As we said in the introduction, it is generally accepted that many late-type fast rotator stars show surface activity phenomena. As a consequence of the quasi-synchronisation of the rotation and orbital periods of many sufficiently evolved late-type binary systems, a variable fraction of the active zones or spots of a star within the system (or both) can be seen throughout each complete period, producing a magnitude variation outside eclipse. Assuming a solar standard model of activity, the position of the spots at different latitudes can give rise to migrations of the associated wave, giving the spots a lower velocity (spots at high latitudes) or higher velocity (spots close to the equator) relative to the average rotation velocity. The latter is the velocity synchronised with the period of the binary system, thus the migration of the spots adds complexity to the task of analysing the light curves. Simultaneously, other specific effects are produced due to the proximity of the two stars, such as deformations due to gravitational interaction, and mutual reflection, that also generate variations in the light curves between minima.
A preliminary inspection of the BH Vir light curves clearly shows a distortion outside the eclipses, called the associated wave, see Fig. 1 (click here). Amplitude variations up to 0.1 magnitudes in the y filter are clearly visible at phases 0.1 an 0.2.
So, before going into the analysis of the light curve it is necessary to clean the associated wave.
Different systems to clean the activity have been used and reported in the literature. Ludington (1978) was the first to use truncated Fourier Series (four terms) to fit the observed points outside eclipse and later to compute the geometrical solution of the binary.
The main problem with this cleaning process is that binary effects between components are also included in the terms of the Fourier Series and the rectified light curve becomes flat between eclipses losing relevant information. Proximity effects are clearly included in the second order terms of the Fourier Series.
In order to avoid this problem, a new approach is presented in this paper. We have used an iterative process that:
To estimate the associated waves from the data of April 1993 and May 1994 in which the light curves taken were well covered, we have used the above referred to iterative process. After 6 iterations the process converged (the differences in the coefficients for the consecutive waves were less than 0.001, and the binary solution variations were less than the accuracy attained).
A main feature is the decrement on the wave amplitudes. It was found that the "real" amplitudes of the associated waves for these two campaigns are only 30% and 50%, in the y filter, of those calculated in the first approximation, measured as the integral of the wave over a whole orbital period. Clearly, in the zero-order solution there is a systematic overestimation of the activity wave by inclusion on the apparent waves of binary effects. Depending on the spot location over the stellar surface, overestimation or underestimation of the activity wave are expected in the zero-order approach. These results for BH Vir activity indicate the necessity to apply a more sophisticated deconvolution process than the zero-order approach.
The waves show the high level of variability of BH Vir with a mean variation amplitude of 0.07 magnitudes in the y filter, measured as the difference between the waves at the point of maximum light. For the other filters, the wave pattern is very much the same. The waves calculated for March 1990 and January 1992 are somehow uncertain because they are based only in 19 and 23 normal points outside eclipses respectively. Due to the small number of points in these two epochs, we could not apply the iterative process to estimate the contribution of the activity. We just estimated the wave adjusting a truncated Fourier series to the residuals obtained with the best binary solution computed for the "clean" light curves of other two epochs, (1543 points). The coefficients found for the u, v, b and y waves respectively are:
| coeff. | March 1990 | January 1992 | April 1993 | May 1994 |
| A0 | -0.015729 | +0.007811 | +0.019489 | +0.020132 |
| A1 | -0.035625 | -0.012013 | +0.005832 | +0.017463 |
| A2 | -0.016264 | +0.023505 | -0.007779 | +0.000981 |
| A3 | +0.008105 | -0.001278 | +0.004047 | +0.015410 |
| A4 | +0.010965 | +0.011224 | -0.016751 | +0.004881 |
| A0 | +0.004730 | +0.010743 | +0.017115 | +0.014607 |
| A1 | -0.044672 | -0.015328 | -0.000763 | +0.013777 |
| A2 | +0.004087 | +0.024259 | -0.005253 | +0.003696 |
| A3 | +0.005884 | -0.008475 | +0.001284 | +0.015283 |
| A4 | +0.022943 | +0.009589 | -0.012563 | +0.003040 |
| A0 | +0.016240 | +0.018524 | +0.016169 | +0.012103 |
| A1 | -0.033652 | -0.010165 | -0.003506 | +0.011396 |
| A2 | -0.002286 | +0.025981 | -0.007304 | +0.004087 |
| A3 | +0.002034 | -0.011804 | +0.000903 | +0.012935 |
| A4 | +0.014351 | +0.007047 | -0.008284 | +0.003509 |
| A0 | +0.014442 | +0.014625 | +0.013222 | +0.013493 |
| A1 | -0.033050 | -0.008445 | -0.004958 | +0.011500 |
| A2 | +0.000443 | +0.020879 | -0.007114 | +0.002866 |
| A3 | +0.000748 | -0.009356 | -0.001119 | +0.010735 |
| A4 | +0.016629 | +0.007618 | -0.004876 | +0.005642 |
Figures 2 (click here), 3 (click here), 4 (click here), 5 (click here) show the estimated waves for each campaign in each filter. The waves corresponding to the first campaign are not reliable because the residual points used for the fitting are not homogeneously distributed, and they are not drawn.
The new method used to estimate the waves is physically correct and is
clearly effective looking at the small dispersion of the points in the
rectified light curve (Fig. 1 (click here)), compared with the dispersion of
points in the observed light curve (Fig. 7 (click here)), specially outside
eclipses, taking into account that they include points from four different
campaigns along four years.
To evaluate the goodness of the method we calculated normal points, binning
the data in intervals of 0.01 of phase, before and after "cleaning" in two
different regions: outside eclipses and in different areas inside the eclipses.
In the rectified light curves the RMS of all the normal points outside eclipses
is equal or less than the accuracy of the photometry (0.009 in the y filter)
while in the non rectified ones is of the order 0.020.
Inside eclipses the scatter gets better. The dispersion
at the bottom of the secondary is of the same order before and after
cleaning. In the primary the dispersion of the 377 points at phases
improves a mean of 0.007 with a maximum at the right end of
the primary [0.07, 0.08) where the 18 points binned go from
RMS 0.033 to 0.015 in the y filter.
However the best global effect of the cleaning occurs from phase 0.06 to
phase 0.23 and from phase 0.61 to 0.68 where the mean RMS change from
0.023, 0.021, 0.023, 0.023 to 0.014, 0.010, 0.010, 0.008
for the 661 points binned in u, v, b and y filters respectively.
Figure 1: Differential light curve BH Vir, y filter
Figure 2: Final waves Jan92, Apr93 and May94, y filter
Figure 3: Final waves Jan92, Apr93 and May94, b filter
Figure 4: Final waves Jan92, Apr93 and May94, v filter
Figure 5: Final waves Jan92, Apr93 and May94, u filter
Figure 6: BH Vir. Evolutionary track, 
.
(Claret & Giménez 1992)
Figure 7: BH Vir rectified light curve + EBOP solution. y filter
The search and fitting of the best geometrical solution of this binary using the EBOP program, has been carried out in two distinct parts.
In every EBOP run (along Phase 1 and Phase 2) we left free the parameters that the code can optimise best:
while we left fixed:
The reflected light is calculated by the program assuming the simple case of
a hemisphere uniformly illuminated (Binnendijk 1960), but
taking into account the eclipse of the reflected light (Etzel
1981). The program calculates that the fraction of the smaller star
covered by the primary at mid-secondary eclipse is 98.82%. The secondary
eclipse appears to be total within the uncertainty of the luminosity
errors, result that is in good agreement with Vincent (1993).
In Table 5 (click here) we list the final EBOP solution.
The geometrical results obtained for the components of BH Vir are consistent
with those obtained by Zhai, although ours improve the accuracy of the radii
of the components relative to the orbit, measured as the RMS of the values
in the four filters. Zhai obtained 
and
, for and , while we obtained
and 
.
Figure 7 (click here) shows the rectified light curve together with the EBOP
best solution.
The decrease in the 
(observed-calculated), and the
disappearance of systematic variations in 
in the final solution
respect to the initial ones, in spite of merging points from four different
campaigns of this active binary,
also comfirms that the iterative method followed is
efficient.
In order to derived the astrophysical parameters of the individual components
of BH Vir from the rectified light curves, after geometrical solution, we
need to estimate the individual magnitudes and colour indices of each star
instead of the integrated values.
First of all we corrected magnitudes and indices of the local interstellar
reddening estimation E(b-y)=0.026.
We can use the photometric values during totality
(the secondary eclipse), together with the 
computed running
EBOP, to get the individual values for each of the components.
To derive the photometric values for the blue component we followed the
usual technique given by Lacy (1977).
In Table 2 (click here) magnitudes and indices are given outside
eclipse (BH Vir) and during totality (blue component), together with
the computed magnitudes and indices for the secondary star
(red component).
| name | y0 | (b-y)0 | m0 | c0 |  |
| BH Vir | 9.504 | 0.369 | 0.185 | 0.312 | 2.592 |
| 7 | 5 | 9 | 12 | 7 | |
| BH Vir Hot | 10.019 | 0.348 | 0.184 | 0.333 | 2.627 |
| 7 | 5 | 9 | 12 | 7 | |
| BH Vir Cold | 10.560 | 0.403 | 0.190 | 0.275 | 2.572 |
| 7 | 5 | 9 | 12 | 7 |
The radiative parameters for the hot and cool components have been derived
by using the well know
standard calibration procedure for F type stars given by Crawford
(1975), and extended by Olsen (1988), for early G
type stars. Effective temperatures, 
, and visual surface
brightness, 
, were computed from the semiempirical calibrations of
Saxner & Hammarback (1985), and Moon (1985),
respectively. For the cool component we have used the standard photometric
calibrations given by Olsen (1984), for late-type stars.
Results are given in Table 3 (click here). Effective temperatures, visual
flux brightness and absolute magnitudes led us to classify the
hot component as a G0 main sequence star. The cool component can be
classified as a G2 main sequence star, both with solar composition.
The distance, 135 pc, computed from the calculated absolute magnitude
Mv, is slightly smaller than the one found previously (Clement et
al. 1993), when we analysed the data for the three first campaigns,
but is inside the indetermination of the Mv value.
As we can see the comparisons are located at a comparable
distance.
| name | Mv | | [Fe/H] |  | d | Sp |
| BH Vir Hot | 4.37 | 3.77 | 0.1 | 6158 | 135 | G0 V |
| 3 | 3 | 2 | 60 | 19 | ||
| BH Vir Cold | 4.41 | 3.74 | -0.3 | 5714 | 170 | G2 V |
| 3 | 3 | 2 | 60 | 22 |
| parameters | number | u | v | b | y |
 | V-1 | 0.895 | |||
|
| V-2 | 0.20 | |||
 | V-3 | 1.0 | |||
| prim. limb dark. coeff. | 4 | 0.85 | 0.82 | 0.76 | 0.67 |
| second. limb dark. coeff. | 5 | 0.88 | 0.84 | 0.78 | 0.69 |
| i | V-6 | 88.0 | |||
 | 7 | 0 | |||
 | 8 | 0 | |||
| prim. gravita. dark. coeff. | 9 | 0.25 | |||
| second. gravita. dark. coeff. | 10 | 0.25 | |||
| prim. reflection (internal calculation) | 11 | 0 | |||
| second. reflection (internal calculation) | 12 | 0 | |||
| masses ratio | 13 | 0.967 | |||
| lead/lag ang | 14 | 0 | |||
| third light | 15 | 0 | |||
| out of phase | V-16 | ||||
| maximum light | V-17 | 0.008 | 0.054 | -0.073 | -0.170 |
| integration ring | 18 | 5 | |||
| period | 19 | 0.81687099 | |||
| initial epoch-2440000 | 20 | 3230.609 | |||
| parameters | number | u | v | b | y |  |
|
| V-1 | 0.59765 | 0.60404 | 0.65071 | 0.68763 | |
|
|  | 0.00421 | 0.00244 | 0.00231 | 0.00230 | |
|
| 2 | 0.2634 | 0.0006 | |||
|
| 3 | 0.891 | 0.002 | |||
|
| 0.2347 | 0.0002 | ||||
| prim. limb dark. coeff. | 4 | 0.85 | 0.82 | 0.76 | 0.67 | |
| second. limb dark. coeff. | 5 | 0.88 | 0.84 | 0.78 | 0.69 | |
| i | 6 | 87.45 | 0.16 | |||
|
| 7 | 0.0 | = | = | = | |
|
| 8 | 0.0 | = | = | = | |
| prim. gravita. dark. coeff. | 9 | 0.25 | = | = | = | |
| second. gravita. dark. coeff. | 10 | 0.25 | = | = | = | |
| prim. reflection (internal calculation) | 11 | 0.00851 | 0.00855 | 0.00904 | 0.00937 | |
| second. reflection (internal calculation) | 12 | 0.01435 | 0.01427 | 0.01404 | 0.01373 | |
| masses ratio | 13 | 0.894 | ||||
| lead/lag ang | 14 | 0.0 | ||||
| third light | 15 | 0.0 | ||||
| out of phase | 16 | 0.01165 | 0.01171 | 0.01183 | 0.01175 | |
| maximum light | 17 | -0.01179 | 0.04064 | -0.08383 | -0.18474 | |
| integration ring | 18 | 5 | ||||
| period | 19 | 0.81687099 | ||||
| initial epoch-2440000 | 20 | 3230.609 | ||||
 | 0.019571 | 0.011258 | 0.010129 | 0.009652 | ||
Once the solution of the binary system was determined with sufficient
precision - that is, , , and i -,
combining these values with the amplitudes K1 and K2, deduced
from the radial velocity curve (Popper 1995), and with the
values of the effective temperature and the flux of the primary star,
and 
determined from the photometric calibration of
the theoretical optimum light curve computed with EBOP, we calculated the
absolute parameters: masses, radii, surface gravity, luminosity and
bolometric magnitude. We used the astrophysical fundamental formula
collected in Schimdt-Kaler et al. (1982). Table 6 (click here) list these
values. The errors are obtained through the formula.
The masses calculated in this paper differ from the ones published by Koch (1967), Guiricin et al. (1984) and Zhai et al. (1990) because they are based in new values of K1 and K2 derived recently by Popper (1995) and new i value.
The radii in this paper are also different from the calculated by other
authors and more precise, Zhai gave 
and
while we obtained 
and
.
The fact that the photometric light curve has been clean with a
more sophisticated method leads to different values of A. Zhai reported
while we found 
.
We used the recent evolutionary model of
Claret & Giménez (1992) to estimate the age of BH Vir and
confirmed that both components lye on the same isochrone.
The initial parameters for the model calculations were the masses given in
Table 6 (click here) and solar metalicity. The temperatures, gravities and
ages predicted with the model are:
, 
and
for the primary star and
, 
and
for the secondary star, which gives an estimated age
of 
for the system. The temperature and gravity
calculated by this model agree with our calculations given in Table 6 (click here).
The evolutionary branch of this model can be seen in Fig. 6 (click here).
| Literature | P | K1 | K2 | |||
| 0.81687099 | 138.4 | 154.9 | ||||
| 0.00000010 | 1.0 | 1.0 | ||||
| Our | EBOP | | | | i | |
| 0.688 | 0.2634 | 0.2348 | 87.45 | |||
| 0.002 | 0.0006 | 0.0002 | 0.16 | |||
| Our | photometry | redd = | 0.026 | |||
| u0 | v0 | b0 | y0 | (b-y)0 | m10 | c10 |
| 11.765 | 10.900 | 10.368 | 10.019 | 0.348 | 0.184 | 0.333 |
| 0.009 | ||||||
| Primary | Star | |||||
| A |  |  |  |  |
 | |
| 4.744 | 1.133 | 1.250 | 4.30 | 3.77 | 6158 | |
| 0.092 | 0.017 | 0.007 | 0.01 | 0.03 | 60 | |
| d |  |  | BC |
 | ||
| 153 | 1.98 | 3.90 | -0.20 | 4.10 | ||
| 18 | 0.02 | 0.05 | 0.27 | 0.26 | ||
| Secondary | Star | |||||
 |  |  |
 |  | ||
| 1.013 | 1.114 | 4.35 | 3.73 | 5607 | ||
| 0.016 | 0.006 | 0.01 | 0.03 | 60 | ||
 |  | BC | 
| |||
| 1.08 | 4.55 | -0.20 | 4.75 | |||
| 0.02 | 0.05 | 0.27 | 0.26 |