On Saturn, solid diamond should exist from about 6,000 to 36,000 km below the cloud tops, reaching more than halfway down into the planet (Saturn's equatorial radius
is 60,000 km).
2] is the radius of the umbra (which term applies to both total and annular eclipses) at a height Z above the fundamental plane, and n is the speed of the shadow (my emphasis), both reckoned in units of the Earth's equatorial radius.
Multiplication by 6378, being the equatorial radius of the Earth in km, followed (if hourly variations are used for the derivatives) by division by 3600, the number of seconds in one hour, gives the result in km per second.
The centre of the lunar umbra in the fundamental plane has the coordinates (x,y) in units of the equatorial radius of the Earth.
2] = 1, where p is the geocentric distance of the poles in units of the equatorial radius of the Earth.