Mon. Not. R. Astron. Soc. 425, 1104—1120 (2012)

Gravitational softening as a smoothing operation

Joshua E. Barnes

Institute for Astronomy, University of Hawaii,
2680 Woodlawn Drive, Honolulu, Hawaii 96822

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.


Supplemental Figures

Mathematica Code

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[].

Table Files

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:

  1. Radius r
  2. Underlying density ρ(r)
  3. Enclosed mass M(r)
  4. Smoothed density ρ(r;ε)
  5. Approximate density ρ(rε)

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
Updated: 27 August 2012
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