Sink particle accretion test

Transcription

1 Sink particle accretion test David A. Hubber & Stefanie Walch 1 Objectives Simulate spherically-symmetric Bondi accretion onto a sink particle for an isothermal gas. Calculate the accretion rate onto a sink particle for cases where the sink radius is greater than, and less than, the sonic radius. Determine how quickly steady state accretion can be achieved, if at all. For steady-state accretion, compare the accretion rate to the expected Bondi accretion rate for all sink sizes. 2 Introduction Sink particles (Bate et al., 1995) have become a vital algorithmic tool for simulators when performing gravitational collapse problems such as star and planet formation. In the simplest case, sink particles model accretion onto a protostar by defining an accretion radius (often simply called the sink radius) where any matter that falls within the sink radius that is bound, is instantaneously accreted to the sink by adding its mass, momentum and angular momentum to that of the sink. However, there are differences between the various implementations of sink particles in both Smoothed Particle Hydrodynamics (SPH) and mesh codes. SPH codes discretise the fluid into discrete masses. Therefore, the simplest approach to modelling accretion is to simply accrete whole SPH particles when they enter the sink radius. Various accretion criteria can be employed (e.g. Bate et al., 1995) to control how particles accrete. However, this method usually results in a void of particles inside the sink leading to inaccurate hydrodynamical forces near the sink. Hubber et al. (2013) attempted to improve the situation by allowing partial accretion of an SPH particle. Instead of accreting the whole particle, a simple sub-grid accretion model was developed using the two limiting cases of purely spherical accretion, and rotational (i.e. disc) accretion and interpolating between the two cases. Mesh codes use a different approach to sink accretion. In the original mesh sink implementation of Krumholz et al. (2004), they calculate any excess mass inside the sink that would lead to the gravitational instability and then accrete only that excess to the sink. Therefore there is always a non-zero density of gas inside the sink radius to ensure continuity of the density field. Other mesh codes (e.g. Federrath et al., 2010) are based on this approach, although contain some variations. 1

2 2.1 Aim of the test Although simple to implement, sink particles can potentially introduce large errors into simulations if their effects are not treated properly. The biggest problem with sink particles is that removing gas from a simulation also removes its hydrostatic support from the surrounding gas. Instead of modelling a contracting protostar with an inwards pressure gradient (and therefore outwards pressure force), simulators could be modelling a density trough and hence an outwards pressure gradient (and therefore inwards pressure force) which could lead to an artificially enhanced accretion rate. For sink particle prescriptions that do not accrete all gas inside them and use some other presciption for removing gas, then the opposite effect is also possible, i.e. too little gas is removed leading to artificially lower accretion rates. The main aim of this test is to determine (i) if sink particles introduce any unwanted side effects, and (ii) if the effect is to artificially increase or decrease the accretion rate. Since disc accretion is highly dependent on the effective numerical viscosity, we model simple spherically symmetric accretion following Bondi (1952). This will help determine if various sink particle implementations are able to consistently model simplistic steady-state accretion. More sophisticated tests modelling disc accretion may be developed in the future. 3 Test The Bondi accretion problem is described in many articles and text books, so we only summarise the main practical results here which are important in setting up the test. Bondi accretion models the steady-state spherically-symmetric accretion of a non-self gravitating gas onto a central star of mass M from a uniform background gas of density ρ 0. For simplicity, we will model an isothermal gas of pure hydrogen with (dimensionless) temperature T = 1 and dimensionless isothermal sound speed a = 1. The accretion rate onto the star is given by Ṁ = e3/2 πg 2 M 2 ρ 0 a 3, (1) where is the constant e = At large distances from the central star, the gas moves subsonically; near to the star, the gas moves super-sonically such that the gas is effectively in freefall towards the star. The transition point between subsonic and supersonic is called the sonic radius, R SONIC = GM 2a 2. (2) Hubber et al. (2013) found that if the sink particle size, R S, was smaller than the sonic radius (R S R SONIC ), then the lack of hydrodynamical forces was not important and accretion was correctly modelled. If the sink particle was larger than the sonic radius (R S R SONIC ), then any problems with the accretion are apparent and the accretion rate may be incorrectly modelled. Figure 1 shows the behaviour of the original sink particle implementation (black dashed line) and the newer improved sinks (red solid line) for SPH. The computational domain will be of size 8 (in dimensionless units). This can be either a spherical region of radius 8 (if spherical coordinates are used) or a cubic region of range 8 < x < +8 in each dimension. 2

3 Figure 1: Accretion rates for isothermal Bondi accretion onto a sink particle for various sink particle sizes using the Hubber, Walch & Whitworth (2013) sink particle algorithm. The black dashed line shows the case for simple accretion (where particles are accreted immediately as they enter the sink); in this case, accretion is accurate for supersonic infall (i.e. R < R SONIC ) but breaks down for subsonic infall (i.e. R > R SONIC ) where the inaccurate hydrodynamical forces cause large errors. For the new accretion model, the accretion rates are accurate for all sink radii, both for supersonic and subsonic infall. We place a sink particle of mass m SINK at the origin which has a sink radius r SINK. The sink radius will take values either side of the sonic radius. Following Hubber et al. (2013), we take the values R S /R SONIC = 1/8, 1/4, 1/2, 1, 2, 4.8. We will then measure the steady-state accretion rate for each sink particle implementation. We note that the sink radius will be the only variable quantity in this test. The simulation run-time will be measured in units of the quantity t 0 = R SONIC /a which is the time for a sound-wave to cross the distance of a single sonic radius. In order to ensure a steady-state is achieved for each simulation, we will run the simulation for t END = 4t 0. In our dimensionless units, R SONIC = 1/2; therefore t 0 = 1/2 so t = 1/10 and t END = Bondi solution The main initial conditions of the test are a spherically symmetric density and velocity field setup to model the steady-state trans-sonic solution of the Bondi accretion problem (Bondi, 1952). The Bondi equations were solved numerically and outputted to a file BONDI SOLUTION.dat which is available on the Starbench web-page. The file is a simple four-column data file with the columns recording (1) radial position, (2) integrated mass (i.e. all mass between origin and current radius), (3) gas density, and (4) inward velocity (N.B. no negative sign in table). Each solution is in dimensionless form with the radial position being in units of the sonic radius, 3

4 R SONIC, masses in units of 4πρ 0 R 3 SONIC /3, densities in units of the background density, ρ 0, and velocities in units of the isothermal sound speed, a. 3.2 Additional caveats The following caveats must be considered when preparing and running the simulations The gas is not self-gravitating and therefore only gravity due to the sink itself should be considered. The gravitational mass of the sink must not change, even when accreting more mass. This is to ensure the validity of the steady-state solution. This might require the user to modify some lines of code in the respective accretion or gravitational routines. Due to inaccuracies in the hydrodynamics method and the initial accretion of mass at the start of the simulation, it may take a short amount of time for the simulation to settle into a steady-state. This time will be of order the sound-crossing time of the sink, R S /a. Therefore the simulation must be run long enough to allow the numerical steady-state to be achieved. 3.3 SPH setup For the SPH simulations, we have prepared files containing particle positions, masses, velocities and internal energies in order to simplify the process of setting up initial conditions. The files were created by first creating a closed hexagonal-packed array of particles, ignoring all particles greater than unity distance from the origin, hence creating a unit sphere. This lattice is then stretched radially in order to reproduce the density distribution from the tabulated numerical solution. For authors who wish to set-up their own initial conditions, or who are required to set-up them up internally in their own code, then the file BONDI SOLUTION.dat contains the numerical solution. Since the SPH particle distribution is finite, the outer edge of the particle distribution will start to leak away due to a rarefaction wave that propagates inwards. Since the gas is isothermal and moving subsonically at large radii, we can estimate the time it takes for the wave to reach the central sink and therefore ensure the simulation time is short enough to prevent this artifact from corrupting the solution. 3.4 AMR setup For setting-up the mesh simulations, the properties of each cell of the density and velocity should be interpolated from the BONDI SOLUTION.dat file that is provided. Note that the radial positions are logarithmically spaced and therefore must be accounted for when interpolating the table properties to the mesh initial conditions. If the mesh code allows continuous/free-flowing boundary conditions, then these should be employed to minimise any boundary effects in the code. 4

5 For the resolution, it is advised to properly resolve the sonic radius, and/or the sink radius, either with the full grid resolution (if using a uniform grid) or with the use of AMR. To resolve the smallest sink particle (R S = 1 8 R SONIC = 1/16) in a region of size 16 across requires an effective resolution of at least If the mesh code allows mirror symmetry such that only one octant needs to be simulated, then this can reduce the run-time and memory constraints. 4 Output Format Each simulation should generate three separate output files. The first should record the mass of the central sink at regular time intervals t = 1 5 t 0 = 1/10 until the simulation end time (t END = 2). This output filename should have the form SINKACC RSXXX CODENAME.dat. RSXXX is the sink radius (e.g. RS0.25) For mesh simulations, the resolution should also be appended to the filename description (e.g. SINKACC RS0.125 RES256 CODENAME.dat). The format of these files should be a simple 2-column format, the first column with the time and the second column with the mass of the sink. The second and third files should contain the radial profiles of the gas properties at the beginning and end of the simulation, i.e. either grid cells or SPH particles. These files should contain the radial positions, densities and radial velocity component (i.e. v ˆr) for each gas element in a simple 3-column format. The filenames should have the form BONDI TFIRST RSXXX CODENAME.dat and BONDI TLAST RSXXX CODENAME.dat for the first and last snapshots respectively. For mesh codes, this should also include the resolution as described above. References Bate, M. R., Bonnell, I. A., and Price, N. M. (1995). Modelling accretion in protobinary systems. MNRAS, 277: Bondi, H. (1952). On spherically symmetrical accretion. MNRAS, 112: Federrath, C., Banerjee, R., Clark, P. C., and Klessen, R. S. (2010). Modeling Collapse and Accretion in Turbulent Gas Clouds: Implementation and Comparison of Sink Particles in AMR and SPH. ApJ, 713: Hubber, D. A., Walch, S., and Whitworth, A. P. (2013). An improved sink particle algorithm for SPH simulations. MNRAS, 430: Krumholz, M. R., McKee, C. F., and Klein, R. I. (2004). Embedding Lagrangian Sink Particles in Eulerian Grids. ApJ, 611: