Introduction

Neutron stars more than a few minutes old are uniformly rotating and satisfy to high accuracy the equation of state that describes cold neutron-star matter. As described in Chapter 5, their 2-dimensional family of equilibria is bounded by four curves: at low central density by a sequence of marginally stable configurations having minimum mass at constant angular momentum; at high density by a sequence of marginally stable configurations having maximum mass at constant angular momentum; by the sequence of nonrotating stars; and by the sequence of stars rotating at the Kepler (mass-shedding) limit on angular velocity. A nonaxisym-metric instability driven by gravitational radiation (the CFS instability) is likely to set an upper limit on rotation more stringent than the Kepler limit, drawing a more restrictive boundary at large rotation on the surface of stable equilibria.

Finding these bounding lines of marginal stability is a primary focus of work on the stability theory of relativistic stars. This chapter is devoted to a detailed presentation of this theory and several of its principal results. The presentation here is restricted to linear stability theory, to finding criteria for stability of first-order perturbations of an equilibrium. Numerical treatments of the nonlinear evolution of stable and unstable modes are discussed in Chapter 10.

The masses of neutron stars are limited by an instability to collapse, and an instability driven by gravitational waves may limit their spin. Their oscillations are relevant to X-ray observations of accreting binaries and to gravitational wave observations of neutron stars formed during the coalescence of double neutron-star systems. This volume pulls together more than 40 years of research to provide graduate students and researchers in astrophysics, gravitational physics, and astronomy with a self-contained treatment of the structure, stability, and oscillations of rotating relativistic stars. Numerical and analytic work are both essential to the subject, and their interplay is emphasized in our treatment.

The book is intended for more than one audience: Readers who need to work through mathematical details of stellar perturbations and stability theory will find them here, in derivations and proofs of principal results. More commonly, a reader working in relativistic astrophysics will want the principal results of the theory but will need only a few of the derivations. The text is also designed to provide a coherent treatment for this second audience, with an exposition of the results preceding the more mathematical derivations. Although our primary concern is with rotating stars, we begin our discussion of oscillations and stability with spherical stars for completeness and to make the presentation accessible to readers with no previous knowledge of relativistic perturbation theory.

The Newtonian approximation describes to extraordinarily high precision the gravitational field of low-mass stars over the course of their evolution, from the instability to collapse that triggers their formation to their death as white dwarfs. In a high-mass star, however, when the nuclear reactions that halted its initial collapse ultimately die out, the core's renewed collapse leads either to a star above nuclear density or to a black hole at whose center is a speck that, at least momentarily, is vastly beyond any known density. In both of these final states of stellar evolution, general relativity plays a fundamental role. The relativistic stars of nature have a complex composition, spanning fifteen orders of magnitude in density. Thought to consist primarily of a gas of neutrons with a gradually varying density of free protons, electrons, and muons, they are surrounded by a crust of ordinary matter, and their cores may hold hyperons, pion or kaon condensates, or possibly free quarks. In fact, our uncertainty about the behavior of matter above nuclear density is (in 2012) great enough to allow what we call neutron stars to be strange-quark stars, collections of up, down, and strange quarks surrounded by a thin normal crust. In the conventional neutron-star model, a much thicker, 1-km crust surrounds an interior in which neutrons and protons form a two-component superfluid.

This chapter describes the spectroscopic techniques that are used in most near-UV, visual, and IR spectrometers. Spectroscopy in this range is particularly important for astrophysics, and instruments for these wavelengths can be constructed using conventional optical elements, such as lenses, prisms, gratings, and normal-incidence mirrors. These instruments share many common features.

Commercially Available Spectrometers

Because optical spectroscopy plays an important role in many different branches of science, medicine, and industry, many manufacturers of optical instrumentation offer commercially produced spectrometers of different types. Most of these devices are not suited for the low light levels from astronomical sources. However, in addition to some instruments that have been designed specifically for astronomical applications (mainly for amateurs), there are commercial “low light level” general-purpose spectrometers on the market, which can be (and are) operated successfully for low-resolution spectroscopy at small telescopes. These commercially available instruments have the advantage of being complete systems, which include high-quality CCD or IR detectors. In many cases they can be conveniently connected to the USB port of a computer for instrument control and data readout. Usually the light from a telescope must be fed to the spectrometer using an optical fiber (see Section 4.5), but some of these spectrometers can be attached directly to a telescope focus. Observers interested in commercially available spectrometers will have no difficulties finding the addresses of potential suppliers by searching for optical spectrometers in the Internet.

So far, we have discussed two basic types of spectroscopic techniques. At high photon energies, the observations were based on the detection of individual photons. Their frequency was determined either by measuring their energy or by measuring their wavelength by means of optical effects. At low (radio) frequencies, the electromagnetic waves were directly recorded, and their frequency distribution was derived using electronic methods. As noted in Chapter 3 (Equation 3.42), under thermal equilibrium conditions, the detection of individual photons requires photon energies hv > kT. Thus, depending on the detector temperature, the transition between photon detection and radio-astronomical methods is expected to take place at FIR or submillimeter wavelengths. In practice, there exists a significant overlap of frequencies at which both types of detection methods can be used, and for which the preferred technique depends on the specific scientific objective. Moreover, in the submillimeter range it is sometimes of advantage to use bolometers, which record light indirectly by measuring the heat that is produced when photons are absorbed. Because of the choice of methods, special techniques have been developed for astronomical observations at these wavelengths, and sometimes combinations of radio and optical techniques are employed. A good example of the diversity of methods used in the FIR/submillimeter range are the three spectroscopic instruments of the Herschel Space Observatory (see Figure 9.4), which (as will be described later) use three different techniques.

The purpose of this chapter is to give a brief introduction to the special methods of the FIR/submillimeter range and to discuss their relative advantages and drawbacks for practical observations.

Although the grating instruments discussed in the preceding chapter dominate present-day astronomical spectroscopy at optical wavelengths, currently there are several other techniques in use at ground-based and space-based observatories. This chapter outlines the principles, the present applications, and the potential of three of these alternative methods.

Fabry-Perot Techniques

Fabry-Perot (FP) devices are based on the interference of light rays that are multiply reflected between partially transmissive mirrors. In astronomy, the main applications of the FP technique are special spectrometers and narrowband interference filters. FP spectroscopy was developed in the final years of the nineteenth century by the French physicists Charles Fabry and Alfred Perot. The technique was soon applied to astronomy. Alfred Perot himself used this method for measuring motions in the solar atmosphere.

The basic principle of the FP technique is outlined in Figure 5.1. The heart of any FP device is a pair of partially transmissive mirrors, separated by the distance l. In the context of FP devices, such a mirror pair is called an etalon (from the French designation of a standard length). In Figure 5.1, the light rays from the telescope are assumed to enter the space between the mirrors at angles γ to the mirror normals from the left (A). As indicated in the figure, the partially transmissive mirrors result in multiple reflections at the two mirror planes. At each reflection a fraction of the light passes through the mirror, while another fraction is returned into the space between the mirrors.

Radio astronomers typically use coherent receivers, which record and analyze the electromagnetic radiation directly. As was pointed out in Section 1.4, this facilitates spectroscopy at radio wavelengths. Making use of electronic frequency filters or of natural resonance effects, receivers can be built that are sensitive to a narrow frequency range only. If such receivers contain electronic components for which the parameters can be varied, the receivers can be “tuned” to specific frequencies. In this case, spectra can be be obtained by tuning narrowband receivers within a certain wavelength range, or by combining receivers that are tuned to different frequencies. As described in Section 1.4, the first radio spectra were obtained in this way, and commercially available radio spectrometers still use this technique. However, if spectra are obtained by tuning a receiver, the different frequencies are measured sequentially and the duty cycle for a given frequency is inversely proportional to the spectral resolution. Therefore, such spectrometers are inefficient and not well suited for the low radiation levels of faint astronomical radio sources. To study the very low signal levels from cosmic sources, more efficient methods must be used, which allow us to record many frequencies simultaneously. Some of these methods are described in the following sections.

The aim of this chapter is to provide an introduction to the technical background of the different types of radio spectrometers at a level that an observer needs to select the optimal instrument for a given task and to assess the potentials and the limitations of the different methods.

The purpose of this book is to provide an introduction to present-day astronomical spectroscopy. Thus, this chapter on the historical development will be restricted to a brief outline of selected milestones that provided the basis for the contemporary techniques and that are helpful for an understanding of the present terminologies and conventions. The reader interested in more details of the historic evolution of astronomical spectroscopy may find an extensive treatment of this topic in two excellent books by John Hearnshaw (1986, 2009). Additional information can be found in older standard works on astronomical spectroscopy, which were published by Hiltner (1964), Carleton (1976), and Meeks (1976). Apart from (still up-to-date) historical sections, these books provide extensive descriptions of methods that have been used in the past, before they were replaced by the more efficient contemporary techniques.

Early Pioneers

Astronomy is known for its long history. Accurate quantitative measurements of stellar positions and motions were already carried out millennia ago. On the other hand, spectroscopy is a relatively new scientific tool. It became important for astronomical research only during the past 200 years. The late discovery of spectroscopy may have been due to the scarcity of natural phenomena in which light is decomposed into its different colors. Moreover, for a long time the known natural spectral effects were not (or not correctly) understood. A prominent example is the rainbow. Reports of rainbows and thoughts about their origin are found in the oldest known written texts, and in most parts of the world almost everybody alive has seen this phenomenon.