NPOI Measurements of Ten Stellar Oscillators
Ellyn K. Baines, J. Thomas Armstrong, Henrique R. Schmitt, James A. Benson, R. T. Zavala, Gerard T. van Belle
aa r X i v : . [ a s t r o - ph . S R ] J a n NPOI Measurements of Ten Stellar Oscillators
Ellyn K. Baines, J. Thomas Armstrong, Henrique R. Schmitt
Remote Sensing Division, Naval Research Laboratory, 4555 Overlook Avenue SW,Washington, DC 20375 [email protected]
James A. Benson, R. T. Zavala
U.S. Naval Observatory, Flagstaff Station, AZ 86001
Gerard T. van Belle
Lowell Observatory, Flagstaff, AZ 86001
ABSTRACT
Using the Navy Precision Optical Interferometer, we measured the angulardiameters of 10 stars that have previously measured solar-like oscillations. Oursample covered a range of evolutionary stages but focused on evolved subgiantand giant stars. We combined our angular diameters with
Hipparcos parallaxesto determine the stars’ physical radii, and used photometry from the literatureto calculate their bolometric fluxes, luminosities, and effective temperatures. Wethen used our results to test the scaling relations used by asteroseismology groupsto calculate radii and found good agreement between the radii measured here andthe radii predicted by stellar oscillation studies. The precision of the relations isnot as well constrained for giant stars as it is for less evolved stars.
Subject headings: visible: stars, stars: fundamental parameters, techniques: in-terferometric)
1. Introduction
Asteroseismology, the study of stellar oscillations, is a powerful tool to infer informa-tion about stellar structure with minimal model dependence (see, e.g., Brown & Gilliland1994; Christensen-Dalsgaard 2004). The frequencies of the observed oscillations depend onthe sound speed inside the star, which in turn is dependent on properties of the interior 2 –such as density, temperature, and gas motion (Carrier et al. 2010). The number of starsobserved using asteroseismology and the quality of the data have increased dramatically inrecent years, thanks to the photometric space missions
MOST ( Microvariability and Oscilla-tions of STars,
Walker et al. 2003),
CoRoT ( Convection, Rotation, and planetary Transits,
Baglin et al. 2006; Auvergne et al. 2009), and
Kepler (Borucki et al. 2010; Koch et al. 2010).The resulting stellar parameters are key to the statistical analysis of fundamental stellarproperties and for testing stellar interior and evolutionary models (see, e.g., Chaplin et al.2011).Interferometry has the potential to make important contributions to asteroseismology,in part through the determinations of the targets’ sizes (Cunha et al. 2007). Using interfer-ometry, we can measure the angular diameters of stars with resolutions down to tenths of amilliarcsecond (see, e.g., Huber et al. 2012a; Baines et al. 2012). Once we know the appar-ent diameter of a star as well as its distance from parallax measurements, we can calculateits physical size. Then we can test the relationships used to derive stellar properties fromasteroseismology observations by comparing the radii estimated using the asteroseismologyrelations to those measured interferometrically.Huber et al. (2012b) presented interferometric diameters of 10 stars that had oscillationmeasurements from
CoRoT and
Kepler . They found an agreement between asteroseismicand interferometric radii of .
4% for dwarf stars and ∼
13% for giant stars. Their sampleincluded five dwarf, one subgiant, and four giant stars. Here we focus on the more evolvedstars: one dwarf, four subgiant, and five giant stars.We observed these stars using the Navy Precision Optical Interferometer (NPOI) inorder to measure their angular diameters. We then calculated their radii and effective tem-peratures, and used spectral energy distribution fits to determine their bolometric fluxes andluminosities. Section 2 discusses the NPOI and our observing process; Section 3 describesthe visibility measurements and how we calculated various stellar parameters; Section 4 ex-plores the relationship between radii determined using asteroseismology observations andradii measured interferometrically; and Section 5 summarizes our findings.
2. Interferometric Observations
The NPOI is an interferometer located on Anderson Mesa, AZ, and consists of twonested arrays: the four stations of the astrometric array (AC, AE, AW, and AN, whichstand for astrometric center, east, west, and north, respectively) and the six stations of theimaging array, of which two stations are currently in operation (E6 and W7) and three more 3 –will be coming online in the near future (E7, E10, and W10). The current baselines, i.e., thedistances between the stations, range from 16 to 79 m, and our maximum baseline will be432 m when the E10 and W10 stations are completed within the next year. We use a 12-cmregion of the 50-cm siderostats and observe in 16 spectral channels spanning 550 to 850 nmsimultaneously (Armstrong et al. 1998).Each observation consisted of a 30–second coherent (on the fringe) scan in which thefringe contrast was measured every 2 ms, paired with an incoherent (off the fringe) scan usedto estimate the additive bias affecting the visibility measurements (Hummel et al. 2003).Scans were taken on five baselines simultaneously. Each coherent scan was averaged to 1–second data points, and then to a single 30–second average. The dispersion of 1–secondpoints provided an estimate of the internal uncertainties.The target list was derived from the sample of stars with stellar oscillations that werebright enough to observing using the NPOI, which has a magnitude limit of V = 6 .
5. Theyalso had to be resolved with the longest existing baseline, which gives a resolution limitof approximately 1 milliarcsecond (mas). This resulted in a list of 10 targets with stellaroscillation observations available to observe using the NPOI.We interleaved data scans of the 10 asteroseismic targets with one to three calibratorstars for each target. Our calibrators are stars that are significantly less resolved on thebaselines used than the targets. This meant that uncertainties in the calibrator’s diameterdid not affect the target’s diameter calculation as much as if the calibrator star had asubstantial angular size on the sky. The calibrator and target scans were measured as closein time and space as possible, which allowed us to convert instrumental target and calibratorvisibilities to calibrated visibilities for the target. Preference was given to calibrators within10 ◦ of the target stars, as was the case for 13 of the 16 calibrator stars used. On rareoccasions, no suitable calibrator stars were within that angular distance so we resorted tostars that were more distant, with a maximum separation of 17 ◦ .We estimated the calibrator stars’ sizes by constructing their spectral energy distribution(SED) fits using photometric values published in Ljunggren & Oja (1965), McClure & Forrester(1981), Olsen (1993), Jasevicius et al. (1990), Golay (1972), H¨aggkvist & Oja (1970), Kornilov et al.(1991), Eggen (1968), Johnson et al. (1966), Cutri et al. (2003), and Gezari et al. (1993)as well as spectrophotometry from Glushneva et al. (1983), Glushneva et al. (1998), andKharitonov et al. (1997) obtained via the interface created by Mermilliod et al. (1997). Theassigned uncertainties for the 2MASS infrared measurements are as reported in Cutri et al.(2003), and an uncertainty of 0.05 mag was assigned to the optical measurements. We de-termined the best fit stellar spectral template to the photometry from the flux-calibratedstellar spectral atlas of Pickles (1998) using the χ minimization technique (Press et al. 1992; 4 –Wall & Jenkins 2003). The resulting calibrator angular diameter estimates are listed in Table1.
3. Results3.1. Angular Diameter Measurement
Interferometric diameter measurements use V , the square of the fringe visibility. For apoint source, V is unity, while for a uniformly-illuminated disk, V = [2 J ( x ) /x ] , where J is the Bessel function of the first order, x = πBθ UD λ − , B is the projected baseline towardthe star’s position, θ UD is the apparent uniform disk angular diameter of the star, and λ isthe effective wavelength of the observation (Shao & Colavita 1992). θ UD results are listed inTable 2. Our data files in OIFITS format are available upon request.A more realistic model of a star’s disk includes limb darkening (LD). If a linear LDcoefficient µ λ is used, V = (cid:18) − µ λ µ λ (cid:19) − × " (1 − µ λ ) J ( x LD ) x LD + µ λ (cid:16) π (cid:17) / J / ( x LD ) x / . (1)where x LD = πBθ LD λ − (Hanbury Brown et al. 1974). We used effective temperature ( T eff )and surface gravity (log g ) values from the literature with a microturbulent velocity of 2 kms − and to obtain µ λ from Claret & Bloemen (2011). These values and the resulting θ LD arelisted in Table 2. Figures 1 and 2 show the θ LD fits for all the stars. The two stars withthe largest percent uncertainties in the θ LD fit (2%) are HD 146791 and HD 181907. This isbecause their visibility curves are less well sampled with respect to spatial frequency thanthe other targets.Seven of the 10 stars measured here had previous interferometric diameter measure-ments. They are listed in Table 3 and are plotted against our values in Figure 3. In allcases but one, the uncertainty on our diameter measurement is smaller than those from theliterature, and they all agree to within 3- σ .The uncertainty for the θ LD fit was derived using the method described in Tycner et al.(2010), who showed that a non-linear least-squares method does not sufficiently account foratmospheric effects on time scales shorter than the window between target and calibratorobservations. They describe a bootstrap Monte Carlo method that treats the observationsas groups of data points because the NPOI collects data in scans consisting of 16 channels 5 –simultaneously. They discovered that when the 16 data points were analyzed individually,a single scan’s deviation from the trend had a large impact on the resulting diameter anduncertainty calculation. On the other hand, when they preserved the inherent structure ofthe observational data by using the groups of 16 channels instead of individual data points,the uncertainty on the angular diameter was larger and more realistic. This method makes noassumptions about underlying uncertainties due to atmospheric effects, which are applicableto all stars observed using ground–based instruments. It should be noted that the number ofcalibrator stars used in the observations have no apparent effect on the θ LD fit uncertainty. For each star, the parallax from van Leeuwen (2007) was converted into a distance,which we then combined with our measured θ LD to calculate the linear radius ( R ). In orderto determine the luminosity ( L ) and T eff , we constructed each star’s SED using the sourcesand technique of fitting spectral templates to observed photometry as described in Section2. The resulting SED gave us the bolometric flux ( F BOL ) and allowed for the calculation ofextinction A V with the wavelength-dependent reddening relations of Cardelli et al. (1989).We combined our F BOL values with the stars’ distances to estimate L using L =4 πd F BOL . We also combined the F BOL with θ LD to determine each star’s effective tem-perature by inverting the relation, F BOL = 14 θ σT , (2)where σ is the Stefan-Boltzmann constant and θ LD is in radians.Because µ λ is chosen based on a given T eff , we checked to see if µ λ and therefore θ LD would change based on our new T eff . In most cases, µ λ changed by 0.0 or 0.01, and thelargest difference was 0.08 for HD 181907. The resulting θ LD values changed at most by 1%,and all but three changed by 0.2% or less. This was well within the uncertainties on θ LD ,and re-calculating T eff with the new θ LD made at most a 26 K difference (for HD 181907,which has an uncertainty of 199 K). These values all converged after this one iteration, andthese are the final numbers listed in Table 2. For every “scan,” 30 seconds of data are collected in each of the 16 wavelength channels with a mea-surement once every 2 milliseconds. During the processing described in Section 2, all the 30-second-scan’sdata points are averaged into one data point for each channel, so we go from 30 seconds of data per channelto one averaged data point per channel.
4. Discussion
Two scaling equations relate observed asteroseismic quantities to fundamental stellarparameters: ∆ ν ∝ M R − , (3)where ∆ ν is the large separation of oscillation modes of the same degree and consecutiveorders and M is the mass of the star (Ulrich 1986), and ν max ∝ M R − T − eff , (4)where ν max is the frequency of maximum oscillation power (Brown et al. 1991; Kjeldsen & Bedding1995). These equations are often used to calculate stellar radii and masses from oscillationobservations. However, when R is measured interferometrically, we can test the relationsthemselves.We used ∆ ν and ν max from the references listed in Table 4 and assumed uncertaintiesof 1% in ∆ ν and 3% in ν max (Huber et al. 2012b) when no uncertainties were provided inthe references. We combined the frequency measurements with effective temperatures fromthe literature ( T lit ) to calculate R from the asteroseismic measurements alone. It shouldbe noted that the T lit used has little impact on ν max . A variation of 100 K causes a 0.9%change in ν max , which is typically on the order of or smaller than the uncertainties in T lit (Huber et al. 2012b).Table 4 and Figure 4 show the results comparing the radii calculated from asteroseismol-ogy ( R a ) and those measured using interferometry ( R i ). The stars with the largest differencebetween the two are HD 153210, HD 161797, and HD 168723. For the latter two stars, pre-viously published angular diameters agree with our measurement to 1% or less. HD 153210has not been previously measured, and our diameter agrees with that predicted by the SEDwithin 2- σ . For the remaining stars, R a and R i agree within 3- σ ( ∼ σ ( ∼ σ (1 to 2%).Barban et al. (2004) quote a range for ν max between 80 and 170 µ Hz for HD 168723,and Stello et al. (2009) lists a value of 130 µ Hz. If we use 130 µ Hz in our calculation for R a ,the result is 12.08 ± R ⊙ , which is approximately twice the value of R i = 5.92 ± R ⊙ .However, if we use the lower end of the range, i.e., 80 µ Hz, R a is 7.43 ± R ⊙ , which is stilla 26% difference from the one presented here. We believe further asteroseismic observations 7 –of this star would be particularly interesting.Some stars have more evenly and completely sampled data along the visibility curvesthan others; for example, HD 146791 and HD 181907 do not have as wide a range of mea-surements as a function of spatial frequency that other stars such as HD 121370 and HD153210 display. We considered whether or not this would have an effect on the scatter inFigure 4 but the stars with sparsely sampled curves do not correspond to the outliers, sothat is not the issue. In general, we do not observe any systematic trend as a function of size.The residual scatter in the giant stars is comparable to what Huber et al. (2012b) found, andshows the relationships between observed ∆ ν and ν max and stellar radii are not as precisefor evolved giant stars as they are for dwarf stars.The scaling relation for ν max is considered to be less robust than the relation for ∆ ν (Huber et al. 2012b), so we wanted to test it. We combined Equation (3) with our R i to calculate stellar masses, and then combined the masses with our new T eff to calculate ν max values and compare them to the measured values. Table 4 lists and Figure 5 showsthe results. The largest outliers are again HD 153210, HD 161797, and HD 168723 dueto the discrepancies in calculated radii described above. HD 150680 also shows a 16%difference between the observed and calculated ν max value. Our angular diameter for thisstar matches those measured by Nordgren et al. (2001) and Mozurkewich et al. (2003) towithin 3- σ , and it only differs from the SED estimate by 3%. We note that the mass listedin Kallinger et al. (2009) is 1.19 M ⊙ while the mass determined by Marti´c et al. (2001) is1.3 to 1.5 M ⊙ . The latter agrees with the mass determined using our interferometric radiusmeasurement: 1.33 ± M ⊙ . In general we observe good agreement between the observedand calculated ν max within the uncertainty bars with no systematics with respect to stellarsize or evolutionary status.
5. Summary
We measured the angular diameters of 10 stars using the NPOI. The combination ofthese observations with other information from the literature allowed us to calculate thestars’ R , T eff , F BOL , and L . We compared our interferometric radius R i values to thosedetermined from asteroseismic scaling relations and found good agreement between the two,particularly for the less evolved stars. Then we also used ∆ ν from the literature and our R i to calculate the stars’ masses and ν max to put that scaling relation to the test as well. Again,the results agreed to within a few σ in general.The relations work best for main-sequence stars and have limited precision for giant 8 –stars. Hopefully future observations with planned spacecraft such as Gaia (Perryman 2003)and
TESS (Transiting Exoplanet Survey Satellite, Ricker et al. 2009) as well as plannedupgrades to existing interferometers such as the addition of longer baselines on the NPOI(increased resolution), new hardware (increased magnitude limit), and the eventual additionof large telescopes to the array (see, e.g., Armstrong et al. 2013) will lead to significantimprovements when combining data from two techniques.The Navy Precision Optical Interferometer is a joint project of the Naval ResearchLaboratory and the U.S. Naval Observatory, in cooperation with Lowell Observatory, and isfunded by the Office of Naval Research and the Oceanographer of the Navy. This research hasmade use of the SIMBAD database, operated at CDS, Strasbourg, France. This publicationmakes use of data products from the Two Micron All Sky Survey, which is a joint project ofthe University of Massachusetts and the Infrared Processing and Analysis Center/CaliforniaInstitute of Technology, funded by the National Aeronautics and Space Administration andthe National Science Foundation.
REFERENCES
Absil, O., Defr`ere, D., Coud´e du Foresto, V., et al. 2013, A&A, 555, A104Ammons, S. M., Robinson, S. E., Strader, J., et al. 2006, ApJ, 638, 1004Armstrong, J. T., Mozurkewich, D., Rickard, L. J, et al. 1998, ApJ, 496, 550Armstrong, J. T., Hutter, D. J., Baines, E. K., et al. 2013, Journal of Astronomical Instru-mentation, in pressAuvergne, M., Bodin, P., Boisnard, L., et al. 2009, A&A, 506Baglin, A., Michel, E., Auvergne, M., & COROT Team 2006, Proceedings of SOHO18/GONG 2006/HELAS I, Beyond the spherical Sun, 624Baines, E. K., White, R. J., Huber, D., et al. 2012, ApJ, 761, 57Barban, C., De Ridder, J., Mazumdar, A., et al. 2004, SOHO 14 Helio- and Asteroseismology:Towards a Golden Future, 559, 113Barban, C., Matthews, J. M., De Ridder, J., et al. 2007, A&A, 468, 1033Bonanno, A., Benatti, S., Claudi, R., et al. 2008, ApJ, 676, 1248 9 –Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science, 327, 977Brown, T. M., Gilliland, R. L., Noyes, R. W., & Ramsey, L. W. 1991, ApJ, 368, 599Brown, T. M., & Gilliland, R. L. 1994, ARA&A, 32, 37Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245Carrier, F., Eggenberger, P., & Bouchy, F. 2005, A&A, 434, 1085Carrier, F., De Ridder, J., Baudin, F., et al. 2010, A&A, 509, A73Chaplin, W. J., Kjeldsen, H., Christensen-Dalsgaard, J., et al. 2011, Science, 332, 213Christensen-Dalsgaard, J. 2004, Sol. Phys., 220, 137Claret, A., & Bloemen, S. 2011, A&A, 529, A75Corsaro, E., Grundahl, F., Leccia, S., et al. 2012, A&A, 537, A9Cunha, M. S., Aerts, C., Christensen-Dalsgaard, J., et al. 2007, A&A Rev., 14, 217Cutri, R. M., et al. 2003, The IRSA 2MASS All-Sky Point Source Catalog, NASA/IPACInfrared Science Archivedi Folco, E., Absil, O., Augereau, J.-C., et al. 2007, A&A, 475, 243Eggen, O. J. 1968, London, H.M.S.O., 1968Gezari, D. Y., Schmitz, M., Pitts, P. S., & Mead, J. M. 1993, UnknownGlushneva, I. N., Doroshenko, V. T., Fetisova, T. S., et al. 1983, Trudy GosudarstvennogoAstronomicheskogo Instituta, 53, 50Glushneva, I. N., Doroshenko, V. T., Fetisova, T. S., et al. 1998, VizieR Online Data Catalog,3207, 0Golay, M. 1972, Vistas in Astronomy, 14, 13H¨aggkvist, L., & Oja, T. 1970, A&AS, 1, 199Hanbury Brown, R., Davis, J., Lake, R. J. W., & Thompson, R. J. 1974, MNRAS, 167, 475Huber, D., Ireland, M. J., Bedding, T. R., et al. 2012a, MNRAS, 423, L16Huber, D., Ireland, M. J., Bedding, T. R., et al. 2012b, ApJ, 760, 32 10 –Hummel, C. A., Benson, J. A., Hutter, D. J., et al. 2003, AJ, 125, 2630Jasevicius, V., Kuriliene, G., Strazdaite, V., et al. 1990, Vilnius Astronomijos ObservatorijosBiuletenis, 85, 50Johnson, H. L., Mitchell, R. I., Iriarte, B., & Wisniewski, W. Z. 1966, Communications ofthe Lunar and Planetary Laboratory, 4, 99Kallinger, T., Weiss, W. W., De Ridder, J., Hekker, S., & Barban, C. 2009, The EighthPacific Rim Conference on Stellar Astrophysics: A Tribute to Kam-Ching Leung,404, 307Kharitonov, A. V., Tereshchenko, V. M., & Knyazeva, L. N. 1997, VizieR Online DataCatalog, 3202, 0Kjeldsen, H., & Bedding, T. R. 1995, A&A, 293, 87Koch, D. G., Borucki, W. J., Basri, G., et al. 2010, ApJ, 713, L79Kornilov, V. G., Volkov, I. M., Zakharov, A. I., et al. 1991, Trudy Gosudarstvennogo Astro-nomicheskogo Instituta, 63, 1Ljunggren, B., & Oja, T. 1965, Arkiv for Astronomi, 3, 439Marti´c, M., Lebrun, J. C., Schmitt, J., Appourchaux, T., & Bertaux, J. L. 2001, SOHO10/GONG 2000 Workshop: Helio- and Asteroseismology at the Dawn of the Millen-nium, 464, 431Mazumdar, A., M´erand, A., Demarque, P., et al. 2009, A&A, 503, 521McClure, R. D., & Forrester, W. T. 1981, Publications of the Dominion Astrophysical Ob-servatory Victoria, 15, 439M´erand, A., Kervella, P., Barban, C., et al. 2010, A&A, 517, A64Mermilliod, J.-C., Mermilliod, M., & Hauck, B. 1997, A&AS, 124, 349Morel, T., & Miglio, A. 2012, MNRAS, 419, L34Mozurkewich, D., Armstrong, J. T., Hindsley, R. B., et al. 2003, AJ, 126, 2502Nordgren, T. E., Germain, M. E., Benson, J. A., et al. 1999, AJ, 118, 3032Nordgren, T. E., Sudol, J. J., & Mozurkewich, D. 2001, AJ, 122, 2707 11 –Olsen, E. H. 1993, A&AS, 102, 89Perryman, M. A. C. 2003, GAIA Spectroscopy: Science and Technology, 298, 3Pickles, A. J. 1998, PASP, 110, 863Pijpers, F. P., Teixeira, T. C., Garcia, P. J., et al. 2003, A&A, 406, L15Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, Numerical recipesin C. The art of scientific computing (Cambridge: University Press, c1992, 2nd ed.)Prugniel, P., Soubiran, C., Koleva, M., & Le Borgne, D. 2007, arXiv:astro-ph/0703658Prugniel, P., Vauglin, I., & Koleva, M. 2011, A&A, 531, A165Ricker, G. R., Latham, D. W., Vanderspek, R. K., et al. 2009, American AstronomicalSociety Meeting Abstracts
This preprint was prepared with the AAS L A TEX macros v5.2.
12 –Table 1. Observing Log and Calibrator Stars’ Angular Diameters.Target Other Calibrator Date Baselines θ LD , cal HD Name HD (UT) Used † Obs (mas)10700 τ Cet 11171 2005/09/17 AC-W7, AE-W7 19 0.682 ± η Boo 122408 2012/05/10 AW-E6, E6-W7 126 0.521 ± ǫ Oph 141513 2010/03/21 AW-W7 30 0.524 ± ζ Her 156164 2005/08/18 AE-W7, E6-W7 50 0.887 ± κ Oph 148112 2013/02/23 AC-AE, AC-AW, AC-E6 18 0.443 ± ± ± θ LD , cal HD Name HD (UT) Used † Obs (mas)161797 µ Her 166014 2010/08/21 AE-AW, AW-E6 270 0.596 ± ξ Dra 159541 2013/04/27 AE-AW, AW-E6 18 0.520 ± ± ± η Ser 161868 2004/05/20 AC-AE, AC-AW 90 0.668 ± ± ± ± θ LD , cal HD Name HD (UT) Used † Obs (mas)2013/05/31 AC-AW, AW-E6 162013/06/03 AW-E6 212013/06/05 AW-E6 10188512 β Aql 195810 2007/05/26 AN-AW, AW-W7 14 0.394 ± † The maximum baseline lengths are AC-AE 18.9 m, AC-AW 22.2 m, AC-E6 34.4m, AC-W7 51.3 m, AE-AW 37.5 m, AE-W7 64.2 m, AN-AW 38.2 m, AN-E6 45.6 m, AW-E653.3 m, AW-W7 29.5 m, and E6-W7 79.4 m. The θ LD , cal estimates were determined using thetechnique described in Section 2. Table 2. Stellar Parameters.
Target Spectral Parallax θ UD θ LD σ LD R linear L F
BOL T eff σ Teff
HD Type (mas) µ λ (mas) (mas) (%) Cals ( R ⊙ ) ( L ⊙ ) (10 − erg s − cm − ) (K) %10700 G8.5 V 273.96 ± ± ± ± ± ± ±
13 0.2121370 G0 IV 87.75 ± ± ± ± ± ± ±
18 0.3146791 G9.5 III 30.64 ± ± ± ± ± ± ±
55 1.1150680 G0 IV 93.32 ± ± ± ± ± ± ±
19 0.3153210 K2 III 35.66 ± ± ± ± ± ± ±
24 0.5161797 G5 IV 123.33 ± ± ± ± ± ± ±
16 0.3163588 K2 III 28.98 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±
199 3.8188512 G9.5 IV 78.00 ± ± ± ± ± ± ±
11 0.2Note. — The parallaxes are from van Leeuwen (2007); the µ λ coefficients are from Claret & Bloemen (2011) in the R -band with a microturbulent velocity of 2 kms − . The sources of T eff and log g used to determine µ λ were the following: Prugniel et al. (2011) for HD 10700, HD 121370, HD 161797, HD 168723, and HD 188512;Wu et al. (2011) for HD 146791 and HD 163588; Prugniel et al. (2007) for HD 150680; Morel & Miglio (2012) for HD 153210; and Ammons et al. (2006) for HD 181907.
16 –Table 3. Angular Diameter Comparison.Target θ LD , thiswork θ LD , SED % θ LD , previous %HD (mas) (mas) diff (mas) Reference diff10700 2.072 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Table 4. Comparing Interferometric and Asteroseismic Results.
Target T i T lit R i R a % ∆ ν ν max M ν max , calc %HD (K) (K) ( R ⊙ ) ( R ⊙ ) diff ( µ Hz) ( µ Hz) Reference ( M ⊙ ) ( µ Hz) diff10700 5301 ±
13 5348 ±
45 0.81 ± ± ± ±
108 10121370 6128 ±
18 5967 ±
45 2.61 ± ± ± ± ±
37 10146791 4907 ±
55 4918 ±
28 10.40 ± ± ± ± ± ±
19 5758 ±
81 2.61 ± ± a Kallinger et al. (2009) 1.34 ± ±
21 16153210 4637 ±
24 4559 ±
116 11.02 ± ± ± ± ±
16 5454 ±
35 1.71 ± ± ± ± ±
22 21163588 4451 ± ±
25 11.56 ± ± ± ± ± ±
63 5.92 ± ± ± ± ±
199 5637 ±
228 12.10 ± ± ± b Carrier et al. (2010) 1.17 ± ± ±
11 5082 ±
69 2.98 ± ± ± ±
72 Corsaro et al. (2012) 1.28 ± ± T i is the interferometrically calculated effective temperature from Table 2; T lit is the effective temperature from the literature as listed in Table2. R i is the interferometrically measured radius; R a is the radius calculated using ν max and ∆ ν from asteroseismic observations as well as T lit ; ν max and ∆ ν are from the references listed; M is the mass calculated using ∆ ν and R i ; and ν max , calc is the ν max calculated using M and T i . a No ν max was listed, so it was calculated using T eff from Marti´c et al. (2001) and M and R from Kallinger et al. (2009). b No ν max was listed, so it was calculated using T eff , M , and R from Carrier et al. (2010).
18 –Fig. 1.— θ LD fits for stars observed with one calibrator. The solid lines represent thetheoretical visibility curve for the best fit θ LD , the points are the calibrated visibilities, andthe vertical lines are the measurement uncertainties. The uncertainty in the θ LD fit is notshown because it largely indistinguishable from the best fit θ LD curve on this scale. See Table2 for the uncertainty. 19 –Fig. 2.— θ LD fits for stars observed with two or three calibrators. The symbols are the sameas in Figure 1. 20 – θ L D , he r e ( m a s ) θ LD,previous (mas)-0.10.00.1 θ h - θ p θ L D , he r e ( m a s ) θ LD,SED (mas)-0.50.00.5 θ he r e - θ SE D Fig. 3.— Comparison between θ LD measured here and previous interferometric measurementsfrom the literature (left panel) and compared to SED fits (right panel). The bottom panelsshow the residuals to the fit. The values used are listed in Table 3. 21 – R I N T E R F ( R S un ) -0.4-0.20.00.20.4 ( R I - R S ) / R S R I N T E R F ( R S un ) SEISM (R Sun )-0.4-0.20.00.20.4 ( R I - R S ) / R S Fig. 4.— Comparison between interferometrically measured radii and those determinedasteroseismologically listed in Table 4. The square represents the dwarf star, the trianglesare subgiant stars, and the circles are giant stars. In the large bottom panel, the targetsfrom Huber et al. (2012b) are added in as stars. The dashed line is the 1:1 ratio. The smallbottom panels show the residuals to the fit normalized to the asteroseismic radii. 22 – ν m a x , c a l c ( µ H z ) -0.4-0.20.00.20.4 ( ν m , c a l c - ν m , ob s ) / ν m , ob s ν m a x , c a l c ( µ H z )
10 100 1000 ν max,obs ( µ Hz)-0.4-0.20.00.20.4 ( ν m , c a l c - ν m , ob s ) / ν m , ob s Fig. 5.— Comparison between the calculated and observed ν maxmax