In Chapter 2 we saw the basic workings of the tree algorithm. Now we will discuss some methods that can be used to optimise the performance of this type of code. Although most of these techniques are not specific to tree codes, they are not always straightforward to implement within a hierarchical data structure. It therefore seems worthwhile to reconsider some of the common tricks of the N-body trade, in order to make sure that the tree code is optimised in every sense – not just in its N log N scaling.

There are basically two points of possible improvement:

• Improvement of the accuracy of the particle trajectory calculation by means of higher order integration schemes and individual timesteps. This is especially important for problems involving many close encounters of the particles, that is, ‘collisional’ problems.

• Speedup of the computation time needed to evaluate a single timestep by use of modern software and hardware combinations, such as vectorisation, and special-purpose hierarchical or parallel computer architecture.

Individual Timesteps

For most many-body simulations one would like the total simulated time T = ntΔt (where nt is the number of timesteps) to be as large as possible to approach the hydrodynamic limit. However, the choice of the timestep Δt has to be a compromise between this aim and the fact that as Δt increases, the accuracy of the numerical integration gets rapidly worse.

The hierarchical tree method can not be adapted only for Monte Carlo applications: It can also be modified to perform near-neighbour searches efficiently. This means that the tree algorithm could also have applications for systems with short-range or contact forces. Hernquist and Katz (1989) first showed how the tree structure can be used to find near neighbours through range searching. Following their method, the near-neighbour search is performed the following way.

Consider a system in which only neighbours lying within a distance h will interact with the particle i in question. For the near-neighbour search this sphere is enclosed in a cube whose sides are of length 2h. The tree is built the usual way and the tree search starts at the root. The tree search is performed in a very similar way to the normal force calculation of Section 2.2 by constructing an interaction list. The main difference is that the s/d criterion is substituted by the question: ‘Does the volume of the search cube overlap with the volume of the pseudoparticle presently under consideration?’

If there is no overlap, this branch of the tree contains no near neighbours and is not searched any further. However, if there is an overlap, the cell is subdivided into its daughter cells and the search continues on the next highest level. If the cell is a leaf – meaning there is only one particle in the cell – one has to check whether it actually lies within the radius h of particle i.

Today, the study of The Structure of the Sun is one of the most exciting and rapidly evolving fields in physics. Helioseismology has provided us with a new tool to measure the physical state of the interior of a star, our Sun. This technique is successful to a depth of 0.7 R⊙ (i.e. 0.3 R⊙ from the centre). Deeper than this, observational data has been scarce. However, data are now becoming available from Earth-bound helioseismic networks (GONG, TON, IRIS, BISON,…) and from experiments on board SOHO (GOLF, MDI, VIRGO). These should allow the spectrum of gravity modes for the Sun to be determined, and thus the physical state of the solar core.

This book provides an up-to-date and comprehensive review of our current understanding of the Sun. Each chapter is written by a world expert. They are based on lectures given at the VIth Canary Islands Winter School on Astrophysics. This timely conference brought together leading scientists in the field, postgraduates and recent postdocs students. The aim was to take stock of the new understanding of the Sun and to focus on avenues for fruitful future research. Eight lecturers, around 60 students, and staff from the IAC met in the Hotel Gran Tinerfe in Playa de las Américas (Adeje, Tenerife) from the 5th to the 16th of December, 1994. It was a fortnight of intense and enjoyable scientific work.

INTRODUCTION

This is almost an impossible task, to summarize the subject of Global Changes in the Sun so I must apologize in advance for limiting the scope of these lectures to issues that have been choosen, in part, because of personal interests. I hope that the references provide the reader with footpoints from which to explore a larger set of questions which bear on this subject.

Here we will not discuss the very long, evolutionary, timescales over which the sun changes, nor will we explore the fast changes associated with flares and other transient phenomena. While these discussions depend on some results from MHD models of the solar magnetic cycle, we will not be concerned with the MHD mechanism. These lectures will not address the questions needed to understand local physical models that describe, for example, granulation. On the other hand we will describe some of the physical problems that connect the small-scale behavior of the sun to its global properties. By “global property” I mean an observable that is connected by physically important timescales to the entire sun: limb shape and brightness, largescale magnetic field, oscillation frequencies, solar luminosity, and solar irradiance are all examples of global properties.

Here we are interested in understanding the deviations of the sun from some standard one-dimensional static stellar model. This is a subject we can hardly approach for other stars, and for the sun it is difficult because the physics of magnetic fields and convection are linked over a wide range of spatial and temporal scales.

ABSTRACT

The X-ray Solar Physics Satellite Yohkoh has provided us with a number of new findings about the high temperature and high energy processes occurring in solar flares, in active regions, and in the background corona. According to these new findings, hot and dense corona above active regions seem to be maintained, at least in part, with the injections of already heated mass along the magnetic loops from the footpoint below. The outermost loops of the magnetic structures of these active regions are expanding away almost continuously in the case of “active” active regions. These give us quite a different and lively picture about the active region corona compared with a previous static picture with steady heating that we had based on the previous low cadence observations. New clues to the mechanism of flares, which were hidden thus-far in the yet fainter and relatively short stages before the start of flares, have been revealed by the wide-dynamic range, high cadence observations with the scientific instruments aboard Yohkoh. Those preflare signatures and their changes containing essential information about the mechanism of flares, now allow us to pursue truer understanding about the flare mechanism. The same merits of Yohkoh (wide-dynamic range and high-cadence observations) have shown us for the first time in its full form the highly dynamical behavior of the faint background corona, together with the influence of the changes in active regions sometimes exerting overwhelming effects on the surrounding corona.

FOREWORD

Broadly speaking, the inverse problem is the inverse of the forward problem. In the case of contemporary helioseismology, the forward problem is usually posed as that of determining the eigenfrequencies of free oscillation of a theoretical model of the sun. That problem is discussed by Christensen-Dalsgaard in this volume. I call inverting that problem the ‘main’ inverse problem. It is the one that I shall be discussing almost exclusively in this chapter. But also included in the forward problem must be the theoretical modelling of the oscillations as they really occur in the sun, forced, we believe, predominantly by the turbulence in the convection zone, and modulated by their nonlinear interactions with other modes of oscillation and by the perturbations they induce to the very convection that drives them, through variations in the turbulent fluxes of heat and momentum. The inverse of that problem is to derive from the fluid motion of the visible layers in the atmosphere of the sun, which I presume to be ‘observed’, estimates of the frequencies that the modes would have had had they not been disturbed by the other forms of motion. The outcome of that prior inversion provides the data for the main inverse problem.

This chapter is entitled: Testing solar models …. By ‘solar models’ is meant any theoretical description of the sun that we might have in mind.

INTRODUCTION

The present chapter addresses the forward problem, i.e., the relation between the structure of a solar model and the corresponding frequencies. As important, however, is the extent to which the frequencies reflect the physics and other assumptions underlying the model calculation. Thus in Section 2 I consider some aspects of solar model computation. In addition, the understanding of the diagnostic potential of the frequencies requires information about the properties of the oscillations, which is provided in Section 3. Section 4 investigates the relation between the properties of solar structure and the oscillations by considering several examples of modifications to the solar models and their effects on the frequencies, while Section 5 considers further analyses of the observed frequencies. Finally, the prospects of extending this type of work to other stars are addressed in Section 6.

A more detailed background on the theory of solar oscillations was given, for example, by Christensen-Dalsgaard & Berthomieu (1991), Gough (1993), and Christensen-Dalsgaard (1994). For other general presentations of the properties of solar and stellar oscillations see, e.g., Unno et al. (1989) and Gough & Toomre (1991).

A little history

The realization that observed frequencies of solar oscillation might provide information about the solar interior goes back at least two decades. Observations of fluctuations in the solar limb intensity (Hill & Stebbins 1975; Hill, Stebbins & Brown 1976), and the claimed detection of a Doppler velocity oscillation with a period close to 160 minutes (Brookes, Isaak & van der Raay 1976; Severny, Kotov & Tsap 1976) provided early indications that global solar oscillations might be detectable and led to the first comparisons of the reported frequencies with those of solar models (e.g. Scuflaire et al. 1975; Christensen-Dalsgaard & Gough 1976; Iben & Mahaffy 1976; Rouse 1977).

ANALYSIS TOOLS AND THE SOLAR NOISE BACKGROUND

When we observe solar oscillations, we are concerned with measuring perturbations on the Sun that are almost periodic in space and time. The periodic waves that interest us are, however, embedded in a background of broadband noise from convection and other solar processes, which tend to obscure and confuse the information we want. Also (and worse), the “almost-periodic” nature of the waves leads to problems in the interpretation of the time series that we measure. Much of the subject of observational helioseismology is thus concerned with ways to minimize these difficulties.

Fourier Transforms and Statistics

A common thread runs through all of the analysis tricks that one plays when looking at solar oscillations data, and indeed through many of the purely instrumental concerns as well: this thread is the Fourier transform. The reason for this commonality is, of course, that we are dealing with (almost) periodic phenomena – either the acoustic-gravity waves themselves, or the light waves that bring us news of them. Since many of the same notions will recur repeatedly, it is worth taking a little time (and boring the cognoscente) to review some of the most useful properties of Fourier transforms and power spectra. In what follows, I shall simply state results and indicate some of the more useful consequences. We shall see below that even when the Big Theorems of Fourier transforms do not apply, (as with Legendre transforms, for instance), analogous things happen, so that the Fourier example is a helpful guide to the kind of problems we may have.

Supernova and supernova remnant research are two of the most active fields of modern astronomy. SN 1987A has given us a chance to observe a supernova explosion and its aftermath in unprecedented detail, a process that continues to unfold today. Meanwhile, thanks to major advances in optical, radio, and X-ray astronomy, we have gained unprecedented views of the populations and spectrum evolution of supernovae of all kinds. These results have spurred a renaissance in theoretical studies of supernovae. Likewise, samples of well-observed supernovae are becoming large enough that we are closing fast on the goal of using supernovae to determine the cosmic distance scale.

Studies of supernovae and supernova remnants are inextricably linked and we are learning fast about the connections. We now recognize that mass loss from the supernova progenitor star can determine the structure of the circumstellar medium with which the supernova ejecta interact. An outstanding example is the ring around SN1987A. There are many supernovae in which much of the early optical, radio and X-ray emission are due to interaction of the ejecta with circumstellar matter rather than radioactivity within the supernova itself. Just in time for this colloquium, nature provided a particularly spectacular example of such an interacting supernova with SN1993J in M81, one of the brightest supernovae of this century. Moreover, the X-ray spectra of supernova remnants provide a powerful new tool to measure supernova nucleosynthesis yields.

Observational selection effects and the lack of accurate distances for most Galactic SNRs pose problems for studies of the distribution of SNRs in the Galaxy. However, by comparing the observed Galactic longitude distribution of high surface brightness SNRs with that expected from simple models – which avoids some of the problems with selection effects and the lack of distances – a Gaussian scale length of ≈ 7 kpc in Galactocentric radius is obtained for SNRs.

Introduction

The distribution of SNRs in the Galaxy is of interest for many astrophysical studies, particularly in relation to their energy input into the ISM and for comparison with the distributions of possible progenitor populations. Such studies are, however, not straightforward. First, current catalogues of SNRs miss objects due to observational selection effects. Second, there are no reliable distance estimates available for most identified remnants. Here I use a sample of 182 Galactic SNRs from a recently revised catalogue (this proceedings), all but one of which have observed radio flux densities and angular sizes, to derive the distribution of SNRs in the Galaxy by comparing the observed distribution of bright remnants with Galactic longitude with that expected from simple models.

The Problems

The Selection Effects

Although, as discussed by Aschenbach (this proceedings), many new SNRs may soon be identified from the ROSAT X-ray survey, the identification of SNRs in existing catalogues has, generally, been made at radio wavelengths.

Explosion calculations of SN 1987A generate pictures of Rayleigh-Taylor fingers of radioactive 56Ni (56Ni → 56Co → 56Fe) which are boosted to velocities of several thousand km s−1. From the KAO observations of the mid-IR iron lines, a picture of the iron in the ejecta emerges which is consistent with the ‘frothy iron fingers’ having expanded to fill about 50% of the metal-rich volume of the ejecta (vm ≤ 2500 km s−1). The ratio of the nickel line intensities I([Ni I]7.5µm)/I([Ni II]6.6µm) yields a high ionization fraction of xNi 0.9 in the volume associated with the iron-group elements at day 415, before dust condenses in the ejecta.

From the KAO observations of the dust's thermal emission (2 µm − 100 µm), it is deduced that when the grains condense their infrared radiation is trapped, their apparent opacity is gray, and they have a surface area filling factor of about 50%. The dust emission from SN 1987A is featureless: no 9.7 µm silicate feature, nor PAH features, nor dust emission features of any kind are seen at any time. The total dust opacity increases with time even though the surface area filling factor and the dust/gas ratio remain constant. This suggests that the dust forms along coherent structures which can maintain their radial line-of-sight opacities, i.e., along fat ringers. The coincidence of the filling factor of the dust and the filling factor of the iron strongly suggests that the dust condenses within the iron, and therefore the dust is iron-rich.