Schiehallion experiment

The Schiehallion was well suited for the experiment due to its symmetrical shape and its location away from other mountains.

The Schiehallion experiment was an experiment to determine the mean mass density of the earth. With financial support from the Royal Society , it was carried out in the summer of 1774 in the area of ​​Mount Schiehallion . The experiment comprised the measurement of the slight deviation from the vertical in the earth's gravitational field due to the gravitational attraction of the mountain. The Schiehallion was considered suitable for this measurement because of its isolated location and almost symmetrical shape. The impetus for the experiment was, among other things, the observation of gravity anomalies when examining the Mason-Dixon line .

The experiment was previously considered by Isaac Newton as possible experimental proof for his theory of gravity , but was later rejected by him. Nevertheless, a group of scientists, including above all the British court astronomer Nevil Maskelyne , were convinced that the effect was measurable and decided to carry out the experiment. The angle of deflection depended on the relative densities and volumes of the earth and the mountain: By determining the density and volume of the Schiehallion, the density of the earth could also be determined. With the knowledge of the density of the earth, in turn, approximate values ​​for the densities of the other planets, their moons and the sun could be calculated, which were previously only known in their relative proportions. In addition, the concept of the isoline was invented to simplify the measurement of the mountain , which later developed into a standard method in cartography .

Experimental rationale

A plumb bob hangs straight down in a symmetrical gravitational field . However, if there is a sufficiently large mass such as a mountain nearby, the perpendicular will slightly deviate from the position otherwise assumed due to its gravitational attraction. The change in the vertical direction relative to a given object - for example a star - is carefully measured on two opposite sides of the mountain. The mass of the mountain is determined in an independent measurement by determining its volume and an estimate of the mean density of its rock. Now these values ​​can be used to determine the density of the earth and thus its mass.

Isaac Newton considered the effect in his Philosophiae Naturalis Principia Mathematica , but assumed that no real mountain could create a deflection large enough to be measured. He believed noticeable effects of gravity to be possible only on the planetary scale. However, it turned out that Newton's pessimism was unfounded: Although his calculations showed a deflection of less than two arc minutes (assuming an idealized three-mile high mountain), this very small angle was within the measurable range for instruments at the time.

An attempt to test Newton's idea could consequently provide both evidence for his law of gravitation and an estimate of the mass and density of the earth. With the help of the measured values, it was also possible to determine the value of Newton's gravitational constant , although this was not the original intention of the scientists who carried out the experiment. A first value for the gravitational constant does not appear in the scientific literature until almost a hundred years later.

Historical considerations for the search for suitable mountains

Mount Chimborazo from the French experiment of 1738

Chimborazo, 1738

The two French astronomers Pierre Bouguer and Charles Marie de La Condamine were the first to attempt to carry out the experiment. In 1738 they carried out their measurements on the 6310 m high Chimborazo volcano in Ecuador . ^a In 1735 they left France with their expedition on the way to South America. The actual goal was to measure an arc of meridians with an extension of one degree in geographical longitude near the equator. However, they took the opportunity to also conduct the diversion experiment. Despite adverse conditions with regard to the terrain and climate, they carried out two measurements in December 1738 at heights of 4680 and 4340 m. Bouguer wrote in a paper from 1749 that they could measure a deflection of 8 arc seconds; however, he judged the informative value of their results to be low and suggested that the experiment should be repeated under more favorable conditions in France or England. He added that the experiment at least showed that the earth could not be hollow , as suggested by some scholars of the time, including Edmond Halley .

Schiehallion, 1774

The symmetrical ridge of Schiehallion, of Loch Rannoch viewed from

The court astronomer Nevil Maskelyne suggested to the Royal Society in 1772 that another attempt should be made. In his eyes the experiment would do credit to the nation on whose territory the experiment would be carried out. He therefore suggested Mount Whernside in Yorkshire and the Blencathra and Skiddaw massifs in Cumberland as suitable locations. The Royal Society then formed a committee to look into the matter and appointed Maskelyne, Joseph Banks and Benjamin Franklin as members. The committee then hired astronomer and surveyor Charles Mason ^b to find a suitable mountain.

After a lengthy search in the summer of 1773, Mason declared the 1083 m high mountain Schiehallion the most suitable place. This is located between Loch Tay and Loch Rannoch in the middle of the Scottish highlands. The mountain lies apart from other elevations that could have disrupted its gravitational influence, and its symmetrical ridge in east-west direction simplified the calculations. Its steep flanks on the north and south sides made it possible for the experiment to be carried out near its center of gravity, thereby increasing the effect of the deflection.

Mason refused, however, to carry out the attempt himself for the offered fee of one guinee per day. Therefore, the task fell to Maskelyne, for the fulfillment of which he was temporarily released from his duties as court astronomer. In his work he was assisted by the mathematician and surveyor Charles Hutton and by Reuben Burrow , a mathematician at the Royal Observatory of Greenwich. Furthermore, some workers were hired to build observatories for the astronomers and to support them in their work. The research group was excellently equipped: among their astronomical measuring instruments there was a quadrant made of brass from the Venus expedition of James Cook of 1769, a zenith telescope and a precise pendulum clock for measuring time during the observations. In addition, a theodolite and a Gunter chain for measuring the mountain were purchased, as well as two air pressure gauges ( barometers ) for measuring altitude. The available funds were large, since the expedition to observe the passage of Venus , with which the Royal Society at the same time of King George III. was commissioned, was less well financed.

Historical measurements and data processing

Astronomical measurements

The deflection is the difference between the true zenith determined by astrometry and the apparent zenith determined by the perpendicular.

Observatories were built on the north and south sides of the mountain, as well as a hut to accommodate the scientists and equipment. ^c The workers, on the other hand, lived mainly in simple tents. First, Maskelyne's astronomical measurements were made. He had to measure the pole distance compared to the vertical direction for various stars at the exact time they were facing south. The weather conditions were often unfavorable due to fog and rain. Nevertheless, he was able to carry out 76 measurements on 34 stars in one of the two directions and 93 on 39 stars in the other from the southern observatory. On the north side he carried out 68 observations on 32 stars and 100 on 37 stars. By alternately pointing the movable axis of the zenithoscope to the east and west, he was able to successfully avoid systematic deviations due to possible collimation errors.

In order to determine the deflection caused by the mountain, it was necessary to include the curvature of the earth's surface: For the observer, the local zenith changes like the degree of latitude on which he is located. After considering other effects such as precession, aberration and nutation, Maskelyne showed that the difference between the locally determined zenith when observing north and south of the Schiehallion was 54.6 arc seconds. During the land survey, there was a difference of 42.94 arc seconds in latitude for the two observation stations, so that the sum of the deflections south and north reported an angle of 11.6 arc seconds.

Maskelyne published his first results in the Philosophical Transactions of the Royal Society of 1775, using preliminary data for the shape of the mountain and thus for his focus. He concluded that a deflection of 20.9 arc seconds would be expected if the mean densities of the Schiehallion and Earth were the same. Since the actual deflection was about half that value, he could estimate the mean density of the earth to be about twice that of Schiehallion. A more precise value could only be determined after the surveying work on the mountain had been completed.

Maskelyne pointed out that the experiment had clearly shown that the Schiehallion had a gravitational attraction and that this had to apply to all mountains. Newton's law of gravity has thus been confirmed. In recognition of his services, the Royal Society awarded Maskelyne the Copley Medal in 1775 . His biographer Alexander Chalmers later wrote that if there were any doubts about the truth of the Newtonian system, they have now been completely dispelled.

Land surveying

The work of the surveyors was severely hampered by the unfavorable weather, which is why the work could only be completed in 1776. In order to determine the volume of the mountain, it had to be broken down into a series of prisms and the volume of each prism determined. The triangulation, which was carried out by Charles Hutton, was associated with a great deal of effort: the surveyors had taken thousands of bearings at over a thousand measuring points around the mountain. In order to be able to process his data efficiently, he came up with the idea of ​​interpolating lines at certain intervals between his measured values, using measuring points at the same height. This not only made it easy for him to determine the height of the prisms, but also gave him an immediate impression of the shape of the terrain through the various lines. Hutton had thus invented the isolines , which have since been used frequently in cartographic relief representations. ^d

Density of the celestial bodies in the solar system according to Hutton

Heavenly bodies	Density (kg m −3 )
	Hutton, 1778	Today's value
Sun	1100	1408
Mercury	9200	5427
Venus	5800	5204
earth	4500	5515
moon	3100	3340
Mars	3300	3934
Jupiter	1100	1326
Saturn	410	687

Hutton had to determine the pull emanating from each one of the many prisms - a job almost as arduous as measuring the mountain itself. This work took him two years until he was finally able to present his results in a hundred-page essay to the Royal Society in 1778. According to his calculations, the attraction of the perpendicular through the earth should be 9933 times as strong as the attraction through the mountain at the respective observation stations, if the density of the earth and the Schiehallion were the same. The measured deflection of 11.6 arc seconds resulted in a ratio of the attractive forces of 17,804: 1, taking into account the effect of latitude on gravity. The mean density of the earth is therefore or about times as high as that of the mountain. Due to the lengthy process of measuring the mountain, the values ​​in Maskelyne's calculations changed only slightly. Hutton took the value of 2500 kg · m ⁻³ for the density of the Schiehallion and added the density of the earth to it, i.e. 4500 kg · m ⁻³ . Compared with the now generally accepted value of the soil density of 5515 kg · m ⁻³ , the error was less than 20%. ${\displaystyle {\tfrac {17.804}{9.933}}}$ ${\displaystyle {\tfrac {9}{5}}}$ ${\displaystyle {\tfrac {9}{5}}}$

The fact that the mean density of the earth should be so clearly above that of the rock on its surface also meant that the material deeper in the earth's interior had to be a lot denser. Hutton correctly speculated that the material in the earth's core was probably metallic and could have a density of around 10,000 kg · m ⁻³ . He estimated that the metallic part takes up around 65% of the earth's diameter.

Follow-up experiments

See also : gravitational balance

Henry Cavendish used a more direct and accurate method to measure the mean density of the earth 24 years after the Schiehallion experiment. In 1798 he used an extremely sensitive gravitational balance to determine the attraction between two masses of lead. The value 5448 kg · m ⁻³ determined by Cavendish was only 1.2% away from the currently recognized value of 5515 kg · m ⁻³ and its result was not significantly improved until much later, in 1895, by Charles Boys. ^e The experiment of the gravitational balance is therefore also called the Cavendish experiment in honor of Cavendish.

John Playfair undertook another survey of the Schiehallion in 1811; regarding the rock layers he came to new results and determined a density of 4560 to 4870 kg · m ⁻³ , whereupon Hutton defended his value in a paper from 1821 against the Royal Society. In the calculations made by Playfair, the density value came closer to today's value, but it was still too low and considerably worse than that determined by Cavendish a few years earlier.

Arthur's Seat , the subject in Henry James' experiment of 1856

An experiment comparable to the Schiehallion experiment was carried out in 1856 by Henry James , the director general of the United Kingdom's national surveying authority ( Ordnance Survey ). This experiment took place at Arthur's Seat in Edinburgh . With funds from the Ordnance Survey , James was able to extend his land survey to within 13 miles of the mountain. He received a value of about 5300 kg · m ⁻³ for the density .

In an experiment from 2005, a modified form of the experiment from 1774 was used: Instead of determining differences at the local zenith, the oscillation times of a pendulum on the Schiehallion and at its foot were compared very precisely. The period of oscillation of a pendulum depends on the force of gravity, i.e. the local gravitational acceleration. The pendulum is expected to swing more slowly in the higher position, but the mass of the mountain reduces this effect. This form of experiment has the advantage that it is considerably easier to carry out than the one from 1774. However, in order to achieve the desired measurement accuracy, the period of oscillation of the pendulum must be measured with a relative accuracy of one millionth. This experiment yielded 8.1 ± 2.4 · 10 ²⁴  kg as the value for the earth mass, which corresponds to an average density of 7500 ± 1900 kg · m ⁻³ . ^f

A modern examination of the geophysical data made it possible to include new influencing factors in the analysis that could not be taken into account in 1774. With the help of a digital elevation model of the surroundings within a radius of 120 kilometers, many new findings on the geology of the Schiehallion and, last but not least, a digital computer, an average earth density of 5480 ± 250 kg m ^{−3 could} be calculated in 2007 . This value can be seen as proof of the great accuracy of Maskelyne's astronomical observations , especially in comparison with the modern value of 5515 kg · m ⁻³ .

Mathematical method of calculation

Representation of the forces as force arrows

Please refer to the schematic representation of the individual forces on the right. The problem has been simplified in that the attraction is only viewed on one side of the mountain. A plumb line of the mass is at a distance from the center of gravity of a mountain of mass or density . It is deflected by the small angle in the direction of the gravitational force , the weight force points in the direction of the earth. The vectorial sum of and creates tensile stress in the plumb line . The earth possesses the mass , the radius and the density . ${\displaystyle \textstyle m}$ ${\displaystyle \textstyle d}$ ${\displaystyle \textstyle P}$ ${\displaystyle \textstyle M_{\mathrm {B} }}$ ${\displaystyle \textstyle \rho _{\mathrm {B} }}$ ${\displaystyle \textstyle F}$ ${\displaystyle \textstyle \theta }$ ${\displaystyle \textstyle P}$ ${\displaystyle \textstyle W}$ ${\displaystyle \textstyle W}$ ${\displaystyle \textstyle F}$ ${\displaystyle \textstyle T}$ ${\displaystyle \textstyle M_{\mathrm {E} }}$ ${\displaystyle \textstyle r_{\mathrm {E} }}$ ${\displaystyle \textstyle \rho _{\mathrm {E} }}$

The two gravitational forces acting on the perpendicular are given by Newton's law of gravitation:

${\displaystyle F={\frac {GmM_{B}}{d^{2}}},\quad W={\frac {GmM_{E}}{r_{E}^{2}}}.}$

Here denote the Newtonian gravitational constant. By looking at the relationship between being able to and being eliminated from the calculation: ${\displaystyle \textstyle G}$ ${\displaystyle \textstyle F}$ ${\displaystyle \textstyle W}$ ${\displaystyle \textstyle G}$ ${\displaystyle \textstyle m}$

${\displaystyle {\frac {F}{W}}={\frac {(GmM_{B})/d^{2}}{(GmM_{E})/r_{E}^{2}}}={\frac {M_{B}}{M_{E}}}{\left({\frac {r_{E}}{d}}\right)}^{2}={\frac {\rho _{B}}{\rho _{E}}}{\frac {V_{B}}{V_{E}}}{\left({\frac {r_{E}}{d}}\right)}^{2}}$

Thereby are and the volume of the mountain and the earth. In the state of equilibrium, the vertical and horizontal components of the tension in the cord are related to the angle of deflection due to the forces of gravity : ${\displaystyle \textstyle V_{\mathrm {B} }}$ ${\displaystyle \textstyle V_{\mathrm {E} }}$ ${\displaystyle \textstyle T}$ ${\displaystyle \textstyle \theta }$

${\displaystyle W=T\cos \theta ,\quad F=T\sin \theta .}$

If you replace now , you get: ${\displaystyle \textstyle T}$

${\displaystyle \tan \theta ={\frac {F}{W}}={\frac {\rho _{B}}{\rho _{E}}}{\frac {V_{B}}{V_{E}}}{\left({\frac {r_{E}}{d}}\right)}^{2}.}$

Since , , and are known and are and have been measured, a value for the ratio can be calculated: ${\displaystyle \textstyle V_{\mathrm {E} }}$ ${\displaystyle \textstyle V_{\mathrm {B} }}$ ${\displaystyle \textstyle d}$ ${\displaystyle \textstyle r_{\mathrm {E} }}$ ${\displaystyle \textstyle \theta }$ ${\displaystyle \textstyle d}$ ${\displaystyle \textstyle \rho _{\mathrm {E} }:\rho _{\mathrm {B} }}$

${\displaystyle {\frac {\rho _{E}}{\rho _{B}}}={\frac {V_{B}}{V_{E}}}{\left({\frac {r_{E}}{d}}\right)}^{2}{\frac {1}{\tan \theta }}.}$

Remarks

^aAt that time the area belonged to the viceroyalty of Peru . Contemporary sources therefore speak of a 'Peruvian expedition'.

^bMason had previously determined the course of the Mason-Dixon Line, which was used as the border between the northern and southern states of the United States, together with Jeremiah Dixon .

^c Remains of these buildings can still be found on the mountain slopes.

^dThis could also be described as a rediscovery: Edmond Halley had already introduced lines of the same misalignment in 1701 , Nicholas Cruquius used lines of the same sea ​​depth in 1727 .

^eIn Cavendish's work, the value 5448 kg · m ^{−3 appears} . However, he made a calculation error: His measurements actually give the value 5480 kg · m ⁻³ . Francis Baily only noticed this mistake in 1821.

^fThe value 1.0832 · 10 ¹²  km ^{3 is} used for the volume of the earth .

Individual evidence

1. ↑ ^{a b c d} R.D. Davies: A Commemoration of Maskelyne at Schiehallion . In: Quarterly Journal of the Royal Astronomical Society . 26, No. 3, 1985, pp. 289-294. bibcode : 1985QJRAS..26..289D .
2. ^ ^{A b} Newton: Philosophiæ Naturalis Principia Mathematica , Volume II, ISBN 0-521-07647-1 , p. 528. Translated: Andrew Motte, First American Edition. New York, 1846
3. ↑ ^{a b c d e f g h i} R.M. Sillitto: Maskelyne on Schiehallion: A Lecture to The Royal Philosophical Society of Glasgow . October 31, 1990. Retrieved December 28, 2008.
4. ^ A. Cornu, Baille, JB: Détermination nouvelle de la constante de l'attraction et de la densité moyenne de la Terre . In: Comptes rendus de l'Académie des sciences . 76, 1873, pp. 954-958.
5. ↑ ^{a b} J.H. Poynting: The Earth: its shape, size, weight and spin . Cambridge, 1913, pp. 50-56.
6. ↑ ^{a b c d e f g h} J. H. Poynting: The mean density of the earth 1894, pp. 12-22.
7. ^ N. Maskelyne: A proposal for measuring the attraction of some hill in this Kingdom . In: Philosophical Transactions of the Royal Society . 65, 1772, pp. 495-499. bibcode : 1775RSPT ... 65..495M .
8. ^ ^{A b c} Edwin Danson: Weighing the World . Oxford University Press, 2006, ISBN 978-0-19-518169-2 , pp. 115-116.
9. ^ ^{A b} Edwin Danson: Weighing the World . Oxford University Press, 2006, ISBN 978-0-19-518169-2 , p. 146.
10. ^ ^{A b} The "Weigh the World" Challenge 2005 . countingthoughts. April 23, 2005. Retrieved December 28, 2008.
11. ↑ ^{a b} J.H. Poynting: The Earth: its shape, size, weight and spin . Cambridge, 1913, pp. 56-59.
12. ^ ^{A b c d} N. Maskelyne: An Account of Observations Made on the Mountain Schiehallion for Finding Its Attraction . In: Philosophical Transactions of the Royal Society . 65, No. 0, 1775, pp. 500-542. doi : 10.1098 / rstl.1775.0050 .
13. ↑ ^{a b c d} J. H. Poynting, Thomson, JJ: A text-book of physics 1909, ISBN 1-4067-7316-6 , pp. 33-35.
14. ↑ AS Mackenzie: The laws of gravitation; memoirs by Newton, Bouguer and Cavendish, together with abstracts of other important memoirs 1900, pp. 53-56.
15. ^ A. Chalmers: The General Biographical Dictionary , Volume 25 1816, p. 317.
16. ↑ ^{a b c d e f} Edwin Danson: Weighing the World . Oxford University Press, 2006, ISBN 978-0-19-518169-2 , pp. 153-154.
17. ↑ ^{a b c d} C. Hutton: An Account of the Calculations Made from the Survey and Measures Taken at Schehallien . In: Philosophical Transactions of the Royal Society . 68, No. 0, 1778, p. 689. doi : 10.1098 / rstl.1778.0034 .
18. ↑ ^{a b} Planetary Fact Sheet . In: Lunar and Planetary Science . NASA. Retrieved January 2, 2009.
19. ^ Russell McCormmach, Jungnickel, Christa: Cavendish . American Philosophical Society , 1996, ISBN 978-0-87169-220-7 , pp. 340-341.
20. ^ ^{A b c} G. Ranalli: An Early Geophysical Estimate of the Mean Density of the Earth: Schehallien, 1774 . In: Earth Sciences History . 3, No. 2, 1984, pp. 149-152.
21. ^ Charles Hutton: On the mean density of the earth . In: Proceedings of the Royal Society . 1821.
22. ↑ James: On the Deflection of the Plumb-Line at Arthur's Seat, and the Mean Specific Gravity of the Earth . In: Proceedings of the Royal Society . 146, 1856, pp. 591-606. JSTOR 108603 .
23. ^ The "Weigh the World" Challenge Results . countingthoughts. Retrieved December 28, 2008.
24. ↑ ^{a b} J.R. Smallwood: Maskelyne's 1774 Schiehallion experiment revisited . In: Scottish Journal of Geology . 43, No. 1, 2007, p. 15 31. doi : 10.1144 / sjg43010015 .

Coordinates: 56 ° 40 '4 "  N , 4 ° 5' 52"  W.