Sphericity

With my recently developed highly skeptical world view, I have been greatly enjoying the revival of discussions on the shape of the Earth for a number of reasons:

1.  It turns out that in reality, every living person, whatever they believe, relies upon the assumption that the surface of a large, undisturbed body of water will lie flat and horizontal whenever they do any practical work.  This is especially true of engineers, such as myself, even on large scale projects covering tens of kilometres such as mines.
2. When one is challenged to make ones own observations on this subject, one finds that ones preconceived ideas have a huge effect on the way we interpret what we see.
3. The technical difficulty is not that high, while the heat of the internet discussion is very high.
4. The use of this topic to provide a dividing line between “normal” people and any person that has a more skeptical view of the way that one is governed.

As my personal investigations have unfolded, the various objections raised has prompted me to organise the investigation into topics:

a)  The default assumption for every practical purpose is that the Earth is flat.

b) The basic geometrical considerations of the spherical model

c)  The effects of atmospheric refraction in modifying the sightings made

d)  How surveyors deal with curvature and refraction

e) The consideration of focal lengths, vanishing points and magnification

f)  The “transmissivity” of the atmosphere, i.e. the measure of how far light reflected from an object wil travel in air before being completely blocked from reaching the observer and in particular, the Fresnel zone.

In this page, I will cover my current thinking on these items.

 

In Reality, EVERYONE Assumes The Earth Is Both Flat and Unmoving

The reader will probably be quite shocked to realise that, despite what they may or may not believe, they personally will always assume that the earth is both flat and unmoving for any practical task they undertake.  Firstly, lets define our terms.

Clearly, the land forms of the earth are not flat in their entirety.  Moreover, the oceans also rise and fall with the tide.  However, the surface of large, undisturbed bodies of water, such as lakes are assumed to lie flat and horizontal.  More importantly, it is assumed that if a pipeline or channel is used to connect two lakes, then the water will flow from “higher” lake to the “lower” lake or remain still if the surface of the lakes is at the same level.  The relative height of these lakes, and all other topographical features of surveyed land, is measured by comparing its height to that of “sea level”.  “Sea Level” is a concept that was originally based on careful measurements of tide movements, with the average height of the sea at any one point becoming the “zero datum”.  The height of a geographic point as measured by typical land surveys is given as a “Reduced Level (RL)”, that is, the height of that point compared to an imaginary “sea level” directly, vertically below it.

It turns out, that for every practical purpose, EVERYONE single one of us ASSUMES that points of equal RL lie on a completely flat, horizontal plane.  There are NO exceptions.  When we use a spirit level, when we dig a drain, build a house, a road, a rail track, a skysraper or a mine, we ALWAYS assume that all points of equal RL lie on a flat, horizontal plane.  What is astonishing is that this assumption has never been the cause of construction errors.  No person, be they drain digger or architect or engineer has EVER had to make a correction to this assumption to allow for the curve in the earth that they BELIEVE is the true shape of the earth.

Even more significantly, all aeronautical engineering and navigation is also based on the assumption that the earth is both flat and unmoving.   The use of critical gyroscopic instruments assumes that all points of equal RL lie on a flat horizontal plane and that, furthermore, that the only movement that needs to be taken into account for navigation purposes is that of the plane.

Similarly, no person is ever required to make a correction for the rotation of the earth in calculating either the route or the flying time when flying from, say, New York to Los Angeles and then back again.

FOR ALL PRACTICAL PURPOSES, EVERYONE ASSUMES THAT THE EARTH IS BOTH FLAT AND UNMOVING…. and this assumption never lets them down.

Basic Geometrical Model of the Spherical Earth

I came across a simple way to calculate the expected distance to the horizon from any observer making an observation to the sea level horizon on a spherical planet.  I neglected to reference the information, so i hope that the originator of this method gets his due recognition somehow.

 

The method used is easily picked up from this diagram.  If an observer places his eyes (or other surveying equipment)  a height “h” above sea level, then the distance to the horizon of calm sea level, “X”, can be easily calculated using the arithmetic shown in the diagram.

The model of the Earth that is considered by convention to be the most accurate is that it is a PLANET which is roughly spherical but slightly bulging in the middle or perhaps slightly pear shaped ball with an average radius of 6,371 kilometres.

In the case, this calculation simplifies to the very close approximation X =3.57 x SQRT(h), where X is in km and h is in metres.   For example, an observer 9 metres above sea level should find that curve of the earth flips over at 3.57 x 3 = 10.7 km away.

So let us plug these values into the equation and see what we should expect with respect to the distance between the ocean horizon point and an observer at a height h above sea level on the planet Earth..

Atmospheric Refraction

In certain circumstances, the path of light travelling from an object to the observers eye or telescope can deviate from a straight line as a result of atmospheric refraction.   The most significant variables in calculating the net affect of refraction are air density, water vapour content and temperature differential with height.  In the case of surveying observations made more or less horizontally, the temperature differential is orders of magnitude more important than that of density and vapour differentials.

Rarely articulated is the knowledge that  refraction can result in 3 quite different outcomes… light curves upward, light goes straight and light bends down, depending on the values of the temperature gradient (dT/dh):

1. Light will curve upwards when the colder ground (or sea) absorbs heat from the atmosphere, causing the atmospheric temperature to increase with distance from the ground and therefore dT/dh > 0.
2. Light will curve downward when the warmer ground emits heat into the atmosphere, resulting in a negative temperature gradient, i.e. dT/dh < 0.
3. Light will travel in straight lines  no heat transfer takes place (dT/dh = 0) or on windy days when turbulence mixes the atmosphere sufficiently to eliminate a temperature gradient.

When refraction for surveying is discussed, the general observation is made that the  atmosphere refracts the horizontal line of sight downward, making the level rod reading smaller and that he typical effect of refraction is equal to about 14% of the effect of earth curvature.  Moreover, it is also said that the effect increases the closer to the ground the light ray get and errors in the region of 5mm/km have resulted.

In fact, it is well understood that when dT/dh > 0, the effect is curve the light ray upwards away from the modelled curve of the earth, which should have the apparent effect of giving the earth a smaller radius and a tighter curve.  Equally, there are many occasions when dT/dh =0, particularly on windy days when turbulent mixing of the atmosphere eliminates the temperature gradient and this circumstance is not limited briefly to evening and morning.

How  Surveyors Account for The Curve of the Earth and Refraction

In everyday practical life, surveyors involved in civil and mining projects undertake “Plane Surveys”.  A Plane Survey is based on the assumption that area being surveyed is built upon a plane surface starting at “sea level”.  The determination of sea level will be taken up later in this note and this level is the “zero point” for what surveyor call the Reduced Level or RL.  That is to say, the “RL” of sea level is assumed to be zero.

Plane Surveys are said to be perfectly valid for distances under 200km.  Empirically, this assumption has been repeatedly demonstrated to be PERFECTLY valid in the case of  an uncountable number of surveys  undertaken over hundreds of years.

While this may seem unremarkable, one should note that the deviation of from the straight and narrow of the putative curve of the earth over a linear distance of 200km is 3,200m, a difference in elevation that is difficult to miss on the basis of reconciling the overlapping of plane surveys.

Clearly, one would expect that these plane surveys that require high levels of precision with respect to RL would need to take into account the curvature of the Earth  over relatively small differences.  For example, in surveying a railway track, a linear distance of 2km requires a correction of 300mm, an amount which cannot be ignored when the gradient of the track is critical.

No doubt the reader will be somewhat surprised to learn that standard techniques employed by professional surveyors simply avoid the need to take curvature/refraction into account.  The need to correct for errors due to curvature and refraction is eliminated by using a combination of:

1. Proper field procedures (taking shorter shots and balancing shots) can practically reduce errors
2. Wherever possible, staff readings should be kept at least 0.5 m above the ground,
3. Using short observation distances (25 m) equalized for backsight and foresight
4. Simultaneous Reciprocal Trigonometrical Heighting
5. Observations made at each station at exactly the same time, cancels the effects of curvature and refraction

The above image is a copy of a slide taken, under fair use rules,  from  http://facstaff.cbu.edu/~gmcginni/classes/CE%20115%20Field%20Measurments/PowerPoint%20Presentations/Leveling%20Theory.pdf

Focal Length and Perspective an Overview

In making optical observations, one can make of use simple rules which apply to the interpretation of what is being observed.

The first is that when humans focus on any particular object, then our eye is fixed on a particular distance.  The net effect of this is that objects which are at different distances are likely to be indistinct and are likely to be unobserved psychologically.

How Far Can Light Travel in Air

One of the stranger comments that I have heard in recent times about the shape of the earth is that “if the earth were flat, then we would be able to see China from California..”

The problem with this argument is that the atmospheric gases and particulates absorb and eventually block the path of light.

The different gases which make up the first layer of the atmosphere above sea level consist of N2 (78%), O2 (21%), CO2, H2O, CO, SO2, etc. Each of these gases absorbs light at different parts of the spectrum and, as a result, various combinations of conditions result in more or less absorbtion.  For observations at sea level, the components which vary the most is carbon dioxide and gaseous water. If you are interested in some more detail, this is an ok source:

http://nature.berkeley.edu/~penggong/textbook/chapter2/html/sect24.htm

H2O is most variable in the atmosphere.
CO2 varies seasonally.

Generally speaking, this means that light reflected off an object in China could not reach the shores of California, even if the Earth was indeed flat.  Nonetheless, some spectacularly long observations have been made, for example:

“What’s the record for visibility without help from the silhouetting effect? I think that might belong to the report of the expedition led by Korzenewsky (1923), who reported seeing snow-capped peaks of a mountain range 750 km away. Conditions were perfect: the lower atmosphere was in shadow at sunset; the peaks were quite high (4650 meters, or over 15,000 feet); they were covered with white snow, increasing their visibility; and there must also have been considerable looming to bring these distant features above the observers’ horizon. As the observation was made on June 1, near the peak of superior-mirage season, the looming is not improbable, though the amount required is hard to believe. The observers themselves were in the deserts of Turkestan [now southeastern Kazakhstan] at a height of nearly a kilometer, where the dryness of the air favored extreme clarity, and looking across a broad, sandy depression. And, of course, much of the air path was in thinner air well above ground level, because of the mountains’ height.”

http://www-rohan.sdsu.edu/~aty/explain/atmos_refr/horizon.html

Testing the Model

To check the spherical model of the world I live in, I went to my local beach at North Shore in Victoria, Australia with a pair of binoculars as the sun was setting.  I stood at water level and looked east with the sun  setting behind me in the west.   I could easily see the coastline of the Bellarine Peninsula at water line up to Portarlington and easily see the channel marker at that point.

As the attached image from Google Earth shows, this distance is well over 20 km.  According to the spherical model, I needed to be around 40 metres or 130′ in the air before I would be able to make the observation that I did.  I have made similar observations using a telescope between Indented Head and Port Melbourne.  These observations would have required that I be 130 metres or 400′ in the air.

The second aspect of this discussion is the calculation of the declination of the horizon line from the horizontal.  The angle “a”,  shown in the diagram, can be calculated from as a = acos(X/(R+h).  The “declination” of the horizon is 90-a.   For a commercial aircraft travelling at 11,500 m above sea level, the distance to the horizon should be 383km and the declination to the horizon should be a clearly observeable 3.4o from the horizontal.  As any person that flies in a plane will observe, once alerted to the possibility, the horizon remains firmly at eye level.

The conclusion here is simply that the Earth is NOT a near spherical planet with a radius of 6,371 km.

This is really only the start of a remarkable situation.  In my very long career as a Mining Engineer, i have often either conducted or made use of plane surveying.  I have never experienced the need to make use of the spherical model in any of the survey work that I have undertaken or used to establish mining reserves, haul roads, rail lines, conveyor belts or any other item, even though the distance often involve tens of kilometres.  A small survey of my engineering colleagues reveals a similar outcome.  In fact, it turns out that there is NO case of any civil or mining engineer applying corrections for sphericity when building a rail line, canal, tunnel, skyscraper, aqueduct or pipeline…. no matter if these projects extend over hundreds of kilometres.

In other words, for the practical world in which we live, there is NO actual application of spherical (i.e. “geodetic”) modelling in any engineering projects of any kind, whatsoever.

By the way, every person can make the simple observations that I have made… and I urge you to do so.

 

A Discussion of “UN Flag” Flat Earth Model

One of the models of the Earth that has been proposed is that it is flat and looks like the UN Flag.  https://en.wikipedia.org/wiki/Portal:United_Nations#/media/File:Flag_of_the_United_Nations.svg

Until quite recently, the routes chosen to fly between continents in the southern latitudes were consistent with this map.  Consider the following examples:

a)  To fly from Melbourne to Nairobi, practically all flights flew first to Bangkok or some other nearby city, and then over India into Nairobi… a straight line on the UNFlag model.

b)  In flying from Perth to Johannesburg, the Qantas flights would stop at the Cocos-Keeling Islands… another straight line on the UNFlag model.

c)  There were no direct flights from Australia or New Zealand to South America

d)  There were no flights over Antarctica

Within the past few years however, a number of changes have taken place, namely:

e)  Qantas now flies directly from Sydney to Santiago and return.  My sister-in-law has taken this return flight twice.   Air New Zealand also states that it has flights from Auckland to Santiago

f)  South African Airways now travel directly from Perth to Johannesburg

However, there are still no flights over the Antarctic.

Google Earth has no useable photographic information inside the the S85 latitude circle.

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