Carbon Isotope Chemistry in Protoplanetary Disks: Effects of C/O Ratios

Carbon isotope fractionation of CO has been reported in the disk around TW Hya, where elemental carbon is more abundant than elemental oxygen ([C/O]elem > 1). We investigated the effects of the [C/O]elem ratio on carbon fractionation using astrochemical models that incorporate isotope-selective photodissociation and isotope exchange reactions. The 12CO/13CO ratio could be lower than the elemental carbon isotope ratio due to isotope exchange reactions when the [C/O]elem ratio exceeds unity. The observed 12CO/13CO and H12CN/H13CN ratios around TW Hya could be reproduced when the [C/O]elem ratio is 2–5. In the vicinity of the lower boundary of the warm molecular layer, the formation of ices leads to the gas-phase [C/O]elem ratio approaching unity, irrespective of the total (gas + ice) [C/O]elem ratio. This phenomenon reduces the variation in the 12CO/13CO ratio across different [C/O]elem ratios.


INTRODUCTION
The isotopic ratio of molecules is a powerful tool for investigating the origin of solar system materials and for revealing the possible chemical link between the solar system and the interstellar medium (ISM) (see, e.g., Ceccarelli et al. 2014;Nomura et al. 2023, for a recent review).The carbon isotope ratios are reported to be roughly constant in the Solar System (Clayton & Nittler 2004).The 12 CO/ 13 CO ratio in the solar photosphere (93.48 ± 0.68, Lyons et al. 2018) is slightly higher than terrestrial carbonates (Craig 1957;Fleisher et al. 2021, VPDB, 88.99), while 12 CO/ 13 CO ratio in comet 67P/Churyumov-Gerasimenko is 86 ± 8 (Rubin et al. 2017).
On the other hand, the carbon isotope ratio of CO, one of the main reservoirs of carseokholee@kasi.re.kr bon, evolves in star-forming regions.In the early stages, no significant carbon fractionation is observed in both the gas and ice phases of CO (Boogert et al. 2002;Pontoppidan et al. 2003;Agúndez et al. 2019;Yoshida et al. 2019).However, the solar-mass young stellar objects exhibit 12 CO/ 13 CO ranging from ∼85 to 160 when using the near-infrared CO absorption lines (Smith et al. 2015).These values notably higher than the 12 CO/ 13 CO ratios of ISM (62 ± 4, Langer & Penzias 1993).Intriguingly, divergent ratios of 31 +17 −10 (TYC8998-760-1b) and 10.2-42.6 (WASP-77Ab) have been reported in the atmosphere of young super-Jupiter (Zhang et al. 2021;Line et al. 2021).
The former is efficient in the surface layer of cloud and disk irradiated by the interstellar and stellar UV radiation, while the latter works in the cold regions (e.g., Röllig & Ossenkopf 2013;Furuya & Aikawa 2018;Visser et al. 2018).
The 12 CO/ 13 CO ratio in the protoplanetary disk differs between observations and previous chemical models.The carbon isotope fractionation of CO has been spatially observed in the disk around TW Hya (Zhang et al. 2017;Yoshida et al. 2022).On the other hand, chemical models exhibit that the 12 CO/ 13 CO ratio is similar to the the elemental carbon isotope ([ 12 C/ 13 C] elem ) ratio (Woods & Willacy 2009;Roueff et al. 2015;Viti et al. 2020).The chemical models used a typical ISM values like elemental carbon abundance relative to total hydrogen ([C] elem ) of ∼10 −4 and elemental carbon to oxygen ([C/O] elem ) ratio lower than unity.However, the observations toward disk sources show an elevated [C/O] elem ratio (≥ 1.0, e.g., Bergin et al. 2016) and the depletion of CO is also observed (e.g., Kama et al. 2016).
The [C/O] elem ratio higher than unity could change the 12 CO/ 13 CO ratio (Yoshida et al. 2022).Therefore, in this work, we will investigate the effect of [C/O] elem ratio as well as [C] elem on the carbon isotope ratios in the disk.We will briefly describe our model in Section 2. In Section 3, we will present our results, and in Section 4, we will compare our results with the observations.Then, we will summarize our conclusions in Section 5.

MODELS
We investigate carbon isotope features in the protoplanetary disk around TW Hya using the axisymmetric two-dimensional (2 D) thermochemical disk model, packages of unified modeling for radiative transfer, gas energetics, and chemistry (PURE-C, Lee et al. 2021).The code calculates the gas and dust temperatures and the chemical abundances self-consistently for given (gas and dust) density profiles and the radiation fields.

Physical parameters of disk
The density profiles of the gas and dust grains are adopted from the model of the TW Hya disk (Kama et al. 2016), which could reproduce the spectral energy distributions and the observed fluxes of multiple lines.The gas mass is set to 2.3 × 10 −2 M ⊙ and the gas to dust mass ratio is set to 200.The small dust grains (0.005 µm-1 µm) are coupled with gas and their mass is only 1 % of the total grain masses, while the large dust grains (r g = 0.005 µm-1 mm) are concentrated near the midplane and their scale height is 0.2 × the scale height of the gas.Representative grain sizes of 0.1 µm and 10 µm, respectively, and density of 2.09 g cm −3 are used for the chemical model (Woitke et al. 2016).The far-ultraviolet (UV) and X-ray luminosities (L UV and L X ) are 0.017 L ⊙ and 1.4 × 10 30 erg s −1 , respectively.The cosmic ray ionization rate is 5 × 10 −19 s −1 (Kama et al. 2016).The gas number density and the resultant UV flux normalized by Draine field (Draine 1978), ionization ra/tes (X-ray and Cosmic ray), and dust temperature are shown in Figure 1.The gas temperature decreases as the initial elemental abundances of carbon and oxygen increase, resulting in more efficient cooling, as presented in Figure A1.

Chemical model
Our chemical network is based on that in Furuya & Aikawa (2018), but extended to include mono-13 C species, relevant isotope exchange reactions (e.g., Watson 1976;Langer et al. 1984;Roueff et al. 2015;Colzi et al. 2020;Loison et al. 2020), and isotope-selective photodissociation of CO (e.g., Bally & Langer 1982;Visser et al. 2009).In our network, we do not distinguish 13 C isotopomer with different 13 C position for simplicity; e.g., 13 CCH and C 13 CH are treated as the same species.As we are interested in only the small species such as CO, HCN, HCO + , we reduce the network by choosing the species used in Lee et al. (2021) with 13 C-bearing species instead of 15 N-bearing species.
Important isotope exchange reactions in our models are: and we adopt the reaction rates in Table 1 of Roueff et al. (2015) and in Table 2 of Loison et al. (2020).Note that the reactions with number of 12, 13, 19, and 20 in Table 2 of Loison et al. (2020) are also included in this work.Forward reactions are exothermic reactions due to the zero point energy differences between products and reactants presented as temperatures in the reactions (1)-( 7), and are faster than backward ones, especially in cold conditions ( 30 K).The most important isotope exchange reaction is the reaction (1) (e.g., Langer et al. 1984), because the CO is a main reservoir of volatile carbon (Pontoppidan et al. 2014), and many C-bearing species, such as CN and HCN, formed from C + .Reaction (1) results in a 13 C-enrichment of CO and the molecules formed from CO, whereas it leads to a 13 Cdeficiency of the molecules formed from C + .The self-shielding of H 2 (Draine & Bertoldi 1996), C (Kamp & Bertoldi 2000), and N 2 (Heays et al. 2014) as well as CO (Visser et al. 2009) are considered in our model.The column densities used in the self-shielding are calculated by averaging the vertical and radial column densities weighted by the UV flux along each direction (see details in Lee et al. 2021).The isotope selective photodissociation of CO is significant only near the CO photodissociation front (near the surface layers of the molecular cloud or the disk).The self-shielding of 12 CO and 13 CO makes the 12 CO/ 13 CO ratio higher than the elemental carbon isotope ratio ([ 12 C/ 13 C] elem ), while the carbon isotope ratio of the molecules formed from C + becomes lower than [ 12 C/ 13 C] elem .
In our models, gas-ice chemistry is described by the two-phase model (Hasegawa et al. 1992).Our model takes into account gas-phase chemistry, interactions between gas and (icy) grain surfaces, and grain surface chemistry.Note that the all chemical reactions on the grain surface work only in the top two layers because man-tle layers (> 2 layers) are assumed to be inert (Aikawa & Herbst 1999;Cuppen et al. 2017).In addition, the photodissociation of ices is included in this work, which is assumed to have the same rates in the gas-phase.In the lower boundary of the warm molecular layer, CO ice can become CO 2 ice on dust grains, which removes the CO in the gas phase (Bergin et al. 2014;Furuya & Aikawa 2014).Therefore, not only the gas phase C/O ratio but also carbon isotope ratios of the gas phase species can be affected.In this work, we use the age of TW Hya (∼10 Myr, e.g., Herczeg et al. 2023) as the chemical evolution time to compare our results with observed values toward TW Hya.Furthermore, we compare the results between the evolution times of 1 Myr and 10 Myr.

Initial abundances
We investigate the effects of the [C/O] elem ratio and the [C] elem on the carbon isotope ratios of observable species in the warm molecular layer.The initial abundances are listed in Table 1 (Cleeves et al. 2015;Lee et al. 2021).The elemental carbon isotope ([ 12 C/ 13 C] elem ) ratio of 69 is adopted (Wilson 1999).The elemental carbon abundances, [C] elem are 1.7 × 10 −6 , 1.7 × 10 −5 , and 1.0 × 10 −4 in the model names beginning with A, B, C, respectively.The numbers following the model names indicate the [C/O] elem ratios.Note that the chemistry in the warm molecular layer tends to approach or reach an equilibrium state around the typical disk age of 1 Myr).Therefore, the results are predominantly determined by the elemental carbon and oxygen abundances rather than the initial abundances of species: whether the initial carbon and oxygen are in molecular forms or atomic forms.Thus, the initial abundances are set using only water ice, CO, and C as shown in Table 1. Figure 2 shows the 2D abundance distributions for CO, HCO + , HCN, and C 2 H in the reference model with [C] elem = 1.7× 10 −6 and [C/O] elem = 2.0, which reproduces well the observed molecular lines (Kama et al. 2016).The gas-phase CO is abundant in the warm molecular layer (e.g., Aikawa et al. 2002), which is surrounded by the CO photodissociation front (the dot-dashed white curves where the CO abun-dance is half of [C] elem ) at the upper boundary and the CO snow surface (∼20-30 K, the solid gray curves) at the lower boundary.
The lower boundary of warm molecular layer in the inner disk (< 30 au) can be modified by the CO 2 ice formation.When the ionization rate is higher than ∼10 −18 s −1 , the gasphase CO is converted into CO 2 ice within a few Myr (Bergin et al. 2014;Furuya & Aikawa   2014; Lee et al. 2021).Therefore, CO is depleted above the CO snow surface (the solid gray curve) as shown in the top left panel of Figure 2, where the X-ray ionization rate is higher than 10 −18 s −1 .In addition, CO ice reacts with OH ice, which is a product of the photodissociation of water ice, and also becomes the CO 2 ice (Ruaud & Gorti 2019;Furuya et al. 2022a).However, in regions with the UV flux (the sum of stellar and external fields) above ∼0.1 Draine field, the CO 2 ice is photodissociated and goes back to the CO ice, which is subsequently thermally desorbed.
Below the CO snow surface in the outer disk, a fraction of HCN and C 2 H as well as CO are in the gas phase, which could contribute to the observed column density.They are photodesorbed by the UV photons when the UV flux is higher than 0.1 Draine field (the dashed curves in Figure 2), although thermal desorption is inefficient and most volatiles are in the ice phase.Note that the binding energies (on water ices) of HCN, C 2 H, and CO 2 are 3700 K, 3000 K, and 2600 K, respectively, which are higher than the CO binding energy of 1150 K (Wakelam et al. 2017;Furuya & Aikawa 2018).
Figure 3 shows the 2D distributions of the carbon isotope ratios for the same molecules depicted in Figure 2, and the vertical distributions of the abundances and the isotope ratios at r = 30 au are displayed in the right panels in Figure 4.In the UV-dominated surface layers of the disk, the isotope selective photodissociation of CO determines 12 CO/ 13 CO ratio (e.g., Visser et al. 2018).All CO molecules are efficiently photodissociated, resulting in a very small abundance of CO, thus, the ratio is close to [ 12 C/ 13 C] elem near the atmosphere surface (z/r 0.4 around r = 30 au).However, this ratio increases as the CO molecules begin to survive from the UV photons (light-blue color in the top left panel) in the vicinity of the CO photodissociation front (e.g., Visser et al. 2009).Nevertheless, the isotope exchange reaction (1) partly mitigates the fractionation caused by isotope selective photodissociation of CO when the gas temperature falls below ∼100 K (indicated by the solid white curves in Figure 3) near the CO photodissociation front (Woods & Willacy 2009).Therefore, even around the CO photodissociation front, CO is enriched in 13 C as shown in the top left panel of Figure 3 (see also Figure A3).
In the warm molecular layer, CO becomes enriched in 13 C (red color) through the isotope exchange reaction (1).The reaction (2) induces greater carbon fractionation in HCO + than in CO.On the other hand, the molecules formed from C + (HCN and C 2 H) are 13 C-poor in the warm molecular layer as shown in the right column of Figure 3.The degree of carbon isotope fractionation of the molecules is more significant below the CO snow surface (solid gray lines) in the outer disk because the reaction (1) is more efficient in the colder region.
The carbon isotope ratios in the disk could change with the age of the disk.As shown Lee et al. in the middle row of Figure 4, the molecules in the lower height of the warm molecular layer mainly contribute to the (vertically integrated) observed column density.The chemistry in the warm molecular layer generally becomes an equilibrium condition within 1 Myr.However, the CO abundance starts to decrease due to CO 2 ice formation after ∼1 Myr (Furuya & Aikawa 2014) in the lower boundary of the warm molecular layer in the inner disk (< 40 au), where the X-ray ionization rate is as low as 10 −18 -10 −17 s −1 .Therefore, the observed carbon isotope ratio as well as the abundances could be evolved between ∼1-10 Myr.

Effects of [C/O] elem
Figure 4 shows vertical distributions of abundance (top) and carbon isotope ratio (bottom) at the distance of 30 au from the central star when [C/O] elem ratios are 0.5, 1.0, and 2.0 from left to right.When [C/O] elem ≤ 1, most elemental carbon exists in the form of CO in the warm molecular layer (0.1 ≤ z/r ≤ 0.35).On the other hand, when [C/O] elem > 1, atomic carbon, carbon chain molecules, and cyanide (e.g., C 2 , C 2 H, HCN) are also abundant due to the presence of a sufficient amount of leftover elemental carbon remaining after the formation of CO (e.g., Bergin et al. 2014;Cleeves et al. 2018;Lee et al. 2021).
The 12 CO/ 13 CO ratio in the warm molecular layer depends on the [C/O] elem ratio, and is primarily determined by the isotope exchange reaction (1).When the [C/O] elem ratio is ≤ 1, the self-shielding is very efficient, making CO the primary reservoir for elemental carbon.Therefore, the isotope exchange reactions cannot induce the CO fractionation, and thus, the 12 CO/ 13 CO ratio is close to [ 12 C/ 13 C] elem (see the blue lines in the left and middle columns of Figure 4, Woods & Willacy 2009).However, when the [C/O] elem ratio exceeds unity (right column of Figure 4), the abundances of atomic carbon gas and ices of HCN and carbon chain molecules, such as C m and C m H n , become comparable to that of CO gas.Consequently, sufficient amount of C + continues to be supplied by carbon-containing species, and the isotope exchange reaction (1) results in the 12 CO/ 13 CO ratio lower than [ 12 C/ 13 C] elem ratio, as presented by the blue line in the bottom right panel of Figure 4.
Figure 5 exhibits the chemical evolution at the vicinity of the lower boundary of the warm molecular layer (z/r ∼0.1) at 30 au in the reference model.Ices of HCN and C m H n are abundant and comparable to the CO gas, and they exhibit 13 C-deficiency due to 13 C-deficient C + , produced by the isotope exchange reaction (1), when [C/O] elem > 1.The ices of HCN and C m H n undergo a transformation to the CO gas after a few 10 3 -10 4 yrs, which is initiated by the photodesorption of ices by stellar UV photons.The newly supplied elemental carbons (as well as elemental oxygens) in the gas phase increase the CO abundance and decrease the 12 CO/ 13 CO ratio through the reaction (1).
Carbon isotope ratios of other molecules are also altered by the [C/O] elem ratio.Woods & Willacy (2009) showed that based on the dominance of reaction (2) (Langer et al. 1984), one can estimate the expected isotope ratio of H 12 CO + /H 13 CO + through the following relation: The dashed black lines in Figures 4 and 6 show the H 12 CO + /H 13 CO + ratio using the above equation along with the 12 CO/ 13 CO ratio (the solid blue lines) while the solid black lines exhibit the H 12 CO + /H 13 CO + ratio in our model.The strong gradients in physical and chemical conditions including both gas temperature, CO abundance, and electron abundance cause our models to deviate from this simple relationship (see Figure 3 in Furuya et al. 2022b), and thus  models will be necessary for interpreting observed H 12 CO + /H 13 CO + ratios in disks.HCN and C 2 H are formed from C + , and thus, their carbon isotope ratios align with 12 C + / 13 C + .However, when the [C/O] elem ratio exceeds unity, the atomic carbon and C 2 are abundant in the warm molecular layer.Thus, isotope exchange reactions 3-7 (Loison et al. 2020) contribute to lower H 12 CN/H 13 CN and 12 C 2 H/C 13 CH ratios compared to the 12 C + / 13 C + ratio as shown in the bottom panel of Figure 4.

Effects of [C] elem
The elemental carbon abundance ([C] elem ) of disks is poorly constrained, and like oxygen, it is thought to vary (e.g., Ansdell et al. 2016).In this section we explore differences in the elemental carbon abundances on the carbon isotope ratios.Figure 6 shows the differences in abundances and carbon isotope ratios for Figure 5.Chemical evolution at 30 au and z/r= 0.1 in the reference model.The solid and dashed lines represent the abundances (on the left axis) and carbon isotope ratios (on the right axis), respectively.The values for CO, C + , and HCN ice are plotted in the red, blue, and green lines, respectively.The C + abundance is depicted as increased by a factor of by 10 9 .the molecules presented in Figure 4, considering variations in elemental carbon abundances ([C] elem ), fixing the [C/O] elem ratio of 2.0.The peak abundances of CO/C/C + increase by a factor of 10 and ∼50 as the [C] elem increases.The height of the CO photodissociation front where the CO abundance is a half of [C] elem , increases with a higher [C] elem .This is because the CO self-shielding is more effective due to the higher CO abundance in the atmosphere.However, it is worth noting that the contribution of this height to the column density, when integrated along the entire height, is negligible as shown in the middle panels of Figure 6.
Carbon isotope ratios of CO, atomic carbon, and C + in the warm molecular layer show a similar trend regardless of [C] elem as shown in the bottom panels of Figure 6.In the warm molecular layer, the ratios are mainly determined by the isotope exchange reaction (1).As the [C] elem increases, the gas temperature could be decreased due to efficient cooling by CO and H 2 O in the upper warm molecular layer (z/r > 0.25; see Figure A1).However, the gas temperature is close to the dust temperature in the lower warm molecular layer.Therefore, CO, atomic C, and C + show similar carbon isotope ratios across different [C] elem in the warm molecular layer.
However, in the atmosphere, the isotope selective photodissociation of CO is important, which is more efficient as [C] elem increases.Furthermore, gas temperature decreases according to [C] elem .Therefore, carbon isotope ratios exhibit complicated trends due to the competition between the isotope exchange reaction (1) and the isotope selective photodissociation of CO.
The carbon isotope ratios of the products are determined by the contribution of the inheritance from CO or C + and the additional isotope reactions: reaction (2) for HCO + , reactions ( 5) and ( 6) for the C 2 H, and reactions (3) and (4) for the HCN, respectively.In the warm molecu-lar layer, the abundances of CO, C, and C + are more sensitive to the elemental carbon abundance compared to those of HCO + , HCN, and C 2 H. Thus, the contribution of the latter isotope reactions increases, and the HCO + , C 2 H, and HCN are 13 C-enriched compared to their mother species (CO and C + ).

Comparison with observations
Previous disk models used the elemental abundances similar to the ISM values with the [C/O] elem ratio lower than unity (Woods & Willacy 2009;Furuya & Aikawa 2014;Visser et al. 2018).In this case, the 12 CO/ 13 CO ratio is similar to the [ 12 C/ 13 C] elem ratio (Roueff et al. 2015;Viti et al. 2020) or close to the initial 12 CO/ 13 CO ratio (Woods & Willacy 2009).On the other hand, molecules formed from C + in the cold region could undergo fractionation through the reaction (1).
However, the elemental abundances of the disk around TW Hya differ from the ISM condition.The bright C 2 H emission is observed toward TW Hya, implying an elevated [C/O] elem ratio (≥ 1.0, e.g., Bergin et al. 2016), and the depletion of CO is also observed (Kama et al. 2016).Furthermore, those features are common in the disk sources (Bergin et al. 2016;Miotello et al. 2019;Bergner et al. 2020;Bosman et al. 2021a,b;Sturm et al. 2022).Therefore, the observed carbon isotope ratios could be explained by elevated [C/O] elem ratios.

CO
Spatially resolved observations of the carbon isotope fractionation have been conducted in the disk around TW Hya.The observationally derived 12 C 18 O/ 13 C 18 O ratio is 40 +9 −6 around 5-20.5 au (Zhang et al. 2017) and the 12 CO/ 13 CO ratio is 23±6 around 60 -100 au and ∼100 in the outer disk (> 100 au, Yoshida et al. 2022).
Figure 7 shows the CO column densities (top panels) and the column density ratios of 12 CO to 13 CO (bottom panels) as functions of a distance from the central star.The elemental carbon abundances, [C] elem are 1.7 × 10 −6 , 1.7 × 10 −5 , and 1.0 × 10 −4 , respectively, from left to right, respectively.The model with [C/O] elem ∼5 reasonably well reproduces the observed 12 CO/ 13 CO ratio.However, the observed 12 CO/ 13 CO ratio is still slightly lower than the 12 CO/ 13 CO ratio predicted by our model with [C/O] elem = 5 (see the solid lines in the left bottom panel of Figure 7), which implies that the [C/O] elem ratio might be even higher than five and/or the gas temperature might be lower in the TW Hya disk than that in our model.When we reran the reference model after artificially reducing the gas temperature by half, the carbon isotope ratio decreases by ∼15-20%.
The [C/O] elem and [C] elem could affect the CO column density and carbon isotope ratio.When [C/O] elem > 1, the CO column density decreases due to an oxygen deficiency relative to carbon for the formation of CO.On the other hand, when [C/O] elem < 1, the column density of CO decreases due to its conversion into CO 2 ice near the CO snow surface.The CO column densities increase as [C] elem increases, and the trends of CO column density with respect to the [C/O] elem ratio are similar in all three cases with varying [C] elem .The observed CO column density in the TW Hya disk (Huang et al. 2018) appears to be similar to the models with [C] elem = 1.7 × 10 −6 .The dotted and solid lines indicate the models with the evolution time of 1 Myr and 10 Myr, respectively.The column densities within r = 30 au slightly decrease because the CO 2 ice still forms after 1 Myr.
When [C/O] elem ≤ 1, the column density ratio of 12 CO/ 13 CO is close to the [ 12 C/ 13 C] elem ratio.However, as the [C/O] elem ratio exceeds unity, the column density ratio decreases below the [ 12 C/ 13 C] elem ratio.The trends of 12 CO/ 13 CO with respect to the [C/O] elem ratio are consistent across all three [C] elem cases.Furthermore, the 12 CO/ 13 CO ratio is unaffected by the [C] elem .
Our results are consistent with a simple approach in Yoshida et al. (2022).In the normal dense ISM condition, where elemental carbon to oxygen ratio ([C/O] elem ) is ∼0.5 and most volatile carbon is locked up in CO, the degree of carbon fractionation in CO by reaction (1) is not significant (deviation of 12 CO/ 13 CO from [ 12 C/ 13 C] elem is ∼30 % at most, Furuya et al. 2011).However, this situation can change when [C/O] elem is 1 because, in that case, volatile carbon carriers as abundant as CO exist.According to Yoshida et al. (2022), the 12 CO/ 13 CO ratio normalized by the elemental carbon isotope ratio [ 12 C/ 13 C] elem is roughly ex-Figure 7. CO column densities (top) and column density ratio of 12 CO/ 13 CO along the distance from the central star.The models with the elemental carbon abundances ([C] elem ) of 1.7 × 10 −6 , 1.7 × 10 −5 , and 1.0 × 10 −4 are plotted in the left, middle, and right columns, respectively.The [C/O] elem ratios of 0.5, 1.0, 1.5, 2.0, and 5.0 are presented in the black, green, blue, red, and cyan lines.The solid and dashed lines indicate the models with the evolution times of 10 and 1 Myr, respectively.The black dotted lines in the top panels indicate the observed CO column density (Huang et al. 2018).The gray boxes in the bottom panels indicate the observed 12 CO/ 13 CO ratios (Zhang et al. 2017;Yoshida et al. 2022).
pressed by assuming that the reaction (1) is in chemical equilibrium, only C + and CO are the carbon carriers, and [C/O] elem is larger than unity.The above equation would give the lower limit of the normalized ratio, because other carbon carriers, such as C atoms and icy species, besides C + and CO, should exist in reality.When [C/O] elem is exp(35 K/T ) (∼33 for 10 K and ∼3 for 30 K), the ratio inversely scales with [C/O] elem .Therefore, 12 CO/ 13 CO can be significantly different from [ 12 C/ 13 C] elem when [C/O] elem > 1.According to Equation 9, the [C/O] elem ratio of 3-10 is needed to reproduce the observed 12 CO/ 13 CO ratio in the inner disk around TW Hya (Yoshida et al. 2022).

HCO +
The observed H 13 CO + and HC 18 O + column densities in the whole disk around TW Hya are (7.8 ± 2.0) × 10 11 cm −2 and (7.5 ± 1.3) × 10 10 cm −2 , respectively, based on the H 13 CO + and HC 18 O + 4-3 lines (Furuya et al. 2022b).When we assume that the H 12 CO + /HC 18 O + ratio is the elemental ratio of 557 (Wilson 1999), the H 12 CO + column density and the H 12 CO + /H 13 CO + ratio are (4.2 ± 0.7) × 10 13 cm −2 and 54 ± 17, respectively (see the horizontal gray boxes in Figure 8).The [C/O] elem ratio and [C] elem do not affect significantly the HCO + column density and the H 12 CO + /H 13 CO + ratio.The H 12 CO + /H 13 CO + ratio is lower than the 12 CO/ 13 CO ratio through the isotope exchange reaction (2).This ratio is by ∼20% more fractionated compared to 12 CO/ 13 CO ratio in the inner disk (<40 au) regardless of [C/O] elem .Most our models exhibit narrow ranges of column density and carbon isotope ratio compared to the observed ones as shown in Figure 8.

HCN
The H 12 CN/H 13 CN ratio is reported as ∼86 ± 4 toward TW Hya using the optically thin HCN 4-3 hyperfine lines and H 13 CN 4-3 line (Hily-Blant et al. 2019).The observed H 12 CN/H 13 CN ratio in the TW Hya disk (open circles in the bottom panels of Figure 9) is well fit with the models with [C/O] elem = 2-5 (the red and cyan lines) when [C] elem is 1.7 × 10 −6 .
Figure 9 shows the HCN column densities (top panels) and the column density ratios of H 12 CN to H 13 CN (bottom panels).The HCN column density increases with the [C/O] elem ratio and is saturated when [C/O] elem ≥ 1.5.The H 12 CN/H 13 CN ratio within r = 80 au increases with the [C/O] elem ratio and reaches its maximum value of ≥ 120 around the [C/O] elem ratio of unity.Then, the ratio decreases with increasing the [C/O] elem ratio when [C/O] elem > 1.For a given [C/O] elem ratio, the H 12 CN/H 13 CN ratio decreases as [C] elem increases via the isotope exchange reaction (5) as mentioned in Section 3.
When [C] elem is higher, the lower [C/O] elem ratio could fit the observations.In addition, the ratio could be lower than [ 12 C/ 13 C] elem ratio when [C/O] elem = 5.The H 12 CN/H 13 CN ratio is higher at 10 Myr (solid lines) compared to 1 Myr (dashed lines), which is the opposite trend shown in the 12 CO/ 13 CO ratio (see Section 4.1.1).However, their differences are too small to distinguish observationally.

C 2 H
The bright C 2 H emission in the disk indicates that the [C/O] elem ratio exceeds unity.The observed C 2 H column density toward the TW Hya disk (Kastner et al. 2014) could be fit with the models with [C/O] elem ≥ 1.5 as shown in the previous works (e.g., Bergin et al. 2014;Lee et al. 2021).However, the 12 C 2 H/C 13 CH ratio toward TW Hya has been not reported yet.Thus, we expected the 12 C 2 H/C 13 CH ratio based on our model.Note that 13 CCH and C 13 CH are treated as the same species in our work.Their carbon isotope ratios could be differ through 13 CCH + H → C 13 CH + H + 8.1 K, thus, their isotope ratios are similar in our work (Furuya et al. 2011).Figure 10 shows the C 2 H column densities (top panels) and the column density ratios of C 2 H and C 13 CH (bottom panels).When [C] elem is 1.7 × 10 −6 , the 12 C 2 H/C 13 CH ratio increases up to ∼180 at the [C/O] elem ratio of 1.5 (blue line) then it decreases with increas-ing the [C/O] elem ratio.The 12 C 2 H/C 13 CH ratio is governed by a balance between the inheritance from 12 C + / 13 C + (resulting in an elevated 12 C 2 H/C 13 CH) and the reactions ( 6) and ( 7) (leading to a reduction in the 12 C 2 H/C 13 CH).Thus, the maximum 12 C 2 H/C 13 CH ratio Lee et al.
(∼200) is achieved at the [C/O] elem ratio of ∼1 when [C] elem is 1.7 × 10 −5 and 1.0 × 10 −4 because of the growing significance of the latter contribution.

Gas phase carbon isotope ratio
The isotope ratio measured in disks is primarily in gas phase carriers.In this section we focus on its gas-phase values for different molecules commonly observed.The 12 CO/ 13 CO ratio depends on the gas-phase [C/O] elem ratio (hereafter [C/O] gas ) while the [C/O] elem ratio mentioned in this work is the total values including both gas and ice phases.Figure 12 shows the gas phase carbon isotope ratio.The gas phase carbon isotope ratio exhibits a similar trend independent of [C] elem .When the [C/O] elem ratio is ≤ 1, the carbon isotope ratio is close to [ 12 C/ 13 C] elem above the UV flux of 0.1 Draine field (the dashed yellow lines).However, when the [C/O] elem ratio ex-ceeds unity, 13 C is enriched in the gas phase near the lower boundary of the warm molecular layer where CO is abundant in the gas phase (above the CO snow surface and the UV flux of 0.1 Draine field).It implies that ice abundances are comparable to the CO abundance and 13 C is deficient in the ice phase there.On the other hand, the gas exhibits a 13 C-deficiency below the CO snow surface in the outer disk where only small portion (∼a few %) of elemental carbon is in the gas phase and dominant gas phase species are atomic carbon.

Effects of Ionization rate
Carbon fractionation mainly occurs in the cold regions through the isotope-exchange reactions (e.g., Nomura et al. 2023).Thus, ionization rates might affect the carbon isotope ratios.In the lower boundary of the warm molecular layer, the formation of C + and 13 C + starts from the reaction of CO + He + and the later is ionized by X-rays and/or cosmic rays in the reference model.Therefore, the efficiency of the isotope exchange reaction (1) might depend on the ionization rates by X-rays and/or cosmic rays.
The reference model could fit the observations of the ionization tracers, HCO + and N 2 H + .The HCO + column density, which is derived by using HC 18 O + 4-3 line (Furuya et al. 2022a), is fitted with the reference model as mentioned in Section 4.1.2although the HCO + column density is insensitive to the ionization rates (Aikawa et al. 2021).Our model could also fit the observed N 2 H + column density of ∼10 13 cm −2 , which is measured using N 2 H + 1-0 and 4-3 lines (Cleeves et al. 2015;Schwarz et al. 2019).Note that Cleeves et al. (2015) determined an ionization rate (X-ray + CR) below 10 −19 s −1 at the disk midplane, derived through the fitting of the observed HCO + and N 2 H + data.In contrast, our model shows an ionization rate exceeding 10 −18 s −1 , as presented in the bottom left panel of Figure 1.We find a reduced ionization rate is needed, however how the [C/O] elem ratio and the isotope ratio of carbon may play an important role that should be explored in future work.
Figure 13 shows the effects of ionization rates to 12 CO/ 13 CO ratio.In the reference model (the black line), the ionization by X-rays is more dominant than that by cosmic rays (with the rate of 5 × 10 −19 s −1 Kama et al. 2016) except for the mid-plane of the inner disk (< 10 au).The red and blue lines indicate the model with X-ray luminosity 10 −3 times that of the reference model (L X ).The latter model has an order of magnitude lower cosmic ray ionization rate compared to the reference model.The ionization rates at z/r = 0.1, where the contribution to the column density is maximum, at r = 30 au are 3.2 × 10 −18 s −1 , 4.7 × 10 −19 s −1 , and 5.0 × 10 −20 s −1 , for the black, red and blue line models, respectively.The column densi- ties of HCO + are similar within a factor of two among the models while the N 2 H + column densities in the red and blue line models are a factor of ∼10 and ∼30 lower than that in the reference model.2. H 12 CO + /H 13 CO + ratio is by ∼20% more fractionated compared to 12 CO/ 13 CO ratio in the inner disk (<40 au) regardless of [C/O] elem .Most models could reproduce the observed H 12 CO + /H 13 CO + ratio of 54±17 within the error range.
3. It is not straightforward to infer the [C/O] elem ratio from the carbon isotope ratios of HCN and carbon chain molecules (e.g., C 2 H).This is because they exhibit an increasing trend followed by a decrease according to the [C/O] elem ratio.In addition, they are also affected by [C] elem .However, their column densi-ties are sensitive to [C/O] elem ratio, and thus, can be used as an indicator of [C/O] elem ratio.Thus, chemical models are essential for interpreting observations.The H 12 CN/H 13 CN ratio observed in the TW Hya disk could be reproduced when the [C/O] elem ratio of 2-5.
4. When [C/O] elem > 1, in the vicinity of the lower boundary of warm molecular layer, the 13 C-enriched CO dominates in the gas phase.In addition, the gas phase [C/O] elem ratio approaches unity because excess elemental carbon in the gas phase is removed through ice formation in the form of HCN and carbon chain molecules (C m H n ), which exhibit 13 C depletion.Thus, 13 C is enriched in the bulk of the gas while being deficient in the bulk of the ice phase.

Figure 2 .
Figure 2. Abundance distributions for CO, HCO + , HCN, and C 2 H in the reference model with [C] elem = 1.7 × 10 −6 and [C/O] elem = 2.0.The solid and dashed gray curves indicate the CO snow surface using the balancing between freeze-out onto dust grains and thermal evaporation (20-30 K), and the dust-attenuated UV flux of 0.1 Draine Field (Draine 1978).The CO photodissociation front where the CO abundance is a half of [C] elem , is presented in the dot-dashed curves.The white solid line indicates the gas temperature of 100 K.
y) means x × 10 y .b The initial abundance of 13 C bearing species is by a factor of 69 lower than that of 12 C bearing species except for C 2 H (34.5).The latter contains two carbon atoms and doubly counted.c The elemental carbon abundances are 1.7 × 10 −6 , 1.7 × 10 −5 , and 1.0 × 10 −4 for the model names beginning with A, B, and C, respectively.The following numbers indicate the [C/O] elem ratio.

Figure 3 .
Figure 3. Carbon isotope distributions for the same molecules in Figure 2. The curves correspond to those in Figure 2. Note that the carbon isotope ratio of C 2 H is derived as 2 × n(C 2 H)/n(C 13 CH).

Figure 4 .
Figure 4. Vertical abundance (top) and isotope ratio (bottom) distributions along the height at 30 au in the [C/O] elem ratio of 0.5 (left), 1.0 (middle), and 2.0 (right) with the [C] elem = 1.7 × 10 −6 .The relative contributions of each height to the vertically integrated column density are presented in the middle layer.The blue, black, red, orange, green, and cyan lines indicates the values for the CO, HCO + , HCN, C 2 H, C, and C + , respectively.The black dashed lines in the bottom panels indicate the H 12 CO + /H 13 CO + ratio using Equation8.The horizontal dotted grey lines present the 0.5, 1, and 2 times [ 12 C/ 13 C] elem ratios.Note that the carbon isotope ratio of C 2 H is derived as 2 × n(C 2 H)/n(C 13 CH).

Figure 9 .
Figure 9. Same as Figure 7 except for HCN.The gray bar and red symbols are the observed column density (Kastner et al. 2014) and H 12 CN/H 13 CN ratios (Hily-Blant et al. 2018).

Figure 10 .
Figure 10.Same as Figure 7 except for C 2 H.The gray boxes indicate the observed C 2 H column density (Kastner et al. 2014).
Figure 11 shows the 2 D distribution of only the [C/O] gas ratios.The [C/O] gas ratio in the warm molecular layer is close to the total value ([C/O] elem ) except for the lower boundary where the [C/O] gas ratio is close to unity.When the [C/O] elem ratio is 0.5 (top rows), CO 2 ice is dominant just above the CO snow surface, thus, the [C/O] gas ratio approaches unity.When the [C/O] elem ratio is higher than unity (bottom rows), HCN and carbon chain ices, like C m H n , attain notable abundances (comparable to CO) near the lower boundary of the warm molecular layer in the inner disk.This occurrence leads to the [C/O] gas ratio approaching unity regardless of the [C/O] elem ratio and [C] elem .However, in mid-plane at the outer disk, gas phase CO freezes out onto dust grains below the CO snow surface, and atomic carbon is dominant in the gas phase.Therefore, the [C/O] gas ratio is higher than the [C/O] elem ratio when the [C/O] elem ratio exceeds unity, while the [C/O] gas ratio is close to the [C/O] elem ratio in other cases regardless of [C] elem .

Figure 11 .
Figure 11.Distribution of gas phase [C/O] elem ratio.Models have [C] elem = 1.7 × 10 −6 (left), 1.7 × 10 −5 (middle), and 1.0 × 10 −4 (right) and the [C/O] elem ratios of 0.5, 1.0, and 2.0 from top to bottom layers.The solid and dashed black curves indicate the CO snow surface using the balancing between freeze-out onto dust grains and thermal evaporation (20-30 K), and the dust-attenuated UV flux of 0.1 Draine Field (Draine 1978).The dot-dashed lines represent the CO-photodissociation front where the CO abundance is a half of [C] elem .The solid white line indicate the gas temperature of 100 K.

Figure 12 .
Figure 12.Same as Figure 11 except for the gas phase 12 C/ 13 C isotope ratio.The colors of the curves are changed for better visibility.
the effects of elemental carbon to oxygen ([C/O] elem ) ratio and elemental carbon abundance to hydrogen ([C] elem ) on the carbon isotope ratio in the disk.1.The 12 CO/ 13 CO ratio monotonically decreases with increasing [C/O] elem ratio, and CO appears 25% and >50% more fractionated in the outer disk (>40 au) when [C/O] elem are 1.5 and 5, respectively.The 12 CO/ 13 CO ratio is close to [ 12 C/ 13 C] elem when [C/O] elem is ≤ 1 except for the inner disk (<20 au) regardless of [C] elem .When [C/O] elem > 1, CO is enriched in 13 C in the warm molecular layer through the isotope exchange reaction (1).The 12 CO/ 13 CO ratio observed in the TW Hya disk could be reproduced when the [C/O] elem ratio is higher than 2-5.

Figure 13 .
Figure13.Effects of ionization rates on the 12 CO/ 13 CO ratio.The reference model is represented in the black line.The red and blue lines indicate the models with a X-ray luminosity 10 −3 times that in the reference model, and additionally, a cosmic ray ionization rate one tenth of the reference model's value, respectively.

Figure A2 .Figure A3 .Figure A4 .Figure A5 .Figure A7 .Figure A9 .
Figure A2.CO abundance distributions for the same models in Figure A1.The gray curves are the same as those in Figure A1.The solid white curve indicates the gas temperature of 100 K.

Table 1 .
Initial abundances relative to the total hydrogen nuclei.Species Abundance a Species Abundance a + , HCN, C 2 H, C, and C + , respectively.The black dashed lines in the bottom panels indicate the H 12 CO + /H 13 CO + ratio using Equation8.The horizontal dotted grey lines present the 0.5, 1, and 2 times [ 12 C/ 13 C] elem ratios.Note that the carbon isotope ratio of C 2 H is derived as 2 × n(C 2 H)/n(C 13 CH).