The flux-weighted gravity-luminosity relation of Galactic classical Cepheids
aa r X i v : . [ a s t r o - ph . S R ] J u l Astronomy & Astrophysicsmanuscript no. GAIA_FGLR c (cid:13)
ESO 2020July 7, 2020
The flux-weighted gravity-luminosity relation of Galactic classicalCepheids ⋆ M. A. T. Groenewegen
Koninklijke Sterrenwacht van België, Ringlaan 3, B–1180 Brussels, Belgiume-mail: [email protected] received: ** 2020, accepted: * 2020
ABSTRACT
The flux-weighted gravity-luminosity relation (FWGLR) is investigated for a sample of 477 classical Cepheids (CCs), including starsthat have been classified in the literature as such but are probably not. The luminosities are taken from the literature, based on thefitting of the spectral energy distributions (SEDs) assuming a certain distance and reddening. The flux-weighted gravity (FWG) istaken from gravity and e ff ective temperature determinations in the literature based on high-resolution spectroscopy.There is a very good agreement between the theoretically predicted and observed FWG versus pulsation period relation that couldserve in estimating the FWG (and log g ) in spectroscopic studies with a precision of 0.1 dex.As was known in the literature, the theoretically predicted FWGLR relation for CCs is very tight and is not very sensitive to metallicity(at least for LMC and solar values), rotation rate, and crossing of the instability strip. The observed relation has a slightly di ff erentslope and shows more scatter (0.54 dex). This is due both to uncertainties in the distances and to the pulsation phase averaged FWGvalues. Data from future Gaia data releases should reduce these errors, and then the FWGLR could serve as a powerful tool in Cepheidstudies.
Key words.
Stars: distances - Stars: fundamental parameters - Stars: variables: Cepheids - distance scale
1. Introduction
Classical Cepheids (CCs) are considered an important standardcandle because they are bright, and thus they comprise a linkbetween the distance scale in the nearby universe and that fur-ther out via those galaxies that contain both Cepheids and SNIa(see Riess et al. 2019 for a determination of the Hubble constantto 1.9% precision, taking into account the new 1.1% precisedistance to the Large Magellanic Cloud from Pietrzy´nski et al.2019).This is the third paper in a series on Galactic CCs based onthe
Gaia second data release (GDR2, Gaia Collaboration et al.2018). Groenewegen (2018) (hereafter G18) started from an ini-tial sample of 452 Galactic CCs with accurate [Fe / H] abun-dances from spectroscopic analysis. Based on parallax data from
Gaia
DR2, supplemented with accurate non-
Gaia parallax datawhen available, a final sample of about 200 FU mode Cepheidswith good astrometric solutions was retained to derive period-luminosity ( PL ) and period-luminosity-metallicity ( PLZ ) rela-tions. The influence of a parallax zeropoint o ff set on the derived PL ( Z ) relation is large and means that the current GDR2 resultsdo not allow to improve on the existing calibration of the relationor on the distance to the LMC (as also concluded by Riess et al.2018). The zeropoint, the slope of the period dependence, andthe metallicity dependence of the PL ( Z ) relations are correlatedwith any assumed parallax zeropoint o ff set. Send o ff print requests to : Martin Groenewegen ⋆ Table A.3 is available in electronic form at the CDS viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp: // cdsweb.u-strasbg.fr / cgi-bin / qcat?J / A + A / . Tables A.1 andA.2, and Figure A.2 are available in the online edition of A&A. In Groenewegen (2020) (hereafter G20) the sample was ex-panded to 477 stars. Using photometry over the widest avail-able range in wavelength (and at mean light when available) thespectral energy distributions (SEDs) were constructed and fittedwith model atmospheres (and a dust component when required).For an adopted distance and reddening these fits resulted in abest-fitting bolometric luminosity ( L ) and the photometricallyderived e ff ective temperature ( T e ff ). This allowed for the deriva-tion of period-radius ( PR ) and PL relations, the construction ofthe Hertzsprung-Russell diagram (HRD), and a comparison totheoretical instability strips (ISs). The position of most stars inthe HRD was consistent with theoretical predictions. Outlierswere often associated with sources where the spectroscopicallyand photometrically determined e ff ective temperatures di ff ered,or with sources with large and uncertain reddenings.In this paper the relation between bolometric absolute mag-nitude and the flux-weighted gravity (FWG), g F ∼ g / T ff , is in-vestigated: the so-called flux-weighted gravity-luminosity rela-tion (FWGLR). The tight correlation between g F and luminos-ity was first demonstrated by Kudritzki et al. (2003, 2008) forblue supergiants, and was then used for extragalactic distancedeterminations in Kudritzki et al. (2016). Anderson et al. (2016)demonstrated that theoretical pulsation models for CCs also fol-lowed a tight FWGLR, in fact tighter than the PL relation, andthat there was a good correspondence between observed g F andperiod for a sample of CCs. The latest calibration of the FWGLRis presented in Kudritzki et al. (2020) based on 445 stars rangingfrom M bol = + . − . g F are derived, and Article number, page 1 of 9 & Aproofs: manuscript no. GAIA_FGLR the correlation with period and luminosity are presented. A briefdiscussion and summary concludes the paper.
2. Theoretical FWGLR for CCs
The FWG is defined as log g F = log g − · log (cid:16) T e ff / (cid:17) (Kudritzki et al. 2003). Kudritzki et al. (2020) present the latestcalibration of the FWG against absolute bolometric magnitudeas M bol = (3 . ± . g F − g ⊙ F ) + (4 . ± .
01) (1)for log g F ≥ log g bF and M bol = (3 . ± . g F − g bF ) + ( − . ± .
09) (2)for log g F < log g bF , with a scatter of 0.17 and 0.29 mag, respec-tively. The break in the relation is set at g bF = .
0, while the FWGof the Sun is g ⊙ F = . ⊙ ), initial rotation rates ( ω ini = . , . , . Z = . , . , . L , T e ff ), and pulsa-tion periods are given at the entry and exit of the IS for variouscrossings. They used these models to show the tight FWGLR forCCs for the first time (Fig. 16 in Anderson et al. 2016).The top panel in Fig. 1 shows the theoretical FWGLR basedon these models for FU pulsators with periods > . > . Z = .
006 and 0 . M bol = (3 . ± . g F − g bF ) + ( − . ± . P = P / (0 . − .
027 log P )following Feast & Catchpole (1997). The best fit islog g F = ( − . ± . P + (3 . ± . g F is desired. Figures andrelations for FU and FO models separately are presented in theAppendix.
3. Sample and observed FWGs.
The sample studied here is the collection of 477 stars consideredin G20. It is based on the original sample of 452 stars compiledin G18, extended by 25 additional stars for which accurate ironabundances have since become available, including five CCs inthe inner disk of our Galaxy (Inno et al. 2019).G20 constructed the SEDs for these stars, considering pho-tometry from the ultraviolet to the far-infrared, and as much aspossible at mean light. Distances and reddening were collected
Fig. 1.
Top panel.
FWGLR based on the pulsation models inAnderson et al. (2016). FU models are shown in red, FO models areshown in blue. For clarity FU (FO) models are plotted with an o ff setof + − .
05) dex in M bol . Symbols indicate the entry point of theIS, the lines connect it to the exit point of the IS. The first, second,and third crossing models are plotted as circles, squares, and triangles,respectively. Solar metallicity models are plotted with open symbols,models with Z = .
006 with filled symbols. The black lines refer toEqs. 1 and 2, the green line to the best fit (Eq. 3).
Bottom panel.
Rela-tion between FWG and period for the same models. The period of theFO models was fundamentalised. The green line refers to the best fit,Eq. 4. from the literature. Distances from GDR2 data was available for232 sources, and from other parallax data for 26 stars.Luck (2018) (hereafter L18) published a list of abundancesand stellar parameters for 435 Cepheids based on the analysisof 1137 spectra. L18 reduced all data in a uniform way usingMARCS LTE model atmospheres (Gustafsson et al. 2008). Ef-fective temperatures were determined in that paper using theline depth ratio (LDR) – e ff ective temperature calibration ofKovtyukh (2007) as updated by Kovtyukh (2010, private com-munication to Luck), while gravities were determined from theionisation balance between Fe I and Fe II lines, and micro-turbulent velocities ( v t ) by forcing there to be no dependencein the per-line Fe I abundances on equivalent width (see L18 foradditional details).Table 1 contains information on the set of 52 CCs for whichfive or more spectra were available in L18 taken at di ff erentphases in the pulsation cycle. FWGs are calculated on the onehand from the mean e ff ective temperatures and mean gravities(as given by L18 in his Table 11), and on the other hand from ananalysis of the FWGs calculated for the individual epochs andplotted versus phase. Using the code P eriod
04 (Lenz & Breger2005) to fit a low-order harmonic, this gives the mean log g F , the Article number, page 2 of 9. A. T. Groenewegen: The flux-weighted gravity-luminosity relation of Galactic classical Cepheids amplitude of the log g F curve, and the rms value. Some log g F phased curves with fits are shown in Fig. 2.These curves show considerable scatter even when the pul-sation cycle is well sampled. This is likely due to the error bar inan individual determination of g F . The error on e ff ective temper-ature generally has a negligible contribution in this. Ninety-fivepercent of individual e ff ective temperature error bars among the1137 spectra in L18 are between 30 and 220 K with a median of65 K. An error of 100 K at T e ff = g F , much smaller than the error on log g , which wasestimated to be ∼ g F val-ues determined from the averages of the e ff ective temperaturesand gravities, and from fitting the log g F curve with phase showessentially the same result, especially when seven of more spec-tra are averaged (with an average di ff erence between Cols. 6 and8 of − . ± .
04 dex).Figure 3 shows the FWGLR for the sample of 52 stars fromTable 1, where the luminosity and error are taken from G20.Equations 1, 2, and 3 are plotted as reference. Using a linearbi-sector fit (using the code SIXLIN from Isobe et al. 1990) thebest fit is M bol = (2 . ± . g F − . + ( − . ± .
08) (5)with an rms of 0.38 mag (blue line in the figure). A standardleast-squares fit has a shallower slope of 2 .
54. The theoreticalfit is shown in Eq. 3, and this fit di ff ers by about 0.4 mag atlog g F = .
5. Alternatively, the observed log g F values are sys-tematically too small by 0 . / . ∼ .
14 dex. At lower FWGor longer periods the di ff erence with the theoretical relation islarger.Table A.3 collects the FWG data for the entire sample of 477stars. Overall, most of the data (435 stars) come from L18, andfor the remaining stars log g and T e ff have been collected fromthe literature in order to calculate log g F . Multiple determinationsof log g F have been averaged and so can di ff er slightly from thevalues in Table 1. The table also includes the period, pulsationtype, distance with error, and luminosity with error from G20.Figure 4 shows the observational equivalent to the bottom panelin Fig. 1, the FWG determined from spectroscopy against pulsa-tion period (fundamentalised for FO pulsators).There is a tight correlation between the two quantities. Re-moving non-CCs (see Table A.3) and applying iterative 3 σ clip-ping results in the fitlog g F = ( − . ± .
03) log P + (3 . ± .
03) (6)with an rms of 0.16 dex, in very good agreement with the the-oretically predicted relation. Interestingly, many of the outlierscome from a single source, Genovali et al. (2014), who derivedvery low log g values for some objects. Some additional infor-mation and fits are provided in Appendix A.Figure 5 is the equivalent to Fig. 3 for the entire sample,using a simple averaging of the available FWGs. The error ondistance is now taken into account in calculating the error on lu-minosity. Following the discussion above and in the Appendix,the data from Genovali et al. (2014) has been excluded, and toreduce the scatter only stars with two or more spectra are consid-ered. A linear bi-sector fit applying iterative 3 σ clipping resultsin M bol = (2 . ± . g F − . + ( − . ± .
06) (7)with an rms of 0.54 mag using 170 stars and is shown as theblue line in the figure. This is currently the best observationaldetermination of the FWGL relation for CCs.
Fig. 2.
FWG vs pulsation phase for four CCs. The typical error bar ineach point is 0.15 dex in FWG, as indicated in the bottom plot. The linesare low-order harmonic fits to the data (see Col. 7 in Table 1).
4. Discussion and summary
The relation between FWG and period, and FWG and bolomet-ric luminosity is investigated for a sample 477 CCs. The FWGsare derived from e ff ective temperatures and log g values avail-able in the literature based on high-resolution spectroscopy. Theoverall majority of parameters have been compiled from a singlesource (L18) that determined log g and T e ff in an uniform man-ner. For a subset of stars multiple-phase data is available. TheFWG-Period and FWGLR are compared to theoretical modelsfrom Anderson et al. (2016)A very good agreement is found between the theoretical andobserved relations between FWG and period. These relationscould serve as a prediction for a reasonable range in log g values(assuming an e ff ective temperature) in a spectroscopic analysis. Article number, page 3 of 9 & Aproofs: manuscript no. GAIA_FGLR
Table 1.
FWGL data for the subsample with more than five spectra.Name Period N spec < T e ff > < log g > < log g F > N h log g F Ampl rms Luminosity(days) (K) (cgs) (cgs) (cgs) (cgs) (cgs) (L ⊙ )V473 Lyr 1.490780 5 6019 2.30 3.18 1 3.205 0.056 0.006 572.3 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Notes.
Column 1: Name. Column 2: Period (as quoted in L18). Column 3: Number of spectra L18. Column 4: Average e ff ective temperature(quoted in Table 11 in L18). Column 5: Average log g (quoted in Table 11 in L18). Column 6: Average log g F based on Cols. 4 and 5. Column 7:Number of harmonics used in the time analysis. Column 8: Mean log g F . Column 9: Amplitude in the log g F curve. Column 10: RMS. Column 11:Luminosity and error (from Table 1 in G20). The error is the fit error, and does not include the error on the distance. The distance and error on thedistance needed to calculate the total error on L are given in Table A.3. The observed FWGLR is found to have a shallower slopethan the theoretical relation. It is not clear at the moment if thisis a significant e ff ect or not. As the observed relation betweenFWG and period agrees with the theoretical relation, one wouldbe inclined to think that there could be a systematic e ff ect in thebolometric magnitudes of the long-period Cepheids. They are rarer and on average at longer distance, likely to be more sus-ceptible to (systematic) errors on parallax. This is qualitativelyconfirmed by repeating the fit of Eq. 7 restricting the sample tostars with σ L / L < .
2. The slope is increased, but has a largererror bar (3 . ± .
19) and the rms is reduced to 0.44 mag.
Article number, page 4 of 9. A. T. Groenewegen: The flux-weighted gravity-luminosity relation of Galactic classical Cepheids
Fig. 3.
FWGLR based on the subsample with more than five spectra.FU mode pulsators are plotted as circles (filled circles for periods over10 days), FO pulsators as open squares, the single second-overtone pul-sator as open triangle. The black lines refer to Eqs. 1 and 2, the greenline to Eq. 3. The blue line is a fit to the data points, excluding Y Oph(Eq. 5).
Fig. 4.
FWG vs fundamental pulsation period. Some outliers are named.The green line refers to the best fit, Eq. 7, which excludes the outliersand non-CCs indicated by a red cross.
Fig. 5.
FWGLR, with some outliers named. The black lines refer toEqs. 1 and 2, the green line to Eq. 3. The blue line refers to the best fit,Eq. 7, which excludes the outliers and non-CCs indicated by a red cross.Outliers located outside the plot window are SU Cru (log g F = . M bol = − . g F = . M bol = + . g F = . M bol = − . On the other hand, although the T e ff determinations basedon the LDR method are precise (as discussed earlier), possiblesystematic e ff ects (which would also a ff ect the determination oflog g and log g F ) could play a role (Mancino et al. 2020). Forthe subsample of 52 stars in L18 with five or more spectra, thecycle averaged T e ff s (as quoted in Table 1) are compared to thephotometrically derived T e ff s based on the SED fitting in G20.The errors on the photometrically derived e ff ective temperatures(the median is 180 K) are larger than those derived from spec-troscopy. There are two outliers Y Oph and S Vul, where the pho-tometrically derived temperatures are considerably lower thanthose quoted in L18 (570 and 830 K; > . σ ). For the otherstars the di ff erence (spectroscopically - photometrically derived T e ff ) is 140 ±
150 K.Systematic errors on the determination of the gravity couldalso play a role. The methodology used by L18 to determinethe stellar parameters, in particular v t and gravity, is the standardone. A non-standard method is sometimes also used in the litera-ture, as introduced by Kovtyukh & Andrievsky (1999). To avoidnon-LTE-sensitive stronger Fe i lines, v t is derived from Fe ii lines and weak Fe i lines alone. This leads to higher v t , which inturn leads to higher gravities when the ionisation balance is en-forced. For δ Cep Kovtyukh & Andrievsky (1999) find that thegravities are higher by 0.5 dex using the non-standard method.The matter is also debated in Yong et al. (2006). They note thatthe non-standard method ‘has merits’, but show that their de-rived gravities using the standard method are self-consistent, oneargument being that this gravity also produces ionisation equilib-rium for Ti i lines that are more susceptible to non-LTE e ff ectsthan Fe. The non-standard method is also used in Takeda et al.(2013). Anderson et al. (2016) excluded the gravities from thatpaper as they di ff ered from other sources they used. Twelve starsoverlap with the sample of stars with multi-epoch data from L18(in Table 1). Takeda et al. (2013) present stellar parameters atbetween 7 and 17 epochs. The mean e ff ective temperatures andmean gravities are calculated, as well are FWGs at these epochsbased on the data in Takeda et al. (2013), and fitted with low-order harmonic sine curves, as before, to give the mean FWG.The di ff erence (min - max (mean)) between the parameters fromthe non-standard method minus those from the standard methodare − − +
444 (167) K in T e ff , + . − + .
72 ( + .
36) dex inlog g , and + . − + .
67 ( + .
34) dex in FWG, with tendenciesthat the di ff erence in all three quantities decreases with increas-ing period.The FWGLR has the potential to be an alternative to the PL relation in distance determination (Anderson et al. 2016). In itscurrent empirically best calibrated version it is not. The scatterof 0.54 mag is larger than the 0.40 mag in the bolometric PL relation determined in G20 using the identical sample of stars,distances, and luminosities.One issue is that the independent variable period is knownwith great precision, while the independent variable FWG hasa non-negligible error associated with it. The fitting of the FWGversus pulsation phase did not provide more precise mean FWGsthan simple averaging. As the slope of the FWGLR is reasonablysteep, any uncertainty on the FWG leads to a three times largeruncertainty in M bol .The discussion above also demonstrates that the stellar pa-rameters should be derived in a uniform way. To exclude theinfluence of data analysis inhomogeneity altogether, Eq. 7 wasre-determined using data only from L18. The usable sam-ple is reduced to 161 stars and the slope and o ff set changemarginally, less than 1 σ . The standard approach used by L18seems to give consistent results when considering the compari- Article number, page 5 of 9 & Aproofs: manuscript no. GAIA_FGLR son to theory and the independent calibration of the FWGLR byKudritzki et al. (2020). Changes in the FWG by ∼ + . − . ff eredby Gaia in mind. Future data releases will provide informationthat will impact and improve on the results obtained here. Pri-marily, improved parallaxes, taking into account binarity in theastrometrical solution, will provide more precise distances andthus bolometric luminosities (e.g. through the SED fitting per-formed in G20).Secondly,
Gaia
RVS spectra and
Gaia Bp / Rp spectro-photometry will provide estimates of the stellar parameters(log g , T e ff , also metallicity) in future releases. Only mean spec-tra in data release 3, and epoch spectra in data release 4 (Brown2019). An older analysis by Recio-Blanco et al. (2016) indicatethat end-of-mission accuracies in log g of 0.1 dex or better can bereached in intermediate-metallicity F and G giants of magnitude G ∼ . − . g down to G = +
18 months until the end of 2020, andlikely until the end of 2022, these numbers should improve. Inconclusion, the FWGLR could prove to become an extremelyuseful tool in Cepheid studies.
Acknowledgements.
I would like to thank Dr. Bertrand Lemasle for interestingdiscussion on the determination of log g and commenting on a draft version ofthis paper. This research has made use of the SIMBAD database and the VizieRcatalogue access tool operated at CDS, Strasbourg, France. References
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Article number, page 6 of 9. A. T. Groenewegen: The flux-weighted gravity-luminosity relation of Galactic classical Cepheids
Appendix A: Additional material
Additional fits for the FWGLR based on the models ofAnderson et al. (2016) are given in Table A.1 for the three di ff er-ent metallicities, and with the slope fixed to the value in Eq. 3.The results for Z = .
006 and 0 .
014 agree within the error andjustify the use of a single relation combining the two metallici-ties (Eq. 3). The Z = .
002 models di ff er by a larger amount,qualitatively in agreement with the remark in Kudritzki et al.(2020) on the fact that low metallicities (below − . ff ect on the FWGLR. Table A.1.
Fits of the type M bol = a · (log g − g bF ) + b . a b rms metallicity3 . ± . − . ± .
014 0.13 Z = . .
35 fixed − . ± .
014 0.143 . ± . − . ± .
017 0.16 Z = . .
35 fixed − . ± .
014 0.163 . ± . − . ± .
016 0.17 Z = . .
35 fixed − . ± .
012 0.18The bottom panel of Fig. 1 and Eq. 4 present the relationbetween FWG and pulsation period based on the models ofAnderson et al. (2016) with the overtone periods converted toFU periods. Figure A.1 and Eqs. A.1 and A.2 give the results forFU and FO pulsators separately. The best fits arelog g F = ( − . ± . P + (3 . ± . g F = ( − . ± . P + (3 . ± . g F values are available the scatter in the rela-tion decreases. Assuming that the intrinsic scatter in the relationis 0.093 dex (Eq. 4) a single determination has an estimated errorof about 0.13 dex (dominated by the error on log g ), while aver-aging six or more spectra leads to an error of about 0.09 dex.As noted in the main text, and illustrated by comparingFig. 4 and the top panel in Fig. A.2, a fair fraction of out-liers are stars with T e ff and log g taken from Genovali et al.(2014). Genovali et al. (2014) also present multiple observationsfor some stars, and XX Sgr and WZ Sgr are in common with thesubsample of stars in L18 with five or more available spectra. Acomparison shows that the di ff erence in log g F is dominated bythe di ff erence in log g , that are of the order 0.5 dex. For some ofthe stars in the present sample the log g F (and log g ) values aretoo low by 1 dex. As they seem to use the same methodology asL18 in deriving the stellar parameters, no simple explanation iso ff ered to explain this discrepancy.Table A.3 compiles the FWG and luminosity data for the entiresample. The full table is available at the CDS. Fig. A.1.
Relation between FWG and period for FU (top panel) andFO (bottom panel) models. The meaning of the symbols and colours isexplained in Fig 1. The green lines refer to the best fits, Eqs. A.1 andA.2. Article number, page 7 of 9 & Aproofs: manuscript no. GAIA_FGLR
Table A.2.
Fits of the type g F = a · log P + b . a b rms N Remarks − . ± .
028 3 . ± .
025 0.159 443 standard, Eq. 7 − . ± .
028 3 . ± .
025 0.158 442 excluding Genovali et al. (2014) − . ± .
036 3 . ± .
032 0.160 275 N sp =
1, excluding Genovali et al. (2014) − . ± .
073 3 . ± .
060 0.172 87 N sp =
2, excluding Genovali et al. (2014) − . ± .
116 3 . ± .
100 0.160 32 N sp = −
5, excluding Genovali et al. (2014) − . ± .
106 3 . ± .
110 0.139 20 N sp = −
10, excluding Genovali et al. (2014) − . ± .
088 3 . ± .
086 0.125 31 N sp ≥
11, excluding Genovali et al. (2014) − . ± .
063 3 . ± .
064 0.127 50 N sp ≥
6, excluding Genovali et al. (2014)
Article number, page 8 of 9. A. T. Groenewegen: The flux-weighted gravity-luminosity relation of Galactic classical Cepheids
Table A.3.
FWG data for the entire sample (first entries only).
Name Type Period d σ d L σ L N spec log g F σ log g F Min-Max Ref.(days) (kpc) (kpc) (L ⊙ ) (L ⊙ ) (cgs) (cgs) (cgs)AA Gem DCEP 11.302 3.400 0.829 3400.0 122.7 2 2.216 0.11 0.04 1AA Mon DCEP 3.938 3.922 0.709 922.8 33.6 1 3.211 0.16 - 1AB Cam DCEP 5.788 4.200 0.966 1463.5 79.3 1 2.754 0.15 - 1AC Mon DCEP 8.014 2.400 0.400 1991.6 42.2 4 2.766 0.08 0.21 1AD Cam DCEP 11.261 4.600 0.756 2048.8 87.0 2 2.301 0.11 0.07 1AD Cru DCEP 6.398 2.994 0.394 1881.9 93.2 1 2.730 0.15 - 1AD Gem DCEP 3.788 2.500 0.673 966.0 32.0 2 2.914 0.11 0.12 1AD Pup DCEP 13.596 4.100 0.946 4650.8 356.8 1 2.103 0.15 - 1AE Tau DCEP 3.897 3.367 0.606 953.2 11.9 1 2.802 0.15 - 1AE Vel DCEP 7.134 2.100 0.187 1842.6 169.2 1 2.663 0.15 - 1AG Cru DCEP 3.837 1.506 0.094 1773.5 49.7 1 2.864 0.15 - 1AH Vel DCEPS 4.227 0.752 0.035 2604.0 37.7 2 2.838 0.11 0.04 1alpha UMi DCEPS 3.970 0.133 0.002 2410.9 622.8 2 2.888 0.11 0.14 2,3AN Aur DCEP 10.291 3.400 0.985 3124.5 58.2 2 2.630 0.11 0.24 1AO Aur DCEP 6.763 3.400 0.839 1620.9 49.3 3 2.728 0.09 0.18 1AO CMa DCEP 5.816 3.600 0.434 1197.9 58.1 1 2.950 0.16 - 1AP Pup DCEP 5.084 1.183 0.051 2579.5 87.4 1 2.869 0.15 - 1AP Sgr DCEP 5.058 0.861 0.041 1651.8 38.7 1 2.780 0.15 - 1AQ Car DCEP 9.769 3.030 0.317 3837.4 289.0 1 2.702 0.15 - 1AQ Pup DCEP 30.149 2.900 0.336 11481.5 330.8 1 2.533 0.15 - 1AS Per DCEP 4.973 1.200 0.087 1042.0 36.7 2 2.810 0.11 0.30 1AT Pup DCEP 6.665 1.637 0.085 2495.6 194.9 1 2.757 0.15 - 1AV Cir DCEPS 3.065 0.944 0.033 2169.7 85.7 1 2.843 0.15 - 1AV Sgr DCEP 15.415 2.100 0.287 4413.1 139.5 1 2.609 0.15 - 1AW Per DCEP 6.464 0.700 0.044 1646.8 76.6 11 2.802 0.05 0.47 1AX Cir DCEP 5.273 0.500 0.151 1854.6 33.1 3 2.782 0.09 0.08 1AX Vel DCEP(B) 2.593 1.517 0.077 1750.2 166.6 2 3.047 0.11 0.05 1AY Cen DCEP 5.310 1.689 0.100 1864.4 303.0 1 2.821 0.15 - 1AZ Cen DCEPS 3.212 2.137 0.158 2017.4 50.1 1 2.986 0.15 - 1BB Cen DCEPS 3.998 3.610 0.363 3100.8 110.7 1 2.888 0.15 - 1BB Gem DCEP 2.308 4.082 0.825 1135.9 49.5 1 3.123 0.16 - 1BB Her DCEP 7.508 3.623 0.759 3122.0 153.2 8 2.684 0.05 0.32 1BB Sgr DCEP 6.637 0.700 0.023 1529.1 30.8 1 2.800 0.15 - 1BC Pup DCEP 3.544 6.500 1.109 938.2 64.4 2 2.760 0.11 0.13 4,(17) Notes.
Column 1: Name. Column 2: Type (from Table 1 in G20). Nomenclature follows that used by the VSX (Watson et al. 2006) a . Column 3:Period (from G20). Column 4: Distance (from G20). Column 5: Error on distance (from G20). Column 6: Luminosity (from G20). Column 7:Error on Luminosity (from G20). The error is the fit error, and does not include the error on the distance. If the total error on L is desired it canbe calculated from q σ + ∆ with ∆ = L · ((1 + σ d / d ) − N spec .Column 9: Average of available log g F values. Column 10: Estimated error on the average log g F value. This includes the error on T e ff (when notgiven in the reference a conservative value of 100 K has been used) and the error on log g (assumed to be 0.15 dex, unless given specifically),divided by p N spec . Column 11: Di ff erence between highest and lowest log g F value. Column 12: References for log g and T e ff values to calculatelog g F and error: (1) Luck (2018), (2) Andrievsky et al. (1994), (3) Boyarchuk & Lyubimkov (1981), (4) Luck et al. (2003), (5) Schmidt et al.(2011), (6) Andrievsky et al. (2002b), (7) Lemasle et al. (2008), (8) Andrievsky et al. (2013), (9) Kovtyukh et al. (2005), (10) Luck et al. (2006),(11) Yong et al. (2006), (12) Lemasle et al. (2015), (13) Romaniello et al. (2008), (14) Andrievsky et al. (2004), (15) Lemasle et al. (2007), (16)Andrievsky et al. (2002a), (17) Genovali et al. (2014), (18) Anders et al. (2019), (19) Martin et al. (2015), (20) Andrievsky et al. (2016), (21)Inno et al. (2019). Numbers in parentheses indicate references not considered. a described in . Article number, page 9 of 9 & Aproofs: manuscript no. GAIA_FGLR
Fig. A.2.
FWG vs period. The data from Genovali et al. (2014) is ex-cluded in all plots. Di ff erent panels show di ff erent selections on thenumber of available g F values. From top to bottom: all, N sp = N sp = N sp = − N sp = − N sp ≥≥