Detecting planet pairs in mean motion resonances via astrometry method
Dong-Hong Wu, Hui-Gen Liu, Zhou-Yi Yu, Hui-Zhang, Ji-Lin Zhou
DDraft version October 18, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
DETECTING PLANET PAIRS IN MEAN MOTION RESONANCES VIA ASTROMETRY METHOD
Dong-Hong Wu, Hui-Gen Liu, Zhou-Yi Yu, Hui-Zhang and Ji-Lin Zhou
School of Astronomy and Space Science, Nanjing UniversityKey Laboratory of Modern Astronomy and Astrophysics in Ministry of Education, Nanjing University, Nanjing, 210046, China;[email protected]
Draft version October 18, 2018
ABSTRACTGAIA leads us to step into a new era with a high astrometry precision ∼ µ as. Under such aprecision, astrometry will play important roles in detecting and characterizing exoplanets. Specially,we can identify planet pairs in mean motion resonances(MMRs) via astrometry, which constrainsthe formation and evolution of planetary systems. In accordance with observations, we consider twoJupiters or two super-Earths systems in 1:2, 2:3 and 3:4 MMRs. Our simulations show the false alarmprobabilities(FAPs) of a third planet are extremely small while the real two planets can be good fittedwith signal-to-nois ratio(SNR) >
3. The probability of reconstructing a resonant system is relatedwith the eccentricities and resonance intensity. Generally, when SNR ≥
10, if eccentricities of bothplanets are larger than 0.01 and the resonance is quite strong, the probabilities to reconstruct theplanet pair in MMRs ≥ >
15% whenSNR ≤
10. Extrapolating from the Kepler planet pairs near MMRs and assuming SNR ∼
3, we willdiscover and reconstruct a few tens of Jupiter pairs and hundreds of super-Earth pairs in 2:3 and 1:2MMRs within 30 pc. We also compare the differences between even and uneven data cadence andfind that planets are better measured with more uniform phase coverage.
Subject headings: stars:planetary systems-astrometry-methods:data analysis-methods:numerical INTRODUCTION
Until April 11, 2016, 1642 planets have been detected,1038 of them are in multiple planet systems , and about41% of planet-host stars have more than one planetarycompanion. Due to the high precision of Kepler mission,many planets in multiple planet systems have been con-firmed by transit timing variation(TTV) (Steffen et al.2012; Ford et al. 2012; Fabrycky et al. 2012; Xie 2013,2014). However, this method is limited when two plan-ets are very close to the resonance center, because theperiod of TTV is quite long and hard to be determinedwell. Yang et al. (2013) shows a TTV signal with period ∼ m sin i . Onlya few planets are detected by both transit and radial ve- exoplanet.org locity methods. Besides, some of the orbital elementsare degenerate and time series of planetary mean lon-gitudes are usually not available in extrasolar systems,which make us hardly know whether they are in MMRsor not. Two of the exceptions are the HD 82943 andHD 45364 systems. Planets detected by radial velocityaround HD 82943 and HD 45364 systems are confirmedto be in 1:2 and 2:3 MMRs by dynamical stability analy-sis (Lee et al. 2006; Correia et al. 2009), i.e. the systemsare stable only if the planets are in MMRs. However,some systems are still not confirmed to be in or nearMMRs even they are very close to the resonance center,for example, the period ratio of EPIC201505350 b and cas displayed in the K2 data is 1.503514, among the clos-est systems to a 2:3 commensurability detected so far(Armstrong et al. 2015).In past days, astrometric measurements with m as pre-cision such as HIPPARCOS (Perryman et al. 1997) doesnot allow the detection of exoplanets. A star with aJupiter at 1 AU located at 30 pc has a periodic astromet-ric signature of about 30 µ as and is hardly detected with1 m as precision. However, with the improvement of thetechnique, many researches have shown that astrometricobservations with µ as-level precision are possible, suchas GAIA which have been launched in 2013 (Lattanziet al. 2000; Sozzetti et al. 2001; Lattanzi et al. 2002, 2005;Sozzetti 2010) and STEP in plan (Chen 2014). GAIA canachieve a single-measurement astrometric error of a fewtens of µ as (Sahlmann et al. 2015), which is sufficient todetect a Jupiter at 1 AU around a solar-like star within 30pc. STEP is designed to have a single-measurement as-trometric precision of 1 µ as, and has potential to detecthabitable super-Earth around solar-like star at 30 pc. a r X i v : . [ a s t r o - ph . E P ] D ec m /m c oun t R /R c oun t . Fig. 1.—
Distribution of mass ratio m /m and radius ratio R /R of the near MMR planet pairs, subscript 1 and 2 representthe the inner and outer planet, respectively. The data (provided byhttp://exoplanet.org) includes all multiple planet systems whichcontain planet pairs in or near MMRs with ∆ < . j − P / ( jP ) − P and P are the periods of the inner andouter planet, respectively). Astrometry can provide more information of the planets,including the six orbital elements and the mass of eachplanet, which are essential to decide whether the planetpairs are in MMRs or not.This paper is arranged as follows. In Section 2, wedescribe the astrometry method in detecting exoplanetsand simulation setup. In Section 3, we investigate thelimits of SNR to detect the planet pairs and analysisthe fitting results of the planets in our simulations. Theresonance-reconstruction probabilities of planet pairs in1:2, 2:3 and 3:4 MMRs are shown in Section 4. In Section5, we calculate the FAPs of a detected planet system inor near MMRs. In Section 6, we estimate how manyplanet pairs in MMRs can be detected and reconstructedin 30 pc. The differences between even and uneven datacadences are present in Section 7. Finally, we concludeour results and discuss how to reconstruct planet pairsin MMRs better in Section 8. DETECTING EXOPLANETS BY ASTROMETRY ANDSIMULATION SETUP
Setups of planet pairs in MMRs
The mass and radius ratios of planets observed to be inor near MMRs are shown in Figure 1. Generally, we con-sider planet systems containing two planets with equalmass in 1:2, 2:3 and 3:4 MMRs in this paper. We onlysimulate planet systems with two Jupiters and two super-Earths separately. Both masses of super-Earths are setas 10 Earth masses. Hereafter, super-Earth means planetwith 10 Earth masses.Planets in MMRs can be produced by migration andrandomly perturbing the orbital elements of the planetsnear MMRs. We simplify the migration model by addinga slow inward semi-major axis migration to the outerplanet, thus the outer orbit will approach to the center of MMRs. For example, giving a number of planet pairsnear ( j −
1) : j MMR ( j =2,3,4), we add a typical migra-tion with timescale of 5 × years to let the planet sys-tems evolve into MMRs. We halt the migration while theplanet pairs are in ( j −
1) : j MMR ( j =2,3,4), thus we canobtain samples of planet pairs in MMRs. With differentmigration time, we can attain different eccentricities ofthe planet pairs in MMRs. There is a positive correlationbetween e and e for resonant planet pairs produced thisway. Besides, after planets migrate in the disk, they areusually locked in MMRs which are very stable (Lee et al.2009), i.e. the resonance intensities of these systems arestrong. However, if there are more planets in the disk,after the disk disappears, the resonance will be disturbedand may be not as strong as they were while migrationhalted. To complete our samples with different resonanceintensities, we also produce resonant planet systems byrandomly perturbing the orbital elements of the planetsnear ( j −
1) : j MMR ( j =2,3,4). We choose planet pairswith initial ∆ = ( j − P / ( jP ) − < . P and P are the periods of the inner and outer planet,initial eccentricities are randomly distributed from 0 and0.4, initial inclinations are randomly distributed from 0to 5 ◦ , other orbital elements: Ω, ω and M are randomlydistributed from 0 to 360 ◦ . ∆ is a measure of nearness toresonance. For 2:3 and 1:2 MMRs, we have two groups ofresonant planet systems produced by two methods men-tioned above. But for 3:4 MMR of two Jupiters, we onlyhave resonant systems by random method because plan-ets are scattered before they migrate to be captured in3:4 MMR.In this paper, all the inner planets are located at 0.8 ∼ × years. There are only two resonance angles φ and φ forthe ( j −
1) : j MMR, i.e., φ = jλ − ( j − λ − (cid:36) and φ = jλ − ( j − λ − (cid:36) . λ i = M i + (cid:36) i is the mean longi-tude, M i is the mean anomaly while (cid:36) i = Ω i + ω i (i=1,2).Subscript 1 and 2 of the orbital elements represent theinner and outer planet, respectively. In general, a planetpair is considered to be in MMR as long as one resonanceangle is in libration (Murray & Dermott 1999; Raymondet al. 2008). To obtain refined samples of planets inMMRs, we only choose planet systems with libration am-plitudes of both φ and φ less than 300 degree in 2 × years. The systems with only one resonance angle in li-bration are not included in our samples. The numbersof planet pairs in each MMR are shown in the first col-umn in Table 3 and Table 4. All planets in MMRs inour samples are nearly face on with inclinations between0 ◦ and 10 ◦ . The mutual inclinations of planet pairs areless than 5 ◦ . The eccentricity distributions are shown inFigure 2. Simulation of astrometric data
Detecting exoplanets by astrometry with µ as precisionhas become possible since the launch of GAIA. Similarto the RV method, astrometry measures projected move-ments of the host star around the barycenter of the sys-tem. By measuring the movement of the star, we canacquire planetary orbits and masses. The astrometricmeasurements in x and y(x and y represent the pro-jected movement in RA and DEC direction, respectively)at time t relative to the reference frame of background : m =m =1 × −3 randommigration m =m =3 × −5 e : : . Fig. 2.—
Distribution of e and e for the samples in MMR, e and e are the eccentricities of the inner and outer planet, respec-tively. The red dots represent samples produced by migration andthe blue ones represent samples produced randomly. We can seethat there is a positive correlation between e and e for samplesfrom migration while samples from random method have a widerdistribution of eccentricities. The top, middle and the lower panelsare samples of 2:3, 1:2 and 3:4 MMRs, respectively. The left panelsshow the samples of the Jupiter pairs and the right panels showthe super-Earth pairs. stars are modeled with (Black & Scargle 1982) : x = x + µ x ( t − t ) − P x π + X + Err x , (1) y = y + µ y ( t − t ) − P y π + Y + Err y , (2)In Equations(1)-(2), x and y are the coordinate off-sets. µ x and µ y are the proper motions of the star. P x and P y are the parallax parameters which will be pro-vided in the observation, π is the annual parallax of thestar. x , y , µ x , µ y and π are taken as stellar parameters.X and Y are the displacements in the star’s position dueto its planetary companion(s), Err x and Err y are single-measurement astrometric errors. In this paper, when wesimulate astrometry data, we fix µ x = 50 m as/year, µ y =-30 m as/year. All the planet systems are set to be 30 pcaway from us. A planet with mass m p and semi-majoraxis a p will lead to an astrometric signature of: S = 3( m p m Earth )( a p m (cid:63) m Sun ) − ( d − µas (3)on the star with a distance of d , m (cid:63) is the stellar mass.We adopt a simple Gaussian measurement error model,i.e. Err x and Err y follow a Gaussian distribution withstandard deviation σ m in our simulations.. The SignalNoise Ratio(SNR) is defined as S/σ m , which is similarwith the definition in Casertano et al. (2008). Note,the SNR defined here is for single measurement. Equa-tions (1)-(2) can be complicated to include aberrationof starlight and perspective acceleration, so we assumethese effects have been perfectly removed from measure-ments. P x and P y can be provided given the orbit ofthe satellite, here we use a one-year circular orbit to sim-plify the parallax model. After generating planet pairsin MMRs in Section 2.1, we use a RKF7(8) (Fehlberg 1968) N-body code which includes the full Newtonianinteraction between the planets to simulate astrometricdata and sample every 0.1 year, each simulation consistsof a time series of coordinate measurements accordingto Equations (1)-(2) with a nominal mission lifetime setas 5 years. Therefore, we have a set of 50 points [ x ( t i ), y ( t i )], i = 1 , , ...,
50, each represents a measurement atobservation time t i . orbital parameter fitting procedure In general, interaction between planets can be ignoredbecause it can hardly affect the motion of stars (Sozzettiet al. 2001). Most of the multiple-planet systems discov-ered by radial velocity techniques can be well modeledby planets on independent Keplerian orbits (Casertanoet al. 2008), such as the 55 Cancri system with five plan-ets around the primary star (Fischer et al. 2008). Whena star host two planetary companions, we also assumethat the astrometric signal of the host star is the super-position of two strictly non-interacting Keplerian orbits.Ignoring the interaction between planets, X and Y areexpressed as (Catanzarite 2010): X = N (cid:88) i =1 (cos E i − e i ) A i + (cid:113) − e i (sin E i ) F i , (4) Y = N (cid:88) i =1 (cos E i − e i ) B i + (cid:113) − e i (sin E i ) G i . (5)In Equations (4)-(5), E i is the eccentric anomaly, e i isthe eccentricity of the planets, A i , F i , B i , G i are ThieleInnes constants, which encode amplitudes and orienta-tions of the orbits such as the inclinations of planets I i ,arguments of pericenter ω i , longitudes of ascending nodesΩ i (i=1,...,N). N is the number of planets.We use a hierarchical scheme to fit the orbits of theplanets, the details of orbit reconstruction have been de-scribed in Catanzarite (2010), here we briefly introducethe concrete process:Step 1: Ignore the planetary influence on the star andinvert the Equations (1)-(2) by linear least squares tocalculate x , y , µ x , µ y and π , then we have the initialvalue of the stellar parameters for the next step.Step 2: Remove the coordinate offsets, proper motionand parallax from data, and then analysis the residualswith the periodogram (Scargle 1982) to see if there is asignificant period( P ) which exist both in x and y direc-tion. If there is one, we obtain an initial guess of theperiod of the most significant planet. We identify a cer-tain orbit when the False Alarm Probability(FAP) of thecorresponding period is less than 1%. As we have two-dimensional time-series astrometric data, we calculatethe joint periodogram defined in Catanzarite et al. (2006)as the sum of the Lomb-Scargle periodogram power fromeach dimension. The calculation of FAP can be found inScargle (1982) and Horne & Baliunas (1986).Step 3: We randomly choose the initial value of ec-centricity e and the moment that the planet pass itsperihelion t of the planet. The stellar parameters x , y , µ x , µ y , π and the period of the planet P with initialvalues obtained in Step 1 and 2 are also fitted. Equa-tions (1)-(2) are easily inverted by linear least squares toyield the 4 Thiele Innes constants A , F , B and G .X and Y can be calculated and we have fitted x and y.We adopt the MCMC algorithm in our fitting procedure.After the MCMC chains converge, we’ll have more pre-cise stellar parameters, P , e and t of the first planet. I , ω , Ω and ( a m ) /m (cid:63) can be calculated accordingto the Thiele Innes constants.Step 4: The projected motion of star due to the firstplanet is then removed from the astrometric data, againwe use the periodogram to search for significant peaksin the residuals. If there is one with FAP smaller than1%, then it provides an initial guess for the period of thesecond planet( P ), the data is then fitted with a two-planet reflex motion model.Step 5: Continue Step 2-4 until no significant signalappears in the periodogram.For each two-planet system, there are 5+2 × µ x , µ y , π , x , y , P , P , t , t , e , e . Other parameters of planets can be de-rived from P , P and the 8 Thiele Innes constants. Ifwe know the mass of the star in advance, we will alsoobtain semi-major axis and the mass of the each planet.Nowadays, with the development of spectrometry andastroseismology, the mass of the star can be measuredwith a precision of 10% (Creevey et al. 2007; Epsteinet al. 2014). Besides, semi-major axis of the planets areobtained through the relation between the mass of thehost star and the orbital period in our fitting procedure, a and a are proportional to m / (cid:63) , masses of the planets m and m are proportional to m / (cid:63) , the derivation canbe found in Catanzarite (2010). We briefly illustrated ithere. The astrometric signature of planet i has on thehost star is S i , with the parallax we calculated, we havean estimation of the semimajor axis of the stellar reflexmotion a (cid:63),i = S i /π (i=1,2). The center of mass equationgives the planets’s mass m i a i = m (cid:63) a (cid:63),i (i=1,2). Togetherwith Kepler’s 3 rd law a i = m (cid:63) P i , we can determinethe ratios of a /a and m /m . Therefore, the preci-sion of the stellar mass won’t affect the characteristicsof MMRs because a /a , m /m and other orbital ele-ments are independent of the stellar mass. As we adoptthe linear function in Equations (4)-(5), we can’t distin-guish the solution of parameters ω i , Ω i from ω i + 180 ◦ ,Ω i +180 ◦ (i=1,2) without the information of position vari-ation in the direction of our sight. However, if the orbitsof planets are face on, the two solutions of parameterswon’t influence the resonance configuration. In this pa-per, to simplify the problem, we assume all the centerstars have the solar mass. We run each MCMC with3 × iterations and statistics are derived on the last1 × elements. We choose the best-fit parameters asthe median of posterior distribution. More details aboutthe MCMC procedure can be seen in the Appendix. DETECTING PLANETS WITH DIFFERENT SNRS
To investegate the detection of planet with differentSNR, we simulate single planet systems with differentmass and semi-major axis around a solar-like star at 30pc. We adopt a detection criterion of a planet as men-tioned in section 2, i.e., the FAP of of the corresponding period is less than 1%. The left and right panels in Fig-ure 3 show the ability we can detect and characterizethe planet with observational errors σ m = 0.3 µ as and10 µ as. Our simulations show planets with SNR > ∼ ∼ < N ≈ −
30 with single signal-to-noise ratio
K/σ ≈ − K is the signal amplitude in ra-dial velocity, detection of signals < σ requires N ≥ ≥ µ as, while a star at 30 pc with a super-Earth at 1.0AU has a periodic astrometric signature of about 1 µ as.As the interaction between the planets are ignored in thefitting procedure, the Jupiter pairs with observational er-rors σ m = 10 µ as and the super-Earth pairs with obser-vational errors σ m = 0 . µ as locate near the line SNR=3,which indicates that they can be detected and character-ized well. The fitting results of two-planet system in thefollowing sections are good, i.e. the reduced chi-squarevalue χ distributed between 0.9 and 1.3 for > ≥ ∼
3, the abso-lute fitting errors of eccentricities largely increased, espe-cially for planet pairs in 1:2 MMR. Other absolute fittingerrors of orbital elements such as I i , ω i + Ω i , M i (i=1,2)are also very sensitive to observational errors. Here wecompare the difference between the fitted ω i +Ω i and true ω i +Ω i (i=1,2) because the Keplerian model we use yieldstwo orbital solutions ω i , Ω i and ω i + 180 ◦ , Ω i + 180 ◦ ,which have been mentioned in Section 2.3. Note that asthere is degeneracy between ω i and M i when the eccen-tricities are very small, the absolute fitting errors of ω i and M i decrease with the increase of eccentricities. Com-pared with 2:3 and 3:4 MMR, planet pairs in 1:2 MMRshave larger relative fitting errors of masses and absolutefitting errors for other orbital parameters. The 1:2 pe-riod ratio makes it hard to fit orbital parameters of both . Fig. 3.—
The fitting errors log (cid:112) ( δ a + δ m ) / σ m = 0 . µ as(the left) and 10 µ as(the right).We simulate a large number of single planet systems with different planetary mass and semi-major axis to check if we can detect andcharacterize them by astrometry method. All the central stars have 1 solar mass and they are 30 pc away from us. The astrometry data aregenerated with an even cadence of 0.1 year. The green line at a = 0 .
341 AU represents a period of 0.2 year, which is the minimum periodcan be found with a sample cadence of 0.1 year. The magenta line at a = 2 .
924 AU represents a period of 5 year. The blue line representsplanet systems with SNR=1 and the dark line represents planet systems with SNR=3. The region between the green and magenta linewith SNR > planets as well as those of the 2:3 and 3:4 MMRs becauseof harmonic. Average orbital parameter fitting errors ofsuper-Earth pairs are similar with those of Jupiter pairswhen they have similar SNRs. The small relative fittingerrors of planetary mass and semi-major axis guaranteeour successful detection and characterization of planetsystems in our simulations. THE PROBABILITY TO RECONSTRUCT PLANETPAIRS IN MMRS
After fitting the orbital parameters of the planets, wecheck the stabilities of the planet systems. Because ifthe fitted orbital parameters deviate far from the trueones, the fitted planet systems will be unstable, espe-cially the Jupiter pairs. We use β and β to indicatethe probability of the fitted resonance angles φ and φ in libration. To obtain the probability of planet pairs inMMRs, we divide the total integral time 2 × yearsinto 5 equal parts and check if the resonance angles sim-ulated in fitted systems are cycling in each 4 × years.The probability of planet pairs in MMRs is defined asthe fraction of time with librating resonance angle. Weuse β , the larger one between β and β to represent theprobability of reconstructing a planet pair in MMR. The stabilities and probabilities in MMRs of thefitted planet systems
For a two-planet system, the separation of the plan-ets should be at least 3.5 R H according to Gladman1993if the planets are Hill stable. R H is the Hill radius of aplanet. For a Jupiter at 1 AU, 3.5 R H is about 0.242 AU,so the outer planet should be outside of 1.242 AU witha period ratio P /P smaller than 0.72. The Hill stabil-ity indicates Jupiter pairs near 1:2 and 2:3 MMRs arelikely to be stable, while those near 3:4 MMR are alwaysunstable unless they are exactly in 3:4 MMR. This anal-ysis is corresponding with our simulations that Jupiterpairs with P /P ∼ / R H for a super-Earth at 1 AU is 0.075AU, so the outer planet should be outside of 1.075 AUwith period ratio P /P < . , the fraction of both plan-ets with 5 M Earth ∼ M Earth are 23 . . M J ∼ M J are 2 . exoplanet data used here are from exoplanets.org TABLE 1average fitting errors for Jupiter pairs. observational (cid:12)(cid:12) a fit − a true (cid:12)(cid:12) /a true (cid:12)(cid:12) m fit − m true (cid:12)(cid:12) /m true (cid:12)(cid:12) e fit − e true (cid:12)(cid:12) (cid:12)(cid:12) i fit − i true (cid:12)(cid:12) (cid:12)(cid:12) ω fit + Ωfit − ω true − Ωtrue (cid:12)(cid:12) (cid:12)(cid:12) M fit − M true (cid:12)(cid:12) . × − . × − × − . × − . × − . × − . ◦ /1 . ◦ . ◦ /14 . ◦ . ◦ /18 . ◦ µ as 1 . × − . × − . .
013 0 . .
01 14 . ◦ /8 . ◦ . ◦ /15 . ◦ . ◦ /20 . ◦ µ as 4 . × − . × − . .
05 0 . .
04 28 . ◦ /16 . ◦ . ◦ /39 . ◦ . ◦ /49 . ◦ . × − . × − . × − . × − . × − . × − . ◦ /0 . ◦ . ◦ /19 . ◦ . ◦ /25 . ◦ µ as 9 . × − . × − . .
01 0 . .
07 13 . ◦ /8 . ◦ . ◦ /66 . ◦ . ◦ /90 . ◦ µ as 3 . × − . × − . .
05 0 . .
13 26 . ◦ /18 . ◦ . ◦ /72 . ◦ . ◦ /103 . ◦ . × − . × − . .
01 3 . × − . × − . ◦ /2 . ◦ . ◦ /6 . ◦ . ◦ /5 . ◦ µ as 1 . × − . × − . .
013 0 . .
018 11 . ◦ /9 . ◦ . ◦ /7 . ◦ . ◦ /12 . ◦ µ as 3 . × − . × − . .
05 0 . .
06 21 . ◦ /19 . ◦ . ◦ /16 . ◦ . ◦ /27 . ◦ Note. The left values near ”/” are fitting errors of the inner planet while the right values are fitting errors of the outer planet.
TABLE 2average fitting errors for super-Earth pairs. observational (cid:12)(cid:12) a fit − a true (cid:12)(cid:12) /a true (cid:12)(cid:12) m fit − m true (cid:12)(cid:12) /m true (cid:12)(cid:12) e fit − e true (cid:12)(cid:12) (cid:12)(cid:12) i fit − i true (cid:12)(cid:12) (cid:12)(cid:12) ω fit + Ωfit − ω true − Ωtrue (cid:12)(cid:12) (cid:12)(cid:12) M fit − M true (cid:12)(cid:12) . × − . × − × − . × − . × − . × − . ◦ /0 . ◦ . ◦ /8 . ◦ . ◦ /6 . ◦ µ as 1 . × − . × − . .
015 3 . × − . × − . ◦ /7 . ◦ . ◦ /12 . ◦ . ◦ /19 . ◦ µ as 4 . × − . × − . .
05 0 . .
03 28 . ◦ /15 . ◦ . ◦ /26 . ◦ . ◦ /36 . ◦ . × − . × − . . × − . × − .
016 1 . ◦ /0 . ◦ . ◦ /55 . ◦ . ◦ /68 . ◦ µ as 1 . × − . × − . .
02 0 . .
09 15 . ◦ /7 . ◦ . ◦ /68 . ◦ . ◦ /87 . ◦ µ as 0 . . × − . .
13 0 . .
23 30 . ◦ /17 . ◦ . ◦ /78 . ◦ . ◦ /108 . ◦ . × − . × − . × − . × − . × − . × − . ◦ /0 . ◦ . ◦ /8 . ◦ . ◦ /3 . ◦ µ as 1 . × − . × − . .
016 0 . .
018 9 . ◦ /8 . ◦ . ◦ /16 . ◦ . ◦ /28 . ◦ µ as 4 . × − . × − . .
05 0 . .
06 19 . ◦ /17 . ◦ . ◦ /36 . ◦ . ◦ /55 . ◦ Note. The left values near ”/” are fitting errors of the inner planet while the right values are fitting errors of the outer planet.
RKF7(8)(Fehlberg 1968) integrator which includesfull Newtonian interactions between the planets to checkif the fitted planet systems are stable in 2 × years.The stable fractions of the fitted planet systems andthe fractions of stable planet systems with β > . ω i and M i (i=1,2) which makes it hard to deter-mine ω i and M i (i=1,2) correctly. In our simulations,super-Earth pairs in 2:3 and 3:4 MMRs with β < . × − whennot considering observational errors. For 1:2 MMRs,the 1:2 period ratio makes it hard to have good fittingresults because of the influences of harmonic. Due tothe reasons above, a small fraction of planet systems arenot well-reconstructed.In Table 3 we can see that when σ m ≤ µ as, morethan 90% of the fitted Jupiter pairs in 2:3 and 1:2 MMRsare stable. For Jupiter pairs in 3:4 MMR, even withoutobservational errors, only half of fitted planet systemsare stable. Although the fitting errors of the Jupiterpairs in 3:4 MMR are similar to those in 2:3 MMR, it’sharder for planets to be locked in 3:4 MMRs than in2:3 MMRs, therefore, stable fractions of Jupiter pairs in3:4 MMR are much less than those in 2:3 MMR. Whenonly considering the MMR-reconstruction probabilitiesin stable fitted systems, more than 80% of Jupiter pairsin 3:4 MMR can be reconstructed with β > .
5. We checkthe long time stabilities of a few systems in 3:4 MMR with β < . TABLE 3Fraction of stable and well-reconstructed Jupiter pairswith different observational errors. even cadenceobservational fraction of stable fraction of stable planeterror systems systems with β > . ± µ as 95% 87% ± µ as 90% 58% ± ± µ as 99% 71% ± µ as 91% 58% ± ± µ as 48% 98% ± µ as 42% 85% ± the MMR-reconstruction prabability Sample number of Jupiter pairs in 2:3 MMR Sample number of Jupiter pairs in 1:2 MMR Sample number of Jupiter pairs in 1:2 MMRNote. The uncertainties are calculated as the difference betweenfraction of β > . β > . N/ N is the sample number of each MMRshown in the parenthesis. N/ N/ For super-Earth pairs, results in Table 4 shows thatthe stable fractions of fitted planet systems are generallymuch larger than those of the Jupiter pairs. As we havementioned above, if two Jupiters are not in MMR, theyare likely to be unstable according to Hill stability. In ob-servations, planets with Jupiter mass observed to be nearMMR are usually confirmed to be in MMR according totheir dynamical stability (Lee et al. 2006; Correia et al.2009). In this paper, we didn’t do such kind of research.So when considering the MMR-reconstruction probabili-ties in stable fitted systems, the fractions of super-Earthpairs with β > . β > . ≥
10. When SNR=3, the fractions largely dropto 40 − TABLE 4Fraction of stable and well-reconstructed super-Earthpairs with different observational errors. even cadenceobservational fraction of stable fraction of stable planeterror systems systems with β > . ± µ as 99% 76% ± µ as 99% 42% ± ± µ as 100% 79% ± µ as 100% 49% ± ± µ as 99% 77% ± µ as 98% 40% ± Sample number of super-Earth pairs in 2:3 MMR Sample number of super-Earth pairs in 1:2 MMR Sample number of super-Earth pairs in 1:2 MMRNote. The uncertainties are calculated similar to those in Table 3.
MMR-reconstruction with different ∆The distributions of ∆ for the Jupiter pairs and super-Earth pairs in our samples are shown in Figure 4 andFigure 5. We can see that ∆ concentrate upon small val-ues. For Jupiter pairs, ∆ ∼ − , while for super-Earthpairs, ∆ ∼ − . As the resonance width increased withthe mass of the planet pairs (Deck et al. 2013), the val-ues of ∆ for Jupiters are much larger than super-Earths.Planet pairs in 2:3 MMR have a much wider distribu-tion of ∆ than those in 1:2 and 3:4 MMRs. Becausein our simulations, planet pairs with large ∆ are gener-ated by migration, so Jupiter pairs in 3:4 MMR tend tohave a small ∆ for a lack of samples from migration. Aswe used a simplified migration model, planet pairs withsmall eccentricities usually have large ∆. From Figure2 we can find that planet pairs from migration in 2:3 c oun t c oun t ∆ ( × −3 ) c oun t . Fig. 4.—
Distribution of the normalized distance ∆ from theresonant center of the Jupiter pairs. The top, middle and thelower panels are samples of the 2:3, 1:2 and 3:4 MMRs, respectively.Samples with large ∆ are not shown here. c oun t c oun t ∆ ( × −4 ) c oun t . Fig. 5.— distribution of the normalized distance ∆ from theresonant center of the super-Earth pairs. The top, middle and thelower panels are samples of the 2:3, 1:2 and 3:4 MMRs. Sampleswith large ∆ are not shown here.
MMR have more samples with small eccentricities thanplanet pairs in 1:2 and 3:4 MMRs, so ∆ distribution isbroader for 2:3 MMR pairs than for 1:2 or 3:4 MMR. Inthis section, we will check the relation between ∆ andthe MMR-reconstruction probabilities.Although relative fitting errors of semi-major axis ofthe planets are very small(Table 1 and 2), the absolutefitting errors of ∆ can be large. As ∆ is calculated ac-cording to average periods of the planet pairs in 2 × years, a small variation on initial semi-major axis willlead to large difference in ∆. We calculate ∆ fit accord-ing to average fitted periods of the planets in 2 × years and find that the average differences between ∆and ∆ fit are around 2 × − without observational er-rors for both Jupiter and super-Earth pairs. When thereare observational errors, the average differences between : without error3 µ as error10 µ as error MM R − r e c on s t r u c t i on p r obab ili t y : : ∆ ( × −3 ) . Fig. 6.—
Relations between the MMR-reconstruction probabil-ities of the Jupiter pairs in MMR and ∆. The top, middle and thelower panels are results of the 2:3, 1:2 and 3:4 MMRs. The red,blue and magenta lines show results with observational errors with σ m =0, 3 µ as, 10 µ as, respectively. ∆ and ∆ fit reach 10 − .To check the correlations between ∆ and MMR-reconstruction probability, we sort the samples in eachMMR with increasing ∆ and divide them into 10 partswith the same number of samples. Define β (∆) as theaverage value of β for planets in each part. Figure 6 andFigure 7 shows β (∆) at different ∆. We don’t show the10 th part with the largest ∆ for extreme large variations.If observations are carried out without any errors, we canreconstruct nearly all the systems in MMRs independentof ∆.For Jupiter pairs in 2:3 and 1:2 MMR, there is a de-crease of β (∆) with the increase of ∆ . With the in-crease of ∆, the resonance becomes fragile and smallvariation of ∆ may destroy the resonance, therefore theMMR-reconstruction probabilities decrease. If the σ m =3 µ as, the MMR-reconstruction probabilities are largerthan 60%, with a slight decrease with ∆. While obser-vational errors are large enough, e.g. σ m = 10 µ as, β (∆)are less than 80% for both the 2:3 and 1:2 MMRs withlarge dispersions. β (∆) of the 3:4 MMR are mostly con-strained by stability and still > β (∆)should decrease with the increase of ∆. However there isno obvious negative correlation. This is because of thelarge absolute fitting errors of ∆ for super-Earth pairs.As we have mentioned in the previous paragraph, the av-erage differences between ∆ and ∆ fit reach 10 − , which ismuch larger than the distribution range of ∆ for Super-Earth pairs, but smaller than that for Jupiter pairs.So there is no positive correlation between ∆ and ∆ fit : without error0.1 µ as error0.3 µ as error0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.400.20.40.60.81 MM R − r e c on s t r u c t i on p r obab ili t y : : ∆ ( × −4 ) . Fig. 7.—
Relations between the MMR-reconstruction probabili-ties of the super-Earth pairs in MMR and ∆. The top, middle andthe lower panels are results of the 2:3, 1:2 and 3:4 MMRs. Thered, blue and magenta lines show results with observational errors0, 0.1 µ as, 0.3 µ as, respectively. for super-Earth, while ∆ fit increases with ∆ for Jupiterpairs. i.e., the negative correlation between ∆ and β (∆)are hidden by large fitting errors of ∆ for Super Earth. β (∆) are smaller than those of the Jupiter pairs withsimilar SNRs. As we calculate β (∆) in the stable sys-tems, Jupiter pairs remain stable are more likely to bein MMR because of Hill stability. When σ m = 0 . µ as, β (∆) for 2:3 MMR drops from 80% to 60% with the in-crease of ∆. For 1:2 and 3:4 MMRs, β (∆) is about 80%.When σ m = 0 . µ as, β (∆) is smaller than 60%.The large dispersions makes the relation between β (∆) and ∆ a little obscure, which also indicates someother factors can influence the MMR-reconstructionprobabilities such as the eccentricity and resonance in-tensity. We’ll analysis the relations between MMR-reconstruction and eccentricity and resonance intensityin the following subsections. MMR-reconstruction with different eccentricities
In addition to ∆, eccentricities also have important in-fluence on the probability of reconstructing a planet pairin MMR. We divide the systems in each MMR into 4parts according to the eccentricities of the planets : I: e > . e > .
1, II: e < . e > .
1, III: e > . e < .
1, IV: e < . e < .
1. We calculate the averagevalues of β (hereafter β ( e )) in each part with different ob-servational errors for the Jupiter pairs and super-Earthpairs, as shown in Figure 8 and Figure 9. To better il-lustrate the relation between MMR-reconstruction prob-ability and eccentricity, we only calculate β ( e ) in eccen-tricity bins with the number of planet pairs larger than I II III IV00.51 β ( e ) µ as3 µ as0 I II III IV00.51 β ( e ) I II III IV00.51 part β ( e ) . Fig. 8.—
Relations between the average resonance-reconstruction probabilities β ( e ) and eccentricities of the Jupiterpairs in MMRs. The top, middle and the lower bar graphs areresults of the 2:3, 1:2 and 3:4 MMRs, respectively. The red, blueand magenta colors represent observational errors =0, 3 µ as, 10 µ as, respectively. I,II, III and IV represent different ranges of ec-centricities, i.e., I: e > . e > .
1, II: e < . e > .
1, III: e > . e < .
1, IV: e < . e < .
1. The number of samplesin each part is larger than 20. The error bars displayed on thebins represent only the uncertainty due to Poisson statistics. For1:2 MMR, there are few samples in part II. For 3:4 MMR, thereare few samples in part III and IV, therefore, we didn’t show theresults in these part here.
20. The uncertainties due to Poisson statistics are shownas error bars displayed on the eccentricity bins. Differentcolors represent different observational errors. Obviously, β ( e ) with large error decreases in all cases.We can see that eccentricities are essential for variationof β ( e ) in different parts. For planet pairs in 2:3 and 1:2MMRs, β ( e ) in part I are the larger than β ( e ) in part IV.The increase of β ( e ) from part IV to part I is obvious forJupiter pairs in 2:3 and 1:2 MMR. Stability constrainsare quite strong in Jupiter pairs in 3:4 MMR, few planetpairs remain in part III and IV, β ( e ) in part I are largerthan those in part II. For super-Earth pairs in 3:4 MMR,the increase of β ( e ) from part IV to part I is not obvi-ous, this is because the average amplitudes of resonanceangles are not well-distributed from part I to part IV, ofwhich the influence on β will be discussed in the followingsection. However, β ( e ) in part IV is still the smallest in3:4 MMR. The positive correlation between β ( e ) and ec-centricities indicates that eccentricities are important toMMR-reconstruction, because the precision of ω i and M i are very sensitive to the precision of e i (i=1,2). Simula-tions show that when eccentricities are smaller than 0.01, ω i, fit + M i, fit may deviate from the true value obviously.Even if ω i, fit + M i, fit (i=1,2) equals to the true value, it’shard to decide both ω i and M i (i=1,2) accurately when e i < . ω i and M i (i=1,2) makes us reconstruct planet pairs with smalleccentricities in MMRs ambiguously. Large e i can avoidthis degeneracy and result in more accurate ω i and M i .Besides, the resonance widths increase with eccentricitiesof the planet(Deck et al. 2013). With similar absolute fit- I II III IV00.51 β ( e ) µ as0.1 µ as0 I II III IV00.51 β ( e ) I II III IV00.51 part β ( e ) . Fig. 9.—
Relations between the average resonance-reconstruction probabilities β ( e ) and eccentricities of the super-Earth pairs in MMR. The top, middle and the lower bar graphsare results of the 2:3, 1:2 and 3:4 MMRs, respectively. The red,blue and magenta colors represent observational errors =0, 0.1 µ as,0.3 µ as, respectively. I,II,III and IV represent different ranges ofeccentricities, i.e., I: e > . e > .
1, II: e < . e > . e > . e < .
1, IV: e < . e < .
1. The number ofsamples in each part is larger than 20. The error bars displayed onthe bins represent only the uncertainty due to Poisson statistics.For Super-Earth pairs in 1:2 MMR, there are few samples in partII, so we didn’t show the results here. ting errors of eccentricities, planet pairs with large eccen-tricities are more probable to remain in MMRs. There-fore, β ( e ) increases with e i (i=1,2). MMR-reconstruction with different resonanceintensities
To check how well we reconstruct the MMRs, wecompare the average amplitudes of φ and φ in fit-ted systems(hereafter A φ i , fit ) with those in real sys-tems(hereafter A φ i (i=1,2) during 2 × years. In Fig-ure 10 and 11, the red crosses represent planet pairswith β i > . β i < . β i > . µ and standard deviation σ for each kind of MMRat different observational errors. As shown in the Fig-ure 10 and 11, we find that with the increase of obser-vational errors, both mean values and standard devia-tions becomes larger and larger, indicating that fittedresonance angles deviate more and more from the truevalues. Besides, the samples with β i < . β i < .
5, the resonance angles φ i onlylibrate in 2 × · β i years, the average amplitudes shouldbe larger than those with β i > . A φ i (i=1,2) which means weak MMRs. To show thedistribution of blue crosses clearly, we plot a dotted linein Figure 10 and 11, which represents A φ i = 30 ◦ (i=1,2),to divide the total samples into two categories. The val-ues on the left and right side of the dotted line representthe fractions of the blue crosses in the two categories0 . Fig. 10.—
The residuals of average amplitudes of resonance angles φ and φ in fitted systems( A φ i fit ) with those in real systems ( A φ i )of the Jupiter pairs. The top histograms are the distributions of the average amplitudes of resonance angles φ and φ , i.e., A φ and A φ .The red crosses represent Jupiter pairs with β i > . β i < . A φ i = 30 ◦ (i=1,2). The values on the left side of the dot lines represent the blue fractions of Jupiter pairs with A φ i < ◦ whilethe values on the right side represent the blue fractions of Jupiter pairs with A φ i > ◦ (i=1,2). µ and σ are the mean value and standarddeviation of residuals of A φ and A φ with β i > . A φ and A φ with observational errors 0, 3 µ as, 10 µ as. with A φ i < ◦ and A φ i > ◦ (i=1,2). We find thatthe blue fractions on the right side are generally largerthan those on the left side for Jupiter pairs in 2:3 and1:2 MMRs. For Jupiter pairs in 3:4 MMR, blues frac-tions on each side are close because the stability willexclude part of systems with large A φ i . For super-Earthpairs, there are much more systems with eccentricities < A φ i is not always larger than the blue fraction withsmall A φ i . To exclude the non-uniform distribution ofsystems with small eccentricities in the two categorieswith different A φ i , we choose samples of the super-Earthpairs with e > .
01 and e > .
01 to recalculate theblue fractions of the two categories, which are shown asblue values in Figure 11. We find that the blue fractionsfor A φ i > ◦ are larger than those for A φ i < ◦ (i=1,2) in 1:2, 2:3 and 3:4 MMRs.Beyond that, there are obvious differences betweenthe blue fractions of A φ and A φ for Jupiter pairs in2:3 and 1:2 MMRs. Compared to A φ , A φ concentratesupon a smaller value for 2:3 MMR. Naturally, withthe same level of observational errors and similardeviation from the original values of φ and φ , φ ismore likely to remain in libration than φ , therefore,the planet pairs with β < . β < .
5. On the contrary, for 1:2 MMR, A φ concentrates upon a much smaller value than A φ , so φ is much easier to be reconstructed in libration than φ .The positive correlation between the blue fraction and A φ i (i=1,2) indicates that the stronger the intensitiesof the MMRs, the easier the MMRs can be reconstructed.Analyses above indicate that the MMR-reconstructionprobabilities are related with eccentricities and resonanceintensities of the planet pairs. To better compare thedifference of MMR-reconstruction probabilities between1 . Fig. 11.—
Similar to Figure 10 but for the super-Earth pairs. Besides, the values with blue color represent the blue fractions of planetpairs with e > .
01 and e > .
01. Every two panels from top down show residuals of the resonance angle φ and φ with observationalerrors 0, 0.1 µ as, 0.3 µ as. Jupiter pairs and super-Earth pairs, we calculate thefraction of planet pairs with β > . ◦ . The results are shown in Table5. Except for planet pairs in 1:2 MMR with SNR=10,Jupiter pairs can be better reconstructed in MMRs thansuper-Earth pairs, especially when SNR=3, because ofthe Hill stability and planet pairs with larger masses havelarger resonance width according to Deck et al. (2013).In fact, it is quite hard to explain all the difference be-tween super-Earth pairs and Jupiter pairs. Although wehave confined the eccentricity and resonance intensity tocompare the difference of MMR-reconstruction betweenJupiter pairs and super-Earth pairs, we can not eliminatethe sample bias between Jupiter and super-Earth pairstotally. A more refined sample control should be helpfulto eliminate the exception. THE FALSE ALARM PROBABILITY OF PLANET PAIRSIN OR NEAR MMRS ANALYSIS
In section 4, we have investigated the probability toreconstruct planet pairs in MMRs. Consequently, the fraction of well reconstructed planet pairs with β > . P − ) can be obtainedaccording to previous section. Actually, some systemsare not in but near MMRs. We are also interested in theFAPs (hereafter denoted as P − ) of mistaking planetpairs near MMRs for planet pairs in MMRs.To obtain the FAPs of mistaking near MMR systemsfor systems in MMR, we simulate 1600 Jupiter pairs with∆ < .
04, and 1600 super-Earth pairs with ∆ < . P − ) in our simulations are stableand not in MMRs in 2 × years. In our simulations, P − is calculated as the fraction of planet pairs whichare fitted to be in MMRs with a probability β > P − for different MMRs at different ranges of ∆ fit (or ∆)are shown in Table 6(Jupiter pairs) and Table 7(super-Earth pairs). ∆ fit is calculated with the fitted averageperiods of the planets in 2 × years. Almost all ∆ fit of Jupiter pairs are smaller than 0.04, and ∆ fit of super-Earth pairs are smaller than 0.02. We do not analysis P − for Jupiter pairs near the 3:4 MMR because of theirweak stabilities, i.e., P − ∼ TABLE 5fraction of stable planet pairs with β > . among planet pairs with e i > . (i=1,2) and A φ ( orA φ ) < ◦ Jupiter pairs super-Earth pairsobservational errors 0 3 µ as 10 µ as 0 0.1 µ as 0.3 µ as2:3 96% ±
1% 92% ±
2% 63% ±
1% 97% ±
2% 85% ±
1% 44% ± ±
1% 79% ±
2% 64% ±
4% 99% ±
1% 97% ±
3% 55% ± ±
1% 97% ±
1% 84% ±
1% 97% ±
1% 84% ±
1% 44% ± super-Earth pairs in each MMR, we divide the samplesinto three ranges according to ∆ fit or ∆, and calculate P − in each range of ∆ fit (outside of the brackets) and∆(in the brackets). P − for all samples are shown inthe fourth row of each MMR in Table 6 and Table 7.The fourth rows of each MMR in Table 6 and Table7 show that there is a positive correlation between P − and observational error when P − is calculated via ∆.As the larger the observational error is, the further thefitted orbital parameters deviate from their true values,thus the fitted planet pairs can arrive some islands ofMMRs far away from the initial position in phase spaceand they are probably in MMRs. Unlike the positivecorrelation between P − and observational error, thereis a negative correlation between P − and ∆ fit , i.e., thelarger the ∆ fit is, the further planet pair is away fromthe MMR center, thus it’s less likely to be mistaken fora planet pair in MMR. However, P − have no obviouscorrelation with ∆. Because ∆ fit in false alarm cases areusually small, while ∆ are widely distributed. Only fewcases with large ∆ fit are mistaken for systems in MMR.When ∆ fit is large enough, i.e. ∆ fit > .
02 for Jupiterpairs, P − decrease to smaller than 10%, and ∆ fit > .
005 for super-Earth pairs, P − decrease to about 1%.When we detect a planet pair with period ratio near1:2, 2:3 or 3:4 MMR, and simulation shows that it is inMMR based on the fitted parameters, the detected planetpair in MMR might be a false alarm. To calculate theFAP F − for a detected planet system in MMR, we needthe values of both P − and P − . If we assume the sameNumber N p of planet pairs in or near MMRs, N p · P − planet pairs near MMRs will be mistaken as planet pairsin MMRs, while N p · P − planet pairs in MMRs can bewell reconstructed. Finally, F − is expressed as: F − = P − P − + P − . (6)Note that the meaning of P − is different with that of F − . From Equation (6), we can see that even if P − =1, F − is greater than 0, but smaller than P − .On the other hand, there is another FAP when we re-construct a planet system near MMR. Take P − as theprobability of reconstructing a system near but not inMMR, and take P − as the probability of mistaking anin MMR system for a near MMR system. Similar withthe derivation of F − , the FAP for a near MMR system F − is expressed as: F − = P − P − + P − . (7)It’s easy to obtain P − = 1 − P − and P − = 1 − P − . In observations, only ∆ fit can be obtained, so it’s suitable to adopt the values of P − , P − , P − and P − calculated via ∆ fit . P − in Table 8 are slightly largerthan values in the last column in Table 3 and Table 4,because they are calculated among planet pairs with ∆ fit in the same range of ∆ fit shown in the third rows in Table6 and Table 7.Table 9 and Table 10 show the final F − and F − of a planet system detected in or near MMRs. Gener-ally, the larger the observational errors are, the larger F − and F − are. For both Jupiter and super-Earthpairs, F − and F − are sensitive to the observationalerrors. F − of Jupiter pairs and super-Earth pairs in1:2 MMR are very similar, which are larger than 20%even without observational errors. With the same obser-vational errors, F − and F − for planet pairs in 2:3and 3:4 MMRs are smaller than those in 1:2 MMR. Notethat the particularity of large FAP for planet pairs in 1:2MMR is mainly induced by the significant large P − . Aswe have mentioned before, the 1:2 period ratio makes ithard to fit planet pairs as well as planet pairs with otherperiod ratio. So it’s likely to mistake planet pairs near1:2 MMR for those in 1:2 MMR. When SNR ∼
3, both F − and F − are larger than 30%, therefore, if we de-tect a planet system in or near MMR with low SNR, thesystem should be checked carefully. THE POTENTIAL OF DISCOVERING PLANET PAIRS INMMRS
After calculating the MMR-reconstruction probabil-ities, we can estimate the number of planet pairs inMMRs( N MMR ) which can be measured by astrometry ifwe know the frequency of Jupiter pairs and super-Earthpairs in MMRs around nearby stars.Based on observations before Kepler Mission, Caser-tano et al. (2008) estimate the number of multiple planetsystems that GAIA can detect. In their paper, they listall the multiple planet systems detected and calculatethe fraction of the multiple planet systems which meetthe condition SNR > µ as. Then they extrapolate the resultsto the planet systems GAIA can detected and finally es-timate the number of multiple planet systems they canfind. However, in this paper, it’s hard to estimate N MMR in the same way due to lack of samples with parallaxmeasurements. Among 415 multiple planet systemsdetected, 76 of them have parallax measurements, andonly 27 systems have planet pairs near MMRs. The sam-ples are very rare and no super-Earth pairs near MMRsappear in the 27 systems, so we choose another way toestimate N MMR . exoplanet.org TABLE 6 P − of the two Jupiter system observational error∆ fit (∆) µ as 10 µ as2:3 < .
01 5% ± ± ± ± ± ± . ∼ .
02 3% ± ± ± ± ± ± . ∼ .
04 0 . ± ± ± ± ± ± ≤ .
04 3% ± ± ± ± ± ± < × − ± ± ± ± ± ± × − ∼ .
01 18% ± ± ± ± ± ± . ∼ .
04 2% ± ± ± ± ± ± ≤ .
04 14% ± ± ± ± ± ± P − is the possibility of mistaking a planet system near MMR for one in MMR. The valuesoutside of the brackets are P − s calculated via ∆ fit and values inside of the brackets are P − scalculated via ∆. P − is different from F − in Table 9 and Table 10. ∆ fit is the normalized distance to resonance center of the fitted planet pairs, ∆ fit = ( j − P fit , / ( jP fit , ) − TABLE 7 P − of the two super-Earth system . observational error∆ fit (∆) 0 0.1 µ as 0.3 µ as2:3 < × − ± ± ± ± ± ± × − ∼ × − ± ± ± ± ± ± × − ∼ .
02 0 . ± ± . ± ± . ± ± ≤ .
02 3% ± ± ± ± ± ± < . × − ± ± ± ± ± ± . × − ∼ × − ± ± ± ± ± ± × − ∼ .
02 1% ± ± ± ± . ± ± ≤ .
02 14% ± ± ± ± ± ± < × − ± ± ± ± ± ± × − ∼ × − ± ± ± ± ± ± × − ∼ .
02 0 . ± ± . ± ± . ± ± ≤ .
02 4% ± ± ± ± ± ± The same with Table 6.Note. The uncertainties are calculated similar to those in Table 3.
TABLE 8 P − of different MMRs at different observational errors. Jupiter pairs super-Earth pairsobservational errors 0 3 µ as 10 µ as 0 0.1 µ as 0.3 µ as2:3 MMR ∆ fit < .
01 99% ±
1% 96% ±
1% 62% ±
1% ∆ fit < × − ±
1% 87% ±
2% 56% ± fit < × − ±
1% 81% ±
4% 69% ±
1% ∆ fit < . × − ±
1% 92% ±
1% 66% ± fit < × − ±
1% 97% ±
1% 87% ±
4% ∆ fit < × − ±
1% 86% ±
1% 51% ± The number of planet pairs in MMR reconstructed byastrometry measurements can be expressed as: N MMR = N (cid:63) × f × f × f × f × f . N (cid:63) is the number of tar-get stars, here we adopt N (cid:63) =3 × based on the factthat there are more than 3 × bright stars(V < f is the probability that a star host planets. f is the prob-ability that the planets are in multiple planet systems. f is the probability that there are planets in MMRs in multiple planet systems. f is the probability of planetsin MMRs with Jupiter-like or super-Earth-like masses. f is the probability that the planets in MMRs can bereconstructed by astrometry.According to Cassan et al. (2012), each Milky Waystar hosts at least one planet, i.e., f is set as 100%. Wecalculate f , f , and f based on the planets discoveredso far. According to observations of the Kepler mission,4 TABLE 9FAP of the two Jupiter system F − F − observational error 2:3(∆ fit < .
01) 1:2 (∆ fit < × − ) 2:3(∆ fit < .
01) 1:2 (∆ fit < × − )without error 5% ±
1% 27% ±
1% 1% ±
1% 1% ± µ as 27% ±
1% 35% ±
1% 6% ±
1% 25% ± µ as 40% ±
1% 42% ±
1% 39% ±
1% 38% ± F − is the false alarm probability when we detect a planet pair in MMR. It is calculated on thebasis of the possibility we mistake a planet pair near but not in MMR for the one in MMR andthe possibility we detect a true planet pair in MMR. F − is the false alarm probability when we detect a planet pair near but not in MMR. It iscalculated on the basis of the possibility we mistake a planet pair in MMR for the one near MMR andthe possibility we detect a true planet pair near but not in MMR.Note. The uncertainties are according to uncertainties in Table 6 and Table 8. As we have F − = P − P − +P − , the uncertainty of F − can be estimated as P − dP − − P − dP − ( P − + P − ) . Similarly, F − = P − P − +P − , the uncertainty of F − is P − dP − − P − dP − ( P − + P − ) . TABLE 10FAP of the two super-Earth system F − F − fit < × −
4) 1:2 (∆ fit < . × −
4) 3:4 (∆ fit < × −
4) 2:3(∆ fit < × −
4) 1:2 (∆ fit < . × −
4) 3:4 (∆ fit < × − ±
2% 26% ±
1% 7% ±
1% 2% ±
1% 7% ±
1% 4% ± µ as 21% ±
1% 39% ±
1% 25% ±
1% 14% ±
2% 16% ±
1% 16% ± µ as 44% ±
1% 50% ±
2% 47% ±
1% 44% ±
1% 52% ±
3% 47% ± F − F − Note. The uncertainties are according to uncertainties in Table 9. f ∼ . There is observational bias in Kepler missionwhich tends to discover planets close enough to the hoststar, the planets far away from the host star have smallerprobability to be detected. Therefore, planet pairs in ornear MMRs in observation mostly have semi-major axis < ≥ logP be-tween 2-2000 days, while smaller planets(1-4 R ⊕ ) have aprobability nearly constant in logP between 10 and 300days. Here, we simply assume that the occurrences ofMMRs far away from the host star(1 AU) are similarto those near the host star. As few planets in MMRshave been confirmed, we set f as the probability of nearMMR planet pairs in multiple planet systems. It is rea-sonable because many researches (Lithwick & Wu 2012;Batygin & Morbidelli 2013; Xie 2014; Chatterjee & Ford2015) hint that planet pairs in MMRs can evolve intothe observed MMR offset due to several mechanisms suchas tidal dissipation and planet-planetesimal disk interac-tion. Currently, 415 multiplanet systems are detected,and 135, 91 and 20 planet systems contain planet pairsnear 2:3, 1:2 and 3:4 MMRs, i.e., f = 21 . , .
5% and4 .
8% for 2:3, 1:2 and 3:4 MMRs, respectively. Besides, exoplanet data used in this section are from exoplanets.org, exoplanet.org among the planet pairs near MMRs, the fraction of bothplanets with masses 5 M Earth ∼ M Earth are 23 . . M J ∼ M J are 2 . f = 2 .
79% for Jupiter pairs and f = 23 .
69% for super-Earth pairs.In our simulations, planet pairs in or near MMRs haveinclinations between 0 and 10 ◦ , however, planet pairs inMMRs with inclination ∼ ◦ can also be reconstructedwith a certain probability. We do simulations for a super-Earth pair in 2:3 MMR with their inclinations increasefrom 0 to 90 ◦ and find that the MMR-reconstructionprobability decrease if inclinations ≥ ◦ . Here we sim-ply assume the MMR-reconstruction probability decreaselinearly with increase of inclinations, i.e., f ( i ) = f ( i (cid:39) ◦ )(1 −| i | / ( π/ i = [ − π/ , π/ f ( i (cid:39) ◦ ) is approxi-mated by the MMR-reconstruction probability of planetpairs with nearly face-on orbits which has been calcu-lated in Section 4, i.e., last columns in Table 3 and 4.Besides, assuming a uniformly distribution of planet’s or-bital angular momentum vector, the probability densityof inclination dP ( i ) /di = sin | i | / i = [ − π/ , π/ f = f ( i (cid:39) ◦ ) (cid:82) π/ − π/ sin | i | / − | i | / ( π/ di (cid:39) . f ( i (cid:39) ◦ ). Although f ( i (cid:39) ◦ ) are obtained by sim-ulation of planet pairs near 1 AU, planet pairs at differ-ent locations will lead to the same results with the sameSNR, if we rescale the observational errors and data sam-plings consistant with the locations of the inner planet.Finally, we estimate the probabilities of discoveringand reconstructing the planet systems by astrometrymethod, as shown in Table 11. As all the planetsystems in our simulations are at 30 pc, the MMR-reconstruction probabilities are the inferior limits. The5 TABLE 11Number of planet pairs in MMRs can bedetect and reconstructed by astrometry in 30 pc.
Jupiter pairs super-Earth pairsSNR ∼ ∼ ∼ ∼ number of Jupiter pairs in MMRs can be detected andreconstructed by us are much less than that of the super-Earth pairs. It’s reasonable, for planet systems contain-ing two giants are rare in observations. With observa-tional SNR= 3, we can find tens of giant planet pairs in2:3 and 1:2 MMRs. The reconstruction of super-Earthpairs in MMRs require higher precision to reach SNR ∼ f ( i (cid:39) ◦ ) ∼ EVEN AND UNEVEN DATA CADENCE
The simulations above are all carried out with evendata cadence. In fact, most observations have unevendata cadence because of lots of realistic observationallimits. To find out the influence of uneven data cadenceon planet detection by astrometry method, we comparethe differences between even and uneven data cadencein this section. Although the uneven data cadences dis-cussed here are not realistic cadences schemes for Gaiaand STEP, it’s important to explore how large the influ-ence is.For single planet systems, we simulate 100 super-Earthsystems and 100 Jupiter systems which are 30 pc fromus. All the planets are 0.9 AU from the host star andtheir eccentricities are distributed from 0.01 to 0.5. Allobservations have a set of 50 data points. We choose 8different data cadences of simulated astrometry data asfollows:c1: 80% data points are randomly distributed near theperigee, i.e., − . ◦ < f < . ◦ , where f is thetrue anomaly (hereafter the same). 20% are ran-domly distributed near the apogee, i.e., 129 . ◦ 80% 20% c1 20% 80% c250% 50% c3 50% 50%c440% 40%10% 10% c5 50 points randomly distributed c6 −36 0 36 144 180 216 50 points sampled every 0.1 yr (e=0.2) c7 −36 0 36 144 180 216 50 points uniformly distributed c8 perigee apogee perigee apogee . Fig. 12.— Diagrammatic sketch of the eight kinds of data ca-dence. c1-c8 represent data cadence c1-c8 described in section7,respectively. c4: 50% are randomly distributed near the mid pointof the apogee and perigee, i.e., 43 . ◦ < f < . ◦ .The others are randomly distributed on the oppo-site side, where 230 . ◦ < f < . ◦ .c5: 40% are randomly distributed near the perigee,while 40% are randomly distributed near theapogee. The other 20% are randomly distributedin the left regions;c6: f of all data points are randomly distributed;c7: Times of all data points are uniformly distributed,i.e., even data cadence adopted before this section.c8: All data points are distributed with uniform orbitalphase coverage, i.e., there is one data point in eachrange of f with a width of 7 . ◦ ;The diagrammatic sketches of the 8 data cadences areshown in Figure 12.To illustrate the non-uniformity of the data points,we divide the whole phase coverage of f into 25parts, each with a width of 14 . ◦ . Then we count thenumber of data points in each parts and calculate thevariance( σ ) of them. The variance represents thephase coverage of the observation, i.e., the smaller thevariance is, the more complete the phase coverage is.For each kind of data cadence, the variance changesslowly with eccentricity, therefore, we calculate theaverage variance in each bin of eccentricities, the binrange is set to be 0.1. Set SNR ∼ 10, i.e, σ m = 0 . µ asfor the super-Earth and σ m = 3 µ as for the Jupiter,we fit the planet parameters with data sets c1-c8. Thedifferences between the true and fitted astrometricsignatures caused by the planets are shown in Fig-ure 13 and Figure 14. The residuals are expressed as (cid:113)(cid:80) Ni =1 (( X fit ( t i ) − X ture ( t i )) + ( Y fit ( t i ) − Y ture ( t i )) ) /N .N=50 is the number of data points.Similar characteristics for single Jupiter or super-Earthsystems are obtained in our simulations, which is rea-sonable, because the simulations are done with similarSNR. The left panels of Figure 13 and 14 show the vari-ance of each data cadence at different eccentricities of6 σ pha s e2 Jupiter @ σ m =3 µ as3 5Residual ( µ as) . Fig. 13.— The left panel shows the variances of the data ca-dence at different eccentricities. The green, blue, cyan, magenta,light grey, dark, red and purple lines represent the data cadence c1-c8, respectively. The right panel shows the difference between thetrue and fitted astrometric signature caused by the Jupiter with astandard deviation of observational error σ m = 3 µ as at differentvariances σ and eccentricities e of the data cadence. Differ-ent colors represent data cadences the same with those in the leftpanel. The symbols dot, cross, asterisk, diamond and left trianglerepresent the mean variance and residuals with mean eccentricitiese=0.05, 0.15, 0.25, 0.35 and 0.45, respectively. The circles are themean values with any eccentricities. Jupiters and super-Earths, respectively. The right pan-els show the corresponding residuals at each variance. Inleft panels, σ increases from c8 to c1. The cases ofc8 have zero variances, while σ for c6 and c7 aresmaller than 3, which have much better phase coveragethan c1-c5. σ of c1 are similar to that of c2, be-cause f of data points near the perigee and apogee areuniformly distributed as shown in the top two panels inFigure 12. The same reason can also explain the similar-ity of σ in c3 and c4. In the right panel of Figure13 and 14, with the increase of average variance, theresidual also increases, indicating that more uniform andcomplete phase coverage will ensure a better orbit fittingof the planets. With the similar σ in c1 and c2, theresiduals are nearly the same, i.e., the residuals are notsensitive to the samplings with more data near perigeeor apogee. The similar residuals of c3 and c4 show thereis no differences for data sampling near perigee/apogeeor not. For even data cadence c7, when eccentricities arelarger, we’ll have more data points near the apogee if wesample every 0.1yr, and the variance increases with theeccentricities. Accordingly, the increase of variance leadsto the increase of residuals with the eccentricities, whilethere are no such obvious correlations for other cadences.Empirically, if σ < 3, e.g. c6-c8, the residuals aresmaller than observational errors σ m for single planetsystems.For single planet systems, even with extremely unevendata cadence, all the planets are detected with precise pe-riods, although the residuals vary a lot. When it comesto two-planet systems, things are quite different. Thelarge fitted residuals of the first planet may contaminatethe signal of the secondary planet, thus the period of the σ pha s e2 super−Earth @ σ m =0.1 µ as0.05 0.1 0.15Residual ( µ as) . Fig. 14.— Similar to Figure 13 but for the super-Earth withobservational error σ m = 0 . µ as. secondary planet is hardly determined accurately. So wecompare the differences between even and uneven datacadence for two-planet systems to see how large the in-fluence is. As we adopt a keplerian orbit for each planet,the motion of the host star will be irregular rather thana keplerian orbit. So it’s hard to clearly choose datapoints near the perigee or apogee for both planets. Fortwo planet systems, the star moves around the commoncenter of mass and locates in different quadrants at dif-ferent times. Define α as the angle of data vectors [ x ( t i ), y ( t i )](i=1,...N) with the x axis. We choose all the simu-lated astrometry data of super-Earth pairs in 2:3 MMRin section 4, and test 4 kinds of data samples as follows:d1: Sample in regions with 45 ◦ < α < ◦ and 245 ◦ <α < ◦ .d2: Sample in regions with 0 ◦ < α < ◦ and 180 ◦ <α < ◦ .d3: Sample in regions with − ◦ < α < ◦ and 135 ◦ <α < ◦ .d4: Sample every 0.1 year, i. e., even data samples.Table 12 shows the results of the 4 kinds of samplingfor super-Earth pairs in 2:3 MMRs with σ m = 0 . µ as.We choose σ m = 0 . µ as in order to ensure a large signal-to-noise ratio ∼ 10. Therefore, planets can be detectedwith large confidence and we can compare the influenceof different sampling schemes on MMR-reconstruction inour simulations. Similar with single planet systems, wecan calculate the variance σ phase , i for each planet. Fromdata cadence d1 to d4, the mean values of σ , i , de-noted as σ , i (i=1,2), for both planets largely drop.The reason is obvious because the larger regions we sam-ple in, the more uniform and complete phase coveragewe’ll have. When sampling only in a very small region,take d1 for example, only about 27% of the results con-verge at χ < . 5, while for d3 and d4, all results canconverge at small χ < . χ < . 5, the averageMMR-reconstruction probabilities β also increase with7the phase coverage in the fourth row. We investigatethe fitting errors of eccentricities, which decrease fromd1 to d4, and lead to the decrease of β . β for d1 issmaller than others, because β becomes very small if theperiod of one planet is determined ambiguously. In oursimulations, about 10% of the fitted super-Earth pairswith small χ have large fitting errors of semi-majoraxis for the secondary planet ( δ a > . 1) while peri-ods of both planets in d2-d4 are determined well with δ a i < . χ > 2, which occurs only in d1and d2, most of them are characterized with false periodsof the secondary planets with δ a > . 1. Therefore, theseplanet pairs in 2:3 MMR can be hardly reconstructed.The mean values of β are all < . 08 for planet pairswith χ > 2. We define the average variance of thetwo planet σ = ( σ , + σ , ) / 2. Consistentwith single planet systems, if σ < 3, e.g. d2-d4, theMMR-reconstructed probabilities are much better thand1 with σ > σ < CONCLUSION AND DISCUSSION Astrometry is an ancient technique to discover aster-oids and planets in solar system. With the improve-ments of technique, astrometry method can be extendedto discover the exoplanets around nearby stars to ob-tain more information about the orbit of planets. Usingthese orbital elements and mass of star, we can recon-struct planet systems in mean motion resonances. InSection 2, we introduce the astrometry methods to de-tect planets and the fitting procedure of planetary pa-rameters. Based on observations about planet pairs nearMMRs(Figure 1), we consider planet pairs with equalmasses, i.e., Jupiter pairs and super-Earth pairs. We alsopresent how to simulate samples of planetary systems in1:2, 2:3 and 3:4 MMRs via migration and random meth-ods. Distribution of eccentricities and ∆ for each MMRof the Jupiter pairs and super-Earth pairs are shown inFigure 2, Figure 4 and Figure 5. In Section 3, planetswith SNR > β with the decrease of SNRas shown in Table 3 and 4.2. With the increase of ∆, there is a decrease in MMR-reconstruction probability β for Jupiter pairs in 2:3 and1:2 MMRs in Figure 6, which is not obvious for super-Earth pairs in Figure 7.3. There is a positive correlation between MMR-reconstruction probability and the eccentricity of theplanets for both Jupiter and super-Earth pairs in Figure8 and Figure 9. Planet pairs with e > . e > . e < . e < . 1. Because large eccentricity can avoid the degen-eracy between ω and M and resonance width increasewith eccentricity (Deck et al. 2013).4. MMR-reconstruction probabilities are larger forplanet pairs with strong resonance intensity with A φ i < ◦ (i=1,2) illustrated in Figure 10 and Figure 11.5. With similar SNR, the MMR-reconstruction prob-abilities of Jupiter pairs are larger than those of super-Earth pairs when considering stability, as shown in Table5. In Section 5, we calculate the FAPs when we recon-struct a planet system in or near MMRs. Our main con-clusions are:1. P − as the probability of mistaking a near MMRsystem for a resonant system has a positive correlationwith observational error, meanwhile, it decreases withthe increases of ∆ fit . The results are presented in Table6 and Table 7.2. The FAPs for planets reconstructed to be in MMR F − are largest for planet pairs in 1:2 MMR. It’s diffi-cult to produce a stable Jupiter pair near 3:4 MMR, thus F − ∼ 0. Both F − and F − are sensitive to obser-vational errors. As shown in Table 9 and Table 10, whenSNR ∼ 3, both F − and F − are larger than 30%, soplanets with small SNR detected to be in MMRs shouldbe checked carefully.In Section 6, we estimate the number of discoveringplanet systems in MMRs via astrometry, as shown inTable 11. There are about 3 × stars with V < 10 within30 pc from the Sun, after assuming the occurrence ofplanet pairs in MMRs, we estimate that with SNR= 3,tens of planet pairs with Jupiter masses in 2:3, 1:2 and 3:4MMRs can be potentially reconstructed, and hundreds ofsuper-Earth pairs in 2:3 and 1:2 MMRs can be detected,planet pairs in 3:4 MMRs are very few because of theirrareness based on observation.In Section 7, we compare the difference between evenand uneven data cadences. Extremely uneven data ca-dence with σ > σ < σ in two planet systems, theMMR-reconstruction probabilities with σ < µ as, which can help us find planets ofJupiter mass. If it can reach a precision of about 10 µ as, the probabilities to reconstruct a Jupiter pair in8 TABLE 12Results of uneven data cadence of the super-Earth pairs in 2:3 MMR data cadence d1 d2 d3 d4 σ , / σ , χ < . . 12% 98 . 37% 100% 100% β of χ < . δ a > . δ a < . 05 when χ < . . 52% 0 0 0fraction of χ > . 26% 1 . 40% 0 0 β of χ > δ a > . δ a < . 05 when χ > . 36% 66 . 67% – –Note. d1-d4 represent the four kinds of data cadence for super-Earth pairs in 2:3 MMR in Section 7, σ , i (i=1,2) is the mean value of σ for each planet, β is the mean MMR-reconstruction probability for all super-Earth pairs in 2:3 MMR. Fraction of δ a > . δ a < . 05 when χ < . χ > 2) represent the fraction of planet pairs with relative fitting errors of semi-major axis δ a > . δ a < . 05 among fitted planet pairs with χ < . χ > > 50% at least (see Table 3). If aJupiter pair with such an SNR is reconstructed in MMR,it should be checked very carefully because of the largeFAP ∼ µ as, for a super-Earth 1 AUfrom the host star and 30 pc from us, the SNR ∼ 1, whichis very hard to identify the super-Earth. However, if theplanets is 10 pc from us, the SNR ∼ 3, which will ensurea probability of 40% with FAP ∼ 40% for 2:3 and 1:2MMR. We expect higher precision of astrometry( ∼ . µ as) in the future, thus we will have chances to detectplanets with masses even smaller than Earth, and theprobability to reconstruct super-Earth pairs in MMRswill be improved to as large as 75%(Table 4). All planetsystems in our simulations are at 30 pc, with similarobservational errors, we can reconstruct planet pairs inMMRs with larger probability and smaller FAPs if theyare closer to us.In this paper, we adopt a mission lifetime of 5 yearscomparable with GAIA and STEP. Thus it’s appropri-ate to detect planet systems around 1 AU from the hoststar. Data cadence and the time allocated for obser-vations will influence our planet detecting via astrom-etry. Shorter data cadence helps us to detect planetscloser to the host star, longer mission lifetime enablesus to detect planets with longer period. Only 50 dataare used in this paper, more data can improve the fittingprecision thus may leads to larger MMR-reconstructionprobability. Besides, recent work (Giuppone et al. 2009,2012) have shown potential of detecting and character-izing planet pairs in MMRs using radial velocity data,together with high precision radial velocity data, we canimprove the precision of eccentricities which can help usdetermine ω + Ω accurately. Consequently, planet pairsin MMR can be reconstructed with large probabilitiesand small FAPs. Additionally, although fitting errors for planets in 1:2 MMRs are larger than those in 2:3 and3:4 MMRs, it’s hard to conclude whether planet pairs in1:2 MMR are harder to be reconstructed than the othertwo MMRs or not according to results of our simulations,because many factors influence the MMR-reconstructionprobabilities. E.g., different MMRs have different reso-nance width and resonance structures in e − e phasediagram, it’s hard to have a large number of samples withexactly same distribution of ∆, eccentricities and ampli-tudes of resonance angles for different MMRs. The samereason fits to comparison between super-Earth pairs andJupiter pairs, which are also difficult to figure out thesignificant differences between them.We only simulate planet pairs with equal masses inthe first order MMRs in this paper, other planet systemsin MMRs with different masses such as a Jupiter and asuper-Earth can also be reconstructed with proper obser-vational precision. Planet pairs in high order MMRs suchas 1:3 and 3:5 are not considered here, as these MMRsare much weaker and have a narrower resonance widththan the MMRs discussed in this paper, they need higherprecision to be reconstructed.This research is supported by the Key Devel-opment Program of Basic Research of China (No.2013CB834900), the National Natural Science Foun-dations of China (Nos. 11503009, 11003010 and11333002), Strategic Priority Research Program TheEmergence of Cosmological Structures of the ChineseAcademy of Sciences (Grant No. XDB09000000), theNatural Science Foundation for the Youth of JiangsuProvince (No. BK20130547), 985 Project of Ministra-tion of Education and Superiority Discipline Construc-tion Project of Jiangsu Province, ”Search for TerrestrialExo-Planets”, the Strategic Priority Research Programon Space Science Chinese Academy of Sciences( GrantNo. XDA04060900). REFERENCESArmstrong, D. 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The dark lines are the true values of each parameter of the Jupiter pair.APPENDIX APPENDIX INFORMATION ABOUT THE MCMC PROCEDURE The MCMC code is written based on theories discussed in Ford (2005, 2006). We run each MCMC with 3 × iterations. First of all, we check the values of the parameters at each iteration in the MCMC procedure and find thateach of them converges to a small range near the true values after 10 iterations (Figure 15). Secondly, the acceptanceof each parameter is around 0.2-0.5 and χ < . 7, we then conclude that the Markov chains are convergent. Statisticsare derived on the last 1 × elements, which can reduce the dependence on the initial parameter values. We choosethe best-fit parameters as the median of posterior distribution. Figure 16 and Figure 17 show the posterior distributionsof planetary orbital parameters and the stellar parameters of one case in our simulation. We can see that distributionsfit well to Gaussian distribution and the true values of the parameters locate within one sigma range of the medianvalues.Generally, when we need to fit a lot of parameters using the MCMC method, we should have as many as iterationswe can or have enough chains to approach a global best solution. However, among the 11 parameters we need to fit,the initial values of the 5 stellar parameters in our procedure are derived by the linear least squares, and the initialperiods of the planets are derived by the periodogram. Stellar parameters and period of planets derived here are veryclose to the true values. Therefore, we only need to set initial values for 4 parameters e , t , e and t randomly.During the fitting procedure, t i is set to change between 0 and P i (i=1,2), while e i is set to change between 0 and 1.With a limited parameter space, the MCMC procedure is more efficient (Ford & Gregory 2007). 3 × iterations areenough to lead a high confidence that we have reached a global solution. To confirm that, we simulate the same case inFigure 15 with 100 different initial values of e , t , e and t . The iteration number is 3 × . The median values ofthe posterior distribution are shown in Figure 18. We can see that the median values of the 100 chains locates withinone sigma range of the median values shown in Figure 16. Therefore, we conclude that the chains are convergent withiterations of 3 × and different initial values lead to similar results.1 (year) P =1.565P =1.553std=0.017 c oun t e =0.363e =0.375std=0.097 −0.2 0 0.2 0.40500010000 t (year) t =0.138t =0.116std=0.078 (year) c oun t P =1.152P =1.147std=0.016 e =0.407e =0.297std=0.118 t (year) c oun t t =0.771t =0.803std=0.096 . Fig. 16.— The distribution of the last 1 × fitted parameters of a Jupiter pair in 3:4 MMR with observational errors(blue) σ m = 10 µ as. The red lines are the Gaussian fit to the distribution of the parameters. P t , e t , t t , P t , e t and t t are the true values while P , e , t , P , e and t are the median value of the Gaussian fit, std represent the corresponding standard deviations of the fitted parameters. 024 x 10 x ( µ as) c oun t x =10000.0x =10004.3std=10.3 024 x 10 y ( µ as) y =10000.0y =10004.9std=11.3 024 x 10 µ x ( µ as/year) c oun t µ xt =50000.0 µ x =50002.2std=1.4 −3.001 −3.0005 −3 −2.9995x 10 024 x 10 µ y ( µ as/year) µ yt =−30000.0 µ y =−30001.2std=1.5 024 x 10 π ( µ as) c oun t π t =33333.3 π =33333.7std=2.4 . Fig. 17.— The distribution of the last 1 × fitted stellar parameters of a Jupiter pair in 3:4 MMR with observational errors σ m = 10 µ as(blue). The red lines are the Gaussian fit to the distribution of the parameters. x t , y t , µ xt , µ yt and π t are the true values while x , y , µ x , µ y and π are the median value of the Gaussian fit, std represent the corresponding standard deviations of the fitted parameters. (year) c oun t P =1.152P =1.147std=0.016 =0.407e =0.297std=0.118 (year) c oun t t =0.771t =0.803std=0.096 (year) P =1.565P =1.553std=0.017 c oun t e =0.363e =0.3750std=0.097 −0.2 0 0.2 0.401020 t (year) t =0.138t =0.116std=0.078 . Fig. 18.— The distribution of the values for the best fit parameters of a Jupiter pair in 3:4 MMR with observational errors σ m = 10 µ as(blue)(the same with that in Figure 15) with 100 different initial values of e , t , e and t . The red lines are the Gaussian fit lines inFigure 16. In order to have a clear comparison, the Gaussian fit values are 500 times smaller than those in Figure 16. P t , e t , t t , P t , e t and t t are the true values while P , e , t , P , e and t02