THE ASTROPHYSICAL JOURNAL, 532 :487496, 2000 March 202000. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

BETELGEUSE AND ITS VARIATIONS

DAVID F. GRAYDepartment of Physics and Astronomy, University of Western Ontario, London, Ontario N6A 3K7, Canada ; dfgray=uwo.ca

Received 1999 May 3; accepted 1999 October 6

ABSTRACTObservations of Betelgeuse show that (1) the photospheric spectral lines are very broad with a 1/e

width of 11 km s~1, but they have stable shapes with time as the star varies in other ways ; (2) all thespectral lines within the observed wavelength region change their depths by essentially the same factorand in phase with the photometric brightness of the star on timescales of months ; (3) small deviationsfrom perfect scaling of the line depths imply temperature variations that to rst order are in phase withthe line depths, i.e., cooler temperatures go with weaker lines ; and (4) to second order, the temperaturevariations lag behind the changes in line depths by D5 days. The hypothesis of changing continuousopacity is put forward as a coherent explanation of all of these spectroscopic and photometric eects. Inaddition, the bright spot observed by others is explained as a region where the opacity does not change ;it is not a giant convection cell. The physical reason for the opacity change has yet to be identied.Structure in the cores of spectral lines varies on the timescale of a day, and this may be the signature ofgiant convection cells, but other interpretations are also possible, including hot spots, prominences, ornonradial oscillations.Subject headings : line : proles stars : individual (a Orionis) stars : late-type stars : spots

supergiants

1. INTRODUCTION

By any standard Betelgeuse (a Ori, HR 2061, HD 39801,M2 Iab, B[V \ 1.86) is a remarkable object, a cool super-giant star some 800 times larger than the Sun but with aradius that depends very much on the opacity in the spec-tral region in which the star is viewed. A bright spot ofunknown origin comes and goes on a backdrop of adynamic photosphere. Variations on timescales from daysto decades are seen in visual magnitude, UV continuum,chromospheric indicators, and radial velocity. Surroundingthe star itself is a disk system apparently composed of ejectafrom the star. The study I present here focuses on thephotosphere and its variations, the mechanism of variabil-ity, and the nature of the bright spots seen on the starssurface.

Hipparcos satellite observations determined the parallaxto be 7.6 ^ 1.6 mas. Although this is a considerableimprovement over previous measurements, this star has sig-nicant surface features that vary with time, possibly alter-ing the photometric centroid of the Hipparcos images andcompromising the parallax study. If a bright spot contrib-uted 25% of the light and resided midway between thecenter of the disk and the limb, and using a visual photo-spheric diameter of 55 mas, the centroid shift would benearly half the size of the parallax. The Hipparcos parallaxgives an absolute magnitude of based on theM

V\ [5.1

characteristic apparent magnitude that is observed to varybetween about V \ 0.3 and 0.8. Light and radial velocityvariations show quasiperiodic behavior (e.g., Dupree et al.1987 ; Smith et al. 1989 ; Krisciunas 1996 ; Morgan et al.1997). Although several characteristic periods have beenfound, most notably one near 6 yr and one near 1 yr, it isdoubtful that the star is periodic in any strict sense. Theamplitude, median magnitude, and phase seem to wander,especially for the shorter timescale variations. The light ispolarized D0.5%, and, as one might expect, the polariza-tion is variable in both amplitude and orientation. The

reason for these variations has been a source of speculation(e.g., Hayes 1984 ; Uitenbroek et al. 1998b).

Because of its large size, the angular diameter of the starhas been measured many times. White (1980) and Cheng(1986) summarize the earlier measurements. More recentobservations have been done in the UV (Uitenbroek et al.1998b), in the visible window (Bruscher et al. 1990 ; Wilsonet al. 1992 ; Tuthill et al. 1997 ; Wilson et al. 1997 ; Burns etal. 1997), at 11 mm (Bester et al. 1996), and at 6 cm (Skinneret al. 1997). Minimum size occurs at visible wavelengthsbecause the absorption coefficient is a minimum there andbecause contribution from surrounding dust has little eect.There is some evidence for temporal variation in theangular size (e.g., White 1980), perhaps on a period of 5.8 yr,but complications arising from the presence of the brightspot on the disk make the situation complex and uncertain.

The eective temperature of Betelgeuse is D3600 K (e.g.,Dyck et al. 1998, based on interferometric measurements ofthe angular size at 2.2 mm; see also Lim et al. 1998 andWhite 1980). Published values apparently ignore the varia-bility of the star and do not combine simultaneous measure-ments of angular size and bolometric luminosity. Inaddition to the uncertainty in limb darkening, the conceptof eective temperature itself begins to be ill-dened in starslike this one, where the radius depends markedly on thewavelength. The mass of the star is unknown, but mostinvestigators show a preference for a fairly large mass in therange of 1020 The position the star had on the mainM

_.

sequence is important for understanding the rotation rateas discussed below in 3.

The bright spot, either a single spot or a tight-knit groupof spots, has been reported on several occasions (e.g., Gillil-and & Dupree 1996 ; et al. 1997 ; Uitenbroek et al.Klu ckers1998b, and others). At other times the spot is absent (Burnset al. 1997). Arguments against rotational modulation as anexplanation of the spot variation have been broughtforward in a number of the cited papers. The strongest is

487

TABLE 1

OBSERVATION LOG

Date Year JD [2,440,000 S/N Weight

1996 Mar 23 . . . . . . 1996.226865 10166.532465 578 0.331996 Mar 26 . . . . . . 1996.235216 10169.589120 223 0.051996 Mar 26 . . . . . . 1996.235248 10169.600694 552 0.301996 Mar 29 . . . . . . 1996.243259 10172.532812 719 0.521996 Mar 30 . . . . . . 1996.246008 10173.539062 616 0.381997 Jan 28 . . . . . . . 1997.076788 10477.527778 538 0.291997 Jan 28 . . . . . . . 1997.076805 10477.533854 667 0.451997 Feb 15 . . . . . . 1997.126132 10495.538310 587 0.341997 Feb 22 . . . . . . 1997.145521 10502.615278 624 0.391997 Mar 8 . . . . . . . 1997.183654 10516.533565 662 0.441997 Mar 11 . . . . . . 1997.191863 10519.529919 677 0.461997 Mar 12 . . . . . . 1997.194601 10520.529225 705 0.501997 Mar 18 . . . . . . 1997.211149 10526.569271 591 0.351997 Mar 23 . . . . . . 1997.224715 10531.520891 635 0.401997 Mar 27 . . . . . . 1997.235647 10535.511227 801 0.641997 Mar 31 . . . . . . 1997.246624 10539.517824 733 0.541997 Mar 31 . . . . . . 1997.246636 10539.521991 727 0.531997 Apr 3 . . . . . . . . 1997.254910 10542.542014 733 0.541997 Apr 13 . . . . . . 1997.282284 10552.533565 764 0.581997 Apr 14 . . . . . . 1997.285021 10553.532812 629 0.401997 Apr 17 . . . . . . 1997.293299 10556.554051 729 0.531997 Apr 20 . . . . . . 1997.301446 10559.527662 597 0.361997 Apr 20 . . . . . . 1997.301456 10559.531424 768 0.591997 Apr 26 . . . . . . 1997.317939 10565.547917 712 0.511997 Apr 28 . . . . . . 1997.323394 10567.538889 700 0.491997 Sep 27 . . . . . . 1997.738098 10718.905787 845 0.711997 Oct 6 . . . . . . . . 1997.762744 10727.901736 849 0.721997 Oct 6 . . . . . . . . 1997.762759 10727.907002 749 0.561997 Oct 7 . . . . . . . . 1997.765559 10728.928935 733 0.541997 Nov 29 . . . . . . 1997.910686 10781.900231 804 0.651997 Dec 16 . . . . . . 1997.957078 10798.833507 850 0.721997 Dec 16 . . . . . . 1997.957092 10798.838715 643 0.411997 Dec 18 . . . . . . 1997.962712 10800.889699 709 0.501998 Feb 3 . . . . . . . . 1998.093299 10848.554282 514 0.261998 Feb 3 . . . . . . . . 1998.093320 10848.561921 648 0.421998 Feb 3 . . . . . . . . 1998.093342 10848.569850 528 0.281998 Feb 3 . . . . . . . . 1998.093383 10848.584722 734 0.541998 Feb 3 . . . . . . . . 1998.093415 10848.596412 738 0.551998 Feb 3 . . . . . . . . 1998.093441 10848.606134 667 0.441998 Feb 5 . . . . . . . . 1998.093465 10848.614873 644 0.411998 Feb 5 . . . . . . . . 1998.098813 10850.566898 553 0.311998 Feb 5 . . . . . . . . 1998.098832 10850.573727 660 0.441998 Feb 5 . . . . . . . . 1998.099066 10850.658970 756 0.571998 Feb 6 . . . . . . . . 1998.101289 10851.470602 746 0.561998 Feb 7 . . . . . . . . 1998.104027 10852.469907 684 0.471998 Feb 8 . . . . . . . . 1998.106823 10853.490336 461 0.211998 Feb 8 . . . . . . . . 1998.106845 10853.498438 831 0.691998 Feb 15 . . . . . . 1998.126418 10860.642708 565 0.321998 Feb 15 . . . . . . 1998.126429 10860.646759 659 0.431998 Mar 7 . . . . . . . 1998.181094 10880.599306 670 0.451998 Mar 15 . . . . . . 1998.202769 10888.510590 572 0.331998 Mar 15 . . . . . . 1998.202800 10888.522164 681 0.461998 Mar 15 . . . . . . 1998.202838 10888.535880 741 0.551998 Mar 16 . . . . . . 1998.205530 10889.518519 592 0.351998 Mar 16 . . . . . . 1998.205540 10889.521991 674 0.451998 Mar 22 . . . . . . 1998.222238 10895.616725 519 0.271998 Mar 22 . . . . . . 1998.222252 10895.621875 607 0.371998 Mar 23 . . . . . . 1998.224700 10896.515625 660 0.441998 Apr 4 . . . . . . . . 1998.257598 10908.523437 453 0.211998 Apr 4 . . . . . . . . 1998.257614 10908.528935 556 0.311998 Apr 4 . . . . . . . . 1998.257627 10908.533796 656 0.431998 Apr 10 . . . . . . 1998.274023 10914.518519 425 0.181998 Apr 10 . . . . . . 1998.274033 10914.522164 616 0.381998 Apr 11 . . . . . . 1998.276755 10915.515451 600 0.361998 Apr 11 . . . . . . 1998.276763 10915.518519 685 0.47

488

TABLE 1Continued

Date Year JD [2,440,000 S/N Weight

1998 Apr 20 . . . . . . 1998.301502 10924.548322 193 0.041998 Apr 23 . . . . . . 1998.309665 10927.527546 685 0.471998 Apr 23 . . . . . . 1998.309678 10927.532581 608 0.371998 Apr 24 . . . . . . 1998.312419 10928.532986 832 0.691998 Apr 24 . . . . . . 1998.312434 10928.538252 724 0.521998 Apr 27 . . . . . . 1998.320699 10931.555093 712 0.511998 Apr 28 . . . . . . 1998.323389 10932.537153 764 0.581998 May 5 . . . . . . . 1998.342582 10939.542593 647 0.421998 Aug 2 . . . . . . . 1998.584668 11027.903646 547 0.301998 Aug 3 . . . . . . . 1998.587413 11028.905671 548 0.301998 Aug 14 . . . . . . 1998.617540 11039.901968 574 0.331998 Aug 15 . . . . . . 1998.620304 11040.910880 781 0.611998 Aug 17 . . . . . . 1998.625765 11042.904398 655 0.431998 Aug 19 . . . . . . 1998.631271 11044.913831 690 0.481998 Aug 20 . . . . . . 1998.633993 11045.907292 737 0.541998 Aug 20 . . . . . . 1998.634007 11045.912500 669 0.451998 Aug 20 . . . . . . 1998.634020 11045.917245 683 0.471998 Aug 26 . . . . . . 1998.650440 11051.910532 618 0.381998 Aug 27 . . . . . . 1998.653212 11052.922338 526 0.281998 Aug 30 . . . . . . 1998.661400 11055.910938 659 0.431998 Aug 31 . . . . . . 1998.664167 11056.920891 679 0.461998 Sep 1 . . . . . . . . 1998.666825 11057.891088 563 0.321998 Sep 4 . . . . . . . . 1998.675088 11060.907118 627 0.391998 Sep 6 . . . . . . . . 1998.680637 11062.932581 648 0.421998 Sep 10 . . . . . . 1998.691439 11066.875116 555 0.311998 Sep 11 . . . . . . 1998.694324 11067.928183 661 0.441998 Sep 11 . . . . . . 1998.694333 11067.931713 738 0.541998 Sep 11 . . . . . . 1998.694362 11067.942303 653 0.431998 Sep 12 . . . . . . 1998.697101 11068.941840 661 0.441998 Sep 17 . . . . . . 1998.710776 11073.933333 592 0.351998 Sep 18 . . . . . . 1998.713546 11074.944387 646 0.421998 Sep 19 . . . . . . 1998.716274 11075.939931 699 0.491998 Sep 29 . . . . . . 1998.743530 11085.888542 651 0.421998 Sep 29 . . . . . . 1998.743549 11085.895255 684 0.471998 Oct 10 . . . . . . 1998.773349 11096.772338 679 0.461998 Oct 10 . . . . . . 1998.773364 11096.777836 547 0.301998 Oct 16 . . . . . . 1998.789938 11102.827431 698 0.491998 Oct 17 . . . . . . 1998.792582 11103.792477 286 0.081998 Oct 20 . . . . . . 1998.800866 11106.815972 783 0.611998 Oct 23 . . . . . . 1998.809226 11109.867361 628 0.391998 Oct 24 . . . . . . 1998.812238 11110.966725 500 0.251998 Oct 24 . . . . . . 1998.812249 11110.971065 598 0.361998 Oct 25 . . . . . . 1998.814984 11111.969213 826 0.681998 Oct 29 . . . . . . 1998.825660 11115.865799 662 0.441998 Oct 31 . . . . . . 1998.831373 11117.951042 803 0.641998 Nov 2 . . . . . . . 1998.836538 11119.836516 653 0.431998 Nov 12 . . . . . . 1998.863727 11129.760417 770 0.591998 Nov 13 . . . . . . 1998.866402 11130.736806 653 0.431998 Nov 13 . . . . . . 1998.866416 11130.741898 740 0.551998 Nov 18 . . . . . . 1998.880180 11135.765741 781 0.611998 Nov 22 . . . . . . 1998.891359 11139.845891 729 0.531998 Nov 24 . . . . . . 1998.896641 11141.773785 731 0.531998 Nov 28 . . . . . . 1998.907634 11145.786574 701 0.491998 Dec 2 . . . . . . . 1998.918560 11149.774363 652 0.421998 Dec 2 . . . . . . . 1998.918573 11149.779051 640 0.411998 Dec 3 . . . . . . . 1998.923733 11151.662384 885 0.781998 Dec 3 . . . . . . . 1998.923771 11151.676273 767 0.591998 Dec 9 . . . . . . . 1998.937693 11156.758102 662 0.441998 Dec 9 . . . . . . . 1998.937705 11156.762269 691 0.481998 Dec 9 . . . . . . . 1998.940231 11157.684144 763 0.581998 Dec 12 . . . . . . 1998.945830 11159.727951 654 0.431998 Dec 13 . . . . . . 1998.948644 11160.755035 620 0.381998 Dec 14 . . . . . . 1998.951270 11161.713542 693 0.481998 Dec 14 . . . . . . 1998.951279 11161.717014 652 0.421998 Dec 14 . . . . . . 1998.953904 11162.675058 774 0.601998 Dec 25 . . . . . . 1998.983786 11173.581829 379 0.141998 Dec 27 . . . . . . 1998.987034 11174.767477 566 0.32

489

230 235 240 245 250

0.7

0.8

0.9

1.0

F/Fc

Diode Steps

Ori 6273.39 Ti I

Gaussian11 km/s dispersion

5 exposures

490 GRAY Vol. 532

simply that the timescale of the spots variability (D100days) is much too short to be the rotation period (D6000days or more). Therefore, other mechanisms have been sug-gested, including giant convection cells, also called super-granulation, and shock fronts.

Radial velocity variations are documented (a summarydiagram can be seen in Uitenbroek et al. 1998a), but theirmeaning is unclear, as will be discussed below. Mass out-ows are indicated by velocity increasing outward throughthe chromosphere to D7 km s~1 (Carpenter & Robinson1997). A dust shell around a Ori extends from to 5A, and0A.3there is evidence for episodes of ejection (Bester et al. 1996 ;Skinner et al. 1997 ; Wilson et al. 1997 ; Rinehart et al. 1998).An unusual triple-star model was also put forward toexplain some of the observations (Karovska et al. 1986 ;Hebden et al. 1986 ; Christou et al. 1988 ; Tuthill et al. 1997).

Sections 25 describe the observations, while 6 presentsan explanation, tying the observations together with aphysical picture and giving a new interpretation of thebright spot. Evidence for giant convection cells is presentedin 7.

2. SPECTROSCOPIC OBSERVATIONS

Spectroscopic exposures were started at the University ofWestern Ontario in 1996 and the data presented here runthrough the end of 1998. Details are given in Table 1. The

spectrograph has been described earlier (Gray 1986).coudeThe resolving power is approximately 100,000, and a spanof some 70 centered near 6250 was recorded with anA A1872 diode Reticon detector. The mean signal-to-noise ratiois about 650. Although some of the exposures were taken inthe waxing or waning twilight, skylight is kept to less thanone part per thousand as judged by exposure-meter read-ings taken on blank sky near the star. Although these obser-vations have high signal-to-noise ratios and high spectralresolving power, making them suitable for studies of spec-tral line widths and shapes, no absolute wavelength cali-bration is available and therefore they yield no radialvelocity information.

3. SPECTRAL LINES : GROSS CHARACTERISTICS

One is immediately struck by the large broadening thatsmears out and blends together the photospheric lines ofa Ori. A comparison is made in Figure 1 between a Ori andb And, a somewhat less luminous M star. Many of the

FIG. 1.Part of the 6250 spectral region for a Ori is compared withAthe corresponding spectrum of the less luminous star b And. The velocityeld in the atmosphere of a Ori broadens the lines and increases theblending, but one can still identify a number of lines.

lines seen in b And appear in a Ori as well, although theblending of lines in a Ori is signicantly more severe.

The sharpest of the a Ori lines in these observations isj6273.39 of Ti I at the right edge of the eld in some expo-sures and just out of the eld in most exposures owing toslightly dierent settings of the grating. Because of its sharp-ness, it seems likely that the line suers from relatively littleblending, but being at the extreme edge of the camera eldmeans less than optimum focus and signicant vignetting.Figure 2 shows a Gaussian compared to the prole of thisline. The line extends over at least ^17 km s~1 ; the Gauss-ians dispersion is 11 km s~1. This broadening is probablycaused by rotation and macroturbulence ; these are nowconsidered.

One might expect rotational broadening to be smallsimply because of the enormous radius of a Ori, D800 R

_.

However, if a Ori is a massive star, it might have had arotation rate as high as 400 km s~1 on the main sequence.Moment of inertia and internal redistribution of angularmomentum during postmain-sequence evolution are notwell understood. Still, for comparison purposes, it is worthconsidering a classical example. If angular momentum wereconserved in shells, a main-sequence radius of 8 R

_,

expanded to 800 would reduce 400 km s~1 to 4 km s~1.R_

,Dierent assumptions concerning the moment of inertiacould change this value by a factor of 2 or more either way.For example, Heger & Langer (1998) suggest that consider-able spin-up may occur, while less extreme assumptionsconcerning the main-sequence rotation rate would imply asignicantly lower rotation rate now. The prole shown inFigure 2 is approximately Gaussian, and it is well known(e.g., Gray 1988, chap. 25) that an equal mixture of

FIG. 2.Ti I 6273.39 line, the narrowest in the observed eld, isAcompared to a Gaussian (crosses). The line is located at the extreme edge ofthe 70 eld, and variable vignetting causes the scatter toward the right-Ahand side of the gure. Considering the massive amount of line blending inthis spectrum, the agreement with the Gaussian is remarkable.

No. 1, 2000 BETELGEUSE AND ITS VARIATIONS 491

Doppler shifts from rotation and radial-tangential macro-turbulence can mimic a Gaussian. If this were the case here,v sin i would be D4.5 km s~1 and the macroturbulencedispersion would be km s~1. This value of macro-fRT D 4.5turbulence dispersion is not totally out of line with data forother cool luminous stars (Gray & Toner 1987 ; Gray 1988,chap. 36) although it may be somewhat low. Alternatively,one might postulate a dominant isotropic Gaussian macro-turbulence, or something close to it, within which one couldhide D23 km s~1 rotational broadening.

Using partially resolved images of the star, Uitenbroek etal. (1998b) nd v sin i\ 5 km s~1 from their ultravioletobservations. This value is for the UV radius that is 2 or 2.5times larger than the classical visual photospheric radius. Ifthere is no dierential rotation with depth, their measure-ments imply a visual-photosphere rotation rate ofv sin iD 2.5 or 2 km s~1. They also nd evidence for theinclination of the rotation axis of only 20, which means theactual equatorial rotation rate would be three times larger,or 67 km s~1. A value this size is not very far from ourrough shells estimate above, and the v sin i is compatiblewith the information deduced from the spectral linebroadening in Figure 2.

Rotation is therefore likely to account for some, but notall, of the line broadening. Macroturbulence thereforeremains a major or dominant component of the linebroadening. Macroturbulence in this context includes alllarge-scale motions such as granulation (convection cells)and nonradial oscillations. Furthermore, as we will see inthe next paragraph, the shapes of the spectral lines arenearly constant through cycles of variation in other param-eters. Consequently, macroturbulence must be stable withtime, at least over the span of these observations.

4. FEW-MONTH TIME VARIATIONThe investigation presented here is limited to the varia-

tions occurring on the few-month timescales. These varia-tions are large and easy to see, as illustrated in Figure 3a,where an exposure taken at the minimum line-depth phaseis compared to one at maximum line-depth phase. Whilethere are small and subtle variations as well, the dominantvariation is in the depths of the spectral lines. To rst order,all spectral lines vary together and by nearly the sameamount, i.e., one can scale the depths of the absorption linesin any given exposure and nearly reproduce any other expo-sure. In Figure 3b, the exposure having the weaker lines is

FIG. 3.(a) 1998 September 19 exposure shows shallower lines and was taken when the star was near minimum brightness ; the 1998 November 24exposure has stronger lines and was taken when the star was near maximum brightness. (b) The line depths in the spectrum with the weaker lines arestrengthened 20%. This simple scaling brings the two spectra into close agreement. (c) An expanded view of a portion of (b) shows more of the detail.Although small variations can easily be seen, the widths of features like the j6243 V I blend do not change. (d) The scaling of line depth is the same as for (c),and here we can see that the deeper lines should have a slightly smaller scaling factor than needed for the shallower lines. This is a curve-of-growth eect, asexplained in 6.

10200 10400 10600 10800 11000 112000.75

0.65

0.55

0.45

0.35

JD - 2440000

0.20

0.25

0.30

0.35

Core

Dep

th o

f 62

53 Ori

1998 19991997

V M

agni

tude

0.25 0.300.20

0.25

0.30

Dep

th o

f Fe

I 6

219

Depth of Fe I 6253

Start

End

Ori Last episode of 1998

a)

0.30 0.350.31

0.36

0.41

Dep

th o

f 62

43 V

I bl

end

Depth of 6251 V I

Start

End

b)

492 GRAY Vol. 532

FIG. 4.Upper panel : The depths of the spectral lines vary several percent on timescales of months. Individual cycles look like they should beperiodic but are not. L ower panel : The recent photometry of K. Krisciunasand K. Luedeke (Krisciunas 1996 ; K. Krisciunas 1999, private com-munication) shows that the brightness varies in phase with the changes inline depths.

strengthened by 20%. (This scaling factor uses the highestpeaks in the exposures as a relative reference. Because of theline blending, the true continuum may be somewhat higher.Consequently, the 20% value should be considered approx-imate and an upper limit.) A portion of Figure 3b isenlarged in Figure 3c to illustrate how closely simple scalingcan match the proles.

Equally important, the shapes of the spectral lines, andespecially their widths, remain constant while the depthschange. This is true for all phases of variability. Thereforemacroturbulence in the photosphere has a constant disper-

sion. It follows that there must be a large number of velocityelements continuously on the surface ; the stars surfacecannot be dominated by a few (D110) stochasticallyoccurring giant convection cells. (But see 7 below.) Inparticular, the bright spot seen from time to time on thesurface cannot be a large convection cell because it wouldgrossly distort the line proles when it appeared and wouldnot contribute to the observed line widths when it was gone.

Small deviations from simple scaling are also seen. Therst of these is a curve-of-growth eect ; weaker lines requireslightly more scaling, stronger lines slightly less (Fig. 3d).The reason for this is explained in 6. Another second-ordereect arises from the diering sensitivities to temperaturefrom line to line. We exploit this in 5 as a temperatureindex.

The precise depths of several spectral features were mea-sured by tting vertically oriented parabolas to the line (orblend) cores. These depths are measured downward from anominal continuum. Figure 4 illustrates the typical timevariation of visual magnitude (photometry by Krisciunasand Luedeke, kindly made available by K. Krisciunas) anddepths of the spectral lines ; the variation of other linedepths is similar. These variations are not periodic eventhough individual episodes look like portions of cyclicbehavior. There is enough overlap to see that the spectrallines are deeper when the visual magnitude is brighter, i.e.,variations in line depths are in phase with variations inbrightness. This is an important point as we will see shortly.

There are basically two ways in which the line depths canbe made to scale in this way. The rst way is illustrated by aspectroscopic binary where the light of an early-type star(which has few spectral lines) is added to the light of a latertype star causing the cooler stars lines to appear weakened.If a Ori were an eclipsing binary, this eect would giveweaker spectral lines outside of eclipse, and vice versa. But aOri is fainter when the spectral lines are weaker, exactlyopposite the eclipsing binary eect.

FIG. 5.All lines go through the cycle of weakening and strengthening and do so approximately together. (a) Iron lines, like these two, track each otherwithin the errors of observation. Errors shown are estimated from the photon count and the readout noise of the light detector ; see Table 1. (b) Vanadiumlines also track each other but dier slightly from iron (see Fig. 6).

0.24 0.26 0.28 0.30 0.32 0.34

1.40

1.35

1.30

1.25

1.20

1.15Li

ne-D

epth

Rat

io

6243

V I

to

6253

Fe

I Ori

Mean Line Depth (6 lines)

Start

End

a)

0.24 0.26 0.28 0.30 0.32 0.34

1.25

1.20

1.15

1.10

Line

-Dep

th R

atio

62

51 V

I to

62

53 F

e I Ori

Mean Line Depth (6 lines)

Start

End

b)

No. 1, 2000 BETELGEUSE AND ITS VARIATIONS 493

FIG. 6.The temperature index of vanadium-to-iron line depths is plotted as a function of the mean depths of ve iron lines. A running mean of threepoints is used for the ordinate. Temperature increases upward. The direction of increasing time is shown by arrows. (a) The V I blend j6243 is used in theratio. (b) The V I line j6251 is used in the ratio. Both line-depth ratios show a loop, and the sense of the loop implies that temperature lags behind the changein line depths.

The second way to weaken spectral lines en masse is tohave a change in the continuous opacity. An increase incontinuous absorption coefficient would reduce the bright-ness of the star as it weakened the spectral lines, and this isprecisely the observed phase relation. This is the root of thehypothesis posed in 6 below.

5. LINE DEPTHS : TEMPERATURE AND TIME LAGS

In order to avoid confusion introduced by small dier-ences from cycle to cycle, we concentrate here on the singlecycle comprising the last big dip in line depths between1998.7 and 1998.9. The lines j6219.29, j6252.57, andj6265.14 are all Fe I lines of nearly identical excitationpotential (2.22.4 eV). The depths of these lines track eachother within the errors of measurement, as shown in Figure5a. Vanadium lines are typically more sensitive than ironlines to temperature changes. The V I lines also track eachother reasonably well, especially considering the heavyblending of the j6243 pair (Fig. 5b).

The line-depth ratio of V I to Fe I has been used as atemperature index in other investigations (e.g., Gray &Johanson 1991 ; Gray 1994 ; Gray et al. 1996). Indeed, onesees the vanadium-to-iron ratio strengthen monotonicallywith declining eective temperature in cool stars, and there-fore it seems plausible to interpret variations in this ratio aschanges in temperature. While we cannot rule out changesin the atmospheric structure as the cause of the variation inline-depth ratio, the fact that the excitation potentials arenot terribly dierent (0.3 eV for the V I lines vs. 2.4 eV forthe Fe I line) and that the line depths themselves do notdier greatly argues against this possibility. Two ratios ofV I to Fe I are shown in Figure 6 as a function of mean linedepth. A rise in line-depth ratio implies a decline in tem-perature. With this interpretation, temperature falls as thedepths of the spectral lines decline. In addition, the loop inFigure 6 shows temperature changes lagging behind the

line-depth changes. Ignoring the ne structure of the loop,the general ellipticity implies a time lag D5 days.

6. OPACITY HYPOTHESIS

The following hypothesis is proposed to explain thephotospheric variations of Betelgeuse. A disturbance risesthrough the outer envelope triggering an increase in contin-uous opacity. The opacity increase extends from below thelower photosphere upward through the line-forming region.The increase in opacity restricts the amount of light gettingout, so the star dims. The increase in opacity weakens thespectral lines without distorting their proles, as demandedby the spectroscopic observations presented above. Further,an increase in continuous opacity weakens weaker linesmore ; stronger lines have to be brought down the satura-tion part of the curve of growth, which means they changeless. This explains why one scaling factor does not workexactly for all strength lines, as illustrated in Figure 3dabove.

If other photospheric parameters remain constant, thecontinuous opacity would have to increase by D20% in thej6250 region to produce a drop in the spectral line depths ofD20% (or somewhat less if the continuum has been placedtoo low because of line blending). The drop in ux in thevisual magnitude band (Fig. 4) is D30%, but exactly howthis relates to the drop in ux in j6250 region is not current-ly known. Wilson et al. (1997) estimate the variation at 7000

to be D2/3 that in the V band (see also Morgan et al.A1997). The more important point however, is that the twoeects are in phase, in accord with the opacity hypothesis.What happens to the blocked energy is an important ques-tion that needs further study. It might be redistributed inwavelength or might be stored within the envelope just as itis in the case of sunspots (Spruit 1994).

With less energy coming from below, the photospherecools by outward radiation, but only after the power sup-

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plied from below has been reduced, hence the drop in tem-perature lags the opacity change and the loops in Figure 6are explained. The 5 day lag deduced above from Figure 6 isindicative of the cooling time for the photosphere, butdetailed modeling would be required to make a more deni-tive statement.

Under the Opacity Hypothesis, the bright spot seen onthe disk is simply a region that is spared the increase inopacity suered by surrounding material. The spot becomes visible when light from the rest of the disk fallsaway with the increase in opacity. The spot is therefore nota hot spot in the sense of a rising convection cell, but rathera region that retains the characteristics of the surface shownby the whole disk during the times of minimum opacity.The spot appears to fade away when the surroundingsurface regains its brightness.

The observations strongly support this interpretation ofthe spot, although they are not conclusive (regular monitor-ing of the stars disk would be needed). For example, theobservations of Wilson et al. (1997) were taken during atime of decline in brightness ; they estimate that during thespan of their observations, the ux of the star dropped10%^ 3%. Across these same observations, the total spotcontribution to the ux rose 16%^ 3%. In other words, thespot ux stayed constant to within the errors of measure-ment. The spot study of Uitenbroek et al. (1998b ; see also

et al. 1997) was based on data taken 1995 MarchKlu ckers3, during a phase of minimum brightness, in accord with theopacity hypothesis. The observations of Burns et al. (1997),taken in 1995 October when the star was near maximumbrightness, failed to show the spot, again in accord with theopacity hypothesis. Because of the timing of the obser-vations, the data of Tuthill et al. (1997) oer more modestsupport for the opacity hypothesis, but support nonetheless.Earlier work (Buscher et al. 1990) shows a strong spot in1989 February when the star was at a deep minimum ofbrightness, oering still more support for the hypothesis.Through all of these observations, when the spot is visible,its contrast is characteristically 10%20%. This implies anopacity change of this same amount, and that is also theamount needed to account for the observed changes in thedepths of the spectral lines.

The bright spot seen on the surface might well be at therotation pole, as suggested by Uitenbroek et al. (1998b).With an equatorial rotation rate of 7 km s~1 (as per 2above), a 10 star would have a centripetal accelerationM

_amounting to 5% of the surface gravity. In other words, therotation would be large enough to support a modest densitygradient with latitude on the star. A disturbance producingopacity changes could therefore have a latitude dependence.A counterargument is found in the observations of Tuthillet al. (1997) who observed two other M supergiants in addi-tion to Betelgeuse and found hot spots on all three stars,whereas the probability of seeing stars near pole-on is small,varying with the sine of the inclination of the rotation axis.However, the validity or otherwise of the opacity hypothesisdoes not depend on the spot occurring at a rotation pole.

The giant convection cell hypothesis for explaining thebright spot should probably be abandoned. The mostdamning evidence is simply that the star should be brighterwhen a hot convection cell comes to the surface, not fainteras observed. The invariance of the shapes of the spectrallines, discussed above, would be very surprising indeed ifsuch giant convection cells occurred. Also arguing against

the convection cell hypothesis is the simple fact that thespot is always observed at the same place (within errors) onthe disk. There is every reason, however, still to expectvigorous convection to play a strong role in the atmo-spheric dynamics of stars like Betelgeuse ; the large macro-turbulence and the extended atmosphere, includingmass-loss events, are likely a result of convection eects(e.g., Lim et al. 1998). It also seems plausible that if thegeneral idea of Schwarzschild (1975) is correct, namely thatthere are only relatively few convection cells on such stars,then their variation in density and opacity should surelyintroduce a degree of chaos into the general opacity varia-tions proposed here, i.e., this may be the very reason whyBetelgeuse is only quasiperiodic. Convection cells mightalso be the reason why the spot has structure upon occasionas in the observations of Wilson et al. (1997), or why a polarspot might appear to wander from the pole.

Bester et al. (1996) concluded that the brightness varia-tion of Betelgeuse is a result of temperature changes and notradius changes because the 11 km ux showed much lessvariation than the ux in the visible region. Temperaturechanges are certainly part of the opacity hypothesis, but thespecic variation of the continuous absorption coefficientwith wavelength strongly aects the relative amplitude ofvariation at dierent wavelengths. Consequently, if theopacity hypothesis is correct, their result may need reinter-pretation. In a similar vein, the interpretation of changes inthe TiO band strength (Morgan et al. 1997) may be alteredby changes in the continuous opacity.

7. VARIATIONS ON THE DAY TIMESCALE

Structure in the cores of the stronger lines is variable on atimescale of a day. Figure 7 illustrates the general level ofthe signal, and Figure 8 shows examples of the variation fora small subset of the observations. This structure is typicallyD2% of the line depth and seems to be composed ofabsorption spikes. Less strong lines show similar variations,but the amplitude is scaled down in proportion to the linedepth. Given the small size of this structure, conrmingobservations will be needed. The features at j6258.11 andj6258.71 are composed of several spectral lines, making theinterpretation more challenging because an individual eventcan be replicated in each of the lines in the blend. The

FIG. 7.Spikelike structure in the cores of the lines varies on timescalesof a day. These two examples are chosen arbitrarily from the data set.

No. 1, 2000 BETELGEUSE AND ITS VARIATIONS 495

FIG. 8.Variations in line core structure are illustrated with thissequence of exposures. Both j6258.11 and j6258.71 are composites of twoor three lines, so the core structure is complex. There is less blending inj6261. The horizontal line connecting the left-hand and right-hand sec-tions of the diagram is at a residual ux level of 0.60 for each exposure. Theamplitude of the variations is D2% and is contained within a D10 km s~1span around the center of the line. Some exposures are shown with dashedlines to reduce visual confusion.

feature at j6261 is less blended, yielding a somewhat moreclear picture of the variations. The structure is similar butnot identical in lines of comparable strength. The individualfeatures of the structure itself are not resolved by thespectrograph. Therefore, the features have internalDoppler-shift spans less than D3 km s~1. The excursionsextend over approximately ^5 km s~1 from nominal linecenter.

This line-core structure cannot arise from spots beingcarried across the disk of the star by rotation, since therotation period is vastly longer than the timescale of thevariations. But the Doppler shift of rotation could positionthe signature of surface features. For example, if a large hotconvection cell moved upward at a fraction of a km s~1, itsown Doppler shift would be very small, but if it occurredaway from the central meridian on the stellar disk, its signalwould be Doppler shifted by the projected rotation velocity,up to ^v sin i.

Therefore, one explanation of the core structure is slow-moving, large, bright convection cells (supergranules). Thestructure shows up primarily because the cells are large andbright, meaning that they contribute D2% of the total uxin this region of the spectrum. Bright cells will producedownward spikes, just the inverse of what is seen in starsshowing dark spots. If a cell fades in a day, another maytake its place at a Doppler shift appropriate for the project-

ed rotation rate at the disk position of the new cell. If rota-tional Doppler shifts do play this role, the bright cells neednot have any velocity of their own; the cell could even be atransparent hole through which we see light from deeperhotter layers, perhaps analogous to faculae on the solarsurface, rather than a rising convection cell. More likely thecell does have a rise velocity and the Doppler shift it pro-duces will be added to that of the rotational displacement.

Alternatively, if rotational Doppler shifts play no role,i.e., v sin i is very small compared to the cell velocities, thenDoppler displacements would arise from the cells them-selves. That would imply cell velocities D^5 km s~1, andsince the spikes are seen on both sides of the core, it impliesthat the cells stay hot on their way back down. This wouldbe rather unorthodox behavior for convection cells, butperhaps the cells result from a arelike event rather thanfrom simple convection. The characteristic distance coveredat 5 km s~1 for a day is about 0.6 or less than 0.1% ofR

_the stellar radius.Other explanations of the line-core structure include

absorption spikes caused by cool absorbing material pro-pelled above the photosphere, perhaps similar to solarprominences or Doppler shifts produced by nonradial oscil-lations (e.g., Vogt & Penrod 1983 ; Hatzes 1996). Under thislater interpretation, the dominant period would correspondto the D1 day timescale of the variations and the velocityamplitude would be B5 km s~1.

This discussion is predicated on the reality of the corestructure, which has yet to be conrmed by independentobservations.

8. ADDITIONAL COMMENTS AND PROSPECTUSAlthough the opacity hypothesis correctly predicts a wide

range of behavior shown by Betelgeuse, intensive monitor-ing is needed to delineate further the phasing and strengthvariations of spectral lines, spot characteristics, and thebrightness, radius, and temperature. It is not clear if mag-netic elds even exist in this type of star, but if they do, theymight be a central part of the story. A study of the timevariation of the ux and the spectral line depths with wave-length might delineate the wavelength dependence of thatcomponent of the continuous absorption coefficient that ischanging with time. If, for example, it is the negative hydro-gen ion, then the blue and UV regions should show progres-sively less variation. There is no reason to expect variationsof chromospheric lines to be explained by this photosphericmechanism.

No attempt has been made to integrate the radial velocityand polarization observations into the opacity hypothesis.Using the brightness variations as an intermediary, radialvelocity apparently varies out of phase with the line depths(Smith et al. 1989 ; Uitenbroek et al. 1998a) ; that is,maximum opacity occurs at times of maximum velocity.These are relative velocities in the sense that the maximumvelocity could be outward relative to the center of massmotion of the star. Some caution should be used herebecause the wide lines of Betelgeuse make determination ofpositions of lines difficult. In addition, the long-term varia-tions, for example, those shown by Uitenbroek et al. (1998a),show a rise D6 km s~1 between 1985 and 1990. If we takeeven half this value and multiply it by the time interval, thedistance is essentially equal to the size of the star. Thisimplies that we should not interpret the long-term varia-tions in radial velocity as wave or pulsation amplitudes. If

496 GRAY

we do a similar calculation using a 400 day timescale, thedistance amounts to a small percent of the size of the starand could imply a real change in the size of the star,although no one has reported such changes.

The nature of the disturbance driving the opacity varia-tion has yet to be identied. Pulsations and waves are notstrangers to Betelgeuse research (Dupree et al. 1987 ; Smithet al. 1989 ; Cuntz 1990) and would seem to be likely candi-dates. Detection of radius variations over time or harderlimits on nonvariation could be important constraints pro-vided we rst understand the nature of the opacity varia-tions and how they might aect the apparent radius.Modeling the photosphere and the subphotosphere wouldbe useful for understanding these points and for exploringopacity sources and mechanisms for their change.

9. SUMMARY

High-resolution, high signal-to-noise ratio observationsof the photospheric spectrum reveal several characteristicsabout Betelgeuse and its variations :

1. The broadening of the spectral lines does not vary andis well matched by a Gaussian with a dispersion of 11 kms~1. These characteristics can reasonably be interpreted as acombination of Doppler shifts from rotation and macro-turbulence.

2. The main spectroscopic variation is a scaling of theline depths by D20% across the full cycle, and this occurson a timescale of months, as does the photometric varia-tion ; the lines are shallower when the star is fainter.

3. Curve-of-growth eects and temperature eects areseen as small deviations from the basic scaling in point (2).The photospheric temperature declines when the linedepths decline, but lags behind the line-depth changes byD5 days.

4. The hypothesis of variable continuous opacityexplains all of the above eects.

5. The opacity hypothesis also oers a new explanationof the nature and behavior of the bright spot seen onBetelgeuse : it is an invariant region ; in contrast, the sur-rounding photosphere varies in brightness as the contin-uous opacity changes there, making the spot visible whenthe surroundings are fainter and invisible when the sur-roundings are of similar brightness to the spot.

6. Evidence is presented against interpreting the spot as asingle giant convection cell : (a) the star is fainter when thespot is present, exactly opposite expectations for a hot risingcell ; (b) the spot always appears at essentially the same diskposition, whereas convection cells should appear at randomaround the disk ; and (c) the overall shapes of the spectrallines are essentially invariant while other parameters vary,whereas rising giant convection cells should produce dis-tinct distortions in the spectral lines that come and go withthe cells.

7. Over a span of D^5 km s~1 in the cores of lines, thereis variable structure with amplitude D2% of the line depthoccurring on a timescale of a day. This has been interpretedas the signature of supergranulation, i.e., those large convec-tion cells misinvoked as an explanation of the spot. But inaddition to alternative explanations oered by promi-nences, hot spots, or nonradial oscillations, the reality of thestructure needs corroboration.

I am grateful to the Natural Sciences and EngineeringResearch Council of Canada for continued support of myresearch and the Elgineld Observatory at the University ofWestern Ontario. I wish to thank two enthusiastic andhard-working observing assistants, Christopher Tycner andKevin Brown, for taking some of the observations used inthis paper. The manuscript was improved by commentsfrom A. K. Dupree, R. L. Gilliland, R. E. Griffin, and E. F.Guinan.

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