I have a new proof half-written, but I’m not going to finish it before I have to leave for the airport in a few hours (I still have to pack!), so it’ll be posted after I get home on the 25th of May. See you then!

## 14. Map projections

Cartography is the science and art of producing maps – most commonly of the Earth (although there are also maps of astronomical bodies and fictional worlds). The best known problem in cartography is that of representing the Earth on a flat map with minimal distortion.

If the Earth were itself flat, then this problem would not exist. A flat map would simply be a constant scale drawing of the flat Earth, and it woud be accurate and distortion-free at all points. But it is well known that such a map cannot be made. The reason, of course, is that the Earth is spherical, and the surface of a sphere cannot be projected onto a flat plane without some sort of distortion.

There are numerous different *map projections*, which render areas of the Earth onto a flat map with varying types and amounts of distortion. In all of these projections, some trade off must be made between the different goals of preservation of distances (i.e. a constant distance scale), preservation of directions (e.g. north is always up, east is always to the right), preservation of shapes (geographical regions look the same shape as they do when viewed from the air or a satellite), and preservation of areas (geographical regions of the same area appear the same area on the map). The familiar Mercator projection preserves directions at the expense of all the others, and is infamous for its large distortions of area between the polar and equatorial regions.

The area distortion is apparent when you consider that Africa has an area of 30.4 million square kilometres, while North America, including Central American, the Caribbean islands, the northern Canadian islands, and Greenland, is only 24.7 million square kilometres. On a Mercator map, Greenland all by itself looks larger than Africa, but it is in reality less than a third the size of Australia.

There are projections which give a better impression of the relative areas, but these necessarily distort shapes and distances. The Gall-Peters projection also maps lines of latitude and longitude to straight lines like Mercator, but preserves areas.

Both of these projections have the disadvantage that distortion becomes extreme at the poles. In the Gall-Peters projection, the North and South Poles are mapped to horizontal lines spanning the width of the map, rather than to points. The Mercator projection cannot even show the poles at all, because the projection puts them at infinity.

A map specifically designed to compromise between all the various distortions is the Winkel tripel projection. This projection was adopted by the National Geographic Society as its standard world map projection in 1998 (replacing the Robinson projection, a similar compromise projection), and many textbooks and educational materials now use it.

In the Winkel tripel projection, the lines of latitude and longitude are both curved, indicating that directions and shapes are not preserved faithfully. Areas are also distorted somewhat – Greenland looks almost the same size as Australia, even though it is less than a third the area. But all of the different distortions are moderate compared to the more extreme distortions visible in some of these features in other projections.

The Hammer projection goes further in rectifying the distance distortion issues with the polar regions, by mapping the North and South Poles to single points, as they are on Earth. However, this distorts the shapes of areas near the poles and away from the central meridian even more.

If you want to minimise distortions in a sort of T-shaped area encompassing, say, the Old World continents of Europe, Asia, and Africa, then you can do a bit better by adopting the Bonne projection. This maps a chosen so-called “standard” parallel of latitude to a circular arc, which reduces distortion along that parallel (since in reality parallels of latitude are circles, not straight lines).

Minimising distortions along the straight central meridian and a parallel in the northern hemisphere naturally increases distortions in the southern hemisphere, making this a good choice for Old World maps, which it has been used for extensively, but pretty bad for South America and Oceania.

Oddly enough, we’re now not so far from the standard map advocated by most Flat Earthers. If you map all the parallels of latitude to *complete* circles (rather than partial circular arcs as in the Bonne projection), increasing in radius by a constant amount per degree of latitude, you end up with an azimuthal equidistant projection, centred on the North Pole.

The result is that the northern hemisphere is moderately distorted, but the distortion grows extreme in the south, and the South Pole is mapped to a large circle encircling the whole map. This map projection is good for showing directions relative to the central point (so a variant centred on Mecca is useful for Muslims who wish to know which direction to face during prayer). It’s not a good projection for much else though, because of the severe distance, shape, direction, and area distortions in the southern hemisphere. If you were plotting a trip from Australia to South America, it would be utterly useless.

If the Earth were in fact flat, then it would be possible—indeed *trivial*—to construct flat maps which accurately show the shapes and distances over large areas of the Earth’s surface at a constant scale. No such maps exist. And the fact that cartographers have struggled for centuries to make flat maps of the world, trading off various compromises with arguable degrees of success, is evidence that it’s not possible, and that the Earth is a globe.

## Admin update: travelling 11-25 May

Sorry there hasn’t been a new post for a few days. I’ve been dealing with some urgent deadlines for stuff that needs to be done before I travel next week. I’m going to Portugal for 2 weeks, from 11-25 May. There won’t be any new posts while I’m away, but I hope to get one or two more done before I leave – after I deal with more urgent stuff this weekend.

## 13. Hydrostatic equilibrium

The theory of gravity is wildly successful in explaining and predicting the behaviours of masses. Isaac Newton’s formulation of gravity (published in his *Principia Mathematica* in 1686) is a simple formula that works very well for most circumstances of interest to people. When the gravitational potential energy or the velocity of a mass is very large, Albert Einstein’s general relativity (published 1915) is required to correctly determine behaviour. Newton’s gravity is in fact an approximation of general relativity that gives almost exactly the correct answer when the gravitational energy per unit mass is small compared to the speed of light squared, and the velocity is much smaller than the speed of light. For almost all calculation purposes, Newton’s law is sufficiently accurate to be used without worrying about the difference.

Newton’s law says that the force of gravitational attraction *F* between two bodies equals the universal gravitational constant *G*, multiplied by the masses of the two bodies *m*_{1} and *m*_{2}, divided by the square of the distance *r* between them: F = *G* *m*_{1} *m*_{2}/(*r*^{2}).

Newton himself had no idea why this simple formula worked. Although he showed that it was accurate to the limits of the measurements available to him, he was deeply concerned about its philosophical implications. In particular, he couldn’t imagine how such a force could occur between two bodies separated by any appreciable distance or the vacuum of space. He wrote in a letter to Richard Bentley in 1692:

“That one body may act upon another at a distance through a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one another, is to me so great an absurdity that, I believe, no man who has in philosophic matters a competent faculty of thinking could ever fall into it.”

Newton was so concerned about this that he added an appendix to the second edition of the *Principia* – an essay titled the *General Scholium*. In this he wrote about the distinction between observational, experimental science, and the interpretation of observations (translated from the original Latin):

“I have not as yet been able to discover the reason for these properties of gravity from phenomena, and I do not feign hypotheses. For whatever is not deduced from the phenomena must be called a hypothesis; and hypotheses, whether metaphysical or physical, or based on occult qualities, or mechanical, have no place in experimental philosophy.”

In other words, Newton was being led by his observations to deduce physical laws and how the universe behaves. He refused to countenance speculation unsupported by evidence, and he accepted that the world behaved as observed, even if he didn’t like it. Commenting on Newton’s words in 1840, the philosopher William Whewell wrote:

“What is requisite is, that the hypotheses should be close to the facts, and not connected with them by other arbitrary and untried facts; and that the philosopher should be ready to resign it as soon as the facts refuse to confirm it.”

This affirms the position of a scientist as one who observes nature and tries to describe it as it is. Any hypothesis formed about how things are or why they behave the way they do must conform to all the known facts, and if any future observation contradicts the hypothesis, then the hypothesis must be abandoned (perhaps to be replaced with a different hypothesis). This is the scientific method in a nutshell, and guides our understanding of the shape of the Earth in these pages.

The universal gravitational constant *G* is a rather small number in familiar units: 6.674×10^{−11} m^{3} kg^{−1} s^{−2}. This means that the force of gravity between two everyday objects is so small as to be unnoticeable. For example, even large objects such as two 1-tonne cars a metre apart experience a gravitational force between them of only 6.674×10^{−5} newtons – far too small to move the cars against rolling friction, even with the brakes off. Also, the distance between the masses in Newton’s formula is the distance between the *centres of mass* of the objects, not the closest surfaces. The centres of mass of two cars can’t be brought closer together than about 2 metres in practice, even with the cars touching each other (unless you crush the cars).

Gravity really only starts to significantly affect things when you gather millions of tonnes of mass together. On Earth, the mass of the Earth (5.9722×10^{24} kilograms) itself dominates our experience with gravity. Removing mass 2 from Newton’s formula, we can calculate the acceleration *a* towards the centre of mass of the Earth, caused by the Earth’s gravity, as experienced at the surface of the planet (*r* = 6370 kilometres): *a* = *G* *m*_{1}/(*r*^{2}) = 6.674×10−11 × 5.9722×1024 / (6370×103)^{2} = 9.82 m/s^{2}. This number matches experimental observations we can make of the gravity on the surface of the Earth (for example, using a pendulum: see also Airy’s coal pit experiment).

So large object like planets or other astronomical bodies experience a significant gravitational force on parts of themselves. Think about a tall mountain, such as Mount Everest. Let’s estimate the mass of Mount Everest – just roughly will do for our purposes. It is 8848 metres tall, above sea level. Let’s imagine it’s roughly a cone, with sides sloping at 45°. That makes the radius of the base 8848 metres, and its volume is *π* × 8848^{3} / 3 = 7.25×10^{11} cubic metres. The density of granite is 2.75 tonnes per cubic metre, so the mass of Mount Everest is roughly 2×10^{15} kg. It experiences a gravitational force of approximately 2×10^{16} newtons, pulling it down towards the rest of the Earth.

Obviously Mount Everest is strong enough to withstand this enormous force without collapsing. But how much higher could a mountain be without collapsing under its own mass? The taller a mountain gets, the more force pulls it down, but the structural strength of the rock making up the mountain does not increase. At some point there is a limit. Our conical Mount Everest model spreads that mass over an area of *π* × 8848^{2} square metres. This means the pressure of the rock above on this area is 2×10^{16} / (*π* × 8848^{2}) = 8×10^{7} pascals, or 80 megapascals (Mpa). Now, the compressive strength of granite is about 200 MPa. We’re pretty close already! Not to mention that rock can also shear and deform plastically, so we probably don’t even need to get as high as 200 MPa before something bad (or spectacular, depending on your point of view!) happens. A mountain twice as high as Everest would almost certainly be unstable and collapse very quickly.

As mountains get pushed up by tectonic activity, their bases spread out under the pressure of the rock above, so that they can’t exceed the limit of the tallest possible mountain. In practice, it turns out that glaciation also has a significant effect on the maximum height of mountains on Earth, limiting them to something not much higher than Everest [1].

Now, compared to the size of the Earth, even a mountain as tall as Everest is pretty insignificant. It is barely a thousandth of the radius of the planet. It’s often said that if shrunk down to the same size, the Earth would be smoother than a billiard ball. In a sense, this is actually true! Billiard and snooker balls are specified to be 52.5 mm in diameter, with a tolerance of 0.05 mm [2]. That is just under a 500th of the radius, so it would be acceptable to have billiard balls for professional play that are twice as rough as the Earth – although in practice I suspect that billiard balls are manufactured smoother than the quoted tolerance.

So, there is a physical limit to the strength of rock that means that Earth can’t have any protruding lumps of any significant size compared to its radius. Similarly, any deep trenches can’t be too deep either, or else they’ll collapse and fill in due to the gravitational stress on the rock pulling it together. The Earth is spherical in shape (more or less) because of the inevitable interaction of gravity and the structural strength of rock. Any astronomical body above a certain size will also necessarily be close to spherical in shape. The size may vary depending on the materials making up the body: rock is stronger than ice, so icy worlds will necessarily be spherical at smaller sizes than rocky ones.

The phenomenon of large bodies assuming a spherical shape is known as hydrostatic equilibrium, referring to the fact that this is the shape assumed by any body with no resistance to shear forces, in other words fluids. For ice and rock, the resistance to shear force is overcome by gravity for objects of size a few hundred to a thousand or so kilometres in diameter. The asteroid Ceres is a hydrostatic spherical shape, with a diameter of 945 km. On the other hand, Saturn’s moon Iapetus is the largest known object to deviate significantly from hydrostatic equilibrium, with a diameter of 1470 km. Iapetus is almost spherical, but has an unusual ridge of mountains running around its equator, with a height around 20 km – about 1/36 of the moon’s radius.

It’s safe to say, however, that any planetary sized object has to be very close to spherical – or spheroidal if rotating rapidly, causing a slight bulge around the equator due to centrifugal force. This is because of Newton’s law of gravity, and the structural strength of rock. Our Earth, naturally, is such a sphere.

Flat Earth models must either conveniently ignore this conclusion of physics, or posit some otherwise unknown force that maintains the mass of the Earth in a flat, non-spherical shape. By doing so, they violate Newton’s principle that one must be guided by observation, and discard any hypothesis that does not fit the observed facts.

**References:**

[1] Mitchell, S. G., Humphries, E. E., “Glacial cirques and the relationship between equilibrium line altitudes and mountain range height”. *Geology*, 43, p. 35-38, 2015. https://doi.org/10.1130/G36180.1

[2] Archived from worldsnooker.com on archive.org: https://web.archive.org/web/20080801105033/http://www.worldsnooker.com/equipment.htm

## 12. The sun

Possibly the most obvious property of our sun is that it is visible from Earth during daylight hours, but not at night. The visibility of the sun is in fact what defines “day time” and “night time”. At any given time, the half of the Earth facing the sun has daylight, while the other half is in the shadow of the Earth itself, blocking the sun from view. It’s trivial to verify that parts of the Earth are in daylight at the same time as other parts are in night, by communicating with people around the world.

The first physical property of the sun to be measured was how far away it is. In the 3rd century BC, the ancient Greek Aristarchus of Samos (who we met briefly in 2. Eratosthenes’ measurement) developed a method to measure the distance to the sun in terms of the size of the Earth, using the geometry of the relative positions of the sun and moon. Firstly, when the moon appears exactly half-illuminated from a point on Earth, it means that the angle formed by the sun-moon-Earth is 90°. If you observe the angle between the sun and the moon at this time, you can determine the distance to the sun as a multiple of the distance to the moon.

In the figure, if you measure the angle *θ*, then the ratio of the distance to the sun *S* divided by the distance to the moon *M* is the reciprocal of the cosine of *θ*. Aristarchus then used the size of the shadow of Earth on the moon during a lunar eclipse to obtain further equations relating the distances to the sun and moon and the size of the Earth.

By combining these results, you can calculate the distances to both the sun and the moon in terms of the radius of the Earth. Aristarchus got the wrong answer, estimating that the sun was only about 19 times further away than the moon, because of the limited precision of his naked eye angle measurements (it’s actually 390 times further away). But Eratosthenes later made more accurate measurements (which were again discussed in Eratosthenes’ measurement), most likely using the same method.

The first rigorous measurement of the absolute distance to the sun was made by Giovanni Cassini in 1672. By this time, observations of all the known celestial bodies in our solar system and some geometry had well and truly established the *relative* distances of all the orbits. For example, it was known that the orbital radius of Venus was 0.72 times that of Earth, while the orbit of Mars was 1.52 times that of Earth. To measure the absolute distance to the sun, Cassini used a two-step method, the first step of which was measuring the distance to the planet Mars. This is actually a lot easier to do than measuring the distance to the sun, because Mars can be seen at night, against the background of the stars.

Cassini dispatched his colleague Jean Richer to Cayenne in French Guiana, South America, and the two of them arranged to make observations of Mars from there and Paris at the same time. By measuring the angles between Mars and nearby stars, they determined the parallax angle subtended by Mars across the distance between Paris and Cayenne. Simple geometry than gave the distance to Mars in conventional distance units. Then applying this to the relative distances to Mars and the sun gave the absolute distance from the Earth to the sun.

Since 1961, we’ve had a much more direct means of measuring solar system distances. By bouncing radar beams off the moon, Venus, or Mars and measuring the time taken for the signal to return at the speed of light, we can measure the distances to these bodies to high precision (a few hundred metres, although the distances to the planets change rapidly because of orbital motions) [1].

The Earth orbits the sun at a distance of approximately 150 million kilometres. Once we know this, we can work out the size of the sun. The angular size of the sun as seen from Earth can be measured accurately, and is 0.53°. Doing the mathematics, 0.53°×(*π*/180°)×150 = 1.4, so the sun is about 1.4 million kilometres in diameter, some 109 times the diameter of the Earth. This is the diameter of the visible surface – the sun has a vast “atmosphere” that we cannot see in visible light. Because of its vast distance compared to the size of the Earth, the sun’s angular size does not change appreciably as seen from different parts of the Earth. The difference in angular size between the sun directly overhead and on the horizon (roughly the Earth’s radius, 6370 km, further away) is only about 6370/150000000×(180°/*π*) = 0.002°.

Our sun is, in fact, a star – a huge sphere composed mostly of hydrogen and helium. It produces energy from mass through well-understood processes of nuclear fusion, and conforms to the observed properties of stars of similar size. The sun appears much larger and brighter than stars, and heats the Earth a lot more than stars, because the other stars are all so much further away.

Like all normal stars, the sun radiates energy uniformly in all directions. This is expected from the models of its structure, and can be inferred from the uniformity of illumination across its visible disc. The fact that the sun’s polar regions are just as bright as the equatorial edges implies that the radiation we see in the ecliptic plane (the plane of Earth’s orbit) is reproduced in all directions out of the plane as well.

NASA’s *Ulysses* solar observation spacecraft was launched in 1990 and used a gravity slingshot assist from Jupiter to put it into a solar orbit inclined at about 80° to the ecliptic plane. This allowed it to directly observe the sun’s polar regions.

Now, I tried to find scientific papers using data from *Ulysses* to confirm that the sun indeed radiates electromagnetic energy (visible light, ultraviolet, etc.) uniformly in all directions. However, it seems that no researchers were willing to dedicate space in a paper to discussing whether the sun radiates in all directions or not. It’s a bit like looking for a research paper that provides data on whether apples fall to the ground or not. What I did find are papers that use data from *Ulysses*’ solar wind particle flux detectors to measure if the energy emitted by the sun as high energy particles varies with direction.

As these figures show, the energy emitted by the sun as solar wind particles is pretty constant in all directions, from equatorial to polar. Interestingly, there *is* a variation in the solar wind energy flux with latitude: the solar wind is slower and less energetic close to the plane of the ecliptic than at higher latitudes. The solar wind, unlike the electromagnetic radiation from the sun, is affected by the structure of the interplanetary medium. The denser interplanetary medium in the plane of the ecliptic slows the wind. The amount of slowing provides important constraints on the physics of how the solar wind particles are accelerated in the first place.

Anyway, given there are papers on the variation of solar wind with direction, you can bet your bottom dollar that there would be hundreds of papers about the variation of electromagnetic radiation with direction, *if it had been observed*, because it goes completely counter to our understanding of how the sun works. The fact that the sun radiates uniformly in all directions is such a straightforward consequence of our knowledge of physics that it’s not even worth writing a paper confirming it.

Now, in our spherical Earth model, all of the above observations are both consistent and easily explicable. In a Flat Earth model, however, these observations are less easily explained.

Why is the sun visible in the sky from part of the Earth (during daylight hours), while in other parts of the Earth at the same time it is not visible (and is night time)?

The most frequently proposed solution for this is that the sun moves in a circular path above the disc of the Flat Earth, shining downwards with a sort of spotlight effect, so that it only illuminates part of the disc. Although there is a straight line view from areas of night towards the position of the sun in the sky, the sun does not shine in that direction.

Given that we know the sun radiates uniformly in all directions, we know this cannot be so. Furthermore, if the sun *were* a directional spotlight, how would such a thing even come to be? Directional light sources do occur in nature. They are produced by synchrotron radiation from a rapidly rotating object: for example, a pulsar. But pulsars rotate and sweep their directional beams through space on a timescale of approximately one second. If our sun were producing synchrotron radiation, its spotlight beam would be oscillating many times per minute – something which is not observed.

Even furthermore, if the sun is directional and always above the plane of the Flat Earth, it should be visible in the night sky, as an obscuration passing in front of the stars. This prediction of the Flat Earth model is not seen – it is easy to show that no object the size of the sun obscures any stars at night.

And yet furthermore, if the sun is directional, there are substantial difficulties in having it illuminate the moon. Some Flat Earth models acknowledge this and posit that the moon is self-luminous, and changes in phase are caused by the moon itself, not reflection of sunlight. This can easily be observed not to be the case, since (a) there are dark shadows on the moon caused by the light coming from the location of the sun in space, and (b) the moon darkens dramatically during lunar eclipses, when it is not illuminated by the sun.

In addition to the directional spotlight effect, typical Flat Earth models state that the distance to the sun is significantly less than 150 million kilometres. Flat Earth proponent Wilbur Glenn Voliva used geometry to calculate that the sun must be approximately 3000 miles above the surface of the Earth to reproduce the zenith angles of the sun seen in the sky from the equator and latitudes 45° north and south.

Aside from the fact that Voliva’s distance does not give the correct zenith angles for any *other* latitudes, it also implies that the sun is only about 32 miles in diameter, given the angular size seen when it is overhead, and that the angular size of the sun should vary significantly, becoming only 0.53°/sqrt(2) = 0.37° when at a zenith angle of 45°. If the sun is this small, there are no known mechanisms than can supply the energy output it produces. And the prediction that the sun would change in angular size is easily disproved by observation.

The simplest and most consistent way of explaining the physical properties of our sun is in a model in which the Earth is a globe.

**References:**

[1] Muhleman, D. O., Holdridge, D. B., Block, N. “The astronomical unit determined by radar reflections from Venus”. *The Astrophysical Journal*, 67, p. 191-203, 1962. https://doi.org/10.1086/108693

[2] Barnes, A., Gazis, P. R., Phillips, J. L. “Constraints on solar wind acceleration mechanisms from Ulysses plasma observations: The first polar pass”. *Geophysical Research Letters*, 22, p. 3309-3311, 1995. https://doi.org/10.1029/95GL03532

[3] Phillips, J. L., Bame, S. J., Barnes, A., Barraclough, B. L., Feldman, W. C., Goldstein, B. E., Gosling, J. T., Hoogeveen, G. W., McComas, D. J., Neugebauer, M., Suess, S. T. “Ulysses solar wind plasma observations from pole to pole”. *Geophysical Research Letters*, 22, p. 3301-3304, 1995. https://doi.org/10.1029/95GL03094

## 11. Auroral ovals

Aurorae are visible light phenomena observed in the night sky, mostly at high latitudes corresponding to Arctic and Antarctic regions. An aurora can appear as an indistinct glow from a distance or as distinct shifting curtain-like formations of light, in various colours, when seen from nearby.

Aurorae are caused by the impact on Earth’s atmosphere of charged particles streaming from the sun, known as the solar wind.

The Earth’s magnetic field captures the particles and deflects them (according to the well-known laws of electromagnetism) so that they spiral downwards around magnetic field lines. The result is that the particles hit the atmosphere near the Earth’s magnetic poles.

The incoming high energy particles ionise nitrogen atoms in the upper atmosphere, as well as exciting oxygen atoms and nitrogen molecules into high energy states. The recombination of nitrogen and the relaxation of the high energy states results in the emission of photons. The light is produced between about 90 km and 150 km above the surface of the Earth, as shown by triangulating the positions of aurorae from multiple observing locations.

Observations of aurorae have established that they occur in nearly-circular elliptical rings of width equivalent to a few degrees of latitude (i.e. a few hundred kilometres), usually between 10° and 20° from the Earth’s magnetic poles. These rings, in the northern and southern hemispheres, are called the *auroral ovals*.

The auroral ovals are not precisely centred on the magnetic poles, but rather are pushed a few degrees towards the Earth’s night side. This is caused by the diurnal deflection of the Earth’s magnetic field by pressure from the charged particles of the solar wind.

The auroral ovals also expand when solar activity increases, particularly during solar storms, when increased particle emission from the sun and the resulting stronger solar wind compresses the Earth’s magnetic field, forcing field lines to move away from the poles.

But despite these variations, the auroral ovals in the northern and southern hemispheres move and change sizes more or less in unison, and are always of similar size.

You can see the current locations and sizes of both the northern and southern auroral ovals as forecast based on the solar wind and interplanetary magnetic field conditions as measured by the Deep Space Climate Observatory satellite at https://www.spaceweatherlive.com/en/auroral-activity/auroral-oval.

Earth is not the only planet to display aurorae. Jupiter has a strong magnetic field, which acts to funnel the solar wind towards its polar regions in the same way as Earth’s field does on Earth. Jupiter we can establish by simple observation from ground-based telescopes is close to spherical in shape and not a flat disc. Auroral ovals are observed on Jupiter around both the northern and southern magnetic poles, exactly analogously to on Earth: of close to the same size and shape.

Similar auroral ovals are also seen on Saturn, in both the northern and southern hemispheres [3][4]. And just for the record, Saturn is also easily shown to be spherical in shape, and not a flat disc.

Now, we have established that auroral ovals appear on three different planets, with the southern and northern ovals of close to the same sizes and shapes on each individual planet. Everything is consistent and readily understandable – as long as you assume that the Earth is spherical like Jupiter and Saturn.

If the Earth is flat, however, then the distributions of aurorae in the north and south map to very different shapes and sizes – with no ready explanation for either the shapes or their differences. In particular, large parts of the southern auroral oval end up being extremely far from the southern magnetic pole, in defiance of the electromagnetic mechanism that causes aurorae in the first place.

So the positions of aurorae on a flat Earth cannot be readily explained by known laws of physics, and they also do not resemble the locations and sizes of auroral ovals as observed on other planets. All of these problems go away and become self-consistent if the Earth is a globe.

**References:**

[1] Safargaleev, V., Sergienko, T., Nilsson, H., Kozlovsky, A., Massetti, S., Osipenko1, S., Kotikov, A. “Combined optical, EISCAT and magnetic observations of the omega bands/Ps6 pulsations and an auroral torch in the late morning hours: a case study”. *Annales Geophysicae*, 23, p. 1821-1838, 2005. https://doi.org/10.5194/angeo-23-1821-2005

[2] Grodent, D.,Clarke, J. T., Kim, J., Waite Jr., J. H., Cowley, S. W. H. “Jupiter’s main auroral oval observed with HST‐STIS”. *Journal of Geophysical Research*, 108, p. 1389-1404, 2003. https://doi.org/10.1029/2003JA009921

[3] Cowley, S. W. H., Bunce, E. J., Prangé, R. “Saturn’s polar ionospheric flows and their relation to the main auroral oval”. *Annales Geophysicae*, 22, p.1379-1394, 2004. https://doi.org/10.5194/angeo-22-1379-2004

[4] Nichols, J. D., Clarke, J. T., Cowley, S. W. H., Duval, J., Farmer, A. J., Gérard, J.‐C., Grodent, D., Wannawichian, S. “Oscillation of Saturn’s southern auroral oval”. *Journal of Geophysical Research*, 113, A11205, 2008. https://doi.org/10.1029/2008JA013444

## Admin: travelling this weekend

I am spending an extended Friday-Monday weekend away down the coast with my wife and our dog, for a mini-vacation – so I won’t be posting a new proof this weekend. The next post will be around the middle of next week.

## 10. The Sagnac effect

Imagine beams of light coming from an emitter and travelling around a circular path in both directions, until they arrive back at the source. Such an arrangement can be constructed by using an optic fibre in a circular loop, injecting light at both ends. The distance travelled by the clockwise beam is the same as the distance travelled by the anticlockwise beam, and the speed of light in both directions is the same, so the time taken for each beam to travel from the source back to the origin is the same. So far, so good.

Now imagine the whole thing is *rotating* – let’s say clockwise. For reference we’ll use the numbers on a clock face and the finer divisions into 60 minutes. The optic fibre ring runs around the edge of the clock, with the light source and a detector at 12. Now imagine that the clock rotates fast enough that by the time the clockwise-going light reaches the original 12 position, the clock has rotated so that 12 is now located at the original 1 minute past 12 position. The light has to travel an extra 60th of the circle to reach its starting position (actually a tiny bit more than that because the clock is still rotating and will have gone a tiny bit further by the time the light beam catches up). But the light going anticlockwise reaches the source *early*, only needing to travel a tiny bit more than 59/60 of the circle. The travel times of the two beams of light around the circle are different.

This is a very simplified explanation, and figuring out the mathematics of exactly what happens involves using special relativity, since the speed of light is involved, but it can be shown that there is indeed a time difference between the travel times of beams of light heading in opposite directions around a rotating loop. The time difference is proportional to the speed of rotation and to the area of the loop (and to the cosine of the angle between the rotation axis and the perpendicular to the loop, for those who enjoy vector mathematics). This effect is known as the Sagnac effect, named after French physicist Georges Sagnac, who first demonstrated it in 1913.

Measuring the minuscule time difference between the propagation of the light beams is not difficult, due to the wave nature of light itself. The wavelength of visible light is just a few hundred nanometres, so even a time difference of the order of 10^{-16} seconds can be observed because it moves the wave crests and troughs of the two beams relative to one another, causing visible interference patterns as they shift out of synchronisation. This makes the device an interferometer that is very sensitive to rotational speed.

The Sagnac effect can be seen not only in a circular loop of optic fibre, but also with any closed loop of light beams of any shape, such as can be constructed with a set of mirrors. This was how experimenters demonstrated the effect before the invention of optic fibres. Because the paths of the two beams of light are the same, just reversed, a Sagnac interferometer is completely insensitive to mechanical construction tolerances, and only sensitive to the physical rotation of the device.

Sagnac actually performed the experiment in an attempt to prove the existence of the luminiferous aether, a hypothetical medium permeating all space through which light waves propagate. He believed his results showed that such an aether existed, but Max von Laue and Albert Einstein showed that Sagnac’s effect could be explained by special relativity, without requiring any aether medium for light propagation.

The interesting thing about the Sagnac effect is that it measures absolute rotational speed, that is: rotation relative to an inertial reference frame, in the language of special relativity. In practice, this means rotation relative to the “fixed” position of distant stars. This is useful for inertial guidance systems, such as those found on satellites, modern airliners and military planes, and missiles. The Sagnac effect is used in ring laser gyroscopes and fibre optic gyroscopes to provide an accurate measure of rotational speed in these guidance systems. GPS satellites use these devices to ensure their signals are correctly calibrated for rotation – without them GPS would be less accurate.

Because the magnitude of the Sagnac effect depends on both the rotational speed and the area of the light loop, by making the area large you can make the interferometer incredibly sensitive to even very slow rotation. Rotations as slow as once per 24 hours. You can use these devices to *measure the rotation of the Earth*.

This was first done in 1925. Albert A. Michelson (of the famous Michelson-Morley experiment that disproved the existence of the luminiferous aether), Henry G. Gale, and Fred Pearson acquired the use of a tract of land in Clearing, Illinois (near Chicago’s Midway Airport), and built a huge Sagnac interferometer, a rectangle 610×340 metres in size [1][2].

With this enormous area, the shift in the light beams caused by the rotation speed of the Earth at the latitude of Chicago was around one fifth of a wavelength of the light used – easily observable. The Michelson-Gale-Pearson experiment’s measurements and calculations showed that the rotation speed they measured was consistent with the rotation of the Earth once every 23 hours and 56 minutes – a sidereal day (i.e. Earth’s rotation period relative to the stars; this is shorter than the average of 24 hours rotation relative to the sun, because the Earth also moves around the sun).

Now the interesting thing is that the Sagnac effect measures the *linear rotation speed*, not the angular rotation rate. The Earth rotates once per day – that angular rotation rate is constant for the entire planet, and can be modelled in a flat Earth model simply by assuming the Earth is a spinning disc, like a vinyl record or Blu-ray disc. But the linear rotation speed of points on the surface of the Earth varies.

In the typical flat Earth model with the North Pole at the centre of the disc, the rotation speed is zero at the North Pole, and increases linearly with distance from the Pole. As you cross the equatorial regions, the rotational speed just keeps increasing linearly, until it is maximal in regions near the “South Pole” (wherever that may be).

On a spherical Earth, in contrast, the rotation speed is zero at the North Pole, and varies as the cosine of the latitude as you travel south, until it is a maximum at the equator, then drops again to zero at the South Pole.

Here is a table of rotation speeds for the two models:

Latitude | Speed (km/h) Flat model |
Speed (km/h) Spherical model |
---|---|---|

90°N (North Pole) | 0.0 | 0.0 |

60°N | 875.3 | 837.2 |

41.77°N (Clearing, IL) | 1407.2 | 1248.9 |

30°N | 1750.5 | 1450.1 |

0° (Equator) | 2625.8 | 1674.4 |

30°S | 3501.1 | 1450.1 |

45.57°S (Christchurch) | 3896.6 | 1213.3 |

60°S | 4376.4 | 837.2 |

90°S (South Pole) | 5251.6 | 0.0 |

In the Michelson-Gale-Pearson experiment, the calculated expected interferometer shift was 0.236±0.002 of a fringe (essentially a wavelength of the light used), and the observed shift was 0.230±0.005 of a fringe. The uncertainty ranges overlap, so the measurement is consistent with the spherical Earth model that they used to calculate the expected result.

If they had used the North-Pole-centred flat Earth model, then the expected shift would have been 1407.2/1248.9 larger, or 0.266±0.002 of a fringe. This is well outside the observed measurement uncertainty range. So we can conclude that Michelson’s original 1925 experiment showed that the rotation of the Earth is inconsistent with the flat Earth model.

Nowadays we have much more than that single data point. Sagnac interferometers are routinely used to measure the rotation speed of the Earth at various geographical locations. In just one published example, a device in Christchurch, New Zealand, at a latitude of 43°34′S, measured the rotation of the Earth equal to the expected value (for a spherical Earth) to within one part in a million [3]. Given that the expected flat Earth model speed is more than 3 times the spherical Earth speed at this latitude—and all of the other rotation speed measurements made all over the Earth consistent with a spherical Earth—we can well and truly say that any rotating disc flat Earth model is ruled out by the Sagnac effect.

**References:**

[1] Michelson, A. A. “The Effect of the Earth’s Rotation on the Velocity of Light, I.” *The Astrophysical Journal*, 61, 137-139, 1925. https://doi.org/10.1086%2F142878

[2] Michelson, A. A.; Gale, Henry G. “The Effect of the Earth’s Rotation on the Velocity of Light, II.” *The Astrophysical Journal*, 61, p. 140-145, 1925. https://doi.org/10.1086%2F142879

[3] Anderson, R.; Bilger, H. R.; Stedman, G. E. “ “Sagnac” effect: A century of Earth‐rotated interferometers”. *American Journal of Physics*, 62, p. 975-985, 1994. https://doi.org/10.1119/1.17656

## 9. The South Pole

One of the most popular models for a flat Earth presumes the Earth to be a circular disc centred at the North Pole, and with Antarctica spanning the rim and providing some form of impassable barrier to simply travelling off the disc.

This raises the question of *where the South Pole is*. In such a disc model, there is no single South Pole – all points on the rim of the disc are equally far south. Yet people have travelled to a single geographic place that matches all of the physical requirements of being the South Pole of a spherical Earth:

1. People heading south always end up at this same spot, no matter which line of longitude they head south along.

2. At this place, the sun circles the horizon once per 24 hour period during the southern hemisphere summer, not setting until the autumn equinox, after which it circles below the horizon throughout the southern winter, only rising again at the spring equinox. This is exactly as expected for a physical South Pole on a spherical planet.

3. The place is directly underneath the point in the sky around which the southern stars appear to revolve, in a manner exactly analogous to the movement of the stars at the North Pole, but in the opposite rotational direction. The stars at the North Pole appear to rotate clockwise in the sky, while at the South Pole they rotate anticlockwise in the sky.

Furthermore, from this place, or areas around it, there are no directions in which one cannot travel. There is no visible edge of the disc, nor any impassable barrier to travel.

The same argument applies to the North Pole if the flat Earth is supposed to be a disc with the South Pole at its centre. And if the disc is some other configuration, then there will always be some point on the spherical Earth that ends up corresponding to all the points around the rim of the flat disc Earth – where it is straightforward to travel to that point on the spherical Earth but which is unreachable or should display bizarre properties (such as a rim or impassable barrier) on the flat Earth.

All of this is difficult to explain in a flat Earth model, but is a natural consequence of a spherical Earth.

## 2.c Eratosthenes and the Flat Earth model

A reader has pointed out that we can also use the data collected in our Eratosthenes experiment to test the hypothesis that the Earth is flat and that the difference in shadows is caused by the sun being a relatively small distance from the flat Earth. And we can compare this test to a test of our round Earth hypothesis.

If the Earth is flat, and we make an observation like Eratosthenes, that a vertical stick in one location casts no shadow, while a vertical stick some distance north (or south) does cast a shadow, then we can use geometry to figure out how far away the sun must be.

If the Earth is indeed flat then when we do this calculation for each of our 19 observations we should get the same answer, to within any experimental error. In particular, there should be no systematic difference in our answers that depends on the distance from the equator. Analogously, in our round Earth model, the circumference we have calculated for the Earth from each of our observations should also be the same to within experimental errors, and show no systematic difference depending on distance from the equator.

So let’s test those things! Here are graphs of the results, on which I’ve included a linear least squares best fit line, showing the line’s equation and statistical *R*^{2} score. The *R*^{2} value, or coefficient of determination, is a measure of how likely the data values (the circumference of the Earth in the round Earth hypothesis, or the distance to the sun in the flat Earth hypothesis) are to be correlated with the fixed values (the distance from the equator in both cases). We’ll discuss that after we see the data.

The first thing to notice is that in the top plot, the circumference of the Earth values look fairly evenly scattered around the true value. In the bottom plot, the calculated values for the distance of the sun from Earth are not evenly scattered; they show a pretty clear trend of giving larger distances for the data points closer to the equator and smaller distances for data points further from the equator. We can quantify this by looking at the straight line fits to the data and in particular the *R*^{2} value.

To do a rigorous statistical test, we need to set up our two possible null hypotheses. These are statements that for the purpose of our statistical test *we assume are true*, and then we calculate the probability that what we observe could happen by random chance. Our two null hypotheses are:

1. For the spherical Earth model, the calculated circumference of Earth is independent of the distance from the equator of our data points.

2. For the flat Earth model, the calculated distance of the sun from Earth is independent of the distance from the equator of our data points.

To test these, we use a probability distribution that tells us how likely our observed *R*^{2} scores are. An appropriate one to use is Student’s *t*-distribution. We calculate Student’s *t*-distribution function for 19 data points and 2 degrees of freedom (the *y*-intercept and the slope of our fitted line), determine a value for the function below which 95% of the probability distribution lies, and convert this to an *R*^{2} value using the known transformation. In simpler terms (TL;DR), we’re working out a number *R*^{2}(P<0.05) which, if our calculated values are independent of distance from the equator, then we would expect 95% of experiments to give an *R*^{2} value less than the number *R*^{2}(P<0.05).

Doing the maths, our value for *R*^{2}(P<0.05) is 0.334. What this means is that if our *R*^{2} value is greater than 0.334, then we should *reject* our null hypothesis – the data are statistically inconsistent with the hypothesis (at the 95% confidence level, for those who like statistical rigour*). On the other hand, if our *R*^{2} value is less than 0.334, we cannot reject our null hypothesis – we haven’t proven it to be true, we have just shown that our data are consistent with it.

Now let’s look at our calculated *R*^{2} values. For the spherical Earth hypothesis, *R*^{2} = 0.1358. This is less than the critical value, so our data are consistent with our hypothesis. In contrast, for the flat Earth model, *R*^{2} = 0.9162. This is greater than the critical value, so we can confidently reject the flat Earth hypothesis as inconsistent with our experiment!

So there you have it. Not only did we successfully measure the circumference of the Earth to within our experimental errors, we have now also shown that our experimental results are consistent with a spherical Earth model, and inconsistent with a flat Earth model.

* Note: Choosing the 95% confidence level is typical for statistical hypothesis testing. You should always choose your confidence level before performing the calculations, to avoid any bias in your reporting. You can choose other levels, such as 99%. If I’d done that, we would have found that our data are also inconsistent with the flat Earth model at the more stringent 99% confidence level. In fact, calculating backwards, the confidence level of our rejection is a bit above 99.7%.