Claire L. Davies Phd Thesis

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REVOLUTIONEVOLUTION: TRACINGANGULARMOMENTUMDURINGSTARAND PLANETARYSYSTEMFORMATION

Claire Louise Davies

A Thesis Submitted for the Degree of PhD at the University of St Andrews

2015

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This item is licensed under a Creative Commons Licence Revolution Evolution Tracing Angular Momentum During Star And Planetary System Formation

Claire Louise Davies

This thesis is submitted in partial fulﬁlment for the degree of Doctor of Philosophy in Astrophysics at the University of St Andrews

April 2015

Declaration

I, Claire Louise Davies, hereby certify that this thesis, which is approximately 60,000 words in length, has been written by me, that it is the record of work carried out by me and that it has not been submitted in any previous application for a higher degree.

Date Signature of candidate

I was admitted as a research student in September 2011 and as a candidate for the degree of PhD in September 2011; the higher study for which this is a record was carried out in the University of St Andrews between 2011 and 2015.

Date Signature of candidate

I hereby certify that the candidate has fulﬁlled the conditions of the Resolution and Regulations appropriate for the degree of PhD in the University of St Andrews and that the candidate is qualiﬁed to submit this thesis in application for that degree.

Date Signature of supervisor

In submitting this thesis to the University of St Andrews we understand that we are giving permission for it to be made available for use in accordance with the regulations of the University Library for the time being in force, subject to any copyright vested in the work not being aﬀected thereby. We also understand that the title and the abstract will be published, and that a copy of the work may be made and supplied to any bona ﬁde library or research worker, that my thesis will be electronically accessible for personal or research use unless exempt by award of an embargo as requested below, and that the library has the right to migrate my thesis into new electronic forms as required to ensure continued access to the thesis. We have obtained any third-party copyright permissions that may be required in order to allow such access and migration, or have requested the appropriate embargo below.

The following is an agreed request by candidate and supervisor regarding the electronic publication of this thesis: Access to Printed copy and electronic publication of thesis through the University of St Andrews.

Date Signature of candidate

Date Signature of supervisor

iii

Collaboration Statement

This thesis is the result of my own work carried out at the University of St Andrews between September 2011 and April 2015. Part of the work presented in this thesis has been published in refereed scientiﬁc journals. In all cases, the text in the chapters has been written entirely by me. All ﬁgures, unless explicitly stated in the text, have been produced by me.

Chapters 2 and 3 feature work presented in: “Accretion discs as regulators of stellar angular momentum evolution in the ONC and Taurus-Auriga”: Davies C. L., Gregory S. G., Greaves J. S., 2014, Monthly Notices of the Royal Astronomical Society, 444, 1157. Figs. 2.2, 3.1, 3.2, 3.4, 3.8 3.11, and 3.12 have been reproduced from this paper. All work in this paper was carried out and written by me, with scientiﬁc advice from the co-authors.

Chapter 4 is based on “The outer regions of protoplanetary discs: evidence for viscous evolution?”: Davies C. L., Gregory S. G., Greaves J. S., Ilee J. D., 2015, Astronomy & Astrophysics, submitted. Figs. 4.1 to 4.7 appear here in the same format as in the submitted manuscript. All work in this paper was carried out and written by me, with scientiﬁc advice from the co-authors.

The work presented in Chapter 5 is being prepared for submission to the Monthly Notices of the Royal Astronomical Society and has benefited from collaboration with the Submillimetre Common-User Bolometer Array-2 Observations of Nearby Stars (SONS) consortium. The targets were observed by me and other members of the consortium between February 2012 and September 2014. The data reduction was performed by Wayne S. Holland. The spectral energy distribution modelling (including retrieval of the additional data required for this) was performed by Grant M. Kennedy. Grant provided me with the results of this fitting procedure which I used to calculate the disc masses. Bruce Sibthorpe provided me with his disc-fitting code which I used to determine the radii of the targets showing evidence of extended emission. Jane S. Greaves provided scientific advice.

Claire Louise Davies April 2015

Abstract

Stars form via the gravitational collapse of molecular clouds during which time the proto- stellar object contracts by over seven orders of magnitude. If all the angular momentum present in the natal cloud was conserved during collapse, stars would approach rotational velocities rapid enough to tear themselves apart within just a few Myr. In contrast to this, observations of pre-main sequence rotation rates are relatively slow (∼ 1 − 15 days) indicating that significant quantities of angular momentum must be removed from the star. I use observations of fully convective pre-main sequence stars in two well-studied, nearby regions of star formation (namely the Orion Nebula Cluster and Taurus-Auriga) to determine the removal rate of stellar angular momentum. I find the accretion disc- hosting stars to be rotating at a slower rate and contain less specific angular momentum than the disc-less stars. I interpret this as indicating a period of accretion disc-regulated angular momentum evolution followed by near-constant rotational evolution following disc dispersal. Furthermore, assuming that the age spread inferred from the Hertzsprung- Russell diagram constructed for the star forming region is real, I find that the removal rate of angular momentum during the accretion-disc hosting phase to be more rapid than that expected from simple disc-locking theory whereby contraction occurs at a fixed rotation period. This indicates a more efficient process of angular momentum removal must operate, most likely in the form of an accretion-driven stellar wind or outflow emanating from the star-disc interaction.

The initial circumstellar envelope that surrounds a protostellar object during the ear- liest stages of star formation is rotationally flattened into a disc as the star contracts. An effective viscosity, present within the disc, enables the disc to evolve: mass accretes inwards through the disc and onto the star while momentum migrates outwards, forcing the outer regions of the disc to expand. I used spatially resolved submillimetre detections of the dust and gas components of protoplanetary discs, gathered from the literature, to measure the radial extent of discs around low-mass pre-main sequence stars of ∼ 1−10 Myr and probe their viscous evolution. I find no clear observational evidence for the radial expansion of the dust component. However, I find tentative evidence for the expansion of

vii the gas component. This suggests that the evolution of the gas and dust components of protoplanetary discs are likely governed by diﬀerent astrophysical processes.

Observations of jets and outflows emanating from protostars and pre-main sequence stars highlight that it may also be possible to remove angular momentum from the circumstellar material. Using the sample of spatially resolved protoplanetary discs, I find no evidence for angular momentum removal during disc evolution. I also use the spatially resolved debris discs from the Submillimetre Common-User Bolometer Array-2 Obser- vations of Nearby Stars survey to constrain the amount of angular momentum retained within planetary systems. This sample is compared to the protoplanetary disc angular momenta and to the angular momentum contained within pre-stellar cores. I find that significant quantities of angular momentum must be removed during disc formation and disc dispersal. This likely occurs via magnetic braking during the formation of the disc, via the launching of a disc or photo-evaporative wind, and/or via ejection of planetary material following dynamical interactions.

For Grandma and Ella Acknowledgements

I would ﬁrst and foremost like to thank my family for their unfaltering support and encouragement. In particular, Mum, Dad, Emily, Grandad, and Sarah: I could not have gotten this far without you. Knowing I have the support of such a loving bunch of people back home has provided me with all the motivation I could possibly need. I’m also grateful that my trip to the IAUS 299 in Victoria, BC enabled me to rekindle old ties with my Canadian family. Immense thanks go to Bev and Darrell (and Molly-Moo, of course) who essentially adopted me while I was out there - I cannot believe how welcome you made me feel. That trip has deﬁnitely been the highlight of the past three and a half years.

From an academic point of view, a number of key people deserve particular recognition. My thanks go to my initial supervisor, Jane Greaves and her ability to “accidentally” hire Ford Mustangs on observing trips; to Stuart Littlefair for highlighting the importance of the stellar side of this thesis early on during my PhD; to the support staff and telescope operators at the JCMT; and to members of the star formation and disc communities for their helpful comments and discussions along the way. I’m also indebted to the STFC who provided the funding for my PhD and the ∼ 74, 000 miles I’ve had the opportunity to travel within that time. Most of all, I would like to extend a sincere thank you to my eventual supervisor, Scott Gregory. It is difficult to sufficiently put into words the level of gratitude I feel for the support, guidance, and encouragement you have given me as well as the self-belief that you have instilled in me. The past two years working with you have been an absolute joy.

My thanks are also given to the North East branch of the Institute of Physics, the University of St Andrews Planetarium team, and the astronomical societies of York, Har- rogate, Dundee, and Leeds who have enabled me to throw myself in at the deep end in an eﬀort to get over my fear of public speaking.

I would also like to say a big thank you to my friends and colleagues in St Andrews’ Astronomy group. In particular to Jack and Joe for making St Andrews such a welcoming place to begin with; to Rim for providing advice when needed and her appreciation of the Yorkshire phrase “ey-up”; to Raphie and her search for “a whole new world”; to Milena

xi for all the oﬃce banter; to David Brown and William and their love of all things LEGO; to Jeremy for the racquetball games; and to Markus for the football chats.

Additional thank yous go to David Bajek, Kyle, Coops, and Badger who have been here for me when I needed them the most and to Jen, Paul, Eve, Dougal, Mark, Rodders, and all the old benchies back in and around Tadcaster: I don’t get to see you guys nearly as often as I would like but you never fail to put a smile on my face when I do.

A ﬁnal word of thanks goes to Scott Sneddon and his “wee to-es”. You have listened to me rant, whinge, and complain when I needed to and have made me laugh so hard that my belly hurts on so many occasions. That trip to Inverness for new year 2015 will live long in my memory. You truly are an amazing person and a wonderful friend.

“It is well known that a vital ingredient of success is not knowing that what you’re attempting can’t be done. A person ignorant of failure can be a half-brick in the path of the bicycle of history.” — Terry Pratchett, Equal Rites Contents

Declaration i

Collaboration Statement v

Abstract vii

Acknowledgements xi

1 Introduction 1 1.1 Initial collapse ...... 2 1.1.1 Large-scale magnetic fields as obstacles to collapse ...... 4 1.1.2 Formation of a protostar ...... 7 1.1.3 Rotation during contraction: formation of a disc ...... 8 1.2 Pre-main sequence evolution ...... 10 1.2.1 Changes inside the star ...... 12 1.2.2 Evolution of the circumstellar environment ...... 15 1.2.3 Disc dispersal ...... 20 1.3 Observing the stages of star formation ...... 24 1.3.1 Evolution of the circumstellar environment ...... 24 1.3.2 Signatures of infall ...... 28 1.3.3 Accretion signatures ...... 30 1.3.4 Jets and outflows ...... 31 1.4 Avoiding break-up: the angular momentum problem ...... 32 1.4.1 Angular momentum evolution: an observational perspective . . . . . 32 1.4.2 The theory of disc-locking ...... 35 1.4.3 Alternatives to disc-locking: magnetised winds and outflows . . . . . 38

xv 1.5 Concluding remarks ...... 41

2 Angular momenta calculation 43 2.1 Stellar angular momentum model ...... 43 2.2 Disc angular momentum models ...... 46 2.2.1 Geometrically thin, Keplerian accretion discs ...... 46 2.2.2 Debris discs as thin rings ...... 47 2.3 Stellar rotation rates ...... 49 2.3.1 Projected rotational velocities ...... 49 2.3.2 Rotation periods ...... 49 2.4 Pre-main sequence stellar masses, ages, and radii ...... 52 2.4.1 Eﬀective temperatures from spectral types ...... 52 2.4.2 Bolometric luminosities from photometry ...... 55 2.4.3 Stellar masses and ages from eﬀective temperatures and bolometric luminosities ...... 58 2.4.4 Age spreads in star forming regions ...... 58 2.5 Disc sizes ...... 61 2.5.1 The gas component of protoplanetary discs ...... 62 2.5.2 The dust component of protoplanetary discs ...... 64 2.5.3 Debris discs ...... 65 2.6 Main sequence stellar masses from spectral energy distribution modelling . 69 2.7 Debris disc masses ...... 70 2.7.1 Collisional cascade and total debris disc mass ...... 73 2.8 Summary ...... 75

3 Accretion discs as regulators of stellar angular momentum evolution 77 3.1 Stellar data ...... 78 3.1.1 Effective temperatures ...... 79 3.1.2 Bolometric luminosities ...... 82 3.1.3 Stellar radii, masses and ages ...... 84 3.1.4 Rotation periods ...... 85 3.1.5 Rotational flattening ...... 87 3.1.6 Disc diagnostics ...... 89 3.1.7 Multiplicity ...... 90 3.2 Results and discussion ...... 91 3.2.1 Evolution of specific angular momentum during pre-main sequence contraction: theory ...... 92 3.2.2 Evolution of specific angular momentum during pre-main sequence contraction: observations ...... 96 3.2.3 Focusing on Class III stars ...... 99 3.2.4 Focusing on Class II stars ...... 105 3.2.5 Rotation period distributions and the relation between stellar mass and rotation ...... 106 3.3 Summary ...... 108

4 Outer regions of protoplanetary discs: evidence for viscous evolution? 111 4.1 Observational data ...... 116 4.1.1 Characterising the thermal dust emission ...... 116 4.1.2 Ages of the star-disc systems ...... 121 4.1.3 Accounting for observational bias ...... 122 4.2 Results ...... 124 4.2.1 Evolution of the disc radius: expectation from theory ...... 124 4.2.2 Evolution of the disc radius: observations ...... 125 4.3 Discussion ...... 127 4.3.1 Is an evolutionary sequence observed? ...... 127 4.3.2 Inﬂuence of neighbouring stars ...... 127 4.3.3 Impact of stellar mass ...... 129 4.3.4 Spatial resolution and grain size ...... 129 4.3.5 Gas-to-dust ratio in the outer disc ...... 130 4.3.6 Is angular momentum conserved within the dust disc? ...... 131 4.4 Summary ...... 134

5 Quantifying the angular momentum retained within planetary systems137 5.1 The Submillimetre Common-User Bolometer Array-2 Observations of Nearby Stars James Clark Maxwell Telescope Legacy Survey ...... 138 5.2 Results ...... 142 5.2.1 Radial extent of the debris discs displaying extended emission . . . . 142 5.2.2 Stellar mass ...... 143 5.2.3 Dust mass ...... 144 5.2.4 Angular momentum components of the planetary systems ...... 146 5.3 Discussion ...... 150 5.3.1 Angular momentum evolution during disc dispersal ...... 150 5.4 Summary ...... 152

6 Summary and future work 155 6.1 Main results ...... 156 6.2 Outlook and future work ...... 160 6.2.1 Independent radius and age estimates ...... 160 6.2.2 Protoplanetary discs with uniform spatial resolution ...... 162 6.2.3 The interplay between stars, discs, and planets ...... 163

A Contraction rate of a fully convective, polytropic pre-main sequence star165

B Online resources 169

C Acronyms 171

Bibliography 175 List of Figures

1.1 Schematic representation of star and planetary system formation, adapted from Shu et al. (1987)...... 3

1.2 Snapshot of the evolution of density and velocity as the circumstellar envelope rotationally ﬂattens into a disc from Suttner & Yorke (2001). c AAS. Reproduced with permission...... 9

1.3 The pre-main sequence of the Hertzsprung-Russell diagram generated from Siess et al. (2000) models...... 11

1.4 Spiral density perturbations, imaged in the disc around SAO 206462, using the High Contrast Instrument for the Subaru Next Generation Adaptive Optics (HiCIAO)...... 19

1.5 The Orion proplyds...... 21

1.6 Atacama Large Millimeter/submillimeter Array image of the disc around HLTau...... 22

1.7 Temporal evolution of the disc fraction, taken from Sicilia-Aguilar et al. (2006). c AAS. Reproduced with permission...... 23

1.8 Evolution of the spectral energy distribution during collapse...... 25

1.9 Example of model spectral energy distribution ﬁts to multi-wavelength observations of a Class II and transitional disc, taken from Robitaille et al. (2007). c AAS. Reproduced with permission...... 26

1.10 Asymmetric line proﬁle as a signature of infall...... 29

1.11 Hubble Space Telescope image of the HH 47 protostellar jet...... 31

1.12 Representation of the magnetic interaction between a pre-main sequence star and the inner regions of its accretion disc under the disc-locking framework...... 35

2.1 Schematic representation of the debris disc geometry deﬁned as a torus of radius, r, width, dr, and opening angle, 2i...... 47

2.2 Example of the analysis undertaken in Davies et al. (2014a) to check for the eﬀects of beats and harmonics in the Orion Nebula Cluster...... 51

2.3 Comparison of spectral type-to-eﬀective temperature scales derived by different studies...... 54

xix 2.4 Mass tracks and isochrones of Tognelli et al. (2011) Pisa models with metallicity, Z = 0.02, initial helium abundance, Y = 0.27, mixing length, 5 α = 1.68, and initial deuterium abundance, XD = 2 × 10 ...... 59 2.5 The debris disc around Fomalhaut...... 65 2.6 Example of the process used to determine how well-resolved a debris disc is. 66 2.7 Distribution of major and minor full width half maxima resulting from the simulated point source ﬁtting in Fig. 2.6, overlaid with the results for the original disc image...... 68 2.8 Examples of thermal dust emission from debris discs which lack the centrally concentrated emission expected for a Gaussian intensity proﬁle...... 69 2.9 Results of smoothing the images of ε Eri and Fomalhaut in Fig. 2.8 with an additional 27 arcsec beam...... 70 2.10 Spectral energy distribution of the debris disc system HD 127821...... 71

3.1 Comparison between the bolometric luminosities from Davies et al. (2014a) and Hillenbrand (1997) for members of the Orion Nebula Cluster...... 83 3.2 Hertzsprung-Russell diagram constructed from Siess et al. (2000) pre-main sequence evolutionary models for the Orion Nebula Cluster and Taurus- Auriga...... 86 3.3 Relation between stellar rotation period, P , and the radius of gyration, k2, 1/2 (normalised to that of a perfect sphere, k0 = (2/3) ) for fully convective members of the Orion Nebula Cluster and Taurus-Auriga with P < 15 days. 88 3.4 Contraction rate of high mass and low mass Class II and Class III stars in the Orion Nebula Cluster and Taurus-Auriga...... 94 3.5 HRD for the sample of fully convective ONC sample for which disc classi- ﬁcations were available, overlaid with lines of constant radius...... 95

3.6 Evolution of speciﬁc stellar angular momentum, j?, for the high mass and low mass Class II and Class III samples in the Orion Nebula Cluster and Taurus-Auriga...... 97 3.7 As Fig. 3.6 but for the Orion Nebula Cluster sources identiﬁed as accreting and non-accreting using the equivalent width of Ca ii...... 100

3.8 Relationship between rotation period, P , and stellar age, tage, for the high mass and low mass Class II and Class III stars in the Orion Nebula Cluster and Taurus-Auriga...... 101

3.9 Distribution of speciﬁc stellar angular momentum, j?, for the high mass and low mass Orion Nebula Cluster samples...... 104 3.10 As Fig. 3.9 but for the high mass Taurus-Auriga sample...... 104 3.11 Distribution of rotation periods, P , for the high mass and low mass Orion Nebula Cluster samples...... 107 3.12 As Fig. 3.11 but for the high mass Taurus-Auriga sample...... 107

4.1 Atacama Large Millimeter/submillimeter Array Cycle 0 continuum image for CIDA 1...... 119 4.2 As Fig. 4.1 but for 2MASS J16142029-1906481...... 120 4.3 Hertzsprung-Russell Diagram constructed from the Siess et al. (2000) pre- main sequence evolutionary models for the stellar hosts...... 122 4.4 Evolution of the radial extent of the dust component of the discs...... 125 4.5 Disc radial extent for the thirteen star-disc systems with measurements of the radial extent of the gas component, Rgas, and the dust component, Rdust.126 4.6 Evolution of the speciﬁc angular momentum contained within the dust disc. 133 4.7 Median speciﬁc angular momenta for pre-stellar cores (data from Curtis & Richer 2011), dust discs (data from Table 4.1), the Solar System (data from Cox 2000) and Jupiter...... 133

5.1 The 15 m James Clark Maxwell Telescope following a successful night of Submillimetre Common-User Bolometer Array-2 Observations of Nearby Stars observations on 11 May 2014...... 139 5.2 Maps of 850 µm emission from the thirteen debris discs which exhibit extended emission at this wavelength...... 140 5.3 Maps of 450 µm emission from the two debris discs which exhibit extended emission at this wavelength...... 141 5.4 Comparison of the speciﬁc angular momentum contained within the protoplanetary disc sample presented in Chapter 4 and that retained within the Solar System and the spatially resolved Submillimetre Common-User Bolometer Array-2 Observations of Nearby Stars debris discs whose stellar hosts are not members of binary or multiple systems...... 151

List of Tables

3.1 Stellar data for all members of the Orion Nebula Cluster not identified as binary or multiple for which a spectral type was available in the literature. 80 3.2 Stellar data for all members of Taurus-Auriga not identified as binary or multiple for which a spectral type was available in the literature...... 81 3.3 Magnitudes of the gradients of the numerical recipe fitexy fitting using equation (3.16) for (i) the Orion Nebula Cluster sample alone, and (ii) the combined Orion Nebula Cluster and Taurus-Auriga samples...... 93 3.4 Results of Spearman rank correlation tests and fitexy fits to equation (3.19) for (i) the Orion Nebula Cluster sample alone, and (ii) the combined Orion Nebula Cluster and Taurus-Auriga samples...... 98 3.5 Results of Spearman rank correlation tests performed on the data from Fig. 3.8 for (i) the Orion Nebula Cluster sample alone, and (ii) the combined Orion Nebula Cluster and Taurus-Auriga samples...... 102

4.1 Data for star-disc systems not identiﬁed as members of binary or multiple systems for which spatially resolved thermal continuum emission, spectral types, and Two Micron All Sky Survey photometry were available...... 113 4.2 Stellar masses and ages determined using Tognelli et al. (2011) pre-main sequence evolutionary models for the 25 stars in the sample which lay inside the bounds imposed by the model isochrones...... 128 4.3 Comparison of the results of the Spearman rank correlation tests performed using the data in Figs. 4.4 and 4.5 using Siess et al. (2000) and Tognelli et al. (2011) models...... 128

5.1 Observations summary for the Submillimetre Common-User Bolometer Array- 2 Observations of Nearby Stars targets which appear extended at 850 µm or 450 µm...... 142 5.2 Results obtained from the spectral energy distribution ﬁtting...... 144 5.3 Angular momenta contained within the spatially resolved debris discs. . . . 145 5.4 Angular momentum of the stellar hosts...... 148 5.5 Orbital angular momenta of the known exoplanets included in the sample of spatially resolved debris discs...... 149

xxiii

1 Introduction

tars, celestial bodies undergoing thermonuclear fusion of hydrogen to helium in their cores, form via the gravitational collapse of molecular clouds. During the S initial stages, an isothermal core is quickly distinguished from a surrounding envelope of gas and dust and continues to contract under gravity at free-fall velocities. The density increase within the core leads to an increase in the internal temperature and the core enters a phase of adiabatic collapse. Meanwhile, the circumstellar material continues to accrete onto the core and rotationally flattens into a disc. The central core becomes optically visible and enters the pre-main sequence (PMS) phase. During this time, the turbulent properties of the disc allow material to migrate inwards and accrete along stellar magnetic field lines onto the central star while angular momentum is transferred outwards, forcing the outer regions of the disc to radially expand. Through the combined effects of irradiation, accretion, tidal interactions, and planet formation, the disc eventually disperses, leaving a disc-less PMS star and a possible planetary system. This PMS star continues to contract until it reaches the main sequence (MS). A schematic representation of this

1 Chapter 1. Introduction evolutionary sequence is illustrated in Fig. 1.1.

The role of angular momentum in star and planetary system formation is already quite apparent from this description. In the absence of rotation, a disc would not form and spherical collapse would continue unabated. Also, torques present in the disc enable it to evolve, allowing accretion to continue and planets to form. It is also important to consider the role of angular momentum in the formation of the central core. The proto- stellar object contracts by over seven orders of magnitude during formation. During this time, the rotation present in the natal molecular cloud core is ampliﬁed dramatically. If all the angular momentum present in the initial molecular cloud was conserved, stars would quickly reach velocities great enough to tear themselves apart. This presents astronomers with an angular momentum problem: the sheer fact that stars can form requires angular momentum to be removed from the system.

This chapter provides the background and context to the work presented in this thesis. Sections 1.1 and 1.2 focus on the theoretical description of star and planetary system formation, outlining the changes to the central stellar object and its circumstellar environment during the proto-stellar and PMS phases, respectively. Section 1.3 then presents an overview of star formation from an observational perspective, focusing on signatures of the diﬀerent stages and the classiﬁcation methods used. Finally, Section 1.4 provides an overview of the angular momentum problem and a discussion of the theoretical mechanisms that have so far been proposed to explain how angular momentum may be removed from the star-disc system.

1.1 Initial collapse

Star formation in the Milky Way occurs in molecular clouds, predominantly located in the spiral arms of the galaxy (e.g. Cohen et al. 1980; Dame et al. 1987). These interstellar clouds, composed mainly of molecular hydrogen (H2) gas, are subject to perturbations caused by e.g. (i) turbulence within the cloud, (ii) interactions with neighbouring clouds, or (iii) propagating shock waves from supernovae (Cameron & Truran 1977; Boss 1995; Klessen et al. 1998; Sato et al. 2000; Tan 2000; Preibisch et al. 2002; Bonnell et al. 2003). These produce density ﬂuctuations across the cloud. Over-dense regions experience a

2 1.1. Initial collapse

Figure 1.1: Schematic representation of star and planetary system formation, adapted from Shu et al. (1987): (a) initial collapse within the molecular cloud produces a central pre-stellar core, in-falling envelope, and molecular outflow; (b) the in-falling material is rotationally flattened into a disc, the core becomes proto-stellar, and the outflow begins to clear out a cavity in the enveloping material; (c) the envelope is cleared out while inner disc material continues to accrete onto the PMS star along magnetic field lines; (d) a gap in the inner disc grows as the disc disperses, producing a transitional disc; (e) the disc fully disperses leaving a naked T Tauri star and (likely) a planetary system, composed of planets and/or a debris disc.

3 Chapter 1. Introduction greater gravitational pressure and, if the gravitational potential energy (GPE),

GM 2 U = −α , (1.1) R exceeds the thermal energy, 3 RTM E = , (1.2) therm 2 µ the region becomes gravitationally unstable, initiating collapse. Here, the gas has been assumed to be isothermal and spherically distributed, α is a scaling constant that depends on the mass distribution (see e.g. Appendix A), G is the gravitational constant, µ is the mean molecular weight, and R is the ideal gas constant. The radius below which an over- dense region of mass, M, and temperature, T , will undergo gravitational collapse (the Jeans radius) is then found by inserting equations (1.1) and (1.2) into the Virial Theorem such that α µGM R = (1.3) J 3 RT

(Jeans 1902). Alternatively, it is possible to use the density,

3M ρ = , (1.4) 4πR3 to deﬁne the mass above which a cloud of a given density and temperature will become unstable to gravitational collapse (the Jeans mass),

3 1/2 3RT 3/2 M = . (1.5) J 4πρ αµG

Typical temperatures and H2 number densities within molecular clouds are ∼ 100 K and −3 4 ∼ 10 cm such that MJ ∼ 10 M . Thus, for a 1 M region to collapse, it must be at higher densities and lower temperatures than the bulk of the molecular cloud.

1.1.1 Large-scale magnetic ﬁelds as obstacles to collapse

If the molecular cloud is sufficiently ionised, the internal magnetic field and the motion of the gas become entwined. In such a scenario, the magnetic field exerts a torque on the gas which acts to inhibit gravitational collapse (c.f. Kirby 2009). For this to occur, the magnetic field must be long-lived and stable. In other words, the electrical conductivity,

4 1.1. Initial collapse

σ, in the region must be substantial enough for Ohmic dissipation to be neglected. Such a case exists if the quantity, σL2 , (1.6) c2 is much greater than any timescales of interest. Here L is the characteristic length over which the magnetic ﬁeld, B, in the cloud varies and c is the speed of light.

An estimate for the value of σ in molecular cloud cores can be garnered from considerations of the motions of electrons and ions1 in the molecular cloud which generate B.

The electrons and ions have velocities ue and ui, respectively, which each comprise of a helical component around B as well as a drift component. It is this drift component that provides the stability against collapse.

The drift velocity is accelerated by the ambient electric ﬁeld but also retarded by collisions with neutral material in the cloud (mainly H2) and steady-state drift velocities are quickly established. In the reference frame of the neutral material (denoted by primes), the equation of motion for the electrons is accordingly given by

u 0 0 = −en E 0 + e × B 0 + n f 0 . (1.7) e c e en

Similarly for the ions, u 0 0 = +en E 0 + i × B 0 + n f 0 , (1.8) e c e in assuming ni = ne. Here, e is the charge on an electron, ne and ni are the number densities of electrons and ions, respectively, and E 0 is the electric ﬁeld. The drag forces exerted by 0 0 the neutral material on the electrons and ions, fen and fin, are related to the momentum 0 0 gained by the electrons and the ions following the collisions, pe and pi , and the collision rate.

The electrons and ions gain momentum,

0 0 pe = −meue (1.9) and 0 0 pi = −mnui , (1.10)

1Although negatively charged dust grains will also be present in the cloud, they have a negligible contribution to the magnetic ﬁeld due to their low number density.

5 Chapter 1. Introduction where me and mn are the masses of electrons and neutrals, respectively. The collision rate is governed by the number density of neutral atoms and molecules, nn, and the collisional cross section, σen (for electron-neutral collisions) or σin (for ion-neutral collisions) such that 0 0 0 fen = −nnmehσenueiue, (1.11) and 0 0 0 fin = −nnmnhσinui iui , (1.12) where the angular braces denote values averaged over magnitude and direction.

0 0 Focusing on the components of ue and ui in the common direction of E and f , the 0 0 0 cross products involving B in equations (1.7) and (1.8) vanish. Substituting for fen and fin using equations (1.11) and (1.12), one obtains the following expression for the conductivity:

2 nee 1 1 σ = 0 + 0 . (1.13) nn hσenueime hσinui imn

Assuming the molecular cloud has solar composition, the quantities contained within 0 −7 3 −1 0 the angular braces are estimated to be hσenuei ≈ 1.0 × 10 cm s and hσinui i ≈ 1.5 × −9 3 −1 3 −3 10 cm s . A typical core in such a cloud has L ∼ 1 pc, nn ∼ 10 cm and ne/nn ∼ 4 × 10−7 (Stahler & Palla 2005) such that σL2/c2 ∼ 1011 Myr. This timescale is huge and, as a result, Ohmic dissipation can be neglected in molecular cloud cores. The magnetic ﬁelds are tethered to the gas (i.e. frozen-in), providing additional stability against gravitational collapse. Consequently, the critical mass required for the cloud to collapse is larger than the Jeans mass, MJ, in equation (1.5):

Mcrit ≈ MJ + MB. (1.14)

1/2 The additional component, MB ∝ Φcl/G , accounts for the magnetic support provided by the total magnetic ﬂux threading the cloud, Φcl.

A cloud of mass, MJ ≤ M ≤ Mcrit may still be able to collapse if the level of ionisation in the cloud is low. In this case, the relative velocities of the charged species and the neutral atoms and molecules are larger and neutral material drifts across magnetic ﬁeld lines under the inﬂuence of gravity without substantial build-up of opposing magnetic

6 1.1. Initial collapse forces. The consequent loss of magnetic ﬂux, known as ambipolar diﬀusion, results in a reduction in Mcrit. If this process continues, and Mcrit falls below M, the core will become unstable to gravitational collapse.

1.1.2 Formation of a protostar

The core in this initial stage of collapse is referred to as pre-stellar. It is cold (∼ 10 K) and the emergent radiation exhibits a Planck spectrum peaking at submillimetre/millimetre (hereafter collectively referred to as submm) wavelengths. Until the pre-stellar core reaches a density of ∼ 10−10 kg m−3, it remains optically thin to its own radiation, enabling eﬃcient cooling and keeping the core in an isothermal state. As contraction continues, the central regions begin to exceed this density and become opaque. The heat generated in the core as a result of gravitational collapse can no longer be removed eﬃciently. The collapse enters an adiabatic stage, increasing the core temperature (Masunaga & Inutsuka 2000; Stamatellos et al. 2007b). This results in a corresponding increase in pressure, slowing the rate of collapse.

As contraction continues, the core density and temperature continue to rise. When the central temperature reaches ∼ 1600 K, the energy released by gravitational collapse goes into dissociating and then ionising the molecular hydrogen, rather than increasing the thermal pressure. This initiates a second stage of collapse which terminates when all the core hydrogen has been ionised. Thermal support is re-established and the core regains approximate hydrostatic equilibrium (HE). In the words of Lyman Spitzer Jr. (Spitzer 1948):

During this stage of gravitational contraction an aggregation of dust and atoms will be stable against most disruptive inﬂuences, it should no longer be regarded as a cloud, and will here be called a ‘protostar’.

While in this proto-stellar phase, the core continues to contract and collapse of the circumstellar envelope continues. The speed at which the enveloping material collapses exceeds the local sound speed close to the core, producing a shock at its outer edge. The energy generated by the accretion of material through this accretion shock is radiated back

7 Chapter 1. Introduction into the envelope and dominates the observed proto-stellar luminosity such that

GMcoreM˙ Lproto? ≈ Lacc = , (1.15) Rcore where M˙ denotes the mass accretion rate, Mcore is the core mass and Rcore is the core radius.

The lifetime of the proto-stellar phase of star formation is determined by the rate of collapse of the circumstellar envelope. As the gas and dust within the envelope remain approximately isothermal, the timescale of its collapse is governed by the free-fall time,

3π 1/2 t = . (1.16) ﬀ 32Gρ

For low- and intermediate-mass stars, tﬀ ∼ 0.1 Myr.

1.1.3 Rotation during contraction: formation of a disc

In the absence of rotation, accretion of the circumstellar envelope would continue through an accretion shock at the surface of the proto-stellar core. However, pre-stellar and proto- stellar envelopes are observed to rotate. Therefore, one must consider the eﬀect that this rotation has on the circumstellar material during infall.

Consider a spherical shell centered around a pre-stellar core which undergoes solid body rotation around a ﬁxed axis that threads through the poles of the core. Each parcel of material in the shell is located at the same distance, r, from the centre of the core and thus experiences the same magnitude of gravitational acceleration. At the same time, the projected distance from the rotation axis, r0, is dependent on the latitude, θ: material at lower latitudes experiences greater centripetal acceleration than material at higher latitudes. Combining these two eﬀects, it is possible to see that the direction of the net acceleration also depends on the magnitude of θ (see Fig. 1.2). Due to the axisymmetric nature of the collapse, material collapsing from positive latitudes collides with material collapsing from negative latitudes, forming a disc (Terebey et al. 1984; Adams et al. 1987, see Fig. 1.1b).

The collapse of the circumstellar envelope proceeds from the inside out as the pressure support provided by interior regions is lost. The radial extent of the resulting circumstel-

8 1.1. Initial collapse

Figure 1.2: Snapshot, at t = 1.03 × 104 yrs from figure 5 of Suttner & Yorke (2001), showing the evolution of density (white contours) and velocity (black arrows) as the circumstellar envelope rotationally flattens into a disc. c AAS. Reproduced with permission. lar disc is determined by the total specific angular momentum, j, contained within the envelope from which it forms and the mass of the central pre- or proto-stellar core, Mcore such that2 j2 ri = . (1.17) GMcore

The material within the disc is in near-HE and one can see clearly from Fig. 1.2 that the resulting structure is ﬂared. However, it is important to make clear that the contours in Fig. 1.2 represent logarithmic changes in the density [each contour is separated by ∆(log ρ) = 0.5]. As a result, the vertical scale height appears exaggerated in this ﬁgure.

Typically, the scale height is . 10% of the radial extent of the disc.

Inﬂuence of magnetic ﬁelds

If the envelope is sufficiently ionised and the magnetic field is frozen-in (see Section 1.1.1), magnetic torques can remove angular momentum from the in-falling material (e.g. Krasnopol- sky & Königl2002). Consequently, the radius ri in equation (1.17) should be treated as a

2 The subscript i in equation (1.17) refers to the fact that ri is the initial disc radius. As is discussed in detail in Section 1.2.2, the disc evolves due to the presence of an eﬀective viscosity. The inner regions of the disc migrate inwards and, to conserve angular momentum, the outer regions of the disc radially expand (Lynden-Bell & Pringle 1974; Pringle 1981).

9 Chapter 1. Introduction maximum initial disc radius. At its limiting case, where the magnetic torques are strong and essentially all the angular momentum is removed before the in-falling material reaches the equatorial regions (in terms of the latitude, θ ≈ 0), a disc does not form. This situation occurs in ideal magneto-hydrodynamic (MHD) simulations with even weak ﬁeld strengths (Hennebelle & Fromang 2008; Mellon & Li 2009).

A number of non-ideal MHD effects have been investigated in an attempt to explain how discs can form in the presence of magnetic fields and solve the so-called “magnetic braking catastrophe”. These include ambipolar diffusion (Duffin & Pudritz 2009; Li et al. 2011), Ohmic dissipation (Machida et al. 2011; Tomida et al. 2013) and the Hall effect (Krasnopolsky et al. 2010; Braiding & Wardle 2012). Several studies have also recently suggested alternative solutions including turbulence (Santos-Lima et al. 2012; Myers et al. 2013) and misalignment between the magnetic field and the rotation axis (Hennebelle & Ciardi 2009; Joos et al. 2012; Li et al. 2013). However, none of these effects has been able to sufficiently explain how discs form in the presence of strong magnetic fields and the problem remains unsolved at present.

1.2 Pre-main sequence evolution

While stars with masses & 8 M generate suﬃciently high pressures and temperatures within their cores to begin thermonuclear fusion of hydrogen into helium before their proto-stellar envelopes have fully dispersed, low- and intermediate-mass stars experience a transitional period of contraction known as the PMS. Transition from the proto-stellar stage to the PMS occurs when the accretion luminosity, Lacc, [see equation (1.15)] no longer exceeds the luminosity generated via gravitational contraction, Lint. This coincides with the dispersal of the circumstellar envelope.

PMS evolution encompasses the optically visible stage of star formation and, as a result, can be illustrated using a Hertzsprung-Russell diagram (HRD; see Fig. 1.3). The region occupied by newly emerged PMS stars is known as the stellar birthline (Stahler 1983). From this point in the HRD, the PMS star undergoes slow, quasi-hydrostatic contraction. As it does so, it traverses a path in the HRD appropriate for its mass. Examples of these tracks are illustrated by the dashed lines in Fig. 1.3 which have been generated from Siess et al. (2000) PMS evolutionary models. The contraction stops when the thermonuclear

10 1.2. Pre-main sequence evolution 100 10 ] 1 n u s L [ L 0.1 0.01

10000 9000 8000 7000 6000 5000 4000 3000

Teff [ K ]

Figure 1.3: The PMS of the HRD generated from Siess et al. (2000) models. The dashed lines illustrate the tracks that particular masses of stars take during their PMS contraction and are shown, from right to left, for masses of 0.1, 0.2, 0.3, 0.4, 0.5, 0.75, 1.0, 1.25, 1.5, 2, 2.5, and 3 M . The near-vertical component of these tracks are the Hayashi tracks (Hayashi 1961) while the components with steadily increasing luminosity and increasing effective temperatures are the Henyey tracks (Henyey et al. 1955). The dotted lines illustrate the positions of PMS stars with different ages and are shown, from upper right to bottom left, for 0.01, 0.2, 0.5, 2, 5, 10, and 60 Myr. The solid black line indicates the location of the ZAMS for stars & 0.7 M . The red line marks the boundary between fully convective PMS stars (located to the right of the line) and those that have formed a radiative core (Gregory et al. 2012). fusion reactions within the stellar core generate sufficient pressure to balance the star against its own self-gravity. The region of the HRD occupied by these newly-formed MS stars is known as the zero-age main sequence (ZAMS).

The length of the PMS phase is determined by the time taken for the GPE generated by contraction to be radiated away. As such, it is estimated using the Kelvin-Helmholtz timescale, 2 |U| GM? tKH = ≈ . (1.18) L? R?L?

For a solar-type star (R? = R , M? = M , and L? = L ), tKH ∼ 30 Myr. By comparison, its MS lifetime is ∼ 104 Myr, indicating that the PMS phase constitutes only a tiny fraction of the total lifetime of a star (∼ 0.3% for a solar-type star).

11 Chapter 1. Introduction

1.2.1 Changes inside the star

PMS stars can be subdivided into two groups: low-mass, T Tauri stars (TTSs; Joy 1945) and intermediate mass, Herbig Ae/Be stars (Herbig 1960). TTSs exhibit strong surface magnetic fields (e.g. Johns-Krull 2007) while those of Herbig Ae/Be stars are weak or absent (Alecian et al. 2013a,b). The reason for this likely lies in the differences between their internal structures. The interiors of TTSs are either fully convective or consist of a radiative core and a convective envelope, depending on their mass and age. The convective motions of ionised material within the stellar interior generates a strong surface magnetic field via e.g. an αΩ or α2 dynamo (Parker 1975; Durney et al. 1993). In comparison, the interiors of Herbig Ae/Be stars are either fully radiative or consist of a convective core and radiative envelope. Consequently, they are unable to produce such strong surface fields.

For each fully convective star of a given mass, radius and composition, a minimum eﬀective temperature exists. Assuming that the star is in HE, it lies on a Hayashi track (Hayashi 1961). Stars with partially or fully radiative interiors lie to the left of these tracks and stars not in HE (e.g. pre-stellar cores) can exist to the right of these tracks.

It is clear from Fig. 1.3 that the track followed by a TTS in the HRD depends on its internal structure. The red line in Fig. 1.3 marks the boundary between fully convective TTSs (located to the right of this line) and those which have formed radiative cores. Three features are immediately clear from Fig. 1.3:

• TTSs are initially fully convective and, during this time, evolve at near-constant eﬀective temperature while the luminosity decreases (following their Hayashi tracks);

• not all TTSs form radiative cores (those . 0.35 M remain fully convective throughout their formation and during their MS evolution);

• following the formation of a radiative core, the surface temperature increases and the TTS evolves at a gently increasing luminosity (following a Henyey track; Henyey et al. 1955).

The formation of a radiative core requires the stellar interior to be stable against convection. To see how changes in the interior during collapse create the conditions for this, consider a parcel of gas of mass, m, at a distance r1 from the centre of the star. The

12 1.2. Pre-main sequence evolution pressure, temperature, and density of this parcel are each dependent on r. At r = r1, the parcel of gas is at pressure, P1, temperature, T1, and density, ρ1. If the density and temperature of the parcel increase slightly to ρ2 = ρ1 + δρ and T2 = T1 + δT , respectively, the parcel will become buoyant and rise to r2 = r1 + δr. Whether the parcel continues to rise, or sinks back to r1 depends on the magnitude of the temperature gradient outside of the parcel, |dT/dr|, between r1 and r2. If

dT dT > , (1.19) dr rad dr ad where the subscripts “rad” and “ad” denote the radiative and adiabatic temperature gradients, the parcel exists in a region of convective instability and will continue to rise 3.

One can use the diﬀusion equation in the form

3 ∂T 3κLint T = − 2 4 , (1.20) ∂Mr 256π σBr to determine the luminosity at which the stellar interior becomes stable against convection. Here, Mr is the mass of a spherical shell at radius, r, κ is the opacity, Lint is the internally generated luminosity, and σB is the Stefan-Boltzmann constant. Combining equation (1.20) with the equation of HE:

∂P GMr = − 4 , (1.21) ∂Mr 4πr it is possible to show that ∂T 3κL T 3 = int . (1.22) ∂P 64πσBGMr If the entropy, s, increases with radius, the region is stable against convection. At the stability limit, ∂s = 0, (1.23) ∂Mr allowing the critical luminosity to be deﬁned as:

3 64πσBT GMr ∂T Lcrit = . (1.24) 3κ ∂P s=const.

Then, if Lint < Lcrit, the stellar interior is stable against convection and a radiative core

3This is the Schwarzschild criterion for convective stability (Schwarzschild 1906).

13 Chapter 1. Introduction forms.

As the star contracts, the internal temperature increases, resulting in an increasing portion of the interior becoming ionised. The opacity becomes dominated by free-free and bound-free transitions such that κ ∝ ρT −7/2, (1.25) and, as the temperature increases, the opacity decreases. It is clear from equation (1.24) that a decrease in opacity leads to an increase in Lcrit. At the same time, the contraction leads to a reduction in the surface area at each layer within the star which, in turn, leads to a reduction in Lint. The reduction of Lcrit occurs at a more rapid rate than that of Lint and, when Lint drops below Lcrit, the interior becomes stable against convection, forming a radiative core.

The diﬀerence between Herbig Ae/Be stars and TTSs is the timescale over which this process occurs. Herbig Ae/Be stars are suﬃciently massive that their interiors become stable against convection before their envelopes have dispersed and they become optically visible. As a result, they arrive at the stellar birthline with fully radiative interiors. In comparison, TTSs emerge onto the stellar birthline whilst fully convective. The interiors of

“low mass” TTSs (. 0.35 M ) remain unstable to convection throughout their formation and MS evolution and, as such, do not form a radiative core. The age at which TTSs

& 0.35 M develop radiative cores is mass-dependent and can be estimated from PMS evolutionary models (e.g. D’Antona & Mazzitelli 1994; Baraﬀe et al. 1998; Siess et al. 2000; Tognelli et al. 2011). For example, using Siess et al. (2000) models, Gregory et al.

(2012) derived the age, in Myr, at which a star of mass, M?,(& 0.35 M ) develops a radiative core: 2.364 M tcore ≈ 1.494 . (1.26) M?

Once formed, the radiative zone grows radially outwards until the TTS reaches the ZAMS. Until the radiative region constitutes ∼ 75% of the total mass of the star, the rate of energy ﬂowing through the surface layers of the TTS is controlled by the convective envelope. As a result, the TTS continues to evolve through the HRD with near-constant eﬀective temperature for a period of time following the formation of its radiative core (see Fig. 1.3). The surface luminosity begins to increase once the mass contained in the

14 1.2. Pre-main sequence evolution

2 4 convective envelope falls below ∼ 0.25M?. As the star is still contracting, and L? ∝ R?Teﬀ , the surface temperature must also increase and the star then follows a Henyey track in the HRD.

1.2.2 Evolution of the circumstellar environment

The circumstellar discs that form during the proto-stellar phase (Section 1.1.3) are found to be ubiquitous around young TTS. The evolution of these discs relies on the presence of an eﬀective viscosity which enables material to migrate inwards and accrete onto the central star. To understand this in more detail, consider a geometrically thin disc with total mass, Mdisc M?, such that a parcel of mass, dm, at a distance, r, from the star undergoes Keplerian rotation, GM 1/2 Ω = ? . (1.27) K r3

Adopting cylindrical coordinates (r, z, φ), the surface density, Σ, depends on the vertical distance of the mass parcel from the disc midplane, denoted by z, and the density of the disc, ρ(z), such that dΣ = ρ(z)dz, (1.28) and dm = 2πrΣdr. (1.29)

The non-uniform rotation in the disc produces a velocity shear and, provided the disc is viscous, a frictional force is exerted by neighbouring disc annuli (the nature of this viscosity is considered in the following subsection). The magnitude of the frictional force per unit length, f, along the circumference of an annulus located at a distance r from the rotation axis is given by ∂Ω f(r) = νrΣ , (1.30) ∂r where ν is the viscosity coeﬃcient. The net torque of an annulus located at r + δr on an annulus at r is then given by

∂Ω G(r) = 2πr2νΣ r . (1.31) ∂r

For a disc in Keplerian rotation, ∂Ω/∂r < 0, so the annulus at r transfers angular mo-

15 Chapter 1. Introduction mentum to the annulus at r + δr and migrates inward.

The rate of change of angular momentum is equal to the net torque such that

D ∂G Dj J˙ = (jdm) = −dr = dm , (1.32) Dt ∂r Dt where j is the speciﬁc angular momentum of the annulus at radius r and D/Dt denotes a Lagrangian derivative. The right hand side of equation (1.32) is also given by

Dj ∂j ∂j dm = dm + v · ∇j = dm v , (1.33) Dt ∂t d ∂r where vd is the radial drift velocity. Then, by combining equation (1.29) with equations (1.32) and (1.33), it is possible to see that

∂j ∂G 2πrΣ v = − . (1.34) d ∂r ∂r

1/2 As the rotation is Keplerian, j = (GM?r) and

∂j 1 = Ω r. (1.35) ∂r 2 K

Combining this with equation (1.27), one can rearrange equation (1.34) in terms of the drift velocity: 3 ∂ v = − (νΣr1/2). (1.36) d Σr1/2 ∂r

Substituting this into the mass continuity equation for the surface density,

∂Σ 1 ∂ + (Σrv ) = 0, (1.37) ∂t r ∂r d one obtains the equation that determines the evolution of the surface density,

∂Σ 3 ∂ ∂ = r1/2 νΣr1/2 (1.38) ∂t r ∂r ∂r

(Lynden-Bell & Pringle 1974; Pringle 1981).

16 1.2. Pre-main sequence evolution

Assuming that the viscosity is a power law in r and is time-independent,

ν ∝ rγ, (1.39) equation (1.38) has a similarity solution (Lynden-Bell & Pringle 1974) of the form

" (2−γ) # C −(5/2−γ)/(2−γ) R Σ(r) = γ τ exp − (1.40) 3πν1R τ

(Hartmann et al. 1998). Here, C is a scaling constant, a radial scale factor, r1, has been introduced such that r R ≡ , (1.41) r1 and

ν1 ≡ ν(r1), (1.42) and τ is a non-dimensional time whereby

t τ = + 1, (1.43) ts where the viscous scaling time, 2 1 r1 ts = 2 . (1.44) 3(2 − γ) ν1

The mass ﬂux in such a disc is then given by

" # " # 2(2 − γ)R(2−γ) R(2−γ) M˙ (r, t) = Cτ −(5/2−γ)/(2−γ) 1 − exp − , (1.45) τ τ and material in the inner regions of the disc migrates inwards and accretes onto the star. At the same time, to conserve angular momentum, the outer regions of the disc must expand. The radius at which the mass ﬂux changes sign is known as the transition radius,

Rtrans, and, from inspection of equation (1.45), is given by

τ 1/(2−γ) R = r (1.46) trans 1 2(2 − γ)

(Hartmann et al. 1998). Thus, the rate at which mass is accreted onto the star and at which the disc expands are dependent on the radial dependence of the viscosity, γ.

17 Chapter 1. Introduction

Possible forms of the viscosity

What produces the viscous forces within the disc is one of the hot topics of disc physics. The molecular viscosity of the gas is unable to provide the observed mass accretion rates (Pringle 1981) and alternative methods of driving turbulence in the disc have been suggested, such as magneto-rotational instabilities (MRIs; Balbus & Hawley 1991; Tout & Pringle 1992) and gravitational instabilities (GIs; e.g. Laughlin & Bodenheimer 1994).

The stability of the disc against the production of GIs is represented by the Toomre Q parameter, κc Q = s , (1.47) πGΣ

(Toomre 1964) where, if Q & 1.3, non-axisymmetric GIs generate spiral density waves (Laughlin & Bodenheimer 1994; see Fig. 1.4) which eﬃciently transport angular momen- 4 tum through the disc. Here, cs is the local sound speed and κ is the epicyclic frequency ,

2Ω d 1/2 κ = (r2Ω) . (1.48) r dr

It is possible to parameterise the stability of the disc against GIs using the mass ratio, 3 1/2 2 q = Mdisc/M?, which enters through κ (= [GM?/R ] if Ω = ΩK) and Σ (≈ Mdisc/[πR ]), and the disc temperature, T , which enters through cs (= RT/µ). For a disc to achieve

Q & 1.3 and generate non-axisymmetric GIs, it must satisfy

c q s . (1.49) & ΩR

For typical discs, a value of q > 0.1 is required to become unstable to GIs. Observations of disc around TTSs have revealed typical values of q ≈ 0.01 (e.g. Andrews et al. 2013). As a result, these low mass discs are probably stable against GI production. However, these discs would have been more massive at earlier times suggesting that GIs may provide eﬃcient angular momentum transport during the proto-stellar phase.

In the case of MRI, the viscosity is provided by the stellar magnetic field threading into the inner regions of the disc or, in the outer disc, by the fossil magnetic field of the disc itself. Consider a uniform vertical field, that penetrates through a Keplerian disc at

4Note that, in the case of Keplerian rotation, κ = Ω.

18 1.2. Pre-main sequence evolution

Figure 1.4: Spiral density perturbations, generated by GIs, imaged in the disc around SAO 206462, using the High Contrast Instrument for the Subaru Next Generation Adaptive Optics (HiCIAO). Credit: NAOJ/Subaru. a distance, r, from the stellar surface. Then, consider two mass elements in the disc, m1 and m2, located at r and separated in the vertical direction by δz. The vertical ﬁeld line initially threads through m1 and m2. Now, consider a small radial displacement of m1 and m2 such that m1 now lies at r − δr and m2 at r + δr. This displacement creates tension in the ﬁeld line threading through m1 and m2 which acts to slow m1 and speed up m2, reducing the angular momentum of the former and increasing that of the latter. As a result, m1 migrates closer to the star and m2 migrates outwards. This further increases the tension and the process continues.

Now, consider the changes to the direction of the vertical field line induced by this process. The radial drift of both m1 and m2 introduces a radial component to the magnetic field. In addition, as m1 and m2 undergo Keplerian rotation, m1 has greater angular velocity than m2. The resulting twisting of the magnetic field line introduces an azimuthal component. This weakens the vertical component of the field and generates MHD turbulence (Balbus et al. 1996).

For MRIs to occur, (i) the disc must be in at least approximate Keplerian rotation; (ii) the disc material must be suﬃciently ionised for it to couple to the magnetic ﬁeld;

19 Chapter 1. Introduction

(iii) the initial magnetic ﬁeld must have a poloidal component; and (iv) the magnetic energy density must be less than the thermal energy density (Gammie 1996; Turner et al. 2014). Ionisation can occur in the inner disc as a result of thermal ionisation and regions further out in the disc can become ionised following irradiation by external cosmic rays (Umebayashi & Nakano 1981). In the mid-regions of the disc (the so-called dead, undead or zombie zones; Mohanty et al. 2013), which are not suﬃciently ionised, the form of the viscosity remains uncertain.

1.2.3 Disc dispersal

As the disc expands, the disc surface density decreases at large radii and the disc transitions from optically thick to optically thin. Radiation from the stellar host and neighbouring stars impacting on the optically thin regions of the disc launches a photo-evaporative wind at the surface as the gas particles have suﬃcient energy to escape. The removal of the shelter provided by the optically thick gas results in sublimation of the dust, which is consequently also removed in the photo-evaporative wind.

Tidal interactions with neighbouring stars can also strip away material from the outer regions of the disc or act as an external stirring mechanism, altering the dynamics of disc material. This can increase or disrupt the accretion process or, alternatively, increase the relative velocities of planetesimals within the disc. In turn, this latter process increases the likelihood of collisions between planetesimals resulting in fragmentation rather than sticking, inhibiting planet formation and possibly resulting in a debris disc (Backman & Paresce 1993).

Stars forming in regions containing O- and B-type stars (e.g. the Orion Nebula Cluster; ONC) are subject to the strong stellar winds emanating from these objects. These can strip away the enveloping and disc material (see Fig. 1.5). For example, Mann et al. (2014) found that discs in the ONC located around stars within a projected distance of 0.03 pc from the massive O star, θ1 Ori C were less massive than those located at a greater distance.

Once initiated, the process of disc clearing is rapid (∼ 0.1 Myr; Alexander et al. 2006a,b). This is highlighted by the comparatively few transitional discs (Fig. 1.1d) which exhibit mid- or far-infrared (hereafter MIR and FIR, respectively) excesses but lack a near-

20 1.2. Pre-main sequence evolution

Figure 1.5: The Orion proplyds. Hubble Space Telescope images of protoplanetary discs in the Trapezium region of the ONC. The discs are observed in silhouette against the bright nebulous background. Some proplyds also display cometary-like tails caused by irradiation by the hot, neighbouring O-type star, θ1 Ori C. Credit: NASA, ESA and L. Ricci (ESO).

21 Chapter 1. Introduction

Figure 1.6: ALMA image of the disc around HL Tau. The dark rings visible in the image may indicate gaps in the disc, cleared out by planets as they accrete gas and dust. Credit: ALMA (NRAO/ESO/NAOJ); C. Brogan, B. Saxton (NRAO/AUI/NSF). infrared (NIR) excess, indicating the presence of an inner hole in the disc (e.g. Gutermuth et al. 2008; Megeath et al. 2012).

Forming planets within the disc

If the disc is suﬃciently massive compared to its stellar host (Mdisc & 0.1M?), it may be possible for planets to form via gravitational collapse due to the presence of GIs (see

Section 1.2.2). If the Toomre parameter, 1 < Q . 1.3, axisymmetric gravitational perturbations can survive within the disc. The disc is likely to fragment, leading to the formation of low-mass binary companions (Stamatellos et al. 2007a) or giant planets (Raﬁkov 2005).

Alternatively, planets may form via core-accretion. In this case, dust grains, sheltered from the stellar radiation by the optically thick gas, grow in size through coagulation (Lissauer 1993; Lissauer & Stewart 1993). As the grains grow, they de-couple from the gas, and settle towards the disc midplane (Chiang & Goldreich 1997). This results in an

22 1.2. Pre-main sequence evolution

Figure 1.7: Temporal evolution of the disc fraction, measured using NIR and MIR excesses, taken from Sicilia-Aguilar et al. (2006). c AAS. Reproduced with permission. increase in the local dust density which, in turn, increases the rate of collisions between particles. Assuming that the increased density of dust does not alter the distribution of collision velocities, this leads to further grain growth. The grains grow into pebbles which themselves collide and form planetesimals. As a planetesimal continues to gain mass through collisions, its escape velocity increases, allowing the planetesimal to acquire a gaseous envelope.

The role of planet formation in dispersing the disc can be seen in Fig. 1.6. Here, dark lanes are observed to be present within the disc of HL Tau, indicating the possible presence of planets which have cleared a gap in the disc via accretion of disc material (ALMA Partnership et al. 2015).

Surveys of disc fractions in various nearby star forming regions have been used to constrain the typical disc lifetime (see Fig. 1.7). In turn, the lifetimes of circumstellar discs set the timescales for planet formation. The accuracy with which one may estimate the disc lifetime ultimately depends on the accuracy with which one is able to assign stellar ages (c.f. Naylor 2009; see Section 2.4) and identify disc-hosting young stellar objects (YSOs). For instance, Haisch et al. (2001) found that half of PMS stars had lost their accretion discs by an age of only 3 Myr, based on NIR disc classiﬁcations. However, studies employing Spitzer IRAC colours to classify their samples ﬁnd typical disc lifetimes

23 Chapter 1. Introduction of ∼ 5 Myr (Padgett et al. 2006; Hillenbrand et al. 2008). The methods used to reliably distinguish between accretion-disc hosting and disc-less stars are discussed in more detail in the following section.

1.3 Observing the stages of star formation

1.3.1 Evolution of the circumstellar environment

The theoretical descriptions of proto-stellar and PMS evolution presented in Sections 1.1 and 1.2 enable astronomers to observationally identify YSOs at various epochs of star formation. YSOs have traditionally been grouped into four evolutionary classes based on the form of their spectral energy distributions (SEDs; Lada & Wilking 1984; Adams et al. 1987; Lada 1987; Andre et al. 1993). The proto-stellar phase is subdivided into two observational stages, denoted Class 0 and Class I whereby a Class 0 YSO is pre-stellar and a Class I YSO is proto-stellar. The PMS phase is also split into two with Class II YSOs harbouring accretion discs and Class III YSOs being disc-less.

Due to the isothermal nature of pre-stellar cores, their SEDs resemble modiﬁed blackbodies at a temperature of ∼ 30 K. The emission peaks at 30-100 µm (Fig. 1.8a), requiring submm observations for detection. As the YSO enters the proto-stellar phase, the circumstellar envelope remains optically thin to infra-red (IR) radiation so the SED still features long wavelength emission. However, the photosphere of the newly-formed quasi- hydrostatic core and its circumstellar disc also contribute signiﬁcantly to the SED at MIR wavelengths (Fig. 1.8b).

The SED of a Class II YSO resembles a blackbody with additional IR and ultra-violet (UV) excess emission (Fig. 1.8c). The UV excess is associated with accretion (see Sec- tion 1.3.3) and the IR excess is produced by circumstellar material re-emitting absorbed stellar radiation at longer wavelengths (Mendoza V. 1968; Cohen 1973). As the stellar photosphere is visible at optical wavelengths, this circumstellar material cannot be spherically distributed: the extinction that would result from the observed IR excess emission is

∼ 100−1000 magnitudes while observed extinction levels are . 3 magnitudes. As such, the observer must have an almost clear view of the stellar photosphere and the circumstellar material must be conﬁned to a disc.

24 1.3. Observing the stages of star formation

Figure 1.8: Evolution of the SED during collapse: (a) a pre-stellar core – the emission resembles that of a blackbody at ∼ 30 K; (b) a protostar – the SED features emission at submm wavelengths from the cool outer envelope and IR emission from the warmer disc; (c) a disc-hosting PMS star – excess emission at UV and IR wavelengths, a result of accretion and reprocessed starlight from the dusty disc, are observed superposed on to the spectrum of the emerging stellar photosphere (dashed line). Observationally, objects that resemble (a), (b), and (c) are known as Class 0, Class I, and Class II objects, respectively.

25 Chapter 1. Introduction

Figure 1.9: Example of model SED fits to multi-wavelength observations of (a) a Class II and (b) a transitional disc, taken from figure 1 of Robitaille et al. (2007). The black circles represent actual data, the black line marks the best-fit model, and the grey lines mark the remaining good-fits to the data. The dashed line shows the SED of the stellar photosphere. c AAS. Reproduced with permission. As the disc dissipates, it enters a period of transition between a Class II and Class III YSO. During this time, the NIR (and possibly MIR) excess emission seen for Class II YSOs disappears while the FIR excess remains (see Fig. 1.9). This indicates that the warmer dust grains present in the inner regions of the disc, closer to the stellar host, are no longer present but the cooler grains in the outer disc are still present. As the IR part of the SED is only sensitive to the dust component of the disc, there may still be gas present within the inner regions of the disc. If the star is still accreting this gas from the disc, the SED may still display a UV excess but this would be weak in comparison to that observed in Class II YSO SEDs.

By the time a PMS star reaches the Class III stage, accretion has ceased, the disc has dispersed, and the SED resembles that of a blackbody (Lada & Wilking 1984; Lada 1987) with peak emission at the eﬀective temperature of the photosphere.

IR diagnostics

If a YSO is observed over a sufficient wavelength range, its SED can be constructed. Comparison of this observed SED to a range of model SEDs can then be used to classify the object (e.g. Robitaille et al. 2007). An example of this process is illustrated in Fig. 1.9. Often, the wavelength coverage is insufficient for this and classification methods focus solely on the IR part of the SED (λ ≈ 2.0 − 25 µm; Lada 1987; Greene et al. 1994).

Most early IR classiﬁcation methods were based on the NIR excess. For example, the

26 1.3. Observing the stages of star formation

(Ic − K) colour was used to distinguish between purely photospheric emission (Class III YSOs) and excess emission associated with the presence of a disc (Class II YSOs). How- ever, due to the low contrast between photospheric and disc emission at NIR wavelengths, identifications based on J-, H-, or K-band emission were found to miss up to 30% of circumstellar discs detected at longer wavelengths (Hillenbrand et al. 1998; Lada et al. 2000). In addition, observations used to measure the NIR excesses were often taken at different epochs and were consequently affected by the intrinsically variable nature of the emission from PMS stars (see Section 2.3.2).

More recently, the Spitzer Infra-red Array Camera (IRAC) has provided high resolution, contemporaneous observations between 3.6 and 8.0 µm, enabling more reliable PMS classiﬁcation. Over this wavelength range, the spectral index, αi−(i+1), (Lada 1987) is deﬁned as log λi+1Fλi+1 − log (λiFλi ) αi−(i+1) = − , (1.50) log (λi+1) − log (λi) where i = 1, 2, 3, 4 and refers to the waveband such that [2.2, 3.6, 4.5, 5.8] µm are wavelengths [λ1, λ2, λ3, λ4].

IRAC ([3.6]−[4.5]) vs. ([5.8]−[8.0]) colour-colour plots are used to reliably distinguish between disc hosting, protostellar, and disc-less YSOs (Allen et al. 2004; Whitney et al. 2004; Gutermuth et al. 2008; Megeath et al. 2012). Class III YSOs display near-zero IRAC colours as their emission is purely photospheric in origin (Allen et al. 2004; Hartmann et al. 2005; Gutermuth et al. 2009). Class II YSOs that lack a NIR (and possibly mid- IR) excess due to an inner hole being present in the disc (so-called transitional discs) may exhibit similar ([3.6] − [4.5]) colours but also display an IR excess at 8.0 µm and/or Spitzer Multiband Imaging Photometer (MIPS) 24 µm (Winston et al. 2007). Following Prisinzano et al. (2008), a YSO may be identiﬁed as Class III if it satisﬁes all of the following criteria:

1. [2.2] − [3.6] < 0.5;

2. [3.6] − [4.5] < 0.2;

3. [4.5] − [5.8] < 0.2;

4. [5.8] − [8.0] < 0.2.

27 Chapter 1. Introduction

Alternatively, a Class III YSO may display measurable emission in R- or I-wavebands but remain undetected at wavelengths longer than 2.2 µm (Two Micron All Sky Survey K-band).

In comparison, Class 0/I and Class II YSOs exhibit redder ([3.6] − [4.5]) and ([5.8] −

[8.0]) colours. Proto-stellar objects display a substantial IR excess with ([3.6]−[4.5]) & 0.7 (Gutermuth et al. 2008; Kryukova et al. 2012), corresponding to a spectral index, α & −0.3 (Greene et al. 1994). Exact definitions of the loci of Class II and Class 0/I YSOs in IRAC ([3.6] − [4.5]) vs. ([5.8] − [8.0]) colour-colour plots differ between studies. For example, Megeath et al. (2004) identify YSOs exhibiting 0 < ([3.6] − [4.5]) < 0.8 and 0.4 < ([5.8] − [8.0]) < 1.1 as Class II and those exhibiting redder colours as Class 0/I. In comparison, Hartmann et al. (2005) employed slightly more restrictive criteria to avoid contamination of their Class II sample by transitional discs and low-density, low-luminosity Class I YSOs (c.f. Allen et al. 2004; Whitney et al. 2004). In Hartmann et al. (2005), Class II YSOs lie in the locus identified by 0.2 < ([3.6]−[4.5]) < 0.7 and 0.6 < ([5.8]−[8.0]) < 1.0 while Class 0/I YSOs display similar ([5.8]−[8.0]) colours but redder ([3.6]−[4.5]) colours.

For some YSOs, reliable classification is often complicated by the viewing geometry. For example, a Class II object viewed near edge-on can exhibit an IRAC spectral index expected of a more embedded object. Similarly, a Class I object viewed pole-on may exhibit a flatter spectrum akin to that expected of a Class II object (Calvet et al. 1994; Robitaille et al. 2007). Rebull et al. (2006) found that a sample of disc-hosting and disc-less YSOs could be distinguished in a robust manner that was little influenced by inclination by using a cut-off at ([3.6] − [8.0]) = 1.0.

1.3.2 Signatures of infall

Radio or submm emission lines sometimes display asymmetric signatures (see Fig. 1.10) which are indicative of spherical infall associated with the pre- and proto-stellar phases. To be observable, these lines must be moderately optically thick and the degree of excitation of the line must increase with cloud depth (i.e. with increasing temperature and density). 18 Examples of typically used molecular lines are those of CS, C O, and H2CO (e.g. Walker et al. 1986; Myers et al. 1995; Zhou 1995). The requirement of a moderately optically thick line can be explained by considering the effects of optical depth on the line profile. If one considers a collapsing envelope with radially decreasing density profile (ρ ∝ r−1/2)

28 1.3. Observing the stages of star formation

Figure 1.10: Asymmetric line profile of a moderately optically thick spectral line, signifying the presence of infall. As the line is moderately optically thick, the near- side components of the near and far hemispheres of the envelope contribute most to the observed intensity. As the line intensity increases with depth, and the near side of the far hemisphere rotates at a greater velocity than the near side of the near hemisphere, the intensity of the blue-shifted component is greater than the red-shifted component, producing an asymmetric line profile. and radially increasing infall velocity, the profile of an optically thin line will consist of a blue-shifted component from the far hemisphere and a red-shifted component from the near hemisphere. The width of the line is related to the layers in the core over which the line is observed. Most of the mass of the cloud is located in the outer regions, where the infall velocity is small. As such, most of the emission will be located close to the rest wavelength, λ0, of the line and the line profile will be symmetric about λ0. However, by the same argument, one can see that an expanding shell would exhibit the same profile, with the blue-shifted and red-shifted components coming from the near and far sides of the envelope, respectively. There then exists a degeneracy and one cannot tell observationally using an optically thin line, whether the envelope is expanding or in-falling.

Now consider instead the line proﬁle for a moderately optically thick line. The near- stationary material at large radii superposes an absorption feature close to λ0. This absorption feature is asymmetric. Most of the blue-shifted component comes from the near side of the hemisphere at the far side of the envelope, which has a greater infall velocity. At the same time, the majority of the red-shifted component comes from the near side of the hemisphere at the near side of the envelope, which has a smaller infall velocity. Then, since the line intensity increases with depth, one can see that the blue

29 Chapter 1. Introduction shifted component from the near side of the far hemisphere will be more intense than the red shifted component from the near side of the near hemisphere (Fig 1.10).

1.3.3 Accretion signatures

The UV excess observed in the SEDs of Class II YSOs is provided by photospheric hotspots associated with magnetospheric accretion. Accretion from the circumstellar disc occurs along magnetic field lines that stem from the stellar surface and thread into the inner regions of the disc (e.g. Collier Cameron & Campbell 1993; Shu et al. 1994; Kenyon et al. 1996; see Fig. 1.1c). The infall of material at nearly free-fall velocities creates hot spots (temperatures ∼ 104 K) on the stellar photosphere. This results in Balmer and Paschen continuum emission (at UV and blue optical wavelengths) and strong emission lines (e.g. Hα, the Ca ii IR triplet, and He i λ5876 A;˚ Kuhi 1974; Kuan 1975; Rydgren et al. 1976; Johns-Krull et al. 1999). The blue Paschen continuum emission veils photospheric absorption lines which makes the lines appear filled-in. In the strongest accretors, this effect can result in the lines being seen in emission rather than absorption.

A PMS star exhibiting these features is identified as a classical TTS (CTTS). PMS stars that lack strong line emission but display strong X-ray emission indicative of chromo- spheric/coronal activity are identified as weak-lined TTSs (WTTSs) and are understood to not be in the process of actively accreting. It is important to recognise that subtle differences exist between the Class II/III and CTTS/WTTS classifications. A disc-hosting PMS star may experience episodic accretion (Enoch et al. 2009; c.f. the recent review by Audard et al. 2014). If it does so, a Class II object may fluctuate between a CTTS and a WTTS. Also, a WTTS may not necessarily be a disc-less YSO. Littlefair et al. (2005) found that some WTTS are weak accretors, indicating that at least some gas must be present in the disc.

Accretion diagnostics

The most common method used to distinguish between CTTSs and WTTSs relies on the equivalent width (EW) of the Hα emission line. PMS stars with EW (Hα) < 10 A˚ were identiﬁed as WTTS and those with EW (Hα) > 10 A˚ were CTTS (e.g. Bouvier 1990). However, the EW (Hα) is highly dependent on spectral type (Mart´ın1998; White & Basri 2003), resulting in unreliable classiﬁcation for late-type PMS stars. To overcome this,

30 1.3. Observing the stages of star formation

Figure 1.11: Hubble Space Telescope Wide Field Planetary Camera 2 image of the HH 47 protostellar jet (c.f. Hartigan et al. 2011). Credit: NASA, ESA, and P. Hartigan (Rice University). certain studies have more closely analysed the shape of the Hα emission line (e.g. Sicilia- Aguilar et al. 2005) or used the full-width at 10% (e.g. Jayawardhana et al. 2003; White & Basri 2003) to classify their samples.

Other studies have favoured alternative emission line diagnostics such as the Ca ii IR triplet (λ8498 A,˚ λ8542 A,˚ and λ8662 A),˚ which displays minimal dependence on spectral type (Hillenbrand et al. 1998). TTSs are then classiﬁed based on EW (Ca ii). CTTSs have Ca ii in emission with EW (Ca ii) < −1 A˚ (Hillenbrand et al. 1998) and WTTS have Ca ii in absorption with EW (Ca ii) > 1 A˚ (Flaccomio et al. 2003).

1.3.4 Jets and outﬂows

The accretion of circumstellar material onto proto-stellar cores produces highly collimated bipolar jets (see Fig. 1.1) with typical velocities of a few hundred km s−1. These transfer momentum to the surrounding molecular gas, producing large-scale outﬂows containing −2 ∼ 10 − 200 M and with typical velocities around a tenth that of the jets (Bally et al. 2007). These outﬂows extend up to ∼ 0.1 − 3 pc and their interaction with the interstellar medium leads to the formation of shocks. These shocks appear as knots of optical emission (e.g. Fig. 1.11) and are referred to as Herbig-Haro objects (Herbig 1951; Haro 1952).

During the PMS, outﬂows are still observed but are less well-collimated. These features are associated with magnetospheric accretion of disc material but whether they originate from the stellar surface (e.g. accretion-powered stellar winds; Matt & Pudritz 2005a) or from the inner disc (e.g. disc winds; Pudritz et al. 2007) remains uncertain due to present resolution capabilities (see Section 1.4.3). What is certain is that these jets and

31 Chapter 1. Introduction outﬂows remove angular momentum from the star-disc system and help to solve the angular momentum problem.

1.4 Avoiding break-up: the angular momentum problem

Molecular cloud cores have measured rotational velocities ∼ 10−13 − 10−14 s−1 (Boden- heimer 2011). During collapse, these cores contract by roughly seven orders of magnitude. If the angular momentum, J ∝ ΩR2, contained within the cores is conserved, the rotation rate, Ω, must increase by fourteen orders of magnitude. The centripetal acceleration of such an object would far exceed the gravitational acceleration, tearing the YSO apart before it reached the MS. The sheer fact that stars exist demonstrates that it must be possible for angular momentum to be removed during star formation.

During the 1980s, observational studies of the rotation rates of PMS stars suggested that significant amounts of angular momentum must be removed within the first few Myr of −7 −1 formation. A solar mass star, accreting at a typical rate of 10 M yr and conserving angular momentum during contraction would reach break-up velocity after just 1 Myr (Hartmann & Stauffer 1989). In comparison, the observed rotation rates of PMS stars were much slower, below around one fifth of the break-up velocity (Vogel & Kuhi 1981; Bouvier et al. 1986; Hartmann et al. 1986).

Based on the Ghosh & Lamb (1979) model for angular momentum removal from accreting neutron stars, Königl(1991) suggested that the magnetic interaction between a star and the inner regions of its disc would introduce a torque sufficient to brake the star. This section first presents a review of the observational evidence for accretion disc-regulated angular momentum removal before discussing the theory of disc-locking as well as more recent theoretical developments.

1.4.1 Angular momentum evolution: an observational perspective

In the early- to mid-1990s, observations of TTSs in the ONC and Taurus-Auriga star forming regions revealed bimodal rotation period distributions (Attridge & Herbst 1992; Bouvier et al. 1993; Edwards et al. 1993; Bouvier et al. 1995; Eaton et al. 1995; Choi & Herbst 1996). These results were later reinforced by larger ONC and Taurus-Auriga samples (e.g. Herbst et al. 2000b, 2002; Rodr´ıguez-Ledesmaet al. 2010). Similar results

32 1.4. Avoiding break-up: the angular momentum problem have also been observed in other young star forming regions such as ρ Oph, IC 348, Lupus, NGC 2264, NGC 6530, and the Upper Scorpius OB association (e.g. Shevchenko & Herbst 1998; Wichmann et al. 1998b; Herbst et al. 2000a; Cohen et al. 2004; Lamm et al. 2005; Cieza & Baliber 2007; Dahm et al. 2012; Henderson & Stassun 2012).

The longer period peak in the bimodal distribution was found to be populated by accreting PMS stars (indicated by e.g. strong Hα; see Section 1.3.3) or disc-hosting PMS stars (indicated by e.g. a signiﬁcant IR excess; see Section 1.3.1) while the shorter period peak consisted mainly of non-accretors or disc-less PMS stars. The peak of slower rotators was interpreted as indicating accretion disc-regulated angular momentum removal. Then, following the dissipation of (at least the inner regions of) its disc, the star spins up as it contracts, conserving angular momentum as it does so, and is observed in the peak of more rapid rotators.

Accretion disc-regulated angular momentum evolution has not found unanimous support, however. Certain studies did not observe a relationship between stellar rotation and accretion disc indicators or found that the distribution of rotation periods in their sample was unimodal, rather than bimodal (e.g. Stassun et al. 1999; Rebull 2001; Makidon et al. 2004). However, these contrasting ﬁndings could be explained in terms of a variety of biases which mask the underlying relationship between the presence of accretion discs and PMS stellar rotation. These include (e.g. Herbst et al. 2002; Littlefair et al. 2005; Lamm et al. 2005; Cieza & Baliber 2007)

• contamination by aliasing and beat periods;

• inclusion of non-member stars at a variety of ages;

• unreliable indicators of accretion discs;

• an underlying relation between rotation and stellar mass.

Beats and aliasing

If the rotation period is measured using ground-based photometry, a sampling interval of 1 day is introduced. If the true rotation period is close to 1 day, one may measure the beat period, rather than the true rotation period. In addition, the variability which allows the rotation period to be measured is due to the presence of accretion hotspots or cool

33 Chapter 1. Introduction starspots on the stellar surface. If multiple spots are present, one may measure a harmonic period rather than the true rotation period. These eﬀects are considered in more detail in Section 2.3.2.

Age dependence

The rotation rates of PMS stars in older star forming regions are observed to be more rapid (e.g. Irwin & Bouvier 2009). As such, inaccurately identifying members of a star forming region can blur the distributions of accretion disc-hosting stars and disc-less stars. Thus, the distribution of rotation periods can appear more unimodal.

Accretion-disc indicators

As discussed in Sections 1.3.1 and 1.3.3, accretion disc indicators based on Spitzer IRAC colours and Ca ii EWs are more reliable than those based on NIR excess and Hα EWs. If one cannot reliably separate accretion disc-hosting YSOs from disc-less YSOs, the distributions of what are believed to be Class II and III YSOs or CTTS and WTTS will be contaminated. As a result, one may not recover the observed relationship between rotation and disc presence.

Mass dependence

An underlying relationship between rotation rate and stellar mass has been uncovered with the rotation period distribution of low mass stars (M? . 0.35 − 0.5 M , depending on the adopted PMS evolutionary model) appearing unimodal (e.g. Herbst et al. 2002; Cieza & Baliber 2007). Studies which failed to separate the “high” and “low” mass TTSs failed to recover the previously observed bimodal rotation period distributions (e.g. Stassun et al. 1999).

This mass eﬀect is partially attributable to the comparative sizes of “high” and “low” mass TTSs. For a sample of stars of a given age and speciﬁc stellar angular momentum, 2 j?, those with lower masses will have smaller radii, R?. Consequently, as j? ∝ R?/P , the rotation periods, P , of the lower mass sample will be shorter than the higher mass sample (Herbst et al. 2001). Thus, the lower mass slow rotators are shifted towards the peak of rapid rotators, blurring the bimodality found for the higher mass sample.

Cieza & Baliber (2007) found the bimodality of the rotation period distributions to be

34 1.4. Avoiding break-up: the angular momentum problem

Figure 1.12: Representation of the magnetic interaction between a PMS star and the inner regions of its accretion disc under the disc-locking framework, based on figure 1 of Matt & Pudritz (2005b). The closed stellar- and disc-magnetic field lines are represented by red and blue lines, respectively. The black lines represent open stellar-magnetic field lines. Accretion of disc material onto the star occurs along closed stellar magnetic field lines, as indicated by the arrow. Three important regions of the disc are also noted: the truncation radius, Rtrunc; the co-rotation radius, Rco; and the radial extent of the closed stellar magnetic field, Rout. severely affected by even a small contamination of stars with spectral types later than M2. The difference in size between the higher and lower mass stars is not sufficient to explain this on its own and the location of this boundary remains poorly understood. The most promising underlying physical explanation relates to changes in the strength and geometry of the large-scale stellar magnetic field around this spectral type (Lamm et al. 2005).

1.4.2 The theory of disc-locking

Accretion disc-regulated angular momentum removal was initially attributed to a magnetic spin-down torque provided by the differential rotation of a PMS star and its Keplerian accretion disc (Ghosh & Lamb 1979; Camenzind 1990; Königl1991; Collier Cameron & Campbell 1993). As illustrated in Fig. 1.12, in models of magnetospheric accretion the stellar magnetic field interacts with the disc at a range of radii. If no angular momentum removal process operates, the star would naturally spin up as it contracted and accreted material from the disc (Matt & Pudritz 2005b). The threading of the stellar magnetic field lines into the disc generates additional torques on the disc. Whether spin-up or spin- down torques are generated depends on the relative position of the disc truncation radius,

35 Chapter 1. Introduction

Rtrunc, and the co-rotation radius, Rco:

• ﬁeld lines penetrating the disc at r > Rco generate a spin-down torque on the star as the disc rotates at a slower rate than the star.

• ﬁeld lines penetrating the disc at r < Rco generate a spin-up torque on the star as the disc rotates faster than the star.

• ﬁeld lines penetrating the disc at r = Rco generate no torque as the disc and the star rotate at the same rate.

To understand what governs the position of Rtrunc, consider, for a moment, the simple scenario of spherically symmetric in-falling material, accreting at near free-fall velocity onto a star with a near-dipolar magnetic ﬁeld. In this case, the density, ρ(r), velocity, v(r), and magnetic ﬁeld, B(r), are given by,

M˙ ρ(r) = , (1.51) 4πr2v(r)

2GM 1/2 v(r) = ? , (1.52) r and −3 r −3 B(r) = B? = µ1r . (1.53) R?

The truncation radius, Rtrunc, is then estimated using the Alfv´ensurface radius, RA, where the energy density of the stellar magnetic ﬁeld is balanced against the kinetic energy density of the in-falling material:

B2 1 = ρv2. (1.54) 8π 2

Solving for r = RA, −1/7 ˙ −2/7 4/7 RA = (2GM?) M µ1 , (1.55) where M˙ is the accretion rate of material through the spherical surface of radius, RA, and 3 µ1 = B?R? is the magnetic moment (Elsner & Lamb 1977).

When considering magnetospheric accretion, rather than spherical accretion, the dependence of RA on stellar mass, accretion rate, and magnetic ﬁeld strength remains the

36 1.4. Avoiding break-up: the angular momentum problem same but a proportionality constant, α, is included to account for the fact that the accretion no longer takes place through a spherical surface5. Thus,

−1/7 ˙ −2/7 4/7 Rtrunc ≈ RA = α(GM?) M µ1 , (1.56) where M˙ now denotes the mass accretion rate onto the star along the magnetic field lines 3 and B? in µ1 = B?R? denotes the strength of the dipole component of the magnetic field at the stellar equator. In reality, the magnetic fields of accreting PMS stars are not dipolar (Donati et al. 2007, 2008, 2010b, 2011b). However, higher order multipole components of the magnetic field fall-off with radius at a faster rate than the dipole component. As a result, as long as the disc is truncated at a sufficient distance from the stellar surface, the dipole component dominates over high order components of the field and equation (1.56) remains valid for calculating Rtrunc (e.g. Adams & Gregory 2012; Johnstone et al. 2014). However, the higher order multipole components must be considered in models of the accretion flow as the accreted material moves closer to the star where their influence is stronger (e.g. Gregory & Donati 2011).

If all else remains equal, it is possible to see from equation (1.56) that (i) more massive stars truncate their discs at smaller distances from their photospheres; (ii) larger accretion rates force the truncation radius closer to the stellar surface as the accretion flow in the disc pinches the field inwards at the equator (Shu et al. 1994); and (iii) stars with stronger magnetic fields truncate their discs at larger radii.

At the co-rotation radius, the rotational velocity of the stellar surface, Ω? = 2π/P , is the same as the Keplerian velocity [equation (1.27)] of the disc such that

GM P 2 1/3 R = ? . (1.57) co 4π2

It is possible to see that an equilibrium situation exists in this relatively simple set-up

(K¨onigl1991). If the net torque provided by material between Rtrunc and Rout is negative and the star spins down, the co-rotation radius increases. This reduces the net torque as more radii near Rco now provide a spin-up torque. Similarly, if the net torque is initially positive, the stellar spin-up results in a smaller Rco which reduces the net torque.

5 Using values of RA calculated from MHD magnetospheric accretion simulations, Long et al. (2005) estimated α ∼ 1/2.

37 Chapter 1. Introduction

For the magnetic torque to sufficiently brake the star during contraction, Rout must be sufficiently larger than Rco. In addition, for the magnetic field to remain connected to the disc and generate a sufficient spin-down torque, it must be stable over multiple rotations. However, differential twisting of the magnetic field lines (e.g. Lovelace et al. 1995), together with the competing processes of accretion and diffusion, limit the size of the connected region in the disc and reduce the extent of the field beyond co-rotation (Shu et al. 1994; Bardou & Heyvaerts 1996; Agapitou & Papaloizou 2000; Matt & Pudritz 2005b; Zanni & Ferreira 2009). As a result, the disc-locking mechanism alone appears insufficient to spin down the star.

1.4.3 Alternatives to disc-locking: magnetised winds and outﬂows

In addition to the influence of the closed stellar magnetic field lines considered above, Fig. 1.12 also highlights the presence of open magnetic field lines which stem from both the star and the disc. These open field lines allow for angular momentum to be removed from the star-disc system as a whole, rather than transfer angular momentum from the star to the disc as in the case of disc-locking.

The open field lines stemming from the disc can drive a disc wind (Black & Scott 1982; Blandford & Payne 1982; Ferreira 1997; Ouyed et al. 1997). This can remove angular momentum from the disc and is understood to be one of the mechanisms that drives the observed outflows from disc-hosting PMS stars discussed in Section 1.3.4 (Pudritz et al. 2007). However, the disc wind does not affect the spin rate of the star. Instead, stellar angular momentum removal mechanisms are driven by open stellar magnetic field lines. These fall into three main categories: accretion-powered stellar winds; X-winds; and magnetospheric ejections. These are discussed individually in more detail below.

The efficiency of angular momentum removal via winds and outflows is determined by the relative strength of the dipole component of the magnetic field as this governs the position of the disc truncation radius [see equation (1.56)] as well as the level of flux from the open magnetic fields (Gregory et al. 2008; Adams & Gregory 2012; Johnstone et al. 2014). Donati et al. (2011a) found outflow-driven angular momentum removal to be most efficient in low mass TTSs (M? ∼ 0.5 − 1.3 M ). More massive stars had already formed a substantial radiative core (see Section 1.2.1) which inhibits the build up of a strong dipole

ﬁeld (Donati et al. 2011a). In addition, lower mass stars (. 0.35 M ), although still

38 1.4. Avoiding break-up: the angular momentum problem fully convective, appear to have weaker large-scale magnetic ﬁelds (Donati et al. 2010b; Gregory et al. 2012; Donati et al. 2013). Thus, the magnetic ﬁelds of stars later than ∼M2 truncate their discs closer to the star, meaning they rotate more rapidly than their higher mass, fully convective counterparts. This may go some way in explaining why stars with

M? . 0.35 M have unimodal rotation period distributions while those of higher mass stars are observed to be bimodal (Section 1.4.1).

X-winds

The Königl(1991) disc-locking model was extended by Shu et al. (1994) to show that excess angular momentum transferred to the disc could drive an outflow from a region in the disc close to the disc truncation radius (the so-called X-wind model after its “extraordinary” nature). In this model, further developed in Ostriker & Shu (1995), a dipole magnetic field, aligned with the stellar and disc rotation axes, is considered. The magnetic pressure provided by the accretion flow in the disc forces the X-region (a narrow annulus straddling the co-rotation radius) to rotate as a solid body with Ω = Ω?. Material interior to this region rotates at a less rapid rate, threads onto pinched field lines, and is accreted onto the star. At the same time, material external to the X-region rotates faster, threads onto outward-leaning field lines, and escapes in a wind. Torques from the loaded stellar magnetic field lines act to transfer angular momentum from the star to the disc. The disc material consequently migrates outwards at which point it experiences a torque from the wind. Angular momentum is transfered from the disc into the wind and the disc material migrates back towards the star.

The model was developed in Mohanty & Shu (2008) to incorporate non-dipolar magnetic field configurations. These models have so far been able to reproduce observed stellar spin rates for PMS stars with well-studied magnetic fields and accretion properties (e.g. Donati et al. 2007, 2008). However, the kinematics of jets launched from young stars (see Section 1.3.4) are not consistent with predictions from the X-wind model (Ferreira et al. 2006; Cabrit 2007). In addition, the X-wind model predicts that the disc be truncated close to the co-rotation, which has not been observed (Najita et al. 2003; Le Blanc et al. 2011).

39 Chapter 1. Introduction

Accretion-powered stellar winds

Provided that the outflow rate from the stellar wind is sufficiently high (∼ 10% of the mass accretion rate), it may be possible for enough stellar angular momentum to be removed to explain the observed rotation rates of 1 Myr PMS stars (Matt et al. 2012). Matt & Pudritz (2005a) suggested that it may be possible to generate such outflow rates if some of the potential energy of the accreted material was used to power a stellar wind. However, it remains unclear how the accretion power would drive such a wind and whether accretion onto PMS stars could generate sufficient power to drive these outflows (Cranmer 2008; Zanni & Ferreira 2011).

From an observational perspective, there does appear to be support for these accretion- powered stellar winds. For instance, the appearance of hydrogen and helium emission lines associated with stellar winds is, in general, correlated with mass accretion rates (e.g. Johns-Krull 2007; Kurosawa et al. 2011).

Magnetospheric ejections

The twisting of the magnetic field, generated by the differential rotation between the star and the disc (Lovelace et al. 1995), limits the effectiveness of disc-locking as an angular momentum removal mechanism. However, the same process can lead to the generation of magnetospheric ejections (Aarnio et al. 2012). These are able to remove angular momentum from the star via a spin-down torque and also remove angular momentum from the accreting material. In their extreme, magnetospheric ejections enter so-called propeller phases (Romanova et al. 2005) whereby the magnetic field disrupts the accretion of material onto the star and drives an outflow from the star-disc system.

No direct observational evidence exists, at present, for a PMS star in the propeller phase. However, Zanni & Ferreira (2013) showed that, assuming dipole magnetic field strengths of several kilo-Gauss, the combined effect of stellar winds and magnetospheric ejections can provide sufficient spin-down torques to brake the star. Whether these field strengths are typical in accreting PMS stars remains uncertain at present due to the small numbers of stars with well-studied magnetic fields (e.g. Donati et al. 2008, 2010a).

40 1.5. Concluding remarks

1.5 Concluding remarks

The primary aim of this thesis is to observationally trace the evolution of angular momentum during the formation of stars and their planetary systems. Chapter 2 outlines the models used in the subsequent chapters to calculate the stellar and disc angular momenta. It also describes each of the observational techniques required for the calculation of the parameters of the angular momentum models. In the chapters that follow, the angular momentum evolution as described in these introductory pages is tested using observational data for stars and their discs at diﬀerent epochs of star formation:

• Chapter 3 focuses on the rate of stellar angular momentum removal during the PMS. In particular, observational data for fully convective PMS members of the ONC and Taurus-Auriga star forming regions, available in the literature, are used to investigate whether the disc-locking hypothesis (whereby an accretion disc-hosting star contracts and accretes at ﬁxed rotation period; see Section 1.4.2) can explain the observed angular momentum evolution.

• Chapter 4 focuses on the evolution of angular momentum in proto-planetary discs. Spatially resolved submillimetre and millimetre observations are used to determine whether the expansion of the dust and gas components of the disc, as expected from viscous disc evolution theory (Section 1.2.2), can be observed.

• Chapter 5 focuses on the angular momentum retained within star and planetary systems following their formation. Spatially resolved debris discs from the Submillime- tre Common-User Bolometer Array 2 Observations of Nearby Stars (SONS) James Clark Maxwell Telescope (JCMT) Legacy Survey are used to quantify the angular momentum contained within the various components of the planetary systems.

Finally, Chapter 6 summarises the ﬁndings of this work and how the results presented herein can be built upon to further understand star-disc-planet interactions.

41 Chapter 1. Introduction

42 2 Angular momenta calculation

he main goal of this thesis is to observationally trace the evolution of angular momentum during the formation of stars and their planetary systems. This T chapter presents the models used to calculate the stellar and disc angular momenta together with a discussion of the methods used to determine the observational parameters required to study their temporal evolution.

2.1 Stellar angular momentum model

Considering pre-main sequence (PMS) and main sequence (MS) stars as polytropic spheres1 of rotation rate, Ω?, mass, M?, and radius, R?, the stellar angular momentum is given by

2 2 J? = k M?R?Ω?, (2.1)

1The gas pressure, P , and density, ρ, of a polytrope are related by P = Kρ(n+1)/n where n is the polytropic index and K is a constant.

43 Chapter 2. Angular momenta calculation

(Chandrasekhar & M¨unch 1950; Krishnamurthi et al. 1997). The radius of gyration, k2, of a polytrope depends on the density distribution2, θn(ξ), where ξ = r/α is the radial scale factor in the Lane-Emden equation,

1 d dθ ξ2 = −θn (2.2) ξ2 dξ dξ

(Lane 1870; Emden 1907), and

n + 1 1/2 α = Kρ1/(n−1) , (2.3) 4πG c where G is the gravitational constant. Following Motz (1952), the radius of gyration is given by Z ξ1 2 2 6 2 k = 1 + 4 ξ θn dξ , (2.4) 3 ξ1(dθn/dξ)ξ=ξ1 0 where ξ1 is the ﬁrst root of the equation θn(ξ) = 0.

The value of n depends on the dominant heat transfer mechanism in operation. As outlined in Section 1.2.1, stars with M? . 0.35 M remain fully convective throughout their formation and MS evolution whilst more massive stars form radiative cores during their PMS evolution (Limber 1958; Chabrier & Baraﬀe 1997; Gregory et al. 2012). There then exist two scenarios: when convection is the dominant heat transfer mechanism in the stellar interior, the stars are best described by a polytropic index, n = 3/2 such that k2 = 0.205 and, when radiation is the dominant mechanism, stellar interiors are described by n = 3, corresponding to k2 = 0.08 (c.f. Motz 1952; Ruci´nski1988).

Equation (2.1) is a simpliﬁed version of the stellar angular momentum as the angular velocity, Ω?, is not always constant throughout the star. Helioseismic observations show that the solar surface rotation rate depends on latitude and rotation rates of material within the solar interior depend on depth (Schou et al. 1998). At present, there are no observational constraints on the levels of diﬀerential rotation in the interiors of PMS stars. Also, model-independent asteroseismic determinations of the interior rotation rates of MS stars other than the Sun have so far been restricted to two A- to F-type stars nearing the end of the MS (KIC 11145123 and KIC 9244992). These stars both show much less

2The more common form of the density distribution, ρ(r), is related to the polytropic density distribution n by ρ(r) = ρc θ (ξ) where ρc is the central density.

44 2.1. Stellar angular momentum model pronounced interior diﬀerential rotation than the Sun (Kurtz et al. 2014; Saio et al. 2015).

It is assumed, as a ﬁrst approximation, that stellar interiors undergo solid body rotation - a result typically found in numerical models (e.g. Kuker & Rudiger 1997) and magneto- hydrodynamic (MHD) simulations (e.g. Browning 2008) for fully convective stars. How- ever, this may lead to the underestimation (if, as for KIC 9244992, the surface rotates slower than the core; Saio et al. 2015) or overestimation (if, as for KIC 11145123, the surface rotates faster than the core; Kurtz et al. 2014) of the true angular momentum of partially radiative stars.

The assumption that a single Ω? describes the surface rotation rate can be assessed using the results of Doppler tomographic imaging. Partially radiative stars are known to exhibit solar-like surface differential rotation (e.g. Barnes et al. 2000; Donati et al. 2000; Waite et al. 2011). However, Barnes et al. (2005) found a decrease in surface differential rotation with increasing convective zone depth which may indicate that fully convective stars do not exhibit surface differential rotation. So far, surface differential rotation rates, dΩ, have only been measured for a handful of fully convective PMS stars due to the telescope-time intensive nature of Doppler tomographic imaging techniques (which typically involve taking high resolution spectra on consecutive nights for > 1 stellar rotation). Fully convective, non-accreting PMS stars (namely TWA 6, LkCa 4, V410 Tau, V819 Tau, and V830 Tau) each have differential rotation rates consistent with solid body rotation (dΩ = 0) to within 1.7 σ (Skelly et al. 2008, 2010; Carroll et al. 2012; Donati et al. 2014; Donati et al. in prep.). In contrast to this, Donati et al. (2010b) found that the fully convective, accreting PMS star V2247 Oph exhibited substantial differential rotation with dΩ = 0.32 ± 0.05 rad day−1. To date, this is the only fully convective PMS star with measured surface differential rotation and further observations are required to determine how common differential rotation like that observed in V2247 Oph is. V2247 Oph is the lowest mass star for which magnetic field topology information has been obtained and has a large-scale magnetic field that is more complex than that of higher mass fully convective PMS stars. It is possible that it may exist in a regime of dynamo bi-stability whereby stars with otherwise similar parameters have drastically different large-scale magnetic topologies and surface differential rates, as has been found for the lowest mass MS M-dwarfs (Morin et al. 2010; Gregory et al. 2012).

45 Chapter 2. Angular momenta calculation

In Chapter 3, it is assumed that the surfaces of fully convective PMS stars rotate as solid bodies (the usual assumption of stellar evolution models e.g. Eggenberger et al. 2012). This is consistent with the observations of TWA 6, LkCa 4 and V410 Tau, as well as the ﬁndings of Barnes et al. (2005). This assumption of uniform surface rotation is also maintained in Chapter 5 which concerns partially and fully radiative MS stars.

The methods used to measure stellar rotation rates are discussed in Section 2.3 and those used to estimate stellar masses and radii are discussed in Sections 2.4 (for PMS stars) and 2.7 (for MS stars).

2.2 Disc angular momentum models

2.2.1 Geometrically thin, Keplerian accretion discs

A protoplanetary disc can be modelled as a ﬂat, axisymmetric disc with a rotation axis perpendicular to the disc plane. A particle of mass, m, residing in the disc at a distance, r, from the rotation axis has angular momentum,

J = mrv(r), (2.5) where v(r) is the Keplerian velocity at r,

GM 1/2 v(r) = ? , (2.6) r where G is the gravitational constant and M? is the mass of the stellar host. To calculate the total angular momentum of the disc, Jdisc, a description of how the mass is distributed within the disc is required. This is provided by the disc surface density,

dm Σ(r) = . (2.7) 2πrdr

Then, Z Rdisc 1/2 3/2 Jdisc = 2π(GM?) r Σ(r) dr, (2.8) Rin where Rin and Rdisc denote the inner and outer edge of the disc, respectively.

Resolved observations of protoplanetary discs reveal that most of the mass in the disc

46 2.2. Disc angular momentum models

Figure 2.1: Schematic representation of the debris disc geometry deﬁned as a torus of radius, r, width, dr, and opening angle, 2i. The top image shows a near face-on view of the torus and the bottom image shows an edge-on view of its cross-section. is contained at large r (Andrews et al. 2009) such that material at Rdisc dominates the angular momentum budget. As such, Rin ≈ 0.

The form of the disc surface density, together with a discussion of the methods used to constrain the radial extent of protoplanetary discs are presented in Section 2.5.1 and 2.5.2. The methods used to estimate the mass of the stellar host are presented in Section 2.4.

2.2.2 Debris discs as thin rings

The dust and planetesimals that constitute a debris disc are assumed to be located within a torus of radius, r, width, dr, and opening angle, 2i, caused by the largest orbital inclinations, i, of the constituent particles (see Fig. 2.1). These particles are formed via catastrophic collisions whereby impact velocities are large enough to result in fragmentation (Backman & Paresce 1993; Section 1.2.3). In this scenario, the particle diameter distribution is given by n(D) = KD2−3q, (2.9)

47 Chapter 2. Angular momenta calculation where K is a scaling parameter, D is the particle diameter, and q relates to the size dependence of the strength of the grains (Dohnanyi 1968). Observations of debris discs ﬁnd values close to q = 11/6 (e.g. Wyatt & Dent 2002), the theoretical expectation for a self-similar inﬁnite collisional cascade (Tanaka et al. 1996). Assuming all of the particles are spherical, the total mass in the cascade is

Z Dmax Mtot = m(D)n(D) dD, (2.10) Dmin where the individual particle mass is given by

π m(D) = D3ρ(D). (2.11) 6

If q < 2 and all grains have the same density, ρ, the contribution from large grains (Dmax) dominates and π M = ρKD6−3q. (2.12) tot 6(6 − 3q) max

The angular momentum of the debris disc, Jdisc, depends linearly on Mtot such that the largest grains also dominate the angular momentum. Assuming that their orbital inclinations are small and r + dr ≈ Rdisc, the debris disc angular momentum can be approximated to that of a thin ring with moment of inertia,

2 Idisc = MtotRdisc. (2.13)

Then, assuming that the rotation is Keplerian [equation (2.6) with r = Rdisc], the angular momentum of a debris disc is given by

1/2 Jdisc = Mtot(GM?Rdisc) . (2.14)

The methods of estimating the radial extent of the disc, the stellar mass, and the disc mass are discussed in Sections 2.5.3, 2.6, and 2.7, respectively.

48 2.3. Stellar rotation rates

2.3 Stellar rotation rates

2.3.1 Projected rotational velocities

A measure of the surface stellar rotation rate can be made from observations of rotation- induced spectral line broadening. As the star rotates, one side of it is moving towards the observer whilst the other side is moving away. Therefore, spectral lines originating from the stellar photosphere exhibit blue- and red-shifted components, respectively. The width of this line broadening is dependent on the equatorial rotational velocity, v, of the star. However, this method is complicated by the inclination, i, of the star3 meaning that an observer actually measures the projected rotational velocity, v sin i. Calculating Ω? requires an independent measure of the stellar inclination or an alternative measurement of the rotational velocity. If the only available indicator of rotation is v sin i, it is not possible to measure J? using equation (2.1). Instead, as Ω? = v/R?, v sin i provides a measure of the “projected” stellar angular momentum,

2 J? sin i = k M?R?v sin i, (2.15) which provides a lower limit to the stellar angular momentum.

2.3.2 Rotation periods

PMS stars exhibit strong surface magnetic ﬁelds (Feigelson et al. 2007; Johns-Krull 2007; see Section 1.2.1) which generate strong surface activity. This produces large starspots, covering up to 10−50% of the stellar surface (Herbst et al. 2002). In addition, the accretion of disc material can produce hot spots on the surfaces of Class II stars. If the lifetimes of these cool starspots and accretion hotspots are longer than the stellar rotation period, the induced optical and near infra-red (NIR) photometric variability can be used to deduce the rotation period, P = 2π/Ω?, of the star using time-series analysis (e.g Herbst et al. 1994; Stassun et al. 1999; Frasca et al. 2009; Rodr´ıguez-Ledesmaet al. 2009).

Rotation-induced variations in the stellar ﬂux are monitored using high-cadence observations. Often, a time-series, X(ti), with i = 1, 2,...,N0, is produced from the Fourier transform of the data to deduce the rotation period. A periodogram, deﬁned as a function

3The stellar inclination is deﬁned such that a star viewed pole-on has i = 0◦.

49 Chapter 2. Angular momenta calculation of the frequency, ω, as

h i2 h i2  PN0 PN0 1 j=1 X(tj) cos ω(tj − τ) j=1 X(tj) sin ω(tj − τ) PX(ω) =  +  , (2.16) 2  PN0 2 PN0 2  j=1 cos ω(tj − τ) j=1 sin ω(tj − τ) where τ is defined by PN0 sin 2ωtj tan(2ωτ) = j=1 , (2.17) PN0 j=1 cos 2ωtj can be fit to the data in Fourier-space (Lomb 1976; Scargle 1982). The stellar rotation period then corresponds to the peak value of PX(ω). Scargle (1982) showed that in the case of the null hypothesis where measured data points are independent Gaussian random variables, the probability that PX(ω) will fall between z(> 0) and z + dz for any given frequency is exp(−z)dz. Consequently, for Ni independent frequencies, the probability of none providing values larger than z is (1 − exp[−z])Ni . The significance of the inferred period is then indicated by the false alarm probability (FAP)4, given by

P (> z) = 1 − (1 − exp[−z])Ni (2.18)

(Scargle 1982; Horne & Baliunas 1986; Attridge & Herbst 1992).

As outlined in Section 1.4.1, issues with this method can arise when multiple spots are present on the stellar surface or, in the case of ground-based observations, when the stellar rotation period is close to 1 day. In the case of the former, the measured rotation period may only be a fraction of the true value (a harmonic) and in the latter case, the day-night cycle of the Earth introduces a sampling interval such that the recorded rotation period may be a beat period, B, rather than the true rotation period. The eﬀect of harmonics and beats may be removed from a sample by favouring studies which take observations over multiple seasons or by comparing the results of multiple studies of the same region (Cieza & Baliber 2007; Rodr´ıguez-Ledesmaet al. 2009). An example of identifying beat periods using 1 1 1 = + , (2.19) B ±P⊕ ±P where P⊕ is the rotation period of the Earth, is shown by the red lines in Fig. 2.2. The blue lines also highlight the presence of harmonic periods in this dataset.

4A small FAP indicates a more signiﬁcant periodic signal.

50 2.3. Stellar rotation rates

● 15

●

● ● ● ● ● ● [ days ] [ days ● ● 10 d ● ● ●●

o ● ● ● i ● ●

r ● ●

e ● ● ● ● ●● ● P

● ●● ● d ●● ●●

n ● ● ● ● ●● ● a ● ● ● ●● 5 ● b ● ● ● − ●●

J ● ● ● ● ●● ● ● ●● J−band Period [ days ] [ days J−band Period ● ● ●● ● ● ● 0

0 5 10 15 I−band Period [ days ]

Figure 2.2: Example of the analysis undertaken in Davies et al. (2014a) to check for the eﬀects of beats (red dotted lines) and harmonics (blue dashed lines) in the Orion Nebula Cluster (ONC). The I-band periods are from Herbst et al. (2002); Parihar et al. (2009) and Rodr´ıguez-Ledesmaet al. (2009) and the J-band periods are from Carpenter et al. (2001). Stars located on the solid black line have the same measured rotation period in both the optical and NIR.

Alternative sources of periodicity

Studies of photometric rotation periods must also be cautious of possible pollution by alternative sources of periodicity. For example, a companion surrounded by an extended dusty disc may be able to produce photometric variability on similar time-scales to stellar rotation periods (Percy et al. 2010). Cross-checking samples against stellar multiplicity studies is required to minimise this eﬀect.

In addition, temperate, opacity, or geometric fluctuations in the inner regions of discs around Class II stars can cause periodic flux changes at NIR and mid infra-red (MIR) wavelengths (Bouvier et al. 2003; Alencar et al. 2010; Morales-Calderónet al. 2011; Arte- menko et al. 2012; Cody et al. 2014). If these variations emanate from regions close to the corotation radius (Rco; see Section 1.4.2), they can be used as indicators of stellar surface rotation rates. However, PMS rotation periods longer than ∼ 15 days likely trace fluctuations emanating from regions of the disc further from the star than Rco. Accordingly, these cannot be used to determine Ω?.

51 Chapter 2. Angular momenta calculation

2.4 Pre-main sequence stellar masses, ages, and radii

Effective temperatures, Teff , and bolometric luminosities, L?, are determined from spectral types and optical/NIR photometry. Through comparison of Teff and L? to theoretical PMS evolutionary models (see Section 1.2), stellar masses, M?, radii, R?, and ages, tage, can be determined. This section outlines the details of this technique.

2.4.1 Eﬀective temperatures from spectral types

Spectral types are assigned based on by-eye comparison of optical spectra to spectral standards and/or using spectral indices designed to utilise the temperature-dependent strengths of emission and absorption features. PMS M-dwarf spectra are characterised by strong metallic oxide TiO and VO absorption features which increase in strength with decreasing temperature and do not appear in stars earlier than K5 (Kirkpatrick et al. 1993; Riddick et al. 2007). The spectra of PMS K-dwarfs are categorised by MgH and Mg b absorption features at ∼ 5150 A˚ (e.g. Herczeg & Hillenbrand 2014) and earlier types by atomic absorption lines. For example,

• the relative strengths of Ca i (λ6162 A)˚ and Na i (λ5892 A)˚ lines can be used to distinguish between early F and late K spectral types;

• the G-band feature and the relative strengths of the ∼ 4300 A˚ and Hγ lines can be used to distinguish between early F and late G spectral types;

• the strengths of Balmer and Ca ii H and K lines can be used to distinguish between B and A spectral types with the Ca ii K line being particularly useful for discerning between B and early A

(e.g. Cannon & Pickering 1912; Hern´andezet al. 2004; Covey et al. 2007; Alecian et al. 2013a; Hillenbrand et al. 2013; Mooley et al. 2013; Herczeg & Hillenbrand 2014).

It is also possible to estimate the spectral types of PMS M-dwarfs using narrow-band photometry to create photometric indices (Hillenbrand & Carpenter 2000; Muench et al. 2002; Lucas et al. 2005; Da Rio et al. 2010; Andersen et al. 2011; Da Rio et al. 2012). Da Rio et al. (2010) determined spectral types between M2 and M6 for a sample of 217 Orion Nebula Cluster (ONC) stars using the relative magnitudes of a 200 A-wide˚ ﬁlter, centered

52 2.4. Pre-main sequence stellar masses, ages, and radii on 6200 A,˚ and an extrapolation of a linear fit to the V - and Cousins Ic-band photometry to the same wavelength. Similarly, Da Rio et al. (2012) used a narrow band filter centered on 7700 A,˚ combined with deeper photometry, to determine spectral types over the same range for a much larger sample of 1280 ONC stars. Hillenbrand et al. (2013) compared the Da Rio et al. (2010) and Da Rio et al. (2012) photometrically determined spectral types to their own spectroscopically determined values, finding significant scatter. However, the authors also noted that the observed level of scatter for stars later than M5 is similar to that seen when comparing their spectral types to other spectroscopically determined values (e.g. Hillenbrand 1997).

Once an estimate of the spectral type is available, effective temperatures are often determined via the application of a spectral type-to-effective temperature conversion scale. Early studies adopted Cohen & Kuhi (1979), Bessell (1979), Bessell (1991), or Kenyon & Hartmann (1995) MS dwarf scales or modified versions of these (e.g. Hillenbrand 1997) but these do not account for the lower surface gravities of PMS dwarfs. An update for PMS M-dwarfs, based on a scale intermediate between those of giants and dwarfs, was derived by Luhman et al. (2003). These M-dwarf scales are typically adopted but synthetic spectra of M-dwarfs determined from stellar atmosphere models have advanced significantly since 2003. Casagrande et al. (2008) and Rajpurohit et al. (2013) obtained new conversions for M-type stars by comparing observed, low resolution spectra to Cond-Gaia synthetic spectra and BT-Settl synthetic spectra, respectively (the latter of which were calculated from the Phoenix code; e.g. Allard & Hauschildt 1995; Allard et al. 2012). Significant differences exist between the two (pink and black lines, respectively in Fig. 2.3), especially for the late M-dwarfs.

PMS stars have low surface gravities and their photospheres often exhibit large, cool starspots (see Section 2.3.2). As a result, typically adopted MS dwarf temperature scales, such as those outlined above, are not ideally suited to determine effective temperatures for these stars (Gullbring et al. 1998; Luhman 1999; Stauffer et al. 2003; Da Rio et al. 2010; Pecaut & Mamajek 2013; Herczeg & Hillenbrand 2014). Pecaut & Mamajek (2013) and Herczeg & Hillenbrand (2014) have derived temperature scales specifically for PMS dwarfs over spectral types F0 to M5 and F5 to M9, respectively. Both of these studies used BT-Settl models to determine synthetic data which were then compared to observations. Pecaut & Mamajek (2013) compared BT-Settl synthetic photometry to observed optical

53 Chapter 2. Angular momenta calculation

●

● 7000

● ● ● ●

●

● ● ● ● ● ● ● ● ● ● ●

6000 ● ● ● ● ● ● ● ● ● [ K ]

f ●

f ● ● e ● T ● ● ● ● ● ● ● ● ●

5000 ●

● ●

●

● ● ●

● ● ● ● ● ● ● ● ● ● ● 4000 ● ● ●

F0 F5 G0 G5 K0 K5 M0 Spectral type

●

4000 ● ●

●

● ●

● ● ● ● ● ● ● ● ● ● ● 3500 ● ● ● ● ●

● ● ● ● ● ● ● ● [ K ]

f ● f

e ● ● ● ● T 3000

● ● ● ● ● ●

● ● ●

● 2500 ● ●

●

M0 M5 M9 Spectral type Figure 2.3: Comparison of spectral type-to-eﬀective temperature scales for spectral types F0 to M0 (top) and M0 to M9 (bottom) derived by diﬀerent studies (red: Cohen & Kuhi 1979; cyan: Bessell 1979 and Bessell 1991; dark green: Kenyon & Hartmann 1995; pink: Luhman et al. 2003; light green: Casagrande et al. 2008; black: Rajpurohit et al. 2013; blue: Pecaut & Mamajek 2013; orange: Herczeg & Hillenbrand 2014).

54 2.4. Pre-main sequence stellar masses, ages, and radii and infra-red (IR) photometry and combined these with newly determined spectral types to determine temperature scales for 5–30 Myr old stars. In comparison, the Herczeg & Hillenbrand (2014) scales were obtained by ﬁtting Phoenix/BT-Settl synthetic spectra to a combination of K5 to M8.5 spectral type templates determined in their study and Pickles (1998) luminosity class IV templates for spectral types F to K3. These PMS temperature scales are plotted for comparison with the Kenyon & Hartmann (1995), Casagrande et al. (2008), and Rajpurohit et al. (2013) MS dwarf scales in Fig. 2.3. The MS and PMS calibrated scales agree for ∼K6 to M4 but, at earlier spectral types (∼G3 to K6) PMS stars are cooler than their MS counterparts by up to ∼ 250 K.

2.4.2 Bolometric luminosities from photometry

Bolometric luminosities can be calculated by applying a bolometric correction to a single distance modulus- and extinction-corrected apparent magnitude (e.g Hillenbrand 1997) whereby L? 2 log = [Mbol, − (Φ − AΦ) + DM − BCΦ(Teﬀ )] . (2.20) L 5

Here, Mbol, = 4.755 mag is the bolometric absolute magnitude of the Sun (Mamajek

2012), DM is the distance modulus, and Φ and BCΦ(Teﬀ ) are the apparent magnitude and temperature-dependent bolometric correction in the chosen waveband (e.g. V , R, or

I). The extinction in waveband Φ, AΦ, is determined via comparison of observed (obs) to intrinsic (int) colours,

AΦ = X [(Φ2 − Φ)obs − (Φ2 − Φ)int]. (2.21)

The value of X depends on the chosen wavebands, Φ and Φ2, and is taken from existing extinction laws where applicable (e.g. Rieke & Lebofsky 1985; Yuan et al. 2013) or determined by transforming these extinction laws to the chosen photometric system.

The choice of wavebands in equations (2.20) and (2.21) is crucial to minimise contamination from accretion-induced ultra-violet (UV) excess emission or IR excess emission from dust within discs for PMS stars (see Section 1.3). As a result, adopted wavebands typically lie between the R and H bands (∼ 650−1630 nm). In addition, as PMS stars are intrinsically variable (see Section 2.3.2), it is important to ensure that contemporaneous photometry is used in calculating the observed colour, (Φ2 − Φ)obs.

55 Chapter 2. Angular momenta calculation

Distances to star forming regions

Calculation of bolometric luminosities using equation (2.20) requires a measure of the stellar distance. In some cases, estimates of the distances to individual stars have been made using astrometry (e.g. Hipparcos; ESA 1997). However, even when individual stellar distance estimates are available, most studies adopt a single distance for all members of a particular star forming region. Chapters 3 and 4 focus on PMS stars within the Lupus clouds, the Taurus-Auriga star forming region, ρ Ophiuchus, Upper Scorpius, and the ONC. The adopted distances to each of these regions are discussed individually below.

Lupus clouds: Most distances for this star forming region have been estimated using foreground MS stars with previously determined distances and reddening. The reddening increases with distance and displays a jump at the distance of Lupus due to the presence of dust grains. Using this method, Franco (1990), Franco (2002) and Alves & Franco (2006) determined distances to two of the individual Lupus clouds (165 ± 15 pc for Lupus IV and 130 to 150 pc for Lupus I). In addition, Hughes et al. (1993), Crawford (2000), and Lombardi et al. (2008) determined average distances of 140 ± 20 pc, 150 ± 10 pc, and 155 ± 8 pc, respectively for the entire Lupus complex. Individual trigonometric parallax measurements are also available for six stars in the region5 (Wichmann et al. 1998a; Bertout et al. 1999; Neuh¨auseret al. 2008) yielding distance estimates between ∼ 100 (RU Lup in Lupus II) and ∼ 240 pc (HR 6000 in Lupus III). In Chapter 4, a distance of 150 ± 10 pc is adopted for Lupus clouds I and II and a distance of 200 ± 20 pc is adopted for Lupus III, as in Comer´on(2008).

Taurus-Auriga star forming region: Very Long Baseline Interferometry (VLBI) trigonometric parallax measurements have produced distance estimates for the Taurus- Auriga star forming region: the far side of the region is located at ∼ 160 pc (Torres et al. 2009) while the near-side of the region is located at ∼ 130 pc (Torres et al. 2012). To reﬂect this, a mean distance of 140 ± 20 pc for Taurus-Auriga (Elias 1978; Loinard et al. 2007; Torres et al. 2009, 2012) is adopted in Chapters 3 and 4.

ρ Ophiuchus: Traditionally, ρ Oph was assumed to be located at 165 pc (Chini 1981) but this value has been revised down in more recent works based on photometric distance estimates (∼ 125 pc; de Geus et al. 1989; Knude & Hog 1998; Lombardi et al.

5These stars are RU Lup, RY Lup, HR 6000, HR 5999, Sz 68, and GQ Lup.

56 2.4. Pre-main sequence stellar masses, ages, and radii

2008). These results have found support from distance estimates using VLBI trigonometric parallax of binary stars in the core of the region. Loinard et al. (2008) measured distances +7.2 +5.8 of 116.9−6.4 pc and 121.9−5.3 pc for S1 and DoAr 21 and estimated a mean distance of +4.5 120.0−4.2 pc to the core of ρ Oph. A distance of 125 ± 5 pc to ρ Oph is adopted in Chapter 4.

Upper Scorpius: Upper Sco is part of the much larger Scorpius-Centaurus-Lupus- Crux Association (Sco OB2, Sco-Cen). An average distance to the region of 145 ± 2 pc has been determined using Hipparcos trigonometric parallax measurements of the Sco-Cen association (de Zeeuw et al. 1999). This distance is adopted in Chapter 4.

ONC : Early estimates of the distance to the ONC6 ranged from 300 to 540 pc based on de-reddened apparent magnitudes of O and B members (e.g. Trumpler 1931; Minkowski 1946; Sharpless 1952), zero-age MS calibration (e.g. Johnson & Hiltner 1956; Walker 1969), and radial velocity measurements of O and B stars with known proper motions (e.g. Strand 1958; Johnson 1965). Distance estimates have also been made using the orbital kinematics of known binaries within the ONC. Stassun et al. (2004) determined a distance of 419 ± 21 pc based on the orbit of an eclipsing binary but the ages determined for this system suggests it may be a foreground object. Kraus et al. (2007) and Kraus & Hillenbrand (2009) determined distance estimates of 434±12 pc, 387±11 pc, and 410±20 pc from orbital solutions for the θ1 Ori C spectroscopic binary. Jeﬀries (2007a) suggested a distance of 392 ± 32 pc based on the statistical assessment of the stellar inclinations of non-accreting members. Trigonometric parallax measurements from water masers in the ONC and the nearby Becklin-Neugebauer/Kleinmann-Low (BN/KL) region (located slightly behind the ONC; Zuckerman 1973; Genzel & Stutzki 1989) measure distances of +24 389−21 pc (Sandstrom et al. 2007) and 437±19 pc (Hirota et al. 2007), respectively. Finally, the most accurately determined measurement of the ONC is that of Menten et al. (2007) who derived a distance of 414±7 pc based on VLBI trigonometric parallax of several ONC members. This value is adopted in Chapters 3 and 4.

6Some authors refer to the ONC as the Orion Nebula or the Trapezium.

57 Chapter 2. Angular momenta calculation

2.4.3 Stellar masses and ages from eﬀective temperatures and bolometric luminosities

Estimates of the masses and ages of PMS stars can be made by comparing eﬀective temperatures and bolometric luminosities (Sections 2.4.1 and 2.4.2) to PMS evolutionary models. These models consist of theoretical mass tracks and isochrones that describe the mass- and age-dependent location of a PMS star in a Hertzsprung-Russell diagram (HRD; see Section 1.2). An example of the mass tracks and isochrones of one such set of PMS models is shown in Fig. 2.4 for 0.2 − 7 M and 1 − 100 Myr. The position of each star is compared to the grid of mass tracks and isochrones and the ones closest to the star’s location provide the estimate of its mass and age.

This method also allows for an estimate of the uncertainty on the determined stellar mass and age to be made. Errors associated with the PMS models themselves related to the choice of input stellar evolution physics (e.g. the treatment of convection and chemical composition; Siess 2001), are not accounted for. Instead, the uncertainties in luminosity and effective temperature (propagated from the error in spectral type) define the major and minor axes of an ellipse in the HRD [ log(Teff ) – log(L?/L ) space]. The major and minor axes of the corresponding ellipse in M? – tage space define the uncertainties in stellar mass and age. These are calculated by iteratively tracing round the circumference of the ellipse in log(Teff ) – log(L?/L ) space and calculating the stellar mass and age at each point. The maximum and minimum values of stellar mass and age calculated in this way then provide an estimate of the uncertainty on the stellar mass and age.

This method results in non-symmetric upper- and lower-bounded uncertainties. The diﬀerence is most apparent for the errors on the age estimates. Contraction occurs on a

Kelvin-Helmholtz time-scale, tKH ∝ 1/R?, (see Section 1.2) such that the rate of contraction slows down with time. This causes the isochrones to “bunch up” in the HRD at older ages (see lower panel of Fig. 2.4). Consequently, the upper bounded errors on stellar age exceed the lower bounded errors, assuming symmetric errors on log(L?/L ).

2.4.4 Age spreads in star forming regions

Variations in L? are observed across members of individual star forming regions with the same spectral type, producing an apparent spread in the inferred radii and ages of the

58 2.4. Pre-main sequence stellar masses, ages, and radii 1000 100 ] n u s 10 L [ L 1 0.1 0.01

20000 10000 8000 7000 6000 5000 4000 3000

Teff [ K ] 1000 100 ] n u s 10 L [ L 1 0.1 0.01

20000 10000 8000 7000 6000 5000 4000 3000

Teff [ K ]

Figure 2.4: Mass tracks (top panel) and isochrones (bottom panel) of Tognelli et al. (2011) Pisa models with metallicity, Z= 0.02, initial helium abundance, Y= 0.27, mix- 5 ing length, α = 1.68, and initial deuterium abundance, XD= 2 × 10 . The mass tracks are plotted from right to left for 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.75, 1, 1.2, 1.4, 1.6, 1.8, 2, 2.2, 2.6, 3, 5, and 7 M . The isochrones are plotted from top to bottom for ages of 1, 2, 3, 4, 5, 6, 10, 15, 20, 50, 75, and 100 Myr.

59 Chapter 2. Angular momenta calculation stellar sample. Whether this spread in radius is real or caused by observational effects is still up for debate (e.g. the recent review by Soderblom et al. 2014). For example, the presence of an unresolved binary component causes a star to appear more luminous, and therefore larger and younger. This effect is most pronounced for members of star forming regions older than ∼ 15 Myrs (Preibisch 2012) as the spacing between isochrones is smaller (see bottom panel of Fig. 2.4). However, significant age spreads are inferred for star forming regions with much younger stellar populations, such as the ONC and Taurus (∼ 1 − 2 Myrs; Hillenbrand 1997; White & Ghez 2001). Systematic errors on

L? associated with diﬀerential extinction, photometric variability (see Section 2.3.2) and inadequate correction for variable accretion (Hartmann 2001; Soderblom et al. 2014) are more problematic for these younger regions of star formation. These observational eﬀects certainly contribute to the observed luminosity spread but cannot explain its full extent (Burningham et al. 2005; Preibisch & Feigelson 2005; Hillenbrand et al. 2008; Slesnick et al. 2008; Da Rio et al. 2010).

Jeﬀries (2007b) used a novel technique to estimate stellar radii for a sample of stars within the ONC independently of L?. Projected stellar rotational velocities were combined with rotation periods to estimate an inclination-dependent stellar radius. After accounting for the random inclinations of the stellar rotation axes, together with any observational uncertainties, a spread in stellar radius was still observed, suggesting that the spread in

L? is caused by a true spread in stellar radius. Under this assumption, individual stellar radii, R?, can be calculated from the eﬀective temperature and the bolometric luminosity,

R L 1/2 T 2 ? = ? eff, , (2.22) R L Teff,? where Teff, = 5771.8 K (Mamajek 2012).

The presence of a radius spread does not necessarily mean that a spread in age is present in the sample, however. Certain studies argue that the spread in R? can be explained by magnetic effects which can reduce the convective efficiency or lead to significant photospheric spot coverage (e.g. Spruit & Weiss 1986; Jackson & Jeffries 2014). However, age spreads are inferred in samples of the youngest late-type stars which are fully convective. For these stars, the level of radius inflation produced by the inhibition of convective efficiency is negligible (∼ 0.1 − 2%; Chabrier et al. 2007; Morales et al. 2010; Feiden &

60 2.5. Disc sizes

Chaboyer 2014). Additionally, by considering spot temperatures for K and early-M stars at 82−90% of their photospheric temperatures (Boyajian et al. 2012) and spot coverage of ∼ 1-40% (O’Neal et al. 1998; Barnes & Collier Cameron 2001; Barnes et al. 2004; O’Neal et al. 2004; Morin et al. 2008; Hackman et al. 2012), Feiden & Chaboyer (2014) found that unattainably high interior magnetic ﬁeld strengths were required for starspots to produce the degree of radial inﬂation inferred from L? spreads.

−5 −1 The occurrence of episodic accretion events with mass accretion rates ≥ 10 M yr during the assembly phase have been proposed as an alternative method of producing the inferred radius spreads (Tout et al. 1999; Baraffe et al. 2002, 2009). Depending on the amount of accretion kinetic energy absorbed by the star during this phase, the star can either contract at a greater rate and then remain at almost constant radius for ∼ 10 Myr or it can inflate to larger radii before quickly contracting back to the non-accreting isochrone expected of its mass and age (Baraffe et al. 2009; Littlefair et al. 2011). Thus, the spread in stellar radii could indicate differences in accretion history rather than age. However, the ability of this mechanism to produce the observed L? spreads at low masses has been contested (Hosokawa et al. 2011) and results are dependent on the assumed initial protostellar mass in “cold accretion” models (Baraffe et al. 2012).

Support for the presence of real age spreads in star forming regions has been found using alternative age diagnostics. Studies of lithium depletion levels in members of the ONC and Taurus reveal that a few percent of stars are consistent with having ages ≥ 10 Myr (Palla et al. 2007; Sacco et al. 2007). In addition, Sergison et al. (2013) determined the ages for members of the ONC and NGC 2264 using lithium depletion and PMS isochrones, ﬁnding a modest correlation between the two age indicators. With this in mind, age spreads in star forming regions are assumed to be real and isochronal ages are used to study the evolution of stellar and disc angular momentum during the PMS in Chapters 3 and 4.

2.5 Disc sizes

Resolved images of protoplanetary and debris discs exist at optical, IR, and submillimetre/millimetre (hereafter collectively referred to as submm) wavelengths. By analysing these images and comparing them to theoretical models of disc evolution, it is possible to determine the radial extent of material in the disc. For protoplanetary discs, these

61 Chapter 2. Angular momenta calculation resolved observations can also reveal how the disc material is distributed. This section details how the size and underlying structure of protoplanetary discs can be estimated using observations of molecular gas (Section 2.5.1) and dust (Section 2.5.2), and how the size of debris discs can be determined from 850 µm thermal continuum emission (Section 2.5.3).

2.5.1 The gas component of protoplanetary discs

The main constituent of protoplanetary discs is molecular hydrogen but its zero dipole moment means that tracers of the less-abundant molecular gas and dust are required to constrain the radial extent of the disc. Submm detections of CO line emission (e.g. 12CO J = 1 → 0 and 13CO J = 2 → 1) are often used (e.g. Dutrey et al. 2008; Schaefer et al. 2009) but can be plagued by contamination from grain freeze-out and large-scale emission within the star forming region, making it difficult to determine the location of the disc outer edge. This effect is greatly reduced by using submm CN line emission (e.g. CN J = 2 → 1) which suffers little from confusion (Guilloteau et al. 2013).

A parametric disc model, ﬁt to the calibrated visibilities, can be used to determine the radial extent of the molecular gas emission (e.g. Dutrey et al. 1994; Pi´etuet al. 2007). Such models typically assume power-law descriptions for the rotation velocity,

r 1/2 v(r) = v , (2.23) 100 100 au where the value of the exponent reﬂects the assumption that the disc is Keplerian; the disc temperature, r −q T (r) = T ; (2.24) 100 100 au the scale height of the disc,

r −h H(r) = H ; (2.25) 100 100 au and the gas surface density,

r −p Σ(r) = Σ . (2.26) 100 100 au

Here, the subscript 100 denotes the value of the scale factor at r = 100 au.

62 2.5. Disc sizes

The disc outer radius, Rdisc, is then one of ﬁfteen free parameters in the model that describes the observed emission:

• the scale parameters v100, T100, H100, and Σ100, together with exponents, q, h, and p describe the rotation, temperature, scale height, and surface density;

• the inner and outer radii, Rin and Rdisc, describe the extent of the disc;

• the position coordinates of the disc centre, (x0, y0), the position angle (P.A.) of the 7 8 rotation axis , the inclination, i, of the disc , and the systematic velocity, vLSR, describe the geometry of the disc;

• the local line width, dv, accounts for any thermal and/or turbulent line broadening.

The power-law form of equations (2.24) and (2.25) are valid for discs in hydrostatic equilibrium (HE), a reasonable approximation for protoplanetary discs (e.g. Chiang & Goldreich 1997). However, the form of equations (2.23) and (2.26) are only valid under particular assumptions. Equation (2.23) relies on the disc self-gravity remaining small such that gravitational instabilities (GIs; see Section 1.2.2) do not form and the disc remains Keplerian. Analysis of the observed molecular line emission can reveal whether this is the case. A Keplerian disc will exhibit a double-peaked line proﬁle and a lobed feature in the channel map corresponding to the systematic velocity (e.g. Dutrey et al. 1998; Guilloteau & Dutrey 1998). However, these features are also present in objects dominated by infall (such as the envelopes of protostars, see Section 1.3.2). The velocity ﬁeld and integrated intensity map must be inspected to distinguish between Keplerian rotation and infall: the former produces a velocity gradient along the major axis of the integrated emission whereas the latter produces a velocity gradient along the minor axis.

The form of the disc surface density in equation (2.26) is not motivated from astrophysical theory (see Section 1.2.2). Instead, it is based on an empirical study of the distribution of material within the minimum mass solar nebula (Hayashi 1981) and is not consistent with disc evolution theory (see below). However, the use of a theoretically motivated form of the surface density would further increase the complexity of the model outlined above, which is already computationally intensive.

7 P.A. is deﬁned so that v100 in equation (2.23) is positive. 8i runs between −90◦ and 90◦ with i = 0 corresponding to a face-on disc.

63 Chapter 2. Angular momenta calculation

2.5.2 The dust component of protoplanetary discs

Attempts to constrain the disc radial extent tend to favour dust tracers. A few discs have been resolved in scattered light (e.g. Ardila et al. 2007; Quanz et al. 2012) but these studies are limited to the closest star-disc systems and brightest stellar hosts due to present resolution and sensitivity capabilities. Others have been resolved using the comparative optical qualities of the disc material and background cluster environment (e.g. the silhouette discs in the ONC; O’Dell & Wong 1996; Bally et al. 1998) but these require favourable viewing conditions for detection.

High-resolution submm interferometers such as the Submillimeter Array (SMA), IRAM Plateau de Bure Interferometer (PdBI), Combined Array for Millimeter-wave Astron- omy (CARMA), Nobeyama Millimeter Array (NMA), and the Atacama Large Millime- ter/submillimeter Array (ALMA) have also been used to constrain the sizes of spatially resolved thermal continuum emission from the dust component of protoplanetary discs. Due to the optically thin nature of the disc at submm wavelengths, the observed intensity proﬁle is directly related to the spatial distribution of material within the disc (Beckwith et al. 1990). As such, the emission proﬁle can be modeled using a suitable form of the disc surface density to determine the disc radius.

Initial attempts at estimating disc radii in this manner were made using the simple power-law parameterisation for the surface density proﬁle, given by equation (2.26) (e.g. Andrews & Williams 2007b). However, as stated above, this form of the surface density is not determined from disc evolution theory. More recently, studies such as Hughes et al. (2008), Andrews et al. (2009), Isella et al. (2009), and Guilloteau et al. (2011) have favoured a parameterisation described by the similarity solution of the surface density of a thin, viscously evolving Keplerian disc surrounding a point mass, M?, as given by equation (1.40): " (2−γ) # C −(5/2−γ)/(2−γ) R Σ(r) = γ τ exp − 3πν1R τ

(Lynden-Bell & Pringle 1974; Pringle 1981; Hartmann et al. 1998; see Section 1.2.2). Here,

C is a normalisation constant, R ≡ r/r1 where r is the cylindrical radius and r1 is a radial scale factor, ν1 is the viscosity evaluated at r1, γ describes the radial dependence of the viscosity [see equation (1.39)], and τ = t/ts + 1 where ts is the viscous scaling time, given

64 2.5. Disc sizes

Figure 2.5: The debris disc around Fomalhaut. The blue part of the image shows scattered light from dust grains at optical wavelengths, observed using the NASA/ESA Hubble Space Telescope. The orange arc shows thermal emission from the dust grains observed in the submm using ALMA. Credit: ALMA (ESO/NAOJ/NRAO). Visible light image: the NASA/ESA Hubble Space Telescope by equation (1.44): 2 1 r1 ts = 2 . 3(2 − γ) ν1 As a result, the disc can then be fully described by ﬁtting for γ, a normalisation factor, and a characteristic radius.

2.5.3 Debris discs

Unlike primordial circumstellar discs, debris discs are almost completely devoid of gas. As a result, estimates of the radial extents of debris discs rely on spatially resolved detections of scattered or reprocessed starlight from dust grains. Fig. 2.5 shows an example of such observations of the disc around Fomalhaut.

Assuming the debris disc is axisymmetric, a two-dimensional Gaussian model of the form

1 f(x, y) = N exp − A(x − x )2 + B(x − x )(y − y ) + C(y − y )2 , (2.27) 2 0 0 0 0

65 Chapter 2. Angular momenta calculation

Figure 2.6: Example of the process used to determine how well-resolved a debris disc is. The top-left panel shows the original image, centered on the disc (η Corvi in this instance). The central position of the simulated maps (denoted by “SIM”) has been shifted compared to the original map and a point source has been inserted. The procedure used to determine the FWHM of the disc in the original image is repeated in each of the simulated maps for the simulated point source. may be ﬁt to the submm emission where

cos2 θ sin2 θ A = 2 + 2 , (2.28) σx σy

1 1 B = 2 sin θ cos θ 2 − 2 , (2.29) σx σy and sin2 θ cos2 θ C = 2 + 2 . (2.30) σx σy

The emission can then be fully described using six parameters9: the amplitude (N), the central point (given by x0 and y0), the degree of spread in the x and y directions (σx and

σy, respectively), and the orientation (θ).

The best fit parameters are found using Levenberg-Marquardt non-linear least-squares fitting. This procedure compares the form of a two-dimensional Gaussian function to an intensity map of each disc using the idl routine mpfit2dfun (Markwardt 2009). Where the two-dimensional Gaussian provides a reasonable fit to the map, the full width half

9The central point of the disc is assumed to be located at the position of the star, deﬁned by its right ascension and declination, and θ is measured east of north.

66 2.5. Disc sizes maximum (FWHM) of the major axis,

p FWHMx = 2 2 ln(2) σx, (2.31) provides an estimate of the angular size of the disc. The physical radius of the disc is then

q 2 2 Rdisc = d/2 FWHMx − FWHMb, (2.32)

10 where FWHMb is the FWHM of the beam and d is the stellar distance.

By comparing the results of this fitting procedure with those of a point source, artificially inserted into the image, it is possible to assess how well resolved the disc emission is. The original intensity map is shifted by a randomly determined RA and dec such that the disc no longer lies at the centre of the image. A two-dimensional Gaussian point source (FWHMx = FWHMb) is then placed at the centre of the new image (see Fig. 2.6). The same fitting procedure outlined above is then performed on this new image and the process is repeated. Finally, the FWHM of the fit to the disc image is compared to the distribution of point source FWHM (which are assumed to be normally distributed) to determine the significance of the result (see Fig. 2.7). This is expressed in terms of the standard deviation, σ, of the distribution of point source FWHM.

In some cases, emission from debris discs is not well-approximated by a Gaussian profile. Fig. 2.8 shows two such examples: ε Eri on the left and Fomalhaut on the right. Neither image features centrally concentrated emission, as expected of a Gaussian intensity profile. Instead, the emission in the image on the left is concentrated in a ring, as expected from a well-resolved, near face-on debris disc, and that in the image on the right is concentrated into two lobes, as expected of a more inclined disc. To estimate the radial extent of discs such as these using Gaussian fitting, it is possible to artificially increase the effective beam size by smoothing the intensity map with a two-dimensional circular Gaus- sian point spread function (PSF). This is equivalent to increasing the distance between the observer and the star-disc system and is achieved using the gausmooth subroutine of the starlink kappa package (Currie & Berry 2013). Each resulting pixel in the smoothed image is then the mean of the pixels within a specified box width, weighted by the PSF. Examples of the results of this process are shown in Fig. 2.9 where the images of ε Eri and

10The telescope beam is assumed to be circular.

67 Chapter 2. Angular momenta calculation

Figure 2.7: Distribution of major and minor FWHM resulting from the simulated point source ﬁtting in Fig. 2.6, overlaid with the results for the original disc image (the red and blue vertical lines). The standard deviation of a Gaussian ﬁt to the distribution of FWHM (σ) is used to determine how well-resolved the disc emission is.

68 2.6. Main sequence stellar masses from spectral energy distribution modelling

Figure 2.8: Examples of thermal dust emission from debris discs which lack the centrally concentrated emission expected for a Gaussian intensity proﬁle. Left panel: ε Eri which features emission concentrated in a ring. Right panel: Fomalhaut which features emission concentrated in two lobes. The colour scale represents varying intensity across the map. Fomalhaut in Fig. 2.8 have been smoothed with an additional PSF with FWHM=27 arcsec, simulating a factor of two increase in stellar distance.

2.6 Main sequence stellar masses from spectral energy distribution modelling

The spectral energy distribution (SED) of a debris disc consists of a stellar photospheric component plus an IR excess as the dust grains within the disc absorb UV and optical stellar photons and re-emit them at longer wavelengths (Figure 2.10). Photometry shortward of ∼ 10 µm is used to constrain the effective temperature, Teff , and radius, R?, of the star11 via least-squares fitting to stellar atmospheric models (e.g. the phoenix Gaia grid; Brott & Hauschildt 2005).

The stellar luminosity, L?, may then be calculated from the best ﬁt eﬀective temperature and stellar radius. An estimate of the stellar mass, M?, can also be made through the application of a suitable mass-luminosity relation of the form

ν L? ∝ M? , (2.33)

11The exact location of the cut-off wavelength depends on the temperature of the IR excess and is varied for each star to ensure the best fit to the photospheric emission and also to make sure that the IR excess does not affect the fit to the photospheric emission.

69 Chapter 2. Angular momenta calculation

Figure 2.9: Results of smoothing the images of ε Eri and Fomalhaut in Fig. 2.8 with an additional 27 arcsec beam. The emission is much more centrally concentrated, as expected for a Gaussian. where ν depends on M? (Prialnik 2009):

• for M?/M . 0.8, ν ∼ 2.5;

• for 0.8 . M?/M . 2, ν ∼ 4.5;

• for M?/M & 2, ν ∼ 3.

2.7 Debris disc masses

At submm wavelengths, disc emission is optically thin such that the ﬂux, Fν,disc, is directly proportional to the dust mass. Assuming the dust is isothermal,

2 d Fν,disc Mdust = , (2.34) κνBν(Tdust)

(Hildebrand 1983; Beckwith et al. 1990) where d is the stellar distance, Bν(Tdust) is the 2 −1 Planck function at dust temperature, Tdust, and κν = 1.7 cm g is the dust opacity (Zuckerman & Becklin 1993). This value of the dust opacity originates from estimates of the typical grain size in debris discs and corresponds to all the grains contributing to the observed ﬂux being ∼ 300 µm in size (Pollack et al. 1994).

In reality, a range of dust temperatures are present within the debris disc and Tdust represents the characteristic temperature that describes the observed continuum emission.

70 2.7. Debris disc masses

Figure 2.10: SED of the debris disc system HD 127821. The photospheric and disc components are clearly visible in the two-humped structure. The results of the parameter ﬁtting detailed in the text are displayed in the top-right corner of the image. Credit: Grant Kennedy and the SCUBA-2 Observations of Nearby Stars (SONS) Consortium.

71 Chapter 2. Angular momenta calculation

This characteristic temperature can be estimated using SED fitting. The NIR photometry of the stellar blackbody fit, described in Section 2.6, is subtracted from the observed fluxes to determine the contribution from the dust. A least-squares minimisation technique is then used to fit the dust component of the debris disc SED using,

−1 Fν = nBν(Tdust)Xλ , (2.35) where  1 λ < λ0 Xλ = (2.36) β (λ/λ0) λ > λ0 is the emission eﬃciency at wavelength, λ, and n is the solid angle of the disc seen by the observer, σ n = tot . (2.37) d2

Here, d is the stellar distance and σtot is the total cross-sectional area of the dust,

2 σtot = 4πr f, (2.38)

2 measured in au where f = Ldisc/L? is the fractional luminosity and the blackbody ra- dius12, 278.32 L 1/2 r = ? , (2.39) Tdust L is measured in au. The debris disc SED is then fully described using six free parameters:

Teff and R? from the fit to the stellar component of the SED together with Tdisc, f, λ0, and β from the disc component fit (e.g. Kennedy et al. 2012). The resulting value of Tdisc is then used to calculate the dust mass using equation (2.34).

For some debris discs, a single temperature modified blackbody model [equation (2.35)] is not sufficient to describe the emission. Instead, the broad nature of the SEDs of these discs are better fit by two different temperatures (e.g. Chen et al. 2006; Panićet al. 2013). This does not necessarily mean that the dust exists in two spatially distinct rings and can similarly be explained by dust grains exhibiting a range of temperatures (Matthews et al. 2007; Kennedy & Wyatt 2014). At 850 µm, the colder dust component dominates the disc emission (Panićet al. 2013) and, for discs fit by two-component modified blackbodies, the

12 This is not the same as Rdisc in Section 2.5.3.

72 2.7. Debris disc masses lower of the two dust temperatures is used in the calculation of Mdust.

2.7.1 Collisional cascade and total debris disc mass

As shown in Section 2.2.2, the total mass of a disc whose size distribution is governed by a collisional cascade is dominated by the largest particles [equation (2.12)]. However, masses calculated using equation (2.34) are only sensitive to the smallest dust grains. To see this, consider the total cross sectional area of the dust in equation (2.38). Assuming all of the particles in the disc are spherical and the distribution of particle sizes is given by equation (2.9), Z Dmax σtot = σ(D)n(D) dD, (2.40) Dmin where π σ(D) = D2, (2.41) 4 then, as long as q > 5/3, the contribution from small grains (Dmin) dominates and

π σ = KD5−3q. (2.42) tot 4(3q − 5) min

An estimate of the total mass of the disc can be made by combining equations (2.12), (2.38), and (2.42). In this case,

6−3q 4(6q − 10)π 2 Dmax Mtot = ρr fDmin . (2.43) 18 − 9q Dmin

The lower end of the size distribution

Particles smaller than Dmin either spiral in towards the host star due to Poynting-Robertson (P-R) and/or stellar wind drag or are blown out of the star-disc system by radiation and/or stellar wind pressure. The drag components may be ignored in studies of discs around A- to K-type stars as the high observed densities mean that the collisional lifetime of a particle is shorter than its P-R lifetime (Wyatt et al. 1999; Wyatt 2005). In these discs, small grains become unbound when Frad/Fgrav ≥ 0.5 where Frad is the force due to radiation pressure and Fgrav is the gravitational force. Then,

L? M 2700 Dmin = 0.8 (2.44) L M? ρ

73 Chapter 2. Angular momenta calculation in µm, where ρ is the dust density, assumed to be that of basalt (2700 kg m−3; Dominik & Decin 2003).

The low luminosities of M-type stars yield Frad/Fgrav < 0.5 for all dust grain sizes and equation (2.44) is not valid. For these stars, P-R and stellar wind drag are understood to be the dominant mechanisms responsible for grain removal such that Dmin depends on the stellar wind speed and mass outﬂow rate (Plavchan et al. 2005). However, these parameters are highly uncertain for M-type stars leading to studies typically adopting

Dmin = 1 µm (e.g. Lestrade et al. 2012; Morey & Lestrade 2014).

Such sharp cut-offs in the size distribution at Dmin are only approximations. A strict lower limit as imposed by equation (2.44) or by setting Dmin = 1 µm would cause a “wave” in the size distribution, leading to a radial dependence of q (Durda et al. 1998). In addition, as grains smaller than Dmin are blown out of the disc, they may catastrophically collide with particles slightly larger than Dmin, thus affecting the lower end of the size distribution (Krivov et al. 2000). However, it is likely that these small grains would have little effect on the largest planetesimals in the cascade which dominate the mass budget.

The upper end of the size distribution

The size of the largest object involved in the cascade, Dmax, is governed by the rate of catastrophic collisions it experiences. This, in turn, depends on the size distribution [equation (2.9)] as a collision is only likely to be catastrophic if the two impacting particles are of roughly equal size (Wyatt et al. 1999). Assuming q = 11/6, the timescale over which such a catastrophic collision is likely to occur is given by,

dr t (D ) = 5.2 × 10−10 r7/3D1/2 D−1/2Q? 5/6e−5/3M −4/3f −1, (2.45) cc max r max min D ?

(Wyatt 2008) where r and dr are in au, Dmax is in km, Dmin is in µm, QD is the planetes- −1 imal strength measured in J kg , e is the average planetesimal eccentricity, M? is in solar units, and tcc(Dmax) is in Myrs. A particle of size greater than Dmax in a disc around a star younger than tcc(Dmax) is unlikely to have experienced a catastrophic collision and cannot be part of the cascade. Thus, an estimate of Dmax may be found by setting tcc(Dmax) equal to the stellar age.

74 2.8. Summary

2.8 Summary

This chapter has outlined the models and observational parameters required to calculate the angular momentum of stars and their protoplanetary and debris discs. These are used in the following chapters to study the evolution of angular momentum during the formation of stars and their planetary systems.

Assuming that a star can be modelled as a rotating sphere of polytropic index, n (corresponding to radius of gyration, k2), its angular momentum can be calculated from estimates of its stellar mass, radius, and rotation rate using equation (2.1):

2 2 J? = k M?R?Ω?.

The value of n (and therefore k2) is determined by the internal structure of the star. For example, fully convective stars have n = 3/2 (corresponding to k2 = 0.205) while fully radiative stars have n = 3 (corresponding to k2 = 0.08; Motz 1952). The methods used to estimate the stellar mass, radius, and rotation rate diﬀer between MS and PMS stars.

For PMS stars, age-dependent stellar masses are determined by comparing eﬀective temperatures, Teﬀ , and bolometric luminosities, L?, to theoretical models of PMS evolution

(Section 2.4.3) while stellar radii, R?, are calculated directly from Teff and L?, assuming the stars emit as blackbodies and, furthermore, that the spread in L? observed within a particular star forming region represents a true spread in stellar radius (Section 2.4.4). In turn, the values of Teff and L? are determined from spectroscopically or photometrically determined spectral types and optical (or near-IR) photometry (Sections 2.4.1 and 2.4.2). On the other hand, MS stellar masses are estimated using mass-luminosity relations applicable for their spectral types. In turn, their luminosities are estimated from Teff and

R?, which result from SED ﬁtting (Section 2.6).

The stellar rotation rate can be measured in one of two ways. If the surface of the star exhibits starspots, or if the star is actively accreting from a disc, periodic variations in the stellar ﬂux, modulated by the stellar rotation, can be used to determine the rotation period (Section 2.3.2). However, caution has to made when employing this method as alternative sources of variability which are not modulated by stellar rotation can con- taminate samples and, for ground-based observations, the day-night sampling interval can

75 Chapter 2. Angular momenta calculation lead to beat periods being measured. Alternatively, one may use rotationally-broadened spectral lines to determine the rotational velocity (Section 2.3.1). However, unless an estimate of the stellar inclination can be made, one can only measure the projected rotational velocity. In this case, only a lower limit can be placed on the stellar angular momentum using equation (2.15): 2 J? sin i = k M?R?v sin i.

By modelling protoplanetary discs as geometrically thin, axisymmetric, Keplerian discs, its angular momentum can be determined from the stellar mass, disc radius, Rdisc, and an appropriate form of the disc surface density, Σ(r), using equation (2.8):

Z Rdisc 1/2 3/2 Jdisc = 2π(GM?) r Σ(r) dr. Rin

On the other hand, debris discs, which are formed via a collisional cascade of planetesimals (Backman & Paresce 1993), are conﬁned to a thin torus-like structure. Calculation of the angular momentum of debris discs using equation (2.14):

1/2 Jdisc = Mtot(GM?Rdisc) requires an estimate of the disc mass, disc radius, and stellar mass.

The form of Σ(r) is assumed to be either a simple power law in radius [equation (2.26)] or follow from the similarity solution of the surface density of a viscously evolving Keplerian disc [equation (1.40)]. The chosen parameterisation of Σ(r) can then be used to model molecular line emission and thermal dust emission at submm wavelengths to determine the disc radius (Sections 2.5.1 and 2.5.2). Debris disc radii can be determined from fitting observed submm intensity distributions with a Gaussian profile (Section 2.5.3). When the emission is not well-approximated by a Gaussian, additional smoothing with an effective

Gaussian beam can be applied in order to estimate Rdisc.

76 3 Accretion discs as regulators of stellar angular momentum evolution

his chapter is a summary of the work presented in Davies et al. (2014a) in which the role of accretion discs in regulating the removal of stellar angular momen- T tum was re-addressed in light of recent substantial updates to spectral type estimations and newly established intrinsic colours, eﬀective temperatures, and bolometric corrections for pre-main sequence (PMS) stars from Pecaut & Mamajek (2013). These account for the lower surface gravities and highly spotted surfaces of PMS dwarfs compared to their main sequence (MS) counterparts and are thus more applicable than the typically adopted MS dwarf scales (e.g. Cohen & Kuhi 1979; Kenyon & Hartmann 1995; Luhman 1999; see Section 2.4.1).

The Davies et al. (2014a) study focuses on T Tauri stars within the Orion Nebula Clus- ter (ONC) and Taurus-Auriga. The well-studied nature of these regions of star formation, together with their youthful stellar populations (∼ 1 Myr and ∼ 2.8 Myr, respectively;

77 Chapter 3. Accretion discs as regulators of stellar angular momentum evolution

Hillenbrand 1997; White & Ghez 2001), enabled the study to concentrate solely on stars which are fully convective. With the exception of V2247 Oph, fully convective stars do not exhibit the surface diﬀerential rotation that their partially radiative counterparts are observed to display (Barnes et al. 2000; Donati et al. 2000; Skelly et al. 2008; Donati et al. 2010b; Skelly et al. 2010; Waite et al. 2011; Carroll et al. 2012; Donati et al. 2014). Addi- tionally, the interiors of fully convective stars are assumed to undergo solid body rotation in numerical models and magneto-hydrodynamic simulations (e.g. Kuker & Rudiger 1997; Browning 2008; see Section 2.1). As such, a measure of the photospheric rotation period at a single latitude can be used as a measure of the rotation rate of the entire star.

Previous studies of PMS rotational evolution had been affected by inclusion of non- members, contamination by beat and harmonic periods, unreliable methods of accretion- disc identification, and the stellar mass-rotation relation (Stassun et al. 1999; Rebull 2001; Herbst et al. 2002; Makidon et al. 2004; Cieza & Baliber 2007), as discussed in Section 1.4.1. Here, every effort was made to create a sample which was clear of all these previously identified sources of bias. This chapter outlines how this sample was compiled (Section 3.1) before discussing the results presented in Davies et al. (2014a) in Section 3.2.

3.1 Stellar data

Estimates of stellar masses, radii, ages, and rotation rates, as well as reliable indicators of accretion disc presence, were required to calculate the specific stellar angular momentum, j? = J?/M?, and study its evolution during the accretion disc-hosting phase of star formation (see Section 2.1). Both the ONC and Taurus-Auriga have well studied stellar populations (e.g. Hillenbrand 1997; Luhman et al. 2010) but a range of different methods had been adopted to calculate their properties. Spectral types for members of both star forming regions had been recently revised or, for some low-mass members, determined for the first time. In light of this, and to achieve a uniform sample across both star forming regions, effective temperatures and bolometric luminosities were re-determined in Davies et al. (2014a). These were then combined with rotation periods taken from the literature to calculate the specific stellar angular momentum. In terms of the rotation period, equation (2.1) becomes 2πk2M R2 j = ? ? . (3.1) ? P

78 3.1. Stellar data

In Davies et al. (2014a), the radius of gyration, k2, was computed separately for each star following the method outlined in Herbst & Mundt (2005) which accounted for rotational flattening (see Section 3.1.5). However, this method is only an approximation for fully convective stars as it models the stellar rotation as that of a rotationally flattened, uniform density, thin spherical shell. Following more recent correspondence with Maria Jaqueline Vasconcelos and JérômeBouvier, the results presented in this chapter adopt a value of k2 which approximates the stellar rotation as that of an n = 3/2 polytropic sphere undergoing solid body rotation (c.f. Moraux et al. 2013). In this case, k2 = 0.205 (Motz 1952), as outlined in Section 2.1. This change in k2 has not affected the conclusions of the Davies et al. (2014a) study.

3.1.1 Eﬀective temperatures

Spectral types are translated into eﬀective temperatures using spectral type-to-eﬀective temperature scales (see Section 2.4.1). Previous studies of PMS stars typically adopted MS dwarf scales (e.g. Cohen & Kuhi 1979; Bessell 1991; Kenyon & Hartmann 1995; Luhman 1999) but these do not fully account for the lower surface gravities and highly spotted surfaces of PMS stars. The Pecaut & Mamajek (2013) temperature scales, empirically derived for 5–30 Myr stars of spectral types F0 to M5, take these factors into account and were adopted in Davies et al. (2014a). Although the members of the ONC and Taurus- Auriga are likely younger than 5 Myr, these temperature scales are still more applicable than MS dwarf scales.

The spectral types of the majority of ONC members were retrieved from the most recent cluster census of Hillenbrand et al. (2013) but this study did not include all members. Da Rio et al. (2010) spectral types, calculated from 7770 A˚ narrow-band photometry, were adopted for the lowest mass stars missing from this study. Hillenbrand et al. (2013) had found a seemingly large scatter between these photometrically-determined spectral types and their own but had also recognised that a similar level of scatter was present between previous spectroscopic spectral types for the lowest masses. Individual references for the remaining ONC members and the full Taurus-Auriga sample are detailed in Tables 3.1 and 3.2 for the ONC and Taurus-Auriga samples, respectively. Where individual errors on spectral types were not published, an estimate of ±1 spectral subtype was adopted.

79 Chapter 3. Accretion discs as regulators of stellar angular momentum evolution 15 days, e – > P Obs Ref Class Acc. Note 30 opt91 H00 opt III H02 — II a A a,d 33 opt G95 II40 — opt70 H00 a opt III H00 — — d N g 50 opt37 G95 MIR09 M11 — MIR III — M11 N III a,e N a a 5599 opt15 opt H00 opt R09 III H0271 III N II opt N H00 a N a II a N — 45 opt H00 III — — 7623 opt opt H00 H00 III — — N — a ...... P ) classiﬁcation (A – accreting, N – not accreting); ii 47 3 5458 0 14 5215 17 6 5940 10 1 44 — — — II A a,f 7162 6 5 9073 17 65 8 10 60 3 61 — — — II N f 6348 6 9 ...... ) (days) 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ?

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± R 90 49 77 96 68 09 28 74 92 31 00 88 30 00 08 18 21 ...... 3 4 4 3 7 4 3 2 4 4 6 4 4 4 4 4 3 6 5 3 5 1 4 9 0 2 4 2 2 2 3 3 2 5 9 6 5 8 1 8 5 4 4 7 2 4 5 6 6 6 1 ...... 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 +0 − +0 − +0 − +0 − +0 − +0 − +1 − +3 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +1 − age 2 8 2 4 3 0 7 2 6 9 4 5 6 7 7 6 1 t ...... 2 1 1 1 0 1 3 4 0 0 0 0 0 0 0 0 1 but unable to classify source as II or III). ? j 28 21 18 27 02 49 25 30 59 57 85 59 49 51 51 41 45 22 27 13 20 49 34 14 23 52 34 57 63 58 49 48 57 40 ...... ) (Myr) (R 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ?

+0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − M 53 73 86 47 51 21 20 00 11 26 08 76 65 63 66 60 54 ...... ?

L L 89 2 79 1 83 1 004 2 128119 2 2 917357 2 2 914 2 698 2 011935 2 2 143964 2 854 1 1 808 1 601 1 . . . ...... log eff T (K) (M 1 H 4655 1 1 Ste1 5210 H 1 5210 1 1 H3 5390 H13 0 5210 0 1 H11 5030 H1 H 1 4920 Ste 4550 4760 0 1 0 1 H11 4760 Ste1 Ste 4550 0 1 Sta 4550 1 Ste 4550 0 4550 0 0 1 Ste 4550 0 1 Ste 4550 0 1 Ste 4550 0 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 5 . (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) SIMBAD SpTLT Ori Ref V1229 OriV1963 Ori K0 G8 V2235 Ori G8 V403 Ori K1 AK Ori K3 K2 V1232 OriV1509 Ori G6 V426 K2 OriAF Ori K2 KM OriV348 Ori G8 V1331 Ori K3 K3 V1444 Ori K3 K3 V2299 Ori K3 V1294 Ori K3 V1333 Ori K3 , f – no reliable rotation period, g – able to calculate crit j > ? Table 3.1: Stellar dataliterature. for The all full membershere table of to is the illustrate available ONC in itsHillenbrand not et electronic content. identified al. form as Column (2013) onluminosity; binary (H13; 1: the or except columns SIMBAD supplementary [D10] multiple 6, identifier; – data foradopted columns Da 7, CD which rotation Rio 2 a at and period et and spectral thecorresponding al. 8: in 3: type back 2010); reference days, was of columns adopted stellar (E93 the available this 4 spectral –H00 in mass waveband and thesis. – of the Edwards type 5: in Herbst et A the and effective solar et al. sample measurement associated– temperature al. units, 1993, (opt is Frasca reference and 2000b, et G95 shown – logarithmic age as C01 al. – bolometric optical,12 in in – 2009, Gagne NIR Carpenter and Myr, P09 et – et 13: – and al. al. near Parihar 1995, 2001, MIR stellar infra-red, et Re01 C96 excess MIR radius al. – – – classification in Rebull 2009, Choi mid (II 2001, R09 solar & infra-red), Rh01 – – Herbst units; and – Rodr´ıguez-Ledesmaet disced, 1996, al. Rhode columns III S99 2009, et 9, – al. – M11 10, Stassun 2001, – disc-less), et H02 and Morales-Calderónet and al. – al. 11: EW 2011); Herbst 1999, et columns (Ca al. 2002, F09 column 14: reasons,outside where the applicable, limits for imposed exclusion by from Siess the et final al. analysis (2000) (a models – (see not Section fully 3.1.3), convective c (see – Section lacking 2.1), required b photometry, – d star – lies j

80 3.1. Stellar data Class Refs Notes ) colour; column 8: adopted c V-I 86 — 15,21,47 a,g 812 — 15,21,47 a,g 661 II 30,11,46,41 a 662 — 15,21,47 a,g 816 — 15,21,47 a,g 736 — 15,21,47 a,g . P 1683 — 15,21,47 a,g 0789 — 15,21,47 a,g ...... 63 — — 21 a,f 4234 2 — — 15,21 a,f 27 1 25 — — 15,47 a,f 23 2 22 3 2121 2 — — 15,47 a,f 22 0 18 0 17 1 ...... ) (days) 0 0 0 0 0 0 0 0 0 0 0 0 ?

± ± ± ± ± ± ± ± ± ± ± ± R 36 16 68 17 11 85 78 69 80 81 55 45 ...... 1 1 1 1 4 3 2 2 2 1 1 1 0 5 3 8 5 0 5 5 ...... 4 0 1 6 5 2 6 7 7 8 4 1 3 6 3 9 ...... 3 3 3 3 0 1 1 1 1 2 2 2 +3 − +4 − +6 − +6 − +0 − +1 − +1 − +2 − +2 − +3 − +3 − +3 − age 0 0 0 5 9 1 1 9 7 9 7 6 . . . . t ...... -band magnitude and ( but not able to classify source as II or III). 0 3 3 4 5 6 7 8 c 14 13 13 15 ? I j 57 25 15 15 16 17 15 16 07 09 16 12 34 21 13 14 15 13 14 13 10 14 13 15 ...... ) (Myr) (R 0 0 0 0 0 0 0 0 0 0 0 0 ?

+0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − M 28 21 95 69 67 51 46 41 37 36 30 23 ...... ?

L L 506 1 205 1 082 1 943823 2 2 580 1 395 1 408 1 256 1 222 1 178571 1 1 ...... log c I 05 0 04 0 09 0 02 0 12 0 05 0 72 0 88 0 88 0 – ...... V ) colour; columns 6 and 7: ); columns 9, 10, and 11: stellar mass in solar units, age in Myr, and stellar radius in solar c 61 0 38 0 68 0 10 1 58 1 69 1 36 1 60 1 60 1 ......

B-V /L ? VI L 89 8 62 —84 8 — 0 94 9 3288 — 9 — 0 93 9 85 9 62 — — 0 61 8 65 9 73 9 – ...... 72 1 03 0 3438 1 0 72 0 65 0 26 0 56 0 15 0 62 0 22 0 33 0 ...... VB eff T (K) (M -band magnitude and ( 2 5210 9 2 5970 9 2 5740 9 2 5210 10 1 4760 10 2 4920 9 2 4920 12 2 50302 11 4920 10 2 4920 10 2 4920 10 2 5030 10 V ± ± ± ± ± ± ± ± ± ± ± ± 15 days, f – no reliable rotation period available, g – able to calculate > (1)HD 282624 G8 (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) SIMBAD SpT HD 282600 K2 HD 283782 K1 HD 285281 K1 HD 283641V1298 K0 Tau K1 HD 285957 K1 HD 281691 K1 HD 31281 G1 V1299 Tau G3 V1319 Tau G8 HD 284266 K0 Table 3.2: Stellarin data the for literature. allshown members The here of full to Taurus-Auriga illustrate table not itscolumns is content. identified 4 available Column as and in 1: binarylogarithmic 5: electronic SIMBAD bolometric or form identifier; luminosity, multiple on log( column for 2: the which adopted supplementary a spectral data type; spectral CD column type at 3: was the effective available back temperature; of this thesis. A sample is units; column 12:([1] adopted – rotation Cohen periodBouvier & in et Kuhi days; al. 1979, column 1993,& [2] 13: [7] Hartmann – SED – 1995, Bouvier classification Edwards [12]– et et (II – Bouvier al. al. – Fernandez et 1986, 1993, & disced, al.Wichmann [3] [8] Eiroa III et 1997, – – 1996, – al. [17] Grankin Herbst [13] disc-less); 2000, 1993,– – & – column [22] [9] Grankin Briceñoet Grankin Koret 14: al. – – 1997, 1996, 1988, Herbst 2002, references Mora[31] [18] [14] [4] et et [27] – – – – al. al. Broeg – Briceñoet Osterloh Beckwith 1994, al. 2001, et Vieira– et et [10] 1998, al. [23] al. et Grosso al. – [19] 2006, – 1996, al. et 1990, Strom – [32] Roberge [15][41] 2003, al. & [5] Luhman – – et – Strom [28] 2007, – Kundurthy & Wichmann al. Luhman 1994, – [37] Bouvier et Rieke et 2001, et– [11] al. Luhman 1990, – 1998, al. [24] al. – Cody 2006, [6] 2004, Chapillon [20] 1996, – Kenyon 2010, [33] et – [29] – [16] et Stassun [42]convective – al. Briceñoet – al. (see et – al. Padgett 2013, Andrews Section 2008, al. Rebull et 1999, 2.1), [47] & 2001, [38] al. etP [21] b – Williams [25] 2006, – al. – – [34] – Grankin 2005, 2010, Grankin star – White [30] 2013); [43] et lies Scholz & – – column al. outside et Ghez Andrews Massarotti the 15: 2008, al. 2001, et limits et [39] 2006, reasons, [26] al. imposed al. [35] – where 2011, by 2005, – Luhman Siess [44] applicable, Xing et et – for et al. al. Furlan al. exclusion (2000) 2009, et 2006, from models, al. [40] [36] the c 2011, – – final [45] Espaillat lacking analysis – et optical Xiao (a al. photometry, et – d 2010, al. – not 2012, fully [46]

81 Chapter 3. Accretion discs as regulators of stellar angular momentum evolution

In general, good agreement was found between published spectral types from diﬀerent studies for the remaining ONC members and full Taurus-Auriga sample. Where multiple spectral types had been published for the same star in multiple studies, the spectral type found by the most recent study was adopted. Where a range of spectral types was quoted for a particular star by a single study, the median spectral type of the published range was adopted. In this case, the error on the spectral type was adjusted to account for the increased range of possible values. For instance, a star with a published spectral type of K2-K7 was assigned a spectral type of K4.5 and an error of ±2 spectral subtypes.

3.1.2 Bolometric luminosities

Bolometric luminosities were determined by applying a bolometric correction to a single apparent magnitude, corrected for extinction and distance modulus. Distance moduli were assumed to be constant for all stars within each star forming region with distances of 414±7 pc (Menten et al. 2007) and 140±20 pc (Elias 1978; Loinard et al. 2007; Torres et al. 2009, 2012) adopted for ONC and Taurus-Auriga members, respectively (see Section 2.4.2).

Cousins Ic-band photometry was used to calculate bolometric luminosities for the bulk of the sample. This choice of waveband should minimise contamination from accretion- induced UV excess emission and disc-related IR excess emission (see Section 1.3). Using equation (2.20), the bolometric luminosity is given by

L? 2 log = [Mbol, − (Ic − AIc ) + DM − BCIc (Teﬀ )] . (3.2) L 5 where Mbol, = 4.755 mag is the bolometric magnitude of the Sun (Mamajek 2012). Spec- tral type-dependent intrinsic colours, (V − Ic)int, and V -band bolometric corrections,

BCV(Teﬀ ), from Pecaut & Mamajek (2013) were used to derive the individual Ic-band bolometric corrections,

BCIc (Teﬀ ) = BCV(Teﬀ ) + (V − Ic)int. (3.3)

Individual extinction values, AIc , were determined using equation (2.21) with Ic- and V -band magnitudes,

AIc = 1.56[(V − Ic)obs − (V − Ic)int]. (3.4)

82 3.1. Stellar data 1.5 ● ● ● ● ●● ● ●●●

1 ● ● ● ●● ● ● ● ● ● ● ● ● ● ●●

0.5 ● ●●● ●●● ●●●

) [ H97 ] ● ●●● ● n ●●

u ●● 0 ● ●●●●●●● ● s ● ●●●● ●●●● ●

L ●●●● ● ●●●●●●● ● ●● ●●●●●●●● ●●●●●●●●● / ● ●●●●●●●●● r ● ●●●●● ● ● ● ●●●●●●● a ● ●● t ●●●● ● ● ●●●●●●●●●● ● s ●● ●●●●●●●●● ● −0.5 ● ●●●●●●●● L ●●● ●● ●●●●●●●●● ● ● ● ●●●●● ● ●●●●●●●●●● ●●●●● ●● ●●●●●● ● ● ●●● ● ●●●● log( ●●●●● −1 ●●●● ●●●● ● ●●●● ●●●●● ● ●● ●●● ●●● ● ●● ●

−1.5 ●

−1.5 −1 −0.5 0 0.5 1 1.5

log( Lstar / Lsun ) [ D14 ]

Figure 3.1: Comparison between the bolometric luminosities from Davies et al. (2014a) and Hillenbrand (1997) for members of the ONC. The Hillenbrand (1997) values have been adjusted to account for the diﬀerent distance modulus and solar bolometric luminosity used in Davies et al. (2014a). The dashed line shows a one- to-one ﬁt to the data. In general, good agreement is found with the main source of spread resulting from updated spectral typing. Here, the value of 1.56 is from the extinction law of Rieke & Lebofsky (1985), transformed from the Johnsons to the Cousins photometric system (Hillenbrand 1997). If the calculated

AIc was negative, AIc was set equal to zero. In these cases, the observed V − Ic colour was too blue for the assigned spectral type. Consequently, the bolometric luminosities calculated for these stars are lower limits and were considered as such in the analysis.

The necessary V - and Ic -band photometry for the ONC members were retrieved from Hillenbrand (1997). Fig. 3.1 compares the Davies et al. (2014a) luminosities to those of Hillenbrand (1997), adjusted to account for the diﬀerent distance modulus and solar bolometric magnitude used in Davies et al. (2014a). For the majority of stars, the luminosities agree well with one another with the main diﬀerence between the old and new values associated with the updated spectral typing.

The individual references for the Taurus-Auriga photometry are detailed in Table 3.2. The size of the resulting Taurus-Auriga sample was much smaller than the ONC due mainly to the comparatively smaller stellar population: ∼ 348 (Luhman et al. 2010) compared

83 Chapter 3. Accretion discs as regulators of stellar angular momentum evolution to > 1000 (Da Rio et al. 2010), but also in part due to a dearth of contemporaneously observed Ic- and V -band photometry in the region. In an attempt to counter this, the Taurus-Auriga sample was supplemented with bolometric luminosities calculated from V - band photometry using equation (2.20). For these stars, the V -band extinction, AV, were determined using equation (2.21) with B- and V -band photometry,

AV = 3.09[(B − V )obs − (B − V )int]. (3.5)

Here, the value of 3.09 is from the Rieke & Lebofsky (1985) extinction law.

As noted above, the choice of wavebands should ensure that contamination by accretion and disc emission is at minimal levels across the range of stellar masses probed. However, the luminosities of the most actively accreting stars, as well as those hosting highly inclined discs may be underestimated due to complications associated with estimating the levels of extinction towards these objects (Hillenbrand 1997; Kraus & Hillenbrand 2009; Da Rio et al. 2010). In the case of the former, the veiling associated with accretion (see Section 1.3.3) leads to the star appearing bluer, reducing the estimated extinction correction and, as a result, causing the star to appear less luminous (c.f. Hartmann 2001; Manara et al. 2013). In an eﬀort to account for these eﬀects, a conservative uncertainty of ±0.1 dex in log(L?/L ) was assumed for all stars.

3.1.3 Stellar radii, masses and ages

Under the assumption that the range of luminosities calculated in Section 3.1.2 indicated a true spread in stellar radii and ages (see Section 2.4.4), the radii, masses and ages of the stars, as well as their respective errors, were calculated from eﬀective temperatures and bolometric luminosities using the methods outlined in Section 2.4. Hertzsprung-Russell diagrams (HRDs) for the ONC and Taurus-Auriga samples, overlaid with Siess et al. (2000) PMS evolutionary models, are displayed in Fig. 3.2. Stars that lay outside of the boundaries imposed by the Siess et al. (2000) mass tracks and isochrones could not be assigned a stellar mass or age. Additionally, an upper mass limit of 3.0 M and an upper age limit of 60 Myr were employed. Stars more massive than this are not T-Tauri stars and those that appear older than this are either not members of the ONC or Taurus-Auriga or, as discussed in Section 2.4.4, likely to be inaccurately positioned in the HRD due to

84 3.1. Stellar data underestimated bolometric luminosities caused by e.g. high levels of extinction resulting from a near edge-on disc.

As this study is only concerned with fully convective stars, those which have formed radiative cores needed to be removed from the sample. The age at which this occurs depends on the stellar mass (see Section 1.2.1). Stars below ∼ 0.35 M remain fully convective throughout their formation and during their MS evolution (Limber 1958; Chabrier & Baraﬀe 1997) whilst more massive stars form radiative cores during their contraction. Based on Siess et al. (2000) evolutionary models, Gregory et al. (2012) derived the mass- dependent age at which a star of mass greater than 0.35 M develops a radiative core, given by equation (1.26): 2.364 M tcore ≈ 1.494 Myr. M? This demarcation is highlighted by the red line in Fig. 3.2: stars to the right of this line are fully convective whereas those that lie to the left have formed radiative cores. Stars with isochronal ages, tage, greater than their individual tcore limit were removed from the analysis.

3.1.4 Rotation periods

Previously published rotation periods were gathered from the literature. Individual references are listed in Tables 3.1 and 3.2 for the ONC and Taurus-Auriga samples, respectively. Where errors for the rotation period had not been reported by the original authors, a conservative estimate of 0.01 days was assumed. Every eﬀort was made to remove rotation periods aﬀected by harmonics and beat phenomena (see Section 2.3.2) and stars with rotation periods longer than 15 days were removed from further analysis as it is unlikely that these trace photospheric rotation. Instead, these are more likely to be caused by occulta- tion of the stellar surface by disc material exterior to the co-rotation radius (Artemenko et al. 2010; Cody et al. 2014).

85 Chapter 3. Accretion discs as regulators of stellar angular momentum evolution

0 M . ● ● ● ● 3000 ● 5, and 3 . ● ● ● ● 0, 2 . ● ● ● ● ● ● ● ● ● 5, 2 . ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ] 2, 1 . K ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● [ f 4000 f 0, 1 ● ● ● ● ● ● ● . e T ● ● ● 5, 2, 5, 10, and 60 Myr. The solid . 8, 1 . ● ● ● ● ● 2, 0 . ● ● ● ● ● 6, 0 . ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 05, 0 ● 5, 0 5000 . . ● ● ● ● 4, 0 . 01, 0 . ● 3, 0 stars and the solid red lines marks the age at which ● . 6000

2, 0

10 1 0.1 0M

n u s L / L 1, 0 . − 7 . ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 3000 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ] ● ● ● K ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● [ ● ● f 4000 f ● ● ● ● ● ● ● ● ● ● ● e ● ● T ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 5000 ● ● ●

6000

10 1 0.1

n u s L / L black line marks the position of the zero-age main sequence for 0 the transition from aremoved fully from convective further to analysis.the a lower partially Individual left radiative errors of interior for each occurs each plot. [equation star (1.26)]. were Stars considered on with the an left average of error this bar red included line for were reference in stars. Isochrones (dotted black lines) are shown (from top to bottom) for ages 0 Figure 3.2: HRDmass constructed from tracks Siess (dashed et black al. lines) (2000) are PMS shown evolutionary (from models for right the to ONC left) (left) for and 0 Taurus-Auriga (right). The

86 3.1. Stellar data

It was imperative to ensure that both the ONC and Taurus-Auriga samples were free from contamination by non-members. As outlined in Section 1.4.1, inclusion of non- members in previous studies of angular momentum evolution had led to a blurring of the distribution of fast and slow rotators. Known members of the Orion ﬂanking ﬁelds (Rebull 2001) and other neighbouring regions such as L1641W, and NGC 1981 (Alves & Bouy 2012; Bouy et al. 2014) were removed from the sample. Only stars that lay within the “traditional” region of the ONC (84.1◦ ≤ RA ≤ 83.0◦ and −5.0◦ ≤ dec ≤ −5.7◦; Cieza & Baliber 2007) were retained in Table 3.1. Even with this cut applied to the right ascension (RA) and declination (dec), the ONC sample could still be contaminated by members of the somewhat overlapping clusters L1641N and NGC 1980 (Alves & Bouy 2012; Bouy et al. 2014). With this in mind, all objects listed in Hillenbrand (1997) as having a membership probability < 98% were removed from the sample and any additional non-members were removed by cross-referencing with the studies of Fang et al. (2013) and Pillitteri et al. (2013).

3.1.5 Rotational ﬂattening

Under the influence of a sufficiently high rotation rate, the shape of the star can deviate from a perfect sphere. The sample was checked for the presence of rotational flattening through comparison of the surface radius of gyration, k2, to that of a uniform density spherical shell (k2 = 2/3). Following Herbst & Mundt (2005), the radius of gyration of a rotationally flattened shell is given by

4 16 8 16 52 2 −1 k2 = a4 + a3b + a2b2 + ab3 + b4 × 2a2 + b2 , (3.6) 3 15 7 105 1155 5 where, for fully convective stars (polytropic index, n = 3/2; Section 2.1),

a = 1.74225ν + 1 (3.7) and b = 3.86184ν. (3.8)

Here, 2π ν = 2 (3.9) GP ρc

87 Chapter 3. Accretion discs as regulators of stellar angular momentum evolution 1.5 1.4 1.3 0 k / k 1.2 1.1 1

0 2 4 6 8 10 12 14 P [ days ]

Figure 3.3: Relation between stellar rotation period, P, and the radius of gyration, k, 1/2 normalised to that of a uniform density spherical shell, k0= (2/3) for fully convective members of the ONC and Taurus-Auriga with P < 15 days. where G is the gravitational constant and ρc is the central density of the star (Chan- drasekhar 1935).

Individual radii of gyration were calculated for each star using central densities from Siess et al. (2000) PMS isochrones. As shown in Fig. 3.3, using equations (3.6) to (3.9), the majority of stars in the sample exhibit k2 close to 2/3. However, a few of the fastest rotators appear to be distorted from a spherical shape. If an object of mass, M?, is substantially rotationally ﬂattened, the centripetal acceleration,

4π2R a = eq , (3.10) cent P 2 of the equatorial regions will exceed the gravitational acceleration,

GM? agrav = 2 , (3.11) Req casting serious doubts on the reliability of the adopted rotation period. To check for this, the shape of each star was investigated.

88 3.1. Stellar data

Following Chandrasekhar (1935), the equatorial radius of the rotating sphere is related to the radius of a non-rotating star, R?,0, according to

Req = a − bP2 (θ = 0) (3.12) R?,0 where a and b are given by equations (3.7) and (3.8), θ is the usual polar angle, and

P2(θ = 0) is the second order Legendre polynomial. This can then be used to deﬁne a critical rotation period, 2πR3/2 P = eq , (3.13) crit 1/2 (GM?) where, if P < Pcrit, the object will break apart.

In turn, this can be used to deﬁne a critical speciﬁc stellar angular momentum, jcrit, by inserting equation (3.13) into equation (3.1). Then, any star with

j 2πR3/2 ? = eq > 1 (3.14) 1/2 jcrit (GM?) P has an unphysical rotation period and, accordingly, was excluded from further analysis.

In total, ten stars were removed from the sample based on this criterion. All of these were at the low mass end of the sample, having spectral types of M3.5 to M5.5. Nine of these had rotation periods measured at a fraction of a day. It is possible that these rotation periods are aﬀected by beat phenomena but were not picked up in the analysis of Section 3.1.4 due to their being only one recorded rotation period for each star. For the other star, 2MASS J05351587-0520404, the rotation period is measured at a more reasonable 1.18 days but its luminosity is very high for its spectral type (M5). Thus, it appears much more inﬂated than other stars in the sample with similar spectral types. It is possible that the luminosity is overestimated for this star, perhaps due to the presence of an unresolved binary component (see Section 2.4.4). These ten stars are noted in Tables 3.1 and 3.2.

3.1.6 Disc diagnostics

Observations of an accretion disc-rotation relation are dependent on the use of a reliable method to identify accretion discs, as discussed in Section 1.4.1. For most of the stars in the Taurus-Auriga sample, the results of detailed spectral energy distribution (SED)

89 Chapter 3. Accretion discs as regulators of stellar angular momentum evolution modelling were available in the literature (Kenyon & Hartmann 1995; Andrews & Williams 2005; Luhman et al. 2010; Rebull et al. 2010). These were used to identify the presence (or absence) of an infra-red (IR) excess and so deﬁne the source as a Class II (or Class III) object (see Section 1.3.1). In addition, resolved submillimetre and millimetre observations of directly imaged discs around individual stars in Taurus-Auriga (Kitamura et al. 2002; Rodmann et al. 2006; Andrews & Williams 2007b; Guilloteau et al. 2011) were used to supplement the Taurus-Auriga Class II sample.

For the ONC sample, the same detailed SED modelling was not available. Instead, measurements of Spitzer IRAC fluxes were used to determine the evolutionary state of the sample. 2.2, 3.6, 4.5, 5.8, and 8.0 µm fluxes were gathered from Rebull et al. (2006), Cieza & Baliber (2007), Prisinzano et al. (2008), and Morales-Calderónet al. (2011). Class III young stellar objects (YSOs) were identified according to the criteria outlined in Prisinzano et al. (2008):

1. [2.2] − [3.6] < 0.5;

2. [3.6] − [4.5] < 0.2;

3. [4.5] − [5.8] < 0.2;

4. [5.8] − [8.0] < 0.2, or, alternatively, if no emission was detected longward of 2.2 µm. YSOs were identified as Class II if ([3.6] − [8.0]) > 1.0 (Rebull et al. 2006) or if 0.2 < ([3.6] − [4.5]) < 0.7 and 0.6 < ([5.8] − [8.0]) < 1.0 (Hartmann et al. 2005; see Section 1.3.1). As the Spitzer IRAC fluxes came from multiple sources, identification required agreement between the four studies from which they were retrieved. Where identifications did not agree, the sources remained unclassified. No attempt was made to classify protostellar objects and these objects are included in Tables 3.1 and 3.2 amongst the unclassified sources.

3.1.7 Multiplicity

To minimise the eﬀects of multiplicity on rotation period and luminosity calculations (see Sections 2.3.2 and 2.4.4, respectively), the ONC and Taurus-Auriga samples were cross-checked against previous studies of multiple stellar systems (Leinert et al. 1993; Nordstrom & Johansen 1994; Mathieu 1994; Osterloh & Beckwith 1995; Duchˆene1999;

90 3.2. Results and discussion

Oh et al. 2006; Reipurth et al. 2007; Kraus & Hillenbrand 2007; F˝urészet al. 2008; Tobin et al. 2009; Luhman et al. 2009, 2010; Rebull et al. 2010; Cieza et al. 2012; Daemgen et al. 2012; Harris et al. 2012; Andrews et al. 2013; Correia et al. 2013). All members previously identified as binary or multiple or whose spectroscopy suggested the existence of an unresolved companion (Morales-Calderónet al. 2011; Hillenbrand et al. 2013) were removed from the sample and are not present in Tables 3.1 or 3.2.

3.2 Results and discussion

A portion of the compiled datasets for the ONC and Taurus-Auriga members are presented in Tables 3.1 and 3.2, respectively1. These include all members (Section 3.1.4) for which a spectral type was available that had not previously been identiﬁed as a binary or multiple system (Section 3.1.7). As outlined in Section 3.1.4, several cuts were applied to the data: all stars with rotation periods longer than 15 days, isochronal ages greater than their individual tcore, or j?/jcrit > 1 are included in the tables but were removed from the analysis.

Equation (3.1) was used to calculate the speciﬁc stellar angular momenta, j?, for all stars for which a measured rotation period, stellar radius and an estimate of its age were available. This resulted in 352 and 32 fully convective stars within the ONC and Taurus-

Auriga, respectively, for which j? was calculated. Of these, it was possible to classify 226 ONC and 24 Taurus-Auriga members as either disc-hosting (Class II) or disc-less (Class III).

The sample was also split into two, mass-segregated groups in an attempt to avoid any bias introduced by the mass-rotation relation observed in previous studies of PMS rotational evolution (Herbst et al. 2000b; Cieza & Baliber 2007; Section 1.4.1). Due to the apparent spread in age across the samples (Fig. 3.2), the mass cut is based on spectral type rather than directly on stellar mass. The “high mass” sample contains stars of spectral types M2 or earlier and the “low mass” sample contains those later than M2. This demarcation is based on the work of Cieza & Baliber (2007) who varied the location of their spectral type cut and investigated the eﬀect this had on the nature of the period distributions in the ONC. The bimodality of their “high mass” sample was severely

1The full form of these tables are available in electronic form in the supplementary data CD at the back of this thesis.

91 Chapter 3. Accretion discs as regulators of stellar angular momentum evolution affected by even a small contamination by lower mass stars. Also, their results were consistent with a sudden change in stellar magnetic field strength or structure between spectral types M2 and M3. Over the range of ages investigated here, this spectral type corresponds to ∼ 0.35 M , the mass below which stars remain fully convective during their MS lifetimes (Section 1.2.1). The final samples considered here consist of 91 ONC and 20 Taurus-Auriga stars of spectral types K0 to M2 and 135 ONC and 4 Taurus-Auriga stars of spectral type later than M2.

Before considering how j? evolves with age for the various samples in Section 3.2.2, it is important to ﬁrst consider the expected temporal evolution based on theoretical considerations.

3.2.1 Evolution of speciﬁc angular momentum during pre-main sequence contraction: theory

2 It is clear from equation (3.1) that, as j? ∝ R?/P , the evolution of speciﬁc angular momentum during the PMS depends on the stellar contraction rate and how the stellar rotation period evolves with time. These quantities are now each considered in turn.

For a fully convective polytrope, descending a Hayashi track in the HRD (Teﬀ ≈ const.), it is straightforward to show that2 the contraction rate is given by

−1/3 R? ∝ tage . (3.15)

Fig. 3.4 shows the rate of stellar contraction in the ONC and Taurus-Auriga samples overlaid on theoretical tracks from Siess et al. (2000) PMS evolutionary models.

The presence of a relation between R? and tage is tested for using the numerical recipe fitexy routine in idl. This routine produces a minimum-χ2 ﬁt to the data using the linear relation,

log(R?) = −β1 log(tage) + γ1, (3.16) while accounting for the heteroscedastic errors on both variables. The routine can account for symmetric errors on R? and tage but, due to the method of their estimation, the errors in stellar age are not symmetric. Instead, symmetric errors in log(L?/L ) will result in

2A derivation of equation (3.15) is given in Appendix A.

92 3.2. Results and discussion

Table 3.3: Magnitudes of the gradients of the numerical recipe fitexy ﬁtting using equation (3.16) for (i) the ONC sample alone, and (ii) the combined ONC and Taurus- Auriga samples. Column 1: sample; columns 2 and 3: value of β1 for the high mass (HM) and low mass (LM) samples, respectively.

Sample β1 HM LM (1) (2) (3) ONC Class II 0.53 ± 0.09 0.42 ± 0.07 ONC Class III 0.53 ± 0.08 0.56 ± 0.14 ONC & Tau Class II 0.53 ± 0.08 0.42 ± 0.07 ONC & Tau Class III 0.53 ± 0.07 0.56 ± 0.14 upper bounded errors on tage that are larger than the lower bounded errors. This is due to the 1/R? contraction timescale during the fully convective stage of star formation [see equation (1.18) of Section 1.2] which causes the isochrones to become more “bunched up” at older ages (see Fig. 3.2). Here, the maximum of the lower and upper bounded error on tage is used in the ﬁtting procedure.

The resulting values of β1 are presented in Table 3.3. Due to the comparatively small size of the Taurus-Auriga samples, the ONC sample is considered alone and compared to the combined ONC-and-Taurus-Auriga (ONC & Tau) sample. The result of combining the Taurus-Auriga and ONC samples does not alter the outcome of the fits. The χ2 probability, q = 1 for each fit, indicating that the linear model is a good fit to the data. Assuming that the observed radius and age spreads in the ONC and Taurus-Auriga samples are real (see Section 2.4.4), the rate of stellar contraction is steeper than, but in rough agreement with, that expected from purely theoretical considerations [equation (3.15)].

The tightness of the correlation between R? and tage is surprising when compared to the average errors on R? and tage. This arises due to a combination of two factors:

• requiring the PMS stars to be fully convective imposes an upper limit to the age at any radius. As a result, stars cannot lie to the right of the solid red line in Fig. 3.4;

• following deuterium burning (illustrated in Fig. 3.4 by near-horizontal tracks 3) the

region of R?-tage space occupied by fully convective PMS stars of masses 0.2 to

∼ 1 M is narrow. The majority of stars in the sample lie in a region of the HRD occupied by stars that have ceased deuterium burning and thus reside in this narrow

region of R?-tage space.

3During this phase, support against gravitational contraction is provided by the release of energy from the fusion of deuterium which leads to evolution at near-constant radius.

93 Chapter 3. Accretion discs as regulators of stellar angular momentum evolution 7 10 later than M2 ● 6 ● 10 Age [ yrs ] 5 ● 10 ONC Class II ONC Class III Class II Tau−Aur Class III Tau−Aur (dashed black lines). An average error bar is included

10 1

n u s r a t s R / R 5, and 3 M . 5, 2, 2 . 7 ﬁts to the data are presented in Table 3.3. These are slightly steeper than, 10 2, 1 . K0 − M2 0, 1 fitexy . ● 75, 1 . 6 ● 10 5, 0 . 4, 0 Age [ yrs ] . 3, 0 . , 0 2 5 . ● 10 1, 0 .

ONC Class II ONC Class III Class II Tau−Aur Class III Tau−Aur

10 1

n u s r a t s R / R Figure 3.4: ComparisonClass of III (crosses) the starsmodels contraction in for rate the masses of ONC oftowards high (blue) 0 the and mass bottom Taurus-Auriga (left (red)expected. left to panel) The theoretical of and gradients tracks eachbut low of from plot. in mass the Siess rough numerical et (right A recipeby al. agreement panel) dotted (2000) with, consistent Class grey PMS those contraction II evolutionary lines; expected rate (open see from is equation squares) purely (3.15)]. observed and theoretical between considerations both of Class contraction II on a and Hayashi Class track III [illustrated objects, as

94 3.2. Results and discussion 10 ] 1 n u s L [ L 0.1

4000 3000

Teff [ K ]

Figure 3.5: HRD for the sample of fully convective ONC sample for which disc classi- ﬁcations were available. Isochrones (dotted black lines) for ages 0.01, 0.05, 0.2, 0.5, 1, 2, 5, 10, and 60 Myr and mass tracks (dashed black lines) for masses 0.1 and 3.0 M from Siess et al. (2000) PMS evolutionary models are shown. The fully convective limit [solid red line; see equation (1.26)] and lines of equal radius (for 1, 2, 3, 4, and 5 R ; blue dashed lines) are also illustrated. For fully convective PMS stars which have ceased deuterium burning, the lines of equal radius run almost parallel to the isochrones, giving rise to the tight correlation between R? and tage seen in Fig. 3.4. This latter factor is illustrated more clearly in Fig. 3.5. Here, the sample of fully convective PMS stars from the ONC with disc classiﬁcations have been overlaid onto Siess et al. (2000) PMS evolutionary model isochrones. Also shown are the lines of constant radius (at 1, 2, 3, 4, and 5 R ). Most of the sample lies in a region of the HRD where the isochrones and lines of constant radius are nearly parallel to one another. Thus, the region of the HRD populated by fully convective PMS stars in which deuterium burning has ceased cover a narrow range of ages over a particular range of radii.

Taking β1 = 1/3 [equation (3.15)], the speciﬁc stellar angular momentum would be −2/3 −1 expected to evolve as j? ∝ tage P . If, averaged over a timescale of a few Myr, the rotation period varies according to a simple power law of the form

n P ∝ tage, (3.17)

95 Chapter 3. Accretion discs as regulators of stellar angular momentum evolution then −2/3−n j? ∝ tage . (3.18)

There then exist three scenarios: a star evolving at a ﬁxed rotation rate would have n = 0; a star that is spinning down would have n > 0; and a star that spins up would have n < 0.

One would expect the evolution of the rotation period (and therefore j?) to diﬀer between Class II and Class III stars as the former are driven by the astrophysics of the star-disc interaction. In PMS rotational models, Class II stars are commonly assumed to be locked to their discs such that accretion and contraction occur without the star spinning up (e.g. Gallet & Bouvier 2013). In this case, the surface rotation rate remains ﬁxed at the disc truncation radius rotation rate, which is assumed to be Keplerian. As such, a Class II star −2/3 is expected to have n = 0 and j? would be expected to reduce with age as j? ∝ tage .

On the other hand, Class III stars, which have lost their accretion discs, would conserve angular momentum as they contracted (neglecting the likely small loss of angular momentum in the stellar wind). If j? = const., one would expect n = −2/3 such that −2/3 Class III stars spin up according to P ∝ tage as they contract. However, as is discussed in the following subsection, this is not observed. Instead, assuming the luminosity spreads inferred from the ONC and Taurus-Auriga HRDs indicate real age spreads in these regions

−β2 (see Section 2.4.4), the results suggest that j? ∝ tage with β2 ≈ 2–2.5 for both the Class II and Class III samples.

3.2.2 Evolution of speciﬁc angular momentum during pre-main sequence contraction: observations

Fig. 3.6 displays the evolution of j? for the high mass and low mass ONC and Taurus- Auriga samples. A Spearman rank correlation test was used to test for the presence of correlation between j? and tage. The results of this are presented in Table 3.4. As before, the ONC sample is considered on its own and compared to the combined ONC & Tau sample. Combining the samples does not alter the outcome of the correlation tests signiﬁcantly, suggesting a consistency between the results in the two regions.

96 3.2. Results and discussion 7 10 later than M2 6 10 Age [ yrs ] 5 10

ONC Class II ONC Class III Class II Tau−Aur Class III Tau−Aur

10 10 10

14 13 12

r a t s ] s m [ j

1 2 − 7 10 K0 − M2 with age is observed in both of the Class II and Class III samples. The results of a ? j 6 10 Age [ yrs ] 5 for the high mass (left panel) and low mass (right panel) Class II and Class III samples in the ONC 10 ? j

ONC Class II ONC Class III Class II Tau−Aur Class III Tau−Aur

10 10 10

14 13 12

r a t s ] s m [ j

1 2 − Spearman rank correlationlevels test (Table indicate 3.4). that the the null hypothesis of zero correlation can be rejected at statistically signiﬁcant Figure 3.6: Evolutionand of Taurus-Auriga. Theright coloured corner symbols of have both the plots. same A meaning reduction as of in Fig. 3.4. An average error bar is included in the upper

97 Chapter 3. Accretion discs as regulators of stellar angular momentum evolution

Table 3.4: Results of Spearman rank correlation tests and fitexy fits to equation (3.19) for (i) the ONC sample alone, and (ii) the combined ONC and Taurus-Auriga samples. Column 1: sample; columns 2 and 3: Spearman rank correlation coefficient, ρ, for the high mass (HM) and low mass (LM) samples; columns 4 and 5: corresponding two- sided probability of finding this value of ρ by chance; columns 6 and 7: magnitudes of the gradients of the fits to equation (3.19) in the cases where the correlation is statistically significant.

Sample Spearman ρ Spearman p–value β2 HM LM HM LM HM LM (1) (2) (3) (4) (5) (6) (7) ONC II −0.54 −0.53 < 10−4 < 10−4 2.53 ± 0.64 1.84 ± 0.50 ONC III −0.76 −0.67 10−4 10−4 1.98 ± 0.37 3.72 ± 1.53 ONC & Tau II −0.53 −0.53 < 10−4 < 10−4 2.34 ± 0.53 1.86 ± 0.50 ONC & Tau III −0.72 −0.67 10−4 10−4 2.08 ± 0.40 3.95 ± 1.71 ONC accretors −0.50 −0.68 1.3 × 10−2 1.8 × 10−4 — 1.67 ± 1.05 ONC non-accretors −0.59 −0.47 < 10−4 3.2 × 10−3 2.24 ± 0.42 4.51 ± 2.97

−β2 Considering the evolution of specific stellar angular momentum, j? ∝ tage , in its logarithmic form, the numerical recipe fitexy is used to find a fit to the linear relation

log(j?) = −β2 log(tage) + γ2. (3.19)

As in Section 3.2.1, the maximum of the upper and lower bounded errors on tage were used as the fitexy routine can only account for symmetric heteroscedastic errors. The resulting values of β2 for the high and low mass Class II and Class III samples are presented in

Table 3.4. Considered at face value, these results appear to show that j? decreases with tage for both the Class II and Class III samples. Furthermore, the stellar angular momentum removal rates are consistent between the Class II and Class III high and low mass samples with β2 ≈ 2–2.5. Initially, the rate of reduction of j? with tage is surprising, especially in the case of the Class III stars which should be conserving angular momentum as they contract (see Section 3.2.1). The reliability of these results was further tested using alternative indicators of the presence of accretion to identify the accretion disc-hosting and disc-less stars.

The equivalent widths (EWs) of emission lines such as Ca ii 8542 A˚ [one of the Ca ii IR triplet lines, denoted EW (Ca ii)] and Hα [denoted EW (Hα)] are used as indicators of active accretion (see Section 1.3.3). The EW (Ca ii) is adopted here due to its minimal dependence on spectral type (Hillenbrand et al. 1998; White & Basri 2003). Measures of EW (Ca ii) were retrieved from Hillenbrand et al. (1998). Stars with Ca ii seen in emission with EW (Ca ii) < −1 A˚ were identiﬁed as accretors, as in Hillenbrand et al.

98 3.2. Results and discussion

(1998), and those with Ca ii seen in absorption with EW (Ca ii) > 1 A˚ were identiﬁed as non-accretors, based on Flaccomio et al. (2003). As shown in Fig. 3.7, the reduction in j? with tage observed in Fig. 3.6 is recovered when using EW (Ca ii) to distinguish between accretion disc-hosting and non-accreting, disc-hosting stars.

The results of the fitexy ﬁt to the accreting and non-accreting high and low mass samples are shown in Table 3.4. The results for the low mass accreting and non-accreting samples and the high mass non-accreting sample are consistent with those found using IR excess measurements to identify the presence of accretion discs. However, a statistically signiﬁcant correlation between j? and tage is not recovered for the high mass accretors. This is due, in part, to the smaller number of stars in the accreting sample as compared to the Class II sample.

The observed reduction in j? with tage for the Class III sample is not an artifact of the adopted method of disc classiﬁcation. Instead, as is argued below, it is likely that individual Class III stars are evolving with j? ≈ const. and that the trend observed in Figs. 3.6 and 3.7, apparent when considering the Class III members of a star forming region together, can be naturally explained by Class II stars rapidly losing their discs at a variety of ages.

3.2.3 Focusing on Class III stars

Considerable overlap is observed between the ages of Class II and Class III stars in Figs. 3.4, 3.6, and 3.8, and between accreting and non-accreting stars in Fig. 3.7. Double-sided Kolmogorov-Smirnov (KS) tests were used to assess the consistency between the ages of the Class II and Class III samples where a probability close to one would indicate that the samples were consistent with being drawn from the same parent population. Probabilities of 0.17, 0.72, and 0.38 were found for the high mass ONC sample, the low mass ONC sample, and the high mass Taurus-Auriga sample, respectively. This result is consistent with previous studies which found that PMS stars lose their discs at a range of ages: analysis of the disc fraction in PMS clusters of various ages revealed a range of inner disc lifetimes between 1–10 Myrs (c.f. Hillenbrand 2008).

99 Chapter 3. Accretion discs as regulators of stellar angular momentum evolution is not recovered in 7 10 age ). The average error is t ii and later than M2 ? j 6 10 Age [ yrs ] 5 10 seen in Fig. 3.6 is recovered for the low mass samples ONC accretors ONC non−accretors

age

10 10 10

14 13 12

r a t s

] s m [

j with 1 2 − statistics are presented in Table 3.4. ? j fitexy 7 10 K0 − M2 6 10 Age [ yrs ] 5 10

ONC accretors ONC non−accretors

10 10 10

14 13 12

r a t s ] s m [ j

1 2 − the case ofclassified the as high disc-hosting. mass accretors. The resulting This Spearman is rank due, and in part, to the smaller number of stars identified as accretors compared to those Figure 3.7: Asincluded Fig. in 3.6 the but upperand for right the corner the high of ONC each mass sources plot. identified non-accreting The as sample. reduction accreting in and However, non-accreting a using statistically EW significant (Ca correlation between

100 3.2. Results and discussion 7 10 later than M2 6 10 Age [ yrs ] 5 10

ONC Class II ONC Class III Class II Tau−Aur Class III Tau−Aur

10 1 0.5 5

[ days ] days [ P 7 10 K0 − M2 for the high mass (left panel) and low mass (right panel) Class II and Class III stars in age 6 t 10 and Age [ yrs ] P 5 10 ONC Class II ONC Class III Class II Tau−Aur Class III Tau−Aur is found using a Spearman rank correlation test (Table 3.5).

age

10 1 0.5 5

[ days ] days [ P and P Figure 3.8: Relationship between the ONC and Taurus-Auriga.between The coloured symbols have the same meaning as in Fig. 3.4. No signiﬁcant evidence for a correlation

101 Chapter 3. Accretion discs as regulators of stellar angular momentum evolution

Table 3.5: Results of Spearman rank correlation tests performed on the data from Fig. 3.8 for (i) the ONC sample alone, and (ii) the combined ONC and Taurus- Auriga samples. Column 1: sample; columns 2 and 3: Spearman rank correlation coeﬃcient, ρ, for the high mass (HM) and low mass (LM) samples; columns 4 and 5: corresponding two-sided probability of ﬁnding this value of ρ by chance.

Sample Spearman ρ Spearman p-value HM LM HM LM (1) (2) (3) (4) (5) ONC Class II −0.23 −0.27 0.11 0.03 ONC Class III −0.07 0.06 0.67 0.65 ONC & Tau Class II −0.30 −0.26 0.02 0.03 ONC & Tau Class III −0.10 0.05 0.51 0.66 The method of identifying stars as either accretion disc-hosting or disc-less stars adopted in this study minimises contamination by so-called transitional discs (see Sec- tion 1.3.1) which are observed to exhibit gaps in their inner regions. The presence of such gaps are thought to indicate that the disc clearing process is underway. Consistency between the ages of the accretion disc-hosting and disc-less stars presented here suggests that disc clearing is a rapid process as the time taken for a star to transition between Class II and Class III is small, in agreement with typical disc dispersal timescales of ∼ 0.1 Myr (Alexander et al. 2006a,b; Section 1.2.3).

Following disc dispersal, a PMS star undergoing gravitational contraction is expected to conserve angular momentum (i.e. j? = const.). Consequently, the rotation rate is 2 −2/3 expected to increase as P ∝ R? ∝ tage (i.e. n = −2/3, Section 3.2.1). Fig. 3.8 displays the relationship between P and tage for the ONC and Taurus-Auriga samples. A Spearman rank correlation test was used to check for the presence of a correlation. The results, presented in Table 3.5, are consistent with the null hypothesis of zero correlation. However, this does not mean that n = 0 [see equation (3.17)] for the individual stars or that the speciﬁc angular momentum of the individual Class III stars is reducing with tage. Instead, the location of a Class III star in Fig. 3.8 is determined by a combination of accretion disc-regulated spin evolution while a Class II object and spin-up at a rate of n ≈ −2/3 following disc dispersal. Without the ability to determine the age at which the individual Class III stars lost their discs, there is no way to separate the Class II rotational evolution from the subsequent Class III rotational evolution. Therefore, the predicted spin-up of Class III stars during their contraction at constant angular momentum is hidden by the range of observed disc lifetimes.

This also explains why a consistent removal rate of stellar angular momentum is ob-

102 3.2. Results and discussion served between the Class III and Class II PMS stars (Table 3.4). If all of the Class II stars were released from their discs at the same age, one would expect to observe β2 = 0 for individual stars during the Class III phase (i.e. j? = const., Section 3.2.1). Thus, one would not expect a relation between j? and tage when considering the sample of Class III stars within the star forming region as a whole. However, the range of possible disc lifetimes means that the location of a Class III star in Fig. 3.6 (and a non-accreting star in Fig. 3.7) is dependent on the eﬃciency of the angular momentum removal mechanism operating during its disc lifetime combined with the evolution at constant j? following disc dispersal.

The diﬀerence between the amount of j? retained within the younger and older Class III stars is an artifact of the increasing upper limit to the range of possible disc lifetimes as a star ages. In other words, the youngest Class III stars must have had very short disc lifetimes to be observed as such, thus allowing less time for angular momentum to be removed during the star-disc interaction (Class II phase). At the same time, the oldest Class III stars do not need to have had such short disc lifetimes, enabling the star-disc interaction to remove angular momentum for a longer period of time before the disc dispersed and the angular momentum then remained constant. Then, on average, the younger Class III stars will contain more angular momentum than their older counterparts.

To see this, consider Figs. 3.9 and 3.10 which compare the distributions of j? for Class

II and Class III stars. The Class II stars are observed to contain less j?, on average, than the Class III stars. The mean j? in the high mass Class II and Class III ONC samples are 5.56 × 1012 m2 s−1 and 1.49 × 1013 m2 s−1, respectively. Similarly, for the low 12 2 −1 mass ONC sample, the mean Class II j? is 6.24 × 10 m s while the mean Class III 12 2 −1 j? is 9.54 × 10 m s . For the high mass Taurus-Auriga sample, the mean Class II j? 12 2 −1 13 2 −1 is 3.94 × 10 m s and the mean Class III j? is 1.01 × 10 m s . Double-sided KS tests indicate that the Class II and Class III samples are drawn from the same parent population at probabilities of 0.00045 (high mass ONC sample), 0.016 (low mass ONC sample), and 0.0086 (high mass Taurus-Auriga sample). The Class III stars contain more j?, on average, as they evolve with j? = const. while j? continues to be removed from the Class II stars. Over time, this would result in a broadening of the distribution of Class III j? as more longer-lived discs continued to disperse (testing this idea further would require extension to older star forming regions; see Chapter 6).

103 Chapter 3. Accretion discs as regulators of stellar angular momentum evolution

Class II 25 Class II Class III Class III 20

20 M3 − M6.5 K0 − M2 15 15 10 10 Number of stars Number of stars 5 5 0 0

12 13 14 12 13 14 2 −1 2 −1 log( jstar [ m s ] ) log( jstar [ m s ] )

Figure 3.9: Distribution of j? for the high mass (left panel) and low mass (right panel) ONC samples. Open dashed columns show the full samples; black and white hatched columns show the Class II stars; blue columns show the Class III stars. For both the high mass and low mass samples, Class II stars contain less j?, on average, than Class III stars. Double-sided KS tests indicate that the probabilities of the Class II and Class III samples being drawn from the same parent population are 0.00045 (high mass) and 0.016 (low mass).

5 Class II Class III

K0 − M2 Number of stars 0

13 2 −1 log( jstar [ m s ] ) Figure 3.10: As Fig. 3.9 but for the high mass Taurus-Auriga sample. The low mass stars are not displayed as they number only 4. As in Fig. 3.9, Class II stars contain less j? than Class III stars. A double-sided KS test reveals that the probability that the samples are drawn from the same parent population is 0.0086.

104 3.2. Results and discussion

3.2.4 Focusing on Class II stars

The rotational evolution of accretion-disc hosting (Class II) stars is governed by the star- disc interaction with the rotation rate depending on the location of the truncation radius,

Rtrunc (see Section 1.4.2). Rtrunc is a function of both the magnetic ﬁeld strength at the inner disc and the mass accretion rate [see equation (1.56); K¨onigl1991; Bessolaz et al. 2008; Johnstone et al. 2014], both of which are known to vary with time (e.g. Donati et al. 2011a; Audard et al. 2014). Thus, as a Class II star contracts, the location of Rtrunc relative to the co-rotation radius in the disc will likely change, leading to periods of spin-up and spin-down (e.g. Romanova et al. 2002; Matt & Pudritz 2005b). If, over a timescale n of a few Myr, the rotation period varies as a power law, on average, with P ∝ tage then,

−β1 assuming no prior knowledge of the stellar contraction rate (i.e. R? ∝ tage ), it follows

2 −β2 from j? ∝ R?/P ∝ tage [equations (3.1) and (3.19)] that

β2 = 2β1 + n. (3.20)

In Section 3.2.1, values of β1 were found to be larger than, but in rough agreement with, purely theoretical considerations of a contracting fully convective star where β1 = 1/3

(Table 3.4). In this case, the value of β2, and therefore the removal rate of angular momentum, is solely dependent on the value of n. A Class II star in a disc-locked state would rotate at the same rate as the Keplerian rotation rate at Rtrunc and would evolve with n = 0 (i.e. constant P ). In this case, β2 = 2/3 (see Section 3.2.1) such that −2/3 j? ∝ tage . If the net eﬀects of the torques in the star-disc system are such that the star spins down (the n > 0 case), the reduction in j? with tage may be more rapid. Conversely, if the net torque causes the star to spin up (the n < 0 case), j? will either decrease (for −2/3 < n < 0), remain constant (for n = −2/3), or increase (for n < −2/3) with age.

The observations discussed in Section 3.2.2 suggest that, in the ONC and Taurus-

−β2 Auriga, j? reduces with tage as j? ∝ tage with β2 ≈ 2–2.5 for Class II stars. This angular momentum removal rate is more rapid than is expected if the stars are locked to their discs. It suggests that Class II stars may be eﬃciently spun down during the star-disc interaction phase, despite their contraction and the accretion of high angular momentum material from the inner disc. The mechanism by which this could occur is likely some

105 Chapter 3. Accretion discs as regulators of stellar angular momentum evolution form of outﬂow, such as one of (or a combination of) those detailed in Section 1.4.3.

In apparent contrast to these results, no clear correlation between the rotation periods and ages of the Class II stars is found when considering the entire sample as a whole (Fig. 3.8). However, this does not rule out individual Class II stars being locked to their discs as there is likely to be a range of disc-locking periods across the sample resulting in a range of rotation periods amongst the Class II PMS stars. This has long been observed (e.g. Herbst et al. 2002; see also Section 1.4.1). Additionally, any change in the relative position of the truncation radius and the co-rotation radius would result in changes in n during the lifetime of the disc which would introduce additional scatter in the Class II sample in Fig. 3.8.

3.2.5 Rotation period distributions and the relation between stellar mass and rotation

Figs. 3.11 and 3.12 show the distributions of rotation periods for the ONC and Taurus- Auriga samples, respectively. A range of rotation periods is observed for both the Class II and Class III samples suggesting that, as n = 0 (Section 3.2.4), a range of disc-locking periods do exist. Also, the bimodal distribution seen previously for high mass ONC stars (Herbst et al. 2002; Cieza & Baliber 2007) is recovered with the Class II stars rotating at slower rates, on average, than the Class III stars. A double-sided KS test reveals a probability of 0.0027 that the samples are drawn from the same parent population. The bimodality is also visible in the smaller high mass Taurus-Auriga sample. However, a double-sided KS test does not reveal a signiﬁcant diﬀerence between the samples with a probability of 0.08.

The bimodal distribution observed for the high mass samples indicates accretion disc- regulated rotation during the Class II phase, followed by spin up during the Class III phase. At the same time, it is clear from Figs. 3.11 and 3.12 that not all of the Class III stars are rapid rotators and, similarly, not all of the Class II stars are slow rotators. It is possible that the slowly rotating Class III stars have only recently been released from their discs and have not yet spun up and that the faster rotating Class II stars have disc truncation radii closer to their photospheres and so are locked to a faster spinning region of their Keplerian discs. The disc truncation radius is related to the mass accretion rate and the dipole component of the large scale stellar magnetic ﬁeld (e.g. Adams & Gregory

106 3.2. Results and discussion

Class II Class II 30 K0 − M2 Class III later than M2 Class III 15 20 10 Number of stars Number of stars 10 5 0 0

0 5 10 15 0 5 10 15 P [ days ] P [ days ]

Figure 3.11: Distribution of rotation periods for the high mass (left panel) and low mass (right panel) ONC samples. Open dashed columns contain the full sample; black and white hatched columns contain the Class II stars; blue columns contain the Class III stars. The bimodal distribution is recovered for the high mass sample with mean rotation periods of 7.07 days (Class II) and 5.11 days (Class III). The unimodal distribution is also recovered for the low mass sample with mean rotation periods of 4.27 days (Class II) and 3.57 days (Class III). Double-sided KS tests indicate that the probabilities of the Class II and Class III samples being drawn from the same parent population are 0.0027 (high mass) and 0.16 (low mass).

5 Class II Class III

K0 − M2 Number of stars 0

0 5 10 15 P [ days ]

Figure 3.12: As Fig. 3.11 but for the high mass Taurus-Auriga sample. Again, the Class II stars have slower rotation rates than the Class III stars, on average with mean rotation periods of 8.30 days and 4.35 days, respectively. However, due to the size of the sample, this result is not statistically signiﬁcant: a double-sided KS test indicates that the probability of the samples being drawn from the same parent population is 0.076.

107 Chapter 3. Accretion discs as regulators of stellar angular momentum evolution

2012) so Class II sources in the peak of rapid rotators may have higher accretion rates and/or weaker dipole components of their magnetic ﬁelds.

This idea can be extended to the low mass ONC sample. The unimodal distribution of rotation periods is recovered (e.g. Herbst et al. 2002; right hand panel of Fig. 3.11) with consistent rotation periods found between the Class II and Class III stars. A double- sided KS test reveals a probability of 0.16 that they are drawn from the same parent population. This suggests that the disc truncation radii of the lowest mass stars may lie closer to their photospheres than those of the higher mass stars as a result of the weaker dipole components of their large-scale magnetic fields. Donati et al. (2010b) and Gregory et al. (2012) argue that the lowest mass PMS stars may have more complex magnetic fields. This would result in smaller disc truncation radii and faster disc-locked stellar spin rates than found for higher mass, fully convective stars. Additional observations of the magnetic field topologies of the lowest mass PMS stars are required to confirm this.

3.3 Summary

This chapter has presented the results of Davies et al. (2014a) in which the evolution of specific stellar angular momentum in fully convective Class II and Class III PMS stars was investigated. To do this, a consistent sample of PMS stars within the ONC and Taurus- Auriga was constructed. Rotation periods were gathered from the literature and checked for the presence of beat and harmonic periods. Recent updates to spectral type assign- ments (e.g. Hillenbrand et al. 2013) were adopted. Effective temperatures were assigned to these spectral types, and bolometric luminosities were calculated from optical photometry, using intrinsic colours, spectral type-to-effective temperature scales, and bolometric corrections from Pecaut & Mamajek (2013). These are an improvement over the typically adopted MS dwarf scales as they account for the lower surface gravities and highly spotted surfaces of PMS stars. The effective temperatures and bolometric luminosities were used to calculate stellar radii, assuming that the stars radiate as blackbodies and the observational spread in stellar luminosity across the sample is real. Stellar masses, ages, and their respective uncertainties were estimated consistently across the ONC and Taurus-Auriga samples using Siess et al. (2000) PMS evolutionary models.

With the spectral type updates and the careful removal of rotation period bias, non-

108 3.3. Summary members, and known binaries, the bimodal distribution of rotation periods seen previously for the high mass stars (e.g. Attridge & Herbst 1992; Edwards et al. 1993; Choi & Herbst 1996; Herbst et al. 2000b) is recovered. In addition, the unimodal distribution seen for the low mass stars (Herbst et al. 2002; Cieza & Baliber 2007) is also recovered. On average, disc-hosting stars typically rotate slower than disc-less stars across all samples. In each sample, there are a range of measured rotation periods. The rapid and slowly rotating populations contain both Class II and Class III sources. The slower rotating Class III stars have likely lost their discs only recently while the faster rotating Class III stars have likely had longer to spin up. If disc-locking operates, the rapidly rotating Class II stars likely have larger mass accretion rates and/or weaker magnetic fields than the slower rotating Class II stars. In addition, the slower rotation rates of the higher mass, fully convective stars compared to the lower mass, fully convective stars is likely a result of lower mass stars having more complex large-scale magnetic fields, as indicated by the analysis of magnetic field topologies in PMS stars (Donati et al. 2010b; Gregory et al. 2012).

Considering the sample as a whole, no correlation between the rotation period and age is observed. However, on average, the Class II stars typically rotate at slower rates, emphasizing that discs do play a role in regulating the rotation of accreting PMS stars. It is likely that a range of disc-locked rotation periods exist due to variations in the mass accretion rate and the magnetic ﬁeld both in the same star over time (e.g. Donati et al. 2011a; Audard et al. 2014 and references therein) and across the sample (Donati et al. 2010b; Gregory et al. 2012).

If the age spreads observed in the ONC and Taurus-Auriga are assumed to be real (Section 2.4.4), the speciﬁc stellar angular momentum is observed to reduce with age in

−β2 the Class II sample with j? ∝ tage and β2 ≈ 2–2.5. These stars appear to be losing angular momentum at a faster rate than would be expected if they were locked to their discs at a ﬁxed rotation rate during contraction. This result suggests a more eﬃcient angular momentum removal process operates within the star-disc system. Individual Class III stars are expected to conserve angular momentum as they contract yet, over the sample presented here, the Class III j? is observed to reduce with tage. Furthermore, the rate at which this occurs is roughly the same rate as for the Class II sample. On average, Class

III stars are observed to contain more j? than Class II stars. This can be explained by Class II stars losing their discs at a variety of ages (indeed, there are a mixture of stars

109 Chapter 3. Accretion discs as regulators of stellar angular momentum evolution with and without discs at any particular age within the ONC and Taurus-Auriga): if one considers two Class II stars with the same initial j? which lose angular momentum at the same rate, the one that loses its disc (and is observed as a Class III star) will evolve with constant j?, while the one that retains its disc (and is observed as a Class II star) will continue to lose angular momentum. This naturally explains why the Class III stars are found, on average, to have larger j? than the Class II stars.

The correlations observed here ultimately depend on the accuracy with which stars can be positioned within the HRD. Throughout this study, the spread in stellar luminosities have been interpreted as a real spread in stellar radii and, in turn, a real spread in age (see Section 2.4.4 for a detailed discussion). However, if this assumption is incorrect and either the stars are coeval or their ages are indicative of diﬀering accretion histories (e.g. Littlefair et al. 2011), the conclusions presented in this chapter will require further examination. For example, diﬀerent stellar age indicators, preferably independent of stellar radius measurements, or more reliable bolometric luminosity calculations could be used. These are addressed in more detail in Chapter 6.

110 4 Outer regions of protoplanetary discs: evidence for viscous evolution?

ccretion discs with well-coupled gas and dust components are expected to radially expand to conserve angular momentum (Lynden-Bell & Pringle 1974; A Pringle 1981; Hartmann et al. 1998), as outlined in Section 1.2.2. Recently, studies of the detailed structure of a small number of individual accretion discs have highlighted a possible inconsistency between the radial extent of the gas and dust components (Pani´cet al. 2009; Andrews et al. 2012; de Gregorio-Monsalvo et al. 2013; Rosenfeld et al. 2013; Zhang et al. 2014), suggesting that the evolution of the dust component of protoplanetary discs may be governed by diﬀerent dynamics. This chapter is a summary of the results presented in Davies et al. (2015, submitted to A&A) which investigated whether the evolution of the outer regions of dust and gas components of spatially resolved protoplanetary discs is consistent with viscous disc evolution theory. Particular focus was placed on searching for observational evidence to support (or refute) the conclusions of

111 Chapter 4. Outer regions of protoplanetary discs: evidence for viscous evolution? modern theoretical disc evolution models (e.g. Birnstiel & Andrews 2014) which predict a continued expansion of the gas component, while the dust component rapidly reaches a ﬁxed, grain size-dependent outer edge.

The Davies et al. (2015, submitted to A&A) study was the first of its kind, made possible by the increase in the number of spatially resolved discs during the last half-decade thanks to sensitivity improvements to existing interferometers and early science results from the Atacama Large Millimeter/submillimeter Array (ALMA). The study focuses on spatially resolved thermal continuum emission from discs present around stars not previously identified as members of binary or multiple systems. The complex dynamics of multiple systems can result in e.g. misalignment between the disc axis and the orbit of the secondary (Williams et al. 2014), increased flaring in the disc (Bouwman et al. 2006), truncation of the outer disc (Rosotti et al. 2014), and/or stripping of disc material (Cieza et al. 2009). Each of these processes increases the already complex nature of protoplanetary disc evolution. In addition, if the separation between the primary and secondary stars is ≤ 15 au, the disc is likely to be circumbinary rather than circumstellar (Artymowicz & Lubow 1994) and the binary pair may be unresolved in the photometry used to determine the stellar bolometric luminosity, L?. In turn, this would affect the placement of the star on the

Hertzsprung-Russell diagram (HRD; see Section 2.4), resulting in an overestimation of L? and a corresponding underestimation of the stellar age (Hartmann 2001; see Section 2.4.4).

The necessary disc and stellar data were retrieved from the literature and resulted in a sample of 32 star-disc systems in nearby star forming regions, namely the Orion Nebula Cluster (ONC), Taurus-Auriga, the Lupus clouds (Lup I, Lup II, and Lup III), ρ Ophiuchus (ρ Oph), and Upper Scorpius (Up Sco). A portion of the discs in this sample had been included in high-resolution surveys of molecular gas in protoplanetary discs (e.g. Koerner & Sargent 1995; Simon et al. 2000; Dutrey et al. 2008; Schaefer et al. 2009; Chapillon et al. 2012; Guilloteau et al. 2014) with the radial extents of the carbon monoxide (CO) or cyanide (CN) components of thirteen of the discs previously determined from analysis of the spatially resolved emission (see Section 2.5.1 for details).

Details of the methods used to compile the sample and estimate stellar ages are presented in Section 4.1. The results are presented in Section 4.2 and discussed in the context of viscous disc evolution theory and angular momentum conservation in Section 4.3.

112 Table 4.1: Data for star-disc systems not identified as members of binary or multiple systems for which spatially resolved thermal continuum emission, spectral types, and 2MASS photometry were available. Column 1: identifier; column 2: star forming region; column 3: spectral type (measurement uncertainties are ±0.5 subtypes except: a – ±0.2; b – ±0.3; c – ±1); columns 4 & 5: 2MASS J and H magnitudes; column 6 & 7: stellar mass and age, determined using Siess et al. (2000) PMS evolutionary models; columns 8, 9, & 10: dust radius, wavelength of observation, and physical resolution; column 11: gas radius; column 12: specific angular momentum contained within the dust component of the disc; column 13: references for spectral types, thermal continuum observations, and molecular line observations [(1) Hillenbrand 1997; (2) Luhman et al. 2000; (3) Preibisch et al. 2002; (4) Eisner et al. 2005; (5) Wilking et al. 2005, (6) Andrews & Williams 2007a; (7) Luhman et al. 2010; (8) Oberg¨ et al. 2011; (9) Hillenbrand et al. 2013; (10) Alcaláet al. 2014; (11) Herczeg & Hillenbrand 2014; (12) Koerner & Sargent 1995; (13) Kitamura et al. 2002; (14) Dutrey et al. 2003; (15) Kessler-Silacci 2004; (16) Andrews & Williams 2007b; (17) Lommen et al. 2007; (18) Dutrey et al. 2008; (19) Pinte et al. 2008; (20) Mann & Williams 2009; (21) Panićet al. 2009; (22) Schaefer et al. 2009; (23) Guilloteau et al. 2011; (24) Ricci et al. 2011; (25) Carpenter et al. 2014; (26) Guilloteau et al. 2014; (27) Mann et al. 2014; (28) Ricci et al. 2014].

ID Region SpT J2M H2M M? Age Rdust λobs Res Rgas jdust Refs 15 2 −1 (M ) (Myr) (au) (mm) (au) (au) (10 m s )

113 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) +0.06 +0.2 +6.2 GW Lup Lup I M2.3 10.07 9.18 0.32−0.02 1.2−0.2 128 ± 50 1.4 380 — 28.5−5.6 11, 17 +0.01 +0.1 +6.2 HD 142560 Lup II K7 8.73 7.82 0.72−0.14 0.6−0.1 107 ± 34 1.4 380 — 39.1−7.3 10, 17 +0.10 +0.4 +14.4 IM Lup Lup II K6.0 8.78 8.09 0.68−0.05 1.0−0.2 240 ± 116 1.3 230 750 ± 50 56.9−13.9 11, 19, 21 +0.07 +0.1 +6.5 V1279 Sco Lup III M0.4 9.53 8.65 0.46−0.06 0.6−0.2 200 ± 53 1.4 500 — 42.7−6.3 11, 17 +0.09 +0.3 +3.9 JW 534 ONC M2 11.25 10.24 0.34−0.05 0.1−0.1 125 ± 5 0.856 200 — 29.0−2.2 1, 27 +0.28 +0.9 +8.4 JW 537 ONC K5 11.73 10.71 0.81−0.20 1.0−0.3 148 ± 3 0.856 200 — 48.8−6.0 9, 27 +0.08 +2.6 +26.6 MLLA 324 ONC M2.5 13.78 12.84 0.31−0.08 3.7−1.1 830 ± 580 0.856 200 — 71.5−26.6 1, 27 +0.38 +1.3 +22.9 LV 3 ONC K3.5 11.36 10.40 1.26−0.31 1.3−0.6 128 ± 96 0.856 200 — 56.6−22.3 2, 27 +0.11 +0.6 +6.0 V2252 Ori ONC K6 11.44 10.64 0.73−0.14 1.3−0.4 116 ± 29 0.880 830 — 41.0−6.5 9, 24 +0.06 +26 +6.2 V2377 Ori ONC K5 14.11 13.15 0.73−0.05 33−15 435 ± 58 0.880 580 — 79.4−6.0 1, 20 +0.13 +0.8 +3.5 2M J16262367-2443138 ρ Oph K5 9.39 8.40 0.83−0.11 1.5−0.5 112 ± 5 0.880 210 — 43.0−3.0 5, 16 Chapter 4. Outer regions of protoplanetary discs: evidence for viscous evolution? 6, 16 Refs 5, 16 8, 16, 12 4, 16 5, 16 11, 28 7, 23, 22 11, 23, 14 11, 16, 15 11, 23, 26 11, 23, 26 11, 23, 26 11, 23, 26 11, 16, 26 11, 13, 12 11, 16 11, 23 11, 23, 18 11, 23 ) 1 − 1 2 8 3 2 8 9 4 2 0 6 5 3 1 8 3 6 9 1 5 0 7 1 5 1 5 5 4 5 7 7 2 8 4 8 1 3 8 ...... s 0 2 2 4 2 0 1 0 2 0 1 0 1 3 4 2 2 4 2 2 +3 − +2 − +3 − +3 − +3 − +1 − +2 − +0 − +3 − +1 − +1 − +1 − +1 − +2 − +2 − +4 − +1 − +2 − +2 − m 8 0 3 1 6 5 7 5 3 7 0 1 1 3 4 4 3 0 4 dust ...... j 15 21 6 32 7 25 9 40 10 24 10 22 60 32 13 33 11 18 19 16 10 29 25 38 ± ± ± ± ± ± ± ± ± ± ± gas R − − − Res 3 80 122 3 60 520 3 60 295 3 70 463 3 70 641 3 60 — 16 3 150 630 3 100 — 49 3 140 300 33 270 33 280 241 33 260 — 38 ...... obs 880 330857 170 — 227 43 880 200 — 32 887 80 — 11 880 220 565 880 200 — 27 ...... λ 5 0 3 0 9 1 5 1 2 0 1 1 20 0 1 1 1 1 1 1 1 1 5 2 140 360 5 0 1 1 1 1 2 1 4 0 11 1 10 1 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± dust R 49 73 54 70 49 99 60 42 85 95 67 44 100 105 56 70 103 148 108 1 1 2 1 1 6 2 2 2 3 2 5 5 1 3 1 7 6 6 3 1 1 6 1 3 4 1 2 3 2 8 9 2 2 2 2 1 1 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 +0 − +0 − +0 − +0 − +0 − +1 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +1 − +1 − 9 1 8 0 0 2 6 5 9 4 5 0 0 0 7 6 5 4 2 Age ...... 0 0 1 0 1 0 1 1 1 1 2 0 0 0 1 2 2 0 0 11 12 20 04 09 02 07 11 09 09 02 07 01 04 01 03 02 06 18 15 10 14 07 08 03 04 02 05 07 06 08 04 08 07 21 04 08 16 ...... ) (Myr) (au) (mm) (au) (au) (10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ?

+0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − +0 − M M 82 64 77 22 78 16 52 51 48 78 32 77 28 46 45 61 31 71 44 ( ...... 57 0 08 0 58 0 12 0 22 0 55 0 43 0 97 0 68 0 76 0 34 0 24 0 80 0 12 0 60 0 17 1 45 0 60 0 09 0 2M ...... H 30 7 72 7 31 10 73 10 85 10 10 8 43 8 48 8 83 8 63 8 44 9 14 8 47 8 85 7 19 9 34 8 62 9 25 10 00 8 2M ...... J 11 a 5 5 9 6 9 3 9 0 10 3 9 3 9 8 10 5 9 5 9 0 8 0 9 ...... Oph K5 8 Oph K8 13 Oph K6Oph M4 8 11 Oph K6 10 ρ ρ ρ ρ ρ – continued V1121 Oph V2508 Oph YLW 58 CIDA 1 Tau M4 Haro 6-33 Tau M0 11 BP Tau Tau M0 AA Tau Tau M0 CI Tau Tau K5 CY Tau Tau M2 DL Tau Tau K5 DM Tau Tau M3 DN Tau Tau M0 DO Tau Tau M0 DR Tau Tau K6 FT Tau Tau M2 GM Aur Tau K6 HL Tau Tau K3 10 (1)2M J16264502-2423077 (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) Table 4.1 ID2M J16262407-2416134 Region SpT

114 Table 4.1 – continued

ID Region SpT J2M H2M M? Age Rdust λobs Res Rgas jdust Refs 15 2 −1 (M ) (Myr) (au) (mm) (au) (au) (10 m s ) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) +0.10 +0.2 +4.6 IQ Tau Tau M1.1 9.42 8.42 0.38−0.01 0.7−0.2 151 ± 11 2 220 225 ± 21 33.8−1.3 11, 13, 26 +0.11 +0.3 +1.3 2M J16142029-1906481 Up Sco K5 10.26 8.89 0.85−0.15 0.9−0.3 23 ± 1 0.880 80 — 19.7−1.8 3, 25 115 Chapter 4. Outer regions of protoplanetary discs: evidence for viscous evolution?

4.1 Observational data

Investigating disc evolution required estimates of both the size of the disc and the age of the stellar host. A list of protoplanetary discs with spatially resolved submillimetre or millimetre (hereafter collectively referred to as submm) continuum emission was compiled from the literature and formed the basis of the sample. Previous multiplicity surveys of nearby star forming regions1 were used to remove circumbinary discs and star-disc systems that had previously been identiﬁed as members of binary of multiple systems. The remaining star-disc systems were then cross-matched against published spectral types and Two Micron All Sky Survey (2MASS; Cutri et al. 2003; Skrutskie et al. 2006) photometry so that an estimate of the stellar age could be determined (see Section 4.1.2). The ﬁnal sample consisted of 32 discs around stars of spectral types K and M and is presented in Table 4.1.

4.1.1 Characterising the thermal dust emission

The intensity of thermal emission at frequency, v, Iv(r), is given by

Iv(r) ∝ (1 − exp[−τ(v, r)])Bv(T (r)) (4.1)

where τ denotes the optical depth and Bv(T (r)) is the Planck function at temperature, T (e.g. Beckwith et al. 1990). At submm wavelengths, the disc is optically thin and the Rayleigh-Jeans approximation is valid. Under these criteria, equation (4.1.1) can be written as

Iv(r) ∝ τ(v, r)T (r) ∝ κ(v, r)Σ(r)T (r), (4.2) where κ and Σ are the dust opacity and surface density, respectively. One then requires parameterisations of κ(v, r), Σ(r), and T (r) to fully describe the observed thermal dust emission.

1Mundt et al. (1983); Hartmann et al. (1986); Rodriguez et al. (1986); Chen & Simon (1988); Leinert & Haas (1989); Chen et al. (1990); Simon & Guilloteau (1992); Ghez et al. (1993); Leinert et al. (1993); Reipurth & Zinnecker (1993); Mathieu (1994); Richichi et al. (1994); Osterloh & Beckwith (1995); Simon et al. (1995); Brandner et al. (1996); Dutrey et al. (1996); Jensen et al. (1996); Ghez et al. (1997); Mathieu et al. (1997); Nürnberger et al. (1998); Bertout et al. (1999); Köhleret al. (2000); White & Ghez (2001); Hogerheijde et al. (2003); Duchêneet al. (2004); White & Hillenbrand (2005); Correia et al. (2006); Andrews & Williams (2007a); Guilloteau et al. (2008); Kastner et al. (2008); Isella et al. (2009); Lommen et al. (2010); Ricci et al. (2010, 2011); Shirono et al. (2011); Daemgen et al. (2012); Ubach et al. (2012); Andrews et al. (2013); Akeson & Jensen (2014); Herczeg & Hillenbrand (2014).

116 4.1. Observational data

The dust opacity is typically assumed to be radially invariant in disc-ﬁtting models (e.g. Beckwith et al. 1990; Isella et al. 2009; Guilloteau et al. 2011). However, the higher densities and velocities of dust in the inner regions of discs compared to the outer regions introduces a radial dependence to the grain growth timescale. In turn, this would lead to a radially-dependent dust opacity with the inner regions of the disc exhibiting a smaller dust opacity than the outer regions (Dullemond & Dominik 2005; Garaud 2007). To date, it has not been possible to place observational constraints on the degree to which the dust opacity varies with radius (c.f. Andrews et al. 2009) and, for consistency with previous studies, a radially invariant dust opacity is assumed here.

Submm emission from protoplanetary discs arises from regions of the disc close to the midplane (Chiang & Goldreich 1997). The temperature of this region depends on the number, and energy, of stellar photons impacting onto the disc surface that are scattered into the disc. As the photon flux decreases with increasing distance from the star, one would, as a first approximation, expect a radial temperature profile of the form T (r) ∝ r−q. From purely theoretical considerations, Dullemond et al. (2001) found that a disc in hydrostatic equilibrium (HE) would exhibit a relatively shallow temperature profile with q = 0.5. From an observational perspective, efforts to constrain the value of q have been possible using fits to the infra-red portion of the spectrum. The disc inner regions reveal typical values of 0.4 . q . 0.6 and outer regions often reveal flatter profiles (e.g. Beckwith et al. 1990; Andrews et al. 2009).

Assuming that the dust and gas components of protoplanetary discs are well-coupled, Σ(r) is governed by the viscous evolution of the gas, parameterised by equation (1.38):

∂Σ 3 ∂ ∂ = r1/2 νΣr1/2 ∂t r ∂r ∂r

(Lynden-Bell & Pringle 1974; Pringle 1981) where r is the cylindrical radius. If one further assumes that the viscosity, ν, is time invariant and is represented by a power law in radius [equation (1.39)], Iv(r) can be fully described by the radial dependence of the temperature proﬁle, q, the radial dependence of the viscosity, γ, a characteristic radius, and a normalisation factor (see Section 2.5.2).

The choice of characteristic radius has diﬀered between studies. For example, Guil- loteau et al. (2011) use the radius containing 63% of the disc mass while Isella et al.

117 Chapter 4. Outer regions of protoplanetary discs: evidence for viscous evolution?

(2009) use the transition radius, Rtrans, which corresponds to the radius at which the mass flux within the disc changes sign. The number of free parameters used in the model fitting procedures has also differed between studies with e.g. Andrews et al. (2011) fixing γ = 1. With this in mind, assessing whether the dust components of the discs included in this study are viscously evolving required re-modelling of the discs using consistent model parameters and assumptions.

The Davies et al. (2015, submitted to A&A) sample is composed of 32 discs around T Tauri stars that have been spatially resolved by the Submillimeter Array (SMA; Andrews & Williams 2007b; Lommen et al. 2007; Pinte et al. 2008; Mann & Williams 2009; Ricci et al. 2011), IRAM Plateau de Bure Interferometer (PdBI; Guilloteau et al. 2011), Nobeyama Millimeter Array (NMA; Kitamura et al. 2002), and ALMA (Carpenter et al. 2014; Mann et al. 2014; Ricci et al. 2014). Although the data for a number of the discs remains proprietary, the original authors each published Gaussian ﬁts to their observations. It was noted that if (i) the dust opacity is radially invariant, (ii) the discs are isothermal, and (iii) γ = 0 in equation (1.40):

" (2−γ) # C −(5/2−γ)/(2−γ) R Σ(r) = γ τ exp − , 3πν1R τ the continuum emission intensity would follow a Gaussian proﬁle of the form,

r2 I (r) ∝ exp − . (4.3) v 2σ2

Under these assumptions, the full width half maximum (FWHM) of the major axis of the Gaussian fit can be used to estimate a characteristic radius in a consistent manner across the sample of discs. If the disc is better described by a different value of γ and/or q (and/or if the dust opacity varies with radius), the emission will be better described by an intensity profile of the form

r2 I (r) ∝ exp − r−m, (4.4) v 2σ2 with m encompassing the combined power-law dependences of Σ(r), κ(r), and T (r). In this instance, a Gaussian ﬁt will underestimate the true radial extent of the disc by a factor depending on m.

118 4.1. Observational data

Figure 4.1: ALMA Cycle 0 continuum image for CIDA 1. The black contour marks the FWHM of the two-dimensional elliptical Gaussian ﬁt prior to deconvolution. The beam size is 0.0065 × 0.0046 at P.A. = 9.12◦ and is shown in the bottom left corner. The deconvolved FWHM has dimensions 0.00452 ± 0.00021 × 0.00235 ± 0.00041 with P.A. = 19.2 ± 3.9◦. The wavelength of observations is indicated in the top right corner.

Radial extent of the dust

Assuming the observed submm emission follows a Gaussian distribution [equation (4.1.1)], the characteristic dust radius was estimated using an ellipse containing 95% of the Gaus- sian emission such that the angular size of the dust disc, θdust ' 1.4 FWHM where FWHM = 2p2 ln(2)σ and σ is the standard deviation. The physical dust disc radius is then

Rdust = 0.5 θdustd, (4.5) where d is the adopted distance to the star-disc system: 414 ± 7 pc for the ONC (Menten et al. 2007), 140 ± 20 pc for Taurus-Auriga (Elias 1978; Loinard et al. 2007; Torres et al. 2009), 145 ± 2 pc for Up Sco (de Zeeuw et al. 1999), 125 ± 5 pc for ρ Oph (Loinard et al. 2008; Lombardi et al. 2008), 150 ± 10 pc for Lup I and II, and 200 ± 20 pc for Lup III (Comer´on2008; see Section 2.4.2).

The results of deconvolved two-dimensional elliptical Gaussian ﬁts2 to continuum emission had previously been published for 30 of the 32 discs in the sample (individual references are presented in Table 4.1). For the other two discs (CIDA 1 and 2MASS J16142029-

2Assuming the discs are circular and geometrically thin, the observed elliptical nature of the disc is purely due to projection eﬀects.

119 Chapter 4. Outer regions of protoplanetary discs: evidence for viscous evolution?

Figure 4.2: As Fig. 4.1 but for 2MASS J16142029-1906481. The deconvolved FWHM has dimensions 0.00224 ± 0.00013 × 0.00184 ± 0.00016 with P.A. = 29 ± 19◦ and the beam size is 0.0068 × 0.0042 at P.A. = −75.08◦. 1906481), calibrated ALMA continuum image ﬁles were obtained from the archive.3 Two- dimensional elliptical Gaussian ﬁts were performed to the continuum emission using the Common Astronomy Software Applications (CASA) package (McMullin et al. 2007). The resulting deconvolved major and minor axis FWHM are shown in Figs. 4.1 and 4.2 for CIDA 1 and 2MASS J16142029-1906481, respectively. For each of the 32 discs, the major axis FWHM was used to calculate Rdust according to equation (4.5). These are presented in Table 4.1.

Radial extent of the gas

Estimates of the radial extent of the spatially resolved molecular gas component of the discs resulting from line emission modelling (Section 2.5.1) were also retrieved from the literature. Those based on CN line emission (Guilloteau et al. 2014) were preferred over CO (Koerner & Sargent 1995; Simon et al. 2000; Dutrey et al. 2003; Kessler-Silacci 2004; Dutrey et al. 2008; Pani´cet al. 2009; Schaefer et al. 2009), where both were available, due to the limited confusion from large-scale molecular gas in the birth cloud (Guilloteau et al. 2013). In total, gas disc radii were available for thirteen of the original sample of 32 discs and are presented, along with their individual references, in Table 4.1.

3https://almascience.nrao.edu/aq/

120 4.1. Observational data

4.1.2 Ages of the star-disc systems

Multiple methods to determine stellar ages have been used across the star forming regions included in this study, often employing different approximations or models. To gain a consistent sample, new mass and age estimates were derived by comparing effective temperatures, Teff , and bolometric luminosities, L?, to Siess et al. (2000) PMS evolutionary isochrones (Z = 0.02 with convective overshooting; Section 2.4). Effective temperatures were determined from spectral types, gathered from the literature (individual references are given in Table 4.1). These included recent updates from Herczeg & Hillenbrand (2014) for seventeen of the Taurus-Auriga and Lupus members in the sample. The Herczeg & Hillenbrand (2014) study was the first to assign spectral types to these stars using modern techniques. As such, their spectral types differ from more commonly cited values, especially those from Cohen & Kuhi (1979), which had insufficient spectral coverage to classify M-type stars.

As in Section 3.1.1, spectral types were translated into eﬀective temperatures using the Pecaut & Mamajek (2013) temperature scales for 5-30 Myr old PMS stars. Although these scales have been calibrated for use on PMS stars older than those included in Davies et al. (2015, submitted to A&A), they are still more applicable than typically adopted main sequence (MS) dwarf scales (e.g. Bessell & Brett 1988; Bessell 1995; Kenyon & Hartmann 1995; Luhman 1999; see Section 2.4.1) as they account for the lower surface gravities and highly spotted surfaces of PMS stars (Gullbring et al. 1998; Luhman 1999; Stauﬀer et al. 2003; Da Rio et al. 2010; Herczeg & Hillenbrand 2014).

Bolometric luminosities were calculated by applying a bolometric correction to extinction- and distance modulus-corrected 2MASS J-band magnitudes, J2M, using

L? 2 log = [Mbol, − (J2M − AJ2M ) + DM − BCJ2M (Teﬀ )]. (4.6) L 5

Here, Mbol, = 4.755 mag is the solar bolometric magnitude (Mamajek 2012) and DM is the distance modulus (see Section 2.4.2). This waveband was chosen to minimise contamination from disc material and accretion across the sample of K and M PMS stars (c.f. Brice˜noet al. 1998; Luhman 1999). Individual values of the 2MASS J-band extinction,

AJ2M , were determined from contemporaneous 2MASS J- and H-band photometry. The

121 Chapter 4. Outer regions of protoplanetary discs: evidence for viscous evolution? 10 ●

● ● ● ● ● ● ● ] ● ● ● n

u ●● ● s ● L ● ● ● ●● [ 1 ●● L ● ● ● ●

●

● ● ● ● 0.1

5000 4000 3000

Teff [ K ]

Figure 4.3: HRD constructed from the Siess et al. (2000) PMS evolutionary models for the stellar hosts. Isochrones (dotted lines) are shown from top right to bottom left for 0.01, 0.05, 0.2, 0.5, 2, 5, 10, and 60 Myr and the zero age main sequence (solid line) is shown for masses greater than 0.7 M . Stellar evolutionary tracks (dashed lines) are shown from bottom right to top left for 0.1, 0.2, 0.3, 0.4, 0.5, 0.75, 1.00, 1.25, 1.5, 2, 2.5, and 3 M . An average error bar is included in the top right corner for reference. extinction laws presented in Yuan et al. (2013) were used to derive

AJ2M = 2.77[(J2M − H2M) − (J2M − H2M)0], (4.7)

where (J2M − H2M)0 is the temperature-dependent intrinsic colour. As in Section 3.1.2, the Pecaut & Mamajek (2013) 2MASS J-band bolometric corrections, BCJ2M (Teﬀ ), and intrinsic colours were adopted.

Stellar masses, ages, and corresponding uncertainties were estimated by comparing Teﬀ and L? to Siess et al. (2000) PMS evolutionary isochrones using the methods outlined in Section 2.4. The corresponding Hertzsprung-Russell diagram (HRD) is shown in Fig. 4.3.

4.1.3 Accounting for observational bias

The sample presented in Table 4.1 is composed of data from a variety of studies with a range of diﬀerent observational parameters. In principle, observational biases that may aﬀect our investigation are as follows:

• observations with shorter integration times may not be sensitive to the full extent

122 4.1. Observational data

of the disc;

• discs observed with coarser beams may only be resolved if large;

• longer wavelength observations may trace material closer to the stellar host.

The submm continuum fluxes of the discs were inspected to assess the likelihood that any emission remained undetected. Thirteen of the discs had flux measurements determined from both spatially resolved and unresolved observations at the same wavelength. For each of these discs, the spatially resolved submm fluxes are within 3σ of the unresolved fluxes. For these discs, all the thermal continuum emission is being observed. However, one cannot rule out the possibility that Rdust may be underestimated for the remaining nineteen discs which lack unresolved observations at the same wavelength.

The star-disc systems included in this study are members of star forming regions located at distances ranging from 125 to 414 pc (Section 4.1.1). The data has been gathered from studies using a range of submm interferometers, each with diﬀerent angular resolutions. The corresponding physical resolutions in au are presented in Table 4.1. Although the range of physical resolution is large (60-830 au), it is important to note that this sample is the best that can be achieved with currently available data. Also, all but three of the discs are resolved at better than 380 au and the physical resolution range is narrower for the subsample with spatially resolved gas components (60-280 au). The possible impact that the variations in resolution across the sample may have on the results are addressed in Section 4.3.4.

Larger grains, which emit more strongly at longer wavelengths than smaller grains, may form only rather close to the stellar host, where sticking times are shorter, as suggested in the pioneering study of Pérezet al. (2012). However, the Pérezet al. (2012) study extends much further into radio wavelengths than the wavelength range probed here (0.856 − 2 mm; see Table 4.1). In addition, figure 3 of Pérezet al. (2012) includes 0.88 and 2.8 mm observations which are fit by two model discs of very similar size. This suggests that different grain physics is a necessary consideration to take into account, but is most important for data obtained at longer wavelengths than those used here.

123 Chapter 4. Outer regions of protoplanetary discs: evidence for viscous evolution?

4.2 Results

4.2.1 Evolution of the disc radius: expectation from theory

Disc evolution is controlled by the viscosity, ν, and is described by the temporal evolution of the gas surface density, given by equation (1.38):

∂Σ 3 ∂ ∂ = r1/2 νΣr1/2 ∂t r ∂r ∂r

(Lynden-Bell & Pringle 1974; Pringle 1981; see Section 1.2.2). Lynden-Bell & Pringle (1974) showed that if the viscosity was time-invariant and represented by ν ∝ Rγ, the surface density asymptotically approaches the similarity solution given by equation (1.40):

" (2−γ) # C −(5/2−γ)/(2−γ) R Σ(r) = γ τ exp − . 3πν1R τ

Here, R ≡ r/r1 where r1 is a radial scale factor, ν1 ≡ ν(r1), τ = t/ts + 1 where ts is the viscous scaling time (equation 1.44), and C is a normalisation constant.

Such a disc is then separated into two regimes (Hartmann et al. 1998): material in the inner regions migrates inward and accretes onto the star while material further out migrates outwards to conserve angular momentum, causing the disc to radially expand.

The expansion rate of the outer radius of the disc is governed by ts. Rearranging equation (1.44): 2 1 r1 ts = 2 3(2 − γ) ν1 and substituting ν ∝ Rγ, the temporal evolution of the outer disc can be described by

R ∝ t1/(2−γ). (4.8)

If one assumes the dust and gas in the disc are well-coupled, the evolution of the gas surface density governs the evolution of the whole disc and the evolution of the dust component is also described by equation (1.40) with Σ(r) = ζΣd(r) where ζ is the gas-to- dust ratio. Typically, it is assumed that ζ is radially and time invariant (Beckwith et al. 1990), such that the outer radius of the dust disc also evolves according to equation (4.8).

To measure Rdust in Section 4.1.1, the value of γ was ﬁxed (γ = 0). As a result, one would

124 4.2. Results 1000 500 [ au ] t s u d R 100 50

105 106 107 108 Age [ yrs ]

Figure 4.4: Evolution of the radial extent of the dust component of the discs. No correlation between Rdust and age is observed. Instead, substantial scatter in the data is found: a Spearman rank correlation test revealed a p-value of 0.52.

1/2 expect Rdisc ∝ t for the sample considered here.

4.2.2 Evolution of the disc radius: observations

Fig. 4.4 shows the dust radius, Rdust, plotted against stellar age. Assuming that the sample traces an evolutionary sequence and that the dust and gas within the disc are well- coupled, no evidence for the expansion of the dust disc, as expected from viscous evolution, is observed (a Spearman rank correlation test revealed ρ = 0.12 and a probability of 52% that the same ρ could be found by chance). Instead, a range of disc radii are observed around stars at a variety of ages.

The thirteen measurements of gas disc radii were also used to investigate the temporal evolution of the radial extent of the molecular gas component. The left-hand and middle panels of Fig. 4.5 show the temporal evolution of the radial extent of the dust and gas components of these thirteen discs, respectively. The results of a Spearman rank correlation test revealed that considerable scatter is still present in the data (ρ = −0.33 with a p-value of 0.27). However, focusing on the gas component of the disc, tentative evidence is found for a viscously spreading gas disc (ρ = 0.61 with a p-value of 0.026).

125 Chapter 4. Outer regions of protoplanetary discs: evidence for viscous evolution? 81 . = 0 ρ . No evidence dust 6 with age ( /R 10 [ yrs ] e gas dust g R A /R gas R . The evolution of (from left to dust

10 1

and

t s u d s a g R / R gas R 026) and an increase in . 26), as for the full sample of discs in Fig. 4.4. Meanwhile, . 6 10 [ yrs ] e g A 61 with a p-value of 0 .

= 0

33 with a p-value of 0

500 100 .

s a g [ au ] au [ R − = ρ ) are observed. 4 6 − 10 [ yrs ] 10 e g × A

100 50

t s u d [ au ] au [ R for a changea in tentative dust increase radius inwith is the a observed gas p-value ( of radius 7 with age ( Figure 4.5: Disc radialright): extent the for radial the extent thirteen of star-disc systems dust; with the measurements radial of extent of gas; and the ratio of the radial extent of gas to dust,

126 4.3. Discussion

The right-hand panel of Fig. 4.5 shows the temporal evolution of the ratio of the gas radius to the dust radius, Rgas/Rdust. The results of a Spearman rank correlation test −4 suggest that Rgas/Rdust increases with age (ρ = 0.81 with a p-value of 7.8 × 10 ). If, as was originally assumed, the dust within the disc is well-coupled to the gas, Rgas/Rdust should be constant with age. Whether this observation is real or a product of observational bias, and what possible implications this may have on viscous disc evolution theory, are addressed below.

4.3 Discussion

4.3.1 Is an evolutionary sequence observed?

Although a degree of age spread is anticipated in star-forming regions, other observational effects can lead to errors in estimating L? and, therefore, the stellar age (Soderblom et al. 2014; Section 2.4.4). The use of contemporaneous J2M- and H2M-band photometry in Section 4.1.2 minimises contamination from accretion- and disc-induced UV and IR excess emission (Briceñoet al. 1998; Luhman 1999). The age estimates presented here are also consistent with previous average age estimates for the star forming regions (de Geus et al. 1989; Hughes et al. 1994; Hillenbrand 1997; White & Ghez 2001; Preibisch et al. 2002; Comerónet al. 2003; Mer´ın et al. 2008; Wilking et al. 2008) and the observed spread in ages is larger than the typical error uncertainty (see Fig. 4.3). Therefore, the results presented in Section 4.2 are unlikely to be caused by errors in assigning stellar ages.

The results presented in Section 4.2 are also independent of the adopted PMS evolutionary model. Masses and ages were re-calculated using Tognelli et al. (2011) Pisa evolutionary models4 and are presented in Table 4.2. The same analysis conducted in Sec- tion 4.2 was performed using the ages in Table 4.2. The results of performing Spearman rank correlation tests on the data presented in Figs. 4.4 and 4.5 using Siess et al. (2000) and Tognelli et al. (2011) models are presented in Table 4.3.

4.3.2 Inﬂuence of neighbouring stars

As noted at the beginning of this chapter, close encounters within binary or multiple systems can lead to changes in the disc structure (Bouwman et al. 2006; Cieza et al. 2009;

4Seven stars lay outside the bounds imposed by the Tognelli et al. (2011) isochrones and so could not be assigned a mass or age using these models.

127 Chapter 4. Outer regions of protoplanetary discs: evidence for viscous evolution?

Table 4.2: Stellar masses and ages determined using Tognelli et al. (2011) PMS evolutionary models for the 25 stars in the sample which lay inside the bounds imposed by the model isochrones.

Source M? Age (M ) (Myr) +0.03 +0.3 GW Lup 0.34−0.03 1.3−1.2 +0.02 +0.3 HD 142560 0.58−0.03 0.6−0.5 +0.11 +0.3 IM Lup 0.62−0.02 1.1−0.1 +0.23 +0.5 JW 537 0.72−0.15 1.0−0.2 +0.14 +3.5 MLLA 324 0.35−0.11 3.9−0.9 +0.37 +0.7 LV 3 1.02−0.22 1.0−0.3 +0.09 +0.4 V2252 Ori 0.69−0.13 1.3−0.3 +0.06 +31 V2377 Ori 0.76−0.07 35−17 +0.11 +0.5 2MASS J16262367-2443138 0.78−0.14 1.4−0.3 +0.03 +0.2 2MASS J16264502-2423077 0.56−0.03 0.7−0.5 +0.13 +0.1 V1121 Oph 0.73−0.10 0.8−0.2 +0.10 +0.5 V2508 Oph 0.61−0.07 0.3−0.2 +0.10 +0.3 Haro 6-33 0.47−0.04 1.2−0.2 +0.03 +0.4 BP Tau 0.43−0.02 0.7−0.6 +0.04 +0.2 AA Tau 0.43−0.02 1.2−0.4 +0.10 +0.3 CI Tau 0.68−0.05 1.0−0.2 +0.03 +0.3 CY Tau 0.34−0.03 1.3−1.2 +0.09 +0.6 DL Tau 0.73−0.08 1.5−0.3 +0.04 +0.7 DM Tau 0.32−0.05 2.9−0.5 +0.04 +0.2 DN Tau 0.45−0.02 1.1−0.3 +0.05 +0.2 DR Tau 0.62−0.06 0.6−0.5 +0.07 +0.7 GM Aur 0.72−0.08 2.0−0.6 +0.19 +0.7 HL Tau 1.22−0.21 1.4−0.4 +0.03 +0.7 IQ Tau 0.39−0.02 0.2−0.1 +0.12 +0.2 2MASS J16142029-1906481 0.71−0.09 0.9−0.1

Table 4.3: Comparison of the results of the Spearman rank correlation tests performed using the data in Figs. 4.4 and 4.5 (a – log(Rdust) versus log(age), b – log(Rgas) versus log(age), and c – log(Rgas/Rdust) versus log(age)) using Siess et al. (2000) and Tognelli et al. (2011) models.

Siess et al. (2000) model Tognelli et al. (2011) model Fig. q p-value q p-value 4.4 0.12 0.52 0.14 0.50 4.5 (a) −0.33 0.27 −0.29 0.36 4.5 (b) 0.61 0.03 0.67 0.02 4.5 (c) 0.81 7.8 × 10−4 0.79 2.4 × 10−3

128 4.3. Discussion

Guilloteau et al. 2011; Rosotti et al. 2014). These effects are not exclusive to binary or multiple systems, however. de Juan Ovelar et al. (2012) found that protoplanetary discs in star forming environments with stellar densities ≥ 103.5 pc−2 were smaller, on average, than those in less crowded fields. By comparison, the densest star forming region included in this study (the ONC) has a stellar density of ∼ 102 pc−2 (Da Rio et al. 2014). In addition, although ONC discs situated in the extreme-UV (EUV) dominated region (within 0.03 pc of the massive O star, θ1 Ori C) have been found to be less massive than those located outside of this region (Mann et al. 2014), no evidence has been found to suggest that these discs are smaller (Vicente & Alves 2005). As such, dynamical interactions and EUV radiation are not expected to affect the results presented in Section 4.2.

4.3.3 Impact of stellar mass

The stars in this sample cover a range of spectral types from K3 to M4.5, corresponding to a mass range of 0.16 − 1.44M (based on Siess et al. 2000 models; see Table 4.1). In a recent study, Andrews et al. (2013) found more massive stars host more massive discs. The possible bias that this could introduce to the results presented in Section 4.2 was investigated. A Spearman correlation test for partial correlation between disc radius and age, given the stellar mass, resulted in a probability of random distribution of 53%. Additionally, if the range of stellar masses probed by the Andrews et al. (2013) study is limited to the range probed here, the statistical signiﬁcance of the correlation between dust mass and stellar mass is not recovered. Given these factors, the range of stellar masses is not expected to aﬀect the results presented in Fig. 4.4.

4.3.4 Spatial resolution and grain size

It is clear that observations using a coarser beam will only be able to detect the largest discs whereas those that have a greater resolving power will be able to resolve smaller discs. Also, this study includes regions of star formation from 125 to 414 pc, and greater resolving power is required to detect the discs located in more distant star forming regions. To identify the impact that the combination of these two factors may have on the results presented in Section 4.2, the analysis was restricted to the ﬁve Taurus discs observed at 1.3 mm at a spatial resolution better than 100 au (namely BP Tau, CI Tau, CY Tau, DL Tau, and DM Tau; see Table 4.1). As for the full sample of 32 discs, no evidence for

129 Chapter 4. Outer regions of protoplanetary discs: evidence for viscous evolution? a linear relation between log(Rdust) and log(age) is found (a Spearman rank correlation test reveals ρ = −0.10 with a p-value of 0.87). However, the previously observed relation between log(Rgas/Rdust) and log(age) is not retained (a Spearman rank correlation test reveals ρ = 0.7 with a p-value of 0.23). Ideally, a larger sample of dust discs observed at similar spatial resolution is necessary to determine the impact of spatial resolution on the results presented here (see Chapter 6).

The sample of dust discs presented in Table 4.1 has been compiled from observations taken at a range of wavelengths (0.856-2 mm). Longer wavelengths trace emission from larger dust grains which are preferentially located closer to the stellar host (Guilloteau et al. 2011; P´erezet al. 2012) due to the size dependence of the drift velocity and the relatively short sticking timescales in the inner regions of the disc (Testi et al. 2014, and references therein). As such, the wavelength dependence of Rdust may introduce an element of bias into the results presented in Section 4.2.

As outlined in Section 4.1.3, the wavelength dependence of Rdust is more pronounced at longer wavelengths than used in this study with P´erezet al. (2012) ﬁnding very similar disc sizes for 0.88 and 2.8 mm observations. Nevertheless, as a test of whether the range of wavelengths included in this study introduces a bias to the results in Section 4.2, the discs observed at the longest wavelengths (2 mm) are removed from the sample. The apparent −3 linear relation between log(Rgas/Rdust) and log(age) remains with a p-value of 1.4 × 10 for ρ = 0.83.

4.3.5 Gas-to-dust ratio in the outer disc

High angular resolution studies have suggested a possible radial dependence of the gas-to- dust ratio in the outer disc. Using detailed modelling of combined high spatial resolution continuum images and CO line emission, Panićet al. (2009), Andrews et al. (2012), de Gregorio-Monsalvo et al. (2013), Rosenfeld et al. (2013), and Zhang et al. (2014) found that a single disc radius was unable to fit both the gas and dust emission. Instead, the gas components of the discs included in these studies was consistently found to extend to larger radii than the dust component. Also, in a recent theoretical study, Birnstiel & Andrews (2014) showed that the surface density of a sufficiently evolved disc may not be controlled solely by gas dynamics. Instead, by incorporating radial drift of dust grains and collision-induced fragmentation into their models of disc evolution, these authors

130 4.3. Discussion showed that dust grains are not well-coupled to the gas over the age ranges probed here. While the gas component in their models expanded at a rate determined by the viscosity [equation (4.8)], the dust component quickly reached a ﬁxed outer radius within 0.1 Myr.

If the gas disc is expanding due to the viscous evolution of the disc, as tentatively suggested by the middle panel of Fig. 4.5, this, combined with the constant dust disc radius (left-hand panel of Fig. 4.5), would result in an increase of Rgas/Rdust with age (right-hand panel of Fig. 4.5). Thus, the results presented in Section 4.2 may indicate a radial dependence to the gas-to-dust ratio in the outer disc.

Assuming that the gas and dust components are well-coupled at an earlier time, as suggested by Birnstiel & Andrews (2014), the spread in Rgas/Rdust observed in Fig. 4.5 may be attributed to a range of initial disc radii. The initial radius of the disc is determined by the amount of angular momentum contained within pre-stellar cores such that discs formed from higher angular momentum cores have larger initial radii (e.g. Pi´etuet al. 2014; see

Section 1.1.3). Hartmann et al. (1998) showed that if the age of the star tage tvisc, the outer radius of the disc is independent of its initial radius. Assuming a form of the viscosity described by Shakura & Sunyaev (1973), typical values of tvisc in the outer disc are ∼ 10 Myr. As the sample of star-disc systems presented here are typically younger than this (see Table 4.1), it is likely that the range of disc radii observed in Fig. 4.4 at each age is due to a spread in initial disc radii following the spherical collapse phase of star formation.

4.3.6 Is angular momentum conserved within the dust disc?

So far, the angular momentum contained within cores has been assumed to be conserved during the disc formation and evolution. However, observations of jets and outflows emanating from young stellar objects (e.g. Bally et al. 2000) show that these assumptions may not be valid (Section 1.3.4). Due to present resolution limits, it is not clear whether these jets and outflows remove angular momentum from the disc, the stellar host, or a combination of both. Current methods attempting to constrain the location of the launch- site through the rotation and morphology of these outflows find that winds originating in the inner disc (∼ 0.1 to a few au) are, in some cases, preferred to those emanating from the stellar surface (Testi et al. 2002; Bacciotti et al. 2002; Ferreira et al. 2006; Launhardt et al. 2009; Ainsworth et al. 2013; Klaassen et al. 2013).

131 Chapter 4. Outer regions of protoplanetary discs: evidence for viscous evolution?

Angular momentum can also be removed during the disc clearing process. As the gas in the disc expands to larger radii, the spatial density decreases and the disc transitions from optically thick to optically thin (Section 1.2.3). The disc is subject to ionising radiation from the stellar host or by nearby members of the same star forming region (Mann et al. 2014) and, if the gas becomes unbound, an evaporative wind can form, removing angular momentum from the disc surface (Hollenbach et al. 1994, 2000; Alexander & Armitage 2007). Further loss of angular momentum can also take place during the latter stages of planetary system formation through dynamical planet-planet interactions which result in ejections (Davies et al. 2014b).

Here, the sample of spatially resolved discs can be used to assess whether the specific angular momentum is conserved within protoplanetary discs. Focusing on the dust component of the disc, which is confined to the midplane (Chiang & Goldreich 1997), the specific angular momentum, jdust, can be modelled as that of a thin, Keplerian disc (Section 2.2.1) whereby

jdust = Rdustv(Rdust) (4.9)

1/2 where v(Rdust) = (GM?/Rdust) is the Keplerian velocity at the outer edge of the disc.

Values of jdust were calculated using the disc radii (Section 4.1.1) and stellar masses (Section 4.1.2) and are presented in Table 4.1.

Fig. 4.6 shows the resulting plot of jdust against stellar age. The results of a Spearman rank correlation test suggest no correlation exists between log(jdust) and log(age)(ρ = 0.20 with a p-value of 0.28). This same result is obtained when using the Tognelli et al. (2011) PMS evolutionary models to estimate the stellar masses and ages (ρ = 0.25 with a p-value of 0.23). Therefore, the sample of spatially resolved dust discs presented in Table 4.1 are consistent with speciﬁc angular momentum being conserved during disc evolution. In addition, the observed spread in jdust is consistent with the idea that these discs formed from cores with a range of angular momenta (e.g. Pi´etuet al. 2014; Section 4.3.5).

It is also possible to use this sample of spatially resolved discs to study the evolution of circumstellar angular momentum during the diﬀerent stages of star and planetary system formation. In Fig. 4.7, the median jdust for the dust discs is compared to that contained within pre-stellar cores (Curtis & Richer 2011), the Solar System (SS5; Cox 2000), and

5This does not include the contribution from the Sun.

132 4.3. Discussion 17 10 ] 1 − s

2 m [ t s u d j 16 10 105 106 107 108 Age [ yrs ]

Figure 4.6: Evolution of the speciﬁc angular momentum contained within the dust disc. Considerable spread is observed in the range of speciﬁc angular momenta and no dependence on the stellar age is observed (ρ = 0.20 with a p-value of 0.28). 6

● 5 ] 1 − s

2 m

4 16 10 ● [ 3 2 specific AM

● ● 1

core dust Solar Jupiter System

Figure 4.7: Median specific angular momenta for pre-stellar cores (data from Curtis & Richer 2011), dust discs (data from Table 4.1), the SS (data from Cox 2000) and Jupiter. The error bars on the core and dust data points are the median absolute deviations. An overall decrease in the specific angular momentum is observed. As- suming that the cores will not fragment and the star-disc systems did not form as binaries or multiple systems, ∼ 36% of specific angular momentum is lost during disc formation. Again, assuming that these discs do not fragment and that the SS did not form with a stellar companion, ∼ 65% of specific angular momentum is lost during disc dispersal and dynamical ejection during planetary system formation.

133 Chapter 4. Outer regions of protoplanetary discs: evidence for viscous evolution? the orbit of Jupiter. Assuming that the cores do not fragment (Section 1.2.2) and that the star-disc systems in our sample were not formed within binary or multiple systems, Fig. 4.7 may be treated as an evolutionary sequence.

Comparing the median specific angular momenta contained within the cores (5.25 × 1016 m2 s−1) and the dust discs (3.36 × 1016 m2 s−1) reveals that ∼ 36% of the specific angular momentum is lost during collapse to form the disc. Similarly, comparing the median angular momentum contained within the dust discs to that contained within the SS (1.17×1016 m2 s−1; Cox 2000), reveals that up to ∼ 65% of the specific angular momentum is lost during disc dispersal and ejection of material during the latter stages of planetary system formation. This latter result is dependent on the SS being a typical planetary system in terms of the angular momentum that it contains. This result is considered further in Chapter 5.

4.4 Summary

In this chapter, the evolution of the outer radius of the dust component of protoplanetary discs around stars not previously identified as members of binary or multiple systems has been investigated. A sample of protoplanetary discs with spatially resolved thermal continuum emission and both contemporaneous 2MASS photometry and a spectroscopically determined spectral type for the stellar host has been compiled from the literature, resulting in a sample of 32 star-disc systems. To enable a consistent assessment of radial expansion, a two-dimensional elliptical Gaussian fit to the continuum emission has been used to constrain the radial extent of dust within the disc. Spectral types and 2MASS photometry have been used to constrain the effective temperatures, Teff , and bolometric luminosities, L?, of the stellar hosts using the methods outlined in Sections 2.4.1 and 2.4.2, respectively. Siess et al. (2000) PMS evolutionary isochrones were then used to estimate stellar masses and ages from Teff and L? in a consistent manner across the sample.

No evidence for an increase in dust radius with age is found, as would be expected for a viscously evolving disc with well-coupled gas and dust components (Lynden-Bell & Pringle 1974; Hartmann et al. 1998). Instead, a variety of disc radii are observed across the age range probed (∼ 0.1 − 10 Myr). The same result is also found if Tognelli et al. (2011) Pisa PMS evolutionary models are used instead of Siess et al. (2000) models. The

134 4.4. Summary removal of binary and multiple systems together with the relatively low stellar densities of the star forming regions included in our study should limit the impact of outer disc truncation via dynamical interactions (de Juan Ovelar et al. 2012; Rosotti et al. 2014).

Thirteen of the star-disc systems in the sample also had previous estimates of the radial extent of molecular gas within the disc obtained from CO or CN line emission (Koerner & Sargent 1995; Dutrey et al. 2003; Kessler-Silacci 2004; Dutrey et al. 2008; Pani´cet al. 2009; Schaefer et al. 2009; Guilloteau et al. 2014). This subsample was used to study the temporal evolution of the gas disc. Again, no evidence for the radial expansion of the dust component of the disc is found for this subsample. However, at the same time, tentative evidence is observed for the radial expansion of the gas component. In addition, the ratio of the gas disc radius to the dust disc radius, Rgas/Rdust, is observed to increase with age. This result is again independent of the chosen PMS evolutionary model.

The possible impact of observational biases, including the range of beam sizes and observed wavelengths, have been taken into account. Pérez et al. (2012) found that longer wavelength observations, which are sensitive to larger grains, trace material closer to the star. However, this effect is more pronounced at longer wavelengths than those probed in this study. Also, smaller discs require greater physical resolutions for detection meaning that the smallest observed discs are preferentially located in the closest star forming regions imaged at the highest resolution. To investigate the possible effects that these observational biases had on the sample, the analysis was limited to the five discs in Taurus from the Guilloteau et al. (2011) study that were spatially resolved at 1.3 mm at better than 100 au resolution. As before, no evidence for the radial expansion of the dust component was observed for these five discs and tentative evidence for an increase in Rgas/Rdust remained.

The observed lack of radial expansion of the dust component combined with the tentative detection of an increase in gas disc radius with age may indicate that the evolution of the gas and dust components of the disc are governed by diﬀerent astrophysical processes, as suggested theoretically by Birnstiel & Andrews (2014). This result is also consistent with previous detailed studies of high spatial resolution continuum imaging and CO line emission for a handful of protoplanetary discs which found that the gas component consistently extends to larger radii than the dust component (Pani´cet al. 2009; Andrews et al.

135 Chapter 4. Outer regions of protoplanetary discs: evidence for viscous evolution?

2012; de Gregorio-Monsalvo et al. 2013; Rosenfeld et al. 2013; Zhang et al. 2014).

The observed scatter in the gas and dust disc radii at each age is attributed to a spread in the initial disc radii, itself resulting from the range of angular momenta contained within progenitor cores (e.g. Pi´etuet al. 2014; see Section 1.1.3). A degree of spread in the amount of angular momentum contained within protostellar cores is expected due to the turbulent properties of, and degree of magnetic braking present within, the parent molecular cloud (Burkert & Bodenheimer 2000; Hennebelle 2013; see Section 1.1.3).

Finally, the measurements of the radial extent of the dust disc and estimates of the stellar masses were used to assess whether the specific angular momentum in the dust component of the disc is conserved during disc evolution. The discs were modelled as thin, axisymmetric, Keplerian discs. No evidence for significant angular momentum removal during disc evolution was observed. The median specific angular momentum in the sample was compared to that contained within a sample of pre-stellar cores in Perseus (Curtis & Richer 2011) and to that contained within the Solar System (Cox 2000). If this is interpreted as an evolutionary sequence, significant amounts of angular momentum must be lost during disc formation, dispersal, and the latter stages of planetary system formation.

Although the sample of star-disc systems compiled for this study represent the best available at present, the small number of resolved protoplanetary discs with well-studied stellar hosts that are currently available ultimately limit the investigations undertaken. The results presented here indicate tentative support for theoretical models (e.g. Birnstiel & Andrews 2014) which predict that the dust component of accretion discs rapidly reaches a ﬁxed, grain size-dependent outer radius while the gas component continues to expand. Further study, requiring larger samples of spatially resolved protoplanetary discs with more uniform spatial resolution is required to conﬁrm this result. This is discussed further in Chapter 6.

136 5 Quantifying the angular momentum retained within planetary systems

hilst the spin rates of main sequence (MS) stars are well-studied (e.g. Barnes 2003, 2007; Mamajek & Hillenbrand 2008; Irwin & Bouvier 2009; W Irwin et al. 2011; Angus et al. 2015), little is presently known about the angular momentum content of their planetary systems. This is predominantly due to the dearth of spatially resolved debris disc detections and the limitations of current exoplanet surveys. The aim of this chapter is to quantify the angular momentum retained within stars and their planetary systems following their formation, concentrating on spatially resolved debris discs in the Submillimetre Common-User Bolometer Array-2 Observations of Nearby Stars (SONS) James Clark Maxwell Telescope (JCMT) Legacy Survey.

Calculation of the debris disc angular momentum, Jdisc, required estimates of the total mass of debris contained within the disc, Mtot; the radius of the disc, Rdisc; and the mass

137 Chapter 5. Quantifying the angular momentum retained within planetary systems of the stellar host, M?. From equation (2.14):

1/2 Jdisc = Mtot(GM?Rdisc) . (5.1)

The radii were estimated from Gaussian ﬁts to the intensity map of each disc (see Sec- tion 2.5.3) while estimates of the masses were gathered from spectral energy distribution (SED) ﬁtting (see Section 2.6).

The work presented in this chapter has been conducted by both myself and other members of the SONS consortium. The work undertaken by my collaborators is included here for completeness. An overview of the observations taken by myself and members of the SONS consortium1, along with the data reduction undertaken by Wayne S. Holland, is presented in Section 5.1. I performed the fitting to estimate the disc radii using a code initially developed by Bruce Sibthorpe (the details and results of which are outlined in Section 5.2.1). The SEDs were produced and fit using debris disc models (see Kennedy et al. 2012 for details) by Grant M. Kennedy. I used the resulting fits (which are detailed in Table 5.2) to determine stellar masses, dust masses, and the total debris disc masses (the results of which are presented in Sections 5.2.2 and 5.2.3). I calculated the angular momentum of the stellar hosts, their debris disc, and any known exoplanets. The results of these calculations are presented in Section 5.2.4.

A discussion of the results in the context of the angular momentum evolution of the circumstellar material during protoplanetary disc dispersal is presented in Section 5.3. This builds upon the work presented towards the end of Section 4.3.6.

5.1 The Submillimetre Common-User Bolometer Array-2 Observations of Nearby Stars James Clark Maxwell Tele- scope Legacy Survey

Originally known as the Submillimetre Common-User Bolometer Array-2 Unbiased Nearby Stars (SUNS) survey, SONS was initially designed to be an unbiased survey of 500 stars of spectral types A to M (Phillips et al. 2010). However, the debris disc key programmes

1Including Mark Booth, Andy G. Gibb, Jane S. Greaves, Wayne S. Holland, Grant M. Kennedy, Jean- Fran¸coisLestrade, Jonathan P. Marshall, Brenda C. Matthews, Olja Pani´c,Neil M. Phillips, Bruce Sibthorpe, Jonathan Tottle, and Mark C. Wyatt.

138 5.1. The Submillimetre Common-User Bolometer Array-2 Observations of Nearby Stars James Clark Maxwell Telescope Legacy Survey

Figure 5.1: The 15 m James Clark Maxwell Telescope following a successful night of SONS observations on 11 May 2014. The blue box housing SCUBA-2 is visible on the left side of the image. of the Herschel space observatory identiﬁed the candidate discs within the original SUNS target sample. As a result, the targets of the SUNS survey were altered and their number reduced to 100. The survey was no longer unbiased and instead focused on targets most likely to result in detection, resulting in the name change (Matthews et al. 2013).

Each of the 100 SONS targets were observed simultaneously at 850 µm and 450 µm using the Submillimetre Common-User Bolometer Array-2 (SCUBA-2; Holland et al. 2003a; Audley et al. 2004) instrument on the JCMT (Fig. 5.1) between February 2012 and September 2014. The observations were conducted in JCMT weather bands 2 and 3, corresponding to precipitable water vapour levels of 1 − 2.4 mm and zenith sky opacities of τ850 µm = 0.21 − 0.53. Individual observations were typically of 30 min duration, while the total integration time on each target ranged between 1 and 4 hours with the aim of reaching an rms noise level of < 1.4 mJy beam−1 at 850 µm for each source (Pani´cet al. 2013; Holland et al. in prep.). The full width half maximum (FWHM) of the primary beam of the JCMT is 14 arcsec at 850 µm and 7.5 arcsec at 450 µm.

The data were reduced using the Dynamic Iterative Map-Maker within the starlink smurf package (Chapin et al. 2013) using a configuration file specially designed for faint, compact sources (c.f. Panićet al. 2013; Holland et al. in prep.). Low frequency noise (e.g. astronomically- or instrumentally-induced large-scale structures) was removed using a frequency-domain, high-pass filter at 1 Hz which resulted in a flat, uniform intensity

139 Chapter 5. Quantifying the angular momentum retained within planetary systems

Figure 5.2: Maps of 850 µm emission from the thirteen debris discs which exhibit extended emission at this wavelength. The HD numbers are displayed in the top right corner of each map.

140 5.1. The Submillimetre Common-User Bolometer Array-2 Observations of Nearby Stars James Clark Maxwell Telescope Legacy Survey

Figure 5.3: Maps of 450 µm emission from the two SONS debris discs which exhibit extended emission at this wavelength. The HD numbers are displayed in the top right corner of each map. map for each target. The data were flux calibrated against the primary calibrator Uranus and also secondary calibrators CRL 618 and CRL 2688 from the JCMT calibrator list (Dempsey et al. 2013). The final maps have estimated flux calibration uncertainties of ∼ 5% and ∼ 10% at 850 µm and 450 µm, respectively. Finally, the images were smoothed with a 7 arcsec FWHM Gaussian to improve the signal-to-noise ratio (Panićet al. 2013; Holland et al. in prep.).

Of the 100 SONS targets, twelve featured > 3σ emission extending outside the 14 arcsec diameter of the primary beam at 850 µm. These extended sources formed the basis of the sample considered here. A further two discs featured > 3σ emission extending outside the 7.5 arcsec diameter of the primary beam at 450 µm and were added to this initial sample. An additional debris disc, ε Eri (HD 22049), observed as part of the SCUBA-2 Guaranteed Time observations also displayed extended emission at 850 µm and was added to the sample. This resulted in a total of ﬁfteen extended debris discs. The 850 µm and 450 µm intensity maps displaying the extended emission of each disc are shown in Figs. 5.2 and 5.3, respectively.

The noise in each of the intensity maps displayed in Figs. 5.2 and 5.3 was estimated using multiple measurements of the flux densities across each image. The flux within circular 14 arcsec beams placed 4 arcsec apart over the central 3 × 3 arcmin2 area was integrated and the resulting distribution of flux was fitted with a Gaussian (c.f. Panić et al. 2013; Holland et al. in prep.). The standard deviation of these fits served as an estimate of the noise level in each image. The observed 850 µm fluxes and noise levels of these fifteen discs are listed in Table 5.1.

141 Chapter 5. Quantifying the angular momentum retained within planetary systems

Table 5.1: Observations summary for the SONS targets which appear extended at 850 µm or 450 µm. Columns 1 & 2: target HD number and alternative names, where applicable; column 3: spectral type; column 4: stellar distance and its reference (a – Butler et al. 2006; b – van Leeuwen 2007); column 5: Integrated ﬂux at 850 µm; columns 6 & 7: Gaussian major and minor axes (c – smoothed with an additional 27 arcsec beam; d – smoothed with an additional 17.5 arcsec beam); column 8: the disc radius.

Target Alt. Name SpT Dist F850 µm Major Minor Rdisc (HD) (pc) (mJy) (arcsec) (arcsec) (au) (1) (2) (3) (4) (5) (6) (7) (8) 22049 ε Eri K2V 3.22 ± 0.01 a 19.0 ± 0.7 45 ± 5 c 31 ± 7 c 90 ± 5 109085 η Crv F2V 18.28 ± 0.06 b 15.5 ± 1.7 25 ± 6 21 ± 4 270 ± 25 216956 Fomalhaut A4V 7.70 ± 0.03 b 90 ± 4 44 ± 1 c 29 ± 1 c 206 ± 2 10647 q1 Eri F9V 17.43 ± 0.08 b 30.3 ± 4.1 23 ± 2 — 240 ± 10 107146 LTT 13439 G2V 27.5 ± 0.4 b 20.5 ± 4.5 — — — 127821 NLTT 37640 F4IV 31.8 ± 0.3 b 21.5 ± 5.0 41 ± 1 c 32 ± 4 c 760 ± 10 161868 γ Oph A0V 31.5 ± 0.2 b 14.2 ± 2.5 33 ± 4 19 ± 6 570 ± 30 170773 HR 6948 F5V 37.0 ± 0.6 b 27.1 ± 3.0 51 ± 3 d — 1100 ± 30 207129 — G2V 16.0 ± 0.1 b 10.5 ± 1.8 28 ± 7 — 260 ± 25 218396 HR 8799 A5V 39 ± 1 b 28.7 ± 3.8 32 ± 3 25 ± 3 700 ± 30 38858 LTT 2380 G4V 15.2 ± 0.1 b 12.5 ± 1.8 38 ± 5 c 36 ± 4 c 370 ± 15 48682 56 Aur F9V 16.72 ± 0.08 b 6.5 ± 1.1 60 ± 5 — 520 ± 20 172167 Vega A0V 7.68 ± 0.02 b 20.4 ± 1.1 24 ± 2 23 ± 2 110 ± 5 197481 AU Mic M1Ve 9.9 ± 0.1 b 14.2 ± 2.2 — — — 9672 49 Ceti A1V 59 ± 1 b 14.8 ± 3.1 — — —

5.2 Results

5.2.1 Radial extent of the debris discs displaying extended emission

The intensity maps displayed in Figs. 5.2 and 5.3 were used to estimate the radial extent of the debris discs. Each map was fit using a two-dimensional Gaussian function using the idl routine mpfit2dfun (Markwardt 2009). To estimate the significance of the extended emission, an artificial point source was inserted across each image and the fitting procedure repeated (see Section 2.5.3). The resulting distribution of FWHM was fit with a

Gaussian and the standard deviation, σg, of these ﬁts provided an estimate of the expected “extension” of a point source in the same image.

Following the Gaussian fitting, the residual maps were visually inspected. The intensity maps of five of the discs (HD 22049, HD 216956, HD 127821, HD 38858, and HD 170773) were not well-approximated by a Gaussian profile as the emission was not centrally concentrated (see Figs. 5.2 and 5.3). The 850 µm intensity maps for HD 22049, HD 216956, HD 127821, and HD 38858 were smoothed with an additional 27 arcsec FWHM Gaussian using the gausmooth subroutine of the starlink kappa package (Currie & Berry 2013;

142 5.2. Results see Section 2.5.3). This has the same effect as increasing the stellar distance by a factor of two. Similarly, the 850 µm intensity map of HD 170773 was smoothed using an additional 17.5 arcsec FWHM Gaussian, effectively artificially increasing the stellar distance by a factor of 1.5.

The resulting FWHM of the major and minor axes of the two-dimensional Gaussian

ﬁts are presented in Table 5.1. Only discs that featured extended emission at > 3σg were identiﬁed as being spatially resolved. In the cases where values are missing from Table 5.1, the extended emission had FWHM < 3σg. The radii of the discs displaying > 3σg extended emission were then calculated using an adapted form of equation (2.32):

q 2 2 2 Rdisc = d/2 FWHMx − (FWHMb + FWHMsmo), (5.2) where the subscripts x, b, and smo refer to the major axes of the two-dimensional Gaus- sian fit, the beam2, and the additional 7 arcsec FWHM Gaussian smoothing component, respectively. For the intensity maps that were ill-fit by a Gaussian profile, the FWHMsmo also accounted for the additional 27 arcsec or 17.5 arcsec FWHM Gaussian smoothing applied. The radial extents of the spatially resolved debris discs are presented in Table 5.1.

5.2.2 Stellar mass

SEDs for the spatially resolved SONS debris discs were constructed by Grant. G. Kennedy using multi-wavelength photometry gathered from the literature.3 The adopted model used to fit the SEDs consisted of a stellar atmosphere component and an additional disc component which describes the infra-red (IR) excess (Kennedy et al. 2012; see Fig. 2.10 in Section 2.6). Photometry shortward of 10 µm was used to fit the stellar component with phoenix Gaia models (Brott & Hauschildt 2005) via least-squares minimisation. This resulted in an estimate of the stellar effective temperature, Teff , and the stellar radius, R?, which are presented in Table 5.2.

2The beam is assumed to be circular. 3Helou & Walker (1988); Bessell (1990); Moshir & et al. (1990); Sylvester et al. (1996); ESA (1997); Hauck & Mermilliod (1998); Holland et al. (1998); Høg et al. (2000); Wilner et al. (2002); Cutri et al. (2003); Holland et al. (2003b); Sheret et al. (2004); Williams et al. (2004); Najita & Williams (2005); Wyatt et al. (2005); Beichman et al. (2006); Mermilliod (2006); Su et al. (2006); Williams & Andrews (2006); Carpenter et al. (2008); Liseau et al. (2008); Roccatagliata et al. (2009); Ishihara et al. (2010); Liseau et al. (2010); Nilsson et al. (2010); Sibthorpe et al. (2010); Wright et al. (2010); Hughes et al. (2011); Marshall et al. (2011); Moóret al. (2011); Acke et al. (2012); Gáspáret al. (2013); Su et al. (2013); Chen et al. (2014); Duchêneet al. (2014); Matthews et al. (2014).

143 Chapter 5. Quantifying the angular momentum retained within planetary systems

Table 5.2: Results obtained from the SED fitting. Column 1: target HD number; column 2: stellar effective temperature; column 3: stellar radius; column 4: dust temperature (or the cool component of the disc if fit by the two-temperature model); column 5: warm component of disc, where applicable; column 6: fractional luminosity, f=Ldust/L?; columns 7 & 8: cut-off wavelength and exponent in equation (2.36) used to describe the emission efficiency at long wavelengths (a – value fixed in the fitting procedure; see text for details).

Target Teﬀ R? Tdust Twarm log (f) λ0 β (HD) (K) R (K) (K) (µm) (1) (2) (3) (4) (5) (6) (7) (8) 22049 5100 ± 12 0.74 ± 0.04 31.0 ± 0.5 120 ± 20 −3.92 ± 0.04 65 ± 8 1.1 ± 0.1 109085 6802 ± 30 1.66 ± 0.08 40.2 ± 3.5 212 ± 18 −3.92 ± 0.11 155 ± 172 0.2 ± 0.1 216956 8600 ± 114 1.86 ± 0.09 22.2 ± 3.1 105 ± 4 −2.81 ± 0.08 266 ± 46 1.4 ± 0.1 10647 6182 ± 32 1.10 ± 0.06 59.3 ± 0.6 — −3.51 ± 0.03 668 ± 67 1a 127821 6616 ± 25 1.33 ± 0.07 48.1 ± 0.7 — −3.80 ± 0.03 557 ± 107 1a 161868 8940 ± 107 2.2 ± 0.1 62.2 ± 4.2 120 ± 10 −2.10 ± 0.12 447 ± 72 1a 170773 6697 ± 27 1.42 ± 0.07 44.4 ± 0.6 — −3.55 ± 0.03 453 ± 52 1a 207129 5977 ± 26 1.06 ± 0.05 48.0 ± 1.6 — −3.95 ± 0.07 78 ± 10 0.6 ± 0.1 218396 7384 ± 35 1.43 ± 0.07 36.4 ± 0.7 129 ± 9 −3.87 ± 0.06 462 ± 142 1.5 ± 0.8 38858 5780 ± 22 0.91 ± 0.05 53.3 ± 1.5 — −2.07 ± 0.05 935 1a 48682 6047 ± 44 1.25 ± 0.06 36.9 ± 6.6 — −2.43 ± 0.47 96 ± 49 0.8 ± 0.2 172167 9170 ± 105 2.9 ± 0.2 47.4 ± 6.0 131 ± 32 −2.73 ± 0.24 73 ± 8 1.7 ± 0.1

The stellar luminosity, L?, was estimated from Teﬀ and R? using the Stefan-Boltzmann equation. Then, using an appropriate mass-luminosity relation (see Section 2.6), an estimate of the stellar mass, M?, was made. The calculated stellar masses are displayed in Table 5.3.

5.2.3 Dust mass

The component of a debris disc SED longward of ∼ 10 µm is provided by thermal emission from dust grains. As the emission is optically thin at submillimetre wavelengths, an estimate of the dust mass, Mdust, may be made using the 850 µm ﬂux, F850 µm. From equation (2.34), 2 d F850 µm Mdust = . (5.3) κνBν(Tdust)

2 −1 Here, d is the stellar distance, κν = 1.7 cm g is the dust opacity (Zuckerman & Becklin

1993), and Bν(Tdust) is the Planck function at dust temperature, Tdust.

Distances to the stellar hosts of each of the spatially resolved SONS debris discs were retrieved from the literature and are presented, along with their individual references, in Table 5.1. The dust temperatures were estimated by fitting the dust component of the debris disc SEDs with a single- or dual-temperature modified blackbody model, as described in Section 2.7. These models account for the reduced emission efficiency, Xλ, of

144 5.2. Results

Table 5.3: Angular momenta contained within the spatially resolved SONS debris discs. Column 1: target HD number; column 2: dust mass; column 3: stellar mass; column 4: stellar age from Rhee et al. (2007) (except for: a – Delgado Mena et al. 2014); column 5: total disc mass; column 6: angular momentum contained within the disc. Mdust Mtot Target log M? tage log log (Jdisc) M⊕ M⊕ 2 −1 (HD) (M ) (Myr) (kg m s ) (1) (2) (3) (4) (5) (6) 0.05 22049 −2.800.03 0.78 ± 0.09 730 0.59 ± 0.09 40.4 ± 0.1 0.05 109085 −1.500.04 1.49 ± 0.14 300 1.64 ± 0.37 41.9 ± 0.4 0.07 216956 −1.230.05 2.25 ± 0.24 220 0.69 ± 0.40 40.9 ± 0.4 0.01 10647 −1.420.01 1.17 ± 0.12 300 1.99 ± 0.30 42.1 ± 0.3 0.01 127821 −0.950.01 1.45 ± 0.14 200 2.24 ± 0.37 42.7 ± 0.4 0.03 161868 −1.250.03 2.48 ± 0.26 200 1.49 ± 0.25 42.0 ± 0.2 0.01 170773 −0.680.01 1.47 ± 0.13 200 2.80 ± 0.30 43.3 ± 0.3 0.02 207129 −1.860.02 1.03 ± 0.10 600 0.91 ± 0.35 41.0 ± 0.4 0.01 218396 −0.530.01 1.67 ± 0.15 30 0.34 ± 0.12 40.8 ± 0.1 0.02 a 38858 −1.870.02 0.94 ± 0.11 8480 0.69 ± 0.40 40.9 ± 0.4 0.10 48682 −1.920.07 1.07 ± 0.12 600 0.44 ± 0.99 40.7 ± 1.0 0.06 172167 −2.200.05 2.65 ± 0.44 220 0.96 ± 0.57 41.1 ± 0.6 large grains by introducing λ0 and β [equation (2.36)]:

 1 λ < λ0 Xλ = . β (λ/λ0) λ > λ0

The goodness of fit for each model was assessed to determine whether each disc was fit better by a single-temperature (Tdust) or dual-temperature (Tdust and Twarm) model. The resulting best-fit model parameters are presented in Table 5.2.

Six of the spatially resolved SONS discs were better ﬁt by a dual-component blackbody. The cool temperature component dominates the emission at 850 µm (Pani´cet al. 2013) and is used in equation (5.3) to estimate Mdust. Dust masses for all of the discs are presented in Table 5.3.

Total debris disc mass

The majority of the mass of the debris disc is contained within more massive particles (e.g. pebbles and planetesimals; Section 2.2.2) which contribute little to the observed ﬂux (Section 2.7.1). Provided an appropriate estimate of the maximum and minimum particle size within the disc can be made, the total mass contained within the disc, Mtot, can be determined.

145 Chapter 5. Quantifying the angular momentum retained within planetary systems

As each of the stellar hosts in the sample of spatially resolved SONS discs presented here have spectral types earlier than M (see Table 5.1), the minimum dust grain size is set by the radiation pressure (Wyatt et al. 1999; Wyatt 2005). From equation (2.44), the size of the smallest grain in the disc, measured in µm, is accordingly given by

L? M 2700 Dmin = 0.8 , L M? ρ where ρ is the dust density, typically assumed to be that of basalt (2700 kg m−3; Wyatt

2008). At the same time, the size of the most massive particle, Dmax, involved in the collisional cascade which produced the debris disc is set by the age of the system, tage (see

Section 2.7.1). A particle larger than Dmax has a catastrophic collision timescale longer than the age of the system and is therefore unlikely to form part of the cascade.

Following Wyatt et al. (2007), the total mass contained within the disc (measured in

Earth masses, M⊕) is then given by,

dr −1 M = 4.16 × 109R−1/3 Q? −5/6e5/3M 1/3t f 2L . (5.4) tot disc r D ? age ?

−1 Here, QD = 200 J kg is the particle strength, e is the average eccentricity of the orbits of particles within the disc4, and (dr/r) is the width of the disc. As the Gaussian ﬁtting procedure used to constrain Rdisc does not provide an estimate of (dr/r), a typical disc width of 0.1 (e.g. Booth et al. 2013) is adopted here.

The fractional luminosities of the discs, f = Ldisc/L?, were estimated as part of the SED ﬁtting and are presented in Table 5.2. Estimates of the ages of the stellar hosts were gathered from Rhee et al. (2007) and, in the case of HD 38858, from Delgado Mena et al.

(2014). These are presented alongside the calculated values of Mtot in Table 5.3.

5.2.4 Angular momentum components of the planetary systems

The debris disc angular momenta, Jdisc, were calculated from the disc radii (Section 5.2.1), stellar masses (Section 5.2.2), and total disc masses (Section 5.2.3) using equation (5.1).

The resulting values of Jdisc for each of the spatially resolved SONS debris discs are presented in Table 5.3.

4The eccentricity, e, is typically assumed to be equal to the inclination, i, and the function, f(e, i) = (1.25e2 + i2)1/2 = 0.1.

146 5.2. Results

Stellar angular momentum

An estimate of the stellar angular momentum, J?, was also made. Modelling a MS star as a polytropic sphere of mass, M?, and radius, R?, rotating at angular velocity, Ω?, the angular momentum is given by equation (2.1):

2 2 J? = k M?R?Ω?,

(Chandrasekhar & M¨unch 1950; Krishnamurthi et al. 1997; see Section 2.1). Here, k2 is the radius of gyration which is related to the polytropic index, n (Motz 1952). The masses of all twelve stellar hosts indicate that their interiors are either fully radiative or consist of a convective core and a radiative envelope. In both of these cases, the radiative component dominates the mass budget and the radius of gyration can be approximated to that of a fully radiative star (n = 3). In this case, k2 = 0.08 (Motz 1952).

A direct measure of Ω? is not possible. Instead, a measure of the projected stellar rotation velocity, v sin i, is made from the rotational broadening of spectral lines (see

Section 2.3.1). As such, the projected stellar angular momentum, J? sin i, is measured, rather than J?. This provides a lower limit to the stellar angular momentum. From equation (2.15): 2 J? sin i = k M?R?v sin i.

Ten of the stellar hosts had v sin i measurements available in Glebocki & Gnacinski (2005) while measurements for HD 216956 were taken from Hadjara et al. (2014). No measurement of v sin i was found in the literature for HD 170773 and, as a result, calculation of J? sin i for this object was not possible. The values of v sin i and calculated J? sin i are presented in Table 5.4.

Exoplanet angular momentum

Five of the twelve spatially resolved SONS debris discs exist around stars hosting known exoplanets. A brief description of the exoplanets considered here is given below.

ε Eri b: The motion of the stellar host, ε Eri (HD 22049), around the centre of mass of the star-planet system has been detected via radial velocity and astrometric methods (Benedict et al. 2006). The former of these methods detects the motion of the star into

147 Chapter 5. Quantifying the angular momentum retained within planetary systems

Table 5.4: Angular momentum of the stellar hosts. Column 1: target HD number; columns 2 & 3: projected rotational velocity and its reference; column 4: projected stellar angular momentum.

Target v sin i Reference log (J? sin i) (HD) (km s−1) (kg m2 s−1) (1) (2) (3) (4) 22049 1.9 ± 0.6 Glebocki & Gnacinski (2005) 41.1 ± 0.1 109085 66.8 ± 12.0 Glebocki & Gnacinski (2005) 43.3 ± 0.1 216956 93.0 ± 16.0 Hadjara et al. (2014) 43.6 ± 0.1 10647 5.4 ± 0.7 Glebocki & Gnacinski (2005) 41.9 ± 0.1 127821 46.8 ± 16.6 Glebocki & Gnacinski (2005) 43.0 ± 0.2 161868 201.0 ± 12.7 Glebocki & Gnacinski (2005) 44.1 ± 0.1 170773 — — — 207129 1.8 ± 2.1 Glebocki & Gnacinski (2005) 41.3 ± 0.5 218396 47.2 ± 7.6 Glebocki & Gnacinski (2005) 43.1 ± 0.1 38858 2.7 ± 0.5 Glebocki & Gnacinski (2005) 41.4 ± 0.1 48682 4.3 ± 0.5 Glebocki & Gnacinski (2005) 41.8 ± 0.1 172167 22.8 ± 4.3 Glebocki & Gnacinski (2005) 43.3 ± 0.1 and out of the plane of the sky, induced by the presence of a companion, by analysing Doppler-shifted spectral lines. The latter method detects motion of the star on the plane of the sky, induced by the presence of a companion, using photometric parallax. On its own, the radial velocity method can only constrain the minimum planet mass, Mp sin i. However, when combined with the astrometric method, an estimate of the line-of-sight inclination of the orbit, i, is possible, thus enabling a direct estimate of the planet mass,

Mp.

Fomalhaut b: The presence of an exoplanet in orbit around HD 216956 has been suggested from detailed inspection of multiple-epoch coronographic observations obtained with the Hubble Space Telescope (HST; Kalas et al. 2008). Assuming the planet is coplanar with the disc, Kalas et al. (2008) estimated a semi-major axis, a = 115 au. These authors also placed a lower limit on the orbital eccentricity of Fomalhaut b (e & 0.13) by comparing the deprojected space velocity to the circular speed of the planet and assuming that the planet is observed at periastron. An upper limit to the mass of the planet (3 MJ) was also made by considering the dynamics of the interaction between the planet and the debris disc.

HD 10647 b: Butler et al. (2006) include this object in their catalogue of exoplanets discovered via transit and radial velocity surveys that lay within 200 pc of the Sun. HD 10647 b itself was discovered using the radial velocity method. These authors calculated Mp sin i for the planet, which provides a lower limit on the mass of the planet.

148 5.2. Results

Table 5.5: Orbital angular momenta of the known exoplanets included in the sample of spatially resolved SONS debris discs. Column 1: exoplanet name; column 2: planet mass; column 3: semi-major axis; column 4: eccentricity; column 5: references (1 – Benedict et al. 2006; 2 – Kalas et al. 2008; 3 – Butler et al. 2006; 4 – Marois et al. 2008; 5 – Soummer et al. 2011; 6 – Marois et al. 2010; 7 – Mayor et al. 2011); column 6: orbital angular momentum.

Target Mp a e Refs log (Jp) 2 −1 (HD) (MJ) (au) kg m s (1) (2) (3) (4) (5) (6) ε Eri b 1.55 ± 0.24 3.39 ± 0.36 0.702 ± 0.039 1 41.68 ± 0.09 Fomalhaut b . 3 115 & 0.13 2 < 43.1 HD 10647 b ≥ 0.93 ± 0.18 2.03 ± 0.15 0.16 ± 0.22 3 ≥ 41.58 ± 0.09 HR 8799 b ∼ 5 24 0 4, 5 ∼ 42.9 HR 8799 c ∼ 7 38 0 4, 5 ∼ 43.2 HR 8799 d ∼ 7 68 0.10 4, 5 ∼ 43.3 HR 8799 e ∼ 7 14.5 ≈ 0 6 ∼ 43.0 HD 38858 b ≥ 0.0961 ± 0.012 1.038 ± 0.019 0.27 ± 0.17 7 ≥ 40.39 ± 0.06 HR 8799 b, c, d, and e: Like Fomalhaut b, the four planetary companions to HD 218396 have been identiﬁed using direct imaging. Marois et al. (2008) used high- contrast observations with Keck and Gemini telescopes to identify HR 8799 b, c, and d. Discovery of a fourth planet, HR 8799 e, was announced by Marois et al. (2010). Planet mass estimates were determined through comparison of measured planet luminosities to cooling tracks based on core-accretion models, using an age estimate of 60 Myr for the planetary system. The masses presented in Table 5.5 have been revised down from those estimated by Marois et al. (2008) and Marois et al. (2010) to reﬂect the younger age adopted here (30 Myr). Marois et al. (2010) and Soummer et al. (2011) placed restrictions on the eccentricities of the four planets in the system by requiring that the planets exist in dynamically stable orbits.

HD 38858 b: Mayor et al. (2011) discovered a planet around HD 38858 using the radial velocity method. A lower limit to the possible mass of the planet is provided by their estimation of Mp sin i.

The orbital angular momentum of these exoplanets dominates over their rotational angular momentum. Therefore, the angular momentum of a planet of mass, Mp, is well- approximated by

Jp = µl. (5.5)

Here, M M µ = ? p , (5.6) M? + Mp

149 Chapter 5. Quantifying the angular momentum retained within planetary systems is the reduced mass and 2 1/2 l = [a(1 − e )G(M? + Mp)] , (5.7) is the speciﬁc relative angular momentum of a planetary orbit with semi-major axis, a, and eccentricity, e. The masses, orbital parameters, and angular momenta of the exoplanets known to exist in the star-debris disc systems considered here are summarised in Table 5.5.

5.3 Discussion

5.3.1 Angular momentum evolution during disc dispersal

In Section 4.3.6, constraints were placed on the amount of angular momentum lost during the latter stages of planet formation and disc dispersal. The median angular momentum contained within the sample of protoplanetary discs was compared to that retained within the Solar System (SS). Here, this analysis is extended to include the sample of spatially resolved SONS debris discs. It is important to bear in mind that the surveys of these extra-solar planetary systems are by no means complete and fainter discs, smaller planets and/or planets on wider orbits may remain undetected. As such, both the debris disc and planetary angular momentum components calculated here are treated as lower limits to the angular momentum contained within the planetary systems.

The analysis was restricted to planetary systems where the host star has not previously been identiﬁed as a member of a binary or multiple system. The dynamical evolution of discs in binary and multiple systems is more complex (see introduction to Chapter 4). Furthermore, the angular momentum of wide-binary systems is dominated by the orbit of the secondary. Ten of the stellar hosts considered here lack stellar companions (Rodriguez & Zuckerman 2012; Ertel et al. 2014) while wide-orbit stellar companions to HD 216956 and HD 207129 have been found (Luyten 1938; Barrado y Navascu´eset al. 1997; Di Folco et al. 2004; Eggleton & Tokovinin 2008; Mamajek 2012; Mamajek et al. 2013). As a result, HD 216956 and HD 207129 were removed from further analysis.

Fig. 5.4 compares the median speciﬁc angular momentum contained within the dust component of protoplanetary discs calculated in Chapter 4 (3.36 × 1016 m2 s−1) with that retained within the SS5 (1.17 × 1016 m2 s−1; Cox 2000), as in Section 4.3.6. Added to

5This does not include the contribution from the Sun.

150 5.3. Discussion 16 10 − − ] − − 1 − − s

− − 2 − −

m − [ j 15

10 −

− −

14 − 10

proto− Solar planetary System discs 22049 10647 38858 48682 109085 127821 161868 170773 218396 172167

Figure 5.4: Comparison of the speciﬁc angular momentum, j, contained within the protoplanetary disc sample presented in Chapter 4 and that retained within the SS and the spatially resolved SONS debris discs whose stellar hosts are not members of binary or multiple systems (labelled by their HD numbers). Data for the SS was taken from Cox (2000). The error bar on the protoplanetary disc data point represents the median absolute deviation. The lower limits placed on j in the planetary systems by the estimates of jdisc are shown by black arrows. The contributions of exoplanets to j, where applicable, are shown by red arrows. The blue region marks the median j (± the median absolute deviation) for the sample of SONS debris discs.

151 Chapter 5. Quantifying the angular momentum retained within planetary systems this are the lower limits on the speciﬁc angular momentum contained within the spatially resolved debris discs, jdisc = Jdisc/Mtot. For the debris discs that exist around known exoplanet-hosting stars, the speciﬁc relative angular momentum of the planet orbit, l, is also shown. In the case of HD 218396, the value plotted is the summed contribution for all four planets (HR 8799 b, c, d, and e).

15 2 −1 The median jdisc for the SONS planetary systems is 3.05 × 10 m s . Considering this value as a lower limit to the angular momentum retained within planetary systems, this is consistent with the results presented in Section 4.3.6 whereby up to ∼ 65% of angular momentum contained within protoplanetary discs is lost during disc dispersal and the latter stages of planet formation.

5.4 Summary

This chapter has presented a quantified discussion of the angular momentum retained in planetary systems following protoplanetary disc dispersal. The sample is based on the SONS survey of debris discs and focuses on the fifteen discs which appear extended at 850 µm or 450 µm. Through comparison of the observed FWHM to that of a point source artifically inserted into the image, twelve of the discs showed evidence of significantly extended emission and were deemed to be spatially resolved. Radii estimates for these spatially resolved discs, ranging from 90 − 1100 au, were obtained from the deconvolved FWHM of the Gaussian intensity profiles fit to the observed intensity maps.

The SONS 450 µm and 850 µm fluxes were combined with optical, IR, and submillimetre photometry obtained from the literature to construct an SED for each of the spatially resolved debris discs. phoenix Gaia stellar atmospheric models (Brott & Hauschildt 2005) were fit to the component of the SED shortward of 10 µm, resulting in an estimate of the stellar effective temperature and stellar radius. From this, an estimate of the stellar mass was made. Single-temperature (Tdust) and dual-temperature (Tdust and Twarm) modified blackbody models were fit to the emission longward of 10 µm. The results of this fitting procedure were used to determine total debris disc masses of 2 − 630 M⊕. In comparison, the mass contained within typical protoplanetary discs is ∼ 100−7000 M⊕ (Andrews et al. 2013).

The stellar masses, disc masses, and disc radii were used to calculate the total disc

152 5.4. Summary angular momenta, Jdisc, using equation (5.1), resulting in Jdisc ranging between 2.5 × 1040 and 2.0 × 1043 kg m2 s−1. The stellar mass and radii estimates resulting from the short-wavelength component of the SED ﬁtting were combined with projected rotational velocities available in the literature to place lower limits on the angular momenta of the 41 stellar hosts, J? sin i. The resulting values of J? sin i range between 1.3 × 10 and 1.3 × 1044 kg m2 s−1. Also, ﬁve of the twelve spatially resolved SONS star-debris disc systems were previously known to harbour exoplanets. The orbital angular momenta, Jp, of these planets were calculated by combining the stellar mass estimates with estimates of the planet masses and orbital parameters, gathered from the literature. Three of the planets only had lower limits on their masses and, as such, their angular momenta is also a 40 lower limit. The resulting values of Jp ranged between a lower limit of 2.5 × 10 and 2.0 × 1043 kg m2 s−1.

Finally, a comparison was made between the speciﬁc angular momentum contained within the spatially resolved protoplanetary discs in Chapter 4 and that retained within the Solar System (not including the solar component) and the spatially resolved SONS planetary systems (not including their stellar hosts). As the surveys of the planetary systems considered here are by no means complete, faint discs, low-mass planets, and/or wide-orbit planets may remain undetected. As a result, the sample of planetary system angular momenta presented here serves as a lower limit to the angular momentum retained in planetary systems following protoplanetary disc dispersal. Considering the sample as such, the speciﬁc angular momentum retained within spatially resolved debris discs is consistent with the result presented in Section 4.3.6, whereby up to ∼ 65% of angular momentum is removed during protoplanetary disc dispersal and the latter stages of planet formation. This likely occurs via a disc or photo-evaporative wind (Black & Scott 1982; Shu et al. 1993; Alexander & Armitage 2007) or through ejection of planetary material through dynamical interactions of planets and planetesimals in the disc (Davies et al. 2014b).

153 Chapter 5. Quantifying the angular momentum retained within planetary systems

154 6 Summary and future work

t the beginning of this thesis, an overview of angular momentum evolution during star and planetary system formation was presented, focusing on disc A formation; disc expansion; jets and outflows emanating from the star-disc system; and the methods proposed to avoid stars reaching break-up velocities as they spin-up during their contraction from molecular cloud cores to the zero-age main sequence (ZAMS). The relatively slow rotation rates observed for pre-main sequence (PMS) stars required large quantities of angular momentum to be removed from the star within the first few Myrs. The bimodality of the rotation period distributions of PMS stars suggested that accretion discs were playing an important role in regulating the rotation rates of their stellar hosts. In addition, the existence of a mass-rotation relation indicated a possible link between the efficiency of the angular momentum removal process and the stellar mass. The work presented in Chapter 3 of this thesis aimed to observationally address the role of accretion discs in regulating the removal of stellar angular momentum during the PMS while paying particular attention to the dependence on stellar mass.

155 Chapter 6. Summary and future work

Observations of the jets and outﬂows emanating from protostars and PMS stars highlighted that angular momentum could also be removed from the circumstellar environment. Meanwhile, if angular momentum is conserved within discs around stars that are actively accreting, viscous disc evolution theory implies that the outer regions of the disc should radially expand (Lynden-Bell & Pringle 1974; Pringle 1981; Hartmann et al. 1998). The work presented in Chapter 4 used spatially resolved submillimetre and millimetre (collectively referred to as submm) observations of the gas and dust components of protoplanetary discs to investigate whether this radial expansion could be observed. The chapter also aimed to address whether angular momentum was conserved during disc formation, evolution, and dispersal through comparison of the protoplanetary disc angular momentum with that contained within cores and the Solar System (SS). This latter point was built upon in Chapter 5 using spatially resolved debris discs from the Submillimetre Common-User Bolometer Array-2 Observations of Nearby Stars (SONS) James Clark Maxwell Telescope (JCMT) Legacy Survey to constrain the angular momentum retained within planetary systems other than the SS.

The main results presented in Chapters 3, 4, and 5 of this thesis are summarised in Section 6.1. An overview of the possible future avenues of research which build upon this work is presented in Section 6.2.

6.1 Main results

Detailed results of the work presented in Chapters 3 to 5 were discussed at the end of each chapter. Here, the main results are summarised as follows:

1. The distribution of rotation periods of fully convective PMS stars of spectral type earlier than M2 is observed to be bimodal while that of fully convective PMS stars of later spectral types is observed to be unimodal, in agreement with previous studies (e.g. Herbst et al. 2002; Cieza & Baliber 2007). This result follows the application of recent updates to spectral types, the adoption of spectral type-to-eﬀective temperature scales empirically derived speciﬁcally for PMS stars, and the careful removal of non-members, binary and multiple systems, and beat and harmonic periods. On average, PMS stars hosting accretion discs (Class II objects) typically rotate slower than their disc-less counterparts (Class III objects) but a range of rotation periods

156 6.1. Main results

exist across the sample. It is likely that discs do play a role in regulating the rotation of accreting PMS stars. The range of rotation periods observed is likely due to variations in the mass accretion rate and magnetic ﬁeld both in a single star over time (Donati et al. 2011a; Audard et al. 2014) and across the sample as a whole (Donati et al. 2010b; Gregory et al. 2012).

2. The populations of both rapid and slow rotators contain both Class II and Class III fully convective PMS stars. If disc-locking operates, the more rapidly rotating Class II stars likely have larger mass accretion rates and/or weaker magnetic ﬁelds such that they interact with a region of their disc closer to the stellar surface which rotates at a faster rate than regions further from the star. The slowly rotating Class III PMS stars have likely only recently been released from their discs and thus have not had suﬃcient time to spin up.

3. Fully convective PMS stars of spectral types later than M2 rotate faster, on average, than those earlier than M2. At the age of the Orion Nebula Cluster (ONC) and Taurus-Auriga star forming regions (∼ 1 − 2 Myrs; Hillenbrand 1997; White & Ghez

2001), a spectral type of M2 corresponds to a stellar mass of ∼ 0.35 M . PMS stars

with masses . 0.35 M have more complex large-scale magnetic ﬁelds and weaker dipole components (Donati et al. 2010b; Gregory et al. 2012). As such, their discs are truncated closer to their photospheres.

4. Assuming that the age spreads observed in the ONC and Taurus-Auriga are real and not introduced by observational eﬀects (e.g. Hartmann 2001) or diﬀering accretion

histories (e.g. Littlefair et al. 2011), the specific angular momentum, j?, of the −β Class II sample is observed to reduce with age with j? ∝ tage where β ≈ 2 − 2.5. This rate is faster than that expected from simple disc-locking theory, whereby the star contracts at a fixed rotation rate, and suggests that more efficient methods of angular momentum removal are in operation within the star-disc system, such as an accretion-driven stellar wind or outflow from the disc or some form of periodic magnetospheric ejection process (e.g. Aarnio et al. 2012; Zanni & Ferreira 2013).

5. The sample of Class III PMS stars contained more angular momentum than the sample of Class II PMS stars. To explain this, PMS stars must lose their discs at a variety of ages. Indeed, the ages of the Class II and Class III samples are found to be

157 Chapter 6. Summary and future work

consistent. This eﬀect can be demonstrated by considering two initially disc-hosting

PMS stars which have the same specific stellar angular momentum, j?,1, and which are subject to the same spin-down torque but which have different disc lifetimes. The one that loses its disc first is observed as a Class III PMS star and evolves at

constant angular momentum, j?,2 < j?,1. At the same time, the other star continues to lose angular momentum and is observed as a Class II PMS star with angular

momentum, j?,3 < j?,2.

6. A decrease in the speciﬁc angular momentum of the Class III PMS stars with age

is also observed. Furthermore, the apparent reduction rate of j? is consistent with that observed for the Class II sample. This is a surprising result at ﬁrst glance but can be explained by considering the age spread within the ONC and Taurus-Auriga as well as the range of disc lifetimes across the sample. For the youngest stars in the sample to be observed as Class III objects, their disc lifetimes must have been short. The range of possible disc lifetimes that an object observed as Class III could have had increases with the age of the object. At the same time, a longer-lived accretion disc can provide a spin-down torque for a longer period of time, allowing more stellar angular momentum to be removed. Therefore, the chance of observing

a low-j? disc-less PMS star increases as one considers older and older stars. As the

disc-less PMS stars evolve at constant j? while they remain fully convective, the plot

of j? versus tage mimics that of the Class II PMS stars.

7. No evidence for an increase in the radius of the dust component of protoplanetary

discs, Rdust, with age, as would be expected from a viscously evolving disc with well coupled gas and dust components (Lynden-Bell & Pringle 1974; Pringle 1981; Hartmann et al. 1998), is observed. This result is independent of the adopted PMS evolutionary model used to determine the stellar ages. The removal of discs around stars previously identiﬁed as members of binary or multiple systems from the sample should also limit the impact of disc truncation via dynamical interactions with neighbouring stars (e.g. de Juan Ovelar et al. 2012; Rosotti et al. 2014). At the same time, tentative evidence for an increase in the radius of the gas component of

protoplanetary discs, Rgas, with age is observed. In addition, an increase in the ratio

of gas radii to dust radii, Rgas/Rdust, is also observed. These results indicate that the evolution of the gas and dust components of protoplanetary discs are likely governed

158 6.1. Main results

by diﬀerent astrophysical processes, as suggested from a theoretical perspective by Birnstiel & Andrews (2014). This is also consistent with the results from previous detailed studies of high resolution CO line emission and continuum imaging of a handful of protoplanetary discs whereby the gas consistently extends to larger radii than the dust grains (Pani´cet al. 2009; Andrews et al. 2012; de Gregorio-Monsalvo et al. 2013; Rosenfeld et al. 2013; Zhang et al. 2014).

8. A range of Rdust and Rgas are observed over the 1 − 10 Myr age range probed by the sample. This is attributable to a spread in the initial radii of the discs. The initial radius of the disc that forms during the rotational collapse of the circumstellar envelope is governed by the angular momentum contained within the progenitor core: those containing more angular momentum form discs with larger initial radii. In turn, a range of core angular momenta are expected due to the turbulent properties of, and degree of magnetic braking present within, the parent molecular cloud (Burkert & Bodenheimer 2000; Hennebelle 2013).

9. No evidence for significant disc angular momentum removal during disc evolution is observed. However, comparing the median specific angular momentum contained within the dust component of protoplanetary discs with that contained within pre- stellar cores within Perseus (Curtis & Richer 2011) and that retained within the SS revealed that significant amounts of angular momentum must be lost from circumstellar material during disc formation and disc dispersal. This likely occurs via magnetic braking (Krasnopolsky & Königl2002) or via the launching of a disc or photo-evaporative wind (Black & Scott 1982; Shu et al. 1993; Alexander & Armitage 2007). It is also possible that angular momentum is removed from the star-disc system via ejection of planetesimals or planets following dynamical interactions (e.g. Davies et al. 2014b).

10. Twelve of the ﬁfteen SONS debris discs which show evidence of extended emission

at 450 µm and/or 850 µm are spatially resolved. The radii, Rdisc, total masses, Mtot,

and angular momenta, Jdisc, of these discs range from 90 − 1100 au, 2 − 630 M⊕, and 2.5 × 1040 − 2.0 × 1043 kg m2 s−1, respectively. Upper limits to the angular momenta contained within their host stars range between 1.3 × 1041 and 1.3 × 1044 kg m2 s−1. Five of the SONS debris discs exist around known exoplanet-hosting stars. The

159 Chapter 6. Summary and future work

orbital angular momentum of these planets ranges between 2.5 × 1040 and 2.0 × 1043 kg m2 s−1.

11. The angular momenta contained within the SONS debris discs serve as lower limits to that retained within planetary systems following protoplanetary disc dispersal. Comparing the median speciﬁc angular momentum contained within the dust com-

ponents of the protoplanetary discs, jdust, to that retained within the SONS debris discs suggests that signiﬁcant amounts of angular momentum must be lost during protoplanetary disc dispersal. This is consistent with the result found when com-

paring jdust to that retained within the SS.

In addition to the main results outlined above, the importance of the dataset built for the work undertaken in Chapter 3 has also been recognised by e.g. Vasconcelos & Bouvier (2015) who have incorporated the data into new theoretical models of stellar rotational evolution. The partially radiative stars within the sample have also been combined with data for the star forming regions NGC 2264, IC 348, NGC 2362, and NGC 6530 to investigate whether radiative core development leads to star-disc interaction spin-up of PMS stars (Dagleish, Gregory & Davies, in prep.).

6.2 Outlook and future work

6.2.1 Independent radius and age estimates

In Chapter 3, both the stellar radii and the stellar ages were estimated from effective temperatures, Teff , and bolometric luminosities, L?: the latter through application of PMS evolutionary models. It would be useful to determine these independently from one another to test the conclusions of Davies et al. (2014a) and further investigate the accretion disc-regulated angular momentum evolution of fully convective PMS stars. There are two possible avenues to explore: determining individual ages for each star using a method independent of Teff and L? or extending the study to include other regions of star formation and using an average age for each region.

Alternative age diagnostics

The ages of fully convective stars (and those which have formed radiative cores but main- tain a sustantial convective envelope) can be determined by analysing their lithium content.

160 6.2. Outlook and future work

−9 Newly formed proto-stellar cores inherit a lithium abundance, nLi, which is ∼ 10 times that of hydrogen. The internal temperature increases during contraction and, when the core temperature reaches ∼ 2.5 × 106 K, lithium burning begins. The fully convective nature of these stars means that the lithium is well-mixed within the interior and the surface layers display a corresponding reduction in nLi. For stars more massive than ∼ 0.35 M , a radiative core develops and the depth of the convection zone decreases. As the radiative core grows, the temperature at the base of the convection zone drops and, in stars between 6 ∼ 0.6 and 1 M , the temperature can drop below the ∼ 2.5 × 10 K required for lithium burning (Baraﬀe et al. 1998; Siess et al. 2000). When this happens, surface lithium depletion ceases. Thus, over a narrow range of masses, nLi can be used to determine the stellar age.

The mass range over which lithium depletion can be used as an age estimator is itself age dependent. For the star forming regions considered within this thesis (i.e. those younger than ∼ 10 Myr), one requires stars in the narrow range of 0.4 < M < 0.6 M (Sergison et al. 2013). Stars less massive than this do not undergo lithium burning by this age and thus do not exhibit lithium depletion while more massive stars develop substantial radiative cores within this timeframe. As a result, their surface lithium reaches a plateau and cannot be used as an age indicator.

To determine age estimates based on lithium depletion, one requires high resolution spectra (with R ≥ 10, 000) and sufficient signal to noise to determine EW(Li). One also requires spectra or photometry to determine the effective temperatures and surface gravities of the stellar sample which are used to translate EW(Li) into nLi. As for the Hertzsprung-Russell diagram ages used in Chapter 3, it should be noted that this process is also both model dependent and affected by the presence of surface inhomogeneities and the mass accretion rate (Soderblom et al. 2014; Somers & Pinsonneault 2015).

Extension to other regions

Due to the uncertainty surrounding the reality of age spreads in individual star forming regions, and the range of stellar properties found in each region, studies which compare the properties of PMS stars in diﬀerent regions often adopt average ages (e.g. Irwin et al. 2011). Extending the study of angular momentum evolution in fully convective stars presented in Chapter 3 to include older star forming regions (e.g. NGC 2264 and IC 348)

161 Chapter 6. Summary and future work would allow for an average age to be used.

Extending the study to older star forming regions would also provide a test of the prediction outlined in Chapter 3 whereby the distribution of specific stellar angular momentum for fully convective Class III stars would broaden with age. The range of possible disc lifetimes that a sample of Class III PMS could have had increases as one includes older Class III PMS stars into the sample. As a result, the spin-down torque provided by the star-disc interaction and/or outflows from the star-disc system would act for longer and, on average, result in older Class III PMS stars harbouring less angular momentum. At the same time, the younger fully convective Class III PMS stars would be expected to evolve with constant angular momentum as they continue their contraction, free from the braking effects of the disc.

6.2.2 Protoplanetary discs with uniform spatial resolution

A larger sample of protoplanetary discs imaged at more uniform resolution is required to conﬁrm the results presented in Chapter 4. This would remove the observational bias associated with using data compiled from observations with a range of beam sizes. Greater resolution is provided by the Atacama Large Millimeter/submillimeter Array (ALMA) which would also allow extension of the study to more distant regions of star formation.

Models of the thermal emission expected from dust grains in protoplanetary discs assume that the dust surface density follows that prescribed for the gas component in viscous disc evolution theory (e.g. Isella et al. 2009; Guilloteau et al. 2011). However, if the results presented in Chapter 4 whereby the dust and gas components of protoplanetary discs are not well-coupled at large radii is confirmed, the applicability of this assumption is brought into question. Further examination of the evolution of the disc at different disc radii is required. One possible avenue for extension of this study is to focus on the evolution of the transition radius, Rtrans, which defines the region of the disc where the mass flux changes sign [see equation (1.46) in Section 1.2.2]. This region migrates outwards through the disc at the same rate as the disc is expected to expand (c.f. Isella et al. 2009) meaning that a measure of the rate of increase of Rtrans would also shed light on the radial dependence of the viscosity, γ.

162 6.2. Outlook and future work

6.2.3 The interplay between stars, discs, and planets

One aspect of angular momentum evolution not so far considered is that of planetary migration in circumstellar discs. Giant gaseous exoplanets have been found to reside close to their host stars, often with signiﬁcant orbital obliquities (Morton & Johnson 2011; Wright et al. 2011). These so-called hot-Jupiters cannot form in such close proximity to their stellar hosts and instead are understood to form further out in the disc and migrate inwards through an exchange of angular momentum with the disc (Goldreich & Tremaine 1980; Ward 1986; Tanaka et al. 2002). The possibility that these exoplanets arrive into short-period orbits while their host star is still accreting material from its disc has prompted theoretical studies to suggest that the observed orbital obliquities may be driven, at least in part, by star-disc interactions (Lai et al. 2011; Batygin 2012; Batygin & Adams 2013). In turn, this predicts primordial misalignment of the orbital axis of the disc and the rotational axis of its host star.

To measure the misalignment between a star and its disc, one requires an estimate of the inclination of both objects. The stellar angular velocity, Ω, is related to the rotation period, P , projected rotational velocity, v sin i, and stellar radius, R?, via

2π Ω = , (6.1) P and v sin i Ω = , (6.2) R? sin i such that, by combining equations (6.1) and (6.2), one may obtain an estimate of sin i and therefore the inclination of the star.

Estimates of disc inclinations require spatially resolved detections of discs seen either in scattered light, thermal emission from dust, or molecular line emission. Also, in the case of the Orion proplyds (see Fig. 1.5), preferential viewing conditions allow resolved discs to be imaged in silhouette against the bright nebulous background. These observations can be modelled using Monte Carlo radiation transfer simulations (e.g. Wood et al. 2001, 2008) and the inclinations deduced from e.g. least-squares ﬁtting. Alternatively, where multi- wavelength observations are available for individual star-disc systems, spectral energy distributions (SEDs) can be constructed and compared to models to estimate the disc

163 Chapter 6. Summary and future work inclination.

The measurements described above are only sensitive to inclinations with respect to the plane of the sky. A statistical method is being developed by Stuart Littlefair and collaborators to account for the “unseen” element of the star-disc geometry. The nature of this method requires a large sample of discs for it to gain meaningful results. At present, the number of systems for which the inclination of the disc and the stellar rotation axis are known is very small (∼ 10), primarily due to the dearth of spatially resolved protoplanetary discs. As such, determining the frequency of misaligned star-disc systems, and establishing whether the star-disc interaction can drive star-planet misalignment, requires a considerably larger sample.

The best targets for a study of this kind lie in NGC 2264 and ρ Ophiuchus (ρ Oph). These regions have been included in high-cadence, space-based photometric surveys1 in recent years, which will result in measured rotation periods for a signiﬁcant portion of their members. In addition, observations of ρ Oph and NGC 2264 with the Very Large Tele- scope Fibre Large Array Multi Element Spectrograph (VLT-FLAMES) are already being undertaken as part of the Gaia-European Southern Observatory (ESO) survey, resulting in high-precision measurements of v sin i and stellar radii. The members of NGC 2264 and

ρ Oph for which measurements of P , v sin i, and R? are made can then form the basis of a submm-wavelength interferometer observing proposal (e.g. ALMA, the Submillimeter Observatory, or the Combined Array for Millimeter-wave Astronomy) to determine the disc inclination.

1NGC 2264 was twice surveyed by CoRoT (COnvection ROtation et Transits plan´etaires;e.g. Alencar et al. 2010; Cody et al. 2014) and ρ Oph featured in the Kepler K2 campaign ﬁeld 2.

164 A Contraction rate of a fully convective, polytropic pre-main sequence star

n Chapter 3, it was stated that the contraction rate of a fully convective, polytropic, −1/3 pre-main sequence (PMS) star follows R? ∝ t [equation 3.15]. Here, a derivation I of this relation is presented. According to the Virial Theorem, half of the energy generated within the stellar interior is converted into thermal energy while the other half is radiated away. This internal energy is generated through gravitational collapse such that the luminosity of a PMS star in approximate hydrostatic equilibrium (HE),

1 dU L = 4πR2σ T 4 = − , (A.1) ? ? B eff 2 dt where R? is the stellar radius, σB is the Stefan-Boltzmann constant, Teff is the effective temperature, and U is the gravitational potential energy (GPE). Thus, using a suitable

165 Appendix A. Contraction rate of a fully convective, polytropic pre-main sequence star expression for U, it is possible to determine the contraction rate of a PMS star.

The GPE is deﬁned as

Z R? Gm Z R? G U = − dm ≡ − d(m2), (A.2) 0 r 0 2r where G is the gravitational constant, m denotes a small mass element, and r denotes the distance from the centre of the PMS star. Integrating by parts and recognising that m(r = 0) = 0 and m(r = R?) = M?,

2 Z R? 2 GM? Gm U = − − 2 dr. (A.3) 2R? 0 2r

One can then use the equation of HE,

dP (r) Gm(r)ρ(r) = − , (A.4) dr r2 where P (r) and ρ(r) are the pressure and density at radius r, to replace dr in equation (A.3): GM 2 Z R? m U = − ? + dP. (A.5) 2R? 0 2ρ

PMS stars are modelled as polytropes with P and ρ related by

P = Kρ(n+1)/n, (A.6) where K is a scaling constant and n is the polytropic index. Diﬀerenting equation (A.6) with respect to ρ gives dP n + 1 = Kρ1/n. (A.7) dρ n

Noticing also that d(P/ρ) 1 = Kρ1/n, (A.8) dρ nρ equations (A.7) and (A.8) can be used to replace dP in equation (A.5):

GM 2 n + 1 Z R? U = − ? + m d(P/ρ). (A.9) 2R? 2 0

Integrating by parts, and recognising that m(r = 0) = 0 and p(r = 0) = 0, equation (A.9)

166 becomes GM 2 n + 1 Z R? P U = − ? − dm. (A.10) 2R? 2 0 ρ Then, using the mass continuity equation,

dm(r) = 4πr2ρ(r), (A.11) dr substituting r2dr = d(r3)/3, and integrating by parts, equation (A.10) becomes

GM 2 n + 1 Z R? U = − ? + 4πr3 dP. (A.12) 2R? 6 0

Equations (A.4) and (A.11) can then be used to replace dP such that equation (A.12) becomes GM 2 n + 1 Z R? Gm U = − ? − dm. (A.13) 2R? 6 0 r Recognising that the integral in equation (A.13) is simply the deﬁnition of the GPE [equation (A.2)], 3GM 2 U = − ? , (A.14) (5 − n)R? where, for fully convective stars, n = 3/2.

Equations (A.1) and (A.14) can then be combined to show that,

4 1 28πσBTeff 4 dR? = − 2 dt, (A.15) R? 3GM? as M? ≈ const. during the PMS. Then, as Teff ≈ const. while PMS stars remain fully convective, 1 R? 28πσ T 4 t = B eff t . (A.16) R3 GM 2 ? R0 ? 0

Introducing R?(t = 0) = R0 where R0 is the radius at the end of the protostellar phase

(the stellar birth line) and noting that R? R0, it is possible to see that

2 1/3 GM? −1/3 R? = 4 t . (A.17) 28πσBTeﬀ

167 Appendix A. Contraction rate of a fully convective, polytropic pre-main sequence star

168 B Online resources

1. The VizieR Catalogue Service: http://vizier.u-strasbg.fr/

2. NASA’s Astrophysics Data System: http://adsabs.harvard.edu/abstract_service.html

3. The SIMBAD data base: http://simbad.u-strasbg.fr/simbad/

4. The Extrasolar Planets Enclyclopaedia: http://exoplanet.eu/

5. The Catalog of Circumstellar Disks: http://circumstellardisks.org/

6. Eric Mamajek’s “Basic Astronomical Data for the Sun”: http://sites.google.com/site/ mamajeksstarnotes/basic-astronomical-data-for-the-sun

7. Atacama Large Millimeter/submillimeter Array (ALMA) archive: https://almascience.

nrao.edu/aq/

169 Appendix B. Online resources

170 C Acronyms

2MASS – Two Micron All Sky Survey ALMA – Atacama Large Millimeter/submillimeter Array BC – Bolometric Correction BN/KL – Becklin-Neugebauer/Kleinmann-Low CARMA – Combined Array for Millimeter-wave Astronomy CASA – Common Astronomy Software Applications CTTS – Classical T Tauri star DM – Distance Modulus ESA – European Space Agency ESO – European Southern Observatory EUV – Extreme ultra-violet EW – Equivalent width FAP – False Alarm Probability FIR – Far infra-red

171 Appendix C. Acronyms

FWHM – Full Width Half Maximum GI – Gravitational instability GPE – Gravitational potential energy HE – Hydrostatic equilibrium HRD – Hertzsprung-Russell Diagram HST – Hubble Space Telescope IR – Infra-red IRAC – Infra-red Array Camera IRAM – Institut de Radioastronomie Millim´etrique JCMT – James Clark Maxwell Telescope KS – Kolmogorov-Smirnov MHD – Magneto-hydrodynamic MIR – Mid infra-red MRI – Magneto-rotational instability MS – Main sequence NAOJ – National Astronomical Observatory of Japan NASA – National Aeronautics and Space Administration NIR – Near infra-red NMA – Nobeyama Millimeter Array NRAO – National Radio Astronomy Observatory ONC – Orion Nebula Cluster P.A. – Position Angle PdBI – Plateau de Bure Interferometer PMS – Pre-main sequence P-R – Poynting-Robertson PSF – Point spread function RA – Right ascension SCUBA-2 – Submillimetre Common-User Bolometer Array 2 SED – Spectral Energy Distribution SMA – Submillimeter Array SONS – SCUBA-2 Observations of Nearby Stars SpT – Spectral type

172 SS – Solar System TTS – T Tauri star UV – Ultra-violet VLBI – Very Long Baseline Interferometry VLT-FLAMES – Very Large Telescope Fibre Large Array Multi Element Spectrograph WTTS – Weak-lined T Tauri star YSO – Young stellar object ZAMS – Zero-Age Main Sequence

173 Appendix C. Acronyms

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