The accretion process on protostars

Abstract

The process of mass accretion onto Young Stellar Objects (YSOs) plays a fundamental role in determining the final stellar mass and setting the initial conditions for planet formation. Despite its critical role, our understanding of accretion remains fragmented, particularly for what concerns the earliest, protostellar phases (Class 0/I). While the community has consolidated a comprehensive knowledge of the accretion process of the later-stage Classical T Tauri Stars (CTTSs), a similar level of understanding is critically lacking for the protostellar phase, where the bulk of the mass is assembled. This work aims to review recent major results, both from the observational and numerical point of view, bridging the gap between the two approaches and providing an updated, complete assessment of accretion in protostellar sources. We present different techniques to measure accretion on protostars, analyze how methodological differences affect parameter estimation, discuss the caveats in comparing with numerical models, and suggest the next steps to take towards an ever more exhaustive picture of the protostellar phase.

1 Introduction

Low-mass star formation is fundamentally regulated by the accretion process, which governs the assembly of the stellar mass. The term “accretion” can refer to both stellar accretion, the transfer of material from the inner disk onto the protostellar surface, or to disk accretion, when the material is moving across or onto the disk at larger radii. In this review, we will focus on the first one, referring to it as accretion for brevity.

The accreting material, stored in a surrounding envelope and/or an accretion disk orbiting the protostar at different evolutionary stages, approaches the protostellar surface as it loses angular momentum and migrates inwards; the actual mechanism leading to the mass incorporation depends on the strength and morphology of the magnetic field. At the protoplanetary disk stage, when the forming star is only surrounded by a disk of gas and dust, observations and theoretical models consistently support the magnetospheric accretion (MA) paradigm (Hartmann et al., 2016; Manara et al., 2023). Alternative magnetospheric configurations, such as the X-wind scenario, have also been proposed within this framework (e.g., Shang et al., 2007). In this context, material from the infalling envelope is first processed through the circumstellar disk, where its inward transport is regulated by either, or a combination of, a macroscopic viscosity (Lynden-Bell and Pringle, 1974; Pringle, 1981; Shakura and Sunyaev, 1973) and magneto-hydrodynamical (MHD) winds (Blandford and Payne, 1982; Lesur, 2021). At the inner disk edge, the flow is typically truncated by the stellar magnetic field and channeled along magnetic field lines onto the stellar surface.

Whether magnetospheric accretion also operates during the earlier protostellar phases, however, remains an open question. MA relies on specific physical conditions, most notably the presence of strong, large-scale stellar magnetic fields of the kilogauss level; when the magnetic field is weak, topologically complex, or unable to effectively truncate the disk, a different accretion regime may operate. The alternative scenario, boundary-layer (BL) accretion, produces high accretion rates where inflow pressure exceeds magnetic field pressure. In this picture, the inner disk is not truncated by the magnetospheric field lines but instead extends down to the stellar surface, allowing disk material to accrete directly onto the star; this mass transfer happens within a narrow boundary layer at the star–disk interface, rather than in magnetically funneled accretion shocks (Lynden-Bell and Pringle, 1974; Pringle, 1981; Popham et al., 1993; Popham, 1997). Such conditions may be common during the earliest protostellar phases, when the magnetic fields are not developed or not strong enough yet and high accretion rates are needed to build the forming star (Hartmann et al., 2016). Nevertheless, current observational constraints do not rule out the presence of dynamically important magnetic fields in protostars - in particular for the less embedded ones; magnetospheric accretion remains therefore a physically viable and widely adopted framework even in the protostellar phases. Discriminating between these regimes is still an open theoretical and observational challenge.

Regardless of the exact physical mechanism, accretion is intrinsically linked to mass ejection phenomena, such as jets and winds, which play a crucial role in extracting angular momentum from the disk–star system and thereby enable continued mass accretion (e.g., Blandford and Payne, 1982; Shu et al., 1994; Pudritz et al., 2007; Frank et al., 2014). In particular, the X-wind model explicitly couples magnetospheric accretion and jet launching at the inner disk truncation radius (Shu et al., 1994); through this coupling, the accretion process regulates not only the transfer of mass onto the forming star, but also the angular momentum budget and the thermal and dynamical evolution of the inner disk. The different accretion regimes, as well as other more distributed disk-mediated inflows (commonly referred to as streamers), are therefore expected to have distinct impacts on the structure, stability, and susceptibility to fragmentation of the inner disk; their influence is concentrated in the region at and within the co-rotation radius (typically a few stellar radii, corresponding to au in low-mass systems). Moreover, the stellar and disk accretion history established during the embedded phases, characterized by gravitational instability and episodic bursts (e.g., Vorobyov and Basu, 2005b; Vorobyov and Basu, 2006; Dunham and Vorobyov, 2012), determines the mass distribution, thermal structure, and stability of young disks (e.g., Kratter et al., 2010; Tsukamoto et al., 2017b); as a consequence, it sets the initial conditions for subsequent planet formation and early disk evolution. Observations indicate that Class 0/I disks can host substantial mass reservoirs (e.g., Tobin et al., 2016; Tychoniec et al., 2018; Sheehan and Eisner, 2017), while theoretical studies show that massive, gravitationally unstable disks may undergo fragmentation, potentially leading to the formation of giant companions (e.g., Kratter et al., 2010; Forgan and Rice, 2011; Vorobyov, 2013). In addition, episodic and potentially violent accretion events can significantly alter the thermal structure of the disk (e.g., Bae et al., 2014; Meyer et al., 2017), thereby influencing both fragmentation conditions, the early growth of massive planetary envelopes, and the physical and chemical conditions of the disk material (Morbidelli et al., 2024).

In this context, identifying which accretion mechanisms operate during the protostellar phase, and quantifying their intensity, geometry, and efficiency, is of paramount importance to understand the stellar mass budget and to link star formation, disk evolution, and the emergence of planetary systems. Over the past decade, several comprehensive reviews have addressed the physics of accretion in young stellar objects. However, most of them have primarily focused on the more evolved young stars, where magnetospheric accretion is observationally well established (e.g., Hartmann et al., 2016; Manara et al., 2023); in these works, the protostellar stages are typically only briefly mentioned, mainly in the context of the early mass assembly or episodic accretion. Conversely, the most recent dedicated review of the protostellar phase (Dunham et al., 2014) provides a broad overview of embedded evolution in the aftermath of the major Spitzer and 2MASS surveys, but devotes limited space to the detailed physics and diagnostics of accretion itself. Since then, the field has undergone substantial progress; new observational facilities and techniques—particularly high-resolution infrared spectroscopy and interferometry with instruments such as at the Very Large Telescope (VLT) and, more recently, the James Webb Space Telescope (JWST) —have enabled direct probes of embedded accretion processes that were previously inaccessible. At the same time, advances in numerical simulations have refined our understanding of disk instability, episodic accretion, magnetic regulation, and disk–envelope interaction during the embedded phases. Given these developments, and in view of the transformative capabilities expected from forthcoming facilities such as the Extremely Large Telescope (ELT), a comprehensive reassessment of low-mass protostellar accretion is timely. This review aims to provide such an update, synthesizing the observational and theoretical progress achieved over the last decade and outlining the key open questions that will shape the next-generation of studies.

With this goal in mind, we have structured this review as follows. We describe the classification of Young Stellar Objects in Section 2; we summarize the current constraints on protostellar accretion from the observational and theoretical perspective in Sections 3 and 4, respectively; we discuss the emerging picture of protostellar accretion in Section 5, and finally outline the open questions and next challenges in Section 6.

2 Classification of accreting young stellar objects

Star formation begins with the gravitational collapse of prestellar cores, overdense structures within molecular clouds often associated with filamentary environments (e.g., Arzoumanian et al., 2011; André et al., 2014; Könyves et al., 2015). When the core mass exceeds the Bonnor–Ebert critical mass, gravitational collapse ensues (e.g., Ebert, 1955; Bonnor, 1956; Keto and Caselli, 2008); during the collapse, the increasing density and opacity of the gas lead to the formation of the first hydrostatic core, a transient, pressure-supported object in which the collapse is temporarily halted (Larson, 1969; Masunaga and Inutsuka, 2000). The onset of the second collapse then gives rise to the formation of a protostar surrounded by a circumstellar disk, marking the beginning of the protostellar phase, during which the forming star remains embedded within its parental envelope and accretes a substantial fraction of its final mass (Stahler et al., 1986). By the time the envelope dissipates, most of the stellar mass has been assembled, the protostellar phase ends, and the young star enters the pre–main-sequence (PMS) phase. In this Section, we summarize the main evolutionary stages of Young Stellar Objects (YSOs), describe their corresponding observational classes, and discuss the limitations of this classification.

Figure 1 provides a summary of the main observational and physical properties of Class 0 and Class I sources. Throughout this review, we use the term protostars to refer to Class 0, Class I, and FS objects, specifying the individual Class when necessary. We frequently compare results obtained for protostars with those derived for the more extensively studied CTTS population, partly because CTTSs are less embedded and suffer lower extinction than protostars, making them more observationally accessible.

FIGURE 1

Ideally, classifications based on these different diagnostics would yield consistent results. In practice, however, discrepancies are common. According to Enoch et al. (2009), classifications based on are more reliable for distinguishing protostars (Class 0 and I) from PMS stars (Class II), as sources with K are classified as Class I based on their bolometric temperature, yet often exhibit infrared spectral indices consistent with Class II. However, Enoch et al. (2009) also show that sources with discrepant classifications predominantly occupy the range , characteristic of flat-spectrum (FS) objects, which were introduced as a transitional class between Class I and II (Greene et al., 1994) (see Figure 2). Note that, instead, discrepancies in the classification become negligible when is computed using 4.5 µm (Furlan et al., 2016).

FIGURE 2

Since the classification described above is based on observational diagnostics, caution is required when applying its vocabulary to numerical simulations or when directly associating observational Classes with physical evolutionary stages. We discuss the caveats and uncertainties in the following Section 2.1.

2.1 Classification uncertainties and evolutionary timescales

The straightforward interpretation of observational Classes in terms of evolutionary stages, while tempting, requires caution: the derivation of evolutionary timescales from the SED diagnostic features is tightly linked to the definition and identification of Classes, which may result in a major cause of systematic error in lifetime estimates. Timescales associated with the different protostellar and PMS phases are a critical ingredient for understanding when and how stellar and planetary masses are assembled: in particular, estimates of the lifetimes of sources in the various Classes underpin our interpretation of mass accretion histories and set the temporal framework for disk evolution and planet formation.

To date, the most widely adopted lifetimes are derived from population statistics based on IR photometric classifications, primarily from Spitzer surveys (e.g., Evans et al., 2009). While this approach provides a homogeneous and statistically robust framework, it implicitly assumes that photometric Class assignments accurately trace evolutionary stage. However, growing observational evidence suggests that photometric diagnostics alone may be insufficient to unambiguously determine the evolutionary status of YSOs. Near-IR (NIR) spectroscopic follow-ups of Spitzer/2MASS-selected samples have revealed that a non-negligible fraction of sources classified as protostars exhibit spectroscopic properties commonly associated with more evolved objects, including detectable photospheric absorption features and accretion signatures typical of CTTSs (Fiorellino et al., 2021; Le Gouellec et al., 2024). A further illustrative case is that of HOPS 315, identified as a Class I source based on its bolometric temperature ( K), but showing a molecular jet typically associated with younger objects, suggesting that it may instead be a Class 0 protostar (Dutta et al., 2022). These findings provide a concrete example of the classification ambiguities and highlight the limitations of relying exclusively on SED-based criteria to obtain reliable evolutionary stage estimates. Another potential source of misclassification is the viewing angle. Geometrical effects can significantly distort the observed spectral energy distribution: for example, edge-on Class II sources may exhibit high extinction due to the disk midplane and be misidentified as Class I objects, while a Class I source viewed through an outflow cavity may appear as a Class II (Masunaga and Inutsuka, 2000; Robitaille et al., 2006); there have also been hints of some Class II PMSs showing evidence for envelopes (Furlan et al., 2016). Accretion variability can further cause YSOs to temporarily cross class definition boundaries (Dunham et al., 2010).

In this context, Fiorellino et al. (2021) proposed that classification schemes based on spectroscopic diagnostics, rather than on the infrared spectral index alone, may offer a more physically meaningful description of YSOs, particularly for Class I and FS sources. This approach naturally links the classification to the dominant physical processes at play, including the emergence of stellar photospheres, accretion activity, and circumstellar obscuration. A complementary perspective is provided by Federman et al. (2023), who demonstrated that the ratio of compact (disk) to extended (envelope) 870 µm emission–measured with the Atacama Compact Array (ACA) and Atacama Large Millimeter Array (ALMA) out to 2,500 au–can serve as an evolutionary indicator that is less sensitive to inclination and extinction effects than traditional SED-based classifications. Importantly, this metric does not redefine the Class 0/I/FS categories, but rather provides an independent structural tracer that statistically correlates with them. In their sample, Class 0 protostars are predominantly envelope-dominated (with a disk-to-envelope flux ratio ), FS sources are largely disk-dominated , and Class I objects span both regimes , suggesting that structural measures of the disk–envelope configuration can robustly trace the evolutionary progression.

Adopting such physically motivated classification metrics would likely lead to a reassessment of the relative populations of protostellar classes derived from Spitzer surveys, and consequently of their inferred lifetimes. For instance, in the NGC 1333 sample analyzed by Fiorellino et al. (2021), 7 out of 17 objects (40%) that were identified as Class I based on their SEDs were found to be more consistent with Class II sources when spectroscopic diagnostics were considered. This shows that class lifetimes could be revised at a comparable level; in this framework, the traditional boundaries between Classes become less rigid, and evolutionary stages are instead defined by the physical structure of the system rather than by photometric slopes alone. Such revisions would directly impact estimates of the duration of the main mass-assembly phases, both for stars and for forming planetary systems.

When observational classes are implicitly interpreted as proxies for age or evolutionary stage, the very definition of “protostar” can be ambiguous. While the classification is formally based on IR spectral indices and bolometric temperatures, quantities that primarily trace the presence and properties of circumstellar envelopes, the terminology is often used interchangeably with chronological age. In this simplified interpretation, objects younger than yr are usually referred to as Class 0; Class I/FS corresponds to ages between and yr; finally, Class II refers to ages older than yr. In order to disentangle observational appearance from physical structure, the term Stage has been introduced to denote a classification based on intrinsic properties–such as the relative mass of the envelope and disk or the envelope accretion rate–rather than on spectral indices (e.g., Andre et al., 1993; Robitaille et al., 2006). In this framework, Stage 0/I sources are defined by the presence of a substantial infalling envelope (e.g., or above a given threshold), independently of their observed SED shape. Although the Stage classification is physically motivated, mapping observational Classes onto physical Stages remains non-trivial. The age-based interpretation is intuitively appealing, however conflating observational class with evolutionary time can be problematic, particularly when comparing objects of very different stellar masses (for example, at a fixed age, a 1 star is significantly more evolved than a 0.1 star, since PMS contraction timescales are strongly mass-dependent).

More issues arise when comparing observations with numerical simulations; in the latter, the definition of “age” is often model-dependent and may refer to different reference points, such as the onset of core collapse, the formation of a protostar, or the attainment of a specific physical condition. Moreover, simulations that begin with pre-existing stellar objects may not explicitly account for the earlier collapse and accretion phases. For these reasons, caution is required when directly mapping observational classes onto simulated evolutionary timelines. A more robust alternative to pinpoint the Class of a simulated YSO is based on the envelope mass fraction , defined as the ratio of the envelope mass over total (envelope + sink) mass; in this framework, Class 0 corresponds to , while Class I is . Classification schemes based on spectral diagnostics or on physically motivated flux ratios may also provide a more direct link to the quantities tracked in numerical models, thereby facilitating a more meaningful comparison between observations and simulations. Observationally, estimates of the central protostellar mass based on kinematic measurements of disks and envelopes are becoming increasingly available for well-studied sources, providing an important empirical anchor for such physically motivated classifications (see Section 3.1). We emphasize this caveat not to discourage such comparisons, but rather to encourage careful and consistent interpretations.

3 Observations

Because protostars are deeply embedded within their natal envelopes, optical observations are severely limited. IR and radio wavelengths, where the radiation from the forming star is reprocessed and re-emitted by dust in the disk and envelope, are better suited for observing these objects; however, not all IR (especially NIR) emission arises from purely reprocessed radiation. At wavelengths around 2 µm part of the observed flux may consist of direct or scattered photospheric emission, as well as hydrogen recombination lines tracing accretion, as a consequence of the local reprocessing of the UV/optical photons or their observation in scattered light through outflow cavities (e.g., Habel et al., 2021). Observations at these wavelengths provide access to a range of diagnostics that can be used to infer the fundamental stellar properties and the characteristics of the accretion process.

In this Section, we present the main stellar and accretion parameters considered in this review; after an overview of the relevant quantities, we describe each of them in depth, outlining the observational diagnostics and techniques commonly employed to measure them.

The accretion luminosity quantifies the energy released as material accretes from the circumstellar disk onto the forming star. For CTTSs, has been constrained for large samples by modeling the ultraviolet (UV) continuum excess emission generated by accretion shocks above the stellar photosphere (Calvet and Gullbring, 1998; Schneider et al., 2020; Pittman et al., 2022; Manara et al., 2023), as well as through empirical correlations between and the U-band excess luminosity (Valenti et al., 1993; Gullbring et al., 1998; Herczeg and Hillenbrand, 2008; Sicilia-Aguilar et al., 2010). Unfortunately, applying these methods to embedded protostars is challenging. In these objects, the U-band emission is severely attenuated by the high visual extinction associated with the dense envelope; furthermore, strong continuum veiling requires the detection of the stellar photosphere, which is often impossible because of the faintness of the photospheric emission compared to the accretion luminosity. As a result, direct modeling of the UV excess becomes impractical, and constraining in the earliest protostellar phases remains highly challenging.

Alternatively, the accretion luminosity can be derived from emission lines luminosity , such as H I, He I, Fe I, Ca I, Na I, and O I, using empirical relations between and (Muzerolle et al., 1998; Herczeg and Hillenbrand, 2008; Alcalá et al., 2014; 2017; Fiorellino et al., 2025). It is important to stress, however, that a tight – correlation does not necessarily imply a direct physical or causal link between the line-emitting region and the accretion shock itself; as discussed by Mendigutía et al. (2015), part of the observed correlations may arise from underlying dependencies on stellar luminosity or other global parameters, rather than from a one-to-one physical connection between line formation and accretion energy release. Moreover, interferometric studies have shown that hydrogen recombination lines such as Br and H do not always originate exclusively in magnetospheric funnel flows, but can also trace disk winds or more extended inner-disk regions, while still preserving the empirical – calibration. Similarly, there has been evidence of empirical relations between and the luminosity of forbidden lines such as [O I]6,300, whose origin is clearly associated with outflows rather than direct accretion. These examples highlight that an empirical calibration should not be automatically interpreted as evidence for a specific accretion geometry or emission mechanism.

Building on this approach, and making the critical assumption that the empirical – relations originally calibrated for CTTSs are applicable to embedded protostars under the MA scenario, the study of accretion in these young objects has progressed significantly in recent years. Notably, White and Hillenbrand (2004) were able to study a sample of 15 very bright Class I protostars at optical wavelengths, concluding that most of the sources in their sample are not in the main accretion phase and are older than previously assumed. This was possible because the targets were among the optically brightest Class I sources, likely characterized by relatively low extinction and favorable viewing geometries. Such optical analyses, however, remain the exception. In most cases, dust in the protostellar disk and envelope absorbs the stellar and accretion luminosity at optical wavelengths and re-emits it at longer wavelengths; as a consequence, protostars are typically very bright in the IR and largely inaccessible at shorter wavelengths. At NIR bands, in particular, emit a number of commonly used tracers which provide a powerful tool to investigate both stellar and accretion properties of protostars. The main ones are (i) strong H I lines tracing accretion, such as and , formed in the hot ( K), dense gas of accretion columns; (ii) CO band-heads in the -band, which probe the warm inner disk region; (iii) wind and jet tracers, such as [Fe II] at 1.64 µm and lines at 2.12 and 2.24 µm, which are also indirectly linked to accretion; (iv) stellar photospheric features, which allow spectral-typing and stellar parameters constraining; and finally (v) the -band magnitude, which can be used to estimate the stellar luminosity .

Under the assumption that accretion onto protostars proceeds via magnetospheric accretion, as in CTTSs, stellar parameters such as the stellar radius and mass are required not only to gain a deeper understanding of the protostellar phase, but also to compute the mass accretion rate:

where is the gravitational constant and is the inner radius, usually assumed to be for CTTSs (Gullbring et al., 1998). Direct NIR interferometric measurements with VLTI/GRAVITY have spatially resolved the emitting regions in several young CTTS systems, finding characteristic sizes of only a few stellar radii. In DoAr 44, the region is constrained to an upper limit of 0.047 au (, Bouvier et al., 2020), and similarly compact sizes are reported for CI Tau in the GRAVITY YSO survey (Gravity Collaboration et al., 2023). These spatial scales are broadly consistent with expectations for magnetospheric truncation radii in low-mass accreting stars. Although these observations trace the spatial extent of the emission rather than the magnetosphere directly, the inferred sizes provide an observational estimate of the inner disk radius , commonly associated with the magnetospheric truncation radius in the magnetospheric accretion framework. In addition, recent spectroscopic modeling suggests that the inner disk radius in CTTSs may lie as close as , smaller than the canonical often assumed in magnetospheric accretion prescriptions (Pittman et al., 2025). Together, these results indicate that magnetospheric truncation radii are likely confined to a few stellar radii, though with significant object-to-object variation. Equation 1 implicitly assumes radiatively efficient (cold) accretion, whereby the gravitational potential energy released by the infalling material is entirely radiated away. If a fraction of the accretion energy is instead absorbed by the protostar (hot accretion), the inferred mass accretion rates would be correspondingly higher. The applicability of this assumption to embedded protostars is discussed in Section 5.4.

Lastly, in protostellar systems the disk is not an isolated structure, but interacts with both the forming star (by providing the material that is accreted) and the surrounding envelope (from which it is continuously replenished). As a result, the accretion process dynamically links the disk mass to the growth of the central protostar; disk and envelope masses and sizes therefore carry key information on the accretion process and on the efficiency and timescales of mass assembly during the embedded phases.

In the following subsections, we describe the main parameters required to study the accretion process and the spectroscopic features linked to them. For a discussion on the applicability of the magnetospheric accretion assumption during the protostellar phase, see Section 5.

3.1 Stellar parameters

We refer to stellar parameters as the quantities describing the intrinsic properties of a protostar, such as the effective temperature , surface gravity , stellar luminosity , radius , and mass . We include in this list also the visual extinction measured along the line of sight of the forming star; this is because, despite not being an intrinsic property of the protostar, its accurate estimate is required to reliably derive these parameters from observations.

Accurate spectral typing, obtained by comparing the photospheric absorption lines of protostars with those of non-accreting Class III templates or with synthetic stellar atmosphere spectra, allows the determination of , , and , provided that spectra of sufficiently high resolution and signal-to-noise are available. From short bands, ideally optical or the band, can be estimated using appropriate bolometric corrections; once placed on the Hertzsprung–Russell diagram, evolutionary tracks can then be used to derive and . While this is not the only method to determine stellar parameters, it is the most commonly adopted in the literature and, unless otherwise stated, it is the method used in the works reviewed here. Indeed, based on this approach, high-resolution NIR spectra from one of the first surveys targeting stellar properties of Class I and FS sources allowed spectral typing for 41 out of 52 objects (Doppmann et al., 2005). That analysis provided estimates of , , and broadly similar to those of CTTSs, but also showed that large uncertainties, primarily due to the difficulty in constraining the extinction, prevented firm conclusions. In addition, higher projected rotational velocities and angular momenta than those observed in Class II sources were reported. In a subsequent study, doubling the sample size to 110 Class I and FS young stellar objects, Connelley and Greene (2010) found a much wide range of extinction values (0–60 mag) and spectral types spanning from A0 to M7, with two main peaks around K2 and M3. These results suggest systematically later spectral types compared to CTTSs.

The protostellar mass can also be inferred using interferometric observations. Reaching angular resolutions of 0.05″, corresponding to 10 AU in nearby star-forming regions, ALMA¹ routinely resolves the velocity fields of young star disks. The stellar mass of protostars can be estimated from ALMA observations by modeling the rotation of molecular gas under the assumption of Keplerian motion, whereby the gas dynamics is dominated by the gravitational potential of the central object. In this framework, the azimuthal velocity of the gas is given by , where is the radial distance from the central source. This approach has proven to be robust for Class II systems, where the circumstellar disk is well defined and largely isolated, and the observed velocity field is typically consistent with purely Keplerian rotation (e.g., Simon et al., 2000; Pinte et al., 2018). In this case, stellar masses can be derived with uncertainties of the order of 5%–10%, increasing to 20% when uncertainties in disk inclination are taken into account.

For more embedded sources, however, the validity of the Keplerian assumption becomes increasingly uncertain. In Class I protostars, the disk is often still embedded within a rotating and infalling envelope, whose contribution to both the gravitational potential and the observed velocity field can lead to deviations from purely Keplerian rotation (Tobin et al., 2012; Harsono et al., 2014; Yen et al., 2015; Aso et al., 2015). As a result, the measured velocities may reflect a combination of disk rotation, envelope infall, and projection effects, and the derived masses are frequently better described as dynamical masses that include contributions from the inner disk and envelope. Reliable estimates in Class I systems therefore require high angular resolution to spatially isolate the disk-dominated region, as well as the use of optically thin molecular tracers. Even under favorable conditions, the determination of stellar masses carries significant uncertainties, with typical values of order 20%–50% (Yen et al., 2017; Segura-Cox et al., 2020).

In Class 0 protostars, the situation is even worse as the circumstellar environment is generally dominated by a massive envelope, and clear Keplerian disks are either absent or confined to very small radii. Even when compact disks are present, identifying Keplerian rotation is challenging not only because of limited spatial resolution, but also because of high optical depths and line of sight contamination from the infalling envelope, which can obscure or distort the kinematic signature of the inner disk. In these systems, the gas kinematics are often governed by infall and angular momentum conservation rather than by Keplerian rotation, making direct stellar mass measurements highly uncertain or unfeasible (Tobin et al., 2012; Ohashi et al., 2014; Yen et al., 2015; Maury et al., 2019), with one of the higher sources of uncertainty being how the velocity of the disk near the protostar is measured (Tobin and Sheehan, 2024). Consequently, stellar masses inferred from ALMA observations of Class 0 sources are often reported as upper or lower limits, or as total dynamical masses rather than as direct measurements of the protostellar mass alone.

Although dynamical stellar masses cannot generally be derived for embedded protostars with the same accuracy and reliability as for Class II systems, they provide independent physical constraints that can be used to reduce the large uncertainties affecting inferred from evolutionary tracks, for instance by ruling out incompatible solutions or by providing meaningful upper limits on .

Alternatively, stellar parameters can be derived by fitting the observed SED with theoretical models (e.g., Robitaille et al., 2006; Hatchell et al., 2007). This approach relies on a strong theoretical framework and requires a well-sampled SED, ideally constructed from simultaneous multi-wavelength observations in order to minimize the effects of variability. In practice, however, SED fitting primarily constrains the total luminosity that heats the disk and envelope, and separating from is highly model-dependent and often degenerate. In many Class 0 applications, the heating luminosity is effectively assumed to be dominated by ; moreover, even with a perfectly sampled SED this method also requires to know what fraction of the luminosity is due to accretion and to adopt a mass-to-radius relationship, which in most cases impacts the accuracy of this methodology. This approach is used when high-quality observations at the appropriate wavelengths are not available, and is particularly common for Class 0 sources, for which direct constraints on stellar parameters from spectroscopy or disk kinematics are often unavailable.

In summary, despite significant progress in the last decade, the accuracy with which stellar and accretion parameters can be constrained in embedded protostars remains limited. For Class I sources, typical uncertainties are of order 1 spectral subclass in spectral type, 1 mag in , and up to 1 dex in both and when derived from evolutionary tracks. While can often be constrained within 0.3 dex under favorable assumptions (see Section 3.4), the propagated uncertainty on may reach dex, owing to the combined uncertainties on extinction, veiling, stellar parameters, and the assumed accretion geometry. For Class 0 protostars, where direct spectroscopic constraints on stellar parameters are generally unavailable and disk kinematics are often dominated by envelope infall, the uncertainties are comparable or larger. In many cases, stellar masses and radii cannot be measured directly and are instead inferred through model-dependent assumptions or reported as upper or lower limits. As a consequence, both stellar and accretion parameters estimates for Class 0 sources remain subject to substantial systematic uncertainties.

3.2 Veiling

Veiling refers to the phenomenon whereby photospheric absorption lines appear weaker than expected in the stellar spectrum. This effect arises because an additional continuum emission is superimposed on the photospheric spectrum, partially filling in the absorption lines and making them appear shallower. Quantitatively, the veiling at a given wavelength is defined aswhere is the excess continuum flux and is the intrinsic photospheric flux at wavelength .

In accreting systems, the excess continuum responsible for veiling is produced by the release of accretion energy. Under the assumption of magnetospheric accretion, gas from the circumstellar disk is funneled along stellar magnetic field lines and impacts the stellar surface at near free-fall velocities. This process generates accretion shocks with characteristic temperatures of K, emitting primarily in the UVB and optical bands, as well as even hotter components contributing to the X-ray emission. At longer wavelengths, particularly in the NIR, a significant fraction of the continuum excess may instead originate from the hot inner disk rim at the dust sublimation radius, which is heated by stellar irradiation and accretion luminosity (e.g., Muzerolle et al., 2003). These accretion hot spots produce a largely featureless continuum; as a result, veiling arises because the accretion-generated emission is broad-band, contains few or no absorption features, peaks in the UV and optical, and adds flux on top of the stellar spectrum, thereby reducing the contrast of photospheric absorption lines. Consequently, higher accretion rates lead to stronger excess continuum emission and increased veiling, whereas lower accretion rates result in weaker or negligible veiling. Strong correlations between veiling and accretion diagnostics have been observed in CTTSs, including H emission (Valenti et al., 1993; Hartigan et al., 1995), UV excess (Calvet and Gullbring, 1998), and emission (Muzerolle et al., 1998; Natta et al., 2004), and the mass accretion rate (Calvet and Gullbring, 1998; Muzerolle et al., 1998; Fiorellino et al., 2022a; Nelissen et al., 2023).

While magnetospheric accretion provides a well-established framework to explain veiling in CTTSs, continuum excess emission, and thus veiling, could in principle also be produced in boundary-layer accretion regimes through viscous dissipation at the disk–star interface (Lynden-Bell and Pringle, 1974; Popham and Narayan, 1995). In boundary-layer accretion, the excess continuum emission arises from the braking and viscous dissipation of Keplerian disk material at the disk–star interface, rather than from localized accretion shocks. As shown by Mendigutía (2020) for more massive stars , the spatial and spectral distribution of BL emission differs from that expected in the MA scenario, and the resulting SED is more closely associated with a hot inner disk than with localized stellar hot spots. A clear comparison between magnetospheric hot-spot emission in CTTSs and disk-dominated emission in FU Orionis–type objects is presented by Hartmann et al. (2016), highlighting how the continuum shape and wavelength dependence differ substantially between the two accretion regimes. These differences in the origin and temperature structure of the emitting region may therefore lead to veiling with spectral properties distinct from those observed in CTTSs dominated by MA (Hartmann et al., 1997).

In Class I protostars, the measured veiling in the band can reach values as high as (Fiorellino et al., 2023). It is likely that even higher values occur in very embedded sources for which veiling cannot be reliably measured. The -band veiling has also been found to correlate with both the mid-IR (MIR) disk luminosity and the equivalent width of the line (corrected for veiling–Doppmann et al., 2003). Part of this correlation may arise from the dependence of the dust sublimation radius on the total luminosity (stellar plus accretion); as the accretion luminosity increases, the dust sublimation radius moves outward, increasing the emitting surface area of the hot inner disk wall and thereby enhancing the NIR continuum excess (e.g., Dullemond et al., 2001; Muzerolle et al., 2003; D’Alessio et al., 2006). Moreover, is systematically higher in protostars than in CTTSs, indicating a larger amount of circumstellar material and higher mass accretion rates in Class I sources (Doppmann et al., 2005). This supports the view that Class I and FS young stellar objects are actively accreting protostars, with accretion rates exceeding those typically observed in CTTSs.

3.3 The CO bands

Accreting YSOs show CO bands both in emission and absorption. CO band absorption arises when cool molecular gas lies along the line of sight to a hotter background continuum source. In protostars, this absorption can originate in several locations, including the stellar photosphere (in late-type stars), the cool disk atmosphere (e.g., the outer disk or disk surface), and the inner envelope in Class I sources. The physical conditions required for CO absorption include (i) temperatures of – K, which are typically needed to significantly populate the vibrational levels responsible for strong band-head emission (e.g., Najita et al., 1996), (ii) a high CO column density, and (iii) a background continuum dominated by the stellar photosphere, hot dust emission, or the accretion continuum. CO absorption occurs because the background continuum is bright, while the intervening CO gas absorbs photons in rovibrational transitions without emitting enough radiation to compensate.

CO absorption or emission depends on the mass accretion rate (Calvet et al., 1991). CO band absorption is typically observed in Class II and Class I protostars, FU Orionis–like objects (hot disk midplan with the disk atmosphere absorbing), in systems with edge-on disks, and in sources with low or moderate accretion rates (Greene and Lada, 1996; Pontoppidan et al., 2005; Hartmann et al., 2016; Connelley and Reipurth, 2018).

CO band emission originates in hot, dense molecular gas, typically located in the inner regions of circumstellar disks. It traces the inner gaseous disk at radii of 0.1–1 AU, inside or close to the dust sublimation radius, and is occasionally associated with dense disk winds. The typical physical conditions required for CO band emission are temperatures of – K and gas densities 10¹⁰ cm⁻³. When observed in emission, CO is thermally excited and the lines are optically thick or moderately optically thick. Emission occurs because the CO gas itself becomes an efficient emitter: it is sufficiently hot to populate high vibrational levels, collisional excitation is effective, and the emitted radiation exceeds the absorption of the background continuum. CO band emission is typically observed in strongly accreting protostars, high-veiling systems, and sources with hot inner disks, and is often detected when the line is observed in emission (Doppmann et al., 2005; Connelley and Greene, 2010). CO emission is reported more frequently in Class 0 (in about 60%–70% of sources) than in Class I protostars (about 15%, Laos et al., 2021), suggesting higher accretion activity in the former compared to the latter (Le Gouellec et al., 2024; Figure 3).

FIGURE 3

It is important to distinguish between the NIR CO overtone band-head emission at 2.3 µm discussed above and the CO fundamental emission at 4.6–5 µm. The latter is commonly detected towards deeply embedded Class 0 protostars and traces warm molecular gas in the inner envelope–disk interface or in disk winds (as revealed by recent JWST observations–Federman et al., 2024; Rubinstein et al., 2024; van Dishoeck et al., 2025). While both transitions probe warm gas, they arise under different excitation conditions and may trace distinct physical components of the embedded system.

Some young stars exhibit variability in their CO band profiles, which may be linked to changes in the accretion rate and the associated thermal and structural evolution of the inner disk (e.g., Eisner et al., 2013; Hein Bertelsen et al., 2016). Variability can arise from accretion-driven changes in disk heating, modifications of the inner disk structure, variations in the continuum veiling level, or variations in disk geometry and inclination. For instance, during strong accretion phases, the enhanced viscous heating of the disk interior and the irradiation of its surface can raise the temperature of the inner disk, promoting CO band emission. During quiescent phases, a cooler disk atmosphere overlying a hotter continuum-emitting region may instead produce CO bands in absorption. In contrast, the CO fundamental emission at 4.6–5 µm may remain in emission even when the overtone bands weaken or transition into absorption, as shown for deeply embedded sources such as I16253 (e.g., Federman et al., 2024; Narang et al., 2026). This difference likely reflects the distinct excitation conditions and emitting regions of the two transitions, with the fundamental one tracing warm molecular gas that is less directly tied to the hottest inner disk layers.

3.4 Empirical relations as a tool to measure the accretion luminosity

UVB, optical, and NIR H I emission lines, as well as lines from other species such as O I, He I, Ca I, Na I, and Fe I, are routinely found to exhibit empirical correlations with the accretion luminosity of CTTSs (e.g., Muzerolle et al., 1998; Alcalá et al., 2014; 2017; Fiorellino et al., 2025). Notably, the luminosity of H I lines is found to correlate with the accretion luminosity even for transitions from high-n atomic levels producing emission in the mid-infrared (Salyk et al., 2013; Rigliaco et al., 2015; Komarova and Fischer, 2020; Rogers et al., 2024; Testi et al., 2025; Tofflemire et al., 2025), consistent with their origin in the hot, dense gas of magnetospheric accretion columns and associated shocks. Thanks to JWST, emission from other species, such as CO, H₂O, OH, HCN, C₂H₂, and CO₂ have also been found to correlate with the accretion luminosity in the MIR (Mallaney et al., 2026), with a similar trend observed for the CO fundamental emission (Rubinstein et al., 2024). However, some studies have shown that empirical relations based on MIR line diagnostics can be affected by contamination from disk winds or jet emission in some systems, potentially leading to an overestimate of the accretion luminosity if such contributions are not properly accounted for (e.g., Bajaj et al., 2024). Lastly, the empirical relation linking the Ly flux to the integrated UV continuum between nm found studying a sample of 16 Class II YSOs (France et al., 2014) is at the basis of a recent attempt to constrain on a Class 0 protostar. This correlation has been proposed as a way to estimate the contribution of Ly emission to the UV flux in the second absorption band of H₂O ( nm), which is responsible for the photodissociation of H₂O into OH. In this Section, we review when and under which assumptions the diagnostics mentioned above have been applied to the analysis of protostars.

Empirical relations linking to the luminosities of the Pa and Br lines in CTTSs were first established by Muzerolle et al. (1998). In the same work, these relations were then applied to a sample of protostars in order to constrain their accretion luminosities. Prior to this study, the accretion luminosity of protostars was often taken to be comparable to their bolometric luminosity , under the assumption that the intrinsic stellar luminosity was negligible. Contrary to the expectations, using empirical relations calibrated on CTTS samples and assuming MA, Muzerolle et al. (1998) and later Beck (2007) found that the accretion luminosities of Class I sources were significantly lower than their bolometric luminosities and comparable to those of CTTSs. This result suggests that a substantial fraction of the disk material is accreted onto the forming star through episodic accretion events (Fischer et al., 2023) or during earlier evolutionary stages, such as the Class 0 phase. Subsequent studies focusing on small samples of Class I protostars tested the – relation and found that it remains valid even for line luminosities up to an order of magnitude higher than those originally calibrated by Muzerolle et al. (1998). These works also showed that the majority of the protostars examined (five out of six) exhibits high accretion levels, with reaching up to of the bolometric luminosity (Nisini et al., 2005; Greene and Lada, 2002); however, the limited sample sizes prevented robust conclusions. More recent investigations have emphasized that the applicability of these empirical relations to deeply embedded protostars remains uncertain. In particular, radiative transfer effects, extinction, and scattering within the envelope may cause NIR tracers to probe only a fraction of the intrinsic magnetospheric emission. For example, Harsono et al. (2023) showed that NIR accretion tracers may underestimate the accretion luminosity in protostars, as most of the magnetospheric emission can be completely obscured from view. In this scenario, the observed line fluxes trace only a small fraction of , arising primarily from scattered light along the less extincted cavity walls. As a consequence, more sophisticated approaches than the direct application of empirical relations are required for protostars in order to properly account for the occultation of the central source and the associated NIR emission (Delabrosse et al., 2024; Fiorellino et al., 2025).

Fiorellino et al. (2023) assembled a sample of 58 Class I and FS sources, including previously published observations. Using updated empirical relations (Alcalá et al., 2017), they estimated the accretion luminosity , stellar parameters, and the mass accretion rate in a homogeneous and self-consistent manner across the entire sample (Antoniucci et al., 2008; Fiorellino et al., 2021). This approach relies on three main assumptions. (i) the total bolometric luminosity is assumed to be the sum of the intrinsic stellar luminosity and the accretion luminosity, with the disk emission treated as reprocessed accretion energy; (ii) the combined disk and envelope contributions, which cannot be disentangled in protostars, are parameterized through the -band veiling; (iii) empirical relations calibrated on CTTSs between H I line luminosities and accretion luminosity are assumed to remain valid for Class I and FS sources. Formally, (i) implies , where the disk luminosity is included in ; (ii) leads to the expression , where accounts for excess continuum emission; (iii) adopts the calibration , with and (Alcalá et al., 2017), supported by interferometric evidence that traces inner-disk accretion and ejection processes². Consequently, given the observed -band magnitude , the luminosity , , the distance, and the -band veiling , and assuming an age for the source (which sets the spectral type and thus the bolometric correction, ), the extinction remains the only free parameter. Its value is determined by requiring consistency between the stellar luminosity obtained from assumptions (i) and (ii). A detailed description of this method is provided in Section 5.1 of Fiorellino et al. (2021).

In this way, Fiorellino et al. (2023) confirmed and generalized the main results previously obtained for the NGC 1333 cluster by Fiorellino et al. (2021). Specifically, Class I and FS protostars exhibit accretion luminosities and mass accretion rates spanning the ranges and , respectively, in both cases systematically higher than those measured in CTTSs (see Figure 4). They also found that only a small fraction of sources (about 25%) have accretion luminosities dominating the bolometric output, that is . This latter result, however, is not confirmed by the subsequent study of Testi et al. (2025), who analyzed a sample of Class I protostars in the MIR using empirical relations involving the Pf and Br lines derived in the same work. They found for about 50% of their sample, a fraction comparable to that reported for NGC 1333 (40%, Fiorellino et al., 2021). By contrast, using stellar masses measured from Keplerian disk rotation and comparing them with , Hartmann et al. (2025) found that many protostars, including Class 0 objects, exhibit low values of . However, they caution that their conclusions may be affected by selection biases, as their sample is dominated by sources with large disks, and emphasize that larger and more homogeneous samples are required to draw firm conclusions on protostellar accretion and mass distributions. Possible explanations for this tension are discussed in Section 5.2.

FIGURE 4

The Br line is also detected in Class 0 protostars, with a frequency similar to that observed in Class I sources and exceeding 60% (Doppmann et al., 2005; Le Gouellec et al., 2024). However, Class 0 objects generally exhibit stronger Br emission, with broader line widths (up to ), suggesting faster and more massive accretion flows (Laos et al., 2021; Le Gouellec et al., 2024, see also Figure 3). More specifically, the velocity profiles appear to differ between the two protostellar classes: Class I sources often show blueshifted line centroids, whereas Class 0 protostars tend to display symmetric profiles with little or no velocity shift. These differences have been suggested to indicate accretion mechanisms of different nature in Class 0 systems (Le Gouellec et al., 2024). Owing to the large uncertainties in the determination of and the possibility of different accretion geometries, measurements of the accretion luminosity in Class 0 protostars are therefore even more challenging than in Class I sources.

Given the severe observational limitations affecting accretion diagnostics in Class 0 protostars, a number of indirect and highly model-dependent approaches have recently been explored. This leads Watson et al. (2025) to suggest an alternative method based on OH spectral features produced by H₂O photodissociation. In this framework, the observed OH rotational ladder, once corrected for extinction, directly traces the rate at which H₂O molecules are dissociated by UV photons from the central source, since each dissociation event produces an excited OH molecule that cascades down the ladder. The derived OH luminosity therefore provides a lower limit to the UV photon production rate in the 115–145 nm band. The main source of uncertainty lies in estimating the fraction of UV photons absorbed by water, which depends on assumptions about the size and inflow geometry of the H₂O-emitting region. After correcting for this effect, the inferred total UV flux is compared with predictions from magnetospheric accretion shock models, originally developed for CTTS systems, to derive the mass accretion rate. Applying this method, the inferred mass accretion rate for the target Class 0 protostar was , well below typical accretion rates reported for protostars. We point out that, as highlighted by the authors themselves, this approach has been applied to a single Class 0 source showing spectral characteristics uncommon for this class. Indeed, the lack of detected water emission enables the assumption that most gas-phase H₂O is being photodissociated, while other classical accretion tracers are not clearly detected, likely owing to the high extinction typical of Class 0 objects. Furthermore, this methodology relies on the assumption that MA regulates accretion in Class 0 as it does in CTTSs, which has not yet been demonstrated. It therefore remains unclear whether this methodology can be straightforwardly extended to more embedded or less peculiar Class 0 protostars.

3.4.1 relationship: applicability and limitations

Empirical relations linking emission line luminosities to accretion luminosity have become a widely employed tool to estimate mass accretion rates in young stellar objects. When applied within a self-consistent framework that simultaneously constrains extinction, veiling, and stellar properties, such relations provide an efficient way to characterize accretion across large samples. In particular, IR surveys have demonstrated the power of empirical relations to probe accretion in embedded sources that are otherwise inaccessible at optical wavelengths.

Nevertheless, a major source of uncertainty is represented by the high challenge of measurement, needed to deredden the H I fluxes to get corrected line luminosities. Moreover, these empirical relations are valid for, and calibrated on, CTTSs. As a consequence, their applicability to protostars requires careful consideration, given the physical differences between these evolutionary stages. Fiorellino et al. (2025) argue that the use of CTTS-based relations for Class I and FS sources is justified if these objects accrete through a magnetospheric process. Under this assumption, empirical relations involving H I recombination lines can be applied to Class I and FS sources to derive accretion luminosities. In support of this view, recent measurements suggest that magnetic field strengths in Class I protostars are comparable to those of CTTSs (Flores et al., 2024), indicating that magnetospheric accretion may already operate at these stages. However, different observational techniques probe distinct components of the magnetic field. While Flores et al. (2024) measure the total surface magnetic field strength through Zeeman broadening, thus being sensitive to both ordered and small-scale components, Drouglazet et al. (2026) employ spectropolarimetric observations that primarily trace the large-scale, organized magnetic field.

In this context, Drouglazet et al. (2026) detect strong large-scale magnetic fields in only 6 out of the 15 Class I sources observed, with the remaining 9 objects showing no clear magnetic signature. These results suggest that, while strong magnetic fields may be present in Class I protostars, the large-scale field component required to efficiently truncate the disk and regulate magnetospheric accretion may not be equally developed in all sources, pointing to a diversity in magnetic field topology and accretion geometry within the Class I population. Furthermore, if Class I protostars host magnetic field strengths comparable to those of CTTSs but exhibit higher mass accretion rates, the balance between magnetic and accretion pressures is expected to shift, leading to smaller truncation radii. This effect should be taken into account when deriving from empirical relations calibrated on more evolved systems.

At the same time, important caveats must be kept in mind. Protostars are characterized by a much higher continuum veiling than CTTSs, with typical values reaching , compared to in more evolved systems; as emphasized by Fiorellino et al. (2025), neglecting or improperly accounting for veiling can lead to severe biases in line luminosities and, consequently, inferred accretion rates. In addition, in deeply embedded sources, part of the observed emission may arise from scattered or reprocessed radiation rather than directly tracing the accretion shock, further complicating the interpretation. These effects are likely to be particularly severe in Class 0 protostars, for which magnetic fields are expected to be absent/weak and measurements are scarce, thus accretion may proceed through boundary-layer–like mechanisms rather than through magnetospheric funnel flows.

Empirical relations are therefore most reliable when applied to Class I and FS YSOs, provided that extinction and veiling are constrained and that the dominant accretion regime is magnetospheric. Outside these conditions, accretion luminosities inferred from empirical relations should be regarded as approximate or as lower limits.

3.4.2 Implications for accretion onto planets

The use of empirical accretion relations has also motivated attempts to characterize accretion onto forming planets, where direct measurements of mass accretion rates are even more challenging. Observations of H and NIR hydrogen emission from a small number of accreting protoplanets suggest that, at least phenomenologically, similar diagnostics may trace accretion-related processes (e.g., Sallum et al., 2015; Haffert et al., 2019; Thanathibodee et al., 2019).

Part of the motivation for applying stellar accretion diagnostics to giant planets stems from the apparent continuity between the brown dwarf (BD) and planetary-mass regimes. Empirical accretion relations originally calibrated for CTTSs have been successfully extended to BDs, yielding accretion luminosities and mass accretion rates over several orders of magnitude in stellar mass (e.g., Muzerolle et al., 2003; Natta et al., 2004; Herczeg and Hillenbrand, 2008; Almendros-Abad et al., 2024). Since BDs have substantially lower masses and are expected to host weaker large-scale magnetic fields than CTTSs (e.g., Reiners and Basri, 2007; Morin et al., 2010), their inclusion has further encouraged the exploratory extension of empirical accretion diagnostics toward even lower-mass objects. Given that the mass boundary between brown dwarfs and giant planets is not sharply defined and may reflect formation history rather than a strict physical threshold (e.g., Whitworth et al., 2007; Chabrier et al., 2014; Gilbert et al., 2026), it is tempting to explore whether similar tools can be applied to giant forming planets.

However, the physical conditions of planetary accretion differ fundamentally from those of stellar and substellar accretion, including the geometry of the accretion flow, the depth of the gravitational potential, and the role of circumplanetary disks. Radiation hydrodynamic simulations indicate that gas accretes onto forming giant planets through a circumplanetary disk and forms a compact, high-temperature accretion shock at the planetary surface, with emission properties that depend sensitively on the shock structure and local opacity (e.g., Szulágyi and Ercolano, 2020). In this configuration, the geometry and thermodynamics of the accretion flow differ substantially from the magnetospheric paradigm commonly invoked for young stars. As a consequence, accretion luminosities and mass accretion rates inferred for protoplanets obtained using relationships remain highly uncertain and model-dependent, relying on assumptions about shock physics, emission efficiency, and radiative transfer (e.g., Aoyama and Ikoma, 2019; Aoyama et al., 2021).

Nonetheless, constraining accretion onto planets is crucial for understanding the final masses of giant planets and for linking disk accretion during the protostellar phase to the early growth of planetary systems. Progress in this area will require both improved theoretical models of planetary accretion and high-sensitivity, high-angular-resolution observations capable of isolating planetary emission from the surrounding disk. In this context, improving our understanding of accretion mechanisms during the protostellar phase, when planet formation starts, is expected to provide valuable insights into planet formation, with direct implications for how accretion luminosities and mass accretion rates are inferred for forming planets.

3.5 Outflows as a proxy of accretion

Outflows and jets are a ubiquitous feature of the earliest stages of star formation and are commonly interpreted as indirect tracers of mass accretion in protostars (e.g., Schwartz, 1977; Reipurth and Bally, 2001; Sicilia-Aguilar et al., 2020; Lee, 2020; Avachat, 2023). In magneto-centrifugal launching scenarios, the ejection of mass and angular momentum through collimated jets and wider-angle molecular outflows is physically linked to disk-mediated accretion onto the central protostar, leading to an approximate proportionality between the mass-loss rate and the mass accretion rate (e.g., Shu et al., 1994; Pudritz et al., 2007; Frank et al., 2014). Observational support for this connection comes from correlations between accretion-related quantities and outflow properties, such as momentum flux and mass-loss rate , measured through molecular tracers (such as CO) and atomic or ionic emission associated with jets (e.g., Cabrit and Bertout, 1992; Bontemps et al., 1996; Hartigan et al., 1995; Bally, 2016). Quantitative correlations between accretion and ejection rates have been explored for more evolved Class II objects (e.g., Rota et al., 2025), while similar analyses are not yet available for embedded Class 0/I protostars because of the scarcity of data.

In deeply embedded protostars, where direct accretion diagnostics are often inaccessible because of high extinction and envelope opacity, outflows provide a valuable, though indirect, probe of accretion activity. [Fe II] and H₂ emission is routinely observed in both Class 0 and Class I sources, with H₂ at 2.12 µm detected in emission in about 90%–100% of Class 0 protostars and in only 40%–50% of Class I objects (Connelley and Greene, 2010; Laos et al., 2021; Le Gouellec et al., 2024; Federman et al., 2024; Narang et al., 2026). However, the interpretation of outflow properties as instantaneous accretion tracers is not straightforward: molecular outflows integrate the accretion history over timescales of – yr, and episodic accretion can produce complex and non-linear signatures in the observed outflow energetics (e.g., Vorobyov and Basu, 2006; Offner et al., 2011). Indeed, different tracers probe distinct temporal baselines: while accretion tracers probe the instantaneous mass accretion rate, millimeter CO observations trace outflow activity on yr timescales, and far-IR CO emission samples shocks on intermediate ( yr) timescales. Far-IR CO line luminosities from outflows have also been found to correlate with , further supporting a statistical link between ejection and accretion in embedded sources (Manoj et al., 2013; Manoj et al., 2016). This temporal integration is further compounded by the intrinsic variability of protostellar systems (see Section 3.7), which decouples short-term accretion fluctuations from the large-scale outflow response. Consequently, while outflows are robust indicators of ongoing or recent accretion in protostars, they primarily trace time-averaged accretion rates rather than instantaneous values. Consistently, using as a proxy for and combining it with [Fe II] and [O I] diagnostics of jets, Watson et al. (2016) estimated that the mass ejection rate in Class 0 protostars corresponds to 10% of the mass accretion rate, in agreement with expectations from magneto-centrifugal launching models. Despite these caveats, recent observational studies have provided further empirical support for a close connection between accretion and ejection in embedded protostars. In particular, Ghosh and Bhattacharyya (2026) found that the empirical correlations between the 3.3 mm continuum flux and the Br line flux, as well as between the mass accretion rate and the ionized mass-loss rate previously established for Class II sources, also hold for Class I protostars. Recent ALMA studies have begun a quantitative exploration of the connection between and on a statistical basis by deriving accretion rates directly from jet properties. For instance, Dutta et al. (2024) estimated jet mass-loss rates from CO emission associated with high-velocity knots and inferred accretion rates by combining the bolometric and jet kinetic luminosities under the assumption that both are powered by accretion. In this framework, they obtained ratios spanning to , significantly broader than the typically used value of (Hartigan et al., 1995). These estimates, however, are affected by substantial uncertainties; the derivation of depends on assumptions on excitation temperature, CO abundance, geometry, and inclination, and is likely to provide a lower limit, as it primarily probes the dense jet axis rather than the full outflow. Moreover, the inferred accretion rates rely on simplified assumptions on the stellar mass and radius, and on the fraction of attributable to accretion. Overall, these factors can introduce uncertainties of at least tens of percent and potentially larger systematic biases. Despite these limitations, such studies demonstrate that, at least in a statistical sense, outflow properties retain a measurable link to accretion even in deeply embedded protostars.

High-angular-resolution observations have also revealed that protostellar jets and outflows are not only powerful but also highly collimated, even at distances smaller than au from the central protostar (e.g., Bjerkeli et al., 2016; Lee et al., 2018). JWST observations are providing unprecedented insight into jets from deeply embedded protostars, revealing both their molecular and atomic structure at high spatial resolution. For instance, studies of Class 0 sources such as HH211 and IRAS 15398-3359 have shown highly collimated jets with layered molecular and atomic components, while recent IFU observations have resolved the kinematics of jets in sources such as B335 and HOPS 153 (e.g., Yang et al., 2022; Narang et al., 2025; Federman et al., 2026). Similarly, ALMA observations of protostellar systems have shown that molecular and atomic jets maintain narrow opening angles close to their launching regions, supporting magneto-centrifugal models in which jets efficiently extract angular momentum from the disk and thereby regulate accretion onto the protostar (e.g., Shang et al., 1998; Lee et al., 2017a; Hirota et al., 2017; Tabone et al., 2017; Bally and Reipurth, 2023). This early collimation implies that jets can interact with the surrounding envelope and natal core in a highly directional manner, carving cavities, entraining molecular gas, and injecting momentum and energy into the immediate environment (She and Orszag, 1991; Mendoza et al., 2004; Shang et al., 2006; Liang et al., 2020). Such feedback can influence both the infall geometry and the efficiency of mass accretion, as well as the fragmentation and turbulence of the parent core (e.g., Arce et al., 2010; Offner and Arce, 2014). Numerical simulations further indicate that collimated outflows play a crucial role in regulating star formation by limiting stellar masses, removing excess angular momentum, and driving turbulence on core and clump scales (e.g., Federrath et al., 2014; Federrath, 2015). Together, these observational and theoretical results suggest that outflow collimation is a key ingredient in the self-regulation of accretion during the embedded phases, linking small-scale jet-launching physics to larger-scale feedback processes in star-forming regions.

3.6 Protostellar disk and envelope mass and size

The disk and envelope contributions are fundamental when studying protostellar systems. In particular, the accretion process directly links the flow of material from the disk into the forming star, and recently discovered streamers may play a role in the accretion process as well. The formation and evolution of these structures are governed by the redistribution of angular momentum during collapse and are strongly influenced by magnetic fields, which regulate disk formation, braking, and mass transport. In this Section, we review methodologies to constrain the disk and envelope properties, such as their mass, radius, and possible sub-structures, as well as how they are linked to the accretion process.

High-angular-resolution interferometric observations with facilities such as ALMA, NOEMA³, and the VLA⁴ have revolutionized the study of protostellar disks and envelopes, enabling direct constraints on disk kinematics and mass distributions at spatial scales of a few tens of astronomical units. Disk masses in protostellar systems are most commonly estimated from (sub-)millimeter continuum observations, under the assumption that the emission is dominated by optically thin dust and can be converted into a total disk mass by adopting a dust opacity, a characteristic dust temperature, and a gas-to-dust mass ratio (e.g., Hildebrand, 1983; Beckwith et al., 1990). While this approach has proven effective for Class II disks (albeit with caveats - see Miotello et al., 2023), its application to embedded protostars is subject to significant uncertainties. First of all, dust emission from the surrounding envelope can contaminate the disk flux; furthermore, parts of the disk may be optically thick even at millimeter wavelengths, and the dust temperature and opacity are poorly constrained in the presence of strong irradiation and accretion heating (Jørgensen et al., 2009; Dunham et al., 2014; Segura-Cox et al., 2018; Maury et al., 2019). Envelope masses are similarly derived from continuum or molecular line observations, but disentangling envelope and disk contributions requires high-angular-resolution data and often relies on parametric modeling of the density and temperature structure, introducing additional degeneracies.

To mitigate these limitations, more recent studies have combined continuum observations with kinematic information from molecular lines to isolate the rotationally supported disk component and to constrain disk radii and masses more robustly. The resulting measurements have mean values of –50 au and –30 , depending on the evolutionary stage. However, the evolutionary trend of disk properties remains debated; while some studies suggest that disk radii and masses decrease from Class 0 to Class I and FS sources (e.g., Tobin et al., 2020), other works employing radiative transfer models with variable dust properties do not find strong evidence for a systematic decline in disk mass along the embedded phases (e.g., Sheehan et al., 2022). Radiative transfer modeling of both dust and gas emission has also been employed to account for optical depth effects and temperature gradients within the disk and inner envelope. Nevertheless, while disk radii estimates have been demonstrated to be accurate and reliable, significant systematic uncertainties remain on disk masses when compared to post-processing results (see Section 5.1.3 for a dedicated discussion). estimates in protostars can vary by factors of a few depending on the adopted assumptions on dust properties, temperature structure, and the treatment of envelope contamination. Tung et al. (2024) showed that optical depth effects, dust evolution, and unresolved substructures can lead to substantial underestimates of the true disk mass when relying on simple continuum-based methods. These results highlight that, while the constraints on disk masses are reaching increasing spatial and kinematic detail, their absolute values in embedded protostars remain intrinsically uncertain.

Taking in mind these uncertainties, recent studies have begun to establish empirical links between disk properties and accretion in embedded systems. Fiorellino et al. (2022b) found a statistically significant correlation between the mass accretion rate and the disk mass in Class I protostars–displayed in Figure 5, extending to earlier evolutionary stages the – relation previously identified in Class II disks (Manara et al., 2016). A linear correlation between the logarithm of these properties is expected to naturally arise in Class II objects in the viscous magnetospheric accretion framework (Lodato et al., 2017); although the correlation in Class I sources appears flatter and exhibits larger scatter, the presence of such a correlation suggests that the disk mechanisms such as viscosity might already play a role in regulating accretion during the embedded phase. At the same time, theoretical and observational considerations indicate that in Class 0/I systems the relevant mass reservoir may not be limited to the disk alone. Mendigutía et al. (2018) argued that a correlation between and the total circumstellar mass, including both disk and envelope , is expected during the embedded phases, reflecting the ongoing interaction between the forming star, its disk, and the surrounding envelope. In this picture, accretion rates are not regulated solely by internal disk viscous evolution, but are also influenced by large-scale mass infall and environmental feeding. A complete analysis of this relation should be performed with models also considering the combined effects of envelope replenishment, disk growth, and time-variable accretion.

FIGURE 5

Beyond the disk itself, high-sensitivity molecular line observations have uncovered increasingly complex kinematics in protostellar envelopes. Several studies have identified large-scale infalling structures, often referred to as accretion streamers, connecting the surrounding molecular core to the disk (e.g., Tokuda et al., 2018; Pineda et al., 2020; Garufi et al., 2021; Gupta et al., 2023; Kuffmeier et al., 2023; Valdivia-Mena et al., 2024). These asymmetric inflow patterns challenge the classical picture of spherical or axisymmetric collapse and indicate that disk growth may be governed by highly structured, time-dependent mass delivery from the envelope (as also found in large scale numerical simulations–see Section 4.4.3). While the detailed connection between these large-scale streamers and the accretion processes operating in the inner disk remains an open question, their presence suggests that envelope dynamics play an important role in setting the boundary conditions for disk accretion.

Chemical diagnostics provide an additional tool to probe the disk–envelope interface. Specific molecular tracers, such as N₂H⁺ and C¹⁸O, are commonly used to isolate dense gas and to identify chemical differentiation across the transition region between the infalling envelope and the forming disk (e.g., Öberg et al., 2011; Harsono et al., 2015; van Terwisga et al., 2019). Spatial variations in molecular abundances and excitation conditions offer complementary constraints on the physical structure and kinematics of the accreting material, helping to disentangle rotationally supported disk gas from infalling envelope components.

3.7 The variability of protostars

Protostellar accretion is increasingly recognized as a highly variable process, yet its temporal behavior remains poorly constrained, particularly during the deeply embedded phases. The main reason is the embedded nature of protostars, which prevents systematic monitoring at optical wavelengths; as a consequence, studies of protostellar variability - and more specifically of variability - rely almost entirely on infrared time-domain surveys.

Mid-infrared (MIR) monitoring campaigns have provided the first systematic view of protostellar variability on timescales from months to years. In particular, time-domain observations from WISE⁵ and NEOWISE⁶ have widespread MIR variability among Class 0 and Class I protostars, with light curves displaying both stochastic fluctuations and long-term trends (e.g., Morales-Calderón et al., 2011; Fischer et al., 2019; Park et al., 2021; Zakri et al., 2022; Kulkarni et al., 2026). Because MIR emission traces warm dust in the inner disk and envelope, such variability is commonly interpreted as reflecting changes in the accretion luminosity, disk heating, or inner disk geometry (Contreras Peña et al., 2020). However, disentangling these contributions remains challenging, limiting a direct interpretation of MIR variability in terms of instantaneous accretion rates. NIR time-domain surveys have further advanced the study of protostellar variability by probing shorter wavelengths while still mitigating the effects of extinction. In particular, the ESO Public Survey Vista Variables in the Vía Láctea (VVV) has provided multi-epoch -band light curves for millions of sources across the Galactic bulge and inner disk (e.g., Minniti et al., 2010; Saito et al., 2012). Despite these advances, NIR monitoring programs specifically designed to characterize non-episodic protostellar accretion variability, to compare protostellar and PMS variability, remain scarce.

More extreme manifestations of protostellar variability are provided by episodic accretion events, such as FU Orionis- and EX Lupi-type outbursts, during which the accretion rate increases by several orders of magnitude for periods ranging from months to decades. Infrared and submillimeter spectroscopy during these events has revealed dramatic changes in disk structure, chemistry, and thermal balance, supporting the idea that episodic accretion plays a major role in the mass assembly of young stars (e.g., Hartmann and Kenyon, 1996; Audard et al., 2014). Growing observational evidence suggests that such bursts are not rare events, but instead represent a recurrent phase of protostellar evolution, consistent with predictions from models invoking disk gravitational instability and envelope-fed accretion (e.g., Vorobyov and Basu, 2015; Fischer et al., 2023). VVV data have enabled systematic searches for variable and eruptive young stellar objects, revealing numerous YSOs - predominantly Class I systems - with variability spanning from days to years (e.g., Guo et al., 2020). Combined spectroscopic and photometric analyses show that extremely strong outbursts (FUors–like) are characterized by distinctive NIR spectral changes correlated with luminosity increases, consistent with enhanced accretion activity. The effort by Contreras Peña et al. (2025) to systematically collect and classify young stars undergoing episodic accretion further indicates that a substantial fraction of embedded protostars may experience accretion bursts that remain undetected at optical wavelengths due to extinction, emphasizing the importance of infrared monitoring.

Quantifying the incidence of episodic accretion during the protostellar phase remains challenging, primarily because accretion bursts are intrinsically rare, transient, and often obscured. Nevertheless, recent infrared and submillimeter time-domain surveys have begun to place statistical constraints on how frequently protostars undergo accretion outbursts. Near-infrared monitoring of large samples of embedded young stellar objects with the VVV and VVVX⁷ surveys indicates that approximately 2%–3% of Class I protostars exhibit high-amplitude ( mag) variability consistent with episodic accretion at any given time (Contreras Peña et al., 2024). After accounting for survey completeness and temporal sampling, these results imply recurrence timescales of order – yr for major, FU Orionis–like outbursts during the Class I phase (Fischer et al., 2019). Complementary constraints from MIR and submillimeter monitoring surveys, including Spitzer, WISE/NEOWISE, and the JCMT Transient Survey, suggest that lower-amplitude accretion variability is more common, with a substantial fraction (10–20%) of protostars exhibiting measurable luminosity changes on timescales of months to years (e.g., Johnstone et al., 2018; Fischer et al., 2019; Lee Y.-H. et al., 2021). While robust incidence estimates for Class 0 and FS sources remain scarce, these observations indicate that episodic accretion is a recurrent, though temporally sparse, component of protostellar mass assembly rather than an exceptional phenomenon.

The prevalence of strong accretion variability during the protostellar phase has important implications for early disk evolution and planet formation. On one hand, the large mass reservoirs available in young disks suggest that planet formation could begin already during the embedded stages (Tychoniec et al., 2020). On the other hand, the highly dynamic accretion environment, characterized by frequent luminosity bursts and rapid structural disk changes, raises questions about the survivability and growth of planet-forming solids (e.g., Vorobyov and Basu, 2006; Audard et al., 2014; Cieza et al., 2016; Kadam et al., 2022; Das et al., 2025). If disk instabilities and episodic accretion are as common as current observations suggest, it remains unclear how these interact with the standard magnetospheric accretion occurring during quiescence and how efficiently planets can form and evolve under such conditions. Understanding the interplay between accretion variability, disk evolution, and early planet formation therefore represents one of the major open questions in the study of protostellar accretion.

4 Models and simulations

Numerical simulations of star formation provide a valid tool to explore the influence of physical mechanisms, initial conditions, and environments on the assembly and evolution of protostars. Since the pioneering mono-dimensional studies of Larson (1969), Penston (1969), and Shu (1977), the field has made tremendous progress: modern simulations in both two- and three-dimensional configuration explore the effects of hydrodynamics, radiative transfer, and magnetic fields both in the ideal and non-ideal approximation, and provide diagnostics for evolutionary and accretion parameter to compare against the observational constraints. In this Section, we aim at summarizing the numerical effort carried out in the last few decades: we give an overview of the three main computational techniques in Section 4.1, and discuss the simulations and their results in Sections 4.2, 4.3, and 4.4, following increasing physical complexity.

4.1 Numerical methods: an overview

Simulations of cloud/core collapse leading to protoplanetary disk formation can be performed with three main numerical techniques: thin disk approximation, Smoothed Particle Hydrodynamics (SPH), and Adaptive Mesh Refinement (AMR). In the following subsections, we briefly present each of these approaches and discuss their strengths and limitations.

4.1.1 Thin disk (two-dimensional models)

Thin-disk simulations assume that vertical force balance is established on timescales shorter than those governing cloud contraction. Isothermal, non-rotating, magnetized cylindrical or spherical clouds quickly reach vertical hydrostatic equilibrium - fast enough to beat the contraction in the direction perpendicular to the field lines caused by ambipolar diffusion (Fiedler and Mouschovias, 1993). This motivates the choice of a geometrically thin disk, with a scale height smaller than its radial extent, as model cloud. This magnetized thin disk approximation, originally proposed by Ciolek and Mouschovias (1993) and Basu and Mouschovias (1994), is one of the simplest numerical frameworks to address the problem of a contracting cloud; the vertical integration reduces the dimensions to two, thereby resulting in an inexpensive setup that can be evolved up to the virtually entire Class I lifetime (of the order of Myr). This perk comes at the expense of a realistic description of the magnetic field geometry, as well as the disk vertical structure; the neglect of vertical stratification and consequently three-dimensional instabilities may lead to different fragmentation thresholds compared to those derived in full 3D simulations.

4.1.2 Smoothed particle hydrodynamics (SPH)

Smoothed Particle Hydrodynamics (SPH), originally formulated by Lucy (1977) and Gingold and Monaghan (1977), is a Lagrangian scheme to solve the equations of hydrodynamics by approximating the continuum dynamics of fluids through the use of particles with properties smoothed over a local kernel. Lagrangian schemes are a class of particle-based numerical frameworks, as opposed to Eulerian schemes, which instead are grid-based; Eulerian codes feature a uniform, fixed Cartesian mesh (i.e., with cells aligned to the axes of a Cartesian coordinate system), while Lagrangian schemes employ a moving mesh that follows the flow of the fluid. This characteristic makes Lagrangian methods intrinsically adaptive, in that changes in the density and flow morphology are accounted for without the need for mesh refinement. This also implies that resolution is automatically concentrated in regions of high particle density, avoiding waste of computational resources outside the regions of interest. Another major perk of the Lagrangian SPH framework is the built-in conservation properties in the equation of motion: linear momentum, angular momentum, and entropy are simultaneously exactly conserved independently of configuration, which makes SPH particularly well suited to treat complex (3D) geometries. The main limitation of SPH simulations is the lack of an explicit treatment of physical viscosity, which is addressed by the addition of an artificial viscosity to capture shocks and dissipative processes. Such artificial viscosity can be implemented in multiple ways, which in turn can impact the treatment of angular momentum transport; however, most artificial viscosities can be directly translated into a combination of Navier-Stokes shear and bulk viscosity terms and can therefore be built in a physically-motivated way (see, e.g., Flebbe et al., 1994; Watkins et al., 1996). For a review on SPH methods and their astrophysical approaches, see Monaghan (1992), Monaghan (2005), Price (2004), Rosswog (2009), Springel (2010).

4.1.3 Adaptive mesh refinement (AMR)

The Adaptive Mesh Refinement (AMR) method, originally proposed by Berger and Oliger (1984) and Berger and Colella (1989), is a Eulerian scheme which covers the computational domain by a hierarchy of nested grids that are dynamically refined in regions where higher resolution is required. The main advantage of AMR is its flexibility, in that it simultaneously resolves larger and smaller scales; furthermore, in contrast to SPH methods, shocks and discontinuities are easily captured without the need for artificial terms, and the Eulerian grid allows for a straightforward implementation of non-ideal MHD terms, radiative diffusion, and chemical cooling. The price to pay for such detailed simulations is a substantial computational cost, which limits the evolutionary timescales that can be probed with AMR codes. For core collapse simulations, this usually leads to an evolution up to a maximum of a few yr. Furthermore, to prevent the timestep to become prohibitively small near the central object, protostars are usually replaced by sink particles - unresolved particles that can accrete and interact with the other particles in the simulation only through gravity. The criteria to transform an over dense gas particle and its neighbors into a sink particle involve not only density, but also the smoothing length, as well as tests to determine whether the particles may be in the process of tidal disruption or bouncing, ensuring that sinks only replace groups of particles that would otherwise continue to collapse (see Bate et al., 1995 for details). The use of sink particles, while extremely functional to keep the computational time under control and allow to proceed past the formation of the first protostars, has the disadvantage of introducing unresolved regions; this impacts the treatment of internal processes, such as the emission of radiation from the accreting gas and radiation feedback, and does not allow to separate the inner disk from the protostar itself, which influences the determination of accretion properties.

4.1.4 Summary

The three numerical approaches described above - thin disk approximation, SPH, and AMR - provide a complementary perspective to the problem of cloud collapse simulations. While none of the three is able to simultaneously (i) resolve the large-scale collapse, (ii) capture the detailed physics of accretion onto the protostars, and (iii) include radiative feedback processes, they all play a crucial role depending on the nature of the science question. Thin disk models are the optimal choice to study the long-term evolution of protostars, virtually up to the end of the protoplanetary disk (Class II) phase, while they do not allow the detailed modeling of accretion and radiative feedback; SPH is a powerful tool for three dimensional simulations, with its adaptive resolution and intrinsic conservation properties, but the presence of an artificial viscosity limits the resolution of shocks and the treatment of radiation and magnetic fields; finally, AMR is excellent at resolving the multi-scale nature of the collapse and capturing the details of non-ideal MHD effects, as well as radiation transport and dynamical feedback, but its extreme computational cost restricts its use to shorter timescales and smaller samples. The choice of the numerical approach is therefore strongly dependent on the goal of the simulation; the current state of the art is a combination of works that have employed all three methods, as we discuss in the following.

4.2 Pure hydrodynamic simulations

The first generation of cloud collapse simulations featured a pure hydrodynamical approach with self-gravity, typically employing the thin disk approximation or SPH.

4.2.1 The Vorobyov Basu series: a historical perspective

Vorobyov Basu were the first to couple the collapse of a rotating pre-stellar core to the formation of protostars and accretion disks around them, leading to a self-consistent accretion history. Their seminal series of papers presented suites of simulations in the thin disk approximation, starting from the collapse of an originally starless, magnetically supercritical core (i.e., with the effect of the magnetic field comparable to, but weaker than, that of gravity).

The early papers (Vorobyov and Basu, 2005a; Vorobyov and Basu, 2005b, Vorobyov and Basu, 2006) focused on the onset of instability in the star formation process, showing how the protostar and protoplanetary disk, formed as a result of the collapse of a single core, quickly become unstable to the formation of a spiral, which originates from the continuous infall of material from the envelope. Within the spiral arms, fragmentation takes place as dense protoplanetary clumps come together and occasionally fall onto the protostar, leading to episodes of vigorous accretion (as high as ) as opposed to the “quiescent” periods (where the accretion rate is of the order of –). Figure 6 shows an example of the highly variable accretion rates resulting from these simulations. Subsequent works (Vorobyov and Basu, 2007; Vorobyov and Basu, 2008) have further evolved the disks well into the Class II stage, following their accretion history as shaped by the gravitational torques produced by low amplitude, non-axisymmetric density perturbations that replace the early spiral structure. The original model was later expanded and gradually complicated to include additional effects, from improved thermal physics introducing a detailed energy balance equation accounting for radiative cooling, viscous and shock heating, as well as heating due to the stellar and background irradiation (Vorobyov and Basu, 2010; Vorobyov and Basu, 2015), to dust (in the two-population approximation of Birnstiel et al., 2012; Vorobyov et al., 2018) and metallicity Vorobyov et al. (2020).

FIGURE 6

This series of thin-disk models of collapsing cores played a major role in understanding the accretion processes in the protostellar phase, tackling both the high time variability and the accretion outbursts dominating the mass budget in the Class 0/I phase.

4.2.2 Early SPH and grid-based simulations

SPH simulations of star formation moved in the direction of a full 3D framework, while still (at least in the early works) maintaining a pure hydrodynamical approach coupled to self-gravity and simple barotropic or isothermal equations of state. The very first 3D calculation to follow the collapse of a molecular cloud core to stellar densities was performed by Bate (1998), followed by several pieces of work focusing on different aspects of the outcome of the star formation process. Within this framework, protostars form in clustered environment (see Figure 7): as opposed to isolated collapse models, these simulations showed that accretion of material is a dynamical, competitive process, often terminated by ejection rather than smoothly declining (Bonnell et al., 1997; 1998; Bonnell et al., 2001; Bonnell et al., 2003; Bate et al., 2003); as a consequence, the final mass of a protostar is determined by far more variables than only its initial core properties. These calculations demonstrated that non-steady accretion is a generic outcome of the hydrodynamic collapse in clustered environments, pinpointing a physical origin for accretion variability beyond the instability in the disk itself. Subsequent extensions of the SPH simulations included radiative transfer and are discussed in Section 4.3. In parallel to the thin-disk and SPH approaches, grid-based pure hydrodynamic simulations have also been employed in the context of molecular cloud core collapse, although to a smaller extent (Matsumoto and Hanawa, 2003; Banerjee et al., 2004). These studies provided an Eulerian validation of the non-steady nature of accretion, in absence of other physical mechanisms, as a natural outcome of gravitational collapse.

FIGURE 7

The main limitation of simple, purely hydrodynamical models is that they neglect radiation (Section 4.3) and magnetic fields (Section 4.4). Most early simulations simply modeled the cloud collapse with either an isothermal or barotropic equation of state; the first one holds until densities of g , after which it would transition to adiabatic (Larson, 1969; Masunaga et al., 1998; Masunaga and Inutsuka, 2000); the adiabatic approximation, however, only works in the assumption that the gas is heating because of compression alone. The barotropic equation of state, on the other hand, is based on the evolution of the temperature at the highest density (as calculated with radiative transfer) during the collapse, and fails to correctly describe the temperature distribution in a complex, three-dimensional simulation. An accurate description of the core fragmentation process (which heavily depends on the gas temperature) however cannot neglect the radiative description of the central protostar: once it forms, the heating of the gas due to accretion vastly exceeds that caused by compression, to the point where it can be sufficient to prevent the fragmentation of low-mass protostellar disks into brown dwarfs (Matzner and Levin, 2005). In this context, the natural improvement of simple hydrodynamical models is the coupling with radiative transfer to what is commonly referred to as radiation hydrodynamics (RHD).

4.3 Radiation hydrodynamics (RHD)

The addition of radiation transport in simulations of molecular cloud collapse was implemented in both SPH and grid-based codes, often in the flux-limited diffusion (i.e., treating radiation like a diffusing quantity and imposing a physically-motivated limit on the flux) and gray (i.e., integrated over the entire frequency range) approximation. One of the first three dimensional implementations by (Whitehouse and Bate, 2006) showed a remarkable dependence in the temperature evolution from the initial conditions, supporting the need for radiative transfer models to obtain an accurate temperature description. Several works (Krumholz et al., 2007; Bate, 2009; Offner et al., 2009) confirmed the role of radiation feedback from accreting protostars in inhibiting fragmentation, thereby resulting in fewer, more massive objects with higher accretion rates; Bate (2009) estimated the protostar formation efficiency to be reduced by a factor four compared to the hydrodynamical simulations with a barotropic equation of state (see Figure 8). Other works have also employed frequency-dependent radiative transfer models (see, e.g., Vaytet et al., 2012, Vaytet et al., 2013, which however confirmed the general findings of the gray approximation models, which manage to capture well the key aspects of thermal evolution relevant for the formation of protostars. Bate (2010) evolved RHD simulations beyond the formation of the stellar core, finding that the energy released in the process is high enough to drive a shock wave through the disk, which dramatically decreases the accretion rate on to the stellar core and launches a bipolar outflow.

FIGURE 8

Bate (2009) with permission, CC BY 4.0.

4.4 Magnetohydrodynamics (MHD)

The applicability of the pure hydrodynamical and RHD framework essentially depends on the ionization of the medium. In principle, only a completely unionized medium can be described in a purely hydrodynamical framework; however, stars form from the condensation of galactic interstellar medium (ISM), where the contribution of magnetic fields cannot be neglected. Magnetic fields are ubiquitously observed at all scales of the star-formation process, from molecular clouds (Crutcher, 1999; Bourke et al., 2001; Soler and Hennebelle, 2017) to pre-stellar cores (Heiles and Crutcher, 2005; Troland and Crutcher, 2008; Maury et al., 2018) to protoplanetary disks (Alves et al., 2018; Ohashi et al., 2025), and observational evidence suggests they play a crucial role in shaping key observables including accretion flows, disk masses and sizes (Maury et al., 2018; Galametz et al., 2020; Cabedo et al., 2023). The interested reader can find dedicated reviews in Crutcher (2012), Hennebelle and Inutsuka (2019), Krumholz and Federrath (2019), Zhao et al. (2020), and Pattle et al. (2023).

Magnetization is fundamental in the collapse of pre-stellar cores in that it impacts the evolution of angular momentum. While angular momentum is essentially constant in an unmagnetized medium (Matsumoto and Hanawa, 2003), the presence of magnetic fields and therefore a magnetic tension makes it possible to exchange angular momentum between the fluid particles through Alfvén waves (Shu et al., 1987; Zhao et al., 2020); as a consequence, the angular momentum of the core is reduced and the rotation slowed down, a process called magnetic braking, which prevents the formation of unrealistically large disks (Hennebelle et al., 2016; Wurster et al., 2018; Zhao et al., 2020; Lee Y.-N. et al., 2021; Lee Y.-N. et al., 2024).

In the following, we describe the physics of magnetic models qualitatively and refer the reader to the review of Hennebelle and Inutsuka (2019) for a detailed mathematical treatment.

4.4.1 Ideal MHD

The easiest assumption we can make when introducing magnetic fields is the so-called ideal MHD, which considers fluids as perfect conductors, i.e., perfectly coupled to the field. The main implication of this assumption is that the dissipative terms in the magnetic equations vanish, therefore resulting in a state of constant entropy; the fluid particles are attached to the field lines, so that they can flow along them but cannot cross them, a phenomenon commonly referred to as flux freezing. In the flux freezing regime, the twisting of the magnetic field lines caused by rotation applies a counter force to the rotation itself, effectively slowing rotation down (hence the name magnetic braking) and increasing the infall of gas in the radial direction.

Magnetic braking plays a fundamental role in reproducing the observed disk sizes; however, in the flux freezing regime, it can be strong enough to fully impede the formation of disks around protostars, the so-called “magnetic braking catastrophe.” Indeed, simulations of collapsing cores in the ideal MHD approximation found magnetic braking to efficiently remove angular momentum during the collapse, while simultaneously launching large-scale bipolar outflows, and hinder the formation of rotationally-supported disks (Allen et al., 2003; Hennebelle and Teyssier, 2008; Joos et al., 2012; Machida et al., 2004; Machida et al., 2005; Shu et al., 2004). 3D simulations confirmed the strong suppression of disk formation in the Class 0 phase, when even allowed at all (Price and Bate, 2007; Mellon and Li, 2008; Hennebelle and Fromang, 2008; Galli et al., 2006).

Even when radiative transfer is included, ideal MHD simulations predict very small or transient disks and efficient angular-momentum extraction (e.g., Commercon et al., 2010; Tomida et al., 2010). The ideal approximation holds when the ionization rate is high enough (precisely, , Wurster et al., 2018); however, the typical ionization rates of molecular clouds are low (i.e., , Redaelli et al., 2021; Pineda et al., 2024) and the commonly assumed value is (Padovani et al., 2009; Neufeld and Wolfire, 2017). The low ionization rate implies a substantial influence of non-ideal MHD effects, which need to be accounted for in star formation simulations, and are able to prevent the magnetic braking catastrophe (Mellon and Li, 2009; Dapp et al., 2012; Tomida et al., 2015; Wurster et al., 2016; Zhao et al., 2016; Vaytet et al., 2018; Tsukamoto et al., 2015; Marchand et al., 2018); however, there are other ways to restore disk formation even in ideal MHD, such as considering misalignment between the rotational axis and the magnetic field (Hennebelle and Ciardi, 2009; Joos et al., 2012; Li et al., 2013; Krumholz et al., 2013; Codella et al., 2014; Lee et al., 2017b) and the influence of turbulence and dynamical environments (Machida and Matsumoto, 2011; Seifried et al., 2012; Santos-Lima et al., 2012; Wurster et al., 2019; He and Ricotti, 2023; 2025; Mayer et al., 2025).

4.4.2 Non-ideal MHD

Moving away from the ideal MHD approximation implies introducing dissipative terms, to describe the fact that not all particles are perfectly coupled to the magnetic field and, therefore, do not have an instantaneous response to changes in the magnetic field itself. The three key effects introduced in non-ideal MHD models are Ohmic dissipation (electron-ion and electron-neutral collisions), ambipolar diffusion (ion-neutral collisions), and the Hall effect (drift velocity between positive and negative ions, dispersive term–Wardle, 2004): these account for the finite resistivity and the inertia between the different charge carriers (Wardle and Ng, 1999; Pandey and Wardle, 2008; Zhao et al., 2018). The relative importance of the non-ideal effects depends on the coupling of the charged particles with the magnetic field, so that they dominate in different regimes; in the context of protoplanetary disk evolution, Tsukamoto et al. (2017a) and Wurster (2021) have shown that a complete treatment of disk formation ought to include all three.

Ambipolar diffusion, also referred to as the ion-neutral drift, has a particularly important role in star formation. In partially ionized fluids, as is the case for molecular clouds, McKee and Ostriker (2007) and Pineda et al. (2024), the magnetic field can have an indirect impact on neutral particles through their collisions with the ions and the consequent exchange of momentum. In general, compared to the unmagnetized case, the collapse of a magnetically supported core is significantly slowed down; as the neutrals cross the field lines, the magnetic flux decreases and eventually becomes low enough that the gravitational collapse can take place. Several simulations (Basu and Ciolek, 2004; Heitsch et al., 2004; Li and Nakamura, 2004; Nakamura and Li, 2008; 2011; Loo et al., 2008; Vazquez-Semadeni et al., 2011; Bailey and Basu, 2014; Bailey et al., 2017) have shown that subcritical magnetic fields (i.e., giving a high enough magnetic support to inhibit the dynamical collapse) drastically reduce the star formation rate.

4.4.3 Large-scale simulations

A realistic treatment of the large- (i.e., clump-) scale magnetic field is also fundamental to produce consistent protoplanetary disk populations. Suites of simulations in both the RHD (Elsender and Bate, 2021), ideal (Kuffmeier et al., 2017; Kuffmeier et al., 2019; Bate, 2018) and non-ideal (Wurster and Lewis, 2020; Lebreuilly et al., 2021; Lebreuilly et al., 2024a; Lebreuilly et al. 2024b) MHD framework have been run to gain a statistical insight, with a zoom-in technique allowing to resolve the smaller disk scales. Figure 9 shows an example of the output of these models (adapted from Lebreuilly et al., 2021): discs form around sink particles within the simulation, with different morphology and characteristics depending on the physics considered. Zooming-in on each sink, it is possible to determine key disc properties such as the mass and radius; moreover, signatures of the accretion process are visible in the flow of material around and within the disc.

FIGURE 9

Recently, Ahmad et al. (2025) have modeled the formation of protostars and disks finding that magnetic fields and turbulence drive highly anisotropic accretion onto the disk via dense streamers. MHD instabilities have also been found to shape the infalling envelope into streamers rather than axially symmetric infall (Machida and Basu, 2025); note that the origin of streamers is not limited to instabilities, but can be the consequence of pre-existing turbulence (Tu et al., 2024) or misalignment between the magnetic and rotation axes (Machida et al., 2026).

5 Protostellar accretion: the emerging picture

5.1 Uncertainties in the retrieval and comparison of observables from simulations

The availability of a large number of numerical simulations enables statistical analyses and the identification of robust trends. A comparison between numerical results and observations is both natural and ultimately desirable; however, such comparisons must be approached with care to ensure that physically equivalent quantities are being contrasted. Establishing a fair correspondence between simulated and observed properties is non-trivial for both physical and numerical reasons, which we discuss below.

5.1.1 Stellar properties and mass accretion rate

The main limitation to accessing the central star in simulations is the numerical resolution. This is especially easy to grasp for simulations employing sink particles, where the undeniable advantage of maintaining a reasonable computational time is exchanged for the introduction of unresolved regions around the protostars⁸. Furthermore, as opposed to the observational determination of accretion rates–which is essentially a luminosity measure, as described in Section 3, numerical simulations often calculate the actual amount of mass transferring from the disk/envelope system onto the innermost region within the resolution limit. The caveat is therefore twofold: for one, the actual diagnostic is different–the result of a calculation based on the stellar parameters and some luminosity on one hand, and an actual measure of accreting mass on the other; furthermore, the accretion rate as measured in simulation is that onto the smallest resolution element in the proximity of the star, not necessarily the star itself. If this is the case, then the accretion rate would be a combination of the material reaching the actual protostar and the surrounding region, within the resolution limit, somewhat corresponding to the inner disk.

5.1.2 Accretion luminosity

The observable of accretion is the accretion luminosity (Section 3), and the mass accretion rate is then computed by combining with the stellar properties and inverting Equation 1. However, protostellar is not directly observed, but rather derived (see Section 3.4).

On the contrary in most numerical simulations, such as those by Bate or Lebreuilly, the path is reversed. The accretion rate is determined from the (radiation-, magneto-)hydrodynamics of the gas, as the flow of material reaching the protostar (or sink particle), and is then computed based on Equation 1. Even accounting for radiative feedback, the caveat persists: if the is derived from , adding a consistent treatment of radiation will “only” ensure a correct impact on the rest of the structure, but will not remove the fundamental assumption that luminosity is a derived quantity. As these large-scales simulations often do not resolve the transport of angular momentum from the inner disk to the stellar surface, the - conversion requires assuming (i) a protostellar radius and (ii) the radiative efficiency (see Section 3); the physics of gas transport in the inner disc and from the disc to the star is also often not represented in large-scale simulations. Finally, determining the confidence level of the derived accretion luminosity is not trivial, and is often based on statistics (for example, it could be defined as the standard deviation in the measurements obtained for a population of objects) or outburst amplitudes within the model rather than an uncertainty in the value per se.

From an observational perspective, the accretion luminosity represents the primary measurable quantity, whereas the mass accretion rate is a derived parameter that inherits and amplifies the uncertainties associated with and with the stellar properties entering Equation 1. As a consequence, observational estimates of typically carry substantially larger uncertainties than those on . This introduces an additional caveat when comparing simulations and observations: an apparent agreement in may partly reflect the large propagated uncertainties affecting the observationally inferred accretion rates, rather than a genuine consistency between models and data.

There are also numerical studies, like in Myers et al., 1998, which instead prescribe an accretion history based on analytical models (note that these are, indeed, closer to models than full large-scale simulations); in this case, the accretion luminosity is not derived from the accretion rate anymore, but is a direct result of the assumed model rather than the output of a self-consistent simulation. Observed and simulated accretion luminosities and mass accretion rates should therefore be compared with particular care, taking into account the different assumptions and sources of uncertainty underlying their derivation.

5.1.3 Synthetic observations

A promising approach to exploit the predictions from simulations is producing synthetic observations. Synthetic observations (see Haworth et al., 2018 for a review) are essentially radiative-transfer post-processed results of numerical simulations, whose outputs can be directly compared with the observational data (such as continuum images or SEDs). The advantage of this method is that going from a physical model to synthetic observables usually require fewer assumptions than converting an observation into a physical model; on the other hand, it is important that the post-processing includes the instrumental effects and observational biases specific to the dataset to compare against (e.g., Koepferl and Robitaille, 2017), to ensure one-to-one correspondence. The production of synthetic observations, and the extraction of physical observables with the same techniques employed with real data, ensures a fair comparison beyond the naive assumption that simulations and observations can access the exact same quantity (although, a fair comparison still does not remove the uncertainties introduced below the synthetic observation overlay). In the last couple of decades, several studies have performed post-processing of simulations across the whole star formation process, from the large scales of the ISM (e.g., Smith et al., 2014; Juvela et al., 2019), to the intermediate clump fragmentation (Nucara et al., 2025), to the smallest cores and disks (e.g., Dipierro et al., 2015; Maury et al., 2022; Redaelli et al., 2024).

Another perk of synthetic observations is that they allow to test the accuracy of the methods employed to derive physical properties from observational data. In the context of Class 0/I objects, Tung et al. (2024) performed synthetic observations of the population of YSOs produced by the non-ideal MHD simulations of Lebreuilly et al. (2024a), with the goal of comparing the retrieved disk radii and masses to the real⁹ values. They found that, while disk sizes can be accurately recovered (with uncertainty of a factor 1.6–2.2), the same does not hold for the disk masses. Indeed, disk masses are systematically underestimated (from a factor to 10 in the most extreme cases), because of the failure of the assumption of optically thin emission - fundamental in the conversion of flux into mass.

Accretion-related quantities pose an even greater challenge. To date, synthetic observations aimed at recovering accretion luminosities or mass accretion rates remain very limited, as both and depend on sub-grid accretion physics and observational diagnostics that are not self-consistently captured in current post-processing frameworks, and are therefore typically prescribed rather than inferred in a forward-modeling sense.

5.2 Discrepancies in accretion luminosity estimates from observations

A long-standing issue in the study of protostellar accretion concerns the wide range of accretion luminosities inferred for Class I sources by different observational studies (see also, Fischer et al., 2023). While it is generally expected that protostars accrete at higher rates than CTTSs, early works based on NIR diagnostics suggested comparable accretion levels for Class I and Class II objects. In particular, Muzerolle et al. (1998) and later Beck (2007) found accretion luminosities for Class I protostars similar to those measured in CTTSs, a result that was interpreted as evidence for episodic accretion or for most of the stellar mass being assembled at earlier evolutionary stages. However, subsequent studies based on larger samples and different diagnostics have challenged this picture, reporting systematically higher and in Class I sources compared to Class II stars (e.g., Nisini et al., 2005; Fiorellino et al., 2021; Fiorellino et al., 2023).

One potential source of discrepancy lies in the treatment of extinction. For instance, Beck (2007) derived extinction values assuming that intrinsic hydrogen line ratios follow Case B recombination predictions (Hummer and Storey, 1987), i.e., optically thick conditions in the Lyman lines and optically thin emission in higher-order transitions. If this assumption does not hold in dense, optically thick protostellar environments, the resulting extinction corrections, and hence the inferred accretion luminosities, may be underestimated. In addition to methodological differences, sample selection effects are likely to play a major role. The Class I population is intrinsically heterogeneous, spanning a wide range of envelope masses, accretion histories, and evolutionary stages. Both observations and theoretical models indicate that several physical properties of Class I sources, including , , and , exhibit dispersions approaching two orders of magnitude (e.g., Jørgensen et al., 2009; Dunham et al., 2014; Tobin et al., 2015; Fiorellino et al., 2023), naturally leading to a broad distribution of accretion luminosities. Differences in the fraction of sources for which the accretion luminosity dominates the bolometric output further complicate direct comparisons between studies. For example, the high fraction (95%) of sources with reported in the mid-infrared–bright sample analyzed by Testi et al. (2025) may reflect the younger age and more embedded nature of these populations. In contrast, the samples analyzed by Fiorellino et al. (2021), Fiorellino et al. (2023), and White and Hillenbrand (2004) show significantly lower fractions of such objects, ranging from 25% to 40%. A plausible explanation for this difference is related to sample selection: Fiorellino et al. (2023) considered a deliberately diverse sample of Class I sources, while the White and Hillenbrand (2004) sample was optically visible and therefore likely biased toward less embedded and, on average, lower-accreting objects, potentially including sources at more advanced evolutionary stages. However, differences in the methodologies adopted to estimate and , particularly between NIR and MIR approaches where the extinction law varies, must also be taken into account when interpreting these results. Recent results further highlight the potential impact of diagnostic choices. For instance, Fiorellino et al. (2025) showed that, for CTTSs, accretion luminosities derived from Br systematically trace the lower end of the distribution, with Br-based estimates representing a lower limit in about 80% of the cases. If a similar effect applies to Class I sources or more deeply embedded protostars, it would imply that current estimates of based on near-infrared tracers may systematically underestimate the accretion luminosity. This effect could contribute to part of the discrepancy between studies based on different tracers.

Both observational biases and analysis techniques can significantly affect inferred accretion properties. Ultimately, only large, homogeneous samples analyzed using multiple, independent diagnostics and consistent methodologies will allow us to disentangle intrinsic source-to-source variations from systematic effects, and to establish robust constraints on protostellar accretion luminosities. Applicability and limitations of empirical relationships are discussed in Section 3.4.1.

5.3 Protostellar mass budget

Notwithstanding the caveats discussed in Section 5.2, comparing the simulated and with the observational constraints is still informative. While there are no accretion surveys of Class 0 YSOs, the typical accretion luminosities and mass accretion rates ranges observed in Class I and FS sources are about and (Fiorellino et al., 2023).

From the numerical perspective, to obtain a fair comparison with observational surveys, it is necessary to consider works focused on producing statistically significant samples. Most of the currently available literature on simulations of protostellar accretion, however, does not satisfy this requirement: the majority of works simulate single sources, or a handful (like the case of Kuffmeier et al., 2018, with six sinks in total), but not enough to derive meaningful statistical properties. Large-scale simulations certainly have a higher computational cost, but remain indispensable to correctly interpret the observational results: at the time of writing, the only published works that can be considered for this scope are those of Bate (2012) and Lebreuilly et al. (2021), Lebreuilly et al. (2024b).

Qualitatively speaking, these models recover the same orders of magnitude and reach higher rates; Figure 10 shows the ranges of derived accretion rates across different simulations, spanning from around to , compared to the observational constraints. A possible interpretation of the higher simulated accretion rates is that they can only be sustained during the Class 0 stage, possibly accreting via non-magnetospheric accretion scenarios. To test this possibility, developing new robust methodologies to constrain accretion rates on Class 0 is fundamental. Similarly, it is currently very challenging to infer during FUors-like bursts, because the hot disk covers the information on the stellar photosphere, making very challenging to determine the stellar radius and mass, needed to compute .

FIGURE 10

If accretion rates of – are confirmed in very embedded sources (e.g., Class 0 and the faintest Class I objects), and if sustained over the duration of the protostellar phase (of order 1 Myr), they would be sufficient to assemble low-mass stars. However, evolutionary models in which the inner envelope mass declines exponentially predict infall rates in the range – during the Class 0 phase to form stars between 0.1 and 2 (e.g., Fischer et al., 2017). These values suggest that extremely high sustained accretion rates (e.g., ), which are not supported by typical observed bolometric luminosities, may not be required. Conversely, if typical accretion rates do not exceed , an additional contribution to the mass budget would likely be required, possibly in the form of episodic high-accretion events such as FU Orionis–like outbursts. Zakri et al. (2022) estimate of burst frequencies in Class 0 protostars of order one event every 400 years indicate that a non-negligible fraction of the final stellar mass could, in principle, be accreted during such episodes. Therefore, the relative contribution of steady versus episodic accretion to the global mass assembly remains an open question, strongly dependent on the true burst duty cycle and on the time-averaged accretion history.

A further caveat in protostellar accretion studies is that many key physical parameters are not directly measured, but inferred from observational tracers through model-dependent assumptions. Accretion luminosities are typically derived from emission-line fluxes via method based on empirical relations, with intrinsic dispersions of order 0.3 dex, and are additionally affected by uncertainties in extinction (often 1 mag), veiling, and the physical origin of the line emission. Stellar masses and radii are commonly estimated by placing sources on PMS evolutionary tracks, which remain uncertain during the protostellar phase owing to ongoing accretion and poorly constrained initial conditions, potentially introducing uncertainties up to 1 dex. These combined effects propagate into the derived mass accretion rates, for which total uncertainties can approach 2 dex.

5.4 How do protostars accrete?

Observationally inferred protostellar accretion rates are often lower than these values (see Section 3.4), but this apparent discrepancy should be interpreted with caution. Current estimates (i) are generally derived under the assumption of magnetospheric accretion, whose applicability during the earliest protostellar stages remains uncertain or are determined by assuming , providing only upper limits; (ii) are available primarily for Class I and FS objects, which trace relatively evolved phases and may accrete at lower rates than Class 0 sources (e.g., Fischer et al., 2017; Laos et al., 2021; Le Gouellec et al., 2024); and (iii) are biased toward the brightest and least embedded objects, preferentially selecting more evolved systems with potentially lower accretion activity. As a consequence, currently measured protostellar accretion rates may not capture the full mass assembly history. An alternative, and not mutually exclusive, explanation is that a substantial fraction of the stellar mass is accreted through episodic bursts, which are largely missed by time-averaged diagnostics (e.g., Audard et al., 2014; Vorobyov and Basu, 2015; Zakri et al., 2022).

From a theoretical standpoint, high accretion rates challenge the standard magnetospheric picture. The truncation radius scales as (Koenigl, 1991), implying that even magnetic field strengths comparable to those measured in CTTSs would yield a substantially reduced at protostellar , potentially approaching . In this regime, magnetospheric funnel flows and high-latitude accretion shocks may become inefficient or observationally indistinguishable from direct disk–star accretion. One plausible alternative is boundary-layer accretion, in which the disk material accretes directly onto the stellar surface without being channeled by a large-scale stellar magnetosphere. Rather than being truncated at several stellar radii, the inner disk extends down to the stellar surface, where the gas dissipates its excess angular momentum and kinetic energy within a narrow boundary layer at the star–disk interface (Lynden-Bell and Pringle, 1974; Pringle, 1981; Popham et al., 1993; Popham, 1997). Such a regime is expected to occur when the stellar magnetic field is too weak, too complex, or overwhelmed by high mass accretion rates to efficiently truncate the disk, conditions that may be common during the earliest protostellar phases (Hartmann et al., 2016).

Taken together, these considerations indicate that the accretion geometry in protostars may differ substantially from that of CTTSs, and that multiple accretion regimes may coexist or operate at different evolutionary stages. One possible scenario is that boundary-layer accretion dominates during the earliest, deeply embedded phases, and gradually transitions to magnetospheric accretion at later stages, with the two modes potentially coexisting between the Class 0 and Class I stages.

6 Conclusions and next steps

Protostellar accretion remains one of the least constrained phases of low-mass star formation, despite its central role in setting stellar masses, disk properties, and the initial conditions for planet formation. Throughout this review, we have shown that this difficulty primarily stems from the intrinsic complexity and time variability of the accretion process, combined with the indirect and often time-averaged nature of the available diagnostics, and with the deeply embedded nature of protostars during their main accretion phase.

A first key result emerging from current observations is that accretion during the embedded phases cannot be described as a steady process. Variability is ubiquitous, and episodic accretion is strongly suggested by both numerical models and time-domain observations. However, while accretion variability is now well established qualitatively, its quantitative role in the protostellar mass budget remains poorly constrained. The frequency, duty cycle, intensity, and contribution of accretion bursts to the total assembled mass are still uncertain, particularly for Class 0 sources, which remain largely inaccessible to direct accretion diagnostics. Ambitious spectroscopic variability survey in the infrared are necessary to quantify the contribution of protostellar variability to the stellar mass budget.

A second major limitation lies in our ability to infer accretion properties from observations. Accretion luminosities and mass accretion rates in protostars are not measured directly, but inferred through empirical relations, assumptions on the accretion geometry, and poorly constrained stellar parameters. Many of these diagnostics were developed for Class II systems and implicitly assume magnetospheric accretion, whose applicability during the protostellar phase remains uncertain. As a result, current estimates of and for embedded sources are subject to large and often difficult-to-quantify systematic uncertainties.

Similarly, disk and envelope properties—key ingredients in regulating accretion—are challenging to measure reliably during the embedded phases. While disk radii can now be constrained with increasing confidence, disk mass estimates remain highly uncertain due to optical depth effects, envelope contamination, and poorly constrained dust properties. These uncertainties directly impact attempts to link accretion rates, disk evolution, and instability-driven variability. Post-processing data analysis specifically applied to the earliest stages could enlight the best measurement strategy.

From a theoretical perspective, numerical simulations robustly predict time-variable and often episodic accretion histories, but comparing these predictions with observations remains non-trivial. Simulated accretion rates, luminosities, and evolutionary timescales do not map straightforwardly onto observationally inferred quantities, and differences in definitions, resolution, and diagnostics complicate direct comparisons. Without a consistent framework to connect physical accretion histories to observable quantities, it remains difficult to assess whether models and observations are genuinely consistent or merely overlapping within large uncertainties. Furthermore, large-scale numerical frameworks are rather limited: with only a handful of studies producing statistically significant samples of synthetic protostars, the derivation of meaningful expected trends and subsequent comparison to observational evidence remains limited.

In this context, progress in understanding protostellar accretion requires a shift in focus. Rather than relying on individual objects or time-averaged diagnostics, future efforts must adopt a population-based approach that explicitly accounts for accretion variability. On the theoretical side, population synthesis models that include episodic accretion, disk evolution, and envelope replenishment are needed to link physical accretion processes to statistically meaningful, observable quantities: such models provide a natural framework to explore how different accretion regimes contribute to stellar mass assembly and to predict the distributions of accretion-related observables. Despite the significant computational cost, this development is as a timely, necessary effort to move towards a comprehensive understanding of protostellar accretion. In complementary manner, on the observational side, constraining protostellar accretion demands larger and more homogeneous samples of Class 0 and Class I sources, combined with systematic time-domain studies capable of quantifying accretion variability and burst frequencies. Infrared and submillimeter monitoring, together with coordinated spectroscopic follow-up, will be essential to bridge the gap between instantaneous diagnostics and long-term accretion histories.

Ultimately, protostellar accretion should be regarded not as a single, well-defined mechanism, but as a time-dependent process shaped by evolving disk–envelope structures, variable mass supply, and potentially multiple accretion regimes. Clarifying how these ingredients combine to assemble stellar mass remains a fundamental challenge. Addressing it will require integrating observations and models within a consistent, population-based framework that treats variability as an essential feature of protostellar accretion rather than as a secondary complication. Because this same process sets the mass reservoir, thermal structure, and dynamical conditions of young disks and may operate through analogous mechanisms during planet growth, understanding protostellar accretion is central not only to stellar assembly, but to the physical origin of planetary systems.

The advent of high-sensitivity observational facilities such as the Square Kilometer Array Observatory (SKAO) and the Extremely Large Telescope (ELT) will enable the detailed and accurate study of the most embedded and further ( kpc) protostars, providing more complete samples and, overall, better statistical constraints. In particular, the ELT will allow us to extend our understanding of the star formation process to environments with metallicities different from that of the solar neighborhood. In this context, extending Gaia-like astrometric capabilities to the infrared, as envisioned for GaiaNIR, will be crucial to probe deeply embedded populations and fully characterize the earliest stages of star formation.

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