Seppo Mikkola

An implementation ofN-body chain regularization

The chain regularization method (Mikkola and Aarseth 1990) for high accuracy computation of particle motions in smallN-body systems has been reformulated. We discuss the transformation formulae, equations of motion and selection of a chain of interparticle vectors such that the critical interactions requiring regularization are included in the chain. The Kustaanheimo-Stiefel (KS) coordinate transformation and a time transformation is used to regularize the dominant terms of the equations of motion. The method has been implemented for an arbitrary number of bodies, with the option of external perturbations. This formulation has been succesfully tested in a generalN-body program for strongly interacting subsystems. An easy to use computer program, written inFortran, is available on request.

Formation of the widest binary stars from dynamical unfolding of triple systems

The formation of very wide binary systems, such as the α Centauri system with Proxima (also known as α Centauri C) separated from α Centauri (which itself is a close binary A/B) by 15,000 astronomical units (1 au is the distance from Earth to the Sun), challenges current theories of star formation, because their separation can exceed the typical size of a collapsing cloud core. Various hypotheses have been proposed to overcome this problem, including the suggestion that ultrawide binaries result from the dissolution of a star cluster—when a cluster star gravitationally captures another, distant, cluster star. Recent observations have shown that very wide binaries are frequently members of triple systems and that close binaries often have a distant third companion. Here we report N-body simulations of the dynamical evolution of newborn triple systems still embedded in their nascent cloud cores that match observations of very wide systems. We find that although the triple systems are born very compact—and therefore initially are more protected against disruption by passing stars—they can develop extreme hierarchical architectures on timescales of millions of years as one component is dynamically scattered into a very distant orbit. The energy of ejection comes from shrinking the orbits of the other two stars, often making them look from a distance like a single star. Such loosely bound triple systems will therefore appear to be very wide binaries.

An asteroidal companion to the Earth

Near-Earth asteroids range in size from a few metres to more than 30 km: in addition to playing an important role in past and present impact rates on the Earth, they might one day be exploited as bases for space exploration or as mineral resources. Many near-Earth asteroids move on orbits crossing that of the Earth, but none has hitherto been identified as a dynamical companion to the Earth. Here we show that the orbit of asteroid 3753 (1986 TO), when viewed in the reference frame centred on the Sun but orbiting with the Earth, has a distinctive shape characteristic of ‘horseshoe’ orbits. Although horseshoe orbits are a well-known feature of the gravitational three-body problem, the only other examples of objects moving on such orbits are the saturnian satellites Janus and Epimetheus—and their behaviour is much less intricate than that of 3753. Moreover, the fact that 3753 exhibits such a dynamical relationship with the Earth shows that, although it is not a satellite of our planet per se, it is, apart from the Moon, the only known natural companion of the Earth.

Practical Symplectic Methods with Time Transformation for the Few-Body Problem

The use of the extended phase space and time transformations for constructing efficient symplectic algorithms for the investigation of long term behavior of hierarchical few-body systems is discussed. Numerical experiments suggest that the time-transformed generalized leap-frog, combined with symplectic correctors, is one of the most efficient methods for such studies. Applications extend from perturbed two-body motion to hierarchical many-body systems with large eccentricities.

Predicting the Next Outbursts of OJ 287 in 2006-2010

In its nearly regular cycle of outbursts the quasar OJ 287 is due for another outburst season in 2006-2010. The prediction for the exact timing depends on the adopted model. In the precessing binary model of Lehto and Valtonen the timing depends on the time delay between the impact on the primary disk and the time when the impacted gas becomes optically thin. The time delay in turn depends on the properties of the accretion disk, the accretion rate, and the viscosity parameter α, which are not exactly known. We study the flexibility in timing provided by the uncertainties. In order to fix the model, two methods are used: the wobble of the jet, induced by the secondary, and the timing of the 1956 outburst, which has not been previously used. As a result, rather definite dates for the outbursts are obtained, which are different from a straightforward extrapolation of the past light curve. A new optical light curve with many new historical as well as recent points of observation have been put together and has been analyzed in order to reach these conclusions. Also, the high-frequency radio observations are found to agree with the jet wobble picture.

Explicit Symplectic Algorithms For Time‐Transformed Hamiltonians

By Hamiltonian manipulation we demonstrate the existence of separable time‐transformed Hamiltonians in the extended phase‐space. Due to separability explicit symplectic methods are available for the solution of the equations of motion. If the simple leapfrog integrator is used, in case of two‐body motion, the method produces an exact Keplerian ellipse in which only the time‐coordinate has an error. Numerical tests show that even the rectilinear N‐body problem is feasible using only the leapfrog integrator. In practical terms the method cannot compete with regularized codes, but may provide new directions for studies of symplectic N‐body integration.

A chain regularization method for the few-body problem

A regularization method for integrating the equations of motion of small N-body systems is discussed. We select a chain of interparticle vectors in such a way that the critical interactions requiring regularization are included in the chain. The equations of motion for the chain vectors are subsequently regularized using the KS-variables and a time transformation. The method has been formulated for any number of bodies, but the most important application appears to be in the four-body problem which is therefore discussed in detail.

Implementing Few-Body Algorithmic Regularization with Post-Newtonian Terms

We discuss the implementation of a new regular algorithm for simulation of the gravitational few-body problem. The algorithm uses components from earlier methods, including the chain structure, the logarithmic Hamiltonian, and the time-transformed leapfrog. This algorithmic regularization code, AR-CHAIN, can be used for the normal N-body problem, as well as for problems with softened potentials and/or with velocity-dependent external perturbations, including post-Newtonian terms, which we include up to order PN2.5. Arbitrarily extreme mass ratios are allowed. Only linear coordinate transformations are used and thus the algorithm is somewhat simpler than many earlier regularized schemes. We present the results of performance tests which suggest that the new code is either comparable in performance or superior to the existing regularization schemes based on the Kustaanheimo-Stiefel (KS) transformation. This is true even for the two-body problem, independent of eccentricity. An important advantage of the new method is that, contrary to the older KS-CHAIN code, zero masses are allowed. We use our algorithm to integrate the orbits of the S stars around the Milky Way supermassive black hole for one million years, including PN2.5 terms and an intermediate-mass black hole. The three S stars with shortest periods are observed to escape from the system after a few hundred thousand years.

A Time-Transformed Leapfrog Scheme

We present a time-transformed leapfrog scheme combined with the extrapolation method to construct an integrator for orbits in N-body systems with large mass ratios. The basic idea can be used to transform any second-order differential equation into a form which may allow more efficient numerical integration. When applied to gravitating few-body systems this formulation permits extremely close two-body encounters to be considered without significant loss of accuracy. The new scheme has been implemented in a direct N-body code for simulations of super-massive binaries in galactic nuclei. In this context relativistic effects may also be included.

Algorithmic regularization with velocity-dependent forces

Algorithmic regularization uses a transformation of the equations of motion such that the leapfrog algorithm produces exact trajectories for two-body motion as well as regular results in numerical integration of the motion of strongly interacting few-body systems. That algorithm alone is not sufficiently accurate and one must use the extrapolation method for improved precision. This requires that the basic leapfrog algorithm be time-symmetric, which is not directly possible in the case of velocity-dependent forces, but is usually obtained with the help of the implicit mid-point method. Here, we suggest an alternative explicit algorithmic regularization algorithm which can handle velocity-dependent forces. This is done with the help of a generalized mid-point method to obtain the required time symmetry, thus eliminating the need for the implicit mid-point method and allowing the use of extrapolation.