1076:-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. 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 and
4434:
51:
1211:, 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:
450:-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.
349:
332:, 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. This is done by using a linear theory approximation or a low-order
880:
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
948:
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
392:
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,
482:
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
298:
376:
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.
564:
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
559:
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-order
156:
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 a
2178:
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 time
1104:
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 during
911:
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.
355:
354:
351:
350:
2174:; 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 the
356:
3028:
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,
192:
3296:
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
531:
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.
2243:, 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
2419:
353:
915:
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 include
4419:
3526:
3429:
1276:
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:
762:
1033:
1268:
904:. 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 the
843:
651:
524:. 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 as
3286:
2469:
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 on
5069:
5016:
4983:
2464:
2308:
352:
3081:
431:
2113:
2032:
1951:
1191:
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. In
878:
322:
184:
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 collisionless
3331:
793:
1055:
3237:
2274:
2241:
3054:
2301:
2208:
2151:
2070:
1989:
1908:
1836:
1764:
681:
1870:
1798:
1726:
1189:
908:, 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.
422:
3104:
3553:
517:, and so direct integration of the differential equations can be prohibitively computationally expensive. Therefore, a number of refinements are commonly used.
936:
Although there are millions or billions of particles in typical simulations, they typically correspond to a real particle with a very large mass, typically 10
944:
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, a
4975:
1080:
would affect the internal structure of dark matter halos. Simulations that model both dark matters and baryons are needed to study small-scale structures.
696:
293:{\displaystyle {\frac {df}{dt}}={\frac {\partial f}{\partial t}}+\mathbf {v} \cdot \nabla f-{\frac {\partial f}{\partial \mathbf {v} }}\cdot \nabla \Phi }
4668:"On the Clustering Tendencies among the Nebulae. II. a Study of Encounters Between Laboratory Models of Stellar Systems by a New Integration Procedure"
1129:
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(
977:
173:). This quantity needs not have any physical significance, but must be chosen as a compromise between accuracy and manageable computer requirements.
3436:
3339:
4966:
1216:
881:
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.
1110:
463:
2466:
is static, however, from the perspective of an observer seeing only position, it will take two time steps to see a change in velocity.
605:
5034:
4707:
Navarro, Julio F.; Frenk, Carlos S.; White, Simon D. M. (December 1997). "A Universal
Density Profile from Hierarchical Clustering".
597:, particles are assumed to be divided between the surrounding 2x2 vertices of the mesh. The potential energy Φ can be found with the
4554:"A comparison of the evolution of density fields in perturbation theory and numerical simulations - II. Counts-in-cells analysis"
2210:, the resultant acceleration of a body due to its neighbouring masses is independent of its velocity, however, for the time step
1203:
is useful for establishing the fundamental mathematical structures as well as data containers required for propagation; namely
4921:
4869:
113:
1148:
reaches the destination ball, the problem is PSPACE-hard. These bounds are based on similar complexity bounds obtained for
4959:
4453:
4931:
Callahan, Paul B.; Kosaraju, Sambasiva Rao (1992). "A decomposition of multidimensional point sets with applications to
940:. This can introduce problems with short-range interactions between the particles such as the formation of two-particle
535:
There are two basic approximation schemes to decrease the computational time for such simulations. These can reduce the
513:
included ten billion) and the number of particle-particle interactions needing to be computed increases on the order of
798:
566:
5164:
5102:
3245:
1208:
4433:
5087:
4490:
536:
170:
5169:
4952:
4468:
498:
5107:
4418:
5149:
1192:
4944:
161:
cannot afford to have a particle for each atom or molecule of gas as this would require on the order of
5001:
4913:
4861:
890:
385:
328:. These two coupled equations are solved in an expanding background Universe, which is governed by the
5112:
4484:
1204:
552:
17:
3059:
2426:
1057:
is the softening parameter. The value of the softening parameter should be set small enough to keep
169:), so a single 'particle' would represent some much larger quantity of gas (often implemented using
5006:
4778:
2076:
1995:
1914:
848:
305:
5092:
5082:
969:
594:
525:
467:
446:(UK) has dedicated his entire scientific life to the development of a series of highly efficient
443:
3307:
2477:
for these bodies must be reserved before the bodies are configured; to allow for scalability, a
769:
58:-body simulation of the cosmological formation of a cluster of galaxies in an expanding universe
5097:
5077:
4773:
2183:
are added, it will gather velocity according to its acceleration. For a given instant in time,
1040:
684:
466:
are significant. This is incorporated in the simulation as an evolving measure of distance (or
5128:
5039:
4996:
4821:
4797:
3209:
2246:
2213:
1149:
510:
427:
325:
4892:
509:
of particles involved is usually very large (typical simulations include many millions, the
5059:
4888:
4837:
4809:
4726:
4679:
4622:
4524:
3032:
2474:
2279:
2186:
2118:
2037:
1956:
1875:
1803:
1731:
666:
660:
521:
478:
of their physical energy). However, the contributions of general relativity and the finite
471:
42:
474:
system, which causes the particles to slow in comoving coordinates (as well as due to the
398:
8:
5159:
5054:
4415:
in the simulation, the trajectories resulting from the above propagation is shown below:
2171:
1842:
1770:
1698:
1168:
905:
333:
329:
125:
4841:
4813:
4730:
4683:
4626:
4528:
3086:
2414:{\displaystyle {\vec {r}}_{t_{n+1}}={\vec {r}}_{t_{n}}+{\vec {v}}_{t_{n}}\cdot \Delta t}
494:-body simulations are simple in principle, because they involve merely integrating the 6
483:
toroidal), so that one edge of the simulation volume matches up with the opposite edge.
5174:
5024:
4742:
4716:
4640:
4612:
4581:
4561:
3538:
941:
917:
520:
Numerical integration is usually performed over small timesteps using a method such as
459:
185:
158:
117:
4917:
4865:
4750:
4648:
4635:
4600:
2478:
502:
166:
565:
particles per cell. For simulations where particles are not evenly distributed, the
5154:
5049:
4900:
4896:
4879:
Bertschinger, Edmund (1998). "Simulations of structure formation in the universe".
4734:
4687:
4630:
4571:
4532:
4479:
2470:
1200:
1106:
1089:
893:
algorithms are used to get fairly accurate estimates of the path of objects in the
688:
598:
479:
389:
78:
38:
4601:"Second-order Lagrangian perturbation theory initial conditions for resimulations"
5044:
5029:
3206:
The positions and velocities established above are interpreted to be correct for
570:
129:
4800:(1960). "Die numerische Integration des n-Körper-Problemes für Sternhaufen. I".
4849:
4576:
4553:
4447:
4439:
925:
593:
in which space is discretised on a mesh and, for the purposes of computing the
439:
86:
4537:
4512:
1083:
5143:
4768:
John H. Reif; Stephen R. Tate (1993). "The
Complexity of N-body Simulation".
4754:
4746:
4652:
4644:
4585:
4473:
921:
590:
31:
3521:{\displaystyle {\vec {r}}_{n+1}={\vec {r}}_{n}+{\vec {v}}_{n}\cdot \Delta t}
3424:{\displaystyle {\vec {v}}_{n}={\vec {v}}_{n-1}+{\vec {a}}_{n}\cdot \Delta t}
4991:
4412:
901:
894:
361:
145:
97:
30:
This article is about a topic in physics. For the automobile platform, see
2115:, the projection of the objects velocity vector in Cartesian space along
2034:, the projection of the objects velocity vector in Cartesian space along
1953:, the projection of the objects velocity vector in Cartesian space along
1872:, the projection of the objects position vector in Cartesian space along
1800:, the projection of the objects position vector in Cartesian space along
1728:, the projection of the objects position vector in Cartesian space along
4974:
4721:
4566:
1273:
795:
is the comoving wavenumber and the hats denote
Fourier transforms. Since
181:
137:
133:
37:
For engineering problems and simulations involving many components, see
4935:-nearest-neighbors and n-body potential fields (preliminary version)".
3532:
1058:
961:
937:
435:
81:
of particles, usually under the influence of physical forces, such as
945:
561:
426:
effort. The first purely calculational simulations were then done by
67:
50:
960:
is a numerical trick used in N-body techniques to prevent numerical
757:{\displaystyle {\hat {\Phi }}=-4\pi G{\frac {\hat {\rho }}{k^{2}}},}
4738:
4692:
4667:
3242:
The extent of a simulation would logically be for the period where
2170:
Commonly, N-body simulations will be systems based on some type of
475:
360:
N-body simulation of 400 objects with parameters close to those of
4617:
4476: – Units of measurement based on universal physical constants
4551:
1077:
968:
goes to infinity). This is obtained by modifying the regularized
82:
63:
4545:
1028:{\displaystyle \Phi =-{\frac {1}{\sqrt {r^{2}+\epsilon ^{2}}}},}
4462:
1138:
1101:
1097:
1093:
1092:, and thus include only the gravitational force. Incorporating
556:
324:
is the velocity, and Φ is the gravitational potential given by
144:-body simulations are used to study the dynamical evolution of
100:, from investigating the dynamics of few-body systems like the
4456: – All of space observable from the Earth at the present
1196:
965:
845:, the gravitational field can now be found by multiplying by
683:
is the density (number of particles at the mesh points). The
101:
124:-body simulations are used to study processes of non-linear
4767:
1084:
Incorporating baryons, leptons and photons into simulations
372:-body simulations, the equations of motion of a system of
105:
2165:
339:
1125:-body reachability problem is defined as follows – given
109:
1263:{\displaystyle {\vec {a}}={\frac {1}{m}}\sum {\vec {F}}}
1165:
The simplest implementation of N-body simulations where
4772:. Lecture Notes in Computer Science. pp. 162–176.
4458:
Pages displaying short descriptions of redirect targets
1144:
On the other hand, if the question is whether the body
4487: – Approximation algorithm for the n-body problem
4465: – Computer software for cosmological simulations
2429:
2079:
1998:
1917:
1845:
1773:
1701:
1171:
458:
Many simulations are large enough that the effects of
4976:
Numerical methods for ordinary differential equations
3541:
3439:
3342:
3310:
3248:
3212:
3089:
3062:
3035:
2311:
2282:
2249:
2216:
2189:
2121:
2040:
1959:
1878:
1806:
1734:
1219:
1043:
980:
851:
801:
772:
699:
669:
608:
401:
308:
195:
4700:
4429:
964:
when a particle comes too close to another (and the
3547:
3520:
3423:
3325:
3280:
3231:
3098:
3075:
3048:
2458:
2413:
2295:
2268:
2235:
2202:
2145:
2107:
2064:
2026:
1983:
1945:
1902:
1864:
1830:
1792:
1758:
1720:
1262:
1183:
1064:
1049:
1027:
872:
837:
787:
756:
675:
645:
539:to O(N log N) or better, at the loss of accuracy.
416:
316:
292:
4706:
4605:Monthly Notices of the Royal Astronomical Society
4558:Monthly Notices of the Royal Astronomical Society
3531:The above can be implemented quite simply with a
838:{\displaystyle {\vec {g}}=-{\vec {\nabla }}\Phi }
646:{\displaystyle \nabla ^{2}\Phi =4\pi G{\rho },\,}
453:
5141:
4930:
1692:contains enough room for a state vector, where:
1133:) bits of accuracy and the target time is poly(
691:where the Poisson equation has the simple form
393:the equations of motion can be integrated with
4907:
4552:C.M.Baugh; E.Gaztañaga; G. Efstathiou (1995).
4960:
4937:STOC '92: Proc. ACM Symp. Theory of Computing
3281:{\displaystyle t_{0}\leq t<t_{\text{end}}}
2303:, as the shift in position is calculated via
884:
464:Friedmann-Lemaitre-Robertson-Walker cosmology
112:system to understanding the evolution of the
27:Simulation of a dynamical system of particles
4878:
3304:calculating the accelerations of each body (
3106:which will progress the simulation forward:
1272:An example of a programmatically stable and
900:People often decide to put a satellite in a
528:can have 4th order accuracy or much higher.
486:
4881:Annual Review of Astronomy and Astrophysics
4820:
4796:
4659:
4493: – Computer simulation of the universe
4450: – Computer simulation of the universe
3433:calculating the new position of each body (
1116:
687:can solve this efficiently by going to the
581:) time per iteration with fixed dimension.
96:-body simulations are widely used tools in
4967:
4953:
4592:
2134:
2130:
2053:
2049:
1972:
1968:
1891:
1887:
1819:
1815:
1747:
1743:
1160:
151:
4777:
4720:
4691:
4634:
4616:
4575:
4565:
4536:
4510:
3336:calculating the velocities of each body (
642:
176:
165:particles for each mole of material (see
4828:-Körper-Problemes für Sternhaufen. II".
4824:(1963). "Die numerische Integration des
4665:
347:
49:
4908:Binney, James; Tremaine, Scott (1987).
4858:-body Simulations: Tools and Algorithms
4848:
4598:
3083:, as well as the incremental time step
2166:Initialisation of simulation parameters
2162:contains enough room for a mass value.
931:
584:
14:
5142:
1155:
384:-body simulations were carried out by
4948:
4504:
2276:is only dependent on its velocity at
114:large-scale structure of the universe
4513:"N-body simulations (gravitational)"
3555:exists in the aforementioned range:
3183:// approximately a decade in seconds
4770:Automata, Languages and Programming
4511:Trenti, Michele; Hut, Piet (2008).
4454:Large-scale structure of the cosmos
3301:calculating the forces on each body
24:
4790:
4417:
3512:
3415:
2473:and mean velocities will is used.
2405:
981:
832:
823:
703:
619:
610:
287:
284:
270:
262:
250:
230:
222:
25:
5186:
4599:Jenkins, Adrian (21 April 2010).
949:a vanishing force on themselves.
567:well-separated pair decomposition
501:defining the particle motions in
5103:Backward differentiation formula
4636:10.1111/j.1365-2966.2010.16259.x
4432:
1121:Reif and Tate prove that if the
310:
274:
243:
4491:Bolshoi cosmological simulation
2459:{\textstyle {\vec {v}}_{t_{n}}}
1088:Many simulations simulate only
542:
499:ordinary differential equations
380:The first direct gravitational
171:Smoothed Particle Hydrodynamics
4901:10.1146/annurev.astro.36.1.599
4761:
4469:Galaxy formation and evolution
3497:
3475:
3447:
3400:
3372:
3350:
3317:
3291:
3076:{\displaystyle t_{\text{end}}}
3017:// a planet similar to neptune
2855:// a planet similar to jupiter
2639:// a planet similar to mercury
2437:
2383:
2354:
2319:
1254:
1226:
864:
826:
808:
779:
734:
706:
454:General relativity simulations
432:Astronomisches Rechen-Institut
411:
405:
13:
1:
4497:
2963:// a planet similar to uranus
2909:// a planet similar to saturn
2108:{\textstyle e_{5}={\dot {z}}}
2027:{\textstyle e_{4}={\dot {y}}}
1946:{\textstyle e_{3}={\dot {x}}}
2747:// a planet similar to earth
2693:// a planet similar to venus
2585:// a star similar to the sun
952:
924:, various measurements from
873:{\displaystyle -i{\vec {k}}}
317:{\displaystyle \mathbf {v} }
7:
5088:List of Runge–Kutta methods
4830:Zeitschrift fĂĽr Astrophysik
4802:Zeitschrift fĂĽr Astrophysik
4425:
4411:Focusing on the inner four
2801:// a planet similar to mars
1193:object-oriented programming
589:Another possibility is the
10:
5191:
4914:Princeton University Press
4862:Cambridge University Press
3326:{\displaystyle {\vec {a}}}
891:gravitational perturbation
885:Special-case optimizations
788:{\displaystyle {\vec {k}}}
505:. In practice, the number
36:
29:
5121:
5068:
5015:
4982:
4709:The Astrophysical Journal
4672:The Astrophysical Journal
4538:10.4249/scholarpedia.3930
3201:// gravitational constant
3023:
2157:
1687:
1050:{\displaystyle \epsilon }
487:Calculation optimizations
92:for other applications).
4577:10.1093/mnras/274.4.1049
3557:
3108:
2483:
1279:
1117:Computational complexity
1109:which could account for
1037:(rather than 1/r) where
569:methods of Callahan and
537:computational complexity
368:In direct gravitational
5165:Cosmological simulation
5093:Linear multistep method
4893:1998ARA&A..36..599B
4666:Holmberg, Erik (1941).
3232:{\displaystyle t=t_{0}}
2269:{\displaystyle t_{n+1}}
2236:{\displaystyle t_{n+1}}
1161:Common boilerplate code
970:gravitational potential
906:oblateness of the Earth
595:gravitational potential
444:University of Cambridge
152:Nature of the particles
5098:General linear methods
5078:Exponential integrator
4822:von Hoerner, Sebastian
4798:von Hoerner, Sebastian
4422:
3549:
3535:which continues while
3522:
3425:
3327:
3282:
3233:
3100:
3077:
3050:
2460:
2423:Without acceleration,
2415:
2297:
2270:
2237:
2204:
2147:
2109:
2066:
2028:
1985:
1947:
1904:
1866:
1832:
1794:
1760:
1722:
1264:
1185:
1051:
1029:
874:
839:
789:
758:
685:fast Fourier transform
677:
647:
418:
365:
318:
294:
177:Dark matter simulation
136:from the influence of
59:
5170:Computational physics
5129:Symplectic integrator
5113:Gauss–Legendre method
4485:Barnes–Hut simulation
4421:
3550:
3523:
3426:
3328:
3283:
3234:
3101:
3078:
3051:
3049:{\displaystyle t_{0}}
2481:command may be used:
2461:
2416:
2298:
2296:{\displaystyle t_{n}}
2271:
2238:
2205:
2203:{\displaystyle t_{n}}
2148:
2146:{\displaystyle \left}
2110:
2067:
2065:{\displaystyle \left}
2029:
1986:
1984:{\displaystyle \left}
1948:
1905:
1903:{\displaystyle \left}
1867:
1833:
1831:{\displaystyle \left}
1795:
1761:
1759:{\displaystyle \left}
1723:
1265:
1186:
1052:
1030:
875:
840:
790:
759:
678:
676:{\displaystyle \rho }
648:
553:Barnes–Hut simulation
511:Millennium simulation
428:Sebastian von Hoerner
419:
359:
340:Direct gravitational
319:
295:
77:is a simulation of a
53:
5070:Higher-order methods
5060:Leapfrog integration
5017:Second-order methods
3539:
3437:
3340:
3308:
3246:
3210:
3087:
3060:
3033:
2427:
2309:
2280:
2247:
2214:
2187:
2119:
2077:
2038:
1996:
1957:
1915:
1876:
1865:{\textstyle e_{2}=z}
1843:
1804:
1793:{\textstyle e_{1}=y}
1771:
1732:
1721:{\textstyle e_{0}=x}
1699:
1217:
1184:{\textstyle n\geq 3}
1169:
1041:
978:
972:of each particle as
932:Two-particle systems
849:
799:
770:
697:
667:
606:
591:particle mesh method
585:Particle mesh method
522:leapfrog integration
417:{\displaystyle O(N)}
399:
306:
193:
43:Multibody simulation
5083:Runge–Kutta methods
5055:Newmark-beta method
5002:Semi-implicit Euler
4984:First-order methods
4842:1963ZA.....57...47V
4814:1960ZA.....50..184V
4731:1997ApJ...490..493N
4684:1941ApJ....94..385H
4627:2010MNRAS.403.1859J
4529:2008SchpJ...3.3930T
2172:equations of motion
1195:languages, such as
1156:Example simulations
526:Runge–Kutta methods
472:comoving coordinate
334:perturbation theory
330:Friedmann equations
126:structure formation
5150:Physical cosmology
5040:Beeman's algorithm
5025:Verlet integration
4850:Aarseth, Sverre J.
4423:
3545:
3518:
3421:
3323:
3278:
3229:
3099:{\displaystyle dt}
3096:
3073:
3046:
2456:
2411:
2293:
2266:
2233:
2200:
2143:
2105:
2062:
2024:
1981:
1943:
1900:
1862:
1828:
1790:
1756:
1718:
1260:
1181:
1047:
1025:
918:Lyapunov stability
889:Several different
870:
835:
785:
754:
673:
643:
462:in establishing a
460:general relativity
414:
366:
326:Poisson's Equation
314:
290:
186:Boltzmann equation
118:physical cosmology
60:
5137:
5136:
5007:Exponential Euler
4923:978-0-691-08445-9
4910:Galactic Dynamics
4871:978-0-521-12153-8
3548:{\displaystyle t}
3500:
3478:
3450:
3403:
3375:
3353:
3320:
3275:
3070:
2440:
2386:
2357:
2322:
2102:
2021:
1940:
1257:
1243:
1229:
1069:-body simulations
1020:
1019:
867:
829:
811:
782:
749:
737:
709:
661:Newton's constant
503:Newtonian gravity
357:
344:-body simulations
279:
237:
214:
167:Avogadro constant
16:(Redirected from
5182:
5035:Trapezoidal rule
4969:
4962:
4955:
4946:
4945:
4940:
4927:
4904:
4875:
4845:
4817:
4784:
4783:
4781:
4765:
4759:
4758:
4724:
4722:astro-ph/9611107
4704:
4698:
4697:
4695:
4663:
4657:
4656:
4638:
4620:
4611:(4): 1859–1872.
4596:
4590:
4589:
4579:
4569:
4567:astro-ph/9408057
4549:
4543:
4542:
4540:
4508:
4480:Virgo Consortium
4459:
4442:
4437:
4436:
4407:
4404:
4401:
4398:
4395:
4392:
4389:
4386:
4383:
4380:
4377:
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4373:orbital_entities
4371:
4368:
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4361:orbital_entities
4359:
4356:
4353:
4350:
4347:
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4329:
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4320:
4317:
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4313:orbital_entities
4311:
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4301:orbital_entities
4299:
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4230:
4227:
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4215:
4212:
4209:
4206:
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4187:orbital_entities
4185:
4182:
4179:
4176:
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4157:orbital_entities
4155:
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4149:
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4137:
4134:
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4128:
4125:
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4119:
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4110:
4107:
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4008:
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3999:
3996:
3993:
3990:
3987:
3984:
3981:
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3975:
3972:
3969:
3966:
3963:
3962:orbital_entities
3960:
3957:
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3942:
3939:
3936:
3933:
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3864:
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3839:orbital_entities
3837:
3834:
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3828:
3827:orbital_entities
3825:
3822:
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3816:
3813:
3810:
3807:
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3803:orbital_entities
3801:
3798:
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3789:
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3765:
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3753:
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2966:orbital_entities
2964:
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2910:
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2898:
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2804:orbital_entities
2802:
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2750:orbital_entities
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2696:orbital_entities
2694:
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2586:
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2565:
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2534:orbital_entities
2532:
2529:
2526:
2523:
2520:
2517:
2514:
2511:
2508:
2505:
2502:
2499:
2496:
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2492:orbital_entities
2490:
2487:
2465:
2463:
2462:
2457:
2455:
2454:
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2267:
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2264:
2242:
2240:
2239:
2234:
2232:
2231:
2209:
2207:
2206:
2201:
2199:
2198:
2161:
2160:
2152:
2150:
2149:
2144:
2142:
2138:
2114:
2112:
2111:
2106:
2104:
2103:
2095:
2089:
2088:
2071:
2069:
2068:
2063:
2061:
2057:
2033:
2031:
2030:
2025:
2023:
2022:
2014:
2008:
2007:
1990:
1988:
1987:
1982:
1980:
1976:
1952:
1950:
1949:
1944:
1942:
1941:
1933:
1927:
1926:
1909:
1907:
1906:
1901:
1899:
1895:
1871:
1869:
1868:
1863:
1855:
1854:
1837:
1835:
1834:
1829:
1827:
1823:
1799:
1797:
1796:
1791:
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1765:
1763:
1762:
1757:
1755:
1751:
1727:
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1502:
1499:
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1433:
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1421:
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1409:
1406:
1403:
1400:
1397:
1394:
1391:
1388:
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1376:
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1364:
1361:
1358:
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1304:
1301:
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1292:
1289:
1286:
1283:
1269:
1267:
1266:
1261:
1259:
1258:
1250:
1244:
1236:
1231:
1230:
1222:
1201:boilerplate code
1190:
1188:
1187:
1182:
1107:galaxy formation
1090:cold dark matter
1056:
1054:
1053:
1048:
1034:
1032:
1031:
1026:
1021:
1018:
1017:
1005:
1004:
995:
991:
879:
877:
876:
871:
869:
868:
860:
844:
842:
841:
836:
831:
830:
822:
813:
812:
804:
794:
792:
791:
786:
784:
783:
775:
763:
761:
760:
755:
750:
748:
747:
738:
730:
728:
711:
710:
702:
689:frequency domain
682:
680:
679:
674:
652:
650:
649:
644:
638:
618:
617:
599:Poisson equation
573:yield optimal O(
480:speed of gravity
425:
423:
421:
420:
415:
390:Lund Observatory
358:
323:
321:
320:
315:
313:
302:In the equation,
299:
297:
296:
291:
280:
278:
277:
268:
260:
246:
238:
236:
228:
220:
215:
213:
205:
197:
164:
130:galaxy filaments
79:dynamical system
75:-body simulation
39:Multibody system
21:
5190:
5189:
5185:
5184:
5183:
5181:
5180:
5179:
5140:
5139:
5138:
5133:
5117:
5064:
5045:Midpoint method
5030:Velocity Verlet
5011:
4978:
4973:
4924:
4872:
4793:
4791:Further reading
4788:
4787:
4766:
4762:
4705:
4701:
4664:
4660:
4597:
4593:
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4546:
4509:
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3453:
3442:
3441:
3440:
3438:
3435:
3434:
3406:
3395:
3394:
3393:
3378:
3367:
3366:
3365:
3356:
3345:
3344:
3343:
3341:
3338:
3337:
3312:
3311:
3309:
3306:
3305:
3294:
3272:
3268:
3253:
3249:
3247:
3244:
3243:
3223:
3219:
3211:
3208:
3207:
3204:
3203:
3200:
3197:
3194:
3191:
3188:
3185:
3182:
3179:
3176:
3173:
3170:
3167:
3164:
3161:
3158:
3155:
3152:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3122:
3119:
3116:
3113:
3110:
3088:
3085:
3084:
3067:
3063:
3061:
3058:
3057:
3040:
3036:
3034:
3031:
3030:
3024:
3020:
3019:
3016:
3013:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2989:
2986:
2983:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2953:
2950:
2947:
2944:
2941:
2938:
2935:
2932:
2929:
2926:
2923:
2920:
2917:
2914:
2911:
2908:
2905:
2902:
2899:
2896:
2893:
2890:
2887:
2884:
2881:
2878:
2875:
2872:
2869:
2866:
2863:
2860:
2857:
2854:
2851:
2848:
2845:
2842:
2839:
2836:
2833:
2830:
2827:
2824:
2821:
2818:
2815:
2812:
2809:
2806:
2803:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2779:
2776:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2740:
2737:
2734:
2731:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2671:
2668:
2665:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2614:
2611:
2608:
2605:
2602:
2599:
2596:
2593:
2590:
2587:
2584:
2581:
2578:
2575:
2572:
2569:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2506:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2471:semi-major axes
2448:
2444:
2443:
2432:
2431:
2430:
2428:
2425:
2424:
2394:
2390:
2389:
2378:
2377:
2376:
2365:
2361:
2360:
2349:
2348:
2347:
2330:
2326:
2325:
2314:
2313:
2312:
2310:
2307:
2306:
2287:
2283:
2281:
2278:
2277:
2254:
2250:
2248:
2245:
2244:
2221:
2217:
2215:
2212:
2211:
2194:
2190:
2188:
2185:
2184:
2168:
2158:
2126:
2122:
2120:
2117:
2116:
2094:
2093:
2084:
2080:
2078:
2075:
2074:
2045:
2041:
2039:
2036:
2035:
2013:
2012:
2003:
1999:
1997:
1994:
1993:
1964:
1960:
1958:
1955:
1954:
1932:
1931:
1922:
1918:
1916:
1913:
1912:
1883:
1879:
1877:
1874:
1873:
1850:
1846:
1844:
1841:
1840:
1811:
1807:
1805:
1802:
1801:
1778:
1774:
1772:
1769:
1768:
1739:
1735:
1733:
1730:
1729:
1706:
1702:
1700:
1697:
1696:
1688:
1684:
1683:
1680:
1677:
1674:
1671:
1668:
1665:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1611:
1608:
1605:
1602:
1599:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1542:
1539:
1536:
1533:
1530:
1527:
1524:
1521:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1491:
1488:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1341:
1338:
1335:
1332:
1329:
1326:
1323:
1320:
1317:
1314:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1284:
1281:
1249:
1248:
1235:
1221:
1220:
1218:
1215:
1214:
1170:
1167:
1166:
1163:
1158:
1119:
1086:
1071:
1042:
1039:
1038:
1013:
1009:
1000:
996:
990:
979:
976:
975:
955:
934:
887:
859:
858:
850:
847:
846:
821:
820:
803:
802:
800:
797:
796:
774:
773:
771:
768:
767:
743:
739:
729:
727:
701:
700:
698:
695:
694:
668:
665:
664:
634:
613:
609:
607:
604:
603:
587:
577: log
545:
489:
456:
400:
397:
396:
394:
348:
346:
309:
307:
304:
303:
273:
269:
261:
259:
242:
229:
221:
219:
206:
198:
196:
194:
191:
190:
179:
162:
154:
46:
35:
28:
23:
22:
15:
12:
11:
5:
5188:
5178:
5177:
5172:
5167:
5162:
5157:
5152:
5135:
5134:
5132:
5131:
5125:
5123:
5119:
5118:
5116:
5115:
5110:
5105:
5100:
5095:
5090:
5085:
5080:
5074:
5072:
5066:
5065:
5063:
5062:
5057:
5052:
5047:
5042:
5037:
5032:
5027:
5021:
5019:
5013:
5012:
5010:
5009:
5004:
4999:
4997:Backward Euler
4994:
4988:
4986:
4980:
4979:
4972:
4971:
4964:
4957:
4949:
4943:
4942:
4928:
4922:
4905:
4887:(1): 599–654.
4876:
4870:
4854:Gravitational
4846:
4818:
4792:
4789:
4786:
4785:
4779:10.1.1.38.6242
4760:
4739:10.1086/304888
4715:(2): 493–508.
4699:
4693:10.1086/144344
4678:(3): 385–395.
4658:
4591:
4544:
4502:
4501:
4499:
4496:
4495:
4494:
4488:
4482:
4477:
4471:
4466:
4460:
4451:
4448:Millennium Run
4444:
4443:
4440:Physics portal
4427:
4424:
3558:
3544:
3529:
3528:
3517:
3514:
3511:
3506:
3499:
3496:
3489:
3484:
3477:
3474:
3467:
3462:
3459:
3456:
3449:
3446:
3431:
3420:
3417:
3414:
3409:
3402:
3399:
3392:
3387:
3384:
3381:
3374:
3371:
3364:
3359:
3352:
3349:
3334:
3319:
3316:
3302:
3293:
3290:
3271:
3267:
3264:
3261:
3256:
3252:
3226:
3222:
3218:
3215:
3109:
3095:
3092:
3066:
3043:
3039:
2484:
2451:
2447:
2439:
2436:
2410:
2407:
2404:
2397:
2393:
2385:
2382:
2375:
2368:
2364:
2356:
2353:
2346:
2339:
2336:
2333:
2329:
2321:
2318:
2290:
2286:
2263:
2260:
2257:
2253:
2230:
2227:
2224:
2220:
2197:
2193:
2167:
2164:
2156:Additionally,
2154:
2153:
2141:
2137:
2133:
2129:
2125:
2101:
2098:
2092:
2087:
2083:
2072:
2060:
2056:
2052:
2048:
2044:
2020:
2017:
2011:
2006:
2002:
1991:
1979:
1975:
1971:
1967:
1963:
1939:
1936:
1930:
1925:
1921:
1910:
1898:
1894:
1890:
1886:
1882:
1861:
1858:
1853:
1849:
1838:
1826:
1822:
1818:
1814:
1810:
1789:
1786:
1781:
1777:
1766:
1754:
1750:
1746:
1742:
1738:
1717:
1714:
1709:
1705:
1280:
1256:
1253:
1247:
1242:
1239:
1234:
1228:
1225:
1180:
1177:
1174:
1162:
1159:
1157:
1154:
1118:
1115:
1085:
1082:
1070:
1063:
1046:
1024:
1016:
1012:
1008:
1003:
999:
994:
989:
986:
983:
954:
951:
933:
930:
926:ergodic theory
886:
883:
866:
863:
857:
854:
834:
828:
825:
819:
816:
810:
807:
781:
778:
753:
746:
742:
736:
733:
726:
723:
720:
717:
714:
708:
705:
672:
641:
637:
633:
630:
627:
624:
621:
616:
612:
586:
583:
544:
541:
488:
485:
455:
452:
440:Sverre Aarseth
413:
410:
407:
404:
345:
338:
312:
289:
286:
283:
276:
272:
267:
264:
258:
255:
252:
249:
245:
241:
235:
232:
227:
224:
218:
212:
209:
204:
201:
178:
175:
153:
150:
26:
9:
6:
4:
3:
2:
5187:
5176:
5173:
5171:
5168:
5166:
5163:
5161:
5158:
5156:
5153:
5151:
5148:
5147:
5145:
5130:
5127:
5126:
5124:
5120:
5114:
5111:
5109:
5106:
5104:
5101:
5099:
5096:
5094:
5091:
5089:
5086:
5084:
5081:
5079:
5076:
5075:
5073:
5071:
5067:
5061:
5058:
5056:
5053:
5051:
5050:Heun's method
5048:
5046:
5043:
5041:
5038:
5036:
5033:
5031:
5028:
5026:
5023:
5022:
5020:
5018:
5014:
5008:
5005:
5003:
5000:
4998:
4995:
4993:
4990:
4989:
4987:
4985:
4981:
4977:
4970:
4965:
4963:
4958:
4956:
4951:
4950:
4947:
4938:
4934:
4929:
4925:
4919:
4915:
4911:
4906:
4902:
4898:
4894:
4890:
4886:
4882:
4877:
4873:
4867:
4863:
4859:
4855:
4851:
4847:
4843:
4839:
4835:
4832:(in German).
4831:
4827:
4823:
4819:
4815:
4811:
4807:
4804:(in German).
4803:
4799:
4795:
4794:
4780:
4775:
4771:
4764:
4756:
4752:
4748:
4744:
4740:
4736:
4732:
4728:
4723:
4718:
4714:
4710:
4703:
4694:
4689:
4685:
4681:
4677:
4673:
4669:
4662:
4654:
4650:
4646:
4642:
4637:
4632:
4628:
4624:
4619:
4614:
4610:
4606:
4602:
4595:
4587:
4583:
4578:
4573:
4568:
4563:
4559:
4555:
4548:
4539:
4534:
4530:
4526:
4522:
4518:
4514:
4507:
4503:
4492:
4489:
4486:
4483:
4481:
4478:
4475:
4474:Natural units
4472:
4470:
4467:
4464:
4461:
4455:
4452:
4449:
4446:
4445:
4441:
4435:
4430:
4420:
4416:
4414:
4413:rocky planets
4139:r_unit_vector
4109:r_unit_vector
4079:r_unit_vector
3998:r_unit_vector
3556:
3542:
3534:
3515:
3509:
3504:
3494:
3487:
3482:
3472:
3465:
3460:
3457:
3454:
3444:
3432:
3418:
3412:
3407:
3397:
3390:
3385:
3382:
3379:
3369:
3362:
3357:
3347:
3335:
3314:
3303:
3300:
3299:
3298:
3289:
3269:
3265:
3262:
3259:
3254:
3250:
3240:
3224:
3220:
3216:
3213:
3107:
3093:
3090:
3064:
3041:
3037:
2510:OrbitalEntity
2486:OrbitalEntity
2482:
2480:
2476:
2472:
2467:
2449:
2445:
2434:
2421:
2408:
2402:
2395:
2391:
2380:
2373:
2366:
2362:
2351:
2344:
2337:
2334:
2331:
2327:
2316:
2304:
2288:
2284:
2261:
2258:
2255:
2251:
2228:
2225:
2222:
2218:
2195:
2191:
2182:
2177:
2173:
2163:
2159:OrbitalEntity
2139:
2135:
2131:
2127:
2123:
2099:
2096:
2090:
2085:
2081:
2073:
2058:
2054:
2050:
2046:
2042:
2018:
2015:
2009:
2004:
2000:
1992:
1977:
1973:
1969:
1965:
1961:
1937:
1934:
1928:
1923:
1919:
1911:
1896:
1892:
1888:
1884:
1880:
1859:
1856:
1851:
1847:
1839:
1824:
1820:
1816:
1812:
1808:
1787:
1784:
1779:
1775:
1767:
1752:
1748:
1744:
1740:
1736:
1715:
1712:
1707:
1703:
1695:
1694:
1693:
1689:OrbitalEntity
1480:OrbitalEntity
1468:OrbitalEntity
1456:OrbitalEntity
1432:OrbitalEntity
1278:
1275:
1270:
1251:
1245:
1240:
1237:
1232:
1223:
1212:
1210:
1206:
1205:state vectors
1202:
1198:
1194:
1178:
1175:
1172:
1153:
1151:
1147:
1142:
1140:
1136:
1132:
1128:
1124:
1114:
1112:
1108:
1103:
1099:
1095:
1091:
1081:
1079:
1075:
1068:
1065:Results from
1062:
1060:
1044:
1035:
1022:
1014:
1010:
1006:
1001:
997:
992:
987:
984:
973:
971:
967:
963:
959:
950:
947:
943:
939:
929:
927:
923:
922:Lyapunov time
919:
913:
909:
907:
903:
898:
896:
892:
882:
861:
855:
852:
817:
814:
805:
776:
764:
751:
744:
740:
731:
724:
721:
718:
715:
712:
692:
690:
686:
670:
662:
658:
653:
639:
635:
631:
628:
625:
622:
614:
601:
600:
596:
592:
582:
580:
576:
572:
568:
563:
558:
554:
550:
540:
538:
533:
529:
527:
523:
518:
516:
512:
508:
504:
500:
497:
493:
484:
481:
477:
473:
469:
465:
461:
451:
449:
445:
441:
437:
433:
429:
408:
402:
391:
387:
386:Erik Holmberg
383:
378:
375:
371:
363:
343:
337:
335:
331:
327:
300:
281:
265:
256:
253:
247:
239:
233:
225:
216:
210:
207:
202:
199:
188:
187:
183:
174:
172:
168:
160:
149:
147:
146:star clusters
143:
139:
135:
131:
127:
123:
119:
115:
111:
107:
103:
99:
95:
91:
90:-body problem
89:
84:
80:
76:
74:
69:
65:
57:
52:
48:
44:
40:
33:
32:GM N platform
19:
4992:Euler method
4936:
4932:
4909:
4884:
4880:
4857:
4853:
4833:
4829:
4825:
4805:
4801:
4769:
4763:
4712:
4708:
4702:
4675:
4671:
4661:
4608:
4604:
4594:
4557:
4547:
4520:
4517:Scholarpedia
4516:
4506:
4410:
4133:acceleration
4103:acceleration
4073:acceleration
3941:acceleration
3530:
3295:
3241:
3205:
3021:
2475:Memory space
2468:
2422:
2305:
2180:
2175:
2169:
2155:
1685:
1271:
1213:
1164:
1145:
1143:
1134:
1130:
1126:
1122:
1120:
1087:
1073:
1072:
1066:
1036:
974:
957:
956:
938:solar masses
935:
914:
910:
902:frozen orbit
899:
895:Solar System
888:
765:
693:
656:
654:
602:
588:
578:
574:
551:, such as a
549:tree methods
548:
546:
543:Tree methods
534:
530:
519:
514:
506:
495:
491:
490:
468:scale factor
457:
447:
381:
379:
373:
369:
367:
362:Solar System
341:
301:
189:
180:
155:
141:
134:galaxy halos
121:
98:astrophysics
93:
87:
72:
71:
61:
55:
47:
4523:(5): 3930.
4283:N_ASTEROIDS
3695:N_ASTEROIDS
3614:N_ASTEROIDS
3292:Propagation
3025:N_ASTEROIDS
2528:N_ASTEROIDS
1207:, and thus
1150:ray tracing
1111:galaxy bias
1061:realistic.
1059:simulations
962:divergences
476:redshifting
438:, Germany.
182:Dark matter
138:dark matter
5160:Simulation
5144:Categories
4498:References
4289:entity_idx
4271:entity_idx
4259:entity_idx
3533:while loop
3011:102.413e24
2975:4495.060e9
2921:2872.463e9
2867:1433.529e9
2849:1898.19e24
2795:0.64171e24
2633:0.33011e24
1686:Note that
1146:eventually
436:Heidelberg
5175:Particles
4774:CiteSeerX
4755:0004-637X
4747:1538-4357
4653:0035-8711
4645:1365-2966
4618:0910.0258
4586:1365-2966
3513:Δ
3510:⋅
3498:→
3476:→
3448:→
3416:Δ
3413:⋅
3401:→
3383:−
3373:→
3351:→
3318:→
3260:≤
2957:86.813e24
2903:568.34e24
2813:778.570e9
2759:227.923e9
2741:5.9724e24
2705:149.596e9
2687:4.8675e24
2651:108.209e9
2438:→
2406:Δ
2403:⋅
2384:→
2355:→
2320:→
2100:˙
2019:˙
1938:˙
1255:→
1246:∑
1227:→
1176:≥
1045:ϵ
1011:ϵ
988:−
982:Φ
958:Softening
953:Softening
865:→
853:−
833:Φ
827:→
824:∇
818:−
809:→
780:→
735:^
732:ρ
722:π
716:−
707:^
704:Φ
671:ρ
636:ρ
629:π
620:Φ
611:∇
562:multipole
288:Φ
285:∇
282:⋅
271:∂
263:∂
257:−
251:∇
248:⋅
231:∂
223:∂
159:gas cloud
140:. Direct
68:astronomy
18:Softening
4852:(2003).
4426:See also
4043:r_vector
4025:r_vector
4007:r_vector
3926:r_vector
3914:r_vector
3902:r_vector
3890:r_vector
3878:r_vector
3866:r_vector
3815:r_vector
3779:r_vector
3743:r_vector
3737:r_vector
3195:6.67e-11
2597:57.909e9
2579:1.989e30
1274:scalable
1137:) is in
946:softened
571:Kosaraju
128:such as
5155:Gravity
5108:Yoshida
4889:Bibcode
4838:Bibcode
4810:Bibcode
4808:: 184.
4727:Bibcode
4680:Bibcode
4623:Bibcode
4525:Bibcode
3995:Vector3
3734:Vector3
3632:Vector3
2783:24.07e3
2729:29.78e3
2675:35.02e3
2621:47.36e3
1333:Vector3
1321:Vector3
1309:Vector3
1285:Vector3
1209:vectors
1199:, some
1102:photons
1098:leptons
1094:baryons
1078:baryons
928:, etc.
470:) in a
442:at the
430:at the
424:
395:
388:at the
364:planets
83:gravity
64:physics
5122:Theory
4939:. ACM.
4920:
4868:
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4463:GADGET
4256:size_t
3938:double
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3719:m2_idx
3701:m2_idx
3683:m2_idx
3671:m2_idx
3668:size_t
3620:m1_idx
3602:m1_idx
3590:m1_idx
3587:size_t
3186:double
3156:double
3141:double
3126:double
3111:double
3022:where
2999:5.43e3
2945:6.80e3
2891:9.68e3
2504:sizeof
2498:malloc
2479:malloc
2176:forces
1540:double
1531:double
1522:double
1513:double
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1495:double
1486:double
1477:inline
1438:double
1429:struct
1357:double
1348:double
1339:double
1330:inline
1291:double
1282:struct
1139:PSPACE
942:binary
766:where
655:where
557:octree
4743:eISSN
4717:arXiv
4641:eISSN
4613:arXiv
4582:eISSN
4562:arXiv
4055:r_mag
4037:r_mag
4019:r_mag
3983:r_mag
3953:BIG_G
3854:r_mag
3572:t_end
3560:while
3189:BIG_G
3165:86400
3159:t_end
3150:86400
2181:steps
1663:->
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1555:->
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1390:->
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966:force
555:, an
116:. In
102:Earth
85:(see
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1369:this
1100:and
663:and
132:and
106:Moon
66:and
41:and
4897:doi
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