Issue 
A&A
Volume 624, April 2019



Article Number  A46  
Number of page(s)  9  
Section  Planets and planetary systems  
DOI  https://doi.org/10.1051/00046361/201834901  
Published online  04 April 2019 
Coorbital exoplanets from closeperiod candidates: the TOI178 case
^{1}
Physikalisches Institut, Universität Bern,
Gesellschaftsstr. 6,
3012
Bern, Switzerland
email: adrien.leleu@space.unibe.ch
^{2}
European Southern Observatory, Alonso de Cordova 3107,
Vitacura Casilla
19001,
Santiago 19, Chile
^{3}
Center for Space and Habitability, University of Bern,
Gesellschaftsstr. 6,
3012
Bern, Switzerland
^{4}
IMCCE, Observatoire de Paris – PSL Research University, UPMC University Paris 06, University Lille 1, CNRS,
77 Avenue DenfertRochereau,
75014
Paris, France
^{5}
CFisUC, Department of Physics, University of Coimbra,
3004516
Coimbra, Portugal
^{6}
Observatoire de Genève, Université de Genève,
51 ch. des Maillettes,
1290
Versoix, Switzerland
^{7}
Blue Marble Space Institute of Science,
1001 4th Ave Suite 3201,
Seattle,
WA 98154, USA
^{8}
Paris Observatory,
LUTh UMR 8102,
92190
Meudon, France
Received:
17
December
2018
Accepted:
1
February
2019
Despite the existence of coorbital bodies in the solar system, and the prediction of the formation of coorbital planets by planetary system formation models, no coorbital exoplanets (also called trojans) have been detected thus far. Here we study the signature of coorbital exoplanets in transit surveys when two planet candidates in the system orbit the star with similar periods. Such a pair of candidates could be discarded as false positives because they are not Hillstable. However, horseshoe or longlibrationperiod tadpole coorbital configurations can explain such period similarity. This degeneracy can be solved by considering the transit timing variations (TTVs) of each planet. We subsequently focus on the threeplanetcandidate system TOI178: the two outer candidates of that system have similar orbital periods and were found to have an angular separation close to π∕3 during the TESS observation of sector 2. Based on the announced orbits, the longterm stability of the system requires the two closeperiod planets to be coorbital. Our independent detrending and transit search recover and slightly favour the three orbits close to a 3:2:2 resonant chain found by the TESS pipeline, although we cannot exclude an alias that would put the system close to a 4:3:2 configuration. We then analyse the coorbital scenario in more detail, and show that despite the influence of an inner planet just outside the 2:3 MMR, this potential coorbital system could be stable on a gigayear timescale for a variety of planetary masses, either on a trojan or a horseshoe orbit. We predict that large TTVs should arise in such a configuration with a period of several hundred days. We then show how the mass of each planet can be retrieved from these TTVs.
Key words: celestial mechanics / planets and satellites: detection / planets and satellites: dynamical evolution and stability
© ESO 2019
1 Introduction
Among the known multiplanetary systems, a significant number contain bodies in (or close to) first and secondorder meanmotion resonance (MMR; Fabrycky et al. 2014). However, thus far no planets have been found in a zeroorder MMR, also referred to as trojan or coorbital configuration, despite several dedicated studies (Madhusudhan & Winn 2009; Janson 2013; Hippke & Angerhausen 2015a), and the TROY project (LilloBox et al. 2018a,b).
Such bodies are numerous in the solar system, such as the trojans of Jupiter and Neptune, or some of the Saturnian satellites. Although most of the time the mass of one of the coorbitals is negligible with respect to the other, Janus and Epimetheus (coorbital moons of Saturn) only have a mass ratio of 3.6. Indeed, trojan exoplanets can be stable on durations comparable to the lifetime of a star as long as (m_{1} + m_{2}) < m_{0}∕27, where m_{1} and m_{2} are the mass of the coorbitals and m_{0} is the mass of the star (Gascheau 1843). This implies that even the two most massive planets of our solar system, Jupiter and Saturn, could share the same orbital period with a difference in mean longitudes librating around 60°. The less massive the two coorbitals are, the larger their amplitude of libration around the L_{4}∕L_{5} equilibria. When both masses are similar to the mass of Saturn or lower, the two coorbitals can also be on a stable horseshoe orbit, in which the difference of the mean longitudes librates with an amplitude of more than 310° (Garfinkel 1976; Érdi 1977; Niederman et al. 2018). Eccentric/inclined orbits offer a wealth of other stable configurations that are extensively studied (Namouni & Murray 2000; Giuppone et al. 2010; Morais & Namouni 2013; Robutel & Pousse 2013; Leleu et al. 2018a). Trojan exoplanets are a byproduct of our understanding of planetary system formation (Cresswell & Nelson 2008) and can form through planet–planet scattering or insitu accretion at the Lagrangian point of an existing planet (Laughlin & Chambers 2002). However, there are currently only few constraints on the expected characteristics (such as the amplitude of libration) of coorbital exoplanets due to the complexity of the evolution of such a configuration in a protoplanetary disc (Cresswell & Nelson 2009; Giuppone et al. 2012; Pierens & Raymond 2014; Leleu et al. 2018b).
The detection of coorbital exoplanets is challenging due to the existence of degeneracies with other configurations across various detection techniques, transit timing variations (TTVs; Janson 2013; Vokrouhlický & Nesvorný 2014), radial velocities; and astrometry (Laughlin & Chambers 2002; Giuppone et al. 2012; Leleu et al. 2015); while the transit of both planets require closein coplanar systems. The multiplanet systems Kepler132, Kepler271; and Kepler730 were first announced to contain closeperiod planets until a more detailed analysis disfavoured the coorbital scenario in favour of the planets orbiting two different stars of a binary (Kepler132), or in a 2:1 MMR (Kepler271 and Kepler730; Lissauer et al. 2011, 2014).
In this study we focus on (quasi)coplanar orbits, as we aim to describe potential signals in the data from past and current transit surveys such as Kepler/K2 and TESS (Borucki et al. 2010; Ricker et al. 2014) and prepare for the analysis of future missions such as CHEOPS and PLATO (Benz et al. 2018; Rauer et al. 2014; Hippke & Angerhausen 2015b). After a brief summary onthe dynamics and stability of closeperiod planets, in Sect. 2 we discuss how TTVs can be used to remove the degeneracy between coorbitals and seemingly similar but distinct orbital periods when candidates do not have a signaltonoise ratio (S/N) good enough to identify each transit individually. In Sect. 3 we consider the case of the TOI178 system, where two of the three announced planet candidates have close orbital periods. We first perform an independent detrending of the light curve and search for transits. Subsequently, assuming that the estimated periods from the TESS data validation report are correct, we perform a stability analysis of this threeplanetsystem as a function of the masses of the planets, and predict the TTVs that should be observed in such a system.
2 Coorbital dynamics and stability
2.1 Coorbital motion
We consider the motion of two planets of mass m_{1} and m_{2} orbiting around a star of mass m_{0} with their semimajor axes and mean longitudes a_{j} and λ_{j}, respectively. When the semimajor axes of the two planets are close enough, in the quasicoplanar quasicircular case, the evolution of the resonant angle ζ = λ_{1} − λ_{2} can be modelled by the secondorder differential equation (Érdi 1977; Robutel et al. 2015): (1)
where μ = (m_{1} + m_{2})∕m_{0}, and η is the average meanmotion, defined as the barycenter of the instantaneous meanmotion of the two planets: (m_{1} + m_{2})η = m_{1}n_{1} + m_{2}n_{2} (Robutel et al. 2011). The phase space of Eq. (1) is shown in Fig. 1. ζ = 180° corresponds to the hyperbolic L_{3} Lagrangian equilibrium, while ζ = ±60° are the stable configurations L_{4} and L_{5}. Orbits that librate around these stable equilibria are called tadpole, or trojan (in reference to the trojan swarms of Jupiter). Examples of trojan orbits are shown in purple in Fig. 1. The separatrix emanating from L_{3} (black curve) delimits trojan orbits from horseshoe orbits (examples are shown in orange), for which the system undergoes large librations that encompass L_{3}, L_{4}, and L_{5}.
The libration of the resonant angle ζ is slow with respect to the average meanmotion η. The fundamental libration frequency ν is proportional to . In the neighbourhood of the L_{4} or L_{5} equilibria, (Charlier 1906). Away from the equilibrium, we compute ν by integrating Eq. (1). Its value is given in Fig. 1 (lower panel) with respect to the initial value of the resonant angle.
Fig. 1 Top: phase space of the coorbital resonance valid for small eccentricities and inclinations. The x axis displays the resonant angle ζ = λ_{1} − λ_{2} while the y axis is its normalised angular frequency, which is proportional to . Bottom: value of the normalised fundamental frequency ν for initial conditions along the line (ζ,) of the top graph. Both graphs were obtained by integrating Eq. (1). 

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2.2 Stability of similarperiod planets
The main parameters we have access to when detecting a planet through transit surveys are the orbital period, the epoch of transit, the radius of the planet, and the impact parameter. As the eccentricities of the planets are generally unconstrained by these observations, the stability of detected multiplanetary systems is estimated through criteria involving the mass and semimajor axis of the planets, such as Hill stability. Such criteria however do not take into account the stability domain of the coorbital 1:1 MMR. For coplanar circular orbits, the width of this domain (i.e. a_{1} − a_{2} or P_{1} −P_{2}) is at its largest for ζ = ±60°; see Fig. 1. As shown in previous studies (Leleu et al. 2018a), the stability domains of these configurations depend mainly on μ = (m_{1} + m_{2})∕m_{0}, and very little on m_{2}∕m_{1}.
To illustrate the stability of circular coplanar coorbitals, we integrate the threebody problem for a grid of initial conditions. Taking initial conditions along the vertical dashed line in Fig. 1 (ζ(0)= 60°) allows us to study all the possible coorbital configurations in the coplanar circular case from the Lagrangian equilibria (P_{1} ∕P_{2} = 1) to horseshoe orbits (stability around L_{5} is obtained as well due to the symmetry of the problem). We therefore integrate a grid of initial conditions along P_{1} ∕P_{2} ∈ [1, 1.09] for various values of μ = (m_{1} + m_{2})∕m_{0} ∈ [2 × 10^{−6}, 2 × 10^{−3}]. The results of these integrations are shown in Fig. 2.
For each set of initial conditions, the system is integrated over 10^{5} orbital periods using the symplectic integrator SABA4 (Laskar & Robutel 2001) with a time step of 0.01001 orbital periods. Trajectories with a relative variation of the total energy above 10^{−7} are considered unstable. Such trajectories are identified with white pixels. These shortterm instabilities are generally due either to the overlap of secondary resonances in the coorbital region (Robutel & Gabern 2006; Páez & Efthymiopoulos 2015, 2018), or to the overlap of firstorder MMRs outside this domain (Wisdom 1980; Deck et al. 2013; Petit et al. 2017). Grey pixels identify the initial conditions for which the diffusion of the mean motion of one of the planets between the first and second half of the integration is higher than 10^{−5.5} (Laskar 1990, 1993; Robutel & Laskar 2001). The integration time of 10^{5} orbital period is not enough to assess the stability on the lifetime of a planetary system. However, from the estimates regarding the diffusion variation versus time given in Robutel & Laskar (2001) and Petit et al. (2018), we deduce that a meanmotion diffusion rate lower than 10^{−7} derived from integrations over 10^{7} orbits enables us to ensure the stability over 10^{10} orbits. This was checked for a lowerresolution grid of initial conditions. Gigayear stable orbits are shown by black dots in Fig. 2. Due to the chosen resolution of initial conditions, the chaotic area in the vicinity of the separatrix between the trojan and horseshoe domains is visible only for large masses. The purple line represents the stability criterion proposed by Deck et al. (2013) for the outer limit of the chaotic area: . The black line indicates P_{1}∕P_{2} ≈ 1.04, which is close to the estimated value for both TOI178 and Kepler132 candidates. Pairs of planets have to be either above the black curve (in the coorbital resonance) or below the purple one (on separated orbits) to be on longterm stable orbits.
Fig. 2 Stability map as a function of P_{1}∕P_{2} and μ = (m_{1} + m_{2})∕m_{0}, for circular coplanar orbits, taking ζ = 60° as initial condition (vertical dashed line in Fig. 1). Coloured pixels are longterm stable with respect to the integration time of 10^{5} orbits. Grey pixels are longterm unstable, and white pixels are shortterm unstable. The black curve represents the mutual Hill radius of the planets, , while the vertical line represents P_{1}∕P_{2} ≈ 1.04 which is close to the estimated value for the candidates of the TOI178 system discussed in Sect. 3. The colour code is the amplitude of libration: black for zero (lagrangian equilibria), orange for horseshoe orbits. The bottomright corner shows stable orbits outside of the coorbital area, in yellow. The purple line is the stability criterion from the overlapping of firstorder MMRs (Deck et al. 2013). Black dots represent orbits that are stable over 10^{10} orbital periods; see the text for more details. 

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2.3 Transit timing variations of similarperiod planets
Using the notations, reference frame, and results of Leleu et al. (2017), planet j transits, at first order in eccentricities and inclinations, when (2)
where ζ(t) is given by Eq. (1). The TTVs induced by the coorbital configuration are detailed in Ford & Holman (2007) and Vokrouhlický & Nesvorný (2014). In this study we simply comment onthe following point: a pair of coorbital exoplanets might be mistaken for two planets on close but distinct nonresonant orbits if either one of the following two situations exists.
Their libration period is significantly longer than the duration of the observation. In this case, the continuous evolution of the instantaneous period of the coorbitals can be retrieved using Eq. (3). This is the case for TOI178, which is discussed in Sect. 3.
They are on horseshoe orbits and have similar radii. This case is discussed in the remainder of this section. In both cases, since the orbits are within the Hill instability region, one of the candidates could be labelled as false positive and therefore rejected, despite the fact that the planets are on stable coorbital orbits.
We illustrate the horseshoe orbit case using the Kepler132 system which has four validated planets (Rowe et al. 2014), two of them with P_{1} = 6.4149 day and P_{2} = 6.1782 day (planetsKepler132 c and Kepler132 b). As the central star of Kepler132 was shown to be a possible wide binary, these two planets wereeach subsequently announced to orbit a different component of the binary, as they would be Hillunstable if they orbited the same star (Lissauer et al. 2014). We refer to this scenario as scenario (i). In this case, no significant TTVs are expected forthese two planets.
We propose an alternative scenario (ii) that we illustrate in Fig. 3. For this figure, we integrated the orbit of two masses m_{x} = 3 × 10^{−5}m_{0}, m_{y} = 3.75 × 10^{−5} m_{0} around a m_{0} = 1.37 solar mass star in a horseshoe orbit. Two planets in such an orbit “exchange” their position every half libration period. We approximate this motion by “jumps” between an upper period: P_{x,+} (resp. P_{y,+}), and a lower period P_{x,−} (resp. P_{y,−}) for planet x (resp. y).
For this example, we chose initial conditions for planets x and y such that the averages of the upper positions (P_{x,+} + P_{y,+})∕2 (resp. lower position (P_{x,−} + P_{y,−})∕2) are the orbital periods announced for Kepler132 c (resp. Kepler132 b), shown as black dashed lines in Fig. 3. As the swapping between higher and lower periods for x and y happens when they are near conjunction, and is quick with respect to the libration timescale, both scenario (i) (planets 1 and 2) and (ii) (planets x and y) yield similar transit timings; the bottom panel of Fig. 3 represents the simulated river diagram of that system, folding the time of transits of the planets x and y with respect to the fixed outer period P_{1}. If we assume that we cannot distinguish the transits of planets x and y, they can be mistaken for planet 1 having moderate libration around the orbit of period P_{1} (black dots in bottom panel of Fig. 3), and planet 2 oscillating around a distinct orbit P_{2} (purple dots), which result in almostvertical lines in this river diagram.
We hence consider the possibility that the announced planets 1 and 2 of the Kepler system could instead be planets x and y represented in blue and orange in Fig. 3. In this case, planets 1 and 2 would be “fictitious” planets that are alternately planets x and y: “planet 1” which has an announced period of P_{1} = 6.4 days is actually planet x half of the time, and planet y the rest of the time; the same goes for planet 2. This scenario is possible because the two announced planets have almost indistinguishable radii (R_{1} = 1.3 ± 0.3 and R_{2} = 1.2 ± 0.2).
However, in the scenario (ii) the fictitious planets 1 and 2 exhibit significant TTVs when the actual planets x and y have different masses (see bottom panel of Fig. 3): the instantaneous period of each planet librates around a mean orbital period (Robutel et al. 2011). The upper and lower positions of the planets x and y read: (4)
with δP = (P_{1} − P_{2}). This jump occurs at every conjunction, . As P_{1} is the averaged value of P_{x,+} and P_{y,+}, the TTVs have an amplitude of (5)
and a period of 2P_{swap}. We note that the amplitude is slightly overestimated as the instantaneous periods are continuously swapping instead of jumping. This smooth evolution also produces TTVs, but they are negligible with respect to those described by Eq. (5) as long as m_{x} and m_{y} differ by more than a few percent. In Fig. 3, m_{x}∕m_{y} = 5∕4, inducing TTVs estimated at 8.37 h by Eq. (5).
In the Kepler132 case, our lightcurve analysis excluded TTVs larger than half an hour on both planets 1 and 2. If we consider scenario (ii), that is, that the transits are instead produced by two planets x and y on horseshoe orbits, Eq. (5) yields m_{x} − m_{y} < 0.0066(m_{x} + m_{y}), implying a mass difference below 1.5% between the two planets. Although scenario (i), where each star of a binary has a similar planet at almost equal orbital periods, seems unlikely, scenario (ii), where two planets on a horseshoe orbit have equal masses down to the percent level, does not seem much likelier.
Fig. 3 Top: instantaneous orbital period of two planets in a horseshoe orbit. Bottom: TTVs river diagram of the transits of both planets with respect to the outer dashed line at ~ 6.4 days. In the background, black (purple) dots give the river diagram of a planet in the isolated orbit P_{1} (P_{2}). 

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Parameters for the three candidates in the TOI178 system, extracted from the DVR of the TESS mission.
3 TOI178
The first release of candidates from the TESS alerts of Sector 2 included three planet candidates in the TESS Object of Interest (TOI) TOI178 (or TYC 6991004751). The candidates TOI178.01, TOI178.02, and TOI178.03 transited 4, 3, and 2 times, respectively, during the 27 days of observation. The TESS pipeline fits converge on a solution where all three candidates are of planetary nature. In the TESS datavalidation report (DVR) all three planets pass all firstorder tests to exclude potential falsepositive signals. The difference image centroid offsets relative to the TESS Input Catalog position and relative to the out of transit centroid are within 2σ for all three planet candidates (except 2.09σ for planet 3), showing no indication for an eclipsingbinary scenario. The pipeline ghost diagnostic test that searches for correlation of the time series of core and halo apertures ruled out optical ghosts of bright eclipsing binaries outside of the target apertures as the source of the transitlike features. The bootstrap test excludes a falsealarm scenario by 10^{−13} or smaller for all three planetary candidates. The only two yellow flags are the 7.6 S/N and slight centroid offset of 2.09σ for the candidate TOI178.03.
During the TESS observation of sector 2, the period ratios of the two outer candidates was close to 1.04 (see Table 1), while the phase between these planets was in the range ζ ∈ [280°:310°] (assuming circular orbits), which is in the vicinity of the L_{5} equilibria (ζ = 300°, see Fig. 1). Furthermore, the radius of the outermost candidate is estimated to be 3.7 ± 1.5 Earth radii, while the radius of the other candidate is poorly constrained. An estimation of the mass of these candidates is difficult, although the outermost appears to be a “Neptunelike” object (Chen & Kipping 2017), and is hence very unlikely to have a subEarth mass. As a result, as shown in Sect. 2.2, these candidates have to be in the coorbital resonance inorder to be on stable orbits; see the vertical line in Fig. 2. This prompted us to study this system in more detail. For consistency with Sect. 2, in the rest of the paper we the two coorbital candidates as planets 1 and 2, while we refer to the planet at 6.5 days as planet 3; see Table 1.
3.1 Independent transit search and orbital fit
We run an independent transit search to confirm the result of the TESS Science Processing Operations Center (SPOC) pipeline (Jenkins et al. 2016), starting from the target pixel file, which we downloaded from the MAST^{1}. We first calculate the centroid position and fullwidthhalfmaximum (FWHM) of the pointspreadfunction (PSF) of the target for each frame. We then generate the light curve by using a circular tophat aperture, tracking the PSF centre in each frame.
Our transit search and detrending pipeline is based on the Gaussian process (GP) pipeline used to detrend K2 light curves in Luger et al. (2017) and Grimm et al. (2018), modified to work with the higher cadence rate of TESS light curves. We find that the systematic noise encountered in TESS light curves has a different source from that found for K2; instead of being correlated with the PSF centroid offset, it has a strong correlation with the x and y FWHM of the target PSF. Therefore, we employ a similar procedure as in Luger et al. (2017), running a GP regression pipeline to simultaneously fit the systematic noise correlated with the x and y FWHM, and the longerterm fluctuations caused by stellar variability. The noise is then subtracted to produce a flattened light curve. The detrended light curve has a similar residual noise level as the PDCSAP light curve produced by the SPOC pipeline (with median absolute deviations of 0.126% and 0.124%, respectively).
This GPbased detrending method is independent from the TESS pipeline, which uses cotrending basis vectors derived from the entire ensemble of light curves in a dataset (Jenkins et al. 2016). As a result, we can provide an independent check that the transit signals are not spuriously caused by data processing.
We run a standard transit search to find the ten strongest transitlike signals, based on similar procedures in Vanderburg et al. (2016), Dressing et al. (2017), and Mayo et al. (2018), for example. First, we perform a series of BLS fits to find periodic dimming events in the light curve, removing each signal for subsequent fits (Kovács et al. 2002). For each of the ten signals, we fit a transit model based on Mandel & Agol (2002). We use the batman package to compute the transit model (Kreidberg 2015), inputting limbdarkening parameters calculated from Claret & Bloemen (2011).
Running the transit search on our detrended light curve, we recover all three transit signals found by the TESS pipeline. However, we also find a strong signal with a 4.96 day period and a firsttransit time of t_{0} = 1458.006 (BTJD). This is an alias of candidate 2. We calculate the loglikelihood difference between the 9.96 period and its alias to be 7.74, in favour of the 9.96 day period. This would correspond to a difference of ~ −15 in the Bayesian information criterion, which is considered significant (Schwarz 1978). However, our limited constraints on the properties of the host star will have an effect on the likelihood ratio, as it is calculated based on a transit model which requires stellar mass and limb darkening.
Given the S/N limit of the TESS observations, we cannot fully confirm the 9.96 day period of candidate 2 based on individual transits. Wethen check if the TTVs can help us discriminate between the two scenarios: (i) The configuration announced by the TESS pipeline, where the orbits are near a 3:2:2 resonant chain; and (ii) the case where candidate 2 is in a 4.96 day period, resulting in a near 4:3:2 resonant chain. The date of individual transits, derived both from the light curve detrended by the TESS pipeline and the result of our own detrending, are given in Table A.1.
We perform the TTV analysis with an ensemble differential evolution Markov chain Monte Carlo method (DEMCMC; Braak 2006; Vrugt et al. 2009), similar to that described in Grimm et al. (2018). This method uses the GPU Nbody code GENGA (Grimm & Stadel 2014) to calculate the orbital evolution of the planets and the transit times for the DEMCMC steps. The estimated masses in each scenario are summarised in Table 2, while the posterior distribution functions can be found in Fig. A.1. We note that these posteriors do not take into account the longterm stability of the fitted orbits. Due to the low number of observed transits, and the short timespan of the observations, the uncertainties on the masses are quite large, and do not allow us to further discriminateone scenario from the other. We note however a difference between the posterior distributions of m_{2} ∕m_{0} between the two scenarios: if candidate 2 is on a 4.94 day orbit then its mass should be significantly smaller than the other two candidates.
Our analysis of the current observations does not allow for the 4:3:2 scenario to be fully discarded. In the following sections we nonetheless consider the case that is favoured by both our light curve analysis and the TESS pipeline: the near 3:2:2 resonant chain. The stability of the orbit of planets 1 and 2 is discussed in Sect. 2.2. However, the presence of planet 3 near a 2:3 MMR with the coorbital pair might further reduce their stability domain (Robutel & Gabern 2006). We analyse the stability of this threeplanet system in the following section.
Orbital fit to the transit timings observed by TESS.
3.2 Stability analysis
As P_{1}∕P_{3} ≈ 1.58 and P_{2} ∕P_{3} ≈ 1.52, the coorbital configuration is just outside the 3:2 MMR with planet 3. As a result, we expect resonances between the libration frequency ν of the coorbitals and the frequency Ψ = 2n_{3} − 3η (called great inequality), where n_{3} is the meanmotion of planet 3 and η the average meanmotion of the coorbitals (Robutel & Gabern 2006). As we have no information on the eccentricities of the planets, we analyse the stability in the circular case. Both ν and Ψ depend on the averaged meanmotion, which in turn depends on the mass of the star and the mass ratio between the two coorbitals. The mass of the star is a common factor to all involved frequencies and can therefore be ignored, as the initial conditions are taken using the relative orbital periods of the planets. Given the constraints we have on the orbits, the three main parameters for the stability of the system are the masses of the three planets.
We therefore check the stability of the coorbital configuration as a function of m_{1} ∕m_{0}, m_{2} ∕m_{0}, and m_{3} ∕m_{0} in two different planes: in Fig. 4 (a) we vary m_{1}∕m_{0} and m_{2}∕m_{0}, taking an arbitrary value for m_{3} = 2.5 × 10^{−5}m_{0} (≈ 5.3 M_{earth}, using m_{0} given in Table 1); while in (b) we fix m_{1}∕m_{2} varying μ and m_{3}∕m_{0}. The initial values of the mean longitudes and semimajor axis are derived from Table 1, and the orbits are initially coplanar and circular. These figures are obtained in the same way and have the same colour code as in Fig. 2; see Sect. 2.2.
In Fig. 4a, we recover the stable domains for horseshoe and trojan orbits, as was the case of Fig. 2. We see however that these stability domains are crossed by disruptive resonances. We identified the main chaotic structures to be in the wake of resonances of the form Ψ = pν + g, where g is one of the three secular frequencies of the system, small with respect to ν. The p = [2, 3, 4] resonances cross the trojan area, while p = [5, 6, 7, 8] disturb the horseshoe domain (for more details, see Robutel & Gabern 2006). For two isolated coorbitals m_{1} ∕m_{2} has close to no impact on the stability of the orbits in the coplanar quasicircular case (Leleu et al. 2018a). Here however, the mass repartition between the coorbitals shift the value of the averaged meanmotion η, displacing the positions of the disruptive resonances. This effect, combined with the evolution of the resonant frequency ν (which is a function of both μ and the amplitude of libration of ζ, see Fig. 1), gives the unstable structures displayed in Fig. 4. Longterm stable areas remain for many values of m_{1} and m_{2}, but m_{1} > m_{2} is overall favoured by this stability analysis.
Fixing the mass ratio to an arbitrary value m_{1}∕m_{2} = 2, and changing μ and m_{3} ∕m_{0}, we show in Fig. 4b that m_{3}∕m_{0} has little effect on the position of the resonant structures, as it does not impact the value of the concerned frequencies beside the secular frequency g. As a result, an increase of m_{3}∕m_{0} only increases the width of the chaotic area near these resonances, further reducing the stability domains. A good estimation of the mass of the planet 3 can hence further constrain the possible coorbital configuration of the planets 1 and 2. It is important to bear in mind, however, that panel b only represents the evolution of the m_{1} = 2m_{2} line on panel a, and hence a more detailed analysis is required once more constraints are obtained on the masses.
Fig. 4 Stability domains for the coorbital candidates of the TOI178 system. Panel a: as a function of the mass of the two coorbitals, fixing m_{3} = 2.5 × 10^{−5}m_{0}. Panel b: as a function of the mass of the two coorbitals μ and the inner planet m_{3}. Dotted lines represent the intersection of the planes of initial conditions (panels a and b). The colour code is the same as Fig. 2. Panel a shows black dots to represent orbits that are stable over 10^{10} orbital periods; see Sect. 2.2 for more details. The numbers displayed in panel a are the value of p for the disruptive Ψ = pν + g resonances. 

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Fig. 5 Example of TTVs for TOI178.02 (m_{1}, orange) and TOI178.03 (m_{2}, blue) for two arbitrary sets of masses, taking as initial conditions the configuration of the system during the observation of sector 2 by TESS. Top: m_{1} = 1 × 10^{−4}m_{0}, m_{2} = 2.5 × 10^{−5}m_{0}, resulting in a horseshoe configuration. Bottom: m_{1} = 4 × 10^{−4}m_{0}, m_{2} = 2.5 × 10^{−5}m_{0}, resulting in a large amplitude trojan configuration. 

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3.3 Predicted TTVs for future observations
As shown in the previous section, depending on m_{1} and m_{2}, the TOI178system could harbour stable coorbital exoplanets, either in a trojan or a horseshoe configuration. Using Eq. (3), we show in Fig. 5 the TTVs that should exhibit such configurations, taking as initial conditions the orbital elements summarised in Table 1, along with two sets of arbitrary masses. In the top panel, m_{1} + m_{2} = 1.25 × 10^{−4}m_{0}, which results in a horseshoe orbit (see Fig. 2), while in the bottom planet m_{1} + m_{2} = 4.25 × 10^{−4}m_{0}, which results in a tadpole orbit with a large amplitude of libration.
Figure 5 also illustrates why planet 1, despite transiting three times during the observation of sector 2 by TESS, would not display significant TTVs during that time: in both examples, the evolution of the TTVs is quasilinear over the first 30 days. As a result, these TTVs can be absorbed by a redefinition of the orbital period of the planet. This linearity is due to the fact that ζ ∈ [280°:310°] during the observations, which corresponds to a global extremum of the instantaneous period of both coorbitals regardless of their amplitude of libration; see Fig. 1.
For this system, the full TTVs induced by the coorbital motion happen over hundreds of days. Figure 6 gives the amplitude and period of the TTVs expected for planet 1 for a grid of masses of the coorbital candidates. The effect of the inner planet 3 is neglected, and should be small compared to the libration in the coorbital resonance. The TTV amplitude of one planet is proportional to the mass of the other planet, and proportional to the resonant angle (Eq. (3)). As a result, theamplitude of the TTVs expected for planet 2 can easily be deduced by swapping the labels of the x and y axis in the lower panel of Fig. 6.
Due to the evolution of the resonant angle of ≈ 30° during the observation of sector 2 by TESS, at least one of the two coorbital candidates should exhibit TTVs of the order of 1 day or more. If these TTVs are detected, it would not only confirm the existence of the coorbital pair, but would also allow for unique and precise determinations of m_{1} ∕m_{0} and m_{2} ∕m_{0}.
Fig. 6 Period (top) and amplitude (here for planet 1, TOI178.02, bottom) of the predicted TTVs induced by the coorbital motion, assuming circular orbits derived from Table 1. The effect of planet 3 is neglected. The amplitude of TTVs predicted for planet 2 (TOI178.03) are obtained by swapping the x and y labels of the bottom panel. 

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4 Summary and conclusions
In Sect. 2 we reviewed the main properties of the 1:1 MMR in the case where both objects are of planetary nature, and we gave the constraints on the TTVs that allowed us to decipher whether or not the two planets on apparently closeperiod orbits are actually in a horseshoe configuration. We applied this method to the Kepler132 system, and concluded that Kepler132 b and Kepler132 c need to have equal masses down to the percent level for their TTVs to be consistent with the coorbital hypothesis.
In Sect. 3 we analysed the case of the TOI178 system in detail, where two planet candidates appear to be on a coorbital orbit. All firstorder analyses described in the TESS data validation report point at transit signals of planetary nature coming from one system in TOI178. One question still to be addressed pertains to whether or not these detected transits were correctly associated to individual planets. Our independent detrending and transit search recovers all three transit signals found by the TESS pipeline at the 6.5, 9.9, and 10.5 day periods, but we cannot exclude an alias for the TOI178.03 (planet 2 in our study) which would put that planet on a 4.9 day orbit, resulting in a configuration in, or close to, a 4:3:2 threebody resonance. Our TTV analysis showed however that in this case, TOI178.03 is expected to be significantly less massive than the other two candidates. Another potential alternative scenario would be that the two transits of the TOI178.03 are indeed two individual transits of outer planets.
Assuming that the orbits summarised in Table 1 are correct, we performed a stability analysis of the system allowing for a large range in mass for each planet. As long as the mass of the inner planet is not much more massive than the sum of the mass of the two coorbital candidates, stable coorbital configurations can exist for a billion years. The stability analysis favoured the case where the TOI178.02 is more massive than TOI178.03. During the time of the observation of the sector 2 by TESS, the phase between the two candidates was λ_{1} − λ_{2} ∈ [280°, 310°]. This allows the bodies to be on a vast range of amplitudes of libration around the L_{5} Lagrangian point (ζ = 300°) or in a horseshoe orbit (see Fig. 1), depending on their mass. The minimal values of the mass of the bodies are also constrained by the stability diagrams Figs. 2 and 4.
TTVs that should be induced by the coorbital motion during the three transit observed of the TOI178.03 cannot be used to further constrain the system because ζ ∈ [280°, 310°] correspond to a local extremum for the instantaneous period of the bodies. Important longterm TTVs, on an observation timespan of hundreds of days, should however be observed on at least one of the two candidates; see Figs. 5 and 6. Such TTVs must be observed if the orbits reported in Table 1 are correct, and would allow for the masses of the TOI178.02 and TOI178.03 to be constrained with great precision.
Alternatively radialvelocity measurements can be used, both on their own (Laughlin & Chambers 2002; Leleu et al. 2015) and in combination with the transit measurements (Ford & Gaudi 2006; Leleu et al. 2017) to confirm the coorbital nature of the system. More constraints on TTVs might be provided by Gaia (Gaia Collaboration 2016), and by the ESA mission CHEOPS to be launched in autumn 2019 (Benz et al. 2018).
Acknowledgements
We thank the anonymous referee for his useful comments and suggestions. The authors acknowledge support from the Swiss NCCR PlanetS and the Swiss National Science Foundation. We thank Juan Cabrera for useful discussions. J.S. is grateful to Françoise Roques for preliminary discussions. P. R. acknowledges financial support from the Programme National de Planétologie (INSUCNRS). A.C. acknowledges support from projects UID/FIS/04564/2019, POCI010145FEDER022217, and POCI010145FEDER029932, funded by COMPETE 2020 and FCT, Portugal. This work was granted access to the HPC resources of MesoPSL financed by the Region Ile de France and the project Equip@Meso (reference ANR10EQPX2901) of the programme Investissements d’Avenir supervised by the Agence Nationale pour la Recherche.
Appendix A TTVs of the TOI178
Posterior of the transit times for the individual transits of each of the three planet candidates in TOI178, both from the light curve sdetrended by the TESS pipeline, and our own analysis.
Fig. A.1 Posterior distribution functions of the planetary masses for the scenario (i) in the top panel, and scenario (ii) in thebottom panel. The heat maps are twodimensional histograms normalised by the total number of points, in logarithmic scale. The histogram subplots on the diagonal show the median (dashed line) and the 1σ uncertainty (thin lines) of the estimated masses. 

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All Tables
Parameters for the three candidates in the TOI178 system, extracted from the DVR of the TESS mission.
Posterior of the transit times for the individual transits of each of the three planet candidates in TOI178, both from the light curve sdetrended by the TESS pipeline, and our own analysis.
All Figures
Fig. 1 Top: phase space of the coorbital resonance valid for small eccentricities and inclinations. The x axis displays the resonant angle ζ = λ_{1} − λ_{2} while the y axis is its normalised angular frequency, which is proportional to . Bottom: value of the normalised fundamental frequency ν for initial conditions along the line (ζ,) of the top graph. Both graphs were obtained by integrating Eq. (1). 

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In the text 
Fig. 2 Stability map as a function of P_{1}∕P_{2} and μ = (m_{1} + m_{2})∕m_{0}, for circular coplanar orbits, taking ζ = 60° as initial condition (vertical dashed line in Fig. 1). Coloured pixels are longterm stable with respect to the integration time of 10^{5} orbits. Grey pixels are longterm unstable, and white pixels are shortterm unstable. The black curve represents the mutual Hill radius of the planets, , while the vertical line represents P_{1}∕P_{2} ≈ 1.04 which is close to the estimated value for the candidates of the TOI178 system discussed in Sect. 3. The colour code is the amplitude of libration: black for zero (lagrangian equilibria), orange for horseshoe orbits. The bottomright corner shows stable orbits outside of the coorbital area, in yellow. The purple line is the stability criterion from the overlapping of firstorder MMRs (Deck et al. 2013). Black dots represent orbits that are stable over 10^{10} orbital periods; see the text for more details. 

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In the text 
Fig. 3 Top: instantaneous orbital period of two planets in a horseshoe orbit. Bottom: TTVs river diagram of the transits of both planets with respect to the outer dashed line at ~ 6.4 days. In the background, black (purple) dots give the river diagram of a planet in the isolated orbit P_{1} (P_{2}). 

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In the text 
Fig. 4 Stability domains for the coorbital candidates of the TOI178 system. Panel a: as a function of the mass of the two coorbitals, fixing m_{3} = 2.5 × 10^{−5}m_{0}. Panel b: as a function of the mass of the two coorbitals μ and the inner planet m_{3}. Dotted lines represent the intersection of the planes of initial conditions (panels a and b). The colour code is the same as Fig. 2. Panel a shows black dots to represent orbits that are stable over 10^{10} orbital periods; see Sect. 2.2 for more details. The numbers displayed in panel a are the value of p for the disruptive Ψ = pν + g resonances. 

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In the text 
Fig. 5 Example of TTVs for TOI178.02 (m_{1}, orange) and TOI178.03 (m_{2}, blue) for two arbitrary sets of masses, taking as initial conditions the configuration of the system during the observation of sector 2 by TESS. Top: m_{1} = 1 × 10^{−4}m_{0}, m_{2} = 2.5 × 10^{−5}m_{0}, resulting in a horseshoe configuration. Bottom: m_{1} = 4 × 10^{−4}m_{0}, m_{2} = 2.5 × 10^{−5}m_{0}, resulting in a large amplitude trojan configuration. 

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In the text 
Fig. 6 Period (top) and amplitude (here for planet 1, TOI178.02, bottom) of the predicted TTVs induced by the coorbital motion, assuming circular orbits derived from Table 1. The effect of planet 3 is neglected. The amplitude of TTVs predicted for planet 2 (TOI178.03) are obtained by swapping the x and y labels of the bottom panel. 

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In the text 
Fig. A.1 Posterior distribution functions of the planetary masses for the scenario (i) in the top panel, and scenario (ii) in thebottom panel. The heat maps are twodimensional histograms normalised by the total number of points, in logarithmic scale. The histogram subplots on the diagonal show the median (dashed line) and the 1σ uncertainty (thin lines) of the estimated masses. 

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In the text 
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