Richard K. Barrett

The conformal group SO(4,2) and Robertson-Walker spacetimes

The Robertson-Walker spacetimes are conformally flat and so are conformally invariant under the action of the Lie group SO (4,2), the conformal group of Minkowski spacetime. We find a local coordinate transformation allowing the Robertson-Walker metric to be written in a manifestly conformally flat form for all values of the curvature parameter k continuously and use this to obtain the conformal Killing vectors of the Robertson-Walker spacetimes directly from those of the Minkowski spacetime. The map between the Minkowski and Robertson-Walker spacetimes preserves the structure of the Lie algebra so (4,2). Thus the conformal Killing vector basis obtained does not depend upon k , but has the disadvantage that it does not contain explicitly a basis for the Killing vector subalgebra. We present an alternative set of bases that depend (continuously) on k and contain the Killing vector basis as a sub-basis (these are compared with a previously published basis). In particular, bases are presented which include the Killing vectors for all Robertson-Walker spacetimes with additional symmetry, including the Einstein static spacetimes and the de Sitter family of spacetimes, where the basis depends on the Ricci scalar R

Balltracking: an highly efficient method for tracking flow fields

We present a method for tracking solar photospheric flows that is highly efficient, and demonstrate it using high resolution MDI continuum images. The method involves making a surface from the photospheric granulation data, and allowing many small floating tracers or balls to be moved around by the evolving granulation pattern. The results are tested against synthesised granulation with known flow fields and compared to the results produced by Local Correlation tracking (LCT). The results from this new method have similar accuracy to those produced by LCT. We also investigate the maximum spatial and temporal resolution of the velocity field that it is possible to extract, based on the statistical properties of the granulation data. We conclude that both methods produce results that are close to the maximum resolution possible from granulation data. The code runs very significantly faster than our similarly optimised LCT code, making real time applications on large data sets possible. The tracking method is not limited to photospheric flows, and will also work on any velocity field where there are visible moving features of known scale length.

Does the isotropy of the CMB imply a homogeneous universe? Some generalized EGS theorems

We demonstrate that the high isotropy of the cosmic microwave background (CMB), combined with the Copernican principle, is not sufficient to prove homogeneity of the universe - in contrast to previous results on this subject. The crucial additional factor not included in earlier work is the acceleration of the fundamental observers. We find the complete class of irrotational perfect fluid spacetimes admitting an exactly isotropic radiation field for every fundamental observer and show that they are Friedmann-Lema?tre-Robertson-Walker (FLRW) if and only if the acceleration is zero. While inhomogeneous in general, these spacetimes all possess three-dimensional symmetry groups, from which it follows that they also admit a thermodynamic interpretation. In addition to perfect fluids models we also consider multi-component fluids containing non-interacting radiation, dust and a quintessential scalar field or cosmological constant in which the radiation is isotropic for the geodesic (dust) observers. It is shown that the non-acceleration of the fundamental observers forces these spacetimes to be FLRW. While it is plausible that fundamental observers (galaxies) in the real universe follow geodesics, it is strictly necessary to determine this from local observations for the cosmological principle to be more than an assumption. We discuss how observations may be used to test this.

An Empirical Method to Determine Electron Energy Modification Rates from Spatially Resolved Hard X-Ray Data

We discuss a technique for determining the energy loss (or gain) rates affecting high-energy electrons from spatially resolved observations of the hard X-ray bremsstrahlung signature that they produce. The procedure involves two main steps—determining the local electron flux spectrum from inversion of the hard X-ray spectrum using a matrix technique, and evaluating the changes (due to energy losses) in the electron flux spectra at different positions in the source via the continuity equation for total electron flux. In order to test the viability of this numerical technique, we generate a set of simulated hard X-ray photon count spectra, corresponding to different models of electron energy loss, characterized parametrically through an exponent α in the energy loss rate equation, including the case α = 1, which corresponds to the electrons losing energy solely through Coulomb collisions in an ionized target. We then add Poisson noise in the hard X-ray count rate spectra, based on a nominal detector area and observation integration interval, and use the above procedure on this simulated noisy data set to determine the energy-loss rate as a function of energy in each model. For count rates associated with large flares, the procedure reproduces well the collisional energy loss profile for electron energies up to about 40 keV, even when no statistical smoothing (regularization) methodology is applied. Above this energy, the method breaks down due to the data noise present, but the method could be extended to higher energies by use of a suitable regularized inversion technique. When other (noncollisional) models of energy loss are used to generate the simulated hard X-ray data, the procedure produces energy loss forms that are demonstrably and quantifiably different from the purely collisional case. This shows that even using a simple, unregularized inversion procedure, spatially resolved hard X-ray spectra can indeed be used to compare models of energy transport in solar flares. We discuss our results with reference to the forthcoming High Energy Solar Spectroscopic Imager mission, which will provide data of the necessary quality for the application of our technique.

Reduction of interpolation errors when using local correlation tracking for motion detection

Local correlation tracking (LCT) is a commonly used motion tracking technique, particularly in solar physics. When used to track motions smaller than one pixel per time sample, interpolation of the original data is required. We demonstrate that it is possible to introduce large systematic errors by using an inappropriate interpolation method, and describe how to avoid these errors. The effect of these errors on the calculated velocity field is demonstrated on simulated solar granulation data.

Inference of hot star density stream properties from data on rotationally recurrent DACs

The information content of data on rotationally periodic recurrent discrete absorption components (DACs) in hot star wind emission lines is discussed. The data comprise optical depths τ(w, φ) as a function of dimensionless Doppler velocity w = (∆λ/λ0)(c/v∞) and of time expressed in terms of stellar rotation angle φ. This is used to study the spatial distributions of density, radial and rotational velocities, and ionisation structures of the corotating wind streams to which recurrent DACs are conventionally attributed. The simplifying assumptions made to reduce the degrees of freedom in such structure distribution functions to match those in the DAC data are discussed and the problem then posed in terms of a bivariate relationship between τ(w, φ) and the radial velocity vr(r), transverse rotation rate Ω(r) and density ρ(r ,φ ) structures of the streams. The discussion applies to cases where: the streams are equatorial; the system is seen edge on; the ionisation structure is approximated as uniform; the radial and transverse velocities are taken to be functions only of radial distance but the stream density is allowed to vary with azimuth. The last kinematic assumption essentially ignores the dynamical feedback of density on velocity and the relationship of this to fully dynamical models is discussed. The case of narrow streams is first considered, noting the result of Hamann et al. (2001) that the apparent acceleration of a narrow stream DAC is higher than the acceleration of the matter itself, so that the apparent slow acceleration of DACs cannot be attributed to the slowness of stellar rotation. Thus DACs either involve matter which accelerates slower than the general wind flow, or they are formed by structures which are not advected with the matter flow but propagate upstream (such as Abbott waves). It is then shown how, in the kinematic model approximation, the radial speed of the absorbing matter can be found by inversion of the apparent acceleration of the narrow DAC, for a given rotation law. The case of broad streams is more complex but also more informative. The observed τ(w, φ) is governed not only by vr(r) and Ω(r) of the absorbing stream matter but also by the density profile across the stream, determined by the azimuthal (φ0) distribution function F0(φ0) of mass loss rate around the stellar equator. When F0(φ0 )i s fairly wide inφ0, the acceleration of the DAC peak τ(w, φ )i nw is generally slow compared with that of a narrow stream DAC and the information on vr(r), Ω(r) and F0(φ0) is convoluted in the data τ(w, φ). We show that it is possible, in this kinematic model, to recover by inversion, complete information on all three distribution functions vr(r), Ω(r )a ndF0(φ0) from data on τ(w, φ )o f sufficiently high precision and resolution since vr(r )a ndΩ(r) occur in combination rather than independently in the equations. This is demonstrated for simulated data, including noise effects, and is discussed in relation to real data and to fully hydrodynamic models.

Inversion of Thick-Target Bremsstrahlung Spectra from Nonuniformly Ionised Plasmas

The effects of non-uniform plasma target ionisation on the spectrum of thick-target HXR bremsstrahlung from a non-thermal electron beam are analysed. In particular the effect of the target ionisation structure on beam collisional energy losses, and hence on inversion of an observed photon spectrum to yield the electron injection spectrum, is considered and results compared with those obtained under the usual assumption of a fully ionised target.The problem is formulated and solved in principle for a general target ionisation structure, then discussed in detail for the case of a step function distribution of ionisation with column depth as an approximation to the sharp coronal–chromospheric step structure in solar flare plasmas. It is found that such ionisation structure has very dramatic effects on derivation of the thick-target electron injection spectrum F0(E0) as compared with the result F*0(E0) obtained under the usual assumption of a fully ionised target: (a) Inferred F*0 contain more electrons than F0 and in some cases include electrons at energies where none are actually present. Although the total (energy-integrated) beam fluxes in the two cases do not differ by a factor of more than Aee/AeH, the spectral shapes can differ greatly over finite energy intervals resulting in the danger of misleading results for total fluxes obtained by extrapolation. (b) The unconstrained mathematical solution for F0 for any photon spectrum is never unique, while that for F*0 is unique. When the physical constraint F0 ≥ 0 is added, for some photon spectra solutions for F0 may not exist or may not be unique. (This is not an effect of noise but of real analytic ambiguity.) (c) For data corresponding to F*0 with a low-energy cut-off, or a cut-off or rapid enough exponential decline at high energies, a unique solution F0 does exist and we obtain a recursive summation for its evaluation.Consequently, in future work on the inversion of HXR bremsstrahlung spectra it will be vital for algorithms to include the effects of target ionisation if spurious results on thick-target electron spectra are not to be inferred. Finally it is pointed out that the depth of the transition zone, and its evaporative evolution during flares may be derivable from its effect on the HXR spectrum.

The classical Kepler problem and geodesic motion on spaces of constant curvature

In this paper we clarify and generalize previous work by Moser and Belbruno concerning the link between the motions in the classical Kepler problem and geodesic motion on spaces of constant curvature. Both problems can be formulated as Hamiltonian systems and the phase flow in each system is characterized by the value of the corresponding Hamiltonian and one other parameter (the mass parameter in the Kepler problem and the curvature parameter in the geodesic motion problem). Using a canonical transformation the Hamiltonian vector field for the geodesic motion problem is transformed into one which is proportional to that for the Kepler problem. Within this framework the energy of the Kepler problem is equal to (minus) the curvature parameter of the constant curvature space and the mass parameter is given by the value of the Hamiltonian for the geodesic motion problem. We work with the corresponding family of evolution spaces and present a unified treatment which is valid for all values of energy continuous...

A non-uniqueness problem in solar hard x-ray spectroscopy

We consider the hard x-ray emission process by interaction between the electrons and the ions in the solar atmosphere. We provide the integral equations describing this process as an inverse problem in the case of uniform ionization of the plasma and of a simple but rather realistic approximation of non-uniform conditions. The singular system of the integral operators is computed analytically in the continuous case for the uniform ionization model and numerically in the case of discrete data for both uniform and non-uniform ionization conditions. By analytical arguments and analysis of the singular spectrum we point out that non-uniform ionization results in an ambiguous interpretation of the solution of the integral equation, this solution not being unique. Finally, we briefly recall that this analysis facilitates methods for recovering unique and regularized solutions from high-resolution hard x-ray spectral data soon to be forthcoming from the HESSI space mission.

Kinematic model inversions of hot star recurrent DAC data - tests against dynamical CIR models

The Discrete Absorption Components (DACs) commonly observed in the ultraviolet lines of hot stars have previously been modelled by dynamical simulations of Corotating Interaction Regions (CIRs) in their line-driven stellar winds. Here we apply the kinematic DAC inversion method of Brown et al. to the hydrodynamical CIR models and test the reliability of the results obtained. We conclude that the inversion method is able to recover valuable information on the velocity structure of the mean wind and to trace movement of velocity plateaux in the hydrodynamical data, though the recovered density profile of the stream is correct only very near to the stellar surface.