The Disk Substructures at High Angular Resolution Project (DSHARP). IX. A High-definition Study of the HD 163296 Planet-forming Disk

As a first step toward characterizing the structure of the dust rings, we fit the crests and troughs of the continuum emission with circles defined by the position of the center (Δx₀, Δy₀), the radius r, the inclination i at which they are observed, and the position angle (PA) of their projected major axis. The fit is performed as discussed in Huang et al. (2018). In brief, we locate the points (x, y) corresponding to the radial maxima (or minima) of each ring (or gap), and use the python emcee package to calculate the ellipses that best reproduce this set of points. The best-fit parameters are listed in Table 1. Bright rings and dark gaps are labeled with the suffixes B and D, respectively, while the number appearing in the feature name corresponds to their radius in au. The uncertainties on the best-fit values correspond to the 16th and 84th percentile of the marginalized posterior probability distribution.

The outermost features B67, D86, B100, D145, and B155, are well described by concentric circles with an average inclination of (46.7 ± 0.1)° and average PA of (133.3 ± 0.1)°. Taken at face value, the circular model fitting indicates that D10 and D45 have lower inclinations. We believe that the lower inclination of D45 is due to the presence of the dust crescent centered at 55 au, which partially fills the gap making it appear more circular (i.e., less inclined), while the lower inclination of D10 is compatible with the effect of beam smearing, which is discussed in more details below.

Figure 2 shows the deprojected map and the azimuthally averaged radial profile of the continuum emission obtained using the values for the disk inclination and PA derived from the outermost rings. In the deprojected map, the outer rings appear as vertical stripes, confirming that these structures do indeed have a similar inclination and PA, and that they are intrinsically circular in shape. The intensity ratio between B14 and D10 varies with the azimuthal angle reaching a maximum of about 1.2 along the disk major axis (θ = ±90°) and a minimum of about 1 along the disk minor axis (θ = 0°). This variation can be explained by the fact that the effective spatial resolution for an inclined disk is higher along the disk major axis compared to the disk minor axis. In practice, B14 and D10 are spatially resolved only along the disk major axis, while they blend together along the disk minor axis.

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Azimuthal variations in the contrast between bright rings and dark gaps are also visible for the outer pairs. For example, along the westward semimajor axis (θ = −90°), the intensity reaches a minimum that is consistent with the noise of the observations at the center of D45, and a maximum of 0.91 mJy beam⁻¹ at the center of the adjacent bright ring B67. This corresponds to an intensity ratio larger than 37. In the opposite direction, i.e., along the eastward disk semimajor axis (θ = 90°), the intensity ratio B67/D45 is 13. As comparison, the intensity ratio along the southward (θ = 180°) and northward (θ = 0°) direction of the disk minor axis is about 7 and 5, respectively (see Table 2 for a summary of the intensity ratio among rings pairs). The difference between intensity ratios measured along the disk major and minor axes is likely due to beam smearing, while the difference between the intensity ratios measured along the disk major axis suggests the presence of azimuthal asymmetries in the dust emission.

The center and right panels of Figure 2 show the azimuthally averaged profile of the deproject continuum emission expressed in units of brightness temperature ${T}_{b}^{{pl}}$ calculated using the full Planck equation as

$$\begin{eqnarray}&&{T}_{b}^{{pl}}=\displaystyle \frac{h\nu }{{k}_{b}}{\left[\mathrm{ln}\left(\displaystyle \frac{2h{\nu }^{3}{\rm{\Omega }}}{{c}^{2}{F}_{\nu }}+1\right)\right]}^{-1},\end{eqnarray} \tag{ 1 }$$

where the F_ν is the flux density integrated over a beam defined by the solid angle Ω = πθ_minθ_max/(4 ln 2), where θ_min and θ_max are the minimum and maximum FWHM of the synthesized clean beam. The brightness temperature varies between about 50 K (corresponding to an intensity of 3.8 mJy beam⁻¹) in the innermost disk regions down to about 3 K (or a flux density of 0.0025 mJy beam⁻¹) at about 180 au. The pairs B67/D45 and B100/D86 have brightness temperature ratios of 4.3 and 2.4, respectively. Note that the difference between brightness temperature and intensity ratios is due to the nonlinearity of Equation (1).

Dullemond et al. (2018) finds that many of the continuum rings revealed by DSHARP observations, including the rings B67 and B100 of the HD 163296 disk, have a radial width that is narrower than the estimated pressure scale height of the gaseous disk, and conclude that this is strong evidence that these rings have formed by radial migration of dust particles toward local maxima of the gas pressure. Dullemond et al. (2018) defines the width of a ring as the dispersion of a Gaussian function that reproduces the deprojected and azimuthally averaged radial profile of the dust emission across the ring itself. This modeling approach is very fast, but it does not account for the observational noise, nor for the discrete sampling of the uv-plane performed by an interferometer, or for the fact that the synthesized beam is not circular.

Here, we compare the widths of the HD 163296 rings measured by Dullemond et al. (2018) with those estimated by a more accurate, but much slower, modeling of the continuum visibilities in the uv domain. We start by assuming that the dust emission is axisymmetric and expressed by a sum of n Gaussian functions

$$\begin{eqnarray}&&I(r)=\displaystyle \sum _{i=1}^{n}{I}_{i}{e}^{-{\left(r-{r}_{i}\right)}^{2}/2{w}_{d,i}^{2}}.\end{eqnarray} \tag{ 2 }$$

We generate synthetic 2D images of the continuum emission that are inclined and rotated to simulate all possible disk viewing angles. The center of emission is allowed to vary with respect to the center of the image (i.e., the phase center). Synthetic visibilities (V^mod) are then calculated by taking the Fourier transform of the image at the spatial frequencies of the observed visibilities (V^obs). The goodness of the model fit is evaluated through a usual χ² test, where ${\chi }^{2}\,=\sum w{({V}^{\mathrm{obs}}-{V}^{\mathrm{mod}})}^{2}$, and w is the weight of each visibility measurement provided by the ALMA pipeline. The calculation of χ² was performed using the CPU-based version of python package galario (Tazzari et al. 2018). Finally, the χ² is used to sample the posterior likelihood using the python package emcee (Foreman-Mackey et al. 2013).

An initial exploration of the parameter space indicates that a minimum of five Gaussian components are required to reproduce the observed continuum emission. In total, the model therefore has 19 free parameters: the coordinates (x₀, y₀) of the center of the emission, the disk inclination i and PA, and the radius r, peak intensity I, and width w_d of five Gaussian rings. We run emcee using 200 walkers initialized around the parameter values obtained by a Gaussian fitting of the radial intensity profile as in Dullemond et al. (2018). Each walker is evolved for an initial burn-in run of 10⁴ steps, after which they all converge toward a stable set of parameters. Following Goodman & Weare (2010), we sample the posterior distribution by letting emcee run until convergence, which is established based on the mean autocorrelation time of all the model parameters. We find that convergence is achieved after about 4 × 10⁴ steps. The optimal model parameters are then calculated as the median of the marginalized probability distribution and are listed in Table 3, while the related uncertainties correspond to the 0.3th and 99.7th percentile of the marginalized posterior probability distribution. This would be equivalent to a 3σ uncertainty for a Gaussian posterior distribution. A comparison between the image of the residual and the map of the continuum emission is presented in the Appendix.

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For B67, we find a width of ∼6.6 au, that is very close to the 6.8 au derived by Dullemond et al. (2018). Conversely, the width of B100 is about 25% larger than that measured by fitting the intensity profile. As noted in Figure 3 of Dullemond et al. (2018), the intensity radial profile of B100 has a peaked central region and bright wings that are not well reproduced by a Gaussian function. The value of the width in Dullemond et al. (2018) results from fitting only the innermost part of the ring between 94 and 104 au, while our value resulted from modeling the global profile including the bright wings of the ring. The fact that the two modeling procedures achieve the same result for rings that are intrinsically Gaussian (i.e., B67) indicates that the results of Dullemond et al. (2018) are independent of the exact shape of the beam, or, more generally, from nonlinear effects caused by the discrete uv-sampling and image deconvolution. In practice, the uv-sampling of the observation is sufficiently uniform and the S/N of the continuum map is sufficiently high to allow to perform the data analysis in the image domain without loss of accuracy. This is a great advantage because modeling the dust rings in the image domain takes only a few seconds, compared to the several days, or weeks, required to perform the model fitting in the Fourier domain.

Our analysis indicates that the ring B14 has a radius of about 15.5 au, slightly larger than the value calculated by fitting the crest of the emission as discussed in Section 3.1 (see also Table 1). The best-fit model also includes a small ring of 4 au in radius, therefore suggesting the presence of a inner circular cavity like those observed in transitional disks. However, as discussed in the next section, our azimuthally symmetric best-fit model poorly matches the dust emission from the innermost disk regions, which instead seems to indicate the presence of a dust crescent.

Finally, the last component of the model is a ring with a radius of about 32 au that is required to reproduce the kink in the radial profile of the emission observed at about 30 au from the center. Similar kinks in the radial profile of the dust continuum emission are observed in other DSHARP sources and are discussed in Huang et al. (2018).

The map obtained by imaging the residual visibilities obtained by subtracting the best-fit Gaussian model from the observations (Figure 3) reveals several features with absolute intensities above 5 times the rms noise and peak intensities as high as 30 times the rms noise. A comparison between the image of the residual and the continuum map is shown in the Appendix. The two most prominent structures have the shape of crescents and were already introduced at the beginning of Section 3 (see panels (b) and (c) of Figure 1). The innermost crescent is located at about 4 au from the center and it overlaps with the innermost ring component of the Gaussian model discussed above. Given its azimuthally asymmetric structure, and following the nomenclature adopted for the rings and gaps, we label this feature as C4 (C meaning "crescent"). The peak of C4 has a PA of −44° and an intensity is 0.55 mJy beam⁻¹, or about 20% of the continuum intensity measured at the same position. C4 extends by about 180° in azimuth and by about one resolution element in radius. C4 resembles the crescents observed in other Herbig Ae disks that have been attributed to the presence of anticyclonic vortices (e.g., Casassus et al. 2013; Isella et al. 2013; van der Marel et al. 2013; Boehler et al. 2018) or to an optically thick inner disk warped with respect to the outer disk (e.g., Marino et al. 2015; Benisty et al. 2017). However, higher angular resolution observations are required to investigate the morphology of HD 163296 innermost disk regions in greater details.

The second crescent, labeled as C55, is centered at an orbital radius of about 55 au and PA of 99°. The peak intensity is 0.64 mJy beam⁻¹, corresponding to a S/N of 28. Modeling C55 intensity as a 2D Gaussian function in the polar plane $I(r,\theta )\propto {e}^{-\left[{\left(r-{r}_{c}\right)}^{2}/2{\sigma }_{r}^{2}+{\left(\theta -{\theta }_{c}\right)}^{2}/2{\sigma }_{\theta }^{2}\right]}$, we find σ_θ = 17 au and σ_r = 3 au. Accounting for beam convolution (Equation (3)), the intrinsic radial and azimuthal widths of C55 are ∼2.2 au and ∼16.9 au, respectively.

In addition to C4 and C55, the image of the residuals shows fainter asymmetries along the B67 and B100 dust rings. If the dust emission is optically thin, intensity variations along the rings might probe the clumpiness of the dust distribution, while if the emission is optically thick, they reveal information about variations in the dust temperature. The intensity profiles along B67 and B100 are shown in the bottom-right panel of Figure 3. In the case of B67, the residual intensity varies between ±0.15 mJy beam⁻¹, to be compared with an azimuthally averaged intensity value measured at the same radius of 1.0 mJy beam⁻¹. The azimuthal intensity profile of B67 shows small amplitude variations characterized by an angular scale of about 4°, and larger variations with angular scales of 30°–50°. Whereas the small-scale variations have the same size of the synthesized beam and are most likely caused by the noise of the observations, the larger-scale structures might trace real asymmetries in the dust emission and, as the emission from B67 is estimated to be optically thin (see Section 5.1), in the dust density.

The CO emission extends far beyond the outer edge of millimeter-wave continuum emission up to an angular distance of 56, or about 560 au, from the central star. As discussed in Isella et al. (2016), the discrepancy between the radial extent of the millimeter wave dust and CO emission indicates a sharp drop in either the dust opacity or dust column density beyond about 200 au.

The CO emission observed at intermediate velocities (e.g., v_lsr = 3.40–5.00 km s⁻¹ and v_lsr = 6.28–8.20 km s⁻¹) shows a characteristic "dragonfly" structure tracing both the front and rear warm CO-emitting layers of the disk (Figure 5). The emission coming from the rear of the disk is fainter than that from the front due to a combination of lower temperature of the emitting gas and absorption from the intervening material. As discussed in Rosenfeld et al. (2013), the fact that the front and rear sides of the disk appear as two distinct emitting regions is explained by the large optical depth of the CO line (whose intensity therefore mostly depends on the gas temperature) and a vertical gradient of the disk temperature characterized by warm surfaces and a cold disk midplane. Furthermore, the separation between the front and rear CO emission measures the vertical geometry of the CO-emitting surfaces.

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We compare the observed geometry of the CO emission with that of a parametric model where the CO line originates from a geometrically thin layers characterized by a distance from the midplane z_co = ±z_co,0 (r/r₀)^q. As a demonstration of the technique, we show in Figure 5 parametric isovelocity curves corresponding to the front side of the disk calculated as presented in the Appendix. The geometry of the isovelocity curves depends on z_co,0, q, as well as on the stellar mass and disk inclination. Assuming the disk inclination derived from the continuum (i = 467) and a stellar mass of 2.0 M_⋆, we obtain a good match between model and observations for q = 0.5 and z_co(100 au) = 30 au.

Our observations reveal that the CO emission coming from the back side of the disk is dimmer at the location of the bright dust rings B67 and B100 compared to the emission observed at the location of the dark gaps D45 and D87 (Figure 6). This dimming, which is likely due to absorption of the line emission as it crosses the dusty rings, provides a tool to constrain the optical depth of continuum emission independently from the assumptions on the dust temperature. The CO emission coming from the back side of the disk can be expressed as ${I}_{\mathrm{co}}^{\mathrm{obs}}={I}_{\mathrm{co}}{e}^{-{\tau }_{d}^{\mathrm{ext}}}$, where ${\tau }_{d}^{\mathrm{ext}}$ is the line-of-sight dust extinction optical depth equal to the sum of the absorption and scattering optical depths (${\tau }_{d}^{\mathrm{ext}}={\tau }_{d}^{\mathrm{abs}}+{\tau }_{d}^{\mathrm{sca}}$), while I_co is the initial unattenuated CO emission. Defining the parameter as $\epsilon \equiv {\tau }_{d}^{\mathrm{abs}}/{\tau }_{d}^{\mathrm{ext}}$, the absorption optical depth of the dust continuum can be written as

$$\begin{eqnarray}&&{\tau }_{d}^{\mathrm{abs}}=\epsilon \mathrm{ln}\displaystyle \frac{{I}_{\mathrm{co}}}{{I}_{\mathrm{co}}^{\mathrm{obs}}}.\end{eqnarray} \tag{ 3 }$$

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The observations of the CO emission directly measure I_co^obs along the bright continuum rings, while the unattenuated emission I_co can be estimated through the procedure illustrated in Figure 6. In practice, in the channels where CO absorption is observed, we measure the intensity of the CO emission along the isovelocity curves corresponding to the back side emission. We then use the values measured within the dust gaps, where, based on the results of the previous section, τ_d ≪ 1, to estimate I_co at the position of the bright rings. We measure the ratio ${I}_{\mathrm{co}}/{I}_{\mathrm{co}}^{\mathrm{obs}}$ for B67 and B100 along 18 different lines of sight (i.e., 18 different azimuthal angles along the dust rings) and find mean values of 1.9 ± 0.1 and 2.1 ± 0.1. We also perform this measurement for B168 finding ${I}_{\mathrm{co}}/{I}_{\mathrm{co}}^{\mathrm{obs}}=1.1\pm 0.2$. The corresponding dust extinction optical depths are ${\tau }_{d}^{\mathrm{ext}}({\rm{B}}67)=0.64\pm 0.05$, ${\tau }_{d}^{\mathrm{ext}}({\rm{B}}100)=0.74\pm 0.05$, ${\tau }_{d}^{\mathrm{ext}}({\rm{B}}168)=0.1\pm 0.2$. It is worth noting that the ratio ${I}_{\mathrm{co}}/{I}_{\mathrm{co}}^{\mathrm{obs}}$ could vary with the azimuthal angle if ${\tau }_{d}^{\mathrm{ext}}$ is not constant along the dust rings. However, the sensitivity of our observations does not allow us to perform such an analysis, and we must instead rely on the azimuthally averaged value of the CO intensity ratio to investigate the dust extinction properties.

As from Equation (3), the absorption optical depth of the dust rings depends on the dust scattering properties through the parameter . For dust grains that are much smaller than the wavelength of the observations (in our case, for particle sizes a ≪ λ/2π ∼ 0.2 mm), scattering is negligible compared to absorption, and  ≃ 1. For grain sizes that are comparable to the wavelength of the observations (a ∼ λ/2π ∼ 0.2 mm), scattering, and absorption properties of spherical grains follow Mie theory, and can vary between about 0.1 and 1, depending on the dust composition and internal structure. Finally, if a ≫ 0.2 mm, scattering and absorption are equal, and  = 0.5 (see, e.g., Bohren & Huffman 1998). Without any prior information about the size and composition of dust grains, the observed absorption of the CO emission constrains the absorption optical depth of B67 and B100 between about 0.07 and 0.7, while the lack of CO absorption at the location of B168 sets an upper limit to the absorption optical depth of this ring at about 0.3. A caveat of this approach is that the emission from the front side of the disk could partially contaminate the emission from the back side, particularly close to the star where the separation between the two surfaces is small. Also, the ratio ${I}_{\mathrm{co}}/{I}_{\mathrm{co}}^{\mathrm{obs}}$ could be affected by beam smearing, implying that our measurements provide only a lower limit for the true contrast between the initial and absorbed CO emission, and consequently, the corresponding dust optical depth. However, the angular resolution of the CO observations is sufficiently high to believe that this effect is small.

If the absorption optical depth of the dust rings is measured from CO absorption, the corresponding continuum intensity I_ν can be used to estimate the temperature T_d of the emitting dust. If dust scattering is negligible (${\tau }_{d}^{\mathrm{ext}}={\tau }_{d}^{\mathrm{abs}}$ and  = 1), the conversion between flux density and dust temperature is

$$\begin{eqnarray}&&{T}_{d}=\displaystyle \frac{h\nu }{{k}_{b}}{\left[\mathrm{ln}\left(\displaystyle \frac{2h{\nu }^{3}{\rm{\Omega }}(1-{e}^{-{\tau }_{d}^{\mathrm{abs}}})}{{c}^{2}{F}_{\nu }}+1\right)\right]}^{-1},\end{eqnarray} \tag{ 4 }$$

or, using Equation (3), as

$$\begin{eqnarray}&&{T}_{d}=\displaystyle \frac{h\nu }{{k}_{b}}{\left[\mathrm{ln}\left(\displaystyle \frac{2h{\nu }^{3}{\rm{\Omega }}(1-{I}_{\mathrm{co}}^{\mathrm{obs}}/{I}_{\mathrm{co}})}{{c}^{2}{F}_{\nu }}+1\right)\right]}^{-1}.\end{eqnarray} \tag{ 5 }$$

From the measured continuum intensities we obtain dust temperatures of 24 K and 15 K for B67 and B100, respectively, while the upper limits for the optical depth of B168 translates in a minimum temperature of about 6 K.

If dust scattering is not negligible (${\tau }_{d}^{\mathrm{ext}}\gt {\tau }_{d}^{\mathrm{abs}}$ and  < 1), the dust temperature can be estimated using the formalism presented in Birnstiel et al. (2018). In this case, the intensity is written as ${I}_{\nu ,d}\sim S({T}_{d},{\tau }_{\nu ,d}^{\mathrm{ext}},\epsilon )(1-{e}^{-{\tau }_{\nu ,d}^{\mathrm{ext}}})$, where the source function S_ν depends on the dust temperature, and on the extinction and scattering optical depth along the line of sight. Adopting in the range between 0.1 and 1, we obtain dust temperatures between 130 K and 24 K for B67, and between 70 K and 15 K for B100. The fact that the dust temperature depends on the dust-scattering properties hampers the application of this technique as an independent measure of disk temperature. However, if the dust temperature is measured through other means, then the dust-scattering properties can by constrained (see Section 5).

At typical densities and temperatures of protoplanetary disks, the peak of the ¹²CO J = 2–1 line emission is optically thick and the CO rotational levels are in local thermodynamic equilibrium (see, e.g., Weaver et al. 2018). The intensity at the peak of the line therefore measures the temperature of the emitting gas. Following Weaver et al. (2018), we generate a CO temperature map from non-continuum-subtracted line emission to avoid underestimating the line intensity along lines of sight where the continuum is optically thick (Figure 7). The CO temperature reaches values as high as 120 K in the innermost disk regions and as low as 20 K at the disk outer edge. The peak intensity map is characterized by an evident asymmetry with respect to the apparent disk major axis: the southwest side of the disk is hotter than the northeast side. This is explained in first approximation by the fact that the CO-emitting layer is a conic surface observed from a direction inclined with respect to the axis of the cone. The orientation of the asymmetry implies that the northern side of the disk is the closest to the observer (see also Rosenfeld et al. 2013).

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Assuming, as in the previous section, that the peak of the line is emitted from a geometrically thin layer at a distance from the disk midplane given by ${z}_{\mathrm{co}}={z}_{{co},0}{(r/{r}_{0})}^{q}$, it is possible to deproject the observed CO temperature and estimate its profile as a function of the cylindrical radius r. Using the values for z_co and q presented above, we derive the CO temperature profile shown in the right panel of Figure 7. Between 30 and 500 au, the radial profile of the CO temperature scales almost linearly with the radius and follows the relation T_co (K) ∼ 87–0.14 (r/au). Within 30 au, the CO brightness temperature drops. This is consistent with the expected drop in brightness temperature as the emitting area becomes smaller than the angular resolution of the observations (beam dilution).

The overall kinematics of the CO gas is well described by Keplerian rotation around a star with a mass of about 2.0 M_⊙ (Figure 8). Both the integrated velocity map and the position–velocity diagram are in first approximation symmetric with respect to the apparent disk minor axis. The systemic velocity inferred from CO observations is 5.8 km s⁻¹ and is in agreement with previous results. The position–velocity diagram also reveal diffuse emission at a velocities between about 12.5 and 14 km s⁻¹ that might trace a redshifted stellar wind (Klaassen et al. 2013) or perhaps emission from the leftover of the HD 163296 parent cloud.

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The CO map registered at a velocity of 6.92 km s⁻¹ (Figures 4 and 5) is characterized by a kink at δR.A. ∼ −1'', δdecl. ∼ 15. This feature was previously reported by Pinte et al. (2018) and has been attributed to perturbations in the gas kinematics caused by a planet with a mass of 2 M_J orbiting at 260 au from the star. Unfortunately, despite the better angular resolution, the observations obtained using the extended ALMA configuration achieve a spectral resolution about 10 times worse than that of those presented by Pinte et al. (2018), and do not allow us to investigate the origin of this feature. Similarly, the coarse velocity resolution of our data might not allow to investigate the small deviations from Keplerian rotation observed within the dust gaps and reported by Teague et al. (2018).

To date, the HD 163296 disk has been the subject of several studies aimed at constraining the density and thermal structure of the disk through forward modeling of the spatially resolved millimeter-wave molecular line and continuum emission (e.g., Isella et al. 2007, 2016; Tilling et al. 2012; Rosenfeld et al. 2013; Flaherty et al. 2015). This approach relies on a large number of assumptions including the choice of the parametric forms for the gas and dust surface density and temperature (when the latter is not calculated using radiative transfer models), the dust and molecular abundances, the dust opacities, the stellar properties (mass, luminosity, distance), and the gas kinematics (e.g., Keplerian motion, turbulent velocity). Because of the large computational time required to generate synthetic images at high spatial and spectral resolution, the comparison between models and observations is generally performed by either manually adjusting the model parameters until obtaining a satisfactory result (e.g., Isella et al. 2016), or by using automatic algorithms to search for best-fit models among a subset of model parameters. Not surprisingly, forward modeling performed by different investigators lead to different results.

Figure 9 compares the temperature inferred from the 1.25 mm dust continuum intensity to the models discussed in Isella et al. (2016), Rosenfeld et al. (2013), and Flaherty et al. (2015). We choose these three studies because they assume a similar parametric form for the disk temperature and are based on high spectral and angular resolution observations of the ¹²CO line emission. If scattering from dust grains is negligible (${\tau }_{d}^{\mathrm{ext}}={\tau }_{d}^{\mathrm{abs}}$,  = 1), the brightness temperature ${T}_{b}^{{pl}}$, the physical dust temperature T_d, and the optical depth ${\tau }_{d}^{\mathrm{abs}}$ are linked by the relation ${B}_{\nu }({T}_{b}^{{pl}})={B}_{\nu }({T}_{d})(1-{e}^{-{\tau }_{d}^{\mathrm{abs}}})$. Within 30 au, the model temperatures are bracketed between Rosenfeld et al. (2013) and Isella et al. (2016), and would imply optical depths of about 1.3 and 0.7, respectively. Beyond 30 au, ${T}_{b}^{{pl}}$ drops well below the model predictions, suggesting that the emission becomes more optically thin. In particular, the model temperatures give ${\tau }_{d}^{\mathrm{abs}}\lt 0.02$ and ${\tau }_{d}^{\mathrm{abs}}\sim 0.04\mbox{--}0.05$ at the center of the dark gaps D45 and D86, respectively, and ${\tau }_{d}^{\mathrm{abs}}\sim 0.5\mbox{--}0.7$ and ${\tau }_{d}^{\mathrm{abs}}\sim 0.3\mbox{--}0.5$ at the bright rings B67 and B100.

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These measurements rely on the assumption that scattering is negligible, but, as discussed in Section 4.1, this is likely not the case if dust grains have sizes comparable or larger than the wavelength of the observations. The inclusion of scattering significantly complicates the interpretation of the observed continuum emission and a simple relation between ${T}_{b}^{{pl}}$, T_d, and the absorption and scattering optical depths, ${\tau }_{d}^{\mathrm{abs}}$ and ${\tau }_{d}^{\mathrm{sca}}$ does not exist. Fortunately, however, it is possible to derive an approximated analytic solution to constrain T_d, ${\tau }_{d}^{\mathrm{abs}}$ or ${\tau }_{d}^{\mathrm{sca}}$ if the other two quantities are known (see Section 5 of Birnstiel et al. 2018).

In the case of HD 163296, the direct measurement of the extinction opacity at the bright rings B67 and B100 discussed in Section 4.1 provide a unique tool to constrain the scattering properties of the circumstellar dust. Using the formalism discussed in Birnstiel et al. (2018), we calculate the physical dust temperature required to reproduce the dust continuum emission measured at B67 (0.77 mJy beam⁻¹) and B100 (0.45 mJy beam⁻¹) for values of the scattering parameter varying from 0.1 (${\tau }_{d}^{\mathrm{sca}}=9{\tau }_{d}^{\mathrm{abs}}$) to 1 (no scattering). The comparison between these temperatures, which are indicated with horizontal segments in the left panel of Figure 9, and those predicted by Flaherty et al. (2015), Isella et al. (2016), and Rosenfeld et al. (2013) constrain , and therefore the dust-scattering properties. For B67, we find that ≳ 0.6 leads to dust temperature that is consistent with the models. We note that the midplane temperature of Flaherty et al. (2015) is slightly below the minimum temperature consistent with the observations, which corresponds to  = 1. Interestingly, for B100, the consistency between models and observations requires between 0.4 and 0.6, while  = 1 would imply a physical dust temperature of only 15 K. The difference in scattering properties between B67 and B100 is intriguing and, taken at face value, might suggest a variation in the dust properties between B67 and B100. However, the difference might also result from the fact that, due to beam smearing, the measured value of ${\tau }_{d}^{\mathrm{ext}}({\rm{B}}67)$ might be a lower estimate of the extinction opacity, and consequently, of the scattering opacity. Future ALMA observations expressly designed to image the molecular gas emission from the HD 163296 at both high angular resolution and sensitivity are required to place more stringent constraints on the dust extinction optical depth.

We conclude this section by comparing the peak brightness temperature of the CO emission measured in Section 4.2 to the temperature of the CO-emitting layer predicted by Flaherty et al. (2015), Isella et al. (2016), and Rosenfeld et al. (2013) models (see the right panel of Figure 9). In Section 4.1 we found that the CO-emitting layer corresponds to the surface defined by z(r) = 30 au (r/100 au)^0.5. Within about 100 au, the measured gas temperature is below the model predictions. This is likely due to the effect of beam dilution as discussed in Weaver et al. (2018). Between 100 and 400 au, the temperature of the CO-emitting layer predicted by Rosenfeld et al. (2013) and Isella et al. (2016) is in good agreement with the observations, while the temperature predicted by Flaherty et al. (2015) is about 30% lower. Also, while the temperature of the CO-emitting layer predicted by Rosenfeld et al. (2013) and Isella et al. (2016) is similar to that of the stellar irradiated disk surface (dashed line), as expected in the case of a very optically thick line, the Flaherty et al. (2015) temperature is closer to that achieved in the disk midplane (dotted lines).

In Section 3, we found that the radial profile of the continuum emission from the HD 163296 disk can be represented by a sum of azimuthally symmetric Gaussian rings. However, significant asymmetric structures appear once the symmetric component of the emission is removed. In this section, we discuss these results in the framework of the planet–disk interaction mechanism, while we refer to Huang et al. (2018) for a discussion of other processes capable of generating dust rings. We also refer the reader to Dullemond et al. (2018) for a discussion of the observed rings properties in the context of dust-trap models.

Massive planets are expected to open gaps in the surface density of protoplanetary disks along their orbits through the strong mutual gravitational interaction with the circumstellar material (see, e.g., Bryden et al. 1999). While the planet minimum mass for this to happen varies as a function of the disk physical properties, most of the simulations agree that for typical disk conditions planets more massive than 20–40 Earth masses should produce gaps observable at the spatial scales probed by ALMA (e.g., Dong et al. 2012; Isella & Turner 2018). In the limit of nearly inviscid disks, simulations have also shown that even planets with masses as low as few Earth masses might generate visible rings (Dong et al. 2018). Given the ubiquity of planets in the universe, it seems therefore reasonable to associate the ring structures observed by ALMA with the presence of planetary systems in the act of forming.

In the case of HD 163296, the planet–disk interaction hypothesis is supported by the detection of gas depletion and deviation from Keplerian rotation within the dark gaps D87 and D143 (Isella et al. 2016; Teague et al. 2018), which imply that the emission gaps correspond to gaps in the gas density. The comparison with hydrodynamic simulations indicates that D45, D87, and D143 might results from the dust clearing operated by planets with masses between 0.5 and 2 M_J (Liu et al. 2018; Teague et al. 2018). However, the ring morphology of the HD 163296 disk might also be consistent with gravitational perturbations by a single planet with a mass as low as 65 M_E orbiting at about 100 au from the star if the disk viscosity is very low (α < 10⁻⁴, Dong et al. 2018). Overall, these results are consistent with the planet masses discussed in Zhang et al. (2018), which have been estimated from the azimuthally averaged profile of the continuum emission shown in Figure 2. In addition, Zhang et al. (2018) find that the newly discovered gap B10 might be generated by a planet with a mass between 0.2 and 1.5 M_J.

Here we want to point out that the detection of asymmetries in the dust emission discussed in Section 3.3 corroborates the hypothesis that the HD 163296 disk is shaped by the interaction with yet-unseen planets. As shown in Figures 6 and 7 of Zhang et al. (2018), features similar to the crescent observed inside the D45 gap (inset b of Figure 1) are naturally produced by the planet–disk interaction whenever the disk viscosity is low ($\alpha \lt {10}^{-3}$) and the planet mass is above about 0.1 M_J. The crescents observed in numerical simulations have two different origins: they either trace dust particles trapped in anticyclonic vortices that form at the edge of the gap opened by the planet (Li et al. 2001), or probe material trapped at the Lagrangian points along the planet orbit. Without any pretense of accuracy, we compare in Figure 10 the map of the 1.25 mm dust continuum emission recorded toward HD 163296 with a synthetic image of a disk model characterized by the presence of a 0.15 M_J planet orbiting at 54 au from the central star. The model is part of the suite of models presented in Zhang et al. (2018) and was generated assuming a viscosity parameter α = 10⁻⁴ and a disk pressure scale height h = 0.05r. This comparison is purely qualitative, but nevertheless shows the resemblance between the corotation features predicted by models and the circular arc observed in HD 163296.

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