Star Spots and Other Phenomena of Active Regions

As is observed on our own Sun (Fig. 3.22 ), stars can have spots. Stellar surface imaging by microlensing [cf. Sasselov (1998a, b)] shows directly that spots are present on other stars as well. A star spot is a region with higher or lower temperature than the surrounding photosphere, and thus it modifies the local flux. By way of physical analogy to the Sun, we should expect magnetic spots to result from convection in the outer envelope and differential rotation. Accompanying phenomena include small "pores,"35 umbral and penumbral dark regions, and spot groups. Additional phenomena are bright flocculi (Latin for tufts of wool) or plages (French for the white sand on a beach) - most easily seen in the lines of Ca II H&K and Ha - and faculae ("little torches"), sometimes called "white-light plages." Finally,

34 The term apsidal motion refers to the rotation of Lines of apsides of an eccentric binary orbit, or the rotation of the periastron. Apsidal motion is caused by perturbations to the 1/r gravitational potential and indicates deviations from Keplerian elliptic motion.

35 Pores are small sunspots. This is a term in solar research. Pores are usually at the resolution limit on visual images of the solar disk. We would need an enormous number of these to make a difference to hemispherical flux values, of course.

Fig. 3.22 Sun spots. This white light image of the Sun was taken in Calgary on September 3, 1988, in the early afternoon (20:05 UT) by Fred M. Babott. It shows spot groups, spot umbrae, penumbrae, pores, and toward the limb, faculae. Courtesy F. M. Babott we mention solar prominences, best seen in Ha, but also seen in "white light," which are bright off the limb but may appear as dark filaments when projected onto the photospheric disk.

In the light curve context, many researchers have included star spots into light curve models. Poe & Eaton (1985) give most of the historical and technical background for the analysis of spotted stars. A more recent review is by Linnell (1993). In this book, we characterize spots, as in the Wilson-Devinney model, by four parameters: latitude 0, longitude 0, angular radius p, and temperature factor tf. Some authors use the colatitude 0c := 90° - 0 instead of 0. The spots subtend circular solid angles at the star centers and, in their ideal form (infinitely fine grid of the surface), are essentially circular areas on the surface, except for the effect of the star's asphericity, and the ellipticity of features, such as round spots, produced by foreshortening, most noticeable at the limbs. 0 is the longitude of the spot-center. The reference direction is the line of centers, and 0 increases as in a right-handed system (set up separately for each component). The spot-center colatitude 0c is zero at the North (+z) pole and increases to 180° at the other pole. As seen from the North pole, the binary orbit is described counterclockwise. The spot angular radius p is half the angle subtended by the spot at the center of the star. The spot temperature factor tf is the ratio between the local surface temperature and the local undisturbed temperature. Due to reflection and gravity effects, the surface temperature across a spot may not be constant. Temperature factors less than and greater than unity correspond to cool spots and hot spots, respectively. As noted above, such a characterization of spot regions can at best be only a rough approximation of the true physical situation. Below, we give a mathematical representation of that simplified approach, as it exists in the Wilson-Devinney model.

Let the angles 0c and y represent a point on the surface of the star. In addition 0s and ys refer to the coordinates of a particular surface spot with radius ps. From the cosine law of spherical trigonometry, the angular distance of the point (0, y) to the center of the spot s follows as cos = cos 0c cos 0CS + sin 0c sin 0CS cos(y - ys), (3.4.3)

where 0c and 0sc denote the colatitudes, and cos(y - ys) can be computed as cos(y - ys) = cos y cos ys + sin y sin ys. (3.4.4)

From (3.4.3) it is easy to check whether the point (0, y) "lies" within the spot. If so, the spot-free local temperature T in (0, y) is modified by the temperature factor tf. This is summarized in the formula

'l l\ 1, if As > Ps, where TlSC denotes the local temperature after the spot correction has been applied.

A slightly more realistic model is that of Hill & Rucinski (1993), which allows for elliptical spot regions, with the major axis as an optionally adjustable parameter. However, elliptical spots may cause problems because the least-squares problem may easily become overparametrized.

A further complication arises from the dynamics of a binary system: Except in the synchronously rotating, circular orbit case, the physical surface of a star moves w.r.t. (model) grid elements, so that the spot longitude can be a function of time. Magnetic spots, as observed on our Sun, take part in the motion of the surface. In contrast, an accretion hot spot on an asynchronously rotating star (in circular orbit) could remain fixed with the grid. In an eccentric orbit the grid rotates at a nonuniform rate so a time-dependent spot longitude transformation36

n needs to be applied, where F is the rotation parameter (3.1.74), 0 is the orbital phase, and u is the true anomaly; subscript 0 refers to conjunction. Note that 0 and u need not be in the ranges 0 and 1 and 0 to 2n, respectively, so that the effects of longitude drift for spots can be followed over many orbital cycles.

36 Some kinds of spots (such as accretion hot spots) do not rotate with a star. For such cases, the transformation is not applied; therefore, = const.

At least some of the difficulties occurring in the analysis of EB light curves which show unequally high consecutive light curve maxima37 might be overcome by including star spots in the model [Yamasaki (1982), Milone et al. (1987), Hill et al. (1989,1990)], although there are probably several physical effects that produce unequal maxima (Davidge & Milone 1984).

As other light curve parameters, spots and their parameters tf) can be estimated by least-squares methods. Budding & Zeilik (1987) and Zeilik et al. (1988) use adjustable spot parameters to represent the light curves of short-period systems with RS CVn-like phenomena. Historically, a Wilson-Devinney spot parameter (tf) was first optimized by Milone et al. (1987), in the light curve analysis of the contact system RW Comae Berenices; by repeated trials with slightly readjusted values, a parabola of Swr2 versus tf was constructed and solved for the minimum. This procedure was extended to all spot parameters by Milone et al. (1991). Spots were automatically adjusted within the Wilson-Devinney program for the first time by Kang & Wilson (1989). All of this work was done with the Wilson-Devinney program of the early 1980s.

Spot parameters differ greatly from other light curve parameters, such as the mass ratio or inclination, in that they can vary significantly on a relatively short timescale (say days).38 A steady change in spot longitude with time, permits, in principle, the determination of spot migration periods. However, the determination of spot parameters requires abundant, accurate, synoptic data and careful analysis. Complete light curves must be obtained before the spot or spot groups achieve perceptible motion in longitude. Spot fitting ideally should be subject to determinacy tests (Banks & Budding 1990).

Must spots be used to model light curve perturbations? Even though most light curve analysts would argue that their presence is likely, and in the interest of achieving a more physically realistic picture of the binary system, they should be modeled, there are some doubts. As we have noted, the O'Connell effect may have more than one origin, and without further substantial evidence, such as molecular absorption features characteristic of M-stars in stars of otherwise higher temperatures [cf. Vogt (1979), or Ramsey & Nations (1980)], or Doppler imaging from line-profile analysis, the assumption of a spot cause usually is not justified. Milone et al. (1987) demonstrated that an analysis following rectification of the light curve produced no significant differences in parameters from those modeled with dark spots placed on either component, except for the parameters T2 and i, where the differences were

37 Sometimes called the O'Connell effect (Davidge & Milone 1984), named after D. J. K. O'Connell (1951), who demonstrated that unequal maxima in light curves was not a "periastron effect" because it is found predominantly in systems with circular orbits. Wesselink suggested the new usage, which has become widely accepted. The effect has also been called the "Kwee effect." Unfortunately, it has become a practice to use these names as catch-all terms for a variety of physical effects. As defined, it is purely phenomenological.

38 Active prominences vary on a scale of hours and flares even over minutes. Modeling such effects would involve some stochasticity and is therefore not covered here.

nevertheless small. Some effects spots have on light curves could also be produced by circumstellar matter clouds (see Sect. 3.4.4.5).

Nevertheless, in some cases stellar surface imaging by microlensing shows spots. In many other cases the following indications in light and color curves support the existence of spots:

• phases of minima agree in, e.g., V and in V-I but not in U-B;

• color amplitudes increase when going to V ^ R ^ I as expected for cool spots; and

As spots have relatively short life times and change in size quickly, one would like to fit curve-dependent spot parameters, and be able to make allowance for differential stellar rotation and latitude migration of spot groups. A disadvantage is the increased number of parameters leading to uniqueness problems.

3.4.3 Atmospheric Eclipses

The term "atmospheric eclipse" refers to an eclipse of a star by one with an extended atmosphere. The classical example is that of Z Aurigae, in which the width of the eclipse is greater in the ultraviolet than in longer wavelengths [see Fig. 1 in Wilson (1960, p. 441), based in turn on Roach & Wood (1952)]. We extend this idea to include eclipses of underlying radiation by overlying material of a different nature than chromospheric and coronal layers in their average, "quiet" state. Readers interested in extended atmospheres themselves are referred to Wehrse (1987) or Wolf (1987).

Atmospheric eclipses occur in binary systems in which at least one component has an extended atmosphere. The EB V444 Cygni (WN5+O6) is such a system, with partial eclipses at both primary (at A = 424.4 nm the depth is 0.225 in light units normalized to 1) and secondary minimum (depth 0.141) and with insignificant reflection and ellipticity effects. Here the WN5 component has an extended atmosphere and is in front at primary minimum, which is an atmospheric eclipse. For simplicity the stars are considered as spheres and the orbit as circular. Light curve modeling of early-type stars with essentially spherical geometry has been carried out by a number of investigators, and Roche geometry modeling of eclipses by translucent plasma clouds in the atmospheres of early-type systems was developed by Kallrath and Milone at the University of Calgary (Milone 1993), and is now in the 2007 WD version. Such clouds might be produced by stellar wind interactions.

The mathematical formalism (see Sect. 6.4) usually includes a term such as

which represents the amount of radiation absorbed by the extended atmosphere of the WN5 component. Here I0 denotes the brightness at the center of the disk of the normal O6 star, and t (f) is the optical depth along the line-of-sight intersecting the

WR component at a distance f from its center when the disk of the WR component is viewed by transillumination. Transillumination means, in addition to the light reflecting and originating from the surface of the WR disk, we see light from the O6 star passing through the extended atmosphere.

In the past light curves of early-type systems such as DQ Cep and V444 Cyg were analyzed by treating atmospheric eclipses as perturbations to photospheric eclipses but now it is possible, and perhaps more appropriate, to use a consistent model including all physical effects in a binary system simultaneously.

3.4.4 Circumstellar Matter in Binaries

Circumstellar matter in binaries may occur in gas streams, rings and disks, clouds, and boundary regions of colliding stellar winds. Distinction among these categories depends on one's focus. Gas streams may establish rings and disks, stellar winds might be considered as special kinds of gas streams, and boundary layers of colliding stellar winds might be considered as clouds with a special geometry. What the categories have in common is that they absorb, reemit, scatter, and thereby redirect the star light.

The existence of circumstellar material was detected first by Wyse (1934) who found Balmer emission in several Algol-type binaries. That this material is often manifested as an accretion disk has been known since the pioneering observations of RW Tauri by Joy (1942) in the early 1940s. Spectra showed double-peaked emission line features that could be explained if they arose from a ring of material circling the hotter and more massive star. Struve painstakingly observed Algols for many years [see Struve (1944), for example], and a large number of high-quality spectra demonstrated that these features were characteristic of Algols, not anomalies in a handful of systems. About the same time, Kuiper (1941), in a paper on p Lyrae, pointed out the importance of the inner Lagrangian point L1 in understanding gas flows in interacting binaries.

Spectroscopic and photometric observations showed another peculiarity of Algols. The hotter, more massive primaries were clearly main sequence stars, but the less massive secondaries had radii much too large for the main sequence (i.e., they were evolved subgiants or giants). The so-called Algol paradox begs the question: How does a binary evolve to the point where the low mass star is evolved, but the high mass star is still on the main sequence?

Roche geometry provides the key that solves the puzzle. See any of the sources Crawford (1955), Hoyle (1955, pp. 197-200), or Pustylnik (2005) for a wonderful explanation of the modern resolution of the Algol paradox. Algol-type systems are stable because the less massive star is filling its Roche lobe. As matter is transferred, the physical size of the lobe increases because of the increasing separation of the stars, which slightly detaches the star from the lobe. The star is expanding on a nuclear timescale, so mass transfer events tend to be sporadic and of small scale.

Hoyle (1955) and Crawford (1955) proposed that Algols were systems that had experienced large-scale mass transfer and that the less massive secondary had originally been the more massive star. It evolved first, overflowed its Roche lobe, and transferred enough mass to the other star to reverse the mass ratio. Morton (1960) showed that a binary system with the more massive star filling its Roche lobe is unstable to mass transfer. When the more massive star loses mass through the L1 point, its Roche lobe shrinks because its mass becomes smaller and the separation of the stars decreases. Although Morton failed to take the latter fact into account, the decreasing separation only increases the rate of mass transfer and creates a classic instability. The mass transfer is very rapid in the early stages and becomes continually slower toward the end of the process. A further important point is that the equilibrium radius of the mass-losing star can change as a result of mass loss. Whether it increases or decreases depends on the star's evolutionary state. Highly evolved stars (that still retain their envelopes) tend to increase their radii upon losing mass. Detailed reviews of the evolution of Algols were published by Plavec (1968) and Paczynski (1971). Batten (1973b, 1976) discusses and reviews observations of the flow of matter within binary systems. Gas streams can become evident in spec-troscopic data and may influence radial velocities, or produce peculiar light curve disturbances. The situation in the detached system VV Orionis is discussed in some detail in Sect. 3.4.4.2. In addition to Batten's reviews, the whole proceedings of the IAU Symposium 73, edited by Batten (1973a), is a useful source and covers many aspects of the topic, e.g., Plavec's (1973) review on the evolutionary aspects of circumstellar matter in binary systems. Finally, we mention the proceedings of the IAU symposium, edited by Appenzeller & Jordan (1987), which cover many aspects of circumstellar matter around single stars as well.

Spectroscopic observations made with the International Ultraviolet Explorer led Plavec (1980) to identify a group of systems (the W Serpentis stars) that appeared to be in the rapid phase of mass transfer. In some of these, such as j3 Lyrae [Wilson (1974); Wilson & Terrell (1992)], the gainer is completely engulfed by the circumstellar material. Wilson (1981) worked out structural models of these thick disks and proposed that the gainer has been spun up by the accreting material to the centrifugal limit, thus preventing the material from settling onto the star. However, viscous and tidal interactions will eventually decrease the angular momentum of the stars, allowing the disk material to be accreted. This leaves the system in a state where both stars fill their limiting lobes. For the less massive secondary, the limiting lobe is the classical Roche lobe. For the rapidly rotating primary, the limiting lobe is bounded by the equipotential that has the equatorial material rotating at the centrifugal limit. Systems in such a configuration, the so-called double-contact binaries, were predicted by Wilson (1979) and analysis of observations shows that these systems do indeed exist (Wilson et al. 1985). Over time, tidal forces will slow the rotation of the primary to synchronism, and it then will become an Algol-type system.

The later stage of mass transfer in an Algol-type system is small scale and sporadic and is understood39 as follows. A key point is that the stars move farther apart

39 A more complete discussion of this problem would also involve the equilibrium radius of the mass-losing star; cf. Plavec (1968).

to conserve angular momentum when flow is from the less to the more massive star. The subgiant star expands on a nuclear timescale (i.e., slowly). Each small burst of transferred matter produces relatively large increases in the critical lobe size as the binary conserves orbital angular momentum, i.e., the separation of the stars increases, and the star slightly detaches from the lobe. Thus the transfer process tends to turn itself off, and proceeds in small episodes as the star undergoes its evolutionary expansion. This is the slow phase of mass transfer. Early in the mass transfer process the situation is quite different. The mass-losing star then is the more massive one, and as mass is transferred the stars come together to conserve angular momentum, and the lobe size decreases. Thus, mass transfer tends to run away and is limited only by the thermal timescale of the envelope (rapid phase of mass transfer). Mezzetti et al. (1980) provide statistics of 55 Algol-type EBs related to mass transfer and mass loss.

3.4.4.1 Gas Streams

Whereas light curve models imply static or quasi-static physics, circumstellar gas streams, if modeled correctly, require a fully dynamical treatment based on the equations of radiation hydrodynamics. A full treatment may be beyond present-day computing power, but the rapid increase in computing power over the last few decades has made it possible to make the models more and more realistic and to model the gas flow in Algol-type systems, as well as the radiation that arises from the gas.

When insufficient computer power was available, gas streams, and also rings and disks, were often modeled with multiple particle trajectories; see, e.g., Gould's (1959) particle path model (for details see Sect. 3.4.4.2), Kruszewski's (1967) analysis on exchange of matter inclose binary systems and ring formation, or Smak's (1978) analysis of the escape of particles from disks in close binary systems. For a review of problems of gaseous motions within binary stars we refer to Huang (1973).

Prendergast (1960) showed that the mean free path of gas particles was much smaller than the separation of the two stars, indicating that a hydrodynamical treatment of the problem was necessary. Due to a lack of computing power at the time, Prendergast's solutions were limited by simplifying assumptions, such as ignoring the pressure gradient terms in Euler's equation and assuming hydrostatic support perpendicular to the orbital plane. An improved treatment was given by Prendergast & Taam (1974), who simulated solutions of the Boltzmann equation rather than solve a set of difference equations. Unfortunately, their scheme had an inherent artificial viscosity which was coupled to the grid resolution, but the results of their application to a system similar to U Cephei were intriguing, indicating, among other things, that mass transfer was nonconservative (i.e., some mass was ejected from the binary).

Another important contribution to understanding mass transfer in Algols was by Lubow & Shu (1975). Using matched asymptotic expansions, they developed a semi-analytical model of the gas flow. Exploiting the existence of a parameter labeled £

wd m with a being the isothermal sound speed w the orbital rotation rate, and d the binary separation, they reduced the parameter space of the equations to that of one parameter: the binary mass ratio. They also treated gas flow near the Lagrangian point, L1.

Lin & Pringle (1976) outlined a two-dimensional, many-body approach to the problem where the gainer is of negligible size compared to the stars' separation, such as in cataclysmic variables and X-ray binaries. Their's was apparently the first fully Lagrangian method for treating gas flows in close binaries. The scheme included viscosity, but did not treat pressure gradients. Based on their simulations, Lin and Pringle concluded that disks can be well defined and comparable in size to the Roche lobe.

Whitehurst (1988b) extended the Lin and Pringle model to include pressure gradients as outlined by Larson (1978). Larson's approach, assuming that the particles are extended, deformable gas clouds, is somewhat simplistic, although not too confining because the disk is dominated by angular momentum transport. Whitehurst's major simplifying assumption was that the energy dissipated in particle interactions was instantaneously radiated away through the disk surface. This makes the pressure calculations somewhat crude, but it was obviously the next step to take in the development of models of mass transfer. Whitehurst (1988a) applied his model to the SU UMa star Z Chamaeleontis and achieved impressive agreement with observed light curves, containing superhumps.40

With the advent of 8-m class telescopes and more sensitive detectors, the next few years promise great progress in the study of circumstellar material in Algols. Newly developed models and observational techniques (especially polarimetry) should greatly improve our understanding of mass transfer in binaries, and therefore binary star evolution.

3.4.4.2 Gas Stream in the VV Orionis System

Based on a particle path model, Duerbeck (1975) has studied gas streams in VV Orionis. He discusses the consequences of circumstellar matter for the light curve, equivalent widths of the hydrogen lines, and the H^ -index. His analysis gives us an illustrative example of how special features can be added to an otherwise standard EB analysis.

The disturbances or irregular features in the light curve during phases 0.6 and 0.7 are interpreted as light loss caused by scattering through particles of a stream. The basic parameter describing this gas stream is its particle density. The geometry

40 Superhumps are periodic increases in brightness of 20-30% that occur during superoutbursts (outbursts lasting 10-14 days as opposed to normal outbursts lasting 2-3 days). The superhump period is somewhat longer than the orbital period of the system and is explained by the precession of the (elliptical) accretion disk.

270*

270*

Fig. 3.23 Trajectories in a binary system. This figure, Fig. 9 in Duerbeck (1975), illustrates the first hypothesis. There is one gaseous stream present; the trajectories are computed according to Gould (1959). Courtesy H. W. Duerbeck

Fig. 3.23 Trajectories in a binary system. This figure, Fig. 9 in Duerbeck (1975), illustrates the first hypothesis. There is one gaseous stream present; the trajectories are computed according to Gould (1959). Courtesy H. W. Duerbeck and its flow pattern are taken from Gould's (1959) particle path model. The model computes trajectories within the framework of the restricted three-body problem and uses an ejection velocity of 400km/s. Figure 3.23 shows typical trajectories or orbits. There exist escape orbits, connecting trajectories, and loops originating and ending on the same component. As shown in Fig. 3.24, neglecting the pressure gradient, the trajectories form a gas stream which can absorb light.

Under reasonable assumptions it is shown that the gas stream is almost completely ionized, which allows concentration on electron scattering. In that case, neglecting multiple light scattering, the transmitted intensity IT after passing the gas stream is related to the incident light intensity I0 by

where x is the path length and ne is the electron number density in cm-3.

Duerbeck's approach was to derive ne directly from comparing the observed light curve with the light intensity I0 derived from the Russell-Merrill model, and independently from the equivalent widths of lines. Two hypotheses were checked against observations by applying (3.4.9) to the "unperturbed" light computation:

1. One gas stream ejected is from that part of the secondary's surface, which is heated by the hotter component. From the observed light curve, i.e., transmitted light IT, Duerbeck estimated that about 10% of the light of the secondary is lost by scattering. This yields x = 10-12 cm and ne = 1.5 • 1011 cm-3 and agrees well with the values 1011 < ne • cm3 < 2.6 • 1011 derived from the equivalent widths of hydrogen lines.

Fig. 3.24 Gas streams in VV Orionis. This figure, Fig. 10 in Duerbeck (1975), shows the two gas streams involved in the second hypothesis. Courtesy H. W. Duerbeck

2. Two gas streams (indicated as I and II in Fig. 3.24) exist, i.e., in addition to the hypothesis above, another gas stream flow from the primary to the secondary should be present. This explains both the disturbance at phase 0.35 and the asymmetry of the primary minimum.

In both cases, the discussion involves only the geometry of the gas stream projected onto the orbital plane. Although the model produces plausible results and qualitatively explains features in the observed light curve, it might be worthwhile to reanalyze the data with a modern light curve program based on Roche geometry including some special features for streams or clouds.

3.4.4.3 Disks and Rings

Algols and other binaries with slow or intermediate mass transfer develop thin disks. To consider the disks and rings in the model requires the computation of the gas flow, radiative properties (in particular, emission line strengths and profiles), and the calculation of the spectral energy distribution by spatial integration taking into account Doppler shifts and eclipses. In some cases, disks can be approximated as non-self-gravitating and governed by celestial mechanics in the first approximation, but with nonnegligible viscous and pressure interactions.

RoZyczka & Schwarzenberg-Czerny (1987) present and solve two-dimensional hydrodynamical models for the stream-disk interaction in cataclysmic binaries, focusing on the collision region. As an example of a fully three-dimensional stream-accretion diskcomputation we refer to Dgani et al. (1989) who computed the time-dependent interaction between the stream from the inner Lagrangian point and the accretion disk, and the response of the disk to an increase in the mass transfer rate.

Terrell (1994) applied the method of Smoothed Particle Hydrodynamics (SPH) to the problem and also made a detailed computation of the spectral energy distribution of the radiation emitted by the gas. Pressure gradients were calculated in the usual manner for SPH [see Monaghan (1992) and references therein], but the viscosity was computed with a newly developed algorithm [Terrell & Wilson (1993); Terrell (1994)]. Viscosity was modeled by allowing particles within a specified distance of one another to exchange momentum, with close encounters being stronger than more distant ones. This scheme avoids problems inherent in earlier models such as artificial acceptance/rejection of interactions based on a Cartesian grid, as in the Lin and Pringle scheme. Terrell also computed Ha line profiles by coupling the radiative transfer code of Drake & Ulrich (1980) to his hydrodynamics code and found

1.0 -0.8 -0.6 -0.4 -0.2 ->- 0.0 --0.2 --0.4 --0.6 -0.8 -1.0 -

i—i—i—i—i—i—i—i—T -0.8-0.6-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Telescopes Mastery

Telescopes Mastery

Through this ebook, you are going to learn what you will need to know all about the telescopes that can provide a fun and rewarding hobby for you and your family!

Get My Free Ebook


Post a comment