Titius–Bode law

Later work by Blagg and Richardson significantly revised the original formula, and made predictions that were subsequently validated by new discoveries and observations. It is these re-formulations that offer "the best phenomenological representations of distances with which to investigate the theoretical significance of Titius–Bode type Laws".^[1]

Richardson formulation[edit]

In a 1945 Popular Astronomy magazine article,^[17] the science writer D.E. Richardson apparently independently arrived at the same conclusion as Blagg: That the progression ratio is 1.728 rather than 2. His spacing law is in the form:


$${\displaystyle \ R_{n}={\bigl (}\ 1.728\ {\bigr )}^{n}\ \varrho _{n}(\theta _{n})\ ,}$$


where ${\displaystyle \varrho _{n}}$ is an oscillatory function with period ${\displaystyle 2\pi }$, representing distances ${\displaystyle \varrho _{n}}$ from an off-centered origin to points on an ellipse.