Thermal Tomography of the Inner Regions of Protoplanetary Disks with thengVLA and ALMA
Satoshi Okuzumi,1 Munetake Momose,2 and Akimasa Kataoka3
1

Tokyo Institute of Technology, 2–12–1 Ookayama,
Meguro, Tokyo 152–8551
2
Ibaraki University, 2–1–1 Bunkyo,
Mito, Ibaraki 310-8512
3
National Astronomical Observatory of Japan, 2–21–1 Osawa,
Mitaka, Tokyo 181–8588
okuzumi@eps.sci.titech.ac.jp

Abstract
Understanding the temperature structure of protoplanetary disks is crucial for answering the fundamental question of when and where in the disks rocky planets like our own form. However, the thermal structure of the inner
few au of the disks is poorly understood not only because of lack of observational constraints but also because of
the uncertainty of accretion heating processes. Here, we propose thermal tomography of the inner regions of protoplanetary disks with the ngVLA and ALMA. The proposed approach is based on the assumption that the inner disk
regions are optically thick at submillimeter wavelengths but are marginally optically thin at longer millimeter wavelengths. By combining high-resolution millimeter continuum images from the ngVLA with submillimeter images atcomparable resolutions from ALMA, we will be able to reconstruct the radial and vertical structure of the inner few
au disk regions. We demonstrate that the thermal tomography we propose can be used to constrain the efﬁciency of
midplane accretion heating, a process that controls the timing of snow-line migration to the rocky planet-forming
region, in the few au regions of protoplanetary disks at a distance of 140 pc.
Key words: dust, extinction — planets and satellites: formation — protoplanetary disks — submillimeter:planetary systems — magnetohydrodynamics (MHD) — telescopes
1. Introduction
The temperature structure of protoplanetary disks dictates
where planets of different compositions form in the disks.
Rocky planets including the Earth are generally believed tohave formed inside of the snow line, where the disk temperature reaches the sublimation temperature of water ice.
Enhancement of solids outside the snow line due to ice condensation may help gas giants form within the lifetime of thegas disks (Kokubo & Ida, 2002). The composition of solids
may even determine how planets form because the stickinessof solid particles generally depends on their surface composition (Chokshi et al., 1993).
Despite its importance in planet formation, the temperature
structure of the inner regions of protoplanetary disks is poorly
understood. Conventionally, the disk temperature structure has
been studied based on the viscous accretion disk model thatassumes vertically uniform viscosity. According to this classical disk model, accretion heating at the midplane pushes
the midplane snow line in ∼ 1 Myr-old disks out to 2–3 au(Davis, 2005; Garaud & Lin, 2007; Oka et al., 2011; Bitsch
et al., 2015). However, the validity of the classical viscousmodel has been questioned by recent magnetohydrodynamical(MHD) simulations of protoplanetary disks showing that accretion heating can only take place near the disk surface, where theionization fraction is relatively high (Hirose & Turner, 2011;
Mori et al., 2019). A recent disk temperature model based on
MHD simulations predicts that the snow line in ∼ 1 Myr-old

disks must lie interior to 1 au (Mori et al., 2021). This raises
the fundamental question of why the Earth is so dry with abulk water content likely well below 1 wt% (Abe et al., 2000;
Marty, 2012), because the solids outside the snow line in thesolar nebula would have contained water ice in excess of 1 wt%
as witnessed by carbonaceous chondrites and comets (e.g., vanDishoeck et al., 2014). In fact, simulations show that rocky
planetary embryos at 1 au are likely to grow into waterworlds
if the snow line migrates inward of the 1 au orbit within 1 Myr
after disk formation (Sato et al., 2016) unless some mechanism stops the inward migration of the icy dust particles (e.g.,
Morbidelli et al., 2016).
To summarize, constraining the thermal structure of the inner regions of protoplanetary disks from astronomical observations is crucial for fully understanding how rocky planets form.
In this article, we propose a methodology for testing disk temperature models by multi-wavelength, high-resolution imaging
with the ngVLA and ALMA.
2. Theoretical Background
In principle, the efﬁciency of midplane heating can be studied by measuring disk temperatures at different vertical depths.
In classical accretion disk models with vertically uniform kinematic viscosity, accretion heating mainly takes place at the
midplane, leading to the midplane temperature higher than thesurface temperature in inner disk regions (e.g., Davis, 2005).
In contrast, in MHD accretion disk models, Joule heating takes

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Fig. 1. Temperature as a function of the cylindrical radius r and distance from the midplane z for the viscous and MHD disks (panels (a) and (b),respectively). The contours mark T = 100, 200, and 300 K. The maps are truncated at three scale heights below and above the midplane.

3. Proof-of-concept Simulations

disks, which means that we neglect possible spatial variationsof the dust-to-gas mass ratio and grain size distribution due todust evolution.
To mimic observations, Gaussian convolution is applied tosmooth the theoretical continuum intensity proﬁles at angularresolutions of ALMA and the ngVLA. Speciﬁcally, we adopt
angular resolutions of 12, 5, and 16 mas for the ALMA 0.87
mm, ngVLA 3 mm, and ngVLA 1 cm bands, respectively, estimated by Ricci et al. (2018) assuming 8 and 20 hr observations
with ALMA and the ngVLA, respectively. The modeled disks
are assumed to be located a distance of d = 140 pc away fromthe Earth, resulting in spatial resolutions of 1.7, 0.7, and 2.2
au for the 0.87 mm, 3 mm, and 1 cm band images, respectively. Based on the sensitivity estimates by Ricci et al. (2018),
the rms noise in the brightness temperature maps for the threebands are estimated to be ≈ 0.6, 2, and 0.8 K. The sensitivities
are high enough for a precise measurement of thermal emission
from around the snow line, where T ≈ 160 K.

3.1. Models

3.2. Results

We examine the feasibility of the disk thermal tomography
described above using two disk temperature models. One isthe classical viscous disk model (Lynden-Bell & Pringle, 1974)
with the viscous alpha parameter α taken to be 10−3 throughout the disk. The other is the MHD accretion disk model recently proposed by Mori et al. (2021), which assumes accretion
heating to occur on thin layers near the disk surfaces. For simplicity, we take the depth of the heating layers to be 0.4 g cm−2in column mass density measured from inﬁnity (Mori et al.,2019, 2021). The radial and vertical temperature proﬁles forthe two accretion disk models are generated using the planeparallel radiative transfer model by Hubeny (1990). In addition to accretion heating, we also include heating by stellar
radiation (Chiang & Goldreich, 1997). For both models, we
assume a uniform accretion rate of Ṁ = 2 × 10−7 M⊙ yr−1 ,a stellar mass of 1M⊙ , a stellar luminosity of 1L⊙ , and a uniform infrared opacity of 5.0 cm2 g−1 . Figure 1 shows the twodimensional temperature maps for the two disk models.
The two-dimensional temperature proﬁles are used to generate maps of dust continuum emission from a face-on disk
at submillimeter to centimeter wavelengths. The disk opacity in this wavelength range is assumed to follow a power law
κ(λ) = 6 × 10−3 (λ/3 mm)−1.7 cm2 g−1 , where λ is the wavelength. Again, the opacity is taken to be uniform throughout the

Figure 2 presents the maps of the brightness temperatures
TB at three wavelengths from the viscous disk model. The upper and lower panels are the maps before and after Gaussiansmoothing, respectively, Comparison between the upper andlower panels shows that the angular resolutions of the ALMA
submillimeter band and ngVLA millimeter bands are highenough to resolve the radial distribution of TB in the inner
few au. It is interesting to note that the brightness temperature of the inner disk region is maximized at λ = 3 mm anddecreases both longward and shortward of 3 mm. This reﬂects
the vertical thermal structure of the uniformly viscous disk, inwhich the temperature decreases as we go away from the midplane. At λ = 0.87 mm, the inner region is optically thick andthe thermal emission comes from ≈ 2 scale heights above themidplane. At λ = 3 mm, the inner region is marginally optically thick and the thermal emission is dominated by the midplane. Longward of 3 mm, the inner region is optically thinner
at longer wavelengths, and the brightness temperature of theregion decreases with increasing λ. This demonstrates the importance of multi-wavelength observations using ALMA andthe ngVLA in studying internal heating in the inner regions ofprotoplanetary disks.
Figure 3 is the same as Figure 2, but for the MHD accretion disk model. The brightness temperatures of the MHD

place well above the midplane, leading to vertically uniform
temperature structure with the midplane no hotter than the surface (Hirose & Turner, 2011; Mori et al., 2019). This difference critically affects when the snow line crosses the 1 au orbit
in the disks (Mori et al., 2021).
Multi-wavelength observations of the disks’ inner regions
using the ngVLA and ALMA will enable us to perform thermal tomography—imaging of the temperature structure in both
the radial and vertical directions—of the rocky planet-forming
regions. The ngVLA will provide access to the midplane temperature structure of the inner few au of the disks at millimeterto centimeter wavelengths, at which the inner regions are likely
to be optically marginally thick to optically thin. ALMA is already capable of mapping the dust continuum emission fromwell above the midplane at comparable resolutions at opticallythick, submillimeter wavelengths.

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Fig. 2. Upper panels: maps of the brightness temperatures TB at λ = 0.87 mm, 3 mm, and 1 cm (left, center, and right panels, respectively) from a face-on
viscous accretion disk. The contours mark TB = 100, 200, and 300 K. Lower panels: synthetic brightness temperature maps after Gaussian smoothing,assuming d = 140 pc and spatial resolutions of 12, 5, and 16 mas for the ALMA 0.87 mm, ngVLA 3 mm, and ngVLA 1 cm bands, respectively (Ricci etal., 2018).

disk are overall lower than those of the viscous disk. Still,
the synthetic map for the ngVLA 3 mm band accurately captures the brightness temperature proﬁle of the MHD disk up
to TB ≈ 200 K and thus allows us to study the thermal structure around the water snow line (T ≈ 160 K). Because of its
vertically uniform temperature structure, the MHD disk model
produces nearly identical brightness temperature maps (except
for the unresolved very inner region) at optically thick wavelengths of λ <∼ 3 mm. Therefore, multi-band observations at
ALMA submillimeter bands and the ngVLA 3 mm band would
allow us to test whether midplane heating is inefﬁcient aroundthe midplane in a disk.

et al., 2010; Ros & Johansen, 2013; Okuzumi et al., 2016;
Schoonenberg & Ormel, 2017). All these dust evolution processes may alter the temperature structure and emission proﬁles in the inner regions of protoplanetary disks (e.g., Banzatti
et al., 2015; Pinilla et al., 2017) and therefore should be included in future modeling.

4. Conclusion

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We have demonstrated that thermal tomography with thengVLA and ALMA will enable us to discriminate between different models for disk thermal structure. Such observations
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5. Acknowledgments
We thank Takahiro Ueda for discussions. This work was
supported by JSPS KAKENHI Grant Numbers JP18H05438,
JP19K03926, JP19K03941, and JP20H00182.

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Fig. 3. Same as Figure 2 but for the MHD disk model.

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