DETECTING THE COMPANIONS AND ELLIPSOIDAL VARIATIONS OF RS CVn PRIMARIES. I. σ GEMINORUM

RS Canum Venaticorum (RS CVn) stars are spotted, active binary systems exhibiting photometric and Ca H and K variability (Hall & Stars 1976). Often tidally locked, these systems are composed of an evolved primary star (giant or subgiant) and a subgiant or dwarf companion (Berdyugina 2005; Strassmeier 2009). With active binaries, not only is there potential to determine the component masses and system evolutionary history, but there is also the opportunity to understand the magnetic field interactions through active longitudes, particular longitudes that are 180° apart with persistent, long-lived starspots (Berdyugina & Tuominen 1998; Berdyugina 2005).

Observing the magnetic phenomena of rapidly rotating evolved stars also sheds light on the magnetic activity of rapidly rotating young stars, such as T Tauri stars. Both T Tauri and RS CVn systems have starspots that are analogous to sunspots—cool starspots resulting from stifled convection in the outer layers of the stars due to strong magnetic fields (Petrov 2003; Berdyugina 2005).

σ Geminorum (σ Gem, HD 62044, HIP 37629, HR 2973) is an RS CVn system known to exhibit starspots, often ascribed to "active longitudes" (e.g., Hall et al. 1977; Henry et al. 1995). The system has been characterized as a single-lined spectroscopic binary (Herbig & Spalding 1955) with a K1III primary (Roman 1952). The orbital period of σ Gem is slightly longer than the primary star's rotation period derived from the fastest rotating spots (${P}_{\mathrm{orb}}=19.60$ days, ${P}_{\mathrm{rot},\mathrm{min}}=19.47$ days; Kajatkari et al. 2014).

Because of its large starspots, σ Gem is a frequent target for understanding starspot evolution. Eberhard & Schwarzschild (1913) first reported σ Gem as active and potentially spotted due to fluctuations in the Ca H and K lines as the star rotated. Decades later, Hall et al. (1977) identified photometric variations that suggested starspots (${\rm{\Delta }}V\sim 0.07$). Initial models of the surface of σ Gem often showed the surface with two starspots oriented on opposite sides of the primary star (Fried et al. 1983). Berdyugina & Tuominen (1998) emphasize that, due to tidal locking, the starspots are located such that one spot constantly faces the companion and the other spot is 180° offset. The majority of spot models applied to the light curves of σ Gem consist of two spots on a spherical star (Eker 1986; Strassmeier et al. 1988; Olah et al. 1989; Henry et al. 1995; Jetsu 1996; Padmakar & Pandey 1999; Kajatkari et al. 2014). Doppler images have suggested the surface is covered with a larger number of smaller spots (Hatzes 1993; Kővári et al. 2001, 2014).

To understand the binary system, we present our analysis of the first detections of the companion in our interferometric and radial velocity data sets, as well as photometric data. In Section 2, we describe the observations for our data sets. In Section 3, we discuss our analysis of the data sets, including the first astrometric and spectroscopic detections of the companion star and orbital parameters. In Section 4, we present evolutionary constraints and a Hertzsprung–Russell (H–R) diagram. In Section 5, we discuss our analysis of the photometric data set, including detected ellipsoidal variations and measured gravity darkening. In Section 6, we present the conclusions of our study of σ Gem.

2.1. Interferometry

2.2. Radial Velocity

To constrain the spectroscopic orbit of σ Gem, we utilized three independent sets of radial velocity data: two sets of single-lined velocities for the primary and a new set of double-lined velocities for both components of the binary.

One set of the radial velocity measurements for the primary star was published in Massarotti et al. (2008). These 39 data points were obtained with two identical CfA Digital Speedometers (Latham 1992) on the 1.5 m Wyeth Reflector (Oak Ridge Observatory) and 1.5 m Tillinghast Reflector (Fred Lawrence Whipple Observatory, FLWO) telescopes (2003 December 30–2007 June 5).

From 2012 October 1 to 2015 January 9, using the Tillinghast telescope with the Tillinghast Reflector Echelle Spectrograph (TRES; Fűrész 2008), we were able to make 15 detections of the secondary spectra for the first time. Along with 16 new primary star measurments, these new radial velocities are presented in Table 2. We add 0.14 $\mathrm{km}\;{{\rm{s}}}^{-1}$ to these sets of radial velocities to account for these data being reported on the CfA native system (Stefanik et al. 1999; note that the correction is inaccurately stated as a subtraction in this reference). For the details of these observations and data analysis, see Appendix A.

Table 2.  Radial Velocity Data of σ Gem (CfA)

HJD−2400000	Primary ($\mathrm{km}\;{{\rm{s}}}^{-1}$)	Secondary ($\mathrm{km}\;{{\rm{s}}}^{-1}$)
56202.0199	10.66	103.44
56230.0436	77.98	−9.80
57002.9567	14.17	94.68
57003.9301	9.71	101.50
57014.8600	79.02	−12.11
57015.8427	75.96	−10.77
57018.8519	51.08	28.91
57019.9181	38.76	60.57
57020.9452	27.84	78.02
57021.8962	19.05	84.20
57024.9812	9.10	104.10
57025.9317	12.59	99.41
57026.8891	18.88	85.69
57028.9057	38.58	...
57029.8670	49.81	27.28
57031.8806	69.76	−2.88

Note. Errors on the primary radial velocities are 0.84 $\mathrm{km}\;{{\rm{s}}}^{-1}$. Errors on the secondary radial velocities are 3.8 $\mathrm{km}\;{{\rm{s}}}^{-1}$. These were then scaled for our orbit fit to have a total ${\chi }^{2}=1.00$. Note that these radial velocities are on the native CfA system. We added 0.14 $\mathrm{km}\;{{\rm{s}}}^{-1}$ for use in our analysis (Stefanik et al. 1999).

A machine-readable version of the table is available.

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The additional radial velocity data set consists of 43 spectrograms of the primary star of σ Gem taken between 2009 January 12 and 2014 December 1 with the Tennessee State University 2 m Automatic Spectroscopic Telescope (AST), a fiber-fed echelle spectrograph, and a CCD detector at Fairborn Observatory, Arizona (see Table 3; Eaton & Williamson 2004, 2007). At first, the detector was a 2048 × 4096 SITe ST-002 A CCD with 15 μm pixels. Eaton & Williamson (2007) discussed the reduction of the raw spectra and wavelength calibration. Those echelle spectrograms have 21 orders that cover the wavelength range 4920–7100 Å with an average resolution of 0.17 Å, corresponding to a resolving power of 35,000 at 6000 Å. Those spectra have a typical signal-to-noise value of 30.

Table 3.  Radial Velocity Data of σ Gem (AST/TSU)

HJD−2400000	Primary ($\mathrm{km}\;{{\rm{s}}}^{-1}$)
54843.9716	38.0
54844.8597	28.4
54845.6849	20.5
54846.6455	14.0
54847.6848	9.2
54848.7712	9.2
54849.7451	12.1
54850.6645	17.6
54856.9772	75.7
54859.7922	73.1
54860.6910	66.7
54861.6447	58.7
54862.6829	48.3
54863.6434	37.3
54864.6443	26.7
54865.6596	17.8
54866.6401	11.7
54867.6456	9.1
54868.9572	10.3
54869.6594	13.8
54870.6346	21.1
54975.6558	77.9
54976.6372	76.4
54981.6497	33.3
54982.6541	23.0
54983.6544	15.2
54984.6546	9.9
54985.6548	8.5
54986.6547	10.7
55060.9627	23.5
55061.9642	16.0
55066.9516	23.3
55069.9622	54.3
55070.9596	64.0
55071.9572	71.9
55072.9375	76.6
56984.7820	9.4
56987.7638	19.7
56988.7240	28.0
56989.7317	38.9
56990.7354	50.0
56991.7355	60.9
56992.8661	70.8

Note. Errors on the primary radial velocities are 0.3 $\mathrm{km}\;{{\rm{s}}}^{-1}$. These data were then scaled for our orbit fit to have a total ${\chi }^{2}=1.00$.

A machine-readable version of the table is available.

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In the summer of 2011 the AST SITe CCD and its dewar were retired and replaced with a Fairchild 486 CCD, a 4096 × 4096 array of 15 μm pixels, that is housed in a new dewar. With the new CCD the wavelength coverage ranged from 3800 to 8600 Å. The resolution was reduced slightly to 0.24 Å, a resolving power of 25,000 at 6000 Å. These more recent spectra have signal-to-noise ratios of about 70.

Fekel et al. (2009) provided an extensive general description of the velocity measurements of the Fairborn AST spectra. In the case of σ Gem, we measured a subset of 63 lines from our solar-type star line list that covers the 4920–7120 Å region. Because the lines of σ Gem have significant rotational broadening, we fit the individual lines with a rotational broadening function. The Fairborn velocities are on an absolute scale. A comparison of our unpublished measurements of several IAU standard stars with those determined by Scarfe et al. (1990) indicates that the Fairborn Observatory velocities from the SITe CCD have a small zero-point offset of −0.3 $\mathrm{km}\;{{\rm{s}}}^{-1}$. Velocities from the Fairchild CCD spectra have a slightly larger zero-point offset of −0.6 $\mathrm{km}\;{{\rm{s}}}^{-1}$ relative to those of Scarfe et al. (1990). Thus, in Table 3 we corrected our measured velocities by either 0.3 or 0.6 $\mathrm{km}\;{{\rm{s}}}^{-1}$, depending on which detector was used.

2.3. Photometry

We used differential photometry of σ Gem and a comparison star from the Tennessee State University T3 0.4 m Automated Photometric Telescope (APT) located at Fairborn Observatory, Arizona. For details on the observational procedure and photometers see Henry (1999) and Fekel et al. (2005).

The differential Johnson B and V light curves cover 1987 November 21–2015 March 13 (see Table 4 and Figure 1). Subsets of these data were analyzed by Henry et al. (1995) and Kajatkari et al. (2014). For the first time, we make the full set of T3 APT photometry of σ Gem available in Table 4.

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Table 4.  Johnson B and V  Differential Magnitudes of σ Gem from the T3 APT

HJD−2400000	${\rm{\Delta }}B$ (σ Gem—HD 60318)a	${\rm{\Delta }}V$ (σ Gem—HD 60318)	${\rm{\Delta }}B$ (υ Gemb—HD 60318)	${\rm{\Delta }}V$ (υ Gem—HD 60318)
47121.0481	−1.006	−1.123	⋯	⋯
47122.0465	−1.010	−1.135	⋯	⋯
47125.0364	−1.030	−1.125	−0.766	−1.303
47126.0364	−1.011	−1.128	−0.757	⋯
47128.0310	−0.975	⋯	−0.759	⋯

Notes. A portion is shown here for guidance regarding its form and content.

^aHD 60318, V = 5.33, B − V = 1.01. ^bυ Gem (HD 60522, 69 Gem), V = 4.06, B − V = 1.54.

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In order to derive the astrometric orbit of σ Gem, we searched for the companion with model fitting. We modeled the system with the resolved primary star and an unresolved secondary. We allowed the primary radius along the major axis, primary major-to-minor axis ratio, primary major axis position angle, primary-to-secondary flux ratio, and secondary position to vary. During the fitting, we weighted the data such that the separate observables (squared visibilities, closure phases, and triple amplitudes) contributed to the final ${\chi }^{2}$ with equal weight. The parameter errors for the primary star size and the primary-to-secondary flux ratio were based on the epoch-to-epoch variation, while the relative positional error of the secondary compared to the primary were based on the residuals to the orbit fit (see the discussion on orbit fitting).

The coordinates of the detections on five nights (UT 2011 December 8; 2012 November 7, 8, 24, and 25 are listed in Table 5). The H-band flux ratio for the primary star to the secondary is 270 ± 70. In addition to detecting the secondary star, we measured the uniform disk diameter of the primary to be ${\theta }_{\mathrm{UD},1}=2.335\pm 0.007$ mas (limb-darkened disk diameter ${\theta }_{\mathrm{LD},1}=2.417\pm 0.007$ mas) with a major-to-minor axis ratio of 1.02 ± 0.03. Our measurements are slightly larger than those in the CHARM2 catalog (uniform disk diameter of ${\theta }_{\mathrm{UD},1}=2.18\pm 0.05$ mas, limb-darkened disk diameter of ${\theta }_{\mathrm{LD},1}=2.31\pm 0.05$ mas; Richichi et al. 2005).

Table 5.  Detections for the Companion of σ Geminorum with Respect to the Primary

UT Date	JD−2400000	Separation	Position	Error Ellipse	Error Ellipse	Error Ellipse	Reduced
		(mas)	Angle (°)a	Major Axis (mas)b	Minor Axis (mas)b	Position Angle (°)a	${\chi }^{2}$
2011 Dec 08	55903.95	2.84	19.1	0.30	0.09	80	4.4
2012 Nov 07	56238.97	4.34	8.6	0.04	0.03	280	2.8
2012 Nov 08	56239.86	4.69	359.7	0.13	0.06	300	2.1
2012 Nov 24	56256.00	2.03	39.0	0.08	0.06	30	1.5
2012 Nov 25	56256.95	3.13	21.6	0.05	0.04	320	1.7

Notes. These detections give an H-band ($1.5-1.8\;\mu {\rm{m}}$) flux ratio for σ Gem primary-to-secondary of 270 ± 70. The uniform disk fit for the primary star is ${\theta }_{\mathrm{UD},1}=2.335\pm 0.006$ mas (limb-darkened disk diameter ${\theta }_{\mathrm{LD},1}=2.417\pm 0.006$ mas) with a 1.02 ± 0.03 major-to-minor axis ratio.

^aEast of north. ^bScaled error bars to ensure a total ${\chi }^{2}=1.00$ as described in Section 3.2.

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To determine the binary orbit, we simultaneously fit our interferometric and radial velocity data with Monte Carlo realizations. The five interferometric points are as described above, and we present the scaled error bars of the major and minor axis in Table 5 to give our fit a total ${\chi }^{2}=1.00$. For the radial velocity data we combine the Massarotti et al. (2008; adding 0.14 $\mathrm{km}\;{{\rm{s}}}^{-1}$ to account for the values reported on the CfA native system) data, new CfA data, and the AST data to fit simultaneously with the astrometry. The radial velocity errors are similarly scaled (${\mathrm{rms}}_{\mathrm{CfA},1}=0.84$ $\mathrm{km}\;{{\rm{s}}}^{-1}$, ${\mathrm{rms}}_{\mathrm{AST},1}\;=$ $0.3\mathrm{km}\;{{\rm{s}}}^{-1}$, ${\mathrm{rms}}_{\mathrm{CfA},2}=3.8$ $\mathrm{km}\;{{\rm{s}}}^{-1}$).

Using the complete radial velocity data sets, we find an eccentricity of e = 0.014 ± 0.004, consistent with the slightly eccentric orbits reported by Harper (1935), Pourbaix et al. (2004), and Massarotti et al. (2008). However, Luyten (1936), Batten et al. (1978), and Dümmler et al. (1997) reported a circular orbit. To investigate this discrepancy, we used the APT light curve to eliminate the primary star's radial velocity data that were obtained when σ Gem presented starspots (${\rm{\Delta }}V\gt 0.04$), as these could cause shifts in the velocities (e.g., Saar & Donahue 1997). The remaining primary star radial velocity data obtained when σ Gem did not exhibit large starspots from the Massarotti et al. (2008)/CfA data set span 2006 December 6–2007 June 5, and those from the AST data set span 2009 January 12–June 4. Using the primary star's truncated data set with 42% of the Massarotti et al. (2008)/CfA and 33% of the AST epochs removed, we find that the orbit is consistent with a circular orbit, e = 0.002 ± 0.002, and we adopt a circular orbit for the rest of this paper.

Requiring eccentricity e = 0 and the argument of periastron for the primary ω = 0° the simultaneous Monte Carlo realizations gave the orbital parameters and their 1σ errors listed in Table 6. The visual orbit is illustrated in Figure 2, and the radial velocity curve is presented in Figure 3. We use the conventions presented by Heintz (1978), where the argument of periastron, ω, and the time of nodal passage (maximum recessional velocity), T₀, are defined by the primary star's orbit. The ascending node, Ω, is independent of definition, being equivalent with respect to either the primary or secondary star.

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Table 6.  Orbital and Stellar Parameters of σ Gem

Measured Parameters	Value
Semimajor axis, a (mas)	4.63 ± 0.04
Eccentricity, e	0
Inclination, i (°)	107.7 ± 0.8
Argument of periastron, ω (°)a	0
Ascending node, Ω (°)	1.2 ± 0.8
Period, Porb (days)	19.6027 ± 0.0005
Time of nodal passage, T0 (HJD)b	2453583.98 ± 0.03
Velocity semi-amplitude, K1 ($\mathrm{km}\;{{\rm{s}}}^{-1}$)	34.62 ± 0.08
Velocity semi-amplitude, K2 ($\mathrm{km}\;{{\rm{s}}}^{-1}$)	60 ± 2
System velocity, γ ($\mathrm{km}\;{{\rm{s}}}^{-1}$)	43.41 ± 0.08
Uniform disk diameter, ${\theta }_{\mathrm{UD},1}$ (mas)	2.335 ± 0.007
Limb-darkened disk diameter, ${\theta }_{\mathrm{LD},1}$ (mas)c	2.417 ± 0.007
Primary major-to-minor axis ratio	1.02 ± 0.03
H-band flux ratio, primary-to-secondary	270 ± 70
Orbital parallax, π (mas)	25.8 ± 0.4
Distance, d (pc)	38.8 ± 0.6
Derived Parameters
Average primary radius, R1 (${R}_{\odot }$)d	10.1 ± 0.4
Primary luminosity, L1 (${L}_{\odot }$)	39 ± 2
Primary surface gravity, $\mathrm{log}$ g1 (cm s−2)	2.54 ± 0.02
Primary mass, M1 (${M}_{\odot }$)	1.28 ± 0.07
Secondary mass, M2(${M}_{\odot }$)	0.73 ± 0.03
System age (Gyr)	5 ∓ 1
Literature Parameters
Primary effective temperature, ${T}_{\mathrm{eff},1}$ (K)e	4530 ± 60
Primary metallicity (iron), Fe/Hf	0.0

Notes.

^aRadial velocity convention for primary with respect to the center of mass. ^bTime of maximum recessional velocity of the primary star. ^cWe applied a 3.5% correction from uniform to limb-darkened disk diameter. This is equivalent to a limb-darkening coefficient $\alpha =0.27$. ^dUsing limb-darkened disk diameter. ^e ${T}_{\mathrm{eff},1}$ is an average of temperatures given by Głȩbocki & Stawikowski (1979), Poe & Eaton (1985), Stawikowski & Głȩbocki (1994), O'Neal et al. (1996), Kővári et al. (2001), and Massarotti et al. (2008). The $1-\sigma$ error is the standard deviation of these values. ^f[Fe/H] = −0.02 (Mallik 1998); approximated as [Fe/H] = 0.00.

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Our orbital parallax, $\pi =25.8\pm 0.4$ mas, can be compared with the Hipparcos parallax of 26.68 ± 0.79 mas (ESA 1997). As an unresolved binary with a variable component, σ Gem does not exhibit the photocenter shifts found to be troublesome for measuring binary system parallax with Hipparcos (ESA 1997; Halbwachs et al. 2005). Assuming that the secondary is negligibly bright, the semimajor axis of the photocentric orbit of the primary is at most 1.71 mas wide, which is at the limit of detectability (Pourbaix 2002) for Hipparcos. Combining Hipparcos data and our visual orbit, the parallax is 26.4 ± 0.8 mas, consistent with our orbital parallax. For our subsequent analysis, we adopt our higher-precision orbital parallax, $\pi =25.8\pm 0.4$ mas.

With a circular orbit and ${P}_{\mathrm{orb}}\sim {P}_{\mathrm{rot}}$ (e.g., Kajatkari et al. 2014), we expect σ Gem to have aligned rotational and orbital axes. Given our orbital and stellar parameters, we can calculate the obliquity of the system. Comparing our calculated value of $v\mathrm{sin}i=(2\pi {R}_{1}/{P}_{\mathrm{orb}})\times v\mathrm{sin}i=24.8\pm 0.4$ $\mathrm{km}\;{{\rm{s}}}^{-1}$ with the observational rotational velocity of $v\mathrm{sin}i=26.7\pm 0.5\mathrm{km}\;{{\rm{s}}}^{-1}$ (from the TRES spectra), we find that the calculation is smaller than the observational value. This discrepancy could be attributed to the estimate of mircroturbulence or the presence of the large spot structures on the surface of σ Gem during the TRES observations instead of a small, non-zero obliquity.

Using our complete orbital fit, we obtain model-independent masses ${M}_{1}=1.28\pm 0.07\ {M}_{\odot }$ and ${M}_{2}=0.73\pm 0.03\ {M}_{\odot }$. With the stellar parameters of the primary star (including ${T}_{\mathrm{eff},1}=4530\pm 60$ K; see Table 6) and the primary-to-secondary H-band flux ratio detected using the CHARA/MIRC data (270 ± 70), we are able to constrain the parameters (luminosity, temperature, and radius) of the secondary star. We use the flux ratio and NextGen stellar atmospheres (Hauschildt et al. 1999) to constrain the stellar flux to calculate a range of luminosities for reasonable effective temperatures (4000– 4700 K) for a $0.73\pm 0.03\ {M}_{\odot }$ main-sequence star (see Figure 4). We obtain a range of luminosities ($0.11-0.15\ {L}_{\odot }$) and radii ($0.70-0.59\ {R}_{\odot }$). We note that our analysis predicts a primary-to-secondary Johnson V-band flux ratio of 290 assuming ${T}_{\mathrm{eff},2}=4500$ K), which is not in agreement with the flux ratio given by the spectroscopic 519 nm light ratio (∼70 primary-to-secondary; see Appendix A for details on this measurement). In order for our flux ratios to be in agreement, the secondary star would have ${T}_{\mathrm{eff},2}=6400$ K, which is not consistent with the spectroscopic observations, nor with a main-sequence star given the location on the H–R diagram. We cannot rule out the effect of starspots on the discrepant flux ratios as these were not accounted for when interferometrically detecting the companion and the spot features present during the interferometric and spectroscopic observations differ as evidenced in the APT light curve. Additionally, Prato et al. (2002) and Lehmann et al. (2013) also reported discrepancies between TODCOR-reported flux ratios and their expected values. Therefore, we use only the H-band flux ratio.

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We plot the location of the components of σ Gem on an H–R diagram, as well as the corresponding evolutionary tracks. We use Dartmouth stellar evolution tracks (Fe/H = 0.0, α/Fe = 0.0, PHOENIX-based models; Dotter et al. 2008) for the interpolated model masses (${M}_{1,\mathrm{model}}=1.28\pm 0.07\ {M}_{\odot }$, ${M}_{2,\mathrm{model}}=0.73\pm 0.03\ {M}_{\odot }$). Our primary falls nearly on the $1.28\ {M}_{\odot }$ evolutionary track with an estimated temperature of 4530 ± 60 K (Głȩbocki & Stawikowski 1979; Poe & Eaton 1985; Stawikowski & Głȩbocki 1994; O'Neal et al. 1996; Kővári et al. 2001; Massarotti et al. 2008). The range of locations for the secondary on the H–R diagram passes through the main sequence for a star of $0.73\ {M}_{\odot }$. We find an age of the system of $5\mp 1$ Gyr. Based upon the masses and age of the stars, we suggest that the primary star is an evolved late F-type star that is now a K giant. The secondary star is a main-sequence early K star.

Henry et al. (1995) and Kajatkari et al. (2014) previously published subsets of the APT light curve data for starspot modeling and measuring differential rotation. Both studies emphasized the presence of active longitudes on opposite sides of σ Gem to explain the quasi-sinusoidal variation appearing at half of the orbital period.

We removed long-term trends, folded the APT photometry over the orbital period (${P}_{\mathrm{orb}}=19.6027$ days), and binned the data (0.025 in phase). The resultant Johnson B and V light curves are presented in Figure 5. The quasi-sinusoidal trend observed in the averaged light curves suggests the possibility of ellipsoidal variations due to distortions of the primary star partially filling its Roche lobe potential. With a Roche lobe radius of $16.5\ {R}_{\odot }$, we obtain ${R}_{1}/{R}_{L}=0.61$ (Eggleton 1983).

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We used the light-curve-fitting software package Eclipsing Light Curve (ELC; Orosz & Hauschildt 2000) to model the ellipsoidal variations using our orbital parameters with no free parameters (gravity darkening assumed to be β = 0.08; Lucy 1967, see Figure 5). The characteristics of the ellipsoidal variations with this model, as compared to the light curve of σ Gem, indicate that the long-term signature is indeed likely due to ellipsoidal variations, in contrast to previous suggestions that the periodicity at ${P}_{\mathrm{orb}}/2$ is due to active longitudes aligned with the orbit (e.g., Henry et al. 1995; Jetsu 1996; Berdyugina & Tuominen 1998; Kajatkari et al. 2014; Kővári et al. 2014). We note that rotation periods derived from the analysis of the light curve (e.g., Kajatkari et al. 2014) suggest the star is rotating slightly faster than the orbital period, further supporting our identification of ellipsoidal variations in σ Gem. It should be noted that removing the effect of ellipsoidal variations from the light curve does not eliminate all starspot signatures (see Appendix B).

The ELC model fit of ellipsoidal variations can be improved to better match our data. We modeled the system again with no free parameters except for the gravity darkening coefficient, β for ${T}_{\mathrm{eff}}\propto {g}^{\beta }$ (von Zeipel 1924), as Espinosa Lara & Rieutord (2012) recently suggested $\beta \sim 0.21$ for convective stars, substantially higher than the canonical $\beta \sim 0.08$ (Lucy 1967) value assumed in our fixed-parameter fit. Although our average light curve is still contaminated by some residual spot modulation, we find that $\beta =0.02\pm 0.02$ with error bars determined by bootstrapping over observing seasons of the 27 years of observation in the APT light curve. This value strongly rules out $\beta \gt 0.1$ for this system (see Figure 5).

In this work, we have made the first visual detections of the secondary star of σ Gem using interferometric and spectroscopic observations. We establish the first visual orbit by combining the interferometric detections with radial velocity data. The determination of orbital parameters has allowed for model-independent mass determinations (${M}_{1}=1.28\pm 0.07{M}_{\odot },{M}_{2}=0.73\pm 0.03\ {M}_{\odot }$).

Folded and binned photometric data have shown evidence of ellipsoidal variations, gravitational distortions of the primary star caused by the close companion. The light curve is comparable to light curve models created only from stellar and orbital parameters (assuming no starspots). Although the ellipsoidal variations are only a small effect, the primary star of σ  Gem is not spherical, partially filling its Roche lobe potential and having a surface temperature gradient. Our establishment of ellipsoidal variations offers a compelling alternative explanation to the previously purported detections of active longitudes, starspots on either side of the primary star (Henry et al. 1995; Jetsu 1996; Berdyugina & Tuominen 1998; Kajatkari et al. 2014; Kővári et al. 2014).

Our new orbital elements, along with the folded light curve, also allow measurements of gravity darkening. We find that $\beta =0.02\pm 0.02$, a value of gravity darkening lower than the values suggested by theory (Lucy 1967; Espinosa Lara & Rieutord 2011, 2012).

Table 7.  σ Gem Orbital Parameters from Radial Velocity Curves^a

Parameter	TRES/TODCOR	TRES/TODCOR	Massarotti et al. (2008)	AST/TSU	AST/TSU
	e free	e fixed	e free	e free	e fixed
Orbital period, Porb (days)	19.6059 ± 0.0020	19.6065 ± 0.0018	19.60437 ± 0.00053	19.6041 ± 0.0002	19.6041 ± 0.0002
Center of mass–velocity, γ ($\mathrm{km}\;{{\rm{s}}}^{-1}$)	43.554 ± 0.077	43.553 ± 0.074	43.043 ± 0.066	43.25 ± 0.07	43.33 ± 0.06
Semi-amplitude, primary, ${K}_{1}$ ($\mathrm{km}\;{{\rm{s}}}^{-1}$)	35.19 ± 0.10	35.192 ± 0.098	34.776 ± 0.100	34.60 ± 0.08	34.60 ± 0.08
Semi-amplitude, secondary, ${K}_{2}$ ($\mathrm{km}\;{{\rm{s}}}^{-1}$)	60.5 ± 1.6	60.5 ± 1.5	...	...	...
Eccentricity, e	0.0018 ± 0.0029	0	0.0143 ± 0.0026	0.002 ± 0.003	0
Time of periastron passage, T (HJD)	2456899.0 ± 5.9	...	2453507.96 ± 0.71	2456985.3 ± 3.7	...
Longitude of periastron, ${\omega }_{1}$ (°)	$38.\pm 109$	...	46 ± 13	181 ± 68	...
Time of maximum velocity T0 (HJD)	...	2456916.571 ± 0.012	...	...	2453564.31 ± 0.02
${M}_{1}{\mathrm{sin}}^{3}i$ (${M}_{\odot }$)	1.126 ± 0.068	1.126 ± 0.065	...	...	...
${M}_{2}{\mathrm{sin}}^{3}i$ (${M}_{\odot }$)	0.655 ± 0.022	0.655 ± 0.021	...	...	...
${a}_{1}\mathrm{sin}i$ (106 km)	9.487 ± 0.027	9.488 ± 0.027	9.374 ± 0.024	9.33 ± 0.02	9.33 ± 0.02
${a}_{2}\mathrm{sin}i$ (106 km)	16.31 ± 0.43	16.32 ± 0.42	...	...	...
$a\mathrm{sin}i$ (${R}_{\odot }$)	37.09 ± 0.62	37.09 ± 0.60	...	...	...
Mass ratio, $q={M}_{2}/{M}_{1}$	0.582 ± 0.016	0.582 ± 0.15	...	...	...
Mass function, $f{(M)=({M}_{2}\mathrm{sin}i)}^{3}/{({M}_{1}+{M}_{2})}^{2}$	...	...	0.0854 ± 0.00066	0.0841 ± 0.0006	0.0841 ± 0.0006
rms velocity residuals, ${\sigma }_{1}$ ($\mathrm{km}\;{{\rm{s}}}^{-1}$)	0.29	0.28	0.45	0.30	0.30
rms velocity residuals, ${\sigma }_{2}$ ($\mathrm{km}\;{{\rm{s}}}^{-1}$)	4.8	4.6	...	...	...
Light ratio at 519 nm	0.0139	0.0139	...	...	...

Note.

^aThese radial velocity data sets are the complete sets and include data contaminated by starspots.

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In this paper, we have demonstrated that precision interferometry at CHARA is now capable of detecting the faint main-sequence companions of bright RS CVn primary stars. We are currently processing new data for other close, bright RS CVn systems and will be publishing these results in a series of follow-up papers.

We thank H.A. McAlister, J.A. Orosz, M. Reynolds, and M. Rieutord for their helpful comments and discussions. The authors also thank Lou Boyd of the Fairborn Observatory for the decades of crucial support he has given to our photometric program. The interferometric data in this paper were obtained at the CHARA Array, funded by the National Science Foundation through NSF grants AST-0908253 and AST-1211129, and by Georgia State University through the College of Arts and Sciences. The MIRC instrument at the CHARA Array was funded by the University of Michigan. The photometric data were supported by NASA, NSF, Tennessee State University, and the State of Tennessee through its Centers of Excellence program. R.M.R. would like to acknowledge support from the NASA Harriet G. Jenkins Pre-Doctoral Fellowship and a Rackham Graduate Student Research Grant from the University of Michigan. J.D.M. and R.M.R. acknowledge support from NSF grant AST-1108963. A.N.A. and J.D.M. acknowledge the support of NSF AST-1311698. S.K. acknowledges support from an STFC Rutherford Fellowship (ST/J004030/1). This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France, and the Jean-Marie Mariotti Center SearchCal service⁹ co-developped by FIZEAU and LAOG/IPAG, and of CDS Astronomical Databases SIMBAD and VIZIER.¹⁰

The CfA radial velocity data were obtained using two identical CfA Digital Speedometers (Latham 1992) on two different telescopes: the 1.5 m Wyeth Reflector at the Oak Ridge Observatory located in the town of Harvard, Massachusetts, and the 1.5 m Tillinghast Reflector at the Fred Lawrence Whipple Observatory (FLWO) on Mount Hopkins, Arizona. In the first set of radial velocity measurements, altogether 78 observations were obtained: 32 with the Tillinghast Reflector and 46 with the Wyeth Reflector. Those radial velocities have all been published (Massarotti et al. 2008), so the details of the procedures and reductions will not be repeated here, except to provide some overall characteristics of the data. Forty of the observations were obtained over a span of less than an hour using the Wyeth Reflector. They have been averaged to give one data point for the analysis in this paper. However, those forty observations provide an opportunity to evaluate the precision of the individual velocity measurements from the CfA Digital Speedometers for an RS CVn primary with line broadening corresponding to a rotational velocity of 25 $\mathrm{km}\;{{\rm{s}}}^{-1}$. The standard deviation of a single velocity from the average of all 40 is 0.40 $\mathrm{km}\;{{\rm{s}}}^{-1}$. This compares favorably with the value of 0.45 $\mathrm{km}\;{{\rm{s}}}^{-1}$ reported by Massarotti et al. (2008) for the rms velocity residuals from their orbital solution using all 78 velocities.

To avoid confusion, the velocities determined using the CfA Digital Speedometers have always been published on the native system of the instruments. For the analysis in this paper we added 0.14 $\mathrm{km}\;{{\rm{s}}}^{-1}$ to the published velocities to put them on an absolute system defined by the IAU Radial Velocity Standard Stars (Stefanik et al. 1999; note that the sign of the correction was given as minus by mistake in that paper).

A new set of velocities for both components of the σ Gem system was obtained using the TRES (Fűrész 2008) at FLWO during the period 2012 October 1–2015 January 9. TRES is a modern fiber-fed CCD echelle spectrograph with a resolution of 6.7 $\mathrm{km}\;{{\rm{s}}}^{-1}$, very similar to that of the CfA Digital Speedometers. However, the free spectral range of the echelle order centered at 519 nm near the Mg b features is 10 nm, compared to 4.5 nm for the CfA Digital Speedometers. For both instruments this is the wavelength window used for the determination of absolute velocities and for TODCOR analyses. Furthermore, the signal-to-noise ratio per resolution element (SNRe) that is achievable with the TRES CCD detector is much higher than was possible with the intensified photon-counting detectors used in the CfA Digital Speedometers. The SNRe values for the old CfA spectra ranged from 30 to 100, while the typical value for the new TRES observations is 500.

Fourteen strong TRES spectra were obtained of σ Gem over a period 30 nights in 2014 December and 2015 January with the goal of detecting the lines of the secondary and deriving a double-lined spectroscopic orbit for the first time. Fortunately, two earlier observations from 2012 October were available in the TRES archive, which provided a two-year baseline for determining a more accurate orbital period. All 16 TRES spectra were analyzed using TODCOR (Zucker & Mazeh 1994) as implemented at CfA by G. Torres, and using the CfA library of synthetic spectra to choose the optimum templates. Only one of the observations failed to give a reliable velocity for the secondary, due to close blending of the lines from the two stars.

The light of the primary dominates the composite spectrum of σ Gem, so it was straightforward to choose the template that gave the highest value for the average peak of the one-dimensional correlations; the parameters for the best template were ${T}_{\mathrm{eff},1}=4500$ K, $\mathrm{log}{g}_{1}=2.5$ (cgs), $v\mathrm{sin}{i}_{1}=25$ $\mathrm{km}\;{{\rm{s}}}^{-1}$, and solar metallicity. Because the secondary is so much fainter, its template parameters were only weakly constrained by the two-dimensional correlations. Therefore, we adopted the mass derived in this paper to guide the choice of secondary template parameters, ${T}_{\mathrm{eff},2}=4250$ K, $\mathrm{log}{g}_{2}=4.5$ (cgs), v $\mathrm{sin}{i}_{2}$ =2 $\mathrm{km}\;{{\rm{s}}}^{-1}$, and solar metallicity; the rotational velocity was selected under the assumption of tidal synchronization of the secondary spin with the orbital period. We also tried TODCOR solutions using the nearest neighbors from our library for the secondary template, and found that the results were not sensitive to the secondary template parameters.

TODCOR has a mode in which the light ratio between the secondary and the primary in the observed wavelength window can be treated as a free parameter and thus can be determined from the spectra. We selected 10 observations where the velocities of the primary and secondary were well-separated and used these to determine a light ratio at 519 nm of 0.0139 ± 0.0025 (standard deviation of the values from the mean). If the errors are well-behaved in this analysis, the uncertainty of the mean light ratio from TODCOR could be formally better by about a factor of 3, or about 6%. For the final TODCOR analysis we fixed the light ratio to 0.0139 for all the observations. The TODCOR velocities and times of observation are reported in Table 2 on the same zero point as the native CfA Digital Speedometer system. For the analysis in this paper, 0.14 $\mathrm{km}\;{{\rm{s}}}^{-1}$ was added to the velocities in Table 2 to put them on the IAU system.

The orbital parameters are reported in Table 7 for two different double-lined spectroscopic orbits based on just the TODCOR velocities from the 16 TRES spectra. For the first solution the eccentricity was allowed to be a free parameter. Since the derived eccentricity was not significant—e =0.0018 ± 0.0029—we also derived a solution for a circular orbit. Note that the mass ratio is well constrained; q = 0.582 ± 0.016 (2.7%). Table 7 also contains the orbit reported by Massarotti et al. (2008) and the orbit fit with only the data from the AST/TSU data set. Like the results from the TRES spectra, the eccentricity from the AST radial velocities, e = 0.002 ± 0.003, is consistent with a circular orbit. Note that these are the entire data sets collected at these facilities and contain radial velocity measurements that were obtained when starspots were present on the stellar surface.apj513756t8

Our double-lined orbital solution is among the most extreme that we have derived using the Mg b region in terms of the light ratio. Indeed, attempts to get good solutions from the neighboring echelle orders on either side of the Mg b order were unsatisfactory. Therefore, we decided it would be prudent to test the reliability of our TODCOR analysis by creating sets of simulated observations using our library of synthetic spectra. For the time of each observed spectrum we simulated that observation by shifting the two synthetic template spectra by the velocities from the orbital solution and coadding after scaling by the light ratio found by TODCOR from the real observations. We found that a TODCOR analysis of the simulated observations reproduced the mass ratio and light ratio of the real data well within the estimated errors, and that the mass ratio was not sensitive to the value chosen for the light ratio. Thus, we have no evidence for a systematic error in our determination of a light ratio of 0.0139 at 519 nm. This is quite different from the light ratio reported for the interferometric observations of 0.0038 at the H-band.

In order to better justify our conclusion that ellipsoidal variations can explain previous claims of "active longitudes" on σ Gem, we have re-plotted some photometry from Kajatkari et al. (2014) in Figure 6, along with our prediction of the expected ellipsoidal variation component using the ELC software and system parameters from Table 6 using gravity darkening parameter β = 0.02. In Figure 6, we include data from two epochs, one showing very little overall variability and one showing high variability. In the first epoch ("Segment 8, Set 45" of Kajatkari et al. 2014), the photometric data clearly showed a double-peaked light curve when phased with the orbital period, previously interpreted as due to active longitudes (see Kajatkari et al. 2014). Here, we now see that by removing the expected ellipsoidal variation, the signature of two spots on opposite sides of the star (the basis for the active longitudes claims) nearly completely disappears (see Figure 6). The second epoch ("Segment 9, Set 1" of Kajatkari et al. 2014) is dominated by one spot and the ellipsoidal variations are not discernible. Nonetheless, future starspot modelers should account for the underlying ellipsoidal variations before performing detailed light curve analysis or surface brightness inversions. We have included our calculation of the expected ellipsoidal variations for Johnson UBVRIJHK in Table 8 to assist future workers—a full re-analysis of the starspot properties is beyond the scope of this paper.

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