The two methods I know to measure Earth's radius — the method of Eratosthenes and the stopwatch method — require the Sun visible during measurement. But in my location such weather is quite rare: it's almost always very cloudy.

What are some alternative methods of measuring the Earth's radius, which don't require clear sky?

  • $\begingroup$ Related: physics.stackexchange.com/q/26427/2451 and links therein. $\endgroup$ – Qmechanic Oct 22 '17 at 20:23
  • $\begingroup$ The stopwatch method you link is a special case of a large class of approaches involving the distance to the horizon. Those that use terrestrial objects as scales and markers require good visibility toward the horizon, but not clear skies. $\endgroup$ – dmckee --- ex-moderator kitten Oct 22 '17 at 20:23
  • $\begingroup$ The way Al-Biruni did: you see the horizon from the top of a mountain of known height. $\endgroup$ – Cosmas Zachos Oct 29 '17 at 21:48

The radius of the earth ($R_{earth}$) can be calculated by measuring how the force of gravity ($g$) falls off with height ($y$): $$R_{earth}=-2g\frac{dy}{dg}$$ This works indoors, but requires a sensitive gravimeter and a tall building, e.g. see this undergraduate experiment. Gravimeters are pretty expensive at present, but cheaper ones may become available in a few years time.

As is noted in the comments, many methods that measure the distance to the horizon will also work on cloudy days as long as the distant objects being observed are not obscured. For example, see this measurement on a large calm lake. If your eye (or camera) is a height $h_1$ above the water and at a distance $d$ you can just see the top of tower with height $h_2$ above the water, then the radius of the earth is (by my calculation) $$R_{earth}=\frac{d^2(h_1+h_2-2\sqrt{h_1 h_2})}{2(h_2-h_1)^2}$$ As implied by the answer to Prove the Earth is round without vision, this method should also work with line-of-sight FM radio or television signals which are not blocked by clouds or fog; if you know the height of the broadcast tower, you can try driving away from it until the signal just disappears.

Any of these measurements can be affected by local deviations from their assumptions. For example, local mass density variations will affect the gravimeter readings, atmospheric refraction can change the lake horizon, and FM signals can be blocked by hills or buildings. Under good conditions, however, these methods can produce a rough result for the radius of the earth.

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  • $\begingroup$ Yeah, atmospheric refraction can be very misleading for this type of experiment. $\endgroup$ – Ruslan Oct 30 '17 at 5:59

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