3. Geopotential

Isaac Newton (1643–1727) was an English mathematician, physicist, and astronomer. In 1687, he stated his three laws of motion in Philosophæ Naturalis Principia Mathematica: (1) the principle of inertia, (2) his famous second law, which says that, in a Galilean frame, the force is equal to mass times acceleration, and (3) the principle of action and reaction. It can be shown that (1) is a special case of (2) and that (3) can be deduced from (2). The fundamental second law (2) was not expressed in exactly this way by Newton. Combined with Kepler’s law of elliptical orbits, the second law can be used to derive Newton’s universal law of gravity [see (4.​115)–(4.​117)]. Newton’s work dominated the eighteenth century, in mathematics (analysis, solution of equations) and in physics, especially in optics, with the publication of Opticks. Regarding Newton’s date of birth, it is interesting to note that 25 December 1642 on the Julian calendar, which was still used in England at the time, corresponds to 4 January 1643 on the Gregorian calendar.

Carl Fiedrich Gauß (1777–1855) was a German astronomer, mathematician, and physicist. He was extremely precocious and interested in astronomy from an early age. He invented a method for calculating the orbital elements of the planets (see the note about Piazzi), then developed powerful methods for handling the problems of celestial mechanics, such as the theory of least squares, in his work Theoria motus corporum coelestium (1809). In mathematics, he invented congruences (modulo) and studied quadratic forms, error analysis (bell-shaped curve, 1821), regular polygons, conformal representations, spherical trigonometry, and the curvature of surfaces (1827). He revolutionised the field of geodesy by introducing and developing novel methods, and he was not afraid to go out into the field, e.g., setting up the cadastral survey of the Hanover region between 1817 and 1821. In physics, he carried out fundamental work on magnetism (Allgemeine Theorie des Erdgeomagnetismus, published in 1839), electricity (Gauss’ theorem), and optics (Gaussian optics). His contemporaries called him the Prince of Mathematicians. So who was the king?

In 1742, MacLaurin showed that the ellipsoid of revolution spinning on its minor axis was the only geometrical shape that could meet requirements. Later, Poincaré showed that, for much faster rotation, there were other possibilities, but they are not relevant to the planets.

The water is in equilibrium so there is no reason why it should flow from left to right, or from right to left!

Jean le Rond d’Alembert (1717–1783) was a French mathematician, physicist, and philosopher. He published Recherche sur la précession des équinoxes et sur la nutation de l’axe de la Terre dans le système newtonien, in 1749. In 1743, he had stated the principle that carries his name in his Traité de dynamique. With Diderot, he wrote the Encyclopédie.

“I make here the same distinction as M. de Maupertuis (La Figure de la Terre déterminée, etc.) between weight and gravity. By weight, I understand the natural force with which all bodies fall, and by gravity, the force with which the body would fall if the Earth’s rotation were not to alter its effect and direction.” Clairaut, in his introduction to Théorie de la Figure de la Terre.

The moment of inertia I _x is defined by \(I_{x} =\,\int\int\int ({y}^{2} + {z}^{2})\mathrm{d}M\), while the product of inertial I _xy is defined by \(I_{xy} =\,\int\int\int xy\,\mathrm{d}M\). In the literature, the moments of inertia are often denoted by A = I _x , B = I _y , and C = I _z , whence (3.25) and (3.26) become \(J_{2} = (C - A)/M{R}^{2}\).

When geodesists realised that the shape of the Earth was not exactly ellipsoidal, they chose to call it the geoid (Listing in 1873), which is tautological: it is like saying that the Earth is Earth-shaped! One sometimes sees the word telluroid, a disharmonious product of Latin–Greek hybridisation that is just as tautological.

For the great majority of geopotential models, C ₀₀ = 1 and \(C_{10}^{{\ast}} = 0\). There are some exceptions, however, such as EIGEN-CHAMP03-S, with C ₀₀ = 1 and \(C_{10}^{{\ast}} = -0.83390966 \times 1{0}^{-9}\) or GRIM5-C1, with \(C_{00} = 1 - 1.14 \times 1{0}^{-10}\) and \(C_{10}^{{\ast}} = 0\).

We use the notation \(\varpi\) only in this chapter. In subsequent chapters, we shall use the notation \(\dot{\varOmega }_{\mathrm{T}}\) for this quantity, and we shall explain why when the time comes. The angular speed \(\varpi\) is equal to one revolution per sideral day, or \(\varpi = 7.292115 \times 1{0}^{-5}\,\) rad s⁻¹.

ESA’s presentation of the satellite GOCE provides an interesting illustration of the levels of accuracy attained in weight measurements at a given location:

Let us indulge in a little science fiction! Imagine a planet just like the Earth, but rotating faster, with angular speed \(\varpi ^{\prime } \). Let us calculate the weight at the equator, assuming that the planet is spherical with radius R. From (3.32), we find where m _a is defined below by (3.34). This term represents the contribution of the centrifugal acceleration to the weight when gravity is taken as unity. For the Earth, \(m_{a} = 1/288\). If m _a  = 1, the weight is zero, and all bodies at the equator on the surface find themselves in weightless conditions. For such a fast-spinning version of Earth, we thus have \({(\varpi ^{\prime }/\varpi )}^{2} = 288\), or \(\varpi ^{\prime } \approx 17\varpi\). With such an angular speed, the day lasts 17 times less than on the real Earth, i.e., a mere 84.5 min. Furthermore, this is equal to the period of rotation of a terrestrial satellite at zero altitude, as we shall see in (5.​9) of Chap. 5.

Alexis Claude Clairaut (1713–1765) was a French astronomer and mathematician. He entered the French Academy of Sciences at the age of eighteen, after astonishing the assembly by his investigation of geometric curves. He soon turned his attention to geodesy and celestial mechanics, publishing Théorie de la figure de la Terre tirée des principes de l’hydrostatique in 1743. This explored the differences in the acceleration due to the weight at the poles and the equator. He then studied the three-body problem and published his Théorie de la Lune in 1752. He was also one of the first to investigate gravitational perturbations (see the historical note on the return of Halley’s comet in Sect. 6.​8.​1).

Carrying out the calculation to second order in the small quantities, we obtain where \(m_{b} {=\varpi }^{2}{a}^{2}b/\mu = m_{a}(1 - f)\). The numerical result is \(J_{2} = 1.082634 \times 1{0}^{-3}\), implying a relative error of 7 × 10⁻⁶ compared with the value of J ₂ given in the text.

Clairaut was pursuing an idea of Newton and Huygens that the Earth’s rotation, weight, and flattening were all related. Indeed, he formulated this idea, but not in the way discussed here. He did not use the concept of potential, invented later by Lagrange, and did not exploit the coefficient J ₂ in this form.

The unit of acceleration in CGS units is the Gal in homage to Galileo. Hence, 1 Gal = 1 cm s⁻². Geodesists use the milliGal, 1 mGal = 10⁻⁵ m s⁻².

Orbital elements for the first few revolutions are: altitude at perigee h _p = 228 km, altitude at apogee h _a = 947 km, inclination i = 65. 128^∘, period T = 96. 17 min (\(\Delta T = 1.80\) s/day), perigee on latitude 41^∘ N. Last signal 26 October 1958. Reentry 4 January 1958.

Sputnik-2 was launched on 4 November 1957. Orbital elements for the first few revolutions are: h _p = 225 km, h _a = 1, 671 km, i = 65. 310^∘, T = 103. 75 min (\(\Delta T = 3.08\) s/day), perigee on latitude 40^∘ N. Reentry 14 avril 1958. Sputnik-3 was launched on 15 May 1958. Orbital elements for the first few revolutions: h _p = 226 km, h _a = 1, 881 km, i = 65. 188^∘, T = 105. 95 min (\(\Delta T = 0.75\) s/day), perigee at latitude 45^∘ N. Reentry 6 April 1960.

At the time, this was considered a scoop: the Earth was pear-shaped. Given the scale of these discrepancies, just 15 m compared with a radius of 6,400 km, this was perhaps slightly exaggerated!

D. King-Hele summed this up in a little refrain of his own invention: When you cut a slice

Through the polar ice

The Earth is like a pear.

But sliced along the equator

She looks like a potato –

Among these, the US series GEOS (Geodetic Earth Orbiting Satellite), GEOS-1 (Explorer-29), GEOS-2 (Explorer-36), PAGEOS, LAGEOS, with passive ranging (PA) or laser rangling (LA), which followed on from the satellites Echo-1 and Echo-2 (balloon-borne), ANNA-1B (Army, Navy, Nasa, Air Force, the first satellite to emit flashes), ADE-A (Atmospheric Density Explorer, Explorer-19), Beacon Explorer-1 (BE-B, Explorer-22, or S-66a, the first satellite equipped with laser reflectors), and Beacon Explorer-2 (BE-C, Explorer-27). After 1970, came the French satellites Starlette and Stella, launched in 1975 and 1993, the Japanese satellite EGP (Experimental Geodetic Payload), also called EGS-1 (Earth Geodetic Satellite or Ajisai, meaning “hydrangea” in Japanese), launched in 1986, and the Russian satellite Fizeau (Meteor-2-21), launched in 1993.

Whence the nickname of Tom and Jerry given by the mission team.

In the space community, GOCE is generally pronounced “go-chay”.

The satellites Navstar/GPS-35 and 36 (or USA-96, 100), launched in 1993 and 1994, are equipped with laser reflectors.

This is another acronym with subtle connotations, recalling the German word Eigenwert introduced into mathematics by Hilbert and transformed to “eigenvalue” in English. The English word corresponding to eigen is “own”, from the Old English āgen.

EIGEN-6C2 is a combined global gravity field of a maximum degree/order 1949 which has been inferred from the combination of the following data:

As an example, here are the values from the EGM96 model. For the time variation:

\(\mathrm{d}C_{20}^{{\ast}}/\mathrm{d}t = +1.162755 \times 1{0}^{-11}\ {\mathrm{year}}^{-1}\,,\quad \mathrm{or}\;\;\mathrm{d}J_{2}/\mathrm{d}t =\dot{ J}_{2} = -2.60 \times 1{0}^{-11}\ {\mathrm{year}}^{-1}\;,\)

\(\mathrm{d}C_{21}^{{\ast}}/\mathrm{d}t = -0.337 \times 1{0}^{-11}\ {\mathrm{year}}^{-1}\,,\quad \mathrm{and}\;\;\mathrm{d}S_{21}^{{\ast}}/\mathrm{d}t = +1.606 \times 1{0}^{-11}\ {\mathrm{year}}^{-1}\;.\)

This variation, particularly clear between 1985 (when measurements began) and 1995, is considered to be the signature of post-glacial rebound. From 1995 to 2013, J ₂ ceased to vary. Post-glacial rebound seems to have been compensated by the melting of ice and other effects due to global warming. The contribution of the tides is 4. 173 × 10⁻⁹ for \(C_{20}^{{\ast}}\).

Since the melting of the polar ice caps, the ground level has been rising by several centimetres per year for the past few thousand years, whether it be in Canada, Scandinavia, or the Antarctic.

Henry Cavendish (1731–1810), the British physicist and chemist, was the first to obtain a precise value for G, which he published in 1798 in a famous paper entitled Experiments to determine the density of the Earth. He used a subtle method: instead of taking advantage of very large masses (like those who, at the time, sought to measure the deviation of a plumb line by a mountain), he used a torsion balance with a very fine thread, suspending two small metal weights (50 g). Bringing two large lead balls (30 kg) to a distance of 15 cm, he measured the torsion of the thread, using a mirror to create a “light lever”, and deduced G from the period of the motion ( ∼ 2 h). He thereby calculated the density of the Earth and found d = 5. 48 (current value 5.52). This density is greater than that of the rocks in the Earth’s crust ( ∼ 2. 7), and Cavendish thus demonstrated that the Earth contained a very dense central part. The method was later refined by Charles Boys (1895) using a very fine quartz thread (2 μm) and still smaller masses (2.7 g, 7.5 kg at 15 cm), over a short period (3 min). This type of experiment is still used to measure G, but the relative accuracy does not exceed \(\updelta G/G = 1.2 \times 1{0}^{-4}\). Other ways are now sought to improve accuracy. Current recommendations (CODATA 2010) give the value

LAGEOS-1 (NASA), launched 4 May 1976, and LAGEOS-2 (NASA–ASI, Italy), launched 22 October 1992, are almost identical satellites, of mass 410 kg and diameter 60 cm, each carrying 426 circular reflectors (422 fused silica glass and 4 germanium). They are often referred to as LAGEOS and LAGEOS-II, respectively. Characteristics of similar geodetic satellites: the Japanese satellite Ajisai, launched on 2 August 1986, mass 685 kg, diameter 2.15 m, 1,436 triangular fused silica reflectors; the Soviet satellites Etalon-1 and 2, launched on 10 January 1989 and on 31 May 1989, identical, mass 1,415 kg, diameter 1.29 m, 2,146 hexagonal reflectors (2,140 fused silica, 6 germanium).

The International Time Bureau (Bureau International de l’Heure, or BIH) was set up in Paris at the beginning of the twentieth century (officially in 1912, but effectively in 1919) to centralise time determinations made around the world and hence to define a universal time scale. It is associated with the Paris Observatory. For these historical reasons, the official acronyms of the various time scales used by astronomers and physicists maintain the order of the words in the French name, e.g., TAI, TCB, etc., with the sole exception of UT. In 1988, the BIH founded the IERS (International Earth Rotation and Reference Systems Service) to monitor the parameters of the Earth’s rotation, while the organisation in charge of International Atomic Time (Temps Atomique International TAI) was transferred to the International Bureau of Weights and Measures (Bureau International des Poids et Mesures BIPM).

At the present time, there are 43 stations, almost all in the northern hemisphere.

The main satellites are as follows. Satellites for geodesy: LAGEOS-1 and 2, LARES, GFO, GFZ-1, Stella, Starlette, Wespac-1, Etalon-1 and 2, Ajisai, CHAMP, GRACE-A and B, GOCE; the Japanese satellite H2A-LRE in geosynchronous transfer orbit (GTO). Environmental and oceanographic satellites: TOPEX/Poseidon, Jason-1 and 2, ERS-1 and 2, Envisat, TerraSAR-X and TanDEM-X, ICESat, CryoSat-2, Meteor-2-21 (Fizeau), HY-2A. Navigation satellites: Navstar/GPS-35 and 36, the Japanese satellite QZS-1, practically all the Russian GLONASS satellites, the European satellites GIOVE-A and B and operational Galileo satellites; from Compass-M1, the Chinese satellites in the Compass-M series, Compass-I and Compass-G. Astronomical satellites: RadioAstron on a very high orbit. -M -I -G

The ratio of the number of photons received to the number of photons emitted, called the link budget, goes as r ⁻⁴, where r is the distance. When the target is the Moon, this ratio is very small indeed, in fact, just a few photons over the whole night! Only two stations have taken data over a long period: the McDonald Observatory in Texas and the OCA in Grasse, France.

The system has been or is still carried by the French satellites SPOT-2, 3, 4, and 5, Pléiades-1A and 1B, the French–Chinese HY-2, the French–US satellites TOPEX/Poseidon, Jason-1 and 2, the French-Indian SARAL, and the European satellites Envisat and CryoSat-2. It is also planned for the French–US Jason-3 and the European Sentinel-3-A.

Adrien Marie Legendre (1752–1833) was a French mathematician. He introduced the polynomials which are now named after him in his Recherches sur la figure des planètes (1784). When put in charge of geodetic measurements (the distance between the Paris and Greenwich meridians) by the revolutionary government known as the Convention, he made significant contributions to spherical trigonometry. He obtained new results in number theory, and also in the study of elliptic functions, the beta and gamma functions, and the Euler integrals. His work Eléments de Géométrie was reprinted thirteen times between 1794 and 1827.