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Erschienen in:

2014 | OriginalPaper | Buchkapitel

2. Geodesy

verfasst von : Michel Capderou

Erschienen in: Handbook of Satellite Orbits

Verlag: Springer International Publishing

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Abstract

When the representation of the Earth passes the sphere of the ellipsoid revolution, we enter the era of geodesy. Latitude must then be defined in different ways.

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Vorheriges Kapitel Geometry of the Ellipse

Nächstes Kapitel Geopotential

If they intersect at all, a sphere and a plane intersect in a circle. If the plane passes through the center of the sphere, the circle in question is a great circle, otherwise it is a small circle.

When it was officially chosen as prime meridian at the International Meridian Conference, held in Washington in 1884, the meridian passing through the Royal Observatory in Greenwich was already being used in most shipping charts (the British delegate declared at the time that, in terms of sheer tonnage, 72 % of world shipping trade was using charts based on Greenwich). On the other hand, when it came to land charts, there was a multitude of different zero meridians. Naturally, France used the Paris meridian set down by Cassini, which passed through the Paris Observatory. Italy referred to the meridian in Rome, Russia to the one in Pulkovo, and so on. The meridian which passes through El Hierro Island (Isla de el Hierro), the westernmost island of the Canaries, and hence the westernmost point known to Europe before the discovery of the Americas, had the advantage that it gave only positive longitudes at the time. It had often been used in Europe in the seventeenth and eighteenth centuries. Even in the nineteenth century it was still to be found on several maps from central Europe.

The word “geodesy” comes from the New Latin geodesia, as attested in the sixteenth century. This in turn came from the Greek, and in particular from the prefix geo-, ἡ γη̃, γη̃ς, meaning “the Earth” or “the country”, and -desy, ἡ δαίς, δαιτός, meaning “share”, in the sense of equal shares distributed at mealtimes.

In the navy, the nautical mile is defined as the distance equivalent to 1′ of arc of latitude, with the relation 1 nautical mile = \(40 \times 1{0}^{6}/(360 \times 60) = 1\;851.851\) m. The second of arc is equivalent to 1^{′ ′} = 1 nm/60 = 30. 864 m. The speed is obtained directly in nautical miles per hour by spacing knots every 15.432 m (equivalent to 0. 5^{′ ′} of arc) and measuring the rate at which the knots go by for a period of 30 s (or 0.5 min).

The nadir is the direction given by the vertical, but in the downward direction. The opposite direction is the zenith. The word “nadir” comes from the Arabic nāḍir, from the root of the verb “to look towards”.

Note that η = 1 only corresponds to h = 0 in the equatorial plane for ψ = 0. If we set η = 1 with \(\psi =\pi /2\) in (2.37), we obtain \(h/R_{\mathrm{e}} = f\), or \(h = R_{\mathrm{e}} - R_{\mathrm{p}}\). At the poles, zero altitude h = 0 corresponds to \(\eta = 1 - f\).

At the exact center of this circular world was the Temple of Apollo in Delphi. The name Delphi, οἱ Δελφοί, ω̃ν in Greek, is closely related to the word ἡ δελφύς, ύος, which means “womb”.

Jean–Félix Picard (1620–1683) was a French astronomer and geodesist. He invented the sighting telescope with crosswires, which allowed him to carry out very accurate surveying, and in particular, levelling. Having determined the Earth’s radius, as explained in the main text (Mesure de la Terre, 1671), he immediately communicated his result to Newton, who was thus able to check the relation between the accelerations and the squares of the distances, and thereby obtained a clear confirmation of his universal theory of gravitation. In another area, Picard was the first to carry out systematic measurements of the diameter of the solar disk. He observed its variations and sought the connection with climate change on Earth. His series of measurements between 1666 and 1682 was continued by La Hire from 1683 to 1718.

Given a straight line distance to be measured in the field, the idea is to consider a chain of adjacent triangles along the straight line. The apices of the triangles are church towers or other features that are clearly visible from afar. The angles of these triangles are then measured, and if the length of one side is known, the lengths of the other two sides can be deduced by trigonometry. This yields the required distance. The side that is actually measured is called the baseline.

In Picard’s own words: “The cobbled road from the mill in Ville–Juive to the pavillion in Juvisy, a straight line with no significant unevenness, was considered ideal as baseline for this undertaking.” Its length was very carefully measured in both directions by juxtaposition of toises. Today, this 11 km section of the D7 (previously the N7), which goes under the landing strips at Orly airport, is still very straight, apart from one or two recent urban adjustments. Each end is commemorated by a pyramidal marker stone.

As the radius of the Earth had been known since the previous year, the Academy sent Jean Richer to Cayenne to observe the parallax of Mars (the angle subtended by the Earth’s diameter as viewed from Mars) in a joint effort with Picard, who had remained in Paris. By thus measuring the distance to Mars, Kepler’s third law allows one to deduce the sizes of the planetary orbits. So the length scale of the whole Solar System was at this point underpinned by the Villejuif–Juvisy baseline!

The name Cassini will come up several times during this section. Indeed, there was a genuine dynasty of outstanding astronomers, often numbered with Roman numerals like the crowned heads of Europe:

Gian Domenico Cassini (1625–1712), known to the French as Jean Dominique Cassini, or Cassini I, was actually of Italian origin (from Nice). He soon became famous for his work on geodesy, and especially on astronomy, with his very accurate observations of the planets Mars and Jupiter. He set up tables of the Galilean moons of Jupiter, a fundamental step in the determination of longitudes, because the eclipses of these satellites constitute instantaneous signals for an Earth-based observer. So it was really the “transfer of the century” when Louis XIV called Cassini to Paris in 1669 to entrust him with the foundation and the running of the Paris Observatory. In 1679, the observatory began to publish La Connaissance des Temps, a publication that is still alive today and which lists the positions of the heavenly bodies to the greatest possible accuracy. Cassini continued his observation of the moons of Jupiter, work which allowed Olaüs Römer to show that the speed of light was not infinite. He also improved observations of the Moon and Saturn. This was the Cassini who gave his name to the Cassini Division of Saturn’s rings and also to the Cassini space probe, designed to explore Saturn and its environment.
Jacques Cassini (1677–1756), or Cassini II, was the son of Jean Dominique. He pursued his father and Picard’s geodetic measurements, but the publication of De la grandeur et de la figure de la Terre (1722), in which he made a mistake over the flattening of the Earth, was later to reduce his scientific status. This was the Cassini who instigated the scientific dispute between the Cassini dynasty and Newton.
César François Cassini de Thury (1714–1784), or Cassini III, was the son of Jacques. After assisting his father with his geodetic measurements, he devoted himself to cartography. See Fig. 2.8. In 1750, Louis XV asked him to map the whole of the kingdom. This is the Cassini who is remembered for the Cassini projection and the Cassini map of France.
Jacques Dominique Cassini (1748–1845), or Cassini IV, was the son of César François. He was to finally publish the map of France in 1790.

From 1669 to 1793, the Cassinis ran the Paris Observatory, either officially or unofficially, each son succeeding his father.

Traditionally, the contrast is illustrated by: mandarin or lemon? There seems to be an anachronism here, since the word “mandarin” did not appear in French until 1773.

Pierre Louis Moreau de Maupertuis (1698–1759), a French physicist, led the expedition to Lapland. In so doing, he earnt these two graceful lines from Voltaire: Vous allâtes vérifier en ces lieux pleins d’ennui

Ce que Newton connut sans sortir de chez lui.

[Inthis soulless landscape you are sure to construe

What Newton in his college lodgings always knew.]

While there is probably no connection with this typical irony, Maupertuis subsequently published, in 1744, his famous Principe de moindre action (Principle of least action).

This map, already mentioned earlier, used a scale of 1/86,400 (one line for 100 toises). It comprised 182 sheets to cover the whole kingdom. Cassini de Thury used a novel projection, now called the Cassini projection, in which he plotted lines perpendicular to the Paris meridian. These lines are not parallels, i.e., they are not the loci of points at constant latitude, but great circles (for a spherical Earth). The perpendicular line passing through the Paris Observatory goes from Granville to Strasbourg. This projection is the transverse aspect of the projection known in French as the plate-carrée (flat square) projection.

In a field with acceleration due to gravity equal to g, the period T of a pendulum of length l is given by \(T = 2\pi \sqrt{l/g}\) in SI units. With T = 2 s, this gives \(l = g{/\pi }^{2}\) numerically. As g varies between the equator and the pole, l varies from 0.991 to 0.996 m. We note that the metre was chosen close to the length of this pendulum, whereas a doubled metre would have been close to the toise.

The angles were measured to an accuracy of 1^{′ ′} of arc using Borda’s repeating circle method, with instruments made by his assistant Lenoir.

It was not a good time to be carrying out this kind of expedition. In the thick of the revolution, hauling strange-looking instruments to the tops of church towers or onto the battlements of castles was unnecessarily intriguing for the local populations. There were many unfortunate incidents, with material being sabotaged and surveyors arrested, among other things.

Given the delays in completing the measurements, a provisional metre had been adopted on 1 August 1793. This metre would give \(\mathcal{L}(\pi /2) = 5, 130, 430\ \mbox{ toises}\).

Once it had been determined relative to the Earth ellipsoid, the metre was then fixed by the General Conference on Weights and Measures (Conférence Générale des Poids et Mesures CGPM). In 1889 (the first CGPM), the metre was defined by the prototype deposited at the Archives de France. In 1960 (the 11th CGPM), it was defined in terms of a particular wavelength of light emitted by krypton 86. Since 1983 (the 17th CGPM), the metre has been defined relative to the speed of light (see Sect. 6.10).

Titel: Geodesy
verfasst von: Michel Capderou
Verlag: Springer International Publishing
Buch: Handbook of Satellite Orbits
Print ISBN: 978-3-319-03415-7

Electronic ISBN: 978-3-319-03416-4

Copyright-Jahr: 2014
DOI: https://doi.org/10.1007/978-3-319-03416-4_2

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