Historical development

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The study of stellar rotation began at the turn of the seventeenth century, when sunspots were observed for the first time through a refracting telescope. Measurements of the westward motion of these spots across the solar disk were originally made by Johannes Fabricius, Galileo Galilei, Thomas Harriot, and Christopher Scheiner. The first public announcement of an observation came from Fabricius (1587-c. 1617), a 24-year old native of East Friesland, Germany. His pamphlet, De maculis in Sole observatis et apparente earum cum Sole conversione, bore the date of dedication June 13, 1611 and appeared in the Narratio in the fall of that year. Fabricius perceived that the changes in the motions of the spots across the solar disk might be the result of foreshortening, with the spots being situated on the surface of the rotating Sun. Unfortunately, from fear of adverse criticism, Fabricius expressed himself very timidly. His views opposed those of Scheiner, who suggested that the sunspots might be small planets revolving around an immaculate, nonrotating Sun. Galileo made public his own observations in Istoria e Dimostrazioni intorno alleMacchie Solari e loro Accidenti. In these three letters, written in 1612 and published in the following year, he presented a powerful case that sunspots must be dark markings on the surface of a rotating Sun. Foreshortening, he argued, caused these spots to appear to broaden and accelerate as they moved from the eastern side toward the disk center. The same effect made the sunspots seem to get thinner and slower as they moved toward the western side of the disk. Galileo also noticed that all spots moved across the solar disk at the same rate, making a crossing in about fourteen days, and that they all followed parallel paths. Obviously, these features would be highly improbable given the planetary hypothesis, which is also incompatible with the observed changes in the size and shape of sunspots.

The planetary hypothesis, championed by Scheiner among others, was thus convincingly refuted by Galileo. Eventually, Scheiner'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 Sun's 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, who sponsored the work. (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.

It is not until 1667 that any further significant discussion of stellar rotation was made. In that year the French astronomer Ismael Boulliaud (1605-1694) suggested that the variability in light of some stars (such as Mira Ceti) might be a direct consequence of axial rotation, with the rotating star showing alternately its bright (unspotted) and dark (spotted) hemispheres to the observer. This idea was popularized in Fontenelle's Entretiens sur la pluralite des mondes - a highly successful introduction to astronomy that went through many revised editions during the period 1686-1742. To be specific, he noted "... that these fixed stars which have disappeared aren't extinguished, that these are really only half-suns. In other words they have one half dark and the other lighted, and since they turn on themselves, they sometimes show us the luminous half and then we see them sometimes half dark, and then we don't see them at all." * Although this explanation for the variable stars did not withstand the passage of time, it is nevertheless worth mentioning because it shows the interest that stellar rotation has aroused since its inception. As a matter of fact, nearly three centuries were to elapse before Boulliaud's original idea was fully recognized as a useful method of measuring the axial rotation of certain classes of stars, that is, stars that exhibit a detectable rotational modulation of their light output due to starspots or stellar plages.

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 and Gustav Sporer. They confirmed, independently, that the outer visible envelope of the Sun does not rotate like a solid body; rather, its period of rotation varies as a function of heliocentric latitude. From his own observations made during the period 1853-1861, Carrington derived the following expression for the Sun's rotation rate:

where $ is the heliocentric latitude. Somewhat later, Herve Faye found that the formula more satisfactorily represented the dependence of angular velocity on heliocentric latitude. Parenthetically, note that Carrington also found evidence for a mean meridional motion of sunspots. Convincing evidence was not found until 1942, however, when Jaakko Tuominen positively established the existence of an equatorward migration of sunspots at heliocentric latitudes lower that about 20° and a poleward migration at higher latitudes.

The spectroscope was the instrument that marked the beginning of the modern era of stellar studies. As early as 1871 Hermann Vogel showed that the Sun's rotation rate can be detected from the relative Doppler shift of the spectral lines at opposite edges of the solar disk, one of which is approaching and the other receding. Extensive measurements were made visually by Nils Duner and Jakob Halm during the period 1887-1906. They showed a rotation rate and equatorial acceleration that were quite similar to those obtained from the apparent motion of sunspots. They concluded that Faye's empirical law

* Bernard le Bovier de Fontenelle, Conversations on the Plurality of Worlds, translation of the 1686 edition by H. A. Hargreaves, p. 70, Berkeley: University of California Press, 1990.

adequately represented the spectroscopic observations also, but their coverage oflatitude was double that of the sunspot measurements. The first spectrographic determinations of solar rotation were undertaken at the turn of the twentieth century by Walter S. Adams at Mount Wilson Solar Observatory, California.

William de Wiveleslie Abney was the first scientist to express the idea that the axial rotation of single stars could be determined from measurements of the widths of spectral lines. In 1877, he suggested that the effect of a star's rotation on its spectrum would be to broaden all of the lines and that"... other conditions being known, the mean velocity of rotation might be calculated."* In 1893, while doubts were still being expressed with regard to measurable rotational motions in single stars, J. R. Holt suggested that axial rotation might be detected from small distortions in the radial velocity curve of an eclipsing binary. Thus, he argued,

... in the case of variable stars, like Algol, where the diminution of light is supposed to be due to the interposition of a dark companion, it seems to me that there ought to be a spectroscopic difference between the light at the commencement of the minimum phase, and that of the end, inasmuch as different portions of the edge would be obscured. In fact, during the progress of the partial eclipse, there should be a shift in position of the lines; and although this shift is probably very small, it ought to be detected by a powerful instrument. t

Confirmation of this effect was obtained by Frank Schlesinger in 1909, who presented convincing evidence of axial rotation in the brightest star of the system S Librae. However, twenty more years were to elapse before Abney's original idea resulted in actual measurements of projected equatorial velocities in single stars. This notable achievement was due to the efforts of Otto Struve and his collaborators during the period 1929-1934 at Yerkes Observatory, Wisconsin.

A graphical method was originally developed by Grigori Shajn and Otto Struve. 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. Comparison with an observed line profile gave the projected equatorial velocity v sin i along the line of sight. These early measurements indicated that the values of v sin i fell into the range 0-250 km s-1 and may occasionally be as large as 400 km s-1 or even more. As early as 1930 it was found that the most obvious correlation between v sin i and other physical parameters is with spectral type, with rapid rotation being peculiar to the earliest spectral classes. This was originally recognized by Struve and later confirmed by statistical studies of line widths in early-type stars by Christian T. Elvey and Christine Westgate. The O-, B-, A-, and early F-type stars frequently have large rotational velocities, while in late F-type and later types rapid rotation occurs only in close spectroscopic binaries. A study of rotational line broadening in early-type close binaries was also made by Egbert Adriaan Kreiken. From his work it is apparent that the components of these binaries have their rotational velocities significantly diminished with respect to single, main-sequence stars of the same spectral type. The following year, 1936, Pol Swings

* Mon. Not. R. Astron. Soc., 37 (1877), p. 278. t Astronomy and Astro-Physics, 12 (1893), p. 646.

properly established that in close binaries of short periods axial rotation tends to be either perfectly or approximately synchronized with the orbital motion.

At this juncture the problem was quietly abandoned for almost fifteen years. Interest in the measurements of axial rotation in stars was revived in 1949 by Arne Slettebak. Extensive measurements of rotational velocities were made during the 1950s and 1960s by Helmut A. Abt, Robert P. Kraft, Slettebak, and others. However, because the only observational technique available was to determine line widths in stars from photographic spectra, these studies were limited almost entirely to stars more massive than the sun (M > 1.5M0) and to main-sequence or post-main-sequence stars. Since appreciable rotation disappears in the middle F-type stars, higher-resolution spectra are therefore required to measure rotational broadening in the late-type stars. In 1967, Kraft pushed the photographic technique to its limit to measure v sin i as low as 6 km s-1 in solar-type stars. Now, as early as 1933, John A. Carroll had suggested the application of Fourier analysis to spectral line profiles for rotational velocity determinations. In 1973, the problem was reconsidered by David F. Gray, who showed that high-resolution data make it possible to distinguish between the Fourier transform profile arising from rotation versus those arising from other broadening mechanisms. Since the late 1970s systematic studies of very slow rotators have been made by Gray, Myron A. Smith, David R. Soderblom, and others. Current techniques limit the measurement accuracy of projected rotational velocities to 2 km s-1 in most stars.

Periodic variations in the light output due to dark or bright areas on some rotating stars have also been used to determine the rotation periods of these stars. Although the principle of rotational modulation was suggested as early as 1667 by Ismael Boulliaud, convincing detection of this effect was not made until 1947, when Gerald E. Kron found evidence in the light curve of the eclipsing binary AR Lacertae for surface inhomo-geneities in its G5 component. The principle was therefore well established when in 1949 Horace W. Babcock proposed the so-called oblique-rotator model for the magnetic and spectrum variations of the periodic Ap stars. Kron's result was forgotten till 1966, when interest in the principle of rotational modulation was independently revived by Pavel Chugainov. A large body of literature has developed since the late 1960s. This work generally divides according to the method used to estimate the rotation periods, with the two types being (i) photometric monitoring of light variations produced by large starspot groups or bright surface areas and (ii) measurements of the periodic variation in strength of some emission lines that are enhanced in localized active regions in the chromosphere. These techniques have the advantage that a rotation period can be determined to much higher precision than v sin i and are free of the sin i projection factor inherent to the spectrographic method. Moreover, very accurate rotation periods can be derived even for quite slowly rotating stars at rates that would be impossible to see as a Doppler broadening of their spectral lines.

A different line of inquiry was initiated by the discovery of the so-called five-minute oscillations in the solar photosphere. The first evidence for ubiquitous oscillatory motions was obtained in the early 1960s by Robert B. Leighton, Robert W. Noyes, and George W. Simon. However, it is not until 1968 that Edward N. Frazier suggested that "... the well known 5 min oscillations are primarily standing resonant acoustic waves."*

Two years later, Roger K. Ulrich presented a detailed theoretical description of the phenomenon, showing that standing acoustic waves may be trapped in a layer beneath the solar photosphere. Thismodel was independently proposed in 1971 by John W. Leibacher and Robert F. Stein. In 1975, Franz-Ludwig Deubner obtained the first observational evidence for these trapped acoustic modes. Soon afterward, it was realized that a detailed analysis of the frequencies of these many oscillatory modes could provide a probe of the solar internal rotation. Indeed, because axial rotation breaks the Sun's spherical symmetry, it splits the degeneracy of the nonradial modes with respect to the azimuthal angular dependence. A technique for measuring the solar internal rotation from these frequency splittings was originally devised by Edward J. Rhodes, Jr., Deubner, and Ulrich in 1979. Since 1984, following the initial work of Thomas L. Duvall, John W. Harvey, and others,* diverse methods have been used to determine the Sun's internal angular velocity.

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