Eduard I. Vorobyov

THE BURST MODE OF PROTOSTELLAR ACCRETION

We present new numerical simulations in the thin disk approximation that characterize the burst mode of protostellar accretion. The burst mode begins upon the formation of a centrifugally balanced disk around a newly formed protostar. It comprises prolonged quiescent periods of low accretion rate (typically 10-7 M☉ yr-1) that are punctuated by intense bursts of accretion (typically 10-4 M☉ yr-1, with duration 100 yr) during which most of the protostellar mass is accumulated. The accretion bursts are associated with the formation of dense protostellar/protoplanetary embryos, which are later driven onto the protostar by the gravitational torques that develop in the disk. Gravitational instability in the disk, driven by continuing infall from the envelope, is shown to be an effective means of transporting angular momentum outward and mass inward to the protostar. We show that the disk mass always remains significantly less than the central protostars mass throughout this process. The burst phenomenon is robust enough to occur for a variety of initial values of rotation rate and frozen-in (supercritical) magnetic field and a variety of density-temperature relations. Even in cases where the bursts are nearly entirely suppressed, a moderate increase in cloud size or rotation rate can lead to vigorous burst activity. We conclude that most (if not all) protostars undergo a burst mode of evolution during their early accretion history, as inferred empirically from observations of FU Orionis variables.

The Burst Mode of Accretion and Disk Fragmentation in the Early Embedded Stages of Star Formation

We revisit our original papers on the burst mode of accretion by incorporating a detailed energy balance equation into a thin-disk model for the formation and evolution of circumstellar disks around low-mass protostars. Our model includes the effect of radiative cooling, viscous and shock heating, and heating due to stellar and background irradiation. Following the collapse from the prestellar phase allows us to model the early embedded phase of disk formation and evolution. During this time, the disk is susceptible to fragmentation, depending upon the properties of the initial prestellar core. Globally, we find that higher initial core angular momentum and mass content favors more fragmentation, but higher levels of background radiation can moderate the tendency to fragment. A higher rate of mass infall onto the disk than that onto the star is a necessary but not a sufficient condition for disk fragmentation. More locally, both the Toomre Q-parameter needs to be below a critical value and the local cooling time needs to be shorter than a few times the local dynamical time. Fragments that form during the early embedded phase tend to be driven into the inner disk regions and likely trigger mass accretion and luminosity bursts that are similar in magnitude to FU-Orionis-type or EX-Lupi-like events. Disk accretion is shown to be an intrinsically variable process, thanks to disk fragmentation, nonaxisymmetric structure, and the effect of gravitational torques. The additional effect of a generic α-type viscosity acts to reduce burst frequency and accretion variability, and is likely to not be viable for values of α significantly greater than 0.01.

The origin of episodic accretion bursts in the early stages of star formation

We study numerically the evolution of rotating cloud cores, from the collapse of a magnetically supercritical core to the formation of a protostar and the development of a protostellar disk during the main accretion phase. We find that the disk quickly becomes unstable to the development of a spiral structure similar to that observed recently in AB Aurigae. A continuous infall of matter from the protostellar envelope makes the protostellar disk unstable, leading to spiral arms and the formation of dense protostellar/protoplanetary clumps within them. The growing strength of spiral arms and ensuing redistribution of mass and angular momentum creates a strong centrifugal disbalance in the disk and triggers bursts of mass accretion during which the dense protostellar/protoplanetary clumps fall onto the central protostar. These episodes of clump infall may manifest themselves as episodes of vigorous accretion (≥10-4 M☉ yr-1), as is observed in FU Orionis variables. Between these accretion bursts, the protostar is characterized by a low accretion rate (<10-6 M☉ yr-1). During the phase of episodic accretion, the mass of the protostellar disk remains less than the mass of the protostar.

OBSERVED LUMINOSITY SPREAD IN YOUNG CLUSTERS AND FU Ori STARS: A UNIFIED PICTURE

The idea that non-steady accretion during the embedded phase of protostar evolution can produce the observed luminosity spread in the Herzsprung–Russell diagram (HRD) of young clusters has recently been called into question. Observations of FU Ori, for instance, suggest an expansion of the star during strong accretion events, whereas the luminosity spread implies a contraction of the accreting objects, decreasing their radiating surface. In this paper, we present a global scenario based on calculations coupling episodic accretion histories derived from numerical simulations of collapsing cloud prestellar cores of various masses and subsequent protostar evolution. Our calculations show that, assuming an initial protostar mass Mi ∼ 1 MJup, typical of the second Larson’s core, both the luminosity spread in the HRD and the inferred properties of FU Ori events (mass, radius, accretion rate) can be explained by this scenario, providing two conditions. First, there must be some variation within the fraction of accretion energy absorbed by the protostar during the accretion process. Second, the range of this variation should increase with increasing accretion burst intensity and thus with the initial core mass and final star mass. The numerical hydrodynamics simulations of collapsing cloud prestellar cores indeed show that the intensity of the accretion bursts correlates with the mass and initial angular momentum of the prestellar core. Massive prestellar cores with high initial angular momentum are found to produce intense bursts characteristic of FU Ori-like events. Our results thus suggest a link between the burst intensities and the fraction of accretion energy absorbed by

Variable Protostellar Accretion with Episodic Bursts

We present the latest development of the disk gravitational instability and fragmentation model, originally introduced by us to explain episodic accretion bursts in the early stages of star formation. Using our numerical hydrodynamics model with improved disk thermal balance and star-disk interaction, we computed the evolution of protostellar disks formed from the gravitational collapse of prestellar cores. In agreement with our previous studies, we find that cores of higher initial mass and angular momentum produce disks that are more favorable to gravitational instability and fragmentation, while a higher background irradiation and magnetic fields moderate the disk tendency to fragment. The protostellar accretion in our models is time-variable, thanks to the nonlinear interaction between different spiral modes in the gravitationally unstable disk, and can undergo episodic bursts when fragments migrate onto the star owing to the gravitational interaction with other fragments or spiral arms. Most bursts occur in the partly embedded Class I phase, with a smaller fraction taking place in the deeply embedded Class 0 phase and a few possible bursts in the optically visible Class II phase. The average burst duration and mean luminosity are found to be in good agreement with those inferred from observations of FU-Orionis-type eruptions. The model predicts the existence of two types of bursts: the isolated ones, showing well-defined luminosity peaks separated with prolonged periods (~ 10^4 yr) of quiescent accretion, and clustered ones, demonstrating several bursts occurring one after another during just a few hundred years. Finally, we estimate that 40\%--70\% of the star-forming cores can display bursts after forming a star-disk system.

Formation of giant planets and brown dwarfs on wide orbits

Context. We studied numerically the formation of giant planet (GP) and brown dwarf (BD) embryos in gravitationally unstable protostellar disks and compared our findings with directly-imaged, wide-orbit (> 50 AU) companions known to-date. Aims. The viability of the disk fragmentation scenario for the formation of wide-orbit companions in protostellar disks around (sub-)solar mass stars was investigated. The particular emphasis was paid to the survivability of GP/BD embryos formed via disk gravitational fragmentation. Methods. We used numerical hydrodynamics simulations of disk formation and evolution with an accurate treatment of disk thermodynamics. The use of the thin-disk limit allowed us to probe the long-term evolution of protostellar disks, starting from the gravitational collapse of a pre-stellar core and ending in the T Tauri phase after at least 1.0 Myr of disk evolution. We focused on models that produced wide-orbit GP/BD embryos, which opened a gap in the disk and showed radial migration timescales similar to or longer than the typical disk lifetime. Results. While disk fragmentation was seen in the majority of our models, only 6 models out of 60 revealed the formation of quasi-stable, wide-orbit GP/BD embryos. The low probability for the fragment survival is caused by efficient inward migration/ejection/dispersal mechanismswhich operate in the embedded phase of star formation. We found that only massive and extended protostellar disks (> 0.2 M⊙), experiencing gravitational fragmentation not only in the embedded but also in the T Tauri phases of star formation, can form wide-orbit companions. Disk fragmentation produced GP/BD embryos with masses in the 3.5–43 MJ range, covering the whole mass spectrum of directly-imaged, wide-orbit companions to (sub-)solar mass stars. On the other hand, our modelling failed to produce embryos on orbital distances < 170 AU, whereas several directly-imaged companions were found at smaller orbits down to a few AU. Disk fragmentation also failed to produce wide-orbit companions around stars with mass < 0.7 M⊙, in disagreement with observations. Conclusions. Disk fragmentation is unlikely to explain the whole observed spectrum of wide-orbit companions to (sub)solar-mass stars and other formation mechanisms, e.g., dynamical scattering of closely-packed companions onto wide orbits, should be invoked to account for companions at orbital distance from a few tens to � 150 AU and wide-orbit companions with masses of the host star � 0.7 M⊙. Definite measurements of orbit eccentricities and a wider sample of numerical models are needed to distinguish between the formation scenarios of GP/BD on wide orbits.Context. We studied numerically the formation of giant planet (GP) and brown dwarf (BD) embryos in gravitationally unstable protostellar disks and compared our findings with directly-imaged, wide-orbit (>∼ 50 AU) companions known to-date. Aims. The viability of the disk fragmentation scenario for the formation of wide-orbit companions in protostellar disks around (sub-)solar mass stars was investigated. The particular emphasis was paid to the survivability of GP/BD embryos formed via disk gravitational fragmentation. Methods. We used numerical hydrodynamics simulations of disk formation and evolution with an accurate treatment of disk thermodynamics. The use of the thin-disk limit allowed us to probe the long-term evolution of protostellar disks, starting from the gravitational collapse of a pre-stellar core and ending in the T Tauri phase after at least 1.0 Myr of disk evolution. We focused on models that produced wide-orbit GP/BD embryos, which opened a gap in the disk and showed radial migration timescales similar to or longer than the typical disk lifetime. Results. While disk fragmentation was seen in the majority of our models, only 6 models out of 60 revealed the formation of quasi-stable, wide-orbit GP/BD embryos. The low probability for the fragment survival is caused by efficient inward migration/ejection/dispersal mechanisms which operate in the embedded phase of star formation. We found that only massive and extended protostellar disks (>∼ 0.2 M⊙), experiencing gravitational fragmentation not only in the embedded but also in the T Tauri phases of star formation, can form wide-orbit companions. Disk fragmentation produced GP/BD embryos with masses in the 3.5–43 MJ range, covering the whole mass spectrum of directly-imaged, wide-orbit companions to (sub-)solar mass stars. On the other hand, our modelling failed to produce embryos on orbital distances <∼ 170 AU, whereas several directly-imaged companions were found at smaller orbits down to a few AU. Disk fragmentation also failed to produce wide-orbit companions around stars with mass <∼ 0.7 M⊙, in disagreement with observations. Conclusions. Disk fragmentation is unlikely to explain the whole observed spectrum of wide-orbit companions to (sub)solar-mass stars and other formation mechanisms, e.g., dynamical scattering of closely-packed companions onto wide orbits, should be invoked to account for companions at orbital distance from a few tens to ≈ 150 AU and wide-orbit companions with masses of the host star ≤ 0.7 M⊙. Definite measurements of orbit eccentricities and a wider sample of numerical models are needed to distinguish between the formation scenarios of GP/BD on wide orbits.

FORMATION AND SURVIVABILITY OF GIANT PLANETS ON WIDE ORBITS

Motivated by the recent discovery of massive planets on wide orbits, we present a mechanism for the formation of such planets via disk fragmentation in the embedded phase of star formation. In this phase, the forming disk intensively accretes matter from the natal cloud core and undergoes several fragmentation episodes. However, most fragments are either destroyed or driven into the innermost regions (and probably onto the star) due to angular momentum exchange with spiral arms, leading to multiple FU-Ori-like bursts and disk expansion. Fragments that are sufficiently massive and form in the late embedded phase (when the disk conditions are less extreme) may open a gap and evolve into giant planets on typical orbits of several tens to several hundreds of AU. For this mechanism to work, the natal cloud core must have sufficient mass and angular momentum to trigger the burst mode and also form extended disks of the order of several hundreds of AU. When mass loading from the natal cloud core diminishes and the main fragmentation phase ends, such extended disks undergo a transient episode of contraction and density increase, during which they may give birth to a last and survivable set of giant planets on wide and relatively stable orbits.

Secular evolution of viscous and self-gravitating circumstellar discs

We add the effect of turbulent viscosity via the α-prescription to models of the self-consistent formation and evolution of protostellar discs. Our models are non-axisymmetric and are carried out using the thin-disc approximation. Self-gravity plays an important role in the early evolution of a disc, and the later evolution is determined by the relative importance of gravitational and viscous torques. In the absence of viscous torques, a protostellar disc evolves into a self-regulated state with the Toomre parameter Q ∼ 1.5 - 2.0, non-axisymmetric structure diminishing with time and maximum disc-to-star mass ratio ξ = 0.14. We estimate an effective viscosity parameter α eff associated with gravitational torques at the inner boundary of our simulation to be in the range 10 -4 -10 -3 during the late evolution. The addition of viscous torques with a low value a = 10 -4 has little effect on the evolution, structure and accretion properties of the disc, and the self-regulated state is largely preserved. A sequence of increasing values of a results in the discs becoming more axisymmetric in structure, being more gravitationally stable, having greater accretion rates, larger sizes, shorter lifetimes and lower disc-to-star mass ratios. For a = 10 -2 , the model is viscous-dominated, and the self-regulated state largely disappears by late times. The axisymmetry and low surface density of this model may contrast with observations and pose problems for planet formation models. The use of α = 0.1 leads to very high disc accretion rates and rapid (within 2 Myr) depletion of the disc, and seems even less viable observationally. Furthermore, only the non-viscous-dominated models with low values of a = 10 -4 -10 -3 can account for an early phase of quiescent low accretion rate M ∼ 10 -8 M ⊙ yr -1 (interspersed with accretion bursts) that can explain the recently observed Very Low luminosity Objects (VeLLOs). We also find that a modest increase in disc temperature caused by a stiffer barotropic equation of state (y = 1.67) has little effect on the disc accretion properties averaged over many disc orbital periods (∼10 4 yr), but can substantially influence the instantaneous mass accretion rates, particularly in the early embedded phase of disc evolution.

Self‐regulated gravitational accretion in protostellar discs

We present a numerical model for the evolution of a protostellar disc that has formed self-consistently from the collapse of a molecular cloud core. The global evolution of the disc is followed for several million years after its formation. The capture of a wide range of spatial and temporal scales is made possible by use of the thin-disc approximation. We focus on the role of gravitational torques in transporting mass inward and angular momentum outward during different evolutionary phases of a protostellar disc with disc-to-star mass ratio of order 0.1. In the early phase, when the infall of matter from the surrounding envelope is substantial, mass is transported inward by the gravitational torques from spiral arms that are a manifestation of the envelope-induced gravitational instability in the disc. In the late phase, when the gas reservoir of the envelope is depleted, the distinct spiral structure is replaced by ongoing irregular non-axisymmetric density perturbations. The amplitude of these density perturbations decreases with time, though this process is moderated by swing amplification aided by the existence of the discs sharp outer edge. Our global modelling of the protostellar disc reveals that there is typically a residual non-zero gravitational torque from these density perturbations, i.e. their effects do not exactly cancel out in each region. In particular, the net gravitational torque in the inner disc tends to be negative during first several million years of the evolution, while the outer disc has a net positive gravitational torque. Our global model of a self-consistently formed disc shows that it is also self-regulated in the late phase, so that it is near the Toomre stability limit, with a near-uniform Toomre parameter Q ≈ 1.5-2.0. Since the disc also has near-Keplerian rotation, and comparatively weak temperature variation, it maintains a near-power-law surface density profile proportional to r -3/2 .

The effect of a finite mass reservoir on the collapse of spherical isothermal clouds and the evolution of protostellar accretion

Motivated by recent observations that detect an outer boundary for starless cores, and evidence for time-dependent mass accretion in the Class 0 and Class I protostellar phases, we re-examine the case of spherical isothermal collapse in the case of a finite mass reservoir. The presence of a core boundary, implemented through a constant-volume approximation in our simulation, results in the generation of an inward-propagating rarefaction wave. This steepens the gas density profile from r -2 (self-similar value) to r -3 or steeper. After a protostar forms, the mass accretion rate M evolves through three distinct phases: (1) an early phase of decline in M, which is a non-self-similar effect due to rapid and spatially non-uniform infall in the pre-stellar phase; (2) for large cores, an intermediate phase of near-constant M from the infall of the outer part of the self-similar density profile, which has low (subsonic) infall speed in the pre-stellar phase; and (3) a late phase of rapid decline in M when accretion occurs from the region affected by the inward-propagating rarefaction wave. Our model clouds of small to intermediate size make a direct transition from phase (1) to phase (3) above. Both the first and second phase (if the latter is indeed present) are characterized by a temporally increasing bolometric luminosity L bol , while L bol is decreasing in the third (final) phase. We identify the period of temporally increasing L bol with the Class 0 phase, and the later period of terminal accretion and decreasing L bol with the Class I phase. The peak in L bol corresponds to the evolutionary time when 50 ± 15 per cent of the cloud mass has been accreted by the protostar. This is in agreement with the classification scheme proposed in the early 1990s by Andre et al.; our model adds a physical context to their interpretation. We show how our results can be used to explain tracks of envelope mass M env versus L bol for protostars in Taurus and Ophiuchus. We also develop an analytic formalism that successfully reproduces the protostellar accretion rate from profiles of density and infall speed in the pre-stellar phase. It shows that the spatial gradient of infall speed that develops in the pre-stellar phase is a primary cause of the temporal decline in M during the early phase of protostellar accretion.