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Direct integration of the Newtonian and Post-Newtonian equations of motion of the gravitational N-body problem

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# pn-nbody
Software for the direct integration of the Newtonian and post-Newtonian equations of motion of the gravitational N-body problem (for more information, see [Heinze, Schäfer & Brügmann 2026](https://arxiv.org/abs/2602.06961)).

## Features
* Direct integration of nonspinning post-Newtonian N-body dynamics through conservative 2PN order, including the leading dissipative 2.5PN radiation-reaction terms
* Complete general-N 2PN dynamics, including optional numerical evaluation of the four-body $U_{\mathrm{TT}}^{(4)}$ contribution for $N \geq 4$
* Multiple integration methods: the adaptive Cash–Karp embedded Runge–Kutta RK5(4) method, the second-order symmetric and symplectic implicit midpoint method, and the classical fourth-order Runge–Kutta method
* Configurable initial-condition presets for binaries, hierarchical triples, binary–single and binary–binary scattering, figure-eight configurations, Newtonian Plummer clusters, and relativistic monoenergetic clusters
* Automatic merger handling with active-body bookkeeping, remnant-generation tracking, and merger-event output
* NR-calibrated analytic prescriptions for the remnant mass, spin, and recoil kick, including Lousto–Zlochower and Barausse-based options
* Optional impulse splitting to reduce the number of expensive $U_{\mathrm{TT}}^{(4)}$ evaluations
* Python tools for static trajectory plots and two- or three-dimensional animations

To our knowledge, `pn-nbody` is the first publicly available code to implement the complete general-$N$ dynamics through 2PN order, including the four-body $U_{\mathrm{TT}}^{(4)}$ contribution.

## Todo
* Performance optimizations (parallelization, approximations for objects further away)
* Regularization / handle close encounters
* Additional ODE integration methods (e.g. higher-order symplectic methods, Hermite integrators)
* Spins and spin evolution

## Requirements
* C compiler (e.g. gcc/clang)
* Linux/macOS
* Optional: Python with numpy and matplotlib (for visualizations)
* Optional: Cuba 4.2.2 (If you want to solve the full 2PN equations of motion for N bodies, with N > 3, it is necessary to compute the four-point correlation function UTT4, see e.g. [Heinze, Schäfer & Brügmann 2026](https://arxiv.org/abs/2602.06961). UTT4 contains an integral that can currently only be solved numerically, which is computationally very expensive. The numerical evaluation requires the [Cuba library](https://feynarts.de/cuba/). If you want to do that, download and install it. You can compile and run the program also without the Cuba library, but then you cannot include UTT4.)

## How To Install
### Option 1: Build without the Cuba library
```bash
git clone https://github.com/fmheinze/pn-nbody.git
cd pn-nbody
make
```

### Option 2: Build with the Cuba library
```bash
git clone https://github.com/fmheinze/pn-nbody.git
cd pn-nbody
make CUBA_DIR=/path/to/Cuba
```

## Quick Start Guide
1. After building the executable, you should be able to run the program. You can try one of the examples provided in the `./test` directory, e.g.:
```
./exe/pn-nbody ./test/test_figure_eight.par
```

2. After the simulation has finished, the output should be located in the `./output` directory.  You can visualize the output with the included python scripts in the `./viz` directory, e.g.:
```
python3 ./viz/trajectory_anim.py ./output/test_figure_eight/output_pos.dat
```

3. You can check out the other files in the `./test` directory to see more examples of what is possible.

## License

This project is licensed under the MIT License. See the LICENSE file for details.

pn-nbody

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