Shocks in dense clouds. IV. Effects of grain-grain processing on molecular line emission
AAstronomy & Astrophysics manuscript no. dustprocessingIV˙accepted c (cid:13)
ESO 2018October 30, 2018
Shocks in dense clouds
IV. Effects of grain-grain processing on molecular line emission
S. Anderl, V. Guillet, G. Pineau des Forˆets , and D. R. Flower Argelander-Institut f¨ur Astronomie, Universit¨at Bonn, Auf dem H¨ugel 71, 53121 Bonn, Germanye-mail: [email protected] Institut d’Astrophysique Spatiale (IAS), UMR 8617, CNRS, Bˆatiment 121, Universit´e Paris Sud 11, 91405 Orsay, France LERMA (UMR 8112 du CNRS), Observatoire de Paris, 61 Avenue de l’Observatoire, 75014 Paris, France Physics Department, The University of Durham, Durham DH1 3LE, UKReceived 4 March 2013; accepted 19 June 2013
ABSTRACT
Context.
Grain-grain processing has been shown to be an indispensable ingredient of shock modelling in high density environments.For densities higher than ∼ cm − , shattering becomes a self-enhanced process that imposes severe chemical and dynamical con-sequences on the shock characteristics. Shattering is accompanied by the vaporization of grains, which can, in addition to sputtering,directly release SiO to the gas phase. Given that SiO rotational line radiation is used as a major tracer of shocks in dense clouds, it iscrucial to understand the influence of vaporization on SiO line emission. Aims.
We extend our study of the impact of grain-grain processing on C-type shocks in dense clouds. Various values of the magneticfield are explored. We study the corresponding consequences for molecular line emission and, in particular, investigate the influenceof shattering and related vaporization on the rotational line emission of SiO.
Methods.
We have developed a recipe for implementing the e ff ects of shattering and vaporization into a 2-fluid shock model, resultingin a reduction of computation time by a factor ∼
100 compared to a multi-fluid modelling approach. This implementation was com-bined with an LVG-based modelling of molecular line radiation transport. Using this combined model we calculated grids of shockmodels to explore the consequences of di ff erent dust-processing scenarios. Results.
Grain-grain processing is shown to have a strong influence on C-type shocks for a broad range of magnetic fields: the shocksbecome hotter and thinner. The reduction in column density of shocked gas lowers the intensity of molecular lines, at the same time ashigher peak temperatures increase the intensity of highly excited transitions compared to shocks without grain-grain processing. ForOH the net e ff ect is an increase in line intensities, while for CO and H O it is the contrary. The intensity of H emission is decreasedin low transitions and increased for highly excited lines. For all molecules, the highly excited lines become sensitive to the value ofthe magnetic field. Although vaporization increases the intensity of SiO rotational lines, this e ff ect is weakened by the reduced shockwidth. The release of SiO early in the hot shock changes the excitation characteristics of SiO radiation, although it does not yield anincrease in width for the line profiles. To significantly increase the intensity and width of SiO rotational lines, SiO needs to be presentin grain mantles. Key words. shock waves – magnetohydrodynamics (MHD) – dust, extinction – ISM: clouds – ISM: jets and outflows – ISM:kinematics and dynamics
1. Introduction
Shocks are ubiquitous in the interstellar medium, occurringwhen matter moves into a more rarefied medium at a veloc-ity that exceeds the local sound speed. Depending on the valueof the local magnetosonic speed, di ff erent types of shocks canbe distinguished. The classical shocks are faster than any sig-nal speed in the shocked medium, so the preshock medium isnot able to dynamically respond to the shockwave before it ar-rives. This type of shock is called ”J-type” (see e.g. Hollenbach& McKee (1979); McKee & Hollenbach (1980)). A di ff erent sit-uation can occur if a low degree of ionization and the presence ofa magnetic field allow magnetosonic waves to precede the shock.Ions then decouple from the neutrals and are already acceleratedin the preshock gas, so that there is no longer a discontinuityin the flow of the ion fluid. In the preshock gas, the ions heatand accelerate the neutrals and broaden the heating region, sothat heating and cooling take place simultaneously. The shocktransition can then become continuous in the neutral fluid (”C- type shocks”) also. Shocks of this type are thicker and less hotthan J-type shocks (see Draine 1980; Draine & McKee 1993).Observations often reveal shocks as bow-shaped structures, withthe ambient material being compressed and pushed aside (e.g.Nissen et al. 2007; Davis et al. 2009).Shocks play an important role in the energy budget of theinterstellar medium by determining the energetic feedback ofevents such as supernovae, stellar winds, cloud-cloud collisions,or expanding HII regions. On the other hand, shocks have amajor influence on the chemistry of the interstellar medium.Among the most characteristic chemical tracers of shock wavesare species that are typically heavily depleted on dust grains,such as Fe, Mg, or Si. Dust processing occurring in the violentenvironment of shocked media is able to release these speciesinto the gas phase (e.g. Li ff man & Clayton 1989; O’Donnell &Mathis 1997). The understanding of dust processing in shocksis therefore intimately linked with the theoretical interpretationof characteristic emission lines in environments where shocksoccur. a r X i v : . [ a s t r o - ph . GA ] A ug . Anderl, V. Guillet, G. Pineau des Forˆets and D. R. Flower: Shocks in dense clouds The processing of dust in shocks can have two di ff erent con-sequences. It either changes the dust-to-gas mass ratio or altersthe dust size distribution. The former occurs in the processes ofsputtering and vaporization, while the latter is also found withshattering. Sputtering denotes energetic impacts of gas particleson grains that can release species from the grain surfaces intothe gas phase. This can happen either at very high temperatures(thermal sputtering) or at high relative gas–grain velocities (in-ertial sputtering). Sputtering of dust grains has been the subjectof many theoretical studies (Barlow 1978; Tielens et al. 1994;Jones et al. 1994; May et al. 2000; Van Loo et al. 2013) and hasbecome an established ingredient of shock models. Shattering,which is the fragmentation of grains due to grain-grain colli-sions, has been identified as a crucial process for determining thegrain-size spectrum of interstellar grains (Biermann & Harwit1980; Borkowski & Dwek 1995). Together with vaporization,which describes the return of grain material to the gas phase fol-lowing grain-grain collisions, shattering needs to be included inorder to account for UV extinction curves (Seab & Shull 1983).Moreover, recent infrared and sub-mm observations hint at anoverabundance of small dust grains relative to the expected size-distribution in parts of the interstellar medium and thereby stressthe importance of dust processing (Andersen et al. 2011; PlanckCollaboration 2011), which has also been associated with MHDturbulence acceleration of dust grains or charge-fluctuation in-duced acceleration (e.g. Ivlev et al. 2010; Hirashita 2010).A rigorous theoretical description of grain-grain collisionsrequires a detailed description of shock waves in solids, whichhad not been undertaken before the work of Tielens et al. (1994),earlier studies having relied on much simplified models (e.g.Seab & Shull 1983). The e ff ect of the microphysics of grainshattering and its e ff ects on the grain size distribution in J-typeshocks in the warm intercloud medium was studied by Joneset al. (1996). Slavin et al. (2004) extended this work by explic-itly following individual trajectories of the grains, considered astest particles.In a series of papers, Guillet et al. (2007, 2009, 2011, here-after Papers I, II and III) have shown that a multi-fluid approachto the dust dynamics, together with a detailed calculation ofthe grain charge distribution, shows shattering and the accom-panying vaporization to be indispensable ingredients of shockmodels. For C-type shocks at preshock densities higher than ∼ cm − , shattering becomes dramatically self-enhanced, dueto feedback processes: electrons are heavily depleted on to smallgrain fragments, and the lack of electrons in the gas phase af-fects the grain dynamics, resulting in even more shattering andproduction of small grains. Owing to the increase of the totalgeometrical grain cross-section, these shocks become much hot-ter and narrower, and vaporization becomes important for therelease of depleted species into the gas phase. Both the dynam-ical and the chemical consequences of shattering a ff ect the pre-dicted observational characteristics, compared with models inwhich shattering and vaporization are neglected. Given that con-ditions favoring C-shocks are frequently found in dense cloudsand Bok globules, it is necessary to evaluate the observationalconsequences of shock models including grain-grain processingfor a proper understanding of massive star formation and interac-tions of supernova remnants with molecular cloud cores (Cabritet al. 2012).The aforementioned studies leave open a series of issues thatwe aim to address in this paper. The three main questions are: – How do the e ff ects of shattering and vaporization change ifthe magnetic field strength is varied? – What are the consequences of shattering for molecular lineemission? – How do shattering and associated vaporization of SiO and Cinfluence the SiO and atomic carbon line emission?In order to answer the last two questions, it is necessary to in-troduce a detailed treatment of radiative transport. However, dueto the numerical complexity of the multifluid treatment of PapersI–III, a self-consistent merging of this multifluid model with anLVG treatment of molecular line emission would be technicallydi ffi cult. It would also prevent the resulting model from beinga practical analysis tool whenever large grids of models are re-quired. Therefore, we have developed a method for implement-ing the e ff ects of shattering and vaporization into a 2-fluid shockmodel that is su ffi ciently general to be applied to any similarmodel. We incorporated the main features, neglecting all theminor details of grain-grain processing; the resulting saving incomputation time amounts to a factor of ∼ ff ect of shattering on the shock structure for di ff erentvalues of the magnetic field in the preshock gas (Sect. 3). In Sect.4, we consider observational diagnostics, with the main empha-sis being on the rotational line emission of SiO. In addition, weinvestigate the influence of vaporization on the [C I] emissionlines. The results are discussed and summarized in Sect. 5.
2. Our model
In order to study the e ff ect of shattering and vaporization onmolecular line emission, we have built on the findings of PapersI - III, where shattering and vaporization are described within amulti-fluid formalism for the dust grains. These results neededto be transferred to a 2-fluid formulation, as used in the modelof Flower & Pineau des Forˆets 2010 (hereafter FPdF10), whosemodel includes a detailed treatment of molecular line radiativetransfer, in the presence of the cosmic microwave backgroundradiation. In the present Section, we first describe the treatmentof dust in the model of FPdF10 and then summarize the multi-fluid treatment of dust and its implications, finally outlining howwe introduced the e ff ects of shattering and vaporization into themodel of FPdF10. The way in which dust is treated in the LVG-model of FPdF10derives from the work of Flower & Pineau des Forˆets 2003(hereafter FPdF03). This one-dimensional, steady-state, 2-fluidmodel of plane–parallel C-type shocks solves the magneto-hydrodynamical equations in parallel with a large chemical net-work, comprising more than 100 species and approximately Although electrons and ions are treated as one dynamical fluid, theirtemperatures are calculated separately.2. Anderl, V. Guillet, G. Pineau des Forˆets and D. R. Flower: Shocks in dense clouds ro-vibrational levels arecomputed at each integration step, along with the populationsof the rotational levels of other important coolants, such asCO, OH and H O. In these cases, the molecular line transferis treated by means of the large velocity gradient approxima-tion (see also Sect. 4). ‘Dust’ is included in the forms of poly-cyclic aromatic hydrocarbons (PAHs), represented by C H ,and large grains from bulk carbonaceous material and silicates(specifically olivine, MgFeSiO ). The large grains are assumedto have a power law size distribution, d n g ( a ) / d a ∝ a − . (Mathiset al. 1977), and radii in the range 100 to 3000 Å (0.3 µ m). In thepreshock medium, the grain cores are covered by ice mantles,consisting of the chemical species listed in Table 2 of FPdF03and including H O, CO and CO . Both PAHs and large grainsexist as neutral, singly positively and singly negatively chargedspecies. The physical treatment of dust comprises: – determination of the grain charge distribution, limited to Z = − , , – sputtering of grain cores and mantles due to grain-gas colli-sions; – removal of mantles by cosmic ray desorption; – build-up of mantles by adsorption of gas-phase species in thepostshock gas. To properly account for the e ff ects of grain-grain collisions inshocks, it is necessary to introduce a multi-fluid description ofthe dust dynamics, in which dust grains are treated as test par-ticles. In Papers I-III the dust size distribution was modeled bythe use of discrete bins, with grain sizes ranging from 5 to 3000Å. PAHs were not included as separate species, distinct from thedust grains (as in the two-fluid model), but were incorporatedinto the dust size distribution. The equations describing the 2-Dgrain dynamics and complete charge distribution were integratedfor each individual bin size, independently for silicate and car-bon grains.While the shock structure is not much a ff ected by this moreaccurate treatment of the grain dynamics and charging (Paper I,Appendix D ), the introduction of dust shattering in grain-graincollisions and the corresponding changes to the grain size dis-tribution have major e ff ects on shocks in dense clouds with alow degree of ionization (Paper III). When grains collide witha velocity greater than 1.2 km s − (carbon grains) or 2.7 km s − (silicate grains), a fraction of their mass, which increases withvelocity, is shattered into smaller fragments. At higher veloci-ties, another fraction is vaporized and released into the gas phase(the numerical treatment follows the models of Tielens et al.(1994) and Jones et al. (1996), where the vaporization thresh-old is ∼
19 km s − ).Paper III demonstrated that there is a marked shift of the dustsize distribution towards smaller grains when the grain dynam-ics, charging and evolution are coupled self-consistently with theshock dynamics and chemistry. The most dramatic consequenceis an increase of the total geometrical grain cross-section – whichgoverns the coupling between the neutral and charged fluids – Although changes to the re-accretion of mantle species in the multi-fluid treatment give rise to a slightly di ff erent temperature profile in thepostshock gas. V ( k m / s ) distance (cm)V i V n T n ( K ) n H / n H T n n H / n H0 Fig. 1.
Temperature and density (upper panel) and velocity pro-files in the shock frame (lower panel) for a 30 km s − C-typeshock with a magnetic field parameter b = . n H = cm − . The full curves show the results ob-tained when grain-grain processing is treated as in Paper III;the broken curves are obtained by following the approach ofFPdF03.by a factor of ∼
10. The shock becomes narrower by a factor of ∼
4, and the peak temperature increases by a factor of 1.5–2 (seeFig. 1). The small grain fragments deplete ions and electronsfrom the gas until grains become the dominant charge carriers,a situation known as a dusty plasma (Fortov et al. 2005). Thestrong e ff ect of shattering, for preshock densities of 10 cm − and higher, can be understood in terms of the feedback processesdescribed in Paper III. At these high preshock densities, it turnsout that vaporization related to shattering becomes important, ifnot dominant, compared to sputtering. As described in Sect. 2.2, there are two ways in which shatter-ing a ff ects the shock wave: first it increases the collisional dustcross-section, and second it lowers the degree of ionization ofthe gas. In the model of FPdF10, dust as a dynamical speciesand dust as a chemically reacting species were treated separately.Our implementation and validation of shattering in the FPdF10model are described in Appendix A.1. Here we only summarizeour basic approach. – The dynamical e ff ects of shattering can be simulated by in-corporating the change in the total grain cross section (cid:104) n σ (cid:105) ,computed with the model of Paper III; this is done by mul-tiplying (cid:104) n σ (cid:105) by an additional factor that varies with thespatial coordinate, z , through the shock wave. This factor ismodelled as an implicit function of the compression of theionized fluid; its value is 1 in the preshock gas, and its maxi-mum value – derived from the multi-fluid model – is attained This factor refers to the total cross-section of grain cores. The re-accretion of grain mantles in the postshock gas leads to a much largerincrease in the total cross-section, by a factor of ∼
100 (see. Fig. A.1).However, this large increase is irrelevant to the shock dynamics, as ithappens where the momentum transfer between the ionized and the neu-tral fluids is almost complete ( T ∼
100 K). 3. Anderl, V. Guillet, G. Pineau des Forˆets and D. R. Flower: Shocks in dense clouds
02 10 w i d t h K ( c m ) T n, max (K) Grid without grain-grain processingGrid with grain-grainprocessing
Fig. 2.
Peak temperature of the neutral fluid and the shock width(in cm), to T =
100 K in the cooling flow, for the grid of mod-els without (broken lines) and with (full lines) grain-grain pro-cessing. The preshock density is 10 cm − , and the shock ve-locities are 20 km s − (blue), 30 km s − (green) and 40 km s − (red) for various values of the magnetic field parameter, b , in B ( µ G) = b (cid:112) n H (cm − ).when shattering is completed. It is possible to use linear fitsof the parameters that are required. – The chemical e ff ects of shattering, i.e. the consequences forgrains as charged species, as determined by grain-chargingreactions, need to be considered separately. In order to modelthe influence of shattering on the abundances of chargedgrains, a shattering source-term is introduced, whose valueis consistent with (cid:104) n σ (cid:105) . Grain-grain collisions lead not only to shattering of grains, andconsequent changes in the shock structure and degree of ion-ization, but also to their vaporization, when the impact velocityis higher than the vaporization threshold of ∼
19 km s − . We as-sume that the silicon released through vaporization is in the formof SiO (Nagahara & Ozawa 1996; Wang et al. 1999). We haveimplemented the e ff ect of SiO vaporization in a similar way asfor shattering ; the details are given in Appendix A.2. Our pro-cedure involves introducing an additional term in the rate of cre-ation of gas-phase SiO from Si and O in the grain cores. Thus,vaporization is incorporated as a new type of pseudo-chemicalreaction. The rate of creation of SiO through vaporization is de-termined by parameterizing the results of the multi-fluid models.Another modification of the FPdF10 model concerns themantle thickness, which is now calculated as described in PaperI, Appendix B. Our simplified, but self-consistent, treatment ofshattering and vaporization reduces the computation time by afactor of more than 100. We emphasize the e ff ects of SiO vaporization because the fine-structure lines of atomic carbon are optically thin and do not requirean LVG treatment.
3. The influence of grain-grain processing on theshock structure
Having implemented shattering and vaporization in the model ofFPdF10, we first study the dependence of grain-grain processingand feedback on the strength of the transverse magnetic field.The analysis of Paper III was restricted to shocks propagatingat the critical velocity, which defines the fastest possible C-typeshock for a given magnetic field (or, conversely, the lowest valueof the magnetic field possible for a given shock velocity). Thiscritical velocity is determined by the condition that the shock ve-locity should be only slightly smaller than the velocity of mag-netosonic waves, V ms = B (cid:112) πρ c where ρ c is the mass density of matter that is strongly cou-pled to the magnetic field. We assumed that the magnetic fieldstrength, B , in dense clouds scales with the total proton density, n H , as B ( µ G) = b (cid:112) n H (cm − ) (Crutcher 1999), which impliesthat the magnetic energy density is proportional to n H . The as-sumed power-law exponent of 0.5 is somewhat lower than themore recent value of 0.65 ± cm − .We consider only this value because, on the one hand, it wasshown that the change in the grain size distribution due to shat-tering is negligible for a preshock density of 10 cm − , and, onthe other hand, the strength of the feedback from shattering athigher densities prevents the multi-fluid model of Paper III fromconverging at a preshock density of 10 cm − , and hence thereare no results with which to compare at this density. We haverestricted our calculations to shocks that do not fully dissociateH (shock velocity V s ≤ − ), in practice to 20, 30 and 40km s − .The values of the magnetic field that we considered were de-termined by two considerations. First, in order to study criticalshocks, we adopted the corresponding (minimum) values of b (see Paper III); and, second, we varied b for a given shock ve-locity in order to study the influence of variations in the magneticfield strength. Thus, we consider three values of the b parameterfor each velocity, in steps of 0.5, where the lowest value is closeto that for a critical shock; the lowest values are b = . − , b = . − , and b = . − . Thecorresponding grid of nine models is summarized in Table 1. Wenote that a comparison of models with di ff erent shock velocitiesand the same magnetic field is possible for the case b = .
4. Anderl, V. Guillet, G. Pineau des Forˆets and D. R. Flower: Shocks in dense clouds
Table 1.
Parameters defining our grid of models. V s [km s − ] b B [ µ G] n H [cm − ]20 1.0 316 10
20 1.5 474 10
20 2.0 632 10
30 1.5 474 10
30 2.0 632 10
30 2.5 791 10
40 2.0 632 10
40 2.5 791 10
40 3.0 949 10 The most dramatic e ff ect of shattering is the change in the over-all shock structure, owing to the increase in the collision cross-section of the dust and the corresponding increase in the cou-pling between the neutral and charged fluids. Figure 2 shows thepeak temperature of the neutral fluid and the shock width (upto the point at which the temperature has fallen to 100 K) forour grid of nine models, as compared with the results obtainedwithout grain-grain processing.The peak temperature of the models that include shatteringincreases by a factor of 1.5–2 for the shocks closest to the criti-cal velocity, as in Paper III. Our models show that this increase issmaller for slower shocks, and for shocks with higher magneticfield strengths, because the charged grains are more stronglybound to and protected by the magnetic field. Thus, shocks with V s =
20 km s − and b = ff ering by a factor of only 1.7 for V s =
20 km s − and b = ff ect on the peak temperature. We returnto this point in Sect. 4, where we consider the di ff erences be-tween the spectra predicted by these two categories of model;but it is already clear that shattering is significant, even for non-critical shocks, and should be included in steady-state C-typeshock models with high preshock densities ( n H ≥ cm − ).
4. Observational consequences
In our model, the molecular line transfer is treated using the largevelocity gradient (LVG) approximation (e.g. Surdej 1977), al-lowing for self-absorption via the escape probability formalism.The LVG method is well adapted to the conditions of shocks,where flow velocities change rapidly. The computation of themolecular energy level populations is performed in parallel withthe integration of the chemical and dynamical rate equations, asintroduced for CO by Flower & Gusdorf (2009). The molecular line transfer of H O, CH OH, NH , OH and SiO has since beenadded (Flower et al. 2010; Flower & Pineau des Forˆets 2010,2012.The modelling of interstellar shock waves requires manymolecular energy levels to be considered. As described inFPdF10, we include levels of H O up to an energy of approx-imately 2000 K above ground. Although this is less than themaximum temperatures that are attained, the high values of theradiative (electric dipole) transition probabilities ensure that thepopulations of higher, neglected levels remain small, and hencethe associated errors in the computed line intensities are modest.Above the maximum temperatures for which the rotational de-excitation coe ffi cients have been calculated, their values are as-sumed to remain constant (cf. Flower & Pineau des Forˆets 2012,Appendix A). Rate coe ffi cients for excitation are obtained fromthe detailed balance relation.The consequences of grain-grain processes for the molecularline radiation of H , H O and OH for representative shocks of V s =
30 km s − are briefly discussed in Sect. 4.1 and in moredetail in Appendix B. In Sect. 4.2, we consider the rotational lineemission of SiO and, in Sect. 4.3, the vaporization of carbon. We have seen that including grain-grain processing leads to anincrease in the peak shock temperature; this a ff ects the chemistryand gives rise to higher fractional abundances of molecules inexcited states. The intensities of lines emitted by highly excitedstates are thereby enhanced. At the same time, the column den-sity of shocked gas decreases, which tends to reduce the intensityof molecular line emission. The net e ff ect depends on the chem-ical and spectroscopic properties of the individual molecules.In Appendix B, we show that the intensities of all transitionsof OH increase in shocks faster than 30 km s − that incorporategrain-grain processing; this is due to the temperature sensitiv-ity of OH formation. On the other hand, the intensities of theemission lines of H , CO, and H O decrease because of the re-duction in the column density of shocked material. Furthermore,the inclusion of grain-grain processing introduces a dependenceof the intensities of lines from highly excited states on the mag-netic field, owing to the variation of the peak shock temperaturewith the field strength (see Sect. 3.2 and Fig. 2).
SiO is a prominent indicator of shock processing in dense cloudsassociated with jets and molecular outflows (e.g. Bachiller et al.1991; Martin-Pintado et al. 1992; Gueth et al. 1998; Nisini et al.2007; Cabrit et al. 2007). The first chemically adequate theoret-ical study of SiO production by sputtering in C-type shocks wasconducted by Schilke et al. (1997); this work was pursued sub-sequently by Gusdorf et al. (2008b,a). These studies consideredthe release, by sputtering, of Si from grain cores and SiO fromgrain mantles, but not the process of vaporization, which couldmodify the predicted SiO rotational line profiles. We note thatCabrit et al. (2012) found that, in the protostellar jet HH212, atleast 10% of elemental Si could be present as gas-phase SiO, ifthe wind is dusty. To explain such a high value, they hint at thepossible importance of grain-grain processing and the release ofSiO by vaporization. The sputtering of Si from the grain coresseems unable to account for the observations in this case, as thedynamical timescale of 25 yr is too short for the chemical con-version, in the gas phase, of the sputtered Si into SiO.
5. Anderl, V. Guillet, G. Pineau des Forˆets and D. R. Flower: Shocks in dense clouds -7 -6 -5 -4 -3 -2 -1 va po r i z e d / s pu tt e r e d f r ac t i on [ % ] bSi V s = 40 km s -1
20 km s -1
30 km s -1 bC20 km s -1
30 km s -1
40 km s -1 Fig. 3.
The fractions of Si (left panel) and C (right panel) re-leased from grain cores in model M1 of Table 2 by vaporiza-tion (full lines) and sputtering (broken lines), for V s =
20 km s − (blue), V s =
30 km s − (green), and V s =
40 km s − (red),as functions of the magnetic field parameter, b , in B ( µ G) = b (cid:112) n H (cm − ). Table 2.
Summary of the three dust processing scenarios inves-tigated in order to study the release of SiO into the gas phase.The percentage refers to elemental silicon in the form of SiO inthe mantles.
Scenario Si in cores SiO in mantles Grain-grain processingM1 yes no yesM2 yes no noM3 yes 10% no
The influence of grain-grain processing on the SiO line emis-sion is twofold. On the one hand, vaporization strongly increasesthe amount of SiO released from grain cores, at su ffi ciently highshock speeds. On the other hand, as the process of vaporiza-tion is necessarily related to shattering of dust grains, shockswith grain-grain processing have a di ff erent structure (cf. Sect.3.2). The consequences of the latter e ff ect for the emission bymolecules whose abundances are not directly influenced by va-porization are summarized in Sect. 4.1.In order to address the question of how the related e ff ects ofshattering and vaporization a ff ect the emission of SiO in C-typeshocks, we compare results of shock models, obtained includingand excluding grain-grain processing, for three di ff erent scenar-ios, summarized in Table 2. Models corresponding to scenario1 (M1) were calculated using the implementation of shatteringand vaporization described in Sect. 2.3 and 2.4 with the param-eters of Table A.1. Models corresponding to scenario 2 (M2) donot include grain-grain processing, but Si is still released by thesputtering of grain cores. Scenario 3 (M3) di ff ers from M2 inthat 10% of the elemental Si is assumed to be in the form of SiOin the grain mantles. In total, there are 9 × =
27 individualmodels to be computed. -12 -11 -10 -9 -8 -7 -6 -5 S i O f r ac t i on a l a bund a n ce distance (cm)T n x(SiO)x(SiO) M1 distance (cm)T n x(SiO)M2 T n ( K ) distance (cm)T n x(SiO)M3 -12 -11 -10 -9 -8 -7 -6 -5 -4 f r ac t i on a l a bund a n ce distance (cm)T n x(OH)x(SiO) M1x(O ) distance (cm)T n M2 x(O )x(OH) T n ( K ) distance (cm)T n M3 x(O )x(OH) Fig. 4.
Upper panel: Evolution of the fractional abundance ofSiO in the gas phase (left ordinate, full curves) and temperatureof the neutral fluid (right ordinate, broken curves). From left toright: dust modelling scenarios M1, M2, and M3. The modelsshown in all three panels are for V s =
20 km s − with b = V s =
30 km s − with b = V s =
40 km s − with b = V s =
20 km s − with b = (left ordinate, full bluecurves) and OH (left ordinate, full red curves), together withthe temperature of the neutral fluid (right ordinate, black bro-ken curves), for V s =
30 km s − and b = In order to compare the relative importance of vaporization andsputtering, we investigate the release of the Si in grain cores intothe gas phase by each of these processes during the passage of ashock wave. The left-hand panel of Figure 3 shows the fractionof Si eroded from grain cores through sputtering and vaporiza-tion in model M1. There is negligible sputtering for V s ≤ − , because the adopted sputtering threshold for refractorygrain material is ∼
25 km s − (May et al. 2000). Both the sputter-ing of Si and the vaporization of SiO are inhibited by the mag-netic field. In the case of sputtering, the maximum ion-neutraldrift velocity decreases with increasing field strength, whereas,for vaporization, the charged grains are more strongly coupled tothe field and hence to the charged fluid (we assume that the mag-netic field is ‘frozen’ in the charged fluid). For models M3 (SiOin mantles) with velocities su ffi cient for mantle sputtering, all theSiO initially in the mantles is released into the gas phase. The ro-tational line emission of SiO depends not only on the amount ofsilicon released from the grains but also on the location of its re-lease and the physical conditions prevailing where SiO is presentwithin the shock wave.The upper panel of Figure 4 shows the fractional abundanceof SiO in the gas phase, for dust models M1–3 of Table 2 and
6. Anderl, V. Guillet, G. Pineau des Forˆets and D. R. Flower: Shocks in dense clouds three di ff erent values of the shock speed. In model M1, vaporiza-tion releases SiO directly into the gas phase in the region wherethe temperature of the neutral fluid is rising steeply. The higherthe shock speed, the more SiO is produced. The large increasein the total grain cross section in the postshock gas enhancesthe rate of formation of mantles and removes SiO from the gasphase.The variation of the fractional abundance of SiO is very dif-ferent if only the sputtering of grain cores is considered (M2),because the Si that is released by sputtering has to be trans-formed into SiO by gas-phase chemical reactions, predominantlyoxidation by O and OH. The corresponding reactions areSi + O −→ SiO + O (1)Si + OH −→ SiO + H . (2)For reaction (1) the rate coe ffi cient (cm s − ) k = . × − ( T / − . exp( − / T )was measured by Le Picard et al. (2001). The rate coe ffi cient forreaction (2), which is not measured, was adopted to be the same.Thus, the fractional abundance of SiO in the gas phase de-pends not only on the amount of Si sputtered from the grain coresbut also on the abundances of O and OH, displayed in the lowerpanel of Fig. 4. The chemical delay in SiO production is apparentin the upper panel of Fig. 4; the abundance of SiO peaks in thecool and dense postshock region. If SiO is initially in the grainmantles (M3), its release is rapid and complete even before thetemperature of the neutral fluid rises significantly. For the shockmodels considered, all the SiO in the mantles is released into thegas phase. We have assumed that 10% of elemental silicon isinitially in the form SiO in the mantles; this is the largest of thevalues considered by Gusdorf et al. (2008b). If the same fractionof silicon is initially in the form of SiH in the mantles, thereoccurs the same chemical delay in the production of SiO in thegas phase as in models M2. The di ff erences in the SiO abundance profiles in the three casesM1–3 – specifically, whether SiO is already present in the hotgas, early in the shock wave, or only in the cold postshock gas– have consequences for the relative intensities of the SiO rota-tional lines. To illustrate this point, we compare, in Fig. 5, thepeak temperatures of the rotational lines of SiO, relative to the J = V s =
30 km s − shock in Fig. 7.The first thing to note is the similarity of the relative peakline temperatures for the V s =
30 km s − and V s =
40 km s − models with only core sputtering (M2); this can be ascribed tothe chemical delay in SiO formation. The rotational lines up tothe 8–9 transition in these models mostly stem from the coldpostshock gas, where almost maximum compression is reached.Therefore, the lines have large optical depths and near-LTE exci-tation conditions, which makes the relative peak temperatures ofthe transitions displayed independent of the shock velocity. Themodels with vaporization (M1), on the other hand, show a clearvariation of excitation conditions with shock speed. The lines are The threshold velocity for sputtering of the mantles is ∼ µ − km s − , where µ is the reduced mass of the colliding species, rel-ative to atomic hydrogen (Barlow 1978). T p eak , r e l a t i ve J up Fig. 5.
The peak temperatures (K) of the rotational emissionlines of SiO, relative to the J = J up . Displayed are the shock models in our grid that are moststrongly influenced by vaporization, i.e. V s =
20 km s − with b = V s =
30 km s − with b = V s =
40 km s − with b = V s =
30 km s − , J up > V s =
40 km s − , J up > ff er-ences in the relative peak line temperatures are much less be-tween shocks of di ff erent velocities than is the case for modelswith vaporization. The same amount of SiO is released for bothvelocities and the column density of the cooling gas is greaterthan in models that include vaporization. Most of the radiationarises in the cooling flow, where the temperature profiles arevery similar and the optical depths are large: for model M2, with V s =
30 km s − and b = J = ∼ < τ ∼ <
6, depending on the transition. We conclude that clearvariations of the relative line intensities with the shock velocityare characteristic of those models in which grain vaporizationoccurs.
Given that the excitation conditions of the SiO lines di ff er in thethree scenarios M1–3, we expect the SiO spectral line profilesdi ff er also, with respect to line width, location of the peak, andintegrated flux. Line profiles for V s =
30 km s − , divided by thepeak temperature of the J = J transitions, but similar widths for high- J transitions, as model M2, and much narrower lines than modelM3. There is a weak variation with J up of the location of the linepeak in models M1 and M2. For M1, the peak in the profile of
7. Anderl, V. Guillet, G. Pineau des Forˆets and D. R. Flower: Shocks in dense clouds
M1M2M3 V s - V n (km s -1 ) T S i O ( - ) / T S i O ( - ) p eak SiO(2-1)
M1M2M3 V s - V n (km s -1 ) T S i O ( - ) / T S i O ( - ) p eak SiO(3-2)
M1M2M3 V s - V n (km s -1 ) T S i O ( - ) ( K ) M3: T
SiO (5-4) / 7
SiO(5-4)
M1M2M3 V s - V n (km s -1 ) T S i O ( - ) / T S i O ( - ) p eak SiO(6-5)
M1M2M3 V s - V n (km s -1 ) T S i O ( - ) / T S i O ( - ) p eak SiO(9-8)
M1M2M3 V s - V n (km s -1 ) T S i O ( - ) / T S i O ( - ) p eak SiO(10-9)
Fig. 6.
Profiles of the SiO rotational transitions (2–1), (3–2), (5–4), (6–5), (9–8), and (10–9), for V s =
30 km s − and b = T n ( K ) distance (cm)T n SiO(2-1)SiO(5-4)SiO(9-8)M1 distance (cm)T n SiO(2-1)SiO(9-8)SiO(5-4)M2 S i O li n e t e m p e r a t u r e ( K ) distance (cm)T n SiO(2-1)SiO(5-4)SiO(9-8)M3
Fig. 7.
Temperature profiles of the neutral fluid (left ordinate,broken curves) and SiO line temperatures (right ordinate, fullcurves) of the rotational transitions J = V s =
30 km s − , b = V n ∼ − , whereas, forhigher transitions, the peak moves to hotter gas, at V n ∼ − .A similar trend is seen for model M2, but, in this case, it is onlythe highest (and very weak) transitions that peak earlier in theshock wave. In contrast, the line peaks for model M3 are alwayslocated at the same velocity of V n ∼
24 km s − . Thus, the di ff er-ences in shock structure and spatial distribution of SiO between models M1 and M2 do not have a strong e ff ect on the SiO ro-tational line profiles. In particular, the release of SiO throughvaporization early in the shock wave does not lead to significantbroadening of the lines . However, the lines are strongly broad-ened if SiO is present in the grain mantles (scenario M3), andthis seems to be the only way of accounting for observed linewidths of several tens of km s − if they are to be explained by onesingle shock. Alternatively, broad SiO lines can be explained bythe existence of several shocks inside the telescope beam, suchthat the individual narrow line profiles appear spread out in ra-dial velocity due to di ff erent velocities and inclination angles, inthe observer’s frame. Similarly, the lines would be broadened bythe velocity profile associated with a bow shock (e.g. Brand et al.1989). An interesting question is whether the release of SiO due tovaporization significantly increases the integrated (along the z -direction) SiO rotational line intensities. Fig. 8 shows, that in-deed there is an increase in the intensities of the highly ex-cited transitions. Furthermore, the slope of the integrated inten-sity curves di ff ers between models M1 and M2 for transitions3 ≤ J up ≤ J up ∼
8. Anderl, V. Guillet, G. Pineau des Forˆets and D. R. Flower: Shocks in dense clouds Td V ( K k m s - ) J up Fig. 8.
Integrated intensities of the rotational transitions J up → J up − V s =
20 km s − , b = V s =
30 km s − , b = V s =
40 km s − , b = -3 -2 -1 Td V / Td V ( - ) J up b = 1.5b = 2.0b = 2.5 Fig. 9.
Integrated intensities of the rotational transitions J up → J up − J up =
5. The shock velocity is V s =
30 km s − , with values of themagnetic field parameter b = b = b = J up ∼ >
10; they peak at J up =
3. On theother hand, the peak of the curve corresponding to model M1 at V s =
40 km s − is displaced to J up =
4, owing to the higher max-imum temperature. As may be seen from Fig. 8, the integratedline intensities decrease by a factor ∼
100 between 5 ≤ J up ≤ ff ect of variations in the magnetic field onthe integrated SiO rotational line intensities. In this Figure, theintegrated intensities are expressed relative to the 5–4 transition,in order to focus attention on the excitation conditions, ratherthan the release of SiO into the gas phase. For model M1, thereis a clear variation with magnetic field strength, which is notpresent for model M2. This di ff erence can be understood fromFig. 2, which shows that the maximum temperature of the neutralfluid hardly varies with the magnetic field strength for modelM2, whereas the maximum value changes for model M1. Thus,Fig. 9 demonstrates that the excitation conditions vary with thestrength of the magnetic field when the e ff ects of grain-graincollisions are incorporated. Our model, and the models of Papers I–III, includes two di ff er-ent populations of dust grains: silicate and carbon grains. Whileemission of SiO could be used as a tracer of vaporization inshocks because of the strong depletion of Si and SiO in quies-cent gas, the e ff ects of vaporization on [C I] emission will not beseen as clearly. Nonetheless, the vaporization of graphite grainsmodifies the emission of atomic carbon. In order to predict themagnitude of this e ff ect, we have used the multi-fluid model ofPaper III.Fig. 3 (right-hand panel) shows the fraction of carbon re-leased from grain cores by sputtering (M2) and vaporization(M1). At low velocities, the destruction of carbon grains is lessthan for silicates because the gyration of large carbon grainsis damped more rapidly, owing to their lower specific density.At higher velocities, for which small grains also contribute sig-nificantly to vaporization, the fraction of carbon released fromgraphite grains is higher than the fraction of silicon from sili-cates; graphite grains are more strongly a ff ected by shattering.Table B.1 shows the intensities of two [C I] forbidden lines, at609.8 µ m and 370.4 µ m. The values in these Tables confirm thatthe e ff ect of vaporization on the [C I] lines is less than on theSiO rotational transitions.
5. Concluding remarks
In this series of papers, the consequences of grain-grain colli-sions in shock waves have been investigated; such processes hadbeen ignored in previous studies of C-type shocks. Shattering isa key factor in the production of the large populations of verysmall grains (mostly carbonaceous: see, for example, Draine &Lee 1984) in the turbulent ISM (Hirashita 2010). It has beenshown, in earlier papers in the series, that C-type shock wavesprovide similar dynamical conditions to turbulence, and so shat-tering needs to be included in shock models.
9. Anderl, V. Guillet, G. Pineau des Forˆets and D. R. Flower: Shocks in dense clouds
Our study relies on the shattering model that was developedin Paper III, and our results inherit the dependence of this modelon parameters such as the size of the smallest fragments (5Å inPaper III), the slope of the size distribution of the fragments,their charge distribution, and the composition of the dust grains.Whilst shattering almost certainly occurs in the ISM, as a directconsequence of the weak coupling of large grains to the mag-netic field in high density clouds, the threshold density at whichshattering starts to have a significant e ff ect on the shock dynam-ics is more uncertain.In our model, the processes of shattering and vaporizationare linked, so that the threshold density for shattering appliesalso to vaporization. The parameters of the shattering model inPaper III were chosen to minimize the amount of shattering andthereby yield a conservative estimate of the threshold density.It was shown that strong feedback on the shock dynamics wasexpected only for densities higher than ∼ cm − . The resultsof the present paper include some observational predictions thatshould help to constrain the grain-shattering model. The mainconclusions of our study are as follows:1. The influence of grain-grain processing on the overall shockstructure was found to be significant for the full range ofmagnetic field strengths that we studied. The maximum tem-perature increases by a factor of 1.5–2, and the shock widthis reduced by a factor 4–5. The inclusion of grain-grain pro-cessing changes the dependence of the shock structure onthe magnetic field strength. While for shocks without grain-grain processing there is only a weak dependence of the peaktemperature on the magnetic field, the peak temperature be-comes strongly dependent on the magnetic field when grain-shattering and vaporization are incorporated. Consequently,the intensities of, in particular, highly excited molecular tran-sitions become dependent on the strength of the magneticfield. While shattering is shown to be important for all mod-els of our grid, the vaporization of SiO from silicate graincores is significant only for fast shocks and low magneticfields.2. There are two consequences of grain-grain processing forthe molecular line emission: the reduced shock width resultsin a lower column density of shock-heated gas, whereas thehigher peak temperature can modify the chemistry and en-hance the fractional abundances of molecules in highly ex-cited states. Which of these tendencies prevails is decidedby the chemical and physical properties of the individualmolecules and their transitions. The intensities of all lines ofOH increase in shocks with velocities greater than 30 km s − ,when grain-grain processing is included, owing to the tem-perature sensitivity of OH formation. On the other hand, theintensities of the emission of CO and H O decrease becauseof the reduced column density of shocked material. In thecase of H , the intensities of highly-excited transitions in-crease, whilst the intensities of lines of lower excitation de-crease.3. The release of SiO through collisional vaporization of sili-cate grain cores enhances the integrated intensities of SiOrotational lines, mainly from highly-excited levels. However,this e ff ect is counteracted by the reduction in shock width. Toobtain significantly higher line intensities, it is necessary tointroduce SiO into the grain mantles. The situation is sim-ilar with respect to the widths of the SiO rotational lines.Although vaporizations releases SiO early, in the hot part ofthe shock wave, the reduction in the shock width preventsthe lines from becoming significantly broader than in mod- els that neglect grain-grain processing. Therefore, vaporiza-tion alone cannot account for broad lines if only one singleshock is considered. To obtain broad profiles, it seems nec-essary that SiO should be present in the grain mantles, suchthat mantle sputtering releases SiO, also in the early part ofthe shock wave. It is essential that SiO is released directly,thereby eliminating the chemical delay that would be asso-ciated with its production, in gas-phase reactions, from Si orSiH .Due to the fact, that most of the SiO emission stems fromthe early part of the shock, where the temperature profile de-pends on the shock velocity, the SiO rotational line ratiosvary with the shock velocity; this variation is not present ifvaporization is ignored. The e ff ect of vaporization on [C I]emission lines was found to be less than for the SiO rota-tional lines. Acknowledgements.
We are grateful to an anonymous referee for useful com-ments that helped to strengthen the paper. S. Anderl acknowledges support bythe DFG SFB 956, the International Max Planck Research School (IMPRS)for Astronomy and Astrophysics, and the Bonn-Cologne Graduate School ofPhysics and Astronomy.
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Appendix A: Shattering and vaporization: theirimplementation and validation
A.1. Shattering
A.1.1. Dynamical effects of shattering
The dynamical e ff ects of shattering on the shock structure canbe simulated by ensuring that the total grain cross section, (cid:104) n σ (cid:105) ,changes in accord with the results of Paper III. We found thatit is possible to model the increase of (cid:104) n σ (cid:105) due to shattering bymultiplying the total grain cross section, in the absence of grain-grain processing, by an additional factor, which varies with thespatial coordinate, z , through the shock wave. This factor is mod-elled as an intrinsic function of the compression of the ion fluid,normalized to the theoretical postshock compression at infinity,as predicted by the Rankine-Hugoniot relations. This normalizedion compression parameter constitutes a function, η ( z ), varyingfrom 0 in the preshock to 1 in the postshock medium.The preshock (medium 1) and postshock (medium 2) kinetictemperatures are approximately equal in the C-type shock mod-els of our grid. Thus, the Rankine-Hugoniot continuity relationsmay be applied across the shock wave, replacing the relation ofconservation of energy flux by the isothermal condition, T = T . Then, setting the ratio of specific heats γ =
1, the expressionfor the compression ratio, ρ /ρ , across the shock wave may bederived (cf. Draine & McKee 1993). Under the conditions of ourmodels, which are such that M s = V s / c (cid:29) M A = V s / V A (cid:29) s is the sonic Mach number of the flow, evaluated inthe preshock gas, c is the sound speed, and M A is the Alfv´enicMach number, also in the preshock gas, the compression ratioreduces to ρ /ρ ≈ √ A , whence V postshock = V A √ = B √ × π . m H n H , (A.1)where V postshock is the flow speed in the postshock gas, m H theproton mass and n H = n (H) + n (H ) the proton density in thepreshock gas. Using η ( z ) = V s / V i ( z ) − V s / V postshock − V s is the shock velocity and V i the velocity of the ionfluid in the shock frame, respectively), the factor by which thetotal grain cross section is enhanced is given by Σ ( z ) = η ( z ) · ( Σ max − + . (A.3)The extent of the increase of (cid:104) n σ (cid:105) due to shattering is givenby the final value of the shattering-factor, Σ max . This value de-pends on the shock velocity and the magnetic field and is an external parameter, which needs to be extracted from the multi-fluid models. However, this simple functional form did not re-produce satisfactorily the onset of shattering in the shock. Wetherefore propose the following refined expression Σ = (cid:18) η β − sin (2 π η β )2 π (cid:19) · (cid:18) Σ max − (cid:19) + β , which needs to be extractedfrom the corresponding multi-fluid model. β describes the delayin the shattering feedback, relative to the ion compression, andis only weakly dependent on the shock velocity. The parameters, β and Σ max , which correspond to the grid of models introducedin Section 3.1, are listed in Table A.1. Linear fits in the shockvelocity V s and the magnetic field parameter b are Σ max = . + . · V s − . · b (A.5)and β = . − . · V s + . · b . (A.6)The increase of the total grain collisional cross-section needs tobe consistent with the mean square radius of the grains and withtheir total number density, following the compression of the ions.Analyzing the multi-fluid computations corresponding to PaperIII, we find a reasonable approximation to the behaviour of thetotal grain number density, n G , and the mean square radius, (cid:104) σ (cid:105) G :the former increases as Σ , the latter decreases as Σ − , relative tothe corresponding values without shattering. These changes af-fect the rates of grain-catalyzed reactions, adsorption to the grainmantles, excitation of H in collisions with grains, and transferof momentum and thermal and kinetic energy between the neu-tral and the charged fluids.Figure A.1 shows a comparison between the multi-fluidmodel of Paper III and our current model for a representativeshock for which n H = cm − , V s =
30 km s − and b = ± ∼
15 %for the entire grid of models incorporating shattering.
A.1.2. Chemical effect of shattering
The change in the total grain cross-section, (cid:104) n σ (cid:105) , owing to shat-tering, has consequences for the rates of grain-charging reac-tions. In order to model the e ff ect of shattering on the abun-dances of charged grains, the corresponding chemical sourceterm needs to be introduced.The charge distribution of the fragments is essentially un-known and cannot be numerically integrated separately from In order to make a direct comparison with the results of Paper III, itis necessary to disable the LVG treatment of molecular line transfer inour current model. 11. Anderl, V. Guillet, G. Pineau des Forˆets and D. R. Flower: Shocks in dense clouds T n ( K ) Paper IIICurrent model110100 100 1000 N o r m a li z e d < n ! > du s t flow time (yr) core + mantlecore Fig. A.1.
Upper panel: temperature profiles of the neutral fluidfor a shock with n H = cm − , V s =
30 km s − and b = (cid:104) n σ (cid:105) , for the same shock models, with and without taking intoaccount the grain mantles and normalized to the preshock valuesof (cid:104) n σ (cid:105) . Table A.1.
Parameters relating to the modification by shatteringof the total grain cross-section, (cid:104) n σ (cid:105) and to SiO vaporization, forthe grid of multi-fluid models introduced in Section 3.1. V s [km s − ] b B [ µ G] Σ max β x (SiO peak )20 1.0 316 11.7 0.8 1.53 · −
20 1.5 474 10.5 1.1 1.99 · −
20 2.0 632 8.5 2.0 2.16 · −
30 1.5 474 14.2 0.8 1.55 · −
30 2.0 632 11.7 0.9 7.92 · −
30 2.5 791 10.5 1.2 5.63 · −
40 2.0 632 18.4 0.8 9.13 · −
40 2.5 791 13.7 0.9 3.43 · −
40 3.0 949 11.4 1.1 1.34 · − that of grains already present in the medium. In the present pa-per , the charge distribution of fragments is designed to ensure The situation was more complicated in Paper III, which dealt witha full grain size distribution. Fragments were allocated to the size binscorresponding to their individual mass, while the corresponding masswas removed from the projectile and target size bins. This proceduredoes not allow for charge conservation, because small grains carry muchmore (negative) charge per unit mass than large grains. To compensatefor this charge excess, the charge distributions of all grain sizes was charge conservation. We fitted the fragment charge distributionby aligning the shock widths (see Fig. A.1, upper panel). Thisprocedure yielded a charge distribution in which 1 / / / (cid:88) j ∈{ G0 , G + , G −} n j · V j · Σ − ( z ) = constant (A.7)where n G0 is the number density of neutral grains, n G + , G − arethe number densities of positively and negatively charged grains,respectively, V G0 = V n is the velocity of the neutral grains, inthe shock frame, and V G + , G − = V i is the velocity of the chargedgrains, in the shock frame. Di ff erentiation of (A.7), subject tothe charge distribution of the fragments, then yieldsd n G0 d z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) shat = · · n G , preshock V s V n · Σ · d Σ d z (A.8)for the neutral grains, andd n G + , G − d z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) shat = · · n G , preshock V s V i · Σ · d Σ d z (A.9)for the charged grains, where n G , preshock is the total (chargedand neutral) number density of grains in the preshock gas. Thederivative of Σ is given byd Σ d z = (cid:18) dd z η β − cos (cid:16) πη β (cid:17) dd z η β (cid:19)(cid:18) Σ max − (cid:19) (A.10)wheredd z η β = β η ( β − d η d z (A.11)andd η d z = − V s / V postshock − V s V d V i d z . (A.12)With these parameters, our model is able to reproduce the maine ff ects of the increase in the grain cross section, reported inPaper III: the e ff ective rate of recombination of electrons andions is enhanced; the fractional abundance of free electrons fallsby three orders of magnitude; and dust grains become the domi-nant charge carriers, with equal numbers of positively and nega-tively charged grains being produced. A.2. Vaporization
The e ff ect of vaporization is modelled as an additional termin the creation rate of gas-phase SiO, from Si and O in graincores (denoted by **), corresponding to a new type of pseudo-chemical reaction: Si** + O** = SiO + GRAIN shifted infinitesimally at each shattering event to ensure charge conser-vation.12. Anderl, V. Guillet, G. Pineau des Forˆets and D. R. Flower: Shocks in dense clouds
Because vaporization sets in suddenly, when the vaporizationthreshold is reached, the function Ω ( z ) = + exp (cid:16) − · (cid:16) V s / V i ( z ) − V s / V postshock (cid:17)(cid:17) (A.13)can be used to approximate the rate of creation of SiO. The func-tion Ω ( z ) is centred at the point where the compression of thecharged fluid reaches 1 / in the exponent,which determines the steepness of the function, derives from afit to the numerical results of Paper III. Using this function, thecreation rate (cm − s − ) can be expressed asd n (SiO)d t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) vapo = d Ω ( z )d t · n H · x (SiO) peak (A.14) = d Ω ( z )d z · V n · n H · x (SiO) peak (A.15) = d Ω ( z )d z · V s · n H , preshock · x (SiO) peak , (A.16)where use is made of the conservation of the flux of n H , andwheredd z Ω ( z ) = − Ω ( z ) · (cid:16) − Ω ( z ) (cid:17) · V s V · d V i d z . (A.17)The spatial change in number density of SiO, owing to vaporiza-tion, is then given byd n (SiO)d z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) vapo = (cid:32) d n (SiO)d t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) vapo − n (SiO) d V n d z (cid:33) V n (A.18)The peak fractional abundance x (SiO) peak needs to be com-puted with the multi-fluid model and constitutes a free parame-ter, given in Table A.1 for our grid of models. As can be seenin Table A.1, vaporization of SiO is relevant only for the shockswith high velocity and low magnetic field; it can be neglectedfor the shocks with V s =
20 km s − and b = b =
2, as well asfor the model with V s =
30 km s − and b = ) graincores. This Figure compares the fractional abundance of SiO,as computed with our current model (incorporating grain-grainprocessing) and as predicted by the multi-fluid model of PaperIII. By construction, the peaks of the fractional abundance ofSiO agree, whereas the timescale for its accretion on to grainsdi ff ers between the two models, as is visible in the plot; this dis-crepancy relates to the imperfect agreement of the total cross-section, (cid:104) n σ (cid:105) , in the postshock medium (see Fig. A.1). However,we have verified that the timescale for SiO accretion is not criti-cal to our analysis: the complete neglect of accretion on to grainsleads to increases in the integrated intensities of the lowest ro-tational transitions of SiO and CO, by factors of ∼ ∼
3, re-spectively. We note that the intensities of these transitions are, inany case, a ff ected by the foreground emission of ambient, non-shocked gas. Appendix B: Molecular line emission
The introduction of shattering leads to a reduction of the shockwidth, and hence to lower column densities of shocked material, -12 -11 -10 -9 -8 -7 -6
100 1000 x ( S i O ) Tn ( K ) neutral flow time (yr) Paper IIICurrent model
Fig. A.2.
The fractional abundance of SiO (full curves), as deter-mined when including grain-grain processing, using the presentmodel (in red) and the multi-fluid model of Paper III (in blue);the shock parameters are n H = cm − , V s =
30 km s − and b = l n N / g E v,J (K) v = 0, J = 2J = 3 J = 4J = 5 J = 6 J = 7 J = 8 J = 9 J = 10 H b = 1.5b = 2.0b = 2.5 Fig. B.1.
Computed H excitation diagrams for rovibrationallevels with energies E v , J ≤
10 000 K and for shocks with n H = cm − , V s =
30 km s − and b = b = b = ff ect the chemistry andenhance the fractional abundances of molecules in excited states.Which of these e ff ects prevails is determined by the chemicaland spectroscopic properties of the individual molecular species. B.1. H The intensities of pure rotational and ro-vibrational lines of H contain key information on the structure of C-type shock waves,
13. Anderl, V. Guillet, G. Pineau des Forˆets and D. R. Flower: Shocks in dense clouds Td V ( K k m s - ) J up CO b = 1.5b = 2.0b = 2.5 Fig. B.2.
Integrated intensities of the rotational transitions J up → J up − n H = cm − , V s =
30 km s − and b = b = b = Td V ( K k m s - ) ortho-H O b = 1.5b = 2.0b = 2.5E up (K) Fig. B.3.
Integrated intensities of selected rotational transitionsof ortho-H O plotted against the excitation energy of the emit-ting level, expressed relative to the energy of the J = = K ground state of para-H O. Results are shown for shocks with n H = cm − , V s =
30 km s − and b = b = b = , by approximately an order ofmagnitude. With increasing energy of the emitting level, the ef-fect of the decrease in the shock width is compensated by the E up (K) OH b = 1.5b = 2.0b = 2.5 Td V ( K k m s - ) Fig. B.4.
Integrated intensities of the rotational transitions of OHfor emitting levels of negative parity, plotted against the excita-tion energy of the upper level. Results are shown for shocks with V s =
30 km s − and b = b = b = ∼ > V s =
20 km s − , ∼ > V s =
30 km s − , and ∼ > V s =
40 km s − , with theexact values depending on the magnetic field strength. The com-puted intensities of selected lines of H are given in Table B.1,together with the intensities of forbidden lines of atomic oxy-gen (63 µ m and 147 µ m), of atomic sulfur (25 µ m) and of [C I](610 µ m and 370 µ m). B.2. CO, H O and OH
Figure B.2 shows the integrated line intensities,
T dV , of the ro-tational transitions of CO, plotted against the rotational quantumnumber, J , of the emitting level for shocks with n H = cm − and V s =
30 km s − . The line intensities, which are listed inTables B.2–B.4, are lower when shattering is included, owing tothe reduction in the shock width; this e ff ect is most pronouncedat low magnetic field strengths. While the intensities computedwith models M2 peak at around J up =
7, those of models M1peak at higher values of J up and exhibit a plateau extending to J up ≈
12. These di ff erences reflect the corresponding excita-tion conditions. As in the case of SiO (see Sect. 4.2), the peaktemperature shows a stronger dependence on the strength of themagnetic field in models that include grain-grain processing.Accordingly, the integrated intensities of highly excited transi-tions vary with b for models M1.Similarly to CO, the intensities of lines of H O also becomeweaker when shattering is included, owing to the reduced shockwidth. Figure B.3 shows the computed intensities of the lines ofortho-H O as a function of the excitation energy of the emittinglevel. The intensities of all the lines of ortho- and para-H O thatfall in the Herschel / PACS / HIFI bands are listed in Tables B.6–B.11.
14. Anderl, V. Guillet, G. Pineau des Forˆets and D. R. Flower: Shocks in dense clouds -22 -21 -20 -19 -18 -17 -16 -15 -14 distance (cm)M1 T n H OCOH OH C oo li ng r a t e ( e r g c m - s - ) T n ( K ) distance (cm)M2, M3T n H OCOH OH Fig. B.5.
The rate of cooling by the principal molecularcoolants, H (mauve), H O (green), CO (blue) and OH (red) for V s =
40 km s − and b = / M3(right panel), which are shown together because the presence ofSiO in grain mantles does not a ff ect the cooling of the shockwave. Note the di ff erent distance intervals on the x -axes.Contrary to the behaviour of CO and H O, the line intensitiesof OH become stronger for the 30 km s − and 40 km s − shockswhen shattering is included, as may be seen for V s =
30 km s − in Figure B.4. The lines displayed fall within the Herschel / PACSband; their emitting levels have negative parity. The intensitiesof all transitions of OH observable with Herschel are listed inTable B.5. The increase in the integrated intensities in modelsM1 is due to the higher peak temperatures, which favour theconversion of the gas-phase oxygen that is not bound in CO intoOH. Again, we see a variation of the integrated line intensitieswith the magnetic field strength in models M1, associated withthe temperature-dependent rate of OH formation.On the basis of these findings, it is interesting to ask how theallowance for grain-grain processing a ff ects the radiative coolingof the medium. Although the narrower shock width, and hencelarger velocity gradients, in scenario M1 might be expected tomodify the optical thickness of the lines, and thereby their rateof cooling, we do not detect such an e ff ect in our models, asis demonstrated by Fig. B.5. Instead, we see an increase in thecontribution of OH to the rate of cooling, owing to the enhancedabundance of gas-phase OH in the hot shocked medium (see thelower panel of Fig. 4).
15. Anderl, V. Guillet, G. Pineau des Forˆets and D. R. Flower: Shocks in dense clouds
Table B.1.
Selected H , [OI], [CI], and [SI] line intensities (erg cm − s − sr − ), for shocks with velocities V s =
20 km s − (top), V s =
30 km s − (middle), and V s =
40 km s − (bottom) and the magnetic field strengths listed in Table 1. Results are given formodels M1, which include grain-grain processing, and M2 (in parentheses), which neglect grain-grain processing. The preshockdensity is n H = cm − . Transition λ ( µ m) v20b1, M1 (v20b1, M2) v20b1.5, M1 (v20b1.5, M2) v20b2, M1 (v20b2, M2)H P → P P → P P → P P → P = →
3P J = λ ( µ m) v30b1.5, M1 (v30b1.5, M2) v30b2, M1 (v30b2, M2) v30b2.5, M1 (v30b2.5, M2)H P → P P → P P → P P → P = →
3P J = λ ( µ m) v40b2, M1 (v40b2, M2) v40b2.5, M1 (v40b2.5, M2) v40b3, M1 (v40b3, M2)H P → P P → P P → P P → P = →
3P J = Table B.2.
Intensities of CO lines (erg cm − s − sr − ) for shocks with velocity V s =
20 km s − and the magnetic field strengthslisted in Table 1. Results are given for models M1, which include grain-grain processing, and M2 (in parentheses), which neglectgrain-grain processing. The preshock density is n H = cm − . Transition Freq. (GHz) λ ( µ m) v20b1, M1 (v20b1, M2) v20b1.5, M1 (v20b1.5, M2) v20b2, M1 (v20b2, M2)CO (1–0) 115.27 2600.7 3.47e-08 (1.85e-07) 9.49e-08 (2.63e-07) 1.47e-07 (2.69e-07)CO (2–1) 230.54 1300.4 6.39e-07 (2.41e-06) 1.28e-06 (3.23e-06) 1.76e-06 (3.14e-06)CO (3–2) 345.80 866.96 2.72e-06 (1.01e-05) 4.69e-06 (1.30e-05) 6.33e-06 (1.22e-05)CO (4–3) 461.04 650.25 7.01e-06 (2.73e-05) 1.15e-05 (3.50e-05) 1.57e-05 (3.24e-05)CO (5–4) 576.27 520.23 1.41e-05 (5.78e-05) 2.36e-05 (7.55e-05) 3.40e-05 (7.07e-05)CO (6–5) 691.47 433.55 2.42e-05 (1.00e-04) 4.22e-05 (1.36e-04) 6.46e-05 (1.30e-04)CO (7–6) 806.65 371.65 3.76e-05 (1.51e-04) 6.71e-05 (2.09e-04) 1.09e-04 (2.06e-04)CO (8–7) 921.80 325.22 5.51e-05 (2.13e-04) 9.84e-05 (2.91e-04) 1.66e-04 (2.94e-04)CO (9–8) 1036.9 289.12 7.68e-05 (2.83e-04) 1.36e-04 (3.76e-04) 2.32e-04 (3.86e-04)CO (10–9) 1152.0 260.24 1.02e-04 (3.54e-04) 1.76e-04 (4.52e-04) 3.01e-04 (4.70e-04)CO (11–10) 1267.0 236.61 1.29e-04 (4.16e-04) 2.17e-04 (5.09e-04) 3.63e-04 (5.31e-04)CO (12–11) 1382.0 216.93 1.56e-04 (4.57e-04) 2.52e-04 (5.38e-04) 4.09e-04 (5.62e-04)CO (13–12) 1496.9 200.27 1.78e-04 (4.72e-04) 2.75e-04 (5.33e-04) 4.32e-04 (5.58e-04)CO (14–13) 1611.8 186.00 1.94e-04 (4.64e-04) 2.87e-04 (5.03e-04) 4.34e-04 (5.25e-04)CO (15–14) 1726.6 173.63 2.00e-04 (4.29e-04) 2.83e-04 (4.47e-04) 4.11e-04 (4.65e-04)CO (16–15) 1841.3 162.81 2.02e-04 (3.94e-04) 2.75e-04 (3.95e-04) 3.84e-04 (4.10e-04)CO (17–16) 1956.0 153.27 1.98e-04 (3.52e-04) 2.59e-04 (3.42e-04) 3.49e-04 (3.52e-04)CO (18–17) 2070.6 144.78 1.89e-04 (3.09e-04) 2.39e-04 (2.91e-04) 3.09e-04 (2.97e-04)CO (19–18) 2185.1 137.20 1.78e-04 (2.67e-04) 2.17e-04 (2.44e-04) 2.70e-04 (2.48e-04)CO (20–19) 2299.6 130.37 1.64e-04 (2.28e-04) 1.94e-04 (2.03e-04) 2.32e-04 (2.04e-04)CO (21–20) 2413.9 124.19 1.47e-04 (1.91e-04) 1.70e-04 (1.66e-04) 1.96e-04 (1.65e-04)CO (22–21) 2528.2 118.58 1.31e-04 (1.59e-04) 1.47e-04 (1.34e-04) 1.63e-04 (1.33e-04)CO (23–22) 2642.3 113.46 1.15e-04 (1.31e-04) 1.26e-04 (1.08e-04) 1.34e-04 (1.06e-04)CO (24–23) 2756.4 108.76 9.98e-05 (1.07e-04) 1.07e-04 (8.63e-05) 1.09e-04 (8.34e-05)CO (25–24) 2870.3 104.44 8.58e-05 (8.66e-05) 8.98e-05 (6.85e-05) 8.83e-05 (6.54e-05)CO (26–25) 2984.2 100.46 7.32e-05 (6.98e-05) 7.50e-05 (5.41e-05) 7.08e-05 (5.10e-05)CO (27–26) 3097.9 96.772 6.20e-05 (5.60e-05) 6.23e-05 (4.25e-05) 5.63e-05 (3.95e-05)CO (28–27) 3211.5 93.348 5.21e-05 (4.46e-05) 5.13e-05 (3.32e-05) 4.44e-05 (3.04e-05)CO (29–28) 3325.0 90.162 4.35e-05 (3.53e-05) 4.20e-05 (2.58e-05) 3.48e-05 (2.33e-05)CO (30–29) 3438.4 87.190 3.59e-05 (2.78e-05) 3.40e-05 (1.99e-05) 2.70e-05 (1.77e-05)CO (31–30) 3551.6 84.410 2.95e-05 (2.17e-05) 2.74e-05 (1.52e-05) 2.08e-05 (1.33e-05)CO (32–31) 3664.7 81.805 2.40e-05 (1.68e-05) 2.19e-05 (1.16e-05) 1.58e-05 (9.96e-06)CO (33–32) 3777.6 79.359 1.92e-05 (1.29e-05) 1.72e-05 (8.71e-06) 1.19e-05 (7.37e-06)CO (34–33) 3890.4 77.058 1.52e-05 (9.72e-06) 1.34e-05 (6.48e-06) 8.87e-06 (5.39e-06)CO (35–34) 4003.1 74.889 1.18e-05 (7.21e-06) 1.02e-05 (4.73e-06) 6.46e-06 (3.87e-06)CO (36–35) 4115.6 72.842 8.96e-06 (5.22e-06) 7.60e-06 (3.37e-06) 4.60e-06 (2.71e-06)CO (37–36) 4228.0 70.907 6.52e-06 (3.64e-06) 5.43e-06 (2.32e-06) 3.15e-06 (1.84e-06)CO (38–37) 4340.1 69.074 4.46e-06 (2.39e-06) 3.65e-06 (1.50e-06) 2.03e-06 (1.17e-06)CO (39–38) 4452.2 67.336 2.72e-06 (1.41e-06) 2.19e-06 (8.72e-07) 1.17e-06 (6.69e-07)CO (40–39) 4564.0 65.686 1.25e-06 (6.22e-07) 9.89e-07 (3.81e-07) 5.10e-07 (2.88e-07) 17. Anderl, V. Guillet, G. Pineau des Forˆets and D. R. Flower: Shocks in dense clouds Table B.3.
Intensities of CO lines (erg cm − s − sr − ) for shocks with velocity V s =
30 km s − and the magnetic field strengthslisted in Table 1. Results are given for models M1, which include grain-grain processing, and M2 (in parentheses), which neglectgrain-grain processing. The preshock density is n H = cm − . Transition Freq. (GHz) λ ( µ m) v30b1.5, M1 (v30b1.5, M2) v30b2, M1 (v30b2, M2) v30b2.5, M1 (v30b2.5, M2)CO (1–0) 115.27 2600.7 4.01e-08 (2.64e-07) 9.93e-08 (3.42e-07) 1.65e-07 (4.14e-07)CO (2–1) 230.54 1300.4 7.45e-07 (3.35e-06) 1.41e-06 (4.14e-06) 2.05e-06 (4.88e-06)CO (3–2) 345.80 866.96 3.19e-06 (1.37e-05) 5.06e-06 (1.65e-05) 7.19e-06 (1.92e-05)CO (4–3) 461.04 650.25 8.29e-06 (3.68e-05) 1.22e-05 (4.39e-05) 1.72e-05 (5.12e-05)CO (5–4) 576.27 520.23 1.68e-05 (7.71e-05) 2.49e-05 (9.47e-05) 3.57e-05 (1.12e-04)CO (6–5) 691.47 433.55 2.89e-05 (1.33e-04) 4.42e-05 (1.71e-04) 6.54e-05 (2.07e-04)CO (7–6) 806.65 371.65 4.49e-05 (2.01e-04) 7.03e-05 (2.69e-04) 1.07e-04 (3.30e-04)CO (8–7) 921.80 325.22 6.56e-05 (2.86e-04) 1.04e-04 (3.88e-04) 1.61e-04 (4.79e-04)CO (9–8) 1036.9 289.12 9.13e-05 (3.90e-04) 1.44e-04 (5.26e-04) 2.26e-04 (6.46e-04)CO (10–9) 1152.0 260.24 1.22e-04 (5.06e-04) 1.92e-04 (6.74e-04) 3.00e-04 (8.15e-04)CO (11–10) 1267.0 236.61 1.55e-04 (6.22e-04) 2.43e-04 (8.13e-04) 3.77e-04 (9.65e-04)CO (12–11) 1382.0 216.93 1.88e-04 (7.22e-04) 2.93e-04 (9.23e-04) 4.48e-04 (1.07e-03)CO (13–12) 1496.9 200.27 2.18e-04 (7.91e-04) 3.35e-04 (9.86e-04) 5.03e-04 (1.12e-03)CO (14–13) 1611.8 186.00 2.42e-04 (8.26e-04) 3.66e-04 (1.00e-03) 5.40e-04 (1.12e-03)CO (15–14) 1726.6 173.63 2.55e-04 (8.14e-04) 3.80e-04 (9.65e-04) 5.50e-04 (1.05e-03)CO (16–15) 1841.3 162.81 2.63e-04 (7.89e-04) 3.86e-04 (9.15e-04) 5.49e-04 (9.77e-04)CO (17–16) 1956.0 153.27 2.62e-04 (7.45e-04) 3.81e-04 (8.46e-04) 5.34e-04 (8.86e-04)CO (18–17) 2070.6 144.78 2.55e-04 (6.87e-04) 3.68e-04 (7.65e-04) 5.07e-04 (7.87e-04)CO (19–18) 2185.1 137.20 2.43e-04 (6.23e-04) 3.48e-04 (6.81e-04) 4.73e-04 (6.89e-04)CO (20–19) 2299.6 130.37 2.27e-04 (5.57e-04) 3.24e-04 (5.98e-04) 4.35e-04 (5.96e-04)CO (21–20) 2413.9 124.19 2.07e-04 (4.88e-04) 2.94e-04 (5.16e-04) 3.91e-04 (5.06e-04)CO (22–21) 2528.2 118.58 1.86e-04 (4.23e-04) 2.64e-04 (4.40e-04) 3.47e-04 (4.25e-04)CO (23–22) 2642.3 113.46 1.65e-04 (3.62e-04) 2.34e-04 (3.72e-04) 3.05e-04 (3.54e-04)CO (24–23) 2756.4 108.76 1.45e-04 (3.08e-04) 2.05e-04 (3.11e-04) 2.65e-04 (2.93e-04)CO (25–24) 2870.3 104.44 1.25e-04 (2.60e-04) 1.78e-04 (2.59e-04) 2.28e-04 (2.40e-04)CO (26–25) 2984.2 100.46 1.08e-04 (2.18e-04) 1.54e-04 (2.14e-04) 1.95e-04 (1.96e-04)CO (27–26) 3097.9 96.772 9.19e-05 (1.81e-04) 1.32e-04 (1.76e-04) 1.66e-04 (1.58e-04)CO (28–27) 3211.5 93.348 7.78e-05 (1.50e-04) 1.12e-04 (1.43e-04) 1.39e-04 (1.27e-04)CO (29–28) 3325.0 90.162 6.53e-05 (1.23e-04) 9.40e-05 (1.16e-04) 1.17e-04 (1.02e-04)CO (30–29) 3438.4 87.190 5.43e-05 (1.00e-04) 7.85e-05 (9.33e-05) 9.66e-05 (8.07e-05)CO (31–30) 3551.6 84.410 4.49e-05 (8.08e-05) 6.51e-05 (7.44e-05) 7.94e-05 (6.35e-05)CO (32–31) 3664.7 81.805 3.67e-05 (6.47e-05) 5.34e-05 (5.89e-05) 6.46e-05 (4.95e-05)CO (33–32) 3777.6 79.359 2.97e-05 (5.12e-05) 4.33e-05 (4.60e-05) 5.19e-05 (3.81e-05)CO (34–33) 3890.4 77.058 2.37e-05 (4.00e-05) 3.45e-05 (3.55e-05) 4.11e-05 (2.90e-05)CO (35–34) 4003.1 74.889 1.85e-05 (3.06e-05) 2.70e-05 (2.68e-05) 3.19e-05 (2.16e-05)CO (36–35) 4115.6 72.842 1.41e-05 (2.29e-05) 2.06e-05 (1.98e-05) 2.42e-05 (1.57e-05)CO (37–36) 4228.0 70.907 1.03e-05 (1.64e-05) 1.51e-05 (1.40e-05) 1.76e-05 (1.10e-05)CO (38–37) 4340.1 69.074 7.11e-06 (1.11e-05) 1.04e-05 (9.36e-06) 1.20e-05 (7.26e-06)CO (39–38) 4452.2 67.336 4.37e-06 (6.68e-06) 6.39e-06 (5.58e-06) 7.31e-06 (4.27e-06)CO (40–39) 4564.0 65.686 2.02e-06 (3.03e-06) 2.95e-06 (2.50e-06) 3.35e-06 (1.89e-06)18. Anderl, V. Guillet, G. Pineau des Forˆets and D. R. Flower: Shocks in dense clouds Table B.4.
Intensities of CO lines (erg cm − s − sr − ) for shocks with velocity V s =
40 km s − and the magnetic field strengthslisted in Table 1. Results are given for models M1, which include grain-grain processing, and M2 (in parentheses), which neglectgrain-grain processing. The preshock density is n H = cm − . Transition Freq. (GHz) λ ( µ m) v40b2, M1 (v40b2, M2) v40b2.5, M1 (v40b2.5, M2) v40b3, M1 (v40b3, M2)CO (1–0) 115.27 2600.7 3.42e-08 (3.41e-07) 1.00e-07 (4.18e-07) 1.73e-07 (4.86e-07)CO (2–1) 230.54 1300.4 6.14e-07 (4.33e-06) 1.45e-06 (5.02e-06) 2.22e-06 (5.69e-06)CO (3–2) 345.80 866.96 2.73e-06 (1.76e-05) 5.30e-06 (1.98e-05) 7.52e-06 (2.21e-05)CO (4–3) 461.04 650.25 7.90e-06 (4.67e-05) 1.30e-05 (5.22e-05) 1.76e-05 (5.82e-05)CO (5–4) 576.27 520.23 1.81e-05 (9.75e-05) 2.68e-05 (1.11e-04) 3.63e-05 (1.25e-04)CO (6–5) 691.47 433.55 3.51e-05 (1.68e-04) 4.79e-05 (1.99e-04) 6.58e-05 (2.30e-04)CO (7–6) 806.65 371.65 6.04e-05 (2.53e-04) 7.70e-05 (3.11e-04) 1.07e-04 (3.66e-04)CO (8–7) 921.80 325.22 9.53e-05 (3.58e-04) 1.14e-04 (4.49e-04) 1.61e-04 (5.35e-04)CO (9–8) 1036.9 289.12 1.40e-04 (4.91e-04) 1.59e-04 (6.17e-04) 2.27e-04 (7.36e-04)CO (10–9) 1152.0 260.24 1.93e-04 (6.46e-04) 2.11e-04 (8.08e-04) 3.03e-04 (9.57e-04)CO (11–10) 1267.0 236.61 2.51e-04 (8.09e-04) 2.67e-04 (1.00e-03) 3.84e-04 (1.18e-03)CO (12–11) 1382.0 216.93 3.11e-04 (9.61e-04) 3.23e-04 (1.18e-03) 4.64e-04 (1.37e-03)CO (13–12) 1496.9 200.27 3.66e-04 (1.08e-03) 3.71e-04 (1.31e-03) 5.32e-04 (1.50e-03)CO (14–13) 1611.8 186.00 4.12e-04 (1.15e-03) 4.09e-04 (1.38e-03) 5.84e-04 (1.56e-03)CO (15–14) 1726.6 173.63 4.39e-04 (1.17e-03) 4.29e-04 (1.38e-03) 6.09e-04 (1.54e-03)CO (16–15) 1841.3 162.81 4.57e-04 (1.16e-03) 4.40e-04 (1.36e-03) 6.22e-04 (1.50e-03)CO (17–16) 1956.0 153.27 4.61e-04 (1.11e-03) 4.39e-04 (1.30e-03) 6.17e-04 (1.42e-03)CO (18–17) 2070.6 144.78 4.52e-04 (1.04e-03) 4.26e-04 (1.21e-03) 5.98e-04 (1.31e-03)CO (19–18) 2185.1 137.20 4.33e-04 (9.61e-04) 4.05e-04 (1.11e-03) 5.68e-04 (1.19e-03)CO (20–19) 2299.6 130.37 4.07e-04 (8.71e-04) 3.79e-04 (1.00e-03) 5.31e-04 (1.07e-03)CO (21–20) 2413.9 124.19 3.74e-04 (7.73e-04) 3.46e-04 (8.86e-04) 4.85e-04 (9.36e-04)CO (22–21) 2528.2 118.58 3.37e-04 (6.76e-04) 3.11e-04 (7.74e-04) 4.37e-04 (8.13e-04)CO (23–22) 2642.3 113.46 3.01e-04 (5.86e-04) 2.76e-04 (6.69e-04) 3.89e-04 (6.99e-04)CO (24–23) 2756.4 108.76 2.65e-04 (5.02e-04) 2.42e-04 (5.74e-04) 3.43e-04 (5.96e-04)CO (25–24) 2870.3 104.44 2.30e-04 (4.27e-04) 2.11e-04 (4.88e-04) 2.99e-04 (5.04e-04)CO (26–25) 2984.2 100.46 1.99e-04 (3.61e-04) 1.82e-04 (4.12e-04) 2.59e-04 (4.23e-04)CO (27–26) 3097.9 96.772 1.70e-04 (3.03e-04) 1.56e-04 (3.45e-04) 2.23e-04 (3.53e-04)CO (28–27) 3211.5 93.348 1.45e-04 (2.52e-04) 1.32e-04 (2.88e-04) 1.90e-04 (2.93e-04)CO (29–28) 3325.0 90.162 1.22e-04 (2.08e-04) 1.11e-04 (2.38e-04) 1.61e-04 (2.41e-04)CO (30–29) 3438.4 87.190 1.03e-04 (1.71e-04) 9.31e-05 (1.95e-04) 1.35e-04 (1.97e-04)CO (31–30) 3551.6 84.410 8.53e-05 (1.39e-04) 7.72e-05 (1.59e-04) 1.12e-04 (1.60e-04)CO (32–31) 3664.7 81.805 7.03e-05 (1.12e-04) 6.34e-05 (1.28e-04) 9.22e-05 (1.28e-04)CO (33–32) 3777.6 79.359 5.73e-05 (8.96e-05) 5.15e-05 (1.02e-04) 7.50e-05 (1.02e-04)CO (34–33) 3890.4 77.058 4.61e-05 (7.05e-05) 4.12e-05 (8.06e-05) 6.01e-05 (7.99e-05)CO (35–34) 4003.1 74.889 3.63e-05 (5.44e-05) 3.23e-05 (6.22e-05) 4.72e-05 (6.13e-05)CO (36–35) 4115.6 72.842 2.79e-05 (4.09e-05) 2.47e-05 (4.68e-05) 3.61e-05 (4.59e-05)CO (37–36) 4228.0 70.907 2.06e-05 (2.96e-05) 1.82e-05 (3.38e-05) 2.66e-05 (3.31e-05)CO (38–37) 4340.1 69.074 1.43e-05 (2.02e-05) 1.26e-05 (2.30e-05) 1.84e-05 (2.24e-05)CO (39–38) 4452.2 67.336 8.88e-06 (1.22e-05) 7.73e-06 (1.40e-05) 1.13e-05 (1.35e-05)CO (40–39) 4564.0 65.686 4.13e-06 (5.58e-06) 3.58e-06 (6.36e-06) 5.22e-06 (6.14e-06) 19. Anderl, V. Guillet, G. Pineau des Forˆets and D. R. Flower: Shocks in dense clouds Table B.5.
Intensities of OH lines (erg cm − s − sr − ) observable with the PACS instrument on the Herschel Space Observatory, forshocks with velocities V s =
20 km s − (top), V s =
30 km s − (middle), and V s =
40 km s − (bottom) and the magnetic field strengthslisted in Table 1. Results are given for models M1, which include grain-grain processing, and M2 (in parentheses), which neglectgrain-grain processing. The preshock density is n H = cm − . Transition λ ( µ m) E upup
40 km s − (bottom) and the magnetic field strengthslisted in Table 1. Results are given for models M1, which include grain-grain processing, and M2 (in parentheses), which neglectgrain-grain processing. The preshock density is n H = cm − . Transition λ ( µ m) E upup (K) v20b1, M1 (v20b1, M2) v20b1.5, M1 (v20b1.5, M2) v20b2, M1 (v20b2, M2)1834.8 GHz 163.39 269.8 1.65e-08 (1.76e-08) 2.58e-08 (2.51e-08) 3.92e-08 (3.03e-08)1839.0 GHz 163.01 270.2 2.42e-08 (2.54e-08) 3.75e-08 (3.58e-08) 5.65e-08 (4.35e-08)2510.0 GHz 119.44 120.5 1.79e-07 (2.71e-07) 2.88e-07 (4.89e-07) 4.38e-07 (4.20e-07)2514.0 GHz 119.23 120.7 9.78e-08 (1.65e-07) 1.60e-07 (3.14e-07) 2.44e-07 (2.50e-07)3035.0 GHz 98.76 415.9 3.11e-08 (3.08e-08) 4.61e-08 (4.01e-08) 6.78e-08 (5.11e-08)3036.0 GHz 98.74 415.5 1.00e-08 (1.00e-08) 1.50e-08 (1.31e-08) 2.22e-08 (1.68e-08)3111.0 GHz 96.36 269.8 4.01e-09 (4.28e-09) 6.28e-09 (6.11e-09) 9.54e-09 (7.36e-09)3114.0 GHz 96.27 270.2 5.86e-09 (6.16e-09) 9.07e-09 (8.67e-09) 1.37e-08 (1.05e-08)3544.0 GHz 84.60 290.5 8.10e-08 (8.30e-08) 1.24e-07 (1.13e-07) 1.86e-07 (1.41e-07)3551.0 GHz 84.42 291.2 1.94e-08 (2.01e-08) 3.00e-08 (2.76e-08) 4.52e-08 (3.44e-08)3786.0 GHz 79.18 181.7 1.01e-07 (1.19e-07) 1.61e-07 (1.88e-07) 2.47e-07 (2.03e-07)3789.0 GHz 79.12 181.9 1.11e-07 (1.27e-07) 1.76e-07 (1.96e-07) 2.68e-07 (2.17e-07)4210.0 GHz 71.22 617.9 1.51e-08 (1.45e-08) 2.13e-08 (1.81e-08) 3.04e-08 (2.31e-08)4212.0 GHz 71.17 617.6 2.92e-09 (2.81e-09) 4.17e-09 (3.53e-09) 5.96e-09 (4.52e-09)4592.0 GHz 65.28 510.9 2.78e-08 (2.71e-08) 4.05e-08 (3.46e-08) 5.87e-08 (4.43e-08)4603.0 GHz 65.13 512.1 4.03e-09 (3.94e-09) 5.90e-09 (5.05e-09) 8.58e-09 (6.47e-09)Transition λ ( µ m) E up (K) v30b1.5, M1 (v30b1.5, M2) v30b2, M1 (v30b2, M2) v30b2.5, M1 (v30b2.5, M2)1834.8 GHz 163.39 269.8 1.33e-07 (5.28e-08) 6.54e-08 (5.38e-08) 5.46e-08 (5.24e-08)1839.0 GHz 163.01 270.2 1.98e-07 (7.76e-08) 9.67e-08 (7.86e-08) 8.00e-08 (7.62e-08)2510.0 GHz 119.44 120.5 4.38e-07 (6.17e-07) 6.95e-07 (6.58e-07) 5.94e-07 (6.68e-07)2514.0 GHz 119.23 120.7 7.47e-07 (3.46e-07) 3.77e-07 (3.76e-07) 3.26e-07 (3.88e-07)3035.0 GHz 98.76 415.9 2.66e-07 (1.00e-07) 1.27e-07 (9.94e-08) 1.02e-07 (9.38e-08)3036.0 GHz 98.74 415.5 8.54e-08 (3.23e-08) 4.08e-08 (3.21e-08) 3.28e-08 (3.04e-08)3111.0 GHz 96.36 269.8 3.24e-08 (1.28e-08) 1.59e-08 (1.31e-08) 1.34e-08 (1.27e-08)3114.0 GHz 96.27 270.2 4.80e-08 (1.88e-08) 2.34e-08 (1.90e-08) 1.94e-08 (1.84e-08)3544.0 GHz 84.60 290.5 6.69e-07 (2.59e-07) 3.24e-07 (2.61e-07) 2.66e-07 (2.50e-07)3551.0 GHz 84.42 291.2 1.59e-07 (6.20e-08) 7.73e-08 (6.26e-08) 6.38e-08 (6.04e-08)3786.0 GHz 79.18 181.7 7.87e-07 (3.27e-07) 3.93e-07 (3.40e-07) 3.34e-07 (3.39e-07)3789.0 GHz 79.12 181.9 8.78e-07 (3.59e-07) 4.36e-07 (3.71e-07) 3.67e-07 (3.67e-07)4210.0 GHz 71.22 617.9 1.37e-07 (4.97e-08) 6.34e-08 (4.83e-08) 4.89e-08 (4.45e-08)4212.0 GHz 71.17 617.6 2.64e-08 (9.61e-09) 1.22e-08 (9.35e-09) 9.50e-09 (8.65e-09)4592.0 GHz 65.28 510.9 2.45e-07 (9.07e-08) 1.15e-07 (8.90e-08) 9.07e-08 (8.31e-08)4603.0 GHz 65.13 512.1 3.53e-08 (1.31e-08) 1.66e-08 (1.29e-08) 1.31e-08 (1.21e-08)Transition λ ( µ m) E up (K) v40b2, M1 (v40b2, M2) v40b2.5, M1 (v40b2.5, M2) v40b3, M1 (v40b3, M2)1834.8 GHz 163.39 269.8 6.25e-06 (2.49e-07) 4.66e-07 (1.67e-07) 2.05e-07 (1.36e-07)1839.0 GHz 163.01 270.2 9.23e-06 (3.69e-07) 6.95e-07 (2.47e-07) 3.05e-07 (2.01e-07)2510.0 GHz 119.44 120.5 6.49e-05 (2.67e-06) 4.84e-06 (1.86e-06) 2.15e-06 (1.58e-06)2514.0 GHz 119.23 120.7 3.68e-05 (1.45e-06) 2.60e-06 (1.02e-06) 1.16e-06 (8.84e-07)3035.0 GHz 98.76 415.9 1.23e-05 (4.90e-07) 9.37e-07 (3.25e-07) 4.06e-07 (2.60e-07)3036.0 GHz 98.74 415.5 4.21e-06 (1.57e-07) 3.01e-07 (1.04e-07) 1.30e-07 (8.37e-08)3111.0 GHz 96.36 269.8 1.52e-06 (6.06e-08) 1.13e-07 (4.07e-08) 4.99e-08 (3.32e-08)3114.0 GHz 96.27 270.2 2.24e-06 (8.94e-08) 1.68e-07 (5.99e-08) 7.38e-08 (4.87e-08)3544.0 GHz 84.60 290.5 3.11e-05 (1.24e-06) 2.35e-06 (8.30e-07) 1.03e-06 (6.70e-07)3551.0 GHz 84.42 291.2 7.93e-06 (2.96e-07) 5.59e-07 (1.98e-07) 2.44e-07 (1.60e-07)3786.0 GHz 79.18 181.7 3.54e-05 (1.49e-06) 2.74e-06 (1.02e-06) 1.22e-06 (8.42e-07)3789.0 GHz 79.12 181.9 3.90e-05 (1.66e-06) 3.06e-06 (1.12e-06) 1.36e-06 (9.26e-07)4210.0 GHz 71.22 617.9 6.52e-06 (2.48e-07) 4.85-07 (1.63e-07) 2.07e-07 (1.29e-07)4212.0 GHz 71.17 617.6 1.29e-06 (4.79e-08) 9.34e-08 (3.15e-08) 4.00e-08 (2.50e-08)4592.0 GHz 65.28 510.9 1.18e-05 (4.79e-08) 8.66e-07 (2.96e-07) 3.73e-07 (2.36e-07)4603.0 GHz 65.13 512.1 1.79e-06 (6.46e-08) 1.25e-07 (4.27e-08) 5.37e-08 (3.40e-08)20. Anderl, V. Guillet, G. Pineau des Forˆets and D. R. Flower: Shocks in dense clouds Table B.6.
Intensities of ortho-H O lines (erg cm − s − sr − ) observable with the PACS (top) and HIFI (bottom) instruments on theHerschel Space Observatory, for shocks with velocity V s =
20 km s − and the magnetic field strengths listed in Table 1. Results aregiven for models M1, which include grain-grain processing, and M2 (in parentheses), which neglect grain-grain processing. Thepreshock density is n H = cm − . Transition λ ( µ m) E upup
20 km s − and the magnetic field strengths listed in Table 1. Results aregiven for models M1, which include grain-grain processing, and M2 (in parentheses), which neglect grain-grain processing. Thepreshock density is n H = cm − . Transition λ ( µ m) E upup (K) v20b1, M1 (v20b1, M2) v20b1.5, M1 (v20b1.5, M2) v20b2, M1 (v20b2, M2)5500.1 GHz 54.506 732.1 1.31e-06 (2.25e-06) 1.56e-06 (2.12e-06) 1.38e-06 (1.44e-06)5437.8 GHz 55.131 1274. 5.49e-06 (8.49e-06) 6.44e-06 (7.07e-06) 5.49e-06 (4.75e-06)5276.5 GHz 56.816 1324 1.97e-05 (3.03e-05) 2.30e-05 (2.43e-05) 1.95e-05 (1.64e-05)5107.3 GHz 58.699 550.4 1.10e-04 (1.98e-04) 1.29e-04 (1.94e-04) 1.16e-04 (1.33e-04)4764.0 GHz 62.928 1553 2.74e-07 (4.08e-07) 3.20e-07 (3.36e-07) 2.67e-07 (2.24e-07)4734.3 GHz 63.323 1071 3.62e-05 (5.86e-05) 4.26e-05 (4.87e-05) 3.72e-05 (3.32e-05)4600.4 GHz 65.166 795.5 2.57e-05 (4.19e-05) 3.03e-05 (3.89e-05) 2.61e-05 (2.61e-05)4535.9 GHz 66.092 1013 9.11e-06 (1.44e-05) 1.07e-05 (1.29e-05) 9.17e-06 (8.64e-06)4512.4 GHz 66.437 410.7 1.54e-04 (2.99e-04) 1.76e-04 (2.47e-04) 1.74e-04 (1.85e-04)4456.6 GHz 67.268 410.7 7.77e-06 (1.24e-05) 1.08e-05 (2.21e-05) 6.45e-06 (1.24e-05)4240.2 GHz 70.702 1274 7.40e-07 (1.14e-06) 8.68e-07 (9.53e-07) 7.39e-07 (6.40e-07)4166.9 GHz 71.946 843.5 6.01e-05 (9.94e-05) 7.09e-05 (8.79e-05) 6.23e-05 (5.96e-05)4000.2 GHz 74.944 1126 2.34e-06 (3.70e-06) 2.75e-06 (3.13e-06) 2.36e-06 (2.11e-06)3977.0 GHz 75.380 305.3 2.90e-04 (4.45e-04) 3.37e-04 (4.79e-04) 2.63e-04 (3.07e-04)3971.0 GHz 75.495 1806 3.32e-07 (4.91e-07) 3.83e-07 (3.59e-07) 3.20e-07 (2.45e-07)3807.3 GHz 78.742 432.2 1.30e-04 (2.33e-04) 1.53e-04 (2.21e-04) 1.36e-04 (1.53e-04)3654.6 GHz 82.031 643.5 1.07e-04 (1.81e-04) 1.26e-04 (1.70e-04) 1.11e-04 (1.15e-04)3536.7 GHz 84.766 1013 1.61e-06 (2.56e-06) 1.90e-06 (2.29e-06) 1.62e-06 (1.53e-06)3167.6 GHz 94.643 795.5 3.28e-06 (5.34e-06) 3.87e-06 (4.97e-06) 3.34e-06 (3.33e-06)3165.5 GHz 94.704 702.3 1.26e-06 (2.24e-06) 1.51e-06 (2.23e-06) 1.34e-06 (1.50e-06)3013.2 GHz 99.492 468.1 1.97e-04 (3.45e-04) 2.33e-04 (3.41e-04) 2.04e-04 (2.31e-04)2970.8 GHz 100.91 574.7 2.80e-05 (4.71e-05) 3.32e-05 (4.70e-05) 2.84e-05 (3.13e-05)2774.0 GHz 108.07 194.1 6.17e-04 (1.30e-03) 7.25e-04 (1.52e-03) 6.04e-04 (1.00e-03)2664.6 GHz 112.51 1340 3.64e-07 (5.72e-07) 4.24e-07 (4.42e-07) 3.66e-07 (3.04e-07)2640.5 GHz 113.54 323.5 4.32e-04 (7.79e-04) 5.14e-04 (8.34e-04) 4.39e-04 (5.55e-04)2462.9 GHz 121.72 550.4 5.88e-06 (9.46e-06) 7.77e-06 (1.16e-05) 5.79e-06 (7.11e-06)2344.3 GHz 127.88 1126 7.24e-07 (1.14e-06) 8.51e-07 (9.68e-07) 7.30e-07 (6.52e-07)2264.1 GHz 132.41 432.2 2.81e-05 (4.18e-05) 3.53e-05 (6.21e-05) 2.45e-05 (3.65e-05)2221.8 GHz 134.93 574.7 1.01e-05 (1.70e-05) 1.20e-05 (1.70e-05) 1.03e-05 (1.13e-05)2196.3 GHz 136.49 410.7 2.79e-05 (4.79e-05) 3.62e-05 (8.13e-05) 2.36e-05 (4.71e-05)1918.5 GHz 156.26 642.4 6.92e-07 (1.17e-06) 8.33e-07 (1.13e-06) 7.12e-07 (7.49e-07)1867.7 GHz 160.51 732.1 9.50e-07 (1.63e-06) 1.13e-06 (1.54e-06) 9.99e-07 (1.04e-06)1716.8 GHz 174.62 196.8 6.14e-04 (1.34e-03) 7.21e-04 (1.31e-03) 6.77e-04 (9.16e-04)1669.9 GHz 179.53 114.4 9.57e-04 (2.22e-03) 1.09e-03 (2.06e-03) 1.05e-03 (1.49e-03)1661.0 GHz 180.49 194.1 1.93e-04 (4.63e-04) 2.30e-04 (5.06e-04) 2.03e-04 (3.41e-04)1162.9 GHz 257.79 305.3 5.65e-05 (1.39e-04) 7.12e-05 (1.64e-04) 6.68e-05 (1.11e-04)1153.1 GHz 259.98 249.4 1.22e-04 (2.54e-04) 1.49e-04 (3.43e-04) 1.12e-04 (2.17e-04)1097.4 GHz 273.19 249.4 8.72e-05 (1.96e-04) 1.04e-04 (1.71e-04) 1.07e-04 (1.25e-04)556.94 GHz 538.28 61.0 5.61e-05 (1.32e-04) 6.01e-05 (1.21e-04) 5.73e-05 (8.89e-05) 21. Anderl, V. Guillet, G. Pineau des Forˆets and D. R. Flower: Shocks in dense clouds Table B.7.
Intensities of ortho-H O lines (erg cm − s − sr − ) observable with the PACS (top) and HIFI (bottom) instruments on theHerschel Space Observatory, for shocks with velocity V s =
30 km s − and the magnetic field strengths listed in Table 1. Results aregiven for models M1, which include grain-grain processing, and M2 (in parentheses), which neglect grain-grain processing. Thepreshock density is n H = cm − . Transition λ ( µ m) E up (K) v30b1.5, M1 (v30b1.5, M2) v30b2, M1 (v30b2, M2) v30b2.5, M1 (v30b2.5, M2)5500.1 GHz 54.506 732.1 2.16e-06 (5.36e-06) 2.80e-06 (6.34e-06) 3.60e-06 (6.74e-06)5437.8 GHz 55.131 1274 8.87e-06 (2.26e-05) 1.21e-05 (2.60e-05) 1.60e-05 (2.68e-05)5276.5 GHz 56.816 1324 3.13e-05 (8.00e-05) 4.30e-05 (8.99e-05) 5.69e-05 (9.14e-05)5107.3 GHz 58.699 550.4 1.75e-04 (4.39e-04) 2.13e-04 (4.97e-04) 2.72e-04 (5.15e-04)4764.0 GHz 62.928 1553 4.44e-07 (1.13e-06) 6.15e-07 (1.32e-06) 8.16e-07 (1.36e-06)4734.3 GHz 63.323 1071 5.78e-05 (1.47e-04) 7.78e-05 (1.66e-04) 1.02e-04 (1.70e-04)4600.4 GHz 65.166 795.5 4.28e-05 (1.05e-04) 5.60e-05 (1.25e-04) 7.23e-05 (1.33e-04)4535.9 GHz 66.092 1013 1.50e-05 (3.76e-05) 2.00e-05 (4.43e-05) 2.62e-05 (4.65e-05)4512.4 GHz 66.437 410.7 2.35e-04 (5.69e-04) 2.82e-04 (5.97e-04) 3.60e-04 (5.97e-04)4456.6 GHz 67.268 410.7 1.89e-05 (3.95e-05) 2.66e-05 (6.85e-05) 3.09e-05 (8.70e-05)4240.2 GHz 70.702 1274 1.20e-06 (3.04e-06) 1.64e-06 (3.51e-06) 2.16e-06 (3.62e-06)4166.9 GHz 71.946 843.5 9.74e-05 (2.45e-04) 1.29e-04 (2.83e-04) 1.68e-04 (2.95e-04)4000.2 GHz 74.944 1126 3.76e-06 (9.59e-06) 5.09e-06 (1.10e-05) 6.71e-06 (1.13e-05)3977.0 GHz 75.380 305.3 5.48e-04 (1.11e-03) 6.56e-04 (1.35e-03) 7.45e-04 (1.49e-03)3971.0 GHz 75.495 1806 5.24e-07 (1.32e-06) 7.29e-07 (1.42e-06) 9.63e-07 (1.41e-06)3807.3 GHz 78.742 432.2 2.22e-04 (5.14e-04) 2.80e-04 (6.19e-04) 3.49e-04 (6.68e-04)3654.6 GHz 82.031 643.5 1.76e-04 (4.35e-04) 2.27e-04 (5.11e-04) 2.92e-04 (5.41e-04)3536.7 GHz 84.766 1013 2.65e-06 (6.66e-06) 3.55e-06 (7.86e-06) 4.64e-06 (8.23e-06)3167.6 GHz 94.643 795.5 5.47e-06 (1.35e-05) 7.15e-06 (1.60e-05) 9.23e-06 (1.69e-05)3165.5 GHz 94.704 702.3 2.11e-06 (5.21e-06) 2.69e-06 (6.29e-06) 3.43e-06 (6.79e-06)3013.2 GHz 99.492 468.1 3.31e-04 (7.96e-04) 4.16e-04 (9.46e-04) 5.23e-04 (1.01e-03)2970.8 GHz 100.91 574.7 4.87e-05 (1.14e-04) 6.22e-05 (1.40e-04) 7.79e-05 (1.52e-04)2774.0 GHz 108.07 194.1 1.01e-03 (2.62e-03) 1.15e-03 (3.14e-03) 1.39e-03 (3.42e-03)2664.6 GHz 112.51 1340 5.73e-07 (1.46e-06) 7.81e-07 (1.58e-06) 1.03e-06 (1.58e-06)2640.5 GHz 113.54 323.5 7.37e-04 (1.76e-03) 8.96e-04 (2.12e-03) 1.11e-03 (2.29e-03)2462.9 GHz 121.72 550.4 1.30e-05 (2.68e-05) 1.89e-05 (4.43e-05) 2.29e-05 (5.58e-05)2344.3 GHz 127.88 1126 1.16e-06 (2.96e-06) 1.57e-06 (3.39e-06) 2.07e-06 (3.49e-06)2264.1 GHz 132.41 432.2 5.71e-05 (1.27e-04) 7.08e-05 (1.80e-04) 8.42e-05 (2.07e-04)2221.8 GHz 134.93 574.7 1.76e-05 (4.13e-05) 2.24e-05 (5.05e-05) 2.81e-05 (5.47e-05)2196.3 GHz 136.49 410.7 5.57e-05 (1.36e-04) 6.77e-05 (1.98e-04) 7.99e-05 (2.33e-04)1918.5 GHz 156.26 642.4 1.23e-06 (2.87e-06) 1.66e-06 (3.67e-06) 2.08e-06 (4.13e-06)1867.7 GHz 160.51 732.1 1.57e-06 (3.89e-06) 2.03e-06 (4.60e-06) 2.62e-06 (4.90e-06)1716.8 GHz 174.62 196.8 9.24e-04 (2.48e-03) 1.09e-03 (2.70e-03) 1.37e-03 (2.79e-03)1669.9 GHz 179.53 114.4 1.38e-03 (3.73e-03) 1.58e-03 (3.81e-03) 1.96e-03 (3.89e-03)1661.0 GHz 180.49 194.1 2.93e-04 (8.45e-04) 3.41e-04 (9.58e-04) 4.19e-04 (1.03e-03)1162.9 GHz 257.79 305.3 8.11e-05 (2.61e-04) 9.72e-05 (3.20e-04) 1.35e-04 (3.48e-04)1153.1 GHz 259.98 249.4 2.09e-04 (5.63e-04) 2.44e-04 (7.36e-04) 2.97e-04 (8.35e-04)1097.4 GHz 273.19 249.4 1.27e-04 (3.43e-04) 1.52e-04 (3.55e-04) 2.00e-04 (3.50e-04)556.94 GHz 538.28 61.0 7.98e-05 (2.06e-04) 8.45e-05 (2.00e-04) 9.89e-05 (2.04e-04)22. Anderl, V. Guillet, G. Pineau des Forˆets and D. R. Flower: Shocks in dense clouds Table B.8.
Intensities of ortho-H O lines (erg cm − s − sr − ) observable with the PACS (top) and HIFI (bottom) instruments on theHerschel Space Observatory, for shocks with velocity V s =
40 km s − and the magnetic field strengths listed in Table 1. Results aregiven for models M1, which include grain-grain processing, and M2 (in parentheses), which neglect grain-grain processing. Thepreshock density is n H = cm − . Transition λ ( µ m) E upup
40 km s − and the magnetic field strengths listed in Table 1. Results aregiven for models M1, which include grain-grain processing, and M2 (in parentheses), which neglect grain-grain processing. Thepreshock density is n H = cm − . Transition λ ( µ m) E upup (K) v40b2, M1 (v40b2, M2) v40b2.5, M1 (v40b2.5, M2) v40b3, M1 (v40b3, M2)5500.1 GHz 54.506 732.1 5.58e-06 (9.15e-06) 3.78e-06 (1.08e-05) 4.89e-06 (1.19e-05)5437.8 GHz 55.131 1274 1.47e-05 (3.87e-05) 1.53e-05 (4.56e-05) 2.14e-05 (4.94e-05)5276.5 GHz 56.816 1324 4.82e-05 (1.35e-04) 5.37e-05 (1.56e-04) 7.51e-05 (1.67e-04)5107.3 GHz 58.699 550.4 3.39e-04 (7.11e-04) 2.75e-04 (7.70e-04) 3.44e-04 (8.04e-04)4764.0 GHz 62.928 1553 7.07e-07 (1.96e-06) 7.73e-07 (2.35e-06) 1.09e-06 (2.55e-06)4734.3 GHz 63.323 1071 9.55e-05 (2.46e-04) 9.84e-05 (2.83e-04) 1.35e-04 (3.03e-04)4600.4 GHz 65.166 795.5 9.22e-05 (1.83e-04) 7.40e-05 (2.17e-04) 9.76e-05 (2.37e-04)4535.9 GHz 66.092 1013 2.84e-05 (6.51e-05) 2.59e-05 (7.73e-05) 3.51e-05 (8.41e-05)4512.4 GHz 66.437 410.7 5.35e-04 (8.84e-04) 3.80e-04 (9.42e-04) 4.63e-04 (9.85e-04)4456.6 GHz 67.268 410.7 1.27e-04 (8.96e-05) 4.90e-05 (1.41e-04) 5.51e-05 (1.74e-04)4240.2 GHz 70.702 1274 1.98e-06 (5.21e-06) 2.07e-06 (6.15e-06) 2.88e-06 (6.65e-06)4166.9 GHz 71.946 843.5 1.83e-04 (4.15e-04) 1.66e-04 (4.81e-04) 2.22e-04 (5.19e-04)4000.2 GHz 74.944 1126 6.26e-06 (1.62e-05) 6.44e-06 (1.89e-05) 8.87e-06 (2.03e-05)3977.0 GHz 75.380 305.3 1.64e-03 (2.05e-03) 9.85e-04 (2.40e-03) 1.10e-03 (2.62e-03)3971.0 GHz 75.495 1806 7.12e-07 (2.20e-06) 8.98e-07 (2.48e-06) 1.27e-06 (2.62e-06)3807.3 GHz 78.742 432.2 6.95e-04 (8.97e-04) 4.06e-04 (1.09e-03) 4.95e-04 (1.21e-03)3654.6 GHz 82.031 643.5 3.82e-04 (7.41e-04) 3.01e-04 (8.65e-04) 3.90e-04 (9.37e-04)3536.7 GHz 84.766 1013 5.03e-06 (1.15e-05) 4.58e-06 (1.37e-05) 6.22e-06 (1.49e-05)3167.6 GHz 94.643 795.5 1.18e-05 (2.34e-05) 9.46e-06 (2.78e-05) 1.25e-05 (3.02e-05)3165.5 GHz 94.704 702.3 5.90e-06 (8.94e-06) 3.70e-06 (1.07e-05) 4.67e-06 (1.18e-05)3013.2 GHz 99.492 468.1 8.28e-04 (1.36e-03) 5.70e-04 (1.59e-03) 7.08e-04 (1.73e-03)2970.8 GHz 100.91 574.7 1.47e-04 (2.03e-04) 8.82e-05 (2.48e-04) 1.10e-04 (2.76e-04)2774.0 GHz 108.07 194.1 2.17e-03 (4.21e-03) 1.56e-03 (4.71e-03) 1.79e-03 (5.06e-03)2664.6 GHz 112.51 1340 8.39e-07 (2.40e-06) 9.72e-07 (2.69e-06) 1.35e-06 (2.85e-06)2640.5 GHz 113.54 323.5 1.78e-03 (3.00e-03) 1.24e-03 (3.45e-03) 1.48e-03 (3.73e-03)2462.9 GHz 121.72 550.4 7.19e-05 (6.04e-05) 3.27e-05 (9.70e-05) 3.95e-05 (1.21e-04)2344.3 GHz 127.88 1126 1.94e-06 (5.02e-06) 1.99e-06 (5.84e-06) 2.74e-06 (6.28e-06)2264.1 GHz 132.41 432.2 1.94e-04 (2.46e-04) 1.06e-04 (3.13e-04) 1.23e-04 (3.53e-04)2221.8 GHz 134.93 574.7 5.05e-05 (7.31e-05) 3.16e-05 (8.86e-05) 3.94e-05 (9.80e-05)2196.3 GHz 136.49 410.7 1.60e-04 (2.56e-04) 9.58e-05 (3.20e-04) 1.10e-04 (3.57e-04)1918.5 GHz 156.26 642.4 5.16e-06 (5.30e-06) 2.55e-06 (7.21e-06) 3.22e-06 (8.64e-06)1867.7 GHz 160.51 732.1 4.07e-06 (6.64e-06) 2.75e-06 (7.87e-06) 3.55e-06 (8.66e-06)1716.8 GHz 174.62 196.8 1.62e-03 (3.74e-03) 1.39e-03 (3.94e-03) 1.69e-03 (4.10e-03)1669.9 GHz 179.53 114.4 2.15e-03 (5.35e-03) 1.98e-03 (5.35e-03) 2.38e-03 (5.47e-03)1661.0 GHz 180.49 194.1 5.39e-04 (1.28e-03) 4.43e-04 (1.37e-03) 5.20e-04 (1.45e-03)1162.9 GHz 257.79 305.3 1.37e-04 (3.89e-04) 1.20e-04 (4.39e-04) 1.52e-04 (4.74e-04)1153.1 GHz 259.98 249.4 5.07e-04 (9.47e-04) 3.38e-04 (1.12e-03) 3.89e-04 (1.24e-03)1097.4 GHz 273.19 249.4 2.20e-04 (5.04e-04) 1.97e-04 (5.22e-04) 2.42e-04 (5.29e-04)556.94 GHz 538.28 61.0 1.14e-04 (2.84e-04) 1.06e-04 (2.71e-04) 1.20e-04 (2.74e-04) 23. Anderl, V. Guillet, G. Pineau des Forˆets and D. R. Flower: Shocks in dense clouds Table B.9.
Intensities of para-H O lines (erg cm − s − sr − ) observable with the PACS (top) and HIFI (bottom) instruments on theHerschel Space Observatory, for shocks with velocity V s =
20 km s − and the magnetic field strengths listed in Table 1. Results aregiven for models M1, which include grain-grain processing, and M2 (in parentheses), which neglect grain-grain processing. Thepreshock density is n H = cm − . Transition λ ( µ m) E upup
20 km s − and the magnetic field strengths listed in Table 1. Results aregiven for models M1, which include grain-grain processing, and M2 (in parentheses), which neglect grain-grain processing. Thepreshock density is n H = cm − . Transition λ ( µ m) E upup (K) v20b1, M1 (v20b1, M2) v20b1.5, M1 (v20b1.5, M2) v20b2, M1 (v20b2, M2)5322.5 GHz 56.325 552.3 2.46e-05 (4.10e-05) 2.91e-05 (4.81e-05) 2.37e-05 (3.10e-05)5280.7 GHz 56.771 1324 4.69e-06 (7.15e-06) 5.49e-06 (5.91e-06) 4.63e-06 (3.96e-06)5201.4 GHz 57.636 454.3 6.05e-05 (1.10e-04) 7.08e-05 (1.04e-04) 6.41e-05 (7.16e-05)5194.9 GHz 57.709 1270 9.33e-07 (1.41e-06) 1.09e-06 (1.25e-06) 9.06e-07 (8.23e-07)4997.6 GHz 59.987 1021 1.76e-06 (2.71e-06) 2.06e-06 (2.59e-06) 1.71e-06 (1.69e-06)4850.3 GHz 61.808 552.3 3.85e-07 (6.35e-07) 4.60e-07 (7.56e-07) 3.68e-07 (4.84e-07)4724.0 GHz 63.457 1070 8.78e-06 (1.40e-05) 1.03e-05 (1.18e-05) 8.87e-06 (8.00e-06)4468.6 GHz 67.089 410.4 6.05e-05 (1.16e-04) 7.04e-05 (1.13e-04) 6.54e-05 (7.90e-05)4218.4 GHz 71.067 598.8 1.54e-05 (2.56e-05) 1.82e-05 (2.67e-05) 1.54e-05 (1.76e-05)4190.6 GHz 71.539 843.8 1.76e-05 (2.95e-05) 2.08e-05 (2.53e-05) 1.85e-05 (1.73e-05)3798.3 GHz 78.928 781.1 3.35e-06 (5.31e-06) 3.95e-06 (5.66e-06) 3.27e-06 (3.66e-06)3691.3 GHz 81.215 1021 2.36e-07 (3.63e-07) 2.77e-07 (3.48e-07) 2.29e-07 (2.27e-07)3599.6 GHz 83.283 642.7 3.16e-05 (5.37e-05) 3.74e-05 (5.04e-05) 3.29e-05 (3.40e-05)3331.5 GHz 89.988 296.8 5.73e-05 (9.59e-05) 6.85e-05 (1.03e-04) 5.74e-05 (6.87e-05)3182.2 GHz 94.209 877.8 2.05e-07 (3.32e-07) 2.44e-07 (3.57e-07) 2.05e-07 (2.32e-07)3135.0 GHz 95.626 469.9 5.68e-05 (9.94e-05) 6.73e-05 (1.01e-04) 5.86e-05 (6.75e-05)2968.7 GHz 100.98 195.9 1.86e-04 (3.60e-04) 2.14e-04 (4.13e-04) 1.68e-04 (2.66e-04)2884.3 GHz 103.94 781.1 7.63e-07 (1.21e-06) 9.01e-07 (1.29e-06) 7.45e-07 (8.34e-07)2685.6 GHz 111.63 598.8 1.77e-06 (2.94e-06) 2.09e-06 (3.08e-06) 1.77e-06 (2.02e-06)2631.0 GHz 113.95 725.1 1.35e-06 (2.40e-06) 1.60e-06 (1.98e-06) 1.47e-06 (1.40e-06)2391.6 GHz 125.35 319.5 1.12e-04 (2.04e-04) 1.33e-04 (2.18e-04) 1.14e-04 (1.46e-04)2365.9 GHz 126.71 410.4 3.25e-06 (5.59e-06) 4.66e-06 (8.41e-06) 3.33e-06 (4.95e-06)2164.1 GHz 138.53 204.7 2.83e-04 (5.69e-04) 3.36e-04 (6.40e-04) 2.82e-04 (4.21e-04)2074.4 GHz 144.52 396.4 1.10e-05 (1.84e-05) 1.31e-05 (2.17e-05) 1.04e-05 (1.39e-05)2040.5 GHz 146.92 552.3 5.41e-07 (8.92e-07) 6.47e-07 (1.06e-06) 5.18e-07 (6.80e-07)1919.4 GHz 156.19 296.8 2.97e-05 (5.47e-05) 3.59e-05 (7.92e-05) 2.50e-05 (4.86e-05)1794.8 GHz 167.03 867.3 1.81e-07 (3.03e-07) 2.16e-07 (3.08e-07) 1.87e-07 (2.04e-07)1602.2 GHz 187.11 396.4 9.43e-06 (1.59e-05) 1.12e-05 (1.86e-05) 8.99e-06 (1.20e-05)1228.8 GHz 243.97 195.9 3.48e-05 (9.50e-05) 4.45e-05 (9.36e-05) 4.62e-05 (6.63e-05)1113.3 GHz 269.27 53.4 2.64e-04 (6.40e-04) 2.97e-04 (5.82e-04) 2.89e-04 (4.25e-04)987.93 GHz 303.45 100.8 2.10e-04 (5.15e-04) 2.42e-04 (4.97e-04) 2.33e-04 (3.55e-04)752.03 GHz 398.64 136.9 6.59e-05 (1.75e-04) 7.72e-05 (1.71e-04) 7.54e-05 (1.21e-04)24. Anderl, V. Guillet, G. Pineau des Forˆets and D. R. Flower: Shocks in dense clouds Table B.10.
Intensities of para-H O lines (erg cm − s − sr − ) observable with the PACS (top) and HIFI (bottom) instruments on theHerschel Space Observatory, for shocks with velocity V s =
30 km s − and the magnetic field strengths listed in Table 1. Results aregiven for models M1, which include grain-grain processing, and M2 (in parentheses), which neglect grain-grain processing. Thepreshock density is n H = cm − . Transition λ ( µ m) E up (K) v30b1.5, M1 (v30b1.5, M2) v30b2, M1 (v30b2, M2) v30b2.5, M1 (v30b2.5, M2)5322.5 GHz 56.325 552.30 4.33e-05 (1.03e-04) 5.31e-05 (1.34e-04) 6.64e-05 (1.48e-04)5280.7 GHz 56.771 1324.0 7.51e-06 (1.93e-05) 1.03e-05 (2.21e-05) 1.38e-05 (2.26e-05)5201.4 GHz 57.636 454.30 9.73e-05 (2.39e-04) 1.19e-04 (2.65e-04) 1.50e-04 (2.74e-04)5194.9 GHz 57.709 1270.3 1.54e-06 (3.88e-06) 2.08e-06 (4.62e-06) 2.74e-06 (4.84e-06)4997.6 GHz 59.987 1021.0 2.97e-06 (7.39e-06) 3.94e-06 (9.09e-06) 5.15e-06 (9.70e-06)4850.3 GHz 61.808 552.30 7.05e-07 (1.62e-06) 8.95e-07 (2.20e-06) 1.11e-06 (2.51e-06)4724.0 GHz 63.457 1070.0 1.41e-05 (3.60e-05) 1.90e-05 (4.10e-05) 2.51e-05 (4.21e-05)4468.6 GHz 67.089 410.40 9.46e-05 (2.37e-04) 1.12e-04 (2.61e-04) 1.42e-04 (2.70e-04)4218.4 GHz 71.067 598.80 2.62e-05 (6.37e-05) 3.32e-05 (7.92e-05) 4.23e-05 (8.59e-05)4190.6 GHz 71.539 843.80 2.82e-05 (7.16e-05) 3.73e-05 (8.08e-05) 4.87e-05 (8.35e-05)3798.3 GHz 78.928 781.10 5.73e-06 (1.41e-05) 7.39e-06 (1.81e-05) 9.56e-06 (1.97e-05)3691.3 GHz 81.215 1021.0 3.98e-07 (9.91e-07) 5.29e-07 (1.22e-06) 6.91e-07 (1.30e-06)3599.6 GHz 83.283 642.70 5.17e-05 (1.29e-04) 6.68e-05 (1.51e-04) 8.63e-05 (1.60e-04)3331.5 GHz 89.988 296.80 1.09e-04 (2.28e-04) 1.33e-04 (2.99e-04) 1.60e-04 (3.30e-04)3182.2 GHz 94.209 877.80 3.47e-07 (8.74e-07) 4.48e-07 (1.12e-06) 5.89e-07 (1.22e-06)3135.0 GHz 95.626 469.90 9.47e-05 (2.32e-04) 1.19e-04 (2.78e-04) 1.51e-04 (2.99e-04)2968.7 GHz 100.98 195.90 3.21e-04 (7.66e-04) 3.69e-04 (8.96e-04) 4.19e-04 (9.80e-04)2884.3 GHz 103.94 781.10 1.31e-06 (3.23e-06) 1.69e-06 (4.12e-06) 2.18e-06 (4.49e-06)2685.6 GHz 111.63 598.80 3.02e-06 (7.33e-06) 3.83e-06 (9.14e-06) 4.87e-06 (9.91e-06)2631.0 GHz 113.95 725.10 2.11e-06 (5.34e-06) 2.74e-06 (5.74e-06) 3.55e-06 (5.85e-06)2391.6 GHz 125.35 319.50 1.92e-04 (4.57e-04) 2.34e-04 (5.55e-04) 2.89e-04 (6.03e-04)2365.9 GHz 126.71 410.40 7.21e-06 (1.60e-05) 1.03e-05 (2.66e-05) 1.25e-05 (3.32e-05)2164.1 GHz 138.53 204.70 4.71e-04 (1.19e-03) 5.54e-04 (1.42e-03) 6.74e-04 (1.54e-03)2074.4 GHz 144.52 396.40 2.05e-05 (4.55e-05) 2.57e-05 (6.10e-05) 3.10e-05 (6.94e-05)2040.5 GHz 146.92 552.30 9.90e-07 (2.28e-06) 1.26e-06 (3.10e-06) 1.56e-06 (3.54e-06)1919.4 GHz 156.19 296.80 5.46e-05 (1.33e-04) 6.41e-05 (1.80e-04) 7.51e-05 (2.08e-04)1794.8 GHz 167.03 867.30 3.01e-07 (7.63e-07) 3.89e-07 (9.46e-07) 5.08e-07 (1.02e-06)1602.2 GHz 187.11 396.40 1.72e-05 (3.89e-05) 2.11e-05 (5.08e-05) 2.56e-05 (5.67e-05)1228.8 GHz 243.97 195.90 4.90e-05 (1.61e-04) 5.99e-05 (1.80e-04) 8.46e-05 (1.84e-04)1113.3 GHz 269.27 53.400 3.72e-04 (1.03e-03) 4.20e-04 (1.01e-03) 5.13e-04 (1.02e-03)987.93 GHz 303.45 100.80 2.97e-04 (8.56e-04) 3.40e-04 (8.95e-04) 4.31e-04 (9.18e-04)752.03 GHz 398.64 136.90 9.10e-05 (2.81e-04) 1.03e-04 (2.98e-04) 1.35e-04 (3.04e-04) 25. Anderl, V. Guillet, G. Pineau des Forˆets and D. R. Flower: Shocks in dense clouds Table B.11.
Intensities of para-H O lines (erg cm − s − sr − ) observable with the PACS (top) and HIFI (bottom) instruments on theHerschel Space Observatory, for shocks with velocity V s =
40 km s − and the magnetic field strengths listed in Table 1. Results aregiven for models M1, which include grain-grain processing, and M2 (in parentheses), which neglect grain-grain processing. Thepreshock density is n H = cm − . Transition λ ( µ m) E upup