Precise Masses and Orbits for Nine Radial-velocity Exoplanets

Stellar magnetic activity is a term that encompasses a range of phenomena observed of a star, including flares, star spots, and any chromospheric and coronal activity. Mid-F type and cooler main-sequence stars experience a decline in magnetic activity levels and rotational velocity as they age. The well-accepted explanation for the weaker activity and the slower rotation rates of old stars is the dynamo theory developed by Schatzman (1962, 1990) and Parker (1955, 1979): late-type stars develop subsurface convection zones that support a magnetic dynamo. The dynamo provides the energy to heat the stellar chromosphere and corona (Kraft 1967; Hall 2008; Testa et al. 2015, and references therein). The chromospheres eject winds in the form of jets or flares. These rotate with the stellar angular velocity out to the Alfvén radius, carrying away angular momentum which, in turn, causes the star to spin down over time. This effect is known as magnetic braking through the magnetic dynamo process (Noyes et al. 1984). The declining rotational velocity due to magnetic braking results in a reduction in the magnetic field generated by the stellar dynamo which, in turn, diminishes the star's Ca ii HK and X-ray emissions. Ca ii HK and X-ray emission weaken fastest at young ages, thus age dating for young stars is easier than for their old field star counterparts (Soderblom 2010).

Ca ii HK emission, via the chromospheric activity index ${R}_{\mathrm{HK}}^{{\prime} }$, has been widely used as an age indicator for solar-type (G and K) stars. Age dating of Sun-like stars with ${R}_{\mathrm{HK}}^{{\prime} }$ was first done by Wilson (1968) at the Mount Wilson Observatory. The age-activity relationship has since been developed and calibrated across spectral types later than mid-F (Skumanich 1972; Noyes et al. 1984; Soderblom et al. 1991; Barnes 2007; Mamajek & Hillenbrand 2008; Angus et al. 2015; David & Hillenbrand 2015; Lorenzo-Oliveira et al. 2016; van Saders et al. 2016). Conventionally, ${R}_{\mathrm{HK}}^{{\prime} }$ is calculated from the Ca ii S-index, a measurement of the strength of the emission. Mamajek & Hillenbrand (2008) calibrated the activity relations to an activity level of $\mathrm{log}{R}_{\mathrm{HK}}^{{\prime} }\geqslant -5.0$ dex or a Rossby number (ratio of rotation period to convective overturn time) of 2.2, roughly corresponding to the solar activity and rotation.

X-ray activity traces magnetic heating of the stellar corona, providing another indirect probe of rotation (Golub 1996; Mewe 1996; Jardine et al. 2002; Güdel 2004; Zhuleku et al. 2020). Brandt et al. (2014) combine ${R}_{\mathrm{HK}}^{{\prime} }$ and X-ray activity to infer a Rossby number and, from this, an age via gyrochronology using the Mamajek & Hillenbrand (2008) calibration. Brandt et al. (2014) further allows for a distribution of times, dependent on spectral type, on the C-sequence where the dynamo is saturated and magnetic braking is inefficient (Barnes 2003). Their model sets a floor for the Rossby number at >2.2 and inflates the error in the activity and rotation by 0.06 dex to account for systematic uncertainties.

A photometric rotation period provides a more direct measure of the Rossby number and a tighter constraint on the stellar age. For stars that have directly measured rotation periods, Brandt et al. (2014) combine this measurement with the indirect probes of coronal and chromospheric emission to derive an age posterior. We refer the reader to that paper for a more detailed discussion.

We adopt the Bayesian inference model described in Brandt et al. (2014) to estimate stellar ages. We take the Ca ii HK S-indices from the catalog of Pace (2013) and references therein and convert them to Mt. Wilson ${R}_{\mathrm{HK}}^{{\prime} }$ using the relations described in Noyes et al. (1984). We extract X-ray activities R_X from the ROSAT all-sky survey bright and faint source catalogs (Voges et al. 1999, 2000). For stars that are not in either ROSAT catalog, we take the nearest detection in the faint source catalog and use five times its uncertainty as our upper limit on X-ray flux.

Here we describe the method we use to estimate the bolometric luminosity for our sample of host stars. We adopt the absolutely calibrated effective temperature scales in Casagrande et al. (2010) for FGK stars across a wide range of metallicities. These authors use the InfraRed Flux Method (IRFM) to derive model-independent effective temperatures T_eff in stars with different spectral types and metallicities. Homogeneous, all-sky Tycho-2 (B_T , V_T ) optical photometry (Høg et al. 2000) was combined with 2MASS infrared photometry in the JHK_s bands (Skrutskie et al. 2006) to calibrate reliable T_eff for a sample of solar analogs via IRFM.

All of our stars are late-G and K Sun-like dwarfs with optical and infrared photometry from Tycho-2 (BV)_T and 2MASS JHK_s , thus we use Table 5 of Casagrande et al. (2010) to derive their bolometric integrated fluxes. In Table 5, the set of Tycho-2 and 2MASS flux calibrations apply to a [Fe/H] range between −2.7 and 0.4, and a color range between 0.19 and 3.29. The standard deviation ${\sigma }_{{\phi }_{\xi }}$ in the Tycho-2 and 2MASS flux calibrations is the smallest for the m_ξ = V_T flux calibrations with color indices V_T − J (0.7%) and V_T − K_s (0.9%). We use the V_T − J colors to calibrate V_T magnitudes for our sample of stars with the exception of HD 221420, which has a large uncertainty in its 2MASS J-band magnitude due to saturation (4.997 ± 0.252 mag). Therefore, for HD 221420, we use the V_T − K_s index instead to calculate the flux calibration in its V_T .

The final bolometric magnitudes can be estimated in terms of the V_T magnitude, the distance d, and the the integrated bolometric fluxes ${{ \mathcal F }}_{\mathrm{Bol}}(\mathrm{Earth})$:

$$\begin{eqnarray}&&{L}_{\mathrm{Bol}}=4\pi {d}^{2}{10}^{-0.4{V}_{T}}{{ \mathcal F }}_{\mathrm{Bol}}(\mathrm{Earth}).\end{eqnarray} \tag{ 1 }$$

We express L_Bol in units of solar luminosity, adopting a value of L_☉ = 3.828 × 10³³ erg s⁻¹. Our parallaxes all have fractional uncertainties ∼10⁻³; they contribute negligibly to our luminosity uncertainties.

We propagate the errors using standard error propagation techniques. We estimate the final uncertainty in the luminosities as

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\sigma }^{2}({L}_{\mathrm{Bol}})}{{L}_{\mathrm{Bol}}^{2}} & = & {\left(\displaystyle \frac{\sigma ({{ \mathcal F }}_{\mathrm{Bol}})}{{{ \mathcal F }}_{\mathrm{Bol}}}\right)}^{2}+{\left(\displaystyle \frac{2\sigma (\varpi )}{\varpi }\right)}^{2}\\ & & +{\left(0.4\mathrm{ln}(10)\sigma ({V}_{T})\right)}^{2}+{\left({\sigma }_{{\phi }_{\xi }}\right)}^{2}\end{array}\end{eqnarray} \tag{ 2 }$$

where the uncertainty in the flux calibration is ${\sigma }_{{\phi }_{\xi }}$ is 0.9% for HD 221420 and 0.7% for the rest of the sample. We summarize in Table 1 the Tycho-2 (BV)_T and the 2MASS JHK_s photometry and our derived luminosity values for the host stars that we study.

Table 1. Adopted Stellar Parameters for Stars Investigated in This Work ^a

Star (HD #)	29021	81040	87883	98649	106252	106515A	106515B b	171238	196067	196068 b	221420
HIP ID	21571	46076	49699	55409	59610	59743A	59743B	91085	102125	102128	116250
$\bar{\omega }$ (mas)	32.385	29.063	54.668	23.721	26.248	29.315	29.391	22.481	25.033	25.038	32.102
$\sigma [\bar{\omega }]$	0.024	0.041	0.030	0.022	0.027	0.030	0.029	0.032	0.021	0.017	0.033
VT (mag)	7.842	7.791	7.660	8.066	7.479	8.063	8.298	8.737	6.510	7.070	5.884
σ[VT ]	0.011	0.012	0.012	0.011	0.011	0.014	0.015	0.020	0.010	0.010	0.009
BT (mag)	8.612	8.527	8.800	8.793	8.158	8.994	9.284	9.589	7.193	7.790	6.643
σ[BT ]	0.016	0.017	0.018	0.017	0.017	0.021	0.022	0.029	0.015	0.015	0.014
Ks (mag)	6.082	6.159	5.314	6.419	5.929	6.151	6.267	6.831	5.080	5.713	4.306
σ[Ks ]	0.020	0.021	0.020	0.020	0.026	0.026	0.017	0.017	0.016	0.018	0.036
J (mag)	6.518	6.505	5.839	6.811	6.302	6.585	6.746	7.244	5.417	6.061	4.997
σ[J]	0.029	0.019	0.020	0.019	0.024	0.024	0.030	0.019	0.021	0.027	0.252
RX	<−5.14	−5.27	<−5.25	<−4.87	<−5.20	<−4.84	<−4.77	<−4.83	<−5.08	<−4.87	<−5.55
${R}_{\mathrm{HK}}^{{\prime} }$	...	−4.71	−4.98	−4.96	−4.85	−5.10	−5.15	−4.96	−5.06	−5.02	−5.05
Prot (days)	...	15.98	...	26.441	...	...	...	...	...	...	...
[Fe/H](dex) c	−0.30 ± 0.10	−0.04 ± 0.05	0.11 ± 0.10	−0.06 ± 0.04	−0.06 ± 0.04	0.03 ± 0.10	0.03 ± 0.10	0.20 ± 0.10	0.23 ± 0.07	0.24 ± 0.10	0.34 ± 0.07
M*,model(M⊙) d	0.86 ± 0.02	0.99 ± 0.02	0.80 ± 0.02	0.97 ± 0.02	1.05 ± 0.02	0.89 ± 0.03	0.86 ± 0.03	0.92 ± 0.03	1.26 ± 0.07	1.12 ± 0.06	1.28 ± 0.08
L*(L⊙) e	0.659 ± 0.017	0.838 ± 0.018	0.338 ± 0.008	0.968 ± 0.019	1.328 ± 0.030	0.686 ± 0.018	0.567 ± 0.017	0.627 ± 0.018	3.435 ± 0.068	2.007 ± 0.045	3.850 ± 0.072
Spectral Type	G5	G2/G3	K0	G3/G5 V	G0	G5 V	G8 V	G8 V	G5 III	G5 V	G2 V
Age (Gyr) f	${5.5}_{-1.0}^{+1.2}$	${1.79}_{-0.26}^{+0.30}$	${7.6}_{-1.8}^{+2.8}$	${4.44}_{-0.58}^{+0.68}$	${3.00}_{-0.60}^{+0.80}$	${8.2}_{-2.1}^{+2.7}$	${8.4}_{-2.1}^{+2.6}$	${7.0}_{-1.6}^{+6.0}$	${5.8}_{-2.2}^{+4.3}$	${6.2}_{-2.2}^{+4.1}$	${6.6}_{-2.0}^{+3.7}$

Notes.

^a References. $\bar{\omega }$ from Gaia eDR3 (Gaia Collaboration et al. 2021);V_T and B_T from Tycho-2 (Høg et al. 2000), K_s from 2MASS (Cutri et al. 2003);R_X from ROSAT (Voges et al. 1999); rotation period for HD 81040 from Reinhold & Hekker (2020), alternatively 15.2579 days from Oelkers et al. (2018);${R}_{\mathrm{HK}}^{{\prime} }$ from (Pace 2013); rotation period for HD 98649 from Oelkers et al. (2018); spectral types from Wright et al. (2003) and Skiff (2014). ^b Secondary stellar companion. ^c We estimate [Fe/H] from the median of all entries in Soubiran et al. (2016) with a default uncertainty of 0.1 dex. The references for each star are: Kim et al. (2016) for HD 29021; Sousa et al. (2006), Gonzalez et al. (2010), Kang et al. (2011), Luck (2017), Rich et al. (2017), Sousa et al. (2018) for HD 81040;Valenti & Fischer (2005), Kotoneva et al. (2006), Mishenina et al. (2008), Maldonado et al. (2012), Brewer et al. (2016), Luck (2017) for HD 87883; Datson et al. (2012), Santos et al. (2013, 2003, 2004, 2005), Porto de Mello et al. (2014), Ramírez et al. (2014), Rich et al. (2017) for HD 98649; Sadakane et al. (2002), Heiter & Luck (2003), Laws et al. (2003), Huang et al. (2005), Valenti & Fischer (2005), Luck & Heiter (2006), Takeda et al. (2007), Fuhrmann (2008), Gonzalez et al. (2010), Kang et al. (2011), Datson et al. (2015), Spina et al. (2016), Luck (2017), Sousa et al. (2018) for HD 106252; Santos et al. (2013) for HD 106515A and HD 106515B; Brewer et al. (2016) for HD 171238; Valenti & Fischer (2005), Bond et al. (2006), Santos et al. (2013) for HD 196067 and HD 196068; and Valenti & Fischer (2005), Bond et al. (2006), Sousa et al. (2006, 2008), Tsantaki et al. (2013), Jofré et al. (2015), Maldonado & Villaver (2016), Soto & Jenkins (2018) for HD 221420. ^d The stellar masses are estimated from PARSEC isochrones (see text). ^e The stellar luminosities are computed based on the absolute calibration of effective temperature scales performed by Casagrande et al. (2010; see text). ^f The stellar ages and their 1σ error bars are approximated using the Bayesian technique of Brandt et al. (2014).

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We infer masses for all of our stars using the PARSEC isochrones (Bressan et al. 2012) by matching the observed luminosity at the spectroscopic metallicity and activity-based age.

We take spectroscopic metallicities from the PASTEL catalog (Soubiran et al. 2016, and references therein). The PASTEL catalog is a large compilation of high-dispersion spectroscopic measurements from the literature. For the metallicities, we take the median of all entries in the PASTEL catalog. We estimate the uncertainty in the metallicity as the standard deviation of the measurements in the PASTEL catalog multiplied by $\sqrt{n/(n-1)}$ to correct for the bias in the estimator. Three targets—HD 171238, HD 29021, and HD 106515 A—have only single entries in the PASTEL catalog. For these stars, we use the single measurement and assume a conservative uncertainty of 0.1 dex.

Once we have a probability distribution for metallicity, we take a fine grid of stellar models and, for each, compute a likelihood using

$$\begin{eqnarray}\begin{array}{rcl}{ \mathcal L } & = & p({\tau }_{\mathrm{model}})\times \exp \left[-\displaystyle \frac{{\left({L}_{\star }-{L}_{\mathrm{model}}\right)}^{2}}{2{\sigma }_{L}^{2}}\right.\\ & & \left.-\displaystyle \frac{{\left({[\mathrm{Fe}/{\rm{H}}]}_{\star }-{[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{model}}\right)}^{2}}{2{\sigma }_{[\mathrm{Fe}/{\rm{H}}]}^{2}}\right].\end{array}\end{eqnarray} \tag{ 3 }$$

In Equation (3), p(τ_model) is our activity-based age posterior at the model age. We then marginalize over metallicity and age to derive a likelihood as a function of stellar mass. We report the mean and standard deviation of this likelihood in Table 1 as M_⋆(model) and adopt it as our (Gaussian) prior for orbital fits. The variances of these distributions are all small; weighting by an initial mass function has a negligible effect.

Here we describe the results of the activity-based age analysis and the background for each individual star in our sample. The stellar distances we refer to in this section are inferred from Gaia EDR3 parallaxes. The fractional uncertainties on parallax are all ≲ 0.1%, so we simply take distance as the inverse of the parallax. The adopted stellar parameters in this paper, including Tycho-2 (Høg et al. 2000) and 2MASS (Cutri et al. 2003) photometry, ${R}_{\mathrm{HK}}^{{\prime} }$, and R_X, are summarized in Table 1.

We combine the chromospheric index ${R}_{\mathrm{HK}}^{{\prime} }$, the X-ray activity index R_X, and in some cases a photometric rotation period, to derive an age posterior for each star. Figure 1 shows the activity-based age posteriors for the whole sample of stars.

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2.4.1. HD 29021 (HIP 21571)

2.4.2. HD 81040 (HIP 46076)

2.4.3. HD 87883 (HIP 49699)

2.4.4. HD 98649 (HIP 55409)

2.4.5. HD 106252 (HIP 59610)

2.4.6. HD 106515A (HIP 59743A)

2.4.7. HD 171238 (HIP 91085)

2.4.8. HD 196067 (HIP 102125)

2.4.9. HD 221420 (HIP 116250)

HD 29021 was observed with the SOPHIE+ survey for giant planets with 66 RV measurements on a 4.5 yr time baseline (Rey et al. 2017). The resulting orbital solution yielded a mass of $M\sin i=2.4\,{M}_{\mathrm{Jup}}$ and an orbital period of 3.7 yr, placing the planet just outside the outer limit of its star's habitable zone. The dispersion of the residuals from this fit is within the expected levels for SOPHIE+ and no long-term drift was observed by these authors. We use all of the 66 RVs from the SOPHIE search for northern exoplanets.

Sozzetti et al. (2006) reported the detection of a massive planetary companion orbiting the young disk star, HD 81040, based on five years of precise RV measurements with the HIRES spectrograph on the 10 m Keck telescope and the ELODIE fiber-fed echelle spectrograph on the 1.93 m telescope at the Observatoire de Haute-Provence in France. HIRES/Keck monitored this star for 8.5 months in 1991, obtaining a total of 3 RV measurements, and 23 follow-up RV data were taken with the ELODIE spectrograph from 2002 to 2005. The orbital fit to all of the data unveiled a massive planet with period P = 1001.7 ± 7.0 days, e = 0.526 ± 0.042 and $M\sin i\,=6.86\pm 0.71\,{M}_{\mathrm{Jup}}$ (Sozzetti et al. 2006). We utilize the combined data from ELODIE and HIRES/Keck for our orbit analysis.

The RV monitoring of HD 87883 began in 1998 December at Lick (Fischer et al. 2013). A total of 44 RV measurements with Lick have a median uncertainty of 5.3 m s⁻¹. An additional 25 higher-quality RV measurements with the HIRES instrument in 2009 on the Keck telescope have a median precision of 4.0 m s⁻¹ (Fischer et al. 2009). A Keplerian fit to RV data from both Lick and Keck revealed a single long-period RV companion with a period of 7.55 ± 0.24 yr, an eccentricity of 0.53 ± 0.12 and a minimum mass of $M\sin i\,=1.78\pm 0.13\,{M}_{\mathrm{Jup}}$ (Fischer et al. 2009). Butler et al. (2017) has published an additional 46 RVs with a median uncertainty of 1.35 m s⁻¹ (Butler et al. 2017) that extends the total time baseline to more than 20 yr. We use all of the above measurements in our orbit analysis.

For HD 98649, a total of 11 radial-velocity measurements were published from CORALIE-98 with a median measurement uncertainty of 4.4 m s⁻¹. Then, 37 data were published from CORALIE-07 with a median error of 3.3 m s⁻¹. Using the combined data, Marmier et al. (2013) uncovered a companion with $M\sin i=6.8\,{M}_{\mathrm{Jup}}$, a period of ${13.6}_{-1.3}^{+1.6}$ yr on an eccentric orbit with e = 0.85. CORALIE measured 15 additional RVs in 2014. We make use of all of the 70 RV measurements acquired with the CORALIE spectrograph in the 16 yr time span.

HD 106252 was observed with the ELODIE high-precision echelle spectrograph at Haute-Provence Observatory as part of the ELODIE survey for northern extra-solar planets (Perrier et al. 2003). A total of 40 high-precision RV measurements with a median uncertainty of 10.5 m s⁻¹ were obtained since 1997 March (Perrier et al. 2003). Using these data, Perrier et al. (2003) performed orbit analysis for the companion and found a well-constrained solution with minimum mass of $M\sin i\approx 7.56\,{M}_{\mathrm{Jup}}$, an eccentricity of 0.471 ± 0.028, and a period of 4.38 ± 0.05 yr. Fischer et al. (2002) independently confirmed the planet using 15 RV measurements (median uncertainty of 11 m s⁻¹) from Hamilton/Lick, but with poorer constrains due to shorter temporal coverage Perrier et al. (2003). We use both the 40 RVs from ELODIE and 15 measurements from Hamilton.

HD 106252 was observed again with both ELODIE and Hamilton after the initial publications in 2002/2003, providing an additional 15 RVs from ELODIE with a median uncertainty of 11.1 m s⁻¹, and 54 RVs from Hamilton with median errors of 10.5 m s⁻¹ in 2006 (Butler et al. 2006). We also use the most recent RV measurements in 2009 that include 12 RVs from CDES-TS2 (median uncertainty 10.35 m s⁻¹) and 43 RVs (median uncertainty 9.3 m s⁻¹) from the High Resolution Spectrograph (HRS) instrument mounted on the 9.2 m Hobby-Eberly Telescope (HET) at the McDonald Observatory (Wittenmyer et al. 2016). Altogether, we use 179 RV data points spanning 12 yr.

In the same CORALIE survey, a massive planet, HD 106515 Ab, was discovered around one star in the binary system HD 106515 (Marmier et al. 2013). A total of 19 and 24 RV measurements have been collected with CORALIE-98 and CORALIE-07 with median measurement uncertainties of 6.3 m s⁻¹ and 4.1 m s⁻¹, respectively (Marmier et al. 2013). Its orbital parameters are well constrained, with $M\sin i\,=9.61\pm 0.14\,{M}_{\mathrm{Jup}}$, P = 3630 ± 12 days, e = 0.572 ± 0.011, and a semimajor axis of 4.590 ± 0.010 au. Similar to HD 196067, with the presence of a stellar companion, the Kozai–Lidov pumping mechanism (Kozai 1962; Lidov 1962) may be a viable explanation for the high eccentricity. We use all the CORALIE RVs.

The RV monitoring of HD 171238 started in 2002 October using the CORALIE spectrograph mounted on the 1.2 Euler Swiss telescope at La Silla Observatory in Chile (Ségransan et al. 2010). Those authors published 32 RV measurements from CORALIE-98 with a median measurement uncertainty of 7.3 m s⁻¹. An additional 65 RV measurements from CORALIE-07 show a median measurement error of 3.7 m s⁻¹. Using these radial velocities, Ségransan et al. (2010) reported a massive companion with $M\sin i=2.60\,{M}_{\mathrm{Jup}}$, a period of 4.17 yrs, a semimajor axis of 2.5 au, and an eccentricity of e = 0.40. They also speculate that the presence of spots on the stellar surface and active chromospheric activity might be the cause of the large RV jitter (10 m s⁻¹) from their single planet Keplerian model fitting. We adopt the 32 RVs from CORALIE-98, 65 RVs from CORALIE-07 and an additional 9 RVs published in 2017 with the HIRES spectrograph on the Keck Telescope (Butler et al. 2017).

Like HD 98649 b, HD 196067 b was also discovered in the CORALIE survey for southern exoplanets. A total of 82 Doppler measurements have been obtained since 1999 September: 30 were from CORALIE-98 with a median measurement uncertainty of 7.5 m s⁻¹ and 52 from CORALIE-07 with a median uncertainty of 5.9 m s⁻¹ (Marmier et al. 2013). The data for C98 are sparsely sampled due to poor coverage near periastron (Marmier et al. 2013). With the caveats from this poor coverage, the companion's $M\sin i$ is constrained between 5.8 and 10.8 M_Jup, the period between 9.5 and 10.6 yr, and the eccentricity between 0.57 and 0.84. We use the RVs from both CORALIE-98 and CORALIE-07.

The companion to HD 221420 was discovered by Kane et al. (2019) using 18 yr of RV data acquired with the Anglo-Australian Telescope ([AAT]; AAPS; Diego et al. 1990). The median uncertainty of the 88 measurements is 1.33 m s⁻¹. Kane et al. (2019) obtain an orbital period of ${61.55}_{-11.23}^{+11.50}$ yr, an RV semi-amplitude of ${54.7}_{-3.6}^{+4.2}$, an eccentricity of ${0.42}_{-0.07}^{+0.05}$, and a $M\sin i$ of ${9.7}_{-1.0}^{+1.1}\,{M}_{\mathrm{Jup}}$; they infer a semimajor axis of 18.5 ± 2.3 au. The HARPS spectrograph (ESO) also observed the target from 2003 to 2015, offering 74 higher precision measurements with a median uncertainty of 0.94 m s⁻¹. Both AAT+HARPS data are recently utilized by Venner et al. (2021) to derive a precise dynamical mass of 22.9 ± 2.2M_Jup. We include all the RV measurements from AAT and HARPS for this target.

Figure A2 shows the fitted posterior distributions for the orbital elements of HD 29021 b. All are Gaussian apart from the inclination, which is bimodal with equal likelihoods for prograde and retrograde orbits: either $33\buildrel{\circ}\over{.} {7}_{-4.9}^{+6.8}$ or $146\buildrel{\circ}\over{.} {3}_{-6.8}^{+4.9}$. Although the RV data only cover one and a half periods, the RV orbit of HD 29021 b is well constrained. For HD 29021 b, we obtain a dynamical mass of ${4.47}_{-0.65}^{+0.67}{M}_{\mathrm{Jup}}$, an eccentricity of ${0.453}_{-0.013}^{+0.014}$, and a semimajor axis of ${74.29}_{-0.62}^{+0.61}$ au. The $M\sin i$ inferred from our fit is ${2.483}_{-0.068}^{+0.070}{M}_{\mathrm{Jup}}$. We compare our orbital solution with the only orbital fit for this system published by Rey et al. (2017) where $M\sin i=2.4\pm 0.2{M}_{\mathrm{Jup}}$, e = 0.459 ± 0.008, $a={2.28}_{-0.08}^{+0.07}$, and $P={3.732}_{-0.012}^{+0.013}$ yr; all of our parameters are consistent with this orbital solution.

HD 81040 b has the shortest period in our sample; the RV orbit is well constrained by the joint HIRES and ELODIE RV data (see Figure A1). The posterior distributions of selected parameters for HD 81040 are displayed in Figure A2. Our orbital inclination indicates that it presents itself as an edge-on system with an orbital inclination of either $73{^\circ }_{-16}^{+12}$ or $107{^\circ }_{-12}^{+16}$. Our solution reveals that the true mass of HD 81040 b is ${7.24}_{-0.37}^{+1.0}\,{M}_{\mathrm{Jup}}$, slightly higher than these previous RV-only $M\sin i$ values. Out of the two $M\sin i$ values reported for this system, our inferred $M\sin i=6.87\pm 0.19\,{M}_{\mathrm{Jup}}$ agrees better with a value of 6.86 ± 0.71 M_Jup from Sozzetti et al. (2006) than a value of 7.27 ± 0.98 from Stassun et al. (2017). The orbit of HD 81040 b is consistent with being edge-on: we find a 0.9% probability that the planet will transit. If it does transit, we predict transit times of 2021 November 11 and 2024 October 8, each with an unfortunately large uncertainty of about 25 days.

The posterior probability distributions of selected parameters and their covariances for HD 87883 b are shown in Figure A2. The posterior for the semimajor axis is bimodal, peaking at 3.7 au and 3.9 au, respectively. Both the prograde and retrograde orbital inclinations are relatively face-on with values of $16\buildrel{\circ}\over{.} {8}_{-1.4}^{+1.7}$ and $163\buildrel{\circ}\over{.} {1}_{-1.7}^{+1.4}$. Figure A1 shows the HIRES and Hamilton RVs, and the corresponding best-fit Keplerian models and the residuals. Two families of orbits are possible; current RV data are insufficient to completely constrain the planet's semimajor axis and eccentricity.

HD 87883 was assessed by Fischer et al. (2009) prior to the 2007 publication of the HIRES/Keck RVs. Their eccentricity was poorly constrained (e ≥ 0.4) due to the incomplete coverage of the orbital phase approaching periastron. Using the extra RV data, we find consistent orbital parameters with Fischer et al. (2009) and Stassun et al. (2017) except for $M\sin i$ where we find a higher value. HD 87883's orbit is face-on and eccentric from our analysis. Our most likely orbit yields $M={6.31}_{-0.32}^{+0.31}\,{M}_{\mathrm{Jup}}$, a = ${3.77}_{-0.094}^{+0.12}$ au, P = ${8.23}_{-0.34}^{+0.32}$ yr, and $e={0.720}_{-0.027}^{+0.038}$. Still, further RV monitoring of target will be required to fully constrain its orbit.

The posterior distributions for HD 98649 b are depicted in Figure A2, and the fits to the RVs and astrometric accelerations are shown in Figure A1. Our solution shows that HD 98649 b is a highly eccentric and massive planet with an eccentricity of ${0.852}_{-0.022}^{+0.033}$ and a true planetary mass of ${9.7}_{-1.9}^{+2.3}\,{M}_{\mathrm{Jup}}$. There are two possible inclinations: either $136\buildrel{\circ}\over{.} {3}_{-13}^{+8.1}$ or $43\buildrel{\circ}\over{.} {7}_{-8.1}^{+13}$. The orbit of HD 98649 b was studied in Rickman et al. (2019) and Marmier et al. (2013). Both studies agree on a $M\sin i$ value of around 6.8 ± 0.5 M_Jup, and an eccentricity e = 0.85 ± 0.05 but digress on the semimajor axis and the period of the system. We obtain a value of $a={5.97}_{-0.21}^{+0.24}$ au, which agrees with a value of a = 5.6 ± 0.4 au found by Marmier et al. (2013), and marginally with the value of ${6.57}_{-0.23}^{+0.31}$ au derived by Rickman et al. (2019).

The posterior probabilities for HD 106252 b are illustrated in Figure A2. The posterior probabilities follow nearly Gaussian distributions except for the orbital inclination. The inclination is slightly bimodal, depending on whether the companion is in retrograde or prograde orbital motion. Figure A1 demonstrates the agreement between the calibrated Hipparcos–Gaia proper motions from the HGCA. As shown in Figure A1, the RV orbit of HD 106252 b is well constrained thanks to full orbital phase coverage from 12 yr of RV data. We obtain relatively tight constraints on the dynamical mass of the system, with $M={10.0}_{-0.73}^{+0.78}\,{M}_{\mathrm{Jup}}$ and an orbital inclination of $46\buildrel{\circ}\over{.} {0}_{-4.1}^{+4.9}$ (or $134\buildrel{\circ}\over{.} {0}_{+4.1}^{-4.9}$). Our derived $M\sin i$ of 7.20 ± 0.13 M_Jup corroborates the dynamical analyses of Wittenmyer et al. (2009), Stassun et al. (2017), and Liu et al. (2014) within 5%. Other orbital parameters such as semimajor axis, period, and eccentricity are also in perfect agreement with results obtained by these authors.

The orbits of both HD 106515 Ab and HD 106515 B in the three-body system HD 106515 Ab are fully constrained (see Figure A1). The induced astrometric acceleration by HD 106515 Ab on HD 106515 A is the most significant in our list. The posterior distributions for HD 106515 Ab and HD 106515 B are shown in Figure A2. Our three-body fit reveals a brown dwarf companion to HD 106515 A with a precise dynamical mass of ${18.9}_{-1.4}^{+1.5}\,{M}_{\mathrm{Jup}}$ on a ≈10 yr period. Our $M\sin i$ value of 9.25 ± 0.25 M_Jup agrees with the 9.08 ± 0.14 M_Jup from Saffe et al. (2019)'s analysis within 2%. All other fitted parameters are also in agreement with Saffe et al. (2019) and Marmier et al. (2013).

Our three-body fit yields a semimajor axis of ${335}_{-42}^{+96}$ au for the HD 106515 A/B binary. This is consistent with a 201 au projected separation given in Gaia EDR3 (see Table 3) and a 329 au semimajor axis value provided in Marmier et al. (2013) based on the positions given by Gould & Chaname (2004). Rica et al. (2017) also found a semimajor axis of 345 au and an eccentricity of about 0.42 that agrees with ours.

Our best-fit orbit for HD 171238 b reveals a dynamical mass of ${8.8}_{-1.3}^{+3.6}\,{M}_{\mathrm{Jup}}$, an $M\sin i={2.599}_{-0.088}^{+0.090}\,{M}_{\mathrm{Jup}}$, a semimajor axis of $a={2.519}_{-0.033}^{+0.032}$ au, and an eccentricity $e={0.358}_{-0.026}^{+0.028}$, all with tight 1σ errors. The MCMC posterior (Figure A2) for the mass of HD 171238 b shows a secondary peak around 15 M_Jup. The RV orbit presented in Figure A1 is constrained by three RV instruments. The inclination ($17\buildrel{\circ}\over{.} {1}_{-5.0}^{+3.1}$ or $162\buildrel{\circ}\over{.} {9}_{-3.1}^{+5.0}$) from our solution indicates a relatively face-on orbit for HD 171238 b. The $M\sin i$, eccentricity, and semimajor axis are more consistent with the orbit analysis of Ségransan et al. (2010; $M\sin i=2.60\pm 0.15{M}_{\mathrm{Jup}}$, a = 2.54 ± 0.06 au, and $e={0.400}_{-0.065}^{+0.061}$) than that of Ment et al. (2018). HD 171238 b's mass is bimodal, possibly either ≈9 M_Jup or ≈15 M_Jup. Future Gaia data releases will confirm the planet's orbit and mass.

HD 196067 b is in a three-body system with HD 196067 and HD 196068, HD 196067 being the host star and HD 196068 being the stellar companion to HD 196067. The wide-orbit stellar companion HD 196068 does accelerate HD 196067, though the >1000 au separation results in a minimal contribution to the astrometric acceleration from HD 196068. Our MCMC posteriors from the three-body fit with orvara are shown in Figure A2 for the inner and outer companions, respectively.

The RV orbit of HD 196067 b (see Figure A1) is not perfectly constrained. CORALIE-98 data did not sufficiently sample the sharp turnaround region near periastron. This lack of data results in a bimodal distribution for the eccentricity in the MCMC posteriors, one near e = 0.6 and the other near e = 0.85 (see Figure A2). For HD 196067 b, our three-body solution favors a dynamical mass of ${12.5}_{-1.8}^{+2.5}\,{M}_{\mathrm{Jup}}$, a semimajor axis of ${5.10}_{-0.17}^{+0.22}$, and an orbital period of ${9.88}_{-0.43}^{+0.63}$ yr. The orbital period is consistent with the minimum period of 9.5 yr found by Marmier et al. (2013) who have studied the system using the same data as us. They also found an eccentricity of $e={0.66}_{-0.09}^{+0.18}$, which is more consistent with the lower peak (e ≈ 0.60) than the higher one in our bimodal eccentricity posteriors. The inclination distribution is again, bimodal: either $41^\circ 2{\,}_{-9.1}^{+28}$ or $138^\circ 8{\,}_{-28}^{+9.1}$. We find that its true mass was underestimated by RV-only works. Our estimate of its true dynamical mass of ${12.8}_{-1.8}^{+2.6}\,{M}_{\mathrm{Jup}}$ is nearly twice as much as the minimum mass found by Marmier et al. (2013). HD 196067 b is interesting as it lies extremely close to the deuterium-burning limit (Spiegel et al. 2011) that divides planets and brown dwarfs. We expect that further RV data near the periastron passage of the companion will enable us to finalize its full orbit and confirm the dynamical nature of this companion.

The posterior distributions for HD 196068 are shown in Figure A2. The inclination of the stellar companion HD 196068, instead of being bimodal, has a single value of $10\buildrel{\circ}\over{.} {6}_{-5.2}^{+6.1}$, suggesting a nearly face-on orbit. This is because the single relative astrometric measurement from Gaia at epoch 2016.0 described in Section 5 successfully differentiated prograde from retrograde orbits of HD 196068. Our three-body solution leads to a projected binary semimajor axis of ${1631}_{-213}^{+208}$ au, significantly higher than a value of 932 au in Marmier et al. (2013). This can be explained by the stellar companion being near apastron in an eccentric orbit where $e={0.731}_{-0.031}^{+0.026}$.

The posterior distributions for selected parameters for HD 221420 b from the joint orbit fit are illustrated in Figure A2. The RV orbits and astrometric proper motions are displayed in Figure A1. The best-fit curves agree with both the RV data and proper motions from the HGCA.

The orbital solution for HD 221420 b was first derived by Kane et al. (2019) using only the AAT RVs. This orbital solution presents a period of ${61.55}_{-11.23}^{+11.50}$ yr, an eccentricity of ${0.42}_{-0.07}^{+0.05}$, a semimajor axis of ${18.5}_{-2.3}^{+2.3}$ au, and a minimum mass of ${9.7}_{-1.0}^{+1.1}{M}_{\mathrm{Jup}}$. This minimum mass is 36.5% discrepant with ours. This companion has recently been revisited by Venner et al. (2021) who use AAT and HARPS RVs and Hipparcos–Gaia (DR2) astrometry to constrain the precise dynamical mass. They find an orbital inclination of $164\buildrel{\circ}\over{.} {0}_{-2.6}^{+1.9}$, a precise dynamical mass of 22.9 ± 2.2 M_Jup, a semimajor axis of ${10.15}_{-0.38}^{+0.59}$ au, and a period of ${27.62}_{-1.54}^{+2.45}$ yr. Using the AAT and HARPS RVs, and absolute astrometry from Hipparcos and Gaia EDR3, we obtain a companion mass of ${20.6}_{-1.6}^{+2.0}{M}_{\mathrm{Jup}}$, an eccentricity of ${0.162}_{-0.030}^{+0.035}$, a semimajor axis of ${9.99}_{-0.70}^{+0.74}\,\mathrm{au}$, and a period of ${27.7}_{-2.5}^{+3.0}$. Our EDR3 solution agrees well with that of Venner et al. (2021), with one distinction being our inclination is again bimodal with equal maximum likelihood: i = $17\buildrel{\circ}\over{.} {85}_{-2.8}^{+2.9}$ or $162\buildrel{\circ}\over{.} {2}_{-2.9}^{+2.8}$. We validate the findings by Venner et al. (2021) that the companion is near the 25M_Jup upper limit for core accretion and disk instability to be considered plausible formation channels. Venner et al. (2021) also identify an M-dwarf stellar companion, HD 221420 B, that may be bound to HD 221420 A, but at a separation of more than 20,000 au it contributes negligibly to HD 221420's acceleration. HD 221420 b has the longest period and highest companion mass among the RV companions we sample. As a result, it is also the most accessible of our substellar companions for future direct imaging.

Direct imaging can probe the outer architecture of a system (Chauvin et al. 2005; Lafrenière et al. 2008; Ireland et al. 2010; Rameau et al. 2013; Lagrange 2014; Bowler 2016). However, only about a dozen wide-orbit giant planets have been discovered at separations of ∼10–150 au of their host stars (e.g., Chauvin et al. 2018; Marois et al. 2008). These surveys have mostly been blind. HD 221420b represents a rare case of a substellar companion whose mass and position we can determine before imaging.

We use our orbital fit to predict the position of HD 221420 b at future epochs. In Figure 3, we show the predicted locations of HD 221420 b at epochs 2022.0, 2023.0, 2024.0, and 2025.0. Despite the fact that HD 221420 b has never been imaged, our 68% confidence interval places it within a box of about 100 × 200 mas, almost sufficient to locate a fiber for the GRAVITY interferometer (Gravity Collaboration et al. 2017). For the next couple of years the brown dwarf will be offset about 04 west of its host star in a slow orbit. Unfortunately, we lack the data to establish whether this orbit is clockwise or counter-clockwise on the sky.

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HD 221420 b is an unusually promising accelerating system for high-contrast imaging follow-up. It consists of a relatively low-mass brown dwarf orbiting a bright (V = 5.8) and nearby (d = 31.2 pc) host star. Depending on the system age (Section 2.4), the ATMO2020 evolutionary models (Phillips et al. 2020) predict an effective temperature of 400–600 K for HD 221420 b. These low temperatures correspond to a late-T or early-Y spectral type. The H-band contrast predicted from ATMO2020 ranges from 16 to 20 mag depending on the brown dwarf's mass and the system age, while the predicted $L^{\prime}$ contrast ranges from 13 to 15 mag. These exceed the typical performance of SPHERE (Beuzit et al. 2008) but may be achievable with long integrations and, as Figure 3 shows, with the knowledge of where to look.