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Table 2

Summary of regression statistics for all triple orbital configurations, for regression equations aio ∕a2, aoi ∕a2 = C + b1μ1 + b2μ2 + b3e1 + b4e2 + b5i2 + b6Ω2 + b7ω2.

Orbit Crit. Higha Motionsb Crit. semi-major axis ratio Regression coefficients Integrations Model fit
type ratio incl. Star 3 Planet Min Mean σ Max C a μ1 μ2 e1 e2 i2 Ω2 ω2 No. Bounds Rate R2 F SE MAPE
found (%) (%)
P1 aio ∕a2 P P 0.131 0.383 0.147 0.892 0.439 0.000 −0.005 −0.020 −0.004 −0.114 0.000 −0.001 −0.001 5 413 1 782 33 0.132 34 0.093 19
R 0.113 0.519 0.108 0.828 0.572 0.000 0.009 −0.043 −0.010 −0.468 0.000 0.000 0.000 5 343 976 18 0.670 241 0.062 10
R P 0.110 0.385 0.077 0.897 0.463 0.000 0.192 −0.037 −0.021 −0.187 −0.001 0.000 0.000 2 122 569 27 0.547 285 0.358 9
R 0.112 0.537 0.132 0.857 0.656 0.000 −0.056 −0.022 0.005 −0.510 0.000 0.000 0.000 2 685 539 20 0.710 162 0.071 12
NK P P 0.049 0.484 0.099 0.837 0.545 0.000 −0.044 −0.018 0.000 0.000 −0.001 0.000 0.000 342 256 75 0.024 2 0.099 18
K P 0.288 0.495 0.100 0.748 0.303 0.001 −0.170 0.017 0.000 0.000 0.002 0.000 0.000 319 47 15 0.177 2 0.095 19
NK R P 0.423 0.666 0.078 0.853 0.685 −0.001 0.066 0.029 0.000 0.000 −0.001 0.000 0.000 273 178 65 0.202 4 0.024 13
K P 0.105 0.531 0.096 0.818 0.738 0.000 −0.038 −0.083 0.000 0.000 −0.001 0.000 0.000 207 102 49 0.194 6 0.088 12
P2 aoi ∕a2 P P 1.253 2.936 1.449 5.197 2.376 −0.003 0.053 0.044 0.090 1.996 0.000 −0.001 −0.002 5 413 5 133 95 0.829 3 097 0.248 6
R 1.184 1.976 0.507 4.916 1.482 −0.004 0.131 −0.008 −0.036 1.727 0.001 0.001 0.001 5 343 5 082 95 0.865 3 910 0.198 6
R P 0.924 2.773 0.891 5.384 2.089 −0.004 0.809 0.114 0.015 1.298 −0.001 0.000 0.000 2 122 1 895 89 0.271 26 0.126 31
R 0.591 1.960 0.571 4.957 1.708 −0.005 0.182 0.023 0.018 1.693 −0.001 0.000 0.000 2 685 2 455 91 0.773 1 039 0.269 8
NK P P 1.755 1.941 0.051 2.242 1.920 −0.002 0.048 0.016 0.000 0.000 0.002 0.000 0.000 342 341 100 0.269 30 0.044 4
K P 1.759 1.933 0.176 2.532 2.345 −0.004 0.066 0.043 0.000 0.000 −0.003 0.000 0.000 319 319 100 0.589 117 0.115 3
NK R P 1.095 1.335 0.127 1.820 1.288 −0.002 0.085 −0.004 0.000 0.000 0.007 0.000 0.000 273 273 100 0.555 83 0.085 4
K P 1.363 1.753 0.195 2.510 1.356 −0.003 0.033 −0.016 0.000 0.000 0.008 0.000 0.000 207 204 99 0.574 64 0.129 5
S1 aio ∕a1 P P 0.015 0.180 0.049 0.760 0.390 0.000 −0.746 0.005 −0.398 −0.006 0.000 0.000 0.000 1 532 819 54 0.583 142 0.081 35
R 0.015 0.185 0.122 0.771 0.392 0.000 −0.667 0.003 −0.381 0.002 0.000 0.000 0.000 1 480 785 53 0.567 131 0.081 37
R P 0.015 0.197 0.059 0.772 0.277 0.000 −0.387 −0.005 −0.125 −0.005 0.000 0.000 0.000 1 586 957 60 0.126 17 0.139 67
R 0.028 0.196 0.157 0.772 0.423 0.000 −0.451 −0.006 −0.056 −0.048 0.001 0.000 0.000 1 455 791 54 0.176 20 0.143 74
S3 aio ∕a2 P P 0.009 0.289 0.098 0.893 0.393 0.000 −0.075 0.037 −0.020 −0.604 0.000 0.000 0.000 1 604 1 599 100 0.712 481 0.084 25
R 0.010 0.361 0.158 0.920 0.444 0.001 0.001 0.113 −0.011 −0.580 0.000 0.000 0.000 1 562 1 501 96 0.790 705 0.071 17
R P 0.007 0.287 0.096 0.853 0.391 0.002 −0.071 0.033 −0.001 −0.597 0.000 0.000 0.000 1 596 1 591 100 0.727 522 0.077 23
R 0.009 0.360 0.156 0.908 0.425 0.000 −0.002 0.114 −0.002 −0.580 0.000 0.000 0.000 1 537 1 489 97 0.790 688 0.071 17
Averagec /totald 0.473 0.944 0.237 1.789 0.921 0.000 0.183 0.036 0.050 0.433 0.001 0.000 0.000 45 760 29 683 65 0.495 – – –

Notes. The directional movements of the stability bounds for planetary orbits are generally as expected, except for retrograde inner planetary orbits, for which the bound increases instead of shrinking. This may be attributable to the small sample, since very few retrograde star/retrograde planet combinations were stable. Italics denote coefficients with signs that were not as expected. Ignoring coefficients smaller than 0.01, only four did not have the expected sign and these were all small (< |0.06|). The semi-major axis ratio a is included as an error check as the critical ratio scales with it and its coefficient should be zero. Its average absolute value of < 0.0005 therefore provides an indication of the intrinsic error of the methodology. (a) NK – non-Kozai, K – Kozai. (b) P – prograde, R – retrograde. (c) Of absolute values for regression coefficients. (d)Of integrations and bounds.

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