Physical Parameters of the Multiplanet Systems HD 106315 and GJ 9827^*†

GJ 9827 (K2-135) is a bright (V = 10.3 mag, K = 7.2 mag), nearby (distance = 30 pc) K6 dwarf star hosting three planets discovered in K2 Campaign 12 (Niraula et al. 2017; Rodriguez et al. 2018). Planets b and c orbit near a 3:1 resonance at 1.2 days and 3.6 days, with planet d at 6.2 days. These three planets span the gap seen in the radius distribution of small planets (Fulton et al. 2017) sized at 1.529 ± 0.058 R_⊕, 1.201 ± 0.046 R_⊕, and 1.955 ± 0.075 R_⊕ respectively. Niraula et al. (2017) additionally collected seven radial velocity observations with the FIbrefed Echelle Spectrograph (FIES; Frandsen & Lindberg 1999; Telting et al. 2014) to vet the system and to derive stellar parameters.

The mass of planet b was first determined with radial velocity observations from the Carnegie Planet Finder Spectrograph (PFS; Crane et al. 2006, 2008, 2010) on Magellan II by Teske et al. (2018; M_b  ∼ 8 M_⊕), who placed upper limits on planets c and d (M_c  < 2.5 M_⊕, M_d  < 5.6 M_⊕). Through additional measurements with the High Accuracy Radial velocity Planet Searcher (HARPS; Mayor et al. 2003) and the HARPS for the Northern hemisphere (HARPS-N), Prieto-Arranz et al. (2018) determined the masses of all three planets (M_b  = 3.74 ± 0.50 M_⊕, M_c  = 1.47 ± 0.59 M_⊕, and M_d  = 2.38 ± 0.71 M_⊕). The masses of planets b and d were further refined by Rice et al. (2019) with new HARPS-N radial velocity measurements and a Gaussian process informed by the K2 light curve (M_b  = 4.91 ± 0.49 M_⊕ and M_d  = 4.04 ± 0.84 M_⊕). Both Prieto-Arranz et al. (2018) and Rice et al. (2019) discuss how the inner planets have high densities and the outer planet has a lower density, suggesting that photoevaporation or migration could have played a role in the evolution of this system; we discuss this possibility further in Section 6.

HD 106315 (K2-109) is a bright (V = 8.97 mag, K = 7.85 mag) F5 dwarf star hosting two planets discovered in K2 Campaign 10 (Crossfield et al. 2017; Rodriguez et al. 2017). Planet b is a small (R_b  = 2.40 ± 0.20 R_⊕) planet with an orbital period of 9.55 days; planet c is a warm Neptune-sized (R_c  = 4.379 ± 0.086 R_⊕) planet with an orbital period of 21.06 days.

This system was further characterized with HARPS radial velocity observations by Barros et al. (2017) to determine the planets' masses (M_b  = 12.6 ± 3.2 M_⊕ and M_c  = 15.2 ± 3.7 M_⊕). They concluded that HD 106315 b likely has a rocky core and a decent water mass fraction whereas HD 106315 c has a substantial hydrogen-helium envelope.

Additional transits of HD 106315 c were observed with two ground-based facilities: the Euler telescope (Lendl et al. 2017) and the Cerro Tololo Inter-American Observatory (Barros et al. 2017). These measurements improved the precision on both the orbital period and the time of transit.

Later Zhou et al. (2018) investigated the system architecture through measuring the obliquity of HD 106315 c using Doppler tomography and constraining the mutual inclination of HD 106315 b through dynamical arguments. They found that these two planets both have low obliquities, consistent with the few other warm Neptunes with measured obliquities (e.g., Albrecht et al. 2013).

We collected radial velocity measurements of GJ 9827 and HD 106315 with the High Resolution Echelle Spectrometer (HIRES; Vogt et al. 1994) on the Keck I Telescope on Maunakea. These exposures were taken through an iodine cell for wavelength calibration (Butler et al. 1996). The HIRES data collection, reduction, and analysis followed the California Planet Search method described in Howard et al. (2010).

We obtained 92 measurements of GJ 9827 with HIRES between 2017 September 22 and 2020 January 8 (Table 3). These data were collected with the C2 decker (14'' × 0861, resolution = 50 k) with a typical signal-to-noise ratio (S/N) of 200/pixel (250k on the exposure meter, median exposure time of 18.5 minutes). We also collected a higher resolution template observation with the B3 decker (14'' × 0574, resolution = 67 k) on 2017 December 30 with an S/N of 200/pixel without the iodine cell. Both the C2 and B3 decker allow for sky subtraction, which is important for the quality of the radial velocities for a V = 10 mag star. We included an additional 142 measurements in our GJ 9827 analysis, for a total of 234 measurements: 7 from FIES (Niraula et al. 2017), 36 from PFS (Teske et al. 2018), 35 from HARPS (Prieto-Arranz et al. 2018), and 64 from HARPS-N (Prieto-Arranz et al. 2018; Rice et al. 2019).

We obtained 352 measurements of HD 106315 with HIRES between 2016 December 23 and 2020 February 1 (Table 4); 53 of these observations were previously published in Crossfield et al. (2017). These data were collected with the B5 decker (35 × 0861, resolution = 50 k) with a typical S/N of 200/pixel (250k on the exposure meter, median exposure time of 4.8 minutes). Data were typically taken in groups of three consecutive observations to mitigate p-mode oscillations; Barros et al. (2017) estimated p-mode periods of ∼20 minutes whereas Chaplin et al. (2019) estimate timescales to be ∼30 minutes. When possible, multiple visits separated by an hour were taken to improve precision due to the high $v\sin i;$ these data were then binned in nightly bins to average over short-timescale activity. We also collected a higher resolution template observation with the B3 decker on 2016 December 24. The template was a triple exposure with a total S/N of 346/pixel (250k each on the exposure meter) without the iodine cell.

We obtained 25 measurements of HD 106315 with PFS between 2017 January 6 and 2018 June 30 (Table 4). Data taken prior to 2018 February were taken with the 05 slit (resolution∼80k); a single observation with an exposure time of 10 to 25 minutes was taken per night. After a PFS upgrade in 2018 February, multiple exposures were taken with the 03 slit (resolution ∼130k). As with the HIRES data, we binned these consecutive observations for our analysis. An iodine-free template, consisting of three 1000 s exposures, was taken with the 03 slit on 2018 June 27. The PFS data were reduced using a custom IDL pipeline and velocities extracted based on the methodology described in Butler et al. (1996).

We additionally include 84 measurements from HARPS (Barros et al. 2017), for a total of 461 measurements (160 binned points) in our HD 106315 analysis. We collected 125 measurements on the Automated Planet Finder (APF; Radovan et al. 2014; Vogt et al. 2014) but do not include them in the analysis due to the high scatter (30 m s⁻¹ nightly rms, 7.3 m s⁻¹ radial velocity uncertainty), listed in Table 4.

We updated the stellar parameters for GJ 9827 and HD 106315 to incorporate the latest measurements, especially the Gaia DR2 parallaxes (Gaia Collaboration et al. 2016, 2018; Luri et al. 2018). We used multiband stellar photometry (Gaia G and Two Micron All-Sky Survey JHK), the Gaia parallax, and a stellar effective temperature and metallicity derived from Keck/HIRES spectra via the SpecMatch-Emp tool (Yee et al. 2017). The SpecMatch-Emp values are T_eff = 6318 ± 110 K and 4195 ± 70 K, and [Fe/H] = −0.21 ± 0.09 and −0.29 ± 0.09 for HD 106315 and GJ 9827, respectively. We input the above values into the isoclassify tool using the grid-mode option (Huber et al. 2017) to derive the stellar parameters listed in Table 5.

The discovery papers for HD 106315 included seeing-limited imaging data and K-band Keck/NIRC2 infrared adaptive optics imaging to rule out nearby stellar companions (Rodriguez et al. 2017; Crossfield et al. 2017). We include here additional high-contrast imaging data to improve the magnitude contrast constraints on nearby companions.

We observed HD 106315 on 2019 June 20 UT using the Zorro speckle interferometric instrument ³⁷ mounted on the 8 m Gemini South telescope located on the summit of Cerro Pachon in Chile. Zorro simultaneously observes in two bands, one centered at 832 nm with a width of 40 nm and the other centered at 562 nm with a width of 54 nm, obtaining diffraction-limited images with inner working angles 0017 and 0026, respectively. Our data set consisted of 3 minutes of total integration time taken as sets of 1000 × 0.06 sec images. All the images were combined and subjected to Fourier analysis leading to the production of final data products including speckle reconstructed imagery (see Howell et al. 2011). Figure 2 shows the 5σ contrast curves in both filters for the Zorro observation and includes an inset showing the 832 nm reconstructed image. The speckle imaging results confirm HD 106315 to be a single star to contrast limits of 5–8.6 magnitudes, ruling out main-sequence companions fainter than HD 106315 itself within the spatial limits of 2 to 125 au.

Zoom In Zoom Out Reset image size

The K2 light curve for GJ 9827 shows quasi-periodic variation with signs of active region evolution between rotation cycles (Figure 3). The K2 photometry shown in this paper was produced using k2phot (Petigura et al. 2015, 2017). A Lomb–Scargle periodogram of the K2 data shows two strong peaks around 15 and 30 days consistent with previous works, one peak is likely the rotation period and the other a harmonic. We consider both peaks since stellar rotation periods often do not appear as the highest peak in a periodogram (Nava et al. 2020). The shorter period is favored by Niraula et al. (2017) from the $v\sin i$ measurement, whereas the longer period is favored by Rodriguez et al. (2018), Teske et al. (2018), Prieto-Arranz et al. (2018), and Rice et al. (2019) from a combination of periodogram, autocorrelation, and Gaussian process analyses on the light curve as well as from the inferred age of GJ 9827.

Zoom In Zoom Out Reset image size

The Keck/HIRES S_HK and radial velocity data shown in Figure 3 both reveal a tenuous stellar rotation signal at 30 days, consistent with the longer peak in the K2 light curve periodogram. In agreement with previous findings, we conclude that this 30 day signal is likely caused by stellar rotation, as it is present in both the S_HK data and the photometry. Since there is power at the same period in our radial velocity data, we need to account for this signal in our radial velocity analysis in order to derive accurate mass measurements for the planets. We mitigated this signal using a Gaussian process, as described below in Section 5.1.

Similar to GJ 9827, we aim to understand the stellar activity component of the radial velocity data through investigating the possible relationships between the K2 light curve, the Calcium II H&K, and Hα stellar lines, and our radial velocity data. The projected rotational velocity measurement ($v\sin i=13.2\,\pm 1$ km s⁻¹) combined with the obliquity measurement (λ = −10.9 ± 3.7; Zhou et al. 2018) suggest a stellar rotation period of 4.78 ± 0.15 days.

HD 106315 was observed in K2 Campaign 10; this campaign had a 14 day data gap resulting in 49 days of contiguous data. With a 4.8 day rotation period, the shorter campaign should not impact our conclusions about stellar activity from this photometry. The K2 light curve (Figure 4) has low photometric variability; the periodogram shows a small peak near the stellar rotation period at 4.8 days and a larger peak at the second harmonic of the rotation period at 9.6 days.

Zoom In Zoom Out Reset image size

We next investigated the potential radial velocity signal from the stellar rotation by examining the S_HK and Hα data in the HIRES spectra (Table 4). We find no significant peaks near 4.8 days or elsewhere in the Lomb–Scargle periodograms of the HIRES activity indicators and radial velocity data (Figure 4). The absence of these signals suggests that the stellar rotation is not contributing a significant stellar activity signal to the radial velocity measurements, potentially attributed to the low spot coverage of this F star (<1%; Kreidberg et al. 2020).

Stellar photometry of both systems was collected from the Fairborn Observatory in Arizona to lengthen the photometry baseline from which to look for stellar variability.

Photometry of GJ 9827 was collected with the Tennessee State University Celestron C14 0.36 m Automated Imaging Telescope (AIT; Henry 1999; Eaton et al. 2003). A total of 74 observations were collected from 2018 September 22 to 2020 January 27 with the Cousins R filter (Table 6). The differential magnitudes were computed by subtracting the average brightness of seven comparison stars in the same field of view. A frequency spectrum of the observations show no significant periodicities between 1 and 100 days; the observations scatter about their mean with a standard deviation of 0.00372 mag.

Photometry of HD 106315 was collected with the T12 0.80 m Automatic Photoelectric Telescope (APT); the T12 APT is essentially identical in construction and operation to the T8 0.8 m APT described in Henry (1999). A total of 43 observations of HD 106315 were collected between 2018 February 9 and 2018 June 7 in both the Stromgren b and y filters by T12's two-channel photometer (Table 7). The two filters were averaged together into the (b + y)/2 filter to increase the data precision. The differential magnitudes were calculated using three comparison stars: HD 105374, HD 105589, and HD 106965. A frequency spectrum of the observations show no significant periodicities between 1 and 100 days; the observations scatter about their mean with a standard deviation of 0.00256 mag.

There is evidence of stellar activity in our radial velocity data from the periodogram analysis in Section 4. We include a Gaussian process with a quasi-periodic kernel to model this activity signal in our radial velocity fit. The kernel has the form

$$\begin{eqnarray}&&k(t,t^{\prime} )={\eta }_{1}^{2}\ \exp \left[-\displaystyle \frac{{\left(t-t^{\prime} \right)}^{2}}{{\eta }_{2}^{2}}-\tfrac{{\sin }^{2}\left(\tfrac{\pi \left(t-t^{\prime} \right)}{{\eta }_{3}}\right)}{2{\eta }_{4}^{2}}\right],\end{eqnarray} \tag{ 1 }$$

where the hyperparameter η₁ is the amplitude of the covariance function, η₂ is the active region evolutionary timescale, η₃ is the period of the correlated signal, and η₄ is the length scale of the periodic component. We explore these hyperparameters for this system by performing a maximum-likelihood fit to the K2 light curve, S_HK, and Hα data with the quasi-periodic kernel (Equation (1)), then determine the errors through an MCMC analysis.

The K2 light curve fit is well constrained by the Gaussian process and produces a stellar rotation period consistent with the periodogram analysis of this data (η₃ = ${28.62}_{-0.38}^{+0.48}$). The Hα data has very low variation; it is not well fit by this kernel and does not produce meaningful posteriors.

The S_HK data is well fit by this quasi-periodic kernel and produces a stellar rotation period (η₃) consistent with our periodogram analysis in Section 4. The photometry and the S_HK data both produce consistent posteriors; we choose to adopt the posteriors from the S_HK fit because these data are taken simultaneously with the radial velocity data and are therefore a direct indicator of the chromospheric magnetic activity. The posteriors on the parameters from our S_HK fit are: ${\gamma }_{{S}_{\mathrm{HK}}}$ = ${0.646}_{-0.026}^{+0.027}$, ${\sigma }_{{S}_{\mathrm{HK}}}$ = ${0.0183}_{-0.0032}^{+0.0035}$, η₁ = ${0.079}_{-0.012}^{+0.017}$, η₂ = ${94}_{-25}^{+50}$ days, η₃ = ${29.86}_{-0.83}^{+0.78}$ days, and η₄ = ${0.587}_{-0.096}^{+0.14}$.

We then performed a Gaussian process fit on the radial velocity data including priors on η₂, η₃, and η₄ equivalent to the S_HK fit posteriors. We tested fits including a trend, curvature, and planet eccentricities but reject all of these models due to their higher AIC values. These tested fits resulted in semi-amplitudes for all three planets consistent to 1σ for planets b and d, and 2σ for planet c with the circular three-planet Gaussian process fit.

We present our GJ 9827 results in Table 8. We list the results from a circular three-planet case with and without a Gaussian process for comparison, and adopt the fit including the Gaussian process shown in Figure 5. We measure masses for these planets to be M_b  = 4.87 ± 0.37 M_⊕, M_c  = 1.92 ± 0.49 M_⊕, and M_d  = 3.42 ± 0.62 M_⊕.

Zoom In Zoom Out Reset image size

Table 8. GJ 9827 Radial Velocity Fit Parameters

Parameter	Name (Units)	Keplerian Fit	Gaussian Process Fit (Adopted)
Orbital Parameters
Pb	Period (days)	${1.2089765}_{-2.3e-06}^{+2.2e-06}$	1.2089765 ± 2.3e − 06
Tconjb	Time of Conjunction (BJD)	2457738.82586 ± 0.00026	2457738.82586 ± 0.00026
Rb	Radius (R⊕)	≡1.529 ± 0.058	≡1.529 ± 0.058
eb	Eccentricity	≡0.0	≡0.0
ωb	Argument of Periapse	≡0.0	≡0.0
Kb	Semi-amplitude (m s−1)	3.5 ± 0.32	4.1 ± 0.3
ab	Semimajor Axis (AU)	0.01866 ± 0.00019	0.01866 ± 0.00019
Mb	Mass (M⊕ )	${4.12}_{-0.38}^{+0.39}$	4.87 ± 0.37
ρb	Density (g cm−3)	${6.32}_{-0.87}^{+1.0}$	${7.47}_{-0.95}^{+1.1}$
Pc	Period (days)	${3.648095}_{-2.4e-05}^{+2.5e-05}$	3.648095 ± 2.4e − 05
Tconjc	Time of Conjunction (BJD)	2457742.19927 ± 0.00071	${2457742.19929}_{-0.00071}^{+0.00072}$
Rc	Radius (R⊕)	≡1.201 ± 0.046	≡1.201 ± 0.046
ec	Eccentricity	≡0.0	≡0.0
ωc	Argument of Periapse	≡0.0	≡0.0
Kc	Semi-amplitude (m s−1)	1.28 ± 0.32	1.13 ± 0.29
ac	Semimajor Axis (AU)	${0.03896}_{-0.0004}^{+0.00039}$	${0.03896}_{-0.0004}^{+0.00039}$
Mc	Mass (M⊕ )	${2.17}_{-0.55}^{+0.54}$	1.92 ± 0.49
ρc	Density (g cm−3)	${6.9}_{-1.8}^{+2.0}$	${6.1}_{-1.6}^{+1.8}$
Pd	Period (days)	6.20183 ± 1e − 05	6.20183 ± 1e − 05
Tconjd	Time of Conjunction (BJD)	2457740.96114 ± 0.00044	${2457740.96114}_{-0.00044}^{+0.00045}$
Rd	Radius (R⊕)	≡1.955 ± 0.075	≡1.955 ± 0.075
ed	Eccentricity	≡0.0	≡0.0
ωd	Argument of Periapse	≡0.0	≡0.0
Kd	Semi-amplitude (m s−1)	1.63 ± 0.31	1.7 ± 0.3
ad	Semimajor Axis (AU)	${0.0555}_{-0.00057}^{+0.00056}$	${0.0555}_{-0.00057}^{+0.00055}$
Md	Mass (M⊕ )	3.29 ± 0.64	3.42 ± 0.62
ρd	Density (g cm−3)	${2.41}_{-0.52}^{+0.58}$	${2.51}_{-0.51}^{+0.57}$
Instrument Parameters
${\gamma }_{\mathrm{HIRES}}$	Mean Center-of-mass (m s−1)	$-{1.87}_{-0.39}^{+0.38}$	$-{2.4}_{-1.4}^{+1.3}$
γHARPS	Mean Center-of-mass (m s−1)	31946.64 ± 0.37	${31947.7}_{-3.6}^{+4.0}$
γHARPS−N	Mean Center-of-mass (m s−1)	${31948.64}_{-0.42}^{+0.43}$	${31950.2}_{-2.6}^{+2.7}$
γPFS	Mean Center-of-mass (m s−1)	0.28 ± 0.86	0.6 ± 1.2
γFIES	Mean Center-of-mass (m s−1)	${31775.5}_{-1.2}^{+1.1}$	31775.6 ± 1.5
${\sigma }_{\mathrm{HIRES}}$	Jitter (m s−1 )	${3.45}_{-0.27}^{+0.32}$	${2.15}_{-0.43}^{+0.49}$
σHARPS	Jitter (m s−1 )	${1.65}_{-0.35}^{+0.39}$	${0.91}_{-0.45}^{+0.44}$
σHARPS−N	Jitter (m s−1 )	${2.79}_{-0.35}^{+0.39}$	${0.74}_{-0.45}^{+0.44}$
σPFS	Jitter (m s−1 )	${4.68}_{-0.62}^{+0.75}$	4.0 ± 1.1
σFIES	Jitter (m s−1 )	${0.0001}_{-0.0001}^{+0.0016}$	${0.035}_{-0.035}^{+2.6}$
GP Parameters
${\eta }_{1,\mathrm{HIRES}}$	GP Amplitude (m s−1 )	N/A	${3.7}_{-1.0}^{+1.2}$
η1,HARPS	GP Amplitude (m s−1 )	N/A	${5.3}_{-2.2}^{+3.5}$
η1,HARPS−N	GP Amplitude (m s−1 )	N/A	${5.1}_{-1.5}^{+2.3}$
η1,PFS	GP Amplitude (m s−1 )	N/A	4.0 ± 1.1
η1,FIES	GP Amplitude (m s−1 )	N/A	${0.035}_{-0.035}^{+2.6}$
η2	Evolutionary Timescale (days)	N/A	${82}_{-14}^{+17}$
η3	Period of the Correlated Signal (days)	N/A	${28.62}_{-0.38}^{+0.48}$
η4	Length Scale	N/A	${0.418}_{-0.065}^{+0.082}$

Note. Derived parameters use M_* = 0.593 ± 0.018, R_* = 0.579 ± 0.019 (this work), R_b /R_* = 0.02420 ± 0.00044, R_c /R_* = 0.01899 ± 0.00036, and R_d /R_* = 0.03093 ± 0.00062 (Rodriguez et al. 2018).

Download table as:  ASCIITypeset image

For HD 106315, the circular two-planet fit is favored by the AIC over fits with a trend, curvature, or planet eccentricities; results are listed in Table 9 and the fit is displayed in Figure 6. In agreement with Barros et al. (2017), we do not see evidence of the trend suggested in Crossfield et al. (2017) with an AIC value 1.25 larger than the circular case. We determine masses for the HD 106315 system to be M_b  = 10.5 ± 3.1 M_⊕ and M_c  = 12.0 ± 3.8 M_⊕.

Zoom In Zoom Out Reset image size

Table 9. HD 106315 Radial Velocity Fit Parameters

Parameter	Name (Units)	Keplerian Fit (Adopted)	Gaussian Process Fit
Orbital Parameters
Pb	Period (days)	9.55288 ± 0.00021	${9.55288}_{-0.00021}^{+0.00019}$
Tconjb	Time of Conjunction (BJD)	${2457586.5476}_{-0.0025}^{+0.0024}$	${2457586.5479}_{-0.0026}^{+0.003}$
Rb	Radius (R⊕)	≡2.40 ± 0.20	≡2.40 ± 0.20
eb	Eccentricity	≡0.0	≡0.0
ωb	Argument of Periapse	≡0.0	≡0.0
Kb	Semi-amplitude (m s−1)	${2.88}_{-0.84}^{+0.85}$	${2.91}_{-0.85}^{+0.79}$
ab	Semimajor Axis (AU)	${0.0924}_{-0.0012}^{+0.0011}$	${0.0924}_{-0.0012}^{+0.0011}$
Mb	Mass (M⊕ )	10.5 ± 3.1	${10.6}_{-3.1}^{+2.9}$
ρb	Density (g cm−3)	${4.1}_{-1.4}^{+1.9}$	${4.1}_{-1.4}^{+1.8}$
Pc	Period (days)	21.05652 ± 0.00012	21.05653 ± 0.00012
Tconjc	Time of Conjunction (BJD)	${2457569.01767}_{-0.00096}^{+0.00097}$	${2457569.0178}_{-0.001}^{+0.0012}$
Rc	Radius (R⊕)	≡4.379 ± 0.086	≡4.379 ± 0.086
ec	Eccentricity	≡0.0	≡0.0
ωc	Argument of Periapse	≡0.0	≡0.0
Kc	Semi-amplitude (m s−1)	2.53 ± 0.79	${2.61}_{-0.87}^{+0.74}$
ac	Semimajor Axis (AU)	${0.1565}_{-0.002}^{+0.0019}$	${0.1565}_{-0.002}^{+0.0019}$
Mc	Mass (M⊕ )	12.0 ± 3.8	${12.4}_{-4.2}^{+3.5}$
ρc	Density (g cm−3)	${0.78}_{-0.25}^{+0.26}$	${0.81}_{-0.27}^{+0.24}$
Instrument Parameters
${\gamma }_{\mathrm{HIRES}}$	Mean Center-of-mass (m s−1)	$-{2.48}_{-0.97}^{+0.96}$	$-{2.7}_{-1.1}^{+1.0}$
γHARPS	Mean Center-of-mass (m s−1)	$-{3462.94}_{-0.71}^{+0.7}$	$-{3462.77}_{-0.87}^{+1.1}$
γPFS	Mean Center-of-mass (m s−1)	$-{2.9}_{-2.7}^{+2.8}$	$-{2.5}_{-3.3}^{+3.2}$
${\sigma }_{\mathrm{HIRES}}$	Jitter (m s−1 )	${8.33}_{-0.79}^{+0.85}$	${6.4}_{-1.1}^{+1.2}$
σHARPS	Jitter (m s−1 )	${2.94}_{-1.0}^{+0.94}$	${2.3}_{-1.4}^{+1.0}$
σPFS	Jitter (m s−1 )	${9.4}_{-2.3}^{+2.6}$	${4.0}_{-2.7}^{+4.6}$
GP Parameters
${\eta }_{1,\mathrm{HIRES}}$	GP Amplitude (m s−1 )	N/A	${5.2}_{-1.7}^{+1.1}$
η1,HARPS	GP Amplitude (m s−1 )	N/A	${2.3}_{-1.4}^{+1.0}$
η1,PFS	GP Amplitude (m s−1 )	N/A	${4.0}_{-2.7}^{+4.6}$
η2	Evolutionary Timescale (days)	N/A	${5.27}_{-0.65}^{+0.54}$
η3	Period of the Correlated Signal (days)	N/A	${4.5}_{-0.65}^{+0.49}$
η4	Length Scale	N/A	${0.56}_{-0.04}^{+0.036}$

Note. Derived parameters use M_* = 1.154 ± 0.043, R_* = 1.269 ± 0.024 (this work), R_b /R_* = 0.01708 ± 0.00135 (Crossfield et al. 2017), and R_c /R_* = 0.031636 ± 0.0001834 (Kreidberg et al. 2020).

Download table as:  ASCIITypeset image

In contrast with our GJ 9827 analysis, we choose not to include a Gaussian process in our HD 106315 fit as we do not see evidence for stellar rotation induced activity contamination in the activity indicators or radial velocity data. We suspect the low spot coverage of HD 106315(<1%; Kreidberg et al. 2020) is why we see a small rotation signal in the photometry and a lack of this signal in our radial velocity data.

Barros et al. (2017) do use a Gaussian process for their analysis of HD 106315. The derived Gaussian process period is 2.8 days and their FWHM measurements also show a similar periodicity leading them to believe that this signal arises from stellar activity. At the time, Zhou et al. (2018) had not yet measured the obliquity, therefore, Barros et al. (2017) hypothesized that this 2.8 day signal was the stellar rotation period or half of the rotation period.

If this signal is associated with stellar activity, it is possible that their high-cadence radial velocity run is more sensitive to this activity than our data collection spanning multiple years. The HARPS measurements were collected on 47 nights over three months, whereas we have 94 nights of HIRES measurements over three years. It is also possible that the Gaussian process used by Barros et al. (2017) had fit spurious noise instead of a stellar activity signal; the 2.8 day signal is too short to be the rotation period or half of the rotation period. Hotter stars (T_eff > 6200 K) often have shallow convective envelopes and inefficient magnetic dynamos which result in fewer spots on the stellar surface (Kraft 1967). Therefore, hotter stars like HD 106315 may not have enough starspots for this type of Gaussian process to be effective.

For completeness, we perform a Gaussian process fit on the HD 106315 radial velocity data. We first fit the K2 data using a Gaussian process as this data set showed periodicity at the stellar rotation period; the posteriors of this fit are: γ_K2 = ${3633710}_{-200}^{+190}$ e⁻ s⁻¹, σ_K2 = ${117}_{-15}^{+16}$ e⁻ s⁻¹, η₁ = ${655}_{-68}^{+84}$ e⁻ s⁻¹, η₂ = ${5.17}_{-0.64}^{+0.66}$ days, η₃ = ${4.49}_{-0.26}^{+0.61}$ days, and η₄ = ${0.55}_{-0.044}^{+0.04}$. We then performed a Gaussian process fit on the radial velocity data including priors on η₂, η₃, and η₄ from the K2 fit posteriors. This fit results in semi-amplitudes consistent to 1σ for both planets; the full results are shown in Table 9. The Gaussian process fit has a higher AIC value (ΔAIC = 7.38) suggesting that Gaussian process parameters do not significantly improve the fit. For this reason, and as we do not see signs of stellar activity in our activity indicators or radial velocity data, we adopt the fit without a Gaussian process.

We explored the range of planet eccentricities consistent with system stability through N-body simulations as including eccentricity was not warranted in our radial velocity fits for either system. The literature on GJ 9827 assumed circular orbits for their fits (Prieto-Arranz et al. 2018; Rice et al. 2019). For HD 106315, Barros et al. (2017) include eccentricity terms in their radial velocity analysis resulting in e_b  = 0.1 ± 0.1 and e_c  = 0.22 ± 0.15, although they do not discuss if including the eccentricity terms improve the fit. Our eccentric radial velocity fit for HD 106315 resulted in e_b  = 0.18 ± 0.17 and e_c  = 0.21 ± 0.24, consistent with Barros et al. (2017). Though our eccentric fit had a higher AIC than the circular fit (ΔAIC = 6.22) suggesting that including eccentricity did not sufficiently improve the fit to justify the additional parameters.

We evaluated the stability of both systems using spock (Tamayo et al. 2020). spock predicts whether a given orbital configuration is stable by using rebound (Rein & Liu 2011) to simulate the first 10⁴ orbits of a system and then calculating the probability that this system is stable for a full 10⁹ orbits by comparing it to a wide sample of full simulations. These full simulations include mean-motion resonance and mutually inclined and eccentric systems; the parameters are drawn from those typically encountered in current multiplanet systems.

We initialized both systems at the maximum-likelihood values for the planet masses, orbital periods, times of conjunction, and stellar masses derived in this paper. We then varied e and ω for all planets to explore the stability of the system. For HD 106315, we varied e₁ and e₂ from 0.0 to 0.9 in steps of 0.1. At each eccentricity pair, we performed a grid of simulations varying ω₁ and ω₂ from zero to 2π in steps of $\tfrac{\pi }{5}$, resulting in 10,000 simulations. We then averaged over the simulated ω grid to calculate the average probability that a given eccentricity pair is stable (Figure 7).

Zoom In Zoom Out Reset image size

HD 106315 b and HD 106315 c are in relatively close orbits at periods of 9.55 and 21.06 days; their orbits are unstable if either planet has a large eccentricity. The system has a probability of stability greater than 50% when e₁ ≤  0.4 and e₂ ≤  0.3; the highest probability of stability is when both planets are in circular orbits.

GJ 9827 b, GJ 9827 c, and GJ 9827 d are in even more closely packed orbits at orbital periods of 1.2, 3.6, and 6.2 days. Therefore, for GJ 9827, we varied e₁, e₂, and e₃ from 0.0 to 0.4 in steps of 0.1 as larger eccentricities for any of the three planets resulted in unstable orbits. At each eccentricity triplet we perform a grid of simulations varying ω₁, ω₂, and ω₃ from zero to 2π in steps of $\tfrac{\pi }{5}$, creating a total of 125,000 simulations. We then averaged over the ω grid to calculate the average probability that a given eccentricity triplet is stable (Figure 8).

Zoom In Zoom Out Reset image size

We find that the GJ 9827 system is unstable if e₃ ≥  0.3 and the system has very low stability at e₃ = 0.2. For e₃ ≤  0.1, the system can be stable with e₂ ≤  0.2 and e₁ ≤  0.4. This system has a smaller range of stable eccentricity values since the planets are packed closer together.

We then convert these eccentricity constraints to secondary eclipse timing constraints (Winn 2010, Equation (33)). We note that the planets in these two systems are not particularly favorable targets for thermal emission spectroscopy based on the emission spectroscopy metric (ESM; Kempton et al. 2018; HD 106315 b: 3; HD 106315 c: 6; GJ 9827 b: 14; GJ 9827 c: 4; and GJ 9827 d: 6).

From our eccentricity constraints and assuming ω = 0, the maximum offsets of the secondary eclipse time for HD 106315 b and HD 106315 c are 2.4 days and 4.0 days respectively. The maximum secondary eclipse timing offsets for the GJ 9827 system are 0.31 days, 0.46 days, and 0.39 days for planet b, c, and d respectively.