DD Draconis - pulsating variable star with rapid period change

Using photoelectric, CCD and visual measurements of RRc type variable star DD Draconis I conclude following: rapid period change determined using O-C diagram described by b = 3.51. Using Fourier decomposition I have derived following physical parameters: M = 0.69 M_¤, L = 56.1 L_¤, T_eff = 7248 K, Y = 0.259 and M_V = 0.88 mag. I compared this values to globular cluster variables and non-linear theoretical models and estimated [Fe/H] ~ -1.7 and Z ~ 0.0004. The evolutionary stage of DD Dra is also discussed owing for explanation of rapid period change. Recent theory cannot explain such high period change rate.

I wanted to derive basic physical and light curve (period, asymetricity...) parameters of DD Dra (= BV 234 = GSC/TYC 4215 1869). I also wanted to examine evoutionary status of DD Dra.

I used visual, photoelectric and CCD measurements of DD Dra both own and adopted.

Variability of DD Dra was discovered by Strohmeier (1958) who classified it as long period variable with magnitude range (11.2 - 12.0) mag (pg). Filatov (1960) stated that it is eclipsing binary and using nine plate minima derived ephemeris which were adopted to GCVS (Kholopov et al. 1985):

Very inaccurate period motivated Agerer and Lichtenknecker (1988) to conduct photoelectric photometry. They obtained three nights of continous observations in B and V bands and concluded that DD Dra is pulsating RRc type variable with amplitudes 0.66 and 0.54 mag in B and V bands, respectively. Following ephemeris were derived:

Kühlenz (1991) investigated Sonnberg patrol plates with DD Dra and because he couldn't find one period, he stated that DD Dra is multi-periodic variable (RRd or RR(b), assuming GCVS classification) with periods P₀ = 0.32679 and P₁ = 0.24631 days, respectively. The difference 1/P₁-1/P₀ is almost 1, though, the P₁ probably is not true period, but only alias of P₀ rising from observations from one place on Earth. Agerer and Dahm (2000) didn't confirm Kühlenz's find and stated very rapid period change written in these two equations:

I have obtained all visual observations through 250 mm (f/6) Dobsonian scope placed at the balcony on the suburb of Brno, Czech Republic. I have obtained 83 estimates spanning 50 days in autumn of 1999. All observations are in table 1, used comparison stars are in table 2.

(I have removed detailed discussion of methods of visual observations and also long discussion of errors common among visual observations.

Equation decribing used Nijland-Blazhke method - included for continuity of numbering equations)

Equation describing shift between visual and V band observations. Coefficients are (Zissel 1998): k1 = 0.182 and k2 = -0.032 mag.)

The largest part of quite inhomogenous photolectric and CCD observations came from F. Agerer et al. (D. Husar, R. Diethelm, D. Lichtenknecker), which were published partly in BAVM or in IBVS. These observations were conducted using B and V filters and have error of about 0.02 mag.

Further observations came from M. Zejda and D. Hanzl from Nicolas Copernicus Observatory and Planetarium in Brno, Czech Republic (Zejda 2000). All frames were taken using SBIG ST-7 camera mounted on 400 mm diameter telescope. Exposition times are ranging from 10 to 90 s. All observations were made with unfiltered camera and therefore cannot be used for Fourier decomposition and comparison with other filtered observations. The errors are here about 0.05 mag. Frames were reduced using MUNIDOS (Novak and Hroch 2000).

P. Sobotka measured DD Dra using same equipment as above but with V and R filters, reduction was the same as above.

The last part of data consists of observations conducted by author at Vyskov observatory (Czech Republic) using 170 mm telescope and CCD camera SBIG ST-7 and V filter with exposition 60 s. Data were reduced using MUNIDOS and individual errors of points are lower than 0.1 mag.

For determination of maxima I have used Kwee-van Worden method implemented in AVE (Barbera 1996).

Maxima timings from Agerer et al. (1988, 2000 and others) were used without any change. Agerer and Hübscher (2000) recently published maximum which is different from others at the same time. Its shift is about 0.2 days, which is very similar to distance between maximum and minimum in DD Dra. This confusion was confirmed by observer (Husar 2000).

Maxima timing from Vyskov observatory should be taken with care because it has insufficient coverage of branches.

Unfortunately, data from Brno observatory don't contain any sufficient part of light curve to reliably determine maxima. From the minima timings I derived maxima using asymetricity parameter Q obtained from high-quality V band light curves of Agerer et al. Value of Q for DD Dra is 0.37. I have used the same algorithm for determination of maxima from visual observations. All maxima used for construction of O-C diagram are in table 3.

For O-C diagram construction I have used following ephemeris derived from latter half of maxima:

In Figure 1 I have plotted O-C diagram of DD Dra computed from all available maxima (description of used symbols is in figure's caption). There are also three functions. Solid lines denote abrupt period change, dashed line shows parabolic fit and example of higher-order fit (6th order) is denoted using dash-and-dotted line.

As seen from Figure 1, the period change can be either abrupt or continous with possible more complex behaviour. The only thing, which could help now could be maximum around JD 2449500, because in this time we can discriminate between these three types of fits. In further discussion, only continous period change will be used, because it is quite common among RR Lyr stars and is probably due to evolutionary effects (Lee 1991, Jurcsik et al. 2000). There are two parameters, which are used for description of continous period change: b and a. These are defined: b = DP/Dt and a = b/P. Using parabolic fit to the O-C diagram (weights same as in description of Figure 1) I obtained: b = 3,51 ± 0,12 d/Myr and a = 10,7 ± 0,4 Myr^-1.

These changes in period can be written is ephemeris form. If abrupt period change is confirmed the equation (3) can be used for description of behaviour between JD 2447271 and JD 2448205 and more precise version of elements (4) after JD 2450719 is here:

One of the basic characteristics of variable star is its light curve. With it we can discriminate between the two subtypes of RR Lyr stars: RRab and RRc. Figure 2 shows typical light curves of these two subtypes. Along with determination of subtypes we can obtain from the light curves also basic physical parameters (mass, luminosity, effective temperature, chemical composition, absolute magnitude, distance).

In the further sections of this work I'll be frequently using phased light curves. Due to the complexity of period changes in DD Dra (as shown in section 3.2) it is impossible to phase all available data without some corrections. In the first sight I tried to fit O-C diagram with parabola (equation 9) and use its formula to substract from each point of the raw measurements actual value of O-C. After this process significant scatter and shift between different epochs remained. Therefore I decided to use only so short baseline of data to ensure that period change can be negligible and linear ephemeris are enough.

Figure 3 shows phased light curve of DD Dra using B and V band data of Agerer et al. along with changes in B-V (constructed employing program Differencer - Brát 2000). The light curve looks rather common, the only unusual thing is double maximum. Figure 4 shows light curve of DD Dra in V and R bands and difference V-R using data of Sobotka (1999). Figure 5 shows light curve of DD Dra constructed from my visual observations. Zero points of phase were chosen arbitrarily, mainly 0.

I have found no hint of so-called Blazhke effect, which with tens to hundreds days changesamplitude and phase of the variable. Although for long-time known only among RRab, it was recently along with non-radial oscillations discovered in RRc variables (Alcock et al. 2000, Olech et al. 1999 and others).

It is generally known that cepheids are used for determination of distances of galaxies. In the last few years RR Lyr stars are used for independent method of measuring distances. Unfortunately, their usefullness is limited by inhomogenous abundation of metals. Without spectra (available only for bright objects) we would stay limited on very rough estimates or even wrong result.

However, there is one method, which can "override" necessity of spectra and uses only light curve in one of the standard passbands (B, V or R). It is based on assumption that light curve of pulsating star depends only on few basic physical parameters. This approach employs so-called Fourier decomposition, which describes any light curve as sum of (co)sinusoids:

where A_i are amplitudes, j_i phases(j_i = wt₀) and w = 2p/period. Amplitude and phase terms are usually combined in the following manner:

The most important equation concernig metallicity is known only for RRab variables. The situation is more complicated among RRc stars, where, of course, equations derived for RRab stars are not valid. Beside that, there arises possbility that light curves of RRc stars are not dependant on the metallicity (Simon and Clement 1993; Jurcsik 2001). Fortunately, we are able to determine some basic other basic parameters as effective temperature (T_eff), luminosity (L/L_¤), mass (M/M_¤) and helium abundance (Y) (from hydrodynamical models of Simon and Clement 1993):

Absolute visual magnitude can be obtained from the empirical relation of Kovács (1998):

One problem with this equation is that it has been derived for sinus rather than cosinus function in equation (10). The correction is: j^*₂₁ = j₂₁ - p/2. The only parameter, which we are not able to determine is abundance of iron defined as:

where N are counts of particles of corresponding elements. As defined, the Sun has [Fe/H] = 0. [Fe/H] is very good indicator of total metallicity and it is often used so.

In case of DD Dra (RRc star) I used following approach. I divided V band data of Agerer et al. into three segments (in which two inosculate to obtain complete phase coverage) and computed phased light curves with actual period. In Period98 (Sperl 1998) I fitted the phased light curves with 10^th order Fourier fit as in equation (10). The program exported semi-amplitude A (mag) and phase G (G = t₀/P) of the each fitted sinusoid. The phase term is then: j = (2p/P)*GP = 2pP; I multipled phase G from Period98 with 2p. Then I continued according to the equations (11) and (12) and the results is set of Fourier parameters in Table 4. Phased light curve with fit is shown in Figure 6. I checked my approach on two variables from OGLE survey, V8 and V10 in field BW2. In V8, all results were the same as in Morgan et al. (1998). In V10 the agreement was worse, but this could be attributed to the very noisy light curve.

For comparison of other variables I have plotted values obtained for DD Dra into Figures 7 and 8. Figure 7 was taken from Kaluzny et al. (2000) based on RR Lyr stars from globular cluster M5, which has [Fe/H] = -1.23. Figure 8 was taken from Morgan et al. (1998) based on RRc stars of Galactic bulge (small stars).



Figure 7: Fourier parameters vs. logP for stars of globular cluster M5 (from Kaluzny et al. 2000). Points left from logP = -0.4 are RRc, rights RRab. DD Dra is designated with green cross (table 4). The size of symbol doesn't represent any kind of error	Figure 8: Fourier parameters vs. logP for RRc stars from OGLE database (Morgan et al. 1998). Stars denote OGLE stars, crosses are field RRc with [Fe/H] = -1.5, triangles are field RRc with [Fe/H] = -2.0. DD Dra is designated with green cross and size of the symbol doesn't represent any kind of error.

Using equations (13) to (17) and average values from Table 4 I derived basic physical parameters of DD Dra. Because it is known (Kaluzny, Olech and Stanek 2000) that equation (17) little bit fainter absolute magnitude values I decided to compute it using indenpendent method:

where M_0bol is absolute bolometric magnitude of Sun (assuming 4.70 mag). Absolute visual magnitude is then M_V = M_bol-BC, where BC is color corection je barevná korekce proportional to metallicity (Sandage a Cacciari 1990):

Assuming position of DD Dra in Figure 7, the star should be somewhat metal poorer (Morgan et al.1998): [Fe/H] ~ -1,7 (maybe smaller). From M_V we get distance to DD Dra:

Table 5 contains mass, effective temperature, luminosity, abundance of helium, absolute visual magnitude and distance of DD Dra. Unfortunately, Period98 is not able to count errors of individual parameters of the fit so I am not able to determine errors of obtained physical parameters (although, phase coverage and quality of observations are very good, resulting errors are thought to be also low). For computing of the distance modulus I used mean magnitude from visual observations. Correction of interstellar extinction was applied using infrared dust maps of Schlegel, Finkbeiner and Davis (1998). I have omitted correction between visual and V passbands (order of hundreths of magnitude). Interstellar reddening towards DD Dra is E_(B-V) = 0.04 mag a corresponding extinction in V band A_V = 0.15 mag (using filter tables of Schlegel, Finkbeiner and Davis 1998).

In section 3.2 I derived the period change rate of DD Dra to be b = 3,51 d/Myr. For example, in set of 48 RRc stars in globular clusterw Centauri, Jurcsik et al. (2000) detected only one variable, which has higher period change rate than DD Dra. There exist three possible explanations of period changes (or changes in O-C) among RR Lyr stars. At first, so called light time effect caused by third body orbiting variable star can be excluded due to high values in O-C, which would cause unacceptably high masses of the third body. At second, these changes can irregular, which doesn't explain anything, but it is sometimes observed (Jurcsik et al. 2000). At third, the most probable are evolutionary changes due to the evolution from HB (Horizontal Branch - place in HR diagram occupied by RR Lyr stars) to AGB (Asymptotic Giant Branch). According to Jurcsik (2001) no generally acceptable theory explaining evolutionary changes has been done yet. Period changes with rates b = 1 to 2 d/Myr can occur at final stage of HB evolution, just before AGB. These stars are required to be very metal poor ([Fe/H] = -1.7 to -2.0) and should be about 0.3 to 0.5 dex of log (L/L_¤) above (ZAHB - Zero Age HB). Periods of these stars should be either very high or very low, i.e. P₁ < 0.2 or P₁ > 0.5 or P₀ < 0.3 or P₀ > 0.8.

The theory doesn't fully explain behaviour of DD Dra. Firstly, the period does not fall into the required region. As was shown above, DD Dra probably meets the metallicity condition. The [Fe/H] quantity doesn't fully describe whole abundance of metals in star (metals mean in astronomy all elements heavier than helium). Total mass-fraction of metals in star denote quantity Z, which is proportional to [Fe/H], but we have to now f, a enrichment factor in respect to iron (obtained from spectra). I estimated value of Z to be ~ 0.0004 (maybe lower) using models of Caput et al. (2000). Figure 9 shows position of DD Dra in HR diagram with plotted instability strips, ZAHB and evolutionary tracks for stars of different mass (Bono et al. 1997). DD Dra lies at blue edge of first overtone instability strip. If we assume that DD Dra has Z = 0.0004, then it is about 0.1 dex of log (L/L_¤) above ZAHB. For Z = 0.001, the distance is only 0.06 dex of log (L/L_¤). The theoretical condition of luminosity is not satisfied by DD Dra.

Using distance and position in sky, I conclude that DD Dra belongs either to population of thick disk or more probably to galactic halo (Baade's population II).

After analysis of visual, photoelectric and CCD measurements of DD Dra I conclude that it is pulsating RRc variable with rapid period change. This change could be either abrupt or more probably continous. Using Fourier decomposition of V band light curve I determined basic physical parameters of DD Dra. Using comparison of both Fourier and physical parameters with theoretical models and stars from globular clusters I derived [Fe/H] ~ -1.7 and Z ~ 0.0004. DD Dra surpasses doesn't fulfil basic ideas about rapid period changes in RRc stars.

I acknowledge with thanks M. Zejda, P. Sobotka and F. Agerer for providing theirs CCD measurements and P. Hájek and K. Koss for help with conducting my observations at Vyskov observatory. I also thank J. Jurcsik for fruitful discussions on RRc stars and Fourier parameters.

Author (pejcha@meduza.info) will be pleased to send you all figures in higher resolution and/or whole paper in other formats.

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log (M/M_¤) = 0.52 log P - 0.11j₃₁ + 0.39	(13)
log (L/L_¤) = 1.04 log P - 0.058j₃₁ + 2.41	(14)
log (T_eff/K) = 3.265 - 0.3026 log P - 0.1777 log (M/M_¤) + 0.2402 log (L/L_¤)	(15)
log Y = -20.26 + 4.935 log (T_eff/K) - 0.2638 log (M/M_¤) + 0,.318 log (L/L_¤).	(16)


Figure 6: Phased V band light curve of Agerer et al. fitted with 10^th order Fourier fit as in equation (10).

R_ij = A_i/A_j	(11)
j_ij = j_i - ij_j.	(12)