Theory of Rotating Stars. (PSA-1)

Theory of Rotating Stars. (PSA-1)

by Jean-Louis Tassoul

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Ever since the first observations of sunspots in the early seventeenth century, stellar rotation has been a major topic in astronomy and astrophysics. Jean-Louis Tassoul synthesizes a large number of theoretical investigations on rotating stars. Drawing upon his own research, Professor Tassoul also carefully critiques various competing ideas.

In the first three


Ever since the first observations of sunspots in the early seventeenth century, stellar rotation has been a major topic in astronomy and astrophysics. Jean-Louis Tassoul synthesizes a large number of theoretical investigations on rotating stars. Drawing upon his own research, Professor Tassoul also carefully critiques various competing ideas.

In the first three chapters, the author provides a short historical sketch of stellar rotation, the main observational data on the Sun and other stars on which the subsequent theory is based, and the basic Newtonian hydrodynamics used to study rotating stars. Following a discussion of some general mechanical properties of stars in a state of permanent rotation, he reviews the main techniques for determining the structure of a rotating star and its stability with respect to infinitesimal disturbances. Since the actual distribution of angular momentum within stars is still unknown, Professor Tassoul considers various models of angular momentum as well as of meridional circulation. He devotes the rest of his study to the problems concerning various groups of stars and stages in stellar evolution.

Originally published in 1979.

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Princeton University Press
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Princeton Series in Astrophysics Series , #1

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Theory of Rotating Stars

By Jean-Louis Tassoul


Copyright © 1978 Princeton University Press
All rights reserved.
ISBN: 978-0-691-08211-0


An Historical Overview


The study of stellar rotation began about the year 1610, when sunspots were observed for the first time through a refracting telescope. The first public announcement of an observation came in June 1611 from Johannes Goldschmidt (1587–1615) — a native of East Friesland, Germany — who is generally known by his Latinized name Fabricius. From his observations he correctly inferred the spots to be parts of the Sun itself, thus proving axial rotation. He does not appear to have appreciated the importance of this conclusion, and pursued the matter no further.

According to his own statement, Galileo Galilei (1564–1642) observed the sunspots toward the end of 1610. He made no formal announcement of the discovery until May 1612, by which time such observations had also been made by Thomas Harriot (1560–1621) in England and by the Jesuit Father Christoph Scheiner (1575–1650) in Germany. A controversy about the nature of sunspots made Scheiner a bitter enemy of Galileo and developed into a quarrel regarding their respective claims to discovery. Since Scheiner had been warned by his ecclesiastical superiors not to believe in the reality of sunspots (because Aristotle's works did not mention it), he announced his discovery in three letters written under the pseudonym of Apelles. He explained the spots as being small planets revolving around the Sun and appearing as dark objects whenever they passed between the Sun and the observer. These views opposed those of Fabricius and Galileo, who claimed that the spots must be on or close to the solar surface.

In the following year, 1613, Galileo replied with the publication of his Istoria e Dimostrazioni intorno alle Macchie Solari e loro Aceidenti. In these three letters he refuted Scheiner's conclusions and, for the first time, publicly declared his adherence to the heliocentric theory of the solar system — thus initiating the whole sad episode of his clashes with the Roman Inquisition. As far as solar rotation is concerned, one of his arguments is so simple — and at the same time so convincing — that it may be worthwhile to reproduce it here. Galileo noticed that a spot took about fourteen days to cross the solar disc, and that this time was the same whether the spot passed through the center of the solar disc or along a shorter path at some distance from the center. However, he also noticed that the rate of motion of a spot was by no means uniform, but that the motion always appeared much slower when near the solar limb than when near the center. This he recognized as an effect of foreshortening, which would result if and only if the spots were near the solar surface.

Scheiner's views were thus crushingly refuted by Galileo. Eventually, the Jesuit father's own observations led him to realize that the Sun rotates with an apparent period of about 27 days. To him also belongs the credit of determining with considerably more accuracy than Galileo the position of the Sun's equatorial plane and the duration of its rotation. In particular, he showed that different sunspots gave different periods of rotation and, furthermore, that the spots farther from the solar equator moved with a slower velocity. Scheiner published his collected observations in 1630 in a volume entitled Rosa Ursina sive Sol. (Dedicated to the Duke of Orsini, the title of the book derives from the badge of the Orsini family, which was a rose and a bear.) This was truly the first monograph on solar physics.

For more than two centuries the problem of solar rotation was practically ignored, and it is not until the 1850s that any significant advance was made. Then, a long series of observations of the apparent motion of sunspots was undertaken by Richard Carrington (1826–1875), a wealthy English amateur, and by Gustav Spörer (1822–1895), a German astronomer. They confirmed, independently, that the outer visible envelope of the Sun does not rotate like a solid body, i.e., its period of rotation varies as a function of heliocentric latitude. They showed that the rotation period is minimum at the equator and increases gradually toward the poles. After correcting for the annual motion of the Earth around the Sun, Carrington derived a mean period of 24.96 days at the solar equator.

Several attempts have been made to represent analytically the dependence of rotational speed on latitude by means of an empirical formula. From his own observations made during the period 1853–1861, Carrington derived the following expression for the daily angle of solar rotation:

[xi](deg/day) = 14[??]42 – 2[??]75 sin7/4 φ, (1)

where φ is the heliocentric latitude. Somewhat later, the French astronomer Hervé Faye (1814–1902) found that the formula

[xi](deg/day) = 14[??]37 – 3[??]10 sin2 φ (2)

more satisfactorily represented the dependence of rotation on heliocentric latitude. Other formulae have been suggested, but Faye's empirical law has remained in use until now.

The next step was taken by Hermann Vogel (1841–1907) in Potsdam. In 1871, by means of a new spectroscope devised by Johann Zöllner (1834–1882), Vogel showed that the solar rotation can be detected from the relative Doppler shift of the spectral lines at opposite edges of the solar disc, one of which is approaching and the other receding. Accurate and extensive observations were made visually by Nils Dunér (1839–1914) in Lund and Upsala, and then by Jakob Halm (1866–1944) in Edinburgh. Dunér and Halm used a relative method of measurement based on the difference in behavior of the solar and terrestrial lines. They concluded that Faye's law adequately represents the spectroscopic observations also, but their coverage of latitude was double that of the sunspot measurements. After these early visual observations made during the period 1887–1906, photography almost completely superseded the human eye. The first spectrographic determinations of solar rotation were undertaken at the turn of the century by Walter S. Adams (1876–1956) and George E. Hale (1868–1938) at Mount Wilson Solar Observatory, California.


In 1877, Sir William de Wiveleslie Abney (1843–1920) suggested that axial rotation might be responsible for the great widths of certain stellar absorption lines. He correctly pointed out that a spectrum is actually a composite of light from all portions of the star's disc. Because of the Doppler effect, wavelengths in the light of the receding edge of the star would be shifted toward the red, those from the approaching edge toward the violet. Abney concluded: "There would be a total broadening of the line, consisting of a sort of double penumbra and a black nucleus. ... More than this, rotation might account for the disappearance of some of the finer lines of the spectrum. ... I'm convinced that from a good photograph much might be determined" (M.N. 37, 278, 1877). These suggestions were severely criticized by Vogel because some stellar spectra contained both broad and narrow lines. Concerning the use of photography, Vogel went further in saying that "even on the most successful photograph only relative measures of lines would be possible in regard to their width, so that no conclusions could be obtained in regard to rotation; the widths of lines on different photographic plates would depend upon the length of exposure, the sensitivity of the plate, and the length of development" (A.N. 90, 71, 1877).

Abney's suggestion found little favor among his contemporaries. The reason may be due to the enormous weight accorded the opinion of Vogel, who, for many years, dominated the field of stellar spectroscopy. In any case, only a few years after his emphatic predictions, Vogel introduced photography into stellar spectroscopy and completely reversed his stand in 1898, expressing himself in favor of the hypothesis that rotation does produce a measurable broadening of stellar lines.

As often happens in the history of science, the discovery of axial rotation in stars was purely accidental. In 1909, convincing evidence of rotation in the eclipsing and spectroscopic binary δ Librae was obtained by Frank Schlesinger (1871–1943), then at Allegheny Observatory, Pittsburgh. He noticed that just before and just after light minimum, the radial velocities he had measured on his spectrograms departed from their expected values. A positive excess was observed just before mid-eclipse, while after mid-eclipse the departure was found to be negative. Schlesinger concluded that this occurrence could be produced if the brightest star rotates around an axis, so that, at the time of partial eclipse, the remaining portion of its apparent disc is not symmetrical with respect to its axis of rotation. One year later, Schlesinger observed a similar phenomenon in the eclipsing system λ Tauri. In 1924, at the University of Michigan, Richard A. Rossiter (1886–1977) positively established the effect in β Lyrae and gave a complete curve of the residuals in velocity during the eclipse, while Dean B. McLaughlin (1901–1965) investigated the effect in the binary star β Persei. These were the first accurate measurements of the axial rotation of stars.

Another approach to the determination of stellar rotation was provided by spectroscopic binaries not known to be variable in light. In 1919, at Mount Wilson, Adams and Alfred H. Joy (1882–1973) studied the binary W Ursae Majoris of spectral type F8 and having a period of 0.334 day. They observed that "the unusual character of the spectral lines is due partly to the rapid change in velocity during even our shortest exposures but mainly to the rotational effect in each star, which may cause a difference of velocity in the line of sight of as much as 240 km/sec between the two limbs of the star" (Ap. J. 49, 190, 1919). Ten years later, a systematic study of rotational line-broadening in spectroscopic binaries was undertaken jointly by Grigori Abramovich Shajn (1892–1956) in the Soviet Union and Otto Struve (1897–1963) in the United States. (Their collaboration took place by mail.) Shajn and Struve positively established that in spectroscopic binaries of short period, at least, line broadening is essentially a result of rotation.

The next step was to extend these measurements to single stars. Indeed, the possibility existed that rapid rotation occurred only in binary stars, perhaps because tidal forces produced synchronization of axial rotation and orbital revolution. Since the Sun has a very slow axial rotation (about 2 km/sec at its equator), most astrophysicists at the time believed that the rotation of all other single stars was probably also small. During 1930–1934, a systematic study of stellar rotation was undertaken by Struve, in collaboration with Christian T. Elvey (1899–1972) and Miss Christine Westgate, at the Yerkes Observatory of the University of Chicago. The measurements were made by fitting the observed contour of a spectral line to a computed contour obtained by applying different amounts of Doppler broadening to an intrinsically narrow line-contour having the same equivalent width as the observed line. They showed that the measured values of the rotational component of the velocity along the line of sight fell in the range 0–250 km/sec, and may occasionally be as large as 400 km/sec or even more. A correlation of rotational velocity with spectral type was originally discovered by Struve and Elvey in 1931: the O-, B-, A-, and early F-type stars frequently have large rotational velocities, while in late F-type stars and later types rapid rotation occurs only in close spectroscopic binaries. They also found that supergiants of early and late types, and normal giants of type F and later, never show conspicuous rotations.

At this juncture the problem was quietly abandoned for fifteen years. Interest in the measurements of axial rotation in stars was revived in 1949 by Arne Slettebak at Ohio State University.


Research into the influence of rotation upon the internal structure and evolution of a star has a long history. Sir Isaac Newton (1643–1727) was the first to realize the importance of the law of gravitation for the explanation of the figures of celestial bodies. He originally discussed the figure of the Earth in the Philosophiae Naturalis Principia Mathematica(Book III, Propositions 18–20, 1687) on the hypothesis that it might be treated as a homogeneous, slightly oblate spheroid rotating with constant angular velocity. Assuming that such a spheroid is a figure of equilibrium, Newton asserted that two perpendicular columns of fluid — one axial and the other equatorial — bored straight down to the Earth's center must have equal weight. From this condition, he was able to derive the causal relationship [Florin] = (5/4)m between the ellipticity [Florin] of a meridional section and the ratio m of the centrifugal force at the equator to the (average) gravitational attraction on the surface.

Further progress was made by Christiaan Huygens (1629–1695) in his Discours de la cause de la pesanteur, which was published at Leiden in 1690. To him we owe a necessary condition for the relative equilibrium of a rotating mass of fluid — that the resultant force of the attraction and the centrifugal force at any point of the free surface must be normal to the surface at that point. Huygens never accepted that adjacent particles of matter attract each other, but he did admit the existence of an attracting force always directed to a fixed point. Accordingly, in one of his models, he assumed that the Earth's gravity is a single, central force varying inversely as the square of the distance from its center. Thence, in making use of Newton's principle of balancing columns, Huygens derived the relation [Florin] = (1/2)m for his model, when departure from sphericity is small. As we now know, this is equivalent to the hypothesis that the density of the Earth is infinite at the center; and it should be contrasted with Newton's work in which it is assumed that the Earth is homogeneous in structure. In actual practice, the measured quantity [Florin] for the Earth is comprised between the values derived by Newton and Huygens for their two extreme models.

While Newton's ideas led ultimately to contemporary mechanics, at this stage they did not gain immediate acceptance in Europe. During the first half of the eighteenth century, most continental scientists were strong advocates of a vortex theory that had been devised by René Descartes (1596–1650) in 1644. The Cartesians rejected Newton's ideas mainly because attraction, regarded as a cause, was unintelligible; and they somehow inferred from their systems of vortices that the Earth should be flatter at the equator. In the 1730s, to settle the dispute between the Cartesians and the Newtonians, geodetic expeditions were sent to different parts of the Earth to measure the length of an arc of meridian. The expedition to Lapland returned first and confirmed that the Earth is indeed flatter at the poles. Pierre Louis Moreau de Maupertuis (1698–1759) — its leader and the first Frenchman with the courage openly to declare himself a Newtonian — was called "the great flattener." At this time the vortex theory was a lost cause.

During 1737–1743, the most significant advances in the subject were made by Alexis-Claude Clairaut (1713–1765) in Paris and by Colin Maclaurin (1698–1746) in Edinburgh. Again, to appreciate the importance of their discoveries, it must be borne in mind that the science of hydrostatics was then in an imperfect stage of its development. Actually, this early work was made without a clear understanding of the concept of internal pressure. Hence, whenever Clairaut and Maclaurin established a proposition, they had to rely upon Newton's principle of balancing columns, Huygens's principle of the plumb line, or both. As a matter of fact, necessary and sufficient conditions of hydrostatic equilibrium remained unknown until 1755, when Leonhard Euler (1707–1783) definitively established the general equations of motion for an in viscid fluid.

As we have seen, Newton assumed without demonstration an oblate spheroid as a possible figure of equilibrium for a slowly rotating mass of fluid. In 1737, Clairaut obtained an expression for the attraction of a homogeneous spheroid at any point of its surface, when the body does not greatly depart from a sphere. To first order in the ellipticity, he then showed that Huygens's principle of the plumb line obtains at every point on the free surface of Newton's model.


Excerpted from Theory of Rotating Stars by Jean-Louis Tassoul. Copyright © 1978 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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