arXiv:1703.01449v1 [astro-ph.SR] 4 Mar 2017
MNRAS cetdXX eevdYY noiia omZZZ form original in YYY; Received XXX. Accepted et r ujc odsotosdet h at’ atmo- ⋆ Earth’s the to due distortions to subject are ments stel methods precise achieve Both finally interferometry. radii. to parallaxes from accurate those require more than are that diameters accurate angular to leads models atmosphere tyo tr rmteSI etGnrto Spectral Generation Next STIS the from (NGSL; stars Library spectrophotom- A flux of absolute of etry comparison the that shown 2015 2009 cuaeprmtr o D295 n t lntfrom planet its and spectrophotometry 209458 HST HD for parameters Accurate ie h otdrc a oacranselrangu- and nearby stellar relatively ascertain to limited to is way (e.g.: it stars bright ac- direct but pro diameters, most determine interferometry lar tran- the to optical/infrared (spectro-)photometric vides required Long-baseline from are radii sits. planetary radii curate stellar Accurate INTRODUCTION 1 c 3 2 1 .dlBurgo del C. eatmnod srfıia nvria eL aua 38 Laguna, La Ten de Laguna, Astrof´ısica, Universidad La de 38205 Departamento Canarias, Astrof´ısica de de Instituto nttt ainld Astrof´ısica, de Nacional Instituto -al [email protected] E-mail: 06TeAuthors The 2016 Recently, esrmnso aalxsfo rudbsdinstru- ground-based from parallaxes of Measurements ). ; a ru ta.2014 al. et Braun van 000 , 1 – 10 led reo&dlBurgo del & Prieto Allende rg ta.2006 al. et Gregg a el o ru 2009 Braun von & Belle van 21)Pern etme 01Cmie sn NA L MNRAS using Compiled 2021 September 9 Preprint (2016) 1 ⋆ n .AlnePrieto Allende C. and ; oaine l 2015 al. et Boyajian epeetudtdprmtr o h trH 048adistr its and diameter 209458 angular HD star stellar the The for parameters updated present We ABSTRACT rvmn npeiino oeta ortmscmae oavailable to compared times radius four diameter stellar than The angular more determinations. This of precision atmosphere. the in model with provement Kurucz observed flux best-fitting absolute the the between tio nua imtrwt the with diameter angular rbtrt t netit,adteeoeterdu cuays accuracy radius the therefore with and uncertainty, improved its to tributor sdrvdfo h orsodn rni et ntelgtcrean effect curve the light determine the accurately in we depth fitting, transit model T the corresponding From the radius. from derived is e words: literature. the Key from those with results the lofidpeievle rmrcn aoaIohoe,sc as ener such spectral Isochrones, Padova integrated recent the from values and precise diameter find angular also the from lated etlprmtr;paesadstlie:fnaetlparame fundamental satellites: and planets parameters; mental T piayEet´nc,Li niu ro1 t.M.Tonantz Ma. Sta. 1, Erro Enrique Electr´onica, Luis y Optica ´ ihaporaestellar appropriate with ) eff eff =6071 =6099 ± ± 0,wihi npretareetwt h au f6070 of value the with agreement perfect in is which 20K, 1.W riea ossetpcuefo hs ehd n com and methods these from picture consistent a at arrive We 41K. ; ( 2016 anse al. et Baines ; tr:amshrs tr:fnaetlprmtr;exopla parameters; fundamental stars: atmospheres; stars: Gaia iie al. et Ligi have ) smaueet.Terdu fteexoplanet the of radius The measurements. ’s lar rf,Spain erife, 0 aLgn,Tnrf,Spain Tenerife, Laguna, La 206 2 - , 3 Hipparcos θ ( olcigatoerc htmti,adsetocpcd 10 about spectroscopic for and photometric, astrometric, collecting 1 planet challenged has that planet ( inflated theories formation period short a is its that finding is interferometer, diameter Array angular limb-darkened CHARA the with sensitivity higher much and accuracy than position in precision ESA’s Astronomy. of areas ent sky. a eue 2007 Leeuwen van n uioiisfrmr hn1800sasbihe than brighter stars 118,000 ( motions mag than V=12.4 proper more for distances, luminosities positions, and precise problem. determine this to overcome space ESA’s from Observations sphere. rniigpae icvrd( discovered planet transiting 4 =0.2254 oaine l 2015 al. et Boyajian ± http://www.cosmos.esa.int/web/gaia y ( Gyr 2 D295 sanab prla Π=20.15 (parallax nearby a is 209458 HD Hipparcos R Hipparcos ⋆ Hipparcos =1.20 rgnmti aalx hc stemi con- main the is which parallax, trigonometric ± eoe l 2006 al. et Melo .07msi bandfo h vrg ra- average the from obtained is mas 0.0017 9 tr ih3 with stars ± S 1997 ESA . a a eakbeipc nmn differ- many in impact remarkable a had has .5R 0.05 a h rtsaemsinflydedicated fully mission space first the was 0 trwt netmtdaeof age estimated an with star G0V ) adig2007 Mardling ubeSaeTelescope Space Hubble .I abr D295 ,tefirst the b, 209458 HD harbors It ). ⊙ ; nl,Pel,Mexico Puebla, intla, a eue 2007 Leeuwen van ∼ < sacrandb obnn the combining by ascertained is .I a enrcnl observed recently been has It ). V hrona ta.2000 al. et Charbonneau ∼ < 0mag 20 Gaia .H 048i looeof one also is 209458 HD ). R ters θ ol esignificantly be hould ⋆ niiggatplanet. giant ansiting LD anhdi 03 is 2013, in launched , =1.20 ydsrbto.We distribution. gy 1 0.225 = R ersnsa im- an represents ihamc higher much a with v temperature, ive p A T =1.41 vrteentire the over ) interferometric E ± tl l v3.0 file style X ± u stellar our d .6R 0.06 es funda- nets: n htof that and 4Kcalcu- K 24 ± ± .0 mas 0.007 ± .0mas, 0.80 0.06 ⊙ .This ). pare and ata R J , 2 C. del Burgo and C. Allende Prieto only two systems with absolute stellar and planetary masses expected resolving power R5 varies slightly with wavelength, derived from high resolution spectroscopy (Snellen et al. as described in the STIS Instrument Handbook6. Accord- 2010). The other is HD 189733 (de Kok et al. 2013). ing with the NICMOS Instrument Handbook7 the resolving Modeling the Spectral Energy Distribution (SED) of power is R ∼ 200 per pixel over the full field of view of the G-type stars is relatively easy in the optical and infrared, camera. We found the FWHM values in the FITS header where the continuum opacity is dominated by H− bound-free of the CALSPEC spectrum for HD 209458 to be smaller and free-free absorption, but becomes increasingly harder than those needed to match this spectrum in the ultraviolet in the ultraviolet due to the accumulation of spectral lines, and visible (R≈560-700). Therefore, we applied our own es- the complex contribution of bound-free opacity from neutral timates of FWHM for each grating to properly smooth our atoms, and the shift of the line formation from the photo- theoretical models before comparing them to the observa- sphere to higher atmospheric layers. tions. HD 209458 was observed with the instruments STIS and NICMOS onboard NASA/ESA Hubble Space Telescope (HST). Observing from space avoids the scintillation noise resulting from air turbulence in the Earth’s atmosphere, yielding an improvement of up to three orders of magni- 2.2 Stellar atmosphere models tude in (spectro-)photometric precision. Moreover, the lim- We computed synthetic spectra covering the wavelength itations owing to the day-night cycle and variable weather range 0.2–2.5 µm with wavelength steps equivalent to 0.3 − conditions are eliminated. km s 1. The calculations are based on Kurucz ATLAS9 In this paper we present results based on the analysis model atmospheres (M´esz´aros et al. 2012) and the synthesis 2 of the HST spectrophotometry of HD 209458 . Section 2 code ASSǫT (Koesterke et al. 2008; Koesterke 2009), oper- describes the observed and modeled spectra, and the evo- ated in 1D (plane-parallel geometry) mode. The reference lutionary models used here. In Section 3, the analysis and solar abundances adopted in the model atmosphere and the results on the properties of this host star and its planet are synthesis are from Asplund et al. (2005). explained, including an accurate effective temperature for The equation of state includes the first 92 elements in HD 209458 and the most accurate to date nearly model- the periodic table and 338 molecules (Tsuji 1964, 1973, 1976, independent radii for the star and its planet. This section with some updates). Partition functions are adopted from − also includes a comparison of our results with those in the lit- Irwin (1981). Bound-free absorption from H, H , HeI, HeII, erature. Section 4 is devoted to a discussion on the potential and the first two ionization stages of C, N, O, Na, Mg, Al, precision and accuracy expected for the applied technique Si, Ca (from the Opacity Project; see Cunto et al. 1993) and a comparison with others. Finally, Section 5 presents and Fe (from the Iron Project; Bautista 1997; Nahar 1995) our conclusions. are included. Line absorption is modeled in detail using the atomic and molecular (H2, CH, C2, CN, CO, NH, OH, MgH, SiH, and SiO) files compiled by Kurucz8. 2 OBSERVED AND MODELED SPECTRA Level dissolution near the Balmer series limit is ac- counted for (Hubeny et al. 1994). The radiative trans- 2.1 Absolute spectrophotometry fer calculations include Rayleigh (H; Lee & Kim 2004) and electron (Thomson) scattering. The damping of H The HST archive contains STIS CCD first order spec- lines are treated in detail using Stark (Stehl´e 1994; troscopy (G430L, G750M, and G750L) and NICMOS slitless grism imaging spectroscopy (G096, G141, and G206) for HD Stehl´e& Hutcheon 1999) and self-broadening (for Balmer 3 Barklem et al. 2000; Ali et al. 1966, for Lyman and Paschem 209458. The reduced data were downloaded from the CAL- 4 lines). SPEC Calibration Database , which contains the composite stellar spectra for the flux standards on the HST system. To ease the derivation of atmospheric parameters from STIS data are fairly homogeneous, even though their the HST spectrophotometry, we computed a grid of model wide spectral coverage relies on using different gratings. The surface fluxes for effective temperatures in the range 5750 ≤ Teff ≤ 10,000 K, surface gravity 1.0 ≤ log g ≤ 5.0 (g in −2 −1 cm s ), microturbulence -0.3 ≤ log ξt ≤ 0.9 (ξt in km s ), 2 We adopted the nominal values of the International Astronomi- metallicity -5 ≤ [Fe/H] ≤ +1.0, and α-element enhancement cal Union (IAU) 2015 Resolution B3 (Prˇsa et al. 2016): solar effec- -1.0 ≤ [α/Fe] ≤ +1.0, in steps of 250 K, 0.5 dex, 0.3 dex, 20 3 −2 tive temperature of 5772 K; GM⊙=1.3271244 10 m s and 0.25 dex, and 0.25 dex, respectively. Spectra were convolved 17 3 −2 GMJ=1.2668653 10 m s , for the products of the gravita- with a series of Gaussian kernels (see §2.1) to account for tional constant by the masses of the Sun and Jupiter, respectively; instrumental broadening. The effect of interstellar reddening 8 7 R⊙=6.957 10 m and RJ=7.1492 10 m for the Sun radius and was considered following Fitzpatrick (1999). the equatorial radius of Jupiter, respectively. We applied the def- inition of the Resolution B2 of the XXVIII General Assembly of the IAU in 2012 for the astronomical unit au=149 597 870 700 m. We also employed G=6.67428 10−11 m3 kg−1 s−2, which is rec- ommended by the IAU Working Group on Numerical Standards 5 λ for Fundamental Astronomy, NSFA, (Luzum et al. 2011). For the R ≡ δλ , where δλ is the full width at half maximum (FWHM) Stefan-Boltzmann constant, we adopted the CODATA 2014 value of a line spread function. σ=(5.670367 ± 0.000013) × 10−8 Wm−2 K−4 (Mohr et al. 2015) 6 http://www.stsci.edu/hst/stis/documents/handbooks/currentIHB/cover.html 3 The FITS file hd209458 stisnic 006.fits. 7 http://www.stsci.edu/hst/nicmos/documents/handbooks/handbooks/current NEW/nicmos ihb.pdf 4 http://www.stsci.edu/hst/observatory/crds/calspec.html 8 kurucz.harvard.edu
MNRAS 000, 1–10 (2016) Accurate parameters for HD 209458 and its planet 3
2.3 Stellar evolution models Table 1. Stellar parameters and 1σ uncertainties of HD 209458 obtained from different methods: 1) BF: Best-fitting Kurucz We employed the PARSEC (stellar tracks and isochrones with model atmosphere to ascertain T , θ (through the comparison the PAdova & TRieste Stellar Evolution Code) Isochrones eff with the observed spectrum), and R⋆ (using θ and Π). L⋆ is then (version 1.2S: Bressan et al. 2012; Chen et al. 2014, 2015; calculated from R⋆ and Teff; 2) Procedure based on the bolo- Tang et al. 2014), with two different grids adequate to derive metric flux to determine Teff using the theoretical model (fbol- the stellar properties of the Sun and our target of interest. mod) and the observed spectrum (fbol-obs) in combination with In the first grid, which is used to calibrate the system (given θ. Also to obtain L⋆ using Π; 3) EM: Search for PARSEC (ver- that HD 209458 is particularly similar to the Sun), the initial sion 1.2S) evolution models using the likelihood function L; 4) metallicity Zini ranges from 0.0151 to 0.0211, in steps of Determination of log g from R⋆ (using BF) and either ρ⋆ derived 0.0005, age (τ) from 4.1 to 5.3 Gyr, in steps of 0.01 Gyr, from the light curve (BF-LC) or M⋆ derived from the PARSEC models (BF-EM). BF-LC and BF-EM also permit to calculate and initial mass (Mini) ranging from 0.09 M⊙ to the highest − mass established by the stellar lifetime, in steps of 10 4 M . M⋆ and ρ⋆, respectively. SynPhot: Synthetic photometry in the ⊙ Johnson-Cousins UBV RI and 2MASS systems extracted from In the second grid, Zini goes from 0.0046 to 0.0246, in steps the CALSPEC spectrum of this star. of 0.005, τ from 0.1 to 12.1 Gyr, in steps of 0.3 Gyr, and Mini spans from 0.09 M⊙ to the highest mass given the stellar −3 Parameter Value±uncertainty Note lifetime, in steps of 2 10 M⊙. The PARSEC models include the evolution of a star from Teff (K) 6071±20 BF; see §3.1 q its formation to the asymptotic giant branch phase for low- 6070±24 fbol-mod; see §3.5 q and intermediate mass stars, or the carbon ignition phase 6064±24 fbol-obs; see §3.5 q for massive stars. The age ruler by these models includes 6099±41 EM; see §3.8 θ (mas) 0.2254±0.0017 BF; see §3.3 the pre-main sequence lifetime, which is about 40 Myr for R⋆ (R⊙) 1.20±0.05 BF; see §3.4 the Sun and slightly lower for HD 209458. These models q 1.20±0.06 EM; see §3.8 provide, among other stellar parameters, actual mass, lu- L⋆ (L⊙) 1.77±0.14 fbol-mod; see §3.6 minosity, effective temperature, surface gravity, bolometric q 1.76±0.14 fbol-obs; see §3.6 magnitude, and magnitudes in a chosen photometric sys- q 1.77±0.14 BF; see §3.6 tem (Johnson-Cousins for this research) as a function of the q 1.79±0.14 EM; see §3.8 stellar age, metallicity, and the initial mass. The radius can Mbol (mag) 4.11±0.09 EM; see §3.8 −2 be trivially calculated from the mass and gravity, and the log g [cm s ] 4.38±0.06 BF-LC; see §3.7 q mean density from the mass and radius. We only used the 4.33±0.04 BF-EM; see §3.7 q 4.33±0.04 EM; see §3.8 photometric bands B and V provided by PARSEC, which M (M ) 1.26±0.15 BF-LC; see §3.7 were obtained with the responses and zero points derived ⋆ ⊙ q 1.12±0.04 EM; see §3.8 by Ma´ız Apell´aniz (2006). The corresponding stellar param- −3 ρ⋆ (g cm ) 0.91±0.11 BF-EM; see §3.7 eters for the regular sampling in Mini were computed from q 0.91±0.14 EM; see §3.8 the tabulated values using linear interpolation. τ (Gyr) 3.5±1.4 EM; see §3.8
U 8.235 SynPhot; see §3.2 B 8.204 q V 7.655 q 3 ANALYSIS AND RESULTS R 7.342 q I 7.037 q 3.1 Spectral analysis J 6.635 q H 6.346 q We searched for the values of Teff, log g, [Fe/H], [α/Fe], and Ks 6.326 q ξt, associated with the model fluxes that best fit the spec- trum of HD 209458 in the wavelength range from 0.29 to 2.5 µm. For this fit we divided both observations and models, code FER R E9 (Allende Prieto et al. 2006) using the Un- resampled on the same wavelength grid, by their own mean constrained Optimization BY Quadratic Approximation fluxes. Interstellar extinction was considered as well, finding (UOBYQA) algorithm (Powell 2002). Model fluxes with any that the optimal solution led to a negative value (E(B − V ) set of atmospheric parameters are derived by cubic Bezier = -0.09 mag), and therefore was neglected given the prox- interpolation (see Auer 2003, and references therein) in the imity of HD 209458 (∼50 pc). This is in agreement with the grid of model fluxes described in §2.2. small value E(B −V )=0.003 mag found from the same HST STIS spectra have been proven to be very useful to as- data by Bohlin et al. (2014), who noted that the CALSPEC certain angular diameters of A-type stars, but a free param- data for this star agrees with its preferred model to ∼2%. eter fitting does not necessarily provide accurate values of We defined three wavelength regions of interest: ultra- e.g., surface gravity (Allende Prieto & del Burgo 2016). For violet (UV: 0.29-0.50 µm), visible (VIS: 0.50-1.0 µm), and HD 209458, the fitting with the lowest value for chi-square near-infrared (NIR: 1.0-2.5 µm). We performed fittings to (χ2) was that of VIS-NIR-all/free, but the extracted param- VIS-NIR, VIS, UV-VIS-NIR, and UV-VIS, allowing all stel- eters (Teff=6125 K, log g=4.78, [Fe/H]=-0.19) have extreme lar parameters to vary (all/free), fixing log g (g/fixed), and values (the highest ones for Teff and log g, and the second fixing all parameters except Teff (Teff/free). In this way, we lowest one of [Fe/H] among all fittings). Since our goal is could gauge the achieved accuracy and precision for deter- mining stellar parameters for the different cases. The optimization was done using the FORTRAN90 9 Available from http://hebe.as.utexas.edu/ferre
MNRAS 000, 1–10 (2016) 4 C. del Burgo and C. Allende Prieto
1.0 1.0
0.8
0.8 0.6
0.4
0.6 flux relative
0.2
0.0 0.4 relative flux relative
300 350 400 450 500 wavelength [nm]
0.2
0.0
500 1000 1500 2000 2500 wavelength [nm]
Figure 1. HST spectrum (solid black line) and best-fitting model (solid red line) for HD 209458. Also, at the bottom the residuals (black line) and the 1σ uncertainties for the flux (red lines), stored in the FITS header of the CALSPEC spectrum. The spectra, residuals, and uncertainties between 290 and 500 nm are also zoomed. to derive appropriate parameters of the star, a fitting was overestimated by ∼50% in the VIS region, which is inferred 10 performed fixing all parameters but Teff, which must be from the comparison of these flux uncertainties with the well constrained given the broad wavelength range. residuals between the observed and modeled spectra. This 2 We indeed argue that VIS-Teff/free is our best fitting explains the low estimate of χred. In most of the UV the (Teff=6071 K, log g=4.38, [Fe/H]=0, [α/Fe]=0, log ξt=0.04) residuals are significantly larger than the estimated uncer- because: 1) The VIS spectrum has a high signal-to-noise and tainties stored in the FITS header of the HD 209458 spec- many absorption lines that are very well modeled, in con- trum, confirming the difficulties to model this region. Con- trast to the complex UV and the fainter nearly-featureless versely, the residuals in the NIR fall within the estimated NIR; and 2) The zero-point of the flux scale is very well flux uncertainties. constrained in the visible (see §3.2). We estimated an un- certainty in Teff of 20 K, which is based on: 1) The differ- ences between VIS-g/fixed and VIS-NIR-g/fixed (increase of ≈20K) and UV-VIS-NIR-g/fixed (same value); and 2) 3.2 Photometry The increment between VIS-Teff/free and VIS-NIR-Teff/free We computed the Johnson-Cousins and 2MASS photometry amounts to ≈15 K, and the decrease with respect to UV- from the CALSPEC spectrum of HD 209458 (see Table 1). VIS-NIR-Teff/free is of ≈50 K. The UV cannot be fitted as We adopted the Optimized Bessel Bandpass Functions pre- well as the optical, mainly due to the crowding of lines and sented by Bohlin & Landolt (2015). These correspond to the continuum absorption by metals. UBVRI response curves of Bessell & Murphy (2012) shifted Therefore, our best fitting model (reduced chi-square 2 by up to -31 A˚ to minimize the differences between the Lan- χ =0.462; N=1096 data points) to the CALSPEC spec- red dolt photometry and the synthetic CALSPEC photometry trum relies on the VIS region, arriving at Teff=6071±20 K of 11 spectrophotometric flux standards. The uncertainty of (see Table 1). Fig. 1 shows the modeled and observed spec- the monochromatic flux at 555.75 nm (555.6 nm in air) is tra. We note the flux uncertainties (systematic and statisti- 0.5% or 0.005 mag (see Bohlin et al. 2014, and the CAL- cal) from the FITS header of the CALSPEC spectrum are SPEC Calibration Database). Table 5 of Bohlin & Landolt (2015) lists the magnitudes and uncertainties for the conver- 10 The value of log g was determined in §3.7 and [Fe/H] is from sion to magnitudes of Vega on the Johnson-Cousing system. Torres et al. (2008); see also §3.8 with very similar values. For example, V =0.028 mag, rms=0.005, and uncertainty in
MNRAS 000, 1–10 (2016) Accurate parameters for HD 209458 and its planet 5 the mean of 0.003 mag. These magnitudes for Vega were con- 0.245 11 firmed with our code , with differences below 0.001 mag. 0.240 The values so obtained for HD 209458, B=8.204±0.010 and V =7.655±0.008 mag, are slightly different from those 0.235 computed using the response curves of Ma´ız Apell´aniz (2006), which lead to B = 8.212 and V = 7.649, but such 0.230 differences are within the 1σ uncertainties. Our value of 0.225 V is similar to that of V =7.65 mag from Bohlin (2010) [mas] and slightly larger than that from Bohlin et al. (2014) θ 0.220 12 of V =7.63 . It is also in agreement with the value of 0.215 V =7.65±0.01 given by Torres et al. (2008). The 2MASS J, H, and Ks photometry was extracted 0.210 from the CALSPEC spectrum using the response curves 0.205 and zero points from Cohen et al. (2003a). Our values of J=6.635, H=6.346, and Ks=6.326 are in fair agree- 0.200 500 1000 1500 2000 2500 ment with those of J=6.591±0.020, H=6.37±0.04, and wavelength [nm] Ks=6.308±0.026 from Cutri et al. (2003). Given the uncer- tainties in the 2MASS passbands, absolute values of differ- ences correspond to ∼2σ, ∼0.5σ, and ∼0.5σ, respectively. Figure 2. Angular diameter θ ≃ 2pf/F versus wavelength. The There are different response curves for the Johnson and vertical dash-dotted lines indicate the wavelength limits used to Cousins systems in the literature, yielding different syn- obtain the mean value of θ and its standard deviation, which are thetic magnitudes extracted from a certain model spec- marked by the continuous line and dashed lines, respectively. Note trum. Apart from the above-mentioned Optimized Bessel that although the fitting is applied to the VIS region, the match Bandpass Functions, we have employed the response curves is very good in the full wavelength range. The total uncertainty from Cohen et al. (2003b), Ma´ız Apell´aniz (2006), and in θ was calculated quadratically adding to the above-mentioned standard deviation the contributions from the error in the zero- Mann & von Braun (2015) to compute the magnitudes of 13 point on the flux scale and the error in the spectral shape due to HD 209458 in the UBVRI filter-bands . The differences be- the uncertainties in the stellar parameters (see §3.3). tween the maximum and minimum extracted values are 0.09, 0.016, 0.02, 0.03, and 0.013 mag for U, B, V , R, and I, re- spectively. The most extreme differences are 0.09 mag in the the 0.5-1.0 µm wavelength range. We used the best fitting U band. Note that bandpass functions may be multiplied by model obtained in §3.1 (Teff=6071 K, log g= 4.38, [Fe/H]=0, a mean atmospheric transmission spectrum appropriate for [α/Fe]=0, log ξt=0.04) and find the angular diameter to a certain observing site (e.g., Cohen et al. 2003b, for CTIO, be θ=0.2254±0.0017 mas (see Table 1). Our value is more Chile). Earth’s atmospheric transmission variations lead to precise than the interferometric one of 0.225±0.007 mas systematic errors in the UV. This has an impact on the re- from Boyajian et al. (2015). It is also better than the value sults from model atmosphere fittings to broad-band photom- of 0.224±0.004 mas determined by Casagrande et al. (2010) etry, together with the reduced number of points used to con- with a methodology similar to ours, probably due to the use strain the solution. Our technique overcomes this problem, of approximate SEDs provided with the model atmospheres since it uses the high signal-to-noise CALSPEC spectrum, adopted in their analysis. Fig. 2 shows θ versus wavelength, which is absolutely calibrated in flux and presents ≃550 re- where it is observed that the fitting based on the VIS region solved elements in the wavelength range between 500 and is very good in the full spectral range. Indeed, we found a 1000 nm (≃1030 resolved elements in the full wavelength very similar value for the stellar angular diameter in the range of STIS and NICMOS). wavelength ranges VIS and VIS-NIR. The uncertainty σ(θ) is quantified by adding in quadra- ture three terms: 1) a random error contribution estimated 3.3 Stellar angular diameter from the scatter of the flux ratio across the selected wave- In a similar way to del Burgo et al. (2010) and length interval (0.0008 mas); 2) the error in the spectral Allende Prieto & del Burgo (2016) the angular diame- shape because of the uncertainties in the stellar parameters, ter θ ≃ 2R/d = 2 f/F for HD 209458 was determined which is computed from a MonteCarlo approach with 100 from the average ratiop
MNRAS 000, 1–10 (2016) 6 C. del Burgo and C. Allende Prieto that study some of the extracted values for log g were lower in Table 1) demonstrate the high precision of the spectral (by up to 0.5 dex) than those ascertained from stellar struc- shape and flux zero-point of the HST data, as well as the ture models. However, metallicity and surface gravity (as excellent match to our best-fitting stellar atmosphere model. well as the effective temperature) have a very limited im- Boyajian et al. (2015) found that Teff=6092±103 K from pact on the spectral energy distribution of A-type stars in their fbol and interferometric limb-darkened angular diame- the wavelength range 0.40-0.80 µm (see also del Burgo et al. ter, and Casagrande et al. (2010) obtained Teff=6113±49 K. 2010). Our analysis for the G-type star HD 209458 relies on Our more precise fbol and θ (see §3.3) lead to a significantly an accurate determination of Teff and log g, which is fixed more precise value for the effective temperature. in our fitting. This leads to a slightly higher value of σ(θ) compared to that derived from an analysis in which all pa- rameters are free to vary. 3.6 Stellar luminosity First, we determined the stellar luminosity from the bolo- metric flux and the Hipparcos parallax (L ∝ fbol ), which 3.4 Stellar radius Π2 leads to L⋆=1.76±0.14 L⊙ and 1.77±0.14 L⊙ using the Combining the angular diameter obtained in the previ- values of fbol obtained from the observed and theoreti- ous section with the trigonometric parallax from Hipparcos cal spectrum, respectively (see §3.5). Note σ(L⋆) is dom- (Π=20.15±0.80 mas, van Leeuwen 2007) for HD 209458 inated by the uncertainty in the parallax. Alternatively, 2 4 we derive the stellar radius R⋆=1.20 ± 0.05 R⊙ (see Ta- it is found that L/L⊙ = R/R⊙ Teff/Teff,⊙ , yielding ble 1). This value is in excellent agreement with that L⋆=1.77±0.14 L⊙ from R ⋆ (see §3.4 ) and Teff(see §3.1). from Boyajian et al. (2015) (1.20±0.06 R⊙), obtained from These values are compiled in Table 1. their limb-darkened angular diameter and the Hipparcos Our results are in good agreement with the val- parallax. But we also found our result to be consistent ues of L⋆=1.79±0.15 L⊙ and L⋆=1.77±0.01 L⊙ from with others from stellar evolution models such as those Boyajian et al. (2015) and Bonfanti et al. (2015), respec- from Bonfanti et al. (2015) (1.20±0.04 R⊙), Torres et al. tively. The former is obtained from a model atmosphere fit- +0.014 (2008) (1.155−0.016 R⊙), and our own analysis described ting to broad-band photometry and the latter is based on in §3.8 (1.20±0.06 R⊙). Note our precision is similar to stellar evolutionary models. Our value of L⋆=1.79±0.14 L⊙ that of Bonfanti et al. (2015), who employed PARSEC (ver- derived from PARSEC models (see §3.8) is also consistent sion 1.0, Bressan et al. 2012), but quite different to that with all these values. Note, however, the very small uncer- of Torres et al. (2008), who used Yonsei-Yale (Y2) series by tainty in the luminosity given by Bonfanti et al. (2015) even a though they used Padova Isochrones (version 1.0). Also from Yi et al. (2001) constrained by [Fe/H], Teff, and R⋆ (a is the semi-major axis of the orbit) using a likelihood function. stellar evolutionary models, Torres et al. (2008) found a rel- atively low value of L⋆=1.62±0.10 L⊙, but it is still in agree- ment with the others. 3.5 Bolometric flux and effective temperature
We determined the bolometric flux fbol = (2.289±0.011) × − − − 3.7 Stellar mass, mean density and gravity 10 8 erg cm 2 s 1 from the integration between 0.29 and 300 µm of the HD 209458 spectrum, extended towards the We applied two different approaches to determine two of UV (λ < 0.29 µm) with a Kurucz model with Teff=6071 three stellar parameters (mean density, mass, and gravity) K. The fluxes beyond 2.5 µm come from a Kurucz model from the stellar radius (see §3.4) combined with either the introduced to complete the SED (Bohlin 2010). The ultra- mass or the mean density from the literature or from the violet and infrared portions added to the SED respectively PARSEC models (see §3.8). First, given the radius and the 3M contribute only 1.6% and 2.9% of the total flux. The use of mass, we found the mean stellar density to be ρ = 4πR3 GM different models (with other stellar parameters values within and gravity g = R2 , where G is the gravitational constant. the uncertainties) to complete the observed STIS+NICMOS There is good agreement among results for the stellar mass spectrum has a negligible effect on the results presented here. M⋆=1.11±0.02 M⊙ (Bonfanti et al. 2015), 1.12±0.03 M⊙ −8 Alternatively, we obtained fbol= (2.298±0.011) × 10 (Torres et al. 2008), and 1.12±0.04 M⊙ (see §3.8). We used −2 −1 erg cm s from the best model derived in §3.1. Any of our values of M⋆ (see §3.8) and R⋆ (see §3.4) to arrive −3 our values of fbol (which are consistent with each other) is at ρ⋆=0.91±0.11 g cm and log g= 4.33±0.04 (with g in −2 ∼5 times more precise than that of Boyajian et al. (2015) cm s ) (see Table 1). −8 −2 −1 with fbol= (2.33±0.05) × 10 erg cm s , which they Second, we used our stellar radius and a value derived from a model fitted to broad-band photometry col- for the mean stellar density ascertained by Torres et al. lected from the literature. Casagrande et al. (2010) arrived (2008) from the (high signal-to-noise) HST/STIS transit −8 −2 −1 at fbol= (2.335±0.025) × 10 erg cm s , adopting light curve (Brown et al. 2001). Torres et al. (2008) deter- −3 [Fe/H] = 0.03±0.02, log g=4.50±0.04, Tycho 2, and 2MASS mined ρ⋆=1.024±0.014 g cm from the expression ρ⋆ = 3 3 photometry. 3π a Rp GP 2 R⋆ − ρp R⋆ , inserting the semi-major to stel- From the bolometric flux we determine the effective a 0.25 lar radius ratio ⋆ and the period P . They neglected the 4fbol R temperature (Teff = σθ2 , where σ is the Stefan- second term, which is generally very small because the plan- Boltzmann constant). We have thus found the effective tem- etary radius Rp ≪ R⋆. From this mean stellar density and perature to be Teff=6064±24 K from the CALSPEC spec- our R⋆ we calculated M⋆=1.26±0.15 M⊙ and more signif- − trum, and 6070±24 K from the best-fitting model, in con- icantly log g=4.38±0.06 (g in cm s 2) (see Table 1). The sonance with that derived in §3.1. These results (compiled estimated value for the uncertainty of log g derived in §3.8
MNRAS 000, 1–10 (2016) Accurate parameters for HD 209458 and its planet 7
then the output values are significantly different to the Sun’s Table 2. Stellar parameters (Teff, log g, τ, L, Mbol, M, R, and ρ) and 1σ uncertainties for the Sun inferred from our method values, in particular Teff=5837 K. In addition, we applied on the PARSEC models for the initial solar metallicity from corrections to be consistent with the IAU 2015 Resolution Bressan et al. (2012), MV=4.862±0.020 from Pecaut & Mamajek B3, on recommended nominal constants (Prˇsa et al. 2016). (2013) and B − V =0.653±0.003 from Ram´ırez et al. (2012). Cor- These corrections renormalize the inferred values from PAR- rections have been applied to be consistent with the IAU 2015 SEC to match the adopted ones for the Sun’s parameters Resolution B3 nominal values for R ,M , and L (Prˇsa et al. 15 ⊙ ⊙ ⊙ at the IAU 2015. The corrections for M⊙ , L⊙, and R⊙ 2016). are respectively of 0.034%, 0.47%, and 0.040%. The results are given in Table 2. Note the good agreement between our Parameter Value±uncertainty inferred values and those approved by the IAU, e.g. the ef-
Teff (K) 5778±8 fective temperature Teff=5772 K. log g [cm s−2] 4.435±0.006 To determine the stellar parameters of HD 209458 using τ (Gyr) 4.7±0.3 PARSEC Isochrones and the grid introduced in §2.3, we used 16 L/L⊙ 1.000±0.014 as input values the metallicity [Fe/H]=0.00±0.05 dex of Mbol (mag) 4.740±0.015 Torres et al. (2008), and our values MV= 4.18±0.09 mag M/M⊙ 0.998±0.004 and B − V = 0.549±0.013 mag, which were calculated from R/R⊙ 1.002±0.007 the Hipparcos parallax and the values of B and V obtained ρ/ρ 0.992±0.022 ⊙ in §3.2. The values of the stellar properties derived from this analysis (Teff, R⋆, L⋆, Mbol, log g, M⋆, ρ⋆, and τ) are given is smaller, but we argue that log g=4.38±0.06 (with g in in Table 1. We found negligible differences in the inferred − cm s 2) is more accurate and nearly model-independent, so stellar parameters when using steps in age of 0.1, 0.2, and we decided to use it as a fixed parameter in the fitting pro- 0.4 Gyr instead of 0.3 Gyr , as well as for steps in Zini as low cess described in §3.1. as 0.0005 instead of 0.005. Also when considering different steps (as low as 0.0001 M⊙) in Mini, which were generated from a linear interpolation. 3.8 Isochrones Bonfanti et al. (2015) also employed Padova Isochrones (v1.0) finding that HD 209458 has Teff=6084±63 K and In order to derive the stellar parameters from PARSEC mod- − log g=4.30±0.10 (with g in cm s 2), which are in good agree- els, we followed a procedure similar to that described by ment with our values Teff=6099±41 K and log g=4.33±0.04 Jørgensen & Lindegren (2005), but using B − V instead of − (with g in cm s 2). We note that we could have sub- Teff, and with flat priors. We estimated the different param- tracted 6 K to the inferred effective temperature since eters from the likelihood function, L, given in their Equation this is the formal difference between the output and in- 4, and integrating that over the model parameters, namely put values for the Sun (see Table 2), but we did not age (τ), initial mass (Mini), and initial metallicity (Zini). do it since the difference is within the uncertainties. L equals the probability of getting the observed data (MV, Bohlin et al. (2014) used the same HST data employed in B−V , and metallicity) for a given set of parameters (τ, Zini, this analysis and Kurucz models to arrive at Teff=6100 K, and Mini). For Teff, for instance, the estimated value and its − log g=4.20 (g in cm s 2), [ M ]=-0.04, and E(B − V )=0.003 variance are: H mag. From an spectroscopic analysis Santos et al. (2004) −2 found Teff=6117±26 K, log g= 4.48±0.08 (g in cm s ), −1 [Fe/H]=0.02±0.03, and ξt=1.40±0.06 km s . Their value E(Teff)= (L× Teff) dMini dτ dZini (1) Z for the effective temperature is higher than ours, 6071±20 K (see §3.1). Torres et al. (2008) derived Teff=6065±50 K, −2 2 log g=4.42±0.04 (g in cm s ), and [Fe/H]=0.00±0.05 dex, V ar(Teff)= (L× [Teff − E(Teff)] ) dMini dτ dZini (2) Z from a combination of different values in the literature. They − also obtained log g=4.361+0.007 (g in cm s 2) from stellar We employed the solar parameters to calibrate our −0.008 evolutionary models. Note the very small estimated uncer- method with the grid mentioned in §2.3. We adopted tainties for the last value. In general, all these values are for the Sun the values MV= 4.862±0.020 mag from consistent with each other. Our derived age τ=3.5±1.4 Gyr Pecaut & Mamajek (2013) and B − V =0.653±0.003 mag is also in good agreement with other values in the litera- from Ram´ırez et al. (2012), together with the initial so- +0.8 ture, such as 3.1−0.7 Gyr (Torres et al. 2008), 4.0±1.2 Gyr lar metallicity Zini=0.01774 given by Bressan et al. (2012). (Bonfanti et al. 2015), and 4±2 Gyr (Melo et al. 2006). See The modeled values for the Sun in the corresponding PAR- also §3.4 and §3.7 for the comparisons of R⋆, M⋆, and ρ⋆. SEC isochrone are MV= 4.841 mag, MB=5.530 mag, and Mbol=4.769 mag. Therefore, we had to introduce corrections to the model variables MB (subtracting 0.015 mag), and MV (adding 0.021 mag), in order to be consistent with the mag- nitudes from Pecaut & Mamajek (2013) and Ram´ırez et al. (2012). We also applied a correction to Mbol (subtracting 0.03 mag) to be in accordance with the IAU 2015 Resolu- 15 The Sun’s mass adopted in PARSEC was provided by Alessan- tion B2 value of 4.74 mag14. If these corrections are not made dro Bressan (private communication). 16 This leads to a solar value for Zini, taking into account the expected evolution of the metallicity for a star similar to the Sun 14 http://www.iau.org/news/announcements/detail/ann15023/ (see Bonfanti et al. 2015).
MNRAS 000, 1–10 (2016) 8 C. del Burgo and C. Allende Prieto
Table 3. Planetary and orbital properties and corresponding 1σ (whose uncertainty is dominated by that in the parallax) uncertainties of HD209458b: radius, mass, mean density, grav- and the surface gravity, computed adopting a precise value ity, and semimajor axis. We used our R⋆ obtained in §3.4 and of the mean density from the HST/STIS transit light curve Rp (Torres et al. 2008). The determination of the planetary pa- the ratios R⋆ from Torres et al. (2008) (R&rT) and Evans et al. (2015) (R&rE) to determine Rp. R&gT: Same radius and log gp rameters also benefits from the high accuracy achieved for from Torres et al. (2008) to determine the mass. ρp was obtained the stellar parameters. from Rp and Mp (M&R). R&aT: a was ascertained from the ratio Our determination of the angular diameter θ is more a R⋆ from Torres et al. (2008) and R⋆. than four times more precise than that from interferometry by Boyajian et al. (2015). These authors warned about their Parameter Value±uncertainty Note angular size for HD 209458 being at the resolution limit of CHARA/PAVO and that they used calibrators up to ∼30% R (R ) 1.41±0.06 R&rT; see §3.9 p J smaller than this object. Nevertheless, their value is consis- q 1.42±0.06 R&rE; see §3.9 tent with ours. Mp (MJ) 0.74±0.06 R&gT; see §3.10 −3 ρp (gr cm ) 0.32±0.05 M&R; see §3.10 While interferometry can be only applied to bright −2 log gp [cm s ] 2.963±0.005 Torres et al. (2008) nearby stars, absolute flux spectrophotometry com- a (au) 0.0490±0.0020 R&aT; see §3.10 pared with stellar atmosphere models yields very pre- cise angular diameters for a wide range of luminosities. Allende Prieto & del Burgo (2016) have shown that while 3.9 Planetary radius and semi-major axis for a bright A star such as Vega the achieved uncertainty of 0.4% in θ is similar to that from interferometry, for fainter We determined the radius of HD 209458b from the radius of A stars (V∼3-6 mag) the precisions from the absolute flux the host star and the transit depth. The latter provides the comparison method are several times better than the respec- planet to star radius ratio. Torres et al. (2008) found that tive interferometric precisions. Rp = 0.12086±0.00010 and a =8.76±0.04 from the STIS R⋆ R⋆ For the faintest A stars in the sample of light curve (Brown et al. 2001). Allende Prieto & del Burgo (2016) (7.1
MNRAS 000, 1–10 (2016) Accurate parameters for HD 209458 and its planet 9 tightly constrained by a precise value of the effective temper- 5 CONCLUSIONS ature. While the metallicity permits to constrain the stellar Absolute flux spectrophotometry can efficiently provide mass and age (see review of Chaplin & Miglio 2013, and nearly model independent stellar angular diameters and ef- references therein). Mean density, surface gravity and ra- fective temperatures for a broad range of spectral types (A- dius determined from this technique are shown to be largely type and sun-like stars). The values of the angular diame- model independent. ters derived from flux ratios are several times more precise Huber et al. (2013) found, for more than half of their than those from interferometry. Combining angular diam- sample, discrepancies greater than 50% between the densi- eters with accurate parallaxes it is possible to obtain very ties derived from asteroseismology and those from transit accurate radii. For transiting systems, the high level of accu- models assuming circular orbits (mostly underestimated). racy in the stellar radius can be translated to the planetary They concluded this was due to systematics in the modeled radius Rp from the transit depth determination, and then impact parameters or the presence of candidates in eccen- to the parameters that are related to Rp. We have illus- tric orbits. It will be eventually possible to compare, for an trated the application of this methodology to the determi- statistically significant sample of stars, the values of their nation of fundamental parameters for the well-known star radii obtained from all the above-mentioned techniques with HD 209458 and its transiting exoplanet, obtaining 1σ un- uncertainties better than 1%. This will allow us to further certainties of 0.3% in Teff, and 4% in the radii of the star search for and reduce systematic errors. and its planet. The parallax to be provided by Gaia will It is well-known that stellar activity (e.g., spots) may allow us to significantly improve the accuracy and precision affect the determination of the planetary parameters from in the stellar/planetary radii, which will be then limited by light curves, such as the planet-to-star radius ratio. Plan- the precision in the stellar angular diameter (0.75%). etary radii values are generally prone to such systematics. However, apart from the presence of transits, no variabil- ity has been reported for HD 209458. We stress the good consistency between the values of the planetary radius ob- ACKNOWLEDGEMENTS tained from our stellar radius and the HST/STIS light curve This work has been supported by Mexican CONACyT re- in the optical, and the Spitzer/IRAC light curve in the in- search grant CB-2012-183007. CAP is thankful to the Span- frared. We conclude that the determination of the radius of ish MINECO for support through grant AYA2014-56359-P. the planet is not affected by this issue. This research has made use of the SIMBAD database, oper- As previously mentioned the parallax uncertainty is the ated at CDS, Strasbourg, France and NASA’s Astrophysics main contributor to that in the stellar radius. If using the Data System. modeled value Π=21.12±0.72 mas by Torres et al. (2008) the stellar radius decreases to R⋆=1.15±0.04 R⊙. From this value and our determination of M⋆ using PARSEC (see §3.8), −3 REFERENCES we found ρ⋆=1.05±0.12 g cm and log g= 4.37±0.03 (g in − cm s 2), instead of the values obtained from the first ap- Ali A. W., Griem H. R., 1966, PhRv, 144, 366 −3 proach presented in §3.7, which are ρ⋆=0.91±0.11 g cm Allende Prieto C., del Burgo C., 2016, MNRAS, 455, 3864 and log g= 4.33±0.04 (g in cm s−2). Regarding the sec- Allende Prieto C., Beers T. C., Wilhelm R., Heidi J., Rockosi C. M., Yanni B., Lee Y. S., 2006, ApJ, 636, 804 ond approach described in §3.7, the new value of R⋆ has Allende Prieto C., 2001, ApJ, 547, 2001 a small impact on log g (4.36 instead of 4.38; with g in −2 Asplund, M., Grevesse, N., & Sauval, A. J. 2005, in Barnes T. cm s ), but it reduces M⋆ from 1.26 to 1.10 M⊙. Note G., II, Bash F. N., eds, ASP Conf. Ser. Vol. 336, COSMIC the latter is closer to the mass derived from PARSEC models ABUNDANCES as Records of Stellar Evolution and Nucle- in §3.8. Concerning Rp, it drops by ∼5% to 1.35±0.05 RJ osynthesis in Honor of David L. Lambert. Astron. Soc. Pac., when using the parallax of Torres et al. (2008). a also drops, San Francisco, p. 25 from 0.0490±0.0020 au to 0.0467±0.0016 au. The latter is Auer L., 2003, in Hubeny I., Mihalas D., Werner K., eds, ASP closer to the value of a=0.0471±0.0005 au from Torres et al. Conf. Ser.Vol. 288, Stellar Atmosphere Modeling. Astron. Soc. 17 (2008) . Finally, Mp = 0.67±0.05 MJ when using the par- Pac., San Francisco, p. 3 allax of Torres et al. (2008). This value is similar to the dy- Baines E. K., McAlister H. A., ten Brummelaar T. A., Sturmann J., Sturmann L., Turner N. H., Ridgway S. T., ApJ, 701, 154 namical mass of 0.64±0.09 MJ obtained by Snellen et al. Barklem P. S., Piskunov N., O’Mara B. J., 2000, A&AS, 142, 467 (2010). Bautista M. A., 1997, A&AS, 122, 167 Our results confirm that the radius of HD 209458b is Bessell M., Murphy S., 2012, PASP, 124, 140 too large for the composition and age of its host star, chal- Bohlin R. C., 2007, The Future of Photometric, Spectrophoto- lenging current theory models of internal structure. Gaia metric and Polarimetric Standardization, 364, 315 will provide a reliable parallax measurement, which can be Bohlin R. C., 2010, AJ, 139, 1515 used to improve on the stellar and planetary parameters, Bohlin R. C., 2014, AJ, 147, 127 in particular the stellar radius, with an expected precision Bohlin R. C., Gordon K. D., Tremblay P.-E., 2014, PASP, 126, better than 1%. This will be useful to check the goodness 711 of the stellar evolutionary models through the comparison Bohlin R. C., Landolt A. U., 2015, AJ, 149, 122 with direct and nearly model-independent values. Bonfanti A., Ortolani S., Piotto G., Nascimbeni V., 2015, A&A, 575, 18 Boyajian T., von Braun K., van Belle G., et al., 2012, ApJ, 757, 112 17 a Note that from the values for R⋆ and R⋆ of these authors we Boyajian T., von Braun K., Feiden G. A., et al., 2015, MNRAS, +0.0006 find a=0.0470−0.0007 au. 447, 846
MNRAS 000, 1–10 (2016) 10 C. del Burgo and C. Allende Prieto
Bressan A., Marigo P., Girardi L., Salasnich B., Dal Cero C., Snellen I. A. G., de Kok R. J., de Mooij E. J.W., Albrecht S., Rubele S., Nanni A. 2012, MNRAS, 427, 127 2010, Science, 465, 1049 Brown T. M., Charbonneau D., Gilliland R. L., Noyes R. W., Stehl´eC., 1994, A&AS, 104, 509 Burrows A., 2001, ApJ, 552, 699 Stehl´eC., Hutcheon R., 1999, A&AS, 140, 93 Casagrande L., Ram´ırez I., Mel´endez J., Bessell M., Asplund M., Tang J., Bressan A., Rosenfield Ph., Slemer A., Marigo P., Girardi 2010, A&A, 512, 54 L., Bianchi L., 2014, MNRAS, 445, 4287 Chaplin W. J., Miglio A., 2013, ARA&A, 51, 353 Torres G., Winn J. N., Holman M. J., 2008, ApJ, 677, 1324 Charbonneau D., Brown T. M., Latham D. W., Mayor M., 2000, Tsuji T., 1964, Ann.Tokyo Astron. Obs., 9, 1 ApJ, 529, 45 Tsuji T., 1973, A&A, 23, 411 Chen Y., Bressan A., Girardi L., Marigo P., Kong X., Lanza A., Tsuji T., 1976, PASJ, 28, 543 2015, MNRAS, 452, 1068 van Belle G. T., von Braun K., 2009, ApJ, 694, 1085 Chen Y, Girardi L., Bressan A., Marigo P, Barbieri M., Kong X., van Leeuwen F., 2007, A&A, 474, 653 2014, MNRAS, 444, 2525 von Braun K., Boyajian T. S., van Belle G. T., et al., 2014, ApJ, Cohen M., Wheaton Wm. A., Megeath S. T., 2003a, AJ, 126, 694, 1085 1090 Yi S. K., Demarque P., Kim Y.-C., Lee Y.-W., Ree C. H., Lejeune Cohen, M., Megeath, S. T., Hammersley, P. L., Mart´ın-Luis, F., T., & Barnes S., 2001, ApJS, 136, 417 & Stauffer, J. 2003b,AJ, 125, 2645 Cunto W., Mendoza C., Ochsenbein F., Zeippen C. J., 1993, Bull. This paper has been typeset from a TEX/LATEX file prepared by Inf. Cent. Donnees Stellaires, 42, 39 the author. Cutri R. M., Skrutskie M. F., van Dyk S., et al., 2003, VizieR Online Data Catalog II/246 del Burgo C., Allende Prieto C., Peacocke T., 2010, J. Instrum., 5, 1006 de Kok R. J., Brogi M., Snellen I. A. G., Birkby J., Albrecht S., de Mooij E. J. W., 2013, A&A, 554, 82 ESA, The Hipparcos and Tycho catalogues. Astrometric and photometric star catalogues derived from the ESA Hippar- cos Space Astrometry Mission, Publisher: Noordwijk, Nether- lands: ESA Publications Division, 1997, Series: ESA SP Series vol no: 1200, ISBN: 9290923997 Evans Th. M., Aigrain S., Gibson N., Barstow J. K., Amundsen D. S., Tremblin P., Mourier P., 2015, MNRAS, 451, 680 Fitzpatrick E. L., 1999, PASP, 111, 63 Gregg M. D., Silva D., Rayner J., et al. 2006, The 2005 HST Calibration Workshop: Hubble After the Transition to Two- Gyro Mode, 209 Hubeny, I., Hummer, D. G., & Lanz, T., 1994, A&A, 282, 151 Huber D., Chaplin, W. J.; Christensen-Dalsgaard, J., et al., ApJ, 767, 127 Irwin A. W., 1981, ApJS, 45, 621 Jones J., White R.J., Boyajian T., et al. 2015, ApJ, 813, 58 Jørgensen B. R. & Lindegren L., 2005, A&A, 436, 127 Koesterke, L., 2009, AIP Conf. Ser., 1171, 73 Koesterke, L., Allende Prieto, C., & Lambert, D. L. 2008, ApJ, 680, 764 Lee, H.-W., & Kim, H. I. 2004, MNRAS, 347, 802 Ligi R., Creevey O., Mourard D., et al., 2015, A&A, 586, 94 Luzum B., Capitaine N., Fienga A., et al. 2011, Celestial Mechan- ics and Dynamical Astronomy, 110, 293 Ma´ız Apell´aniz J., 2006, AJ, 131, 1184 Mann A. W., von Braun K., 2015, PASP, 127, 102 Mardling R. A., 2007, MNRAS, 382, 1768 Melo C., Santos N. C., Pont F., Israelian G., Mayor M., Queloz D., Udry S., 2006, A&A, 460, 251 M´esz´aros Sz., Allende Prieto C., Edvardsson B., et al., 2012, AJ, 144, 120 Nahar S. N., 1995, A&A, 293, 967 Mohr P. J., Newell D. B., Taylor B. N. 2015, arXiv:1507.07956 Pecaut M. J., Mamajek E. E., 2012, ApJS, 208, 9 Powell, M. J. D. (2002). ”UOBYQA: unconstrained optimization by quadratic approximation”. Mathematical Programming, Series B (Springer) 92: 555-582 Prˇsa A., Harmanec P., Torres G., et al., 2016, AJ, accepted, arXiv:1605.09788 Ram´ırez I., Michel R., Sefako R., Tucci Maia M., Schuster W. J., van Wyk F., Mel´endez J., Casagrande L., Castilho B. V., 2012, ApJ, 752, 5 Santos N. C., Israelian G., Mayor M., 2004, A&A, 415, 1153
MNRAS 000, 1–10 (2016)