity, and taking the
feedback of the grains on the disc structure into account. This is not yet
computationally feasible. However, different studies have looked at some of
these aspects (e.g. Drażkowska et al., 2019; Savvidou et al., 2020; Chachan et
al., 2021; Binkert et al., 2021; Szulágyi et al., 2021; Krapp et al., 2021),
with some properties emerging. We highlight briefly four of them.

Firstly, since the density in the accretion flow decreases outwards, the
layers in the dusty part of the flow (where $T\lesssim 1500$ K) that are
closest to the planet will likely contribute the most. For $T\gtrsim 700$ K,
only silicate along with carbon, iron, or troilite remain (Semenov et al.,
2003; Woitke et al., 2016). Secondly, in 2D simulations, the pressure
perturbation of the planet keeps the large grains outside of its orbit
(“filters them out”; e.g. Paardekooper & Mellema, 2004; Rice et al., 2006; Zhu
et al., 2012; Bae et al., 2019). For example, Drażkowska et al. (2019) found a
maximum grain size $a_{\rm max}\sim 0.03$ cm around the planet instead of
$a_{\rm max}\sim 3$ cm at larger orbital radii in theirglobal-disc
hydrodynamical simulations combined with a dust evolution model. Also the size
distribution near the planet is different, with $n(a)\propto a^{-q}$ with
$q\approx 4$, where $n(a)$ is the number density of grains of size $a$ (J.
Drażkowska 2020, priv. comm.). This is steeper than the commonly used Mathis
et al. (1977) ISM distribution with $q=3.5$.

In 3D, meridional circulation could bring large grains towards a forming
planet (Bi et al., 2021; Szulágyi et al., 2021), bridging the pressure bump.
Realistically, this however depends on the amount of turbulence and settling
(Dullemond & Dominik, 2004) and the strength, for instance, of the vertical
shear instability (VSI; Flock et al., 2020), or the viscosity and the gap
depth (Kanagawa et al., 2018).

Thirdly, a range of different minimum grain sizes $a_{\rm min}$ is used in the
literature: $a_{\rm min}=0.1$–1 $\upmu$m (e.g. Okuzumi et al., 2012; Kataoka
et al., 2014; Bae et al., 2019; Stammler et al., 2019; see brief review in
Xiang et al., 2020); $a_{\rm min}=1\leavevmode\nobreak\ \upmu$m in Drażkowska
et al. (2019) but with some pile-up near $a_{\rm min}$ in the resulting
distribution; Flock et al. (2016) used a smaller value: $a_{\rm
min}=5\leavevmode\nobreak\ $nm202020Not $a_{\rm min}=5\leavevmode\nobreak\
\upmu$m as their Appendix A states (M. Flock 2020, priv. comm.). That their
opacity curve has features at $\lambda\ll 5$ $\upmu$m suggests this, since at
$\lambda\ll 2\pi a_{\rm min}$ the geometric, $\lambda$-independent limit for
the cross-section per particle $\sigma=2\pi a^{2}$ must hold (e.g. Mordasini,
2014)..

Fourthly, concerning the dust abundance, the simulations of Pinilla et al.
(2012) or Drażkowska et al. (2019) find depletions by a factor of
$10^{3}$–$10^{4}$ relative to the global disc abundance. In a sample of seven
discs, Powell et al. (2019) found with a method independent of a tracer-to-
mass conversion a global212121Although $f_{\textrm{d/g}}$ likely varies on
global disc scales (Soon et al., 2019). dust-to-gas ratio of order
$f_{\textrm{d/g}}\sim 10^{-4}$–$10^{-3}$. Recent global-disc simulations of
grain growth and drift (e.g. Savvidou et al., 2020; Chachan et al., 2021) lend
support to this. Finally, if planets form early (e.g. (Manara et al., 2018)),
the dust particles may be locked up in macroscopic objects (pebbles,
planetesimals, planetary cores), also reducing $f_{\textrm{d/g}}$.

We assume that the dust flowing onto the planet has the same size distribution
as in the gap around the planet on scales of ${R_{\textnormal{Hill}}}$. Based
on the previous discussion, to estimate the dust opacity we consider the
following parameter values: $a_{\rm min}\in\\{0.01,1\\}\leavevmode\nobreak\
\upmu$m, $a_{\rm max}\in\\{0.1,1\\}$ mm, $q\in\\{3.5,4\\}$, the fraction of
(amorphous) carbon $f_{\mathrm{C}}\in\\{0.2,0.8\\}$ by mass, with the rest
made up of the usual astrophysical silicate (pyroxene: Mg0.7Fe0.3SiO3), and
the porosity $P\in\\{0.1,0.2,0.3\\}$, the vacuum fraction by volume (and not
by mass).

As shown in Woitke et al. (2016), $\kappa_{\bullet,\,\mathrm{H}\,\alpha}$
increases with increasing $q$ (more small grains) and $f_{\mathrm{C}}$ (more
carbon relative to silicate) or with decreasing $a_{\rm min}$ (smaller grains
included) or $a_{\rm max}$ (larger grains excluded). The dependence of the
opacity on the size can be understood from the fact that small grains have a
larger cross-section-to-mass (area-to-volume) ratio than large grains. For the
parameter values here, the opacity is almost insensitive to the porosity $P$,
varying at most (at the highest carbon fraction) by 30 % for the range of $P$
covered here. Therefore, we fix $P=0.2$ hereafter.

####  5.2.2 Resulting monochromatic dust opacity

To calculate the dust opacity, we use with the convenient
OpTool222222Available at https://github.com/cdominik/optool. (Dominik et al.,
2021), which uses Toon & Ackerman (1981) and, amongst others, the
“distribution of hollow spheres” (Min et al., 2005). Optical constants are
from Zubko et al. (1996) and Dorschner et al. (1995) for the carbon and
silicates, respectively. Other parameters are as preset.

We plot in Figure 9b the opacity from the UV to the mid-infrared (MIR) for
different dust models. We label the curves by $q$, $a_{\rm min}$, and $a_{\rm
max}$. At H $\alpha$, this leads to a similar or larger range of opacities
compared to including other materials such as iron (troilite) or water (ice)
or even CHON organic material, and varying their abundance. Silicate-rich
grains ($f_{\mathrm{C}}=0.2$) lead to strong absorption at 10 $\upmu$m (the
“ten-micron bump”).

For these opacity curves, the logarithmic average opacity slope
$p_{\textnormal{UV}}=\Delta\log\kappa/\Delta\log\lambda$ between H $\beta$
($\lambda_{\mathrm{H}\,\beta}=486$ nm) and H $\alpha$ is roughly
$p_{\textnormal{UV}}\approx-0.5$ to $-1$. Larger dust grain size distribution
powerlaw exponent values $q$ and smaller values of $a_{\rm min}$ make the
slope steeper, that is, $p_{\textnormal{UV}}$ more negative, while $a_{\rm
max}$ essentially does not change the slope much (see Figure 3 of Woitke et
al., 2016) and only decreases the absolute amount of extinction. For
comparison, the extinction law of Cardelli et al. (1989), valid at H $\alpha$
and H $\beta$, has $p_{\textnormal{UV}}=-1.2$ ($p_{\textnormal{UV}}=-0.9$) for
$R_{V}=3.1$ ($R_{V}=5$). Wang & Chen (2019) recently adjusted the Cardelli et
al. (1989) extinction law with $R_{V}=3.1$ and obtained
$p_{\textnormal{UV}}=-1.4$. We note that the fundamental reason why our curves
are flatter than the empirical ISM curve is not yet understood.

For reference, we indicate in Figure 9b the position of some hydrogen lines
and of the Balmer jump (H $\infty$ at 3646 Å) as well as the flanking $U$-band
F336W and F390W filters of the Hubble Space Telescope (HST). This is motivated
by the recent detection of PDS 70 b by Zhou et al. (2021) at F336W, a first
for an accreting planet. We also diplay NIR and MIR filters from VLT/NACO
($J$, $H$, $K_{\mathrm{s}}$, $L^{\prime}$, $M^{\prime}$). Finally, we also
show a scaled spectrum with ${T_{\textnormal{eff}}}=1200$ K and $\log g=4$
from BT-Settl, which approximately fits the data for PDS 70 b (Stolker et al.,
2020a; Wang et al., 2021b).

Figure 9c displays the opacity $\kappa_{\mathrm{dust,\,H}\,\alpha}$ as a
function of $f_{\textrm{d/g}}$. We consider that a range of
$f_{\textrm{d/g}}\sim 10^{-5}$–$3\times 10^{-4}$ is plausible given the
relative depletion by a factor of 100–1000 found by Drażkowska et al. (2019)
and spatial variations (on scales larger than planets’ Hill spheres, as the
results of Soon et al., 2019 suggest).

We obtain that the material opacity of the dust can be anywhere between
$\kappa_{\bullet,\,\mathrm{H}\,\alpha}\sim 300$ and $3\times 10^{4}$ cm2
g${}^{-1}_{\mathrm{dust}}$, which translates into
$\kappa_{\mathrm{dust,\,H}\,\alpha}\sim 10^{-2}$–$10$ cm2
g${}^{-1}_{\textrm{gas}}$ at $f_{\textrm{d/g}}\sim 10^{-5}$–$3\times 10^{-4}$.
Uncertainties in both the size distribution of the grains and the material
properties (including composition and porosity) lead to this spread of three
orders of magnitude. The large uncertainty in
$\kappa_{\bullet,\,\mathrm{H}\,\alpha}$ justifies our simplification that it
is constant in the accretion flow.

###  5.3 Comparison with other estimates of dust opacity

We compare to the opacity used by Sanchis et al. (2020). (See also the
discussion of their work in Section 7.6.) They took the ISM opacity in the $V$
band $\kappa_{V}=107$ cm2 g${}^{-1}_{\textrm{gas}}$ $\kappa_{V}=107$ cm2
g(gas)-1 from Güver & Özel (2009)232323For comparison, the commonly used law
of Bohlin et al. (1978) gives an opacity only 10 % smaller than in Güver &
Özel (2009). assuming $f_{\textrm{d/g}}=0.01$ and a mean molecular weigth
$\mu=2.353$, and used the ISM dust extinction law from Cardelli et al. (1989)
with $R_{V}=3.1$ (E. Sanchis 2020, priv. comm.) to scale to other bands up to
$K$, and the one from Chiar & Tielens (2006) for longer wavelengths. Doing
this for H $\alpha$, we obtain $\kappa_{\mathrm{dust,\,H}\,\alpha}=88$ cm2
g${}^{-1}_{\textrm{gas}}$. This is much higher than the upper range of our
suggested values. Nevertheless, the intrinsic opacity (cross-section per gram
of dust) is similar to that of some of our curves.

In Figure 9c we also include for reference the dust opacities used in Szulágyi
& Ercolano (2020) at their assumed $f_{\textrm{d/g}}=0.01$. We computed them
with OpTool for their pure-silicate, silicate–water–carbon (“mixture”), and
pure-graphite cases (respectively from weakest to strongest). The lowest dust
opacity of Szulágyi & Ercolano (2020), their “silicate” case
($\kappa_{\mathrm{dust,\,H}\,\alpha}=2.9$ cm2 g${}^{-1}_{\textrm{gas}}$), is
consistent with our opacities, whereas the two other cases are 10–20 times
higher than ours. Here again, however, the range of intrinsic opacities is
almost identical to ours, with the real difference only in the adopted dust-
to-gass mass ratio.

Flock et al. (2016), who assumed $f_{\mathrm{C}}=37.5$ % and silicate for the
rest (see above for the other parameters), used the MieX code (Wolf &
Voshchinnikov, 2004) to obtain $\kappa_{\bullet,\,\mathrm{H}\,\alpha}=1300$
cm2 g${}_{\mathrm{dust}}^{-1}$. With their reference global mass ratio
$f_{\textrm{d/g}}=10^{-2}$, their opacity as a cross-section per gram of gas
fits within our range. The same holds for the opacity of Rab et al. (2019)
(see above), which we computed with OpTool to be
$\kappa_{\bullet,\,\mathrm{H}\,\alpha}=280$ cm2 g${}_{\mathrm{dust}}^{-1}$. It
is however at the lower edge of the range of intrinsic opacities because of
their large maximal grain size ($a_{\rm max}=3$ mm). Such large grains are not
expected in our case, in the accretion flow onto a growing planet.

###  5.4 Final estimate of the absorption by the dust

Comparing the estimated dust opacity with the minimum opacity for absorption
to be important (Figure 9), we find that the dust optical depth in the
accretion flow is likely less than unity ($A_{R}<1$ mag) for planets more
massive than a few $M_{\mathrm{J}}$ accreting at $\dot{M}\lesssim 3\times
10^{-6}\leavevmode\nobreak\ M_{\mathrm{J}}\,\mathrm{yr}^{-1}$ and even at a
very high rate $\dot{M}\lesssim 3\times 10^{-5}\leavevmode\nobreak\
M_{\mathrm{J}}\,\mathrm{yr}^{-1}$ if we consider the middle of the opacity
range ($\kappa_{\mathrm{dust,\,H}\,\alpha}\approx 0.3$ cm2
g${}_{\mathrm{gas}}^{-1}$). This does not depend much on the geometry
(SpherAcc or Polar), and even less on the choice of the hot- or cold-
population radii (not shown). For the highest accretion rates that we consider
this would lead to some absorption (optical depth of a few), similar to the
contribution from the gas to the absorption (Figure 5). If the dust opacity is
at the lower end of the estimated range
($\kappa_{\mathrm{dust,\,H}\,\alpha}\approx 10^{-2}$ cm2
g${}_{\mathrm{gas}}^{-1}$), there is no accretion rate for which the dust can
reduce the H $\alpha$ flux by more than a percent.

At low masses ($M_{\mathrm{p}}\lesssim 3\leavevmode\nobreak\ M_{\mathrm{J}}$)
and for opacity values at the high end
($\kappa_{\mathrm{dust,\,H}\,\alpha}\sim 10$ cm2 g${}_{\mathrm{gas}}^{-1}$),
however, there can be a large amount of extinction reaching
$A_{\mathrm{H}\,\alpha}\sim 10$ mag for $\dot{M}\sim
10^{-5}\leavevmode\nobreak\ M_{\mathrm{J}}\,\mathrm{yr}^{-1}$.

In Section 7.6, we compare these results to the work of Sanchis et al. (2020)
and Szulágyi & Ercolano (2020), whose density structures are very different
from ours due to their smoothing of the gravitational potential.

##  6 Observational consequences and example applications

Here, we present an absorption-modified
${L_{\textnormal{H}\,\alpha}}$–$\dot{M}$ relationship (Section 6.1), and apply
the models to Delorme 1 (AB)b and PDS 70 b (Section 6.2).

###  6.1 Accretion rate from H $\alpha$ luminosity

Figure 10:  Absorption-modified relationship (solid curves) between H $\alpha$
luminosity and accretion rate for the warm- (left panel) and cold-population
(right panel) radius fits for the three accretion geometries (rows; see
labels). We consider only absorption by the gas,and also show the case of no
absorption (dotted curves). Curves are for masses of 2–16 $M_{\mathrm{J}}$
(bottom to top, but with an inversion in MagAcc-Cold at very high $\dot{M}$).
Horizontal bands are for the luminosity of PDS 70 b (Zhou et al., 2021) and
PDS 70 c (Hashimoto et al., 2020). The fit of Ingleby et al. (2013) for CTTSs
is also shown (pink dashed line). For reference, one panel shows extinction
arrows for $A_{\mathrm{H}\,\alpha}=4$ and 8 mag.

Several authors have studied the empirical correlation between H $\alpha$
luminosity and accretion rate for CTTSs (e.g. Natta et al., 2004; Rigliaco et
al., 2012; Ingleby et al., 2013). Recently, this has been extended
theoretically to a mass of 6 $M_{\mathrm{J}}$ for the scenario in which the H
$\alpha$ is generated by magnetospheric accretion columns (Thanathibodee et
al., 2019). In Aoyama et al. (2021), we presented the
$\dot{M}$–${L_{\textnormal{H}\,\alpha}}$ correlation for our models for
accreting planets, looking also at other hydrogen lines. Here, we briefly
present this correlation again but including the effects of absorption. Given
the large uncertainty about the dust opacity (Section 5.2), we consider only
the gas opacity here.

Aoyama et al. (2020) explained that self-absorption, which occurs in the
postshock region (depicted in Figure 2c), lets the scaling of
${L_{\textnormal{H}\,\alpha}}$ with $\dot{M}$ become sub-linear. We find that
absorption by the matter flowing onto the planet strengthens this trend and,
depending on $f_{\mathrm{fill}}$, can lead to a maximum (“saturation”)
luminosity, that is, a flattening and turning over of
${L_{\textnormal{H}\,\alpha}}$ as a function of $\dot{M}$. At low $\dot{M}$,
the H $\alpha$ luminosity is intrinsically low, while a high $\dot{M}$ leads
both to a higher H $\alpha$ luminosity and to a stronger extinction, with the
second effect dominating.

Our $\dot{M}$–${L_{\textnormal{H}\,\alpha}}$ correlation is shown in Figure 10
for $M_{\mathrm{p}}=2$ to 16 $M_{\mathrm{J}}$ for all geometries and both
radius fits. For objects with $M_{\mathrm{p}}\lesssim 15\leavevmode\nobreak\
M_{\mathrm{J}}$, the maximum line-integrated luminosity is
${L_{\textnormal{H}\,\alpha,\,\textnormal{max}}}\approx 3\times
10^{-4}\leavevmode\nobreak\ {L_{\odot}}$ in the SpherAcc geometry and
${L_{\textnormal{H}\,\alpha,\,\textnormal{max}}}\approx
10^{-4}\leavevmode\nobreak\ {L_{\odot}}$ in the Polar and MagAcc geometries,
reached over a range of accretion rates around $\dot{M}\gtrsim 3\times
10^{-5}\leavevmode\nobreak\ M_{\mathrm{J}}\,\mathrm{yr}^{-1}$. This is
relatively insensitive to the choice of the radius fit. Including the
extinction by the dust would only lead to a stronger downturn because its
importance increases with $\dot{M}$ (see Figure 9).

Interestingly, this implies that for a range of ${L_{\textnormal{H}\,\alpha}}$
values, especially in non-spherically symmetric geometries, there are two
solutions to explain a given ${L_{\textnormal{H}\,\alpha}}$ observation: a low
$\dot{M}$ without extinction, or a high $\dot{M}$ with extinction. The
luminosities of PDS 70 b and c fall precisely in this range of
${L_{\textnormal{H}\,\alpha}}$. We return to this in detail in Section 6.2.
The high-$\dot{M}$ solution might be statistically not preferred because
observing a planet in a (presumably short) phase of massive accretion is
unlikely. Assuming additional amounts of absorption by dust implies accretion
rates intermediate between the low and the high solutions.

Could this maximum on the luminosity be responsible for the non-detections of
dedicated recent surveys (Cugno et al., 2019; Zurlo et al., 2020; Xie et al.,
2020)? Roughly, these surveys typically reached sensitivities down to
${L_{\textnormal{H}\,\alpha}}\sim 10^{-7}$–$10^{-6}\leavevmode\nobreak\
{L_{\odot}}$ beyond 15 au. For most extinction geometries (see Figure 10),
this is a few to several orders of magnitude below the peak luminosity. Only
for low masses and small filling factors could extinction push a planet into
non-detectability, at high accretion rates (in Figure 10, where the dotted
lines differ from the solid ones). Closer in to the surveyed stars, where
planets are expected to be more numerous (e.g. Bowler, 2016; Fernandes et al.,
2019; Nielsen et al., 2019; Vigan et al., 2021), the sensitivity curves become
quickly much less constraining; that is, the ${L_{\textnormal{H}\,\alpha}}$
upper limits are higher. In that case, extinction affects only the higher-mass
planets, but less than lower-mass planets. The low-mass planets242424For
example $M_{\mathrm{p}}\lesssim 2\leavevmode\nobreak\ M_{\mathrm{J}}$, but the
steepness of the sensitivity curves implies that no single value is
representative. are not detectable in any case. Altogether, extinction by the
accreting gas does not appear to be the reason why so few accreting planets