The efficiency of dust trapping in ringed proto-planetary discs

When imaged at high-resolution, many proto-planetary discs show gaps and rings in their dust sub-mm continuum emission profile. These structures are widely considered to originate from local maxima in the gas pressure profile. The properties of the underlying gas structures are however unknown. In this paper we present a method to measure the dust-gas coupling $\alpha/St$ and the width of the gas pressure bumps affecting the dust distribution, applying high-precision techniques to extract the gas rotation curve from emission lines data-cubes. As a proof-of-concept, we then apply the method to two discs with prominent sub-structure, HD163296 and AS 209. We find that in all cases the gas structures are larger than in the dust, confirming that the rings are pressure traps. Although the grains are sufficiently decoupled from the gas to be radially concentrated, we find that the degree of coupling of the dust is relatively good ($\alpha/St \sim 0.1$). We can therefore reject scenarios in which the disc turbulence is very low and the dust has grown significantly. If we further assume that the dust grain sizes are set by turbulent fragmentation, we find high values of the $\alpha$ turbulent parameter ($\alpha \sim 10^{-2}$). Alternatively, solutions with smaller turbulence are still compatible with our analysis if another process is limiting grain growth. For HD163296, recent measurements of the disc mass suggest that this is the case if the grain size is 1mm. Future constraints on the dust spectral indices will help to discriminate between the two alternatives.


INTRODUCTION
The Atacama Large Millimeter/submillimeter Array (ALMA) is revolutionising our understanding of protoplanetary discs thanks to its unprecedented angular resolution. When imaged at high resolution, most (though not all, Facchini et al. 2019;Long et al. 2019) discs show a rich morphology of structures, in terms of crescents (van der Marel et al. 2013), spirals (Pérez et al. 2016) and rings (ALMA Partnership et al. 2015;van der Plas et al. 2017;Fedele et al. 2017Fedele et al. , 2018Dipierro et al. 2018;Clarke et al. 2018). This latter category in particular is the one occurring most frequently, as shown spectacularly by the high-resolution DSHARP campaign (Andrews et al. 2018) rosotti@strw.leidenuniv.nl and by other efforts with large disc samples (Long et al. 2018;van der Marel et al. 2019).
These rings are interesting for many reasons. Firstly, they are thought to be dust traps, where the dust stops drifting towards the star and accumulates. In this sense, they could be the solution to the long-standing problem of how to reduce the importance of radial drift, which if unimpeded would deplete discs on a very short timescale (Takeuchi & Lin 2005;Brauer et al. 2007), leaving little solid mass to form the rocky planetary cores (Greaves & Rice 2010;Manara et al. 2018;Rosotti et al. 2019). Secondly, the most likely interpretation for the origin of these rings is that a population of young planets is already present at these early stages; the rings are therefore a tool to study the masses and locations of these young planets (Rosotti et  Independently from their formation mechanisms, there is another sense in which the commonly imaged rings are important: they provide us with new windows to probe disc physics. One example is the magnitude of the turbulence, another long standing problem in planet formation (Lynden-Bell & Pringle 1974), typically parametrised through the dimensionless α parameter (Shakura & Sunyaev 1973). The amount of turbulence is a crucial parameter regulating, just to name a few examples, the efficiency of gas accretion onto the star and forming planets (Bodenheimer et al. 2013), how discs responds to planets (Kley & Nelson 2012;Zhang et al. 2018), the vertical mixing of molecular species (Semenov & Wiebe 2011), the importance of fragmentation for dust evolution (Ormel & Cuzzi 2007;Birnstiel et al. 2012), and many other processes. Because turbulence in proto-planetary discs is expected to be highly sub-sonic, degeneracies with the disc temperature mean however that the turbulence is proving very difficult to constrain directly (Teague et al. 2016;Flaherty et al. 2017;Teague et al. 2018b) from broadening of emission lines. This is where rings come to the rescue, since the dust is also subject to turbulence and can also be employed as an observational tracer of turbulence. To be more precise, in this way it is only possible to measure α/St rather than α, where St is the so-called Stokes parameter quantifying the aerodynamic coupling between gas and dust. Pinte et al. (2016) showed that turbulence in the vertical direction can be measured by quantifying the degree of smoothing of the emission profile along the disc semi-minor axis. With a complementary method, Dullemond et al. (2018, hereafter D18) used the radial width of dust rings to put constraints on the turbulence in the radial direction. Unfortunately, with their methodology a turbulence measurement requires comparing the radial width of features in the dust and gas surface densities, but only data regarding the former were available. Therefore, they were able only to identify a range of permitted values and not to measure the value of α/St.
Thankfully, there is a way forward to improve on the analysis of D18. In addition to the continuum emission, ALMA is also revolutionising our view of the gas disc. Thanks to the combination of ALMA extreme sensitivity and spatial resolution, plus new techniques developed to make use of these innovative data, the gas rotational velocity can now be studied with high precision. As highlighted by a few spectacular examples (Teague et al. 2018a;Pinte et al. 2018a;Teague et al. 2019;Dullemond et al. 2020), there is now a growing realisation that most discs are not in perfect Keplerian rotation, with deviations amounting to a few percent. In our current understanding of disc dynamics, these deviations in the gas velocity are the very reason why we observe rings in the dust distribution (Whipple 1972).
Applying these techniques to the DSHARP data opens up the possibility of measuring the turbulence by combining information about the dust and the gas. In addition, measuring the width of gas structures directly confirms that these structures are dust traps if the gas width is larger than the dust width (see discussion in D18). Performing these measurements is the goal of this paper. As a proof-of-concept, we focus here on the two discs with most prominent structures in gas and continuum, namely HD163296 and AS 209.
The paper is structured as follows. Section 2 introduces the method we use to measure the dust-gas coupling and the width of gas structures. We then present our results and discuss possible caveats in section 3. Finally, we draw our conclusions in section 4.

METHODS
Our analysis is based on the publicly available data from the DSHARP ALMA large programme 1 , focusing on the discs of HD 163296 and AS 209 (Andrews et al. 2018;Isella et al. 2018;Guzmán et al. 2018). Our goal is to measure the dustgas coupling α/St. The new aspect of this paper is that we use the 12 CO data-cubes to measure the slope of the deviation from Keplerian rotation of the gas in the proximity of the continuum peaks. As we will show, in combination with the width of the dust rings, this can be used to yield a measurement of α/St. Additionally, from the same data, we can also measure the width of the rings in the gas distribution.

Calculating the rotation curve
To calculate the rotation curve we follow the method described in Teague et al. (2018c) which is broken into two aspects: the measurement of the 12 CO emission surface in order to correctly deproject the data into annuli, and secondly the inference of v φ in each annulus.
Measuring the emission surface is done by fitting the map of line centres, made using bettermoments (Teague & Foreman-Mackey 2018), which fits a quadratic curve to the pixel of peak intensity and two neighbouring pixels, with a Keplerian rotation pattern including a correction for the 3D geometry of the disk, as described in Keppler et al. (2019) using the eddy Python package (Teague 2019). As AS 209 has cloud contaminated regions, we additionally use the method described in Pinte et al. (2018b) which is less sensitive to cloud-contamination to verify the emission surfaces we obtain. We find excellent agreement with previous determinations in these sources (Teague et al. 2018c;Isella et al. 2018).
Using these emission surfaces we deproject the data into disk-centric coordinates, (r, φ), and divide them into annuli with a width of 1/4 of the beam major axis. We stress that this binning does not remove the spatial correlation between nearby annuli, however minimises the impact of Keplerian shear across the beam when measuring v φ . Within each annulus the projected component of v φ is assumed to vary as a function of azimuth, v φ, proj = v φ · cos φ · sin i, where φ is measured from the red-shifted major axis of the disk and i is the disk inclination. Using eddy (Teague 2019), v φ is inferred by finding the value of v φ which allows for the spectra to be shifted back to a common line center (the systemic velocity). More details of the exact fitting procedure can be found in Teague et al. (2018c). This procedure is repeated for each annulus, yielding v φ as a function of radius. In order to measure the deviation δv φ = v φ − v K from the Keplerian value v K , we fit the v φ profiles with a double power-law profile. While a single power-law would be sufficient for a purely Keplerian rotation profile, the inclusion of radial pressure gradients and changes in the emission height with radius result in systematic deviations from a pure v K Keplerian profile. An important fact to stress is that in reality we do  not know the true Keplerian value, because we do not know precisely enough the stellar mass. As a surrogate, we employ the deviation from the double power-law fit, implying that there might be a constant, unknown offset (with magnitude of a few per cent) between our δv φ and the deviation from Keplerian -but for simplicity, in the rest of the paper we will often refer to δv φ as "deviation from Keplerian". Our analysis is not affected by this offset since we will show that it relies on the derivative.

Using the rotation curve to measure α/St and gas widths
By studying δv φ , we can now determine α/St. Assuming that the disc is razor-thin, we show in appendix A that, to first order in r − r 0 , the dust surface density is a Gaussian with width w d . The following expression (see Equation A12) links the width w d with the dust-gas coupling and the observables: As we derive in appendix A1, the same observables we use to measure α/St can also be used to measure the width of the gas rings w g , using the following expression (see Equation A15): (2) Figure 1 illustrates graphically this method, showing how Gaussians of different width produce a different gradient in the deviation from Keplerian rotation. In the expressions above, r 0 is trivially obtained as the location of the dust ring. We already discussed in the previous section how we derive the rotation curve and we discuss more in detail in section 3.1 how we measure the slope. For what concerns the dust width w d , we measure it from the continuum images as in D18. We discuss in the next paragraph the last parameter, the gas temperature.
The razor-thin model should be considered only as pedagogical since it is very well known that the CO emission comes from an elevated surface. Therefore, a proper modelling should take into account the disc vertical structure. We show in appendix B that in practice this can be accounted for using the gas temperature at the emitting layer, instead of the midplane temperature, for computing c 2 s in Equation 1 and Equation 2. At the emitting layer, the temperature can be estimated with high precision from the 12 CO data using the peak brightness temperature given the high optical depth of 12 CO; we therefore use directly these values from the data.

Derived values
We show in the middle panel of Figure 2 the rotation curves extracted from the data. For comparison we show also the continuum profiles on the top panels. We note that in the vicinity of the continuum peaks δv φ decreases, as expected in the case of a pressure maximum. The curves are also reasonably well described by a constant slope, with the most spectacular example in the vicinity of the B100 peak of HD 163296, with a relatively extended radial range. Overall there is therefore reasonable agreement between the dust structure and the rotation curve. That being said, we note that in the B100 peak of HD 163296 and in the B120 peak of AS 209 the region with a decreasing δv φ is not symmetrical with respect to the continuum peak, as one might instead expect. We discuss possible explanations for this in section 3.4.
To measure the slope, we find that the raw derivative (bottom panel) can be relatively noisy, given the data spatial resolution and signal to noise; we thus perform a linear fit which is more robust towards the noise and correctly accounts for uncertainties. We list in Table 1 the measured gradients ∂δv φ /∂r.
In Table 1 we list also the gas temperature T B close to the peak, measured using the peak brightness temperature (see section 2), and the Keplerian velocity v kep (estimated via the double power-law fit). To compute the sound speed, we assume a mean molecular weight µ = 2.37. We also list the dust widths w d measured by fitting Gaussians in the azimuthally averaged continuum profiles (Andrews et al. 2018;Huang et al. 2018), which agree in all cases with those measured by D18, except for B155 which was not analysed by them (in this case we fitted the continuum emission between 150 and 158 au following the procedure in D18). As already argued by D18, the finite radial extent of the dust rings implies that some mechanism is stirring the dust in the radial direction.  Table 1. Values derived from the observations for the five pressure traps analysed in this paper with 1σ uncertainties. (1) (2) (3) (4) (5) (6) (7) Ring We now quantify the efficiency of this stirring mechanism. We report in Table 1 the resulting α/St values, deduced using Equation 1, and we plot graphically the constraints in the α − St plane in Figure 3. In general, we find that the degree of coupling of the dust is relatively good, with an average α/St ∼ 0.1. We discuss the implications of these results in section 4.
Finally, we also measure the gas width w g using Equation 2. Although D18 did not analyse the gas data to measure gas widths, they identified possible lower and upper limits based on different physical criteria. Our values fall in-side this range except for B100, in which case we find a wider gap than their upper limit -this might be because they employed the full width half-maximum to set a constraint on the possible width. We note that in all cases the gas widths are larger than the dust ones, providing support to the idea that these structures are pressure traps.

Deriving an α value
Because the dust dynamics depends only on the ratio α/St, so far we have been unable to measure individually the two  We mark with the stars the fragmentation limit. The solid line marks the region where fragmentation due to turbulence never operates and therefore another process must limit the grain size. In the region marked with the dashed line, fragmentation due to turbulence is possible, but because of the lower St another process is limiting the grain size more efficiently than fragmentation. For HD163296, we use the disc surface density profile of Booth et al. (2019) to set a constraint on St, assuming a grain size of 1 mm. 8 × 10 −3 6 × 10 −4 68 2 × 10 −3 6 × 10 −4 B100 4 × 10 −3 7 × 10 −4 57 3 × 10 −3 10 −4 B155 4 × 10 −3 10 −3 53 3 × 10 −3 10 −4 AS 209 B74 10 −2 10 −3 0.3 B120 10 −2 2 × 10 −3 0.3 Notes (1) Value of α if the grain size is set by fragmentation (2) Value of α below which fragmentation does not limit grain size (3) Gas surface density derived from observations (4) Stokes number with the previous value of the surface density, assuming a grain size of 1 mm. (5) α value obtained combining the previous constraint on St and our measurement of α/St.
parameters. To break the degeneracy between them, we need some information on St. As shown by Birnstiel et al. (2012), in models of dust coagulation the dust grain size is limited by either fragmentation or radial drift. Since the rings are pressure maxima, the dust is not rapidly drifting in their vicinity; therefore, it is plausible to assume that the grain size should be set by fragmentation (e.g., see Bae et al. 2018). In this case, using Equation 3 of Birnstiel et al. (2012): where u f is the fragmentation velocity and c s the sound speed in the midplane D18. We report in Table 2 the resulting α frag values when assuming a value of the fragmentation velocity of 10 m s −1 (note the linear dependence on this parameter). We also added these values as the stars markers in Figure 3. Conventionally, α is assumed to lie in the approximate range [10 −4 , 10 −2 ]; the values we find are towards the upper end of this range.
We cannot know if the grain size is indeed set by fragmentation, but we argue that α frag is an upper limit on the value of α. This is because, if α was greater than α frag , fragmentation would limit the grain size to a St not compatible with our measurement of α/St. Instead, it is acceptable that α is lower than α frag if we invoke the presence of another process (e.g., bouncing, or a residual level of radial drift) setting the grain size, and that this process limits the grain size to a value smaller than what would be set by fragmentation. This is marked with the solid and dashed lines in Figure 3 (see next paragraph for the difference between solid and dashed).
Lastly, it should be noted that the relative velocity in grain collisions increases with St but only until St = 1 (Ormel & Cuzzi 2007); increasing St further decreases the relative velocity. This has two important practical consequences. Firstly, Equation 3 assumes that the relative velocity always increases with St and therefore is only valid for the case St < 1; we have verified a posteriori that in all cases we obtain an acceptable solution, i.e. with St < 1. Secondly, if α is sufficiently low, even for St = 1 the relative velocity is lower than the fragmentation velocity and therefore the fragmentation limit never applies. We define α min,frag as this critical value of α; its value is α min,frag = 2/3 u 2 f /c 2 s (note that in this case the dependence on u f is quadratic); as previously, we compute these values assuming a value of the fragmentation velocity of 10 m s −1 and report them in Table 2. These values are the separation between the solid and dashed line in Figure 3. The significance of the solid region is that, even without our measurements of α/St, in this region we must invoke another process limiting grain growth, or growth would proceed unimpeded.
To summarise, there are two possible scenarios; in the first, the grain size is set by fragmentation and α takes the value we report as α frag . It is interesting to note that for HD163296 these values are incompatible with the upper limit reported by Flaherty et al. (2017) of 3 × 10 −3 , especially for B67, while for B100 and B155 a slight reduction in fragmentation velocity could still make the two measurements compatible. In the second scenario, another process is setting the grain size and α can take any value smaller than α frag . In this case, depending on the value of α, we can also further argue that if α min,frag < α < α frag this process must be more efficient than fragmentation.

Which Stokes numbers are compatible with grain growth?
Because St is linked to the grain size, it is worth asking what values are compatible with the well known results of dust grain growth in proto-planetary discs (see Testi et al. 2014 for a review); in turn, this sets a constraint on α given the measurements of α/St that we present in this paper. The Stokes number in the Epstein regime can be expressed as: where a is the grain size and we have assumed a dust bulk density of 1 g cm −3 . To put some constraint on St, we thus need measurements of the gas surface density. For HD163296, such a measurement is provided by the recent detection of 13 C 17 O . Given the non-detection of this disc in the HD 1-0 transition (Kama et al. 2019), the gas surface density of this disc is reasonably well constrained, since increasing it would make it incompatible with the non-detection of HD and gravitationally unstable (in contrast with the lack of observed spiral arms), while lowering it would make it incompatible with the detection of 13 C 17 O. We list in Table 2 the surface density Σ gas at each ring location from the disc model of Booth et al. (2019); we then use this surface density to compute the resulting Stokes number St Σ . These values are plotted as the triangles in Figure 3. The resulting α (which we list as α Sigma in the table), once combined with our measurements of α/St, seem to exclude the possibility that α is high and that the grain size is set by fragmentation. In order to make fragmentation the process setting grain size, we would need a grain size larger by a factor 10 to increase α Sigma , or a fragmentation velocity smaller by a similar amount to decrease α frag (or a combination of both).
For AS209 instead, Favre et al. (2019) reports significantly lower values of the surface density from CO isotopologues observations. At face value, this would point towards the need for much larger St (∼ 0.5), that are not compatible with our constraints since they would imply a value of α greater than α frag . Additionally, this result would be at odds with attempts at modelling the dust structure in AS 209 as due to disc-planet interaction, that consistently highlighted the need for low viscosity values in this particular disc (Fedele et al. 2018;Zhang et al. 2018). However, it is well known that due to carbon depletion CO-derived disc masses are generally underestimated in T Tauri stars (e.g., Miotello et al. 2017) and therefore it is likely that the true disc mass is significantly higher than the estimate of Favre et al. (2019). For this reason, we do not plot these constraints in Figure 3, nor indicate them in Table 2.
A caveat of this analysis is that we have simply assumed that the grains are 1 mm. A better estimate would be needed, but we note that, even if ALMA has now been in operation for a few years, very few discs have been studied with sufficient spatial resolution at multiple wavelengths to study the grain properties in the rings, and therefore we have limited information on the grain size. For example, for HD163296 the spatial resolution of Guidi et al. (2016) was not enough to resolve the rings; Dent et al. (2019) was not able to place constraints on the grain size due to the degeneracy between grain growth and optical depth, and the limited difference in wavelength between band 6 and band 7. While polarisation could in principle be an alternative way of placing constraints on the grain size, the analysis of Ohashi & Kataoka (2019) shows that the potentially high optical depth of the rings in the sub-mm makes the grain size unconstrained, highlighting the need for data at longer wavelengths. Future high-resolution studies will provide constraints on the grain size, in this way further breaking the degeneracy between α and St.

Caveats
As we highlighted when describing Figure 2, the rotation curve we derive from the data is broadly consistent with the dust continuum structure. However, we wish to discuss in this section two effects that are in partial tension with the dust continuum. The first one has been already introduced in section 3.1, namely that for B100 in HD163296 and B120 for AS 209 the continuum peak is not located at the center of the radial range over which δv φ decreases. The problem is particularly severe for AS 209, in which case δv φ starts decreasing only at the location of the peak in the continuum. For AS 209, this effect has already been noted by Teague et al. (2018c). We note that, because we do not know the true value of the Keplerian velocity, there is some uncertainty in the exact location of the pressure maximum (i.e., a constant vertical offset in δv φ would shift radially the location where the pressure gradient crosses zero, see Keppler et al. 2019 for an example). While this could be enough to explain the inconsistency for B100 in HD163296, it does not appear to be the case for B120 for AS 209, since a vertical offset would not change the fact that δv φ increases (i.e., has a positive derivative) at radii smaller than inside the continuum peak.
There are two reasons that could explain this discrepancy. The first one is what we discuss in appendix B, namely the effect of a local variation in the height of the emission surface. Figure B1 shows an example where the deviation from Keplerian starts decreasing only outside the location of the pressure maximum. The second reason is the possibility that the gap structure is intrinsically not symmetrical. Within the framework of this paper, we cannot account for an asymmetry because, to first order in r − r 0 , the gap structure is symmetrical by construction, but this is a possibility we plan to investigate in future papers. The issue is of interest because hydro-dynamical models of disc-planet inter-action tend to predict a steeper pressure profile inside the pressure maximum than outside. It is also suggestive that observational studies of transition discs show  a similar difference in the dust distribution on the two sides of the pressure maximum. Therefore, both effects go in the same direction. With the current data, it is not currently possible to disentangle between them.
The second caveat we wish to discuss concerns the magnitude of the deviation from Keplerian. We can hypothesise that at sufficient distance from the pressure bump the pressure profile goes back to some smooth, negative slope and therefore the rotation curve is sub-Keplerian. As already discussed in this paper, we cannot measure this unperturbed slope because of the uncertainties in the mass of the star, as well as the height of the emission surface. However, we can write that in the unperturbed region the deviation from Keplerian should be of order δv ), where we have called γ the logarithmic slope of the unperturbed surface density profile, and in the last passage we have ignored factors of order unity. For the pressure bump to be a pressure maximum, the change in v φ induced by the bump needs to be high enough for the rotation curve to transition from sub-to super-Keplerian rotation. Given a radial range ∆r of the variation and a slope m = ∆δv φ /∆r, this means that the total variation m∆r induced by the pressure bump needs to be larger (in absolute value) than the unperturbed value (c s /v K ) 2 (note that, because of the analysis of appendix B, it does not matter whether we perform this comparison in the midplane or at the emission surface). We list in Table 1 the ∆r we employ and the ratio between the total variation and (c s /v K ) 2 . It can be seen that, whereas in AS 209 the total variation is comfortably higher than what is needed to produce pressure maxima, mostly because of the larger slopes measured from the data, for HD163296 the total variation is barely larger than the constraint; for B155, the slope we measure is not large enough to produce a pressure maximum. Even for the other two rings, this does not leave much free room to have a pressure maximum. This could be because the pressure bumps in this disc are indeed only shallow maxima, or because γ is small (i.e., the unperturbed surface density is very shallow), or it could be because of additional physics we are missing in our analysis.
It should be remarked that the datasets we analysed were not designed to study gas kinematics; for example they have a quite limited spectral resolution (0.35 km/s native resolution, with the actual resolution roughly two times worse due to Hanning smoothing). Ultimately, separate datasets explicitly designed to study kinematics are needed to re-assess in future works the caveats we describe here.

DISCUSSION AND CONCLUSIONS
In this paper we presented a unique approach to analyse the disc kinematics and, in comparison with the sub-mm continuum emission, measure the α/St ratio at the ring centres, in this way providing constraints on the level of turbulence and the dust-gas coupling. Moreover, our method also measures the width of gas pressure bumps. Our results confirm that the structures in the gas are larger than in the dust and that α/St < 1, thereby providing evidence that the rings now ubiquitously imaged are dust traps, at least for the two discs studied here.
At the same time, our results also imply a relatively large value of α/St, with a typical value of 0.1. Our constraints are illustrated in Figure 3 and they imply that we can reject a scenario in which the disc is characterised by low turbulence (e.g., α = 10 −4 ) and the grains have large Stokes numbers (e.g., St = 0.1). On the contrary, our results imply that if the grains have large Stokes numbers then the disc must also be very turbulent (at least in the radial direction), for example in the case limited by fragmentation (see α frag values in Table 1). This case also constitutes an upper limit on the value of α. On the other hand, such high values of the turbulence appear to be in tension with the lack of a direct detection (Flaherty et al. 2017), at least for the case of HD 163296. We note that the analysis carried by Flaherty et al. (2017) assumed homogeneous, isotropic turbulence, and it is possible that this discrepancy may be solved by relaxing this assumption. The other possibility is instead that the discrepancy points to a different physical regime, namely that the grain size in these discs is not set by fragmentation (α smaller than α frag in Table 2). This possibility is more in line with recent theoretical work proposing that accretion is mostly driven by winds launched by the magnetic field. This option is compatible with our data and, as we discuss in section 3.3, also with recent measurements of the disc mass  for HD163296. For AS 209 instead, our results are in tension with the low disc mass inferred from C 18 O observations (Favre et al. 2019), although it is likely that carbon depletion is severely affecting those measurements.
Lastly, it should be noted that future high-resolution gas observations of optically thin lines (which for the two discs we analysed will be conducted by the approved Large Programme MAPS) will test our measurements of gas widths and will provide an independent constraint. We remark that the method we propose here is cheaper in terms of observing time since it requires brighter, optically thick lines. A validation of our method would then allow to apply it to a larger disc sample. The blue line depicts the case of a constant z/r of the emission surface (note that this means that height of the emission surface, measured in scale-heights, slightly decreases with radius because the scale-height increases with radius), while in the orange line we consider a local increase in the height of the emission surface. This leads to a morphological change in the shape of the rotation curve, but it does not affect the average value.