Numerics

Truelove Klein Mc Kee et al. (‘98) - AMR

​Self-Gravitational hydrodynamics with 3d AMR: methodolgy and applications to molecular cloud collapse and fragmentation.

Summary

Develops methods for AMR, dynamic grids that allot finer resolution over many length scales that is imperative for studying problems of gravitational collapse. The criteria for refinement is crucial, and it is the jeans condition. This sets the resolution smaller than the local Jean’s condition, allowing new benchmarks in the probing of the dynamics. They find uniformly rotating spherical clouds to collapse along the equatorial plane. When perturbed, these form ‘filamentary singularties’ that don’t fragment when isothermal.

Methodology

This paper goes through in extensive detail on the 3 components of their code methodology: the hyperbolic solvers that employ the Gudonov method for solution of the hydro equations, elliptic solvers that utilize AMR multigrid method to solve Poisson’s equation, and finally these two methods operation within an AMR framework. The use of stencils as different layers of cell-centered quantities used for averaging the node centered quantities (for self - gravity) is detailed. Paper refers to Almgren for discussion on AMR multigrid cycle procedures.

AMR and refinement criteria

Paper refers to Pember et al. (‘98) for a summary of the procedure for advancing grids. Their methods are based on Berger and Oliger (‘84), Berger and Collela (‘89), and Bell et al. (‘94).

To trigger refinement from level 0 to level 1, the code employs a density criterion to cells that contain gas of the original cloud. Such cells are identified if they contain a density greater than or equal to ½ the original cloud density at the outer edge. This results in the level 0 cells effectively removing the boundary of the computational volume from the surface of the cloud. Jean's condition is than used for refinement to 2nd and higher levels (discussed below).

The Truelove criteria

Jeans' analysis of the linearized 1D gravitohydrodynamic (GHD) equations of a medium of infinite extent lead to the expression for the jeans length:

\lambda_j = (\frac{\pi c_s^2}{G \rho})^{1/2}$

which showed that perturbations that are larger than this are gravitationally unstable and collapse. Truelove ('97) showed that these instabilities can be triggered from numerical errors of the GHD solvers. Errors introduced on the cell scale at coarser grids can be transmitted to finer levels and can lead to 'artificial fragmentation'. A way to avoid this is to maintain resolution of the local jeans length. By defining:

J = \frac{\triangle x}{\lambda_j} $

Truelove ('97) showed that keeping

J \leqq 0.25$

prevented artificial fragmentation of an isothermal cloud spanning 7 decades of density. It is expected to be necessary albeit not necessarily sufficient for isothermal collapse. By not having the resolution of gradients can trigger artificial viscosity (used for numerical stability), which can lead to an incorrect formulation of the problem originally deemed inviscid.

Poisson's boundary conditions

The boundary condition they employ for self-gravity is periodic. They describe this as mimicking an infinite region of repeating boxes. Thus, when they model a cloud in the box using these boundary conditions, the solver for the gravity within this box, takes into account the gravitational effects of an infinite number of other spheres in boxes. To minimize the effect of these other spheres, they suggest making the distance between the center-to-center images d=4R. By increasing the d=8R, results in less gravitational pull by the mirrored clouds that hastens the collapse by only 1%.

Fedderath et al. (‘10) - Sink Particles

​Modeling Collapse and Accretion in Turbulent Gas Clouds: Implementation and Comparison of Sink Particles in AMR and SPH

Background

The numerical difficulty with modeling the collapse of a clump, while keeping track of the entire cloud, is given by the fact that the free-fall time, Tff, where

where

Tff = (3\pi / 32G\rho)^{1/2} $

decreases with increasing density and so resolution of the subgrids is demanded over many dynamical time scales. The two methods thus far designed to deal with this matter are 'Jeans heating' and 'sink particles'. Sink particles are a more realistic methodology and was first developed for SPH by Bate et al. ('95). The algorithm was later adapted by Krumholz et al. ('04) for Eulerian AMR. This paper describes a more rigorous series of checks for sink formation.

Sink Implentation

Sink particles enable the star formation rate/star formation efficiency, and mass distribution, to be addressed in a robust and quantitative way.

Sink algorithms originated from the notion of a density threshold. In earlier work, such a threshold was defined, and once surpassed, a sink particle was placed in the grid.

The present work, however, has added in addition to this criterion, a series of checks to insure that sinks are formed only in gravitationally bound and collapsing structures. These are listed as follows:

• Converging flow
• Bound system
• Jean's unstable
• Gravitational potential minima

Else, under conditions such as shear, a sink may erroneously form.

Tests

Tests included collapse of Bonner Ebert and singular isothermal spheres.

Self Similarity Solutions for Collapse

Shu (‘77)

​Self-similar collapse of isothermal spheres and star formation

Hunter (‘77)

​The collapse of unstable isothermal spheres

Larson & Penston et al. (‘04)

[​http://adsabs.harvard.edu/abs/1969MNRAS.144..425P Dynamics of self-gravitating gaseous spheres-III. Analytical results in the free-fall of isothermal case] [

Bonnor Ebert Spheres

Banerjee & Pudritz et al. (‘04)

​The formation and evolution of protostellar discs; three-dimensional adaptive mesh refinement hydrosimulations of collapsing, rotating Bonnor-Ebert spheres

Summary

3d hydro, AMR simulations of rotating Bonnor Ebert spheres with varying degrees of initial rotation. Final collapse structures depend on the degree of rotation, with the general trend being slower rotation leads to bars/filaments, faster rotation leads to rings. Radiative cooling is incorporated into the model and a more complicated EOS.

Methods

Both high and low mass BE spheres were studied. All runs had an m=2 density perturbation. The EOS was more sophisticated than earlier works in that it was a function of density and time.

Results

No fragmentation/non-central collapse in the isothermal regime.

Bars or rings form after the 'non-isothermal core' accretes surrounding gas and forms a disc.

Temperature shocks form after the isothermal phase; as the core begins to hit up and halt the collapse, cold material from the envelope falls in and forms a shock front of temperature discontinuity. The inner shock front sets the conditions that control subsequent disc structure and dynamics.

Foster and Chevalier (‘93)

​Gravitational Collapse of an Isothermal Sphere

Motivation

While the collapse problem of an isothermal sphere is well studied, there remains some conflict on the correct analytic solution for the collapse. Larson and Penston (1969) produced a solution for the collapse problem that was later criticized by Shu (1977) as being ad hoc in some of the mathematical formulations. Later numerical studies by Hunter (1977), however, showed that the collapse does follow the LP solution. Despite this, the Shu solution remains the prominent treatment of the problem. The authors here set out to check which solution indeed describes the collapse problem.

Summary

The Shu solution begins from core formation (born out of a singular isothermal sphere) and initiates collapse into the medium through an 'expansion wave'. In contrast, the LP solution follows the collapse of a uniformly dense, static sphere prior to and up to core formation. Hunter, however, found that the LP solution can in fact be extended past core formation. The Shu solution evolves to a solution similar to homologous inflow like the LP solution.

Foster and Chevalier ran simulations of a Bonner Ebert sphere with an outer density of 1/1000 the density at the outermost edge of the clump (my simulations used 1/100). The ambient therefore had a large sound speed throughout the constant pressure medium, that remained nearly constant throughout. Their results are robustly similar to the LP solution.

Stability Tests

The 'marginally stable sphere' had a xi=xiCrit=6.451. A density enhancement of 1% above equilibrium value was needed to bring out expected stability behavior:

• xi = 3, flow oscillated about v=0 for ~ 15 free fall times
• xi = 6.451, collapse happened in 5 ff times
• xi = 20, collapse happened in 6 ff times

Collapse Results

• Inflow is initially marked by velocity proportional to r for small r, peaking at xi = 2, and turning back to 0 velocity out toward edge of sphere.

• Simulation approaches LP solution in center region.

• Velocity smoothly transitions from homologous inflow to an inflow rate of ~ -3Cs after core formation.

• Inner region evolves fast, outer evolves slow.

• Core formation is not a subsonic adjustment of density with radius as Shu said. The inner regions collapse to a core supersonically.

• Varying the boundary conditions/initial conditions did not change the behavior from tending to the LP solution. Likewise, neither did changing the perturbation conditions. The mass accretion rate, however, if it became constant, was proportional to the magnitude of the perturbation as Shu found.

• Should compare anything to mass accretion rate?

Hennebelle Whitworth Gladwin Andre (‘2002)

​Protostellar collapse by compression

The paper begins by providing some background information on different phases of the pre-stellar and protostellar phases. Definitions of class 0, 1, 2 are provided as well as their relative properties (flat inner core regions, envelope properties that differ in high density star forming regions compared to more sparse regions, infall velocity maps, etc.) and lifetimes (class 0 < class I < class II)- as inferred by statistical arguments. The observations seem to tell a coherent story of star formation that is at odds with the "standard theory of star formation" of Shu, Adams, and Lizano (1987), which describes collapse as occurring by way of the similarity solution of the SIS, predicting a prestellar cloud that is centrally peaked and static (vrad ~ 0 for all r), that collapses from the inside out in an expansion wave, and forms a central protostar that accretes surrounding material at a CONSTANT rate. However, these predictions of the SIS similarity solution are not supported by observation. While, the standard theory may be an okay 0th order approximation in quiescent, sparse regions of star formation such as Taurus, certainly more dynamic models in densely packed areas like rho-Ophiuchus are needed.

The paper then begins to set up its motivation - namely that research on flat topped isothermal spheres have begun to be pursued, such as F&C. They say F&C gets more accurate results, but that the collapse is triggered artificially. They plan to take the studies further on non-singular spheres, but by initiating collapse in a more physically relevant way (increasing Pext as suggested by Myers et al). They claim that most of the observational constraints (such as decreasing accretion rate, velocity fields, initial condition) are recovered by this model (although in my opinion - they also seem to have been recovered in F&C model too… so maybe they find MORE accurate lifetimes for their collapsing BE spheres). I find it interesting that they do not call the sphere a Bonnor Ebert sphere, but rather an isothermal sphere… is there any significance to this?

In their results section they present very weak quantitative results/explanations/insight of their plots (interestingly they spend very little time developing equations of BE sphere and also do not mention the critical values of the BE sphere such as Pcrit, Mcrit). Their different runs are broken down into the rate of compression, in how much time (measured in units of sound crossing time — ostensibly in terms of the BE sphere) does the Pext double. They initiate the collapse of the critical BE sphere by increasing Pext (note - any increase in Pext would exceed Pcrit as their initial sphere is already critical), but then they continue to increase Pext throughout the course of the simulation (this potentially is similar to what we did in practice by just allowing the simulation to evolve "naturally" with no forced increase in Pext, simply Pext at the sphere's outer edge increased naturally by the infall of material onto that outer edge). In the first couple of cases they show, compression is slow. In the first 3 panels of Fig. 1, a sink has not yet formed, and so they identify this phase as the pre-stellar (aka the pre-protostellar) phase. It appears to me that they get a nice outside in flow beginning to develop in these plots, although the radius of the BE sphere is unclear. No compression wave seems evident in these plots, but instead, it seems to be a re-equilibration of matter in terms of the language I use in my paper. This could be due to early subsonic adjustment of the material into a modified BE profile that exceeds the critical mass, and hence collapses in the canonical fashion. They use no such language, and make no such identification which is curious. They describe this, as well as all other cases, as being a compression wave solution. This might make sense if you follow Whitworth language that all collapse problems are due to compression waves, but some of the waves have 0 amplitude..? This case looks most like some of our lighter ambient runs, where the Pext increases very slowly due to a slow accumulation of matter.. In the next 2 panes of the plot, a central sink has formed that accretes matter in a free-fall manner (vrad~r^½) that they say moves outward in time. I can't tell from this plot that this type of flow is moving outward in time. They talk about the mean cruising velocity (Whitworth language) for the different runs, but it is unclear if this is an average over the entire sphere at a given time, or if it is at a given radius over time, or what.. They say that the mean cruising velocity is set up by the compression wave, but how do they know when it penetrated the sphere if they can't even see it in the density plot? They do say that in this case the compression wave converges on the center, at which time the sink forms, so maybe in this limiting case of the Whitworth solution, the compression wave looks like the classic result?

Skipping to figure 3, for which the Pext doubles in a crossing time (aka strong compression). I have a hard time understanding their description in the text, p 874, right column, 2nd paragraph. They say a small compression wave moves in (I see this in plot), but they go on to say that before the wave converges on the center to form the sink (and class 0 phase - last 2 panes in figure), the central density has hardly changed "for the inner regions were unaware of Pext increasing". First of all, in the plot the central density is clearing increasing in panes 1-3. Second of all, they would be unaware only for a supersonic compression wave — for which they did not establish exists. They say again that around the sink, a free fall v field is set up, but in the outer regions a uniform sonic field is established (v~0.12 km/s). Again, I do not see a uniform v field in the outer regions.. I assume what they mean is uniform in r… not for a fixed r over time.. This plot does however, seem to show a change in the velocity field from the early "compression wave phase" to the later "classic phase", in the language of my paper. Although I think these plots overall do a poor job showing the evolution of the flow since they only sample 3 time states before sink formation, they don't capture the dynamics very well.

In Figure 4 for even stronger compression, the compression wave is now evident in the plots. I see in their velocity plots a turn-over similar to wee see for the different phases of evolution. The last plot they show before the sink forms, seems to be indicating a compression wave is about to "converge on the center". However, we also saw this type of evolution, but between this time and sink formation in our runs, the density profile evened out to a collapsing BE profile. I wonder if Hennebelle et al found similar results, and if so, why they did not choose to report it… This case seems most like our Matched case.

In Figure 5, the compression wave seems to completely rip through the BE sphere, although no mention of the speed of the incoming compression wave is given (which would have been interesting to compare to the triggered star formation stuff Shule has been looking at, which for mach > 20 has been shown to disrupt the BE sphere from being able to form a sink). It seems like the compression wave keeps building mass because of the abrupt increasing change in Pext that adds density to the wave in an isothermal fashion.

The results section concludes with the authors saying that when Pext is increased but then kept steady the results are different, citing figure 6. They do not ellaborate, so the reader is left to guess at what features they may be referring to. Looking at these plots, I am curious why the turn-up in vrad at large radii… We also saw a turn up in vrad in some cases…

Molecular Clouds

Turbulence