Non-relativistic Mergers

April 2026 · Fluid Dynamics &Gravitation

One of the most striking objects in the universe is neutron stars: they are the densest objects in the universe, since anything denser turns into a black hole.

If two neutron stars end up close to each other, they can merge together turning into either a black hole or a new, more massive, neutron star. Such events are remarkable because they produce a vast amount of electromagnetic radiation and gravitational waves that can be observed. Since we have only been able to observe gravitational waves for a few years, there is currently only one main event that led to precise new measurements: GW170817.

A complete simulation of such events requires a complex solution to general relativity. As a first approximation to this type of events, one can perform a non-relativistic simulation. The end results for two self-bounded stars then look something like below.

There, two planets are initialized with a finite velocity that causes them to orbit each other. Then, they quickly get attracted to and deformed by each other, ultimately merging in a single star.

Euler equations

The main equations to solve are the compressible self-gravitating Euler equations of fluid dynamics, which describe the behavior of a compressible fluid neglecting any friction. They express the local conservation of mass, momentum and energy of a fluid subjected to its own gravity. Introducing the space and time dependent mass density $\rho(t,\vec{x})$ , velocity $\vec{v}(t,\vec{x})$ and total energy $E(t,\vec{x})$ of the fluid, they read

\begin{align} \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) & = 0 \\ \frac{\partial \rho \vec{v}}{\partial t} + \nabla \cdot (\vec{v} \rho \vec{v}) + \nabla p & = - \rho \nabla \phi \\ \frac{\partial E}{\partial t} + \nabla \cdot \left[ (E + p) \vec{v} \right] & = - \rho \vec{v} \cdot \nabla \phi \end{align}

where we also introduced the pressure field $p(t,\vec{x})$ . The total energy $E$ is the sum of internal and kinetic energy

E = \rho e + \frac{1}{2} \rho \vec{v}^2

Self-gravitation is realized through the gravitational potential $\phi(t,\vec{x})$ in $(2)$ and $(3)$ , which is found as a solution of the Poisson equation

\begin{equation} \nabla^2 \phi = 4\pi G \rho \end{equation}

with Newton’s gravitation constant $G$ . As a consequence, lower density parts of the fluid are attracted to higher density ones.

To close the system $(1)$ to $(4)$ , we need to provide a relation between the pressure $p$ and the internal energy $e$ . Such a relation defines the equation of state of the system, and implement the microscopic details of the fluid we want to simulate. Here I choose

p = \rho e (\gamma - 1)

with $\gamma=5/3$ . This corresponds to an ideal fluid and is good enough to obtain qualitatively correct behavior of fluids. In a more realistic treatment, the equation of state has to be computed from a microscopic model of the star interior.

More pictures

The Euler equations involves more than just the density. Here are the different quantities for the merger simulation shown at the beginning.

Head-on collision

This is how the results look for two stars that collide head-on with a strong velocity towards each other. This shows nicely how the matter gets ejected by the strong pressure produced during the collision.

The Julia code I used to solve the Euler equations and produce the videos is accessible at this github repository.