A resonant sextuplet of sub-Neptunes transiting the bright star HD 110067

Abstract

Planets with radii between that of the Earth and Neptune (hereafter referred to as ‘sub-Neptunes’) are found in close-in orbits around more than half of all Sun-like stars^1,2. However, their composition, formation and evolution remain poorly understood³. The study of multiplanetary systems offers an opportunity to investigate the outcomes of planet formation and evolution while controlling for initial conditions and environment. Those in resonance (with their orbital periods related by a ratio of small integers) are particularly valuable because they imply a system architecture practically unchanged since its birth. Here we present the observations of six transiting planets around the bright nearby star HD 110067. We find that the planets follow a chain of resonant orbits. A dynamical study of the innermost planet triplet allowed the prediction and later confirmation of the orbits of the rest of the planets in the system. The six planets are found to be sub-Neptunes with radii ranging from 1.94R_⊕ to 2.85R_⊕. Three of the planets have measured masses, yielding low bulk densities that suggest the presence of large hydrogen-dominated atmospheres.

Main

HD 110067 (TIC 347332255) is a bright K0-type star in the constellation of Coma Berenices with mass and radius of approximately 80% of the Sun’s. The Transiting Exoplanet Survey Satellite (TESS) monitored HD 110067 as part of its observations of Sector 23 (ref. ⁴). The data, processed by the TESS Science Processing Operations Center (SPOC)⁵, exhibited several dips that could be associated with transiting planets. The SPOC reported two candidates: one with an orbital period of 5.642 days based on three dips with apparently similar depth and duration, and a second with an unconstrained orbital period based on a single event. The TESS reobserved the star 2 years later in Sector 49, revealing nine further transits incompatible with the previously announced candidates (Fig. 1).

Fig. 1: Space-based photometry from the TESS and the CHEOPS of HD 110067.

Transits identified for each planet after our analyses are each associated with a different colour. a–f, Photometric time series of TESS Sector 23 (a,b, 2-min cadence), TESS Sector 49 (c,d, 20-s cadence) and CHEOPS (e,f). Red points in a show the reprocessed data affected by scattered light and high levels of sky background. Time units are in TESS Julian Date (TJD ≡ BJD − 2457000), in which BJD is the Barycentric Julian Date in units of days. Detrended light curves and the final model are shown in b,d,f. g–l, TESS phase-folded transits of each planet in the system. The transit model and binned photometry are colour-coded following the same convention. For HD 110067 b (g) and d (i), CHEOPS photometry is shown atop the TESS photometry with an arbitrary offset for clarity. m, Top-down view of the planetary system. Axes show the distance from the central star as a function of the semimajor axis and equilibrium temperature of the planet.

Combining TESS Sectors 23 and 49, we could associate a fraction of the transits with two new planet candidates: HD 110067 b, a planet with an orbital period of 9.114 days, and HD 110067 c, a planet with an orbital period of 13.673 days. For the remaining unidentified transit events (two in Sector 23 and four in Sector 49), we attempted to match them by modelling them individually using a purely shape-based transit model agnostic to the orbital period (Methods and Extended Data Fig. 1). Then we compared them in duration–depth space, allowing us to identify two ‘duo transits’ (planets seen to transit only once in each of the two widely separated sectors) and two single transits (solitary transit events seen only in Sector 49). Duo transits have orbital periods limited to a finite number of harmonics or aliases that are constrained at the short-period end by the extent of continuous photometry observed before/after transit and at the long end by the distance between transits. Targeted observations with the CHaracterising ExOPlanets Satellite (CHEOPS)⁶ allowed us to rule out many of these aliases and confirm a third planet in the system, HD 110067 d, with an orbital period of 20.519 days (Fig. 1).

The orbital periods of planets HD 110067 b, c and d (9.114, 13.673 and 20.519 days, respectively) have ratios very close to 3/2 (P_c/P_b = 1.5003 and P_d/P_c = 1.5007). Mean-motion resonances (MMRs) are orbital configurations in which the period ratio of a pair of planets is oscillating near a rational number of the form (k + q)/k, in which k and q are integers. First-order MMRs (in which q = 1) are the most common among planetary systems, as well as resonant chains. For the two innermost pairs of planets (bc and cd), we define two resonant angles ϕ₁ = 2λ_b − 3λ_c + ϖ_c and ϕ₂ = 2λ_c − 3λ_d + ϖ_c, in which λ is the mean longitude of the planets and ϖ its longitude of periastron. Because dλ_x/dt = 2π/P_x, and given the aforementioned period ratios between b, c and d, a generalized Laplace relation links the triplet of planets through 2/P_b − 5/P_c + 3/P_d ≈ 0 (refs. ^7,8,9). This relation ensures that the associated Laplace angle Ψ_bcd = ϕ₁ − ϕ₂ = 2λ_b − 5λ_c + 3λ_d evolves slowly (dΨ_bcd/dt ≈ 0), indicating that the three planets might indeed be trapped in a Laplace chain of 3/2 resonances.

We explored the possibility that the remaining ‘unmatched’ dips correspond to planets that continue the first-order generalized Laplace resonant chain of HD 110067 b, c and d. Using a Laplace relation similar to that above, we can predict the possible orbital period of the planets, in a similar fashion to the discoveries of TOI-178 f and TRAPPIST-1 h (refs. ^10,11). We assume that the next planet in the system must continue the generalized Laplace chain of first-order resonances. Considering the most common first-order MMRs (2/1, 3/2, 4/3, 5/4, 6/5), the only continuation of the chain that matches the spare duo transit supports the presence of a fourth planet in the system, HD 110067 e, with an orbital period of 30.7931 days (in a 3/2 MMR with planet d).

We are then left with two mono transits in TESS Sector 49, which we assume to correspond to two separate planets f and g that continue the chain. With single transits, we cannot use the same method as above and must introduce a new argument. In nature, all known three-body resonant chains are close to an equilibrium, that is, an equilibrium value of their angles Ψ (refs. ^10,12,13,14) (Extended Data Fig. 2). Assuming low eccentricities, only the period and a single transit of each planet are required to estimate the value of Ψ. We can therefore try out different periods for planets f and g, then use their single transits to estimate Ψ_def and Ψ_efg and see if it lands close to an equilibrium of the chain. Here we try the same set of first-order MMRs (2/1, 3/2, 4/3, 5/4, 6/5) between e and f and between f and g (50 combinations in total; see Methods). Of those, the only combination not excluded by existing data that can be close to an equilibrium of the chain is that in which P_f/P_e = 4/3 and P_g/P_f = 4/3, yielding the three outer generalized Laplace angles at less than 20° from the closest equilibrium (Methods, Extended Data Table 4 and Extended Data Fig. 3).

According to this prediction, planets HD 110067 f and g would have orbital periods of 41.0575 and 54.7433 days, respectively. If so, both planets would have transited during TESS Sector 23 observations, but during a time when the effect of scattered light and the relative contributions of the Earth and Moon to the image backgrounds were highly relevant. These frames are typically discarded, but a detailed reprocessing of the TESS Sector 23 observations triggered by this prediction showed two further transit events at 1943.6 TJD (TESS Julian Date, TJD ≡ BJD − 2457000) and 1944.1 TJD, matching exactly our model based on the resonant dynamics of the system (Fig. 1). Furthermore, a ground-based multi-instrument photometric campaign to catch a predicted transit of HD 110067 f (Methods and Extended Data Fig. 4) recovered a statistically significant detection (ΔWAIC = 9.5, in which WAIC is the Watanabe–Akaike information criterion) with a consistent depth and duration at the expected time, confirming its orbital period.

Also, we collected high-precision radial velocities of the star with the CARMENES (ref. ¹⁵) and HARPS-N (ref. ¹⁶) instruments. The dominant signals in the data correspond to the magnetic activity of the star rather than the planetary companions but, by applying state-of-the-art analyses to model stellar activity (Methods), we could independently confirm the detection of HD 110067 f, measuring an orbital period and phase (using agnostic priors on both quantities) that matched the transits. We used the radial velocities of the system to measure precise masses for three of the planets (HD 110067 b, d and f) and place upper limits on the remaining ones (Fig. 2). Although transit timing variations caused by the mutual gravitational interactions of the planets are expected in resonant systems¹⁷, our photometric analysis did not measure any substantial deviation from the linear ephemeris (below 5 min for the inner triplet), probably because of the low number of individual transits for each planet. Further monitoring of the system will enable an independent measurement of the planetary masses using this technique.

Fig. 2: Radial velocity data from CARMENES and HARPS-N of HD 110067.

Time units are the same as Fig. 1. a–e, Radial velocity (RV) and full-width at half-maximum (FWHM) time series after being corrected by inferred offsets. Each panel shows: RV data together with full, stellar and planetary inferred models (a); RV data with the stellar model subtracted (b); RV residuals (c); FWHM data together with the inferred stellar model (d); and FWHM residuals (e). HARPS-N (blue) and CARMENES (orange) measurements are shown by solid points with 1σ error bars, with a semitransparent error bar extension accounting for the inferred jitter. The solid lines show the inferred full model coming from our multidimensional Gaussian process model (Methods) and the lightly shaded areas show the 1σ and 2σ credibility intervals of the corresponding model. For the RV time series in a, we also show the inferred stellar (red) and planetary (green) recovered signals with an offset for clarity. f–k, Phase-folded RV signals for all the planets following the subtraction of the systemic velocities, stellar signal and other planets. Nominal RV observations are shown as light-grey points. Solid points show data binned to a tenth of the orbital phase. The inferred model is shown with a solid line following the colour-coding of Fig. 1. The planets with mass-measurement uncertainties smaller than 3σ (f,h,j) are marked with thicker lines.

The HD 110067 planetary system thus comprises at least six transiting planets orbiting in a chain of first-order MMRs (3/2—3/2—3/2—4/3—4/3). The planets have radii ranging between 1.94 and 2.85 times the radius of the Earth, orbital periods between 9 and 55 days and equilibrium temperatures between 440 and 800 K (Fig. 3). Thanks to the constraints on the planetary masses and their location above the radius valley^18,19, our internal composition modelling concludes that all the planets in the system (with the exception perhaps of planet e, which remains undetected in the radial velocity data) must have large hydrogen-dominated atmospheres to explain their relatively low bulk densities (see Supplementary Information). A summary of the most relevant properties of the system is presented in Table 1. We note that for both planets e and g, their orbital period measurement relies on a prediction based on the dynamical properties of the system, but an independent third transit observation confirming each orbit has not yet been obtained. Given the low mutual inclination of the system (less than 1°), further transiting planets may yet be found at periods longer than 70 days, which would correspond to orbits within or beyond the habitable zone of the star^20,21.

Fig. 3: Properties of the HD 110067 system compared with the known sub-Neptune-sized planet population.

a, Period–radius diagram. b, Mass–radius diagram. Mass–radius relationships for different planet compositions are taken from ref. ²⁴. The 3σ mass upper limits are shown for planets HD 110067 c, e and g, with 1σ error bars for the rest. c, Host-star brightness in J-band magnitude as a function of planet equilibrium temperature showing systems with five or more transiting planets. Lower magnitudes indicate brighter stars. In all plots, point size is proportional to the planet radius, whereas the colour represents a proxy of the expected atmospheric scale height in transmission spectroscopy using the JWST following the metric of ref. ²⁵. When mass is not known, we use the empirical mass–radius relation of ref. ²⁶ to compute this metric. Planet population properties were retrieved from the NASA Exoplanet Archive on May 2023.

Table 1 Summary of the planetary parameters of the HD 110067 system

From an observational point of view, HD 110067 is the brightest star found to host more than four transiting exoplanets. The current delicate configuration of the planetary orbits in HD 110067 rules out any violent event over the billion-year history of the system²², making it a rare ‘fossil’²³ to study migration mechanisms and the properties of its protoplanetary disk in a pristine environment. The combination of host-star brightness and the inferred presence of extended atmospheres in most of its planets makes HD 110067 the most favourable multiplanetary sub-Neptune system to be observed in transmission spectroscopy with the James Webb Space Telescope (JWST) (Fig. 3). HD 110067 offers a chance to gain insight into the nature of sub-Neptune planets and where, how and under what conditions resonant chains form and survive.

Methods

Data

TESS photometry

HD 110067 was observed in TESS Sectors 23 and 49 (18 March 2020 to 16 April 2020 and 26 February 2022 to 26 March 2022, respectively) (ref. ⁴). The star was included on the TESS Candidate Target List (CTL)²⁷ and therefore the target was observed at 120-s cadence in both sectors. The target images are processed by the TESS SPOC pipeline at NASA Ames⁵, which calibrates the pixels, performs simple aperture photometry (SAP), flags poor-quality data and removes systematic trends to create the so-called ‘Presearch Data Conditioning’ light curve (PDCSAP)^28,29,30. Finally, the SPOC pipeline runs a wavelet-based transiting planet search for periodic exoplanets^31,32,33, which—in the case of the Sector 23 data—revealed two threshold-crossing events (TCEs)—that is, candidate planets—which passed data validation checks^34,35. Manual vetting of these two candidates resulted in the assignment of two TESS Objects of Interest (TOI), namely, TOI-1835.01 and TOI-1835.02 (ref. ³⁶).

The first TCE was alerted by SPOC as TOI-1835.01 with a period of 5.641 days. Although two of the three transits associated with this ephemeris seemed to be of similar depth and duration, the third did not seem to be associated with a clear transit. However, this could have been because of the proximity to a systematic dip caused by a momentum dump. A second TCE, TOI-1835.02, was alerted as a single transit at 1948.98 TJD, as opposed to the period of 11.107 days proposed by the SPOC owing to an apparent discrepancy in the transit depth between the two transits purportedly linked by the transiting planet search. On the other hand, a peak in the background flux ruled out the planetary origin of an apparent transit feature at TJD = 1940.5. Both the depth and shape of this feature were strongly dependent on the detrending method used in the light curve, which is indicative of its spurious nature.

HD 110067 was later reobserved by TESS in Sector 49. To confirm the TCEs and plan immediate follow-up, we downloaded the TESS Image CAlibrator Full Frame Images (TICA FFI)³⁷ only a week after downlink for each of the first and then second orbits. We computed a light curve from the 10-min cadence TICA FFI cut-outs using SAP and clipped regions of high brightness because of the Earth and Moon, as well as parts of the light curve affected by systematics such as momentum dumps. The first orbit alone revealed at least five new clear-transit features. These all seemed to have varying depths and durations and none were compatible with the 5.64-day period implied by TOI-1835.01. The second orbit showed three more clear transit events, making a total of eight in Sector 49 and five in Sector 23. A 20-s cadence target pixel file was later made available for HD 110067, which resulted in higher-precision photometry.

Manual vetting of both Sectors 23 and 49 PDCSAP light curves revealed unexpectedly large systematic uncertainties (Gini’s mean difference, that is, the average point-to-point absolute difference, of 215 ppm in 3-h bins) that were absent from the uncorrected SAP flux. This has been seen in many bright stars that have stellar variability (for example, refs. ^38,39). To better correct these systematics, we performed a custom extraction of the TESS light curves for both sectors using the quaternion detrending technique against spacecraft motion developed in ref. ⁴⁰. This involved fitting a model consisting of a linear combination of a basis spline (with breakpoints spaced every 1.5 days to model long-timescale stellar or instrumental variability) and decorrelation with parameters linked to systematic flux changes, namely, the means and standard deviations of the spacecraft quaternion time series (and the squared time series) within each exposure. Using a linear least squares technique (matrix inversion), we solved for the best-fit coefficients of our free parameters while iteratively excluding 3σ outliers from the fit until convergence was reached. After calculating the best-fit model for the systematics of each aperture, we then subtracted it from the uncorrected light curve and identified the aperture that produced the light curve with the lowest photometric scatter. The final light curves used in our subsequent analyses (Gini’s mean difference of 130 ppm in 3-h bins) are shown in Fig. 1.

Finally, as discussed above, a large portion of the data was missing from the PDCSAP light curves because of high levels of scattered light and sky background from the Earth and Moon. The dates affected coincide with the potential transit events of planets f and g based on our dynamical model prediction. Therefore, to recover data affected by scattered light, we performed a custom extraction of the TESS light curves for both sectors using a pixel level decorrelation (PLD) method^41,42,43 implemented in the PLDCorrector class of the community Python package lightkurve (ref. ⁴⁴). This method uses: (1) a spline polynomial fit to describe stellar variability; (2) principal component analysis (PCA) eigenmodes to model the background light; and (3) the PLD technique to account for pointing and mechanical effects. Before applying the PLDCorrector, we add the background flux and errors estimated by the TESS SPOC pipeline back onto the SAP light curve. Flux level, fraction and crowding adjustments are then applied to the corrected light curve. To automatically optimize the selection of parameter values for the PLDCorrector, we evaluate the resulting light curve using the Savitzky–Golay combined differential photometric precision (sgCDPP) proxy algorithm^45,46 implemented in lightkurve. For a grid of PLDCorrector parameter values, we calculate the harmonic mean of these quantities and select the corrected light curve that minimizes it. We use these data for the cadences missing in the quaternion-detrended light curve in our final analyses (marked with a different colour in Fig. 1).

CHEOPS photometry

The CHEOPS mission is a European Space Agency small-class mission dedicated to studying bright, nearby exoplanet host stars for the purpose of making high-precision photometric observations of transiting planets⁶. We collected 19 separate visits of HD 110067 with the CHEOPS between 11 April 2022 and 17 May 2022 under Guaranteed Time Observing programmes ID-048 and ID-031. The goal of these observations is (1) to confirm the true orbital period of single-transiting and duo-transiting planet candidates and (2) to improve the planetary radius precision and ephemeris of confirmed planets. This has been done for large planets producing deep eclipses from the ground^47,48 and for small planets from space^39,49,50. An observing log summarizing the duration of each visit, its average observing efficiency (considering the gaps produced by Earth occultations or passages over the South Atlantic Anomaly along the low-Earth orbit of the spacecraft) and photometric precision are presented in Extended Data Table 1.

To provide the highest-quality photometric precision, we opted to perform custom photometric extraction of the CHEOPS imagettes using point-spread-function (PSF) photometry as implemented by the PIPE package^51,52. For bright targets such as HD 110067, light curves generated with PIPE exhibit lower median absolute differences than those generated by the CHEOPS Data Reduction Pipeline⁵³. The shorter cadence of the CHEOPS imagettes allows a higher cadence light curve and PSF detrending is also better at removing trends owing to systematic factors and background stars. As various PSF models have already been generated and vary as a function of stellar temperature, we opted to use a PSF generated using the star HD 189733, with a similar spectral type as HD 110067. To preserve inter-visit flux differences, we normalized the entire CHEOPS data together instead of individually. This revealed clear visit-to-visit flux differences owing to stellar rotation with an amplitude larger than that of the TESS (as stellar activity is typically more pronounced at bluer bandpasses). The final light curves used in our subsequent analyses are shown in Fig. 1 and Supplementary Fig. 1.

Ground-based photometric campaign

We carried out a campaign on the night of 23 May 2022 to attempt to confirm the 41.05-day period orbit of HD 110067 f as predicted by our resonance chain analysis. Photometric observations were taken using 14 telescopes using seven different filters, which observed from various locations to continuously cover a temporal baseline of more than 11 h (between 22:52:55UT 23 May 2022 and 10:01:33UT 24 May 2022). This window is long enough to catch the 5-h transit expected from 02:52UT to 07:12UT. However, no single location was able to cover both ingress and egress. A summary of the observations is shown in Extended Data Table 2. Details from each individual observation are shown below. Extended Data Fig. 4 shows the data and best-fit models as discussed in the section ‘Confirming the predictions’.

Teide Observatory

We observed HD 110067 on 23 May 2022 using the MuSCAT2 instrument installed at the 1.5-m Telescopio Carlos Sánchez (TCS) located at the Teide Observatory, Spain⁵⁴. The images were taken simultaneously in g, r, i and z_s filters with the telescope heavily defocused and with short exposure times of 3–5 s, depending on the band, to avoid saturation. Relative light curves for each band and instrument of HD 110067 were extracted by aperture photometry using a custom pipeline⁵⁵ with optimal aperture radii of 8.1″ to 11.3″, depending on the band. Note that there was a technical problem on the dome of the TCS between BJD-2459723 = 0.488 and 0.526; we discarded the data taken during this period.

We also observed HD 110067 on 23 May 2022 with one of the 1-m telescopes from Las Cumbres Observatory (LCO) global network located at the Teide Observatory, Spain⁵⁶. The observations were obtained through Director’s Discretionary Time programme 2022A-005 (PI: Wilson). We collected 181 frames with an exposure time of 20 s, covering 2.5 h, using the 4,096 × 4,096-pixel SINISTRO camera. The images were calibrated by the standard LCO BANZAI pipeline⁵⁷. Differential photometric data were extracted using AstroImageJ (AIJ)⁵⁸.

Paranal Observatory

We observed HD 110067 on 23 May 2022 using the Next-Generation Transit Survey (NGTS) facility located at the European Southern Observatory’s (ESO) Paranal Observatory in Chile⁵⁹. The NGTS consists of twelve 20-cm, f/2.8 telescopes with Andor cameras and red-sensitive (600–900 nm) deep-depletion e2v CCDs. Nine NGTS telescopes observed from 23:14UT to 04:35UT, covering a predicted transit ingress of HD 110067 f and spanning an air mass range of 1.7–2.5. Two telescopes started observing 2 h late owing to a technical issue. All nine telescopes were defocused to avoid saturating the bright target star during the 10-s exposures. The NGTS camera shutters were not functional and so were kept open during the entire observing block. That caused the stars to streak during the 1.5-s readout sequences but without any apparent detrimental effect on the photometry. Observing without using the shutters is now the standard operation mode of the NGTS. We performed standard differential aperture photometry, using large aperture radii of 6.5–8.0 pixels, and carefully selecting comparison stars to avoid those that exhibited variability. The light curve of each telescope was normalized individually and no detrending was performed.

F. L. Whipple Observatory

We observed HD 110067 on 24 May 2022 using the Tierras instrument installed at the refurbished 1.3-m telescope located at the F. L. Whipple Observatory atop Mount Hopkins, Arizona, USA. The instrument is designed to regularly achieve a photometric precision of 250 ppm on timescales of both 10 min and a complete observing season. The design choices that permit this precision include a four-lens focal reducer and field flattener that increase the field-of-view of the telescope, a custom narrow-bandpass filter centred around 863.5 nm to minimize precipitable water vapour errors and a fully automated mode of operation⁶⁰. A total of 1,262 4-s exposures were gathered with Tierras for HD 110067. Astrometric calibrations were done in real time during data gathering and were stored in WCS headers in the FITS files. The FITS files were then passed through the Tierras image-reduction pipeline to perform bias corrections and image stitching (the CCD chip is read out through separate amplifiers). AIJ was used for photometric extraction. These observations were gathered shortly after Tierras started science operations and the data were not flat-fielded because knowledge of the flat field was incomplete at the time. The r.m.s. of the 15-min binned data is 323 ppm. The photometric precision on this target is ultimately limited by scintillation, as the target was observed down to an air mass of 2.37. The observations were mildly affected by cirrus.

San Pedro Mártir Observatory

We observed HD 110067 on 24 May 2022 with the 1-m SAINT-EX telescope at the Observatorio Astronómico Nacional de la Sierra de San Pedro Mártir in Baja California, Mexico⁶¹. SAINT-EX is equipped with a deep-depleted and back-illuminated Andor IKON CCD and a filter wheel. The observations were defocused and acquired in the ‘zcut’ filter, a custom filter optimized to reduce the systematic uncertainties in the light curves of red stars resulting from precipitable water vapour, with an exposure time of 10 s. The data were reduced with AIJ using the standard corrections for bias, flat-fielding and dark current. AIJ was also used to carry out the aperture photometry of the time series, producing the light curves and relevant metadata. The observations were mildly affected by high-altitude cirrus.

Haleakala Observatory

We observed HD 110067 on 24 May 2022 using the MuSCAT3 instrument mounted on the 2-m Faulkes Telescope North (FTN) at Haleakala Observatory on Maui, Hawaii, USA⁶². The images were taken simultaneously in g, r, i and z_s filters with the telescope heavily defocused and with short exposure times of 3–5 s, depending on the band, to avoid saturation. Relative light curves for each band and instrument were extracted by aperture photometry using a custom pipeline⁶³ with optimal aperture radii of 8.1″ to 11.3″, depending on the band. There was a guiding issue on the FTN around BJD-2459723 = 0.795, which caused a large shift of the stellar positions on the detectors; we treated the MuSCAT3 data as two independent datasets separated by that time.

High-resolution imaging

As part of our standard process for validating transiting exoplanets, and to assess the possible contamination of bound or unbound companions on the derived planetary radii⁶⁴, we observed HD 110067 with near-infrared adaptive optics (AO) imaging at Palomar Observatory and with optical speckle imaging at Gemini North. Gaia DR3 is also used to provide further constraints on the presence of undetected stellar companions and wide companions. No close-in (≲1″) stellar companions were detected by either the near-infrared AO or optical speckle imaging.

Palomar Observatory

The Palomar Observatory observations of HD 110067 were made with the PHARO instrument⁶⁵ behind the natural guide star AO system P3K (ref. ⁶⁶) on 8 January 2020 in a standard five-point quincunx dither pattern with steps of 5″ in the narrow-band Br-γ filter. Each dither position was observed three times, offset in position from each other by 0.5″ for a total of 15 frames, with an integration time of 1.4 s per frame for total on-source times of 21 s. PHARO has a pixel scale of 0.025″ per pixel for a total field of view of approximately 25″. The sensitivities of the final combined AO image were determined by injecting simulated sources azimuthally around the primary target every 20° at separations of integer multiples of the central source’s full width at half maximum (FWHM)⁶⁷. The Palomar data have a sensitivity Δmag = 2 at 0.1″ and Δmag = 9 at 1″; the final sensitivity curve is shown in Supplementary Fig. 2.

Gemini Observatory

We observed HD 110067 with the Alopeke speckle imaging camera at Gemini North on 10 June 2020 (ref. ⁶⁸). We obtained five sets of 1,000 frames, each frame having an integration time of 60 ms, obtaining images in each of the instrument’s two bands (centred at 562 nm and 832 nm). The observations were reduced using our standard software pipeline⁶⁹ and reached a 5σ sensitivity of Δmag = 7 (blue channel) and Δmag = 6.8 (red channel) at separations of 0.5″. The reconstructed speckle images show no evidence of other nearby point sources. The final sensitivity curve is shown in Supplementary Fig. 2.

Gaia Space Observatory

As well as the high-resolution imaging, we have used Gaia to identify any wide stellar companions that may be bound members of the system^70,71. There are no further widely separated companions identified by Gaia that have the same distance and proper motion as HD 110067. Furthermore, the Gaia DR3 astrometry provides extra information on the possibility of inner companions that may have gone undetected by either Gaia or the high-resolution imaging data. The Gaia renormalized unit weight error (RUWE) is a metric, similar to a reduced chi-square, in which values that are ≲1.4 indicate that the Gaia astrometric solution is consistent with the star being single, whereas RUWE values ≳1.4 may indicate an astrometric excess noise, possibly caused by the presence of an unseen companion (for example, ref. ⁷²). HD 110067 has a Gaia EDR3 RUWE value of 0.94, indicating that the astrometric fit is consistent with a single-star model.

Radial velocity monitoring

Calar Alto Observatory

We observed HD 110067 using the CARMENES instrument¹⁵ installed at the 3.5-m telescope of Calar Alto Observatory in Almería, Spain, between 3 July 2020 and 4 July 2021. We collected 39 high-resolution spectra under the observing programmes F20-3.5-011 (PI: Nowak) and H20-3.5-013 (PI: Luque). Radial velocities and further spectral indicators were derived using raccoon (ref. ⁷³) and serval (ref. ⁷⁴). Although the mean internal precision of the template-matching serval radial velocities is 3.1 m s⁻¹, the precision of the cross-correlation method raccoon radial velocities is 2.9 m s⁻¹, so we used the latter in our analyses.

Roque de los Muchachos Observatory

We observed HD 110067 with the HARPS-N spectrograph mounted at the 3.6-m Telescopio Nazionale Galileo¹⁶ of Roque de los Muchachos observatory in La Palma, Spain, between 30 May 2020 and 4 May 2022. We collected 72 high-resolution spectra under the observing programmes CAT19A_162 (PI: Nowak), CAT21A_119 (PI: Nowak) and ITP19_1 (PI: Pallé) that were used to measure the photospheric properties of the star and precise radial velocities. Radial velocities and further spectral indicators were derived using an online version of the DRS pipeline⁷⁵, the YABI tool and serval⁷⁴. Both the YABI-derived and serval-derived radial velocities have a median internal precision of 1.0 m s⁻¹, but we used the YABI ones (based on the cross-correlation method) in our final analyses for consistency with the CARMENES dataset.

Stellar parameters

Photospheric parameters and abundances

To properly characterize the planetary system around HD 110067, we first conduct a series of analyses to determine the properties of the host star. We derive the stellar spectral parameters by applying the widely used ARES+MOOG tools to our co-added HARPS-N spectra^76,77,78. ARES (refs. ^79,80) measures the equivalent widths of iron lines in the spectrum that are converted into stellar atmospheric parameters using the MOOG radiative transfer code⁸¹ applied to Kurucz model atmospheres⁸². In Extended Data Table 3, we report the effective temperature T_eff, surface gravity log g and metallicity [Fe/H] obtained on convergence of ionization and excitation equilibria using this method. Furthermore, we measure the stellar vsin i from the HARPS-N spectra using ZASPE (ref. ⁸³).

We further study the photospheric parameters by conducting a classical curve-of-growth analysis on our co-added HARPS-N spectrum using our aforementioned spectral parameters to obtain [Mg/H] and [Si/H] abundances for HD 110067. Using the ARES+MOOG framework detailed above, we obtain the equivalent widths⁸⁰ for these elements, which are converted to abundances assuming local thermodynamic equilibrium^81,82. The specific details of this analysis are beyond the scope of this paper and can be found in refs. ^84,85. We report the stellar abundances in Extended Data Table 3.

Physical parameters

Using our spectral parameters and the ATLAS^82,86 and PHOENIX⁸⁷ catalogues, we build spectral energy distributions of HD 110067 that we compare to optical and infrared broadband photometry of the star (see Extended Data Table 3) to derive the stellar angular diameter and effective temperature by means of the infrared flux method⁸⁸. This is conducted in a Markov chain Monte Carlo (MCMC) approach^89,90, with which we convert the angular diameter to the stellar radius using the Gaia EDR3 offset-corrected parallax⁹¹ with model uncertainties accounted for using a Bayesian modelling averaging. We report the stellar radius R_⋆ in Extended Data Table 3.

Last, we complete our stellar characterization by determining the mass and age of HD 110067. We constrain two sets of stellar evolutionary models with the help of our derived values for T_eff, log g and R_⋆ (ref. ⁹²). On the one hand, we use an isochrone placement algorithm^93,94 and interpolate over precomputed grids of PARSEC v1.2s (ref. ⁹⁵) isochrones. On the other hand, we use the Code Liègeois d’Évolution Stellaire (CLES)⁹⁶ combined with a Levenberg–Marquardt minimization scheme⁹⁷ to optimize the best-fitting evolutionary track. The results from the two methods are combined to determine the mass and age of the star that is reported in Extended Data Table 3.

The [Fe/H] and age of HD 110067 indicate that this star could belong to the galactic thick disk stellar population or be an older member of the galactic thin disk. The values of [Mg/H] and [Si/H], being within 1σ of [Fe/H], show that the star is not enhanced in α-capture elements and are indicative of a typical thin disk chemical composition. We determined the kinematic properties of HD 110067 by using the Gaia EDR3 astrometry to compute the local standard of rest space velocities of this star following ref. ⁹⁸. From these velocities, we compute that the probability of kinematic membership in the galactic thin disk is 0.9911 ± 0.0029. Thus, we conclude that HD 110067 is on the older, more metal-poor end of the distribution of the galactic thin disk stellar population.

Analysis

Space-based photometry modelling

We performed simultaneous modelling of the space-based photometry. We used the quaternion-detrended TESS data combined with the PLD-detrended data for the missing Sector 23 gaps and the PIPE-detrended CHEOPS data for the three visits containing transits. We built transit models for the six planets with exoplanet (ref. ⁹⁹). Owing to its nature as a rotating telescope on a near-Earth orbit, even PSF-detrended CHEOPS photometry can include systematic trends. However, these typically correlate with other measurements, for example, roll angle, background or contaminant flux. To not bias the transit model and to better propagate uncertainties on the derived parameters, we performed CHEOPS decorrelation alongside our photometric transit modelling. We first fitted each CHEOPS transit individually alongside several possible decorrelation factors, allowing us to assess which decorrelation factors are most useful. This also enabled us to test whether such decorrelation is shared among all CHEOPS visits or individual to a single light curve. From this analysis, we included the following parameters in the linear correlation: position centroids, the second harmonic of the cosine of the roll angle, cos 2Φ, the change in telescope temperature and quadratic trends with the x–y centroids. CHEOPS data have also been known to contain flux trends that vary stochastically as a function of roll angle over shorter frequencies (see, for example, ref. ¹⁰⁰). These are not well removed using simple trigonometric functions, hence we also modelled a flexible spline shared between all visits to model shorter-timescale variation. To incorporate stellar variability, a floating mean and flux trend were also fitted to each CHEOPS visit, as well as an individual jitter term.

Informative priors were used on limb-darkening parameters using the theoretical quadratic limb-darkening parameters for TESS (ref. ¹⁰¹) and CHEOPS (ref. ¹⁰²), with uncertainties inflated to 0.1 in all cases to guard against systematic offsets. The impact parameter and radius ratio are fitted from a broad uniform and log-normal prior, respectively, whereas the period and mid-transit epoch are fitted using broad normal priors from the transits identified and modelled above. Stellar parameters from Extended Data Table 3 were used as inputs to the model with Gaussian priors. Orbits were assumed to be circular in all cases, which is a good approximation for planetary systems with several transiting planets^103,104,105. The prior and posterior distributions of each parameter in the model are shown in Supplementary Table 1.

Properties of the unmatched transits

Our first modelling of the TESS space-based photometry was able to account for a total of five transits of planet b (two in TESS Sector 23 and three in TESS Sector 49) and four transits of planet c (two each in TESS Sectors 23 and 49). However, this analysis left six ‘unmatched’ transits in the original TESS light curves. To pair the transits, we fitted each transit individually using a purely shape-based transit model agnostic to the orbital period using MonoTools¹⁰⁶. From this analysis, we then compared each transit in duration–depth space, allowing us to clearly see that both transits from Sector 23 shared unique regions of this parameter space with two more transits seen in Sector 49 (duo transits), whereas the two longest-duration transits seen only in Sector 49 were solitary (mono transits). Extended Data Fig. 1 shows this result.

We then modelled both duo transits and single transits using MonoTools fitting. This allows long-period planets to be modelled in a way that the transit model is agnostic of the orbital period with the implied period distributions being manipulated using priors. This technique works for single transits or duo transits. In the case of two transit events separated by a long gap, the planetary transit is fitted leaving the orbital period open and the implied transit shape is used to calculate the probability for each of the possible period aliases. For single transits, potential orbital period windows are computed. In both cases, the period probability distribution comes from a combination of a simple period prior (longer-period planets are geometrically disfavoured)¹⁰⁷, an eccentricity prior (eccentric orbits are disfavoured in multitransiting systems)¹⁰⁸ and a stability prior using the orbits of other planets in the system (orbit-crossing is disallowed) (further details in refs. ^39,109). The resulting marginalized period predictions for planets HD 110067 d, e, f and g are shown in Supplementary Fig. 3, with posterior values of ${21.6}_{-1.6}^{+2.9},\,{29.9}_{-3.3}^{+4.6},\,{40.1}_{-5.1}^{+7.1}$ and 47.0 ± 8.0 days, respectively.

Continuing the resonant chain

In this section, we expand the analyses that led to the prediction of the orbits of planets HD 110067 e, f, and g based on the generalized Laplace resonant configuration of the three inner transiting planets in the system. We assume that all events mentioned in the previous section are transits that belong to planets that continue the resonant chain.

For transiting systems, generalized three-body Laplace angles can be estimated in the zeroth order in eccentricity, defined as Ψ_e=0, from the times of mid-transit and the orbital period of the planets (see, for example, ref. ¹¹⁰). This estimation differs from the actual generalized three-body Laplace angle proportionally to the eccentricities (equation (15) of ref. ¹¹¹). Notably, for known systems with a chain of three-body resonances, all Ψ_e=0 lie close to an equilibrium of the chain, as seen in Extended Data Fig. 2. The largest distance is about 43° for the inner triplet of K2-138 (ref. ¹¹²). For HD 110067, the estimated angle Ψ_e=0,bcd is also at about 44° from its theorized 180° equilibrium. Through the study of transit timing variations over several years, we can obtain constraints on the underlying generalized three-body Laplace angles. In known cases, we can see that these angles oscillate with amplitudes of a few tens of degrees at most around their equilibrium value (see Fig. 2 of ref. ¹² for Kepler-60 and Fig. 25 of ref. ¹³ for TRAPPIST-1).

As shown above, the two events at 2646.088 TJD and 1937.851 TJD have fully consistent shapes. Among the probable periods computed with MonoTools (Supplementary Fig. 3), P_e = 30.7931 days is the only one that continues the resonant chain, with P_e/P_d = 1.5007, landing inside the common 3:2 MMR (see Supplementary Fig. 4). We compute the observed value of the associated generalized three-body Laplace angle Ψ_e=0,cde = 169.995°, which is at only 10° from the expected 180° equilibrium. We hence predict a period of 30.7931 days for planet HD 110067 e if it is in the resonant chain.

For the remaining two mono transits, we try a set of first-order MMRs (2/1, 3/2, 4/3, 5/4, 6/5) between planets 4 and 5 and the same between planets 5 and 6 (hence 25 combinations). Each of these combinations has to be tested assuming that the transit at TJD = 2641.5778 belongs to the fifth planet and TJD = 2656.0944 belongs to the sixth planet (case A), and vice versa (case B). Fortunately, many of these 50 possibilities are excluded by existing data. We end up with four possibilities for case A and nine for case B. As seen in Fig. 2, all known chains of Laplace resonances have either their estimated generalized three-body Laplace angle Ψ_e=0 or their actual generalized three-body Laplace angle Ψ close to an equilibrium of the chain. We will hence favour the configurations that are closest to an equilibrium of the chain. For each case, the distance of each estimated angle to its closest equilibrium ΔΨ = |Ψ_e=0 − Ψ_eq| is given in Extended Data Table 4. The case A2, with P_f/P_e = 4/3 and P_g/P_f = 4/3, comes out as a favourite, with the three outer generalized three-body Laplace angles at less than 20° from the closest equilibrium. Also, one can note that 4/3 MMRs are relatively common in resonant chains (see Supplementary Fig. 4).

For completeness, we study the role that the eccentricity of the orbit plays in the prediction. To estimate the generalized three-body Laplace angle

$$\varPsi =l{\lambda }_{1}-(l+m){\lambda }_{2}+m{\lambda }_{3}$$

(1)

at a given epoch, we estimate the value of the λ_j as follows. Transits occur when the true longitude of the planet is equal to l₀ = −π/2. At first order in the eccentricity,

$${\lambda }_{0}={l}_{0}-2e\sin ({l}_{0}-\varpi )=-\frac{{\rm{\pi }}}{2}+2e\cos (\varpi ).$$

(2)

We then assume the planet to be in a circular, unperturbed orbit to compute the value of its mean longitude at the time of transit t₀, λ₀ = −π/2. We hence obtain

$$\lambda (t)=-\frac{{\rm{\pi }}}{2}+\frac{(t-{t}_{0})}{P}2{\rm{\pi }}.$$

(3)

The error on Ψ made by assuming zero eccentricity is hence, at first order¹¹¹

$$| \varPsi -{\varPsi }_{e=0}| =| 2l{e}_{1}\cos {\varpi }_{1}-2(l+m){e}_{2}\cos {\varpi }_{2}+2m{e}_{3}\cos {\varpi }_{3}| .$$

(4)

This error can thus be substantial (several tens of degrees) if the eccentricities are on the order of several parts per hundred, as is the case for Kepler-223 (ref. ¹¹⁰). Therefore, we study whether a given combination of the eccentricities and longitudes of periastron can make Ψ_e=0 closer to the equilibrium than Ψ actually is, or vice versa. We check this for the cases presented in Extended Data Table 4.

Each case sets the orbital period of the planets and their mid-transit time. We estimate the planetary masses using the mass–radius relation from ref. ²⁶. Then, varying the remaining parameters ${k}_{i}={e}_{i}\cos {\varpi }_{i}$ and ${h}_{i}={e}_{i}\sin {\varpi }_{i}$, we minimize the cost function

$$C={\mathcal{A}}({\varPsi }_{{\rm{b}}{\rm{c}}{\rm{d}}})+{\mathcal{A}}({\varPsi }_{{\rm{c}}{\rm{d}}{\rm{e}}})+{\mathcal{A}}({\varPsi }_{{\rm{d}}{\rm{e}}{\rm{f}}})+{\mathcal{A}}({\varPsi }_{{\rm{e}}{\rm{f}}{\rm{g}}})$$

(5)

over 200 years, in which ${\mathcal{A}}(\varPsi )=2{\rm{\pi }}$ if Ψ circulates and ${\mathcal{A}}(\varPsi )$ is the peak-to-peak amplitude of libration of Ψ otherwise. For each case, 40 MCMC runs are conducted to minimize C, using REBOUND¹¹³ for the N-body integration and samsam¹¹⁴ for the MCMC. For each run, the k_i and h_i parameters are randomly initialized in the [−0.05, 0.05] range, which are also their boundaries during the MCMC runs. This allows eccentricities that are comparable with those of Kepler-60 (ref. ¹¹⁵) and Kepler-223 (ref. ¹¹⁰), which are other known chains for which the inner planets are far enough from the star to not have their eccentricities damped by tides.

The best solution of each fit is shown in Extended Data Fig. 3. Case A2 is the only one for which the best solutions consistently have a peak-to-peak amplitude of the generalized three-body Laplace angles below 50° on average across the four angles. In all other cases, we were not able to find values of the k_i and k_j parameters below 85° of amplitude on average, with the exception of case A0, for which an average of about 66° was reached. The best solutions found across all MCMC runs for the A2 and A0 cases integrated for 1,000 years of evolution are shown in Supplementary Fig. 5. This analysis shows that the A2 case remains the one with the highest potential of being close to an equilibrium, while showing that all other cases cannot have an amplitude of libration smaller than 66° on average across their generalized three-body Laplace angles, regardless of the values of the k_i and k_j parameters. The case A2, with P_f/P_e = 4/3 and P_g/P_f = 4/3, is hence our prediction for the outer architecture of the HD 110067 system.

Confirming the predictions

Recovering the missing cadences of TESS Sector 23 observations

On the basis of the dynamical analysis presented above, the probable orbital periods associated with the two mono transits observed in TESS Sector 49 are approximately 41.05 and 54.74 days, respectively. According to this prediction, both planets transited their host star during TESS Sector 23 observations, but at a time when the photometry was highly affected by scattered light and sky background contamination. The Earth was a notable source of scattered light at the beginning of both Sector 23 orbits (TJD = 1928.09 and TJD = 1941.83) and the Moon was a notable source of scattered light for a few days after the beginning of the second orbit (between 1942 and 1947 TJD). The cadences affected were flagged by SPOC, thus not leaving enough valid data to derive cotrending basis vectors and missing in the PDCSAP light curve.

Our custom extraction using the PLD method from the section ‘TESS photometry’ was able to recover the missing data, showing two mono transits at 1943.6 and 1944.1 TJD (Fig. 1 and also Supplementary Fig. 6). Using MonoTools, we confirmed that the transits were consistent in duration–depth space with the two mono transits from TESS Sector 49 and separated by an integer number of orbits that matched the orbital periods predicted in our dynamical analysis for planets f and g. With this data reduction, all six planets in the HD 110067 system have been detected in transit at least twice, allowing a precise orbital period determination if we impose priors based on the hypothesis that all planets are trapped in a chain of first-order MMRs. Furthermore, we recovered an extra transit of planet b at the beginning of TESS Sector 23 observations.

Modelling of the ground-based photometric campaign

Targeted observations of HD 110067 were carried out on the night of 23 May 2022 to attempt to confirm the 41.05-day orbit of HD 110067 f as predicted through our resonance chain analysis. To reveal whether a transit was present in the combined dataset, we built a combined photometric model using all ground-based observations. To remove spurious systematic trends in a way that does not bias any transit fit, we opted to perform simultaneous linear decorrelation of each photometric dataset using the various metadata time series available. In all cases, for example, we included an air mass term in the decorrelation as well as a measure of the FWHM width. We also included two position-centroid terms (for MuSCAT2 and MuSCAT3), information on comparison star total counts and FWHM width (for LCO, Tierras and SAINT-EX), and interpolated colour time series derived from the relative shift in flux across bands in the MuSCAT2 and MuSCAT3 filters (as used in ref. ⁴⁹). In all cases, the metadata were normalized to a time series with μ = 0, σ = 1 and modelled using a single scaling parameter with a normal prior of μ = 0, σ = 0.5. Quadratic limb-darkening parameters were also constrained using normal priors dictated by theoretical limb-darkening parameters as computed for each of the nine passbands using LDTK¹¹⁶ and with inflated uncertainties following the methodology of the space-based photometric analysis. Each of the four time series observed by MuSCAT3 was split into two around an observing gap that occurred owing to the star passing close to the zenith at 06:57UT 24 May 2022. Individual time series were used for each of the nine NGTS telescopes, which were decorrelated independently. The final result is 24 individual photometric time series. An offset was also applied to each light curve, as well as a single global slope parameter to include the possibility of stellar activity.

The transit parameters were constrained on the basis of those found in a fit of the TESS Sector 49 mono transit. The predicted period used was 4/3 × P_e = 41.051 ± 0.1 days, with the uncertainty implying a divergence from the perfect integer period of 2.4 × 10⁻³—larger than those values found for the inner three planets. We limited the period to 41.0 ± 0.2 days to ensure a transit fit that could be explored with the temporal baseline of the photometry. Owing to the non-continuous nature of the photometry, the probability density function of the observed transit time is probably asymmetric and could potentially have several minima. Therefore, analyses using classical sampling techniques (MCMC, Hamiltonian Monte Carlo etc.) may not reveal the full picture. To initially test this, we kept all other parameters equal but split the range of periods covered by the time series into 36 bins across the anticipated period range (40.8 < P < 41.2) and fitted a constrained model for each. This would allow us to see the variation in the goodness-of-the-fit as a function of transit epoch. The transit model was built with exoplanet⁹⁹ and optimized using pymc3 (ref. ¹¹⁷), specifically with the pymc3-ext sampling, which enabled correlated parameters for each time series to be grouped together, speeding up the computation. To assess whether or not a transit model was justified over a flat model, we used the WAIC^118,119, as implemented in arviz¹²⁰.

Our results show a preference for a transit at the expected period P = 41.04 ± 0.01 days, with a ΔWAIC of 9.5 over a transit-free model, as can be seen in Extended Data Fig. 4. Most instruments showed a weak preference for a peak at P ≈ 41.05 days, with the exception of LCO (which observed no in-transit data) and SAINT-EX (which is the most affected by cirrus). This is only equivalent to moderate evidence for a roughly 41.051-day period of HD 110067 f. The lower two panels of Extended Data Fig. 4 show that both models (with or without transit) fit reasonably well. This is in part because systematic effects dominate over astrophysical signals for transits with depths below 1,000 ppm, especially when the target star is observed at a low air mass. A further peak in the WAIC is seen at about 40.9 days, but this hypothetical transit is covered only by the initial 1 h of MuSCAT2 photometry and does not fit our predicted period, hence we consider it spurious. This campaign shows that such transits are at the very limit of what is possible with ground-based observations. However, observations during a more favourable observing season and without technical issues such as the meridian flip of MuSCAT3 (which, unluckily, coincided with the expected egress) may have constrained better the presence of a transit.

Modelling of the radial velocity data

We carried out an initial frequency-based exploration of the CARMENES and HARPS-N spectroscopic datasets to see which substantial signals are present and those related to stellar activity using periodograms¹²¹. Supplementary Figs. 7 and 8 show that the dominant signal in the generalized Lomb–Scargle periodograms of both the radial velocities and main activity indicators (CCF-FWHM, differential line width, Mount Wilson’s S-index, Hα emission) is attributable to the rotational period of the star, measured photometrically to be approximately 20 days using TESS and CHEOPS data. As well known, stellar activity induces spurious radial velocity signals (for example, refs. ^{122,123,124,125}), which should be properly removed to unveil induced Keplerian motions in the star. We followed two independent approaches to model the data and minimize the impact of stellar activity effects on the detection and mass determination of the planets in the system.

Method I: SN-fit and breakpoint algorithm

Spurious radial velocity signals induced by stellar activity come from the line-shape variations in stellar spectra. Those can be quantified through the FWHM and the asymmetry of the cross-correlation function (CCF) computed from the spectra (for example, refs. ^126,127,128). Following ref. ¹²⁹, we first fit skew normal (SN) functions to the CCFs available from HARPS-N and CARMENES. An SN function is not only characterized by a location and a scale parameter (which are the counterparts of the mean and standard deviation of a Gaussian) but it has a further free parameter that expresses its skewness (hereafter denoted with γ). For each observation, through the SN-fit, we were able to retrieve the stellar radial velocity ($\overline{{\rm{RV}}}$, quantified through the SN median), the FWHM_SN, the contrast A and the asymmetry γ. The errors σ_RV of the $\overline{{\rm{RV}}}$ measurements were inferred using a bootstrap approach. Denoting with f_CCF the flux of a CCF data point, each point was perturbed by sampling values from a normal distribution whose standard deviation is equal to $\sqrt{{f}_{{\rm{CCF}}}}$, as the errors affecting the CCF data points are expected to be Poissonian.

After that, we applied the breakpoint method¹³⁰ to both the HARPS-N and CARMENES radial velocity time series. The algorithm has been designed to detect those locations along the radial velocity time series at which the correlation changes against the vector [FWHM_SN, A, γ] are statistically significant. The goal is to then detrend the radial velocity time series by applying a piecewise interpolation to each segment found by the breakpoint algorithm rather than performing an overall correction to the whole time series. In this way, we are able to better correct for the contamination of stellar variability as shown by refs. ^130,131. Finally, we jointly analysed the radial velocity time series using the MCMCI code¹³², in which we switched off the interaction with stellar evolutionary models to speed up the computations. We set up the detrending function on each piecewise stationary segment found by the breakpoint algorithm as a polynomial of the following form

$$\begin{array}{r}{{\rm{RV}}}_{\star }={\beta }_{0}+\mathop{\sum }\limits_{k=1}^{{k}_{t}}{\beta }_{k,t}{t}^{k}+\mathop{\sum }\limits_{k=1}^{{k}_{F}}{\beta }_{k,F}{{\rm{FWHM}}}_{{\rm{SN}}}^{k}+\mathop{\sum }\limits_{k=1}^{{k}_{A}}{\beta }_{k,A}{A}^{k}+\\ \,+\mathop{\sum }\limits_{k=1}^{{k}_{\gamma }}{\beta }_{k,\gamma }{\gamma }^{k}+\mathop{\sum }\limits_{k=1}^{{k}_{R}}{\beta }_{k,R}\log {{R}_{{\rm{HK}}}^{{\prime} }}^{k},\end{array}$$

(6)

in which (k_t, k_F, k_A, k_γ, k_R) is the vector of the polynomial orders whose optimal value has been established by launching several MCMC preliminary runs and selecting the combination that produces the minimum Bayesian information criterion (BIC)¹³³.

After performing a longer MCMCI run composed of four independent runs (300,000 steps each), which successfully converged as checked through the Gelman–Rubin test¹³⁴, we retrieved the posterior distributions of the system parameters. Their median values along with their error bars at the 1σ level are reported in Supplementary Tables 2 and 3. The full radial velocity time series and the phase-folded radial velocities of those planets whose detection is above the 3σ level (planets d and f) are shown in Supplementary Fig. 9.

Method II: multidimensional GP

On the other hand, we also perform a multidimensional Gaussian process (GP) approach to characterize the stellar and planetary signals in our radial velocity time series, as in refs. ^135,136. This approach has proven useful to disentangle stellar and planetary signals in multiplanet systems (for example, refs. ^137,138). We create N-dimensional GP models, including N time series ${{\mathcal{A}}}_{i}$, as

$$\begin{array}{c}{{\mathcal{A}}}_{1}={A}_{1}G(t)+{B}_{1}\dot{G}(t)\\ \vdots \\ {{\mathcal{A}}}_{N}={A}_{N}G(t)+{B}_{N}\dot{G}(t),\end{array}$$

(7)

in which the variables A₁, B₁,…, A_N, B_N, are free parameters that relate the individual time series to G(t) and $\dot{G}(t)$. In this approach, G(t) is assumed to be a latent (unobserved) variable that represents the projected area of the visible stellar disk that is covered by active regions as a function of time.

We model the stellar signal using a GP whose covariance between two times t_i and t_j is given by

$${\gamma }_{{\rm{QP}},i,j}=\exp \left[-\frac{{\sin }^{2}\left[{\rm{\pi }}({t}_{i}-{t}_{j})/{P}_{{\rm{GP}}}\right]}{2{\lambda }_{{\rm{p}}}^{2}}-\frac{{({t}_{i}-{t}_{j})}^{2}}{2{\lambda }_{{\rm{e}}}^{2}}\right],$$

(8)

in which γ_QP,i,j is the quasiperiodic kernel, whose hyperparameters are, P_GP, the GP characteristic period, λ_p, the inverse of the harmonic complexity and λ_e, the long-term evolution timescale.

We perform a two-dimensional GP model between the radial velocities and the FWHM. We note that these quantities are equivalent in the HARPS-N and CARMENES data. The multidimensional covariance matrix was created using the kernel given in equation (8) and its derivatives^135,136. We assume that radial velocities can be described as ${{\mathcal{A}}}_{i}={A}_{i}G(t)+{B}_{i}\dot{G}(t)$, whereas the FWHM time series is described as ${{\mathcal{A}}}_{i}={A}_{i}G(t)$. The planetary signals were included in the model as the mean function of the radial velocity time series. We use N Keplerian signals (in which N is the number of planetary signals), each one of them depending on the time of minimum conjunction t₀, orbital period P and Doppler semiamplitude K. All orbits are fixed to be circular, so the eccentricity and angle of periastron are fixed. For the FWHM, the mean function was treated as an offset, noting that we include a different offset per instrument. We also include a jitter term per time series and per instrument to account for unaccounted systematic errors.

We perform MCMC samplings of the parameter space using the code pyaneti (refs. ^136,139). We sample the parameter space with 250 walkers and create the posterior distributions with the last 5,000 iterations of converged chains with a thin factor of 10. This leads to posterior distributions of 125,000 points for each sampled parameter. Figure 2 shows the spectroscopic time series resulting from this joint analysis. Median values along with their 1σ uncertainties are reported in Supplementary Table 4.

Modelling techniques using GP are particularly subject to overfitting, given their flexibility to reproduce (see, for example, refs. ^140,141). To test the robustness of our GP model, we carried out a cross-validation analysis. We repeat the two-dimensional GP model described in this section but applied only to the HARPS-N data. Then, we create a predictive model with the inferred parameters and overlay the CARMENES data with median offsets subtracted (similar to a training/evaluation set for machine learning algorithms). Supplementary Fig. 10 shows this analysis, zoomed in to the 2021 observing campaign. The plot shows that the radial velocity data are in agreement with the predictive model, suggesting that our assumption that the stellar signal imprinted in the radial velocities can be described with a two-dimensional GP is valid for the time span of our observations. For the CCF FWHM CARMENES data, the correlation with the radial velocity measurements is not as strong as for the HARPS-N data, thus the prediction is less accurate in this case.

Both of these methods clearly detect planets HD 110067 d and f, and method II also detects planet b. However, the detection levels differ slightly. Recalling that the two different techniques are based on different radial velocity extraction methodologies and on different treatments of stellar activity, on the one hand, the slight output tension suggests that the radial velocity data alone do not strongly constrain all six Keplerian signals. We tested further stellar mitigation approaches, such as sinusoid fitting at the stellar rotation period and its harmonics or GP decorrelation as a function of time only, but they were all unsuccessful at constraining the masses of any of the planets (only the time-dependent GP model could recover the signal of planet f but with a much larger uncertainty, K_f = 2.0 ± 1.0 m s⁻¹). On the other hand, the radial velocity semiamplitudes inferred from the two methods are compatible within about 1.5σ, with the statistical tension Δ_I-II below 1σ for planets b, c, d and g. Extended Data Fig. 5 shows the pairs of posterior density functions for the radial velocity semiamplitudes of each planet for comparison.

For HD 110067 f, after imposing a Gaussian prior centred around 41.05 days in both methods I and II radial velocity models, we indeed recover a substantial radial velocity signal with a detection level of approximately 3σ. The planet transits only twice in the TESS data and the ground-based photometric campaign hints at moderate evidence for a planetary transit compatible with this value. Therefore, to secure an independent detection of planet f from spectroscopy, we performed a radial-velocity-only analysis imposing a uniform unbounded prior (between 30 and 54 days) to the orbital period. The MCMC converged and detected a clear Keplerian signal with a period P_f,uni = 40.2 ± 0.2 days. Thus, the radial velocity data independently suggest the presence of a putative planet having a period close to 41.05 days. The tension at the approximately 4σ level with the predictions from the resonant chain model and the transit observations prove that the current radial velocity dataset cannot fully constrain the entire architecture of the planetary system.

Finally, we checked whether there are further Keplerian signals in the radial velocity time series. In particular, given that both planets e and g were not detected through our previous radial velocity analyses in which model-dependent values of the orbital periods were imposed as priors, we investigated the presence of potential planetary signals at different periods that could be attributed to planets e or g. To this end, we performed an MCMC run, in which we modelled planets b and c (which are clearly confirmed by the transit events) along with planets d and f (which are clearly detected also in the radial velocity time series). As a result, we produced the generalized Lomb–Scargle periodograms¹²¹ of the residuals, obtained after subtracting the Keplerian signals of all four planets from the activity-cleaned time series (Supplementary Fig. 11). The high false-alarm probability level of the highest peak in both the HARPS-N (18%) and CARMENES (10%) residuals suggests that there are no signals left in the radial velocity data that could be associated with other planets or a misidentification of the orbital periods of planets e and g.

Final model

We computed a final model of the photometric and spectroscopic datasets of the HD 110067 system. On the basis of the analyses above, neither the light curves nor the radial velocities are precise enough to constrain the eccentricity of the planets. Assuming circular orbits, the photometry and radial velocity thus only constrain jointly the period and phase of a given planet in the system. However, the transit data dominate the precision of these two quantities (by several orders of magnitude). Therefore, for our final model, we opt to perform an independent analysis of the photometry and radial velocity datasets, in which priors inform the planet periods and phases in the radial velocity model based on the posterior distributions of the photometry-only fit. Besides, the large number of free parameters in each of the models makes it computationally expensive to run a joint fit, not to mention the complications for numerical samplers to explore the vast multidimensional parameter space. Table 1 shows the most relevant planetary parameters of the system based on the photometric fit from Supplementary Table 1, the radial velocity fit using method II from Supplementary Table 4 and the stellar parameters from Extended Data Table 3. A corner plot with the posterior distribution of the fitted transit parameters is shown in Supplementary Fig. 12. The resulting best-fit models and corresponding credibility bands are presented in Fig. 1 for the TESS and CHEOPS photometry and in Fig. 2 for the radial velocities.

Planetary internal structures

Using a Bayesian analysis^142,143, we computed the possible internal structures of the six planets of the system, using the results provided in Table 1, Extended Data Table 3 and the planetary masses from methods I and II. The forward model used to compute the likelihood is based on a four-layer structure: a central core (iron and sulfur), a silicate mantle (containing Si, Mg and Fe), a water layer and a gas layer (H and He). The equation of state (EOS) of water is the one in ref. ¹⁴⁴, the core EOS is the one in ref. ¹⁴⁵ and we use the EOS in ref. ¹⁴⁶ for the silicate mantle. The thickness of the gas envelope, which depends on the planetary age, mass etc., is derived from ref. ¹⁴⁷. Note that the influence of the gas layer on the innermost planet (compression and thermal effect) is not included in our model, as the mass of the gas layer for the six planets is small (see below). The planetary Si/Mg/Fe molar ratio in all planets is assumed to be equal to the stellar one. The prior distribution of the mass fractions of the three innermost layers (core, mantle and water layer) is assumed to be uniform on the simplex—the surface defined by the sum of the three mass fractions equal to one. Also, the mass fraction of the water layer is assumed to be 50% at most^148,149, and for the gas mass, we use a uniform log prior.

The results from this analysis are shown in Extended Data Fig. 6. Our model shows that the gas mass content of all planets is on the order 10⁻³M_⊕ to 10⁻¹M_⊕ (median value, see Supplementary Table 5), with the notable exception of HD 110067 e (median value of about 10⁻⁷M_⊕ using the masses from method I, approximately 10⁻³M_⊕ using method II). The apparent lack of an atmosphere of planet e (located just outside planet d, which is the most gas-rich of the system, according to the internal structure models) is puzzling. If confirmed by future better determination of its density, the origin of the peculiar internal structure of planet e will have to be understood in the context of the very fragile architecture of the whole HD 110067 system. On the other hand, the water fraction for all planets is essentially unconstrained, owing to the still large uncertainty in the planetary masses. However, according to simulations of combined planetary formation and evolution, independently of the accretion mechanism (planetesimal-based or pebble-based), all the planets in the system have masses and radii consistent with a formation beyond the ice line^150,151,152. Therefore, it is possible that, even though the water content is unconstrained in our model, the cores of the planets are rich in volatiles. JWST observations of some atmospheric trace gases (particularly ammonia, methane and/or methanol) could be used as a proxy for the presence of a deep or shallow surface that could break the degeneracies from internal composition models using bulk density measurements alone^153,154.

Extended data figures and tables

Extended Data Fig. 1 Transit duration versus transit depth for all unassigned transits in the TESS data.

TESS Sector 23 and Sector 49 are shown as different colours. The numbers above each transit denote the mid-transit time in TJD. Contours represent percentile levels, the innermost one corresponding to the 50th percentile and the outermost to the 99th percentile by increments of 10%. The transit of planet f in PLD photometry is marked with * to indicate that its properties are heavily affected by pretransit systematic noise.

Extended Data Fig. 2 Generalized three-body Laplace angles for known systems in resonant chains.

Included are the Galilean satellites Kepler-60 (refs. ^12,115), Kepler-80 (ref. ¹⁵⁹), K2-138 (ref. ¹¹²), Kepler-223 (ref. ¹¹⁰), TRAPPIST-1 (ref. ¹³) and TOI-178 (ref. ¹⁰). Measurements belonging to the same system are marked with the same colour. The line marks the observed distance to the theorized equilibrium (marked with a circle). The distances are estimated at the zeroth order in eccentricity^110,111. For most systems, a single estimation of the generalized Laplace angle is made, whereas ref. ¹¹⁰ made an estimation for each Kepler quarter.

Extended Data Fig. 3 Observed distance from the equilibrium for all the simulated scenarios in which planets f and g continue the resonant chain.

The y axis is converted to the mean peak-to-peak amplitude from the generalized three-body Laplace angle using the following expression: mean $({\mathcal{A}}({\varPsi }_{i}))=C/4$. Case A2 remains the one that has the potential to be the closest to an equilibrium.

Extended Data Fig. 4 Results from the ground-based campaign to detect HD 110067 f.

a, ΔWAIC for each of the constrained period bins when compared with a transit-free model. b,c, Best-fit decorrelated photometry with (b) and without (c) a transit model. Each light curve from each telescope has been offset for clarity. Error bars represent 1σ uncertainties.

Extended Data Fig. 5 Results from the two radial velocity analyses to measure the mass of each of the planets in the HD 110067 system.

Each histogram represents the posterior density function (pdf) of the radial velocity semiamplitudes as inferred from method I (red) and method II (blue). The area underneath each histogram is normalized to unity.

Extended Data Fig. 6 Gas mass fraction of the HD 110067 planets as a function of their equilibrium temperature.

We infer two values per planet by assuming the different planetary masses from our method I (red) and method II (blue) radial velocity analyses. The boxes, orange lines, green triangles and red stars represent, respectively, the 25th and 75th percentiles, medians, means and modes of the posterior distributions. The opacity of the vertical lines is proportional to the posterior distribution.

Extended Data Table 1 CHEOPS observing log

Extended Data Table 2 Ground-based photometric campaign observing log

Extended Data Table 3 Stellar parameters of HD 110067

Extended Data Table 4 Distance of the estimated generalized three-body Laplace angle Ψ_e=0 to the closest equilibrium for all period ratios that are not excluded by available observations