Abstract
If the calculations of the motion are made in a Galilean frame (Earth-centered space-fixed), in most practical cases, one needs to know the position of the satellite relative to the Earth (Earth-centered Earth fixed). The satellite track is well defined by its geographical coordinates, longitude and latitude. We take this opportunity to make a brief statement on map projections.
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Notes
- 1.
The word “precession”, meaning “the action of preceding”, was coined by Copernicus around 1530 (præcessio, nis in Latin) to speak about the precession of the equinoxes, i.e., the retrograde motion of the equinoctial points. This term was then taken up in mechanics to describe the corresponding Euler angle. In the motion of the satellite orbital plane, the word “precession” clearly refers to a motion that may actually be prograde, as well as retrograde.
- 2.
The three Euler angles are traditionally denoted by ψ, θ, \(\varphi\), or by \(\varphi\), θ, ψ, in those orders. To avoid confusion with the latitudes denoted by ψ and \(\varphi\) elsewhere in the book, we have chosen to use the notation α i , i = 1, 2, 3.
- 3.
It is no accident that the word “month” is so similar to “Moon”, and the same is true of these words in German and related languages. The Indo-European root∗ men, ∗ mes refers to the Moon, lunation (lunar month), and measurement or mensuration (of time). Many languages in this family still use similar terms, but this is not the case in Greek or Latin. These two languages called the Moon “the bright one” (ἡ σελήνη, ης ; luna, æ), which gives the present French name “Lune”. See also the note on Chandrasekhar.
- 4.
The noun ἡ σύνοδος, ου, “synod”, is made up of σύν, meaning “with” or “together”, and ἡ ὁδός, ου, meaning “path” or “journey”. In Ancient Greek, it already had the double meaning of “meeting” and “conjunction of heavenly bodies”, both of which illustrate the idea of “things happening at the same time”.
- 5.
The expression for q G00 contains four terms:
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The first gives the position of the Greenwich meridian at the date taken as origin, viz., J2000.0.
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In the second, the coefficient of T u is equal to the number of seconds in one day (86,400) multiplied by the number of days in one Julian century (36,525), divided by the number of days in the tropical year (N tro = 365. 2421897).
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The third is related to nutation.
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The fourth accounts for the precession of the equinoxes.
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- 6.
Unlike the sphere, a cylinder is developable. If the body of a big cat could be assimilated to a cylinder, one could understand how the tiger might change into a bedside rug without deformation.
- 7.
Jean Henri Lambert (1728–1777) was a Swiss and German astronomer, mathematician, and physicist, with French ancestry. In astronomy, he calculated the trajectories of comets and understood that the Milky Way was just a modest galaxy in the Universe. In physics, he discovered the fundamental law of photometry. In his many mathematical works, among which he demonstrated the irrationality of π (1766), he attributed great importance to problems of perspective and cartographic projections. He defined a great many projections, several of which bear his name today. The best known is the conformal conical projection, used in France for the map of France since 1922 and the cadastral survey since 1938.
- 8.
The older literature is full of different adjectives describing these properties, e.g., autogonal and orthomorphic for conformal, and authalic, homolographic, equiareal, and equivalent for equal-area, while an aphylactic projection is one that is neither equal-area nor conformal. Such terms have now fallen into disuse.
- 9.
Certain cylindrical projections, such as the Arden–Close or Miller projections, are intended to “improve” Mercator, representing the poles by means of mathematical tricks. However, they thereby stray from the basic motivation for the Mercator projection, namely, the property of being conformal.
- 10.
Gerardus Mercator (1512–1594) was a Flemish mathematician and geographer. (At this time, university members commonly Latinised their names. In this case, Gerhard Kremer changed his name to Gerardus Mercator, since mercator means “merchant” in Latin, just as kremer does in Flemish.) He made globes, maps, and astronomical instruments for Charles Quint. Realising that accurate maps were not only useful for navigation but had strategic and commercial importance, he established the first conformal cylindrical projection and thereby founded modern mathematical cartography. His main work was the Atlas sive cosmographicæ meditationes de fabrica mundi et fabricati figura, a huge collection of maps, not all of which use the Mercator projection. The Atlas was expanded and re-edited many times. After his death, his son Rumold, then the geographer Jocodus Hondius, continued this work of cartography and edition. For his first collection of maps in 1583, Mercator chose the name “Atlas” and illustrated the frontispiece of the work with the picture in Fig. 8.7 of the Greek god Atlas, who holds up the sky and observes the Earth from above. Later, the word “atlas” would become the standard name for this type of geographical work.
- 11.
One first determines the shortest path between two points (orthodromy). This is then plotted on the Mercator map and approximated by a succession of straight-line segments (loxodromy). The Mercator map indicates the course to follow and hold, since the projection is conformal.
- 12.
We exemplify with two extracts to this effect:
Any projection inevitably distorts, none is completely innocent. The classic projection in our old school atlas is the Mercator projection, based on a factitious graticule which plots not only the “parallels” but also the meridians as parallel straight lines. The polar regions are stretched out of all proportion, while the “temperate” (= white) regions occupy a much greater space than their actual area would justify. The equator is pushed right down to the bottom of the map, which gives a completely false impression of the ratio of land to sea.
Jean Chesneaux, L’état du Monde 1982. Annuaire économique et géopolitique mondial, François Maspero éd., Paris 1982.
The distortions of Mercator’s map did not seem strange to Europeans in the sixteenth century, an era of expanding colonial empires. Yet today, although European colonialism belongs largely to the past, Mercator’s sixteenth-century map still retains much of its grip.
United Nations Development Programme. Maps and Map-Makers, UNESCO Courier, June 1991.
- 13.
This adjective comes from the Greek στερεός, meaning solid, and γράφειν, meaning to engrave or write. Ptolemy reports that the stereographic projection was established by Hipparchos of Nicaea.
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Capderou, M. (2014). Ground Track of a Satellite. In: Handbook of Satellite Orbits. Springer, Cham. https://doi.org/10.1007/978-3-319-03416-4_8
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DOI: https://doi.org/10.1007/978-3-319-03416-4_8
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