- Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
The formation of protostars and their disks has been understood as the result of the gravitational collapse phase of an accumulation of dense gas that determines the mass reservoir of the star-disk system. Against this background, the broadly applied scenario of considering the formation of disks has been to model the collapse of a dense core assuming spherical symmetry. Our understanding of the formation of star-disk systems is currently undergoing a reformation though. The picture evolves from interpreting disks as the sole outcome of the collapse of an isolated prestellar core to a more dynamic picture where disks are affected by the molecular cloud environment in which they form. In this review, we provide a status report of the state-of-the-art of spherical collapse models that are highly advanced in terms of the incorporated physics together with constraints from models that account for the possibility of infall onto star-disk systems in simplified test setups, as well as in multi-scale simulations that cover a dynamical range from the Giant Molecular Cloud environment down to the disk. Considering the observational constraints that favor a more dynamical picture of star formation, we finally discuss the challenges and prospects in linking the efforts of tackle the problem of star-disk formation in combined multi-scale, multi-physics simulations.
1 Introduction
The by-now classical approach of modeling the formation of a star dates back to more than half a century to the pioneering work of (Larson, 1969), who started from the assumption of an isolated spherical core that collapses due to its own gravity. This assumption has become the standard approach in modeling the formation of individual stars and their disks that form as a result of conservation of angular momentum during the collapse phase. However, in recent years it has become more and more clear that the morphology of the precursors of stars, namely, prestellar cores, often deviates significantly from spherical symmetry in turbulent filamentary Giant Molecular Clouds (André et al., 2014). In addition, asymmetric features (‘streamers’) (see review by Pineda et al., 2023), as well as strong indications for late infall (e.g., SU Ginski et al., 2021) challenge our traditional view on disk formation. This review is an attempt to concisely summarize the developments made in developing state-of-the-art multi-physics models of spherical collapse, and put those in context to multi-scale models that account for the larger scale dynamics of the molecular cloud environment. We emphasize that the scope of the review is the formation process of disks. These disks evolve over time and disperse. An overview of important effects regarding disk dispersal through binary interactions (e.g., Kuruwita and Federrath, 2019; Offner et al., 2023), stellar flybys (e.g., Cuello et al., 2023; Smallwood et al., 2023) or external photoevaporation (e.g., Winter and Haworth, 2022) can be found in the respective references. Note that developments that later marked important advancements for modeling disk formation were initially included in studies that focused on binary/multiple formation. Examples are the use of adaptive mesh refinement (AMR) (Truelove et al., 1998; Kratter et al., 2010) or nested grids (Burkert and Bodenheimer, 1993; Matsumoto and Hanawa, 2003) in spherical collapse simulations. The focus on multiplicity rather than disk properties is not surprising considering that disks easily form in hydrodynamical simulations.
This review focuses on results obtained through modeling of (proto-)star formation. The presented results are derived using various codes that adopt different methodology. Traditionally, star formation has either been modeled following a Lagrangian approach, i.e., smoothed particle hydrodynamics (SPH), or an Eulerian approach, i.e., a mesh/grid codes. In SPH codes, the gas distribution is represented through particles and the properties of the gas is computed through averaging over the nearest neighbors of interacting particles. SPH codes that are used for modeling disk formation are dragon (Goodwin et al., 2004), phantom (Price et al., 2018), optimized versions of sphNG (Benz et al., 1990), as well as Godunov SPH methods (Iwasaki and Inutsuka, 2011). In Eulerian codes, the gas is discretized in form of a mesh or grid consisting of cells, and the evolution of the gas is simulated by calculating the flux through the cell boundaries to update the cell quantities. The grid is typically assumed to be cartesian or spherical in star formation models. Examples of grid codes used for disk modeling are athena (Stone et al., 2008), athena++ (Stone et al., 2020), dispatch (Nordlund et al., 2018), enzo (Bryan et al., 2014), flash (Fryxell et al., 2000), Pluto (Mignone et al., 2012), Orion (Klein, 1999), ramses (Teyssier, 2002; Fromang et al., 2006), sfumato (Matsumoto, 2007) and zeus (Stone and Norman, 1992). While SPH codes intrinsically adapt to resolve the higher densities during the star formation process, many grid codes offer the possibility to resolve the process by the use of a grid with flexible cell sizes. The resolution of the grid can either be fixed at the beginning of the simulation in form of static or flexible through AMR. Static grids are often used for spherical or nested grids, where the forming star is located at the center such that by construction the gas close to the star is resolved with higher resolution than the gas at larger radial distances. Nowadays, the traditional distinction between SPH and grid codes has become increasingly softened by the development of methods that contain properties of both approaches such as moving mesh codes (arepo Springel, 2010a; Weinberger et al., 2020) and codes allowing for the use of meshless methods (gizmo Hopkins, 2015). Each method has its benefits and disadvantages. On the one hand for instance, it is by construction straight-forward to handle advection of flows with SPH algorithms, while more care is required in grid codes. On the other hand, the use of constrained transport in grid codes (Evans and Hawley, 1988; Balsara and Spicer, 1999; Londrillo and del Zanna, 2004) guarantees the absence of unphysical magnetic monopoles
For more details on the methods that are commonly used in star formation modeling, we refer the reader to dedicated reviews. For the technicalities of SPH, see for instance (Tricco, 2023, in this volume) or previous reviews (Rosswog, 2009; Springel, 2010b; Price, 2012). For an overview of the numerical methods in grid-based codes, we refer to Teyssier (2015); Teyssier and Commerçon (2019) as well as to the respective papers corresponding to the individual codes. Alternatives to the more traditional methods are presented and discussed in Hopkins (2015).
Section 2 of this review provides an overview of disk formation in spherical collapse with a focus on how to solve the magnetic braking problem. Section 3 focuses on the dynamics beyond the prestellar core and how modeling can be used to account for the effect of infall onto the disk formation process. Section 4 summarizes the results and provides an outlook on how multi-scale, multi-physics models can help to understand the small-scale subtleties of disk formation that is important for planet formation in the context of the larger-scale dynamics.
2 Formation of disks in models of isolated spherical collapse
In general, the assumption of spherical collapse is the obvious first choice for modeling gravitational collapse. The assumption of spherical symmetry simplifies the three-dimensional spatial problem to a one-dimensional problem. Moreover, it allows to study the effect of various parameters in a well-defined setup. Over the years, more and more parameters were added to the setup to test their effect on the collapse phase. An important additional parameter of the setup was to initialize the sphere with a net rotation rate. Theory predicts that during the collapse, the vertical velocity components cancel out each other, but the net rotational component remains such that the collapse leads to the formation of a rotationally supported disk as a consequence of angular momentum conservation. Purely hydrodynamical models (i.e., models without magnetic fields) demonstrated the formation of disks with smaller/larger disk sizes for initially weak/strong rotation of the sphere using smoothed particle hydrodynamics (SPH) (Bate, 1998; 2010; Walch et al., 2009; 2010), adaptive mesh refinement (AMR) (Truelove et al., 1998; Banerjee et al., 2004) or nested grid codes (Yorke et al., 1993; Saigo et al., 2008; Machida et al., 2010).
To prevent very low time steps during the collapse, it is common practice in many models to introduce a sink particle that is created at the center once a critical density is exceeded and possible additional criteria are fulfilled. During the further evolution, the sink accretes mass from the surrounding gas according to a prescribed recipe (for a detailed overview on numerical methods including sink particles, please refer to the review by Teyssier and Commerçon, 2019).
As the next step, models started to account for the presence of magnetic fields in molecular clouds (Crutcher, 2012) by carrying out magnetohydrodnamical simulations. To account for magnetic fields in the setup, the sphere is typically initialized with a magnetization defined by the ratio of enclosed mass
with a numerical constant
According to theory, cores are magnetically supported against collapse for
but in this section we focus on spherical collapse and therefore refer to a scenario for which Equation 3 is not applicable. The first generations of these collapse simulations considered the ideal MHD case of sufficiently ionized gas such that the magnetic field is well coupled to the bulk neutral gas. The induction equation corresponding to ideal MHD is
with bulk velocity of the gas velocity
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Figure 1. Illustration of the effect of initial conditions on the collapse phase shown face-on (top panels) and edge-on (bottom panels). From left to right: hydrodynamical case (no
Until the late 2000s, including magnetic fields into the setup was therefore considered catastrophic for disk formation, which was at tension with the growing evidence of Keplerian disks in observations (Brinch et al., 2007; Lommen et al., 2008; Jørgensen et al., 2009; Lee, 2011; Murillo et al., 2013; Codella et al., 2014; Harsono et al., 2014). At the same time, magnetic fields are the most promising candidate for producing these fast jets and low-velocity disk winds (Donati et al., 2010; Bjerkeli et al., 2016; Lee et al., 2018; Moscadelli et al., 2022) though winds especially at later stages of disk evolution might also be launched by photoevaporation (Alexander et al., 2014). While part of the material in the disk subsequently accretes onto the protostar, a substantial amount of the mass is ejected vertically from the system in narrow protostellar jets (Guszejnov et al., 2021; 2022), or in disk winds that have wider opening angles (Watson et al., 2016). Constraining the role of magnetic fields is challenging because they are difficult to observe. To get an idea of the magnetic field structure in star forming regions, one can measure the linear polarization of non-spherical dust grains as they tend to align with the underlying magnetic field due to radiative torques (Sadavoy et al., 2019; Le Gouellec et al., 2020). However, dust polarization in disks seems to be dominated by effects of self-scattering within about 100 AU from the star (Kataoka et al., 2016), (though dichroic extinction may be responsible for parts of the polarization signal in disks, too) (Lee et al., 2021). That means dust polarization is a good tracer of the structure (though not on the strength) of magnetic fields on scales of the protostellar environment beyond the disk, which allows us to at least obtain loose constraints on the role of magnetic fields in the star formation process (Pattle et al., 2023). As these observations inform us about the presence of magnetic fields in the star-forming regions (Le Gouellec et al., 2020), there is consensus that they are the important ingredient of magnetically driven disks (Lesur et al., 2023).
The challenging question for modelers was therefore: ‘how to avoid this catastrophe during spherical collapse?’ As of today, there is consensus that magnetic braking can be sufficiently reduced to allow disk formation even for spherical collapse setups with high magnetization of
• non-ideal MHD, or
• misalignment between magnetic field orientation and rotational axis, or
• turbulence.
We emphasize that there are already reviews (e.g., Wurster and Li, 2018; Zhao et al., 2020b; Tsukamoto et al., 2023b) in the literature available that summarize the progress made in overcoming the magnetic braking catastrophe during the last
2.1 Non-ideal MHD
One way of overcoming the magnetic braking catastrophe is by accounting for resistive effects (commonly referred to as ‘non-ideal MHD’) such that the magnetic field lines are no longer tightly coupled to the bulk motion of the gas. In fact, dense cores in molecular clouds are weakly ionized (Bergin and Tafalla, 2007) and according to models the ionization rates can reach values of
Moreover, the energy equation for the time evolution of internal energy
If expressed in terms of total energy
Note that the Hall resistivity does not affect the internal energy equation because in contrast to ohmic dissipation and ambipolar diffusion, it is a dispersive, non-dissipative process. Using the modified induction equation (Equation 5), and a modified equation for internal (Equation 6) or total energy (Equation 7), one can carry out single-fluid non-ideal MHD simulations.
2.1.1 Ohmic resistivity
Reflecting the complexity in accounting for the effects numerically, the first non-ideal effect included in models was ohmic dissipation, followed by ambipolar diffusion and at last the Hall effect. In the early approaches, ohmic resistivity was set to constant values (Shu et al., 2006; Krasnopolsky et al., 2010) with the goal to test which values of
2.1.2 Ambipolar resistivity
The low probability of obtaining the right conditions for circumventing strong magnetic braking solely through ohmic resistivity gave rise to a revival of seriously accounting for ambipolar diffusion as it is the dominant process at lower densities present in prestellar cores. The idea of considering ambipolar diffusion as an important process of individual star formation dates back to the 1970s (Mouschovias, 1976; 1977; 1979), when it was studied in the context of redistributing the magnetic field to avoid the pile-up of magnetic pressure at the center due to magnetic-flux freezing that would prevent protostellar collapse (see Das and Basu, 2021, for a more recent linear instability analysis of magnetized sheets with ambipolar diffusion). Against this background, Hennebelle et al. (2016) estimated that disks form with typical disk sizes around 18 au based on analytical calculations that account for ambipolar diffusion.
Early 2D models by Mellon and Li (2009) did not find evidence for disk formation induced by including ambipolar diffusion, but their setup disallowed disk formation on radii less than
As anticipated in previous models (Tomida et al., 2013), a follow-up study by Tomida et al. (2017) that introduced a sink particle, demonstrated the formation of significantly larger disks
2.1.3 Hall effect
The Hall effect is the most difficult resistivity to implement and therefore the one that has been less studied although its possible influence on protostellar collapse was previously pointed out and studied (semi-)analytically (Wardle and Ng, 1999; Wardle, 2004; Braiding and Wardle, 2012). Today there is a decent number of papers presenting the results of spherical collapse simulations that incorporate the Hall effect in 2D grid simulations using zeus (Krasnopolsky et al., 2011; Li et al., 2011; Zhao et al., 2020a; Zhao et al., 2021), 3D SPH simulations using a Godunov SPH code (Tsukamoto et al., 2015a; 2017) as well as (Wurster et al., 2016; Wurster et al., 2018a; Wurster et al., 2018b; Wurster et al., 2019; Wurster et al., 2022), 3D AMR simulations with ramses (Marchand et al., 2018; 2019), and 3D moving mesh simulations using arepo (Zier et al., 2024a) that incorporate Hall resistivity. In theory, the Hall resistivity can adopt positive or negative values, but in practice it is typically negative during the protostellar collapse as shown in SPH (Tsukamoto et al., 2015a; Wurster et al., 2016) as well as in cartesian grid simulations (Marchand et al., 2016). In contrast to ohmic dissipation and ambipolar diffusion, the Hall effect depends on the direction of the magnetic field. In the case of anti-parallel alignment of magnetic field and angular momentum, it can cause spin up of the gas and lead to the formation of a larger disk. Contrary, in the case of parallel alignment of
2.1.4 Misalignment between magnetic field and rotational axis
The relative orientation of initial rotational axis and
2.1.5 Turbulence
While it is consensus that magnetic braking quenches disk formation in ideal MHD models of spherical collapse with alignment between rotational and
However, it has also been pointed out by Seifried et al. (2012), who considered a larger (
2.1.6 The role of the ionization rate
Today, it is consensus that non-ideal MHD effects can resolve the magnetic braking problem of disk formation in spherical collapse simulations. The bigger question today is how much they affect the collapse and disk formation phase. Especially, early models assumed resistivities relatively crudely only roughly accounting for the dependency on density. The individual resistivities depend on the underlying physical conditions and current state-of-the-art models account for this dependency by using pre-computed values that are assigned to the local density, magnetic field strength and temperature. The computation of the table of resistivities is done by using chemical equilibrium models. For instance, Marchand et al. (2016) illustrated the differences compared to the values assumed in previous studies by Duffin and Pudritz (2009) and Machida et al. (2007). Apart from that there are some differences depending on the chemical model that was used to compute the tables. In the table by Zhao et al. (2016) the Hall resistivity is lower than the ambipolar resistivity for a cosmic-ray ionization rate of
While these differences between the tables are generally more subtle, another physical effect is more crucial. It is common practice to assume non-ideal MHD coefficients that were computed for a fixed cosmic-ray ionization rate
Wurster et al. (2018c) carried out a parameter study of the very early protostellar collapse phase with all three non-ideal MHD effects using resistivities corresponding to cosmic-ray ionization rate in the range of
2.2 Dust
2.2.1 The role of dust on the resistivities
The resistivities are also affected by the assumed dust distribution (Zhao et al., 2018; Koga et al., 2019; Marchand et al., 2020). For a distribution with a large number of small grains (
2.2.2 Incorporation of dust dynamics and growth in collapse models
In the context of planet formation, essential ingredients to incorporate in the models are dust dynamics and dust growth. This is particularly important considering the growing evidence for an early onset of planet formation. Some (magneto-)hydrodynamical models started to include the dynamics of dust in young forming disks in spherical collapse simulations. Vorobyov et al. (2019) and Bate (2022), independently carried out hydrodynamical simulations of disk formation in which they included the dynamics of dust particles during disk formation. Consistent with analytical predictions, they find size-dependent radial drift of dust particles. Lebreuilly et al. (2020) demonstrated that dust drifts toward the inner part of the disks with larger grains accumulating in the inner parts of the disk, where they cause an enhanced dust-to-gas ratio. More recently (Vorobyov et al., 2024), followed up on their earlier work that already included dust growth and drift, by also accounting for the back-reaction of the dust on the gas (Stoyanovskaya et al., 2018). They concluded that the dust-to-gas ratio becomes high enough for the onset of planetesimal formation after only
Tsukamoto et al. (2021) also modeled the drift of dust particles during the collapse, but also considered grain growth. In contrast to Lebreuilly et al. (2020), they find that a fraction of the dust particles can be elevated by an outflow in the inner part of the disk, become entrained in the envelope, and eventually fall back onto the outer part of the disk. This idea is conceptually similar to earlier scenarios suggested to explain the transport of the oldest solids in the Solar System, namely, CAIs and chondrules (Shu et al., 1996; 1997). Tsukamoto et al. (2021) envision a scenario of multiple cycles that contribute to grain growth. The dust grains drift through the disk, grow during the drift phase, are ejected through an outflow in the inner disk and fall back onto the outer disk as a larger grain. Considering multiple cycles of this mechanism, this could lead to a grain size distribution in the disk that is shifted towards larger grains and which should be imprinted in the dust opacity spectral index of the envelope around the forming star-disk system. Considering the meteoritic record, the transport mechanism could potentially explain the imprints of reprocessing reported for some chondrules. Recently, Cacciapuoti et al. (2024) followed up on this transport scenario by measuring the dust opacity spectral index in a few cores with outflows. Their results do not reveal an unambiguous correlation of dust growth with outflow strength though and future observations are required to further test the scenario.
3 Beyond isolated spherical collapse
While variations in disk sizes between different models with non-ideal MHD are often solely explained as a sign of the underlying non-ideal MHD resistivities, another aspect tends to be overlooked, namely, the role of the initial and boundary conditions. It is a challenge to draw conclusions about the importance of individual effects from results that were obtained using different model setups codes. Some groups start from initial conditions assuming a density profile according to a Bonnor-Ebert (BE) sphere, whereas others start with a uniform density distribution. Considering this general issue, the parameter study by Machida et al. (2014) stands out in terms of constraining the role of various effects. They adopted several model setups that were previously used by groups (Hennebelle and Ciardi, 2009; Li et al., 2011; Machida et al., 2011; Joos et al., 2012; Krasnopolsky et al., 2012; Seifried et al., 2012) and recomputed those models with their own code. Activating the same non-ideal MHD effects and starting from similar initial mass-to-flux ratios, they found significant differences in disk formation depending on the assumption of the initial density profile. Models starting with a uniform density distribution led to disk sizes of the order of 10 au, while models starting with a density distribution of a Bonnor-Ebert sphere yield larger disk sizes of about 100 au. They also emphasized the role of the accretion recipe onto a sink. Under identical initial conditions, allowing the sink to accrete from a smaller region in its vicinity favors the formation of larger disks compared to a sink that is allowed to accrete from a larger region.
These results point to a more fundamental issue of modeling individual star formation as the outcome of the collapse of an isolated sphere. In fact, Larson (1969) stated already that they assumed ‘the simplest assumptions’ on the boundary condition and they ‘again adopted the simplest assumptions’ for the initial conditions in the spherical collapse scenario. While the assumption of spherical collapse has proven to be very helpful in constraining the effect of various physical parameters during collapse, it is important to keep in mind that reality can differ significantly from this idealized approximation. It seems that the assumption of spherical symmetry is a fair assumption for the earliest phase of protostellar collapse corresponding to the formation of the first and second core, and model predictions about the properties of protostars during the first few thousands of years are likely to be very accurate - except for the uncertainty of the ionization rate at these early stages.
However, (almost) all of the observed disks are significantly older than
We know, however, from observations that the morphology of prestellar cores is significantly affected by the underlying dynamics in filamentary Giant Molecular Clouds (André et al., 2014; Kainulainen et al., 2017; Hacar et al., 2023; Pineda et al., 2023). This implies to investigate the process of disk formation with a different approach than the spherical core setup has become timely. As of today, there are relatively few studies compared to the large number of classical collapse models that account for larger-scale dynamics, such as infall or binary interaction, in the context of disk formation. Both stellar encounters as well as infall have been modeled in computing-intense zoom-in simulations as well as in simplified model setups that allow to carry out cheaper models in terms of computing-time. In the following, we will summarize the efforts made in recent years distinguishing between the progress made in models that were configured to model specific scenarios through parameter studies and compute-intense models that aim for resolving the processes in multi-scale simulations.
3.1 The role of infall
Considering the presence of accretion streamers even around presumably evolved stars, it is becoming increasingly evident that disk formation models need to take into account for the possibility of infall that is often anisotropic (Kuffmeier et al., 2017; Kuznetsova et al., 2019; 2020). Currently most of the numerical constraints on infall are from models that model the cloud dynamics, but do not resolve the formation of disks. Padoan et al. (2014) demonstrated that infall through Bondi-Hoyle accretion can induce accretion bursts that are a prominent explanation for the luminosity problem. Moreover, in line with the interpretation of the inertial flow model proposed by Padoan et al. (2020), Pelkonen et al. (2021) showed that a significant amount of the material accreting onto the star was initially not gravitationally bound to the core, and the relative mass fraction of accreting material scales with the final mass of the star. This is also consistent with earlier results by Smith et al. (2011), who reported a prolonged accretion history of the more massive stars in their hydrodynamical simulations of a turbulent cloud. Following up on these results, Kuffmeier et al. (2023) also showed that a Class II young stellar object can return to Class I or even Class 0 phase in the event of massive infall. It remains to be self-consistently modeled how the disk reacts to such events. However, there are parameter studies that considered the effect of individual infall events on the properties of star-disk systems.
Starting from spherical collapse models, several groups considered the case of infall from the envelope onto an existing disk (Bae et al., 2015; Lesur et al., 2015; Vorobyov et al., 2015; Kuznetsova et al., 2022). They all find that the infall can trigger instabilities in the disk. Carrying out hydrodynamical simulations, Vorobyov et al. (2015) showed that infall can trigger gravitational instabilities that trigger accretion bursts of the star. Bae et al. (2015) and Kuznetsova et al. (2022) find that these infall events trigger Rossby-wave instabilities, which Kuznetsova et al. (2022) attribute as possible seeds for gap, ring and structure formation in disks. Considering the possibility of misaligned infall from the envelope, Thies et al. (2011) demonstrated in hydrodynamical simulations of a collapsing sphere with differences in the angular momentum orientation of the outer infalling radial layer that such infall can induce the formation of an outer disk that is misaligned with respect to the primordial inner disk. Considering the indications for late, post-collapse infall, Dullemond et al. (2019) developed a model of cloudlet capture, in which a low-mass gaseous cloudlet encounters a star modeled as a point mass with an impact parameter. As a result of the encounter the hydrodynamical simulations carried out with the Pluto code showed the formation of extended arm features similar to structures seen around Herbig stars such as for instance AB Aurigae. The formation of such an extended arm (streamer) was also reported in the cloudlet capture models by Hanawa et al. (2022, 2024). Kuffmeier et al. (2020) followed up on the cloudlet capture scenario by carrying out simulations with the arepo code, in which they showed that such encounters cannot only lead to the formation of extended arms, but also to the formation of a second-generation disk. In the presence of an already (or still) existing primordial disk, such an event can lead to the formation of a system consisting of misaligned inner and outer disk (Kuffmeier et al., 2021), which is observable as shadows in the outer disk in scattered light observations (Krieger et al., 2024). SU Aur is the most prominent candidate that might undergo such an event today (Ginski et al., 2021). The cloudlet capture scenario has also been adopted by Unno et al. (2022), who considered an encounter of a magnetized cloudlet. They showed that the event can lead to magnetic acceleration of the existing inner disk for a favorable orientation of the magnetic field of the cloudlet compared to the field orientation in the disk.
3.2 Clump to disk
Bate (2018) carried out 3D hydrodynamical simulations with
In the case of Bate (2018), simulating the extended range of scales came at the cost of reduced physics compared to state-of-the-art simulations of spherical collapse simulations. Radiative transfer was included, but magnetic fields were not taking into account such that magnetic braking is not accounted. Lebreuilly et al. (2021) followed a similar approach of modeling the formation of stars dynamics of a massive turbulent sphere with an initial mass of 1,000
The authors caution nonetheless state that non-ideal MHD is important for disk formation. Based on the results of collapse simulations with higher resolution (Joos et al., 2013) they expect that magnetic braking will be more efficient and the disks therefore smaller if the resolution at the highest levels was higher than
Comparing the ambipolar diffusion runs with mass-to-flux ratios of
3.3 Giant Molecular Cloud to disk
While the clump-models by Bate (2018); Elsender and Bate (2021) as well as Lebreuilly et al. (2021, 2024a) allowed to derive disk population synthesis, the models starting from an initial spherical clump of
Kuffmeier et al. (2018) also carried out a zoom-in simulations for one of the stars with a resolution as high as 0.06 au for a time interval of 1,000 years at about 50 kyr after star formation. These simulations resolved the infall of a gas blob onto the disk that triggered a gravitational instability that was responsible for an accretion burst. While the dynamical range of the simulation allowed to take into account for the larger scale environment, important physical effects such as non-ideal MHD and/or a more realistic treatment of the thermodynamics was lacking in these models compared to state-of-the-art models of spherical collapse. The challenge for the upcoming decade will be to fill the gap between spherical collapse models with advanced multi-physics treatment, but highly idealized initial and boundary conditions, and multi-scale models that self-consistently account for the Giant Molecular Cloud dynamics, but lack advanced treatment of relevant physical processes. The improvements of infrastructure of more powerful supercomputers together with optimized numerical codes that enable more efficient computing (e.g., Hopkins, 2017; Nordlund et al., 2018; Price et al., 2018; Stone et al., 2020; Weinberger et al., 2020) allow us to accept this challenge. This also includes the possibility to account for stellar feedback mechanisms that can shape and affect disk formation. For instance, UV radiation feedback plays an important role for the overall cloud dispersal as shown in radiation hydrodynamics models (Kim et al., 2018; Fukushima et al., 2020). Together with implementations of protostellar outflows, e.g., Guszejnov et al. (2021) or Lebreuilly et al. (2024b), these are potentially important effects shaping the accretion process of stars and thereby their disks. More recently, as part of the STARFORGE initiative (Grudić et al., 2021), Guszejnov et al. (2022) already demonstrated the possibility to study supernova feedback, stellar radiation, protostellar jets and winds together in one simulation. Continuing these efforts while applying high enough resolution will allow us to test the frequency and properties of various outcomes of infall-induced features such as misaligned disks or disk instabilities that are possible outcomes of a heterogeneous star formation process happening in Giant Molecular Clouds (see illustration in Figure 2).
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Figure 2. Illustration of heterogeneous star formation and possible outcomes of disk formation in a Giant Molecular Cloud. The upper panel shows the density fluctuations in a filamentary cloud based on data from (Kuffmeier et al., 2017). The remaining part sketches different stages of the disk. The left inset shows a young disk with a strong magnetically-induced outflow enclosing a jet, the middle inset shows infall through a streamer solely delivering material, while the right inset shows the scenario where infall leads to the formation of a misaligned outer disk during a late infall event. The cartoons were made by Martine Lützen, a former Master student at the Niels Bohr Institute.
4 Reflections and outlook
Since the first models of protostellar collapse of a spherical core (Larson, 1969), there have been a lot of successful efforts in improving our understanding of protostellar formation. It is possible to resolve the collapse with a resolution that even allows to model the dynamics within the inner one au of the forming protostar, while there also as has been systematical incorporation and testing of additional physical effects. Starting from the crucial assumption for disk formation of initial rotation, models have improved to a degree where it is possible to account for more subtle effects such as the role of the ionization rate or the dust size distribution on the resistivities, which modify the efficiency of magnetic braking of the forming disk.
Regarding initial and boundary conditions, the assumption of spherical collapse of an isolated prestellar core has proven to be successful as a the commonly used, fiducial assumption to test and compare the parameter space. In fact, the early ALMA images of dust continuum (Partnership et al., 2015; Andrews et al., 2018), despite their substructures and diversity, gave the impression as if the disks are indeed isolated entities that can be considered detached from the environment in which they are embedded in. However, today we know that there is a severe observational bias in observing dust emission at relatively large wavelength of about 1 mm. Gas observations of disks, reveal a different picture (Öberg et al., 2021) than dust continuum images obtained at
As elaborated in Section 3.1, the late addition of gas with substantial angular momentum even offers an additional path to disk formation beyond the initial protostellar collapse phase. In agreement with earlier suggestions by Padoan et al. (2005), Kuffmeier et al. (2023) demonstrated that the larger contribution of late infall can explain the subtle trend of larger disk sizes for increasing mass of the corresponding host stars seen in CO observations (Long et al., 2022). Furthermore, the infall of material is typically misaligned with respect to the orientation of the star-disk system (Kuffmeier et al., 2024), especially at later stages (Pelkonen et al., 2024), leading to a more chaotic pathway of star-disk formation (Bate et al., 2010). Most recently, conceptual papers followed up on the possibility of post-collapse disk formation arguing that a significant amount of observed Class II disks are in fact the result of prolonged Bondi-Hoyle like accretion in the turbulent interstellar medium (Padoan et al., 2024; Winter et al., 2024). If this mode of second-generation disk formation proves to be significant, it also implies that interpreting surveys of stellar age-dependent disk fractions (Haisch et al., 2001; Mamajek, 2009; Richert et al., 2018) solely as an outcome of the evolution of either viscous or wind-driven disks (e.g., review by Manara et al., 2023, and references therein) is misleading. In the post-collapse infall picture, the lower fraction of disks around more evolved stars instead reflects the decreasing probability of experiencing mass replenishment through late gas encounters. Disks around older stars should instead be considered as either disks that experience prolonged mass replenishment or as second-generation disks that formed several million years after the initial protostellar collapse phase through post-collapse gas encounters.
Considering the compelling observational evidence, we have recently started to consider filamentary accretion via streamers as part of the disk formation process. The effect of infall on the processes in the disk and their impact on planet formation has only been poorly investigated yet, though infall through streamers may be of key importance (Bae et al., 2015; Lesur et al., 2015; Kuffmeier et al., 2018; Kuznetsova et al., 2022). In particular, infall likely plays an important role in.
• regulating the disk size,
• triggering instabilities in young disks and thereby initiating substructures,
• inducing misaligned disks visible as shadows in scattered light observations,
• seeding finite amplitude pressure traps,
• potentially modifying the chemical composition of planetary systems,
• resetting the disk entirely.
Also we commonly use the term ‘streamer’ regardless whether it describes filamentary infall on scales of
What is lacking at the current stage is a model that allows us to distinguish between the relevance and occurrence of the different modes without an intrinsic bias in the model setup that favors or even excludes possibly important scenarios. A self-consistent model of the accretion process of star-disk systems with high resolution together with the molecular cloud dynamics for long enough times (i.e., beyond
From a technical point of view, the challenge for the upcoming years will be to connect state-of-the-art multi-physics models with multi-scale models that currently lack parts of the physics that is important for disk formation on smaller scales. Future models will allow us to fill the currently existing gap in understanding to what extent the properties of disks are governed by the protostellar environment and how frequent various outcomes such as infall-induced misaligned disks and instabilities are. The prospects in succeeding with this task are bright. Some models have already succeeded in resolving young disks with a resolution of less than 0.1 au in the context of a magnetized Giant Molecular Cloud that is
Author contributions
MK: Writing–original draft, Writing–review and editing.
Funding
The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. The research of MK is supported by a H2020 Marie Skłodowska-Curie Actions grant (897524) and a Carlsberg Reintegration Fellowship (CF22-1014).
Acknowledgments
MK thanks the two referees for constructive feedback and useful suggestions that helped to improve the quality of the manuscript. MK also thanks the organizers of the workshop Simulating Physics in Celestial Ecosystems (SPiCE) in Sendai, Japan, for an inspiring week full of exciting science and discussions.
Conflict of interest
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
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Keywords: star formation, (magneto-)hydrodynamics, disk formation phase, non-ideal MHD, infall, accretion
Citation: Kuffmeier M (2024) Magnetohydrodynamical modeling of star-disk formation: from isolated spherical collapse towards incorporation of external dynamics. Front. Astron. Space Sci. 11:1403075. doi: 10.3389/fspas.2024.1403075
Received: 18 March 2024; Accepted: 26 June 2024;
Published: 28 August 2024.
Edited by:
Kengo Tomida, Tohoku University, JapanCopyright © 2024 Kuffmeier. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Michael Kuffmeier , kueffmeier@nbi.ku.dk