On the patterns observed in Kepler multi-planet systems
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On the patterns observed in
Kepler multi-planet systems
Wei Zhu ( 祝 伟 ) Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, ON M5S 3H8, Canada
ABSTRACTRecent studies claimed that planets around the same star have similar sizes and masses and regularspacings, and that planet pairs usually show ordered sizes such that the outer planet is usually thelarger one. Here I show that these patterns can be largely explained by detection biases. The
Kepler planet detections are set by the transit signal-to-noise ratio (S / N). For different stellar propertiesand orbital period values, the same S / N corresponds to different planetary sizes. This variation inthe detection threshold naturally leads to apparent correlations in planet sizes and the observed sizeordering. The apparently correlated spacings, measured in period ratios, between adjacent planetpairs in systems with at least three detected planets are partially due to the arbitrary upper limit thatthe earlier study imposed on the period ratio, and partially due to the varying stability threshold fordifferent planets. After these detection biases are taken into account, we do not find strong evidence forthe so-called “intra-system uniformity” or the size ordering effect. Instead, the physical properties of
Kepler planets are largely independent of the properties of their siblings and the parent star. It is likelythat the dynamical evolution has erased the memory of
Kepler planets about their initial formationconditions. In other words, it will be difficult to infer the initial conditions from the observed propertiesand the architecture of
Kepler planets.
Keywords: methods: statistical — planetary systems — planets and satellites: general INTRODUCTIONAlmost all planets reside in multi-planet systems, andthe physical and orbital properties of planets in the samesystem convey important clues about their formationand evolution. To date the majority of the multi-planetsystems were found by
Kepler (Borucki et al. 2010). However,
Kepler could only detect those planets thattransited their hosts and had transit signals above somecertain noise level. This limits our knowledge about thedetected multi-planet systems and complicates the the-oretical interpretation.As
Kepler observations provide directly the transitplanet-to-star radius ratio, the relative sizes of planetsinside the same system can be easily examined. Basedon data from the first four months of observations, Lis-sauer et al. (2011) pointed out that adjacent planetswere likely to have very similar radii, as most of thepairs show R p , in /R p , out ≈
1. Here R p , in and R p , out are Corresponding author: Wei [email protected] See a list of all known multi-planet systems at NASA Exo-planet Archive (Akeson et al. 2013). the radii of the inner and the outer transiting planets(tranets), respectively. This feature was further stud-ied in Ciardi et al. (2013), and the authors reportedthat most ( > Kepler appeared to have this size-location correla-tion: the outer planet was larger than the inner one. In both studies, the authors compared the observed andthe simulated radius ratio distributions to determine thestatistical significance of their finding. Their simulateddistributions were produced by randomly drawing radiifrom the observed radius distribution and going throughcustomized signal-to-ratio (S/N) cuts. As I will explainlater, this approach does not capture all the detectionbiases.Later follow-up efforts that lead to better charac-terizations of
Kepler stars allow comparisons of plan-etary parameters across systems. Recently, Weiss etal. (2018a, hereafter W18) claims that planetary sys- Ciardi et al. (2013) argued that this result only applied toplanet pairs in which at least one was approximately Neptune-sized or larger. However, without this constraint the results arequalitatively similar in the statistical sense. See their Figures 4and 11. a r X i v : . [ a s t r o - ph . E P ] J a n Wei Zhu tems are like “peas in a pod,” namely that the planetsorbiting around the same host have similar sizes andregular spacings. They took the large sample of
Ke-pler multi-planet systems whose parameters were refinedby the California-
Kepler
Survey (CKS, Petigura et al.2017; Johnson et al. 2017), sorted the
CKS planets inthe same systems according to their orbital periods, andcomputed the correlation between sizes of neighboringCKS planets. They then quantified the significance ofthis correlation through bootstrap tests and found thatthe observed correlation could not be explained by ran-domly resampling the observed size distribution. Theprocedure was similar for the spacings between planets.A later work by Millholland et al. (2017) adopteda similar statistical approach and further claimed thatthe masses of planets inside the same system, given byHadden & Lithwick (2017) from analyzing the transittiming variations (TTVs, Agol et al. 2005; Holman &Murray 2005), should also be similar. This, togetherwith the aforementioned trends about radius and spac-ing, was summarized as the “intra-system uniformity”(Millholland et al. 2017).However, an issue that was overlooked in these stud-ies (Lissauer et al. 2011; Ciardi et al. 2013; Weiss et al.2018a; Millholland et al. 2017) is the detection thresh-old. Below I use the transit detection as an example, butthe idea applies to TTV mass measurements as well (seeSection 7). The Kepler transit search pipeline requires anominal minimum S/N of 7 .
1; for the planet sample usedin W18, a higher (S/N = 10) threshold was used. Thedetection completeness depends strongly on the S/N atlow end (e.g., Fressin et al. 2013; Thompson et al. 2018).Given the dominating contribution from smaller planetsand weaker transits (e.g., Hsu et al. 2019), the S/N val-ues of
Kepler transit detections as a result appear topile up toward the detection threshold (e.g., Figure 1 ofCiardi et al. 2013) and are not affected by the variationsof stellar parameters or noise levels. For a certain S/Nthreshold, the resulting planet radius threshold dependson the orbital period as well as the host properties. Thisvariation was not fully taken into account by those ref-erenced works in generating simulated parameters.A more proper way is fully forward modeling the de-tection and selection processes from the intrinsic plane-tary (radius or mass) distribution. Instead of randomly A “CKS planet” is a planet that transits the host, is detectableby
Kepler , and is included in the CKS sample. CKS planets arealmost certainly valid planets, but because of the geometric transitprobability and
Kepler detection sensitivity, the CKS planets arenot necessarily all the planets in those systems. The W18 work was posted on the arXiv pre-print server beforethe Millholland et al. (2017) work. drawing parameters from the observed distribution, oneshould draw from the intrinsic distribution and then ap-ply the same detection criteria (e.g., S/N cut) on thesesimulated planets. This process requires knowing theintrinsic planet distribution function and having accessto the automated
Kepler detection pipeline. It is fur-ther complicated by the fact that the planet distributionfunction is period dependent (e.g., Dong, & Zhu 2013;Hsu et al. 2019) and possibly multiplicity dependent,and that the
Kepler detection efficiency is weakly mul-tiplicity dependent (Zink et al. 2019).There is a shortcut that circumvents these problems.In the full forward modeling approach, one generatessynthetic planetary systems, adds stellar noises, passesthem to the
Kepler detection pipeline, decides whichplanets are detectable, and finally performs statisticalanalyses on the simulated detections. Through thiswhole process planetary physical parameters are con-verted into transit observables and the detectability ofindividual planet is controlled by the transit S/N. Giventhe central role of transit S/N, we can directly start fromthis parameter as a shortcut to the full forward mod-eling approach. As shown in Figure 1 and the upperright panel of Figure 2, the S/N distribution is indepen-dent of the transit multiplicity and the stellar properties.Therefore, we can randomly draw detections from thisuniversal S/N distribution, derive planetary parameters,and perform the same statistical analysis as we do on thereal data. As the relation between S/N and planetaryradius indicates (Equation 1), when the minor contri-bution from orbital period is ignored, correlated sizeswill definitely lead to correlated S/N values, but corre-lated S/N values does not necessarily mean correlatedsizes because of the same stellar size and stellar noiselevel that two adjacent planets share. Therefore, by ran-domly sampling the S/N distribution and thus assumingno correlation in S/N values, I am being more generousthan just assuming no size correlation.In this work, I apply this more robust statistical ap-proach to study the patterns observed in
Kepler multi-planet systems. I describe the planet sample in Sec-tion 2 and explain the basic idea in section 3. Thenin Sections 4 and 5 I discuss the issues involved in the“peas in a pod” claim. The size-location correlation isre-evaluated in Section 6. Finally, I briefly comment onthe intra-system mass uniformity claim and then discussthe results in Section 7. SAMPLEI use the same multi-planet sample as in W18. Thissample includes 909 CKS planets in 355 multi-planetsystems. The parameters of the individual planets and atterns in
Kepler multi-planet systems Transit S/N0.00.20.40.60.81.0 C D F All tranet2-tranet3-tranet4-tranet
Figure 1.
Cumulative distributions of transit S/N. Differ-ent transit multiplicities are shown in different colors, andthe overall sample is shown in black. There is no clear de-pendence of S/N on the transit multiplicity. of their hosts were provided in Table 1 of W18. Ofrelevance to this study are the stellar mass M (cid:63) , stel-lar radius R (cid:63) , 6 hr Combined Differential PhotometricPrecision (CDPP , a measure of the stellar noise level,Christiansen et al. 2012), impact parameter b , planetaryradius R p , and orbital period P . This table did not in-clude a column of S/N, but it can be easily computedby S / N = ( R p /R ⊕ ) (cid:112) . /P CDPP (cid:112) /T . (1)Here T is the transit duration, given by T = 13 hr (cid:18) P (cid:19) / (cid:18) ρ (cid:63) ρ (cid:12) (cid:19) − / (cid:112) − b , (2)where ρ (cid:63) is the stellar mean density. Note that the Equa-tion (2) of W18 did not have the factor √ − b , whichmade their S/N overestimated, although this would onlyhave a minor effect on the results.I show in Figure 2 the stellar noise level CDPP ver-sus three chosen parameters: planet radius R p , periodratio of adjacent CKS planets, and the transit detec-tion S/N. For demonstration purposes I divide the wholesample into two at the median CDPP : the quiet sam-ple and the noisy sample .The S/N value dictates the significance of a tran-sit detection and is a more fundamental observable insignal detections than planetary radius, which is noteven a direct observable in transit light curve. BecauseS/N already takes into account variation of stellar noise level CDPP (Equation (1)), one does not expect theS/N distribution to be different between the quiet andthe noisy samples. Indeed, a two-sample Kolmogorov-Smirnov (KS) test between the S/N distributions fromtwo samples gives p = 0 .
18, confirming that the S/Ndistribution is invariant to the variations of stellar prop-erties. However, the radius distributions and the periodratio distributions from the two samples are different:the two-sample KS test gives p < − and p < − respectively. These differences are most prominent atsmall values of R p and period ratio. The difference in R p is due to the projection of the same S/N distributioninto different stellar samples: smaller planets are moreeasily detected around more quiet stars. Then throughthe dynamical stability requirement, the difference in R p distributions propagates into the difference in period ra-tio distributions. See Section 5 for more details.Therefore, across the whole sample there is a universalS/N distribution but no universal radius distribution orperiod ratio distribution. The latter two were used inthe bootstrap test by W18. THE IDEA OF THIS PAPERFigure 3 illustrates how the variation in detectionthreshold originating from a fixed S/N can lead to sizeand spacing correlations in observed planets.First of all, one should know that there are moresmaller planets than larger ones, at least down to
Ke-pler ’s sensitivity limit, after the correction of detectionbias (e.g., Hsu et al. 2019). Therefore, if only planetsabove a certain size are detectable, then the detectedplanets will tend to pile up toward the detection thresh-old. In realistic missions such as
Kepler , the detectionthreshold is usually fixed in S/N because of its centralrole in signal detection. However, given the relation be-tween planetary radius and transit S/N (Equation (1)) afixed S/N threshold will lead to different radius thresh-olds for stars with different noise levels. Such a varyingradius threshold will naturally lead to a correlation be-tween the sizes of neighboring
Kepler planets. See theleft panel of Figure 3 for a simple illustration.W18 used bootstrap tests on planetary radii to studythe significance of the size correlation. The underlyingassumption behind their radius bootstrap test is that theradius distribution, P ( R p ), should be universal acrossdifferent stellar subsamples. I have shown in the previ-ous section that this assumption does not hold for thesample under investigation. Instead, the radius distri-bution depends on the stellar properties, specifically thenoise level CDPP . I denote this conditional radiusdistribution as P ( R p | CDPP ) and note that P ( R p ) (cid:54) = P ( R p | CDPP ) . (3) Wei Zhu C D F Quiet sampleNoisy sampleKS p < 10 KS p < 10 KS p = 0.18 R p ( R )10 C D PP h r Period ratio 10 S/N
Figure 2.
Lower panels show the noisy level of the planet host, characterized by CDPP , versus planet radius R p (lower left),planetary period ratio (lower middle), and signal-to-noise ratio (S/N) of the transit detection (lower right). The gray dashedhorizontal lines mark the median CDPP , based on which the sample is divided into two. In the upper panels we compare thecumulative distribution functions (CDF) of the individual parameter ( R p , period ratio, and S/N) from the two samples, andthe two-sample KS test p values are indicated. The radius and period ratio distributions of the two samples are statisticallydifferent. In the case of period ratio (lower middle panel), we also mark with the black dashed line the smallest CDPP forvarious period ratios. The S/N distributions are very similar between the two samples. Nearly 50% of planet detections haveS / N <
30, i.e., only a factor of three above the detection threshold (S / N = 10, as marked by the vertical dashed line).
A proper radius bootstrap test should therefore be ran-domly drawing radii from the conditional radius dis-tribution P ( R p | CDPP ). Unfortunately, such a con-ditional radius distribution cannot be easily specified,but one can use the relation between R p and S / N tofurther simplify the procedure. With other parame-ters the same (as is required by the bootstrap test), R p uniquely determines S / N. Thus randomly sam-pling P ( R p | CDPP ) is equivalent as randomly sam-pling S / N from P (S / N | CDPP ) and then deriving R p from S / N. Recalling that the CDPP distribution isuniversal across different stellar subsamples, P (S / N | CDPP ) = P (S / N) , (4)one can further simplify the proper radius bootstrap testas randomly resampling the S / N distribution P (S / N).This is another justification of the S / N-resamplingmethod that is used in this work.Similar to the claimed size correlation, the claimedspacing correlation is also affected by the variation indetection threshold. With decreasing stellar noise, the minimum period ratio between adjacent planets also de-creases, as shown in the lower middle panel of Figure 2.This is probably because of the stability boundary, inthe measure of period ratio, decreases with decreasingplanet size and we refer to Section 5 for a more detailedexplanation. As the transit probability strongly biasestoward small period values and thus small period ratios,the detectable planet pairs tend to pile up toward thesmallest period ratio (i.e., the stability boundary). Thevariation of this stability boundary in an inhomogeneousstellar sample naturally leads to a spacing correlation.See the right panel of Figure 3 for an illustration. ON THE RADIUS UNIFORMITYIn the left panel of Figure 4 I show the radius of oneCKS planet, R j , versus the radius of the outer adjacentCKS planet, R j +1 . This is very similar to Figure 2 ofW18. The correlation coefficient (Pearson r ) betweenlog R j and log R j +1 of all planet pairs is r = 0 .
65, con-sistent with the value reported in W18. atterns in
Kepler multi-planet systems Figure 3.
A schematic view of how an inhomogeneous stellar sample will lead to the size (left panel) and spacing (right panel)correlations. For stars with a certain noise level, only planets above a certain size are detectable. This can be seen in thelower left panel of Figure 2. Given that there are more smaller planets than larger ones, detected planet pairs tend to clusteraround the corner that is defined by the vertical and horizontal sensitivity limits. For different stellar samples (as measured bytheir noise levels), these clusterings appear at different locations along the diagonal line, and thus a collection of these planetswill naturally show a size correlation (left panel). Similarly, the stability boundary in the measure of period ratio decreaseswith decreasing planet sizes, as seen in the lower middle panel of Figure 2. Because of the strong bias in orbital period oftransit detections, observable planet triplets will tend to cluster around the corner of the stability boundary. For different stellarsamples (as measured by the noise level), these clusterings appear at different locations along the diagonal line, and thus acollection of these planet triplets will naturally show a spacing correlation (right panel). R j ( R )0.250.5124816 R j + ( R ) Pearson r = 0.65 Quiet sampleNoisy sample
Data R j ( R )0.250.5124816 Pearson r = 0.05 Resampling R p R j ( R )0.250.5124816 Pearson r = 0.48 Resampling S/N
Figure 4.
Comparisons between radii of adjacent planets ( R j and R j+1 ). In the left panel is the distribution of planet pairsin the CKS multi-planet sample. In the middle and the right panels are the synthetic planet pairs generated in two differentapproaches: resampling R p and resampling S/N. The former was used in W18. Planet pairs from two stellar samples (quietand noisy) are plotted with different colors to highlight the difference. The correlation coefficient between log R j and log R j +1 is also indicated at the lower right corner of each panel. Wei Zhu r N u m b e r o f t r i a l s Resampling R p ResamplingS/N
Figure 5.
The distributions of the Pearson r coefficientsfrom statistical tests. The blue and orange histograms areresults from two different approaches, and the black dashedline indicates the measured correlation value ( r = 0 . To assess the importance of this correlation W18 gen-erated synthetic planet systems in which the radius ofeach planet was randomly drawn from the overall ra-dius distribution. An example realization following theprocedure of W18 is shown in the middle panel of Fig-ure 4. Note that in this plot there appears to be fewersub-Earth-sized planets. This is because, following W18,simulated planets with S/N <
10 are excluded. This stepreduces the number of planets by nearly 20%.However, as discussed in Section 2 and illustrated inFigure 2, the S/N distribution is more fundamental anduniversal than the radius distribution in transit signaldetections. In particular, the radius distribution ap-pears different for stars with different noise levels. Thiscan also be seen in the left panel of Figure 4, where Ihave differentiated the planet pairs from quiet and noisysamples. Note the similarity between this plot and theleft panel of Figure 3. Following the reasoning in Sec-tion 3, I therefore modify the bootstrap test of W18.Instead of resampling the R p distribution, I resamplethe S/N distribution and then, with other parametersunchanged, derive R p from Equation (1). Note thatwhile I am bootstrapping transit S/N, I am essentially In practice, this can be easily achieved with the relation R newp = R p (cid:112) (S / N) new / (S / N). performing a forward modeling (see Section 1 for thedetailed explanation). The result from one random testis shown in the right panel of Figure 4. In this simu-lated sample the planet pairs from two stellar samplesshow a systematic offset, a feature that is similar to thedata (left panel). The radii of adjacent planets also showsignificant correlation, r = 0 . r coefficients, and show their his-togram in Figure 5. For comparison purposes I alsoproduce the histogram of r coefficients from 1000 boot-strap tests following the W18 procedure (i.e., resampling R p ), and the resulting histogram peaks at r ≈
0, similarto what W18 had (see their Figure 5). By resampling onthe more fundamental parameter S / N, I almost alwaysreproduce, at least qualitatively, the observed size corre-lation, although with an average correlation coefficient r ≈ . Kepler can only detect transiting planets abovecertain S/N threshold, it is very likely that many of the
Kepler multi-planet systems may contain additional un-detectable planets. Outside the period limit that
Kepler can probe ( ∼ Kepler period domain, there are also signsfor additional planets. Dynamical studies have sug-gested that majority of the
Kepler multi-planet sys-tems are not fully packed if the detected planets are allthe planets in the system. According to Fang, & Mar-got (2013) ∼
55% of systems with at least four detectedplanets can contain additional intervening planets with-out leading to dynamical instability, and the fractionis even higher for systems with two or three detectedplanets (see also Pu & Wu 2015). Furthermore, the factthat
Kepler planet detections pile up toward the de-tection threshold is also suggesting that smaller andundetectable planets do exist.An intervening undetectable planet between two de-tectable ones likely also transits and has a smaller (com- The detection efficiency of
Kepler pipeline depends on thetransit S/N. As Ciardi et al. (2013) have shown with some ear-lier Kepler sample (see their figure 1), Kepler detection is onlycomplete for SNR (cid:38)
25. Therefore, when the incompleteness of thedetection pipeline is taken into account, S/N of ∼
20 is still at theedge of the detection threshold. atterns in
Kepler multi-planet systems R j ( R )0.250.5124816 R j + ( R ) Pearson r = 0.65 Threshold S/N=10 R j ( R )0.250.5124816 Pearson r = 0.74 Threshold S/N=20 R j ( R )0.250.5124816 Pearson r = 0.81 Threshold S/N=30
Figure 6.
The size correlation plots for different choices of threshold S/N values. Increasing the threshold S/N is equivalent todecreasing
Kepler sensitivity, which leads to increased size correlation. Extrapolating to lower S/N thresholds, this test suggeststhat a much better
Kepler -like mission which is sensitive to smaller planets will find a much weaker size correlation. R j ( R )0.250.5124816 R j + ( R ) Pearson r = 0.69 Quiet sampleNoisy sample r N u m b e r o f t r i a l s Pairs from high-multiples
Figure 7.
The left panel shows the comparison between radii of adjacent planets ( R j and R j+1 ) from compact systems, definedas systems with at least 4 transiting planets. planet pairs from two stellar samples are marked with different colors. Thecorrelation coefficient is given in the lower right corner. For these planet pairs, 1000 statistical tests is performed, in whichthe S/N (rather than R p ) is resampled, and the resulting Pearson r values are shown as the histogram in the right panel. Theobserved correlation coefficient is not much different from the correlation we get in the statistical test. pared to the detection threshold) size. What the addi-tion of such a smaller intervening planet does to theradius correlation (e.g., left panel of Figure 4) is two-fold. First, one planet pair that shows strong correla-tion is removed and then, two planet pairs that showmuch weaker correlation are added. The combined con-sequence is that the size correlation is reduced signif-icantly. To demonstrate this effect, one would like to increase Kepler ’s sensitivity to recover the smaller plan-ets. This is obviously not practical, so I turn to theopposite direction. I increase the S/N threshold usedin the statistical test, which is equivalent as increasingstellar noises and thus lowering
Kepler ’s sensitivity, andthen measure the size correlation in the same way. As This test was originally suggested by Xi Zhang.
Wei Zhu shown in Figure 6, the size correlation becomes strongerin such “down-graded”
Kepler missions. This is alignedwith our speculation and suggests that, in a superior
Kepler mission which can detect much smaller planetsand thus is less affected by detection biases, the sizecorrelation should be much weaker.Another way to show the influence of additional plan-ets on the size correlation is to restrict to high-multiplesystems that are less likely to contain additional planetsbecause of stability requirement. I only include systemswith at least four CKS planets and repeat the same sta-tistical tests. The results are shown in Figure 7. Thistime the distribution of the correlation coefficients fromstatistical tests is statistically closer to the observedvalue. This again confirms that the missing planets dohave an effect on the size correlation. ON THE PERIOD RATIO UNIFORMITYW18 also claimed that the spacings between planets,measured by the period ratios, are correlated in sys-tems with at least three CKS planets. To reach thisconclusion, they first identified all CKS planet triplets,which consisted of all the CKS planets in three-planetsystems and three consecutive planets in higher-multiplesystems. For each planet triplet they computed the pe-riod ratio between the inner two planets, P in , and theperiod ratio between the outer two planets, P out . Con-sidering the incomplete sensitivity to large period ratios,W18 only included the planet triplets whose P in < P out <
4. Then they computed the Pearson r co-efficient between the two variables log P in and log P out and found r = 0 .
46. To assess the significance of thiscorrelation, they generated synthetic systems, in whichthe period ratios were randomly drawn from the over-all period ratio distribution, and found that the corre-lations in these simulated samples were systematicallymuch smaller than the observed one.This approach may produce biased results in twoways. First, the cut at period ratio P = 4 is fairly ar-bitrary and not physically motivated. In the left panelof Figure 8 I show the P out versus P in for all planettriplets in the W18 sample. One can see that the detec-tion limit at large period ratios is diagonal rather thanflat. This can be well explained by the detectability ofthese multi-planet systems. If the planets in the samesystem have coplanar or nearly coplanar orbits, the de-tectability of all planets only depends on the orbital pe-riod of the outermost planet, P outermost . For a planettriplet, this detection threshold scales as ∝ P in P out . Inthe left panel of Figure 8, I plot the line that corre-sponds to P in P out = 25, and it roughly agrees with theupper boundary of all data points. Adopting this phys- ically motivated detection threshold, I find the Pearson r = 0 .
21 between log P in and log P out . Restricting to P in P out <
16 gives r = 0 .
15. Both correlations aremuch weaker than that given by W18.The second issue is the varying detection thresholdof transit. At first glance, the S/N, as given in Equa-tion (1), has only weak dependence on orbital periodand no explicit dependence on the period ratio. Theperiod ratio comes into play via the dynamical stabilityrequirement. The stability boundary is typically mea-sured in the number ( K ) of mutual Hill radii, r H , a − a = K · r H , r H ≡ a + a (cid:18) m + m (cid:19) / , (5)where a i and m i are the semi-major axis and mass ofthe inner ( i = 1) and outer ( i = 2) planets, respec-tively. For simplicity, we further assume m ≈ m ≈ M ⊕ ( R p /R ⊕ ) . Note that this is not valid in general,but it is acceptable for the planet pairs that are justabove the detection threshold and close to the instabil-ity limit. Then with Kepler’s third law we can have arough scaling between the planetary size and the criticalperiod ratio for dynamical stability P = P P ≈ . K (cid:18) R p R ⊕ (cid:19) . (6)Below we adopt K = 20, although in reality the thresh-old on K also depends on many factors, such as indi-vidual planet masses, eccentricities and mutual inclina-tions, etc (e.g., Chambers et al. 1996; Zhou et al. 2007;see Pu & Wu 2015 for a detailed discussion). As Figure 2shows, the smallest planet detectable around a typical“noisy” star is 1 R ⊕ , for which the stability threshold is P crit , ≈ .
4. The smallest detectable around a typi-cal “quiet” star, in contrast, is 0 . R ⊕ , with a stabilitythreshold of P crit , ≈ .
2. This varying threshold is vis-ible in the lower middle panel of Figure 2 as well asthe right panel of Figure 8. Again for demonstrationpurposes I have differentiated the planet triplets fromthe quiet sample and noisy sample with different col-ors. No planet triplets from the noisy sample are belowthe red dashed line, which denotes P in P out = 2 ≈ P , ,whereas planet triplets from the quiet sample can extendfurther down to P , . This varying stability thresholdwas not taken into account in W18.The varying stability threshold applying to an inho-mogeneous stellar sample naturally leads to a spacingcorrelation, as is illustrated in the right panel of Fig-ure 3. Note the similarity between this plot and theright panel of Figure 8. Generating synthetic planetarysystems that meet all detection thresholds of individual atterns in Kepler multi-planet systems in O u t e r p e r i o d r a t i o o u t Quiet sampleNoisy sample i n o u t = in O u t e r p e r i o d r a t i o o u t Figure 8.
The outer period ratio versus the inner period ratio for all triplets (left panel) and the subset with both period ratios < ≥ < P in P out . The gray dashed line in the left panelshows an example. We also label with different colors the planet triplets from two stellar samples to highlight their differentdistributions. In particular, there is no planet triplet with P in P out < planets and of the triplet as well as the stability thresh-old is not trivial, so I cannot assess quantitatively theimpact of this effect on the Pearson r coefficient. How-ever, as a qualitative check, if only planet triplets fromthe noisy sample are used, I have r = 0 .
25 even with thesquare cut at P = 4. This is much smaller than whatone has ( r = 0 .
46) if both noisy and quiet samples areused. ON THE SIZE ORDERINGLissauer et al. (2011) and Ciardi et al. (2013) firstnoticed that the
Kepler multi-planet systems show asize-location correlation. Specifically, the larger planetin any planet pair is most often the one with the longerperiod. To check the statistical significance against ob-servation biases, these authors compared the radius ra-tio distributions between observation and simulation. Ingenerating simulated planet pairs, they randomly drewradii from the observed radius distribution and then, tomimic their selection procedure, eliminated those whichwould not be detected if either of the planets at theorbital period of the other one fell below the specifiedS/N threshold. Their simulated radius ratio distributionshowed equal number of planet pairs with R p , in < R p , out and R p , in > R p , out . Ciardi et al. (2013) also performedseveral other tests, including using different S/N thresh-olds and maximum periods. The size ordering was al- ways observed. Therefore, it was concluded that thesize-location correlation has a physical origin.This statistical approach suffers the same issue asthe W18 one. Using the CKS multi-planet sample, Ishow in Figure 9 the cumulative distributions of transitS/N ratios and radius ratios between planets in pairs.Similar to what Lissauer et al. (2011) and Ciardi etal. (2013) found, here I also have more than 60% ofplanet pairs showing the so-called size-location corre-lation: R p , in < R p , out . However, the S/N ratio is onaverage unity, suggesting that the transit signal of theinner planet is as strong as that of the outer one. This iswhat one expects if randomly pairs up the transit S/Nvalues from the observed S/N distribution. See the greencurve in the left panel of Figure 9. Given the relationbetween S / N and radius (Equation (1)), one can derivethe radius ratio from the S / N ratio(S / N) in (S / N) out ≈ (cid:18) R p , in R p , out (cid:19) (cid:18) P in P out (cid:19) − / . (7)In the above approximation we ignore the contributionfrom impact parameters. Because of the term involvingthe period ratio, two transit signals with equal S/N nat-urally lead to a pair of planets with R p , in < R p , out , thatis, the size-location correlation.The distributions from randomly sampling theS/N distribution do not match the observed distri-0 Wei Zhu (S/N) in /(S/N) out C D F AdjacentAllRandom 10 R p, in / R p, out C D F Figure 9.
Cumulative distributions of the transit S/N ratio (left panel) and the radius ratio (right panel) between planets ina pair. The ratio is specified as the property of the inner to the property of the outer. The two ways of forming planet pairsare studied. The blue curves use only pairs from adjacent planets, whereas the orange curves include pairs from non-adjacentplanets. The green curves are results from randomly sampling the S/N distribution. The vertical dashed lines mark the boundarywhere the quantity of the inner equals the quantity of the outer, and the horizontal solid lines with colors mark the values atwhich the curves meet these equalities. The green regions mark the 1 σ confidence interval, derived from 1000 realizations ofthe random sampling, and the horizontal dashed lines mark the median. On average the inner tranet has the same S/N as theouter one, which is what one expects from the random sampling. This naturally leads to an excess of pairs with R p , in < R p , out (i.e., the size-location correlation). butions perfectly, in particular in the range 1 < (S / N) in / (S / N) out <
4, or equivalently 1 < R p , in /R p , out <
2. There are two possible reasons. First, transitingplanets in some pairs do show a weak size correlation.This is also suggested in Section 4, as the random S / Nsampling cannot fully explain the observed correlationstrength (Figure 5). However, as is also suggested inSection 4, this can possibly be explained by transit bias,as we are not detecting all the planets in the same sys-tem. Regardless, the fraction of planet pairs that showthe size correlation is likely a small fraction ( (cid:46) / N ratio to almostexactly unity.Another possible reason that can account for the devi-ation is some subtle detection bias in the planet searchpipeline. When S/N values are randomly paired up, itis implicitly assumed that the detection efficiency doesnot depend on parameters other than S / N. This is notentirely true in reality. For the same value of S/N, thedetection efficiency decreases gradually with the orbitalperiod (see, e.g., Figure 9 of Thompson et al. 2018). Thiseffect biases against planet pairs with large R p , in /R p , out ,the type of pairs that are in short for the observational distribution to match the simulated one. Future detailedstudies are needed to quantify this effect.In short, the observed size-location correlation in Ke-pler multi-planet systems can be mostly, if not fully,explained by detection biases. It is possible that someplanet pairs do have ordered sizes, but they only consistof a small fraction ( (cid:46) DISCUSSIONIn this work, I re-examine several claims about the rel-ative sizes and spacings between
Kepler planets aroundthe same host. I make use of the observed transit S/Nvalues, because they are observationally more funda-mental than other parameters such as the planet radius.I present several findings: • The apparently similar sizes of planets in the same
Kepler system can be largely explained by the pro-jection of the same S/N cut onto different stellarproperties. • The apparently correlated spacings, measured inperiod ratios, between adjacent planet pairs in sys-tems with at least three detected planets are par-tially due to the arbitrary upper limit that W18 atterns in
Kepler multi-planet systems Planet mass ( M )01234567 D i s t a n c e t o c o mm e n s u r a b ili t y ( × ) R R R R Figure 10.
An illustration of the planet pairs used in Mill-holland et al. (2017) for claiming the intra-system mass uni-formity. For each pair, I show on the x -axis the masses ofindividual planets and on the y -axis the distance to exactperiod commensurability, ∆, given by Table 2 of Hadden &Lithwick (2017). The size of the symbol reflects the radiusof the planet. As ∆ decreases, lower masses can be detectedvia the TTV technique. This detection bias is not taken intoaccount by Millholland et al. (2017) in constructing syntheticsystems. imposed on the period ratio and partially due tothe varying stability threshold in different stellarsamples. • The observed size-location correlation can be ex-plained by the projection of the same S/N ontodifferent values of orbital period. As far as thetransit detection is concerned, the inner and theouter transiting planets on average have similarS/N values.The claim of intra-system mass uniformity by Mill-holland et al. (2017) suffers very similar issues. Below Idraw the analogy between Millholland et al. (2017) andW18 studies and defer a more quantitative analysis forfuture works. The analysis of Millholland et al. (2017)was performed on a sample of 89 planets from 37 Keplersystems, whose masses were constrained by Hadden &Lithwick (2017) through TTV. Whether or not a TTV Millholland et al. (2017) had 8 planets with only mass upperlimits, as indicated in Hadden & Lithwick (2017), in their sample:Kepler-23 d, Kepler-24 e, KOI-115.03, Kepler-105 b, Kepler-114 b,Kepler-114 d, Kepler-310 c, and KOI-427.01. It is not appropriate mass measurement can be made is more directly relatedto the TTV amplitude than to the planet mass. For apair of planets, the TTV amplitude is generally depen-dent on the distance from the period commensurability,∆, which is defined as (Lithwick et al. 2012)∆ ≡ (cid:12)(cid:12)(cid:12)(cid:12) P out P in J (cid:48) J − (cid:12)(cid:12)(cid:12)(cid:12) . (8)Here P in and P out are the orbital periods of the innerand outer planet in a TTV pair, respectively, and J (cid:48) /J is the closest small integer ratio for P in /P out . Otherthings being equal, a smaller ∆ means that a lowerplanet mass can be measured from TTV. This makesthe ∆ − m p relation (Figure 10) somewhat analogous tothe CDPP − R p relation (lower left panel of Figure 2).Consequently, Millholland et al. (2017) reshuffling theplanet mass is similar to W18 bootstrapping the planetradius. As shown in Section 4, this approach leads tobiased results.Therefore, the so-called intra-system uniformity andthe size ordering effect that appear in the Kepler multi-planet systems can be mostly, if not entirely, explainedby observational biases. As far as the data is able toinform, the physical properties of one
Kepler planet arelargely independent of the properties of both its siblingsand the parent star.So far the analysis has been done on the
Kepler multi-planet systems, but the same conclusion likely appliesto all
Kepler planets. It is true that over half of thetransiting planets were found in systems with only onetransiting planets (i.e., single-tranet systems). However,this is most likely a result of selection effect, as
Kepler only detects planets that transit the host star. The or-bital properties, such as eccentricity and mutual incli-nation, of single-tranets and multi-tranets are different(e.g., Xie et al. 2016; Zhu et al. 2018a; Van Eylen etal. 2019), but this does not necessarily mean that theirphysical properties are different as well. In fact, studieshave shown that the planetary properties and the stellarproperties of single-tranet and multi-tranet systems aresimilar (e.g., Munoz Romero, & Kempton 2018; Zhu etal. 2018a; Weiss et al. 2018b), suggesting they are likelythe same population. Nevertheless, future ground-basedradial velocity observations will be able to tell whetheror not this is true.A recent work by He et al. (2019) applied the full for-ward modeling method to study the
Kepler multi-planet to treat mass upper limits and mass measurements in the sameway. The removal of these planets reduces the number of transit-ing planets in four systems (Kepler-105, 114, 310, 549) down toone, so these systems are excluded from the sample. In the end Ihave 77 planets from 33 systems. Wei Zhu systems. The authors generated multi-planet systemsfollowing specific prescriptions, passed the simulatedsystems through simplified
Kepler detection pipeline,and compared their simulated planet catalogs to the realcatalog. The authors find a better match in a combina-tion of selected observables between the simulated andreal catalogs once the periods and radii of planets aroundthe same host are assumed to be correlated. However, itis unclear whether the improvement in the fit is due tophysical correlations or some artifacts in the model. Infact, as their best-fit models show (Figures 3-5 of He etal. 2019), the match to transit depth and transit depthratio distributions, the most relevant ones for the sizecorrelation, is not improved once the size correlation isintroduced. This is in agreement with the conclusionof the current paper that there is no evidence for thesize correlation. The spacing correlation is a much morecomplicated issue in such a full forward modeling ap-proach. To give a specific example, how to generate sta-ble multi-planet systems remains an unsolved problem.The critical spacing for long-term stability depends onmany factors, including the number of planets (i.e., mul-tiplicity, Funk et al. 2010) and orbital properties (i.e.,eccentricity and mutual inclination, Pu & Wu 2015), thelatter of which have also been shown to be multiplicity-dependent (Xie et al. 2016; Zhu et al. 2018a). Using afixed K value for all systems, as was done in He et al.(2019), is not realistic. More work is needed.The conclusion that the properties of Kepler planetsare largely independent of the properties of their sib-lings and the parent star has theoretical implications.Either the formation of
Kepler planets had almost norequirement for their birth environment, or the (likely)chaotic evolution erased their memory of the initial con-dition. This latter scenario is more likely once severalother pieces of evidence are put together.
Kepler multi- planet systems are shown to be dynamically compact(Pu & Wu 2015) and have very diverse compositions(Wu, & Lithwick 2013; Marcy et al. 2014; Hadden &Lithwick 2017). The nearly flat period ratio distribu-tion, arisen from either in situ formation (Petrovich etal. 2013; Wu et al. 2019) or breaking the chain of reso-nances after migration (Izidoro et al. 2017), also pointsto a stage of dynamical instability. Finally,
Kepler plan-ets are shown to be strongly correlated with outer giantplanets (Zhu & Wu 2018; Bryan et al. 2019; Herman etal. 2019). The planet-planet scatterings that are respon-sible for the large eccentricities of the cold giant planets(Chatterjee et al. 2008; Juri´c, & Tremaine 2008) caneasily drive dynamical instabilities in the inner system.As the orbital velocity far exceeds the escape velocityfor the majority of
Kepler planets, the encounters be-tween planets during the dynamical evolution can sig-nificantly revise their physical properties, thus removingthe imprint of the initial formation conditions. In otherwords, it will be difficult to infer the initial conditionsfrom the properties of current
Kepler planets. A dif-ferent conclusion was reached in Kipping (2018) fromstudying the size orderings of
Kepler planets. Kipping(2018) assumed that the observed orderings are physicaland free from observational biases. This is not true, asthe present study has shown.I would like to thank Subo Dong, Cristobal Petro-vich, Yanqin Wu, Norm Murray, and Eve Lee for dis-cussions, and particularly Xi Zhang for the suggestionabout the new test in the size correlation section. I alsothank the anonymous referees for comments and sugges-tions on the manuscript. I also thank Lauren Weiss forkindly sharing her code which helps reproduce their re-sults. W.Z. was supported by the Beatrice and VincentTremaine Fellowship at CITA.REFERENCES
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