4. Photometric analysis

The strong curve asymmetry appeared as the brightness difference between the light maxima in the light curves of XY UMa in Fig. 1 (click here). Such asymmetric light curves of the system observed by Geyer (1976) and are interpreted as stellar star-spot activity of the primary component of this binary. Geyer proposed the following model for the interpretation of this phenomena. The spotted area of the G2V component of XY UMa was on the far side of the observer during primary minimum. Thus a larger light loss was caused by the transit eclipse of the smaller component. Under these conditions the observer, outside of the primary eclipse, always sees parts of the spotted area, and this phenomenon can be well observed shortly after the secondary component is occulted. This causes the plateau shaped part of the light curve, and at the same time the system brightness is at its lowest value. In this work, we took into consideration this model proposed by Geyer.

The synthetic light curve technique of Wilson & Devinney (1971) was applied to the system XY UMa. This method has been described by many authors. The modern version of the method developed by Wilson (private communication, 1992) was used for our solutions. The major revision is that of 1992, which has detalied reflection and non-linear limb darkening (both optional), adjustment of spot parameters, an optional provision for spots to move with the rotating surface, capability for following light curve development over large numbers of orbits, and greater speed. The method assumes the star surfaces to be equipotentials and computes the light curve as a function of the following parameters: the orbital inclination i, surface potentials , flux-weighted average surface temperatures , mass ratio , unnormalized monochromatic luminosities , limb-darkening coefficients , gravity-darkening exponents , bolometric albedos . Throughout this paper the subscripts h and c refer to the primary (hotter) and secondary (cooler) component, respectively. Also, this model can computing optionally the following star-spot parameters: the latitude of a star-spot center , the longitude of a star-spot center , the angular radius of a star-spot , and the temperature factor of a spot .

For the solution, the 197 observational points were combined into 50 normal points in each colour, and weighted directly according to the number of individual observations included in a point. The normal points were given in Table 6 (click here). The light curves were analyzed independently for B and V colours. The temperature of primary component was taken from Budding & Zeilik (1987) as equal to 5700 K corresponding to the G5 spectral class given in literature. The limb-darkening coefficients were taken from Jassur (1986), the gravity-darkening exponents from Lucy (1967). The bolometric albedos and were set to be equal to 0.5 from Rucinski (1969) for convective atmospheres. These parameters were kept constant during all the iterations.

As a first step in the photometric analysis of the system, the value of the q parameter was taken to be 0.75 which is very close to the one given by Geyer (1977). The purpose here is to determine the parameters of the spot with the Wilson-Devinney code, which is supposed to take place on the component causing an asymmetry in the light curves of the system XY UMa (see Fig. 1 (click here)). Wilson-Devinney code needs spot parameters predicted closely. Otherwise, to converge to a meaningful solution is very hard to achieve. Therefore, we have calculated unspotted light curve for V colour of XY UMa by using geometrical and physical parameters of the system from Geyer (1977) and Jassur (1986). This unspotted V light curve has been subtracted from observed V light curve. This difference provided as a rough light curve which has only the effects of the spots. The spot parameters are then predicted roughly by fitting a synthetic light curve to this difference curve. The synthetic light curve which is used to predict rough spot parameters is computed according to Eker (1994). This rough spot parameters were then adopted for Wilson-Devinney code. This is analyzed for maculation (star-spot) effects. Only one spot group was used. So, the search was made in the observed V light curve by choosing , , , , i, , , , and as adjustable parameters in Wilson-Devinney program. The analysis was made with detached configuration (i.e. Mode 2). The parameters thus obtained were given in Table 7 (click here). The probable error of each adjustable parameter given in Table 7 (click here) was calculated while all the other parameters were fixed to values given in the table.

  
Figure 6: The behaviour as a function of mass ratio q

  
Figure 7: Normal points of XY UMa in magnitudes, and theoretical light curves corresponding to the parameters obtained from the solutions B and V light curves. Dots and solid lines denote normal points and theoretical light curves, respectively

  
Figure 8: The configuration of XY UMa for q=0.828

Using the Wilson-Devinney (1992 version) program, we found that a circular dark spot which covers of stellar photosphere of primary component (see Table 7 (click here)) could account for the intrinsic light variations of the system. The temperature factor of the spot, , computed by this program shows that the spot temperature equals to about 3815 K for a primary star temperature of 5700 K.

At later steps, the spot parameters given in Table 7 (click here) were considered to be fixed in order to make things easier and then the light curves were freed off the disturbing effects of the spots.

There is no spectroscopic mass ratio for XY UMa in the literature and in order to test the photometric mass ratio given in the literature, a photometric q-search was performed. The search was made in the V light curve by choosing i, , , , and as adjustable parameters. The analysis was made with detached configuration (i.e. MODE 2). The weighted sum of the squared residuals, , for the corresponding mass ratios are shown in Fig. 6 (click here). As can be seen from the figure, the variation of the weighted sum of the squared residuals versus mass ratio gives a minimum around q = 0.80. Therefore, we used this value of q as a starting input parameter in the final solution. The convergent solutions were obtained with the free parameters by iterating, as usual, until the corrections on the parameters became smaller than the corresponding probable errors. The mass ratio was found as from solution of V light curve. But in the successive iterations in the B light curve, the value of q increased continuously and a meaningful solution could not be obtained. Therefore, the value of q obtained from the solution of V light curve was adopted and kept fixed in the solution of B light curve, then the other parameters were calculated. The results of final solutions are given in Table 8 (click here). The theoretical light curves calculated with the final elements given in Table 8 (click here) are shown in Fig. 7 (click here). As it seen, they agree quite well. But, the theoretical B light curve slightly differs from observations around the mid-primary.

The configuration of the XY UMa calculated with the Roche model is shown in Fig. 8 (click here). Accordingly, the system XY UMa is a detached binary.

   

Phase l_B l_V

0.7065 0.8971 0.9186 1

0.7359 0.4411 0.0795 0.9220 0.9317 10

0.7813 0.4364 0.0759 0.9260 0.9348 7

0.8316 0.4466 0.0926 0.9175 0.9206 11

0.8789 0.4659 0.1065 0.9013 0.9090 8

0.9039 0.4865 0.1115 0.8845 0.9048 2

0.9155 0.5225 0.1295 0.8558 0.8900 2

0.9230 0.5250 0.1537 0.8538 0.8705 3

0.9343 0.5910 0.2080 0.8037 0.8282 1

0.9419 0.6233 0.2467 0.7802 0.7994 3

0.9560 0.7380 0.3610 0.7024 0.7198 1

0.9566 0.7290 0.3400 0.7082 0.7338 1

0.9646 0.8085 0.4195 0.6584 0.6822 2

0.9753 0.9030 0.5000 0.6038 0.6337 1

0.9823 0.9342 0.5250 0.5868 0.6194 5

0.9972 0.9710 0.5790 0.5673 0.5894 1

0.9975 0.9785 0.5795 0.5634 0.5892 2

0.0052 0.9750 0.5903 0.5652 0.5834 3

0.0159 0.9560 0.5623 0.5751 0.5986 4

0.0228 0.9207 0.5420 0.5941 0.6098 3

0.0334 0.8335 0.4575 0.6435 0.6589 4

0.0407 0.7705 0.4113 0.6817 0.6874 4

0.0496 0.7035 0.3275 0.7249 0.7423 2

0.0581 0.6502 0.2747 0.7612 0.7791 6

0.0685 0.5650 0.2260 0.8231 0.8147 2

0.0748 0.5475 0.1793 0.8364 0.8503 4

0.0820 0.5100 0.1390 0.8656 0.8823 2

0.0976 0.4464 0.0974 0.9175 0.9165 5

0.1246 0.4192 0.0692 0.9407 0.9406 10

0.1764 0.3714 0.0253 0.9829 0.9792 10

0.2244 0.3538 0.0024 0.9988 1.0000 11

0.2743 0.3526 0.0057 1.0000 0.9970 9

0.3187 0.3619 0.0190 0.9915 0.9849 7

0.3757 0.3864 0.0302 0.9695 0.9748 5

0.4129 0.4223 0.0528 0.9381 0.9549 4

0.4321 0.4363 0.0733 0.9261 0.9370 3

0.4511 0.4840 0.1180 0.8865 0.8994 1

0.4673 0.4945 0.1360 0.8780 0.8847 2

0.4808 0.5010 0.1430 0.8728 0.8790 1

0.4969 0.5155 0.1590 0.8613 0.8663 2

0.5166 0.5255 0.1645 0.8534 0.8619 2

0.5397 0.5335 0.1780 0.8472 0.8513 2

0.5572 0.5165 0.1580 0.8605 0.8670 2

0.5659 0.5225 0.1490 0.8558 0.8742 2

0.5848 0.5133 0.1275 0.8630 0.8916 4

0.6034 0.5040 0.1150 0.8704 0.9019 1

0.6191 0.5063 0.1221 0.8686 0.8960 7

0.6490 0.4925 0.1160 0.8796 0.9011 4

0.6797 0.4801 0.0987 0.8896 0.9155 7

0.6978 0.4730 0.0930 0.8955 0.9203 1

^*:
weights directly according to the number of individual
observations included in a point.

   

latitude of longitude of angular radius of temperature factor PSHIFT i

spot center spot center the spot of the spot

0.670 0.0028 3871 K q

3.5981 4.0073 0.75 0.9430 0.0570 0.0009 (fixed)

Table 7: Geometrical and physical elements of the star-spot of primary component of XY UMa and the other photometric parameters obtained from solution of V light curve

   

Parameters Blue (B) Yellow (V)

0.609 0.670 a

e 0.00 0.00 i

0.750 0.600

0.900 0.800

0.500 0.500

0.500 0.500

0.320 0.320

0.320 0.320

5700 K 5700 K

3873 K 3845 K

3.6897 3.7287

4.2279 4.2047

q 0.828 0.828

0.9663 0.9413

0.0337 0.0587

0.3440 0.3396

0.4013 0.3919

0.3580 0.3528

0.3780 0.3715

0.2608 0.2627

0.2795 0.2820

0.2661 0.2681

0.2747 0.2770

0.0015 0.0009