Exoplanets Prediction in Multiplanetary Systems
PPublications of the Astronomical Society of Australia (PASA)doi: 10.1017/pas.2021.xxx.
Exoplanets Prediction in Multiplanetary Systems
M. Mousavi-Sadr , , G. Gozaliasl and D. M. Jassur Department of Theoretical Physics and Astrophysics, Faculty of Physics, University of Tabriz, Tabriz, Iran Department of Physics, University of Helsinki, P. O. Box 64, FI-00014, Helsinki, Finland Email: [email protected]
Abstract
We present the results of a search for additional exoplanets in all multiplanetary systems discovered todate, employing a logarithmic spacing between planets in our Solar System known as the Titius-Bode(TB) relation. We use the Markov Chain Monte Carlo method and separately analyse 229 multiplanetarysystems that house at least three or more confirmed planets. We find that the planets in ∼
53% ofthese systems adhere to a logarithmic spacing relation remarkably better than the Solar System planets.Using the TB relation, we predict the presence of 426 additional exoplanets in 229 multiplanetarysystems, of which 197 candidates are discovered by interpolation and 229 by extrapolation. Altogether,47 predicted planets are located within the habitable zone (HZ) of their host stars, and five of the 47planets have a maximum mass limit of 0.1-2 M ⊕ and a maximum radius lower than 1.25 R ⊕ . Our resultsand prediction of additional planets agree with previous studies’ predictions; however, we improve theuncertainties in the orbital period measurement for the predicted planets significantly. Keywords:
Planets and satellites: detection – planets and satellites: dynamical evolution and stability –planets and satellites: formation – planets and satellites: general
The number of detected exoplanets is growing rapidly, sothat over four thousand exoplanets have been detectedand confirmed to date. There are also thousands ofother candidate exoplanets that require further follow-up observation. The Kepler spacecraft plays a key rolein detecting these systems. The Kepler space mission’smain goal was to discover Earth-size exoplanets in ornear the habitable zone (HZ) of solar-like stars anddetermine the fraction of the hundreds of billions ofstars in our galaxy that might possess such planets(Koch et al., 1998).Photometry with the transit method is the most suc-cessful exoplanet discovery method, which has been usedby Kepler space mission, Transiting Exoplanet SurveySatellite (TESS), and many ground-based observatories(Borucki et al., 2010; Ricker et al., 2015; Deeg & Alonso,2018). However, this method has its own difficulties. Forexample, transits are detectable only when the planet’sorbit happens to be almost exactly aligned with theobserver’s line-of-sight. This covers only a small fractionof exoplanets. Furthermore, the planet’s transit lasts fora small fraction of its total orbital period. As a result, itis not very likely to detect planets’ transits, especiallythose with long orbital periods. This study sets out to predict the existence of additional undetected planetsin multiple exoplanet systems.In our Solar System, there is a simple logarithmicspacing between planets, which has been known for overtwo centuries as the Titius-Bode (TB) law. Its classicalrelation is: a n = 0 . . × n , (1)where a n represents the semi-major axis of the n th planetin AU. The planet Mercury corresponds to n = −∞ ,Venus to n = 0, Earth to n = 1 and so on (Nieto &Davis, 1974). After the discovery of the planet Uranusin 1781 by Frederick William Herschel, it was recognisedthat the TB law predicted this planet’s semi-major axis(Nieto & Davis, 1974).The discovery of the TB relation motivated manyobservation programs to investigate and detect the lostfifth planet, which eventually led to the discovery of theasteroid Ceres (Sawyer Hogg, 1948). The predictionsmade using the TB relation also played a key role inexploring the planet Neptune, but not as accuratelyas Uranus. Interestingly, the satellite systems of thegiant planets also follow a TB relation (Lyttleton, 1960;Brookes, 1970).The TB relation was used effectively to predict lostundetected objects in our Solar System. It was believed1 a r X i v : . [ a s t r o - ph . E P ] F e b Mousavi-Sadr, Gozaliasl, and Jassur that this relation could help make similar predictions indetected multiple exoplanet systems, too. The five-planet55 Cnc system was one of the first multiple- exoplanetsystems where Poveda & Lara (2008) applied TB to pre-dict the undetected planets. They found that a simpleexponential TB relation reproduces the semi-major axesof the five observed planets. They also predicted twoadditional planets at distances of 2 and 15 AU. Usingthe 55 Cnc system, Chang (2008) also checked whetherthe TB relation is enforceable on exoplanetary systemsby statistically analysing the distribution of the ratio ofperiods of two planets in the 55 Cnc system, by compar-ing it with that derived from the TB relation. Chang(2010) again repeated this calculation for 31 multipleexoplanet systems and concluded that the adherence ofthe Solar System’s planets to the TB relation might notbe fortuitous; thus, we could not ignore the possibilityof using the TB relationship in exoplanetary systems.Moreover, Lara et al. (2012) showed that like 55 Cnc,ten other planetary systems (ups And, GJ 876, HD160691, GJ 581, Kepler-223, HR 8799, Kepler-20, Kepler-33, HD 10180, and Kepler-11) host four or more planetsthat also obey a similar (but not identical) TB relation.Bovaird & Lineweaver (2013) (hereafter, BL13) useda sample of 68 multiple exoplanet systems with at leastfour planets, including samples of both confirmed andcandidate systems. They identified a sample of exoplanetsystems that are likely to be more complete and testedtheir adherence to the TB relation. They found thatmost of these exoplanetary systems adhere to the TB re-lation better than the Solar System. Using a generalizedTB relation, they predicted 141 additional exoplanets,including a planet with a low radius (
R < . R ⊕ ) andwithin the HZ of the Kepler-235.Using the predictions made by BL13, Huang & Bakos(2014) analysed Kepler’s long-cadence data to search for97 of those predicted planets and obtained a detectionrate of ∼ ± Figure 1.
The mass-radius distribution of the exoplanets inour sample, separated into five groups based on their detectionmethods: imaging (black stars), radial velocity (red circles), transit(blue squares), and transit timing variations (green triangles). inclined orbits, and the detection techniques. Thus theydid not expect all predicted planets to be detected. Theyestimated the geometric probability to transit for all 228predicted planets and highlighted a list of 77 planetswith high transit probability, resulting in an expecteddetection rate of ∼
15 percent, which was about threetimes higher than the detection rate measured BL13.Recently, Lara et al. (2020) used data from 27 exo-planetary systems with at least five planets and appliedtheir proposed method to find the reliability of the TBrelation and its predictive capability to search for plan-ets. They removed planets from the system one by oneand used TB relation to recover them, where they wereable to recover the missing one 78% of the time. Thisnumber was much higher than when they tried this withrandom planetary systems, where 26% of planets wererecovered. Using statistical tests, they showed that theplanetary orbital periods in exoplanetary systems werenot consistent with a random distribution and concludedit to be an outcome of the interactions between trueplanets.The purpose of this study is to test the adherenceof 229 multiple exoplanet systems with at least threedetected planets to the TB relation, and compare theiradherence rate with the Solar System’s using Markovchain Monte Carlo (MCMC) (see Goodman & Weare,2010; Foreman-Mackey et al., 2019) as a precise methodfor regression while considering the reported uncertain-ties of physical values. We also aim to predict the exis-tence of additional undetected exoplanets and estimatetheir physical properties, either maximum mass or max-imum radius. Moreover, we highlight exoplanets locatedwithin the HZ of their host stars. Using a sample of sevenmultiple exoplanet systems with detected planets afterpredictions made by BL15, we also aim to determine rediction of Exoplanets
We used physical parameters of exoplanets, includingplanetary orbital period, radius, and mass, and the stel-lar mass and radius from two exoplanet databases: theNASA Exoplanet Archive and the Extrasolar PlanetsEncyclopedia . We also used conservative and optimisticlimits of the HZ of available stars from the HabitableZone Gallery .The total number of multiple exoplanet systems withat least three confirmed planets available is 230 systemsto date, hosting 818 planets. 81.5% of these planets havebeen detected using the transit method and 16.4% usingthe radial velocity method. The remaining 2.1% of plan-ets have been identified using transit timing variations,imaging, pulsar timing, and orbital brightness modu-lation methods. Figure 1 represents the mass-radiusdistribution of exoplanets for five groups of planets, sep-arated according to their detection methods. Of the 230systems, 142 systems host three planets, 59 systemshost four planets, 20 systems include five planets, andseven systems host six planets. TRAPPIST-1 is the onlysystem that contains seven planets, and KOI-351 theonly one that contains eight. Figure 2 represents thedistribution of the orbital period of member exoplanetsin various exoplanetary systems used in this study.The exoplanetary systems 55 Cnc, GJ 676 A, HD125612, K2-136, Kepler-132, Kepler-296, Kepler-47,Kepler-68, and ups And consist of binary stars. GJ667 C and Kepler-444 are also triple star systems. Thesystem Kepler-132 possesses four planets such that twoof them (Kepler-132 b and Kepler-132 c) have roughlysimilar orbital periods of 6.1782 days and 6.4149 days.After more detailed studies, it was found that Kepler-132b and Kepler-132 c cannot orbit the same star (Lissaueret al., 2014). Consequently, we exclude Kepler-132 fromour analyses; our final sample of exoplanetary systemsthus contains 229 systems. http://exoplanetarchive.ipac.caltech.edu/ http://exoplanet.eu/ http://hzgallery.org/ Figure 2.
The distribution of the orbital period of 813 memberexoplanets in 229 multiple-exoplanet systems hosting at least threeplanets (gray bars), four planets (red bars), and five (or more)planets (blue bars).
Figure 3. σ as a function of the average log period spacingbetween planets, S p (Equation 5), of exoplanet systems. The blackline goes through two points: the origin (0,0) and ( S p , σ ), where S p is the compactness of the Solar System and σ is the value requiredfor the Solar System to yield χ /dof = 1 in Equation 4. Thecyan triangle shows the Solar System, and the black dots showthe exoplanet systems with no planet insertions (systems with χ /dof ≤ S p , σ ) of the systems after insertionshave been made. Mousavi-Sadr, Gozaliasl, and Jassur
Figure 4.
The TB relation and steps of linear regressions are applied to the data of system GJ 667 C . Detected and predicted planetsare shown with black and red dots, respectively. For each step, the value of n ins represents the number of inserted planets. The valuesof γ and χ /dof are also shown where the highest value of γ is in the fourth step as the best combination of detected and predictedplanets. The two black dashed lines show ± σ uncertainties around the best scaling relation (black solid line). The blue lines are a set of100 different realisations, drawn from the multivariate Gaussian distribution of the parameters (for the fourth step: m=0.1924, b=0.856and ln< σ >=-3.73), and the scatter co-variance matrix is estimated from the MCMC chain. rediction of Exoplanets We use the TB relation to predict additional undetectedplanets for all multiple exoplanet systems, with at leastthree confirmed planets.The TB relation can be written in terms of the orbitalperiods as follows: P n = P α n , n = 0 , , , ..., N − . (2)Where P n is the orbital period of the n th planet, and Pand α are fitting parameters. In analysing the exoplane-tary systems, the Solar System is used to guide how welleach system adheres to the TB relation. In logarithmicspace, the TB relation is written as follows:log P n = log P + n log α = b + m × n, n = 0 , , , ..., N − . (3)Where b=log P and m=log α are intercept and slope ofthe relation, respectively. For a system with N planets,the χ /dof value can be calculated by: χ ( b, m ) N − N − N − X n =0 [ ( b + m × n ) − log P n σ ] , (4)N-2 is the number of degrees of freedom (N planets - 2 fit-ted parameters (b and m)), and σ represents the system’ssparseness or compactness. The sparseness/compactnessof a system is calculated from σ = 0.273 S p , where S p rep-resents the average log period spacing between planetsas defined by: S p = log P N − − log P N , (5)where N is the number of planets in the system and P N − ,and P are the largest and smallest orbital periods in thesystem, respectively. We plot the σ values (see Equation4) as a function of sparseness/compactness ( S p ; seeEquation 5) in Fig. 3. The black line goes through twopoints: the origin and the specific S p and σ values for theSolar System ( σ is the value required for Solar Systemto yield χ /dof = 1 in Equation 4). The black pointsshow the exoplanet systems with no planet insertions,gray points indicate systems before planet insertion, andred points indicate systems after insertions have beenmade.Using a proper value for σ , χ /dof = 1 is adjustedfor the Solar System case. If the detected planets in asystem adhere to the TB relation better than the planetsof the Solar System ( χ /dof ≤ χ /dof > χ /dof value calculated for each possibility. The new specificinserted planet is chosen from 5,000 cases when produc-ing the minimum value of χ /dof . Similarly, insertingup to 10 new specific planets for each system step bystep (two planets for the second step, three planets forthe third step, etc.) covers all possible locations andcombinations between two adjacent planets. The perioduncertainty of an inserted planet ( e ins ) is calculatedusing the uncertainties of detected planets in the samesystem (e.g., e , e , e ,...) as follows: e ins = q e + e + e + .... (6)We adopt the highest value of the parameter γ , whichis the improvement in the χ /dof per inserted planet,for identifying the best combination of detected andpredicted planets. γ is defined by: γ = ( χ i − χ j χ j ) n ins , (7)where χ i and χ j are the χ values before and afterinserting of n ins planets.We analyse each system’s data separately and usethe MCMC method to quantify the uncertainties of thebest-fit parameters.By applying the lowest signal-to-noise ratio (SNR) ofthe detected planets in the same system to the predictedplanet’s orbital period, the maximum mass or maximumradius of the predicted planets is calculated.For transiting detected planets, the maximum radiusis calculated by: R max = R minSNR ( P predicted P minSNR ) . , (8)and for radial velocity detected planets, the maximummass is calculated by: M max = M minSNR ( P predicted P minSNR ) , (9)where R minSNR , M minSNR , and P minSNR are the ra-dius, mass, and orbital period of the detected planetwith the lowest SNR, respectively. After calculating themaximum radius or maximum mass of the predictedplanets, using the mass-radius relationship establishedby Bashi et al. (2017) (hereafter, B17) ( R p ∝ M . and R p ∝ M . for the small and large planets, respec-tively), we convert the radius values to mass values, andvice versa. Mousavi-Sadr, Gozaliasl, and Jassur
Table 1
Data corresponding to Fig. 4. n insa χ /dof γ b Period (days) ON c a Number of the inserted planet. b γ = ( χ i − χ j ) / ( χ j × n ins ), where χ i and χ j are the χ valuesbefore and after inserting of n ins planets, respectively. c The orbital number of the inserted planet.
We use the dynamical spacing criterion (∆) to analysehow our predicted objects could be stable in their posi-tions in the exoplanetary system. The dynamical spacing∆ between two adjacent planets with masses M and M and orbital periods P and P orbiting a host starwith a mass of M ∗ was defined by Gladman (1993) andChambers et al. (1996) as follows:∆ = 2 M / ∗ ( P / − P / )( M + M ) / ( P / + P / ) . (10)In a planetary system, two adjacent planets are less likelyto be stable in their positions when the value of theirdynamical spacing is small (∆ ≤ x ≤ x = | N j N i − P n +1 P n | N j N i , (11) N i and N j are positive integers with N i < N j ≤
5, and P n and P n +1 are the orbital periods of two adjacentplanets.The transit phenomenon of a planet is only observableif the planetary orbit plane is close to the line-of-sightbetween the observer and the host star. In other words,the pole of the planetary orbit must be within the angle d s /a p , where d s is the stellar diameter, and a p is theplanetary orbital radius. Then, the geometric transitprobability ( P tr ) of a planet can be estimated using d s / a p , where P tr = 0 .
5% for an Earth-size planet at 1AU orbiting a Solar-size star (Borucki & Summers, 1984;Koch & Borucki, 1996). According to the models of plan-etary systems, multi-planetary systems, like the SolarSystem, are assumed to be formed out of common pro-toplanetary disks (Winn & Fabrycky, 2015). Therefore,the orbital planes should have small relative inclinationsso that the Kepler multi-planetary systems are highlycoplanar (BL15). We use this criterion to estimate thetransit probability of predicted exoplanets and prioritisehow soon the predicted planets can be detected. rediction of Exoplanets Table 2
Predicted exoplanets within the HZ of host stars in multi-planetary systems. Columns 1, 2, and 3 present the id,host star name, and discovery method (Dis.). Columns 4, 5, and 6 present the orbital period in days, distance from theparent star in AU, and the orbital number (ON). The estimated maximum radius ( R Max ) and maximum mass ( M Max ) inthe Earth unit are presented in columns 7 and 8. Column 9 lists the transit probability ( P tr ). The conservative ( HZ Cons )and optimistic ( HZ Opt ) HZ limits in AU are presented in columns 10 and 11, respectively.Panel Host name Dis. a Period(days) a(AU) ON b R Max ( R ⊕ ) M Max ( M ⊕ ) P tr (%) HZ Cons (AU) HZ Opt (AU)1 GJ 163 RV 72 . +2 . − . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +8 . − . . +0 . − . . +1 . − . . +0 . − . . +0 . − . . +0 . − . . +1 . − . . +0 . − . . +3 . − . . +0 . − . . +1 . − . . +0 . − . . +11 . − . . +0 . − . . +18 − . . +0 . − . . +14 . − . . +0 . − . . +8 . − . . +0 . − . . +12 . − . . +0 . − . . +14 . − . . +0 . − . . +79 . − . . +0 . − . . +13 . − . . +0 . − . . +58 . − . +0 . − . . +71 . − . . +0 . − . . +75 . − . . +0 . − . . +19 . − . . +0 . − . . +2 . − . . +0 . − . . +13 . − . +0 . − . . +36 . − . . +0 . − . . +106 . − . . +0 . − . . +39 . − . +0 . − . . +8 . − . . +0 . − . . +6 . − . . +0 . − . . +45 . − . . +0 . − . . +50 . − . . +0 . − . . +41 . − . . +0 . − . . +232 . − . . +0 . − . . +155 − . . +0 . − . . +325 . − . . +0 . − . . +40 . − . . +0 . − . . +108 . − . . +0 . − . . +206 . − . . +0 . − . . +32 . − . . +0 . − . . +124 . − . . +0 . − . . +317 . − . . +0 . − . . +160 . − . . +0 . − . . +83 . − . . +0 . − . . +70 . − . . +0 . − .
10 E 3.2 11.2 0.42 1.24-2.17 0.98-2.2944 Kepler-401 Tr 640 . +299 . − . . +0 . − . . +78 . − . . +0 . − .
11 E 5.5 30.2 0.38 1.44-2.51 1.14-2.6546 Kepler-603 Tr 527 . +1408 . − . . +1 . − . . +462 . − . . +0 . − . a Discovery method of the system: ’Tr’ and ’RV’ represent transit and radial velocity, respectively. b Orbital numbers (ON) followed by ’E’ indicate the extrapolated planets, and followed by ’C’ indicate that the corresponding orbitalperiods have been flagged as "Planetary Candidate" in the NASA Exoplanet Archive.
Mousavi-Sadr, Gozaliasl, and Jassur
Figure 5.
The one- and two-dimensional marginalized posteriordistributions of the scaling relation parameters for the highest γ value corresponding to the fourth step (( n ins = 4)) of the linearregression of the GJ 667 C system, as shown in Fig. 4. Figure 6.
Dynamical spacing ∆ and the total number of adjacentexoplanet pairs that are in orbital resonance with each other. Thesolid blue line shows the number of resonance pairs consideringour predicted planets, and the dotted black line shows the numberbefore inserting any predicted planets into systems. The values forthe Solar System are also shown for reference via the orange dashedline. The vertical dash-dotted red line corresponds to ∆ = 10 andseparates the less and more stable adjacent planet pairs regimes.
To investigate the probability of the existence of addi-tional planets in exoplanetary systems, we apply the TBrelation and the MCMC method to analyse the dataof a sample of 229 systems that contain at least threeconfirmed exoplanets. We find that 122 systems adhereto the TB relation better than the Solar System withoutany need for interpolation. For those 107 systems thatadhere to the TB relation worse than the Solar System,we insert up to 10 new additional planets. For example,Fig. 4 shows the best linear regression (TB relation) forup to 10 additional planet inserts in the GJ 667 C sys-tem. For GJ 667 C, the highest γ value is in the fourthstep, where the number of inserted planets ( n ins ) is four.Black dots represent the detected planets and red dotsthe predicted planets. The black line shows the best(mean) scaling relation, and the two dashed lines showthe ± σ uncertainties around this relation. The bluelines are a set of 100 different realisations, drawn fromthe multivariate Gaussian distribution of the parame-ters, where for the highest γ value: m=0.1924, b=0.856,and ln< σ >=-3.73. The scatter covariance matrix is alsoestimated from the MCMC chain. Figure 5 illustratesthe one- and two-dimensional marginalised posteriordistributions of the scaling relation parameters for thefourth step of the linear regression for the GJ 667 Csystem.Table 1 represents the data corresponding to Fig. 4and lists the number of inserted planets ( n ins ) (column1), the values of parameters χ /dof (column 2), γ foreach step of interpolation (column 3), orbital periods(column 4), and orbital numbers (ON) of inserted planets(column 5).After interpolating all 107 systems, we find that thesesystems adhere to the TB relation better than the So-lar System or to approximately the same extent. Ofthese 107 interpolated systems, 50 systems need one, 33systems need two, and the remaining 24 systems needmore than two additional planets to be inserted. We alsopredict an extrapolated planet beyond the outermostdetected planet for all systems in our sample. We pre-dict the existence of 426 possible additional exoplanetsin these systems, of which 197 are predicted by inter-polation. It should be noted that six of the predictedplanets in Kepler-1388, Kepler-1542, Kepler-164, Kepler-374, Kepler-402, and Kepler-403 have been flagged as"Planetary Candidates" in the NASA Exoplanet Archive.To verify whether some of the predicted exoplanetsare in the HZ of their parent stars, we use conservativeand optimistic definitions of the HZ (Kane & Gelino,2012). Among the predicted exoplanets, 47 exoplanetslie within the HZ of their host stars. 27 exoplanets outof 47 are located within the conservative HZ. Further- rediction of Exoplanets Figure 7.
Orbital periods and scaled radii of exoplanets in multiple-planet systems including the predicted exoplanets from extrapolations.The cyan and red circles indicate detected and predicted planets in systems, respectively. The green circles also indicate the predictedplanets located within the HZ of their parent stars. The estimated radius of the predicted planet in GJ 676 A is higher than themaximum possible limit of a typical planet. Furthermore, due to the discovery method of HR 8799, the predicted planet’s radius is notcalculated. Therefore, GJ 676 A and HR 8799 are excluded. Mousavi-Sadr, Gozaliasl, and Jassur
Figure 7. continued rediction of Exoplanets Figure 7. continued Mousavi-Sadr, Gozaliasl, and Jassur
Figure 8.
Orbital periods and scaled radii of exoplanets in multiple-planet systems with interpolated and extrapolated planet predictions.The cyan and red circles indicate detected and predicted planets in systems, respectively. The green circles also indicate the predictedplanets within the HZ of their parent stars. Due to the discovery methods of KIC 10001893 and PSR B1257+12, the predicted planets’radii are not calculated. Therefore, KIC 10001893 and PSR B1257+12 are excluded. rediction of Exoplanets Figure 8. continued Mousavi-Sadr, Gozaliasl, and Jassur
Figure 8. continued rediction of Exoplanets Figure 9.
The radius vs. period of detected (blue dots) and pre-dicted (red dots) exoplanets of 229 multi-planetary systems usedin this study. The dashed horizontal and vertical lines correspondto the radius and orbital period of the Earth respectively. Thevast majority of the detected and predicted planets are in largerradii and tighter orbits than Earth. more, 14 exoplanets in the HZ have been predicted byinterpolation, and the remaining 33 exoplanets havebeen predicted by extrapolation. Table 2 presents the47 predicted exoplanets that lie within the HZ of thehost star. Columns 1, 2, and 3 present the id, the hoststar name, and discovery method (Dis.), respectively.Columns 4, 5, and 6 present the orbital period in days,the distance from the parent star (a) in AU, and theorbital number (ON). The estimated maximum radiusand maximum mass in the Earth unit are presented incolumns 7 and 8. Column 9 lists the transit probability(
P tr ). The conservative and optimistic HZ limits forAU are presented in columns 10 and 11, respectively.Kepler-167 is a four-planet system where we predictthree interpolated additional exoplanets, in which twoplanets have an orbital period of 157.0 and 373.5 days,located within the conservative HZ. The host star Kepler-186 is also a five-planet system including two predictedexoplanets with orbital periods of 41.2 and 73.6 days;these two additional planets were found by interpolationand are located within the optimistic and conservativeHZ, respectively.Borucki et al. (2011) classified exoplanets into thefollowing class sizes: Earth-size ( R p < . R ⊕ ), super-Earth-size (1 . R ⊕ ≤ R p < R ⊕ ), Neptune-size (2 R ⊕ ≤ R p < R ⊕ ) and Jupiter-size (6 R ⊕ ≤ R p < R ⊕ ). Fol-lowing this classification, five of our predicted exoplan-ets within HZ have maximum radii within the Earth-size range, 11 super-Earth-size, 27 Neptune-size, threeJupiter-size, and one with a maximum radius largerthan twice that of Jupiter’s. Using the proposed cate-gories based on planet mass by Stevens & Gaudi (2013),our five predicted planets have maximum masses within the range of Earth (0 . M ⊕ − M ⊕ ). In addition, 22and 19 predicted planets have maximum masses withinthe range of super-Earth (2 M ⊕ − M ⊕ ) and Neptune(10 M ⊕ − ⊕ ), respectively. As a result, among our 47predicted exoplanets within HZ, there are only five exo-planets whose estimated maximum mass and radius arewithin the mass and radius range of Earth: the fourthand fifth planets of Kepler-186, the second planets ofGJ 3138 and Wolf 1061, and the extrapolated planet ofYZ Cet.We use the dynamical spacing criterion (∆) to investi-gate our predicted objects’ stability at HZ. We calculatethe ∆ values for all adjacent planet pairs in these 45systems, which host 47 predicted exoplanets within theirHZ. We find that when inserting our predicted plan-ets into systems, the average percentage of pairs with∆ ≤
10 increases from ∼
25% to ∼ ≤ ∼ . ≤ χ /dof , slope (m), intercept (b), and predicted orbitalperiod, respectively. Column 8 reports whether the pre-dicted period values in this paper and BL15 (or BL13)are consistent within error (Y) or not (N). Columns 9 to12, respectively, list the orbital number (ON), estimatedmaximum radius ( R Max ), and maximum mass ( M Max )in the Earth radius and mass unit, and the transit prob-ability ( P tr ). The columns have been sorted based onthe transit probability in descending order.Table 4 lists the systems with interpolated and ex-trapolated planet predictions. In this table, we present6 Mousavi-Sadr, Gozaliasl, and Jassur the χ /dof before and after interpolation in the fourthand fifth columns. γ and ∆ γ (where ∆ γ =( γ - γ )/ γ ; γ and γ are the highest and second-highest γ valuesfor system, respectively) are listed in columns 6 and 7,respectively. The definitions of other columns in thistable are the same as in Tab. 3. Table 4 is also sortedby the transit probability in descending order.In Fig. 7, we show the new systems containing thedetected planets (cyan circles) and the predicted (redcircles) planets by extrapolation, while the predictedplanets within the HZ are also shown as green circles.The sizes of the symbols are scaled based on the planet’sradius. Similarly, Fig. 8 presents the detected exoplanets(cyan circles) and the predicted exoplanets (red circles)by interpolation and extrapolation.Figure 9 illustrates the radius versus the orbital periodof the detected (blue points) and predicted (red points)exoplanets. We find similar trends for both detectedand predicted exoplanets. As seen, the vast majority ofexoplanets have larger radii and shorter orbital periodsthan Earth (the radius and orbital period of the Earthhave been shown with vertical and horizontal dottedblack lines) due to observational limitations.We note that the estimated masses (and radii) of GJ676 A, Kepler-56, and WASP-47 are excluded becausethey are higher than the maximum possible limit of atypical planet. Furthermore, due to the lack of stellarparameters, the transit probabilities of HD 31527, HD136352, and Kepler-402 are not calculated. To examine the reliability of the predictions made bythe TB relation, we look for some planetary systems inthe literature that have had new exoplanets detectedrecently, notably those systems that have been predictedto have additional planets by BL15. We find the followingseven systems with new planet detection: Kepler-1388,Kepler-150, Kepler-1542, Kepler-20, Kepler-80, Kepler-82, and KOI-351.The detected planets in Kepler-150 and Kepler-82 werenot predicted by BL15 (Schmitt et al., 2017; Freuden-thal et al., 2019), while the rest of the five planets havebeen detected with orbital periods that agree with theirpredicted periods. For Kepler-1388 and Kepler-1542, twoplanetary candidates with orbital periods of 75.73 and7.23 days have been detected, the detected periods ofwhich are consistent with their predicted orbital periods,73 . ± . . ± . . ± . . ± . . ± . This study uses the TB relation to predict the probabilityof finding additional planets in multiplanet systems withat least three member planets, similar to the methodused by BL15. In principle, we update the BL15 studyand utilise the Markov Chain Monte Carlo (MCMC) Sim-ulation. The MCMC method is used to analyse systemsand quantify the uncertainties of the best-fit parameters(Foreman-Mackey et al., 2019). The main feature of ourmethod is inserting thousands of hypothetical planetswith random orbital periods into each of the systems andusing the highest γ value to achieve the most accuratepredictions.To compare our method and that of BL15, we removeCeres and the planet Uranus (two objects predicted bythe TB law to exist) from the Solar System and insertthousands of random planets into the Solar System,covering all possible locations and combinations betweenthe two adjacent planets, to achieve the minimum valueof χ /dof . This leads us to recover the actual SolarSystem with highest γ value, where the predicted orbitalperiods’ calculated errors are 20% for Ceres and 23% forUranus. On the other hand, the BL15 method results inan error value of 23% for both Ceres and Uranus. Weremove the planets Earth and Mars, which are withinthe HZ range of the Sun, and apply the TB relation tothe system to predict their orbital periods. This recoversthe combination of the actual Solar System where thecalculated errors of orbital periods are 18% for Earthand 41% for Mars. We perform this process once againusing the BL15 method, which gives us the error valuesof 18.5% for Earth and 42% for Mars. For a bettercomparison, we repeat this process for other planets inthe Solar System and exoplanetary systems, and withdifferent combinations of planet removal.We also reapply our method to Kepler-1388, Kepler-1542, Kepler-20, Kepler-80, and KOI-351, regardless ofthe successfully prediction of planets by BL15, to com- rediction of Exoplanets Figure 10.
Comparison of uncertainty on the predicted orbitalperiods calculated by BL15 and this study. We remove planetsfrom systems in various combinations and apply the TB relationto recover the orbital periods of removed planets. Each blue datapoint belongs to a specific combination of removed planets fromsystems, and five red data points represent those detected planetsafter the predictions (see Tab. 5) made by BL15. pare the validity of the methods used in predicting theadditional planets. As shown in Tab. 5, our predictionshave fewer errors, and sometimes the predicted periodshave smaller error bars. These fewer error bars can beinterpreted as advantages of using the MCMC methodin predicting exoplanets based on the TB relation. Usingthe paired samples t-test, the p-value is estimated toequal 0.025, which is less than 0.05 and statistically sig-nificant. Therefore, we conclude that, on average, thereis evidence that applying the MCMC method does leadto more precise predictions than BL15’s method. Thedifference between the calculated errors with our methodand BL15’s is illustrated in Fig. 10, where each blue datapoint belongs to a planet recovery. Five red data pointsrepresent those detected planets after the predictionsmade by BL15.
Using the Markov Chain Monte Carlo method, we applythe TB relation to all available exoplanetary systemswith three or more confirmed exoplanets, a total of229 systems, to examine their adherence to the TBrelation in comparison to the Solar System and to predictthe existence of possible additional planets. For thosesystems that adhere to the TB relation better than theSolar System, we extrapolate one additional planet, andfor each of the remaining systems that adhere worse,we interpolate up to 10 specific planets between thedetected planets and identify new possible additionalplanets in the systems.We present a list of 229 analysed exoplanetary systems(of which 123 of them have not been previously analysedby either BL13 or BL15) containing their unique TBrelations and a total of 426 additional predicted exo-planets, of which 47 are located within the HZ of theirparent stars. We also estimate that five of the predictedplanets in HZ have maximum mass and radius limitswithin the Earth’s mass and radius range.As an important result, the planets of ∼
53% of oursample system adhere to a logarithmic spacing relationbetter than the planets of the Solar System. Therefore,there is a need to work with more comprehensive data ofmultiplanetary systems to reveal the probable dynamicalor gravitational aspects of the TB relation.We find new planet detection for seven exoplanetarysystems after the predictions made by BL15 and com-pare the detected and predicted orbital periods. We findthat both the detected and predicted orbital periodsagree very well within errors. Our predictions also agreeroughly better than those made by BL15, indicatingthat using a precise modeling method and measure-ments could improve our predictions and uncertaintiesas well. However, to claim the (un)reliability of the TBrelation in predicting the presence of additional planetsin exoplanetary systems, we require much more data andfurther follow-up observations of the exoplanetary sys-tems. Thus, we must wait for upcoming new exoplanetsurveys or ongoing surveys such as TESS.
This research has used theNASA Exoplanet Archive, Ex-trasolar Planets Encyclopedia, and Habitable Zone Gallery.We wish to thank Kenneth P. K. Quek for the English lan-guage editing. We also thank Ashkan M. Jasour and ThomasHackman for their useful comments on the manuscript. Theauthor acknowledges the usage of the following python pack-ages, in alphabetical order: astropy (Astropy Collaborationet al., 2013, 2018), chainConsumer (Hinton, 2019), emcee (Foreman-Mackey et al., 2019), matplotlib (Hunter, 2007), numpy (van der Walt et al., 2011), and scipy (Virtanen et al.,2020). Mousavi-Sadr, Gozaliasl, and Jassur
Table 3
Systems with only extrapolated planet predictions. Columns 1 and 2 present the host star name and discoverymethod (Dis.). Column 3 reports a flag that defines whether the system has already been analysed by BL15 (or BL13) (Y)or not (N). Columns 4 to 7 present χ /dof , slope (m), intercept (b), and predicted orbital period, respectively. Column 8reports whether the predicted period values in this paper and BL15 (or BL13) are consistent within error (Y) or not (N).Columns 9 to 12, respectively, list the orbital number (ON), estimated maximum radius ( R Max ), and maximum mass ( M Max )in the Earth radius and mass unit, and the transit probability ( P tr ).Host name Dis. a F χ dof m b Period(days) F ON b R Max ( R ⊕ ) M Max ( M ⊕ ) P tr (%)Solar System - Y 1.000 0 . +0 . − . . +0 . − . - - - - - -Kepler-207 Tr Y 0.001 0 . +0 . − . . +0 . − . . +0 . − . Y 3 2.1 5.2 7.31Kepler-217 Tr Y 0.676 0 . +0 . − . . +0 . − . . +5 . − . Y 3 1.8 3.9 6.90Kepler-374 Tr Y 0.320 0 . +0 . − . . +0 . − . . +2 . − . N 3 C 1.4 2.5 5.23Kepler-60 Tr Y 0.327 0 . +0 . − . . +0 . − . . +2 . − Y 3 2.1 5.2 4.83Kepler-23 Tr Y 0.135 0 . +0 . − . . +0 . − . . +2 . − . Y 3 2.4 6.6 4.46Kepler-223 Tr Y 0.318 0 . +0 . − . . +0 . − . +3 . − . Y 4 4.2 18.3 4.23Kepler-431 Tr Y 0.306 0 . +0 . − . . +0 . − . . +2 . − . Y 3 0.8 0.9 4.18HR 858 Tr N 0.234 0 . +0 . − . . +0 . − . . +5 . − . N 3 2.5 7.1 4.02Kepler-256 Tr Y 0.236 0 . +0 . − . . +0 . − . . +5 . − Y 4 2.8 8.7 3.99Kepler-107 Tr Y 0.410 0 . +0 . − . . +0 . − . . +5 . − . Y 4 1.1 1.6 3.92K2-219 Tr N 0.010 0 . +0 . − . . +0 . − . . +0 . − . N 3 1.9 4.3 3.91Kepler-226 Tr Y 0.474 0 . +0 . − . . +0 . − . . +2 . − . Y 3 1.3 2.2 3.68Kepler-271 Tr Y 0.001 0 . +0 . − . . +0 . − . . +0 . − . N 3 0.9 1.1 3.64Kepler-444 Tr Y 0.227 0 . +0 . − . . +0 . − . . +1 . − . Y 5 0.6 0.5 3.56Kepler-208 Tr Y 0.599 0 . +0 . − . . +0 . − . . +7 . − . Y 4 1.5 2.8 3.54Kepler-203 Tr Y 0.591 0 . +0 . − . . +0 . − . +11 . − . Y 3 1.7 3.5 3.40Kepler-758 Tr Y 0.215 0 . +0 . − . . +0 . − . . +6 − . Y 4 2.2 5.6 3.11Kepler-339 Tr Y 0.193 0 . +0 . − . . +0 . − . . +2 . − Y 3 1.3 2.2 3.07Kepler-272 Tr Y 0.177 0 . +0 . − . . +0 . − . . +6 . − . Y 3 2.4 6.6 3.06Kepler-350 Tr Y 0.207 0 . +0 . − . . +0 . − . . +7 . − . Y 3 2.5 7.1 2.93K2-148 Tr N 0.411 0 . +0 . − . . +0 . − . . +4 − . N 3 1.8 3.9 2.90Kepler-1254 Tr Y 0.171 0 . +0 . − . . +0 . − . . +3 . − . Y 3 1.7 3.5 2.88K2-138 Tr N 0.007 0 . +0 . − . . +0 . − . . +0 . − . N 5 2.7 8.2 2.83Kepler-24 Tr Y 0.737 0 . +0 . − . . +0 . − . . +10 . − . Y 4 2.8 8.7 2.83Kepler-18 Tr Y 0.129 0 . +0 . − . . +0 . − . . +9 . − . Y 3 2.7 8.2 2.69Kepler-114 Tr Y 0.097 0 . +0 . − . . +0 . − . . +2 . − . Y 3 1.7 3.5 2.57Kepler-305 Tr Y 0.610 0 . +0 . − . . +0 . − . . +9 . − . Y 5 3.0 9.9 2.50Kepler-304 Tr Y 0.568 0 . +0 . − . . +0 . − . . +7 − Y 4 2.2 5.6 2.44Kepler-206 Tr Y 0.052 0 . +0 . − . . +0 . − . . +6 . − Y 3 1.4 2.5 2.38Kepler-197 Tr Y 0.390 0 . +0 . − . . +0 . − . . +10 . − . Y 4 1.0 1.3 2.34Kepler-398 Tr Y 0.001 0 . +0 . − . . +0 . − . . +0 . − . Y 3 1.1 1.6 2.34Kepler-338 Tr Y 0.580 0 . +0 . − . . +0 . − . . +21 . − . Y 4 2.6 7.6 2.29Kepler-450 Tr Y 0.126 0 . +0 . − . . +0 . − . . +14 . − . Y 3 1.4 2.5 2.27Kepler-446 Tr Y 0.257 0 . +0 . − . . +0 . − . . +3 . − . Y 3 1.5 2.8 2.21Kepler-301 Tr Y 0.185 0 . +0 . − . . +0 . − . . +12 . − . Y 3 2.1 5.2 2.18K2-239 Tr N 0.818 0 . +0 . − . . +0 . − . . +4 . − . N 3 1.2 1.9 2.07L 98-59 Tr N 0.624 0 . +0 . − . . +0 . − . +6 . − . N 3 1.2 1.9 2.04Kepler-191 Tr Y 0.074 0 . +0 . − . . +0 . − . . +4 − . Y 3 1.8 3.9 2.02YZ Cet RV N 0.012 0 . +0 . − . . +0 . − . . +0 . − . N 3 H 1.2 1.9 1.96Kepler-85 Tr Y 0.106 0 . +0 . − . . +0 . − . +2 . − . Y 4 1.4 2.5 1.95Kepler-292 Tr Y 0.396 0 . +0 . − . . +0 . − . . +7 . − . Y 5 2.5 7.1 1.91Kepler-221 Tr Y 0.135 0 . +0 . − . . +0 . − . . +6 . − . Y 4 3.1 10.5 1.89K2-198 Tr N 0.007 0 . +0 . − . . +0 . − . . +2 − . N 3 2.6 7.6 1.75Kepler-334 Tr Y 0.253 0 . +0 . − . . +0 . − . . +23 . − . Y 3 1.7 3.5 1.70 rediction of Exoplanets Table 3 continuedHost name Dis. a F χ dof m b Period(days) F ON b R Max ( R ⊕ ) M Max ( M ⊕ ) P tr (%)Kepler-102 Tr Y 0.898 0 . +0 . − . . +0 . − . . +10 . − . Y 5 0.8 0.9 1.66Kepler-92 Tr Y 0.032 0 . +0 . − . . +0 . − . . +10 − . Y 3 2.4 6.6 1.64Kepler-445 Tr N 0.013 0 . +0 . − . . +0 . − . . +0 . − . N 3 H 1.4 2.5 1.61Kepler-127 Tr N 0.604 0 . +0 . − . . +0 . − . . +58 . − . N 3 2.2 5.6 1.60TOI-270 Tr N 0.427 0 . +0 . − . . +0 . − . . +8 . − . N 3 H 2.0 4.7 1.59Kepler-54 Tr Y 0.451 0 . +0 . − . . +0 . − . . +11 . − . Y 3 1.6 3.2 1.58Kepler-164 Tr Y 0.257 0 . +0 . − . . +0 . − . . +31 . − . N 3 C 2.7 8.2 1.56Kepler-224 Tr Y 0.215 0 . +0 . − . . +0 . − . . +8 . − . Y 4 2.3 6.1 1.56Kepler-257 Tr Y 0.353 0 . +0 . − . . +0 . − . . +52 − Y 3 6.2 37.1 1.54Kepler-244 Tr Y 0.083 0 . +0 . − . . +0 . − . +9 . − . Y 3 1.2 1.9 1.53Kepler-295 Tr Y 0.131 0 . +0 . − . . +0 . − . +8 . − . Y 3 1.5 2.8 1.53Kepler-104 Tr Y 0.026 0 . +0 . − . . +0 . − . . +12 . − . Y 3 4.3 19.1 1.49Kepler-172 Tr Y 0.044 0 . +0 . − . . +0 . − . . +9 . − . Y 4 3.4 12.4 1.46Kepler-84 Tr Y 0.548 0 . +0 . − . . +0 . − . . +24 . − . Y 5 2.6 7.6 1.45Kepler-247 Tr Y 0.435 0 . +0 . − . . +0 . − . . +39 . − . Y 3 3.3 11.8 1.42Kepler-327 Tr Y 0.520 0 . +0 . − . . +0 . − . . +27 . − . Y 3 1.6 3.2 1.41Kepler-770 Tr N 0.726 0 . +0 . − . . +0 . − . . +116 . − . N 3 2.8 8.7 1.41Kepler-238 Tr Y 0.822 0 . +0 . − . . +0 . − . . +62 . − . Y 5 5.3 27.9 1.40Kepler-249 Tr Y 0.001 0 . +0 . − . . +0 . − . . +0 . − . Y 3 H 1.9 4.3 1.38Kepler-81 Tr Y 0.313 0 . +0 . − . . +0 . − . . +13 . − . Y 3 1.4 2.5 1.36Kepler-83 Tr Y 0.079 0 . +0 . − . . +0 . − . . +9 . − . Y 3 2.8 8.7 1.36K2-16 Tr N 0.069 0 . +0 . − . . +0 . − . . +11 . − . N 3 2.2 5.6 1.28K2-58 Tr N 0.105 0 . +0 . − . . +0 . − . . +22 . − . N 3 2.2 5.6 1.28Kepler-106 Tr Y 0.331 0 . +0 . − . . +0 . − . . +21 . − . Y 4 1.3 2.2 1.26Kepler-122 Tr Y 0.907 0 . +0 . − . . +0 . − . . +47 − . Y 5 2.1 5.2 1.25Kepler-299 Tr Y 0.066 0 . +0 . − . . +0 . − . . +13 . − . Y 4 2.3 6.1 1.25KOI-94 Tr N 0.185 0 . +0 . − . . +0 . − . . +45 . − . N 4 4.2 18.3 1.25K2-32 Tr N 0.960 0 . +0 . − . . +0 . − . . +40 . − . N 4 1.5 2.8 1.23Kepler-222 Tr Y 0.037 0 . +0 . − . . +0 . − . . +16 . − . Y 3 4.7 22.4 1.21Kepler-282 Tr Y 0.501 0 . +0 . − . . +0 . − . . +22 . − . Y 4 1.7 3.5 1.18Kepler-53 Tr Y 0.065 0 . +0 . − . . +0 . − . . +13 . − Y 3 3.5 13.1 1.17Kepler-332 Tr Y 0.003 0 . +0 . − . . +0 . − . . +2 . − . Y 3 1.4 2.5 1.07K2-233 Tr N 0.141 0 . +0 . − . . +0 . − . . +42 . − N 3 2.4 6.6 1.06Kepler-215 Tr Y 0.893 0 . +0 . − . . +0 . − . . +75 . − . Y 4 2.0 4.7 1.05Kepler-245 Tr Y 0.073 0 . +0 . − . . +0 . − . . +13 . − Y 4 3.2 11.1 1.05Kepler-325 Tr Y 0.027 0 . +0 . − . . +0 . − . . +21 . − . Y 3 3.6 13.8 1.05K2-72 Tr N 0.980 0 . +0 . − . . +0 . − . . +18 − . N 4 H 1.5 2.8 1.03Kepler-79 Tr Y 0.631 0 . +0 . − . . +0 . − . . +62 . − . Y 4 4.1 17.5 1.00Kepler-1388 Tr Y 0.477 0 . +0 . − . . +0 . − . . +14 . − . Y 4 C,H 2.9 9.3 0.97Kepler-154 Tr Y 0.968 0 . +0 . − . . +0 . − . . +68 − . N 5 2.9 9.3 0.96Kepler-20 Tr Y 0.421 0 . +0 . − . . +0 . − . . +33 . − . Y 6 1.6 3.2 0.92Kepler-52 Tr Y 0.039 0 . +0 . − . . +0 . − . . +12 . − . Y 3 H 2.4 6.6 0.89Kepler-176 Tr Y 0.170 0 . +0 . − . . +0 . − . +28 . − . Y 4 1.8 3.9 0.87Kepler-171 Tr Y 0.210 0 . +0 . − . . +0 . − . . +63 . − Y 3 2.5 7.1 0.86HD 20781 RV N 0.261 0 . +0 . − . . +0 . − . . +79 . − . N 4 H 6.3 38.0 0.85Kepler-31 Tr Y 0.001 0 . +0 . − . . +0 . − . . +1 . − . Y 3 4.7 22.4 0.85Kepler-229 Tr Y 0.001 0 . +0 . − . . +0 . − . . +0 . − . Y 3 4.5 20.7 0.84Kepler-331 Tr Y 0.101 0 . +0 . − . . +0 . − . . +13 . − . Y 3 H 1.9 4.3 0.83Kepler-357 Tr Y 0.072 0 . +0 . − . . +0 . − . . +32 − . Y 3 3.9 16.0 0.83Kepler-288 Tr Y 0.024 0 . +0 . − . . +0 . − . . +31 . − . Y 3 3.5 13.1 0.80 Mousavi-Sadr, Gozaliasl, and Jassur
Table 3 continuedHost name Dis. a F χ dof m b Period(days) F ON b R Max ( R ⊕ ) M Max ( M ⊕ ) P tr (%)Kepler-55 Tr Y 0.900 0 . +0 . − . . +0 . − . . +58 . − Y 5 H 2.8 8.7 0.77K2-3 Tr N 0.850 0 . +0 . − . . +0 . − . . +71 . − . N 3 H 1.8 3.9 0.74K2-155 Tr N 0.516 0 . +0 . − . . +0 . − . . +75 . − . N 3 H 2.4 6.6 0.74Kepler-399 Tr Y 0.276 0 . +0 . − . . +0 . − . +44 . − . Y 3 1.6 3.2 0.70TRAPPIST-1 Tr N 0.574 0 . +0 . − . . +0 . − . +5 . − . N 7 0.8 0.9 0.68Kepler-19 Tr N 0.659 0 . +0 . − . . +0 . − . +145 . − . N 3 4.6 21.5 0.67Kepler-235 Tr Y 0.032 0 . +0 . − . . +0 . − . . +13 . − Y 4 H 2.5 7.1 0.66Kepler-130 Tr Y 0.002 0 . +0 . − . . +0 . − . . +12 . − . Y 3 2.2 5.6 0.62Kepler-166 Tr Y 0.019 0 . +0 . − . . +0 . − . . +39 . − Y 3 H 3.5 13.1 0.61Kepler-296 Tr Y 0.033 0 . +0 . − . . +0 . − . . +8 . − . Y 5 H 2.1 5.2 0.61Kepler-289 Tr N 0.001 0 . +0 . − . . +0 . − . . +1 . − . N 3 3.5 13.1 0.59Kepler-51 Tr Y 0.825 0 . +0 . − . . +0 . − . . +124 . − . Y 3 10.7 100.0 0.58Kepler-251 Tr Y 0.891 0 . +0 . − . . +0 . − . . +232 . − . Y 4 H 3.5 13.1 0.53HD 20794 RV N 0.702 0 . +0 . − . . +0 . − . . +155 − . N 4 H 3.3 11.9 0.52Kepler-30 Tr Y 0.169 0 . +0 . − . . +0 . − . . +108 . − . Y 3 H 7.0 46.2 0.49Kepler-298 Tr Y 0.966 0 . +0 . − . . +0 . − . . +317 . − . Y 3 H 3.2 11.1 0.45Kepler-401 Tr Y 0.088 0 . +0 . − . . +0 . − . . +299 . − . Y 3 H 3.1 10.5 0.40Kepler-603 Tr Y 0.754 0 . +0 . − . . +0 . − . . +1408 . − . Y 3 H 4.0 16.7 0.36GJ 3293 RV N 0.871 0 . +0 . − . . +0 . − . . +132 − . N 4 6.9 45.5 0.33HD 69830 RV N 0.607 0 . +0 . − . . +0 . − . . +1496 − . N 3 10.7 99.8 0.25Kepler-174 Tr Y 0.824 0 . +0 . − . . +0 . − . . +1567 . − . Y 3 3.1 10.5 0.18tau Cet RV N 0.481 0 . +0 . − . . +0 . − . . +1300 . − . N 4 3.5 13.1 0.14HD 10180 RV Y 0.617 0 . +0 . − . . +0 . − . . +4366 − . Y 6 12.1 209.7 0.08GJ 676 A RV N 0.669 1 . +0 . − . . +0 . − . . +347768 . − . N 4 - - 0.01HR 8799 Im Y 0.114 0 . +0 . − . . +0 . − . . +841068 . − . Y 4 - - 0.01HD 31527 RV N 0.764 0 . +0 . − . . +0 . − . . +2046 . − . N 3 7.8 55.9 -HD 136352 RV N 0.989 0 . +0 . − . . +0 . − . . +462 . − . N 3 H 5.4 28.6 -Kepler-402 Tr Y 0.952 0 . +0 . − . . +0 . − . . +4 . − . N 4 C 1.6 3.2 - a Discovery method of the system: ’Tr’, ’RV’, and ’Im’ represent transit, radial velocity, and imaging, respectively. b Orbital numbers (ON) followed by ’H’ indicate the predicted planets within the HZ, and ON followed by ’C’ indicate that thecorresponding orbital periods have been flagged as "Planetary Candidate" in NASA Exoplanet Archive. rediction of Exoplanets T a b l e S y s t e m s w i t h i n t e r p o l a t e d a nd e x t r a p o l a t e dp l a n e t p r e d i c t i o n s . C o l u m n s nd p r e s e n tt h e h o s t s t a r n a m e a ndd i s c o v e r y m e t h o d ( D i s . ) . C o l u m n r e p o r t s a fl ag t h a t d e fin e s w h e t h e r t h e s y s t e m h a s a l r e a d y b ee n a n a l y s e db y B L ( o r B L )( Y ) o r n o t( N ) . C o l u m n s nd p r e s e n t χ / d o f b e f o r e a nd a f t e r i n t e r p o l a t i o n . C o l u m n s t o10 , r e s p e c t i v e l y ,li s tt h e γ , ∆ γ , s l o p e ( m ) ,i n t e r c e p t( b ) , a ndp r e d i c t e d o r b i t a l p e r i o d . C o l u m n r e p o r t s w h e t h e r t h e p r e d i c t e dp e r i o d v a l u e s i n t h i s p a p e r a nd B L ( o r B L ) a r e c o n s i s t e n t w i t h i n e rr o r ( Y ) o r n o t( N ) . C o l u m n s t o15 , r e s p e c t i v e l y ,li s tt h e o r b i t a l nu m b e r ( O N ) , e s t i m a t e d m a x i m u m r a d i u s ( R M a x ) , a nd m a x i m u mm a ss ( M M a x ) i n t h e E a r t h r a d i u s a nd m a ss un i t , a nd t h e t r a n s i t p r o b a b ili t y ( P t r ) . H o s t n a m e D i s . a F ( χ d o f ) i ( χ d o f ) j γ ∆ γ b m b P e r i o d ( d a y s ) F O N c R M a x ( R ⊕ ) M M a x ( M ⊕ ) P t r ( % ) K - T r N . . . . . + . − . − . + . − . + . − . N . . .
31 2 . + . − . N . . .
87 4 . + . − . N . . .
11 49 . + . − . N E . . . K e p l e r - T r Y . . . . . + . − . . + . − . . + . − . Y . . .
91 4 . + . − . Y . . .
27 8 . + . − . N . . .
80 73 . + . − . Y E . . . K - T r N . . . . . + . − . − . + . − . . + . − . N . . .
28 28 . + . − . N E . . . K - T r N . . . . . + . − . − . + . − . . + . − . N . . .
10 2 . + . − . N . . .
73 4 . + . − . N . . .
44 42 . + . − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
26 3 . + . − . N . . .
70 17 . + . − . N E . . . K e p l e r - T r Y . . . . . + . − . − . + . − . . + . − . Y . . .
22 2 . + . − . Y . . .
79 21 . + . − . N E . . . K e p l e r - T r Y . . . . . + . − . − . + . − . . + . − . Y . . .
19 40 . + . − . Y E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
96 45 + . − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
42 6 + . − . N . . .
79 9 . + . − . N . . .
62 15 . + . − . N E . . . K I C O B M N . . . . . + . − . − . + . − . . + . − . N -- .
27 1 . + . − . N E -- .
27 55 C n c R VY . . . . . + . − . − . + . − . . + . − . Y . . .
81 1037 . + − . Y . . .
22 19597 . + . − . Y E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
63 27 . + . − . N E . . . K e p l e r - T r Y . . . . . + . − . . + . − . . + . − . Y . . .
59 4 . + . − . Y . . .
11 6 . + . − . Y E , C . . . Mousavi-Sadr, Gozaliasl, and Jassur T a b l e c o n t i nu e d H o s t n a m e D i s . a F ( χ d o f ) i ( χ d o f ) j γ ∆ γ b m b P e r i o d ( d a y s ) F O N c R M a x ( R ⊕ ) M M a x ( M ⊕ ) P t r ( % ) K e p l e r - T r Y . . . . . + . − . . + . − . . + . − . N . . .
47 10 . + . − . N . . .
37 17 . + . − . N . . .
98 54 + . − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
30 7 . + . − . N . . .
77 115 . + . − . N E . . . H D T r N . . . . . + . − . − . + . − . . + . − . N . . .
01 88 . + . − N E . . . K e p l e r - T r N . . . . . + . − . − . + . − . . + . − . N . . .
78 3 . + . − . N E . . . G J T r N . . . . . + . − . . + . − . . + . − . N . . .
20 10 . + . − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
98 9 . + . − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
83 8 . + . − . N . . .
52 90 . + . − N E . . . K - T r N . . . . . + . − . . + . − . . + . − . N . . .
79 5 . + . − . N . . .
59 17 . + . − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
70 18 . + . − . N . . .
90 25 . + . − . N . . .
03 48 . + . − . N E . . . K - T r N . . . . . + . − . . + . − . + . − . N . . .
27 21 . + . − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
23 13 . + . − . N . . .
59 21 . + . − . N . . .
62 90 . + . − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
14 106 . + . − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
11 152 . + − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
10 103 . + − . N . . .
12 247 . + . − . N . - .
30 508 . + . − . N . - .
42 2324 . + . − . N E , H . - . rediction of Exoplanets T a b l e c o n t i nu e d H o s t n a m e D i s . a F ( χ d o f ) i ( χ d o f ) j γ ∆ γ b m b P e r i o d ( d a y s ) F O N c R M a x ( R ⊕ ) M M a x ( M ⊕ ) P t r ( % ) K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
64 18 + . − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
49 19 . + − . N . . .
17 89 . + . − N E . . . K e p l e r - T r Y . . . . . + . − . . + . − . . + . − . Y . . .
13 30 + . − . Y E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
86 29 . + . − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
83 14 . + . − . N . . .
37 42 . + . − N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
67 25 . + . − . N . . .
47 76 . + . − . N E . . . K - T r N . . . . . + . − . . + . − . . + . − . N . . .
26 25 . + . − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
23 27 . + . − . N E . . . K e p l e r - T r Y . . . . . + . − . . + . − . . + . − . Y . . .
22 26 . + . − . Y . . .
86 107 . + − . Y E . . . K e p l e r - T r Y . . . . . + . − . . + . − . . + . − . Y . . .
14 13 . + . − . Y . . .
54 20 + . − Y . . .
37 65 . + . − . Y E . . . H D R VN . . . . . + . − . . + . − . + . − . N . . .
13 41 . + . − . N E . . . K - T r N . . . . . + . − . . + . − . . + . − . N . . .
00 12 . + . − . N . . .
00 43 . + . − . N E . . . K e p l e r - T r Y . . . . . + . − . . + . − . . + . − . Y . . .
99 7 . + . − . Y . . .
72 27 . + . − . Y . . .
56 66 . + . − . Y E , H . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
89 22 . + − . N . . .
73 121 . + . − . N E . . . K e p l e r - T r Y . . . . . + . − . . + . − . . + . − . Y . . .
69 18 . + . − . Y . . .
61 64 . + . − . Y E . . . Mousavi-Sadr, Gozaliasl, and Jassur T a b l e c o n t i nu e d H o s t n a m e D i s . a F ( χ d o f ) i ( χ d o f ) j γ ∆ γ b m b P e r i o d ( d a y s ) F O N c R M a x ( R ⊕ ) M M a x ( M ⊕ ) P t r ( % ) K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
64 437 . + . − . N E , H . . . K - T r N . . . . . + . − . . + . − . . + . − . N . . .
60 20 . + . − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
60 11 . + . − . N . . .
42 35 . + . − . N E . . .
71 61 V i r R VN . . . . . + . − . . + . − . . + . − . N . . .
42 368 . + . − . N E , H . . . K e p l e r - T r Y . . . . . + . − . . + . − . . + . − . Y . . .
40 17 . + . − . Y . . .
04 51 . + . − Y E . . . K e p l e r - T r N . . . . . + . − . . + . − . + . − . N . . .
39 62 . + . − . N E . . . H D R VN . . . . . + . − . . + . − . . + . − . N . . .
36 13 . + . − . N . . .
36 16 . + . − . N . . .
92 20 . + . − . N . . .
51 31 + . − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
23 28 . + . − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
22 31 . + . − . N E . . . G J R VY . . . . . + . − . . + . − . . + . − . Y . . .
18 8 . + . − . Y . . .
56 16 + . − . Y . . .
63 245 . + . − . Y E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
14 19 . + . − . N . . .
76 52 . + . − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
10 39 . + . − . N . . .
45 83 . + . − N . . .
49 160 . + . − . N . . .
96 297 . + . − . N . . .
64 1245 . + . − . N E . . . K e p l e r - T r Y . . . . . + . − . . + . − . . + . − . N C . . .
90 107 + . − . Y E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
82 47 . + . − . N . . .
34 110 . + . − . N . . .
34 524 + . − . N E , H . . . rediction of Exoplanets T a b l e c o n t i nu e d H o s t n a m e D i s . a F ( χ d o f ) i ( χ d o f ) j γ ∆ γ b m b P e r i o d ( d a y s ) F O N c R M a x ( R ⊕ ) M M a x ( M ⊕ ) P t r ( % ) H I P T r N . . . . . + . − . − . + . − . . + . − . N . . .
80 44 . + − . N . . .
37 82 . + . − . N . . .
57 266 . + . − . N . . .
72 301 . + . − N . . .
67 701 . + . − . N E , H . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
75 19 . + . − . N . . .
60 97 . + . − . N E . . . K e p l e r - T r Y . . . . . + . − . . + . − . . + − . Y . . .
73 54 . + . − . Y E . . . K e p l e r - T r Y . . . . . + . − . . + . − . . + . − . Y . . .
60 27 . + − Y . . .
97 68 . + . − . Y E . . . K - T r N . . . . . + . − . . + . − . . + . − . N . . .
53 17 . + . − . N . . .
09 40 . + . − . N E , H . . . H D R VN . . . . . + . − . . + . − . + . − N . . .
38 53 . + . − . N . . .
20 122 + . − . N . . .
27 1524 . + . − . N . . .
24 8830 . + . − . N E . . . K - T r N . . . . . + . − . . + . − . . + . − . N . . .
37 37 . + . − . N E . . . up s A nd R VN . . . . . + . − . . + . − . . + . − . N . . .
34 9096 . + . − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
34 39 . + . − . N E . . . K e p l e r - T r Y . . . . . + . − . . + . − . . + − Y . . .
25 62 . + − . Y E . . . K O I - T r Y . . . . . + . − . . + . − . . + . − . Y . . .
25 35 . + − . Y . . .
40 499 . + . − . Y E , H . . . H D R VN . . . . . + . − . . + . − . . + . − . N . . .
22 110 . + . − . N . . .
05 14576 . + − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
22 73 . + . − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
17 58 . + − N . . .
94 258 . + . − . N E . . . Mousavi-Sadr, Gozaliasl, and Jassur T a b l e c o n t i nu e d H o s t n a m e D i s . a F ( χ d o f ) i ( χ d o f ) j γ ∆ γ b m b P e r i o d ( d a y s ) F O N c R M a x ( R ⊕ ) M M a x ( M ⊕ ) P t r ( % ) K e p l e r - T r Y . . . . . + . − . . + . − . . + . − . Y . . .
03 25 . + . − . Y . . .
27 32 . + . − . Y . . .
91 62 . + . − . Y E . . . K e p l e r - T r N . . . . . + . − . . + . − . + . − . N . . .
75 58 . + . − . N E . . . G J C R VN . . . . . + . − . . + . − . . + . − . N . . .
75 17 . + . − . N . . .
12 106 . + . − . N . . .
64 174 . + . − . N . . .
46 387 . + . − N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
70 35 . + . − . N . . .
02 147 + . − . N E . . . V T a u T r N . . . . . + . − . . + . − . . + . − . N . . .
69 109 . + . − . N E . . . P S R B + P T N . . . . . + . − . . + . − . . + . − . N -- .
50 159 . + . − . N E -- . G J R VN . . . . . + . − . . + . − . . + . − . N . . .
47 20 . + . − . N E . . . K e p l e r - T r Y . . . . . + . − . . + . − . . + − . Y . . .
44 34 . + . − . Y . . .
75 55 . + . − . Y . . .
26 136 . + . − . Y E , H . . . H I P R VN . . . . . + . − . . + . − . + . − . N . . .
44 4568 . + . − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
39 44 . + . − . N . . .
01 100 . + . − N E . . . K e p l e r - T r N . . . . . + . − . . + . − . + . − . N . . .
37 212 . + . − . N E . . . K e p l e r - T r Y . . . . . + . − . . + . − . . + . − . N . . .
34 10 . + . − . Y . . .
82 16 . + . − . N . . .
61 127 . + . − . Y E . . . H D R VN . . . . . + . − . . + . − . . + . − . N . . .
27 84 . + . − . N . . .
14 261 . + . − . N H . . .
53 8240 . + . − . N E . . . rediction of Exoplanets T a b l e c o n t i nu e d H o s t n a m e D i s . a F ( χ d o f ) i ( χ d o f ) j γ ∆ γ b m b P e r i o d ( d a y s ) F O N c R M a x ( R ⊕ ) M M a x ( M ⊕ ) P t r ( % ) W A S P - T r N . . . . . + . − . − . + . − . . + . − . N . . .
14 161 . + . − . N . . .
89 2179 . + . − . N E . - . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
05 208 . + . − . N E . . . G J R VN . . . . . + . − . . + . − . . + . − N H . . .
05 214 . + . − . N . . .
99 1778 . + . − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
84 30 . + . − . N E . . . K e p l e r - T r Y . . . . . + . − . . + . − . . + . − . Y . . .
43 170 . + . − . Y E . . . G J R VN . . . . . + . − . . + . − . . + . − . N H . . .
42 133 . + . − N E . . . K e p l e r - T r Y . . . . . + . − . . + . − . . + . − . Y . . .
39 148 . + . − . N . . .
80 298 . + . − . N H . . .
50 1336 . + . − . N E . . . K e p l e r - T r Y . . . . . + . − . . + . − . . + − . Y . . .
34 71 . + . − . Y . . .
89 465 + . − . Y E . . . K e p l e r - T r Y . . . . . + . − . . + . − . . + . − . Y H . . .
26 73 . + . − . Y H . . .
85 234 . + . − . Y E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
25 119 . + . − . N . . .
92 469 . + . − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
22 157 + . − . N H . . .
64 373 . + . − . N H . . .
36 2511 . + . − . N E . . . H I P R VN . . . . . + . − . . + . − . . + − . N . . .
16 1665 . + . − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
13 285 . + . − . N E , H . . . G J R VN . . . . . + . − . . + . − . + . − . N H . . .
12 1470 . + . − . N E . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
03 223 + . − . N E , H . . . K e p l e r - T r N . . . . . + . − . . + . − . . + . − . N . . .
93 333 . + . − . N H . . .
45 2952 . + . − N E . . . Mousavi-Sadr, Gozaliasl, and Jassur T a b l e c o n t i nu e d H o s t n a m e D i s . a F ( χ d o f ) i ( χ d o f ) j γ ∆ γ b m b P e r i o d ( d a y s ) F O N c R M a x ( R ⊕ ) M M a x ( M ⊕ ) P t r ( % ) H D R VY . . . . . + . − . . + . − . + . − . Y H . . .
87 472 + . − . N E . . . H D R VN . . . . . + . − . . + . − . . + . − . N . . .
80 288 . + . − . N . . .
41 780 . + . − . N . . .
21 1695 . + − N . . .
12 12768 . + . − . N E . . . W o l f R VN . . . . . + . − . . + . − . . + . − . N H . . .
70 773 . + . − . N E . . . H D R VN . . . . . + . − . . + . − . . + . − . N H . . .
55 2343 . + . − . N . . .
19 11999 . + . − . N E . . . H D R VN . . . . . + . − . . + . − . + . − . N . . .
45 4578 + . − . N E . . . H D R VN . . . . . + . − . . + . − . . + . − . N H . . .
42 845 . + . − . N . . .
22 5246 . + . − . N E . . . H D R VN . . . . . + . − . . + . − . . + . − . N . . .
16 12242 . + − . N E . . .
06 47 U m a R VN . . . . . + . − . . + . − . . + . − . N . . .
09 26359 . + . − . N E . . . a D i s c o v e r y m e t h o d o f t h e s y s t e m :’ T r ’,’ R V ’,’ O B M ’, a nd ’ P T ’ r e p r e s e n tt r a n s i t , r a d i a l v e l o c i t y , o r b i t a l b r i g h t n e ss m o du l a t i o n , a ndpu l s a r t i m i n g , r e s p ec t i v e l y . b ∆ γ = ( γ - γ ) / γ , w h e r e γ a nd γ a r e t h e h i g h e s t a nd s ec o nd - h i g h e s t γ v a l u e s f o r t h e s y s t e m , r e s p ec t i v e l y . c O r b i t a l nu m b e r s ( O N ) f o ll o w e db y ’ E ’i nd i c a t e t h ee x t r a p o l a t e dp l a n e t s , f o ll o w e db y ’ H ’i nd i c a t e t h e p r e d i c t e dp l a n e t s w i t h i n t h e H Z , a nd f o ll o w e db y ’ C ’i nd i c a t e t h a t c o rr e s p o nd i n g o r b i t a l p e r i o d s h a v e b ee nfl agg e d a s " P l a n e t a r y C a nd i d a t e " i n NA S A E x o p l a n e t A r c h i v e . rediction of Exoplanets Table 5
Systems with detected planets since predictions made by BL15.
Host name Detected period(days) BL15 predicted period(days) Our predicted period(days) BL15 error(%) Our error(%)Kepler-1388 75.73 73 . ± . . +14 . − . . ± . . +0 . − . . ± . . +5 . − . . ± . . +0 . − . . ± . . +1 . − . Mousavi-Sadr, Gozaliasl, and Jassur