Disks around young stars
Star formation in a nutshell
The idea of a rotating flattened distribution of material around young stars has a long history. At the end of the XVIII century, the German philosopher Immanuel Kant (1755) and the French scholar Pierre-Simon Laplace (1798) independently observed that the orbits of the celestial bodies which compose the Solar System were nearly coplanar4,5. They concluded that this peculiarity had origin in the process itself that gave birth to the Solar System, and supposed that, at the beginning, the matter which would eventually produce the Sun, the planets and the minor bodies was distributed in a low-density rotating nebula, which extended beyond the actual size of the solar system, that would progressively collapse under the eﬀect of gravity and flatten along its spin axis due to the centrifugal force, determining a disk-like distribution of material around the center of gravitational collapse (the Sun-to-be), on the equatorial plane of this latter (the nebular hypothesis for the origin of the Solar System; Wolf 1884b,a). This rotating disk, in the Kant-Laplace model, was considered to be the birth site of planets orbiting around the central star.
This idea contains in embryo the main traits of the currently accepted picture of star for-mation. While nearly two centuries elapsed before final evidence for disks around pre-Main Sequence (PMS) stars was gained from direct imaging (Fig. 1.1a), circumstellar disks are now considered a ubiquitous product of the earliest stages of star formation, with a major impact on early stellar evolution.
Stars are born in dense cores inside molecular clouds (Stahler & Palla 2005; Lada 2005). Dense cores are the densest regions of the interstellar medium, and are found at the smallest scales in the hierarchical structure of dark cloud complexes in the Galaxy. They have diameters of the order of ∼ 0.1 pc, masses of the order of 10 M⊙, AV > 10, temperatures of around 10 K and densities of ∼ 104 particles/cm3. Zeeman eﬀect measurements in dark cloud regions indicate magnetic fields of the order ∼30 µG (Goodman et al. 1989).
At the beginning, dense cores are held in equilibrium by the force balance between thermal and magnetic pressure of the gas, on one side (outward push that would tend to disperse the concentration of particles), and self-gravity, on the other side (inward push that would drive the core to collapse on itself). This balance can be disrupted by the action of external radiation onto the molecular cloud, or by other triggering events (e.g., the interaction of the cloud with supernova shock waves or spiral density waves), which determine an enhancement in the external pressure. If the cloud is suﬃciently massive6, it may become unstable under the eﬀect of these processes, and gravitational collapse may be ignited.
If conditions for collapse are met, the cloud begins to contract; gravitational potential energy is released and transformed into thermal energy of the gas and then into radiation. The infall starts at large radii, where pressure support is less important. Because of the low initial density, at this stage the cloud is optically thin; hence the internal energy gained from the process can be eﬀectively radiated away and the collapse is isothermal. Local increase in density implies that individual fragments of the parental cloud may become gravitationally unstable on their own (see expression for the Jeans mass MJ 6) and continue their dynamical collapse as separate condensation nuclei. This dynamical cloud fragmentation is enhanced by the centrifugal force exerted on individual cloud fragments as a result of the increase in angular velocity due to angular momentum conservation during cloud contraction; such fragments may be ejected from the cloud and continue the collapsing process on their own. A single cloud may give birth to hundreds of stars, as suggested by the fact that young stars are typically found in star clusters or associations, rather than isolated; this scenario is illustrated in numerical simulations (e.g., Bate et al. 2003). Even at the scales of dense cores, at least two stars are usually formed out of the material which compose a single core (Stahler & Palla 2005).
Inside each fragment, the collapse is non-homologous: the inner region collapses on shorter timescales than the outer region. As a result, the density increases rapidly in the core center, which then becomes opaque to its own radiation and starts heating up under the eﬀect of further external compression. This determines an increase in pressure in the innermost regions, which counteracts and delecerates the inward drift of material from the surrounding envelope of material. As the outer gas layer of the inner core radiates freely in the infrared, internal pressure is released and core contraction is re-enhanced. The inner region of the initial cloud fragment has given birth to a protostar surrounded by an infalling envelope. The process of gravitational collapse of dense cores and subsequent formation of protostars takes place on timescales of roughly 105 years, or free-fall time (Garcia et al. 2011).
Since the collapse phase occurs on a dynamical timescale, the angular momentum the cloud core possesses at the beginning of the rapid collapse cannot be transferred eﬃciently to the external medium; due to the large size of the initial cloud core, even small initial rotational velocities will be significantly amplified during the gravitational collapse due to angular momen-tum conservation, thus preventing infalling material from accreting directly, i.e. radially, onto the protostellar core. This determines the formation of a rotationally supported circumstellar disk (Hartmann 2009).
Overall properties can be derived by assuming a fixed central mass equal to M. If mate-rial with specific angular momentum h falls on a circular orbit around the central mass while maintaining its angular momentum, then the radius R of the circular orbit is given by h2 R = (1.1) GM
If we assume spherical symmetry for the gravitational collapse process, all the material which arrives at the centre at a given instant of time comes from the same initial radius r0 within the cloud; if the protostellar cloud core is initially in uniform rotation with angular velocity Ω, the specific angular momentum at r0 varies with the angle θ from the rotation axis as h = Ωr02 sin θ.
Thus, infalling material from diﬀerent directions will have diﬀerent angular momenta and arrive at the equatorial plane at diﬀering radii. Material near the rotation axis will fall in close to the central star, while material coming from regions at θ ∼ π/2 will fall in to a maximum centrifugal radius given by Ω2 rc = r04 (1.2) GM
In the assumptions of a spherically symmetric initial cloud and of axisymmetric rotation, the process has complete symmetry above and below the equatorial plane θ = π/2. Thus, the momentum fluxes of the infalling material on either side of the disk plane perpendicular to the disk have the same magnitude and opposite directions, resulting in a shock layer at the equator, which dissipates the kinetic energy of motion perpendicular to the equatorial plane. This leads to an accumulation of matter in a thin structure in the equatorial plane, i.e. to the formation of a rotating disk, whose diameter can vary between a few hundred and a few thousand AU (Hartmann 2009).
At the end of the collapse phase in the star formation process, a system composed of a central protostar surrounded by a disk is thus born. The system then enters the phase of disk accretion. This has a longer duration than the previous phase, typically extending over a few Myr, as deduced from the analysis of the disk frequency rate as a function of age in young clusters and star-forming regions (Fig. 1.2). At the beginning of the accretion phase, the mass of the protostar is about 5 × 10−2 M⊙ (Stahler & Palla 2005) and the central temperature of the core is not suﬃcient to ignite nuclear reactions. Most of the mass of the star is built during the earliest stages of the accretion phase, prior to the T Tauri evolutionary phase, when the protostar is still embedded in its envelope (Calvet & D’Alessio 2011).
In the following sections, I will focus on the structure of circumstellar disks around young stars and on the physics which governs disk accretion. To complete the overview on the star formation process and its interconnection with the formation of planetary systems, I shall sum-marize here in a few lines the ultimate fate of disks around protostars.
Dust disks within ∼20 AU distance to the central star tend to “clear” (i.e., to become undetectable in infrared excess studies) on timescales of a few Myr (Hartmann 2009). Dusty component tends to settle into the disk midplane due to the drag forces resulting from interaction with the gas (Lada 2005). In this process the micron-sized dust grains collide and can merge together forming increasingly massive solid particles. Iteration of this process and merging with condensates produced in cooling processes in the disks lead to the formation of kilometer-sized planetesimals, which may produce terrestrial planets by accretion with other planetesimals or accrete gaseous atmospheres and thus constitute the cores of giant planets (Lada 2005). Gas removal is thought to be caused by photoevaporation by the extreme ultraviolet radiation of the central star (Hartmann 2009).
As the protostellar core begins the accretion phase, its luminosity is dominated by accretion. Once a protostar reaches a mass of about 0.2 – 0.3 M⊙, the central temperature reaches 106 K and nuclear reactions of deuterium burning are ignited, thus providing the protostar with another source of luminosity. Accretion can continue until the central temperature of the protostar reaches 107 K and hydrogen fusion is ignited (Lada 2005). The ignition of hydrogen burning marks the beginning of the Main Sequence phase of stellar lifetime. The stage of accretion and final condensation by which a protostar finally evolves in a Main Sequence star lasts on average several tens of Myr, but this length of time can strongly vary depending on the stellar mass (from tens of thousands of years for a ∼15 M⊙ star to hundreds of Myr for a ∼0.1 M⊙ star).
Structure and physics of circumstellar disks
Recent measurements of the Zeeman broadening of photospheric absorption lines (e.g., Johns-Krull 2007) have shown that mean magnetic fields of a few kG are typically found at the surface of CTTS. The origin of such strong large-scale magnetic fields is likely attributable to dynamo processes (Donati & Landstreet 2009). Spectropolarimetric studies of magnetic field topology at the surface of CTTS have illustrated that, while some TTS host simple large-scale, dipolar field structure, other exhibit complex, multipolar magnetic fields (Gregory et al. 2012). Large-scale stellar field lines permeate the stellar surface; these may have a complex geometry (Donati & Landstreet 2009), although the dipolar component, which falls oﬀ more slowly with distance than higher-order components, appears to be dominating in the star-disk coupling (Gregory et al. 2012; see Sect. 1.3.1). The magnetic field may have a strength of ∼1 G in the inner regions of the disk; values of a few mG are instead measured at typical distances of ∼ 103 AU from the central object (Donati & Landstreet 2009).
Magnetic fields in CTTS have a most important role in the dynamics of disk accretion, as illustrated in the following sections.
Angular momentum transport inside the disk
In order for disk accretion to occur, a net inward flow of material must take place in the disk. This is possible if angular momentum is extracted from the inner region of the disk and transferred to a small fraction of disk particles at large radial distances.
In a first approximation, we can describe the disk as a series of concentric rings in Keplerian rotation around the central object of mass M, with angular velocity depending on the distance R to the protostar as ΩK (R) = GM/R3 1/2. The expression for ΩK indicates that an inner ring rotates faster than its neighboring outer ring. Hence, friction between two neighboring rings in the disk will tend to spin up the external ring and spin down the internal ring, i.e., to transfer angular momentum outward. Since the gravitational potential constrains the orbital motion in the Keplerian description, the decelerated material from the inner ring will be carried inwards, while material from the outer ring will move to larger radii. Since energy is lost in the process due to frictional dissipation, the net gravitational potential energy of the system must decrease, which is achieved with a net inward motion of disk mass (as illustrated in Lynden-Bell & Pringle 1974).
A classic mechanism considered for the coupling and mixing of adjacent rings in a gaseous disk is turbulent viscosity (e.g., Frank et al. 2002). In this picture, single particles in the disk are subject to chaotic motion with typical scale λ (corresponding to the mean free path) and random velocity v,˜ and can diﬀuse from one ring to another, thus yielding angular momentum exchange (left panel of Fig. 1.3). The product of λ by v˜ defines the kinematic viscosity ν associated with the shearing motion between two rings of the disk: ν = λ v˜. In the assumption of a geometrically thin accretion disk, composed of particles which essentially move on circular orbit in a single plane, the time evolution of the ring is governed by the equations of mass conservation and of angular momentum conservation in a viscous fluid, and by the diﬀerential torque exterted at its inner and outer boundaries due to the diﬀerential rotation in the disk. In the case of a constant viscosity ν, Lynden-Bell & Pringle (1974) showed that, starting with an initial density distribution representing a ring at radius R1, the surface density of the disk as a function of radial distance and time is described by −1/4 −1 2 td 2πR12 2td Σ(x, t ) = x td exp −(1 + x ) I 1/4 x , (1.3) d where x = R/R1, td = 6νt/R12 and I1/4 is the modified Bessell function of fractional order. This solution is illustrated in Fig. 1.4; the net eﬀect of the viscosity is to spread the ring of material, ultimately concentrating mass at smaller radii, while small amounts of matter are pushed to large distances to conserve angular momentum (Hartmann 2009). The quantity R12/ν which appears in the definition of the time coordinate corresponds to the average time for a particle in the ring to diﬀuse a distance R1 in a random walk. In a steady disk, where the inward mass flux is constant in time, the expression for the surface density distribution given by the conservation of the angular momentum flux is (Hartmann 2009) Σ = 3πν » 1− R⋆ # , (1.4) ˙ R 1/2 M where R⋆ is the stellar radius and it is assumed that the shear goes to zero close to the stellar surface.
The viscosity ν is often described using the parametric α prescription introduced by Shakura & Sunyaev (1973), originally in the context of disks around black holes. In this prescription, ν = αcsH, where cs is the sound speed in the medium, H is the scale height and α is a parameter that describes the eﬃciency of the angular momentum transport mechanism. In the assumption that, for turbulence in the disk, the scale λ of the eddies is less than the disk scale height H and the turbulent velocity v˜ is less than the sound speed cs, α ≤ 1 (Pringle 1981). In this description, Eq. 1.4 becomes Σ = 3παcsH » 1− R⋆ # . (1.5) ˙ R 1/2 M
It can be noted from Eq. 1.5 that ˙ M ∝ νΣ ∝ αcsHΣ; hence, higher viscosity in the disk leads to larger mass accretion rate for a given surface density.
Numerical simulations of pure hydrodynamic mixing by turbulent convection between adja-cent rings in the disk have shown that the mechanism is not eﬀective in transporting angular momentum outwards; in fact, the resulting transport is small and on average directed inward (e.g., Stone & Balbus 1996). This can be understood by looking at the schematic picture in Fig. 1.3: particles in the inner ring have lower angular momentum than particles in the outer ring, as a result of the Keplerian velocity distribution; hence, diﬀusion of particles from the inner to the outer ring has the eﬀect of moving lower angular momentum outward. Similarly, particles which diﬀuse from the outer to the inner ring bring higher angular momentum inward. Additional mechanisms are then needed to enhance the viscosity of the disk and carry angular momentum outward.
Magnetorotational instability (MRI) is one possible mechanism to produce an α-type viscos-ity. This is schematically illustrated in the middle panel of Fig. 1.3. Assume two neighboring disk rings are initially connected by a magnetic field line in the radial direction. The diﬀerential rotation between the two rings will have the eﬀect of stretching the field line; the restoring force of the “spring” will tend to spin up the outer ring and spin down the inner ring. This will deter-mine an inward movement of the decelerated material in the inner ring, which fall in deeper to the potential well and increases its angular velocity further; conversely, the accelerated material in the outer ring will be pushed outward and slow down. This enhances the stretching of the magnetic field line, resulting in longer and longer loops of magnetic fields. The development of this instability is contingent upon the presence of ionized gas in the disk, whose particles can ef-fectively couple with the magnetic field. It is believed that the energetic (X-ray) irradiation from the central young object may act as an eﬃcient mechanism for producing non-thermal ionization of disks. This ionizing source is likely eﬀective only on the external layers of the circumstellar disk; the innermost regions, close to the midplane of the disk, are not penetrated by stellar flux. Assuming that angular momentum transport in disks is driven by magnetic fields, these internal regions will then be inactive in this respect (the “dead zones” described in Gammie 1996).
Another mechanism that may contribute to the picture of angular momentum transport in disks is gravitational instability (illustrated in the right panel of Fig. 1.3). This may take place when the disk is massive enough relative to the central object (e.g., in the earliest stages of disk evolution). In this condition, it might be self-gravitating. Assume a radially extended concentration of mass is present in the disk. Diﬀerential rotation between adjacent disk rings will cause it to be elongated in a trailing spiral arm. If the gravitational instability is strong enough to keep the mass concentration bound together in spite of the shearing, the inner part of the spiral arm will pull on the outer part of the spiral arm, tending to accelerate the outer region. This causes a net flow of angular momentum outward. When a small region in a self-gravitating disk is compressed, its self-gravity is increased, but the process also determines an increase in its spin velocity, which produces a centrifugal force to oppose gravity. The balance of these two eﬀects leads to the instability criterion. Numerical simulations indicate that gravitational instabilities can generate turbulence which appears as unsteady, spiral wave density structures. This type of exchange mechanism is eﬀective on larger scales in the disk compared to viscous friction, which has a local eﬀect.
Table of contents :
1.1 On the origin of the T Tauri case
1.1.1 A new class of young, variable stars
1.1.2 The T Tauri phenomenon across the spectrum
1.1.3 T Tauri phase in the paradigm of low-mass early stellar evolution
1.2 Disks around young stars
1.2.1 Star formation in a nutshell
1.2.2 Structure and physics of circumstellar disks
1.3 Disk accretion in T Tauri stars
1.3.1 Magnetospheric accretion
1.4 The manifold variability of T Tauri stars
1.4.1 Variability on mid-term (days to weeks) timescales
1.4.2 Variability on shorter and longer timescales
1.4.3 The space-borne revolution in YSO variability studies
1.5 Open issues in disk accretion from an observational perspective
1.5.1 Aim and outline of this thesis
2 The Coordinated Synoptic Investigation of NGC 2264
2.1 The young open cluster NGC 2264
2.2 The CSI 2264 project
2.2.1 Overview of the observing campaign
2.2.2 CFHT dataset
2.3 CSI 2264: a synthesis
2.4 Specific contribution from this thesis
3 Mapping the different accretion regimes in NGC 2264
3.1 The colors of young stars at short wavelengths
3.1.1 The color loci of field stars in the SDSS system
3.1.2 Colors and UV excess of young stars
3.1.3 UV excess vs. different accretion diagnostics
3.2 A UV census of the NGC 2264 young stellar population
3.2.1 New CTTS candidates in NGC 2264
3.2.2 Field contaminants in the NGC 2264 sample
3.3 Derivation of individual stellar parameters
3.3.1 Individual AV estimates
3.3.2 Spectral types and effective temperatures
3.3.3 Bolometric luminosities
3.3.4 Stellar masses and radii
3.4 UV excess and mass accretion rates
3.4.1 Measuring the UV flux excess
3.4.2 From u-band excess luminosity to total accretion luminosity
3.4.3 Mass accretion rates
3.5 Accretion regimes in NGC 2264
3.5.1 The ˙Macc −M⋆ relationship
3.5.2 Accretion variability
3.5.3 Different accretion regimes/mechanisms
3.5.4 Evolutionary spread across the cluster
4 The UV variability of young stars in NGC 2264
4.1 A closer look at photometric variability in the CFHT sample
4.1.1 Measuring the variability of CTTS and WTTS: light curve rms
4.1.2 Measuring the variability of CTTS and WTTS: Stetson’s index J
4.2 The imprints of disk accretion in UV variability
4.2.1 A comparison between UV excess and u-band variability
4.2.2 Time evolution on the r vs. u − r diagram of the cluster
4.2.3 Exploring the color signatures of different physical scenarios
4.2.4 A global picture of color variability for different YSO types
4.3 A spot model description of YSO variability
4.3.1 Formulation of the spot model
4.3.2 Implementation of the model
4.3.3 A global picture of spot properties for TTS in NGC 2264
4.3.4 The different nature of modulated variability for CTTS vs. WTTS
4.4 Timescales of variability for the accretion process
5 The accretion–rotation connection in young stars
5.1 Photometric period determination
5.1.1 Period-search methods used in this study
5.1.2 Implementation of the period-search routine
5.2.1 Period distribution for NGC 2264: CTTS vs. WTTS
5.2.2 Mass dependence?
5.2.3 Are CTTS periods similar in nature to WTTS periods?
5.3 The accretion–rotation connection
6 Conclusions and perspectives
6.1 Case of study: a brief recap
6.2 Main points of this work
6.2.1 Different accretion regimes coexist within the cluster
6.2.2 ˙Macc reflect a diversity in accretion mechanisms and cluster evolution
6.2.3 Variability in young stars has a broadly assorted nature
6.2.4 Timescales of days dominate the variability of WTTS and CTTS
6.2.5 Disks have an impact on the rotation properties of young stars
A List of referred publications
B Mapping accretion and its variability in the young open cluster NGC 2264: a study based on u-band photometry
C UV variability and accretion dynamics in the young open cluster NGC2264 179