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.

Preprints

Single-spaced with low-resolution figures.
Single-spaced with full-resolution figures.
Double-spaced with full-resolution figures.
http://arxiv.org/abs/1205.2729: astro-ph page.

Supplemental Figures

Density profiles of Hernquist models
Compare with Fig. 12.
Evolution of Jaffe profiles
Results for self-consistent and smoothed DFs are shown on left and right, respectively. In each panel, lowest curve shows earliest time.
Integration parameter tests
Various panels show results of varying Δt, θ, and N.

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[].

Update: 19 June 2012 — fixed error in massNFW[] (changed a^2 to a^3); took Re[] part of rhoGammaTaperSoft2[] and rhoNFWTaperSoft2[] to suppress occasional complex results.

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:

Radius r
Underlying density ρ(r)
Enclosed mass M(r)
Smoothed density ρ(r;ε)
Approximate density ρ(r_ε)

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