KepVsSolSys  =  Kepler Exoplanet Systems vs. Solar System

The vast majority of transiting planets (candidates and confirmed ones) uncovered by the Kepler space telescope orbit quite close to their host stars compared to planetary distances found in the Solar System.  Even though the completeness of exoplanet-detection is (and may always be) at issue, it still seemed worth searching the data for any relationship between the relative orbital distances of multiple-planet Kepler systems and the Solar System.

All orbital data used was taken from the NASA Exoplanet Archives (NEA) as of 6-Dec-2013.  Exoplanet systems [candidates + confirmed] were separated by multiplicity after all False Positives were removed.  No attempt was made to remove other "outliers" even though a small percentage of exoplanet semi-major axes were well outside a normal distribution (> 3 sigma).  This resulted in the following:
1-Exoplanet systems: 2126
2-Exoplanet systems: 774
3-Exoplanet systems: 399
4-Exoplanet systems: 200
5-Exoplanet systems: 85
6-Exoplanet systems: 2
7-Exoplanet systems: 1 [NEA lists this (KOI-351.0x, Kepler-90x, KIC-11442793) as a 6-planet system but two very recent papers (Schmitt2013 and Cabrera2013) each disclosed a 7th planet with a orbital period = 124.9 days.]

Then each grouping's Positional Average Semi-Major Axes (SMA) were determined along with their standard deviations:

1-ExoP (2126 cases):    all planets:        0.1662 ± 0.2061 AU

2-ExoP (774 cases):    all 1st planets:    0.0867 ± 0.0670 AU
                                     all 2nd planets:   0.2021 ± 0.1847 AU

3-ExoP (399 cases):    all 1st planets:    0.0646 ± 0.0417 AU
                                     all 2nd planets:   0.1171 ± 0.0775 AU
                                     all 3rd planets:    0.2233 ± 0.1727 AU

4-ExoP (200 cases):    all 1st planets:    0.0510 ± 0.0225 AU
                                     all 2nd planets:   0.0826 ± 0.0333 AU
                                     all 3rd planets:    0.1295 ± 0.0561 AU
                                     all 4th planets:    0.2280 ± 0.1384 AU

5-ExoP (85 cases):    all 1st planets:    0.0429 ± 0.0154 AU
                                   all 2nd planets:   0.0651 ± 0.0230 AU
                                   all 3rd planets:    0.0913 ± 0.0324 AU
                                   all 4th planets:    0.1296 ± 0.0485 AU
                                   all 5th planets:    0.2135 ± 0.1018 AU

6-ExoP (2 cases):    all 1st planets:    0.0706 ± 0.0302 AU
                                 all 2nd planets:   0.0997 ± 0.0118 AU
                                 all 3rd planets:    0.1520 ± 0.0057 AU
                                 all 4th planets:    0.1999 ± 0.0054 AU
                                 all 5th planets:    0.2815 ± 0.0417 AU
                                 all 6th planets:    1.0450 ± 0.8146 AU

7-ExoP (1 case): not used in the analysis to follow; only one case known.

Solar System (data from NASA's Solar System Exploration web pages):
            1st (Mercury):               0.3870 AU
            2nd (Venus):                 0.7230 AU
            3rd (Earth):                   1.0000 AU
            4th (Mars):                    1.5240 AU
            "5th" (Avg.Asteroids):    2.8600 AU
            "6th" (Jupiter):               5.2030 AU
            "7th" (Saturn)                9.5820 AU
            "8th" (Uranus):              19.2290 AU
            "9th" (Neptune):            30.1040 AU

Graphs of these data are shown below in Figures 1. through 5. with or without error bars, with various y-axes, and as a semi-log plot.  In each case, "9" on the x-axis represents the Solar System where the Asteroid Belt has been designated as the "5th planet", Jupiter as the "6th planet", etc.

BLUE data points: best-fit binomial equation [y = 0.0143x^2 - 0.1161x + 0.2710; R^2 = 0.9902] through all average SMA's of 1st exoplanets in each multiple system, + MERCURY's SMA.

RED data points: best-fit binomial equation [y = 0.0309x^2 - 0.2687x + 0.6343; R^2 = 0.9922] through all average SMA's of 2nd exoplanets in each multiple system, + VENUS's SMA.

GREEN data points: best-fit binomial equation [y = 0.0501x^2 - 0.4739x + 1.2051; R^2 = 0.9989] through all average SMA's of 3rd exoplanets in each multiple system, + EARTH's SMA.

PURPLE data points: best-fit binomial equation [y = 0.0908x^2 - 0.9213x + 2.4623; R^2 = 1.0000] through all average SMA's of 4th exoplanets in each multiple system, + MARS's SMA.

ORANGE data points: best-fit binomial equation [y = 0.1979x^2 - 2.1088x + 5.8105; R^2 = 1.0000] through all average SMA's of 5th exoplanets in each multiple system, + ALL ASTEROID'S AVERAGE SMA.

BROWN data points: showing the average of the SMA's of 6th exoplanets in the two known 6-ExoP systems, + JUPITER's SMA.


While some of the errors are considerable, these plots point to a continuum that may exist for all multiple planetary systems including the Solar System.  If, indeed, the numerous phenomena that can affect the evolution of planetary systems (see Davies'13) occur randomly, perhaps the continuum of averages discussed here reflects some original conditions.  Long-term, it will be interesting to see if these observed trends will be further supported as additional 7-, 8-, 9-, and higher-exoplanet systems are uncovered, and how these observations bear on planet-formation and orbital-evolutionary theory.

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KOI's (Blue)  or
K2 Objects (Green)

• NASA Exoplanet Archives:
• Schmitt et al., 2013, arXiv:1310.5912.
• Cabrera et al., 2013, arXiv:1310.6248.
• NASA Solar System Exploration:
• Davies, Adams, Armitage, Chambers, Ford, Morbidelli, Raymond, & Veras, 2013, arXiv:1311.6816.

13-Dec-2013; 15-Dec-2013; last updated 17-Dec-2013.
Lastly, the standard deviations of each of the SMA averages presented in Figures 1. through 5. are "semi-log-plotted" in Figure 7.  It is not surprising that these diminish in sequential order from outermost toward innermost planets in each multiple system; exceptions are seen in the 6-ExoP system but these include only two known examples.
But for a given planetary position in the SERIES of multiple systems [i.e., all firsts (innermosts): BLUE circles; all seconds: RED squares; all thirds: GREEN triangles; all fourths: PURPLE diamonds; all fifths: ORANGE circles; and all sixths (outermost): BROWN squares], the standard deviations also diminish from lower to higher multiplicity systems (the only exception being the 1st planets of the two 6-ExoP systems).  This was not anticipated since the number of lower multiplicity systems substantially exceeds the number of higher ones.  This result points to less and less architectural variation in progressing from lower to higher multiplicity exoplanet systems.
All the Solar System planets [+ the asteroid belt], having very small SMA errors, virtually overlap in this semi-log plot at x = 9.
The close relationship within the above family of best-fit binomial curves is evidenced from plots of the coefficients against one another.  Given the general form: y = ax^2 + bx + c, and plotting the "a"s vs. "b"s, the "b"s vs. "c"s, and the "c"s vs. "a"s gave strong linear relationships, i.e.: with very high coefficients-of-determination of 0.9997, 0.9998, and 0.9991, respectively (Figure 6.).
Figure 5. is the semi-log version of Figures 1.-4. to which the one known Kepler 7-planet system (KOI-351) has also been added.  The curves have not been redetermined to include the 7-ExoP system points; using SMA averages of just two examples of 6-ExoP systems and only one 7-ExoP system is far from satisfying but, among the Kepler systems, those are the only ones known.  (It appears that the combined examples of Kepler and non-Kepler systems may improve this situation a bit…soon to be uploaded.)  For now, it appears that the non-monotonic nature of the curves is "suggested" by the 6-ExoP, 7-ExoP, and Solar systems.  Specifically for the 7-ExoP system (KOI-351), while the 1st and 2nd planet distances are low, the 3rd, 4th, and 5th are either close to their respective positional lines or at least distinctly higher than the corresponding positional 6-ExoP cases.