HYDROSTATIC EQUILIBRIUM

In the last two chapters, many of the observational details of stellar spectra were discussed, along with the basic physical principles behind the production of the observed lines.Analysis of that light, collected by ground-based and space-based telescopes, enables astronomers to determine a variety of quantities related to the outer layers of stars, such as effective temperature, luminosity, and composition. However, with the exceptions of the ongoing detection of neutrinos from the Sun (which will be discussed later in this chapter and in Chapter 11) and the one-time detection from Supernova 1987A (Section 15.3), no direct way exists to observe the central regions of stars.

Determining the Internal Structures of Stars

To deduce the detailed internal structure of stars requires the generation of computer models that are consistent with all known physical laws and that ultimately agree with observable surface features. Although much of the theoretical foundation of stellar structure was understood by the first half of the twentieth century, it wasn't until the 1960s that sufficiently fast computing machines became available to carry out all of the necessary calculations. Arguably one of the greatest successes of theoretical astrophysics has been the detailed computer modeling of stellar structure and evolution. However, despite all of the successes of such calculations, numerous questions remain unanswered. The solution to many of these problems requires a more detailed theoretical understanding of the physical processes in operation in the interiors of stars, combined with even greater computational power.

The theoretical study of stellar structure, coupled with observational data, clearly shows that stars are dynamic objects, usually changing at an imperceptibly slow rate by human standards, although they can sometimes change in very rapid and dramatic ways, such as during a supernova explosion. That such changes must occur can be seen by simply considering the observed energy output of a star. In the Sun, 3.839 × 1026 J of energy is emitted every second. This rate of energy output would be sufficient to melt a 0◦C block of ice measuring 1 AU × 1 mile × 1 mile in only 0.3 s, assuming that the absorption of the energy was 100% efficient. Because stars do not have infinite supplies of energy, they must eventually use up their reserves and die. Stellar evolution is the result of a constant fight against the relentless pull of gravity.

In this chapter a systematic derivation of the Cotton tensor is presented following García et al. (2004). In 3D spaces the Cotton tensor is prominent as the substitute for the Weyl tensor. It is conformally invariant and its vanishing is equivalent to conformal flatness. However, the Cotton tensor arises in the context of the Bianchi identities and is present in any dimension n. In this text, its irreducible decomposition is performed and the number of independent components as n(n2−4)/3 is determined. Subsequently, its characteristic properties are exhibited. An algebraic classification of the Cotton tensor in three dimensions is accomplished. This classification is used throughout this text; for each class of the derived here solutions the Cotton tensor is evaluated and classified.

The nonlinear coupling of gravity to matter in general relativity presents difficult technical problems in attempts to understand the gravitational interaction of elementary particles and strings or to investigate details of the gravitational collapse; see Witten (1991). Progress in the former area has come mainly from treating quantum fields as propagating on fixed background geometries: Polchinski (1998), whereas much of the progress in the latter has come from detailed numerical works: Choptuik (1993); Abrahams and Evans (1993); Gundlach (1999).

Exact solutions of the relevant matter–gravity equations can play an important role by shedding light on questions of interest in both general relativity and string theory. One is often interested in certain classes of solutions with specified asymptotic properties; the most common of them are the asymptotically flat spacetimes. Recent work in string theory has, via the AdS/CFT conjecture, highlighted the importance of the asymptotically AdS spacetimes, see Maldacena (1998). The AdS/CFT correspondence relates a quantum field theory in d dimensions to a theory in d + 1 dimensions that includes gravity Gubser et al. (1998), and Witten (1998). This is the motivation for looking at the conformally flat spaces and at the spaces of constant curvature. For this reason we decided to review the subject and to collect some old and new results that are nowadays important in the context of anti-de Sitter spacetimes and to present them in modern language. These results seem presently not to be too well known in the community.

Galaxies come in a wide range of forms. Among these are some especially beautiful examples with prominent ring structures. These galactic exceptions have varied causes.

THE MORPHOLOGY OF RING GALAXIES

Like the irregular and interacting galaxies, ring galaxies are galactic exceptions whose history has played a deciding role. They are the result of collisions of two galaxies which led to something new – a ring galaxy. In addition to a bright nucleus, this type of galaxy has a ring, which is usually symmetrically oriented about the centre. The size of the ring greatly exceeds the size of the central component. This ring is very prominent in photos since it contains bright starforming regions with young, massive stars. One usually recognizes a blue ring which contrasts in colour with the yellowish central component. The colours of these components remind one of a spiral galaxy with its bulge and the spiral arms, thus a ring is comparable to a closed spiral arm. This morphological peculiarity was explicitly taken into account by de Vaucouleurs in his classification scheme: “(R)” is placed at the beginning of the description when a ring is seen. When the morphology of the ring is observed in more detail, differences from galaxy to galaxy become apparent. One thus distinguishes two cases: a) the ring formed via a genuine collision of two partner galaxies, b) the ring structure is the result of wound-up tidal tails of two mutually orbiting partner galaxies.

In the latter case, perspective also plays an important role, since tidal tails result in a sufficiently bright ring only when seen one on top of the other. Ring galaxies formed by a collision are often referred to in the literature as “true colliding galaxies”. But in this case as well, a small angle between the plane of the ring and the line of sight can mean that one doesn't recognize the ring, since it is covered by dust regions or parts of the spiral arm. It is assumed that only about 30 per cent of all ring galaxies can be reliably identified.