1)The third version doesn't work. When we talk about the height above average surface level regarding the horizon we are talking about sea level (
The distance to horizon calculator points this out since all the formulas are adjusted to sea level) and considering that the distance between the pov of Vivi and the other side of the river is covered in water what we care about is the height above sea level.
2)The deck to sea level height can't be .23m as in chapter 99 Oda provided us with
a diagram for the going merry and below the deck there are the men's and women's quarters. We can just calculate
the height of the women's quarters which would be 3.52m (
We know that's the room because of another of the diagram's provided) and then use that and assume that's the full height of the deck above sea level. We should add this value to Vivi's eyesight height in the other 2 versions since our surface of reference is sea level. That relatively small height on its own changes the result for the planet's radius drastically from
806,643 km to
246,717km, so yeah it can't be ignored.
3)Another major problem is that the objects we're looking at from the other side of the river are mountains which are elevated above sea level. These tall objects can be viewed from beyond our horizon distance. In
this calculator we can put the height of the observation point and the height of the object beyond the horizon to find the maximal distance at which the top of this object will be visible using earth parameters. We can use this calculator to approximate the minimal height needed for an object to be visible from a certain distance, for example the minimal height needed for an object to be visible from 50km away at the height we have is
115.3m (
137.9m for the height of Vivi's eyes), obviously a taller mountain even more visible from 50km away (what we'll be able to view is the difference between the mountain's height and the minimal height required that we calculated earlier from the top, if the mountain is a good amount larger than the minimal height then you'd see most of it and won't notice the missing part at the base). Considering
Alabasta is a pretty mountainous kingdom it's not that ridiculous to assume that these mountains that we see in the other side of the Sandora River are much taller than these minimal values, so the fact that they are visible from 50km away doesn't really give us the info needed to find the diameter of the planet.
And to illustrate this even further since the previous calculator uses earth parameters this time let's use the parameters of a planet arbitrarily larger than earth and smaller than the one found in the blog for example one with a diameter of 40000km, to find the apparent radius of the planet due to atmospheric refraction we
will use this formula/
This formula, while the formula for the maximal distance between two points we can use
This which is
d = sqrt(2*R*h1) + sqrt(2*R*h2) we can then use this to find the formula for the minimum height of an object to be visible from a certain distance to find
h2 = (d - sqrt(2*R*h1))^2 / (2*R). For our 40000km diameter planet we'll use in this example the radius will be 20000km and the apparent radius due to atmospheric refraction will be 20000*7/6=23333.33km=23333330m for 50 km away we'll have h2 = (50000 - sqrt(2*23333330*(5.071)))^2 / (2*23333330) = 25.68m ((50000 - sqrt(2*23333330*(1.551)))^2 / (2*23333330)=37.9m if we were to use Vivi's eye height rather than the full height which still gives us a short minimal height), so in this planet with 40000km diameter you can see objects as short as 37.9m from 50km away and as such the mountains will be easily visible from that distance since 37.9m is a negligible value compared to their heights, so we really can't deduce the planet's diameter with the method used in the blog which gets a 1,611,864 km diameter planet (a very large difference from the 40000km used here) for viewing these mountains from 50km away. (For a planet with a diameter of 90000km (50000 - sqrt(2*52500000*(1.551)))^2 / (2*52500000)=13.2m which is pretty much nothing when we're talking about mountains)
4)This is more so just a nitpick but I think another problem with using the distance to horizon to calculate the planet's diameter is that Oda is inconsistent in drawing the distance to horizon. For example, the coup de burst which we
canonically know launches the Sunny 1km away made the ship
go to the edge of the horizon being barely visible. So I think opting for other methods would be better.