Individual Differences in Children’s Paths to Arithmetical Development

Cross-sectional and longitudinal data consistently indicate that mathematical difficulties are more prevalent in older than in younger children (e.g. Department of Education, 2011). Children’s trajectories can take a variety of shapes such as linear, flat, curvilinear and uneven, and shape has been found to vary within children and across tasks (J Jordan et al. 2009). There has been an increase in the use of statistical methods which are specifically designed to study development, and this has greatly improved our understanding of children’s mathematical development. However, the effects of many cognitive and social variables (e.g. working memory and verbal ability) on mathematical development are unclear. It is likely that greater consistency between studies will be achieved by adopting a componential approach to study mathematics, rather than treating mathematics as a unitary concept.

Individual and couple decision behavior under risk: evidence on the dynamics of power balance

This article reports results of an experiment designed to analyze the link between risky decisions made by couples and risky decisions made separately by each spouse. We estimate both the spouses and the couples' degrees of risk aversion, we assess how the risk preferences of the two spouses aggregate when they make risky decisions, and we shed light on the dynamics of the decision process that takes place when couples make risky decisions. We find that, far from being fixed, the balance of power within the household is malleable. In most couples, men have, initially, more decision-making power than women but women who ultimately implement the joint decisions gain more and more power over the course of decision making.

The Unification of Asymmetry Signatures of Type Ia Supernovae

We present a compilation of the geometry measures acquired using optical and IR spectroscopy and optical spectropolarimetry to probe the explosion geometry of Type Ia supernovae (SNe Ia). Polarization measurements are sensitive to asymmetries in the plane of the sky, whereas line profiles in nebular phase spectra are expected to trace asymmetries perpendicular to the plane of the sky. The combination of these two measures can overcome their respective projection effects, completely probing the structures of these events. For nine normal SNe Ia, we find that the polarization of Si II ?6355 at 5 days before maximum (p Si II ) is well correlated with its velocity evolution (\dot{v}_Si II), implying that \dot{v}_Si II is predominantly due to the asymmetry of the SNe. We find only a weak correlation between the polarization of Si II and the reported velocities (v neb) for peak emission of optical Fe II and Ni II lines in nebular spectra. Our sample is biased, with polarization measurements being only available for normal SNe that subsequently exhibited positive (i.e., redshifted) v neb. In unison these indicators are consistent with an explosion in which the outer layers are dominated by a spherical oxygen layer, mixed with an asymmetric distribution of intermediate-mass elements. The combination of spectroscopic and spectropolarimetric indicators suggests a single geometric configuration for normal SNe Ia, with some of the diversity of observed properties arising from orientation effects.

Detonations in sub-chandrasekhar-mass C+O white dwarfs

Explosions of sub-Chandrasekhar-mass white dwarfs (WDs) are one alternative to the standard Chandrasekhar-mass model of Type Ia supernovae (SNe Ia). They are interesting since binary systems with sub-Chandrasekhar-mass primary WDs should be common and this scenario would suggest a simple physical parameter which determines the explosion brightness, namely the mass of the exploding WD. Here we perform one-dimensional hydrodynamical simulations, associated post-processing nucleosynthesis, and multi-wavelength radiation transport calculations for pure detonations of carbon-oxygen WDs. The light curves and spectra we obtain from these simulations are in good agreement with observed properties of SNe Ia. In particular, for WD masses from 0.97 to 1.15 Msun we obtain 56Ni masses between 0.3 and 0.8 Msun, sufficient to capture almost the complete range of SN Ia brightnesses. Our optical light curve rise times, peak colors, and decline timescales display trends which are generally consistent with observed characteristics although the range of B-band decline timescales displayed by our current set of models is somewhat too narrow. In agreement with observations, the maximum light spectra of the models show clear features associated with intermediate-mass elements and reproduce the sense of the observed correlation between explosion luminosity and the ratio of the Si II lines at ?6355 and ?5972. We therefore suggest that sub-Chandrasekhar-mass explosions are a viable model for SNe Ia for any binary evolution scenario leading to explosions in which the optical display is dominated by the material produced in a detonation of the primary WD. © 2010. The American Astronomical Society.

Double-detonation sub-Chandrasekhar supernovae: Synthetic observables for minimum helium shell mass models

In the double-detonation scenario for Type Ia supernovae, it is suggested that a detonation initiates in a shell of helium-rich material accreted from a companion star by a sub-Chandrasekhar-mass white dwarf. This shell detonation drives a shock front into the carbon-oxygen white dwarf that triggers a secondary detonation in the core. The core detonation results in a complete disruption of the white dwarf. Earlier studies concluded that this scenario has difficulties in accounting for the observed properties of Type Ia supernovae since the explosion ejecta are surrounded by the products of explosive helium burning in the shell. Recently, however, it was proposed that detonations might be possible for much less massive helium shells than previously assumed (Bildsten et al.). Moreover, it was shown that even detonations of these minimum helium shell masses robustly trigger detonations of the carbon-oxygen core (Fink et al.). Therefore, it is possible that the impact of the helium layer on observables is less than previously thought. Here, we present time-dependent multi-wavelength radiative transfer calculations for models with minimum helium shell mass and derive synthetic observables for both the optical and ? -ray spectral regions. These differ strongly from those found in earlier simulations of sub-Chandrasekhar-mass explosions in which more massive helium shells were considered. Our models predict light curves that cover both the range of brightnesses and the rise and decline times of observed Type Ia supernovae. However, their colors and spectra do not match the observations. In particular, their B - V colors are generally too red. We show that this discrepancy is mainly due to the composition of the burning products of the helium shell of the Fink et al. models which contain significant amounts of titanium and chromium. Using a toy model, we also show that the burning products of the helium shell depend crucially on its initial composition. This leads us to conclude that good agreement between sub-Chandrasekhar-mass explosions and observed Type Ia supernovae may still be feasible but further study of the shell properties is required.

Double-detonation sub-Chandrasekhar supernovae: Can minimum helium shell masses detonate the core?

The explosion of sub-Chandrasekhar mass white dwarfs via the double detonation scenario is a potential explanation for type Ia supernovae. In this scenario, a surface detonation in a helium layer initiates a detonation in the underlying carbon/oxygen core leading to an explosion. For a given core mass, a lower bound has been determined on the mass of the helium shell required for dynamical burning during a helium flash, which is a necessary prerequisite for detonation. For a range of core and corresponding minimum helium shell masses, we investigate whether an assumed surface helium detonation is capable of triggering a subsequent detonation in the core even for this limiting case. We carried out hydrodynamic simulations on a co-expanding Eulerian grid in two dimensions assuming rotational symmetry. The detonations are propagated using the level-set approach and a simplified scheme for nuclear reactions that has been calibrated with a large nuclear network. The same network is used to determine detailed nucleosynthetic abundances in a post-processing step. Based on approximate detonation initiation criteria in the literature, we find that secondary core detonations are triggered for all of the simulated models, ranging in core mass from 0.810 up to 1.385 M? with corresponding shell masses from 0.126 down to 0.0035 M?. This implies that, as soon as a detonation triggers in a helium shell covering a carbon/oxygen white dwarf, a subsequent core detonation is virtually inevitable.

Light curves for off-centre ignition models of Type Ia supernovae

Motivated by recent models involving off-centre ignition of Type Ia supernova explosions, we undertake three-dimensional time-dependent radiation transport simulations to investigate the range of bolometric light-curve properties that could be observed from supernovae in which there is a lop-sided distribution of the products from nuclear burning. We consider both a grid of artificial toy models which illustrate the conceivable range of effects and a recent three-dimensional hydrodynamical explosion model. We find that observationally significant viewing angle effects are likely to arise in such supernovae and that these may have important ramifications for the interpretation of the observed diversity of Type Ia supernova and the systematic uncertainties which relate to their use as standard candles in contemporary cosmology. © 2007 RAS.

2D simulations of the double-detonation model for thermonuclear transients from low-mass carbon-oxygen white dwarfs

Thermonuclear explosions may arise in binary star systems in which a carbon-oxygen (CO) white dwarf (WD) accretes helium-rich material from a companion star. If the accretion rate allows a sufficiently large mass of helium to accumulate prior to ignition of nuclear burning, the helium surface layer may detonate, giving rise to an astrophysical transient. Detonation of the accreted helium layer generates shock waves that propagate into the underlying CO WD. This might directly ignite a detonation of the CO WD at its surface (an edge-lit secondary detonation) or compress the core of the WD sufficiently to trigger a CO detonation near the centre. If either of these ignition mechanisms works, the two detonations (helium and CO) can then release sufficient energy to completely unbind the WD. These 'double-detonation' scenarios for thermonuclear explosion of WDs have previously been investigated as a potential channel for the production of Type Ia supernovae from WDs of ~ 1 M . Here we extend our 2D studies of the double-detonation model to significantly less massive CO WDs, the explosion of which could produce fainter, more rapidly evolving transients. We investigate the feasibility of triggering a secondary core detonation by shock convergence in low-mass CO WDs and the observable consequences of such a detonation. Our results suggest that core detonation is probable, even for the lowest CO core masses that are likely to be realized in nature. To quantify the observable signatures of core detonation, we compute spectra and light curves for models in which either an edge-lit or compression-triggered CO detonation is assumed to occur. We compare these to synthetic observables for models in which no CO detonation was allowed to occur. If significant shock compression of the CO WD occurs prior to detonation, explosion of the CO WD can produce a sufficiently large mass of radioactive iron-group nuclei to significantly affect the light curves. In particular, this can lead to relatively slow post-maximum decline. If the secondary detonation is edge-lit, however, the CO WD explosion primarily yields intermediate-mass elements that affect the observables more subtly. In this case, near-infrared observations and detailed spectroscopic analysis would be needed to determine whether a core detonation occurred. We comment on the implications of our results for understanding peculiar astrophysical transients including SN 2002bj, SN 2010X and SN 2005E. © 2012 The Authors Monthly Notices of the Royal Astronomical Society © 2012 RAS.

A theoretical color-velocity correlation for supernovae associated with gamma-ray bursts

We carry out the first multi-dimensional radiative transfer calculations to simultaneously compute synthetic spectra and light curves for models of supernovae driven by fast bipolar outflows. These allow us to make self-consistent predictions for the orientation dependence of both color evolution and spectral features. We compare models with different degrees of asphericity and metallicity and find significant observable consequences of both. In aspherical models, we find spectral and light curve features that vary systematically with observer orientation. In particular, we find that the early-phase light curves are brighter and bluer when viewed close to the polar axis but that the peak flux is highest for equatorial (off-axis) inclinations. Spectral line features also depend systematically on observer orientation, including the velocity of the Si II 6355 Å line. Consequently, our models predict a correlation between line velocity and color that could assist the identification of supernovae associated with off-axis jet-driven explosions. The amplitude and range of this correlation depends on the degree of asphericity, the metallicity, and the epoch of observation but we find that it is always present and acts in the same direction. © 2012. The American Astronomical Society. All rights reserved..

Normal type Ia supernovae from violent mergers of white dwarf binaries

One of the most important questions regarding the progenitor systems of Type Ia supernovae (SNe Ia) is whether mergers of two white dwarfs can lead to explosions that reproduce observations of normal events. Here we present a fully three-dimensional simulation of a violent merger of two carbon-oxygen white dwarfs with masses of 0.9 M and 1.1 M combining very high resolution and exact initial conditions. A well-tested combination of codes is used to study the system. We start with the dynamical inspiral phase and follow the subsequent thermonuclear explosion under the plausible assumption that a detonation forms in the process of merging. We then perform detailed nucleosynthesis calculations and radiative transfer simulations to predict synthetic observables from the homologously expanding supernova ejecta. We find that synthetic color light curves of our merger, which produces about 0.62 M of Ni, show good agreement with those observed for normal SNe Ia in all wave bands from U to K. Line velocities in synthetic spectra around maximum light also agree well with observations. We conclude that violent mergers of massive white dwarfs can closely resemble normal SNe Ia. Therefore, depending on the number of such massive systems available these mergers may contribute at least a small fraction to the observed population of normal SNe Ia. © 2012 The American Astronomical Society. All rights reserved.

On the γ-ray emission of Type Ia supernovae

A multidimension, time-dependent Monte Carlo code is used to compute sample ?-ray spectra to explore whether unambiguous constraints could be obtained from ?-ray observations of Type Ia supernovae. Both spherical and aspherical geometries are considered and it is shown that moderate departures from sphericity can produce viewing-angle effects that are at least as significant as those caused by the variation of key parameters in 1D models. Thus, ?-ray data could, in principle, carry some geometrical information, and caution should be applied when discussing the value of ?-ray data based only on 1D explosion models. In light of the limited sensitivity of current ?-ray observatories, the computed theoretical spectra are studied to revisit the issue of whether useful constraints could be obtained for moderately nearby objects. The most useful ?-ray measurements are likely to be of the light curve and time-dependent hardness ratios, but sensitivity higher than currently available, particularly at relatively hard energies (~2-3 MeV), is desirable. © 2008 The Authors. Journal compilation © 2008 RAS.

Time-dependent three-dimensional spectrum synthesis for Type Ia supernovae

A Monte Carlo code (artis) for modelling time-dependent three-dimensional spectral synthesis in chemically inhomogeneous models of Type Ia supernova ejecta is presented. Following the propagation of ?-ray photons, emitted by the radioactive decay of the nucleosynthesis products, energy is deposited in the supernova ejecta and the radiative transfer problem is solved self-consistently, enforcing the constraint of energy conservation in the comoving frame. Assuming a photoionization-dominated plasma, the equations of ionization equilibrium are solved together with the thermal balance equation adopting an approximate treatment of excitation. Since we implement a fully general treatment of line formation, there are no free parameters to adjust. Thus, a direct comparison between synthetic spectra and light curves, calculated from hydrodynamic explosion models, and observations is feasible. The code is applied to the well-known W7 explosion model and the results tested against other studies. Finally, the effect of asymmetric ejecta on broad-band light curves and spectra is illustrated using an elliptical toy model. © 2009 RAS.