3 Photometric variation outside the eclipse

The mean light and colour variations obtained outside eclipse in 1998 and 1999 are shown in Figs. 1 and 2, respectively. From these figures it is clearly seen that the wave-like distortion generally has two minima and is asymmetrical in shape rather than being truly sinusoidal as it was in 1997 (Paper I). Therefore, it emerges that MM Her had at least two separate active regions in those years. In 1998 the first spot gives rise to the deeper minimum in the outside-of-eclipses light curve at phase 0.10, while the second spot is facing the observer at phase 0.62. Similar effects appeared around phases 0.05 and 0.43 in 1999, respectively. The amplitude of the light variation is about  $0\hbox{$.\!\!^{\rm m}$ }04$ in 1998, it increased to about  $0\hbox{$.\!\!^{\rm m}$ }05$in 1999. The new light and colour curves also show anti-correlation as it appeared in the 1997 light and colour curves, i.e. the system is bluer at the phase of the wave minimum in the light curve. In Paper I, we tried to explain this behavior as the contribution to the total light and colour of the photospheric facular structures that surround starspots.

Long-term photometric observations allow us to search for active permanent longitudes on the RS CVn type stars. Long-lived active-longitude structures have been observed on several spotted stars like RS CVn, EI Eri, II Peg, $\sigma$ Gem, HR 7275 and HK Lac (Rodonò et al. 1995; Berdyugina & Tuominen 1998; Berdyugina et al. 1999; Olah et al. 1991; Henry et al. 1995). The deepest minimum outside eclipses (in eclipsing binaries) or in the light curve (in non-eclipsing binaries) is caused by the largest active region (Spots Group 1 or Spot 1) on the cooler component. The shallower (secondary) minimum should correspond to smaller active region (Spots Group 2 or Spot 2). The displacement of the light minima in time gives us useful information about the migration of the active regions, spots or spot groups. The previous values given in Table 2 of Paper I and the new phases of the light minima ( $\theta_{\rm min}$, as described in Paper I) for Spots 1 and 2 found in this study plotted against epoch are shown in Fig. 3. In the system's light curve obtained in 1997 (Paper I) the distortion effect of the second spot is not seen. The phase change of maximum visibility of the second spot in 1998 and 1999 provides an improvement of the migration period of Spot 2. Since no photometric observation of the system was obtained between 1985-1997, and the 1997 light and colour curve shows only the first spot, the new observations are very important to detect the migration period of the second spot. Applying a linear fit to the phase-shift, the following equations were re-obtained:

where t is in years. The mean time intervals required for Spot 1 and Spot 2 to sweep the light curve once were computed using Eqs. (2) and (3). If the  $\theta_{\rm min}$s are replaced with integers corresponding to the start of the cycles, such as 3 and 4, the time differences between the corresponding years give the values for migration periods. Therefore, we found migration periods of $5.8\pm0.1$ years for Spot 1 and $5.9\pm0.1$ years for Spot 2. These values for the migration periods are in good agreement with that given in Paper I. The arrangement of the spots in two permanent strips can be easily interpreted as two long-lived active longitudes rotating practically at the same velocity.

The brightnesses outside the eclipses of the four different special phases (0.00, 0.25, 0.50 and 0.75) for data sets from Paper I and this study are presented in Table 2. To obtain the brightnesses at phases 0.00 and 0.50, the effects of the eclipses were first removed from the observed light curves, and then, the light and colour curves were represented with free-hand curves. V magnitudes at these special phases were read from the free-hand mean light curves which represent all the observed magnitudes quite well. The data presented in Table 2 are plotted in Fig. 4. The variations of the brightness at each special phase show an almost cyclic change with a period of about 6 years. This value is in good agreement with the migration periods of the spots previously estimated by us.

 

 

Table 2: The V magnitudes of MM Her at four special phases for different data sets.

References	Mean Epoch	Phase 0.00	Phase 0.25	Phase 0.50	Phase 0.75
Sowell et al. (1983)	1976.49	0.900	0.900	0.900	0.895
Sowell et al. (1983)	1976.67	0.880	0.894	0.900	0.890
Sowell et al. (1983)	1977.45	0.860	0.889	0.850	0.852
Sowell et al. (1983)	1977.58	0.890	0.880	0.870	0.865
Sowell et al. (1983)	1978.40	0.880	0.885	0.880	0.870
Sowell et al. (1983)	1979.64	0.880	0.872	0.880	0.886
Sowell et al. (1983)	1980.24	0.900	0.878	0.870	0.885
Sowell et al. (1983)	1980.61	0.880	0.869	0.890	0.890
Evren (1986)	1983.54	0.840	0.870	0.840	0.860
Evren (1986)	1984.52	0.840	0.844	0.830	0.861
Evren (1986)	1985.56	0.860	0.838	0.830	0.851
Tas et al. (1999)	1997.63	0.880	0.859	0.870	0.891
This study	1998.52	0.863	0.871	0.881	0.876
This study	1999.53	0.862	0.884	0.878	0.875

Figure 4: The long-term variation of MM Her brightnesses at four special phases.

The amplitude of the brightness variations at each special phase ranges between  $0\hbox{$.\!\!^{\rm m}$ }10$ and  $0\hbox{$.\!\!^{\rm m}$ }15$ (see Fig. 4). The apparent visual magnitude of the system did not exceed  $9\hbox{$.\!\!^{\rm m}$ }45$ at any special phase. However, it diminishes up to  $9\hbox{$.\!\!^{\rm m}$ }60$ at phases 0.50 and 0.75. In the years in which the luminosities corresponding to each special phase are lowest, the wave-like distortion has a minimum at that phase. Therefore, the cyclic variations belonging to these phases are related to the spot migration.

Figure 5: The amplitudes of the light variations against the minimum phases of the wave-like distortions.

The amplitudes of the light variations taken from Table 2 in Paper I and found in this study are plotted in Fig. 5 against the minimum phases of the wave-like distortions for each data set. When the variation is examined, a maximum is seen at phase $\approx$0.50. So if the spots seen at phase $\approx$0.50 are located on the cooler component of MM Her, they are on the hemisphere facing the companion. In principle there is no reason why these variations taking place in the vicinity of the secondary minimum should be more conspicuous than at other phases.

A similar behaviour was found in the light variation of II Peg by Berdyugina et al. (1999). They reported that the largest active region on the surface of II Peg shows the tendency to be closer to the secondary component of the binary and a long-standing concentration of spots on the hemisphere facing the companion has been found from light curve analysis of AR Lac (Lanza et al. 1998). Such variation may be caused by the effect of a secondary component in the system, i.e. the secondary seems to affect the magnetic activity of the primary and the spot parameters.