Using the inclinations of Kepler systems to prioritize …...Titius–Bode-based exoplanet predictions T. Bovaird, 1 ,2‹ C. H. Lineweaver 3 and S. K. Jacobsen4 1Research School of

MNRAS 448 3608ndash3627 (2015) doi101093mnrasstv221

Using the inclinations of Kepler systems to prioritize newTitiusndashBode-based exoplanet predictions

T Bovaird12lsaquo C H Lineweaver123 and S K Jacobsen4

1Research School of Astronomy and Astrophysics Australian National University Canberra ACT 2611 Australia2Planetary Science Institute Australian National University Canberra ACT 2601 Australia3Research School of Earth Sciences Australian National University Canberra ACT 2601 Australia4Niels Bohr Institute University of Copenhagen Copenhagen 1017 Denmark

Accepted 2015 January 30 Received 2015 January 30 in original form 2014 December 19

ABSTRACTWe analyse a sample of multiple-exoplanet systems which contain at least three transitingplanets detected by the Kepler mission (lsquoKepler multiplesrsquo) We use a generalized TitiusndashBode relation to predict the periods of 228 additional planets in 151 of these Kepler multiplesThese TitiusndashBode-based predictions suggest that there are on average 2 plusmn 1 planets in thehabitable zone of each star We estimate the inclination of the invariable plane for each systemand prioritize our planet predictions by their geometric probability to transit We highlight ashort list of 77 predicted planets in 40 systems with a high geometric probability to transitresulting in an expected detection rate of sim15 per cent sim3 times higher than the detection rateof our previous TitiusndashBode-based predictions

Key words planets and satellites detection ndash planets and satellites formation ndash planets andsatellites terrestrial planets ndash planet-disc interactions

1 IN T RO D U C T I O N

The TitiusndashBode (TB) relationrsquos successful prediction of the periodof Uranus was the main motivation that led to the search for anotherplanet between Mars and Jupiter eg Jaki (1972) This search ledto the discovery of the asteroid Ceres and the rest of the asteroidbelt The TB relation may also provide useful hints about the pe-riods of as-yet-undetected planets around other stars In Bovairdamp Lineweaver (2013 hereafter BL13) we used a generalized TBrelation to analyse 68 multiplanet systems with four or more de-tected exoplanets We predicted the existence of 141 new exoplanetsin these 68 systems Huang amp Bakos (2014 hereafter HB14) per-formed an extensive search in the Kepler data for 97 of our predictedplanets in 56 systems This resulted in the confirmation of five ofour predictions (Fig 4 and Table 1)

In this paper we perform an improved TB analysis on a largersample of Kepler multiple-planet systems1 to make new exoplanetorbital period predictions We use the expected coplanarity ofmultiple-planet systems to estimate the most likely inclination ofthe invariable plane of each system We then prioritize our originaland new TB-based predictions according to their geometric prob-ability of transiting Comparison of our original predictions withthe HB14 confirmations shows that restricting our predictions to

E-mail timothybovairdanueduau1 Accessed on 2014 November 4 httpexoplanetarchiveipaccaltecheducgi-binTblViewnph-tblViewapp=ExoTblsampconfig=cumulative

those with a high geometric probability to transit should increasethe detection rate by a factor of sim3 (Fig 8)

As in BL13 our sample includes all Kepler multiplanet systemswith four or more exoplanets but to these we add three-planetsystems if the orbital periods of the systemrsquos planets adhere betterto the TB relation than the Solar system (equation 4 of BL13) Usingthese criteria we add 77 three-planet systems to the 74 systems withfour or more planets We have excluded three systems KOI-284KOI-2248 and KOI-3444 because of concerns about false positivesdue to adjacent-planet period ratios close to 1 and close binary hosts(Lissauer et al 2011 Fabrycky et al 2014 Lillo-Box Barrado ampBouy 2014) We have also excluded the three-planet system KOI-593 since the period of KOI-59303 was recently revised excludingthe system from our three-planet sample Thus we analyse 151Kepler multiples with each system containing 3 4 5 or 6 planets

11 Coplanarity of exoplanet systems

Planets in the Solar system and in exoplanetary systems are believedto form from protoplanetary discs (eg Winn amp Fabrycky 2014) Theinclinations of the eight planets of our Solar system to the invariableplane are (in order from Mercury to Neptune) 63 22 16 1703 09 10 07 (Souami amp Souchay 2012) Jupiter and Saturncontribute sim86 per cent of the total planetary angular momentumand thus the angles between their orbital planes and the invariableplane are small 03 and 09 respectively

In a given multiple-planet system the distribution of mutual in-clinations between the orbital planes of planets is well described by

Ccopy 2015 The AuthorsPublished by Oxford University Press on behalf of the Royal Astronomical Society

at Australian N

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Invariable planes of Kepler systems 3609

Table 1 Systems with candidate detections by HB14 (in bold) plusKOI-1151 (Petigura Howard amp Marcy 2013a) and KOI-1860b afterplanet predictions were made by BL13

System Predicted Detected Predicted Detectedperiod (d) period (d) radius (Roplus) radius (Roplus)

KOI-719 14 plusmn 2 1577 le 07 042KOI-1336 26 plusmn 3 2751 le 24 104KOI-1952 13 plusmn 2 1327 le 15 085KOI-2722a 168 plusmn 10 1653 le 16 116KOI-2859 52 plusmn 03 543 le 08 076KOI-733 NA 1511 NA 30KOI-1151a 96 plusmn 07 1043 le 08 07KOI-1860b 25 plusmn 3 2484 le 27 146

Notes aPredicted by preprint of BL13 (draft uploaded 2013 Apr 11httparxivorgpdf13043341v1pdf) detected planet reported by Ke-pler archive and included in analysis of BL13b2014 October Kepler Archive update during the drafting of this paper

a Rayleigh distribution (Lissauer et al 2011 Fang amp Margot 2012Figueira et al 2012 Ballard amp Johnson 2014 Fabrycky et al 2014)For the ensemble of Kepler multiplanet systems the mode of theRayleigh distribution of mutual inclinations (φj minus φi) is typicallysim1ndash3 (Appendix A1 and Table A1) Thus Kepler multiple-planetsystems are highly coplanar The Solar system is similarly coplanarFor example the mode of the best-fitting Rayleigh distribution ofthe planet inclinations (φj) relative to the invariable plane in theSolar system is sim1 (see Figs 1 and 2)2

The angle θ j is a Gaussian distributed variable with a mean of 0(centred around 〈θ〉) and standard deviation σθ Based on previousanalyses (Table A1) we assume the typical value σθ = 15 Weuse this angle to determine the probability of detecting additionaltransiting planets in each system

Estimates of the inclination of a transiting planet come from theimpact parameter b which is the projected distance between thecentre of the planet at mid-transit and the centre of the star in unitsof the starrsquos radius

b = a

Rlowastcos i (1)

where Rlowast is the radius of the star and a is the semimajor axis of theplanet For edge-on systems typically 85 lt i lt 95 However sincewe are unable to determine whether b is in the positive z directionor the negative z direction (Figs 1b and B1) we are unable todetermine whether i is greater than or less than 90 By conventionfor transiting planets the sign of b is taken as positive and thus thecorresponding i values from equation (1) are taken as i le 90

The impact parameter is also a function of four transit light-curve observables (Seager amp Mallen-Ornelas 2003) the period Pthe transit depth F the total transit duration tT and the total transitduration minus the ingress and egress times tF (the duration wherethe light curve is flat for a source uniform across its disc) Thus theimpact parameter can be written as

b = f (P F tT tF) (2)

2 See Appendix A for an explanation of why the distribution of mutualinclinations is on average a factor of

radic2 wider than the distribution of the

angles φj in Fig 1 between the invariable plane of the system and the orbitalplanes of the planets

Eliminating b from equations (1) and (2) yields the inclination i asa function of observables

i = cosminus1

[Rlowasta

f (P F tT tF)

] (3)

From equation (1) we can see that for an impact parameter b = 0(a transit through the centre of the star) we obtain i = 90 anlsquoedge-onrsquo transit

The convention i le 90 is unproblematic when only a singleplanet is found to transit a star but raises an issue when multipleexoplanets transit the same star since the degree of coplanaritydepends on whether the actual values of ij (j = 1 2 N where Nis the number of planets in the system) are greater than or less than90 For example the actual values of ij in a given system could beall gt90 all lt90 or some in-between combination Although wedo not know the signs of θ j = 90 minus ij for individual planets wecan estimate the inclination of the invariable plane for each systemby calculating all possible permutations of the θ j values for eachsystem (see Appendix A2) In this estimation we use the plausibleassumption that the coplanarity of a system should not depend onthe inclination of the invariable plane relative to the observer

12 The probability of additional transiting exoplanets

We wish to develop a measure of the likelihood of additional transit-ing planets in our sample of Kepler multiplanet systems The moreedge-on a planetary system is to an observer on Earth the greater theprobability of a planet transiting at larger periods Similarly a largerstellar radius leads to a higher probability of additional transitingplanets (although with a reduced detection efficiency) We quantifythese tendencies under the assumption that Kepler multiples havea Gaussian opening angle σθ = 15 around the invariable planeand we introduce the variable acrit Planets with a semimajor axisgreater than acrit have less than a 50 per cent geometric probabilityof transiting More specifically acrit is defined as the semimajor axiswhere Ptrans(acrit) = 05 (equation B1) for a given system

In a given system a useful ratio for estimating the amount ofsemimajor axis space where additional transiting planets are morelikely is acritaout where aout is the semimajor axis of the detectedplanet in the system which is the furthest from the host star Thelarger acritaout the larger the semimajor axis range for additionaltransiting planets beyond the outermost detected planet Values forthis ratio less than 1 mean that the outermost detected planet isbeyond the calculated acrit value and imply that additional transitingplanets beyond the outermost detected planet are less likely Fig 3shows the acritaout distribution for all systems in our sample Thefact that this distribution is roughly symmetric around acritaout sim 1strongly suggests that the outermost transiting planets in Keplersystems are due to the inclination of the system to the observer andare not really the outermost planets

In Section 2 we discuss the follow-up that has been done on ourBL13 planet detections In Section 3 we show that the sim5 per centfollow-up detection rate of HB14 is consistent with selection effectsand the existence of the predicted planets In Section 4 we extendand upgrade the TB relation developed in BL13 and predict the peri-ods of undetected planets in our updated sample We then prioritizethese predictions based on their geometric probability to transit andemphasize for further follow-up a subset of predictions with hightransit probabilities We also use TB predictions to estimate the av-erage number of planets in the circumstellar habitable zone (HZ) InSection 5 we discuss how our predicted planet insertions affect theperiod ratios of adjacent planets and explore how period ratios are

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3610 T Bovaird C H Lineweaver and S K Jacobsen

Figure 1 Panel (a) our coordinate system for transiting exoplanets The x-axis points towards the observer Lj is the 3D angular momentum of the jth planetand is perpendicular to the orbital plane of the jth planet 〈L〉 is the sum of the angular momenta of all detected planets (equation A2) and is perpendicular tothe invariable plane of the system We have chosen the coordinate system without loss of generality such that 〈L〉 has no component in the y direction φj isthe angle between Lj and 〈L〉 Let Lj be the projection of Lj on to the xndashz plane θ j is the angle between 〈L〉 and Lj 〈θ〉 is the angle between the z-axisand 〈L〉 ij is the inclination of the planet (equation 3) θ j = 90 minus ij and is the angle between the z-axis and Lj such that θ j = 〈θ〉 + θ j (equation A4)Panel (b) shows the xndashz plane of panel (a) with the y-axis pointing into the paper The observer is to the right The grey shaded region represents the lsquotransitregionrsquo where the centre of a planet will transit its host star as seen by the observer (impact parameter b le 1 see equation 1) Four planets b c d and j arerepresented by blue dots The intersection of the orbital plane of the jth planet with the xndashz plane is shown (thin black line) All angles shown in panel (a) (withthe exception of φj) are also shown in panel (b) The thick line is the intersection of the invariable plane of the system with the xndashz plane Because we are onlydealing with systems with multiple transiting planets all these angles are typically less than a few degrees but are exaggerated here for clarity

Figure 2 The coplanarity of planets in the Solar system relative to theinvariable plane With the exception of Mercury the angles between theorbital planes of the planets and the invariable plane are well represented bya Rayleigh distribution with a mode of sim1

tightly dispersed around the mean period ratio within each systemIn Section 6 we summarize our results

2 FO L L OW-U P O F B L 1 3 P R E D I C T I O N S

BL13 used the approximately even logarithmic spacing of exo-planet systems to make predictions for the periods of 117 additionalcandidate planets in 60 Kepler detected systems and 24 additionalpredictions in 8 systems detected via radial velocity (7) and di-rect imaging (1) which we do not consider here NASA ExoplanetArchive data updates confirmed our prediction of KOI-272205(Table 1)

HB14 used the planet predictions made in BL13 to search for97 planets in the light curves of 56 Kepler systems Within these56 systems BL13 predicted the period and maximum radius thelargest radius which would have evaded detection based on thelowest signal-to-noise of the detected planets in the same system

Figure 3 Histogram of the acritaout values for our sample of Kepler mul-tiples The distribution peaks for a ratio just below 1 Approximately half ofthe systems lie to the right of 1 For these systems if there are planets withsemimajor axes a such that aout lt a lt acrit then the geometric probabilityof them transiting is greater than 50 per cent Note that this does not accountfor the detectability of these planets (eg they could be too small to detect)The majority of the predicted planets that are insertions (a lt aout) havegeometric transit probabilities greater than 50 per cent when acritaout lt 1Systems on the right have more room for detections and in general pre-dicted planets in these systems have higher values of Ptrans The blue curveis the expected distribution of our sample of Kepler multiples if they all haveplanets at TB-predicted semimajor axes extrapolated out to sim4 times acrit Theblue curve is consistent with the observed distribution indicating that ouracritaout distribution is consistent with the system in our sample containingmore planets than have been detected

Predicted planets were searched for using the Kepler Quarter 1 toQuarter 15 long-cadence light curves giving a baseline exceeding1000 d Once the transits of the already known planets were detectedand removed transit signals were visually inspected around thepredicted periods

Of the 97 predicted planets searched for by HB14 5 candidateswere detected within sim1σ of the predicted periods (5 planets of the

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Figure 4 Exoplanet systems where an additional candidate was detected after a TB relation prediction was made (see Table 1) The systems are shown indescending order of acritaout Previously known planets are shown as blue circles The predictions of BL13 and their uncertainties are shown by the red filledrectangles if the Ptrans value of the predicted planet is ge 055 (equation B1 and Fig 8) or by red hatched rectangles otherwise The new candidate planets areshown as green squares The critical semimajor axis acrit (Section 12) beyond which Ptrans(acrit) lt 05 is shown by a solid black arc

Figure 5 Same as Fig 4 except here our TB predictions are based on γ with n2ins (equation 9) rather than on the γ with nins of equation 5 of BL13 Comparing

Fig 4 with this figure in the KOI-719 system the number of predicted planets goes from 4 to 2 while in KOI-1151 the number of predicted planets goes from3 to 2 In both cases the detected planet is more centrally located in the predicted region The uncertainties (width of red rectangles) in Fig 4 and Table 1 areslightly wider than Fig 5 due to excessive rounding of predicted period uncertainties for some systems in BL13

6 planets in bold in Table 1 see also Fig 4) Notably all new planetcandidates have Earth-like or lower planetary radii One additionalcandidate was detected in KOI-733 which is incompatible with thepredictions of BL13 This candidate is unique in that it should

have been detected previously based on the signal to noise of theother detected planets in KOI-733 In Table 1 the detected radiiare less than the maximum predicted radii in each case The newcandidate in KOI-733 has a period of 1511 d and a radius of

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Figure 6 Reported dip significance parameter (DSP) from Table 1 of HB14for previously known exoplanets in the five systems with a new detectionA linear trend can be seen for the DSP and the SNR as reported by theKepler team (Christiansen et al 2012) HB14 required planet candidatesto have a DSP gt 8 to survive their vetting process The sizes of the bluedots correspond to the same planetary radii representation used to makeFigs 11ndash14

3 Roplus At this period the maximum radius to evade detection shouldhave been 22 Roplus With the possible exception of KOI-1336 wherea dip significance parameter (DSP Kovacs amp Bakos 2005) was notreported all detected candidates have a DSP of ge 8 which roughlycorresponds to a Kepler signal-to-noise ratio (SNR Christiansenet al 2012) of 12 (see Fig 6) HB14 required DSP gt 8 forcandidate transit signals to survive their vetting process

3 IS A 5 PE R C E N T D E T E C T I O N R AT ECONSISTENT W IT H SE LE C TIO N E F F ECTS

From a sample of 97 BL13 predictions HB14 confirmed 5 How-ever based on this sim5 per cent detection rate HB14 concluded thatthe predictive power of the TB relation used in BL13 was question-able Given the selection effects how high a detection rate shouldone expect We do not expect all planet predictions to be detectedThe predicted planets may have too large an inclination to transitrelative to the observer Additionally there is a completeness fac-tor due to the intrinsic noise of the stars the size of the planetsand the techniques for detection This completeness for Kepler datahas been estimated for the automated light-curve analysis pipelineTERRA (Petigura et al 2013a) Fig 7 displays the TERRA pipelineinjectionrecovery completeness After correcting for the radius andnoise of each star relative to the TERRA sample in Fig 7 the planetdetections in Table 1 have an average detection completeness in theTERRA pipeline of sim24 per cent That is if all of our predictionswere correct and if all the planets were in approximately the sameregion of period and radius space as the green squares in Fig 7and if all of the planets transited we would expect a detection rateof sim24 per cent using the TERRA pipeline It is unclear how thistranslates into a detection rate for a manual investigation of the lightcurves motivated by TB predictions

We wish to determine from coplanarity and detectability ar-guments how many of our BL13 predictions we would have ex-pected to be detected An absolute number of expected detectionsis most limited by the poorly known planetary radius distributionbelow 1 Roplus (Howard et al 2012 Dong amp Zhu 2013 Dressing ampCharbonneau 2013 Fressin et al 2013 Petigura Marcy amp Howard2013b Foreman-Mackey Hogg amp Morton 2014 Morton amp Swift2014 Silburt Gaidos amp Wu 2015) Large uncertainties about theshape and amplitude of the planetary radius distribution of rockyplanets with radii less than 1 Roplus make the evaluation of TB-basedexoplanet predictions difficult Since the TB relation predicted theasteroid belt (Masteroid lt 10minus3MEarth) there seems to be no lowermass limit to the objects that the TB relation can predict Thismakes estimation of the detection efficiencies strongly dependenton assumptions about the frequency of planets at small radii

Figure 7 The simulated detection completeness of the new candidate plan-ets in the TERRA pipeline (modified from fig 1 of Petigura et al 2013a)Here we overplot as green squares the eight planets listed in Table 1 Thecompleteness curves are averaged over all stellar noise and stellar radii in thePetigura et al (2013a) sample (the 42 000 least noisy Kepler stars) Greencircles indicate the lsquoeffective radiusrsquo of the new candidates based on thenoise and radius of their host star in comparison to the median of the quietest42 000 sample From the signal to noise the effective radius can be calculatedby Rp effRp = (RlowastR median) times (CDPPCDPPmedian)12 where CDPP isthe combined differential photometric precision defined in Christiansen et al(2012) Taking a subset of 42 000 stars from the Kepler input catalogue withthe lowest 3-h CDPP (approximately representative of the sample in thefigure) we obtain CDPPmedian asymp 60 ppm and R median asymp 115 R13 Usingthe effective radius and excluding the outlier KOI-733 the mean detectioncompleteness for the seven candidate planets in Table 1 is sim24 per cent

Let the probability of detecting a planet Pdetect be the productof the geometric probability to transit Ptrans as seen by the observer(Appendix B) and the probability PSNR that the planetary radius islarge enough to produce an SNR above the detection threshold

Pdetect = Ptrans PSNR (4)

The geometric probability to transit Ptrans is defined in equation(B1) and illustrated in Figs 1 and B1 The five confirmations fromour previous TB predictions are found in systems with a muchhigher than random probability of transit (Fig 8) This is expectedif our estimates of the invariable plane are reasonable

To estimate PSNR we first estimate the probability that the radiusof the planet will be large enough to detect In BL13 we estimatedthe maximum planetary radius Rmax for a hypothetical undetectedplanet at a given period based on the lowest signal-to-noise ofthe detected planets in the same system We now wish to estimatea minimum radius that would be detectable given the individualnoise of each star We refer to this parameter as Rmin which is theminimum planetary radius that Kepler could detect around a givenstar (using a specific SNR threshold) For each star we used themean CDPP (combined differential photometric precision) noisefrom Q1ndashQ16 When the number of transits is not reported we usethe approximation Ntrans asymp Tobsf0P where Tobs is the total observingtime and f0 is the fractional observing uptime estimated at sim092for the Kepler mission (Christiansen et al 2012)

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Figure 8 Histogram of Ptrans (equation B1) the geometric probability oftransit for the 97 predicted planets from BL13 that were followed up byHB14 The blue histogram represents the five new planets detected by HB14(Table 1) As expected the detected planets have high Ptrans values comparedto the entire sample The red and blue solid lines represent the empiricalcumulative distribution function for the two distributions A KndashS test of thetwo distributions yields a p-value of 18 times 10minus2 Thus Ptrans can be used toprioritize our predictions and increase their probability of detection

The probability PSNR depends on the underlying planetary radiusprobability density function We assume a density function of theform

f (R) = df

dlog R=

k(log R)α R ge 28 Roplusk(log 28)α R lt 28 Roplus

(5)

where k = 29 and α = minus192 (Howard et al 2012) The discon-tinuous distribution accounts for the approximately flat number ofplanets per star in logarithmic planetary radius bins for R 28 Roplus(Dong amp Zhu 2013 Fressin et al 2013 Petigura et al 2013b Silburtet al 2015) For R 10 Roplus the distribution is poorly constrainedFor this paper we extend the flat distribution in log R down to aminimum radius Rlow = 03 Roplus It is important to note that for theSolar system the poorly constrained part of the planetary radiusdistribution contains 50 per cent of the planet population For ref-erence the radius of Ceres a lsquoplanetrsquo predicted by the TB relationapplied to our Solar system has a radius RCeres = 476 km = 007Roplus

The probability that the hypothetical planet has a radius thatexceeds the SNR detection threshold is then given by

PSNR =int Rmax

Rminf (R)dRint Rmax

Rlowf (R)dR

(6)

We do not integrate beyond Rmax since we expect a planet witha radius greater than Rmax would have already been detected Wedefine Rmax by

Rmax = RminSNR

(Ppredict

PminSNR

)14

(7)

where RminSNR and PminSNR are the radius and period respectivelyof the detected planet with the lowest signal-to-noise in the systemPpredict is the period of the predicted planet Rmin depends on theSNR in the following way

Rmin = Rlowastradic

SNRth CDPP

(3 h

ntr tT

)14

(8)

where SNRth is the SNR threshold for a planet detection ntr is thenumber of expected transits at the given period and tT is the transitduration in hours See Fig 9 for an illustration of how the integralsin PSNR (equation 6) depend on the planet radii limits Rmin and Rlow

While Ptrans is well defined PSNR is dependent on the SNR thresh-old chosen (SNRth) the choice of Rlow and the poorly constrained

Figure 9 The assumed distribution of planetary radii described in Section 3The distribution is poorly constrained below 1 Roplus indicated by the greydashed line For low-mass stars the planetary radius distribution may de-cline below 07 Roplus (Dressing amp Charbonneau 2013 Morton amp Swift 2014)Alternative estimations show the planetary radius distribution continuingto increase with smaller radii (continuing the flat logarithmic distribution)down to 05 Roplus (Foreman-Mackey et al 2014) For our analysis we haveextrapolated the flat distribution (in log R) down to Rlow We indicate threeregions for a hypothetical system at a specific predicted period The lsquoal-ready detectedrsquo region refers to the range of planetary radii which shouldalready have been detected based on the lowest SNR of the detected planetsin that system Rmin is the smallest radius which could produce a transitsignal that exceeds the detection threshold and is the boundary between theundetectable and detectable regions

Figure 10 The mean detection rate Pdetect (equation 4) of the BL13 pre-dictions is dependent on the transit signal-to-noise threshold (SNRth inequation 8) used in light-curve vetting (x-axis) and on the probability oflow-radii planets ie on how far the flat logarithmic planetary radius distri-bution in Fig 9 should be extrapolated (Rlow on the y-axis) For example inthe denominator of equation (6) integrating down to a radius Rlow of 06 Roplusand setting an SNR threshold of 7 (which sets Rmin in the numerator) givesan expected detection fraction of sim25 per cent (blue values in the upperleft of the plot) Integrating down to a radius of 02 Roplus and having a DSPthreshold of 8 (converted from SNR according to Fig 6) gives an expecteddetection fraction of sim5 per cent (red values in the lower right)

shape of the planetary radius distribution below 1 Roplus This isdemonstrated in Fig 10 where the mean Pdetect from the predic-tions of BL13 (for DSP gt 8) can vary from sim2 to sim11 per centPerforming a KndashS test on PSNR values (analogous to that in Fig 8)indicates that the PSNR values for the subset of our BL13 predictionsthat were detected are drawn from the same PSNR distribution asall of the predicted planets For this reason we use only Ptrans thegeometric probability to transit to prioritize our new TB relationpredictions We emphasize a subset of our predictions which have

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a Ptrans value ge 055 since all of the confirmed predictions of BL13had a Ptrans value above this threshold Only sim13 of the entiresample have Ptrans values this high Thus the sim5 per cent detectionrate should increase by a factor of sim3 to sim15 per cent for our newhigh-Ptrans subset of planet period predictions

4 U P DAT E D P L A N E T PR E D I C T I O N S

41 Method and inclination prioritization

We now make updated and new TB relation predictions in all 151systems in our sample If the detected planets in a system adhere tothe TB relation better than the Solar system planets (χ2dof lt 10equation 4 of BL13) we only predict an extrapolated planet be-yond the outermost detected planet If the detected planets adhereworse than the Solar system we simulate the insertion of up to ninehypothetical planets into the system covering all possible locationsand combinations and calculate a new χ2dof value for eachpossibility We determine how many planets to insert and whereto insert them based on the solution which improves the systemrsquosadherence to the TB relation scaled by the number of inserted plan-ets squared This protects against overfitting (inserting too manyplanets resulting in too good a fit) In equation 5 of BL13 we intro-duced a parameter γ which is a measure of the fractional amountby which the χ2dof improves divided by the number of planetsinserted Here we improve the definition of γ by dividing by thesquare of the number of planets inserted

γ =

(χ2

i minusχ2f

χ2f

)n2

ins

(9)

where χ2i and χ2

f are the χ2 of the TB relation fit before and afterplanets are inserted respectively while nins is the number of insertedplanets

Importantly when we calculate our γ value by dividing by thenumber of inserted planets squared rather than the number of plan-ets we still predict the BL13 predictions that have been detectedIn two of these systems fewer planets are predicted and as a resultthe new predictions agree better with the location of the detectedcandidates This can be seen by comparing Figs 4 and 5

We compute Ptrans for each planet prediction in our sample of151 Kepler systems We emphasize the 40 systems where at leastone inserted planet in that system has Ptrans ge 055 Period predic-tions for this subset of 40 systems are displayed in Table C11 andFigs 11ndash13 As discussed in the previous section we expect a de-tection rate of sim15 per cent for this high-Ptrans sample Predictionsfor all 228 planets (regardless of their Ptrans value) are shown inTable C2 2 (where the systems are ordered by the maximum Ptrans

value in each system)

42 Average number of planets in circumstellar HZ

Since the search for earth-sized rocky planets in circumstellar HZ isof particular importance in Fig 14 for a subset of Kepler multipleswhose predicted (extrapolated) planets extend to the HZ we haveconverted the semimajor axes of detected and predicted planets intoeffective temperatures (as in fig 6 of BL13) One can see in Fig 14that the HZ (shaded green) contains between 0 and 4 planets Thusif the TB relation is approximately correct and if Kepler multiplanetsystems are representative of planetary systems in general there areon average sim2 HZ planets per star

More specifically in Table 2 we estimate the number of planetsper star in various lsquoHZrsquo namely (1) the range of Teff between Marsand Venus (assuming an albedo of 03) displayed in Fig 14 as thegreen shaded region (2) the Kopparapu et al (2013) lsquooptimisticrsquo and(3) lsquoconservativersquo HZ (lsquorecent Venusrsquo to lsquoearly Marsrsquo and lsquorunawaygreenhousersquo to lsquomaximum greenhousersquo respectively) We find onaverage 2 plusmn 1 planets per star in the lsquoHZrsquo almost independently ofwhich of the three HZs one is referring to Using our estimates of themaximum radii for these predominantly undetected (but predicted)planets as well as the planetary radius distribution of Fig 9 weestimate that on average sim16 of these sim2 planets or sim03 arelsquorockyrsquo We have assumed that planets with R le 15 Roplus are rocky(Rogers 2014 Wolfgang amp Lopez 2014)

5 A D JAC E N T PL A N E T PE R I O D R AT I O S

HB14 concluded that the percentage of detected planets(sim5 per cent) was on the lower side of their expected range (sim5ndash20 per cent) and that the TB relation may overpredict planet pairsnear the 32 mean-motion resonance (compared to systems whichadhered to the TB relation better than the Solar system without anyplanet insertions ie χ2dof le 1) There is some evidence that apeak in the distribution of period ratios around the 32 resonance isto be expected from Kepler data after correcting for incompleteness(Steffen amp Hwang 2014) In this section we investigate the periodratios of adjacent planets in our Kepler multiples before and afterour new TB relation predictions are made

We divide our sample of Kepler multiples into a number of sub-sets Our first subset includes systems which adhere to the TBrelation better than the Solar system (where we only predict an ex-trapolated planet beyond the outermost detected planet) Systemswhich adhere to the TB relation worse than the Solar system wedivide into two subsets before and after the planets predicted bythe TB relation were inserted Adjacent planet period ratios can bemisleading if there is an undiscovered planet between two detectedplanets which would reduce the period ratios if it was includedin the data To minimize this incompleteness we also construct asubset of systems which are the most likely to be completely sam-pled (unlikely to contain any additional transiting planets within therange of the detected planet periods)

Systems which adhere to the TB relation better than the Solarsystem (χ2dof le 1) were considered by HB14 as being the sampleof planetary systems that were most complete and therefore had adistribution of adjacent planet period ratios most representative ofactual planetary systems However the choice of BL13 to normalizethe TB relation to the Solar systemrsquos χ2dof is somewhat arbitraryThe Solar systemrsquos χ2dof is possibly too high to consider all thosewith smaller values of χ2dof to be completely sampled

We want to find a set of systems which are unlikely to hostany additional planets between adjacent pairs due to the systembeing dynamically full (Hayes amp Tremaine 1998) We do this byidentifying the systems where two or more sequential planet pairsare likely to be unstable when a massless test particle is insertedbetween each planet pair (dynamical spacing lt 10 Gladman1993 BL13)

The dynamical spacing is an estimate of the stability of adja-cent planets If inserting a test particle between a detected planetpair results in either of the two new values being less than 10we consider the planet pair without the insertion to be completeThat is there is unlikely to be room between the detected planetpair where an undetected planet could exist without making theplanet pair dynamically unstable Therefore since the existence of

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Figure 11 The architectures and invariable plane inclinations for Kepler systems in our sample which contain at least one planet with a geometric probabilityto transit Ptrans ge 055 There are 40 such systems out of the 151 in our sample The 14 with the highest Ptrans values are plotted here The remaining 26 areplotted in the next two figures The order of the systems (from top to bottom) is determined by the highest Ptrans value in each system The thin horizontal dottedline represents the line of sight to Earth ie where the i value of a planet would be 90 The thick grey line in each system is our estimate of the invariant planeangle 〈θ〉 (Appendix A2) The value of 〈θ〉 is given in degrees to the right of each panel (see also Fig A1) The green wedge has an opening angle σθ = 15and is symmetric around the invariable plane but is also limited to the grey region where a planet can be seen to transit from Earth (b le 1 equation 1) Thethick black arc indicates the acrit value beyond which less than 50 per cent of planets will transit (equation B1) Predicted planets and their uncertainties areshown by solid red rectangles if the Ptrans value of the predicted planet is ge 055 or by red hatched rectangles otherwise Thus the 77 solid red rectangles in the40 systems shown in Figs 11ndash13 make up our short list of highest priority predictions (Table C11) Our estimate of the most probable inclination ambiguitiesin a system are represented by vertically separated pairs of blue dots connected by a thin black line (see Appendix A2)

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Figure 12 The same as Fig 11 but for the next 16 systems in our sample which have at least one planet with Ptrans ge 055 Note that some of the new detectedplanets from the predictions of BL13 have been included in the Kepler data archive (see Table 1) and that these planets are included in our analysis

an undetected planet between the planet pair is unlikely we referto the planet pair as lsquocompletely sampledrsquo Estimating complete-ness based on whether a system is dynamically full is a reason-able approach since there is some evidence that the majority of

systems are dynamically full (eg Barnes amp Raymond 2004) ForKepler systems in particular Fang amp Margot (2013) concluded thatat least 45 per cent of four-planet Kepler systems are dynamicallypacked

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Figure 13 The same as Figs 11 and 12 but for the remaining 10 systems in our sample which have at least one planet with Ptrans ge 055

If at least two sequential adjacent-planet pairs (at least threesequential planets) satisfy this criteria we add the subset of thesystem which satisfies this criteria to our lsquomost completersquo sampleWe use this sample to analyse the period ratios of Kepler systemsThe period ratios of the different samples described above are shownin Fig 15

One criticism from HB14 was that the TB relation from BL13inserted too many planets To address this criticism we have rede-fined γ to be divided by the number of inserted planets squared(denominator of equation 9) This introduces a heavier penaltyfor inserting planets Fig 15 displays the distributions of pe-riod ratios when using the γ from BL13 and the new γ ofequation (9) (panels b and c respectively) When using our newlydefined γ the mean χ2dof (displayed on the left-hand side ofthe panels) more closely resembles that of our lsquomost completesamplersquo (panel e)

Since each panel in Fig 15 represents a mixture of planetarysystems with different distributions of period ratios Fig 16 may bea better way to compare these different samples and their adherenceto the TB relation For each planetary system in each panel in Fig 15we compute the mean adjacent-planet period ratio Fig 16 shows thedistribution of the offsets from the mean period ratio of each systemHow peaked a distribution is is a good measure of how well thatdistribution adheres to the TB relation A delta function peak at anoffset of zero would be a perfect fit The period ratios of adjacent-

planet pairs in our dynamically full lsquomost complete samplersquo (greenin Fig 16) display a significant tendency to cluster around themean ratios This clustering is the origin of the usefulness of the TBrelation to predict the existence of undetected planets The proximityof the thick blue curve to the green distribution is a measure of howwell our TB predictions can correct for the incompleteness in Keplermultiple-planet systems and make predictions about the probablelocations of the undetected planets

6 C O N C L U S I O N

HB14 investigated the TB relation planet predictions of Bovairdamp Lineweaver (2013) and found a detection rate of sim5 per cent (5detections from 97 predictions) Apart from the detections by HB14only one additional planet (in KOI-1151) has been discovered inany of the 60 Kepler systems analysed by BL13 ndash indicating theadvantages of such predictions while searching for new planetsCompleteness is an important issue (eg Figs 7 and 10) Somelarge fraction of our predictions will not be detected because theplanets in this fraction are likely to be too small to produce SNRabove some chosen detection threshold Additionally the predictedplanets may have inclinations and semimajor axes too large to transittheir star as seen from Earth All new candidate detections based onthe predictions of BL13 are approximately Earth-sized or smaller(Table 1)

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Figure 14 The effective temperatures of planets within the 31 systems from our sample which extend out to the green habitable zone (HZ) after our planetpredictions are made For the purpose of estimating the number of HZ planets per star (see Table 2) we extrapolate additional planets (grey squares) beyondthe HZ The sizes of the red hashed squares represent the Rmax of the predicted planet

Table 2 The estimated number of planets per star within various lsquohabitable zonesrsquo

All planets Rocky planets (R le 15 Roplus)Sample MarsndashVenus K13 lsquooptimisticrsquo K13 lsquoconservativersquo MarsndashVenus K13 lsquooptimisticrsquo K13 lsquoconservativersquo

All 151 systems 20 plusmn 10 23 plusmn 12 15 plusmn 08 015 015 010Least extrapolationb 16 plusmn 09 17 plusmn 08 13 plusmn 07 040 035 030

Notes aK13 lsquooptimisticrsquo and lsquoconservativersquo habitable zones refer to the lsquorecent Venusrsquo to lsquoearly Marsrsquo and lsquorunaway greenhousersquo to lsquomaximumgreenhousersquo regions from Kopparapu et al (2013) respectivelybThe 31 systems in the sample shown in Fig 14 are those which need the least extrapolation (red hashed squares) to extend out to (or beyond) thegreen MarsndashVenus HZ

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Figure 15 The period ratios of adjacent-planet pairs in our sample ofKepler multiples which can be compared to fig 4 of HB14 Panel (a) repre-sents systems which adhere to the TB relation better than the Solar system(χ2dof lt 1) Panel (b) represents those systems which adhere to the TBrelation worse than the Solar system and where BL13 inserted planets Thispanel shows the period ratios of adjacent-planet pairs after planets are in-serted Panel (c) is similar to panel (b) except that the γ value which is usedto determine the best TB relation insertion for a given system is divided bythe number of inserted planets squared In BL13 and panel (b) γ was dividedby the number of inserted planets Panel (d) shows the period ratios betweenadjacent pairs of the same systems from panels (b) and (c) except before theadditional planets from the predictions of BL13 have been inserted Panel(e) represents our most complete sample and contains the systems whichare more likely to be dynamically full (as defined in Section 5) The meanχ2dof value for each subset is shown on the left-hand side of each panelKeplerrsquos bias towards detecting compact systems dominated by short periodplanets may explain why the Solar systemrsquos adjacent period pairs (blackhatched histogram in panel (e) are not representative of the histogram inpanel (e) The periods of predicted planets are drawn randomly from theirTB relation predicted Gaussian distributions (Tables C11 and C12)

Figure 16 Each system has a mean value for the adjacent-planet periodratios within that system This figure shows the distribution of the periodratios offset from the mean period ratio of the system The colours of thedistributions correspond to subsets in Fig 15 The green distribution is fromthe lsquomost complete samplersquo in panel (e) and is our best estimate of what adistribution should look like if an appropriate number of planets have beeninserted The grey distribution indicates that the sampling of these systemsis highly incomplete The thick and thin blue lines represent panel (c) in twodifferent ways The thick blue line uses planet periods drawn randomly froma normal distribution centred on the periods predicted by the TB relationwith the width set to the uncertainty in the period predictions (Tables C11and C12) The thin blue line uses periods at their exact predicted value

For a new sample of Kepler multiple-exoplanet systems contain-ing at least three planets we computed invariable plane inclinationsand assumed a Gaussian opening angle of coplanarity of σθ = 15For each of these systems we applied an updated generalized TBrelation developed in BL13 resulting in 228 predictions in 151systems

We emphasize the planet predictions which have a high geometricprobability to transit Ptrans ge 055 (Fig 8) This subset of predictionshas 77 predicted planets in 40 systems We expect the detectionrate in this subset to be a factor of sim3 higher than the detectionrate of the BL13 predictions From the 40 systems with planetpredictions in this sample 24 appeared in BL13 These predictionshave been updated and reprioritized We have ordered our list ofpredicted planets based on each planetrsquos geometric probability totransit (Tables C11 and C12) Our new prioritized predictions shouldhelp ongoing planet detection efforts in Kepler multiplanet systems

AC K N OW L E D G E M E N T S

TB acknowledges support from an Australian Postgraduate AwardWe acknowledge useful discussions with Daniel Bayliss MichaelIreland George Zhou and David Nataf This research has madeuse of the NASA Exoplanet Archive which is operated by theCalifornia Institute of Technology under contract with the NationalAeronautics and Space Administration under the Exoplanet Explo-ration Program

R E F E R E N C E S

Ballard S Johnson J A 2014 ApJ preprint (arXiv14104192)Barnes R Raymond S N 2004 ApJ 617 569Bovaird T Lineweaver C H 2013 MNRAS 435 1126Christiansen J L et al 2012 PASP 124 1279Dong S Zhu Z 2013 ApJ 778 53Dressing C D Charbonneau D 2013 ApJ 767 95Fabrycky D C Winn J N 2009 ApJ 696 1230Fabrycky D C et al 2014 ApJ 790 146Fang J Margot J-L 2012 ApJ 761 92Fang J Margot J-L 2013 ApJ 767 115Figueira P et al 2012 AampA 541 A139Foreman-Mackey D Hogg D W Morton T D 2014 ApJ 795 64Fressin F et al 2013 ApJ 766 81Gladman B 1993 Icarus 106 247Hayes W Tremaine S 1998 Icarus 135 549Howard A W et al 2012 ApJS 201 15Huang C X Bakos G A 2014 MNRAS 681 674 (HB14)Jaki S L 1972 Am J Phys 40 1014Johansen A Davies M B Church R P Holmelin V 2012 ApJ 758 39Kopparapu R K et al 2013 ApJ 765 131Kovacs G Bakos G 2005 preprint (astro-ph0508081)Lillo-Box J Barrado D Bouy H 2014 AampA 556 A103Lissauer J J et al 2011 ApJS 197 8Morton T D Swift J 2014 ApJ 791 10Petigura E A Howard A W Marcy G W 2013a Proc Natl Acad Sci

110 19273Petigura E A Marcy G W Howard A 2013b ApJ 770 69Rogers L A 2014 ApJ preprint (arXiv14074457)Seager S Mallen-Ornelas G 2003 ApJ 585 1038Silburt A Gaidos E Wu Y 2015 ApJ 799 180Souami D Souchay J 2012 AampA 543 A133Steffen J H Hwang J A 2014 MNRAS preprint (arXiv14093320)Tremaine S Dong S 2012 AJ 143 94Watson G S 1982 J Appl Probab 19 265Weissbein A Steinberg E 2012 preprint (arXiv12036072)Winn J N Fabrycky D C 2014 ARAampA preprint (arXiv14104199)Wolfgang A Lopez E 2014 ApJ preprint (arXiv14092982)

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APPENDIX A ESTIMATION O F THEI N VA R I A B L E P L A N E O F E X O P L A N E TSYSTEMS

A1 Coordinate system

In Fig 1(a) and this appendix we set up and explain the coordinatesystem used in our analysis The invariable plane of a planetarysystem can be defined as the plane passing through the barycentreof the system and is perpendicular to the sum 〈L〉 of all planets inthe system

〈L〉 =sum

j

Lj (A1)

where Lj = (Lx Ly Lz) is the orbital angular momentum of thejth planet One can introduce a coordinate system in which the x-axis points from the system to the observer (Fig 1a) With an x-axisestablished we are free to choose the direction of the z-axis Forexample consider the vector Lj in Fig 1(a) If we choose a varietyof zprime-axes all perpendicular to our x-axis then independent of the

zprime-axis the quantityradic

L2yprime + L2

zprime is a constant Thus without loss of

generality we could choose a zprime-axis such that Lyprime = 0 In Fig 1(a)we have chosen the z-axis such that the sum of the y-componentsof the angular momenta of all the planets is zero

〈L〉 =sum

j

Lj = ( 〈L〉x 0 〈L〉z ) (A2)

In other words we have choosen the z-axis such that the vectordefining the invariable plane 〈L〉 is in the xndashz plane We definethe plane perpendicular to this vector as the invariable plane of thesystem

The angular separation between Lj and 〈L〉 is φj φj is a positive-valued random variable and can be well represented by a Rayleighdistribution of mode σφ (Fabrycky amp Winn 2009) For the jth planetLj is the projection of Lj on to the xndashz plane The angle betweenLj and the z-axis is θ j The angle between 〈L〉 and the z-axis is 〈θ〉where

〈θ〉 =sum

j θjLjsumLj

(A3)

The angular separation in the xndashz plane between Lj and 〈L〉 is θ jIn the xndashz plane we then have the relation (Fig 1a)

〈θ〉 + θj = θj (A4)

where θ j is a normally distributed variable centred around 〈θ〉with a mean of 0 In other words θ j can be positive or negative Apositive definite variable such as φj is Rayleigh distributed if it canbe described as the sum of the squares of two independent normally

distributed variables (Watson 1982) ie φj =radic

θ 2j + θ 2

j(yminusz)

where θ j (yminusz) is the unobservable component of φj in the yndashz

plane perpendicular to θ j (see Fig 1a) From this relationship theGaussian distribution of θ j has a standard deviation equal to themode of the Rayleigh distribution of φj σθ = σφ

We can illustrate the meaning of the phrase lsquomutual inclinationrsquoused in the literature (eg Fabrycky et al 2014) For example inFig 1(a) imagine adding the angular momentum vector Lm ofanother planet And projecting this vector on to the xndashz plane andcall the projection Lm (just as we projected Lj into Lj ) Now wecan define two lsquomutual inclinationsrsquo between the orbital planes ofthese two planets ψ3D is the angle between Lj and Lm and ψ isthe angle in the xndashz plane between L j and Lm (ie |θ j minus θm|)

Since both θ j and θm are Gaussian distributed with meanμ = 0 their difference θ j minus θm is Gaussian distributed with

σ(θj minusθm) =radic

σ 2θj

+ σ 2θm

= radic2σθ and μ(θj minusθm) = 0 Hence

ψ = |θ j minus θm| is a positive-definite half-normal Gaussian withmean μψ = radic

2π σ(θj minusθm) = 2radicπσθ

For ψ3D the angle between Lj and Lm we have

ψ3D =radic

(θj minus θm)2 + (θj(yminusz) minus θm(yminusz))2 (A5)

From above (θ j minusθm) is Gaussian distributed with σ(θj minusθm) =radic2σθ Since we expect σ(θj(yminusz)minusθm(yminusz)) = σ(θj minusθm) ψ3D is

Rayleigh distributed with mode σ i (reported in Table A1 ) Thatis σψ3D = σi = radic

2 σθ The mean of the Rayleigh distribution ofψ3D is μψ3D = radic

π σθ On average we will have μψ3Dμψ = π2

A2 Exoplanet invariable planes permuting planet inclinations

An N-planet system has N different values of θ j (see Fig 1) Sinceobservations are only sensitive to |θ j| we do not know whether

Table A1 Comparison of exoplanet coplanarity studies

Reference i distribution Observables Modea of Rayleigh distributed mutual inclinations Sample (quarter multiplicity)

Lissauer et al (2011) Rayleigh N bp σ i sim 20 Kepler (Q2 1ndash6)

Tremaine amp Dong (2012) Fisher Np σci lt 40 RV amp Kepler (Q2 1ndash6)

Figueira et al (2012) Rayleigh Np σdi sim 14 HARPS amp Kepler (Q2 1ndash3)

Fang amp Margot (2012) Rayleigh R of R Np ξ e σci sim 14 Kepler (Q6 1ndash6)

Johansen et al (2012) Uniform i + rotation f Np σ i lt 35 Kepler (Q6 1ndash3)Weissbein amp Steinberg (2012) Rayleigh Np No fit Kepler (Q6 1ndash6)Fabrycky et al (2014) Rayleigh Np ξ σ i sim 18 Kepler (Q6 1ndash6)

Ballard amp Johnson (2014) Rayleigh Np σi = 20 +40minus20

Kepler M-dwarfs (Q16 2ndash5)

Notes aThe mode σ i is equal to the σψ3D discussed at the end of Appendix A1 Thus σi = σψ3D = radic2 σθ Assuming σθ = 15 is equivalent to assuming

σ i = 21bNp is the multiplicity vector for the numbers of observed n-planet systems ie Np = ( of one-planet systems of two-planet systems of three-planetsystems )cConverted from the mean μ of the mutual inclination Rayleigh distribution σi = radic

2π μdConverted from Rayleigh distribution relative to the invariable plane σi = radic

2 σθ eξ is the normalized transit duration ratio as given in equation 11 of Fang amp Margot (2012)fEach planet is given a random uniform inclination between 0 and 5 This orbital plane is then rotated uniformly between 0 and 2π to give a randomlongitude of ascending node

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we are dealing with positive or negative angles To model thisuncertainty we analyse the 2Nminus1 unique sets of permutations forpositive and negative θ j values For example in a four-planet systemconsider the planet with the largest angular momentum We set ourcoordinate system by assuming its inclination i is less than 90 Wedo not know whether the i values of the other three planets are on thesame side or the opposite side of 90 The permutations of the +1sand minus1s in equation (A6) represent this uncertainty There will bekmax = 23 = 8 sets of permutations for the θ j of the remaining threeplanets defined by θ jMj k where Mj k is the permutation matrix

Mjk =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 1 11 1 minus11 minus1 11 minus1 minus1

minus1 1 1minus1 1 minus1minus1 minus1 1minus1 minus1 minus1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A6)

For each permutation k we compute a notional invariant plane bytaking the angular-momentum-weighted average of the permutedangles (compare equation A3)

〈θ〉k =sum

LjθjMjksumLj

(A7)

which yields eight unique values of 〈θ〉k each consistent with thebj θ j and ij values from the transit light curves (see equations 1ndash3)For each of these 〈θ〉k we compute a proxy for coplanarity which isthe mean of the angular-momentum-weighted angle of the orbitalplanes around the notional invariant plane

σ〈θ〉k =sum

Lj |〈θ〉k minus θjMjk|sumLj

(A8)

Figure A1 Top panel the red histogram represents the final 〈θ〉 anglesfor the systems in our sample The blue line represents the normalized areaout to acrit as a function of 〈θ〉 (equation B5) The bottom panel shows thedispersion for our Kepler multiples about the calculated 〈θ〉 value for eachsystem The smaller values of σ 〈θ〉 at 〈θ〉 sim 0 are probably due to theoverrepresentation of i = 90 in the Kepler data

The smaller the value of σ〈θ〉k the more coplanar that permutationset is This permutation procedure is most appropriate when thesystem is close to edge-on since in this case the various planetsare equally likely to have actual inclinations on either side of 90By contrast when 〈θ〉 is large these permutations exaggerate theuncertainty since most planets are likely to be on the same sideof 90 as the dominant planet Thus using this method close-to-edge-on systems with 〈θ〉 05 will yield the smallest andmore appropriate dispersions which we find to be in the range0 σ〈θ〉k 15 Since the coplanarity of a system should notdepend on the angle to the observer the values of σ〈θ〉k should notdepend on 〈θ〉 We find that this condition can best be met when wereject permutations which yield values of σ〈θ〉k less than 04 andgreater than 15 (see Fig A1) When no permutations for a givensystem meet this criteria we select the single permutation which isclosest to this range Since the sign of 〈θ〉 is not important whenmore than one permutation meets this criterion we estimate 〈θ〉by taking the median of the absolute values of the 〈θ〉k for which04 σ〈θ〉k 15 These permutations are used in Figs 11ndash13where the most probable inclination ambiguities are indicated bytwo blue planets at the same semimajor axis one above and onebelow the i = 90 dashed horizontal line

A P P E N D I X B C A L C U L AT I N G T H EGEOMETRI C PROBABI LITY TO TRANSITPtrans

We assume σθ = 15 is constant over all systems (Section 11)such that the geometric probability for the jth planet to transit is thefraction of a Gaussian-weighted opening angle within the transitregion where the standard deviation of the Gaussian is σ j = ajσθ

(Fig B1 and equation B1)

Ptrans(〈θ〉 aj ) = 1

σj

radic2π

int u2

u1

eminus u2

2σ2j du (B1)

Figure B1 Illustration of the variables required to calculate Ptrans

(equation B1) the fraction of a Gaussian within the transit region (b 1)We assume that a planet with semimajor axis aj in a system whose invari-able plane is the thick line will have a position θ j whose probability willbe described by a normal distribution along the red line perpendicular tothe invariant plane The integration variable u of equation (B1) is along thisperpendicular red line The mean of the Gaussian is at the point (xo zo)where u = 0 The standard deviation σ j = ajσθ is also the half-width ofthe green region at a distance aj from the host star

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where the integration limits u1 and u2 are defined by (see Fig B1)

u1 = Rlowast + zo

cos〈θ〉 (B2)

u2 = Rlowast minus zo

cos〈θ〉 (B3)

z0 = minusaj sin〈θ〉 (B4)

with the axes in Fig B1 being the same as in panel (b) of Fig 1 wehave zo lt 0 And since 〈θ〉 sim 0 we have cos 〈θ〉 sim 1

In the panels of Figs 11ndash13 the green area is a function of the〈θ〉 of the system We can integrate Ptrans to get the size of the green

area that is closer to the host star than acrit This yields an area thatis an estimate of the amount of parameter space in which planetscan transit

Area(〈θ〉) =int acrit(〈θ〉)

0Ptrans da (B5)

In the top panel of Fig A1 the blue curve is a normalized versionof equation (B5) and represents the statistical expectation of thenumber of systems as a function of 〈θ〉 (ignoring detection biases)The histogram shows the values we have obtained

A P P E N D I X C TA B L E S O F P L A N E TP R E D I C T I O N S

Table C1 77 planet predictions with a high geometric probability to transit (Ptrans ge 055) in 40 Kepler systems

System Number γ γ a(

χ2

dof

)i

(χ2

dof

)f

Inserted Period a Rmaxb Teff Ptrans

inserted planet (d) (au) (Roplus) (K)

KOI-1198 2 22 03 719 074 1 21 plusmn 04 003 13 1642 1002 43 plusmn 07 006 15 1297 100

KOI-1955 4 34 02 1023 019 1 26 plusmn 03 004 09 1568 1002 41 plusmn 05 005 10 1347 1003 64 plusmn 07 007 12 1157 0994 101 plusmn 11 010 13 994 094

KOI-1082 2 230 50 502 006 1 18 plusmn 02 003 10 1184 1002 28 plusmn 03 004 11 1029 1003 E c 147 plusmn 14 011 16 588 072

KOI-952 1 22 52 236 076 1 15 plusmn 03 002 09 904 100KOI-500 2 76 19 547 018 1 15 plusmn 02 002 12 1091 100

2 21 plusmn 02 003 13 960 100KOI-4032 2 09 19 120 027 1 34 plusmn 02 004 08 1347 100

2 45 plusmn 02 005 09 1224 0993 E 69 plusmn 03 007 10 1061 091


2 251 plusmn 25 017 24 679 064KOI-2859 1 102 minus10 169 016 1 24 plusmn 01 003 06 1242 098KOI-250 5 07 11 226 014 1 48 plusmn 04 005 12 686 096

2 66 plusmn 06 006 13 616 0913 92 plusmn 08 007 14 553 083

KOI-168 0 ndash ndash 014 ndash 1 E 226 plusmn 12 017 26 909 0952 E 332 plusmn 14 021 29 800 0873 E 486 plusmn 17 028 32 704 0764 E 713 plusmn 22 036 35 620 064

KOI-2585 0 ndash ndash 055 ndash 1 E 148 plusmn 07 012 11 967 0952 E 207 plusmn 08 015 12 865 0873 E 289 plusmn 09 019 13 774 0784 E 404 plusmn 11 024 14 692 0675 E 564 plusmn 13 030 16 619 057


KOI-505 3 16 01 820 056 1 219 plusmn 25 015 45 788 0922 347 plusmn 39 020 51 676 0783 55 plusmn 7 027 57 580 062

KOI-1831 1 13 02 254 111 1 82 plusmn 11 008 09 739 091KOI-248 1 183 808 248 013 1 42 plusmn 05 004 14 633 089KOI-880 1 40 224 135 028 1 118 plusmn 20 010 21 761 089KOI-1567 2 59 448 151 007 1 114 plusmn 12 010 20 668 089



2 239 plusmn 25 017 16 702 0743 371 plusmn 39 023 18 607 060

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Table C1 ndash continued


χ2

dof

)i

(χ2

dof

)f



KOI-701 6 180 74 404 001 1 84 plusmn 08 007 06 621 087KOI-1306 1 57 04 412 062 1 137 plusmn 21 011 15 756 085KOI-2722 0 ndash ndash 054 ndash 1 E 234 plusmn 15 017 14 774 078

2 E 330 plusmn 18 021 15 690 0673 E 465 plusmn 23 027 16 615 056

KOI-1358 0 ndash ndash 001 ndash 1 E 136 plusmn 10 010 16 522 0742 E 210 plusmn 13 014 18 451 060








KOI-719 1 11 15 136 066 1 148 plusmn 20 011 08 514 069KOI-1060 0 ndash ndash 022 ndash 1 E 327 plusmn 26 022 21 703 068KOI-3083 0 ndash ndash 056 ndash 1 E 132 plusmn 05 011 07 751 066

2 E 169 plusmn 05 013 07 692 057KOI-156 0 ndash ndash 010 ndash 1 E 179 plusmn 10 012 16 476 061KOI-137 0 ndash ndash 013 ndash 1 E 312 plusmn 31 019 31 539 060KOI-1151 0 ndash ndash 085 ndash 1 E 330 plusmn 22 020 10 564 059KOI-1015 0 ndash ndash 062 ndash 1 E 361 plusmn 35 022 23 590 058KOI-2029 0 ndash ndash 031 ndash 1 E 237 plusmn 16 015 09 514 056KOI-664 0 ndash ndash 006 ndash 1 E 403 plusmn 30 023 15 618 056

Notes aγ = (γ 1 minus γ 2)γ 2 where γ 1 and γ 2 are the highest and second highest γ values for that system respectively (seeBovaird amp Lineweaver 2013)bRmax is calculated by applying the lowest SNR of the detected planets in the system to the period of the inserted planet Seeequation (7)cA planet number followed by lsquoErsquo indicates the planet is extrapolated (has a larger period than the outermost detected planet inthe system)

Table C2 All 228 planet predictions in 151 systems (Table C1 1 is a high Ptrans subset of this table)


χ2

dof

)i

(χ2

dof

)f

Inserted Period a Rmaxb Teff

e Ptransd



KOI-1955 4 34 02 1023 019 1 26 plusmn 03 004 09 1568 1002 41 plusmn 05 005 10 1347 1003 64 plusmn 07 007 12 1157 0994 101 plusmn 11 010 13 994 0945 E 62 plusmn 7 033 20 541 042

KOI-1082 2 230 50 502 006 1 18 plusmn 02 003 10 1184 1002 28 plusmn 03 004 11 1029 1003 E 147 plusmn 14 011 16 588 072

KOI-952 1 22 52 236 076 1 15 plusmn 03 002 09 904 1002 E 400 plusmn 62 019 21 299 036


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KOI-2859 1 102 minus10 169 016 1 24 plusmn 01 003 06 1242 0982 E 51 plusmn 03 005 08 967 054

KOI-250 5 07 11 226 014 1 48 plusmn 04 005 12 686 0962 66 plusmn 06 006 13 616 0913 92 plusmn 08 007 14 553 0834 241 plusmn 19 014 18 401 0535 332 plusmn 26 017 20 360 0446 E 633 plusmn 50 027 23 290 029




KOI-505 3 16 01 820 056 1 219 plusmn 25 015 45 788 0922 347 plusmn 39 020 51 676 0783 55 plusmn 7 027 57 580 0624 E 140 plusmn 20 051 72 426 036







KOI-701 6 180 74 404 001 1 84 plusmn 08 007 06 621 0872 266 plusmn 26 015 08 423 0523 391 plusmn 37 019 09 372 0414 57 plusmn 6 025 10 327 0335 84 plusmn 8 032 11 288 0256 180 plusmn 20 054 13 223 0157 E 390 plusmn 40 090 16 172 009


KOI-2722 0 ndash ndash 054 ndash 1 E 234 plusmn 15 017 14 774 0782 E 330 plusmn 18 021 15 690 0673 E 465 plusmn 23 027 16 615 056

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KOI-156 0 ndash ndash 010 ndash 1 E 179 plusmn 10 012 16 476 061KOI-137 0 ndash ndash 013 ndash 1 E 312 plusmn 31 019 31 539 060KOI-1151 0 ndash ndash 085 ndash 1 E 330 plusmn 22 020 10 564 059KOI-1015 0 ndash ndash 062 ndash 1 E 361 plusmn 35 022 23 590 058KOI-2029 0 ndash ndash 031 ndash 1 E 237 plusmn 16 015 09 514 056KOI-664 0 ndash ndash 006 ndash 1 E 403 plusmn 30 023 15 618 056KOI-2693 0 ndash ndash 001 ndash 1 E 191 plusmn 14 012 10 430 053KOI-1590 0 ndash ndash 064 ndash 1 E 287 plusmn 33 016 18 452 053KOI-279 0 ndash ndash 013 ndash 1 E 56 plusmn 6 030 13 586 053KOI-1930 0 ndash ndash 060 ndash 1 E 72 plusmn 7 035 23 541 052KOI-70 1 74 29 351 042 1 391 plusmn 54 022 12 498 052

2 E 130 plusmn 20 049 17 331 024KOI-720 0 ndash ndash 014 ndash 1 E 348 plusmn 36 020 29 477 051KOI-1860 0 ndash ndash 002 ndash 1 E 495 plusmn 56 027 17 512 049KOI-1475 0 ndash ndash 094 ndash 1 E 246 plusmn 30 014 17 377 048KOI-1194 0 ndash ndash 052 ndash 1 E 290 plusmn 25 016 18 374 047KOI-2025 0 ndash ndash 021 ndash 1 E 405 plusmn 24 024 23 647 047KOI-733 0 ndash ndash 022 ndash 1 E 355 plusmn 35 020 28 437 046KOI-2169 0 ndash ndash 087 ndash 1 E 76 plusmn 04 007 07 868 046KOI-2163 0 ndash ndash 006 ndash 1 E 463 plusmn 31 026 17 532 044KOI-3319 0 ndash ndash 001 ndash 1 E 459 plusmn 53 025 21 517 044KOI-2352 0 ndash ndash 028 ndash 1 E 203 plusmn 12 016 11 845 044KOI-1681 0 ndash ndash 015 ndash 1 E 127 plusmn 11 009 13 415 044KOI-1413 0 ndash ndash 014 ndash 1 E 562 plusmn 38 028 18 458 044KOI-2597 0 ndash ndash 013 ndash 1 E 177 plusmn 10 014 18 791 043KOI-2220 0 ndash ndash 019 ndash 1 E 191 plusmn 16 014 12 695 042KOI-1161 0 ndash ndash 018 ndash 1 E 214 plusmn 19 014 21 574 042KOI-582 0 ndash ndash 008 ndash 1 E 303 plusmn 23 018 18 466 041KOI-82 0 ndash ndash 092 ndash 1 E 383 plusmn 29 021 09 408 041KOI-157 1 37 69 315 069 1 75 plusmn 8 034 27 439 041

2 E 170 plusmn 20 060 34 334 024KOI-864 0 ndash ndash 009 ndash 1 E 440 plusmn 46 024 29 453 040KOI-939 0 ndash ndash 024 ndash 1 E 203 plusmn 21 015 19 640 040KOI-898 0 ndash ndash 008 ndash 1 E 391 plusmn 36 020 28 360 040KOI-841 2 71 04 435 015 1 63 plusmn 11 031 27 409 039

2 130 plusmn 30 051 33 320 0253 E 580 plusmn 100 135 47 196 009

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KOI-408 0 ndash ndash 051 ndash 1 E 59 plusmn 7 029 26 461 039KOI-1909 0 ndash ndash 026 ndash 1 E 55 plusmn 6 029 19 500 038KOI-2715 0 ndash ndash 062 ndash 1 E 261 plusmn 29 014 39 379 038KOI-1278 0 ndash ndash 052 ndash 1 E 73 plusmn 7 035 20 455 038KOI-1867 0 ndash ndash 053 ndash 1 E 313 plusmn 36 016 17 323 037KOI-899 0 ndash ndash 001 ndash 1 E 331 plusmn 35 016 19 293 037KOI-1589 0 ndash ndash 056 ndash 1 E 82 plusmn 9 037 24 440 037KOI-884 0 ndash ndash 045 ndash 1 E 53 plusmn 7 025 29 362 037KOI-829 0 ndash ndash 007 ndash 1 E 76 plusmn 8 036 35 442 037KOI-94 0 ndash ndash 019 ndash 1 E 130 plusmn 20 055 42 452 036KOI-2038 0 ndash ndash 011 ndash 1 E 370 plusmn 23 021 18 494 036KOI-1557 0 ndash ndash 058 ndash 1 E 182 plusmn 19 012 20 499 035KOI-571 2 199 52 487 007 1 409 plusmn 56 019 08 294 035

2 73 plusmn 10 028 09 242 0243 E 230 plusmn 40 061 12 164 011

KOI-1905 0 ndash ndash 001 ndash 1 E 72 plusmn 8 032 17 374 034KOI-116 0 ndash ndash 034 ndash 1 E 86 plusmn 10 038 13 425 034KOI-2732 0 ndash ndash 022 ndash 1 E 100 plusmn 20 044 13 426 034KOI-665 0 ndash ndash 001 ndash 1 E 112 plusmn 10 010 15 893 034KOI-1931 0 ndash ndash 020 ndash 1 E 152 plusmn 08 012 15 661 033KOI-886 0 ndash ndash 046 ndash 1 E 332 plusmn 22 016 16 298 032KOI-1432 0 ndash ndash 007 ndash 1 E 87 plusmn 13 038 20 397 032KOI-945 0 ndash ndash 004 ndash 1 E 107 plusmn 7 046 26 424 032KOI-869 0 ndash ndash 008 ndash 1 E 84 plusmn 12 035 38 349 031KOI-111 0 ndash ndash 003 ndash 1 E 110 plusmn 20 042 28 376 030KOI-1364 0 ndash ndash 041 ndash 1 E 349 plusmn 33 020 30 508 030KOI-1832 0 ndash ndash 003 ndash 1 E 110 plusmn 20 044 36 381 030KOI-658 0 ndash ndash 060 ndash 1 E 207 plusmn 18 015 14 649 030KOI-1895 0 ndash ndash 011 ndash 1 E 64 plusmn 6 027 26 289 030KOI-2926 0 ndash ndash 049 ndash 1 E 73 plusmn 8 027 28 260 029KOI-1647 0 ndash ndash 007 ndash 1 E 74 plusmn 9 034 20 460 029KOI-941 0 ndash ndash 036 ndash 1 E 75 plusmn 12 032 47 343 029KOI-3741 0 ndash ndash 001 ndash 1 E 353 plusmn 31 021 22 709 028KOI-700 0 ndash ndash 091 ndash 1 E 120 plusmn 20 048 21 388 028KOI-1563 0 ndash ndash 062 ndash 1 E 269 plusmn 25 017 36 491 028KOI-2135 0 ndash ndash 012 ndash 1 E 140 plusmn 20 053 19 414 028KOI-2433 0 ndash ndash 055 ndash 1 E 150 plusmn 20 057 29 395 028KOI-2086 0 ndash ndash 033 ndash 1 E 152 plusmn 06 013 27 933 027KOI-1102 0 ndash ndash 075 ndash 1 E 324 plusmn 27 020 28 630 027KOI-3097 0 ndash ndash 031 ndash 1 E 156 plusmn 06 013 13 1033 027KOI-1445 0 ndash ndash 001 ndash 1 E 150 plusmn 30 059 14 391 027KOI-232 0 ndash ndash 093 ndash 1 E 110 plusmn 20 046 21 391 026KOI-520 0 ndash ndash 018 ndash 1 E 110 plusmn 20 042 18 314 026KOI-2707 0 ndash ndash 029 ndash 1 E 110 plusmn 20 045 23 335 025KOI-152 0 ndash ndash 064 ndash 1 E 160 plusmn 20 060 36 381 025KOI-1332 0 ndash ndash 003 ndash 1 E 170 plusmn 30 063 38 337 025KOI-2485 0 ndash ndash 017 ndash 1 E 164 plusmn 12 012 18 575 025KOI-877 0 ndash ndash 032 ndash 1 E 400 plusmn 34 020 16 343 024KOI-775 0 ndash ndash 004 ndash 1 E 78 plusmn 9 030 24 253 024KOI-757 0 ndash ndash 001 ndash 1 E 110 plusmn 20 041 42 295 024KOI-510 0 ndash ndash 005 ndash 1 E 78 plusmn 11 034 35 413 024KOI-117 0 ndash ndash 042 ndash 1 E 233 plusmn 20 017 14 692 023KOI-935 0 ndash ndash 003 ndash 1 E 180 plusmn 30 068 41 374 023KOI-285 0 ndash ndash 004 ndash 1 E 94 plusmn 9 043 22 490 022KOI-671 0 ndash ndash 062 ndash 1 E 264 plusmn 20 017 15 614 022KOI-834 0 ndash ndash 084 ndash 1 E 120 plusmn 20 047 25 351 022KOI-509 0 ndash ndash 022 ndash 1 E 120 plusmn 20 046 25 334 022KOI-904 0 ndash ndash 093 ndash 1 E 100 plusmn 20 037 22 252 022KOI-723 0 ndash ndash 004 ndash 1 E 74 plusmn 10 033 42 357 022KOI-1436 0 ndash ndash 019 ndash 1 E 313 plusmn 36 019 23 555 021

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KOI-2073 0 ndash ndash 008 ndash 1 E 130 plusmn 20 046 28 274 020KOI-812 0 ndash ndash 004 ndash 1 E 110 plusmn 20 038 25 224 020KOI-474 0 ndash ndash 054 ndash 1 E 230 plusmn 40 077 38 336 020KOI-907 0 ndash ndash 091 ndash 1 E 250 plusmn 50 076 45 309 019KOI-1422 0 ndash ndash 004 ndash 1 E 110 plusmn 20 037 19 198 019KOI-710 0 ndash ndash 069 ndash 1 E 125 plusmn 07 012 17 998 019KOI-623 0 ndash ndash 040 ndash 1 E 422 plusmn 35 023 13 592 019KOI-282 0 ndash ndash 001 ndash 1 E 280 plusmn 50 083 15 299 019KOI-620 0 ndash ndash 084 ndash 1 E 230 plusmn 20 074 76 298 018KOI-3925 0 ndash ndash 038 ndash 1 E 179 plusmn 16 013 35 779 018KOI-2167 0 ndash ndash 005 ndash 1 E 260 plusmn 50 082 18 303 017KOI-1426 0 ndash ndash 002 ndash 1 E 290 plusmn 30 088 53 294 016KOI-1127 0 ndash ndash 081 ndash 1 E 142 plusmn 11 012 20 681 016KOI-191 0 ndash ndash 099 ndash 1 E 180 plusmn 40 061 34 301 016KOI-1430 0 ndash ndash 098 ndash 1 E 200 plusmn 30 057 32 215 015KOI-806 0 ndash ndash 016 ndash 1 E 310 plusmn 40 089 25 247 015KOI-612 0 ndash ndash 010 ndash 1 E 290 plusmn 40 079 36 235 015KOI-481 0 ndash ndash 002 ndash 1 E 160 plusmn 40 056 40 282 014KOI-2714 0 ndash ndash 009 ndash 1 E 640 plusmn 120 155 34 272 013KOI-1258 0 ndash ndash 096 ndash 1 E 430 plusmn 70 111 51 218 010KOI-564 0 ndash ndash 077 ndash 1 E 530 plusmn 110 128 43 222 010KOI-1922 0 ndash ndash 002 ndash 1 E 790 plusmn 220 169 29 199 008KOI-2183 0 ndash ndash 070 ndash 1 E 770 plusmn 190 164 30 188 007KOI-518 0 ndash ndash 084 ndash 1 E 940 plusmn 190 159 31 124 005KOI-2842 0 ndash ndash 027 ndash 1 E 95 plusmn 08 007 32 559 000

Notes aγ = (γ 1 minus γ 2)γ 2 where γ 1 and γ 2 are the highest and second highest γ values for that system respectively (seeBL13)bRmax is calculated by applying the lowest SNR of the detected planets in the system to the period of the inserted planet Seeequation (7)cA planet number followed by lsquoErsquo indicates the planet is extrapolated (has a larger period than the outermost detected planet inthe system)dPtrans values ge 055 are shown in bold indicating a higher probability to transiteTeff values between Mars and Venus (206 to 300 K assuming an albedo of 03) are shown in bold

This paper has been typeset from a TEXLATEX file prepared by the author

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Table 1 Systems with candidate detections by HB14 (in bold) plusKOI-1151 (Petigura Howard amp Marcy 2013a) and KOI-1860b afterplanet predictions were made by BL13

System Predicted Detected Predicted Detectedperiod (d) period (d) radius (Roplus) radius (Roplus)

KOI-719 14 plusmn 2 1577 le 07 042KOI-1336 26 plusmn 3 2751 le 24 104KOI-1952 13 plusmn 2 1327 le 15 085KOI-2722a 168 plusmn 10 1653 le 16 116KOI-2859 52 plusmn 03 543 le 08 076KOI-733 NA 1511 NA 30KOI-1151a 96 plusmn 07 1043 le 08 07KOI-1860b 25 plusmn 3 2484 le 27 146

Notes aPredicted by preprint of BL13 (draft uploaded 2013 Apr 11httparxivorgpdf13043341v1pdf) detected planet reported by Ke-pler archive and included in analysis of BL13b2014 October Kepler Archive update during the drafting of this paper

a Rayleigh distribution (Lissauer et al 2011 Fang amp Margot 2012Figueira et al 2012 Ballard amp Johnson 2014 Fabrycky et al 2014)For the ensemble of Kepler multiplanet systems the mode of theRayleigh distribution of mutual inclinations (φj minus φi) is typicallysim1ndash3 (Appendix A1 and Table A1) Thus Kepler multiple-planetsystems are highly coplanar The Solar system is similarly coplanarFor example the mode of the best-fitting Rayleigh distribution ofthe planet inclinations (φj) relative to the invariable plane in theSolar system is sim1 (see Figs 1 and 2)2

The angle θ j is a Gaussian distributed variable with a mean of 0(centred around 〈θ〉) and standard deviation σθ Based on previousanalyses (Table A1) we assume the typical value σθ = 15 Weuse this angle to determine the probability of detecting additionaltransiting planets in each system

Estimates of the inclination of a transiting planet come from theimpact parameter b which is the projected distance between thecentre of the planet at mid-transit and the centre of the star in unitsof the starrsquos radius

b = a

Rlowastcos i (1)

where Rlowast is the radius of the star and a is the semimajor axis of theplanet For edge-on systems typically 85 lt i lt 95 However sincewe are unable to determine whether b is in the positive z directionor the negative z direction (Figs 1b and B1) we are unable todetermine whether i is greater than or less than 90 By conventionfor transiting planets the sign of b is taken as positive and thus thecorresponding i values from equation (1) are taken as i le 90

The impact parameter is also a function of four transit light-curve observables (Seager amp Mallen-Ornelas 2003) the period Pthe transit depth F the total transit duration tT and the total transitduration minus the ingress and egress times tF (the duration wherethe light curve is flat for a source uniform across its disc) Thus theimpact parameter can be written as

b = f (P F tT tF) (2)

2 See Appendix A for an explanation of why the distribution of mutualinclinations is on average a factor of

radic2 wider than the distribution of the

angles φj in Fig 1 between the invariable plane of the system and the orbitalplanes of the planets

Eliminating b from equations (1) and (2) yields the inclination i asa function of observables

i = cosminus1

[Rlowasta

f (P F tT tF)

] (3)

From equation (1) we can see that for an impact parameter b = 0(a transit through the centre of the star) we obtain i = 90 anlsquoedge-onrsquo transit

The convention i le 90 is unproblematic when only a singleplanet is found to transit a star but raises an issue when multipleexoplanets transit the same star since the degree of coplanaritydepends on whether the actual values of ij (j = 1 2 N where Nis the number of planets in the system) are greater than or less than90 For example the actual values of ij in a given system could beall gt90 all lt90 or some in-between combination Although wedo not know the signs of θ j = 90 minus ij for individual planets wecan estimate the inclination of the invariable plane for each systemby calculating all possible permutations of the θ j values for eachsystem (see Appendix A2) In this estimation we use the plausibleassumption that the coplanarity of a system should not depend onthe inclination of the invariable plane relative to the observer

12 The probability of additional transiting exoplanets

We wish to develop a measure of the likelihood of additional transit-ing planets in our sample of Kepler multiplanet systems The moreedge-on a planetary system is to an observer on Earth the greater theprobability of a planet transiting at larger periods Similarly a largerstellar radius leads to a higher probability of additional transitingplanets (although with a reduced detection efficiency) We quantifythese tendencies under the assumption that Kepler multiples havea Gaussian opening angle σθ = 15 around the invariable planeand we introduce the variable acrit Planets with a semimajor axisgreater than acrit have less than a 50 per cent geometric probabilityof transiting More specifically acrit is defined as the semimajor axiswhere Ptrans(acrit) = 05 (equation B1) for a given system

In a given system a useful ratio for estimating the amount ofsemimajor axis space where additional transiting planets are morelikely is acritaout where aout is the semimajor axis of the detectedplanet in the system which is the furthest from the host star Thelarger acritaout the larger the semimajor axis range for additionaltransiting planets beyond the outermost detected planet Values forthis ratio less than 1 mean that the outermost detected planet isbeyond the calculated acrit value and imply that additional transitingplanets beyond the outermost detected planet are less likely Fig 3shows the acritaout distribution for all systems in our sample Thefact that this distribution is roughly symmetric around acritaout sim 1strongly suggests that the outermost transiting planets in Keplersystems are due to the inclination of the system to the observer andare not really the outermost planets

In Section 2 we discuss the follow-up that has been done on ourBL13 planet detections In Section 3 we show that the sim5 per centfollow-up detection rate of HB14 is consistent with selection effectsand the existence of the predicted planets In Section 4 we extendand upgrade the TB relation developed in BL13 and predict the peri-ods of undetected planets in our updated sample We then prioritizethese predictions based on their geometric probability to transit andemphasize for further follow-up a subset of predictions with hightransit probabilities We also use TB predictions to estimate the av-erage number of planets in the circumstellar habitable zone (HZ) InSection 5 we discuss how our predicted planet insertions affect theperiod ratios of adjacent planets and explore how period ratios are

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f (R) = df

dlog R=


(5)



PSNR =int Rmax

Rminf (R)dRint Rmax

Rlowf (R)dR

(6)


Rmax = RminSNR

(Ppredict

PminSNR

)14

(7)


Rmin = Rlowastradic

SNRth CDPP

(3 h

ntr tT

)14

(8)






MNRAS 448 3608ndash3627 (2015)

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γ =

(χ2

i minusχ2f

χ2f

)n2

ins

(9)

where χ2i and χ2














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MNRAS 448 3608ndash3627 (2015)

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R E F E R E N C E S



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〈L〉 =sum

j

Lj (A1)



L2yprime + L2



〈L〉 =sum

j

Lj = ( 〈L〉x 0 〈L〉z ) (A2)



〈θ〉 =sum

j θjLjsumLj

(A3)


〈θ〉 + θj = θj (A4)



θ 2j + θ 2

j(yminusz)






σ 2θj

+ σ 2θm





ψ3D =radic





















MNRAS 448 3608ndash3627 (2015)

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Mjk =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣



⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A6)


〈θ〉k =sum

LjθjMjksumLj

(A7)


σ〈θ〉k =sum


(A8)







σj

radic2π

int u2

u1

eminus u2

2σ2j du (B1)



MNRAS 448 3608ndash3627 (2015)

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Dow

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u1 = Rlowast + zo

cos〈θ〉 (B2)


cos〈θ〉 (B3)






0Ptrans da (B5)





χ2

dof

)i

(χ2

dof

)f








2 45 plusmn 02 005 09 1224 0993 E 69 plusmn 03 007 10 1061 091



2 66 plusmn 06 006 13 616 0913 92 plusmn 08 007 14 553 083








2 239 plusmn 25 017 16 702 0743 371 plusmn 39 023 18 607 060

MNRAS 448 3608ndash3627 (2015)

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χ2

dof

)i

(χ2

dof

)f




2 E 330 plusmn 18 021 15 690 0673 E 465 plusmn 23 027 16 615 056














χ2

dof

)i

(χ2

dof

)f


e Ptransd







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(χ2

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e Ptransd




















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(χ2

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e Ptransd
















2 130 plusmn 30 051 33 320 0253 E 580 plusmn 100 135 47 196 009

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χ2

dof

)i

(χ2

dof

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e Ptransd



2 73 plusmn 10 028 09 242 0243 E 230 plusmn 40 061 12 164 011


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χ2

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(χ2

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e Ptransd






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f (R) = df

dlog R=


(5)



PSNR =int Rmax

Rminf (R)dRint Rmax

Rlowf (R)dR

(6)


Rmax = RminSNR

(Ppredict

PminSNR

)14

(7)


Rmin = Rlowastradic

SNRth CDPP

(3 h

ntr tT

)14

(8)






MNRAS 448 3608ndash3627 (2015)

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γ =

(χ2

i minusχ2f

χ2f

)n2

ins

(9)

where χ2i and χ2














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R E F E R E N C E S



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〈L〉 =sum

j

Lj (A1)



L2yprime + L2



〈L〉 =sum

j

Lj = ( 〈L〉x 0 〈L〉z ) (A2)



〈θ〉 =sum

j θjLjsumLj

(A3)


〈θ〉 + θj = θj (A4)



θ 2j + θ 2

j(yminusz)






σ 2θj

+ σ 2θm





ψ3D =radic





















MNRAS 448 3608ndash3627 (2015)

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Mjk =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣



⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A6)


〈θ〉k =sum

LjθjMjksumLj

(A7)


σ〈θ〉k =sum


(A8)







σj

radic2π

int u2

u1

eminus u2

2σ2j du (B1)



MNRAS 448 3608ndash3627 (2015)

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u1 = Rlowast + zo

cos〈θ〉 (B2)


cos〈θ〉 (B3)






0Ptrans da (B5)





χ2

dof

)i

(χ2

dof

)f








2 45 plusmn 02 005 09 1224 0993 E 69 plusmn 03 007 10 1061 091



2 66 plusmn 06 006 13 616 0913 92 plusmn 08 007 14 553 083








2 239 plusmn 25 017 16 702 0743 371 plusmn 39 023 18 607 060

MNRAS 448 3608ndash3627 (2015)

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Dow

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χ2

dof

)i

(χ2

dof

)f




2 E 330 plusmn 18 021 15 690 0673 E 465 plusmn 23 027 16 615 056














χ2

dof

)i

(χ2

dof

)f


e Ptransd







MNRAS 448 3608ndash3627 (2015)

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χ2

dof

)i

(χ2

dof

)f


e Ptransd




















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χ2

dof

)i

(χ2

dof

)f


e Ptransd
















2 130 plusmn 30 051 33 320 0253 E 580 plusmn 100 135 47 196 009

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd



2 73 plusmn 10 028 09 242 0243 E 230 plusmn 40 061 12 164 011


MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd






MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from







MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from












MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




f (R) = df

dlog R=


(5)



PSNR =int Rmax

Rminf (R)dRint Rmax

Rlowf (R)dR

(6)


Rmax = RminSNR

(Ppredict

PminSNR

)14

(7)


Rmin = Rlowastradic

SNRth CDPP

(3 h

ntr tT

)14

(8)






MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from






γ =

(χ2

i minusχ2f

χ2f

)n2

ins

(9)

where χ2i and χ2














MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from





MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from









MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from







MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from








R E F E R E N C E S



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from





〈L〉 =sum

j

Lj (A1)



L2yprime + L2



〈L〉 =sum

j

Lj = ( 〈L〉x 0 〈L〉z ) (A2)



〈θ〉 =sum

j θjLjsumLj

(A3)


〈θ〉 + θj = θj (A4)



θ 2j + θ 2

j(yminusz)






σ 2θj

+ σ 2θm





ψ3D =radic





















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



Mjk =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣



⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A6)


〈θ〉k =sum

LjθjMjksumLj

(A7)


σ〈θ〉k =sum


(A8)







σj

radic2π

int u2

u1

eminus u2

2σ2j du (B1)



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



u1 = Rlowast + zo

cos〈θ〉 (B2)


cos〈θ〉 (B3)






0Ptrans da (B5)





χ2

dof

)i

(χ2

dof

)f








2 45 plusmn 02 005 09 1224 0993 E 69 plusmn 03 007 10 1061 091



2 66 plusmn 06 006 13 616 0913 92 plusmn 08 007 14 553 083








2 239 plusmn 25 017 16 702 0743 371 plusmn 39 023 18 607 060

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f




2 E 330 plusmn 18 021 15 690 0673 E 465 plusmn 23 027 16 615 056














χ2

dof

)i

(χ2

dof

)f


e Ptransd







MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd




















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd
















2 130 plusmn 30 051 33 320 0253 E 580 plusmn 100 135 47 196 009

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd



2 73 plusmn 10 028 09 242 0243 E 230 plusmn 40 061 12 164 011


MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd






MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from












MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




f (R) = df

dlog R=


(5)



PSNR =int Rmax

Rminf (R)dRint Rmax

Rlowf (R)dR

(6)


Rmax = RminSNR

(Ppredict

PminSNR

)14

(7)


Rmin = Rlowastradic

SNRth CDPP

(3 h

ntr tT

)14

(8)






MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from






γ =

(χ2

i minusχ2f

χ2f

)n2

ins

(9)

where χ2i and χ2














MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from





MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from









MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from







MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from








R E F E R E N C E S



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from





〈L〉 =sum

j

Lj (A1)



L2yprime + L2



〈L〉 =sum

j

Lj = ( 〈L〉x 0 〈L〉z ) (A2)



〈θ〉 =sum

j θjLjsumLj

(A3)


〈θ〉 + θj = θj (A4)



θ 2j + θ 2

j(yminusz)






σ 2θj

+ σ 2θm





ψ3D =radic





















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



Mjk =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣



⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A6)


〈θ〉k =sum

LjθjMjksumLj

(A7)


σ〈θ〉k =sum


(A8)







σj

radic2π

int u2

u1

eminus u2

2σ2j du (B1)



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



u1 = Rlowast + zo

cos〈θ〉 (B2)


cos〈θ〉 (B3)






0Ptrans da (B5)





χ2

dof

)i

(χ2

dof

)f








2 45 plusmn 02 005 09 1224 0993 E 69 plusmn 03 007 10 1061 091



2 66 plusmn 06 006 13 616 0913 92 plusmn 08 007 14 553 083








2 239 plusmn 25 017 16 702 0743 371 plusmn 39 023 18 607 060

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f




2 E 330 plusmn 18 021 15 690 0673 E 465 plusmn 23 027 16 615 056














χ2

dof

)i

(χ2

dof

)f


e Ptransd







MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd




















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd
















2 130 plusmn 30 051 33 320 0253 E 580 plusmn 100 135 47 196 009

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd



2 73 plusmn 10 028 09 242 0243 E 230 plusmn 40 061 12 164 011


MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd






MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




f (R) = df

dlog R=


(5)



PSNR =int Rmax

Rminf (R)dRint Rmax

Rlowf (R)dR

(6)


Rmax = RminSNR

(Ppredict

PminSNR

)14

(7)


Rmin = Rlowastradic

SNRth CDPP

(3 h

ntr tT

)14

(8)






MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from






γ =

(χ2

i minusχ2f

χ2f

)n2

ins

(9)

where χ2i and χ2














MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from





MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from









MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from







MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from








R E F E R E N C E S



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from





〈L〉 =sum

j

Lj (A1)



L2yprime + L2



〈L〉 =sum

j

Lj = ( 〈L〉x 0 〈L〉z ) (A2)



〈θ〉 =sum

j θjLjsumLj

(A3)


〈θ〉 + θj = θj (A4)



θ 2j + θ 2

j(yminusz)






σ 2θj

+ σ 2θm





ψ3D =radic





















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



Mjk =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣



⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A6)


〈θ〉k =sum

LjθjMjksumLj

(A7)


σ〈θ〉k =sum


(A8)







σj

radic2π

int u2

u1

eminus u2

2σ2j du (B1)



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



u1 = Rlowast + zo

cos〈θ〉 (B2)


cos〈θ〉 (B3)






0Ptrans da (B5)





χ2

dof

)i

(χ2

dof

)f








2 45 plusmn 02 005 09 1224 0993 E 69 plusmn 03 007 10 1061 091



2 66 plusmn 06 006 13 616 0913 92 plusmn 08 007 14 553 083








2 239 plusmn 25 017 16 702 0743 371 plusmn 39 023 18 607 060

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f




2 E 330 plusmn 18 021 15 690 0673 E 465 plusmn 23 027 16 615 056














χ2

dof

)i

(χ2

dof

)f


e Ptransd







MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd




















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd
















2 130 plusmn 30 051 33 320 0253 E 580 plusmn 100 135 47 196 009

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd



2 73 plusmn 10 028 09 242 0243 E 230 plusmn 40 061 12 164 011


MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd






MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from






γ =

(χ2

i minusχ2f

χ2f

)n2

ins

(9)

where χ2i and χ2














MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from





MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from









MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from







MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from








R E F E R E N C E S



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from





〈L〉 =sum

j

Lj (A1)



L2yprime + L2



〈L〉 =sum

j

Lj = ( 〈L〉x 0 〈L〉z ) (A2)



〈θ〉 =sum

j θjLjsumLj

(A3)


〈θ〉 + θj = θj (A4)



θ 2j + θ 2

j(yminusz)






σ 2θj

+ σ 2θm





ψ3D =radic





















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



Mjk =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣



⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A6)


〈θ〉k =sum

LjθjMjksumLj

(A7)


σ〈θ〉k =sum


(A8)







σj

radic2π

int u2

u1

eminus u2

2σ2j du (B1)



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



u1 = Rlowast + zo

cos〈θ〉 (B2)


cos〈θ〉 (B3)






0Ptrans da (B5)





χ2

dof

)i

(χ2

dof

)f








2 45 plusmn 02 005 09 1224 0993 E 69 plusmn 03 007 10 1061 091



2 66 plusmn 06 006 13 616 0913 92 plusmn 08 007 14 553 083








2 239 plusmn 25 017 16 702 0743 371 plusmn 39 023 18 607 060

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f




2 E 330 plusmn 18 021 15 690 0673 E 465 plusmn 23 027 16 615 056














χ2

dof

)i

(χ2

dof

)f


e Ptransd







MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd




















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd
















2 130 plusmn 30 051 33 320 0253 E 580 plusmn 100 135 47 196 009

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd



2 73 plusmn 10 028 09 242 0243 E 230 plusmn 40 061 12 164 011


MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd






MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from





MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from









MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from







MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from








R E F E R E N C E S



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from





〈L〉 =sum

j

Lj (A1)



L2yprime + L2



〈L〉 =sum

j

Lj = ( 〈L〉x 0 〈L〉z ) (A2)



〈θ〉 =sum

j θjLjsumLj

(A3)


〈θ〉 + θj = θj (A4)



θ 2j + θ 2

j(yminusz)






σ 2θj

+ σ 2θm





ψ3D =radic





















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



Mjk =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣



⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A6)


〈θ〉k =sum

LjθjMjksumLj

(A7)


σ〈θ〉k =sum


(A8)







σj

radic2π

int u2

u1

eminus u2

2σ2j du (B1)



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



u1 = Rlowast + zo

cos〈θ〉 (B2)


cos〈θ〉 (B3)






0Ptrans da (B5)





χ2

dof

)i

(χ2

dof

)f








2 45 plusmn 02 005 09 1224 0993 E 69 plusmn 03 007 10 1061 091



2 66 plusmn 06 006 13 616 0913 92 plusmn 08 007 14 553 083








2 239 plusmn 25 017 16 702 0743 371 plusmn 39 023 18 607 060

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f




2 E 330 plusmn 18 021 15 690 0673 E 465 plusmn 23 027 16 615 056














χ2

dof

)i

(χ2

dof

)f


e Ptransd







MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd




















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd
















2 130 plusmn 30 051 33 320 0253 E 580 plusmn 100 135 47 196 009

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd



2 73 plusmn 10 028 09 242 0243 E 230 plusmn 40 061 12 164 011


MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd






MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from





MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from









MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from







MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from








R E F E R E N C E S



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from





〈L〉 =sum

j

Lj (A1)



L2yprime + L2



〈L〉 =sum

j

Lj = ( 〈L〉x 0 〈L〉z ) (A2)



〈θ〉 =sum

j θjLjsumLj

(A3)


〈θ〉 + θj = θj (A4)



θ 2j + θ 2

j(yminusz)






σ 2θj

+ σ 2θm





ψ3D =radic





















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



Mjk =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣



⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A6)


〈θ〉k =sum

LjθjMjksumLj

(A7)


σ〈θ〉k =sum


(A8)







σj

radic2π

int u2

u1

eminus u2

2σ2j du (B1)



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



u1 = Rlowast + zo

cos〈θ〉 (B2)


cos〈θ〉 (B3)






0Ptrans da (B5)





χ2

dof

)i

(χ2

dof

)f








2 45 plusmn 02 005 09 1224 0993 E 69 plusmn 03 007 10 1061 091



2 66 plusmn 06 006 13 616 0913 92 plusmn 08 007 14 553 083








2 239 plusmn 25 017 16 702 0743 371 plusmn 39 023 18 607 060

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f




2 E 330 plusmn 18 021 15 690 0673 E 465 plusmn 23 027 16 615 056














χ2

dof

)i

(χ2

dof

)f


e Ptransd







MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd




















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd
















2 130 plusmn 30 051 33 320 0253 E 580 plusmn 100 135 47 196 009

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd



2 73 plusmn 10 028 09 242 0243 E 230 plusmn 40 061 12 164 011


MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd






MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from









MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from







MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from








R E F E R E N C E S



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from





〈L〉 =sum

j

Lj (A1)



L2yprime + L2



〈L〉 =sum

j

Lj = ( 〈L〉x 0 〈L〉z ) (A2)



〈θ〉 =sum

j θjLjsumLj

(A3)


〈θ〉 + θj = θj (A4)



θ 2j + θ 2

j(yminusz)






σ 2θj

+ σ 2θm





ψ3D =radic





















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



Mjk =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣



⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A6)


〈θ〉k =sum

LjθjMjksumLj

(A7)


σ〈θ〉k =sum


(A8)







σj

radic2π

int u2

u1

eminus u2

2σ2j du (B1)



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



u1 = Rlowast + zo

cos〈θ〉 (B2)


cos〈θ〉 (B3)






0Ptrans da (B5)





χ2

dof

)i

(χ2

dof

)f








2 45 plusmn 02 005 09 1224 0993 E 69 plusmn 03 007 10 1061 091



2 66 plusmn 06 006 13 616 0913 92 plusmn 08 007 14 553 083








2 239 plusmn 25 017 16 702 0743 371 plusmn 39 023 18 607 060

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f




2 E 330 plusmn 18 021 15 690 0673 E 465 plusmn 23 027 16 615 056














χ2

dof

)i

(χ2

dof

)f


e Ptransd







MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

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(χ2

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e Ptransd




















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

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(χ2

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e Ptransd
















2 130 plusmn 30 051 33 320 0253 E 580 plusmn 100 135 47 196 009

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

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e Ptransd



2 73 plusmn 10 028 09 242 0243 E 230 plusmn 40 061 12 164 011


MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd






MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from







MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from








R E F E R E N C E S



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from





〈L〉 =sum

j

Lj (A1)



L2yprime + L2



〈L〉 =sum

j

Lj = ( 〈L〉x 0 〈L〉z ) (A2)



〈θ〉 =sum

j θjLjsumLj

(A3)


〈θ〉 + θj = θj (A4)



θ 2j + θ 2

j(yminusz)






σ 2θj

+ σ 2θm





ψ3D =radic





















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



Mjk =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣



⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A6)


〈θ〉k =sum

LjθjMjksumLj

(A7)


σ〈θ〉k =sum


(A8)







σj

radic2π

int u2

u1

eminus u2

2σ2j du (B1)



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



u1 = Rlowast + zo

cos〈θ〉 (B2)


cos〈θ〉 (B3)






0Ptrans da (B5)





χ2

dof

)i

(χ2

dof

)f








2 45 plusmn 02 005 09 1224 0993 E 69 plusmn 03 007 10 1061 091



2 66 plusmn 06 006 13 616 0913 92 plusmn 08 007 14 553 083








2 239 plusmn 25 017 16 702 0743 371 plusmn 39 023 18 607 060

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

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)i

(χ2

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2 E 330 plusmn 18 021 15 690 0673 E 465 plusmn 23 027 16 615 056














χ2

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(χ2

dof

)f


e Ptransd







MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

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e Ptransd




















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

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χ2

dof

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(χ2

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e Ptransd
















2 130 plusmn 30 051 33 320 0253 E 580 plusmn 100 135 47 196 009

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

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e Ptransd



2 73 plusmn 10 028 09 242 0243 E 230 plusmn 40 061 12 164 011


MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd






MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from








R E F E R E N C E S



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from





〈L〉 =sum

j

Lj (A1)



L2yprime + L2



〈L〉 =sum

j

Lj = ( 〈L〉x 0 〈L〉z ) (A2)



〈θ〉 =sum

j θjLjsumLj

(A3)


〈θ〉 + θj = θj (A4)



θ 2j + θ 2

j(yminusz)






σ 2θj

+ σ 2θm





ψ3D =radic





















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



Mjk =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣



⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A6)


〈θ〉k =sum

LjθjMjksumLj

(A7)


σ〈θ〉k =sum


(A8)







σj

radic2π

int u2

u1

eminus u2

2σ2j du (B1)



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



u1 = Rlowast + zo

cos〈θ〉 (B2)


cos〈θ〉 (B3)






0Ptrans da (B5)





χ2

dof

)i

(χ2

dof

)f








2 45 plusmn 02 005 09 1224 0993 E 69 plusmn 03 007 10 1061 091



2 66 plusmn 06 006 13 616 0913 92 plusmn 08 007 14 553 083








2 239 plusmn 25 017 16 702 0743 371 plusmn 39 023 18 607 060

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f




2 E 330 plusmn 18 021 15 690 0673 E 465 plusmn 23 027 16 615 056














χ2

dof

)i

(χ2

dof

)f


e Ptransd







MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd




















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd
















2 130 plusmn 30 051 33 320 0253 E 580 plusmn 100 135 47 196 009

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd



2 73 plusmn 10 028 09 242 0243 E 230 plusmn 40 061 12 164 011


MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd






MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from





〈L〉 =sum

j

Lj (A1)



L2yprime + L2



〈L〉 =sum

j

Lj = ( 〈L〉x 0 〈L〉z ) (A2)



〈θ〉 =sum

j θjLjsumLj

(A3)


〈θ〉 + θj = θj (A4)



θ 2j + θ 2

j(yminusz)






σ 2θj

+ σ 2θm





ψ3D =radic





















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



Mjk =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣



⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A6)


〈θ〉k =sum

LjθjMjksumLj

(A7)


σ〈θ〉k =sum


(A8)







σj

radic2π

int u2

u1

eminus u2

2σ2j du (B1)



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



u1 = Rlowast + zo

cos〈θ〉 (B2)


cos〈θ〉 (B3)






0Ptrans da (B5)





χ2

dof

)i

(χ2

dof

)f








2 45 plusmn 02 005 09 1224 0993 E 69 plusmn 03 007 10 1061 091



2 66 plusmn 06 006 13 616 0913 92 plusmn 08 007 14 553 083








2 239 plusmn 25 017 16 702 0743 371 plusmn 39 023 18 607 060

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f




2 E 330 plusmn 18 021 15 690 0673 E 465 plusmn 23 027 16 615 056














χ2

dof

)i

(χ2

dof

)f


e Ptransd







MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd




















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd
















2 130 plusmn 30 051 33 320 0253 E 580 plusmn 100 135 47 196 009

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd



2 73 plusmn 10 028 09 242 0243 E 230 plusmn 40 061 12 164 011


MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd






MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



Mjk =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣



⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A6)


〈θ〉k =sum

LjθjMjksumLj

(A7)


σ〈θ〉k =sum


(A8)







σj

radic2π

int u2

u1

eminus u2

2σ2j du (B1)



MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



u1 = Rlowast + zo

cos〈θ〉 (B2)


cos〈θ〉 (B3)






0Ptrans da (B5)





χ2

dof

)i

(χ2

dof

)f








2 45 plusmn 02 005 09 1224 0993 E 69 plusmn 03 007 10 1061 091



2 66 plusmn 06 006 13 616 0913 92 plusmn 08 007 14 553 083








2 239 plusmn 25 017 16 702 0743 371 plusmn 39 023 18 607 060

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f




2 E 330 plusmn 18 021 15 690 0673 E 465 plusmn 23 027 16 615 056














χ2

dof

)i

(χ2

dof

)f


e Ptransd







MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd




















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd
















2 130 plusmn 30 051 33 320 0253 E 580 plusmn 100 135 47 196 009

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd



2 73 plusmn 10 028 09 242 0243 E 230 plusmn 40 061 12 164 011


MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd






MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from



u1 = Rlowast + zo

cos〈θ〉 (B2)


cos〈θ〉 (B3)






0Ptrans da (B5)





χ2

dof

)i

(χ2

dof

)f








2 45 plusmn 02 005 09 1224 0993 E 69 plusmn 03 007 10 1061 091



2 66 plusmn 06 006 13 616 0913 92 plusmn 08 007 14 553 083








2 239 plusmn 25 017 16 702 0743 371 plusmn 39 023 18 607 060

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f




2 E 330 plusmn 18 021 15 690 0673 E 465 plusmn 23 027 16 615 056














χ2

dof

)i

(χ2

dof

)f


e Ptransd







MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd




















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd
















2 130 plusmn 30 051 33 320 0253 E 580 plusmn 100 135 47 196 009

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd



2 73 plusmn 10 028 09 242 0243 E 230 plusmn 40 061 12 164 011


MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd






MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f




2 E 330 plusmn 18 021 15 690 0673 E 465 plusmn 23 027 16 615 056














χ2

dof

)i

(χ2

dof

)f


e Ptransd







MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd




















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd
















2 130 plusmn 30 051 33 320 0253 E 580 plusmn 100 135 47 196 009

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd



2 73 plusmn 10 028 09 242 0243 E 230 plusmn 40 061 12 164 011


MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd






MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd




















MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd
















2 130 plusmn 30 051 33 320 0253 E 580 plusmn 100 135 47 196 009

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd



2 73 plusmn 10 028 09 242 0243 E 230 plusmn 40 061 12 164 011


MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd






MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd
















2 130 plusmn 30 051 33 320 0253 E 580 plusmn 100 135 47 196 009

MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd



2 73 plusmn 10 028 09 242 0243 E 230 plusmn 40 061 12 164 011


MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd






MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd



2 73 plusmn 10 028 09 242 0243 E 230 plusmn 40 061 12 164 011


MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd






MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from




χ2

dof

)i

(χ2

dof

)f


e Ptransd






MNRAS 448 3608ndash3627 (2015)

at Australian N



Dow

nloaded from

Documents

Using the inclinations of Kepler systems to prioritize …...Titius–Bode-based exoplanet predictions T. Bovaird, 1 ,2‹ C. H. Lineweaver 3 and S. K. Jacobsen4 1Research School of