Identifying Anticyclonic Vortex Features Produced by the Rossby Wave Instability in Protoplanetary Disks

Several nearby protoplanetary disks have been observed to display large scale crescents in the (sub)millimeter dust continuum emission. One interpretation is that these structures correspond to anticyclonic vortices generated by the Rossby wave instability within the gaseous disk. Such vortices have local gas over-densities and are expected to concentrate dust particles with Stokes number around unity. This process might catalyze the formation of planetesimals. Whereas recent observations showed that dust crescent are indeed regions where millimeter-size particles have abnormally high concentration relative to the gas and smaller grains, no observations have yet shown that the gas within the crescent region counter-rotates with respect to the protoplanetary disk. Here we investigate the detectability of anticyclonic features through measurement of the line-of-sight component of the gas velocity obtained with ALMA. We carry out 2D hydrodynamic simulations and 3D radiative transfer calculation of a protoplanetary disk characterized by a vortex created by the tidal interaction with a massive planet. As a case study, the disk parameters are chosen to mimic the IRS 48 system, which has the most prominent crescent observed to date. We generate synthetic ALMA observations of both the dust continuum and 12CO emission around the frequency of 345 GHz. We find that the anticyclonic features of vortex are weak but can be detected if both the source and the observational setup are properly chosen. We provide a recipe for maximizing the probability to detect such vortex features and present an analysis procedure to infer their kinematic properties.


INTRODUCTION
Protoplanetary systems such as IRS48 (van der Marel et al. 2013), LkHα 330 (Isella et al. 2013, HD 142527 (Muto et al. 2015), MWC 758 (Isella et al. 2010), and SAO 206462 (Pérez et al. 2014) exhibit large cavities and prominent crescents in the (sub-)millimeter dust continuum emission. In systems where spatially resolved multi-wavelength observations of both dust and gas emission exist, namely IRS 48, HD 142527, and MWC 758, the continuum crescents appear to originate from the azimuthal and radial concentration of solid particles toward local maxima of the gas pressure (Casassus et al. 2015;Boehler et al. 2017;van der Marel et al. 2015). Such concentration is believed to result from the decoupling between dust and gas, and subsequent mi-gration of solid particles in the direction of the gas pressure gradient (see, e.g., Weidenschilling 1977;Birnstiel et al. 2013). This interpretation is supported by the fact that gas-dust decoupling is most efficient for grains with Stokes number around unity, which, for typical densities of protoplanetary disks, correspond indeed to grains that emit mostly at (sub-)millimeter wavelengths.
Whereas the motion of dust particles in gaseous disks is relatively well understood, the origin of gas pressure maxima is not. A commonly accepted explanation for the existence of asymmetric pressure maxima in protoplanetary disks is that they originate from large scale vortices resulting from hydrodynamic instabilities. Specifically, vortices can be generated by planets (Li et al. 2005;Zhu et al. 2014;Fu et al. 2014b) or viscosity transitions (Regály et al. 2013;Miranda et al. 2017). A massive planet or a sharp radial viscosity transition create a steep radial gradient in the gas density which triggers the Rossby Wave Instability (RWI, Lovelace et al. 1999;Li et al. 2000) which at nonlinear stage can produce large-scale vortices (Li et al. 2001(Li et al. , 2005Ou et al. 2007). Numerical simulations show that the formation and evolution of a vortex strongly depend on the disk viscosity. In particular, a vortex can form and survive for hundreds to thousands orbits only if the viscosity is low (α 10 −3 ∼ 10 −5 , Fu et al. 2014a). Taken at face value, this result is in conflict with the typical α 0.01 required to explain mass accretion rates found in protoplanetary disks (Hartmann et al. 1998). However, theoretical models suggest that disk viscosity might vary with the distance from the star, and that the innermost disk regions (<1 AU), those responsible for the accretion onto the star, might be more viscous than the disk regions probed by current spatially resolved observations (>10 AU). Lacking direct measurements (but see Flaherty et al. 2017), the level of viscosity in protoplanetary disks remains highly debated.
In addition to informing about disk viscosity, vortices might be pivotal in the formation of planets: they can speed up the dust coagulation and the formation of planetesimals (Barge & Sommeria 1995), stop the radial inward drift of dust particles (Weidenschilling 1977), and help overcome the fragmentation barrier (Brauer et al. 2008). Vortices generated at the edges of low ionization regions could perhaps lead to the formation of the planetesimals needed by core accretion model (Pollack et al. 1996) to form giant planets. Moreover, vortices generated by giant planets could perhaps control the formation and migration of rocky planets in their region of influence (see, e.g., Liu et al. 2018). Studying the presence of vortices in protoplanetary disks is therefore important to understand the architecture of the Solar systems and of the multitude of planetary systems discovered so far.
In this paper, we study the direct detectability of vortices from spatially and spectrally resolved observations of the molecular line emission at millimeter-wavelengths. Observations of dust continuum crescents support the vortex hypothesis and, when combined with molecular line data, provide indication of the efficiency of the dust concentration process. However, direct measurements of the gas vorticity are necessary to i) confirm that the observed dust crescents correspond to gas vortices and ii) study the mechanisms related to the vortex formation and evolution. Here, we focus on the scenario in which a vortex is created by a massive planet and simulate its density and velocity structure through high resolution 2D hydrodynamic calculations of the disk-planet interaction. We then perform radiative transfer calculation to generate synthetic images of the dust and CO emission, and process them through the ALMA simulator to properly account for observational noise. Synthetic observations are then analyzed to search for the signature of the vortex in the intensity moments of the gas intensity. We find that kinematic signatures of a vortex in a disk like IRS 48 can be detected by ALMA. We provide the observational set up required to achieve such a detection. Our analysis is complementary and extends the recent study of the effect of planet-disk interaction on the CO line emission presented by Pérez et al. (2018). Note that our analysis should be applicable to vortices excited via different mechanisms as described above, because the typical velocity variations associated with vortices are similar, with their magnitudes reaching a fraction of the local sound speed (Li et al. 2001).
The outline of this paper is as follows. In Section 2, we describe our numerical model setup and methods. In Section 3, we present the results of the numerical simulations and the synthetic images obtained. In section 4, we discuss and summarize our results.

Planet-disk interaction model
We use the two-fluid hydrodynamic 2D code LA-COMPASS (Li et al. 2005(Li et al. , 2008Fu et al. 2014b) to simulate the formation and evolution of a large scale vortex in a protoplanetary disk caused by the interaction with a massive planet. The hydrodynamic model is initialized using properties similar to that of the IRS 48 system, though our results apply to protoplanetary disks in general. We adopt a stellar mass M = 2 M (van der Marel et al. 2013) and a stellar effective temperature T = 9400 K (Follette et al. 2015). The initial gas surface density profile is Σ gas (r) = Σ gas,0 (r/r 0 ) −1 where Σ gas,0 = 1.73 g cm −2 and r 0 = 35 au. The inner and outer boundaries of the disk are 0.4r 0 = 14 au and 6.68r 0 = 233.8 au, respectively. Modeling by Bruderer et al. (2014) have indicated that the inner gaseous disk of IRS 48 is depleted between 0.4 − 20 au. Here we have chosen the initial inner gas disk boundary at 14 au and, with disk-planet interaction, the inner gas disk will be significantly depleted over time. Overall, we find that our results on vortices do not depend on the choice of the inner disk boundary. At the beginning of the simulation, gas and dust are well coupled with a gas-to-dust ratio of 100. Dust grains are modeled as compact spheres with a diameter of s d = 0.20 mm and internal density of ρ d = 1.2 g cm −3 . The corresponding Stokes number at r 0 is 0.022. The choice of the grain size has little effect on our results and is motivated by the fact that these are the grains that emit most the thermal radiation observed at λ = 2πs d ≈ 1 mm. In particular, we find that our specific choice of 0.2 mm grain size does not affect the gas density and kinematics of the vortex.
Following the results of van der Marel et al. (2013), we introduce a 10 M J mass planet at r 0 = 35 au from the star, so that µ = M p /M * = 5 × 10 −3 and the planet orbital period equals to about 146 years. We perform hydrodynamic simulations by assuming a locally isothermal equation of state, where the radial profile of the temperature is T (r) = T 0 (r/r 0 ) −1/2 and the corresponding radial profile of the sound speed is c s = c s,0 (r/r 0 ) −1/4 , with c s,0 = Ω 0 h 0 , h 0 = 0.085r 0 , and Ω 0 = GM /r 3 0 . This assumed disk temperature is consistent with that derived from the radiative transfer calculation discussed in the next section. We adopt a 2D uniform polar grid with 3072 cells in both the radial and azimuthal direction. Each cell has therefore a radial size of about 0.07 au, or h 0 /30. We adopt a Shakura-Sunyaev viscosity α parameter of 7 × 10 −5 (Shakura & Sunyaev 1973), which is the optimal value for generating vortices (Fu et al. 2014a). In this paper, we do not consider the feedback of dust on the vortex structure (Fu et al. 2014b) nor the planet migration. The viscous heating is also neglected in our simulation because the viscous dissipation rate D(R) = 9 8 νΣ GM R 3 (Frank et al. 2002;Chiang & Youdin 2010), Σ is gas surface density of disk. If the disk radiates as a blackbody σT 4 = D(R), the fluctuation of temperature caused by the viscous heating is small so that we can ignore the viscous heating even at the vortex region. Figure 1 shows the surface density profile of gas and dust in our two-fluid hydrodynamic simulations after 720 orbits of the planet, or about 10 5 years. The planet carves a deep circular gap in the gas distribution from about 25 au to 45 au (left panel). At the outer edge of the gap, the gas surface density sharply increases by more than a factor 10 within 5 au. This sharp increase causes the formation of a vortex that extends radially from 55 au to 75 au, and azimuthally from 163 • to 251 • (0 • is along the x axis direction). The center of vortex is located at about 65 au from the central star and at an azimthal angle of 207 • .
The dust distribution is strongly affected by the planet as well. It features a circular gap that is much wider than that is in the gas and extends out to a radius of about 85 au, i.e., well beyond the location of the vortex. Within this gap, our model predicts a strong accumulation of dust caused by the radial and azimuthal trapping of solid particles toward the center of the vortex. As a result of the disk-planet interaction and of dust migration toward gas pressure maxima, the dust-to-gas ratio increases by about a factor of ten at the outer edge on the dust gap, and by almost a factor of 100 within the vortex. Whereas at the center of the vortex the model predicts a dust-to-gas ratio close to unity, it is important to note that this value is likely overestimated due to the lack of dust feedback which might quench the concentration of solid particles (Fu et al. 2014b). Figure 2 shows the gas velocity v g along the r and φ directions relative to the Keplerian motion v K , as well as the Rossby number of the gas, The main perturbation induced by the planet is an anticyclonic rotation (clockwise, Ro < 0) centered at the position of the vortex and extending azimuthally for almost 1/4 of the disk between about 160 • and 250 • . In this region, the radial and azimuthal velocity of the gas varies between ±0.5 km s −1 and ±0.3 km s −1 , respectively. The velocity deviations from Keplerian motion are comparable to the local sound speed and, as discussed in Li et al. (2001), might produce shocks capable of forming spiral density waves themselves. For the purpose of this paper, a key aspect is that the velocity perturbations associated with the vortex are sufficiently large to be detected with spatially resolved spectroscopic observations of molecular line emission from protoplanetary disks. The detectability of the kinematic signatures of a vortex, and, in general, of the planet-disk interaction, is discussed in the following sections.

Obtaining Disk Temperature Distributions
We use the radiative transfer code RADMC-3D (Dullemond 2012) to calculate the temperature and emission of the disk perturbed by the planet following a procedure similar to that described in Jin et al. (2016).
In brief, we first convert the 2D gas surface density distribution generated by LA-COMPASS into a 3D cylindrical density structure defined as ρ(r, φ, z) = Σ(r,φ) √ 2πh(r) e −z 2 2h(r) 2 , where, as above, h(r) = c s (r)/Ω(r) = h 0 (r/r 0 ) 5/4 . Since the disk opacity at the wavelengths at which disk heating and cooling occur (i.e., optical and infrared) is controlled by (sub)micron grains that are coupled to the gas, it is appropriate to use the gas density for the calculation of the disk temperature. To calculate the disk temperature we assume a disk opacity proper of interstellar dust made of astronomical silicates (Weingartner & Draine 2001), organic carbonates (Zubko et al. 1998), water ice, with fractional abundances as in Pollack et al. (1994), and a grain size distribution n(a) ∝ a −3.5 between 0.01 − 1 µm. We include instead grains as large as 1 mm to calculate the millimeter-wave dust emission. In both cases, the gas-to-dust ratio is set to 100. We then perform a radiative transfer Monte Carlo simulation which leads to a Monte Carlo noise on the disk midplane temper-  ature below 3% as required to properly solve for the hydrostatic equilibrium (see, e.g. Isella & Turner 2018). Since the disk density and temperature are mutually dependent, a few iterations are required to reach a stable solution. We find that the disk reaches stable hydrostatic and thermal equilibrium at a midplane temperature T (r) ≈ 73K(r/r 0 ) −1/2 , or h(r) = 0.085(r/r 0 ) 5/4 . The disk temperature is to the first approximation az-imuthally constant (Figure 3), although the higher gas and dust density at the vortex causes a small azimuthal temperature variations (by a few degrees), whose effects might be difficult to detect. A more detailed discussion on the radial and azimuthal temperature perturbation caused by the disk-planet interaction can be found in Isella & Turner (2018).
We note that the 3D disk temperature obtained via RADMC-3D gives the typical two-layered structure as shown in Figure 3 (e.g., Chiang & Goldreich 1997). The radial temperature profile in the midplane, however, is lower than the initial temperature profile used in the 2D hydrodynamical simulations by about 20 − 30%. Furthermore, because the 3D disk temperature distributions were obtained via post-processing radiative transfer calculations, these quantities are presumably not selfconsistent with the realistic dynamic radiation hydrodynamic calculations. The exact impact of these simplifications will need to be understood in future studies.

Synthetic images and mock-up ALMA observations
Once the disk temperature is set, we calculate the synthetic disk emission in the 12 CO J = 3 − 2 line emission at 345.796 GHz and in the surrounding dust continuum emission. To calculate the dust emission we assume that millimeter grains are settled toward the midplane with respect to the gas. Their vertical distribution is assumed to be Gaussian with a pressure scale height h dust (r) = 0.1h(r) (Isella et al. 2016;Pinte et al. 2016). The CO emission is calculated assuming LTE (see, e.g., Weaver et al. 2018, for a discussion on the applicability of LTE to protoplanetary disks) and a fractional abundance χ( 12 CO) = 1.4 × 10 −4 relative to H 2 . The Doppler shift of the line emission is calculated using the gas velocity v g provided by the hydrodynamic simulation discussed above (see Figure 2). The spectral profile of the emission line is calculated accounting only for thermal broadening, i.e., no turbulence is included in the model. This choice is motivated by the low values of turbulence measured in some protoplanetary disks (Flaherty et al. 2017(Flaherty et al. , 2018Teague et al. 2018). We generate synthetic images for disk inclinations of 25 • and 50 • in a velocity interval between ±7 km s −1 .
As the final step in our modeling procedure, we use the 'simobserve' task in the Common Astronomy Software Applications package (CASA 1 McMullin et al. 2007) to generate mock-up ALMA observations that account for the effect of finite angular resolution and sensitivity. Our goal is to investigate the detectability of the kinematic signatures of the vortex in the line emission and the factors that affect such signatures. Since the vortex has a spatial extent of approximately 30 AU, or 0.25 at a source distance of 120 pc, we adopt an array configuration that provides a FWHM beam size of about 0.1 with natural weighting (CASA antenna file number 17). Lower resolution images would not spatially resolve the vortex, while observing the CO emission at higher resolution with good sensitivity would require very long observations.
The azimuthal position of the vortex is important because observations of the line emission measure the component of gas velocity along the line of sight. As a consequence, if the vortex is located along the apparent minor axis of the disk (θ = 90 • ), the v φ component of the gas velocity will mostly be perpendicular to the line of sight and will not generate any Doppler shift. In this model, the kinematic signature of the vortex would depend on the radial component of the gas velocity. Vice versa, when the vortex is close to θ = 180 • , its observable kinematic signature will depend on v φ , while the v r component along the line of sight would be close to zero. As such, we generate mock-up observations by rotating the synthetic models so that the vortex is either close to the projected minor axis of the disk (θ = 128 • ) or to its major axis (θ = 198 • and θ = 205 • ).
Beside the angular resolution of the observations and the location of the vortex, the detectability of the vortex kinematics would depend on the sensitivity and velocity resolution of the observations. Here, we generate mockup observations that provide velocity resolutions of 0.1 km s −1 and 0.05 km s −1 , and sensitivities between 3.5 K and 5.5 K.
In total, we generated a set of 5 mock-up observations whose observational parameters are listed in Table 1. Each mock-up observation consists of an image of the dust continuum and a continuum subtracted channel maps of the 12 CO emission.
A subset of 12 CO channel maps for model A are shown in Figure 4, while spatially integrated spectra for all models are presented in Figure 5. The peak of the line emission is optically thick (τ co > 500) across most of the disk including the vortex, while the dust emission reaches an optical depth of about 5 only at the center of the vortex. The line spectra present the characteristic double peak shape caused by the disk rotation. The line emission of model B is narrower and more intense because of the smaller inclination. In general, the blueshifted peaks have slightly higher intensities than the red-shifted peaks. As discussed in the next section, this is caused by the enhanced gas surface density in the vor- tex region. It is worth mentioning that we have investigated the observability of the vortex also in the 13 CO J=3-2 line, but given the much lower signal-to-noise ratio of the simulated observations (or, the much longer telescope time required to achieve the same signal-tonoise ratio of 12 CO observations), they do not provide any advantage in detecting the kinematic signature of the vortex, and therefore are not discussed here.
In addition to channel maps and spatially integrated spectra, we produced images of the zeroth, first, and second moment of the line emission, corresponding to the spectrally integrated line intensity, intensity weighted velocity centroid, and intensity weighted line width, respectively. These maps are shown in Figure 6 together with the image of the dust continuum emission.
The center of the vortex in the moment maps is marked as the intersection between the black dashed line and the black dashed ellipse. The zeroth and second moment maps of 12 CO look like butterfly with cross-shape feature, because the inclination of disk lets the channels with larger velocity gradient along the line of sight have better relative velocity resolutions (Adam 1990).
In the following section we discuss the observability of the vortex in each of these models.

KINEMATIC SIGNATURE OF THE VORTEX IN ALMA MOCK-UP OBSERVATIONS
At first glance, Figures 4-6 do not show any obvious signature of the vortex in the molecular line emission, despite the fact that it significantly perturbs the gas velocity. The first result of our investigation is therefore that vortices lead to rather small perturbations in the line emission and can hardly be identified by visually inspecting channel maps.
Here we search for the faint vortex signature based on the following idea. In the model of an unperturbed Keplerian disk, channel maps with the same absolute value of the radial velocity are symmetric with respect to the apparent minor axis of the disk (which corresponds to the projection of the angular moment vector on the plane of the sky). This means that, besides the opposite sign of the radial velocity, the blue and red shifted sides of the disk should be similar. We can therefore search for kinematic perturbations from Keplerian rotation by folding the line emission along the disk minor axis and subtracting, pixel by pixel, the red shifted emission from the blue shifted line intensity. Figures 7 shows this procedure applied to the moment maps of the ALMA mock up observations. Below we discuss the signature of the vortex in each of the five models listed in Table 1.

Model A: Standard Run
In this models, the disk has an inclination of 50 • similar to IRS 48 (Geers et al. 2007) and the center of vortex is located at a position angle of 198 • . The observations have a velocity resolution of 0.1 km s −1 and achieve a sensitivity of 5.2 K. In the zeroth moment map of 12 CO, the intensity of gas emission in center of vortex is lower than in the nearby region. This is in conflict with the fact that the surface density of gas peaks at the center of the vortex (see Figure 1). This phenomenon was also reported by Boehler et al. (2017) in the HD 142527 CO gas line observations and is caused by the fact that the optically thick line emission arising from the disk surface absorbs the continuum emission originating from deeper layers of the disk. Continuum subtraction then leads to artificial removal of line emission ). When both dust and gas emission are optically thick (as it happens at the center of the vortex), this situation become more severe and continuum subtraction might result in the removal of most of the line emission. In our model, however, the intensity of the 12 CO emission only drops slightly due to continuum subtraction and the detectability of the vortex itself is not affected. We have tested this by generating images of the non-continuum subtracted line emission. But since the results are very similar, we are not showing them here.
The difference between the blue shifted side and red shifted side of zeroth moment map of the 12 CO J = 3−2 line for model A is shown in the top row of Figure 7    negative outside the radius of the vortex and positive inside. This is due to the fact that the gas and dust density enhancement within the vortex changes how the disk is illuminated by the star, and consequently its temperature. In practice, the inner side of the vortex is more directly illuminated by the star and it is slightly hotter than the unperturbed disk. Vice versa, the outer part of the vortex is illuminated at a more slanted angle and it is slightly colder. The kinematic signature of the vortex can be seen by looking at the difference between the blue and red shifted sides of the first and second intensity moment. The vortex in the protoplanetary disk is an anticyclone, so it rotates clockwise compared with the anticlockwise rotation of the whole disk. This effect results in a detectable gradient in the velocity of the gas along the line of sight across the radial and azimuthal extent of the vortex. We can see that the blue shifted side of disk has a larger value of line of sight velocity than the red shifted side outside 75 au in all models, as shown by the second row of Figure 7 with the overall reddish background over the whole domain. (The implications of this effect will be explored in future studies.) On top of the reddish velocity difference background, however, there is a blue deviation(∼ 0.1km s −1 ) between 55 au to 75 au, 169 • to 198 • . This blue deviation is caused by the anticyclone vortex. The similar feature was also seen in the first moment maps of model B, D and E. In particular, gas located outward of the vortex center rotates slower than Keplerian and its velocity along the line of sight is about -0.15 km/s lower than the velocity at the center of the vortex. Vice versa, gas located inside the radius of the vortex center rotates faster than Keplerian, and its line of sight velocity is about 0.15 km/s faster than that of the center of the vortex. As a result, the velocity gradient along the vortex is about 0.3 km/s, i.e., about three times bigger than the velocity resolution of the observations. Note that such a velocity gradient can manifest itself both in the radial and azimuthal directions.
The vortex also has a signature in the line width as described by the second moment of the line intensity, as shown in the bottom panel of Figure 7. In this model, regions close to the center of the vortex have higher ve-locity dispersions which is caused by the fact that the observations do not spatially resolve the innermost part of the vortex and therefore the emission from gas regions moving at different velocities are all observed within one resolution element. At the resolution of 0.1 this effect is however rather minor (the velocity dispersion increases by only 0.05 km/s).
In order to further illustrate how the velocity fields affect the moment maps, we run another 5 models, which are marked as A' to E'. The surface density and temperature profile are kept the same as model A to E and we just changed the velocity input as purel Keplerian velocity field when we generate the synthetic images using RADMC-3D.    Figure 7. The surface density and temperature profiles of A' to E' are the same as A to E in Figure 7, but the velocity field input of RADMC-3D is now purely Keplerian velocity field. Compared with Figure 7, the blue and red deviations between 55 au to 75 au are much weaker in the middle row. Differences between the two figures indicate that the anticyclonic velocity field of a vortex can be observationally detected. they are quite a bit weaker and not as systematic. 2) Features in the second momentum maps of Figure 8 become too weak to discern. These comparisons indicate that velocity fields containing a vortex made by a planet can indeed be "visibly" different from a purely Keplerian velocity field, lending further support of detecting such features in actual observations.
In conclusion, the combination of the perturbation signatures observed in the moment zeroth, first, and second, can pinpoint the position of the vortex and yield the magnitude of its rotational velocity.

Model B: Effect of Disk Inclination
In the model B, we lower the disk inclination from 50 • to 25 • . Because of the lower disk inclination, the disk is more circular and the gas velocity along the line of sight is smaller. As a result, the signature of the vortex in the gas moment maps becomes less significant than in model A. It is obvious that the blue and red deviations are less significant than the model A (∼ 0.75km s −1 ). Lowering further the disk inclination would lead to a very weak and undetectable kinematic signature of the vortex. In the limiting model of a face-on disk, the component of the gas rotational velocity along the line of sight vanishes and the vortex becomes invisible. We compare the deviation of moment maps in B and B', we find that it is hard to see the the vortex features in B' because of the low inclination and the Keplerian velocity field in B'. This confirms that the low inclination is not favorable to find vortex features in the 2D hydrodyanmic models. In reality, vortices might cause a vertical circulation of the gas and perhaps they might be observed even at face-on inclination. However exploring this scenario would require 3D hydrodynamic models. In the opposite situation when the disk inclination is larger than 60 • , the molecular line emission arising from the inner disk regions might be blocked by the outermost flaring disk regions. This would also hamper the investigation of the vortex kinematics. Our analysis shows therefore that the best inclination range to detect kinematics signatures of the vortex spans between 30 • and 60 • .

Model C: Effect of the Azimuthal Position of Vortex
Because observations measure the component of the gas velocity along the line of sight, the azimuthal position of the vortex with respect to the apparent minor axis of the disk has an effect on its observability. In particular, velocity perturbations with respect to the Keplerian motion of vortices located close to the disk major axis (PA∼ 0 • ,180 • ) are proportional to V φ sin(i), while velocities perturbations of vortices located close to the disk minor axis (PA∼ 90 • ,270 • ) depend on V r sin(i), where V φ and V r are the tangential and radial velocities, respectively (Figure 2). In model C, we rotate the viewing angle so that the center of the vortex is at a position angle of 128 • . Despite the fact that part of the vortex now extends in the red-shifted part of the disk, its kinematic signature remains clearly visible in the first moment difference. Actually, the amplitude of the velocity difference is larger compared to model A and varies between ±0.25 km/s.

Model D: Effect of Thermal Noise
The thermal noise level of the observation is proportional to 1/ √ t, where t is the observation time. To investigate the effect of thermal noise on the observations, in model D we increase the integration time on source from 2 hr to 4 hr. The rms noise per channel consequently drops from 2.9 mJy beam −1 in model A to about 2.1 mJy beam −1 . Because the noise level has dropped by about 30%, the dust continuum and gas moment maps in model D achieve higher signal-to-noise and appear smoother compared to model A. However, the difference between blue shifted and red shifted part of the first and second moments changes only slightly. The comparison between D and D' is similar to the cases between A and A', but just less noisy.

Model E: Effect of Velocity Resolution
In model E, we increase the velocity resolution from 0.1 km/s to 0.05 km/s. Since the rms noise is proportional to 1/ √ ∆v, we extend the total observation time to 4 hours in order to compare with our standard run. The dust continuum and moment maps are shown in the fifth column of Figure 6, while differences between the blue and red shifted sides of the disk are presented in the fifth column of Figure 7. The comparison of E and E' in Figure 7 and Figure 8 are similar with A and A'. Overall, the kinematics signature of the vortex in model E is very similar to that observed in model A and D. The higher velocity resolution results in a smoother map of the second moment but the improvement is marginal and perhaps not worth the longer integration time.

DISCUSSION AND CONCLUSIONS
ALMA observations have revealed azimuthal asymmetries in several protoplanetary disks that are thought to trace large scale vortices produced by the Rossby Wave Instability excited by the planet-disk interactions. If this interpretation is correct, such vortices should perturb the motion of the gas, and consequently, the spectral profile of the gas emission from the affected regions. To investigate whether these perturbations could be de-tected, we have performed high-resolution 2D hydrodynamic and 3D radiative transfer simulations of a protoplanetary disk characterized by the presence of an anticyclonic vortex generated by the gravitational interaction between the disk and a 10 Jupiter mass planet located at 35 au from the central star. The stellar and planet properties were chosen to roughly match those of the IRS 48 system which is characterized by the most extreme azimuthal asymmetry discovered to date. The vortex traps the dust as previously discussed by Fu et al. (2014a) and produces a compact azimuthal asymmetry characterized by an azimuthal density variation larger than three orders of magnitude. As a reference, the azimuthal dust density variation measured in IRS 48 is larger than a factor of 100. By comparison, the gas density varies azimuthally only by a about a factor of two ( Figure 1). In addition to the dust trapping, the vortex induces variations in the gas velocity of the order of the gas sound speed (Figure 2) which can be probed be observing the molecular line emission.
We used CASA to generate images of dust continuum and 12 CO J=3-2 emission around the frequency of 345 GHz, that could be observed in ALMA band 7. We chose this band because it provides the best sensitivity to thermal gas and dust emission. We explored the observability of the vortex kinematic signature by varying observational parameters such as the disk inclination and position angle, the integration time, and the velocity resolution.
Overall, we found that despite the rather large perturbations in the gas velocity (Figure 2), the signature of the vortex on the 12 CO line emission is weak, and it can be easily overlooked by visually inspecting the channel maps or the intensity moments. This is due to the fact that (i) the azimuthal variation in the gas density at the radius of the vortex is much weaker than the azimuthal variation in the dust density, and (ii) the 12 CO line is mostly optically thick and therefore only weakly depends on the gas density. Nevertheless, we show that the velocity perturbations due to the vortex can be recovered by comparing the blue and red-shifted sides of the disk. We find, in particular, that the rotation of the vortex is clearly visible by folding the first moment map along the disk apparent minor axis and subtracting the blue-shifted emission from the red-shifted emission, or vice versa. This is a simple approach that does not require performing any detailed and time consuming forward modeling.
In order to be more concrete, we have used the disk parameters of IRS 48 system as a case study to investigate the detection feasibility by ALMA. By exploring different observational conditions, we established the follow-ing recipe to maximize the probability to directly detect vortices in protoplanetary disks using ALMA observations.
1. To detect the kinematic signature of a vortex, it is necessary to spatially resolve it across a few resolution elements. Although a careful analysis of the spatially integrated line profile might also reveal velocity perturbations, our simulations show that the effect on spatially integrated spectra is generally too weak to be unambiguously attributed to a vortex. Given the angular size of the dust crescents observed so far, we find that the observations should achieve an angular resolution of about 0.1 , corresponding to spatial scales of about 10-20 au at the distance of nearby star forming regions.
2. To measure the velocity gradient across the vortex, the observations should detect the line emission with a peak signal-to-noise ratio larger than 15. Because the temperature of protoplanetary disks at a few tens of au from the central star ranges between 20-100 K, the observations should achieve a brightness temperature sensitivity of 1-7 K if the line under consideration is optically thick and it fills the beam entirely (i.e., there is no beam dilution). Achieving such a sensitivity with a beam size of 0.1 and a velocity resolution of 0.1 km/s would require between about 1 hr (to reach an rms of 7 K) and 48 hr (to reach an rms of 5 K) of integration time on source with ALMA. Observing disks around hot stars is therefore highly preferable. Furthermore, increasing the angular resolution to 0.08 would imply incrementing the observing time by a factor of 2.5, therefore making such an observational program hardly feasible. Observations of more optically thin lines, such as, for example, 13 CO J= 3 − 2 could potentially better trace the perturbations in the gas density caused by the vortex. However, we find that achieving the required s/n ratio on this line would require more than 15 hr of integration time on source.
3. To measure the velocity gradient across the vortex, the observations should have a velocity resolution at least 3× higher than the amplitude of the velocity perturbations induced by the vortex. We find that the velocity variations associated with the vortex can be comparable to the sound speed in the gas, which, for typical temperatures values, varies between 0.3 -0.6 km/s. Our simulations show that a velocity resolution of 0.1 km/s provides the best compromise between the need to spatially resolve the vortex kinematics and those of achieving a sufficiently high S/N ratio.
4. Finally, we find that the disk inclination plays an important role in the capability to investigate the vortex kinematics. A low disk inclination (i < 30 • ) will cause a smaller projection of the gas velocity along the line of sight and weaker vortex signatures. A high disk inclination (i > 60 • ) might cause the emission from the vortex to be absorbed by the outermost disk regions. The best target candidates should therefore be those disks characterized by moderate inclinations. In the same time, we find that the azimuthal position of vortex in the moment maps is related to the detection the asymmetric feature in the gas line moment and the effect is still unclear. This may be figured out in our future work.
One future study is to investigate the signatures of vortices in a background disk flow which is eccentric. Such configurations can be produced if the planet is quite massive or the disk contains a binary system.