Classical Cepheids: Yet another version of the Baade-Becker-Wesselink method
aa r X i v : . [ a s t r o - ph . GA ] N ov , Classical Cepheids: Yet another version of theBaade–Becker–Wesselink method
A. S. Rastorguev ⋆ , A. K. Dambis Sternberg Astronomical Institute, Universitetskii pr. 13, Moscow, 119992 Russia
Submitted to: Astrophysical Bulletin, 2011. Received: 2010
ABSTRACT
We propose a new version of the Baade–Becker–Wesselink technique, which allowsone to independently determine the colour excess and the intrinsic colour of a radiallypulsating star, in addition to its radius, luminosity, and distance. It is consideredto be a generalization of the Balona approach. The method also allows the function F ( CI ) = BC ( CI ) + 10 × log ( T eff ( CI )) for the class of pulsating stars consideredto be calibrated. We apply this technique to a number of classical Cepheids withvery accurate light and radial-velocity curves and with bona fide membership in openclusters (SZ Tau, CF Cas, U Sgr, DL Cas, GY Sge), and find the results to agree wellwith the reddening estimates of the host open clusters. The new technique can alsobe applied to other pulsating variables, e.g. RR Lyraes. Key words:
Cepheids; luminosities; radii; color excess.
Classical Cepheids are the key standard candles, which areused to set the zero point of the extragalactic distance scale(Freedman et al. 2001) and also serve as young-populationtracers of great importance (Binney and Merrifield 1998).They owe their popularity to their high luminosities andphotometric variability (which make them easy to identifyand observe even at large distances) and the fact that theluminosities, intrinsic colours, and ages of these stars areclosely related to such an easy to determine quantity as thevariability period.It would be best to calibrate the Cepheid period-luminosity (PL), period-colour (PC), and period-luminosity-colour relations via distances based on trigonometric par-allaxes, however, the most precisely measured parallaxesof even the nearest Cepeheids remain insufficiently accu-rate and, more importantly, they may be fraught with sofar uncovered systematic errors. Here the Baade–Becker–Wesselink method (Baade 1926; Becker 1940; Wesselink1946) comes in handy, because it allows the Cepheid dis-tances (along with the physical parameters of these stars)to be inferred, thereby providing an independent check forthe results based on geometric methods (e.g., trigonometricand statistical parallax).However, all the so far proposed versions of theBaade–Becker–Wesselink method (surface brightness tech-nique (Barnes and Evans 1976), maximum-likelihood tech- ⋆ E-mail: [email protected]. nique (Balona 1977)) depend, in one way or another, onthe adopted reddening value. Both techniques are based onthe same astrophysical background but make use somewhatdifferent calibrations (limb-darkened surface brightness pa-rameter, bolometric correction – effective temperature pair)on the normal colour. Here we propose a generalization ofthe Balona (1977) technique, which allows one to indepen-dently determine not only the star’s distance and physicalparameters, but also the amount of interstellar reddening,and even calibrate the dependence of a linear combinationof the bolometric correction and effective temperature onintrinsic colour.
We now briefly outline the method. First, the bolometricluminosity of a star at any time instant is given by the fol-lowing relation, which immediately follows from the Stefan–Boltzmann law:
L/L ⊙ = ( R/R ⊙ ) × ( T /T ⊙ ) . (1)Here L , R , and T are the star’s bolometric luminosity, ra-dius, and effective temperature, respectively, and the ⊙ sub-script denotes the corresponding solar values. Given that thebolometric absolute magnitude M bol is related to bolometricluminosity as M bol = M bol ⊙ − . × log ( L/L ⊙ ) , (2)we can simply derive from Eq. (1): A. S. Rastorguev, A. K. Dambis M bol − M bol ⊙ = − × log ( R/R ⊙ ) − × log ( T /T ⊙ ) (3)Now, the bolometric absolute magnitude M bol can be writ-ten in terms of the absolute magnitude M in some pho-tometric band and the corresponding bolometric correction BC : M bol = M + BC, (4)and the absolute magnitude M can be written as: M = m − A − × log ( d/ pc ) . (5)Here m , A , and d are the star’s apparent magnitude and in-terstellar extinction in the corresponding photometric band,respectively, and d is the heliocentric distance of the star inpc. We can therefore rewrite Eq. (3) as follows: m = A + 5 × log ( d/ pc ) + M bol ⊙ + 10 × log ( T ⊙ ) − × log ( R/R ⊙ ) − BC − × log ( T ) . (6)Let us introduce the function F ( CI ) = BC + 10 × log( T ),the apparent distance modulus ( m − M ) app = A + 5 × log ( d/ pc ), and rewrite Eq. (7) as the light curve model: m = Y − × log ( R/R ⊙ ) − F. (7)where constant Y = ( m − M ) app + M bol ⊙ + 10 × log ( T ⊙ ) . We now recall that interstellar extinction A can be de-termined from the colour excess CE as A = R λ × CE ,where R λ is the total-to-selective extinction ratio for thepassband-colour pair considered, whereas M bol ⊙ , R ⊙ , and T ⊙ are rather precisely known quantities. The quantity F ( CI ) = BC + 10 × log ( T ) is a function of intrinsic colourindex CI = CI − CE . Balona (1977) used a very crudeapproximation for the effective temperature and bolomet-ric correction, reducing the right-hand of the light curvemodel (Eq. 7) to the linear function of the observed colour,with the coefficients containing the colour excess in a latentform. It should be noted that Sachkov et al. (1998); Sachkov(2002) used non-linear approximation in Eq. (7) to calculateCepheid radii.The key point of our approach is that the values of func-tion F are computed from the already available calibrationsof bolometric correction BC ( CI ) and effective tempera-ture T ( CI ) (Flower 1996; Bessell et al. 1998; Alonso et al.1999; Sekiguchi and Fukugita 2000; Ramirez and Melendez2005; Biazzo et al. 2007; Gonzalez Hernandez and Bonifacio2009). These calibrations are expressed as high-order powerseries in the intrinsic colour: F ( CI ) = a + N X k =1 a k · CI k , (8)with known { a k } and N amounting to 7; in some cases thedecomposition also includes the metallicity ([ F e/H ]) and/orgravity ( log g ) terms.As for the star’s radius R , its current value can be de-termined by integrating the star’s radial-velocity curve overtime ( dt = ( P/ π ) · dϕ ): R ( t ) − R = − pf · Z ϕϕ ( V r ( t ) − V γ ) · ( P/ π ) · dϕ, (9) where R is the radius value at the phase ϕ (we use meanradius, < R > = ( R min + R max ) / V γ , the systemic radialvelocity; ϕ , the current phase of the radial velocity curve; P ,the star’s pulsation period, and pf is the projection factorthat accounts for the difference between the pulsation andradial velocities. Given the observables (light curve – appar-ent magnitudes m , colour curve – apparent colour indices CI , and radial velocity curve – V r ) and known quantitiesfor the Sun, we end up with the following unknowns: dis-tance d , mean radius < R > , and colour excess CE , whichcan be simply found by the least-squares or maximum like-lihood technique.In the case of Cepheids with large amplitudes of lightand colour curves (∆ CI > . m ) it is also possible to apply amore general technique by setting the expansion coefficients { a k } in Eq. (8) free and treating them as unknowns. Weexpanded the function F = BC + 10 × log ( T ) in Eq. (7)into a power series about the intrinsic colour index CI st ofa well-studied “standard” star (e.g., α Per or some otherbright star) with accurately known T st : F = BC st + 10 × log ( T st ) + N X k =1 a k · ( CI − CE − CI st ) k (10)The best fit to the light curve is provided with theoptimal expansion order N ≃ −
9. We use this modifi-cation to calculate the physical parameters and reddening CE of the Cepheid, as well as the calibration F ( CI ) = BC ( CI ) + 10 × log ( T eff ( CI )) for the given metallicity[ F e/H ]. Our sources of data include Berdnikov’s extensive mul-ticolor photoelectric and CCD photometry of classi-cal Cepheids (Berdnikov 1995, 2008) and very ac-curate radial-velocity measurements of 165 northernCepheids (Gorynya et al. 1992, 1996, 1998, 2002) taken in1987-2009 (about 10500 individual observations) with aCORAVEL-type spectrometer (Tokovinin 1987). These datasets are nearly synchronous, to prevent any systematic errorsin the computed radii and other parameters due to the evo-lutionary period changes resulting in phase shifts betweenlight, colour and radial velocity variations (Sachkov et al.1998). We adopt T ⊙ = 5777 K, M bol ⊙ = +4 . m (Gray2005). We proceeded from ( V, B − V ) data and found thebest solutions for the V -band light curve and ( B − V ) colorcurve to be those computed using the F (( B − V ) ) func-tion based on two calibrations (Flower 1996; Bessell et al.1998) of similar slope (see Fig. 2 e); the poorer results ob-tained using the other cited calibrations can be explainedby the fact that the latter involved insufficient number ofsupergiant stars. There is yet no consensus concerning the projection factor(PF) value to be used for Cepheid variables (Nardetto et al.2004; Groenewegen 2007; Nardetto et al. 2007, 2009). Dif-ferent authors use constant values ranging from 1.27 to lassical Cepheids: Yet another version of the Baade–Becker–Wesselink method p = ( − . ± . × log ( P, days ) + (1 . ± . , (11)though we repeated all calculations with other variants ofPF dependence on the period and pulsation phase to assurethe stability of the calculated colour excess. To test the new method, we used the maximum likelihoodtechnique to solve Eq. (7) for the V -band light curve and B − V colour curve for several classical Cepheids resid-ing in young open clusters: SZ Tau (NGC 1647), CF Cas(NGC 7790), U Sgr (IC 4725), DL Cas (NGC 129), GY Sge(anonymous OB-association (Forbes 1982)) as well as forapproximately 30 field Cepheids from our sample. We foundtwo log ( T eff ) calibrations – those of Flower (1996) andBessell et al. (1998) – combined with the BC ( V ) calibrationas a function of normal colour ( B − V ) proposed by Flower(1996) – to yield the best fit to the observed V -band lightcurve via Eq. (7). A weak sensitivity of calculated reddening, E B − V , to the adopted PF value (constant or period/phase–dependent) and to the derived < R > value can be explainedby very strong dependence of the light curve’s amplitude onthe effective temperature, ∼ × log ( T ), and, as a conse-quence, on the dereddened colour.Though the internal errors of the reddening E B − V seemto be very small, the values determined using the two bestcalibrations, Flower (1996) and Bessell et al. (1998), maydiffer by as much as 0 . − . m , due to the systematicshift between these two calibrations (Fig. 2 e). Table 1 sum-marizes the inferred parameters for the cluster Cepheidsstudied. Fig. 1 shows the observed and smoothed data andthe final fit to the V -band light curve for U Sgr Cepheid.Our reddenings seem to agree well with the correspondingWEBDA values, particularly if we remember that the er-rors of the adopted cluster reddening estimates are as highas ± . m . Our next step will be to make use of the cali-brations of T eff and BC as a function of red and infraredcolours ( V − R, V − I, V − K ) and to compare derivedreddening ratios with the conventional extinction laws.Note that the inferred radius and luminosity of SZ Tauare too large for its short period; this Cepheid probably pul-sates in the 1 st or even in the 2 nd overtone, as may be indi-rectly evidenced by its low colour amplitude (about 0 . m ).Fig. 2 shows the observed data, the fit to the V -band light curve, and the inferred calibration F = 10 × log ( T eff ) + BC ( V ) vs ( B − V ) ) calculated for TT AqlCepheid (as a 5 th -order expansion in the normal colour). The inferred calibration is very close to that of Flower(1996). We used α Per as the “standard” star, with T st ≈ (6240 ± K , [ F e/H ] ≈ − . ± .
06 (Lee et al. 2006),( B − V ) st ≈ . m and E B − V ≈ . m (WEBDA, for α Per cluster). To take into account the effect of metal-licity on the zero-point F ( CI ) st , we estimated the gra-dient dF ( CI ) st /d [ F e/H ] ≈ +0 .
24 from the calibrationsby Alonso et al. (1999); Sekiguchi and Fukugita (2000);Gonzalez Hernandez and Bonifacio (2009). For TT Aql, E B − V ≈ (0 . ± . m . In some cases (with large ampli-tude of color variation) the “free” calibration (Eq. 10) canmarkedly improve the model fit to the observed light curveof the Cepheid variable. Fig. 2 f shows the example of cali-brations of the F functions derived from nine Cepheids withdifferent metallicity and surface gravity values. Temperaturedifference at T eff ∼ − K is amounted to 3 − We grateful to M.V. Zabolotskikh for her assistance indata preparation and to L.N. Berdnikov, Yu.N. Efremov,M.E. Sachkov, V.E. Panchuk and A.B. Fokin for commentsand helpful discussions. This research has made use of theWEBDA database operated at the Institute for Astronomyof the University of Vienna. Our work is supported by theRussian Foundation for Basic Research (projects nos. 08-02-00738-a, 07-02-00380-a, and 06-02-16077-a).
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90 0.40 ± ± ± ±
87 0.54 ± ± ± ±
25 0.50 ± ± ± ±
58 0.47 ± ± ± ±
163 1.44 ± ± ±
11 -6.27 ± Phase V r − V γ , k m / s (b) Phase B − V (c) Phase ∆ R / R O (d) Phase V Figure 1.
Panel (a): Observed and fitted radial-velocity curve of U Sgr. Standard deviation σ V r = 1 . km/s . Panel (b): Observed andsmoothed colour curve. Panel (c): Radius variation with phase. Panel (d): Observed and fitted light curve. A. S. Rastorguev, A. K. Dambis (a) V r − V γ , k m / s Phase (b)
Phase B − V (c) Phase ∆ R / R O (d) Phase V (e) (B−V) l g T e ff + B C ( V ) Bessel et al. 1998Flower 1996TT Aql α Per (f) (B−V) l g T e ff + B C ( V ) BB SgrFM AqlTT AqlX CygCD CygRU SctT MonWZ SgrSV Vul
Figure 2.
Panel (a): Observed and fitted radial-velocity curve of TT Aql. Standard deviation σ V r = 1 . km/s . Panel (b): Observedand smoothed colour curve. Panel (c): Radius variation with phase. Panel (d): Observed and fitted light curve. Panel (e): Calculatedcalibration for TT Aql (function F = 10 × log ( T eff ) + BC ( V ) vs ( B − V ) )) and calibrations by Flower (1996) and Bessell et al. (1998).Also shown is the position of the standard star α Per corrected for metallicity difference. Panel (f): Calculated calibration (function F = 10 × log ( T eff ) + BC ( V ) vs ( B − V )0