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A&A
Volume 686, June 2024
Article Number A246
Number of page(s) 12
Section Interstellar and circumstellar matter
DOI https://doi.org/10.1051/0004-6361/202449154
Published online 18 June 2024

© The Authors 2024

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This article is published in open access under the Subscribe to Open model. Subscribe to A&A to support open access publication.

1 Introduction

While the classical picture of star formation assumes the gravitational collapse of an isolated spherical stellar core similar to a Bonnor-Ebert or singular isothermal sphere (Larson 1969; Penston 1969; Shu 1977; Foster & Chevalier 1993), it is now well supported that most star-forming cores lie within filaments of molecular clouds (André et al. 2014). Consequently, stars do not form in isolation but can accrete actively from their environment. This picture has also been established by observations with the emergence of streamers, defined as velocity coherent non-isotropic accretion structures that have been detected during all stages in the evolution of young stellar objects (Pineda et al. 2023).

During the disk build-up in embedded protostellar class-0/I objects, non-isotropic accretion streams have been observed in dust (Le Gouellec et al. 2019) and molecular emission (Cabedo et al. 2021; Murillo et al. 2022; Valdivia-Mena et al. 2022; Kido et al. 2023). Typically, the measured sizes are around a few thousand AU with accretion rates of around 1.0 × 10−6 M⊙ yr−1. However, there have been cases of streamers extending up to 10 000 AU (Pineda et al. 2020) and accretion rates exceeding 1.0 × 10−5 M⊙ yr−1 (Valdivia-Mena et al. 2023). Most observations show one major accretion stream onto a single core. Nonetheless, there have been detections of more complex structures such as multiple streamers (Thieme et al. 2022) and streamers accreting onto binary system (Hsieh et al. 2023).

Non-isotropic infall has a major impact on the forming pro-toplanetary disk, potentially leading to various substructures and instabilities (Bae et al. 2015; Lesur et al. 2015; Kuffmeier et al. 2021; Kuznetsova et al. 2022). This raises the fundamental question of whether planet formation already begins during the build-up of the disk in the class-0 phase in contrast to an isolated unperturbed disk in the class-II phase which is often assumed (Testi et al. 2014; Lesur et al. 2022). Not only was it shown by Drażkowska & Dullemond (2018) that planetesimals may form during infall, but there is even growing observational evidence that planet formation may start at an earlier stage. Submillimetre observations of disks in the class-II phase show insufficient dust masses to explain the formation of exoplanetary systems (Najita & Kenyon 2014; Ansdell et al. 2017; Manara et al. 2018) in contrast to dust masses of disks in the class-0/I phase (Greaves & Rice 2011; Tychoniec et al. 2018, 2020; Tobin et al. 2020). Additionally, there are indications of dust growth in the early phases of disk formation (Kwon et al. 2009; Harsono et al. 2018; Hsieh et al. 2019; Segura-Cox et al. 2020). Finally, the isotopic distribution in meteorites (Kruijer et al. 2014; Van Kooten et al. 2016; Kruijer et al. 2017) and lack of water in the inner Solar System (Morbidelli et al. 2016) suggest the early formation of a dynamical barrier such as Jupiter.

Several numerical studies have also shown that the large scales play an essential role in the distribution of mass accretion onto the disk. Simulations of collapsing turbulent cores usually lead to the formation of accretion channels onto the disk, regardless of the inclusion of magnetic fields (Walch et al. 2010; Joos et al. 2013; Seifried et al. 2013, 2015; Matsumoto et al. 2017; Lam et al. 2019). Even without initial turbulence, a misalignment of the magnetic field in comparison to the global angular momentum can result in the formation of overdensities (Hennebelle et al. 2020). These can have a major impact on the accretion source function that specifies where accreted material enters the disk. For example, Lee et al. (2021) detected larger accretion rates towards the centre than expected from angular momentum conservation of solid body rotation. Some studies were also able to self-consistently resolve the dynamical range from molecular clouds down to the disk using either adaptive mesh refinement (Kuffmeier et al. 2017, 2018, 2019, 2023; Lebreuilly et al. 2021; Pelkonen et al. 2021) or smoothed-particle hydrodynamics methods (Bate 2018). Despite the different setups and physics used, all of the simulations stress that the inhomogeneous accretion via streamers is highly dependent on the environment and leads to instabilities, disk misalignment and even late replenishment of the disk (Kuffmeier et al. 2023).

While star formation can take place in molecular clouds in various different environments, the purpose of this paper is to explore the case of a collapsing core within a turbulent filament. The turbulent environment of the core is generated self-consistently by gas accretion onto the filament. As we do not resolve the full physics needed to explore the disk evolution, we concentrate on the analysis of the accretion onto the disk itself. The paper is organised as follows: first we present the numerical simulation and initial condition in Sect. 2. Thereafter, we present the results in Sect. 3 and its various subsections. We compare our simulation to other numerical studies in Sect. 4 and conclude in Sect. 5.

2 Numerical set-up

The simulation was run with the code RAMSES by Teyssier (2002) using the MUSCL scheme (Monotonic Upstream-Centred Scheme for Conservation Laws, van Leer 1977), a second-order Godunov scheme for solving the discretised Euler equations in conservative form on a Cartesian grid. As Riemann solver we used the HLLC-Solver (Harten-Lax-van Leer-Contact, Toro et al. 1994) together with the multidimensional MC slope limiter (monotonized central-difference, van Leer 1979).

We set up a forming filament in a radially converging flow as defined in Heigl et al. (2020) as an large-scale initial condition. The simulation consists of a 3D box of size 0.4 pc with periodic boundary conditions in the filament direction, in this case the x-axis. We use an isothermal equation of state with a temperature of 10.0 K and a molecular weight of µ = 2.36 which results in a sound speed of around 0.19 km s−1. We define a cylindrical inflow in a cylindrical shell at the edge of the box onto the central x-axis with a fixed density and radial velocity which are constantly reset at every time-step. We show a schematic diagram of the initial condition in Fig. 1. In order to break the symmetry, the density in the inflow region is varied on a cell-by-cell basis with a random value which varies around the mean with a maximum of 50%. The inflow density and the density inside the domain is set to 3.92 × 10−22 g cm−3 which corresponds to a number density of about 1.0 × 102 cm−3 and the inflow velocity is set to Mach 6.0 for the isothermal gas which equals a velocity of 1.13 km s−1. The mass accretion rate per length onto the filament therefore corresponds to Ṁ/L = 8.4 M⊙ pc−1Myr−1. This flow not only acts as an external pressure to confine the filament material but also drives turbulence inside the filament which quickly settles to a constant level. In Heigl et al. (2018), we showed that the total three-dimensional turbulent velocity within the filament is around 0.23 km s−1, or Mach 1.18, for an inflow velocity of Mach 6.0. This value is independent of the accretion flow density itself as well as the strength of its random variation as kinetic energy is converted to turbulent energy. However, the turbulent velocity does show an anti-correlation to the density profile with the centre of the filament showing lower values than the surface.

The minimum resolution of the simulation is 2563 cells which is equivalent to 1.56 × 10−3 pc or about 323 AU. However, we heavily make use of adaptive mesh refinement in order to resolve the scales down to the creation of an accretion disk with a maximum resolution of 163833 cells corresponding to 2.44 × 10−5 pc or 5.04 AU. In order to minimise dissipation across refinement levels and to resolve thin streamer structures in a large region around the forming disk, we employ an aggressive refinement criterion where the Truelove criterion for the maximum density has to be fulfilled by a factor of 128 (Truelove et al. 1997).

As soon as the maximum refinement level reaches a density of 1.83 × 10−13 g cm−3 where the Jean’s length is not resolved by at least four cells, a sink particle is placed in the centre if the region fulfils several conditions. In the standard version of RAMSES these conditions consist mainly of the gravitational potential dominating over the thermal and kinetic support. We also allow mass accretion onto the sinks by using the implemented Bondi accretion algorithm with the standard setting of the accretion radius of four cells. The amount and distribution of sink particles strongly depend on the formation conditions and the subsequent accretion. However, as this paper’s main focus is the creation of streamers and as we do not include enough physics to properly describe the processes inside of the disk, we will explore the sink population in a future paper. In this study, the main purpose of the sink particles is to take the role of a mass sink in order to keep the simulation running. Doing so, their impact on the streamers are minimal.

thumbnail Fig. 1

Isometric sketch of the initial condition. We define a converging flow onto the central x-axis injected in a cylindrical inflow boundary. The x-dimension itself is periodic. Furthermore, we perturbed the inflow density on a cell-by-cell basis in order to brake the symmetry of the simulation.

3 Results

3.1 Core formation

Over time a turbulent filament forms inside the box constrained by the accretion pressure of the radial flow. As soon as the filament’s line-mass reaches close to critical values given by (ML)crit =2cs2G≈1.06×1016 g cm−1≈16.4 M⊙ pc−1,${\left( {{M \over L}} \right)_{{\rm{crit }}}} = {{2c_s^2} \over G} \approx 1.06 \times {10^{16}}{\rm{g}}{\rm{c}}{{\rm{m}}^{ - 1}} \approx 16.4{M_ \odot }p{c^{ - 1}},$(1)

with cs being the isothermal sound speed and G being the gravitational constant, a core begins to form as shown in Fig. 2. We do not impose any overdensity in the initial condition or the accretion flow and the location where the core forms is only determined by the initial seed for the random number generator. The core itself shows turbulent motions and substructure. Before its collapse, the core’s density profile closely follows a Bonnor-Ebert distribution as shown in the upper panel of Fig. 3. The measured mean density in each radial bin is given by the solid line together with its standard deviation shown as the grey area. We compare it to the isothermal Bonnor–Ebert profile shown as the dashed line as well as the turbulent Bonnor–Ebert profile where we take into account the support of a turbulent pressure component to the sound speed as σtot=cs2+σ1D2${\sigma _{{\rm{tot}}}} = \sqrt {c_s^2 + \sigma _{1{\rm{D}}}^2} $(2)

where σ1D=σ3D/3${\sigma _{1{\rm{D}}}} = {\sigma _{3{\rm{D}}}}/\sqrt 3 $ is the turbulent velocity in one dimension which we calculate from the mean of the three-dimensional turbulent velocity of radial bins around the density maximum. This velocity is shown as solid line in the bottom panel of Fig. 3. One can see that the turbulent velocity is relatively constant throughout the Bonnor–Ebert sphere at a value with a mean of 0.096 km s−1. We use the mean value for our calculation of the turbulent Bonnor–Ebert profile in the upper panel given by the dotted line. The actual density seems to match the turbulent profile better, the error bars however are too large to distinguish between the isothermal and turbulent profile. Comparing the three-dimensional turbulent velocity within the filament on a global scale with a value of 0.23 km s−1 to the three-dimensional turbulent velocity within the core with a mean value of 0.17 km s−1, one can see that the latter is slightly lower. In Heigl et al. (2020), we showed that accretion drives turbulent motions on the surface of the filament which subsequently decay at higher densities towards the centre of the filament where we measured very similar values of turbulence as in the core.

In addition to determining the turbulent velocity, we also plot the kinetic energy power spectrum of the core in order to verify if it follows a turbulent cascade which we would expect if it is dominated by turbulent motions. Figure 4 shows the calculated power as solid line together with a Kolmogorov scaling of −5/3 used for subsonic turbulence given by the dashed line and a Burger’s scaling of -2 which is characteristic of supersonic turbulence shown by the dashed-dotted line. One can see that turbulent motions are driven on the scale of the filament radius as indicated by the vertical dotted line and decays towards smaller scales. Compared to the theoretical models, the slope is close to either prediction but seems to drop off faster at smaller scales. There could be several reasons for this behaviour. On the one hand, we have a centrally concentrated density profile which is far from the incompressible fluid where overdensities equilibrate rapidly as assumed by Kolmogorov. On the other hand, simulations tend to loose power at small scales due to numerical diffusion which can influence the slope. Nevertheless, one can see that the kinetic energy does follow a cascade within the core itself.

Furthermore, we investigate if there is an overall rotation of the core which provides the angular momentum for the formation of a disk and which can potentially influence the formation of streamers. For a core with solid body rotation there are analytic models which predict the accretion rate onto the disk as a function of time and radius (Ulrich 1976; Hueso & Guillot 2005). Therefore, we also plot the total specific angular momentum in the upper panel of Fig. 5. We calculate the total specific angular momentum as jtot =∑ ρυ×r∑ ρ,${j_{{\rm{tot }}}} = {{\sum \rho \upsilon \times {\bf{r}}} \over {\sum \rho }},$(3)

where we use the density ρ, the velocity υ and the radial position r of each cell. We determine the total specific angular momentum of each radial shell individually, shown as the solid line, as well as cumulatively for all cells within a given radius, shown by the dashed line. As the total angular momentum is nonzero, there is some bulk rotation which, however, is very small compared to overall turbulent motion.

Both curves, the one calculated per radius and the cumulative one, seem to scale close to linearly with the radius as indicated by the fit to the former value given by the dotted line with a slope of 1.10 ± 0.02. This is due to the fact that the bulk velocity is relatively constant with radius, similar to the turbulent velocity. Observations of the cumulative total angular momentum rather suggest a scaling with radius to the power of 1.5 (Goodman et al. 1993; Caselli et al. 2002; Pirogov et al. 2003; Chen et al. 2007; Tobin et al. 2011; Yen et al. 2015). This scaling was proposed by Burkert & Bodenheimer (2000) as a consequence of sampling the turbulent velocity field at different scales and we also show it in the plot as dashed-dotted line. One can see, that it does not seem to match the total profiles as good as the linear fit. However, if we limit the fitted area to typical scales used in observations of above 1000 AU, the slope is slightly steeper with a slope of 1.21 ± 0.02 which matches both scalings similarly well.

While there seems to be some bulk rotation, the core shows no signs of solid body rotation which would suggest the total angular momentum to be proportional to the radius squared. Moreover, the bulk motion is not necessarily rotating in the same direction throughout the core as shown by the lower panel. Here, we plot the direction of the angular momentum vector for each radial shell. As one can see the azimuthal and polar angle are constantly changing throughout the core which proves that there is no ordered overall rotation.

thumbnail Fig. 2

Projection of the core at 93.2 kyr before its collapse and sink particle creation. The left hand side shows the projection of the whole box with a size of 0.4 pc. The right hand side shows a zoom onto the core with a radius of 7500 AU. One can see that the core shows substructure as it inherits turbulent motions from the large-scale filament.
thumbnail Fig. 3

Radial density and velocity profiles of the forming core 93.2 kyr before its collapse. The upper panel shows the mean density profile as solid black line together with Bonner–Ebert sphere profiles where the dashed line is an isothermal model and the dotted line has an additional turbulent term. For this, we use the one-dimensional turbulent velocity and its mean given in the lower panel as solid and dotted black lines.
thumbnail Fig. 4

Kinetic energy power spectrum centred around the core. The dashed line shows the expectation for subsonic turbulence given by Kolmogorov’s theory and the dashed-dotted line the expectation for supersonic turbulence given by Burger’s turbulence. The vertical dotted line shows the scale of the radial extent of the filament at the core position.
thumbnail Fig. 5

Total specific angular momentum of the forming core and its orientation 93.2 kyr before collapse. The upper panel shows the angular momentum of each radial shell as solid line as well as the cumulative total angular momentum up to that radius as dashed lines. We fitted a slope to the former which shows that they both follow a close to linear trend given by the dotted line. We also plot the predicted slope for turbulent cores by Burkert & Bodenheimer (2000) as the dashed-dotted line. The lower panel shows the orientation of the total specific angular momentum vector of each radial shell.

3.2 Core collapse and disk formation

After a short time, the core begins to collapse and a sink particle forms in its centre. Over time, more and more material falls onto the sink particle and, as not all of it is accreted directly, a disk forms around it. This situation is shown in Fig. 6 which shows a face and edge-on projection of the density within a box of 4000 AU centred around the central sink particle at 24.32 kyr after its formation. Here, the central sink has reached a mass of 0.68 M⊙ and the disk a mass of 0.09 M⊙. In addition, several sinks have formed throughout the disk which usually merge with larger sink particles, especially the central sink, at later stages.

On large scales, depending on the column density contrast, one can clearly make out denser streams of material of lengths of up to several thousand AU flowing onto the disk. They do not necessarily settle to the disk plane before hitting the disk, but also seem to be accreted within the disk. For the further analysis of the streamer properties, we concentrate on this snapshot.

In order to investigate the geometry of the streamers, we cut a spherical shell out of the three-dimensional density distribution at a radius of 400 AU from the central sink particle and project it on a map using a Hammer projection as shown in Fig. 7. One can see, that most of the area is filled by low density material of around 1.0 × 10−18 g cm−3 and that the denser material of more than 1.0 × 10−16 g cm−3 forms concentrated, distinct structures. If the streamers would be filamentary by nature, they would appear as point-like density enhancements where they cross the shell. However, most of the higher density structures show a large extent in one direction which is typical of sheet-like structures. This also becomes apparent when looking at the density projection around the disk in different angles which show vastly different configurations of density enhancements which is caused by the projection of sheets along the line-of-sight. The sheet-like geometry of the overdensities was already noted in Kuffmeier et al. (2017) and is consistent to large radii, however the density contrast also becomes weaker which can be seen below in Fig. 12 where we show the density maps for larger distances. We also note, that the Hammer projections are created from an interpolation on a grid of regular angle spacing. However, the statistical analysis shown in the next sections is performed on a Healpix map (Górski et al. 2005) which uses an equal-area and iso-latitude pixelisation in order to circumvent an oversampling of high latitudes.

thumbnail Fig. 6

Face-on and edge-on density projections of the forming disk at around 24.32 kyr after the sink formation. The box has a size of 4000 AU around the central sink and is rotated to align the disk. The colour bar is adjusted to highlight the denser streamers flowing onto the disk. The density enhancements are accreted from all directions, as well as from above and below the disk, and extend to several thousand astronomical units.
thumbnail Fig. 7

Hammer projection of a spherical shell of the density at a distance of 400 AU from the central sink. The overdensities are visible as elongated density enhancements.
thumbnail Fig. 8

Radially averaged density and velocity profiles during collapse. The mean density is given by the solid blue and the mean radial velocity by the solid red line. We also show the expected density and velocity profile assuming a free-fall model given by the blue and red dashed lines, respectively.

3.3 Infall velocities and spherical accretion

In order to measure the infall, we calculate the mean density and mean radial velocity at every radius which we show in Fig. 8 as the solid blue and red line together with the standard deviation of the radial velocity shown as the shaded red area. The standard deviation of the density is quite large, especially for small radii as the density varies over three orders of magnitude. Therefore, we do not show it in order to keep the plot legible. We compare the resulting profiles to the expected density and velocity of free-fall collapse given by the dashed blue and red line, respectively.

The free-fall velocity is given by vff=2G (M*+Mdisk )r, ${v_{{\rm{ff}}}} = \sqrt {{{2G\left( {{M_*} + {M_{{\rm{disk }}}}} \right)} \over r}} {\rm{, }}$(4)

with M* and Mdisk being the mass of the central sink particle and the disk and r being the radius. The free-fall density is then calculated assuming that the mass accretion rate M˙=4πr2ρυff$\dot M = 4\pi {r^2}\rho {\upsilon _{{\rm{ff}}}}$(5)

is constant, at least for small radii, as there is no mass build-up in a radial shell. We also observe that the mass accretion rate is roughly constant at around 2.6 × 10−5 M⊙ yr−1 which we use to derive the density profile of ρff=M˙4π 2GM*r3.${\rho _{{\rm{ff}}}} = {{\dot M} \over {4\pi \sqrt {2G{M_*}{r^3}} }}.$(6)

One can see that both the density and velocity closely follow the free-fall model. For small radii below 280 AU, the density shows a strong increase which is the accretion shock onto the outer radius of the disk where the infall velocity also reaches a maximum. For larger radii above 3000 AU, we see that the free-fall model is not a good description of the infall anymore which is where pressure effects start to play a larger role and where we have a transition to the filament material.

An interesting aspect of the collapse is whether the density enhancements have a different radial and non-radial velocity compared to the low density regions. Non-radial in this context means the velocity calculated by combining the azimuthal and polar velocity components. As material is concentrated into denser structures, its angular momentum could add up and force it onto more non-radial dominated orbits or the angular momentum could cancel out, forcing it onto more radial orbits. However, there does not seem to be a strong trend which we show in Fig. 9. At a distance of 400 AU, we calculate the mean radial and absolute non-radial velocity within sixteen different logarithmic density bins, given by the solid red and orange line, respectively. In addition, we also calculate the median value as the distribution of the velocity within each density bin is not necessary Gaussian. As one can see the radial velocity is relatively independent of the density, despite it spanning three orders of magnitude. Overall, it is slightly lower but very close to the free-fall velocity given by the dashed red line of about 1.8 km s−1. The offset could be due to the reason that the disk at this distance cannot be approximated as a point-like gravitational source which should lead to a slightly lower free-fall velocity at small radii. There is a small decrease of the mean radial velocity in the very low density bin and a very shallow trend to lower velocities at the very high density bins. The reason for this at very low densities could be that the material can be more easily diverted onto non-radial orbits due to the gravitational attraction of the denser gas. We investigate this by looking at the mean absolute non-radial velocity component given in each density bin by the orange solid line in Fig. 9 with a standard deviation depicted by the orange shaded area. The non-radial motions are much slower than the radial collapse with values of around 0.75 km s−1. And indeed, the lowest density bin shows a slight increase in non-radial motions indicating that there is a process forcing material onto non-radial orbits. We also see a general trend to slower velocities for larger densities which could be caused by the collimation of material into sheets where there is some canceling of angular momentum.

A consequence of a relatively uniform infall velocity is that the mass accretion rate scales with the density. This means that most of the mass which lands in the disk is accreted via the denser structures. We visualise this fact in Fig. 10 where we plot the surface fraction in dark blue and the mass accretion rate in light blue in the same density bins as in Fig. 9. While most cells show a low density, with the maximum being at around 6.0 × 10−18 g cm−3, it is the high density cells above 2.0 × 10−17 g cm−3 where most mass is accreted. Focussing only on the very dense structure above 1.0 × 10−16 g cm−3 shows that close to half of the total mass accretion rate through the shell, namely 1.24 × 10−5 M⊙ yr−1 of 2.58 × 10−5 M⊙ yr−1, is accreted in only a fraction of 0.1 of the total surface. Therefore, the dense accretion structures play an essential role in bringing in material to small scales. Moreover, given that Fig. 9 shows that the non-radial velocity and thus the specific angular momentum is relatively constant in each density bin. Therefore, the accretion of absolute angular momentum shows the same distribution as the mass accretion rate.

thumbnail Fig. 9

Velocity statistics as function of density at a distance of 400 AU. The mean radial velocity is shown by the solid red and the mean absolute non-radial velocity by the solid orange line together with their standard deviation given by their respective shaded area. In addition, we also plot the median velocities given by the dotted lines and the free-fall velocity for the given central mass and distance shown by the dashed line.
thumbnail Fig. 10

Mass accretion distribution as function of density at a distance of 400 AU from the central sink particle. The dark blue bars show the distribution of the surface fraction of each density bin and the light blue bars show their respective total accretion rate. As one can see, most mass is accreted at high densities and low surface fractions.

3.4 Accretion onto the disk

Additionally, it is vital to understand where the material enters the disk as not all material is accreted at the outer radius. We measure the mass accretion rate onto the disk at each radius by azimuthally summing up the flux through the surfaces located ten cells above and below the central plane using the velocity perpendicular to the disk. We show the resulting total mass accretion rate in the top panel of Fig. 11 as the black line. Additionally, we split-up the total accretion rate into the side of the disk it is accreted onto. The accretion rate from the top is shown as red and the accretion rate from the bottom as teal line. Moreover, we also split the total accretion rate into low and high density material with a threshold density of 1.0 × 10−16 g cm−3 and plot the corresponding rates as light and dark blue lines. As one can see, most mass is accreted between 30 and 100 AU with a steep drop towards smaller and larger radii up to the maximum extent of the disk of about 280 AU. The integral over all radii gives 1.53 × 10−5 M⊙ yr−1 which is similar to the total mass accretion rate of which we already measured on larger scales.

The accretion itself is highly asymmetrical which can be seen in the separate accretion rates for the different sides of the disk. While it is dominated at large radii by accretion coming from below the disk, the accretion at lower radii is mainly coming from above. This can also be observed in the projection plots of Fig. 6 and implies that the angular momentum is not distributed isotropically. Looking at the split-up between low and high density, one can also see that the bulk of the mass accretion consists of high density material. This could already be seen from Fig. 10, but shows that the density enhancements continue down onto the disk itself. Interestingly, the low density regime does not display the same strong tendency for a preferred accretion radius but shows a more distributed, albeit negligible, accretion with a similar rate at all radii.

The middle panel shows the so-called source function, the mass accretion rate per azimuthal surface area, for the same mass accretion rate as given in the top panel. We also include predictions from two different models for the mass accretion rate of a core under solid body rotation. Although we already showed that the initial core does not rotate as a solid body, this is the only model for which an analytic description of the mass accretion rate exist. The dashed line shows the model by Ulrich (1976) which assumes that every mass element of a radial shell hits the disk under a parabolic orbit according to its angular momentum with the central sink being in its focus. This leads to the functional form SU(r,t)=M˙(t)4πrc(t)2(rrc(t))−1(1−rrc(t))−1/2${S_U}(r,t) = {{\dot M(t)} \over {4\pi {r_c}{{(t)}^2}}}{\left( {{r \over {{r_c}(t)}}} \right)^{ - 1}}{\left( {1 - {r \over {{r_c}(t)}}} \right)^{ - 1/2}}$(7)

with rc being the centrifugal radius, the maximum impact radius for a given angular momentum, given by rc=Ω2r04GM*${r_c} = {{{\Omega ^2}r_0^4} \over {G{M_*}}}$(8)

which depends on the shells angular speed Ω and its radius r0. The dashed-dotted line uses the model by Hueso & Guillot (2005) which assumes that the material is accreted directly onto its respective Keplerian radius leading to a slightly more centrally dominated source function: SH(r,t)=M˙(t)8πrc(t)2(rrc(t))−3/2(1−rrc(t))−1/2.${S_H}(r,t) = {{\dot M(t)} \over {8\pi {r_c}{{(t)}^2}}}{\left( {{r \over {{r_c}(t)}}} \right)^{ - 3/2}}{\left( {1 - \sqrt {{r \over {{r_c}(t)}}} } \right)^{ - 1/2}}.$(9)

In principle, the mass accretion rate and the centrifugal radius both depend on time. However, in the case of free-fall collapse, the mass accretion rate is given by the density distribution and the evolution of the centrifugal radius can be described analytically for solid body rotation. As we do not start with solid body rotation, we cannot calculate the centrifugal radius as function of time. Therefore, we use the respective radial extent of the disk, in this case 280 AU, as an estimate for the models. As expected, the measured mass accretion rate deviates strongly from the case of solid body rotation which predicts larger accretion rates towards the centre and close to the centrifugal radius which is shown as the vertical dotted line in all panels of Fig. 11. The deficit at small radii is a consequence of the accreted material lacking very low angular momentum. For solid body rotation this is the material close to the rotational axis. However, due to the turbulent initial condition, there is no ordered rotation and therefore not much gas with very low angular momentum. The same argument is valid inversely for the mass accretion rate at very large radii. For solid body rotation this is given by the material in the plane of rotation. However, in our case, the maximum angular momentum is limited by the random motion of the gas which is statistically more unlikely to reach large values consistently.

We also check the time dependence of the mass accretion rate by looking at other snapshots of the simulation. The bottom panel of Fig. 11 shows the source function for a very late time step of 63.34 kyr. At this point in time, the disk has a radius of about 240 AU and the mass accretion rate onto the disk equals 1.28 × 10−5 M⊙ yr−1. As one can see the pattern of the mass accretion has not changed in a qualitative manner. Compared to the analytic predictions of solid body rotation, there still is a lack of accretion at very small and very large radii. We therefore conclude, that the accretion pattern is indeed formed by the turbulent initial condition.

During the core collapse, each shell is accreted consecutively. As not all the mass of a shell is combined into one single accretion stream, the accretion should show a main accretion radius, set by the total angular momentum of the original shell, and a distribution around this radius, set by the spread in angular momentum of the incoming material. As each shell is increasing in total angular momentum, the disk and equally the main accretion radius should grow over time. We can test this assumption of angular momentum conservation by comparing the radius of the original shell derived in two different ways. On the one hand, we can compare the angular momentum of the current main accretion radius to the initial angular momentum distribution of Fig. 5. On the other hand, we can also calculate the distance the currently accreting material travelled assuming free-fall collapse as: Rff=(8G(M*+Mdisk )t2π2)1/3${R_{{\rm{ff}}}} = {\left( {{{8G\left( {{M_*} + {M_{{\rm{disk }}}}} \right){t^2}} \over {{\pi ^2}}}} \right)^{1/3}}$(10)

where t is the time after the start of the collapse.

We first use the snapshot at 24.32 kyr with a peak in mass accretion at around 80 AU and with a non-radial velocity of the dense gas of around 0.6 kms−1. This gives a rough value of the incoming total angular momentum of 6–8 × 1019 cm2 s−1. Comparing this value to Fig. 5 means that the material originates from a shell of roughly two to four thousand AU. The distance the gas would travel due to the free-fall time calculated for the mass contained in the sink-disk system of 0.82 M⊙ also equals around four thousand AU. In comparison, the main accretion radius at 63.34 kyr has not changed much and shows a value of around 80–100 AU. The mass of the sink-disk system has increased to 1.57 M⊙ which results in a free-fall radius of around ten thousand AU. According to Fig. 5, this would imply a total angular momentum of around 2–4 × 1020 cm2 s−1 . The reason why the main accretion radius did not change much is that the non-radial velocity of the dense gas also increased to about 1.0 kms−1. Therefore, the angular momentum contained in the dense streams again matches well with the total angular momentum of the shell. This means both snapshots show a good agreement with the total angular momentum of the respective shell calculated before the collapse. The main accretion radius does not necessarily grow over time, but is counterbalanced by an increased non-radial velocity. Nevertheless, this idealised picture does not always apply because the angular momentum of each individual shell is not necessarily aligned to all other shells. Therefore, accretion is also able to heavily disrupt the disk. We also observe this in our simulation where the disk size can change quite drastically over time.

thumbnail Fig. 11

Mass accretion onto the disk measured at ten cells above and below the central plane. The x-axis is normalised by the disk radius as indicated by the vertical dotted line. The top panel shows the total mass accretion rate given by the black line. In addition, the red and teal line split the total rate into the side of the disk the material is accreted onto. Moreover, the light and dark blue lines divide the total rate into low and high density material by setting a density threshold. The middle panel shows the source function, that is the mass accretion rate normalised by its azimuthal surface area, together with its analytic predictions for solid body rotation. In comparison, the bottom panel shows the same as the middle panel but for a snapshot at a late time step in the protostellar disk accretion.

3.5 Residual velocities

From the analysis of the radial velocity it is clear that essentially all material is falling onto the central sink in a free-fall manner. This only leaves the residual motions perpendicular to the infall as cause of the collimation of material into denser sheets. Therefore, we analyse the residual velocities in more detail. In the top left panel of Fig. 12, we show the same map as in Fig. 7 with the non-radial motions displayed on top where we use the azimuthal and polar velocity components to determine the direction and magnitude of the flow. As one can see, the density enhancements coincide either with strong changes in magnitude of the flow or where two flows converge. This means that the main locations of the formation of accretion sheets are given by where gas with different angular momentum collides and forms shock-like density enhancements.

In order to emphasise this point, we also calculate the velocity divergence of the residual motions which we show on the right hand side of Fig. 12. These maps only take into consideration the azimuthal and polar velocities and neglects the radial velocity component. Negative values of the velocity divergence shown in red are areas where material is being compressed and positive values in grey show areas where material is being diluted. Comparing the map to the density map, one can clearly see that the areas of compression coincide with overdensities which shows again that it is the flows of the residual motions during free-fall which bring material together into denser accretion sheets.

During infall, there will be a wide range of angular momentum. Gas with more eccentric orbits will at some point cross the orbits of the more radially dominated orbits of the denser accretion sheets. This means the density enhancements will accumulate more and more material along their way, leading to fewer but even denser structures. From top to bottom of Fig. 12 we plot the density and velocity divergence at further and further distances. As one can see, as the size and contrast of structures visible in the density goes down the farther out we go, the residual motions define more and more small-scale convergence zones. Even at large distances, where radial collapse has not yet collimated a lot of material and the residual motions are still dominated by the turbulent velocity field of the filament, one can already observe convergence zones. This suggests that the initial seeds of overdensities are given by the turbulent motions on large scale. In order to verify that it is the turbulent velocity field which creates this pattern, we also plot the velocity divergence around the core of Fig. 2 at 93.2 kyr before the collapse in Fig. 13. Here, the forming core only consists of an overdensity of a factor of two. As one can see, even at this early stage where their is no radial collapse yet, a very similar pattern is visible. This means that the initial turbulence on large scales indeed sets the formation pattern of the dense accretion sheets.

One can also measure the collimation of material during infall in the density probability distribution at different distances as shown in Fig. 14. From light to dark blue, we plot the density probability distribution of spherical shells starting from large distances down to close to the disk. One can see a systematic trend where the density at large distances follows a log-normal distribution with a dispersion of σ = 0.62 ± 0.02, whereas if one goes closer to the disk, one can see the development of a power-law tail. In that regard, the evolution from larger to smaller radii follows a similar process as the density in a turbulent box under gravitational collapse. The longer gravity is able to compress the material, the more pronounced the powerlaw tail becomes. This suggests the same process is happening during the infall onto the central sink. Indeed, if we fitted a power-law to the high-density tail we get a similar but slightly smaller index of −1.33 ± 0.03 as in simulations of turbulent boxes which vary around −1.5 up to −2.5 (Kritsuk et al. 2011; Federrath & Klessen 2013). This suggests that gravity plays an essential role for the evolution of the overdensities which we explore in the following subsection.

thumbnail Fig. 12

Hammer projections of the density and velocity divergence at different distances for the same time step. From top to bottom: we show larger and larger radii. The top left panel shows the same density as given in Fig. 7 together with the respective velocity vectors indicating the direction and magnitude of the motions perpendicular to the infall. This velocity field defines the velocity divergence on the right hand side which shows structures where material is compressed due to converging velocities in red. As one can see, during infall the gas forms less but more condensed structures the closer it gets to the central sink particle.
thumbnail Fig. 13

Velocity divergence at 3200 AU at around 93.2 kyr before the sink formation. The velocity field is dominated by turbulent motions and already shows the typical structures visible during collapse in Fig. 12 which lead to the formation of the dense accretion sheets.
thumbnail Fig. 14

Density probability distribution of shells at different distances as given by the legend. The distribution follows a log-normal function with gas closer to the central sink particle showing the development of a powerlaw.

3.6 Gravitational focussing of the overdensities

In order to explore the question if the overdensities gravi-tationally dominate their local environment, we calculate the divergence of the acceleration field analogously to the velocity field and show the results in Fig. 15. Negative values coloured in red show areas where material is gravitationally compressed while positive values coloured in blue show areas where material is being gravitationally pulled apart. Very prominent is the gravitational attraction of the disk itself, pulling material into the central plane. It dominates over the gravitational pull of the streamers except for the area towards the right hand side of the viewing direction. Here it is interrupted by the gravitational field of one of the densest streamers and a close-by sink particle which pull the gas stronger into their direction, turning their vicinity blue. Compared to Fig. 7, one can see that there is a one-to-one correspondence of the highest overdensities to areas with large negative divergence of the acceleration. However, not all structures are able to gravitationally pull material together. Some overdensities visible in the density map have no correspondence in the acceleration divergence map. We investigate this further in Fig. 16 where we show the mean and median acceleration divergence as function of the density using the same bins as in Fig. 9. As one can see, only the very dense structures above around 1.0 × 10−16 g cm−3 show a significant negative acceleration divergence which means only the densest fraction of the overdensities are able to gravitationally concentrate material at this radius.

This pattern remains the same going to larger distances until the gravitational pull onto the central axis of the filament dominates as can be seen in Fig. 17. Compared to the small-scale velocity divergence at large distances visible in Fig. 12, the missing gravitational divergence of small-scale structure means that overdensities are mainly initialised by the velocity field. During infall, they are collimated and accrete enough material to gravi-tationally dominate their vicinity and gather even more material. This explains the increase in non-radial orbits of low density material which can be easily pulled towards the dense streamers seen in Fig. 9 as well as the development of a power-law tail in the density probability function observed in Fig. 14.

thumbnail Fig. 15

Hammer projection of the acceleration divergence at 400 AU from the central sink particle. Negative values show areas where material is gravitationally compressed and positive values where it is pulled apart. The gravitational compression of the disk is visible as the large distributed area together with the most massive accretion structures.

4 Comparison to related work

Our main finding that the accretion is set by the large-scale environment agrees with other simulations such as Kuffmeier et al. (2017), Bate (2018) and Lebreuilly et al. (2021). We show that the formation of overdensities occurs naturally in a turbulent environment due to the initial velocity field. We find that this has a strong influence on where mass is accreted onto the disk. However, this effect may depend strongly on the included physics. For example, the simulation of Lee et al. (2021) which includes magnetic fields finds a centrally concentrated mass accretion.

Although the general outcome of overdensity formation during collapse is robust for different initial conditions, we made several simplifications in our simulations. We conducted this work in order to set a baseline to compare future parameter studies to. Of major importance would be the inclusion of magnetic fields. Depending on the strength of the magnetic field, it can have a significant impact on the gas dynamics, potentially leading to a reduction of overdensities. Magnetic fields also have a strong effect particularly on the sizes of disks due to magnetic braking and outflows (Allen et al. 2003; Galli et al. 2006; Price & Bate 2007; Hennebelle & Fromang 2008; Duffin & Pudritz 2009; Commerçon et al. 2010; Seifried et al. 2011, 2012b) which typically leads to rather small disk radii. While we deliberately did not analyse the disk properties in this study, we do find a rather large disk. However, it is still in a very dynamic stage where the size can vary quite drastically. Moreover, depending on dust properties and cosmic-ray ionisation rates, non-ideal MHD effects can reduce the impact of strong magnetic fields, which again leads to more frequent disk formation and larger disk sizes (Shu et al. 2006; Tsukamoto et al. 2015a; Masson et al. 2016; Zhao et al. 2016, 2018; Wurster & Bate 2019). Furthermore, there is also evidence that the effect of magnetic breaking can be reduced by non-idealised initial conditions where turbulence plays an essential role, not necessarily for magnetic reconnection, but by misaligning the angular momentum from the magnetic field vector (Hennebelle & Ciardi 2009; Seifried et al. 2012a, 2013; Joos et al. 2013; Li et al. 2013; Wurster et al. 2019; Hirano et al. 2020). A comparison of the accretion including MHD effects will be the focus of a future study.

In addition, while many studies include a barotropic equation-of-state in order to mimic thermal effects of the disk above the opacity limit, radiative feedback from the star has been shown to affect the temperature structure and therefore the fragmentation of the disk (Commerçon et al. 2010; Tsukamoto et al. 2015b). Moreover, thermal pressure is able of driving outflows even in the complete absence of magnetic fields (Bate 2011; Bate et al. 2014). To which extent radiative feedback has an effect on overdensities in the accretion flow has to be explored in future simulations

thumbnail Fig. 16

Mean acceleration divergence per density bin at a distance of 400 AU from the central sink together with its standard deviation. While for low densities the divergence is positive and small, one can see that the gravitational compression dominates for densities larger than 1.0 × 10−16 g cm−3 where there is a steep drop to large negative values.
thumbnail Fig. 17

Hammer projection of the acceleration divergence at 3200 AU from the central sink particle. The centre and the border of the image point in the direction of the filament. The plot shows the large-scale gravitational acceleration onto the central axis due to the filament potential.

5 Conclusions

In this study, we simulated the collapse of a core which forms self-consistently within a turbulent filament in order to analyse the origin and properties of overdensities in the accretion flow. We analysed the morphological, kinematic and gravitational properties of the core before and during the collapse in great detail and are able to draw the following conclusions:

  • 1.

    Overdensities in the accretion flow are created naturally as a consequence of the initial turbulent velocity field. This velocity field is set on large scales and is already present before the collapse;

  • 2.

    Instead of having a filamentary morphology, we find that the overdensities in our simulation are rather sheet-like where the observable accretion pattern depends on the line-of-sight projection. This morphology follows from the shock-like convergence of residual motions during collapse;

  • 3.

    The collapse is essentially in free-fall with no significant difference in the radial velocity of the high and low-density gas despite it spanning several orders of magnitude. We only see a slight tendency for lower non-radial velocities and a more direct radial infall of the high density gas compared to a higher non-radial velocity and more eccentric infall of the low density gas;

  • 4.

    The mass accretion onto the disk differs drastically from an idealised initial condition of a core with solid body rotation. This is due to the fact that the core inherits a turbulent velocity distribution from its environment with no generally ordered rotation;

  • 5.

    The turbulent velocity field defines a log-normal probability density function at large radii which develops a power-law tail at smaller radii, similar to the gravitational collapse in a turbulent box;

  • 6.

    Both the increase in non-radial velocity at low densities and the development of a power-law tail at large densities, are a consequence of the gravitational pull of the most massive overdensities which are able to gravitationally dominate their local environment. This leads to more and more collimated structures over time during the infall.

Acknowledgements

We thank the anonymous referee for constructive comments that improved the quality of the paper. We also thank Til Birnstiel and Jaime Pineda for useful discussions. This research was supported by the Excellence Cluster ORIGINS which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy -EXC-2094-390783311.

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All Figures

thumbnail Fig. 1

Isometric sketch of the initial condition. We define a converging flow onto the central x-axis injected in a cylindrical inflow boundary. The x-dimension itself is periodic. Furthermore, we perturbed the inflow density on a cell-by-cell basis in order to brake the symmetry of the simulation.
In the text
thumbnail Fig. 2

Projection of the core at 93.2 kyr before its collapse and sink particle creation. The left hand side shows the projection of the whole box with a size of 0.4 pc. The right hand side shows a zoom onto the core with a radius of 7500 AU. One can see that the core shows substructure as it inherits turbulent motions from the large-scale filament.
In the text
thumbnail Fig. 3

Radial density and velocity profiles of the forming core 93.2 kyr before its collapse. The upper panel shows the mean density profile as solid black line together with Bonner–Ebert sphere profiles where the dashed line is an isothermal model and the dotted line has an additional turbulent term. For this, we use the one-dimensional turbulent velocity and its mean given in the lower panel as solid and dotted black lines.
In the text
thumbnail Fig. 4

Kinetic energy power spectrum centred around the core. The dashed line shows the expectation for subsonic turbulence given by Kolmogorov’s theory and the dashed-dotted line the expectation for supersonic turbulence given by Burger’s turbulence. The vertical dotted line shows the scale of the radial extent of the filament at the core position.
In the text
thumbnail Fig. 5

Total specific angular momentum of the forming core and its orientation 93.2 kyr before collapse. The upper panel shows the angular momentum of each radial shell as solid line as well as the cumulative total angular momentum up to that radius as dashed lines. We fitted a slope to the former which shows that they both follow a close to linear trend given by the dotted line. We also plot the predicted slope for turbulent cores by Burkert & Bodenheimer (2000) as the dashed-dotted line. The lower panel shows the orientation of the total specific angular momentum vector of each radial shell.
In the text
thumbnail Fig. 6

Face-on and edge-on density projections of the forming disk at around 24.32 kyr after the sink formation. The box has a size of 4000 AU around the central sink and is rotated to align the disk. The colour bar is adjusted to highlight the denser streamers flowing onto the disk. The density enhancements are accreted from all directions, as well as from above and below the disk, and extend to several thousand astronomical units.
In the text
thumbnail Fig. 7

Hammer projection of a spherical shell of the density at a distance of 400 AU from the central sink. The overdensities are visible as elongated density enhancements.
In the text
thumbnail Fig. 8

Radially averaged density and velocity profiles during collapse. The mean density is given by the solid blue and the mean radial velocity by the solid red line. We also show the expected density and velocity profile assuming a free-fall model given by the blue and red dashed lines, respectively.
In the text
thumbnail Fig. 9

Velocity statistics as function of density at a distance of 400 AU. The mean radial velocity is shown by the solid red and the mean absolute non-radial velocity by the solid orange line together with their standard deviation given by their respective shaded area. In addition, we also plot the median velocities given by the dotted lines and the free-fall velocity for the given central mass and distance shown by the dashed line.
In the text
thumbnail Fig. 10

Mass accretion distribution as function of density at a distance of 400 AU from the central sink particle. The dark blue bars show the distribution of the surface fraction of each density bin and the light blue bars show their respective total accretion rate. As one can see, most mass is accreted at high densities and low surface fractions.
In the text
thumbnail Fig. 11

Mass accretion onto the disk measured at ten cells above and below the central plane. The x-axis is normalised by the disk radius as indicated by the vertical dotted line. The top panel shows the total mass accretion rate given by the black line. In addition, the red and teal line split the total rate into the side of the disk the material is accreted onto. Moreover, the light and dark blue lines divide the total rate into low and high density material by setting a density threshold. The middle panel shows the source function, that is the mass accretion rate normalised by its azimuthal surface area, together with its analytic predictions for solid body rotation. In comparison, the bottom panel shows the same as the middle panel but for a snapshot at a late time step in the protostellar disk accretion.
In the text
thumbnail Fig. 12

Hammer projections of the density and velocity divergence at different distances for the same time step. From top to bottom: we show larger and larger radii. The top left panel shows the same density as given in Fig. 7 together with the respective velocity vectors indicating the direction and magnitude of the motions perpendicular to the infall. This velocity field defines the velocity divergence on the right hand side which shows structures where material is compressed due to converging velocities in red. As one can see, during infall the gas forms less but more condensed structures the closer it gets to the central sink particle.
In the text
thumbnail Fig. 13

Velocity divergence at 3200 AU at around 93.2 kyr before the sink formation. The velocity field is dominated by turbulent motions and already shows the typical structures visible during collapse in Fig. 12 which lead to the formation of the dense accretion sheets.
In the text
thumbnail Fig. 14

Density probability distribution of shells at different distances as given by the legend. The distribution follows a log-normal function with gas closer to the central sink particle showing the development of a powerlaw.
In the text
thumbnail Fig. 15

Hammer projection of the acceleration divergence at 400 AU from the central sink particle. Negative values show areas where material is gravitationally compressed and positive values where it is pulled apart. The gravitational compression of the disk is visible as the large distributed area together with the most massive accretion structures.
In the text
thumbnail Fig. 16

Mean acceleration divergence per density bin at a distance of 400 AU from the central sink together with its standard deviation. While for low densities the divergence is positive and small, one can see that the gravitational compression dominates for densities larger than 1.0 × 10−16 g cm−3 where there is a steep drop to large negative values.
In the text
thumbnail Fig. 17

Hammer projection of the acceleration divergence at 3200 AU from the central sink particle. The centre and the border of the image point in the direction of the filament. The plot shows the large-scale gravitational acceleration onto the central axis due to the filament potential.
In the text

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