4. Discussion

4.1. Activity of ZZ UMa

One of the main problems when analysing photometric light curves for late-type binary systems is the modulation that can be seen outside eclipses, due to the effect of intrinsic surface activity phenomena in one or both components on stars of spectral type later than F. The contribution of activity mixes with the proximity effects, all of them being observed outside eclipses. The light curves of ZZ UMa show variations outside eclipses that can be attribute to this effect. So, before going into the geometrical and radiative analysis we need to clean the light curves of stellar activity contribution.

To determine the associated wave due to activity we used a new iterative method that approximates the wave at the same time that computes the EBOP solution. This method has been used in the analysis of BH Vir, a late-type binary system that shows a high level of activity which is variable with time, and proved to be very efficient, see Paper II.

For ZZ UMa this new method takes into account the fact that the modulation observed outside eclipses is produced by different physical processes: tidal distortion, mutual reflection, and the presence of spots on the surface of the stars. Quasi-synchronic rotation of the stars in evolved systems generate an overlapping of the stellar activity phenomena with the binarity effects, both having the same period.

The first step of the iterative process is to adjust a second order truncated Fourier series,

to the points outside eclipses, obtaining the first approximation to the wave.

The second step is to subtract this wave to the light curve and find the first binary solution for the rectified light curve of the system by using the EBOP code.

The third step is to estimate the new approximation to the activity wave by adjusting a Fourier series to the residuals of the theoretical curve, that we assumed are mainly due to activity effects.

We have iterated this cleaning process until the wave and the binary solution became stable, which happened in the fourth iteration. The coefficients found for the waves are:

Figures 1 (click here) and 2 (click here) show the results of our first estimations and final waves in the u, v, b and y filters. Both figures present differential magnitudes versus phase; continuous line corresponds to the waves in the y filter, triangules, squares, and crosses, correspond to the b, v, and u filters respectively.

  
Figure 1: Initial associated activity waves for ZZ UMa

  
Figure 2: Final associated activity waves for ZZ UMa

We found that the variation of the waves in the iterative process was small. The mean of the differences between the coefficients of the first and the final waves was only , A₃ being the coefficient that concentrated most of the variation, as expected.

The amplitude of the waves measured as the difference between the points of maximum and minimum light is 0.053, 0.048, 0.042 and 0.040 for u, v, b, and y respectively. More representative of the real contribution of the activity is the integral of the waves along one complete phase, that represents the stellar filling factor, which was found to be 0.040, 0.039, 0.033 and 0.026 for u, v, b, and y respectively. That is, assuming a black-spot model we estimate that 4% of the total area is covered by spots.

Although the activity contribution to the light curve of ZZ UMa is small and it does not show a high level of variability with time, it would have been better to estimate the activity waves independently for each period, in order to take into account the possible migration of the waves with time, but we do not have complete light curves for any of the observing campaigns.

The shape and (b-y) colour of the waves suggest the existence of two active centres of similar intensity, at phase 0.2 and 0.6 isolated by a less perturbed photosphere.

4.2. Geometrical parameters of ZZ UMa

To estimate the geometrical parameters (inclination of the orbit, ratio of radii, etc.), we solved the light curve by using the EBOP code (Etzel 1975). This program is designed for the solution of detached binary systems with small side-effects due to proximity. Thus, it is adequate for ZZ UMa, for which our estimation of the radii relative to the orbit is 0.16 and 0.12 for the primary and secondary components respectively.

The process of optimisation involved in the search for the EBOP solution must always be constrained by a priori bounds imposed on the astrophysical parameters being optimised. Otherwise, as pointed out by some authors, (Popper 1984, 1993; Andersen 1991), it is possible to find different sets of solutions which fit the observed light curve with equal precision. The solution is not uniquely determined.

To get realistic first approximation to some of the values that we attempt to optimise with EBOP,( and ), we did a photometric pre-calibration of the light curves (Clement et al. 1993). Limb darkening coefficients were taken from Claret & Giménez (1990), assuming the initial values for the effective temperatures and gravity. Gravity darkening coefficient was calculated with the black body formula of Martinov (1973) using our initial estimation of effective temperatures. For the radius of the primary star relative to the orbit, and the inclination of the orbit, we assumed and i=88.7 degrees as a first approximation, after a preliminary analysis. From new spectroscopy data (Popper 1995) and , we quoted the mass ratio, q=0.81. However, it should be noted that these values are still preliminary. The period and the initial epoch are from Mallama (1980).

In Table 5 (click here) we list the initial parameters adopted to run EBOP. "V" marks the six free parameters in every EBOP run. Those are the parameters that EBOP can optimise better. All the others remained fixed.

After subtracting the final waves from the raw data we searched for the EBOP solution following the recommendations of Etzel in the EBOP users guide (Etzel 1981).

  

name y₀ (b-y)₀ m₀ c₀

ZZ UMa 9.798 0.385 0.201 0.317 2.596

0.007 0.008 0.014 0.016 0.008

Hot 10.110 0.373 0.180 0.337 2.603

0.007 0.006 0.011 0.017 0.008

Cold 11.307 0.422 0.275 0.250 2.594

0.035 0.054 0.065 0.085 0.016

   

parameters number u v b y

V-1 0.613

V-2 0.16

V-3 0.74

prim limb dark coef 4 0.88 0.83 0.77 0.68

second limb dark coef 5 0.94 0.88 0.81 0.73

i V-6 88.7

7 0.0

8 0.0

prim gravita dark coef 9 0.25

second gravita dark coef 10 0.25

prim reflection (internal calculation) 11 0.0

second reflection (internal calculation) 12 0.0

mass ratio 13 0.81

lead/lag ang 14 0.0

third light 15 0.0

out of phase V-16 0.0

maximum light V-17 -0.113 -0.177 -0.232 -0.270

integration ring 18 5

period 19 2.29926000

initial epoch 20 2441499.59530

   

Parameter number u v b y mean

1 0.440 0.462 0.528 0.556

0.009 0.008 0.007 0.006

2 0.158 0.161 0.164 0.160 0.161 0.002

3 0.766 0.761 0.759 0.766 0.763 0.003

0.121 0.123 0.124 0.122 0.123 0.001

prim limb dark coef 4 0.88 0.83 0.77 0.68

second limb dark coef 5 0.94 0.88 0.81 0.73

i 6 87.99 88.05 87.90 88.09 88.01 0.07

7 0.0

8 0.0

prim gravita dark coef 9 0.25

second gravita dark coef 10 0.25

prim reflection (internal calculation) 11 0.002 0.002 0.002 0.002

second reflection (internal calculation) 12 0.005 0.005 0.005 0.005

mass ratio 13 0.81

lead/lag ang 14 0.0

third light 15 0.0

out of phase 16 0.002 0.002 0.003 0.003

maximum light 17 -0.170 -0.227 -0.274 -0.297

integration ring 18 5

period 19 2.29926

initial epoch 20 1499.59530

L(primary) 0.79976 0.79395 0.79620 0.75822

L(secondary) 0.20024 0.20605 0.23080 0.24178

0.018 0.016 0.014 0.012

   

Literature P K₁ K₂

2.29926 93.2 114.5

0.0001 0.5 0.5

Ours EBOP i

0.556 0.161 0.123 88.01

0.007 0.002 0.001 0.07

Ours photometry redd = 0.006

Primary star (b-y)₀

0.373 2.603

0.008 0.005

Primary star A

9.45 1.182 1.519 4.147 3.742 5903

0.04 0.013 0.023 0.017 0.026 60

d BC M_v (p)

172 2.472 3.66 -0.27 3.93

21 0.023 0.05 0.27 0.26

Secondary Star

0.962 1.159 4.293 3.678 5097

0.010 0.013 0.015 0.026 60

BC M_v (s)

0.800 4.88 -0.27 5.15

0.023 0.06 0.27 0.26

Table 6 (click here) lists the definitive solution. Last column () is the RMS dispersion of the calculated values in every filter.

The program calculates internally the mutual reflection assuming the simple case of a hemisphere uniformly illuminated (Binnendijk 1960), but taking into account the eclipse of the reflected light (Etzel 1981). The small values obtained for the reflection confirm that this simple model is good enough.

The synthesised light curve presents total secondary eclipse (EBOP computes 10 identical values, from phase 0.494 to phase 0.501, in every filter), confirming our hypothesis. Thus, the deconvolution process from the light observed at quadrature and at the bottom of the secondary star is adequate.

So we can conclude that the binary system undergoes total eclipse and that both components have small deformation effect due to proximity.

The values for the radii and the orbit inclination obtained,





are accurate enough to allow us to calculate the fundamental astrophysical parameters, masses and luminosities, precisely and fulfil the final objective of this work.

4.3. Radiative and absolute parameters of ZZ UMa. Evolutionary status

Following the same procedure used for BH Vir, described in Paper II, we have

1. Decoupled both stars, from the EBOP values at quadrature and at the bottom of the secondary eclipse (that is total).
2. Estimated the radiative parameters for the hot and cold components.
3. Computed the absolute parameters for both stars, combining the binary solution (, , , i), with the photometric calibrations (, ), and the semi-amplitudes (K₁, K₂) of the radial velocity curve.

In Table 4 (click here) we list the rectified intrinsic magnitudes and indices outside eclipse (ZZ UMa), and during totality (hot component), as well as the computed values for the cold component. The errors for ZZ UMa and the hot star are the rms of the rectified observed points at quadrature (10 points), and at the bottom of the secondary eclipse (10 points). For ZZ UMa Cold the errors are calculated through the formula used in the process of deconvolution.

Table 7 (click here) shows the absolute parameters calculated for both stars inside the system. A is the semi-mayor axis of the orbit calculated from the masses of both stars and the orbital period, using the fundamental formula given in Schmidt-Kaler et al. (1982). In the same table, and are the flux brightness and effective temperature for the primary star calculated from our photometry ((b-y)₀, ), using the calibrations of Moon (1984) and Saxner & Hammarbäck (1985). The flux brightnes for the cold component, , was calculated from and computed with EBOP. The effective temperature for the secondary star was deduced as well from and , assuming that the bolometric corrections for both stars are the same.

From the radiative parameters, the primary star can be classified as G0, while the secondary would be G5, both being main sequence stars. An estimation of their metallicity using the calibrations of Nissen (1981) gives a metal content of [Fe/H] and [Fe/H] for the hot and cold components. Both stars have solar metallicities.

The distance (172 pc) computed from the absolute magnitude, M_v, locates ZZ UMa at a distance similar to the comparisons. So the reddening estimation from the comparisons, can be assumed for the system.

We have used the evolutionary model of Claret & Giménez (1992) to estimate the age of the system. The initial parameters for the model calculations were the masses given in Table 7 (click here), and solar abundances. This model covers a wide range of masses (1 to 40 M₀), but the cold component of ZZ UMa lies out of that range and it would be necessary to extrapolate the model. For the Hot component the values predicted inside the model were:

which are in good agreement with our results.

For completeness we used the evolutionary model of Schaller et al. (1992) and found that both components lie on the isochrone , which is in good agreement and within the uncertainty of the result given above.

Figure 4 (click here) shows the rectified light curve together with the EBOP solution. We believe that the spreading of points observed in some parts of the light curve (entrance of the primary eclipse, and phase 0.9), is due mainly to residual activity effects, coming from the mixing of data from eight different epochs.

  
Figure 3: Differential light curve for ZZ UMa. y filter

  
Figure 4: ZZ UMa + EBOP definitive solution. y filter