We investigate magnetic activity properties of 21 stars via medium resolution optical spectra and long-term photometry. Applying synthetic spectrum fitting method, we find that all targets are cool giant or sub-giant stars possessing overall [M/H] abundances between 0 and $-0.5$ . We find that six of these targets exhibit only linear trend in mean brightness, while eight of them clearly shows cyclic mean brightness variation. Remaining seven target appear to exhibit cyclic mean brightness variation, but this cannot be confirmed due to the long timescales of the predicted cycle compared to the current time range of the photometric data. We further determine seasonal photometric periods and compute average photometric period of each target. Analysed sample in this study provides a quantitative representation of a positive linear correlation between the inverse of the rotation period and the cycle period normalised to the rotation period, on the log-log scale. We also observe no correlation between the activity cycle length and the relative surface shear, indicating that the activity cycle must be driven by a parameter rather than the differential rotation. Our analyses show that the relative surface shear is positively correlated with the rotation period and there is a noticeable separation between main sequence stars and our sample. Compared to our sample, the relative surface shear of a main sequence star is larger for a given rotation period. However, dependence of the relative surface shear on the rotation period appears stronger for our sample. Analysis of the current photometric data indicates that the photometric properties of the observed activity cycles in eight targets seem dissimilar to the sunspot cycle.

中文翻译：

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21 个色球活动变量的光谱和光度分析：活动周期和差分旋转

我们通过中等分辨率光谱和长期光度法研究 21 颗恒星的磁活动特性。应用合成光谱拟合方法，我们发现所有目标都是冷巨星或亚巨星，总体[M/H]丰度在0到 $-0.5$ . 我们发现，其中 6 个目标的平均亮度仅呈现线性趋势，而其中 8 个目标明显显示出循环平均亮度变化。其余七个目标似乎表现出周期性平均亮度变化，但由于与光度数据的当前时间范围相比，预测周期的时间尺度较长，因此无法证实这一点。我们进一步确定季节性光度周期并计算每个目标的平均光度周期。本研究中的分析样本在对数对数尺度上提供了旋转周期的倒数与归一化为旋转周期的循环周期之间的正线性相关性的定量表示。我们还观察到活动周期长度和相对表面剪切之间没有相关性，表明活动周期必须由参数驱动，而不是差速旋转。我们的分析表明，相对表面切变与自转周期呈正相关，主序星与我们的样本之间存在明显的分离。与我们的样本相比，主序星的相对表面切变在给定的自转周期内更大。然而，对于我们的样本，相对表面剪切对旋转周期的依赖性似乎更强。对当前光度数据的分析表明，在八个目标中观察到的活动周期的光度特性似乎与太阳黑子周期不同。我们的分析表明，相对表面切变与自转周期呈正相关，主序星与我们的样本之间存在明显的分离。与我们的样本相比，主序星的相对表面切变在给定的自转周期内更大。然而，对于我们的样本，相对表面剪切对旋转周期的依赖性似乎更强。对当前光度数据的分析表明，在八个目标中观察到的活动周期的光度特性似乎与太阳黑子周期不同。我们的分析表明，相对表面切变与自转周期呈正相关，主序星与我们的样本之间存在明显的分离。与我们的样本相比，主序星的相对表面切变在给定的自转周期内更大。然而，对于我们的样本，相对表面剪切对旋转周期的依赖性似乎更强。对当前光度数据的分析表明，在八个目标中观察到的活动周期的光度特性似乎与太阳黑子周期不同。