Incorporation of stochastic chemistry on dust grains in the PDR code using moment equations I. Application to the formation of H2 and HD
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Context. Unlike gas-phase reactions, chemical reactions taking place on interstellar dust grain surfaces cannot always be modeled by rate equations. Due to the small grain sizes and low flux, these reactions may exhibit large fluctuations and thus require stochastic methods such as the moment equations. Aims. We evaluate the formation rates of H2, HD and D2 molecules on dust grain surfaces and their abundances in the gas phase under interstellar conditions. Methods. We incorporate the moment equations into the Meudon PDR code and compare the results with those obtained from the rate equations. Results. We find that within the experimental constraints on the energy barriers for di usion and desorption and for the density of adsorption sites on the grain surface, H2, HD and D2 molecules can be formed e ciently on dust grains. Conclusions. Under a broad range of conditions, the moment equation results coincide with those obtained from the rate equations. However, in a range of relatively high grain temperatures, there are significant deviations. In this range, the rate equations fail while the moment equations provide accurate results. The incorporation of the moment equations into the PDR code can be extended to other reactions taking place on grain surfaces.Keywords:
Rate equation
Reaction rate
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Aims. We investigate the stability of nonisothermal Bonnor-Ebert spheres in the context of a model that includes a self-consistent calculation of the gas temperature. In this way, we can discard the assumption of equality between the dust and gas temperatures and study the stability as the gas temperature changes with the chemical evolution of the cooling species.Methods. We use a gas-grain chemical model to calculate the chemical evolution. The model includes a time-dependent treatment of depletion onto grain surfaces, which strongly influences the gas temperature as the main coolant molecule CO depletes from the gas. The dust and gas temperatures are solved with radiative transfer calculations. For consistent comparison with previous work, we assume that the cores are deeply embedded in a larger external structure, corresponding to visual extinction A V ext = 10 mag at the core edge. We also study the effect of lower values of A V ext .Results. We find that the critical nondimensional radius ξ 1 derived here, which determines the maximum density contrast between the core center and the outer boundary, is similar to our previous work where we assumed T dust = T gas . Here, the ξ 1 values lie below the isothermal critical value ξ 0 ~ 6.45, but the difference is less than 10 %. We find that chemical evolution does not notably affect the stability condition of low-mass cores ( ⊙ ), which have high average densities and a strong gas-grain thermal coupling. In contrast, for higher masses the decrease in cooling due to CO depletion causes substantial temporal changes in the temperature and in the density profiles of the cores. In the mass range 1−2 M ⊙ , ξ 1 decreases with chemical evolution, whereas above 3 M ⊙ , ξ 1 instead increases with chemical evolution. We also find that decreasing A V ext strongly increases the gas temperature, especially when the gas is chemically old, and this causes ξ 1 to increase with respect to models with higher A V ext . However, the derived ξ 1 values are still close to ξ 0 . The density contrast between the core center and edge derived here varies between 8 and 16 depending on core mass and the chemical age of the gas, compared to the constant value ~14.1 for the isothermal BES.
Isothermal process
Extinction (optical mineralogy)
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Dust and gas energetics are incorporated into a cluster-scale simulation of star formation in order to study the effect of heating and cooling on the star formation process. We build on our previous work by calculating separately the dust and gas temperatures. The dust temperature is set by radiative equilibrium between heating by embedded stars and radiation from dust. The gas temperature is determined using an energy-rate balance algorithm which includes molecular cooling, dust–gas collisional energy transfer, and cosmic-ray ionization. The fragmentation proceeds roughly similarly to simulations in which the gas temperature is set to the dust temperature, but there are differences. The structure of regions around sink particles has properties similar to those of Class 0 objects, but the infall speeds and mass accretion rates are, on average, higher than those seen for regions forming only low-mass stars. The gas and dust temperature have complex distributions not well modeled by approximations that ignore the detailed thermal physics. There is no simple relationship between density and kinetic temperature. In particular, high-density regions have a large range of temperatures, determined by their location relative to heating sources. The total luminosity underestimates the star formation rate at these early stages, before ionizing sources are included, by an order of magnitude. As predicted in our previous work, a larger number of intermediate-mass objects form when improved thermal physics is included, but the resulting initial mass function (IMF) still has too few low-mass stars. However, if we consider recent evidence on core-to-star efficiencies, the match to the IMF is improved.
Fragmentation
Energetics
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The rate equation approach to the chemistry occurring on grain surfaces in interstellar clouds has been criticized for not taking the discrete nature of grains into account. Indeed, investigations of simple models show that results obtained from rate equations can be significantly different from results obtained by a Monte Carlo procedure. Some modifications of the rate equations have been proposed that have the effect of eliminating most of the differences with the Monte Carlo procedure for simplified models of interstellar clouds at temperatures of 10 K and slightly above. In this study we investigate the use of the modified rate equations in more realistic chemical models of dark interstellar clouds with complex gas-grain interactions. Our results show some discrepancies between the results of models with unmodified and modified rate equations; these discrepancies are highly dependent, however, on the initial form of hydrogen chosen. If the initial form is mainly molecular, at early stages of cloud evolution there are some significant differences in calculated molecular abundances on grains, but at late times the two sets of results tend to converge for the main components of the grain mantles. If the initial form is atomic hydrogen, there are essentially no differences in results between models based on the unmodified rate equations and those based on the modified rate equations, except for the abundances on grains of some minor complex molecules. Thus, the major results of previous gas-grain models of cold, dark interstellar clouds remain at least partially intact.
Rate equation
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Observations of spiral galaxies show a strong linear correlation between the ratio of molecular to atomic hydrogen surface density R_mol and midplane pressure. To explain this, we simulate three-dimensional, magnetized turbulence, including simplified treatments of non-equilibrium chemistry and the propagation of dissociating radiation, to follow the formation of H_2 from cold atomic gas. The formation time scale for H_2 is sufficiently long that equilibrium is not reached within the 20-30 Myr lifetimes of molecular clouds. The equilibrium balance between radiative dissociation and H_2 formation on dust grains fails to predict the time-dependent molecular fractions we find. A simple, time-dependent model of H_2 formation can reproduce the gross behavior, although turbulent density perturbations increase molecular fractions by a factor of few above it. In contradiction to equilibrium models, radiative dissociation of molecules plays little role in our model for diffuse radiation fields with strengths less than ten times that of the solar neighborhood, because of the effective self-shielding of H_2. The observed correlation of R_mol with pressure corresponds to a correlation with local gas density if the effective temperature in the cold neutral medium of galactic disks is roughly constant. We indeed find such a correlation of R_mol with density. If we examine the value of R_mol in our local models after a free-fall time at their average density, as expected for models of molecular cloud formation by large-scale gravitational instability, our models reproduce the observed correlation over more than an order of magnitude range in density.
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We have designed an improved algorithm that enables us to simulate the chemistry of cold dense interstellar clouds with a full gas-grain reaction network.The chemistry is treated by a unified microscopic-macroscopic Monte Carlo approach that includes photon penetration and bulk diffusion.To determine the significance of these two processes, we simulate the chemistry with three different models.In Model 1, we use an exponential treatment to follow how photons penetrate and photodissociate ice species throughout the grain mantle.Moreover, the products of photodissociation are allowed to diffuse via bulk diffusion and react within the ice mantle.Model 2 is similar to Model 1 but with a slower bulk diffusion rate.A reference Model 0, which only allows photodissociation reactions to occur on the top two layers, is also simulated.Photodesorption is assumed to occur from the top two layers in all three models.We found that the abundances of major stable species in grain mantles do not differ much among these three models, and the results of our simulation for the abundances of these species agree well with observations.Likewise, the abundances of gas-phase species in the three models do not vary.However, the abundances of radicals in grain mantles can differ by up to two orders of magnitude depending upon the degree of photon penetration and the bulk diffusion of photodissociation products.We also found that complex molecules can be formed at temperatures as low as 10 K in all three models.
Interstellar ice
Kinetic Monte Carlo
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Condensation of H 2 in the interstellar medium (ISM) has long been seen as a possibility, either by deposition on dust grains or thanks to a phase transition combined with self-gravity. H 2 condensation might explain the observed low efficiency of star formation and might help to hide baryons in spiral galaxies. Our aim is to quantify the solid fraction of H 2 in the ISM due to a phase transition including self-gravity for different densities and temperatures in order to use the results in more complex simulations of the ISM as subgrid physics. We used molecular dynamics simulations of fluids at different temperatures and densities to study the formation of solids. Once the simulations reached a steady state, we calculated the solid mass fraction, energy increase, and timescales. By determining the power laws measured over several orders of magnitude, we extrapolated to lower densities the higher density fluids that can be simulated with current computers. The solid fraction and energy increase of fluids in a phase transition are above 0.1 and do not follow a power law. Fluids out of a phase transition are still forming a small amount of solids due to chance encounters of molecules. The solid mass fraction and energy increase of these fluids are linearly dependent on density and can easily be extrapolated. The timescale is below one second, the condensation can be considered instantaneous. The presence of solid H 2 grains has important dynamic implications on the ISM as they may be the building blocks for larger solid bodies when gravity is included. We provide the solid mass fraction, energy increase, and timescales for high density fluids and extrapolation laws for lower densities.
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Fraction (chemistry)
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This series of papers investigates the early stages of planet formation by modeling the evolution of the gas and solid content of protostellar disks from the early T Tauri phase until complete dispersal of the gas.In this first paper, I present a new set of simplified equations modeling the growth and migration of various species of grains in a gaseous protostellar disk evolving as a result of the combined effects of viscous accretion and photoevaporation from the central star.Using the assumption that the grain size distribution function always maintains a power-law structure approximating the average outcome of the exact coagulation/shattering equation, the model focuses on the calculation of the growth rate of the largest grains only.The coupled evolution equations for the maximum grain size, the surface density of the gas and the surface density of solids are then presented and solved self-consistently using a standard 1+1 dimensional formalism.I show that the global evolution of solids is controlled by a leaky reservoir of small grains at large radii, and propose an empirically derived evolution equation for the total mass of solids, which can be used to estimate the total heavy element retention efficiency in the planet formation paradigm.Consistency with observation of the total mass of solids in the Minimum Solar Nebula augmented with the mass of the Oort cloud sets strong upper limit on the initial grain size distribution, as well as on the turbulent parameter α t .Detailed comparisons with SED observations are presented in a following paper.
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The aim of this thesis is to predict the column densities of various neutral and ionised molecular species that are formed, or are likely to be formed, in the neutral envelope of a planetary nebula (PN). To this end a computer program has been constructed based on existing code (Abgrall et al. 1992) that considers a large set of chemical reactions covering the formation and destruction of the chemical species considered in the model. The rate coefficient of a chemical reaction will either depend on the local gas kinetic temperature if it is a gas phase reaction, or on the local radiation field spectrum if it is a photoreaction. To model the reaction network it is therefore also required to model the heating and cooling processes in the nebula to determine the kinetic temperature and also to solve the radiative transfer equation to determine the energy spectrum of ultraviolet radiation. Formation of the H2 molecule on the surface of dust grains and cosmic ray interactions are also considered. The ultraviolet absorption spectrum of the dominant molecules H2 and CO and the photodissociation rates are both functions of the rotational population. Rate coefficients for collisional cooling also depend on the rotational state. For these reasons, to model the thermal processes and the radiative transfer accurately it is also required to model the processes contributing to rotational excitation and de-excitation of H2 and CO to determine the distribution amongst their various rotational levels. Dust grains play a significant role in much of the physics occurring in the nebula, not least because they represent the catalyst for the formation of molecular hydrogen. Dust also represents the most important source of opacity for the ultraviolet radiation field and hence a significant part of the thesis is devoted to a consideration of the probable dust composition and optical properties. The results of the model are shown and a comparison is made between the predictions of the model and recent computations of molecular column densities based on astronomical observations of planetary nebulae. The probable sources of large discrepancies are discussed within the context of assumptions and possible omissions in the physical model.
Rotational energy
Mean kinetic temperature
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Chemistry plays an important role in the interstellar medium (ISM), regulating heating and cooling of the gas, and determining abundances of molecular species that trace gas properties in observations. Although solving the time-dependent equations is necessary for accurate abundances and temperature in the dynamic ISM, a full chemical network is too computationally expensive to incorporate in numerical simulations. In this paper, we propose a new simplified chemical network for hydrogen and carbon chemistry in the atomic and molecular ISM. We compare results from our chemical network in detail with results from a full photo-dissociation region (PDR) code, and also with the Nelson & Langer (1999) (NL99) network previously adopted in the simulation literature. We show that our chemical network gives similar results to the PDR code in the equilibrium abundances of all species over a wide range of densities, temperature, and metallicities, whereas the NL99 network shows significant disagreement. Applying our network in 1D models, we find that the $\mathrm{CO}$-dominated regime delimits the coldest gas and that the corresponding temperature tracks the cosmic ray ionization rate in molecular clouds. We provide a simple fit for the locus of $\mathrm{CO}$ dominated regions as a function of gas density and column. We also compare with observations of diffuse and translucent clouds. We find that the $\mathrm{CO}$, $\mathrm{CHx}$ and $\mathrm{OHx}$ abundances are consistent with equilibrium predictions for densities $n=100-1000~\mathrm{cm^{-3}}$, but the predicted equilibrium $\mathrm{C}$ abundance is higher than observations, signaling the potential importance of non-equilibrium/dynamical effects.
Atomic carbon
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In this thesis, I have tackled two seemingly unrelated problems in the modeling of the neutral interstellar medium (ISM).
The first is the description of H2 formation on interstellar dust grains under realistic conditions. The precise determination of the H2 formation rate and abundance is crucial, as it controls most of the subsequent development of the chemical complexity in the ISM, as well as part of its physics. The temperature of small grains (less than 10 nm) fluctuates constantly as those grains are sensitive to the energy of individual UV photons, and the surface mechanisms of H2 formation, which are sensitive to the grain temperature, are kept out of equilibrium by the fluctuations. I have developed an exact resolution formalism for the statistical equilibrium of this system, and implemented its numerical resolution. Among other results, taking the fluctuations into account leads to large differences for the Langmuir-Hinshelwood mechanism, whose efficiency is increased in atomic gas and decreased inside molecular gas.
The second problem is related to the ubiquitous presence of molecules such as CH+, whose formation is highly endothermic, in the diffuse ISM where the observed gas temperature (less than 100 K) is insufficient to trigger their formation. It has been proposed that the intermittent dissipation of turbulence could inject the necessary energy, creating hot spots, which could also explain the observed rotational excitation of H2 in such regions. At small scales, the gas is thus perturbed by strong fluctuations of the energy injection rate. I propose a model for the Lagrangian evolution of the local physico-chemical state of the gas based on stochastic processes, and apply it to derive the distribution of the gas temperature in the diffuse atomic medium, and the average excitation of H2 in the diffuse molecular gas.
Both problems are thus similar and can be described in a more abstract way as systems whose state is perturbed by strong fluctuations of their environment. In order to derive the statistical equilibrium of the system in our two cases, similar methods are used based on the framework of (markovian) stochastic processes. The results obtained here demonstrate the usefulness of this approach, and possible developments for other applications are discussed.
Finally, I also present the modeling of the PDR NGC7023 Northwest with the Meudon PDR Code in comparison to Hershel observations, showing the excellent capacity of the Meudon PDR Code to reproduce the observables in dense and intense PDRs.
Endothermic process
Formalism (music)
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