Chapter 5 – Solar System Orbits (Bode’s Law)

Bode’s Law

In 1772, the German astronomer Johann Bode demonstrated that a simple mathematical progression — eg 0 + 4, 3 + 4, 6 + 4, 12 + 4,… — could explain the average distance of successive planets from the Sun — where Earth had a value of 10 (6 + 4). At the time, only six planets were known to exist, not surprisingly, they fitted reasonably well with what came to be known as ‘Bode’s Law’.

The discovery of Uranus beyond Saturn in 1781 and the minor planet Ceres, between Mars and Jupiter, in 1801 confirmed the Law by falling nicely into place. However, the discovery of Neptune in 1846 and Pluto in 1930 did not uphold the Law — Pluto occupying the orbit allocated by Bode for Neptune as the following short program demonstrates. Perhaps you can improve on Bode to provide a better explanation?

The program
The upper half of the screen LISTs and DRAWs semi-orbital arcs calculated from the variable ‘bode’ (z + 4); the lower half the actual values as READ from DATA. The display is scaled to maximum size for the DRAW routine where PI = semi-circle, and the program stops with an error message (integer out of range) in attempting to DRAW the final arc for Pluto according to Bode’s Law. Thus Neptune’s actual orbit, in the lower display, fits the space between Uranus and Neptune as allocated by Bode and shown in the upper display. Study of the numerical values displayed confirms this as Figure 5.3 shows.

Figure 5.3

Note: This program, in demonstrating Bode’s Law, refers to average distances from the Sun. Most planetary orbits are not neat circles but slightly elliptical — highly so in Pluto’s case causing it to overlap Neptune’s orbit. The program Pluto’s Orbit, later in this chapter, fully explains this situation.

10 REM Titius-Bode’s Law
20 LET x=185: LET y=88
30 LET z=0: DIM a\$(32*11)
40 PRINT PAPER 6;a\$; PAPER 1;a\$: FLASH 1
50 PRINT AT 0,1;”Titius-Bode’s Law”
60 PRINT AT 11,1;”Actual dist”
70 FLASH 0: PAPER 6: INK 9
80 FOR n=1 TO 10
90 LET bode=z+4