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,

Classical Cepheids: Yet another version of the Baade­Becker­Wesselink metho d

arXiv:1011.3305v2 [astro-ph.GA] 19 Nov 2010

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 Baade­Becker­Wesselink technique, which allows one to independently determine the colour excess and the intrinsic colour of a radially pulsating star, in addition to its radius, luminosity, and distance. It is considered to be a generalization of the Balona approach. The method also allows the function F (C I0 ) = B C (C I0 ) + 10 â log (Tef f (C I0 )) for the class of pulsating stars considered to be calibrated. We apply this technique to a number of classical Cepheids with very accurate light and radial-velocity curves and with bona fide membership in open clusters (SZ Tau, CF Cas, U Sgr, DL Cas, GY Sge), and find the results to agree well with the reddening estimates of the host open clusters. The new technique can also be applied to other pulsating variables, e.g. RR Lyraes. Key words: Cepheids; luminosities; radii; color excess.

1

INTRODUCTION

Classical Cepheids are the key standard candles, which are used to set the zero p oint of the extragalactic distance scale (Freedman et al. 2001) and also serve as young-p opulation tracers of great imp ortance (Binney and Merrifield 1998). They owe their p opularity to their high luminosities and photometric variability (which make them easy to identify and observe even at large distances) and the fact that the luminosities, intrinsic colours, and ages of these stars are closely related to such an easy to determine quantity as the variability p eriod. It would b e b est to calibrate the Cepheid p eriodluminosity (PL), p eriod-colour (PC), and p eriod-luminositycolour relations via distances based on trigonometric parallaxes, however, the most precisely measured parallaxes of even the nearest Cep eheids remain insufficiently accurate and, more imp ortantly, they may b e fraught with so far uncovered systematic errors. Here the Baade­Becker­ Wesselink method (Baade 1926; Becker 1940; Wesselink 1946) comes in handy, b ecause it allows the Cepheid distances (along with the physical parameters of these stars) to b e inferred, thereby providing an indep endent check for the results based on geometric methods (e.g., trigonometric and statistical parallax). However, all the so far prop osed versions of the Baade­Becker­Wesselink method (surface brightness technique (Barnes and Evans 1976), maximum-likelihood tech

nique (Balona 1977)) dep end, in one way or another, on the adopted reddening value. Both techniques are based on the same astrophysical background but make use somewhat different calibrations (limb-darkened surface brightness parameter, b olometric correction ­ effective temp erature pair) on the normal colour. Here we prop ose a generalization of the Balona (1977) technique, which allows one to indep endently determine not only the star's distance and physical parameters, but also the amount of interstellar reddening, and even calibrate the dep endence of a linear combination of the b olometric correction and effective temp erature on intrinsic colour.

2

THEORETICAL BACKGROUND

We now briefly outline the method. First, the b olometric luminosity of a star at any time instant is given by the following relation, which immediately follows from the Stefan­ Boltzmann law: L/L = (R/R )2 â (T /T )4 . (1)

Here L, R, and T are the star's b olometric luminosity, radius, and effective temp erature, resp ectively, and the subscript denotes the corresp onding solar values. Given that the b olometric absolute magnitude Mbol is related to b olometric luminosity as Mbol = Mbo
l

- 2.5 â log (L/L ),

(2)

E-mail: rastor@sai.msu.ru.

we can simply derive from Eq. (1):


2

A. S. Rastorguev, A. K. Dambis
l

Mbol - Mbo

= -5 â log (R/R ) - 10 â log (T /T )

(3)

Now, the b olometric absolute magnitude Mbol can b e written in terms of the absolute magnitude M in some photometric band and the corresp onding b olometric correction BC : Mbol = M + B C, and the absolute magnitude M can b e written as: M = m - A - 5 â log (d/10 pc). (5) (4)

Here m, A, and d are the star's apparent magnitude and interstellar extinction in the corresp onding photometric band, resp ectively, and d is the heliocentric distance of the star in p c. We can therefore rewrite Eq. (3) as follows: m = A + 5 â log (d/10 pc) + Mbo
l

+ 10 â log (T ) (6)

- 5 â log (R/R ) - B C - 10 â log (T ).

Let us introduce the function F (C I0 ) = B C + 10 â log(T ), the apparent distance modulus (m - M )app = A + 5 â log (d/10 pc), and rewrite Eq. (7) as the light curve model: m = Y - 5 â log (R/R ) - F. where constant Y = (m - M )
a pp

where R0 is the radius value at the phase 0 (we use mean radius, < R >= (Rmin + Rmax )/2 ); V , the systemic radial velocity; , the current phase of the radial velocity curve; P , the star's pulsation p eriod, and pf is the pro jection factor that accounts for the difference b etween the pulsation and radial velocities. Given the observables (light curve ­ apparent magnitudes m, colour curve ­ apparent colour indices C I , and radial velocity curve ­ Vr ) and known quantities for the Sun, we end up with the following unknowns: distance d, mean radius < R >, and colour excess C E , which can b e simply found by the least-squares or maximum likelihood technique. In the case of Cepheids with large amplitudes of light and colour curves (C I 0.4m ) it is also p ossible to apply a more general technique by setting the expansion coefficients {ak } in Eq. (8) free and treating them as unknowns. We expanded the function F = B C + 10 â log (T ) in Eq. (7) s into a p ower series ab out the intrinsic colour index C I0 t of a well-studied "standard" star (e.g., Per or some other bright star) with accurately known T st :
N

F = BC

st

+ 10 â log (T st ) +
k =1

s ak · (C I - C E - C I0 t )k (10)

(7)

+ Mbo

l

+ 10 â log (T ).

We now recall that interstellar extinction A can b e determined from the colour excess C E as A = R â C E , where R is the total-to-selective extinction ratio for the passband-colour pair considered, whereas Mbol , R , and T are rather precisely known quantities. The quantity F (C I0 ) = B C + 10 â log (T ) is a function of intrinsic colour index C I0 = C I - C E . Balona (1977) used a very crude approximation for the effective temp erature and b olometric correction, reducing the right-hand of the light curve model (Eq. 7) to the linear function of the observed colour, with the coefficients containing the colour excess in a latent form. It should b e noted that Sachkov et al. (1998); Sachkov (2002) used non-linear approximation in Eq. (7) to calculate Cepheid radii. The key p oint of our approach is that the values of function F are computed from the already available calibrations of b olometric correction B C (C I0 ) and effective temp erature T (C I0 ) (Flower 1996; Bessell et al. 1998; Alonso et al. 1999; Sekiguchi and Fukugita 2000; Ramirez and Melendez 2005; Biazzo et al. 2007; Gonzalez Hernandez and Bonifacio 2009). These calibrations are expressed as high-order p ower series in the intrinsic colour:
N

The b est fit to the light curve is provided with the optimal expansion order N 5 - 9. We use this modification to calculate the physical parameters and reddening C E of the Cepheid, as well as the calibration F (C I0 ) = B C (C I0 ) + 10 â log (Tef f (C I0 )) for the given metallicity [F e/H ].

3

OBSERVATIONAL DATA, CONSTANTS, AND CALIBRATIONS

F (C I 0 ) = a 0 +
k =1

k ak · C I0 ,

(8)

with known {ak } and N amounting to 7; in some cases the decomp osition also includes the metallicity ([F e/H ]) and/or gravity (log g ) terms. As for the star's radius R, its current value can b e determined by integrating the star's radial-velocity curve over time (dt = (P /2 ) · d):


Our sources of data include Berdnikov's extensive multicolor photoelectric and CCD photometry of classical Cepheids (Berdnikov 1995, 2008) and very accurate radial-velocity measurements of 165 northern Cepheids (Gorynya et al. 1992, 1996, 1998, 2002) taken in 1987-2009 (ab out 10500 individual observations) with a CORAVEL-typ e sp ectrometer (Tokovinin 1987). These data sets are nearly synchronous, to prevent any systematic errors in the computed radii and other parameters due to the evolutionary p eriod changes resulting in phase shifts b etween light, colour and radial velocity variations (Sachkov et al. 1998). We adopt T = 5777 K, Mbol = +4.76m (Gray 2005). We proceeded from (V , B - V ) data and found the b est solutions for the V -band light curve and (B - V ) color curve to b e those computed using the F ((B - V )0 ) function based on two calibrations (Flower 1996; Bessell et al. 1998) of similar slop e (see Fig. 2 e); the p oorer results obtained using the other cited calibrations can b e explained by the fact that the latter involved insufficient numb er of sup ergiant stars.

4

THE PROJECTION FACTOR

R(t) - R0 = -pf ·
0

(Vr (t) - V ) · (P /2 ) · d,

(9)

There is yet no consensus concerning the pro jection factor (PF) value to b e used for Cepheid variables (Nardetto et al. 2004; Groenewegen 2007; Nardetto et al. 2007, 2009). Different authors use constant values ranging from 1.27 to


Classical Cepheids: Yet another version of the Baade­Becker­Wesselink method
1.5, as well as PFs dep ending on the pulsation p eriod and other parameters. Different approaches lead to small systematic differences in the inferred Cepheid parameters, first of all, in the radii. Based on geometrical considerations, Rastorguev (2010) derived phase-dep endent PFs as simple three-parametric analytic expressions dep ending on the pulsation velocity, limb darkening coefficient, and sp ectral line broadening, adjusted to CORAVEL radial velocities of Cepheids. We susp ect that the p eriod dep endence reflects mainly the dep endence of the PF on limb darkening. To compare our results with other calculations, we finally adopted a moderate dep endence of PF on the p eriod advocated by Nardetto et al. (2007): p = (-0.064 ± 0.020) â log (P, day s) + (1.376 ± 0.023), (11) though we rep eated all calculations with other variants of PF dep endence on the p eriod and pulsation phase to assure the stability of the calculated colour excess.

3

5

PRELIMINARY RESULTS

The inferred calibration is very close to that of Flower (1996). We used Per as the "standard" star, with T st (6240 ± 20) K , [F e/H ] -0.28 ± 0.06 (Lee et al. 2006), (B - V )st 0.48m and EB-V 0.09m (WEBDA, for Per cluster). To take into account the effect of metallicity on the zero-p oint F (C I0 )st , we estimated the gradient dF (C I0 )st /d[F e/H ] +0.24 from the calibrations by Alonso et al. (1999); Sekiguchi and Fukugita (2000); Gonzalez Hernandez and Bonifacio (2009). For TT Aql, EB-V (0.65 ± 0.03)m . In some cases (with large amplitude of color variation) the "free" calibration (Eq. 10) can markedly improve the model fit to the observed light curve of the Cepheid variable. Fig. 2 f shows the example of calibrations of the F functions derived from nine Cepheids with different metallicity and surface gravity values. Temp erature difference at Tef f 6600 - 5100 K is amounted to 3 - 5%. When applied to an extensive sample of Cepheid variables with homogeneous photometric data and detailed radial velocity curves, the new method is exp ected to give a completely indep endent scale of reddenings, a new Period Colour - Luminosity relation, and a new distance scale for the Milky-Way Cepheids.

To test the new method, we used the maximum likelihood technique to solve Eq. (7) for the V -band light curve and B - V colour curve for several classical Cepheids residing in young op en clusters: SZ Tau (NGC 1647), CF Cas (NGC 7790), U Sgr (IC 4725), DL Cas (NGC 129), GY Sge (anonymous OB-association (Forb es 1982)) as well as for approximately 30 field Cepheids from our sample. We found two log (Tef f ) calibrations ­ those of Flower (1996) and Bessell et al. (1998) ­ combined with the B C (V ) calibration as a function of normal colour (B - V )0 prop osed by Flower (1996) ­ to yield the b est fit to the observed V -band light curve via Eq. (7). A weak sensitivity of calculated reddening, EB-V , to the adopted PF value (constant or p eriod/phase­ dep endent) and to the derived < R > value can b e explained by very strong dep endence of the light curve's amplitude on the effective temp erature, 10 â log (T ), and, as a consequence, on the dereddened colour. Though the internal errors of the reddening EB-V seem to b e very small, the values determined using the two b est calibrations, Flower (1996) and Bessell et al. (1998), may differ by as much as 0.03 - 0.05m , due to the systematic shift b etween these two calibrations (Fig. 2 e). Table 1 summarizes the inferred parameters for the cluster Cepheids studied. Fig. 1 shows the observed and smoothed data and the final fit to the V -band light curve for U Sgr Cepheid. Our reddenings seem to agree well with the corresp onding WEBDA values, particularly if we rememb er that the errors of the adopted cluster reddening estimates are as high as ±0.05m . Our next step will b e to make use of the calibrations of Tef f and B C as a function of red and infrared colours (V - R, V - I , V - K ) and to compare derived reddening ratios with the conventional extinction laws. Note that the inferred radius and luminosity of SZ Tau are too large for its short p eriod; this Cepheid probably pulsates in the 1st or even in the 2nd overtone, as may b e indirectly evidenced by its low colour amplitude (ab out 0.15m ). Fig. 2 shows the observed data, the fit to the V band light curve, and the inferred calibration F = 10 â log (Tef f ) + B C (V ) vs (B - V )0 ) calculated for TT Aql Cepheid (as a 5th -order expansion in the normal colour).

6

ACKNOWLEDGEMENTS

We grateful to M.V. Zab olotskikh for her assistance in data preparation and to L.N. Berdnikov, Yu.N. Efremov, M.E. Sachkov, V.E. Panchuk and A.B. Fokin for comments and helpful discussions. This research has made use of the WEBDA database op erated at the Institute for Astronomy of the University of Vienna. Our work is supp orted by the Russian Foundation for Basic Research (pro jects nos. 08-0200738-a, 07-02-00380-a, and 06-02-16077-a).

REFERENCES Alonso, A., Arribas, S., Martinez-Roger, C., 1999, AsApSuppl, 140, 261 Baade, W., 1926, AN, 228, 359 Barnes, T.G., Evans, D.S., 1976, MNRAS, 174, 489 Balona, L.A., 1976, MNRAS, 178, 231 Becker, W., 1940, ZA, 19, 289 Berdnikov, L. N., 1995, Astrophysical applications of stellar pulsation. Proceedings of IAU Colloquium 155 held in Cap e Town, South Africa, 6-10 February 1995; eds. Stobie, R. S. and Whitelock, P. A., ASP Conference Series, 83, 349 Berdnikov, L. N., 2008, VizieR On-line Data Catalog: CDS I I/285 Bessell, M. S., Castelli, F., Plez, B., 1998, AsAp, 333, 231 Biazzo, K., Frasca, A., Catalano, S., Marilli, E., 2007, AN, 328, 938 Binney, J. and Merrifield, M., 1998, Galactic astronomy, Princeton, NJ : Princeton University Press Flower , Ph. J., 1996, ApJ, 469, 355 Forb es, D., 1982, AJ, 87, 1022 Freedman, W. L., Madore, B. F., Gibson, B. K., et al. 2001, ApJ, 553, 47 Gonzalez Hernandez, J. I., Bonifacio, P. 2009, AsAp, 497, 497


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A. S. Rastorguev, A. K. Dambis

Gorynya, N. A., Irsmamb etova, T. R., Rastorguev, A. S., Samus', N. N., 1992, SvAL, 18, 316 Gorynya, N. A., Samus', N. N., Rastorguev, A. S., Sachkov, M. E., 1996, AstL, 22, 175 Gorynya, N. A., Samus', N. N., Sachkov, M. E., Rastorguev, A. S., Glushkova, E. V., Antipin, S. V., 1998, AstL, 24, 815 Gorynya, N. A., Samus, N. N., Sachkov, M. E., Rastorguev, A. S., Glushkova, E. V., Antipin, S. V., 2002, VizieR Online Data Catalog: I I I/229 Gray, C. D. F., 2005, The Observation and Analysis of Stellar Photospheres, Cambridge: Cambridge University Press Groenewegen, M. A. T.,2007, AsAp, 474, 975 Lee, B.-C., Galazutdinov, G. A., Han, I., Kim, K.-M., Yushchenko, A. V., Kim, J., Tsymbal, V., Park., M.-G., 2006, PASP, 118, 636 Nardetto, N., Fokin, A., Mourard, D., Mathias, Ph., Kervella, P., Bersier, D., 2004, AsAp, 428, 131 Nardetto, N., Mourard, D., Mathias, Ph., Fokin, A., Gillet, D., 2007, AsAp, 471, 661 Nardetto, N., Gieren, W., Kervella, P., Fouque, P., Storm, J., Pietrzynski, G., Mourard, D., Queloz, D., 2009, AsAp, 502,951 Ramirez, I., Melendez, J., 2005, ApJ, 626, 465 Rastorguev, A. S., 2010, Variable Stars, the Galactic halo and Galaxy Formation, Proceedings of an international conference held in Zvenigorod, Russia, 12-16 Octob er 2009; eds. Sterken, Chr., Samus, N., Szabados, L., Published by Sternb erg Astronomical Institute of Moscow University, Russia, 225; arXiv:1001.1648 Sachkov, M. E., Rastorguev, A. S., Samus', N. N., Gorynya, N. A., 1998, AstL, 24, 377 Sachkov, M. E. 2002, AstL, 28, 589 Sekiguchi, M., Fukugita, M., 2000, AJ, 120, 1072 Tokovinin, A. A., 1987, SvA, 31, 98 Wesselink, A. J., 1946, BAN, 10, 91


Classical Cepheids: Yet another version of the Baade­Becker­Wesselink method
Table 1. Physical parameters, distances, analysed using the new version of the (http://www.univie.ac.at/webda/ ) are also (1982). Distances are calculated using RV = Star SZ Tau CF Cas U Sgr DL Cas GY Sge Cluster NGC 1647 NGC 7790 IC 4725 NGC 129 Anon OB Period (d) 3.149 4.875 6.745 8.001 51.78 and interstellar reddening values for the cluster Cepheids BW method. Reddening values from WEBDA data base shown for comparison; asterisk: EB-V adopted from Forbes 3.3. E
B -V

5

Distance (pc) 796±90 3585±87 613±25 2067±58 2136±163

E 02 02 03 05 05

B -V

(WEBDA)

< R > /R 57.0±7. 46.7±0. 54.2±1. 69.3±1. 208±11 0 9 8 6

M -

V

0. 0. 0. 0. 1.

40± 54± 50± 47± 44±

0. 0. 0. 0. 0.

0. 0. 0. 0. 1.

370 531 475 548 29±0.06 (*)

4. 3. 3. 4. 6.

32± 41± 90± 12± 27±

0. 0. 0. 0. 0.

25 05 08 06 15

25 20
1.3

15 10
1.2

V -V , km/s

B-V

5 0 -5 -10

1.1

r



1

0.9

-15 -20
0.8

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

(a)
3

Phase

(b)
6.2 6.3

Phase

2

6.4 6.5

1

R/R

O

6.6

V

0

6.7 6.8

-1

6.9 7 7.1

-2

-3

0

0.2

0.4

0.6

0.8

1

7.2

0

0.2

0.4

0.6

0.8

1

(c)

Phase

(d)

Phase

Figure 1. Panel (a): Observed and fitted radial-velocity curve of U Sgr. Standard deviation V r = 1.1 k m/s. Panel (b): Observed and smoothed colour curve. Panel (c): Radius variation with phase. Panel (d): Observed and fitted light curve.


6

A. S. Rastorguev, A. K. Dambis
30
1.6

20

V - V , km/s

10
1.4

0

r

B-V
1.2 1 0.9 0



-10

-20

-30

0

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

(a)
8 6 4 2

Phase

(b)
6.4

Phase

6.6

6.8

O

R/R

7

0 -2

V
7.2 7.4 7.6 0

-4 -6 -8

0

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

(c)
38.6 38.4 38.2

Phase

(d)
Bessel et al. 1998 Flower 1996 TT Aql Per

Phase

38.2 38 37.8 37.6 37.4 37.2 37 36.8 36.6

10 lg Teff + BC(V)

38 37.8 37.6 37.4 37.2 37 36.8

10 lg T

eff

+ BC(V)

BB Sgr FM Aql TT Aql X Cyg CD Cyg RU Sct T Mon WZ Sgr SV Vul

0.4

0.5

0.6

0.7

0.8

0.9

1

0.4

0.5

0.6

0.7

0.8
0

0.9

1

1.1

(e)

(B-V)0

(f )

(B-V)

Figure 2. Panel (a): Observed and fitted radial-velocity curve of TT Aql. Standard deviation V r = 1.3 k m/s. Panel (b): Observed and smoothed colour curve. Panel (c): Radius variation with phase. Panel (d): Observed and fitted light curve. Panel (e): Calculated calibration for TT Aql (function F = 10 â log (Tef f ) + B C (V ) vs (B - V )0 )) and calibrations by Flower (1996) and Bessell et al. (1998). Also shown is the position of the standard star Per corrected for metallicity difference. Panel (f ): Calculated calibration (function F = 10 â log (Tef f ) + B C (V ) vs (B - V )0 )) for 9 Cepheids with large amplitudes of the colour curves and different metallicities.