Examining Differential Rotation on Stars Using the Matrix

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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. In actuality, this division is artiﬁcial, as there are no regions of the sun that are truly quiet, i.e., contain no phenomena of activity. Further, one cannot study the active Sun independently of the quiet Sun, and vice versa (Wilson,

1994).

3Unless the dark hemisphere of the star faces exactly towards or away from the other, which in general it would not, then the starspot variability causes the two maxima to be unequal in height and causes the rising and falling eclipse branches to be asymmetric with respect to each other. This causes the brightness during totality, if any, to slant either up or down (Hall, 1994).

4 The earliest high-resolution photographs of the Sun exhibited a fine mottled structure known as photospheric granulation. This pattern is compromised of bright granules of differing polygonal shapes separated by narrow dark lanes, that appear to show an average center-to-center spacing of 1400 km, but with a range of ∼300 km to over 2000 km, corresponding to temperature variations of ∼200 K. These techniques also appeared to show the presence of patterns of outflowing horizontal fields, of speeds ∼0.5 km s−1, and di- mensions 20,000-40,000 km. These patterns, called supergranulation, are used as evidence to argue that if a convective zone extends some 200,000 km below the surface, several classes of cell sizes should be expected.

Thus it is inferred that this supergranulation is evidence of subsurface convection. The surface velocity field also appears to exhibit a strong oscillatory component, labelled the five-minute oscillations4 due to their characteristic period, was eventually interpreted as standing acoustic waves trapped in resonant cavities below the photosphere. Thus when the oscillatory component is filtered out of the velocity data, granulation is convective (Wilson, 1994).

The essential example of an active region is at least one but more generally several sunspots. In addition, regions brighter than the surrounding photosphere can be found near the limb, and these are known as photospheric faculae. In chromospheric observations, these white-light faculae are seen to be overlaid by bright areas called plages which are visible across the disk. Narrow-band observations have also revealed elongated dark filaments in the vicinity of the spots and plages. Well away from these, narrow-band chromospheric observations show prominences, which are intricately structured filamentary bands of relatively cooler gas that extend to hundreds of thousands of kilometers or more above the limb. Collectively, these phenomena are referred to as solar activity, of which the underlying cause is understood to be the solar magnetic field

(Wilson, 1994).

This magnetic field is thought to be driven by the interaction of the rotation and convection in the stellar interior, known as the “dynamo.” The most readily observable component of the solar magnetic field is that which is compressed in intrinsically strong flux concentrations, and has a field strength of 1-2 Kg

(Schrijver, 1996). The ﬂux concentration is compressed such that its magnetic ﬁeld is strongly enhanced until the magnetic pressure balances the external gas pressure from the surrounding photospheric plasma

4This was discovered by Leighton, Noyes and Simon in 1962 (as cited in Wilson (1994)

5 (Zwaan, 1991). These concentrations, referred to as ﬂux tubes, are embedded in turbulent convection ﬂows

that lash and displace them. Being subjected to a continual folding and twisting around themselves and

other tubes surrounding them cause the ﬂux tubes to break up into smaller fragments and merge with

other concentrations of magnetic ﬂux. These merges result in the apparently random positioning of ﬂux

concentrations on length scales below ∼2000 km. The larger collections of magnetic flux that build up this network, can be of sizes up to several thousand kilometers. These collections also appear to be part of this continuously breaking down and merging of smaller concentrations. Thus, most of the magnetic field is collected into regions of strong down flows, or “sinks.” Here upflows appear to occur only where there is no intrinsically strong field, and vice versa. The dispersal of flux is a combined result of diffusion, differential rotation and meridional flow.5 The diffusion is large enough to allow some of the flux to cancel with opposite polarities at the equator. Much of the flux however, appears to be transported towards the poles of the

Sun, which results in the formation of a pattern of poleward arches (Schrijver, 1996). Schrijver (1996) also states that the solar magnetic field surfaces in bipolar active regions with a large range of sizes, where the area of the regions is proportional to their flux content. The combination of flux emergence, dispersal and cancellation together determine the distribution of the magnetic field over the surface of the Sun.

The Sun rotates diﬀerentially, the equatorial rotation period being ∼25 days, while the higher latitudes

have rotation periods of ∼35 days. Superimposed on diﬀerential rotation are torsional oscillations that

denote latitude zones of slightly faster and slightly slower than average rotation rates as compared to the

local diﬀerential rate. The signiﬁcance of this is that bands progress from high latitudes to the equator, and

may bear an important relationship to the sunspot cycle (Wilson, 1994).

4 Sunspots

A typical sunspot observed in white light consists of a dark central region, the umbra which is surrounded by

a lighter region of what appear to be alternately bright and dark radial ﬁlaments, the penumbra. However

sunspots vary vastly in appearance. The diameter of a typical umbra is 10,000 km, though the diameters of

5A type of atmospheric circulation pattern in which the meridional (north and south) component of motion is unusually pronounced.

6 the largest spots may be greater than 20,000 km. Penumbral widths are on the order of 10,000-15,000 km.

Thermal proﬁles of sunspots with optical depth6 have shown that the umbra is about 2600 K cooler than

the photosphere, when compared at an equal optical depth, and when compared to the convection zone at

the same geometrical depth, i.e., at the same depth on the photosphere, the temperature diﬀerence is close

to 10,000 K. An explanation for this phenomena seems to be due to the blocking of convection by intense

vertical magnetic ﬁelds. This idea, put forth by Biermann in 1941, (as cited in Foukal (1990)), is based on

the horizontal motions of overturning convection which is inhibited by the magnetic volume force j x B in

the presence of a strong vertical magnetic ﬁeld. In the limit where the ﬁeld energy density B2/8π is much

larger than the kinetic energy density 1/2ρv2 of the convection, the normal overturning should be inhibited.

Properties of sunspot groups such as their east-west orientation, and Hale’s polarity rules, indicate that

they are formed due to the conﬁguration of a coherent magnetic structure from a source of well-ordered

toroidal (azimuthally oriented) magnetic ﬂux within the solar interior (Schussler, 2002). This toroidal ﬁeld

is simply caused by the simple dynamo process which invokes a nonuniform rotation of the Sun to shear a

weak poloidal ﬁeld (∼10 gauss). In regions of the convective zone the convection produces local Ω shaped

and U shaped bulges in the toroidal ﬁeld that when rotated into the meridional planes, provide a magnetic

circulation that makes up the weak poloidal ﬁeld (Parker, 1991). This is due to the magnetic ﬁeld appearing

on the surface in ﬁbril form, as a collection of magnetic ﬂux bundles, in an apparent updraft from the ‘dynamo

region.’ This ﬁeld moves up to the surface in the form of Ω loops, with a magnetic ﬁeld strength as much

as 3 × 1022 Maxwells emerging at an active site in a couple of days, which is repeated at the same location

at intervals of a week or so, providing an “activity nest.” This emergent ﬂux, which leads to the formation

of sunspots, have a tendency to bundle together, a curious feature as individual ﬂux bundles observed in

the photosphere tend to expand to ﬁll out the available volume; thus when when crowded together, the

expanded ﬁeld in the chromosphere is compressed in opposition to its pressure which corresponds to the

energy density of B2/8π. These newly-formed ﬂux bundles soon compress as a result of the downﬂow of gas

within the interior of the ﬂux bundle and/or as a result of radiative cooling at the surface (Parker, 1991).

These accumulated ﬂux bundles then become dark, or cool, and form a pore when its overall diameter reaches

6A given thickness of plasma may present diﬀerent opaqueness characteristics to incident radiation which depend on the properties of the plasma and the frequency of radiation. For an element of plasma of thickness dz and opacity κv(z), an element of optical thickness dτ maybe deﬁned as dτ = −κv(z)dz (Wilson, 1994).

7 about 1500 km (Parker, 1991). A spot is then born by the darkening and growth in diameter of such a pore,

which turns it into an umbra (Foukal, 1990). Parker (1991) mentions that an astonishing property of sunspot

formation is that this clustering tendency continues as long as there is still fresh magnetic ﬂux emerging at

the center of this region. If emergence ceases, clustering also ceases, thus resulting in the disintegration of

the sun spot or sunspots.

Evolution of longer-lived spot groups proceeds through a series of steps described by the Zurich Classiﬁ-

cation System. Class ‘A’ describes either a small single spot or a group of spots spanning a few degrees. This

spot or spots have no penumbrae or any kind of structure as seen in other classes described below. Class

‘B’ describes a bipolar group of spots lacking penumbrae. These spots are generally described structurally

as being concentrated at the opposite ends of a bipolar group that is roughly oriented east-west. Classes

‘C’ and ‘D’ describe groups of spots that are similar to ‘B’, but while Class ‘C’ has at least one spot with

penumbra, in D the largest spots show penumbrae. Classes ‘E’ and ‘F’ are of large bipolar groups of spots

that extend over 10◦ and 15◦ in longitude respectively with the larger spots having penumbrae and with many smaller spots in between them. These classes represent the largest maximum size and complexity of group; In Classes ‘G’, ‘H’ and ‘J’ the sunspot groups decrease in size and structure, sort of a reversal of the classes that led up to the Classes ‘E’ and ‘F’, such that ﬁnally a single roundish spot is that which is deﬁned. Only a small fraction of spots live long enough to complete a full evolutionary cycle through stages

‘A’ through ‘J’. A large fraction, (over half) of spots/spot groups pass through stages A-F within 10-14 days and subsequently spend a longer period of around 2 or more months decaying to stage ‘J’ (Foukal, 1990).

The Sun’s magnetic activity has mainly been recorded via daily observations of the number of sunspots on the disk, where a measure of sunspot activity can be found using Wolf’s sunspot index number, deﬁned

R = k(10g + f) (1) where f is the number of individual spots, g the number of recognizable spot groups, and k is a correction factor that is intended to adjust for diﬀerences between observers, telescopes and site conditions (Foukal,

1990). The most fundamental aspects of the Sun’s magnetic oscillation can be described as the remarkably

8 regular 11-year variation of sunspot numbers, which is accompanied by equally regular oscillations in the

latitude distribution of spots and the polarity of their ﬁelds, together with the 22-year polarity reversal at

high latitudes. Here, it was found that if the poloidal ﬁeld were to complete a full cycle and return to its

original conﬁguration after 22 years, the global ﬁeld periodicity and the sunspot magnetic periodicity would

be the same, though ∼90◦ out of phase, with the poloidal ﬁeld peaking at the sunspot minimum. Daily maps

of the sunspot positions on the disk have shown that the latitude band occupied by spots drifts towards

the equator as the solar cycle progresses from minimum through maximum, through the minimum of the

next cycle. This latitude drift, known as Sporer’s law is best illustrated by the plot known as the Maunder butterﬂy diagram, which shows the zone occupied by the spots, typically spanning 15-20 degrees of latitude, moves steadily over 11 years; the ﬁrst spots of a new cycle are typically centered around 25◦-30◦, while the last spots of the cycle, which appear 11 years later, are found around 20◦ closer to the equator (Foukal,

1990).

When studying long-term behavior of solar activity cycles, long-term irregularities in this behavior as observed, could have possible inﬂuences on the climate. Findings indicate that extended periods of depressed solar activity, such as the Maunder7 and Sporer8 Minima, correspond to periods of colder climate as judged by weather records kept in Europe and North America (Foukal, 1990).

5 Models of solar activity

Several models have been proposed and some subsequently supported by mathematical analyses to explain the diverse phenomena associated with the solar activity cycle. These models maybe classiﬁed as (i) relaxation models, (ii) dynamo wave models and (iii) forced oscillator models. Here both (i) and (ii) are particular examples of dynamo theory. Of the relaxation models the best known is that proposed by Horace Babcock

(1961), known as the ‘Babcock Model’ (Wilson, 1994) (see Figure 1). Babcock’s model proposes the 22-year magnetic cycle in ﬁve stages as described by Wilson (1994): an initially global poloidal magnetic ﬁeld (stage

1), shown by Figure 1(a), which is then wound by diﬀerential rotation into a spiral ﬁeld with a strong toroidal

7This is the period of a genuine reduction of sunspots during the period 1640-1705, which corresponded to the climatic excursion in Europe known as the ‘Little Ice Age’ (Wilson, 1994). 8A similar activity decrease in the activity of the Sun in the late 15th and early 16th Centuries.

9 Figure 1: The Babcock Model (Carroll, 1996).

component (stage 2), shown by Figure 1(b). As the winding is increased, Ω-loops or kinks form in the toroid and float to the surface, where they emerge as active regions (stage 3) shown by Figure 1(c), giving rise to sunspot groups and other manifestations of activity before decaying. An essential feature of this model is that it establishes a new global poloidal field of opposite polarity at or before the next solar minimum, prior to the winding process of the new cycle. Babcock postulated that as bipolar region activity decayed, a flux loop that had formed to connect a leader and follower spot regions expanded into the corona and the foot-

10 points9 moved apart. As, on average, the magnetic axis of a new active region from follower to leader is tilted towards the equator by ∼12◦, Babcock said that if this tilt were maintained as the foot-points moved apart, the follower flux should tend to move polewards, while the leader flux would tend to move equatorwards, where both fluxes would then cancel with existing polar fields, and a trans-equatorial poloidal loop would then form, connecting follower flux from one hemisphere with that of the other (stage 4) shown in Figure

1(d). The accumulation of such loops would result in the cancellation of the old and the development of the new poloidal field (stage 5). This model appeared to explain the reversed magnetic polarities of sunspot pairs in opposite hemispheres and consecutive cycles, the initial tilt of the magnetic axis towards the equator, the reversal of the polar fields and their close involvement with the cycle and to a lesser extent the butterfly diagram. This model showed that the magnetic tension, and thus the buoyancy of the toroidal field was the greatest at mid-latitudes, and thus the first active regions of a new cycle emerged there. However, this model does not explain why subsequent regions emerged along the equator-ward branch of the toroid but not along the pole-ward branch of the toroid. There also was no evidence of decaying leader flux moving towards, or far less, cancelling across the equator. Observations show both leader and follower flux drifting towards higher latitudes before decaying (Wilson, 1994).

6 Starspots

Compared to the current knowledge of sunspots, knowledge of starspots remains poor. Most of what is known about starpots is based on their brightness contrast; very little is known about their magnetic features, and questions are posed as to whether these are magnetic features similar to sunspots. Evidence supporting the magnetic nature of starspots is wholly indirect, and Solanki (2002) lists some of this evidence:

1) Magnetic suppression of convective energy transport is the most eﬃcient means of producing signiﬁcant, localized darkenings on the surface of a cool star; in particular, the evolution of the size, shape and number of starspots on the stellar surface is best understood in terms of magnetic features.

2) The detected starspots are much larger than even the largest sunspots. This larger size can be explained with a larger amount of magnetic ﬂux on these stars. However, due to the limited spatial resolution achievable

9these are the points of the ﬂux loop that connects the bipolar active region with preceding and following polarity

11 with the employed detection techniques (e.g., Doppler Imaging) it is generally not possible to resolve sunspot- sized features. Thus, what appears as a single starspot may or may not be composed of multiple sunspot-sized spots.

3) Another major difference between starspots and sunspots, namely the high latitudes of the starspots compared with the near-equatorial location of sunspots, has been explained in two different ways: a) The magnetic field generated at the base of the convection zone is susceptible to the enhanced influence of the

Coriolis force on rapidly rotating stars exhibiting high-latitude spots. b) Meridional circulation causes the magnetic flux to concentrate increasingly closer to the stellar pole as the total amount of flux increases. A combination of both effects may well be acting, but this requires further studies.

Two general properties of stars feature prominently in the study of stellar cycles: the rotation rate (or period), and the presence and depth of a convection zone. High-resolution spectroscopy, the main source of data on stellar rotation rates, yields estimates of the projected rotation speed about a rotation axis in the sky plane, v sin i, from the analysis of spectral line shapes. Rotation rates may also be determined from the following: short-period variations in luminosity, CaII emission, or any other indicator of activity arising from the non-uniform distribution of regions of activity on the star’s surface (Wilson, 1994).

Using stellar modelling techniques, the existence and locations of stellar convection zones maybe inferred.

Although not numerically accurate, results of these models are qualitatively important and show that very hot early-type stars of spectral types O and B have radiation as the dominant transport mechanism, while shallow convection zones that begin just below the surface occur in models of stars of spectral types F, and these zones increase with depth (with respect to radius) in classes through F, K, and M, where mid- to late

M-type stars are almost entirely convective (Wilson, 1994). In the years of monitoring stellar activity, the main features of solar activity have been detected in other stars when calibrated against observations of the sun as a star, most strikingly, the detection of cycles in chromospheric emission similar to the sun’s 11-year cycle. It has been found after more than 20 years of monitoring approximately 100 F, G and K stars, that roughly 40% of these stars exhibit cycles of lengths between 7-15 years. Relatively little has been deduced about the plage atmospheres in main sequence dwarfs similar to the sun, except that their chromospheric, transition and coronal emissions exhibit similar emissions to those seen in solar plages, but are generally

12 much brighter (Foukal, 1990).

Applications of the method where the Zeeman signature in the intensity proﬁle of a magnetically sensitive

line is compared to that of a magnetically insensitive line is used to observe stellar magnetic ﬁelds directly.

These have confirmed the presence of magnetic fields in a number of stars, where field strengths of ∼ 1-2 kG with area coverage by the magnetic fields of the visible surface of 20-80% are generally reported. The higher degree of activity exhibited on these stars, characterizes the differences between highly active stars and relatively quiet ones such as the Sun, and appears to involve the fractional area covered by magnetic fields on the stellar surface also known as the filling factor (Wilson, 1994).

The starspot model which usually invokes circular spots is the only hypothesis which can successfully account for the wide diversity of photometric variations in these stars. Detailed studies of the shape and size of both light and color variations have yielded eﬀective spot temperatures to be near 3400 ± 200 K, generally

1000-1200 K cooler than unspotted regions in young BY Dra stars. The degree of spottedness of stars is variable with time, and with hints that spot activity on some late-type dwarfs is cyclic. The clearest evidence of this is on the star BD +26◦730, which seems to show smooth, well-deﬁned variations with a periodicity that indicates a 60-year spot cycle, found through slow changes in the mean luminosity indicative of cycles in the spot formation rate (Wilson, 1994).

7 Imaging Starspots

Hall (1996) explains three approaches to starspot mapping: Doppler imaging, light curve inversion and eclipsing binaries. Doppler imaging uses the Doppler component of temperature or magnetically sensitive spectral line proﬁles in rapidly rotating stars to locate and track individual features. In a Doppler imaging line proﬁle, the location of a given feature in longitude is determined from its displacement from the line center. Here, latitude is determined, on a lesser scale, from the radial velocity amplitude of such features

(Byrne, 1990). A light curve is regarded as a poorer source of information than Doppler imaging since the information given by Doppler imaging consists of intensity versus radial velocity, whereas input from a light curve is simply intensity versus time. In addition, a light curve cannot reveal the presence of a polar spot, as

13 this produces no rotational modulation. Another drawback is that most modelling of light curves has been restricted to determining the size, latitude and longitude of two or three round spots. Eclipsing binaries, where a spotted star is eclipsed by another, can be used to provide an independent handle on the mapping problem. However, this technique alone will provide an incomplete map as only one hemisphere is scanned, and only a strip of restricted width in latitude (Hall, 1996).

Direct imaging of stellar surfaces is possible using interferometric techniques. As described by Mozurkewich

(1996), an interferometer works to coherently bring together two separated sections of the wavefront (see

Figure 2). The afocal telescopes intercept sections of the wavefront and direct them back toward the beam container. As light is not monochromatic, portions of the same wave front intercepted by the telescope must arrive at the beam container simultaneously, and this is accomplished by having a variable delay inserted into the path from at least one of the telescopes. It can be assumed that this delay is sinusoidal, and the amplitude and phase of the interference fringe may be measured to check the intensity distribution of the source.

Figure 2: A two-element interferometer (Mozurkewich, 1996).

14 Photometric techniques for indirectly imaging a stellar surface generally use a form of “spot modelling,” wherein assumptions are made regarding the number and shapes of the spots. Then, the sizes, locations and brightnesses of the spots relative to the stellar photosphere are varied in order to obtain the best possible ﬁt between the light curve implied by the spot distribution and the actual light curve. However, there is a more advanced technique of determining the features of a stellar surface based on the light curve that does not rely on any assumptions regarding the number, shapes and locations of the spots.10 The method is called

Matrix light-curve inversion (Harmon (2000) and references therein).

8 The MLI method

The Matrix light-curve inversion (MLI) method is an indirect way of inverting photometric light curves of a given star such that its surface features may be discerned (Harmon, 2000). Thus the basic premise of MLI is to deduce the appearance of the star’s surface from its light curve. An artifice that is used for this purpose are patches on the stars’ surface which permit scrutiny over the entire surface. These patches are formed by subdividing the stellar surface into a series of spherical rectangles that are bounded by circles of latitude and meridians of longitude. Each patch is assumed to radiate uniformly across its surface. The individual contribution of a patch to a star’s brightness is given by multiplying the solid angle by the specific intensity, where the solid angle is the area projected along the line of sight divided by r2, where r is the distance from the earth. It is impossible to obtain data points for every moment in time for the light-curve, thus it must be sampled at specific times, i.e., the light-curve will consist of a series of discrete data points. By repeatedly obtaining data points as the star rotates, the assumed patch brightness may then be summed up. This is known as the “forward calculation.” This gives the light curve that the model of the star will produce. As the objective of the MLI method is modelling the surface of the star rather than modelling light-curves, what may be done instead is finding the light-curve that corresponds to the patch intensities, to fit the light-curve obtained through observational data.

One may think that this can be accomplished by ﬁnding the surface which minimizes the residual, a measure of how much the model light curve diﬀers from the data. If we consider an n-dimensional space in

10It does, however, require knowledge of (or a reasonable guess as to) the brightnesses of the spots relative to the photosphere.

15 which each axis is the intensity of a patch, every point in the positive region of this space will correspond to a possible surface appearance, in turn a possible solution. A computer program can be then used to crawl around this space, and find the point which best gives the lowest possible residual value. Many different algorithms can be used to find this lowest residual value, one being the “conjugate gradient algorithm” which uses the gradient to find the ‘bottom.’ Real data have scatter in their corresponding light curves, which is due to noise. If the surface of a star is peppered over with tiny spots, the variations due to the little spots will cause its light curve to look like noise. If the computer were instructed to minimize the residual, it will try to mimic the noise, and put a lot of spots on the surface. Thus, it is not ideal to minimize residual. An alternative method must be used in order to keep the surface from becoming ‘salt and peppery.’ Instead of minimizing the residual, constrained minimization is employed. This process will assume the “smoothest” possible solution and thus will suppress noise artifacts. This can be done using the following objective function, E(J~; λ):

E(J~; λ) = G(J~) + λS(J~) (2) where G(J~) is the goodness-of-ﬁt, i.e., the residual between the data light curve and the model light curve,

S(J~) is the penalty function, i.e., a measure of how smooth the model surface is with a smoother surface

~ having a smaller value of S, and J is a vector whose components are the Jbij, which is a set of calculated speciﬁc intensities of the patches. Thus, instead of minimization of G, minimization of E, is what is required here. λ is known as the smoothing function. In order to minimize E, λ must be varied to get the “best” possible value. This is done by choosing a λ such that G is equal to the estimated noise level in the data.

Then the minimization of E has yielded the smoothest surface for which G is the estimated noise level. Thus, the goal is to choose the smoothest surface and not the minimum residual (Harmon, 2000).

From whole-disk white-light images of the Sun, sunspots appear as dark spots on an almost uniformly bright photosphere. There is no mix of bright and dark spots, and the same may be assumed for a star in question that is being analyzed for starspot modelling. In order to do this, it is advantageous to choose a penalty function, S that biases the solution to dark spots on a uniformly bright background. This may be

16 obtained by deﬁning S using the following equation:

PNs PMi 2 cij(hJˆi) S(Jˆ) = i=1 j=1 (3) PNs i=1 Mi

Where the cij are the weighting factors that allow the solution to be biased in favor of either bright or dark

spots. Ns is the value of i, the latitude band in which the patch lies, for the southernmost latitude band for

which the centers of at least some patches are visible. Mi is the number of patches in longitude for the ith

latitude band. hJˆi is the average value of all the Jˆij for this set of patches, where:

PNs PMi Jˆij hJˆi = i=1 j=1 (4) PNsMi i=1

It can be noted that S takes on its minimum value of zero for a featureless surface, i.e., a uniformly bright

surface. In order to make the penalty bigger for bright patches than for dark patches cij is deﬁned as:

  ˆ ˆ  1, Jij ≤ hJi, cij = (5)  ˆ ˆ  B, Jij > hJi

Here B is deﬁned as the bias parameter, and by choosing B > 1 the solution is biased in favor of dark spots.

This is due to the fact that when B > 1, a patch brighter than hJˆi by some certain amount increases S by a factor of B more than a patch dimmer than hJˆi by the same amount (Harmon, 2000).

Thus, as explained previously, the advantage of the MLI over other techniques used to determine stellar

surface features is that it makes no a priori assumptions in regard to the number of spots on a star’s surface,

or their location. One signiﬁcant validation of the MLI technique was through light curve inversions of

Pluto made by Drish et al(1995, as cited in Harmon (2000)) that were later found to compare favorably

to maps of Pluto obtained through direct imaging via the Hubble Space Telescope as produced by Stern,

Buie, & Trafton (1997, as cited in Harmon (2000)). However, in the case of Pluto, there were light curves

available for diﬀerent angles of the rotation axis for surface imaging by MLI, which is not possible for stellar

imaging as the aforementioned light curves are not available for the diﬀerent angles of the rotation axis. This

difficulty is overcome however, by the use of different photometric filters. Here, the photometric filters show

17 differences in photospheric limb darkening, where the limb higher photospheric layers are systematically seen as being less bright. These differences in photospheric limb darkening along with variations in the spot-to- photosphere intensity ratios result in variances in the amplitudes of the light curves taken through different

ﬁlters. These variations of amplitudes may be used to improve the quality of solutions and the ability to signiﬁcantly improve the reconstruction of spot latitudes accurately using multicolor photometry is possible.

In particular, simultaneous inversions of multiple light curves obtained using different filters allow for the increase in latitude resolution. Consider available light curves for a certain star in two wavelength bands, where there is no limb darkening in the first band and strong limb darkening in the second band such that regions near the limb appear almost invisible, due to them being so dark. A small black spot on the surface of this star will cause the largest effect on both light curves when it lies near the equator; Conversely, it will produce a minimal effect on both light curves if it lies near the pole. But, if this star is such that its rotation axis lies perpendicular to the line of sight, then as seen on the hemisphere visible from Earth, the spot when near the pole will always appear near the limb. Thus as seen in the second passband, it will appear essentially invisible due to the strong limb darkening. If a dip in the light curve is shown in the first passband, but a corresponding dip is not shown in the second passband, then it is possible to infer that the spot is at a higher latitude. Validation of this is shown in Harmon (2000), which also shows that latitude discrimination is further enhanced by four-color photometry.

9 Data

Due to inclement weather, we were unable to obtain data through our own observations in order to carry out analysis. Thus the data for the purpose of analysis for this research project was obtained from the

Automatic Telescope Site at Fairborn Observatory in Southern Arizona, where Tennesssee State University operates seven automatic photoelectric telescopes. Data was obtained using a 0.4-m Automatic Photometric

Telescope (APT) which is a 16-inch Cassegrain telescope (Henry, 2000). The 0.40m APT is a joint project of the Vanderbilt University Department of Physics and Astronomy and the Tennessee State University Center of Excellence in Information Systems, and is primarily dedicated to long-term photometric monitoring of

18 chromospherically active single and binary stars (Henry, 2000). Here the Young Solar Analogs catalog was used to gather stars for stellar surface imaging (see Gaidos (2000)). The stars analyzed in this research were HD 1835 (constant period), HD 20630, HD 206860, HD 30495, HD 97334 and HD 72905 (all variable period). These data were categorized according to B and V magnitudes (of the star) and Julian dates.11

10 Analysis

For analysis purposes, the text ﬁles containing the Julian dates and B and V magnitudes for each star were imported as Microsoft Excel ﬁles, and were separated according to the ‘season’ observed, obtained from

Gaidos (2000). These were then used to determined the phase angle and the relative intensities, and initial light-curves were constructed in order to check for a significant decrease and then increase in magnitude, which could signify the presence of a spot on the surface of the star during that given season. Seasons containing these significant variances were then converted back into text files in order to be read by MLI.

Using the effective temperatures T for the stars as given by Gaidos (2000), log g values, were looked up for these effective temperatures, and these were then used to locate the limb darkening coefficients for the respective B and V magnitudes. g is the acceleration due to gravity at the surface given in cgs units (Harmon,

2000). Here, limb darkening is modelled using the coeﬃcients computed from ATLAS model atmospheres

(Kurucz (1991), as cited in Harmon (2000)) by Van Hamme (1993); these coeﬃcients depend on both the temperature and acceleration due to gravity on the surface of the model, where log g for the models vary in steps of 0.5. Once values for T and log g were obtained, these were entered into an initial shell script with estimates for the root mean squared error (rms) for noise for the separate B and V light-curves, and initial inclination angles of 45◦. These initial parameters were then run by the MLI program in order to generate the surfaces of the stars, such that the rms levels and inclination angles could be determined. Once the rms values for the separate B and V light-curves were determined, these values were entered into another shell script with both B and V parameters included, and this was then run by the MLI program. Once surfaces were generated, various conﬁgurations of the star were examined in order to determine the inclination angle and the best possible rms values, which would be the lowest possible value without the spots “falling apart”,

11This is deﬁned to be the number of days since noon, January 1, 4713 BC

19 i.e., becoming stretched out rather than showing circular shapes.

11 Results

The main purpose of this project is to detect differential rotation of a star using MLI. This would be done by examining the starspot models constructed by MLI and using these models to deduce the location of a spot on the surface of a star. If the spot appears at different latitudes for different seasons, then differential rotation would be evident.

Preliminary inversions for HD 20630 do not appear to show diﬀerential rotation (see Figures 3 and 4).

The spots resolved on the surface of the star appear to be at the same latitude for the diﬀerent seasons.

Figure 3: At an inclination angle of 45◦ for season 1.

Figure 4: At an inclination angle of 45◦ for season 2.

The appearance of the spot at the same latitude as the inclination angle entered into the program is not

20 a coincidence. The program tends to place spots at the given inclination angles, thus evidence for differential rotation is not shown. This is due to the program trying to search for the smoothest possible solution to fit the generated light curve to the observed light curve, and in order to obtain the smoothest possible solution, it seeks to generate the smallest possible spot that will give a good fit to the light curve. This would occur when the spot passes directly through the line of sight. As a larger spot near the the pole would produce a similar light curve variation as a smaller spot at a latitude near the inclination angle, the solution process may tend to model a larger spot a different latitude with a smaller spot whose latitude is equal to the inclination. Thus a smaller spot at the wrong latitude will be favored by the minimization algorithm, as long as it fits the light curve well enough (Harmon, 2000). In addition, the spots on this star appear to be small in size, and the program is not as sensitive to resolving smaller spots as it would be for larger spots.

Simulations that have been carried out by Harmon (1999) show that for larger spots on the surface of a star, the diﬀerences in latitude may be resolved (see Figures 5 and 6).

Figure 5: Simulation of a spot of radius 10◦ as seen from the center of the star using B,V,R and I filters. The figure on the right shows the MLI reconstruction while the figure on the left shows the artificial star (Harmon, 1999).

21 Figure 6: Simulation of similar spot of radius 20◦ using B and V ﬁlters (Harmon, 1999)

Its is also important to note that in the simulations using the B, V, R and I filters, the spot is resolved better than when using only the B and V filters, and the latitude differences are shown more clearly. By modifying the program so as to better resolve latitude differences, differential rotation may be successfully resolved in the future. It is apparent however that the program does find a reasonably good fit to the light curve data observed, as can be seen in the figures below (see Figures 7 and 8).

One way to do this would be to modify the latitude-dependent weighting parameters wi for patches of different latitudes to account for the different latitudes of spots. This weighting factor wi for the ith latitude band is defined as being proportional to the differences between the maximum and minimum projected area times the limb darkening for the patches in that band. Harmon (2000) found that results for inversions using weighting when compared to results for inversions that used no such weighting showed only a modest improvement in the latitude resolution while significantly increasing the time required for the solutions to converge. Thus, latitude-dependent weighting was not employed in successive applications of MLI. For this research however, latitude-dependent weighting may play a vital role in determining differential rotation and

22 Figure 7: Light curve for spot at an inclination angle of 60◦ for season 1.

Figure 8: Light curve for spot at an inclination angle of 60◦ for season 2.

thus the author of the program would need to modify the program and take latitude weighting into account.

Unfortunately, the author of the program will not be able to make such modiﬁcations until the summer.

23 References

Byrne, P. B. 1990, in Surface Inhomogeneities on Late Type Stars, ed P. B. Byrne & D. J. Mullen (Berlin: Springer-Verlag), 3. Byrne, P. B. 1991, in NATO ASIC Proc. 375, Sunspots: Theory and observations, ed J. H Thomas & N. O. Weiss (Dordrecht: Kluwer Academic), 63. Carroll, B. W., & Ostlie, D. A. 1996, An Introduction to Modern Astrophysics (Reading, MA: Addison- Wesley). Foukal, P. 1990, Solar Astrophysics (New York: Wiley). Gaidos, E. J., Henry, G. W., & Henry, S. M. 2000 AJ, 120, 1006. Hall, D. S. 1994, IAPPP, 54, 1. Hall, D. S. 1996, in IAU Symp. 176, ed K. G. Strassmeier & J. L. Linsky (Dordrecht: Kluwer Academic), 217. Harmon, R. O. 1999, Ph.D. thesis, Univ. of Chicago. Harmon, R. O., & Crews, L. J. 2000, AJ, 120, 3274. Henry, G. 2000, T3 0.40m APT Speciﬁcations Tennessee State University, 2005. Accessed 2005.04.22 (http://schwab.tsuniv.edu/t3.html). Mozurkewich, D. 1996, in IAU Symp. 176, ed K. G. Strassmeier & J. L. Linsky (Dordrecht: Kluwer Aca- demic), 131. Parker, E. N. 1991, in NATO ASIC Proc. 375, Sunspots: Theory and observations, ed J. H Thomas & N. O. Weiss (Dordrecht: Kluwer Academic), 413. Schrijver, C. J. 1996, in IAU Symp. 176, ed K. G. Strassmeier & J. L. Linsky (Dordrecht: Kluwer Academic), 1. Schussler, M. 2002, AN, 323, 377. Solanki, S. K. 2002, AN, 323, 165. Thomas, J. H., & Weiss, N. O. 1991, in NATO ASIC Proc. 375, Sunspots: Theory and observations, ed J. H Thomas & N. O. Weiss (Dordrecht: Kluwer Academic), 3. Van Hamme, W. 1993, AJ, 106, 2096. Wilson, P. R. 1994, Solar and Stellar Activity Cycles (Cambridge: Cambridge Univ. Press). Zwaan, C. 1991, in NATO ASIC Proc. 375, Sunspots: Theory and observations, ed J. H Thomas & N. O. Weiss (Dordrecht: Kluwer Academic), 75.