S. Mikkola obtained major results in symplectic integration of eccentric orbits by introducing regularization with time transformation. He also discovered a more rigorous way of treating drag and other non-conservative forces in connection with symplectic integration.

With S. Aarseth (IOA, Cambridge, UK), Mikkola improved the treatment of motions of binaries in star cluster simulations by use of so called Stumpff-functions which give accurate analytical solutions in case of negligible perturbations.

S. Mikkola has started a collaboration with Surrey University researchers P. Palmer and Y. Hashida on the dynamics of artificial satellites. The objectives of this work are to develop fast, concise and accurate software for autonomous orbit determination of micro-satellites. Thus far three papers have been submitted as a result of this collaboration

P. Nurmi, S. Mikkola, J.Q. Zheng and M. Valtonen have developed methods of calculating small body orbits in the Solar System. In addition to normal orbit integration, a method has been used where the perturbations over a whole orbit are calculated, and the orbital parameters are updated at the apocentre. This is essentially a single step calculation per orbit. A faster Monte Carlo code evaluates orbital changes only after close encounters, typically once after 10³ revolutions. Different methods have been evaluated in order to study the credibility of the fast methods.

Asteroid and Comet Dynamics

S. Mikkola studied the orbital dynamics of asteroids mainly in collaboration with K.A. Innanen and P. Wiegert (York University, Canada). One major work in this field was the identification of the asteroid 3753 (1986 TO) as an `Earth companion': [Wiegert, Innanen and Mikkola, Nature, 12 Jun 1997]. This object has an unusual orbit in which it has essentially two types of motion: most of the time the object moves in a so called horseshoe orbit in the neighborhood of the Earth's orbit, while occasionally the orbit type is what is termed a quasi-satellite orbit. In the latter orbit type the object appears to be circling the Earth at large distance.

The Monte Carlo method was applied to the problem of exchange of comets and minor planets between the Solar Systems and planetary systems around other stars by J.Q. Zheng and M. Valtonen. It is shown that a certain amount of exchange of material between planets in two different planetary systems is possible, in particular during the early stages of evolution. Together with C.Mileikowski et al., Zheng and Valtonen have also considered the possibility of transfer of life between different planetary systems.

Three body problem and multiple stars

The dynamics of binary systems are well understood. The evolution of a restricted three body gravitating system, where small bodies orbit one or two dominant bodies is also reasonably well understood. Hierachial three body systems with one very compact binary and a relatively very distant third body can be approximated in most cases as two binary systems. Complications arise when one considers three bodies of equal mass. These kinds of systems show high sensitivity to initial conditions.

P. Heinämäki, H.J. Lehto, A. Chernin and M. Valtonen have been studying the evolution of a family of triple systems. The investigation has been confined to systems initally at rest. The first part of the investigation was to characterize the evolution of individual orbits. Even in quasi-regular hierarchial states, chaotic intermittency is significantly present indicating non-predictability of orbits. A strange attractor of dimension slightly above 2 was detected. During the second part of this investigation the evolution of one hundred close-by orbits located in four different regions of the homology map were followed. The homology map is a scale independent way of describing the configuration of a triple system. Two separate concepts were studied. The first was Kolmogoroff-Smirnoff entropy, which descibes the rate of expansion of an infinitesimal drop in the homology map. The second parameter studied was the Lyapunov exponent which measures the divergence rate of nearby orbits. A close dependence between these two parameters and high sensitivity to initial conditions was found.

M. Valtonen has applied the known statistical distributions of the breakup of three-body systems to the case of triple stars and quadruple stars. He finds that the observed distributions of orbital elements in binary stars, both in isolated binaries and in hierarchical binaries, agree well with the assumption that the observed binaries are remnants of three-body interactions and ejections of single stars. In particular, the mass ratio distributions which vary with the spectral type of the primary, are well explained.

A. Chernin, S. Mikkola and M. Valtonen have reviewed the three- body problem and have presented the complete solution of the General three-body problem in a statistical sense. This work is based on earlier work by D.C. Heggie and J. Monaghan.

The stellar system CH-Cygni has long been enigmatic in its behavior. This system is believed to be a triple one in which there occurs mass flows which, however, began only recently (about 40 years ago). S. Mikkola developed, in collaboration with K. Tanikawa, a model which consistently explains all the thus far available observations in terms of the Kozai resonance. The Kozai resonance is a dynamical phenomenon in which the eccentricity and inclination of a binary experience large amplitude variations due to the presence of a third star circling the binary at a greater distance.

Solar Research

Using the Metsähovi radio maps of the Sun, A. Riehokainen, E.Valtaoja and S. Urpo (HUT) have studied the enhanced temperature regions (ETR), point-like microwave sources sometimes visible at high solar latitudes. The nature of these small structures is completely unknown; in radio maps they resemble sunspots, which also are visible as microwave enhancements, but the polar ETRs occur at high latitudes where no sunspots are detected. Analyzing the Metsähovi maps, it has been shown that the ETR participate in the overall solar rotation and their activity exhibits the usual 12-year solar cycle, but in antiphase with the sunspot cycle.

Comparisons with the optical polar faculae data from the Kislovodsk Solar Station (Russia) indicate that the ETRs may be related to the faculae. Simultaneous radio and optical observations in 1997 (together with V. Makarov, V. Makarova and A. Tlatov, Russia) show that the ETRs and the polar faculae preferentially occur in the same areas. Further comparisons with, e.g., SOHO data will hopefully clarify the nature of this mysterious phenomenon.

Magnetic cataclysmic variables (Am Her stars)

S. Katajainen, V. Piirola and H. Lehto have studied magnetic Cataclysmic Variables (polars) using the 5-band (UBVRI) photopolarimetric Turpol-instrument at the NOT during 1996 (Piirola), 1997 (Katajainen and Piirola) and 1998 (Katajainen and Lehto). The data from the polars BY Cam, V884 Her, QQ Vul and V1309 Ori have been analyzed and modelled with constant temperature cyclotron emission models. By fitting the polarization and the brightness variations over the orbital period, the inclination angle of the spin axis of the white dwarf (WD), the strength of the magnetic field, the number and the sizes and locations of different emission areas (extended arcs) on the surface of the WD have been estimated.

The V1309 Ori NOT-data have combined with the data obtained at the CASLEO (2.1 metre) telescope, Argentina, by Dr. F. Scaltriti (Turin Observatory, Italy). V1309 Ori is an enigmatic object. The synchronization of the white dwarf rotation with the binary orbital motion by magnetic braking is difficult to be explained with the long orbital period (7.98 hours). With the orbital period almost four times longer than in most polars, this source is expected to have a very strong magnetic field. Modelling of the circular polarization data in the UBVRI-bands, however, suggests a relatively low magnetic field of only 40 MG, instead of the expected field of hundreds of MGs. The inclination angle of this system was calculated to be between 80 and 85 degrees. Circularly polarized emission radiates from two different poles, located almost at diametrically opposite sites. During the orbital period there is a deep eclipse in the lightcurve (see phase 0 in the figure). This eclipse is almost 4 magnitudes deep in the B- band, where the object fades to about 19.5 magnitudes.

The BY Camelopardalis data were modelled by fitting three accretion poles: one dominating negative pole and two weak positive poles. An accretion geometry with a quadrupole field is suggested. The fourth pole is assumed to have almost negligible cyclotron emission. The magnetic field is between 25 and 30 MG and the inclination angle of the spin axis about 50 degrees. H. Lehto has applied also the wavelet analysis techniques to study short timescale variability in the lightcurves of BY Camelopardalis.

Lightcurves of V1309 Ori (left) and the simulation (line) and the observed points of circular polarization (right). The data obtained at the NOT is from phase 0 to phase 0.5 and data between phases 0.5 and 0 is from CASLEO. The circular polarization observations were performed only at the NOT. Numbers of the cyclotron harmonics used for the model fitting are given (from 3 to 7).

The observed (points) and the simulated (lines) lightcurves of BY Camelopardalis. The numbers of the cyclotron harmonics used for the model fitting are given (from 5 to 10).

The circular (observed & simulated) and linear polarization curves of BY Camelopardalis. Harmonic numbers are the same as in the previous figure.