The sinking ship effect and the flat earth theory

As pointed out in my previous article there are people out there that firmly believe the earth is flat. According to them the "sinken ship effect" is just a vision. While most conspiracy theories can be very hard to disprove (no second hand information can be trusted if there really is some great conspiracy going on), the effects of the round earth should be easy to see by almost anyone.

As a sailor, the "sinking ship effect" is easily observable - but I've never actually sat down and observed this effect carefully. Today I had the perfect chance - so I did.

In a scientific experiment, one should first predict the outcome of the experiment and then actually do the experiment. Unfortunately I didn't have the chance to look up facts and do the calculations in advance this time.

There were three ships at anchor within a zone marked as "anchorage" on the map. The third was quite far from the other two.

At 5 nautic miles, I could clearly see the two ships that were near to each other. I could apparently see all of the ships; the dark hull and the white overbuilding and white cranes. Around 6 nautic miles, the deck of the ship was visible while we were at the top of the waves and invisible while we were low. At 7 nm, the deck of the ship was fully below the horizon at all times. At 8 nm, the deck of the neighbour ship also couldn't be seen, and the cranes of the first ship was not visible. I didn't check the third ship. No, this was not just my imagination and poor vision. Yes, I used binocular.

So, let's do the math again. The elevation between two relatively nearby points A and B due to the earths curvature can be be approximated to:

tan(sector/2)*distance/2

One nautic mile is the same as one minute of longtitude, and also the same as 1852 metres. Let d be the distance in nautic miles, and we get the elevation in metres e to be:

e(d) = tan(2*pi*d/360/60)*d*1852/2

Actually, the tan part there can be left out, as tan(q) -> q for small values of q.

To make this article up to SteemSTEM standards I should of course come up with the proof that my formula is right; however, I'm sitting in a boat here and I'm soon running out of batteries. I decided to do some quick quality assurance ... I found an earth curve calculator on the internet. It's slightly more advanced than my formula, but the numbers matches (if I put 1 nautic mile into my formula, I get 0.2694. If I put 1852 metres into the curve calculator and eye height 0, I get 0.2692 metres of the target hidden due to the curvature of the earth. Close enough.

According to my formula, we get (approximated):

e(5) = 7
e(6) = 10 
e(7) = 13
e(8) = 17

if one has the eyes right by the sea level. We have a pretty high cockpit, my eyes are probably ~2 metres above the sea level while sitting in the cockpit, in addition the wave height is probably around a metre ... ignoring that, we get that the deck of the smaller of the two ships should be around 10 metres above the sea level, while the deck of the bigger ship should be around 15 metres above sea level.

A quick check on MaritimeTraffic shows that the smaller ship was Kyrakatingo

While the bigger ship was Furevik

I'm not able to get the deck height of those vessels before I'm running out of battery - anyone else up for the challenge?

(Keep in mind that there may be several metres difference between a loaded and an unloaded ship)

Here is a photo from when we were passing those ships:

(own work, CC-BY-SA 4.0, full photo at https://ipfs.io/ipfs/QmdXbaLTBjVfHvYobd15WeV3CZyuWnGEyD5XA88tzE9ez7)