Chromospheric Activity of HAT-P-11: An Unusually Active Planet-hosting K Star

We reduce the raw ARCES spectra with IRAF methods to subtract biases, remove cosmic rays, normalize by the flat field, and determine the wavelength calibration with exposures of a thorium-argon lamp.⁹ We fit the spectrum of an early-type star with a high-order polynomial to measure the blaze function, and we divide the spectra of HAT-P-11 and the MWO stars by the polynomial fit to normalize each spectral order.

Next, the normalized spectra must be shifted in wavelength into the rest-frame by removing their radial velocities. We remove the radial velocity by maximizing the cross-correlation of the ARCES spectra with PHOENIX model atmosphere spectra (Husser et al. 2013).

To calibrate the ARCES spectra, we follow the calibration procedure developed in Isaacson & Fischer (2010) for HIRES. We collect 51 spectra of 30 stars in the Duncan et al. (1991) MWO sample with the ARC 3.5 m Telescope at APO and the ARCES spectrograph, including 22 K stars, 7 G stars, and one M star—see Figure 1 for example spectra.

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We measure the S-index for these stars by taking the sum of the flux in the cores of the H and K features at 3968.47 and 3933.66 Å, weighted with a triangular weighting function with FWHM = 1.09 Å. We normalize the weighted emission by the flux in pseudocontinuum regions R and V, which are 20 Å-wide bands centered on 3900 and 4000 Å, respectively. Then S on the MWO-calibrated scale is

$$\begin{eqnarray}&&{S}_{\mathrm{APO}}=\displaystyle \frac{a\,H+b\,K}{c\,R+d\,V},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{S}_{\mathrm{MWO}}={C}_{1}{S}_{\mathrm{APO}}+{C}_{2},\end{eqnarray} \tag{ 2 }$$

where $a,\,b,\,c,\,d,\,{C}_{1}$, and C₂ are parameters that can be tuned to make ARCES S-indices match the scale of S_MWO (Duncan et al. 1991). Following the example of Isaacson & Fischer (2010), we chose values of $a,b,c,d$ so that S has roughly equal flux contributions from the H and K emission lines and roughly equal flux from the R and V pseudocontinuum regions in the APO spectra. Thus, we set $a=c=1$, and we choose b = 2 and d = 1, so that $H\sim b\,K$ and $K\sim d\,V$.

Because S varies over time for each star in the sample, the linear correlation between the S_APO and S_MWO will have some intrinsic spread. To incorporate this into our model, we adopt the $\langle S\rangle$ and the standard deviation of S from Duncan et al. (1991) as the measurement and uncertainty of the MWO values. We solve for the constants C₁ and C₂ and their uncertainties with Markov Chain Monte Carlo (MCMC; Goodman & Weare 2010; Foreman-Mackey et al. 2013); the results are shown in Figure 2. We find ${C}_{1}={21.26}_{-0.83}^{+0.99}$ and ${C}_{2}={0.009}_{-0.009}^{+0.011}$. The S-indices for each target are enumerated in Table 1. The software tools used to calculate calibrated S-indices with spectra from ARCES are publicly available.¹⁰

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With the transformation computed in the previous section, we can compare the APO S-index measurements with those from the archival Keck/HIRES spectra. HAT-P-11 was the target of several Keck/HIRES programs since the planet's discovery, with the aims of measuring the Rossiter–McLaughlin effect and searching for a third body.¹¹ We measure S-indices from these observations with the HIRES activity pipeline described in Isaacson & Fischer (2010). A complete description of the HIRES pipeline developed for these measurements is beyond the scope of this paper; we refer the reader to Isaacson & Fischer (2010) for the full description. The result is an unevenly sampled series of 239 S measurements spanning 2007–2016, with typical ${\rm{S}}/{\rm{N}}\sim 20$.

Figure 3 shows the S-index of HAT-P-11 between 2008 and 2017, including both the APO and Keck spectra. The APO S-indices are enumerated in Table 2. The black circles are measurements from Keck/HIRES, and the red circles are measurements from APO. The chromospheric activity of HAT-P-11 appears to ramp up in the years preceding the Kepler mission, stabilize near a maximum throughout the Kepler years from 2009 to 2014, and then decline rapidly from 2014 to 2016 before rising at a similar rate through the present.

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Activity indices like S vary on two timescales—one driven by the rotation period of the star (29.2 days) as active regions rotate into and out of view, the other being the longer timescale throughout the activity cycle. It appears from this limited time series that HAT-P-11 may have entered activity maximum near the beginning of the Kepler observations. It is interesting to note that Morris et al. (2017) found that the starspots of HAT-P-11 are distributed in active latitudes centered on $\bar{{\ell }}=16^\circ \pm 1^\circ$, similar to the distribution of sunspots near solar activity maximum. The S-index appears to corroborate that if HAT-P-11's activity cycle is similar to the Sun's, it was at active maximum during the Kepler years.

Further spectroscopic monitoring will be required to determine the period of the activity cycle of HAT-P-11. During the summer of 2017, we measure ${\langle S\rangle }_{2017}=0.54\pm 0.04$, still less than the earliest available measurements from the summer of 2007 ${\langle S\rangle }_{2007}=0.569\pm 0.006$—it appears that one complete activity cycle has not yet elapsed. This gives a lower limit on the activity cycle period of ∼10 years. The solar activity cycle is not perfectly periodic; the mean cycle duration and standard deviation are 10.9 ± 1.2 years (Hathaway et al. 2002). This lower limit on the activity cycle of HAT-P-11 observed thus far is longer than some solar activity cycles, but not longer than the mean. In principle, one could attempt to measure the period of HAT-P-11's cycle by invoking empirical models of S inspired by the solar activity cycle. However, as we will discuss below, HAT-P-11's S-index distribution with time departs from the typical S distribution of the Sun—HAT-P-11 spends more time near active maximum than minimum. Therefore, we can only constrain a lower limit on the period with the observations thus far and refrain from attempting to fit for the cycle period.

The activity minimum of HAT-P-11 was short compared to its maximum. We note the marked difference between HAT-P-11 and the Sun in this respect in Figure 3. As we have not yet measured a full activity cycle of HAT-P-11 and the time sampling is very uneven, it is difficult to make concrete statements about the true duration of its maximum or minimum. One quantity that might be robust against these biases is the fraction of the activity cycle spent above or below the mean of S_min and S_max, assuming the minimum and maximum observed S are close to the true minimum and maximum of S throughout the cycle. The Sun spends 25% of the cycle above $0.5({S}_{\min }+{S}_{\max })=0.174$. The same quantity for HAT-P-11 is $0.5({S}_{\min }+{S}_{\max })=0.53$, and its S was above that value from around or before 2008 until mid-2015, and stayed below that value until mid-2017. It appears that HAT-P-11 spent 63% of its cycle above the mean of S_min and S_max, assuming an ∼11 year cycle.

If HAT-P-11 has an ∼11 year activity cycle, it falls in a sparse region of rotation period-cycle period parameter space. Böhm-Vitense (2007) showed that stars typically fall into one of two categories: the young, active A-sequence, and old, inactive I-sequence (see Böhm-Vitense 2007, Figure 1). A-sequence stars typically have ratios of cycle periods to rotation periods ${P}_{\mathrm{cyc}}/{P}_{\mathrm{rot}}\sim 360$, and I-sequence stars have ${P}_{\mathrm{cyc}}/{P}_{\mathrm{rot}}\sim 80$. The Sun is a notable outlier residing in between the sequences with ${P}_{\mathrm{cyc}}/{P}_{\mathrm{rot}}\sim 160$. If HAT-P-11's activity cycle is roughly ∼11 years, it is interesting to note that it has ${P}_{\mathrm{cyc}}/{P}_{\mathrm{rot}}\sim 139$, and it falls between the A and I sequences, like the Sun.

The APO spectra in Figure 4 demonstrate that the activity has increased significantly in just the last two years. If we assume the cycle has a period of ∼11 years, we expect the next maximum to begin near 2019–2020. We predict that TESS photometry of HAT-P-11 will show spot occultations similar to those observed during the Kepler mission.

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Is the chromospheric activity of HAT-P-11 typical for K4 dwarfs? The photometric analysis by Morris et al. (2017) cannot be reproduced for many other Kepler stars. HAT-P-11 is exceptionally bright (V = 9.59), and the highly inclined orbit of its planet reveals the latitude distribution of spots during transit events. Because very few similar targets are available for comparison in the Kepler sample, we seek to compare the spectrum of HAT-P-11 to spectra of other stars.

The S-index varies with stellar effective temperature, so the large S of HAT-P-11 compared to the Sun does not imply that HAT-P-11 has more chromospheric activity than the Sun. The ${R}_{{HK}}^{{\prime} }$ index is often used instead to compare activity levels across the main sequence by normalizing the flux in the Ca ii H & K emission features by the basal flux from the photosphere (Noyes et al. 1984). For stars of any effective temperature, ${R}_{{HK}}^{{\prime} }$ increases with chromospheric activity.

We compute the ${R}_{{HK}}^{{\prime} }$ index for a large sample of main sequence stars, following the procedure of Mittag et al. (2013). We gathered published S-indices from Duncan et al. (1991), Wright et al. (2004), and Isaacson & Fischer (2010). As in Mittag et al. (2013), we select only main sequence stars with color and absolute magnitude cuts using Hipparcos parallaxes (Perryman et al. 1997), yielding a sample of 4677 main sequence stars with measured S-indices. We then solve for ${R}_{{HK}}^{{\prime} }$ for each star. We also compute ${R}_{{HK}}^{{\prime} }$ for HAT-P-11 using its mean $\langle S\rangle =0.58\pm 0.04$.

The ${R}_{{HK}}^{{\prime} }$ index for each star in the literature is shown in Figure 6. The characteristic rise in chromospheric activity for later type stars is visible in the right half of the figure. HAT-P-11 is the red square with $B-V=1.19$ and $\mathrm{log}{R}_{{HK}}^{{\prime} }=-4.35$ (evaluated for $\langle S\rangle$). The Sun is represented with the ⊙ symbol and resides in the inactive sequence, blueward of HAT-P-11.

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HAT-P-11 has typical chromospheric activity among field stars of similar color. It lies near the lower envelope of ${R}_{{HK}}^{{\prime} }$ at $B-V\sim 1.2$, which one might expect given its rotation period ${P}_{\mathrm{rot}}=29.2$ days, as ${R}_{{HK}}^{{\prime} }$ declines with increasing age and rotation period (Noyes et al. 1984). HAT-P-11 also hosts a close-in planet, which one might speculate could affect its activity, and in general, we do not know the planet occurrence frequency in this sample of stars. As a result, it is perhaps more interesting to compare the chromospheric activity of HAT-P-11 to planet-hosting stars of similar color and age.

Is the chromospheric activity of HAT-P-11 typical for planet-hosting K stars with long rotation periods? We build a sample of stars with similar effective temperatures to HAT-P-11 from the California-Kepler Survey (CKS), in which Petigura et al. (2017) published Keck/HIRES spectra and Johnson et al. (2017) published precise stellar parameters for 1305 planet-hosting Kepler stars. We gather the CKS spectra of 107 planet-hosting stars that meet the following criteria: they have (1) effective temperatures in the range $4600\lt {T}_{\mathrm{eff}}\lt 5200$ K (Petigura et al. 2017), bracketing HAT-P-11 at ${T}_{\mathrm{eff}}=4780\pm 50$ K (Bakos et al. 2010), (2) innermost planets with orbital periods $\lt 20$ days, and (3) ${\rm{S}}/{\rm{N}}\gt 50$, where we approximate S/N by dividing the median flux in the blue spectrum by the median uncertainty in the public CKS spectra.

We replicate the procedure in Section 2.1 to calculate S-indices for the CKS stars using the HIRES correction coefficients from Isaacson & Fischer (2010) and use the relations of Mittag et al. (2013) to compute ${R}_{{HK}}^{{\prime} }$ indices for each star, as in Section 4.1. We also gather rotation periods for each star from Mazeh et al. (2015), and we compute the Rossby number for each star with convective turnover times computed from the relations of Wright et al. (2011).

We are particularly interested in whether or not HAT-P-11 experiences unusual planet-induced tides, which could provide a coupling between the planet's orbit and stellar activity. To quantify the relative tidal force on each star at the seat of stellar activity, the base of the convective zone, we use the simple dimensionless parameter ,

$$\begin{eqnarray}&&\epsilon =\displaystyle \frac{{M}_{p}}{{M}_{\star }}{\left(\displaystyle \frac{{R}_{c}}{a}\right)}^{3},\end{eqnarray} \tag{ 3 }$$

where M_p and ${M}_{\star }$ are the masses of the planet and star, respectively, R_c is the radius of the convective zone, and a is planet's orbital semi-major axis (Ogilvie 2014). The above equation is the ratio of the tidal force of the planet at the base of the convective zone ${F}_{\mathrm{tide}}\propto {M}_{p}{R}_{c}/{a}^{3}$ to the local force of gravity within the star ${F}_{g}\propto {M}_{\star }/{R}_{c}^{2}$. This simple ratio does not take into account other likely important factors like the stellar obliquity. However, in the vast majority of cases, the stellar obliquity is not known, so we continue to investigate tides with the imperfect index , and discuss the implications of this caveat below.

We evaluate the tidal for the innermost planet in each system. We adopt the stellar radii and semi-major axes from the isochrone fits of Johnson et al. (2017). We estimate the radius of the convective zone for each star by interpolating between the convective zone radii of model stars by van Saders & Pinsonneault (2012), having solar metallicity with $4600\lt {T}_{\mathrm{eff}}\lt 5200$ K. Finally, we adopt the mass measurement for HAT-P-11 b, ${M}_{p}=0.081{M}_{J}$ from Bakos et al. (2010) and estimate the most probable mass for all other planets using the forecaster tool by Chen & Kipping (2017), given the measured planetary radii from Johnson et al. (2017).

We recover the canonical result that ${R}_{{HK}}^{{\prime} }$ generally decreases with rotation rate (or age) in Figure 7. It appears that HAT-P-11 (large square) has more chromospheric activity than other planet-hosting stars of similar rotation periods and effective temperatures. The gray box in Figure 7 circumscribes 31 stars with rotation periods $25\lt {P}_{\mathrm{rot}}\lt 35$ days. The median activity level within the box is only 40% of HAT-P-11's, $\mathrm{log}{R}_{{HK}}^{{\prime} }=-4.75$.

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Could the relatively high level of chromospheric activity on HAT-P-11 (for stars with its age and color) be related to the tides raised by the close-in planet? We show the –${R}_{{HK}}^{{\prime} }$ distribution of stars with $25\lt {P}_{\mathrm{rot}}\lt 35$ days in Figure 8. We note that in this range of rotation periods, HAT-P-11 (red square) has the most chromospheric activity and the second-greatest . However, the lack of correlation between ${R}_{{HK}}^{{\prime} }$ and suggests: (1) the tidal force is not the only important factor in setting the level of chromospheric activity, and (2) the simple parameterization of does not capture all of the relevant physics.

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Significant tides are raised by HAT-P-11 b, even when compared to solar system objects. We can compare it's ${\mathrm{log}}_{10}\epsilon =-7.7$ with more familiar systems. at the base of the solar convective zone due to tides raised by Mercury gives ${\mathrm{log}}_{10}\epsilon =-13$, much less than any of the systems in Figure 8. The Earth–Moon system experiences more similar tides with ${\mathrm{log}}_{10}\epsilon =-7.3$, and the dimensionless tidal force on HAT-P-11 is significantly less than the Jupiter-Io tides, ${\mathrm{log}}_{10}\epsilon =-6.6$. The tides raised by HAT-P-11 b are significant and may have non-negligible effects on the interior dynamics of the star.

Why is there no correlation between tidal force and chromospheric activity in this sample? We investigate the system with the greatest in more detail. That system is KOI-1050 with ${\mathrm{log}}_{10}\epsilon =-7.2$. Its star has mass ${M}_{\star }=0.83{M}_{\odot }$, effective temperature ${T}_{\mathrm{eff}}=5068$ K, and rotation period ${P}_{\mathrm{rot}}=29.9$ days. It is orbited by planet with radius ${R}_{p}=1.85{R}_{\oplus }$, orbital period 1.27 day, and forecasted mass ${M}_{p}=4.4{M}_{\oplus }$. ${R}_{{HK}}^{{\prime} }$ of KOI-1050 is less than half of HAT-P-11's despite its larger .

The dimensionless tidal force does not account for the transfer of energy due to obliquity tides—the result of additional torques in star–planet systems with planetary orbits that are highly misaligned with respect to the stellar spin. The amplitude of energy transfer due to obliquity tides is proportional to ${\sin }^{2}\psi$, where ψ is the angle between the orbital angular moment vector and the stellar spin axis (Wisdom 2004). The high obliquity of HAT-P-11 $\psi =106^\circ {}_{-11}^{+15}$ suggests that obliquity tides on HAT-P-11 will contribute additional dissipation of orbital energy, which is not reflected by . Lacking measurements of the obliquity of the other systems in Figure 8 such as KOI-1050, the observations presented here are insufficient to comment on whether or not the high obliquity of HAT-P-11 is responsible for its greater ${R}_{{HK}}^{{\prime} }$ activity index compared to KOI-1050, for example. Future measurements of spin–orbit misalignment of close-in planets with active host stars may allow us to link stellar activity to the tides raised by planets.

It is also possible that a weak correlation between and ${R}_{{HK}}^{{\prime} }$ is obscured by the large uncertainties in the forecasted planet masses. The parameter with the largest uncertainty in the calculation for is the planet mass. The forecasted mass estimates from Chen & Kipping (2017) have large uncertainties that reflect both measurement uncertainty and intrinsic spread in the observed planetary mass–radius relation. For the planets in the sample considered here, the median uncertainty in the forecasted planet mass is 78%, which contributes to uncertainty on the order of 1 dex in epsilon. Thus, the typical uncertainty in our estimates of is significant compared to the range observed. Follow-up radial velocity or transit timing variation measurements are needed to better constrain planet masses, and thus their tidal influence on their host stars. Alternative activity indicators have been used to search for evidence that tidal interactions stoke stellar activity (see reviews by Wright & Miller 2015; Poppenhaeger 2017). For example, Poppenhaeger & Wolk (2014) and Miller et al. (2015) found enhanced stellar coronal X-ray emission from host stars that are expected to have strong tidal interactions with their hot Jupiters. Saar & Cuntz (2001) measured the chromospheric emission in the Ca ii infrared-triplet over time for stars with hot Jupiters and found that the emission did not fluctuate on the orbital period of the planet. In addition to tidal interactions, close-in planets can also interact magnetically with their host stars. Cohen et al. (2010) and Lanza (2010) showed that star–planet magnetic interactions can affect the angular momentum evolution of host stars. These studies examined planets with orbit normals that were aligned or anti-aligned with the stellar spin. The magnetic interactions between the star and planet in the HAT-P-11 system—with its planet in a polar orbit—are worthy of further study and beyond the scope of this paper.