Calibrating the α parameter of convective efficiency using observed stellar properties
X.S. Wu, S. Alexeeva, L. Mashonkina, L. Wang, G. Zhao, F. Grupp
aa r X i v : . [ a s t r o - ph . S R ] A p r Astronomy & Astrophysicsmanuscript no. Alpha_CMA_I_00_017 c (cid:13)
ESO 2018July 13, 2018
Calibrating the α parameter of convective efficiency usingobserved stellar properties X.S. Wu , , , S. Alexeeva , L. Mashonkina , L. Wang , , G. Zhao , and F. Grupp , Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, 20 Datun Road,Chaoyang District, Beijing 100012, China Max-Planck-Institut für Extraterrestrische Physik, Giessenbachstrasse, D-85748 Garching, Germany University of the Chinese Academy of Sciences, 19A Yuquan Road, Shijingshan District, Beijing, 100049, China Institute of Astronomy, Russian Academy of Sciences, RU-119017 Moscow, Russia Universitäts Sternwarte München, Scheinerstr. 1, D-81679 München, GermanyReceived ; accepted
ABSTRACT
Context.
Synthetic model atmosphere calculations are still the most commonly used tool when determining precise stellar parame-ters and stellar chemical compositions. Besides three-dimensional models that consistently solve for hydrodynamic processes, one-dimensional models that use an approximation for convective energy transport play the major role.
Aims.
We use modern Balmer-line formation theory as well as spectral energy distribution (SED) measurements for the Sun andProcyon to calibrate the model parameter α that describes the e ffi ciency of convection in our 1D models. Convection was calibratedover a significant range in parameter space, reaching from F-K along the main sequence and sampling the turno ff and giant branchover a wide range of metallicities. This calibration was compared to theoretical evaluations and allowed an accurate modeling ofstellar atmospheres. Methods.
We used Balmer-line fitting and SED fits to determine the convective e ffi ciency parameter α . Both methods are sensitive tothe structure and temperature stratification of the deeper photosphere. Results.
While SED fits do not allow a precise determination of the convective parameter for the Sun and Procyon, they both favorvalues significantly higher than 1.0. Balmer-line fitting, which we find to be more sensitive, suggests that the convective e ffi ciencyparameter α is ≈ . ≈ . ffi ciency as stars evolve to the giant branch is more dramatic than predicted by models. Key words.
Stars: atmospheres – Stars: fundamental parameters – Stars: late-type – Stars: Balmer lines
1. Introduction
One-dimensional (1D) stellar model atmospheres are widelyused for determining basic physical parameters, such as e ff ectivetemperatures, surface gravities, luminosities, masses, and chem-ical compositions of stars at di ff erent evolutionary stages. Theyalso serve as boundary conditions for stellar evolution codes.While codes for 3D model atmospheres self-consistentlysolve for convection in the atmospheric layers, 1D codes applya simplified approach such as described by Böhm-Vitense 1958(BV hereafter) or Canuto & Mazzitelli 1991, 1992 (CMA here-after). As these formulations do not self-consistently determinethe convective flux, at least one calibration parameter is required:the α -parameter, which sets the e ffi ciency of convection.Recent calibrations of the 1D α -parameter based on 3D the-oretical model atmospheres have been presented by Magic et al.2014 (hereafter, Magic14) and earlier by Ludwig et al. (1999).The primary goal of this paper is to compare observationalconstraints on α CMA , the α -parameter in the formulation ofCanuto & Mazzitelli (1992), to theoretical predictions of 3Dmodels. This will allow us to use a proper α CMA for 1D modelatmospheres across a wide range of stellar parameter space.
Send o ff print requests to : F. Grupp; e-mail: [email protected] We use the approach as Fuhrmann et al. 1993 (FAG93 here-after), using the dependence of Balmer line profiles on the at-mospheric structure, which we call the α CMA -parameter. For thistest we use a sample of stars (including the Sun) with stellarparameter start values determined independently of model at-mospheres, for example, by astrometrical methods. As anothertool for determining α CMA , we use stellar spectral energy dis-tributions (SED) in the visible and near-UV spectral range. Thismethod is of course limited to objects with good absolute cali-bration of the SED. We here use the Sun and Procyon.On the model side, the adopted model atmosphere code is theplane-parallel, chemically homogeneous, local thermodynami-cal equilibrium (LTE) code MAFAGS-OS presented by Grupp(2004a). Stellar convection is treated in MAFAGS-OS accord-ing to the formalism of CMA. The authors take into account aspectrum of turbulent eddies with di ff erent sizes, which is moreadvanced than the one single eddy in the mixing-length the-ory (MLT; Böhm-Vitense 1958). The molecular weight withinthe convective zone is variable, a detail neglected by MLT.Nevertheless, it is a parametric and simplified approach, not aself-consistent numerical formalism. MLT furthermore assumesthat the turbulence is incompressible and sets the mixing-length Λ = α H p , where H p denotes the local pressure scale height,and leaves the so-called mixing-length parameter α (hereafter, Article number, page 1 of 14 & Aproofs: manuscript no. Alpha_CMA_I_00_017 α MLT ) as a free parameter. Therefore, calibrations by comparingthe models with the observational data are necessary.Some previous works suggest that α MLT is greater than unityfor the Sun and varies with stellar parameters. For example,Miglio & Montalbán (2005) constrained α MLT of the α Cen-tauri A + B systems using a method from asteroseismology andfound an α MLT for component B higher by 10% than that of A.Moreover, α MLT was found to be strongly dependent on stel-lar mass (e.g., Yıldız et al. 2006), but not on metallicity (e.g.,Ferraro et al. 2006). In contrast, using the solar-like stars inthe Kepler field, Bonaca et al. (2012) indicated that α MLT corre-lates more significantly with metallicity than T e ff or log g . With2D hydrodynamical models of dwarf stars, Ludwig et al. (1999)showed that α MLT decreases with increasing T e ff and log g , andthis was confirmed by 3D simulations of Trampedach (2007).Canuto & Mazzitelli (1992) introduced Λ = α z by takinginto account local e ff ects, where z is the distance to the top of theconvective zone, and its value is close to the polytrope H p . α = Λ = z in Canuto & Mazzitelli (1991).In previous versions of the MAFAGS-OS program, the α accord-ing to the CMA theory (hereafter, α CMA ) is fixed to 0.82, becausethe evolutionary stage of the Sun, as determined by its internalstructure, can be well reproduced by this value (Bernkopf 1998).However, this is only one calibration point in parameter space,there is no evidence that stars throughout the whole H-R diagramnecessarily have the same α CMA values.In this paper, we analyze a sample of extensively studied"standard stars" with di ff erent chemical compositions and evo-lutionary stages. All of them have angular diameter data ob-tained with modern interferometers, which – together with dis-tance measurements from H ipparcos – enable direct measure-ments of stellar radii, e ff ective temperature, and surface gravi-ties. The variation of α CMA across the H-R diagram can thereforebe constrained.
2. Stellar samples and basic stellar parameters
Our sample consists of ten dwarf and turno ff stars as well asone giant, with e ff ective temperatures ranging from 4600 K to6600 K and metallicities ([Fe / H]) from − + ff er et al. 1998) on the 2.2mtelescope at Calar Alto Observatory. The optical spectra havesignal-to-noise ratios of >
200 and resolving powers ( R = λ/ ∆ λ )of ≈ ff er et al. 1998). Normalizationwas performed manually. The sample is shown in the physicalHertzsprung-Russell-Diagram (HRD) in Fig. 1. Interferometricdiameter determination for eight of the program stars is avail-able (see Table 1).To minimize the di ff erent instrumental response e ff ects, thespectroscopic parameters ( T e ff , log g , [Fe / H] and ξ ) were takenfrom Fuhrmann (1998, 2004), who derived them with the sameFOCES spectra as we used in this study. The available in-terferometric observations provide accurate bolometric fluxes( F BOL ) with typical uncertainties of ∼
2% (e.g., Boyajian et al.2012). Combined with the interferometrically measured stellarangular diameter ( θ ) measurements with typical relative uncer-tainties of 1 ∼
3% for our samples, the uncertainties of T e ff = FG would like to suggest defining a "standard star" as a star that -as soon as it is studied in detail - is found to be pretty "non-standard".Procyon and HD 140283 are stars that fit this category very well. T eff (K) l og g HD 6582HD 10700 HD 22049HD 39587 HD 103095HD 122563HD 217014HD 45282HD 140283Procyon Sun -2.0-1.00.0 [ F e / H ] Fig. 1.
HRD of the program stars. Color encodes metallicity, size en-codes stellar diameter measurements. F BOL /θ ) / are lower than 1.5%, which is precise enoughas cross-validations of the spectroscopic data.Therefore, whenever possible, we used interferometric T e ff and θ given by recent measurements with VLTI or CHARA tocompose interferometric parameters. The spectroscopic and in-terferometric parameters are summarized in Table 1.The log g of stars that have interferometric measurementsparameters was derived using the direct angular diameter ofthe star, its distance from the Hipparcos parallax (van Leeuwen2007), and its mass, either from the astrometric measurementsof a binary companion (for Procyon A), or by fitting the stellarevolutionary track in the H-R diagram. The parameters metallic-ity ([Fe / H]) and micro-turbulence velocity ( ξ ), which cannot beobtained by interferometric means, were kept the same as thosein the spectroscopic group. The solar abundance mixture wastaken from Lodders et al. (2009). For the five low-metallicitystars in our sample with [Fe / H] < − / H] = − / H] = − / H] = − / H] = − / H] = − α -elements enhancements of [ α / Fe] = +
3. Model atmospheres
For each set of stellar parameters, model atmospheres were cal-culated using the model atmosphere code MAFAGS-OS (Grupp2004a,b; Grupp et al. 2009). The opacity-sampling (OS) codeis based on the opacity distribution function (ODF) version ofT. Gehren (1979). In the reprogrammed version of Reile (1987),the code relies on the following basic assumptions: – Plane-parallel 1D geometry. – Chemical homogeneity throughout the atmosphere. – Hydrostatic equilibrium. – Convection is treated according to the formalism ofCanuto & Mazzitelli (1991, 1992). – No chromosphere or corona. – Local thermodynamical equilibrium. – Flux conservation throughout all 80 layers.While these assumptions can break down for hot stars, stars withvery extended atmospheres, and the coolest stars, they are valid
Article number, page 2 of 14u et al.: CMA calibration
Table 1.
Best-fitting α CMA HD T e ff (K) θ LD Ref. log g a [Fe / H] a T ∗ e ff (K) The best α CM χ (d.o.f)H α H β H γ H α H β H γ Interferometric and IRFMSun 5780 ±
40 – 4.44 0.00 5780 2.0 ≥ ≥ ±
50 5.448 ± − ≥ ≥ ±
50 2.078 ± − ≥ ≥ ±
100 0.679 ± − ≥ ±
50 1.051 ± − ≥ ≥ ±
85 0.972 ± − ≥ ±
55 0.748 ± + ≥ ≥ ±
75 2.148 ± − ≥ ±
50 – Casagrande et al. (2010) 3.70 − ±
50 0.948 ± − ±
50 – Casagrande et al. (2010) 3.07 − − ≥ ≥ − ≥ − ≥ ≥ − ≥ + ≥ ≥ − ≥ Notes. a References of the adopted log g and [Fe / H] are in the text. T ∗ e ff denotes the best e ff ectivetemperatures obtained from the fitting procedure. for temperatures from 4000 through 15000 K and for gravitiesfrom the main sequence down to log g ≈ < . Figure 2 shows the influence of the convective e ffi ciency param-eter α CMA on temperature stratification, gas pressure, and elec-tron pressure of a solar atmospheric model. While a low valueof α CMA results in low convective flux, high α CMA leads to anincrease of the convective flux and - as the total flux is preservedthrough all layers - lower radiative flux in the convection zoneof the stellar atmosphere. Low radiative flux leads to lower tem-perature gradients. Therefore, the high- α CMA case shows a lowertemperature in the inner region of the stellar atmosphere.For our program stars, increasing α CMA from 0.5 to 2.5 leadsto decreasing temperatures in the inner atmosphere by up to1000–1300 K at log( τ ) ≃ . ff ect of a variation of α CMA on the solar SED is shownin Fig. 3. Flux changes of up to 5% of the solar SED in the UVand of 1% in the near-IR region are notable. There are somenode points at 550, 1350, and 2050 nm around which the fluxesare insensitive to mixing-length parameters, while the continuumof the SED between each two shows opposite tendencies with α CMA .The trends in model structures and fluxes are similar for otherstars in our sample, with the only exception of the A0V starVega. As Vega has no convection zone reaching the stellar at-mosphere, no change with α CMA is present in the stellar flux.
4. Spectral energy distributions
In the following section we directly compare the computed ab-solute fluxes to the observational data to calibrate α CMA . Abso-lute flux data are only available for a few stars. For our sample,reliable absolute SED data are only available for the Sun andProcyon.
Neckel & Labs (1984) measured solar irradiance between 3300and 12500 Å by combining the absolute fluxes of high-resolutionspectra obtained with the Fourier Transform Spectrometer (FTS)at the McMath Solar Telescope on Kitt Peak. The spectra wereaveraged every 10 Å between 3305 and 6295 Å, and every 20 Åbetween 6310 and 8690 Å.Thuillier et al. (2003) presented the solar irradiance from2000 Å to 24000 Å obtained with the SOLSPEC and SOSP spec-trometers mounted on space missions. However, for wavelengthranges above 8700 Å, their data are sampled with a rate di ff erentfrom that in visible and UV domains. Therefore we only adoptedthe spectra below 8700 Å to compare with our solar fluxes.Figure 4 shows the data and absolute deviations of solar ir-radiance of Neckel & Labs (1984) and Thuillier et al. (2003) to-gether with the computed solar fluxes for three di ff erent α CMA values of 0.5, 1.5, and 2.5. A visual check prefers a fit betweenblue ( α CMA = .
5) and green ( α CMA = . α CMA from these data. Neverthe-less, data points with values below 1.0 can be excluded from the
Article number, page 3 of 14 & Aproofs: manuscript no. Alpha_CMA_I_00_017 F λ ( W m − ◦ A − ) ◦ A)0.000.050.10 a b s o l u t e d e v i a t i o n F λ ( W m − ◦ A − ) ◦ A)−0.050.000.050.100.150.20 a b s o l u t e d e v i a t i o n Fig. 4.
Comparison of solar absolute fluxes ( solid dots ) to synthetic flux levels with α CMA = . red ), 1.5 ( green ), and 2.5 ( blue ).Observational data were taken from Neckel & Labs (1984) ( upper panel ) and Thuillier et al. (2003) ( lower panel ). The absolute deviation ( F model − F obs ) is plotted in the lower part of each panel. O ff sets of 0.05 were added to improve the visualization. Neckel data. The data of Thuillier et al. (2003) would only allowfor very low convective e ffi ciency if the blue region is ignored. Ingeneral, we find an overestimation of the solar flux in the regimebelow 5000 Å. This can be caused by missing atomic or molec-ular opacity. Variations of convective e ffi ciency do not changethis region significantly, thus can be excluded as the cause ofthis shortcoming. The ultraviolet spectrum of Procyon A has been obtained withthe Space Telescope Imaging Spectrograph (STIS) onboard theHubble Space Telescope (HST) with high-quality calibration in the range from 220 nm up to 405 nm. As the STIS spectra spanseveral echelle orders and overlap between each pair of orders,we averaged bins of 2 nm width for the STIS measured data andMAFAGS-OS model calculations to allow a most direct com-parison. This comparison is shown in Fig. 6 for α CMA rang-ing from 0.5 to 2.5 for a model with a T e ff / log g / [Fe / H] /ξ = / / − . / ∼
35% for Mg ii H & K at 280 nm. The average devi-ation is within 4%.Figure 7 displays the 2D plots of χ versus α CMA (x axes)and the scaling factor of computed flux (y-axes) for four modelswith di ff erent temperatures. The di ff erence of ∼
70 K in T e ff causes a di ff erence of ∼
4% in the total fluxes. The scaling factors
Article number, page 4 of 14u et al.: CMA calibration CM χ Neckel & Labs, 1984 CM χ Neckel & Labs, 1984 CM χ Thuillier et al. 2003 CM χ Thuillier et al. 2003
Fig. 5.
Relations of χ of observational and computed fluxes versus α CMA for the solar irradiance of Neckel & Labs (1984) and Thuillier et al.(2003). Observational data between 4500 and 8700 Å (left plots) and between 5500 and 8700 Å (right plots) were taken into account. were introduced to account for the o ff sets between the computedfluxes and the observational data to first-order approximation.The relative uncertainty ( ∼ ff set of up to 2% of the observational SED. Therefore the ±
3% of the scaling factor on the total flux is reasonable. Thederived α CMA for each model was estimated by minimizing the χ with the observational SED over the 2D plane. We found thatmodel (c) and (d) led to smaller χ with the observational data,which favors higher T e ff for Procyon A. Models (b), (c), and (d)led to quite consistent α CMA ≃ . χ , while model(a) favors smaller α CMA ≃ .
5. NLTE calculations of Balmer-line profiles
Our analysis is based on the nonlocal thermodynamical equilib-rium (NLTE) line formation for H i using the method describedin Mashonkina et al. (2008). In brief, the model atom includeslevels with principal quantum numbers up to n ≤
19. We checkedthe influence of inelastic collisions with neutral hydrogen atoms,as computed following Steenbock & Holweger (1984), on thestatistical equilibrium (SE) of H i and resulting profiles of theBalmer lines in the models. The di ff erences in normalized fluxesbetween including and neglecting H + H collisions are smallerthan 0.0005 of the continuum flux for the core-to-wing transi-tion region that is most sensitive to T e ff variations. Therefore allthe NLTE calculations were performed with pure electronic col-lisions with the exception of a very metal poor (VMP) giant at- mosphere 4600 / / − . / / − H = ffi cients were then used by the codeSIU (Reetz 1991) to calculate the synthetic line profiles. Theabsorption profiles of Balmer lines were computed by convolv-ing the profiles resulting from the thermal, natural, and Starkbroadening, as well as self-broadening. The Stark profile wasbased on the unified theory as developed by Vidal et al. (1970,1973, VCS). The calculations of Stehle (1994) based on a dif-ferent theory agreed reasonably well with the data of VCS.For self-broadening, we applied the Lorentz profile with a half-width computed using the cross-section and velocity parameterfrom Barklem et al. (2000). As shown by Barklem et al. (2000),"the complete profiles obtained from overlapping line theory arevery closely approximated by the p-d component of the relevantBalmer line". This gives ground to apply the impact approxima-tion to describe the self-broadening e ff ect on H α , H β , and H γ .We selected the model 6530 / / − .
05 to show the behaviorof the departure coe ffi cients b i = n NLTE i / n LTE i of the relevant lev-els of H i (Fig. 8). Here, n NLTE i and n LTE i are the SE and thermal(Saha-Boltzmann) number densities, respectively. Very similarbehavior of b i was found for all the model atmospheres investi-gated in this study. The ground state and the n = n = Article number, page 5 of 14 & Aproofs: manuscript no. Alpha_CMA_I_00_017 T ( K ) α CM =0.5α CM =1.5α CM =2.5 −6 −5 −4 −3 −2 −1 0 1 2 log(τ ) −1000−5000 ∆ T ( K ) P α CM =0.5α CM =1.5α CM =2.5 −6 −5 −4 −3 −2 −1 0 1 2 log(τ ) ∆ P -3 -2 -1 P e α CM =0.5α CM =1.5α CM =2.5 −6 −5 −4 −3 −2 −1 0 1 2 log(τ ) −6000−4000−20000 ∆ P e Fig. 2.
Temperature ( T ), pressure ( P ), and electron pressure ( P e ) versusoptical depth log( τ ) with three di ff erent mixing-length parameters α CMA in the MAFAGS-OS models of Sun. trolled by H α . In the layers where the continuum optical depthdrops below unity, H α serves as the pumping transition, whichresults in an overpopulation of the upper level. For instance, b ( n = ≃ .
017 around log τ = − .
57, where the H α core-to-wing transition is formed. Starting from log τ ≃ − . α core cause an un-derpopulation of the n = α , NLTE leads to a weak-ening of the core-to-wing transition compared to the LTE case(Fig. 9).To compare them with observations, the computed syn-thetic profiles were convolved with a profile that combines in-strumental broadening with a Gaussian profile of 3.2 km s − to F λ ( × e r g s − c m − ◦ A − ) α CM =0.50α CM =0.82α CM =1.50α CM =2.50 R e s i d u a l ( % ) Fig. 3.
Upper panel: Computed solar flux F λ with di ff erent α CMA . Thespectrum is sampled every 50 Å along the wavelength. The transmissioncurves of U, B, V, R, I in Bessell photometric system, and J, H, K s in2MASS system are also overplotted (from left to right). The residualsto fluxes with α CMA = .
82 are plotted in the lower panel. F λ ( × − e r g s − c m − n m − ) STISα CM =0.50α CM =1.50α CM =2.50
200 250 300 350 400 450Wavelength (nm)0.00.51.0 a b s o l u t e d e v i a t i o n Fig. 6.
Comparison of absolute fluxes of Procyon A ( solid lines ) ob-tained with HST / STIS with MAFAGS-OS model fluxes with α CMA = . red ), 1.5 ( green ), and 2.5 ( blue ) in model (a). The fluxes weresummed every 2 nm. The absolute deviation ( F model − F obs ) were plottedin the lower panel. O ff sets of 1.0 in the upper panel and 0.5 in the lowerpanel were added for clarity. − for di ff erent stars and broadening caused by macro-turbulence with a radial-tangential profile of 4 km s − . Rota-tional broadening was only taken into account for Procyon,with V sin i = − (Fuhrmann 1998). All the remainingdwarf and subgiant stars were assumed to be slow rotators, with V sin i ≤ − . It is worth noting that the thermal broaden-ing of Balmer lines, with the most probable velocity of about V t =
10 km s − in the atmospheres of the investigated stars,is more pronounced than the broadening caused by rotation,macroturbulence, and instrumental profile together. Therefore,the uncertainty in the external broadening parameters only hasan minor e ff ect on the computed Balmer-line wings. Article number, page 6 of 14u et al.: CMA calibration
Fig. 7. χ versus α CMA and the scaling factor for the HST / STISdata and the computed absolute fluxes between 220 and 405 nmwith models of di ff erent T e ff for Procyon A. From (a) to (d): T e ff = / / / Fig. 8.
Departure coe ffi cients log b i for the five lowest levels of H i as afunction of continuum optical depth τ referring to λ = / / − .
05. Tick marks indicate the locationsof core-to-wing transition (0.85 to 0.95 in normalized flux) formationdepths for H α and H β .
6. Fitting procedure of Balmer-line wings
Our analysis of H α is based on profile fitting in the wavelengthrange from 6473 to 6653 Å for Procyon and from 6520 to 6608Å for the remaining stars. We analyzed H β in the wavelengthrange 4815 – 4908 Å only for the metal-poor stars HD 103095,HD 122563, HD 45282, and HD 140283. For HD 122563 andHD 140283 we also analyzed H γ profiles in the wavelength rangefrom 4315 to 4365 Å . Before fitting, we made the followingpreparations of the observed profiles for all stars.We excluded the Balmer line cores up to 0.9 of the rela-tive flux (F li / F ci ), because the core half-width of the hydrogenline cannot be described in cool stars with classical model atmo-spheres. Thus only the pressure-broadened wings were analyzed.As our aim is to fit the profiles of Balmer lines, all blended lines R e l a t i v e f l u x Fig. 9.
Synthetic NLTE (continuous curve) and LTE (dashed curve)flux profile of H α computed with the MAFAGS-OS 6530 / / − . α CMA = .
3) model compared to the observed spectrum of Procyon(bold dots). should be removed from the observed profiles. The high resolu-tion of the employed spectra allows us to assume that unblendedspectral windows do exist and can be used for the fitting. Weidentified spectral windows across the line that are expected to befree from blends in the stars of interest. This was done by meansof calculating Balmer line profiles with and without metal lines.Then we created a mask that was free from any blends. Basedon the examination of the spectra, this is reasonable at H α forall the reference stars, and at H β for HD 122563, HD 45282, andHD 140283, and at H γ only for HD 122563 and HD 140283. It isimpracticable to define spectral windows without any blends atH β and H γ in solar metallicity stars.The mask consists of a spectral window of pure H α , H β , andH γ (when available) lines with noise alone. The same mask wasapplied for all stars, but minor modifications were made for eachstar individually. Modifications of the mask were necessary toreject stars in some specific cases, for example, if there was aningress of atmospheric lines in the spectral window or for Pro-cyon, which has much stronger macroturbulence and V sin i thanother stars. It is very important to clearly distinguish betweennoise fluctuations and weak metal lines in the spectra. In thiscontext, we relied upon the spectral line database and inspectedall Balmer line profiles.We also investigated the e ff ect of spectral resolution on thechoice of which spectral windows to mask. It is necessary tohave a proper number of spectral windows between blended linesbecause only by using such a mask are we able to derive the sameresult regardless of whether we use high-resolution spectra or thespectra degraded to the resolution of our observations. This maskis quite reliable in the case of H α for all stars, but for H β and H γ we were unable to define such a mask for all stars. This makesthe results obtained from H β and H γ profiles less reliable.We also took into account the e ff ect of rotational broaden-ing and broadening caused by macroturbulence on the choiceof spectral windows of mask. While all our sample stars areslow rotators with V sin i ≤ − except for Procyon with V sin i = . − , we were able to find a proper number ofspectral windows between blending lines around H α profiles. Inthe case of H β and H γ profiles, this e ff ect becomes importantbecause there are many blending lines in the two profiles. This Article number, page 7 of 14 & Aproofs: manuscript no. Alpha_CMA_I_00_017 prevented us from defining a reliable mask for all stars, exceptfor some metal-poor stars (HD 103095, HD 122563, HD 45282,and HD 140283), which only contain few blending lines in theirH β and H γ profiles.In this work we used a method based on a reduced χ statis-tics. Statistical measurement of the goodness-of-fit was per-formed by comparing the theoretical profile and an observedprofile through a χ minimization procedure (Nousek & Shue1989). The estimator of the fit goodness is defined by χ = n − p − n X i = ( O i − C i ) σ i , (1)where O i is the relative flux measured at the wavelength λ i , C i is the relative flux computed from the model at the same wave-length λ i , σ i is the variance of the data point i that is defined asthe (S / N) − ratio at the same wavelength λ i , n is the number ofdata points, and p is the number of free parameters of the modelso that n − p − σ i at the wavelength λ i . Although it is known that theS / N depends on the number of recorded photons I ( ∼ √ I ), weadopted the mean constant S / N ratio for these stars because weonly fit the wings of each line up to 0.9 where the S / N ratiois not changed dramatically. For Procyon, for example, the rel-ative flux at a S / N of 0.85 is 10% lower than that with a S / Nof 0.995. Apart from the inner wings, which have less stronglyblended lines because they are saturated by the hydrogen line,equal weights were assigned for all spectral windows of a lineprofile.There were two free parameters, α CMA and the e ff ective tem-perature, while the other parameters (log g , [Fe / H], ξ ) were fixed.Assuming the Poissonian error distribution, which approachesthe normal distribution at high S / N, the minimization of χ pro-vides the maximum-likelihood estimate of the parameters, in ourcase for α CMA and T e ff .We also compared a grid of Balmer-line profiles for a givenmodel grid with a 25 K interval in T e ff and a 0.1 interval in α CMA .By varying the free parameters of the model, we computed thevalue of χ at each step. First we fixed one of the two free pa-rameters, T e ff , and analyzed the function of χ , which smoothlychanges with varying α CMA and follows an asymmetric parabolicform with one single minimum. The best α CMA and T e ff werechosen from the requirement of the lowest χ value. The vari-ation of temperature within the error bars allows us to estimatethe upper and lower values for our α CMA . An example of the fit-ting for H α profile in the metal poor star HD 103095 is shown inFig.10. Estimated errors are presented in Table 2 for the programstars. The strength of the hydrogen line depends only weaklyon the surface gravity or the metallicity. We considered howthe typical errors (0.1 dex) for either of them may a ff ect theresult. Microturbulence does not a ff ect the formation of theBalmer-line wings (Fuhrmann et al. 1993), therefore we did notconsider the influence of microturbulence. Only the tempera-ture can strongly a ff ect the wings of hydrogen lines. With in-creasing temperature the wings of hydrogen lines tend to bestronger. The e ff ective temperature is the main source of er-rors in our results. The influence of other sources of error, such R e l a t i v e f l u x Fig. 10.
Example of fitting the H α wings in the metal-poor starHD 103095. Bold dots correspond to the observed data. The theoreti-cal H α profile was calculated with (dashed curve) and without (solidcurve) blending lines. Only bold dots were used for determining the χ statistic. as Stark-broadening, self-broadening, and He-broadening, wereconsidered in Fuhrmann et al. (1993, 1994) and Barklem et al.(2002). The latter showed that the influence of Stark- and self-broadening on the behavior of Balmer lines is insignificant forsolar metallicity stars, but tends to be stronger for metal-poor at-mospheres. Moreover, the determination of the continuum place-ment is one of the most significant uncertainties in this work.For the sample spectra, the error in the continuum placementof the high-quality spectra is approximately 0.5% of the contin-uum flux, depending on the S / N ratio. As a result of the inter-dependency of errors, it is di ffi cult to precisely account for thecombined e ff ect, therefore we divided all errors into two groups:those propagated from the stellar parameters ( T e ff , log g , and[Fe / H]), and those caused by observational aspects (continuumplacement and fitting error).
7. Results
In Fig. 11 we plot the profiles of the Balmer lines for three ofour program stars (HD 122563, HD 103095, and the Sun). Theprofiles are only slightly di ff erent under LTE or NLTE assump-tions for dwarfs, but exhibit significant variations with α CMA ontheir wings. The changes with α CMA from 0.5 to 2.5 are simi-lar (for H α ) and even larger (for H β and H γ ) to that of T e ff of afew hundreds of Kelvin. For the metal-poor giant HD 122563,the NLTE e ff ect plays a more important role than α CMA for H α profile, but is less sensitive than α CMA parameter. Figure 11 alsoindicates that if the α CMA parameter in the model atmosphere isnot correct, the T e ff may be biased by up to a few hundreds ofKelvin by fitting H β or H γ line profiles. Therefore, caution mustbe exercised when choosing α CMA in 1D model atmospheres.
The Sun is the best-known star with high-quality observationsand accurate physical parameters. Following Fuhrmann et al.(1993), we attempted to calibrate α CMA for the solar atmo-sphere. Our work di ff ers from that of Fuhrmann et al. (1993) inthat we have used the self-broadening theory of Barklem et al. Article number, page 8 of 14u et al.: CMA calibration
Table 2.
Estimated errors in CMA factor for H α profiles. Error Sun Procyon HD 10700 HD 103095 HD 39587 HD 6582 HD 217014 HD 22049 HD 122563 HD 45282 HD 140283 ∆ T e ff ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ∆ log g = ± . ± . ± . ± . ± . ∆ [Fe / H] = ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ∆ λ ( ◦ A) HD122563 ∆ λ ( ◦ A) HD103095 ∆ λ ( ◦ A) sun α CM =0 .
50 LTE α CM =1 .
50 LTE α CM =2 .
50 LTE α CM =0 .
50 NLTE α CM =1 .
50 NLTE α CM =2 .
50 NLTE
Fig. 11.
Comparison of the line profiles of H α , H β , and H γ (from bottom to top) for HD 122563, HD 103095, and the Sun. The x-axis is thewavelength shift relative to the center of the Balmer lines. The colors are coded with di ff erent α CMA (from 0.5 to 1.5), while the solid lines anddashed lines represent profiles using LTE and NLTE line formations, respectively. The dotted lines are profiles with LTE models of α CMA = . T e ff +
100 K. For clarity, o ff sets of 0.2 and 0.4 are added to the H β and H γ , respectively. (2000). For investigations of the Sun we employed the solar at-las (Kurucz et al. 1984). We adopted an e ff ective temperature of5777 K with an error bar of ±
40 K. By fitting theoretical H α pro-files to reduced observational profiles, the two parameters T e ff and α CMA were varied within error bars and reasonable physicallimits, respectively. The results are shown in Fig. 12, where the χ minimum indicates the best T e ff and α CMA .Figure 13 compares the left wing of H α line profile for theSun with the theoretical spectrum for α CMA = . T e ff = ±
20 K and α CMA = . ± . α profiles. Nevertheless, it is importantto note that the value of χ is still high, which suggests that ouragreement between the observational and the theoretical H α pro-files is not perfect. This means that the quality of the observedsolar flux spectra is higher than the quality of theoretical surveys.From comparison with SED, a low value for α CMA < . Article number, page 9 of 14 & Aproofs: manuscript no. Alpha_CMA_I_00_017 not allow precisely determining the convective e ffi ciency, but themeasurements are consistent with an α CMA of 2.0.
Procyon (HD 61421, HR 2943) is one of the brightest stars in thenight sky ( V = . . ± . M ⊙ and a faintwhite dwarf (Procyon B) with a mass of 0 . ± . M ⊙ , or-biting each other with a period of 40.8 years (Girard et al. 2000).Spectroscopic analyses gave rather discrepant e ff ective tempera-tures, from 6470 K (by fitting the Balmer lines; Zhao & Gehren2000) to 6850 K (by measuring the equivalent widths of alarge set of iron lines; Heiter & Luck 2003). The proximityof Procyon ( π = . ± .
26 mas; van Leeuwen 2007) tothe Earth allows us to directly measure its angular diame-ter (e.g., Hanbury Brown et al. 1974; Mozurkewich et al. 1991;Aufdenberg et al. 2005; Chiavassa et al. 2012). Kervella et al.(2004) found T e ff = ±
50 K and log g = . ± .
2, whichagrees well with other works (e.g., Allende Prieto et al. 2002;Fuhrmann et al. 1997). We used a high-quality spectrum of H α for Procyon and applied the same fitting method as described inthe previous sections.The results are displayed in Fig. 12, where the χ minimumindicates the best T e ff and α CMA . For this star we find T e ff = ±
50 K and α CMA = . ± .
7. The error of α CMA is notsymmetric in this case, but we adopted the largest. The left wingof the H α line profile for Procyon is shown in Fig. 13, where twotheoretical spectra with α CMA = . α CMA = . α CMA ≈ . The final results of our fitting are given in Table 1, includingthe names of the program stars, their parameters, the best α CMA along with the limits, and the χ values that indicate the best fit ineach case. We also present the best-fit T e ff within the error barsthat was obtained during the χ minimization procedure. Tem-peratures with asterisks are the best-fit temperatures. The resultsfrom H β and H γ for some stars are less reliable. Even thoughgood-quality observations of H β and H γ are available for all re-maining stars, these lines were not adopted because choosing areliable mask was problematic.The observed spectra and the best-fitting line profiles fromthis method are plotted in Fig. 14. When applying the above-mentioned method, we have to keep in mind that weak blend-ing lines may, nevertheless, be presented in a chosen windows.Therefore, it would be better to inspect the best profiles visuallyto give preference to the case where all observed points are be-low the theoretical line profile, but the χ – method minimizesthe deviations without considering whether the observed pointsare above or below the theoretical profile. In this context, forthe χ method, α CMA will be slightly underestimated, while thebest-fit T e ff will be overestimated.The derived T e ff for stars HD 10700, HD 39587, HD 217014,HD 22049, HD 122563, HD 45282, and HD 140283 agree wellwith those derived from astrometric and IRFM methods. How-ever, for HD 103095 we were unable to find an appropriate T e ff T e ff [ K ] T e ff [ K ] Fig. 12.
Survey of the reduced χ obtained from H α profile fitting forthe observed solar flux spectra (top panel) and spectra of Procyon (bot-tom panel) with di ff erent α CMA and T e ff . The preferred parameter areais located where χ is minimal. All χ are not normalized and indicatea good fit in each case. in the temperature range T e ff = ±
100 K. Thus, for this starwe adopted a higher spectroscopic temperature of T e ff = µ Cas, 30 Cas, HR 321) is a close binary sys-tem that consists of a slightly metal-deficient subdwarf (com-ponent A, V = .
17) and a component B fainter by 5.3 mag-nitudes ( V = . ∼ T e ff = g = .
45, [Fe / H] = − . ξ = .
89 km s − ) were taken from Fuhrmann (2004).Boyajian et al. (2012) measured its limb-darkened angular diam-eter θ LD = . ± .
009 mas with CHARA, corresponding tothe stellar radius of 0 . ± . R ⊙ , combining with the Hip-parcos parallax ( π = . ± .
82 mas; van Leeuwen 2007).This is agrees well with the spectroscopic radius ( R = . R ⊙ ),but the interferometric T e ff = ∼
120 K lower than thatof Fuhrmann (2004). By comparing the H α profile, we found thebest α CMA = . α CMA = . Article number, page 10 of 14u et al.: CMA calibration
Fig. 13.
Left wings of H α line profiles for the Sun with α CMA = . α CMA = . α CMA = . χ and supports a higher T e ff = T e ff of 5339 K derived by the photometric b − y = .
437 and c = .
213 (Hauck & Mermilliod 1998) withthe color- T e ff relation of Alonso et al. (1996).HD 10700 ( τ Cet, 52 Cet, HR 509) is a nearby and inac-tive G8 V star, with a distance of only 3 . ± .
002 pc ( π = . ± .
17 mas; van Leeuwen 2007). It is one of the mostfrequently observed targets in several precise radial velocitysurveys for extrasolar planets, and Tuomi et al. (2013) reportedfive planets with minimum masses of up to 6.6 M ⊕ orbiting it.Spectroscopic analyses indicate that its T e ff ranges from 5283K (Valenti & Fischer 2005) to 5420 K (Takeda et al. 2005),and [Fe / H] range from − − θ LD = . ± .
031 mas (Di Folco et al. 2004), and T e ff was perfectly consistent with the spectroscopic parameters( T e ff = g = .
53, [Fe / H] = − .
49, and ξ = . − ; Mashonkina et al. 2011). With a least χ fitting, wefound the best α CMA = .
8, and the best T e ff = T e ff = b − y and c color(Hauck & Mermilliod 1998; Alonso et al. 1996).HD 22049 ( ǫ Eri, 18 Eri, HR 1084) is a nearby ( π = . ± .
16 mas; van Leeuwen 2007), chromospherically active K2 Vstar orbited by a dusty ring (Greaves et al. 1998). Atmosphericparameter determinations based on high-resolution spectra gavevery consistent results, with a T e ff ranging from 5054 K(Fuhrmann 2004) to 5200 K (Luck & Heiter 2005), log g rangingfrom 4.40 (Mishenina et al. 2013) to 4.72 (Takeda et al. 2005),and [Fe / H] ranging from − .
18 (Ghezzi et al. 2010) to + T e ff = g = / H] = − ξ = − ). Di Folco et al. (2004) suggested an interferomet-ric angular diameter of θ LD = . ± .
029 mas and an ef-fective temperature of T e ff = α profile fitting withspectroscopic and interferometric methods result in quite con-sistent α CMA = .
2. The best-fit T e ff = T e ff = b − y and c (Hauck & Mermilliod 1998).HD 39587 ( χ Ori, 54 Ori, HR 2047) is a variable star of RSCVn type, orbiting by a faint ( V A = V B = .
8) companion with a mass of 0 . ± . M ⊙ (Irwin et al. 1992). Boyajian et al.(2012) measured its angular diameter θ LD = . ± .
009 maswith a precision of 0.9%. This yield a radius of 0 . ± . R ⊙ with a Hipparcos parallax of 115 . ± .
27 mas (van Leeuwen2007). Boyajian et al. (2012) also gave its e ff ective temperatureas T e ff = α CMA = . α CMA = . / H] ∼ − .
6) that can be seen by the naked eye ( V = . T e ff = g = .
60, [Fe / H] = − ξ = − ). Recent inter-ferometric measurements with CHARA and PTI (Creevey et al.2012) gave its angular diameter as θ LD = . ± .
012 mas, and T e ff = ±
41 K, which is consistent with its spectroscopictemperature. We found the best-fit α CMA = . α profile,while the fitting of H β and H γ gave satisfactory agreement with α CMA = .
5. The α CMA is lower than that of the dwarfs, implyingthat the stellar convective energy transport become less e ffi cientafter stars evolved o ff the main sequence.HD 217014 (51 Peg, HR 8729) is a nearby ( π = . ± . ff ective temperaturesas ranging from 5710 K (Maldonado et al. 2012) to 5832 K(Ramírez et al. 2009). The narrow-band photometric color in-dices b − y = .
416 and c = .
371 (Hauck & Mermilliod 1998)indicate a rather low T e ff = T e ff = g = .
33, [Fe / H] =+ ξ = − , Fuhrmann 1998) agree for the e ff ec-tive temperature with the interferometric results of T e ff = θ LD = . ± .
027 mas (Baines et al. 2008). We founda best-fit of α CMA = . / H] = − .
5) at lowGalactic latitude ( b = − . ◦ ). Previous spectroscopic analysesgave its T e ff as ranging from 5150 K (Fulbright 2000) to 5344 K(Gratton et al. 2000). Considering its high reddening with E ( B − V ) ≃ .
82 at this direction (Schlegel et al. 1998), we used theIRFM T e ff from Casagrande et al. (2010), which is thought tobe less a ff ected by interstellar medium. The Balmer profile χ fitting gives α CMA = . α and α CMA = . β withthe best-fit T e ff = E ( B − V ) causesan uncertainty of ∼
300 K on T e ff , therefore the photometrictemperature is not reliable.For the Sun we used the widely adopted parameters of T e ff = g = .
44, [Fe / H] = ξ = .
90 km s − andfor Vega T e ff = g = .
95, [Fe / H] = − .
50, and ξ = .
00 km s − (Castelli & Kurucz 1994). The solar α CMA from theH α Balmer line is 2.0, while a determination based on the solarflux suggests a value of 1.5–1.75. Although this is consistentwithin the error bars, it has to be noted that, for both the Sunand Procyon, SED fitting leads to a notably lower value of α CMA than Balmer-line fitting. α CMA
Figure 15 shows our final results for the e ffi ciency parameter α CMA as determined from Balmer-line fitting. Except for evolvedstars, α CMA is consistent with a value of about 2.0 for all objects
Article number, page 11 of 14 & Aproofs: manuscript no. Alpha_CMA_I_00_017 T eff (K) l og g +2.30+1.80 +2.20+1.90 +2.40+1.00+2.00+1.00+1.70+1.90 +2.00 -2.0-1.00.0 [ F e / H ] Fig. 15.
Determined α CMA for the program stars. An error of α CMA of ≈± . . . . . T e ff = g = .
70 where the error reaches ≈ ± . within the given insecurities. There is no obvious correlationwith metallicity and a weak trend of increasing α for the coolestmain-sequence objects in the investigated range of F- and G-typestars. This confirms the behavior and numerical values predictedby Magic et al. (2014) for this part of the HRD.For the evolved stars of our sample α CMA is of about 1.0.Although Magic et al. (2014) predicted the same trend of lowerconvective e ffi ciency for objects with lower log g , the magni-tude of the change is much greater in our sample than that sug-gested by the 3D simulation of Magic et al. (2014). Our samplemainly consists of main-sequence stars and only contains threeevolved objects with well-determined parameters. Nevertheless,the stars clearly show a behavior di ff erent from that suggested byMagic et al. (2014). The three objects span a wide range of tem-peratures, therefore the evolutionary state seems to be the maindiscriminator that separates them from the rest of the sample.
8. Discussion and implication
Figure 16 shows the di ff erence between the suggested α CMA andthat from Magic et al. (2014), who adopted a calibration usingthe entropy jump (Fig. 3 in their work). As described above, thedi ff erences are small along the main sequence and increase dra-matically later in the evolutionary sequence, which suggests thatthere is either a stronger variation of α CMA as predicted by theSTAGGER grid, or that our modeling of the Balmer lines is in-correct. The latter is excluded because we generally determineda temperature that agreed well with astrometric and IRFM data.Based on the similar method of FAG93, our analysis indi-cates that if the theory of Barklem et al. (2000) for Balmer-linebroadening is adopted instead of that of Vidal et al. (1970, 1973),while the old ODF version is replaced by the MAFAGS-OS ver-sion, it is very unlikely to derive a low value of α CMA = . α CMA ≈ . ffi ciency naturally influencesthe temperatures determined by the Balmer-line method. It fur-thermore changes stellar SEDs and in turn filter fluxes and colordeterminations. All stellar lines that have contribution functions T eff (K) l og g −0.33−0.24 +0.09−0.06 +0.32−0.90+0.05−0.87−0.20+0.17 +0.02 -2.0-1.00.0 [ F e / H ] Fig. 16. Di ff erence between our determined α CMA and the one predictedby Magic et al. (2014). down to the convection scone (very strong lines) are subject topotential changes as well.
Acknowledgements.
We are grateful to K.Fuhrmann for providing some spectralobservations. X.S.W. thanks R.Wittenmyer and H.N. Li for their useful com-ments and suggestions. X.S.W., L.W., and G.Z. are supported by the NationalNatural Science Foundation of China under grant No. 11390371 and 11233004.S.A. and L.M. are supported by the Russian Foundation for Basic Research(grants 14-02-31780 and 14-02-91153). F.G. would like to thank T. Gehren forhis support, knowledgable comments, and so many helpful discussions.
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Fig. 14.
Best fits of the H α lines in our stellar sample. Solid lines are the theoretical spectra corresponding to the best fit and dotted lines represent theobserved spectra that were used for determining the χ2