Titius–Bode law

The Titius–Bode law (sometimes termed just Bode's law) is a hypothesis that the bodies in some orbital systems, including the Sun's, orbit at semi-major axes in a function of planetary sequence. The formula suggests that, extending outward, each planet would be approximately twice as far from the Sun as the one before. The hypothesis correctly anticipated the orbits of Ceres (in the asteroid belt) and Uranus, but failed as a predictor of Neptune's orbit and has eventually been superseded as a theory of solar system formation. It is named for Johann Daniel Titius and Johann Elert Bode.

The law relates the semi-major axis ${\displaystyle a}$ of each planet outward from the Sun in units such that the Earth's semi-major axis is equal to 10:

where ${\displaystyle x=0,3,6,12,24,48...}$ with the exception of the first step, each value is twice the previous value. There is another representation of the formula: ${\displaystyle a=2^{n}\times 3+4}$ where ${\displaystyle n=-\infty ,0,1,2...}$. The resulting values can be divided by 10 to convert them into astronomical units (AU), resulting in the expression

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