Numerical modelling of heating in porous planetesimal collisions

Collisions between planetesimals at speeds of several kilometres per second were common during the early evolution of our Solar System. However, the collateral effects of these collisions are not well understood. In this paper, we quantify the efficiency of heating during high-velocity collisions between planetesimals using hydrocode modelling. We conducted a series of simulations to test the effect on shock heating of the initial porosity and temperature of the planetesimals, the relative velocity of the collision and the relative size of the two colliding bodies. Our results show that while heating is minor in collisions between non-porous planetesimals at impact velocities below 10 km s⁻¹, in agreement with previous work, much higher temperatures are reached in collisions between porous planetesimals. For example, collisions between nearly equal-sized, porous planetesimals can melt all, or nearly all, of the mass of the bodies at collision velocities below 7 km s⁻¹. For collisions of small bodies into larger ones, such as those with an impactor-to-target mass ratio below 0.1, significant localised heating occurs in the target body. At impact velocities as low as 5 km s⁻¹, the mass of melt will be nearly double the mass of the impactor, and the mass of material shock heated by 100 K will be nearly 10 times the mass of the impactor. We present a first-order estimate of the cumulative effects of impact heating on a porous planetesimal parent body by simulating the impact of a population of small bodies until a disruptive event occurs. Before disruption, impact heating is volumetrically minor and highly localised; in no case was more than about 3% of the parent body heated by more than 100 K. However, heating during the final disruptive collision can be significant; in about 10% of cases, almost all of the parent body is heated to 700 K (from an initial temperature of ∼300 K) and more than a tenth of the parent body mass is melted. Hence, energetic collisions between planetesimals could have had important effects on the thermal evolution of primitive materials in the early Solar System.     

Numerical modelling of shock waves in the solar wind

This review briefly surveys some recent progress in the numerical simulation study of magnetohydrodynamic (MHD) shock waves in the solar wind.     

Numerical modelling of permafrost in bedrock in northern Fennoscandia during the Holocene

The occurrence of permafrost in bedrock in northern Fennoscandia and its dependence on past and presently ongoing climatic variations was investigated with one- (1D) and two-dimensional (2D) numerical models by solving the transient heat conduction equation with latent heat effects included. The study area is characterized by discontinuous permafrost occurrences such as palsa mires and local mountain permafrost. The ground temperature changes during the Holocene were constructed using climatic proxy data. This variation was used as a forcing function at the ground surface in the calculations. Several versions of the present ground temperature were applied, resulting in different subsurface freezing–thawing conditions in the past depending on the assumed porosity and geothermal conditions.Our results suggest that in high altitude areas with a cold climate (present mean annual ground temperature between 0°C and −3°C), there may have been considerable variations in permafrost thickness (ranging from 0 to 150 m), as well as periods of no permafrost at all. The higher is the porosity of bedrock filled with ice, the stronger is the retarding effect of permafrost against climatic variations.Two-dimensional models including topographic effects with altitude-dependent ground temperatures and slope orientation and inclination dependent solar radiation were applied to a case of mountain permafrost in Ylläs, western Finnish Lapland, where bedrock permafrost is known to occur in boreholes to a depth of about 60 m. Modelling suggests complicated changes in permafrost thickness with time as well as contrasting situations on southern and northern slopes of the mountain.Extrapolating the climatic warming of the last 200 years to the end of the next century when the anticipated increase in the annual average air temperature is expected to be about 2 K indicates that the permafrost occurrences in bedrock in northern Fennoscandia would be thawing rapidly in low-porosity formations. However, already a porosity of 5% filled with ice would retard the thawing considerably.     

Numerical investigation of motions of resonant asteroids in the three-dimensional space

Motions of asteroids in mean motion resonances with Jupiter are studied in three-dimensional space. Orbital changes of fictitious asteroids in the Kirkwood gaps are calculated by numerical integrations for 10⁵ – 10⁶ years. The main results are as follows: (1) There are various motions of resonant asteroids, and some of them are very complicated and chaotic and others are regular. (2) The eccentricity of some asteroids becomes very large, and the variation of the inclination is large while the eccentricity is large. (3) In the 3:1 resonance, there is a long periodic change in the variation of the inclination, when (7 : ) is a simple ratio (7: longitude of perihelion, : longitude of node). (4) In the 7:3 resonance, the variation of the inclination of some resonant asteroids is so large that prograde motion becomes retrograde. Some asteroids in the 7:3 resonance can collide with the Sun as well as with the inner planets.     

Numerical studies of the transport of solar protons in interplanetary space

Numerical solutions of the Fokker-Planck equation governing the transport of solar protons are obtained using the Crank-Nicholson technique with the diffusion coefficient represented by K_r=K₀r^b where r is radial distance from the Sun and b can take on positive or negative values. As b ranges from +1 to ?3, the time to the observation of peak flux decreases by a factor of 5 for 1 MeV protons when $\text{V}\text{K}{}_0\text{=}\text{3 AU}{}^{\mathrm{b?1}}$ where V is the solar wind speed. The time to peak flux is found to be very insensitive to assumptions concerning the solar and outer scattering boundary conditions and the presence of exponential time decay in the flux does not depend on the existence of an outer boundary. At $\text{V}\text{K}{}_0\text{? 15 AU}{}^{\mathrm{b?1}}$, 1 MeV particles come from the Sun by an almost entirely convective process and suffer large adiabatic deceleration at b?0 but for b=+1, large Fermi acceleration is possible at all reasonable $\text{V}\text{K}{}_0$ values. Implications of this result for the calculation and measurement of particle diffusion coefficients is discussed. At $\text{b?0}$, the pure diffusion approximation to transport overestimates by a factor 2 or more the time to peak flux but as b becomes more negative, the additional effects of convection and energy loss become less important.     

Libration dynamics and stability of electrodynamic tethers in satellite deorbit

This paper studies libration dynamics and stability of deorbiting nano-satellites by short and bare electrodynamic tethers. A critical aspect of satellite deorbit by an electrodynamic tether is to maintain the tether aligned with the local vertical and stable while subjected to external perturbations. The dynamics of electrodynamic tether system in deorbit application is divided into the orbital motion of the center of system’s mass and the tether libration motion relative to that center. Major space environmental perturbations including the current-induced electrodynamic force, atmospheric drag, oblateness effect of the Earth, irregularity of geomagnetic field, variable plasma density, solar radiation pressure, and lunisolar gravitational attractions are considered in the dynamic analysis. Quantitative analyses are provided in order to characterize the order of the perturbative torques during the deorbit process. A single index is derived from the libration energy to stabilize the libration motion by regulating the current in the tether through simple on-off switching. Numerical results show that the libration dynamics of an electrodynamic tether has significant impacts on the deorbit process and the electrodynamic tether cannot effectively deorbit satellites without libration stability control. The proposed current regulation strategy is simple and very effective in stabilizing libration motion of an electrodynamic tether.     

Numerical modeling of the formation,orbital evolution,and distribution of fragments of space debris in near-Earth space

This paper describes the numerical models designed by the authors to investigate the formation, orbital evolution, and spatial distribution of fragments of space debris emerging in orbits as a result of space-craft fragmentation. It cites the results of the testing of the models and the data of their use.     

Numerical modelling of the impact crater depth-diameter dependence in an acoustically fluidized target

2D numerical modelling of impact cratering has been utilized to quantify an important depth-diameter relationship for different crater morphologies, simple and complex. It is generally accepted that the final crater shape is the result of a gravity-driven collapse of the transient crater, which is formed immediately after the impact. Numerical models allow a quantification of the formation of simple craters, which are bowl-shaped depressions with a lens of rock debris inside, and complex craters, which are characterized by a structural uplift. The computation of the cratering process starts with the first contact of the impactor and the planetary surface and ends with the morphology of the final crater. Using different rheological models for the sub-crater rocks, we quantify the influence on crater mechanics. To explain the formation of complex craters in accordance to the threshold diameter between simple and complex craters, we utilize the Acoustic Fluidization model. We carried out a series of simulations over a broad parameter range with the goal to fit the observed depth/diameter relationships as well as the observed threshold diameters on the Moon, Earth and Venus.     

Numerical modelling of the vertical structure and dark halo parameters in disc galaxies

The non‐linear dynamics of bending instability and vertical structure of a galactic stellar disc embedded into a spherical halo are studied with N‐body numerical modelling. Development of the bending instability in stellar galactic disc is considered as the main factor that increases the disc thickness. Correlation between the disc vertical scale height and the halo‐to‐disc mass ratio is predicted from the simulations. The method of assessment of the spherical‐to‐disc mass ratio for edge‐on spiral galaxies with a small bulge is considered. Modelling of eight edge‐on galaxies: NGC 891, NGC 4738, NGC 5170, UGC 6080, UGC 7321, UGC 8286, UGC 9422 and UGC 9556 is performed. Parameters of stellar discs, dark haloes and bulges are estimated. The lower limit of the dark‐to‐luminous mass ratio in our galaxies is of the order of one within the limits of their stellar discs. The dark haloes dominate by mass in the galaxies with very thin stellar discs (NGC 5170, UGC 7321 and UGC 8286) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Use of multi-point analysis and modelling to address cross-scale coupling in space plasmas: Lessons from Cluster

The properties of plasmas (in space) are fundamentally governed by both ‘cross-scale’ coupling and comparative temporal behaviour operating over the micro-, meso-, and (MHD-) fluid regimes: for example, under conditions of turbulence, during magnetic reconnection and in shocks and other plasma boundaries. These themes map to a number of related and overlapping, phenomena, where known phenomena play different roles in each theme. Detailed understanding of fundamental plasma processes therefore requires analysis of both theoretical models (to distinguish the collisionless from the collisional regimes) and multi-scale measurements (suitable to address issues of stationarity). In particular, the investigation of phenomena requires analysis techniques which can distinguish and quantify temporal behaviour and the multi-scale spatial behaviour. The analysis of existing, multi-point data sets has led to a number of data co-ordination methods, such as the four spacecraft analysis tools developed for cluster, and we consider examples here. Advanced analysis concepts may be investigated with suitable considerations of measurement quality:adequate sampling of phenomena (for example, to extract the necessary information on the mechanisms operating) requires suitable spacecraft configurations and directly relates to the measurement quality achievable. A particular issue is how to resolve temporal behaviour across the spatial regimes, so that the data set is suitably coordinated. With the addition of theoretical modelling (in the context of particular phenomena) both the space and laboratory plasma regimes may be compared and we give an example of nonlinear wave coupling across spatial scales in this context.     

Atomic modelling of nova shells

Atomic models developed by the author and used to interpret the spectra of cool shells from old classical novae are reviewed. Some of the remaining problems relating to the spectra are discussed briefly. The use of these data in photoionization models is described, and the evolution of nova shells is discussed.     

CO modelling of CW leo

Preliminary results are presented of fitting the spectral energy distribution and newly obtained¹²CO and¹³CO J =1-0, 2-1 and 3-2 line profiles of CW Leo. This is done with a model which includes a self-consistent calculation of the gas temperature.     

Shock modelling of Planetary Nebulae

The kinematics of Planetary Nebulae are analyzed in terms of the solutions to the equations of hydrodynamic equilibrium developed by J. Cantó. We apply our analysis to the Planetary Nebulae NGC 6905 and NGC 6537. A detailed spectroscopic study of these objects reveals the existence of high nuclear velocities, together with complex kinematic structures and unusual emission line intensities. Shock ionization clearly plays a key role in these nebulae. Remarkably good agreement is obtained when comparing the synthetic maps and spectra resulting from the shock solutions with the observational data.     

Regression modelling in selenodesy

Earth, Moon, and Planets - The existing models of different orders that describe surface and gravitational fields of planets contain a redundant number of members with no useful information, which...     

Reorientation of planets with elastic lithospheres

The orientation of a planet is controlled by the positions of the principal axes of the inertia tensor relative to the planetary surface. Using the theory for the deflection of thin elastic shells the principal axes are computed after emplacement of an arbitrary axisymmetrical load. The partial compensation of the load and the partial relaxation of rotational flattening are included in the computation. It is found that the amount of reorientation is independent of lithosphere thickness. The parameters controlling the amount of reorientation are the location of the load and the size of the load compared to the rotational flattening. The results indicate that the Tharsis rise has probably reoriented Mars by only 3 to 9° and certainly less that 18°. The position of the Caloris Basin on Mercury indicates that if the surrounding lava sheet controls the planetary orientation then the lava sheet is probably less than 2000 m thick.     

Gaussian process modelling of asteroseismic data

The measured properties of stellar oscillations can provide powerful constraints on the internal structure and composition of stars. To begin this process, oscillation frequencies must be extracted from the observational data, typically time series of the star's brightness or radial velocity. In this paper, a probabilistic model is introduced for inferring the frequencies and amplitudes of stellar oscillation modes from data, assuming that there is some periodic character to the oscillations, but that they may not be exactly sinusoidal. Effectively, we fit damped oscillations to the time series, and hence the mode lifetime is also recovered. While this approach is computationally demanding for large time series (>1500 points), it should at least allow improved analysis of observations of solar-like oscillations in subgiant and red giant stars, as well as sparse observations of semiregular stars, where the number of points in the time series is often low. The method is demonstrated on simulated data and then applied to radial velocity measurements of the red giant star  ξ Hydrae  , yielding a mode lifetime between 0.41 and 2.65 d with 95 per cent posterior probability. The large frequency separation between modes is ambiguous, however we argue that the most plausible value is 6.3 μHz, based on the radial velocity data and the star's position in the Hertzsprung–Russell diagram.     

Mathematical modelling of the sunspot cycle

The sunspot record for the time interval 1749–1977 can be represented conveniently by an harmonic model comprising a relatively large number of lines. Solar activity can otherwise be considered as a sequence of partly overlapping events, triggered periodically at intervals of the order of 11 years. Each individual cycle is approximated by a function of the Maxwell distribution type; the resulting impulse model consists of the superposition of the independent pulses. Application of these two models for the prediction of annual values of the Wolf sunspot numbers leads to controversial results. Mathematical modelling of the sunspot time series does not give an unambiguous result.     

Theoretical modelling of SiO maser lineshapes

Various factors affecting the lineshapes of SiO masers have been considered: saturation, competitive gain, velocity redistribution and properties of the circumstellar envelope supporting the masers. The lineshape of a single maser spot is likely to be dominated by conditions of complete velocity redistribution (CVR) for low to moderate amplification, changing to NVR (no velocity redistribution) only for strongly saturating masers. Both CVR and NVR regimes can produce peculiar lineshapes. When many maser cells were considered, distributed at random in the stellar envelope, the individual lineshapes were combined to produce synthetic spectra. Maser action is predicted in higher rotational transitions than those observed to date, for example the J=7-6 transition of v=2.