planet: mercury venus earth mars jupiter saturn R max 69.7 109 152 249 815.7 1507 R-mean 57.91 108.21 149.6 227.94 778.34 1427.01 R min 45.9 107.4 147 206 740.9 1347
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R max = maximum distance planet to sun, in millions of kilometers (Mkm)
R min = minimum distance planet to sun, in Mkm R-mean = semi-major axis of ellipse (a = R max + R min / 2), in Mkm
Ratios for Gap #1: Saturn/JupiterKepler's strategy in Ch. 21 was to calculate nine ratios and choose the best fit. For example: R-max, R-mean, or R-min for Saturn, divided by R-max, R-mean, or R-min for Jupiter, produces the following nine ratios: |
Ratios Saturn-max Saturn-mean Saturn-min Jupiter-max 1.8475 1.7494 1.6513 Jupiter-mean 1.9361 1.8333 1.7306 Jupiter-min 2.0340 1.9260 1.8181
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These nine ratios
are to be compared with the harmonic ratio for the cube, ~1.7321.
Based on the Copernican data available to him, Kepler chose
the ratio in the center for his model (Saturn-mean/Jupiter-mean),
but with this modern data, the ratio to the right of center
(Saturn-min/Jupiter-mean) is the best fit.
As this strategy produces a spectrum of possible ratios to choose, spanning densely enough the interval [1.6, 2.5], it does seem a bit like cheating. We suggest that only four of the nine ratios make sense for Kepler's model (max/max, mean/mean, min/min, and min/max). And further, the same ratio ought to be chosen for each of the five gaps. Following the actual model illustrated in the frontispiece of the Mysterium Cosmographicum, we will settle on the upper right corner ratio (min/max). This means that the cube is to fit between the inside of the larger shell of Saturn, and the outside of the smaller shell of Jupiter. Given this choice of ratio and our modern data, the error between Kepler's model and the geometry of the solar system is: (1.7321 - 1.6513)/1.6513 ~= 4.9% Kepler's choices for the other gapsAs indicated above, Kepler chose mean/mean for Gap #1. For Gap #2 (Jupiter/Mars) and Gap #3 (Mars/Earth), he chose our favorite, min/max. For Gaps #4 and #5 he chose max/max.Revised 20 jan 2002 by ralph abraham |