We discuss the action of a generic group on a complex vector space, representing the elements of the group as matrices.

Progress in astronomy is associated with the construction of new telescopes

and new instruments. This chapter only mentions a few selected initiatives of interest,

to give a flavor of the tools that astronomers are considering for optical observations from the ground and from space. Similarly, on the side of science, this chapter examines only one major set of observations from space, the so-called Hubble Deep Fields, and then proceeds to outline a landmark discovery made at the turn of the century, that is the observations of distant supernovae that have led to convincing evidence that the universe is not only expanding, but, at the present epoch, is actually accelerating. A large investment, not only in the field of optical astronomy, is being made in placing telescopes at special locations very far from Earth. These special sites correspond to Lagrangian points, that is, equilibrium points of the restricted three-body problem for the Sun-Earth system. At the end of this chapter, a digression is made on these concepts, which also allows us to introduce the tidal radius, one concept frequently used in dynamical astronomy.

Electromagnetic radiation is the primary source of astronomical information.

In particular, until the early 1930s astronomy was all based on the use of telescopes

that extended the power of the human eye, but were restricted to the

collection of visible light. In general, the sources of astronomical electromagnetic radiation and other sources of astronomical information are what we call visible matter. This chapter introduces some key concepts and notation that characterize light and the collection of light for astronomical purposes. It addresses the main types of information that we may extract from the observations, by means of imaging and spectroscopy, recalling the difference between apparent and intrinsic properties of the astronomical sources and the fact that the light from distant sources is often a mixture of photons from different stars or different

components. This serves as an excuse for a quick introduction to important

concepts, such as stellar populations, mass-to-light ratios, mean motions, and

velocity dispersions. In closing the chapter, a method is described to measure the distance to a stellar system based on the application of a very simple dynamical model to a suitable set of observations.

The tracking of the orbit of a star around SgrA* is one of the most beautiful achievements of astronomy. It is the focus of this chapter. This measurement has led to the most convincing evidence for the existence of a supermassive black hole and to an accurate measurement of its mass. In addition, this is also a simple example of the general dynamical paradigm used to determine whether a system contains some form of invisible matter coexisting with the visible matter. Here, the invisible matter (a black hole) has nothing to do with the concept of dark matter as is commonly envisaged in modern astrophysics. After a section on the dynamical paradigm that leads to declare a discrepancy between mass present and visible mass, some observations are recalled that for decades have suggested that our Galaxy should host a central supermassive black hole. Then the main characteristics of the more recent study of star orbits close to the source SgrA* are described, with additional comments on the detection of supermassive central black holes in other galaxies. The final dynamical section is devoted to some general concepts about orbits; it also includes a short description of quasi-circular star orbits in spherical or axisymmetric time-independent potentials.

In regular, normal spiral galaxies the overall kinematics of the disk can be described in terms of a mean axisymmetric rotation around the center, following a fluid model presented earlier in the book. Making reliable measurements of galaxy rotation curves is an art that requires great expertise. This chapter explains how the study of the rotation curves of spiral galaxies led to the discovery of dark matter halos: decisive evidence was eventually obtained in the mid-1980s, by referring to radially extended radio rotation curves. The decomposition of a rotation curve in the relative contributions of dark and visible matter to the total gravitational field is a step that still remains largely ambiguous. In a conservative approach, the role of dark matter is often thought to be minimal and to become dominant only in the outer parts of the galaxy, but there remain several unexplained aspects and unresolved questions. Touching upon a nontrivial dynamical issue, the problem of making self-consistent decompositions is briefly addressed. Finally, two dynamical arguments are examined that go beyond the direct inspection of the properties that characterize the observed basic state of spiral galaxies and call for the presence of a dark halo as a solution to otherwise unexplained stability properties of galaxy disks.