Although the two-body problem in Newtonian dynamics is easily solved, the motion of three or more gravitating objects has no analytical solution. To explore the behavior of clusters (the N-body problem) we must approach them computationally. In this chapter we focus on the Hamiltonian approach, as Mathematica is especially suited for obtaining practical solutions of Hamiltonian equations. We will start with a simple Newtonian problem and show how it is transformed into Hamiltonian form. We will then move to a description of the many ways Mathematica and its available tools support the study of differential equations in general and Hamiltonian equations in particular. A good foundation to the Hamiltonian approach can be found in Abraham and Marsden (1980), which covers analytical dynamics, Hamiltonian dynamics, and its application to celestial mechanics.

As we will see, the N-body problem often leads to chaotic motion, specifically deterministic chaos. Deterministic chaos has its roots in the celestial mechanics of the solar system, which is one example we will explore. The first text to hint at chaos (as we now understand it) was in Poincaré's book Les méthodes nouvelles de la Méchanique celeste. We will here follow its English translation (Poincaré, 1993). A modern perspective of the subject can be found in Diacu and Holmes (1996) and Peterson (1993), although references on the subject are vast and a comprehensive list is not covered here.

In the first part of the twentieth century, the study of the cosmos (hence cosmology), meaning the study of the universe, was mainly theory. Much of the observational material was beyond the grasp of the telescopes of the time, photography was the only detection medium, and spectroscopy was very primitive. Astrophysics had to thrive by observational studies of the sun, the major planets, and bright stars. Nebulae and clusters had been cataloged by Messier, William Herschel, Caroline Herschel, and John Herschel by the thousands, but few astronomers of the time ever thought they would have the technology to truly study such things in detail.

Lack of observational data has historically been the achilles heel of cosmology. As recently as 1990 Kolb and Turner complained that astrophysics in general and cosmology in particular needed more observations. The present situation is in marked contrast to this view, such that there are now so much data publicly available that it strains our computational ability to analyze it. So many people are currently involved with data reduction and analysis that in the last decadal survey there was a call for the formation of a new area of astronomy and astrophysics called astroinformatics (Borne, 2009).

We start this chapter with several topics that are active parts of this frontier subfield of astronomy and astrophysics, as much of the data in the large databases are galactic and extragalactic. As we shall see, our examination will be limited not by the available data, but by our ability to analyze these data on a personal computer. We begin with an examination of the Hubble law, including evidence of universal acceleration. We then examine galaxies, including radio galaxies and quasars. Finally we look at cosmic structure and evolution.

In this chapter we explore several aspects of dynamics within the Milky Way galaxy, including spiral density waves, the effects of dark matter, and the region near the central black hole. We also examine the effects of galactic rotation on stellar clusters by returning to the Orion Trapezium cluster studied in Chapter 6. Finally we look at the role of gas and dust within the galaxy on astrochemistry.

The existence of dark matter

Because of dust obscuration along the galactic equator, radial velocity studies of bright O and B stars in the 1930s and 1940s were able to discern only three or possibly four spiral arm structures in the part of the galaxy nearest the sun. At the time it was assumed that the derived circular orbit velocities fit Kepler's laws, and from that a “reasonable” mass for the galaxy was obtained. That assumption was accepted without much question. After World War II, with the prediction and subsequent discovery of the atomic hydrogen ground state emission called the 21 cm radiation, a new tool was available for measuring the distribution of matter in the Milky Way and other galaxies.

The proton and electron in an ordinary hydrogen atom are both fermions with a spin of ½. As such they both have magnetic moments that can be either parallel or antiparallel. The energy of the bound electron is slightly higher when parallel rather than antiparallel. This creates an energy difference within the ground state, an effect known as hyperfine splitting. When the electron flips a photon is emitted with a frequency of 1420.4 MHz, which corresponds to a wavelength of 21.1 cm. Unlike visible light, these 21 cm radio waves can penetrate cold (∼10 K) dust clouds, which have predominantly molecular rather than atomic hydrogen, that are found within spiral arms. The atomic hydrogen that produces the 21 cm radiation are found in abundance in the “warmer” (∼100 K) space between the cold molecular clouds. When the Dutch radio astronomer Hendrik C. van de Hulst predicted and subsequently found this radiation, the full extent of the galaxy could be observed and characterized.

It is evident from observation that most of the interstellar medium is permeated with charged particles and permanent magnetic fields. If the ionization in a particular region is complete (no neutral particles) the gas is called a plasma. Thus, interstellar space is always a low density, nearly collisionless, environment where particles may go centuries without encountering a particle of the same kind. It is most certainly dominated by a plasma or at least a highly ionized gas. In this chapter we consider the behavior of light and electric charges in such plasmas in a variety of situations. We will typically use protons and electrons as test particles.

Many treatments of plasmas in astrophysics consider only “cold,” virtually collisionless plasmas, but there are a number of instances, particularly in the vicinity of stars and protostars, where one must examine higher density, higher temperature situations. A cold plasma is one in which the kinetic motion of the protons and electrons generally can be ignored. For warm plasmas, electron and ion temperature becomes a contributing factor and we must take kinetic theory into account via the Maxwell–Boltzmann equation (MBE). Finally we look at two diverse, but actually closely related, applications of plasma theory in astrophysics, the first using pulsars to map the electron density and magnetic field within the Milky Way galaxy, the second being a model of solar wind.