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H2 FORMATION IN THE ISM Ben Waghorn, APRIL 2003 H2 FORMATION IN THE INTERTELLAR MEDIUM BEN WAGHORN Department of Physics and Astronomy, Johns Hopkins University [email protected] ABSTRACT The formation of molecular hydrogen in the ISM is not well understood. A widely accepted method for estimating the formation rate of H2 was through the modeling of the recombination of hydrogen atoms on dust surfaces. Hollebach et al. (1971) were the first to use this model by calculating the sticking and mobility of H atoms on the surface of grains, yielding the steady state rate equation for H2 formation. It was later discovered that hydrogen recombination takes place on small dust grains, limiting the validity of these earlier rate equations. Ofer at al. (2001) proposed at master equation for hydrogen recombination on grain surfaces taking into account both the discrete nature of the H atom and the fluctuations in the number of atoms on the grain. The equation can then calculate the recombination rate of H as a function of the grain size and the temperature. During these developments in theories to understand the formation of H2 other methods for the formation were identified and examined, in particular the negative ion route (Field 2000). These gas phase reaction were further investigated by Glover (2003) under the relevant conditions for diffuse molecular clouds in the ISM and were found to be dominated by H2 formation on dust grain surfaces. In this review article I will be summarizing the above mentioned theories for molecular hydrogen’s formation rate in the ISM and will be comparing the results to observational data, primarily obtained using FUSE obsrervations. Earlier attempts to answer the question of the formation rate were made by considering dust grains that consisted of perfectly regular grain surfaces, made of a dirty ice material (Gould and Salpeter 1963). Their method contained an error by not including the zero-point energy of the absorbed energy, the results of which found that the recombination would be efficient for temperatures below 10˚20˚K. This error was corrected finding temperatures below 5˚-8˚K but this method overestimated the zero-point effect. A more accurate rate equation was calculated using accurate wavefunctions and binding energies for the adsorbed states of a hydrogen atom on a perfect surface. Hollenbach and 1. INTRODUCTION Molecular hydrogen is the most abundant molecule in the universe and is the dominant mass component of gas in regions of star formation. It also forms important intermediate stages in the formation of larger molecules and acts as a coolant in certain conditions, especially in the early universe. However the process of H2 formation is still not completely understood. It is widely believed that the formation process for molecular hydrogen in the ISM is by the recombination of pairs of absorbed hydrogen atoms on the surfaces of dust grains. 1 H2 FORMATION IN THE ISM Ben Waghorn, APRIL 2003 Salpeter (1971) addressed a complication to this method namely that physical-adsorption cannot hold the hydrogen atoms on the grains for long enough for them to combine at the relevant temperatures. The binding energies (divided by k) to the edges of graphite flakes are of the order of 50,000˚K compared to the physical adsorption energies to the order of 200˚500˚K. However, the activation energy for formation of molecules is too large to be formed in the ISM. This problem was avoided by Hollenbach and Salpeter (1971) by considering defect sites on the surfaces of realistic interstellar dust grains, enhancing the binding energies giving the reactive atoms greater binding energies than the molecules. The results, as are shown later in this paper (§2), show that H2 recombination is efficient over a wide range of temperatures. the grain leads to the calculation of recombination rates. The master equation uses experimentally and computationally obtained surface parameters to explore the recombination and other chemical reactions on small grains, for example reactions involving oxygen and hydrogen on the grain surface. Using far-UV absorption spectra obtained with FUSE H2 formation rates can be calculated and compared to the theoretical values (Gry et al. 2002) (§5). This data is taken for three late B stars that were shown not to interact with the interstellar matter responsible for the absorption and that do not contribute to the incident radiation field. By taking previously observed column densities along the relevant lines of sight estimates of the H2 formation rates were found. This however does not complete the picture for the formation of molecular hydrogen in the ISM. These rate equations describe the diffusion, reaction and desorption processes on the grain surface giving a solution for the time evolution of the average densities of atoms and molecules on the surface, neglecting fluctuations. These results are probably accurate for macroscopic surfaces but their validity was later questioned for this study due to the small grain size and low flux within the ISM. The numbers of H atoms on the grain surfaces are expected to be small and fluctuations become significant. Biham and Furman (2001) and Biham and Lipshtat (2002) take the master equation approach for the formation of H2 (§3). This equation takes into account both the discrete nature of the H atoms as well as the fluctuations. Here the time evolution of the probability of n atoms occupying A final formation mechanism that is addressed in this paper is that of the relevance of gas-phase H2 formation (§4). Most of the molecular hydrogen that forms in the gas phase does so via the formation of an intermediate H- ion, as shown later in this paper. Gas-phase formation is the dominant mechanism for a number of cases, dominating for a dust-to-gas ratio less that a critical value. The relative effects of this process for the conditions within the ISM are addressed in this paper. 2. THE RATE EQUATION MODEL The rate of molecular hydrogen formation on dust grains can be written, 1 RG v H n g n H g , (1) 2 2 H2 FORMATION IN THE ISM Ben Waghorn, APRIL 2003 2.2 Surface Mobility where vH is the average thermal velocity of hydrogen atoms in the neutral hydrogen gas, g is the mean Now that we know the likelihood of an atom becoming adsorbed by the grain we need to look at how the H atom can move about the surface. By quantum mechanical methods the diffusion time, tD, required for an atom to move from one site on the grain surface to a neighboring site is finite, even at zero temperature. For a perfectly regular surface, the adsorbed-state wavefunctions are nonlocalized and form an energy band. If the difference between the adsorption binding energy, D, and H binding energy, D′ is much greater than this energy band the wavefunction is fully localized. Interstellar dust grains must therefore have surface irregularities, due to the number of dislocations from UV radiation, soft X-rays and low-energy cosmic rays. Such irregularities will increase the binding, showing that for the enhanced sites the binding energy for H is larger than the binding energy for H2. geometric cross-section of a dust grain, n H and n g are the number densities of hydrogen atoms and dust grains, and γ is the recombination coefficient, defined as that fraction of atoms, striking a grain, which eventually forms a molecule on that grain. 2.1 Sticking Coefficient Firstly, in calculating the recombination efficiency, γ, for gas atoms on grain surfaces (Hollenbach and Salpeter 1971), the number of incident hydrogen atoms that become adsorbed by the grain, namely the sticking coefficient S, needs to be calculated. In general, S is a function of the grain temperature T, the gas temperature Tgas, and the adsorption binding energy D. In practice the gas temperature is greater than the dust temperature, both of which are much smaller than D. Let N be the number of regular sites, N′ the number of enhanced “irregular” sites, and N′s the number of scattering sites on the grain. Most of the atoms that are adsorbed onto the grain are done so onto a patch of the regular, smooth, surface. It takes a transient time ttr for the hydrogen atom to find and become bound to an enhanced site, covering a patch of about (N/N′) sites. The transient time was found to be, 1/ 2 NN s N 1 t tr tD 0 , N N where ν0 is the characteristic latticevibrational frequency of the solid. It is shown that if these circumstances hold, S is independent of T, with S being approximated by, 2 0.8 3 S (Tgas ) , 1 2.4 2 0.8 3 E where c , with Ec being the kTgas characteristic total energy being transferred to the surface. For given temperatures that are comparable to that of regions of atomic hydrogen, Г is of order unity making the sticking coefficient of order 0.3. These values are obtained neglecting quantum effects but for a first order approximation yield reliable results. For a hydrogen atom on the regular surface (having a limited probability of 3 H2 FORMATION IN THE ISM Ben Waghorn, APRIL 2003 recombination) the evaporation time is roughly; 1 t ev 0 exp D / kT small fraction of time on regular sites but the probability of evaporation is much greater during this small fraction of time. The evaporation time is found to be sufficiently large so that evaporation occurs even on regular sites. With the assumption that the enhanced sites help in the recombination, the necessary condition is that the atoms reach an enhanced site before they evaporate, namely t tr t ev . Below this temperature Hollenbach and Salpeter found that for high recombination efficiency specific temperatures and binding energies must be achieved so that atoms will have much longer evaporation times that the time ts between successive striking events. Also the atom must be either able to reach equilibrium with an enhanced site or form a molecule with another atom in a time less that the evaporation time from the regular surface. 2.3 Specific Grain Examples Examining this situation for the unrealistic case of a perfect surface, results were found as follows. For a region with neutral hydrogen densities, n 1 10 3 cm-3 the mean time for a new atom to stick is t s 10 4 10 sec. The transient time for N 10 6 is 6 given t tr 10 sec. This shows that recombination will take place almost immediately if a second atoms sticks to the grains whilst the first atom is adsorbed. However, on a perfect lattice with the grain temperature T greater that a critical temperature Tc Tc 13K the adsorbed atom will evaporate before the second atom can stick to form a molecule. For T between 6.5˚ and 11˚K this effect will occur. Following all of the above discussions for molecular hydrogen formation the conclusion can be made that recombination is almost 100% efficient for H atoms, once they are thermalized on the grain surface. This assumes that grain temperatures are below about 25˚K. 3. MASTER EQUATION The rate equation described in §2 forms a good background for this topic but has a number of flaws. It had been noted (Charnley, Tielens & Rodgers 1997) that hydrogen recombination takes place on small grains so the rate equation only has limited validity. This is due to the fact that these equations only take into account average concentrations of H atoms on the surface, ignoring fluctuations and the discrete nature of the H atoms. For the case of very small grains and low flux, as in diffuse clouds in the ISM, this property becomes Now taking the more realistic case containing enhanced site, there are a number of possible scenarios. Two such cases that can be considered are firstly the high temperature case where the occupation of N regular surface sites can be neglected, and secondly the low temperature case where some irregularities are required for high recombination efficiencies. For the high temperature case Tgr 11K the atom spends only a 4 H2 FORMATION IN THE ISM Ben Waghorn, APRIL 2003 significant. The numbers of H atoms striking the surface of the grain could be as few as 0, 1 or 2. Clearly recombination cannot occur without at least two H atoms simultaneously occupying the surface so any previous averages no longer suffice. This is where the master equation for hydrogen recombination on grain surfaces (Biham et al. 2001) becomes important. This covers the two main problems of the rate equation, namely the discrete nature of the H atoms and also fluctuations. The master equation uses probabilities that there are a particular number of H atoms, N H 0,1,2,..... occupying the grain at a time t, PH(NH). The master equation provides the time evolution of PH(NH) from which the recombination rate can be calculated. This can be used alongside surface parameters that have been experimentally determined to investigate the recombination process on microscopic grains, taking into account grain size, flux and surface temperatures relevant to the ISM. where n H 2 (t ) is the coverage of H2 molecules at time t. The flux term represents the flux of incoming atoms multiplied by the LangmuirHinshelwood rejection term, where H atoms deposited on top of H atoms or H2 molecules are rejected. This term is equivalent to the above sticking coefficient. A second rate equation shows the change of the number of H2 molecules on the surface as a function of μ, the fraction of H2 molecules remaining on the surface upon formation, and WH 2 , the H2 desorption coefficient, dn H 2 a H n H2 WH 2 n H 2 dt , The H2 production rate, rH 2 , is thus given by, rH 2 (1 )aH nH2 WH 2 nH 2 . 3.2 Master Equation for H2 formation on small grains 3.1 Rate Equations for macroscopic surfaces I will now show the master equation (Biham et al 2001) which is required for conditions in the ISM that are failed by the rate equations. Due to the small flux of H atoms it is now more convenient to use the total number of H atoms and H2 molecules on the grain, NH and N H 2 , rather that the amount per unit area previously. The expectation value for the number of H atoms on the grain, N H is In order to follow the theory behind the master equation it is first necessary to restate the rate equation in terms of effective flux of atoms, f H , the desorption coefficient, WH , and depletion of H atoms due to recombinations into H2 molecules. Therefore the rate of change of H atom coverage, nH (t ) , on the surface is given by, given as Sn H , where S is the number of adsorption sites on the grain, and similarly for H2. The incoming flux is now given as FH Sf H and a H is replaced by AH aH / S , which is approximately the inverse of the time dnH f H (1 nH nH 2 ) WH nH 2a H nH2 dt 5 H2 FORMATION IN THE ISM Ben Waghorn, APRIL 2003 required for an atom to visit nearly all the adsorption sites on the surface. The H2 production rate is given by RH 2 SrH 2 . Thus the new rate equations can be written as, d NH dt d N H2 FH WH N H 2 AH N H PH(NH). The final term describes the recombination effect on the number of adsorbed H atoms. The rate of formation of H2 on the surface (in units of molecules s-1) is thus given by, 2 AH FH 2 AH N H WH 2 N H 2 dt where the term, FH 2 , takes into account the flux of H2 formed from the gas phase as shown later in this paper (§4). These equations hold, as in the previous case, when the grain size is large but when the number of atoms on the grain is reduced another method needs to be taken. 2 N N H N H 2 H 1PH N H The first two equations are ignored as neither zero nor one H atoms on the surface of the grain are sufficient to recombine to form hydrogen. In order to complete the analysis and gain an expression for the recombination rate we now need to find the probability that there are N H 2 molecules on the This new method, the master equation, uses the probability, PH(NH), that there are NH hydrogen atoms on the grain. Clearly, summing the probability over all possible values of occupation yields unity. The time derivatives of these probabilities, PH N H , are calculate by Biham et al with the generic result, grain, PH 2 N H 2 . The time evolution of these probabilities is given by, [N 1P N 1 P N ] P N 1 P N PH 2 N H 2 FH 2 PH 2 N H 2 1 PH 2 N H 2 WH 2 N H2 + RH 2 H2 H2 H2 H2 H2 PH N H FH PH N H 1 PH N H WH N H 1PH N H 1 N H PH N H AH [N H 2N H 1PH N H 2 N H N H 1PH N H ] where FH 2 is the flux of H2 molecules Each of the above equations for N H 0,1,2,... includes three terms. The first term describes how the probabilities vary with incoming flux. The probability of NH occupants increases when an H atom is adsorbed onto a grain already containing NH-1 adsorbed atoms, and decreases when is already contains NH atoms. The next term shows that an atom desorbed from a grain with NH adsorbed atoms decreases the probability The final step in calculating the recombination rate is to write the expectation values for NH and N H 2 in terms of the respective probabilities, and then to obtain the new rate equations. H2 H2 H2 H2 that stick on the grain and WH 2 is the desorption rate. Again μ is the fraction of molecules that remain on the surface after formation. 6 H2 FORMATION IN THE ISM Ben Waghorn, APRIL 2003 Clearly the expectation value for the number of H atoms on the grain is given by, NH N N H 0 H dt PH N H dt 4. COMPARING GAS-PHASE AND GRAIN CATALYZED H2 FORMATION. Thus far we have only considered the idea of dust grain catalysis for the formation of molecular hydrogen in the ISM. This isn’t however the only possible scenario so this section will be taken to examine another possible route for production, namely via the gas-phase (Field 2000). Firstly I will introduce this idea and will then look at the conditions for which this process is relevant. FH W H N H 2 AH N H N H 1 d N H2 FH 2 AH N H N H 1 FH / 2 for the case of the master equation. and likewise for the expectation value of number of molecules. We can know rewrite the rate equations using the time evolution probabilities for H atoms and H2 molecules giving, d NH RH 2 2 WH 2 N H 2 The net rate at which H2 molecules desorb into the gas-phase is given by the recombination rate, R H 2 (molecules s-1), H2 can be formed in the gas phase, primarily through the reactions, RH 2 1 AH N H N H 1 WH w N H 2 FH 2 H e H H H H2 e It is seen from the rate equations listed above for the master equation that the 2 previous N H has been replaced (1) (2) with some reaction occurring via; H H H 2 by N H2 N H . This is the key factor H 2 H H 2 H distinguishing the two approaches. For macroscopically large grains the two factors are comparable so either method would yield accurate results. However, for the more likely case of a small grain where N H will be small the two factors will be considerably different and the improved master equation is required to obtain accurate recombination results. although this second formation process is slower. Most of the dust formed in the gas phase does so by the first set of reactions, with the first of the two reactions (1) occurring more slowly, thus achieving a small equilibrium abundance of H-. Due to this equilibrium there are two factors that are important to the formation rate of H2. Firstly we have the rate at which the H- forms and secondly the fraction of H- ions that survive to form H2. The recombination efficiency, that is the fraction of adsorbed H atoms that desorb in the form of H2 molecules, η, is given by, 7 H2 FORMATION IN THE ISM Ben Waghorn, APRIL 2003 The rate at which H- forms H2 depends predominantly on the rate at which H- is destroyed by mutual neutralization with H+ ions, H H 2 H (3) or by photodetachment by the incident radiation field, H H e (4) relative to the H2 formation rate. Below is an example calculation (for more details and more examples see Glover 2003) of the H2 formation rate via the Hstate, RH , H . If the formation of H2 via mentioned H- route and also the rate of formation due to the H 2 reaction, RH 2 , gas RH , H RH , H 2 2 2 , and also for the grain-catalyzed rate, written in a different form from §2 but containing the same information, D RH 2 ,dust k dust ntot n H DMW Here k dust is the temperature dependant formation rate, ntot the total particle number density, D the dust-to-gas ratio, and DMW the D value in the Milky Way. 2 reaction (2) occurs much faster that an destruction of H-, the formation rate is given by, RH , H k1 ne n H This equation yields a critical dust-togas ratio, Dcr, for which the two formation rate are equal. Dcr is the ratio required for grain catalyzed formation to overtake the gas-phase formation, RH 2 , gas Dcr DMW k dust ntot n H . 2 where k1 is the reaction rate of reaction (1). This shows that the H2 formation rate is approximately equal to the Hformation rate, as expected. If on the other hand H- destruction via mutual neutralization (3) dominates over H2 formation or photodetachment (4), then the reaction rate equation becomes, kk RH , H 1 2 nH2 2 k3 Further analysis of this function (Glover 2003) shows a strong temperature dependence to Dcr. At low temperatures the formation of H2 by dust grain catalysis is relatively efficient and D needs only be small (little dust) before catalysis dominates. This situation changes above a few hundred ˚K as the efficiency of grain catalysis drops significantly and gas-phase formation of H2 continues to grow. The ISM contains diffuse molecular clouds with temperature around 80˚K and low fractional ionizations x n H / n H of order 10-7. In these conditions grain catalyzed formation dominates by many orders of magnitude. assuming ne n H , where k i is the rate, cm 3 s 1 , of reaction i , expressed in terms of temperature T. These limiting cases are specific examples of the general H2 negative ion formation rate give by, k 2 nH RH , H k1 ne n H 2 k 2 nH k3 nH k 4 For the sake of comparison of gas-phase formation against dust grain catalysis it is useful to write the rate equations in the following form, firstly for the gas-phase formation including the above This summarized analysis of the gasphase formation method yields the conclusion that for the case of the ISM the conditions are such to greatly favor 8 H2 FORMATION IN THE ISM Ben Waghorn, APRIL 2003 as nH , nH 2 and n nH 2nH 2 . Without shielding the photo-dissociation rate in the Solar Neighborhood is 0 , G the radiation field value and S the shielding factor including H2 selfshielding and dust extinction. H2 formation by the method of dust grain catalysis. The master equation in §3 takes into account the gas-phase formation of H2 via the term FH 2 . 5. H2 FORMATION OBSERVATIONS RATE Now nR can be calculated by integrating the above equation over the line of sight, for constant n , 1 f nR 0 S . 2 1 f Here f is the molecular hydrogen fraction: f 2 N H 2 / N total where N total is the total column density and can be derived from the extinction and S the mean shielding factor. Now that I have looked at a number of theoretical predictions for the formation rates of H2 in the ISM it is useful to use observational data, from which formation rates can be calculated, as a comparison. Gry et al. (2002) addressed this issue by using far-UV absorption spectra obtained with FUSE looking at three late type B stars. H2 formation and excitation had previously been studied by observing mid-infrared transitions between rotational levels of the vibrational ground state of H2 but only for warm photo-dissociation regions. Gry et al. readdress the question of H2 formation, now within the diffuse ISM, by analyzing FUSE observations of three stars. IRAS maps show that these stars do not heat the matter responsible for absorption. In practice a model in needed to determine the abundance and distribution of H2 molecules over it rovibrational levels as a function of depth into the cloud. With the model f can be calculated, and therefore values of nR found for each star. Now all that is needed to get an estimate of the formation rate R is an estimate of the gas density n . Seeing as the assumption for this procedure is that clouds are in thermal balance, the fact that gas density and temperature are uniquely related can be used. Tracing the temperature observationally therefore also traces the density. A temperature diagnostic used in this case is to obtain the column density ratio of the first two levels, N J 1 / N J 0 , from which the gas density can be determined. The first method in the process to calculate, observationally, H2 formation rates is to determine the product nR . R is the formation rate that is ultimately being searched for and n is the mean effective H2 density along each line of sight. To effectively calculate the rate of formation of H2 we need to assume an equilibrium environment where the H2 formation balances its photo-dissociation. Such a situation can be expressed as, Gry et al. (2002) evaluate this data with their results displayed below in table 1. The values of R for the three lines of sight are close to each other and with significantly lower uncertainties than the nH nR nH 2 0 GS The atomic, molecular and total hydrogen densities are given 9 H2 FORMATION IN THE ISM Ben Waghorn, APRIL 2003 similar previous values found by Jura (1975). These values can be converted directly into an H2 formation timescale of 1 / nR 2 x10 7 yrs . This model tells us that the photodissociation timescale is larger that the H2 formation timescale, especially in the shielded layers of the absorbing cloud. Table 1. H2 formation rate R from the product nR and the density n estimated from N H 2 , J 1 / N H 2 , J 0 . Gry et al. 2002. ncm Rcm s nR s 1 3 3 1 HD 102065 2.3 x10 15 50 HD 108927 0.87 x10 15 28 HD 96675 2.0 x10 15 50 4.5 x10 17 3.1x10 17 4.0 x10 17 requirements on the regular surface of a dust grain and on the surface of irregularities to produce the observable high recombination efficiency. 6. CONCLUSIONS In this paper I have addressed the question of the formation rate of molecular hydrogen in the ISM. The two primary processes, namely dust-grain catalysis and gas-phase production, were both examined and compared for the relevant diffuse cloud conditions. It was found that the gas-phase formation process via the negative ion route, and to a lesser extent via H 2 , was important only at temperature greater than expected in the ISM. Results from gasphase formation analysis show that, in practice, it is slower than dust-grain catalysis, either because of a shortage of free electrons and protons, or because the incident radiation field destroys the ions before they have a chance to form H2. This method however fails to account for the microscopic size of grains and small incident flux of H atoms striking the grain. The master equation, summarized in §3, introduces a new approach for the simulation of hydrogen recombination on microscopic dust grains in the ISM. This approach uses probabilities of N H hydrogen atoms occupying a grain surface, thus accounting for fluctuations in numbers. The master equation can be extended to describe more complex situation involving chemical reactions with multiple species, for example oxygen. Calculations for finding the formation rate of OH using the master equation are shown in Biham et al (2001). More details into solving the master equation were emitted from this paper but have been reviewed (Biham and Lipshtat 200) in detail giving expressions for Therefore, the dominant effect for H2 formation is via dust-grain catalysis. This paper summarized two such models, initially showing the important effects in calculating the recombination rates. Such earlier methods discuss the 10 H2 FORMATION IN THE ISM Ben Waghorn, APRIL 2003 probability function and recombination efficiencies in terms of flux and hopping rate. uncertainties of conditions within the ISM, such as dust grain sizes and shapes, there is still work to be done as uncertainties are contained in the present models. The master equation successfully models H2 formation in the ISM but due to REFERENCES Biham, O., and Furman, I. 2001, Ap.J., 553, 595. Biham, O., and Lipshtat, A. 2002, Physical Review E, 66. Cazaux, S., and Tielens, G.G.M. 2002, Ap.J. (Letters), 575, L29. 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