The radius and effective temperature of the binary Ap star beta CrB from CHARA/FLUOR and VLT/NACO observations
H. Bruntt, P. Kervella, A. Merand, I. M. Brandao, T. R. Bedding, T. A. ten Brummelaar, V. Coude du Foresto, M. S. Cunha, C. Farrington, P. J. Goldfinger, L. L. Kiss, H. A. McAlister, S. T. Ridgway, J. Sturmann, L. Sturmann, N. Turner, P. G. Tuthill
aa r X i v : . [ a s t r o - ph . S R ] D ec Astronomy&Astrophysicsmanuscript no. bruntt˙betacrb c (cid:13)
ESO 2018June 11, 2018
The radius and effective temperature of the binary Ap star β CrBfrom CHARA/FLUOR and VLT/NACO observations ⋆ H. Bruntt , , P. Kervella , A. M´erand , I. M. Brand˜ao , , T. R. Bedding , T. A. ten Brummelaar , V. Coud´e du Foresto ,M. S. Cunha , C. Farrington , P. J. Goldfinger , L. L. Kiss , , H. A. McAlister , S. T. Ridgway , J. Sturmann ,L. Sturmann , N. Turner , and P. G. Tuthill LESIA, CNRS UMR 8109, Observatoire de Paris-Meudon, 5 place Jules Janssen, F-92195 Meudon Cedex, France Sydney Institute for Astronomy, School of Physics, The University of Sydney, NSW 2006, Australia European Southern Observatory, Alonso de C´ordova 3107, Casilla 19001, Santiago 19, Chile Universidade do Porto, Centro de Astrof´ısica, Rua das Estrelas, PT 4150-762 Porto, Portugal Departamento de Matem´atica Aplicada, Faculdade de Ciˆencias, Universidade do Porto, 4169 Porto, Portugal Center for High Angular Resolution Astronomy, Georgia State University, PO Box 3965, Atlanta, Georgia 30302-3965, USA Konkoly Observatory of the Hungarian Academy of Sciences, Budapest, Hungary National Optical Astronomy Observatory, PO box 26732, Tucson, AZ 85726, USA.Received ??-?? 2009 ; Accepted ??-?? ????
ABSTRACT
Context.
The prospects for using asteroseismology of rapidly oscillating Ap (roAp) stars are hampered by the large uncertainty infundamental stellar parameters. Results in the literature for the e ff ective temperature ( T e ff ) often span a range of 1000 K. Aims.
Our goal is to reduce systematic errors and improve the T e ff calibration of Ap stars based on new interferometric measurements. Methods.
We obtained long-baseline interferometric observations of β CrB using the CHARA / FLUOR instrument. In order to disen-tangle the flux contributions of the two components of this binary star, we additionally obtained VLT / NACO adaptive optics images.
Results.
We determined limb darkened angular diameters of 0 . ± .
017 mas for β CrB A (from interferometry) and 0 . ± .
017 mas for β CrB B (from surface brightness-color relations), corresponding to radii of 2 . ± . ⊙ (3.4 % uncertainty) and1 . ± .
07 R ⊙ (4.5 %). The combined bolometric flux of the A + B components was determined from satellite UV data, spectropho-tometry in the visible and broadband data in the infrared. The flux from the B component constitutes 16 ± J and K magnitudes. Combining the flux of the Acomponent with its measured angular diameter, we determine the e ff ective temperature T e ff (A) = ±
180 K (2.3 %).
Conclusions.
Our new interferometric and imaging data enable a nearly model-independent determination of the e ff ective temperatureof β CrB A. Including our recent study of α Cir, we now have direct T e ff measurements of two of the brightest roAp stars, providing astrong benchmark for an improved calibration of the T e ff scale for Ap stars. This will support the use of potentially strong constraintsimposed by asteroseismic studies of roAp stars. Key words. techniques: interferometric, stars: chemically peculiar, stars: fundamental parameters, stars: individual: β CrB, α Cir, γ Equ, 10 Aql
1. Introduction
Photometric and spectroscopic determinations of the e ff ectivetemperatures of Ap stars are a ff ected by systematic errors. Thishas been corroborated by asteroseismic data of rapidly oscillat-ing Ap (roAp) stars in general and, more recently, by the first in-terferometric determination of the angular diameter of the roApstar α Cir (Bruntt et al. 2008). Unfortunately, the intriguing as-teroseismic potential o ff ered by roAp stars is strongly compro-mised by the presence of these systematic errors. We thereforeseek to make direct measurements of the radii and e ff ective tem-peratures of a number of Ap stars using interferometric and spec-trophotometric measurements. We will first give a brief sum-mary of the properties of β CrB before describing our obser-vations, data reduction (Sect. 2) and analysis (Sect. 3).
Send o ff print requests to : H. Bruntt ⋆ Based on observations made with ESO telescopes at the La SillaParanal Observatory, under ESO DDT program 281.D-5020(A).
Correspondence to : [email protected] β CrB β CrB (HD 137909, HIP 75695) is one of the brightest, coolest,and best-studied magnetic Ap stars. The literature on the staris extensive and we only mention a few of the most impor-tant results here. β CrB has been classified as type A9 Sr Eu Crstar by Renson & Manfroid (2009). Its binary nature was firstsuggested by Campbell & Moore (1907), and recent determina-tions of its orbital elements were obtained by Tokovinin (1984)and North et al. (1998). From speckle interferometric measure-ments using narrow-band filters, Horch et al. (2004) measuredthe magnitude di ff erence to be 2.37 mag at 551 nm and 1.99 magat 503 nm. In the analysis presented in Sect. 3 we retain theorbital elements obtained by Tokovinin (1984), as they are insignificantly better agreement with our NACO astrometry thanNorth et al. (1998).Neubauer (1944) suggested that a third body could bepresent in the system, causing radial velocity variationswith a period of ≈
321 days, but Oetken & Orwert (1984),Kamper et al. (1990) and S¨oderhjelm (1999) excluded this pos-
H. Bruntt et al.: FLUOR and NACO observations of β CrB A and B sibility. Recently, Muterspaugh et al. (2006) established an up-per limit of ≈
10 to 100 M J (depending on the orbital period)for a possible substellar tertiary from di ff erential interferomet-ric astrometry. Trilling et al. (2007) searched for 24 and 70 µ minfrared excess around β CrB using
Spitzer but did not find any.Interestingly, the Spitzer flux they obtained is significantly belowthe expected flux at 24 µ m, and slightly lower (although compat-ible) at 70 µ m. This result could be due to the chosen physicalparameters for their stellar atmosphere model. In the following,we will therefore consider that β CrB is a binary system. β CrBa pulsatingstar?
Early photometric searches for pulsation in β CrB (e.g.Weiss & Schneider 1989; Kreidl 1991) gave null results and thiscontributed to the discussion of the existence of non-oscillatingAp stars (“noAp”; Kurtz 1989). This has changed since the ad-vent of large telescopes and ultra-stable spectrographs. Based onspectroscopic time series of a single Fe line, Kochukhov et al.(2002) claimed the first possible detection of a pulsation modein β CrB with a period of 11.5 min. This result was questionedby Hatzes & Mkrtichian (2004) and was also not confirmedby Kurtz et al. (2007). However, the good agreement betweenthe independent spectroscopic studies of Hatzes & Mkrtichian(2004), Kurtz et al. (2007) and Kochukhov et al. (2008) con-firmed that β CrB is indeed an roAp star with a single knownlow-amplitude mode with period 16.2 min. The most robust re-sult was found by Kurtz et al. (2007), who used 2 hours of high-cadence time-series spectra obtained with VLT / UVES. Theydetected a single oscillation frequency at 1 .
031 mHz ( P = . . ± . − in the H α lineand a higher amplitude in the cesium lines. Unlike most roApstars, variation was observed only in singly-ionized rare-earthelements, but not doubly ionized lines. The abundance analy-sis done by Kurtz et al. (2007) on their averaged spectrum con-firmed earlier investigations by Ryabchikova et al. (2004). Theseanalyses show that β CrB has an overabundance of rare-earth el-ements but only by about 1 dex, contrary to the 2–3 dex seen inmost roAp stars.
2. Observations and data reduction
We observed β CrB on 16 June 2008 using the NasmythAdaptive Optics System (NAOS; Rousset et al. 2003) of theVery Large Telescope (VLT), coupled to the CONICA infraredcamera (Lenzen et al. 1998), abbreviated as NACO. We se-lected the smallest available pixel scale of 13 . ± .
03 mas / pix(Masciadri et al. 2003), giving a field of view of 13.6 ′′ × ′′ .This small scale resulted in good sampling of the Point SpreadFunction (PSF). We employed the J and K filters of NACO, withrespective bandpasses of 1 . ± . µ m and 2 . ± . µ m, to-gether with a neutral density filter (labeled “ ND2 short ”, trans-mission ≈ . .We obtained 20 images in the J band, and 40 images in the K band, each with an exposure time of 0.35 s. This is the min-imum full-frame integration time of CONICA. The J band im-ages were collected during ≈ K images in ≈ http: // / instruments / naco / inst / filters.html NE 5 AU−0.2 0.0 0.2−0.20.0 arcseconds a r cs e c ond s Fig. 1.
Average NACO image of β CrB A and B in the K band,together with the binary orbit from Tokovinin (1984). The po-sitions of β CrB A and B measured on our NACO image aremarked with a “ ⋆ ” and “ + ” symbol, respectively. The scale ofthe two axes are in arc seconds, relative to β CrB A. The linearscale in AU is based on the
Hipparcos parallax.observations the DIMM seeing at Paranal in the visible was good(0 . − . ′′ ), resulting in a high Strehl ratio ( ≈ − . On the NACO images,we measured both the di ff erential photometry and the di ff eren-tial astrometry of β CrB B relatively to β CrB A taken as thereference.To measure the relative astrometry, we treated each imageseparately using the Yorick software package. We used a clas-sical χ minimization to fit an extracted sub-image of β CrB A(with a size of 9 × d (RA) and d (Dec), the flux ratio and the background level,although we used only the relative separations for our astro-metric analysis. In order to estimate the associated error bars,we used the bootstrapping technique described by Kervella et al.(2004a). This technique is also called “sampling with replace-ment” and consists of constructing a hypothetical, large pop-ulation derived from the original measurements and estimatethe statistical properties from this population. The techniqueallows us to compute meaningful confidence intervals withoutany assumption on the properties of the underlying population(e.g. a Gaussian distribution). We validated the adopted Fourierinterpolation method by comparing the results with a simpleGaussian fit of the two PSF cores. The two methods yield ex-actly the same relative positions (within 150 µ as), although theGaussian fit has a slightly larger dispersion due to the mismatchof the slightly seeing-distorted PSF and the Gaussian function. IRAF is distributed by the NOAO, which is operated by theAssociation of Universities for Research in Astronomy, Inc., under co-operative agreement with the National Science Foundation. http: // yorick.sourceforge.net / . Bruntt et al.: FLUOR and NACO observations of β CrB A and B 3
Table 1.
Interferometric calibrators selected from M´erand et al.(2005).
Spectral K UD diameterHD type [mag] in K band [mas]147266 G8III 3 . . ± . . . ± . . . ± . We obtained the following vector separations along the RA andDec directions of B relatively to A, for the epoch of the observa-tions (MJD 54633.08): d (RA) = . ± . ± .
40 mas , (1) d (Dec) = − . ± . ± .
47 mas . (2)The two stated error bars are the statistical and systematic un-certainties, respectively. The latter includes the pixel scale un-certainty and the detector orientation uncertainty. These valuescorrespond to a separation r and position angle α (east of north)of: r = . ± .
63 mas , (3) α = . ± .
13 degrees . (4)These measurements were done on the K images since they havethe best Strehl ratio. In the J band, the Strehl ratio was lower andmore unstable, resulting in a significantly variable backgroundfrom A to B. Although its average value is not a concern, itsslope tends to slightly shift the average apparent position of B ,by − . − . K band is presented in Fig. 1, together with the orbit by Tokovinin(1984). Our astrometric measurement falls on the predicted orbitwithin only 7 mas. Note that the reference epoch of the orbitalelements by North et al. (1998) appears to be late by approxi-mately 300 days.The photometry was obtained in two steps: (1) we obtainedthe combined photometry of the two stars; (2) we then measuredthe di ff erential flux of B relative to A. We will discuss these stepsin the following.(1) The combined ADU count was computed from theNACO images by using a large aperture enclosing the full PSFsof the two stars. It was then converted to magnitude using theZero Points obtained routinely by the observatory on the samenight: ZP ( J ) = . ± .
048 and ZP ( K ) = . ± . . ± .
03 mag for the neutral densityfilter. These zero points are not corrected for atmospheric ab-sorption, but as they were obtained at low airmass ( ≈ . ≈ .
01 mag in J and K .We corrected the atmospheric absorption using the standard val-ues by Nikolaev et al. (2000), namely 0.092 mag / AM (relative tounit airmass) for J and 0.065 mag / AM for K , for our observationairmass of 1.71. We obtain: m J (A + B) = . ± . , (5) m K (A + B) = . ± . . (6)(2) The di ff erential photometry was obtained slightly di ff er-ently, since the di ff use background of β CrB A tends to contam-inate the flux of star B (but the reverse e ff ect is negligible). Wefirst computed aperture photometry of A on the average J and K Table 2.
Journal of observations.
MJD
B PA V % V ( ⋆ ) % − .
388 324 .
09 6 .
17 33 . ± . . ± .
7A 604 .
408 323 .
61 1 .
40 31 . ± . . ± .
3A 604 .
433 323 . − .
70 29 . ± . . ± .
0A 604 .
488 327 . − .
40 34 . ± . . ± .
9A 605 .
296 245 . − .
57 45 . ± . . ± .
1A 605 .
326 248 . − .
62 44 . ± . . ± .
7B 604 .
398 323 .
76 3 .
65 3 . ± .
25 70 . ± .
2B 605 .
254 238 . − .
58 4 . ± .
38 97 . ± . Notes: B is the projected length of the interferometric baseline, PA is the projection angle of the baseline, V is the observedsquared visibility and V ( ⋆ ) is the squared visibility correctedfor the presence of the other component (see Eq. 12). images using very small aperture radii of 3 pixels in the J bandand 4 pixels in the K band (contamination is lower in K ). Wecalculated the median background value in concentric rings cen-tered on A. This value was then subtracted from component B’sflux. This allowed us to subtract the di ff use light from the PSFwings of A at the position of B. We checked that the residualbackground around B was negligible. We then integrated the fluxof B on the ring-median-subtracted image using the same aper-ture radius as for A. We obtain the following flux ratios of eachstar relative to the total of the two, i.e. ρ ( ⋆ ) = f ( ⋆ ) / f (A + B): ρ J (A) = . ± . , ρ J (B) = . ± . , (7) ρ K (A) = . ± . , ρ K (B) = . ± . . (8)Note that the quoted uncertainties are statistical errors and do notinclude possible flat-fielding errors. From the combined magni-tudes determined above, we obtain the individual magnitudes of β CrB A and B: m J (A) = . ± . , m J (B) = . ± . , (9) m K (A) = . ± . , m K (B) = . ± . . (10)The individual J , K magnitudes have large uncertainties, but westress that we only use the values of ρ for the interpretation ofour interferometric data (Sect. 2.2), and they are known with amuch higher accuracy. Our interferometric observations of β CrB took place on 17–18 May 2008 in the near infrared K ′ band (1 . ≤ λ ≤ . µ m) at the CHARA Array (ten Brummelaar et al. 2005) us-ing FLUOR (the Fiber Linked Unit for Optical Recombination;Coud´e du Foresto et al. 2003). We used the FLUOR DataReduction Software (DRS; Coud´e Du Foresto et al. 1997;Kervella et al. 2004a; M´erand et al. 2006) to extract the squaredinstrumental visibility of the interference fringes. We used threedi ff erent interferometric calibrators in order to calibrate the visi-bilities on sky. Their properties are listed in Table 1. We note thatthe angular diameters of the calibrator stars are comparable to orlarger than the target star. Therefore they contribute significantlyto the uncertainty of the angular diameter measurement. The cor-responding systematic uncertainties were propagated into the fi-nal angular diameter uncertainties.The light from both stars of β CrB is injected simultane-ously in the FLUOR fibers, since the acceptance angle is 0 . ′′ H. Bruntt et al.: FLUOR and NACO observations of β CrB A and B on the sky. However, due to their on-sky separation of ≈ . ′′ (cf. Eq. 3), two fringe packets are formed at di ff erent opticalpath di ff erences.For this reason, we have to correct our measured visibilityfor this e ff ect. The monochromatic visibility of the binary is: V = ρ K (A) V (A) + ρ K (B) V (B) e i π B · Γ / λ , (11)where ρ K (A) and ρ K (B) are the relative fluxes of A and B, V (A)and V (B) are the individual visibilities, B the baseline vector, Γ the angular separation between A and B, and λ is the wave-length. Because we observed over a relatively broad wavelengthrange with FLUOR, and since the binary is well-resolved by ourbaselines, the fringes appeared as two distinct fringe packets.Moreover, FLUOR has a limited window in terms of optical pathdi ff erence, corresponding to a limited field of view. For β CrBA + B, the two fringe packets are not present in a single fringescan. Hence, the squared visibility measured by FLUOR in thecase of A is V = ρ K (A) V (A) . (12)In the case of the observations of B, this multiplicative factor ρ K (B) is small (due to the faintness of the star) and causes anamplification of the error bars on the true visibility V (B) forour second observation of this star. After reducing and calibrat-ing the data with the DRS pipeline for each component, we havecorrected the visibilities for this e ff ect and then derived the an-gular diameters using limb-darkened models from Claret (2000).This leads to the following angular diameters: θ LD (A) = . ± .
017 mas (2 . , (13) θ LD (B) = . ± .
054 mas (10 . . (14)The angular diameter of β CrB B is significantly more uncertainthan that of A, due to the very low apparent visibility (“ V ob-served” in Table 2) of its interference fringes “on top” of the in-coherent flux from A. The large multiplicative factor ρ K (B) ≈ V error bar. Only twomeasurements of B could be derived from our data, and one ofthese points (MJD 54604.398) dominates the fit. In the case of A,our six data points are in fair agreement with the best-fit modelwith a reduced χ of 2 . We can compare the measured angular diameters of β CrB A andB with the predictions from the surface brightness-colour (here-after SBC) relations calibrated by Kervella et al. (2004b) usingtheir ( V , V − K ) relation. The K band magnitudes were obtainedin Sect. 2.1. We derive the V band magnitudes from the totalmagnitude of the system of m V (A + B) = . ± .
05 (Rufener1988), and the magnitude di ff erence ∆ m = . ± .
10 measuredby Horch et al. (2004) by speckle interferometry at 503 nm. Wehave adopted estimated uncertainties on m V and ∆ m since theyare not given explicitly in the references. This gives the compo-nent magnitudes in V : m V (A) = . ± . , m V (B) = . ± . , (15)and the predicted photospheric angular diameters: θ LD (A) = . ± .
028 mas (4 . , (16) θ LD (B) = . ± .
017 mas (4 . . (17) The predicted and measured angular diameters of β CrB A aretherefore in excellent agreement, while they are in satisfactoryagreement for B (di ff erence of 1.7 σ ). We note than increas-ing the uncertainty on m V by a factor two does not significantlychange the uncertainty of the angular diameters. The original
Hipparcos parallax of β CrB is π = . ± .
69 mas(Perryman & ESA 1997), consistent with the new reduction byvan Leeuwen (2007) of π = . ± .
76 mas. However, as thenew reduction is not corrected for binarity e ff ects, we adopt theoriginal Hipparcos parallax.For β CrB A, the angular diameter measurement presentedin Sect. 2.2 represents a significant improvement in accuracy,by a factor of 1 .
6, over the surface brightness-colour estimateof Sect. 2.3. For the B component, this is not the case, as thevisibility measurement is made particularly di ffi cult by the pres-ence of the brighter A component. For the subsequent analysispresented in Sect. 3, we therefore choose to adopt for β CrB Aour direct interferometric angular diameter measurement, whilefor B we will use the SBC estimate computed from our K bandNACO photometry. This gives the following linear radii: R interf (A) = . ± .
09 R ⊙ (3 . , (18) R SBC (B) = . ± .
07 R ⊙ (4 . . (19)
3. The effective temperatures and masses of β CrB
In the following we will determine the e ff ective temperaturesand luminosities of the components of β CrB using two methods.The first method (Sect. 3.1) relies on the bolometric correction(model-dependent), while the second method (Sect. 3.2) is onlyweakly model-dependent. We will then compare the radius and T e ff of the components with a grid of isochrones to determinetheir approximate age and evolutionary masses (Sect. 3.3). + BC + parallax We used the bolometric corrections ( BC V ) from Bessell et al.(1998). For the measured V − K values of 0 . ± .
09 and0 . ± .
09, for A and B we get BC V (A) = .
04, BC V (B) = . K (A) = .
39, and BC K (B) = .
01 (with log g = V band photometry: m bol (A) = . ± . , m bol (B) = . ± . . (20)The same computation with the K band magnitudes gives iden-tical values within 0.01 mag. We thus obtain the followingbolometric luminosities, assuming the Hipparcos parallax and M bol = .
75 for the Sun (recommendation by IAU 1999): L BC (A) = . ± . ⊙ , L BC (B) = . ± . ⊙ . (21)We can now use the radii determined in Sect. 2.4 to de-rive the e ff ective temperatures of the two stars through L / L ⊙ = ( R / R ⊙ ) ( T e ff / T e ff ; ⊙ ) , where we use the solar value T e ff ; ⊙ = T BCe ff (A) = ±
200 K , T BCe ff (B) = ±
230 K . (22) . Bruntt et al.: FLUOR and NACO observations of β CrB A and B 5
Fig. 2.
The solid black line shows the flux distribution for β CrB A + B, obtained by combining IUE satellite UV data, spectropho-tometry from Breger (1976) (triangles) and Alekseeva et al. (1996) (thick curve), and broad band fluxes from Morel & Magnenat(1978) (box symbols). The curve peaking at ≃ . × − erg / s / cm / Å is the ATLAS9 model spectrum fitted to the B component,scaled by fitting the V and NACO J , K magnitudes (filled circles). The open circles indicate the assumed 3-sigma error bars on thecombined flux of the components at six di ff erent wavelengths. The inset shows the details from 2000–8000 Å on a linear flux scale. + flux The above method for the determination of T e ff has the caveatthat it relies on the bolometric correction being valid for thesestars. The BCs from Bessell et al. (1998) are calculated from at-mospheric models and do not depend on the metallicity. As acheck, we will determine the T e ff of the A component by a di-rect method, meaning it will only be weakly dependent on theassumed model atmosphere. This is done by calculating the in-tegrated bolometric flux of A + B and subtracting the flux fromthe B component using an ATLAS9 model with the T e ff deter-mined above.The bolometric flux of the combined system, β CrB A + B,was obtained by combining data from the literature from the UVto the near IR as shown in Fig. 2. In the UV range we used fivespectrograms from the Sky Survey Telescope (Jamar et al. 1976)obtained at the
IUE “Newly Extracted Spectra” data archive ,covering the wavelength interval 1150 Å < λ < < λ < V JKL from Morel & Magnenat (1978).We interpolated the points between the broadband data and madea linear extrapolation at the end points (UV and IR ranges), al-though in practice the contribution is negligible. Finally, we cal-culated the weighted average flux, which is shown as the solid http: // sdc.lae ff .inta.es / cgi-ines / IUEdbsMY / black line in Fig. 2. For the relative flux uncertainties we as-sumed 15 % in the UV, 6 % in the optical, and 10 % in the nearIR. For the ranges where extrapolations were made, we doubledthese errors. The adopted uncertainties are larger than the origi-nally published values. We adjusted them based on the disagree-ment between di ff erent sources of data in the same wavelengthranges, i.e. the spectrophotometric data from Breger (1976) andAlekseeva et al. (1996) in Fig. 2.Since β CrB is a binary system, extra care must be takenwhen computing the bolometric flux of the primary star. Thebinary has a maximum angular separation of 0.3” and all avail-able flux data contain the combined light of the two components.Since our main interest is the A component, we have to estimateand subtract the flux of the B component. To accomplish this, wefitted Kurucz models to the m J , m K and m V magnitudes of the B component, taking into account the statistical errors on the mag-nitudes (solid circles in Fig. 2). For the m J and m K magnitudeswe adopted the measurements from NACO, while the m V magni-tude used was that derived in Eq. 15. The spectra for the Kuruczmodels were obtained using the IDL routine kurget1 (ATLAS9models) and the corresponding database of models available inthe IUE reduction and data analysis package IUEDAC . Westarted by converting the m V magnitude of β CrB B into flux us-ing the relation (6) of Rufener & Nicolet (1988) and used thisresult to calibrate the models. We then converted the Kuruczfluxes at the NACO J and K central wavelengths (12,650 Å and21,800 Å) into magnitudes using m = − . × log ( f / f ). Here m is the magnitude in a given filter, f is the flux at the central wave- http: // archive.stsci.edu / iue / iuedac.html H. Bruntt et al.: FLUOR and NACO observations of β CrB A and B length of that filter and f is the standard zeroth-magnitude fluxfor the same filter. The values of f were computed by integratingthe flux of Vega through each of the J and K filters of the NACOinstrument, and assuming that Vega has zero magnitude in allbands . We then generated 100 values for m J and m K of β CrB Bby adding random fluctuations consistent with the uncertainties.For each set we determined the Kurucz model that best fittedeach set of magnitudes. The average integrated flux of the 100fitted Kurucz models was found to be f B = (1 . ± .
3) 10 − erg / s / cm . We adopted a rather large uncertainty (25 %) sincewe only have three broadband flux measurements of the B com-ponent.The observed flux from the combined system was computedby integrating the black curve shown in Fig. 2, from which weobtained f A + f B = (7 . ± .
4) 10 − erg / s / cm and the bolometricflux of the primary component is thus found to be f A = . ± . − erg / s / cm . We then obtained the e ff ective temperatureusing the relation, σ T ff = f bol θ , (23)where f bol is the bolometric flux and θ LD is the limb-darkenedangular diameter. Inserting the values for f A and the angular di-ameter from Sec. 2.2 we obtain an e ff ective temperature of T θ e ff (A) = ±
180 K . (24)Combining this with the radius we get the luminosity L θ (A) = . ± . ⊙ . (25)These values are in agreement with those in Sect. 3.1 wherewe used the (model dependent) bolometric correction. Sincethe calculation using Eq. 23 is nearly model-independent (limb-darkening coe ffi cients depend on atmosphere models), we adoptthese values as our final estimates of T e ff and the luminosity. Wenote that if we neglect the contribution to the flux from the Bcomponent, T θ e ff (A) becomes 350 K higher.Several determinations of T e ff are found in the literature andwe mention a few here. Kochukhov & Bagnulo (2006) used pho-tometric indices to determine T e ff and found 7430 ±
200 K, whichis significantly lower than our value. Netopil et al. (2008) havedetermined the T e ff of β CrB from three photometric systems(Str¨omgren, Geneva and Johnson) and compared these with val-ues in the literature. The mean value for the photometric in-dices is 7710 ±
260 K and the mean of the literature values is8340 ±
360 K. This is a typical example of the large scatter foundfor chemically peculiar A stars. However, since the rms scatteris large, the results summarized by Netopil et al. (2008) are inacceptable agreement with our new determination. It is worthstressing that our determination is the first that is not a ff ectedby photometric calibration errors or interstellar reddening, andis only weakly dependent on the adopted limb darkening. β CrB
To investigate the evolutionary status of the two componentsof β CrB we have compared the observed T e ff and radius withisochrones from the BASTI grid (Pietrinferni et al. 2004) asshown in Fig. 3. To transform the mass fraction of heavy el-ements ( Z ) of the isochrones to spectroscopic [Fe / H] valueswe used the solar value Z ⊙ = . ff au et al. 2009), i.e. We note that Bohlin & Gilliland (2004) recently found V = . ± .
008 for Vega.
Table 3.
Measured quantities and derived fundamental parame-ters.
Parameter A B π [mas] 28 . ± .
69 Same as A θ LD [mas] 0 . ± .
017 0 . ± . f bol [10 − erg / s / cm ] 6 . ± . . ± . R / R ⊙ . ± .
09 1 . ± . L / L ⊙ . ± . . ± . T e ff [K] 7980 ±
180 6750 ± M / M ⊙ . ± .
15 1 . ± . . ± .
10 Same as A
Fig. 3.
Radius– T e ff diagram showing the location of the twocomponents of β CrB. The thin lines are BASTI isochrones for[Fe / H] = + .
10 for ages 0.5, 0.6 and 0.7 Gyr. The thick lines arefor the sames ages but for higher metallicity [Fe / H] = + .
28. Onsome of the isochrones the box symbols mark the masses at 1.3–1.5 M ⊙ and 2.0–2.2 M ⊙ in steps of 0.1 M ⊙ .[Fe / H] ≃ log ( Z BASTI / Z ⊙ ). We assumed an uncertainty of ± .
002 on Z ⊙ , which corresponds to ± .
05 dex on [Fe / H].In Fig. 3 we show two sets of isochrones with [Fe / H] =+ .
10 and + .
28. The higher metallicity appears to be in bet-ter agreement with the location of the B component. Kurtz et al.(2007) found [Fe- ii / H] = + . ± .
22 from 11 lines of singlyionized Fe. This is the metallicity in the photosphere but we as-sume it represents the entire star. Since radial stratification of Feis well-known to be present in roAp stars, our assumed metal-licity is an approximation, but it seems to be supported by theagreement with the location of the stars in the radius– T e ff dia-gram in Fig. 3. With this assumption we determine the age tobe 0 . ± .
10 Gyr and the masses of the components to be M A / M ⊙ = . ± .
15 and M B / M ⊙ = . ± .
10. These “evo-lutionary masses” are in good agreement with the dynamicalmasses determined by North et al. (1998): M A / M ⊙ = . ± . M B / M ⊙ = . ± . . Bruntt et al.: FLUOR and NACO observations of β CrB A and B 7
4. Discussion and conclusion
We have determined the e ff ective temperature of the primarycomponent of the binary β CrB using a technique that is onlyweakly model-dependent. We used interferometric data to mea-sure the angular diameter and the fluxes were constrained usingNACO J , K measurements of each individual component in thebinary.We determined the primary component of β CrB to have T e ff = ±
180 K. In comparison, literature values for the com-bined star span 7230–8700 K (considering 1- σ uncertainties).From a similar analysis of flux data and interferometric data theroAp star α Cir, we found T e ff = ±
170 K (Bruntt et al.2008). For that star the literature values also span a large rangefrom 7470–8730 K. It is interesting that for β CrB A, our re-sult is in the middle of the range of previous estimates while for α Cir the estimate is at the low end of the range. If we comparesolely with T e ff estimates from the same photometric system,Kochukhov & Bagnulo (2006) found 7430 ±
200 K for β CrB and7670 ±
200 K for α Cir. We must remember that the photometricindex of β CrB includes both components, and will always givea systematically low temperature. Taking this into account, thephotometric values from Kochukhov & Bagnulo (2006) seem toagree with our fundamental (i.e. model-independent) T e ff valueswithin about ±
300 K. It will be necessary to make interferomet-ric measurements of several more of the brightest Ap stars to beable to improve the T e ff scale of these peculiar stars.Accurate determinations of T e ff have important impact forthe asteroseismic modelling in future work. α Cir was observedfor 84 days with the 52 mm star tracker on the now defunctWIRE satellite. From the light curves Bruntt et al. (2009) de-tected five frequencies of which two had not been observed be-fore. These two lie symmetrically around the well-known dom-inant mode at 2442 µ Hz to form a triplet. Bruntt et al. (2009)interpreted the equidistant separation as half the large separa-tion. Combining this with the new T e ff , the properties of thestar could be constrained based on preliminary theoretical mod-elling of the observed pulsation modes. To obtain similar re-sults for β CrB would be worthwhile now that all ingredientsfor the modelling of the star are available, making it the sec-ond roAp star with well-established fundamental atmosphericparameters. This would require an ambitious asteroseismic cam-paign (Kurtz et al. 2007) using a network of telescopes with sta-ble spectrographs like the Stellar Observations Network Group(SONG; Grundahl et al. 2008).Our understanding of roAp stars would benefit from obtain-ing interferometric angular diameters of more targets. However,even with the most sensitive beam combiners currently available,only a handful are bright enough to yield a radius measurementto better than 2 %. Having now measured α Cir and β CrB, wenext propose to observe γ Equ and 10 Aql.
Acknowledgements.
The authors would like to thank all the CHARA Arrayand Mount Wilson Observatory daytime and nighttime sta ff for their sup-port. The CHARA Array was constructed with funding from Georgia StateUniversity, the National Science Foundation, the W. M. Keck Foundation,and the David and Lucile Packard Foundation. The CHARA Array is oper-ated by Georgia State University with support from the College of Arts andSciences, from the Research Program Enhancement Fund administered by theVice President for Research, and from the National Science Foundation underNSF Grant AST 0606958. STR acknowledges partial support from NASA grantNNH09AK731. MSC acknowledges the support of the Portuguese MCTES andof the FSE, of the European Union, through the programme POPH. IMB wouldlike to acknowledge the support from the Fundac¸ ˜ao para a Ciˆencia e Tecnologia(Portugal) through the grant SFRH / BD / / References
Alekseeva, G. A., Arkharov, A. A., Galkin, V. D., et al. 1996, Baltic Astronomy,5, 603Bessell, M. S., Castelli, F., & Plez, B. 1998, A&A, 333, 231Bohlin, R. C. & Gilliland, R. L. 2004, AJ, 127, 3508Breger, M. 1976, ApJS, 32, 7Bruntt, H., Kurtz, D. W., Cunha, M. S., et al. 2009, MNRAS, 396, 1189Bruntt, H., North, J. R., Cunha, M., et al. 2008, MNRAS, 386, 2039Ca ff au, E., Maiorca, E., Bonifacio, P., et al. 2009, A&A, 498, 877Campbell, W. W. & Moore, J. H. 1907, Lick Observatory Bulletin, 4, 162Claret, A. 2000, A&A, 363, 1081Coud´e du Foresto, V., Borde, P. J., M´erand, A., et al. 2003, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, ed. W. A.Traub, Vol. 4838, 280Coud´e Du Foresto, V., Ridgway, S., & Mariotti, J.-M. 1997, A&AS, 121, 379Cox, A. N. 2000, Allen’s astrophysical quantities (Springer-Verlag)Grundahl, F., Christensen-Dalsgaard, J., Arentoft, T., et al. 2008,Communications in Asteroseismology, 157, 273Hatzes, A. P. & Mkrtichian, D. E. 2004, MNRAS, 351, 663Horch, E. P., Meyer, R. D., & van Altena, W. F. 2004, AJ, 127, 1727Jamar, C., Macau-Hercot, D., Monfils, A., et al. 1976, Ultraviolet bright-starspectrophotometric catalogue. A compilation of absolute spectrophotomet-ric data obtained with the Sky Survey Telescope (S2 /
68) on the EuropeanAstronomical Satellite TD-1Kamper, K. W., McAlister, H. A., & Hartkopf, W. I. 1990, AJ, 100, 239Kervella, P., S´egransan, D., & Coud´e du Foresto, V. 2004a, A&A, 425, 1161Kervella, P., Th´evenin, F., Di Folco, E., & S´egransan, D. 2004b, A&A, 426, 297Kochukhov, O. & Bagnulo, S. 2006, A&A, 450, 763Kochukhov, O., Landstreet, J. D., Ryabchikova, T., Weiss, W. W., & Kupka, F.2002, MNRAS, 337, L1Kochukhov, O., Ryabchikova, T., Bagnulo, S., & Lo Curto, G. 2008,Contributions of the Astronomical Observatory Skalnate Pleso, 38, 423Kreidl, T. J. 1991, MNRAS, 248, 701Kurtz, D. W. 1989, MNRAS, 238, 261Kurtz, D. W., Elkin, V. G., & Mathys, G. 2007, MNRAS, 380, 741Lenzen, R., Hofmann, R., Bizenberger, P., & Tusche, A. 1998, in Societyof Photo-Optical Instrumentation Engineers (SPIE) Conference Series, ed.A. M. Fowler, Vol. 3354, 606Masciadri, E., Brandner, W., Bouy, H., et al. 2003, A&A, 411, 157M´erand, A., Bord´e, P., & Coud´e Du Foresto, V. 2005, A&A, 433, 1155M´erand, A., Coud´e du Foresto, V., Kellerer, A., et al. 2006, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 6268Morel, M. & Magnenat, P. 1978, A&AS, 34, 477Muterspaugh, M. W., Lane, B. F., Kulkarni, S. R., et al. 2006, ApJ, 653, 1469Netopil, M., Paunzen, E., Maitzen, H. M., North, P., & Hubrig, S. 2008, A&A,491, 545Neubauer, F. J. 1944, ApJ, 99, 134Nikolaev, S., Weinberg, M. D., Skrutskie, M. F., et al. 2000, AJ, 120, 3340North, P., Carquillat, J.-M., Ginestet, N., Carrier, F., & Udry, S. 1998, A&AS,130, 223Oetken, L. & Orwert, R. 1984, Astronomische Nachrichten, 305, 317Perryman, M. A. C. & ESA, eds. 1997, ESA Special Publication, Vol. 1200,The HIPPARCOS and TYCHO catalogues. Astrometric and photometric starcatalogues derived from the ESA HIPPARCOS Space Astrometry MissionPietrinferni, A., Cassisi, S., Salaris, M., & Castelli, F. 2004, ApJ, 612, 168Renson, P. & Manfroid, J. 2009, A&A, 498, 961Rousset, G., Lacombe, F., Puget, P., et al. 2003, in Society of Photo-OpticalInstrumentation Engineers (SPIE) Conference Series, ed. P. L. Wizinowich &D. Bonaccini, Vol. 4839, 140Rufener, F. 1988, Catalogue of stars measured in the Geneva Observatory pho-tometric system : 4 : 1988 (Sauverny: Observatoire de Geneve, 1988)Rufener, F. & Nicolet, B. 1988, A&A, 206, 357Ryabchikova, T., Nesvacil, N., Weiss, W. W., Kochukhov, O., & St¨utz, C. 2004,A&A, 423, 705S¨oderhjelm, S. 1999, A&A, 341, 121ten Brummelaar, T. A., McAlister, H. A., Ridgway, S. T., et al. 2005, ApJ, 628,453Tokovinin, A. A. 1984, Pisma Astronomicheskii Zhurnal, 10, 293Trilling, D. E., Stansberry, J. A., Stapelfeldt, K. R., et al. 2007, ApJ, 658, 1289