[Paper Review] N-MODY: a code for collisionless N-body simulations in modified Newtonian dynamics

P. Londrillo, Carlo Nipoti|ArXiv.org|Mar 31, 2008

Scientific Research and Discoveries3 citations

TL;DR

N-MODY is a parallel particle-mesh N-body code in spherical coordinates for simulating collisionless systems under Modified Newtonian Dynamics (MOND), solving the non-linear MOND field equation via a relaxation method with spherical harmonics. It reveals that dynamical processes like violent relaxation, merging, and dynamical friction differ significantly in MOND compared to Newtonian gravity with dark matter, with longer timescales for relaxation and merging and stronger frictional forces.

ABSTRACT

We describe the numerical code N-MODY, a parallel particle-mesh code for collisionless N-body simulations in modified Newtonian dynamics (MOND). N-MODY is based on a numerical potential solver in spherical coordinates that solves the non-linear MOND field equation, and is ideally suited to simulate isolated stellar systems. N-MODY can be used also to compute the MOND potential of arbitrary static density distributions. A few applications of N-MODY indicate that some astrophysically relevant dynamical processes are profoundly different in MOND and in Newtonian gravity with dark matter.

Motivation & Objective

• To develop a numerical code for collisionless N-body simulations in Modified Newtonian Dynamics (MOND), an alternative to dark matter in explaining galactic kinematics.
• To solve the non-linear MOND field equation efficiently in spherical coordinates using a relaxation method with spherical harmonics expansion.
• To enable simulations of isolated stellar systems and arbitrary static density distributions under MOND, providing a tool for testing MOND against Newtonian dynamics.
• To investigate how fundamental dynamical processes—such as violent relaxation, phase mixing, merging, and dynamical friction—behave differently in MOND compared to Newtonian gravity with dark matter.
• To deliver a publicly available, parallel FORTRAN 90 code that supports both serial and MPI-parallel execution for high-performance computing.

Proposed method

• N-MODY uses a particle-mesh scheme in spherical coordinates, reconstructing the density field via linear or quadratic shape functions on a grid.
• The MOND potential is computed on the grid by solving the non-linear field equation ∇·[μ(|∇ϕ|/a₀)∇ϕ] = 4πGρ using a relaxation method with spherical harmonics expansion.
• The code employs a relaxation parameter (ω⁻¹ ≈ 0.3–0.5) and convergence tolerance (ε = 10⁻³) to achieve numerical solutions with ∥M̂(ϕ)∥∞ ≈ 0.1 and ∥M̂(ϕ)∥rms ≈ 0.01.
• Particle accelerations are interpolated from the grid using the same shape functions and transformed from spherical to Cartesian components for time integration.
• Time integration is performed using a second- or fourth-order leapfrog scheme, ensuring energy and momentum conservation in simulations.
• The code is parallelized via MPI, with particles uniformly distributed among processors and domain decomposition applied to the θ-coordinate for partial parallelization of the potential solver.

Experimental results

Research questions

• RQ1 How do violent relaxation and phase mixing timescales differ in MOND compared to Newtonian gravity with dark matter?
• RQ2 What is the efficiency of dynamical friction in MOND systems relative to Newtonian systems with dark matter?
• RQ3 How does the merging timescale of galaxies differ in MOND versus Newtonian gravity?
• RQ4 To what extent does the non-linear MOND field equation alter the structure and evolution of isolated stellar systems compared to Newtonian potentials?
• RQ5 Can a spherical-coordinate-based particle-mesh code efficiently and accurately solve the MOND field equation for astrophysically relevant systems?

Key findings

• The MOND potential solver in N-MODY converges in 5–10 iterations with a tolerance of ε = 10⁻³, achieving numerical accuracy with ∥M̂(ϕ)∥∞ ≈ 0.1 and ∥M̂(ϕ)∥rms ≈ 0.01.
• In MOND simulations, the potential solver consumes ~10% of the total CPU time in serial mode when N ≈ 10Ng, but becomes a major bottleneck in parallel runs due to limited parallelization.
• Parallelization efficiency for the MOND solver is ~70%, with the computational cost scaling roughly with the number of grid points Ng = (Nr+1)NθNφ.
• The total CPU time per particle per timestep in N-MODY remains below 10⁻⁵ s for N = 10⁷ particles and Ng = 128×64×128, even with 16 processors.
• Violent relaxation, phase mixing, and galaxy merging occur significantly more slowly in MOND than in Newtonian gravity, indicating a suppression of dynamical processes.
• Dynamical friction is stronger in MOND systems than in equivalent Newtonian systems with dark matter, suggesting enhanced orbital decay rates for massive objects in MOND.