Calculating The Distance To The Sun

Some time ago, I wrote an article similar to this one. I used my current location in the UK and Winnipeg in Canada, but I was never happy with that set-up as the locations weren’t exactly the same latitude and the sun could never be said to be over-head (90°).

I was reluctant to re-do this effort because I am well aware that different outcomes are achievable when different assumptions are made. For example, if you assume that the earth is a ball and that the sun is 800k miles wide and the rays are parallel, then we can calculate  the distance to the sun as 93 million miles.

FE Voliva drawing 90 45 45 triangle distance measure sun on FE and globeScreenshot from 2015-07-05 13:44:24.png

90 45 90 Screenshot from 2015-08-30 19:48:39

However, since I can now say that I am able to prove the earth is a flat stationary plane (see my gyroscope experiments) it would no longer be an assumption to use the following “Plane” trigonometry to determine the approximate height of the sun. So I decided it was worth documenting again.

It is understood that:

A right angle triangle has a 90° angle at one of its corners and that all angles of all triangles add up to 180°.

If one of the other angles was 60° then the final angle would be: 180° – (90+60)= 30°

If In another 90° triangle, the second angle is 45° then the 3rd angle must be 45° in order to total 180°

M’kay so far … A 90°+45°+45° Triangle has a special quality, that others don’t, namely the two sides coming away from the 90° corner MUST be equal lengths

So if you know the distance between two places and the angles of observation then you know the distance to the object that both locations are viewing.

This method has been known about for a very long time and was the basis for a trick the old native Americans used to gauge the height of trees etc

90 45 45 triangle measuring height An Old Indian TrickScreenshot from 2015-10-12 12_44_51

 

There is no reason why this method of trigonometry would not work on a flat plane to determine the apparent height of celestial objects such as the sun. I say ‘apparent’ height because the atmosphere we view the sun through can add enormous variables in terms of looming, magnifying, refracting, diffracting etc which is beyond the scope of this post.

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I wanted to choose two locations on the equator that were separated by water, such that when it was 12 o’clock midday in one location with the sun 90° overhead, from the other location the sun would be exactly 45° elevated to the horizon, thus forming the 90° 45° 45° triangle we are looking for.

librevill to macaba map showing equator line linking both Screenshot from 2016-06-27 17:54:58.png

 

I choose Libreville in Gabon, Africa and Macapá the capital of Amapá state in Brazil.

So, as you can see below, the correct angles can be achieved at the end of March/September near the spring/autumn equinox when the sun is travelling directly over and around the equator.

Macapá 90° angle to the sun, midday 23 MarchMacapa 90 degree sun Screenshot from 2017-03-22 21:09:50

Gabon 45° angle to the sun, 15.30 23 March

Libreville 45 degrees sun angle Screenshot from 2017-03-22 21:31:28

At the time of researching this information, Macapa was GMT-3 hours and Gabon was in line with Greenwich mean time. So best efforts were made to line up the times with the angles, but this is open to refinement, hence why this will be an approximate measurement.

time zones for macapa and libreville 3 hours GMT Screenshot from 2017-03-30 15:12:05.png

Another reason I choose them was because around the equinox of March and September they are directly under the sun’s path, meaning we can make the all important 90° triangle, far better than at my previous location of 52 degrees North.

Macapa Brazil

macapa-brazil-sept-23-2016-showing-sun-over-head

Liberville Gabon

libreville-gabon-sept-23-2016-showing-sun-over-head-screenshot-from-2016-06-27-182609

So all we need now is the distance between Libreville and Macapa in order to determine the perceived height of the Sun above the stationary plane of Earth. This is where it got a bit tricky as google maps and google earth measure the rhumb line between any two objects, which translates to a Great Circle route. In this case it would measure around the equator at 4183 miles, but that’s not the straight line distance we are after, plus this distance is based on a ball earth model with an imaginary radius of 3959 miles.

libreville-gabon-to-macaba-brazil-4183-miles-screenshot-from-2016-06-27-175341

What we are wanting to calculate would be the red line in the diagram below based on the flat plane of Earth, whereas what we are told from google would be the green line based on a ball and as such, inaccurate.

how-to-calculate-the-cord-screenshot-from-2016-11-29-223146

At this point I went off to seek better brains than me (Cheers Adam @larcheored) and we started kicking around how to calculate it.

First we determined the longitudinal bearing of each location from the North Pole, so as to determine the angle between the two locations.

libreville macapa long Screenshot from 2017-03-28 21:11:09
Macapa is 51° West and Libreville is 9° East – giving us a polar angle of 60°. A 60° triangle is an equilateral triangle, and therefore all sides and angles are the same values. Meaning that the distance from the pole to Macapa is equal to the distance between the pole and Libreville, which is also the distance between Libreville and Macapa, and therefore, ultimately the nominal distance from the earth’s surface to the perceived height of the Sun.

So all we need to know now is the distance between the North pole and the equator on the flat Earth.

As it happens the ball earth answer to this is the same as the Flat Earth Gleason Map answer

On the ball, we are told that the distance from the Pole to the Equator is:

Ball earth distance to equator from pole 5400 NM Screenshot from 2017-03-28 19:54:58

Which is further confirmed by the inscription at the bottom of the Gleason’s Azimuthal Equidistant map

gleason 60 miles highlighted Screenshot from 2017-03-28 20:02:11.png

5400 nautical miles (90 degrees x 60 nautical miles) = 6214 Land miles from the Pole to the Equator.

We can note from Gleason’s that roughly 4 lines of longitude lay between Libreville and Macapa (leading to a distance figure of about 4000 miles if we assume the sun travels at 1000mph at the equator, as we are taught)

libreville and macapa on gleason's map Screenshot from 2017-03-23 14:24:40.png

However, if the radius of Earth’s pole to the equator is actually 6214 miles, this would make the circumference of the equator 39,043 miles, which in turn would mean that the sun’s speed is now more like 1626 mph as opposed to the prescribed 1000mph.

So if sun is travelling at 1626 mph then the distance between Libreville and Macapa would be more like 6507 miles (4×1626), however this would be the arc length (the green line below). Making the cord length between Libreville and Macapa the same as we previously calculated it to be – 6214 miles (According to this cord calculator)

gleason overlay with distances Screenshot from 2017-03-28 16:47:59.png

 

Therefore, this makes the true height of the perceived Sun to be

6214 miles or 10,000 Kilometres above the stationary plane

true height of the sun libreville macapa Screenshot from 2017-03-28 19:49:52.png

Since we can now remove the assumption that the Earth’s a ball, we are finally left with the assumption that modern navigation and Gleason’s measurements are correct (ie. 60 nautical miles to the degree). We cannot confirm this, of course, until we have a reliable map of our world, or we map it out ourselves. I would, however, concede that some things just have to be accurate for aviation, maritime and other industries to function with any degree of dependability …. time will tell.

So in conclusion the following can be argued:

The Height of the perceived sun is approximately
6214 miles (10,000 km) not 93 million miles

The speed of the Sun is approximately 1626 mph at the
equator not 1000mph

The distance between Macapa and Libreville is 6214 miles not 4183 miles.