🪐 Orbital Dynamics & Solar System Spacing
Titius-Bode Law Planetary Distance Calculator
Compute theoretical orbital semi-major axis distances ($a = 0.4 + 0.3 \times 2^m\text{ AU}$) and compare historical geometric predictions against modern observed astronomical orbits.
Select Planet / Orbit Index
Earth Orbital Semi-Major Axis
1.000 AU (Predicted)
Predicted Dist (AU)
1.000 AU
Actual Dist (AU)
1.000 AU
Distance in Million km
149.60 M km
💡 Formula Calculation:
$$a = 0.4 + 0.3 \times 2^1 = 0.4 + 0.6 = 1.00\text{ AU}$$
The Historical Significance of the Titius-Bode Law
Formulated by Johann Daniel Titius in 1766 and popularized by Johann Elert Bode in 1772:
- Empirical Sequence: $$a_m = 0.4 + 0.3 \times 2^m \text{ AU}$$
- Discovery of Uranus (1781): William Herschel discovered Uranus at 19.2 AU, matching Bode's prediction ($19.6\text{ AU}$) with astonishing 2% accuracy.
- Discovery of Ceres (1801): Giuseppe Piazzi discovered dwarf planet Ceres in the gap at $m=3$ ($2.77\text{ AU}$ vs $2.8\text{ AU}$ predicted).
- Failure at Neptune (1846): Neptune was found at 30.1 AU instead of the predicted 38.8 AU (29% error), showing the rule is a resonant geometric coincidence.