absolute properties of the eclipsing binary star v396 ... - IOPscience

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The Astronomical Journal, 128:3005–3011, 2004 December # 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A.

ABSOLUTE PROPERTIES OF THE ECLIPSING BINARY STAR V396 CASSIOPEIAE Claud H. Sandberg Lacy1 Department of Physics, University of Arkansas, Fayetteville, AR 72701; [email protected]

Antonio Claret Instituto de Astrofı´sica de Andalucı´a, CSIC, Apartado 3004, E-18080 Granada, Spain; [email protected]

Jeffrey A. Sabby1 Department of Physics, University of Arkansas, Fayetteville, AR 72701; [email protected]

Ben Hood School of Physics and Astronomy, University of St. Andrews, St. Andrews, Fife KY16 9SS, UK; [email protected]

and Florin Secosan Department of Physics, University of Arkansas, Fayetteville, AR 72701; [email protected] Receivved 2004 July 19; accepted 2004 September 8

ABSTRACT We present 6450 differential observations in the V filter measured by a robotic telescope, as well as 28 pairs of radial velocities from high-resolution spectroscopic observations of the detached EA-type, 5.505 day period, double-lined eclipsing binary star V396 Cas. Absolute dimensions of the components are determined with good precision (better than 1% in the masses and radii) for the purpose of testing various aspects of theoretical modeling. We obtain 2:398
0:022 M and 2:592
0:013 R for the hotter, larger, more massive and more luminous photometric primary (star A), and 1:901
0:016 M and 1:779
0:010 R for the cooler, smaller, less massive and less luminous photometric secondary (star B). The effective temperatures and interstellar reddening of the stars are accurately determined from uvby
photometry: 9225
150 K for the primary and 8550
120 K for the secondary, corresponding to spectral types of A1 and A3, and 0.119 mag for Eby . The metallicity of the stars appears to be significantly less than solar. The stars are located at a distance of about 560 pc near the plane of the Galactic disk. The orbits of the stars are apparently circular, and spectral line widths give observed rotational velocities that are not synchronous with the orbital motion for both components. The components of V396 Cas are main-sequence stars with an age of about 420 Myr according to models. Key words: binaries: eclipsing — binaries: spectroscopic — stars: evolution — stars: fundamental parameters — stars: individual ( V396 Cassiopeiae) Online material: machine-readable table

1. INTRODUCTION

2003 September, it had obtained 6450 differential observations in the V filter ( Table 3) with a standard error of 0.008 mag. In this paper we present the analysis of these photometric and spectroscopic data to determine accurate measurements of the absolute properties of this main-sequence binary star system. The results are among the most accurate determinations to date for any eclipsing binary. We then compare our results to those of theory, where we find good agreement with models having an age of about 420 Myr.



The discovery of V396 Cas ( BD +55 2920, HD 240229, SAO 35242, GSC 04006-01219; V ¼ 9:56 10:08, A1 + A3 V) as a variable star is due to Strohmeier (1962); it was originally classified as an Algol type ( EA). The first eclipse ephemeris was given by Strohmeier (1962) in the discovery paper: Min I ¼ 2;425;883:850 þ 11:12576E. We show below that this original orbital period is entirely wrong, beginning in the first decimal place! Another wrong period (15.28 days) is listed by Nha et al. (1991)! The star was little studied until Lacy (1984) announced the discovery of double lines in high-resolution spectra. Lacy continued his spectroscopic program at Kitt Peak National Observatory ( KPNO), sustained after 1998 by Sabby, and Lacy also obtained absolute photometry in the UBV and uvby
systems while he was a visitor at Mount Laguna Observatory in the autumn of 1989. In 2000 mid-November, an automated photometric telescope at Kimpel Observatory on the campus of the University of Arkansas at Fayetteville (the URSA telescope) began operation with V396 Cas as one of its targets. By

2. TIMES OF MINIMUM AND THE ORBITAL PERIOD A number of photographic times of eclipse for V396 Cas have been measured since its discovery (Strohmeier 1962; Strohmeier & Knigge 1962). These dates appear to be mostly spurious, corresponding to predicted times of eclipse based on the initial (wrong) eclipse ephemeris. Beginning in 2000 mid-November, Lacy began photometric observations in V with the URSA robotic telescope at Kimpel Observatory on the University of Arkansas at Fayetteville campus. After a few months, it became clear that the published ephemeris was very wrong. In order to determine the true period, rough radial velocities were measured from 28 spectrograms obtained at Kitt Peak National Observatory and McDonald

1 Visiting Astronomer, Kitt Peak National Observatory, National Optical Astronomy Observatories, operated by the Associated Universities for Research in Astronomy under cooperative agreement with the National Science Foundation.

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Vol. 128

TABLE 1 Dates of Minima of V396 Cas

Year

HJD  2,400,000

Uncertainty (days)

Epocha

Typeb

O  Cc (days)

2001.463.............................. 2001.523.............................. 2001.742.............................. 2002.021.............................. 2002.933.............................. 2003.505.............................. 2003.536.............................. 2003.626.............................. 2003.686..............................

52,078.8571 52,100.8771 52,180.70689 52,282.5588 52,615.63940 52,824.84687 52,835.8586 52,868.8911 52,890.9123

0.0003 0.0003 0.00012 0.0003 0.00020 0.00020 0.0004 0.0004 0.0005

18.4997 14.5001 0 18.5001 79 117 119 125 129

2 2 1 2 1 1 1 1 1

+0.0009 0.0010 0.00003 +0.0001 +0.00021 0.00025 +0.0005 +0.0002 0.0005

a

The epoch number is relative to the mean primary ephemeris: Min I ¼ 2;452;180:70692(12) þ 5:5054718(16)E. 1 = primary eclipse; 2 = secondary eclipse. c Residuals are relative to separate fits to the dates of primary and secondary eclipses: Min I ¼ 2;452;180:70692(12) þ 5:5054718(16)E, Min II ¼ 2;452;078:85621(22) þ 5:5054725(105) E. b

Observatory (see x 3 below). The Macintosh computer application Mac.Period,2 written by Lacy, was then applied to the radial velocities to estimate the spectroscopic period, initially found to be 5.50545(2) days based on an average of the primary and secondary radial velocity results. The uncertainty in the last decimal place is given in parentheses. This correct period is almost exactly 50/101 times the initial published value of Strohmeier. By using the correct period, Lacy was then able to predict and observe successfully both the primary and secondary eclipses. Dates of eclipses were measured by using the method of Kwee & van Woerden (1956) as adapted to a Macintosh computer, and are listed in Table 1. The mean of weighted least-squares solutions for the dates of primary and secondary eclipse gives the following linear ephemeris: Min I (HJD) ¼ 2;452;180:70692(12) þ 5:5054718(16)E: The uncertainties in the last decimal place are given in parentheses. There is no significant difference between the orbital period based on the primary eclipses and that based on the secondary eclipses. The value for the orbital period is consistent with the initial estimate based on the rough radial velocities, but is much more accurate. The linear ephemeris for Min I above was adopted for use in our spectroscopic and light-curve solutions below. The phase of secondary eclipse is nearly half, 0.50009(5), so the orbit is apparently close to circular. 3. SPECTROSCOPIC OBSERVATIONS, REDUCTIONS, AND ORBIT V396 Cas was observed spectroscopically at McDonald Observatory with the 2.7 m reflector in 1983 August, and at KPNO beginning in 1983 December. A variety of gratings and detectors were used with the 2.1 m reflector or the coude´ feed telescope at KPNO and the coude´ spectrometer between then and the last observation in 2001 January. These detectors were CCDs with fixed geometry, manufactured by the RCA, TI, and Loral companies. The resolution (2 pixels) was typically 0.025 nm with coverage between 10 and 32 nm centered near a wavelength of 450 nm. The signal-to-noise ratio was typically 50–100. Wavelength calibration errors contributed less than 0.5 km s1 to the radial velocity uncertainties. The standard error of the radial velocities is shown below to be about 1.5 km s1 2

See http://www.uark.edu/misc/clacy/BinaryStars.

for the photometric primary. A total of 28 spectrograms yielded usable radial velocity pairs. Projected rotational velocities (v sin i ) were determined by comparing line widths of the binary components’ unblended 4481 8 Mg ii line and 4550 8 Fe ii + Ti ii with artificially broadened features in the spectra of comparison stars with known small values of v sin i. These comparison stars were chosen to be of nearly the same spectral type as V396 Cas and were observed with the same instrumental configuration as the binary. The reference stars used were 68 Tau ( HR 1389, A2 IV–V, v sin i ¼ 11 km s1; Fekel 2003) and HR 8404 (21 Peg, B9.5 V, v sin i ¼ 4 km s1; Fekel 1999). From these comparisons in the 28 spectrograms, we find v sin i values of 16
2 and 21
2 km s1 for the photometric primary (star A, the hotter, more luminous star) and secondary (star B) components of V396 Cas, respectively. The photometric primary star A is the hotter star, the one in back during primary eclipse; for V396 Cas, this turns out to be the larger, more massive, and more luminous star. Later, these directly measured v sin i values will be shown to be significantly different from the theoretical circular synchronous values for the components. Line strengths (equivalent widths) of the 4481 and 4550 8 lines were estimated on the 28 high-quality spectrograms of V396 Cas. These lines were the only ones where both binary components could be reliably measured from our spectrograms. The mean line strength ratio of the cooler star to the hotter star was 0:431
0:010 for 4481 8 and 0:615
0:015 for 4550 8. These line strength ratios should be similar to the photometric solution’s luminosity ratio, but perhaps not quite exactly the same since the two components differ slightly in temperature and we are comparing an equivalent width ratio in the blue to a light ratio in the yellow-green. In x 3 below we show that neither of these values agrees with that of the fitted photometric orbit. Both ratios are significantly larger than the photometric solution’s. We speculate that the abundance of heavy elements in the atmosphere of the cooler star is larger than that in the hotter star’s and that there is a different abundance ratio for Mg and for Fe. Radial velocities of the components were carefully measured by cross-correlation with the FXCOR task in IRAF3 using standard stars (68 Tau and HR 8404) with known radial 3 IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation.

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TABLE 2 KPNO Radial Velocities of V396 Cas and Residuals from the Final Spectroscopic Orbit

HJD  2,400,000

RVA ( km s1)

RVB ( km s1)

(O  C )A ( km s1)

(O  C )B ( km s1)

Orbital Phase

45,575.9523................ 45,685.6731................ 45,246.9700................ 46,247.9770................ 46,398.7170................ 50,942.9525................ 51,449.8512................ 51,451.9520................ 51,452.6835................ 51,454.8407................ 51,548.5733................ 51,549.6491................ 51,552.5692................ 51,553.6962................ 51,554.6355................ 51,554.6682................ 51,708.8969................ 51,791.8988................ 51,792.9587................ 51,792.9952................ 51,793.8647................ 51,793.9503................ 51,908.5645................ 51,909.6219................ 51,909.7294................ 51,910.5662................ 51,911.5919................ 51,911.6774................

83.4 93.5 95.7 66.2 +73.5 89.2 97.7 +54.2 +75.4 81.4 90.3 71.4 +39.6 65.7 96.0 95.7 93.4 72.5 +26.9 +30.7 +74.9 +75.7 +25.6 +75.9 +75.4 +23.4 70.5 75.9

+86.4 +101.0 +94.9 +55.3 121.1 +86.2 +98.7 88.5 118.3 +81.9 +87.6 +63.7 73.0 +59.8 +96.6 +95.1 +99.0 +65.0 56.5 60.3 120.0 118.6 56.5 119.4 118.4 54.5 +62.4 +67.8

+3.4 +3.7 0.9 2.1 1.5 0.7 0.4 +1.2 0.3 +1.2 1.0 0.1 +2.4 +0.9 0.4 0.9 0.6 1.8 +0.3 +0.9 0.8 0.3 0.2 +0.1 +1.0 1.6 0.9 0.4

+0.5 +2.3 0.4 2.3 2.3 1.1 0.1 +1.7 +1.2 +2.2 0.7 2.9 1.4 +0.5 0.1 0.8 +5.7 0.8 +0.2 +0.5 0.7 +1.3 0.8 +0.3 0.4 +1.8 0.7 2.8

0.329 0.259 0.211 0.394 0.774 0.177 0.249 0.631 0.764 0.156 0.181 0.376 0.907 0.111 0.282 0.288 0.302 0.378 0.570 0.577 0.735 0.751 0.569 0.761 0.780 0.932 0.119 0.134

velocities as templates (these velocities are taken from Fekel 1999). Standard star spectra were synthetically broadened by convolution with the broadening function of Gray (1992, p. 374) to match the appropriate binary star component before crosscorrelation with the binary star spectra. Adopted KPNO velocities and residuals from the final spectroscopic orbit are listed in Table 2. A spectroscopic orbit was fitted by using the algorithm of Daniels (1966). The orbit and the observations are shown in Figure 1. Some of the spectroscopic orbital elements were consistent with, but of lower precision than, the photomet-

rically derived values (x 4 below), so for the final spectroscopic orbit, these values were fixed at these photometrically determined values. In particular, the spectroscopic orbits produced values of eccentricities that were not significantly different from zero (mean of 0:003
0:005), while the initial photometric orbital fit ( below) yielded a value of 0:007
0:003, marginally different from zero. As discussed below, further investigation showed that the photometric eccentricity actually is not significantly different from zero, so a circular orbit is assumed to be the case. We have adopted this photometric eccentricity value of zero in the final spectroscopic fit. 4. PHOTOMETRIC OBSERVATIONS AND ORBITAL SOLUTION

Fig. 1.—Spectroscopic observations and adopted orbit for V396 Cas. The open circles correspond to observations of the photometric primary ( by definition, the hotter star). Phase 0 is primary eclipse.

Absolute photometry on both the UBV and uvby
photometric systems was obtained by Lacy (1992, 2002b). The components of V396 Cas are early A type, so both UBV and uvby
techniques were used to estimate the reddening and intrinsic color of the stars. The absolute photometric observations were obtained while Lacy was visiting Mount Laguna Observatory in the autumn of 1989. Details of the observing program are reported elsewhere ( Lacy 1992, 2002b). Six sets of observations of V396 Cas on the standard UBV system were made ( Lacy 1992). The value of the reddeningfree parameter Q ¼ (U  B)  0:72(B  V ) was calculated and compared with the standard stars of Johnson & Morgan (1953). The value matched for a mean spectral type of A2, with an intrinsic color index of (B  V )0 ¼ 0:06
0:03, which, through the calibration of Flower (1996), corresponds to an effective temperature of 8815
330 K.

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Five sets of observations of V396 Cas on the standard uvby
system were made. Mean values and estimated standard errors outside eclipse are as follows: V ¼ 9:555
0:007, b  y ¼ 0:168
0:006, m1 ¼ 0:105
0:009, c1 ¼ 1:088
0:012, and
¼ 2:892
0:008. The values of the standard errors are estimated from those of standard stars observed on the same nights. The value of V is an average with that of Lacy (1992). Applying the prescriptions of Crawford (1978, 1979), we find that the intrinsic color index is (b  y)0 ¼ 0:041
0:005 and Eby ¼ 0:127
0:005, which corresponds to EBV ¼ 0:171
0:007 and (B  V )0 ¼ 0:024
0:008. We adopt an average value from the two different determinations: Eby ¼ 0:119
0:008, EBV ¼ 0:160
0:010, and (B  V )0 ¼ 0:035
0:011. The rather large color excess is due to the fact that the star is only 3 from the Galactic plane and is located at a large distance of about 560 pc (see below). According to Table 1 of Popper (1980), the adopted (B  V )0 color index corresponds to a mean spectral type A2 star with an effective temperature of 9080
120 K, in fair agreement with the result above from Flower’s (1996) independent calibration, but with higher accuracy. As a result of the analysis of the differential photometry in V below, we know that the secondary surface brightness value strongly constrains the difference in surface brightness parameters of the two components to be
FV ¼ 0:026, which, through Table 1 of Popper (1980), leads to
log T ¼ 0:033 and log Tp ¼ 3:965
0:007 (Tp ¼ 9225
150 K), log Ts ¼ 3:932
0:006 (Ts ¼ 8550
120 K). We adopt spectral types of A1 V + A3 V. The uvby
indices can be used to estimate the metallicity of stellar atmospheres with the calibration of Stro¨mgren (1966), ½ Fe=H ¼ 0:3  12m1 , which gives for V396 Cas a value ½ Fe=H ¼ 0:34, about half the solar metallicity in the surface layers, at least. This result is consistent with the metal content adopted in the theoretical stellar models (see x 5). It is also consistent with the fact that the measured equivalent widths, converted to intrinsic equivalent widths by using the photometric light ratio, are smaller than for comparable nearby stars. For the primary these values are 0.188 and 0.149 8 for the k4481 and k4550 lines, respectively, and 0.219 and 0.249 8 for the secondary. These equivalent width values are significantly weaker than those of field stars of the same type, as measured by Lacy (2002a). Kimpel Observatory4 consists of a Meade 10 inch f /6.3 LX-200 telescope with a Santa Barbara Instruments Group ST8 CCD camera ( binned 2 ; 2 to produce 765 ; 510 pixel images with 2B3 square pixels) inside a Technical Innovations RoboDome, and controlled automatically by an Apple Macintosh G4 computer. The observatory is located on top of Kimpel Hall on the Fayetteville campus, with the control room directly beneath the observatory inside the building. Sixty-second exposures through a Bessel V filter (2.0 mm of GG 495 and 3.0 mm of BG 39) were read out and downloaded by ImageGrabber (camera control software written by Sabby) to the control computer over a 30 s interval, then the next exposure was begun. The observing cadence was therefore about 90 s per observation. The variable star would frequently be monitored continuously for 4–8 hr. V396 Cas was observed on 55 nights during parts of three observing seasons from 2000 November 14 to 2003 September 8. The images were analyzed by a virtual measuring engine application written by Lacy that flat-fielded the images, automatically located the variable and comparison stars in the 4

See http://ursa.uark.edu.

Vol. 128 TABLE 3 Differential Magnitudes for V396 Cas in V Orbital Phase


V

HJD  2,400,000

0.20706................. 0.20727................. 0.20749................. 0.20769................. 0.20791.................

0.355 0.355 0.350 0.360 0.357

51,862.52954 51,862.53070 51,862.53186 51,862.53301 51,862.53418

Note.—Table 3 is presented in its entirety in the electronic edition of the Astronomical Journal. A portion is shown here for guidance regarding its form and content.

image, measured their brightnesses, subtracted the corresponding sky brightness, and corrected for the differences in air mass between the stars. Extinction coefficients were determined nightly from the comparison star measurements. They averaged 0.25 mag per air mass. The three comparison stars were GSC 04006-01337 (10.4 mag), GSC 04006-01296 (11.4 mag), and GSC 04006-01110 (11.2 mag). All comparison stars are within 80 of the variable star. The comparison star magnitude differences were constant at the level of 0.014 mag for the standard deviation of the magnitude differences of the first two comparison stars, and 0.008 mag for the standard deviation of the nightly means. An effective comparison star magnitude was formed from the sum of all the comparison star intensities. The resulting 6450 V magnitude differences (variable minus comparison stars) are listed in Table 3, and normal points are shown in Figures 2, 3, and 4. The light-curve fitting was done with the NDE model as implemented in the code EBOP ( Etzel 1981; Popper & Etzel 1981), and the ephemeris adopted is that of x 2. The main adjustable parameters are the relative surface brightness of the secondary star (Js) in units of that of the primary, the relative radius of the primary (rA) in units of the separation, the ratio of radii (k ¼ rB =rA ), the inclination of the orbit (i), and the geometric factors e cos ! and e sin !, which account for the orbital eccentricity. Auxiliary parameters needed in the analysis include the gravity-brightening coefficient, which we adopt as 1.00 for the primary and 0.97 for the secondary based on their temperatures (Claret 1998). Linear limb-darkening coefficients (u) were included as variables to be fitted, but were constrained

Fig. 2.—Differential light curve of V396 Cas in the V filter. Sixteen-point normals are shown. The solid curve is the fitted photometric model. Residuals from the fitted orbit have a standard deviation of 0.004 mag.

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Fig. 3.—Differential light curve of V396 Cas in the V filter near primary eclipse. Sixteen-point normals are shown. The solid curve is the fitted photometric model.

to differ by 0.05, based on the theoretical values of Dı´azCordove´s et al. (1995). The mass ratio (q ¼ MB =MA ¼ 0:7928) was adopted from the spectroscopic analysis in x 3. Other adjusted parameters were the magnitude at quadrature and the phase of primary eclipse. The amount of ‘‘reflected light’’ was calculated from bolometric theory. The secondary eclipse is a total eclipse according to the model. The fitting procedure converged to a solution, but detailed examination of the fit showed that the light ratio was significantly different from the spectroscopic equivalent width ratios in x 3. This solution is listed in Table 3 and shown in Figures 3, 4, and 5. Because the values of the equivalent width ratios for the two lines measured do not agree with each other, we have chosen to rely on the ratio as determined by the light-curve model. A test for third light gave a value not significantly different from zero, and there is no sign of it in the images or spectroscopy, so we assume that there is none. We have examined residuals (O  C ) from the fit and find a significant trend between the outside-eclipse levels before and after secondary eclipse (see Fig. 4). The observed light level just before the secondary eclipse is slightly fainter

Fig. 4.—Differential light curve of V396 Cas in the V filter near secondary eclipse. Sixteen-point normals are shown. The solid curve is the fitted photometric model.

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Fig. 5.—Theoretical evolutionary tracks (solid curves) for the components of V396 Cas and the observed values of log g and log TeA (shown with error bars). The dashed curves on either side of the tracks correspond to varying the assumed masses by 1 . The best fit occurs at an age of about 420 Myr for both components.

( by 0.007 mag) than the fitted model, and slightly brighter (0.003 mag) after the eclipse. An examination of the original data values shows that the depression before secondary eclipse in the normal points of Figure 4 is the result of a relatively small number of significantly fainter observations in this region of the light curve. We can find no a priori reasons to reject these fainter observations, but believe that the values are anomalous because of some unknown problem with our measurement techniques. We consider this explanation more likely than the alternative, that there is some intrinsic variation in the eclipsing binary star itself. This type of observational problem is not unique to this binary and can be seen, for example, in the photoelectric observations of YY Sgr ( Lacy 1993). We were concerned that this asymmetry would have a significant effect on the fitted orbital parameters, so we tried a number of numerical experiments to test various ideas, such as checking the sensitivity of the solution to slightly different values of limbdarkening coefficients. An unexpected finding was that the eccentricity parameters were rather sensitive to the assumed limb darkening, and in most of these cases the fitted eccentricity values were not significantly different from zero according to the estimated parameter uncertainties. Other parameters were not sensitive to such variations. For this reason we have adopted circular orbits for the final orbit fittings. We also tried excluding the apparently anomalous points in the phase region 0.460–0.475. The fitted orbital parameters were not changed significantly by this omission, although some of the error estimates were larger, as one might expect. Torres et al. (2000) have discussed the problem of the correlation between the fitted parameters rA , the radius of the primary star, and k, the ratio of radii, and the proper determination of the true uncertainty in the secondary radius, rB . We have used these methods in estimating the uncertainty of the secondary radius. The EBOP program does not directly give the correlation matrix among parameters. Popper (1984) found that accurate information about the limb-darkening coefficients in cases similar to this requires data sets containing a couple of hundred observations at our precision within minima. Our data set contains a few thousand observations in eclipse, so it is reasonable to expect the limbdarkening coefficients to be well-determined by the fit, as is

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TABLE 4 Orbital Elements of V396 Cas Element

Star A

Adjusted quantities:  ( km s1) ..................................... K ( km s1)..................................... e...................................................... Js (star B) ....................................... r ...................................................... k...................................................... i (deg)............................................. u...................................................... e cos ! ........................................... e sin ! ............................................ LB /LA in V...................................... Derived quantities: a sin i (R)..................................... M sin3 i (M) ................................. q = MB /MA ..................................... Other quantities pertaining to the fit: P (days).......................................... Teff ( K) ...........................................  ( km s1) .....................................  (mag) .......................................... Nobs , spectroscopic......................... Nobs , photometric ........................... Time span, spectroscopic (yr)........ Time span, photometric (yr) ..........

indicated by the formal errors of the fit. The fitted limbdarkening coefficients of 0:45
0:07 and 0:50
0:07 may be compared to theoretical values for solar composition by AlNaimy (1978) of 0.45 and 0.51, by Wade & Rucinski (1985) of 0.435 and 0.491, by Dı´az-Cordove´s et al. (1995) of 0.528 and 0.577, and by Claret (2000), as calculated for our particular value of metallicity, of 0.540 and 0.581. The fact that the fitted values of the limb-darkening coefficients are somewhat smaller than some of the theoretical values cited might point to a problem with the theories, or it might be the result of the surface abundance anomalies implied by the line strength ratios discussed in x 3. If the values of the limb-darkening coefficients are fixed at those derived from Claret (2000), however, the fit within primary eclipse during the annular phases is clearly not acceptable.

Star B

10.6
0.3 86.7
0.4

10.6
0.4 109.3
0.5 0.000
0.006 0.801
0.011 0.1215
0.0005 0.0834
0.0004 0.687
0.005 88.82
0.07 0.45
0.07 0.50
0.07 0.00000
0.00006 0.0000
0.0030 0.368
0.007 21.33
0.11 2.396
0.022 1.900
0.016 0.793
0.005 5.5054718
0.0000016 9225
150 8550
120 1.47 1.86 0.008405 28 6450 17.35 2.81

dimensions and masses for V396 Cas shown in Table 5. Table 1 of Popper (1980) has been used for the radiative quantities. He adopted the bolometric corrections of Hayes (1978). The absolute visual magnitudes have been calculated from equation (2) of Popper (1980), based on the radii and visual surface brightness parameters. The absolute bolometric magnitudes are computed from the radii and effective temperatures, adopting the bolometric absolute magnitude of the Sun as 4.75. We have assumed a normal reddening law (Av ¼ 4:3Eby ), estimating the color excess from the Stro¨mgren and Johnson photometry given above. The masses are determined to an accuracy of 0.9%

5. ABSOLUTE DIMENSIONS The combination of the spectroscopic orbital elements and light-curve orbital elements in Table 4 leads to the absolute TABLE 5 Absolute Properties of V396 Cas Parameter Mass (M)............... Radius (R)............. log g (cm s2).......... vsync (km s1) .......... Semimajor axis (R) log Teff ..................... log L (L)................ Mbol (mag)............... MV (mag)................. Distance (pc)...........

Star A 2.398 2.592 3.990 23.8







3.965 1.642 0.65 0.67







Star B

0.022 1.901
0.016 0.013 1.779
0.010 0.004 4.216
0.005 0.1 16.4
0.1 21.33
0.11 0.007 3.932
0.006 0.028 1.183
0.024 0.06 1.79
0.06 0.05 1.74
0.05 560
23

Fig. 6.—Theoretical evolutionary tracks (solid curves) for radii of the components of V396 Cas. The dashed curves on either side of the tracks correspond to varying the assumed masses by 1 . The pairs of horizontal lines represent the range of radii actually measured (
1 ). The upper pair of horizontal lines corresponds to the hotter, more massive star, which evolves more quickly. The best fit occurs in an overlap region at a log age of about 8.62 (420 Myr) for both components.

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and the radii are good to about 0.6%. The observed values of v sin i for both the primary and secondary appear to be not synchronous with the mean orbital motion. The distance that we derive for the system corresponds to a parallax of  ¼ 1:80
0:07 mas, but the star is not listed in the Hipparcos catalog. 6. DISCUSSION The absolute dimensions of the components of V396 Cas (masses and radii) are formally determined to better than 1% accuracy. It is of interest, therefore, to compare them with predictions from recent models of stellar evolution. Claret has produced a number of evolutionary models of stars on the basis of an updated version of the code described by Claret (1995), with the exact masses determined above, for a variety of chemical compositions in order to try to fit the components of V396 Cas. The best fit was obtained for a chemical composition of Z ¼ 0:020, X ¼ 0:70, and with a mixing-length parameter of  ¼ 1:68 and moderate core-overshooting, 0.2Hp . The fit to the evolutionary tracks is shown in Figures 5 and 6. Both components match the theory at a log age of 8:62
0:04 (420
40 Myr). A comparison with the predictions of tidal theory was made by computing the circularization time using the damping formalism by Zahn (1992 and references therein) as described by Claret & Cunha (1997). According to this model the system is expected to achieve orbital circularization at an age of 770 Myr (log age = 8.89), which is not consistent with the observations since the evolutionary age that we derive for the system, which has a circular orbit, is younger, as seen above. The theory by Zahn predicts that synchronization of the rotation of star A with the orbital motion also occurs at an age of 770 Myr ( log age = 8.89), while star B will synchronize its rotation later at an age of 1.3 Gyr ( log age = 9.1). These estimates for the times of rotational synchronization are consistent with the observations, given that the values of v sin i that we measure (mean values of 16
2 and 21
2 km s1) are not the same as the expected circular synchronous values (see Table 5). The hydrodynamic theory by Tassoul & Tassoul (1997 and references therein) often gives predictions for the circularization and synchronization times that are much shorter than the other tidal theory, suggesting that it may be too efficient (e.g.,

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Claret et al. 1995). In this case the difference is more than an order of magnitude for the synchronization times, but is almost identical for the circularization time (log age = 8.87). The Tassouls’ synchronization times are at a log age = 7.40 (star A) and 7.41 (star B). The circularization time of the Tassouls’ theory is not consistent with the observations for V396 Cas, and the synchronization times are both too short since the observed rotation rates are still asynchronous with the orbital motion. 7. CONCLUSIONS New photometric and spectroscopic observations of the eclipsing binary V396 Cas combined with a reanalysis of data from the literature have allowed us to derive definitive orbital parameters and physical properties of the component stars. Our determinations have formal errors smaller than 1% in the masses and radii. V396 Cas thus joins the elite of stars with wellmeasured absolute properties. At an age of about 420 Myr according to models, the system is a middle-aged main-sequence pair of stars with a somewhat lower surface abundance of heavy elements compared to the solar composition. The two currently favored mechanisms that describe the tidal evolution of binary star properties disagree in their prediction of whether the rotations of the V396 Cas stars should be synchronous with their orbits, but agree that the stars’ orbits should be eccentric, which they are not. Our own estimate of the age of this system would seem to favor the damping formalism of Zahn (1992) because of the asynchronous rotations observed in this particular case, but this same theory also predicts that the orbits should be eccentric, which they are not.

We are grateful to Daryl Willmarth at KPNO for assistance with the spectroscopic observations there. The authors wish to thank the anonymous referee for his or her useful suggestions, which improved the quality of this paper significantly. This research has made use of the SIMBAD database, operated at the CDS, Strasbourg, France, and of NASA’s Astrophysics Data System Bibliographic Services. This research was partly supported by a grant to C. H. S. L. from the National Science Foundation.

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