Saturday, June 15, 2024

Solar system means

Computer scienceSolar system means


Yesterday I stumbled on the fact that the size of Jupiter is roughly the geometric mean between the sizes of Earth and the Sun. That’s not too surprising: in some sense (i.e. on a logarithmic scale) Jupiter is the middle sized object in our solar system.

What I find more surprising is that a systematic search finds mean relationships that are far more accurate. The radius of Jupiter is within 5% of the geometric mean of the radii of the Earth and Sun. But all the mean relations below have an error less than 1%.

The radius of Mercury equals the geometric mean of the radii of the Moon and Mars, within 0.052%.

The radius of Mars equals the harmonic mean of the radii of the Moon and Jupiter, within 0.08%.

The radius of Uranus equals the arithmetic-geometric mean of the radii of Earth and Saturn, within 0.0018%.

See the links below for more on AM, GM, and AGM.

Now let’s look at masses.

\begin{eqnarray*} M_\Earth &=& \mbox{GM}\left(M_\Mercury, M_\Neptune\right) \\ M_\Pluto &=& \mbox{HM}\left(M_\Moon, M_\Mars\right) \\ M_\Uranus &=& \mbox{AGM}\left(M_\Moon, M_\Saturn\right) \\ \end{eqnarray*}

The mass of Earth is the geometric mean of the masses of Mercury and Neptune, within 2.75%. This is the least accurate approximation in this post.

The mass of Pluto is the harmonic mean of the masses of the Moon and Mars, within 0.7%.

The mass of Uranus is the arithmetic-geometric mean of the masses of of the Moon and Saturn, within 0.54%.

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