Sphericity

Ellipsoidal objects

The sphericity, ${\displaystyle \mathrm{\Psi }}$, of an oblate spheroid (similar to the shape of the planet Earth) is:

${\displaystyle \mathrm{\Psi }=\frac{{\pi }^{\frac{1}{3}}(6V_p{)}^{\frac{2}{3}}}{A_p}=\frac{2\sqrt[3]{a{b}^2}}{a+\frac{b^2}{\sqrt{a^2-b^2}}\ln \left(\frac{a+\sqrt{a^2-b^2}}{b}\right)},}$

where a and b are the semi-major and semi-minor axes respectively.

Derivation

Hakon Wadell defined sphericity as the surface area of a sphere of the same volume as the particle divided by the actual surface area of the particle.

First we need to write surface area of the sphere, ${\displaystyle A_s}$ in terms of the volume of the object being measured, ${\displaystyle V_p}$

${\displaystyle A_s^3={\left(4\pi r^2\right)}^3=4^3{\pi }^3{r}^6=4\pi \left(4^2{\pi }^2{r}^6\right)=4\pi \cdot 3^2\left(\frac{4^2{\pi }^2}{3^2}{r}^6\right)=36\pi {\left(\frac{4\pi }{3}{r}^3\right)}^2=36\pi V_p^2}$

therefore

${\displaystyle A_s={\left(36\pi V_p^2\right)}^{\frac{1}{3}}={36}^{\frac{1}{3}}{\pi }^{\frac{1}{3}}{V}_p^{\frac{2}{3}}=6^{\frac{2}{3}}{\pi }^{\frac{1}{3}}{V}_p^{\frac{2}{3}}={\pi }^{\frac{1}{3}}{\left(6V_p\right)}^{\frac{2}{3}}}$

hence we define ${\displaystyle \mathrm{\Psi }}$ as:

${\displaystyle \mathrm{\Psi }=\frac{A_s}{A_p}=\frac{{\pi }^{\frac{1}{3}}{\left(6V_p\right)}^{\frac{2}{3}}}{A_p}}$

Sphericity of common objects