## 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).

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.

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.

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 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. 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%).

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).

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

## 35. The Eötvös effect

In the opening years of the twentieth century, scientists in the field of geodesy (measuring the shape and gravitational field of the Earth) were interested in making measurements of the strength of gravity all over the Earth’s surface. To do this, they trekked to remote regions of the world with sensitive gravimeters, to take the readings. On land this was straightforward enough, but they also wanted measurements taken at sea.

Around 1900, teams from the Institute of Geodesy in Potsdam took voyages into the Atlantic, Indian, and Pacific Oceans on ships, and made measurements using their gravimeters. The collected data were brought back to Potsdam for analysis. There, the readings fell under the scrutinising eyes of the Hungarian physicist Loránd Eötvös, who specialised in studying the variation of Earth’s gravitational field with position on the surface. He noticed an odd thing about the readings.

Because of the impracticality of stopping the ship every time they wanted to take a reading, the scientists measured the Earth’s gravity while the ships were moving. There was no reason to suppose this would make any difference. But Eötvös found a systematic effect. Gravity measurements taken while the ship was moving eastward were lower than readings taken while the ship was moving westward.

Eötvös realised that this effect was being caused by the rotation of the Earth. The Earth’s equatorial circumference is 40,075 km, and it rotates eastward once every sidereal day (23 hours, 56 minutes). So the ground at the equator is rotating at a linear speed of 465 metres per second. To move in a circular path rather than a straight line (as dictated by Newton’s First Law of Motion), gravity supplies a centripetal force to any object on the Earth’s surface. The necessary force is equal to the object’s mass times the velocity squared, divided by the radius of the circular path (6378 km). This comes to m×4652/6378000 = 0.0339m. So per kilogram of mass, a force of 0.0339 newtons is needed to enforce the circular path, an amount easily supplied by the Earth’s gravity. (This is why objects don’t get flung off the Earth by its rotation, a complaint of some spherical Earth sceptics.)

What this means is that the effective acceleration due to gravity measured for an object sitting on the equator is reduced by 0.0339 m/s2 (the same units as 0.0339 N/kg) compared to if the Earth were not rotating. But if you’re on a ship travelling east at, say, 10 m/s, the centripetal force required to keep you on the Earth’s surface is greater, equal to 4752/6378000 = 0.0354 N/kg. This reduces the apparent measured gravity by a larger amount, making the measured value of gravity smaller. And if you’re on a ship travelling west at 10 m/s, the centripetal force is 4552/6378000 = 0.0324 N/kg, reducing the apparent gravity by a smaller amount and making the measured value of gravity greater. The difference in apparent gravity between the ships travelling east and west is 0.003 m/s2, which is about 0.03% of the acceleration due to gravity. For a person of mass 70 kg, this is a difference in apparent weight of about 20 grams (strictly speaking, a difference in weight of 0.2 newtons, which is 20 grams multiplied by acceleration due to gravity).

Eötvös set out these theoretical calculations, and then organised an expedition to measure and test his results. In 1908, the experiment was carried out on board a ship in the Black Sea, with two separate ships travelling east and west past one another so the measurements could be made at the same time. The results matched Eötvös’s predictions, thus confirming the effect.

In general (if you’re not at the equator), your linear speed caused by the rotation of the Earth is equal to 465 m/s times the cosine of your latitude, while the radius of your circular motion is also equal to 6378 km times the cosine of your latitude. The centripetal force formula uses the square of the velocity divided by the radius, so this results in a cosine(latitude) term in the final result. That is, the size of the Eötvös effect also varies as the cosine of the latitude. If you measure it at 60° latitude, either north or south, the difference in gravity between east and west travelling ships is half that measured at the equator.

The Eötvös effect is well known in the field of gravimetry, and is routinely corrected for when taking measurements of the Earth’s gravitational strength from moving ships[1], aircraft[2], or submarines[3]. The reference on submarines refers to a gravitational measurement module for use on military submarines to enhance their navigation capability as undersea instruments of warfare. This module includes an Eötvös effect correction for when the sub is moving east or west. You can bet your bottom dollar that no military force in the world would make such a correction to their navigation instruments if it weren’t necessary.

One paper I found reports measurements made of the detailed structure of gravitational anomalies over the Mariana Trough in the Pacific Ocean south of Japan. It states:

Shipboard free-air gravity anomalies were calculated by subtracting the normal gravity field data from observed gravity field data, with a correction applied for the Eötvös effect using Differential Global Positioning System (DGPS) data.[4]

The results look pretty cool:

Map of gravitational anomalies in the Mariana Trough region of the Pacific Ocean, as obtained by shipboard measurement, corrected for the Eötvös effect. (Figure reproduced from [4].)

Another paper shows the Eötvös effect more directly:

Graph showing measurements of Earth’s gravitational field strength versus distance travelled by a ship in the South Indian Ocean. In the leftmost section (16), the ship is moving slowly westward. In the central section (17) the ship is moving at a faster speed westward, showing the increase in measured gravity. In the right section (18) the ship is moving eastward at slow speed, and the gravity readings are lower than the readings taken in similar positions while moving westward. (Figure reproduced from [5].)

If the Earth were flat, on the other hand, there would be no Eötvös effect at all. If the flat Earth is not rotating (as most models posit, with the sun moving above it in a circular path), obviously there is no centripetal acceleration happening at all. Even if you adopt a model where the flat Earth rotates about the North Pole, the centripetal acceleration at every point on the surface is parallel to the surface, towards the pole, not directed downwards. So an Eötvös-like effect would actually cause a slight deflection in the angle of gravity, but almost zero change in the magnitude of the gravity.

The Eötvös effect shows that not only is the Earth rotating, but that it is rotating about a central point that is underneath the ground, not somewhere on the surface. If you stand on the equator and face east, the surface of the Earth is rotating in the direction you are facing and downwards, not to the left or right. Furthermore, the cosine term shows that at equal latitudes both north and south, the rotation is at the same angle relative to the surface, which can only be the case if the Earth is symmetrical about the equator: i.e. spherical.

References:

[1] Rousset, D., Bonneville, A., Lenat, J.F. “Detailed gravity study of the offshore structure of Piton de la Fournaise volcano, Réunion Island”. Bulletin of Volcanology, 49(6), p. 713-722, 1987. https://doi.org/10.1007/BF01079822

[2] Thompson, L.G., LaCoste, L.J. “Aerial gravity measurements”. Journal of Geophysical Research, 65(1), p. 305-322, 1960. https://doi.org/10.1029/JZ065i001p00305

[3] Moryl, J., Rice, H., Shinners, S. “The universal gravity module for enhanced submarine navigation”. In IEEE 1998 Position Location and Navigation Symposium, p. 324-331, April 1996. https://doi.org/10.1109/PLANS.1998.670124

[4] Kitada, K., Seama, N., Yamazaki, T., Nogi, Y., Suyehiro, K., “Distinct regional differences in crustal thickness along the axis of the Mariana Trough, inferred from gravity anomalies”. Geochemistry, Geophysics, Geosystems, 7(4), 2006. https://doi.org/10.1029/2005GC001119

[5] Persson, A. “The Coriolis Effect: Four centuries of conflict between common sense and mathematics, Part I: A history to 1885”. History of Meteorology, 2, p.1-24, 2005. https://www.semanticscholar.org/paper/The-Coriolis-Effect%3A-Four-centuries-of-conflict-and-Persson/c9e72567af65e44384fba048bbf491d3ac3a30ff

## 25. Planetary formation

Why does Earth exist at all?

The best scientific model we have for understanding how the Earth exists begins with the Big Bang, the event that created space and time as we know and understand it, around 14 billion years ago. Scientists are interested in the questions of what possibly happened before the Big Bang and what caused the Big Bang to happen, but haven’t yet converged on any single best model for those. However, the Big Bang itself is well established by multiple independent lines of evidence and fairly uncontroversial.

The very early universe was a hot, dense place. Less than a second after the Big Bang, it was essentially a soup of primordial matter and energy. The energy density was so high that the equivalence of mass and energy (discovered by Albert Einstein) allowed energy to convert into particle/antiparticle pairs and vice versa. The earliest particles we know of were quarks, electrons, positrons, and neutrinos. The high energy density also pushed space apart, causing it to expand rapidly. As space expanded, the energy density reduced. The particles and antiparticles annihilated, converting back to energy, and this process left behind a relatively small residue of particles.

Schematic diagram of the evolution of the universe following the Big Bang. (Public domain image by NASA.)

After about one millionth of a second, the quarks no longer had enough energy to stay separated, and bound together to form the protons and neutrons more familiar to us. The universe was now a plasma of charged particles, interacting strongly with the energy in the form of photons.

After a few minutes, the strong nuclear force could compete with the ambient energy level, and free neutrons bonded together with protons to form a few different types of atomic nuclei, in a process known as nucleosynthesis. A single proton and neutron could pair up to form a deuterium nucleus (an isotope of hydrogen, also known as hydrogen-2). More rarely, two protons and a neutron could combine to make a helium-3 nucleus. More rarely still, three protons and four neutrons occasionally joined to form a lithium-7 nucleus. Importantly, if two deuterium nuclei collided, they could stick together to form a helium-4 nucleus, the most common isotope of helium. The helium-4 nucleus (or alpha particle as it is also known in nuclear physics) is very stable, so the longer this process went on, the more helium nuclei were formed and the more depleted the supply of deuterium became. Ever since the Big Bang, natural processes have destroyed more of the deuterium, but created only insignificant additional amounts – which means that virtually all of the deuterium now in existence was created during the immediate aftermath of the Big Bang. This is important because measuring the abundance of deuterium in our universe now gives us valuable evidence on how long this phase of Big Bang nucleosynthesis lasted. Furthermore, measuring the relative abundances of helium-3 and lithium-7 also give us other constraints on the physics of the Big Bang. This is one method we have of knowing what the physical conditions during the very early universe must have been like.

Diagrams of the nuclei (and subsequent atoms) formed during Big Bang nucleosynthesis.

The numbers all point to this nucleosynthesis phase lasting approximately 380,000 years. All the neutrons had been bound into nuclei, but the vast majority of protons were left bare. At this time, something very important happened. The energy level had lowered enough for the electrostatic attraction of protons and electrons to form the first atoms. Prior to this, any atoms formed would quickly be ionised again by the surrounding energy. The bare protons attracted an electron each and become atoms of hydrogen. The deuterium nuclei also captured an electron to become atoms of deuterium. The helium-3 and helium-4 nuclei captured two electrons each, while the lithium nuclei attracted three. There were two other types of atoms almost certainly formed which I haven’t mentioned yet: hydrogen-3 (or tritium) and beryllium-7 – however both of these are radioactive and have short half-lives (12 years for tritium; 53 days for beryllium-7), so within a few hundred years there would be virtually none of either left. And that was it – the universe had its initial supply of atoms. There were no other elements yet.

When the electrically charged electrons became attached to the charged nuclei, the electric charges cancelled out, and the universe changed from a charged plasma to an electrically neutral gas. This made a huge difference, because photons interact strongly with electrically charged particles, but much less so with neutral ones. Suddenly, the universe went from opaque to largely transparent, and light could propagate through space. When we look deep into space with our telescopes, we look back in time because of the finite speed of light (light arriving at Earth from a billion light years away left its source a billion years ago). This is the earliest possible time we can see. The temperature of the universe at this time was close to 3000 kelvins, and the radiation had a profile equal to that of a red-hot object at that temperature. Over the billions of years since, as space expanded, the radiation became stretched to longer wavelengths, until today it resembles the radiation seen from an object at temperature around 2.7 K. This is the cosmic microwave background radiation that we can observe in every direction in space – it is literally the glow of the Big Bang, and one of the strongest observational pieces of evidence that the Big Bang happened as described above.

Map of the cosmic microwave background radiation over the full sky, as observed by NASA’s WMAP satellite. The temperature of the radiation is around 2.7 K, while the fluctuations shown are ±0.0002 K. The radiation is thus extremely smooth, but does contain measurable fluctuations, which lead to the formation of structure in the universe. (Public domain image by NASA.)

The early universe was not uniform. The density of matter was a little higher in places, a little lower in other places. Gravity could now get to work. Where the matter was denser, gravity was higher, and these areas began attracting matter from the less dense regions. Over time, this formed larger and larger structures, the size of stars and planetary systems, galaxies, and clusters of galaxies. This part of the process is one where a lot of the details still need to be worked out – we know more about the earlier stages of the universe. At any rate, at some point clumps of gas roughly the size of planetary systems coalesced and the gas at the centre accreted under gravity until it became so massive that the pressure at the core initiated nuclear fusion. The clumps of gas became the first stars.

The Hubble Extreme Deep Field. In this image, except for the three stars with visible 8-pointed starburst patterns, every dot of light is a galaxy. Some of the galaxies in this image are 13.2 billion years old, dating from just 500 million years after the Big Bang. (Public domain image by NASA.)

The first stars had no planets. There was nothing to make planets out of; the only elements in existence were hydrogen with a tiny bit of helium and lithium. But the nuclear fusion process that powered the stars created more elements: carbon, oxygen, nitrogen, silicon, sodium, all the way up to iron. After a few million years, the biggest stars had burnt through as much nuclear fuel in their cores as they could. Unable to sustain the nuclear reactions keeping them stable, they collapsed and exploded as supernovae, spraying the elements they produced back into the cosmos. The explosions also generated heavier elements: copper, gold, lead, uranium. All these things were created by the first stars.

Supernova 2012Z, in the spiral galaxy NGC 1309, position shown by the crosshairs, and detail before and during the explosion. (Creative Commons Attribution 4.0 International image by ESA/Hubble, from Wikimedia Commons.)

The interstellar gas cloud was now enriched with heavy elements, but still by far mostly hydrogen. The stellar collapse process continued, but now as a star formed, there were heavy elements whirling in orbit around it. The conservation of angular momentum meant that elements spiralled slowly into the proto-star at the centre of the cloud, forming an accretion disc. Now slightly denser regions of the disc itself began attracting more matter due to their stronger gravity. Matter began piling up, and the heavier elements like carbon, silicon, and iron formed the first solid objects. Over a few million years, as the proto-star in the centre slowly absorbed more gas, the lumps of matter in orbit—now large enough to be called dust, or rocks—collided together and grew, becoming metres across, then kilometres, then hundreds of kilometres. At this size, gravity ensured the growing balls of rock were roughly spherical, due to hydrostatic equilibrium (previously discussed in a separate article). They attracted not only solid elements, but also gases like oxygen and hydrogen, which wrapped the growing protoplanets in atmospheres.

Protoplanetary disc of the very young star HL Tauri, imaged by the Atacama Large Millimetre Array. The gaps in the disc are likely regions where protoplanets are accreting matter. (Creative Commons Attribution 4.0 International image by ALMA (ESO/NAOJ/NRAO), from Wikimedia Commons.)

Eventually the star at the centre of this protoplanetary system ignited. The sudden burst of radiation pressure from the star blew away much of the remaining gas from the local neighbourhood, leaving behind only that which had been gravitationally bound to what were now planets. The closest planets had most of the gas blown away, but beyond a certain distance it was cold enough for much of the gas to remain. This is why the four innermost planets of our own solar system are small rocky worlds with thin or no atmospheres with virtually no hydrogen, while the four outermost planets are larger and have vast, dense atmospheres mainly of hydrogen and hydrogen compounds.

But the violence was not over yet. There were still a lot of chunks of orbiting rock and dust besides the planets. These continued to collide and reorganise, some becoming moons of the planets, others becoming independent asteroids circling the young sun. Collisions created craters on bigger worlds, and shattered some smaller ones to pieces.

Saturn’s moon Mimas, imaged by NASA’s Cassini probe, showing a huge impact crater from a collision that would nearly have destroyed the moon. (Public domain image by NASA.)

Uranus’s moon Miranda, imaged by NASA’s Voyager 2 probe, showing disjointed terrain that may indicate a major collision event that shattered the moon, but was not energetic enough to scatter the pieces, allowing them to reform. (Public domain image by NASA.)

The left over pieces of the creation of the solar system still collide with Earth to this day, producing meteors that can be seen in the night sky, and sometimes during daylight. (See also the previous article on meteor arrival rates.)

The process of planetary formation, all the way from the Big Bang, is relatively well understood, and our current theories are successful in explaining the features of our solar system and those we have observed around other stars. There are details to this story where we are still working out exactly how or when things happened, but the overall sequence is well established and fits with our observations of what solar systems are like. (There are several known extrasolar planetary systems with large gas giant planets close to their suns. This is a product of observational bias—our detection methods are most sensitive to massive planets close to their stars—and such planets can drift closer to their stars over time after formation.)

One major consequence of this sequence of events is that planets form as spherical objects (or almost-spherical ellipsoids). There is no known mechanism for the formation of a flat planet, and even if one did somehow form it would be unstable and collapse into a sphere.

## 24. Gravitational acceleration variation

When you drop an object, it falls down. Initially the speed at which it falls is zero, and this speed increases over time as the object falls faster and faster. In other words, objects falling under the influence of gravity are accelerating. It turns out that the rate of acceleration is a constant when the effects of air resistance are negligible. Eventually air resistance provides a balancing force and the speed of fall reaches a limit, known as the terminal velocity.

Ignoring the air resistance part, the constant acceleration caused by gravity on the Earth’s surface is largely the same everywhere on Earth. This is why you feel like you weigh the same amount no matter where you travel (excluding travel into space!). However, there are small but measurable differences in the Earth’s gravity at different locations.

It’s straightforward to measure the strength of the acceleration due to gravity at any point on Earth with a gravity meter. We’ve already met one type of gravity meter during Airy’s coal pit experiment: a pendulum. So the measurements can be made with Georgian era technology. Nowadays, the most accurate measurements of Earth’s gravity are made from space using satellites. NASA’s GRACE satellite, launched in 2002, gave us our best look yet at the details of Earth’s gravitational field.

Being roughly a sphere of roughly uniform density, you’d expect the gravity at the Earth’s surface to be roughly the same everywhere and—roughly speaking—it is. But going one level of detail deeper, we know the Earth is closer to ellipsoidal than spherical, with a bulge around the equator and flattening at the poles. The surface gravity of an ellipsoid requires some nifty triple integrals to calculate, and fortunately someone on Stack Exchange has done the work for us[1].

Given the radii of the Earth, and an average density of 5520 kg/m3, the responder calculates that the acceleration due to gravity at the poles should be 9.8354 m/s2, while the acceleration at the equator should be 9.8289 m/s2. The difference is about 0.07%.

So at this point let’s look at what the Earth’s gravitational field does look like. The following figure shows the strength of gravity at the surface according to the Earth Gravitational Model 2008 (EGM2008), using data from the GRACE satellite.

Earth’s surface gravity as measured by NASA’s GRACE and published in the Earth Gravitational Model 2008. (Figure produced by Curtin University’s Western Australian Geodesy Group, using data from [2].)

We can see that the overall characteristic of the surface gravity is that it is minimal at the equator, around 9.78 m/s2, and maximal at the poles, around 9.83 m/s2, with a transition in between. Overlaid on this there are smaller details caused by the continental landmasses. We can see that mountainous areas such as the Andes and Himalayas have lower gravity – because they are further away from the centre of the planet. Now, the numerical value at the poles is a pretty good match for the theoretical value on an ellipsoid, close to 9.835 m/s2. But the equatorial figure isn’t nearly as good a match; the difference between the equator and poles is around 0.6%, not the 0.07% calculated for an ellipsoid of the Earth’s shape.

The extra 0.5% difference comes about because of another effect that I haven’t mentioned yet: the Earth is rotating. The rotational speed at the equator generates a centrifugal pseudo-force that slightly counteracts gravity. This is easy to calculate; it equals the radius times the square of the angular velocity of the surface at the equator, which comes to 0.034 m/s2. Subtracting this from our theoretical equatorial value gives 9.794 m/s2. This is not quite as low as 9.78 seen in the figure, but it’s much closer. I presume that the differences are caused by the assumed average density of Earth used in the original calculation being a tiny bit too high. If we reduce the average density to 5516 kg/m3 (which is still the same as 5520 to three significant figures, so is plausible), our gravities at the poles and equator become 9.828 and 9.788, which together make a better match to the large scale trends in the figure (ignoring the small fluctuations due to landmasses).

Of course the structure and shape of the Earth are not quite as simple as that of a uniformly dense perfect ellipsoid, so there are some residual differences. But still, this is a remarkably consistent outcome. One final point to note: it took me some time to track down the figure above showing the full value of the Earth’s gravitational field at the surface. When you search for this, most of the maps you find look like the following:

Earth surface gravity residuals, from NASA’s GRACE satellite data. The units are milligals; 1 milligal is equal to 0.00001 m/s2. (Public domain image by NASA, from [3].)

These seem to show that gravity is extremely lumpy across the Earth’s surface, but this is just showing the smaller residual differences after subtracting off a smooth gravity model that includes the relatively large polar/equatorial difference. Given the units of milligals, the largest differences between the red and blue areas shown in this map are only different by a little over 0.001 m/s2 after subtracting the smooth model.

We’re not done yet, because besides Earth we also have detailed gravity mapping for another planet: Mars!

Surface gravity strength on Mars. The overall trend is for lowest gravity at the equator, increasing with latitude to highest values at the poles, just like Earth. (Figure reproduced from [4].)

This map shows that the surface gravity on Mars has the same overall shape as that of Earth: highest at the poles and lowest at the equator, as we’d expect for a rotating ellipsoidal planet. Also notice that Mars’s gravity is only around 3.7 m/s2, less than half that of Earth.

Mars’s geography is in some sense much more dramatic than that of Earth, and we can see the smaller scale anomalies caused by the Hellas Basin (large red circle at lower right, which is the lowest point on Mars, hence the higher gravity), Olympus Mons (leftmost blue dot, northern hemisphere, Mars’s highest mountain), and the chain of three volcanoes on the Tharsis Plateau (straddling the equator at left). But overall, the polar/equatorial structure matches that of Earth.

Of course this all makes sense because the Earth is approximately an ellipsoid, differing from a sphere by a small amount of equatorial bulge caused by rotation, as is the case with Mars and other planets. We can easily see that Mars and the other planets are almost spherical globes, by looking at them with a telescope. If the structure of Earth’s gravity is similar to those, it makes sense that the Earth is a globe too. If the Earth were flat, on the other hand, this would be a remarkable coincidence, with no readily apparent explanation for why gravity should be stronger at the poles (remembering that the “south pole” in most flat Earth models is the rim of a disc) and weaker at the equator (half way to the rim of the disc), other than simply saying “that’s just the way Earth’s gravity is.”

References:

[1] “Distribution of Gravitational Force on a non-rotating oblate spheroid”. Stack Exchange: Physics, https://physics.stackexchange.com/questions/144914/distribution-of-gravitational-force-on-a-non-rotating-oblate-spheroid (Accessed 2019-09-06.)

[2] Pavlis, N. K., Holmes, S. A., Kenyon, S. C. , Factor, J. K. “The development and evaluation of the Earth Gravitational Model 2008 (EGM2008)”. Journal of Geophysical Research, 117, p. B04406. https://doi.org/10.1029/2011JB008916

[3] Space Images, Jet Propulsion Laboratory. https://www.jpl.nasa.gov/spaceimages/index.php?search=GRACE&category=Earth (Accessed 2019-09-06.)

[4] Hirt, C., Claessens, S. J., Kuhn, M., Featherstone, W.E. “Kilometer-resolution gravity field of Mars: MGM2011”. Planetary and Space Science, 67(1), p.147-154, 2012. https://doi.org/10.1016/j.pss.2012.02.006

## Pendulum experiment

With my Science Club class of 7-10 year olds, I did an experiment to test what factors influence the period of swing of a pendulum, and to measure the strength of Earth’s gravity. I borrowed some brass weights and a retort stand from my old university Physics Department and took them to the school. Then with the children we did the experiment!

We set up pendulums with different lengths of string, measuring the length of each one. With each pendulum length, we tested different numbers of brass weights, and pulling the weight back by a different distance so that the pendulum swung through shorter or longer arcs. For each combination of string length, mass, and swing length, I got the kids to time a total of 10 back and forth swings with a stopwatch. I recorded the times and divided by 10 to get an average swing time for each pendulum.

Here’s a graph showing the pendulum period (or “swing time” as I’m calling it with the kids), plotted against the mass of the weight at the end.

Pendulum period versus mass. The different colours indicate different pendulum lengths.

Here’s a graph showing the pendulum period (or “swing time” as I’m calling it with the kids), plotted against the swing distance (i.e. the amplitude).

Pendulum period versus swing distance. The different colours indicate different pendulum lengths.

These first two graphs show pretty clearly that the period of the pendulum is not dictated by either the mass of the pendulum or the amplitude of the swing. If you look at the different colours showing the pendulum length, however, you may discern a pattern.

And here’s a graph showing the pendulum period plotted against the length of the pendulum.

Pendulum period versus length. The line is a power law fit to all the points.

In this case, all the points from different pendulum masses and swing amplitudes but the same length are clustered together (with some scatter caused by experimental errors in using the stopwatch). This indicates that only the length is important in determining the period. This matches the first order approximation theoretical formula for the period of a pendulum, T:

T = 2π√(l/g),

where l is the length and g is the acceleration due to gravity. To calculate g from the experimental data, I squared the period numbers and calculated the slope of the best fit line passing through zero to (iT2). Then g = 4π2 divided by the slope… which gives g = 10.0 m/s2.

The true value is 9.81 m/s2, so we got it right to a little better than 2%. Which I consider pretty good given the fact that I had kids as young as 7 making the measurements!

Although this is an “other science” entry on this blog and not a proof of the Earth’s roundness, I’m planning to combine the results of this experiment with our ongoing measurement of the sun’s shadow length of a vertical stick at the end of the year, to calculate not only the size of the Earth, but also its mass. It’ll be interesting to see how close we can get to that!

## 13. Hydrostatic equilibrium

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

Newton’s law says that the force of gravitational attraction F between two bodies equals the universal gravitational constant G, multiplied by the masses of the two bodies m1 and m2, divided by the square of the distance r between them: F = G m1 m2/(r2).

Newton’s law of gravitation describes the force F between two bodies m1 and m2 separated by a distance r between their centres of mass.

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

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

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

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

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

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

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

The universal gravitational constant G is a rather small number in familiar units: 6.674×10−11 m3 kg−1 s−2. This means that the force of gravity between two everyday objects is so small as to be unnoticeable. For example, even large objects such as two 1-tonne cars a metre apart experience a gravitational force between them of only 6.674×10−5 newtons – far too small to move the cars against rolling friction, even with the brakes off. Also, the distance between the masses in Newton’s formula is the distance between the centres of mass of the objects, not the closest surfaces. The centres of mass of two cars can’t be brought closer together than about 2 metres in practice, even with the cars touching each other (unless you crush the cars).

Gravity really only starts to significantly affect things when you gather millions of tonnes of mass together. On Earth, the mass of the Earth (5.9722×1024 kilograms) itself dominates our experience with gravity. Removing mass 2 from Newton’s formula, we can calculate the acceleration a towards the centre of mass of the Earth, caused by the Earth’s gravity, as experienced at the surface of the planet (r = 6370 kilometres): a = G m1/(r2) = 6.674×10−11 × 5.9722×1024 / (6370×103)2 = 9.82 m/s2. This number matches experimental observations we can make of the gravity on the surface of the Earth (for example, using a pendulum: see also Airy’s coal pit experiment).

So large object like planets or other astronomical bodies experience a significant gravitational force on parts of themselves. Think about a tall mountain, such as Mount Everest. Let’s estimate the mass of Mount Everest – just roughly will do for our purposes. It is 8848 metres tall, above sea level. Let’s imagine it’s roughly a cone, with sides sloping at 45°. That makes the radius of the base 8848 metres, and its volume is π × 88483 / 3 = 7.25×1011 cubic metres. The density of granite is 2.75 tonnes per cubic metre, so the mass of Mount Everest is roughly 2×1015 kg. It experiences a gravitational force of approximately 2×1016 newtons, pulling it down towards the rest of the Earth.

Approximating Mount Everest as a cone of rock to calculate the pressure on the base.

Obviously Mount Everest is strong enough to withstand this enormous force without collapsing. But how much higher could a mountain be without collapsing under its own mass? The taller a mountain gets, the more force pulls it down, but the structural strength of the rock making up the mountain does not increase. At some point there is a limit. Our conical Mount Everest model spreads that mass over an area of π × 88482 square metres. This means the pressure of the rock above on this area is 2×1016 / (π × 88482) = 8×107 pascals, or 80 megapascals (Mpa). Now, the compressive strength of granite is about 200 MPa. We’re pretty close already! Not to mention that rock can also shear and deform plastically, so we probably don’t even need to get as high as 200 MPa before something bad (or spectacular, depending on your point of view!) happens. A mountain twice as high as Everest would almost certainly be unstable and collapse very quickly.

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

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

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

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

Iapetus, one of the moons of Saturn, photographed by NASA’s Cassini space probe. (Public domain image by NASA.)

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

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

References:

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

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

## 4. Airy’s coal pit experiment

Gravity is the force that causes objects to fall towards the ground. Observations of the movements of the planets led Isaac Newton in 1687 to publish his formulation of the force between two objects caused by gravity, stating that the force is proportional to the masses of the objects and the reciprocal of the square of the distance between them. This simple relationship has been wildly successful, although it was superseded in 1915 when Albert Einstein published his general theory of relativity. Einstein’s model differs from Newton’s only by imperceptible amounts, except when extremely large masses or speeds close to the speed of light are involved. For something the size of Earth, Newton’s law of gravity works just fine.

Given Newton’s law, some relatively straightforward vector calculus can be used to prove Gauss’s law for gravity, which gives a relationship between the gravitational flux of an enclosed surface and the amount of mass inside that surface. For symmetrical cases like spherical objects, the gravitational flux is just the gravitational field strength multiplied by the surface area of the sphere. The details are not as important here as the result: For a spherical object, the gravitational force of the object at any point—outside or inside the object—depends only on the distance from the centre of the object and the amount of mass within a sphere of that radius.

So consider the Earth – assuming it’s spherical. If you are on the surface or above it, the gravitational force you feel is produced by the entire mass of the Earth. However, if you are beneath the surface of the Earth, all of the mass of the Earth at shallower depths has no effect on you – the gravitational pull in all different directions cancels exactly to zero. You only feel the gravity from the part of the Earth that is deeper than you are. This means that as you burrow deeper into the Earth, the gravitational force you feel decreases, until eventually, if it were possible to reach the centre of the Earth, it would be zero. On the other hand, if the Earth is flat there’s no a priori reason to think that gravity should get progressively less strong as you go deeper underground.

Gauss’s Law for gravity. (Human not to scale.)

Gravitational force, it turns out, is fairly easy to measure. The period of a swinging pendulum depends on the force of gravity, and we’ve been able to measure small changes in the period of a pendulum fairly precisely for hundreds of years. Since before Newton’s time, in fact. The Elizabethan-era philosopher Francis Bacon first suggested taking pendulums up mountains to see if gravity varied with altitude in 1620. This experiment was actually carried out in 1737 by the French mathematician Pierre Bouguer, in the Peruvian Andes. (And perhaps more about that particular experiment another day.)

But in the 1820s the British astronomer George Biddell Airy realised that if you measured the force of gravity at the surface of the Earth, and also down a deep mine, you should get two different values. Not only that, but the size of the difference and the depth of the mine could be used to calculate the density of the Earth. He began experimenting in 1826, but unfortunately his first attempt failed due to a mine flood. Airy was a busy guy, accepting the post of Astronomer Royal in 1835 and discovering and inventing a whole bunch of other stuff. But finally in 1856 he tried the gravity experiment again.

George Biddell Airy. (Public domain image)

Airy used a coal pit at the Harton Colliery, near Harton in the county of Tyne and Wear in north-eastern England. The pit was 1260 feet (384 metres) deep, and at the bottom Airy built a sophisticated pendulum and time measurement system. He compared timing measurements made at the surface and the bottom of the pit over a period of 60 hours with the same length pendulum, and discovered that the pendulum at the bottom of the pit ran slower by 2.24 seconds per day.

Airy’s pendulum apparatus at the bottom of the Harton coal pit. Figure reproduced from [1].

For our purposes, this difference is the evidence we need that the Earth is spherical. We predicted that if the Earth is spherical then gravity should be lower at the bottom of a pit than on the surface, and Airy showed that is indeed true. But he didn’t stop there, because of course he already knew that the Earth was round, and its circumference. With that piece of data and his pendulum measurement, he could calculate the density of the Earth, finding a figure of 6.62 times the density of water.

As it turns out, modern measurements give a density of 5.51, about 17% less. Airy’s coal pit experiment was very fiddly, and it’s a credit that he got so close to the correct answer.

Schematic diagram of Airy’s pendulum apparatus. Figure reproduced from [2].

Now remember that previously we’ve shown that Eratosthenes measured the size of the Earth, simply using sticks and shadows. Airy’s experiment shows that once you know the size of Earth, you can get a decent measurement of the density of the planet using something as simple as a pendulum. And once you know the size and the density of something, its mass is simply the volume multiplied by the density.

In other words, if you’re clever enough you can measure the mass of the Earth with a stick, a length of string, and a weight.

References:

[1] Airy, G. B., “Lecture on the Pendulum-Experiments at Harton Pit”, lecture delivered at Central Hall, South Shields, 24 October 1854, Longman & Co., London. https://books.google.com/books?id=JRZcAAAAQAAJ

[2] Airy, G. B., “Account of Pendulum Experiments Undertaken in the Harton Colliery, for the Purpose of Determining the Mean Density of the Earth”, Philosophical Transactions of the Royal Society of London, 146, p. 297-355, 1856. https://doi.org/10.1098/rstl.1856.0015