Hydrodynamic simulation of the formation of protostars during molecular cloud collapse and the chemical evolution in these processes

Thesis submitted for the degree of Doctor of Philosophy (Science) of the University of Calcutta

Ankan Das Indian Centre for Space Physics 43 Chalantika, Garia Station Road Kolkata 700084, India i ABSTRACT

In this thesis, the hydrodynamics of a proto-star formation and the chemical evolution of the star forming region are explored. This is an important branch of astrophysics, where one studies how the interstellar molecules are produced during the molecular cloud collapse. An Interstellar Medium (ISM) is the space between the stars within a galaxy. In the Introductory Chapter, we discuss the composition of the ISM in detail. There are increasing observational evidences of complex interstellar molecules, a theoretical understanding of which is essential. Armed with the observational results and with theoretical models, scientists are now building experimental set up to explore the Universe more accurately especially to look for complex pre-biotic molecules in space. This could ultimately resolve the age-old puzzle of the Origin of Life on earth. In Chapter 1, a brief introduction on these aspects and the current knowledge of the outcome of the observational, theoretical and experimental studies are presented. In Chapter 2, the general processes of a proto-star formation are discussed with an emphasize to the criteria of collapse. Theoretical approaches used in studying the formation of proto-stars are also presented. A brief discussion about the self- similar collapse, following the technique used by Shu (1977) has been done. To mimic a proto-star formation scenario, a time dependent, spherically symmetric, hydrodynamic collapse of a self-gravitating flow has been studied. Two models are used to study the hydrodynamic evolution: in the first model, matters are injected into an initially low density region and in the second model, initially a constant density cloud is taken and it is allowed to collapse due to self-gravity. Evolution of the central core is studied for both the cases. In the next attempt, the angular motion and shock formation in the proto-star formation process are included to study the dynamic behaviour of the self-gravitating cloud.

H2 is the most abundant molecular species in the Universe. Therefore, to explore the chemical universe, an accurate knowledge of the recombination efficiency of H2 is necessary. It is now well accepted that the H2 molecules are mainly formed on the grains. Normally, the rate equation approach is used to study the H2 formation, ii because this is very economic as far as the computational time is concerned. But it is found that a correction is needed in the Rate equation to calculate the recombination efficiency properly. In Chapter 3, a discussion about the formation of H2 molecule in the interstellar medium is carried out. A Monte-Carlo simulation is performed to study the recombination of hydrogen on grain surfaces in a variety of cloud conditions. A comparison of our results, after modifying the Rate equation has been made and it is noted that the result is consistent with that obtained from our studies. In Chapter 4, the formation of simple and complex molecules in the gas phase has been discussed. A large sized gas phase chemical network is used. To obtain the chemical evolution of a collapsing cloud, the chemical model is coupled with the previously discussed hydrodynamic models. A comparison of our results with those obtained from observation has been made and it is noted that for the lighter molecules the agreement is generally very good. For complex molecules our results tend to under-predict the abundances. In Chapter 5, a Monte Carlo study corresponding to the formation of a complex molecule, Methanol, and a very abundant species, namely, water, are discussed. This studies are to find the dependence of their production rates on the binding energies, reaction mechanisms, temperatures, and the grain site number. Effective grain surface area available for the chemical reaction is calculated accurately and the effective recombination timescales are shown as functions of grain and gas parame- ters. It is noted that the formation rate of various molecules is strongly dependent on the binding energies. When the binding energies are high, it is very difficult to produce significant amounts of the molecular species. Instead, the grain is found to be full of atomic species. The production rates are found to depend on the num- ber density in the gas phase. When the density is high, the production of various molecules on the grains is small as grain sites are quickly filled up by atomic species. If both the Eley-Rideal and Langmuir-Hinselwood mechanisms are considered, then the production rates are maximum and the grains are filled up relatively faster. Thus, if allowed, the Eley-Rideal mechanism can also play a major role and more so, when the grain is full of immobile species. It is shown that the concept of the effective grain surface area, which has been introduced in Chapter 3, plays a sig- nificant role in the molecule formation process on the grain surface. A comparison of our results with the observational data has been carried out and a very good iii agreement between them is found, especially for relatively lighter molecules. Finally, in Chapter 6, a brief conclusion on the whole scenario and work to be done in near future is discussed. iv ACKNOWLEDGMENTS

I would like to express my gratitude to all of them who made it possible to complete my thesis. This thesis could not have been written without Prof. Sandip K. Chakrabarti who not only served as my supervisor but also encouraged and motivated me throughout my PhD period. He introduced me into this beautiful world of astrophysics. I want to thank my co-supervisor Dr. Sonali Chakrabarti for her constant help and inspiration. She raised a number of interesting points at the time of discussions, which enriched me a lot. I also want to thank my group mate Dr. Kinsuk Acharyya for his valuable suggestions and discussions. I thank from the core of my heart, to all my colleagues at the Indian Centre for Space Physics. I had a very beautiful time during this period with Dr. Samir Mondal, Dr. Anuj Nandi, Mr. Dipak Debnath, Mr. Ritabrata Sarkar, Mr. Prasad Basu, Mr. Broja Gopal Dutta, Mr. Partha Sarathi Pal, Mr. Sudipta Sasmal, Mr. Indrajit Laha, of Indian Centre For Space Physics, and Mr. Himadri Ghosh, Dr. Soumen Mondal and Dr. Santabrata Das of S.N. Bose National Centre For Basic Sciences. Among the all individuals of my academic circles, I want to especially thank to Dr. Samir Mondal for his immense help throughout my PhD carrier. I had a very useful academic and non academic discussions with him. Among the senior colleagues of the Centre, I would thank Prof. J. N. Chakravorty, Vice president of the Centre for his valuable suggestions and caring guidance. I would also like to acknowledge the Abdus Salam International Centre For Theoretical Physics, Trieste, Italy, for giving me an opportunity to participate in a school in May 2006. I thank my loving parents, Mr. Basudeb Das and Mrs. Pratima Das, for their continuous support and making me absolutely independent in taking all the decisions in my life. I want to dedicate this thesis in the name of my father, who constantly inspired me throughout my carrier. He always asks me to follow one famous phrase “Do not put off till tomorrow what you can do today”. May be I was not a hundred percent honest in following this, but it gives me the much required kick to complete my job in time! I want to thank Mr. Atanu Das for his caring guidance, who is not only my elder brother but also my class mate and a friend. A special thank goes v to my wife Mrs. Atasi Das for her constant inspiration. Among the other family members I want to thank my grand mother Mrs. Durga Das, my nephew Master Ayush Das, my father in law Mr. Palan Chandra Paramanick and my mother in law Mrs. Kalpana Paramanick. At the end I want to thank my child Master Arnesh Das, who was born during the period of my writing this thesis. Finally, I want to thank the Department of Science and Technology (DST) and Indian Space Research Organization (ISRO) for their financial assistance to the projects in which I was a research fellow. vi PUBLICATIONS IN REFEREED JOURNALS

1. Formation of Water and Methanol in Star Forming Molecular Clouds, Das, A., Acharyya, K., Chakrabarti, S., Chakrabarti, S. K., As- tronomy and Astrophysics 486, 209, (2008).

2. Time evolution of simple molecules during proto-star collapse, Das, A., Chakrabarti, S. K., Acharyya, K., Chakrabarti, S., New Astronomy 13, 457, (2008).

3. Effective grain surface area in the formation of molecular hydro- gen in interstellar clouds, Chakrabarti, S. K., Das, A., Acharyya, K., Chakrabarti, S., Astronomy and Astrophysics 457, 167, (2006).

4. Recombination efficiency of molecular hydrogen on interstellar grains- II. A numerical study, Chakrabarti, S. K., Das, A., Acharyya, K., Chakrabarti, S., Bull. Astr. Soc. India 34, 299, (2006). vii PUBLICATIONS IN PROCEEDINGS

1. Formation of Water and Methanol in Star Forming Molecular Clouds, Chakrabarti, S., Das, A., Acharyya, K., Chakrabarti, S. K., in Origin of Life and Evolution of Biosphere proceedings of ‘XV International Confer- ence on the Origin of Life’, (in press), (2008).

2. Methanol Formation: A Monte Carlo Study, Das, A., Acharyya, K., Chakrabarti, S., Chakrabarti, S. K., in ‘Organic Matter in Space’ Proceedings of IAU Symposium no. 251, (Ed.) Sun Kwok, Cambridge University Press, (2008), p. 2132.

3. Time dependent chemical evolution of molecular clouds, Das, A., Chakrabarti, S. K., Acharyya, K., Chakrabarti, S., in Book of Abstract of Complex molecules in space and the Present status and prospects with ALMA, (2006), p. 59.

4. Time dependent chemical evolution of molecular clouds, Das, A., Chakrabarti, S. K., Acharyya, K., Chakrabarti, S., in Book of Abstract of ‘Chemical Evolution of The Universe’, Faraday Discussions 133, (Ed.) Ian R. Sims, Royal Society of Chemistry, (2006).

5. Monte-Carlo simulation of Production of Hydrogen Molecule on Grain Surfaces, Chakrabarti, S. K., Das, A., Acharyya, K., Chakrabarti, S., in Book of Abstract of ‘Chemical Evolution of The Universe’, Faraday Discussions 133, (Ed.) Ian R. Sims, Royal Society of Chemistry, (2006).

6. Monte-Carlo simulation of Production of Hydrogen Molecule on Grain Surfaces, Chakrabarti, S., K., Acharyya, K., Chakrabarti, S., Das, A., in Book of Abstract of Complex molecules in space and the Present status and prospects with ALMA, (2006), p. 57.

7. Can amino acids be formed during the evolution of molecular cloud?, Acharyya, K. Chakrabarti, S. K., Chakrabarti, S.,Das, A., in Book of viii

Abstract of ‘Chemical Evolution of The Universe’, Faraday Discussions 133, (Ed.) Ian R Sims, Royal Society of Chemistry, (2006).

8. Recombination efficiency of molecular hydrogen on interstellar grains- II. A numerical study, Chakrabarti, S. K., Das, A., Acharyya, K.,Chakrabarti, S. K., ICTP Preprint , (2006), IC2006030.

9. Average recombination time of atomic hydrogen on grain surfaces: A Monte Carlo study, Das, A., Chakrabarti, S. K., Acharyya, K., Chakrabarti, S., in Book of Abstract of 36th COSPAR Scientific Assembly, (2006), p. 623.

10. Production of complex bio-molecules in collapsing interstellar cloud, Acharyya, K., Chakrabarti, S. K., Chakrabarti, S., Das, A., Astrochemistry Throughout the Universe: Recent Successes and Current Challenges Proceed- ings of IAU Symposium no. 231, (Eds) Dariusz C. Lis, Geoffrey A. Blake, Eric Herbst, Cambridge University Press, (2005), p. 155.

11. Monte-Carlo simulation of Molecular Hydrogen Formation on Grain Surfaces, Das, A., Chakrabarti, S. K., Chakrabarti, S., Acharyya, K., ‘In- ternational Meeting on Star Clusters’ Proceedings of The 23rd Meeting of the ASI, (Ed.) G.C. Anupama, Bull. Astr. Soc. India 33, (2005), p. 390. ix

Contents

1 Introduction 1 1.1 The History of Understanding of the Interstellar Space ...... 2 1.2 InterstellarMatter ...... 3 1.2.1 InterstellarClouds ...... 5 1.3 ComponentsofISM...... 8 1.3.1 TheNeutralInterstellarGas ...... 8 1.3.2 IonizedInterstellarGas...... 11 1.3.3 InterstellarDust ...... 12 1.3.4 SupernovaRemnants ...... 15 1.3.5 PlanetaryNebula ...... 16 1.4 EnergySourcesofISM ...... 17 1.4.1 Radiationfield ...... 17 1.4.2 MagneticField ...... 18 1.4.3 CosmicRay ...... 18 1.5 InterstellarChemistry ...... 20 1.5.1 Gas-phaseChemicalReaction ...... 20 1.5.2 Grain-SurfaceChemicalReaction ...... 25 1.6 Observational Study of the Interstellar Molecule ...... 29 1.7 Experimental Study of the Interstellar Molecule ...... 39 1.8 Theoretical Study of the Interstellar Molecule ...... 45

2 Formation of Protostars 49 2.1 ConditionforCollapse ...... 49 2.2 Similarity Solutions for Self Gravitating Isothermal Flow ...... 51 2.3 Collapse of a Spherically Symmetric Gas cloud ...... 54 2.4 CollapseofaRotatingGasCloud ...... 65

3 H2 Formation 70 x

3.1 Procedures to Handle the H2 Formation ...... 71 3.1.1 RateEquationMethod ...... 72 3.1.2 MasterEquationMethod...... 75 3.1.3 Monte-CarloApproach ...... 78 3.2 Results...... 80 3.2.1 OlivineGrain ...... 84 3.2.2 AmorphousCarbonGrain ...... 86 3.2.3 Comparison ...... 88

4 Formation of Molecules in the Gas Phase 91

5 Formation of Molecules on the Grain Surface 102 5.1 Mechanisms ...... 102 5.2 ProceduresofSimulation...... 107 5.2.1 RateEquationMethod ...... 107 5.2.2 MonteCarloMethod ...... 110 5.3 Results...... 115 5.3.1 Abundances of Various Species on the Grain Surface . . . . . 115 5.3.2 TemperatureDependence ...... 122 5.3.3 Variation in γ and α ...... 123 5.3.4 A Comparison Between LH and ER Schemes ...... 130 5.3.5 Comparison of Results with the Effective Rate Equations . . .131 5.3.6 Comparison of Results with the Observation ...... 131

6 Concluding Remarks 135

7 Collection of Published Papers in Refereed Journal 144 List of Figures

1.1 Witch Head reflection nebula (courtesy: NASA)...... 4

1.2 North America Nebula NGC 7000 (courtesy: NASA)...... 5

1.3 Horse Head nebula (courtesy: NASA)...... 6

1.4 Orion nebula (courtesy: NASA)...... 12

1 1.5 Extinction curve as a function of λ− (Mathis, 1990)...... 14

1.6 NGC 6543, The Cat’s Eye Nebula (courtesy: NASA)...... 17

1.7 A schematic diagram of the formation of molecules on the grain surfaces. Four major steps accretion, diffusion, reaction, and evaporation is shown. . 28

1.8 Electromagnetic spectrum...... 29

1.9 Rate of discovery of the observed molecules...... 31

1.10 Sub-millimeter emission spectrum observed toward NGC 6334 IRS1 (Biss- chop, 2007)...... 35

1.11 Molecular abundance of the interstellar ice is shown as the percentage of

solid H2O (Boogert and Ehrenfreund, 2004). The molecules shown in the square brackets are uncertain for the identification. For each molecule the abundance in three lines of sight is given: NGC 7538 : IRS9 (black), W 33A (dark grey), and Elias 29 (light grey). Note the large variations of CO,

CH3OH, and OCN abundances and the much more stable CO2 abundance

as a function of sight-line. Solid H2O column densities are taken from Whittet et al. (1996); 8 1018 cm2, Gibb et al. (2000); 11 1018 cm2, × × and Boogert et al. (2000a); 3.4 1018 cm2 for these respective sources. .. 37 × xi xii

1.12 SWS spectrum of NGC7538 IRS9, covering the full SWS spectral range from 2.4 to 45 µm. Species labeled in the Figure are reliably detected, but the uncertain or ambiguous assignations are in the brackets (Whittet et al., 1996)...... 38

1 2.1 Density distribution for A=2.001 and for sound speed a=0.2 km s− . ... 52

1 2.2 Velocity distribution for A=2.001 and for sound speed a=0.2 km s− . ... 53 2.3 Evolution of velocity for Model A is shown. At the beginning, the cloud has a constant velocity. At a later stage it assumes an almost steady state with v(r) r 4/5...... 57 ∼ − 2.4 Density distribution for Model A is shown. From a constant density cloud, it assumes a power-law density distribution. As the core becomes massive enough, it starts to evacuate the matter of the cloud very rapidly, as a result the density over the entire cloud gradually decreases as is shown by the dot-dashed curve...... 58 2.5 Evolution of densities at the central grid and the middle grid for Model A. Towards the end of simulation, the density starts to decrease as the cloud is evacuated by the massive core...... 59 2.6 Evolution of core mass for Model A is shown. After getting enough mass in its core it starts to draw the mass rapidly from the cloud and as a result it grows up very fast...... 60 2.7 Velocity distribution for Model B is shown...... 62 2.8 Density distribution for Model B is shown. Towards the end of the simu- lation the density of the entire cloud drops, due to the high rate of mass accretion at the core...... 62 2.9 Evolution of densities at the central grid and the middle grid for Model B. 63 2.10 Evolution of the core mass for Model B is shown, here core mass increases rapidly at the beginning, but the rate of evolution slows down due to the depletion of the cloud mass...... 64 2.11 Evolution of density of a layer is shown. Initially there was a negligible amount of mass inside the cloud. The density of the inner region increases rapidly as the collapse progresses (Chakrabarti et al., in prep.)...... 68 xiii

4 3.1 Evolution of the H and H2 on an Olivine grain having 10 sites is shown.

The grain is kept at temperature 8 ◦K and facing an effective accretion rate per site of H 7.98 10 8 s 1...... 71 × − − 3.2 In (a-b) the pictures of the grain is shown at two intermediate times (a) 8 108s and at (b)109s. An Olivine grain having 900 sites has been chosen × for this simulation. The grain is kept at 8 ◦K and is bombarded with an accretion rate per site of H 3.02 10 7 per sec. No spontaneous desorption × − has been assumed here. In (c-d) spontaneous desorption has been included

and plotted for the same time as before. Thus, numbers of H2 residing on the grain at any instant are lesser...... 73

3.3 α0 as a function of as is shown for various Olivine grains kept at 8 ◦K. The dashed, dot-dashed and solid curves are for S = 106, 9 104, and 104 × respectively...... 74

3.4 α0 as a function of as is shown for various Olivine grains kept at 10 ◦K. The dashed, dot-dashed and solid curves are for S = 106, 9 104, and 104 × respectively...... 75

3.5 Variation of β0 as a function of as, for various Olivine grains kept at 8 ◦K. The dashed, dot-dashed and solid curves are for S = 106, 9 104, and 104 × respectively...... 77

3.6 Variation of β0 as a function of as, for various Olivine grains kept at 10 K. The dashed, dot-dashed and solid curves are for S = 106, 9 104, and ◦ × 104 respectively...... 79

3.7 Temperature dependence of α0 for the Olivine grains at 10 ◦K (solid), 9

◦K (dot-dashed) and 8 ◦K (dashed). The deviation is highlighted using dotted curves by extrapolating at very low accretion rates...... 82

3.8 A comparison between the recombination efficiency obtained from the Rate equation (solid) and that obtained from our simulation (dashed). We use the accretion rate per site 1.8 10 9 per second for a grain of diameter × − 0.1µm...... 83

3.9 Variation of α0 as a function of as, the effective accretion rate per site,

for various amorphous carbon grains kept at 14 ◦K. The solid, dot-dashed and dashed curves are for S = 104, 9 104, 106 respectively...... 84 × xiv

3.10 Variation of α0 as a function of as, the effective accretion rate per site,

for various amorphous carbon grains kept at 14 ◦K. The solid, dot-dashed and dashed curves are for S = 104, 9 104, 106 respectively...... 86 ×

3.11 Variation of β0 as a function of as, the effective accretion rate per site, for

various amorphous carbon grains kept at 14 ◦K. The solid, dot-dashed and dashed curves are for S = 104, 9 104, 106 respectively...... 88 ×

3.12 Variation of β0 as a function of as, the effective accretion rate per site, for

various amorphous carbon grains kept at 20 ◦K. The solid, dot-dashed and dashed curves are for S = 104, 9 104, 106 respectively...... 89 × 3.13 A comparison of the simulation results (dark circles) with those obtained from analytical considerations (dashed curves) when suitable modification of the average recombination rate is made. An Olivine grain of 104 sites at

a temperature of 8 ◦K has been chosen in this comparison. Dotted curves

are drawn using analytical results for α0 extrapolated to very low accretion rates...... 90

4.1 Evolution of the mass fractions of H and H2 are shown. Mass fraction of

H atom decreases due to the production of H2 & other hydrogenated species. 92

4.2 Time evolution of the mass fractions of some simple molecules like O2,

H2O, CO...... 93

4.3 Time evolution of the mass fractions of HCN and NH3...... 94

4.4 Time evolution of the mass fractions of some complex molecules like C2H5OH,

CH3OH, CH3CHO...... 95

4.5 Time evolution of some bio-molecules like, glycine and alanine ,which are the good precursor of the life formation...... 95

4.6 Time variation of the mass fractions of O2, H2O, CO are shown for the Set 2 initial abundances...... 96

4.7 Time variation of the mass fractions of O2, H2O, CO are shown for the Model B hydrodynamic simulation and Set 1 initial abundances...... 96

4.8 This is also for Model B hydrodynamic simulation, where Set 1 initial abundances are used...... 97 xv

4.9 Evolution of the mass fraction of CO is shown when the freeze-out effect is absent. The results in the 1st, 4th, 7th and 10th shells are displayed. The global average does not shift very much as it has a dominant contribution from the outer shell...... 98

4.10 Time variation of the mass fraction of CO is shown by taking the freeze-out effect into account. Note that in the 1st and the 4th shell, the freeze-out effect is prominent. However, the global averages do not differ by any large margin since they are dominated by the outer shells...... 99

5.1 Different reaction schemes (C1-C4) are shown by a cartoon diagram. Black and shaded circles represent the reactive species and the clear circles rep- resent non-reactive species. A circle around two such species represent the formation of a new species by ER mechanism...... 108

5.2 Evolution of some species for various Models and different reaction mech- anisms are shown for an Olivine grain having 104 number of sites kept at

10 ◦ K and facing the high abundances (Table 5.6) of the gas phase species.116

5.3 Same as in Fig. 5.2, but the low abundances of the accreting species have been used...... 118

5.4 The rate of production of H2O is shown against the surface coverage for C1-C4. The production is higher when the Eley-Rideal mechanism is taken into account, while the production is lower when the reaction takes place at the next time step. The rate is computed using the total production and total time taken at a given instant...... 119

5.5 Rate of production of H2O is shown with the surface coverage for the four different schemes. Production is higher where ER is considered and the reaction takes place at the same time steps. Here the change in the number

of H2O is for a fixed change in percentage of the surface coverage (10%). . 120

5.6 Rate of production of a few selected species as a function of the surface coverage at different temperatures are shown for a grain having 104 sites for Model 2 and C2 scheme. These simulations were carried out for low abundances (Table 5.6) of the accreting species on an Olivine grain. . . . .121 xvi

5.7 Time taken to build a mono-layer is plotted as a function of temperature for a grain having 104 sites and for Model 2 by considering the C2 scheme. These simulations were carried out for low abundances (Table 5.6) of the accreting species on an Olivine grain...... 122

5.8 Variations in γH2O(t) with surface coverage for different reaction schemes for Model 2. Note that when the Eley-Rideal scheme is included (C1 and C4) the production efficiency increases rapidly as the grain is filled up. The simulation were carried out for an Olivine grain having 104 sites and

was kept at temperature 10 ◦K...... 123

5.9 The variation in γH2O(t) with the surface coverage for different grain sizes.

Here, the binding energies of Model 2 are used. Note that γH2O(t) is decreased with increasing grain size since the probability of capturing an incoming species increases with the grain size. Here C2 scheme has been chosen. Here also the simulation were carried on an Olivine grain having 104 sites and was kept at temperature 10K...... 124

5.10 Variation in αH2O(t) with surface coverage for different reaction schemes as

in Fig. 5.8 The formation rate of the new species goes down (i.e., αH2O(t) goes up) for C2 and C3 since the species is created in the next time step. . 125

5.11 Same as in Fig. 5.9 except that αH2O(t) has been plotted. As the grain

size gets smaller, αH2O increases, i.e., the recombination efficiency decreases.126

5.12 Variation in α with the surface coverage for various molecular species that form on the grain surface. All of them deviate from unity. This case is for Model 2 with the C2 scheme...... 127

5.13 Variations in α for water and methanol (averaged over the surface cover- age) as functions of the grain radius and number of sites on the grain. The

filled circles are for H2O and the filled boxes are for methanol. The cor- responding fitted curves are also shown. They are extrapolated to larger grain sizes...... 129

5.14 Fractional production of H2O through ER and LH schemes is plotted.

For low surface coverage, H2O production is dominated by the hopping process and when surface coverage is higher, the Eley-Rideal mechanism

contributes in a major way towards the H2O production...... 130 xvii

5.15 A comparison of the abundances of two of the important species, namely,

H2O and CH3OH in different methods; Rate equation method, Monte- Carlo method, and the Effective Rate equation method. The Monte Carlo results matche very well with the Effective Rate equation method. . . . .132 5.16 Solid state methanol abundances with respect to the water is plotted with the number density of the accreting gas. These simulation were carried out for Model 2 by considering the C2 method. The observed abundances are in between the range 5% to 30% which lies in the range 2.6 104 to × 8 104...... 133 × List of Tables

1.1 Typeofreactions ...... 21 1.2 Reactionsandtheirrates...... 22 1.3 The 151 molecular species detected in space, ordered by mass. . . . . 32

3.1 Comparison of H and H2 abundances in various methods ...... 87

4.1 Comparison of our Results with the observed abundances ...... 100

5.1 SurfaceReactionsintheH,O,andCOmodel ...... 103 5.2 Energy barriers against diffusion and desorption(in degree Kelvin) for theOlivinegrain ...... 104 5.3 Energy barriers against diffusion and desorption(in degree Kelvin) for theOlivinegrain(fromexperiment) ...... 105 5.4 Time scales for the B.E. listed in Table 5.2...... 105 5.5 Time scales for the B.E. listed in Table 5.3...... 106 5.6 Gas-phaseabundancesused ...... 107 5.7 Surface coverage of the major species when one mono layer is built . . 115 5.8 Time taken to build one mono-layer ...... 117 5.9 Calculated value of α, for different species, using low abundances of the

accreting species for Model 2 and keeping the grain at 10 ◦K. These simu- lations were carried out for the C2 scheme by considering an Olivine grain having 104 number of sites...... 128

xviii Chapter 1

Introduction

The universe is rather large with roughly 1084 atoms spreading all over. The event that started the universe is called the Big Bang. At this point in time all matter and energy of the observable universe was concentrated in one point of virtually infinite density. After the Big Bang, the universe started to expand and reached its present form. To our present knowledge, it is still expanding at an accelerated rate. The universe is full of galaxies, stars and very tenuous gas with the number density of about 1 atom/cc in between. A gas collapses to form the stars which light up the sky. Till then the clouds are dark. How to detect these dark clouds and what are their properties? In 1904 Hartmann observed that some of the light coming from the star Delta Orionis was being absorbed before it reached to the Earth. He reported that ab- sorption from the “K” line of Calcium appeared to be extraordinarily weak. He also reported that quite surprisingly this line did not shift. From this, he concluded that the absorption could not have been produced by the orbiting companion of the star, it is produced by an isolated cloud of matter residing somewhere along the line-of-sight to this star. It was concluded that the space between the stars is not empty, it consists of an extremely dilute (by terrestrial standards) mixture of ions, atoms, molecules, larger dust grains, cosmic-rays, and (galactic) magnetic fields. The matter consists of about 99% gas and 1% dust by mass. This is called the Interstellar Medium (ISM). After Big-Bang, some heavier isotopes of hydrogen were produced. No heavier elements, known as metals, were formed since the universe expanded rapidly and became too cold. The heavier elements were produced through various stages of

1 Chapter 1. Introduction 2 the stellar evolution. The interstellar gas is primarily composed of hydrogen and helium, with traces of other species, such as carbon, nitrogen and oxygen. These are the basic elements of the ISM from which the complex molecules are formed via chemical evolution. To understand the chemical evolution of a molecular cloud it is not sufficient to study only the cloud chemistry; it is also necessary to understand the physical properties of the cloud. For this, we need to have a brief knowledge about the different components of the ISM. In this Chapter, we will discuss about all the components needed for our study.

1.1 The History of Understanding of the Interstellar Space

Since ancient times, humans have natural curiosities about the stars and the sky. It has been proved that a number of civilizations in the ancient world were studying the sky. This kind of study mainly belonged to the mystical and practical purposes, not so much for the scientific reasons. One of the major practical advantages for this kind of study was for agriculture. The ancient and mystical study of the space, especially that of Astrology, set the stage for astronomy and astrophysics. The effort began with the aim to understand the position of our Sun in our own galaxy (Milky Way). One of the first known efforts to know the position of the Sun in the galaxy was undertaken in the late 18th century by William Herschel. He incorrectly assumed that by counting the number of stars in different regions of the sky, he could easily identify the center of the galaxy by noting the location of the highest concentration of stars. Since Herschel did not find any region of the sky that had a higher concentration of stars than any other region, he concluded that we must be in the center of the galaxy. In 1906, Jacobus Kapteyn began a similar project, to map the size and the shape of the Milky Way Galaxy. The method he used was similar to Herschels method. He surveyed 206 stars in specific areas of the sky, analyzing their apparent brightness and proper motion. In 1922, the results of his study were finally published. He concluded that our galaxy was 30,000 light years across, 6, 000 light years thick, and the solar system was situated in the middle of it. Kapteyns model of the Milky Way was commonly accepted as accurate for many years. Harlow Shapley is the first one to correctly anticipate the size and structure of our galaxy. He began observing globular clusters which consist of a huge collection of Chapter 1. Introduction 3 stars. He discovered a type of Cepheid variables from which he accurately measured the distances. From his model the calculated diameter of our galaxy becomes to be 300, 000 light years. More importantly he determined that the Galactic center to be located in the constellation of Sagittarius. Edwin Hubble is considered to be one of the foremost astronomers of the modern era, who developed a 100 inch telescope. Through his observations and using existing laws of astronomy he was able to determine the distance to the Andromeda galaxy. He calculated that the distance was about one million light years (latest estimates are 2.5 million light years). Before this, most astronomers believed that the Milky Way was the entire Universe. But the measured distance of Andromeda was large enough to place this well beyond the confines of the Milky Way as defined by Shapley. This huge increase in the size of the Universe led to many new areas and objects to explore. Hubble also devised a classification system for galaxies which is still in use today. He classified galaxies into spiral galaxies (like our Milky way and the Andromeda galaxy), which contain a lot of interstellar gas and dust; and elliptical galaxies, which are almost void of material between their stars.

1.2 Interstellar Matter

Compared to the size of an entire galaxy, stars are virtually points, so that the region occupied by the interstellar matter constitutes nearly all the physical volume of a galaxy. Although the density of interstellar matter is far lower than in the best laboratory vacuum, the total mass contained between the stars is about 5% of the mass of the universe. On an average, the interstellar matter in our region of the galaxy consists of about one atom of gas per cubic centimeter and 25 to 50 microscopic solid particles per cubic kilometer. In contrast, the air at sea level on Earth contains about 1019 molecules of gas per cubic centimeter. This gives an idea about the density of the Interstellar matter. Interstellar matter is mostly gaseous, but about 1% (in mass) is in the form of interstellar grains or dust. Some of the interstellar material is visible, sometimes through small telescopes, as nebulae. Normally there are three types of nebulae. Chapter 1. Introduction 4

Figure 1.1: Witch Head reflection nebula (courtesy: NASA).

1. Reflection Nebulae

Reflection nebulae are clouds of dust which are simply reflecting the light of a nearby star or stars. The light is bright enough to give sufficient scattering to make the dusts visible. Example of this kind of nebula is the Witch Head reflection nebula (IC2118) (Fig. 1.1). It is about 1000 light years away from us and is located with the bright star Rigel in the Orion constellation. The colour of this nebula is blue. This is due to the fact that the dust grains reflect blue light more efficiently than the red colour.

2. Emission Nebula

When a gas cloud is positioned nearby a very hot star or stars and causes the ion- ization in the gas, the gas starts to emit light of various colour. This kind of gas is known as the emission nebula. The North America Nebula NGC 7000 (Fig. 1.2) is an example of an emission nebula in the constellation Cygnus, close to Deneb. This nebula is very often called The North America Nebula because it resembles Earth’s continent of North America. Chapter 1. Introduction 5

Figure 1.2: North America Nebula NGC 7000 (courtesy: NASA).

3. Dark Nebula

A dark nebula is a type of interstellar cloud that is so dense that it obscures the light from the background emission. The extinction of the light is caused by the in- terstellar dust grains located in the coldest, densest part of larger molecular clouds. The example of this kind of nebula is the Horsehead Nebula (Fig. 1.3).

1.2.1 Interstellar Clouds

Interstellar cloud is the generic name given to an accumulation of gas, plasma and dust in our and other galaxies. In other words, an interstellar cloud is a denser- than-average region of the interstellar medium. Depending on the density, size and temperature of a given cloud, the hydrogen in it can be neutral (H I regions), ionized (H II regions) (i.e. a plasma), or molecular (molecular clouds). Neutral and ionized clouds are sometimes also called diffuse clouds, while molecular clouds are sometimes also referred to as a dense clouds. The interstellar cloud contains material of the 4 2 8 3 temperature ranges from 10 to 10 ◦K and densities from 10 to 10 H atoms cm− . Chapter 1. Introduction 6

Figure 1.3: Horse Head nebula (courtesy: NASA).

Diffuse Atomic Cloud

Interstellar radiation field (ISRF) is strong enough to keep the species in the atomic form. Temperature in this region is of the order of 30 100 ◦K, density is of a few − tens of particles per cm3, having visual extinction is less than 0.1. The absorption lines are observed mainly in the UV and the visible wavelengths. Due to the low density, the chemistry does not depend much upon the presence of grains. The contribution of the grain is to form the molecular hydrogen. The fraction of atomic hydrogen in the molecular form is less than 0.1. So the chemistry is due to the pure gas-phase chemistry. The abundances of the molecules in these regions are well determined as the region is optically thin.

Diffuse Molecular Cloud

The ISRF is quite attenuated here and the molecular hydrogen formation rate is significant. Due to the presence of sufficient amount of H2, it is called the Diffuse molecular cloud. Due to the ISRF Carbon remains almost in the ionized state and supplies the free electrons. Presence of the diatomic molecules like, H2, CO, CN, + CH, CH , OH, C2, CS and SiO is discovered in this region. But surprisingly some Chapter 1. Introduction 7

+ + complex molecules like HCO , N2H , HCN, HNC, C2H, C3H2, and H2CO are also found in these region.

Translucent Clouds

This a region of cloud, which is surrounded by the diffuse molecular cloud. When we observe the different region of the cloud some light may have lost due to the absorption or scattering by the cloud. This loss is defined by a parameter called visual extinction. This region has the visual extinction of the order of one, which means that the density of this region is quite high. Density of the region is of the order of 500 to 5000 particles per cm3 and temperature is quite low (15 50 − ◦K). Due to the attenuation of the ISRF significantly, a very rich chemistry takes place here. The conversion of C+ occurs with a substantial rate and C and O are channelized to form CO at the first preference. This type of cloud was defined by the Vandeshoeck and Black (1989). Almost all the atomic hydrogens are converted into the molecular hydrogen. Rest are utilized for the formation of other species. The presence of molecular hydrogen and the absence of the C+ in this region, makes the chemistry much different from the diffuse clouds.

Dense Cloud

The ISRF is completely attenuated. Visual extinction is 5 10, density is very − high 104 to 107 particles per cm3. Here chemistry is completely different from ∼ the previous regions. Here hydrogen is essentially molecular, and there is little UV radiation so that the ionization of the carbon is almost negligible. This region is best studied by the IR. Grain chemistry plays a dominant role in this region. Increasing observational evidences suggest that there are a number of molecules which are being formed on the grains. Many complex molecules have been observed in this region. A trace amount of prebiotic molecules have also been observed. Chapter 1. Introduction 8

1.3 Components of ISM

1.3.1 The Neutral Interstellar Gas

We can divide the neutral Interstellar gas in two parts, atomic component and the molecular component.

The Neutral Atomic Gas

This component contains most of the mass of the ISM. There are 3 main observables to study the neutral gas (Lequeux 2005): 1. 21-cm lines of the atomic hydrogen, 2. The fine structure lines in the far IR and, 3. The interstellar absorption lines.

1. 21-cm Line of Atomic Hydrogen

Most of the hydrogen gas in the ISM is in cold atomic form or molecular form. In 1944 Hendrik van de Hulst predicted that the cold atomic hydrogen (H I) gas should emit a particular wavelength of radio energy from a slight energy change in the hydrogen atoms. The wavelength is 21.1 centimeters (frequency = 1420.4 MHz). This radiation is popularly called the 21-cm line radiation. The electron moving around the proton can have spin parallel or anti-parallel with the direction of the proton’s spin. The energy state of an electron spinning anti-parallel is slightly lower than the energy state of a parallel-spin electron. Since an atom always wants to be in the lowest possible energy state, the electron will eventually flip to the anti- parallel spin direction if it was somehow knocked to the parallel spin direction. Since the energy difference is very small, transition is highly forbidden with an extremely 15 1 small probability of 2.9 10− s− . This means that the time for a single atom of × neutral hydrogen to undergo this transition is around 10 million (107) years and so is unlikely to be seen in a laboratory on the Earth. However, as the total number of atoms of neutral hydrogen in the interstellar medium is very large, this emission line is easily observed by radio telescopes. The advantage of this line is that it is not blocked by dust due to its long wavelength. The 21-cm line radiation provides the best way to map the structure of the Galaxy. An external magnetic field shifts the energy levels of the H atom (called the Zeeman effect) and so 21-cm radiation can be used to estimate the strength of the magnetic field in the ISM also. Chapter 1. Introduction 9

The neutral medium is organized in (a) cold diffuse HI clouds (cold neutral medium, CNM) and (b) Warm inter cloud gas (Warm neutral medium, WNM)

(a) Cold Neutral Medium

This region consists of typically 1 to 5% volume of the ISM. Temperature is of the 3 order of 50 to 100 ◦K, density varies from 20 to 50 atoms per cm and the hydrogen belongs to the neutral state. This region is best studied with the help of the HI 21-cm absorption lines.

(b) Warm Neutral Medium

By volume it is 10 to 20% of the ISM. Temperature in this region varies from 6 103 4 × to 10 ◦K, which is much higher than the CNM region but Density (0.2 to 0.5 atoms per cm3) is one order less than the CNM region. This region is best studied by the HI 21-cm emission lines.

2. Fine Structure Lines

Fine structure lines in the far-IR are the main cooling source for the ISM. This process is dominant in most regions of the ISM, except regions of hot gas and regions deep in molecular clouds. Due to the collision, atoms are excited to the upper levels, which eventually release their energy through the photon emission, resulting in the decrease in energy. For example, the collisional excitation of the n = 2 level of hydrogen release a Lyα photon and comes to the lower energy state.

3. Absorption Lines

Interstellar absorption lines supply important information on the chemical composi- tion and the physical conditions in the ISM. Many interstellar absorption lines have been observed in the spectra of stars. In the visible and near UV, we observe lines + + + from atoms (Na, K, Ca), ions (Ca , Ti ) and molecules (CN, CH, CH , C2). Chapter 1. Introduction 10

Molecular Component

Many molecules are observed in the ISM. A significant number of molecules are also found in the comets. Molecules have been discovered in external galaxies as well. The molecules are identified by noting 3 types of transitions: 1. Electronic transition, 2. Vibrational transition, and 3. Rotational transition.

1. Electronic Transition

The energy of this kind of transition is of the order of several eV and they are generally in the UV range. An electronic transition of a molecule is resolved into a series of different lines originated from the vibrational transition. Each of these lines can itself be decomposed into several lines corresponding to the rotational sub-levels. Thus an electronic spectrum is very complex. Several molecules are discovered by + noting this type of transition such as, H2, CO, OH in the far UV, CH, CH and CN in near UV.

2. Vibrational Transition

The vibrational transitions of molecules occur between energy levels that result from the quantization of the possible modes of vibration. Generally, this is due to the stretching modes. But for the complex molecules, bending and deformation modes are also present. This type of transitions have energies typically a fraction of an eV, and corresponding wavelengths are in the infrared range. The vibrational transitions of H2 have been observed from the ground and from the space. This kind of transitions are also noticed from CO and H2O.

3. Rotational Transitions

The rotational transitions occur due to the quantization of the rotation of molecules. The energies associated with the rotation are of the order of meV and the wave- lengths are normally in the sub-millimeter to centimeter range. Most of the detected molecules (diatomic to polyatomic) in the ISM are discovered through their rota- tional transitions. Chapter 1. Introduction 11

1.3.2 Ionized Interstellar Gas

The interstellar gas can be ionized via various mechanisms. The gas can be ionized by the far UV radiation which is emitting from the hot stars, it may be ionized due to the collisions or X-ray ionization or ionization by high energy charged Particles. In general we can divide this into three parts: HII regions, diffused ionized medium, and hot interstellar medium.

HII region

HII regions are actually one type of emission nebula. They are created by the young, massive stars which ionize nearby gas clouds with high energy UV radiation. They are composed primarily of hydrogen, hence the name (astronomers use the term HII to refer to ionized hydrogen), and have temperatures of around 10, 000 ◦K. They can extend over several hundred light years or they can be so compact that they do not even stretch 1 light year across. Correspondingly, they have a large range of densities, from a few atoms per cm3, to millions of atoms per cm3 for the most compact regions. The average lifespans of HII regions are only a few million years, during which time they play a key role in the propagation of star formation through molecular clouds. The most famous example of a HII region is the Orion Nebula (Fig. 1.4), where the very luminous stars of the Trapezium cluster are ionizing the surrounding hydrogen cloud. HII region chemically composed of about 90% hydrogen. Due to the the strongest emission line at 656.3 nm HII regions appear to be red. Rest part of the HII region consists of helium, with trace amounts of heavier elements.

Diffuse Ionized Gas

This region is situated outside the HII region. It originates either from leaks of ionized gas out of HII regions or from the ionization by the UV radiation of isolated hot stars. In our galaxy, diffuse ionized gas contains more mass than the HII regions. Its total mass is of the order of 1/3 of that of the neutral atomic gas region. The temperature of this region is of the order of 8000 ◦K. The best method to study this region is to observe the optical recombination and fine structure lines. The existence of a diffuse Hα emission from outside the HII region has long been known. This Chapter 1. Introduction 12

Figure 1.4: Orion nebula (courtesy: NASA). region is mainly observed by faint emissions of H+, N+, and S+ (positive hydrogen, nitrogen, and sulfur ions) detectable in all directions.

Hot gas

6 In 1956 Spitzer suggested the existence of the hot gas (T > 10 ◦K) in the galaxy. In 1968 Bowyer et al. discovered a diffuse emission in soft X-rays ( < 1 keV), Jenkins et al. in 1974 observed absorption lines of OVI. Interstellar absorption lines of NV and CIV were also detected. Emission of Interstellar X-ray lines from OVII,OVIII and other ions were detected. These observations suggest the existence of the hot diffuse gas. This gas comes mainly from the supernovae remnant.

1.3.3 Interstellar Dust

It has been realized that the dusts are mainly formed via two mechanisms (a) old sun like near death stars generate dust (b) infrared space missions have revealed the dust is produced in supernovae explosions. The first process takes several billion years, by contrast Supernovae explosions produce dust in much less time, only about 10 million years. Dust particles in the interstellar medium are composed mainly of Chapter 1. Introduction 13

silica (SiO2), magnesium and iron silicates (e.g. Olivine, orthopyroxine, forsterite), amorphous carbon (do not have the crystalline structure) or water ice. Although by mass the dust consists of only 1% of the total Interstellar medium, it plays an extremely important role in the physics and chemistry of the Interstellar medium.

For instance, it is well established that H2, the most abundant molecule in the interstellar medium, forms on dust grains (Spitzer, 1978). Probably grain chemistry plays a significant role for the formation of other chemical compounds in the ISM. The dust contains significant amount of heavy elements and have a large opacity. It obscures all but the relatively nearby regions at visual and ultraviolet wavelengths and re-radiates the absorbed energy in the far-infrared part of the spectrum. It provides a major part ( 30%) of the total luminosity of the Galaxy (Mathis, 1990). ∼ According to Mathis (1990), the far-infrared radiation from the dust removes the gravitational energy of collapsing clouds and allow the star formation to occur.

Interstellar Reddening and Extinction

We see a decrease in the luminosity of a star when we look it through a dust cloud. This happens mainly due to two reasons: absorption of photon by the materials of the grains and the scattering of photons towards the directions other than the incoming photon. This phenomenon is called extinction. Extinction depends on the grain composition, shape and size distribution and also upon the incoming wave- length. Because of the size of the dust particles, scattering of the blue light is favoured. Therefore, the light that reaches us appear more red than it would have been without the interstellar dust. This effect is known as the interstellar reddening. This process is similar to that which makes the sun red at the sunset. Because the infrared and radio lights have much longer wavelengths than the visible light, the interstellar clouds are nearly transparent to these types of light. Due to that reason, infrared and radio observations are used to study the distant regions of our galaxy which are hidden from view at visible wavelengths. Similarly, Titan’s atmosphere is filled with obscuring particulate grains. The visible-wavelength images of Titan show only the haze while infrared-wavelength images penetrate the haze to show surface features. Chapter 1. Introduction 14

1 Figure 1.5: Extinction curve as a function of λ− (Mathis, 1990).

Interstellar Dust Emission

Interstellar grains are in general heated by the absorption of UV and visible ra- diation, and cooled by the thermal emission of infrared photons. There are other heating and cooling mechanisms which can also be be efficient; these are molecule- grain collisions deep inside the molecular clouds, which cool the grains because they are generally warmer than the gas, and electron grain collisions in the hot gas of supernovae remnants. The bigger grains are mainly responsible for most of the ex- tinction in the visible and in the infrared, and for most of the emission at wavelength longer than 60µm.

Sizes of the Interstellar Grain

The size of a interstellar dust grain varies from nanometers to micrometers. These are intimately mixed with the interstellar gas. If we look at the extinction curve (Fig. 1.5) we find that the extinction keeps rising throughout infrared, visible and near and far UV wavelength range. Extinction of the ISM gets saturated when the size of the grain becomes comparable with the wavelength. So the extinction we have 7 in the visible wavelength are from the grain of size 2 10− meter. In the Ultraviolet × Chapter 1. Introduction 15

9 8 the extinction occurs due to the 5 10− to 2 10− meter grains. As there are × × different types of grains available in different parts of the ISM, much effort was given previously to explain the observed extinction curves. The most widely used model is the Mathis-Rumpal-Nordsieck model (MRN model). This model consists of graphite and silicate grains with a power-law size distribution. The mathematical form of the size distribution is,

3.5 ni(a)da = AinH a− da, (1-1)

9 7 This relation is valid between 5 10− m size grain to 2.5 10− meter grain. × × MRN adopted spherical grains, for which Mie theory can be used to compute ex- tinction cross sections, and we shall do the same; in this case a is the grain radius. Draine and Lee (1984) extended the wavelength coverage of the MRN model, con- structed dielectric functions for astronomical silicates and graphites, and found the 25.13 following normalizations for the size distribution: Ai = 10− for graphites and 25.11 2.5 10− cm for silicates. It has been noted that the total dust volume is dominated by the large grains while the number density and surface area are dominated by the small grains. After the derivation of the above expression Weingrater and Draine (Weingartner and Draine 2001a, 2001b) revised the MRN distribution for the whole range of grain sizes, which is applicable to the Milky way. They constructed the size distributions for carbonaceous and silicate grain populations in different regions of the Milky Way. The size distributions include larger number of very small carbonaceous grains (including polycyclic aromatic hydrocarbon [PAH] molecules) to account for the observed infrared and microwave emission from the diffuse interstellar medium.

1.3.4 Supernova Remnants

Supernova remnants are produced due to the violent explosion of massive stars at the end of their life. This explosion, called the supernova explosion, is the most energetic event in the universe. This explosion causes a single stars to shine the entire surrounding region. It is observed that the supernova occur on an average one or two per 100 years per galaxy. But this incident plays a crucial role in the ISM. When a first generation star is born it is made up of 90% hydrogen and 10% helium. Due to the nuclear fusion that occurs in the center of the star, hydrogen nuclei Chapter 1. Introduction 16 combine to from helium nuclei. The released energy is utilized during the most of the life time. Once the hydrogen at the center is exhausted, the helium starts to fuse to form species with higher atomic number like carbon, nitrogen, and oxygen. The core begins to fuse once the star contracts at the end of the helium-burning stage. The cores of these massive stars become layered like onions as progressively heavier atomic nuclei build up at the center, with an outermost layer of hydrogen gas, surrounding a layer of hydrogen fusing into helium, surrounding a layer of helium fusing into carbon, surrounding layers that fuse to progressively heavier elements. Once the inner core is converted to iron, no more energy is available from the fusion process and the inner core collapses catastrophically to form a neutron star or black hole. In the resulting explosion, the outer layers of the star are blown out into space with a velocity of up to 15,000 km/s. It expands until it reaches up to several hundred light years in diameter. During this process the matter which are produced inside the stars are distributed in the ISM.

1.3.5 Planetary Nebula

Stars having mass more than 8 solar mass will end their life by supernova explosions, but for the medium and low mass stars of the order of our sun’s mass will end their journey by forming planetary nebulae. The process is much like the supernova ex- plosion. When a star like our Sun comes to age, having burnt away all the hydrogen to helium in its core in its main sequence phase and also the helium to carbon and oxygen, its nuclear reactions come to an end in its core, while helium burning goes on in a shell. This process makes the star expanding, and causes its outer layers to pulsate, which becomes more and more unstable, and loses mass in strong stellar winds. The instability finally causes the ejection of a significant part of the star’s mass in an expanding shell. The stellar core remains as an extremely hot central star, which emits high energetic radiation. Chapter 1. Introduction 17

Figure 1.6: NGC 6543, The Cat’s Eye Nebula (courtesy: NASA).

1.4 Energy Sources of ISM

1.4.1 Radiation field

Besides the diffuse matter, the interstellar space is filled with the electromagnetic radiation field produced by stars and interstellar matter. The radiation field spectrum is dominated by the emission from stars of late spectral classes, which have a large peak in the near infrared, at wavelength of about 1 micron. Another stellar component of the radiation field is due to the OB class stars, which have a peak in the UV range (at 0.1micron). The energy density spectrum varies along the galaxy. It depends on the distribution of the stars of spectral type. The stellar radiation field also have the contribution from the early type of stars which have a peak near far-ultraviolet. The strength of this region is expressed in terms of the Habing field. Another component of the interstellar radiation field (ISRF) is due to the emis- sion in the far infrared of the dust associated with the interstellar matter and heated by absorption of star light. Depending on the temperature, it is possible to distin- guish three components [Cox et al., 1986]; Chapter 1. Introduction 18

Cold Dust: with temperature of about 15-25 ◦K, it is associated to the HI regions and to the molecular clouds. It may be heated by both old and young stellar population.

Warm Dust: with temperature of about 30-40 ◦K, it is associated to HII regions and is heated by stars of spectral class O and B.

Hot dust: with temperature of about 250-500 ◦K, it constitutes of very small (radius 5 A˚) grains heated by general ISRF and normal grains (radius 0.1 µm) heated ∼ ∼ by M giants. There are a number of photon fields available at the ISM, which influences the physics and chemistry of the ISM.

1.4.2 Magnetic Field

Dynamics of a large part of the ISM is governed by the magnetic field. They con- tribute significantly to the total pressure which balances the ISM against gravity. Magnetic field plays a crucial role in the star formation process. Magnetic recon- nection is the most probable heating source of the ISM. Magnetic fields also control the density and distribution of cosmic-rays in the ISM. Galactic magnetic fields can be observed in the optical range via starlight which is polarized by interstellar dust grains in the foreground. Total magnetic field strengths can be determined from the intensity of total synchrotron emission, assuming en- ergy balance (equipartition). It has been observed that the average strength of the magnetic field for a galaxy having a spiral shape is 10µG. Our earth has a very ∼ strong magnetic field 0.5G. Magnetic filed strength varies significantly in different ∼ regions of the galaxy.

1.4.3 Cosmic Ray

Cosmic-rays (CR) are not really rays at all, but particles. These are the energetic charged particles that travel at nearly the speed of light. Cosmic-rays were discov- ered in 1912 by Victor Hess. They are produced by a number of different sources, such as the stars, supernovae and their remnants, neutron stars and black holes, as well as active galactic nuclei and radio galaxies. The energy of cosmic-rays is usually measured in the units of MeV or GeV. Most of the cosmic-rays have the en- ergies between 100MeV to 10 GeV. In this context it is believed that most galactic Chapter 1. Introduction 19 cosmic-rays derive their energy from supernovae explosions. It consists of almost 90% the nuclei of the hydrogen (protons), about 9% are helium nuclei (α particles) and about 1% heavier elements (like carbon, oxygen, magnesium, silicon and iron). Because cosmic-rays are electrically charged they are deflected by magnetic fields, and their directions have been randomized. Due to this reason it is impossible to tell where they originated. However, the cosmic-rays in other regions of the Galaxy can be traced by the electromagnetic radiation they produce. There are different types of cosmic-rays:

Solar Cosmic Rays

The solar cosmic-rays are the solar energetic particles which originate mostly from the solar flares. Coronal mass ejections and shocks in the interplanetary medium can also produce some of the energetic particles. Solar cosmic-rays have energies upto several hundred MeV/nucleon (it crosses sometimes up to few GeV/nucleon). It is noticed that the solar cosmic-rays increase with the increase in strength of the solar flares. It is also noticed that the increase in the intensity of the solar cosmic-rays resulted the decrease in the all other cosmic-rays. This decrease is due to the solar wind with its magnetic field, that sweeps some of the galactic cosmic-rays outwards, away from the sun and earth.

Galactic Cosmic Rays

Galactic cosmic-rays are the energetic charged particles which enter into the so- lar system from the outside. They consist of atomic nuclei (mostly protons) and electrons. Most galactic cosmic-rays have energies too low to penetrate the earth’s atmosphere. The magnetic field of earth tends to channel the energetic charge par- ticles to the poles.

Extragalactic Cosmic Rays

This type of cosmic-rays are very energetic particles and come from beyond our galaxy. The energies of these particles are of the order of 1015 eV. Due to the ∼ very low flux of extragalactic cosmic particles received on the Earth, very little is known about their composition. Chapter 1. Introduction 20

Ultra-High-Energy Cosmic-Ray

This kind of cosmic-rays consist of particle which have an extremely large kinetic energy. The energies may be around 1015 eV. ∼

Anomalous Cosmic-Rays

These types of cosmic-rays unexpectedly have very low energies. Electrically neutral atoms are unaffected by the magnetic fields but when they are subsequently ionized, they are accelerated into low-energy cosmic-rays by the solar winds. The low energy particle may also be resulted due to the fact that hitting of the galactic cosmic-ray on to the solar wind which decelerate the energetic particles.

1.5 Interstellar Chemistry

The chemistry inside the ISM, which determines the formation and destruction of interstellar molecules is of great interest for understanding the ISM. Interstellar chemistry is very much dependent on the properties of the interstellar cloud. There are a number of processes which lead to the formation of molecules in the ISM. But in general it can be separated into two classes: Gas-phase chemical reaction and Grain-surface chemical reaction.

1.5.1 Gas-phase Chemical Reaction

Depending on the general properties of the Gas-phase reactions it can be divided into the categories listed in Table 1.1. It is noted in Table 1.1 that we can classify the generic gas phase reactions into three categories: bond formation processes, which link atoms into simple or more complex species; bond destruction processes, which fragment species into smaller species; and the bond-rearrangement reactions, which transfer parts of one co-reactant into another one. Generic gas-phase reaction types and their typical reaction rates are summarized in Table 1.2. We can estimate the rate of a typical reaction. For two or three body reactions, the rate coefficient is given by, β 3 1 κ = α(T/300) exp( γ/T ) cm s− . (1-2) − Chapter 1. Introduction 21

Table 1.1: Type of reactions Bond formation Bond destruction Bond rearrangement Radiative association Phototdissociation Ion-molecule Collisional association Dissociative recombination Charge-transfer Associative detachment Neutral-neutral

where, α,β,γ are three constants and T is the gas temperature. For direct cosmic- ray ionization, 1 κ = α s− , (1-3) whereas for cosmic-ray induced photo reactions,

β 1 κ = α(T/300) γ/(1 ω) s− , (1-4) − where, α is the cosmic-ray ionization rate, γ is the probability per cosmic-ray ioniza- tion that the appropriate photo reaction takes place, and ω is the dust grain albedo in the far UV (typically 0.6 at 150 nm). For interstellar photo reactions, the rate is given by: 1 κ = αexp( γA ) s− , (1-5) − V where, α is the rate in the unshielded interstellar radiation fields, AV is the extinction at visible wavelength due to the interstellar dust, and γ is the increased extinction of dust at UV wavelengths. The following are the details of the generic gas-phase reactions:

Photodissociation

The UV photons present in the diffuse ISM are a dominant destruction agent for small molecules. Inside a cloud, the radiation field will be attenuated by the dust. 9 In the unshielded radiation field this type of reactions have a typical rate of 10− 1 s− . These reactions mainly occur in the outer edge of the molecular cloud where UV photons are profuse in number. Deep inside the cloud some UV photons can be created due to radiative association, which could also lead to this type of reactions. The rate is calculated by, κ = a exp[ bA ], (1-6) pd − v Chapter 1. Introduction 22

Table 1.2: Reactions and their rates Type of reaction Typical rate comments Photo dissociation 9 1 AB+hν A+B 10− s− (a) → Neutral-neutral 11 3 1 A+B C+D 4 10− cm s− (b) → × Ion-molecule + + 9 3 1 A + B C + D 2 10− cm s− (c) → × Charge-transfer + + 9 1 A + B A + B 10− s− (c) → Radiative association A + B AB + hν (d) → Dissociative recombination + 7 3 1 A + e C + D 10− cm s− → Collisional association 32 6 1 A + B + M AB + M 10− cm s− (c) → Associative detachment 9 3 1 A− + B AB + e 10− cm s− (c) → (a) Rate in the unshielded radiation field. (b) Rate in the exothermic direction with no activation barrier energy. (c) Rate in the exothermic direction. (d) Rate highly reaction specific. Chapter 1. Introduction 23

where, Av is the visual extinction due to the dust, a is the unshielded rate and b is the self-shielded rate. An example of this kind of reaction is,

CH + hν C + H. →

Neutral-neutral

This type of reaction often possesses appreciable activation barriers because of the necessary bond breaking associated with the molecular re-arrangement. This type of reaction is important when the gas is warm, e.g., in stellar ejecta, in hot cores associated with proto-stars, in dense photo-dissociation regions associated with lu- 11 minous stars, or in the post-shock regions. The rates are generally around 4 10− 3 1 × cm s− when no activation barrier is present. Reaction rates are calculated as,

κ = α(T/300)βexp( γ/KT ), (1-7) − where, α, β, γ are constants. One example of this type of reaction is,

H +O OH+H. 2 → In general, the only neutral-neutral reactions that occur in the cold conditions of the dark clouds are those involving atoms or radicals, often with non-singlet electronic ground states that do not have activation barriers. One example of this type of reaction is, C+OH CO + H. →

Ion-molecule

Exothermic ion-molecule reaction occur rapidly because the strong polarization- induced interaction potential can be used to overcome any activation barrier energy involved. In the exothermic direction, this kind of reactions have a typical rate of 9 3 1 2 10− cm s− and the reaction rates are in the form, ×

κ = α. (1-8) One example of this kind of reaction is,

H+ + H H+ + H. 2 2 → 3 Chapter 1. Introduction 24

Charge Transfer

This type of reaction is of great importance for setting the ionization balance in the HII region. The charge exchange between O and H+ is a very important reaction in the ISM because it ionizes the oxygen which is then able to participate in the 9 3 1 chemistry of the ISM. This type of reaction has a typical rate of 10− cm s− in the exothermic direction. An example is:

H+ +O H+O+. →

Radiative Association

In this type of reaction, the product after the collision of two species is stabilized through the emission of a photon. The reaction of this type have highly reaction specific rates. The reaction rates are given by,

κ = α. (1-9)

One example of this type of reaction is,

H + C CH + hν. →

Dissociative Electron Recombination

This type of reaction involves in the capture of an electron by an ion to form a neutral in an excited electronic state that can dissociate. Typical rate of such a 7 3 1 reaction is around 10− cm s− and the rate coefficient is calculated by,

κ = α(T/300)β. (1-10)

One example of this kind of reaction is,

OH+ + e O + H. →

Collisional Association

In laboratory settings, three-body reaction generally dominates chemistry,

A + B + M AB + M, → Chapter 1. Introduction 25

32 6 1 with rates 10− cm s− . These reactions generally have very little importance ≈ in the astrophysical environment except for dense gas near stellar photosphere or in 11 3 dense ( 10 cm− ) circumstellar disks. ≈

Associative Detachment

In this case, an anion and an atom collide and the neutral product stabilizes through 9 3 1 an electron emission. This type of reactions has a rate of around 10− cm s− in the exothermic direction.

H + H− H + e. → 2

1.5.2 Grain-Surface Chemical Reaction

It is now clear that the surface chemistry plays a very important role in the formation and destruction of the interstellar molecules. In this section we will discuss the basic processes on the grain surfaces which govern the interstellar chemistry. Four steps govern the grain surface chemistry: accretion, diffusion, reaction, and evaporation (Fig. 1.7).

Accretion

In this process, the gas phase species are deposited on the grain surface. At the time of accretion a gas phase molecule feels a weak attraction because of the Van der Waals forces. With physical adsorption (physisorption) there is only an increase in surface concentration during adsorption. The heat of adsorption for this process is about 1 10 kcal/mole of adsorbed gas. With chemical adsorption (chemisorption) − the heat of adsorption is about 20-200 kcal/mole of adsorbed gas. A chemical bond is formed between the molecule and the surface. Like the heat of adsorption, the activation energy of the adsorption process is also higher in the chemisorption process than the physisorption process. As a consequence physisorption is relevant at lower temperatures and chemisorption is relevant at higher temperatures. The interaction potential depends on the distance to the surface and the location of the sites on the surfaces. For a perfect crystal there should be a regular variation of the potential energy across the surface but for a disordered material this regularity is lost. For the theoretical understanding we use an average potential energy. Chapter 1. Introduction 26

The accretion rate of a species on grains is given by,

1 kacc(i)= siσdv(i)N(i) s− , (1-11)

th where, si is the sticking coefficient of the i species (it is close to unity for the neutral species), N(i) is the concentration of the ith species on the gas phase, v(i) is the thermal velocity, and σd is the geometrical dust-grain cross section.

Diffusion

The accreted species will bind to the grain surface with a binding energy for physical adsorption, Ed. Ed strongly depends on the composition of the grains. Depending upon the potential energy barriers (Eb) between adjacent surface potential energy wells species will migrate from one surface site to the another. Normally Eb is chosen to be 0.3ED, although some other estimates are also exist. The migration time scale due to the thermal hopping is denoted by,

1 thop = ν0− exp(Eb/kTd) s, (1-12) where, Td is the dust temperature, ν0 is the characteristic vibration frequency for the adsorbed species, which is assumed to be isotropic. Although the characteristic frequency is very often assumed to be the same for all the species, for more accurate estimation following relation is used,

2 1/2 1 ν0 = (2nsED/π m) s− . (1-13) where m is the mass of the adsorbed particle. ns is the surface density of the sites. 15 2 So far, different values of ns are used by several scientists. ns 1.5 10 cm− ∼ ×14 2 used by Hasegawa et al., (1992). Biham et al., (2001) used ns 2 10 cm− for 13 2 ∼ × Olivine grains and n 5 10 cm− for the amorphous carbon grains. s ∼ × The diffusion time tdiff required for an adsorbed particle to sweep over a number of sites equivalent to the whole grain surface is given by

tdiff = Sthop s (1-14) where, S is the total number of sites on a grain. Chapter 1. Introduction 27

Reaction

There are two reaction schemes, the Langmuir-Hinshelwood (LH) mechanism and the Eley-Rideal (ER) mechanism. In the LH scheme, the gas phase species accretes onto a grain and becomes equilibrated with the surface before it reacts with another atom or molecule, and in the ER reaction scheme, the incident gas phase species collides directly with an adsorbed species on the surface and reacts with that species. In order to react the adsorbed species require sufficient mobility. The surface reaction rate Ri,j between surface species i and j occurring due to classical diffusion can be expressed as (Hasegawa et al., 1992)

Ri,j = ki,j(Rdiff,i + Rdiff,j)ninjnd, (1-15) where, ni and nj are the number of species i and j respectively, on an average grain, Rdiff,i and Rdiff,j are the diffusion rate (defined as inverse of the diffusion time), ki,j is the probability for the reaction to happen upon an encounter. The parameter ki, j is in general unity for the exothermic reaction without activation energy. For an exothermic reaction with activation energy Ea and at least one light reactant (H,H2), ki,j can be approximated by the exponential portion of the quantum mechanical probability for tunneling through a rectangular barrier of thickness a:

k = exp[( 4πa/h)(2µE )1/2], (1-16) i,j − a where, µ is the reduced mass and a is taken as 1A˚.

For the light reactive species H and H2, surface migration via tunneling is much faster than due to the classical hopping. The time scale for tunneling is given by,

1 1/2 ttun = ν0− exp[(4πa/h)(2mEb) ]. (1-17)

Evaporation

The residence time of a species on a grain surface is given by,

1 tevap = ν0− exp(ED/kTd) s. (1-18)

This time is the characteristic time scale for a species to acquire sufficient energy through thermal fluctuations to evaporate. Chapter 1. Introduction 28

Figure 1.7: A schematic diagram of the formation of molecules on the grain surfaces. Four major steps accretion, diffusion, reaction, and evaporation is shown. Chapter 1. Introduction 29

Figure 1.8: Electromagnetic spectrum.

1.6 Observational Study of the Interstellar Molecule

After hydrogen and helium, oxygen and carbon are the most abundant elements in the Universe. Molecules are formed through the interactions of these atoms, along with the more abundant hydrogen atom. This chemistry in cloud interiors set the energy balance of the interstellar clouds. Ultimately, the chemical interactions of carbon and oxygen not only govern the formation of various molecular species but also play an important and uncertain role in the collapse of cloud material to form stars and planets. The chemistry of molecular clouds is quite different from the chemistry we see in a laboratory on our Earth. For example, the gas in molecular clouds which can be considered dense by interstellar standards is still more rare than the best vacuum we can achieve in the laboratory. The visible spectrum is only a tiny portion of the total electromagnetic (EM) spectrum (Fig. 1.8), however all are the light waves, they simply lie at wavelengths (energies) that our eye doesn’t respond to. When we look at the Universe in a Chapter 1. Introduction 30 different light, i.e., at non-visible wavelengths, we can see new kinds of objects. For example, high-energy gamma-ray and X-ray telescopes tend to see the most energetic dynamos in the cosmos, such as active galaxies, the remnants from massive dying stars, accretion of matter around black holes, and so forth. Visible light telescopes best probe light produced by stars. Longer-wavelength telescopes best probe dark, cool, obscured structures in the Universe: dusty star-forming regions, dark cold molecular clouds, the primordial radiation emitted by the formation of the Universe shortly after the Big Bang. Only through studying astronomical objects at many different wavelengths astronomers are able to figure out a comprehensive picture of how the Universe works. The observed molecules in the Interstellar medium is now over 150. The rate of discovery of the observed molecules has maintained a steady state (about 5 per year) from the past 35 years (Fig 1.9, Thaddeus, 2006). In Table 1.3 the ob- served molecules are listed according to the number of atoms. Among the ob- served molecules, the complexity varies from molecular hydrogen (H2) to HC11N.

The molecular hydrogen H2 is the commonest molecule in space. The other fa- miliar ones are, water, hydrogen cyanide (HCN), nitrous oxide (N2O), and ethanol

(CH3CH2OH). Among the more complex molecules, esoteric carbon-chains known as cyano-polyynes and the biggest known molecules HC11N are observed. In addi- tion, there is a strong evidence for even larger aromatic molecules called polycyclic aromatic hydrocarbons, or PAHs. Some molecules, such as HCO+, were not even known on Earth at the time of their detection in space. Molecules having simple structure are easy to detect than the complex ones. This is because the simple molecules have fewer symmetries and therefore, a fewer number of transitions than the complex molecules. Molecules with several symmetries and having a large number of transitions can be difficult to identify since the line strength in any particular line will be low. Also, these complex molecules with many low intensity lines contribute significantly to confusion. Molecules in the ISM are observed by observing different transitions. There are mainly three types of transitions: (1) electronic transition with energy differences of a few eV, corresponding to UV/optical wavelengths, (2) vibrational transition with energy difference 0.1 0.01 eV which corresponding to near infrared/infrared − 3 wavelengths, and (3) rotational transitions with energy difference 10− eV, corre- sponding to mm/sub mm wavelengths. There are also some less frequent modes, Chapter 1. Introduction 31

140

120

100

80

Number 60

40

20

0 1930 1940 1950 1960 1970 1980 1990 2000 Year

Figure 1.9: Rate of discovery of the observed molecules. such as bending transitions. The first detection of the molecules in the ISM was done through the observations of both narrow and broad absorption lines seen toward bright stars (Merrill 1934, 1936). In 1940, McKellar identified that the lines originated due to the presence of the molecules CH+, CH and CN. Meanwhile more diffuse lines were detected by Beals and Blanchet (1938). By the early 1940s a few simple diatomic molecules were known to exist in the interstellar space. Through the observations of electronic transitions of adjacent rotational levels of CN, it was established that the effective temperature of the interstellar space was around 3 ◦K. After this discovery not much work was done until the interstellar OH was discovered (Weinreb et al., 1963). This time the observations were made by using radio astronomy. During the next 10 years after this discovery, a huge number of attempts were made for the search of some more complex molecules. Some complex molecules, such as ammonia (NH3) (Cheung et al., 1969), formaldehyde (H2CO) (Snyder et al., 1969), Carbon monoxide (CO) (Wilson, Jefferts, and Penzias 1970), as well as several radio lines of CH (Rydbeck, Elder, Irvine 1973) were detected. Technology is now very advanced and with the help of this, one can estimate the abundances of the simple molecules very confidently but for the larger molecules the Chapter 1. Introduction 32

Table 1.3: The 151 molecular species detected in space, ordered by mass. Species Mass Species Mass Species Mass Species Mass + H2 2 NO 30 HOCO 45 CH3CONH2 59 + + H3 3 CF 31 NH2CHO 45 HNCS 59

CH 13 CH3NH2 31 PN 45 C5 60 + + CH 13 H3CO 31 AlF 46 CH2OHCHO 60

CH2 14 HNO 31 C2H5OH 46 CH3COOH 60

CH3 15 CH3OH 32 CH3OCH3 46 HCOOCH3 60

NH 15 SiH4 32 H2CS 46 OCS 60

CH4 16 HS 33 HCOOH 46 SiS 60 + NH2 16 HS 33 NS 46 C5H 61

NH3 17 H2S 34 CH3SH 48 AlCl 62 + OH 17 H2S 34 SO 48 HOCH2CH2OH 62 + + OH 17 C3 36 SO 48 HC4N 63

H2O 18 HCl 36 C4H 49 CH3C4H 64 + H2O 18 c-C3H 37 C4H− 49 S2 64 + NH4 18 l-C3H 37 NaCN 49 SiC3 64 + H3O 19 c-C3H2 38 C3N 50 SO2 64

HF 20 H2CCC 38 H2CCCC 50 CH2CCHCN 65

C2 24 HCCN 39 HCCCCH 50 CH3C3N 65

C2H 25 C2O 40 MgCN 50 C3S 68

C2H2 26 CH2CN 40 MgNC 50 FeO 72

CN 26 CH3CCH 40 HC3N 51 C6H 73 + CN 26 SiC 40 HCCNC 51 C6H− 73

HCN 27 CH3CN 41 HNCCC 51 C5N 74

HNC 27 CH3NC 41 c-SiC2 52 C6H2 74

C2H4 28 H2CCO 42 C3O 52 HCCCCCCH 74 + CO 28 NH2CN 42 H2C3N 52 HC5N 75 CO+ 28 SiN 42 AlNC 53 KCl 75

H2CN 28 CP 43 CH2CHCN 53 NH2CH2COOH 75 + HCNH 28 HNCO 43 c-H2C3O 54 SiC4 76 + N2 28 HNCO− 43 HC2CHO 54 C6H6 78

CH2NH 29 c-C2H4O44 SiCN 54 C7H 85

HCO 29 CH3CHO 44 SiNC 54 CH3C6H 88 + HCO 29 CO2 44 CH3CH2CN 55 C8H 97 + + HN2 29 CO2 44 C2S 56C8H− 97 + HOC 29 CS 44 C3H4O 56 HC7N 99

SiH 29 N2O 44 CH3CH2CHO 58 HC9N 123

CH3CH3 30 SiO 44 CH3COCH3 58 HC11N 147 + H2CO 30 HCS 45 NaCl 58 Source: astrochemistry.net Chapter 1. Introduction 33 confidence limit is very low. The modern radio telescopes and sensitive cryogenic receivers employing superconducting detectors are able to operate near the temper- ature of liquid helium. Now it is remarkably easy to detect many of the simpler astronomical molecules. Rotational lines in a strong source can sometimes be ob- served in an observation of only a millisecond. However, the larger molecules are much harder to observe. For example, the detection of HC11N required, some 30 hour of observation time for each spectral line.

As the cosmic abundances of the hydrogen is very high, H2 is very abundant in the interstellar medium. But the molecular hydrogen has a large excitation temperature and does not have a permanent electric dipole moment (only quadrupole transitions are possible, which is very weak). Due to that reason the emission line is not usually observable; hence, determining the exact amount of H2 relative to the other molecules has been difficult. In order to detect the abundances of the various species in the ISM the emission line from the rotational transition of CO is most widely used. It was thought for a long time that the water is one of the possible reservoirs of elemental oxygen in the gas phase, but recent Sub-millimeter Wave Astronomy Satellite (SWAS) observation found surprisingly a low abundance of water. Snell et al. (2000) found that the abundance of water relative to H2 in Orion and M17 cloud 10 10 is between 10− to 8 10− . The water abundance in the hot cores range from 6 4 × 10− to 10− (van Dishoeck and Helmich 1996, Helmich et al. 1996; Boogert and Ehrenfreund, 2004). The abundance of water on grains with respect to the total H 4 column density is typically 10− and is the most abundant component (Tielens et al. 1991).

SWAS observation made a serious effort to determine the gaseous O2. Their campaigns put an upper limit of the O2 abundance in cold dark clouds in the range 6 7 of 3 10− 10− (Goldsmith et al. 2000, Pagani et al. 2003, Liseau et al. 2005). × − This low abundance along with the low abundances of gaseous H2O, raises the seri- ous question about the total oxygen budget when compared with the well observed 4 atomic oxygen abundance of 3 10− in diffuse clouds (Meyer et al. 1998, Ehren- × freund and Van Dishoeck 1998). One possible explanation for the ‘missing’ oxygen is that it is frozen out onto grains in the coldest regions. In Fig. 1.10, the sub- millimeter spectrum is shown towards the NGC 6334 IRS1 (Bisschop, 2007). It is clear from the Figure that there are completely different transitions for the different Chapter 1. Introduction 34 species. The measured line intensities are different for the different molecules. The measured line intensities are useful for the determination of the abundances of the species. Interstellar methanol has three types of abundance profile: for the flat profile 9 and for the coldest sources, methanol abundance with respect to water is 10− , 9 7 ∼ profiles with a jump in its abundance from 10− to 10− are for the warmer 8 ∼ sources, and flat profiles at a few 10− for the hot cores (van der Tak et al. 2000). ∼ On the grain surface, the methanol abundance varies from 5% to 30% with respect to H2O. In some sources, such as SgrA and Elias 16, the abundance is even less (Gibb at al., 2000). The observed abundance for methanol along with the line of sight towards high mass and low mass proto-stars is between 0.2 2 105 (Gibb et − × al. 2004; Pontoppidan et al. 2003, 2004). Much higher methanol abundances are found to be associated with the outflows in the regions of low mass star formation, L1157-MM and NGC1333- IRAS2 2 105 and 2 106 (Bachiller and Perez Gutierrez, × × 1997; Bachiller et al. 1998). In other words, either on the grain surfaces or in the hotter region, abundances of theses species are high. This correlation suggests that these species perhaps originate from grains and their productions in the gas phase are inadequate. Glycine is the one of the 20 amino acids found in the proteins and it is the simplest biologically important amino acid. A number of attempts have been made to identify the interstellar glycine (NH2CH2COOH). It is a good precursor of the life formation. In 1994, Snyder et al. claimed that they found glycine molecules in space. Nine years latter, in 2003, Kuan et al. claimed that they detected interstellar glycine toward three sources (Orion KL, W51 e1/e2, and Sgr B2(N-LMH)) in the ISM. They calmed that they have identified 27 spectral lines of glycine utilizing a radio telescope. In 2005 Snyder and his collaborators re-investigated the glycine claim in Kuan et al. (2003). They rigorously attempted to confirm the detection, but it was reported that the glycine was not detected in any of the three claimed sources. Researchers from the Max Planck Institute for Radio Astronomy (MPIfR) in Bonn have detected for the first time a molecule closely related to an amino acid: amino acetonitrile. This organic molecule was found near the galactic center in the constellation Sagittarius. Since the search for glycine has turned out to be extremely difficult, it is necessary to look for the chemically link molecules from which glycine Chapter 1. Introduction 35

Figure 1.10: Sub-millimeter emission spectrum observed toward NGC 6334 IRS1 (Biss- chop, 2007).

may form. Since amino acetonitrile (NH2CH2CN) is probably a direct precursor of glycine, detection of this species may give some hints about the presence of glycine in the interstellar medium. The interstellar dust plays a very significant role in the formation of several species in the ISM. There are increasing observational evidence that selected gas phase molecules in the ISM are produced at least in part on the surfaces of the dust particles and then desorbed to the gas phase. Dense molecular clouds represent very special environments because they are the birth sites of new stars and planetary systems. Because these clouds typically contain enough material in them they are “optically thick”, this is, they absorb or scatter most of the starlight that falls on them from outside they look dark (example; horsehead nebula Fig. 1.3), thus they are also called dark clouds. This has some very important implications for the chemistry of the interstellar medium. Unlike the diffuse medium where intense radiation rapidly destroys all but the most stable molecules, in dense clouds the general galactic radiation field is screened out and a much wider variety of chemical compounds can survive. The temperatures in the dense clouds are extremely low. Typical cloud temperatures are only 10 50 ◦K. − Chapter 1. Introduction 36

As a result, most of the molecules in dense clouds are frozen into icy grain man- tles rather than free floating gas phase molecules. These interstellar ices are often comprised primarily of water ice, although they also contain simple molecules like carbon monoxide (CO), carbon dioxide (CO2), methanol (CH3OH), and ammonia

(NH3). While these clouds look dark to the human eye, they are not so in the infra-red (IR). At IR frequencies, the chemical compounds in the dust selectively absorb spe- cific wavelengths of light that depend on their molecular bonding and composition. The positions of the resulting infrared absorption features can be measured using special infrared telescopes. Comparisons between the infrared absorption bands seen through the telescope and measured in the laboratory can then be used to constrain the composition of the cloud. As a result, IR astronomers now know the composition of clouds located many light years away. Currently, 36 different absorption bands have been detected in the infrared spectra of cold, dense interstellar and circumstellar environments. These are at- tributed to the vibrational transitions of 17 different molecules frozen on dust grains

(Boogert and Ehrenfreund, 2004) H2O is the most common interstellar ice compo- 4 nent at an abundance of typically 10− with respect to the total H column density

(N[H] + 2N[H2]; e.g. Tielens et al. 1991). As summarized in Fig. 1.11, CO2 is the second most common ice component with an abundance of on average 17% with respect to H2O ice over many lines of sight (Gerakines et al. 1999). Reports are present along three lines of sight (Nummelin et al. 2001), the CO2 abundance correlates best of all ice bands with H2O, revealing important information on the formation history of these species. In contrast, it is now well established that the

CH3OH abundance varies by an order of magnitude between different line of sights (Dartois et al. 1999; Pontoppidan et al. 2003). Large abundance variations are also seen for CO and the species responsible for the 4.62 and 6.85 µm bands (likely OCN + and possibly NH4 ), and weak evidence for NH3 variations exists as well. No corre- lation between these variations is apparent, suggesting that they may have different origins, such as most likely evaporation of CO, energetic production of OCN and possibly a different atomic H density at the time of CH3OH production. NGC7538 IRS9 has the richest solid state infrared spectrum of all sources with extensive Short Wavelength Spectrometer(SWS) data from Infrared Space Obser- vatory (ISO). The spectrum illustrated in Fig. 1.12 covers the full spectral range Chapter 1. Introduction 37

Figure 1.11: Molecular abundance of the interstellar ice is shown as the percentage of solid H2O (Boogert and Ehrenfreund, 2004). The molecules shown in the square brackets are uncertain for the identification. For each molecule the abundance in three lines of sight is given: NGC 7538 : IRS9 (black), W 33A (dark grey), and Elias 29 (light grey). Note the large variations of CO, CH3OH, and OCN abundances and the much more stable CO2 abundance as a function of sight-line. Solid H2O column densities are taken from Whittet et al. (1996); 8 1018 cm2, Gibb et al. (2000); 11 1018 cm2, and Boogert et al. (2000a); × × 3.4 1018 cm2 for these respective sources. × Chapter 1. Introduction 38

Figure 1.12: SWS spectrum of NGC7538 IRS9, covering the full SWS spectral range from 2.4 to 45 µm. Species labeled in the Figure are reliably detected, but the uncertain or ambiguous assignations are in the brackets (Whittet et al., 1996). available with the SWS (2.4 to 45 µm). The most striking aspect of the spectrum in Fig. 1.12 is its apparent simplicity. Besides, the strong 9.7 µm silicate feature, it is dominated by the 3.0 µm H2O stretching mode. The absence of other strong features suggests that H2O is the dominant component of interstellar ices (Whittet et al., 1996). The Green Bank Telescope (GBT), the world’s largest fully steerable radio tele- scope, took a leading role in exploring the origin of bio-molecules in the interstellar clouds. GBT discovered acetamide (CH3CONH2), cyclopropenone (H2C3O), prope- nal (CH3CH2CHO) and ketenimine (CH2CNH) towards Sagittarius B2(N), which is near the center of our galaxy Milky Way. The molecules methyl-cyano-diacetylene

(CH3C5N), methyl-triacetylene (CH3C6H), and cyanoallene (CH2CCHCN) were found in the Tauras Molecular cloud (TMC-1), which is relatively nearby at a distance of 450 light years. The discovery of these large molecules in the coldest region of the interstellar medium changed the belief that the large organic molecules have their origin only in the hot cores. GBT forced the scientists to rethink the process of the interstellar chemistry. Chapter 1. Introduction 39

Previously astronomers discovered a series of spectra in the unidentified infrared bands. They suspected that these spectra came from simple molecules on inter- stellar dust particles, but it wasn’t until the late 1990s that scientists confirmed the source of these bands are polycyclic aromatic hydrocarbons (PAHs). These ex- tremely stable organic molecules contain rings of carbon atoms and are widespread on the Earth. PAH molecules are thought to be widely present in many interstellar and circumstellar environments in our Galaxy as well as in other galaxies. Origi- nally, scientists thought that PAHs in outer space existed only around the edges of dense clouds or dying stars. Recent observations suggest that these molecules also reside in the diffuse interstellar medium and in galaxies beyond the Milky Way. For example, NASA’s Spitzer Space Telescope has revealed PAHs in the Milky Way and more-distant regions. Using observations made with the telescopes in Chile and in

Arizona, characteristic of ultraviolet spectra of PAHs (such as anthracene (C14H10) and pyrene (C16H10) were identified (Vijh, Witt, and Gordon, 2004). However, no specific PAH molecule has yet been identified in a source outside the solar system. A new door of science will soon be opened up with the facility of Atacama Large Millimeter/Sub-millimeter Array (ALMA). It will be, by far, the worlds most powerful radio telescope operating at mm and sub-mm wavelengths, both in terms of sensitivity (ability to detect extremely faint sources) and angular resolution (ability to see the fine detail of structure in those same sources). ALMA will be able to detect cosmic sources up to a thousand times fainter than is possible with any existing mm telescope. It will also produce images of these sources with at least ten times higher resolution (down to one hundredth of an arc second) than either the Hubble Space Telescope (HST) or the Very Large Array (VLA) in New Mexico.

1.7 Experimental Study of the Interstellar Molecule

Observational astronomy both from the ground and from space has provided a very detailed information on the interstellar molecules. In order to exploit these truly unique data and to prepare new observations, it is not sufficient to identify the molecules in various regions via their spectral fingerprints but it is necessary to understand, on a fundamental level, the physical and chemical interaction of elec- trons, atoms, molecules and nanoparticles among each other and with radiation. The Laboratory studies in this subject gives us a tool to carry forward to advance Chapter 1. Introduction 40 the science in this interesting branch of astrophysics. Laboratory study offers the facility to simulate the interstellar and circumstellar conditions in a completely con- trolled environment. With the help of the laboratory facility we may address the following questions: 1) How the molecules are identified in space? 2) How the molecules are formed and destroyed? 3) What is the relationship between the gas and grains? 4) What is the role of the interstellar ice in the formation of simple and complex molecules?

Now a days, it is anticipated that among the observed molecules in the space there are more than 100 molecules which are formed on the surfaces of the grain. In- terstellar dust grains are thought to consist of an amorphous silicate or carbonaceous core surrounded by a molecular ice layer (Draine 2003; Gibb et al. 2004). Presently, very little is known about the morphology or chemistry of these grain surfaces, or the porosity of the grains themselves. The densities in interstellar space, even in the densest interstellar clouds, are so low that they exceed the capabilities of the best vacuum systems on the Earth. It is very difficult to set up an experiment in the laboratory to mimic the astrophysical condition. But in order to predict the astrophysical results, we are forced to make some assumptions. Laboratory exper- iments under simulated-space conditions have shown that the environment within these clouds can have the potential for the chemical reactions which result in organic molecules even more complex than those so far observed in space. Such experiments not only serve to confirm what astronomers have found in our galaxy but also provide clues as to what other molecules might be present in the interstellar medium. Ion molecular reactions are the most dominant one in the gas phase. But they are not the only class of chemical reactions that are going on in the ISM. A growing body of laboratory experiments suggest that there are a number of possible types of reactions which can also occur at very low temperatures in the interstellar clouds.

But the reactions that require significant activation energy goes so slowly at 10 ◦K that they won’t have much effect on the overall cloud chemistry. But not every reaction between two neutral radicals or molecules requires activation. Their exper- iments have identified about 50 of these reaction that actually go faster when the temperature gets very low. They noticed that some of the measured rates in these Chapter 1. Introduction 41 low-temperature reactions are so fast that they approach the rates that one would expect if essentially every collision were to result in reaction. Reaction Kinetics in Uniform Supersonic Flow known as CRESU is a very ef- ficient method to study the interstellar chemistry in the laboratory. This is an experiment which is investigating the chemical reactions taking place at very low temperatures. Reactant molecules and a carrier gas are cooled to very low tem- peratures by expansion through a specially designed nozzle. The molecules drop to temperatures as low as 7 ◦K under conditions that are uniform enough to study reaction rates. Pulsed lasers are used to generate free radicals and to observe the rate at which they are removed. Other researchers are attempting to replicate the chemistry taking place on the dust grains in interstellar clouds. Initial efforts have focused on the formation of molecular hydrogen, the simplest molecule occurring in interstellar clouds. Theo- reticians have proposed since the early 1960s that almost all molecular hydrogen in interstellar clouds form on the surface of dust particles. In 1997, Pirronello et al., succeeded in performing this synthesis in the laboratory under conditions that are very close to the interstellar clouds. They performed their experiment on the Olivine grain. Latter this experiment is performed on two other low-temperature surfaces: amorphous carbon and frozen water. They also studied how carbon dioxide might form in interstellar clouds. Carbon dioxide is an unusual interstellar molecule be- cause it is observed only in the ices that form under suitable conditions around dust grains; it’s not present in the gas phase. Gas-grain interactions play a key role in the chemical evolution of star-forming regions. During the first stages of star formation virtually all species accrete onto grains in dense cold cores. At the late stage of the star formation sequence, when so-called hot cores are formed, grains are warmed upto certain temperatures, where molecules can desorb again. In order to characterize this astrophysical process quan- titatively it is necessary to understand the underlying molecular physics by study- ing interstellar ice analogs under laboratory controlled conditions. The appropriate sticking probabilities and surface binding energies of species on dust analogues under interstellar conditions are not known properly. So far, some theoretically predicted values are used in this literature. Several groups throughout the world are trying to do experiments to correctly predict those values. Chapter 1. Introduction 42

For the sake of the completeness of the introduction, we present some of the lab- oratory experiments which try to mimic the interstellar conditions. Before going to the outcome of the experimental details, some basic terminologies are to be clarified. What is TPD? Full form of TPD is Temperature Programmed Desorption. TPD experiments begin with a gas or mixture of gases adsorbed onto a cold crystal surface (often a metal crystal). This surface is then heated at a controlled rate (programmed rate). The adsorbates will then react as they are heated and the reaction products desorb from the surface. A mass spectrometer is used to monitor the desorption products. The results of the experiment are the desorption rate of each product species versus the temperature of the surface which is called the TPD spectrum.

What is RAIRS? Full form of RAIRS is Reflection-Absorption IR Spectroscopy. As a molecule sits on a surface, it will vibrate. Such vibrations can be studied by shining infrared light on to the surface. If the molecule has a dipole moment, that is one end of the molecule has a positive charge and the other end a negative charge, then the molecule can absorb infrared light, but only at certain fixed frequencies. Hence, an infrared spectrum of light reflected from the surface will show absorption peaks which are characteristic of the molecule and its method of bonding to the surface. This is the basis of the RAIRS technique. Vibrations can only be detected if the vibration is perpendicular to the surface. Detection of the infrared spectrum is gen- erally acquired using the FTIR technique.

What is FTIR? Full form of FTIR is Fourier Transform Infrared Spectrometer. FTIR is most useful for identifying chemicals that are either organic or inorganic. It can be utilized to quantize some components of an unknown mixture. It can be applied to the analy- sis of solids, liquids, and gasses. FTIR spectra of pure compounds are generally so unique that they are like a molecular “fingerprint”. While organic compounds have very rich, detailed spectra, inorganic compounds are usually much simpler. For most common materials, the spectrum of an unknown can be identified by comparison to a library of known compounds. Chapter 1. Introduction 43

The Nottingham Surface Astrophysics Experiment is designed to meet the need for the chemical evolution of the ice. The Nottingham facility is a Ultra High 11 Vacuum (UHV) chamber with base pressure 6 10− Torr. They took a cold × ( 10 ◦K) gold finger for sample deposition, a quadrupole mass spectrometer for ∼ TPD, a FTIR for RAIRS, and a quartz crystal micro balance for thin film mass determination. They carried out TPD studies for different ices. They concluded that H O ice should survive to higher temperatures (110 120 ◦K) than previously 2 − understood (90 100 ◦K). They also noted that CO molecules are deposited on − amorphous ice at 10 ◦K. As the temperature is increased around 25 ◦K, CO ∼ diffuses through the pores in the ice. TPD spectra suggest that at slightly higher temperature some CO desorbed from the weakly bound positions. If the temperature is further increased around 70 ◦K a gradual phase change occur from high to low density amorphous ice, in which some pore collapses which closes off pathways to the surface and traps CO within the ice. A second phase change happen at 140 ∼ ◦K, this time, the phase changes to crystalline. Finally 160 ◦K the ice itself, along ∼ with any remaining trapped CO, is desorbed. They noticed that the desorption of

CO from H2O ice is clearly much more complicated than it has been assumed up to now in any astronomical model. The Raymond and Beverly Sackler laboratory for astrophysics in Leiden was established in 1975 and was the first of its kind in the world. The laboratory specializes in spectroscopic experiments, both in the gas phase and in the solid state. With their ultra-high vacuum machines they are also able to control the growth of pure or mixed interstellar ice analogs and are able to simulate the chemical processes that occur in and on top of the ice. The Leiden group have a long record of achievement in the study of interstellar ices, including the processing of simple ices to create more complex species. The data from the recent space mission, the Infrared Space Observatory (ISO) have provided new insights into the nature of interstellar ices and have created a demand for the intensive study of the IR spectra of ices and their response to UV irradiation. It is clear that the astronomical IR spectra can reveal both the present detailed chemistry and the history of material in star-forming environments (Williams and Viti 2002). Present understanding of ice formation is that it is deposited in cold, dark and quiescent regions; H2O appears first, then CO, both in polar (probably H2O) and non polar (probably CO, and possibly N2, O2) environments. The observed ice in Chapter 1. Introduction 44

the ISM is composed of different layers of H2O, CO. The dust, which are closer to a newly formed bright star, is subjected to UV processing which leads to chemical changes and loss of volatiles. This kind of ice mainly consists of H2O, CH3OH, and

CO2.

The existence of CO2 in interstellar ice was suspected from earlier observations of the bending mode at 15.2µm but the ISO mission confirmed that CO2 is widely seen towards embedded massive stars through observations of the stretching mode at 4.27µm, and that the CO2 abundance is high relative to the H2O ice. Since CO2 in the gas phase normally has a very low abundance, it must be formed in the ice. The compound vinyle alcohol has been found in an interstellar cloud of dust and gas near the center of the Milky Way galaxy by radio astronomers using the National Science Foundation’s 12 meter Telescope at Kitt Peak, Arizona. The discovery of this molecule gives us an important tool to understand the formation of complex organic compounds in interstellar space. The major components of interstellar ices are water, carbon dioxide, methanol, formaldehyde, and formic acid. In the laboratory experiment it has been noticed that the hydrogen atoms seem to add to atoms or molecules already in the ice to form water, ammonia, methane, also formaldehyde and methanol. In one school of thought, specially led by Miller, it is believed that some of the prebiotic amino acids could be been generated in the atmosphere of the early earth. This was shown by Miller’s experiment. Miller showed the generation of a variety of amino acids and other organic molecules in the laboratory. Munoz Caro et al. (2002) reported the detection of amino acids. They simulated the ice photo 7 processing by means of a vacuum set up, P 10− mbar, at low temperature, ∼ T 12 ◦K. During the deposition of the gas mixture freezing onto a cold finger, ∼ the mixture was irradiated with a hydrogen flow discharge lamp. After warm up, a small amount of material remains at room temperature. This residue was analyzed and reported the detection of 16 amino acids at room temperature. Bernstein et al. (2002) reported a laboratory demonstration that the complex bio-molecules like glycine, alanine etc. are naturally formed from the ultraviolet photolysis of the interstellar grains. Allamandola and his colleagues have been re-creating molecular clouds in the laboratory to simulate the different chemical reactions that might occur within that Chapter 1. Introduction 45 extremely frigid environment. Their objective was to study the formation of the organic molecules such as PAHs. One of his group’s first findings was that, inside a cold vacuum chamber, photochemistry took place within the tiny ice mantles that form on microscopic grains of dust. When the researchers create ice particles containing simple molecules and irradiate them with ultraviolet light, the molecules begin to interact. Using this irradiation technique, the NASA team and scientists at the University of California, Santa Cruz converted simple molecules like water, methanol, ammonia, and carbon monoxide into compounds that form vesicles with cell like membranes. When the scientists exposed the vesicles to ultraviolet light, the membranes glowed. The researchers speculate that the glowing molecules guard the membrane against ultraviolet radiation, a protection that would be required for a cell to survive. In simulated molecular clouds, the NASA team has also created amino acids, the main components of proteins. The researchers trapped three simple compounds methanol, ammonia, and hydrogen cyanide inside an ice particle and exposed it to ultraviolet light. After warming the particles to room temperature, the researchers detected three different amino acids: alanine, serine, and glycine. The process might underlie the origins of amino acids that have been found in meteorites that have landed on the Earth.

1.8 Theoretical Study of the Interstellar Molecule

Many theoretical approaches have been taken by several scientists to explain the evolutionary history of the ISM. Considerable progresses have been made over a decade to establish the observational data through the theoretical modeling. The interstellar chemistry differs a lot from cloud to cloud. So different chemical com- position is needed for different kinds of molecular clouds. The steps, which are introduced in different models, are generally over ambitious. However, we do take some crude assumptions always, because our knowledge of the interstellar chemistry is still incomplete. The gas-phase chemistry in interstellar clouds is mainly based on the ion-molecule reactions. Cosmic radiation could produce energetic ions to initiate these reactions. This is due to the fact that this kind of reactions require little or no activation energy and it can take place even at temperatures as low as 10 ◦K. The model provides Chapter 1. Introduction 46 an explanation for the unusual mix, by terrestrial standards, of molecules seen in interstellar clouds, and it still provides the basics of current understanding of the gas-phase chemistry of the clouds. Although the gas-phase chemistry is important, reactions on the surfaces of parti- cles as well as in the ice layers that form around them are have a great importance in the interstellar chemistry. The most abundant molecule in interstellar space, molec- ular hydrogen, wouldn’t be the most abundant if it were made only in gas-phase reactions. It has been noticed that the molecules formed in surface chemical reac- tion is different from the products of gas-phase chemistry. The molecules that form through surface chemistry are more hydrogenated, more saturated in the chemical sense, than the molecules formed in the gas phase. One of the distinctive features of gas-phase interstellar molecules is that they are highly unsaturated, with many multiple bonds, particularly between carbon atoms. It is necessary to choose the proper initial composition for the theoretical mod- eling of the interstellar chemistry. Therefore, it is to be noted that the good approx- imation of the initial abundances will reflect the correct abundances of the various interstellar species. Dalgarno and McCray (1972) and Pagel and Edmunds (1981) approximated the initial composition of the solar elements. The resulting Chemical composition of the interstellar cloud very strongly depends upon the different initial values for the carbon:oxygen ([C]/[O]) abundance ratio (Watt 1984). Primarily all the carbon and oxygen is channeled into CO. If excess Carbon remains after the formation of CO, is used to form the hydrocarbons and if excess oxygen remains this results the high abundances of OH and H2O. Both the oxygen chemistry and the carbon chemistry depend on the type of the cloud (i.e., the number density of

H2). Serious efforts have been made over the years to investigate the formation of such molecules in cool interstellar clouds in frigid conditions (Prasad and Huntress, 1980a, b; Leung et al., 1984; Hasegawa et al., 1992; Hasegawa and Herbst, 1993). It is now quite certain that the most important building block, namely, the molec- ular hydrogen (H2) and some of the other lighter molecules must be produced in the presence of grains (Gould and Salpeter, 1963; Hollenbach and Salpeter, 1971; Hollenbach et al., 1971). Several analytical and numerical works have successfully shown how the molec- Chapter 1. Introduction 47 ular hydrogen may have been produced (Biham et al., 2001, Stantcheva et al., 2002, Chakrabarti et al., 2006). A number of results are presented in the literature where hydrodynamic and chemical evolutions have been attempted simultaneously. For ex- ample, Shalabiea and Greenberg (1995) used the pseudo as well as partially real time- dependent models for the hydrodynamical evolution. In the pseudo time-dependent method, they assumed a constant density and temperature of the cloud using which the chemical evolution was computed. In their time-dependent model, they in- cluded the density and temperature variations throughout the cloud. However, in their initial approach to the time-dependent modeling, they assumed a constant temperature but allowed only the density to vary. Ceccarelli et al. (1996) used the “insideout”, isothermal, spherical collapse model of Shu (1977) and coupled it with a time-dependent chemical evolution code. They included the heating and the cooling process with an emphasis on the line emission. Shematovich et al. (1997) used Zeus 2D code which included the heating and the cooling. They present 1D hydrodynamic and chemical evolution of the proto-stellar cloud illuminated by the diffused interstellar UV radiation. They solved the equations of chemical kinetics, hydrodynamics and thermal balance simultaneously. In Lim et al. (1999) 2D nu- merical code was developed using the adaptive grid technique. Here, 454 reactions among 42 atomic and molecular chemical species were taken including the basic elements like H, He, C, N, O and a representative low ionization potential metal Na. At each grid point, the chemical evolution was followed by a calculation of the reaction rates using the local conditions obtained from the hydrodynamical flow. They primarily concentrated on the diffused clouds and emphasized the interfaces of the interstellar media and resulting dynamical mixing. Aikawa et al. (2005) stud- ied time-dependent evolution of Bonner-Ebert spheres by assuming clouds having a specific parameter α which is the ratio of the gravitational force to the pressure force. Recently, Acharyya et al. (2005) solved the Master equations and Rate equa- tions of Biham et al. (2001) for various cloud parameters and followed the evolution of H2 as a function of time. Both of this works and the earlier works of Chakrabarti and Chakrabarti (2000a) employed steady state matter distribution and assumed that the density and the temperature distributions at a given radial distance do not change with time. Chakrabarti and Chakrabarti (2000a) used a large number of species and the reaction rates were taken from the UMIST data base. Some of the reaction rates which were not available in the literature were assumed to be similar Chapter 1. Introduction 48 to other two body reactions. Subsequently, these new and assumed reaction rates were parametrized, with reaction rates up to a thousand times smaller compared to Chakrabarti and Chakrabarti (2000a) to include the effect of the size of the reac- tant molecules (Chakrabarti and Chakrabarti, 2000b). It was shown that even under frigid and tenuous conditions of the interstellar media, a significant and perhaps a detectable amount of simple amino acids and even important ingredients of DNA molecule (such as adenine) may form. Ceccarelli et al. (2000) estimated the upper 10 9 limit of the abundance of glycine to be about 10− (cooler outer cloud) to 7 10− × (hot core). Kuan et al. (2003) estimated the fractional abundance of glycine to be 10 9 10 2.1 10− for Sgr B2, 1.5 10− for Orion, and 2.1 10− for W51. These num- × × × bers are comparable to what was predicted in Chakrabarti and Chakrabarti (2000a), however, there are clearly some debate on the possible pathways for the formation of glycine with the route followed in the Chakrabarti and Chakrabarti (2000a,b). Similarly, there are also some debate on whether glycine is actually observed (Hollis et al., 2003; Snyder et al., 2005). Tarafdar et al. (1985) presented a model of chemical and dynamical evolution of isolated, initially diffused and quiescent interstellar clouds. A semi-empirically derived dependence of the observed cloud temperatures on the visual extinction and density was used in this work. Sorrell (2001) outlined a theoretical model for the formation of the interstellar amino acids and sugars. In this model, first ultraviolet photolysis creates a high concentration of free radicals in the mantles and the heat input due to the grain grain collision causes radicals to react chemically with another to build complex organic molecules. Armed with the present scenario of the subject discussed in this Chapter we have carried out the study of chemical evolution of the molecular cloud. These are discussed in the next Chapters of this Thesis. From the observational knowledge and the experimental results we modeled the interstellar dynamics. Chapter 2

Formation of Protostars

Stars are mainly formed in the relatively densest part of the interstellar cloud.

These regions are extremely cold (temperatures are about 10 to 20 ◦K). At these temperatures and densities, gases are mainly in the molecular form. A proto-star is the central region of a collapsing cloud fragment which is in the process of formation of a star. This has not yet become hot enough and does not have enough mass in the 7 core to initiate the process of nuclear fusion (10 ◦K) in order to halt its gravitational collapse. When the density reaches above a critical value, stars are formed. Since the star formation regions are dense and are opaque to the visible light, we use IR and radio telescopes to investigate them. Star formation begins when the dense parts of the cloud core collapse under their own gravity. These cores typically have masses of around 104 solar masses. The cores are more dense than the outer cloud. As a result, the cloud collapses rapidly.

2.1 Condition for Collapse

The Jean’s instability causes the collapse of a cloud and leads to the star formation. It occurs when the thermal pressure is not sufficient enough to prevent the gravi- tational collapse of a region filled with matter. To maintain the static condition, a cloud requires a certain balance between the kinetic energy and the gravitational energy. Virial theorem provides a general equation which relates the average total kinetic energy T of a system with its average total potential energy V . In h i h T OT i 49 Chapter 2. Formation of Protostars 50 mathematical form, the Viral theorem can be expressed as,

2 T = V . (2-1) h i h T OT i For the sake of simplicity, if we assume that the cloud is spherically symmetric then the total gravitational potential energy can be estimated by integration,

R 4πr2G Vtot = M(r) ρ(r) dr, (2-2) h i − Z0 r where M(r) is the mass within a radius r and ρ(r) is the stellar density at radius r; G represents the gravitational constant and R the radius of the cloud. Assuming a constant density throughout the cloud, this integration yields the formula,

3GM 2 V = . (2-3) |h toti| 5R Total Kinetic energy of the system is, 3 T = NKT (2-4) |h toti| 2 Now, if the gravitational potential energy exceeds twice the kinetic energy then the cloud will start to collapse. So,

3GM 2 > 3Nk T. (2-5) 5R B Assuming a constant density ρ throughout the cloud, we can express the mass of the cloud in the following form: 4 M = πR3ρ. 3 We replace N by, M N = , mp where, mp is the proton mass. Putting the values of N and R in Eqn. 2-5 we can obtain the mass for stellar contraction (Jeans mass). After some algebra, the Jeans mass (MJ ) takes the following form,

3/2 1/2 5kBT 3 MJ = . (2-6)  Gmp  4πρ Chapter 2. Formation of Protostars 51

So far, many approaches have been taken to study the formation of proto-stars. In our approaches towards the study of proto-stars, we have carried out the simula- tions in several ways. In the first approach, we studied the dynamical behaviour of the collapsing cloud by the Similarity Solution method. In the second approach, a spherically symmetric gas cloud is considered and the collapse due to its self-gravity is studied. In the third approach, we included the angular motion in the hydrody- namic flow and solved the hydrodynamic equations by Total Variation Diminishing (TVD) scheme.

2.2 Similarity Solutions for Self Gravitating Isothermal Flow

We consider an isothermal spherical gas cloud which is collapsing due to its self- gravity. Here, we follow the technique used by Shu (1977). The mass conservation equation for the spherically symmetric flow can be expressed as,

∂M ∂M + u =0, (2-7) ∂t ∂r

∂M =4πr2ρ, (2-8) ∂r where, M(r, t) is the total mass inside the radius the r, at time t. Above equations are equivalent to the usual continuity equation,

∂ρ 1 ∂(r2ρu) + =0. (2-9) ∂t r2 ∂r For an ideal isothermal flow, the force equation can be expressed as,

∂u ∂u a2 ∂ρ GM + u = . (2-10) ∂t ∂r − ρ ∂r − r2

In the above equations G is the gravitational constant, a is the isothermal sound speed, r is the local radius r, and t is the instantaneous time. The non-dimensional similarity variable is defined as, r x = . (2-11) at Chapter 2. Formation of Protostars 52

12 1 x 10 sec 12 -16 2 x 10 sec 12 4 x 10 sec 12 8 x 10 sec (|u|) 10

log -18 a = 0.2 km/s A=2.001

-20 14 15 16 17 18 log10(r)

1 Figure 2.1: Density distribution for A=2.001 and for sound speed a=0.2 km s− .

We now get the similarity solutions of the following form,

α(x) a3t ρ(r, t)= , M(r, t)= m(x), u(r, t)= av(x). (2-12) 4πGt2 G For the collapse problem, at t = 0, the mass of the core is assumed to be zero. For the wind problem it corresponds to the final instant when finally the central mass has blown away. Hence for the inflow problem t, x, and m must be positive while v is negative; for outflow, t, x, and m must be negative while v is positive. Replacing Eqns. 2-7 and 2-8 by using Eqn. 2-12, dm dm dm m x + v =0, = x2α. (2-13) − dx dx dx From the above relation, the value of m takes the following form,

m = x2α(x v). (2-14) − Equations 2-9 and 2-10 can be written as, dv 2 [(x v)2 1] = [α(x v) ](x v), (2-15) − − dx − − x − Chapter 2. Formation of Protostars 53

7

12 1 x 10 sec 12 2 x 10 sec 6 12 4 x 10 sec 12 8 x 10 sec

5 (|u|) 10

log 4 a = 0.2 km/s A=2.001

3

2 14 15 16 17 18 log10(r)

1 Figure 2.2: Velocity distribution for A=2.001 and for sound speed a=0.2 km s− .

1 dα 2 [(x v)2 1] = [α (x v)](x v). (2-16) − − α dx − x − − An exact solution of Eqns. 2-14, 2-15, and 2-16 is given by the static state, 2 v =0, α = , m =2x. (2-17) x2 In terms of the dimensional quantities it may be written as,

a2 2a2 u(r, t)=0, ρ(r, t)= , M(r, t)= r. (2-18) 2πGr2 G Now at v 0 as x . From Eqns. 2-14, 2-15 one may obtain the following → → ∞ solutions, A α , (2-19) ∼ x2

v (A 2)/x, (2-20) ∼ − −

m Ax, as x , (2-21) ∼ →∞ Chapter 2. Formation of Protostars 54 where, A is a constant. When we consider the inflow (i.e., v negative), we choose A to be grater than 2. The initial density distribution in the self-similar form is the following, 2 a A 2 ρ(r, 0) = r− . (2-22) 4πG This corresponds to a singular isothermal sphere. Now the collapse of this isothermal sphere occurs if A > 2. From Eqns. 2-19 and 2-20, A > 2. This implies that the 2 envelope density distributions is in the r− form. For small x, i.e., near the origin, Eqns. 2-15 and 2-16 take the following form: m 2m m m , α 0 , v 0 as x . (2-23) → 0 → r2x3 → −r x →∞ In Figs. 2.1 and 2.2 the density and velocity profile is shown respectively for the 1 sound speed a =0.2 km s− and A =2.001. As the time passes by, density evolves towards the higher density. In Fig. 2.2 the absolute value of velocity is plotted. Since we are studying the collapsing cloud, the sign of the velocity is negative. From Figs. 2.1 and 2.2, it is clear that the inner part of the cloud exhibits a power-law 3/2 1/2 dependence, ρ r− , u r− . → →

2.3 Collapse of a Spherically Symmetric Gas cloud

The time dependent simulations have been carried out by various authors using both the Lagrangian and Eulerian methods. For example, Prasad et al. (1991) and Tarafdar et al. (1985) used a Lagrangian scheme with a semi-empirical approach for the energy equation. Shalabiea and Greenberg (1995) used a partially time dependent model in which only the densities are allowed to be time dependent during collapse of the cloud. We have used a finite difference Eulerian scheme (upwind scheme) to solve the Euler equations on a spherical grid. We consider a self-gravitating, collapsing, spherically symmetric gas cloud. We choose the co- ordinate system to be (r,θ,φ) with origin at the center of the proto-star. The Eulerian equations of hydrodynamics written in spherical coordinates are given by: ∂ρ + .(ρV )=0, (2-24) ∂t ∇

∂(ρv ) ∂Φ ∂P ρ r + .(ρv V )= (ρ + )+ (v 2 + v 2), (2-25) ∂t ∇ r − ∂r ∂r r θ φ Chapter 2. Formation of Protostars 55

∂(ρv ) 1 ∂Φ ∂P ρ θ + .(ρv V )= (ρ + ) (v v v 2cotθ), (2-26) ∂t ∇ θ −r ∂θ ∂θ − r r θ − φ

∂(ρA) ∂Φ ∂P + .(ρAV )= (ρ + ), (2-27) ∂t ∇ − ∂φ ∂φ where Eqn. 2-24 is the continuity equation and Eqns. 2-25, 2-26, 2-27 describe momentum transfer. Here ρ is the mass density, V =(vr, vθ, vφ) is the fluid velocity, and A = rsinθvφ is the specific angular momentum. The gravitational potential Φ is determined by the Poisson’s equation,

2Φ=4πGρ. (2-28) ∇ Since we are considering the spherically symmetric case, we can neglect θ and φ component of motions. By considering only the radial component, the above Eqns. (2-24 to 2-27) take the following forms:

∂ρ 1 ∂(ρv r2) + r =0, (2-29) ∂t r2 ∂r

∂(ρv ) 1 ∂(ρv2r2) ∂Φ ∂P r + r = (ρ + ). (2-30) ∂t r2 ∂r − ∂r ∂r Since we are considering only the radial component, we do not need to solve the Poisson’s equation (Eqn. 2-28). The gravitational potential for this case can be calculated by,

Φ= GM(r)/r, (2-31) − where, M(r) is the mass of the cloud inside radial distance r. We assume the ideal gas equation to be,

P = ρkT/µmp, (2-32) where, k is the Boltzmann constant, T is the local temperature, µ is the mean molecular weight, mp is the proton mass. Chapter 2. Formation of Protostars 56

Solution Procedure

We solve the above equations on a logarithmically equal spaced grids along the radial directions. The code is developed to study the collapsing spherical hydrodynamical flow which is strictly one dimensional. Since this is a one dimensional flow, no back-flow is possible. We use the first order upwind differencing method here. After splitting, Eqns. 2-29 and 2-30 look like the following:

dt ρj+1 = ρj (ρj v j r2 ρjv jr2), (2-33) i i − r 2(r r ) i+1 ri+1 i+1 − i ri i i i+1 − i

dt 2 2 2 2 ρj+1v j+1 = ρjv j (ρj v j r2 /ρj ρj v j r2/ρj), i ri i ri − r2(r r ) i+1 ri+1 i+1 i+1 − i ri i i i i+1 − i dt [ρj(Φj Φj)+(P j P j)]. (2-34) −(r r ) i i+1 − i i+1 − i i+1 − i Here, i denotes the index for the radial grid and j denotes the index for the time. To avoid the instability in the code, we chose the operating time step by using the Courant-Friedrichs-Lewy stability criterion, which gives,

v ∆t/∆r 1, (2-35) | | ≤ i.e.,

∆t ∆r/ v , (2-36) ∼ | | where, v is the magnitude of the maximum velocity, ∆t is the time step, and | | ∆r is the grid spacing along the radial direction. We always advance the time step after ensuring that the Courant condition is satisfied. To be on the safer side, we chose time step dt = ∆t/2. The potential at any point is computed as a sum of two terms, one coming from the cloud itself and other is due to the cloud core. The contribution from the cloud is,

Φ = GM (r)/r, cloud − cloud where, Mcloud(r) is the mass between rin and rout. rin is the inner edge of the computational grid and rout is the outer edge of the grid. The cloud mass (Mcloud) Chapter 2. Formation of Protostars 57

1e+08

1e+06

30 year 83 year 10000 633246 year 3165693 year 6333528 year

Velocity (cm/sec) 100

1 1e+14 1e+16 1e+18 Radius (cm)

Figure 2.3: Evolution of velocity for Model A is shown. At the beginning, the cloud has a constant velocity. At a later stage it assumes an almost steady state with v(r) r 4/5. ∼ − at jth time step is calculated by adding contributions from each spherical shell,

j 2 j Mcloud = Σi4πri driρi , (2-37) where, i =1, 2, .....N and the densities at the jth time step are used. We put r = rout to get the potential at the outer boundary. The contribution from the cloud core is,

Φ = GM /r, core − core where, Mcore is the mass of the core, which can be obtained by calculating the mass within the grid rin. Thus the value of Φ(r) that is required during the simulation is dynamically calculated from the mass within r, i.e.,

Φ= G(M + M )/r. − cloud core We use two models to study the hydrodynamic evolution (Das et al., 2008): in the first model, we inject matter into an initially low density region and in the second model, we start with a constant density cloud and let it to collapse due to self-gravity. We study the evolution of the central core for both the cases. Chapter 2. Formation of Protostars 58

1e-15

30 year 156254 year 385483 year 633246 year 1e-18 3165693 year 6333528 year

1e-21

1e-24 Density (gm/cc)

1e-27 1e+14 1e+16 1e+18 Radius (cm)

Figure 2.4: Density distribution for Model A is shown. From a constant density cloud, it assumes a power-law density distribution. As the core becomes massive enough, it starts to evacuate the matter of the cloud very rapidly, as a result the density over the entire cloud gradually decreases as is shown by the dot-dashed curve. Chapter 2. Formation of Protostars 59

1e-15

1e-18

Central grid 1e-21 Inner grid

1e-24 Density (gm/cc)

1e-27 1e+08 1e+10 1e+12 1e+14 Time (sec)

Figure 2.5: Evolution of densities at the central grid and the middle grid for Model A. Towards the end of simulation, the density starts to decrease as the cloud is evacuated by the massive core.

Model A

In this case, we start with a grid of size rout. Initially, the cloud contains a negligible amount of mass. Using this model, we mimic the formation of the cloud itself from a supply of matter from a large reservoir of diffused gas. From the outer boundary, ˙ 2 we inject matter at a constant rate of M =4πρoutvoutrout, where, ρout is the injection density and vout is the injection velocity at the outer boundary (r = rout).

Here, we choose 100 logarithmically equal spaced grid points in between rinandrout. At each time step, we follow the amount of matter which is going inside the grid rin. The rate of increase of Mcore is,

dM core =4πρ v r2 , (2-38) dt in rin in where, ρin is the density at the inner grid point, vrin is the velocity at the inner grid point, rin is the radial distance of the inner grid point from the center of the molecular cloud. Similarly, the mass of the cloud is also dynamically updated. The Chapter 2. Formation of Protostars 60

1e+35

1e+30

1e+25

1e+20 Core mass (gm)

1e+15

1e+08 1e+10 1e+12 1e+14 Time (sec)

Figure 2.6: Evolution of core mass for Model A is shown. After getting enough mass in its core it starts to draw the mass rapidly from the cloud and as a result it grows up very fast.

flow is assumed to be isothermal (T = 10 ◦K) throughout the simulation. In our calculation. we choose, r =3.98 1013cm and r to be 3.54 1018cm. in × out × We assume that anything going inside rin will increase the mass of core while the 18 radius of the core remains rin. We choose rout to be 3.54 10 cm. We have 27 3 × 1 started our simulation with ρint = 10− gm cm− and vint = 50 cms− in all the grids (except on the boundary). Thus, we start with a core of a negligible mass of 3 14 4 1 (4/3)πrinρint =2.64 10 gm. We inject matter at vout = 10 cm s− from the outer × 22 3 4 1 boundary. Here, we choose ρout = 10− gm/cm , vout = 10 cm s− . Thus, the rate of injected matter is given by,

˙ 2 M =4πρoutvoutrout. (2-39)

20 1 Putting the values in the Eqn. 2-39, M˙ comes out to be 1.57 10 gm s− . × The injected matter moves in due to self-gravity of the cloud and the attraction of the core. The simulation was carried out till a few times 106 yr. Fig. 2.3 shows how the velocity profile of the flow changes with time. Time (in years) is marked Chapter 2. Formation of Protostars 61 on each curve. From the Fig. 2.3 it is realized that the flow which began with α a constant velocity, eventually, assumes an almost steady state (v(r) r− with ∼ α 4/5). The evolution of the density distribution is shown in Fig. 2.4. Here ∼ too, an initially constant density distribution assumes a power-law distribution of 3/2 ρ(r) r− toward the end of the simulation. Time (in years) is marked on each ∼ curve. Note that once the core becomes massive, it starts to evacuate the grids and thus the density over the entire cloud gradually decreases, as is shown by the dot-dashed curve drawn at 6.33 106 yr. After about t = 1013 s, i.e., about ∼ × 3 105 yr, densities on each grid started increasing after the empty grids are filled × in by the inflowing matter. This is shown in Fig. 2.5 where the time evolutions of the densities on the 50th grid (solid curve) and the innermost grid (dotted curve) are shown. Towards the end, we note that the densities at these grids gradually decrease as the cloud starts to become empty. Meanwhile, the core starts to grow because of the mass accretion during this period. Fig. 2.6 shows how the mass of the core is evolving.

Model B

We assume that we have a finite sized r = rres reservoir of matter and we are considering only the inner region of size rout for computational purpose. The matter is coming from the reservoir and it is evacuated to form the star itself. The inward velocity is computed self-consistently from the gravitational pull between the matter inside rout and that outside of rout. The whole cloud is assumed to be isothermal

(with T = 10 ◦K) and has a constant density ρint throughout. The size of the whole cloud is rres >> rout. We assume that Mtotal = Mcore + Mcloud + Mout, where, Mcore is the core mass within r = rin which will increase at the same rate as in Eqn. 2.38,

Mcloud is the mass of the cloud within the computational grid, i.e., in between rin and rout and Mout is the mass in between the radius rres and rout. Thus, Mout is the reservoir mass which is depleted as the matter is injected within r = rout. The injection velocity is dynamically calculated in the following way. First, we note that an approximate expression for the acceleration of matter at the outer edge of the computational grid could be taken as,

G(Mcloud + Mcore) fout 2 , (2-40) ∼ rm Chapter 2. Formation of Protostars 62

1e+10

1e+08

190 year 1e+06 3834 year 74124 year 2501220 year Velocity (cm/sec) 10000

100 1e+14 1e+16 1e+18 Radius (cm)

Figure 2.7: Velocity distribution for Model B is shown.

1e-16

1e-18

1e-20

Density (gm/cc) 190 year 3834 year 1e-22 74124 year 2501220 year

1e-24 1e+14 1e+16 1e+18 Radius (cm)

Figure 2.8: Density distribution for Model B is shown. Towards the end of the simulation the density of the entire cloud drops, due to the high rate of mass accretion at the core. Chapter 2. Formation of Protostars 63

1e-16

1e-18

central grid inner grid

Density (gm/cc) 1e-20

1e-22 1e+10 1e+12 1e+14 Time (Sec)

Figure 2.9: Evolution of densities at the central grid and the middle grid for Model B.

where, rm =(rres + rout)/2. Hence the injection velocity is,

vout foutrm, (2-41) ∼ p and the average density of the injected matter is,

Mout ρout = , (2-42) Vout where, V =4/3π(r3 r3 ). By this process, the large cloud would be gradually out res − out evacuated as the star is formed. The cloud mass Mcloud inside rin

dM cloud =4π(ρ v r2 ρ v r2 ), (2-43) dt out out out − in in in where, vin and ρin are respectively the velocity and density at the inner boundary rin. As in the Model A, here also we choose, r = 3.98 1013cm and use 100 grid in × points, which are spaced equally in the logarithmic scale along the radial direction. Chapter 2. Formation of Protostars 64

1e+35

1e+34

1e+33

Mass (gm) 1e+32

1e+31

1e+30 1e+10 1e+12 1e+14 Time (sec)

Figure 2.10: Evolution of the core mass for Model B is shown, here core mass increases rapidly at the beginning, but the rate of evolution slows down due to the depletion of the cloud mass.

20 18 We assume that rres = 10 cm, and rout = 3.16 10 cm. At the start of the 22 3 × 1 simulation we assume that ρint = 10− gm cm− and vint = 10 cm s− at each grid point. Initially, the core mass was (4/3)πr3 ρ =2.64 1019 gm. in int × The velocity assumes almost a steady state after some initial transient time as before. The evolution of velocity is shown in Fig. 2.7. In Fig. 2.8, the evolution of density is shown. Initially, the density of the cloud increases and a new matter is injected from the cloud external to the grid, but as the time proceeds, the core grows while the external cloud is evacuated. This results in eventual decrease in density of the cloud as shown by the dot-dashed curve in Fig. 2.8. Similarly, from Fig. 2.9, it is clear that the density at the middle grid (solid curve) and the innermost grid (dotted curve) will also go down due to the growth of the core mass. In Fig. 2.10, the evolution of core is shown. Unlike in the previous model, the core mass starts growing from the very beginning of the simulation, but the rate of growth is slowed down when the cloud mass is depleted. Model A and Model B, which were used in our simulations were two complimen- Chapter 2. Formation of Protostars 65 tary models. In Model A, we started with an almost empty cloud. The mass of the cloud and the core grew up rapidly due to the accretion of the matter from the outer boundary. In Model B we started with a cloud having sufficient amount of mass. Once the mater is evacuated by the core, matter from the outer reservoir starts to accrete. As a result the core grew very rapidly during the first phase compare to the last phase.

2.4 Collapse of a Rotating Gas Cloud

So far, we have done the hydrodynamic simulation for a spherically symmetric gas cloud. The code was developed to study the dynamic behaviour of a spherically symmetric collapsing cloud. But in order to study the chemical evolution properly it is necessary to study the dynamic behavior of the cloud more accurately. We have started to improve our the hydrodynamic model by including the angular motion and shock formation in the flow. During the hydrodynamic flow the jets and outflows would form and a part of this will fall back on the disk and the matter would be recycled (Chakrabarti et al., in prep.). We solve the hyperbolic system of the following conservation equations for a molecular cloud:

∂ρ ∂ + (ρuk)=0, (2-44) ∂t ∂xk

∂(ρui) ∂ + , (ρuiuk + pδik)=0 (2-45) ∂t ∂xk

∂E ∂ + [(E + p)uk]=0, (2-46) ∂t ∂xk where, E = p/(γ 1)+ ρu 2 is the total energy per unit volume and the rest of the − k variables have their usual meanings. We use TVD (total variation diminishing) schemes to solve the above hydro- dynamic equations. Harten’s TVD scheme (Harten, 1983) is an explicit, second order Eulerian finite difference scheme which solves a hyperbolic system of the con- servation equations. The key merit of this scheme is to achieve a high resolution. Chapter 2. Formation of Protostars 66

This scheme is relatively simple to program compared to the other high accuracy numerical schemes and require less CPU time. The Eqns. 2-44, 2-45, and 2-46 can be written in the vector form as,

∂tq + ∂xFx + ∂yFy + ∂zFz =0, (2-47)

ρ   ρvx q = ρv  . (2-48)  y   ρvz     E    The flux functions are,

ρvx ρvy ρvz  2      ρvx + p ρvxvy ρvxvz F =  ρv v  , F =  ρv 2 + p  , F =  ρv v  , (2-49) x  x y  y  y  z  y z       2   ρvxvz   ρvyvz   ρvz + p        (E + p)vx (E + p)vy (E + P )vz       where, the equation of state is given by E = p/(γ 1)+ρ(v 2 +v 2 +v 2)/2. With the − x y z vector q and the flux functions, Fx(q), Fy(q), Fz(q), the Jacobian matrices, Ax(q)= ∂Fx ∂Fy ∂Fz ∂q , Ay(q) = ∂q , andAz(q) = ∂q , are formed. The corresponding eigenvalues of Ax(q) are, a = v c, a = v , a = v , a = v , a = v + c, 1 x − 2 x 3 x 4 x 5 x γp where c is the sound speed √ρ , The corresponding right eigenvectors are,

1 0 1  v c   0   v  x − x R =  v  , R =  1  , R =  v  , 1  y  2   3  y         vz   0   vz        H vxc vy Θ/2  −      Chapter 2. Formation of Protostars 67

0 1     0 vx + c R =  0  , R =  v  , 4   5  y       1   vz      vz H + vxc     2 2 2 where H =(E + p)/ρ is the enthalpy and Θ = vx + vy + vz . The left eigenvectors which are orthonormal to the right eigenvector, Li.Rm = δlm are obtained similarly. (γ 1)Θ/2+ cv (γ 1)v + c (γ 1)v (γ 1)v γ 1 L = [ − x , − x , − z , − z , − ], 1 2c2 − 2c2 − 2c2 − 2c2 2c2 L =( v , 0, 1, 0, 0) 2 − y (γ 1)Θ (γ 1)v (γ 1)v (γ 1)v γ 1 L = [1 − , − x , − y , − z , − ], 3 − 2c2 2c2 2c2 2c2 − c2 L =( v , 0, 0, 1, 0), 4 − z (γ 1)Θ/2 cv (γ 1)v c (γ 1)v (γ 1)v (γ 1) L = [ − − x , − x − , − y , − z , − ]. 5 2c2 − 2c2 − 2c2 − 2c2 2c2 In the TVD scheme which is based on the Eulerian grid, the flux is computed on the grid boundary while the physical quantities are defined in the grid center. We use the Roe approximate Riemann solution (Roe, 1981) to get the averaged values of the physical quantities at the grid boundary:

√ρivx,i + √ρi+1vx,i+1 vx,i+1/2 = , √ρi + √ρi+1

√ρivy,i + √ρi+1vy,i+1 vy,i+1/2 = , √ρi + √ρi+1

√ρivz,i + √ρi+1vz,i+1 vz,i+1/2 = , √ρi + √ρi+1

√ρiHi + √ρi+1Hi+1 Hi+1/2 = , √ρi + √ρi+1

1 c = [(γ 1)[H (v2 + v2 + v2 )]]1/2. i+1/2 − i+1/2 − 2 x,i+1/2 y,i+1/2 z,i+1/2 To incorporate the self-gravity in this code, we introduce Poisson’s equation. We solve the Poisson’s equation by using SOR (successive over-relaxation) scheme. Chapter 2. Formation of Protostars 68

30 35 30 25 25 20 20 15 10 15 5 0 10 5 1e12 sec 5e12 sec 1e13 sec

30 35 30 25 25 20 20 15 10 15 5 0 10 5 5e13 sec 5e14 sec 1.5e14 sec

30 35 30 25 25 20 20 15 10 15 5 0 10 5 2e14 sec 2.5e14 sec 3e14 sec

Figure 2.11: Evolution of density of a layer is shown. Initially there was a negligible amount of mass inside the cloud. The density of the inner region increases rapidly as the collapse progresses (Chakrabarti et al., in prep.). Chapter 2. Formation of Protostars 69

This is an iterative method used to speed up the convergence of the Gauss-Seidel method. In order to get much convergent results we use Chebyshev Acceleration (CA) in SOR. Introduction of the CA gives the asymptotic rate of convergence in SOR. The significance of CA is that the error always decreases with the each iteration. 18 We take a molecular cloud having size 10 cm, temperature 10 ◦K and let it to collapse due to its self gravity. We divide the region of molecular cloud in 32 equal 20 spaced grids in the x and z direction. We are injecting matter of density 10− gm 3 cm− from the outer boundary of the cloud. We take a sink inside the cloud and assume that any mass going inside will increase the mass of the core. In Fig. 2.11 density evolution of the gas in the meridional cross-section is shown. As the time evolves, the density in the inner region of the cloud is growing up. We plot the density distribution using different colours. Yellow is of highest density while violet is of lowest density. The colour code tells us the density in some of the regions are 20 3 35 times higher than the injection density (10− gm cm− ). In the next Chapters, we will study the chemical evolution on the grains. The hy- drodynamics of the gas presented here and the grain chemistry described next would be combined to study the chemical evolution of the cloud. Chapter 3

H2 Formation

It is well recognized that the dust grains play a major role in the formation of molecular hydrogens in the Interstellar Medium (ISM) (Gould and Salpeter, 1963). The formation of molecular hydrogen is a key process which affects the thermal and density structure of the ISM. Several approaches have been taken to understand the real physical processes which are taking place both theoretically (e.g., Hollenbach, Werner and Salpeter, 1971; Takahashi, Matsuda and Nagaoka, 1999; Biham et al. 2001) as well as experimentally (e.g., Pirronello et al. 1997a,b, 1999). A significant production of H molecule is possible in cooler ( 10 25 ◦K) clouds. Cazaux 2 ∼ − and Tielens (2002, 2004) use both physisorption and chemisorption, to demonstrate that the H production is possible at high temperatures ( 200 400 ◦K) also. 2 ∼ − Biham et al. (2001), and Green et al. (2001) have computed H2 production rate by physisorption. Hollenbach and Salpeter (1970) were the first to introduce the grain surface reactions. Since then it has been used very extensively by several authors (Watson and Salpeter 1972ab; Allen and Robinson 1975, 1976, 1977; Tielens and Hagen 1982; Hasegawa and Herbst 1992; Charnley 2001; Stantcheva et al. 2002, Green et al. 2001; Biham et al. 2001; Stantcheva et al. 2002). These studies mainly belong to two categories – one is the deterministic approach and the other is the stochastic approach. In the deterministic approach, one can completely determine the time evolution of the system, once the initial conditions are known. The Rate equation method belongs to this category. This method is very extensively used by several authors to study the grain surface chemistry (Hasegawa and Herbst 1992; Roberts et al. 2002; Acharyya et al. 2005).

70 Chapter 3. H2 Formation 71

1000

H2 H

100 Number 10

1 100 10000 1e+06 1e+08 1e+10 Time (sec)

4 Figure 3.1: Evolution of the H and H2 on an Olivine grain having 10 sites is shown.

The grain is kept at temperature 8 ◦K and facing an effective accretion rate per site of H 7.98 10 8 s 1. × − −

However, this method is applicable only when there are a large number of reactants on the grain surface. Given that the interstellar medium is very dilute, very often this criteria is not fulfilled and this method cannot be applied. But this method is computationally faster and can very easily be coupled with the gas phase reactions. In the stochastic approach, the fluctuations in the surface abundance due to the statistical nature of the grain is preserved. The Monte-Carlo method and the Master equation methods belong to this category. Both these methods are used by several authors (Charnley 2001; Stantcheva et al. 2002, Green et al. 2001; Biham et al. 2001; Stantcheva et al. 2002).

3.1 Procedures to Handle the H2 Formation

There are different approaches to handle the H2 formation on the grain. Followings are different methods which are used to study the H2 formation on the grain. Chapter 3. H2 Formation 72

3.1.1 Rate Equation Method

If nH be the number of H atoms on a grain at time t and nH2 be the number of

H2 molecules at that instant, then the following equation gives the rate at which number of H changes:

dn H = φ W n 2(A /S)n2 , (3-1) dt H − H H − H H where, φ =F (1 f f ). F is the accretion rate of H. F is calculated by H H − grh − grh2 H H using the relation 1-11, i.e.,

1 FH = scσdvHNH s− where, NH is the number density of hydrogen, vH is the thermal velocity in the gas phase, σd is the cross section of the dust grain, sc is the sticking coefficient (assumed to be 1 for this case). ∼ The term in the parenthesis, known as the Langmuir-Hinshelwood (LH) rejection term, which have a great importance at the higher accretion rate limit. fgrh and fgrh2 are the fraction of the sites occupied by atomic hydrogen and molecular hydrogen respectively. As a whole, the first term of Eqn. 3-1 represents the effective accretion rate at any given time. The second term of Eqn. 3-1 refers to the desorption of hydrogen from the grain surface. WH is the desorption co-efficient of hydrogen =ν exp( E /k T), where, E is the activation barrier energy for desorption of H − 1 b 1 atom, kb is the Boltzmann’s constant and T is the temperature of the grain, assumed to be the same as the gas. The second term of Eqn. 3-1 causes a reduction of the number of hydrogen atoms on the grain, hence the minus sign. On the grain surface, mainly due to the diffusive processes, two H atoms combine to form a single H2 molecule. A = ν exp( E /k T), the hopping rate, gives the probability of this to H − 0 b happen. In the third term of Eqn. 3-1 this recombination is included. The factor 2 on the third term of Eqn. 3-1 represents the production of a H2 molecule is subject to the decrease in the number of two H atoms.

The rate of production of H2 on the grain surface is given by,

dn H2 = µ(A /S)n2 W n , (3-2) dt H H − H2 H2 where, WH2 is the desorption co-efficient of hydrogen molecule and it is calculated by, ν exp( E /k T). The parameter µ represents the fraction of H molecules that − 2 b 2 Chapter 3. H2 Formation 73

(a) (b)

(c) (d)

Figure 3.2: In (a-b) the pictures of the grain is shown at two intermediate times (a) 8 108s and at (b)109s. An Olivine grain having 900 sites has been chosen for this × simulation. The grain is kept at 8 ◦K and is bombarded with an accretion rate per site of H 3.02 10 7 per sec. No spontaneous desorption has been assumed here. In (c-d) × − spontaneous desorption has been included and plotted for the same time as before. Thus, numbers of H2 residing on the grain at any instant are lesser. Chapter 3. H2 Formation 74

1.17

1.14

0 1.11 α

1.08

1.05

1e-14 1e-12 1e-10 1e-08 1e-06 -1 as(s )

Figure 3.3: α0 as a function of as is shown for various Olivine grains kept at 8 ◦K. The dashed, dot-dashed and solid curves are for S = 106, 9 104, and 104 respectively. × remains on the surface upon formation while (1-µ) fraction is desorbed due to the energy released in the recombination process. The H2 production rate RH2 in the gas due to grain is then given by,

R = (1 µ)(A /S)n2 + W n . (3-3) H2 − H H H2 H2 For a fixed flux and a fixed temperature a steady state is reached after some transient dnH time steps. The steady state solution is easily obtained by putting dt = 0 and dnH2 dt = 0. So from Eqns. 3-1 and 3-2 we can have,

2 2 µaHFH [(WH +FH) + 8(aHFH + )] (FH + WH) 2WH2 − nH = q 2 , (3-4) µaHFH 4(aHFH + ) 2WH2

2 µaHnH nH2 = . (3-5) WH2 The recombination efficiency (η) is defined as the fraction of the adsorbed H atoms that evaporate in the form of H2 molecules. Mathematically it can be expressed as the following: Chapter 3. H2 Formation 75

Figure 3.4: α0 as a function of as is shown for various Olivine grains kept at 10 ◦K. The dashed, dot-dashed and solid curves are for S = 106, 9 104, and 104 respectively. ×

R η = H2 . (3-6) FH/2 The recombination efficiency is highly dependent on the temperature. For astro- physically relevant flux range, Olivine type of grain may produce molecular hydro- gen very efficiently in the temperature range 7 ◦K to 9 ◦K. For Amorphous carbon the temperature window shifts towards 12 ◦K to 16 ◦K. If we increase the flux the window shifts towards much higher temperature.

3.1.2 Master Equation Method

The Rate equation method gives very good estimation of the grain surface chemistry when there are large numbers of reactants on the grain surface. But in general, ISM is very dilute in nature and most of the time this criteria is not fulfilled. But this method is computationally faster and can very easily be coupled with the gas phase reactions. To do the exact calculations in the dilute medium, we often use the Master equation method (Biham et al., 2001). This method rely on probabilistic method and is suitable for the study of the H2 formation on the small grain. Suppose the Chapter 3. H2 Formation 76 grain be exposed to an accretion rate FH of H atoms and suppose at any given time the number of H atoms adsorbed on the grain is nH. The probability that there are nH hydrogen atoms on the grain is given by PH(nH), where,

Σn∞H=0PHnH =1. (3-7)

We use the same type of equations as Eqns. 3-1 and 3-2, except that the values of nH and nH2 are denoted by their expectation values. After modification, the Eqns. 3-1 and 3-2 become

d < n > H =F W < n > 2(A /S) < n >2, (3-8) dt H − H H − H H and d < n > H2 =F + µ(A /S)< n >2 W < n >. (3-9) dt H2 H H − H2 H2

Here, FH2 is the accretion rate of H2 molecule. In Eqn. (3-2), we did not consider the effect of FH2 because it was assumed that the molecular flux is negligible. Here, we neglect the LH term because this method is valid in the region of the low accretion rate, where LH term does not have much importance.

The time derivatives of these probabilities, P˙ H(NH) is given by (Biham et.al. 2001),

P˙ (n ) = F [P (n 1) P (n )] H H H H H − − H H + W [(n + 1)P (n + 1) n P (n )] H H H H − H H H + (A /S)[(n + 2)(n + 1)P (n + 2) n (n 1)P (n )]. H H H H H − H H − H H (3-10)

Similarly, the probability that there are nH2 number of hydrogen molecules on the grain is given by PH2 (nH2 ). The time evolution of these probabilities is given by (Biham et. al. 2001),

P˙ (n ) = F [P (n 1) P (n )] H2 H2 H2 H2 H2 − − H2 H2 + W [(n + 1)P (n + 1) n P (n )] H2 H2 H2 H2 − H2 H2 H2 + µR [P (n 1) P (n )]. H2 H2 H2 − − H2 H2 (3-11) Chapter 3. H2 Formation 77

1.5

1.2 0

β 0.9

0.6

0.3 1e-14 1e-12 1e-10 1e-08 1e-06 -1 as(s )

Figure 3.5: Variation of β0 as a function of as, for various Olivine grains kept at 8 ◦K. The dashed, dot-dashed and solid curves are for S = 106, 9 104, and 104 respectively. ×

From these probabilities, we can obtain the expectation values for the number of hydrogen atoms on the grain as,

< nH >= Σn∞H =0nH PH (nH ), (3-12) and the expectation value for the number of hydrogen molecule is,

< n 2 >= Σ∞ n 2 P (n 2 ). (3-13) H nH2 =0 H H H

The time dependence of these exception values, obtained from Eqns. 3-8, 3-9, is given by

d < n > H =F W < n > 2(A /S) < n (n 1) >, (3-14) dt H − H H − H H H −

d < n > H2 =F + µ(A /S) < n >< n 1 > W < n >. (3-15) dt H2 H H H − − H2 H2 This expression is analogous to the Eqns. 3-8 and 3-9 apart from one important difference: the recombination term < n >2 is replaced by < n 2 > < n >. On H H − H Chapter 3. H2 Formation 78 a macroscopically large grain the difference between these terms will be small, and Eqns. 3-8 and 3-9 will provide a good approximation. However, for the small grains where < NH > is small, the difference between these terms is significant and need to be considered to avoid the overestimation. The number of hydrogen molecules which are released back into the gas is given by,

R = (1 µ)(A /S) < n >< n 1 > +W < n > F . (3-16) H2 − H H H − H2 H2 − H2 Similarly, the recombination efficiency which is defined as the fraction of adsorbed

H atoms that desorb in the form of H2 is given by,

η =RH2 /(FH/2). (3-17)

Master equation is normally used when < nH > is of the order of unity. When

< nH >> 1 master equation should be replaced by the Rate equation to save the computational time.

3.1.3 Monte-Carlo Approach

Monte Carlo method is a very efficient method to study the grain chemistry ac- curately. Much like the Master equation method, here also the fluctuations of the surface abundances are also taken into account. For the sake of simplicity, we assume each grain to be square in shape having S = n2 number of sites (square lattice). We use periodic boundary condition to reflect the behaviour of a closed surface.

Hopping time step (th =1/AH) is chosen to be the minimum time step for the sim- ulation (here we do not consider the tunneling effect). We advance the global time by this minimum time step. If φH is the effective accretion rate (as defined above), after every time step, i.e., after every 1/AH seconds, φH/AH number of hydrogen atoms are dropped on the grain. If φH is too low so that φH/AH < 1, then, clearly, one H is dropped after every AH/φH steps. The exact site at which one atom is dropped is obtained by a pair of random numbers (Rx, Ry, Rx, Ry < 1) obtained by a random number generator. This pair would place the incoming hydrogen at (i, j)th grid, where, i and j are the nearest integers obtained using the Int function: i = int(nRx +0.5) and j = int(nRy +0.5). Each atom starts hopping with equal probability in all four directions, which was decided by another random number. Chapter 3. H2 Formation 79

Figure 3.6: Variation of β0 as a function of as, for various Olivine grains kept at 10 ◦K. The dashed, dot-dashed and solid curves are for S = 106, 9 104, and 104 respectively. ×

During the hopping process when one atom enters a site which is already occupied by another atom, we assume that a molecule is formed and increase the number of

H2 by unity. However, when the atom enters a site occupied by an H2, another ran- dom number is generated to decide which one of the other neighbouring sites, it is going to occupy. Thermal evaporation of H and H2 from a grain surface are handled in the following way. Since WH is the desorption rate for H, one atom is supposed to be released to the gas phase after every 1/WH seconds. We generate a random number Rt for every H present on the grain and release (at each time step, i.e., 1/AH seconds) only those for which Rt < WH/AH. A similar procedure is followed for the evaporation of H2 for which the criterion for evaporation was Rt < WH2 /AH.

Due to the spontaneous desorption, a factor of (1-µ) of nH2 is lost to the gas phase. Here too, a random number Rs is generated for each newly formed (within that time step) H present on the grain. Those which satisfy R < (1 µ) are 2 s − removed to the gas phase. Chapter 3. H2 Formation 80

3.2 Results

H2 is the the most abundant species in the ISM. More careful theoretical treatment is necessary to study the formation of H2. Since the over or under production of the H2 may seriously affect the other hydrogen bearing species, which in turn completely change the chemical composition of the simulated molecular cloud. It has been noticed that the recombination time seems to be functions of the grain parameters such as the activation barrier energy, temperature etc. The average time that a pair of atomic hydrogens will take to produce one molecular hydrogen depends on how heavily the grain is already populated by atomic and molecular hydrogens and how fast the hopping and desorption times are. So far, in the literature the diffusion time is defined by Eqn. 1-14. The diffusion rate is just inverse of this time scale, i.e.,

1 AH rdiff = = . Sthop S The argument for reducing the rate by a factor of S is this: on an average, there are S1/2 number of sites in each direction of the grain. Since the hopping is random, it would take square of this, i.e., S number of hopping to reach a distance located at S1/2 sites away, where, on an average, another H is available. Thus, the effective recombination rate was chosen to be AH/S. But in general this is not true. Because the diffusion rate is very much dependent upon the grain parameters. We show that if we write the average recombination rate as A /Sα, where, α is a correction ∼ H term for the recombination rates. It is an empirical factor and needs more careful treatment. We propose that α may not be unity always (Chakrabarti et al., 2006a, 2006b), it is very much dependent upon the grain properties. α So, replacing AH/S in Eqns. 3-1, 3-2 and 3-3 by AH/S , the equations have the following forms,

dn H = φ W n 2(A /Sα)n2 , (3-18) dt H − H H − H H

dn H2 = µ(A /Sα)n2 W n , (3-19) dt H H − H2 H2

R = (1 µ)(A /Sα)n2 + W n . (3-20) H2 − H H H2 H2 Chapter 3. H2 Formation 81

4 In Fig. 3.1, the variation of the number of H and H2 on an Olivine grain of 10 sites 8 1 kept at 8 ◦K and exposed to an effective accretion rate per site of H 7.98 10− s− is × shown. We carried out the simulation 1010 sec. After the initial transient period ∼ is over (2 106 sec), we took time average at every 5.5 106s before plotting the × × numbers. From Fig. 3.1, it is clear that after some transient time step the number of H and H2 have reached a steady state on the grain surface. This steady state is achieved due to the balance between the accretion of the H atom from the gas phase and the evaporation of H and H2 from the grain surface.

We assume, α = α0 at the steady state. So, by Eqn. 3-18 α0 comes to be,

2A n2 α = log( H H )/log(S). (3-21) 0 φ W n H − H H To realize the situation more accurately we introduce one parameter β. This is defined as the ‘catalytic capacity’ of a grain which measures the efficiency of the formation of H2 on that grain surface for a given pair of H residing on it. Let δnH2 be the number of H2 produced in δt time. Since two hydrogen atoms are required to create one H2, the average rate of creation of one H2 per pair of H atom would be given by,

1 δnH2 < AH1 >= . (3-22) 2nH δt We identify the inverse of this rate with the average formation rate given by,

β(t) Tf (t) = S /AH. (3-23)

Thus,

β S (t) = AH/< AH1 >. (3-24) This yields β(t) as a function of time as,

β = log(AH/< AH1 >)/log(S). (3-25) As a steady state is achieved after some transient period. We may assume at t , →∞ β0 = β (Chakrabarti et al., 2006a, 2006b). Our goal is to find the dependence of this parameter on the grain parameters, accretion rate and temperatures. When the Chapter 3. H2 Formation 82

1.4

0 10 K 1.3 0 9 K 0

α 1.2 0 8 K

1.1

1 1e-14 1e-12 1e-10 1e-08 1e-06 -1 as(s )

Figure 3.7: Temperature dependence of α0 for the Olivine grains at 10 ◦K (solid), 9

◦K (dot-dashed) and 8 ◦K (dashed). The deviation is highlighted using dotted curves by extrapolating at very low accretion rates.

accretion rate is very low, finding a second H on the grain would be difficult and it can take several sweeps of the grain surface. Thus β0 > 1 would be a possibility. On the other hand, when the rate is very high, many sites would be occupied by the atomic H and one H would react with another H after a less number of hopping. and β0 could come down to 1/2 or even less. Indeed, this is what we see as well. We have carried out the simulation with and without spontaneous desorption taken into account. In presence of the spontaneous desorption, we remove H2 by generating a random number as soon as one H2 is formed and checking if it is less or more compared to µ. If less, H2 remains on the grain, else it is taken out to the gas phase.

The values of the activation barrier energies E0, E1 and E2 are taken from Katz et al. (1999) and are given by, E0 = 24.7 meV (287 ◦K), E1 = 32.1 meV (372 ◦K) and E2 = 27.1 meV (315 ◦K) for Olivine and E0 = 44 meV (510 ◦K), E1 = 56.7 meV

(657 ◦K) and E2 = 46.7 meV (541 ◦K) for amorphous carbon grains. For Olivine, µ =0.33 and for amorphous carbon µ =0.413 was used. Chapter 3. H2 Formation 83

1 η

0.5

0 6 8 10 12 14 Temperature (Kelvin)

Figure 3.8: A comparison between the recombination efficiency obtained from the Rate equation (solid) and that obtained from our simulation (dashed). We use the accretion rate per site 1.8 10 9 per second for a grain of diameter 0.1µm. × − Chapter 3. H2 Formation 84

Figure 3.9: Variation of α0 as a function of as, the effective accretion rate per site, for various amorphous carbon grains kept at 14 ◦K. The solid, dot-dashed and dashed curves are for S = 104, 9 104, 106 respectively. ×

3.2.1 Olivine Grain

In Fig. 3.2(a-b), the snapshots of the grain surface is shown for two intermediate times(a) 8 108s and (b) 109s. Here, no spontaneous desorption is assumed. We × carry out this simulation for an Olivine grain having 900 sites. The grain is kept at a temperature 8 ◦K. The hollow squares represents the atomic hydrogens and the filled squares represents the molecular hydrogens. This grain is facing an accretion 7 1 rate per site of 3.02 10− s− . Fig. 3.2(c-d) are for the same simulation and for × the same time as before when the spontaneous desorption has been included. The numbers of H2 are fewer since after formation of H2 molecules some part of the H2 are spontaneously desorbed into the gas phase.

α0 as a function of as (the effective accretion rate per site) is shown in Fig. 3.3. The solid, dot-dashed and the dashed curves are for S = 104, 9 104 and for 106 × sites respectively. No spontaneous desorption has been included here (i.e., µ = 1). In the low accretion rate, Monte Carlo method requires a very large computational time. To avoid this huge computational time we have extrapolated our curve to very Chapter 3. H2 Formation 85 low accretion rates just to show the trend of the result under extreme conditions.

We note that α0 is generally higher than unity in the region of our interest. This is because one H takes a longer time (generally more than one sweeping) to find another H. α0 monotonically drops as the accretion rate goes up. In Fig. 3.4 we plot α0 for Olivine grains at T = 10 ◦K as a function of the accretion rate per site a . The solid, dot-dashed and dashed curves are for 104, 9 104 and 106 sites s × respectively. For very low rates, it becomes impossible to carry out the simulations in a computationally viable timescale. We thus extrapolated the curves (dotted) for very low as values. We find that as the accretion rate per site goes down, α0 deviates from unity significantly.

β0 as a function of as is shown for the same case. Here too, we can see that β0 is very high compared to unity for low rates, but becomes 0.5 or lower for higher ∼ rates as expected. In Fig. 3.6 we show β0 as a function of the accretion rates per site for various grain sites and for Olivine at T = 10 ◦K grains respectively. The solid, dotted and dashed curves are for 104, 9 104 and 106 sites respectively as × before.

α0 and β0 is going down with the increase in the site number S also. Since for a smaller grain, the possibility of getting it filled at a high rate that is higher, one would have expected an opposite result. However, it is to be remembered that for a larger grain, the accretion rate itself (φH = Sas) is also large. Hence the plots are to 12 1 4 be compared carefully. For instance, the result of as =5 10− s− for 9 10 sites 11 1 ×4 × is to be compared with that of a =4.5 10− s− for 10 sites in order to make a s × meaningful comparison. In any case, for reasonable φH values with number densities 6 3 6 up to 10 cm− , the relevant as would be below 10− where α0 > 1 in general. When the temperature of the grain is increased, all the rates go down expo- nentially. As a result, we expect α0 to rise with temperature for a given accretion rate. In Fig. 3.7 we show this behaviour for T = 8, 9, 10 ◦K respectively for Olivine grains. We choose S = 104 in this case. This behaviour affects also the recombination efficiency (η). In Fig. 3.8, we compare the temperature dependence of η as obtained from the Rate equations (solid) curve with that obtained from our simulations. The accretion 9 1 rate per site of 1.8 10− sec− was used (same in both the cases). We note that < × > for T 7.5 ◦K the simulation results are higher and for T 7.5 ◦K the simulation ∼ ∼ Chapter 3. H2 Formation 86

Figure 3.10: Variation of α0 as a function of as, the effective accretion rate per site, for various amorphous carbon grains kept at 14 ◦K. The solid, dot-dashed and dashed curves are for S = 104, 9 104, 106 respectively. ×

results are lower. This is because α0 itself is strongly temperature dependent as shown in Fig. 3.7.

3.2.2 Amorphous Carbon Grain

The barrier energies for the Amorphous carbon are higher than those for the Olivine grain and it has been noticed that at higher temperatures the production of H2 is significantly higher than that for the Olivine grain. In Fig. 3.9, we plot the variation of α0 with as at temperature 14 ◦K. The nature of variation of α0 remains the same, namely, α0 goes down with as. A similar variation of α0 is shown in Fig. 3.10 for an amorphous carbon grain kept at 20 ◦K. The solid, dot-dashed and the dashed curves are for S = 106, 9 104 and for 104 sites respectively. In Figs. 3.11 and Figs. 3.12 × we show the variation of β0 for an amorphous carbon grain kept at temperature 14

◦K and 20 ◦K respectively and as it is clear that its value can become as low as 0.5 for very large accretion rate. Chapter 3. H2 Formation 87

Table 3.1: Comparison of H and H2 abundances in various methods

Accretion Rate α0 H with H2 with 1 per site A (S− ) simulation α =1 α =1 simulation α =1 α =1 s 0 6 0 0 6 0 7 6.79 10− 1.04 403.11 407.80 340.57 361.71 383.82 377.48 × 7 2.72 10− 1.05 275.78 278.02 219.31 149.96 158.16 156.53 × 8 2.72 10− 1.07 99.03 99.31 70.35 15.40 16.14 16.11 × 8 1.36 10− 1.08 72.27 72.42 49.80 7.71 8.08 8.07 × 9 5.43 10− 1.09 47.30 47.37 31.52 3.11 3.23 3.23 × 10 1 1 1 2.72 10− 1.11 11.62 11.62 7.01 1.49 10− 1.59 10− 1.60 10− × 10 × 2 × 2 × 2 1.09 10− 1.12 7.46 7.48 4.42 5.81 10− 6.24 10− 6.35 10− × 11 × 2 × 2 × 2 2.72 10− 1.13 3.86 3.82 2.18 1.31 10− 1.49 10− 1.55 10− × 11 × 3 × 2 × 2 ∗2.04 10− 1.13 3.36 3.33 1.89 9.7 10− 1.11 10− 1.17 10− × 12 × 3 × 3 × 3 ∗2.92 10− 1.14 1.32 1.30 0.70 1.3 10− 1.5 10− 1.6 10− × 13 × 4 × 4 × 4 ∗4.16 10− 1.16 0.52 0.51 0.26 2.00 10− 2.00 10− 2.00 10− × 14 × 5 × 5 × 5 ∗5.95 10− 1.17 0.20 0.20 0.10 2.11 10− 2.81 10− 3.13 10− × × × × represents the extrapolated value. ∗ Chapter 3. H2 Formation 88

Figure 3.11: Variation of β0 as a function of as, the effective accretion rate per site, for various amorphous carbon grains kept at 14 ◦K. The solid, dot-dashed and dashed curves are for S = 104, 9 104, 106 respectively. ×

3.2.3 Comparison

In Fig. 3.13 we compare our results with the analytically obtained results from the effective Rate equation method. However we replace S by Sα. The simulation results are shown by the dark circles and those obtained from the analytical considerations are shown by the dashed curves. An Olivine grain of 104 sites at a temperature of

8 ◦K has been chosen in this comparison. Dotted curves are drawn using analytical results for α0 extrapolated to very low accretion rates. We compare our results with those obtained from the analytical considerations with and without our α0 factor. In Table 3.1, we tabulate this comparison. We take 4 an Olivine grain of 10 sites at 8 ◦K and vary the accretion rates. The accretion rate per site of the grain is tabulated on Column 1. In Column 2 we present the exponent α0, which is derived from Monte-Carlo simulations. In Columns 3-5, we present the number of H as obtained by our simulation and the modified equation (Eqns. 3-18 & 3-19) and the standard equations respectively. In Columns 6-8, we present similar results for H2. We find that our simulation matches more accurately Chapter 3. H2 Formation 89

Figure 3.12: Variation of β0 as a function of as, the effective accretion rate per site, for various amorphous carbon grains kept at 20 ◦K. The solid, dot-dashed and dashed curves are for S = 104, 9 104, 106 respectively. × with the analytical results provided S is replaced by Sα. If the standard equation is used, the deviation is very significant. Indeed, the number of H on the grain could be roughly half as much when simplistic analytical model is used. What observe is that on the grains we tend to have more H and less H2 than what analytical work suggests.

H2 is the simplest molecular species in the ISM and it is very abundant also. It is necessary to predict the exact abundances of the H2 accurately. The excess or less production of H2 seriously affects the abundances of the other hydrogen bearing species. We hope that the inclusion of our αs will improve the results of surface chemistry by calculating the exact abundances of the molecular hydrogen and other species in the ISM. Chapter 3. H2 Formation 90

1000

H 1 H2 Number

0.001

1e-06 1e-12 1e-10 1e-08 1e-06 -1 as(s )

Figure 3.13: A comparison of the simulation results (dark circles) with those obtained from analytical considerations (dashed curves) when suitable modification of the average 4 recombination rate is made. An Olivine grain of 10 sites at a temperature of 8 ◦K has been chosen in this comparison. Dotted curves are drawn using analytical results for α0 extrapolated to very low accretion rates. Chapter 4

Formation of Molecules in the Gas Phase

In Chapter 2, the proto-star formation procedure from a collapsing cloud has already been discussed. To obtain the chemical evolution of a molecular cloud, we couple our chemical evolution code with the hydrodynamic code. We used same basic chemical evolution code which was used in Chakrabarti and Chakrabarti (2000a, 2000b). This code was generated to study the evolution of some selected species due to the gas phase only. However, we modify this code by incorporating production of H2 molecules on the grain. Here, we follow the techniques used by Acharyya et al., (2005). We used our code for different grain size distributions, as mentioned by Weingartner and Draine (2001a and 2001b). We used only Olivine kind of grains for simplicity. To have a computer friendly code we divided the grains into three major types (a) 5 A˚, (b) 75 A˚, and (c) 0.2µ. We used Rate equation method or Master equation method depending upon the accretion rate of the atomic hydrogen on the grain surface. Reaction rates were taken from the Miller et al. (1997)(UMIST data base). The reactions for which the rates were not available in the UMIST data 10 3 1 base, we take a conservative value of α = 10− cm− s− , β = γ = 0. This is much like any other typical two body (open shell) reactions even though we may be using neutral-neutral reactions. The chemistry of complex bio-molecule formation in space is very much unknown and it is difficult to quantify the rate with any more certainty. Due to the limited knowledge about the rates of different reactions, our estimation about the abundances of different complex species in the ISM may be over or under estimated, though for lighter species the results are better. We used two models, depending upon the initial abundances of the carbon. We name ‘Set 1’ initial composition, where the initial composition was kept as used in

91 Chapter 4. Formation of Molecules in the Gas Phase 92

0.6

0.5

0.4 H H 0.3 2

Mass Fraction 0.2

0.1

0 1e+10 1e+11 1e+12 1e+13 1e+14 Time (sec)

Figure 4.1: Evolution of the mass fractions of H and H2 are shown. Mass fraction of H atom decreases due to the production of H2 & other hydrogenated species.

Miller et al. (1997). Following the method used by Chakrabarti and Chakrabarti 2000a, we express the initial composition in terms of the mass fractions. In terms of the mass fraction the ratio of the initial composition is,

H:He:C:N:O:Na:Mg:Si:P:S:Cl:Fe

4 4 3 8 =0.64:0.35897 : 5.6 10− :1.9 10− :1.81 10− :2.96 10− × × × × 8 8 8 7 8 8 :4.63 10− :5.4 10− :5.79 10− :4.12 9− :9 10− :1.08 10− . × × × × × × In the ‘Set 2’ initial composition, we use higher amount of carbon to mimic the condition of a collapsing cloud in the environment of a evolved stars. In our chemical network, we use 422 species and near about 4000 chemical re- actions in between them. We already have mentioned in Chapter 2 that we divided our radial grid in 100 logarithmically equal space grids for hydrodynamic evolution. To save the computational time in pursuing the chemical evolution: we divide the cloud in ten logarithmically equal spaced zones along the radial direction instead of 100 logarithmically equal spaced zones and use weighted average density in each zone. To achieve more faster code, we use long time step instead of running the Chapter 4. Formation of Molecules in the Gas Phase 93

1e-05

1e-10

O2 CO 1e-15 H2O Mass Fraction

1e-20

1e-25 1e+10 1e+11 1e+12 1e+13 1e+14 Time (sec)

Figure 4.2: Time evolution of the mass fractions of some simple molecules like O2, H2O, CO. code for the each hydrodynamic time steps. The time step in chemical evolution is totally dictated by the fastest reaction in the network. In every time step, we note the amount of matter which is advecting in or out of a particular zone and update the matter of that zone accordingly. We follow the evolution process till the end of the hydrodynamic simulation. We take a weighted average of the abundances over the entire zone. We follow the evolution of this weighted averages of the abundances (global averages of the abundances) to study the chemically evolving cloud. First, we study the the chemical evolution of different species for Model A hy- drodynamic simulation and for the Set 1 initial abundances. In Fig. 4.1 we show the evolution of H and H2. From the Figure it is clear that as the time evolves, the abundances of the atomic hydrogen is going down due to the production of molecular hydrogen and other hydrogen bearing species. Since atomic hydrogen is a very reactive agent, at the end of the simulation most of the hydrogen atoms are converted into the molecular hydrogen. Fig. 4.2 shows the time evolutions of O2, 5 7 H2O and CO. The final mass fraction of these species are around 10− , 10− and 3 10− respectively. Sudden increase in the mass fraction occurs due to increase of the densities of the collapsing clouds. In Fig. 4.3, the time dependencies of the Chapter 4. Formation of Molecules in the Gas Phase 94

1e-10

1e-15

1e-20 HCN NH3 Mass Fraction

1e-25

1e-30 1e+10 1e+11 1e+12 1e+13 1e+14 Time (sec)

Figure 4.3: Time evolution of the mass fractions of HCN and NH3.

abundances of HCN and NH3 are shown. The final mass fractions of these species 7 are around 10− . In Fig. 4.4, we show the variations of C2H5OH, CH3CHO and ∼ 15 10 9 CH OH the final mean mass fractions are 10− , 10− and 10− respectively. 3 ∼ ∼ ∼ Finally, in Fig. 4.5, we show the evolutions of some of the simple bio-molecules, such 13 as glycine and alanine whose final mass fractions are found to be around 10− to 15 ∼ 10− respectively. ∼ As we mentioned earlier, we use ‘Set 2’ initial composition to study the chemical evolution in the environment of an evolved stars. Here, we choose initial mass fraction of the carbon to be 0.001, which is approximately twice the mass fraction used in ‘Set 1’. We adjust the mass fraction of He accordingly to keep the total mass fraction to be fixed. We noticed that the carbon bearing species become more abundant in the simulations with Set 2 initial composition. In Fig. 4.6, we present the variations of H2O and CO with time for these model. If we compare this Figure with Fig. 4.2 we can easily found more abundant CO molecule. In Figs. 4.7 to 4.8 the time variation of the average abundances of several is shown. Here, we use the Set 1 initial composition and use Model B (hereafter Model B1). We also consider the freeze out effect into the consideration. In this Chapter 4. Formation of Molecules in the Gas Phase 95

1e-10

1e-20

1e-30

1e-40

CH3OH

CH3CHO Mass Fraction 1e-50 C2H5OH

1e-60

1e-70 1e+10 1e+11 1e+12 1e+13 1e+14 Time (sec)

Figure 4.4: Time evolution of the mass fractions of some complex molecules like C2H5OH,

CH3OH, CH3CHO.

1e-18

1e-36 Alanine Glycine 1e-54

Mass Fraction 1e-72

1e-90

1e+11 1e+12 1e+13 1e+14 Time (sec)

Figure 4.5: Time evolution of some bio-molecules like, glycine and alanine ,which are the good precursor of the life formation. Chapter 4. Formation of Molecules in the Gas Phase 96

1e-05

1e-10

O2 1e-15 CO H O

Mass Fraction 2

1e-20

1e-25 1e+10 1e+11 1e+12 1e+13 1e+14 Time (sec)

Figure 4.6: Time variation of the mass fractions of O2, H2O, CO are shown for the Set 2 initial abundances.

1e-05

1e-10

O2 CO H2O Mass Fraction 1e-15

1e-20 1e+10 1e+12 1e+14 Time (sec)

Figure 4.7: Time variation of the mass fractions of O2, H2O, CO are shown for the Model B hydrodynamic simulation and Set 1 initial abundances. Chapter 4. Formation of Molecules in the Gas Phase 97

1e-07

1e-14

1e-21

CH3OH

CH3CHO

C2H5OH

Mass Fraction 1e-28

1e-35

1e+10 1e+12 1e+14 Time (sec)

Figure 4.8: This is also for Model B hydrodynamic simulation, where Set 1 initial abun- dances are used. effect the gas phase species depleted due to the accretion on the grain surface. From the Figs. 4.7 to 4.8, this effect is not very clear since we are plotting the global averages. We calculate the global averages by taking the weighted average. Since the volume of the outermost zone is the largest, its contribution towards the global average is maximum. As the matters are evolving very slowly at the outermost zone, the global averages do not represent any significant difference between the case with a freeze out effect and the case without. In Figs. 4.9-4.10, the evolutions of the mass fraction of CO in the 1st, 4th, 7th and 10th shells are shown. In Fig. 4.9 we do not consider the freeze out effect and in Fig. 4.10 the freeze out effect is taken into account. When no freeze-out is considered (Fig. 4.9), all the shells eventually reach a saturation value of the mass- fraction. This is reached when the formation rate of CO is equal to the destruction rate of CO through hydrogenation. This saturation value is different for the different shells because it depends on the average density of each shell. When we assume a freeze-out, the mass fractions in the innermost four shells start to decrease after reaching the saturation while the other shells do not decrease within the evolution time of the cloud. The evolution of the global average follows initially the evolution Chapter 4. Formation of Molecules in the Gas Phase 98

0.01

0.0001

1e-06

1e-08 1st shell 4th shell 7th shell

Mass Fraction of CO 10th shell global average 1e-10

1e-12 0 2e+13 4e+13 6e+13 8e+13 1e+14 1.2e+14 Time (sec)

Figure 4.9: Evolution of the mass fraction of CO is shown when the freeze-out effect is absent. The results in the 1st, 4th, 7th and 10th shells are displayed. The global average does not shift very much as it has a dominant contribution from the outer shell. Chapter 4. Formation of Molecules in the Gas Phase 99

0.01

0.0001

1e-06

1e-08 1st shell 4th shell

Mass Fraction of CO 7th shell 10th shell 1e-10 global average

1e-12 0 1e+13 2e+13 3e+13 Time (sec)

Figure 4.10: Time variation of the mass fraction of CO is shown by taking the freeze- out effect into account. Note that in the 1st and the 4th shell, the freeze-out effect is prominent. However, the global averages do not differ by any large margin since they are dominated by the outer shells. Chapter 4. Formation of Molecules in the Gas Phase 100

Table 4.1: Comparison of our Results with the observed abundances Molecules Observed Model A Model B References aHCN 6 10−9 1.4 10−9 2.4 10−9 1.9 10−9 Irvine and Hjalmarson (1983) × −8 × −9 × −9 × −9 aNH3 5 10 4.5 10 1.2 10 3.9 10 T¨olle et al. (1981) × × × × aCO 3 10−5 1.8 10−5 6.6 10−5 2.6 10−5 Allen and Knapp (1978) × −8 × −8 × −8 × −8 aH2O 7 10 4 10 2.2 10 1.6 10 Snell et al. (2000) ×−6 × −7 × −7 × −7 aO2 10 3.3 10 1.9 10 2.7 10 Gold et al. (2000) −10 × −11 × −11 × −11 aCH3OH 5 10 2.5 10 4.4 10 1.6 10 Friberg et al. (1988) × −10 × −13 × −13 × −13 aCH3CHO 3 10 1.3 10 9.1 10 1.9 10 Matthews et al. (1985) × −9 × −17 × −16 × −15 bC2H5OH 1.5 10 5.8 10 3.6 10 1.1 10 Ohishi et al. (1995) × × × × Alanine - 2.3 10−17 8.3 10−17 6 10−17 - × × × Glycine 10−10 c 1.7 10−14 2.9 10−14 1.7 10−14 Ceccarelli et al. (2000) × × × 7 10−9 d and × (0.21 1.5) 10−9 Kuan et al. (2003) − × a Observed near TMC-1 (10 circK) b Observed near Orion KL (70 circK) c upper limit (cold cloud) ; d upper limit (hot core) of the 10th shell, but after wards, from around t = 2 1013s, the deviation occurs × as the global average is dominated by denser intermediate shells. In Table 4.1, we present the results of Model A with Sets 1 and 2 chemical com- positions, and the result from Model B1. We also compare the observed abundances of some of the species. For comparison, we converted mass fraction of a species into ratio of the number density of that species and that of H2 so that our results are directly comparable with observables. The references from where the observed abundances have been obtained are provided in Column 6. From the Table we notice that the observed abundances match very well for the molecules having simpler structure. As the molecules get more complex, our results deviate significantly from the observational results. Production of complex molecules by our method does not seem to be very efficient. For instance, for

C2H5OH, glycine, alanine, abundances obtained by our simulations are very low compared to the observed or ‘upper limits’ obtained from observations. We choose following pathways for the formation of the glycine(C2H5NO2) & alanine(C3H7NO2),

C H O+ + e C H OH+H 2 7 → 2 5 C H N + 2H O C H NO + NH 2 4 2 2 → 2 5 2 3 C H N + 2H O+ C H NO + NH . 3 6 2 2 → 3 7 2 3 These may not be the only ways by which these molecules are formed. According to Chapter 4. Formation of Molecules in the Gas Phase 101

Ohishi et al. (1995), C2H5OH is difficult to form in the gas phase and this is thought to be synthesized on the grain surfaces and are then evaporated to the gas phase. For instance, Elsila (2007), used isotopic labeling techniques to test various mechanisms on interstellar ice analogues for the formation of the amino acids. He observed that the formation of each acid can occur via multiple pathways, with potentially different mechanisms for different acids. Thus we believe that there could be several mechanisms than the method adopted by us to form complex molecules. We used 10 3 1 the reaction rate of the order of 10− cm− s− for all the stages of the production. We do not consider the size of the species. As the species gets complex, the cross- section of that species will increase, which could increase the reaction rate by the factor of a few. This may be the another reason of mismatch with the observational results. Chapter 5

Formation of Molecules on the Grain Surface

We already discussed that the complex molecules are not very efficiently formed in the gas phase. They are expected to be formed on the grain surface and are then released to the gas phase. Water and Methanol are the two very important complex molecules observed in the ISM. This have different abundances in the different re- gions. Oxygen is very abundant in the ISM. Looking at the total budget of Oxygen, previously it was thought that water was the one of the possible reservoirs of oxygen. But surprisingly observational evidences are suggesting a low abundance of water in the gas phase. Presence of water and methanol in the solid state are indicating that these kinds of molecules may have formed on the grain. We have carried out a Monte Carlo simulation as well as a simulation using the Rate equation to find out the formation of water and methanol in the dense cloud conditions. We find that the production of these molecules are highly dependent on various grain parameters.

5.1 Mechanisms

We take H, O and CO as the accreting species and consider ten reactions among them. The reactions, which are employed in our network, are listed in Table 5.1. Both type of reaction schemes namely Langmuir-Hinshelwood (LH) mechanism and the Eley-Rideal (ER) mechanism are incorporated in our study. We assume that the molecules are trapped after reactions due to their high binding energies with the grain surface. Binding energy is very much dependent on the incoming species with the grain

102 Chapter 5. Formation of Molecules on the Grain Surface 103

Table 5.1: Surface Reactions in the H, O, and CO model

Number Reactions Ea (◦K) 1 H+H H → 2 2 H+O OH → 3 H+OH H O → 2 4 H+CO HCO 2000 → 5 H+HCO H CO → 2 6 H+H CO H CO 2000 2 → 3 7 H+H CO CH OH 3 → 3 8 O+O O → 2 9 O+CO CO 1000 → 2 10 O+HCO CO +H → 2 surface. The interaction is mainly due to mutually induced dipole moments or it might also form a strong covalent bond with the surface. The incoming species might get trapped in a shallow potential well in a physisorbed site. Recent studies have found evidences of both physisorption and chemisorption processes taking place on a grain surface. In our case, we consider only the physisorbed atoms and molecules since we are dealing with the weakly bound species only. If Ed denotes the binding energy for physical adsorption and Eb the potential energy barrier, then Ed must be greater than Eb for the species to diffuse from one site to the other. Reactions on the grain surface occur due to thermal hopping or tunneling. Tun- neling time is very much dependent on the mass of the particle. For the lighter species, tunneling is much faster than the hopping. So, one need to consider the tunneling effect for the lighter species. Tunneling time is not widely used in the astrophysical models. Pirenello 1997a,b and 1999 explained that the mobility of the hydrogen is primarily due to the thermal hopping. The classical papers like Tie- lens and Hagen (1982) and Hasegawa and Herbst (1992) considered the tunneling for hydrogen and thermal hopping for other simple atoms and molecules. In our simulation, we considered both tunneling time and the thermal hopping time. In our simulations, we used three different sets of barrier energies. First set of binding energies were taken from the earlier works (Allen and Robinson, 1977; Tielens and Allamandola, 1987; Hasegawa and Herbst, 1993). We tabulate this set Chapter 5. Formation of Molecules on the Grain Surface 104 of binding energies in Table 5.2. Here, we did not consider the tunneling effect. Here

Eb = 0.3Ed is used for all the species except for atomic hydrogen. We constructed our second set by using same set of binding energies as tabulated in Table 5.2. The only difference between the first and second set is the diffusion procedure of atomic hydrogen. In the first set we considered the diffusion of atomic hydrogen by thermal hopping but in second set we considered the diffusion of atomic hydrogen by tunneling. We constructed our third set by following the recent experimental findings of Pirronello et al. (1997, 1999) and as interpreted by Katz et al. (1999). They showed that the mobility of atomic hydrogen is much less than what is used in various simulation. Here, we used the binding energies of hydrogen as noted by Katz et al. (1999). For the other species, we increased the binding energies proportionately (Table 5.3). This means that the ratio between the barrier energies of these two tables for different species are exactly the same as for H. The desorption energies of Tables 5.2 and 5.3 was also kept fixed for simplicity.

Table 5.2: Energy barriers against diffusion and desorption(in degree Kelvin) for the Olivine grain

Species Eb Ed H 100 350 O 240 800 OH 378 1260

H2 135 450

O2 363 1210

H2O 558 1860 CO 363 1210 HCO 453 1510

H2CO 528 1760

CH3O 651 2170

CH3OH 618 2060

CO2 750 2500

If we look at the typical time scales for the hopping and evaporation for the binding energies listed in Tables 5.4 and 5.5, we get an idea about the chemical scenario of the grain. Chapter 5. Formation of Molecules on the Grain Surface 105

Table 5.3: Energy barriers against diffusion and desorption(in degree Kelvin) for the Olivine grain (from experiment)

Species Eb Ed H 287 373 O 689 853 OH 1085 1343

H2 387 479

O2 1042 1290

H2O 1601 1982 CO 1042 1290 HCO 1300 1609

H2CO 1515 1876

CH3O 1868 2313

CH3OH 1774 2195

CO2 2153 2664

Table 5.4: Time scales for the B.E. listed in Table 5.2. Species Hopping time Evaporation time Tunneling time (sec) (sec) (sec) 9 2 11 H 7.43 10− 5.35 10 1.99 10− × 2 × 22 × 2 O 2.36 10− 4.95 10 8.31 10− × × × OH 1.91 104 3.86 1042 1.13 102 × 7 × 7 × 10 H 3.07 10− 1.47 10 3.41 10− 2 × × × O 5.97 103 3.64 1040 1.23 107 2 × × × H O 1.06 1012 3.73 1068 3.41 105 2 × × × CO 5.59 103 3.40 1040 6.73 105 × × × HCO 4.13 107 3.31 1053 1.7 108 × × × H CO 7.03 1010 2.25 1064 1.56 1010 2 × × × CH O 1.41 1016 1.32 1082 1.06 1013 3 × × × CH OH 5.44 1014 2.29 1077 6.14 1012 3 × × × CO 3.13 1020 3.14 1096 1.2 1020 2 × × × Chapter 5. Formation of Molecules on the Grain Surface 106

Table 5.5: Time scales for the B.E. listed in Table 5.3. Species Hopping time Evaporation time Tunneling time (sec) (sec) (sec) 1 3 10 H 9.52 10− 5.17 10 3.26 10− × × × O 7.24 1017 9.6 1024 3.33 106 × × × OH 9.38 1034 1.5 1046 7.79 1011 × 4 × 8 × 8 H 2.62 10 2.59 10 3.43 10− 2 × × × O 1.78 1033 1.05 1044 2.09 1020 2 × × × H O 2.04 1057 7.19 1073 6.73 1017 2 × × × CO 1.67 1033 9.83 1043 1.6 1018 × × × HCO 2.43 1044 6.4 1057 1.98 1022 × × × H CO 4.99 1053 2.38 1069 4.35 1025 2 × × × CH O 9.77 1068 2.07 1088 2.88 1030 3 × × × CH OH 8.43 1064 1.62 1083 1.14 1030 3 × × × CO 2.59 1082 4.03 10103 2.45 1042 2 × × ×

In Tables 5.4 and 5.5, the typical time scales are noted for an Olivine grain kept at 10 ◦K and having binding energies as mentioned in Table 5.2 and 5.3 respectively. From Table 5.4 it is clear that only some lighter species have significant mobility. The H atom has a very short hopping time scale, so the surface chemistry is mainly dominated by the hydrogenation reactions. O and H2 also have sufficient mobility. All other species look to be immobile in respect of the mobile species. For H and

H2, tunneling is faster than the hopping so one needs to consider the tunneling for these species. From Table 5.4 it is clear that only H and H2 can thermally evaporate from the grain surface. Until other evaporation mechanism is considered, all other species will remain on the grain surface throughout the simulation. From Table 5.5, only atomic hydrogens have a significant mobility. Tunneling time scales for H and H2 are much shorter than their hopping time scales. Here, we do not consider the tunneling because experimentally there are no evidences of tunneling. Only H atom can thermally evaporate from the surface. Other species will remain on the grain unless other evaporation mechanism is specified. Using the binding energies listed in Tables 5.2 and 5.3, we constructed three models. Thus in Model 1, we used the slowest diffusion rate of hydrogen and hence Chapter 5. Formation of Molecules on the Grain Surface 107

Table 5.6: Gas-phase abundances used Species high (Number density of low (Number density of 5 3 3 3 hydrogen=10 cm− ) hydrogen 10 cm− ) H 1.10 1.15 O 7.0 0.09 CO 7.5 0.075 the binding energy of Table 5.3. In Model 2, we used the binding energy from Table 5.2 but consider thermal hopping. Finally, in Model 3, the binding energy is taken from Table 5.2 but tunneling is considered. We carry out our calculations usually for two different sets of gas phase number densities, which is noted in Table 5.6 (Stantcheva et al. 2002).

5.2 Procedures of Simulation

5.2.1 Rate Equation Method

For this simple system of reaction network (Table 5.1) the governing equations for the number of species on the grain surface are:

dn H =a (H)N W n a n n a n n dt acc H − H H − H,H H H − H,O H O a n n a n n − H,OH H OH − H,CO H CO a n n a n n − H,HCO H HCO − H,H2CO H H2CO a n n +a n n , (5-1) − H,H3CO H H3CO O,HCO O HCO

dn O =a (O)N a n n a n n a n n dt acc O − O,O O O − H,O O H − O,CO O CO a n n W n , (5-2) − O,HCO O HCO − O O

dn CO =a (CO)N a n n a n n W n , (5-3) dt acc CO − H,CO CO H − O,CO CO O − CO CO

dn H2 =0.5a n n W n , (5-4) dt H,H H H − H2 H2 Chapter 5. Formation of Molecules on the Grain Surface 108

In the Same In the Same time step time step

C1 C2

In the next In the next time step time step

C3 C4

Figure 5.1: Different reaction schemes (C1-C4) are shown by a cartoon diagram. Black and shaded circles represent the reactive species and the clear circles represent non-reactive species. A circle around two such species represent the formation of a new species by ER mechanism. Chapter 5. Formation of Molecules on the Grain Surface 109

dn O2 =0.5a n n W n , (5-5) dt O,O O O − O2 O2

dn OH =a n n W n , (5-6) dt H,O H O − OH OH

dn H2O =a n n W n , (5-7) dt H,OH H OH − H2O H2O

dn HCO =a n n W n , (5-8) dt H,CO H CO − HCO HCO

dn H2CO =a n n W n , (5-9) dt H,HCO H HCO − H2CO H2CO

dn H3CO =a n n W n , (5-10) dt H,H2CO H H2CO − H3CO H3CO

dn CH3OH =a n n W n , (5-11) dt H,H3CO H H3CO − CH3OH CH3OH

dn CO2 =a n n W n +a n n (5-12) dt O,CO O CO − CO2 CO2 O,HCO O HCO where, aacc(j) is the accretion rate coefficient defined as,

3 1 aacc(j)= sjσv(j) cm s− , where, sj is the sticking coefficient, σ is the geometrical cross section of the grain, v(j) is the thermal velocity, and N(j) is the abundances of the species in the gas 3 th phase in the unit of cm− , nj is the number of j species on the grain. W (j) is rate of evaporation, which is defined by 1 . a is the diffusion rate in the unit of tevap(j) i,j 1 s− . We calculate ai,j by,

1 1 ai,j = ai + aj = + . thop(i) thop(j)

ri,j Diffusion rate ai,j can also be written as, S , where ri,j is the diffusion rate coefficient and S is the number of sites on a grain. Chapter 5. Formation of Molecules on the Grain Surface 110

5.2.2 Monte Carlo Method

The Monte Carlo procedure used here is the similar to as mentioned in section 3.13. For the sake of simplicity, we assume that a square shaped grain having S number of 15 2 sites. We used Olivine grains having the surface density of sites s =1.5 10 cm− × (Hasegawa et al. 1992) for our case. We assumed that we are dealing with an fcc[100] plane and considered the nearest-neighbour interactions only. We further assumed a periodic boundary condition to mimic the spherical grain structure. According to the periodic boundary condition one species may leave the boundary of the grain from one side and can enter back into the grain from the opposite side. We considered the back diffusion (Chang et al., 2005) in our simulation. Self- crossing paths of randomly walking atoms are automatically included in our model, as we are solving the problem by Monte Carlo method. The minimum time step of our simulation is assumed to be the time taken for hopping or tunneling (whichever is smaller) of an H atom from site to site. If racc is the astrophysically relevant accretion rate under consideration, the ith species will have to be dropped after every th 1/racc seconds. The locations of the accreting i species is obtained from a pair of random numbers (Rx, Ry), themselves obtained by a random number generator. This pair would place the incoming species at (j, k)th site of the grain, where j and k are the nearest integers obtained using Int function: j = int(nRx +0.5) and k = int(nRy +0.5). Here, n = √S, being the number of sites on a square. Now during the hopping or tunneling process, when one species enters the site that is already occupied by another reactant surface species, it will form a molecule, if the activation barrier energy permits. Thermal evaporation of a species is also handled by random numbers. After each hopping and tunneling time, we generate a random th number (Rt) for i species. If Rt is less than Wi/ahi or Wi/ati , then we allow that species to evaporate. Here, Wi is the desorption rate and ahi and ati are the rates of hopping and tunneling of the ith species, respectively. We carried out our simulations for different random number generators and checked that our results do not depend on any specific random number generator. We continue our simulations up to the formation of one mono-layer. We assumed LH and ER both type of reactions in our consideration. Depending on the landing of a gas phase atom or molecule we constructed four different models (Fig. 5.1). The incoming species can land either on a vacant site or on an occupied Chapter 5. Formation of Molecules on the Grain Surface 111 site. When the landing is on a vacant site, the species will simply start scanning the surface for a reactive partner. When it lands on an occupied site, there could be two possibilities; (i) the incoming species can land directly on a reactive species but it may or may not combine with it. If not, it will look for an alternate site for itself. (ii) the incoming species can land directly on a non-reactive species and look for an alternate site. However, looking for an alternate site itself could be achieved in two ways: in the same time step (which is more likely) or in the next time step. These are illustrated in a cartoon diagram C1-C4 in Fig.5.1 (Das et al., 2008). In C1, both the LH and ER schemes are considered, i.e., the incoming atom/molecule is allowed to land on both the unoccupied and occupied sites. With the reactive species, it combines to form a new species (marked with a common circle). It skips the non- reactive species and looks for a new site in the same time step. C4 is otherwise the same as C1, but the incoming species looks for a new site at the next time step after it lands on a non-reactive species. In C2, the incoming species lands on a reactive species, but does not combine and instead finds a new site at the same time step (LH mechanism). C3 is the same as C2, the only difference is that in C3 we place the new incoming species at the next time step.

Let us consider the production of a stable species. H2O is a stable species, since it does not dissociate easily and has a small evaporation probability. So for the sake of simplicity, we can neglect that part from Eqn. 5-7 and it becomes,

dn H2O =r n n /S. (5-13) dt H,OH H OH

The production rate rH,OH/S, is approximately the inverse of the time required for an H atom to visit nearly all the adsorption sites on the grain surface. This is because, in two dimensions, the number of distinct sites visited by a random walker is linearly proportional to the number of steps, up to a logarithmic correction. But in reality the situation is different due to the following two reasons; the first is that the number of hoppings to find a reactant partner depends on the number of atoms or molecules already occupying the grain. When the grain is almost empty, a species has to hop several times, and when the grain is full, it requires less of hops. So the ‘size’ of a grain is relative as far as the hoping element is concerned. The second reason is the blocking effect. This is very common, when we are considering a grain having a mono-layer. We are considering fcc[100] plane, where the interaction Chapter 5. Formation of Molecules on the Grain Surface 112 between the four nearest neighbor sites are possible. Very often one reactive species is blocked in all the four directions due to the presence of various species with which no reaction is possible. For instance, if a H atom lands on a site surrounded by other molecules, with which no reaction is possible, it cannot react and will wait there itself for its desorption unless the blocked species themselves desorb and free the sites for the landed species to move around. The effect becomes more prominent as the grains start to fill up. In the latter case, the blocking effect will be weakened when the blocked element overcomes the energy barrier. As the surface is populated, these types of effects become more and more prominent. To understand the influence of these processes on the recombination, we introduce two parameters, namely, γ and α.

Let tH2O is the “average time” required to form a species H2O after the species H and OH are simultaneously present in our simulation, then dn H2O n n /t , dt ∼ H OH H2O where tH2O =rH,OH/S. The rate with which the species H2O is formed is 1/tH2O. Because of the two effects mentioned above, the actual rate may be assumed to be γ rH,OH/SH2O instead of rH,OH/S. From the above equation,

log[r t ] γ = H,OH H2O . (5-14) H2O logS A hydrogen atom has a very small desorption time scale compared to the other species. It will desorb from the grain if it is not able to find a suitable reaction partner in between its desorption time scales. If H gets another OH to form H2O, then we calculate dtH2O by noting down the time required to form an H2O after the accretion of an H. At the beginning of the simulation, most of the sites are empty and one H will have to more or less scan the whole of the grain to find its reactant partner and hence will take longer time to form a molecule. On the contrary, when most of the sites are filled with the different species, it will take less time to form a molecule and γH2O should decrease. When a molecule is formed due to direct hitting, i.e., through the ER mechanism, γH2O is clearly zero, since production rate is independent of the population of the species on the grain. Thus, in the ER mechanism, as the grain is populated, γH2O tends to become negligible.

So, the general formula to calculate γ for a stable species like H2O is, Chapter 5. Formation of Molecules on the Grain Surface 113

log[r t ] γ = a,b c , (5-15) c logS where, c species is formed due to the reaction between the species of types a and b. Here γ is derived from the time interval between the release of the reactants and the actual reaction to take place. We define another quantity α, to consider the total effect. It depends on the total time and not on the time taken by the species to react. Because of the two competing processes mentioned at the beginning of this section, there will be a net change in the Rate equation itself with time, since

α itself will be time dependent. To quantify this, we replace rH,OH/S in Eq. 5-13 by α(t) rH,OH/S . Thus, the Eq. 5-13, which is rewritten as

dnH2O rH,OHnHnOH log = log[ WH2OnH2O], (5-16) dt SαH2O(t) − can be used to obtain αH2O(t) as,

rH,OHnHnOH dn log[ H2O ] +WH OnH O α (t) = dt 2 2 . (5-17) H2O log(S) Calculations of α for the other species have to be done similarly. We note down the formula to calculate the αs: α for OH is,

rH,OnHnO r n n log[ dnOH H,OH H OH ] dt + αH O +WOHnOH α = s 2 . (5-18) OH log(S)

α for O2 is,

0.5rO,OnOHnO dn log[ O2 ] +WO nO α = dt 2 2 . (5-19) O2 log(S)

α for H2 is, 0.5rH,HnHnH dn log[ H2 ] +WH nH α = dt 2 2 . (5-20) H2 log(S)

α for CH3OH is,

rH,H3COnHnH3CO dn log[ CH3OH ] +WCH OHnCH OH α = dt 3 3 . (5-21) CH3OH log(S) Chapter 5. Formation of Molecules on the Grain Surface 114

α for H3CO is,

rH,H2COnHnH2CO dn r n n log[ H3CO H,H3CO H H3CO ] dt +WH3COnH3CO+ αCH OH α = S 3 . (5-22) H3CO log(S)

α for H2CO is,

rH,HCOnHnHCO dn r n n log[ H2CO H,H2CO H H2CO ] dt +WH2COnH2CO+ αH CO α = S 3 . (5-23) H2CO log(S)

α for CO2 is (neglecting the contribution from reaction number 10 of Table 5.1),

rO,COnOnCO dn log[ CO2 ] +WCO nCO α = dt 2 2 . (5-24) CO2 log(S)

α for HCO is (neglecting the contribution from reaction number 10 of Table 5.1),

rH,COnHnCO r n n log[ dnHCO H,HCO H HCO ] dt +WHCOnHCO+ αH CO α = S 2 . (5-25) HCO log(S) Some of the above equations have a factor of 0.5, which were used to avoid the possibility of the double counting. As the surface is populated by a number of reactant species, the recombination time scale would vary with time and thus α will also vary. As mentioned before, the time t used in computing α is the total time taken since the grain starts populating. On the other hand, γ is an average quantity that is computed instantaneously. We run the simulations to show that α(t) and γ are not necessarily unity. This means that the surface chemistry is not just a matter of random hopping, since the degree of randomness is seriously modified with the population and the abundance of species on the grain surface. The Monte-Carlo simulation reflects a more realistic computation, but it is very time-consuming. In a large-scale computation, it is impractical to do such a simulation. Indeed, we also show that only when α as given above is used in the effective Rate equation α (where every S is replaced by Si , where αi is different for different species), then the abundances of each species obtained from such an equation agree with the results of the Monte-Carlo simulation. Thus, one may as well solve the effective Rate equation using α obtained from our calculations. We have made a table for α for some major species in Table 5.9 for future usage. We ran twelve different cases. For each set Chapter 5. Formation of Molecules on the Grain Surface 115

Table 5.7: Surface coverage of the major species when one mono layer is built Major Used Eb & Ed from Table 5.3 Eb & Ed from Table 5.2 Eb & Ed from Table 5.2 species abundances Tunneling not allowed (Model 1) Tunneling not allowed (Model 2) Tunneling allowed (Model 3) C1 C2 C3 C4 C1 C2 C3 C4 C1 C2 C3 C4 O high 4.98 32.97 33.29 0 1.8 13.13 12.74 0 1.72 13.17 12.83 0 (in %) low 0 12.56 12.61 0 0 13.82 10.24 0 0 13.67 10.68 0 O2 high 19.32 0 0 22.49 25.26 19.77 19.76 28.0 25.39 19.32 19.43 28.16 (in %) low 0.24 0 0 0.38 0.67 0.39 0.76 1.05 0.56 0.35 0.56 0.6 OH high 8.92 11.13 10.97 0 2.89 2.35 2.32 0 2.92 2.19 2.34 0 (in %) low 0 0.2 0.22 0 0 0.15 0.27 0 0 0.15 0.2 0 H2O high 16.33 11.15 11.19 32.97 13.36 10.57 11.23 18.82 13.23 11.34 11.17 18.84 (in %) low 61.04 48.59 48.71 61.43 60.41 46.81 49.69 60.27 60.55 47.02 49.25 60.47 CO high 50.45 44.75 44.54 44.54 48.60 44.99 45.38 44.19 48.61 45.42 45.69 42.47 (in %) low 38.58 38.57 38.38 38.1 12.66 12.41 9.67 12.31 12.16 12.68 9.72 12.12 H2CO high 0 0 0 0 3.86 4.07 3.86 4.46 3.96 3.75 4.07 4.65 (in %) low 0.13 0.07 0.07 0.09 0.16 0.15 0.15 0.15 0.17 0.12 0.12 0.16 CH3OH high 0 0 0 0 0.54 0.64 0.53 1.06 0.58 0.53 0.53 1.16 (in %) low 0 0 0 0 26.01 26.16 29.03 26.14 26.41 25.87 27.2 26.26 CO2 high 0 0 0 0 3.0 1.29 1.15 3.47 2.97 1.3 1.03 4.72 (in %) low 0 0 0 0 0.08 0.06 0.11 0.08 0.14 0.05 2.18 0.39 of binding energies, all the four C1, C2, C3, and C4 cases are considered. Due to the complexity of the system, we restricted ourselves only up to the formation of a mono-layer on the grain surface.

5.3 Results

We carried out our simulation for an Olivine grain for two different sets of gas-phase abundances (Table 5.6) in our study, one with a high density and the other with a 5 3 low density. High density corresponds to the number density of hydrogen 10 cm− 3 3 and low density correspond to 10 cm− .

5.3.1 Abundances of Various Species on the Grain Surface

In Fig. 5.2, we show the results in case of the high gas-phase abundances. Here, 4 we take an Olivine grain having 10 number of sites and kept at 10 ◦K. In Table 5.7 we tabulate the surface coverage of the major species after one mono layer is formed. The first column of the Fig. 5.2 corresponds to the Model 1, i.e, slowest diffusion rate of atomic hydrogen. We noted in Table 5.7 only 16% and 11% of the grain deposition is water. This low production of water is due to the slow diffusion Chapter 5. Formation of Molecules on the Grain Surface 116

Model 1 Model 2 Model 3 1e+04 1e+03 1e+02 C1 1e+01 C1 C1 1e+04 1e+03

1e+02 O C2 1e+01 C2 C2 O2

1e+04 CO

Number 1e+03 OH

1e+02 H2O 1e+01 C3 C3 C3 CH3OH 1e+04 CO2 1e+03 1e+02 1e+01 C4 C4 C4

0 20 40 60 80 1000 20 40 60 80 1000 20 40 60 80 100 Surface Coverage (%)

Figure 5.2: Evolution of some species for various Models and different reaction mecha- 4 nisms are shown for an Olivine grain having 10 number of sites kept at 10 ◦ K and facing the high abundances (Table 5.6) of the gas phase species. Chapter 5. Formation of Molecules on the Grain Surface 117

Table 5.8: Time taken to build one mono-layer Used Eb & Ed from Table 5.3 Eb & Ed from Table 5.2 Eb & Ed from Table 5.2 accretion Tunneling not allowed (Model 1) Tunneling not allowed (Model 2) Tunneling allowed (Model 3) C1 C2 C3 C4 C1 C2 C3 C4 C1 C2 C3 C4 Time high 1.6(3) 1.3(3) 1.3(4) 3.6(4) 1.7(3) 1.6(3) 8.3(3) 3.2(4) 1.7(3) 1.6(3) 5.7(3) 3(4) in Year low 1.1(5) 1.1(5) 9.4(5) 1.1(6) 1.1(5) 1.1(5) 2.4(5) 1.0(6) 1.1(5) 1.1(5) 3.2(5) 1(6) rate and due to the absence of sufficient number of hydrogen atoms to react with the oxygen atom. In C1, the excess of production of water by 4% is mainly due to ER mechanism. There is hardly any methanol produced due to the slow diffusion rate and due to the presence of an activation barrier energy. In C1, around 19% of molecular oxygen is produced, but its absence in C2 suggests that it is produced due to the ER mechanism. Around 50% of the grain surface abundance is CO. The time required to grow one mono layer for C1 and C2 is 1600 years and 1300 years respectively. The grain is very quickly filled up by the accreted species. The results for the cases C3 and C4 differ only slightly in comparison to C2 and C1, respectively. Only a small increase in the molecular species and decrease in the atomic species is observed. The times required to grow a mono layer in C3 and C4 are around 13000 years and 36000 years, respectively. This longer timescale is due to the fact that in these cases atoms/molecules are not bound to accrete on the same time step. In the case of C4 we continue our simulation up to the time when all O and OH are converted to H2O, hence the time taken is much longer. The second Column shows the plots corresponding to Model 2, where thermal hopping of atomic hydrogen is considered. In all the four cases, the most dominant species are CO, O2, and water. Water abundance is very similar, varying between 11% to 19%. A very small amount of methanol has been produced due to the very fast mobility of atomic hydrogen. In all the cases, the production of a significant amount of molecular oxygen suggests that the ER mechanism might have an active role to play. The time required to grow a mono-layer is around 1500 years. The third column also shows a similar trend of the results, where the simulation is carried forward for Model 3. In Fig. 5.3, we show the results for low gas-phase abundance. A significant amount of water is produced in all the cases, since the H/O and H/CO are rela- tively high. The presence of the atomic oxygen and carbon monoxide is also much lower, especially for Models 2 and 3. This is because, in the last cases the mobility Chapter 5. Formation of Molecules on the Grain Surface 118

Model 1 Model 2 Model 3 1e+04 1e+03 C1 1e+02 C1 C1 1e+01 1e+04 1e+03 1e+02 C2 C2 C2 O O 1e+01 2

1e+04 CO

OH Number 1e+03

C3 H O 1e+02 C3 2 1e+01 CH OH C3 3 1e+04 CO2 1e+03 C4 1e+02 C4 C4 1e+01

0 20 40 60 80 1000 20 40 60 80 1000 20 40 60 80 100 Surface Coverage (%)

Figure 5.3: Same as in Fig. 5.2, but the low abundances of the accreting species have been used. Chapter 5. Formation of Molecules on the Grain Surface 119

2e-09

1.5e-09 O 2

H 1e-09 R

C1 C2 5e-10 C3 C4

0 0 20 40 60 80 100 Surface Coverage(%)

Figure 5.4: The rate of production of H2O is shown against the surface coverage for C1-C4. The production is higher when the Eley-Rideal mechanism is taken into account, while the production is lower when the reaction takes place at the next time step. The rate is computed using the total production and total time taken at a given instant. of H being many orders of magnitude higher, the hydrogenation process is more efficient and that uses up O and CO to create water and methanol. This makes the production of methanol significant for Models 2 and 3 and insignificant for Model 1. Compared to the high abundance cases (Fig. 5.2), the accretion rate for low abun- dance is much lower, so the grain is not full of the same atomic species as before. The time required to grow a mono-layer is around 0.1Myr for both the cases. A similar trend is found for C3 and C4, but it takes around 1Myr to fill up the grain. The second and third columns of Fig. 5.3 show a similar result. In all the cases, the most dominant species is water. The abundance of water varies between 46% to 61%. The next dominant species is methanol with an abundance between 26% to 33%. But only a trace amount of molecular oxygen is produced. Table 5.8 shows the time required to grow a mono-layer for different accretion rates.

Figs. 5.4 & 5.5 shows the rate RH2O of production of H2O for various mechanisms. This has been computed in two ways. Figs. 5.4 shows the cumulative average rate Chapter 5. Formation of Molecules on the Grain Surface 120

2e-09

1.5e-09 O 2

H 1e-09 R

C1 C2 5e-10 C3 C4

0 20 40 60 80 100 Surface Coverage(%)

Figure 5.5: Rate of production of H2O is shown with the surface coverage for the four different schemes. Production is higher where ER is considered and the reaction takes place at the same time steps. Here the change in the number of H2O is for a fixed change in percentage of the surface coverage (10%). Chapter 5. Formation of Molecules on the Grain Surface 121

10000

1000 T=10K 100

10

O 10000

O2 1000 CO 100 T=15K OH

H2O Number 10

CH3OH 10000 CO2 1000 T=20K 100

10

1 0 20 40 60 80 100 Surface Coverage(%)

Figure 5.6: Rate of production of a few selected species as a function of the surface coverage at different temperatures are shown for a grain having 104 sites for Model 2 and C2 scheme. These simulations were carried out for low abundances (Table 5.6) of the accreting species on an Olivine grain.

where RH2O =H2O(t)/t. We clearly see that, unless the production rate is very high (e.g. in case of C1), the cumulative rate generally falls off very rapidly with time.

In the C1 scheme, the abundance of H2O is higher because the formation of H2O is dH2O dominated by direct hitting rather than hopping. In Fig. 5.5, we plot RH2O = dt , where dH2O is the production of H2O in dt time in which a fixed percentage dp% of the surface coverage is increased. Thus, in effect, it is the instantaneous rate. For simplicity, we choose dp = 10% so as to get smooth curves. In the case of the LH scheme, i.e., for C2 and C3, as the surface starts to populate, the rejection of the incoming species become important and thus the rate of production decreases with the surface coverage. Chapter 5. Formation of Molecules on the Grain Surface 122

4e+12

3.5e+12 Time (sec)

3e+12

2.5e+12 8 12 16 20 Temperature (K)

Figure 5.7: Time taken to build a mono-layer is plotted as a function of temperature for a grain having 104 sites and for Model 2 by considering the C2 scheme. These simulations were carried out for low abundances (Table 5.6) of the accreting species on an Olivine grain.

5.3.2 Temperature Dependence

Fig. 5.6 shows the production of various species as a function of surface coverage at 10 ◦K, 15 ◦K and 20 ◦K. We carried out this computation only for Model 2 and taking low abundances of the accreting species. In all the three temperatures, the water is the most abundant species. But CH3OH is only efficiently formed in 10

◦K and 15 ◦K. At 20 ◦K, the methanol production is significantly lower because the desorption time scale of atomic hydrogen is much shorter and, at the same time, the activation barrier energy is much higher to produce methanol. In Fig. 5.7 we plot the time required to build a mono-layer on a grain of 104 sites as a function of the temperature of the grain. At higher temperatures, the time taken is lower because the grain is full of nonreactive CO (Fig. 5.7). Chapter 5. Formation of Molecules on the Grain Surface 123

1 Ο 2 Η

γ 0.5

C1 C2 C3 C4

0 0 20 40 60 80 100 Surface Coverage(%)

Figure 5.8: Variations in γH2O(t) with surface coverage for different reaction schemes for Model 2. Note that when the Eley-Rideal scheme is included (C1 and C4) the production efficiency increases rapidly as the grain is filled up. The simulation were carried out for 4 an Olivine grain having 10 sites and was kept at temperature 10 ◦K.

5.3.3 Variation in γ and α

In Fig. 5.8, the variation in ‘catalytic capacity’ for water, namely, γH2O(t) (Eq.

5-14) is shown as a function of the surface coverage. The Figure shows that γH2O(t) gradually decreases with the surface coverage. This is because an increase in the surface coverage will decrease the number of vacant sites and will decrease the time for searching its reactant partner to recombine. The blocking effect also shows its effect by reducing γH2O very rapidly. The lower the value of γH2O, the more efficient the recombination. This parameter is dependent on the number of sites on the grain.

Figure 5.9 shows the change of γH2O with the number of grain sites. Naively, one would have expected that γH2O to remain unity always. In that case, γH2O would be independent of number of sites in the grain. However, we find that γH2O goes down when S is increased. From Eq. 5-14 we note that, when S is increased from 104 to 5 10 , the value of γH2O should go down by about 20%. This is roughly what is seen Chapter 5. Formation of Molecules on the Grain Surface 124

1 Ο 2 Η γ 0.8

4 10 site 4 5 x10 site 5 10 site

0.6 0 20 40 60 80 100 Surface Coverage(%)

Figure 5.9: The variation in γH2O(t) with the surface coverage for different grain sizes.

Here, the binding energies of Model 2 are used. Note that γH2O(t) is decreased with increasing grain size since the probability of capturing an incoming species increases with the grain size. Here C2 scheme has been chosen. Here also the simulation were carried on an Olivine grain having 104 sites and was kept at temperature 10K. Chapter 5. Formation of Molecules on the Grain Surface 125

3.6

3.5

3.4

3.3 Ο 2 Η α 3.2 C1 C2 C3 3.1 C4

3

2.9 0 20 40 60 80 100 Surface Coverage (%)

Figure 5.10: Variation in αH2O(t) with surface coverage for different reaction schemes as in Fig. 5.8 The formation rate of the new species goes down (i.e., αH2O(t) goes up) for C2 and C3 since the species is created in the next time step.

in the Figure.

In Fig. 5.10, the variation in αH2O (Eq. 5-17) for water is shown as a function of time, which is represented here as the surface coverage. Here 104 sites are chosen. This exponent deviates significantly from unity. This deviation is mainly due to blocking the reacting species by the nonreactive species and the population of various species on the grain surface. This is particularly true for cases C2 and C3, since the ER mechanism was not assumed. For those with an ER mechanism (C1 and C4), the numbers actually go down, so the formation rate goes up.

In Fig. 5.11, we plot the variation in αH20 as a function of surface coverage for a different number of sites on the grain. As the number of sites increases, αH2O is seen to decrease as in the case of γH2O. We thus observe that the production rate of any species goes up for larger grains. In Fig. 5.12, we plot α for various species as a function of surface coverage when 104 sites and the Model 2 with C2 mechanism were chosen. The most important Chapter 5. Formation of Molecules on the Grain Surface 126

3.5 Ο 2 Η

α 3

4 10 site 4 5x10 site 5 10 site

2.5 0 20 40 60 80 100 Surface Coverage (%)

Figure 5.11: Same as in Fig. 5.9 except that αH2O(t) has been plotted. As the grain size gets smaller, αH2O increases, i.e., the recombination efficiency decreases. observation from this is that α is not unity for any of these species. Second, up to about 30% of the surface coverage α roughly remains constant, but after that all of them are going up monotonically as the surface gets filled up. Here the ER scheme is not used, and the formation rate goes down with coverage. 3.5 The typical relation between the number density and the radius is r− ∼ (Mathis et al. 1977), while the dependence of the production rate is rn with n 1. ∼ Production is dominated by the smaller grains. But the problem with the smaller grains are that they are lesser stable and could easily evaporate. This aspect clearly requires more thorough study. Since we generally see that α deviates from unity for smaller grains, it is pertinent to ask at, roughly what grain sizes α becomes unity. This would give us a region in which the classical Rate equation would be valid. In Fig. 5.13, we show the variations in α for water and methanol are shown as functions of the grain radius and number of sites on the grain. The corresponding fitted curves are also shown. They are extrapolated to larger grain sizes. We note that, α 1 at around r 3 5µm. Since the computational time to actually carry ∼ ∼ − out the simulation at this size is totally impossible, it could not be independently Chapter 5. Formation of Molecules on the Grain Surface 127

5

4

OH H2O O

α 3 2 HCO H2CO

H3CO

CH3OH 2

1 0 20 40 60 80 100 Surface Coverage (%)

Figure 5.12: Variation in α with the surface coverage for various molecular species that form on the grain surface. All of them deviate from unity. This case is for Model 2 with the C2 scheme. Chapter 5. Formation of Molecules on the Grain Surface 128

Table 5.9: Calculated value of α, for different species, using low abundances of the ac- creting species for Model 2 and keeping the grain at 10 ◦K. These simulations were carried out for the C2 scheme by considering an Olivine grain having 104 number of sites. Surface coverage in (%) Species Site 10 20 30 40 50 60 70 80 90 100 104 2.92 3.08 3.32 3.53 3.68 3.79 3.88 3.98 4.05 4.12 OH 5 104 2.71 2.78 2.98 3.12 3.23 3.32 3.4 3.47 3.53 3.59 × 105 2.61 2.69 2.83 2.98 3.09 3.18 3.25 3.31 3.37 3.42 104 2.95 2.97 3.05 3.27 3.34 3.46 3.47 3.48 3.48 3.47 4 H2O 5 10 2.67 2.69 2.76 2.92 3.04 3.1 3.12 3.13 3.11 3.1 × 105 2.58 2.6 2.68 2.81 2.92 2.97 2.994 2.997 2.997 2.994 104 3.24 3.34 3.61 3.83 4.03 4.17 4.3 4.43 4.55 4.67 4 O2 5 10 2.55 2.64 3.04 3.29 3.5 3.66 3.8 3.92 4.03 4.12 × 105 2.36 2.52 2.79 3.07 3.3 3.46 3.6 3.71 3.81 3.89 104 1.23 1.24 1.28 1.47 1.64 1.78 1.89 2.04 2.25 2.36 HCO 5 104 1.08 1.14 1.29 1.43 1.55 1.68 1.78 1.89 2.04 2.18 × 105 1.07 1.11 1.25 1.41 1.54 1.64 1.74 1.85 1.98 2.14 104 3.25 3.25 3.25 3.29 3.4 3.47 3.49 3.55 3.71 3.77 4 H2CO 5 10 2.77 2.78 2.81 2.95 3.06 3.11 3.13 3.15 3.22 3.32 × 105 2.64 2.66 2.73 2.84 2.95 3.0 3.02 3.06 3.13 3.24 104 1.39 1.38 1.4 1.44 1.55 1.64 1.69 1.81 1.99 2.06 4 H3CO 5 10 1.2 1.21 1.24 1.37 1.49 1.57 1.63 1.69 1.79 1.89 × 105 1.17 1.19 1.26 1.37 1.47 1.54 1.59 1.65 1.75 1.88 104 3.33 3.32 3.33 3.39 3.49 3.54 3.56 3.66 3.79 3.864 4 CH3OH 5 10 2.86 2.87 2.9 3.01 3.12 3.18 3.21 3.24 3.33 3.43 × 105 2.71 2.72 2.78 2.87 3.0 3.04 3.08 3.11 3.19 3.31 Chapter 5. Formation of Molecules on the Grain Surface 129

Number of sites 1.88e4 1.88e6 1.88e8 1.88e10

H2O

CH3OH

H2O

CH3OH α

1 1e-06 1e-05 1e-04 1e-03 Radius (cm)

Figure 5.13: Variations in α for water and methanol (averaged over the surface coverage) as functions of the grain radius and number of sites on the grain. The filled circles are for H2O and the filled boxes are for methanol. The corresponding fitted curves are also shown. They are extrapolated to larger grain sizes. Chapter 5. Formation of Molecules on the Grain Surface 130

1

O 0.8 2 Hopping

0.6

0.4

Eley-Rideal 0.2 Fractional Production of H

0 0 20 40 60 80 100 Surface Coverage(%)

Figure 5.14: Fractional production of H2O through ER and LH schemes is plotted. For low surface coverage, H2O production is dominated by the hopping process and when surface coverage is higher, the Eley-Rideal mechanism contributes in a major way towards the H2O production. verified if this conclusion is rigorously valid. On the other hand, the number of grains at this size is expected to be negligible, and thus the exact knowledge of this limit may not be essential.

5.3.4 A Comparison Between LH and ER Schemes

In Fig. 5.14, we plot the fractional production of H2O for two different schemes at a given surface coverage to show the relative importance between the ER and LH schemes. At the initial stages, when most of the sites are empty, the production of H2O is mainly due to the hopping mechanism i.e., the LH scheme. As the grain starts to populate, the ER scheme starts to play an important role. In the Figure, the uppermost and the lowermost curves are obtained using the low abundance set with C1 model. The other pair of curves is for a higher abundance set in the gas phase. Here the effect is more prominent. In general, the abundance of all the Chapter 5. Formation of Molecules on the Grain Surface 131 species is always a few % higher if the ER scheme is allowed.

5.3.5 Comparison of Results with the Effective Rate Equations

We incorporated our findings into the Rate equation method and named it effective α Rate equation method. We replaced rab/S by rab/S in the Rate equation, where α is coming from the Monte Carlo simulation. In Fig. 5.15, we made a comparison of the abundances of two of the important species, namely, H2O and CH3OH in different methods. We carried out the simulation for an Olivine grain having 104 number of sites and kept at 10 ◦K. Here we used the Model 2 energies and C2 with the low abundances case. The results of our simulation clearly agree with those from the Effective Rate equation very well, but not with the original Rate equation. This indicates that the αs we introduced are very important and their effects are to be taken into account to accurately estimate the production of various species on the grain surface. Using the well-known MRN model of grain size distribution and by the Effective Rate equation method, we find that the production of water and methanol will be 25% and 33%, lesser respectively, than the normal Rate ∼ ∼ equation method. The α that we introduced depends on both the gas and the grain parameters in such a way that it is quite impossible to obtain any analytical formula. We provided a table (Table 5.9), where we tabulated the list of α for Olivine grains at 10K for Model 2 binding energies and C2 scheme of interaction as a function of the surface coverage and number of sites. By the interpolation method, one can obtain α for the other sites and coverages.

5.3.6 Comparison of Results with the Observation

10 10 The abundance of water relative to H2 in cold clouds is between 10− and 8 10− 6 4 × and between 10− to 10− in hot cores. The abundance of methanol with respect to 9 9 7 H2 is 10− , for coldest clouds and between 10− to 10− for warmer clouds and ∼ 8 ∼ a few 10− in hot cores. Observed solid state methanol with respect to water ∼ × is upto 30%. From Table 5.7 it is clear that for the case of Model 1, production of methanol is not significant. For high abundance cases for the Models 2 and 3 are also not very efficient for the production of methanol ( 4% - 6% with respect to ∼ ∼ Chapter 5. Formation of Molecules on the Grain Surface 132

6000

5000 Monte Carlo Rate equation Modified Rate equation H O 4000 2

3000 Number

2000 CH3OH

1000

0 0 20 40 60 80 100 Surface Coverage (%)

Figure 5.15: A comparison of the abundances of two of the important species, namely,

H2O and CH3OH in different methods; Rate equation method, Monte-Carlo method, and the Effective Rate equation method. The Monte Carlo results matche very well with the Effective Rate equation method. Chapter 5. Formation of Molecules on the Grain Surface 133

80

60

40

20 Solid state methanol w.r.t. water

0 2e+04 4e+04 6e+04 8e+04 1e+05 Number Density

Figure 5.16: Solid state methanol abundances with respect to the water is plotted with the number density of the accreting gas. These simulation were carried out for Model 2 by considering the C2 method. The observed abundances are in between the range 5% to 30% which lies in the range 2.6 104 to 8 104. × ×

H2O). For the low abundance cases with Models 2 and 3 methanols are produced in significant amount ( 43% - 57% with respect to H O). In Fig. 5.16 we show ∼ ∼ 2 the ratio of methanol and water as a function of the number density of the gas phase, i.e., accretion rate. Here, we carried out the simulation for Model 2 and by considering the C2 method. The observed ratio is in between the range 5% to 30%. We clearly see that the observed ratio of the number density should be in some 4 3 intermediate range (around 2.6 8 10 cm− ). − × Some methanol is released to the gas phase if the temperature is high enough. in the gas phase. Let nl layers were formed on the grains and were all assumed to evaporate to the gas phase, the gas phase abundances of methanol would be given by

ngnlsS 6 R = =3 10− f 12n10si,0.3S6. nH2 × −

12 Here, ng nH2 1.33 10− = nH2 f 12 the number density of grains, nH2 is the ∼ × − Chapter 5. Formation of Molecules on the Grain Surface 134

number density of H2 in the gas phase, si,0.3 the fractional surface coverage of ith species (e.g., methanol or water) in units of 0.3, S6 the number of sites in the grain 6 in units of 10 , and n10 in units of 10 mono-layers. Now from Table 5.8, time taken for production of a mono-layer is 0.1 Myr for low abundance region. We assumed ∼ that 10 mono-layers are produced in between that time and the grains are deeply inside in higher abundance region. Here the mono-layer production time is lower but the production efficiency of formation of methanol is lower as well. Thus R 7 computed for methanol above, for nl = 1 to 10 varies from 10− (si,0.3 0.03) to 5 ∼ ∼ 3 10− (s 1), generally agrees with the observational results. If, we assume × i,0.3 ∼ that a fraction of grains are sublimated, say, f 12 0.1, the abundance could be − 6 5 ∼ even lower. For water, R 10− to 6 10− , where we have put n = 1 to 10, ∼ × 10 si,0.3 = 0.33 for high abundance, and si,0.3 = 2 for low abundance. This is also in the observed range. Chapter 6

Concluding Remarks

In this thesis we carried out numerical simulations to study the chemical evolution of some simple as well as some complex molecules. A time dependent hydrodynamic code has been developed to explore the dynamic behaviour of a collapsing cloud. The outcome of this code is used to couple with the chemical code, to obtain the chemical evolution of a collapsing cloud. The models used here are distinctly different from the other existing models. We do not consider the heating and cooling processes here. It is found that in the outer edge of the cloud the heating and cooling time scales are comparable with the infalling time scales, so one need to consider this. But deep inside the cloud the infall time scales are much shorter and thus cooling can be ignored. On the other hand, at the outer edge of the cloud, density is very low and thus heating is low. The cooling is also negligible as the reaction rates are low. Thus it may be concluded that the inclusion of the heating and cooling in this code, may not change the result significantly. In our simulations, the effects of grains are considered only for the formation of molecular hydrogen. We compared our results with the observations and found that our results match very well with the observational results except when the molecules are very complex. This mismatch may be due to the reason that there may be other pathways for the formation of these molecules or the reaction rate chosen may be low as compared to the actual rate in the ISM. An average rate is estimated for those reactions for which the reaction rates are not available in the UMIST data base. Now, if for example, this rate is actually higher by a factor of 10, the resulting glycine abundance would have been roughly 103 times higher. Recent observational evidences are suggesting that these kind of complex molecules are mainly formed on the grain surface (at least a part of it). So one need to consider the effect of grain also for the formation of these

135 Chapter 6. Concluding Remarks 136 molecules. A Monte Carlo simulation is carried out for the formation of hydrogen molecule on the grain. Behaviour of the average recombination time of atomic hydrogens on grain surfaces as a function of time, temperature, accretion rate and grain parame- ters were studied. So far, in the literature, recombination time is assumed to be the grain site number divided by the diffusion rate. However, it has been noticed that this implistic assumption is not valid especially when the accretion rate of H on the grain is very low or very high. For very low rates, an atomic hydrogen may have to sweep a grain several times before meeting another atom to form a molecule. For very high rates, the grain surfaces could be partially filled with H and H2 and search for another H need not take S number of hops. It is found that on an average, α0, the index which determines the average recombination time, could be greater than

1 for very low rates. It is also found that α0 depends very strongly on the nature of the grains as well as the temperature and grain site numbers. A new index β has been defined, which is a measure of the average formation rate of H2. This index also is a strong function of the accretion rate per site. In fact, for very high rates the index may go down to 0.5 or even lower. A comparison has been made for the recombination efficiency between the method used here and the Rate equation method and it is been found that it deviate significantly. So we think that our approach with the correction factor α, is much more realistic as compared to the normal Rate equation approach. A similar analysis was carried out for the formation of water, methanol, and other related species on a grain surface using a Monte-Carlo method. Three different sets of binding energies are used for two different gas-phase abundances. Both types of reaction mechanisms namely, ER and LH are considered. Depending on the landing of the incoming species, we considered four models (C1-C4). Besides the formation of these species, it is also shown that the recombination efficiencies are very much dependent on the grain properties. It is found that the formation of various molecules is very much dependent on the binding energies. When the higher binding energies are used, it is very difficult to produce a significant amount of the molecular species. The grain is found to be full of atomic species in this case. The formation of these species is also dependent on the gas phase density. It is noted that, for the high density case, the production of various molecules is small and the grains are filled up relatively quickly by atomic species. If both the ER Chapter 6. Concluding Remarks 137 and LH mechanisms are considered, then the production is always high and the grain is filled up very quickly. As expected, it is also found that, when the grain is more or less empty, the LH scheme is most important. The ER scheme starts dominating more and more as the grains are filled up. A comparison is been made between the results obtained from the Monte Carlo simulation and with that from the effective rate equation (correction term was included) and found that it matches very well. One of the important findings here is the dependency of the correction factor α(t) upon the population of the reactant species on the grain surface. This deviates significantly from unity. This seems to be a very important parameter, because in the usual rate equation it is assumed that α(t) is always 1, which is an overestimation. Another parameter γ is defined, which is named as the ‘catalytic capacity’ and it is noted that it goes down with the increased surface population. This shows that the rate of production indeed increases as the grain is filling up. A strong temperature dependency of α(t) and γ has been found. In our study of grain chemistry, some restrictions have been made: (a) Only a dense cloud is considered where hydrogen is already in the molecular form. Thus accreting gas composition produces primarily water and methanol as is truly the case for dense clouds. In diffused clouds, on the other hand, the composition is such that mostly molecular hydrogen is formed and are desorbed into the gas phase. (b) Only the formation of a mono-layer is studied. Formation of multilayers are not considered here due to the unavailability of sufficient data. (e.g., the binding energies at different surfaces). In near future, we plan to improve the hydrodynamic model by including the angular motion and shock formation in the flow. We expect that jets and outflows would form and a part of this will fall back on the disk and the matter would be recycled. We are anticipating that the chemical abundance will be strongly affected by such recycling of matter. So far, in our model we took the effect of grains only for the formation of hydrogen molecules. In our future studies we are planing to improve our model by taking the effect of grains for the formation of several complex molecules. We are also planning to study the grain chemistry thoroughly including the formation of multilayer. Bibliography

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