Measured Diameters of 2 F-stars in the Beta Pic Moving Group
aa r X i v : . [ a s t r o - ph . S R ] S e p Measured Diameters of Two F-stars in the Beta Pic MovingGroup
M. Simon and G.H. Schaefer Received ; accepted Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY [email protected] The CHARA Array of Georgia State University, Mount Wilson Observatory, MountWilson, CA 91023, USA [email protected] 2 –
ABSTRACT
We report angular diameters of HIP 560 and 21547, two F spectral type pre-main sequence members of the β Pic Moving Group. We used the East-West314-m long baseline of the CHARA Array. The measured limb-darkened angulardiameters of HIP 560 and 21547 are 0 . ± .
032 and 0 . ± .
009 mas,respectively. The corresponding stellar radii are 2.1 and 1.6 R ⊙ for HIP 560 andHIP 21547 respectively. These values indicate that the stars are truly young.Analyses using the evolutionary tracks calculated by Siess, Dufour, and Forestiniand the tracks of the Yonsei-Yale group yield consistent results. Analyzing themeasurements on an angular diameter vs color diagram we find that the ages ofthe two stars are indistinguishable; their average value is 13 ± . ± .
02 and 1 . ± .
05 M ⊙ , respectively. However,analysis of the stellar parameters on a Hertzsprung-Russell Diagram yields agesat least 5 MY older. Both stars are rapid rotators. The discrepancy between thetwo types of analyses has a natural explanation in gravitational darkening. Stellaroblateness, however, does not affect our measurements of angular diameters. Subject headings: stars: Pre-Main Sequence—- stars: Beta Pic Moving Group
1. Introduction
The age and mass of a pre-main sequence (PMS) star are usually estimated from itslocation in the Hertzsprung-Russell diagram (HRD) relative to theoretical calculationsof stellar evolution. Unfortunately, differences among the theoretical calculations canproduce mass and age estimates discrepant by factors of 2 to 3 (Hillenbrand and White,2004; Simon, 2006). The consequences are that the mass-spectrum of stars producedin a star-forming region, the region’s star-forming history, and the chronology of planetformation are imprecisely known. The latter point has taken on a pressing importance asastronomers are poised to detect exoplanets associated with young stars.Measurement of the diameters of young stars as they contract to the main sequencecan provide their ages and a new independent test of the theoretical PMS evolutionarytracks (Simon, 2006). Once a reliable set of evolutionary tracks is established, measurementof the diameters of PMS stars will yield measurements of their absolute ages. Remarkably,members of the β Pic Moving Group (BPMG) are young enough (10-20 MY) and nearenough ( <
50 pc; e.g. Torres et al., 2006) that some are resolvable with the optical/IRinterferometric array of the Center for High Angular Resolution Astronomy (CHARA).The brighter members of the BPMG have parallaxes measured by
HIPPARCOS . The agesof individual stars in the BPMG are determined by HRD fitting and the strength of theLi-line. An age for the group as a whole may be estimated dynamically by tracing thegroup’s motion back to a common origin. The age estimates range from 11 . ± . ±
12 MY (see Table 2 in Fernadez et al. 2008; Yee and Jensen, 2010). Whether thespread represents uncertainties of the dating methods or a range in individual formationtimes is unknown. Astronomers already have evidence of planet formation associated withthe members of the BPMG. Lagrange et al (2010) imaged the exoplanet of β Pic, an A-star.Debris disks are understood as indicators of planet formation (Quillen et al. 2007); β Pic 4 –and AU Mic, a BPMG member of M1 spectral type, are known to have debris disks (Smithand Terrile 1984; Lagrange et al 2010; Liu 2004).We measured the angular diameters of two stars in the BPMG , HIP 560 and 21547,using the CHARA Array. § § §
2. Interferometric Measurements of HIP 560 and HIP 21547
We observed at the CHARA Array located on Mt. Wilson, CA, on the nights of UT11, 12, and 13 Sept. 2010. Our observing assignment was made through an allotmentmade available competitively to the astronomical community by CHARA and administeredby the National Optical Astronomical Observatory through its observing time applicationprocess. The array consists of six 1-m telescopes in a Y-configuration on baselines from 34to 331 m (ten Brummelaar et al. 2005). We used the CLASSIC beam combiner operatingin the H and K ′ bands to observe with the two telescopes on the 314 m E1-W1 baseline.At the CHARA Array these bands are centered at 1.673 and 2.133 µm wavelengths, withbandwidths, 0.285 and 0.349 µm respectively.Table 1 lists the properties of HIP 560 and HIP 21547 we need for this paper; theparallactic distances are van Leeuwen et al.’s (1997) revised HIPPARCOS values. Neitherstar is known to be a spectroscopic binary. HIP 21547 is one component of a commonproper motion binary; its companion, GJ 3305, is at about 1 ′ angular separation. Wechecked the near-IR and thermal-IR fluxes of the stars as given by and WISE .Neither shows evidence of excess emision in the near-IR. HIP 560 is not included in the WISE
Preliminary Data Release. For HIP 21547, the
WISE magnitudes at 3.6, 12, and 22 http://irsa.ipac.caltech.edu 5 – µ m are consistent with the K-band magnitude, ∼ . ∼ WISE band 2 at 4.5 µ m; on a color-color diagram this still places thestar in the region of normal stars without a debris disk. Both stars are rapid rotators withvalues of vsin i in the range typical of F stars (Abt and Hunter 1962).The essential observational datum of an interferometer measurement is the fringecontrast or visibility, V , of the target or calibrator. An observation of a program star or itscalibrators took about 10 minutes each and consisted of a series of data scans through thecentral interferometer fringe and a sequence of shutters. Each program star observation wasbracketed by observations of one or two stellar calibrators with angular diameters smallerthan 0.3 mas and located within 10 ◦ of the target (Table 1). An observation of HIP 560 or21547 and their calibrators required about 2 hours to obtain good S/N.The calibrators were chosen to be unresolved by the interferometer, but theirdiameters are nonetheless finite (Table 1). S. Ridgway (priv. comm.) kindly estimatedthe limb-darkened diameters, φ LD , of the calibrators, using Kervella et al’s (2004) angulardiameter fits for main sequence stars as a function of their V-K color, K-band magnitude,the parallactic distance of the star, and an estimate of its extinction (always small, A V ≤ . V cal of the targets, and their uncertainties σ V cal . The σ V cal include the uncertainties of themeasured visibilities of target and calibrators and an assumed ±
10% uncertainty in thecalculated diameters of the calibrators.Figures 1 and 2 show the calibrated visibilities at H and K vs spatial frequency(projected baseline/wavelength). Fig. 1 for HIP 560 indicates a systematic difference in 6 –angular diameters measured at H and K. We do not think this can be attributed to aproperty of the star because HIP 21547, also an F spectral type star, does not show theeffect. The difference is probably measurement error. At angular diameters as small asthose of HIP 560 and HIP 21547 the curves for a uniform disk and limb-darkened diskare nearly indistinguishable . Nonetheless, we did analyze the visibilities with respect tolimb-darkened models following Hanbury-Brown et al.’s (1974) analysis. Their expressionfor the visibility of a limb-darkened star can be written V LD ( x ) = [ 1 − u λ u λ − [ 12 (1 − u λ ) V UD + u λ x ( sinxx − cosx )]where V UD is the visibility of a uniform stellar disk, V UD = 2 J ( x ) x , in which J is the Besselfunction of order 1, and x = πφBλ with φ the star’s angular diameter, B the projectedbaseline during the scan, and λ the wavelength of observation. We used values of the limbdarkening parameter, u λ , derived by Claret et al (1995) appropriate to stars that haveeffective temperatures of 1.5 to 1.7 M ⊙ stars 10 to 20 MY old, u λ = 0.24 and 0.20, at Hand K respectively. We included a ±
10% uncertainty in the values of u λ in the calculationof the uncertainties of the measured stellar diameters.We fit each calibrated visibility to a limb-darkened diameter, averaged the individualvalues, and present the results, φ Diam ( mas ), for HIP 560 and 21547 in Table 3. Theuncertainties are standard deviations of the mean. The uncertainties are dominated by thescatter of the individual visibility measurements. Table 3 also lists Φ Diam the “absolute”angular diameter, the value of φ scaled to a common distance of 10 pc and the correspondingstellar radii. The uncertainties in Φ Diam include the uncertainties in the HIPPARCOS This would not be true for stars more strongly limb-darkened than F-stars (van Belle etal. 2001). 7 –distances (van Leeuven et al. 2007).
3. Analysis and Discussion3.1. Analysis in the Φ vs ( V − K ) and the HR Diagrams Figure 3 shows Φ
Diam compared with diameters predicted by PMS evolution modelscalculated by Siess et al (2000) (SDF) and the Yonsei-Yale (Y2) models calculated by Yi etal (2003). The Y2 calculations provide the luminosity, L , and effective temperature, T eff at 5 MY intervals during the contraction; we calculated the corresponding radii throughthe defining relation L = 4 πR σT eff where σ is the Stefan-Boltzmann constant. The SDFwebsite provides the photospheric radii directly and connects T eff to magnitudes usingKenyon and Hartmann’s (1995) Table A5. To present the Y2 results with ( V − K ) as theindependent variable, we interpolated their T eff to the Kenyon and Hartmann scale. Thediameter measurements indicate, for both stars, ages in the range 10-15 MY and masses 1.6to 1.8 M ⊙ . The left-hand portion of Table 3 lists more precise values obtained by placingthe observed diameters on a finer mesh in the Φ vs ( V − K ) color diagram (henceforth theΦ CD ); the values lie in the columns designated Φ SDF and Φ Y . The age difference betweenHIP 560 and 21547 is not statistically significant and the SDF and Y2 tracks yield valuesthat are in good agreement. The average age of the two stars is 13 ± . ± .
02 M ⊙ and that of HIP 21547, 1 . ± .
05 M ⊙ .If we did not have angular diameter measurement of HIP 560 and 21547, the only wayto estimate their age and mass would be by their positions on an HRD relative theoretical ∼ siess/ 8 –isochrones. Figure 4 shows such an analysis, here on an HRD plotted as M K vs ( V − K ).It is seen that both the SDF and Y2 tracks indicate ages at least 5 MY older, and masses ∼ . ⊙ smaller than those indicated by the Φ CD (the columns designated HRD SDF andHRD Y in Table 4).The observational inputs, angular diameters, distances, and photometry, and hencethe stellar radii, luminosities, and effective temperatures give essentially the same agesand masses whether the SDF or Y2 models are used. The results are however consistentlydiscrepant when interpreted on the Φ CD or the HRD. This suggests that the source of thediscrepancy lies in the application of theoretical calculations of non-rotating stars to starsthat rotate with high angular velocities. The rotation of stars less massive than ∼ ⊙ slows over their lifetimes becausestellar winds powered by the convective outer layers carry away their angular momentum.The convective zone in main-sequence F spectral type stars disappears toward the earlierF spectral type sub-classes and energy transport becomes entirely radiative. This explainsthe rapid decrease of vsin i from F0V to F9V (e.g. Abt and Hunter 1962; Kraft 1967). Theobservation and theoretical analysis of stellar rotation have a rich history (e.g. Tassoul 2000)and its effects are revealed beautifully now that stars can be imaged interferometrically(e.g. Peterson et al. 2006 a,b; Monnier et al. 2007; Zhao et al. 2009; Che et al. 2011).The effective surface gravity, g eff in a rotating star decreases from the pole to theequator. This produces oblateness and a brightness variation with latitude known as gravitydarkening. The oblateness of a rotating star in radiative equilibrium is given by 9 – o = R e − R p R e = 0 . ω πGρ m where R e and R p are the equatorial and polar radii, ω is the angular velocity, and ρ m is themean density (von Zeipel 1924; Chandrasekhar 1933). Inserting numerical values, o = 0 . R p R ⊙ )( M ⊙ M ∗ )( V eq km/s ) where R p is the polar radius of the star, and M ∗ and V eq are its mass and equatorialvelocity. Sackmann (1970) showed that the decrease of R p from its non-rotating valueis only a few percent even when the star is rotating nearly at break-up. At break-up, R eq = 3 / R p , o = 0 .
33 and V eq,bk = 2 GM ∗ / R p ( eg Collins, 1963). It is safe to apply theseresults to HIP 560 and 21547 because Demarque and Roeder (1967) showed that the outerconvective layers disappear at F early spectral type.To make numerical estimates of the expected oblateness for HIP 560 and 21547, weuse values from SDF for the radius and mass of an early F spectral type star at age 13MY, R ∗ = 1 . ⊙ , M ∗ = 1 . ⊙ . We use these values because, in principle, our measured valuescould be affected by oblateness. We also use the vsin i values, 171 and 95 km/s, measuredfor HIP 560 and 21547, respectively (Table 1). Since the inclinations are not known wecannot calculate the equatorial velocities. We can, however, estimate them by calculatingthe expectation values h vsin i i . The upper bound on the inclination is i = π/
2. If the lowerbound on i were 0, h vsin i i = V eq π/
4. Here, however, a lower bound, i crit , is set by thevalue at which vsin i crit would imply rotation at the breakup velocity. For the adoptedradius and mass, V eq,bk = 357 km/s. Hence, sin i crit = (observed v sini)/357 and i crit = 8 . ◦ and 15 . ◦ for HIP 560 and 21547 respectively. Using these values as a lower bound onthe inclination, the expectation values of the equatorial velocities are 202 and 118 km/s,and of the oblateness, 0.104 and 0.036, for HIP 560 and 21547 respectively. These values 10 –could be measured only if the stars’ rotation axes were positioned for best resolution ofthe oblateness with the east-west baseline of the CHARA array. The measured oblatenessvalues are therefore likely to be smaller. In the case of HIP 21547 our observations probablywould not have measured o ≤ .
036 because the precision of the diameter measurementis ∼
2% (Table 3). For HIP 560, in the unlikely case that a ∼
10% oblateness accounts forthe measured value of its angular diameter, a 10% smaller value of Φ would still place itbetween the 10 and 20 MY isochrones. We conclude that stellar oblateness probably doesnot enter into the interpretation of our results .In a rotating star in radiative equilibrium, the radiative flux at a point on thephotosphere is proportional to g eff (von Zeipel 1924). Since g eff decreases from the pole tothe equator, the corresponding decrease in radiative flux can be characterized as a decreasein the local effective temperature and is called gravity darkening. In the A spectral typestars that have been imaged (Peterson et al. 2006a,b; Monnier et al. 2007; Zhao et al.2009), the temperature difference is large, ∼ i = 0) will appear brighter and its photometric color will behotter than one seen equator-on ( i = π/ ⊙ at various values of V eq and i . Typically the valuesof M V vary by a few tenths mag and of ( B − V ) by a few hundreths. Details depend on thestellar mass and inclination; Maeder and Peytremann’s calculations show that for i & ◦ the rotating star appears less bright and redder than a non-rotating star.The effects of gravity darkening suggest that the discrepancies summarized in Table4 are attributable to rotation of HIP 560 and 21547. If their rotational velocities and Also, the on-sky angle of the projected baseline was essentially the same throughout ourobservations (Table 2). Thus baseline rotation during our observations is not an issue. 11 –inclinations are such that their absolute magnitudes at K are depressed by a few tenthsrelative to a non-rotating star and their ( V − K ) are slightly redder, the ages and massesdeduced from the Φ CD and HRD could be in agreement. A detailed test of this hypothesiswill be possible when the inclinations of these stars are measured. This can be accomplishedby either measuring their light fluctuations and thus rotational periods or by interferometricimaging of their photospheric emission of these stars is mapped. The first approach ispossible now as Garcia-Alvarez et al. (2011) demonstrated by measuring the periods andinclinations of two other stars in the BPMG. If we accept the results of the Φ CD analysis and attribute the discrepancy of resultsfrom the HRD to stellar rotation, we conclude that HIP 560 and 21547 are roughly coevalat an average age of about 13 MY. Their galactic (X,Y,Z) coordinates (Table 1) place themabout 49 pc apart with each moving at speed ∼
22 km/s given by their (U,V,W) velocitycomponents. It is too soon to attribute the results for this pair to all the stars in the BPMGbut our results suggest that the members were born together about 13 MY ago, and thusare at the younger end of the age span described in §
4. Summary and Future Directions
1) The measured angular diameters of HIP 560 and HIP 21547 in the H and K bands are0 . ± .
032 and 0 . ± .
009 mas, respectively. Scaled to a common distance 10pc, theirangular diameters are Φ = 1 . ± .
13 and 1 . ± .
03 mas.2) Analyzing these results in Φ vs ( V − K ) diagram with SDF and Y2 isochrones calculatedyields ages of the two stars that are not different at a statistically significant level. Their 12 –average age is 13 ± . ± .
02 M ⊙ forHIP 560 and 1 . ± .
05 M ⊙ for HIP 21547.3) Analyzing the stellar parameters with the SDF and Y2 isochrones in a conventional HRdiagram yields ages at least 5 MY older and masses ∼ . ⊙ smaller.4) HIP 560 and 21547 are rapid rotators (Table 1). The discrepancy in ages and masses canbe accounted for by gravitational darkening of rotating stars in radiative equilibrium. Adetailed test of this hypothesis will be possible when their their inclinations are determinedeither by measuring their rotational periods or by interometric images of their photospheres.5) Taken together, our results suggest that HIP 560 and 21547 formed coevally about 13MY ago.The most pressing task is to determine whether the 13 MY age applies to the BPMGas a whole. Although most BPMG are in the southern hemisphere, at least 3 other F andG spectral type stars are bright and near enough for measurement with the CHARA array.Stars of later spectral type are too faint for observation at the present time unless, by goodluck, they are close to the sun. Many more of the BPMG members will be observable whenthe planned sensitivity improvements at the CHARA array are realized.We thank the referee for a helpful and unusually warm report. We are grateful toSteve Ridgway for advice and help with the observations. We thank Deane Peterson forreminding us that F-stars can rotate rapidly, for advice about stellar rotation, and forpointing out the Garcia-Alvarez paper. We thank the CHARA staff at Mt. Wilson forthorough support. The CHARA Array is operated by Georgia State University’s Center forHigh Angular Resolution Astronomy on Mount Wilson, California. Access to the CHARA 13 –Array, which is operated with funding from the National Science Foundation and GeorgiaState University, was obtained through a competitive TAC process administered by theNational Optical Astronomy Observatory. Our work was supported in part by NSF GrantAST-09-08406. We used data products from the Two Micron All Sky Survey, which is ajoint project of the University of Massachusetts and the Infrared Processing and AnalysisCenter/California Institute of Technology, funded by the National Aeronautics and SpaceAdministration and the National Science Foundation. Our research has also used of theSIMBAD database operated at CDS, Strasbourg, France. 14 – References
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HIP 560 HIP 21547Alternate Name HD 203 HD 29391Spectral Type F3V F0VV(mag) 6.19 5.22K(mag) 5.24 4.54vsin i (km/s) 170.7 95.0Distance (pc) 39 . ± . . ± . φ LD ( mas ) 0.27 0.25HD 223884 φ LD ( mas ) 0.26Data for β Pic targets from Torres et al. (2006)
17 –Table 2. Calibrated Visibilities
Object λ MJD B (m) PA ( ◦ ) V cal σ V cal
HIP 560 K 55450.286 308.62 69.69 0.828 0.018HIP 560 K 55450.300 311.91 71.87 0.783 0.016HIP 560 K 55450.314 313.44 73.94 0.751 0.015HIP 560 K 55450.332 312.66 76.38 0.866 0.017HIP 560 K 55450.347 309.38 78.47 0.863 0.016HIP 560 K 55452.314 313.51 74.74 0.839 0.015HIP 560 K 55452.324 312.91 76.10 0.814 0.015HIP 560 K 55452.334 311.39 77.39 0.856 0.015HIP 560 H 55451.328 312.81 76.22 0.853 0.028HIP 560 H 55451.342 310.18 78.08 0.897 0.025HIP 560 H 55451.355 305.80 79.82 0.893 0.025HIP 560 H 55451.370 298.79 81.72 0.866 0.026HIP 21547 K 55450.515 313.26 75.68 0.849 0.013HIP 21547 K 55450.525 313.44 75.86 0.819 0.012HIP 21547 K 55450.535 312.45 75.96 0.860 0.014HIP 21547 H 55451.501 311.41 75.40 0.769 0.021HIP 21547 H 55451.515 313.41 75.73 0.776 0.022HIP 21547 H 55451.533 312.20 75.97 0.766 0.021HIP 21547 H 55451.544 310.15 76.02 0.775 0.021HIP 21547 H 55452.496 311.11 75.37 0.773 0.022HIP 21547 H 55452.513 313.43 75.73 0.719 0.020HIP 21547 H 55452.521 313.31 75.88 0.791 0.022HIP 21547 H 55452.533 311.69 75.99 0.729 0.020
Table 3. Measured Sizes
HIP 560 HIP 21547 φ ( mas ) 0 . ± .
032 0 . ± . mas ) 1 . ± .
13 1 . ± . R/ R ⊙ . ± .
14 1 . ± .
18 –Table 4. Measured Ages and Masses
Star Φ
SDF Φ Y HRD
SDF
HRD Y Age (MY)HIP 560 12 . ± . ± +25 − ± . ± . ± +30 − ± ⊙ )HIP 560 1 . ± .
03 1 . ± .
02 1 . ± .
05 1 . ± . . ± .
05 1 . ± .
05 1 . ± .
05 1 . ± .
19 –Fig. 1.— Calibrated visibilities of HIP 560 at H and K vs spatial frequency, the projectedbaseline divided by the wavelength. The dashed curves are for uniform disk models (UD)with the diameters indicated and the solid curves are limb-darkened models. 20 –Fig. 2.— Same as Fig. 1 but for HIP 21547. 21 –Fig. 3.— HIP 560 and 21547 angular diameters scaled to 10pc distance vs (V-K) comparedto isochrones at 10, 20, and 50 MY calculated by SDF (top) and Y2 (bottom). The crossesindicate masses 0.4 to 2.0 M ⊙ at 0.1 M ⊙ intervals for the SDF isochrones and 0.5 to 2.0M ⊙ for the Y2 isochrones. 22 –Fig. 4.— HIP 560 and 21547 on a conventional Hetrzsprung-Russell diagram here presentedas M K vsvs