# Discovery and characterization of two Neptune-mass planets orbiting HD 212729 with TESS

David Armstrong, Ares Osborn, Vardan Adibekyan, Elisa Delgado-Mena, Saeed Hojjatpanah, Steve Howell, Sergio Hoyer, Henrik Knierim, Sérgio Sousa, Keivan Stassun, et al.

<figure><img src="image_1.png" /></figure>

## To cite this version:

David Armstrong, Ares Osborn, Vardan Adibekyan, Elisa Delgado-Mena, Saeed Hojjatpanah, et al. "Discovery and characterization of two Neptune-mass planets orbiting HD 212729 with TESS." Monthly Notices of the Royal Astronomical Society, 2023, 524 (4), pp.5804-5816. 10.1093/mnras/stad2183. hal-04505721

HAL Id: hal-04505721

https://hal.science/hal-04505721v1

Submitted on 16 Mar 2024

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<figure><img src="image_2.png" /></figure>

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# Discovery and characterization of two Neptune-mass planets orbiting HD 212729 with TESS

David J. Armstrong<sup>1,2*</sup> Ares Osborn<sup>1,2</sup> Vardan Adibekyan,<sup>3</sup> Elisa Delgado-Mena,<sup>3</sup> Saeed Hojjatpanah,<sup>4</sup> Steve B. Howell,<sup>5</sup> Sergio Hoyer<sup>5</sup>,<sup>4</sup> Henrik Knierim,<sup>6</sup> Sérgio G. Sousa<sup>1,5</sup>,<sup>3</sup> Keivan G. Stassun<sup>1,7</sup>,<sup>7</sup> Dimitri Veras<sup>1,2,8</sup> David R. Anderson,<sup>1,2</sup> Daniel Bayliss,<sup>1,2</sup> François Bouchy,<sup>9</sup> Christopher J. Burke,<sup>10</sup> Jessie L. Christiansen,<sup>11</sup> Xavier Dumusque,<sup>9</sup> Marcelo Aron Fetzner Keniger,<sup>1,2</sup> Andreas Hadjigeorghiou,<sup>1,2</sup> Faith Hawthorn,<sup>1,2</sup> Ravit Helled,<sup>6</sup> Jon M. Jenkins,<sup>5</sup> David W. Latham,<sup>12</sup> Jorge Lillo-Box,<sup>13</sup> Louise D. Nielsen,<sup>14</sup> Hugh P. Osborn,<sup>15</sup> José Rodrigues,<sup>3</sup> David Rodriguez,<sup>16</sup> Nuno C. Santos,<sup>3,17</sup> Sara Seager,<sup>10,18,19</sup> Paul A. Strøm,<sup>1,2</sup> Guillermo Torres,<sup>12</sup> Joseph D. Twicken,<sup>5,20</sup> Stephane Udry,<sup>9</sup> Peter J. Wheatley<sup>1,2</sup> and Joshua N. Winn<sup>21</sup>

*Affiliations are listed at the end of the paper*

Accepted 2023 July 14. Received 2023 July 12; in original form 2023 June 27

## ABSTRACT

We report the discovery of two exoplanets orbiting around HD 212729 (TOI 1052, TIC 317060587), a $T_{eff}$ = 6146 K star with $V = 9.51$ observed by TESS in Sectors 1 and 13. One exoplanet, TOI-1052b, is Neptune-mass and transits the star, and an additional planet TOI-1052c is observed in radial velocities but not seen to transit. We confirm the planetary nature of TOI-1052b using precise radial velocity observations from HARPS and determined its parameters in a joint RV and photometry analysis. TOI-1052b has a radius of $2.87_{-0.24}^{+0.29} R_⊕$, a mass of $16.9 \pm 1.7 M_⊕$, and an orbital period of 9.14 d. TOI-1052c does not show any transits in the TESS data, and has a minimum mass of $34.3_{-3.7}^{+4.1} M_⊕$ and an orbital period of 35.8 d, placing it just interior to the 4:1 mean-motion resonance. Both planets are best fit by relatively high but only marginally significant eccentricities of $0.18_{-0.07}^{+0.09}$ for planet b and $0.24_{-0.08}^{+0.09}$ for planet c. We perform a dynamical analysis and internal structure model of the planets as well as deriving stellar parameters and chemical abundances. The mean density of TOI-1052b is $3.9_{-1.3}^{+1.7} \text{ g cm}^{-3}$ consistent with an internal structure similar to Neptune. A nearby star is observed in Gaia DR3 with the same distance and proper motion as TOI-1052, at a sky projected separation of ~1500 au, making this a potential wide binary star system.

**Key words:** planets and satellites: detection – planets and satellites: individual: (TOI-1052, TIC 317060587).

## 1 INTRODUCTION

The Transiting Exoplanet Survey Satellite (TESS), launched in April 2018, has discovered more than 6000 exoplanet candidates and more than 300 confirmed exoplanets to date (Ricker et al. 2015), (see NASA Exoplanet Archive, Akeson et al. 2013). TESS has observed nearly the entire sky, in a series of 27-d sectors, with many stars observed for two or more sectors.

This large number of candidates is made up of both real exoplanets and false positives. High resolution and stable spectroscopy for radial velocity measurements is essential not only to confirm the candidate exoplanets but also to measure the mass of the planet precisely. Knowing both mass and radius of a planet provides us an estimation of the bulk density, internal structure, and planetary composition. High resolution spectroscopy allows us to fully derive stellar parameters and chemical abundances allowing studies of planetary origin, formation, and evolution (Alibert et al. 2010, 2015).

The discovery of thousands of exoplanets over recent years has revealed a diverse population of exoplanets ranging from Earth-mass to Jupiter-mass, which also can be categorized in various sub-types including hot-Jupiters and super-Earths. A less well known group is the Neptunian exoplanets, raising questions about their overall occurrence, and an apparent lack of short period Neptune-mass exoplanets – the so-called Neptunian desert (Szabó & Kiss 2011; Mazeh, Holczer & Faigler 2016) which could be due to tidal disruption and photoevaporation (Beaugé & Nesvorný 2013; Mazeh et al. 2016), combined with potential migration of evaporating planets (Boué et al. 2012). Different programs including our NCORES and NOMADS HARPS large programs aim at a more detailed investigation of Neptunian planets, and have successfully discovered several such exoplanets (e.g. Nielsen et al. 2020; Hoyer et al. 2021; Otegi et al. 2021). The number of discoveries in this parameter space remains low and more exoplanets are needed to better understand the population.

In this paper we report the discovery and confirmation of TOI-1052 b, a Neptune-like planet orbiting HD 212729, a G0 high-proper motion southern star with a visual magnitude of 9.51 (TESS mag of

<footnote>* E-mail: d.j.armstrong@warwick.ac.uk</footnote>
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Full *TESS* PDCSAP light curve of TOI-1052 at 2-min cadence with the best fit model overplotted in red. A typical error bar is shown in the top left. Binned data points of width 0.7d are shown.</figcaption></figure>

9.02), and a non-transiting additional planet. We use high-precision radial velocity (RV) measurements from the High Accuracy Radial velocity Planet Searcher spectrograph (HARPS, Pepe et al. 2002) mounted at the ESO La Silla 3.6-m telescope, in the framework of the NCORES program (e.g. Armstrong et al. 2020; Nielsen et al. 2020). Simultaneous analysis of the HARPS RV measurements using high resolution spectroscopy and *TESS* photometry enable us to confirm the nature of the planets as well as determine the stellar parameters of the host star.

The paper is organized as follows: the observations and data of the system are described in Section 2. The stellar parameters and signal analysis are described in Section 3. In Section 4, we describe the joint model and the discussion is presented in Section 5.

## 2 OBSERVATIONS

### 2.1 *TESS* photometry

*TESS* observed TOI 1052 (TIC 317060587) in Sectors 1 and 13, obtaining data from 2018 July 25 to August 22 and from 2019 June 19 to 17 July. The data were reduced in the *TESS* Science Processing Operations Centre (SPOC, Jenkins et al. 2016) pipeline. This pipeline is adapted from Kepler mission pipeline at NASA Ames Research Centre (Jenkins et al. 2010a). Transit events with 9.1 d orbital period were detected in the SPOC search of the 2-min cadence light curve for Sector 1 on 2018 August 28 and for Sector 13 on 2019 July 27 with an adaptive, noise-compensating matched filter (Jenkins 2002; Jenkins et al. 2010b, 2020). A limb-darkened transit model was fitted (Li et al. 2019) and a suite of diagnostic tests were conducted to assess the planetary nature of the signal (Twicken et al. 2018). The *TESS* Science Office (TSO) reviewed the SPOC data validation reports and issued an alert for TOI 1052.01 following the Sector 13 transit search on 2019 August 16 (Guerrero et al. 2021). In a search of the combined data from both sectors, the SPOC reported a signal-to-noise ratio (SNR) of 10.62 and transit depth value of 358.4 + -33.7 parts-per-million (ppm) for the 9.14-d period transit event. The transit signal passed all diagnostic tests, and the source was localized within 2.12 + / -4.55" of the target star.

We used the publicly available Pre-search Data Conditioning Simple Aperture Photometry (PDC-SAP, Twicken et al. 2010; Smith

<figure><img src="image_4.png" /><figcaption>**Figure 2.** Fig. 1 data phase-folded on the best-fitting period for TOI-1052 b with the best fit model overplotted. Binned data points of width 0.002 in phase are shown.</figcaption></figure>

et al. 2012; Stumpe et al. 2012, 2014) light curves provided by the SPOC for the transit modelling. Fig. 1 shows the 2 min cadence *TESS* light curve, and Fig. 2 shows the phase-folded curve for TOI 1052b. We also identified the 9.14-d transit event independently using the Transit Least Squares (TLSs) algorithm Hippke & Heller (2019) with a Signal detection efficiency (SDE) of 19.67. No further significant periodic signal was detected in the light curve.

Following Aller et al. (2020) we searched for sources of flux contamination by over-plotting the *Gaia* DR3 catalogue to the *TESS* Target Pixel Files (TPFs), shown in Fig. 3. According to *Gaia* DR3 (Gaia Collaboration et al. 2022), one star exists inside the *TESS* aperture in addition to TOI-1052 (*Gaia* DR3 6357524189130821376), with a magnitude contrast in the *Gaia* pass-band of 5.38, leading to negligible dilution of the transit. The nearby star is separated from TOI-1052 by 11.51", is at a distance of 128.9 pc consistent with the distance to TOI-1052, and has similar proper motion to TOI-1052 as shown by the proper motion vectors in Fig. 3. As such, the stars potentially form a bound system with a projected sky separation of ~1500 au. The secondary star has a temperature of 3600 K derived from the *Gaia* passbands (Fouesneau et al. 2022).
<figure><img src="image_5.png" /><figcaption>**Figure 3.** TESS pixel data with Gaia DR3 data sources overplotted in sector 1 (top) and 13 (bottom). TOI-1052 is marked with a white cross and the magnitude contrast is shown as red circles. Arrows show the proper motion of each star. Aperture pixels are highlighted in red. Star 2 is a potential bound companion to TOI-1052 with consistent distance and proper motion.</figcaption></figure>

## 2.2 High-resolution imaging

TOI-1052 was observed on 2021 July 07 UT using the Zorro speckle instrument on the Gemini South 8-m telescope (Scott et al. 2021; Howell & Furlan 2022). Zorro provides simultaneous speckle imaging in two bands (562 nm and 832 nm) with output data products including a reconstructed image with robust contrast limits on companion detections. Five sets of 1000 × 0.06 s images were obtained at 832 nm only and processed in the standard reduction pipeline (see Howell et al. 2011). TOI-1052 was found to have no close companions within the angular and $5\sigma$ contrast limits (5–7 magnitudes below the target star) achieved by the observations (see Fig. 4). The angular limits from the 8-m Gemini telescope range from the diffraction limit (20 mas) out to 1.2". At the distance of TOI-1052 (d = 129.8 pc) these angular limits correspond to spatial limits of 2.6 – 155.8 au.

<figure><img src="image_6.png" /><figcaption>**Figure 4.** $5\sigma$ contrast curve for high-resolution imaging observations with Zorro/Gemini. The 832 nm reconstructed image is shown in the upper right.</figcaption></figure>

## 2.3 HARPS follow-up

We collected 53 HARPS high-resolution spectra of TOI-1052 in three observation programmes, between 2021 May 24 and 2021 September 22. The spectrograph is mounted at the ESO 3.6-m telescope at La Silla Observatory, Chile (Mayor et al. 2003) and optimized to measure high-precision RVs. The observations were carried out as part of the NCORES large program (46 obs, ID 1102.C-0249, PI: Armstrong), with supplementary observations from the NGTS-HARPS Program (5 obs, ID 0105.C-0773(A), PI: Wheatley) and the Small planets inside and out program (2 obs, ID: 1106.C-0597(A), PI: Gandolfi).

We used the HARPS Data Reduction Software (DRS) to reduce the data, using a G0 template in order to measure RVs using a cross-correlation function (CCF) (Baranne et al. 1996; Pepe et al. 2002). The spectrum signal-to-noise ratio (SNR) is approx. 40 per pixel leading to a mean photon-noise uncertainty of $2.06 \text{ m s}^{-1}$. The DRS was used to measure the full width half maximum (FWHM), the line bisector, and the contrast of the CCF, as well as several activity indicators. The mean $log(R'_{HK})$ of the star is $-5.32 \pm 0.02$ implying a relatively low-magnetic activity level.

The full RV timeseries is shown in Fig. 5.

# 3 HOST STAR FUNDAMENTAL PARAMETERS

## 3.1 Spectroscopic analysis

The stellar spectroscopic parameters ($T_{eff}$, log g, microturbulence, [Fe/H]) were estimated using the ARES + MOOG methodology. The methodology is described in detail in Santos et al. (2013); Sousa (2014); Sousa et al. (2021). To consistently measure the equivalent widths (EWs) we used the latest version of ARES<fnref n="1" /> (Sousa et al. 2007, 2015). The list of iron lines is the same as the one presented in Sousa et al. (2008). For this we used a co-added HARPS spectrum of TOI-1052. In this analysis we use a minimization process to

<footnote n="1">¹The latest version, ARES v2, can be downloaded at https://github.com/sousasag/ARES</footnote>
<figure><img src="image_7.png" /><figcaption>**Figure 5.** The full RV HARPS timeseries showing the combined best fit model from planets b and c in red. Residuals after subtracting the model are shown further.</figcaption></figure>

find the ionization and excitation equilibrium to converge on the best set of spectroscopic parameters. This process makes use of a grid of Kurucz model atmospheres (Kurucz 1993) and the radiative transfer code MOOG (Sneden 1973). We also derived a more accurate trigonometric surface gravity using recent Gaia data following the same procedure as described in Sousa et al. (2021) which provided a consistent value when compared with the spectroscopic surface gravity. The resulting spectroscopic parameters are given in Table 1. The derived temperature of 6146 ± 62 K is indicative of a late F star as opposed to the G0 type specified in the literature (Houk & Cowley 1975), but not different at high enough confidence for us to reclassify the spectral type.

The abundances of the following elements were also derived using the same tools and models as for the stellar spectroscopic parameters: Mg, Al, Ti, Si, and Ni (detailed in e.g. Adibekyan et al. 2012, 2015), neutron capture elements (used later to obtain ages) as explained in Delgado Mena et al. (2017), and C and O (following Bertran de Lis et al. 2015; Delgado Mena et al. 2021). Although the EWs of the spectral lines were automatically measured with ARES, we performed careful visual inspection of the EWs measurements.

## 3.2 Stellar mass, radius, and age

The stellar mass, radius, and age were estimated from the spectroscopically derived parameters using the PARAM 1.3 web-interface<fnref n="2" /> (da Silva et al. 2006), leading to $R_*= 1.264 \pm 0.033 R_⊙$, $M_*= 1.204 \pm 0.025 M_⊙$, and $Age = 2.3 \pm 1.0$ Gyr. As an alternative we also estimated the stellar mass from the PARAM 1.3 values using the calibration presented in Torres, Andersen & Giménez (2010a) which provided a consistent result ($M_*= 1.19 \pm 0.03 M_⊙$).

As an independent determination of the basic stellar parameters, we also performed an analysis of the broad-band spectral energy distribution (SED) of the star together with the Gaia EDR3 parallax (with no systematic offset applied; see, e.g. Stassun & Torres 2021), in order to determine an empirical measurement of the stellar radius, following the procedures described in Stassun & Torres (2016); Stassun, Collins & Gaudi (2017); Stassun et al. (2018). We pulled the $B_TV_T$ magnitudes from Tycho-2, the JHKs magnitudes from 2MASS,

the W1–W4 magnitudes from WISE, and the $GG_{BP}G_{RP}$ magnitudes from Gaia. Together, the available photometry spans the stellar SED over the wavelength range 0.4–22 µm.

We performed a fit using Kurucz stellar atmosphere models, with the main parameters being the effective temperature ($T_{eff}$), surface gravity (log g), and metallicity ([Fe/H]), for which we adopted the spectroscopically determined values. The remaining free parameter is the extinction $A_V$, which we limited to the maximum line-of-sight value from the Galactic dust maps of Schlegel, Finkbeiner & Davis (1998). The resulting fit has a reduced $\chi^2$ of 1.0 and best fit $A_V = 0.04 \pm 0.04$. Integrating the model SED gives the bolometric flux at Earth, $F_{bol} = 4.123 \pm 0.096 \times 10^{-9} \text{ erg s}^{-1} \text{ cm}^{-2}$. Taking the $F_{bol}$ and $T_{eff}$ together with the Gaia parallax, gives the stellar radius, $R_*= 1.293 \pm 0.030 R_⊙$. In addition, we can again estimate the stellar mass from the empirical relations of Torres, Andersen & Giménez (2010b), giving $M_*= 1.20 \pm 0.07 M_⊙$.

All of our methods of stellar parameter estimation produce consistent results. We adopt the PARAM 1.3 values going forwards, and these are listed in Table 1.

## 3.3 Rotational period and age

Our PARAM fit led to an estimated isochrone age of $=2.3 \pm 1.0$ Gyr. We are also able to estimate the stellar age via the chemical clocks method (see Delgado Mena et al. 2019). The ages estimated from different chemical clocks (together with the Teff and [Fe/H]) are listed in Table 2, giving a weighted average of $1.9 \pm 0.3$ Gyr consistent with the PARAM age. This small error bar just reflects the good agreement between the ages obtained from different chemical clocks. We conservatively adopt the PARAM age with its larger error bar, given the uncertainties associated with stellar age estimation.

We do not find any evidence of periodic variability indicative of rotation in the TESS light curves.

Through the FWHM of the HARPS spectra CCF we are able to estimate the projected rotation velocity $v\sin i$ for the star. The mean FWHM across the spectra is $9.12 \text{ km s}^{-1}$. Using a calibration similar to the one presented in Santos et al. (2002); Maldonado et al. (2017); Hojjatpanah et al. (2019, and references therein) this FWHM implies a $v\sin i$ of $5.63 \pm 0.5 \text{ km s}^{-1}$. We also re-derived the $v\sin i$ by performing spectral synthesis with MOOG on 36 isolated iron lines and by fixing all the stellar parameters, macroturbulent velocity, and

<footnote n="2">²http://stev.oapd.inaf.it/cgi-bin/param_1.3</footnote>
<table><caption>**Table 1.** Stellar parameters.</caption><thead><tr><th>Parameter</th><th>Value</th><th>Source</th></tr></thead><tbody><tr><td colspan="3"><strong>Identifying information</strong></td></tr><tr><td>Identifier</td><td>HD 212 729</td><td></td></tr><tr><td>TOI</td><td>TOI-1052</td><td>TESS</td></tr><tr><td>TIC ID</td><td>317 060 587</td><td>TESS</td></tr><tr><td>2MASS ID</td><td>22300272-7538476</td><td>2MASS</td></tr><tr><td>Gaia ID</td><td>6357524189130820992</td><td>Gaia DR3</td></tr><tr><td colspan="3"><strong>Astrometric parameters</strong></td></tr><tr><td>RA (J2000, deg)</td><td>337.51023851341</td><td>Gaia DR3</td></tr><tr><td>Dec. (J2000, deg)</td><td>-75.64656089247</td><td>Gaia DR3</td></tr><tr><td>Parallax (mas)</td><td>7.74 ± 0.01</td><td>Gaia DR3</td></tr><tr><td>Distance (pc)</td><td>128.7 ± 0.2</td><td>Gaia DR3</td></tr><tr><td colspan="3"><strong>Photometric parameters</strong></td></tr><tr><td>B</td><td>10.09 ± 0.04</td><td>Tycho</td></tr><tr><td>V</td><td>9.51 ± 0.02</td><td>Tycho</td></tr><tr><td>T</td><td>9.02 ± 0.01</td><td>TESS</td></tr><tr><td>G</td><td>9.40 ± 0.01</td><td>Gaia DR3</td></tr><tr><td>J</td><td>8.56 ± 0.02</td><td>2MASS</td></tr><tr><td>H</td><td>8.25 ± 0.04</td><td>2MASS</td></tr><tr><td>K</td><td>8.25 ± 0.03</td><td>2MASS</td></tr><tr><td>W1</td><td>8.160 ± 0.023</td><td>WISE</td></tr><tr><td>W2</td><td>8.255 ± 0.021</td><td>WISE</td></tr><tr><td>W3</td><td>8.190 ± 0.020</td><td>WISE</td></tr><tr><td>W4</td><td>8.119 ± 0.174</td><td>WISE</td></tr><tr><td colspan="3"><strong>Abundances</strong></td></tr><tr><td>[Fe/H](dex)</td><td>0.140 ± 0.013</td><td>Section 3</td></tr><tr><td>[Mg/H](dex)</td><td>0.08 ± 0.04</td><td>Section 3</td></tr><tr><td>[Al/H](dex)</td><td>0.10 ± 0.02</td><td>Section 3</td></tr><tr><td>[Si/H](dex)</td><td>0.11 ± 0.04</td><td>Section 3</td></tr><tr><td>[Ti/H](dex)</td><td>0.12 ± 0.03</td><td>Section 3</td></tr><tr><td>[Ni/H](dex)</td><td>0.13 ± 0.03</td><td>Section 3</td></tr><tr><td colspan="3"><strong>Bulk parameters</strong></td></tr><tr><td>Mass (M<sub>☉</sub>)</td><td>1.204 ± 0.025</td><td>Section 3 (PARAM)</td></tr><tr><td>Radius (R<sub>☉</sub>)</td><td>1.264 ± 0.033</td><td>Section 3 (PARAM)</td></tr><tr><td>T<sub>eff</sub> (K)</td><td>6146 ± 62</td><td>Section 3</td></tr><tr><td>log g (cm s<sup>-2</sup>)</td><td>4.30 ± 0.02</td><td>Section 3 (Gaia)</td></tr><tr><td>log g (cm s<sup>-2</sup>)</td><td>4.39 ± 0.11</td><td>Section 3 (spec)</td></tr><tr><td>v<sub>mic</sub> (km s<sup>-1</sup>)</td><td>1.28 ± 0.02</td><td>Section 3</td></tr><tr><td>v sin i (km s<sup>-1</sup>)</td><td>5.0 ± 0.9</td><td>Section 3</td></tr><tr><td>P<sub>rot</sub>/sin i (d)</td><td>12.8 ± 2.3</td><td>Section 3</td></tr><tr><td>Age(Gyrs)</td><td>2.3 ± 1.0</td><td>Section 3 (PARAM)</td></tr><tr><td>mean log(R'<sub>HK</sub>)</td><td>-5.32 ± 0.02</td><td>HARPS</td></tr></tbody></table>

Note: sources: TESS (Stassun et al. 2019); 2MASS (Skrutskie et al. 2006); Tycho (Høg et al. 2000); WISE (Wright et al. 2010); and Gaia (Gaia Collaboration et al. 2016, 2022)

<table><caption>**Table 2.** Chemical clock age estimates (see Delgado Mena et al. 2019, Table 10).</caption><thead><tr><th>Clock</th><th>Value (Gyr)</th></tr></thead><tbody><tr><td>[Y/Zn]</td><td>2.1 ± 0.5</td></tr><tr><td>[Y/Ti]</td><td>1.6 ± 0.7</td></tr><tr><td>[Y/Mg]</td><td>1.4 ± 0.6</td></tr><tr><td>[Sr/Ti]</td><td>2.3 ± 1.3</td></tr><tr><td>[Sr/Mg]</td><td>2.0 ± 1.1</td></tr><tr><td>[Y/Si]</td><td>1.6 ± 0.7</td></tr><tr><td>[Sr/Si]</td><td>1.8 ± 1.2</td></tr><tr><td>[Y/Al]</td><td>2.7 ± 0.3</td></tr><tr><td>Weighted mean</td><td>1.9 ± 0.3</td></tr></tbody></table>

TOI-1052 l1 periodogram

<table><thead><tr><th>Peaks at (days)</th><th>log<sub>10</sub> FAPs</th></tr></thead><tbody><tr><td>36.5545931</td><td>-1.9000145994448543</td></tr><tr><td>9.13864829</td><td>-9.35135112839019</td></tr><tr><td>6.06181696</td><td>-0.001356950824660627</td></tr><tr><td>22.3389180</td><td>-1.243296493237299e-07</td></tr><tr><td>241.260315</td><td>-1.9286549331065747e-16</td></tr></tbody></table>

<figure><img src="image_8.png" /><figcaption>**Figure 6.** *l*₁ Periodogram as discussed in Section 3.4 showing significant peaks at 9.14 and 36.6 d. Peak values and false alarm probabilities (FAPs) are shown above in the periodogram.</figcaption></figure>

limb-darkening coefficient (Costa Silva, Delgado Mena & Tsantaki 2020), leading to a consistent value of vsin *i* = 5.0 ± 0.9 km s⁻¹, which we adopt. The linear limb-darkening coefficient (0.7) was determined using the ExoCTK package (Bourque et al. 2021) using the determined stellar parameters. The macroturbulent velocity (4.4 km s⁻¹) was determined using the temperature and gravity dependent empirical formula from Doyle et al. (2014).

We estimated the (projected) rotation period directly via the spectroscopic vsin *i* and the *R*<sub>*</sub> determined earlier, which gives *P*<sub>rot</sub>/sin *i* = 12.8 ± 2.3 d. Assuming the stellar orbital inclination is *i* ≈ 90°, then this would represent approximately the true rotation period.

## 3.4 Signal identification

We computed the *l*₁ periodogram (e.g. Hara et al. 2017, 2020) to find periodicities in the RV data. The *l*₁ periodogram uses the theory of compressed sensing adapted for handling correlated noise to analyse the RV without the estimation of the frequency iteratively, see Hara et al. (2017, 2020) for more information. A fundamental difference of the *l*₁ periodogram over the typically used Lomb–Scargle (Lomb 1976; Scargle 1982; VanderPlas 2018) is that all possible frequencies are tested simultaneously. This method reduces aliases in the periodogram. Fig. 6 shows two significant signals, considering the model noise with a 1.5 m s⁻¹ jitter noise, consistent with the eventual jitter found by our best fit model in Table 4. Both the 9.14-d transiting planet period and an additional 36.6-d period were found to be significant with a False Alarm Probability (FAP) < 1.0 percent for the 9.14-d and 1.2 percent for the 36.6-d, as opposed to the other shown peaks which have FAP > 99 percent. Lomb–Scargle periodograms of the RVs with and without the planet signals, FWHM, bisector span (BIS), log RHK, and CCF contrast, calculated with *astropy* (Astropy Collaboration et al. 2022), are shown in Fig. 7, to investigate the planet peaks further and consider a potential activity source for the significant signals. No significant power is found at the 9-d or 36-d periods in any of the indicators. Fig. 7 also demonstrates that two periodic components are required to model the RVs, with no further periodic signals found once the two planets are removed. Note the the initially most significant peak seen in the RVs in the Lomb–Scargle periodogram is at 22 d, which
<table><caption>**Table 3.** Prior distributions used in our joint fit model, fully described in Section 4. The priors are created using distributions in *PyMC3* with the relevant inputs to each distribution described in the table footer. Fit results and derived parameters can be found in Table 4.</caption><thead><tr><th>Parameter</th><th>(unit)</th><th>Prior distribution</th></tr></thead><tbody><tr><td colspan="3"><strong>Planet b</strong></td></tr><tr><td>Period P<sub>b</sub></td><td>(days)</td><td>N(9.13966, 0.001)</td></tr><tr><td>Ephemeris t<sub>0, b</sub></td><td>(BJD-2457000)</td><td>N(1332.9448, 0.02)</td></tr><tr><td>Radius log (R<sub>b</sub>)</td><td>(log R<sub>☉</sub>)</td><td>N(-3.733*, 1.0)</td></tr><tr><td>Impact parameter b<sub>b</sub></td><td></td><td>U(0, 1 + R<sub>b</sub>/R<sub>*</sub>)</td></tr><tr><td>e<sub>b</sub>sin ω<sub>b</sub></td><td></td><td>U(Unit disc)</td></tr><tr><td>e<sub>b</sub>cos ω<sub>b</sub></td><td></td><td>U(Unit disc)</td></tr><tr><td>K<sub>b</sub></td><td>(m s<sup>-1</sup>)</td><td>U(0.0, 50.0)</td></tr><tr><td colspan="3"><strong>Planet c</strong></td></tr><tr><td>Period P<sub>c</sub></td><td>(days)</td><td>N(35.97306, 4.0)</td></tr><tr><td>Ephemeris t<sub>0, c</sub></td><td>(BJD-2457000)</td><td>N(2423.3168, 20.0)</td></tr><tr><td>e<sub>c</sub>sin ω<sub>c</sub></td><td></td><td>U(Unit disc)</td></tr><tr><td>e<sub>c</sub>cos ω<sub>c</sub></td><td></td><td>U(Unit disc)</td></tr><tr><td>K<sub>c</sub></td><td>(m s<sup>-1</sup>)</td><td>U(0.0, 50.0)</td></tr><tr><td colspan="3"><strong>Star</strong></td></tr><tr><td>Mass M<sub>*</sub></td><td>(M<sub>☉</sub>)</td><td>N<sub>B</sub>(1.204, 0.025, 0.0, 3.0)</td></tr><tr><td>Radius R<sub>*</sub></td><td>(R<sub>☉</sub>)</td><td>N<sub>B</sub>(1.264, 0.033, 0.0, 3.0)</td></tr><tr><td colspan="3"><strong>Photometry</strong></td></tr><tr><td>TESS mean</td><td></td><td>N(0.0, 1.0)</td></tr><tr><td>log (Jitter)</td><td>(m s<sup>-1</sup>)</td><td>N(-7.40<sup>†</sup>, 10)</td></tr><tr><td colspan="3"><strong>HARPS RVs</strong></td></tr><tr><td>Offset</td><td>(m s<sup>-1</sup>)</td><td>N(54945.0, 10.0)</td></tr><tr><td>log (Jitter)</td><td>(m s<sup>-1</sup>)</td><td>N(0.37<sup>†</sup>, 5.0)</td></tr></tbody></table>

Notes. Distributions:

* $N(\mu, \sigma)$: a normal distribution with a mean $\mu$ and a standard deviation $\sigma$;
* $N_B(\mu, \sigma, a, b)$: a bounded normal distribution with a mean $\mu$, a standard deviation $\sigma$, a lower bound $a$, and an upper bound $b$ (bounds optional);

* $U(a, b)$: a uniform distribution with a lower bound $a$, and an upper bound $b$.

* equivalent to $0.5(\log(D)) + \log(R_*)$ where $D$ is the transit depth (ppm multiplied by $10^{-6}$) and $R_*$ is the mean of the prior on the stellar radius ($R_{\odot}$);

* equivalent to the log of the minimum error on the HARPS data (m s⁻¹), or the mean error on the *TESS* data. We fit a log value to enforce an broad, non-zero prior covering several orders of magnitude.

is seen with low significance in the $l_1$ periodogram. The 22-d peak is an artefact arising from both the 9.14-d and 36-d planet peaks and vanishes when both planets are removed. Similarly, the 6-d signal seen in both periodograms is an artefact of the planet b peak.

Given the robust detection of the transiting planet in the radial velocities, we are able to confirm the known planetary candidate as TOI-1052 b. Given that there is no sign of stellar activity in any indicator at the 36.6-d additional period, and this period does not match the estimated stellar rotation period or its harmonics, we claim this as an additional planet in the system, TOI-1052 c. The joint fit of photometry and spectroscopy in Section 4 finds a period for planet c of 35.81±0.38d. TOI-1052 c is just within the 4:1 resonance of planet b. The system dynamics are discussed in Section 5.1.

## 4 JOINT MODELLING

The photometry from *TESS* and spectroscopy from HARPS were combined in a joint fit using the *exoplanet* (Foreman-Mackey et al. 2021) code framework. This package also makes use of *starry* (Luger et al. 2019) and *PyMC3* (Salvatier, Wieckiâ &

Fonnesbeck 2016). The photometric model is adjusted to account for the *TESS* exposure time of 2 min.

The model constructs two Keplerian orbits, one for each planet, with orbital period *P*, epoch $t_0$, impact parameter *b*, eccentricity *e* and angle of periastron *ω* as free parameters determining the orbit. The orbital period and epoch are drawn from Gaussian prior distributions with a mean drawn from initial fits and a standard deviation of 0.001 and 4 d for planets *b* and *c*, respectively, approximately 10 times larger than the eventual errors on those parameters. The impact parameter is drawn uniformly between 0 and $1 + R_p/R_*$, where $R_p$ is the planetary radius. *e* and *ω* are drawn via sampling *esin* *ω* and *ecos* *ω* from a unit disc distribution then deriving *e* and *ω*. Additionally the stellar mass and radius are allowed to vary in a Gaussian distribution according to their values from Section 3.

Once the orbit is defined, the planet to star radius ratio $R_p/R_*$ and RV semi-amplitude *K* are drawn from wide uniform distributions. Limb darkening parameters are drawn from the quadratic limb darkening parametrization of Kipping (2013). We introduce a systematic RV offset, a *TESS* photometry offset, and instrumental jitter parameters for both instruments as extra parameters. Jitter is drawn from a broad Gaussian distribution in log-space to allow for a wide range of orders of magnitude, and is then added to the measured instrumental noise in quadrature. All model priors are detailed in Table 3.

We do not include a model for the stellar noise, apart from the jitter term in the RVs, as no significant periodic signal was found in either the RVs, stellar activity indicators or photometry aside from the two planetary signals.

We use a No U-Turn Sampler (NUTS) variant of the Hamiltonian Monte Carlo (HMC) algorithm to draw samples from the posterior chain, for which we use 12 chains each with 5000 steps for a total of 60 000 iterations. We treat the first 1500 samples drawn from each chain as burn-in and subsequently discard them. The resulting Gelman–Rubin statistics (Brooks & Gelman 1998) for each variable are <<1.05, demonstrating the chains have converged.

Our initial fits revealed a marginally significant eccentricity for both planets (at $2.5\sigma$ for planet b and $2.9\sigma$ for planet c). We present fit posterior values with eccentricity, and with both planets fixed at zero eccentricity, in Table 4. The resulting planet parameters are consistent in both models. To compare the models we calculate the WAIC (widely applicable information criterion), which estimates the expected log pointwise predictive density (elpd) of the models (for details on the criterion see Watanabe 2010; Vehtari, Gelman & Gabry 2017). The eccentric model is slightly favoured with a higher elpd, with a difference of 4.0, although this difference is not large enough to be considered significant. We adopt the free eccentricity results going forwards.

Through the results of this analysis, we determine that TOI-1052 b is a mini-Neptune of radius $2.87_{-0.24}^{+0.29} R_⊕$ and mass $16.9 \pm 1.7 M_⊕$. From these values we infer a bulk density of $3.9_{-1.3}^{+1.7} \text{ g cm}^{-3}$. The non-transiting planet TOI-1052 c is found to have $M \sin i_c = 34.3_{-3.7}^{+4.1} M_⊕$. No evidence of transits is seen for planet c. Fig. 10 shows the two planets in the context of the exoplanet population.

The near 4:1 ratio of the orbital periods, and potential eccentricity, invite questions as to whether there is a third planet in between planets *b* and *c*, forming a 1:2:4 ratio. Two planets may mimic a single planet with an eccentric orbit in RV observations, although TOI-1052 c is below the $3\sigma$ significant eccentricity criterion for this issue found in Wittenmyer et al. (2019). We could not find any evidence of such a
<table><caption>**Table 4.** TOI-1052 fit and derived parameters: median and 68 per cent confidence interval.</caption><thead><tr><th>Model</th><th></th><th>With eccentricity (adopted)</th><th>Fixed eccentricity</th></tr></thead><tbody><tr><td colspan="4"><strong>System parameters:</strong></td></tr><tr><td>u<sub>1</sub>, TESS</td><td></td><td>0.80<sup>+0.61</sup><sub>-0.55</sub></td><td>0.85<sup>+0.59</sup><sub>-0.57</sub></td></tr><tr><td>u<sub>2</sub>, TESS</td><td></td><td>-0.07<sup>+0.52</sup><sub>-0.49</sub></td><td>-0.12<sup>+0.54</sup><sub>-0.48</sub></td></tr><tr><td>TESS<sub>offset</sub></td><td>ppm</td><td>-7.9<sup>+3.3</sup><sub>-3.3</sub></td><td>-7.9<sup>+3.2</sup><sub>-3.2</sub></td></tr><tr><td>σ<sub>TESS</sub></td><td>ppm</td><td>61.0<sup>+2.3</sup><sub>-2.3</sub></td><td>61.0<sup>+2.3</sup><sub>-2.3</sub></td></tr><tr><td>σ<sub>HARPS</sub></td><td>m s<sup>-1</sup></td><td>0.79<sup>+0.61</sup><sub>-0.65</sub></td><td>1.62<sup>+0.42</sup><sub>-0.42</sub></td></tr><tr><td>Systemic RV</td><td>m s<sup>-1</sup></td><td>54946.02<sup>+0.38</sup><sub>-0.39</sub></td><td>54945.75<sup>+0.38</sup><sub>-0.39</sub></td></tr><tr><td colspan="4"><strong>Planetary parameters:</strong></td></tr><tr><td>P</td><td>days</td><td>9.139703<sup>+0.000190</sup><sub>-0.000197</sub></td><td>35.806<sup>+0.453</sup><sub>-0.381</sub></td></tr><tr><td>T<sub>0</sub></td><td>BJD - 2457000</td><td>1332.9442<sup>+0.0057</sup><sub>-0.0055</sub></td><td>2425.13<sup>+1.63</sup><sub>-1.64</sub></td></tr><tr><td>K</td><td>m s<sup>-1</sup></td><td>4.70<sup>+0.46</sup><sub>-0.46</sub></td><td>6.11<sup>+0.80</sup><sub>-0.69</sub></td></tr><tr><td>e</td><td></td><td>0.180<sup>+0.090</sup><sub>-0.071</sub></td><td>0.237<sup>+0.090</sup><sub>-0.082</sub></td></tr><tr><td>ω</td><td>rad</td><td>-2.07<sup>+0.24</sup><sub>-0.42</sub></td><td>-0.79<sup>+0.47</sup><sub>-0.53</sub></td></tr><tr><td>a/R<sub>*</sub></td><td></td><td>15.51<sup>+0.43</sup><sub>-0.41</sub></td><td>38.6 ± 1.1</td></tr><tr><td>i</td><td>deg</td><td>87.53<sup>+0.24</sup><sub>-0.20</sub></td><td>-</td></tr><tr><td>R<sub>p</sub>/R<sub>*</sub></td><td></td><td>0.0194<sup>+0.0018</sup><sub>-0.0015</sub></td><td>0.0184<sup>+0.0013</sup><sub>-0.0014</sub></td></tr><tr><td>b</td><td></td><td>0.808<sup>+0.052</sup><sub>-0.079</sub></td><td>0.727<sup>+0.052</sup><sub>-0.085</sub></td></tr><tr><td colspan="4"><strong>Derived parameters:</strong></td></tr><tr><td>M<sub>p</sub></td><td>M<sub>⊕</sub></td><td>16.9<sup>+1.7</sup><sub>-1.7</sub></td><td>-</td></tr><tr><td>R<sub>p</sub></td><td>R<sub>⊕</sub></td><td>2.87<sup>+0.29</sup><sub>-0.24</sub></td><td>-</td></tr><tr><td>ρ<sub>p</sub></td><td>g cm<sup>-3</sup></td><td>3.93<sup>+1.7</sup><sub>-1.3</sub></td><td>-</td></tr><tr><td>a<sub>p</sub></td><td>au</td><td>0.09103<sup>+0.00062</sup><sub>-0.00063</sub></td><td>0.2263<sup>+0.0024</sup><sub>-0.0023</sub></td></tr><tr><td>S</td><td>S<sub>⊕</sub></td><td>246 ± 13</td><td>39.7<sup>+2.3</sup><sub>-2.2</sub></td></tr><tr><td>T<sub>eq</sub>†</td><td>K</td><td>1135</td><td>719</td></tr><tr><td>M<sub>p</sub>sin i</td><td>M<sub>⊕</sub></td><td>-</td><td>34.3<sup>+4.1</sup><sub>-3.7</sub></td></tr></tbody></table>

Notes.* For TOI 1052c $T_0$ corresponds to the time when the planet would have transited.

†Assuming a Bond Albedo of 0.3.

hidden planet, which might be expected to show in the RV residuals if the model is forced to be circular. Absent that evidence, we proceed with the two-planet model but note this possibility in case future observations can probe the system further.

# 5 DISCUSSION

## 5.1 Dynamical analysis

The wide range of allowed eccentricities for both planets raise questions as to their stability and dynamical interactions. Further, the approximate 4:1 ratio of the planets' orbital periods invites a more detailed analysis of a potential resonant interaction and how it affects the stability of the system.

First, because this system is a two-planet system, we can determine if it is Hill stable analytically. By using the procedure outlined in Veras et al. (2013), which is based on the equations in Donnison (2006, 2011), we find that the TOI-1052 is Hill stable for all planetary eccentricities in the ranges $e_b = 0.0 - 0.3$ and $e_c = 0.0 - 0.3$. In fact, the system is comfortably Hill stable: even in the scenario where $e_b = e_c = 0.3$, the system would be Hill stable for $a_c/a_b \gtrsim 2.12$, whereas actually $a_c/a_b \approx 2.49$.

Hence, residence in a strong mean-motion resonance is not necessarily required to stabilize the system. Nevertheless, the

system's proximity to a strong mean-motion resonance is of interest, particularly in context of the entire exoplanet population. Fig. 5 of Weiss et al. (2022) illustrates a statistically significant asymmetry in the population of two-planet pairs which reside just interior versus just exterior to the strongest (first-order) mean-motion resonances, first noted in Fabrycky et al. (2014). The observed population of 4:1 planetary pairs might not yet be high enough for a 4:1 asymmetry to be detectable. In this respect, the TOI-1052 system might provide a valuable data point, although it cannot be excluded that another planet lies between the two detected planets.

In order to explore the system's proximity to resonance, we employ the semi-analytic libration width prescription of Gallardo, Beaugé & Giuppone (2021). This prescription effectively computes bounds within which mean-motion resonant behaviour is possible through a numerical procedure mixed in with analytical theory.

We plot the libration width curves for four cases ($e_b = 0.0, 0.1, 0.2, 0.3$) in Fig. 11 by using an eccentricity resolution of 0.025 in the numerical integration. Superimposed are the uncertainties for the current location of TOI-1052 c. Comparing these uncertainties with the libration width locations indicates that the TOI-1052 system is definitely not in resonance (at least a third-, second-, or first-order resonance), and resides just interior to the 4:1 resonance.
<figure><img src="image_9.png" /><figcaption>**Figure 7.** Lomb–Scargle periodogram of HARPS RVs and activity indicators. The orbital period of each planet is shown as vertical dashed lines. Horizontal dashed lines show the FAP 1 per cent level. A 22-d artefact is seen in the raw RVs at the top, but vanishes when both planet models are removed.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Radial velocities phase-folded at the best-fitting period of TOI-1052 b, with best fit model overplotted in red.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 9.** Radial velocities phase-folded at the best-fitting period of TOI-1052 c, with best fit model overplotted in red.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Figure 10.** Period-Mass diagram showing both planets, with TOI-1052c plotted at its $M_p \sin i$ value. The background density distribution of known exoplanets is shown in green.</figcaption></figure>

The system's close proximity to resonance is characteristic of many exoplanetary systems, although the proximity to a relatively high-order resonance is noteworthy. Proximity to resonance can be a marker for the differential migration rates of planets in their nascent protoplanetary disc, although to date this has primarily been investigated in depth for first-order resonances (Huang & Ormel 2023).

## 5.2 Internal structure

TOI-1052 b is similar to Uranus or Neptune in mass, but has a considerably smaller radius and therefore a denser interior. Fig. 12,
<figure><img src="image_13.png" /><figcaption>**Figure 11.** Proximity of the two planets in the TOI-1052 to the 4:1 mean-motion resonance. The four pairs of curves are libration widths for this resonance. These curves, moving outwards, correspond to $e_b = 0.0, 0.1, 0.2, 0.3$. The planet TOI-1052 c is nearly outside all of these curves, adding to the asymmetry seen around mean-motion commensurabilities in the exoplanet population. This system is also Hill stable, with the critical limit off the scale of the plot.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 12.** Mass–radius diagram showing various internal structure lines from our model labelled in the legend. TOI-1052b can be explained by an Earth-like composition with either 50 per cent water or a H-He atmosphere as described in the text.</figcaption></figure>

which shows the Mass–Radius relation, demonstrates that TOI-1052 b is located between the water line and Earth-like compositional line, suggesting a significant fraction of refractory materials. In comparison, both Uranus and Neptune are located above the pure-water line.

For TOI-1052b, two limiting cases come to mind: a planet with a rocky interior and a substantial hydrosphere and a refractory-rich planet with a primordial H-He atmosphere. We investigate these two scenarios with a layered interior model, consisting of up to four layers: a H-He atmosphere, a water layer, a silicate mantle, and an iron core (see Dorn et al. 2017). Using the inferred age and elemental abundances of TOI-1052, we solve the standard structure equations for two models: 1.) a model where we assume that TOI-1052 b contains no water (no-water model) and 2.) a model where we conversely assume that TOI-1052 b contains no H-He atmosphere

(no-atmosphere model). We put no constraints on the compositions, that is, the elemental ratios, of the other layers. As a result, the iron-to-rock ratio can take any value. For both models, we apply a nested sampling algorithm (Buchner et al. 2014) to explore the permitted parameter ranges that reproduce the measured masses and radii of TOI-1052 b.

We find that the no-water model favours a core-to-mantle mass fraction of nearly unity: $0.96 \pm 0.17$ with a H-He envelope of $2^{+1.4}_{-0.8} \%$. In the case of the no-atmosphere model, while the core-to-mantle mass fraction is poorly constrained ($0.6 \pm 0.5$), this model predicts a water mass fraction of $0.43 \pm 0.12$. The larger uncertainties are caused by the wide range of possible core, mantle, and water layer masses that can reproduce the observed radius and mass compared to the no-water model. Assuming a fixed iron-to-rock ratio, for example, similar to the host star's elemental ratios, decreases the model's uncertainty significantly.

We note that at high planetary masses, layers might not be as distinct as assumed here (e.g. Helled & Stevenson 2017; Bodenheimer et al. 2018). Moreover, the atmospheric mass fraction may be underestimated due to pollution of the H-He envelope by heavier elements, leading to further contraction of the atmosphere (Lozovsky et al. 2018). The interior model also neglects any water that is dissolved deep in the interior, which could increase the overall water mass fraction (Dorn & Lichtenberg 2021). Nevertheless, while these details could change the exact values inferred here, it is clear that TOI-1052 b is enriched with refractory materials and any H-He atmosphere is likely to be minimal.

Additionally, the planet's elemental abundances could differ from its host star, which can change the mantle and the temperature structure. We therefore also considered structure models with varying elemental abundances to investigate this effect. We find that the inferred possible compositions and their error for TOI-1052 b do not change significantly.

## 6 CONCLUSIONS

We report the discovery and characterization of two new planets just outside the 4:1 mean-motion resonance in the bright, $V = 9.5$ TOI-1052 system, using TESS mission data and HARPS RV measurements. We used high-resolution imaging from the Zorro speckle imaging instrument in order to investigate the presence of any nearby companions and find none within the detector limits. We estimated the projected stellar rotation period to be around 12.8 d from measuring line broadening in the spectra, and derived stellar parameters, chemical abundances and an age estimate to reveal the system in more detail.

TOI-1052b is a Neptune-mass planet with a sub-Neptune radius, with a potentially eccentric 9.13-d orbit. The planet's density of $3.93_{-1.3}^{+1.7} \text{ g cm}^{-3}$ implies a composition denser with more heavy elements than Neptune. Limiting case layered interior models show a degeneracy between a rocky planet with a 2 per cent H-He atmosphere and a water-rich planet with a water mass fraction of 0.43.

The companion planet TOI-1052c shows an $M_p \sin i$ of $34.3_{-3.7}^{+4.1} M_{\oplus}$, approx. double the mass of planet b, and orbits on a 35.8-d period. Given its presence near the 4:1 mean-motion resonance, and the potential eccentricity of both planets, the system provides an interesting case study for dynamical interactions.
## ACKNOWLEDGEMENTS

Based on observations collected at the La Silla Observatory, ESO (Chile), with the HARPS spectrograph at the 3.6-m telescope for programs 1102.C - 0249(*F*), 106.21*TJ*.001, and 105.20*G9*.001.

DJA is supported by UKRI through the STFC (ST/R00384X/1) and EPSRC (EP/X027562/1). AO and FH are funded by an STFC studentship. Co-funded by the European Union (ERC, FIERCE, 101052347). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. This work was supported by FCT – Fundação para a Ciência e a Tecnologia through national funds and by FEDER through COMPETE2020 – Programa Operacional Competitividade e Internacionalização by these grants: UIDB/04434/2020; UIDP/04434/2020. VA acknowledges the support from FCT through the following grant number: 2022.06962.PTDC. SGS acknowledges the support from FCT through Investigador FCT contract nr. CEECIND/00826/2018 and POPH/FSE (EC). EDM acknowledges the support from FCT through Investigador FCT contract nr. 2021.01294.CEECIND. HK and RH carried out this work within the framework of the NCCR PlanetS supported by the Swiss National Science Foundation under grants 51NF40_182901 and 51NF40_205606. CD acknowledges support from the Swiss National Science Foundation under grant number PZ00P2_174028. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement SCORE Number 851555). JLB is partly funded by grants LCF/BQ/PI20/11760023, Ramón y Cajal fellowship with code RYC2021-031640-I, and the Spanish MCIN/AEI/10.13039/501100011033 grant number PID2019-107061GB-C61. SH acknowledges CNES funding through the grant number 837319.

This work made use of *tpfplotter* by J. Lillo-Box (publicly available in [www.github.com/jlillo/tpfplotter](https://www.github.com/jlillo/tpfplotter), which also made use of the python packages *astropy*, *lightkurve*, *matplotlib*, and * numpy*.

Funding for the *TESS* mission is provided by NASA's Science Mission Directorate. We acknowledge the use of public *TESS* data from pipelines at the *TESS* Science Office and at the *TESS* Science Processing Operations Centre. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Centre for the production of the SPOC data products. This research has made use of the Exoplanet Follow-up Observation Program website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.

Some of the observations in this paper made use of the High-Resolution Imaging instrument Zorro and were obtained under Gemini LLP Proposal Number: GN/S-2021A-LP-105. Zorro was funded by the NASA Exoplanet Exploration Program and built at the NASA Ames Research Centre by Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett Quigley. Zorro was mounted on the Gemini South telescope of the international Gemini Observatory, a program of NSF's OIR Lab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation on behalf of the Gemini partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional

de Investigación y Desarrollo (Chile), Ministerio de Ciencia, Tecnología e Innovación (Argentina), Ministério da Ciência, Tecnologia, Inovações e Comunicações (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea).

This work has made use of data from the European Space Agency (ESA) mission *Gaia* (https://www.cosmos.esa.int/gaia), processed by the *Gaia* Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the *Gaia* Multilateral Agreement.

## 7 DATA AVAILABILITY

*TESS* data are accessible via the MAST (Mikulski Archive for Space Telescopes) portal at https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html. Imaging data from Zorro are accessible via the ExoFOP-TESS archive at https://exofop.ipac.caltech.edu/tess/target.php?id = 317060587. The exoplanet modelling code and associated python scripts for parameter analysis and plotting are available upon reasonable request to the author. RV data are presented in Table A1.

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## APPENDIX A: HARPS SPECTROSCOPY
<table><caption>Table A1. HARPS spectroscopy.</caption><thead><tr><th>Time<br>[BJD - 2457000]</th><th>RV</th><th>σ<sub>RV</sub></th><th>FWHM<br>[m s<sup>-1</sup>]</th><th>σ<sub>FWHM</sub></th><th>Contrast</th><th>Bis span</th><th>S<sub>MW</sub></th><th>Ha</th></tr></thead><tbody><tr><td>2358.879</td><td>54939.1291</td><td>1.7636</td><td>9116.2436</td><td>1.6357</td><td>33.8180</td><td>28.8576</td><td>0.0939</td><td>0.1816</td></tr><tr><td>2359.889</td><td>54942.2963</td><td>1.8403</td><td>9120.0475</td><td>1.6576</td><td>33.8104</td><td>39.9679</td><td>0.0952</td><td>0.1776</td></tr><tr><td>2360.895</td><td>54941.4163</td><td>1.9099</td><td>9116.7254</td><td>2.3186</td><td>33.7439</td><td>28.4139</td><td>0.0929</td><td>0.1771</td></tr><tr><td>2361.823</td><td>54947.1419</td><td>1.9352</td><td>9119.1708</td><td>1.9099</td><td>33.7593</td><td>39.5333</td><td>0.0945</td><td>0.1767</td></tr><tr><td>2368.779</td><td>54936.1522</td><td>1.4481</td><td>9116.7558</td><td>1.7636</td><td>33.7817</td><td>35.7092</td><td>0.0961</td><td>0.1808</td></tr><tr><td>2368.892</td><td>54938.6397</td><td>1.4729</td><td>9112.4784</td><td>1.8159</td><td>33.7737</td><td>35.3361</td><td>0.0976</td><td>0.1771</td></tr><tr><td>2369.822</td><td>54941.4063</td><td>1.7946</td><td>9118.8143</td><td>1.5630</td><td>33.7737</td><td>31.2242</td><td>0.0950</td><td>0.1767</td></tr><tr><td>2369.901</td><td>54943.1930</td><td>2.7717</td><td>9111.5158</td><td>1.5019</td><td>33.7755</td><td>32.1908</td><td>0.0965</td><td>0.1773</td></tr><tr><td>2370.791</td><td>54946.6359</td><td>2.1584</td><td>9119.6451</td><td>1.7115</td><td>33.8064</td><td>30.9872</td><td>0.0948</td><td>0.1759</td></tr><tr><td>2370.916</td><td>54948.0868</td><td>1.7343</td><td>9112.1726</td><td>2.0355</td><td>33.8194</td><td>38.4706</td><td>0.0926</td><td>0.1768</td></tr><tr><td>2371.810</td><td>54947.3120</td><td>2.1132</td><td>9119.1244</td><td>1.5953</td><td>33.7795</td><td>33.1299</td><td>0.0950</td><td>0.1812</td></tr><tr><td>2371.890</td><td>54945.3488</td><td>2.4833</td><td>9123.6366</td><td>1.5901</td><td>33.8088</td><td>34.4833</td><td>0.0929</td><td>0.1815</td></tr><tr><td>2372.771</td><td>54948.3852</td><td>2.9147</td><td>9118.4646</td><td>1.9352</td><td>33.8020</td><td>40.1501</td><td>0.0922</td><td>0.1812</td></tr><tr><td>2372.896</td><td>54946.2718</td><td>2.7934</td><td>9117.4986</td><td>1.6977</td><td>33.8151</td><td>33.1816</td><td>0.0969</td><td>0.1818</td></tr><tr><td>2376.863</td><td>54947.9693</td><td>1.5630</td><td>9119.9399</td><td>1.7149</td><td>33.7617</td><td>41.4132</td><td>0.0975</td><td>0.1820</td></tr><tr><td>2378.821</td><td>54951.3024</td><td>2.0355</td><td>9107.8579</td><td>1.6013</td><td>33.7624</td><td>37.8135</td><td>0.0971</td><td>0.1769</td></tr><tr><td>2405.829</td><td>54938.4662</td><td>2.2880</td><td>9124.1644</td><td>1.8716</td><td>33.7738</td><td>39.6036</td><td>0.0950</td><td>0.1820</td></tr><tr><td>2409.794</td><td>54946.2449</td><td>1.8159</td><td>9102.3543</td><td>2.4664</td><td>33.8562</td><td>25.0392</td><td>0.1001</td><td>0.1834</td></tr><tr><td>2412.796</td><td>54945.5650</td><td>1.7984</td><td>9126.1584</td><td>1.6767</td><td>33.7726</td><td>28.3786</td><td>0.0964</td><td>0.1818</td></tr><tr><td>2419.683</td><td>54954.3765</td><td>1.8323</td><td>9122.0819</td><td>2.1584</td><td>33.6817</td><td>37.0738</td><td>0.0968</td><td>0.1827</td></tr><tr><td>2419.789</td><td>54956.1376</td><td>1.6042</td><td>9113.9300</td><td>1.7343</td><td>33.7186</td><td>37.8870</td><td>0.0959</td><td>0.1820</td></tr><tr><td>2421.707</td><td>54943.2344</td><td>1.4407</td><td>9126.5836</td><td>2.1132</td><td>33.7816</td><td>34.4838</td><td>0.0946</td><td>0.1771</td></tr><tr><td>2421.806</td><td>54947.2486</td><td>1.7179</td><td>9131.7791</td><td>2.4833</td><td>33.7808</td><td>36.0406</td><td>0.0952</td><td>0.1770</td></tr><tr><td>2422.712</td><td>54945.2644</td><td>2.5884</td><td>9132.1542</td><td>2.9147</td><td>33.7533</td><td>35.6213</td><td>0.0899</td><td>0.1826</td></tr><tr><td>2422.800</td><td>54945.6395</td><td>1.9274</td><td>9134.2216</td><td>2.7934</td><td>33.7735</td><td>43.7364</td><td>0.0956</td><td>0.1815</td></tr><tr><td>2422.935</td><td>54942.7393</td><td>2.7407</td><td>9114.8738</td><td>2.5884</td><td>33.7706</td><td>35.6914</td><td>0.0915</td><td>0.1817</td></tr><tr><td>2423.842</td><td>54941.5699</td><td>1.5777</td><td>9116.5795</td><td>1.9274</td><td>33.7454</td><td>37.4534</td><td>0.0947</td><td>0.1819</td></tr><tr><td>2423.902</td><td>54942.5696</td><td>2.0993</td><td>9133.7634</td><td>2.7407</td><td>33.7231</td><td>33.5343</td><td>0.0921</td><td>0.1775</td></tr><tr><td>2424.785</td><td>54948.8373</td><td>1.6977</td><td>9117.0484</td><td>2.0096</td><td>33.7747</td><td>26.9491</td><td>0.0998</td><td>0.1783</td></tr><tr><td>2425.677</td><td>54953.2174</td><td>2.0096</td><td>9119.6364</td><td>2.1990</td><td>33.7880</td><td>32.6183</td><td>0.0971</td><td>0.1787</td></tr><tr><td>2425.794</td><td>54948.1586</td><td>2.1990</td><td>9109.2590</td><td>1.8323</td><td>33.7758</td><td>35.9231</td><td>0.0987</td><td>0.1820</td></tr><tr><td>2429.803</td><td>54944.2671</td><td>2.4664</td><td>9119.3548</td><td>1.6042</td><td>33.7349</td><td>30.1954</td><td>0.0972</td><td>0.1804</td></tr><tr><td>2443.632</td><td>54948.4897</td><td>1.5019</td><td>9133.8345</td><td>2.0077</td><td>33.7390</td><td>38.6013</td><td>0.0981</td><td>0.1761</td></tr><tr><td>2443.730</td><td>54946.2780</td><td>1.7115</td><td>9120.9396</td><td>1.5777</td><td>33.7542</td><td>28.5068</td><td>0.0941</td><td>0.1802</td></tr><tr><td>2444.647</td><td>54947.6991</td><td>2.8436</td><td>9122.2819</td><td>2.0993</td><td>33.7997</td><td>29.5741</td><td>0.0958</td><td>0.1811</td></tr><tr><td>2444.727</td><td>54952.9344</td><td>3.0458</td><td>9109.2394</td><td>1.7946</td><td>33.7825</td><td>27.8161</td><td>0.0935</td><td>0.1818</td></tr><tr><td>2449.604</td><td>54943.1374</td><td>2.0077</td><td>9104.2840</td><td>2.7717</td><td>33.7045</td><td>36.4253</td><td>0.0930</td><td>0.1806</td></tr><tr><td>2460.746</td><td>54941.9471</td><td>1.6576</td><td>9118.0781</td><td>2.2880</td><td>33.7604</td><td>32.3109</td><td>0.0967</td><td>0.1798</td></tr><tr><td>2461.767</td><td>54949.5535</td><td>1.7149</td><td>9125.4479</td><td>3.1523</td><td>33.7601</td><td>35.3355</td><td>0.0987</td><td>0.1807</td></tr><tr><td>2462.793</td><td>54945.9223</td><td>1.6357</td><td>9124.2665</td><td>2.2858</td><td>33.7449</td><td>27.4409</td><td>0.0952</td><td>0.1812</td></tr><tr><td>2463.802</td><td>54948.8167</td><td>1.8716</td><td>9122.9704</td><td>3.3019</td><td>33.7807</td><td>45.4873</td><td>0.0962</td><td>0.1752</td></tr><tr><td>2464.744</td><td>54946.3708</td><td>1.6013</td><td>9111.8036</td><td>2.6251</td><td>33.8372</td><td>31.8717</td><td>0.0916</td><td>0.1789</td></tr><tr><td>2467.665</td><td>54939.5406</td><td>2.5859</td><td>9114.6408</td><td>1.4407</td><td>33.7794</td><td>33.4307</td><td>0.0974</td><td>0.1811</td></tr><tr><td>2471.654</td><td>54941.8031</td><td>1.9095</td><td>9120.8601</td><td>1.7179</td><td>33.7775</td><td>37.4413</td><td>0.0975</td><td>0.1815</td></tr><tr><td>2473.559</td><td>54941.1851</td><td>1.5953</td><td>9108.0420</td><td>1.4481</td><td>33.7786</td><td>29.5206</td><td>0.0994</td><td>0.1817</td></tr><tr><td>2473.609</td><td>54944.5064</td><td>1.5901</td><td>9122.1321</td><td>1.4729</td><td>33.7577</td><td>28.3784</td><td>0.0995</td><td>0.1818</td></tr><tr><td>2474.613</td><td>54944.8060</td><td>2.3186</td><td>9130.5444</td><td>1.7984</td><td>33.7461</td><td>30.6980</td><td>0.0982</td><td>0.1825</td></tr><tr><td>2475.583</td><td>54940.0604</td><td>1.8033</td><td>9111.5859</td><td>1.8403</td><td>33.7598</td><td>24.7552</td><td>0.0992</td><td>0.1776</td></tr><tr><td>2476.602</td><td>54938.6456</td><td>2.2858</td><td>9128.9792</td><td>2.8436</td><td>33.5903</td><td>39.6079</td><td>0.0982</td><td>0.1829</td></tr><tr><td>2477.579</td><td>54935.1412</td><td>3.3019</td><td>9116.9772</td><td>3.0458</td><td>33.6390</td><td>30.4956</td><td>0.0962</td><td>0.1800</td></tr><tr><td>2477.694</td><td>54933.4389</td><td>2.6251</td><td>9124.6195</td><td>1.8033</td><td>33.7387</td><td>30.3916</td><td>0.0986</td><td>0.1819</td></tr><tr><td>2478.739</td><td>54939.0504</td><td>3.1523</td><td>9112.7792</td><td>1.9095</td><td>33.7201</td><td>40.3934</td><td>0.0980</td><td>0.1817</td></tr><tr><td>2479.654</td><td>54941.7172</td><td>1.6767</td><td>9124.1326</td><td>2.5859</td><td>33.7251</td><td>37.7955</td><td>0.1023</td><td>0.1830</td></tr></tbody></table>

<footnote n="1">¹Department of Physics, University of Warwick, Coventry CV4 7AL, UK</footnote>
²Centre for Exoplanets and Habitability, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK

³Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal

⁴Laboratoire d'Astrophysique de Marseille, Pole de l'Etoile Site de Chateau-Gombert, 38 Rue Frederic Joliot-Curie, Marseille 13338, France

⁵NASA Ames Research Center, Moffett Field, CA 94035, USA

⁶Institute for Computational Science, Center for Theoretical Astrophysics & Cosmology, University of Zurich, Winterthurerstr. 90, 8057 Zurich, Switzerland

⁷Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235, USA

⁸Centre for Space Domain Awareness, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK

⁹Department of Astronomy of the University of Geneva, Geneva Observatory, Chemin Pegasi 51, 1290 Versoix, Switzerland

¹⁰Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

¹¹Caltech/IPAC, Mail Code 100-22, 1200 E. California Blvd. Pasadena, CA 91125, USA

¹²Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

¹³Centro de Astrobiología (CAB, CSIC-INTA), Depto. de Astrofísica, ESAC campus, 28692, Villanueva de la Cañada (Madrid), Spain

¹⁴European Southern Observatory, Karl-Schwarzschild-Straße 2, 85748 Garching bei München, Germany

¹⁵Physikalisches Institut, University of Bern, Gesellsschaftstrasse 6, 3012 Bern, Switzerland

¹⁶Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA

¹⁷Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 4169-007 Porto, Portugal

¹⁸Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

¹⁹Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

²⁰SETI Institute, Mountain View, CA 94043, USA

²¹Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA

This paper has been typeset from a TEX/LATEX file prepared by the author.