Carbon Isotope Fractionation of Complex Organic Molecules in Star-Forming Cores

Recent high-resolution and sensitivity ALMA observations have unveiled the carbon isotope ratios ($^{12}$C/$^{13}$C) of Complex Organic Molecules (COMs) in a low-mass protostellar source. To understand the $^{12}$C/$^{13}$C ratios of COMs, we investigated the carbon isotope fractionation of COMs from prestellar cores to protostellar cores with a gas-grain chemical network model. We confirmed that the $^{12}$C/$^{13}$C ratios of small molecules are bimodal in the prestellar phase: CO and species formed from CO (e.g., CH$_{3}$OH) are slightly enriched in $^{13}$C compared to the local ISM (by $\sim$ 10 $\%$), while those from C and C$^{+}$ are depleted in $^{13}$C owing to isotope exchange reactions. COMs are mainly formed on the grain surface and in the hot gas ($>$ 100 K) in the protostellar phase. The $^{12}$C/$^{13}$C ratios of COMs depend on which molecules the COMs are formed from. In our base model, some COMs in the hot gas are depleted in $^{13}$C compared to the observations. Thus, We additionally incorporate reactions between gaseous atomic C and H$_{2}$O ice or CO ice on the grain surface to form H$_2$CO ice or \ce{C2O} ice, as suggested by recent laboratory studies. The direct C-atom addition reactions open pathways to form \ce{^13C}-enriched COMs from atomic C and CO ice. We find that these direct C-atom addition reactions mitigate $^{13}$C-depletion of COMs, and the model with the direct C-atom addition reactions better reproduces the observations than our base model. We also discuss the impact of the cosmic ray ionization rate on the $^{12}$C/$^{13}$C ratio of COMs.


INTRODUCTION
In the central (< 100 AU) and hot (> 100 K) regions in low-mass protostellar cores, rotational transition lines from various species have been observed.Some species are referred to as complex organic molecules (COMs), defined as organic molecules with six or more atoms (Herbst & van Dishoeck 2009).
According to astrochemical models, the formation of COMs requires a sequence of grain surface processes during star formation.This includes freeze-out of gaseous species onto the grain surface, and grain surface chemistry at low temperature (∼ 10 K) to produce simple icy molecules such as CH 4 and CH 3 OH.After protostar formation, the gas and dust temperatures increase (above Corresponding author: Ryota Ichimura ryotaichimura.astrolife@gmail.com∼ 25 K), radicals form and COMs are produced via radical-radical reactions on the grain surface via diffusion(e.g., Garrod & Herbst 2006;Herbst & van Dishoeck 2009).Additionally, recent laboratory experiments suggest that the atomic C insertion or addition reaction on the grain surface may be important for the production of the COMs and increasing their complexity (Tsuge et al. 2023;Ferrero et al. 2024).Also, nondiffusive grain surface chemistry may be important for the formation of COMs under low-temperature conditions (Shingledecker et al. 2018;Jin & Garrod 2020;Garrod et al. 2022).However, it is still challenging to reveal the formation pathways of COMs.
The measurement of isotope ratios and their fractionation is useful to investigate the local chemical process such as the formation pathways of COMs in star-forming regions.The carbon isotope ratio of different COMs has been measured with high-resolution ALMA observations towards the Class 0 low-mass protostellar binary system IRAS 16293-2422.The values were found to be comparable with the local ISM value ( 12 C/ 13 C = 68; Milam et al. 2005) or to show lower 12 C/ 13 C ratios towards source B (Jørgensen et al. 2016(Jørgensen et al. , 2018)), while are consistent within the error with the local ISM value for source A (Manigand et al. 2020).In the Class I low-mass young outbursting star V883 Ori, COMs are enriched in 13 C ( 12 C/ 13 C ∼ 20-30) (Yamato et al. 2024).
Carbon isotope fractionation in the ISM is thought to occur via exothermic isotope exchange reactions, isotope selective photodissociation, and desorption from grain surface with the different binding energies of isotopologues (Nomura et al. 2023).Especially, several studies of astrochemical models focus on exothermic isotope exchange reactions to reproduce the observed carbon isotope ratios of fractionated simple molecules (e.g.C 2 H/ 13 CCH > 250 and HC 3 N/H 13 CCCN = 79) in dense interstellar clouds (Langer et al. 1984;Furuya et al. 2011;Roueff et al. 2015;Colzi et al. 2020;Loison et al. 2020;Sipilä et al. 2023).Isotope exchange reactions can fractionate carbon-bearing species because of the zero-point vibrational energy difference between 13 C isotopologues and 12 C isotopologues.One of the most important carbon isotope exchange reactions is where the vibrational energy difference is ∆ E = 35 K (Watson et al. 1976).Langer et al. (1984) have introduced this exchange reaction into their gas-phase astrochemical model.They concluded that at a low temperature (∼20 K), the forward reaction of the reaction in Eq.( 1) is efficient compared to the backward reaction, and the small carbon-bearing species are divided into two groups; 13 C-rich species formed from CO (e.g., CO 2 and CH 3 OH) and 13 C-poor species formed from C + (e.g., CH 4 ).In addition to the reaction in Eq.( 1), other carbon isotope exchange reactions have been suggested based on quantum chemical calculations such as 13 C + C 3 ⇌ 12 C + 13 CC 2 + 28 K and lead to carbon isotope fractionation (Roueff et al. 2015;Colzi et al. 2020;Loison et al. 2020).
Isotope selective photodissociation and the difference in binding energies among different isotopologues also cause carbon isotope fractionation.For example, selfshielding of CO against UV photodissociation can cause the fractionation (van Dishoeck & Black 1988;Visser et al. 2009).This effect is however only pronounced at visual extinction of Av ∼1 mag.Moreover, 12 CO have a slightly lower binding energy than 13 CO because it is lighter (Smith et al. 2015(Smith et al. , 2021)).Consequently, carbon-bearing species in the solid phase may be en-riched in 13 C, while those in the gas phase may be depleted in 13 C. Jørgensen et al. (2016Jørgensen et al. ( , 2018) ) suggested that the observed carbon isotope fractionation of COMs in IRAS16293-2422B might be affected by isotope selective photodissociation of CO and/or difference in binding energies between 12 CO and 13 CO.In this paper, we report gas-grain chemical reaction network calculations, systematically investigating the carbon isotope fractionations of COMs.This calculation includes their formation before star formation and their sublimation into the gas phase following star formation.In section 2, we describe a physical model of a star-forming core and the gas-grain chemical model.In Section 3, we present the results of time variation of molecular abundances and 12 C/ 13 C ratios in the base model.The effects of the atomic C insertion or addition reactions on the grain surface and difference in binding energy between 12 CO and 13 CO and the initial condition of carbon and 12 C/ 13 C ratio of C + are also studied.In Section 4, we compare our calculation results with the observations in the IRAS 16293-2422B source.We also discuss the effect of cosmic ray (CR) ionization rates on 12 C/ 13 C ratios of COMs.Our findings are summarized in Section 5. We adopt one-dimensional radiation hydrodynamics simulations by Masunaga & Inutsuka (2000) as a phys-ical model.This model traces the evolution of a dense, starless core to a protostellar core.A fluid parcel is traced in the hydrodynamics simulation, and we calculate the evolution of molecular abundances with a chemical reaction network along the fluid parcel as done in Aikawa et al. (2008Aikawa et al. ( , 2020)).The model is divided into two phases: the static phase and the collapse phase.In the static phase, the prestellar core is in hydrostatic equilibrium assuming that turbulence prevents the contraction of the core.In the calculations, we adopt the gas and dust temperature of 10 K, the number density of hydrogen nuclei of 2.28×10 4 cm −3 , and the visual extinction of 4.51 mag, which correspond to the value of the fluid parcel at 1×10 4 au from the core center (Masunaga & Inutsuka 2000;Aikawa et al. 2008).In the subsequent collapse phase, a protostar is formed due to gravitational collapse and further grows through envelope accretion.Following the first collapse and the formation of an adiabatic core, the protostar is born at 2.5×10 5 yr when the second collapse stops (Masunaga & Inutsuka 2000;Aikawa et al. 2008).This model further proceeds to track the physical evolution for 9.3×10 4 yr until the fluid parcel reaches 30.6 au from the core center.Figure 1 shows the temporal variation of the number density of hydrogen nuclei, gas temperature, and visual extinction in a fluid parcel.We define the moment of the protostar formation as t core = 0 and the end of calculation as t core = t final = 9.3×10 4 yr.To increase the visibility of the evolution around the final stage, we adjust the horizontal axis to the logarithmic scale of t final -t core (Aikawa et al. 2008).At t core = t final in the collapse phase, which is the final time step with both gas and dust temperatures reaching ∼ 198 K, the total H 2 density reaching ∼ 1.49×10 8 cm −3 , and visual extinction ∼ 70 mag.The infalling timescale inside 1000 au is shorter than the lifetime of the protostar in this model so that we can use the time evolution of abundances in a single fluid parcel to represent the true radial distribution (Aikawa et al. 2020).For the sake of simplicity, we set a minimum temperature of 10 K throughout our simulations.

The base model
We utilize the gas-ice astrochemical code based on the rate equation approach (Rokko code; Furuya et al. 2015) and incorporate two hydrogenation reactions on grain surface; CH 2 CO ice + H ice → CH 3 CO ice and CH 3 CO ice + H ice → CH 3 CHO (Ruaud et al. 2015).We also expand to include mono-13 C species and carbon isotope exchange reactions (see Table A1).The chemistry is described by a three-phase model (Furuya et al. 2016).
This model makes a distinction between the surface of the ice mantle and the rest of the ice mantle (Hasegawa & Herbst 1993).Like in Hasegawa & Herbst (1993), we assume that the ice mantle phase remains chemically inert, while the ice surface phase considers chemical reactions.Species in our chemical network consist of carbon skeletons up to three carbons to reduce computational time.We consider the species with a different position of 13 C as the same species (e.g., 12 C 13 CH and 13 C 12 CH are the same species) for simplicity.We have omitted the multiple fractionations, such as two or more 13 C for simplicity.Isotope exchange reactions are taken from Roueff et al. (2015) and Loison et al. (2020), and they are considered in the gas phase. 13C-bearing species, however, are taken into account in ice surface chemistry through the adsorption of 13 C-bearing species onto the grain surface and the subsequent grain surface reactions, such as hydrogenation of species.We neglect 13 C fractionation via the isotope selective photodissociation in the following sections because we only consider the dense region.Our reaction network consists of 733 gas and grain species and 20360 gas phase and grain surface reactions.
We assume the Langmuir-Hinshelwood mechanism to describe two-body reactions on grain surfaces.In this mechanism, species on the grain surface diffuse by thermal hopping and react with each other when they meet.The set of adsorption energies is adopted from Furuya et al. (2015).The barrier for thermal diffusion of atoms and molecules, excluding hydrogen (H), is set to be 40 % of the adsorption energy (E des ).The barrier is set at 30 % of E des for hydrogen.We assume that the sticking probability of colliding gaseous species except for hydrogen onto the grain is unity.For hydrogen, the sticking probability is calculated based on Hollenbach & McKee (1979).We adopt the low-metal elemental abundances taken from Aikawa et al. (2001) through our calculations.The species are assumed to be initially atoms or atomic ions, except for hydrogen, which is in H 2 .The dust grain is spherical with a 0.1 µm radius with the material density of 2.5 g/cm 3 .The dust-togas ratio is set to 0.01.The CR ionization rate of H 2 is set to be 1.3×10 −17 s −1 (Terzieva & Herbst 1998).In this paper, we define "abundance" as the fractional abundance of species to hydrogen nuclei. 12C/ 13 C ratios are measured by the statistical factor corresponding to the two or three indistinguishable carbon atoms.For example, C 2 H has two kinds of carbon isotopologues of 13 C 12 CH and 12 C 13 CH, but they are not distinguished in our chemical reaction network.Therefore, the 12 C/ 13 C ratio of C 2 H is calculated as 'n(C 2 H)/n( 13 C-containung C 2 H)' multiplied by 2 in this work.

The direct C-atom addition reactions
Besides the Langmuir-Hinshelwood process, complex molecules could be formed via nondiffusive grain surface reactions at low temperature (Jin & Garrod 2020;Garrod et al. 2022).Given atomic carbon's high binding energy of 10,000 K, its diffusion on grain surfaces at the low temperatures of the ISM is challenging.However, recent laboratory experiments and quantum chemical calculations reveal that atomic C reacts with icy light species (e.g., H 2 O, and CO) to form H 2 CO and C 2 O ice on the grain surface at 10K, respectively (Molpeceres et al. 2021;Fedoseev et al. 2022;Ferrero et al. 2023).Under ISM condition, H 2 O and CO, with binding energies of 5,600K and 833K respectively, show slow diffusion on the grain surface.Therefore, we characterize the low-temperature reactions involving atomic C with H 2 O and CO chemistries as Eley-Rideal (ER) reactions, which are the direct collision of a gaseous species with adsorbed species on a grain surface.We incorporate the subsequent two ER reactions involving atomic C (hereafter direct C-atom addition reaction) into the model presented in Section 3.2. (2) These direct C-atom addition reactions are assumed to occur when atomic C is adsorbed from the gas phase onto its reaction partner on the grain surface.Therefore, the reaction coefficient (s −1 ) of the direct C-atom addition reaction (k Cadd ) is given by k acc is the accretion rate coefficient of atomic C onto a grain surface.n grain is the number density of dust grains, S is the sticking probability, σ is the geometrical cross-section of a dust grain, ⟨ v ⟩ is the thermal velocity of atomic C, and n s (X) is the number density of species X on a grain surface.For Eq.(4), n s (total) is the number density of total species present on a grain surface, and A is a branching ratio.According to quantum chemical calculations, for the reaction in Eq.( 2), approximately 30% of adsorbed atomic C on H 2 O converts to H 2 CO via a barrierless pathway, while the remaining atomic C is present as it is (Molpeceres et al. 2021;Tsuge et al. 2023), so the branching ratio of the reaction in Eq.( 2) is set to be 0.3 (A = 0.3).For the reaction in Eq.( 3), all adsorbed atomic C on CO converts to C 2 O in a barrierless way (Ferrero et al. 2023), so the branching ratio is set to be 1 (A = 1.0).Some gaseous atomic C take part in these direct C-atom addition reactions instead of the adsorption, so the adsorption rate coefficient of atomic C is adjusted by subtracting the rate coefficient of the direct C-atom addition reaction as where k acc,adj is the adjusted accretion rate coefficient.
The impact of direct C-atom addition reactions on the 12 C/ 13 C ratios of COMs is discussed in Sect.3.2.

The Static Phase
Figure 2 shows the temporal variation of abundances and 12 C/ 13 C ratios of some selected species in the static phase with a fixed temperature (10 K), density (2.28×10 4 cm −3 ) and CR ionization rate (1.3×10 −17 s −1 ).With the exception of considering different positions for 13 C or differences in the number of isotope exchange reactions the models in Colzi et al. (2020) and Loison et al. (2020) are similar to our model in the static phase.Nitrogen-bearing species related to isotope exchange reactions are discussed in Appendix B.2. Dominant carriers of carbon in the gas phase change as time goes on.Initially, C + is dominant.C + is gradually converted to atomic C and then to CO at ∼ 10 5 yr.These small gas-phase species freeze out onto the grain surface and contribute to surface reactions.Adsorbed atomic C is converted to CH 4 ice via a sequence of hydrogenation reactions on the grain surfaces before 10 5 yr, while adsorbed CO is converted to CH 3 OH ice.During the static phase, 13 C fractionation occurs via isotope exchange reactions (Furuya et al. 2011;Colzi et al. 2020;Loison et al. 2020).For example, considering 13 C + + 12 CO ⇌ 12 C + + 13 CO + 35 K (Eq.( 1)) or 13 C + C 3 ⇌ 12 C + 13 CC 2 + 28 K, these isotope exchange reactions lead to the depletion of 13 C in atomic C and C + , while CO and C 3 become enriched in 13 C until 1 × 10 5 yr.Around 2 × 10 5 yr, CO becomes the dominant carrier of gas-phase carbon, so the 12 C/ 13 C ratio of CO gets closer to the local ISM value (Furuya et al. 2011).The abundance of gaseous atomic C decreases with time due to the conversion to CO and CH 4 ice.Around 2 × 10 5 yr, gaseous atomic C is produced from C 3 and C 2 via photodissociation or reaction with atomic O. Therefore, the 12 C/ 13 C ratio of atomic C temporally approaches to the local ISM value due to the low 12 C/ 13 C ratio of C 3 and C 2 .After that, C + , which is produced through  the destruction of CO by He + , is converted to gaseous atomic C or C 3 , so these species become depleted in 13 C. Around 10 3 yr, some species such as C 2 are depleted in 13 C as they are formed from C + or atomic C, and after that they are enriched in 13 C since their major reactants change into 13 C-enriched C 3 or CO.
Icy molecules (e.g.CH 4 and CH 3 OH) are formed via adsorption of these fractionated small species (e.g.CO or atomic C) and surface reactions, and thus the time variation of 12 C/ 13 C ratios of the icy molecules follow those of gas-phase species.For example, before 10 5 yr, icy molecules such as CH 4 and CH 3 OH are formed from atomic C and CO, therefore the 12 C/ 13 C ratios of these icy molecules reflect the ratios of atomic C or CO.After 10 5 yr when CO becomes the main carbon reservoir, the carbon isotope ratio of CH 3 OH ice becomes closer to the local ISM value, reflecting that of CO in the gas phase.Meanwhile, the formation of CH 4 ice becomes negligible since most atomic C is converted to CO after 10 5 yr.Therefore, CH 4 ice keeps the high carbon isotope ratio in the same as before 10 5 yr.Consequently, the 12 C/ 13 C ratios of dominant icy molecules exhibit a bimodal pro-file: depletion of 13 C or similar to or slightly enriched in 13 C (by ∼ 10 %) compared to the local ISM value.
COMs are formed from simple molecules.Before around 10 5 yr when C + and atomic C are the main carbon reservoir, COMs (e.g.CH 3 CHO) and their ices are mainly formed from 13 C-poor molecules, while after around 10 5 yr when CO is main carbon reservoir, COMs (e.g.CH 3 OCH 3 ) and their ice are formed from slightly 13 C-rich CO or CH 3 OH on the grain surface or adsorbed on grains after the formation in the gas phase.The 12 C/ 13 C ratios of COMs follow those of simple molecules and exhibit a bimodal profile.Some molecules including COMs (e.g.CH 3 OCHO and HCOOH) are formed from both 13 C-poor species and 13 C-rich species.For example, CH 3 OCHO is formed from H 2 CO.H 2 CO is formed from CO ice on the grain surface and CH 3 in the gas phase, so the 12 C/ 13 C ratios of these molecules including COMs show the intermediate value between the bimodal profile.

The Collapse Phase
In the collapse phase (see Fig. 1), icy molecules on the grain surface sublimate into the gas phase at their sublimation temperatures, which depend on their binding energies, and the rest in the bulk ice mantle are trapped in water ice and sublimate at ∼ 120 K together with water ice.Moreover, some species are additionally produced by grain surface reactions.Figure 3 shows the temporal variation of abundances and 12 C/ 13 C ratios of selected species in the collapse phase.For CH 4 , the binding energy is set to be 1300 K, CH 4 ice on the surface sublimates around 20 K (t final -t core ∼ 2 × 10 4 yr) and the gas phase CH 4 abundance increases to almost 1 × 10 −6 .After that, whole CH 4 ice sublimates at ∼ 120 K (t final -t core ∼ 10 2 yr) together with water ice.The 12 C/ 13 C ratios of abundant molecules (molecular abundance of ∼ 10 −5 ) molecules such as CH 4 and CH 3 OH after water ice sublimation (t final -t core ∼ 10 2 yr) reflect those of ice formed during the static phase.After water ice sublimates at ∼ 120 K, the 12 C/ 13 C ratio of sublimated CH 4 becomes significantly depleted in 13 C as well as that of CH 4 ice while that of sublimated CH 3 OH becomes enriched in 13 C as well as that of CH 3 OH ice.
On the other hand, some icy COMs (e.g.CH 3 CHO and CH 3 OCHO) are produced via radical-radical reactions on the grain surface during the collapse phase.The abundances of the additionally formed ices are equivalent to or exceeding the abundances of the icy COMs formed during the static phase.Therefore, the 12 C/ 13 C ratios of some sublimated COMs (e.g.CH 3 CHO and CH 3 OCHO) are different from those of their ice formed in the static phase.The 12 C/ 13 C ratios of icy molecules formed during the collapse phase depend on the 12 C/ 13 C ratios of reactants such as radicals.For example, the CH 3 OCHO ice is formed from 13 C-enriched HCO radical ( 12 C/ 13 C ∼ 30) during the collapse phase, so the 12 C/ 13 C ratio of CH 3 OCHO ice decreases, and eventually the 12 C/ 13 C ratio of sublimated CH 3 OCHO is different from that of ice formed in the static phase.Consequently, the 12 C/ 13 C ratios of some COMs after water ice sublimation are essentially a mixture of those formed during the static phase and the collapse phase, sometimes closer to the 12 C/ 13 C ratios of icy COMs formed during the collapse phase rather than those in the static phase.Moreover, the formation of a part of COMs proceeds even in the gas phase via ion-neutral reactions after the sublimation, resulting in distinct carbon isotope ratios compared to those in ice before sublimation.The detailed explanation for other complex molecules is presented in Section 4.1.

The effect of the direct C-atom addition reactions
Based on recent laboratory experiments and quantum chemical calculations, we incorporate the two direct Catom addition reactions by the ER mechanism; Eq.( 2)  ( Molpeceres et al. 2021;Potapov et al. 2021) and Eq.(3) (Fedoseev et al. 2022).Figure 4 shows the temporal variation of abundances and 12 C/ 13 C ratios of some icy species in the static phase without and with the direct C-atom addition reactions.In the model with the direct C-atom addition reactions, the abundances of icy CH 3 OH and CH 3 CHO increase compared to the base model.On the other hand, the abundance of CH 4 ice, which is mainly formed via hydrogenation reactions of atomic C on the grain surfaces, slightly decreases compared to the base model (by ∼ 15 %).The 12 C/ 13 C ratios of icy CH 3 OH and CH 3 CHO in the model with the direct C-atom addition reactions are closer to the ISM value compared to those in the base model.CH 3 OH ice is formed via hydrogenation reactions of 13 C-enriched CO ice on the grain surfaces in the base model.In contrast, in the direct C-atom addition model CH 3 OH ice is additionally formed from slightly 13 C-depleted atomic C via the reaction in Eq.( 2) and the subsequent hydrogenation reactions on the grain surfaces.CH 3 CHO ice is formed from 13 C-depleted C + and atomic C via gasphase reaction with HCO and the subsequent hydrogenation reactions on grain surfaces in the base model.In contrast, in the model with the direct C-atom addition reactions CH 3 CHO ice is additionally formed from slightly 13 C-depleted atomic C and 13 C-enriched CO via the reaction in Eq.( 3) followed by a sequence of hydrogenation reactions on the grain surfaces.These formation processes make differences in carbon isotope ratios of CH 3 OH and CH 3 CHO between the models with and without the direct C-atom addition reactions.
Figure 5 shows formation pathways of CH 3 OH and CH 3 CHO in the models with the direct C-atom addition reactions which suggests that due to these reactions, the isotope ratios of some species could change.Figure 6 shows the temporal variation of abundances and 12 C/ 13 C ratios of some icy species in the collapse phase without and with the direct C-atom addition reactions.Some icy complex molecules related to the direct C-atom addition reactions (e.g.CH 3 CHO) have lower 12 C/ 13 C ratios compared to those in the base model.After water ice sublimation, molecules that are formed via the direct C-atom addition reactions and following reactions, also have lower 12 C/ 13 C ratios compared to those in the base model (see Figure 9).We note that Molpeceres et al. (2021) treated water clusters as the representative ice on grain surfaces, so variations in the composition of interstellar ice could affect the branching ratio A. Therefore, we additionally run models changing the branching ratio A, and find that the results obtained with A = 0.15 are consistent with those derived when A is set to 0.3.We discuss the effect of the direct C-atom addition reactions for selected individual species in Section 4.1 and we show the results of the model with the direct C-atom addition reactions in Appendix B.

The effect of the difference in Binding Energy
between 12 CO and 13 CO If the binding energies of CO isotopologues on the dust surface are mass-dependent and thus 13 CO is larger than that of 12 CO, 12 CO can sublimate at a lower temperature than 13 CO.Consequently, CO in the gas phase would become depleted in 13 C, while CO ice would become enriched in 13 C. Smith et al. (2015) theoretically investigated the mass-dependence of thermal desorption of CO and dust temperature for segregation between 12 CO and 13 CO in the ices.They considered the balance between the adsorption rates and desorption rates for 12 C 16 O and 13 C 16 O.As a result, assuming a 10 K difference in binding energy, the 12 CO/ 13 CO gas ratio reaches twice as high as the elemental 12 C/ 13 C ratio at a gas temperature of 10 K.Moreover, Smith et al. (2021) derived binding energies of pure 12 CO ice and pure 13 CO ice to be 833 ± 5 K, and 846 ± 6 K respectively based on laboratory experiments.We additionally run models assuming that the binding energy of 12 CO is 833 K and that of 13 CO is 846 K for both the static phase and the collapse phase.We find that the difference in binding energy does not affect the 12 C/ 13 C ratios of COMs although the difference leads to the desorption rate of CO to be slightly increased, nearly 10 %.

Dependence on the elemental 12 C/ 13 C ratio and Initial Condition of Carbon
In our models presented in the previous section, we assume the elemental 12 C/ 13 C ratio is 68, which corresponds to the local ISM value (Milam et al. 2005).Here, we explore how our results depend on the assumed value of the elemental 12 C/ 13 C ratio. Figure 7 shows the temporal variation in the collapse phase with the elemental 12 C/ 13 C ratio = 89 which is the average value in the Solar System.We find that the 12 C/ 13 C ratios of molecules are scaled with assumed 12 C/ 13 C ratio.
So far we assumed that carbon is initially present as 12 C + and 13 C + .Here, we investigate the dependence of the initial form of carbon on the carbon isotope ratios of molecules.We additionally run models with and without the direct C-atom addition reactions during the static phase and collapse phase where initially half of the carbon is present as 12 CO and 13 CO with 12 CO/ 13 CO = 68, while the rest is present as 12 C + and 13 C + with 12 C + / 13 C + = 68.Figure 8 shows the temporal variation of abundances and 12 C/ 13 C ratios of some selected species in the static phase without the direct C-atom addition reactions in this case.The abundance of C + is half of that in the base model, so the abundances of molecules formed from C + such as CH 4 become smaller compared to the base model, and their 12 C/ 13 C ratios become significantly elevated compared to the base model.For example, around 10 6 yr in the static phase, the abundance of CH 4 ice is 1.0 × 10 −5 , that is about 2 times smaller and the 12 C/ 13 C ratio of CH 4 is around 225, that is about 1.5 times larger compared to those in the base model, in which the abundance is 2.1 × 10 −5 and the ratio is 133.On the other hand, CO is more abundant than in the base model, but as shown in the base model (see left panel of Fig. 2) almost carbon eventually transforms into CO at 10 6 yr in the static phase.Therefore, the effect of initial carbon form on the molecules formed from CO is smaller than that on the molecules formed from C + .Around 10 6 yr in the static phase, the abundance of CH 3 OH ice is 3.0 × 10 −5 and the 12 C/ 13 C ratio of CH 3 OH is 61, those values are similar to those in the base model, in which the abundance is 3.7 × 10 −5 and the ratio is 60.Some complex molecules formed from both 13 C-poor and 13 C-rich species via radical-radical reactions or the direct C-atom addition reactions are not significantly affected by an ini-Figure 8. Same as left panel of Figure 2 but with the initial condition in which half of the carbon is in the form of CO, while the remaining half is in the form of C + tial form of carbon.However, complex molecules mainly formed from 13 C-poor species such as C 3 H 4 show depletion in 13 C compared to the base model.Therefore, if we consider these molecules, we need to account for the influence of the initial form of carbon on the carbon isotope ratios.In our base model, CH 2 CO gas is depleted in 13 C, and 12 C/ 13 C ∼ 95 after water ice sublimation (Fig. 3).This value is set by the sublimation of CH 2 CO ice.CH 2 CO ice is mainly formed from C 2 ice on the grain surface via a reaction with atomic O followed by a sequence of hydrogen addition reactions on the grain surfaces in early time (t final -t core ∼ 10 4 yr) of the collapse phase.C 2 ice is mainly formed from C 2 H 2 ice or via the adsorption of gaseous C 2 , which become depleted in 13 C ( 12 C/ 13 C ∼ 100) due to the reaction in Eq.( 1), around 10 6 yr in the static phase.Therefore, the 12 C/ 13 C ratio of sublimated CH 2 CO is depleted in 13 C as well.In the model with the direct C-atom addition reactions, the formation of CH 2 CO ice is much more efficient in the static phase compared to the base model.CH 2 CO ice is formed from C 2 O ice on the grain surface via a sequence of hydrogen addition reactions.C 2 O ice is formed by the direct C-atom addition reaction, Eq.(3).So, the 12 C/ 13 C ratio of CH 2 CO ice ( 12 C/ 13 C ∼ 80) is lower compared to the base model ( 12 C/ 13 C ∼ 180) at 10 6 yr in the static phase as well as CH 3 CHO ice discussed in Section 3.2.Additionally, 13 C-depleted CH 2 CO ice is also formed in the collapse phase as in the base model.At the CH 2 CO sublimation temperature (∼ 40 K, t final -t core ∼ 2 × 10 3 yr), the CH 2 CO ice on the grain surface are desorbed into the gas phase and CH 2 CO gas becomes depleted in 13 C.At water sublimation temperature (∼ 120 K, t final -t core ∼ 10 2 yr), CH 2 CO ice in the mantle phase, which is mainly formed in the static phase and enriched in 13 C, is evaporated into the gas phase and CH 2 CO gas eventually turned into slightly enriched in 13 C ( 12 C/ 13 C = 89, Fig. B2).

CH3CHO
Our base model shows that the 12 C/ 13 C ratio of sublimated CH 3 CHO is significantly depleted in 13 C ( 12 C/ 13 C ∼ 115).CH 3 CHO ice is formed from 13 Cdepleted C + and atomic C before 10 5 yr in the static phase as shown in Fig. 5.In addition, in early time (t final -t core ∼ 10 3 yr, Fig. 2) of the collapse phase, CH 3 CHO ice is formed from CH 2 CO ice via a sequence of hydrogenation reactions or a radical-radical reaction between CH 3 and HCO on the grain surfaces.These reactants are slightly depleted in 13 C ( 12 C/ 13 C ∼ 100).Therefore, sublimated CH 3 CHO has lower 12 C/ 13 C ratio compared to the CH 3 CHO ice formed in the static phase (Fig. 3).In the model with the direct C-atom addition reactions, CH 3 CHO ice is mainly formed in the static phase due to the direct C-atom addition reaction, Eq.( 3), and has lower 12 C/ 13 C ratio compared to the base model without the direct C-atom addition reactions (see Section 3.2).Unlike CH 2 CO, CH 3 CHO ice mainly formed in the static phase, so the 12 C/ 13 C ratio of sublimated CH 3 CHO is equal to that of CH 3 CHO ice which is formed in the static phase and has lower 12 C/ 13 C ratio compared to the base model without the direct C-atom addition reactions.Therefore, the direct C-atom addition reactions decrease the 12 C/ 13 C ratio of CH 3 CHO to ∼ 80.
In this work, we introduced two hydrogenation reactions on grain surfaces; CH 2 CO ice + H ice → CH 3 CO ice and CH 3 CO ice + H ice → CH 3 CHO ice, following Ruaud et al. (2015).The hydrogen addition to CH 2 CO ice is the dominant pathway for CH 3 CHO ice formation.This pathway also plays an important role in keeping low carbon isotope ratio of CH 3 CHO in the direct C-atom addition model since C 2 O is formed from 13 C-enriched CO and then the subsequent hydrogenation reactions of C 2 O on the grain surfaces forms CH 3 CHO.Without this hydrogenation pathway of CH 2 CO, the 12 C/ 13 C ratio of CH 3 CHO gas is significantly depleted in 13 C ( 12 C/ 13 C ∼ 120) after water is sublimated even though we incorporate the direct C-atom addition reactions.

CH3OCH3
In our base model, CH 3 OCH 3 ice is mainly formed in the collapse phase from 13 C-poor CH 3 and 13 C-rich CH 3 O radical, so the 12 C/ 13 C ratio of CH 3 OCH 3 ice is intermediate value, ∼ 75 before the sublimation.In addition, CH 3 OCH 3 is formed in the hot gas phase via proton transfer.The gaseous CH 3 OCH 3 is formed from the reaction between CH 3 OH and CH 3 OH 2 + via proton transfer.The 12 C/ 13 C ratio gradually gets close to that of CH 3 OH after the sublimation, and decreases to ∼ 68. (a) Jørgensen et al. (2018).The uncertainties on the ratios are 30% from error propagation of the noise in the observations. (b)  carbon isotope ratios of gas-phase molecules at 30.6 au from the core center in our models are listed. (c) The ISM value is used, which is consistent with the carbon isotope ratios derived using a few optically thin lines of main isotopologues of CH2CO and HCOOH.
In the model with the direct C-atom addition reactions, the 12 C/ 13 C ratio of CH 3 OCH 3 is similar to that in the base model because CH 3 OCH 3 is mostly formed during the collapse phase and the 12 C/ 13 C ratio changes with time.Both in the models with and without the direct C-atom addition reactions, the 12 C/ 13 C ratio of CH 3 OCH 3 becomes closer to the ISM value after water ice is sublimated.

HCOOH
In our base model, HCOOH ice is formed from CO in the static phase, so the ice is slightly enriched in 13 C (by ∼ 10 %) at t core ∼ t final .The 12 C/ 13 C ratio of sublimated HCOOH is also slightly enriched in 13 C.After the sublimation gaseous HCOOH is additionally formed from H 2 CO, which originates from CO, via OH + H 2 CO → HCOOH.Thus, the 12 C/ 13 C ratio of HCOOH doesn't change and remains close to that of CO.In the model with the direct C-atom addition reactions, the 12 C/ 13 C ratio of HCOOH is similar to that in the base model since HCOOH ice is mainly formed from CO during the later static phase (∼ 10 5 yr).

C2H5OH
In our model, the 12 C/ 13 C ratio of C 2 H 5 OH gas is ∼ 69 after water ice sublimation.This value is set by the sublimation of C 2 H 5 OH ice which is mainly formed via radical-radical reaction on the grain surface during the collapse phase.The formation of C 2 H 5 OH ice involves the rection between 13 C-poor CH 3 ice and slightly 13 C-rich CH 2 OH ice.As a result, the 12 C/ 13 C ratio of C 2 H 5 OH aligns with the local ISM value.This remains the case even after including the direct C-atom addition reactions.

CH3OCHO
In our model, the 12 C/ 13 C ratio of CH 3 OCHO gas is ∼ 60 after water ice sublimation.This value is set by the sublimation of CH 3 OCHO ice which is mainly formed via radical-radical reaction on the grain surface during the collapse phase.CH 3 OCHO ice is formed via radical-radical reaction between 13 C-poor CH 3 O ice and 13 C-rich HCO ice, so the 12 C/ 13 C ratio of CH 3 OCHO is the intermediate value (∼ 60) regardless of whether the direct C-atom addition reactions are included.

Comparisons with Observations of IRAS16293-2422B
IRAS 16293-2422B is a low-mass protostar harboring a hot corino.Icy grains accrete towards the central hot region surrounding the protostar, and the bulk water ice sublimates at 100 -200 K.As a result, the composition of the hot corino region is thought to be determined by the ice sublimation.In our model, the region where water ice sublimates, located inside approximately 100 au from the core center, is larger than the angular resolution of the PILS survey, which is a 60 au diameter.Therefore, this region is distinctly resolved.We compare our modeling results to the observational data for the IRAS 16293-2422B, which are obtained by the ALMA-PILS line surveys (Jørgensen et al. 2016(Jørgensen et al. , 2018)).
In Figure 9 and Table 1, our results are compared with the observations.We note that the physical model adopted in our work reproduces moderately well the observations of IRAS 16293-2422 (e.g., Figure 1 of Wakelam et al. 2014).Table 1 compares the observations and the results of our calculations at t core ∼ t final for the base model and the model with the direct C-atom addition reactions.In the base model, sublimated CH 2 CO, CH 3 CHO, and CH 3 OCH 3 are more depleted in 13 C than the observations while HCOOH is similar to the observations.In the model with the direct C-atom addition reactions, CH 2 CO and CH 3 CHO have lower 12 C/ 13 C ratios compared to those in the base model and are similar to the observational data.This suggests that the direct C-atom addition reactions could play a significant role in the formation of observed organic molecules.However, regardless of the direct C-atom addition reactions, CH 3 OCH 3 remains close to the ISM value and the calculated carbon isotope ratio is larger than the observation (∼ 34).We need more investigation to reproduce the observed carbon isotope ratio of CH 3 OCH 3 .
As noted in Jørgensen et al. (2018) many of the observed lines of main isotopologues of CH 2 CO and HCOOH are optically thick, so the column densities of these main isotopologues were derived from the optically thin 13 C isotopologues lines assuming a standard 12 C/ 13 C ratio (= 68).The derived column densities of CH 2 CO and HCOOH are consistent with the observations of a few optically thin transition lines of main isotopologues.

The Effect of Cosmic Ray
Since the calculated carbon isotope ratio of CH 3 OCH 3 gas does not reproduce the observed value, we investigate the effect of cosmic ray on the 12 C/ 13 C ratio of CH 3 OCH 3 and other molecules.CH 3 OCH 3 gas is formed from ion-molecule reaction via proton transfer induced by cosmic-ray ionization after water ice sublimation in the collapse phase.Therefore, the 12 C/ 13 C ratio of CH 3 OCH 3 can be affected by the cosmic ray ionization rate.In the models presented in previous sections, we assume that the cosmic-ray ionization rate of H 2 is constant and is 1.3×10 −17 s −1 .Here we additionally run the astrochemical models with various CR ionization rates.

Interstellar Cosmic Ray Flux
We adopt the CR ionization rate model as a function of gas column density in Padovani et al. (2009Padovani et al. ( , 2018)), in which the detailed processes of energy loss and propagation of CR are taken into account.Padovani et al. (2018) considered two models; the L (Low spectrum) model and the H (High spectrum) model for the interstellar CRs proton spectrum.L model comes from the recent data from the extrapolation of the Voyager missions (Cummings et al. 2016), while the H model is from a measurement of H 3 + in the diffuse medium (Indriolo & McCall 2012).Figure 10 shows the temporal variation of the CR ionization rate for the model with the constant CR ionization rate of 1.3 × 10 −17 s −1 , the H model, and L model.We run the chemical reaction network calculations using the H model and L model for CR ionization rate with and without the direct C-atom addition reactions.
Figure 11 shows the temporal variation of molecular abundances and 12 C/ 13 C ratios in the static phase in the H model.The higher CR ionization rate leads to the shorter timescale of ion-neutral chemical reactions including isotope exchange reactions.Gaseous CO already becomes a dominant carbon reservoir around 10 4 yr.Thus, the degree of the isotope exchange reactions decreases, and then the 12 C/ 13 C ratio of CO gets closer to the local ISM value.The molecules formed from CO, such as CH 3 OH, HCOOH, and CH 3 OCH 3 and ices thereof mainly after 10 5 yr, have the 12 C/ 13 C ratio closer to the local ISM value compared to those in the base model.The 12 C/ 13 C ratios of other species (e.g.CH 3 CHO) and their ices formed from 13 C-depleted C + or atomic C are smaller compared to those in the base model.Around 10 6 yr higher CR ionization rate leads to atomic C being depleted in 13 C because ionized and atomic C is produced by secondary photons, coming from CR-induced H 2 electronic excitation and the efficiency of the isotope exchange reactions increases (Colzi et al. 2020).Therefore, the molecules formed from C + or atomic C, such as gaseous CH 4 also become more depleted in 13 C than in the base model at 10 6 yr.
Figure 12 shows the temporal variation of molecular abundances and 12 C/ 13 C ratios in the collapse phase with the H model.In the early time (t final -t core ∼ 10 4 yr) some icy COMs (e.g.CH 3 OCH 3 ) are produced more efficiently by the higher CR ionization rates, and these are depleted in 13 C.The CR-induced UV photons dissociate stable molecules producing radicals on grain surfaces, and then stimulate the formation of COMs on the grain surface.For example, the CH 3 ice on the grain surface is also depleted in 13 C since the gaseous CH 4 is depleted in 13 C at the end of the static phase.As a result, COMs formed from CH 3 ice by radical-radical reactions on warm grains become more depleted in 13 C after water ice sublimation than in the base model.Moreover, for a higher CR ionization rate, the 12 C/ 13 C ratio of CH 3 OCH 3 gas approaches that of CH 3 OH more quickly, because the CR ionization rate promotes the formation of CH 3 OCH 4 + via ion-molecule reactions and then the formation of CH 3 OCH 3 by the following dissociative recombination, CH 3 OCH 4 + + e − → CH 3 OCH 3 form CH 3 OCH 3 in the warm gas phase.However, the 12 C/ 13 C ratio of CH 3 OCH 3 at t core ∼ t final is ∼ 80, 2) because the 12 C/ 13 C ratio of CH 3 OCH 3 after water ice sublimation is depleted in 13 C ( 12 C/ 13 C ∼ 95).
We incorporate the direct C-atom addition reactions together with the H model. Figure 13 shows the temporal variation of molecular abundances and 12 C/ 13 C ratios in the collapse phase with the H model and the direct C-atom addition reactions.In the H model, the suppression of 12 C/ 13 C ratios of COMs (e.g.CH 3 CHO) does not occur even when considering the direct C-atom addition reactions.Atomic C becomes CO and is depleted in the gas phase at ∼ 10 4 yr in the static phase, so the degree of the direct C-atom addition reactions become smaller, and the abundances of icy COMs pro-duced via the direct C-atom addition reactions in the static phase becomes smaller compared to the model with the constant CR ionization rate of 1.3 × 10 −17 s −1 and the direct C-atom addition reactions.These icy COMs are much more formed via radical-radical reactions in the collapse phase in the same as the model without the direct C-atom addition reactions.As a result, the 12 C/ 13 C ratios of COMs in the warm gas trace those formed on warm dust grains during the collapse phase.For example, the abundance of CH 3 CHO ice at 10 6 yr in the static phase decreases by a factor of 100 (the molecular abundance is ∼ 10 −9 ) (see also Figure 4).On the other hand, the abundance of CH 3 CHO ice formed from 13 C-depleted CH 3 during the collapse phase is ten times larger than that formed in the static phase.As a result, the 12 C/ 13 C ratio of CH 3 CHO gas after water ice sublimation is depleted in 13 C despite incorporating the direct C-atom addition reactions.Therefore, the high CR ionization rate during the static phase is not suitable for the formation of COMs (such as CH 3 CHO) in the viewpoint of their carbon isotope ratios.For the L model, the result is similar to that of the model with the constant CR ionization rate of 1.3 × 10 −17 s −1 .

Cosmic Ray Acceleration after protostar formation
Following the results in Sect.4.3.1, in this section, we consider the effect of possible variations of CR ionization rate during the star formation.CRs are suggested to be accelerated by the local shocks around the protostar (Padovani et al. 2015(Padovani et al. , 2016) ) such as strongly magnetized shock along the outflow or by the accretion shocks.Cabedo et al. (2023) measured abundances of molecular ions in a solar-type protostellar object and suggest high CR ionization rates of 10 −16 -10 −14 s −1 , possibly locally accelerated at shocks.We run the additional model in which we set the CR ionization rate to be 1.3×10 −14 s −1 after the protostellar formation.We note that the CR ionization rate higher than 1.3×10 −14 s −1 destroys COMs via proton transfer by H 3 O + after sub- limation as suggested by Nomura & Millar (2004) and eventually abundance of some molecules (e.g.CH 3 CHO and CH 3 OCH 3 ) decrease to around 1 × 10 −11 at t core = t final .Figure 14 shows the temporal variation of molecular abundances and 12 C/ 13 C ratios in the collapse phase with CR ionization rate of 1.3×10 −14 s −1 and the direct C-atom addition reactions.CH 3 OCH 3 is formed from CH 3 OH and CH 3 OH 2 + via proton transfer after water ice sublimation and then the 12 C/ 13 C ratio of CH 3 OCH 3 approaches that of CH 3 OH and becomes enriched in 13 C (∼ 60).Consequently, the 12 C/ 13 C ratio of CH 3 OCH 3 decreases if the CR ionization rates become high only in the collapse phase.However, the ratio is still higher than the observation towards IRAS16293-2422B (Jørgensen et al. 2018).For other selected molecules, the 12 C/ 13 C ratios of HCOOH, CH 2 CO, CH 3 CHO are similar to those in the base model with the direct C-atom addition reactions (see a right panel of Fig. B2).

SUMMARY
We investigated the carbon isotope fractionation of COMs from prestellar cores to protostellar cores by combining the chemical network model and the radiation hydrodynamical simulation.The temporal variation of the molecular abundances and 12 C/ 13 C ratios were calcu-APPENDIX

B.2. Nitrogen-bearing molecules
Figures B4 and B5 show the temporal variation of abundances and 12 C/ 13 C ratios of nitrogen-bearing species in the static phase and the collapse phase, respectively.These results in the static phase in our model are similar to the results in Colzi et al. (2020) and Loison et al. (2020).The 12 C/ 13 C ratios are affected by carbon isotope exchange reactions.Some molecules (e.g.HNCO, H 2 CN, NH 2 CHO and CH 3 NH) show different time variations compared to our base model at around 10 4 yr in the static phase due to the direct C-atom addition reaction 2. These molecules are formed from H 2 CO ice.In the base model HNCO, H 2 CN, and NH 2 CHO are partly formed from 13 C-enriched CO and H 2 CO, but in the model with the direct C-atom addition reactions H 2 CO is formed via Reaction 2 and is depleted in 13 C, similar to atomic C. Therefore, the 12 C/ 13 C ratios of these molecules increase when the direct C-atom addition reactions are incorporated.CH 3 NH is formed from CH 3 or H 2 CN, so the 12 C/ 13 C ratio changes by the direct C-atom addition reactions.Around 10 6 yr, however, these differences are no longer evident because H 2 CO ice is formed from CO ice via a sequence of hydrogenation reactions on the grain surface.In the collapse phase, there is no difference between our base model and the model with the direct C-atom addition reactions.For HC 3 N, the 12 C/ 13 C ratio depends on the position of 13 C within the molecules.Therefore, we will address these aspects in future work.

Figure 1 .
Figure 1.Temporal variation of the number density of hydrogen nuclei (black line), gas temperature (red line), and visual extinction (blue line) of a fluid parcel along a streamline in a gravitationally collapsing core.

Figure 2 .
Figure 2. Temporal variation of the molecular abundances and 12 C/ 13 C ratios for gaseous species (solid lines) and icy species (dashed lines) during the static phase in the base model.The horizontal black dashed line represents the average 12 C/ 13 C ratio of local ISM.

Figure 3 .
Figure 3. Same as Figure 2 but during the collapse phase.

Figure 4 .
Figure 4. Temporal variation of the molecular abundances and 12 C/ 13 C ratios of CH4 ice, CH3OH ice, and CH3CHO ice during the static phase for the base model (dotted lines) and the model with the direct C-atom addition reactions (solid lines).

Figure 5 .
Figure 5. Schematic formation pathways of CH3OH and CH3CHO with the direct C-atom addition reaction.Gas phase species (g) link to the icy species on the grain surface (s).The dashed arrow indicates the isotope exchange reaction.Red-colored species are depleted in 13 C, while bluecolored species are enriched in 13 C. Purple arrows indicate the direct C-atom addition reaction.Species 2H indicate consecutive hydrogenation reactions on the grain surfaces.

Figure 6 .
Figure 6.Same as Figure 4 but during the collapse phase.

Figure 7 .
Figure 7. Same as right panel of Figure 3 but adopting the direct C-atom addition reactions and initial 12 C/ 13 C = 89.The horizontal black dotted line in the lower panel represents the average 12 C/ 13 C ratio of the Solar System.

Figure 9 .
Figure 9. Temporal variation of 12 C/ 13 C ratios of some organic molecules for the base model (dotted lines) and the model with the direct C-atom addition reactions (solid lines).The black horizontal dashed line is for the average 12 C/ 13 C ratio of local ISM.The observations in IRAS16293-2422B are represented in light gray shaded regions.The observations of C2H5OH and CH3OCHO have large uncertainties, so we refrain from comparing our results with the observations.

Figure 10 .
Figure 10.Temporal variation of the CR ionization rate per H2 for the base model (black line), H model (orange line), and L model (blue line) during the collapse phase.

Figure 11 .
Figure 11.Same as Figure 2 but for the H model for CR ionization rate

Figure 13 .
Figure 13.Same as right panel of Figure 3 but adopting the H model for CR ionization rate and the direct C-atom addition reactions.

Figure 14 .
Figure 14.Same as right panel of Figure 3 but with the direct C-atom addition reactions, and 1.3×10 −14 s −1 as CR ionization rate after the protostar birth.
FigureB1and B2 show the temporal variation of abundances and the 12 C/ 13 C ratios of some molecules with the direct C-atom addition reactions in the static phase and collapse phase, respectively.

Figure B1 .
Figure B1.Same as Figure 2 but adopting the direct C-atom addition reactions

Figure B2 .
Figure B2.Same as Figure 3 but adopting the direct C-atom addition reactions.

Figure B3 .
Figure B3.The temporal variation of abundance and the 12 C/ 13 C ratio of HC2O gas for the base model (dotted lines) and the model with the direct C-atom addition reactions (solid lines).The left panel is for the static phase and the right panel is for the collapse phase.

Figure B4 .
Figure B4.Temporal variation of the molecular abundances and 12 C/ 13 C ratios for gaseous species during the static phase in the model with the direct C-atom addition reactions (solid lines) and the base model (dashed lines).The horizontal black dashed line represents the average 12 C/ 13 C ratio of local ISM.

Figure B5 .
Figure B5.Same as Figure B4 but for during the collapse phase.

Table 1 .
12C/ 13 C ratio for some organic molecules in IRAS16293-2422B and our modeling results.