High-resolution ALMA Observations of Richly Structured Protoplanetary Disks in σ Orionis

High-resolution ALMA Observations of Richly Structured Protoplanetary Disks in σ
Orionis
Jane Huang1
, Megan Ansdell2
, Tilman Birnstiel3,4
, Ian Czekala5
, Feng Long6,10
, Jonathan Williams7
,
Shangjia Zhang1,8,9,10
, and Zhaohuan Zhu8,9
1
Department of Astronomy, Columbia University, 538 West 120th Street, Pupin Hall, New York, NY 10027, USA; jane.huang@columbia.edu
2
NASA Headquarters, 300 E Street SW, Washington, DC 20546, USA
3
University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität München, Scheinerstr. 1, 81679 Munich, Germany
4
Exzellenzcluster ORIGINS, Boltzmannstr. 2, D-85748 Garching, Germany
5
School of Physics & Astronomy, University of St. Andrews, North Haugh, St. Andrews KY16 9SS, UK
6
Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721, USA
7
Institute for Astronomy, University of Hawai‘i at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA
8
Department of Physics and Astronomy, University of Nevada, Las Vegas, 4505 South Maryland Parkway, Las Vegas, NV 89154, USA
9
Nevada Center for Astrophysics, University of Nevada, Las Vegas, Las Vegas, NV 89154, USA
Received 2024 August 11; revised 2024 September 28; accepted 2024 September 30; published 2024 November 18
Abstract
The Atacama Large Millimeter/submillimeter Array (ALMA) has detected substructures in numerous
protoplanetary disks at radii from a few to over 100 au. These substructures are commonly thought to be
associated with planet formation, either by serving as sites fostering planetesimal formation or by arising as a
consequence of planet–disk interactions. Our current understanding of substructures, though, is primarily based on
observations of nearby star-forming regions with mild UV environments, whereas stars are typically born in much
harsher UV environments, which may inhibit planet formation in the outer disk through external photoevaporation.
We present high-resolution (∼8 au) ALMA 1.3 mm continuum images of eight disks in σ Orionis, a cluster
irradiated by an O9.5 star. Gaps and rings are resolved in the images of five disks. The most striking of these is
SO 1274, which features five gaps that appear to be arranged nearly in a resonant chain. In addition, we infer the
presence of gap or shoulder-like structures in the other three disks through visibility modeling. These observations
indicate that substructures robustly form and survive at semimajor axes of several tens of au or less in disks
exposed to intermediate levels of external UV radiation as well as in compact disks. However, our observations
also suggest that disks in σ Orionis are mostly small, and thus millimeter continuum gaps beyond a disk radius of
50 au are rare in this region, possibly due to either external photoevaporation or age effects.
Unified Astronomy Thesaurus concepts: Protoplanetary disks (1300); Planet formation (1241); Radio
interferometry (1346)
1. Introduction
The high detection rates of substructures such as gaps and
rings in high-resolution Atacama Large Millimeter/submilli-
meter Array (ALMA) millimeter continuum observations of
Class II protoplanetary disks have profoundly altered our
understanding of planet formation (e.g., ALMA Partnership
et al. 2015; S. M. Andrews et al. 2018b; J. Huang et al. 2018a;
F. Long et al. 2018; L. A. Cieza et al. 2021; Y. Shi et al. 2024).
Models have long predicted that massive protoplanets can open
gaps in disks (e.g., D. N. C. Lin & J. Papaloizou 1986;
G. Bryden et al. 1999; R. P. Nelson et al. 2000), but prior to the
advent of ALMA, most disks were assumed to have smooth
surface density profiles, with a small fraction of older disks
featuring inner cavities (e.g., P. D’Alessio et al. 1998;
S. M. Andrews et al. 2009). ALMA’s millimeter continuum
observations are used to probe thermal emission from roughly
millimeter-sized dust grains in disk midplanes. The substruc-
tures observed by ALMA at a range of disk radii and ages are
commonly hypothesized to be the result of planet–disk
interactions, implying that giant planet formation begins early
and can occur readily even at semimajor axes of tens to more
than 100 au (e.g., K. D. Kanagawa et al. 2015; R. Dong et al.
2018a; S. Zhang et al. 2018). This interpretation has been
bolstered by direct imaging detections of several giant
protoplanets and protoplanet candidates and detections of
non-Keplerian gas motion inside disk gaps (e.g., M. Keppler
et al. 2018; C. Pinte et al. 2019; R. Teague et al. 2019;
T. Currie et al. 2022; I. Hammond et al. 2023). Alternatively,
some works have explored whether instead of being the
outcome of planet formation, disk rings are sites that
concentrate solids sufficiently to trigger planetesimal formation
(e.g., A. Morbidelli 2020; A. Izidoro et al. 2022; E. J. Lee et al.
2022). In this case, these substructures may arise due to
processes such as dust accumulation at snowlines (e.g.,
K. Zhang et al. 2015; S. Okuzumi et al. 2016; P. Pinilla
et al. 2017), zonal flows (e.g., A. Johansen et al. 2009;
X.-N. Bai & J. M. Stone 2014), surface density enhancements
at the edges of dead zones (e.g., M. Flock et al. 2015; W. Lyra
et al. 2015), disk winds (e.g., S. S. Suriano et al. 2018), or
infall-driven instabilities (e.g., J. Bae et al. 2015; A. Kuznets-
ova et al. 2022). Irrespective of their origins, substructures are
thought to play an essential role in disk evolution through their
influence on the radial transport of material through the disk,
chemistry, and temperature (e.g., P. Pinilla et al. 2012;
F. Alarcón et al. 2020; A. Banzatti et al. 2023).
The Astrophysical Journal, 976:132 (23pp), 2024 November 20 https://doi.org/10.3847/1538-4357/ad84df
© 2024. The Author(s). Published by the American Astronomical Society.
10
NASA Hubble Fellowship Program Sagan Fellow.
Original content from this work may be used under the terms
of the Creative Commons Attribution 4.0 licence. Any further
distribution of this work must maintain attribution to the author(s) and the title
of the work, journal citation and DOI.
1

However, one of the significant biases of published high-
resolution millimeter-wavelength disk observations is that they
have principally targeted nearby (d < 200 pc) star-forming
regions, such as Taurus, Ophiuchus, Lupus, Chamaeleon, and
Upper Sco (see, e.g., J. Bae et al. 2023 and references therein).
The disks targeted in these nearby regions are not necessarily
representative of typical planet formation environments. Stars
are often born in close proximity to O stars and thus exposed to
external far-ultraviolet (FUV) radiation fields of order
103
−104
G0 (M. Fatuzzo & F. C. Adams 2008; A. J. Winter
et al. 2020), where G0 = 1.6 × 10−3
erg s−1
cm−2
is the
H. J. Habing (1968) field. Consequently, the solar system is
thought to have been likely to form in such an environment
(F. C. Adams 2010). In contrast, estimates for the external FUV
radiation fields of disks in nearby star-forming regions are
generally on the order of 100
−102
G0 (e.g., P. F. Goldsmith
et al. 2010; L. I. Cleeves et al. 2016; L. Trapman et al. 2020).
Upper Sco is an OB region that has undergone expansion
(V. Squicciarini et al. 2021), so it is possible that its disks were
exposed to higher external UV radiation in the past. However,
since high-resolution ALMA studies of Upper Sco have
focused on the most massive disks (S. M. Andrews et al.
2018a; J. Stadler et al. 2023), they have likely still been biased
against disks that have experienced strong external UV
radiation in the past.
Models have demonstrated that external UV radiation can
exert a significant influence on disk structure and therefore
shape the properties of the resulting planetary systems.
A. J. Winter et al. (2022) found that external FUV radiation
fields as low as ∼100 G0 can have a significant effect on
planetary growth and migration. Strong external UV radiation
can drive disk mass loss through external photoevaporation,
leading to smaller disks and shorter lifetimes (e.g.,
D. Johnstone et al. 1998; H. Störzer & D. Hollenbach 1999).
Consequently, the occurrence rate of giant planets is expected
to be lower around stars exposed to stronger external UV fields
(e.g., P. J. Armitage 2000; A. J. Winter et al. 2022). The
migration behavior of protoplanets is also sensitive to external
photoevaporation (e.g., D. Veras & P. J. Armitage 2004;
A. J. Winter et al. 2022).
The star-forming regions that feature a large number of stars
currently exposed to high (‹103
G0) external UV radiation are
located at distances of 400 pc or beyond (see, e.g., review by
A. J. Winter & T. J. Haworth 2022). As the nearest one of
these, the Orion molecular cloud complex (d ∼ 400 pc) has
frequently been targeted for studies of the influence of external
photoevaporation on disk populations. Hubble Space Telescope
optical images of proplyds in the Trapezium showed ionization
fronts due to radiation from the nearby O star θ1
Ori C
(C. R. O’Dell et al. 1993; C. R. O’Dell & Z. Wen 1994).
ALMA observations indicate that the disk size distributions in
the Orion Nebula Cluster (ONC) and the OMC1 cloud are
shifted toward smaller radii compared to nearby star-forming
regions, with some combination of external photoevaporation
and dynamical truncation possibly setting disk sizes
(J. A. Eisner et al. 2018; J. Otter et al. 2021). Millimeter-
wavelength surveys have also shown that disk masses tend to
decrease as local FUV field strengths increase in the ONC, σ
Orionis, L1641, and L1647, consistent with the behavior
expected from external photoevaporation (R. K. Mann et al.
2014; M. Ansdell et al. 2017; S. E. van Terwisga &
A. Hacar 2023).
The diminished masses and sizes of disks close to massive
stars in Orion raise the question of whether they still commonly
harbor the kinds of millimeter continuum disk substructures
that are widespread in nearby star-forming regions. Substruc-
tures have been detected in a couple of individual high-
resolution ALMA studies of massive disks in Orion, including
V1247 Ori and GW Ori (S. Kraus et al. 2017, 2020; J. Bi et al.
2020). Most ALMA surveys targeting Orion, though, have had
spatial resolutions coarser than 20 au (e.g., M. Ansdell et al.
2017, 2020; J. A. Eisner et al. 2018; S. E. van Terwisga et al.
2020; N. P. Ballering et al. 2023; S. E. van Terwisga &
A. Hacar 2023), which is wider than the typical scales of disk
substructures resolved in nearby star-forming regions (e.g.,
J. Huang et al. 2018a; F. Long et al. 2018; L. A. Cieza et al.
2021). The highest-resolution ALMA survey of disks in Orion
published thus far has been a 0 03 (∼12 au) survey of the ONC
and OMC1 by J. Otter et al. (2021). Some of their disk images
hint at the presence of substructures, but the bright large-scale
emission in these regions poses a challenge to characterizing
disk morphology.
The σ Orionis cluster presents a prime opportunity to
investigate the properties of disks exposed to strong external
UV radiation. The cluster is generally estimated to be about 3
−5 Myr old, albeit with large uncertainties (J. M. Oliveira et al.
2002, 2004; M. R. Zapatero Osorio et al. 2002; J. A. Caballero
et al. 2019). The cluster is strongly irradiated by an O9.5 star
within the eponymous multiple star system σ Orionis
(H. L. Johnson & W. W. Morgan 1953; R. F. Garrison 1967;
F. M. Walter et al. 1998). (To avoid confusion, we will
henceforth refer to the cluster as σ Orionis and the star as σ
Ori.) With AV values generally <1, the extinction toward this
region is low compared to other parts of Orion (T. A. Lee 1968;
V. J. S. Béjar et al. 2004), so contamination from large-scale
emission does not present a problem for millimeter continuum
disk imaging. To examine the structure of disks in this cluster
in greater detail, we used ALMA to observe a sample of eight
disks at a resolution of 0 02 (∼8 au), improving upon previous
observations by an order of magnitude. Section 2 provides an
overview of the selected targets. The observations and data
reduction are described in Section 3, while modeling and
analysis are presented in Section 4. The results are discussed in
Section 5 and summarized in Section 6.
2. Sample Overview
Our eight targets were selected from the M. Ansdell et al.
(2017) ALMA 1.3 mm continuum survey of Class II disks in σ
Orionis, which imaged disks at a resolution of ∼0 25
(∼100 au). Disk classifications were based on the Spitzer
survey by J. Hernández et al. (2007). While one of the targets,
SO 1153, was categorized by J. Hernández et al. (2007) as
Class I, M. Ansdell et al. (2017) included it in their survey
because its colors were borderline between Class I and Class II.
M. Ansdell et al. (2017) and K. Maucó et al. (2023) did not
note any obvious envelope emission in their millimeter
continuum and 12
CO observations of SO 1153, although the
disk is only moderately resolved.
The sample was restricted to stars with estimated stellar
masses between ∼0.4 and 1 Me, corresponding to the mass
range for which disks have been best characterized with high-
resolution ALMA observations in nearby star-forming regions
(e.g., S. M. Andrews et al. 2018a; F. Long et al. 2018;
L. A. Cieza et al. 2021). Because disks in σ Orionis had not
2
The Astrophysical Journal, 976:132 (23pp), 2024 November 20 Huang et al.

previously been observed at high resolution, we set a
conservative flux cutoff of 1.5 mJy to ensure a reasonable
signal-to-noise ratio (S/N). This cutoff was determined by
generating synthetic ALMA images of disks based on the best-
fit models of the Taurus disks in F. Long et al. (2018) but at a
distance of 400 pc to test the detectability of analogous
substructures in σ Orionis.
Given the flux and stellar mass constraints, we then chose
our targets to span a range of projected separations from σ Ori.
There is some uncertainty in the literature regarding the
distance to σ Ori. The O9.5 star does not have a Gaia parallax
due to its extreme brightness. Using infrared interferometry,
G. H. Schaefer et al. (2016) measured a distance of
387.5 ± 1.3 pc to σ Ori. However, the median Gaia distance
to members of σ Orionis is 402 pc (M. Žerjal et al. 2024).
M. Žerjal et al. (2024) commented that there may be some
systematic offset between distances measured from interfero-
metry and from Gaia, since σ Ori is thought to be at the center
of its cluster. Given this uncertainty, we refer to projected
rather than absolute separations from σ Ori in this work. The
projected separations of the targets range from 0.68 to 2.73 pc,
which would correspond to FUV fields of ∼200−2600 G0 (see
Table 1 and K. Maucó et al. 2023). The FUV field estimates
from K. Maucó et al. (2023) are based on projected separations,
so they should be considered upper bounds. The host star
properties of the selected targets are listed in Table 1.
No obvious substructures are visible in the millimeter
continuum images in M. Ansdell et al. (2017) or K. Maucó
et al. (2023), which reobserved some of the targets from
M. Ansdell et al. (2017) at similar resolution but higher
sensitivity. P. G. Valegård et al. (2024) used the Very Large
Telescope/SPHERE instrument to image the SO 1274 disk in
infrared scattered light, which traces micron/submicron-sized
dust grains in the upper layers of disks. In the SPHERE image,
the disk appears faint, with no apparent substructure. However,
among our targets, J. Hernández et al. (2007) identified SO 897
as a transition disk candidate based on its infrared colors.
The fluxes and projected separations of the targets are plotted
relative to the rest of the disks detected in σ Orionis in Figure 1.
While our flux cutoff biases our sample toward brighter
sources, our target fluxes still span an order of magnitude. A
two-dimensional plot of the positions of the targets with respect
to σ Ori is also shown in Figure 1. The targets are primarily to
the east of σ Ori, which is a consequence of members of the
cluster being preferentially located to the east of σ Ori
(J. A. Caballero 2008). J. A. Caballero (2008) speculated that
the asymmetric distribution of stars in σ Orionis resulted from
variations in the dust surface density in the molecular cloud at
the start of star formation. K. Maucó et al. (2023) measured the
AV and 1.3 mm continuum disk fluxes for 50 stars in σ Orionis;
the AV values as a function of R.A. are plotted in Figure 2. The
AV values are <1 for most sources and exhibit a similar spread
to the east and west of σ Ori. However, given the intermediate
age of the cluster, the present-day spatial variation (or lack
thereof) in AV may not necessarily reflect the relative extinction
levels between the east and west sides of σ Orionis early on in
the cluster’s history.
3. Observations and Data Reduction
Long-baseline 1.3 mm continuum observations of the eight
disks were obtained by ALMA program 2022.1.00728.S (PI:
J. Huang). For all observations, the correlator was configured
with four spectral windows (SPWs) centered at 224, 226, 240,
and 242 GHz. Each SPW had a bandwidth of 2 GHz divided
into 128 channels. The time on target for each execution block
(EB) was 29 minutes 40 s. Three EBs were used to observe
SO 1274, while other targets were observed for two EBs each.
The quasars J0423–0120, J0532–0307, and J0529–0519 were
used as calibrators for all observations. Table 2 lists the
observing date, baseline lengths, and number of antennas for
each EB. Calibrated measurement sets were produced by
ALMA staff with the standard ALMA pipeline in CASA
6.4.1 (CASA Team et al. 2022).
To provide uv-coverage at shorter distances, we retrieved
lower-resolution 1.3 mm observations of our targets from
program 2016.1.00447.S (PI: J. Williams) through the ALMA
archive. These observations were first published in K. Maucó
et al. (2023), who describe them in more detail. In brief, each
target was observed with baselines ranging from ∼15 m to
2.6 km over the course of eight EBs, with a total time on target
of ∼9 minutes. The raw data were calibrated with the CASA
4.7.2 pipeline. Three of the SPWs covered 12
CO, 13
CO, and
C18
O J = 2−1; the 12
CO images are presented in K. Maucó
et al. (2023).
Subsequent processing of the new and archival data was
performed with CASA 6.5. First, we flagged channels where
CO line emission might be present in the archival short-
baseline data and averaged the channels to create
Table 1
Host Star Properties
Name 2MASS Designation SpT M* L* Distance Projected Separation from σ Ori FUV Field
(Me) (Le) (pc) (pc) (logG0)
SO 662 J05384027–0230185 K7 0.64 0.68 394.9 ± 3.1 0.68 3.41
SO 844 J05390136–0218274 M1 0.44 0.62 408.4 3.6
3.4
-
+
2.11 2.42
SO 897 J05390760–0232391 K6 0.7 0.85 375.5 6.8
8.0
-
+
0.77 3.29
SO 984 J05391883–0230531 K7 0.64 0.72 403.5 3.4
3.0
-
+
1.16 2.93
SO 1036 J05392519–0238220 M0 0.59 0.53 388.7 3.4
2.9
-
+
1.21 2.91
SO 1152 J05393938–0217045 M0 0.58 0.61 391.3 3.3
4.2
-
+
2.73 2.20
SO 1153 J05393982–0231218 K5 0.9 0.33 390.4 3.6
3.7
-
+
1.70 2.61
SO 1274 J05395465–0246341 K7 0.64 0.68 400.1 2.9
3.7
-
+
2.39 2.31
Note. Spectral type, stellar mass, stellar luminosity, and external FUV radiation field values come from K. Maucó et al. (2023). Our FUV field values are rescaled from
K. Maucó et al. (2023) because they calculate projected separations using the distances to the individual sources, whereas we use the median distance to the cluster of
402 pc (M. Žerjal et al. 2024) so that the projected distances scale linearly with the angular separations. Stellar distances come from C. A. L. Bailer-Jones et al. (2021),
which is based on data from Gaia Collaboration et al. (2021).
3

pseudocontinuum visibilities for each target. No self-calibra-
tion was applied to SO 897 (the faintest disk in the sample) due
to its low S/N. For the other disks, phase self-calibration was
first performed separately on each EB from the archival short-
baseline data set with a solution interval spanning all scans.
Images of each EB were produced with the Högbom CLEAN
algorithm (J. A. Högbom 1974) as implemented in the
tclean task. (Since the disks are either marginally resolved
or unresolved in the short-baseline observations, multiscale
CLEAN is not necessary.) As in S. M. Andrews et al. (2018a),
the disk centers were determined by fitting each image with a
two-dimensional Gaussian using the imfit task. The EBs
were then aligned with one another using the phaseshift
and fixplanets tasks such that the disk emission was
centered at the phase center. For each disk except SO 897 and
SO 844 (which had insufficient S/N), the EBs were then
imaged together and phase self-calibration was performed with
scan-length solution intervals. Finally, amplitude self-calibra-
tion was performed with scan-length solution intervals on the
combined EBs for all disks except SO 662, SO 844, and
SO 897, which are the three faintest disks in the sample.
Figure 1. Left: a comparison of the 1.3 mm fluxes and projected separations of the sources observed in this work (light blue diamonds) and other disk detections
(purple dots) and upper limits (gray triangles) in σ Orionis from M. Ansdell et al. (2017). The flux values used are all from M. Ansdell et al. (2017) for consistency.
Right: a plot of the coordinates of the σ Orionis sources observed by M. Ansdell et al. (2017). Light blue dots correspond to sources observed in this work, while
purple dots correspond to other disk detections and gray dots correspond to nondetections from M. Ansdell et al. (2017). The position of σ Ori is marked by an orange
star. The sizes of the markers for the detected sources are scaled by the 1.3 mm flux. The dashed circles mark projected separations of 1 and 2 pc, respectively.
Figure 2. AV of stars in σ Orionis observed by K. Maucó et al. (2023), plotted
as a function of R.A. The vertical orange line marks the R.A. of σ Ori. Light
blue dots correspond to sources observed in this work, while purple dots
correspond to other disk detections and gray dots correspond to nondetections.
The sizes of the markers for the detected sources are scaled by the 1.3 mm flux
measured by K. Maucó et al. (2023).
Table 2
Description of EBs in ALMA Program 2022.1.00728.S
Source Date Baseline Lengths Number of Antennas
SO 662 2023 Jul 26 256 m–15.2 km 43
2023 Jul 27 230 m–16.2 km 45
SO 844 2023 Jul 31 230 m–16.2 km 43
2023 Aug 2 230 m–16.2 km 46
SO 897 2023 Jul 28 230 m–16.2 km 48
2023 Jul 31 230 m–16.2 km 46
SO 984 2023 Aug 6 230 m–16.2 km 44
2023 Aug 7 230 m–16.2 km 48
SO 1036 2023 Aug 4 230 m–16.2 km 43
2023 Aug 5 230 m–16.2 km 44
SO 1152 2023 Aug 2 230 m–16.2 km 46
2023 Aug 3 230 m–16.2 km 46
SO 1153 2023 Aug 1 230 m–16.2 km 43
2023 Aug 4 230 m–16.2 km 43
SO 1274 2023 Jul 19 230 m–15.2 km 46
2023 Jul 27 230 m–16.2 km 45
2023 Jul 29 230 m–16.2 km 44
4

We then created channel-averaged measurement sets from
each EB of the long-baseline observations and imaged them
separately using multiscale CLEAN (T. J. Cornwell 2008). Phase
self-calibration was performed on the individual EBs for
SO 1153, the brightest disk in the sample. Self-calibration was
tested on the other sources but did not improve the images. For
SO 1274, SO 1153, and SO 1036, the EBs were aligned with the
same procedure as the short-baseline observations. The disk
emission appeared to be well centered for the other sources, so no
phase shift was applied, but the phase centers were relabeled with
fixplanets to match the short-baseline observations. The
short-baseline and long-baseline observations were then com-
bined and imaged together with multiscale CLEAN and a robust
value of 0.5. For SO 1153, phase self-calibration was performed
on the combined observations with a scan-length solution
interval. Finally, all of the disk images were primary-beam-
corrected. The resulting image properties are listed in Table 3.11
4. Analysis
4.1. Overview of Disk Emission Morphology
Continuum images are shown in Figure 3, while azimuthally
averaged, deprojected radial profiles are shown in Figure 4.
Annular gaps and rings are visible in the SO 1152, SO 1274,
SO 844, and SO 1153 disks, while a small inner cavity is
detected in the SO 897 disk. Some of the substructures appear
to be slightly hexagonal due to the shape of the point-spread
function (see, e.g., S. M. Andrews et al. 2018a). No gaps are
immediately evident in the SO 1036, SO 984, and SO 662
disks, but the radial profiles of SO 1036 and SO 984 exhibit
subtle slope changes.
The disks generally appear to be axisymmetric. However,
SO 897 is slightly brighter (by ∼15%) on its eastern side
compared to its western side (Figure 5). Given the high
inclination of the disk, this brightness asymmetry is likely a
consequence of the viewing geometry. For disks at higher
inclination, the far side can appear brighter due to either a
puffed-up cavity wall (e.g., C. P. Dullemond & J. D. Monnier
2010; Á. Ribas et al. 2024) or a geometrically thick disk (e.g.,
N. Ohashi et al. 2023).
4.2. Parametric Intensity Profile Modeling
Modeling disk emission in the uv-plane is often an effective
technique for inferring the presence of substructures that are
not readily visible in CLEAN images (e.g., K. Zhang et al.
2016; J. Jennings et al. 2020; A. Michel et al. 2023). We thus
fit the observations with parametric intensity models in order to
infer the radial intensity profiles of our targets. For all disks
except SO 897, we assumed that their radial intensity profiles
can be described as the sums of a Gaussian component centered
at r = 0 and N additional Gaussian components with offsets in
the radial directions (these are commonly referred to as
Gaussian rings, but N does not necessarily correspond to the
number of rings visible in the total intensity profile if two or
more of the Gaussian components are closely overlapping):
⎜ ⎟ ⎜ ⎟
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
( )
( )
( )
I r A
r
A
r r
exp
2
exp
2
, 1
i
N
i
i
i
0
2
0
2
1
2
2
å
s s
= - + -
-
=
where r is the disk radius in au in cylindrical coordinates.
Expressions of this form have been shown to reproduce high-
resolution ALMA observations of disks well (e.g., V. V. Guzmán
et al. 2018; A. Isella et al. 2018). The value of N for each disk
was initially chosen based on the number of rings or extended
emission tails visible in the radial profiles extracted from ALMA
images, and higher values of N were tested to determine whether
they better reproduce the disk emission. Since adding more
parameters can result in overfitting the data, we calculated the
Bayesian information criterion (BIC; G. Schwarz 1978) to
determine which model to select (Appendix A).
Given a model radial intensity profile, we then used the
mpol package (I. Czekala et al. 2023; B. Zawadzki et al. 2023)
to generate a model disk image with some position angle (P.
A.), inclination (i), east–west offset from the phase center (Δx),
and north–south offset from the phase center (Δy), under the
assumption that the disk is geometrically thin. Thus, 3N + 6
free parameters are required to specify a disk model fully.
For SO 897, which has an inner cavity, we adopted the
following radial intensity profile:
⎜ ⎟
⎛
⎝
⎞
⎠
( )
( )
( )
I r A
r r
exp
2
. 2
1
1
2
1
2
s
= -
-
Given the observed emission asymmetry (Figure 5), we did
not assume that the emission comes from a flat surface (z = 0 in
disk coordinates). Instead, we assumed that the emission comes
from a flared surface of the form ( )
( )
z z0 1
r =
r f

, where ρ is
the disk radius in arcseconds and z0 and f are free parameters.
We then used the eddy package (R. Teague 2019) to transform
disk coordinates to sky coordinates and mpol to generate a
model image for some P.A., i, Δx, and Δy. Thus, a total of nine
free parameters are required to specify the model for SO 897.
For all disks, we then used mpol to generate model
visibilities mod
 from the model images at the same uv-
coordinates as the observed visibilities obs
 . The log likelihood
(up to a constant) is
∣ ∣ ( )
w
ln
1
2
, 3
i
n
i
1
obs,i mod,i
2
å
= - -
=
  
where w denotes the visibility weights and n is the total number
of visibilities. In general, the absolute scaling of weights in
calibrated measurement sets delivered by ALMA may not be
correct.12
Hence, we used the procedure described in B. Zaw-
adzki et al. (2023) and implemented in the visread package
Table 3
Image Properties
Source Synthesized Beam rms Peak Intensity
(mas × mas (deg)) (μJy beam−1
) (mJy beam−1
)
SO 662 22 × 20 (72.8) 10 0.42
SO 844 22 × 19 (−77.1) 10 0.20
SO 897 24 × 20 (77.4) 10 0.33
SO 984 22 × 20 (−85.0) 11 0.38
SO 1036 24 × 19 (−60.8) 12 0.46
SO 1152 23 × 21 (−82.1) 9 0.27
SO 1153 23 × 20 (−70.5) 11 0.84
SO 1274 23 × 21 (87.0) 9 0.41
11
The continuum visibilities and images can be downloaded from Zenodo:
doi:10.5281/zenodo.13821034.
12
See, for example, https://casaguides.nrao.edu/index.php/DataWeightsAnd
Combination and Y. D. Hezaveh et al. (2013).
5

(I. Czekala et al. 2021) to correct the scaling. For each disk, the
CLEAN model was subtracted from the visibilities, and then a
Gaussian was fit to the scatter in the residual visibilities
normalized by V w
1
s = . The weights were then rescaled by a
factor of f
1
2, where f was the standard deviation of the best-fit
Gaussian. The value of f was typically ∼2.
Gaussian priors were specified for log A0, log 0
s , A
log i,
log i
s, ri, i, z0, f, P.A., Δx, and Δy. Their means and standard
Figure 3. 1.3 mm ALMA continuum images of the disks, sorted in descending order of projected separation from σ Ori. The synthesized beam is shown in the lower
left corner of each panel. An asinh stretch is used for some of the disks in order to show the faint emission at larger radii more clearly.
Figure 4. Azimuthally averaged, deprojected radial profiles of the disks produced from the CLEAN images. The blue shaded ribbon shows the 1σ error, which is
estimated by taking the standard deviation of the pixels in each radial bin and dividing by the square root of the number of beams spanned by the corresponding
annulus. The gray bar in the upper right corner of each panel denotes the major axis of the synthesized beam.
6

deviations were set based on visual inspection of the observed
images and radial profiles.
The posterior distributions were estimated with the pyro
(E. Bingham et al. 2019) implementation of the stochastic
variational inference (SVI) algorithm (M. D. Hoffman et al.
2013). By using parametric expressions to approximate the
posterior distributions, SVI can be used to estimate the
posterior much faster than Markov Chain Monte Carlo
(MCMC) methods typically can and is therefore particularly
advantageous for high-dimensional models, where MCMC
often struggles.13
However, it should be kept in mind that
MCMC methods are theoretically guaranteed to converge to the
true posterior for sufficiently long runs, whereas SVI is an
approximate method. We assumed that the posterior can be
approximated as a multivariate normal distribution (a “guide”
in the parlance of pyro). This assumption appears reasonable
based on the posterior distributions derived from MCMC
modeling of the radial profiles of circumstellar disks (e.g.,
M. A. MacGregor et al. 2015; A. G. Sepulveda et al. 2019).
More generally, it is oftentimes the case that posterior
distributions are approximately normal. Stochastic gradient
descent was used to optimize the evidence lower bound
(ELBO) in order to estimate the parameters of the guide. Using
the Adam optimizer (D. P. Kingma & J. Ba 2017) with a
learning rate of 0.02, we found that 15,000 iterations were
sufficient for the ELBO values to converge (note that these
iterations are steps in the optimization routine and should not
be confused with samples of the posterior).
For each disk, we then generated 2000 samples from the
estimated posteriors and calculated the median value of each
parameter and the 68% confidence intervals. The model values
are given in Appendix B. To check that the parametric intensity
models reasonably describe the observations, we generated
model visibilities from the median parameter values and then
imaged them in the same manner as the observations. A
comparison between the models and observations is shown in
Figure 6. The models reproduce the data well, with maximum
residual levels at ∼4σ.
4.3. Substructure Properties
4.3.1. Locations, Widths, and Depths
With the posterior samples from above, we then generated
2000 model radial intensity profiles for each disk (Figure 7).
Substructures were identified in the following manner. A ring is
defined to be a local maximum in the radial intensity profile
(other than at the disk center), with its location corresponding
to the radius at which the maximum occurs. Likewise, a gap is
a local minimum in the profile, with the location determined by
the radius where the minimum occurs. We follow the
nomenclature of J. Huang et al. (2018a), such that a ring is
labeled by the letter “B” followed by the radial location
rounded to the nearest au, and a gap is labeled by the letter “D”
followed by the radial location rounded to the nearest au. A
cavity is identified if the intensity at r = 0 is less than the peak
intensity. However, given that inner disks have now been
detected in a number of cavities imaged at high resolution (e.g.,
L. Francis & N. van der Marel 2020), the distinction between a
cavity and a gap in some cases may be a matter of resolution. A
shoulder is identified when a radial profile features three
consecutive inflection points without a local maximum or
minimum occurring between the first and last inflection point.
In other words, the radial profile changes from being concave
down to concave up and then concave down again without a
ring or gap being present. The radial location is defined to be
that of the middle inflection point. This definition is
diagrammed in Appendix C. The features that we label
shoulders have sometimes been classified as gaps and rings
in the literature (e.g., J. Huang et al. 2018a) because unresolved
gaps and rings can create shoulder-like features in radial
profiles (e.g., K. Zhang et al. 2016). However, for this work,
we elect to be more conservative in our definitions. The
measured locations, widths, and depths of the substructures are
provided in Table 4.
In addition to the substructures visible in the radial profiles
of the CLEAN images (Figure 4), we infer the presence of a
gap in the SO 662 disk and shoulders in the SO 984, SO 1036,
SO 1152, SO 1153, and SO 1274 disks. To check that the
inferred substructures are not merely an artifact from our choice
of parameterization, we also modeled the visibilities with the
nonparametric modeling code frank (J. Jennings et al. 2020;
Appendix D).
Histograms of the radial locations of the gaps and rings are
shown in Figure 8. The locations have a wide spread, from <10
to >100 au, although SO 1274 accounts for all the gaps and
rings identified outside 62 au. In the DSHARP survey of disks
in nearby star-forming regions, the distribution of detected disk
substructures peaks at radii of ∼30−40 au (J. Huang et al.
2018a), whereas the σ Orionis distributions peak at r < 20 au.
One possible reason for this difference is that the disks in our
sample are on average smaller than those targeted by
DSHARP. For the combined sample of disks from DSHARP
and the Taurus survey (F. Long et al. 2018, 2019), S. Zhang
et al. (2023) found that the peak of the distribution of
substructures occurred at smaller radii for disks with effective
radii less than 50 au compared to those greater than 50 au.
However, these various studies have been performed at
different resolutions and used different analysis techniques,
Figure 5. A zoomed-in view of SO 897, showing the emission asymmetry
across the major axis. Contours are drawn at 0.6, 0.7, and 0.8× the peak
intensity.
13
A demonstration of the application of SVI to parametric visibility modeling
of the AS 209 disk and comparison to MCMC results can be found at https://
github.com/MPoL-dev/examples/blob/main/AS209-pyro-inference/pyro.
ipynb.
7

Figure 6. A comparison of the parametric intensity models to the observations. First column: observations (on the same color scale as Figure 3). Second column:
CLEAN images generated from the model visibilities, on the same color scale as the ﬁrst column. Third column: residual images, with the color scale ranging from
−5σ (blue) to 5σ (red), where σ is the rms listed in Table 3. Dashed contours correspond to the [−4, −3]σ level and solid contours to the [3, 4]σ level. Fourth column:
a comparison of the deprojected, binned observed visibilities to the models. Fifth column: a comparison of the deprojected, azimuthally averaged proﬁles made from
the observed and model CLEAN images.
8

so more disk observations as well as a more homogeneous
analysis will be required to understand what factors affect the
radii at which substructures are most prevalent.
Figure 8 also plots the period ratios of all combinations of gap
pairs and ring pairs in the disks with multiple gaps and rings (SO
844, SO 1152, and SO 1274) under the assumption that the disk
mass is negligible compared to the stellar mass. (D. Tamayo et al.
2015 estimated that the actual resonance locations in a disk are
offset from simple integer period ratios by
M
M
disk
~
*
.) Nonadjacent
pairs are included because it may be the case that a non-planet-
related substructure occurs between two planet-related substruc-
tures. The SO 844 and SO 1152 substructure pairs are not generally
near low-order mean-motion resonances. However, SO 1274
presents a more interesting case. Its inner three gaps (D57, D42,
and D27) are close to a 3:2:1 period ratio (more precisely,
3.08:1.95:1). Meanwhile, D76:D57, B66:B50, B88:B66, and D97:
D76 are close to 3:2 period ratios (1.53, 1.51, 1.56, and 1.46,
respectively). In other words, the five gaps of SO 1274 appear to be
arranged in a nearly resonant chain (3:2, 3:2, 3:2, and 2:1 from
outermost to innermost gap pair), as are B88:B66:B50 (3:2, 3:2).
4.3.2. Optical Depths
We then estimated the optical depths using the expression
⎜ ⎟
⎛
⎝
⎞
⎠
( )
( )
( ( ))
( )
r
I r
B T r
ln 1 , 4
d
t = - -
n
n
n
Figure 7. Radial intensity profiles derived from parametric modeling and labeled with the locations of the substructures. The black curves correspond to profiles
generated with the median values of the marginalized posterior distributions, while the blue curves show the results of 2000 draws from the posterior.
9

where Td is the dust temperature. This expression neglects
scattering, which may lead to underestimates of the optical
depth. However, use of this expression allows for direct
comparison with estimates made for other disks. Following
C. P. Dullemond et al. (2018), we approximate Td with
⎜ ⎟
⎛
⎝
⎞
⎠
( ) ( )
T r
L
r
8
, 5
d 2
SB
0.25
j
p s
= *
where j = 0.02 is the chosen flaring angle. This expression
assumes that the disk is heated through irradiation from its
stellar host, but a nearby massive star can also contribute to
heating the outer regions of a disk. Through radiative transfer
modeling, T. J. Haworth (2021) found that the midplane
temperature of a disk at a separation of 1 pc from a θ1
Ori C–
like system begins to diverge significantly from that of an
isolated disk at radii beyond ∼20 au. θ1
Ori C1 has a
luminosity of 204,000 Le (S. Simón-Díaz et al. 2006),
compared to 41,700 Le for σ Ori Aa (S. Simón-Díaz et al.
2015), so the impact of external heating on the σ Orionis disks
should be weaker. In addition, the larger disks in our sample
(R90 > 45 au; see Section 4.5) have relatively large projected
separations from σ Ori ( 1.7 pc).
The optical depths are plotted in Figure 9. In general, the
profiles dip within the inner disk due to beam dilution
(although SO 897, of course, has a cavity). Some of the
shoulders identified in the radial intensity profiles manifest as
gap–ring pairs in the optical depth profiles. In most disks, the
peak optical depths of the rings and shoulders range from
∼0.25 to 0.8, comparable to the values found for disks in
nearby regions (e.g., C. P. Dullemond et al. 2018; S. Facchini
et al. 2020; J. Huang et al. 2020). S. M. Stammler et al. (2019)
suggested that the apparent tendency for ring optical depths to
fall in this range is a consequence of ongoing planet formation,
while Z. Zhu et al. (2019) found that optically thick rings with
high-albedo dust grains (which lead to significant scattering)
could yield apparent optical depths of ∼0.6.
However, the SO 897 and SO 1153 disks appear to have
anomalously high estimated optical depths. Equation (5) likely
underestimates the temperature at SO 897ʼs ring because
depletion of dust inside the cavity would result in strong
irradiation of the cavity wall. On the other hand, the
anomalously high optical depths of the SO 1153 disk are likely
due at least in part to the estimated L* being too low. K. Maucó
et al. (2023) estimated that L* = 0.33 Le for SO 1153, which is
lower than that of the other sources even though its star is more
massive. K. Maucó et al. (2023) commented that the spectral
type and therefore the L* value for SO 1153 is challenging to
derive due to veiling. If the optical depths are recomputed using
the median L* of the other sources (0.68 Le), they fall more in
line with the rest of the sample. In addition, if SO 1153 is
embedded, which its classification by J. Hernández et al. (2007)
as a Class I young stellar object (YSO) would imply, then
Equation (5) may not be appropriate, since Class I disks tend to
be warmer than Class II disks (e.g., M. L. R. Hoff et al. 2020).
4.4. Disk Fluxes and Sizes
For all disks except SO 897, we used the model radial
intensity profiles to calculate the flux with the following
equation:
( ) ( )
i
d
rI r dr
cos
2 , 6
r
2 0
ò p
¢
where d is the distance to the source and r¢ is chosen to be some
large value (usually a few hundred au) such that the flux
integral has reached its asymptotic value. Because SO 897 was
not modeled as a flat disk, we instead measure its flux from the
model images of the projected disk (although the value is
similar to that derived from using Equation (6)). For SO 662,
SO 844, and SO 984, our derived flux values differ from those
in M. Ansdell et al. (2017) and K. Maucó et al. (2023) by
∼40%−70%. This discrepancy arises because they fit point-
source models to their lower-resolution data.
We then measured the effective radii using the metrics from
A. Tripathi et al. (2017). R68 is defined as the radius that
encloses 68% of the total flux, while R90 is the radius that
encloses 90%. Table 5 lists the median fluxes and effective
radii as well as their 68% confidence intervals. For all disks,
including SO 897, the flux percentages are computed with
respect to the flux calculated with Equation (6).
In models of disks with dynamic pressure bumps, J. Stadler
et al. (2022) found that the outermost bump was generally
Table 4
Substructure Properties
Source Feature Radial Location Width Depth
(au) (au)
SO 662 D9 8.9 ± 0.3 3.1 0.5
0.4
-
+
0.85 ± 0.07
B12 12.46 0.49
0.45
-
+
3.5 0.6
0.4
-
+
L
SO 844 D10 9.7 ± 0.4 12.6 ± 0.3 0.0011 0.0005
0.0008
-
+
B20 20.48 0.12
0.11
-
+
6.6 ± 0.3 L
D29 28.7 ± 0.3 8.7 ± 0.2 0.049 0.009
0.010
-
+
B37 36.8 ± 0.2 6.8 ± 0.3 L
SO 897 Cavity L 7.4 ± 0.2 0.008 ± 0.003
B12 11.9 ± 0.1 9.0 ± 0.3 L
SO 984 S25 25.3 ± 0.2 L L
SO 1036 S18 18.4 0.5
0.4
-
+
L L
SO 1152 S19 19.2 ± 0.7 L L
D34 34.0 ± 0.6 11.2 ± 0.6 0.52 ± 0.03
B41 41.3 ± 0.5 6.3 ± 0.3 L
D49 49.0 ± 0.5 9.3 ± 0.3 0.53 ± 0.02
B62 61.8 ± 0.4 14.2 ± 0.4 L
SO 1153 D8 8.01 0.14
0.15
-
+
6.10 0.14
0.13
-
+
0.38 ± 0.02
B17 17.48 0.18
0.16
-
+
11.3 ± 0.2 L
S34 34.3 ± 0.4 L L
SO 1274 S12 11.6 ± 0.3 L L
D27 26.9 ± 0.2 9.1 ± 0.2 0.21 ± 0.02
B33 32.8 ± 0.2 5.0 ± 0.2 L
D42 42.1 ± 0.3 10.6 0.4
0.3
-
+
0.0030 0.0011
0.0018
-
+
B50 49.7 ± 0.2 4.9 ± 0.3 L
D57 57.0 ± 0.4 9.9 ± 0.4 0.007 0.003
0.004
-
+
B66 65.6 ± 0.2 6.0 ± 0.4 L
D76 75.6 0.4
0.5
-
+
15.2 0.5
0.4
-
+
0.0046 0.0014
0.0018
-
+
B88 88.2 ± 0.3 7.4 ± 0.6 L
D97 97.3 0.7
0.6
-
+
26.2 ± 0.6 0.070 0.007
0.008
-
+
B119 119.3 ± 0.3 6.0 ± 0.5 L
S128 128.47 0.79
0.84
-
+
L L
10

located near R68. For SO 1152, SO 1274, SO 844, SO 897, and
SO 662, the radial location of the outermost identified ring is
within several au of the disk’s R68 value, providing some
support to the idea that disk sizes are often controlled by the
locations of their pressure bumps. For SO 1153, the R68 value
of 52 au is well outside the ring detected at 18 au or the
shoulder detected at 34 au, but it is possible that the extended
emission tail at larger radii may harbor unresolved rings.
Alternatively, given that J. Hernández et al. (2007) categorized
SO 1153 as a Class I YSO, the relationship between disk size
and pressure bump location may differ for Class I and II disks.
4.5. Size–Luminosity Relationship
Several observational studies have found a correlation
between disk sizes and luminosities, although the scalings
vary between different star-forming regions (e.g., A. Tripathi
et al. 2017; S. M. Andrews et al. 2018b; N. Hendler et al.
2020). Following these previous works, the disk luminosity
Lmm is defined as the disk flux rescaled to a distance of 140 pc.
We first sampled flux values from a normal distribution with a
mean and standard deviation corresponding to the values in
Table 5. To account for the systematic flux calibration
uncertainty, we then multiplied the flux samples by scaling
factors randomly drawn from a normal distribution with a mean
of 1 and standard deviation of 0.1. To account for the
uncertainty in distance, we generated posterior samples for the
distances using the Interactive Distance Estimation tool,14
which implements the method described in C. A. L. Bailer-Jo-
nes et al. (2021) to estimate distances from the Gaia catalog
(Gaia Collaboration et al. 2021). The flux values were then
rescaled using randomly drawn samples from the distance
posteriors. Similarly, the distribution of R68 values was
generated by sampling from a normal distribution with a mean
and standard deviation corresponding to the values in Table 5
and rescaling with randomly drawn distance values. We
calculated the medians of these Lmm and R68 distributions as
point estimates and the 16th and 84th percentiles to obtain 1σ
uncertainties.
As in the aforementioned works, we model the relationship
between R68 and Lmm with the following equation:
( ) ( ) ( )
R L
log au log Jy . 7
68 mm
a b
= +
The scatter of R
log 68 about the regression line is assumed to be
Gaussian with standard deviation σ. The data are fit using a
Python port15
of linmix, a Bayesian linear fitting code by
B. C. Kelly (2007) that accounts for uncertainties in both the
independent and dependent variables. The median and 1σ
uncertainties of the posterior distributions are reported in
Figure 8. Top left: radial distribution of all gaps identified in the σ Orionis sample. Top right: radial distribution of all rings. Bottom: a stacked histogram of period
ratios for pairs of gaps and pairs of rings in disks with more than one gap/ring.
14
https://github.com/ElisaHaas25/Interactive-Distance-Estimation/
tree/main
15
https://github.com/jmeyers314/linmix
11

Table 6. The linear fit is plotted with the data points in
Figure 10.
As in nearby star-forming regions (e.g., A. Tripathi et al.
2017; S. M. Andrews et al. 2018b; N. Hendler et al. 2020), we
find a strong correlation between L
log mm and R
log 68, with a
linear correlation coefficient of 0.91. Our slope, β = 0.8 ± 0.2,
is slightly steeper than the values derived for various nearby
regions (0.22−0.6) but consistent within the uncertainties.
Likewise, our intercept, α = 2.6 ± 0.3, is somewhat higher than
the values for nearby regions (1.7−2.2) but also consistent
within the uncertainties. Observing additional disks in σ
Orionis and measuring disk sizes from higher-resolution
observations in other regions will be needed to assess whether
there is a genuine difference in the size–luminosity
Figure 9. Plots of the disk optical depth as a function of radius. Substructures identified from the model radial intensity profiles are labeled. The gray bar in the
SO 1153 plot shows where an optical depth cannot be derived because the estimated dust temperature is less than the brightness temperature, indicating that the dust
temperatures for this source are underestimated.
12

relationship. On the other hand, J. A. Eisner et al. (2018) and
J. Otter et al. (2021) found weak or no evidence for a
correlation in the ONC and a combined ONC/OMC1 sample,
respectively. The former measured a slope of 0.09 ± 0.07, and
the latter measured 0.17 ± 0.05. These works observed disks
with projected separations within a few tenths of a parsec from
θ1
Ori C, so the different size–luminosity relationships in the
ONC and OMC1 compared to our σ Orionis sample may be a
reflection of the more extreme effects of external photoeva-
poration. However, the presence of intracluster material in the
ONC and OMC1 and the accompanying uv cut employed by
J. A. Eisner et al. (2018) and J. Otter et al. (2021) may also
have introduced greater uncertainty.
S. M. Andrews et al. (2018b) tentatively identified R68-L*
and R68-M* correlations. Because we selected targets in a
relatively narrow M* range (and therefore in a relatively
narrow L* range), we do not have the dynamic range to test for
correlations.
4.6. Dust Trapping Analysis
Grain size estimates derived from multifrequency observa-
tions and measurements of gas pressure profiles suggest that
millimeter continuum rings are often dust traps (e.g., F. Long
et al. 2020; G. P. Rosotti et al. 2020; E. Macías et al. 2021;
A. Sierra et al. 2021). In the absence of these kinds of data,
C. P. Dullemond et al. (2018) argued that dust rings with
smaller widths wd than the pressure scale height hp were likely
to be dust traps because a gas pressure bump that is narrower
than hp would not be stable (however, a dust ring being wider
than hp does not necessarily imply that it is not a dust trap).
To assess whether the rings we detect in σ Orionis might be
dust traps, we calculate the ratio of wd to hp for the high-
contrast rings (those with depth values <0.5 in Table 4). The
pressure scale height is given by
( )
( )
( )
h r
k T r r
m GM
, 8
p
d
p
B
3
m
=
*
where Td(r) is estimated using Equation (5) and μ = 2.37 is the
mean molecular weight. We set wd equal to the value of σi
inferred for the corresponding ring in the parametric model (see
Table 10). (Note that this is different from the ring widths
quoted in Table 4, which would be equivalent to the FWHM of
a single Gaussian ring.) The estimates for Tdust, wd, hp, and
wd/hp are listed in Table 7.
Provided that the rings are resolved, the dominant source of
uncertainty in the calculation of wd/hp is likely the dust
temperature estimate for hp. While it is not straightforward to
quantify the temperature uncertainty, one can obtain some idea
by comparing the dust temperature estimates from C. P. Dulle-
mond et al. (2018) using Equation (5) to those derived either
from thermochemical modeling (A. Sierra et al. 2021;
K. Zhang et al. 2021) or from multifrequency continuum
modeling (A. S. Carvalho et al. 2024) for the Elias 24, HD
Table 5
Disk Fluxes and Sizes
Source Fluxa
R68
b
R90
b
(mJy) (au) (au)
SO 662 2.09 ± 0.03 14.2 ± 0.2 17.7 ± 0.3
SO 844 4.95 ± 0.05 36.5 ± 0.2 40.4 ± 0.3
SO 897 1.99 ± 0.02 14.8 ± 0.2 17.9 ± 0.2
SO 984 8.46 0.08
0.09
-
+
34.3 ± 0.2 43.7 ± 0.3
SO 1036 6.34 ± 0.06 28.9 ± 0.2 38.1 0.4
0.3
-
+
SO 1152 9.32 ± 0.06 65.3 ± 0.3 81.9 ± 0.6
SO 1153 16.1 ± 0.1 52.0 ± 0.2 67.7 ± 0.3
SO 1274 15.5 ± 0.2 113.2 0.8
0.7
-
+
126.7 0.6
0.7
-
+
Notes.
a
Error bars do not include ∼10% systematic flux calibration uncertainty.
b
Error bars do not include the uncertainty in distance.
Table 6
Size–Luminosity Regression Results
Parameter Value
α 2.6 ± 0.3
β 0.8 ± 0.2
σ 0.17 0.05
0.10
-
+
r̂ (correlation coefficient) 0.91 0.15
0.06
-
+
Figure 10. A plot of R68 vs. Lmm for the observed σ Orionis disks. Following
A. Tripathi et al. (2017), Lmm is defined as the flux scaled to a distance of
140 pc. The black line corresponds to the median parameters from the linear fit,
while the gray shaded region shows the 68% confidence interval. (Note that the
error bars on R68 are too small to be visible.)
Table 7
Estimated Properties of High-contrast Rings
Source Ring ID Tdust
a
wd hp wd/hp
(K) (au) (au)
SO 844 B20 24.4 2.8 1.4 2.0
B37 18.2 3.0 2.8 1.1
SO 897 B12 34.6 3.9 0.6 6.5
SO 1153 B17 27.0 5.2 0.8 6.5
SO 1274 B33 19.7 2.4 2.1 1.2
B50 16.0 2.1 3.5 0.6
B66 14.0 2.6 4.9 0.5
B88 12.0 3.1 7.1 0.4
B119 10.3 2.3 10.4 0.2
Note.
a
Tdust is calculated with L = 0.68 Le for SO 1153 and with the L* values in
Table 1 for the other disks.
13

163296, and AS 209 disks. In these cases, the empirically
derived temperatures at the rings were within ∼50% of the
estimates from the analytic model. hp also depends on M*, the
values of which were derived by K. Maucó et al. (2023) using
the I. Baraffe et al. (2015) evolutionary models for SO 844 and
the G. A. Feiden (2016) magnetic models for SO 897, SO 1153,
and SO 1274. For the relevant stellar mass ranges,
T. A. M. Braun et al. (2021) found that the mass estimates
from evolutionary models usually agreed with dynamical mass
estimates within 5%−10%, so the uncertainty in Td dominates
over the uncertainty in M*. Thus, we estimate an uncertainty on
hp of ∼25%.
In the SO 1274 disk, wd/hp < 1 for B50, B66, B88, and
B119. Taking into account our rough estimate for the
uncertainty of hp, we consider these rings likely to be dust
traps. Strong dust trapping may be responsible for maintaining
the large size of the SO 1274 disk (R90 = 127 au). For the other
high-contrast rings identified, wd/hp > 1. However, they are all
located at smaller radii than the SO 1274 rings, and thus the
scale heights at those locations would likely be smaller. The
larger wd/hp > 1 values may be a consequence of the identified
rings being underresolved or possibly even composed of
multiple rings, as seen in disks such as HD 169142 (S. Pérez
et al. 2019). The values of wd/hp are slightly larger than 1 for
B37 in SO 844 and B33 in SO 1274, but given the large
uncertainties in hp, they warrant further examination with
improved dust temperature estimates and higher-resolution
observations.
4.7. Planet Mass Estimates
If the disk gaps are due to planets, then one can use the gap
widths to estimate the masses of the embedded planets. We use
the fitting relations introduced in S. Zhang et al. (2018), which
are implemented in code presented in S. Zhang et al. (2022). In
brief, we assume that each observed gap corresponds to one
protoplanet. The semimajor axis rp of the protoplanet is set to
the radial location of its gap (from Table 4). The fractional gap
width Δ is measured from the model radial intensity profile.
The average dust surface density Σdust, avg is estimated from the
model optical depth profile between 1.1rp and ( )
r R
min 2 ,
p 90
using Σdust(r) = τν(r)/κν,abs. (For SO 1153, we assume that
L* = 0.68 Le.) Following S. Zhang et al. (2018), the dust
absorption opacity is set to 0.43 cm2
g−1
, corresponding to the
standard DSHARP dust composition (T. Birnstiel et al. 2018)
with a maximum grain size of a 0.1 mm
max = and a size
distribution of n(a) ∝ a−3.5
(S. Zhang et al. 2018 refer to this as
the “DSD1” distribution). Given a value of Σdust, avg, the
maximum Stokes number Stmax is estimated from the grid of
hydrodynamical simulations in S. Zhang et al. (2018) for the
DSD1 distribution and the aspect ratio hp/r (0.05, 0.07, and
1.0) closest to the value estimated at the gap. However, if Stmax
corresponds to a gas surface density that exceeds the limit for
gravitational stability (Toomre Q = 1, with the thermal profile
from Equation (5)), then the next largest value of Stmax from
the model grid is used. Then, a planet mass Mp can be
calculated from the scaling relations in S. Zhang et al. (2018)
for a given Δ, Stmax,
h
r
p
, and viscosity parameter α. We assume
that α = 10−3
. All other things held equal, changing α by an
order of magnitude changes the Mp estimate by a factor of ∼2.
The planet mass estimates are listed in Table 8. The
estimates are compared for those of the DSHARP survey,
which targeted nearby star-forming regions, in Figure 11. The
overall ranges of inferred planet masses and semimajor axes are
similar, although the σ Orionis values tend to be concentrated
more at somewhat smaller masses and semimajor axes.
The planet mass uncertainties listed in Table 8 and plotted in
Figure 11 are the formal errors calculated from the S. Zhang
et al. (2018) linear fitting method, but the process of inferring
planet masses from disk gap properties has other sources of
uncertainty. Planet mass estimates are sensitive to the disk
temperature, gas surface density, and grain size distribution, the
values of which are crudely approximated. Furthermore, while
our calculations assumed that every gap has a planet,
simulations have shown that in low-viscosity disks, a single
planet may be responsible for multiple gaps (e.g., J. Bae et al.
2017; R. Dong et al. 2017; S. Zhang et al. 2018). In addition,
other studies have illustrated that including effects such as
radiative cooling, self-gravity, magnetized disk winds, or
migration in simulations of planet–disk interactions can affect
the morphology of substructures created by a given planet (e.g.,
F. Meru et al. 2019; R. Miranda & R. R. Rafikov 2019;
S. Zhang & Z. Zhu 2020; Y. Aoyama & X.-N. Bai 2023).
Table 8
Estimated Planet Masses
Source Gap ID Δ Σdust,avg hp/r Σgas,Q = 1 Stmax,used Mp Uncertainty
(g cm−2
) (g cm−2
) (×5.23e−4) (MJup) ( M
log p)
SO 662 D9 0.29 1.3 0.05 1044 0.3 0.46 0.16
0.13
-
+
SO 844 D10 0.74 0.53 0.06 736 0.3 7.9 0.16
0.13
-
+
D29 0.26 1.2 0.07 110 0.3 0.23 0.16
0.13
-
+
SO 1152 D34 0.29 1.3 0.07 94 1 0.36 0.14
0.16
-
+
D49 0.17 1.2 0.07 50 1 0.04 0.14
0.16
-
+
SO 1153 D8 0.5 1.4 0.04 1490 0.3 3.9 0.16
0.13
-
+
SO 1274 D27 0.30 0.43 0.06 151 0.3 0.52 0.16
0.13
-
+
D42 0.22 0.23 0.07 69 1 0.13 0.14
0.16
-
+
D57 0.16 0.22 0.07 40.5 1 0.035 0.14
0.16
-
+
D76 0.18 0.40 0.08 24.7 3 0.07 0.17
0.14
-
+
D97 0.22 0.56 0.08 16 3 0.17 0.17
0.14
-
+
14

5. Discussion
5.1. Implications for Disk Evolution and Planet Formation in σ
Orionis
Our observations suggest that disk substructures are able to
form and survive across a range of external UV environments,
from a few G0 to 103
G0. Substructures are detected even in
very compact disks (R68 < 15 au) in σ Orionis. However,
multiringed systems are only detected at dproj > 2.1 pc. This
may be a matter of resolution, since larger disks tend to be
located at greater separations. If the substructures are due to
mechanisms such as zonal flows or planet–disk interactions,
they should be more readily detectable at larger radii because
the scale height sets their characteristic width, and the scale
height increases with radius (e.g., G. Bryden et al. 1999;
A. Johansen et al. 2009). Nevertheless, SO 844 presents an
interesting contrast with SO 984 and SO 1036. Despite having
similar millimeter continuum disk sizes, SO 844 (dproj = 2.1 pc)
has two deep, wide gaps, while SO 1036 and SO 984
(dproj = 1.2 pc for both) do not have clearly detected disk gaps
(although they each have shoulders).
If the detected substructures are due to planet–disk
interactions, their widths and locations suggest that ice and
gas giants can form in σ Orionis disks on solar system scales.
However, whereas more than half of the DSHARP systems
have gaps detected outside 50 au (J. Huang et al. 2018a),
SO 1274 is the only one of the eight disks in our σ Orionis
survey that does (although SO 1152 has a gap at 49 au). Of
course, neither sample was chosen in an unbiased manner, but
we can use our derived size–luminosity relation (Section 4.5) to
estimate how many other disks in σ Orionis might have
millimeter continuum gaps outside 50 au. We assume that R68
must be greater than 50 au in order for a disk to have a
continuum gap outside 50 au (which holds for our sample and
is expected based on the J. Stadler et al. 2022 models of
dynamical pressure bumps). Our size–luminosity relation
predicts that a disk of this size has a 1.3 mm flux of 9 mJy at
a distance of 402 pc. Of the disks not in our sample, only
SO 540 meets this threshold (M. Ansdell et al. 2017). The
extent of the millimeter continuum emission, though, does not
necessarily denote an upper bound on where protoplanets
might be present. Gaps in scattered light and molecular line
emission, as well as velocity “kinks,” have been detected
outside the millimeter continuum in some disks and in some
cases have been hypothesized to be due to planet–disk
interactions (e.g., H. Avenhaus et al. 2018; C. Pinte et al.
2018; C. J. Law et al. 2021).
The apparent preponderance of substructure pairs near 3:2
period ratios in the SO 1274 disk is intriguing given that the
period ratios of the Kepler multiplanet systems show a peak
near 3:2 as well (D. C. Fabrycky et al. 2014; J. H. Steffen &
J. A. Hwang 2015). The Kepler planets have relatively small
periods (generally less than a couple hundred days), so their
architectures cannot be compared directly with disks imaged by
ALMA. Nevertheless, SO 1274 is potentially a useful system to
model to understand how multiplanet systems in resonant
configurations arise.
A. D. Sellek et al. (2020) modeled the effects of external
photoevaporation on disks with smooth surface density profiles
and found that disks exposed to stronger external FUV
radiation fields experienced more rapid dust depletion via
radial drift, thereby shortening their lifetimes. However,
M. Gárate et al. (2024) found that including substructures in
the models could prolong dust disk lifetimes to a few Myr for
Figure 11. A comparison of the planet masses inferred in the σ Orionis sample in this work (turquoise dots) and those inferred from well-resolved disk gaps in the
DSHARP survey of nearby star-forming regions (orange dots; S. Zhang et al. 2018). The DSHARP values correspond to α = 10−3
and the DSD1 dust distribution.
The error bars show the uncertainties calculated by the linear fitting method. The gray dots correspond to confirmed exoplanets (retrieved from https://
exoplanetarchive.ipac.caltech.edu/ on 2024 July 10). The solar system planets are plotted in dark blue.
15

external FUV radiation levels up to 103
G0. The detection of
substructures in our disk targets, which have an estimated age
of 3−5 Myr (J. M. Oliveira et al. 2002, 2004; M. R. Zapatero
Osorio et al. 2002; J. A. Caballero et al. 2019) and external
FUV fields of 102
−103
G0 (K. Maucó et al. 2023), supports the
need to consider the role of substructures when modeling the
impact of external photoevaporation on disk evolution.
The observed correlation between disk size and luminosity in
nearby star-forming regions has been hypothesized to be due
to dust trapping by substructures (A. Tripathi et al. 2017;
L. Delussu et al. 2024). The prevalence of substructures in our
σ Orionis sample and its strong size–luminosity correlation
broadly fits within this picture. The absence of a clear size–
luminosity correlation in the ONC (J. A. Eisner et al. 2018;
J. Otter et al. 2021) may be a consequence of the higher
external UV radiation fields ( 104
G0; R. K. Mann &
J. P. Williams 2010) either destroying substructures or
inhibiting their formation in the first place.
σ Orionis seems to be similar to other star-forming regions
observed so far in that (approximately) axisymmetric gaps and
rings are the dominant forms of substructures detected.
However, it is worth commenting on the kinds of substructures
that have been observed in other star-forming regions
(S. M. Andrews et al. 2018a; S. M. Andrews 2020) but do
not appear in our sample. No large cavities (r > 20 au), spiral
arms, or vortex-like crescents are visible in our data (although
SO 897 features a small cavity). Besides SO 897, eight
additional members of σ Ori (SO 299, SO 411, SO 540,
SO 587, SO 818, SO 908, SO 1267, and SO 1268) have been
classified as transition disks or transition disk candidates based
on their infrared spectral energy distributions (J. Hernández
et al. 2007; K. Maucó et al. 2016). While most of these disks
are fainter than the ones we observed, SO 411 and SO 540 have
relatively large 1.3 mm fluxes (5.2 and 10.7 mJy, respectively,
as measured by M. Ansdell et al. 2017). S. M. Andrews et al.
(2018a) found that transition disks and nontransition disks
followed similar size–luminosity relationships in nearby star-
forming regions. If the same holds true in σ Ori, then it would
be worthwhile to image SO 411 and SO 540 to determine
whether they have large cavities. Meanwhile, spiral arms that
have been detected in millimeter continuum observations of
disks in nearby star-forming regions have generally been much
lower contrast than rings (e.g., L. M. Pérez et al. 2016; R. Dong
et al. 2018b; J. Huang et al. 2018b; N. T. Kurtovic et al. 2018).
Thus, we would expect similar spiral arms to be challenging to
detect in the σ Orionis disks. Vortex-like crescents have thus
far disproportionately been detected in disks hosted by stars
greater than solar mass (J. Bae et al. 2023), whereas all of our
targets are roughly solar mass or below.
5.2. Nonplanetary Explanations for σ Orionis Substructures
While the substructures detected are plausibly explained by
planet–disk interactions, a number of models have been
presented to explain substructures without invoking planet–
disk interactions. For recent reviews of substructure formation
models, see J. Bae et al. (2023) and G. Lesur et al. (2023). We
comment briefly on some commonly studied alternatives.
One of the most popular alternative explanations has been
that the substructures result from dust evolution at or near
molecular snowlines (e.g., K. Zhang et al. 2015; S. Okuzumi
et al. 2016; P. Pinilla et al. 2017). Several studies have argued
that snowlines are unlikely to account for the majority of
substructures, given that they do not systematically appear at
the estimated radial locations of snowlines (J. Huang et al.
2018a; F. Long et al. 2018; N. van der Marel et al. 2018). That
said, the locations of molecular snowlines are often contested
even in systems that have been much more extensively
observed and modeled (e.g., C. Qi et al. 2013; L. A. Cieza
et al. 2016; M. L. R. Hoff et al. 2017; K. Zhang et al. 2017;
M. L. R. van’t Hoff et al. 2018). The models of substructure
formation via snowlines alone, though, have not yielded a
simple way to reproduce the heterogeneity in the number of
gaps and rings observed in different disks. K. Chen et al.
(2024) suggested that planet–disk interactions could shift
snowline locations by altering the disk thermal structure, and
thus a combination of planet–disk interactions and snowlines
could explain the lack of an obvious pattern in the substructure
numbers or locations. However, there is debate over whether
molecular snowlines can create the high-contrast dust rings that
have been imaged by ALMA. For example, S. M. Stammler
et al. (2017) modeled the effect of CO freeze-out on dust
evolution and found that it results in only a minor decrease in
the dust surface density just interior to the CO snowline.
SO 897 features a disk with a small inner cavity (outer ring
peaks at r ∼ 12 au). Models from G. Picogna et al. (2019) show
that cavities of this size can be produced through photo-
evaporation due to X-rays from the stellar host, but
photoevaporation alone would most likely lead to an accretion
rate significantly lower than SO 897ʼs measured value of
4.7 × 10−9
Me yr−1
(K. Maucó et al. 2023). M. Gárate et al.
(2021) found that models with both dead zones and X-ray
photoevaporation could allow high accretion rates to be
sustained while opening a disk cavity. A telltale feature of
their models is a small disk within the cavity, but our angular
resolution is insufficient to determine whether SO 897ʼs cavity
has an inner disk.
Magnetohydrodynamic winds and zonal flows can generate
substructures on scales similar to those produced by planet–
disk interactions (e.g., X.-N. Bai & J. M. Stone 2014; A. Riols
& G. Lesur 2018; S. S. Suriano et al. 2018; X. Hu et al. 2023).
As discussed in J. Bae et al. (2023), observationally
distinguishing these mechanisms is not straightforward in the
absence of a direct detection of a planet. A particular difficulty
is that only upper limits have so far been derived for the
magnetic field strengths of protoplanetary disks (e.g.,
W. H. T. Vlemmings et al. 2019; R. E. Harrison et al. 2021).
In most cases, the effects of the external FUV radiation field
have not been considered in substructure formation models.
Testing the effects of a range of external FUV levels on
substructure development may offer an additional way to
discriminate between models.
5.3. Caveats and Future Prospects
The external FUV radiation fields impinging on our targets
could be significantly lower than the estimated values listed in
Table 1. First, the true separations from σ Ori could be
significantly larger than the projected separations. Second,
while extinction is currently low in σ Orionis, higher amounts
of intracluster material in the past may have protected the disks
from external FUV radiation (e.g., L. Qiao et al. 2022). Finally,
these disks could have been exposed to lower external FUV
radiation in the past if they migrated from the outskirts of their
cluster (e.g., A. J. Winter et al. 2019).
16

M. Ansdell et al. (2017) and K. Maucó et al. (2023) showed
that disk millimeter continuum fluxes tend to decrease as
projected separations from σ Ori decrease. Given that disk sizes
and luminosities appear to be well correlated in this region
(Section 4.5), this implies that millimeter continuum disk sizes
(and thus the radial range over which millimeter continuum
substructures can occur) also tend to decrease as projected
separations decrease. We noted earlier in the discussion that
disks with millimeter continuum gaps beyond a radius of 50 au
are likely rare in σ Orionis. While external photoevaporation is
plausibly responsible for the apparent absence of large disks
(and by extension, wide-separation millimeter continuum
substructures) at smaller projected separations from σ Ori,
age effects may also contribute. Disks in the 5–11 Myr old
Upper Sco region are systematically smaller than those in the
1–3 Myr old Lupus or 2–3 Myr old Cha I regions (S. A. Bare-
nfeld et al. 2017; N. Hendler et al. 2020). N. Hendler et al.
(2020) suggested that this trend could be the consequence of
either older disks having experienced more radial drift or Upper
Sco having stronger external UV fields compared to the other
regions studied. With an estimated age of 3−5 Myr (J. M. Oli-
veira et al. 2002, 2004; M. R. Zapatero Osorio et al. 2002;
J. A. Caballero et al. 2019), σ Orionis appears to be somewhat
older than the nearby regions with a significant population of
large disks. Observations of the western population of
NGC 2024, which has an estimated age of ∼1 Myr and shows
evidence of mass loss due to external photoevaporation
(S. E. van Terwisga et al. 2020), may help to disentangle the
effects of age and external radiation fields. In addition,
K. L. Luhman (2018) suggested that σ Orionis has smaller
disk fractions compared to some other regions that are several
Myr old (such as Cha I) due to higher stellar density in σ
Orionis. Higher stellar densities increase the probability of
close stellar encounters, which can lead to the truncation or
destruction of disks (e.g., C. J. Clarke & J. E. Pringle 1993;
A. Breslau et al. 2014; A. J. Winter et al. 2018). However,
models from A. J. Winter et al. (2018) suggest that for σ
Orionis–like conditions, external photoevaporation should play
a larger role than dynamical truncation in setting disk sizes.
Our observations only included stars between 0.4 and
0.9 Me. Models indicate that mass loss due to external
photoevaporation is more severe around lower-mass stars
(e.g., T. J. Haworth et al. 2018). Counterintuitively, K. Maucó
et al. (2023) found that disk masses in σ Orionis for stars above
0.4 Me showed a trend with projected separation from σ Ori,
whereas those below 0.4 Me did not. Given that surveys of
very low-mass stars in nearby regions are detecting disk
substructures (N. T. Kurtovic et al. 2021; Y. Shi et al. 2024), it
is plausible that the lower-mass stars in σ Orionis also have
disks with substructures. However, this needs to be confirmed
with observations.
Some of the observational tests used to probe the origins of
substructures in nearby regions, such as direct imaging searches
for protoplanet emission (e.g., M. Keppler et al. 2018;
R. Asensio-Torres et al. 2021; S. Jorquera et al. 2021;
I. Hammond et al. 2023), molecular line kinematics (e.g.,
C. Pinte et al. 2018, 2023; R. Teague et al. 2018), and resolved
spectral index measurements (e.g., T. Tsukagoshi et al. 2016;
C. Carrasco-González et al. 2019; J. Huang et al. 2020; F. Long
et al. 2020), will be difficult or impossible to apply to σ Orionis
with existing facilities due to the larger distance to the cluster
and its relatively small disks. However, such studies may
become feasible with Extremely Large Telescope–class facil-
ities, the Next Generation Very Large Array, or the proposed
ALMA × 10 upgrade.
6. Summary
We used ALMA to image the 1.3 mm continuum of eight
disks in the σ Orionis cluster, producing the highest-resolution
observations to date of disks in this region. Our key findings
are as follows.
1. Gaps, rings, or cavities are visible in the images of five of
the eight disks. Through visibility modeling, we infer that
a sixth disk also has a gap and the remaining two have
shoulder-like structures. The disks appear to be largely
axisymmetric, which is also the case for most disks
observed in nearby star-forming regions.
2. Three of the disks with gaps or cavities have R90 values
less than 50 au, illustrating the diversity of structures in
compact disks.
3. The large SO 1274 disk exhibits an especially rich set of
substructures, with at least five deep, wide gaps located
from 27 to 97 au. The substructures appear to be arranged
nearly in a resonant chain. The outer rings in this system
are narrow and likely dust traps.
4. Disk sizes and luminosities are well correlated within the
sample, a characteristic that has also been observed in
nearby star-forming regions. In most of the sample, the
outermost detected ring is located near the measured
value of R68, supporting the proposition that substructures
play a role in setting apparent dust disk sizes.
5. Given the disk fluxes measured in moderate-resolution
surveys of σ Orionis (M. Ansdell et al. 2017; K. Maucó
et al. 2023), the size–luminosity relationship measured
from our high-resolution observations implies that most
disks in σ Ori are small; thus, few disks in the cluster are
likely to have millimeter continuum gaps beyond a radius
of 50 au, in contrast to the DSHARP sample of disks in
nearby star-forming regions. The small disk sizes may be
a consequence of either external photoevaporation or the
intermediate age of the region.
Our observations suggest that substructures are common not
only in disks in the mildly irradiated nearby star-forming
regions but also in disks exposed to intermediate levels of
external UV radiation (∼102
−103
G0). If these substructures
trace planet–disk interactions, ice and gas giants may still be
forming on solar system scales in σ Orionis, but giant planet
formation at significantly larger semimajor axes (∼50−100 au)
may be rarer compared to nearby star-forming regions. These
observations motivate high-resolution imaging of disks in more
extreme UV environments to investigate the universality of
disk substructures.
Acknowledgments
We thank our contact scientist Ryan Loomis for his
assistance with the project. We also thank the anonymous
referee for comments improving the manuscript. This paper
makes use of the following ALMA data: ADS/JAO.
ALMA#2016.1.00447.S, ADS/JAO.ALMA#2022.1.00728.S.
ALMA is a partnership of ESO (representing its member states),
NSF (USA) and NINS (Japan), together with NRC (Canada),
MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in
17

cooperation with the Republic of Chile. The Joint ALMA
Observatory is operated by ESO, AUI/NRAO and NAOJ. This
material is based upon work supported by the National Science
Foundation under grant No. 2307916. T.B. acknowledges
funding from the European Union under the European Unionʼs
Horizon Europe Research and Innovation Program 101124282
(EARLYBIRD) and funding by the Deutsche Forschungsge-
meinschaft (DFG, German Research Foundation) under grant
325594231 and Germany’s Excellence Strategy—EXC-2094-
390783311. Views and opinions expressed are, however, those
of the authors 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. Support for F.L. was provided by NASA
through the NASA Hubble Fellowship grant #HST-HF2-
51512.001-A awarded by the Space Telescope Science Institute,
which is operated by the Association of Universities for
Research in Astronomy, Inc., under NASA contract NAS5-
26555. S.Z. acknowledges support through the NASA FINESST
grant 80NSSC20K1376. Support for S.Z. was provided by
NASA through the NASA Hubble Fellowship grant #HST-
HF2-51568 awarded by the Space Telescope Science Institute,
which is operated by the Association of Universities for
Research in Astronomy, Inc., for NASA, under contract
NAS5-26555.
Facility: ALMA
Software: analysisUtils (T. R. Hunter et al. 2023),
AstroPy (Astropy Collaboration et al. 2013, 2018, 2022), CASA
(CASA Team et al. 2022), cmasher (E. van der Velden 2020),
eddy (R. Teague 2019), frankenstein (J. Jennings et al.
2020), Interative Distance Estimation (https://
github.com/ElisaHaas25/Interactive-Distance-Estimation/
tree/main), linmix (https://github.com/jmeyers314/
linmix, ported from B. C. Kelly 2007), matplotlib
(J. D. Hunter 2007), mpol (I. Czekala et al. 2023;
B. Zawadzki et al. 2023), pandas (The pandas develop-
ment team, 2022), pyro (E. Bingham et al. 2019),
pytorch (A. Paszke et al. 2019), visread (I. Czekala
et al. 2021), SciPy (P. Virtanen et al. 2020)
Appendix A
Model Selection
The BIC is a metric that is used to select between different
models for a given set of observations (G. Schwarz 1978). It is
defined as (ˆ)
k n L
BIC ln 2 ln q
= - , where k is the number of
free parameters in the model, n is the number of data points,
and (ˆ)
L q is the likelihood evaluated at the posterior median
values. A model is favored if its BIC value is lower than that of
another model. Thus, the BIC penalizes more complex models
unless the likelihood substantially increases.
The expression that we use for our radial profiles, given in
Equation (1), has N + 1 terms, where N can vary. While the
value of N we initially chose to model each disk is based on
visual inspection of the observed radial profile, for systems
where a discrepancy between the model and observations is
apparent, we then ran models for higher values of N and
calculated the corresponding BIC. For each of the following
disks, we report the BICs for the different values of N tested in
Table 9. Since it is the relative rather than the absolute BIC
value that matters for model selection, we subtract a constant
for each disk such that the lowest scoring model has a BIC of 0.
For most disks, the final value of N we selected was
determined by whichever yields the lowest BIC. However, for
SO 1152, we selected the model with N = 4 over the model
with N = 2 because the latter does not reproduce the rings
(Figure 12). The N = 4 model overpredicts the peak intensity
compared to the N = 2 model, but since we are primarily
interested in characterizing the gaps and rings, we used the
N = 4 model for analysis.
Appendix B
Parametric Model Values
Table 10 lists the medians of the marginalized posteriors and
the 68% confidence intervals for the parametric radial intensity
models from Section 4.
Table 9
BIC Values
Source N BIC
SO 662 0 69
1 0
SO 844 2 26
3 0
SO 984 0 371
1 0
SO 1036 0 281
1 0
SO 1152 2 0
3 143
4 100
SO 1153 2 90
3 0
SO 1274 5 278
6 98
7 0
Figure 12. A comparison of the N = 2 and N = 4 model radial profiles
(calculated from the CLEAN images made from the model visibilities) to the
observed profile of the SO 1152 disk.
18

Appendix C
Identifying Shoulders
In this work, we define a gap as a local minimum and a ring
as a local maximum in the model radial intensity profile.
Inward of the gap and outward of the ring, there are inflection
points marking transitions in the profile from downward to
upward concavity. These can be identified as local minima in
Table 10
Inferred Parameters for Radial Intensity Profiles
Source Parameter Value
SO 662 Δx (mas) 1.3 ± 0.3
Δy (mas) −0.1 ± 0.4
P.A. (deg) 109.6 1.2
1.3
-
+
i (deg) 53.4 ± 1.1
A0 (Jy arcsec−2
) 1.6 ± 0.1
σ0 (au) 4.0 0.2
0.3
-
+
A1 (Jy arcsec−2
) 0.50 ± 0.02
σ1 (au) 3.8 ± 0.2
r1 (au) 12.8 ± 0.4
SO 844 Δx (mas) 0.7 ± 0.2
Δy (mas) 0.6 ± 0.3
P.A. (deg) 109.0 1.5
1.8
-
+
i (deg) 30.7 0.6
0.7
-
+
A0 (Jy arcsec−2
) 1.6 ± 0.2
σ0 (au) 2.29 0.14
0.17
-
+
A1 (Jy arcsec−2
) 0.44 ± 0.01
σ1 (au) 2.8 ± 0.1
r1 (au) 20.5 ± 0.1
A2 (Jy arcsec−2
) 0.272 0.009
0.008
-
+
σ2 (au) 3.0 ± 0.1
r2 (au) 36.8 ± 0.2
A3 (Jy arcsec−2
) 0.016 0.004
0.006
-
+
σ3 (au) 3.4 1.1
1.5
-
+
r3 (au) 46 ± 2
SO 897 Δx (mas) −6.3 ± 0.5
Δy (mas) −4.4 ± 0.4
P.A. (deg) 171.8 ± 0.6
i (deg) 62.5 ± 0.6
A1 (Jy arcsec−2
) 0.84 ± 0.02
σ1 (au) 3.9 ± 0.1
r1 (au) 11.9 ± 0.1
z0 (arcsec) 0.09 ± 0.03
f 1.12 0.07
0.06
-
+
SO 984 Δx (mas) −2.4 ± 0.3
Δy (mas) 0.4 0.3
0.2
- -
+
P.A. (deg) 75.0 ± 0.6
i (deg) 51.5 ± 0.4
A0 (Jy arcsec−2
) 0.89 ± 0.02
σ0 (au) 10.9 ± 0.3
A1 (Jy arcsec−2
) 0.319 ± 0.005
σ1 (au) 10.4 ± 0.2
r1 (au) 29.7 0.3
0.4
-
+
SO 1036 Δx (mas) 0.3 ± 0.3
Δy (mas) 1.1 ± 0.4
P.A. (deg) 16.2 1.1
1.0
-
+
i (deg) 47.2 ± 0.6
A0 (Jy arcsec−2
) 1.04 ± 0.04
σ0 (au) 6.1 0.3
0.2
-
+
A1 (Jy arcsec−2
) 0.320 ± 0.006
σ1 (au) 11.3 ± 0.3
r1 (au) 20.3 ± 0.5
SO 1152 Δx (mas) 3.1 ± 0.4
Δy (mas) 0.3 ± 0.4
P.A. (deg) 144.9 ± 0.3
i (deg) 65.8 ± 0.2
A0 (Jy arcsec−2
) 0.78 ± 0.05
σ0 (au) 6.2 ± 0.3
A1 (Jy arcsec−2
) 0.213 0.008
0.009
-
+
σ1 (au) 12.1 ± 0.6
r1 (au) 17.8 ± 1.0
Table 10
(Continued)
Source Parameter Value
A2 (Jy arcsec−2
) 0.156 ± 0.009
σ2 (au) 3.2 0.2
0.3
-
+
r2 (au) 41.4 ± 0.5
A3 (Jy arcsec−2
) 0.125 ± 0.003
σ3 (au) 7.9 ± 0.3
r3 (au) 61.6 0.4
0.5
-
+
A4 (Jy arcsec−2
) 0.063 ± 0.002
σ4 (au) 19.8 0.7
0.8
-
+
r4 (au) 64.3 ± 1.5
SO 1153 Δx (mas) 0.3 ± 0.1
Δy (mas) 0.3 ± 0.1
P.A. (deg) 162.0 ± 0.5
i (deg) 47.4 ± 0.3
A0 (Jy arcsec−2
) 5.9 ± 0.2
σ0 (au) 2.46 ± 0.06
A1 (Jy arcsec−2
) 0.41 ± 0.01
σ1 (au) 5.2 ± 0.1
r1 (au) 17.0 ± 0.2
A2 (Jy arcsec−2
) 0.235 ± 0.002
σ2 (au) 17.1 ± 0.2
r2 (au) 31.1 ± 0.6
A3 (Jy arcsec−2
) 0.084 ± 0.002
σ3 (au) 10.2 ± 0.3
r3 (au) 62.0 0.6
0.5
-
+
SO 1274 Δx (mas) −0.8 ± 0.2
Δy (mas) −0.3 ± 0.3
P.A. (deg) 135.6 ± 2.7
i (deg) 16.0 ± 0.7
A0 (Jy arcsec−2
) 1.15 ± 0.06
σ0 (au) 3.7 ± 0.2
A1 (Jy arcsec−2
) 0.308 ± 0.008
σ1 (au) 7.2 ± 0.2
r1 (au) 12.2 ± 0.4
A2 (Jy arcsec−2
) 0.228 ± 0.009
σ2 (au) 2.4 ± 0.1
r2 (au) 32.89 ± 0.15
A3 (Jy arcsec−2
) 0.134 0.007
0.008
-
+
σ3 (au) 2.1 ± 0.1
r3 (au) 49.7 ± 0.2
A4 (Jy arcsec−2
) 0.103 ± 0.006
σ4 (au) 2.6 ± 0.2
r4 (au) 65.6 ± 0.2
A5 (Jy arcsec−2
) 0.066 ± 0.004
σ5 (au) 3.1 ± 0.2
r5 (au) 88.1 ± 0.3
A6 (Jy arcsec−2
) 0.081 ± 0.005
σ6 (au) 2.3 ± 0.2
r6 (au) 119.3 ± 0.3
A7 (Jy arcsec−2
) 0.0229 ± 0.0009
σ7 (au) 15.0 ± 0.5
r7 (au) 120.9 ± 0.8
19

the
dI
dr
profile. In between these two inflection points, there is an
inflection point separating the gap and ring. This inflection
point marks the transition from upward to downward concavity
and can be identified as a local maximum in the
dI
dr
profile. This
behavior is illustrated in the left column of Figure 13. Along
similar lines, we define a shoulder to be present if there is a
series of three consecutive inflection points without a gap or
ring occurring between them (see the right column of
Figure 13). In this case, the
dI
dr
profile around the shoulder
has a similar shape as the
dI
dr
profile near a gap–ring pair, but the
former never crosses 0 before the first and last inflection point.
Our definition of a shoulder is similar to that used by
M. Yamaguchi et al. (2024) in their analysis of disks in
Taurus, but they also introduce an additional category, “disk
skirt,” which we do not use for the sake of simplicity.
Appendix D
Comparison of Parametric Model Profiles to
Nonparametric Model Profiles from frank
Using the P.A., inclination, and phase center offset derived
from the parametric models in Section 4, we generated radial
intensity profiles through nonparametric visibility modeling
with frank (J. Jennings et al. 2020). The peak S/Ns in the
CLEAN images of our targets range from ∼20 to 80, whereas
frank has commonly been used to model data sets with peak
S/Ns an order of magnitude higher (e.g., L. A. Cieza et al.
2021; J. Jennings et al. 2022a, 2022b). Consequently, our
model intensity profiles are more sensitive to the choice of
hyperparameters α (which determines the maximum baseline
that will be included in the fit based on its S/N) and w (which
determines how much the power spectrum will be smoothed).
The recommended values in the code documentation range
from 1.05 to 1.3 for α and 10−4
to 10−1
for w, with larger
values being more conservative. We thus ran models with two
pairs of values, (α = 1.05, w = 10−4
) and (α = 1.3, w = 0.1).
The fits were performed in logarithmic brightness space
because we found that fitting in linear space tended to produce
either stronger oscillatory artifacts or negative intensities in
some of the deep gaps.
A comparison of the frank models to the parametric
models is shown in Figure 14. By and large, the parametric
models and frank models recover substructures at the same
radial locations. For the deep gaps (i.e., in SO 1274 and
SO 897), the parametric models tend to yield deeper gaps
compared to the frank models, and the frank model depths
are sensitive to the hyperparameter choices. In the SO 1274,
SO 1153, and SO 1036 profiles using α = 1.05 and w = 10−4
,
small-scale oscillations are visible, suggesting that the data are
being overfit.
Whereas the (α = 1.05, w = 10−4
) model for SO 662 has a
gap (similar to the parametric model), the (α = 1.3, w = 0.1)
model only has a shoulder. We imaged the model visibilities
generated by frank and compared the radial profiles extracted
from the model images to the observed radial profile
(Figure 15). The (α = 1.05, w = 10−4
) model better reproduces
the radial profile. Thus, both the parametric and nonparametric
modeling approaches suggest that a gap is present in the
SO 662 disk.
Neither set of frank models reproduces the gaps and rings
in the SO 1152 disk well (Figure 15). The previous section
noted the difficulty in reproducing the SO 1152 radial profile
with the parametric approach as well. This may be a
consequence of the vertical structure influencing the observed
emission due to the disk’s high inclination (∼66°), but deeper
observations will be needed to clarify the disk’s emission
structure.
Figure 13. Left: intensity and
dI
dr
profiles for a disk with a gap and a ring. The locations of inflection points in the radial intensity profile are marked with dotted black
lines. Right: similar to the left column but for a disk with a shoulder.
20

ORCID iDs
Jane Huang https:/
/orcid.org/0000-0001-6947-6072
Megan Ansdell https:/
/orcid.org/0000-0003-4142-9842
Tilman Birnstiel https:/
/orcid.org/0000-0002-1899-8783
Ian Czekala https:/
/orcid.org/0000-0002-1483-8811
Feng Long https:/
/orcid.org/0000-0002-7607-719X
Jonathan Williams https:/
/orcid.org/0000-0001-5058-695X
Shangjia Zhang https:/
/orcid.org/0000-0002-8537-9114
Zhaohuan Zhu https:/
/orcid.org/0000-0003-3616-6822
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