Embracing Integrability in Stellar and Planetary Dynamics

Hamiltonian systems in stellar and planetary dynamics are typically near integrable. For example, Solar System planets are almost in two-body orbits, and in simulations of the Galaxy, the orbits of stars seem regular. For such systems, sophisticated numerical methods can be developed through integrable approximations. Following this theme, we discuss three distinct problems. We start by considering numerical integration techniques for planetary systems. Perturbation methods (that utilize the integrability of the two-body motion) are preferred over conventional "blind" integration schemes. We introduce perturbation methods formulated with Cartesian variables. In our numerical comparisons, these are superior to their conventional counterparts, but, by definition, lack the energy-preserving properties of symplectic integrators. However, they are exceptionally well suited for relatively short-term integrations in which moderately high positional accuracy is required. The next exercise falls into the category of stability questions in solar systems. Traditionally, the interest has been on the orbital stability of planets, which have been quantified, e.g., by Liapunov exponents. We offer a complementary aspect by considering the protective effect that massive gas giants, like Jupiter, can offer to Earth-like planets inside the habitable zone of a planetary system. Our method produces a single quantity, called the escape rate, which characterizes the system of giant planets. We obtain some interesting results by computing escape rates for the Solar System. Galaxy modelling is our third and final topic. Because of the sheer number of stars (about 10^11 in Milky Way) galaxies are often modelled as smooth potentials hosting distributions of stars. Unfortunately, only a handful of suitable potentials are integrable (harmonic oscillator, isochrone and Stäckel potential). This severely limits the possibilities of finding an integrable approximation for an observed galaxy. A solution to this problem is torus construction; a method for numerically creating a foliation of invariant phase-space tori corresponding to a given target Hamiltonian. Canonically, the invariant tori are constructed by deforming the tori of some existing integrable toy Hamiltonian. Our contribution is to demonstrate how this can be accomplished by using a Stäckel toy Hamiltonian in ellipsoidal coordinates.

Use of wetland buffer areas to reduce nitrogen transport from forested catchments: Retention capacity, emissions of N2O and CH4 and vegetation composition dynamics

The use of buffer areas in forested catchments has been actively researched during the last 15 years; but until now, the research has mainly concentrated on the reduction of sediment and phosphorus loads, instead of nitrogen (N). The aim of this thesis was to examine the use of wetland buffer areas to reduce the nitrogen transport in forested catchments and to investigate the environmental impacts involved in their use. Besides the retention capacity, particular attention was paid to the main factors contributing to the N retention, the potential for increased N2O emissions after large N loading, the effects of peatland restoration for use as buffer areas on CH4 emissions, as well as the vegetation composition dynamics induced by the use of peatlands as buffer areas. To study the capacity of buffer areas to reduce N transport in forested catchments, we first used large artificial loadings of N, and then studied the capacity of buffer areas to reduce ammonium (NH4-N) export originating from ditch network maintenance areas in forested catchments. The potential for increased N2O emissions were studied using the closed chamber technique and a large artificial N loading at five buffer areas. Sampling for CH4 emissions and methane-cycling microbial populations were done on three restored buffer areas and on three buffers constructed on natural peatlands. Vegetation composition dynamics was studied at three buffer areas between 1996 and 2009. Wetland buffer areas were efficient in retaining inorganic N from inflow. The key factors contributing to the retention were the size and the length of the buffer, the hydrological loading and the rate of nutrient loading. Our results show that although the N2O emissions may increase temporarily to very high levels after a large N loading into the buffer area, the buffer areas in forested catchments should be viewed as insignificant sources of N2O. CH4 fluxes were substantially higher from buffers constructed on natural peatlands than from the restored buffer areas, probably because of the slow recovery of methanogens after restoration. The use of peatlands as buffer areas was followed by clear changes in plant species composition and the largest changes occurred in the upstream parts of the buffer areas and the wet lawn-level surfaces, where the contact between the vegetation and the through-flow waters was closer than for the downstream parts and dry hummock sites. The changes in the plant species composition may be an undesired phenomenon especially in the case of the mires representing endangered mire site types, and therefore the construction of new buffer areas should be primarily directed into drained peatland areas.