Stellar Properties of Active G and K Stars: Exploring the Connection between Starspots and Chromospheric Activity

We used Spectroscopy Made Easy (SME; Valenti & Piskunov 1996; Valenti & Fischer 2005; Piskunov & Valenti 2017) version 522 to compute the effective temperatures (T_eff), surface gravities (log g), metallicities ([Fe/H]), and projected rotational velocities (v sin i) of all stars in our sample. The SME algorithm fits selected spectral regions with synthetic spectra generated with a particular atmospheric model (ATLAS12; Kurucz 2013) and line list. We followed the basic procedure and adopted the wavelength range and line list from Valenti & Fischer (2005), and analyzed the 5164–5190 Å interval covering the pressure-sensitive Mg i b triplet, and the 6000–6070 and 6100–6180 Å intervals split into 7 segments.

We normalized the Mg i b segment by comparing a solar spectrum (observed with ARCES, reflected off the Moon) to the Wallace et al. (2011) solar atlas, where we matched the velocity offset and instrumental broadening between spectra, then fit a polynomial to the ratio of the two spectra to derive a blaze model for this order. For the other spectral segments, this procedure also works, but the absorption lines in the regions are sparse enough that polynomials fit directly to the spectra are also effective. All spectra were cross-correlated against the solar atlas during normalization. SME also fits for small radial-velocity offsets and residual linear trends across spectral segments.

We initialized SME with properties taken from the literature; when spectroscopic results were not available, we used Gaia DR2 T_eff values; when other parameters were unavailable, we adopted log g = 4.6 dex, [Fe/H] = 0.0 dex, and v sin i = 2 km s⁻¹. Following Valenti & Fischer (2005), microturbulence was fixed to 0.85 km s⁻¹, and we adopted their macroturbulence relation. We did not solve for any additional abundances besides the metallicity parameter for this work, which therefore assumes a scaled-solar composition. After running the initial fit, we perturbed the resulting T_eff by ±200 K and refit the spectra. Fits typically converged to within 10 K of the first solution for stars warmer than T_eff > 4600 K.

The spectra of cooler stars are more complicated, particularly in the Mg i b order. For stars cooler than T_eff < 4600 K, we removed this order from our analysis, then fixed log g to a value drawn from a 3 Gyr (i.e., middle-aged) Dartmouth isochrone with [Fe/H] = +0.10 dex (this choice was arbitrary and made purely for our convenience) at the initial T_eff, and iterated the fit once. We verified convergence by varying the initial T_eff by ±500 K, and resetting values each time to [Fe/H] = 0.0 dex and log g = 4.6 dex.

Ryabchikova et al. (2016) assessed the stellar parameter accuracies attainable with SME for FGK dwarfs with high-quality spectra (e.g., S/N = 200, R = 40,000–110,000), and found uncertainties on T_eff of 50–70 K, on log g of 0.1 dex, and on metallicity of 0.05−0.06 dex (see also Brewer et al. 2016; Piskunov & Valenti 2017). While their spectroscopy methodology (including wavelength regions analyzed) differs from ours, these values provide error floors that are consistent with our results. Results for stars with multiple observations and results from varying the initial conditions typically yielded dispersions much lower than this error floor. For now, we will assume that our uncertainties for T_eff are 50 K, and will later compare our values to the photometric Gaia DR2 and literature values to assign final uncertainties. The results are shown in Figure 4.

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For most of the stars, T_eff, log g, microturbulent velocity (ξ, in km s⁻¹), and [Fe/H] were also determined with a standard model atmospheres approach. Equivalent width (EW) of Fe i and Fe ii lines were first measured in the normalized spectra using the Gaussian profile fitting code DAOSPEC (Stetson & Pancino 2008). DAOSPEC has difficulty measuring lines with EW > 100 mÅ; as a result, those lines were remeasured by hand. The lines from Fulbright et al. (2006) and McWilliam et al. (2013) were utilized. Lines blueward of 5000 Å were not included, to avoid issues with continuum normalization. Strong lines (with reduced EWs⁷ REW > −4.8) were not included, to minimize uncertainties due to, for instance, damping constants (McWilliam et al. 1995).

The EWs were then input into the 2017 version of the Local Thermodynamic Equilibrium (LTE) line analysis code MOOG (Sneden 1973). ATLAS model atmospheres were adopted (Castelli & Kurucz 2004), using an interpolation scheme for temperatures and surface gravities that lie between the grid points. For each Fe line, [Fe/H] was calculated relative to the solar abundance for that line (using the solar EWs from Fulbright et al. 2006 and McWilliam et al. 2013). The appropriate T_eff and ξ were determined by minimizing trends in [Fe i/H] with excitation potential (EP, in eV) and REW, respectively. The appropriate log g was determined with the BaSTI isochrones (Pietrinferni et al. 2004), utilizing the spectroscopic T_eff. This means that the surface gravity could be mildly dependent on the adopted age and metallicity; however, the effect on main-sequence stars is small. The [Fe i/H] and [Fe ii/H] ratios are averages from the individual lines. The surface gravities were not determined by requiring ionization equilibrium (i.e., by requiring that the Fe i and Fe ii abundances be equal), both due to larger uncertainties in the Fe ii abundances and possible non-LTE effects (Kraft & Ivans 2003). While there are ∼10–120 acceptable Fe i lines per star (depending on metallicity, T_eff, and S/N), there are many fewer Fe ii lines; as a result, the random errors in [Fe ii/H] are much larger than those in [Fe i/H].

Note that none of these parameters are independent; the adopted temperatures are therefore dependent upon the other parameters, particularly the microturbulent velocity. The errors in T_eff were determined by assuming errors in ξ of ±0.3 km s⁻¹ (errors in log g and [Fe/H] have a negligible effect on the temperatures).

The MOOG effective temperatures are listed alongside the others in Table 1; the rest of the MOOG results (log g, [Fe i/H], [Fe ii/H], and microturbulence) are enumerated in Appendix C. The temperatures generally agree with the Gaia values, though they may be slightly higher in the coolest stars (see Figure 4). The cooler stars have fewer suitable Fe i lines for an EW analysis; as a result, the temperatures are likely to be more sensitive to line-to-line variations as a result of, for example, continuum placement or atomic data.

The European Space Agency's Gaia mission recently delivered astrometric, photometric, and radial-velocity data for a large number of stars (Gaia Collaboration et al. 2018). The second data release (DR2) included an analysis of the three-band photometry (G_BP, G_RP, G), with the Apsis pipeline (Bailer-Jones et al. 2013), and derived effective temperatures for 161 million sources brighter than G < 17 with 3000 < T_eff < 10,000 K (Andrae et al. 2018). The Apsis pipeline consists of a machine-learning algorithm trained on large T_eff data sets from the literature (e.g., LAMOST, RAVE, the Kepler Input Catalog), and was validated using data from GALAH, nearby solar twin stars, and other sources.

We performed additional tests to assess the accuracy and precision of the DR2 catalog effective temperatures by comparing results for well-characterized nearby stars, including the FGK stars observed by CPS with Keck/HIRES and characterized with SME by Brewer et al. (2016), the K and M dwarfs characterized with interferometry and bolometric fluxes by Boyajian et al. (2012), and the M dwarfs characterized with optical and NIR spectroscopy by Mann et al. (2015). The Gaia DR2 T_eff values are consistent with this joint benchmark sample for T_eff > 4100 K, but diverge at cooler temperatures. The warmer stars show remarkable agreement, having an rms of only 70 K, which is much lower than the typical T_eff accuracy of 324 K quoted by Andrae et al. (2018). This is due to the fact that these stars are all nearby and likely appear unreddened. We fit a polynomial to the DR2 color (G_BP − G_RP) versus the benchmark sample T_eff values to recalibrate the DR2 photometric temperatures and place the cool stars on the benchmark sample's temperature scale. We will use this recalibration later in the next subsection. Table 1 reports the DR2/Apsis values (Andrae et al. 2018).

Jódar et al. (2013) identified a companion to GJ 4099 with high-angular-resolution optical imaging and measured a contrast of ${\rm{\Delta }}i=0.27$ mag (Sloan), a proper motion indicating that the stars are co-moving, and an angular separation of 300–400 mas, which Gaia DR2 would not resolve (see Figure 1 from Ziegler et al. 2018).

Blended photometry will bias the photometric estimation of T_eff. Instead, we model the photometry as a binary system. First, we assembled the K and M dwarf sample from Mann et al. (2015), which provides T_eff and metallicity measurements derived from optical and NIR spectroscopy as well as synthetic $J,H,{K}_{S}$ magnitudes computed by integrating the absolutely flux-calibrated optical spectra over each filter. Next, we cross-matched this catalog with DR2, which provided parallax and the DR2 photometric magnitudes. To model the binary, we randomly drew primary and secondary stars with replacement from the sample, combined the absolute magnitudes, and assembled the 6-band SEDs (Gaia DR2 G, G_BP, G_RP, 2MASS J, H, K_S), then minimized χ² between the empirical binary SEDs and the observed SED for GJ 4099. This procedure yielded temperatures for the primary and secondary of T_eff = 3743 K and 3668 K, respectively. HD 209290 has a DR2 T_eff similar to GJ 4099—we performed a similar fit, assuming it is single, and found T_eff = 3875 K (only 7 K cooler than its DR2 value). Whereas DR2 listed this star as only 36 K warmer than GJ 4099, we find it to be 132 K warmer than the primary and 207 K warmer than the secondary. This is consistent with the SME analysis that found T_eff = 3876 K for HD 209290, T_eff = 3682 K for GJ 4099, and a difference of 194 K. Our photometric analysis of GJ 4099 indicates that the primary is 75 K warmer than the secondary, which is consistent with the Δi measurement from Jódar et al. (2013). Given this small difference in T_eff, and the fact that the components are well-separated and non-interacting, we will assume that our spectrum for GJ 4099 is representative of a quiescent 3700 K photosphere.

We apply a prior to penalize very large spot filling factors, by enforcing that the filling factor is never large enough to skew the broadband colors of the star significantly toward the red. For each star, we compute a Gaussian prior penalty p that a star has an observed Gaia DR2 (G_BP − G_RP) color c_O, compared to the computed two-component mixture model color c_C,

$$\begin{eqnarray}&&\mathrm{ln}p\propto -\displaystyle \frac{1}{2}{\left(\displaystyle \frac{{c}_{O}-{c}_{C}}{{\sigma }_{c}}\right)}^{2},\end{eqnarray} \tag{ 2 }$$

where σ_c = 0.006 is a typical uncertainty on (G_BP − G_RP) for stars in our sample. Since the signal we are searching for is similar in amplitude to the uncertainties of the observations, fits without this prior tend to prefer non-zero spot filling factors simply because the noisy data can be better fit with two templates than one. The color prior has the important effect of ensuring that non-zero spot filling factors yield a sufficient improvement in the model likelihood to offset the penalty incurred by the color prior.

Spot covering fractions on Sun-like stars are typically small, but span orders of magnitude. In Morris et al. (2017b) we show that the Sun has typical spot coverage near 0.03%, while the K4V star HAT-P-11 has a typical spot coverage near 3%. It has been speculated that the wide range in activity could be due in part to the presence of the hot-Jupiter planet in the HAT-P-11 system (Morris et al. 2017a).

Several efforts exist in the literature to search for correlations between the effective temperatures of stars and the temperatures of their spots (e.g., Berdyugina 2005; Mancini et al. 2014). The data sets and analyses are highly heterogeneous and apparent correlations often have several exceptions. Therefore, in order to simplify the analysis in this work, we choose plausible combinations of stellar atmospheres, informed primarily by the temperature contrast of sunspots.

Sunspots have detailed substructure dominated by the cool penumbra, and the much cooler umbra. Since the ratio of penumbral to umbral area of sunspots is roughly a factor of four, and since the intensity of the penumbra is more than a factor of two greater than the intensity of the umbra in sunspots (Solanki 2003), in this work we will approximate starspots as being entirely composed of penumbra. On the Sun, penumbra typically have temperatures 250–400 K cooler than the mean photosphere. In most cases, we choose combinations of stellar spectra with effective temperature differences of a few hundred Kelvin, manually selecting pairs seeking to minimize the differences in metallicity and surface gravity between the template stars and the target star. Therefore one should not interpret the temperature differences listed in Table 2 as detections; they were adopted a priori in order to measure the spot covering fraction under the assumption that starspots on G and K stars generally act like sunspots. The temperatures listed in Table 2 are taken from the SME analysis in Section 2.4.1.

Table 2.  Each Target Star, and the Hotter (T₁) and Cooler (T₂) Components Used to Model Its Spectrum, with Spot Covering Fractions f_S and Model Comparison with the Null Hypothesis Δχ²

Target	Teff	Comp. 1	T1	Comp. 2	T2	ΔT	fS	Δχ2
HD 151288	4337 ± 100	Gl 705	4515 ± 100	HD 209290	3876 ± 100	639	0.31 ± 0.06	197.1
GJ 702 B	4359 ± 100	Gl 705	4515 ± 100	GJ 4099	3682 ± 100	833	0.00 ± 0.06
EQ Vir	4359 ± 100	Gl 705	4515 ± 100	GJ 4099	3682 ± 100	833	0.26 ± 0.06	573.3
HD 175742	4980 ± 40	HD 145742	4801 ± 40	Gl 705	4515 ± 100	286	0.52 ± 0.08	21.8
EK Dra	${5584}_{-193}^{+115}$	HD 210277	5524 ± 40	HD 38230	5077 ± 40	447	0.31 ± 0.06	40.1
Kepler-63	5581 ± 40	HD 10697	5666 ± 40	HD 221639	5091 ± 40	575	0.27 ± 0.06
HD 45088	${4786}_{-36}^{+39}$	HD 145742	4801 ± 40	Gl 705	4515 ± 100	286	0.53 ± 0.07	21.4
HD 113827	4235 ± 100	Gl 705	4515 ± 100	GJ 4099	3682 ± 100	833	0.20 ± 0.06	177.5
HD 134319	5762 ± 40	HD 68017	5620 ± 40	HD 145675	5300 ± 40	320	0.00 ± 0.06
HD 127506	4651 ± 100	HD 145742	4801 ± 100	Gl 705	4515 ± 100	286	0.49 ± 0.08	14.9
HD 200560	5013 ± 40	HD 145742	4801 ± 100	Gl 705	4515 ± 100	286	0.00 ± 0.06
HD 220182	5363 ± 40	HD 145675	5300 ± 40	HD 145742	4801 ± 40	499	0.35 ± 0.06	101.8
HD 41593	5339 ± 40	HD 145675	5300 ± 40	HD 145742	4801 ± 40	499	0.35 ± 0.06	102.6
HD 82106	4726 ± 100	HD 145742	4801 ± 100	Gl 705	4515 ± 100	286	0.00 ± 0.06
HD 79555	4744 ± 100	HD 145742	4801 ± 100	Gl 705	4515 ± 100	286	0.00 ± 0.06
HD 87884	5047 ± 40	HD 221639	5091 ± 40	HD 5857	4689 ± 100	402	0.04 ± 0.10
HD 222107	4693 ± 100	HD 6497	4568 ± 40	GJ 4099	3682 ± 100	886	0.18 ± 0.06	146.0
HD 39587	6027 ± 40	HD 50692	5919 ± 40	HD 62613	5489 ± 40	430	0.02 ± 0.06
HD 149957	4389 ± 100	Gl 705	4515 ± 100	GJ 4099	3682 ± 100	833	0.12 ± 0.06	60.5
HD 122120	4452 ± 100	Gl 705	4515 ± 100	GJ 4099	3682 ± 100	833	0.14 ± 0.06	79.1
HD 47752	4613 ± 100	HD 145742	4801 ± 40	Gl 705	4515 ± 40	286	0.02 ± 0.06
HD 148467	4253 ± 100	Gl 705	4515 ± 100	GJ 4099	3682 ± 100	833	0.23 ± 0.06	245.8
61 Cyg A	4336 ± 100	HD 6497	4568 ± 100	GJ 4099	3682 ± 100	886	0.38 ± 0.06
HAT-P-11	${4757}_{-111}^{+154}$	HD 5857	4689 ± 100	Gl 705	4515 ± 40	174
GJ 9781 A	4391 ± 100	Gl 705	4515 ± 100	GJ 4099	3682 ± 100	833	0.21 ± 0.06	194.8
HD 110833	5004 ± 40	HD 145742	4801 ± 40	Gl 705	4515 ± 100	286	0.00 ± 0.06
σ Draconis	5215 ± 40	HD 42250	5344 ± 40	GJ 4099	3682 ± 100	1662	0.22 ± 0.06	156.2
HD 68017	5620 ± 40	HD 10697	5666 ± 40	HD 221639	5091 ± 40	575	0.00 ± 0.06
HD 67767	5264 ± 40	HD 182488	5367 ± 40	GJ 4099	3682 ± 100	1685	0.15 ± 0.06	61.7
HD 73667	4800 ± 40	HD 221639	5091 ± 40	Gl 705	4515 ± 100	576	0.00 ± 0.06
Kepler-17	5913 ± 40	HD 50692	5919 ± 40	HD 62613	5489 ± 40	430	0.36 ± 0.06
HD 62613	5489 ± 40	HD 10697	5666 ± 40	HD 221639	5091 ± 40	575	0.16 ± 0.06	10.0
HD 14039	${5072}_{-73}^{+98}$	HD 221639	5091 ± 40	HD 5857	4689 ± 100	402	0.02 ± 0.06
51 Peg	5817 ± 40	HD 86728	5816 ± 40	GJ 4099	3682 ± 100	2134	0.10 ± 0.06	24.9
HD 34411	5916 ± 40	HD 86728	5816 ± 40	HD 145675	5300 ± 40	516	0.00 ± 0.06

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We construct a simple test to ensure that the two-component model is not over-fitting the spectra and produces reliable uncertainties. We fit a sample of inactive stars with a linear combination of two stars: the spectrum of the same star and the spectrum of a cooler star. In this control run, we expect that the maximum-likelihood model should choose a spot covering fraction ${f}_{S}\approx 0$, as the spectrum of the target star should be a perfect fit to itself. For a sample of 10 control stars, we find typical f_S < 0.06 (68% confidence). We interpret this result to mean that the smallest reliable uncertainty on f_S should be ∼0.06; so throughout this work, the uncertainty on f_S is the maximum of the formal uncertainty or 0.06.

We also analyze spectra of the twilight sky as a proxy for solar spectra, on a day with no sunspots. We model the solar spectrum as a linear combination of HD 10697 (T_Q = 5654⁺³⁶₋₅₆ K) and HD 221639 (${T}_{S}={5102}_{-106}^{+139}$ K), and measure f_S ≲ 0.001. This measurement is small, due largely to the very high signal-to-noise of the solar observations in comparison with the stellar observations.

EQ Virginis is a K5Ve flare star in the nearby cluster IC 2391 (50 Myr; Barrado y Navascués et al. 2004) with rotation period P_rot = 3.9 ± 0.1 days (Montes et al. 2001; Paulson & Yelda 2006; Suárez Mascareño et al. 2016). Saar et al. (2001) assumed T_Q = 4380 K and T_S = 3550 K, and found a spot covering fraction f_S = 0.43 ± 0.05. Subsequently, O'Neal et al. (2004) adopted 61 Cyg A as the quiescent template (T_eff = 4374 K; Heiter et al. 2015), and measured T_S = 3350 ± 115 K with f_S = 0.34–0.45.

We model the spectrum of EQ Vir as a linear combination of the spectra of Gl 705 (K2V, ${T}_{Q}={4465}_{-66}^{+102}$ K) and GJ 4099 (M2V, T_S = 3682 K), which is a warmer spot temperature than the prior two studies. We recover spot covering fractions f_S = 0.25 ± 0.06, smaller than the spot covering fractions measured by earlier observers. In large part, the discrepancy is due to our use of a strong prior on the color of the star (see Section 3.1.1), which places strict upper limits on the plausible spot covering fractions.

To compare our results with Saar et al. (2001) and O'Neal et al. (2004), and to understand the effect of the color prior, we compute the expected Gaia DR2 (G_BP − G_RP) color of EQ Vir, given the spot properties of O'Neal et al. (2004), and compare it to its observed color. Its photospheric temperature measured via SME is T_eff = 4519 ± 200 K, and synthetic photometry of a PHOENIX model atmosphere with this temperature has color (G_BP − G_RP)_Q = 1.4772. O'Neal et al. (2004) assume the spotted component has temperature T_S = 3350 ± 115 K, which from synthetic photometry should have color (G_BP − G_RP)_S = 2.9739. Combining these colors with spot covering fraction f_S = 0.43 using Equation (6) yields the expected composite color (G_BP − G_RP) = 1.8135. The observed Gaia EQ Vir color (G_BP − G_RP) = 1.4960 ± 0.0054 is 58σ discrepant with the Saar et al. (2001) observation, and up to 62σ discrepant with the range reported by O'Neal et al. (2004). We conclude that the spot covering fraction must be f_S < 0.43, in order not to redden the star significantly beyond its actual color, assuming the effective temperature measured by SME is the correct photospheric temperature.

To verify that the PHOENIX model atmospheres are not responsible for this large discrepancy, we also empirically estimated the DR2 colors by constructing a color–temperature relation based on the Boyajian et al. (2012) K/M benchmark sample, which covers 3054 < T_eff < 5407 K.⁸ This predicts (G_BP − G_RP)_S = 2.608 for 3350 K and 1.324 for 4515 K, and we find a 20σ discrepancy with the observed color. This confirms that there is a significant discrepancy between the predicted and observed colors of EQ Vir using the O'Neal et al. (2004) spot coverage and temperatures.

For completeness, we carry out the same calculation using our maximum-likelihood model for EQ Vir. We adopt Gl 705 for T_Q with (G_BP − G_RP)_Q = 1.4388, and GJ 4099 for T_S with (G_BP − G_RP)_S = 2.1969. Our maximum-likelihood spot covering fraction is f_S = 0.24, which gives a predicted color G_BP − G_RP = 1.4976, which is <1σ consistent with the observed color of EQ Vir.

EK Draconis is a G1.5V BY Dra variable generally thought to resemble the young Sun (Montes et al. 2001). Järvinen et al. (2007) used atomic line profiles to detect both high and low-latitude spots with ΔT = 500 K. Zeeman Doppler imaging from Waite et al. (2017) reveals P_rot = 2.766 ± 0.002 days, and an amazing variety of spots—sometimes at mid-latitudes, sometimes at the pole, covering 2%–4% of the stellar surface. O'Neal et al. (2004) measured a much larger spot coverage f_S = 0.25–0.40 for T_Q = 5830 K and T_S ≳ 3800 K.

We model the spectrum of EK Dra with a linear combination of spectra from HD 210277 (T_Q = 5535⁺⁴²₋₄₈ K) and HD 38230 (${T}_{S}={5253}_{-89}^{+61}$ K). The only TiO band that yields informative constraints on the spot covering fraction is at 7054 Å, predicting a spot covering fraction of f_S = 0.28 ± 0.06, consistent with the lower end of the coverage measured by O'Neal et al. (2004). The higher spot covering fractions are excluded by the color prior (see previous section for a detailed discussion).

HAT-P-11 is an active K4V dwarf in the Kepler field with a transiting hot Jupiter. Transits of the stellar surface reveal frequent starspot occultations (Deming et al. 2011; Sanchis-Ojeda & Winn 2011), which yield spot covering fractions f_S = 0.0–0.1 within the transit chord, whereas rotational modulation predicts ${f}_{S}\gtrsim {0.03}_{-0.01}^{+0.06}$ (Morris et al. 2017b). HAT-P-11 appears to have a ∼10 yr activity cycle, and may be modestly more chromospherically active than planet hosts of similar rotation periods (Morris et al. 2018a). Recent ground-based photometry of spot occultations within the transit chord yielded spot coverage f_S = 0.14 (Morris et al. 2018d).

It is a valuable validation exercise to apply the TiO molecular band modeling technique to HAT-P-11 and to compare with the results of spot occultation analysis. We model the spectrum of HAT-P-11 as a linear combination of the spectra of HD 5857 (${T}_{Q}={4712}_{-71}^{+67}$ K) and Gl 705 (${T}_{S}={4465}_{-66}^{+102}$ K), giving the spots ΔT_eff = 250 K, similar to typical sunspot penumbra. We find a consistent spot covering fraction between observations separated by a year near f_S ≈ 0.15 ± 0.06 (see Figure 7). This is broadly consistent with the spot covering fractions measured in transit by Morris et al. (2017b), if on the high end, and consistent with the more recent observations of Morris et al. (2018d). The TiO estimate might be slightly larger than the typical transit chord crossing spot filling factor because of the small temperature difference that we have assumed here (250 K)—a larger temperature difference would produce a correspondingly smaller spot covering fraction. Perhaps this is evidence that the correct spot temperature should be a bit cooler than our assumed T_S.

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λ Andromedae is a single-lined spectroscopic binary (SB1) with an old disk giant primary identified as an RS CVn variable with rotation period P_rot = 54 days, which has at least two large starspots (Calder 1938; Landis et al. 1978; Bopp 1980; Poe & Eaton 1985; Padmakar & Pandey 1999; Drake et al. 2011; Pandey & Singh 2012; Parks et al. 2014). BV photometry of the rotational modulation predicts one large spot covering 8% of the stellar surface, with an estimated temperature T_S = 4000 ± 300 K (Donati et al. 1995). O'Neal et al. (1998) modeled TiO molecular band absorption ranging f_S = 0.14–0.21. Mirtorabi et al. (2003) developed a TiO photometric index to measure the spot covering fraction over several years and found spot coverage varying from a few to $\lesssim 40 \%$. Frasca et al. (2008) measured simultaneous photometry and spectroscopy of λ And and found ${f}_{S}={0.076}_{-0.014}^{+0.023}$. Using the CHARA array, Parks et al. (2014) found f_S = 0.096.

We model λ And with a combination of HD 6497 (${T}_{Q}={4514}_{-71}^{+82}$ K) and GJ 4099 (${T}_{S}={3846}_{-190}^{+181}$ K). We find f_S = 0.18 ± 0.06, consistent with Donati et al. (1995, 2003), O'Neal et al. (1998), Frasca et al. (2008), and Parks et al. (2014).

Due to the RS CVn nature of λ And, it appears as an outlier in Figure 10, having a very long rotation period but significant, non-zero spot coverage. The apparently high spot coverage at its slow rotation period may be an effect of the low surface gravity of this sub-giant primary star. The low log g of λ And implies a longer convective timescale τ_c, and therefore the Rossby number of the star Ro = P_rot/τ_c may still be small even though its rotation period is long.

This is a double-lined spectroscopic binary (SB2), with an orbital period P_orb = 6.99 days (Griffin & Emerson 1975; Bopp 1980). Glazunova et al. (2014) solved its orbit, characterized the components, and found the following properties for the primary and secondary, respectively: T_eff = 5025 and 4508 K, log g = 4.3 and 4.5 dex, and [Fe/H] = − 0.2 dex.

O'Neal et al. (2001) accounted for the secondary in their analysis and fixed the primary and secondary temperatures to 4925 and 4550 K, then fit for a third component assumed to have a temperature of 3525 K and a spectrum from Gl 96, and found f_S = 0.20−0.35. Glazunova et al. (2014) also noted temporal changes in chemical abundances consistent with substantial spot coverage.

We modeled the TiO features of HD 45088 as a combination of HD 145742 (${T}_{Q}={4851}_{-75}^{+13}$ K) and Gl 705 (${T}_{S}={4465}_{-66}^{+102}$ K), ignoring the binarity, found apparent "spot covering fraction" f_S = 0.52 ± 0.14, which in this case can be instead interpreted as the fractional flux contributed by the secondary component. According to a 3 Gyr PARSEC isochrone model with [Fe/H] = −0.2 dex (Bressan et al. 2012), the stellar radii at our component temperatures are ${R}_{A}=0.69\,{R}_{\odot }$ and R_B = 0.63 R_⊙; the ratio of the cooler area from the secondary to the total area of both stars is then 0.48, which is consistent with the spot covering fraction we inferred. The difference between O'Neal et al. (2001) and our procedure is that we did not account for the secondary's flux, but instead fit its photosphere and returned the relative stellar areas instead of the spot covering fraction. This exercise validates our TiO fitting procedure, at least in this "heavily spotted" regime, as the spot area fractions are consistent with the stellar flux contributions that are expected for this stellar binary.

HD 175742 (K0V) is also an RS CVn variable, with rotation period P_rot = 2.88 days (Rutten 1987). Alekseev & Kozlova (2000) estimated the spectral type of the companion to be M3V (ΔT ≈ 1900 K), then measured the spot covering fraction via rotational modulation and estimated f_S ≲ 0.42 with ΔT = 900 K.

We model HD 175742 with a combination of HD 145742 (${T}_{Q}={4851}_{-75}^{+13}$ K) and Gl 705 (${T}_{S}={4465}_{-66}^{+102}$ K) and find f_S = 0.54 ± 0.13, consistent with Alekseev & Kozlova (2000).

Désert et al. (2011), Bonomo & Lanza (2012), and Davenport (2015) measure positions and differential rotation of spots for this star with ${P}_{\mathrm{rot}}=12.2$ days. Estrela & Valio (2016) measure activity cycle P_cyc = 1.12 ± 0.16 yr. We measure f_S = 0.16 ± 0.06, qualitatively consistent with the relatively large spot covering fraction necessary to produce spot occultations in nearly every transit, though we could not find a precise spot coverage number in the literature.

Messina et al. (1998) use rotational modulation to predict a spot covering fraction of at least ${f}_{S}\gtrsim 0.16$ for this BY Dra-type variable. Adaptive optics imaging revealed that HD 134319 has a M4.5V companion (Mugrauer et al. 2004), but its late type and large separation from the host star of (55) makes it unlikely that it is a significant source of contaminating flux. We model HD 134319 with a combination of HD 210277 (${T}_{Q}={5535}_{-48}^{+42}$ K) and HD 38230 (${T}_{S}={5253}_{-89}^{+61}$ K), and find f_S = 0.0 ± 0.06.