ARCHITECTURE OF KEPLER'S MULTI-TRANSITING SYSTEMS. II. NEW INVESTIGATIONS WITH TWICE AS MANY CANDIDATES

The most closely spaced pair of new candidates are 2248.01 and 2248.04 in KOI-2248 with a period ratio of 1.065. This pair is unlikely to be stable if both these planets are orbiting the same star (due to the separation in terms of Hill radii is likely small; see below). The same situation is discussed in Paper I for KOI-284, where candidates 284.02 and 284.03 have a period ratio of 1.038. In systems such as this, where transits are detected with a low S/N, we must consider the possibility that some subset of the transits were not detected, or spurious transits were detected, thus the observed period is an alias of the true one. We checked aliases at periods 1/4, 1/3, 1/2, 2, 3, and 4 times the nominal period by measuring the depth of the signal at locations implied by those periods. For the KOI-2248 system, the signals are consistent with the reported periods, and inconsistent with these possible aliases.

Likely alternative explanations for this system are (1) one or both candidates is actually a blended eclipsing binary or (2) the two are true planets, but orbiting different members of a wide binary star (Lissauer et al. 2012, 2014). Following Steffen et al. (2010), we can consider the ratio of orbital-velocity normalized transit durations:

$$\begin{equation} \xi \equiv \frac{T_{\rm dur,in} / P_{\rm in}^{1/3}}{T_{\rm dur,out} / P_{\rm out}^{1/3}}, \end{equation} \tag{ 4 }$$

where T_dur is the transit duration and P is the orbital period.²¹ If this quantity is near unity, it implies that the planets are orbiting stars of roughly equal density (perhaps the same star; Lissauer et al. 2012). For the unstable pairs in KOI-284 and KOI-2248, the value of ξ is 0.96 and 0.97, respectively; this is sufficiently close to 1 that this test fails to give evidence of the pairs orbiting different stars. However, it suggests that if the planets are orbiting different stars in a physical binary, then the two stars may be similar and resolvable with high quality imaging–this has already been achieved for KOI-284 (Lissauer et al. 2012).

The next closest pair is KOI-2174, with a period ratio 1.1542 between 2174.03 and 2174.01. We performed the same alias check described above. In this case, every other transit of the smallest planet 2174.03 (at period 7.725 days) is shallower and is marginally consistent with zero (509 ± 57 ppm versus 105 ± 63 ppm). Therefore we adopt an ephemeris with the period doubled (BJD = 15.4502 × E +245509.8024, where E is an integer), in Table 1.

As we continue to wider period ratios, we no longer find reason on stability grounds to question the hypothesis that the systems are truly multiple planets orbiting an individual star. We now discuss other dynamically interesting systems that are closely packed. KOI-1665 has a period ratio 1.17219 between 1665.01 and 1665.02. These are small candidates (1.2 and 1.0 R_⊕) around a solar-type star, so the alias check above is not as powerful. Nevertheless, it raises no suspicion of the measured periods being incorrect. Given the planets' small sizes, they are likely to have low masses and even this extreme period ratio is stable in the Hill sense (see Section 3.3 below). KOI-262 (Kepler 50) has a nearly exact 6:5 commensurability, with a period ratio of 1.20010 ± 0.00003. The planetary nature of this system was confirmed by transit timing variations (Steffen et al. 2012). All other planet pairs have period ratios >1.25. In fact, in Section 4, we note that there may be a significant excess of planet pairs just wide of that period ratio, with planet sizes between Earth and Neptune. Kepler-11b and c (Lissauer et al. 2011a) are confirmed examples of this variety.

We also checked for potential three-body resonances among planets in systems of higher multiplicity. Though we do not investigate whether these resonances are overabundant relative to a random distribution, we point them out because they have a dynamical effect on the systems. Following Quillen (2011), we searched for small values of the frequency

$$\begin{equation} f_{\rm 3\hbox{-}body} = p f_{\rm in} - (p + q) f_{\rm mid} + q f_{\rm out}, \end{equation} \tag{ 5 }$$

where f_in, f_mid, and f_out are the orbital frequencies (inverse periods) of the innermost, middle, and outermost planets, respectively, and p and q are integers. We recovered the possible resonant chain of KOI-730, four planet candidates with period ratios near 4:3, 3:2, and 4:3, as described by Paper I. We also found KOI-2086 (Kepler-60; Steffen et al. 2012), whose three planets are in or near an even more closely packed chain of first-order resonances, 5:4 and 4:3, and where both neighboring pairs of planets orbiting KOI-2086 are offset by the same amount from the two-body resonances:

$$\begin{eqnarray} &&4 f_{\rm in} - 5 f_{\rm mid} = -0{.\!\!^\circ }10 \pm 0{.\!\!^\circ }03\ {\rm day}^{-1}, \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} &&3 f_{\rm mid} - 4 f_{\rm out} = -0{.\!\!^\circ }09 \pm 0{.\!\!^\circ }02\ {\rm day}^{-1}, \end{eqnarray} \tag{ 7 }$$

such that the combined three-body frequency f_3-body, with (p, q) = (1, 1), is −0004 ± 0009 day⁻¹. This is considerably closer to zero than its pair of two-body equivalents, which suggests that this resonance chain could have dynamical significance. This fact places this system, in terms of its proximity to a multibody resonance chain, between KOI-730 and KOI-500. In the latter, the outer four planets are more significantly offset from the two-body resonances, yet are consistent with a three-body resonance (as described in Paper I). A final case of a possible three-body resonance is KOI-720 with planet pairs that are relatively far from two-body resonances, yet the planets 720.04, 720.01, and 720.03 have f_3-body = 2f_in − 5f_mid + 3f_out, of 000 ± 002 day⁻¹. This is despite another candidate planet (720.02) orbiting among them, with an orbital period greater than that of 720.01 but less than that of 720.03. We numerically integrated Newton's equations to model the four planets of KOI-720, starting them on circular orbits with the periods and phases inferred from the transit observations (Table 1), a stellar mass of 0.72 M_☉, and with Equation (1) giving planet masses of 2.0, 7.5, 6.8, 8.0 M_⊕, from the inner to outer planet. The combination of mean motions 2λ_in − 5λ_mid + 3λ_out librated around 180°, with a period of 300 yr, and with an amplitude of 30°. Thus, this three-body resonance has dynamical significance for this system, and a dedicated study of these effects seems warranted.

Next, we investigate stability of the candidate systems. As noted in Paper I, for two-planet systems there exists an analytic Hill-stability criterion, where the planet orbits are unable to cross (e.g., Marchal & Bozis 1982). If the two planets begin on circular orbits with an orbital separation,

$$\begin{equation} \Delta > 2 \sqrt{3}, \end{equation} \tag{ 8 }$$

then they are Hill stable (Gladman 1993). Values of Δ are given for the observed pairs in Table 1. Only KOI-284 and KOI-2248 (see Section 3.1) host pairs of planet candidates that contradict this criterion. In particular, all two-planet candidate systems obey this stability criterion, so we judge them to be plausibly stable.

We are aware of no analytic stability criterion for the systems with more than two planets. However, in systems of three or more planets, the instability timescale generally increases with separation, as in the two-planet case (Chambers et al. 1996; Smith & Lissauer 2009). In Paper I, we numerically integrated all of the systems with more than two planets for 10¹⁰ orbits of the innermost planet using MERCURY (Chambers 1999). We started from circular, coplanar orbits and used our power-law mass–radius relationship. In addition to each pair obeying two-planet stability criteria, we suggested a conservative heuristic criterion,

$$\begin{equation} \Delta _{\rm in} + \Delta _{\rm out}>18, \end{equation} \tag{ 9 }$$

where the "in" and "out" subscripts pertain to the inner pair and the outer pair of three adjacent planets. This latter criterion does not assure stability (particularly if planets are eccentric), though it suggests there is no reason to suspect the system would be unstable, based on the planet sizes and periods sensed by transits. In Figure 2, we plot the Δ's for inner and outer pairs of threesomes. Systems satisfying criteria 8 and 9 we do not analyze further, but the other systems may be unstable and call for further analysis. Most of these systems were already examined in either Paper I or Lissauer et al. (2012). We numerically integrated the remaining three, KOI-620 (Kepler-51; Steffen et al. 2012), KOI-1557, and KOI-2086 (Kepler-60), as described in Paper I. We found them to be plausibly stable. That is, starting on circular, coplanar orbits matching the phase and periods of the data, they suffered neither ejection, nor collision, nor a close encounter within three mutual Hill radii over a time span of 10¹⁰ orbits of the innermost planet (usually of order 10⁸ yr). We also integrated KOI-961 for the same duration using the masses of Muirhead et al. (2012), and found them to be similarly stable.

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The only new system that is unstable was KOI-2248, discussed in Section 3.1. Using the Burlisch–Stoer integrator in MERCURY, the planets began violent gravitational scattering in several synodic timescales. Clearly, this system needs a qualitatively different understanding for its architecture, as noted above. One final system where at least one new planet appears close to instability is KOI-707 = Kepler-33. An analysis of the stability of this system was carried out in the discovery paper (Lissauer et al. 2012), so we performed no additional analysis here.

These outcomes of our stability analysis are for the power-law M_p–R_p relationship (Equation (1)) with α = 2.06. To see how many systems would be unstable if the planets were denser, we considered a larger α value for planets below 2 R_⊕ (an estimate of the Super-Earth/mini-Neptune boundary). We looked for $\Delta < 2 \sqrt{3}$ for any adjacent pair. Below 2 R_⊕, for any α below 6.9, no additional systems violate Hill's stability given circular orbits. Therefore all these planets may have an Earthlike composition, for which α ≃ 3.7 (Valencia et al. 2006), and not violate stability limits.

For planets larger than 2 R_⊕, we may also consider denser planetary structures, by varying α. No additional systems display instability for α ⩽ 2.6 (i.e., M_p = M_⊕(R_p/R_⊕)^2.6). For the slightly higher value of α = 2.7, the pair of planets of KOI-523 and the outer two planets of KOI-620 would be unstable, according to our numerical integrations. In KOI-523, the planets would have (mass, radius) of (0.99 M_Nep, 0.72 R_Nep) and (1.99 M_Sat, 0.74 R_Sat), i.e., a Neptune-mass planet that is 2.6 times as dense as Neptune, and a planet twice as massive as Saturn but that occupies less than half of Saturn's volume. In KOI-620, the planets would have (mass, radius) of (1.16 M_Sat, 0.60 R_Sat) and (1.44 M_Jup, 0.86 R_⊕): more massive, yet much smaller versions of the solar system's gas giants. Such a large α would imply an extreme density for gas-giant planets, even exceeding that of the core-heavy transiting planet HD 149026 (Sato et al. 2005). From this exercise, we see that our conclusions about stability are not sensitive to our adopted masses. Conversely, stability considerations give us little useful constraint on these planets' physical structure.

To summarize this stability study, for all the pairs of planet candidates, only two are suspected to be unstable on million-year timescales given low eccentricities and inclinations: KOI-284 and KOI-2248. Higher multiplicities do not appear unstable either, based on numerical integrations. Using a mass–radius relationship favoring high density, a few more systems could be unstable. We repeat the caveat that we have only considered instability while using initially planar, circular orbits; if these systems contain planets in eccentric orbits, they would likely be less stable.

Morton & Johnson (2011) have emphasized that planet candidates from Kepler, once properly vetted using indicators in the data itself, tend to be highly reliable (>90%), and Lissauer et al. (2012) extended and strengthened this statement for candidate multiple-planet systems. The density of background eclipsing binaries is so low, and the small depth and detailed shape of transits is unlikely to be mimicked because of the photometric precision of Kepler, that more than one pattern of transit signals on a single target are unlikely to occur via a blending of stellar eclipses with additional stellar light. Moreover, Kepler's photometric precision and centroid analyses means transit events occurring on background stars must lie very near the target star, in projection, which is unlikely.

We can now address the statistical reliability of Kepler's multiplanet candidates from a new and independent angle. With so few candidate planetary systems showing instability (2 out of 761 pairs, including the higher-order multiples), we expect most of these candidate systems are real planetary systems. Consider the possibility that pairs are "split multis," defined as a system that appears to be a pair of planets around a star, but the events are actually split into more than one system. The most likely alternatives are (1) one or both members of the pair of planetary candidates is a blended eclipsing binary, or (2) both planet candidates are planets, but they orbit different stars (Lissauer et al. 2012). Such cases need not obey stability constraints. Therefore we can estimate an expected fraction of apparently unstable systems, given the hypothesis that all these candidate systems are split multis.

If we draw two planets from the (P, M_p/M_⋆) values of all the planet candidates in multis, and consider whether that pair would be stable if in the same system, $\Delta \le 2 \sqrt{3}$ occurs in 25,867/403,651 ≃ 6.4% of the draws (the integer numbers are computed from sampling all the possible pairs). That is, one would expect 48.8 pairs to be unstable over the whole set of 761 pairs. Using the Poisson distribution, to have found two or fewer unstable pairs given the expectation value λ = 48.8 has a tiny probability of 8 × 10⁻¹⁹.

On the other hand, if only a fraction f of the systems are split multis, then the expected value of apparently unstable systems falls to λf. Given that only two systems in our sample appear to be unstable, we can place Bayesian constraints on the fraction f. Let us take a prior probability distribution of f which is uniform from 0 to 1: p(f) = 1. Then we can apply Bayes' theorem to estimate the probability of f given the observations:

$$\begin{equation} P(f | {\rm data}) = \frac{ P({\rm data} | f) }{ \int _0^1 df^{\prime } P({\rm data} | f^{\prime })}, \end{equation} \tag{ 10 }$$

where P(data|f) is the probability of the data given f and f' is an integration variable used to determine the normalization. The only information we use (i.e., the "data") is that there are two apparently unstable systems. This probability distribution is given in Figure 3, which shows a mode of 4.1% and a wide range of possible fractions: the 95% credible interval is 1.3%–14.7%. These estimate are marginally larger than the ≲ 2% of the candidates in multiple systems not being true planets estimated by Lissauer et al. (2012). We note that for the present estimate, we are counting planets that are around two different stars in a physically bound binary as a split multi, which as discussed in Section 3.1, is likely to account for one of our unstable pairs, KOI-284, and may well account for the other, KOI-2248.

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These estimates are based on drawing an ensemble of P and M_p/M_⋆ values, which were in turn assumed to follow certain distributions, so let us examine the robustness of the conclusions to varying those assumptions. First, we chose a period distribution P matching the planet candidates in multiple systems. This distribution nearly matches the single-candidate period distribution, so this is appropriate if the split multi hypothesis is that a pair of planets are actually singles around hosts that are blended together. However, this distribution is narrower than the detached eclipsing binary distribution, which may be blended into some of the targets to produce the split-multi signal. To explore this, we selected M_p/M_⋆ as above (a reflection of the distribution of observed depths) but replaced the periods by two draws from the list of eclipsing binaries labeled "detached" by Slawson et al. (2011). This was done in a Monte Carlo fashion, resulting in an unstable fraction λ/761 = 5.07% ± 0.04%. Given that this expectation is lower than above, the fraction of split multis would need to be higher in order to produce two apparently unstable systems: assuming the periods are drawn from the eclipsing binary distribution, the 95% credible interval of f spans 1.7%–18.6%. The second assumption is the particular mass–radius relationship we adopted, which gave M_p/M_⋆. If the planets are actually denser than assumed, more systems would be deemed unstable. Above we tested the sensitivity of our stability results to varying the mass–radius relationship for the known systems, and we found that extreme densities are needed for any additional planetary systems to be unstable. For these random pairs, the number of unstable systems expected would vary smoothly with their assumed masses, and hence the range of f would vary as well.

By considering stability, we have seen that ∼96% of the pairs of multi-transiting candidates are actually planets around the same star. Recall that this estimate is independent of that by Lissauer et al. (2012), who used binary statistics to estimate that in fully vetted systems, ≳ 98% are real planets. In the following sections we rely on such high fidelity, assuming that all the systems are real as we characterize their architectures. Because of their apparent instability, from this point on we cull KOI-284.02, KOI-284.03, KOI-2248.01, and KOI-2248.04. KOI-284 becomes a single-planet system and is not included in the analysis, and KOI-2248 becomes a two-planet system and is analyzed as such.

Here we assume the planetary candidates are in true systems orbiting the same star, and we investigate how the distribution of duration ratios depends on coplanarity. In the limit that all the planetary orbits within a system are circular and coplanar, the impact parameters b and semi-major axes a have the relationship:

$$\begin{equation} b_{\rm out} / b_{\rm in} = a_{\rm out} / a_{\rm in} \quad [{\rm coplanar, circular}], \end{equation} \tag{ 12 }$$

where "in" signifies an interior planet and "out" signifies an outer planet. Thus we expect that b_out will be larger than b_in in systems where both planets are close to coplanar and both transit. At the other extreme, planets around the same star may be sufficiently misaligned to destroy such correlations, which requires a typical mutual inclination i ≳ R_⋆/a, where R_⋆ is the host's radius and a is a typical semi-major axis. In that case, both b_in and b_out would be drawn from the same distribution, and b_in would be larger than b_out half the time. Such a cases have been observed, for instance, in the Kepler-11 e/g and Kepler-10 b/c pairs, the observed b_out is smaller than that given by Equation (12), and the orbits must deviate from coplanarity by at least 1° and 5°, respectively, for these particular pairs. Thus we expect the distribution of impact parameters can help us determine the distribution of mutual inclinations.

We do not have sufficiently accurate stellar properties or good knowledge of impact parameters themselves to perform such comparisons directly. However, transit durations T_dur, from first to fourth contact, are generally well measured and are ≃ 2((1 + r)² − b²)^1/2R_⋆/v_orb, where r ≡ R_p/R_⋆ and v_orb∝P^−1/3. Therefore for each planet in a system, the function ((1 + r)² − b²)^1/2 is proportional to T_dur/P^1/3. The ratio of this latter quantity for the pair of planets, ξ (Equation (4)), is the quantity that is precisely measured and is sensitive to the mutual inclination of the orbits through their relative impact parameters. Given the distribution of b_in of inner planets will be biased toward smaller values if the systems are nearly coplanar, and in most cases r ≪ 1, we expect the ξ to be greater than 1 for nearly coplanar systems. In the limit that misalignment removes impact-parameter correlations, ξ and ξ⁻¹ will have the same distribution.

To simulate the observed ξ distribution, we should take into account photometric noise and eccentricities. Photometric noise typically introduces a relative uncertainty of ∼1% in a duration measurement: σ_dur ≃ T_dur(2r)^1/2/S/N (Carter et al. 2008). We add a Gaussian-random deviate with this standard deviation to the simulated durations. Eccentricity has two effects on the duration: (1) at a given inclination, due to a different star–planet separation, it results in a different impact parameter and transit chord and (2) the projected orbital speed differs from a circular orbit, as does the speed projected on the sky plane; we model both of these effects with Keplerian orbits with uniform-random periastron angle ω. With these effects in place, the population model assumes that mutual inclinations of planetary orbits are excited to a scale δ, and eccentricities of both planets are excited to a scale a factor n times δ. That is, the energy in the eccentricity epicycles is a certain number times the energy in the inclination epicycles.

Both the mutual inclination and eccentricity distributions are modeled as Rayleigh distributions, such that the Rayleigh widths are σ_i = δ (in radians) and σ_e = nδ. This allows us to use a Monte Carlo method to create simulated distributions of ξ as a function of δ and n. To evaluate this distribution, we make 250 mock transit systems for each observed pair of planets, where we have taken only the pairs where both planets are detected at S/N >7.1, the nominal detection limit.

For each mock system we first draw the eccentricities esin ω and ecos ω from Gaussian distributions of width nδ (resulting in a uniform distribution of ω values and a Rayleigh distribution of e values). We discard a trial if either planet's eccentricity is above 1 or if the inner planet's apocenter distance exceeds the outer planet's pericenter distance, given their period ratio—a simple stability criterion. Step two is to draw b_out uniformly within [0, b_{out, max}], where b_{out, max} is the impact parameter the planet would need for the total S/N of the outer planet to drop to 7.1. This modeled S/N is taken as the observed S/N times the square root of the ratio between the modeled duration and the observed one. Step three is to draw b_in from a distribution centered on b_out(a_in/a_out) (Equation (12)) and having a Gaussian σ = δa/(R_⋆ + R_p) (resulting in a Rayleigh distribution of width σ_i = δ in mutual inclination). If |b_in| > b_{in, max}, as above, this planet would not be detected in transit. If the conditions for acceptance are not met at each of these three steps, the process begins anew at step one. If accepted, the mock system's ξ value contributes to the simulated ξ distribution. We compare these models for the distribution of ξ to the data with statistical tests.

Let us first test the null hypothesis that planets around the same star are sufficiently misaligned to destroy the normalized-duration ratio signature discussed above. Assuming their impact parameters are drawn from the same distribution, $T_{\rm dur,in}/P_{\rm in}^{1/3}$ and $T_{\rm dur,out}/P_{\rm out}^{1/3}$ would be distributed in the same manner, therefore their ratios ξ and ξ⁻¹ should also be from the same distribution. We test that in Figure 6, panel (a), where the null hypothesis is that the black and red histograms are equivalent. These histograms are not equal, with the center-of-mass of ξ lying at a significantly larger value than ξ⁻¹, a K-S p-value of 5 × 10⁻¹⁵. This is the signature of planetary orbits lying in nearly the same plane.

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There are potential biases that could affect this conclusion. First, the outer planet's radius is typically larger than the inner one (perhaps due to detection limits; see Paper I), but this would bias ξ to values slightly less than 1, and we observe the opposite. Another aspect is that the box least-squares search that found most of these candidates (B13) was run over a range of durations 0.003–0.05P. Planets outside this range were still found, but not with an optimal matched filter. The search algorithm is less sensitive to the very shortest durations (largest impact parameters) at long period and the very longest durations (smallest impact parameters) at short period. Therefore this effect should bias ξ downward, again against the observed trend. We have not identified other instrumental or analysis biases that could push the distribution to ξ > 1 values, as observed.

Although this test shows the observed ξ distribution is asymmetric, it is indeed quite broad. An ideal model distribution consisting of perfectly coplanar and circular systems would lie entirely above 1, due to the relation in Equation (12). Measurement error introduces additional spread at the few-percent level; modeling this effect gives the green curve in Figure 6, panel (b), which departs only slightly from this ideal.

Therefore, some mutual inclination or eccentricity is clearly needed. To model these, we computed a grid of models (described in Section 5.1) with steps of 0.002 in δ and 1 in n, and we show in Figure 7 the p-value from the K-S test for these models. The peak (best-fit) value has a probability 0.033 and lies at δ = 0.032, n = 1, corresponding to inclinations of ∼18 and eccentricities of 0.032. The typical mutual inclination lies firmly in the range 10–22, showing that planetary systems tend to be quite flat.

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In contrast to this narrow range of mutual inclination, the ξ distribution can be acceptably matched (p-value > 0.01) over a wide range of eccentricities. The geometrical reason for the different dependence on inclination and eccentricity is that if inclinations change by 1%, the duration may change by order unity, but if eccentricities change by 1%, the duration usually changes by ∼1%. Good fits to the ξ distribution can be obtained both for circular orbits and for eccentricities in energy equipartition with the mutual inclination (i.e., σ_e = 2σ_i), and indeed up to several times equipartition. Therefore the ξ distribution is insensitive to the eccentricity distribution, and our conclusion about mutual inclination does not depend on knowing the eccentricity distribution precisely.

Our conclusion that Kepler's planetary systems are flat supports our inference of Paper I, which used the number of planets of each multiplicity to show that the mutual inclination in systems is typically just a few degrees. However, it was possible that mutual inclinations are larger than 10°, provided that planetary systems have many more planets (≳ 10) than expected. To rule out this latter possibility, the radial-velocity (RV) sample was used to place limits on planet multiplicity (Tremaine & Dong 2012; Figueira et al. 2012), breaking the degeneracy and preferring small planetary inclinations of just a few degrees. This conclusion requires significant overlap between the current RV sample and the Kepler sample, which has not been independently demonstrated. Having reached the same conclusion from the Kepler sample alone, we have increased confidence that most planetary systems within 0.5 AU of their stars are flat.

By simulating all planet configurations and only comparing the doubly transiting simulated pairs to the data, our determination of σ_i is unbiased. However, a caveat is that the distribution of inclinations may not be well-characterized by a single Rayleigh distribution, and high-inclination components of the actual distribution would contribute less statistical weight because transits of both planets would be seen only rarely. Thus, as with all applications of parameter-fitting, the limits given on the parameter σ_i hold only to the extent that a member of the family of model distributions describes the actual distribution. Another caveat is that we have used all pairs of planet candidates throughout, such that the N-planet systems are represented more, by a total of N(N − 1)/2 pairs. Therefore the architectures of larger-N systems carry more statistical weight in this analysis.