Snow-lines as probes of turbulent diffusion in protoplanetary discs
aa r X i v : . [ a s t r o - ph . S R ] J un Draft version February 26, 2018
Preprint typeset using L A TEX style emulateapj v. 04/17/13
SNOW-LINES AS PROBES OF TURBULENT DIFFUSION IN PROTOPLANETARY DISCS
James E. Owen
Canadian Institute for Theoretical Astrophysics, 60 St George Street, Toronto, M5S 3H8, ON, CANADA
Draft version February 26, 2018
ABSTRACTSharp chemical discontinuities can occur in protoplanetary discs, particularly at ‘snow-lines’ wherea gas-phase species freezes out to form ice grains. Such sharp discontinuities will diffuse out due to theturbulence suspected to drive angular momentum transport in accretion discs. We demonstrate thatthe concentration gradient - in the vicinity of the snow-line - of a species present outside a snow-linebut destroyed inside is strongly sensitive to the level of turbulent diffusion (provided the chemical andtransport time-scales are decoupled) and provides a direct measurement of the radial ‘Schmidt number’(the ratio of the angular momentum transport to radial turbulent diffusion). Taking as an example thetracer species N H + , which is expected to be destroyed inside the CO snow-line (as recently observedin TW Hya) we show that ALMA observations possess significant angular resolution to constrain theSchmidt number. Since different turbulent driving mechanisms predict different Schmidt numbers, adirect measurement of the Schmidt number in accretion discs would allow inferences about the natureof the turbulence to be made.
Subject headings: accretion, accretion disks - protoplanetary disks - turbulence - astrochemistry INTRODUCTIONAstrophysical discs are observed to transport angularmomentum. It has been hypothesised that such discsare transporting angular momentum through a turbu-lent process (Shakura & Sunyaev 1973). Despite decadesof theoretical research we still lack a sufficient under-standing of turbulence to make quantitative predictions.Several candidate processes exist to sustain turbulencein these discs: the magneto-rotational instability (MRI)(e.g. Balbus & Hawley 1991), gravitational instability(e.g. Lodato & Rice 2004, 2005) the vertical shear insta-bility (Nelson et al. 2013) and the baroclinc instability(e.g. Klahr & Bodenheimer 2003). The MRI is still theleading candidate for angular momentum transport in ac-cretion discs (see Turner et al. 2014 for a recent review).However, at the low temperatures and ionization frac-tions expected in protoplanetary discs, non-ideal MHDeffects become important, qualitatively changing the na-ture of the turbulence and its associated transport prop-erties, or rendering it ineffective resulting in ‘dead-zones’(e.g. Gammie 1996).A corollary to the angular momentum transportproblem in protoplanetary discs is the transport ofdust particles. There is large amounts of astrophys-ical (e.g Bouwman et al. 2001; Dullemond et al. 2006;Hughes & Armitage 2010; Owen et al. 2011b) and cos-mochemical (e.g Gail 2001; Bockel´ee-Morvan et al. 2002;Jacquet et al. 2012; Jacquet & Robert 2013) evidenceto suggest large scale radial transport of dust parti-cles occurs. In particular, the level of diffusion thatresults from the turbulence is an unknown parame-ter. The turbulent diffusion coefficient ( D g ) is oftenparametrised in terms of the turbulent kinematic viscos-ity ( ν ) as D g = ν/Sc where Sc is the Schmidt number.Assumptions of isotropic, Kolmogorov-like turbulencelead to the inference that Sc ≈ Sc = 0 . −
10 (Prinn 1990; Dubrulle & Frisch 1991; Lathrop et al.1992; Carballido et al. 2005; Johansen et al. 2006;Youdin & Lithwick 2007; Zhu et al. 2014).Direct observational measurements of the strength ofturbulent diffusion in the continuum will be compli-cated by dust-drag, grain growth/fragmentation andoptical depth effects. However, it is not only dustparticles that will experience turbulent diffusion; anyrare, gas-phase species (tracer species) where changesin temperature results in a concentration gradient willalso experience turbulent diffusion. If, for example, atracer species is predominately produced at a given ra-dius, the tracer will then diffuse away from this radius,with a distribution that is strongly dependent on thevalue of the Schmidt number (Clarke & Pringle 1988).Jacquet & Robert (2013) considered the case of deuter-ated water and argued the D/H water distribution inChondrites implies the Schmidt number is smaller thanunity.Snow-lines, where a gas phase tracer (for example H O,CO , CO) condenses out of the gas to form ice belowsome temperature represents a scenario where a sharpconcentration gradient can occur. Species such as H O,CO & CO are relatively abundant, such that the surfacelayers can be optically thick. Additionally, model degen-eracies make it difficult to constrain snow-lines directly(even with optically thin isotopologues). However, if theproduction/destruction of yet rarer tracers are regulatedby the presence or absence of gas phase H O, CO or COthen these rare tracers could be used as proxies to de-tect snow-lines. For the CO snow-line it is expected thatN H + and H CO will only be abundant when CO freezesout (Jørgensen et al. 2004; Walsh et al. 2012; Qi et al.2013a,b). Such an expectation has been born out in ob-servations of star-forming cores (Friesen et al. 2010), andsimilar results were obtained in the DISCS
SMA surveyof several nearby protoplanetary discs (Qi et al. 2013a).Recently, Qi et al. (2013b) imaged a hole in N H + using ALMA at a radius of ∼
30 AU, co-incident with the ex- Owen, J. E.pected location of the CO snow-line (based on a free-outtemperature of ∼
17 K). The sharpness of the inner edgewill depend strongly on the strength of the turbulent dif-fusion, with weaker diffusion resulting in a sharper hole.In this letter we demonstrate how the distribution ofa tracer species that is only abundant outside a snow-line (being destroyed inside) is strongly dependant on theSchmidt number, and that
ALMA observations could beable to constrain the Schmidt number in protoplanetarydiscs. DISC MODELWe consider a 1D axis-symmetric disc, where the evo-lution of the gas surface density (Σ g ) and the surfacedensity of any (gas-phase) tracer species (Σ i ) is givenby (e.g. Lynden-Bell & Pringle 1974; Clarke & Pringle1988; Birnstiel et al. 2010; Owen et al. 2011a; Owen2014): ∂ Σ g ∂t = 3 R ∂∂R (cid:20) R / ∂∂R (cid:16) ν Σ g R / (cid:17)(cid:21) (1) ∂ Σ i ∂t + 1 R ∂∂R (cid:20) R Σ i u g − D gR R Σ g ∂X i ∂R (cid:21) = X j S i (Σ g , Σ j , R, t ) (2)where X i is the concentration of the tracer species, u g isthe net radial gas velocity, D gR is the radial gas turbulentdiffusion co-efficient and S i (Σ g , Σ j , R, t ) is a source/sinkterm that represents the production and destruction ofthe tracer species.2.1. Conditions at the snow-line
The following general model can be applied to anysnow-line (H O, CO , CO, etc.) where the chemicaland transport time-scales decouple. Here we specifi-cally consider the case of N H + destruction at the COsnow-line as observed in TW Hya (Qi et al. 2013b). In-side the CO snow-line N H + is destroyed by gas phaseCO, outside the CO snow line this destruction chan-nel is no-longer dominant and it instead is destroyedat a much slower rate by dissociative recombination(Jørgensen et al. 2004). Simulations without transportsuggest the N H + abundance drops by several ordersof magnitude inside the CO snow-line (e.g. Walsh et al.2012) with an abundance that depends on the square ofthe gas-phase CO abundance (Jørgensen et al. 2004).2.1.1. Relevant Time-scales
In order for chemical tracers at the snow-line to beuseful in terms of probing the strength of the turbulentdiffusion, we must de-couple the chemical and dynamicaltime-scales. Namely, the desorption time-scale must befaster than the transport time-scales in-order to create asharp snow-line; furthermore, the destruction time-scaleof the tracer species inside the snow-line must also befast. The transport time-scales of interest are the timewith which to move a radial distance H (where H isthe disc’s scale height which is of order the radial scale We note since the snow-line is sharp, the relevant length scalefor computing time-scales is H , not R . length). Therefore, the advection time-scale ( t adv ) is: t adv ≈ Hu g = 23 α − (cid:18) RH (cid:19) Ω − ≈ × years (cid:16) α . (cid:17) − (cid:18) H/R . (cid:19) − × (cid:18) R SL
30 AU (cid:19) / (cid:18) M ∗ ⊙ (cid:19) − / (3)and the diffusive time-scale ( t dif ) is: t dif ≈ RHD gR (cid:18) ∂ log X∂ log R (cid:19) − (4)It is well known (e.g. Clarke & Pringle 1988;Jacquet et al. 2012; Jacquet & Robert 2013) thelogarithmic concentration gradient rapidly approaches3 ν/ D Rg , thus the diffusive time-scale is identical to theadvection time-scale (this is somewhat unsurprising asthey are both driven by the same process). Thus, therelevant time-scale for the movement of a individualtracer molecule over a radial scale H is ( t adv + dif ) ∼ years.We want to compare this transport time-scale to thetime-scale for the desorption of the snow-line species,along with the destruction of the tracer species insidethe snow-line. Considering our example of N H + andthe CO snow-line then the desorption time-scale is ob-tained by balancing desorption with absorption (withrate constant k S (CO)), so the desorption time-scale be-comes (Takahashi & Williams 2000): t dorb = 1 k S (CO) n CO ≈
10 years µ − (cid:16) a (cid:17) (cid:18) X d . (cid:19) − (cid:18) X CO − (cid:19) − × (cid:18) Σ1 g cm − (cid:19) − (cid:18) H/R . (cid:19) (cid:18) R
30 AU (cid:19) (5)where a is the dust-grain size, X d is the dust-to-gasmass ratio and X CO is the CO abundance. Addition-ally, the destruction time-scale for N H + is (from theJørgensen et al. 2004 simplified network): t des = 1 k des X CO n g ≈
100 years µ − (cid:18) X CO − (cid:19) − (cid:18) Σ1 g cm − (cid:19) − × (cid:18) H/R . (cid:19) (cid:18) R
30 AU (cid:19) (6)where k des is the destruction rate constant .Therefore, the CO/N H + system clearly satisfies t des+dorb ≪ t adv+dif , for conditions experienced in proto-planetary discs. As such we may ignore the details of thechemical rate equations and simply model the destruc-tion of any remaining N H + to occur instantaneously at We will derive this dependence in Section 2.2 for a steady discmodel, but it is a more general result - see Clarke & Pringle (1988). Estimated from Figure 16 of Jørgensen et al. (2004). easuring Turbulent Diffusion 3the snow-line radius; although, we emphasise that thefollowing analysis can be applied to any tracer specieswith similar destruction time-scales. Therefore, in thissituation the source function P j S i (Σ g , Σ j , R, t ) is dras-tically simplified to: S i ( R, t ) = − ˙ M X ∞ i π δ ( R − R SL ( t )) R (7)where ˙ M is the mass-accretion rate, X ∞ i is the concen-tration at large radius, δ ( R ) is the Dirac delta-functionand R SL is the radius of the snow-line. This source func-tion represents the instantaneous destruction of any re-maining tracer species at R = R SL . We note it ignoresany possible vertical structure of the snow-line, whichwill necessarily spread out the destruction region of thetracer species (Walsh et al. 2012) and we discuss the im-plications of this caveat in Section 4.2.2. Steady-disc models
We will now restrict ourselves to a steady disc prob-lem. In that case Equation 1 becomes ˙ M = 3 πν Σ g :where ˙ M = − πRu g Σ g is the accretion rate, and wehave neglected the very small contribution due to an in-ner boundary at finite radius. Furthermore, the equationfor the gas tracer becomes: ∂∂R (cid:20) R Σ i u g − νSc R R Σ g ∂∂R (cid:18) Σ i Σ g (cid:19)(cid:21) = RS i ( R ) (8)where Sc R is the radial Schmidt number. Therefore, ra-dially integrating Equation 8 and using our expressionfor ˙ M we find: − ˙ M π X i − ˙ M πSc R R ∂X i ∂R = Z R d R ′ R ′ S i ( R ′ ) (9)Using Equation 7 we can integrate Equation 9 to find theradial concentration distribution: X i R Sc R / = Z R d R ′ Sc R X ∞ R ′ Sc R / − Θ( R ′ − R SL )(10)where Θ( R ) is the Heaviside step function. Setting X i =0 at R = R SL we find the solution: X i ( R ) X ∞ i = − (cid:16) RR SL (cid:17) − Sc R / if R > R SL R ≤ R SL (11)Therefore, we see that the radial profile of the concen-tration is strongly sensitive to the value of the Schmidtnumber. In Figure 1 we show how the concentrationvaries with radius and Schmidt number. It is impor-tant to emphasise that concentration distribution is inde-pendent of assumptions of the (unknown) viscosity, andmass-accretion rate and can in principle provide a ‘clean’measurement of the Schmidt number. OBSERVABLE CHARACTERISTICSUnfortunately it is not possible to directly observe theconcentration gradient. What is directly observed is thesurface-brightness distribution of the relevant species.The surface-brightness distribution is sensitive to the sur-face density of the tracer species rather than its concen-tration. Thus, we must multiply the concentration of
Radius [R/R SL ] X i / X i ∞ Sc R =0.1Sc R =0.3Sc R =1Sc R =3Sc R =10 Fig. 1.—
The normalised concentration gradient as a function ofradius for different values of Sc R in the range 0.1-10. our tracer species by the gas surface density. Adopt-ing a power-law gas distribution of the form Σ g / Σ SL =( R/R SL ) − γ with a cut-off radius R out , the surface densityof the tracer species is:Σ t = X ∞ Σ SL "(cid:18) RR SL (cid:19) − γ − (cid:18) RR SL (cid:19) − / Sc R − γ (12)in the range R SL < R < R out and Σ t = 0 elsewhere.3.1. Brightness distribution and Visibility profiles
The actual brightness distribution from rotationalemission lines (such as N H + J=4-3), as well as beingsensitive to the surface density distribution is also sen-sitive to the background gas temperature and excitationtemperature of the molecule (which in turn is a func-tion of density and temperature) along with the opticaldepth.However, in the case that the tracer species is opticallythin, and the gas density is far above the critical density( n cr ) for the rotational transition, then the molecularline is thermalised. Thus, if the temperature gradient isweak compared to the scales of interest and n g ≫ n cr then we can approximate the surface brightness as beingdirectly proportional to the surface density of the tracerspecies, provided the line remains optically thin.For the gas at 30 AU the gas density is typically: n g = 5 . × cm − µ − (cid:18) Σ1 g cm − (cid:19) × (cid:18) H/R . (cid:19) − (cid:18) R
30 AU (cid:19) − (13)where µ is the mean molecular weight of the gas. Com-paring this to the critical density of the J = 4 − H + which has a critical density of n cr ∼ cm − (e.g. Friesen et al. 2010), we see that n g ≫ n cr . Wenote, since temperature and density are expected to bepower-laws with radius (e.g. Chiang & Goldreich 1997;Hartmann et al. 1998) then the small additional correc-tions due to temperature and density effects of convertingΣ i to the surface brightness distribution ( B t ) will man-ifest themselves as changes in the power-law index γ inEquation 14.Assuming the disc to be axis-symmetric and observedface-on, we may write the brightness distribution on the Owen, J. E.sky as: B t ( θ ) ≈ B "(cid:18) θθ SL (cid:19) − γ − (cid:18) θθ SL (cid:19) − / Sc R − γ (14)in the range θ SL <θ < θ out and B t = 0 elsewhere, where B is a constant, θ is the angular size on the sky and θ SL is the angular size of the snow-line given by: θ SL = 0 . (cid:18) R SL
30 AU (cid:19) (cid:18) d
150 pc (cid:19) − (15)Thus, we see that the brightness distribution stillretains the strong sensitivity to the Schmidt number.Since the angular resolution required to probe the valueof the Schmidt number is only available through mm-interferometry, we can use our brightness distributionto calculate synthetic visibilities. The visibilities canobtained by a Hankel transform of the axis-symmetricbrightness-distribution such that: V t ( η ) = 2 π Z ∞ d θθB t ( θ ) J (2 πηθ ) (16)where η is a radial baseline co-ordinate defined as η = √ u + v where u & v are the usual baseline co-ordinates.Following the observed N H + J = 4 − R SL = 30 AU, R out = 150 AU and γ = 2.We calculate our synthetic observations assuming thesource is observed face-on at a distance of 150pc. InFigure 2 we show our simulated visibility curves in theleft-hand panels and the surface density profiles in theright hand panel. In the top panels we vary the Schmidtnumber between 0.1-10, in the middle panels we vary γ from 1-3 and in the bottom panels we vary R SL from15-60 AU.The visibility curves in Figure 2 clearly show that vari-ations in the Schmidt number give rise to significantdifferences. Furthermore, comparisons between varyingthe index of the gas surface density γ , snow-line radiusand the Schmidt number are not too degenerate. Thesimplified model presented here contains four free pa-rameters { R SL , R out , Sc R , γ } , which would all need tobe constrained by fitting the visibilities. Inspection ofFigure 2 suggests that sensitivities .
10 % at base-lines ∼ . − DISCUSSIONWe have shown that observations of snow-lines in pro-toplanetary discs using a tracer species (for exampleN H + in the case of the CO snow line Qi et al. 2013a,b)can be used to probe the Schmidt number, a unknownparameter in studies of turbulent transport in accretiondiscs, where current estimates span a range of two-ordersof magnitude ( Sc R = 0.1-10).4.1. Detectability with ALMA
The
ALMA telescope is a sub-mm/mm interferome-ter; once completed it will have ∼
50 individual an-tennas, offering & ∼ N H + J=4-3 line) this providesa maximum spatial resolution of ∼ .
01 arcsec. Figure 2clearly shows that
ALMA posses the required numberof base-lines ( &
50) with separations in the range 0.1-1Km to constrain Schmidt values within the current rangeof uncertainty, provided the observations are sensitiveenough. Taking the TW Hya N H + observations as ref-erence (Qi et al. 2013b) - a source brightness ∼
200 mJybeam − km s − with an rms noise of 8.1 mJy beam − km s − and beam size ∼ . - a similar level of sensitivity, but with a beamsize of . . ALMA of ∼
50 antennas. Thus, such obser-vations are feasible and sufficiently high resolution to al-low constraints to be placed on the Schmidt number atthe distance to TW Hya. Longer integration times andlarger base-lines would be required to reach similar levelsof sensitivity at distances of 150pc.4.2.
Uncovering properties of the turbulence
We have argued that the sharpness of the hole in tracerspecies at the snow-line probes the value of the Schmidtnumber independent of the assumed properties of theturbulence (e.g. assumed value of the viscous ‘ α ’ pa-rameter). Since several snow-lines are expected to occurat different radii, then measurements at different snow-lines would allow the radial dependence of the Schmidtnumber to be probed. In particular, recent simulationsuggest that different non-ideal MHD effects (which dom-inate at different radii Turner et al. 2014) lead to differ-ent Schmidt numbers (Zhu et al. 2014). Thus, compar-ing the simulation predictions of the Schmidt number forvarious turbulent driving mechanisms with the observedvalue would allow inferences about the nature of the tur-bulence to be made.Furthermore, independently measuring the rms turbu-lent velocity ( h v R i ) (e.g. Hughes et al. 2011) then com-bining it with a measurement of the Schmidt num-ber would allow the viscosity (including estimates of α where ν = αH Ω) to be calculated, since D gR = αH Ω /Sc R ≈ h v R i / Ω.4.3.
Caveats & Limitations
We have constructed an idealised model to investigatewhether snow-lines could begin to probe the strengthof turbulent diffusion. As such there are several modelimprovements that must be made before fitting to realdata. Therefore, our model presented in this letter isa ‘proof-of-concept’ rather than a road map for observa-tional modelling. For example, a real protoplanetary discis not one-dimensional. As such the vertical temperaturestructure is not constant and passively heated discs coolas one approaches the mid-plane (Chiang & Goldreich1997). Therefore, the snow-line is unlikely to occur ex-actly at a fixed radius, but is more-likely to be an ex-tended structure with a scale variation of ∼ H , with istime-varying position (Martin & Livio 2012, 2013, 2014). Since the beam size of the current TW Hya observation possesa resolution similar to θ SL then using the current observation toconstrain the Schmidt number seems unlikely; however, using thevelocity channels separately can be used to increase the effectiveresolution (e.g. Qi et al. 2013a). easuring Turbulent Diffusion 5 Baseline [arcsec − ] k R e { V }k [ N o r m a li s e d U n i t s ]
10 1000.010.1
Radius [AU] Σ t [ N o r m a li s e d U n i t s ] Sc R =0.1Sc R =1.0Sc R =10 γ =1 γ =2 γ =3R SL =15 AUR SL =30 AUR SL =60 AU Fig. 2.—
Predicted visibility profiles assuming a source distance of 150 pc (left panels) and tracer surface density profiles. In the toppanel we vary the Schmidt number between 0.1-10; in the middle panels we vary the background gas profile exponent ( γ ) between 1-3; andin the bottom panels we vary the snow-line radius between 15-60 AU. The visibilities are normalised to 1 at ξ = 0 and the surface densityprofiles are normalised so that the peak value has Σ = 1. Additionally the conversion of the gas phase to ice parti-cles at the snow-line will result in turbulent diffusion ofthe gas and ice particles away from the snow-line (in iden-tical manner to that discussed for the snow-line tracerdiscussed here). As such, there is unlikely to be a verysharp change in the gas abundance at the snow-line butrather a smoother change. Furthermore, the chemicaltime-scales may not fully decouple from the transporttime-scales. In the N H + case considered here we haveargued that the time-scales are likely to be decoupled;this may not be the case for all snow-line tracer species,thus dynamical modelling which includes turbulent diffu-sion is needed to determine the importance of this effect.The model presented here should provide a stringent up- per limit of the Schmidt number, and good measurementif it is small ( < ∼ H , then the 1Dmodel would only provide an order of magnitude esti-mate and a better model is need to constrain the Schmidtnumber. Finally, if the snow-line resides in a dead-zone,where there is limited or no turbulence then it is unlikelythis method can be used to cleanly probe the Schmidtnumber; but, dead-zones are not expected at the largeradius of the CO/N H + system discussed here. SUMMARYIn this letter we have shown that the recent observa-tions of snow-lines through tracer species (e.g. N H + or Owen, J. E.H CO in the case of the CO snow-line Qi et al. 2013a,b)could allow direct observational measurements of theSchmidt number in astrophysical accretion discs. In thecase that the chemical time-scale is suitably de-coupledfrom the transport time-scales then the concentrationgradient of the tracer outside the snow-line directly de-pends on the Schmidt number in a power-law fashion( ∼ R − / Sc R ) independent of the choice of the turbulent α parameter.We argue that the effect of turbulent diffusion on sur-face brightness distribution of such a snow line tracer isdetectable with ALMA observations of discs in nearbystar-forming regions which can possess high enoughangular resolution to constrain the current theoreti-cally/numerically estimated values of the Schmidt num- ber ( Sc R ∼ . − O, CO & CO) at different radii in the discs would allow theradial dependence of the Schmidt number to be probed.Coupling these snow-line observations with observationalestimates of the gas-surface density and turbulent line-widths would allow direct estimates of the strength andnature of the turbulence in astrophysical accretion discs.We thank the anonymous referee for a helpful com-ments on the manuscript. JEO is grateful to Karin ¨Obergfor a discussion that sparked this investigation and toRachel Friesen for helpful advice.& CO) at different radii in the discs would allow theradial dependence of the Schmidt number to be probed.Coupling these snow-line observations with observationalestimates of the gas-surface density and turbulent line-widths would allow direct estimates of the strength andnature of the turbulence in astrophysical accretion discs.We thank the anonymous referee for a helpful com-ments on the manuscript. JEO is grateful to Karin ¨Obergfor a discussion that sparked this investigation and toRachel Friesen for helpful advice.