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Handbook of Satellite Orbits
We first explain the detailed calculation for determining the position and velocity of the GPS receiver. We then give the characteristics of different constellations of navigation satellites. We conclude with an appendix on GPS and relativity.
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The velocity of a Navstar/GPS satellite is 3.9 km/s in a Galilean frame. In a terrestrial frame, the relative velocity
\(\\boldsymbol{V }_{i}^{\mathrm{rel}}\\) is of this order, whence we find
\(V _{i}^{\mathrm{rel}}/c \sim 1{0}^{5}\).
Let
ρ′
_{ i } and
ρ
_{ i }
^{ ′ ′ } be the pseudoranges measured for the frequencies
f′ and
f
^{ ′ ′ }, respectively, and let
r
_{ i } be the geometric distance between the satellite
S
_{ i } and the receiver
P :
where
\(\Delta r_{i}\) is the sum of the delays due to causes other than the ionosphere and where
\(\delta r_{i}^{{\prime}}\) and
\(\delta r_{i}^{{\prime\prime}}\) are the delays due to crossing the ionosphere, which depend on the frequency. (These are generally between 1 and 40 m, depending on the time, the day, the location, and the viewing configurations.) It can be shown that these discrepancies
\(\delta r_{i}\) are proportional to
f
^{−2}. Therefore,
whence
By measuring
ρ′
_{ i } and
ρ
_{ i }
^{ ′ ′ }, and using the known ratio of the frequencies,
\(\delta r_{i}^{{\prime}}\) can be determined from
so that, with our notation,
\(\delta r_{i}^{{\prime}} = c\delta _{1,i}t\).
$$\displaystyle{\rho ^{\prime}_{i} = r_{i} +\delta r_{i}^{{\prime}} + \Delta r_{ i}\;,\qquad \rho _{i}^{{\prime\prime}} = r_{ i} +\delta r_{i}^{{\prime\prime}} + \Delta r_{ i}\;,}$$
$$\displaystyle{\delta r_{i}^{{\prime}} = \frac{A_{i}} {{f}^{{\prime}2}}\;,\quad \delta r_{i}^{{\prime\prime}} = \frac{A_{i}} {{f}^{{\prime\prime}2}}\;,}$$
$$\displaystyle{\delta r_{i}^{{\prime\prime}} ={ \left ( \frac{f^{\prime}} {{f}^{{\prime\prime}}}\right )}^{2}\delta r_{ i}^{{\prime}}\;.}$$
$$\displaystyle{\rho ^{\prime}_{i} \rho _{i}^{{\prime\prime}} =\delta r_{ i}^{{\prime}}\delta r_{ i}^{{\prime\prime}} =\delta r_{ i}^{{\prime}}\left [1 {\left ( \frac{f^{\prime}} {{f}^{{\prime\prime}}}\right )}^{2}\right ]\;,\qquad \delta r_{ i}^{{\prime}} = \frac{\rho ^{\prime}_{i} \rho _{i}^{{\prime\prime}}} {1  {(f^{\prime}/{f}^{{\prime\prime}})}^{2}}\;,}$$
When the “civilian” GPS signal was downgraded to deliberately limit positioning accuracy, DGPS provided a way to get around this jamming. Accuracy could be improved from 100 to 22 m.
In the USA, the NDGPS (Nationwide DGPS) uses a 100 or so base stations and covers almost the whole territory. The following system HANDGPS (High Accuracy NDGPS) should provide submeter positioning accuracy, i.e., of the order of a few decimeters. The system known as RTK (Real Time Kinetic) requires a costly base station to be set up close to the operating center, but provides accuracies of the order of 10 cm.
Navstar is run by the US Air Force, while Transit was managed by the US Navy.
The two experimental Time Navigation (Timation) satellites were part of a trial by the US Navy to find an alternative to Transit. These satellites followed a very specific orbit, with
h = 900 km,
i = 70. 0
^{∘}: Timation1, launched on 31 May 1967, Timation2, on 30 September 1969. The two Navstar Technology Satellites (NTS) of the US Air Force followed MEO orbits which prefigured the Navstar orbit: NTS1 (or Timation3, OPS/7518, P733), launched on 14 July 1974,
h = 13, 610 km,
i = 125. 2
^{∘}, NTS2 (P764), launched on 23 June 1977 on a neighbouring orbit to the one adopted for the first GPS missions, with
h = 20, 186 km,
i = 63. 9
^{∘}. NTS2 is discussed further in the appendix of Sect.
14.11.
List of Navstar/GPS satellites:

The 11 satellites of Block I, on the circular orbit h = 20, 020 km, i = 63. 0 ^{∘}, from Navstar1 (GPS1, OPS/5111), launched on 22 February 1978, to Navstar11 (GPS11, USA10) on 9 October 1985.

The 28 satellites of Block II and IIA, h = 20, 020 km, i = 55. 0 ^{∘}, from Navstar21 (GPS14, USA35), launched on 14 February 1989, to Navstar2A19 (GPS38, USA135) on 6 November 1997.

The 21 satellites of Block IIR and IIRM, on the same orbit, from Navstar2R2 (GPS43, USA132), launched on 23 July 1997, to Navstar2R21 (GPS50, USA206, Navstar2RM8) on 17 August 2009.

The 12 satellites of Block IIF, from Navstar2F1 (GPS62, USA213), launched on 28 May 2010, to Navstar2F12, for after 2015.

The eight satellites of Block IIIA, still on the same orbit, from Navstar3A1 to Navstar3A8, from 2014.
Block II: A (advanced), R (replenishment), RM (R modernized), F (followon).
The satellites of Block II and IIA are equipped with two cesium and two rubidium clocks, and the satellites of Block IIR with three rubidium clocks.
Colorado Springs has a monitor station as well as the master station. The other four monitor stations are on islands, distributed more or less equidistantly along the equator. Three of these stations, the ground antennas, are not on US territory and the American government has negotiated unassailable agreements to maintain their control of these highly strategic installations:

Ascension, a small volcanic island in the middle of the Atlantic, is a British Overseas Territory (Saint Helena, Ascension, Tristan da Cunha). The USA set up an airport and a large military base.

Diego Garcia, an atoll in the Chagos Archipelago, is another British territory, belonging to the British Indian Ocean Territory (BIOT). The US army leased the island in 1966 on an extendible 50 year lease, the condition being that the island be cleared of all its inhabitants at the outset. The British government duly deported 1,600 Chagossians to Mauritius and the Seychelles.

Kwajalein is an atoll which, like Bikini with its explosive reputation, belongs to the Republic of the Marshall Islands (RMI). This tiny country, formally independent since 1986, has a “free alliance” with the USA, which guarantees security and defence.

Hawaii is the only station on American territory, since this group of islands became the fiftieth state of the United States in 1959.
List of Glonass satellites:

The first 50 experimental then first generation satellites, from Uragan1 (Kosmos1413), launched on 12 October 1982, to Uragan50 (Kosmos2206), on 30 July 1992.

The 37 satellites of the Glonass series, from Glonass773 (Uragan51, Kosmos2234), launched on 17 February 1993, to Glonass798 (Uragan87, Kosmos2417), on 25 December 2005.

The satellites of the GlonassM series, from Glonass711 (UraganM1, Kosmos2382), launched on 1 December 2001, then Glonass736 to 738 (UraganM27 to M29, Kosmos2644 to 2466), launched on 2 September 2010.

The satellites of the new GlonassK series, beginning with GlonassK111 (UraganK11, Kosmos2471).
The Russian word
zemli means “the Earth”. It comes from the IndoEuropean root *
ghyōm, *
ghemō(n), of the same meaning, which also gives the Latin word
humus, i, meaning “earth” or “soil”. The Latin
homō, hominis comes from the same root. For example, the French word “homme” literally means “born of the earth”.
Launch dates: GIOVEA on 28 December 2005 and GIOVEB on 27 April 2008. These satellites, placed on the Galileo orbit, serve to test the clocks and emitted signal, as well as occupying the frequency bands from an administrative point of view in the eyes of the relevant international organisations. Originally named GSTBv2A and v2B (Galileo System Test Bed Version 2), they were renamed GIOVE (Galileo InOrbit Validation Element). Jupiter is
Giove in Italian, and this is therefore an affectionate allusion to Galileo, who discovered the four largest moons of this planet.
Launch dates: GalileoIOV1 and 2 on 21 October 2011, GalileoIOV3 and 4 on 12 October 2012.
Originally, a recurrence cycle of five revolutions in 3 days was adopted. This corresponds to the orbit denoted by Galileo [0] in Table
14.1. But this orbit was abandoned because the 5:3 recurrence creates resonance effects with the Earth’s gravitational field. In addition, the effects of the Sun and Moon create major instabilities in the Galileo constellation.
Bei Dou is the Chinese name for the constellation
Ursa Major, the Great Bear. In fact, in Chinese astronomy, there are 28 constellations, and they do not correspond exactly to the constellations of Western astronomy, which features many more than this.
Bei means “north” and
Dou is an instrument for measuring grain that looks like a large scoop, just as this constellation is sometimes referred to as the Saucepan or the Big Dipper, owing to its shape. Note that the Beidou satellites were also called Big Dipper at the beginning. The word “compass” comes from the Old French (fourteenth century)
compas, from the verb
compasser, meaning “to measure out in strides”. This in turn derives from the Latin verb composed of
cum, “with”, and
passus, “stride”. The English noun took its present meaning in the fifteenth century. A single idea brings together the Great Bear, which provides an easy way to identify the Pole Star in
Ursa Minor (the Little Bear), and the compass, an instrument used for orientation. That was before the advent of GPS, of course!
Launch dates: Beidou2M1 (CompassM1) on 13 April 2007, Beidou2M3 and M4 on 29 April 2012, Beidou2M2 and M5 on 18 September 2012.
Launch dates for BeiDouG (CompassG): Beidou2G1 on 16 January 2010, Beidou2G2 on 14 April 2009, Beidou2G3 on 2 June 2010, Beidou2G4 on 31 October 2010, Beidou2G5 on 24 February 2012, Beidou2G6 on 25 October 2012. Launch dates for BeiDouI (CompassI): Beidou2I1 on 31 July 2010, Beidou2I2 on 17 December 2010, Beidou2I3 on 9 April 2011, Beidou2I4 on 26 July 2011, Beidou2I5 on 1 December 2011. These satellites Beidou2I
n are also denoted by Beidou2IGSO
n.
It is surprising to find that Beidou1 uses such an antiquated reference system. And it is older than its name would suggest, since it is in fact a carbon copy of the Soviet model Krasovsky1940.
Launch dates: Beidou1A (BNTS1A, DFH51) on 30 October 2000, Beidou1B (BNTS1B, DFH52) on 20 December 2000, Beidou1C (BNTS1C, DFH56) on 24 May 2003.
This is an acronym with a particularly successful double meaning, since
gagan means “sky” in Sanskrit.
The apsidal precession rate is nevertheless very low, viz.,
\(\dot{\omega }= 0.0102{5}^{\circ }\)/day, or less than 4
^{∘} per year, which is easy to compensate.
Launch date: QZS1 on 11 September 2010. QZS1 also has the Japanese name
Michibiki, meaning “guide”.
The Transit satellites fall into five categories:

The five experimental satellites, from Transit1B, launched on 13 April 1960, to Transit4A, launched on 29 June 1961, on similar orbits, with i = 66. 7 ^{∘}, h _{p} = 625 km, h _{a} = 1, 080 km.

The ten prototype satellites in strictly polar orbit, from Transit5A1 on 18 December 1962 to Transit5C1 (OPS/4412) on 4 June 1964, with i = 90. 0 ^{∘}, h = 1, 100 km. Transit5B1, launched on 28 September 1963, was the first satellite actually used by the US Navy.

The 24 operational satellites in the Oscar series (O for “operational”), from TransitO1 (or NSS30010, OPS/5798) on 6 October 1964 to TransitO25 (NSS30250, NIMS25, SOOS4A) and TransitO31 (NSS30310, NIMS31, SOOS4B), launched on 25 August 1988. The satellite TransitO13 (NSS30130,OPS/7218) operated from September 1967 to January 1989.

The three Triad satellites of the Transit Improvement Program (TIP): Triad1 (or TIP1), launched on 2 September 1972, Triad2 (TIP2), on 11 October 1975, and Triad3 (TIP3), on 1 September 1976, still on strictly polar orbits. These demonstrated the feasibility of the dragfree technique.

The three Nova satellites, launched in this order: Nova1 (NSS30480) on 15 May 1981, Nova3 (NSS30500) on 12 October 1984, and Nova2 (NSS30490) on 16 June 1988, all on strictly polar orbits, with i = 90. 0 ^{∘}, h = 1, 180 km.
Even after the end of the Transit programme, certain satellites of the Navy Ionospheric Monitoring System (NIMS) continued to operate, providing data on signal transmission through the ionosphere which could be used by the Navstar/GPS system.
The Soviet system can be divided into three families of satellites:

The 99 military satellites Parus ( parus means “sail”), or TsikadaM (M for “military”), from Parus1 (Kosmos700), launched on 26 December 1974, to Parus99 (Kosmos 2463), on 27 April 2010.

Around 40 Tsikada satellites (this is the insect, the cicada), from Kosmos883, launched on 15 December 1976, to Kosmos2315, on 5 July 1995, on slightly elliptical nearpolar orbits, i = 83 ^{∘}, h _{p} = 960 km, h _{a} = 1, 020 km.

The 8 Nadezhda satellites (the word means “hope”), from Kosmos1383 on 29 June 1982, then Nadezhda1 on 4 July 1989 to Nadezhda7 (NadezhdaM1) on 26 September 2002.
The word “tectonic” was coined in Germany in 1850 from the Greek
tekton, ὁ τέκτων, ονος, meaning “carpenter”. This IndoEuropean root *
tek, “to produce”, appears in the Greek
technê, ἡ τέχνη, ης, “handicraft” (giving “technical”) and the Latin
textor, oris, “weaver”,
textus, us, “tissue”, then “text”. This root is also found textually in the scientific formatting software TE X , created by D. Knuth, and its successor LaTeX.
We wish to thank Pierre Sagnou, Stephen Lyle, and Richard Kerner for helpful discussions.
One also speaks of gravitational redshift, which simply reflects a change of perspective, swapping the emitter and the receiver.
Setting
ψ = 0 in the expansion of the geopotential [see (
3.45)], we obtain
$$\displaystyle{U_{\oplus } = \frac{\mu } {R}\left [1 \left (\frac{1} {2}J_{2} + \frac{3} {8}J_{4}  \frac{5} {16}J_{6} + \frac{35} {128}J_{8}  \frac{63} {256}J_{10}\right )\right ]\;.}$$
The coordinated time in the geocentric spacetime system (TCG,
Temps coordonnée géocentrique) differs from terrestrial time (TT) by a secular term:
where
\(\Delta D\) is the number of days elapsed since the date chosen as origin [see (
6.147)].
$$\displaystyle{\mbox{ TCG} \mbox{ TT} =\varTheta _{\oplus }\times \Delta D \times 86,400\,\mbox{ s}\;,}$$
Neil Ashby tells the following story:
At the time of launch of the NTS2 GPSprecursor satellite (23 June 1977), which contained the first cesium atomic clock to be placed in orbit, there were some who doubted that relativistic effects were real effects that had to be accounted for. A frequency synthesizer was built into the satellite clock system so that after launch, if in fact the rate of a clock in its final orbit were predicted by general relativity, then the synthesizer could be turned on, bringing the clock to the coordinate rate necessary for operation. After one of the cesium atomic clocks was turned on in NTS2, it was operated for about 20 days to measure its clock rate before turning on the synthesizer. The frequency measured during that period was + 442. 4 parts in 10 ^{12} compared to the ground, while relativity theory predicted + 446. 47 parts in 10 ^{12}. The discrepancy was only about four parts in 10 ^{12}, well within the accuracy capabilities of the orbiting clock.
The orbit of NTS2 was described in section “Main Dates”.
Ever since N. Ashby took
e = 0. 02 as an example, which gives
\(\Delta t = 46\) ns, all documents treating this question have reproduced this result of 46 ns. We note that the eccentricity
e never reaches such a value for navigation satellites placed in orbit since the historic period of the first Navstar/GPS satellite launches.
 Titel
 Global Positioning Systems (GPS)
 DOI
 https://doi.org/10.1007/9783319034164_14
 Autor:

Michel Capderou
 Sequenznummer
 14
 Kapitelnummer
 Chapter 14