The origin of the Earth's magnetic field constitutes one of the most fascinating problems of modern physics. Ever since the works of the physicist W. Gilbert in 1600, we have known that the magnetic field detected with a compass has a terrestrial origin, but a precise understanding of its production (together with the good regime of parameters) remains elusive. The generation of the Earth's magnetic field by electric currents inside our planet was proposed by Amp`ere just after the famous experiment done by Ørsted in 1820 (a wire carrying an electric current is able to move the needle of a compass). These currents cross the Earth's outer core, which is made of liquid metal (mainly iron) at several thousand degrees. If it were not maintained by a source these currents would disappear within several thousand years through Ohmic dissipation. Indeed, in the absence of any regenerative mechanism the Earth's magnetic field would decay in a time τdiff that can be estimated with the simple relation τdiff ~ l2/η. Since the Earth's metallic envelope is characterized by a thickness of ~ 2000 km and a magnetic diffusivity η ~ 1m2 s-1, we obtain τdiff ~ 30 000 years. Also, in order to explain the presence of a large-scale magnetic field on Earth since several million years ago, it is necessary to introduce the dynamo mechanism. It was Sir J. Larmor in 1919 who first suggested that the solar magnetic field could be maintained by what he called a self-excited dynamo, a theory explaining the formation of sunspots. Generally speaking, the dynamo effect explains the solar, stellar, and even galactic magnetic fields.

Geophysics, Astrophysics, and Experiments

Experimental Dynamos

The simplest experiment concerning a self-excited dynamo is Bullard's dynamo as shown in Figure 5.1: it is made of a conducting disk which rotates in a medium

where an axial magnetic field B0 is present. An electric wire going around the axis is connected on one side to the axis and on the other side to the disk (the electric contacts do not prevent the rotation). Because of the Lorentz force an electric potential is induced between the center and the side of the disk. The induced electromotive force e1 generates a current i1 in the loop, which in turn generates an axial induced magnetic field B1, which adds to the initial field.

Gravitational lensing offers a unique tool to study dark matter on a broad range of scales, from galaxies to clusters, to the large-scale matter distribution in the Universe. Density fluctuations on large scales have a small amplitude, hence their lensing effects are weak. In this chapter, after introducing the concepts of weak gravitational lensing – which are quite different from those used in strong lensing – I will concentrate on three main topics: the mass distribution of galaxy clusters as obtained from combining strong and weak lensing results, the lensing effects of the large-scale matter distribution in the Universe and lessons to be learned from them, and the weak lensing studies of the mass profiles in the outskirts of galaxies, together with the correlation of galaxies and the underlying dark matter distribution.

Introduction

Since gravitational light deflection is independent of the nature and state of matter, and in particular is equally sensitive to luminous and dark matter, it provides a unique tool for studying the total mass distribution of objects in the Universe. The study of the mass properties in the inner part of galaxies is covered by other chapters in this volume, as well as that of stellar- and planetary-mass objects. In my chapter, the mass distribution on larger scales is treated, namely that of galaxy clusters (Section 5.3), the statistical properties of the mass of galaxies, groups and clusters on large scales (Section 5.5), and that of the large-scale structure in the Universe in Section 5.4.

In preparing this chapter, I have assumed that the reader has read the contribution by Sherry Suyu (Chapter 1, this volume), where the basic theory of gravitational lensing is treated thoroughly, or is otherwise familiar with the concepts of gravitational lensing. Furthermore, I assume a certain familiarity with the cosmological standard model. It should also be mentioned here that I have written about these topics before, with extended lecture notes published as Schneider (2006). For more details on many issues discussed below, I refer the reader to this work. For the same reason, only a few papers from before 2005 are cited. I have made no efforts to present a balanced, or even complete presentation of the subject; therefore, I apologize to all those colleagues whose work has not been discussed or cited.

Magnification maps are an essential tool in microlensing studies. Their calculation is based on very simple principles and it is therefore quite straightforward to implement. This tutorial is intended to show how these calculations are done by using a basic rayshooting procedure. The tutorial assumes some basic knowledge of any programming language, but no previous knowledge of the specific language used here or experience with gravitational lensing computations is needed. The very basics of gravitational lensing are also implicitly assumed at some points. From the computational point of view, the tutorial covers topics ranging from the simplest ray-shooting program for generating images of an object through a simple lens system to the production of magnification maps for quasar microlensing. Source size effects and how to deal with them are also briefly discussed. We finish by also briefly discussing the main improvements that have been introduced into this technique to make calculations faster.

Introductory remarks

It is particularly difficult to put into writing what was intended as a couple of practical sessions on inverse ray-shooting techniques during the Winter School. Unfortunately, there is not much choice but to illustrate the ideas with some code. At this point, a choice has to be made whether to use pseudocode or to choose a given programming language. I have chosen the latter option here in the hope that the reader may actually use the code snippets presented here straight away and be able to produce some useful programs from it. During the lectures, the Python programming language was used for the tutorial. I chose it for a couple of reasons that are enumerated below. I have therefore also used Python here for presenting the techniques introduced in this tutorial. Nevertheless, the reader is not expected to know any Python in advance as I shall introduce all the required information on Python syntax in Section 8.2. Readers should therefore find no problem in following the explanations and/or in translating the code to their favourite programming language. Finally, these lectures were intended as practical sessions and, as such, the focus is on producing some useful code and on understanding the key steps in that process. Performance will be dealt with at some point in order introduce the necessary Python ingredients to make the programs fast enough to produce results in a reasonable time.

We are on the verge of an explosion in data volume owing to recently started or upcoming surveys of the skies. One of the benefits of these new programmes will be the vastly increased number of known strong gravitational lens systems. In this chapter I will discuss three main topics: lens discovery in these surveys; the use of lensing to determine the mass distribution in galaxies, and in particular substructure in massive galaxy haloes; and cosmological measurements with large lens samples.

Introductory remarks

The next few decades will present an especially exciting time for strong gravitational lensing. This is because a combination of new instrumentation and, in some cases, brand new telescopes have come online, or are at an advanced stage of planning. With the enhanced observing capability enabled by these new facilities, a number of large-scale astronomical surveys are planned. These surveys will provide unprecedented combinations of depth, area, angular resolution and, in some cases, will open up poorly explored wavelength regimes. As a result, they should lead to orders of magnitude increases in the number of known strong lens systems. Indeed, although dedicated observational surveys for lenses have proved productive in the past, it is likely that the vast majority of future lenses will be discovered by mining the data produced by the new large surveys. The resulting large samples of lenses will lead to two major advantages: (1) improved statistics for investigations of galaxy properties, evolution in these properties and cosmology etc., and (2) the discovery of rare lens systems that are especially interesting and useful. As a complement to the large surveys, the planned construction of significantly larger ground- and space-based telescopes will provide enhanced follow-up capabilities of the new discoveries. Furthermore, advances in modelling and analysis codes will allow researchers to exploit more of the information available in observations of lens systems.

As is obvious at this point, this chapter has a focus on the field of strong gravitational lensing, and how it can be affected by large recently started and upcoming astronomical surveys. Taking this approach necessarily ignores other aspects of lensing that also have an exciting future. The new surveys will have a strong impact on investigations that utilize weak lensing or microlensing.

This book gathers together the lectures and practical sessions imparted during the XXIVth Canary Islands Winter School of Astrophysics, held at Puerto de la Cruz, Tenerife (4–16 November 2012).

The basic phenomena of gravitational lenses, light deflection and time dilation by gravitational fields, are two essential predictions of Einstein's General Theory of Relativity. Both effects played a prominent role in the classical tests of General Relativity through famous experiments such as the deflection of light by the Sun measured by Eddington during the 1919 solar eclipse and the radar time delays first measured by Shapiro from the echoes of planets and space probes in the Solar System. Owing to rapid developments in technology, these once exotic and difficult-to-measure effects can nowadays be tested millions of times per second with a very popular device, the GPS (gravitational lensing in everyday life). The present and future importance of gravitational lenses is therefore no longer related exclusively to fundamental General Relativity but also (this is our motivation) to its use in probing the properties of astrophysical objects and of the Universe itself.

The optical bench is one of the most common pieces of laboratory apparatus in modern physics. A source emits photons, alpha particles, neutrons or some other kind of ‘bullets’ that interact with a test object (the target) and are subsequently detected by the observer. This set-up enables the researcher to change and move at will any of the components of the experiment to check hypotheses being tested. Astrophysical sources and targets (planets, stars, galaxies, etc.) are too big and distant for the astronomer to be able to manipulate them. Nevertheless, in certain rare cases a distant source (a star, galaxy, or quasar) appears to be almost aligned with an intervening target (a planet, star, galaxy, or galaxy cluster), thus allowing the observer to measure the deflection of the light rays caused by the gravitational field of the target. This is a gravitational lens system (or simply ‘gravitational lens’), an astronomical optical bench that can be used as a tool to study both the source and the deflecting target.

Magnetic reconnection is a fundamental process in plasma physics that allows the transfer of energy from the magnetic field to the plasma in the form of kinetic energy, thermal energy, or particle acceleration (Yamada et al., 2010). The basic process of magnetic reconnection is the following: when two magnetic field lines of opposite directions are close enough, an intense current sheet is created between the two and a topological reorganization of the magnetic field lines occurs. Basically, this mechanism involves a violation of Alfvén's theorem whose origin is the magnetic diffusivity in standard MHD, or for example the Hall effect in collisionless plasmas. In this chapter, we present the elementary mechanism of magnetic reconnection whose main applications range from solar flares – the most violent events in the solar system – to magnetic substorms in planetary magnetospheres which produce spectacular aurorae (see Figure 7.1). Magnetic reconnection is also invoked for stellar coronae, accretion disks, dynamos, and tokamaks, and laboratory experiments have been designed specifically to study this phenomenon, such as the Magnetic Reconnection Experiment (MRX) built in 1995 at the Princeton Plasma Physics Laboratory. Reconnection is actually a fairly general term that is also used in fluid mechanics when the topology of the vorticity lines is modified. Nowadays, the reconnection process is even observed between quantized vorticies in superfluid helium (Bewley et al., 2008).

A Current Sheet in Ideal MHD

We will first consider the two-dimensional stationary magnetic configuration of Figure 7.2 where magnetic field lines of opposite directions are separated by a distance 2l. For example, we can think about two close solar magnetic loops (see Figure 3.4). This external configuration being imposed, one wants to know the properties of the inner region of thickness 2l. We will assume that the norm of

the magnetic field is constant in the external region. From Maxwell's equations, we have

hence for (we assume that the magnetic field varies linearly in the inner region; see Figure 7.2). In other words, a current sheet of thickness 2l appears between the two regions of different magnetic polarity. This current is even more intense given that the sheet is thin and therefore the regions of different magnetic polarity are close.

The Universe is a great laboratory for studying natural plasmas. In the case of the solar system, the Sun is the source of the interplanetary plasma that spreads at a rate between 300 km/s and 1000 km/s. This plasma may encounter several obstacles during its trip: asteroids, comets, or planets. The most interesting obstacles for a physicist are the magnetized planets. With their magnetosphere, these planets significantly increase their cross-section and therefore their interaction with the solar wind; for example the Earth's magnetosphere is about 150 times larger than the Earth.

The system constituted by the solar wind plus the magnetosphere is naturally in a state of dynamic equilibrium, with a relatively thin interface between them which is called bow shock (see Figure 6.1). Behind this shock, there is a turbulent area called the magnetosheath which serves as a transition to the magnetosphere that is reached by crossing a discontinuity called the magnetopause. There is another type of interface for the solar wind: the terminal shock at the edge of the solar system (~100AU) when the wind speed becomes subsonic. Beyond the terminal shock, we have the heliosheath and then the heliopause (the interface where the solar wind is stopped by the interstellar medium). To understand the nature of these shocks and discontinuities, it is necessary to study the evolution of a thin interface in a plasma; that is the subject of this Chapter. To do this, we will use the macroscopic description of the standard compressible MHD.

Rankine–Hugoniot Conditions

The method generally used to get the conditions of a plasma around a discontinuity is to integrate the conservation laws – that we established in

Chapter 3 – around the discontinuity, in the ideal and inviscid limit. In this situation, we recall that, for standard MHD,

In the case of a thin discontinuity,2 in practice a surface S, the only measurable local changes in the plasma are perpendicular to the discontinuity, i.e. along the normal n of S.