A numerical investigation of the one-dimensional Newtonian three-body problem

The one-dimensional Newtonian three-body problem is known to have stable (quasi-)periodic orbits when the masses are equal. The existence and size of the stable region is discussed here in the case where the three masses are arbitrary. We consider only the stability of the periodic (generalized) Schubart's (1956) orbit. If this orbit is linearly stable it is almost always surrounded by a region of stable quasi-periodic orbits and the size and shape of this stable region depends on the masses. The three-dimensional linear stability of the periodic orbits is also determined. Final results show that the region of stability has a complicated shape and some of the stable regions in the mass-plane are quite narrow. The non-linear three-dimensional stability is studied independently by extensive numerical integrations and the results are found to be in agreement with the linear stability analysis. The boundaries of stable region in the mass-plane are given in terms of polynomial approximations. The results are compared with a similar work by Héenon (1977).We thank the referee for pointing out this reference to us.     

A cubic approximation for Kepler's equation

We derive a new method to obtain an approximate solution for Kepler's equation. By means of an auxiliary variable it is possible to obtain a starting approximation correct to about three figures. A high order iteration formula then corrects the solution to high precision at once. The method can be used for all orbit types, including hyperbolic. To obtain this solution the trigonometric or hyperbolic functions must be evaluated only once.     

Computations on the influence of changing climate on the soil moisture and productivity in scots pine stands in southern and northern Finland

A model computation on the evaporative demand in relation to precipitation indicated that, under a changing climate with elevating temperatures, evapotranspiration could exceed the concurrent precipitation during the growing period in southern Finland (61° N), but not in northern Finland (66° N). This could reduce the supply of soil water to enable tree growth on sites with soil of low water holding capacity. This in turn could reduce the productivity of Scots pine more in southern Finland than in northern Finland. In northern Finland, the reduction in growth due to a limited supply of water was partly compensated by the enhanced growth due to a rise in temperature outside dry periods.     

Origin of oort cloud comets in the interstellar space

Comets must form a major part of the interstellar medium. The solar system provides a flux of comets into the interstellar space and there is no reason to suspect that many other stars and their surrounding cometary systems would not make a similar contribution. Occasionally interstellar comets must pass through the inner solar system, but Whipple (1975) considers it unlikely that such a comet is among the known cases of apparently hyperbolic comets. Even so the upper limit for the density of unobserved interstellar comets is relatively high.In addition, we must consider the possibility that comets are a genuine component of interstellar medium, and that the Oort Cloud is merely a captured part of it (McCrea, 1975). Here we review various dynamical possibilities of two-way exchange of comet populations between the Solar System and the interstellar medium. We describe ways in which a traditional Oort Cloud (Oort, 1950) could be captured from the interstellar medium. However, we note that the so called Kuiper belt (Kuiper, 1951) of comets cannot arise through this process. Therefore we have to ask how necessary the concept of the yet unobserved Kuiper belt is for the theory of short period comets.There has been considerable debate about the question whether short period comets can be understood as a captured population of the Oort Cloud of comets or whether an additional source has to be postulated. The problem is made difficult by the long integration times of comet orbits through the age of the Solar System. It would be better to have an accurate treatment of comet-planet encounters in a statistical sense, in the form of cross sections, and to carry out Monte Carlo studies. Here we describe the plan of action and initial results of the work to derive cross sections by carrying out large numbers of comet — planet encounters and by deriving approximate analytic expressions for them. Initially comets follow parabolic orbits of arbitrary inclination and perihelion distance; cross sections are derived for obtaining orbits of given energy and inclination after the encounter. The results are used in subsequent work to make evolutionary models of the comet population.     

Carbon balance in the forest sector in Finland during 1990–2039

In this study we estimated the amount of carbon (C) stored in the forest growing stock and in wood-based products, and the C-sequestration capacity of the forest sector in Finland. Comparison of different management and utilization options for forest resources over the period 1990-2039 indicates that C is stored more efficiently in standing timber than in wood-based products. This implies that an appropriate increase in the length of the rotation in forestry could be optimal for balancing the needs of forest resources for C sequestration and timber production. Increased use of wood, based on sustainable use of forest resources, to substitute for fossil fuels and materials, could decrease the overall C emissions. Release of sequestered C back to the atmosphere can be delayed by prolonging product lifespans, by increasing recycling, or by disposing of discarded products in landfills. To delay C release, and affect the C balance, however, these changes should be substantial.In 1990, the net C balance of the growing stemwood stock was 5.5 Tg C/a, which increased to 16.3 Tg C/a by 2039 if in the future the use of wood would be at the level of the late 1980s. Increased use of wood resulted in a balance of 6.6 Tg C/a or -0.2 Tg C/a, depending on the extent of the use of wood. The average C balance in wood products for the whole period was 3.9, 5.6 or 6.6 Tg C/a, respectively. Changes in production capacity, and consideration of timber and product import and export decreased the average balance from 6.6 Tg C/a to 0.9–1.3 Tg C/a. By comparison, emission from the use of fossil fuels in 1990 C was 14.5 Tg C/a.     

Wet Canopy Evaporation Rate of Three Stands in Western Sichuan, China

The wet canopy evaporation rate (Er) was calculated by Penman-Monteith combination model based on three assumptions and with meteorological variables 2 m above the canopy in three stands, dominated by spruce (SF), fir (FF) and birch (BF) trees, respectively, in the subalpine forests in western Sichuan, China over a growing season. The total amount of the E was 44.5 mm for SF, 88.5 mm for FF and 57.8 mm for BF, accounting for 9.2%, 16.6% and 10.2% of the gross rainfall, respectively, in the measuring period. There was the highest average monthly Er and percentage of E to gross rainfall for FF compared with SF and BF.Mean Er was 0.097 mm h^-1 (ranging from 0.028 to0.487 mm h^-1), 0.242 mm h^-1 (from 0.068 to 0.711 mm h^-1) and 0.149 mm h^-1 (from 0.060 to 0.576 mm h^-1) for SF, FF and BF, respectively. The highest average monthly Er occurred in June was 0.120 mm h^-1 for SF, 0.317 mm h^-1for FF and 0.169 mm h^-1 for BF, and the lowest value in October was 0.083 mm h^-1 for SF, 0.187 mm h^-1 for FF and 0.101 mm h^-1 for BF, respectively. The averages of Er from 8:00 to16:00 were significantly higher than those from 0:00 to 8:00 and from 16:00 to 0:00 for the three stands. The marked daily and monthly differences of Er were contributable to the variations of solar radiation, air temperature and relative humidity above the canopy.     

Monte Carlo Simulations of Oort Cloud Comets: the Influence of Mantle Blow-off on the Inclination Distribution of Jupiter Family

We have developed an efficient Monte Carlo method by which we can evaluate the evolution of comets. There are many poorly known evolutional parameters, and we have investigated the influence of these parameters on the final populations and the inclination distributions of short-period comets. We compare the observed and calculated inclination distributions of different comet populations and obtain a good fit for the inclinations of the Jupiter family comets by assuming a mantle blow-off and a sudden brightening of the comet when its perihelion distance is lowered in a major jump. This revised version was published online in July 2006 with corrections to the Cover Date.     

Mapping the three-body system – decay time and reversibility

In this paper we carry out a quantitative analysis of the three-body systems and map them as a function of decaying time and initial configuration, look at this problem as an example of a simple deterministic system and ask to what extent the orbits are really predictable. We have investigated the behaviour of about 200 000 general Newtonian three-body systems using the simplest initial conditions. Within our resolution these cover all the possible states where the objects are initially at rest and have no angular momentum. We have determined the decay time-scales of the triple systems and show that the distribution of this parameter is fractal in appearance. Some areas that appear stable on large scales exhibit very narrow strips of instability and the overall pattern, dominated by resonances, reminds us of a traditional Maasai warrior shield. Also an attempt is made to recover the original starting configuration of the three bodies by backward integration. We find there are instances where the evolution to the future and to the past lead to different orbits, in spite of time symmetric initial conditions. This implies that even in simple deterministic systems there exists an arrow of time.     

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.     

Orbits of short period comets captured from the Oort cloud

Oort cloud comets occasionally obtain orbits which take them through the planetary region. The perturbations by the planets are likely to change the orbit of the comet. We model this process by using a Monte Carlo method and cross sections for orbital changes, i.e. changes in energy, inclination and perihelion distance, in a single planet-comet encounter. The influence of all major planets is considered. We study the distributions of orbital parameters of observable comets, i.e. those which have perihelion distance smaller than a given value. We find that enough comets are captured from the Oort cloud in order to explain the present populations of short period comets. The median value of cos i for the Jupiter family is 0.985 while it is 0.27 for the Halley types. The results may explain the orbital features of short period comets, assuming that the active lifetime of a comet is not much greater than 400 orbital revolutions.