Correct Hidden Formula

Flat-Earthers like to use the erroneous “8 inches per mile squared” to calculate the height hidden by Earth’s curvature. The following is the correct equation for the purpose, accounting for the observer’s height & atmospheric refraction

h_\mathrm{h} = { R \over \cos\left[ { s \over R } - \arccos\left( { R \over R + h_\mathrm{O} } \right) \right] } - R

Or if the line of sight distance is known as opposed to the distance along Earth’s surface, the correct formula is:

h_\mathrm{h} = \sqrt{ { \left[ d - \sqrt{ { ( R + h_\mathrm{O} ) }^2 - { R }^2 } \right] }^2 + { R }^2 } - R

To account for atmospheric refraction, we need to arrive at the figure of R:

R = { R_\mathrm{E} \over k }

Where:

  • hh = hidden height
  • d = line of sight distance between the observer and the object
  • s = the distance between the observer and the object along Earth’s surface
  • h0 = the height of the observer from Earth’s surface
  • R = radius of the curved light ray due to refraction
  • RE = radius of the Earth without refraction
  • k = refraction coefficient, the exact value depends on the atmospheric condition, 0.13, 0.143 or 0.17 are common nominal values.

References