7. Motion of Orbit, Earth and Sun

Over a few years, there is a slight precessional motion for these orbits, but less than a degree per year. This is due to other perturbations, such as the attraction of the Moon or Sun, radiation pressure, and so on.

To study the magnetosphere, NASA launched two satellites on 3 August 1981 in a single launch: Dynamics Explorer-1 and -2 (Explorer-62 and -64, also called DE-A and DE-B), the first in a high-altitude orbit (h _p = 468 km, h _a = 23, 322 km), the second in a low-altitude orbit (h _p = 304 km, h _a = 1, 002 km). To ensure that they can make joint observations of the same phenomena, it was essential that they move in the same orbital plane, and this was only possible by choosing a polar plane, i = 90^∘, since otherwise nodal precession would have between different for DE-A and DE-B.

This orbit, free of nodal precession, remains fixed in a Galilean frame and the satellite views in a direction orthogonal to the orbital plane. In fact, it views the same region continually for a period of 180 days, then turns round to view in the opposite direction, thus avoiding the Sun. The two regions viewed are at the intersection of the galactic plane and the celestial equator.

The satellite is equipped with gyroscopes and the aim is to measure effects predicted by general relativity: the geodesic effect (6,606 milliarcsec per year, or 1. 8 × 10⁻³ degree/year) and the Lense–Thirring effect, or frame dragging (39 mas/year, or 1. 1 × 10⁻⁵ degree/year). These two effects are maximally decoupled when the axis of rotation of the satellite on its orbit is orthogonal to the axis of rotation of the Earth, i.e., i = 90^∘.

The civil year aims to be as close as possible to the tropical year. The Julian (civil) year, introduced in 45 bc under the auspices of Julius Caesar, with one leap year every four years, has an average duration of It thus differs from the tropical year by 0.78 days per century. In 325 ad, the Council of Nicaea specified how to calculate the date of Easter given the date of the spring equinox, which was 21 March at this time (it had been 25 March at the time of Caesar). In 1582, the discrepancy was therefore and the equinox occurred on 11 March (Easter being calculated from 21 March). To bring the vernal equinox back to 21 March and keep it there, (a) 10 days had to be removed from the calendar and (b) the average length of the civil year had to be slightly modified. This is what was decreed by Pope Gregory XIII in the papal bull known as Inter gravissimas (Inter gravissimas pastoralis officii nostri curas …, “among the most noble tasks of our pastoral ministry …”), which gave the Gregorian calendar: (a) the day following Thursday 4 October will be Friday 15 October 1582; (b) 3 days will be suppressed every 400 years (years that are multiples of 100, but not 400, will not be leap years). The Gregorian (civil) year thus has an average duration of It differs from the tropical year by only 0.3 days per millennium. The Julian year was defined by \(N_{\mathrm{jul}} = N_{\mathrm{civ(j)}}\) in Sect. 6.​10.

If we compare these definitions of the year with the definitions of the different periods of a satellite discussed in Chap. 6, the anomalistic year corresponds to the anomalistic period T _a, while the tropical year corresponds to the nodal or draconitic period T _d. The draconitic year is defined in terms of the lunar motion.

The word “meridian” comes from the Latin adjective meridianus, meaning “relating to noon”, derived from meridies, ei, or “noon”. This noun is constructed from the locative form *mediei die, “in the middle of the day”. The remainder d - d transformed to r - d by a well-known linguistic process known as dissimilation.

The sidereal day is the time taken by the Earth, it is daily rotation, to return to a given direction which is not exactly fixed, since it follows the precessional motion of the equinoxes. The duration of the sidereal day is therefore or 23 h 56 min 04.0905 s. The duration of the stellar day is or 23 h 56 min 04.0989 s. We thus see that the sidereal day is related to the tropical year and the stellar day to the sidereal year.

The word polhodie was coined by the French mathematician L. Poinsot in 1851 from the Greek ὁ πόλος, ου, meaning “pivot” and ἡ ὁδός, ου, meaning “way”. In principle, a Greek word like hodos only retains its initial aspiration (rough breathing) when it combines with the previous letter to form an aspirated letter, e.g., in the words “anode” and “cathode”. One should write “polode”.

This quantity is perhaps more meaningful than \(\dot{\varOmega }\) expressed in radians per second. To avoid any confusion over units, we have used P for this quantity, expressed in round trips per year, whereas other quantities will be expressed in SI units, unless otherwise stated. The quantities P and ν, like ν _a and κ a little later on in our discussion, are ratios of angular frequencies, so they are indeed dimensionless numbers.

One commonly represents directions in space by means of points on a sphere with arbitrary center and radius, called the celestial sphere. With any particular direction, one associates the point of intersection of the celestial sphere and the straight line in that direction with origin at the center of the sphere.

The word “vernal” comes from the Latin vernalis, the adjective derived from ver, veris, meaning “spring”.

Keplerian elements with respect to the mean ecliptic and equinox of J2000.0, for Earth–Moon Barycentre [JPL-DE405]: Ω = 0, \(\omega = 102.93768 - 180 = -77.06232\), \(v_{\gamma } = -\omega \approx 7{7}^{\circ }\).

The old name of Greenwich Mean Time (GMT) is judged inappropriate by astronomers and has been out of use for several decades. It should be avoided, even though it still turns up in certain contexts.

To these irregularities in the apparent solar time correspond variations in the length of the apparent solar day, i.e., the time elapsed between two consecutive solar noons. This varies between 23 h 59 min 39 s and 24 h 00 min 30 s.

Joseph Juste Scaliger (1540–1609), the French scholar, proposed a new chronology in his De emendatione temporum (On the correction of time) in 1583. His idea was to produce a continuous count of the years in such a way as to cover all the great civilisations. He called this the Julian system, by analogy with the Julian calendar (introduced by Julius Caesar). The Julian numbering system, quoted by Kepler, was used by astronomers from 1860. They then added the idea of Julian day and Julian date. Scaliger considered the cycles involved in calculating the date of Easter, which was a major concern for astronomers in the Christian world:

A total of 2,400,000 days are removed from the Julian date, giving an origin on 17 November 1858, and the origin is taken at midnight rather than at midday.

In ℜ, the satellite is synchronous, whilst in ℜ _T, it is stationary. The word geosynchronous, meaning “synchronised with the Earth”, takes its origins from the Greek roots and is more satisfying than the word “geostationary”, which is a Greek–Latin hybrid.

Launch dates: Syncom-1 on 14 February 1961, Syncom-2 on 26 July 1963, and Syncom-3 on 19 August 1964.

Apart from two failures, for ATS-2 and -4, all the satellites were placed on slightly inclined orbits. Launch dates: ATS-1 on 7 December 1966, i = 14. 5^∘, remained operational for 18 years, until April 1985, ATS-3 on 5 November 1967, i = 14. 5^∘, ATS-5 on 12 August 1969, i = 14. 5^∘, and ATS-6 on 30 May 1974, i = 13. 1^∘.

The first in the series was Intelsat-1 F-1, also known as Early Bird, launched on 6 April 1965, i = 14. 7^∘ (stationed over the Atlantic to establish “fixed” telephone links between Europe and the United States). Since then the Intelsat satellites have been launched on a regular basis and placed over the Atlantic, Indian, and Pacific oceans. An intergovernmental consortium set up in 1964, Intelsat became a private company in 2001. After taking over PanAmSat in 2006, Intelsat has 55 satellites in operation at the time of writing (2013).

The speed of the transfer does have an energy cost. The faster one needs to go, the lower the transfer orbit should be. For each transfer maneuver of METEOSAT-5 in the above example, which required two burns, one on the starting orbit and one on the final orbit, EUMETSAT indicate that 300 g of fuel were burnt. The satellite was carrying 6 kg of propellant before the move.

For the satellite TDF-1, north–south control represents 95 % of its consumption. Launched in 1988, this satellite was held in position throughout its period of use. It was then placed in a graveyard orbit, where it was allowed to drift.

The three satellites Sirius-1, -2, and -3 (also called SD-Radio-1, -2, and -3), launched from Kazakhstan on 30 June, 5 September, and 30 November 2000, are on a geosynchronous orbit: e = 0. 2700, h _p = 24, 400 km, h _a = 47, 170 km. They are operational for North America between longitudes 60^∘W and 140^∘W, and broadcast paying music programmes for car radios. Between 2001 and 2006, this music-loving private operator sent the four satellites Rock and Roll, then Rythm and Blues, into geostationary orbit.

Satellites devoted to magnetospheric studies are often placed in elliptical Sun-synchronous orbits. Examples are MagSat, Ørsted, or the two German satellites Aeros-1 and -2. One should also mention those satellites whose orbits, originally intended to be circular, have become elliptical owing to launch errors, e.g., Nimbus-1, mentioned below.

Here we use HS as subscript for Sun-synchronous. It refers to the occasionally heard “heliosynchronous”, a word made up only of Greek roots, hence more satisfying from a linguistic standpoint.

Books are classified in sections according to the main themes covered in this work, and arranged chronologically within each section.