In this chapter, we concern ourselves primarily with the spectroscopic effects of stellar rotation, including the line broadening caused by rotation, techniques for extracting the rotation rate from the broadening, and some of the results. Many of the analysis tools are the same as those discussed in Chapter 17.

Stellar rotation is the driving force for diverse phenomena in stellar atmospheres such as circulation currents, mass loss, and magnetic field generation and its offshoots: starspots, flares, chromospheres and coronae, and activity cycles. The existence of stellar rotation comes as no surprise. Since we believe stars form out of interstellar clouds, and clouds collide with one another and are subject to other torques such as those induced by galactic rotation, we would predict rather large stellar rotation on the basis of the conservation of angular momentum. Moment of inertia scales roughly as the square of the linear dimension. With 44 million solar radii per parsec, cloud rotation should be magnified by something like 15 orders in coming to stellar dimensions. Some stars rotate near the “break-up” velocity, where the gravitational acceleration is comparable to the centripetal force at the equator, but even these rates are small compared to what is expected. Apparently dissipation of angular momentum is an integral part of star formation.

When we look at stellar spectra, we soon recognize the presence of many chemicals. The lines of hydrogen dominate the photospheric spectra of hot stars, but with declining temperature, thousands of lines from other species grow stronger while the lines of hydrogen weaken. One of the tasks of the stellarphotosphere analyst is to disentangle the effects of temperature, pressure, turbulence, and so on from the effects of chemical composition. In broad strokes, the results of chemical analyses tell us that hydrogen is overwhelmingly the most abundant element, comprising ≈90% of the atoms in a normal stellar photosphere, and helium comes next with ≈10%. The remaining elements, often referred to as “metals,” comprise a sprinkling – like salt in a bowl of broth – adding savor and interest. Most of the spectral lines are caused by these low-abundance elements.

Besides our natural curiosity to know what stars are made of, we are motivated to perform chemical composition studies on other fronts. To be more specific, almost any study in stellar atmospheres presupposes some chemical composition. Chemical evidence tells us about the nuclear reactions taking place in stars, internal mixing of material, penetration depths of convection zones, diffusion and gravitational settling, or accretion of material from interstellar space. Lithium and carbon isotope abundances are signatures of the evolutionary stage of a star. The evolution of our galaxy is traceable in part through the differences in chemical composition among stars, since we expect stars formed at different times and in different places to retain information concerning the composition of the material from which they formed.

Observations of stellar photospheres require instruments for collecting and analyzing light. Low-resolution spectrographs are used for measuring the continuous spectrum, while high-resolution spectrographs are needed to measure the spectral lines. In this chapter, we delve into several basic aspects of spectrographs and diffraction gratings, then turn our attention briefly to interferometers, and finally to telescopes. Special features and application of spectrophotometric equipment are discussed separately in Chapter 10 (continua) and Chapter 12 (lines). Chapter 4 is about light detectors.

Spectrographs: some general relations

The basic astronomical spectrograph is shown in Fig. 3.1. It consists of an entrance slit placed at the focus of the telescope, a collimator that intercepts the divergent telescope beam, a dispersing element (prism or grating), and a camera that focuses the dispersed light onto a detector. Since the purpose of the collimator is to make the divergent beam parallel, the distance between the slit and the collimator is the focal length of the collimator, Fcoll. Similarly, the distance between the camera and the focused spectrum is the focal length of the camera, Fcam. The distances between the collimator, disperser, and camera affect the detailed design and optimization of the spectrograph, but do not matter for the basics we are considering at the moment. We concentrate on plane diffraction gratings as the dispersing element since these are almost universally used in astronomical spectrographs.

The remarkable nature of stars is transmitted to us by the light they send. The light escapes from the outer layers of the star – called, by definition, the atmosphere. The complete atmosphere of a star can be viewed comprehensively as a transition from the stellar interior to the interstellar medium. And yet almost the whole visible stellar spectrum comes from a relatively thin part called the photosphere. Obviously we cannot disconnect the photosphere from the adjacent portions of the atmosphere, but in actual fact it is the only region we can study extensively for most stars. It is for this reason that the photosphere has taken its place as the central theme of this book.

Several books have appeared during the last decade dealing with the theory of stellar atmospheres. These works are for the most part excellent. It is to the material largely omitted by these books that the present treatise is directed. My students and I have felt for some time the need of a book that presents the basics of the field through the eyes of an observer and analyzer of stellar atmospheres.

An introduction to a subject, in my opinion, should be presented in a way that can be understood by a reader who has not studied the topic before. It follows that the material should be presented in as simple and straightforward a manner as possible. The Fourier transform (as covered in Chapter 2) is a unifying theme helping to accomplish this aim.

There is hardly a phase of modern astrophysics to which Fourier techniques do not lend some insight or practical advantage. Fourier concepts prove useful in the context of line absorption coefficients, the analysis of line profiles, spectrograph resolution, telescope diffraction, and the study of noise. In these and other applications, convolutions appear in the physics of the situation. Usually it is much easier to visualize a product of functions in place of their convolution and this can be done with Fourier transforms through the convolution theorem.

This chapter forms an introduction to Fourier transforms for those unfamiliar with them and a useful refresher for those who have studied them in past years. The treatment is highly abbreviated, but covers all the concepts used in the remainder of the book. Those wishing to learn the material in a more rigorous and extensive way are referred to the books dealing specifically with Fourier transforms, for instance, the books of Jennison (1961), Bracewell (1965), and Gaskill (1978).

The definition

The Fourier transform of a function is a specification of the amplitudes and phases of sinusoidals which, when added together, reproduce the function. Only one-dimensional functions are treated here. Expansion to two or higher dimensions can be done by the reader with modest effort.

Wonderful growth has occurred in our understanding of stellar photospheres during the 15 years since the appearance of the first edition of “Photospheres.” I have managed to retain the same chapter names and the general plan of the first edition, and many of the equation numbers are also the same. But a significant portion of the material is new or revised. A revolution in light detectors has given us hundreds of times greater efficiency in measuring stellar spectra; Chapter 4 on detectors has been re-done. The astronomical literature is burgeoning with new results on the structure of photospheres, chemical abundances, radius measurements, stellar rotation, and photospheric velocity fields. Many of these results have been incorporated in this second edition, of course. At the same time, I stayed with my original purpose of making this volume an introduction to the subject. Unhappily, this means leaving out numerous exciting topics. My book Lectures (Gray 1988) takes up some of these, and it is recommended as a second installment, after the material in “Photospheres” has been mastered.

More than ever, the reader should keep in mind the fundamental nature of the stellar photosphere: of interest in its own right, with marvelous and intriguing physics, yet the link between the interior and chromospheres, coronae, and interstellar surroundings, and the source of most of our basic information about stars and stellar systems.

The dominant mechanism of energy transport through the surface layers of a typical star is radiation, i.e., photons. Transport by convection is often important below the surface, but rarely carries a significant fraction of the flux in the photosphere. Conduction comes into play in extreme cases such as white dwarfs. So it is radiative transfer that is our main focus here. It is in the domain of radiative transfer where the physical parameters of the material comprising the star are coupled to the spectrum we see and measure. We start by setting up the differential equation describing the flow of radiation through an infinitesimal volume. The integration of the equation can then be accomplished for the geometry of the situation. Unfortunately the step from a differential to an integral equation is not a physical solution to the problem because the integrand still depends on the atomic excitation of the material, which itself depends on the temperature of the material, and the radiation field in the material. Both the thermal (collisional) and the non-thermal (radiative) excitation vary with depth in the photosphere. In most applications of the theory to real stars, a numerical model of the star's photosphere is formed from which the integrand and then the spectrum of the star can be calculated. This chapter, along with Chapters 8 and 9, develops the tools for this modeling.

Motions of the photospheric gases introduce Doppler shifts that shape the profiles of most spectral lines. Doppler shifts arising from the rotation of the star are equally significant, and often dominate the shaping process. (We concentrate on rotation in the next chapter.) In a very general sense, the shape of the distribution of Doppler shifts will be the shape taken on by the spectral lines. Our job is to make this quantitative, to work backward from the line profiles and deduce the nature of the velocity fields. The first task is to separate rotational broadening from the photospheric-velocities broadening. For the photosphere, we would like to know the geometry of the motion, whether we are dealing with wave motion, convective motion, prominence-like geysers, or some other kinds of flow. Measurements of the characteristic sizes and any temperature heterogeneities would help us discern what physics we are dealing with and whether or not the dynamics affects the ionization equilibrium, the excitation, and the transfer of radiation.

For the Sun we have a wealth of information with considerable spatial resolution, and there the velocity fields are dominated by granulation (the top of the convection zone) with a smaller contribution from non-radial oscillations. Since we have no spatial resolution for most other stars, we are fundamentally handicapped in having to deal with disk-integrated line profiles.

The strengths and shapes of spectral lines contain a great deal of information about the stars, and the line absorption coefficient plays a fundamental role here. The situation in this regard is similar to the effect the continuous absorption coefficient has on the shape of the continuum. Lines are more interesting, however, because several different physical effects can enter the structuring of the final absorption coefficient. Each of these has its own variation with wavelength across the line, that is, its own absorption coefficient. The main processes we consider in this chapter are: (1) natural atomic absorption, (2) pressure broadening, of which there are several, and (3) thermal Doppler broadening. The final combined absorption coefficient is the multiple convolution of these individual absorption coefficients. One of the remarkable results of these studies is that the natural atomic absorption and all the significant pressure broadenings have the same wavelength dependence in their individual absorption coefficients (with the notable exception of the hydrogen lines), namely the dispersion profile. This leads to a tremendous simplification, as we shall see, since the convolution of dispersion profiles is a dispersion profile with a half-width equal to the sum of the individual half-widths of the contributing processes. The thermal broadening, on the other hand, reflects the Maxwellian velocity distribution of the absorbing atoms and ions via the Doppler effect, so its wavelength shape is a Gaussian. The final absorption coefficient is therefore a convolution of the dispersion profile with the Gaussian profile.

Stars higher above the main sequence in the HR diagram (refer back to Fig. 1.6) have higher luminosity and lower photospheric pressures. Indeed, pressure effects are often referred to as luminosity effects. Those pressure effects that are large enough to see at classification dispersion are used to assign the luminosity class to the spectrum. Generally speaking, spectroscopic pressure effects are more subtle than temperature effects. On the other hand, while temperature spans a factor of ten or so from O to M stars, pressure ranges over six orders from dwarfs to supergiants, and so even subtle effects are quite measurable in stellar spectra.

Continuum measurements across the Balmer jump can be used with the A and F stars to measure the electron pressure. A comparison of lines formed by neutral atoms to lines of ions gives a measure of electron pressure through the ionization equilibrium. Lines strong enough to show wings (including hydrogen) often have a wing strength that is dependent on the pressure through the van der Waals and Stark broadening. In each case there is also a temperature dependence, so this chapter is closely tied to Chapter 14. Either the temperature must be established securely before executing the pressure analysis or a simultaneous pressure–temperature solution must be made. There are also empirical relations between photospheric pressure and spectroscopic parameters such as macroturbulence.

We study stellar spectra, including both lines and continua, because we are interested in the nature of the star's atmosphere. The behavior of the atmosphere is controlled by the density of the gases in it and the energy escaping through it. These in turn depend on the mass and age of the star and to a lesser extent on chemical composition and angular momentum. Stellar atmospheres are the connecting links between the observations and the rest of stellar astrophysics. In this way two philosophies arise. One is the study of the atmosphere for its own sake and the other is the use of the atmosphere as a tool to connect our observations to other parameters of interest. This book should be useful for students of both philosophies.

The topics brought together to form this chapter are background material which the reader will need. The more advanced reader can profitably skim through to Chapter 2.

What is a stellar atmosphere?

A stellar atmosphere is a transition region from the stellar interior to the interstellar medium. One way to quantify this description is to look at the change in average kinetic temperature with height as observed in the Sun. Figure 1.1 shows the solar temperature profile with the four basic sections labeled, sub-photosphere, photosphere, chromosphere, and corona. For an observer of stellar atmospheres, one concept is very important: the major portion of the visible stellar spectrum originates in the region marked “photosphere.” A study of the visible-light spectrum is essentially a study of the photosphere.

Light detectors have the important job of converting stellar photons into recordable signals. Remarkable and wonderful developments in detectors have occurred over the last two decades, resulting in dramatic increases in speed and improved signal-to-noise ratios. Most notable for stellar-photospheric studies are the silicon array detectors such as the charge-coupled devices (CCDs) and self-scanned arrays. To make quality observations of stellar spectra, the observer must be cognizant of the basic detector parameters: quantum efficiency, spectral response, linearity, noise, and spatial resolution. These properties can be defined for any detector, and the reader can extend the concepts of this chapter to the detector of choice for the job to be done.

Detectors can be grouped into “integrating” and “pulse-counting” types. In the first class, photons are accumulated for some integration time, and then the total signal is measured. Examples are (unintensified) silicon-diode detectors and the photographic process. In the second class, electrical pulses for individual photons are recorded. Examples include photomultipliers and various kinds of electron image tubes. Integrating-silicon devices tend to be used for higher signal-to-noise ratio work and pulse-counting ones for lower signal-to-noise ratios. A high signal-to-noise ratio is needed for much of our work on stellar photospheres.

In this chapter, we consider the acquisition of spectral line data and the integrity of such data. Several types of measurements can be made on a spectral line: total absorption, characteristic width, detailed shape, asymmetry, wavelength position, and polarization. These measurements are not all independent, but it helps to think of them separately when measurements are to be performed. We shall consider all of these except polarization.

Compared to continuum photometry, studies of spectral lines often require very high spectral resolution. How high depends on the width and structure of the spectral lines to be measured, but λ/Δλ in the range of 100 000 or even higher is what we are talking about. This kind of resolving power is obtained with low-order diffraction gratings, echelle gratings, and interferometers. The emphasis here is on grating spectrographs, an elaboration on the basics discussed in Chapter 3. (For details on interferometers, see Vaughan 1967, Connes 1970, and Ridgway & Brault 1984.) High resolution means a small wavelength interval per detector pixel, perhaps 15 or 20mÅ. Contrast this to ∼1000 Å for a wide-band photometric system, a ratio of ≈60 000 or 12 magnitudes. High-resolution spectroscopy, where we want to get at the physics of spectral lines, is the domain of bright stars. Large-aperture telescopes are just as important for high-resolution spectroscopy as they are for faint galaxy work. In practical terms, one can expect a 30 minute exposure of a sixth magnitude star to yield a signal-to-noise ratio of 100 at a resolving power of 105 using a telescope of 1m aperture.