Context. The Earth is strongly depleted in carbon compared to the dust in the ISM implying efficient removal of refractory carbon before parent body formation. It has been argued that grains get rid of their carbon through oxidation and photolysis in the exposed upper disk layers. BR  Aims We assess the efficacy of these Cremoval mechanisms while accounting for the vertical and radial transport of grains. BR  Methods We obtained the carbon and carbon free mass budget of solids by solving two 1D advectiondiffusion equations accounting for the dust grain size distribution and radial transport. The carbon removal acts on the fraction of the grains that are in the exposed layer and requires efficient vertical transport. BR  Results In models without radial transport oxidation and photolysis can destroy most of the refractory carbon in terrestrial planet formation region. But it only reaches the observed depletion levels for extreme parameter combinations and requires that parent body formation was delayed by 1 Myr. Adding radial transport of solids prevents the depletion entirely leaving refractory carbon equally distributed throughout the disk. BR  Conclusions It is unlikely that the observed carbon depletion can ultimately be attributed to mechanisms operating on small grains in the disk surface layers. Other mechanisms of removing carbon quickly and deeply must be studied for example flash heating events or FU Ori outbursts. However a sustained drift barrier or strongly reduced radial grain mobility is necessary to prevent replenishment of carbon from the outer disk.
['protoplanetary disks', 'planets and satellites: composition', 'astrochemistry', 'Astrophysics - Earth and Planetary Astrophysics']
Radial and vertical dust transport inhibit refractory carbon depletion in protoplanetary disks
{'No Header': 'L etter to the E ditor', 'Radial and vertical dust transport inhibit refractory carbon depletion in protoplanetary disks': 'L. Klarmann 1 , C. W. Ormel 1 , and C. Dominik 1  \nAstronomical Institute Anton Pannekoek, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands e-mail: l.a.klarmann@uva.nl  \nReceived June 26, 2018; Accepted September 5, 2018', 'ABSTRACT': 'Context. The Earth is strongly depleted in carbon compared to the dust in the ISM, implying e GLYPH<14> cient removal of refractory carbon before parent body formation. It has been argued that grains get rid of their carbon through oxidation and photolysis in the exposed upper disk layers.  \nAims. We assess the e GLYPH<14> cacy of these C-removal mechanisms accounting for the vertical and radial transport of grains.  \nMethods. We obtain the carbon and carbon free mass budget of solids by solving two 1D advection-di GLYPH<11> usion equations, accounting for the dust grain size distribution and radial transport. The carbon removal acts on the fraction of the grains that are in the exposed layer and requires e GLYPH<14> cient vertical transport.  \nResults. In models without radial transport, oxidation and photolysis can destroy most of the refractory carbon in terrestrial planet formation region. But it only reaches the observed depletion levels for extreme parameter combinations and requires that parent body formation was delayed by 1 Myr. Adding radial transport of solids prevents the depletion entirely, leaving refractory carbon equally distributed throughout the disk.  \nConclusions. It is unlikely that the observed carbon depletion can ultimately be attributed to mechanisms operating on small grains in the disk surface layers. Other mechanisms need to be studied, for example flash heating events or FU Ori outbursts in order to remove carbon quickly and deeply. However, a sustained drift barrier or strongly reduced radial grain mobility are necessary to prevent replenishment of carbon from the outer disk.  \nKey words. Protoplanetary disks - Planets and satellites: composition - Astrochemistry', '1. Introduction': 'The Earth is significantly depleted in carbon (Allègre et al. 2001). Its silicon to carbon ratio is a factor of 10 GLYPH<0> 4 lower than in the Sun (Grevesse et al. 2010) or in the ISM, the base material it formed from (Bergin et al. 2015). The picture is di GLYPH<11> erent for objects formed further out in the solar nebula. The silicon to carbon ratio of carbonaceous chondrites is only a factor 100 lower than in the ISM (Wasson & Kallemeyn 1988), and many comets are not carbon depleted at all (Wooden 2008). This seems to show a gradient of carbon depletion in the solar system, with depletion getting stronger for objects that formed closer to the Sun (e.g. Pontoppidan et al. 2014; Lee et al. 2010; Geiss 1987).  \nHowever, more than half of the carbon in the ISM is expected to be refractory material (Zubko et al. 2004). Several processes have been suggested to remove this refractory carbon from the inner regions of a protoplanetary disks. Gail & Trielo GLYPH<11> (2017) investigate the destruction of refractory carbon species within their radial transport models (see also Gail 2001), but find that oxidation in the disk mid-plane via OH molecules (Finocchi et al. 1997) is not su GLYPH<14> cient to deplete the inner disk region of carbon. Anderson et al. (2017) employ carbon oxidation via atomic oxygen in the hot, upper disk layer (following Lee et al. 2010) and the photolysis of carbon grains directly via UV photons (Alata et al. 2014). In the inner disk regions they reach a carbon depletion comparable to the Earth, but only if all refractory material is in small grains and without taking into account radial dust transport.  \nIn this work, we investigate how the presence of large grains and vertical and radial dust transport influence the refractory carbon in a protoplanetary disk, and the viability of depleting the inner disk region via oxidation and photolysis in the upper disk layers.', '2.1. Disk model': 'For the star we take M ? = 1 M GLYPH<12> , L ? = 1 L GLYPH<12> and the UV field is set to L UV = 0 : 01 L ? . The total disk mass is set to 0.039 M GLYPH<12> . We adopt a dust-to-gas ratio of 0.01, similar to Anderson et al. (2017) and Kamp et al. (2017). The disk surface density follows a power-law profile with GLYPH<6> / r GLYPH<0> 1 and is exponentially cut-o GLYPH<11> at 200 au. The carbon mass fraction fc ( r ) is defined as the ratio of the solid carbon surface density and the total dust surface density, f c = GLYPH<6> c = GLYPH<6> tot. We assume an initial carbon-to-hydrogen abundance of 2 GLYPH<1> 10 GLYPH<0> 4 , which agrees with solar (Asplund et al. 2009) and ISM abundances (Jenkins 2009). Like Anderson et al. (2017), we divide carbon equally between volatiles and refractory grains (see also Zubko et al. 2004). This leads to an initial refractory carbon mass fraction of f c GLYPH<25> 0 : 25.  \nWe use a 1 + 1 D approach to describe the dust movement and composition. The radial dust transport, together with grain growth and fragmentation, is modelled using the twopoppy code by Birnstiel et al. (2012, 2015) and using a fragmentation velocity v f = 10 m s GLYPH<0> 1 throughout the disk (see App. B).  \nFig. 1. Height of the exposed layer divided by the disk radius z 1 = r against disk radius r . Silicate grains are plotted in gray, carbon grains in black. Large grains are settled close to the midplane. In the exposed layer above z 1, carbon grains can be oxidised, indicated in red. The range of vertical dust movement shown by blue arrows. Grains stay for t res in the exposed layer, and within t mix, material from the exposed layer is well-mixed with material from the midplane.  \n<!-- image -->  \nWe refer to the surface layer of the disk where the refractory carbon gets destroyed as the exposed layer , because that layer is exposed to UV photons from the star. This is the layer where carbon reacts with free oxygen and where most of the photolysis occurs. We denote the vertical coordinate of the exposed layer as z 1 and its dust surface density as GLYPH<6> GLYPH<3> .  \nFigure 1 shows the model setup, depicting the exposed layer above a height z 1 as a function of the disk radius r . The blue arrows indicate the dust mixing between the midplane and the exposed layer ( t mix) and the removal and replacement of dust in the exposed layer ( t res) (see Sec. 2.4). In the exposed layer, carbon can be destroyed via oxidation or photolysis, indicated in red.', '2.2. Calculation of the location of the exposed layer': 'The exposed layer is the layer that can be reached by stellar photons. Determining the height z 1 of the exposed layer comes down to determining the height where small grains still coupled to the gas at that height do provide the required optical depth. Because of the flaring geometry of the disk, a radial optical depth GLYPH<28> r = 1 corresponds to a vertical depth GLYPH<28> z = GLYPH<8> where GLYPH<8> = 0 : 05 is the disk flaring angle. The value of z 1 depends on the total dust surface density GLYPH<6> tot, the grain size distribution, the grain opacity GLYPH<20> , and grain settling. We assume that the grains in the exposed layer are in the Rayleigh regime and take GLYPH<20> = GLYPH<20> 0 = 2 GLYPH<1> 10 4 cm 2 = g independent of grain radius s . 1 The surface density GLYPH<6> GLYPH<3> of the exposed layer then follows, GLYPH<6> GLYPH<3> = GLYPH<8> =GLYPH<20> 0. For simplicity, we assume that large grains in the optically geometrical limit, s > s geo = GLYPH<21>= 2 GLYPH<25> GLYPH<25> 0 : 1 GLYPH<22> m for GLYPH<21> = 0 : 55 GLYPH<22> m, do  \nFig. 2. Height z against grain radius s where GLYPH<28> r = 1 at r = 1 au. The point where the z coordinate of GLYPH<28> r peaks defines the height of the exposed layer z 1 and the radius s 1 up to which grains contribute to the opacity in the exposed layer (red arrows). Grains larger than s 1 are too settled to contribute. The size s geo is an upper limit for s 1 as the opacity decreases with larger grain sizes.  \n<!-- image -->  \ns  \ngeo  \nnot contribute to the optical opacity. Then,  \nGLYPH<28> z ( s ; z ) = f GLYPH<20> s f GLYPH<21> z GLYPH<6> GLYPH<20> 0 (1)  \nwhere f GLYPH<20> s = ( s = s max) 4 GLYPH<0> p is the fraction by mass of grains smaller than radius s , assuming a power-law size distribution with exponent with p = 3 : 5 and maximum grain radius s max determined by drift and fragmentation (see App. B). Similarly f GLYPH<21> z is the fraction of the surface density above height z  \nf GLYPH<21> z ( s ; z ) = 1 2 erfc 0 B B B B B @ z p 2 h gr( s ; z ) 1 C C C C C A : (2)  \nIn calculating this fraction we use the dust scale height h gr based on the local Stokes number St corresponding to z and the turbulent GLYPH<11> parameter, h gr = H p GLYPH<11>= ( GLYPH<11> + St( z ; s )). Hence f GLYPH<20> s increases with s while f GLYPH<21> z (for a fixed z ) decreases with s .  \nFigure 2 illustrates these points, plotting the height z where GLYPH<28> r = 1 as function of grain radius s . Grains contribute to the opacity build up in the exposed layer up to a size s 1. Ignoring their settling, all grains would contribute to the exposed layer which would therefore lie very high in the disk. In reality settling causes the largest grains to drop out of the exposed layer. We identify the point where GLYPH<28> r = 1 peaks as the size s 1 and the height z 1. 2', '2.3. Carbon removal': 'Carbon removal in the exposed layer occurs via oxidation or photolysis (see App. A). The oxidation time for one carbon grain in the exposed layer is  \nt ox = 4 3 s 1 GLYPH<26> c n ox v ox Y ox m c (3)  \nwhere GLYPH<26> c = 2 g / cm 3 is the specific density of the carbon grain material, m c the mass of a carbon atom, n ox the number density of oxygen atoms at z 1, v ox the oxygen thermal velocity and Y ox the yield of the oxidation (Draine 1979). The photolysis rate is  \n<!-- image -->  \nd GLYPH<6> c dt = GLYPH<8> F UV Y ph m c f c ; (4)  \nwhere F UV is the UV flux and m c the mass of a carbon atom. The flaring angle GLYPH<8> corrects for the fact that the UV photons do not hit the disk surface perpendicular. The yield is Y ph = 8 GLYPH<1> 10 GLYPH<0> 4 (Alata et al. 2014, 2015). A more detailed description of these processes can be found in Lee et al. (2010) and Anderson et al. (2017) and in Appendix A.', '2.4. Effects of vertical dust transport on removal rates': "The e GLYPH<14> cacy of carbon removal is limited by the ability to vertical transport (cycle) the dust. There are two important timescales regarding to the vertical motions of grains. The first is the overall mixing time t mix, indicating on what timescale material from the midplane and the exposed layer become well-mixed. We obtain t mix from the turbulent di GLYPH<11> usivity and the gas scale height H :  \nt mix = H 2 GLYPH<23> t = 1 GLYPH<10> GLYPH<11> = 100 yr GLYPH<18> GLYPH<11> 10 GLYPH<0> 2 GLYPH<19> GLYPH<0> 1 GLYPH<18> r au GLYPH<19> 3 = 2 (5)  \nwhere the turbulent di GLYPH<11> usivity is assumed equal to the gas viscosity GLYPH<23> t and is parametrised using the GLYPH<11> prescription ( GLYPH<23> t = GLYPH<11> H 2 GLYPH<10> ; Shakura & Sunyaev 1973) with GLYPH<10> the Keplerian frequency. After one mixing timescale, carbon-depleted material  \nFig. 3. All panels show the carbon fraction f c = GLYPH<6> c = GLYPH<6> tot as function of disk radius r . Coloured lines show the time evolution of the carbon fraction f c in the fiducial model from 10 4 yr to 10 6 yr. The vertical line indicates Earth's position. The horizontal dashed line shows depletion by a factor ten. Left: carbon removal by oxidation. Right: carbon removal by photolysis. Top panels: the oxidation (photolysis) fc GLYPH<6> GLYPH<3> = t ox ( fc GLYPH<6> GLYPH<3> ph = t ph) is applied to the carbon in the exposed layer unrestricted by vertical or radial transport. Middle panels: vertical dust transport is taken into account as described in Eq. 7 (Eq. A.9). Bottom panels: vertical and radial dust transport as described in App. B are included. The scale of the y-axis has been changed because of the low level of carbon depletion in these models.  \nfrom the exposed layer and carbon-rich material from the midplane are well mixed.  \nThe other relevant timescale is the residence time 3 of a grain in the exposed layer  \nt res = H z 1 ! 2 1 GLYPH<10> GLYPH<11> = 11 yr GLYPH<18> z 1 3 H GLYPH<19> GLYPH<0> 2 GLYPH<18> GLYPH<11> 10 GLYPH<0> 2 GLYPH<19> GLYPH<0> 1 GLYPH<18> r au GLYPH<19> 3 = 2 : (6)  \nThis time is shorter than t mix because the local pressure scale height at height z GLYPH<29> H is given by H 2 = z . A long t res would limit carbon destruction, because the exposed layer will become depleted in refractory carbon. In that case no carbon will be burned, because carbon-free solids build up the opacity in the exposed layer. Hence, carbon removal becomes ine GLYPH<14> cient when t res is longer than the time to burn a single grain t ox (see Eq. (3)).  \nAccounting for these vertical transport e GLYPH<11> ects, we obtain a carbon destruction rate of:  \nd GLYPH<6> c dt = 2 f c GLYPH<1> min GLYPH<6> GLYPH<3> t ox ; GLYPH<6> GLYPH<3> t res ; f GLYPH<20> s 1 GLYPH<6> tot t mix ! (7)  \nwhere the factor 2 accounts for the two sides of the disk and GLYPH<6> c is the surface density of carbon grains with s < s 1. This expression applies to C-burning; in App. A.2 a similar expression is derived for photolysis.  \nGLYPH<1> Table 1. Carbon fraction f c = GLYPH<6> c = GLYPH<6> tot after 1 Myr at 1 au for models limited by vertical transport and models limited by vertical and radial transport. Initial carbon fraction is f c = 0 : 25. Only one parameter is varied with respect to the fiducial model.", '3. Results': 'Our results are presented in Fig. 3 for the oxidation (left) and the photolysis (right) models.', '3.1. Unrestricted models': 'The upper row plots present the carbon fraction f c without including any transport-limiting factors, i.e., by using only the first term in Eq. (7) ( fc GLYPH<6> GLYPH<3> = t ox). As can be seen, oxidation depletes carbon by a factor of 10 GLYPH<0> 4 out to 1.1 au. Beyond this point carbon burning is essentially shut o GLYPH<11> , because of the exponential dependence of the oxidation yield Y ox on temperature. The photolysis rate, on the other hand, does not depend on temperature ( F UV / r GLYPH<0> 2 but Y ph is constant). After 1 Myr, the disk is depleted by a factor of 10 GLYPH<0> 4 out to 1.3 au and by a factor of 0.1 out to 7 au.', '3.2. Vertical dust transport': 'Accounting for vertical transport e GLYPH<11> ects - i.e., including all three regimes in Eq. (7) - we see that oxidation (Fig. 3c) and photolysis (Fig. 3d) become less e GLYPH<11> ective. Carbon oxidation inward of 1.3 au becomes now limited by the residence time t res. Grains that make it into the exposed layer burn their carbon atoms completely, rendering the overall burning ine GLYPH<14> cient. Similarly, the photolysis rate equals f c GLYPH<6> GLYPH<3> ph = t res GLYPH<0> ph everywhere. The photolysis rate tends to be larger than the oxidation rate, because the UV photons penetrate more deeply, resulting in a larger exposed layer (see App. A.2). However, the photolysis run is just short of reaching depletion levels of 10 GLYPH<0> 4 at 1 au.', '3.3. Vertical and radial dust transport': 'Accounting in addition for radial transport (bottom panels of Fig. 3) further reduces the e GLYPH<14> cacy of carbon destruction. Carbon-rich solids from the outer disk simply drift into the inner region to replenish any carbon depleted material. The carbon destruction becomes drift-limited: carbon will only be depleted when the local destruction time ( t destr = GLYPH<6> c = ( d GLYPH<6> c = dt )) becomes shorter than the drift timescale t drift of the (mass-dominating) pebbles. Since t drift tends to be rather short (200 yr at 1 au) depletion is minimal with little variation throughout the disk.', '3.4. Parameter variation': 'In Table 1 we list the results from additional photolysis runs, quantifying the level of carbon depletion at 1 au after 1 Myr. A higher opacity, a lower GLYPH<11> or a shallower grain size distribution ( p = 3; fewer small grains) only reduce the carbon destruction,  \nbecause GLYPH<6> GLYPH<3> becomes lower or t res increases. For the runs without radial transport, a stronger UV field enhances the depletion as the UV photons penetrate deeper. It can be argued that the low GLYPH<20> 0 run is more appropriate for the opacity in the exposed layer, when the grains lose most of their carbon. This will increase GLYPH<6> GLYPH<3> , and completely remove all carbon interior to 1 au - but only when there is no radial replenishment. Similarly, reducing the fragmentation velocity to 3 m / s leads to more small grains and therefore a stronger depletion at 1 au in the case of only vertical transport. When radial transport is included, the carbon fraction is only reduced by a factor of two. Since the fragmentation velocity outside of the snowline is expected to be much higher than 3 m / s, a more realistic case where the fragmentation velocity depends on the disk radius would lead to even less carbon depletion.  \nIn general, the replenishment of carbon via radial transport renders the depletion independent of the adopted parameters.', '4. Discussion and Conclusion': "Our findings regarding unrestricted C-burning are in line with the study of Anderson et al. (2017). They, too, find that photolysis is the more significant C-depletion mechanism and that the inner disk can become devoid in carbon. Like us, Anderson et al. (2017) account for the limited total time grains spend in the exposed layers. However, they have overestimated the removal by incorrectly assuming that each stay in the exposed layer is short enough to ensure a continuous supply of carbon in the exposed layer. Instead, we found (even in the case of high turbulence) that carbon-depleted grains stay longer in the exposed layer than it takes to remove their carbon. This makes both oxidation and photolysis ine GLYPH<14> cient. Furthermore, adding radial dust transport to the model makes it impossible to deplete the inner disk of carbon even under extreme assumptions about the UV field of the early Sun or the grain opacities.  \nTherefore, we conclude that the only way to ensure the C depletion factors as observed in the Solar System is to invoke an early hot or intense inner disk environment to ensure rapid C-destruction before parent body formation. This needs to happen in combination with a sustained barrier for drift to prevent C-replenishment, for example by the formation of a giant planet. FU Orionis events can lead to inner disk temperatures of several  \nIt is unlikely that other C-destruction mechanisms can change this outcome. Adding oxidation of carbon by OH in the midplane does not increase the carbon depletion su GLYPH<14> ciently (Gail & Trielo GLYPH<11> 2017). For high accretion rates, the midplane region can be heated to roughly 1500 K out to 2 au (Min et al. 2011). However, these accretion rates must then be sustained over a significant time to allow the Earth's building blocks to form.  \nthousand Kelvin (Hartmann & Kenyon 1996; Zhu et al. 2007) over several decades. The composition of Chondrules indicate that several flash heating events happened in the solar nebula (Ciesla 2005; Jones et al. 2000), reaching temperatures of around 2000 K nearly instantly and cooling again within days. To prevent the fast replenishment of carbon, these events need to happen at a high frequency, comparable to the drift timescale. Events at a lower rate can also cause a sustained refractory carbon depletion in the inner disk region, but this requires radial grain mobility to be strongly reduced or halted.  \nAcknowledgements The authors thank the referee for valuable feedback. The authors thank Inga Kamp for sharing her ProDiMo model, Christian Rab for sharing prodimopy , Michiel Min for sharing OpacityTool and Til Birnstiel for sharing twopoppy . We thank the above-mentioned and Rens Waters and Kaustubh Hakim for helpful discussions and Sebastiaan Krijt for his thoughtful comments on the manuscript. L.K. is supported by a grant from NOVA. C.W.O. acknowledges funding by the Netherlands Organization for Scientific Research (NWO; VIDI project 639.042.422). C.D. acknowledges funding by the NWO, project number 614.001.552.", 'Appendix A.1: Oxidation': "The exposed layer is characterized by a steep vertical gradient in gas temperature. However, most of the dust in the exposed layer will be found just above z 1, corresponding to a radial optical depth of unity. Since the penetration of UV photons is also responsible for the heating of the gas, this location can be very well characterized by a single temperature. Based on previous observations and thermo-chemical disk modelling (Fedele et al. 2016; Kamp et al. 2017), we describe the gas temperature in the exposed layer by  \nT g = T i r r i ! GLYPH<0> q : (A.1)  \nwith T i = 750 K, r i = 1 au and q = 0 : 6. This temperature leads to a mean thermal velocity of oxygen atoms of  \nv ox = s 8 k B T g GLYPH<25> m ox ; (A.2)  \nwhere k B is Boltzmann's constant and m ox is the mass of an oxygen atom.  \nThe gas number density at z 1 is then  \nn g = GLYPH<6> g p 2 GLYPH<25>GLYPH<22> m p H exp 0 B B B B @ GLYPH<0> z 2 1 2 H 2 1 C C C C A ; (A.3)  \nwhere GLYPH<6> g is the gas surface density, GLYPH<22> = 2 : 35 the mean molecular weight and m p the proton mass. We use the oxygen number density at z 1, n ox = GLYPH<15> n g, with GLYPH<15> = 10 GLYPH<0> 4 as found in ProDiMo models by Meijerink et al. (2012) and similar to the value of GLYPH<15> GLYPH<25> 2 GLYPH<1> 10 GLYPH<0> 4 shown in Lee et al. (2010).  \nThe probability of removing a carbon atom when a carbon grain is hit by an oxygen atom is given by the yield  \nY ox = A exp ( GLYPH<0> B = T g) ; (A.4)  \nwith A = 2 : 3 ; B = 2580 for T g < 440 K and A = 170 ; B = 4430 for T g > 440 K (Draine 1979).  \nThe rate at which carbon is removed from a single grain by oxidation is  \nk ox = n ox v ox GLYPH<27> Y ox ; (A.5)  \nwhere GLYPH<27> = GLYPH<25> s 2 is the grain cross section. For grains with radius s 1 this leads to a carbon destruction time of:  \nt ox = m gr m c k ox = 4 3 s 1 GLYPH<26> c n ox v ox Y ox m c (A.6)  \nwith m gr the mass of a carbon grain. Grains lose all their carbon when they reside for a time t GLYPH<29> t ox in the exposed layer. The change in grain radius during oxidation is not taken into account in our model.", 'Appendix A.2: Photolysis': 'In the case of photolysis by UV photons, a fraction f c Y ph of the absorbed photons will remove a carbon atom directly, resulting in a destruction rate of:  \nd GLYPH<6> c dt ! ph GLYPH<0> unrestricted = GLYPH<8> F UV Y ph m c f c  \n(A.7) s drift = f d 2 GLYPH<25> GLYPH<6> d GLYPH<26> s ( GLYPH<10> r ) 2 c 2 s GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> d ln P d ln r GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<0> 1 ; (B.3)  \nwhere F UV is the UV field. The flaring angle GLYPH<8> corrects for the fact that the UV photons do not hit the disk surface perpendicular. The yield is Y ph = 8 GLYPH<1> 10 GLYPH<0> 4 (Alata et al. 2014, 2015; Anderson et al. 2017).  \nAnalogous to oxidation, photolysis can also be limited by the residence timescale. However, in the case of photolysis UV photons can reach disk layers below the optical GLYPH<28> r = 1 line, due to forward scattering into the disk (van Zadelho GLYPH<11> et al. 2003). We calculate the height z 1 of the layer exposed to UV radiation by equating t res, the residence time, with t ph, the time to destroy a carbon grain entirely by photolysis:  \nH z 1 ! 2 1 GLYPH<10> GLYPH<11> = 4 3 s 1 GLYPH<26> c F UV Y ph m c exp ( GLYPH<28> r ( s 1 ; z 1)) (A.8)  \nHere, the exponential factor expresses the attenuation of the UV field within the disk. Analogous to the GLYPH<28> r = 1 constraint for the oxidation case, we use this equation to find the height of the exposed layer z 1 ; ph, the value of the optical depth at z 1 ; ph, GLYPH<28> r ; ph (now generally larger than unity) and the amount of exposed material GLYPH<6> GLYPH<3> ph = GLYPH<28> r ; ph GLYPH<8> =GLYPH<20> . Once the layer that is exposed to photolysis is thus characterised, the carbon destruction is calculated using the rates as shown in Eq. (7). Using Eq. A.7 for the unrestricted photolyis rate, the carbon removal rate then becomes:  \nd GLYPH<6> c dt ! ph = 2 f c GLYPH<1> min 0 B B B B @ GLYPH<6> GLYPH<3> ph t ph ; GLYPH<6> GLYPH<3> ph t res GLYPH<0> ph ; f GLYPH<20> s 1 GLYPH<6> tot t mix 1 C C C C A (A.9)  \nwhere t res GLYPH<0> ph now follows from the solution to Eq. (A.8).', 'Appendix B: Radial dust transport': 'We use the twopoppy code by Birnstiel et al. (2012, 2015) to model the radial movement of dust grains. In this code, the dust mass is assigned to two grain sizes, small and large grains. The small grain radius is chosen so that the grains are well coupled to the gas. The radius of the large grains (as well as the ratio of the mass distribution) depends on the local disk conditions, is limited (mainly) by drift and fragmentation and is updated after each timestep. This makes it possible to describe the dust evolution of the disk based on two surface densities. We give here only a short overview over the most important concepts and formulas.  \nAssuming that the Epstein limit of the drag law applies to all relevant grain sizes in the entire disk, compact spherical grains, a self-similar gas surface density profile and an eddy turn over time of t L = 1 = GLYPH<10> , the Stokes number Stmid of a grain with radius s and specific density GLYPH<26> s at the midplane can be written as  \nStmid = s GLYPH<26> s GLYPH<6> g GLYPH<25> 2 : (B.1)  \nThe small, well-coupled grains have a fixed radius, which we assume to be s s = 0 : 1 GLYPH<22> m. The size of the large grains is limited by fragmentation and drift. The size limit due to fragmentation is given by  \ns frag = f f 2 3 GLYPH<25> GLYPH<6> g GLYPH<26> s GLYPH<11> v 2 f c 2 s ; (B.2)  \nwhere c s is the isothermal sound speed and f f is a calibration factor of order unity, leading to grain size slightly below the the fragmentation limit. The size limit due to drift is given by  \nwhere P is the gas pressure and f d is a calibration factor similar to f f . The radius of the large grains is than chosen as s l = min GLYPH<16> s drift ; s frag GLYPH<17> .  \nIn the fragmentation limited regime, the surface density fraction in large grains is f l = 0 : 75, and f l = 0 : 97 in the drift limited regime. We emphasise that these mass distributions and grain sizes are used for the radial and not the vertical dust transport. Only the maximum possible grain size s max = sl is used in the calculation of s 1 and z 1 in Section 2.2.  \nThe radial transport of the dust can now be described by solving two advection-di GLYPH<11> usion equations:  \n@ GLYPH<6> i @ t + 1 r @ @ r " r GLYPH<6> i v i GLYPH<0> Di GLYPH<6> g @ @ r GLYPH<6> i GLYPH<6> g !!# = 0 (B.4)  \nwhere GLYPH<6> i is the surface density of the dust grain, v i the dust velocity due to drift and gas drag, Di the di GLYPH<11> usivity of the species and the index i refers to the small (s) and the large (l) grains. For the small dust component v i will be equal to the gas accretion velocity while for the large component the drift velocity and the gas accreation velocity both contribute. Since the Stokes number of a grain is always smaller than unity in this model, the dust di GLYPH<11> usivity is assumed to be equal to the gas di GLYPH<11> usivity D g which is considered the same as the gas viscosity.  \nTo follow not only the dust mass and size but also the dust composition, we have modified this code. We use four instead of two types of grains, small and large carbon grains ( s c ; s, s c ; l) and small and large silicate grains ( s s ; s, s s ; l). As in the original code, f l is used to distribute the surface density between the particles sizes, and we use f c to distribute the surface density between the di GLYPH<11> erent grain species. The advection-di GLYPH<11> usion equation for small carbon grains now includes a sink term and reads:  \n@ GLYPH<6> c ; s @ t + 1 r @ @ r " r GLYPH<6> c ; s v s GLYPH<0> D s GLYPH<6> g @ @ r GLYPH<6> c ; s GLYPH<6> g !!# = GLYPH<0> d GLYPH<6> c dt (B.5)  \nThe advection-di GLYPH<11> usion equation for large carbon grains has no sink term:  \n@ GLYPH<6> c ; l @ t + 1 r @ @ r " r GLYPH<6> c ; l v l GLYPH<0> Dl GLYPH<6> g @ @ r GLYPH<6> c ; l GLYPH<6> g !!# = 0 (B.6)  \nAs in the case of large carbon grains, the advection-di GLYPH<11> usion equations for small as well as large silicate grains do not include a sink term.  \nThrough all these calculations, we assume a constant, selfsimilar gas surface density profile. The surface density of the destroyed carbon is not added to the gas surface density, but completely removed from the system.  \nOnce the advection-di GLYPH<11> usion equation has been solved, f c is again calculated, since the radial transport of large and small grains changes the carbon distribution within the disk. By recalculating a new common carbon fraction for both grain sizes after each timestep, we assume that coagulation and fragmentation have been e GLYPH<14> cient enough to re-distribute carbon between large and small grains. This maximises the carbon removal efficiency. This carbon fraction value is now the base for the calculation of the carbon destruction. When used in Eq. (7) and Eq. (A.9), it does not indicate the carbon fraction in the exposed layer, but is used to calculate the total amount of carbon available in the exposed layer at the onset of oxidation or photolysis.  \nTable B.1. List of notations'}