Schiehallion experiment
Encyclopedia
Experiment
An experiment is a methodical procedure carried out with the goal of verifying, falsifying, or establishing the validity of a hypothesis. Experiments vary greatly in their goal and scale, but always rely on repeatable procedure and logical analysis of the results...
to determine the mean
Arithmetic mean
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...
density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
of the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
. Funded by a grant from the Royal Society
Royal Society
The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...
, it was conducted in the summer of 1774 around the Scottish
Scotland
Scotland is a country that is part of the United Kingdom. Occupying the northern third of the island of Great Britain, it shares a border with England to the south and is bounded by the North Sea to the east, the Atlantic Ocean to the north and west, and the North Channel and Irish Sea to the...
mountain
Mountains and hills of Scotland
Scotland is the most mountainous country in the United Kingdom. The area north and west of the Highland Boundary Fault is known as the Highlands, and contains the country's main mountain ranges. Scotland's mountain ranges, in a rough north to south direction are: The Highlands & Islands, The Hills...
of Schiehallion
Schiehallion
Schiehallion is a prominent mountain in Perth and Kinross, Scotland. Schiehallion has a rich botanical life, interesting archaeology, and a unique place in scientific history for an 18th-century experiment in 'weighing the world'...
, Perthshire
Perthshire
Perthshire, officially the County of Perth , is a registration county in central Scotland. It extends from Strathmore in the east, to the Pass of Drumochter in the north, Rannoch Moor and Ben Lui in the west, and Aberfoyle in the south...
. The experiment involved measuring the tiny deflection of a pendulum
Pendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position...
due to the gravitational attraction of a nearby mountain. Schiehallion was considered the ideal location after a search for candidate mountains, thanks to its isolation and almost symmetrical shape. One of the triggers for the experiment were anomalies noted during the survey of the Mason–Dixon Line.
The experiment had previously been considered, but rejected, by Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
as a practical demonstration of his theory of gravitation
Newton's law of universal gravitation
Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them...
. However, a team of scientists, notably Nevil Maskelyne
Nevil Maskelyne
The Reverend Dr Nevil Maskelyne FRS was the fifth English Astronomer Royal. He held the office from 1765 to 1811.-Biography:...
, the Astronomer Royal
Astronomer Royal
Astronomer Royal is a senior post in the Royal Household of the Sovereign of the United Kingdom. There are two officers, the senior being the Astronomer Royal dating from 22 June 1675; the second is the Astronomer Royal for Scotland dating from 1834....
, were convinced that the effect would be detectable and undertook to conduct the experiment. The deflection angle depended on the relative densities and volumes of the Earth and the mountain: if the density and volume of Schiehallion could be ascertained, then so could the density of the Earth. Once this was known, then this would in turn yield approximate values for those of the other planets, their moons, and the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
, previously known only in terms of their relative ratios. As an additional benefit, the concept of contour line
Contour line
A contour line of a function of two variables is a curve along which the function has a constant value. In cartography, a contour line joins points of equal elevation above a given level, such as mean sea level...
s, devised to simplify the process of surveying the mountain, later became a standard technique in cartography
Cartography
Cartography is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively.The fundamental problems of traditional cartography are to:*Set the map's...
.
Background
A pendulumPendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position...
hangs straight downwards in a symmetrical gravitational field
Gravitational field
The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...
. However, if a sufficiently large mass such as a mountain is nearby, its gravitational attraction should pull the pendulum's plumb-bob
Plumb-bob
A plumb-bob or a plummet is a weight, usually with a pointed tip on the bottom, that is suspended from a string and used as a vertical reference line, or plumb-line....
slightly out of true. The change in plumb-line angle against a known object—such as a star—could be carefully measured on opposite sides of the mountain. If the mass of the mountain could be independently established from a determination of its volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
and an estimate of the mean density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
of its rocks, then these values could be extrapolated to provide the mean density of the Earth, and by extension, its mass.
Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
had considered the effect in the Principia, but pessimistically thought that any real mountain would produce too small a deflection to measure. Gravitational effects, he wrote, were only discernible on the planetary scale. Newton's pessimism was unfounded: although his calculations had suggested a deviation of less than 2 minutes of arc
Minute of arc
A minute of arc, arcminute, or minute of angle , is a unit of angular measurement equal to one sixtieth of one degree. In turn, a second of arc or arcsecond is one sixtieth of one minute of arc....
(for an idealised three-mile high mountain), this angle, though very slight, was within the theoretical capability of instruments of his day.
An experiment to test Newton's idea would both provide supporting evidence for his law of universal gravitation
Newton's law of universal gravitation
Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them...
, and estimates of the mass and density of the Earth. Since the masses of astronomical objects were known only in terms of relative ratios, the mass of the Earth would provide reasonable values to the other planet
Planet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...
s, their moons
Natural satellite
A natural satellite or moon is a celestial body that orbits a planet or smaller body, which is called its primary. The two terms are used synonymously for non-artificial satellites of planets, of dwarf planets, and of minor planets....
, and the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
. The data were also capable of determining the value of Newton's gravitational constant
Gravitational constant
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
G, though this was not a goal of the experimenters; references to a value for G would not appear in the scientific literature until almost a hundred years later.
Finding the mountain
Chimborazo, 1738
A pair of French astronomers named Pierre BouguerPierre Bouguer
Pierre Bouguer was a French mathematician, geophysicist, geodesist, and astronomer. He is also known as "the father of naval architecture"....
and Charles Marie de La Condamine
Charles Marie de La Condamine
Charles Marie de La Condamine was a French explorer, geographer, and mathematician. He spent ten years in present-day Ecuador measuring the length of a degree latitude at the equator and preparing the first map of the Amazon region based on astronomical observations.-Biography:Charles Marie de La...
were the first to attempt the experiment, conducting their measurements on the 6268 metres (20,564.3 ft) volcano Chimborazo
Chimborazo (volcano)
Chimborazo is a currently inactive stratovolcano located in the Cordillera Occidental range of the Andes. Its last known eruption is believed to have occurred around 550 AD....
in Ecuador
Ecuador
Ecuador , officially the Republic of Ecuador is a representative democratic republic in South America, bordered by Colombia on the north, Peru on the east and south, and by the Pacific Ocean to the west. It is one of only two countries in South America, along with Chile, that do not have a border...
in 1738. Their expedition had left France for South America in 1735 to try to measure the meridian arc
Meridian arc
In geodesy, a meridian arc measurement is a highly accurate determination of the distance between two points with the same longitude. Two or more such determinations at different locations then specify the shape of the reference ellipsoid which best approximates the shape of the geoid. This...
length of one degree of latitude near the equator
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....
, but they took advantage of the opportunity to attempt the deflection experiment. In December 1738, under very difficult conditions of terrain and climate, they conducted a pair of measurements at altitudes of 4,680 and 4,340 m. Bouguer wrote in a 1749 paper that they had been able to detect a deflection of 8 seconds of arc, but he downplayed the significance of their results, suggesting that the experiment would be better carried out under easier conditions in France or England. He added that the experiment had at least proved that the Earth could not be a hollow shell
Hollow Earth
The Hollow Earth hypothesis proposes that the planet Earth is either entirely hollow or otherwise contains a substantial interior space. The hypothesis has been shown to be wrong by observational evidence, as well as by the modern understanding of planet formation; the scientific community has...
, as some thinkers of the day, including Edmond Halley
Edmond Halley
Edmond Halley FRS was an English astronomer, geophysicist, mathematician, meteorologist, and physicist who is best known for computing the orbit of the eponymous Halley's Comet. He was the second Astronomer Royal in Britain, following in the footsteps of John Flamsteed.-Biography and career:Halley...
, had suggested.
Schiehallion, 1774
Royal Society
The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...
in 1772 by Nevil Maskelyne
Nevil Maskelyne
The Reverend Dr Nevil Maskelyne FRS was the fifth English Astronomer Royal. He held the office from 1765 to 1811.-Biography:...
, Astronomer Royal
Astronomer Royal
Astronomer Royal is a senior post in the Royal Household of the Sovereign of the United Kingdom. There are two officers, the senior being the Astronomer Royal dating from 22 June 1675; the second is the Astronomer Royal for Scotland dating from 1834....
. He suggested that the experiment would "do honour to the nation where it was made" and proposed Whernside
Whernside
Whernside is a mountain in the Yorkshire Dales and is one of the Yorkshire Three Peaks, the other two being Ingleborough and Pen-y-ghent. It is the highest point in the ceremonial county of North Yorkshire,Mickle Fell in southern Teesdale is the highest point within the boundaries of the historic...
in Yorkshire
Yorkshire
Yorkshire is a historic county of northern England and the largest in the United Kingdom. Because of its great size in comparison to other English counties, functions have been increasingly undertaken over time by its subdivisions, which have also been subject to periodic reform...
, or the Blencathra
Blencathra
Blencathra, also known as Saddleback, is one of the most northerly mountains in the English Lake District. It has six separate fell tops, of which the highest is the Hallsfell Top.-Name:...
-Skiddaw
Skiddaw
Skiddaw is a mountain in the Lake District National Park in England. With a summit at 931 m above sea level it is the fourth highest mountain in England. It lies just north of the town of Keswick, Cumbria, and dominates the skyline in this part of the northern lakes...
massif in Cumberland
Cumberland
Cumberland is a historic county of North West England, on the border with Scotland, from the 12th century until 1974. It formed an administrative county from 1889 to 1974 and now forms part of Cumbria....
as suitable targets. The Royal Society formed the Committee of Attraction to consider the matter, appointing Maskelyne, Joseph Banks
Joseph Banks
Sir Joseph Banks, 1st Baronet, GCB, PRS was an English naturalist, botanist and patron of the natural sciences. He took part in Captain James Cook's first great voyage . Banks is credited with the introduction to the Western world of eucalyptus, acacia, mimosa and the genus named after him,...
and Benjamin Franklin
Benjamin Franklin
Dr. Benjamin Franklin was one of the Founding Fathers of the United States. A noted polymath, Franklin was a leading author, printer, political theorist, politician, postmaster, scientist, musician, inventor, satirist, civic activist, statesman, and diplomat...
amongst its members. The Committee despatched the astronomer and surveyor Charles Mason
Charles Mason
Charles Mason was an English astronomer who made significant contributions to 18th-century science and American history, particularly through his involvement with the survey of the Mason-Dixon line, which came to mark the division between the northern and southern United States...
to find a suitable mountain.
After a lengthy search over the summer of 1773, Mason reported that the best candidate was Schiehallion
Schiehallion
Schiehallion is a prominent mountain in Perth and Kinross, Scotland. Schiehallion has a rich botanical life, interesting archaeology, and a unique place in scientific history for an 18th-century experiment in 'weighing the world'...
(then spelled Schehallien), a 1083 metres (3,553.1 ft) peak lying between Loch Tay
Loch Tay
Loch Tay is a freshwater loch in the central highlands of Scotland, in the district of Perthshire.It is a long narrow loch of around 14 miles long, and typically around 1 to 1½ miles wide, following the line of the valley from the south west to north east...
and Loch Rannoch
Loch Rannoch
Loch Rannoch is a large body of fresh water in Perth and Kinross, Scotland.The loch is over long in a west-east direction with an average width of about . The River Tummel begins at its eastern end. The Tay Forest Park lies along its southern shore...
in the central Scottish Highlands
Scottish Highlands
The Highlands is an historic region of Scotland. The area is sometimes referred to as the "Scottish Highlands". It was culturally distinguishable from the Lowlands from the later Middle Ages into the modern period, when Lowland Scots replaced Scottish Gaelic throughout most of the Lowlands...
. The mountain stood in isolation from any nearby hills, which would reduce their gravitational influence, and its symmetrical east-west ridge would simplify the calculations. Its steep northern and southern slopes would allow the experiment to be sited close to its centre of gravity, maximising the deflection effect.
Mason however declined to conduct the work himself for the offered commission of one guinea
Guinea (British coin)
The guinea is a coin that was minted in the Kingdom of England and later in the Kingdom of Great Britain and the United Kingdom between 1663 and 1813...
per day. The task therefore fell to Maskelyne, for which he was granted a temporary leave of his duties as Astronomer Royal. He was aided in the task by mathematician and surveyor Charles Hutton
Charles Hutton
Charles Hutton was an English mathematician.Hutton was born at Newcastle-on-Tyne. He was educated in a school at Jesmond, kept by Mr Ivison, a clergyman of the Church of England...
and Reuben Burrow
Reuben Burrow
Reuben Burrow was an English mathematician and orientalist. Initially a teacher, he was appointed assistant to Maskelyne, then astronomer-royal, at the Royal Greenwich Observatory and was involved in the Schiehallion experiment...
, a mathematician of the Royal Greenwich Observatory. A workforce of labourers was engaged to construct observatories for the astronomers, and assist in the surveying. The science team was particularly well-equipped: its astronomical instruments included a 12 inches (30.5 cm) brass quadrant
Quadrant (instrument)
A quadrant is an instrument that is used to measure angles up to 90°. It was originally proposed by Ptolemy as a better kind of astrolabe. Several different variations of the instrument were later produced by medieval Muslim astronomers.-Types of quadrants:...
from Cook's 1769 transit of Venus expedition
First voyage of James Cook
The first voyage of James Cook was a combined Royal Navy and Royal Society expedition to the south Pacific ocean aboard HMS Endeavour, from 1768 to 1771...
, a 10 feet (3 m) zenith sector, and a regulator (precision pendulum clock
Pendulum clock
A pendulum clock is a clock that uses a pendulum, a swinging weight, as its timekeeping element. The advantage of a pendulum for timekeeping is that it is a resonant device; it swings back and forth in a precise time interval dependent on its length, and resists swinging at other rates...
) for timing the astronomical observations. They also acquired a theodolite
Theodolite
A theodolite is a precision instrument for measuring angles in the horizontal and vertical planes. Theodolites are mainly used for surveying applications, and have been adapted for specialized purposes in fields like metrology and rocket launch technology...
and Gunter's chain
Gunter's chain
Gunter's chain is a measuring device used for land survey. It was designed and introduced in 1620 by English clergyman and mathematician Edmund Gunter long before the development of the theodolite and other more sophisticated equipment, enabling plots of land to be accurately surveyed and plotted,...
for surveying the mountain, and a pair of barometer
Barometer
A barometer is a scientific instrument used in meteorology to measure atmospheric pressure. Pressure tendency can forecast short term changes in the weather...
s for measuring altitude. Generous funding for the experiment was available due to underspend on the transit of Venus
Transit of Venus
A transit of Venus across the Sun takes place when the planet Venus passes directly between the Sun and Earth, becoming visible against the solar disk. During a transit, Venus can be seen from Earth as a small black disk moving across the face of the Sun...
expedition, which had been turned over to the Society by the King
George III of the United Kingdom
George III was King of Great Britain and King of Ireland from 25 October 1760 until the union of these two countries on 1 January 1801, after which he was King of the United Kingdom of Great Britain and Ireland until his death...
.
Astronomical
Observatories were constructed to the north and south of the mountain, plus a bothyBothy
A bothy is a basic shelter, usually left unlocked and available for anyone to use free of charge. It was also a term for basic accommodation, usually for gardeners or other workers on an estate. Bothies are to be found in remote, mountainous areas of Scotland, northern England, Ireland, and Wales....
to accommodate equipment and the scientists. Most of the workforce was however housed in rough canvas tents. Maskelyne's astronomical measurements were the first to be conducted. It was necessary for him to determine the zenith distances with respect to the plumb line for a set of stars at the precise time that each passed due south. Weather conditions were frequently unfavourable due to mist and rain. However, from the south observatory, he was able to take 76 measurements on 34 stars in one direction, and then 93 observations on 39 stars in the other. From the north side, he then conducted a set of 68 observations on 32 stars and a set of 100 on 37 stars. By conducting sets of measurements with the plane of the zenith sector first facing east and then west, he successfully avoided any systematic errors arising from collimating the sector.
To determine the deflection due to the mountain, it was necessary to account for the curvature of the Earth: an observer will see the local zenith
Zenith
The zenith is an imaginary point directly "above" a particular location, on the imaginary celestial sphere. "Above" means in the vertical direction opposite to the apparent gravitational force at that location. The opposite direction, i.e...
shift by the same angle as any change in latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...
. After accounting for observational effects such as precession
Precession
Precession is a change in the orientation of the rotation axis of a rotating body. It can be defined as a change in direction of the rotation axis in which the second Euler angle is constant...
, aberration of light
Aberration of light
The aberration of light is an astronomical phenomenon which produces an apparent motion of celestial objects about their real locations...
and nutation
Nutation
Nutation is a rocking, swaying, or nodding motion in the axis of rotation of a largely axially symmetric object, such as a gyroscope, planet, or bullet in flight, or as an intended behavior of a mechanism...
, Maskelyne showed that the difference between the locally-determined zenith for observers north and south of Schiehallion was 54.6 arc seconds. Once the surveying team had provided a difference of 42.94" latitude between the two stations, he was able to subtract this, and after rounding to the accuracy of his observations, announce that the sum of the north and south deflections was 11.6".
Maskelyne published his initial results in the Philosophical Transactions of the Royal Society
Philosophical Transactions of the Royal Society
The Philosophical Transactions of the Royal Society is a scientific journal published by the Royal Society of London. It was established in 1665, making it the first journal in the world exclusively devoted to science, and it has remained in continuous publication ever since, making it the world's...
in 1775, using preliminary data on the mountain's shape and hence the position of its center of gravity
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
. This led him to expect a deflection of 20.9" if the mean densities of Schiehallion and the Earth were equal. Since the deflection was about half this, he was able to make a preliminary announcement that the mean density of the Earth was approximately double that of Schiehallion. A more accurate value would have to await completion of the surveying process.
Maskelyne took the opportunity to note that that Schiehallion exhibited a gravitational attraction, and thus all mountains did; and that Newton's inverse square law of gravitation had been confirmed. An appreciative Royal Society presented Maskelyne with the 1775 Copley Medal
Copley Medal
The Copley Medal is an award given by the Royal Society of London for "outstanding achievements in research in any branch of science, and alternates between the physical sciences and the biological sciences"...
; the biographer Chalmers
Alexander Chalmers
Alexander Chalmers was a Scottish writer.He was born in Aberdeen.Trained as a doctor, he gave up medicine for journalism, and was for some time editor of the Morning Herald...
later noting that "If any doubts yet remained with respect to the truth of the Newtonian system, they were now totally removed".
Surveying
The work of the surveying team was greatly hampered by the inclemency of the weather, and it took until 1776 to complete the task. To find the volume of the mountain, it was necessary to divide it into a set of vertical prismPrism (geometry)
In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...
s and compute the volume of each. The triangulation
Triangulation
In trigonometry and geometry, triangulation is the process of determining the location of a point by measuring angles to it from known points at either end of a fixed baseline, rather than measuring distances to the point directly...
task falling to Charles Hutton was considerable: the surveyors had obtained thousands of bearings to more than a thousand points around the mountain. Moreover, the vertices of his prisms did not always conveniently coincide with the surveyed heights. To make sense of all his data, he hit upon the idea of interpolating
Interpolation
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....
a series of lines at set intervals between his measure values, marking points of equal height. In doing so, not only could he easily determine the heights of his prisms, but from the swirl of the lines one could get an instant impression of the form of the terrain. Hutton had invented contour line
Contour line
A contour line of a function of two variables is a curve along which the function has a constant value. In cartography, a contour line joins points of equal elevation above a given level, such as mean sea level...
s, in common use since for depicting cartographic relief
Cartographic relief depiction
Terrain or relief is an essential aspect of physical geography, and as such its portrayal presents a central problem in cartography, and more recently GIS and 3D Visualization....
.
| Hutton's solar system density table | ||
|---|---|---|
| Body | Density, kg·m−3 | |
| Hutton, 1778 | Modern value | |
| Sun Sun The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields... |
1,100 | 1,408 |
| Mercury Mercury (planet) Mercury is the innermost and smallest planet in the Solar System, orbiting the Sun once every 87.969 Earth days. The orbit of Mercury has the highest eccentricity of all the Solar System planets, and it has the smallest axial tilt. It completes three rotations about its axis for every two orbits... |
9,200 | 5,427 |
| Venus Venus Venus is the second planet from the Sun, orbiting it every 224.7 Earth days. The planet is named after Venus, the Roman goddess of love and beauty. After the Moon, it is the brightest natural object in the night sky, reaching an apparent magnitude of −4.6, bright enough to cast shadows... |
5,800 | 5,204 |
| Earth Earth Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets... |
4,500 | 5,515 |
| Moon Moon The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more... |
3,100 | 3,340 |
| Mars Mars Mars is the fourth planet from the Sun in the Solar System. The planet is named after the Roman god of war, Mars. It is often described as the "Red Planet", as the iron oxide prevalent on its surface gives it a reddish appearance... |
3,300 | 3,934 |
| Jupiter Jupiter Jupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with mass one-thousandth that of the Sun but is two and a half times the mass of all the other planets in our Solar System combined. Jupiter is classified as a gas giant along with Saturn,... |
1,100 | 1,326 |
| Saturn Saturn Saturn is the sixth planet from the Sun and the second largest planet in the Solar System, after Jupiter. Saturn is named after the Roman god Saturn, equated to the Greek Cronus , the Babylonian Ninurta and the Hindu Shani. Saturn's astronomical symbol represents the Roman god's sickle.Saturn,... |
410 | 687 |
Hutton had to compute the individual attractions due to each of the many prisms that formed his grid, a process which was as laborious as the survey itself. The task occupied his time for a further two years before he could present his results, which he did in a hundred-page paper to the Royal Society in 1778. He found that the attraction of the plumb-bob to the Earth would be 9,933 times that of the sum of its attractions to the mountain at the north and south stations, if the density of the Earth and Schiehallion had been the same. Since the actual deflection of 11.6" implied a ratio of 17,804:1 after accounting for the effect of latitude on gravity, he was able to state that the Earth had a mean density of
, or about 
that of the mountain. The lengthy process of surveying the mountain had not therefore greatly affected the outcome of Maskelyne's calculations. Hutton took a density of 2,500 kg·m−3 for Schiehallion, and announced that the density of the Earth was 
of this, or 4,500 kg·m−3. In comparison with the modern accepted figure of 5,515 kg·m−3, the density of the Earth had been computed with an error of less than 20%.That the mean density of the Earth should so greatly exceed that of its surface rocks naturally meant that there must be more dense material lying deeper. Hutton correctly surmised that the core material was likely metallic, and might have a density of 10,000 kg·m−3. He estimated this metallic portion to occupy some 65% of the diameter of the Earth. With a value for the mean density of the Earth, Hutton was able to set some values to Jérôme Lalande
Jérôme Lalande
Joseph Jérôme Lefrançois de Lalande was a French astronomer and writer.-Biography:Lalande was born at Bourg-en-Bresse...
's planetary tables, which had previously only been able to express the densities of the major solar system objects in relative terms.
Repeat experiments
A more direct, and more accurate, measurement of the mean density of the Earth was made 24 years after Schiehallion, when in 1798 Henry CavendishHenry Cavendish
Henry Cavendish FRS was a British scientist noted for his discovery of hydrogen or what he called "inflammable air". He described the density of inflammable air, which formed water on combustion, in a 1766 paper "On Factitious Airs". Antoine Lavoisier later reproduced Cavendish's experiment and...
used an exquisitely sensitive torsion balance to measure the attraction between large masses of lead
Lead
Lead is a main-group element in the carbon group with the symbol Pb and atomic number 82. Lead is a soft, malleable poor metal. It is also counted as one of the heavy metals. Metallic lead has a bluish-white color after being freshly cut, but it soon tarnishes to a dull grayish color when exposed...
. Cavendish's figure of 5,448 ± 0.033 kg·m−3 was only 1.2% from the currently accepted value of 5,515 kg·m−3, and his result would not be significantly improved upon until 1895 by Charles Boys. The care with which Cavendish conducted the experiment and the accuracy of his result has led his name to since be associated with it.
John Playfair
John Playfair
John Playfair FRSE, FRS was a Scottish scientist and mathematician, and a professor of natural philosophy at the University of Edinburgh. He is perhaps best known for his book Illustrations of the Huttonian Theory of the Earth , which summarized the work of James Hutton...
carried out a second survey of Schiehallion in 1811; on the basis of a rethink of its rock strata, he suggested a density of 4,560 to 4,870 kg·m−3, though the then elderly Hutton vigorously defended the original value in an 1821 paper to the Society. Playfair's calculations had raised the density closer towards its modern value, but was still too low and significantly poorer than Cavendish
Henry Cavendish
Henry Cavendish FRS was a British scientist noted for his discovery of hydrogen or what he called "inflammable air". He described the density of inflammable air, which formed water on combustion, in a 1766 paper "On Factitious Airs". Antoine Lavoisier later reproduced Cavendish's experiment and...
's computation of some years earlier.
Henry James (Ordnance Survey)
Major General Sir Henry James FRS MRIA RE was the director-general of the Ordnance Survey, the British Government mapping agency, from 1854 to 1875. Sir Henry was described by the agency itself as "perhaps Ordnance Survey's most eccentric and egotistical Director General"...
, director-general of the Ordnance Survey
Ordnance Survey
Ordnance Survey , an executive agency and non-ministerial government department of the Government of the United Kingdom, is the national mapping agency for Great Britain, producing maps of Great Britain , and one of the world's largest producers of maps.The name reflects its creation together with...
, who instead used the hill Arthur's Seat
Arthur's Seat, Edinburgh
Arthur's Seat is the main peak of the group of hills which form most of Holyrood Park, described by Robert Louis Stevenson as "a hill for magnitude, a mountain in virtue of its bold design". It is situated in the centre of the city of Edinburgh, about a mile to the east of Edinburgh Castle...
in central Edinburgh
Edinburgh
Edinburgh is the capital city of Scotland, the second largest city in Scotland, and the eighth most populous in the United Kingdom. The City of Edinburgh Council governs one of Scotland's 32 local government council areas. The council area includes urban Edinburgh and a rural area...
. With the resources of the Ordnance Survey at his disposal, James extended his topographical survey to a 21-kilometre radius, taking him as far as the borders of Midlothian
Midlothian
Midlothian is one of the 32 council areas of Scotland, and a lieutenancy area. It borders the Scottish Borders, East Lothian and the City of Edinburgh council areas....
. He obtained a density of about 5,300 kg·m−3.
An experiment in 2005 undertook a variation of the 1774 work: instead of computing local differences in the zenith, the experiment made a very accurate comparison of the period of a pendulum at the top and bottom of Schiehallion. The period of a pendulum is a function of g
G-force
The g-force associated with an object is its acceleration relative to free-fall. This acceleration experienced by an object is due to the vector sum of non-gravitational forces acting on an object free to move. The accelerations that are not produced by gravity are termed proper accelerations, and...
, the local gravitational acceleration
Gravitational acceleration
In physics, gravitational acceleration is the acceleration on an object caused by gravity. Neglecting friction such as air resistance, all small bodies accelerate in a gravitational field at the same rate relative to the center of mass....
. The pendulum is expected to run more slowly at altitude, but the mass of the mountain will act to reduce this difference. This experiment has the advantage of being considerably easier to conduct than the 1774 one, but to achieve the desired accuracy, it is necessary to measure the period of the pendulum to within one part in one million. This experiment yielded a value of the mass of the Earth of 8.1 ± 2.4×1024 kg, corresponding to a mean density of 7,500 ± 1,900 kg·m−3.
A modern re-examination of the geophysical data was able to take account of factors the 1774 team could not. With the benefit of a 120-km radius digital elevation model
Digital elevation model
A digital elevation model is a digital model or 3-D representation of a terrain's surface — commonly for a planet , moon, or asteroid — created from terrain elevation data....
, greatly improved knowledge of the geology of Schiehallion, and in particular a computer, a 2007 report produced a mean Earth density of 5,480 ± 250 kg·m−3. When compared to the modern figure of 5,515 kg·m−3, it stood as a testament to the accuracy of Maskelyne's astronomical observations.
Mathematical procedure
Consider the force diagram to the right, in which the deflection has been greatly exaggerated. The analysis has been simplified by considering the attraction on only one side of the mountain. A plumb-bob of mass m is situated a distance d from P, the centre of gravity of a mountain of mass MM and densityDensity
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
ρM. It is deflected through a small angle θ due to its attraction F towards P and its weight
Weight
In science and engineering, the weight of an object is the force on the object due to gravity. Its magnitude , often denoted by an italic letter W, is the product of the mass m of the object and the magnitude of the local gravitational acceleration g; thus:...
W directed towards the Earth. The vector sum of W and F results in a tension T in the pendulum string. The Earth has a mass ME, radius rE and a density ρE.
The two gravitational forces on the plumb-bob are given by Newton's law of gravitation:
Where G is Newton's gravitational constant
Gravitational constant
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
. G and m can be eliminated by taking the ratio of F to W:
Where VM and VE are the volumes of the mountain and the Earth. Under static equilibrium, the horizontal and vertical components of the string tension T can be related to the gravitational forces and the deflection angle θ:
Substituting for T:
Since VE, VM, d and rE are all known, and θ has been measured, then a value for the ratio ρE : ρM can be obtained:
