When observed as astronomical objects, stellar systems appear projected on the plane of the sky. As a consequence, it is important to set the framework for a correct understanding of the relation between intrinsic dynamics and projected properties. Unfortunately, while it is always possible (at least in principle) to project a model and then compare the results with observational data, the operation of inversion (i.e., the recovery of three-dimensional information starting from projected properties) is generally impossible due to obvious geometric degeneracies. Spherical and ellipsoidal geometries are among the few exceptional cases that will be discussed in some depth in Chapter 13. Here, instead, the reader is provided with some of the general concepts and tools needed for projecting the most important properties of stellar systems on a projection plane.

In this chapter, we introduce one of the fundamental and most far-reaching concepts of stellar dynamics (and of plasma physics for the case of electric forces): that of “gravitational collisions." As an application of the framework developed, the two-body relaxation time is derived (in the Chandrasekhar approach) by using the so-called impulsive approximation. The concepts of the Coulomb logarithm and of infrared and ultraviolet divergence are elucidated, with an emphasis on the importance of the correct treatment of angular momentum for collisions with small impact parameters, an aspect that is sometimes puzzling for students due to presentations in which the minimum impact parameter appears as something to be put into the theory “by hand." On the basis of the quantitative tools devised in this chapter, we will show that large stellar systems, such as elliptical galaxies, should be considered primarily as collisionless, while smaller systems, such as small globular clusters and open clusters, exhibit collisional behavior. These different regimes are rich in astrophysical consequences, both from the observational and the theoretical points of view.

Dynamical friction is a very interesting physical phenomenon, with important applications in astrophysics. At the simplest level, it can be described as the slowing down (“cooling”) of a test particle moving in a sea of field particles due to the cumulative effects of long-range interactions (no geometric collisions are considered). Several approaches have been devised to understand the underlying physics (which is intriguing, as the final result is an irreversible process produced by a time-reversible dynamics; e.g., see Bertin 2014; Binney and Tremaine 2008; Chandrasekhar 1960; Ogorodnikov 1965; Shu 1999; Spitzer 1987). In this chapter, the dynamical friction time is derived in the Chandrasekhar approach by using the impulsive approximation discussed in Chapter 7.

In this chapter, we discuss the complementary approach to that presented in Chapter 12 for the construction of stationary, multicomponent collisionless stellar systems. The Abel inversion theorem is introduced, and then a selection of density–potential pairs of spherical, axisymmetric, and triaxial shapes commonly used in modeling/observational works are presented. We finally discuss the solution of the Jeans equations for spherical and axisymmetric systems, and among other things we show how to compute the various quantities entering the virial theorem. For illustrative purposes, we use some of the derived results to investigate the possible physical interpretations of the fundamental plane of elliptical galaxies.

This chapter explores observations and properties of quasars, which were first observed in the 1960s as point-like sources that emit over a wide range of energies from the radio through the IR, visible, UV, and even extending to the X-ray and gamma-rays. They are now known to be a type of active galactic nucleus thought to be the result of matter accreting onto a supermassive black hole (SMBH) at the center of the host galaxy.

Earth’s Moon is quite distinct from other moons in the solar system, in being a comparable size to Earth. We explore the theory that a giant impact in the chaotic early solar system led to the Moon’s formation, and bombardment by ice-laden asteroids provided the abundant water we find on our planet. Further, we find that Earth’s magnetic field shields us from solar wind protons, that protect our atmosphere from being stripped away. The icy moons of Jupiter and Saturn are the best targets for exploring if life exists elsewhere in the solar system.

In our everyday experience, there is another way we sometimes infer distance, namely by the change in apparent brightness for objects that emit their own light, with some known power or luminosity. For example, a hundred watt light bulb at close distance appears a lot brighter than the same bulb from far away. Similarly, for a star, what we observe as apparent brightness is really a measure of the flux of light, i.e. energy emitted per unit time per unit area.

It turns out that stellar binary (and even triple and quadruple) systems are quite common. In Chapter 10 we show how we can infer the masses of stars, through the study of stellar binary systems. For some systems, where the inclination of orbits can be determined unambiguously, we can infer the masses of the stellar components, as well as the distance to the system. Together with the observed apparent magnitudes, this also gives the associated luminosities of their component stars.

In reality., stars are not perfect blackbodies, and so their emitted spectra don’t depend solely on temperature, but instead contain detailed signatures of key physical properties like elemental composition. For atoms in a gas, the ability to absorb, scatter, and emit light can likewise depend on the wavelength, sometimes quite sharply. We find that the discrete energies levels associated with atoms of different elements are quite distinct. We introduce the stellar spectral classes (OBAFGKM).

As a star ages, more and more of the hydrogen in its core becomes consumed by fusion into helium. Once this core hydrogen is used up, how does the star react and adjust? Stars at this post-main-sequence stage of life actually start to expand, eventually becoming much brighter giant or supergiant stars, shining with a luminosity that can be thousands or even tens of thousands that of their core hydrogen-burning main sequence. We discuss how such stars reach their stellar end-points as planetary nebulae or white dwarfs.

This chapter considers stellar ages. Just how old are stars such as the Sun? What provides the energy that keeps them shining? And what will happen to them as they exhaust various available energy sources? We show that the ages and lifetimes of stars like the Sun are set by long nuclear burning timescales and the implications that high-mass stars should have much shorter lifetimes than low-mass stars.

Mass is clearly a physically important parameter for a star, as it will determine the strength of the gravity that tries to pull the star’s matter together. We discuss one basic way we can determine mass, from orbits of stars in stellar binaries, and see the range of stellar masses. This leads us to the Virial Theorem, which describes a stably bound gravitational system.

As a basis for interpreting observations of binary systems in terms of the orbital velocity of the component stars, we review the astrometric and spectrometric techniques used to measure the motion of stars through space. Nearby stars generally exhibit some systematic motion relative to the Sun, generally with components both transverse (i.e., perpendicular to) and along (parallel to) the observed line of sight.

We conclude our discussion of stellar properties by considering ways to infer the rotation of stars. All stars rotate, but in cool, low-mass stars such as the Sun the rotation is quite slow. In hotter, more-massive stars, the rotation can be more rapid, with some cases (e.g., the Berillium stars) near the “critical” rotation speed at the star’s surface.