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Sphere with Tunnel

Find the tunnel between two points $A$ and $B$ on a gravitating Sphere which gives the shortest transit time under the force of gravity. Assume the Sphere to be nonrotating, of Radius $a$, and with uniform density $\rho$. Then the standard form Euler-Lagrange Differential Equation in polar coordinates is

r_{\phi\phi}(r^3-ra^2)+{r_\phi}^2(2a^2-r^2)+a^2r^2 = 0,
\end{displaymath} (1)

along with the boundary conditions $r(\phi=0)=r_0$, $r_\phi(\phi=0)=0$, $r(\phi=\phi_A)=a$, and $r(\phi=\phi_B)=a$. Integrating once gives
{r_\phi}^2={a^2r^2\over {r_0}^2}{r^2-{r_0}^2\over a^2-r^2}.
\end{displaymath} (2)

But this is the equation of a Hypocycloid generated by a Circle of Radius ${1\over 2}(a-r_0)$ rolling inside the Circle of Radius $a$, so the tunnel is shaped like an arc of a Hypocycloid. The transit time from point $A$ to point $B$ is
T=\pi\sqrt{a^2-{r_0}^2\over ag},
\end{displaymath} (3)

g={GM\over a^2}={\textstyle{4\over 3}}\pi\rho G a
\end{displaymath} (4)

is the surface gravity with $G$ the universal gravitational constant.

© 1996-9 Eric W. Weisstein