Tides in clouds: control of star formation by long-range gravitational force

Gravity drives the collapse of molecular clouds through which stars form, yet the exact role of gravity in cloud collapse remains a complex issue. Studies point to a picture where star formation occurs in clusters. In a typical, pc-sized cluster-forming region, the collapse is hierarchical, and the stars should be born from regions of even smaller sizes ($\approx 0.1\;\rm pc$). The origin of this spatial arrangement remains under investigation. Based on a high-quality surface density map towards the Perseus region, we construct a 3D density structure, compute the gravitational potential, and derive eigenvalues of the tidal tensor ($\lambda_{\rm min}$, $\lambda_{\rm mid}$, $\lambda_{\rm max}$, $\lambda_{\rm min}<\lambda_{\rm mid}<\lambda_{\rm max}$), analyze the behavior of gravity at every location and reveal its multiple roles in cloud evolution. We find that fragmentation is limited to several isolated, high-density ``islands''. Surrounding them, is a vast amount of gas ($75 \%$ of the mass, $95 \%$ of the volume) that stays under the influence of extensive tides where fragmentation is suppressed. This gas will be transported towards these regions to fuel star formation. The spatial arrangement of regions under different tides explains the hierarchical and localized pattern of star formation inferred from the observations. Tides were first recognized by Newton, yet this is the first time its dominance in cloud evolution has been revealed. We expect this link between cloud density structure and role gravity to be strengthened by future studies, resulting in a clear view of the star formation process.


INTRODCUCTION
Molecular clouds are complex objects that exhibit significant fluctuations on all scales (Williams et al. 2000).These cloud surface densities often exhibit variations of multiple orders of magnitudes, and are often quantified using log-normal or power-law models (Kainulainen et al. 2009(Kainulainen et al. , 2014)).The collapse of the molecular cloud is a complex, multi-scale process that involves an interplay between turbulence, gravity, magnetic field, and ionization (Dobbs et al. 2014).The newly-formed stars are also organized in hierarchies: Up to 85% of the young stars should form in clusters (Megeath et al. 2016).Stars in one cluster are likely to be assembled from stars formed in groups (Vázquez-Semadeni et al. 2017) which contain small numbers of stars.By modeling the distribution of orbital parameters, a study (Marks & Kroupa 2012) find that stars in clusters are born in regions of surprisingly high stellar mass densities ( = 3 × 10 3  ⊙ pc −3 for a 300  ⊙ cluster).What controls the evolution of these density structures through which stars form remains an open question.
Gravity can play multiple roles in cloud evolution: Inside dense regions, it drives fragmentation and collapse, and outside these regions, it can suppress the low-density gas from collapsing through extensive tidal forces and drive accretion.Past studies focus on the role of gravity in individual, isolated parts, neglecting long-range gravitational interactions.This negligence is partly caused by the picture people have in mind: In the prevailing paradigm for star formation, gravity drives the collapse, balanced by supports from turbulence and magnetic fields (Krumholz et al. 2019;Krause et al. 2020).This picture focuses people's attention on the interplay between e.g.turbulence ★ E-mail: ligx.ngc7293@gmail.com,gxli@ynu.edu.cn and gravity, neglecting the fact that role of gravity can be different depending on the condition.Another reason is methodological, as the most widely-adopted way to quantify the importance of gravity is to evaluate the virial parameter (Bertoldi & McKee 1992), which is the ratio between kinetic and gravitational energy, where gravitational interactions between matter from the inside and the outside of these boundaries are neglected.
The tidal tensor, defined as    =    , where (, , ) is the gravitational potential, provides a detailed description to the local structure of a gravitational potential.The eigenvalues of the tidal tensor contain information on how a region would evolve under gravity.Under self-gravity,  min ≈  max ≈  mean ≈ 4/3 , where a Jeanslike fragmentation should occur (Jeans 1902).Adding gravity from external bodies,  min becomes smaller.When 0 ≲  min <<  mean , the fragmentation is slowed down, accompanied by an increased Jeans mass (Jog 2013), and when  min < 0, Jeans-like fragmentation becomes impossible.Although the cloud collapse is also under the influences from e.g.turbulence (Mac Low & Klessen 2004) and magnetic field (Li et al. 2014), their role is to provide support against gravity and is secondary compared to that from gravity.Thus, a solid understanding of the behavior of gravity should provide direct insights on how collapses occur and help to establish the connection between the density structure of the cloud and star formation.
Deriving the tidal tensor for observed clouds has been a challenging task, as this would require 3D density distributions, whereas current observations only provide 2D information in high resolutions.Taking advantage of a density reconstruction method developed from our previous papers, we perform a first study of the structure of the gravitational field towards a real molecular cloud.
We use the surface density map constructed by combining Her- schel and Planck observations (Zari et al. 2016).The map has a spatial resolution of 0.03 pc.We construct a 3D density distribution (Appendix A), which shares many features with the original density structure.Using results from a numerical simulation performed by (Clark et al. 2019), we verify that these differences are small (Appendix C), and this reconstruction allows us to perform the analysis presented below.

RESULTS
We compute gravitational potential by solving the Poisson equation and derive the tidal tensor (Appendix B) The tidal tensor is a 3x3 symmetric matrix, which have three eigenvalues  min ,  mid ,  max , where  min <  mid <  max .The three eigenvalues controls the gas evolution along three orthogonal directions.The tide is fully compressive if and is extensive if one of the eigenvalues is smaller than zero, e.g.
The maps are plotted in Figs. 1 2. Since our observations have a limited resolution, small-scale density fluctuations are unresolved.This limitation, which is hard to overcome, can be quantified.We adopted the criterion that gas in a voxel is resolved if the linear resolution is larger than the Jeans length (Jeans 1902) ( >  Jeans ), and analyze the unresolved part separately.In our case, the map has an angular resolution of 36", and a linear resolution of  = 0.05 pc.A voxel is expected to contain significant amount of unresolved structure if the resolution larger than the Jeans mass, and gas with  H 2 > 10 5 cm −3 are unresolved, where a temperature of 20 K was assume.When the resolution is limited the amount of density fluctuations is under-estimated.In most cases, this leads to over-estimations of the amount of gas contained in regions under compressive tides.

Suppressing fragmentation by extensive tides
When the tides are extensive, fragmentation can be suppressed.This effect has been overlooked in the past, but can be conveniently studied using our approach.The extensive tides are dominant in the regions where  min < 0 (Fig. 3).To make our values well-defined, we focus on regions where  H 2 > 1000 cm −3 which contains to 40% of the gas in the Perseus region.We further divide the cloud into a few pc- sized "clumps" and study the composition of gas within.In a typical clump like the B1 (Fig. 3), 75% (Method C) of the mass is under extensive tides, and this gas occupy 95% of the volume.These mass fractions are accurate to around 10% (Appendix C).To our surprise, Long-range gravity can influence the evolution of a large majority of the gas.

Hierarchical and localized star formation
To link the density structures of the gas to the spatial organization of the newly-formed stars, using maps of  min , we identify coherent regions where  min < 0, and study their properties.For each region, we derive a mass  core and a size  core .We removed regions whose masses are smaller than the Jeans mass, as they would not be able to collapse under self-gravity.Among the massive, super-Jeans regions, most have sizes that we can barely resolve  ≲ 0.1 pc, and they contain a wide range of masses (0.3 − 30 M ⊙ ) (Fig. 3).We call these regions "islands under compressive tides", which reflects their compactness in space.We find that a clump typically consists of a few tens of these "islands".We use the following equation to estimate the number of stars formed in one of these islands where  * ,total =  gas  SF  −1 * ,mean is the total number of stars,  gas ≈ 1000  ⊙ is the mass of the clump,  SF =  stars / gas is the star formation efficiency of a cluster, and  * ,mean = 0.3  ⊙ is the IMFaveraged stellar mass (Kroupa 2001). island is the number of islands.On average, around 20 stars will be formed in one island.These are called stellar groups.
This spatial arrangement appears to be consistent with observational constraints of the initial condition of star formation.It has been well-established that binaries born in massive star clusters has shorter periods.By modeling the orbital parameter distribution of binaries, one can infer the density of the environment where stars are born.Through this approach, studies (Marks & Kroupa 2012) finds that stars in clusters form in dense regions of small sizes.A 1000  ⊙ clumps such as the B1 region should from a 250  ⊙ cluster.According to previous estimates (Marks & Kroupa 2012), the stars should be born in regions of 3000  ⊙ pc −3 ( H 2 = 5 × 10 5 cm −3 ).We can relate this high stellar density to the high gas densities of these "islands" found in our observations.From our maps, most of these islands are small ( ≲ 0.1 pc), barely resolved, and contain different masses.A typical, 0.1 pc-sized "island" of 10  ⊙ has a gas density of 2500  ⊙ pc −3 , and this is very similar to the stellar density inferred from observations.Although what we measure is the gas density, and the corresponding stellar density can be lower, our "islands" are still growing in mass.The compactness of these "islands" can be related to the high stellar density of clusters stars.In our case, the extensive tides confine star formation to these small, dense regions.
The existence of these islands agrees with predictions of simulations (Vázquez-Semadeni et al. 2017) carried out under the of the Global Hierarchical Collapse scenario (Vázquez-Semadeni et al. 2009).To perform these simulations, the authors used supersonic turbulence to create a hierarchy of density structures, which collapse to produce groups of stars.They related the stellar groups to the hierarchical density structure produced by turbulence.This spatial arrangement is very similar to what was seen from our maps, and the high densities of the gas clumps where groups form are in agreement with the high stellar density of stellar nurseries inferred from obser- vations.In addition, we find that gravity from the dense structures can suppress the ambient gas of lower densities from fragmenting, which limits the range of scales that can collapse, confining star formation to localized regions.We propose that it is this tidal-driven confinement plays a critical role in sustaining the high-density condition where stars form.

Accretion towards dense regions
Another consequence of this spatial arrangement is that to explain the observed star formation efficiency, transport of gas from regions under extensive tides to the dense "islands" is necessary.We use the following equation to describe the transport of gas from the clumps to the stars: where  * is the stellar mass,  gas is the gas mass, (1 −  outflow ) represents the effect of outflow which can transport the infalling gas outwards (Matzner & McKee 2000), and  infall is the fraction of gas clump-scale gas that can contribute to the infall.Adopting values consistent with literature results ( * / gas = 0.25 (Pfalzner et al. 2016),  outflow ≈ 0.5 (Matzner & McKee 2000)), we require  infall ≈ 50%.To explain the current conversion efficiency, roughly half of the gas in a clump is expected to be to be accreted.Since only 25% of the gas in found in the islands under compressive tides, the remaining 25 % likely contributed by the transport of gas from diffusion regions towards these islands.
Using our maps, we can find out which part of the gas will be accreted.We note that the spatial distribution of  max has a stratified structure, where gas located at the cluster centers have larger  max , which implies shorter evolution times ( acc ≈  1/2 max ).This corresponds to the picture where gas located at cluster centers have shorter evolution times and will be accreted first.We can relate  infall to  max by solving where d( max ) d max is the mass-weighted distribution of  max .After this, the locations of gas with  max >  max,crit can be mapped.This approach gives insight on which part of the gas is more likely to end up in stars.In reality, turbulence can leads to some additional mixing which is not accounted for.The results at  max,crit = [0.2,0.5] are plotted (Fig 4).The very first part (  infall = 0.2) of the gas to be converted into stars are those located in the cluster center under compressive tides, followed by gas with 0.2 <  infall < 0.5, which contains both gas at the central regions and gas contained in filamentary structures.To explain the observe star formation efficiency, accretion of gas located where tides are extensive is necessary at the late stage of the evolution.

CONCLUSIONS
One prevailing paradigm about star formation is that the collapse is driven by gravity, balanced by processes such as turbulence and magnetic fields (Krause et al. 2020;Krumholz et al. 2019).This scenario recognizes the role of gravity as the driver of the cloud evolution, but is over-simplified as the various ways gravity can act on the gas are not differentiated.The long-range nature of gravity is also overlooked.
By analyzing the structure of the gravitational field using the tidal tensor, we reveal the multiple ways through which gravity can act on the gas.In dense regions, gravity derives collapse, and in their surroundings, gravity suppresses fragmentation through extensive tides.We find that the vast majority of the gas study is under the influence of the extensive tides -a crucial fact that has been overlooked so far.In addition, our analyses indicate that transport of gas towards the dense regions is necessary.
Our results provide crucial insights into the collapse process.In the turbulent fragmentation theory, (Padoan & Nordlund 2002;Hennebelle & Chabrier 2008;Hopkins 2012), star formation arises from the collapse of the density fluctuations created by turbulence.In the competition accretion scenario (Bonnell et al. 2001), stars in a cluster share a gas reservoir, and they acquire mass by accreting the gas competitively.In our view, turbulence can be responsible for some initial density fluctuations, but gravity quickly takes over, dictating the interactions between the high-density regions and their surroundings through tides.Accretion onto protostars can occur, but the competition is necessarily limited to those staying on the same islands.This picture where star formation occurs in a few more discrete "islands" is consistent with the picture from the Global Hierarchical Collapse scenario (Vázquez-Semadeni et al. 2009).We further discovered the compactness of these islands is the consequence of the mechanism of tidal-induced confinement, and their high densities are maintained by accretion.(Bonnell et al. 2001).The picture of stellar groups accrete gas as whole entities is reminiscent of the "collaborative accretion" scenario (Elmegreen et al. 2014).
Although tides were known since the time of Newton (Newton 1846), this is the first time where its dominant importance in controlling cloud evolution is revealed in detail.We expect the link between the density structure of the gas the role of gravity to be strengthened by future studies through which a clear view of the star formation process be established.

Figure 1 .
Figure 1.Maps of the eigenvalues of the tidal tensor  min towards the Perseus molecular cloud.Seen from the map of  min , for the majority of the volume, the tides are extensive and fragmentation is suppressed.Thin while lines:  H 2 = 3.1 × 10 3,4,5 cm −3 .Thick lines in the  max map: Boundaries of some subregions.The plots are taken at the central plane of a 3D density distribution reconstructed from observations (Appendix A).

Figure 2 .
Figure 2. Maps of the eigenvalues of the tidal tensor  max towards the Perseus molecular cloud.Thin while lines:  H 2 = 3.1 × 10 3,4,5 cm −3 .Thick lines in the  max map: Boundaries of some subregions.The plots are taken at the central plane of a 3D density distribution reconstructed from observations (Appendix A).

Figure 3 .
Figure 3. Importance of extensive tides in the B1 region in the Perseus molecular cloud.(a) A map of  min towards the B1 region.Blue stand for extensive tides ( min < 0) (b) a breakdown of the volume by the properties of tides.(c) a breakdown of mass.(d) mass spectrum of coherent regions under compressive tides.

Figure 4 .
Figure 4. Maps and distributions of  max .(a) maps of  max towards the B1 region.Thick white contour: boundary of the region.Red contour: boundary of the infalling regions which contains 20% of the mass; blue contour: boundary of the infalling region which contains 50 % of the mass.(b) mass-weighted distribution of  max .Red line:  max,crit where  acc = 0.2.Blue line:  max,crit where  acc = 0.5.