Parallax of the moon, experiment update

With Proof 45 I asked readers if they would help perform an experiment to measure the parallax of the moon, and so derive a figure for the radius of the Earth. The experiment was more demanding than when we used Eratosthenes’ method, requiring helpers to stay up for much of the night and take specific photos of the moon and stars. Unfortunately this meant that only two readers contacted me to help perform the experiment, which simply isn’t enough to do it successfully, given the vagaries of weather. (The planned day was rainy here so I couldn’t be a third observer.)

So we don’t have any results to report. I may try this experiment again some time in the future, but I’ll need a significantly higher response from readers if we’re to get it to work.

45. Parallax of the moon

If you hold your arm outstretched before you with a finger raised in front of some object in the distance, and you close one eye and align your finger so that it looks directly in front of a particular object, then close that eye and open the other eye, then your finger will appear to have moved sideways. This is because your left eye and your right eye are not in the same position, so lining up your finger with an object using one eye means that it is not aligned with the other eye. This effect is known as parallax.

Parallax is a purely geometric effect, caused by the divergence of straight lines drawn from a common object to two different viewpoints. We make use of parallax unconsciously, our brains using the different visual information from our two eyes to give us the sensation of viewing a three dimensional world around us.

We’ve discusses parallax briefly in Proof 28: Stereo imaging. Here’s a repeat of the diagram illustrating parallax (with a dog instead of a finger).

Illustration of parallax

Illustration of parallax. The dog is closer than the background scene. Sightlines from your left and right eyes passing through the dog project to different areas of the background. So the view seen by your left and right eyes show the dog in different positions relative to the background. (The effect is exaggerated here for clarity.)

Parallax can be observed wherever you look at two objects at different distances. If you measure the distance between the two viewpoints and the angle between the two different sightlines, and you know the distance to one of the objects, then by simple geometry you can calculate the distance to the other object. A handy approximation is that if one object is much further away than the near object, then you can assume it is infinitely far away, which simplifies the geometry and produces only a negligibly small error.

Parallax can be used in this way to measure the distance to some astronomical objects, namely those objects close enough to Earth to have a measurable parallax angle when compared to the positions of distant stars. This includes objects in our solar system, as well as the very nearest stars. The European Space Agency satellite Hipparcos measured parallaxes of around 100,000 stars, using observations made six months apart so that the two different viewpoints are on opposite sides of the Earth’s orbit, a viewpoint separation baseline of 300 million km. Even with this enormous baseline, the parallax angle of stars is less than one second of arc, requiring precise instruments to even detect. However, there is a much closer celestial object for which the parallax is readily visible with a smaller baseline: the moon.

The moon is not close enough to see parallax from your left and right eyes. It is however close enough that photos of the moon taken from different locations on the Earth at the same time show the moon significantly shifted against the background of stars. With viewpoints on opposite sides of the Earth, the parallax angle of the moon is about one degree – it varies slightly due to the moon’s elliptical orbit.

Illustration of parallax

Illustration of parallax of the moon, as seen from four different points on the Earth’s surface at the same time. Viewed from the North Pole, the moon occults stars in the constellation of Pleiades, but viewed from other locations the moon does not occult the same stars. (Public domain image from Wikimedia Commons.)

Using such observations of the moon, and knowing how far apart the viewpoints are, you can calculate the distance to the moon with high accuracy. But you can also do the reverse calculation: if you know the distance to the moon, you can calculate how far apart the viewpoint locations are. As mentioned in Proof 32: Satellite laser ranging, we have an independent method of measuring the distance to the moon, to an accuracy of better than a millimetre, by bouncing lasers off reflectors left on the moon by the Apollo astronauts. Before 1969, we could also measure the distance to the moon using radar ranging, to an accuracy of about 1 km.[1][2]

Incidentally, the 1965 paper by Yaplee, et al.[2], also includes a calculation of the radius of the Earth from the radar ranging data, as it comes out as a variable that can be solved for if you know the acceleration due to gravity at Earth’s surface and the ratio of the Earth’s mass to the moon’s mass, which were known at the time. Given their radar results, the authors calculate the Earth’s equatorial radius to be 6378.167 km, only 30 metres different to the current reference value of 6378.137 km. (Obviously a result unobtainable if you assume the Earth is flat.)

Illustration of parallax

Diagram from Yaplee, et al., showing the geometry of the Earth-moon system for the purpose of determining the distance to the moon using radar ranging. (Figure reproduced from [2].)

So even without using parallax we know the distance to the moon. This means we can do the reverse calculation, and figure out how far apart different viewpoints (such as the North Pole and South Pole) are. Any two antipodal points, on opposite sides of the Earth, are about 20,000 km apart – as the crow flies, or as a traveller would have to cover to go from one point to the other. However if you calculate how far apart those points are using lunar parallax, you find that they are actually about 12,730 km apart. Why such a big discrepancy?

The simplest explanation for this is that the Earth’s surface is curved, with 20,000 km representing the semi-circumference and 12,730 km being the diameter (twice the radius) of curvature. If the Earth were flat, then the calculated distance between two points for which the travel distance is 20,000 km would also have to be 20,000 km.

Experiment:

With your help, we can perform a lunar parallax experiment to see if the Earth is flat or not, and to measure its radius if we assume it’s spherical. All we need to do is take some photos of the moon and the background stars at the same time, from different places around the world. The best time to do this is when the moon is up most of the night, which means a full moon. The next full moon after this is published occurs on 31 October 2020 at 14:49 UTC. This means the night of 31 October-1 November is perhaps the best time to do this. However I will be travelling that night and unable to participate myself, and perhaps many people will be busy with Halloween activities. So I propose we do it on the following night: the evening of 1 November into the morning of 2 November. This has the slight advantage that the moon rises and sets a bit later, meaning it will be easier for people to get a photo just before sunrise, which is vitally important as I shall explain.

I could set one specific time for everyone to take photos, but that would not be ideal, because half the planet won’t be able to see the moon at the given time, and there will also be places where the moon is up, but it will be too light to see stars. So instead, we will take photos on the hour, every hour, for as much of the night as you can manage. For this to provide useful data, the photos must satisfy the following conditions:

1. We absolutely must have some photos taken in the midnight-to-sunrise period, and preferably some taken just before sunrise (e.g. 4 or 5 a.m.). This is so they can be matched with photos taken in the evening in different time zones. If everyone just takes a few photos in the evening and goes to bed, this won’t work. It’s better if you take a shot before going to bed, and then drag yourself out of bed for another shot before sunrise. I realise not everybody will want to do this, but at least some of you will need to. If you really can’t manage it, please stay up as late as possible and shoot every hour until you do go to bed.

2. Stars must be visible in the photos. Enough stars that we can recognise the constellations, not just 3 or 4 stars. This means you will need to overexpose the moon quite severely. That’s fine, we’re not interested in seeing details of the moon, as long as we can figure out where the middle of the moon is. This means you shouldn’t zoom right into the moon – a moderate focal length lens will work fine, around 50-100 mm for a 35 mm camera. Go out some night beforehand and practice taking a photo of stars to figure out the correct exposure.

Some example photos that would be fine for this experiment:

If you’re interested in taking photos for this experiment, please contact me (dmm [at] dangermouse.net) to register your interest, with your location and a list of hours that you would be willing to be awake to take photos. Maybe send me a sample photo you’ve taken showing stars. I’ll coordinate the list and make sure we have enough people to make the experiment worthwhile, and will let everyone know a few days beforehand whether we will go ahead, or if we should postpone the date until we get more people. And if we do go ahead, what hours will be most valuable for you to take photos.

References:

[1] Yaplee, B. S., Bruton, R. H., Craig, K. J., Roman., N. G. “Radar echoes from the moon at a wavelength of 10 cm.” Proceedings of the Institute of Radio Engineers, 46(1), p. 293-297, 1958. https://doi.org/10.1109/JRPROC.1958.286790

[2] Yaplee, B. S., Knowles, S. H., Shapiro, A., Craig, K. J., Brouwer, D. “The mean distance to the Moon as determined by radar.” In Symposium-International Astronomical Union, 21, p. 81-93, Cambridge University Press, 1965. https://doi.org/10.1017/S0074180900104826

Audio files now available for first six proofs

I’ve still been a bit too busy and distracted to write more proofs recently. However, I mentioned this project to my mother, and she expressed a keen interest in learning more. Unfortunately her eyesight has deteriorated and her lifelong avid reading habit has necessarily been converted to audiobooks.

So I decided to record the proofs to audio files and send them to her. She is thoroughly enjoying them! And now I’m sharing the audio files here on the site. I’ve recorded an Introduction, and the first six Proofs so far – you can access the audio files from the main Index page.

Some of them are verbatim readings of the article. Others that rely strongly on diagrams I have edited slightly so that they flow without requiring reference to the diagrams. I hope that some of you find these files useful.

And I do expect to have a bit more free time to be able to start writing some more proofs in the next week or two.

44. Magnetic striping

We’ve discussed continental drift and plate tectonics in Proof 22. Plate tectonics. There’s another aspect of plate tectonics that was mentioned in passing there, which deserves some further attention. Proof 22 stated:

But as technology advanced, detailed measurements of the sea floor were made beginning in the late 1940s, including the structures, rock types, and importantly the magnetic properties of the rocks.

That last one, the “magnetic properties”, was the piece of evidence that really cemented continental drift as a real thing.

We pick up the history in 1947, when research expeditions led by American oceanographer Maurice Ewing established the existence of a long ridge running roughly north-south down the middle of the Atlantic Ocean. They also found that the crust beneath the ocean was thinner than that beneath the continents, and that the rocks (below the seafloor sediment) were basalts, rather than the granites predominantly found on continents. There was something peculiar about the Earth’s crust around these mid-ocean ridges. And over the next few years, more ridges were found in other oceans, revealing a network of the structures around the globe. The system of mid-ocean ridges had been discovered, but nobody yet had an explanation for it.

Meanwhile, from 1957, the Russian-American oceanographer Victor Vacquier took World War II surplus aerial magnetometers that had been used to detect submerged submarines from reconnaissance aircraft, and adapted them for use in submarines to examine the magnetic properties of the sea floor. It was well known that basalt contained the mineral magnetite, which is rich in iron and can be strongly magnetised.

What Vacquier found was unexpected and astonishing. In a survey of the Mendocino Fault area off the coast of San Francisco, Vacquier discovered that the sea floor basalt was not uniformly magnetised, but rather showed a distinctive and striking pattern. The magnetism appeared to be relatively constant along north-south lines, but to vary rapidly along the east-west direction, causing “stripes” of magnetism running north-south.[1]

Magnetic field measurements on the sea floor near Mendocino Fault

Map of magnetic field measurements on the sea floor near Mendocino Fault, showing strong north-south striping of the magnetic field. (Figure reproduced from [1].)

Follow up observations showed that the stripes were not localised, but extended over large regions of the ocean.[2][3].

Magnetic anomalies on the sea floor off California

Map of magnetic anomalies on the sea floor off California. Shaded areas are positive magnetic anomaly, unshaded areas are negative. (Figure reproduced from [2].)

In fact, these magnetic “zebra stripes” were present pretty much everywhere on the floor of every ocean. They weren’t always aligned north-south though – it turned out that they were aligned parallel to the mid-ocean ridges. The early discoverers of this odd phenomenon had no explanation for it.

Returning to the mid-ocean ridges, American oceanographer Bruce Heezen wrote a popular article in Scientific American in 1960 that informed readers of the recent discoveries of these enormous submarine geological features.[4] In the article, he speculated that perhaps the ridges were regions of upwelling material from deep within the Earth, and the sea floors were expanding outwards from the ridges. Heezen was not aware of any mechanism for regions of Earth’s crust to disappear, so he suggested that the Earth might slowly be expanding, through the creation of new crust at the mid-ocean ridges.

Although Heezen’s idea of an expanding Earth didn’t take hold, his idea of upwelling and expansion along the mid-ocean ridges was quickly combined with existing proposals (that Heezen had overlooked) that crust could be disappearing along the lines of deep ocean trenches, as parts of the Earth moved together and were subducted downwards. The American geologists Harry Hammond Hess and Robert S. Dietz independently synthesised the ideas into a coherent theory of continental drift, combining the hypotheses of seafloor spreading and ocean trench subduction to conclude that the Earth was not changing size, but rather it was fractured into crustal plates that slowly moved, spreading apart in some places, and colliding and subducting in others.[5][6]

Proposed mantle convection by Hess

Earliest diagrams of proposed mantle convection cells causing continental drift, with upwelling at mid-ocean ridges causing seafloor spreading, by Harry Hammond Hess. Figure 7 (top) shows the detailed structure of a mid-ocean ridge, with measured seismic velocities (the speed of seismic waves in the rock) in various regions. Hess proposed that the observed lower speeds in the central and upper zones were caused by fracturing of the rock as it deforms during the upwelling, plus higher residual temperature of the upwelled material. Figure 8 (bottom) shows Hess’s proposed mantle convection cells. (Figures reproduced from [6].)

So by the 1960s, most of the observational pieces of this puzzle were in place. However, the unifying theory that would explain it all still required some synthesis, and acceptance of some unestablished hypotheses. This synthesis was again put together independently by two different groups of geologists: the Canadian Lawrence Morley, and the English Ph.D. student Frederick Vine and his supervisor Drummond Matthews. Morley wrote two papers and submitted them to Nature and the Journal of Geophysical Research in 1963, but both journals rejected his work as too speculative. Vine and Matthews thus received publication priority when Nature accepted their paper later in 1963.[7]

The geologists pointed out that if new rock was being created at the mid-ocean ridges and then spreading outwards, then the seafloor rocks should get progressively older the further away they are from the ridges. Each one of the magnetic zebra stripes running parallel to the ridges then corresponds to rocks of the same age. If, they conjectured, the rocks record the direction of the Earth’s magnetic field when they were formed, and for some reason the Earth’s magnetic field reversed direction periodically, that would explain the existence of the magnetic stripes.

Observed and modelled sea floor magnetic fields

Diagram by Vine and Matthews showing the observed magnetic field strength of the sea floor rocks measured across the Carlsberg Ridge in the Indian Ocean, showing positive and negative regions (solid lines), computed magnetic field strength under conventional (for the time) assumptions (dashed lines), and computed field strength assuming 20 km wide bands in which the Earth’s magnetic field has been reversed. The periodic field reversal matches the observed magnetism much better. (Figure reproduced from [7].)

As with Morley’s rejected papers, this paper was treated with scepticism initially, because it relied on two unproven conjectures: (1) that the rocks maintain magnetism aligned with the Earth;’s magnetic field at the time of solidification and, much more unbelievably, (2) that the Earth’s magnetic field direction reverses periodically. For some geologists, this was too speculative to be believe.

One test of Vine and Matthews’ seafloor spreading hypothesis would be to measure the age of the sea floor using some independent method. If the rocks were found to get progressively older the further away they are from the mid-ocean ridges, then that would be strong evidence in favour of the theory. As it happens, it’s possible to date the age of rocks formed from magma, using a method of radiometric dating known as potassium-argon (K-Ar) dating. Potassium is a fairly common element in rocks, and the isotope potassium-40 is radioactive, with a half-life of 1.248×109 years. Most of the potassium-40 decays to calcium-40 via beta decay (see Proof 29. Neutrino beams for a recap on beta decay), but just over 10% of it decays via electron capture to the inert gas argon-40. Argon is not present in newly solidified rock, but the argon produced by decaying potassium-40 is trapped within the crystal structure. Since the decay rate is known very precisely, we can use the measured ratio of potassium to argon in the rock to determine how long it has been since it formed, for timescales from several million to billions of years.

In the mid-1960s, oceanographers and geologists began drilling cores and taking basalt samples from the sea floor and measuring their ages.[8] And what they found matched the prediction from seafloor spreading: the youngest rocks were at the ridges and became progressively older towards the edges of the oceans.

Age of oceanic crust

Diagram of the age of oceanic crust. The youngest rocks are red, and found along the mid-ocean ridges. Rocks are progressively older further away from the ridges. (Figure reproduced from [9].)

This was exactly what Vine and Matthews predicted. Belatedly, Morley also received his due credit for coming up with the same idea, and their proposal is now known as the Vine-Matthews-Morley hypothesis. The magnetic striping of the ocean floors is caused by the combination of the spreading of the ocean floors from the mid-ocean ridges, and the periodic reversal of Earth’s magnetic field.

Generation of magnetic striping

Generation of magnetic striping on the sea floor. As the sea floor spreads, and the Earth’s magnetic field reverses from time to time, stripes of different magnetic polarity are created and spread outwards. (Public domain image by the United States Geological Survey, from Wikimedia Commons.)

Odd reversals of magnetic fields in continental rocks had been noticed since 1906, when the French geologist Bernard Brunhes found that some volcanic rocks were magnetised in the opposite direction to the Earth’s magnetic field. In the 1920s, the Japanese geophysicist Motonori Matuyama noticed that all of the reversed rocks found by Brunhes and others since were older than the early Pleistocene epoch, around 750,000 years ago. He suggested that the Earth’s field may have changed direction around that time, but his proposal was largely ignored.

With the impetus provided by the seafloor spreading idea, geologists began measuring magnetic fields and ages of more rocks, and found that they matched up with the ages of the field reversals implied by the sea floor measurements. Progress was rapid and the geological community turned around and developed and adopted the whole theory of plate tectonics within just a few years. By the end of the 1960s, what had been ridiculed less than a decade earlier was mainstream, brought to that status by the confluence of multiple lines of observational evidence.

It had been established that the Earth’s magnetic field must reverse direction with periods of a few tens of thousands to millions of years. The remaining question was how?

Up until the development of plate tectonics, the origin of the Earth’s magnetic field had been a mystery. Albert Einstein even weighed in, suggesting that it might be caused by an imbalance in electrical charge between electrons and protons. But plate tectonics not only raised the question – it also suggested the answer.

The core of the Earth was known to be mostly metallic (see Proof 43. The Schiehallion experiment). If there are convection currents in the mantle, then heat differentials at the boundary should also cause convection within the core. The convecting metal induces electrical currents, which in turn produce a magnetic field. In short, the core of the Earth is an electrical dynamo. And because the outer core is liquid, the currents are unstable. Modern computer simulations of convection in the Earth’s core readily produce instabilities that act to flip the polarity of the magnetic field at irregular intervals – exactly as observed in the record of magnetic striped sea floor rocks.

Simulations of magnetic field reversal in Earth's core

Computer simulations of convection currents in the Earth’s core and resulting magnetic field lines. Blue indicates north magnetic polarity, yellow south. The left image indicates Earth in a stable state, with a magnetic north pole at the top and a south at the bottom. The middle image is during an instability, with north and south intermingled and chaotic. The right image is after the unstable period, with the north and south poles now flipped. (Public domain images by NASA, from Wikimedia Commons.)

So we have a fully coherent and self-consistent theory that explains the observations of magnetic striping, along with many other features of the Earth’s geophysics. It involves several interlocking components: convection in the metallic core producing electric currents that generate a magnetic field that is unstable over millions of years and flips polarity at irregular intervals; convection in the mantle producing upwellings of material along mid-ocean ridges, leading to seafloor spreading and continental drift; rocks that record both their age and the direction and strength of the Earth’s magnetic field when they are formed, leading to magnetic striping on the ocean beds.

Of course, this only holds together and makes sense on a spherical Earth. We’ve already seen in Proof 8. Earth’s magnetic field, that simply generating the shape of the planet’s magnetic field only works on a spherical Earth, and is an inexplicable mystery on a flat Earth model. It would be even more difficult to explain the irregular reversal of polarity of the magnetic field without a spherical core dynamo system. And plate tectonics just doesn’t work on a flat Earth either (Proof 22. Plate tectonics). Combining the fact that neither of these explanations work on a flat Earth, there is no explanation for the observed magnetic striping of the sea floors either. So magnetic striping provides another proof that the Earth is a globe.

References:

[1] Vacquier, V., Raff, A.D., Warren, R.E. “Horizontal displacements in the floor of the northeastern Pacific Ocean”. Geological Society of America Bulletin, 72(8), p.1251-1258, 1961. https://doi.org/10.1130/0016-7606(1961)72[1251:HDITFO]2.0.CO;2

[2] Mason, R.G. Raff, A.D. “Magnetic survey off the west coast of North America, 32 N. latitude to 42 N. Latitude”. Geological Society of America Bulletin, 72(8), p.1259-1265, 1961. https://doi.org/10.1130/0016-7606(1961)72[1259:MSOTWC]2.0.CO;2

[3] Raff, A.D. Mason, R.G. “Magnetic survey off the west coast of North America, 40 N. latitude to 52 N. Latitude”. Geological Society of America Bulletin, 72(8), p.1267-1270, 1961. https://doi.org/10.1130/0016-7606(1961)72[1267:MSOTWC]2.0.CO;2

[4] Heezen, B.C. “The rift in the ocean floor”. Scientific American, 203(4), p.98-114, 1960. https://www.jstor.org/stable/24940661

[5] Dietz, R.S. “Continent and ocean basin evolution by spreading of the sea floor”. Nature, 190(4779), p.854-857, 1961. https://doi.org/10.1038%2F190854a0

[6] Hess, H.H. “History of Ocean Basins: Geological Society of America Bulletin”. Petrologic Studies: A Volume to Honour AF Buddington, p.559-620, 1962. https://doi.org/10.1130/Petrologic.1962.599

[7] Vine, F.J. Matthews, D.H. “Magnetic anomalies over oceanic ridges”. Nature, 199(4897), p.947-949, 1963. https://doi.org/10.1038/199947a0

[8] Orowan, E., Ewing, M., Le Pichon, X. Langseth, M.G. “Age of the ocean floor”. Science, 154(3747), p.413-416, 1966. https://doi.org/10.1126/science.154.3747.413

[9] Müller, R.D., Sdrolias, M., Gaina, C. Roest, W.R. “Age, spreading rates, and spreading asymmetry of the world’s ocean crust”. Geochemistry, Geophysics, Geosystems, 9(4), 2008. https://doi.org/10.1029/2007GC001743

43. The Schiehallion experiment

The Ancients had the technology and cleverness to work out the shape of the Earth and its diameter (see 2. Eratosthenes’ measurement). However, they had no reliable method to measure the mass of the Earth, or equivalently its density, which gives the mass once you know the volume. You could assume that the Earth has a density similar to rock throughout, but there was no way of knowing if that was correct.

In fact we had no measurement of the density or mass of the Earth until the 18th century. Perhaps surprisingly, there wasn’t even any observational evidence to decide whether the Earth was actually a solid object, or a hollow shell with a relatively thin solid crust. As late as 1692 the prominent scientist Edmond Halley proposed that the Earth might be composed of a spherical shell around 800 km thick, with two smaller shells inside it and a solid core, all separated by a “luminous” atmosphere (which could escape and cause the aurora borealis).

Edmond Halley's hollow Earth model

Structure of the Earth as proposed by Edmond Halley in 1692, with solid shells (brown) separated by a luminous atmosphere, shown in cross section.

In his 1687 publication of Philosophiæ Naturalis Principia Mathematica, Isaac Newton presented his theory of universal gravitation. Although this provided explicit equations relating the physical properties of gravitational force, mass, and size, for the cases of astronomical objects there were still more than one unknown value, so the equations could not be solved to determine the absolute masses or densities of planets. The best astronomers could do was determine ratios of densities of one planet to another.

But Newton not only proposed his formulation of gravity as a theoretical construct – he also suggested a possible experiment that could be done to test it. As observed by common experience, objects near the surface of the Earth fall downwards – they are attracted towards the centre of the Earth (more precisely, the Earth’s centre of mass). But if the attractive force of gravity is generated by mass as per Newton’s formulation, then unusual concentrations of mass should change the direction of the gravitational pull a little.

We’ve already seen in Proof 24. “Gravitational acceleration variation” that the strength of Earth’s gravitational pull varies across the Earth’s surface due to differences in altitude and density within the Earth. Now imagine a large concentration of mass on the surface of the Earth. If Newton is correct, then such a mass should pull things towards it. The attraction to the centre of the Earth is much stronger, so the direction of the overall gravitational pull should still be almost downwards, but there should be a slight deflection towards the large mass.

There are some convenient large masses on the surface of the Earth. We call them mountains. Newton conceived that one could go somewhere near a large mountain and measure the difference in angle between a plumb line (which indicates the direction of gravity, and is commonly called “vertical”) and a line pointing towards the Earth’s centre of mass (which does not have a well-defined name, since it is more difficult to measure and differs from a plumb line by an amount too small to be significant in engineering or construction – for the purposes of this proof only, I shall abbreviate it to “downwards”). However, Newton believed that any such difference would be too small to measure in practice. He writes in the Principia, Book 3: On the system of the world:

Hence a sphere of one foot in diameter, and of a like nature to the earth, would attract a small body placed near its surface with a force 20,000,000 times less than the earth would do if placed near its surface; but so small a force could produce no sensible effect. If two such spheres were distant but by 1/4 of an inch, they would not, even in spaces void of resistance, come together by the force of their mutual attraction in less than a month’s time; and lesser spheres will come together at a rate yet slower, namely in the proportion of their diameters. Nay, whole mountains will not be sufficient to produce any sensible effect. A mountain of an hemispherical figure, three miles high, and six broad, will not, by its attraction, draw the pendulum two minutes out of the true perpendicular; and it is only in the great bodies of the planets that these forces are to be perceived.[1]

Here is where Newton’s lack of experience as an experimentalist let him down. Two minutes of arc was already within the accuracies of stellar positions claimed by Tycho Brahe some 80 years earlier. If Newton had merely asked astronomers if they could measure a deflection of such a small size, they would likely have answered yes.

Tycho Brahe in his observatory at Uraniborg

Engraving of Tycho Brahe observing in his observatory at Uraniborg, Sweden. (Public domain image from Wikimedia Commons.)

If you can measure how big the deflection angle is with sufficient accuracy, then you can use that measurement to calculate the density of the Earth in terms of the density of the mountain:

ρE/ρM = (VM/VE) (rE/d)2 / tan θ

where:

ρE is the density of the Earth,
ρM is the density of the mountain,
VE is the volume of the Earth,
VM is the volume of the mountain,
rE is the radius of the Earth,
d is the horizontal distance from the centre of the mountain to the plumb bob, and
θ is the angle of deflection of the plumb line from “downwards”.

The volume of the mountain can be estimated from its size and shape, and the density may be assumed to be that of common types of rock. All the other values were known, leaving the as yet unknown density of the Earth as a function of the deflection angle.

Two French astronomers, Pierre Bouguer (who we met in 4. Airy’s coal pit experiment) and Charles Marie de La Condamine, were the first to attempt to make the measurement. In 1735 they led an expedition to South America to measure the length of an arc of one degree of latitude along a line of longitude near the equator. This was part of an experiment by the French Academy of Sciences—along with simultaneous expedition to Lapland to make a similar measurement near the North Pole—to measure the shape of the Earth. Not whether it was spherical; any difference between the measurements would show if it was more accurately a prolate or an oblate ellipsoid.

Bouguer and La Condamine spent ten years on their expedition, making many other physical, geographical, biological, and ethnographical studies. One experiment they tried in 1738 was measuring the deflection of a plumb bob near the 6263 metre high volcano Chimborazo, in modern day Ecuador.

Chimborazo in Ecuador

The volcano Chimborazo in Ecuador. (Creative Commons Attribution 2.0 image by David Ceballos, from Flickr.)

They climbed to an altitude of 4680 m on one flank of the mountain and 4340 m on the other side, battling harsh weather to take the two measurements. Taking two measurements on opposite sides of the mountain allows a subtraction to remove sources of error in locating the “downwards” direction, leaving behind the difference in angle between the two plumb bob directions, which is twice the desired deflection. Bouguer and La Condamine measured a deflection of 8 seconds of arc, however they considered the circumstances so difficult as to render it unreliable. But they did state that this measurement gave a large value for the Earth’s density, thus disproving the hypothesis that the Earth was hollow.

A more precise measurement of the gravitational deflection of a mountain had to wait until 1772, when Astronomer Royal Nevil Maskelyne made a proposal to repeat the experiment to the Royal Society of London.[2] The Society approved, and appointed the quaintly named Committee of Attraction to ponder the proposal. The committee (counting Joseph Banks and Benjamin Franklin among its members) despatched astronomer and surveyor Charles Mason (of Mason-Dixon line fame) to find a suitable mountain. He came back with Schiehallion, a 1083 m peak in central Scotland.

Schiehallion in Scotland

Schiehallion in central Scotland. (Creative Commons Attribution 3.0 Unported image by Wikipedia user Andrew2606, from Wikimedia Commons.)

Schiehallion had several advantages for the measurement. It’s conveniently located for a British expedition. It’s an isolated peak, with no other mountains nearby that could substantially complicate the effects of gravity in the region. It has a very symmetrical shape, making it easy to estimate the volume with some accuracy. And the northern and southern slopes are very steep, which means that by doing the experiment on those sides, the plumb bob can be positioned relatively close to the centre of mass of the mountain, increasing the deflection and making it easier to measure.

Maskelyne himself led the expedition, taking temporary leave from his post as Astronomer Royal. The party built temporary observatories on the northern and southern flanks of Schiehallion, from which they made frequent observations of overhead stars to determine the zenith line (marking the “downwards” direction), so they could compare it to the direction of the hanging plumb line. Maskelyne and his team spent 6 weeks at the southern observatory, followed by 10.5 weeks at the northern one, battling inclement weather to take the required number of observations.[3]

Map of Schiehallion and surrounds

Map of Schiehallion and surrounds. The mountain forms a short ridge running approximately east-west. The positions of the north and south observatories can be seen. (Reproduced from [6].)

Maskelyne had calculated that if the Earth as a whole had the same density as the mountain (i.e. that of quartzite rock), then they should have observed a deflection of the plumb line relative to “downwards” of 20.9 seconds of arc. Preliminary calculations showed a deflection of about half that, meaning the Earth was roughly twice as dense as the mountain.

To mark the successful conclusion of the observations, the expedition celebrated with a rollicking good party. Plenty of alcohol was imbibed (quite possibly Scotch whisky). In the revelry, unfortunately someone accidentally set fire to the northern observatory and it burnt to the ground. The fire claimed the violin of one Duncan Robertson, a junior member of the expedition who had helped to pass the long cold nights of observation by entertaining the other members with his playing. Later, a grateful Maskelyne sent Robertson a replacement violin – not just any violin, but one made by the master craftsman Antonio Stradivari.[4][5]

The mathematician and surveyor Charles Hutton was charged with doing the detailed calculations of the result. He published them in a mammoth 100-page paper in 1778.[6] His final conclusion was that the density of the Earth was 1.8 times the density of the quartzite in Schiehallion, or about 4.5 g/cm3. Since this was so much higher than the densities of various types of rock (typically between 2 and 3 g/cm3), Hutton concluded (correctly) that much of the core of the Earth must be metal, and he calculated that about 65% of the Earth’s diameter must be a metallic core (a little higher than current measurements of 55%).

Hutton's conclusions on the structure of Earth and density of planets

Extract of Hutton’s paper, where he states that roughly 2/3 of the diameter of the Earth must be metallic to account for the measured density. This page also shows Hutton’s calculations of the densities of solar system bodies. (Reproduced from [6].)

This was the very first time that we had any estimate of the density/mass of the Earth, and Hutton also used it to calculate the densities of the Sun, the Moon, and the planets (out to Saturn) based on their known astronomical properties, mostly to within about 20% of the modern values. So the Schiehallion experiment was groundbreaking and significantly increased our fundamental understanding of the Earth and the solar system.

Later experiments confirmed the general nature of the result and refined the figures for the density and structure of the Earth. In particular, Henry Cavendish—a chemist who 20 years earlier had discovered the elemental nature of hydrogen and made several other discoveries about air and elemental gases—turned his attention to physics and performed what has become known as the Cavendish experiment in 1797-98. He constructed a finely balanced mechanism with which he could measure the tiny gravitational attractive force between two balls of lead, which allowed the measurement of the (then unknown) value of Newton’s gravitational constant. Knowing this value, it becomes possible to directly plug in values for the size of the Earth and the acceleration due to gravity and determine the mass of the Earth. Cavendish’s result was accurate to about 1%, confirming Hutton’s conclusion that the Earth must have a core denser than rock. And then in the 20th century, seismology allowed us to confirm the existence of discrete layers within the Earth, with the central core made primarily of metal (a story for a future Proof).

Drawing of Henry Cavendish

Drawing of Henry Cavendish. (Creative Commons Attribution 4.0 International image by the Wellcome Collection of the British Library, from Wikimedia Commons.)

Of course, the conclusions of the Schiehallion experiment—consistent with later experiments using independent methods—depend on the fact that the Earth is very close to spherical, and the fact that gravity works as Newton said (disregarding the later refinement by Einstein, which is not significant here). One of the more popular Flat Earth models assumes that gravity does not even exist as a force, and that objects “fall” to Earth because the Flat Earth is actually accelerating upwards. In such a model, objects always fall directly “downwards” and there is no deflection caused by large masses such as mountains. The Schiehallion experiment directly and simply disproves this Flat Earth model.

If we suppose that a Flat Earth somehow manages to exist with Newtonian gravity (in itself virtually impossible, see 13. Hydrostatic equilibrium), we could posit something like the 859 km thick flat disc mentioned in 34. Earth’s internal heat. Firstly, Newtonian gravity on such a disc would not always pull perpendicular to the ground – inhabitants near the circumference would be pulled at a substantial angle towards the centre of the disc. Ignoring this, if you managed to do the Schiehallion experiment (say at the North Pole), the distance rE in the equation would be effectively 430 km (the distance to the centre of mass of the disc) rather than the radius of the spherical Earth, 6378 km. This should make the observed deflection angle approximately (6378/430)2 = 220 times smaller! Then the observed deflections would imply that the density of the Earth is 220 times higher, or around 990 g/cm3, about 6 times as dense as the core of the Sun. Which is then inconsistent with the assumed density being the same as the spherical Earth (among other problems).

On the other hand, if we allow the density to be a free parameter, we can solve the gravitational and geometric equations simultaneously to derive the thickness of the Flat Earth disc in a “consistent” manner. This produces a thickness of 3020 km, and a density of 92 g/cm3. Which is over 4 times as dense as osmium, the densest substance at non-stellar pressures. So we’ve shown that the Schiehallion experiment proves that this “Newtonian Flat Earth” model cannot possibly be composed of any known material.

Basically, the observations of the Schiehallion experiment cannot be made consistent with a flat Earth, thus providing evidence that the Earth is a globe.

References:

[1] Newton, I. Philosophiae Naturalis Principia Mathematica (1687). Trans. Andrew Motte, 1729.

[2] Maskelyne, N. “A proposal for measuring the attraction of some hill in this Kingdom”. Philosophical Transactions of the Royal Society, 65, p. 495-499, 1772. https://doi.org/10.1098/rstl.1775.0049

[3]. Sillitto, R.M. “Maskelyne on Schiehallion: A Lecture to The Royal Philosophical Society of Glasgow”. 1990. http://www.sillittopages.co.uk/schie/schie90.html

[4] Davies, R. D. “A Commemoration of Maskelyne at Schiehallion”. Quarterly Journal of the Royal Astronomical Society, 26, p. 289, 1985. https://ui.adsabs.harvard.edu/abs/1985QJRAS..26..289D

[5] Danson, E. Weighing the World. Oxford University Press, 2005. ISBN 978-0-19-518169-2.

[6] Hutton, C. “An Account of the Calculations made from the Survey and Measures taken at Schehallien, in order to ascertain the mean Density of the Earth”. Philosophical Transactions of the Royal Society. 68, p. 689-788, 1778. https://doi.org/10.1098/rstl.1778.0034

Posting slowdown

I haven’t posted a new proof in a while, and that will probably continue for a few more weeks. I currently have a bandaged up left hand due to a kitchen knife accident, which is making it slower for me to work on the computer. And I also have my wife working from home due to COVID-19 travel restrictions, which breaks up my days with many more distractions than normal, making it very hard to concentrate in the long blocks of time necessary to research and write up a new proof.

Rest assured that the site is not abandoned! I’ll be back with more new proofs, hopefully within a few weeks.

42. Schumann resonances

A waveguide is a structure that restricts the motion of waves, disallowing propagation in certain directions, and thus concentrating the energy of the wave to propagate in specific other directions. An example of a waveguide is an optical fibre, which is basically a long, thin string of flexible glass or transparent polymer. Light entering one end is channelled along the fibre, unable to escape from the sides, and emerges at almost the same brightness from the far end.

Normally light and other electromagnetic waves, as well as other waves such as sound, spread out in three dimensions. As the energy spreads out to cover more space, conservation of energy causes the wave amplitude to fall off according to the inverse square law: wave amplitude falls as the reciprocal of the square of the distance from the source.

With a waveguide, propagation of the wave can be restricted to a single dimension so the energy doesn’t spread out, resulting in all of the energy being transmitted to the far end (minus a small fraction that may be absorbed or otherwise lost along the way). Sound waves, for example, can be guided by simple hollow tubes, the sound preferring to propagate along the interior air channel than penetrate the tube walls. This is the principle behind medical stethoscopes and old fashioned speaking tube systems.

Another type of waveguide is a transmission line, which is a pair of electrical cables used to transmit alternating current (AC) electrical power. The cables can simply be parallel wires in close proximity, or a coaxial cable, in which an insulated wire runs down the core of tubular conductor. Domestic AC power has a frequency of 50 to 60 hertz, which is low compared to the kilohertz range of radio frequencies. Transmission lines can carry electromagnetic waves up to frequencies of around 30 kHz. Above this, paired wires start to radiate radio waves, so they become inefficient and a different type of waveguide is used.

Radio waveguides are commonly hollow metal tubes. Radio waves travel along the tube, and the conductive metal prevents the waves from leaking to the outside. Such waveguides are used to transmit radio power in radar systems and microwaves in microwave ovens. Anywhere there is a cavity bounded by regions that waves cannot pass through, a waveguide effect can be generated.

A microwave waveguide

A microwave waveguide, which is essentially a hollow metal tube, but precisely machined to optimal dimensions and with high precision connector joints. (Creative Commons Attribution 2.0 image by Oak Ridge National Laboratory, from Flickr.)

Radio waves travel easily through the Earth’s atmosphere, to and from transmission towers and the various wireless devices we use. However the bulk of the Earth is opaque to radio waves; you generally need a mostly unobstructed line of sight, barring relatively thin obstructions like walls.

But there is another region of the Earth that is opaque to (at least some) radio waves. The ionosphere is the region of the atmosphere in which incoming solar radiation ionises the atmospheric gases (mentioned previously in 31. Earth’s atmosphere). It lies between approximately 60 to 1000 km altitude. Since ionised gas conducts electricity, low frequency radio waves cannot pass through it (higher frequencies oscillate too rapidly for the ionised particles to respond).

Opacity of atmosphere vs wavelength

Opacity of the Earth’s atmosphere as a function of electromagnetic wavelength. Long wavelength (low frequency) radio waves are blocked by the ionosphere (right). Other parts of the electromagnetic spectrum are blocked by other aspects of the atmosphere. (Modified from a public domain image by NASA, from Wikimedia Commons.)

Radio waves with wavelengths longer than about 30 metres—or frequencies below about 10 MHz—are thus trapped in the atmosphere between the Earth’s surface and the ionosphere. This forms a waveguide which can carry so-called shortwave radio signals around the world, alternately bouncing off the ionosphere and the Earth’s surface.

There are also natural sources of low frequency radio waves. Lightning flashes in storm systems produce huge discharges of electrical energy, and the sudden release of this energy generates radio waves. If you’ve ever listened to a radio during a thunderstorm you’ll be familiar with the bursts of static caused by strokes of lightning. Lightning generates broadband radio emissions, meaning it covers a wide range of radio frequencies, including the very low frequencies that are guided by the ionospheric waveguide.

Atmospheric scientists measure the amount of lightning around the world by monitoring tiny changes in the Earth’s magnetic field, of the order of picoteslas, caused as these radio waves pass by. The sensitive detectors they use can detect lighting strikes anywhere on the planet. There are a few specific radio frequencies at which the lightning strikes turn out to be especially strong. The following plot shows the intensity of magnetic field fluctuations as a function of radio frequency.

Measurements of magnetic field fluctuation amplitude vs radio frequency

Measurements of magnetic field fluctuation amplitude versus radio wave frequency, averaged over a year of observation, at Maitri Research Station, Antarctica. (Figure reproduced from [1].)

The first peak in the observed radio spectrum is at 7.8 Hz, followed by peaks at 14.3 Hz, 20.8 Hz, and roughly every 6.5 Hz thereafter. People familiar with wave theory will recognise from the pattern that these are likely resonance frequencies, with a fundamental mode at 7.8 Hz, followed by overtones. A wave resonance occurs when an exact number of wavelengths fits into a confined cavity. The wave propagates and bounces around and, because of the precise match with the cavity size, reflected waves end up with peaks and troughs in the same physical position, reinforcing one another. So at the specific resonance frequency, the wave builds up in intensity, while at other frequencies the waves self-interfere and rapidly die down. These resonance frequencies, which are measured at many research stations around the world, are known as Schumann resonances.

The Irish physicist George Francis FitzGerald first anticipated the existence of Schumann resonances in 1893, but his work was not widely circulated. Around 1950, the German physicist Winfried Otto Schumann performed the theoretical calculations that predicted the resonances may be observable, and made efforts to observe them. But it was not until 1960 that Balser and Wagner made the first successful observations and measurements of Schumann resonances.[2]

What causes the radio waves produced by lightning flashes to have a resonance at 7.8 Hz? Well, radio waves travel at the speed of light, so let’s divide the speed of light by 7.8 to see what the wavelength is: the answer is 38,460 km. If you’ve been paying attention to many of these articles, you’ll realise that this is very close to the circumference of the Earth.

Radio waves with a frequency of 7.8 Hz are travelling around the world in the waveguide formed by the Earth and the ionosphere, and returning one wavelength later to constructively interfere and reinforce themselves, producing a measurable peak in Earth’s magnetic field fluctuations at 7.8 Hz. The resonance peak is broad and a little different to 7.5 Hz (the speed of light divided by the circumference of the Earth) because the geometry of a spherical cavity is more complicated than a simple circular loop – effectively some propagation paths are shorter because the waves don’t all take a great circle route.

Schumann resonances diagram

Illustration of Schumann resonances in the Earth’s atmosphere. The ionosphere keeps low frequency radio waves confined to a channel between it and the Earth. Waves propagate around the Earth. At specific frequencies the peaks and troughs line up, producing a resonance that reinforces those frequencies. The blue wave fits six wavelengths around the Earth, the red wave fits three. The fundamental frequency Schuman resonance of 7.8 Hz fits one wave. Not to scale: the ionosphere is much closer to the surface in reality. (Public domain image by NASA/Simoes.)

So Schumann resonances are an observed phenomenon that has a natural explanation – if the Earth is a globe.

If the Earth were flat, then any ionosphere above it would be flat as well, and would still form a waveguide for low frequency radio waves. However it would not be a closed waveguide. Radio waves would propagate out the edges and be lost to space, meaning there would be no observable magnetic field resonances at all. And even if there were an opaque radio wall of some sort at the edge of the flat Earth, the size and geometry of the resulting cavity would be different, resulting in a different set of resonance frequencies, more akin to the frequencies of a vibrating disc, which are not evenly spaced like the observed Schumann resonances.

And so Schumann resonances provide another proof that the Earth is a globe.

References:

[1] Shanmugam, M. “Investigation of Near Earth Space Environment”. Ph.D. Thesis, Manonmaniam Sundaranar University, 2016. https://www.researchgate.net/publication/309209580_Investigation_of_Near_Earth_Space_Environment

[2] Balser, M., Wagner, C. “Observations of Earth–Ionosphere Cavity Resonances”. Nature, 188, p. 638-641, 1960. https://doi.org/10.1038/188638a0

41. Cosmic rays

The French physicist Henri Becquerel discovered the phenomenon of radioactivity in 1896, while performing experiments on phosphorescence – the unrelated phenomenon that causes “glow in the dark” materials to glow for several minutes after being exposed to light. He was interested to see if phosphorescence was related to x-rays, discovered only a few months earlier by Wilhelm Roentgen. In his experiments, Becquerel noticed that uranium salts could darken photographic film, even if wrapped in black paper so that no light could fall on the film, and even from non-phosphorescent uranium samples. The conclusion was that some sort of penetrating rays were being emitted by the uranium itself, without being excited by external energy.

Henri Becquerel in his lab width=

Henri Becquerel in his lab. (Public domain image from Wikimedia Commons.)

Marie and Pierre Curie quickly discovered other radioactive elements, and Becquerel himself discovered by experimenting with magnets that there were three different types of radioactive radiation: two deflected in different directions by a magnetic field and one not deflected at all. In 1899, Ernest Rutherford characterised the first two types, naming them alpha and beta particles, with positive and negative electric charges. Becquerel measured the mass/charge ratio of beta particles in 1900 and determined that they were the same as the electrons discovered by J. J. Thomson in 1897. In 1907 Rutherford showed that alpha particles were the nuclei of helium atoms. And in 1914, he showed that the third type of radiation, named gamma rays, were a form of electromagnetic radiation.

Ernest Rutherford with Hans Geiger

Ernest Rutherford (right), in his lab with Hans Geiger (left), inventor of the Geiger counter. (Public domain image from Wikimedia Commons.)

This was an exciting time in physics, and our understanding of atomic structure was revolutionised within the space of two decades. Besides discovering the basic structure of the atom and how it related to the phenomenon of radioactive decay, several peripheral phenomena also came to the attention of scientists.

One observation was that atoms in the atmosphere were sometimes ionised, or “electrified” as the scientists of the time described it. Ionisation is the process of electrons being stripped off neutral atoms, to form negatively charged free electrons and positively charged atomic ions (consisting of the atomic nucleus and a less-than-full complement of electrons). It was clear that radioactive rays could ionise atoms in the air, and so scientists assumed that it was radiation from radioactive elements in the ground that was ionising the air near ground level.

Father Theodor Wulf

Except strangely the amount of ionising radiation in the atmosphere seemed to increase with increasing altitude. German physicist and Jesuit priest Theodor Wulf invented in 1909 a portable electroscope capable of measuring the ionisation of the atmosphere. He used it to investigate the source of the ionising radiation by measuring ionisation at the base and the top of the Eiffel Tower. He found that the ionisation at the top of the 300 metre tower was a bit over half that at ground level, which was higher than he expected, since theoretically he expected the ionisation to drop by half every 80 metres, so to be less than one tenth the ionisation at ground level. He concluded that there must be some other source of ionising radiation coming from above the atmosphere. However, his published paper was largely ignored.

In 1911, the Italian physicist Domenico Pacini measured the ionisation rates in various places, including mountains, lakes, seas, and underwater. He showed that the rate dropped significantly underwater, and concluded that the main source of radiation could not be the Earth itself. Then in 1912, Austrian physicist Victor Hess took some Wulf electroscopes up in a hot air balloon to altitudes as high as 5300 metres, flying both in daylight, night time, and during an almost complete solar eclipse.

Victor Hess in a hot air balloon flight

Victor Hess (centre), after one of his balloon flight experiments. (Public domain image from Wikimedia Commons.)

Hess showed that the amount of ionising radiation decreased as one moved from ground level up to about 1000 metres, but then increased again rapidly. At 5300 metres, there was approximately twice as much ionising radiation as at ground level.[1] And because the effect occurred at night, and during a solar eclipse, it wasn’t due to the sun. Hess had proven that there was a source of this radiation outside the Earth’s atmosphere. Further unmanned balloon flights as high as 9 km showed the radiation increased even higher with altitude.

Atmospheric radiation readings recorded by Victor Hess

Readings of ionising radiation level (columns 2 to 4) at different altitudes (column 1, in metres), as recorded by Victor Hess. (Figure reproduced from [1].)

What this mysterious radiation was remained unknown until the late 1920s. It was initially thought to be electromagnetic radiation (i.e. gamma rays and x-rays). Robert Millikan named them cosmic rays in 1925 after proving that they originated outside the Earth. Then in 1927 the Dutch physicist Jacob Clay performed measurements while sailing from Java to the Netherlands, which showed that their intensity increased as one moved from the tropics to mid-latitudes.[2] He correctly deduced that the intensity was affected by the Earth’s magnetic field, which implied the cosmic rays must be charged particles.

Atmospheric radiation readings recorded by Jacob Clay

Data recorded by Jacob Clay showing change in ionising radiation with latitude during his voyage from Java to Europe. (Figure reproduced from [2].)

In 1930, the Italian Bruno Rossi realised that if cosmic rays are electrically charged, then they should be deflected either east or west by the Earth’s magnetic field, depending on whether they are positively or negatively charged, respectively.[3] Experiments found that at all locations on the Earth’s surface there are more cosmic rays coming from the west than from the east, showing that most (if not all) cosmic ray particles are positively charged. This observation was called the east-west effect.

Illustration showing incoming cosmic rays deflected to the east

Illustration of the east-west effect. In the space around the Earth (shown as black in this diagram), the Earth’s magnetic field is directed perpendicularly out of the diagram. Incoming cosmic rays are shown in red. When they encounter Earth’s magnetic field, charged particles are deflected perpendicular to the field direction. Positively charged particles are deflected to the right, as shown, meaning that from the surface of the Earth, cosmic rays tend to preferentially come from the west.

Subsequent experiments determined that around 90% of cosmic rays entering our atmosphere are protons, 9% are helium nuclei (or alpha particles), and the remaining 1% are nuclei of heavier elements, with an extremely small number of other types of particles. And in 1936, for his crucial part in the discovery in cosmic rays, Victor Hess was awarded the Nobel Prize for Physics.

The origin of cosmic rays is however still not entirely clear. Our sun produces energetic particles that reach Earth, but cosmic rays are generally defined as coming from outside our own solar system. Our Milky Way Galaxy produces some of the lower energy particles, mostly from the direction of the galactic core, however at very high energies there is a deficit of cosmic rays in that direction, implying a shadowing effect on rays whose origin lies outside our galaxy. Known sources of cosmic rays include supernova explosions, supernova remnants (such as the Crab Nebula), active galactic nuclei, and quasars. But there are some very high energy cosmic rays whose source is still a mystery.

The fact that high energy cosmic rays originate from outside our galaxy, means that they should be isotropic – uniform in intensity distribution, independent of the direction from which they approach Earth. However, the Earth is not a stationary observation platform. The Earth orbits the sun at a speed of almost 30 km/s. But on a galactic scale this motion is dwarfed by the sun’s orbital speed around the core of our galaxy, which is 230 km/s, roughly in the direction of the star Vega, in the constellation Lyra. So relative to extragalactic cosmic rays, Earth is moving at an average speed of approximately 230 km/s. This speed adds to the energy of cosmic rays coming from the direction of Vega, and subtracts from the energies of cosmic rays coming from the opposite direction.

Diagram showing our sun's orbit about the galaxy

Our sun orbits the centre of the Milky Way Galaxy, at a speed of 230 km/s. This speed modifies the speed and direction of incoming cosmic rays from outside the Galaxy. (Modified from a public domain image by NASA, from Wikimedia Commons.)

This difference is known as the Compton-Getting effect after the discoverers Arthur Compton and Ivan Getting.[4] It produces about a 0.1% difference in the energies of cosmic rays coming from the opposite directions, which can be observed statistically. The effect was confirmed experimentally in 1986.[5]

So we have two different observational effects that have been experimentally confirmed in the distribution of cosmic rays arriving at Earth. The Compton-Getting effect shows that the Earth is moving in the direction of the star Vega. Vega is of course above the Earth’s horizon as seen from half the planet’s surface at any one time, and below the horizon (behind the planet) from the other half of the Earth’s surface. By measuring cosmic ray distributions, you can show that the direction defined by the Compton-Getting anisotropy relative to the ground plane varies depending on your position on Earth. In other words, by measuring cosmic rays, you can prove that the Earth’s direction of motion through the galaxy is upwards from the ground in one place, while simultaneously downwards into the ground from a point on the opposite side of the planet, and at intermediate angles in places in between. Which is perfectly consistent for a spherical planet, but inconsistent with a Flat Earth.

The second effect, the east-west effect, is also readily explained with a spherical Earth, with the addition of a simple dipole magnetic field. As can be seen in the diagram above (“Illustration of the east-west effect”), incoming positively charged cosmic rays are uniformly deflected to the right (as viewed from above Earth’s North Pole), resulting in more rays arriving from the west than from the east, independent of location or time of day. The same observed east-west effect could in theory be produced on a Flat Earth, but only if the magnetic field is flattened out as well, holding the same relative orientation to the Earth’s surface as it does on the globe.

Magnetic field as required for the east-west effect, on spherical and flat Earths

Shape of magnetic fields to produce the observed east-west effect in incoming cosmic rays. The required magnetic field for a spherical Earth is very close to a simple dipole, easily generated with known physical principles. The required magnetic field shape for a flat Earth is severely flattened, and cannot be produced with a simple magnetic dynamo model.

This would result in the field being grossly distorted from that of a simple dipole, and thus requiring some exotic method of generating such a complex field – a complex field that just happens to mimic exactly the field of a straightforward dipole if the Earth were spherical. In another application of Occam’s razor (similar to its use in article 8. Earth’s magnetic field), it is more parsimonious to conclude that the Earth is not flat, but spherical.

References:

[1] Hess, V.F. “Über Beobachtungen der durchdringenden Strahlung bei sieben Freiballonfahrten” (Observation of penetrating radiation in seven free balloon flights). Physikalische Zeitschrift, 13, p.1084-1091, 1912. http://inspirehep.net/record/1623161/

[2] Clay, J. “Penetrating radiation”. Proceedings of the Royal Academy of Sciences Amsterdam, 30, p. 1115-1127, 1927. https://www.dwc.knaw.nl/DL/publications/PU00011919.pdf

[3] Rossi, B. “On the magnetic deflection of cosmic rays”. Physical Review, 36(3), p. 606, 1930. https://doi.org/10.1103/PhysRev.36.606

[4] Compton, A. H., Getting, I. A. “An Apparent Effect of Galactic Rotation on the Intensity of Cosmic Rays”. Physical Review, 47, p. 817-821, 1935. https://doi.org/10.1103/PhysRev.47.817

[5] Cutler, D., Groom, D. “Observation of terrestrial orbital motion using the cosmic-ray Compton–Getting effect”. Nature, 322, p. 434-436, 1986. https://doi.org/10.1038/322434a0

40. Falling objects

What could be more simple than dropping an object and watching it fall to the ground? Our everyday experience shows that if you drop something, it falls straight down.

The Ancient Greek philosopher Aristotle used this observation to show that the Earth cannot possibly be moving or rotating. If the Earth were rotating from west to east, as some rival philosophers argued, then when you drop something, the Earth will move eastwards underneath it as it falls, and the object should land some distance west of where you dropped it! We don’t see this happening, ergo, the Earth cannot possibly be moving. Q.E.D.

Aristotle of course got many things wrong in his proposed system of the mechanics of motion and how the cosmos worked – a system known now as Aristotelian physics. Among his other contentions were that heavier objects fall faster than lighter ones, and that an object cannot undergo “unnatural motion” unless acted on by a force (falling is a “natural motion” and therefore requires no force).

One consequence of strict Aristotelian physics is that as soon as an object is released, it can have no sideways motion, and must fall straight down. This is obviously false if you observe objects such as arrows or cannonballs in flight, and natural philosophers of the Middle Ages developed the concept of impetus to explain this. The basic idea is that when a force propels an object, it implants in that object an impetus, which acts as an inherent force within the object itself, pushing it onwards. The concept took several centuries to mature, and was formalised by the French philosopher Jean Buridan in the 14th century (perhaps more famous for Buridan’s ass, which was not his backside, but a philosophical paradox).

Archer firing an arrow

An arrow fired by an archer. The arrow does not drop straight to the ground as soon as it leaves the bow, contrary to Aristotelian physics. (Public domain image from Wikimedia Commons.)

Impetus is a forerunner of how Isaac Newton eventually solved the problem, with his idea of momentum and his Laws of Motion. Newton’s First Law says that an object at rest, or in a state of uniform motion, maintains that state unless acted on by an external force. So an object like an arrow that is fired from a bow at some speed maintains that speed unless acted on by external forces. In practice there are two forces acting on a flying arrow: air resistance, which slows down its horizontal motion, and gravity (another of Newton’s cool ideas), which causes it to begin falling towards the ground. The combination of these two results in the arrow slowing down and dropping in altitude, until eventually it hits something (either a target or the ground).

Newton’s First Law also explains why a dropped object falls straight to the ground, instead of falling to the west as the Earth rotates beneath it. An object held in your hand has the same rotational velocity as the Earth at the point where you’re standing. For example, if you’re on the equator, the rotational speed of the Earth’s surface is about 460 metres per second, so you and anything you’re holding, are moving eastwards at that speed. When you let the object go, it continues moving at 460 m/s eastward as it falls – the same speed as the Earth is moving. And so it falls at your feet, what appears to be directly downwards to you, with no sideways deflection.

Or does it?

Dropping a tennis ball

Dropping a tennis ball. Where does it land? (Creative Commons Attribution 2.0 image by Marco Verch, from Flickr.)

If the Earth were flat and non-rotating, then none of this would be an issue. Objects dropped fall downwards and the non-motion of the Earth doesn’t change anything. You don’t even need Newton’s First Law.

If the Earth were flat and rotating like a record (or optical disc) about a central North Pole, things get a little more complicated. Let’s say the equator is 10,000 km from the North Pole (the same as on our Earth), and the Flat Earth rotates once per 24 hours. Then points on the equator are moving with a speed of 2π×10000/(24×60×60) km/s = 727 m/s (faster than our spherical Earth because the geometry is different). If you drop an object (and believe Newton’s laws), the object is moving east at 727 m/s and maintains that motion in a straight line as it falls. Let’s say you drop it from a height of 2 metres. It takes 0.64 seconds to hit the ground. In 0.64 seconds, the object moves east a distance of 465 metres. The ground is also moving east at 727 m/s, however, the ground is not moving in a straight line – it’s moving in a circle about the North Pole. In 0.64 seconds the ground moves through an angle of 0.0027°. Doing the trigonometry, this means the ground moves 465×cos(0.0027°) m east (which is slightly less than, but so close to 465 m that it’s not worth writing the difference) and 465×sin(0.0027°) m north, which equals 0.022 metres. So, if we live on a rotating Flat Earth, and you drop an object from 2 metres height at the equator, you should see it land 22 millimetres south of straight down.

Dropping objects on a rotating flat Earth

On a rotating Flat Earth, if you drop an object on the equator (green dot, left), both the Earth and the object at that point are moving west to east. As the object falls, the Earth rotates (right). The location on the Earth where you dropped the object moves in a circle (green dot), but the falling object moves in a straight line and lands at the red dot, south of the starting location.

This is a prediction of the rotating Flat Earth model. Repeating the above calculations for different latitudes (assuming distances from the North Pole equal to our round Earth), we would expect a southward deflection of 11 mm at 45° north, or 33 mm at 45° south. Do we observe such a southward deflection of falling objects? No, we don’t.

Now let’s think about our spherical Earth, because things are not quite as straightforward as they might appear. The equator is rotating at a speed of 460 m/s. It’s not moving in a straight line – it’s rotating about the Earth’s axis, once every sidereal day: T = 23 hours, 56 minutes, 4 seconds (see 36. The visible stars). The radius of rotation is the equatorial radius of the Earth: 6378.1 km. If you hold an object 2 metres above the ground, its radius of rotation is 2 metres larger, making the distance it has to travel in a sidereal day 4π metres larger. So the speed of rotation 2 metres above the ground is 4π/T = 0.000145 m/s faster than the ground. In 0.46 seconds, this means that an object 2 metres above the ground moves eastward by 0.067 millimetres greater distance than the ground does.

So if you stand on the equator and drop an object from 2 metres, it should land 0.067 mm east of straight down. If you increase the height and drop an object from 100 metres, it takes 4.52 seconds to fall and by this calculation should land 33 mm east of straight down. The German language Wikipedia has an article on this calculation (there is no English Wikipedia article on it), and derives the same result, but then it says that “a more precise calculation” produces an additional factor 2/3, citing Carl Friedrich Gauss’s collected works without any further explanation. So this gives a deflection of 22 mm for an object dropped from 100 m. There is also an adjustment for latitude, being the usual cosine(latitude) term that we have seen in many of these discussions.

This is a prediction of the rotating spherical Earth model. Do we observe such an eastward deflection of falling objects?

There is in fact a long history of scientists investigating this effect and trying to measure it. In 1674, the French Jesuit priest and mathematician Claude François Milliet Dechales published his Cursus seu Mondus Matematicus, which included a diagram showing the fall of an object from a tower on the rotating Earth. It’s not clear if he ever performed the experiment.

Diagram of dropping object from a tower

Diagram from Dechales’s Cursus seu Mondus Matematicus showing an object F falling from a tower FG. As the Earth rotates the tower FG moves to HI, but the object does not land at I, it lands further east at L. (Public domain image from Wikimedia Commons.)

Isaac Newton himself wrote about the effect in a letter to Robert Hooke, dated 28 November, 1679, just five years later[2].

Newton's letter to Robert Hooke

Newton’s letter of 28 November 1679 to Robert Hooke. Larger version. (Pages reproduced from [2].)

In the letter, Newton drew a diagram of an object falling, not just from a height to the ground, but continuing to fall towards the centre of the Earth (as if the object could pass through the Earth):

Diagram from Newton's letter

Enlargement of the diagram from Newton’s letter.

In the text accompanying the diagram Newton writes:

Then imagine this body be let fall and its gravity will give it a new motion towards the centre of the Earth without diminishing the previous one from west to east. Whence the motion of this body from west to east, by respect that before its fall it was more distant from the centre of the Earth than the parts of the Earth at which it arrives in its fall, will be greater than the motion from west to east of parts of the Earth at which the body arrives in its fall, and therefore it will not descend the perpendicular AC, but outrunning the parts of the Earth will shoot forward to the east side of the perpendicular, describing in its fall a spiral line ADEC.

Newton goes on to suggest that a “descent of but 20 to 30 yards” may be enough to observe the eastward deflection. Being a theoretician, Newton doesn’t seem to have done the experiment, but Hooke tried to measure the eastward deflection of an object falling from a height of 8.2 metres. From this height the expected deflection is about a quarter of a millimetre at the latitude of London—very difficult to measure—and Hooke’s results were inconclusive.

The first positive result was achieved in 1791 by Italian scientist Giovanni Battista Guglielmini. He dropped a total of 16 balls from the top of the Asinelli Tower of Bologna, a height of 78 m, comparing the landing positions to a vertical defined by a plumb-bob line. He concluded that the average eastward deflection of the balls was about 18 mm, compared to a predicted deflection of 11 mm.[3] Of course, these early experiments faced many difficulties, such as air currents, the difficulty of releasing the balls without any sideways motion, and measuring a vertical plumb line accurately.

In 1802, Johann Benzenberg dropped 32 balls from the tower of St Michael’s Church in Hamburg, 76 m high. Being at a higher latitude than Bologna, the expected eastward deflection was 8.7 mm, and Benzenberg recorded an average value of 9 mm. In 1831, Ferdinand Reich dropped lead balls 158  metres down the Drei Brüders (Three Brothers) mine shaft near Freiberg, measuring 28 mm eastward deflection, with a predicted value of 29.4 mm. In 1902 Edwin Hall performed the experiment with 948 separate drops from a height of 23 m at Harvard University, measuring an eastward deflection of 1.5 mm, compared to the predicted 1.8 mm. And Camille Flammarion dropped 144 balls from 68 m in the Pantheon in Paris, measuring a deflection of 6.3 mm, compared to the theoretical 8.1 mm.[3]

This is not an easy experiment to perform with sufficient accuracy. It is sensitive to a lot of complicating factors, particularly air currents, but the overall agreement of observation with the predictions is good. And so the measurable eastward deflection of falling objects provides us with another proof that the Earth is a (rotating) globe.

Note: I’ve talked only about the eastward deflection of falling objects. There is also a smaller predicted deflection in the north-south direction for latitudes away from the equator. That will be discussed in a future Proof.

References:

[1] Dechales, C. F. M. Cursus seu mundus mathematicus (Vol. 1), 1674.

[2] Gunther, R. T. Early Science in Oxford, Volume X, Oxford University Press, Oxford, 1920-1937. https://archive.org/details/earlyscienceinox10gunt/page/52/mode/2up

[3] Tiersten, M., Soodak, H. “Dropped objects and other motions relative to the noninertial earth”. American Journal of Physics, 68, p. 129-142, 2000. https://doi.org/10.1119/1.19385

39. Seismic wave propagation

Our planet is made largely of rocks and metals. The composition and physical state varies with depth from the core of the Earth to the surface, because of changes in pressure and temperature with depth. The uppermost layer is the crust, which consists of lighter rocks in a solid state. Immediately below this is the upper mantle, in which the rocks are hotter and can deform plasticly over millions of years.

Slow convection currents occur in the upper mantle, and the convection cells define the tectonic plates of the Earth’s crust. Where mantle material rises, magma can emerge at mid-oceanic ridges or volcanoes. Where it sinks, a subduction zone occurs in the crust.

The plate boundaries are thus particularly unstable places on the Earth. As the plates shift and move relative to one another, stresses build up in the rock along the edges. At some point the stress becomes too great for the rock to withstand, and it gives way suddenly, releasing energy that shakes the Earth locally. These are earthquakes.

Lisbon earthquake engraving

Engraving of the effects of the 1755 Lisbon earthquake. (Public domain image from Wikimedia Commons.)

The point of slippage and the release of energy is known as the hypocentre of the earthquake, and may be several kilometres deep underground. The point on the surface above the hypocentre is the epicentre, and is where potential destruction is the greatest. Most earthquakes are small and go relatively unnoticed except by the seismologists who study earthquakes. Sometimes a quake is large and can cause damage to structures, injuries, and loss of life.

The energy released in an earthquake travels through the Earth in the form of waves, known as seismic waves. There are a few different types of seismic wave.

Primary waves, or P waves, are compressional waves, like sound waves in air. The rock alternately compresses and experiences tension, in a direction along the axis of propagation. In fact P waves are essentially sound waves of very large amplitude, and they propagate at the speed of sound in the medium. Within surface rock, this is about 5000 metres per second. Primary waves are so called because they are the fastest seismic waves, and thus the first ones to reach seismic recording stations located at any distance from the epicentre. They travel through the body of the Earth. And like sound waves, they can travel through any medium: solid, liquid, or gas.

Secondary waves, or S waves, are transverse waves, like light waves, or waves travelling along a jiggled rope. The rock jiggles from side to side as the wave propagates perpendicular to the jiggling motions. S waves travel a little over half the speed of P waves, and are the second waves to be detected at remote seismic stations. S waves also travel through the body of the Earth, but only within solid material. Fluids have no shear strength, and so cannot return to an equilibrium position when a transverse wave hits it, so the energy is dissipated within the fluid.

Seismic wave types

Illustrations of rock movement in different types of seismic waves. (Figure reproduced from [1].)

Besides these two types of body waves, there are also surface waves, which travel along the surface of the Earth. One type, Rayleigh waves (or R waves, named after the physicist Lord Rayleigh), are just like the surface waves or ripples on water, and causes the surface of the Earth to heave up and down. Another type of surface wave causes side to side motion; these are known as Love waves (or L waves, named after the mathematician Augustus Edward Hough Love). These waves propagate more slowly than S waves, at around 90% of the speed. Love waves are generally the strongest and most destructive seismic waves.

The P and S waves are thus the first two waves detected from an earthquake, and they are easily distinguishable on seismometer recordings.

Seismogram of P and S waves

Seismogram recording of arrival of P waves and S waves at a seismology station in Mongolia, from an earthquake 307 km away. (Figure reproduced from [2].)

The P waves arrive first and produce a pulse of activity which slowly fades in amplitude, then the S waves arrive and cause a larger amplitude burst of activity. Because the relative speeds of the two waves through the same material are known, the time between the arrival of the P and S waves can be used to determine the distance from the seismic station to the earthquake hypocentre, using a graph such as the following:

Seismic wave travel-time curves

Seismic wave travel-time curves for P, S, and L waves. Also shown are three seismograms detected at seismic stations at different distances from an earthquake. (Public domain image from the United States Geological Survey.)

The graph shows the travel times of P, S, and also L waves, plotted against distance from the earthquake on the vertical axis. As you can see, the time between the detection of the P and S waves increases steadily with the distance from the quake.

If you have three seismic stations, you can triangulate the location of the epicentre (using trilateration, as we have previously discussed).

Triangulating the location of an earthquake

Triangulating the location of an earthquake using distances from three seismic stations. (Public domain image from United States Geological Survey.)

Of course, if you have more than three seismic stations, you can pinpoint the location of the earthquake much more reliably and precisely. According to the International Registry of Seismograph Stations, there are over 26,000 seismic stations around the world.

Location of seismic stations

Location of seismic stations recorded in the International Registry of Seismograph Stations. (Figure reproduced from [3].)

Interestingly, notice how the world’s seismic stations are concentrated along plate boundaries, where earthquakes are most common, particularly around the Pacific rim, as well as heavily in the developed nations of the US and Europe.

As shown in the travel-time curve graph, you can also use the propagation time of L waves to estimate distance to the earthquake. Did you notice the difference between the shapes of the P and S wave curves, and the L wave curve? L waves travel along the surface of the Earth. The distance from an earthquake to a detection station is measured conventionally, like everyday distances, also along the surface of the Earth. Since the L waves propagate at a constant speed, the graph of distance (along the Earth’s surface) versus time is a straight line.

But the P and S waves don’t travel along the surface of the Earth. They propagate through the bulk of the Earth. The distance that a P or S wave needs to travel from earthquake to detection site increases more slowly than the distance along the surface of the Earth, because of the Earth’s spherical shape. The S waves are only about 10% faster than the L waves, and you can see that near the epicentre, they arrive only around 10% earlier than the L waves. But the further away the earthquake is, the more of a shortcut they can take through the Earth, and so the faster they arrive, resulting in the downward curve on the graph. Similarly for the P waves.

This is in fact not the only cause of the P and S waves appearing to get faster the further away you are from an earthquake. They actually do get faster as they travel deeper, because of changes to the rock pressure. Deep in the Earth they can travel at roughly twice the speed that they do near the surface. The combination of these effects causes the shape of the curves in the travel-time graph.

If we consider the propagation of seismic waves from an earthquake, they spread out in circles around the epicentre, like ripples in a pond from where a stone is dropped in. The arrival times of the waves at seismic stations equidistant from the epicentre should be the same, since the speeds in any direction are the same. And this is of course what is observed. The following figures show the predicted spread of P waves across the Earth from earthquake epicentres in Washington State USA, near Panama, and near Ecuador, as plotted by the US Geological Survey.

P wave propagation times from Washington

Predicted P wave propagation time in minutes from an earthquake epicentre in Washington State, USA. (Public domain image from United States Geological Survey.)

P wave propagation times from Panama

Predicted P wave propagation time in minutes from an earthquake epicentre near Panama. (Public domain image from United States Geological Survey.)

P wave propagation times from Ecuador

Predicted P wave propagation time in minutes from an earthquake epicentre near Ecuador. (Public domain image from United States Geological Survey.)

These maps are shown on an equirectangular map projection, which of course distorts the shape of the surface of the Earth (as discussed in 14: Map projections). To get a better idea of how the seismic waves propagate, we need to project these maps onto a sphere.

P wave propagation times from Washington, globe

Predicted P wave propagation time in minutes from an earthquake epicentre in Washington State, USA, projected onto a globe.

P wave propagation times from Panama, globe

Predicted P wave propagation time in minutes from an earthquake epicentre near Panama, projected onto a globe.

P wave propagation times from Ecuador, globe

Predicted P wave propagation time in minutes from an earthquake epicentre near Ecuador, projected onto a globe.

In these projections, you can see that the seismic wave travel time isochrones are circles, spreading out around the globe from the epicentres.

At least, the waves spread out in circles on a spherical Earth. In a flat Earth model, such as the typical “north pole in the middle” one, the spread of seismic waves produces elongated elliptical shapes or kidney shapes (such as the ones drawn in 23: Straight line travel), for no apparent or explicable reason.

P wave propagation times from Washington, flat Earth

Predicted P wave propagation time in minutes from an earthquake epicentre in Washington State, USA, projected onto a flat Earth.

P wave propagation times from Panama, flat Earth

Predicted P wave propagation time in minutes from an earthquake epicentre near Panama, projected onto a flat Earth.

P wave propagation times from Ecuador, flat Earth

Predicted P wave propagation time in minutes from an earthquake epicentre near Ecuador, projected onto a flat Earth.

Why should seismic waves propagate more slowly towards or away from the North Pole, and faster along tangential arcs? Why would they take longer to reach an area in the middle of the opposing half of the disc than to reach the far edge of the disc, which is further away? There is no a priori reason, and any proposed justification is yet another ad hoc bandage on the model.

So the propagation speeds of the various seismic waves and the travel times to recording stations provide another proof that the Earth is a globe.

Note: There is more to be said about the propagation of seismic waves, which will provide another, different proof that the Earth is a globe. Some readers no doubt have a good idea what it is already. Rest assured that I haven’t overlooked it, and it will be covered in detail in a future article.

References:

[1] Athanasopoulos, G., Pelekis, P., Anagnostopoulos, G. A. “Effect of soil stiffness in the attenuation of Rayleigh-wave motions from field measurements”, Soil Dynamics and Earthquake Engineering, 19, p. 277-288, 2000. https://doi.org/10.1016/S0267-7261(00)00009-9

[2] Quang, P. B., Gaillard, P., Cano, Y. “Association of array processing and statistical modelling for seismic event monitoring”, Proceedings of the 23rd European Signal Processing Conference (EUSIPCO 2015), p. 1945-1949, 2015. https://doi.org/10.1109/EUSIPCO.2015.7362723

[3] International Seismological Centre (2020), International Seismograph Station Registry (IR). https://doi.org/10.31905/EL3FQQ40