Theory of Stellar Atmospheres: An Introduction to Astrophysical Non-equilibrium Quantitative Spectroscopic Analysis

Theory of Stellar Atmospheres: An Introduction to Astrophysical Non-equilibrium Quantitative Spectroscopic Analysis

by Ivan Hubeny, Dimitri Mihalas

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This book provides an in-depth and self-contained treatment of the latest advances achieved in quantitative spectroscopic analyses of the observable outer layers of stars and similar objects. Written by two leading researchers in the field, it presents a comprehensive account of both the physical foundations and numerical methods of such analyses. The book is ideal

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This book provides an in-depth and self-contained treatment of the latest advances achieved in quantitative spectroscopic analyses of the observable outer layers of stars and similar objects. Written by two leading researchers in the field, it presents a comprehensive account of both the physical foundations and numerical methods of such analyses. The book is ideal for astronomers who want to acquire deeper insight into the physical foundations of the theory of stellar atmospheres, or who want to learn about modern computational techniques for treating radiative transfer in non-equilibrium situations. It can also serve as a rigorous yet accessible introduction to the discipline for graduate students.

  • Provides a comprehensive, up-to-date account of the field
  • Covers computational methods as well as the underlying physics
  • Serves as an ideal reference book for researchers and a rigorous yet accessible textbook for graduate students
  • An online illustration package is available to professors at

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"It is an excellent guide for anyone interested in radiation transport and spectral analyses in astrophysics."—Claudia-Veronika Meister, Zentralblatt MATH

"A magisterial work that will surely be the definitive reference for many years to come."—Ian D. Howarth, The Observatory

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Princeton University Press
Publication date:
Princeton Series in Astrophysics Series
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6.30(w) x 9.40(h) x 2.30(d)

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Theory of Stellar Atmospheres

An Introduction to Astrophysical Non-equilibrium Quantitative Spectroscopic Analysis

By Ivan Hubeny, Dimitri Mihalas


Copyright © 2015 Princeton University Press
All rights reserved.
ISBN: 978-1-4008-5273-4


Why Study Stellar Atmospheres?

A central objective of astrophysics is to use physical theory to simulate the conditions in astrophysical objects. Today's astrophysicist must, in principle, be familiar with just about all of physics: elementary particle theory and nuclear physics; quantum mechanics and atomic/molecular physics; classical electrodynamics, quantum electrodynamics, and plasma physics; hydrodynamics and magnetohydrodynamics; classical gravitational mechanics, special relativity, and general relativity. The goal of this book is decidedly more modest: (1) to present the physical and mathematical tools needed to make models of stellar atmospheres that are realistic enough to fit closely the observed spectrum of a star, and (2) to show how they can be used to make a reliable quantitative spectroscopic analysis of the physical structure and chemical composition of its outer layers. The problems to be faced in achieving it are developing correct formulations of the physics of spectral line and continuum formation and the transport of radiation.

The complex systems of nonlinear equations that describe cosmic objects can be solved using the extremely fast computers now available. In addition, we now have access to most of the electromagnetic spectrum, ranging from radio wavelengths to gamma rays. These observational data provide a solid basis upon which we can construct theoretical structures to interpret them. It is a reasonable metaphor to say that today our picture of the Universe is developing as rapidly, and as radically, as if Galileo and Newton had lived and worked at the same time.

But given the crowded curriculum faced by astrophysics students one may ask, "Why take time to study the outer layers of stars? Is this work merely a 'cottage industry' of no great relevance to the rest of astrophysics?" This question was put pointedly to one of us (D. M.) nearly 50 years ago by E. Salpeter: "Why in the world would anyone want to study stellar atmospheres? They contain only about 10-10 of the mass of a star. Surely such a negligible fraction of its mass cannot affect its overall structure and evolution!" His query is reasonable and deserves a good answer.


Salpeter's question must be put in the perspective of his own seminal work on stellar structure, stellar evolution, and nucleosynthesis. Relative to those studies, the theory of stellar atmospheres at that time had not produced spectacular results, and did not yet have a sound theoretical basis. To provide some context, we outline below a few milestones in the development of today's observational techniques; the theory of astrophysical radiative transfer and quantitative spectroscopy; and the theory of stellar evolution. Other authors would doubtless select different topics than we have chosen here.


At the beginning of the 20th century, the most powerful telescopes were long-focus refractors, the largest being the 40" diameter telescope at Yerkes Observatory, built under the direction of G. E. Hale. They were used for visual observations of the orbits of binary stars (which give data for the determination of stellar masses); photographic observations of radial velocities (the component of a star's velocity, relative to the Sun, along the line of sight); proper motions (which are proportional to a star's velocity, relative to the Sun, perpendicular to the line of sight); and stellar parallaxes (which give a star's distance from the Sun).

After Hale left Yerkes to found the Mt. Wilson Observatory, he directed the construction of the 60" and 100" reflectors on Mt. Wilson and the 200" reflector and 48" wide-angle Schmidt camera on Palomar Mountain. Each increase in aperture permitted fainter objects to be observed. The instrumentation for these large reflectors was at the cutting edge of technology then available. For example, their spectrographs contained large blazed gratings made with interferometrically controlled grating-ruling engines at the headquarters of the observatory in Pasadena. These had higher efficiency and resolution, and better freedom from ghosts, than any previously made. But until the 1950s astronomical telescopes had little automation other than drives to track the apparent motion of stars across the sky. After about 1950, the sophisticated electronic measurement and control techniques of physics laboratories began to invade the mountaintop redoubts of astronomers.

In the past two to three decades observational astrophysics has enjoyed unparalleled growth. We now have telescopes with apertures of several meters, which have thin mirrors whose optical figures can be adjusted to minimize the effects of turbulence in the Earth's atmosphere and corrected for flexure and other transient defects, in real time, using high-speed computers. Soon, it will be possible to restore images to nearly the diffraction limit of a telescope. With such instruments, we can observe objects at low light levels that were previously inaccessible. The far infrared, ultraviolet, X-ray, and gamma-ray regions of the spectrum can now be observed with space telescopes.

Today's echelle spectrographs can capture spectra at high resolution in many orders simultaneously. Photographic plates have been replaced with CCDs (charge-coupled devices), which have very high light detection efficiency and a linear response that permits precise calibration and measurement of images of stars, extended objects, and echelleograms. With them we can now obtain high signal-to-noise spectra of extremely faint objects.

At radio wavelengths, interferometric techniques developed for the Very Large Array (VLA) and the Very Long Baseline Interferometer (VLBI) have produced exquisite pictures showing that many galaxies have massive black holes at their centers, which spew immense jets of material racing outward at relativistic speeds. These data imply the existence and continuing modification of an intergalactic medium from which new galaxies form. In short, it is now possible to observe, and begin to model, phenomena totally unknown only a few years ago.


The two most important observational activities in astrophysics at the start of the 20th century were photometry and spectral classification. Strenuous efforts were made to set up accurate photometric brightness scales for stars, but they were thwarted by the nonlinear response of photographic emulsions to exposure. Through the 1940s, the apparent brightnesses of stars could not be measured to much better than 10%, and often only to 20%–30%. Brightness measured in different wavelength bands could be used to make estimates of stellar colors, which give a low-resolution measure of the distribution of light in their spectra.

The arrival of photomultipliers in the early 1950s gave astronomers very sensitive linear receivers accurate to 1% and revolutionized stellar photometry. With them, standard photometric systems were established and used for incisive analyses of stellar properties (see § 2.4–§ 2.6). By the mid-1960s, photoelectric spectrophotometers calibrated against standard sources having known energy distributions put spectrophotometry on an absolute energy scale (see § 2.3). Today, spectrophotometric measurements of the continua of stars can be matched with high precision to results obtained from physically realistic theoretical model atmospheres that allow for the effects of many thousands of spectral lines. With arrays of CCDs containing thousands of individual detectors one can measure simultaneously the apparent brightness of huge numbers of stars in large fields of the sky.

The state of stellar spectroscopy was better. In the early 1900s, it was found that stellar spectra could be arranged in a sequence that correlates closely with a star's effective temperature. This classification scheme was quickly supplemented with additional criteria based on subtle effects that correlate with the average density in a star's atmosphere. These phenomena could be interpreted using theoretical work in statistical mechanics. By 1914, E. Hertzsprung and H. Russell showed that stars fall in definite sequences in the Hertzsprung-Russell diagram (or HR diagram), a plot of their luminosity versus effective temperature. This discovery had profound implications for the development of a theory of stellar structure and evolution.

Measurement of stellar spectra progressed from mere eye estimates of "line intensities" in the early 1900s to photographic measurements of line profiles and equivalent widths in the 1930s. In contrast to the absolute photometry needed to determine stellar magnitudes, these measurements require only relative photometry, i.e., comparing the light at several wavelengths in a line to the local continuum. Hence the results were more accurate. Today, with linear receivers such as CCDs, very precise measurements can be made simultaneously for a large range of wavelengths.

A breakthrough came in 1944 when W. Baade (see especially the excellent review) made the seminal discovery of two stellar populations in the Galaxy, whose properties were determined by, and give information about, the formation and subsequent development of the Galaxy. His work integrated our observational picture of the stellar content of galaxies and also provided critical guidance to stellar evolution theory. He found that the distributions of the two populations of stars in the HR diagram are distinctively different. HR diagrams for Population I and Population II stars are shown in figures 2.6 and 2.7, respectively.

Population I objects in our Galaxy are typified by (1) galactic clusters (loose clusters containing ~ 102-103 stars), which include (a) a main sequence extending from massive, hot, very luminous blue dwarfs to cool, less massive, much less luminous red dwarfs (see table 2.4); (b) very luminous, cool, red supergiants in young clusters; and (c) subgiants and red giants in older clusters; (2) Cepheid pulsating variables; and (3) interstellar material. These objects are located near the central plane of the disk of the Galaxy. In other spiral galaxies, they are found in bright spiral arms bordered by dark dust lanes.

The great majority of stars in the solar neighborhood belong to Population I and are on the main sequence. In their cores, these stars are converting hydrogen to helium in thermonuclear reactions. This process releases the largest amount of energy per reaction, and hydrogen is the most abundant element in stellar material. Therefore, a star spends more time "burning" hydrogen to helium than in any other stage in its evolutionary history, so most stars will be found in this phase. Population I stars have relatively small velocities with respect to the Sun. They move on high angular momentum, nearly circular orbits around the Galactic center. They have near-solar abundances of "metals" (astrophysical jargon for elements with Z ≥ 6; Z being the atomic number).

Typical Population II objects in our Galaxy are (1) globular clusters (gravitationally bound spherical systems containing ~ 105 - 106 stars); (2) individual halo stars, weak-lined high-velocity stars, and subdwarfs in the solar neighborhood; (3) RR Lyrae pulsating variables; and (4) planetary nebulae and their nuclei, which are old stars in late stages of their life. These objects are not strongly concentrated in the Galactic plane. Indeed, the distribution of globular clusters is roughly isotropic around the center of the Galaxy. Population II stars make up the central bulge in other spiral galaxies and most elliptical galaxies. They have low angular momentum and low velocities around the center of the Galaxy; hence they have high velocities relative to the Sun. The most extreme of these stars, and globular clusters, move on almost radial "plunging" orbits with respect to the Galactic center, [chapter 10]. This kinematic behavior gives clues about the formation of the Galaxy. In 1951, the extreme subdwarfs HD 19445 and HD 140283 were found to have "metal" abundances smaller by at least a factor of 25 to 40 than the Sun (a much too conservative estimate; seethe discussion in [938, p. 433]). Since then, spectroscopic analyses of many Population II stars have been made, which give metal abundances factors of 10 to 105(!) smaller than solar; see, e.g., [85, 206, 258, 273, 347, 635, 802, 1022, 1024, 1036, 1117, 1130, 1131, 1132, 1133, 1136]. These stars presumably represent a primeval population.

At the opposite extreme, some (quite young?) super-metal-rich (SMR) Population I stars have a higher metal abundance than the Sun; see, e.g., [204, 293, 327, 1020, 1070, 1134]. It is clear that as a function of time there has been a progressive enrichment of the metal abundance of the material from which stars form.

The conventional notation used to indicate a star's metal abundance relative to the Sun is

[Fe/H] = log [(Fe*/H*) / (Fe[??]/H[??])]. (1.1)

Here iron is used as a proxy for all elements with Z ≥ 6. This notation is oversimplified because there are variations from star to star in the ratio of the abundance of any chosen element to that of iron. Using criteria based on their distribution, kinematics, and abundances, the idea of stellar populations has been elaborated into a picture having several groups intermediate between the extremes represented by the most metal-poor globular clusters and youngest galactic clusters [800, 805].

The earliest generation of stars was composed of about 90% hydrogen by number; about 10% helium by number; and very small amounts of some isotopes of Li and Be, which were formed from primeval hydrogen in the Big Bang. Modern work shows that the He/H ratio is about the same in the interiors of both Population I and Population II stars. This fact implies that essentially all He was formed in the Big Bang. On the other hand, the existence of young and very old stars, having high and low metal abundances, respectively, shows that there has been a progressive enrichment of elements with Z ≥ 6 in the interstellar material from which stars form. The heavy elements in the interstellar medium are created by nucleosynthesis, i.e., by thermonuclear processing of material in the cores of highly evolved, massive stars. This material is deposited into the interstellar medium by supernovae.

Stellar Structure and Evolution

The earliest models of the internal structure of stars were based on Ritter and Emden's theory of polytropic gas spheres with self gravity. By assuming that the gas pressure pgas = Kρ(n+1)/n, where ρ is the mass density [gm/cm3], the equations for hydrostatic pressure balance can be combined with Poisson's equation for the gravitational field to get a single second-order differential equation for ρ(r), the variation of density with radius in the star. It can be solved analytically for n = 0,1, and 5 and by numerical integration for other values of n.


Excerpted from Theory of Stellar Atmospheres by Ivan Hubeny, Dimitri Mihalas. Copyright © 2015 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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