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# Seeing the unseen: a method to detect unresolved rings in protoplanetary
disks

Chiara E. Scardoni  Institute of Astronomy, University of Cambridge, Madingley
Road, Cambridge, CB3 0HA, UK  Dipartimento di Fisica, Università degli Studi
di Milano, Via Celoria 16, Milano, 20133, Italy  Richard A. Booth
Astrophysics Group, Imperial College London, Blackett Laboratory, Prince
Consort Road, London, SW7 2AZ, UK  School of Physics and Astronomy, University
of Leeds, Leeds, UK  Cathie J. Clarke  Institute of Astronomy, University of
Cambridge, Madingley Road, Cambridge, CB3 0HA, UK  Giovanni P. Rosotti
Dipartimento di Fisica, Università degli Studi di Milano, Via Celoria 16,
Milano, 20133, Italy  Álvaro Ribas  Institute of Astronomy, University of
Cambridge, Madingley Road, Cambridge, CB3 0HA, UK

(Received MONTH DD, YYYY; Revised MONTH DD, YYYY; Accepted MONTH DD, YYYY)

###### Abstract

While high resolution ALMA observations reveal a wealth of substructure in
protoplanetary discs, they remain incapable of resolving the types of small
scale dust structures predicted, for example, by numerical simulations of the
streaming instability. In this Letter, we propose a method to find evidence
for unresolved, optically thick dusty rings in protoplanetary disks. We
demonstrate that, in presence of unresolved rings, the brightness of an
inclined disc exhibits a distinctive emission peak at the minor axis.
Furthermore, the azimuthal brightness depends on both the geometry of the
rings and the dust optical properties; we can therefore use the azimuthal
brightness variations to both detect unresolved rings and probe their
properties. By analyzing the azimuthal brightness in the test-case of ring-
like substructures formed by streaming instability, we show that the resulting
peak is likely detectable by ALMA for typical disc parameters. Moreover, we
present an analytic model that not only qualitatively but also quantitatively
reproduces the peak found in the simulations, validating its applicability to
infer the presence of unresolved rings in observations and characterize their
optical properties and shape. This will contribute to the identification of
disk regions where streaming instability (and thus planet formation) is
occurring.

accretion, accretion disks — protoplanetary disks — planets and satellites:
formation — planets and satellites: terrestrial planets

††journal: ApJ

f

##  1 Introduction

High resolution continuum observations with the Atacama Large Millimeter/sub-
millimeter Array (ALMA) revealed a plethora of substructures in protoplanetary
disks (Brogan et al. 2015; Andrews et al. 2018b, see also the recent reviews
by Drazkowska et al. 2023; Bae et al. 2023). As the data resolution increases,
additional substructures may be uncovered (see, e.g., Facchini et al., 2020),
suggesting that, even at current high resolution, some substructures might not
be resolved. Recent efforts focused on enhancing data analysis techniques;
besides the standard deconvolution operated via the clean algorithm (Högbom,
1974; Clark, 1980; Cornwell, 2008), higher resolution techniques were
developed, working both in the Fourier plane (e.g. Perkins et al., 2015;
Tazzari et al., 2018; Jennings et al., 2020) and in the image plane (e.g.
Akiyama et al., 2017; Nakazato et al., 2019; Zawadzki et al., 2023). Also some
global disks observations suggest the presence of unresolved, optically thick
substructures, e.g. the correlation between millimeter luminosity and the disk
radius identified by Tripathi et al. (2017) and Andrews et al. (2018a). These
works demonstrated that unresolved substructures may hide in existing data,
and that not only the intrinsic data resolution, but also the subsequent data
analysis, can help detailing further the disk dust distribution.

A very natural way to produce sub-au, optically thick substructures is dust
filament formation via streaming instability (SI; Youdin & Goodman 2005;
Johansen et al. 2007; Johansen & Youdin 2007; Youdin & Johansen 2007). SI is a
powerful hydrodynamic instability driven by the dust-gas interaction; the
difference in azimuthal velocity between the dust and the gas components gives
rise to a drag force that slows down the dust grain motion, which thus loses
some angular momentum and move to orbits closer to the central star (radial
drift). The strongest radial drift is obtained for dust grains characterized
by Stokes number $St\sim 1$ and results in the rapid loss of grains as they
spiral toward the central star – this process gives rise to an impediment to
grain growth often referred to as the ’radial drift barrier’ (Adachi et al.,
1976; Weidenschilling, 1977; Takeuchi & Lin, 2002; Brauer et al., 2008; Pinte
& Laibe, 2014). However, the dust backreaction can be strong enough to cause
an increase in the gas azimuthal velocity (and thus a reduced radial drift) to
the point of triggering SI. When triggered, SI promotes the formation of dusty
dense filaments, tangential to the azimuthal direction, which can later
collapse under the action of self-gravity and form planetesimals (Johansen et
al., 2007; Youdin & Johansen, 2007; Carrera et al., 2015; Yang et al., 2017).
SI is therefore at the core of our planet formation theories, but the SI-
induced substructures are too small ($\lambda\ll H$) to be directly resolved
with our current observations. Indirect techniques to study this mechanism can
thus greatly aid in characterizing this effect and its impact in the planet
formation process.

In this context, we aim at exploring whether unresolved substructures can
leave a detectable imprint in the observations. We specifically focus on
dense, optically thick rings over an optically thin background, and analyze
the azimuthal brightness variation in this configuration. Due to the combined
effect of the rings’ geometry and optical depth, we expect maximum (minimum)
emission at the minor (major) axis, when the disk is observed at an
inclination; we develop a qualitative understanding and simplified theoretical
model of such emission in section 2. In section 3.1 we present simulations of
SI, and apply our model to the resulting tangential substructures, confirming
that the model successfully reproduces the emission obtained from the
simulations. We further show that, under realistic physical conditions, the
difference between the major and minor axis is higher than the $10\%$; this
implies that unresolved tangential substructures are potentially detectable in
the disk, even if their size is below ALMA resolution limits. Our findings are
then summarized in section 4.

###  1.1 Azimuthal brightness asymmetries

##  2 Toy model

Figure 1: Upper panel: sketch of a protoplanetary disk containing two
unresolved dense rings. The drawings on the right illustrate the local view of
the unresolved rings at $\varphi=0^{\circ}$ and $\varphi=90^{\circ}$ for a
face-on (boxes on the left), inclined (center), edge-on (right) disk. Lower
panel: 3D view of the box showing the model key parameters (as an example, we
consider the cylinder model).

As an illustration we consider an apparently smooth disk and assume that it
contains two unresolved, optically thick cylindrical rings with a finite
vertical extent at a given radial location $R$ (see the sketch in Figure 1);
at any azimuthal location we therefore expect to measure the emission from
both the optically thick rings and the optically thin background, averaged
over the observable area. To introduce a qualitative understanding of the
azimuthal profile of the disk emission, we focus on two disk patches located
at $\varphi=0^{\circ}$ and $\varphi=90^{\circ}$ (corresponding to the major
and minor axis, respectively) and analyze their emission in both the face-on
and inclined case, as illustrated in Figure 1 (in the sketch the rings are
locally represented by the yellow bands).

For a face-on disk, at different azimuthal angles the local orientation of the
rings varies within the observable area; the relative area of the disk patch
occupied by the optically thick substructures, however, remains the same at
all azimuthal angles. This implies that the averaged emission is independent
of the azimuthal angle.

If we instead observe the disk at an inclination, the projected lengths along
the vertical axis (i.e. the minor axis) are shortened with respect to the
face-on case. At $\varphi=0^{\circ}$ this does not impact the relative area of
the disk patch occupied by the rings, as both the rings and the background
projection are subject to the same shortening effect. Here, the inclination
only has the effect of increasing the column density, leading to an increase
of the overall optical depth. The increase in the column density is also
present at $\varphi=90^{\circ}$. On top of that the shortening of the
projected lengths along the vertical axis determines the reduction of the
optically thin area between the two rings; this happens because the optically
thick rings have a finite thickness which causes the optically thick areas to
infill the optically thin areas when the inclination is sufficiently large.
This effectively increases the relative area occupied by the optically thick
rings. In the limiting case of a edge-on disk, we thus expect the emission at
$\varphi=90^{\circ}$ to be completely optically thick (unless the rings are
infinitesimally geometrically thin). As a consequence the local average
optical depth is higher than that at $\varphi=0^{\circ}$. The overall effect
is that the disk observed at an inclination is characterized by higher (lower)
emission at the minor (major) axis.

###  2.1 Clump covering factor

We model the effect illustrated above by defining a local box associated with
each azimuthal location and computing the averaged emission within the box.
Each box contains an optically thin background and a number of optically thick
elongated substructures (’walls’), resembling resembling sections of
unresolved rings. We consider two models for the geometry of the
substructures: cylinders with elliptic cross-section (semi-axes $r_{\rm c}$
and $z_{\rm c}$) and cuboids with rectangular cross-section (semi-sides
$r_{\rm r}$ and $z_{\rm r}$).111In the rest of the Letter, we use the
subscript ‘c’/‘r’ to refer to the cylinders/rectangular cuboids model. Note
that in both cases $r$ represents half of the face-on width of the
substructures. If $r_{\rm c}=r_{\rm r}$ these models are equivalent when the
disk is face-on, but their 3D structure determines distinct effects when
observed at an inclination. We define as $d_{\rm c,r}$ the distance between
the projected center of two consecutive substructures and $L$ as the face-on
width of the box, thus the number of rings contained in each box is given by
$N_{\rm c,r}=L/d_{\rm c,r}$.

For a given inclination angle $i$, the projection on the sky of the box area
is $A(i)=A_{0}\cos(i)$, where $A_{0}$ is the face-on box area. When the
substructures are modeled as cylinders, their area varies with $\varphi$ and
$i$ as follows

$A_{\rm c}=2(N_{\rm c}-1)h_{\rm c,0}\sqrt{r_{\rm c}^{2}\cos^{2}i+z_{\rm
c}^{2}\sin^{2}i\sin^{2}\varphi},$ (1)

where $h_{\rm c,0}$ is the face-on cylinder height. In the rectangular cuboids
model, the area occupied by substructures is

$\displaystyle A_{\rm r}=$ $\displaystyle 2(N_{\rm r}-1)h_{\rm
r,0}\sqrt{\cos^{2}\varphi\cos^{2}i+\sin^{2}\varphi}\cdot$ (2)
$\displaystyle\cdot\left(r_{\rm
r}\sqrt{\cos^{2}\varphi+\cos^{2}i\sin^{2}\varphi}+z_{\rm r}\sin
i|\sin\varphi|\right).$

The area of the background is then determined as $A_{\rm bg}=A_{\rm
box}-A_{\rm c,r}$. The rings’ covering factor is given by the substructure-to-
box area ratio $A_{\rm c,r}/A_{\rm box}$; this parameter determines the
portion of the box occupied by the rings and, therefore, it regulates the
average optical depth at any $\varphi$.

###  2.2 Optical depth and averaged emission

To find the average emission at each $\varphi$, we start by associating a
face-on value for the optical depth to both the substructures ($\tau_{\nu;\rm
c,r}^{0}$) and the background ($\tau_{\nu,\rm bg}^{0}$). The optical depth of
the background for the inclined system is

$\tau_{\nu,\rm bg}=\frac{\tau_{\nu,\rm bg}^{0}}{\cos i},$ (3)

The optical depth of the inclined substructures is given by

$\tau_{\nu;\rm c,r}=\tau_{\nu;\rm c,r}^{0}\frac{r_{\rm c,r}}{\sqrt{r_{\rm
c,r}^{2}\cos^{2}i+z_{\rm c,r}^{2}\sin^{2}i\sin^{2}\varphi}},$ (4)

where we account for the variations in column density at different azimuthal
locations.222We assume that the substructures are sufficiently optically thick
that we do not need to consider variations of optical depth within them.
However, as we will show later, as long as the substructures are optically
thick ($\tau_{\nu;\rm c,r}^{0}>5$), the exact value of $\tau_{\nu;\rm c,r}$ is
unimportant for the purposes of this Letter.

By defining a temperature profile as in Tazzari et al. (2020b, a)

$T(R)=120{\rm K}\left(\frac{R}{\rm au}\right)^{-3/7},$ (5)

we can find the emission of both the cylinders and the background as

$I_{\nu}=B_{\nu}(T)(1-{\rm e}^{-\tau_{\nu}}),$ (6)

where $B_{\nu}(T)$ is the Plank function. We finally obtain the observable
emission as the emission averaged over the box (i.e. the observable area)

$\langle I_{\nu}\rangle_{\rm box}=\frac{I_{\nu;\rm c,r}A_{\rm c,r}+I_{\nu,\rm
bg}A_{\rm bg}}{A_{\rm box}}.$ (7)

###  2.3 Impact of the model parameters on the azimuthal brightness

Figure 2: Brightness temperature at $50\ \rm au$ as a function of the
azimuthal angle for disks with $60^{\circ}$ inclination containing unresolved,
optically thick rings. The plots in the upper (lower) row refer to the case
where rings are locally modeled as cylinders (rectangular cuboids). In each
column the effect of varying a given system parameter (indicated in the
legend) is analyzed. The black dashed line corresponds to the local
temperature, while the vertical red dashed lines indicate the location of the
minor axis.

Our model comprises 4 parameters, 2 geometric parameters ($2r_{\rm c}/d_{\rm
c}$, $r_{\rm c}/z_{\rm c}$) and 2 optical parameters ($\tau_{\nu,\rm c}$,
$\tau_{\nu,\rm bg}$). Note that $2r_{\rm c}/d_{\rm c}$ is limited by $0\leq
2r_{\rm c}/d_{\rm c}\leq 1$, with $0$ ($1$) corresponding to the case where
the rings occupy no (all) portion of the emitting area. Additionally, we must
define a temperature profile (Equation 5) and the disk inclination.

We analyze the impact of the model parameters on the brightness temperature
azimuthal profile (which probes the azimuthal emission of the disk)

$T_{\rm B}=\frac{h\nu}{k_{\rm
B}}\left[\log\left(1+\frac{2h\nu^{3}}{c^{2}\langle I_{\nu}\rangle_{\rm
box}}\right)\right],$ (8)

where $h$ and $k_{\rm B}$ are the Plank and Boltzmann constants, respectively;
$\nu$ is the frequency of the observation (we use the frequency corresponding
to ALMA band 6); $c$ is the light speed. The parameter analysis is shown in
Figure 2, when the dense rings are locally modeled as cylinders (upper row)
and rectangular cuboids (lower row). We take $2r_{\rm c}/d_{\rm c}=0.25$,
$r_{\rm c}/z_{\rm c}=1$, $\tau_{\nu,\rm bg}=0.1$, $\tau_{\nu,\rm c}=10$ as
reference parameters (gray curve in all the plots), and compute the emission
at the reference radius $50\ \rm au$. The black dashed line shows the local
temperature of the disk. In each panel we show the effect of varying one of
the parameters, as indicated in the legend above each panel column. In the
following we will refer to the brightness temperature at the minor (major)
axis as $T_{\rm min}$ ($T_{\rm maj}$).

The parameter $2r_{\rm c}/d_{\rm c}$ (first column) influences the covering
factor of the rings, and therefore the extent of the optically thick portion
of the emitting area. The higher is $2r_{\rm c}/d_{\rm c}$, the bigger is the
portion occupied by the optically thick substructures and, thus, $T_{\rm
maj}$. In some cases the brightness temperature saturates at the local
temperature; this happens when $2r_{\rm c}/d_{\rm c}$ is high enough to have
the substructures covering the entire emitting area for some azimuthal
locations and it is favored by high $2r_{\rm c}/d_{\rm c}$.

The parameter $r_{\rm c}/z_{\rm c}$ (second column) determines the geometry of
the rings’ section; high (low) values of the parameter correspond to
substructures wider (taller) than tall (wide). As noted previously lower
$r_{\rm c}/z_{\rm c}$ (narrow tall rings) lead to a more pronounced increase
in optical depth along the minor axis. When the disc is inclined, indeed, the
substructure thickness contributes to the projected area, with the biggest
contribution given at $\varphi=90^{\circ}$; this implies that for lower values
of $r_{\rm c}/z_{\rm c}$ the increase of optical depth at the minor axis (and
thus $T_{\rm min}$) is more pronounced. On the other hand, $T_{\rm maj}$ is
independent of $r_{\rm c}/z_{\rm c}$ because at that location we can only see
the width of the substructure, regardless of the disc inclination.

We now consider the optical parameters (third and fourth columns). As evident
from the plots in the third column, the value of $\tau_{\nu,\rm bg}$
determines $T_{\rm maj}$, with higher $\tau_{\nu,\rm bg}$ corresponding to
higher values of $T_{\rm maj}$ (this is because the emission is overall
optically thicker in this case). Provided that the rings are optically thick
($\tau_{\nu,\rm c}>1$) and the background is optically thin ($\tau_{\nu,\rm
bg}<1$), the contrast between the brightness temperature at the minor and
major axis is roughly independent of $\tau_{\nu,\rm bg}$. Since we are
considering optically thick rings ($\tau_{\nu,\rm c}>1$) $\tau_{\nu,\rm c}$
can be discarded, leaving us with a 3 parameter model.

Finally, by comparing the upper and lower row we can notice that modeling the
substructures as cylinders produces a $T_{\rm B}$ oscillation smaller than the
rectangular cuboids’ case; furthermore, the curves associated with the
rectangular cuboids tend to show a flattening in the area around the minor
axis. Both these differences are due to the different contribution given by
the component of the substructures that become visible when the system is
inclined (see Equation 1 and Equation 2), which determines a sharper
transition in the case of the rectangle (since it is characterized by
$90^{\circ}$ angles). In real cases, we can expect an intermediate behavior
between the two illustrated cases; with the $T_{\rm B}(\varphi)$ curve
determined by the specific 3D shape of substructures.

##  3 Streaming instability

Since SI is a natural way to obtain unresolved, optically thick substructures,
we use SI-induced substructures as a case-study for our model.

###  3.1 Local simulations of streaming instability

Figure 3: Dust density obtained from a 3D local simulation of a system
undergoing SI. The yellow overdense filaments are the SI induced dust clumps.

To analyze systems undergoing SI we employ the 2D (radial and vertical
directions) local simulations333Global SI simulations are computationally
prohibitive; however, previous studies (e.g. Yang & Johansen, 2014; Li et al.,
2018) demonstrated that dust concentration in filaments in the non-linear
phase of SI remains robust for bigger boxes. presented in Scardoni et al.
(2021), performed with the hydro-code Athena (Bai & Stone, 2010a, b, c). The
simulated boxes’ size is $L_{r}\times L_{z}=0.05\times 0.15$, with resolution
$N_{r}\times N_{z}=256\times 768$. We consider $N_{\rm tot}=7\cdot 10^{5}$
particles (equally distributed among 28 Stokes numbers) for which we adopt
(shear) periodic boundary conditions in the radial direction, and reflecting
boundary conditions in the vertical direction. We refer to Scardoni et al.
(2021) for details about the simulation set; in the following we show results
for the simulation characterized by: local dust-to-gas mass ratio $Z=0.03$;
particle number distribution $n(a)=n_{0}a^{-3.5}$; pressure support parameter
$\Pi=0.025$; Stokes numbers between $\tau_{\rm s}=10^{-4}-10^{-1}$. The
parameters choice relies on previous studies where the combinations of
parameters triggering SI were identified (e.g. Bai & Stone, 2010b; Carrera et
al., 2015; Johansen et al., 2006, 2007).

To analyze the azimuthal emission properties, we need the 3D dust
distribution. In principle, the azimuthal direction is unimportant as all the
parameters regulating SI only depend on the radius; therefore, we built 3D
boxes from the 2D simulations by assuming that the SI-induced substructures
are azimuthally symmetric. Practically, the 2D boxes are converted into 3D
boxes by replicating the 2D surface density in $R-z$ for $N_{\varphi}=256$
times along the $\varphi$ direction; this is equivalent to actual 3D boxes
with perfectly symmetric azimuthal evolution.

To ensure that the 3D-equivalent simulations are appropriate for our purposes,
we perform two 3D test-simulations, characterized by box of size $L_{r}\times
L_{\varphi}\times L_{z}=0.05\times 0.05\times 0.15$ and resolution
$N_{r}\times N_{\varphi}\times N_{z}=256\times 256\times 768$. We consider the
following parameters: $Z=0.03$, $n(a)=n_{0}a^{-3.5}$, $\Pi=0.025$, $\tau_{\rm
s}=10^{-4}-10^{-1}$ (first simulation) and $\tau_{\rm s}=10^{-3}-1$ (second
simulation). Figure 3 illustrates the typical substructures obtained in the 3D
simulations (converted in physical units, see section 3.2): the dust filaments
are dense, sub-au, tangential substructures, thus they have the same
properties as the system modeled in section 2; we also notice that the SI-
induced substructures are approximately azimuthally symmetric. Due to the high
computational cost, however, these simulations only run for $300\
\Omega^{-1}$; by that timestep dust filaments have been formed, but their
properties have not fully reached steady state, which is expected to require
$\sim 1500-2000\ \Omega^{-1}$ for this choice of parameters (Scardoni et al.,
2021). For this reason in the Letter we use the 3D-equivalent simulations and
demonstrate the appropriateness of this choice in the right panel of Figure 5.

###  3.2 Physical and radiative transfer model

To analyze the emission from the simulated systems, we need to convert the
dimensionless simulations to physical units; for this purpose we take as
reference model the MMSN in Chiang & Youdin (2010). The length unit is defined
by the local gas height $H_{\rm g}(R)=h_{0}R\left({R}/{\rm au}\right)^{2/7}$,
where $h_{0}=0.04$ is the disk aspect ratio at $1\ \rm au$. The mass unit is
defined by the local gas density $\Sigma_{\rm g}(R)=2200\ \rm g/cm^{2}\
\left({R}/{\rm au}\right)^{-1.5}$.

We define an integrated disk model extending from $1-100\ \rm au$, and split
it into $N_{R}\times N_{\varphi}=250\times 1000$ cells. We apply the local
physical units to map the simulated box to each cell, and ‘build’ the disk,
then we incline it by a given angle (we take $i=60^{\circ}$) and compute its
emission at all $(R,\varphi)$ coordinates.

To find the locally averaged emission from the $(R,\varphi)$ locations, we
adopt a simple analytic radiative transfer model. Within the simulated box, we
compute the optical depth as

$\tau_{\nu}=\frac{k_{\nu}^{\rm avg}\Sigma_{\rm d}}{\cos{i}},$ (9)

where $k_{\nu}^{\rm avg}$ is the averaged opacity (which includes information
from all the particles included in the simulation),444We use Birnstiel et al.
2018 code, assuming spherical compact grains with the following composition
(and volume fractions): water (0.6), silicates (0.1) (Warren & Brandt, 2008;
Draine, 2003) and organics (0.3) (Zubko et al., 1996). $\Sigma_{\rm d}$ is the
dust density, $i$ is the disk inclination angle. We then obtain the specific
intensity within the box as

$I_{\nu}=B_{\nu}(T)(1-{\rm e}^{-\tau_{\nu}}),$ (10)

and we average this throughout the simulated box as in Equation 7, obtaining
the observable emission.

###  3.3 Analysis of the azimuthal brightness

Figure 4: 2D maps of a protoplanetary disk undergoing SI obtained by mapping
the local simulation to different locations in the disk. On the left we show
the face-on view, on the right the same disk is seen at $60^{\circ}$
inclination.

In Figure 4 we show the 2D emission maps at ALMA band 6 of the simulation
described in section 3.1. As expected, the face-on system (plot on the left)
is azimuthally symmetric; looking at the isophotes of the inclined system (on
the right), we can notice that they are not elliptical (as expected for a
smooth disc), but curve towards smaller radii along the major axis, indicating
the low brightness temperature on that axis.

As a preliminary test of detectability, we analyze the $T_{\rm B}$ profiles
overall the disk, and select the radii where $\Delta T>10\%\langle T_{\rm
B}\rangle$ (where $\Delta T=|T_{\rm min}-T_{\rm maj}|$) – we chose this
threshold because the amplitude of azimuthal variations measured by Doi &
Kataoka (2021) is $10\%$, showing that a deviation of this kind would be
detectable. In the considered simulation, this condition is satisfied for
radii from $\sim 25-65\ \rm au$. For smaller radii, the high local density
makes the background optically thick, reducing the contrast between
$\tau_{\nu,\rm bg}$ and $\tau_{\nu,\rm c}$, and thus reducing the $T_{\rm B}$
oscillation. Conversely, at larger radii the local density is lower and the
filaments become optically thinner, thus the contrast between $\tau_{\nu,\rm
bg}$ and $\tau_{\nu,\rm c}$ is reduced and the $T_{\rm B}$ oscillation is
milder (see Appendix A). This analysis confirms that, for typical disk
parameters, unresolved optically thick rings would be detectable at a wide
range of radii.

Figure 5: Left panel: brightness temperature azimuthal profile at 50 au for
the SI simulation (black line); the magenta and cyan lines are obtained from
our model in the cylinder and rectangular cuboid configurations, respectively.
Right panel: ratio between the brightness temperature between the 3D
simulations and the 3D-equivalent version of the 2D simulations, considering a
disc inclination of $60^{\circ}$.

We select an arbitrary reference radius within the range where the $\Delta
T>10\%\langle T_{\rm B}\rangle$, namely $R=50\ \rm au$, to test whether our
analytic model can reproduce the simulated emission. For this purpose, we
estimate from the simulation the values for the 3 model parameters: the
geometric parameters $2r_{\rm c}/d_{\rm c}\sim 0.2$ and $r_{\rm c}/z_{\rm
c}\sim 0.33$ are found by analyzing the dust density distribution in the
simulation; the background optical depth $\tau_{\nu,\rm bg}\sim 0.1$ is
determined by taking the opacity model and finding the value corresponding to
the size of the group of small grains contained in the simulation (we consider
a $10\%$ error on the estimated $\tau_{\nu,\rm bg}$ value). Since we
demonstrated that the filaments’ optical depth is unimportant as long as they
are optically thick, we arbitrarily take $\tau_{\nu,\rm c}=10$. The azimuthal
brightness temperature profiles are shown in Figure 5, where the magenta/cyan
lines are the results for the cylinder/rectangular cuboids model, while the
black line shows the brightness temperature resulting from the simulations at
the reference radius. The simulation results are located between the two
models, potentially implying that the specific shape of the filaments is more
complex than that modeled; indeed, from Figure 3, we see that the filaments
are not perfectly represented by either considered model. This suggests that
the $T_{\rm B}-\varphi$ curve shape can be used to constrain the shape of the
unresolved rings.

We caution that in the model we do not account for the observation resolution.
To test the resolution effect, we convolved the image with a Gaussian beam
with ${\rm FWHM}=15\ \rm au$ (resolution of $0.1$ arcsec at $\sim 150\ \rm
pc$), finding that the $T_{\rm B}$ oscillation is reduced of $\sim 0.2$ K.
This suggests that the signature is still observable for favorable conditions
of the system.

Finally, we test that using the 3D-equivalent simulations is appropriate for
our purposes by computing the ratio between the brightness temperature of the
3D simulations and the 3D-equivalent simulations (right panel of Figure 5),
computed at $300\ \Omega^{-1}$. Since the difference between the two
simulations is $\sim 5\%$, we confirm that the 3D-equivalent simulations are a
good approximation of the 3D evolution of the system.

##  4 Discussion and conclusion

In this Letter we proposed a new method to detect unresolved, optically thick
rings. As illustrated in section 2, when the disk is inclined, the specific
combination of geometry and optical properties of the rings causes a
characteristic emission peak at the minor axis, and we argue that this effect
is uniquely related to unresolved optically thick rings. Indeed, when all the
system is optically thick, we expect $T_{\rm B}$ to be constant and equal to
the local temperature; conversely, Doi & Kataoka (2021) demonstrated the
opposite effect in the case of optically thin (but geometrically thick) rings,
where the emission is stronger at the major axis. No other azimuthal emission
asymmetries analyzed so far found the same effect as that presented in this
Letter; for example, D’Alessio et al. (2004) suggested that a wall (i.e. a
location where the emission transits from optically thin to optically thick)
in the emission might determine a maximum in the emission at
$\varphi=90^{\circ}$ and a minimum at $\varphi=270^{\circ}$. The shadowing
from a misaligned inner disk might produce a similar signature for very
specific system’s parameters (Min et al., 2017), this would however require
the shadow to be aligned with the major axis (which depends on the orientation
of the disc with respect to the line of sight). This event is a random chance
and would therefore require a fine tuning of the parameters, we thus expect
this situation to be unlikely. If the predicted signature is found in the
observations, we can check whether the effect is caused by a misaligned inner
disk by using information from GRAVITY observations (see, e.g., Bohn et al.,
2022). The peak (minimum) at the minor (major) axis is therefore peculiar to
optically thin systems containing annular optically thick substructures
(except for specific misaligned inner disks) and, therefore, can be used as a
detection method for such structures.

We demonstrated that the parameters regulating the brightness temperature
profiles are the ring size-to-distance ratio $2r/d$, the ring width-to-height
ratio $r/z$ and the background optical depth $\tau_{\nu,bg}$ (see section
2.3). We linked these parameters to the 3 main observables

  1. 1.

The brightness temperature oscillation between the minor and major axis
$\Delta T$ is primarily dependent of $2r/d$ and $r/z$, with $\Delta T$
increasing for higher $2r/d$ and lower $r/z$.

  2. 2.

The brightness temperature at the major axis $T_{\rm maj}$ is mainly
influenced by $\tau_{\rm bg}$ and $2r/d$, with higher $T_{\rm maj}$ for higher
values of these parameters.

  3. 3.

The detailed shape of the azimuthal profile is dependent on the 3D geometry of
the ring (i.e. whether the ring cross-section is more elliptical or
rectangular).

Therefore, if we have an independent measure of the disk inclination, we can
gain information on the dust optical depth and the 3D structure of the
unresolved rings by fitting the brightness temperature azimuthal profile.

This method is thus a valuable tool to search for observational signatures of
sub-au dusty ring-like substructure, which would otherwise remain undetected.
We demonstrated the applicability of our model to systems undergoing SI
(expected to form dense, elongated, sub-au substructures), showing that: (i)
for typical disc parameters, the brightness temperature oscillation is
$>10\%$, thus likely detectable by ALMA (Doi & Kataoka 2021 measured $T_{\rm
B}$ variations of $\sim 10\%$); (ii) the brightness temperature profile from
simulations is successfully reproduced by the model. This confirms the model
capability to detect and characterize substructures formed in the intermediate
stages of planet formation; more generally, we expect our model to be useful
to test all the processes that predict the formation of elongated dust
substructures.

We highlight that we neglected the scattering, whose effect would make the
optically thick substructures to appear fainter, determining a lower $\Delta
T$. However, the effect is likely small: for optically thick emission
($\tau_{\nu,\rm c}\geq 10$), we would need albedo $>0.8$ to have
$I_{\nu}/B_{\nu}<0.5$ (Zhu et al., 2019). Thus, although a factor $\sim 2$
reduction in $\Delta T$ might happen, it is unlikely to have a complete
removal of the signature due to scattering.

We finally caution that in this Letter we considered symmetric optically thick
rings in otherwise smooth disks. Even in this simplified case, it is
relatively hard to see the signature ‘by eye’ from the 2D maps (see Figure 4).
We thus anticipate that, when considering real observations, only a careful
analysis of $T_{\rm B}-\varphi$ profiles may reveal the peak at the minor
axis. If the signature is found in observations, the model can probe the
unresolved substructures; otherwise, it can still provide upper limits on the
properties of any unresolved substructure that might be present.

We thank the anonymous referee for the helpful comments and suggestions that
improved the quality of this letter. CES acknoledges support from Peterhouse
Cambridge and the Institute of Astronomy (University of Cambridge). CES and
GPR acknowledge support from the European Union (ERC Starting Grant DiscEvol,
project number 101039651) and from Fondazione Cariplo, grant No. 2022-1217.
Views and opinions expressed are, however, those of the author(s) only and do
not necessarily reflect those of the European Union or the European Research
Council. Neither the European Union nor the granting authority can be held
responsible for them. RAB acknowledges support from the Royal Society in the
form of a University Research Fellowship. AR has been supported by the UK
Science and Technology research Council (STFC) via the consolidated grant
ST/W000997/1. This work has also been supported by the European Union’s
Horizon 2020 research and innovation programme under the Marie Sklodowska-
Curie grant agreement number 823823 (DUSTBUSTERS). This work was in part
performed using the Cambridge Service for Data Driven Discovery (CSD3), part
of which is operated by the University of Cambridge Research Computing on
behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The DiRAC component
of CSD3 was funded by BEIS capital funding via STFC capital grants
ST/P002307/1 and ST/R002452/1, and STFC operations grant ST/R00689X/1.

##  Appendix A Results at different radii

As discussed in section 3.1, the brightness temperature variation is stronger
at intermediate radii because the contrast between the background and
filaments’ optical depth is stronger. We illustrate the optical depth radial
variation in the left panel of Figure 6 for our toy model using typical disc
parameters. At small radii, the background is characterized by $\tau_{\rm
bg}>1$, making thus the $T_{\rm B}$ oscillation is weak. At intermediate
radii, the contrast between $\tau_{\rm bg}$ and $\tau_{\rm c}$ is the
strongest, generating the strongest $T_{\rm B}$ oscillation. At large radii,
the contrast between $\tau_{\rm bg}$ and $\tau_{\rm c}$ reduces (due to the
non-linear relation between the opacity and the grain size), reducing the
$T_{\rm B}$ oscillation. We illustrate the normalized $T_{\rm B}$ azimuthal
profiles for 3 selected radii in the right panel of Figure 6, confirming that
the strongest oscillation is obtained at intermediate radii (50 au).

Figure 6: Left panel: optical depth of the background (pink) and the filaments
(blue) as modeled. Right panel: normalized brightness temperature at 10 au
(blue), 50 au (gray), 90 au (pink).

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