Hands-on exercise: measuring the lengths of lines

Physics is about describing the physical world. In physics courses we get used to doing this using mathematics, and sometimes it can seem as if the mathematics is the physics. But our goal is to learn about the physical world, and so sometimes we have to just put the math aside and let the physical world be our teacher. It is in this spirit that we begin this chapter with a hands-on exercise that requires measuring the physical world with your hands. To complete this exercise you will need the following supplies:

• Three cloth or paper measuring tapes, preferably from computer printouts of the file measures.html included on the CD that comes with this book.

• Some Scotch tape.

• One large spherical object such as a large melon, a beach ball or a globe, with a diameter roughly between 15 and 20 cm.

• One flat table or desk.

• A pencil and some graph paper.

If you have printed out the page with the measuring tapes on them from the CD, cut them out with the edges of the paper aligned with the measuring edges of the printed tapes. Tape measures A and B should be taped together at a right angle to one another with the measuring edges facing one another. We will call this taped-together object the Side Measurer.

With this chapter we start the study of a number of important classical solutions of GR. There is no doubt that the most important solution is Schwarzschild's, that describes the static, spherically symmetric gravitational field in the absence of matter that one finds outside any static, spherically symmetric object (star, planet …). It is this, the simplest nontrivial solution that leads to the concept of a black hole (BH), which affords a privileged theoretical laboratory for Gedankenexperimente in classical and quantum gravity.

It is, in fact, a firmly established belief in our scientific community that macroscopic BHs (of the size studied by astrophysicists) are the endpoints of gravitational collapse of stars, which, after a long time, gives rise to Schwarzschild BHs if the stars do not rotate. There should be many macroscopic Schwarzschild BHs in our Universe, since many stars have enough mass to undergo gravitational collapse and there is evidence of supermassive BHs in the centers of galaxies. It has been suggested that smaller BHs could have been produced in the Big Bang. Here we are going to be interested in BHs of all sizes, independently of their origin (primordial, quantum-mechanical, astrophysical …).

We begin by deriving the Schwarzschild solution and studying its classical properties in order to find its physical interpretation. The physical interpretation of vacuum solutions of the Einstein equations is a most important and complicated point (see [168, 169]) since the source, located by definition in the region in which the vacuum Einstein equations are not solved, is unknown. In the case of the Schwarzschild solution, we will be led to the new concepts of the event horizon and BHs.

Hands-on exercise: wave and particle properties

The purpose of this exercise is for you to observe some basic wave and particle properties. To complete this exercise you will need the following:

• Tub of water, or access to a quiet pond, lake or swimming pool.

• Things to float on the surface of the water.

• Pen or pencil and some drawing paper.

• Small projectile such as a stone.

Disturb the middle of the tub just until you are able to make a visible wave on the surface. Watch how the wave propagates. Wait until the surface of the water returns to being flat and make another wave. Keep doing this as many times as necessary to be able to draw what you see on the paper and answer the following questions:

• Does the wave have a definite location at any one moment in time?

• Does the wave have a definite direction as it propagates?

• Approximately how far does the wave travel in 1 s?

• Describe the motion of the water in which the wave moves.

Throw your small projectile in the air at various angles, letting it drop down (not in the water). Keep doing this as many times as necessary to be able to draw what you see and answer the following questions:

• Does the object have a definite location at any one moment in time?

• […]

Lie groups are continuous groups in which the group elements are labeled by continuous coordinates. Hence the groups are infinite. For example, rotations in three dimensions can be labeled by three coordinates. In this case one says the group is three-dimensional. The entire group forms a smooth space called the group manifold.

One can introduce a measure on the group manifold, which is invariant under either left or right multiplication by a fixed group element, and use it to define integration over the group manifold in a way that is compatible with the group structure. Denoting the measure by dg, one can define the volume of the group to be ∫Gdg. If the volume is finite, the group is said to be compact and if it is infinite the group is said to be noncompact.

A simple example of a compact Lie group is the group U(1), which consists of numbers (1 × 1 matrices) of the form eiα, where α is real. The group manifold is a circle in this case, and the invariant measure is (up to normalization) given by dα. Clearly, this is a compact group. Another Lie group, which could be denoted GL(1, R)+, consists of numbers of the form ex, where x is real. In this case the group manifold is the real line, the invariant measure is dx, and the group is noncompact.

In this series of lectures I discuss the basic principles and the modelling of the chemical evolution of galaxies. In particular, I present models for the chemical evolution of the Milky Way galaxy and compare them with the available observational data. ¿From this comparison one can infer important constraints on the mechanism of formation of the Milky Way as well as on stellar nucleosynthesis and supernova progenitors. Models for the chemical evolution of elliptical galaxies are also shown in the framework of the two competing scenarios for galaxy formation: monolithic and hierachical. The evolution of dwarf starbursting galaxies is also presented and the connection of these objects with Damped Lyman-α systems is briefly discussed. The roles of supernovae of different type (I, II) is discussed in the general framework of galactic evolution and in connection with the interpretation of high redshift objects. Finally, the chemical enrichment of the intracluster medium as due mainly to ellipticals and S0 galaxies is discussed.

Basic parameters of chemical evolution

Galactic chemical evolution is the study of the evolution in time and space of the abundances of the chemical elements in the interstellar gas in galaxies. This process is influenced by many parameters such as the initial conditions, the star formation and evolution, the nucleosynthesis and possible has flows.

In these lectures I present a highly opinionated review of the observed patterns of metallicity and element abundance ratios in nearby spiral, irregular, and dwarf elliptical galaxies, with connection to a number of astrophysical issues associated with chemical evolution. I also discuss some of the observational and theoretical issues associated with measuring abundances in H II regions and gas and stellar surface densities in disk galaxies. Finally, I will outline a few open questions that deserve attention in future investigations.

Introduction

The measurement of element abundances in galaxies other than our own has a roughly forty-year history, beginning with early attempts to measure helium abundances in giant H II regions in the Magellanic Clouds and M33 (Aller & Faulkner 1962, Mathis 1962) and pioneering studies of heavy element abundances from forbidden lines in extragalactic H II regions (e.g. Peimbert & Spinrad 1970, Searle 1971, Searle & Sargent 1972). Since then this field has grown tremendously, with high quality oxygen abundance data in some 40 nearby spiral galaxies and more than 100 irregular and compact dwarf galaxies. The amount of data for other elements (C, N, Ne, S, and Ar) has also improved tremendously, thanks largely to improvements in visible-wavelength detectors and the launching of spacecraft observatories, such as IUE, HST, and ISO, which have opened up the UV and IR spectral regions for spectroscopy.

The methods of abundance determinations in H ii regions and planetary nebulae are described, with emphasis on the underlying assumptions and inherent problems. Recent results on abundances in Galactic H ii regions and in Galactic and extragalactic Planetary Nebulae are reviewed.

Introduction

H ii regions are ionized clouds of gas associated with zones of recent star formation. They are powered by one, a few, or a cluster of massive stars (depending on the resolution at which one is working). The effective temperatures T* of the ionizing stars lie in the range 35 000 – 50 000 K. The nebular geometries result from the structure of the parent molecular cloud. Stellar winds, at evolved stages, may produce ring-like structures, but the morphology of H ii regions is generally rather complex on all scales. Typical hydrogen densities n are 103 – 104 cm–3 for compact H ii regions. The average densities in giant extragalactic H ii regions are lower, typically 102 cm–3 since giant H ii regions encompass also zones of diffuse material. The total supply of nebular gas is generally large, so that all (or at least a significant fraction) of the ionizing photons are absorbed.

Planetary nebulae (PNe) are evolutionary products of so-called intermediate mass stars (initial masses of 1 – 8 M⊙) as they progress from the asymptotic giant branch (AGB) to the white dwarf stage.

The distribution of elements in the cosmos is the result of many different physical processes in the history of the Universe, from Big Bang to present times. Its study provides us with a powerful tool for understanding the physical conditions of the primordial cosmos, the physics of nucleosynthesis processes that occur in different objects and places, and the formation and evolution of stars and galaxies. Cosmochemistry is a fundamental topic for many different branches of Astrophysics as Cosmology, Stellar Structure and Evolution, Interstellar Medium, and Galaxy Formation and Evolution.

The advances made in the last decade of the XXth century in the study of the chemical evolution of the Universe have been really spectacular. On one hand, they have been brought by the availability of large-aperture ground-based telescopes and space borne telescopes (working in both the visible and other regions of the electromagnetic spectrum), and on the other hand by advances in theory and numerical modelling techniques in many fields of astrophysics such as stellar evolution stellar atmospheres, the physics of ionised plasmas and atomic and molecular physics.

According to the predictions of the most commonly accepted cosmological models, most of the light elements, especially deuterium and helium, were produced during the first minutes after the Big Bang. Comparison between observed and predicted lightelement abundances is one of the classical fundamental tests of cosmological models. Stellar evolutionary models have advanced considerably in recent years.

Of the light nuclides observed in the universe today, D, 3He, 4He, and 7Li are relics from its early evolution. The primordial abundances of these relics, produced via Big Bang Nucleosynthesis (BBN) during the first half hour of the evolution of the universe provide a unique window on Physics and Cosmology at redshifts ∼ 1010. Comparing the BBN-predicted abundances with those inferred from observational data tests the consistency of the standard cosmological model over ten orders of magnitude in redshift, constrains the baryon and other particle content of the universe, and probes both Physics and Cosmology beyond the current standard models. These lectures are intended to introduce students, both of theory and observation, to those aspects of the evolution of the universe relevant to the production and evolution of the light nuclides from the Big Bang to the present. The current observational data is reviewed and compared with the BBN predictions and the implications for cosmology (e.g., universal baryon density) and particle physics (e.g., relativistic energy density) are discussed. While this comparison reveals the stunning success of the standard model(s), there are currently some challenges which leave open the door for more theoretical and observational work with potential implications for astronomy, cosmology, and particle physics.

Introduction

The present universe is expanding and is filled with radiation (the 2.7 K Cosmic Microwave Background - CMB) as well as “ordinary” matter (baryons), “dark” matter and, “dark energy”.

The horizon for studies of element abundances has expanded dramatically in the last ten years. Once the domain of astronomers concerned chiefly with stars and nearby galaxies, this field has now become a key component of observational cosmology, as technological advances have made it possible to measure the abundances of several chemical elements in a variety of environments at redshifts up to z ≃ 4, when the universe was in its infancy. In this series of lectures I summarise current knowledge on the chemical make-up of distant galaxies observed directly in their starlight, and of interstellar and intergalactic gas seen in absorption against the spectra of bright background sources. The picture which is emerging is one where the universe at z = 3 already included many of the constituents of today's galaxies—even at these early times we see evidence for Population I and II stars, while the ‘smoking gun’ for Population III objects may be hidden in the chemical composition of the lowest density regions of the intergalactic medium, yet to be deciphered.

Introduction

One of the exciting developments in observational cosmology over the last few years has been the ability to extend studies of element abundances from the local universe to high redshifts.

After recalling general knowledge about nuclear reactions and stellar evolution, we highlight aspects of stellar nucleosynthesis and the underlying physics of stellar evolution where progress has been achieved during the last years. In §2, we discuss the bulk nucleosynthesis in massive stars, especially of oxygen which is the most prominent massive star tracer, before we outline effects of rotation in those stars. §3 describes some recent developments in the field of s-process nucleosynthesis, §4 deals with the relevance of close binary systems for nucleosynthesis, and §5 is concerned with the most massive stars.

Introduction

We know 290 stable isotopes. With the exception of the nine lightest ones, they are all synthesised in the deep interior of stars. In order to study the evolutionary history of the abundance of all these nuclei, it is most efficient to group them such that the formation of the isotopes in each group can be understood through the same process. Following the legendary approach of Burbidge et al. (1957), one can break down the nucleosynthesis into half a dozen processes, which can be split further considering more details, but which leave only very few nuclei unexplained. While in what follows we will connect nucleosynthesis processes with evolutionary stages of stars, it is worth pointing out that Burbidge et al.

Introduction

The origins of the chemical elements must rank highly in any intelligent citizen's list of questions about the natural world. Thanks to the efforts of observers and theoreticians over the last half-century, the citizen may now be provided with answers to ‘Where, when, and how were the elements made?’ This remarkable achievement of astrophysics provides one focus for this set of lectures. It is impossible to tell in the available space the complete story of nucleosynthesis from hydrogen to uranium (and beyond) with full justice to the observational and theoretical puzzles that had to be addressed.

Nucleosynthesis began with the Big Bang (see Steigman's contribution to this volume). According to the standard model of this event, nucleosynthesis completed in the first few minutes of the Universe's life resulted in gas composed of 1H, and 4He with 1H/4He ≃ 0.08 by number of atoms, and trace amounts of 2H, 3He, and 7Li. The inability of the rapidly cooling low density Big Bang to synthesise nuclides beyond mass number 7 is due to the fact that all nuclides of mass number 5 and 8 (i.e., potential products from 1H + 4He and 4He + 4He) are highly unstable.

Ashes of the Big Bang cooled. The photons of the cosmic microwave background radiation were set free to roam the Universe. Then came what is known as ‘The Dark Ages’ before galaxies were formed.