$\sigma$ Geminorum (75Gem, HR2973, HD62044) is a bright and moderately-rotating K1 giant in a long-period RS CVn-type binary (Linsky 1984) with $V\approx 4\thinspace\hbox{$.\!\!^{\rm m}$ }15$ and $P_{\rm rot}\approx P_{\rm orb}\approx 20$ days. The photometric light variation is due to spots on the K1III primary component. The secondary component is probably a cool main-sequence star of G or K spectral type (Ayres et al. 1984) but is not seen in the optical spectrum nor in the photometry.

The system shows all signs of solar-like magnetic activity. $\sigma$ Gem is included in the Catalog of Chromospherically Active Binary Stars (Strassmeier et al. 1993, hereafter CABSII) because of its strong Ca II H&K emission originally discovered by Eberhard & Schwarzschild (1913). From early IUE observations in the far UV, Ayres et al. (1984) reported the presence of center-to-limb modulation effects in the spectra. This was further strenghtened by the finding that the H $\alpha$ core emission also varied with the rotational phase (Eker 1986; Bopp et al. 1988) and thus established the star's chromospheric activity due to plage-like regions. The soft X-ray emission also showed such rotational modulation, as reported by Engvold et al. (1988) and Singh et al. (1987), which even suggests an inhomogeneous active corona. Long-term cyclic variation with a probable period of 8.5 years was found by Henry et al. (1995) who analyzed the $\approx$ 16 years long photometric history of $\sigma$ Gem. Variations on time scales from hours to years were detected in the ROSAT PSPC bandpass by Yi et al. (1997). Furthermore, $\sigma$ Gem is long known as a radio source (Spangler et al. 1977; Gibson 1980, CABSII and references therein).

Orbital and rotational period determinations (Strassmeier et al. 1988; Henry et al. 1995; Jetsu 1996; Strassmeier et al. 1999a) obtained very similar values differing by less than 1%. Updated orbital elements were obtained by Bopp & Dempsey (1989), and recently by Dümmler et al. (1997) who concluded, that the secondary component could be a compact object or, alternatively, the primary is not a normal K1III star. Assuming a circular orbit, an orbital period of 19.604 days was determined by Dümmler et al. (1997). The optical spectral lines are rotationally broadened by $v\sin i=27\pm2$ kms^-1 (Eaton 1990). Dümmler et al. (1997) found the same value and also argued for $\log g=2.5$ as the most likely gravity in agreement with canonical values for K giants listed, e.g. in Gray (1992).

The detailed photometric studies with starspot-modelling techniques by Fried et al. (1983), Strassmeier et al. (1988), Oláh et al. (1989), Dempsey et al. (1992), Henry et al. (1995), and, more recently, Jetsu (1996) and Berdyugina & Tuominen (1998) found the presence of at least two long-living large active regions (spots) on the primary. These studies are qualitatively in agreement with Eker's (1986) early spectroscopic study, where an estimation of the inclination angle of $45\hbox{$^\circ$ }\leq i\leq 73\hbox{$^\circ$ }$ was also given.

Finally, Hatzes (1993) presented the first Doppler image of $\sigma$ Gem and found five spots in a band centered at a medium latitude which he interpreted as an active-latitude belt. No evidence of a polar cap-like spot was seen. The $v\sin i$ of 27 kms^-1 and the inclination of i=60 $^\circ$ adopted by Hatzes (1993) yield a radius of $\approx$ 12 $R_\odot$ , which is consistent with a spectral type of a normal K1III giant.

The present paper is the 16th paper in our series of stellar surface structure. The aim of this series is to find evidence for systematic changes of the surface temperature distribution that are related to basic astrophysical parameters like age, mass, and rotation, and to detect specific surface phenomena like differential rotation or meridional flows. In this paper, we present a Doppler-imaging analysis with new high-resolution spectra and by photometry that covered 3.6 consecutive stellar rotations of $\sigma$ Gem.

**Table 1:** Observing log and radial velocities.
HJD	phase	$v_{\rm r}$	$\sigma_{\rm vr}$
(2400000+)	(Eq. (1))	(kms^-1)	(kms^-1)
50389.017	0.033	76.4	1.2
50391.024	0.135	64.1	1.3
50391.920	0.181	59.6	1.7
50392.885	0.230	43.4	1.4
50393.875	0.281	40.6	1.5
50394.838	0.330	27.9	0.7
50395.899	0.384	16.9	1.4
50396.895	0.435	9.4	1.8
50399.057	0.545	7.6	1.4
50400.004	0.594	15.2	1.5
50400.993	0.644	23.8	0.9
50401.988	0.695	34.8	1.4
50404.938^a	0.845	60.8	1.9
50405.862^a	0.892	66.7	2.1
50406.942^a	0.947	75.8	1.5
50408.886^a	1.047	77.9	1.7
50411.960	1.203	55.2	1.7
50412.945	1.254	42.8	1.3
50415.893	1.404	15.2	1.4
50416.864	1.454	9.4	1.2
50417.952	1.509	9.4	1.1
50418.949	1.560	12.0	1.7
50419.890	1.608	16.8	1.4
50420.931	1.661	25.1	1.5
50421.926	1.712	35.2	1.9
50422.912	1.762	45.6	1.7
50423.880	1.811	55.3	1.7
50424.999	1.869	66.1	1.6
50425.980	1.919	75.2	1.5
50426.985	1.970	78.7	1.8
50428.959	2.071	75.0	1.6
50429.973	2.122	68.9	1.4
50430.972	2.173	61.8	1.7
50431.846	2.218	48.4	1.4
50432.878	2.270	38.4	1.9
50433.841	2.320	28.7	1.6
50434.898	2.373	19.2	1.4
50435.934	2.426	13.1	1.3
50436.913	2.476	7.8	1.2
50437.868	2.525	10.2	1.2
50438.833	2.574	14.7	1.2
50439.923	2.630	19.3	1.7
50440.865	2.678	26.4	1.7
50441.942	2.733	41.2	1.9
50444.893^b	2.883	68.0	2.2
50446.900	2.986	78.7	2.6
50447.914	3.037	77.2	1.5
50450.015	3.145	65.7	1.9
50450.937	3.192	57.1	1.3
50451.926	3.242	46.6	1.8
50456.902	3.496	9.0	1.5
50457.896	3.547	12.0	1.3

1 Introduction