We are revisiting Bode's Law as both Ceres and 2003 UB 313 'Eris' are having their status changed by the International Astronomical Union. Essentially Bode's Law discovered in 1766 by Daniel Titius and published in 1772 by Johann Elert Bode, says that the semi-major axis of the planet graphed against its planet number is exponential in nature. That is, the logarithm of the semi-major axis versus the planet number (now 1 to 11) is a straight line. But is this true? Below are the planetary distances and their logarithm values.
Name Number Distance Logarithm Least Square Recomputed Standard
N kilometers Distance of Logarithm distances Error
Mercury 1 57909175 17.87438639 17.83735485 55803930 -0.0364
Venus 2 108208930 18.49957445 18.36764570 94834678 -0.1236
Earth 3 149597890 18.82346152 18.89793655 161164566 0.0773
Mars 4 227936640 19.24457825 19.42822740 273887338 0.2016
Ceres 5 413937362 19.84122522 19.95851826 465451406 0.1244
Jupiter 6 778412020 20.47276653 20.48880911 791000463 0.0162
Saturn 7 1426725400 21.07864773 21.01909996 1344247163 -0.0578
Uranus 8 2870972200 21.77791656 21.54939082 2284449278 -0.2043
Neptune 9 4498252900 22.22695491 22.07968167 3882253686 -0.1369
Pluto 10 5906380000 22.49929896 22.60997252 6597604871 0.1170
Eris 11 10123139618 23.03808969 23.14026337 11212144685 0.1076
Restated Bode's law is Distance in Km = e ^(0.5303 N + 17.3071) Below is the graph of the logarithm of the distance versus planet number. It looks close to a straight line, so I would say that Titius and Bode got it right.
