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An example of the usage of the radius calculator
I have done measurements and calculations like this on numerous occasions, and I got very close to the official value of the radius of the Earth every time. Then I got the idea of making a piece of JavaScript code to put here on my homepage to let other people try to find the radius of the Earth themselves.
I have a country house near Öregrund, Sweden. In the sea, north of Öregrund, we have a lighthouse, Engelska grundet (the English Shallow). It has a height of
21.5 m up to the light. This is a photo of that lighthouse. The photo is from the page linked to above.
I took a photo of it on the 15th of March at 12:25. The distance to the lighthouse from where I took the photo is 15.31 km.
The weather was calm, and the temperature of the water and the air was about the same. I held the camera at eye-level, and placed myself so that my feet were the same height above the waterline as the distance from my eyes to the top of my head. Since I am 1.82 m tall, that meant that the camera lens was now 1.82 m above the water.
This is my photo, quite zoomed in. I could not see more of the lighthouse as I zoomed in than if not zoomed in.
Next, I put the cropped parts of the images next to each other.
Then I simply counted pixels. From the waterline to the middle of the lamp, 332 pixels, or 0.06476 m/pixel. Then, from the waterline on the left picture to the horizon on the lift picture = 87 pixels => 5.634 m.
Next to the Radius of the Earth calculator. I entered observer height 1.82 m, a distance to the object 15.31 km. I also guessed that the Coefficient of refraction was about 0.2, mainly because the water and the air had about the same temperature, and the Coefficient of refraction depends on the temperature gradient. I used the formula
to determine that value.
Next, I entered 5.634 m for the hidden height in the bottom part of the page, then Enter. This gave me a radius of 6281.6 km, which is about 6% bigger than the average radius of the Earth. Even if I kept the default value, 0.13, for the Coefficient of refraction, I got a radius quite close to the actual one, now off by about 15%. And that is quite usual – to get within +/– 20 %. The main reasons are the difficulty in estimating how much is hidden, as seen in the photo, and the problem of finding the correct value for the Coefficient of refraction. It might even vary over the distance to the object.
Given a Coefficient of refraction of 0.2 we would expect to see a hidden height of about 6.2 m, so quite close to the measured 5.6 m, and quite a bit away from the 0 m expected on a flat earth.
A closer lighthouse
There is a smaller lighthouse much closer to Öregrund than the other one is. It is Bellonagrundet, which is 4.03 km away from where I stood. Entering the values 1.82 m for the observer height, 4.03 km for the distance and 0.2 for the Coefficient of refraction. This gives a value of -0.115 m for the”hidden height”. The negative value indicates that the lighthouse is in front of the horizon, about 0.115 m below, but in front of the horizon. This is a cropped part of a photo of that lighthouse taken on the same occasion.
The lighthouse is 6.5 m tall from the waterline to the light. That distance is about 318 pixels in the image. The height of 0.115 m will now be about 4 pixels, and that is close to what we can observe in the image.
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