The Astronomical Journal, 135:1450–1458, 2008 April doi:10.1088/0004-6256/135/4/1450c© 2008. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

MASS MOTIONS IN THE PHOTOSPHERE OF BETELGEUSE

David F. GrayDepartment of Physics & Astronomy, University of Western Ontario, London, Ontario N6A 3K7, Canada; dfgray@uwo.ca

Received 2007 November 27; accepted 2008 February 2; published 2008 March 10

ABSTRACT

Spectroscopic observations of Betelgeuse, taken at the Elginfield Observatory, show velocity and temperaturevariations delineating the systematic, but generally chaotic, rise and fall of photospheric material. The characteristictimescale of the variations is 400 days, while velocities cover a range of ∼9 km s−1. Macroturbulence is constantto about 13% and does not result from motions on the largest scales, but from motions within the large structures.The line bisectors have predominantly a reversed-C shape. Variations in the shapes of bisectors occur on the1 km s−1 level and are not obviously connected to their shifts in wavelength. A likely explanation of the observationsis granulation and giant convection cells accompanied by short-lived oscillations they trigger. Random convectionevents may account for radial-velocity jitter seen in many highly evolved stars.

Key words: convection – stars: individual (Betelgeuse) – stars: oscillations – stars: variables: other – supergiants –techniques: spectroscopic

Online-only material: color figures

1. BACKGROUND

Betelgeuse is a red supergiant star (α Ori, HR 2061, HD39801, M2 Iab, B − V = 1.86). The star varies in brightness,surface features, radial velocity, and line depths, but showsonly small variation in the shapes of the spectral lines. Adetailed description of the star’s characteristics and photosphericline-strength variations was given in an earlier publication(Gray 2000; G1 hereafter). It was shown there that the star’sbrightness is often in phase with the changes in line depths:deeper lines, brighter star. Bright spots that frequently appearin interferometric imaging of the star are sometimes takenas evidence of large granulation cells (e.g., Wilson et al.1997; Young et al. 2000; Freytag et al. 2002; but see G1for an alternative interpretation). Naturally one presumes longtimescales and large convection cells based on dimensionalarguments applied to a star like Betelgeuse, with its radius∼800 solar radii and the low surface gravity of a supergiantstar (Schwarzschild 1975; Boesgaard 1979). I will use the terms“granulation,” “supergranulation,” and “giant convection cell”in analogy with the solar usage. Solar granules have dimensions∼1 Mm, supergranules ∼10 Mm, and giant convection cellsspan the dimension of the convection zone or ∼200 Mm. Thedimensions of each of these will be scaled up in supergiantssuch as Betelgeuse, but the hierarchy remains the same. In the1975 paper, Schwarzschild considered mainly the scaling ofsolar granulation and supergranulation to giant and supergiantstars. Some hydrodynamical models of Betelgeuse showed 3–5 cells on the visible disk (Freytag et al. 2002). However,previous spectroscopic evidence indicated ∼600 cells on thevisible surface of Betelgeuse (Gray 2001; G2 hereafter). Thiswas deduced from the ∼4% variation seen in the widths ofthe spectral lines (but see updated material below). Hundredsof granules are also predicted by the scaling relation givenby Freytag et al. (2002). While such cells would certainlybe large, ∼40 Mm across or about 40 times the dimensionsof solar granules, they fall short of being the predicted largegranulation cells by an order of magnitude. Past investigationswere therefore inconclusive on the existence or the behavior oflarge granulation cells.

Many, perhaps a third of, highly evolved stars show periodiclight variations (e.g., Percy et al. 2003), while others are irregu-lar or semi-regular (e.g., Lebzelter et al. 2005), and Betelgeusehas been placed in the latter category. Semi-regulars seem toshow two dominant periods, one of a few hundred days anda second of a few thousand days (Wood et al. 2004). Sincethe documentation of the unstable radial velocity of Betelgeuseby Plummer (1908), periods have been sought in radial veloc-ity measurements as well (e.g., Sanford 1933; Goldberg 1984;Dupree et al. 1987; Smith et al. 1989; Uitenbroek et al. 1998a).There is consternation in the literature as investigators strug-gle to understand pulsations with a variable period, variableamplitudes and phases, and non-unique connections among ob-served variables. The “long” period for Betelgeuse is ∼2100 ±400 days, while a “short” period of ∼400 days is frequentlyquoted, although values half as large and twice as large havealso been found or suspected. Radial pulsation, orbital motion(Karovska et al. 1986), shock waves, rotational modulation, andconvection cells have been suggested as explanations.

More generally, a number of radial-velocity investigationshave documented excess scatter or random variability in themeasurements of stars at the tip of the red-giant branch, aphenomenon termed “jitter” (Gunn & Griffin 1979; Pryor et al.1988: Carney et al. 2003; among several others). The Betelgeusemeasurements presented below document wavelength shifts andrelated characteristics, and their similarity to velocity jitter isstriking.

An interesting step was taken by Bedding (2003) and Kisset al. (2006) who found evidence supporting the idea thatthe semi-regular light variations of red supergiants arise fromoscillations excited by convection cells. In their analyses andfor some stars, the power peaks are resolved into closely spacedfrequencies analogous to what is seen in the solar 5 minoscillations. This implies that the oscillations are powered by theconvection and not by the ionization/opacity process that drivesoscillations in the instability strip of the HR diagram. With thisinterpretation, the chaotic nature of the variations stems fromthe relatively few convection cells that excite the oscillations,i.e., we see only one or two modes at any one time. Theobservations presented below amplify and confront this idea and

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Figure 1. This typical exposure of Betelgeuse illustrates the shallow broad lines and identifies the specific lines used in the analysis. F/Fc denotes flux normalized tonominal continuum flux.

(A color version of this figure is available in the online journal)

give some information about the physical characteristics of theprocesses.

This paper also expands on G1 and G2, giving additionalinformation on the shapes and broadening of line profiles,their bisectors, and their wavelength positions. Line shifts orradial-velocity-type motions are found to be several times largerthan profile shape variations. The observed behavior requireslarge macroturbulence within the material showing the velocityexcursions.

2. OBSERVATIONS AND MEASUREMENTS

The coude spectrograph at the Elginfield Observatory wasused to acquire the 338 exposures reported on here. Twodetector systems were used. From 1996 March 23 to the endof 1999, the observations were recorded on a Reticon self-scanned array mounted in a Schmidt camera of focal length559 mm (see Gray 1986). From 1999 October 6 onward (three-month overlap with the Reticon), a CCD detector was used on asimple camera having a focal length of 2080 mm. The resolvingpower is ∼100,000. The dispersion is ∼0.039 Å per pixel on theReticon and ∼0.013 Å per pixel on the CCD detector. Signal-to-noise ratios in the continuum are estimated from the photoncount and range from 154 to 885 with a median of ∼500.Figure 1 shows a sample CCD spectrum. The three lines, V i6251.83 Å, Fe i 6252.57 Å, and Ti i 6261.11 Å identified inthe figure, are those used for most of the work described here.The excitation potentials of the first two lines are 0.29 eV and2.40 eV respectively, making the ratio of their line depths a goodindicator of temperature (Gray & Brown 2001). The Ti line isused to study line bisectors and line-width variations. All otherlines are either blended or too shallow. As can be discerned inFigure 1, the spectral lines are shallow and highly broadened,indicating substantial macroturbulence with a dispersion of∼11 km s−1 if an isotropic Gaussian velocity distribution isassumed (G1). Rotational broadening probably contributes 2.0–2.5 km s−1 to the broadening (Uitenbroek et al. 1998b; G1).

An important part of these observations from 2002September onward is their absolute wavelength scale, estab-lished using the telluric absorption inside the spectrograph. Tel-luric measurements are taken both before and after the stellar ex-posure. Details are described in Gray & Brown (2006). Barycen-tric corrections are made using the precepts of Stumpff (1979,1980). The final absolute scale given here is generally good to

50 m s−1 or better, which is more precise than one can mea-sure the positions of broad spectral lines such as those ofBetelgeuse.

3. TEMPERATURE AND BRIGHTNESS VARIATIONS

The ratio of the line depths of V i 6251.83 to Fe i 6252.57(Figure 1) is a measure of the temperature, with larger ratiosindicating cooler temperatures (Gray & Brown 2001). Althoughno calibration exists in this range to convert the line-depth ratiosto temperature, one can make an order-of-magnitude estimate. Ifthe lines were weak, the ratio of their strengths would vary withthe ratio of their Boltzmann excitation factors. For these lines,χ = 0.29 and 2.40 eV respectively, and for the observed rangein line-depth ratios, a temperature change ∼100 K would beimplied. Since the lines are not weak and the light is integratedover the stellar disk, this is a lower limit to the real temperaturevariations.

Figure 2 shows the line-depth ratio as a function of time.Measurement errors are ∼0.005, smaller than the symbol size,as estimated from repeated exposures taken minutes apart. Ascan be seen in the figure, variations occur on many timescales.In some time intervals, there is a hint of oscillation, but there isno one period that stands out. A period of 388 days (±30) wasdeduced by Kiss et al. (2006) from photometric observationsobtained from the American Association of Variable Star Ob-servers (AAVSO). Some of the AAVSO observations (Henden2007, private communication) are compared to the line depthsin Figure 2. The magnitudes were binned in 10 day intervalsand these averages, aside from the occasional erratic point, typ-ically have errors of ∼0.1 mag or less. Also shown are themore precise measurement of Krisciunas & Luedeke (1996),where the typical error is a few thousandths of a magnitude. Theagreement between these two independent photometric studiesis very good, with differences in overall trends, rates of change,and amplitudes of variation rarely exceeding 0.1 mag. Earliermeasurements by Krisciunas (1990, 1992, 1994) also agree withAAVSO data to this level.

In a rough sense, the photometric variation is seen to mimicthe line-depth ratio variation. There are times when the photome-try tracks the line-depth ratio rather well, as in the 11000–11300and 12650–12750 intervals in Figure 2. But more often there aresignificant differences, for example near 10500, 10750, 10850,11600, 12000, and 12550. Similar conclusions have been drawn

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Figure 2. The line-depth ratio of V i 6251.83 to Fe i 6252.57, a temperature index, is shown as a function of time (◦). Superimposed are the AAVSO visual magnitudes,binned in 10 day intervals (+). The photometric observations of Krisciunas & Luedeke (1996) are also shown (×).

(A color version of this figure is available in the online journal)

in the past (e.g., Sanford 1933). Apparently the temperaturevariations account for the bulk of the brightness variations, butthe effective photospheric radius is frequently an independentvariable, i.e., the phenomenon is not simply pulsation on a globalscale.

A Fourier analysis or periodogram of the line-depth ratiois shown in Figure 3, panel (a). The window patterns haverelatively small sidelobes so each large peak is a true frequency.For frequencies higher than ∼0.02 cycles day−1, there is onlynoise. At lower frequencies, there is signal. In particular, thehighest signal corresponds to a period of 427 days, but thereare also several other frequencies present. The second panel(b) in Figure 3 shows the periodogram of the AAVSO dataover the same time interval used in panel (a). Here againseveral frequencies show up, with one corresponding to 418days, essentially the same as for the line-depth ratio. Boththese periods are somewhat longer than the 388 days foundby Kiss et al. (2006), but as they emphasize, a range of periodis expected if oscillations are driven by convection with itsstochastic characteristics.

To probe a step further, panel (c) in Figure 3 shows peri-odograms of AAVSO data for eight independent time spans,each having a duration approximately the same as for panels(a) and (b). There are clumps of peaks near 0.0025 (400 days),the variations being emphasized in this paper, 0.0005 (2000days), and possibly 0.0001 (10,000 days). The last set is longenough to correspond to rotational modulation. One of theseeight spans (JD2433997 to 2438048) shows no peak at all in the0.0025 region. In fact, the middle half of the data, JD2429801to JD2446199, shows no peak in this region. Another interest-ing result from this figure is the Gaussian envelope shown by

the peak heights. A Gaussian envelope in the frequency domainimplies convolution with a Gaussian, i.e., Gaussian smooth-ing, in the time domain. The e−1 width in Figure 3 is 0.0062cycles day−1. The corresponding e−1 width of the smoothing is(0.0062 × π)−1 = 51 days. Perhaps this gives an indication ofthe thermal relaxation time of the material.

4. LINE PROFILE SHAPE AND POSITION VARIATIONS

The line profiles appear to be highly variable, as shown inFigure 4, left-hand panels (a) and (c), where the data are plottedon an absolute wavelength scale and with no renormalizationof line depths. However, the variations are primarily in thepositions of the lines and their depths, with only small changesin shape and broadening, in agreement with G2. For example,if the profiles are repositioned in wavelength and re-scaled indepth (but not width), the variations appear much smaller, asshown in right-hand panels (b) and (d) of Figure 4. Rescalingin depth amounts to changing the continuous opacity, and thisthesis was presented in G1. A variation on this hypothesis is thata light-scattering shell surrounds the photosphere and its opacityvaries with time, but given the tight correspondence shown inG1 between the line depth (which might be affected by a shell)and line-depth ratio (which is highly unlikely to be affected bya shell), this hypothesis can be ruled out for the timescales weare considering.

The small variation in the widths of the lines is illustratedin Figure 5, where the full half width of the λ6261 line isshown as a function of time. Measurement errors are estimatedto be ±0.3 km s−1 or approximately the size of the symbols in

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Figure 3. (a) Periodogram of the line-depth ratio (LDR) shows several peaks at these low frequencies. The observations cover a time window of 4059 days.(b) Periodogram of AAVSO magnitudes over the same time interval used in (a). (c) Periodograms of AAVSO magnitudes for eight time windows having a similarlength to the one used in (a). The peaks lie under a Gaussian envelope.

(A color version of this figure is available in the online journal)

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Figure 4. The left-hand panels, (a) and (c), show spectral lines on an absolute wavelength scale with only continuum normalization. The right-hand panels, (b) and(d), show the same profiles shifted in wavelength and scaled to the same central depth. In panel (b), the V i λ6251.83 line was used for the depth normalization; theFe i λ6252.57 line differs in depth owing to temperature differences. Any of the recorded spectral lines could have been used for this illustration.

(A color version of this figure is available in the online journal)

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Figure 5. The full half width of Ti i λ6261.11 is shown as a function of time. The outlying panels have expanded time dimensions so more detail can be seen. Thevertical scale of the outlying panels is the same as for the main panel (d) but are offset in some cases to center the data in the panel.

(A color version of this figure is available in the online journal)

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Figure 6. Mean core velocity of V i 6251.83, Fe i 6252.57, and Ti i λ6261.11 (plus an unknown constant) is shown as a function of time.

(A color version of this figure is available in the online journal)

the plots. The variations are, therefore, many times larger thanthe measurement errors. The half width varies from about 27 to30

12 km s−1, or some 13% over the course of these observations.

Apparently the large macroturbulence of this star is constant tothat level. A similar stable and isotropic macroturbulence wasfound in the chromospheric study by Lobel & Dupree (2001),but see Carpenter & Robinson (1997) who found evidence fornon-isotropic velocity fields. Freytag et al. (2002) also foundlarge velocities in their hydrodynamical models, “. . . oftenexceeding 20 km s−1. . . .” As can be seen in the various panels ofFigure 5, changes occur on many timescales. A periodogramshows no outstanding peaks, but for frequencies below about0.005 cycles day−1, or periods longer than about 200 days,there is a signal.

The wavelength positions of the line cores of the three lines,λ6251.83, λ6252.57, and λ6261.11, were estimated by eye.Within the uncertainty of measurement and variable blending,

estimated to be 100 m s−1, all three lines show the samepattern of shifts. Their mean is shown as a function of time inFigure 6. The wavelength shifts were converted to velocitiesusing nominal rest wavelengths, but since the center-of-massvelocity of the star is not known to the precision being discussed,true zero velocity remains unknown. Notice the shorter timebase compared to previous figures; no absolute wavelength scalewas available for earlier observations. Naturally, this severelyrestricts the study of velocity variations on a many-year scale.One should be particularly careful not to interpret the twominima near 12700 and 13800 as defining a cycle since anytwo randomly placed minima separated by more than half thetime base will produce such a suggestive plot. Further, noticethe points near 14200 that are as low as those at 12700.

The velocities show the characteristic chaotic behavior withvariations on many timescales. The observed range in velocityis ∼9 km s−1 with an root-mean-square scatter of 2.4 km s−1.

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Figure 7. Bisectors of the Ti i λ6261.11 line are shown for individual observing seasons. Both the shape and the shifts have meaning, but the position of zero velocityis unknown. The numbers indicate the time order of the exposures, so the time progression of positions and shape changes can be seen.

(A color version of this figure is available in the online journal)

This is consistent with observations in earlier epochs (e.g.,Sanford 1933; Boesgaard 1979; Goldberg 1984). A periodogramanalysis again shows signal at low frequencies. The largest peakcorresponds to 365 days, which one is immediately tempted toattribute to the seasonal sampling in the observations. However,characteristic timescales of variation in the range of 300–400days are seen within the 2002–2003, 2005–2006, and the 2006–2007 seasons, and these have nothing to do with the yearlysampling times. It should also be noted that periods of this sizeare typical for semi-regular variables.

There is no simple correlation between the line width and thewavelength shift.

5. BISECTOR SHAPES AND VARIATIONS

As seen in Figure 1, the λ6261 line is one of the very few inthese exposures that is sufficiently deep and possibly unblendedenough to yield a meaningful line bisector. Bisectors for thisline were computed for all exposures in the last five observingseasons where absolute wavelength information is available.They are shown in Figure 7 on an absolute velocity scale withan unknown but constant offset owing to motion of the starin space. Therefore both the shapes of the bisectors and theirrelative shifts have significance. The bisector numbers indicatethe time order.

Bisector errors can be assessed empirically. The noise onthe individual bisector points can be seen by looking closely atindividual bisectors in Figure 7 and noting the wiggles they showbetween the ∼1% F/Fc ordinate steps. This noise is ∼0.1 km s−1

or less except at the very top and bottom, where larger values canoccur (see Gray 1983, 1988). Further, by comparing bisectorsfrom different exposures taken relatively close together in timeone can see the degree of consistency. For example, bisectors

2 & 3 and 10 & 11 in the 2002–2003 season are two cases takenone day apart. Both pairs show the same basic shapes and differby ∼0.1 km s−1 in position, and any variation of the star isincluded in these differences.

Inspection of the figure shows that (1) bisectors taken onsuccessive nights are almost identical, (2) shape variations occuron the 1 km s−1 level, including change in curvature, reversal ofcurvature, and slope, (3) there is no consistent relation betweenthe shape and shift, (4) shifts are much larger than shapevariations, being on the 5 km s−1 scale, and (5) the predominantshape is like a reversed C, mimicking what is seen for starshotter than the granulation boundary (Gray & Toner 1986; Gray& Nagel 1989; Gray 2005; near G0 for the more luminousstars). Given these variations in the bisector shape, classicalradial-velocity measurements, where one number is assignedto the star’s line shifts, clearly loses meaning on scales below∼1 km s−1.

As seen in Figure 7, a C shape, or at least a positive slope,occasionally occurs, but both the C and reversed-C shapes canand do occur at similar velocity displacements. The shapechanges are much larger than those would be induced byclassical radial pulsation of a few km s−1, i.e., the observedshifts. For example, the asymmetry for a radial pulsation thatshifts the profile 3 km s−1 amounts to less than 0.1 km s−1,whereas the observed shape changes are ten times this. Normalvariable-star pulsation is therefore inadequate as an explanationof the bisector variations, whereas large convection cells orsimilar mass motions are compatible with the size of the shifts,their chaotic nature, and the bisector contortions.

The occurrence of reversed-C bisectors on the cool side ofthe granulation boundary is surprising and of some significance.One might be tempted to dismiss the reversed-C shape as arising

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Figure 8. Line-depth ratio is shown as a function of the mean core velocity of V i 6251.83, Fe i 6252.57, and Ti i λ6261.11 for an individual observing season. Arrowsindicate the direction of increasing time. Panel (f) illustrates the full inferred generic behavior, including the directions of the rising temperature and rising velocity.The symbol size indicates the full half width of Ti i λ6261.11.

(A color version of this figure is available in the online journal)

from blends in the line. There are at least two reasons to think thatthe reversed-C shape is not spurious. First, there is no connectionbetween shape and the temperature variations discussed above.In other words, any blending line would have to have the sametemperature dependence as the main λ6261 line; not impossible,but unlikely. Although the Fe i 6219.29 Å line is badly blended,the lower portion shows the reversed-C shape, mimicking λ6261and supporting the reality of the λ6261 shape. Second, a recentstudy of metal-poor red-giant branch stars (Gray et al. 2008,in preparation) reveals that most of them cooler than 4100 Kshow reversed-C bisectors (Betelgeuse is ∼3600 K). For thesered-giant branch stars, because of their low metallicity, thereis little blending and essentially all the available lines showthe same result. Betelgeuse is apparently consistent with otherstars in its region of the HR diagram. Further, since Betelgeuseis a Population I star, the existence of the reversed-C shapeapparently does not require the low metallicity of the starsstudied in D. F. Gray et al. (2008, in preparation).

6. EXCURSIONS OF THE PHOTOSPHERE

The timescales of the larger variations are typically manymonths so one can follow only a portion of the full variationduring any one observing season. Figure 8 gives a summary ofthe temperature index (line-depth ratio) plotted as a function ofcore velocity (as per Figure 6). The general pattern is an increasein temperature followed by a rise of the material followed bya cooling and finally a descending phase. Each season hascaptured only a portion of this behavior, but from five seasons,the general pattern seems clear, as constructed in panel (f).

Apparently large portions of the surface are seen rising andfalling, and this material dominates the star’s spectrum.

The size of the symbols in Figure 8 indicates the relativehalf width of the λ6261 line, which I take to be a proxy for thestrength of the macroturbulence. Although there is no consistentchange in half width during any one cycle, more vigorousexcursions, such as the one in the 2005–2006 season, are hotter,rise faster, and have stronger macroturbulence. The excursionin the 2003–2004 season illustrates the opposite case, where thematerial reaches less hot temperatures, rises more slowly, hassmaller macroturbulence, and eventually peters out.

The difference between the largest fall velocity (2003–2004season) and the largest rise velocity (2005–2006 season) is 7–8 km s−1. Since the absolute zero on the radial velocity scaleis unknown, this cannot be split between rise and fall portions.Further, the angle of rise to the line of sight is not known, so thisvelocity difference is a lower limit. The variation in the line-depth ratio during these episodes ranges from ∼1.05 to 1.30,which is ∼100 K or more, as indicated in Section 3. Any lightfrom other portions of the stellar disk will dilute the true velocityand temperature variations of the moving material. Projectionand dilution factors can be expected to vary from one excursionto the next and cannot be separated from stochastic variationsof the excursion itself.

An estimate of the size of the displacements can be madeby approximating a typical velocity excursion in Figure 6 by asinusoid and integrating over half a cycle. With a semi-amplitudeof ∼2.5 km s−1, and using a characteristic time of 400 days, thematerial would move ∼40 R� or ∼5% of the radius of the star.

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7. ADDITIONAL DISCUSSION AND SUMMARY

The term “photosphere” has been used here to mean thelayers of the atmosphere from which the light we record arises.This is no different from what is commonly meant, but inthe case of Betelgeuse, the geometry of the photosphere isundoubtedly irregular, permeated with structure, and dynamic.Some of the largest of these structures will move the photosphereseveral percent of the stellar radius, as noted in the previoussection. Since the line profiles are dominated by velocity shiftswith much smaller alterations in shape and width, apparentlyone large, or at least bright, feature dominates the spectrumat most times. On the one hand, we might speculate thatthe variations are caused by enormous convection cells withthe excursions discussed in Section 6 being the rising andfalling of the cells. On the other hand, we might imaginethe excursions are the surface trying to pulsate, but beingstochastically disrupted by other pulsation modes or largeconvection cells. It may be that both processes are acting,perhaps interacting. Do the observations allow us to identify thedominant process?

The behavior in Figure 8, especially when coupled with thebisector variations of Figure 7, is highly suggestive of largeconvection cells seen rising through the photosphere, indeed,becoming the photosphere, cooling and falling back down. Atthe same time, it is easy to imagine the material “ringing” tothe eruption of such a large cell. If the ∼400 day timescaleof the excursions is comparable to the “echo” time from thebottom of the convection zone, radial oscillations are likely.Kiss et al. (2006) found mode lifetimes for Betelgeuse to be1140 days, or about three cycles of ringing. By the time threecycles have passed, another major convection cell erupts andchaotic behavior is created in the parameters we observe. Thetimescales for convection-cell episodes and pulsation of thefundamental mode may both lie near 400 days, assuring powerpeaks in the 400 day region of a periodogram, but the stochasticdriving of the convection produces a wandering period, variableamplitude, and variable phase, i.e., the characteristics of theobservations.

It should also be noted that the phase relation between velocityand temperature (Figure 8) is very different from those ofinstability-strip pulsating stars, where the temperature varies inphase, or nearly so, with velocity (see, for example, Wesselink1946; Walhraven et al. 1958). Nor do the Betelgeuse velocityvariations resemble the much larger ones of the cool-supergiantMira stars, which show a monotonic rise over the full periodfollowed by a discontinuous drop seen as line doubling (e.g.,Hinkle et al. 1982; Querci 1986; Jorissen & Udry 1998; Alverezet al. 2001). Variations of the semi-regular variable WZ Caswere studied by Lebzelter et al. (2005). Their results showvelocity to be in anti-phase with brightness, although with pooramplitude tracking. Even though the translation of line-depthratio into magnitude (Figure 2) is imperfect, Betelgeuse doesnot show an anti-phase relation during the time span of myvelocity measurements. Apparently the mechanisms proposedto explain pulsation of other evolved stars are not immediatelyapplicable to Betelgeuse. Recall, however, the basic variationin the depths of photospheric absorption lines (see G1) on thesame ∼400 day timescale that is most obviously explained bychanges in the continuous opacity. It may be that the kappapulsation mechanism (e.g., Bedding et al. 2005) is interactingwith the convection, resulting in transient pulsations.

The quadrature phase relation seen in Figure 8 is at leastpotentially compatible with convection. That is, heating is

followed by the material rising; cooling followed by it falling.Since cooling occurs over most of the rise-velocity phases,considerable overshoot may be implied, depending on where the(unknown) zero velocity lies in Figure 8. The opposite portionof the cycle may be harder to understand, namely, why do we seethe temperature increase prior to seeing rise velocities? Takenat face value, it implies that we see the material heating upbefore it rises, as if a thermal pulse entered the material ratherthan the cell acquiring its velocity below the surface, priorto its becoming visible. Perhaps the observational coverageduring heating phases is actually too incomplete to draw thisconclusion. Furthermore, since we do not know the position ofzero velocity, we cannot be certain that all the velocities wesee are not rise velocities. This could occur if most of the light,most of the time, comes from bright rising cells, the falling cellsbeing so cool and faint as to contribute negligible light.

The lack of close correspondence between visual magnitudeand temperature, discussed in Section 3, is expected with giantconvection cells simply because cells will vary in size, temper-ature, and location on the disk. In addition, limb darkening fora star like Betelgeuse is extreme, so a cell away from the diskcenter would perturb the brightness more than the same cell atthe disk center. A behavior more coherent than the observationsindicate would be expected from radial pulsations.

There is also circumstantial evidence supporting convectioncells as the source of the variations. For example, the radialvelocity of metal lines, as published by Uitenbroek et al.(1998a), shows a systematic rise (along with many smallervariations) of ∼6 km s−1 over a 3500 day interval. Thiscorresponds to a radius change of ∼1300 solar radii or ∼1.6stellar radii, which is hardly likely and implies that such asimple interpretation of the radial velocity is naive. Furthermore,the AAVSO magnitude (again ignoring shorter-term variations)has remained essentially constant over this same time interval.This kind of inconsistency is unlikely to find an explanationin pulsation. Convection velocities, on the other hand, mightwell introduce systematic displacements of this type because(1) the lines are asymmetric and the asymmetry changes, whichcould result in a different radial velocity measurement even ifthe overall position of the line has not significantly changed,and (2) the Doppler shifts of the bright gas might reasonably beexpected to dominate the radial velocity, producing a systematicoffset from the true space motion of the star. Variation of theoffset might vary from one epoch to another, in particular ondecadal timescales.

Polarization measurements (Hayes 1984) show variations instrength and position angle on timescales of months to years, i.e.,the same timescales shown by the excursions, that do not repeatfrom one episode to the next. Not only is periodicity lacking,but fundamental-mode oscillations are not expected to producepolarization by virtue of geometrical symmetry, whereas nosuch symmetry constraints exist with giant convection cells.Hayes concluded that giant granulation cells offered the bestexplanation of his polarization data.

Further, consider solar non-radial oscillations that are be-lieved to be powered by the granulation. These oscillations havecharacteristic velocities of ∼0.4 km s−1 compared to granula-tion velocities an order of magnitude larger. If the excursionsdisplayed in Figure 8 were the comparable oscillations, thenwhere are the larger motions that drive them? The only othervelocities detected are those producing line broadening, andthese are of the same order of magnitude as the excursions. Un-less there is some (unknown) process that transfers the energy

1458 GRAY Vol. 135

of non-resonant modes into the resonant mode, the efficiencywould have to be near unity. It therefore seems more likely thatthe excursions are the driving mechanism. If there are oscilla-tions with ∼400 day periods, rather than this time simply beingthe characteristic eruption time of the cells, they may occur as afollow-up ringing of the excursions in a manner that is naturaland expected.

For all these reasons, the conclusion seems to be that largeconvection cells dominate the ∼400 day variations, while os-cillations are secondary. But is it possible that only one con-vection cell essentially always dominates the spectrum? Thatis, following the argument of G2, since motions of severalkm s−1 are involved, how is it we never see split or bumpyline profiles of the type one might expect if 2–10 convectioncells appeared at the same time? The hierarchy of convection-cell dimensions seen in the Sun may give us a clue. Granulation(∼1 Mm), supergranulation (∼30 Mm), and giant convectioncells (∼200 Mm) have been identified (e.g., Beck et al. 1998;Lisle et al. 2004). The largest of these involves dimensions thesize of the convection zone itself. If this were transferred toBetegeuse, the giant convection cells would have dimensionsessentially equal to the size of the star. Perhaps the observedexcursions are the giant cell motions, while the line broadeningreflects the Doppler-shift distribution of the combined super-granulation, granulation, and smaller scale motions. The downside of this argument is that the solar giant-cell velocities areextremely small, perhaps ∼100 m s−1, two orders smaller thangranulation velocities. To invoke giant cells in Betelgeuse im-plies scaling up the giant-cell velocities to the same order asthe granulation velocities. The 400 day timescale would be thetypical time between cells surfacing, not the turnover time ofone cell.

Even though these arguments suggest that the main variationsarise from convection, the result is not yet definitive.

In any case, the dominant motions associated with theexcursions are not a primary contributor to macroturbulence.Instead, motions within the large structures form the classicalmacroturbulence of the type seen in lower-luminosity stars. Inother words, the motions of the excursions produce mainlydisplacements of spectral lines, while the motions within themoving material dominate the broadening of spectral lines. Is itpossible that these characteristics are an integral part of all starsshowing reversed-C bisector shapes?

The similarity of the chaotic velocity variations of Betelgeuseand the radial-velocity phenomenon termed “jitter” should notbe overlooked. Jitter or real, but noise-like, variations in radialvelocity measurements of high-luminosity stars, after being firstseen by Gunn & Griffin (1979), has remained a puzzle. Jitter hasbeen noted in many investigations dealing with evolved stars.A summary is given by Carney et al. (2003). It seems to makeits appearance near absolute magnitudes ∼0.5 and increases toa few km s−1 at the tip of the red-giant branch. Since such aradial-velocity variation is precisely the type of behavior seen inFigure 6, it seems likely that jitter is caused by stochasticconvective motions and any oscillations they engender. If so,a study of jitter will help map out the increasing vigor ofconvection, perhaps giant convection cells, as a function ofposition in the H–R diagram.

Bear in mind that only a limited range of time variationshas been included in this discussion. Events on longer andshorter timescales are no doubt present. Whether or not detailedhydrodynamical modeling can reproduce these observations

remains to be seen, and such modeling is in progress (Piskunovet al. 2008, in preparation).

I am grateful to the Natural Sciences and EngineeringResearch Council of Canada for continued financial support.I thank M. Debruyne for technical support at the observatory,the observers who contributed to the data over many years, andto Kevin I. T. Brown, Bruce W. Carney, and a referee for helpfulsuggestions. My thanks also go to the work of the AAVSO andits many contributors around the world.

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