Energy conserving schemes for the simulation of musical instrument contact dynamics

Collisions are an innate part of the function of many musical instruments. Due to the nonlinear nature of contact forces, special care has to be taken in the construction of numerical schemes for simulation and sound synthesis. Finite difference schemes and other time-stepping algorithms used for musical instrument modelling purposes are normally arrived at by discretising a Newtonian description of the system. However because impact forces are non-analytic functions of the phase space variables, algorithm stability can rarely be established this way. This paper presents a systematic approach to deriving energy conserving schemes for frictionless impact modelling. The proposed numerical formulations follow from discretising Hamilton׳s equations of motion, generally leading to an implicit system of nonlinear equations that can be solved with Newton׳s method. The approach is first outlined for point mass collisions and then extended to distributed settings, such as vibrating strings and beams colliding with rigid obstacles. Stability and other relevant properties of the proposed approach are discussed and further demonstrated with simulation examples. The methodology is exemplified through a case study on tanpura string vibration, with the results confirming the main findings of previous studies on the role of the bridge in sound generation with this type of string instrument.

Electron impact excitation of Mg VIII: Collision strengths, transition probabilities and theoretical EUV and soft X-ray line intensities for Mg VIII

Context: Mg VIII emission lines are observed in a range of astronomical objects such as the Sun, other cool stars and in the coronal line region of Seyfert galaxies. Under coronal conditions Mg VIII emits strongly in the extreme ultraviolet (EUV) and soft X-ray spectral regions which makes it an ideal ion for plasma diagnostics.

Aims. Two theoretical atomic models, consisting of 125 fine structure levels, are developed for the Mg VIII ion. The 125 levels arise from the 2s(2)2p, 2s(2)p2, 2p(3), 2s(2)3s, 2s(2)3p, 2s(2)3d, 2s2p3s, 2s2p3p, 2s2p3d, 2p(2)3s, 2p(2)3p and 2p(2)3d configurations. Electron impact excitation collision strengths and radiative transition probabilities are calculated for both Mg VIII models, compared with existing data, and the best model selected to generate a set of theoretical emission line intensities. The EUV lines, covering 312-790 angstrom, are compared with existing solar spectra (SERTS-89 and SUMER), while the soft X-ray transitions (69-97 angstrom) are examined for potential density diagnostic line ratios and also compared with the limited available solar and stellar observational data.

Methods. The R-matrix codes Breit-Pauli RMATRXI and RMATRXII are utilised, along with the PSTGF code, to calculate the collision strengths for two Mg VIII models. Collision strengths are averaged over a Maxwellian distribution to produce the corresponding effective collision strengths for use in astrophysical applications. Transition probabilities are also calculated using the CIV3 atomic structure code. The best data are then incorporated into the modelling code CLOUDY and line intensities generated for a range of electron temperatures and densities appropriate to solar and stellar coronal plasmas.

Results. The present effective collision strengths are compared with two previous calculations. Good levels of agreement are found with the most recent, but there are large differences with the other for forbidden transitions. The resulting line intensities compare favourably with the observed values from the SERTS-89 and SUMER spectra. Theoretical soft X-ray emission lines are presented and several density diagnostic line ratios examined, which are in reasonable agreement with the limited observational data available.

Modelling the brightness increase signature due to asteroid collisions

We have developed a model to predict the post-collision brightness increase of sub-catastrophic collisions between asteroids and to evaluate the likelihood of a survey detecting these events. It is based on the cratering scaling laws of Holsapple and Housen (2007) and models the ejecta expansion following an impact as occurring in discrete shells each with their own velocity. We estimate the magnitude change between a series of target/impactor pairs, as- suming it is given by the increase in reflecting surface area within a photometric aperture due to the resulting ejecta. As expected the photometric signal increases with impactor size, but we find also that the photometric signature decreases rapidly as the target aster- oid diameter increases, due to gravitational fallback. We have used the model results to make an estimate of the impactor diameter for the (596) Scheila collision of D = 49 − 65m depending on the impactor taxonomy, which is broadly consistent with previous estimates. We varied both the strength regime (highly porous and sand/cohesive soil) and the tax- onomic type (S-, C- and D-type) to examine the effect on the magnitude change, finding that it is significant at early stages but has only a small effect on the overall lifetime of the photometric signal. Combining the results of this model with the collision frequency estimates of Bottke et al. (2005), we find that low-cadence surveys of ∼one visit per luna- tion will be insensitive to impacts on asteroids with D < 20km if relying on photometric detections.

Radiative hydrodynamic modelling and observations of the X-class solar flare on 2011 March 9

Aims. We investigated the response of the solar atmosphere to non-thermal electron beam heating using the radiative transfer and hydrodynamics modelling code RADYN. The temporal evolution of the parameters that describe the non-thermal electron energy distribution were derived from hard X-ray observations of a particular flare, and we compared the modelled and observed parameters.

Methods. The evolution of the non-thermal electron beam parameters during the X1.5 solar flare on 2011 March 9 were obtained from analysis of RHESSI X-ray spectra. The RADYN flare model was allowed to evolve for 110 s, after which the electron beam heating was ended, and was then allowed to continue evolving for a further 300 s. The modelled flare parameters were compared to the observed parameters determined from extreme-ultraviolet spectroscopy.

Results. The model produced a hotter and denser flare loop than that observed and also cooled more rapidly, suggesting that additional energy input in the decay phase of the flare is required. In the explosive evaporation phase a region of high-density cool material propagated upward through the corona. This material underwent a rapid increase in temperature as it was unable to radiate away all of the energy deposited across it by the non-thermal electron beam and via thermal conduction. A narrow and high-density (ne ≤ 1015 cm-3) region at the base of the flare transition region was the source of optical line emission in the model atmosphere. The collision-stopping depth of electrons was calculated throughout the evolution of the flare, and it was found that the compression of the lower atmosphere may permit electrons to penetrate farther into a flaring atmosphere compared to a quiet Sun atmosphere.