Examining Differential Rotation on Stars using the Matrix Light Curve Inversion Method Charini D. Perera May 4, 2005 1 Introduction Starspots are defined as being local regions on a stellar surface that are cooler and hence appear darker than the surrounding photosphere and occur on those stars whose outer envelope is convective (Hall, 1994). Starspot imaging is important for various reasons. Even though astrophysics is a science where direct physical access is not available for the objects it studies, this is compensated for by the Universe producing a large number of different environments where certain types of phenomena take place. Thus it is possible to understand different processes that take place in the Universe by studying a variety of objects. So, if one were trying to understand the phenomena of sunspots, one could look to observing starspots, which has been one motivation for the field of activity known as the solar-stellar connection. The Sun allows for a two-dimensional study of its activity. However, as it appears fixed at a point of time, it exhibits only one set of stellar parameters such as mass, size, composition and state of evolution. Stars however, though being viewed as essentially one-dimensional objects, offer a wide range of physical parameters that allow various theories to be tested more thoroughly than with the Sun alone. Thus the solar-stellar connection serves to coalesce these two lines of study to allow for further understanding of the Sun and other late-type stars (Wilson, 1994). The focus of this paper is understanding a currently used method of observing starspots using the matrix light-curve inversion (MLI) method. The aim of my research is to map starspots on a star using the MLI technique to examine for evidence of differential rotation and to report these results. 2 History Even though Chinese astronomers recorded naked-eye observations of sunspots even before the birth of Christ, sunspots were first observed telescopically in 1611 by four astronomers, Johann Goldsmid in Holland (1587-1650), Galileo Galilei in Italy (1564-1642), Christopher Scheiner in Germany (1575-1650) and Thomas Harriot in England (1560-1621), though priority of publication belongs to Goldsmid, known by his latinized name Fabricius (Wilson, 1994). Fabricius made these observations of sunspots and used them to infer that the Sun must rotate. Galilio also inferred the Sun’s rotation about a fixed axis with a rotation period of 1 about a month, and noticed that spots within a single group moved relative to one another. Scheiner, who pursued sunspot observations for nearly two decades before finally publishing his work Rosa Ursina sive Sol in 1630, noted that spots occurred within zones of low latitude at either side of the equator but not near the poles, an observation that Galilio also made. He also observed that spots at higher latitudes rotated more slowly and that the axis of rotation was tilted with respect to the ecliptic, and additionally, made detailed drawings of clearly distinguishable umbrae and penumbrae, defined as a dark central regions surrounded by a lighter regions respectively (Wilson, 1994). The German, Henry Schwabe (1789-1875) recorded the occurrence of sunspots for forty-three years, and his table, which clearly showed the 11-year periodicity of annually averaged sunspot numbers, was included in Humboldt’s famous treatise, Kosmos in 1851. Richard Carrington, an English astronomer, deduced through observations that the Sun rotated differentially, where a point at the equator rotates more rapidly than one at higher or lower latitudes. He defined an arbitrary point on latitude 16◦ as longitude zero, and a rotation completed by this point became known as a Carrington Rotation (CR)1 (Wilson, 1994). Carrington also discovered the solar flare, a catastrophic and localized release of energy in the form of electromagnetic radiation which leads to particle acceleration. Another important investigator of sunspot nature and probably the most well known, George Ellery Hale (1868-1938), who concentrated on solar physics, made revolutionary discoveries that not only changed our understanding of sunspot nature, but also astrophysics as a whole. His first result showed that sunspots are cooler (∼4000 K) than the surrounding plasma (∼5800 K). He was also the first person to obtain spectroheliograms in Hα.2 From these he found evidence of hydrogen vortices in sunspots, and reasoned that a whirling mass of free electrons in a sunspot vortex should set up a magnetic field, that if sufficiently intense should split lines in the spectrum of spot vapors into two or more components, known as the Zeeman effect. This led to the important discovery of the presence of magnetic fields on the order of several kilogauss, and the explanation of the presence of these kilogauss fields led him to discover the fascinating sequence of patterns that are usually associated with the sunspot cycle. He noticed that sunspots tended to occur 1The ‘Carrington longitude’ of any point on the Sun is the longitude of the intersection of the meridian through that point with the parallel of latitude at 16◦, relative to the reference point (Wilson, 1994). 2This is the strong Hydrogen line which originates in the chromosphere, the more tenuous part of the atmosphere where the temperature is increasing outwards (Wilson, 1994). 2 in groups, generally containing a few larger spots and a number of smaller spots. By convention the Sun is said to rotate counterclockwise, and it was found that the largest spot of a group tended to be on the western side of the group, and the next largest on the eastern boundary. These were known as the ’leader’ and ’follower’ spots. In addition, the following were discovered to be associated with the sunspot cycle: (i) The leader spots in each hemisphere are generally of one polarity, with the follower spots being of the opposite polarity. (ii) If the leader and follower spots are regarded as magnetic bipoles, then the orientation of these bipoles are opposite in the opposite hemispheres. (iii) The magnetic axes of these leader-follower bipoles are usually inclined slightly towards the equator, the leader spots being closer. (iv) Towards the end of a cycle, while spots with the normal polarity occur close to the equator, spot groups of the new cycle appear at higher latitudes with reversed polarity. (v) Following the minimum between cycles, the polarities of the bipoles have reversed sign. From his observations, Hale was able to associate the 11-year sunspot cycle as being part of a 22-year magnetic cycle. With the help of colleagues, Hale was able to find that in addition to the magnetic fields of sunspots, the Sun also possesses a global magnetic field. Horace Babcock discovered that the global poloidal field, where the north and south hemispheres of the sun exhibit opposite polarities, is oscillatory with a half-period that is equal to the sunspot number period, i.e., the polarities in the hemispheres reverse every 11 years, such that the poloidal field would complete a full cycle after 22 years, and return to its original configuration (Wilson, 1994). The first model of a starspot was suggested by Ismael Boulliau in 1667, who observed the star o Ceti, the first variable star observed. Boulliau mentioned in his observation that this star had one hemisphere darker than the other and varied in brightness as the star rotated about its axis. After this however, many astronomers believed that starspots were the physical mechanism that caused variability between stars, and this belief continued for more than two centuries after Boulliau’s original observation. The belief that the mechanism causing variability between stars was then discounted as being due to starspots and instead being due to pulsation; thus there were no variable stars for a starspot model to describe. However, as the 3 variability due to pulsation was discounted, variability due to starspots was observed, but not recognized. Kron (1952) identified that starspot variability was an additional variability superimposed on the variability normally witnessed in eclipsing binaries, and this unusual behavior was termed “light curve distortion.” Here one of the two stars will have its hemisphere darker than the other, due to starspots being more on one side than the other. Rotation of these starspots, which are almost synchronous with orbital rotation, results in a nearly sinusoidal light variation which appears superimposed on other sources of variability.3 However, starspots were not recognized until 1972. Hoffmeister (1965) discovered starspot like features on four T-Tauri type stars, where he accounted for the quasi-periodic light changes as being due to rotation caused by the non-uniform distribution of light over the surface of the star. Though not clearly stating the word starspot, he did cautiously draw analogy to similar features on the solar surface. Chugainov (1966), after observations of the variable star BY Draconis proposed the cause of observed light variations being due to the existence of a spot on the surface of the rotating star. In 1971, in the first supplement to the third edition of the General Catalog of Variable Stars, a new type of variable star was formally introduced, with the prototype being BY Draconis, where in stars of this type “the light variability is caused probably by the axial rotation of a star with surface brightness anisotropy,” the latter being simplified as ‘starspot’ in the fourth edition of the GCVS, in 1985 (Hall, 1994). 3 Solar Activity Historically, investigations of the sun have been compartmentalized into two sections, studies of the quiet Sun and studies of the active Sun. The term active Sun pertains to regions of the Sun where the phenomena of activity such as sunspots are found, and The quiet Sun is described as regions of no activity.