The Effect of Carbon Grain Destruction on the Chemical Structure of Protoplanetary Disks

We use physical models of PPDs generated using the methodology described in Nomura & Millar (2005) with the addition of X-ray heating as described in Nomura et al. (2007). We model an axisymmetric Keplerian disk, and two types of central stars are considered. The first is a typical T Tauri star with mass M_* = 0.5 M_☉, radius R_* = 2.0 R_☉, and effective temperature T_* = 4000 K (Kenyon & Hartmann 1995). The second is a Herbig Ae star with mass M_* = 2.5 M_☉, radius R_* =2.0 R_☉, and effective temperature T_* = 10,000 K (e.g., Palla & Stahler 1993).

The gas density profiles are determined by assuming vertical hydrostatic equilibrium, the balance between gravity and the pressure gradient force. In order to obtain the gas surface density profiles, we adopt the viscous α-disk model (Shakura & Sunyaev 1973), with a viscous parameter α = 10⁻² and a mass accretion rate $\dot{M}={10}^{-8}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$. The temperature profile of the gas is calculated self-consistently with the gas density profiles by assuming local thermal balance between gas heating and cooling. The thermal processes of heating from X-ray ionization of hydrogen, grain photoelectric heating induced by far-ultraviolet photons, and cooling via gas–grain collisions and line transitions are taken into account. The dust temperature profiles are obtained by assuming local radiative equilibrium between the blackbody emission of grains and the absorption of radiation from the central star, as well as the surrounding dust grains.

Regarding dust, we assume that the dust and gas are well coupled and that the dust-to-gas mass ratio is constant (0.01) throughout the disk. The dust size distribution model in Weingartner & Draine (2001), which reproduces the observational extinction curve of dense clouds, is adopted. The dust properties are important factors for both gas and dust temperatures. The adopted dust opacity is described in Appendix D of Nomura & Millar (2005).

For the UV radiation field two radiation sources are taken into account: radiation from the central star and the ISM. In the case of the T Tauri star, we use a UV-excess model that reproduces the observational data toward TW Hydrae. The components of the stellar UV radiation are blackbody emission, hydrogenic thermal bremsstrahlung emission, and Lyα line emission (see Appendix C of Nomura & Millar 2005 for details). For the Herbig Ae star, the UV excess contributes a significantly smaller fraction to the total UV luminosity compared with that from the stellar blackbody radiation (e.g., Dent et al. 2013). Therefore, we use only the blackbody emission of the central star as the source of the UV radiation field for the Herbig Ae model. For the X-ray radiation, we adopt a TW-Hydrae-like spectrum with a total luminosity ${L}_{X}\sim {10}^{30}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ for the T Tauri star (Preibisch et al. 2005). Due to the nonconvective interiors of Herbig Ae stars, the typical X-ray luminosities are ≳10 times lower than those of T Tauri stars (Güdel & Nazé 2009). Therefore, we assume an X-ray spectrum with ${L}_{X}\sim 3\times {10}^{29}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ and T_X ∼ 1.0 keV for the Herbig Ae star based on observations (e.g., Zinnecker & Preibisch 1994; Hamaguchi et al. 2005).

For full details on the generation of the physical models, see Nomura & Millar (2005) and Nomura et al. (2007). The total grid numbers are 8699 and 12,116 for the disks around T Tauri and Herbig Ae stars, respectively, where 129 radial steps are taken logarithmically for the disk radius from 0.04 to 305 au. Figure 2 shows the physical structure of the disk around the T Tauri star and the Herbig Ae star (see also Walsh et al. 2015; Notsu et al. 2016, 2017). From the top, the gas number density decreases as a function of height and radius. The densest region reaches about 10¹⁵ cm⁻³ near the disk midplane, falling to 10⁵ cm⁻³ in the surface layer, showing the diversity of the environment in the PPDs. The gas and dust temperatures decouple in the diffuse region owing to inefficient collisions between the gas and dust grains. As the density increases toward the disk midplane, the gas and dust temperatures become well coupled. The UV flux from the central star is much stronger in the case of the Herbig Ae star (right panels) than in the T Tauri star (left panels). However, both UV and X-ray photons are absorbed by dust and gas and do not penetrate to the midplane. Since the Herbig Ae star has a higher luminosity, its enhanced gas and dust temperatures make line emission easier to detect than in the T Tauri case (see Section 3.3). Note that different radial ranges are shown in Figure 2 for each type of disk.

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We use a chemical reaction network, containing both gas-phase reactions and gas–grain interactions, to study the chemical evolution of the disks. In order to examine the effect of carbon grain destruction (Lee et al. 2010; Anderson et al. 2017), we focus on those small molecules that are not much affected by grain surface reactions and consider two initial values for the elemental abundance of carbon. Except the abundance of carbon, for both models without and with carbon grain destruction, we use the typical low-metal value often adopted to model the chemistry of dark clouds such as TMC-1 (Table 8 of Woodall et al. 2007; see also Table 1). In the model without carbon grain destruction, the elemental abundance of carbon in gas is the same as that in diffuse clouds, in which the gas-phase elemental abundance of carbon is lower than that of oxygen. For the model with carbon grain destruction, we consider an extreme case assuming that all the carbon in the grains is destroyed and released into the gas phase, and we set the abundance for ionized carbon to be the solar abundance of carbon, ${{\rm{C}}}^{+}=2.95\times {10}^{-4}$ (e.g., Asplund et al. 2009; Draine 2011). In this case, the gas-phase elemental abundance of carbon is larger than that of oxygen. We note that the C/O ratio is larger than unity in the gas phase in this extreme case, although the C/O ratio of the solar photosphere is around 0.5 (Allende Prieto et al. 2002). This is because a large fraction of available elemental oxygen is incorporated into silicate grains, which are more difficult to destroy than carbon grains. According to Lee et al. (2010), if the grain size is small enough, carbon grains could be sputtered into the gas phase in the surface layer of the disks where the gas temperature is high. If turbulent mixing in the vertical direction of the disk is efficient enough, a significant amount of the small carbon grains could be sputtered inside a disk radius of around 10 au in the case of T Tauri disks. Therefore, in this paper we only consider chemistry in the very inner region and treat the T Tauri disk up to 10 au and the Herbig Ae disk, for which the UV irradiation is stronger, up to 50 au. We calculate the chemical evolution up to 10⁶ yr, which is the typical lifetime for PPDs and at which point the chemical structure has almost reached a steady state in the disk. We note that for a more realistic model, the evolution of physical conditions, such as density, temperature, and radiation field, should be treated together with the time evolution of molecular abundances. This is beyond the scope of this work, and for simplicity we use the chemical composition of diffuse clouds as an initial condition and use the fixed physical conditions of each PPD for the calculation of chemical reactions. Also, and again for simplicity, we ignore the change in the dust mass and surface area caused by the carbon grain destruction and the freeze-out of gas-phase molecules on grains during the calculation of chemical reactions.

Table 1.  The Initial Gas-phase Elemental Abundances Relative to Total Hydrogen Nuclei in the Model without and with Carbon Grain Destruction

Element		Abundance
	Without C Grain Destruction		With C Grain Destruction
$\mathrm{He}$	⋯	0.14	⋯
${{\rm{C}}}^{+}$	7.30 × 10−5	⋯	2.95 × 10−4
${\rm{N}}$	⋯	2.14 × 10−5	⋯
${\rm{O}}$	⋯	1.76 × 10−4	⋯
${\mathrm{Na}}^{+}$	⋯	3.00 × 10−9	⋯
${\mathrm{Mg}}^{+}$	⋯	3.00 × 10−9	⋯
${\mathrm{Si}}^{+}$	⋯	3.00 × 10−9	⋯
${{\rm{S}}}^{+}$	⋯	2.00 × 10−8	⋯
${\mathrm{Cl}}^{+}$	⋯	3.00 × 10−9	⋯
${\mathrm{Fe}}^{+}$	⋯	3.00 × 10−9	⋯

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2.2.1. Gas-phase Reactions

We use the RATE06 release of the UMIST Database for Astrochemistry, for the gas-phase chemistry (Woodall et al. 2007). In this chemical network, there are 375 species and 4336 reactions, including 3957 two-body reactions, 214 photoreactions, 154 X-ray/cosmic-ray-induced photoreactions, and 11 reactions of direct X-ray/cosmic-ray ionization. Since the reactions including fluorine, F, and phosphorus, P, have a low impact on the chemistry (Walsh et al. 2010), we remove reactions containing these two elements to reduce the computation time. For the photoreactions, we calculate the UV radiation at each point in the disk as

$$\begin{eqnarray}&&{G}_{\mathrm{FUV}}(r,z)={\int }_{912\,\mathring{\rm A} (13.6\,\mathrm{eV})}^{2068\,\mathring{\rm A} (6\,\mathrm{eV})}{G}_{\mathrm{FUV}}(\lambda ,r,z)d\lambda .\end{eqnarray} \tag{ 1 }$$

The UV radiation is scaled by the interstellar UV flux, ${G}_{0}=2.67\times {10}^{-3}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$ (van Dishoeck et al. 2006), and photoreaction rates at each point in the disk can be approximated as

$$\begin{eqnarray}&&{k}^{\mathrm{ph}}(r,z)=\displaystyle \frac{{G}_{\mathrm{FUV}}(r,z)}{{G}_{0}}{k}_{0}\quad {{\rm{s}}}^{-1},\end{eqnarray} \tag{ 2 }$$

where k₀ is the unshielded photoreaction rate due to the interstellar UV radiation field as compiled in RATE06 (see also Millar et al. 1997).

2.2.2. Gas–grain Interaction

The gas–grain interactions included are the adsorption (freeze-out) and desorption of molecules on and from dust grain surfaces, respectively. When the gas temperature is low enough, the adsorption rate becomes larger than the desorption rate, and therefore molecules freeze onto dust grains. Otherwise, molecules sublimate from dust grains and into the gas phase, so-called "thermal desorption." The rate of thermal desorption, and thus the temperature at which sublimation happens, will depend on the binding energy of each molecule to dust grains. The binding energies of several important molecules adopted in their work are listed in Table 2. In addition to thermal desorption, we consider nonthermal desorption mechanisms: cosmic-ray-induced thermal desorption (Leger et al. 1985; Hasegawa & Herbst 1993) and UV photodesorption (Westley et al. 1995; Willacy & Langer 2000; Öberg et al. 2007). The details of reaction rates for these processes can be found in the Appendix (see also Walsh et al. 2010; Notsu et al. 2016).

Table 2.  The Locations of the Snowlines in T Tauri and Herbig Ae Disks Determined by the Temperature Profiles Given in Figure 2

Molecule	Binding Energy (K)	Snowline in T Tauri Disk (au)	Snowline in HAe Disk (au)
CO	855	>10	>50
CH4	1080	>10	>50
C2H2	2400	9.33	>50
CO2	2990	5.34	37.65
C5	3220	2.66	32.75
C6	3620	2.48	24.77
C6H	3880	2.31	21.54
HCN	4170	2.31	18.74
C7	4430	2.15	17.48
C8	4830	2.01	14.18
H2O	4820	2.01	14.18
HCOOH	5000	2.01	13.22
C9	5640	1.75	10.00
C10	6000	1.52	8.70

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Considering the processes of freeze-out, thermal desorption, cosmic-ray-induced desorption, and photodesorption, the differential equation for the number density of species i on the dust grain is written as (e.g., Hasegawa et al. 1992)

$$\begin{eqnarray}&&\displaystyle \frac{{{dn}}_{i,\mathrm{ice}}}{{dt}}={n}_{i}{k}_{i}^{a}-{n}_{i,\mathrm{ice}}^{\mathrm{desorb}}({k}_{i}^{d}+{k}_{i}^{\mathrm{crd}}+{k}_{i}^{\mathrm{pd}}),\end{eqnarray} \tag{ 3 }$$

where k_i^a is the adsorption rate, k_i^d is the thermal desorption rate, k_i^crd represents the cosmic-ray-induced thermal desorption rate, and k_i^pd denotes the photodesorption rate of species i. n_i is the gas-phase number density of species i, and ${n}_{i,\mathrm{ice}}^{\mathrm{desorb}}$ is the number density of species i located in the uppermost active layers of the ice mantle. The value of ${n}_{i,\mathrm{ice}}^{\mathrm{desorb}}$ is given by Aikawa et al. (1996) as

$$\begin{eqnarray}{n}_{i,\mathrm{ice}}^{\mathrm{desorb}}=\left\{\begin{array}{ll}{n}_{i,\mathrm{ice}} & ({n}_{\mathrm{ice}}\lt {n}_{\mathrm{act}})\\ {n}_{\mathrm{act}}\tfrac{{n}_{i,\mathrm{ice}}}{{n}_{\mathrm{ice}}} & ({n}_{\mathrm{ice}}\geqslant {n}_{\mathrm{act}})\end{array}\right.,\end{eqnarray} \tag{ 4 }$$

where n_ice is the total ice number density of all species (see also Walsh et al. 2014), n_act = $4\pi {r}_{d}^{2}{n}_{d}{n}_{s}{N}_{\mathrm{LAY}}$ represents the number of active surface sites in the ice mantle per unit volume, and N_LAY is the number of surface layers considered as active, assumed to be 2. r_d is the dust grain radius, n_s represents the surface density of sites, and n_d is the number density of dust grains. For more details, please refer to the Appendix.

In order to predict observational signatures of carbon grain destruction in the inner region of PPDs, we perform ray-tracing calculations to estimate the intensity of molecular line emission and obtain intensity maps for models both with and without carbon grain destruction. The line data are taken from the Leiden Atomic and Molecular Database (Schöier et al. 2005), the Cologne Database for Molecular Spectroscopy (Müller et al. 2005), or the Jet Propulsion Laboratory molecular spectroscopic database (Pickett et al. 1998). We modify the original 1D code, RATRAN (Hogerheijde & van der Tak 2000), to calculate the ray-tracing using an axisymmetric 2D disk structure under the assumption of local thermodynamic equilibrium.

The intensity of the line profile at a frequency ν, I_ul(ν), is obtained by solving the radiative transfer equation along the line of sight s in the disk

$$\begin{eqnarray}&&\displaystyle \frac{{{dI}}_{{ul}}(\nu )}{{ds}}=-{\chi }_{{ul}}(\nu )({I}_{{ul}}(\nu )-{S}_{{ul}}(\nu )),\end{eqnarray} \tag{ 5 }$$

where ${\chi }_{{ul}}(\nu )$ is the total extinction coefficient of dust grains and molecular lines and S_ul(ν) is the source function given by

$$\begin{eqnarray}&&{\chi }_{{ul}}(\nu )={\rho }_{d}{\kappa }_{{ul}}+\displaystyle \frac{h{\nu }_{{ul}}}{4\pi }({n}_{l}{B}_{{lu}}-{n}_{u}{B}_{{ul}}){{\rm{\Phi }}}_{{ul}}(\nu )\end{eqnarray} \tag{ 6 }$$

and

$$\begin{eqnarray}&&{S}_{{ul}}(\nu )=\displaystyle \frac{1}{{\chi }_{{ul}}(\nu )}\displaystyle \frac{h{\nu }_{{ul}}}{4\pi }{n}_{u}{A}_{{ul}}{{\rm{\Phi }}}_{{ul}}(\nu ),\end{eqnarray} \tag{ 7 }$$

respectively, where A_ul and B_ul are the Einstein A- and B-coefficients for spontaneous and stimulated emission and B_lu is the Einstein B-coefficient for absorption. n_u and n_l are the number densities of the molecules in the upper and lower levels, respectively. ρ_d is the mass density of dust grains where we simply apply the dust-to-gas mass ratio of ρ_d/ρ_g = 1/100. κ_ul is the dust absorption coefficient at a frequency ν_ul, and h is Plank's constant.

Here Φ(ν) is the line profile function, which is affected by thermal broadening and Keplerian rotation in the disk. As a result, Φ(ν) is given by

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{ul}}(\nu )=\displaystyle \frac{1}{{\rm{\Delta }}{\nu }_{D}\sqrt{\pi }}\exp \left[\displaystyle \frac{-{\left(\nu +{\nu }_{K}-{\nu }_{{ul}}\right)}^{2}}{{\rm{\Delta }}{\nu }_{D}^{2}}\right],\end{eqnarray} \tag{ 8 }$$

where ${\rm{\Delta }}{\nu }_{D}=({\nu }_{{ul}}/c)\sqrt{2{k}_{{\rm{B}}}{T}_{g}/{m}_{i}}$ is the Doppler width, c is the speed of light, T_g represents the temperature of gas, and m_i is the mass of the species, i. ν_K denotes the Doppler shift due to the projected Keplerian velocity along the line of sight and is given by

$$\begin{eqnarray}&&{\nu }_{K}=\displaystyle \frac{{\nu }_{{ul}}}{c}\sqrt{\displaystyle \frac{{{GM}}_{* }}{r}}\sin \phi \sin \theta ,\end{eqnarray} \tag{ 9 }$$

where G is the gravitational constant, M_* is the mass of the central star, r is the distance of the line-emitting region from the central star, ϕ is the azimuthal angle between the semimajor axis and the line linking the point in the disk along the line of sight and the center of the disk, and θ is the inclination angle of the disk.

In order to predict the observable flux density, we integrate Equation (5) along the line of sight s. The intensity at (x, y) in the projected plane is given by

$$\begin{eqnarray}&&I(x,y,\nu )={\int }_{-{s}_{\infty }}^{{s}_{\infty }}{j}_{{ul}}(s,x,y,\nu )\exp (-{\tau }_{{ul}}(s,x,y,\nu )){ds},\end{eqnarray} \tag{ 10 }$$

where j_ul is the emissivity at the position $(s,x,y)$ and frequency ν, given by

$$\begin{eqnarray}&&{j}_{{ul}}(s,x,y,\nu )={n}_{u}(s,x,y){A}_{{ul}}\displaystyle \frac{h{\nu }_{{ul}}}{4\pi }{{\rm{\Phi }}}_{{ul}}(s,x,y,\nu ),\end{eqnarray} \tag{ 11 }$$

and ${\tau }_{{ul}}(s,x,y,\nu )$ is the optical depth from the line-emitting point s to the disk surface ${s}_{\infty }$ at the frequency ν, expressed by

$$\begin{eqnarray}&&{\tau }_{{ul}}(s,x,y,\nu )={\int }_{s}^{{s}_{\infty }}{\chi }_{{ul}}(s^{\prime} ,x,y,\nu ){ds}^{\prime} .\end{eqnarray} \tag{ 12 }$$

The observable line flux integrated all over the disk is given by

$$\begin{eqnarray}&&{F}_{{ul}}(\nu )=\displaystyle \frac{1}{4\pi {d}^{2}}\iint I(x,y,\nu ){dxdy},\end{eqnarray} \tag{ 13 }$$

where d is the distance between the observer and the target object.

In this subsection, we show the effect of carbon grain destruction on carbon-bearing species (HCN, ${\mathrm{CH}}_{4}$, ${{\rm{C}}}_{2}{{\rm{H}}}_{2}$, carbon-chain and hydrocarbon molecules) and oxygen-bearing species (${{\rm{H}}}_{2}{\rm{O}}$, OH, ${{\rm{O}}}_{2}$, and ${\mathrm{CO}}_{2}$), some of which have been detected in PPDs at infrared wavelengths (e.g., Carr et al. 2004; Lahuis et al. 2006; Carr & Najita 2008; Salyk et al. 2008; Pascucci et al. 2009; Gibb et al. 2007). Here we mainly focus on the T Tauri disk model as an analog of our solar system.

First, in order to see how the physical properties affect the molecular abundance profiles, the top panels of Figure 3 show the number density of hydrogen nuclei, the gas temperature, and the dust temperature as a function of height divided by radius (Z/r) at a disk radius of 1 au (left), 3 au (middle), and 10 au (right). The bottom panels show similar plots for the UV flux, the X-ray ionization rate, and the cosmic-ray ionization rate.

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Since gas-phase molecular abundances change dramatically across their snowlines, we estimate the location of the snowlines of some molecules by equating the adsorption rate (Equation (17)) with the desorption rate (Equation (18)). Because thermal desorption rate is related to the binding energy, Table 2 presents the location of the snowline of each species together with binding energies adopted in the model.

Figures 4, 6, 8, and 10 show molecular abundances relative to total hydrogen nuclei in both the gas phase and ice mantle at 10⁶ yr as a function of (Z/r) at the disk radii of 1, 3, and 10 au. The dashed and solid lines represent the cases without and with carbon grain destruction, respectively. Figures 5, 7, 9, and 11 display the 2D fractional abundances of gas-phase and ice mantle molecules as a function of radius and height, out to a maximum radius of 10 au since the destruction of carbon grains mainly occurs in the inner hot region only. In the latter set of figures, the left panels represent the case without carbon grain destruction (C/O < 1), and the right panels represent those with carbon grain destruction (C/O > 1).

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Since the abundance of gas-phase CO has a large effect on molecular abundance differences between the models with and without carbon grain destruction, as is shown below, we present first the gas-phase and ice mantle abundances of CO (Figure 4). In the disk surface, CO gas is dissociated by the far-UV (FUV) photons so that oxygen and carbon are not locked in CO but are mostly in the form of atoms and ions since the synthesis of molecules is difficult owing to the high flux of FUV photons. Figure 5 shows the global CO abundance distribution, and CO is abundant (x(CO) = 10⁻⁴) throughout the whole region near the midplane, since it is easily desorbed inside 10 au owing to its low binding energy. The abundance of CO increases slightly when carbon grains are destroyed, due to the additional elemental carbon now available in the gas phase.

The abundances of other molecules, however, are significantly affected by the gas-phase elemental C/O abundance ratio, especially in the region where CO gas is abundant. In general, carbon-bearing molecules are not expected to be very abundant in an oxygen-rich environment (C/O < 1) since most of the carbon is incorporated into CO. However, in the carbon grain destruction regions, excess carbon exists (C/O > 1), and carbon-bearing molecular abundances can increase dramatically. This type of chemistry is similar to that observed and successfully modeled around carbon-rich AGB stars, such as IRC +10216 (e.g., Millar & Herbst 1994; Glassgold 1996).

Figure 6 shows 1D plots of the abundances of some carbon-bearing species (HCN, ${\mathrm{CH}}_{4}$, ${{\rm{C}}}_{2}{{\rm{H}}}_{2}$, and c-${{\rm{C}}}_{3}{{\rm{H}}}_{2}$). The left and right panels show the molecular abundances in gas and ice, respectively. Figure 7 shows the 2D abundance distribution of the carbon-bearing species. The effects of carbon grain destruction, and the resulting chemistry, are different between the surface and the midplane. Near the midplane, carbon-bearing molecules can form efficiently via gas-phase reactions in the carbon-rich case. The most significant difference between the two cases appears in HCN, especially inside ∼2 au, near the HCN snowline. Due to its high binding energy (see Table 2), HCN can remain on dust grains beyond 2 au in the T Tauri disk. In the carbon-rich case, the peak gas-phase abundance of HCN is ∼10⁻⁵ near the midplane of the inner region. Near the midplane inside 2 au, the maximum differences between the two cases can reach 8 orders of magnitude. In the oxygen-rich case, the peak abundance is ∼10⁻⁷ in the molecular layer. Therefore, HCN appears to be a good tracer of the effect of carbon grain destruction within its snowline, and we will focus on it in Section 3.3. We note, however, that the abundance distribution of HCN could be affected by the initial abundance of species and the ingredients adopted in the chemical model (see, e.g., Figure 14 of Walsh et al. 2015).

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Similar behavior can be seen in ${\mathrm{CH}}_{4}$ and ${{\rm{C}}}_{2}{{\rm{H}}}_{2}$, but the differences are less significant. The snowlines of ${\mathrm{CH}}_{4}$ and ${{\rm{C}}}_{2}{{\rm{H}}}_{2}$ are located outside 10 au and around 9.3 au, respectively. The gas-phase abundances of both molecules increase near the midplane inside the snowlines in the carbon-rich case. However, inside 2 au, most of the carbon is incorporated into HCN and long carbon-chain species in the gas phase (see Figures 8 and 9), and the abundances of ${\mathrm{CH}}_{4}$ and ${{\rm{C}}}_{2}{{\rm{H}}}_{2}$ gas decrease here. We note that ${\mathrm{CH}}_{4}$ is more abundant than HCN and carbon-chain molecules in the gas phase near the midplane inside 2 au in the oxygen-rich case. In the oxygen-rich case ${\mathrm{CH}}_{4}$ gas has its peak abundance (∼10⁻⁷) in the molecular layer. In the case of c-${{\rm{C}}}_{3}{{\rm{H}}}_{2}$, it shows a differences of about 7 orders of magnitude between the two cases. Though it is less abundant than HCN, we treat c-${{\rm{C}}}_{3}{{\rm{H}}}_{2}$ as a possible tracer in Section 3.3 as well. We note that though the reaction network used here treats c-${{\rm{C}}}_{3}{{\rm{H}}}_{2}$ and l-${{\rm{C}}}_{3}{{\rm{H}}}_{2}$ separately, we should be cautious that there are some uncertainties since it is rarely possible to identify the specific isomeric products in a laboratory reaction and we need to rely on some calculations about energetics of the product and so on. Nevertheless, the model calculations show quite good agreement with observations of c- and l-${{\rm{C}}}_{3}{{\rm{H}}}_{2}$ in IRC +10216 and TMC-1 (McElroy et al. 2013).

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Figure 8 presents the 1D plots of C_m(m = 5–10), C_mH(m = 5–9), and C_mH₂(m = 5–9) molecules, where we sum up the abundances of the molecules for m = 5–10 or m = 5–9. Figure 9 shows 2D gas and ice distributions of ${{\rm{C}}}_{6}{\rm{H}}$ and ${{\rm{C}}}_{9}$. We chose these two species as representative of carbon-chain molecules since they are abundant on grain surfaces (Table 4) and, indeed, most of the carbon-chain molecules have spatial distributions similar to these two species and to HCN. We note that we treat the carbon-chain molecules together since grain surface reactions, which may reduce the abundances of carbon-chain molecules, for example, through hydrogenation, are not included here. From our calculations, we find that carbon-chain molecules with a larger number of carbon atoms have larger abundances because the destruction of carbon-chain molecules by molecular ions is inefficient near the midplane of the disk, where the density is high and the ionization degree is very low (∼10⁻¹²). In the carbon-rich case, the gas-phase abundances increase inside the snowline near the midplane. Beyond the snowline, the ice mantle abundances increase significantly. The binding energies of carbon-chain molecules are relatively large, and thus the molecules can remain on the dust grains until very close to the parent star (∼1.5–3 au).

We note that the gaseous carbon-chain molecules are abundant in the molecular layers beyond their snowlines in both cases. This region corresponds to the CO gas depletion layer. Figures 4 and 5 show that CO gas is depleted at Z/r ∼ 0.15, r = 3–10 au. Similar CO gas depletion can be seen in other disk chemical models (e.g., Walsh et al. 2010; Furuya & Aikawa 2014). Though the self and mutual shielding of CO photodissociation are not included in the model, the shielding factors are not so effective especially in the inner region of the disks (Walsh et al. 2012). In our model, the decrease is caused by photodissociation of CO due to FUV irradiation from the central star and the subsequent formation and freeze-out of carbon-chain molecules onto grains. If the FUV field is strong enough, FUV photons dissociate CO into atomic species. Above the CO gas depletion area, the UV destruction is fast, and due to the low density, the adsorption rate onto grains is low. Therefore, any synthesized molecules are difficult to stick on the grains and will eventually convert to CO gas or, nearer the surface, remain as atomic and ionized carbon. Moving deeper toward the disk midplane, where UV photons are extinguished by grains in the high-density molecular layers, CO can remain in the gas phase. A fraction of the carbon-chain molecules in the ice mantle in the CO-gas-depleted layer is photodesorbed into gas, which produces the gaseous carbon-chain molecule rich layer seen in Figures 8 and 9.

Figure 10 shows the 1D plots of oxygen-bearing species (${{\rm{H}}}_{2}{\rm{O}}$, $\mathrm{OH}$, ${{\rm{O}}}_{2}$, and ${\mathrm{CO}}_{2}$) in the gas phase and ice mantle. Figure 11 shows the 2D distribution of gas-phase oxygen-bearing species, ${{\rm{H}}}_{2}{\rm{O}}$ and ${\mathrm{CO}}_{2}$. The results are opposite in behavior to those of carbon-bearing species. They are less abundant when carbon grains are destroyed. In Figure 11, in the oxygen-rich case, the volatile water abundance is enhanced inside its snowline around 2 au. In the carbon-rich case, the gas-phase water abundance is very low inside 10 au, and there is no clear water ice snowline present. The distributions of ${\mathrm{CO}}_{2}$ gas show clearly the snowline around 5 au in both cases. In the oxygen-rich case the abundance peak of ∼10⁻⁴ appears near the midplane inside 2 au, whereas in the carbon-rich case the peak value can only reach ∼10⁻⁶ close to the central star. We note that the abundance of ${{\rm{O}}}_{2}$ gas can be affected by the initial abundance setting, atomic or molecular (Eistrup et al. 2016; Eistrup & Walsh 2019).

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Finally, we briefly summarize in which molecules the carbon is mainly incorporated in both cases. Excluding CO gas, in the oxygen-rich case, carbon mainly resides in ${\mathrm{CO}}_{2}$ gas inside the HCN snowline and as ${\mathrm{CO}}_{2}$ gas and HCN ice beyond the HCN snowline. Beyond the ${\mathrm{CO}}_{2}$ snowline, carbon mainly resides in ${\mathrm{CO}}_{2}$ and HCN ice. A fraction of carbon forms ${\mathrm{CH}}_{4}$ gas throughout the inner disk. Meanwhile, in the carbon-rich case, carbon mainly forms HCN gas and long, gas-phase, carbon-chain molecules inside their snowlines. Beyond it, carbon mainly ends up in long, carbon-chain ice and gaseous ${\mathrm{CH}}_{4}$, while inside the ${\mathrm{CO}}_{2}$ snowline some carbon is converted to gas-phase ${\mathrm{CO}}_{2}$.

To compare the observed carbon depletion gradient in the inner solar system (Figure 1) with the modeled values, we calculate the carbon fraction in grains relative to the solar abundance of silicon as a function of the disk radius,

$$\begin{eqnarray}&&\begin{array}{l}\mathrm{Solid}\ \mathrm{Carbon}\ \mathrm{Relative}\ \mathrm{to}\ \mathrm{Solar}\ \mathrm{Silicon}\\ \quad =\,\displaystyle \frac{\mathrm{Refractory}\ \mathrm{Carbon}+\mathrm{Carbon}\ \mathrm{in}\ \mathrm{Ice}\ \mathrm{Mantle}}{\mathrm{Solar}\ \mathrm{Abundance}\ \mathrm{of}\ \mathrm{Silicon}}.\end{array}\end{eqnarray} \tag{ 14 }$$

We assume that the solar abundance of silicon is $3.55\times {10}^{-5}$ with respect to hydrogen nuclei (Asplund et al. 2009; Draine 2011). The value for the refractory carbon is assumed to be the difference between the solar abundance of carbon, 2.95 × 10⁻⁴ with respect to hydrogen nuclei (Draine 2011), and the gas-phase carbon abundance in the local diffuse ISM, 7.30 × 10⁻⁵ (Table 1). In the model with carbon grain destruction, all the carbon has been released from the refractory material, and thus there is no carbon left in this form. The released refractory carbon will be incorporated into species either in the gas phase or in the ice mantle, and the abundance of ice mantle carbon is obtained from our model calculation described in Section 3.1. Table 3 shows the ratio of the carbon in the solid phase relative to the solar silicon at each disk radius.

In the case without carbon grain destruction, the fraction does not change very much at different radii (Table 3), since it is imposed that the majority of the carbon is locked in refractory form. Figure 12 shows the percentages of the form of carbon for the models with and without carbon grain destruction at a disk radius of 3 au as an example. It shows that most of the carbon (∼75%) is locked in refractory form in the case without carbon grain destruction. In this case the elemental abundance of oxygen is larger than that of carbon (C/O < 1) in the gas phase, and thus most of the remaining carbon (∼24%) is stored in the form of CO gas, with less than 1% in the form of larger organic species. Since the refractory carbon and CO gas do not change significantly in abundance over 1 au < r < 10 au in this model, there is almost no fractional variation in this case. On the other hand, in the case with carbon grain destruction, the fraction varies slightly at 2 au < r < 10 au and suddenly drops at r = 1 au. In this case, all the carbon in refractory form released to the gas phase reacts to form a range of species. Since the elemental abundance of carbon is larger than that of oxygen in gas phase (C/O > 1), oxygen is mainly stored in CO in this case, capturing 1.76 × 10⁻⁴ (see Table 1), the corresponding amount of carbon, that is, 1.76 × 10⁻⁴/2.95 × 10⁻⁴ ≃ 60% of the total carbon as CO gas (Figure 12). The remaining carbon is mainly in the form of carbon-chain molecules in the ice mantle at r ≥ 2 au, as is shown in Section 3.1 (see, e.g., Figures 9 and 10). Table 4 shows the most abundant carbon-bearing species in the ice mantle in the disk midplane at different radii. The carbon-bearing molecules in the ice mantle evaporate into the gas phase inside their snowlines, and the locations of the snowlines depend on their binding energies (see Table 2). Therefore, the fraction of carbon in the solid phase changes across these snowlines of carbon-bearing species. Since the most abundant carbon-bearing molecules in the ice mantle are carbon-chain molecules (see Table 4) whose snowlines are located around the disk radii of ∼2 au (see Table 2), the solid carbon fraction does not decrease until r < 2 au. As we mentioned before, there are some uncertainties in the abundances of carbon-chain molecules in the ice mantle because grain surface reactions are not included in our calculation. Once grain surface reactions are considered, hydrogenation reactions on the surface may lead to the formation of alkanes.

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While the model results reproduce qualitatively the trend of carbon depletion observed in the solar system, the comparison between our result (Table 3) and the solar system data (Figure 1) shows some quantitative discrepancies. There is a relatively high carbon fraction in the asteroid belt (a few to an order of magnitude larger) and too much depletion at 1 au (∼3 orders of magnitude smaller) compared to the observation values. This is possibly caused by the omission of, for example, grain surface reactions and/or turbulent mixing, which we have not yet taken into account in our chemical model and will alter the chemical structure of the disk and the partitioning of carbon between solid and gaseous forms. With grain surface reactions, a higher carbon fraction in the ice mantle might be produced by large carbon-bearing molecules formed from species stuck on the grain surface that react with other species (e.g., Hasegawa et al. 1992; Hasegawa & Herbst 1993; Garrod & Herbst 2006; Garrod et al. 2008; Walsh et al. 2014). On the other hand, turbulent mixing can bring the dust grains from the midplane up to the warm surface layer, where the ice mantle can be photodissociated or thermally desorbed. This leads to a decrease in the solid carbon fraction, perhaps to a level consistent with that in the asteroid belt (e.g., Semenov & Wiebe 2011; Furuya et al. 2013; Furuya & Aikawa 2014). We note that the timescale of vertical mixing is ${\tau }_{\mathrm{mix}}\sim 6\times {10}^{4}{(r/10\mathrm{au})(\alpha /0.01)}^{-1}$ yr, where α represents the parameter for vertical mixing, similar to the α parameter in the viscous disk (e.g., Furuya et al. 2013), and that the timescale is shorter in the inner region. In addition, uncertainties in the binding energies of molecules on grains will affect the solid carbon fraction in grains. Desorption rates depend exponentially on binding energies (see Equation (18)). However, due to the lack of appropriate data for many binding energies, the values are still very uncertain. We therefore enlarged the value of binding energies by 25% to investigate how the solid carbon fraction changes as binding energies change. We found that the value at 1 au becomes comparable to the observed value on Earth when binding energies increase by about 25% (see Table 3). We note that the increased binding energies of carbon-chain molecules are close to the data compiled for the UMIST Database for Astrochemistry⁷ (McElroy et al. 2013).

Our results in Section 3.1 suggest that if carbon grains are destroyed in the inner region of PPDs, it will affect the molecular abundance profiles, a process that could be tested by observing the molecular line emission. In this subsection, we perform ray-tracing calculations, using the molecular abundance profiles obtained from our model calculation, and determine whether or not carbon grain destruction in the inner disk can be tested observationally.

While molecular lines have been observed toward PPDs at infrared and (sub)millimeter wavelengths, we choose lines that are observable with ALMA. As we have seen in Section 3.1, significant differences in molecular abundances between the models with and without carbon grain destruction appear only near the disk midplane, where CO is dominant and not photodissociated. Therefore, (sub)millimeter lines are more suitable to trace potential differences because infrared lines are optically thick and trace only the disk surface layer. In addition, ALMA enables us to spatially resolve the inner region of the disk, which allows us to see the difference between the models more clearly. Among the molecular lines observable at (sub)millimeter wavelengths, we choose the HCN lines because (1) the HCN abundance distribution is very much affected by the carbon grain destruction (see Section 3.1) and (2) they are known to be strong toward PPDs (e.g., Dutrey et al. 1997; Qi et al. 2008; Öberg et al. 2010, 2011; Chapillon et al. 2012; Huang et al. 2017). Three HCN isotopologues, DCN, ${{\rm{H}}}^{13}\mathrm{CN}$, and ${\mathrm{HC}}^{15}{\rm{N}}$, have been detected toward PPDs (e.g., Qi et al. 2008; Guzmán et al. 2015, 2017; Huang et al. 2017). We concentrate here on ${{\rm{H}}}^{13}\mathrm{CN}$ because the DCN/HCN ratio is known to be sensitive to the temperature profile (e.g., Aikawa & Herbst 2001; Willacy 2007; Cleeves et al. 2016; Aikawa et al. 2018), while ${{\rm{H}}}^{13}\mathrm{CN}$ is less affected (e.g., Woods & Willacy 2009). Meanwhile, ${{\rm{H}}}^{13}\mathrm{CN}$ lines are stronger than ${\mathrm{HC}}^{15}{\rm{N}}$ lines (Guzmán et al. 2017), and it is easier to analyze their spatial distribution. The combination of HCN and its isotopologue lines is useful to test the carbon grain destruction model since HCN lines are optically thick and trace mostly the surface layer of the disk, where CO is photodissociated, and the carbon grain destruction does not affect molecular abundances significantly. The HCN isotopologue lines are less optically thick and trace the lower layer of the disk, where CO is not photodissociated and the effect of carbon grain destruction is significant.

Figure 13 displays the profiles of the HCN and ${{\rm{H}}}^{13}\mathrm{CN}$ J = 4–3 lines at 354.5 and 345.3 GHz, respectively, with the spectral resolution of $0.4\,\mathrm{km}/{{\rm{s}}}^{-1}$, for the T Tauri disk, calculated by performing ray-tracing calculations (see Section 3.2) using the obtained physical structure (Figure 2) and the HCN abundance profile (Figure 7). The HCN/${{\rm{H}}}^{13}\mathrm{CN}$ abundance ratio is simply assumed to be the typical interstellar value of 70 (Qi et al. 2011). According to Woods & Willacy (2009), it could be larger by a factor of ≤2 in the disk surface, where the effect of self-shielding is significant and different CO isotopologues are selectively photodissociated. But the assumption is reasonable in the region closer to the disk midplane. We assume that the distance to the T Tauri disk is 70 pc and its inclination angle is 30°. The left and right panels of Figure 13 show the line profiles of HCN and ${{\rm{H}}}^{13}\mathrm{CN}$, respectively. The line profiles are calculated only inside a radius of 5 au since the significant differences between the two models appear mainly inside 2 au (Figure 7). Solid and dashed lines represent models with and without grain destruction, respectively. Figure 13 indicates that the line emission of ${{\rm{H}}}^{13}\mathrm{CN}$ shows more obvious differences between the two cases. However, the peak flux density is less than 1 mJy for the ${{\rm{H}}}^{13}\mathrm{CN}$ line for the model without the carbon grain destruction, and it is too weak to use it for testing the carbon grain destruction model even with ALMA.

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For this reason, we have also modeled a Herbig Ae disk since the radiation from the central source is stronger and the hot region is larger (Figure 2). Thus, the carbon grain destruction region spreads to larger radii, and the observational test will be easier. Figure 14 shows the distribution of HCN in the Herbig Ae disk as a function of radius up to 50 au and the disk height divided by the radius (Z/r). Gas-phase reactions produce HCN efficiently only in the very hot region inside a disk radius of r ≤ 5 au for the model without carbon grain destruction, while gas-phase HCN can be very abundant in the whole region inside the HCN snowline at ∼20 au for the model with carbon grain destruction. Therefore, a dramatic change appears at radii of 2–30 au between the two models. In contrast, the significant differences in the T Tauri disk appear only very close to the central star (inside 2 au), and the flux density is too low to be observed.

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Figure 15 is the zeroth-moment map, that is, the integrated intensity map of the HCN and ${{\rm{H}}}^{13}\mathrm{CN}$ lines, calculated as

$$\begin{eqnarray}&&{M}_{0}(x,y)=\int I(x,y,v){dv},\end{eqnarray} \tag{ 15 }$$

where $I(x,y,v)$ is the intensity as a function of position, (x, y), and velocity, v, given by Equation (10).

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The physical structure (Figure 2) and the HCN abundance profile (Figure 14) of the Herbig Ae disk are used. We assume that the distance to the disk is 140 pc and its inclination angle is 30°. The ${{\rm{H}}}^{13}\mathrm{CN}$ line intensity map shows the difference more clearly than the HCN line. This is because the ${{\rm{H}}}^{13}\mathrm{CN}$ line traces a relatively lower layer of the disk owing to relatively lower optical depth than the HCN line. Figure 16 shows the optical depth of the HCN and ${{\rm{H}}}^{13}\mathrm{CN}$ lines along the vertical direction of the disk for the models without and with carbon grain destruction. The HCN line becomes optically thick up to a disk radius of ∼30 au even for the model without the carbon grain destruction, that is, the HCN line only traces the disk surface, where the difference in the HCN abundance is not significant between the models (Section 3.1). On the other hand, the isotopologue ${{\rm{H}}}^{13}\mathrm{CN}$ line is optically less thick and can trace the HCN abundance difference near the midplane at disk radii of r < 30 au. Therefore, the difference in the emitting regions of the intensity map between the HCN and ${{\rm{H}}}^{13}\mathrm{CN}$ lines is a good tracer for testing the carbon grain destruction.

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Even in those cases for which spatially resolved observations are difficult, the profiles of the HCN and ${{\rm{H}}}^{13}\mathrm{CN}$ lines can be a good tracer as well. Figure 17 shows the line profiles of the HCN line (left panel) and the ${{\rm{H}}}^{13}\mathrm{CN}$ line (right panel). In both profiles, the solid line represents the model with carbon grain destruction, and the dotted line is the model without. Like the T Tauri disk model (Figure 13), the line emission of ${{\rm{H}}}^{13}\mathrm{CN}$ shows more substantial differences between the two models. The differences in the ${{\rm{H}}}^{13}\mathrm{CN}$ line range from high velocity (∼10 km s⁻¹, tracing the inner region) to low velocity (tracing the outer region). In contrast, the differences in HCN emission mainly exist at velocities of −5 to +5 km s⁻¹.

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Figure 18 is the normalized cumulative line flux as functions of disk radius (left panel) and the velocity (right panel). The former is calculated from the disk center to the disk radius of 25 au, and the latter is calculated over the range 0 km s⁻¹ < v < 20 km s⁻¹. Each cumulative flux is normalized using the following equations:

$$\begin{eqnarray}\begin{array}{rcl}{F}_{\mathrm{cum}}(r) & = & \displaystyle \frac{{\displaystyle \int }_{0}^{r}2\pi r^{\prime} {dr}^{\prime} {\displaystyle \int }_{-a}^{+a}I(r^{\prime} ,v){dv}}{{\displaystyle \int }_{0}^{b}2\pi r^{\prime} {dr}^{\prime} {\displaystyle \int }_{-a}^{+a}I(r^{\prime} ,v){dv}},\\ {F}_{\mathrm{cum}}(v) & = & \displaystyle \frac{{\displaystyle \int }_{0}^{v}{dv}^{\prime} {\displaystyle \int }_{0}^{b}I(r,v^{\prime} )2\pi {rdr}}{{\displaystyle \int }_{0}^{+a}{dv}^{\prime} {\displaystyle \int }_{0}^{b}I(r,v^{\prime} )2\pi {rdr}},\end{array}\end{eqnarray} \tag{ 16 }$$

where a is defined as 20 km s⁻¹ and b as 25 au. The blue and green lines are models with and without carbon grain destruction, respectively, with HCN represented by the solid lines and ${{\rm{H}}}^{13}\mathrm{CN}$ by the dashed lines. Comparing the results in the same color, the differences between the two models can be easily distinguished by the ratio of HCN and ${{\rm{H}}}^{13}\mathrm{CN}$. In the left panel, the green dashed line reaches ∼60% of the cumulative flux inside 5 au and the green solid line reaches the same level inside ∼15 au. This can be explained by the intensity map. For the model without carbon grain destruction, the intensity of the ${{\rm{H}}}^{13}\mathrm{CN}$ line is strong only in the very compact region (r < 5 au) as shown in Figure 15. Meanwhile, the HCN line is strong out to r ∼ 30 au, and thus the normalized cumulative flux increases smoothly with radius. On the other hand, for the model with carbon grain destruction, both HCN and ${{\rm{H}}}^{13}\mathrm{CN}$ lines are strong out to r ∼ 30 au and the cumulative fluxes increase smoothly, with no significant difference between them, so that differences between the two models can be tested by the cumulative flux as a function of radius if it can be spatially resolved. Even in the case for which the emission is not spatially resolved, we can see these differences in the line profiles, that is, cumulative flux as a function of velocity (e.g., Zhang et al. 2017). The green lines in Figure 18 (right panel) show the differences between the two models clearly. Comparing the normalized cumulative flux with the line profiles (Figure 17), the shape and the distribution of the model without carbon grain destruction are very different from HCN and ${{\rm{H}}}^{13}\mathrm{CN}$. However, the line shape and the distribution in the model with carbon grain destruction are very similar for both HCN and ${{\rm{H}}}^{13}\mathrm{CN}$. Both figures show significant differences between the HCN and ${{\rm{H}}}^{13}\mathrm{CN}$ lines for the model without the carbon grain destruction, while the difference is not large for the model with carbon grain destruction. Therefore, the differences between the two models can be easily distinguished by the ratio of the HCN and ${{\rm{H}}}^{13}\mathrm{CN}$ line.

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HCN and ${{\rm{H}}}^{13}\mathrm{CN}$ lines have been detected toward some Herbig Ae stars by ALMA (e.g., Guzmán et al. 2015; Huang et al. 2017). Guzmán et al. (2015) have reported the observations of the HCN J = 4–3 and ${{\rm{H}}}^{13}\mathrm{CN}$ J = 3–2 lines with relatively low spatial resolution of >07. Mapping the inner region of the disk (around the disk radius of 10–20 au), taking advantage of ALMA's high spatial resolution and high sensitivity, would be useful to diagnose carbon grain destruction in the inner disk.

Because the abundance distribution of molecules has some uncertainty depending on the chemical model, here we suggest c-${{\rm{C}}}_{3}{{\rm{H}}}_{2}$ 6_1,6–5_0,5 (217.882 GHz) as the other target line. This line has been detected in a Herbig Ae disk (Qi et al. 2013). The line is blended with the c-${{\rm{C}}}_{3}{{\rm{H}}}_{2}$ 6_0,6–5_1,5 line, but we treat only the ${{\rm{C}}}_{3}{{\rm{H}}}_{2}$ 6_1,6–5_0,5 line since it is slightly stronger. We note that other transition lines of c-${{\rm{C}}}_{3}{{\rm{H}}}_{2}$ have been detected toward T Tauri disks (Bergin et al. 2016). Figure 19 shows the distribution of c-${{\rm{C}}}_{3}{{\rm{H}}}_{2}$ in the Herbig Ae disk as a function of radius up to 50 au and the disk height divided by the radius (Z/r), which indicates that the significant difference appears at r < 30 au between the models, similar to the case of HCN. The difference in the abundance distribution results in the clear differences in both the intensity map (Figure 20) and the line profile (Figure 21). To sum up briefly, we suggest $\mathrm{HCN}$, ${{\rm{H}}}^{13}\mathrm{CN}$, and c-${{\rm{C}}}_{3}{{\rm{H}}}_{2}$ as possible tracers of testing the carbon grain destruction effect in PPDs.

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