Inphysics andastronomy, anN-body simulation is a simulation of adynamical system of particles, usually under the influence of physical forces, such asgravity (seen-body problem for other applications).N-body simulations are widely used tools inastrophysics, from investigating the dynamics offew-body systems like theEarth-Moon-Sun system to understanding the evolution of thelarge-scale structure of the universe.^[1] Inphysical cosmology,N-body simulations are used to study processes of non-linearstructure formation such asgalaxy filaments andgalaxy halos from the influence ofdark matter. DirectN-body simulations are used to study the dynamical evolution ofstar clusters.

The 'particles' treated by the simulation may or may not correspond to physical objects which are particulate in nature. For example, an N-body simulation of a star cluster might have a particle per star, so each particle has some physical significance. On the other hand, a simulation of agas cloud cannot afford to have a particle for each atom or molecule of gas as this would require on the order of10²³ particles for each mole of material (seeAvogadro constant), so a single 'particle' would represent some much larger quantity of gas (often implemented usingSmoothed Particle Hydrodynamics). This quantity need not have any physical significance, but must be chosen as a compromise between accuracy and manageable computer requirements.

Dark matter plays an important role in the formation of galaxies. The time evolution of the density f (in phase space) of dark matter particles, can be described by the collisionlessBoltzmann equation

$${\displaystyle \frac{df}{dt}=\frac{\mathrm{\partial }f}{\mathrm{\partial }t}+\mathbf{v}\cdot \mathrm{\nabla }f-\frac{\mathrm{\partial }f}{\mathrm{\partial }\mathbf{v}}\cdot \mathrm{\nabla }\mathrm{\Phi }}$$

In the equation,${\displaystyle \mathbf{v}}$ is the velocity, and Φ is the gravitational potential given byPoisson's Equation. These two coupled equations are solved in an expanding background Universe, which is governed by theFriedmann equations, after determining the initial conditions of dark matter particles. The conventional method employed for initializing positions and velocities of dark matter particles involves moving particles within a uniform Cartesian lattice or a glass-like particle configuration.^[2] This is done by using a linear theory approximation or a low-orderperturbation theory.^[3]

In direct gravitationalN-body simulations, the equations of motion of a system ofN particles under the influence of their mutual gravitational forces are integrated numerically without any simplifying approximations. These calculations are used in situations where interactions between individual objects, such as stars or planets, are important to the evolution of the system.

The first direct gravitationalN-body simulations were carried out byErik Holmberg at theLund Observatory in 1941, determining the forces between stars in encountering galaxies via the mathematical equivalence between light propagation and gravitational interaction: putting light bulbs at the positions of the stars and measuring the directional light fluxes at the positions of the stars by a photo cell, the equations of motion can be integrated with⁠${\displaystyle O(N)}$⁠ effort.^[4] The first purely calculational simulations were then done bySebastian von Hoerner at theAstronomisches Rechen-Institut inHeidelberg, Germany.Sverre Aarseth at theUniversity of Cambridge (UK) has dedicated his entire scientific life to the development of a series of highly efficientN-body codes for astrophysical applications which use adaptive (hierarchical) time steps, an Ahmad-Cohen neighbour scheme and regularization of close encounters. Regularization is a mathematical trick to remove the singularity in the Newtonian law of gravitation for two particles which approach each other arbitrarily close. Sverre Aarseth's codes are used to study the dynamics of star clusters, planetary systems and galactic nuclei.^{[citation needed]}

Many simulations are large enough that the effects ofgeneral relativity in establishing aFriedmann-Lemaitre-Robertson-Walker cosmology are significant. This is incorporated in the simulation as an evolving measure of distance (orscale factor) in acomoving coordinate system, which causes the particles to slow in comoving coordinates (as well as due to theredshifting of their physical energy). However, the contributions of general relativity and the finitespeed of gravity can otherwise be ignored, as typical dynamical timescales are long compared to the light crossing time for the simulation, and the space-time curvature induced by the particles and the particle velocities are small. The boundary conditions of these cosmological simulations are usually periodic (or toroidal), so that one edge of the simulation volume matches up with the opposite edge.

N-body simulations are simple in principle, because they involve merely integrating the 6Nordinary differential equations defining the particle motions inNewtonian gravity. In practice, the numberN of particles involved is usually very large (typical simulations include many millions, theMillennium simulation included ten billion) and the number of particle-particle interactions needing to be computed increases on the order ofN², and so direct integration of the differential equations can be prohibitively computationally expensive. Therefore, a number of refinements are commonly used.

Numerical integration is usually performed over small timesteps using a method such asleapfrog integration. However all numerical integration leads to errors. Smaller steps give lower errors but run more slowly. Leapfrog integration is roughly 2nd order on the timestep, other integrators such asRunge–Kutta methods can have 4th order accuracy or much higher.

One of the simplest refinements is that each particle carries with it its own timestep variable, so that particles with widely different dynamical times don't all have to be evolved forward at the rate of that with the shortest time.

There are two basic approximation schemes to decrease the computational time for such simulations. These can reduce thecomputational complexity to O(N log N) or better, at the loss of accuracy.

Intree methods, such as aBarnes–Hut simulation, anoctree is usually used to divide the volume into cubic cells and only interactions between particles from nearby cells need to be treated individually; particles in distant cells can be treated collectively as a single large particle centered at the distant cell's center of mass (or as a low-ordermultipole expansion). This can dramatically reduce the number of particle pair interactions that must be computed. To prevent the simulation from becoming swamped by computing particle-particle interactions, the cells must be refined to smaller cells in denser parts of the simulation which contain many particles per cell. For simulations where particles are not evenly distributed, thewell-separated pair decomposition methods of Callahan andKosaraju yield optimal O(n log n) time per iteration with fixed dimension.

Another possibility is theparticle mesh method in which space is discretised on a mesh and, for the purposes of computing thegravitational potential, particles are assumed to be divided between the surrounding 2x2 vertices of the mesh. The potential energy Φ can be found with thePoisson equation

$${\displaystyle {\mathrm{\nabla }}^2\mathrm{\Phi }=4\pi G\rho ,}$$

whereG isNewton's constant and${\displaystyle \rho }$ is the density (number of particles at the mesh points). Thefast Fourier transform can solve this efficiently by going to thefrequency domain where the Poisson equation has the simple form

$${\displaystyle \hat{\mathrm{\Phi }}=-4\pi G\frac{\hat{\rho }}{k^2},}$$

where${\displaystyle \overrightarrow{k}}$ is the comoving wavenumber and the hats denote Fourier transforms. Since${\displaystyle \overrightarrow{g}=-\overrightarrow{\mathrm{\nabla }}\mathrm{\Phi }}$, the gravitational field can now be found by multiplying by${\displaystyle -i\overrightarrow{k}}$ and computing the inverse Fourier transform (or computing the inverse transform and then using some other method). Since this method is limited by the mesh size, in practice a smaller mesh or some other technique (such as combining with a tree or simple particle-particle algorithm) is used to compute the small-scale forces. Sometimes an adaptive mesh is used, in which the mesh cells are much smaller in the denser regions of the simulation.

Several differentgravitational perturbation algorithms are used to get fairly accurate estimates of the path of objects in theSolar System.

People often decide to put a satellite in afrozen orbit.The path of a satellite closely orbiting the Earth can be accurately modeled starting from the 2-body elliptical orbit around the center of the Earth, and adding small corrections due to theoblateness of the Earth, gravitational attraction of the Sun and Moon, atmospheric drag, etc.It is possible to find a frozen orbit without calculating the actual path of the satellite.

The path of a small planet, comet, or long-range spacecraft can often be accurately modeled starting from the 2-body elliptical orbit around the Sun, and adding small corrections from the gravitational attraction of the larger planets in their known orbits.

Some characteristics of the long-term paths of a system of particles can be calculated directly. The actual path of any particular particle does not need to be calculated as an intermediate step. Such characteristics includeLyapunov stability,Lyapunov time, various measurements fromergodic theory, etc.

Although there are millions or billions of particles in typical simulations, they typically correspond to a real particle with a very large mass, typically 10⁹solar masses. This can introduce problems with short-range interactions between the particles such as the formation of two-particlebinary systems. As the particles are meant to represent large numbers of dark matter particles or groups of stars, these binaries are unphysical. To prevent this, asoftened Newtonian force law is used, which does not diverge as the inverse-square radius at short distances. Most simulations implement this quite naturally by running the simulations on cells of finite size. It is important to implement the discretization procedure in such a way that particles always exert a vanishing force on themselves.

Softening is a numerical trick used in N-body techniques to prevent numericaldivergences when a particle comes too close to another (and theforce goes to infinity). This is obtained by modifying the regularizedgravitational potential of each particle as

$${\displaystyle \mathrm{\Phi }=-\frac{1}{\sqrt{r^2+{ϵ}^2}},}$$

(rather than 1/r) where${\displaystyle ϵ}$ is the softening parameter. The value of the softening parameter should be set small enough to keepsimulations realistic.

N-body simulations give findings on the large-scale dark matter distribution and the structure of dark matter halos. According to simulations of cold dark matter, the overall distribution of dark matter on a large scale is not entirely uniform. Instead, it displays a structure resembling a network, consisting of voids, walls, filaments, and halos. Also, simulations show that the relationship between the concentration of halos and factors such as mass, initial fluctuation spectrum, and cosmological parameters is linked to the actual formation time of the halos.^[5] In particular, halos with lower mass tend to form earlier, and as a result, have higher concentrations due to the higher density of the Universe at the time of their formation. Shapes of halos are found to deviate from being perfectly spherical. Typically, halos are found to be elongated and become increasingly prolate towards their centers. However, interactions between dark matter andbaryons would affect the internal structure of dark matter halos. Simulations that model both dark matters and baryons are needed to study small-scale structures.

Many simulations simulate onlycold dark matter, and thus include only the gravitational force. Incorporatingbaryons,leptons andphotons into the simulations dramatically increases their complexity and often radical simplifications of the underlying physics must be made. However, this is an extremely important area and many modern simulations are now trying to understand processes that occur duringgalaxy formation which could account forgalaxy bias.

Reif and Tate^[6] prove that if then-body reachability problem is defined as follows – givenn bodies satisfying a fixed electrostatic potential law, determining if a body reaches a destination ball in a given time bound where we require a poly(n) bits of accuracy and the target time is poly(n) is inPSPACE.

On the other hand, if the question is whether the bodyeventually reaches the destination ball, the problem is PSPACE-hard. These bounds are based on similar complexity bounds obtained forray tracing.

The simplest implementation of N-body simulations where$n\ge 3$ is a naive propagation of orbiting bodies; naive implying that the only forces acting on the orbiting bodies is the gravitational force which they exert on each other. Inobject-oriented programming languages, such asC++, someboilerplate code is useful for establishing the fundamental mathematical structures as well as data containers required for propagation; namelystate vectors, and thusvectors, and some fundamental object containing this data, as well as the mass of an orbiting body. This method is applicable to other types of N-body simulations as well; a simulation of point masses with charges would use a similar method, however the force would be due to attraction or repulsion by interaction of electric fields. Regardless, acceleration of particle is a result of summed force vectors, divided by the mass of the particle:

$${\displaystyle \overrightarrow{a}=\frac{1}{m}\sum \overrightarrow{F}}$$

An example of a programmatically stable andscalable method for containing kinematic data for a particle is the use of fixed length arrays, which in optimised code allows for easy memory allocation and prediction of consumed resources; as seen in the following C++ code:

structVector3{doublee[3]={0};Vector3(){}~Vector3(){}inlineVector3(doublee0,doublee1,doublee2){this->e[0]=e0;this->e[1]=e1;this->e[2]=e2;}};structOrbitalEntity{doublee[7]={0};OrbitalEntity(){}~OrbitalEntity(){}inlineOrbitalEntity(doublee0,doublee1,doublee2,doublee3,doublee4,doublee5,doublee6){this->e[0]=e0;this->e[1]=e1;this->e[2]=e2;this->e[3]=e3;this->e[4]=e4;this->e[5]=e5;this->e[6]=e6;}};

Note thatOrbitalEntity contains enough room for a state vector, where:

• $e_0=x$, the projection of the objects position vector in Cartesian space along${\displaystyle \left[100\right]}$
• $e_1=y$, the projection of the objects position vector in Cartesian space along${\displaystyle \left[010\right]}$
• $e_2=z$, the projection of the objects position vector in Cartesian space along${\displaystyle \left[001\right]}$
• $e_3=\dot{x}$, the projection of the objects velocity vector in Cartesian space along${\displaystyle \left[100\right]}$
• $e_4=\dot{y}$, the projection of the objects velocity vector in Cartesian space along${\displaystyle \left[010\right]}$
• $e_5=\dot{z}$, the projection of the objects velocity vector in Cartesian space along${\displaystyle \left[001\right]}$

Additionally,OrbitalEntity contains enough room for a mass value.

Commonly, N-body simulations will be systems based on some type ofequations of motion; of these, most will be dependent on some initial configuration to "seed" the simulation. In systems such as those dependent on some gravitational or electric potential, the force on a simulation entity is independent on its velocity. Hence, to seed theforces of the simulation, merely initial positions are needed, but this will not allow propagation- initial velocities are required. Consider a planet orbiting a star- it has no motion, but is subject to gravitational attraction to its host star. As a time progresses, and timesteps are added, it will gather velocity according to its acceleration. For a given instant in time,${\displaystyle t_n}$, the resultant acceleration of a body due to its neighbouring masses is independent of its velocity, however, for the time step${\displaystyle t_{n+1}}$, the resulting change in position is significantly different due the propagation's inherent dependency on velocity. In basic propagation mechanisms, such as the symplectic euler method to be used below, the position of an object at${\displaystyle t_{n+1}}$ is only dependent on its velocity at${\displaystyle t_n}$, as the shift in position is calculated via

$${\displaystyle {\overrightarrow{r}}_{t_{n+1}}={\overrightarrow{r}}_{t_n}+{\overrightarrow{v}}_{t_n}\cdot \mathrm{\Delta }t}$$

Without acceleration,${\overrightarrow{v}}_{t_n}$ is static, however, from the perspective of an observer seeing only position, it will take two time steps to see a change in velocity.

A solar-system-like simulation can be accomplished by taking average distances of planet equivalent point masses from a central star. To keep code simple, a non-rigorous approach based onsemi-major axes and mean velocities will be used.Memory space for these bodies must be reserved before the bodies are configured; to allow for scalability, amalloc command may be used:

OrbitalEntity*orbital_entities=malloc(sizeof(OrbitalEntity)*(9+N_ASTEROIDS));orbital_entities[0]={0.0,0.0,0.0,0.0,0.0,0.0,1.989e30};// a star similar to the sunorbital_entities[1]={57.909e9,0.0,0.0,0.0,47.36e3,0.0,0.33011e24};// a planet similar to mercuryorbital_entities[2]={108.209e9,0.0,0.0,0.0,35.02e3,0.0,4.8675e24};// a planet similar to venusorbital_entities[3]={149.596e9,0.0,0.0,0.0,29.78e3,0.0,5.9724e24};// a planet similar to earthorbital_entities[4]={227.923e9,0.0,0.0,0.0,24.07e3,0.0,0.64171e24};// a planet similar to marsorbital_entities[5]={778.570e9,0.0,0.0,0.0,13e3,0.0,1898.19e24};// a planet similar to jupiterorbital_entities[6]={1433.529e9,0.0,0.0,0.0,9.68e3,0.0,568.34e24};// a planet similar to saturnorbital_entities[7]={2872.463e9,0.0,0.0,0.0,6.80e3,0.0,86.813e24};// a planet similar to uranusorbital_entities[8]={4495.060e9,0.0,0.0,0.0,5.43e3,0.0,102.413e24};// a planet similar to neptune

whereN_ASTEROIDS is a variable which will remain at 0 temporarily, but allows for future inclusion of significant numbers of asteroids, at the users discretion. A critical step for the configuration of simulations is to establish the time ranges of the simulation,${\displaystyle t_0}$ to${\displaystyle t_{\text{end}}}$, as well as the incremental time step${\displaystyle dt}$ which will progress the simulation forward:

doublet_0=0;doublet=t_0;doubledt=86400;doublet_end=86400*365*10;// approximately a decade in secondsdoubleBIG_G=6.67e-11;// gravitational constant

The positions and velocities established above are interpreted to be correct for${\displaystyle t=t_0}$.

The extent of a simulation would logically be for the period where.

An entire simulation can consist of hundreds, thousands, millions, billions, or sometimes trillions of time steps. At the elementary level, each time step (for simulations with particles moving due to forces exerted on them) involves

• calculating the forces on each body
• calculating the accelerations of each body (${\displaystyle \overrightarrow{a}}$)
• calculating the velocities of each body (${\displaystyle {\overrightarrow{v}}_n={\overrightarrow{v}}_{n-1}+{\overrightarrow{a}}_n\cdot \mathrm{\Delta }t}$
• calculating the new position of each body (${\displaystyle {\overrightarrow{r}}_{n+1}={\overrightarrow{r}}_n+{\overrightarrow{v}}_n\cdot \mathrm{\Delta }t}$

The above can be implemented quite simply with awhile loop which continues while${\displaystyle t}$ exists in the aforementioned range:

while(t<t_end){for(size_tm1_idx=0;m1_idx<9+N_ASTEROIDS;m1_idx++){Vector3a_g={0,0,0};for(size_tm2_idx=0;m2_idx<9+N_ASTEROIDS;m2_idx++){if(m2_idx!=m1_idx){Vector3r_vector;r_vector.e[0]=orbital_entities[m1_idx].e[0]-orbital_entities[m2_idx].e[0];r_vector.e[1]=orbital_entities[m1_idx].e[1]-orbital_entities[m2_idx].e[1];r_vector.e[2]=orbital_entities[m1_idx].e[2]-orbital_entities[m2_idx].e[2];doubler_mag=sqrt(r_vector.e[0]*r_vector.e[0]+r_vector.e[1]*r_vector.e[1]+r_vector.e[2]*r_vector.e[2]);doubleacceleration=-1.0*BIG_G*(orbital_entities[m2_idx].e[6])/pow(r_mag,2.0);Vector3r_unit_vector={r_vector.e[0]/r_mag,r_vector.e[1]/r_mag,r_vector.e[2]/r_mag};a_g.e[0]+=acceleration*r_unit_vector.e[0];a_g.e[1]+=acceleration*r_unit_vector.e[1];a_g.e[2]+=acceleration*r_unit_vector.e[2];}}orbital_entities[m1_idx].e[3]+=a_g.e[0]*dt;orbital_entities[m1_idx].e[4]+=a_g.e[1]*dt;orbital_entities[m1_idx].e[5]+=a_g.e[2]*dt;}for(size_tentity_idx=0;entity_idx<9+N_ASTEROIDS;entity_idx++){orbital_entities[entity_idx].e[0]+=orbital_entities[entity_idx].e[3]*dt;orbital_entities[entity_idx].e[1]+=orbital_entities[entity_idx].e[4]*dt;orbital_entities[entity_idx].e[2]+=orbital_entities[entity_idx].e[5]*dt;}t+=dt;}

Focusing on the inner fourrocky planets in the simulation, the trajectories resulting from the above propagation is shown below:

• Millennium Run – Computer simulation of the universe
• Large-scale structure of the cosmos – All of space observable from the Earth at the presentPages displaying short descriptions of redirect targets
• GADGET – Computer software for cosmological simulations
• Galaxy formation and evolution – Subfield of cosmology
• Natural units – Units of measurement based on universal physical constants
• Virgo Consortium
• Barnes–Hut simulation – Approximation algorithm for the n-body problem
• Bolshoi cosmological simulation – Computer simulation of the universe

• von Hoerner, Sebastian (1960). "Die numerische Integration des n-Körper-Problemes für Sternhaufen. I".Zeitschrift für Astrophysik (in German).50: 184.Bibcode:1960ZA.....50..184V.
• von Hoerner, Sebastian (1963). "Die numerische Integration desn-Körper-Problemes für Sternhaufen. II".Zeitschrift für Astrophysik (in German).57: 47.Bibcode:1963ZA.....57...47V.
• Aarseth, Sverre J. (2003).GravitationalN-body Simulations: Tools and Algorithms.Cambridge University Press.ISBN 978-0-521-12153-8.
• Bertschinger, Edmund (1998). "Simulations of structure formation in the universe".Annual Review of Astronomy and Astrophysics.36 (1):599–654.Bibcode:1998ARA&A..36..599B.doi:10.1146/annurev.astro.36.1.599.
• Binney, James; Tremaine, Scott (1987).Galactic Dynamics.Princeton University Press.ISBN 978-0-691-08445-9.
• Callahan, Paul B.; Kosaraju, Sambasiva Rao (1992). "A decomposition of multidimensional point sets with applications tok-nearest-neighbors and n-body potential fields (preliminary version)".STOC '92: Proc. ACM Symp. Theory of Computing. ACM..

• Physical cosmology
• Gravity
• Simulation
• Cosmological simulation
• Computational physics
• Particles

• Articles with short description
• Short description matches Wikidata
• Use dmy dates from May 2020
• All articles with unsourced statements
• Articles with unsourced statements from November 2017
• Pages displaying short descriptions of redirect targets via Module:Annotated link
• CS1 German-language sources (de)
• Articles containing video clips