The connection between warm carbon chain chemistry and interstellar irradiation of star-forming cores
aa r X i v : . [ a s t r o - ph . GA ] F e b Draft version February 5, 2021
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The connection between warm carbon chain chemistry and interstellar irradiation ofstar-forming cores
Juris Kalv¯ans Ventspils University of Applied SciencesEngineering Research Institute “Ventspils International Radio Astronomy Center”In ˇz enieru 101, Ventspils, LV-3601, Latvia (Received September 19, 2020; Revised January 1, 2021; Accepted February 2, 2021) Submitted to ApJABSTRACTSome observations of warm carbon chain chemistry (WCCC) cores indicate that theyare often located near the edges of molecular clouds. This finding may suggest thatWCCC is promoted in star-forming cores exposed to radiation from the interstellarmedium. We aim to investigate the chemistry of carbon chains in such a core. Achemical simulation of a gas parcel in a low-mass star-forming core with a full levelof irradiation by interstellar photons and cosmic rays was compared to a simulationof a core receiving only one-tenth of such irradiation. In the full irradiation model,the abundances of carbon chains were found to be higher by a factor of few to fewhundred, compared to the model with low irradiation. Higher carbon-chain abundancesin the prestellar stage and, presumably, in the extended circumstellar envelope, arisebecause of irradiation of gas and dust by interstellar photons and cosmic rays. Afull standard rate of cosmic-ray induced ionization is essential for a high carbon-chainabundance peak to occur in the circumstellar envelope, which is heated by the protostar(the “true” WCCC phenomenon). The full irradiation model has lower abundances ofcomplex organic molecules than the low-irradiation model. We conclude that WCCCcan be caused by exposure of a star-forming core to interstellar radiation, or even justto cosmic rays. The Appendix describes an updated accurate approach for calculatingthe rate of cosmic-ray induced desorption.
Keywords: astrochemistry — ISM: clouds, cosmic rays, dust, molecules — stars: for-mation INTRODUCTION
Corresponding author: Juris Kalv¯[email protected]
Warm carbon-chain chemistry (WCCC)in low-mass young stellar objects (YSOs)was first discovered by Sakai et al. (2008a).The envelope in the L1527 molecular cloudhosting an embedded Class 0 or 0/I proto-star IRAS 04368+2557 (Ohashi et al. 1997;
Kalv¯ans
Jørgensen et al. 2002; Sakai & Yamamoto2013) was found to be rich in carbonaceousspecies, such as C H, c-C H , C H, and others.The beam-averaged temperature was foundto be 14 K, while the density is ≈ cm − .Thus, the observations of protostellar car-bon chains were associated with dense “luke-warm” (gas temperature .
40 K) circumstellarenvelopes. WCCC is distinct from the hotcorino chemistry, characterized by abundantand warm ( >
100 K) oxygen-containing com-plex organic molecules (COMs; Cazaux et al.2003; Bottinelli et al. 2004), such as methanolCH OH.The discovery of additional sources, suchas IRAS 15398–3359 in Lupus and IRAS18148–0440 in L483, established WCCC asa separate type of YSOs (Hirota et al. 2009;Sakai et al. 2009a, 2016; Cordiner et al. 2011;Graninger et al. 2016; Higuchi et al. 2018).Other observations revealed a diverse vari-ety of carbon chains (Sakai et al. 2007, 2008b,2009b; Hirota et al. 2010; Tokudome et al.2013; Ag´undez et al. 2015; Araki et al. 2016;Yoshida et al. 2019). The sources were foundto consist of a WCCC core, rich in smallermolecules, such as c-C H , and an extended cir-cumstellar envelope, more rich in longer chains,such as C H and C H, probably remnants fromthe prestellar cloud core. Other molecules, suchas C H and C H, are present in both com-ponents. The WCCC molecules were foundto peak slightly offset relative to the proto-star, i.e., in the lukewarm regions of the cir-cumstellar envelope (Sakai et al. 2009a, 2010;Araki et al. 2017). ALMA observations showthat the very center of the core may containCOMs (Jørgensen et al. 2013; Imai et al. 2016;Oya et al. 2017).WCCC or similar sources have been foundto be associated with very low luminosity ob-jects (YSOs in the earliest star-forming stages;Takakuwa et al. 2011; Cordiner et al. 2012) as well as evolved protostars (Oya et al. 2017).Thus, the carbon-chain rich chemistry can beattributed to starless clouds, such as TMC-1 (e.g., Takano et al. 1990; Sakai et al. 2007),first hydrostatic cores, and Class 0 and I pro-tostars. Later evolutionary stages are likelyto show hot-corino type chemistry (i.e., rich inhot COMs, Graninger et al. 2016). Even mas-sive star-forming regions, rich in carbon chains,have been found (Mookerjea et al. 2010, 2012;Saul et al. 2015; Taniguchi et al. 2018b). Aspecial case of a WCCC object might be theL1489 molecular cloud, which is heated ex-ternally by the nearby protostar L1489 IRS(Wu et al. 2019a). Warm carbon chains havealso been observed in shocked regions of theinterstellar medium (ISM; Lefloch et al. 2018;Wu et al. 2019b).Observational surveys of protostars show thathot corinos and WCCC cores probably are twoextremes and that many YSOs are intermedi-ate in their content of COMs and carbon chains(Graninger et al. 2016; Lindberg et al. 2016;Higuchi et al. 2018; Law et al. 2018). A num-ber of high-resolution interferometric observa-tions have studied the inner workings of the cen-tral engines of WCCC cores (e.g., Sakai et al.2014a,b, 2017; Aso et al. 2015; Oya et al. 2015),revealing that infalling matter undergoes a mildshock at the transition zone between the enve-lope and the protostellar disk. There is a lackof medium spatial resolution interferometric ob-servations that are sensitive to the spatial scaleof a circumstellar envelope (Oya 2020).The working hypothesis for the origin of low-mass WCCC protostellar cores is gas-phase pro-cessing of methane CH , evaporated from ices inthe lukewarm regions of the circumstellar enve-lope (Sakai et al. 2008a, 2009a). This picturewas immediately supported by several astro-chemical modeling studies (Aikawa et al. 2008;Harada & Herbst 2008; Hassel et al. 2008) andis supported also by detections of methane and CCC and irradiation ice is unnecessary. Their in-terpretation of observations also support theidea that carbon chains have been left overfrom the prestellar phase (Charnley & Cordiner2010; Cordiner et al. 2012).In line with observational evidence, modelsshow that WCCC and hot corino chemistry (i.e.,abundant COMs in temperatures above 100 K)can coexist in a single source (Hassel et al.2011; Garrod et al. 2017; Aikawa et al. 2020).The WCCC is initiated by the warm-up ofthe protostellar envelope, inducing evapora-tion of methane, which is converted to car-bon chains and similar species via gas phasechemistry. As the protostar heats the enve-lope and the temperature rises, these species aresynthesized, frozen-out and evaporated again(Taniguchi et al. 2019). Not all of the CH icemight be available for immediate sublimationand participation in WCCC because the wateryice matrix may partially lock CH inside themantle (Wang et al. 2019; Aikawa et al. 2020).A less clear question is the source of CH ice, necessary to produce the WCCC cores.Sakai et al. (2008a, 2009a,b) suggested that afaster collapse of the prestellar core would notallow CO molecules to become the dominantform of carbon, instead carbon would accu-mulate onto grains as CH ice. This ideawas apparently disproved by the model of Aikawa et al. (2020). Astrochemical models donot indicate such a possibility of rapid CH synthesis, as CO always seems to dominatethe C budget in a gas with hydrogen numeri-cal density n H exceeding about 10 cm − , espe-cially when CO self-shielding and mutual shield-ing by H from the interstellar radiation field(ISRF) are considered (e.g., Herbst & Leung1989; Aikawa et al. 2001; Rawlings et al. 2002;Garrod & Herbst 2006). The process that gen-erates methane ice must be uncommon, as onlyrelatively few star-forming cores are affected byit.Observations give clues to the origin ofWCCC. Carbon-chain rich cores in various evo-lutionary stages reside in specific regions in theISM, such as Heiles and Lupus I clouds. Ad-ditionally, the surveys by Higuchi et al. (2018)and Lefloch et al. (2018) indicate that WCCCcores tend to be located outside dense con-centrations and thus are likely to be exposedto higher radiation levels than non-WCCCand hot corino sources. The possible asso-ciation of WCCC cores with higher radiationis also evidenced by WCCC observations nearmassive star-forming regions (Taniguchi et al.2018a; Sicilia-Aguilar et al. 2019) and Her-big Ae/Be stars (Lindberg & Jørgensen 2012;Lindberg et al. 2016; Watanabe et al. 2012).Exposure to an increased radiation flux causeshigh abundances of carbon chains also in star-less and prestellar cores (e.g., Li et al. 2016;Spezzano et al. 2016; Pan et al. 2017). The lat-ter observation gives the possibility that highcarbon chain abundances simply are the resultof synthesis in irradiated cold gas. However,conditions in the WCCC cores have not beenthoroughly measured and their connection to el-evated radiation is not clear, meaning that irra-diation is only one possible cause or prerequisitefor WCCC.The aim of this study is to qualitatively in-vestigate the chemistry of carbon chains in star- Kalv¯ans forming cores irradiated from outside by com-paring the chemistry of carbon chains in ashielded cloud core and in a core exposed toradiation. In this way, we aim to understand, ifoutside irradiation can facilitate the formationof a WCCC core.Isolated cores outside larger dense structuresare more exposed to the ISRF and cosmicrays (CRs). This results in an overall greaterflux of ionizing radiation, even deep withinthe core. Moreover, CR-induced desorption(CRD), caused by whole-grain heating, has re-cently been found to be potentially very effi-cient (Kalv¯ans 2018a). Thus, CRD may sig-nificantly delay the freeze out of both, COand CH (Kalv¯ans & Kalnin 2019), affectingthe possibility of WCCC. The observations byBouvier et al. (2020) show that intense irradia-tion by nearby newborn massive stars can turnthe star-forming cloud into a photodissociationregion (PDR), changing the balance betweenCOMs and carbon-chains. In such a case, nei-ther hot corino, nor WCCC activity was ob-served; the observed CCH and CH OH emissionoriginated in the PDR. This means that we needto consider only mild increase of radiation, notthe orders of magnitude increase experienced byPDRs.In order to fulfill the above aim, we performtwo simulations of chemistry in star-formingcores. One of these was assumed to have a mod-est irradiation, relevant to a shielded dense core,while the other was assumed to be exposed to ahigher level of interstellar radiation. The astro-chemical model is described below in Section 2,while the results of the simulations are analyzedin Section 3. The final conclusions were drawnin Section 4. METHODSWe employ the astrochemical model
Alchemic-Venta , initially developed byKalv¯ans (2015a). The latest version of thismodel is described by Kalv¯ans & Kalnin (2019). The model follows the chemistry in a gas parcelwith conditions relevant to a collapsing densecloud core that transforms into an initial, stillspherical, protostellar envelope. In the enve-lope stage, the parcel undergoes heating by thecentral engine, while other conditions remainunchanged. A comprehensive description of themodel follows.2.1.
Physical model N H N out N H,r - N H r Figure 1.
Schematic geometrical setting of themodel (not to scale). The filled circle indicates thelocation of the modeled cloud parcel, while the solidblack arrows indicate the segments of core diame-ter, along which the hydrogen column density N H was calculated. The empty circle indicates core cen-ter. The task of the physical model is to simulatethe evolution of the specific gas parcel in a low-mass star-forming core that eventually hostsWCCC. The model is neither intended nor isable to simulate realistic evolution and geome-try of the whole star-forming core. The macro-physical model considers a 1D (shell) structureof a spherical star-forming core in Lagrangianmass coordinates. We assumed a core mass of M core =1.0 M ⊙ . Chemistry was calculated onlyat one point of the model, in order to obtainclearly interpretable results. This point, orparcel of infalling gas, was chosen so that it CCC and irradiation ≈ cm − , in line with observations of WCCCcores and other WCCC models. The parcel islocated in a shell with a constant relative masscoordinate of X m =0.55. The core collapse stageends with the assumed formation of the proto-star at t =1.6 Myr. A 0.84 Myr long protostellarstage follows, where the parcel density and col-umn density towards the parcel were held con-stant, while temperature gradually rises, simu-lating the heating by the protostar.The physical model for the prestellar core wasbuilt as follows. The hydrogen number density n H, at the center of the core ( X m =0) evolvesaccording to a free-fall collapse, increasing bythree orders of magnitude (see Table 1). Thecollapse is delayed by a factor of 0.7, as as-sumed by a number of authors, starting withNejad et al. (1990). With an initial densityof 2000 cm − , such a delayed collapse occursover an integration time of t ≈ . n H, now known, the density n H,r at a core radius r was then calculated witha Plummer-like approximation (Plummer1911; Whitworth & Ward-Thompson 2001;Taquet et al. 2014) n H,r = n H, (1 + (cid:16) rr (cid:17) )] η/ , (1)where r is the radius of the central densityplateau, proportional to n − / H, (Keto & Caselli2010) and η was taken to be 3. Equation (1) wasemployed to calculate the gas density and massof core shells with radius r , ranging from 0 to r . The latter value corresponds to X m =1.0.In other words, the total radius of the corewas maintained such that the core mass al- ways is 1.0 M ⊙ . The core was assumed to besurrounded by gas with a column density of N out = 2 × cm − , a feasible value for solar-mass cores (Launhardt et al. 2013). n H , cm -3 t , Myr (0)(0.55) A V , mag t , Myr (0)(0.55) Figure 2.
Evolution of key parameters for the sim-ulation of chemistry in a star-forming cloud core.Top: n H at core center ( X m =0) and at the parcellocated at X m =0.55. Bottom: the A V values forthe core center and the parcel. The known 1D macroscopical structure of thecloud core was used to calculate column den-sity N H and ISRF extinction A V , assuming that A V = N H / . × . With the initial con-ditions listed in Table 1, the initial visual ex-tinction at X m =0.55 is A V, . =1.5 mag, i.e., be-low the value when interstellar water ice hasbeen observed accumulating on grain surfacesWhittet et al. (2001, at a line-of-sight A V of3.2 mag). Figure 1 shows that the parcel wasassumed to be irradiated from two opposite di-rections along the diameter of the core. Themacrophysical model was designed so that gasdensity n H, . at X m =0.55 is 10 cm − at theend of the prestellar phase and during the pro-tostellar phase. Figure 2 shows the resulting Kalv¯ans
Table 1.
Physical parameters at the start and the assumed end of the 1.0 M ⊙ prestellar core collapse stage.The indices 0, 0.55, and 1 indicate the relative mass coordinate X m , except for r , which indicates the radiusof the density plateau. t (Myr) n H, (cm − ) n H, . (cm − ) A V, (mag) A V, . (mag) r . (AU) r (AU) r (AU)0 2 . × . × . × . × . × . × . ×
68 9.0 2 . × . × . × ζ, s -1 t , Myr (irr.)(ref.) T gas , K t , Myr (irr.)(ref.) Figure 3.
Evolution of the CR ionization rate ζ and gas temperature for the irradiation and ref-erence models. Dust temperature follows a curvesimilar to that of T gas . evolution of n H, and n H, . . WCCC probablycannot be reproduced for much higher densitiesbecause of a rapid freeze-out of carbon chains,preventing high gas-phase abundances. Thus,many simulations of WCCC cores consider den-sities of ≈ cm − (see references in Section 1).As for a model that considers a single parcelthat is not in the center of the core, irradia-tion is mostly regulated by A V and, thus, N H ,between the parcel and the closest edge of thecore. Compared to radiation coming from thenear edge (see A V, . in Table 1), the radiationcoming from the opposite side (through the cen- ter of the core) is reduced by 2.4 mag at t =0and 127 mag in the protostellar stage at t =1.6–2.4 Myr (cf. Figures 1 and 2). Thus, the far-sideradiation affects chemistry in the initial stagesand becomes increasingly unimportant in theadvanced stages. Irradiation coming from theopposite side of the cloud was considered for apresumably more realistic chemical history forthe considered parcel.In the envelope, the gas parcel, for which wecalculated chemistry, is located at a distanceof 2640 AU from the protostar. This value ishigher than those known for WCCC cores; ob-servations indicate radius of 500–1000 AU forL1527 (Sakai et al. 2010) and 1500–2500 AU forthe protostar in the Lupus 1 cloud (Sakai et al.2009a). However, as stated above, we do notaim to simulate the collapsing cloud core, in-stead limiting the model to providing an evolu-tionary history for a gas parcel that eventuallyproduces WCCC in the protostellar envelope.The irradiation intensity of our consideredparcel of the cloud core is the key changing pa-rameter. The reference model was assumed tobe irradiated by a tenth of the standard ISRFphoton and CR flux (see Section 2.2), assumingthat the core is partially shielded by its parentcloud complex. In other words, the standard in-tensity of these radiation types was reduced bya factor of 0.1. Hence the abbreviation of thismodel is “1REF0.1”.Figure 3 shows how the CR-induced ionizationrate and temperature differs for the referenceand irradiation models. In the prestellar stage, CCC and irradiation A V . Gas temperature T gas was cal-culated from the data of Hocuk et al. (2016), asin Kalv¯ans et al. (2017): log ( T gas ) =3 . − . . − log ( A V )) +1 . . − log ( A V )) +1 . . − log ( A V )) . (2)Dust temperature T dust was calculated with theapproach of Hocuk et al. (2017), which, in ad-dition to A V , also depends on the ISRF ( G ),and thus differs for the reference and irradiationmodels. The value of T gas was not allowed to belower than that of T dust .For the protostellar stage, T gas and T dust were calculated with the T profile ofGarrod & Herbst (2006) and Garrod et al.(2008). We adopted the long and slow heat-ing time-scale, relevant for low-mass protostars,where a temperature of 200 K is reached in1 Myr. From t = 1 .
62 to 2.45 Myr, the tem-perature rises from ≈
10 K at the end of theprestellar stage to 142 K (the maximum temper-ature in the model) at the end of the protostel-lar envelope stage. The simulations end shortlyafter sublimation of all water ice. Higher tem-peratures are irrelevant for this investigation,which focuses on the lukewarm WCCC cores.2.2.
Chemical model
Alchemic-Venta is a modified kinetic rate-equation model, based on the ALCHEMIC code(Semenov et al. 2010). We employ the UMISTdatabase for astrochemistry 2012 (UDfA12) asthe gas-phase reaction network (McElroy et al.2013). This network includes ionization anddissociation by CR protons, CR-induced pho-tons, and the ISRF, as well as binary reac-tions of neutral and ionic species, including re-combination of cations. The UDfA12 networkmakes our model different from previous WCCCsimulations, which employ networks based on
Table 2.
Initial abundances of chemical speciesrelative to hydrogen.Species Abundance ReferenceH + . × −
1N 2 . × −
1O 1 . × −
1F 6 . × − + . × − + . × − + . × − + . × − + . × − . × − + . × − the OSU database (Aikawa et al. 2008, 2020;Harada & Herbst 2008; Hassel et al. 2008, 2011;Wang et al. 2019). Compared to OSU, UDfA12has recalculated photoreaction rates for the in-terstellar (Draine field) and CR-induced pho-tons. These differences may cause some dis-agreement with the results of the previoussimulations. To prevent the accumulation oflarge amounts of hydrogen at low tempera-tures, the H contact desorption was employed(Hincelin et al. 2015).Table 2 shows the initial elemental abun-dances of chemical species. We employ low-metal abundances, which result in an ice mantlethickness of about 85 monolayers (MLs), whentotal freeze-out occurs. The simulation takes afew 10 years for the chemistry to relax to quasi-equilibrium. Real chemical equilibrium is neverattained for any species because of the changingphysical conditions.The ISRF was assumed to correspond to aphoton flux of 1 . × cm − s − (note, the radi- Kalv¯ans ation comes from two opposite sides, as shownin Figure 1). Self-shielding of H , and self-and mutual shielding (by H ) of CO and N from the ISRF photons was considered with thehelp of the tabulated data by Lee et al. (1996)and Li et al. (2013). Shielding of icy surfacemolecules was not considered because their UVabsorption bands are shifted in relation to thoseof gaseous species. The model includes desorp-tion of products from photodissociation (ISRFand CR-induced) of icy species on the outer sur-face layer of icy grains, according to the generalapproach outlined in Appendix C of Kalv¯ans(2018b).UDfA12 has more than sufficient chemi-cal diversity for considering both carbon-chainspecies and COMs for the aims of this study.The network for surface reactions was adaptedfrom the COMs network by Garrod et al. (2008,based on the OSU database) with changes fromLaas et al. (2011) and Kalv¯ans (2015b). Thisnetwork includes more organic species than theUDfA12 network, thus the COMs surface net-work was reduced accordingly. The surface bi-nary reaction rate was adjusted by reaction-diffusion competition (Garrod & Pauly 2011).Photodissociation rates of icy species wereadopted from UDfA12 and thus are updatedand considerably different from the originalCOMs-OSU network data. Following Kalv¯ans(2018b), surface photoreaction rates were re-duced by a factor of 0.3 in relation to theirgas-phase counterparts. Reactions on the outersurface of the icy grain and within the bulk-icewere included.The icy mantle of molecules adsorbedonto grain surfaces was described with themulti-layer approach (see Kalv¯ans 2015a;Furuya et al. 2017). In addition to the 1–2 ML thick surface layer, three bulk-ice layerswith active chemistry were considered. Thisis sufficient to resolve polar (H O-dominated)and non-polar (CO-dominated) components of evolved icy mantles (Sandford et al. 1988). Forbinary surface reactions, the ratio betweenmolecule binding (or diffusion) energy E b,s and desorption energy E D was taken to be0.50 (Garrod et al. 2008). Reactions can oc-cur either via molecule hopping or tunneling,whichever is faster (Garrod 2013). The ra-tio between the absorption energy of bulk-icespecies E B and E D was taken to be E B /E D = 2and, consequently, the binding energy of man-tle species is E b,m = 1 . E D . E b,m was used tocalculate the rate of bulk ice reactions with theapproach outlined in Kalv¯ans (2015a) and therate of diffusion between the four ice mantle lay-ers (Kalv¯ans & Kalnin 2019).The model considers olivine grains with radius a = 0 . µ m, density of 3 g cm − constituting 1 %of cloud mass. When adsorption occurs, thegrains become covered with molecules occupy-ing an assumed average volume of a cube witha size of 3 . × − cm. Such a molecule vol-ume corresponds to water ice with a density of0.9 g cm − . The surface density of adsorptionsites is 9 . × cm − . The number of adsorp-tion sites in successively adsorbed MLs increasesin accordance with the increasing grain size. Allneutral gaseous species were allowed to becomeadsorbed on grain surfaces. The molecule stick-ing coefficient was taken to be unity, with theexception of hydrogen, for which the coefficientwas calculated following Thi et al. (2010).The model considers several desorption mech-anisms of icy surface species. Desorption byISRF photons regulates the onset of ice accu-mulation at low A V values. CR-induced pho-tons ensure that some desorption occurs even athigh extinctions. For CH , CO, CO , CH OH,N , NH , and H O, experimental photodesorp-tion yields Y ph for both processes were adopted(see Table 3 of Kalv¯ans 2018b, and referencestherein). For all other icy species, Y ph was takento be 0.001. Another important mechanism issimple thermal evaporation. It is the princi- CCC and irradiation a was taken to be 0.03.The above desorption processes were sup-plemented by CR-induced whole-grain heat-ing, resulting in CRD. This desorption mecha-nism was considered in particular detail, thanksto improvements made for the Alchemic-Venta model made in our previous study(Kalv¯ans & Kalnin 2019). In addition, recentadvances in the understanding of evaporativecooling of icy grains have led to an improvedCRD rate calculation. The CRD mechanism isdescribed in detail in Appendix A.The rate of hydrogen ionization by CR pro-tons ζ was calculated following Ivlev et al.(2015), model “High”. This approach gives N H -dependent curve of ζ (Figure 3). Theflux of CR-induced photons, used for CR-induced photodesorption, was made propor-tional to ζ (Cecchi-Pestellini & Aiello 1992;Kalv¯ans & Kalnin 2019). RESULTSAs discussed in Section 2.1, two main astro-chemical simulations of a star-forming cloudcore were run for this study: the reference modeland the irradiation model. The reference cloudcore (Model 1REF0.1 in Table 3) was assumedto be shielded by its parent cloud complex andirradiated by ISRF and CRs at 0.1 of their stan-dard intensity. The irradiation model 2IRR1.0considers a core subjected to standard radiationfields without shielding.In interpreting the modeling results, we usethe notion by Garrod et al. (2008) that the tem-poral chemical evolution of a star-forming core(see Figure 4) also qualitatively represents its 1D spatial structure, with less evolved regionslocated near the outer rim of the core.WCCC regions are defined by their density( ≈ − cm − ) and temperature ( ≈ t interval of 0.8–2.0 Myr, encom-passing the major features in the evolution ofthe carbon- chain abundances, up to a temper-ature of 40 K. Given the variety of the observedWCCC molecules, each figure displays differ-ent species, whenever reasonable. The calcu-lated abundances were compared to those ob-served in L1527, which is the first detectedand probably the most studied WCCC core(e.g. Yoshida et al. 2019). Compared to otherWCCC sources, L1527 has relatively high abun-dances of carbon chains. However, these abun-dances are not outside the realm of those ob-served in other cores, thus L1527 may serve asan example of a ‘classic’, pronounced WCCCsource.3.1. Comparison of reference and irradiationmodels
Figure 4 shows the evolution of the abundancefor a few of the most important gaseous speciesthat affect production of carbon chains. For theirradiation model, CO becomes the main carbonspecies at t =0.81 Myr.After (or behind) the CO photodissocia-tion layer, the irradiation model maintainsa higher abundance of atomic carbon, whichbenefits the synthesis of methane and car-bon chains. WCCC has been associated withgaseous methane, evaporated from icy grains inthe circumstellar envelope (Section 1). In theirradiation model, methane ice abundance in-creases by a factor of up to four, relative tothe reference model. CRD prevents more ef-fective accumulation of CH ice in the irradi-ation model 2IRR1.0, where CRD efficiency isten times higher than in 2REF0.1 (Table 3).In the time interval of interest, the 2IRR1.0model produces abundances of carbon chains0 Kalv¯ans
Table 3.
List of considered models.Flux multiplierNo. Name (type) Abbrev. ISRF CR Description1 Reference 1REF0.1 0.1 0.1 Cloud core, shielded by the parent cloud complex2 Irradiation 2IRR1.0 1.0 1.0 Cloud core exposed to full irradiation from outside3 ISRF irradiation 3ISRF1.0 1.0 0.1 Core exposed to interstellar photons only4 CR irradiation 4CR1.0 0.1 1.0 Core exposed to CRs5 ζ test 5ZETA1.0 0.1 0.1 Model “1REF0.1” with ζ and CR-induced photon fluxincreased by a factor of 10(i.e., the same ζ as in “2IRR1.0” and “4CR1.0” models)6 CRD test 6CRD1.0 0.1 0.1 Model “1REF0.1” with CRD rate increased by a factor of 10(i.e., the same CRD rate as in the “2IRR1.0” “4CR1.0” models) that are, typically, from a few to a few hundredtimes higher than those of the 1REF0.1 model.This result generally confirms that the WCCCphenomenon can be caused by locally increasedirradiation or exposure of a star-forming coreto the ISM. Figure 5 shows that, in the irradia-tion model, a broad plateau of elevated carbon-chain abundances is followed by two abundancepeaks in the 1.5–2.0 Myr interval. The plateaustarts with a small peak at ≈ ≈ − and is not counted as rel-evant to WCCC observations that sample gaswith n H ≥ cm − .The first (or the outer) peak occurs at ≈ t =1.602 Myr), theabundance of gaseous CO, n CO is higher by a factor of 3.1 in the irradiation model, reaching1 . × − relative to H . The abundance ofgaseous C atoms is 94 times higher in the irra-diation model, where it is 5 . × − relative toH , cf. Figure 4. More abundant C benefits tothe formation of carbon chains. The abundanceof atomic O in the 2IRR1.0 model is increasedonly by a factor of 2.7. Carbon-chain formationtriggered by oxygen depletion was proposed al-ready by Cordiner & Charnley (2012).After the first peak, the relative abundanceof gaseous carbon chains drops by one or moreorders of magnitude. During this intermediatetime ( ≈ CCC and irradiation T , K n / n ( H ) t , Myr C+ ref C+ irrC ref C irrCH4 ref CH4 irrCO ref CO irr 0204060801001201401E-091E-081E-071E-061E-051E-04 0.0 0.5 1.0 1.5 2.0 2.5 T , K n / n ( H ) t , Myr H2O(s) ref H2O(s) irrCO(s) ref CO(s) irrCO2(s) ref CO2(s) irrCH4(s) ref CH4(s) irr
Figure 4.
Calculated abundances relative to molecular hydrogen for the reference 1REF0.1 (“ref” in thefigure) and irradiation 2IRR1.0 (“irr”) models for important carbon gaseous species (top) and major icymolecules (bottom). The red curve indicates the temperature of the gas parcel in the 2IRR1.0 model,measured with the right-hand vertical axis. The abbreviation (s) indicates solid surface icy species. also has ten times higher flux of CR-inducedphotons.The second peak at t ≈ . It canbe regarded as the “true” WCCC phenomenonin the model. Interestingly, the gas-phase peakabundance for methane is higher in the 1REF0.12 Kalv¯ans n / n ( H ) t , Myr C H ice ref ice irrgas ref gas irr n / n ( H ) t , Myr C H ice ref ice irrgas ref gas irr n / n ( H ) t , Myr C H ice ref ice irrgas ref gas irr n / n ( H ) t , Myr C H ice ref ice irrgas ref gas irr elevated plateau first peak second peak Figure 5.
Calculated abundances relative to molecular hydrogen for typical carbon-chain species in reference(1REF0.1, “ref”) and irradiation (2IRR1.0, “irr”) models. The vertical gray line indicates the protostarformation time, while the dotted horizontal line indicates abundances observed in L1527 (Sakai et al. 2008a;Araki et al. 2017). The model does not discern between c-C H, c-C H , and l-C H, l-C H , i.e., chain andcyclic isomers are treated together as single species with a single molecular formula. model compared to 2IRR1.0 (1 . × − versus7 . × − , relative to H , respectively, cf. Fig-ure 4). This is because the CRs and CR-inducedphotons in the irradiation model rapidly destroyCH , transferring its carbon to the chains.3.2. Comparison with observations
In accordance with the aims of this study, wepresent only a qualitative comparison of calcu-lation results with observational data. A de-tailed quantitative comparison requires consid-ering a full line of sight, for which at least one-dimensional chemical model is necessary. Such a 1D study will likely be performed in the fu-ture.Observations show coexisting central peakand an extended component for carbon chains inWCCC cores, with no clear boundary betweenthem (Sakai et al. 2010; Araki et al. 2017).Such a picture can be consistent with the re-sults of the 2IRR1.0 model, if we assume thatthe elevated plateau, the first, and the secondpeaks are intermixed along the complex line-of-sight through a real star-forming core. Inthe elevated plateau phase, gas has a notably
CCC and irradiation n / n ( H ) t , Myr C H ice ref ice irrgas ref gas irr n / n ( H ) t , Myr HC N ice ref ice irrgas ref gas irr n / n ( H ) t , Myr C H ice ref ice irrgas ref gas irr n / n ( H ) t , Myr C H ice ref ice irrgas ref gas irr Figure 6.
Calculated abundances relative to molecular hydrogen for selected carbon-chain species, alongwith their observed abundances in L1527, when available (Sakai et al. 2007, 2008a). Details as in Figure 5. lower density than n H in observations ( ≈ versus ≈ cm − ), and thus might not be rele-vant for WCCC. The elevated plateau and thefirst peak can be considered as “remnant”, com-ponents, occurring in outer regions not yet sig-nificantly affected by protostellar activity, whilethe second peak may explain the centrally con-centrated WCCC component, where methane issublimated from ices.In this qualitative study that does not con-sider a full line of sight, a “successful” repro-duction of observational data can be assumedwhen the calculated abundance of a species ex-ceeds the observed value. Figures 5–6 show thatfor most carbon-chain species at least one of the peaks exceeds the observed abundance. In ob-servations, C H has been detected only in thecentral component, apparently corresponding tothe second peak in the model. On the otherhand, C H and C H have an extended struc-ture (Sakai et al. 2010), corresponding to the el-evated plateau and the first peak in the 2IRR1.0model. Such a behavior is approximately repro-duced in the 2IRR1.0 model results, where theextended component for C H is relatively muchweaker than that of C H and C H , while thesecond peak is strong for all three molecules (cf.Figures 5 and 6).Another issue is the presence of COMs inWCCC cores. The correlation between the4 Kalv¯ans abundances of carbon chains and COMs iscomplex (Graninger et al. 2016; Lindberg et al.2016; Higuchi et al. 2018) but it is expected thata “classical” WCCC source, such as the one weare attempting to simulate with the 2IRR1.0model, is abundant in carbon chains and lowon COMs. Surveys comparing the abundancesof only C H and C H to those of CH OHhave been done by Graninger et al. (2016) andLindberg et al. (2016).Figure 7 shows a comparison of abundancesfor typical COMs – methanol and acetaldehydeto those of example carbon chains in the latterstages of the model. At the temperatures rele-vant for hot corinos ( T gas >
100 K, t > . ≥
100 K in thecircumstellar envelope. Quantification of thesefeatures would require a more adequate macro-physical model.Figure 8 shows calculation results for the2IRR1.0 model at relative mass coordinates 0.2, 0.4, 0.55, and 0.7 (in the direction from core cen-ter to its outer edge). These X m values corre-spond to radius from 1600 to 3000 AU, althoughwe emphasize again that this simple model doesnot adequately represent the structure of thestar-forming core. The simulations at the four X m values all have similar temperature curves,although in a real star-forming core it can be ex-pected that circumstellar envelope shells closerto the star are heated faster and reach highertemperatures. However, a qualitative trend canbe obtained from the data in Figure 8 – the car-bon chains reach higher abundances in the outerlayers of the circumstellar envelope. This is pri-marily because of a lower density in the model( n H, . = 8 × , while n H, . = 2 × ), whichdoes not allow rapid freeze-out of carbon chainsthat have just been formed in the gas phase.3.3. The causes of WCCC
To determine the cause of the elevated car-bon chain abundances in the irradiation model,two additional simulations were run. Model3ISRF1.0 has interstellar photon intensity at itsstandard value (1.0) as in the irradiation model,but has a reduced (0.1 of standard) CR inten-sity, similar to that in the reference model. Onthe other hand, model 4CR1.0 has standard in-tensity of CRs and a intensity for ISRF reducedby a factor of 0.1, as in the reference model (seeTable 3).Figure 9 shows that CRs in the 4CR1.0 modelhave a significantly greater effect than inter-stellar photons in producing high carbon-chainabundances, compared to those in the 1REF0.1model. The first and the second peaks hap-pen only due to CRs, while the ISRF and CRscan both contribute to the elevated abundanceplateau before the peaks. The contribution ofISRF is reducing the dominance of CO as thedominant gas-phase carbon species, which al-lows for more carbon chains to form.The effects of CRs are twofold, arising fromionizing radiation (CR protons and their in-
CCC and irradiation n / n ( H ) t , Myr C H OH ice ref ice irrgas ref gas irr n / n ( H ) t , Myr CH CHO ice ref ice irrgas ref gas irr n / n ( H ) t , Myr C H ice ref ice irrgas ref gas irr n / n ( H ) t , Myr HC N ice ref ice irrgas ref gas irr Figure 7.
Calculated abundances relative to molecular hydrogen for examples of COMs and carbon chainsduring the late stage of the 1REF0.1 and 2IRR1.0 models. Details as in Figure 5, observed abundancesshown for L1527 (Sakai et al. 2008a; Yoshida et al. 2019). duced photons) and from grain heating by heavyCR nuclei, such as Si or Fe, which inducesCRD (Appendix A). Figure 10 separates theeffects from these two phenomena, comparinggas phase abundances for some carbon chainscalculated with the reference model 1REF0.1,model 5ZETA1.0, and model 6CRD1.0 (see Ta-ble 3). Model 5ZETA1.0 is similar to the ref-erence model, having the ISRF intensity andCRD frequency reduced by a factor of 0.1, com-pared to their standard values (Section 2.2),while the CR-ionization rate ζ , and, thus, alsoCR-induced photon intensity are at their stan-dard values (flux multiplier is equal to 1.0). On the other hand, model 6CRD1.0 has its CRDfrequency at standard value, while ISRF and ζ are reduced by a factor of 0.1 from their stan-dard values.The plots show that neither a higher CRD in-tensity, nor ζ are able to reproduce the elevatedplateau. Both phenomena contribute to the firstcarbon-chain abundance peak, while only ele-vated ζ is able to reproduce the second peak,which is the actual WCCC region, not a rem-nant from the prestellar core.From the above discussion, the following pic-ture emerges. The early carbon-chain abun-dance plateau arises because of ionization and6 Kalv¯ans n / n ( H ) t , Myr HC N Xm=0.2Xm=0.4Xm=0.55Xm=0.7 n / n ( H ) t , Myr C H Xm=0.2Xm=0.4Xm=0.55Xm=0.7
Figure 8.
Comparison of calculated example carbon-chain gas-phase abundances in the 2IRR1.0 modelsat four different relative mass coordinates X m . Details as in Figure 5; observed abundances in L1527 fromSakai et al. (2008a) and Sakai et al. (2009b). n / n ( H ) t , Myr HC N n / n ( H ) t , Myr CH CCH
Figure 9.
Comparison of calculated gas-phase abundances of examples of carbon-chains in the 1REF0.1(“ref”), 3ISRF1.0 (“isrf”), and 4CR1.0 (“cr”) models. Observed abundances in L1527 from Sakai et al.(2008a). The vertical gray line indicates the protostar formation time. dissociation of molecules by the interstellar pho-tons and CRs, which maintain higher abun-dances of C and C + later in the cloud evolu-tion, allowing more carbon chains to form. Thefreeze-out of oxygen (in the form of water ice) inthe late, dense stages of the prestellar core re-move some destruction paths for carbon-chains, resulting in the first peak. CRs are essential forthis peak, as the CRD process maintains ele-vated abundance of carbon in the gas phase inthe form of the CO molecule, while CR-inducedionization detaches the carbon atoms from CO,allowing carbon chains to form. This same pro-cess allows also a few times more (compared CCC and irradiation n / n ( H ) t , Myr C H - n / n ( H ) t , Myr C H Figure 10.
Comparison of calculated gas-phase abundances of examples of carbon-chains in the 1REF0.1,5ZETA1.0, and 6CRD1.0 models. Observed abundances in L1527 from Sakai et al. (2007) and Sakai et al.(2008b). The vertical gray line indicates the protostar formation time. to the reference model) of CH ice to form ongrain surfaces. Finally, the icy methane evapo-rates and is processed in the gas, producing thesecond peak. Because gas-phase chemistry isdriven by CR-induced ionization, a sufficientlyhigh value for ζ is essential for the second peak.The carbon chains synthesized in the gas arethen dissociated by CR-induced photons, whichmakes the second peak a temporal phenomenon. CONCLUSIONSObservations indicate that WCCC may arisein star-forming cores exposed to interstellarradiation. We have compared simulations ofchemistry in a shielded core and in a core ex-posed to full irradiation by ISRF and CRs. Themain findings are listed below. • The irradiated core has carbon-chain gas-phase abundances higher by a factor of upto 10 , compared to the reference model.The exact increase depends on species andthe evolutionary stage of the core and of-ten results in molecules’ calculated abun-dances being similar to or higher than ob-served abundances. This means that theirradiation model may better reproduce observations. The primary cause of thehigher abundances is a higher ionizing ra-diation flux. • Three general features can be discernedin the evolution of carbon-chain abun-dances – an elevated plateau, a first peakat star-formation time and a second, the“true” WCCC, peak at the temperatureof methane evaporation. The plateau andthe first peak can be considered as “rem-nants” from the prestellar stage. • The elevated plateau occurs at a relativelylow density of ≈ cm − , which is lowerby a factor of ≈
100 than the densitiesassociated with WCCC, and thus mightbe unobservable for some species. Theplateau is caused by the ionization by theISRF and CRs. • The first peak is a product of the com-bined effects of CR-induced ionizationand CRD. • For the second “true” WCCC peak, onlyCR-induced ionization ( ζ ) is essential.It induces an active gas-phase chemistry8 Kalv¯ans that is able to convert the carbon inevaporated methane molecules into car-bon chains. • At lower densities, newly formed carbonchains freeze out slower and the WCCCphenomenon is more pronounced. • Unlike carbon chains, the abundances ofCOMs have no clear correlation with ra-diation, at least up to temperatures of ≈
140 K.We conclude that WCCC is possible in star-forming cores that are sufficiently irradiated byCRs with ζ & − s − . This value can be dif-ferent for models employing chemical networksother than UDfA12. A star-forming core witha CR irradiation decreased by a factor of 0.1(compared to the “standard” irradiation in the model) shows only weak WCCC features and ismore abundant with COMs.ACKNOWLEDGMENTSThis research has been funded by ERDFpostdoctoral grant No. 1.1.1.2/VIAA/I/16/194‘Chemical effects of cosmic ray induced heatingof interstellar dust grains’ being implemented inVentspils University of Applied Sciences. I amalso grateful to Ventspils City Council for itssupport. This research has made use of NASA’sAstrophysics Data System. I thank the anony-mous referees for the thorough review and manyvaluable comments that greatly improved themanuscript. Software:
ALCHEMIC (Semenov et al.2010),
Tcool (Kalv¯ans & Kalnin 2020a)APPENDIX A. COSMIC-RAY INDUCED DESORPTION (CRD)The current version of the
Alchemic-Venta model takes into account the stochastic aspect ofwhole-grain heating by CRs by considering the ices in the heated grains as a separate physical “warm”phase. The ambient grains are converted to hot grains with a rate k warm and converted back (cooldown) with a rate k cool . Such an approach is valid because even a relatively small parcel of the cloudcore with a mass of, e.g., 10 − M ⊙ contains a huge number of heated grains (septillion or more), evenwith the low whole-grain heating rate employed by Hasegawa & Herbst (1993).The warm ice phase differs from ambient icy grains with its temperature T CR , which is higher thanthe temperature T dust of ambient grains. Chemical reactions, reactive desorption, inter-layer diffusionrates, and sublimation all occur at T CR . Photoprocessing was not considered for the warm phasebecause of its unimportance relative to icy molecule photoprocessing from ambient grains. On theother hand, chemical reaction (and thus, reactive desorption) and, especially, sublimation rates arestrongly dependent on temperature. From all these processes, sublimation (i.e., CRD) is the one,which can significantly affect ice composition during the cloud core collapse stage (Kalv¯ans & Kalnin2019).Unlike our previous studies, in the present research we have the tools and data to acquire realisticvalues for T CR , k warm , and k cool . The first task is to choose a simple, yet justified and realistic,approach in calculating these parameters as functions of N H and amount of ices adsorbed on the0.1 µ m grains. First, a single, constant T CR can be employed. This is because the proportionsof different sublimated species remain similar in 40–70 K (as shown by Kalv¯ans & Kalnin 2020a),which is our T CR range of interest (see below). Grain heating rate k warm depends on CR intensity CCC and irradiation t cool , which is inversely proportional to k cool depends onthe properties of the icy mantle – thickness and amount of volatiles.The heating of icy interstellar grains by CRs is a complex phenomenon, with grains carryingvarying amounts of adsorbed ices being heated to various temperatures T CR with various frequencies f T . To obtain a single, characteristic T CR , we calculated the weighed average heating temperaturefrom the complete grain heating energy spectra provided by Kalv¯ans (2018a). The T CR values wererecalculated with the grain heat capacity C derived by Leger et al. (1985) because Kalv¯ans (2018a)partially used the simple Debye approach on C , which is inadequate for grain temperatures exceeding ≈
30 K (Kalv¯ans & Kalnin 2020b). The weighed average ¯ T CR is different for grains with differentice layers: 64 K for bare grains, 58 K for grains with a 0.01 µ m thick ice mantle, 54 K for grainswith 0.02 µ m ice, and 50 K for grains with 0.03 µ m ice. Interestingly, because of changes in CRspectra with increasing column densities (see Padovani et al. 2009), ¯ T CR for grains with 0.03 µ mice decreases from 50 K to 44 K for N H of 2 . × to 5 . × cm − , respectively. All thesetemperatures exceed the crucial 40 K threshold, where the total energy loss becomes dominated byradiative, not evaporative cooling (Kalv¯ans & Kalnin 2020a). We chose ¯ T CR =54 K as a compromisevalue. It corresponds to an ice mantle thickness representing partial freeze out of heavy moleculesonto grains, i.e., representing a stage of ongoing accumulation of molecules in ices, when CRD mostsignificantly affects ice composition. E e v a p / E c oo l , % T CR , K f ' K , s - N H , cm -2 t c oo l , s CO ice, MLs ( a ) ( b ) ( c ) Figure 11. ( a ): percentage of energy that a heated grain loses via sublimation of volatile species ( E subl ),relative to the total energy E cool released during the cooling of the grain. ( b ): CR-induced grain heatingfrequency to the assumed average heating temperature of 54 K. The trendline drawn through the calculationpoints corresponds to Equation A2. ( c ): cooling time of a 0.1 µ m icy grain as a function of CO ice abundanceon the grain, expressed in MLs, Equation (A3). Knowing the chosen grain temperature allows calculating f T . When a single ¯ T CR is used, it mustrepresent all the energy, received by a grain from CRs, that is used for the sublimation of icy molecules.Naturally, this energy arises from impacts by CRs of various types, elevating the temperature of theicy grains to different values. The Tcool program, developed by Kalv¯ans & Kalnin (2020a), wasused to calculate, how much energy E evap goes away with sublimation of layered ices at differentgrain temperatures, compared to the total grain thermal energy lost during cooling E cool . Figure 11 a shows these data graphically. The remainder of the energy goes away via radiative cooling, which in Tcool is calculated with the method of Cuppen et al. (2006).0
Kalv¯ans
The energy, used for sublimation at each temperature T CR , multiplied by the corresponding f T from the data of Kalv¯ans (2018a), gives us the flux of CR energy F E, CR , evap (eV s − ) received by thegrain and inducing sublimation of the interstellar ices. When all this energy is assumed to come fromgrain heating events with ¯ T CR =54 K, the corresponding frequency is f ′ = F E, CR , evap E CR , X subl , , (A1)where E CR , is the total energy imparted by a CR hit, heating a grain to 54 K from the ambient T dust ≈
10 K and X subl , is the part of that energy used for sublimation. E CR , is equal to the energylost by the grain, when it cools from 54 K to ≈
10 K, which is about (2–4) × eV, depending on thethickness of the ice layer. X subl , was taken to be 0.74 from the data of Kalv¯ans & Kalnin (2020a).Finally, the rate coefficient for the transition of icy molecules into the warm ice phase is defined tobe equal to f ′ . Using the corresponding data, the CRD heating frequency numerically transformsinto f ′ = 4 . × N − . H (s − ) , (A2)where N H is expressed in cm − . Figure 11 b shows graphically the resulting heating frequency.The reversal of warm ice species to the ambient ice phase occurs with a rate coefficient k cool . Theestablished practice in astrochemistry is that k cool is inversely proportional to the time of cooling t cool , specific for each given T CR . Hasegawa & Herbst (1993) estimated t cool as the characteristicevaporation time of the CO molecule, the primary volatile species in ices. Here, we retained thesame general approach but defined t cool as the time grain spends in T CR , during which the numberof sublimated volatile molecules equals that sublimated during a realistic cooling of the same grainfrom T CR to 10 K. Only volatile species, such as CO and N are able contribute to the cooling, beforethe grain cools down radiatively (Kalv¯ans & Kalnin 2020a).The thickness and composition of the icy mantle varies as the cloud core evolves. In the initial stages,the layer is thin and consists of mostly of non-volatiles, such as H O. In later stages, the proportionof CO ice reaches 20–40 % relative to water ice (Whittet et al. 2007), while in the densest, coldest,and most shielded parts of the core this proportion may exceed 60 %, as shown by astrochemicalmodels. The volatile-poor grains cool significantly longer, allowing more of their volatiles to escape,when compared to a cooling time that is similar to the CO evaporation time-scale.The cooling times were calculated with the
Tcool program. We considered a 0.1 µ m grain, coveredby an icy mantle with a composition estimated from observations, when possible, or the results ofKalv¯ans & Kalnin (2019). For example, a grain covered with 10 MLs of ice containing 2% of volatilemolecules, relative to water ice, cools down from 54 K to 10 K releasing 2 . × molecules (CO,N , O , and CH ), meaning that the grain lost 2 . × eV via sublimation. The same amount ofsublimated energy and molecules can be lost, when the grain is held at a constant temperature of54 K for 9.5 s. For a grain with 60 MLs of ice and 30 % volatiles, these numbers are 2 . × molecules,2 . × eV, and 5 . × − s. For grains with a thick mantle and abundant (68 %) volatiles thesenumbers level out at ≈ × molecules, ≈ × eV, and ≈ . × − s. The exact values dependon the exact ice thickness, which affects grain heat capacity. When a sufficient amount of volatilesis present, the cooling time-scale remains fairly constant and about twice the characteristic COevaporation time-scale, simply because the evaporative cooling requires the sublimation of moleculesfrom two CO-dominated MLs. CCC and irradiation t cool (s) is to express it as afunction of the amount of volatiles in the icy mantle of the grain: t cool = 0 . b CO − . ≥ t cool ≥ . × − , s, (A3)where b CO is abundance of CO ice, expressed in MLs. The result is shown in Figure 11 c .Summarizing the above, we have derived an N H -dependent grain heating frequency f T and a t cool value that is high for grains with low content of volatiles. This realistic approach effectively meansthat CRD is reasonably efficient only for volatile-poor grains at relatively low column densities, whereit prevents an early accumulation en masse of CO and CO ices before their threshold A V values,which have been derived from observations (4.3 and 6.7 mag, respectively, Whittet et al. 2007). CRDrapidly becomes inefficient when the first few layers of adsorbed icy CO appear, while the gas parcelis shifting to higher column densities.For models that describe CRD of surface species with the simpler approach devised byHasegawa & Herbst (1993), the “duty cycle” or time fraction spent by a grain in 54 K is f ′ × t cool ,and the CRD rate coefficient for species i is k CRD ( i ) = f ′ t cool k evap ( i,
54 K) , (A4)where k evap ( i,
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