An investigation into maser variability in high-mass star forming regions

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in high-mass star forming regions

JP Maswanganye

24827142

Thesis submitted for the degree

Philosophiae Doctor

in

Space

Physics

Potchefstroom Campus of the North-West University

Promoter: Prof DJ van der Walt

Co-promoter: Dr S Goedhart

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I, Maswanganye Jabulani Paul, declare that this thesis titled, AN INVESTIGATION INTO MASER

VARIABILITY IN HIGH-MASS STAR FORMING REGIONS and the work presented in it are my

own. I confirm that:

This work was done wholly or mainly while in candidature for a research degree at this

University.

Where any part of this thesis has previously been submitted for a degree or any other

qualification at this University or any other institution, this has been clearly stated.

Where I have consulted the published work of others, this is always clearly attributed.

Where I have quoted from the work of others, the source is always given. With the

excep-tion of such quotaexcep-tions, this thesis is entirely my work.

I have acknowledged all main sources of help.

Where the thesis is based on work done by myself jointly with others, I have made clear

exactly what was done by others and what I have contributed myself.

Signed:

Date: 4 October 2017

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beauty, the deepest beauty, of nature ... If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.”

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Abstract

Faculty of Natural Sciences Department of Physics

Philosophiae Doctor in Space Physics

An investigation into maser variability in high-mass star forming regions

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with the early phases of high-mass star formation. This implies that studying these maser species is an indirect probe of the dynamics and physical properties of a specific epoch in high-mass star formation processes. Goedhart, Gaylard & van der Walt(2003) investigated the nature of vari-ability in the 6.7- and 12.2-GHz methanol masers associated with G9.62+0.19E and found that they show periodic variations with a 246 day period. Since then, fourteen new methanol masers have been found to show periodic variability, which is a significantly small number compared to known methanol maser regions. The total known population of the 6.7-GHz methanol masers in our Galaxy is 1032 although the total Galactic population of these masers is expected to_∼ 1290 (Green et al.,2017). However, their periods have been found to range from 29.5 to 509 days. Some of the light curves of these sources show remarkable diversity. Also, the origin of the observed periodicity in the class II methanol masers is not yet confirmed, although some authors have proposed possible hypotheses to explain the periodicity in methanol masers.

Considering some of the challenges above, in this thesis, we investigate maser variability in high-mass star-forming regionsby searching for more periodic masers, conducting time-dependent numerical modelling of OH masers and investigate the stability of the determined periods.

In the first part of our investigation, eighteen methanol masers from the 6.7-GHz MMB sur-vey catalogues I, II, III, and IV were selected for a long-term monitoring programme using the 26m HartRAO radio telescope. Two of the eighteen methanol masers were found to be periodi-cally variable in our monitoring window. These sources are - G358.460-0.391, at 6.7-GHz, and G339.986-0.425, at both 6.7- and 12.2-GHz. The periods were searched using the three stan-dard methods - viz. the Lomb-Scargle, epoch-folding and Jurkevich, and fitting an appropriate analytical function using the Levenberg-Marquardt method. It was the latter method which gave the better estimate of periods and their corresponding uncertainties, which are 242_{± 1 and 221}

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0.425 show the strong correlations and time delays. The time delays in G339.986-0.425 across the channels show remarkable structure. However, they could not be correlated with the maser spot map from ATCA interferometric data due to their low spectroscopic and angular resolu-tion. Using the samples from our investigation andGoedhart, Gaylard & van der Walt(2004), the probability of finding a periodic maser in a sample was estimated to be 0.13_{± 0.04. From} this estimate, it was proposed that there may be 38± 15 periodic masers from the remaining 292 maser sources from the 6.7-GHz MMB catalogues I, II, III, IV and V.

In our second investigation, seven periodic methanol masers fromGoedhart, Gaylard & van der Walt (2003), Goedhart, Gaylard & van der Walt (2004), Goedhart et al. (2009) andGoedhart et al.(2013) were used to investigate evolution of the spectra, light curves and periods. These class II methanol masers were in a long-term monitoring programme for 17yr, using the 26m HartRAO radio telescope. These sources are associated with G12.890+0.490, G338.93-0.06, G339.62-0.12, G328.24-0.55, G9.62+0.19E, G188.95+0.89 and G331.13-0.24, and their up-dated periods are 29.42_{± 0.08, 133 ± 1, 201 ± 2, 221 ± 5, 243 ± 3, 393 ± 11 and 506 ± 11} days, respectively. Over the 17yr of monitoring these sources, some maser features did turn on and other faded away, meaning there was an evolution of the spectra. On the other hand, the light curves and periods did not show a significant evolution, and we argued that this ties into the stability of the driving mechanism of the observed periodicity and the light curve.

In our third investigation, the statistical rate equations with line overlap are used to investigate a one-to-one correspondence of the OH masers variability with the time-dependent variability of dust temperature. The investigation is restricted to only 24 levels of OH. Also, only four dust temperature profiles were considered. It was noted that in some cases, it is possible for masers to follow the dust temperature variability, and in other cases, there was no one-to-one correspondence between dust temperature and maser variability. Therefore, it is concluded from

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the dust temperature variations will have a one-to-one correspondence with maser brightness variability. It was also noted that different maser transitions of the same molecule could respond differently to the same dust temperature variability. From this, it can be deduced that it is reasonable to argue that different maser species could also respond differently to the same dust temperature changes.

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This work would not have been possible without many individuals and institutions who invest their finances and time. The list of individuals who contributed to my life for me to get this far could be very long, and I am short of words to show my appreciation for whatever they had sow into my life. My gratitude starts with God for creating this Universe and giving me time to explore it. It is followed by my special thanks to my parents, Mr S. M. Maswanganye and Mrs N. M. Maswanganyi, for giving birth to me and for their support from the very beginning of my life to date. I highly appreciate it.

I owe my uttermost thanks or gratitude to my three Supervisors, Prof. D. J. van der Walt, Dr S. Goedhart and Dr M. J. Gaylard (Deceased: 2014 August 14). Dr M. J. Gaylard found me as the National Astrophysics and Space Science Programme student, and moulded me, although he never lived to see the end of this project, his contribution to my life will be remembered forever. Prof. D. J. van der Walt and Dr S. Goedhart, you both believed in me and supported me throughout. You sacrificed your time to better my life. You were conscious of my weakness which needed a very strong heart and care. I appreciate it so much.

Many thanks to Ms M. E. West for critically reviewing my thesis. Thank you very much. Your contributions to my life and my thesis will be forever be remembered.

Thanks to all my siblings and friends for their support throughout. To name few, Amukelani and Rodney, you guys were always there for a chat to lift my spirit when I was down. Sunelle, thanks for the support in my academics and in all social activities which planned while I was at HartRAO. One not to forget is you Ntukulu Marvelous. You were always there for a chat, you and your husband (Vusi) always welcome me to your house and supported me a lot, and sometimes you helped me get access to some of the journals. Know that GrandPa appreciates everything very much. Thanks to Gingi as well for the signature picture at the end.

I also what to extend my gratitude to Hartebeesthoek Radio Astronomy Observatory (HartRAO), Square Kilometre Array radio telescope (SKA) South Africa and Centre of Space Research in North West University for their financial support, in a form of bursaries, to cover my tuition fees, travel costs, equipment grants and living expenses. My thanks also go to all the staff for their support. To mention few, Ms M. P. Sieberhagen, Ms Lee-Ann van Wyk and Ms Elanie van Rooyen, thank very you. Without HartRAO’s 26m radio telescope, part of this project would not have existed. So, thank you very much for allocating the time for me to conduct masers’ long-term monitoring programmes.

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Declaration of Authorship i

Abstract iii

Acknowledgements vii

Contents viii

List of Figures xii

List of Tables xvi

Abbreviations xvii

Physical Constants xviii

Physical Constants xix

1 Introduction 1

1.1 Motivation. . . 1

1.2 Layout . . . 8

2 High-mass star formation 10 2.1 Giant Molecular Clouds. . . 10

2.1.1 GMCs formation . . . 11

2.1.2 Criterion for a molecular cloud to collapse. . . 13

2.1.3 Stellar initial mass function . . . 16

2.2 Observations: Sequence of events in high-mass star formation . . . 17

2.2.1 Infrared dark clouds (IRDCs) . . . 17

2.2.2 High-mass protostellar objects (HMPOs) . . . 18

2.2.3 Hot molecular cores (HMCs) . . . 19

2.2.4 Ionised hydrogen (HII) regions . . . 20

2.2.5 Summary: Observations toward high-mass star-forming regions . . . . 21

2.3 High-mass star-forming processes . . . 21

2.3.1 Monolithic collapse to form a high-mass star . . . 22

2.3.1.1 Compression . . . 23

2.3.1.2 Collapse . . . 23

2.3.1.3 Accretion . . . 24 viii

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2.3.1.4 Disruption . . . 24

2.3.2 Formation of a cluster of high-mass stars . . . 25

3 Astronomical masers 26 3.1 Observations: masers associated with high-mass star-forming regions . . . 27

3.1.1 Silicon monoxide and formaldehyde masers . . . 27

3.1.1.1 Silicon monoxide masers . . . 27

3.1.1.2 Formaldehyde masers . . . 28

3.1.2 Water vapour masers . . . 29

3.1.3 Hydroxyl masers . . . 31

3.1.4 Methanol masers . . . 32

3.1.4.1 Class I methanol masers . . . 33

3.1.4.2 Class II methanol masers . . . 34

3.1.4.3 Summary of methanol masers . . . 36

3.1.5 Masers as high-mass YSO evolution phases tracers . . . 36

3.2 Maser variability . . . 37

3.2.1 Stochastic variations and flares . . . 37

3.2.2 Periodic variation . . . 39

3.2.3 Origin of the periodic variability in methanol masers . . . 40

3.2.4 Summary on maser variability . . . 42

3.3 Molecular structure . . . 43 3.3.1 Hydroxyl . . . 43 3.3.2 Methanol . . . 45 3.4 Maser model . . . 47 3.4.1 Rate equation . . . 47 3.4.1.1 Line overlap . . . 53

3.4.2 Conditions for forming a maser . . . 56

4 Observational and time series analysis techniques 59 4.1 Single-dish spectral line observations. . . 59

4.1.1 Technical description of the 26m HartRAO radio telescope . . . 59

4.1.2 Fundamentals of a single dish radio telescope . . . 62

4.1.3 26m HartRAO radio telescope calibration . . . 65

4.1.3.1 Pointing offset correction . . . 65

4.1.3.2 Flux density calibration . . . 67

4.1.3.3 Point source sensitivity for the 6.7 and 12.2 GHz receivers. . 70

4.1.4 Bandpass calibration . . . 72

4.1.5 Summary of the single-dish radio telescope observations . . . 74

4.2 Interferometric observations . . . 75

4.2.1 The fundamentals of radio astronomy interferometry . . . 75

4.2.2 Interferometric data calibration. . . 79

4.2.3 Imaging the complex visibilities . . . 80

4.2.4 Summary of radio astronomy interferometry . . . 81

4.3 Time series analysis techniques . . . 82

4.3.1 Period search methods . . . 82

4.3.1.1 The Lomb-Scargle method . . . 82

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4.3.1.3 Epoch-folding using the L-statistic method . . . 86

4.3.1.4 Summary on period search methods. . . 88

4.3.2 Correlation and time-lag test . . . 88

5 Observational results and data analysis 90 5.1 Source selection criteria. . . 90

5.2 Observation strategy . . . 93

5.3 Spectra and time series . . . 94

5.4 Time series analysis . . . 102

5.4.1 Period searches . . . 102

5.4.1.1 The periods and their uncertainties from analytical function fitting techniques . . . 105

5.4.2 Time delays and correlations . . . 120

5.5 Interferometric results. . . 125

5.5.1 Interferometric data collection . . . 125

5.5.2 Maser spot morphology . . . 125

6 Discussion of observational results 129 6.1 Possible origin of periodicity in methanol masers . . . 129

6.1.1 G339.986-0.425 . . . 130

6.1.2 G358.460-0.391 . . . 132

6.2 Time delays . . . 137

6.3 Probability of detecting periodic masers . . . 139

6.4 Period stability of methanol masers. . . 141

7 Results and discussion of the numerical modelling of the masers 145 7.1 Motivation for the investigation . . . 145

7.2 Implementation of OH maser numerical modelling . . . 146

7.2.1 Numerical integration . . . 146

7.2.2 Numerical modelling of the masers . . . 147

7.3 Testing the OH maser numerical modelling code. . . 149

7.4 Searching for conditions under which OH masers are formed . . . 156

7.5 Time-dependent variability of the dust temperature . . . 162

7.5.1 Conclusion: Time-dependent variability of the dust temperature . . . . 168

7.6 Summary of the time-dependent OH maser modelling . . . 169

8 Summary and future research 171 8.1 Summary . . . 171

8.1.1 Long-term monitoring programmes . . . 171

8.1.2 Hydroxyl maser numerical modelling . . . 173

8.2 Future research prospects . . . 174

A Periodically variable methanol masers 189 A.1 Data collection technique . . . 189

A.2 Updated spectra and time series . . . 189

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1.1 Example of methanol maser morphologies . . . 2

1.2 Example of the stochastic variable and flares methanol masers . . . 4

1.3 Example of the periodic variable methanol masers . . . 6

3.1 Diagram of OH molecular structure . . . 44

3.2 Diagram of OH energy levels . . . 44

3.3 Diagram of the CH3OH molecular structure . . . 46

3.4 Possible transitions in a two-level model . . . 48

4.1 Basic microwave radiometer diagram . . . 61

4.2 Pointing correction observation . . . 67

4.3 Driftscan toy model . . . 69

4.4 PSS evolution for the 6.7 GHz receiver . . . 70

4.5 PSS evolution for the 12.2 GHz receiver . . . 71

4.6 Left and right shifted spectra . . . 73

4.7 Two element interferometer model . . . 76

4.8 Example ofuv coverage . . . 78

4.9 Example of the point spread function (P SF ) from uv coverage . . . 78

5.1 Methanol maser spectra of G339.986-0.425 . . . 95

5.2 Methanol maser spectra of G358.460-0.391(3) . . . 96

5.3 Methanol maser time series of G339.986-0.425 at 6.7-GHz . . . 97

5.4 Methanol maser time series of G339.986-0.425 at 12.2-GHz . . . 98

5.5 Methanol maser time series of G358.460-0.391(3) at 6.7-GHz . . . 99

5.6 Methanol maser 2D time series map of G339.986-0.425 at 6.7-GHz . . . 100

5.7 Methanol maser 2D time series map of G339.986-0.425 at 12.2-GHz . . . 100

5.8 Methanol maser 2D time series map of G358.460-0.391(3) at 6.7-GHz . . . 101

5.9 The 6.7-GHz methanol maser Lomb-Scargle periodograms in the G339.986-0.425103 5.10 The 12.2-GHz methanol maser Lomb-Scargle periodograms in the G339.986-0.425 . . . 104

5.11 The 6.7-GHz methanol maser Lomb-Scargle periodograms in the G358.460-0.391105 5.12 The 6.7-GHz methanol maser epoch-folding results in the G339.986-0.425. . . 106

5.13 The 12.2-GHz methanol maser Lomb-Scargle periodograms in the G339.986-0.425 . . . 107

5.14 The 6.7-GHz methanol maser epoch-folding results in the G358.460-0.391. . . 108

5.15 The 6.7-GHz methanol maser epoch-folding results in the G339.986-0.425. . . 109

5.16 The 12.2-GHz methanol maser Lomb-Scargle periodograms in the G339.986-0.425 . . . 110

5.17 The 6.7-GHz methanol maser epoch-folding results in the G358.460-0.391. . . 112 xii

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5.18 Asymmetric cosine fit to the time series of G339.986-0.425 at 6.7-GHz . . . . 113

5.19 asymmetric cosine fit to the time series of G339.986-0.425 at 12.2-GHz . . . . 114

5.20 Absolute cosine fit to the time series of G358.460-0.391 at 6.7-GHz . . . 115

5.21 Evolution of period errors with increasing sample size. . . 116

5.22 Evolution of periods with increasing sample size . . . 117

5.23 Evolution of period errors with increasing sample size. . . 118

5.24 Evolution of periods with increasing sample size . . . 119

5.25 Evolution of period errors with increasing sample size for G358.460-0.391 . . . 120

5.26 Evolution of periods with increasing sample size for G358.460-0.391. . . 121

5.27 The time delays between the 6.7- and 12.2-GHz methanol masers in G339.986-0.425 . . . 123

5.28 The time delays across the velocity channels for methanol masers in G339.986-0.425 . . . 124

5.29 G339.986-0.425 methanol maser spot at 6.7 GHz . . . 126

5.30 G339.986-0.425 maser spots at 6.7 GHz velocity gradient . . . 127

5.31 G358.460-0.391 methanol maser spot map at 6 GHz. . . 128

6.1 G338.93-0.06 time series around the minimum . . . 133

6.2 G358.460-0.391 time series around the minimum . . . 134

6.3 Eclipsing binary toy model . . . 135

6.4 Formation of circular shadow model . . . 136

6.5 Monte Carlo simulation detection rate distribution . . . 140

6.6 Test for Methanol masers period stability . . . 143

7.1 The Boltzmann distribution . . . 148

7.2 Toy model of box line profiles for a partially overlap lines. . . 150

7.3 Evolution of rotational ground state level populations . . . 152

7.4 Evolution of the excitation temperatures . . . 153

7.5 The initial and final distribution of level populations . . . 154

7.6 Intensity from the dust radiation field. . . 156

7.7 Optical depth behaviour as the specific column density is varied . . . 158

7.8 Optical depth as a function of specific column density . . . 160

7.9 Optical depths with . . . 161

7.10 Simulation of the maser light curves . . . 163

7.11 Dust temperature profiles . . . 163

7.12 Evolution of optical depths for different profiles of dust temperatures . . . 164

7.13 Relative amplitudes for different dust temperature profiles . . . 166

7.14 Optical depth variations and relative amplitude with for a pulse dust temperature profile at 50K kinetic temperature . . . 167

7.15 Optical depth and relative amplitude from a pulse dust temperature profile variation168 A.1 Methanol maser Spectra of G9.62+0.19E . . . 190

A.2 Time series of G9.62+0.19E at 6.7-GHz . . . 190

A.3 Time series of G9.62+0.19E at 12.2-GHz . . . 191

A.4 Methanol maser spectra of G12.890+0.490 . . . 192

A.5 Time series of G12.890+0.490 at 6.7-GHz . . . 192

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A.10 Methanol maser spectra of G328.240-0.550 . . . 196

A.11 Time series of G328.240-0.550 at 6.7-GHz. . . 197

A.12 Time series of G328.240-0.550 at 12.2-GHz . . . 198

A.14 Time series of G331.130-0.240 at 6.7-GHz. . . 199

B.1 Methanol maser spectra of G0.092+0.663 . . . 206

B.2 Methanol maser spectra of G6.189-0.358 . . . 206

B.16 Time series of G0.092+0.663 at 6.7-GHz . . . 215 B.17 Time series of G6.189-0.358 at 6.7-GHz . . . 216 B.18 Time series of G8.832-0.028 at 6.7-GHz . . . 217 B.19 Time series of G8.832-0.028 at 12.2-GHz . . . 218 B.20 Time series of G8.872-0.493 at 6.7-GHz . . . 219 B.21 Time series of G312.071+0.082 at 6.7-GHz . . . 220 B.22 Time series of G312.071+0.082 at 12.2-GHz. . . 221 B.23 Time series of G320.780+0.248 at 6.7-GHz . . . 222 B.24 Time series of G320.780+0.248 at 12.2-GHz. . . 223 B.25 Time series of G329.719+1.164 at 6.7-GHz . . . 224 B.26 Time series of G329.719+1.164 at 12.2-GHz. . . 225 B.27 Time series of G335.426-0.240 at 6.7-GHz. . . 226 B.28 Time series of G335.426-0.240 at 12.2-GHz . . . 227 B.29 Time series of G337.052-0.226 at 6.7-GHz. . . 228 B.30 Time series of G337.153-0.395 at 6.7-GHz. . . 229 B.31 Time series of G337.388-0.210 at 6.7-GHz. . . 230 B.32 Time series of G338.925+0.634 at 6.7-GHz . . . 231 B.33 Time series of G338.925+0.634 at 12.2-GHz. . . 232 B.34 Time series of G343.354-0.067 at 6.7-GHz. . . 233 B.35 Time series of G348.617-1.162 at 6.7-GHz. . . 234

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B.36 Time series of G348.617-1.162 at 12.2-GHz . . . 235

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1.1 List of periodic methanol masers . . . 5

2.1 Types of Molecular clouds . . . 11

2.2 HII regions properties . . . 20

4.1 The 26m HartRAO radio telescope specifications . . . 60

4.2 The point source sensitivity values . . . 72

5.1 Southern hemisphere MMB survey parameters. . . 91

5.2 Periodic methanol masers . . . 92

5.3 Methanol masers from the 6.7-GHz MMB catalogues I, II, III and IV. . . 93

5.4 The 6.7 and 12.2 GHz observations specifications . . . 94

5.5 The Lomb-Scargle, epoch-folding and Jurkevich results . . . 111

5.6 Periods for G339.986-0.425 from the asymmetric cosine fit . . . 111

5.7 Periods for G358.460-0.391 from the absolute cosine fit . . . 111

5.8 Periods from three Monte-Carlo simulations . . . 122

6.1 Input of the investigation of period instability algorithm . . . 143

A.1 Determined periods from the Lomb-Scargle, epoch-folding and Jurkevich methods204

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yr year

IRAS Infrared Astronomical Satellite

HII Ionised hydrogen

HartRAO Hartebeesthoek Radio Astronomy

YSO Young Stellar Object

MMB Methanol MultiBeam

ATCA Australia Telescope Compact Array

MERLIN Multi Element Radio Linked Interferometer Network Jodrell Bank Observatory JBO

Australia Telescope National Facility ATNF

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Constant Name Symbol Constant Value (with units) Speed of Light c = 2.99792458_{× 10}10cm s−1 Planck constant h = 6.6260755(40)_{× 10}−27_{erg s}−1

Gravitational constant G = 6.67259(85)_{× 10}−8cm3g−1s−2 Electron charge e = 4.8032068(14)_{× 10}−10_esu

Mass of electron me = 9.1093897(54)× 10−28g

Mass of proton mp = 1.6726231(10)× 10−24g

Mass of neutron mn = 1.6749286(10)× 10−24g

Mass of hydrogen mH = 1.6733× 10−24g

Atomic mass unit amu = 1.6605402(10)× 10−24_g

Boltzmann constant kB = 1.380658(12)× 10−16erg K−1

Stefan-Boltzmann constant σ = 5.67051(19)× 10−5_{erg cm}−2_K−4_s−1

Rydberg constant R_∞ = 1.0974_{× 10}5 _cm−3

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NAME SYMBOL NUMBER IN CGS UNITS Astronomical unit AU 1.496_{× 10}13_cm Parsec pc 3.086_{× 10}18cm Light year ly 9.463× 1017_cm Solar mass M_◦ 1.99_{× 10}33g Solar radius R_◦ 6.96× 1010_cm

Solar luminosity L_◦ 3.9_{× 10}33erg s−1 Solar Temperature T_◦ 75.780× 103 _K

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Introduction

In this introductory chapter, the motivation for investigating maser variability in high-mass star-forming regions is discussed, followed by a brief of the layout of the thesis.

1.1 Motivation

The two brightest class II methanol masers are signposts of high-mass star-forming regions (e.g.

Minier et al.,2005,Pestalozzi, Minier & Booth,2005, and references therein). These methanol maser transitions are described as20− 3−1E and 51 − 60A+, which occur at 6.7- and

12.2-GHz, respectively (Batrla et al., 1987, Menten,1991). For a molecule to mase, a specific set of physical conditions, which cause level population inversion and velocity coherence along the line of sight, are required. The changes in some physical conditions around a maser could change the maser brightness or even quench it. High-mass star-forming regions are dynamic, which implies that the physical conditions in the regions are likely to change with time, resulting in changing or quenching the maser brightness. For example, the lifetime of the 6.7-GHz methanol masers in high-mass star-forming regions is estimated to be between 2.5_×104 and 4.5_×104 yr (van der Walt,2005), meaning that the environment around masers has significantly changed to either quench or destroy them. Thus, studying these masers is, therefore, an indirect probe of the physical properties and dynamics of a specific epoch in the early phase of high-mass star formation.

In some cases, the 6.7- and 12.2-GHz methanol masers have been found to be associated with hydroxyl (OH) and water (H2O) masers, ultracompact HII (UCHII) regions and hot molecular

cores (HMCs) (e.g.Minier, Conway & Booth,2001,Phillips et al.,1998,Walsh et al., 1998). For example, high-resolution imaging of the continuum (HII region) at 8.6-GHz and class II methanol masers show that methanol masers are in most cases offset from the peak emission

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FIGURE 1.1: Examples of the ring-like (G34.75-00.093 and G33.980-00.019), linear (G35.793-00.175) and arched (G33.641-00.228) morphologies of the 6.7-GHz methanol masers by (Bartkiewicz et al.,2009). The spectrum for each source is shown in each subplot,

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of the UCHII region (e.g. Minier, Conway & Booth,2001,Phillips et al.,1998, Walsh et al.,

1998). In high-mass star-forming regions, the class II methanol masers had been proposed to trace accelerated bipolar outflows (Minier et al.,2005,Minier, Conway & Booth,2001). How-ever, other observations suggest that they may trace circumstellar disks (e.g. Norris et al.,1998,

Pestalozzi, Elitzur & Conway,2009, Pestalozzi et al.,2004). In Figure1.1, three examples of methanol maser morphologies are shown. The linear morphologies (similar to those observed in G35.793-00.175, Figure1.1) in high-resolution maser spot maps and the observed velocity gra-dients (e.g. Phillips et al.,1998,Walsh et al.,1998), could be explained by the model presented by Norris et al. (1998). The model argues that we are looking at the masers projected in an edge-on circumstellar disk. Some of the maser spot maps have shown ring-like structures (sim-ilar to what was observed in G34.75-00.093 and G33.980-00.019, Figure1.1), and they were argued to trace circumstellar disks seen face-on (Bartkiewicz, Szymczak & van Langevelde,

2005, Bartkiewicz et al., 2009). However, de Buizer, Bartkiewicz & Szymczak(2012) using their high spatial resolution images in the near- and mid-infrared in a sample of sources with ring-like maser spot morphologies, could not prove nor disprove that the ring-like structure of maser spots traces circumstellar disks seen face-on. From multi-epoch interferometric observa-tions of the masers, we can get information about the internal proper motion of the cloud, which is related to the dynamics of the environment. It has been found in some high-mass star-forming regions that the internal proper motion of methanol masers suggests that the associated UCHII regions are expanding (e.g. Sanna et al.,2010,Sugiyama et al.,2011).

Furthermore, a long-term monitoring programme of the maser flux density might give infor-mation such as the dynamics of the specific set of physical conditions which affect the maser directly or indirectly. The variability of maser flux density can be stochastic, flaring or periodic, which is superimposed on the long-term maser flux density variability. Figure 1.2 shows the time series with examples of flares and stochastic variability of methanol masers in high-mass star-forming regions. The maser flux density of the two maser features of the same source can have different long-term variability, for example, see the time series of G35.20-1.74 in Figure

1.2. The flux density of the maser feature at 46.205 km s−1 decays to zero, whereas for the feature at 44.493 km s−1, its flux density increases monotonically and also underwent a flare. For G351.42+0.64, there is a flare for the -9.692 km s−1 maser feature, but not for the maser feature at -10.510 km s−1. Such behaviour, where the flux density of one maser feature varies differently compared to other maser features, might be suggesting that the changes arise within either the masing gas itself or the masers are exposed to different background sources.

In some cases, the flares in the time series can be argued to occur stochastically in time as shown by the time series of G351.78-0.54, in Figure1.2. In some instances, a new flare starts while another is still decaying. The origin of this complex phenomenon is not easy to explain. It might be associated with an epoch in which there are activities such as accretion and sporadic outflows. In contrast, some masers show slight or no short-term variations, e.g. see the time

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53000 53500 54000 54500 150 300 450

_{G35.20-1.74 (44.493)}

53000 53500 54000 54500 08 16

G35.20-1.74 (46.205)

53800 54000 54200 54400 54600 800 900 1000

Fl

ux de

ns

ity (J

y)

G351.42+0.64 (-10.510)

53800 54000 54200 54400 54600 40 6080 100

G351.42+0.64 (-9.692)

49000 49500 50000 50500

MJD (days)

80 160 240

G351.78-0.54 (1.291)

FIGURE1.2: Example of the stochastically variable masers and flares which are superimposed on the long-term variations. The time series for the G351.78-0.54 at 6.7-GHz, G351.42+0.64 at 12.2-GHz, and G35.20-1.74 at 6.7-GHz were published respectively byMacLeod & Gaylard

(1996),Goedhart et al.(2009), andMaswanganye & Gaylard(2012). The number in the round

brackets in the legend of each panel is the velocity channel (in km s−1_{) of the time series.}

series of G35.20-1.74 in Figure1.2. The variabilities in Figure1.2can be argued to be expected in a dynamic environment such as high-mass star-forming regions, as different epochs are likely to have different activities. Some epochs could be characterised by violent activities, whereas others could be quiet.

Many methanol masers in high-mass star-forming regions have been monitored and have been reported to be variable by, e.g. Caswell et al.(1993) andCaswell et al.(1995). Caswell, Vaile & Ellingsen (1995) investigated the nature of the variabilities of methanol masers in a long-term monitoring programme. These authors concluded that some methanol masers show quasi-periodic variability on the timescale of months and years. Conclusive evidence of quasi-periodic variability was first reported byGoedhart, Gaylard & van der Walt(2003) in G9.62+0.19E for both 6.7- and 12.2-GHz masers, with a 246 day period. The periodic behaviour of methanol masers associated with high-mass star-forming regions was the least expected form of variabil-ity. A further six methanol masers from the remaining 53 of the 54 sources in the monitoring programme ofGoedhart, Gaylard & van der Walt(2004) were confirmed to show periodic vari-ability (Goedhart, Gaylard & van der Walt,2004, Goedhart et al., 2009, 2013). Independent

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TABLE 1.1: List of the currently known periodic methanol masers with their determined periods at either 6.7-GHz or at both 6.7- and 12.2-GHz.

Source name Frequency Period Published (GHz) (days) by

(1) (2) (3) (4)

(1) G12.89+0.49 6.7 & 12.2 29.5 Goedhart et al.(2009) (2) G107.298+5.639 6.7 34.6 Fujisawa et al.(2014)

(3) G75.76+0.34 6.7 119.9 Szymczak, Wolak & Bartkiewicz(2015) (4) G338.93-0.06 6.7 & 12.2 132.8 Goedhart et al.(2013)

(5) G73.06+1.80 6.7 160 Szymczak, Wolak & Bartkiewicz(2015) (6) G45.473+0.134 6.7 195.7 Szymczak, Wolak & Bartkiewicz(2015) (7) G22.357+0.066 6.7 178.1 Szymczak, Wolak & Bartkiewicz(2015) (8) G339.62-0.12 6.7 200.3 Goedhart et al.(2013)

(9) G328.24-0.55 6.7 & 12.2 220.5 Goedhart et al.(2013) (10) G358.460-0.391† 6.7 220.0 Maswanganye et al.(2015) (11) G37.55+0.20 6.7 237 Araya et al.(2010)

(12) G9.62+0.19E 6.7 & 12.2 243.3 Goedhart et al.(2013)

(13) G25.411+0.105 6.7 245 Szymczak, Wolak & Bartkiewicz(2015) (14) G339.986-0.425† 6.7 & 12.2 246 Maswanganye et al.(2016)

(15) G188.95+0.89 6.7 & 12.2 395 Goedhart et al.(2013) (16) G331.13-0.24 6.7 509 Goedhart et al.(2013)

†_{More detailed analysis on the source will be given later.}

searches also confirmed the periodic variability in a further nine sources, which bring the cur-rent total to sixteen periodic methanol masers in our Galaxy (e.g. Araya et al.,2010,Fujisawa et al.,2014,Maswanganye et al.,2015,2016,Szymczak, Wolak & Bartkiewicz,2015,Szymczak et al.,2011). The determined periods of these periodic variable methanol masers were found to be between 29.5 to 509 days. All currently known periodic variable methanol masers and their determined periods are listed in Table1.1, in an ascending order of the determined periods. Some of the shapes of the light curves of periodic masers are significantly different, whereas others can be argued to be similar. Figure1.3shows examples of the light curves of five periodic methanol masers. The diversity of the light curves ledMaswanganye et al.(2015) to classify the light curves in four groups.

The first group has light curves that show a rapid rise followed by an exponential-like decay to a local minimum, which will be called the periodic flare methanol masers group. These light curves have a well-defined quiescent state. Nine sources can be classified into periodic flaring light curves. These are G9.621+0.196E, G328.24-0.55, G331.13-0.24, G22.357+0.066, G45.473+0.134, G73.06+1.80, G75.76+0.34, G37.55+0.20 and IRAS 22198+6336.

A quick decay to a local minimum which is followed by a fast rise characterises the second group. Around the local maximum of these light curves, the flux density of maser increases, at a slower rate, to the local maximum, this is followed by the decay, at an increasing rate. The waveform resembles that of an absolute cosine function, _{|cos ωt|. This group will be called the}

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53200 53400 53600 53800 54000 54200 54400 54600 200 400 600

G9.62+0.20 (1.25)

53880 53900 53920 53940 53960 53980 8 16 24

G12.89+0.49 (37.81)

53000 53500 54000 54500 0 60 120

Fl

ux de

ns

ity (J

y)

G188.95+0.89 (10.659)

51500 52000 52500 53000 53500 54000 54500 0 20 40

G331.13-0.24 (-90.759)

53000 53500 54000 54500

MJD (days)

15 30 45

G338.93-0.06 (-41.859)

FIGURE 1.3: Example of the periodic variable methanol masers. The time series of the G9.62+0.20, G188.95+0.89, G331.13-0.24 and G338.93-0.06 methanol masers were published

byGoedhart et al.(2013), and G12.89+0.49 were fromGoedhart et al.(2009).

Bunny-hop or absolute cosine-like light curves. Two sources have been reported to show this behaviour, viz. G338.93-0.06 (see Figure1.3for the light curve) and G358.460-0.391 (more on the source will be discussed later).

The third group of light curves shows near to sinusoidal variations. Four sources fit the properties of this group, and are G12.89+0.49 (see Figure1.3), G188.95+0.89 (see Figure1.3), G339.986-0.425 (more analysis on the source is reserved for later) and G25.411+0.105. The fourth group is characterised by triangular-shaped periodic variations, and G339.62-0.12 fits into this category. The origin of the periodic methanol maser variations is yet to be confirmed, but there are five hypotheses which attempt to explain this periodicity. These hypotheses argue that the periodic variations are the result of changes in either the maser pumping rate or of the flux of seed photons from the background radio continuum source. The variations in the maser pumping rate are postulated to arise from either a rotating spiral shock in a protobinary system (Parfenov & Sobolev, 2014), periodic accretion in a circumbinary system (Araya et al.,2010) or pulsating high-mass YSO (Inayoshi et al.,2013). On the other hand, periodic variations in the flux of the seed photons are postulated to be due to either a colliding wind binary (CWB) system (van der Walt,2011,van der Walt, Goedhart & Gaylard,2009) or an eclipsing bloated binary companion in a protobinary system (Maswanganye et al.,2015).

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Figures 1.2 and 1.3 suggest that the variability of the flux density of masers is not a simple phenomenon to explain. We noted earlier that some of the time series of methanol masers show short-term variability coupled with long-term variability. Therefore, we can argue that variability of the maser flux density is not only due to changes in pumping rate and the flux of seed photons but perhaps also due to variations in the surroundings of the masing cloudlets. Our brief discussion of the methanol masers in high-mass star-forming regions can be sum-marised in few statements of facts, which are: (i). Some maser spot maps show the ring-like structures, which are thought to trace a circumstellar disk. Others with linear morphology are considered to confirm either a disk or outflows. (ii). The shapes of the light curves of periodic masers are diverse. (iii) The determined periods of the methanol masers range between 29.5 and 509 days. (iv) The origin of periodicity in methanol masers is still to be confirmed. These state-ments of facts led to the present investigation into maser variability in high-mass star-forming regions. From the above, some questions can be raised, such as the following:

• Are the observed shapes of the light curves the only ones or are there more?

• Some proposals or models use the shape of the light curves to support their arguments on the possible origin of the periodicity in methanol masers, e.g. the colliding wind binary system model (van der Walt,2011, van der Walt, Goedhart & Gaylard, 2009), periodic accretion in a circumbinary system proposal (Araya et al.,2010), and the eclipsing bloated binary companion in a protobinary proposal (Maswanganye et al.,2015). It can then be asked, whether the discovery of a new shape of the light curve can lead to a new proposal or model to explain the origin of maser periodicity or result in a unified model to explain all the shapes of light curves.

• It has been noted that the determined periods of the periodic methanol masers range be-tween 29.5 and 509 days (see the list of periods in Table1.1). Thus, the question can be asked, are there sources with periods outside this current range? Also, is this range related to the origin of periodicity? (e.g. Inayoshi et al. (2013) argue that periods less than ten days are not possible as a protostar with small accretion rate can become unstable, but it will not pulsate.)

• From the sixteen periodic variable masers, it is noted that about 63 per cent have periods between 150 and 250 days, could it be suggesting a distribution? Does the determine periods follow a Gaussian distribution with a mean and standard deviation of 210 and 118 days, respectively? If so, why is it the case?

• Is there an evolution in the determined periods or shapes of the light curves?

• Can all the shapes of the light curves be explained by the variability of the maser pumping rate?

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In order to attempt to answer some of the questions raised above, seven methanol masers re-ported to be periodic at either 6.7-GHz or both 6.7- and 12.2-GHz were monitored. These seven sources were from the sample of 54 sources, which were monitored byGoedhart, Gaylard & van der Walt (2003), Goedhart, Gaylard & van der Walt (2004), Goedhart et al. (2009), and

Goedhart et al.(2013). These sources can address questions such as the evolution of the peri-ods, the shape of the light curves and spectra, as they had been monitored for 17 yr, from 1999 to 2016. The second sample had 18 methanol masers from the 6.7-GHz MMB survey catalogues I, II, III, and IV (Caswell et al., 2010, 2011, Green et al., 2010, 2012). The second sample could address some of the questions stated above, such as, e.g. expanding the period range for methanol masers, find new shapes of light curves or even lead to a new proposal of the origin of periodic variability in methanol masers. The Australia Telescope Compact Array interfero-metric data for the selected few sources, especially those found to be varying periodically, could be used to check the spatial distribution of maser features. The maser spot map can be used to investigate the theory about the ring-like and linear morphologies proposed to be tracing the face-on and edge-on circumstellar disks, respectively.

In addition to our observational investigation of maser variability in high-mass star-forming regions, a numerical model of a hydroxyl maser was used to investigate the effects of changing dust temperature with time on a maser. This could be used to address the question about the maser pumping rate producing all the observed shapes of the light curves. Although OH and CH3OH masers are different species, they are both radiatively pumped, and Goedhart et al.

(in prep.) found a strongly correlated periodic flaring behaviour between the OH and CH3OH

masers in G9.621+0.196E.

1.2 Layout

This thesis is comprised of seven chapters excluding the introduction and the two appendixes. These chapters are about high-mass star formation, astronomical masers, observational and time series analysis techniques, observational results and data analysis, discussions of observational results, results and discussions of the maser numerical modelling, and summary and future re-search. The two appendixes show the periodic variable methanol masers and non-periodic vari-able methanol masers, respectively.

The chapter on High-mass star formation gives a brief discussion of the formation and the criterion for a molecular cloud to collapse to form a star. This discussion is followed by what can be learned from observations toward active high-mass star-forming regions. The two final discussions are the processes by which a single high-mass star and a protobinary system are formed.

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In chapter three, titled Astronomical masers, different maser species which are associated with high-mass star-forming regions and their associated astrophysical objects, are discussed first. The maser brightness variability is then discussed, which is followed by a brief review of the molecular structure of methanol and hydroxyl molecules. The last section gives a detailed dis-cussion of a maser model using the statistical rate equations.

The fundamentals and calibration techniques for both single-dish and an interferometer obser-vations are discussed in the chapter on Observational and time series analysis techniques. In the last section of this chapter, three independent period searching methods and the method used to search for correlations and time delays between two time series or light curves, are discussed. The single-dish radio telescope monitoring programme and interferometric results of the two new periodic variable methanol masers are shown, analysed and briefly discussed in the Obser-vational results and data analysischapter. These sources are G339.986-0.425, at both 6.7- and 12.2-GHz, and G358.460-0.391, at 6.7-GHz.

In the Discussion of observational results chapter, the possible origin of the maser variability associated with G339.986-0.425 at both 6.7- and 12.2-GHz, and G358.460-0.391 at 6.7-GHz, measured time delays in masers associated with G339.986-0.425, and the probability of detect-ing a periodic variable methanol maser in a given sample are discussed. This is followed by the investigation of the stability of the periods in the seven periodic methanol masers from Goed-hart, Gaylard & van der Walt(2003),Goedhart, Gaylard & van der Walt(2004),Goedhart et al.

(2009), andGoedhart et al.(2013). These seven sources have been monitored for at least 17 yr. The Results and discussion of the numerical modelling of the masers chapter discusses the im-plementation of the numerical modelling of OH masers, investigation of the correctness of the outputs and searches the specific set of physical parameters for OH to mase. The last part of the chapter investigates how the time-dependent variable dust temperature will affect the brightness of masers. The results were then analysed and discussed.

The last chapter, titled Summary and future research, summarises the findings of our search for new periodic methanol masers, and the results of numerical modelling of hydroxyl masers. Future recommendations, to attempt to address some of the challenges which we could not answer in this project are then proposed.

This thesis is accompanied by two appendices, which are: (i) Appendix A, which shows and discusses the updated spectra and time series of the seven methanol masers reported to be pe-riodically variable byGoedhart, Gaylard & van der Walt(2003), Goedhart, Gaylard & van der Walt(2004),Goedhart et al.(2009) andGoedhart et al.(2013). (ii) Appendix B exhibits and dis-cusses the spectra and time series of the methanol maser sources, which did not show periodic variability over a monitoring window within our sampling intervals.

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High-mass star formation

In our studies, we refer to high-mass stars as OB-type stars with enough mass to produce a type II supernova, meaning that star massM_∗ > 8 M_⊙(Zinnecker & Yorke,2007). In the studies made byPalla & Stahler(1993), it was shown that stars withM_∗ > 8 M_⊙do not go through a pre-main sequence phase. We also adopt this definition of a high-mass star.

High-mass stars play a vital role in the evolution of their host galaxy because they produce, among other things, intense ultraviolet radiation, have strong stellar winds and end their stellar evolution in supernova explosions. For example, supernovae create elements heavier than iron which are injected into the interstellar medium (ISM). High-mass stars are formed in optically thick environments and are embedded inside giant molecular clouds (GMCs) with high extinc-tion in the V-band, AV,& 10 mag. These GMCs are found typically at distances & few kpc

from the Sun, except Orion and Cepheus A, which are at a distance of 437_{± 19 (}Hirota et al.,

2007) and700+31₋₂₈pc (Dzib et al.,2011) from the Sun, respectively. Amongst other things, the great distances to and high V-band extinction in high-mass star-forming regions, make it difficult to observe the formation of high-mass stars directly.

In this chapter, the molecular clouds and the processes of high-mass star formation, are dis-cussed. We start off with the physical properties and formation processes of molecular clouds, after which will follow a discussion on the sequence of events on how the formation of a high-mass star occurs, deduced from observations. The remaining part of the chapter describes the theories of high-mass star and binary system formation.

2.1 Giant Molecular Clouds

Molecular clouds are dense and cold condensation of matter. They are composed of atomic and molecular gases (about 99 per cent), and dust (about 1 per cent). The material is generally

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TABLE 2.1: Classes of Galactic molecular clouds and their corresponding physical properties (Obtained from Stahler & Palla,2004, in Ch. 3.). The physical properties are typical visual extinction along the line of sightAV, total number densityntot, sizeL, temperature T and mass

M .

Cloud type AV ntot L T M

(mag) cm−3 (pc) (K) (M_⊙) (1) (2) (3) (4) (5) (6) Diffuse 1 500 3 50 50 Giant Molecular 2 100 50 15 105 Dark (complexes) 5 500 10 10 104 Dark (Individual) 10 103 2 10 30 Dense cloud 10 104 0.1 10 10

compressible, magnetised, turbulent and fluid. Some of the physical properties of the molecu-lar clouds can be determined from the gas and column number densities. The column number density is important in shielding the cloud from UV radiation of OB-type stars, which can com-pletely photodissociate the atoms and molecules in the cloud.

The physical properties of the molecular cloud such as temperature, size, average number den-sity and the total mass are determined from the molecular and atomic transitions. The electronic, vibrational and rotational transitions emit (or absorb) photons respectively at the UV/visible, in-frared and radio (including submillimetre and millimetre) wavelengths. Therefore, a complete study of molecular clouds requires multi-wavelength observations.

Molecular clouds can be divided into four groups as given by Stahler & Palla (2004). These groups are diffuse, giant molecular, dark and dense core/Bok Globule clouds. Table2.1presents a list of the physical properties of these four groups of molecular clouds, in the order of increas-ing visual extinction. We will limit our discussion to giant molecular clouds (GMCs) as almost all star formation occurs in GMCs, in particular, high-mass stars, which are our specific interest. GMC formation is thus directly related to high-mass star formation. Hence, we will discuss GMC formation and the criteria for a molecular cloud to collapse and form a protostar, which will result in forming a star.

2.1.1 GMCs formation

GMCs are constantly being formed and destroyed in the Miky Way, our Galaxy. Their lifespan is expected to be less than 3.0×107 _{yr (}_{Stahler & Palla}_,₂₀₀₄_{). They are destroyed by the}

OB-type stars which are formed in their cores. Their rate of formation has a direct link to the star formation rate in the Galaxy, especially high-mass stars.

There are several mechanisms proposed to explain the formation of GMCs. However, we will only give a brief description on a few selected proposals, which are:

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• Random coalescence model (Cowie,1980, Norman & Silk,1980), where randomly dis-tributed small clouds collide inelastically and agglomerate into a GMC.

• Parker instability model (Parker,1966). In theParker(1966) instability, the magnetic field and cosmic ray pressure give substantial support to the gravitational collapse of the gas perpendicular to the galactic plane. A perturbation of the mass-to-flux ratio (note that this term will be explained in detail later) is buoyant, which implies it will rise to the surface of the disk plane causing the mass to drift from the regions of high to low magnetic field strength. The regions with weak magnetic fields condense to form GMCs.

• Pressurised accumulation in the shock model (Blitz & Williams, 1999). The OB stars, isolated supernovae and spiral density shocks condense the filaments, sheets and shells of a cloud into GMC.

• Gravitational instability model (Dobbs et al., 2014, Elmegreen, 1990). In this model, the rotation of a galaxy stabilises the stellar component of a galactic disk, leaving the magnetised gas gravitationally unstable, which implies that it will collapse to form GMCs. A density wave is not necessary for initiating the collapse, but it can increase the instability growth. This is a top-down scenario, where large clouds are formed first and fragment into GMCs due to the gravitational instability described byToomre(1964).

• Thermal instability model (Balbus & Soker,1989,Dobbs et al.,2014,Field,1965). The system is isobaric, meaning that the pressure, P , is constant. A thermal instability in a homogeneous medium arises when the net energy loss per gram L is ∂L/T∂T

P <

0, where T is the temperature. _{L is the difference between energy loss and gain. The} thermally unstable region, which is denser and cooler than its surroundings, cools and condenses to form dense, cold giant molecular complexes.

• Spiral arm induced collision model (Dobbs et al., 2014, Elmegreen, 1990). The spiral arms in the Galaxy induce the collision of clumps in the disk to agglomerate to form the GMCs.

Each of these mechanisms has its shortfalls in explaining the results of observations. For exam-ple, the formation of a GMC via smaller cloud collisions and agglomeration is slow, of the order of 108yr (Larson,1994), compared to the GMC lifespan, an order of few 107yr. In a review of the possible mechanisms for forming molecular clouds byElmegreen(1990), the gravitational, plus thermal instability model was argued to be the most likely mechanism by which GMCs are formed. One of the main reasons for gravitational plus thermal instability being the most likely process for forming GMCs is that it suggests that the mass spectrum takes the form of a power law. Larson (1994) argues that gravitational instability is likely to be the process by which GMCs are formed.

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2.1.2 Criterion for a molecular cloud to collapse

A stable molecular cloud is supported against gravitational collapse by magnetic fields, turbu-lent motions of the molecular clumps, rotational motion of the clumps and thermal pressure. The interaction between these forces in a molecular cloud can be summarised using the Virial theorem in the Lagrangian form whichStahler & Palla(2004) derived as,

1

2I = 2¨ T + 2U + W + M, (2.1)

whereI (I = _{R ρ|r|}2d3x, for a massdm), T , U, W and M are: the moment of inertia of a molecular cloud, total kinetic energy of the internal motion, thermal motion, gravitational poten-tial and energy associated with the magnetic field, respectively. ¨I is related to the acceleration of contraction or expansion of the molecular cloud. For a gravitationally stable molecular cloud, the right-hand side of equation2.1is zero, which implies ¨I = 0.

In a stable molecular cloud, which is supported only by thermal pressure against gravitational collapse, the Virial theorem in Lagrangian form in equation2.1becomes

0 = 2_{U + W.} (2.2)

The total internal kinetic energy due thermal pressure of the cloud is _{U =} 3MclkBT

2µmH , where

Mcl, kB, T , µ and mH are respectively the mass of a molecular cloud, Boltzmann constant,

uniform temperature of the cloud, the mean molecular weight and the mass of a hydrogen atom. Also,W =−3GM 2 cl 5 4πρo 3Mcl 1₃

, withρoandG being the uniform density of the cloud and the

universal gravitational constant, respectively. Using _{U and W, as given earlier, and equation}

2.2, the Jean’s massMJ of a molecular cloud can be derived as

MJ ≃ 5kBT GµmH 3/2 3 4πρo 1/2 . (2.3)

If the mass of the cloud exceeds its Jean’s mass, the cloud is gravitational unstable, which implies it will collapse. It follows that if a stable molecular cloud is sufficiently perturbed, either2_{U > W or 2U < W, it could either expand or collapse, respectively.}

During the isothermal collapse, the temperature stays constant but the density of the core in-creases, meaning thatMJ decreases, which could result in further fragmentation of the core.

The temperature stays constant because the cloud cools efficiently by converting the kinetic en-ergy of the atoms and molecules into far-infrared radiation, which is transparent to the cloud. This is a free-fall collapse phase and is characterised by the free-fall timescale,tff (Smith,1995,

Stahler & Palla,2004, in Ch. 10 and 3, respectively). In a self-gravitating spherical molecular cloud, the change in radius for infalling matter is governed by Newton’s second law of motion

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(Smith,1995,Tohline,1982), from which the free-fall timescale can be determined as tff = r 3π 32Gρo ≃ 2.1 × 105 ρo 10−19_{g cm}−3 −1/2 yr. (2.4)

The free-fall timescale in equation 2.4 depends on the density of the cloud. Increasing the density of the cloud decreases the free-fall timescale. At the end of free-fall collapse, a near hydrostatic equilibrium stellar object called a protostar is formed in the core of the molecular cloud.

For a protostar to evolve onto the main sequence in the Hertzsprung–Russell (HR) diagram, it needs to accrete the material from the natal cloud. The time required for a protostar to accrete the natal cloud in order to evolve onto a main sequence star in the HR diagram is called the Kelvin-Helmholtz timescale,tKH. It is also referred as the time for a protostar to reach internal

thermal equilibrium (Shu, Adams & Lizano,1987). It can also set the timescale for the main sequence star to evolve into a giant. However, this is not within our scope of discussion as we are interested in the formation of stars.

The Kelvin-Helmholtz timescale can be derived from the gravitational energy of a near hydro-static equilibrium and its luminosityL. From the Virial theorem, half of the gravitational energy of a near hydrostatic equilibrium object is used to heat the object, and the rest is radiated into space. The timescale for the radiation of gravitational potential energy into space is given as the ratio of half of the gravitational potential energy, and the luminosity of the protostarL, which is the Kelvin-Helmholtz timescaletKH and it is

tKH = 3 10 GM2 RL , ≃ GM 2 RL . (2.5)

Equation2.5gives an estimation of the time in which a protostar evolves onto the main sequence of the HR diagram. This is the time before nuclear or hydrogen fusion commences in the core, after a protostar has been formed (Zinnecker & Yorke,2007).

The Jean’s mass (equation2.3) can be used to investigate the gravitational stability of molecular clouds. The free-fall (equation 2.4) and Kelvin-Helmholtz (equation 2.5) timescales can be compared for the low- and high-mass protostars.

For example, Carey et al.(1998) found that the mass of molecular clouds ranges between 102 and 105 M_⊙,nH2 ∼ 10

5_cm−3 _and_T _{≤ 20 K. Thus ρ}_o _{= 2n}_H

2mH ∼ 2.0 · 10

5_{· 1.7 × 10}−24

∼ 3.3 × 10−19 _{g cm}−3_{. With} _{µ = 2, T = 20 K, and using equation} _2.3_{, the Jean’s mass}

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unstable because the Jean’s mass is less than the dense core mass. This is consistent with the fact that these are regions where stars are forming. In general, GMCs are in pressure equilibrium with the surrounding interstellar medium, which may require sufficient perturbation to initiate collapse.

The Kelvin–Helmholtz timescale and time for accreting massM_∗at a specific accretion rate ˙M_∗, which istacc = M∗/ ˙M∗ (Beuther et al.,2002), can be compared for both low- and high-mass

stars. Consider a solar-like star, with 1M_⊙, 1L_⊙, 1R_⊙ and a typical mass accretion rate of ∼ 10−7_M

⊙yr−1. The accretion time for 1M⊙is∼ 107yr. The Kelvin–Helmholtz timescale

(using equation2.5) is_{∼ 3.0 × 10}7yr. The accretion and Kelvin–Helmholtz timescales are of the same order of magnitude∼ 107 _{yr. For a high-mass star with M = 10 M}

⊙,L = 3.0× 103

L_⊙, R = 54 R_⊙ and typical mass accretion rate of _{∼ 10}−4 M_⊙yr−1 (accretion rate from

Beuther et al.,2002), the accretion and Kelvin–Helmholtz timescale are 105 and 2.0_×104 yr, respectively. The accretion timescale is greater than the Kelvin–Helmholtz timescale by a factor of ten. This implies that only a fraction of mass is accreted before the main sequence phase and a large fraction of mass is accreted while hydrogen is burning in the core (Kahn, 1974). These calculations show that high-mass star formation processes are not just scaled up version of low-mass stars.

Thus far, only the gravitational and thermal parts of the Virial theorem have been considered. If a molecular cloud is supported against gravitational collapse by the magnetic field only, the Virial theorem in equation 2.1will be_{W + M = 0. The energy associated with the magnetic} field is given asM = Φ 2 B 6π2_R orM = | B_|2 R3

6 (Stahler & Palla,2004), where ΦBis the magnetic

flux. The Virial theorem can then be reduced to 0 = |B| 2_R3 6 − 3GM_cl2 5R , = 3G 5R 5Φ2_B 18π2_G − M 2 cl , = 3G 5RM 2 Φ− Mcl2 , (2.6)

whereM_Φ is the critical mass (Crutcher, 2012). The mass-to-flux ratio, Mcl

Φ_B, can be derived from equation2.6as Mcl ΦB =r 5 18 1 π√G. (2.7)

In the case where Mcl

Φ_B > q

5

18_π√1_G or Mcl/MΦ < 1, the magnetic field will prevent the

molecular cloud from gravitational collapse and the cloud is said to be magnetically subcritical. When Mcl

Φ_B < q

5

18_π√1_G orMcl/MΦ> 1, the cloud is gravitationally unstable, implying it will

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2.1.3 Stellar initial mass function

The stellar Initial Mass Function (IMF), defined as the number of stars per cubic parsec per unit logarithmic mass (Miller & Scalo, 1979, Rana, 1987, Richtler, 1994, Salpeter, 1955), is also referred as the distribution of initial stellar masses in a population of stars at birth. The IMF is characterised by a power law for the stars with masses greater than 5M_⊙and is flat for the masses less than 1M_⊙(Meyer et al.,2000). At lower masses (0.1-0.5M_⊙), there are still debates on whether the IMF is flat, continues to rise or turns. It is related to the fragmentation of a GMC into star-forming clumps.

One of the fundamental constraints in star formation in the Galaxy is the birthrate and the initial mass or stellar IMF (Rana,1987,Richtler,1994,Salpeter,1955), which are the average spatial rate of star formation and the frequency distribution of the stellar masses at birth (Miller & Scalo, 1979), respectively. The IMF can be determined using field stars, or different types of stellar clusters (e.g.Chabrier,2003). For the field stars the IMF is determined from the present-day mass function (PDMF), i.e. the number of stars in the main sequence per unit logarithmic mass per cubic parsec. For stars with main sequence lifetimes greater than the age of the Galaxy, the PDMF and IMF are equivalent, whereas for stars with main sequence lifetimes less than the age of the Galaxy, the stellar birthrate history of the Galaxy is required to scale the PDMF to be equivalent to the IMF. The stellar mass is an unobservable quantity but is important to determine the spectrum, lifetime and death of a star (Williams, Blitz & McKee,2000). Thus, the PDMF is determined from the present day luminosity function, as mass is an unobservable quantity (Ch. 4 in Stahler & Palla,2004). On the other hand, embedded stellar clusters are used to sample instantaneously the stellar IMF, in which the stars are at equal distances and ages, and have the same chemical evolution (Lada,2005). The stellar IMF from a young cluster and field stars from the galactic disk are in good agreement with each other (Chabrier,2003).

The mass spectrum of clumps in GMCs is a power law,dN/d ln M _{∝ M}−a_{, where}_a_{∼ 0.6−0.8}

(e.g. Williams, Blitz & McKee, 2000). The core mass spectrum is also characterised by the power law witha_{∼ 1.35 (e.g.}Williams, Blitz & McKee,2000). For example,Salpeter(1955) found the power law part of the stellar IMF to bea∼ 1.35. The core mass spectrum resembles the stellar IMF. If the core mass spectrum is indeed similar to the stellar IMF, then the fraction of the core mass which goes into star formation is approximately independent of the mass, and the cloud fragmentation process determines the stellar IMF.

The stellar IMF is critical, for determining the star formation rate in a galaxy and the chemical evolution of the galaxy, as the different elements that can be synthesised by stars in a mass range ofm and m + dm is proportional to the slope of the stellar IMF. It is also important in determining the mass-to-light-ratio for stars within a mass range ofm and m+dm of a galaxy as this also depends on the slope and the corresponding mass range of the stellar IMF. The power

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law part of the stellar IMF can be used to explain the large separations for the OB-type stars forming in our Galaxy. The high-mass star region in the IMF decreases with increasing mass, as first noticed bySalpeter(1955). This implies that in a small volume of space (few cubic parsec) there are fewer high-mass stars.

2.2 Observations: Sequence of events in high-mass star formation

High-mass stars are formed inside GMCs, which are optically thick, meaning that optical and UV radiation cannot escape the core. Therefore, the submillimetre, millimetre and centimetre astronomical observations are ideal for studying these regions.

A brief summary of mid-infrared observations, through radio wavelengths toward regions where high-mass stars are forming or are still to be formed, was given byvan der Tak & Menten(2005). These authors categorised the observations into five groups, namely: (i) infrared dark clouds (IRDCs), (ii) high-mass protostellar objects (HMPOs), (iii) hot molecular cores (HMCs), (iv) ultracompact HII (UCHII) regions and (v) compact and classic HII regions. This categorisation was also adopted byZinnecker & Yorke(2007). These five groups form the sequence of events for how high-mass stars are formed, based on the observations.

Of the five groups, the first four - which are the IRDC, HMPO, HMC and UCHII regions, are of interest to us. In the survey byGerner et al. (2014) of 59 high-mass star-forming regions at different evolutionary stages, the duration of the chemical evolution stages of the IRDC, HMPO, HMC and UCHII region were estimated to be_{∼ 1.0×10}4, 6.0_×104, 4.0_×104 and 1.0_×104 yr, respectively. From the duration of these four stages, the total chemical evolution time can be estimated to be of the order of 105 yr. The chemistry of the IRDC, HMPO, HMC and UCHII region stages can give us some insight into the initial conditions for forming high-mass stars, and the environments in which such stars form.

In this section, we will discuss some of the molecular species found in the first four groups mentioned earlier and information that can be deduced from studies of these objects. We will also discuss the continuums in submillimetre and millimetre wavelengths which are associated with these astrophysical objects.

2.2.1 Infrared dark clouds (IRDCs)

The IRDCs are considered to have the ideal initial conditions for high-mass star formation. The physical properties of IRDCs can be probed using submillimetre and millimetre molecular line observations. For example,Carey et al.(1998) studied ten Mid-course Space Experiment (MSX) dark clouds, and found that the clouds were cold (T < 20 K ) and dense (nH2 > 10

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The diameters ranged between 0.4 and 15.0pc. Formaldehyde (H2CO) was detected in all ten

dark clouds, which confirms that they are all very dense (Carey et al., 1998). The kinematic distances to these regions were estimated to be between 1 to 8 kpc, which implies that they are not local. Pillai et al.(2006) used ammonia (NH3) to study the physical properties of the

IRDCs, and found that the average temperatures were between 10 and 20K. IRDCs had large linewidths, between 1 and 3.5 km s−1, implying that they are supported by turbulence, and high densities, which correspond to extremely high visual extinction values,AV, ranging between 55

and 450mag. Most studies of IRDCs have shown filamentary morphology, which corresponds to an elongation of a cylinder (Menten, Pillai & Wyrowski,2005).

In most cases, the magnetic field in the high-mass star-forming regions is measured through the Zeeman splitting (Foot,2005) of OH lines and the 21cm hydrogen line. The Zeeman splitting method is used on the actively forming stars. However,Pillai et al.(2015) studied the IRDCs as-sociated with high-mass star-forming regions (G11.11-0.12 and G0.253-0.016). These regions had been argued to be ideal for studying the importance of the magnetic field before high-mass star formation commences. It was found that the mass-to-flux ratio(Mcl/ΦB) is approximately

less than the critical mass-to-flux ratio,(Mcl/ΦB)cric, which implies(Mcl/ΦB)/(Mcl/ΦB)cric≪

1. This means that magnetic field is important in the IRDCs evolution, and that all high-mass star formation simulations, which start with(Mcl/ΦB)/(Mcl/ΦB)cric ≫ 1, have been set up

with incorrect initial conditions. The simulations should start in a subcritical magnetic state, stable against gravitational collapse, and then evolve into a supercritical magnetic state, which would be unstable against gravitational collapse. It has also been argued that the magnetic field channels the gas into the main filaments.

2.2.2 High-mass protostellar objects (HMPOs)

The carbon monoxide, COJ = 2→ 1, mapping observations toward 26 regions with HMPOs showed that 21 regions have bipolar molecular outflow morphology (Beuther et al.,2002). The detection rate of bipolar molecular outflows was approximately 80 per cent, suggesting that the outflows are a very likely phenomenon in high-mass star formation. The accretion rates were derived to be between _{∼ 10}−4 and 10−3 _M

⊙yr−1, and the estimated bolometric luminosity

of∼ 104 _L

⊙. Other CO mapping observations toward HMPOs have been conducted by, e.g.

Henning et al.(2000) andZhang et al.(2001). Zhang et al.(2001) found that about 50 per cent of observed HMPOs have bipolar molecular outflows, with the luminosities ranging from 103 to 105L_⊙and the masses in the outflows were> 10 M_⊙. The bipolar molecular outflows were also highly collimated. To explain the high collimations found in outflows of HMPOs, which were even higher than young low-mass star collimations, Devine et al. (1999) argue that the outflows in the HMPOs are powered by jets during the first 103 to 104 yr, and after this phase, they are powered by wide-angle winds. In the observations made byBeuther et al.(2002), the