Mon. Not. R. Astron. Soc. 425, 1104—1120 (2012)
In self-consistent N-body simulations of collisionless systems, gravitational interactions are modified on small scales to remove singularities and simplify the task of numerically integrating the equations of motion. This `gravitational softening' is sometimes presented as an ad-hoc departure from Newtonian gravity. However, softening can also be described as a smoothing operation applied to the mass distribution; the gravitational potential and the smoothed density obey Poisson's equation precisely. While `softening' and `smoothing' are mathematically equivalent descriptions, the latter has some advantages. For example, the smoothing description suggests a way to set up N-body initial conditions in almost perfect dynamical equilibrium.
Compare with Fig. 12.
Results for self-consistent and smoothed DFs are shown on left and right, respectively. In each panel, lowest curve shows earliest time.
Various panels show results of varying Δt, θ, and N.
listModel.math: Mathematica source code to calculate smoothed gamma and NFW models. Both standard and tapered models are supported. The key routines to generate table files are listGammaModel[] and listNFWModel[].
Each table file begins with a comment line showing the procedure call used in Mathematica to create the file. The next line provides headers for the columns, as follows:
Standard models do not include tapering — the outer power-law continues to the last radius listed.
Model | ε = 1 | ε = 1/2 | ε = 1/4 | ε = 1/8 | ε = 1/16 | ε = 1/32 | ε = 1/64 | ε = 1/128 | ε = 1/256 |
Hernquist | modH_001 | modH_002 | modH_004 | modH_008 | modH_016 | modH_032 | modH_064 | modH_128 | modH_256 |
Jaffe | modJ_001 | modJ_002 | modJ_004 | modJ_008 | modJ_016 | modJ_032 | modJ_064 | modJ_128 | modJ_256 |
NFW | modNFW_001 | modNFW_002 | modNFW_004 | modNFW_008 | modNFW_016 | modNFW_032 | modNFW_064 | modNFW_128 | modNFW_256 |
Tapered gamma models follow the outer power law out to radius b = 100; thereafter the underlying density drops fairly rapidly.
Model | ε = 1/16 | ε = 1/64 | ε = 1/256 |
Hernquist | modHt_016 | modHt_064 | modHt_256 |
Jaffe | modJt_016 | modJt_064 | modJt_256 |
Tapered NFW models follow the standard NFW profile out to radius b, then cut off using a fast exponential taper (Springel & White 1998).
ε = 1/16 | ε = 1/64 | ε = 1/256 | |
b = 5 | modNFW_05_016 | modNFW_05_064 | modNFW_05_256 |
b = 10 | modNFW_10_016 | modNFW_10_064 | modNFW_10_256 |
b = 15 | modNFW_15_016 | modNFW_15_064 | modNFW_15_256 |
Joshua E. Barnes
(barnes at ifa.hawaii.edu)
Updated:
27 August 2012
http://www.ifa.hawaii.edu/~barnes/research/smoothing/index.html |