Detection of coorbital planets by combining transit and radialvelocity measurements
^{1} Physikalisches Institut & Center for Space and Habitability, Universitaet Bern, 3012 Bern, Switzerland
email: adrien.leleu@space.unibe.ch
^{2} IMCCE, Observatoire de Paris − PSL Research University, UPMC Univ. Paris 06, Univ. Lille 1, CNRS, 77 Avenue DenfertRochereau, 75014 Paris, France
^{3} CIDMA, Departamento de Física, Universidade de Aveiro, Campus de Santiago, 3810193 Aveiro, Portugal
^{4} European Southern Observatory, Alonso de Cordova 3107, Vitacura Casilla 19001, Santiago 19, Chile
Received: 16 November 2016
Accepted: 30 January 2017
Coorbital planets have not yet been discovered, although they constitute a frequent byproduct of planetary formation and evolution models. This lack may be due to observational biases, since the main detection methods are unable to spot coorbital companions when they are small or near the Lagrangian equilibrium points. However, for a system with one known transiting planet (with mass m_{1}), we can detect a coorbital companion (with mass m_{2}) by combining the time of midtransit with the radialvelocity data of the star. Here, we propose a simple method that allows the detection of coorbital companions, valid for eccentric orbits, that relies on a single parameter α, which is proportional to the mass ratio m_{2}/m_{1}. Therefore, when α is statistically different from zero, we have a strong candidate to harbour a coorbital companion. We also discuss the relevance of false positives generated by different planetary configurations.
Key words: planets and satellites: detection / celestial mechanics / planetary systems / techniques: radial velocities / techniques: photometric
© ESO, 2017
1. Introduction
Coorbital planets consist of two planets with masses m_{1} and m_{2} orbiting with the same mean motion a central star with mass m_{0}. In the quasicircular case, as long as the mutual inclination remains smaller than a few tens of degrees, the only stable configurations are the Trojan (like Jupiter’s trojans) and the Horseshoe (like Saturn’s satellites Janus and Epimetheus). Stable Trojan configurations arise for (m_{1} + m_{2}) /m_{0} ≲ 4 × 10^{2} (Gascheau 1843), and Horseshoe configurations for (m_{1} + m_{2}) /m_{0} ≲ 2 × 10^{4} (Laughlin & Chambers 2002). We note that, at least when no dissipation is involved, the stability of a given configuration does not depend much on the mass distribution between m_{1} and m_{2}.
Coorbital bodies are common in the solar system and are also a natural output of planetary formation models (Cresswell & Nelson 2008, 2009). However, so far none have been found in exoplanetary systems, likely owing to the difficulty in detecting them. For small eccentricities, there is a degeneracy between the signal induced by two coorbital planets and a single planet in an eccentric orbit or two planets in a 2:1 mean motion resonance (e.g. Giuppone et al. 2012). In favourable conditions, both coorbital planets can eventually be observed transiting in front of the star, but this requires two large radii and small mutual inclination. A search for coorbital planets was made using the Kepler Spacecraft^{1} data, but none were found (Janson 2013; Fabrycky et al. 2014). We hence conclude that coorbitals are rare in packed multiplanetary systems (like those discovered by Kepler), that they are not coplanar, or that one coorbital is much smaller than the other. For larger semimajor axes, we expect that at least one of the coorbitals cannot be observed transiting. When the libration amplitude of the resonant angle is detectable (either by transittime variations or with radialvelocity modulations), we can still infer the presence of both planets (Laughlin & Chambers 2002; Ford & Holman 2007). These effects have not been detected so far, at least not with sufficient precision to rule out other scenarios. However, we cannot conclude that no coorbitals are present in the observed systems: transit timing variation (TTV) and radialvelocity methods will both miss a coorbital companion if the amplitude of libration is not large enough or if its period is too long.
Ford & Gaudi (2006) noticed that for a single planet in a circular orbit, the time of midtransit coincides with the instant where the radialvelocity reaches its mean value. However, if the planet that is transiting has a coorbital companion located at one of its Lagrangian points, there is a time shift ΔT between the midtransit and the mean radialvelocity, that depends on the properties of the coorbital companion. Therefore, when we combine transit and radialvelocity measurements, it is possible to infer the presence of a coorbital companion. This method was developed for circular orbits and for a companion at the exact Lagrangian point (without libration). Although it remains valid for small libration amplitudes (which would just slightly modify the determined mass), coorbital exoplanets can be stable for any amplitude of libration. Moreover, for a single transiting planet in a slightly eccentric orbit, we can also observe the same time shift ΔT, without requiring the presence of a coorbital companion.
In this Letter, we generalise the work by Ford & Gaudi (2006) to eccentric planets in any Trojan or Horseshoe configuration (any libration amplitude). When a planet is simultaneously observed through the transit and radialvelocity techniques, we propose a simple method for detecting the presence of a coorbital companion that relies on a single dimensionless parameter α ∝ m_{2}/m_{1}. Therefore, when α is statistically different from zero, we have a strong candidate to harbour a coorbital companion and we get an estimation of its mass. Moreover, if the secondary eclipse of the transiting planet is also observed, our method further constrains the uncertainty in α. We also discuss the possibility of false positive detections due to other effects.
2. Radialvelocity
Fig. 1 Reference angles for the orbit of a given planet with respect to an arbitrary frame Oxyz, where O is the centre of the star and z the line of sight. 

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In a reference frame where the zaxis coincides with the observer’s line of sight (Fig. 1), the radialvelocity of the star induced by the planet k with mass m_{k} is given by (Murray & Correia 2010) (1)with (2)where M = m_{0} + ∑ _{k}m_{k}, a is the semimajor axis, n is the mean motion, e is the eccentricity, I is the inclination angle between the plane of the sky and the orbital plane, ω is the argument of the pericentre, and f is the true anomaly.
For small eccentricities, we can simplify v_{k} by expanding cosℓ_{k} in powers of e_{k} (Murray & Dermott 1999) (3)where λ_{k} = n_{k}t + ϕ_{k}, and ϕ_{k} is a phase angle. At first order in eccentricity, the radialvelocity induced by a single planet on a Keplerian orbit is thus of the form (4)with (5)If we sum the contribution of two planets on Keplerian orbits, the total radialvelocity of the star becomes (6)where γ is the velocity of the system’s barycentre. In the coorbital quasicircular case, the semimajor axes of the planets librate around their mean value with a frequency , where μ = (m_{1} + m_{2}) /M and n is the meanmotion associated with . The amplitude of the libration goes from 0 at the Lagrangian equilibrium up to in the tadpole domain, and to in the horseshoe domain (Erdi 1977; Robutel & Pousse 2013). We note that horseshoe coorbitals are stable only for μ lower than ≈2 × 10^{4}. For a pair of coorbital planets we hence have where β ≥ 1/2 for tadpole coorbitals and 1/2 ≥ β ≥ 1/3 for the horseshoe configuration.
There are two possible scenarios for which we can consider that n_{1} = n_{2} = n:
when the time span is short with respect to the librationfrequency ν and we do not have the frequency resolution to distinguish n_{1} from n_{2};
when the time span is longer than 2π/ν, and the harmonics of the radialvelocity signal are located at pn + qν with (p,q) ∈ Z^{2}. The harmonics for q ≠ 0 have larger amplitudes if the coorbitals librate with a large amplitude and if their masses are similar. If we can distinguish the effect of the libration in the radialvelocity signal, we can identify coorbitals from radialvelocity alone (see Leleu et al. 2015). If not, the assumption n_{1} = n_{2} = n holds, and the mean longitudes simply read (7)
For the radialvelocity induced by two coorbitals, we hence sum cosines that have the same frequency. At order one in the eccentricities, we obtain an expression which is equivalent to (4), (8)with A = A_{1} + A_{2}, and similar expressions for B, C, and D. The radialvelocity induced by two coorbitals is thus equivalent to the radialvelocity of a single planet on a Keplerian orbit with mean motion n, and orbital parameters given by (9)These expressions are similar to those obtained by Giuppone et al. (2012). We note that this equivalence is broken at order 2 in eccentricity: the next term in the expansion (8) is Ecos3nt + Fsin3nt. In the single planet case, we have (10)which is only also satisfied for eccentric coorbitals for very specific values of the orbital parameters (λ_{1}−λ_{2} = ω_{1}−ω_{2} and e_{1} = e_{2}). Therefore, in most cases, if we can determine , we can solve the degeneracy between a single planet and two coorbitals.
3. Time of midtransit
We now assume that the planet with mass m_{1} is also observed transiting in front of the star. We consider that the planet transits when its centre of mass passes through the cone of light (we do not consider grazing eclipses because of the difficulty in estimating the time of midtransit). For simplicity, we set the origin of the time t = 0 as the time of midtransit. The true longitude ℓ_{1} of midtransit is (Winn 2010) (11)The inclination I_{1} has to be close to π/ 2 because the planet is transiting. Denoting , we have that , which is a negligible quantity. We thus conclude that for t = 0, (12)We can now express the phase angles ϕ_{k}, involved in expressions (5) and (7), in terms of e_{1} and ω_{1}. Since (13)it turns out that (using t = 0) (14)For moderate mutual inclination and at order one in eccentricity we additionally have (Leleu et al. 2015) (15)where ζ = λ_{2}−λ_{1} is the resonant angle. If we cannot see the impact of the evolution of ζ in the observational data, either because its amplitude of libration is negligible or because the libration is slow with respect to the time span of the measurements, we can consider ζ to be constant.
4. Radialvelocity and transit
In Sect. 2, we saw that, at first order in e_{k}, the radialvelocity induced by a pair of coorbital planets is equivalent to that of a single planet. However, the phase angle ϕ_{1} of the observed planet can be constrained by the transit event (Eq. (14)). Thus, assuming that we are able to measure the instant of midtransit for the planet with mass m_{1}, we can replace the phase angles (14) and (15) in the expression of the radialvelocity (8) to obtain (16)where k_{k} = e_{k}cosω_{k} and h_{k} = e_{k}sinω_{k}.
A striking result is that the quantity (17)is different from 0 only if K_{2} ≠ 0, that is only if the transiting planet m_{1} has a coorbital companion of mass m_{2}. Therefore, the estimation of this quantity provides us invaluable information on the presence of a coorbital companion to the transiting planet.
5. Detection methods
We assume that we are observing a star with a transiting planet, and that we are able to determine the orbital period (2π/n) and the instant of midtransit with a very high level of precision. We assume that radialvelocity data are also available for this star, and are consistent with the signal induced by a single planet on a slightly eccentric Keplerian orbit (Eq. (8)).
Setting t = 0 at the time of midtransit, we propose a fit to the radialvelocity data with the following function: (18)The parameters to fit correspond to γ, K = −B, c = C/K, d = D/K, and α = (A + 2C) /K. We fix n because it is usually obtained from the transit measurements with better precision. The dimensionless parameter α is proportional to the mass ratio m_{2}/m_{1} (Eq. (17)). Whenever α is statistically different from zero, the system is thus a strong candidate to host a coorbital companion.
In general^{2}α ≪ 1, which implies that ε = K_{2}/K_{1} ≪ 1, i.e. m_{2} ≪ m_{1}. Making use of this assumption, we obtain simplified expressions for all fitted quantities:
All the fitted parameters are directly related to the physical parameters that constrain the orbit of the observed planet, and they additionally provide a simple test for the presence of a coorbital companion (α ≠ 0). For Trojan orbits, α < 0 (resp. α > 0) corresponds to the L4 (resp. L5) point.
5.1. Antitransit information
Whenever it is possible to observe the secondary eclipse of the transiting planet at a time t = t_{a}, we can access directly the quantity k_{1} by comparing the duration between the primary and secondary transit to half the orbital period computed from the two primary transits (Binnendijk 1960) (20)In this case, since we can get the c = k_{1} parameter from the secondary eclipse (usually with much greater precision than the radialvelocity measurements), we can fix it in expression (18), and thus fit the only four remaining parameters. This allows us to achieve a better precision for α, and thus confirm the presence of a coorbital companion.
5.2. Duration of the transits
The observation of the secondary eclipse of the transiting planet can also constrain the quantity h_{1} by comparing the duration of the primary transit and the secondary eclipse, Δt and Δt_{a}, respectively. We have (Binnendijk 1960): (21)In this case, we also get an estimation for the d = h_{1} parameter before the fit, which can further improve the determination of α. We note, however, that unlike for k_{1}, the precision of this term is not necessarily better than the radialvelocity constrain (Madhusudhan & Winn 2009).
6. False positives
There are other physical effects that can also provide nonzero α, and thus eventually mimic the presence of a coorbital companion. The main sources of error could be due to nonspherical gravitational potentials, the presence of orbital companions, or the presence of an exomoon.
The main consequence of most of the perturbations (general relativity, the J_{2} of the star and/or of the planet, tidal deformation of the star and/or of the planet, secular gravitational interactions with other planetary companions) is in the precession rate of the argument of the pericentre, . However, the mean motion frequency that is determined using the radialvelocity and the transits technique is given by (Eq. (7)), which already contains . Thus, in all these cases our method is still valid.
For closein companions, cannot be considered constant, and we can observe a nonzero α value that could mimic the presence of a coorbital companion. However, strong interactions require large mass companions whose trace would be independently detected in the radialvelocity data and through TTVs. The only exceptions are exomoons, which have the exact same mean motion frequency as the observed planet, or the 2:1 meanmotion resonances with small eccentricity, whose harmonics of the radialvelocity data coincide with the coorbital values.
In the case of exomoons, the satellite switches its orbital position with the planet rapidly, so α oscillates around zero with a frequency ν ~ n that is not compatible with a libration frequency of a coorbital companion. For most of coorbital configurations, the libration frequency is comparable with the libration frequency at the L_{4} equilibrium, , and the average of α is around ζ = ± π/ 3, not zero. Therefore, our method also provides a tool for detecting exomoons.
For the 2:1 meanmotion resonance, we must distinguish which planet transits. If the transiting planet is the inner one, α is impacted by the eccentricity of the outer planet. However, if the outer planet is massive enough to impact the value of α, its harmonic of frequency n/ 2 must be visible in the radialvelocity measurement. If the transiting planet is the outer one, the inner planet impacts α indirectly by modifying the value of the parameter c. This is not a problem if this parameter is well constrained by the antitransit of the transiting planet. Moreover, the inner planet would induce TTV on the transiting planet of the order of m_{2}/m_{0} (Nesvorný and Vokrouhlický 2014). If the semimajor axis of the transiting planet is not too large, the TTVs should be observed, and here again their frequency allows to distinguish the coorbital case from the 2:1 resonance.
7. Conclusion
In this Letter, we have generalised the method proposed by Ford & Gaudi (2006) for detecting coorbital planets with null to moderate eccentricity and any libration amplitude (from the Lagrangian equilibrium to Horseshoe configurations). For highly eccentric orbits this method is not needed because it is possible to use radialvelocity alone to infer the presence of the coorbital companion (Eq. (10)).
Our method is based in only five free parameters that need to be adjusted to the radialvelocity data. Moreover, when it is also possible to observe the secondary eclipse, we have additional constraints which reduce the number of parameters to adjust. One of the free parameters, α, is simply a measurement for the presence of a coorbital companion, which is proportional to the mass ratio m_{2}/m_{1}. As discussed in section 6, other dynamical causes can produce a nonzero α. However, alternative scenarios would also significantly impact the TTV and/or the radialvelocity, and allow us to discriminate between them.
Therefore, if α is statistically different from zero and the TTV and radialvelocity do not show any signature of other causes, the observed system is a strong candidate to harbour a coorbital companion. We additionally get an estimation of its mass. Inversely, if α is compatible with zero, our method rules out a coorbital companion down to a given mass, provided that sinζ is not too close to zero (Eq. (20)). This is unlikely because ζ = 0 corresponds to a collision between the two planets, and ζ = π can only occur in the Horseshoe configuration, hence when (m_{1} + m_{2}) /m_{0} ≲ 2 × 10^{4} (Laughlin & Chambers 2002).
Except when sinζ tends to 0. However, this cannot happen when the sum of the mass of the coorbital is higher than 10^{3} the mass of the star, for stability reasons (Leleu et al. 2015).
Acknowledgments
The authors acknowledge financial support from the Observatoire de Paris Scientific Council, CIDMA strategic project UID/MAT/04106/2013, and the Marie Curie Actions of the European Commission (FP7COFUND). Parts of this work have been carried out within the frame of the National Centre for Competence in Research PlanetS supported by the SNSF.
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All Figures
Fig. 1 Reference angles for the orbit of a given planet with respect to an arbitrary frame Oxyz, where O is the centre of the star and z the line of sight. 

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In the text 