How this calculator works
The horizon calculation uses a tangent from the observer to a sphere. For an object beyond that horizon, the line of sight hides its lower part. Simultaneous solar zenith angles at sites on the same meridian give a central-angle difference; dividing measured separation by that angle estimates Earth radius. Sunset delay comes from horizon dip and the solar hour-angle relation. These are idealised geometric predictions. Terrain, atmospheric refraction, timing and site alignment matter in real observations.
Formula
Horizon dip = acos(R/(R+h)); horizon surface distance = R times dip. Shadow experiment: circumference = 2 pi d/abs(theta_B-theta_A). Sunset hour angle satisfies cos(H) = [sin(elevation)-sin(latitude)sin(declination)]/[cos(latitude)cos(declination)].
Worked example
Shadow angles differing by 7.2 degrees across 800 km imply circumference 40000 km. On a 6371 km sphere, an eye height of 2 m gives a geometric horizon about 5.05 km away.
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Frequently asked questions
Does the model include refraction?
No. Refraction can shift an observed horizon and sunset, sometimes substantially.
Must the shadow measurements be simultaneous?
Yes, and the simple separation method assumes the sites lie on the same meridian with the relevant zenith-angle difference.
Why can sunset disappear from the result?
At some high-latitude seasons the Sun never crosses the selected horizon, so a repeated sunset time is undefined.