THE ASTRONOMICAL JOURNAL, 118 :1831È1844, 1999 October1999. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

ABSOLUTE DIMENSIONS OF THE A-TYPE ECLIPSING BINARY V364 LACERTAE1GUILLERMO TORRES

Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 ; gtorres=cfa.harvard.edu

CLAUD H. SANDBERG LACY2Department of Physics, University of Arkansas, Fayetteville, AR 72701 ; clacy=comp.uark.edu

ANTONIO CLARET

Instituto de de CSIC, Apdo. Postal 3004, E-18080 Granada, Spain ;Astrof•� sica Andaluc•� a, claret=iaa.es

MAMNUN M. ZAKIROV, G. C. ARZUMANYANTS, N. BAYRAMOV, AND A. S. HOJAEV

High-Altitude Maidanak Observatory, Astronomical Institute, Tashkent 700052, Uzbekistan ; mamnun=astro.gov.uz

ROBERT P. STEFANIK AND DAVID W. LATHAM

Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 ; rstefanik=cfa.harvard.edu, datham=cfa.harvard.edu

AND

JEFFREY A. SABBY2Department of Physics, University of Arkansas, Fayetteville, AR 72701

Received 1999 May 5; accepted 1999 June 15

ABSTRACTWe present photoelectric observations in B and V , as well as spectroscopic observations of the 7.3 day

period double-lined eclipsing binary V364 Lacertae. From the analysis of the light curves and the radialvelocity curves we have determined the absolute dimensions of the components with high precision

The masses for the primary and secondary are and([1%). MA

\ 2.333^ 0.015 M_

MB

\ 2.296^ 0.025respectively, and the radii are and We derive alsoM

_, R

A\ 3.307^ 0.038 R

_R

B\ 2.985^ 0.035 R

_.

e†ective temperatures of K and K, and projected rotational veloci-T effA \ 8250 ^ 150 T effB \ 8500 ^ 150ties of km s~1 and km s~1. Evolutionary tracks from current stellarv

Asin i\ 45 ^ 1 v

Bsin i \ 15 ^ 1

evolution models are in good agreement with the observations for a system age of log t \ 8.792(6.2] 108 yr) and for solar metallicity. Hints of a lower metallicity from spectroscopy and photometryappear to be ruled out by these models, but a deÐnitive comparison must await a more accurate spectro-scopic abundance determination. Analysis of all available eclipse timings along with our radial velocitiesof this moderately eccentric system (e\ 0.2873^ 0.0014) has revealed a small but signiÐcant motion ofthe line of apsides of deg cycle~1, corresponding to an apsidal period ofu5 \ 0.00258^ 0.00033U \ 2810 ^ 360 yr. The contribution from general relativity e†ects is signiÐcant (D17%). A comparisonwith predictions from interior structure models shows the real stars to be less concentrated in mass thanexpected. Our measurements of the projected rotational velocities indicate that the primary star is essen-tially pseudosynchronized (synchronized at periastron), while the secondary is spinning 3 times moreslowly and is not yet synchronized. Both the rotational status of the stars and the nonzero eccentricity ofthe orbit are consistent with the predictions from tidal theory, speciÐcally for the radiative dampingmechanism.Key words : binaries : eclipsing È binaries : spectroscopic È stars : evolution È

stars : fundamental parameters È stars : individual (V364 Lacertae)

1. INTRODUCTION

Detached double-lined eclipsing binaries provide anopportunity to determine the physical properties of starswith high accuracy and high precision, most importantlythe mass and the radius. When those determinations aresuch that the uncertainties are and when supplement-[1%ed with accurate measurements of the metal abundance, theobservations allow for critical tests of stellar evolutiontheory that o†er important insight into issues such as theopacities and the treatment of convection, among others(Andersen 1991, 1998). There are currently a few dozen

ÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈ1 Some of the observations reported in this paper were obtained at the

Multiple Mirror Telescope Observatory, a facility operated jointly by theUniversity of Arizona and the Smithsonian Institution.

2 Visiting astronomer, Kitt Peak National Observatory, NationalOptical Astronomy Observatories, which is operated by the Association ofUniversities for Research in Astronomy, Inc., under cooperative agreementwith the National Science Foundation.

cases suitable for this type of test, distributed across muchof the H-R diagram. Additional information is available ineccentric systems that show apsidal motion. These casesallow one to probe the interior structure of the stars and tocompare Ðndings with predictions from theory regardingthe degree of mass concentration. In favorable systems boththe classical terms of the apsidal motion and the generalrelativistic contributions can be tested. Another aspect oftheory that can be confronted with observations is that oftidal evolution. Tidal forces tends to circularize the orbitsand to synchronize the rotation of the components with theorbital motion, and they are very sensitive to the dimen-sions and the internal structure of the stars. The binarysystem discussed in this paper is particularly interesting inthat it can serve to test each of the above-mentioned areasof theoretical modeling.

V364 Lac (also HD 216429, HIP 112928, BD]37 4713,SAO 72799 ; V \ 8.3È9.0, d \ ]38¡44@45A,a \ 22h52m14s.8,epoch and equinox J2000.0) was discovered as a photo-

1831

1832 TORRES ET AL.

metric variable by Frank (1980) in the course of a photogra-phic program to monitor another eclipsing binary (SWLac). V364 Lac was actually one of the comparison stars.Subsequent photoelectric observations conÐrmed theeclipsing nature of the object, with a period of 7.35 days,and established that the orbit is eccentric and the minimaare of similar depth (D0.7 mag). Double lines in the spectrawere reported by Lacy (1984). The spectral type is A3(Guetter 1976), although there are hints of chemical anom-alies (Levato & Abt 1976) that have led to the occasionalclassiÐcation of the object as a metallic-line A star.

Several times of minima for V364 Lac have appeared inthe literature since, but no complete light curves or radialvelocity measurements have been published. In this paperwe present the results of our intensive spectroscopic andphotometric monitoring of the system, and with those datawe determine the absolute dimensions of both componentswith high precision. We also present an analysis of theeclipse timings, which reveals the small but signiÐcantapsidal motion for the Ðrst time. The stars are considerablyevolved o† the zero-age main sequence (ZAMS), with themore massive component being slightly cooler. We use ourdeterminations to test current models of internal structureand stellar evolution.

In addition, our measurements of the projected rotationalvelocities of V364 Lac A and B show that the hotter and lessmassive star is rotating much more slowly than the othercomponent, by a factor of 3. This makes it a particularlyinteresting system to confront with recent theories of tidalevolution (e.g., Zahn 1989 ; Tassoul 1987).

2. SPECTROSCOPIC OBSERVATIONS AND REDUCTIONS

Spectroscopic observations of V364 Lac were collected ata variety of telescopes over a period of more than 15 years.Observations in 1983 August were obtained on the 2.7 mreÑector at the McDonald Observatory, with the Digicondetector. Those spectra cover a 12 nm window centered at451 nm, with a resolution of 0.04 nm. We also used the 2.1m and telescopes at Kitt Peak National Obser-coude� -feedvatory (KPNO) from 1984 to 1998. The observations from1984 to 1990 cover about 10 nm centered at 450 nm, with aresolution of 0.02 nm (2 pixels). The spectra obtained in1998 cover about 32 nm centered at 445 nm and have aresolution of 0.03 nm.

Rotational velocities (v sin i) were measured from threeKPNO spectra of V364 Lac obtained in 1998 that had highsignal-to-noise ratios. Measured line widths were compared

with the corresponding features in spectra of HR 8404 (21Peg, B9.5 V), for which v sin i \ 4 km s~1 (Fekel 1998,private communication). The spectrograms of HR 8404were synthetically broadened with the rotational proÐle ofGray (1992) for a range of values of v sin i until a match wasfound for the binary star features. The resultant values ofv sin i are 15 ^ 1 km s~1 for the photometric primary(hotter) star in V364 Lac, which is actually the spectro-scopic secondary (less massive) star, and 44 ^ 1 km s~1 forthe photometric secondary (cooler) star. To avoid confu-sion, for the remainder of the paper we will refer to the moremassive (cooler) star as the ““ primary ÏÏ (A).

Radial velocities were determined from the McDonaldObservatory and KPNO spectrograms by cross-correlationwith narrow-lined stars of known radial velocities. Spectro-grams of the narrow-lined stars were Ðrst syntheticallybroadened with the rotational proÐle of Gray (1992) tomatch the width of the appropriate binary star component.The IRAF3 task FXCOR was used in the radial velocityanalysis. The two stars used as velocity standards were HR8641 (o Peg, A1 IV) and HR 8404. Fekel (1998, privatecommunication) has found that HR 8641 is, in fact, a low-amplitude single-lined spectroscopic binary, and he hasdetermined an accurate orbit from which its radial velocitywas computed for the time of each of our observations. Theadopted radial velocity of HR 8404 is ]0.2 km s~1 (Fekel1998, private communication). Table 1 presents the velocitymeasurements based on the McDonald and KPNO spectra.

In addition to these observations, V364 Lac was moni-tored with the Center for Astrophysics (CfA) echellespectrographs from 1995 June through 1998 December. Atotal of 68 spectra were obtained using three essentiallyidentical instruments on the 1.5 m Wyeth reÑector at theOak Ridge Observatory (Harvard, Massachusetts), the 1.5m Tillinghast reÑector at the F. L. Whipple Observatory(Mount Hopkins, Arizona), and the Multiple Mirror Tele-scope (also on Mount Hopkins, Arizona). A single echelleorder centered at about 518.7 nm was recorded usingphoton-counting Reticon detectors, providing a spectralwindow of 4.5 nm and a resolving power of j/*j B 35,000.

The zero point of the CfA velocity system was monitoredby means of sky exposures at dusk and dawn, and small

ÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈ3 IRAF is distributed by the National Optical Astronomy Observa-

tories, which are operated by the Association of Universities for Researchin Astronomy, Inc., under cooperative agreement with the NationalScience Foundation.

TABLE 1

MCDONALD]KPNO RADIAL VELOCITY MEASUREMENTS OF V364 LAC AND RESIDUALS

FROM THE FINAL SPECTROSCOPIC ORBIT

HJD RVA

RVB

(O-C)A

(O-C)B

(2,400,000]) (km s~1) (km s~1) (km s~1) (km s~1) Orbital Phasea

45573.8631 . . . . . . ]62.9 [83.9 ]0.40 ]0.89 0.56945867.9852 . . . . . . ]66.5 [89.1 [0.02 [0.23 0.57746071.5876 . . . . . . [96.5 ]82.9 ]4.64 ]1.35 0.27247777.8691 . . . . . . [98.9 ]81.8 ]3.07 [0.59 0.37048148.8231 . . . . . . ]45.0 [66.0 ]1.35 [0.38 0.83048151.7976 . . . . . . [90.6 ]69.9 ]0.79 [1.74 0.23450938.9516 . . . . . . [99.4 ]84.9 ]4.94 ]0.10 0.35950941.9481 . . . . . . ]62.7 [84.3 ]0.62 ]0.05 0.76750944.9501 . . . . . . [74.0 ]50.6 [1.06 [2.28 0.175

a Computed from epoch of deeper eclipse as given in Table 3.

TABLE 2

CFA RADIAL VELOCITY MEASUREMENTS OF V364 LAC AND RESIDUALS FROM THE FINAL

SPECTROSCOPIC ORBIT

HJD RVAa RV

Ba (O-C)

A(O-C)

B(2,400,000]) (km s~1) (km s~1) (km s~1) (km s~1) Orbital Phaseb

49891.7881 . . . . . . ]20.31 [37.65 ]5.96 [0.32 0.91849903.7940 . . . . . . ]48.76 [74.42 [1.96 [0.12 0.55149909.7494 . . . . . . [108.09 ]84.71 [3.28 ]0.92 0.36149918.7494 . . . . . . ]72.64 [90.78 ]3.06 ]2.69 0.58549940.8192 . . . . . . ]65.97 [95.29 [4.42 [1.00 0.58749947.7698 . . . . . . ]37.07 [57.78 ]0.66 ]1.97 0.53349971.7220 . . . . . . ]49.57 [78.96 [4.95 [0.80 0.79149990.7310 . . . . . . [110.07 ]78.29 [9.51 [1.19 0.37750000.6635 . . . . . . ]71.19 [94.09 ]0.32 ]0.68 0.72850007.6566 . . . . . . ]78.25 [103.51 [0.94 [0.27 0.67950021.6641 . . . . . . ]60.52 [91.89 [8.69 ]1.20 0.58450038.6078 . . . . . . ]24.19 [47.53 ]0.33 [0.53 0.88950056.5936 . . . . . . [112.24 ]86.71 [4.40 [0.16 0.33650079.4917 . . . . . . [46.02 ]27.50 ]4.11 [0.71 0.45050111.4996 . . . . . . ]49.55 [75.61 [1.06 [1.42 0.80450294.9507 . . . . . . ]65.27 [86.09 ]1.77 ]1.20 0.75850358.7234 . . . . . . [65.22 ]44.97 ]1.29 ]0.10 0.43350359.8095 . . . . . . ]68.91 [92.51 ]1.17 [0.91 0.58150374.5725 . . . . . . ]67.71 [96.56 [3.33 [1.61 0.58950379.5352 . . . . . . [105.73 ]78.13 [6.26 [0.23 0.26450389.5596 . . . . . . ]75.43 [105.48 [4.51 [1.48 0.62850407.6278 . . . . . . [41.53 ]22.62 ]1.61 ]1.51 0.08550408.6362 . . . . . . [91.00 ]66.56 [2.47 [0.69 0.22350409.7138 . . . . . . [106.27 ]82.15 [3.45 ]0.38 0.36950412.6764 . . . . . . ]60.64 [81.90 ]0.80 ]1.67 0.77250439.6162 . . . . . . [58.88 ]41.82 ]4.53 ]0.10 0.43750442.6176 . . . . . . ]35.38 [60.41 [2.75 ]1.09 0.84550472.5808 . . . . . . ]18.89 [34.97 ]5.51 ]1.37 0.92150621.9307 . . . . . . [96.78 ]69.69 [4.42 [1.45 0.23650622.9240 . . . . . . [107.34 ]81.10 [5.12 [0.06 0.37150624.9071 . . . . . . ]75.49 [106.18 [5.43 [1.19 0.64150653.8373 . . . . . . ]59.95 [90.92 [5.68 [1.47 0.57650654.8740 . . . . . . ]65.32 [98.15 [7.84 [1.04 0.71750710.6757 . . . . . . [103.10 ]86.07 ]3.69 ]0.26 0.30850712.6419 . . . . . . ]65.71 [89.50 ]0.59 [0.57 0.57550739.6237 . . . . . . [98.27 ]74.29 [3.43 ]0.63 0.24550740.7180 . . . . . . [94.70 ]71.57 [1.46 [0.47 0.39450741.8672 . . . . . . ]46.53 [74.50 [3.82 [0.58 0.55150742.6032 . . . . . . ]80.41 [103.52 [0.67 ]1.63 0.65150743.6726 . . . . . . ]54.94 [75.49 ]1.90 ]1.16 0.79650762.6510 . . . . . . [106.18 ]78.84 [6.02 [0.23 0.37850769.5839 . . . . . . [108.08 ]86.96 [0.39 ]0.24 0.32150771.5727 . . . . . . ]71.51 [94.21 [0.38 ]1.60 0.59150771.6507 . . . . . . ]77.13 [98.47 ]1.92 ]0.72 0.60250772.5848 . . . . . . ]68.69 [94.82 [2.00 [0.23 0.72950794.6751 . . . . . . ]67.99 [92.74 [1.60 ]0.74 0.73450798.6218 . . . . . . [102.65 ]79.94 [1.80 ]0.17 0.27150798.6261 . . . . . . [105.24 ]81.25 [4.26 ]1.35 0.27150798.6305 . . . . . . [105.87 ]80.18 [4.77 ]0.15 0.27250799.6257 . . . . . . [86.60 ]63.50 [0.60 [1.17 0.40750799.6300 . . . . . . [85.91 ]64.35 [0.27 ]0.05 0.40850799.6343 . . . . . . [86.59 ]63.59 [1.32 [0.34 0.40850820.5933 . . . . . . [100.32 ]77.08 [2.01 [0.11 0.25950821.5747 . . . . . . [98.32 ]73.55 [4.39 ]0.82 0.39350821.5808 . . . . . . [98.30 ]72.39 [4.77 ]0.07 0.39450821.5860 . . . . . . [96.44 ]73.35 [3.26 ]1.38 0.39450823.5671 . . . . . . ]81.03 [104.29 ]0.40 ]0.41 0.66450823.5771 . . . . . . ]76.50 [105.70 [4.05 [1.09 0.66550967.9696 . . . . . . [105.73 ]86.24 ]0.88 ]0.62 0.30650969.9444 . . . . . . ]63.19 [88.53 [1.82 ]0.29 0.57550970.9584 . . . . . . ]75.47 [96.89 ]1.40 ]1.14 0.71350997.9169 . . . . . . [102.02 ]78.00 [2.66 [0.25 0.38051058.7938 . . . . . . ]76.13 [104.68 [4.71 ]0.23 0.66151087.7840 . . . . . . ]71.29 [100.27 [4.56 [0.43 0.604

1834 TORRES ET AL. Vol. 118

TABLE 2ÈContinued

HJD RVAa RV

Ba (O-C)

A(O-C)

B(2,400,000]) (km s~1) (km s~1) (km s~1) (km s~1) Orbital Phaseb

51088.6333 . . . . . . ]73.85 [97.35 ]1.14 [0.70 0.72051092.7382 . . . . . . [101.22 ]81.47 ]1.10 ]0.21 0.27851122.6111 . . . . . . [111.36 ]87.98 [4.02 ]1.61 0.34251151.5518 . . . . . . [106.11 ]81.98 [3.77 ]0.70 0.278

a Includes corrections described in the text.b Computed from epoch of deeper eclipse as given in Table 3.

run-to-run corrections were applied as described in moredetail by Latham (1992).

Radial velocities from these spectra were derived with thetwo-dimensional cross-correlation technique TODCOR(Zucker & Mazeh 1994). The one-dimensional correlationfunctions used by this algorithm were computed using theIRAF task XCSAO (Kurtz & Mink 1998). The templateswere selected from an extensive grid of synthetic spectrabased on the latest model atmospheres by Kurucz, calcu-lated speciÐcally for the wavelength region of our obser-vations (Morse & Kurucz 1999). They are available for arange of e†ective temperatures, metallicities, rotationalvelocities, and surface gravities.

To set the template parameters we proceeded by iter-ations, starting with temperatures and surface gravitiesfrom preliminary light curve and velocity curve solutions,and adopting the solar value for the metallicity. We deter-mined v sin i for both components directly from our CfAspectra by testing a large number of template combinationsand seeking the best match to our observations. The resultsfor the primary and secondary are km s~1v

Asin i\ 48^ 2

and km s~1, which are similar to the deter-vB

sin i\ 13^ 2minations above using the KPNO observations. The uvbybobservations from the literature suggest a metal abundancesomewhat lower than solar for V364 Lac (see ° 6). Conse-quently, we experimented with a range of metallicities forour templates with the reasonable assumption that bothcomponents have the same composition. Compared withthe results for the solar abundance, we obtain a slightlyhigher correlation value (averaged over all the exposures)for a metal content of [m/H]\ [0.4, which is consistentwith the photometric determination described later.However, this value is rather sensitive to the e†ective tem-peratures adopted, and may also be a†ected by the rapidrotation of the primary star. We estimate the uncertainty tobe at least 0.3 dex. The radial velocities we derive changevery little with metallicity. In the absence of a more accuratespectroscopic determination, we adopt the solar abundancein the following. The Ðnal temperatures and gravities usedfor our templates are K, K,T effA \ 8250 T effB \ 8500

and from the light curve analysislog gA

\ 3.8, log gB

\ 3.9,below. Synthetic spectra for these parameters were calcu-lated by interpolation in our grid.

For the highest accuracy in the mass determinations, caremust be taken to avoid systematic e†ects that could bias thevelocities, such as those produced by line blending.Although TODCOR is largely immune to this problem,residual e†ects could still be present because of the narrowspectral window of the CfA spectra. Following Latham etal. (1996), we examined our velocities for such e†ects bysimulating binary-star observations using synthetic spectra,and processing these artiÐcial observations with TODCOR

in exactly the same way as the real spectra. The correctionswe derive are small but signiÐcant (¹1.5 km s~1), and wehave therefore applied them to our velocities. The Ðnalvalues are listed in Table 2.

3. APSIDAL MOTION ANALYSIS AND SPECTROSCOPIC

ORBITAL SOLUTION

In order to derive the best possible ephemeris for ourspectroscopic solution, as well as for our light curve solu-tions in ° 5, we collected all available times of eclipse fromthe literature, of which there are 17 photoelectric timingsand 31 older photographic minima. We solved for a linearephemeris, adopting errors for the observed times of eclipseas published, when available. For the timings with nopublished errors we estimated the uncertainties iterativelyfrom the residuals : days for the photoelectricppe\ 0.008minima and days for the photographic minima.pph\ 0.05We obtained

Min I \ HJD 2,449,947.40908(90)] 7.3515258(57)E . (1)

The epoch corresponds to the slightly deeper eclipse ofthe hotter and less massive component (photometricprimary). We maintain this as the reference epoch mainlyfor historical reasons. From our Ðt the eclipse of the coolerstar occurs at phase which shows/II\ 0.5163^ 0.0016,that the orbit is eccentric.

With the velocities from the previous section we derivedindependent spectroscopic solutions for the CfA andMcDonald]KPNO observations in order to comparethem, using the ephemeris given above. Table 3 presentsthose results. Aside from the trivial o†set between the veloc-ity zero points, some of the other elements seem slightlydi†erent. For example, the velocity amplitude of theprimary, is about 2 p smaller in the McDonald]K

A,

KPNO solution. We note, however, that the primaryhappens to be the component with the intrinsically largerradial velocity uncertainties because of the considerablylarger rotational broadening. The minimum masses andmass ratio are comparable within their errors. In view of thesmall number of McDonald]KPNO observations (onlynine), we do not consider these di†erences to be statisticallysigniÐcant, and for the Ðnal solution described below wehave merged the two data sets.

Upon further examination of the timing residuals fromour ephemeris solution we noticed patterns that suggestedslightly di†erent periods for the primary and secondaryminima. This prompted us to solve for apsidal motion, forwhich we used the method described by Lacy (1992). Forthis analysis we relied only on the photoelectric minima,since the photographic timings have a very large scatter bycomparison and contribute negligibly because of their much

-1000 -500 0 500

-0.1

0.0

+0.1

Eclipse cycle number

Eph

emer

is d

evia

tion

(day

s)

No. 4, 1999 ABSOLUTE DIMENSIONS OF V364 LAC 1835

TABLE 3

SPECTROSCOPIC ORBITAL SOLUTIONS FOR V364 LAC

Element CfA McDonald]KPNO Combined

Ps(days)a . . . . . . . . . . . . . . . . . . . . . . . . 7.3515258 7.3515258 7.3515458 ^ 0.0000043

c (km s~1) . . . . . . . . . . . . . . . . . . . . . . [11.54 ^ 0.14 [9.56 ^ 0.38 [11.28 ^ 0.12*RV (km s~1)b . . . . . . . . . . . . . . . . . . . . . . . [0.74 ^ 0.37K

A(km s~1) . . . . . . . . . . . . . . . . . . . . 94.92 ^ 0.49 93.40 ^ 0.49 94.47 ^ 0.49

KB

(km s~1) . . . . . . . . . . . . . . . . . . . . 96.04 ^ 0.15 95.79 ^ 1.01 96.02 ^ 0.14e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.2883 ^ 0.0015 0.2796 ^ 0.0044 0.2873 ^ 0.0014u

A(deg) . . . . . . . . . . . . . . . . . . . . . . . . . 264.38 ^ 0.29 267.25 ^ 0.85 265.265 ^ 0.043

u5 (deg cycle~1) . . . . . . . . . . . . . . . . . . . . . . . 0.00258 ^ 0.00033Min I (HJD[2,400,000)a . . . . . . 49,947.40908 49,947.40908 49,947.41198 ^ 0.00060

Pa(days)c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3515984 ^ 0.0000095

U (yr) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810 ^ 360T (HJD[2,400,000)d . . . . . . . . . . 49,947.348 ^ 0.022 49,947.379 ^ 0.048 49,947.3607 ^ 0.0075aA

sin i (Gm) . . . . . . . . . . . . . . . . . . . . 9.189 ^ 0.016 9.07 ^ 0.12 9.148 ^ 0.049aB

sin i (Gm) . . . . . . . . . . . . . . . . . . . . 9.297 ^ 0.016 9.30 ^ 0.12 9.298 ^ 0.049a sin i (R

_) . . . . . . . . . . . . . . . . . . . . . . 26.559 ^ 0.076 26.38 ^ 0.19 26.502 ^ 0.074

MA sin3 i (M_

) . . . . . . . . . . . . . . . . . 2.342 ^ 0.015 2.311 ^ 0.060 2.332 ^ 0.015MB sin3 i (M

_) . . . . . . . . . . . . . . . . . 2.315 ^ 0.025 2.254 ^ 0.041 2.295 ^ 0.025

q 4 MB/M

A. . . . . . . . . . . . . . . . . . . . 0.9883 ^ 0.0055 0.975 ^ 0.013 0.9838 ^ 0.0054

Nobs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 9 77NminI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9NminII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Time span (yr)e . . . . . . . . . . . . . . . . . 3.4 14.7 19.2pA

(km s~1) . . . . . . . . . . . . . . . . . . . . . 3.44 1.19 3.81/2.64pB

(km s~1) . . . . . . . . . . . . . . . . . . . . . 1.03 2.47 1.00/1.16

a Sidereal period and epoch Ðxed from eq. (1) for the CfA and McDonald]KPNO solutions, and adjusted forthe combined solution.

b Velocity o†set in the sense SCfA[ ÏÏMcDonald]KPNO ÏÏT.c Anomalistic period.d Time of periastron passage.e Includes times of minima for the combined solution.

lower weight. Also, since the eccentricity is only weaklyconstrained by the observations in this case, we adopted anaverage spectroscopic value of e\ 0.28 from theMcDonald]KPNO and CfA solutions in Table 3. For theinclination angle we used i\ 89¡, from preliminary lightcurve solutions. We obtained a relatively small but sta-tistically signiÐcant apsidal motion rate of u5 \ 0.00279

degrees per cycle, corresponding to an apsidal^ 0.00040period of U \ 2600 ^ 600 yr. The ephemeris curve is shownin Figure 1, along with the residuals from the observedprimary and secondary minima.

Over the interval of the CfA radial velocity observations(D3.5 yr) the rotation of the line of apsides amounts toslightly under which is not likely to a†ect very much the0¡.5,spectroscopic solution based on those data. TheMcDonald]KPNO spectra, on the other hand, wereobtained over a period of 14.7 yr, and the expected changein the longitude of periastron u in this case is about 2¡. Toavoid possible biases in the solution, it seems advisable,therefore, to account for apsidal motion in the derivation ofthe spectroscopic elements.

Although rarely considered for this purpose, the radialvelocity measurements contain information on u that canbe used to strengthen the determination of the apsidalmotion rate. This is particularly true in a case such as thatof V364 Lac, in which all the spectroscopic observationsconsidered together span 15.3 yr, which is comparable tothe 17.9 yr coverage for the photoelectric eclipse timings.The velocity observations also contain valuable informa-tion on the period, of course. By the same token, the times

of minima can serve to constrain the orbital eccentricity atsome level, although not as strongly as the velocities do forV364 Lac, and, in addition, they are most useful to deter-mine the period very accurately.

The ideal approach, therefore, is to take advantage of allthe observations in a global least-squares solution including

FIG. 1.ÈEphemeris curve for V364 Lac and photoelectric times ofeclipse for the photometric primary ( Ðlled circles) and secondary (opencircles).

0.0 0.2 0.4 0.6 0.8 1.0

-100

0

+100

Phase

1836 TORRES ET AL. Vol. 118

FIG. 2.ÈRadial velocity curves from our Ðnal spectroscopic solutionthat includes times of minima in the least-squares Ðt. The Ðlled circles arethe CfA observations of the primary, the open circles are for the secon-dary, and the plus signs are the velocity measurements fromMcDonald]KPNO.

TABLE 4

ECLIPSE TIMINGS AND RESIDUALS FOR V364 LAC

HJD (2,400,000]) Eclipsea Epochb (O-C) (day) Reference

44143.4913 . . . . . . . . . 2 [789.5 [0.0212 144257.2789 . . . . . . . . . 1 [774.0 [0.0176 144518.4573 . . . . . . . . . 2 [738.5 ]0.0179 144632.2236 . . . . . . . . . 1 [723.0 [0.0029 144849.2483 . . . . . . . . . 2 [693.5 [0.0090 244867.4795 . . . . . . . . . 1 [691.0 ]0.0027 144878.6625 . . . . . . . . . 2 [689.5 [0.0008 144908.0849 . . . . . . . . . 2 [685.5 ]0.0156 144911.5875 . . . . . . . . . 1 [685.0 ]0.0013 144915.4233 . . . . . . . . . 2 [684.5 ]0.0025 145231.5414 . . . . . . . . . 2 [641.5 ]0.0057 249223.4042 . . . . . . . . . 2 [98.5 [0.0002 349278.4195 . . . . . . . . . 1 [91.0 ]0.0004 349594.5363 . . . . . . . . . 1 [48.0 [0.0003 349947.4106 . . . . . . . . . 1 0.0 [0.0014 450013.5782 . . . . . . . . . 1 ]9.0 ]0.0021 550686.3545 . . . . . . . . . 2 ]100.5 ]0.0001 4

a 1 \ photometric primary ; 2 \ photometric secondary.b Computed from epoch of deeper eclipse as given in Table 3.REFERENCES.È(1) Fernandes & Frank 1981 ; (2) Niarchos 1984 ; (3) Lacy

et al. 1995 ; (4) Lacy et al. 1998 ; (5 ) Agerer & 1996.Hu� bscher

both the radial velocities and the times of eclipse, where thespectroscopic elements and the apsidal motion rate aresolved for simultaneously. In this way the rotation of theline of apsides and its e†ect on the radial velocities areimplicitly accounted for. We have done this for V364 Lac,incorporating both of our radial velocity data sets andsolving also for a zero-point o†set between the two, alongwith the other adjustable quantities. The apsidal motionwas treated again using LacyÏs (1992) method, whichinvolves no approximations. The relative weights of the twovelocity sets were determined by iterations (based on theresiduals), separately for the primary and secondary com-ponents because they have di†erent intrinsic precisions dueto their considerably di†erent rotational velocities. Therelative weights between the photoelectric minima and thevelocities were also determined iteratively, using the photo-metric errors quoted above as a starting point.

For the Ðnal global solution we adopted an inclinationangle of as derived below in ° 5. The elements wei \ 89¡.19,obtain are listed in the last column of Table 3. In addition tothe mean sidereal period (average interval between eclipses)

we give also the anomalistic period which is thePs, P

a,

interval between periastron passages, and the derived timeof periastron passage T . The apsidal motion rate is similarto the preliminary determination mentioned above, and theminimum masses for the components are also not far fromthe values in Ðrst two columns of Table 3. The observedvelocities and computed orbit are shown graphicallyin Figure 2, with di†erent symbols for the CfAand McDonald]KPNO measurements. Tables 1 and 2give the velocity residuals from the Ðt. The photoelectriceclipse timings and corresponding residuals are presentedin Table 4.

4. PHOTOMETRIC OBSERVATIONS

Light curves in the Johnson B and V Ðlters were obtainedwith the 0.6 m telescope at the Maidanak Observatory(Uzbekistan). We collected a total of 1011 measurements inB and 1012 in V , which are given in Tables 5 and 6. Nearbyphotometric standards were selected from the list byLandolt (1983). The comparison star for these observationswas BD]37 4716 (HD 216575, SAO 72816 ; F0, V \ 7.8).We note that BD]37 4716 has been suspected of being avariable star (Kholopov 1982). However, other studiesusing this same star as a comparison Hobart, &(Pen8 a,

1993) show no evidence of variability, nor do theRodr•� guezmeasurements made by the Hipparcos satellite (ESA 1997),

TABLE 5

PHOTOMETRIC OBSERVATIONS OF V364 LAC IN THE B BAND

HJD B HJD B HJD B HJD B(2,400,000]) (mag) (2,400,000]) (mag) (2,400,000]) (mag) (2,400,000]) (mag)

49517.4356 . . . . . . 9.162 49572.4306 9.078 49583.4384 8.772 49591.2855 8.60649517.4446 . . . . . . 9.173 49572.4345 9.084 49583.4429 8.795 49591.2873 8.59849517.4490 . . . . . . 9.172 49572.4392 9.121 49583.4447 8.790 49591.2917 8.59649524.3139 . . . . . . 8.516 49572.4432 9.137 49583.4517 8.803 49591.2937 8.59049524.3178 . . . . . . 8.578 49572.4479 9.164 49583.4534 8.805 49591.2986 8.57749524.3271 . . . . . . 8.566 49572.4518 9.170 49583.4576 8.808 49591.3003 8.57549524.3327 . . . . . . 8.552 49572.4567 9.205 49583.4592 8.816 49591.3046 8.58949524.3374 . . . . . . 8.555 49572.4670 9.247 49584.1590 8.558 49591.3065 8.57549525.3902 . . . . . . 8.567 49572.4708 9.260 49584.1633 8.570 49591.3114 8.56949525.3940 . . . . . . 8.562 49572.4766 9.262 49584.1679 8.571 49591.3129 8.560

NOTE.ÈTable 6 is presented in its entirety in the electronic edition of the Astronomical Journal. A portion isshown here for guidance regarding its form and content.

No. 4, 1999 ABSOLUTE DIMENSIONS OF V364 LAC 1837

TABLE 6

PHOTOMETRIC OBSERVATIONS OF V364 LAC IN THE V BAND

HJD V HJD V HJD V HJD V(2,400,000]) (mag) (2,400,000]) (mag) (2,400,000]) (mag) (2,400,000]) (mag)

49517.4356 . . . . . . 8.912 49572.4345 8.866 49583.4429 8.559 49591.2873 8.34649517.4446 . . . . . . 8.914 49572.4392 8.883 49583.4447 8.555 49591.2917 8.36149517.4490 . . . . . . 8.955 49572.4432 8.918 49583.4517 8.555 49591.2937 8.34249524.3178 . . . . . . 8.323 49572.4479 8.954 49583.4534 8.573 49591.2986 8.33949524.3271 . . . . . . 8.347 49572.4518 8.937 49583.4576 8.586 49591.3003 8.31949524.3327 . . . . . . 8.322 49572.4567 8.992 49583.4592 8.581 49591.3046 8.37349524.3374 . . . . . . 8.288 49572.4670 8.987 49584.1590 8.308 49591.3065 8.34649525.3902 . . . . . . 8.315 49572.4708 9.033 49584.1633 8.355 49591.3114 8.33249525.3940 . . . . . . 8.344 49572.4766 9.051 49584.1679 8.343 49591.3129 8.32849525.3996 . . . . . . 8.323 49572.4819 8.965 49584.1720 8.321 49591.3169 8.359

NOTE.ÈTable 6 is presented in its entirety in the electronic edition of the Astronomical Journal. A portion isshown here for guidance regarding its form and content.

which give a scatter of only 0.0145 mag (rms deviation ofthe individual data points from the epoch photometry).

5. LIGHT CURVE SOLUTIONS

The photoelectric measurements for V364 Lac wereobtained over a period of about 3 years. From our results in° 3, the expected change in the line of apsides in this intervalis only approximately which is small enough that it can0¡.4,be ignored for the purpose of phasing the light curves.Therefore, we have adopted in this section the epoch andalso the sidereal period from eq. (1), which is based oneclipse timings that overlap in time with the B and V photo-metry. This period represents the instantaneous siderealperiod, as opposed to the mean sidereal period given inP

sTable 3. The di†erence between the two, of course, is causedprecisely by the apsidal motion.

The light curves were analyzed with the NDE codeEBOP (Etzel 1981 ; Popper & Etzel 1981). With the lessmassive and hotter star (star B) as the photometric primary,the quantities solved for are the central surface brightness ofthe cooler star in terms of that of the hotter star(J

A) (J

B4

the relative radius of the photometric primary in1), (rB)

terms of the semimajor axis, the ratio of the radii (k 4 rA/r

B),

and the inclination angle (i). The mass ratio was Ðxed to thespectroscopic value (Table 3). Theq@\ M

A/M

B\ 1.016

gravity-brightening coefficient, appropriate for stars withradiative envelopes, was adopted from the work by Claret(1998a). The value we used is b \ 0.98. The linear limb-darkening coefficients, and were taken initially fromu

AuB,

Wade & Rucinski (1985) and were improved by iterativeadjustments during the light curve solutions, always main-taining the initial di†erence between the primary and sec-ondary coefficients until the minimum residuals to the Ðtswere obtained.

Independent solutions were obtained in the B and VÐlters solving for all radiative and geometric quantities. Wethen held the geometric elements Ðxed to the average valuesand solved again for the radiative quantities.

Tests were performed to detect third light, but they didnot yield values signiÐcantly di†erent from zero. However,the Hipparcos mission has found indirect evidence thatV364 Lac does have a close companion and classiÐes theeclipsing system as a ““ variability-induced mover ÏÏ (ESA1997). The photocenter of the visible object has been foundby the satellite to move in phase with the light curve, which

is interpreted as being caused by the presence of an unre-solved companion. The periodic displacement of the photo-center from maximum light to minimum light has beenmeasured to be about 10 mas. From this, a lower limit tothe angular separation can be estimated, and is found to be28 mas. Using our more complete ground-based light curve,we have reÐned this estimate to 36 mas, which correspondsto a minimum period of about 26 yr at the distance of V364Lac (see ° 6), if this companion is physically associated4. Theposition angle measured by Hipparcos is 168¡ ^ 40¡. Theactual separation remains unknown, but is likely to besmall. The relative brightness of the third star cannot beestimated directly from the Hipparcos measurements, butmust also be small since it does not seem to a†ect our lightcurves. We have examined our CfA spectra carefully with anextension of the two-dimensional cross-correlation tech-nique TODCOR to three dimensions (Zucker, Torres, &Mazeh 1995), but we see no evidence of light from anotherstar in the system. We conclude that this close astrometriccompanion is below our detection threshold, and thereforewe assume no third light in the photometric solutions.

The elements we obtain are listed in Table 7, where thelast two columns contain the solutions with the geometryÐxed to the average of the B and V results. In these solu-tions we have solved also for the quantities e cos u ande sin u. The resulting eccentricity, e\ 0.276^ 0.003, isabout 3 p smaller than the spectroscopic value(e\ 0.2873^ 0.0014). The main di†erence is in the quantitye sin u, which is related to the relative widths of the minima.The spectroscopic and photometric values of e cos u, on theother hand, are essentially the same, and this has to do withthe displacement of the secondary eclipse from phase 0.5. Asa test we repeated the calculations with e cos u and e sin uÐxed to the spectroscopic values. The results were verysimilar, and the Ðtted light curves are virtually indistin-guishable. Figures 3 and 4 show the Ðnal Ðts in B and V ,respectively, with the horizontal scale expanded near theprimary and secondary eclipse.

The eclipse of the hotter component (the photometricprimary) is an occultation and is nearly total, with 99.7% of

ÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈ4 Careful examination of the velocity residuals from our spectroscopic

orbital solution shows no evidence of perturbations on the visible stars, butthis does not necessarily rule out the physical association, since the trueperiod of the companion could be very long.

1838 TORRES ET AL. Vol. 118

TABLE 7

LIGHT CURVE SOLUTIONS FOR V364 LAC

GENERAL FITS ADOPTED FITS

PARAMETER B V B V

JA

. . . . . . . . . . . . . 0.910 ^ 0.010 0.917 ^ 0.010 0.918^ 0.004 0.910 ^ 0.004rB

. . . . . . . . . . . . . . 0.1125 ^ 0.0009 0.1128 ^ 0.0010 0.1126^ 0.0007k 4 r

A/r

B. . . . . . 1.109 ^ 0.012 1.106 ^ 0.014 1.108^ 0.009

i (deg) . . . . . . . . . 89.29 ^ 0.10 89.09 ^ 0.10 89.19^ 0.07e cos u . . . . . . . . . 0.0236 ^ 0.0002 0.0238 ^ 0.0002 0.0237^ 0.0002e sin u . . . . . . . . . 0.279 ^ 0.004 0.272 ^ 0.004 0.275^ 0.003uA

. . . . . . . . . . . . . 0.75 0.63 0.75 0.63uB

. . . . . . . . . . . . . 0.70 0.60 0.70 0.60LB

. . . . . . . . . . . . . 0.475 ^ 0.009 0.472 ^ 0.010 0.473^ 0.006 0.474 ^ 0.006p (mag) . . . . . . . . 0.0184 0.0194 0.0184 0.0194

the light of that star blocked at phase 0.0. The eclipse of thelarger and cooler star is partial, and about 84% of its light iseclipsed.

6. ABSOLUTE DIMENSIONS

The photometric properties of the components of V364Lac follow from the results of our light curve solutions andthe absolute photometry for the combined light(V \ 8.342^ 0.004, B[V \ 0.189^ 0.002 ; Lacy 1992).The B[V color di†erence between the stars is determinedmost accurately from the relative surface brightness andJ

Ais *(B[V )\ 0.036 using the calibration by Popper (1980),where the more massive star is redder, consistent with itsmore evolved status. The interstellar reddening was deter-mined from available photometry outsideStro� mgrenof eclipse (b \ 2.875^ 0.005, b [ y \ 0.107^ 0.005,

Crawford 1961 ;m1 \ 0.168^ 0.006, c1 \ 1.061^ 0.009 ;Crawford & Warren 1976). We obtain E(b [ y) \]0.050^ 0.007 using CrawfordÏs (1979) calibration forA-type stars, from which E(B[V )\ ]0.068^ 0.010(Crawford 1973). With this, the individual dereddenedcolors are and(B[V )

A\ 0.138^ 0.010 (B[V )

B\ 0.102

where the error in the reddening contributes most^ 0.010,of the uncertainty.

These indices correspond to e†ective temperatures ofK and K, using theT effA \ 8250 ^ 150 T effB \ 8500 ^ 150

calibrations by Popper (1980) and by Mart•� nez-Roger,Arribas, & Alonso (1992), for solar metallicity. The uncer-tainties given above account for all photometric errors aswell as di†erences between the two color/temperature cali-brations. The spectral types corresponding to these tem-peratures are approximately A4 and A3 for the primary andsecondary, respectively (Gray 1992). Chemical peculiaritiesin the spectrum of V364 Lac were reported by Levato &Abt (1976), who classiÐed it as a metallic-line A star, with acombined type of A2 from the Ca II K line, A7 from theBalmer lines, and A8 based on the metallic lines.

An accurate determination of the metal content of V364Lac is unfortunately not available. Our rough estimate in° 2 suggests that the system may be slightly metal-poor([m/H]\ [0.4^ 0.3). Additional information on metal-licity from uvbyb photometry seems consistent with ourestimate. We obtain which corresponds todm1 \ ]0.032,[m/H]\ [0.6 (Edvardsson et al. 1993), with an uncertaintyof about 0.2 dex based on the photometric errors. We note,however, that the stellar evolution models discussed belowseem to rule out a metal abundance as low as this. We point

out also that the temperatures we derive are somewhat sen-sitive to the adopted metallicity. For example, if we use[m/H]\ [0.5, the calibration by et al.Mart•� nez-Roger(1992) predicts temperatures some 200 K lower than thevalues given above for the solar abundance.

The projected rotational velocities adopted for the com-ponents are the weighted average of our two determinationsin ° 2 : km s~1 and kmv

Asin i \ 45^ 1 v

Bsin i \ 15 ^ 1

s~1.The spectroscopic and photometric solutions in ° 3 and

° 5 lead to absolute dimensions for the components of V364Lac as listed in Table 8. The masses and radii are formallygood to about 1% or better. Of particular interest is the factthat the rotation of the primary star appears to be nearlysynchronized with the orbital motion at periastron (v

erot \

km s~1), whereas the secondary is rotating con-42.9^ 0.5siderably more slowly, at less than half of the pseudo-synchronous value, which is expected to be 38.8^ 0.5 kms~1.

Table 8 includes the luminosities as well as the bolo-metric and absolute visual magnitudes of the components,for which we have used and andMbol_ \ 4.72 BC

A\ ]0.02

(Flower 1996). The distance modulus hasBCB

\ ]0.00been corrected for extinction according to A

V\ 3.1] E(B

The corresponding parallax, n \ 2.2^ 0.1 mas, is[V ).consistent with the less precise value obtained by the Hip-parcos mission mas), which could be(ntrig \ 0.9^ 1.2a†ected to some extent by the presence of the close compan-ion described in the previous section.

TABLE 8

PHYSICAL PARAMETERS OF V364 LAC

Parameter Primary Secondary

Mass (M_

) . . . . . . . . . . 2.333 ^ 0.015 2.296 ^ 0.025Radius (R

_) . . . . . . . . 3.307 ^ 0.038 2.985 ^ 0.035

log g . . . . . . . . . . . . . . . . . 3.767 ^ 0.010 3.849 ^ 0.011v sin i (km s~1) . . . . . . 45^ 1 15 ^ 1vsyncrot (km s~1) . . . . . . . 22.8 ^ 0.3 20.5 ^ 0.2verot (km s~1) . . . . . . . . 42.9 ^ 0.5 38.8 ^ 0.5a (R

_) . . . . . . . . . . . . . . . 26.506^ 0.075

log Teff . . . . . . . . . . . . . . 3.9165 ^ 0.0080 3.9294 ^ 0.0077log L/L

_. . . . . . . . . . . . . 1.657 ^ 0.033 1.620 ^ 0.032

Mbol . . . . . . . . . . . . . . . . . 0.577 ^ 0.083 0.670 ^ 0.081M

V. . . . . . . . . . . . . . . . . . 0.557 ^ 0.086 0.670 ^ 0.083

LB/L

A. . . . . . . . . . . . . . . 0.918^ 0.095

V [MV

. . . . . . . . . . . . . 8.281^ 0.071Dist (pc) . . . . . . . . . . . . . 453^ 15

No. 4, 1999 ABSOLUTE DIMENSIONS OF V364 LAC 1839

FIG. 3.ÈLight curves in the B band along with the Ðtted solution fromEBOP. The middle and bottom panels expand the region near the primaryand secondary minima.

V364 Lac has been listed as a member of the Lac OB1association (Hardie & Seyfert 1959 ; Crawford 1961),although other studies have considered it a nonmemberbased on its distance modulus or on its proper motion(Crawford & Warren 1976 ; Levato & Abt 1976). The binarynature of the object was not known at the time. The radial

FIG. 4.ÈSame as Fig. 3 for the V band

velocity we determine for the center of mass isc\ [11.28^ 0.12 km s~1 (Table 3), in excellent agreementwith the mean for the association of [11.2^ 1.7 km s~1, asreported by MelÏnik & Efremov (1995) on the basis of mea-surements for Ðve of the brighter members. We concludefrom this that V364 Lac is indeed a likely member of theassociation. The distance modulus of Lac OB1 as deter-mined by Crawford & Warren (1976) is V [ M

V\ 8.4

2.0 2.2 2.4 2.62

3

4

5

8.68.8

9.0a

4.10 4.05 4.00 3.95 3.90 3.854.4

4.2

4.0

3.8

3.6

3.4

Log

g

Z = 0.020

Z = 0.004b

1840 TORRES ET AL. Vol. 118

which is similar to the value we derive here for the^ 0.3,eclipsing binary (V [ M

V\ 8.28^ 0.07).

7. DISCUSSION

Our precise determination of the absolute dimensions ofV364 Lac allows us to test a number of aspects of theoreti-cal modeling as indicated in ° 1. We describe in this sectionthe comparison with stellar evolution theory, with the pre-dictions on the internal structure of the stars, and also withthe models for tidal evolution of the axial spin of the com-ponents and of the eccentricity of the orbit.

For this purpose we have computed evolutionary tracksfor the exact masses established above for the components,using the evolutionary code by Claret (1995 ; for furtherdetails on the input physics, see also Claret & Gime� nez1992). In addition to using the solar composition (X

_\

0.700, we have computed models for a repre-Z_

\ 0.020),sentative lower metallicity (X \ 0.744, Z\ 0.004) in view ofthe hint that the abundance of V364 Lac could be subsolar.Convection is treated with the standard mixing-length pre-scription, with a Ðxed mixing-length parameter of 1.52H

pthat provides the best Ðt between a solar model and theobserved properties of the Sun. A moderate amount of coreovershooting is also assumed which has(aov \ 0.20H

p),

been found to give a good representation of the observedproperties of binaries with well established absolute dimen-sions (Claret & 1992, 1993 ; see also Stothers &Gime� nezChin 1991). Tests with a lower value of are describedaovbelow.

7.1. Stellar EvolutionThe three fundamental physical properties of V364 Lac

that we wish to compare with theory are the mass, theradius, and the e†ective temperature of each component.We Ðrst compare the observations with stellar evolutiontheory in the mass/radius diagram, since those two quan-

tities are perhaps the most accurately determined for thissystem and, more importantly, they are independent ofexternal calibrations. Figure 5a illustrates this comparison,where we show the determinations for V364 Lac A and B,with their corresponding error bars, and also three iso-chrones computed for the solar metallicity corresponding tolog t of 8.6, 8.8 and 9.0 (age in yr). The isochrone for log t of8.8 provides a very good Ðt to both components. Based onthe measured values of the radii, the ages we determineseparately for each star from the evolutionary tracks are

and wherelog tA

\ 8.799^ 0.010 log tB

\ 8.786^ 0.015,the uncertainties include the errors in both the radii and themasses. The two determinations di†er by only 3%, and theaverage for the system is log t \ 8.792, corresponding to6.2] 108 yr.

In Figure 5b we show the evolutionary tracks in adiagram of surface gravity versus e†ective temperature,where the abscissae (temperatures) now depend on addi-tional photometry (beyond that provided by our lightcurves) and external color/temperature calibrations. Thesolid curves (one for the primary and one for the secondary)represent the tracks for solar metallicity, and the dashedlines correspond to Z\ 0.004 (or [m/H] \ [0.7). Theuncertainty in the placement of each track that comes fromour mass errors is very small, approximately half of theseparation between the tracks for components A and B.Once again, the agreement is very good for solar composi-tion. This would seem to rule out metallicities as low as ourrough estimates in ° 6 ([m/H]D [0.5). The componentsdi†er by less than 2% in mass, and the di†erence in thee†ective temperatures predicted by the models for theaverage age of the system is 85 K, whereas we determine

K.*Teff \ 250^ 200Calculations with a smaller amount of core overshooting

indicate that the match between the(aov \ 0.10Hp)

observed radii and temperatures of V364 Lac and the theo-retical values for a single age is not as good as with aov \

FIG. 5.ÈComparison with stellar evolution models : (a) mass/radius diagram showing three isochrones corresponding to log t \ 8.6, 8.8, and 9.0. Theprimary and secondary of V364 Lac are indicated, with error bars. (b) plane showing two sets of evolutionary tracks for solar metallicity (solidlog g/ log Tefflines) and Z\ 0.004 (dashed lines). There is good agreement of the observations with the solar-abundance models.

8.0 8.2 8.4 8.6 8.8 9.0-2.8

-2.6

-2.4

-2.2

-2.0

Log age

No. 4, 1999 ABSOLUTE DIMENSIONS OF V364 LAC 1841

This is in agreement with results for other systems0.20Hp.

(see Claret & 1992, 1993).Gime� nezAmong the eclipsing binaries with well-determined absol-

ute dimensions (Andersen 1991 and references therein), atleast three have one component that di†ers in mass by lessthan 0.5% from one of the stars in V364 Lac and thereforeprovide a useful comparison. The most interesting case isthat of V1031 Ori, in which the secondary is virtually identi-cal to V364 Lac B not only in mass, but also in radius and ine†ective temperature (Andersen, Clausen, & Nordstro� m1990). The primary of V624 Her also has the same mass andis only marginally cooler and larger (Popper 1984). WX CepB has essentially the same mass as the primary of V364 Lac(Popper 1987) but is less evolved (smaller and hotter).Nevertheless, it agrees perfectly with the evolutionary trackshown in Figure 5 for the more massive star in V364 Lac. Itis unfortunate that none of these three systems has had itsmetallicity determined accurately, in which case they couldprovide an indirect constraint on the abundance of V364Lac.

7.2. Internal StructureAlthough the relative radii of the components of V364

Lac are not very large (about 12% of the semimajor axis),evidence that tidal forces are in e†ect is seen in the apsidalmotion that we have detected through the analysis of theeclipse timings and the radial velocities in ° 3. The observedrate is deg cycle~1, correspondingu5 \ 0.00258 ^ 0.00033to an apsidal period of about 2800 yr. Together with theabsolute dimensions of each component and the measuredrotation rates, this provides valuable information on thestellar interiors through the internal structure constant k2.In the case of V364 Lac, we Ðnd that the contribution of thetidal distortions to the apsidal motion is about 4 timeslarger than that due to rotational distortions. In addition,the general relativistic contribution to the apsidal motion isfairly signiÐcant, amounting to about 17% of the observedvalue. After correcting for this, as well as for the higherorder terms and we derive an average internal struc-k3 k4,ture constant of log k2obs\[2.42 ^ 0.07.

In Figure 6 we compare this value with the predictionsfrom the models using same the evolutionary tracks foreach component as described above, and assuming solarmetallicity. The decrease in as the stars evolve on thelog k2main sequence reÑects the higher mass concentration as thehelium core grows. The mean theoretical value for thecurrent age of the system is whichlog k2theo\ [2.58^ 0.05,seems signiÐcantly di†erent from the observed value, in thesense that the real stars appear to be less concentrated inmass than expected from the models. Most eclipsingbinaries with accurate absolute dimensions available thatshow apsidal motion are well explained by theory (seeClaret & 1993 ; Claret 1997). There are a fewGime� nezexceptions, but those cases are typically discrepant in theopposite direction, being more concentrated in mass thanpredicted. The F-type eclipsing binary BW Aqr, however, isdiscrepant in the same way as V364 Lac, as shown byClausen (1991) and conÐrmed by Claret (1997). The com-ponents di†er in mass by about 7%, the orbit has a moder-ately high eccentricity of 0.17, and as in V364 Lac, therelativistic contribution is signiÐcant.

There are several possible reasons for the disagreement inthe case of V364 Lac. Perhaps the most important is the factthat the time span of the observations used to derive the

FIG. 6.ÈComparison with interior structure models for solar metal-licity. The data point is based on the measured apsidal motion and theabsolute dimensions of the components. The theoretical curve predicts ahigher mass concentration (longer apsidal period) than is observed for theage of the system (log t \ 8.792).

rate of apsidal motion (times of minima and radialvelocities) is only 0.7% of the apsidal period, and thenumber of eclipse timings is relatively small (see Claret1998b). Also, because of the small number of observations,we Ðnd that is somewhat sensitive to the relative weigh-u5ting of the eclipse timings. For example, by changing theerrors assigned to the times of minima with no publisheduncertainties by factors of 10 relative to those that do havepublished errors, the apsidal motion constant changes byabout 1 p, or roughly half of the di†erence with the models.V364 Lac would certainly beneÐt from additional accuratetimes of minima over the next decade or so to help reducepossible systematic errors.

Claret & (1991) have shown that the internalGime� nezstructure constant depends to a certain extent on theamount of convective core overshooting adopted in themodels. Less overshooting will tend to make the star lessconcentrated in mass, which is apparently the result neededfor V364 Lac. Numerical experiments with show,aov \ 0.10however, that the e†ect for V364 Lac is small. We obtain

which hardly improves the agreementlog k2theo\ [2.55,with the observations.

Finally, the presence of a third star in the system detectedby the Hipparcos mission could conceivably be a†ecting theapsidal motion as well, through long-term modulations ofthe inner orbital elements (e.g., Mazeh & Shaham 1979).

7.3. T idal EvolutionAs mentioned earlier, V364 Lac is interesting in that it

displays a peculiarity regarding the rotational rates of thecomponents. The primary star appears to be synchronizedwith the orbital motion at periastron (pseudosynchronized),while the secondary is rotating some 2.6 times slower thanthe pseudosynchronous velocity and also slower than thesynchronous velocity (mean orbital motion). We point out,however, that this assumes the rotational axes of the two

1842 TORRES ET AL. Vol. 118

stars are aligned with each other, as well as with the axis ofthe orbit. Although this is expected as a result of tidal forces,we cannot rule out, for example, that the tilt of the axis ofthe secondary is di†erent from that of the primary and thusstar B shows a smaller rotational rate in projection.

The subject of tidal evolution theory has undergone alively debate in recent years. Two main mechanisms havebeen proposed to explain the various degrees of synchro-nization and circularization observed in close binaries. Inone of them (Tassoul & Tassoul 1997 and referencestherein), tidal distortions cause large-scale hydrodynamicalcurrents that lead to the rotational braking of the stars andthe circularization of the orbit. Zahn (1992 and referencestherein) proposed instead that turbulent dissipation andradiative damping are responsible for tidal friction in late-type and early-type binaries, respectively, and this tends tosynchronize them and to make the eccentricity smaller.Some aspects of the hydrodynamical theory are still contro-versial, and in general this mechanism has been shown to betoo efficient (see, e.g., Claret & Cunha 1997). On the otherhand, a few cases disagree also with the tidal torque theory,which is somewhat less efficient than required in thosesystems. Thus, further tests such as that provided by V364Lac are very important, and they are particularly inter-esting in asynchronous cases like the one discussed here. Wepoint out also that the di†erential equations that govern theevolution of the rotational rates and the eccentricity accord-ing to these theories are derived under the assumption ofsmall eccentricities and near-synchronous rotations. Theseapproximations will limit the conclusions one can drawfrom the comparison with the observations, but they do notinvalidate them.

We focus here mainly on the tidal torque theory, and thebasic procedure we follow for the comparison is essentiallythe same as described in detail by Claret & Cunha (1997).BrieÑy, we integrate the di†erential equations that describethe evolution of the orbital eccentricity and the axial rota-tion in a binary system until those quantities decay to0.05% of their initial values. The times at which this occursare referred to as the synchronization and circularizationtimes to distinguish them from the usual timescales (q). Thetimescales depend on the moment of inertia, on the tidaltorque constant and on the tidal coefficient (orE2, j2 k2).Each of these time-dependent quantities was computed foreach point along the evolutionary tracks for both com-ponents. For completeness we give below the di†erentialequations for the changes in the eccentricity and the axialrotation. For stars with radiative envelopes the timescalesfor synchronization and circularization are

(qsync)rad~1\ [ 1)orb [ )rot

d)rotdt

, (qsync)rad

\ 2.03bg2M7@3 (1] q)2

q2 E2~1 P17@3R7 (2)

and

(qcirc)rad~1\ [ 1e

dedt

, (qcirc)

\ 1.71] 101M3 (1] q)5@3q

E2~1 P7R9 , (3)

where and are the orbital and rotational angular)orb )rotvelocities, respectively, is the radius of gyration, and thebgother symbols have their usual meaning. In the case of stars

with convective envelopes, the corresponding equations are

(qsync)turb~1 \ [ 1)orb[ )rot

d)rotdt

, (qsync)turb

\ 3.95] 102bg2M7@3 (1] q)2

q2 L~1@3j2~1 P4R16@3 (4)

and

(qcirc)turb~1 \ [ 1e

dedt

, (qcirc)turb

\ 1.99] 103M3 (1] q)5@3q

L~1@3j2~1 P16@3R22@3 , (5)

in which L is the luminosity of each star in solar units. Thequantities and are described in full detail by Claret &E2 j2Cunha (1997). Even for relatively hot stars such as V364 Lacboth sets of equations are used, in general, since convectionin the outermost layers becomes important in the laterstages of evolution. In order to consider the contribution ofboth components to the circularization of the orbit, eqs. (3)and (5) are adjusted to give an e†ective timescale, and thedi†erential equation becomes

[ 1e

dedt

\ 1qcirc,A

] 1qcirc,B

, (6)

where the subscripts A and B refer to the primary andsecondary components.

The integration of these di†erential equations was carriedout using a fourth-order Runge-Kutta algorithm, and thevalues of all time-dependent stellar parameters were inter-polated from the corresponding evolutionary tracks foreach star. As in previous sections we have adopted the solarmetallicity.

The circularization time was found to be log tcirc \ 8.880,consistent with the observed eccentricity and the age of thesystem (log t \ 8.792). The synchronization times for theprimary and secondary are andlog tsyncA \ 8.879 log tsyncB \

respectively. Figure 7 illustrates the predicted tidal8.898,evolution of V364 Lac using ZahnÏs (1989) theory, where thevertical dotted lines and the arrows indicate the relevantstages. We show this in the radius/age diagram, since tidalforces depend on a high power of the radii (see eqs. [2]È[5]).Stage 1 represents the observed radii for the current age oflog t \ 8.792. The primary is expected to become synchro-nized at stage 2 when the star expands to(log tsyncA \ 8.879),about twice its current size as it reaches the base of the giantbranch. Shortly afterward the orbit becomes circular (stage3), and Ðnally, the secondary is predicted to become syn-chronized only after circularization of the orbit is achieved(stage 4), and its own radius has increased by a factor oftwo.

Once again the case of BW Aqr is somewhat similar toV364 Lac regarding tidal evolution, in that the less massivesecondaries in both systems are rotating more slowly thanthe mean orbital velocity (asynchronous rotators). Themore massive star in BW Aqr is essentially synchronizedwith the mean orbital motion, while V364 Lac A has notquite reached that stage, although it does appear to bealready synchronized at periastron.

No. 4, 1999 ABSOLUTE DIMENSIONS OF V364 LAC 1843

FIG. 7.ÈComparison with tidal evolution theory. The solid curves arethe evolutionary tracks for each component from the models described in° 7, and the predicted synchronization and circularization times are rep-resented with vertical lines. The measured radii at the current age of thesystem are indicated by arrow 1. The radiative damping mechanism (Zahn1989) predicts synchronization of the primary with the orbital motion atpoint 2, followed by circularization of the orbit at point 3. Synchronizationof the secondary occurs later (arrow 4). This is consistent with the observedeccentricity and rotational rates (see text).

We conclude from the above that the current eccentricityand the rotational state of the components of V364 Lac arewell explainedÈat least qualitativelyÈby the radiativedamping mechanism, including the asynchronous rotationof the secondary.

On the other hand, numerical experiments with thehydrodynamical theory of Tassoul & Tassoul (1997) follow-ing a similar formalism as above (see Claret, &Gime� nez,Cunha 1995) indicate that this mechanism is too efficient, asmentioned earlier. For example, it predicts that both com-ponents of V364 Lac should already be synchronized, whichis clearly not the case.

8. CONCLUSIONS

Precise determinations of the absolute dimensions ofdouble-lined eclipsing binaries can provide stringent tests oftheoretical models that are important to improve ourunderstanding of the Ðne details of how stars operate. Thisis, in fact, the main purpose of observing these systems,whose study might otherwise be considered to be old fash-ioned or pedestrian. In this paper we have presented newlight curves and radial velocity curves for V364 Lac thatyield absolute masses and radii good to about 1% or better,as well as precise measurements of the projected rotationalvelocities of the components.

The case of V364 Lac is an interesting example of howdi†erent aspects of theory can be tested. We Ðnd that thereis good agreement between the observations and evolution-ary tracks for the speciÐc masses of the components, if solarmetallicity is assumed. The lack of an accurate spectro-scopic abundance for the system is perhaps the weakestpoint in this comparison. Our analysis of the times ofminima combined with the velocities has revealed apsidalmotion in the system, which allows us to probe the interiorstructure of the components. The models predict stars witha somewhat more highly concentrated mass distributionthan observed. This could be explained in part by obser-vational errors stemming from the small time coverage ofthe data compared with the 2800 yr apsidal period or by thepresence of a third star in the system, which was detected byHipparcos. Finally, we Ðnd that tidal evolution theory(speciÐcally, the tidal torque mechanism) is able to explainthe noncircular orbit of V364 Lac given the age of thesystem, as well as the asynchronous rotation of thesecondary.

We thank P. Berlind, M. Calkins, J. Caruso, A. Milone, J.Peters, and J. Zajac, who obtained many of the spectro-scopic observations used here, and R. Davis, who maintainsthe CfA database of radial velocities. We are also grateful toDaryl Willmarth at KPNO for assistance with the spectro-scopic observations there. R. Yu. Ishankulov and V. V.Ignatova obtained many of the photometric observations ofV364 Lac. We thank the referee for a number of helpfulcomments on the manuscript. This research has made use ofthe SIMBAD database, operated at CDS, Strasbourg,France.

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