Since the first edition of An Introduction to Modern Astrophysics and its abbreviated companion text, An Introduction to Modern Stellar Astrophysics, first appeared in 1996, there has been an incredible explosion in our knowledge of the heavens. It was just two months before the printing of the first editions that Michel Mayor and Didier Queloz announced the discovery of an extrasolar planet around 51 Pegasi, the first planet found orbiting a main-sequence star. In the next eleven years, the number of known extrasolar planets has grown to over 193. Not only do these discoveries shed new light on how stars and planetary systems form, but they also inform us about formation and planetary evolution in our own Solar System.

In addition, within the past decade important discoveries have been made of objects, within our Solar System but beyond Pluto, that are similar in size to that diminutive planet. In fact, one of the newly discovered Kuiper belt objects, currently referred to as 2003 UB313 (until the International Astronomical Union makes an official determination), appears to be larger than Pluto, challenging our definition of what a planet is and how many planets our Solar System is home to.

Explorations by robotic spacecraft and landers throughout our Solar System have also yielded a tremendous amount of new information about our celestial neighborhood. The armada of orbiters, along with the remarkable rovers, Spirit and Opportunity, have confirmed that liquid water has existed on the surface of Mars in the past. We have also had robotic emissaries visit Jupiter and Saturn, touch down on the surfaces of Titan and asteroids, crash into cometary nuclei, and even return cometary dust to Earth.

Missions such as Swift have enabled us to close in on the solutions to the mysterious gamma-ray bursts that were such an enigma at the time An Introduction to Modern Astrophysics first appeared.We now know that one class of gamma-ray bursts is associated with core-collapse supernovae and that the other class is probably associated with the merger of two neutron stars, or a neutron star and a black hole, in a binary system.

BASIC OPTICS

From the beginning, astronomy has been an observational science. In comparison with what was previously possible with the naked eye, Galileo's use of the new optical device known as the telescope greatly improved our ability to observe the universe (see Section 2.2). Today we continue to enhance our ability to “see” faint objects and to resolve them in greater detail. As a result, modern observational astronomy continues to supply scientists with more clues to the physical nature of our universe.

Although observational astronomy now covers the entire range of the electromagnetic spectrum, along with many areas of particle physics, the most familiar part of the field remains in the optical regime of the human eye (approximately 400 nm to 700 nm). Consequently, telescopes and detectors designed to investigate optical-wavelength radiation will be discussed in some detail. Furthermore, much of what we learn in studying telescopes and detectors in the optical regime will apply to other wavelength regions as well.

Refraction and Reflection

Galileo's telescope was a refracting telescope that made use of lenses through which light would pass, ultimately forming an image. Later, Newton designed and built a reflecting telescope that made use of mirrors as the principal optical component. Both refractors and reflectors remain in use today.

To understand the effects of an optical system on the light coming from an astronomical object, we will focus first on refracting telescopes. The path of a light ray through a lens can be understood using Snell's law of refraction. Recall that as a light ray travels from one transparent medium to another, its path is bent at the interface. The amount that the ray is bent depends on the ratio of the wavelength-dependent indices of refraction of each material, where represents the speed of light within the specific medium. If

is the angle of incidence, measured with respect to the normal to the interface between the two media, and is the angle of refraction, also measured relative to the normal to the interface (see Fig. 6.1), then Snell's law is given by

If the surfaces of the lens are shaped properly, a beam of light rays of a given wavelength, originally traveling parallel to the axis of symmetry of the lens (called the optical axis of the system) can be brought to a focus at a point along that axis by a converging lens [Fig. 6.2(a)].

In the theory of black holes, an important place is assigned to the problem of their interior: understanding it as the matter that generates the external gravitational field. Commonly, the matter is modeled through a perfect fluid. It is worth recalling that in (3 + 1) gravity the interior solution to the Schwarzschild black hole is modeled by an interior Schwarzschild perfect fluid solution with constant energy density, but the question still remains open as to what is the interior solution for the rotating Kerr black hole? In (2+1) gravity one may ask the same question with respect to the rotating BTZ black hole. As we shall see in this chapter, there is an interior solution to the BTZ black hole, modeled by a perfect fluid with constant energy density. Moreover, we succeeded in deriving perfect fluid solutions, such that their fluid velocity possesses differential rotation. On the other hand, rigidly rotating exact perfect fluid solutions derived by Rooman and Spindel (1998) and Lubo et al. (1999) are also reported. Those papers deal with stationary circularly symmetric gravitational sources of the perfect fluid type, with a cosmological constant with focus on restrictions on the physical parameters of the solutions due to the matching conditions between the interior and exterior geometries. In particular, it is established there that finite sources and absence of closed timelike curves privilege negative values of the cosmological constant. Moreover, for stationary configurations, the field equations for constant energy densities have been explicitly solved; if, additionally, the pressure vanishes, interior Gödel-like stars arise.

It is worth pointing out that the literature on stationary perfect fluid solutions of (2+1) gravity is rather scarce; among it one may cite Rooman and Spindel (1998); Lubo et al. (1999); García (2004); Cataldo (2004); Gürses (1994); Obukhov (2003). From Section 8.1 to Section 8.2.5 I follow the presentation given in García (2004), while for the remaining two sections I follow Rooman and Spindel (1998), and Lubo et al. (1999).

Orbit is a computer program designed to calculate the position of a planet orbiting a massive star (or, alternatively, the orbit of the reduced mass about the center of mass of the system). The program is based on Kepler's laws of planetary motion as derived in Chapter 2. References to the relevant equations are given in the comment sections of the code.

The user is asked to enter the mass of the parent star (in solar masses), the semimajor axis of the orbit (in AU), and the eccentricity of the orbit. The user is also asked to enter the number of time steps desired for the calculation (perhaps 1000 to 100,000) and the frequency with which the time steps are to be printed to the output file (Orbit.txt). If 1000 time steps are specified with a frequency of 10, then 100 evenly spaced (in time) time steps will be printed.

The output file can be imported directly into a graphics or spreadsheet program in order to generate a graph of the orbit. Note that it may be necessary to delete the header information in Orbit.txt prior to importing the data columns into the graphics or spreadsheet program.

The source code is available in both Fortran 95 and C++ versions. Compiled versions of the code are also available.

The code may be downloaded from the companion website at