Probing the protoplanetary disk gas surface density distribution with 13 CO emission
Anna Miotello, Stefano Facchini, Ewine F. van Dishoeck, Simon Bruderer
AAstronomy & Astrophysics manuscript no. rad_profiles_Miotello_arxiv c (cid:13)
ESO 2018September 5, 2018
Probing the protoplanetary disk gas surface density distributionwith CO emission
A. Miotello , , S. Facchini , E. F. van Dishoeck , , and S. Bruderer European Southern Observatory, Karl-Schwarzschild-Str 2, D-85748 Garching, Germany Leiden Observatory, Leiden University, Niels Bohrweg 2, NL-2333 CA Leiden, The Netherlands Max-Planck-institute für extraterrestrische Physik, Giessenbachstraße, D-85748 Garching, GermanySeptember 5, 2018
ABSTRACT
Context.
How protoplanetary disks evolve is still an unsolved problem where di ff erent processes may be involved. Depending onthe process, the disk gas surface density distribution Σ gas may be very di ff erent and this could have diverse implications for planetformation. Together with the total disk mass, it is key to constrain Σ gas as function of disk radius R from observational measurements. Aims.
In this work we investigate whether spatially resolved observations of rarer CO isotopologues, such as CO, may be goodtracers of the gas surface density distribution in disks.
Methods.
Physical-chemical disk models with di ff erent input Σ gas ( R ) are run, taking into account CO freeze-out and isotope-selectivephotodissociation. The input disk surface density profiles are compared with the simulated CO intensity radial profiles to checkwhether and where the two follow each other.
Results.
For each combination of disk parameters, there is always an intermediate region in the disk where the slope of the COradial emission profile and Σ gas ( R ) coincide. In the inner part of the disk the line radial profile underestimates Σ gas , as CO emissionbecomes optically thick. The same happens at large radii where the column densities become too low and CO is not able to e ffi cientlyself-shield. Moreover, the disk becomes too cold and a considerable fraction of CO is frozen out, thus it does not contribute to theline emission. If the gas surface density profile is a simple power-law of the radius, the input power-law index can be retrieved withina ∼
20% uncertainty if one choses the proper radial range. If instead Σ gas ( R ) follows the self-similar solution for a viscously evolvingdisk, retrieving the input power-law index becomes challenging, in particular for small disks. Nevertheless, it is found that the power-law index γ can be in any case reliably fitted at a given line intensity contour around 6 K km s − , and this produces a practical methodto constrain the slope of Σ gas ( R ). Application of such a method is shown in the case study of the TW Hya disk. Conclusions.
Spatially resolved CO line radial profiles are promising to probe the disk surface density distribution, as they directlytrace Σ gas ( R ) profile at radii well resolvable by ALMA. There, chemical processes like freeze-out and isotope selective photodisso-ciation do not a ff ect the emission, and, assuming that the volatile carbon does not change with radius, no chemical model is neededwhen interpreting the observations.
1. Introduction
Protoplanetary disk evolution adds to the construction of the hostyoung star and simultaneously provides the material for the for-mation of planets and large rocky bodies. Mass accretion ontothe central star and planet formation are therefore connected pro-cesses where the gaseous and solid disk components play impor-tant and complementary roles (see Armitage 2015, for a review).It is generally assumed that gas is the main disk component ac-counting for 99% of its mass, while only 1% is in the form ofdust. Furthermore, the gaseous component is also what drivesdisk evolution and thus the disk structure at di ff erent stages.An open question is what the relative importance is of di ff erentmechanisms transporting the material onto the star and dispers-ing the disk within few Myr. Disk evolution leads to mass accre-tion onto the central star (Hartmann et al. 2016), but part of thedisk content can be dissipated by high-energy radiation drivenwinds or magnetic torques (Alexander et al. 2014; Gorti et al.2016; Bai 2016) and external processes such as encounters (e.g.Clarke & Pringle 1993; Pfalzner et al. 2005) and external pho-toevaporation (e.g. Clarke 2007; Anderson et al. 2013; Facchiniet al. 2016). Depending on which process is dominant, the diskstructure will be considerably di ff erent. In particular this will im-pact the distribution of the gas as a function of the distance fromthe star, which is described by the surface density distribution, Σ gas . Morbidelli & Raymond (2016) discuss the di ff erence in theresulting disk structures between disk wind models and the stan-dard viscously evolving α -disk model. Suzuki et al. (2016) findthat the inner disk can be heavily depleted as winds e ffi cientlyremove angular momentum from the disk and cause large accre-tion onto the star. On the other hand, a di ff erent disk wind modelby Bai (2016) predicts a surface density profile that scales in-versely with the radius in the innermost 10 au, similarly to whatis expected in the α -disk case. Reliable observational measure-ments of Σ gas as function of radius would therefore be key to un-derstand disk evolution and the relative importance of di ff erentprocesses, as well as how planet formation occurs (Morbidelli &Raymond 2016).Successful attempts have been made to constrain the diskdust surface density Σ dust from high S / N spatially resolved mm-continuum observations. The thermal emission of mm-sizedgrains is largely optically thin in protoplanetary disks and allowsto directly trace the column density of the bulk of the dust (e.g.Andrews 2015; Tazzari et al. 2017). Dust and gas surface den-sities, however, are not necessarily the same. Even though theyare strongly coupled, di ff erent mechanisms are thought to af-fect their evolution, such as radial drift of millimeter-sized grainsmodifying the dust surface density on short timescales. This isthe case e.g., in HD 163296 where the carbon monoxide (CO)emission extends much further than the continuum dust emission Article number, page 1 of 14 a r X i v : . [ a s t r o - ph . S R ] S e p de Gregorio-Monsalvo et al. 2013; Isella et al. 2016), althoughthe much higher optical depth of the CO lines versus millime-ter continuum also contributes significantly to this di ff erence inappearance (Dutrey et al. 1998; Facchini et al. 2017). Anotherexample comes from the Herbig disk HD 169142, where ALMAobservations show that the dust appears to be concentrated intwo rings between 20–35 au and 56–83 au, whereas gas seenin CO isotopologues is still present inside the dust cavity andgap (Fedele et al. 2017). More extreme examples are transi-tional disks with large cavities severely depleted in dust content,but less in gas, with gas cavities smaller than dust cavities (vander Marel et al. 2013, 2016a; Bruderer et al. 2014; Zhang et al.2014). Taken together, it is clear that gas and millimeter dustemission do not follow each other and that a more direct tracerof the gas surface density profile is needed.The main gaseous component is molecular hydrogen (H ),whose emission lines at near- and mid-infrared wavelengths arevery weak and are superposed on strong continuum emission(e.g. Thi et al. 2001; Pascucci et al. 2006). In contrast, CO purerotational transitions at millimeter wavelengths are readily de-tected (e.g. Dutrey et al. 1996; Thi et al. 2001; Dent et al. 2005;Pani´c et al. 2008). CO is the second-most abundant molecule,with a chemistry that is very well studied and implemented inphysical-chemical codes. Owing to their optically thin emissionlines, less abundant CO isotopologues are considered better trac-ers of the gas column than CO (Miotello et al. 2014, 2016;Schwarz et al. 2016). This paper investigates whether spatiallyresolved observations of such lines may be good probes of thegas surface density distribution, once freeze-out and isotope se-lective photodissociation are taken into account.Thanks to the unique sensitivity of ALMA, large surveys ofdisks in di ff erent star-forming regions are being carried out totrace gas and dust simultaneously (Ansdell et al. 2016; Barenfeldet al. 2016; Eisner et al. 2016; Pascucci et al. 2016). In partic-ular, CO isotopologues pure rotational lines have been targetedbut lines are weaker than expected and at the short integrationtime of typically one minute the line data have low S / N. Thismay be interpreted as lack of gas due to fast disk dispersal, oras lack of volatile carbon that leads to faint CO lines (Favre etal. 2013; Ansdell et al. 2016; Schwarz et al. 2016; Miotello etal. 2017). As a consequence, CO lines are optically thin to-wards a larger part of the disk and become much better tracers ofthe gas columns than C O lines, which were thought to be pre-ferred pre-ALMA, but are often undetected. Furthermore, COis less a ff ected by isotope selective photodissociation than C O(Miotello et al. 2014). The focus of this paper is thus on CO,rather than on other isotopologues. Recently, Zhang et al. (2017)have proposed the optically thin lines of C O, successfully de-tected in the TW Hya disk, to be a good probe of Σ gas ( R ) at smallradii. We briefly investigate this possibility from our modelingperspective using another low abundance isotopologue C O.Williams & McPartland (2016) have presented a method toextract disk gas surface density profiles from CO ALMA ob-servations. Assuming a self-similar density distribution as givenby viscous evolution theory, the authors create a large galleryof parametrized models, which they use to extract the disk pa-rameters with a MCMC analysis. Although we share the samescientific question, this paper follows a di ff erent approach, i.e.,we fit a power-law profile to our simulated CO radial profilesas if they were observations, in order to recover the power-lawindex γ . Our analysis shows that it may be di ffi cult to infer both γ and the critical radius independently for small disks. In Section 2, the physical-chemical modeling is presented.The results of the modeling investigation are presented in Sec-tion 3 and their implications are discussed in Section 4.
2. Modeling
The physical-chemical code DALI is used for the disk modeling(Dust And LInes, Bruderer et al. 2012; Bruderer 2013). DALIcalculates the dust temperature, T dust , and local continuum radi-ation field from UV to mm wavelengths with a 2D Monte Carlomethod, given an input disk physical structure and a stellar spec-trum as source of irradiation. Then, a time-dependent (1 Myr)chemical network simulation is run. Subsequently, the gas tem-perature, T gas , is calculated through an iterative balance betweenheating and cooling processes until a self-consistent solution isfound and the non-LTE excitation of the molecules is computed.In this paper, the final outputs are spectral image cubes of COisotopologues computed by a ray-tracer module. Throughoutthis paper, the J = J = − , rather than Jy beam − km s − , in order to not depend on thebeam size. The line intensity can be converted to Jy beam − kms − using the relation in Appendix B of Bruderer (2013). For theray-tracing, a distance of 150 pc is assumed.DALI has been extensively tested and benchmarked againstobservations (see e.g., Fedele et al. 2013) and a detailed descrip-tion of the code is presented by Bruderer et al. (2012) and Brud-erer (2013). For this work, a complete treatment of CO freeze-out and isotope-selective processes is included (for more detail,see Miotello et al. 2014, 2016). The volatile carbon abundanceis assumed to be high, [C] / [H] = . × − , and constant acrossthe disk. The disk surface density distribution is often parametrized byan exponentially tapered power-law function, following the pre-scription proposed by Andrews et al. (2011). Physically thisrepresents a viscously evolving disk, where the viscosity is ex-pressed by ν ∝ R γ (Lynden-Bell & Pringle 1974; Hartmann etal. 1998). The self-similar surface profile is expressed by: Σ gas = Σ c (cid:32) RR c (cid:33) − γ exp − (cid:32) RR c (cid:33) − γ , (1)where R c is the critical radius and Σ c is the surface density at thecritical radius. This parametrization has often been employed tomodel disks with DALI (see e.g., Miotello et al. 2014, 2016).Adopting an exponential taper to the power-law profile of thesurface density distribution (see Eq. 1), as suggested by viscousevolution theory, has the inconvenience that the profile slope de-pends on two free parameters, R c and γ , which can be di ffi cultto disentangle with the current low S / N of the data in the outerregions of disks. A simplification comes from the assumptionthat Σ gas has a pure power-law dependence with radius. In thiscase the power-law index γ is left as a single free parameter: Σ gas = Σ c (cid:16) RR c (cid:17) − γ R ≤ R out , R > R out . (2)In this work both parametrizations have been employed to designthe input density structures. Article number, page 2 of 14. Miotello et al.: Probing the protoplanetary disk gas surface density distribution with CO emission (g cm -2 ) (a) (b) (c) (g cm -2 ) Fig. 1.
Panel a – CO line intensity radial profiles (solid lines) obtained with three representative disk models with input surface densitydistribution Σ gas (dashed lines) chosen to be a simple power-law (see Eq. 2). The model parameters are R out =
200 au, M disk = − M (cid:12) and γ = . , , . Panel b – CO line intensity radial profiles compared with input surface density profilesas in panel a, but convolved with a 0.2" beam.
Panel c –
Column densities of gas-phase CO calculated from the surface to the midplane shown asfunction of the disk radius for the three representative models with simple power-law surface density. The dotted black line indicates the columndensity at which CO self-shielding becomes ine ffi cient ( N = cm − ). Ice-phace CO column densities are not shown in the plot as they arebelow the values shown her. Table 1.
Parameters of the disk models.
Parameter ValueSelf similar Power-law γ R c
30, 60, 200 au – R out
500 au 100, 200 au M gas − , − , − , − , − , − M (cid:12) − , − M (cid:12) First, some of the model results presented in Miotello et al.(2014) using the self similar profile have been analyzed. The freeparameters are then R c , γ , and M disk , whose values are reportedin column 2 of Table 1. A value for R out is also reported, but thisis simply needed for the simulation and does not have any e ff ecton the disk structure, as the exponential cut-o ff truncates the diskat smaller radii. Parameters defining the disk vertical structureare reported in Table 1 of Miotello et al. (2014) and simulate amedium-flared disk with large dust grains settled. From here on,such disk models where the input surface density distribution isset by Eq. 1 will be called self-similar disk models. Additional models have then been run with the simplepower-law surface density structure (see eq. 2). In this case thefree parameters are R out , γ , and M disk as shown in column 3 ofTable 1. From here on, disk models where the input surface den-sity distribution is set by Eq. 2 will be called simple power-law disk models.In order to investigate di ff erent disk mass regimes and thencompare with recent observations of CO isotopologues in pro-toplanetary disks (Ansdell et al. 2016, 2018), a range of modelsfrom less massive to high mass disks have been run (see Tab.1). These observations also show moderately extended emis-sion, therefore the outer radius in the models has been set to 200au. The raytracing of all models presented here is carried outassuming the disk to be at a distance of 150 pc, representative ofthat of the nearby star-forming regions. Furthermore, the outputsynthetic images have either been left unconvolved or convolvedwith a moderate resolution beam of 0.2 (cid:48)(cid:48) . In the first case, theresolution is that of the radial grid assumed for the simulation.The calculation is carried out on 75 cells in the radial direction(logarithmically spaced up to 30 au and then linearly spaced fora radial resolution of ∼
10 au) and 60 in the vertical directionmodels (Miotello et al. 2014). In the new models with the sim-
Article number, page 3 of 14 le power-law surface density structure, a denser grid with 95cells in the radial direction (improved radial resolution in theouter disk of ∼ T e ff = L bol = L (cid:12) (see Miotello et al. 2014,2016). In order to explore the e ff ect of di ff erent stellar proper-ties on the results of this work, we have compared the self similardisk models with some of the Herbig-like disk models presentedin Miotello et al. (2016), where T e ff = L bol = L (cid:12) .We find that stellar temperature and luminosity do not quantita-tively a ff ect the results found with the T-Tauri like disk models,except for pushing the preferred fitting region further out in thedisk for massive disk models (see Sec. 3.4). Such massive Her-big disk models ( M disk = − M (cid:12) ) show radial intensity profileswhich are a factor of 1.6 brighter than those found for the re-spective T-Tauri models.
3. Results
The first test is to compare the input disk surface density pro-files for the DALI models with the simulated CO profiles, asif they were observed at infinite resolution, i.e. not convolvedwith any beam. In practice, the image resolution is the physi-cal grid resolution in our models (see Sec. 2.2). The next stepis then to convolve the simulated CO profiles with a typicalALMA observation beam. The results obtained in the two cases,where Σ gas is parametrized by a simple power-law or followingthe self-similar solution are presented below. The input disk surface density profiles, assumed to be a power-law function of the radius (see Eq. 2), are compared with thesimulated CO profiles for three representative models ( M disk = − M (cid:12) , R out =
200 au, γ = . , , .
5) and are shown in panel(a) of Fig. 1. The surface density profiles (in g cm − ) have beenrescaled by a constant factor in order to visually match the lineintensity profiles. The disk can be divided in three regions: aninner part in which the CO radial profile underestimates Σ gas , acentral zone where the two coincide, and an outer region wherethe line emission drops and deviates from the surface densitydistribution (see case with γ = .
5, for clarity). At small radiithe divergence is caused by the fact that CO emission becomesoptically thick as the columns are very high. At large radii the CO column densities become too low, lower than 10 cm − ,and CO is not able to e ffi ciently self-shield (van Dishoeck &Black 1988; Visser et al. 2009).There are two ways to further investigate these cases. Thecolumn densities of gas-phase CO calculated from the surfaceto the midplane are shown as function of the disk radius in panel(c) of Fig. 1 in blue, light blue and purple for models with γ = . , , . CO column densities drop below 10 cm − .This illustrates that ine ffi cient CO self-shielding a ff ects CO iso-topologues intensity profiles and needs to be considered in thedisk outer regions.As reported in Table 1, less and more massive models havebeen run for the simple power-law case. We observe similar trends, but the radial location of the three regions describedabove are radially shifted to smaller or larger radii, dependingon disk mass (see Fig. A.2). Compared with the representativemodel with mass M disk = − M (cid:12) , for M disk = − M (cid:12) the sim-ulated radial intensity profiles better follow Σ gas at smaller radiias the line is less optically thick. On the other hand CO columndensities smaller than N = cm − are reached at smaller radii.The opposite happens for M disk = − M (cid:12) , where the line emis-sion is optically thick up to almost 100 au, but CO self-shieldingbecomes ine ffi cient much further out, around 200 au. γ We now address the question of whether it is possible to retrievethe underlying surface density power-law index γ by fitting theline intensity profiles. We refer to the power-law index used asinput in DALI as γ input and we label the fitted value γ fit .The fit of a power-law profile to the simulated intensity pro-files is performed as a linear fit in the log I -log R space. As itis clear from Fig. 1 that CO radial intensity profiles follow Σ gas ( R ) only in a particular region, the fit is not carried out overthe whole extent of the disk, but over a radial range that spansfrom R start to R cut . The starting point R start is kept fixed and cho-sen to be just outside the inner region where the emission is opti-cally thin, typically 30–40 au. The range of radii over which thefitting is performed is then varied by changing the value of R cut .The results of the fitting are presented in Fig. 2 for the singlepower-law case. The dashed lines illustrate the value of γ input ,while the colored dots show the value of γ fit if di ff erent R cut arechosen for the fitting. The starting radius for the fitting procedure R start is indicated by the dotted vertical line.For low mass disk models ( M disk = − M (cid:12) , panel a of Fig.2), the power-law index γ input can be retrieved within 20% ofuncertainty in a range of radii that goes from 30 – 40 au to ∼
100 au, the typical scales probed with ALMA. Furthermore, theretrieved γ fit usually overestimates the input value, except forthe γ = . R start to either 30 au or 40 au doesnot change qualitatively the outcome of the fit. For R cut largerthan 100 au, the fitted values deviate significantly from γ input asone enters the region where self-shielding becomes ine ffi cientand the intensity profiles deviate from Σ gas . For higher mass diskmodels ( M disk = − M (cid:12) , panel b of Fig. 2) one needs to chose alarger R start of 100 au, as CO is optically thick much further outthan for the low mass disk case. Furthermore, the uncertaintieson γ fit are larger.In order to choose the correct radial range over which to per-form the fit, one would need to know in which mass regimethe observed disk is. More precisely, one would need to con-strain the total CO gas mass. For faint observed CO fluxes,Miotello et al. (2016) have found a linear relation between lineemission and total disk mass. This could be used to constrain M disk . For brighter observed CO fluxes, this relation is lessreliable and one could employ the dust masses obtained fromcontinuum emission, multiplied by the canonical factor of 100.This would provide an upper limit to the disk mass (see Sec. 4for discussion on carbon depletion). Another possibility is tocombine CO and C O line fluxes (Miotello et al. 2016), if thelatter are available from the same observation.
Finally, the simulated line intensity radial profiles need to beconvolved with a synthesized beam to check how the trends
Article number, page 4 of 14. Miotello et al.: Probing the protoplanetary disk gas surface density distribution with CO emission (a) (b)
Fig. 2.
Results of the power-law fitting of the simulated CO line intensity profiles obtained with the simple power-law models ( M disk = − M (cid:12) and M disk = − M (cid:12) in panel a and b respectively). The values of the fitted power-law index γ fit are shown by the filled dots as a function of theradial range over which the fitting was carried out for the high resolution images (0.01" beam). The empty symbols in panel (a) show the fittedpower-law index γ fit when the simulated images are convolved with a 0.2" beam. The model input power-law indexes are shown by the dashedlines. The dotted line shows the starting point of the fitting R start . are a ff ected by this operation. Observations taken from recentALMA disk surveys have moderate angular resolution, between0.2" and 0.3", corresponding to 30 – 40 au diameter at 150 pc(Ansdell et al. 2016, e.g.). To mimic such observations we con-volve our simulated profiles with a 0.2" beam. The convolved CO line intensity radial profiles are shown in panel b of Fig. 1.As in Sec. 3.1.1, a power-law fitting of the convolved radialintensity profiles has been carried out. The starting point R start has been fixed to 30 au and the results are shown in panel (a) ofFig. 2. The values of γ fit do not qualitatively di ff er from thosefound with the unconvolved radial profiles, shown in panel (a)of Fig. 2. Even with a beam convolution of 0.2" the power-law index can be retrieved to an accuracy of ∼ ∼
30 au to ∼
100 au.
Similar trends as those presented in Sec. 3.1 have been foundfor the self-similar case. The input disk surface density pro-files compared with the simulated CO profiles for three rep-resentative self-similar models ( M disk = − M (cid:12) , R c =
200 au, γ = . , , .
5) are presented in panel (a) of Fig. 3. The sur-face density profiles have been rescaled by a constant factor inthe plots to visually match the line intensity profiles. As for thesimple power-law case it is possible to identify three regions:the inner disk in which CO lines are optically thick and theemission profile is flatter than the Σ gas profile, a central region inwhich the two profiles match, and the outer disk where the lineintensity underestimates the surface density distributionIn the self-similar models a secondary e ff ect adds to inef-ficient CO self-shielding to reduce the emission in the outerdisk. As shown in Fig. A.1, the simple power-law representa-tive model (panel b) for a less massive disk ( M disk = − M (cid:12) ) iswarmer than the self-similar representative disk model (panel a, M disk = − M (cid:12) ). In this second case, the dust temperature T dust drops below 20 K in the midplane at radii larger than 50 au anda significant fraction of CO is frozen out, thus not contributing to the line emission (see panel a of Fig. A.1). Moreover, the in-ner region where the emission lines are optically thick is smallercompared to the power-law case, and the radial range over whichthe line intensity profile follows Σ gas is shifted toward smallerradii.The column density, reported in panel b of Fig. 3, can becompared with the line intensity profiles presented in panel a.The radius where the line intensity profiles decrease and deviatefrom the surface density distribution is similar to that at which CO column densities drop below 10 cm − (at ∼
300 au for γ = . ∼
250 au for γ = .
0, and ∼
200 au for γ = . CO ra-dial intensity profiles have been fitted with an exponentially ta-pered power-law (see Equation 1) to try and retrieve the power-law index γ input and the critical radius R c , input . This procedure isapplied to models where M disk = − M (cid:12) , in order to directlycompare with the results found in the simple power-law case,instead of M disk = − M (cid:12) , as discussed above. The intensityprofiles for the lower mass ( M disk = − M (cid:12) ) self-similar diskmodels are reported in Fig. A.3 in Appendix A. Fit parame-ters are much more ambiguous when fitting the intensity profilesobtained with the self-similar disk models. As shown in Fig.4, the retrieved γ fit and R c , fit can significantly deviate from theoriginal γ input . If R c , input is 200 au, it is still possible to distin-guish γ fit between models with di ff erent γ input (panel a of Fig.4). This holds in particular if R cut >
100 au, in which case γ input is retrieved within 20% and R c , input within 12%, consistent withWilliams & McPartland (2016) large HD163296 disk. If instead R c is smaller, i.e., 30 au or 60 au (panel b and c of Fig. 4 re-spectively), this is not true as γ input cannot be reliably retrieved.It is still possible to obtain a good estimate for R c , which can beretrieved within 13% and 30% if R c , input = ,
30 au respectively.The complication is given by the fact that for smaller values of R c the line intensity radial profile follows the Σ gas profile over avery limited radial range and the fitting does not perform wellenough. The e ff ects of either optical thickness or ine ffi cient self-shielding dominate throughout most of the disk’s extent and the Article number, page 5 of 14 a) (b) (g cm -2 ) Fig. 3.
Panel (a) – CO line intensity radial profiles (solid lines) obtained with three representative disk models with input surface densitydistribution Σ gas (dashed lines) given by the viscously evolving disk model (see Eq. 1). The model parameters are R c =
200 au, M disk = − M (cid:12) and γ = . , , . Panel (b) –
Column densities ofgas-phase (solid lines) and ice-phase (dashed lines) CO calculated from the surface to the midplane shown as function of the disk radius for thethree representative self-similar disk models. The dotted black line indicates the column density at which CO self-shielding becomes ine ffi cient( N = cm − ). radial line intensity profile does not resemble an exponentially-tapered power-law. In summary, the obtained γ fit has a poor re-lation with γ input if R c , input is small (i.e., 30, 60 au). On the otherend, it is always possible to retrieve R c with moderate uncer-tainty.Recent ALMA disk surveys (e.g. Ansdell et al. 2016; Pas-cucci et al. 2016) have shown that typical protoplanetary disksare usually fainter and less extended than previously thought.Therefore, the typical sensitivity of these observations does notallow to detect the regions where the exponential taper woulddominate, if Σ gas were described by Eq. 1. Moreover, our re-sults show that in such external regions the ine ffi ciency of COself-shielding prevents CO isotopologues to be used to constrainthe surface density distribution. Therefore, we consider thepower-law prescription as a simplification of the self-similar diskmodel. Both behave in the same way, i.e. as a simple power-lawdisk model, in the region where CO isotopologue can be reliablyemployed to trace Σ gas . O line intensityradial profiles
Our models show that CO is not a good tracer of Σ gas in the in-ner 30–40 au. Recently, Zhang et al. (2017) have used the rarer C O isotopologue to try determining the surface density pro-file in the inner regions of the well studied TW Hya disk, insidethe CO snowline. Another rather low abundance CO isotopo-logue is C O, expected to be around 24 times more abundantthan C O based on isotope ratios. We thus analyze the inten-sity profiles obtained for C O to check how they can be used toinfer the surface density profile in the disk inner regions.The simulated C O intensity profiles and column densityprofiles for six power-law disk models ( R out =
200 au, M disk = − , − M (cid:12) and γ = . , , .
5) are presented in Fig. 7. Forthe 10 − M (cid:12) mass disk models, the intensity profiles follow theshape of Σ gas in the very inner disk ( R <
10 au, see panel a of Fig.7), apart for the steeper γ = . R < O columndensities drop below 10 cm − very early, at R <
10 au. Thus,any information about Σ gas from C O intensity profiles is lostfor the outer disk regions.
Article number, page 6 of 14. Miotello et al.: Probing the protoplanetary disk gas surface density distribution with CO emission
Self-similar disk (a)(b)(c) (d)(e)(f)
Fig. 4.
Results of the power-law fitting of the simulated CO line intensity profiles obtained with the self-similar disk models with M disk = − M (cid:12) . The values of the fitted power-law index γ fit are shown by the filled dots as a function of the radial range in which the fitting was carriedout. The model input power-law indexes are shown by the dashed lines. The dotted line shows the starting point of the fitting R start . The picture changes if the results of the more massive( M disk = − M (cid:12) ) disk model are analyzed. C O emission isoptically thick out to 50 au, except for the model with γ = . Σ gas is less steep in the inner disk (see panel b in Fig. 7).Therefore C O is not a good tracer of the surface density dis-tribution in the inner disk in massive disks. An even rarer COisotopologue, as C O, should be used in these cases, as wasdone for TW Hya (Zhang et al. 2017). A possible complica-tion comes from the fact that its faint emission may be shieldedby optically thick continuum emission, and this would prevent C O from being a good tracer of the gas column density.As for CO, it is possible to fit the simulated intensity pro-files with a power-law and check for which conditions γ fit resem-bles γ input well enough. γ : the "slope-pivot-region The fitting method of the surface density power-law indexpresented in Sec. 3.1 is not straightforward to be applied toobservations, as this approach involves the choice of two param-eters, R start and R cut , which wold be di ffi cult to determine. On theother hand, there is a relation between the "slope-pivot-point"(see Fig. 5), where CO radial intensity profile starts to deviatefrom the surface density profile due to optical depth, and thecolumn density. The slope-pivot-point will always be close to where τ =
1, thus around the same intensity modulo temperaturee ff ects, which however are not significant for low- J CO lines.One can then perform the fit of γ between the slope-pivot-pointand the point where the CO column density is larger than 10 cm − . Such radial region will always be around the same narrowrange of line intensity.More specifically, at the radius where τ CO =
1, the COintensity is roughly 6 K km s − (see Fig. 5). This results fromthe line intensity, which is controlled by a combination of theline opacity and the temperature. The intensity at the frequency ν of a gas column with line opacity τ ν and uniform temperature T is I ν = B ν ( T ) · (1 − e − τ ν ) (3)where B ν ( T ) is the Planck function. Thus, for τ ν = I ν dependsonly on the temperature. In the Rayleigh-Jeans approximation, I ν ∝ T . Protoplanetary disks have strong gradients in physicalconditions both in radial and vertical direction and thus cannotbe taken as at uniform temperature. However, the bulk of thedisks mass is close to the midplane, where the physical condi-tions change less than in the warm molecular layer of the upperdisk (van Zadelho ff et al. 2001; Bruderer et al. 2012). Mass trac-ers such as CO trace mostly regions close to the midplane. Inthe midplane, the temperature is a weak function of the radius
Article number, page 7 of 14 ith T ∼ r − q with 0 < q <
1. In the simple power-law case thepivot-point is at radii between 60 and 100 au (see Fig. 5) and thetemperature at the radius of the pivot point varies by less thana factor of two. Thus, the intensity also changes by less than afactor of two. This explains the similar intensity at the pivot-point. The explanation may also apply to other molecules withemission mostly from regions close to the midplane (e.g. COor C O).We have compared our results in the simple power-law caseand self-similar disk case, to test that one can always retrieve γ safely around the same range or line intensity. In Fig. 5 the CO emission radial profiles are compared with the total lineoptical depth τ and the CO column density. The line emis-sion can be used to trace the gas distribution if τ < CO column densityneeds to be larger than 10 cm − where the self shielding is ef-fective (shaded region in panels b, and e), i.e. for radii smallerthan those shown by the dashed lines. Combining these two re-quirements one is restricted to a radial range, the "pivot-region",where the CO intensity is always around 6 K km s − , for eachvalue of γ and no matter if the underlying Σ gas is given by asimple power-law or a self-similar disk model. For all modelsanalyzed in this work, the minimum and maximum flux valuesrelative to the slope pivot-regions are 3 K km s − and 8 K kms − . For the Herbig models with M disk = − M (cid:12) in particular,if one computes the line optical depth and the column density,as shown in Fig. 5, the pivot-point shifts toward larger radii.The slope pivot-region is however still located around the valueof 6 K km s − as for the T Tauri models. This means that, inpractice, one would only need to measure the slope of the COradial intensity over a narrow range of radii close to this contourto retrieve the surface density slope (see shaded region in panelsc, and f). The pivot-region will always be very small, as the lineintensity that corresponds to N( CO) = cm − is very close -within a factor of a few - to that where the line becomes opticallythin. The models presented in Fig. 5 have a total disk mass of10 − M (cid:12) and therefore CO emission lines are still optically thinin the interested region of the disk. If the disk was more massive,for example with M disk = − M (cid:12) , the same argument describedabove would be valid at larger radii or for less abundant CO iso-topologues, e.g. for C O. Also for C O, one could retrieve γ by fitting the radial profile at a given contour at 6 K km s − . Infact, in principle one could put together the surface density pro-file over the entire radial range by putting together piecemeal theslopes at the pivot points of each of the CO isotopologues, from C O in the innermost part of the disk, to C O, C O, and CO moving outward in the disk, and finally CO itself in theoutermost region. If the observed disk is inclined by more thanabout 30 ◦ , the data should be deprojected in order to apply thismethod.This finding is promising as it relates the observable line in-tensity radial profile directly with the column density, with noneed to know a priori the radial range for the fit of γ or diskmass. On the other hand, the observer would need good angu-lar resolution and sensitivity to perform this fit at a given con-tour. Typically, one would need a resolution of ∼ . − km s − for the CO J = − − km s − for the CO J = − − km s − (and around 14 mJybeam − km s − for the CO J = − / N ∼ ffi cientS / N in less than one hour observation, as the data do not need tobe spectrally resolved.
We test the slope pivot-region fitting on the TW Hya disk ob-servations ( i ∼ ◦ , d = . O J = − CO J = − O is needed in order toconstrain Σ gas , as this is found to be optically thin everywhere.The observation beam size is ∼ . ”5 × . ”3 and, at such reso-lution, the slope pivot-point is expected to be at 0.15 Jy beam − km s − .Fig. 6 shows the the observed C O ( J = −
2) radial profileof TW Hya in black, with the 3 σ level of uncertainty shown bythe shaded region. If a power-law is fitted to the observed radialprofile between ∼ − km s − , correspondingto radii between ∼
15 and 30 au (see red line in Fig. 6) the fittedpower-law index is γ = .
85, which is within 15% of the modelvalue γ = / N >
20 in the slope pivotregion, the error on gamma is negligible, less than ± .
05. Theslight mismatch between the data and the power-law profile onthe left side of the slope-pivot region is due to the optical thick-ness of the emission that becomes more important. On the rightside of the slope-pivot region, instead, the mismatch is probablydue to the fact that C O is being selectively photodissociated.For less massive disks, one should prefer CO over C O totrace Σ gas , as the first is less a ff ected by isotope-selective pho-todissociation compared with the latter (as shown by Miotello etal. 2014).High angular resolution and sensitivity are however neededto apply this method, but ALMA can achieve them. In the par-ticular case of TW Hya, where our method could be applied, therms was 12 mJy beam − in a 0.1 km s − channel.
4. Discussion
The results presented in Sec. 3 indicate that constraining Σ gas from spatially resolved observations of CO is a challengingtask. However, if the underlying surface density distribution is apower-law, it is possible to constrain its steepness by fitting theemission coming from the correct portion of the disk, in particu-lar at large enough distances from the star that the observed lineis optically thin, but at small enough radii that photo-dissociationand freeze-out are not too significant. Moreover, in these inter-mediate regions, the radial gradient of gas temperature is small,and thus does not a ff ect the gradient of the radial intensity profileof the CO line. From the models shown in this paper, it is pos-sible to retrieve the dependence of the surface density with radiuswith good accuracy (20% on the power-law exponent) especiallyif one has an estimate of the CO total mass or if one has highenough resolution data to fit around the pivot-point-region at 6K km s − .However, in the approach presented in Sec. 3.1.1 and in Sec.3.2, there is the very strong assumption that the surface densityis either a power law function of the radius or it is given by the Article number, page 8 of 14. Miotello et al.: Probing the protoplanetary disk gas surface density distribution with CO emission pivot-regionpivot-point (c) (f)(a)(d) (b)(e) (f)
Fig. 5.
Total CO line optical depth τ (panels a, and d), CO column density (panels b, and e), and CO intensity radial profiles for simplepower-law disk models (top panels) and self-similar disk models (bottom panels). The total disk mass is set to M disk = − M (cid:12) and γ =
1. Theshaded regions in panels (a) and (d) show the range where τ <
1, which holds for radii larger than those shown by the dotted lines. The shadedregion in panels (b) and (e) shows the region where CO column densities are larger than 10 cm − , i.e. for radii smaller than those shown bythe dashed lines. The shaded regions in panels (c) and (f) show the range of CO line intensity for which the conditions τ < N CO > cm − hold. The fit of γ is performed over the "slope-pivot-region" between the dotted and dashed lines, shown by the red arrow in panel (c).
20 40 60 80 R (au) C O ( - ) I ( J y b e a m k m s ) Observed intensity profilePower-law fittingat the slopepivot-region
TW Hya
Fig. 6.
The radial intensity profile of the C O (3-2) line observedwith ALMA (Schwarz et al. 2016) is shown in black and the 3 σ levelof uncertainty is presented by the shaded region. The red line shows thepower-law function used to fit the data at the slope pivot-region. self-similar model. However, the actual underlying surface den-sity in real protoplanetary disks is of course not known. More-over, the main di ffi culty in the fitting is due to the relatively nar-row radial range where CO traces well the underlying surfacedensity, with this radial range limited by optical depth e ff ects inthe inner disk and ine ffi cient self-shielding in the outer regions. Determining additional free parameters, such as the characteris-tic radius of the self-similar profile, leads to a large degeneracybetween the power-law index and the new parameters. To over-come the issue of CO being optically thick at small radii, onepossibility is to probe Σ gas in the inner disk by observing veryoptically thin isotopologues, such as C O and C O. The lat-ter was successfully detected with ALMA in the closest and beststudied protoplanetary disk, TW Hya, by Zhang et al. (2017).The inconveniences of this emission line are its faintness com-bined with the possibility that the continuum becomes opticallythick shielding C O emission at very small radii. Moreoverthe gas temperature structure has a strong impact on the lineemission in the inner disk, therefore possibly independent con-straints of the thermal structure would be needed.An much simpler approach is to fit the power-law index γ atthe slope-pivot-region as explained in Sec. 3.4. This allows areliable estimate of γ both in the case of a simple power-law andself similar disk, by only fitting the intensity profile at the rightintensity contour. The case study presented in Sec. 3.5 showsthe applicability of this method to resolved observations of diskswith ALMA.An additional caveat is volatile carbon depletion, a processthat may be happening in a large fraction of protoplanetarydisks (Favre et al. 2013; Kama et al. 2016; Schwarz et al. 2016;Miotello et al. 2017). In our models, the volatile carbon abun-dance is assumed to be high and constant throughout the disk.If carbon depletion takes place, but it is constant throughout thedisk especially around the pivot-region, this should not add ma-jor complications to retrieving the surface-density profile. How- Article number, page 9 of 14 ver, the nature of this depletion process is not yet known, butit is possible that carbon is sequestered from CO into CO andmore complex ice species in the outer disk, and then drifted in-ward following the large dust grains. If this is true, one would ex-pect a radially dependent decrease of CO abundances in the outerdisk, on top of the reduction due to ine ffi cient self-shielding andfreeze-out. As the ices reach the inner disk then carbon shouldbe liberated into gas phase and quickly return into CO, whichwould then present an increased emission (Du et al. 2015). Thiswould introduce a new degree of degeneration in the employ-ment of CO isotopologues as tracers of the disk surface densitydistribution. Interestingly, the expected enhancement of C Oin the inner disk of TW Hya has not been found by Zhang et al.(2017), showing that much is still to be understood about volatiledepletion in protoplanetary disks.
5. Summary and conclusion
In this work we have addressed the issue of determining thegas surface density distribution in protoplanetary disks by us-ing resolved high signal-to-noise observations of CO. Simu-lated CO intensity radial profiles have been produced usingthe physical-chemical code DALI (Bruderer et al. 2012; Brud-erer 2013), with two di ff erent input surface density profiles: asimple power-law with radius, and the self-similar solution givenby viscously evolving disk theory. By comparing the input Σ gas profiles with the output intensity profiles one always finds thefollowing: – CO emission follows the slope of Σ gas ( R ), but only in a spe-cific disk region. For very small radii the low- J CO linesbecome optically thick and underestimate the surface den-sity, while in the outer disk this happens because the COcolumn density drops below 10 cm − and self-shielding be-comes ine ffi cient. – When fitting a power-law to the simulated intensity profiles,it is possible to recover the input power-law index γ input onlyby performing the fit over the appropriate radial range. Thisholds in particular for simple power-law disk models, where γ input can be retrieved within 20% uncertainty between 30–100 au, even when the profiles are convolved with a 0.2"beam. – In the self-similar case it is not always possible to reliablyretrieve γ input by fitting a self-similar model to the intensityprofile. R c can instead be always retrieved within 30 %. – Fitting the power-law index γ in a narrow range around theslope-pivot-region of the intensity profile allows a reliableestimate of γ both in the case of a simple power-law andself-similar disk. The slope-pivot-region is always locatedaround 6 K km s − . Application of such a method is shownin the case study of the TW Hya disk. – If carbon depletion were constant throughout the disk, thiswould not introduce an additional uncertainty in the employ-ment of CO isotopologues as tracers of the disk surface den-sity distribution. C O may be a better tracer of Σ gas ( R ) in the inner regions formassive disks, circumventing the CO optical depth issue, assuggested by Zhang et al. (2017). For lower mass disks, C Oand C O may be more appropriate. However, both dust opticaldepth and gas temperature may limit the analysis. Thus, com-bining observations of optically thin tracers with CO may bethe best option after-all.
Acknowledgements
The authors thank the referee J. P. Williams, S. Andrews, L.Testi, I. Pascucci, and I. Kamp for the comments which helpedto improve the paper, and E. Bergin, and K. Schwarz for sharingtheir data. Astrochemistry in Leiden is supported by the Nether-lands Research School for Astronomy (NOVA), by a RoyalNetherlands Academy of Arts and Sciences (KNAW) profes-sor prize, and by the European Union A-ERC grant 291141CHEMPLAN. AM acknowledges an ESO Fellowship.
References
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Here some ancillary figures are reported.
Article number, page 12 of 14. Miotello et al.: Probing the protoplanetary disk gas surface density distribution with CO emission (a)$ (b)$(au)$ (au)$ ( a u ) $ Self-similar Simple power-law
Fig. A.1.
Abundance of CO in one quadrant the disk for two representative models. Panel (a) shows the self-similar disk model ( R c =
200 au, M disk = − M (cid:12) and γ = R out =
200 au, M disk = − M (cid:12) and γ = T dust =
20 K surface below which CO freeze-out becomes important.
Simple power-law (a) (b)
Fig. A.2. CO line intensity radial profiles (solid lines) obtained with input surface density distribution Σ gas (dashed lines) given by the simplepower-law disk models (see Eq. 1). The model parameters are M disk = − M (cid:12) (panel a) and M disk = − M (cid:12) (panel b), γ = . , , . elf-similar disk L (b) (c)(a) L Fig. A.3. CO line intensity radial profiles (solid lines) obtained with input surface density distribution Σ gas (dashed lines) given by theself-similar disk model (see Eq. 1). The model parameters are M disk = − M (cid:12) and γ = . , , . R c = , ,,