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.

Aristarchus's method 1

Geometry of the sun, moon, and Earth when the moon appears half-illuminated.

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.

Aristarchus's method 2

A medieval copy of Aristarchus’s drawing of the geometry of the sun-Earth-moon system during a lunar eclipse. (Public domain image.)

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.

The Sun

Our sun, observed in the ultraviolet as a false colour image by NASA’s Solar Dynamics Observatory satellite. (Public domain image by NASA.)

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.

Ulysses' orbit

Polar orbit of Ulysses around the sun, giving it views of both the sun’s north and south poles. (Public Domain image by NASA.)

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 Ulyssessolar wind particle flux detectors to measure if the energy emitted by the sun as high energy particles varies with direction.

Solar proton flux versus latitude

Proton flux density observed by Ulysses at various heliographic (sun-centred) latitudes. -90 is directly south of the sun, 0 would be in the ecliptic plane. The track shows Ulysses’ orbit, changing in distance and latitude as it passes under the sun’s south polar regions. Figure reproduced from [2].

Solar wind energy flux versus latitude

Various solar wind plasma component energy fluxes observed by Ulysses at various heliographic latitudes. Figure reproduced from [3].

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.

Voliva's distance to the sun calculation

Wilbur Glenn Voliva’s calculation that the sun is 3000 miles above the Flat Earth. Reproduced from Modern Mechanics, October 1931, p. 73.

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.

An aurora

An aurora, observed near Eielson Air Force Base, near Fairbanks, Alaska. (Public domain image by Senior Airman Joshua Strang, United States Air Force.)

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

Solar wind and Earth's magnetosphere

Schematic representation of the solar wind streaming from the sun and interacting with the Earth’s magnetic field. The dashed lines indicate paths of solar particles towards Earth. The solid blue lines show Earth’s magnetic field. (Public domain image by NASA.)

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.

Solar wind interacting with Earth's magnetosphere

Diagram of the solar wind interacting with Earth’s magnetic field (field lines in red). The magnetic field deflects the incoming particles around the Earth, except for a fraction of the particles that enter the magnetic polar funnels and spiral down towards Earth’s magnetic poles. (Public domain image by NASA. modified.)

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.

Northern auroral oval

Northern auroral oval observed on 22 January 2004. Figure reproduced from [1].

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.

Northern auroral oval seen by DE-1

Northern auroral oval observed in 1983 by Dynamics Explorer 1 satellite. The large bright patch at left is the daylight side of Earth. (Public domain image by NASA.)

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.

Southern auroral oval

Southern auroral oval observed in 2005 by IMAGE satellite, overlaid on a Blue Marble image of Earth. (Public domain image by NASA.)

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.

Northern and southern auroral ovals

Current northern and southern auroral ovals as forecast by spaceweatherlive.com on 21 April, 2019. The auroral ovals are the same size and shape.

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.

Northern auroral ovals on Jupiter

Auroral ovals on Jupiter observed in the northern and southern polar regions by the Hubble Space Telescope, using the Wide Field Planetary Camera (1996) and the Space Telescope Imaging Spectrograph (1997-2001). Figure reproduced from [2].

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.

Auroral ovals on a flat Earth

Auroral ovals in their observed locations, mapped onto a flat disc Earth. The ovals are vastly different sizes.

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

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.

A light loop

A loop with light travelling in both directions.

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.

A rotating light loop

Now the loop is rotating. By the time the light has travelled around the loop, the exit from the loop has moved a little bit clockwise. So the light travelling clockwise has to travel further to reach the exit, while the light travelling anticlockwise reaches the exit sooner.

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].

Michelson’s Sagnac interferometer

Diagram of Michelson’s Sagnac interferometer in Clearing, Illinois. The Sagnac loop is defined by the mirrors ADEF. The smaller rectangle ABCD was used for calibration measurements. Light enters from the bottom towards the mirror A, which is half-silvered, allowing half the light through to D, and reflecting half in the other direction towards F. The beams complete circuits ADEF and AFED, returning to A, where the half-silvering reflects the beam from D and lets through the beam from F towards the detector situated outside the loop at the left. The light paths are inside a pipe system, which is evacuated using a pump to remove most of the air. (Figure reproduced from [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).

Rotation speed on a flat disc Earth

Rotation speeds at different places on a flat rotating disc Earth (top view of the disc).

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.

Rotation speed on a flat disc Earth

Rotation speeds at different places on a spherical Earth.

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.

Disc shaped flat Earth

A common Flat Earth model: a flat circular disc centred at the North Pole. (Public domain image from Wikimedia Commons.)

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.

Star trails in the southern hemisphere

Star trails in the southern hemisphere, photographed in the Atacama Desert in Chile. (Creative Commons Attribution 4.0 International image by the European Southern Observatory.) This photo shows the motion of the southern hemisphere stars through several hours of a night, moving in apparent circles around the South Celestial Pole (the point directly above the Earth’s South Pole). I tried to find a copyright or royalty-free photo of star trails photographed from the South Pole itself, but could not find any, however Astronomy Picture of the Day has such a photo taken at the South Pole, showing complete circles over a 24-hour period of darkness.

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.

Distance to the sun in a flat Earth model

Using similar triangles to determine the distance from a flat Earth to the sun, given an observation of the shadow of a vertical stick, and knowing the distance from a point where the sun is overhead.

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 R2 score. The R2 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.

Earth's circumference versus distance from equator

Plot of 19 measurements of the Earth’s circumference, assuming the round Earth model, versus distance from the equator.

Distance to sun versus distance from equator

Plot of 19 measurements of the distance of the sun from Earth, assuming the flat Earth model, versus distance from the equator.

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 R2 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 R2 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 R2 value using the known transformation. In simpler terms (TL;DR), we’re working out a number R2(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 R2 value less than the number R2(P<0.05).

Doing the maths, our value for R2(P<0.05) is 0.334. What this means is that if our R2 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 R2 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 R2 values. For the spherical Earth hypothesis, R2 = 0.1358. This is less than the critical value, so our data are consistent with our hypothesis. In contrast, for the flat Earth model, R2 = 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%.

8. Earth’s magnetic field

Magnetic fields have both a strength and a direction at each point in space. The strength is a measure of how strong a force a magnet feels when in the field, and the direction is the direction of the force on a magnetic north pole. North poles of magnets on Earth tend to be pulled towards the Earth’s North Magnetic Pole (which is in fact a magnetic south pole, but called “the North Magnetic Pole” because it is in the northern hemisphere), while south poles are pulled towards the South Magnetic Pole (similarly, actually a magnetic north pole, called “the South Magnetic Pole” because it’s in the south). Humans have used this property of magnets for thousands of years to navigate, with magnetic compasses.

The simplest magnetic field is what’s known as a dipole, because it has two poles: a north pole and a south pole. You can think of this as the magnetic field of a simple bar magnet. The magnetic field lines are loops, with the field direction pointing out of the north pole and into the south pole, and the loops closing inside of the magnet.

A magnetic dipole

Illustration of magnetic field lines around a magnetic dipole. The north and south poles of the magnet are marked.

It’s straightforward to measure both the strength and the direction of the Earth’s magnetic field at any point on the surface, using a device known as a magnetometer. So what does it look like? Here are some contour maps showing the Earth’s magnetic field strength and the inclination – the angle the field lines make to the ground.

Earth's magnetic field intensity

Earth’s magnetic field strength. The minimum field strength occurs over South America; the maximum field strengths occur just off Antarctica, south of Australia, and in the broad patch covering both central Russia and northern Canada. (Public domain image by the US National Ocean and Atmospheric Administration.)

Earth's magnetic field inclination

Earth’s magnetic field inclination. The field direction is parallel to the ground at points along the green line, points into the ground in the red region, and points out of the ground in the blue region. The field emerges vertically at the white mark off the coast of Antarctica, south of Australia – this is the Earth’s South Magnetic Pole. The field points straight down at the North Magnetic Pole, north of Canada – not shown in this Mercator projection map, which omits areas with latitude greater than 70° north or south. (Public domain image by the US National Ocean and Atmospheric Administration.)

Now, how can we explain these observations with either a spherical Earth or flat Earth model? Let’s start with the spherical model.

You may notice a few things about the maps above. The Earth’s magnetic field is not symmetrical at the surface. The lowest intensity point over South America is not mirrored anywhere in the northern hemisphere. And the South Magnetic Pole is at a latitude about 64°S, while the North Magnetic Pole is at latitude 82°N. As it happens, this observed magnetic field is to a first approximation the field of a magnetic dipole – just not a dipole that is centred at the centre of the Earth. The dipole is tilted with respect to Earth’s rotation, and is offset a bit to one side – towards south-east Asia and away from South America. This explains the minimum intensity in South America, and the asymmetry of the magnetic poles.

A magnetic dipole

The Earth’s magnetic field is approximated by a dipole, offset from the centre of the Earth. The rotational axis is the light blue line, with geographic north and south poles marked. The red dots are the equivalent magnetic poles. The North Magnetic Pole is much closer to the geographic north pole than the South Magnetic Pole is to the geographic south pole. (As stated in the text, the “North Magnetic Pole” of the Earth is actually a magnetic south pole, and vice versa.)

Models of the interior of the Earth suggest that there are circulating electrical currents in the molten core, which is composed mostly of iron. These currents are caused by thermal convection, and twisted into helices by the Coriolis force produced by the Earth’s rotation, both well understood physical processes. Circulating electrical currents are exactly what causes magnetic fields. The simplest version of this so-called dynamo theory model is one in which there is a single giant loop of current, generating a simple magnetic dipole. And in fact this dipole fits the Earth’s magnetic field to an average deviation of 16% [1].

This is not a perfect fit, but it’s not too bad. The adjustments needed to better fit Earth’s measured field are relatively small, and can also be understood as the effects of circulating currents in the Earth’s core, causing additional components of the field with smaller magnitudes. (The Earth’s magnetic field also changes over time, but we’ll discuss that another day.)

If the Earth is flat, however, there is no such relatively simple way to understand the strength and direction of Earth’s magnetic field using standard electromagnetic theory. Even the gross overall structure—which is readily explained by a magnetic dipole for the spherical Earth—has no such simple explanation. The shape of the field on a flat Earth would require either multiple electrical dynamos or large deposits of magnetic materials under the Earth’s crust, and they would have to be fortuitously arranged in such a way that they closely mimic a dipole if we assumed the Earth to be a sphere. For any random arrangement of magnetic field-inducing structures on a flat Earth to happen to mimic the field of a spherical planet so closely is highly unlikely. Potentially it could happen, but the Earth actually being a sphere is a much more likely explanation.

That the simpler model is more likely to be true than the one requiring many ad-hoc assumptions is a case of Occam’s razor. In science, particularly, a simpler theory is more easily testable than one with a large number of ad-hoc assumptions. Occam’s razor will come up a lot, and I should probably write a sidebar article about it.

References:

[1] Nevalainen, J.; Usoskin, I.G.; Mishev, A. “Eccentric dipole approximation of the geomagnetic field: Application to cosmic ray computations”. Advances in Space Research, 52, p. 22-29, 2013. https://doi.org/10.1016/j.asr.2013.02.020

2.b Eratosthenes’ measurement results

Thank you to everyone who participated in our measurement of the Earth using Eratosthenes’ method! And thank you to those who tried but were frustrated by the weather – I received several reports of bad weather from the UK, France, and parts of the USA. But we have collected 19 successful observations, from 7 countries: New Zealand, Australia, Israel, Germany, Norway, USA, and Canada. I’ve plotted the locations of the observations on the following map.

Map of measurement locations

Map of observation locations. 16 locations are plotted; 3 of the 19 measurements were taken in the same city as another measurement.

The reason we did this experiment on the date of the equinox (20/21 March) is because that is when the sun is directly over the equator. Rather than use ancient Syene in Egypt as our reference point, where the sun is directly overhead on the summer solstice, we’re doing our calculations based on distance from the equator.

Ideally, I’d have asked all of you to measure the length of your vertical stick, the length of its shadow, and the distance you are away from the nearest point where your stick cast no shadow (which would be on the equator either due south or north of your location). Although in principle you can measure your distance from this point by travelling due south or north to the equator and keeping an accurate log of distance travelled, this would almost certainly have lowered the participation numbers! So as a proxy, I asked for your locations – either the city, or an accurate reading of your latitude from a GPS unit or Google Maps or something similar. This is really just a shortcut so that I can calculate your distance from the equator. And yes, although the existence of “degrees of latitude” is based on the premise that the Earth is spherical, the simple multiplicative relationship between distance and the numbers we call “latitude” still holds in the real world (even if the Earth is flat).

Some summary statistics:

  • Number of data points: 19
  • Shortest distance from equator: 3196 km (Geraldton, Australia)
  • Longest distance from equator: 6662 km (Oslo, Norway)
  • Shortest stick used: 31.5 cm
  • Longest stick used: 250.2 cm

The calculations proceeded as follows:

1. For each location, I calculated the distance from the equator, using the provided latitude.

2. I calculated the angle of the stick’s shadow from the vertical: shadow angle = arctangent(shadow length / stick length).

3. I calculated the circumference of the Earth for each measurement: circumference = 4 × distance from equator × 90°/(shadow angle). Here is a graph of the resulting 19 measurements of the Earth’s circumference, plotted against the length of the stick used in each case.

Map of measurement locations

Plot of 19 measurements of the Earth’s circumference, versus shadow stick length. As the sticks get longer, the results tend to get more accurate, because it is easier to measure the length of the shadow to a smaller percentage error.

4. I calculated the average of the 19 different measurements of circumference, as well as the standard error of the mean, a statistical measure of the expected uncertainty in the average value. (In experiments like this, where we take multiple independent measurements of the same value, we expect there to be some random errors in each result, caused by slight inaccuracies in measuring the lengths of the sticks and shadows. Our best overall estimate is the average of the results, and the amount of scatter in the results can be used to estimate the likely size of any error in the average.)

The result we achieved is that we measured the circumference of the Earth to be 39926 km, with a standard error of 163 km, or (39926 ± 163) km. What this means is that statistically we expect the true value to lie somewhere between 39763 km and 40089 km.

The polar circumference of the Earth is in fact 40008 km, which lies neatly within this range. So we did it! We measured the circumference of the Earth, and we got the right answer to within the statistical uncertainty of our method!

In one small wrinkle, when everyone was reporting their measurements to me, one person reported that his measurement might be a little bit wrong, because he didn’t have access to a level or any other means of ensuring that his stick was exactly vertical when he took the measurement. So he was unsure whether his data should really be included or not. As it turns out, his data produced the measurement with the largest error, the lowest data point on the graph. If we remove his measurement, our average and standard error become: (40012 ± 147) km. Our average is now even closer to the correct answer, a mere 4 km different. If we made many more measurements, being careful to minimise our random errors, we could expect our result to be even better.

So thank you again to all who participated. Now you can honestly brag that you have measured the size of the Earth!

7. Supernova 1987A

Stars produce energy from nuclear fusion reactions in their cores, where the light elements making up the bulk of the star are compressed and heated by gravity until they fuse into heavier elements. There is a limit to this, however, because once iron is produced in the core no more energy can be extracted from it. Fusing iron requires an input of energy. As iron accumulates, the layers near the core collapse inwards, because not enough energy is being produced to hold them up. At a certain point, the collapse speeds up suddenly and catastrophically, the whole core of the star collapsing in a few seconds. This releases an enormous amount of gravitational energy, fusing heavier elements and initiating nuclear reactions in the outer parts of the star, which blow off in a vast explosion. The star has turned into a supernova, one of the most energetic phenomena in the universe. A supernova can, briefly, shine brighter than the entire galaxy of 100 billion (1011) stars containing it.

Historically, supernovae were detected visually, when a “new star” suddenly appeared in the night sky, shining brightly for a few weeks before fading away from sight. We have reliable records of visible supernovae appearing in the years 1006, 1054, 1181, 1572, and 1604, as well as unconfirmed but probable events occurring in 185 and 393. These supernovae all occurred within our own Milky Way Galaxy, so were close enough to be visible to the naked eye. Since 1604, there have been no supernovae detected in our Galaxy – which is a bit of a shame because the telescope was invented around 1608, just too late to observe the most recent one.

Astronomers have used telescopes to observe supernovae in other galaxies since the late 19th century. Almost none of these are visible to the naked eye. But in 1987 a supernova occurred in the Large Magellanic Cloud, a dwarf galaxy satellite of our own, making it the nearest supernova ever observed in the telescopic era. It reached magnitude 3, making it as bright as a middling star in our sky. It was first seen by independent observers in Chile and New Zealand on 24 February 1987.

The Large Magellanic Cloud is visible from the southern hemisphere of Earth, and in the north up to a latitude around 21°N. It is never visible from any point further north. And so supernova 1987A (the first supernova detected in 1987) was never visible from any point further north than 21°N.

Supernova 1987A

Supernova 1987A and the Large Magellanic Cloud. SN 1987A is the bright star just right of the centre of the image. (Photo: Creative Commons Attribution 4.0 International by the European Southern Observatory.)

When a supernova explosion occurs, the collapsing star emits vast quantities of matter and radiation into the surrounding space. Visible light is just one part of the radiation. SN 1987A also emitted gamma rays, x-rays, and ultraviolet light, the latter two of which were detected by space-based telescopes. And it also blasted particles into interstellar space: heavy element nuclei, neutrons, electrons, and other subatomic particles. One of the types of particles produced was neutrinos. Neutrinos have such a small mass that so far we’ve been unable to perform any experiment that can distinguish their mass from zero. And this means that they move at close to the speed of light – so close that we’ve never made any observation that shows them to move any slower.

At the moment of collapse, SN 1987A emitted a huge burst of neutrinos. These travelled through intergalactic space and some of the neutrinos made it to Earth, where some of them were detected. This neutrino burst was detected almost simultaneously at three different neutrino observatories in different parts of the world:

While a total of 24 neutrinos might not sound like a lot, this is significantly higher than the background detection rate of neutrinos from other sources such as our sun and general cosmic rays from random directions in space. And all 24 of these neutrinos were detected within a single 13-second time window – if corrected for the differences in light travel time from SN 1987A to each observatory caused by their locations on the spherical Earth.

You might notice that all three of the detectors listed are in the northern hemisphere. In fact, the southernmost of them is Kamioka, at 36° 20′ 24″ N. This means that the Large Magellanic Cloud, and SN 1987A in particular, are not visible in the sky at any of these detector locations. This fact by itself provides fairly convincing evidence to most people that the Earth cannot be flat, but Flat Earth enthusiasts propose various solutions for the limited visibility of celestial objects from different parts of the Earth. In Flat Earth theory, all visible stars and galaxies are above the plane of the Earth, and obscured from some parts by distance or intervening objects. This obviously requires SN 1987A to be above the plane of the Flat Earth.

In fact, at this point it might seem that the spherical Earth has a problem: If SN 1987A is not visible from the locations of the neutrino detectors, then how did they detect neutrinos from it? The answer is that neutrinos are extremely elusive particles – they barely interact with matter at all. Neutrinos are known to pass right through the Earth with ease. So although the spherical Earth blocked the light from SN 1987A from reaching the neutrino observatories, it did not stop the neutrinos. The neutrinos passed through the Earth to reach the observatories.

Astronomers estimate SN 1987A released around 1058 neutrinos. The blast was 168,000 light years away, so at the distance of Earth, the number of neutrinos passing through the Earth would be approximately 3×1020 neutrinos per square metre. The Kamiokande-II detector is a cylinder of water 16 metres high and 15.6 metres in diameter, so nearly 1023 SN 1987A neutrinos would have passed through it, leading to just 11 detections. This matches the expected detection rate for neutrinos very well.

Additionally, the Kamioka and Irvine-Michigan-Brookhaven detectors are directional – they can determine the direction from which observed neutrinos arrive. They arrived coming up from underground, not down from the sky. The observed directions at both detectors correspond to the position of the Large Magellanic Cloud and SN1987A on the far side of the spherical Earth [1][2].

Kamiokande-II results

Distribution of SN 1987A neutrino detections at Kamiokande-II in energy of produced electrons and angle relative to the direction of the Large Magellanic Cloud (LMC). Detected electrons are produced by two different processes, the first is rapid and highly aligned with neutrino direction, while the second is a slower secondary particle generation process and randomises direction uniformly. Neutrinos 1 and 2 (the earliest in the burst) are aligned directly with the LMC, and the remainder are distributed uniformly. This is statistically consistent with the burst having originated from the LMC. Figure reproduced from [1].

In a flat Earth model, SN 1987A would have to be simultaneously above the plane of the Earth (to be visible from the southern hemisphere) and below it (for the neutrino burst to be visible coming up from under the plane of the Earth). This is self-contradictory. However the observations of SN 1987A are all consistent with the Earth being a globe.

References:

[1] Hirata, K.; Kajita, T.; Koshiba, M.; Nakahata, M.; Oyama, Y.; Sato, N.; Suzuki, A.; Takita, M.; Totsuka, Y.; Kifune, T.; Suda, T.; Takahashi, K.; Tanimori, T.; Miyano, K.; Yamada, M.; Beier, E. W.; Feldscher, L. R.; Kim, S. B.; Mann, A. K.; Newcomer, F. M.; Van, R.; Zhang, W.; Cortez, B. G. “Observation of a neutrino burst from the supernova SN1987A”. Physical Review Letters, 58, p. 1490-1493, 1987. https://doi.org/10.1103/PhysRevLett.58.1490

[2] Bratton, C. B.; Casper, D.; Ciocio, A.; Claus, R.; Crouch, M.; Dye, S. T.; Errede, S.; Gajewski, W.; Goldhaber, M.; Haines, T. J.; Jones, T. W.; Kielczewska, D.; Kropp, W. R.; Learned, J. G.; Losecco, J. M.; Matthews, J.; Miller, R.; Mudan, M.; Price, L. R.; Reines, F.; Schultz, J.; Seidel, S.; Sinclair, D.; Sobel, H. W.; Stone, J. L.; Sulak, L.; Svoboda, R.; Thornton, G.; van der Velde, J. C. “Angular distribution of events from SN1987A”. Physical Review D, 37, p. 3361-3363, 1988. https://doi.org/10.1103/PhysRevD.37.3361

6. Gegenschein

If you shine a light into a suspension of fine particles, the particles will scatter the light. This is easy enough to show with a little bit of flour stirred into a glass of water, or with a dilute solution of milk in water, in which case the particles are small globules of fat. You can see a beam of light passing through such a medium because of the scattering, which is known as the Tyndall effect.

We can model the interaction of light with the scattering particles using Mie scattering theory, named after German physicist Gustav Mie. This is essentially a set of solutions of Maxwell’s equations for the propagation of electromagnetic radiation (in this case, light) in the presence of the scattering objects. If you solve these equations for diffuse particles a bit bigger than the wavelength of light, you can derive the angular distribution of the scattered light. The scattering is far from uniform in all directions. Rather, it has two distinct lobes. Most of the light is scattered by very small angles, emerging close to the direction of the original incoming beam of light. As the scattering angle increases, less and less light is scattered in those directions. Until you reach a point somewhere around 90°, where the scattering is a minimum, and then the intensity of scattered light starts climbing up again as the angle continues to increase. It reaches its second maximum at 180°, where light is reflected directly back towards the source.

This bright spot of reflected light back towards the source is called backscatter. It can be seen when shining light into smoke or fog. It’s the reason why some cars have special fog lights, angled down to illuminate the road, rather than shine straight into the fog and reflect back into the driver’s eyes. Backscatter is also the reason for the bright spot you might have noticed on clouds around the shadow of a plane that you’re flying in (at the centre of the related optical phenomenon of glories).

Another place where there is a collection of smoke-sized particles is in interplanetary space. In the plane of the planets’ orbits around the sun, there is a considerable amount of left over material of sizes around 10 to 100 micrometres, constantly being replenished by asteroid collisions and outgassing from comets. This material is called the interplanetary dust cloud, or the zodiacal dust cloud, because it is densest in the ecliptic—the plane of the planets—which runs through the zodiac constellations. This dust has been sampled directly by several deep space probes: Pioneers 10 and 11, Ulysses, Galileo, Cassini, and New Horizons.

The brightest source of light in the solar system is the sun. As it shines through this interplanetary dust cloud, some of the light is scattered. Most of the scattered light is deflected only by small angles, in accordance with Mie theory. But some is backscattered, and in the backscatter direction there is a peak in brightness of the scattered light directly back towards the sun. S. S. Hong published a paper in 1985, with calculations of the scattering angles of light by the interplanetary dust cloud [1]. Here’s the pertinent plot from the paper:

Scattering intensity v. angle for interplanetary dust

Scattering intensity versus scattering angle for interplanetary dust. Figure reproduced from [1].

The different curves correspond to different choices of a power law to model the size distribution of the dust particles. In each case you can see that most of the scattering occurs at small angles, there is a minimum of scattering intensity around 90°, and the scattering increases again to a second maximum at 180°, the backscattering angle.

As an aside, this backscattering also occurs in interstellar dust, and here’s a figure from a paper by B. T. Draine showing scattering intensity versus angle for the measured dust distributions of the Small Magellanic Cloud, Large Magellanic Cloud, and Milky Way galaxy, plotted for several wavelengths of light [2]. The wavelengths are shown in Angstroms, and in these units visible light occurs between 4000 and 7000 Å (lower being ultraviolet and higher infrared). In these cases the models show minima in scattering around 130°, with the backscattering again being maximal at 180°.

Scattering intensity v. angle for interstellar dust

Scattering intensity versus scattering angle for interstellar dust. Figure reproduced from [2].

We’re not concerned with interstellar dust here, but it shows the general principle that there is a peak in scattered light directly back towards the light source, from fog, smoke, and space dust.

We are concerned with backscatter from the interplanetary dust cloud. Given that this phenomenon occurs, it implies that if we could look into space in the direction exactly opposite the direction of the sun, then we should see backscatter from the interplanetary dust.

If the Earth is spherical, then night time corresponds to the sun being behind the planet. You should see, in the night sky, the point exactly opposite the direction of the sun. You should be able to see, in that direction, the backscattered light of the sun from the interplanetary dust cloud.

Now let’s imagine the Earth is flat. The sun shines on some part of the Earth at all times, so therefore it must be above the plane of the Earth at all times. (How some parts of that plane are in the dark of night is a question for another time. Some Flat Earth models propose a sort of cosmic lampshade for the sun, which makes it more like a spotlight.) At any rate, it should never be possible to look into the sky in the exact opposite direction to the sun. So there should be no point in the night sky with a peak of backscattered sunlight.

Now that we’ve made the predictions from our models, what do we actually see? It turns out that the backscattered sunlight is a visible phenomenon, and it can be seen exactly where predicted by the spherical Earth model. It’s a faint glow in the night sky, centred on the point in the exact opposite direction to the sun. The earliest recorded description of it comes from 1730, by the French astronomer Esprit Pézenas. The German explorer Alexander von Humboldt wrote about it around 1800 on a trip to South America, and gave it the name Gegenschein (German for “counter-shine”).

Unfortunately, in our modern industrial society light pollution is so bad that it’s almost impossible to see the gegenschein anywhere near where people live. You need to go somewhere remote and far away from any settlements, where it is truly dark at night. If you do that, you can see something like this:

Gegenschein

Gegenschein, as seen from the Very Large Telescope site, Cerro Paranal, Chile. (Photo: Creative Commons Attribution 4.0 International by the European Southern Observatory.)

The gegenschein is the glow in the sky just above the centre of the image. Heck, it’s so beautiful, here’s another one:

Gegenschein

Gegenschein, as seen from the Very Large Telescope site, Cerro Paranal, Chile. (Photo: Creative Commons Attribution 4.0 International by the European Southern Observatory.)

This is a fisheye image, with the band of the Milky Way and the horizon wrapped around the edge of the circle. Here the gegenschein is the broad glow centred around a third of the way from the centre, at the 1 o’clock angle.

The visibility of the gegenschein shows that, in places where it is night time, the sun is actually behind the Earth. On a flat Earth, the sun can never be behind the Earth, so the gegenschein would never be visible. And so the optical effect of backscatter provides evidence that the Earth is a globe.

References:

[1] Hong, S. S. “Henyey-Greenstein representation of the mean volume scattering phase function for zodiacal dust”. Astronomy and Astrophysics, 146, p. 67-75, 1985. http://adsabs.harvard.edu/abs/1985A%26A…146…67H

[2] Draine, B. T. “Scattering by Interstellar Dust Grains. I. Optical and Ultraviolet”. The Astrophysical Journal, 598, p. 1017-1025, 2003. https://doi.org/10.1086/379118