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u3a STEM Interest Group��#6, Navigation, its history, science and technology

The early history of navigation relates to the science of directing ships at sea through the establishment of latitude and course by means of geometry, astronomy and instruments such as the astrolabe (200 BC), sextant (John Bird, 1759) and compass (200 AD). Later, with the introduction of increasingly accurate clocks (Harrison’s H5 in 1770), it was possible to measure longitude as well. In the 20th century, radar (Loran, WWII) and later satellite systems such as Transit (1964) and GPS (1993) revolutionised navigation. Finally we will cover the use of latitude and longitude and other measures of location (e.g. Great Britain Grid References) in conjunction with the various styles and projections of terrestrial maps.

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Physical principles of navigation

The earth rotates at a constant speed
Its axis always points in the same direction
Force = mass x acceleration; force = GMm/r²
It rotates round the sun in a predictable ellipse but not at constant speed
The moon rotates round the earth in a predictable ellipse
Satellites rotate round their planets in predictable ellipses
The speed of electromagnetic waves is constant
Light travels in a straight line, distance = speed * time
Frequencies change with relative motion (Doppler effect)
The angles of a triangle add up to 180° (but not on a sphere)
Compasses align with a magnetic field
Plumb lines align with a gravitational field
Corrections to clocks due to relativistic effects

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Useful numbers

1 minute of longitude ≡ 1 nautical mile
earth is an ellipsoid, 1/f ≈ 300
earth's circumference = 40,075 km ≡ 360°
earth's rotational speed = 15° per hour or 460 m/s
speed of light ≈ 300,000 km/s
angular diameter of the Sun and Moon ≈ ½°
atomic clock frequency 9,192,631,770 Hz

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Navigation at sea

Voyages of Zheng He
1421, The Year China Discovered the World (allegedly)
Compasses, anciant and modern
Migration of magnetic pole
Triple axis compass magnetometer
Measuring distance and speed
Determination of Latitude - astrolabes and sextants
Local time - sun dials and the equation of time
Greenwich time - astronomical observations
Determination of Longitude

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Voyages of Zheng He, 1405 to 1433

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1421, The Year China Discovered the� World, Gavin Menzies�

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Dead reckoning�Compass

Han Dynasty (2nd century BC)

Silva compass (1932)

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Migration of magnetic pole�1590 to 2025

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Triple axis compass magnetometer�calibration

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Dead reckoning�Distance and speed

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Speed - pitot tube

Bernoulli equation

dynamic pressure = Pt - Ps

= .5 * density * V²

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Determination of Latitude�Astrolabe

��

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Local time�Sun dial and the equation of time

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Calculating south using �an analogue watch

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Celestial navigation - Lunar distance

moon moves 13.2° per day or ½° per hour

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Celestial navigation - Moons of Jupiter

Saturday, February 4, 2023

00:32 UT, Ganymede begins transit of Jupiter.

03:22 UT, Ganymede ends transit of Jupiter.

05:00 UT, Ganymede's shadow begins to cross Jupiter.

07:32 UT, Ganymede's shadow leaves Jupiter's disk.

16:10 UT, Io enters occultation behind Jupiter.

19:28 UT, Io exits eclipse by Jupiter's shadow.

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Nautical almanac

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Coordinated Universal Time (UTC)�Harrison's clocks H1 and H2

H1, 1735

H2, 1737 - 1739

https://en.wikipedia.org/wiki/Longitude_by_chronometer

Longitude by chronometer is a method, in navigation, of determining longitude using a marine chronometer, which was developed by John Harrison during the first half of the eighteenth century. It is an astronomical method of calculating the longitude at which a position line, drawn from a sight by sextant of any celestial body, crosses the observer's assumed latitude.[1] In order to calculate the position line, the time of the sight must be known so that the celestial position i.e. the Greenwich Hour Angle (Celestial Longitude - measured in a westerly direction from Greenwich) and Declination (Celestial Latitude - measured north or south of the equational or celestial equator), of the observed celestial body is known. All that can be derived from a single sight is a single position line, which can be achieved at any time during daylight when both the sea horizon and the sun are visible. To achieve a fix, more than one celestial body and the sea horizon must be visible. This is usually only possible at dawn and dusk.

The angle between the sea horizon and the celestial body is measured with a sextant and the time noted. The Sextant reading is known as the 'Sextant Altitude'. This is corrected by use of tables to a 'True Altitude'. The actual declination and hour angle of the celestial body are found from astronomical tables for the time of the measurement and together with the 'True Altitude' are put into a formula with the assumed latitude. This formula calculates the 'True Hour Angle' which is compared to the assumed longitude providing a correction to the assumed longitude. This correction is applied to the assumed position so that a position line can be drawn through the assumed latitude at the corrected longitude at 90° to the azimuth (bearing) on the celestial body. The observer's position is somewhere along the position line, not necessarily at the found longitude at the assumed latitude. If two or more sights or measurements are taken within a few minutes of each other a 'fix' can be obtained and the observer's position determined as the point where the position lines cross.

The azimuth (bearing) of the celestial body is also determined by use of astronomical tables and for which the time must also be known.

From this, it can be seen that a navigator will need to know the time very accurately so that the position of the observed celestial body is known just as accurately. The position of the sun is given in degrees and minutes north or south of the equational or celestial equator and east or west of Greenwich, established by the English as the Prime Meridian.

The desperate need for an accurate chronometer was finally met in the mid 18th century when an Englishman, John Harrison, produced a series of chronometers that culminated in his celebrated model H-4 that satisfied the requirements for a shipboard standard time-keeper.

Many nations, such as France, have proposed their own reference longitudes as a standard, although the world’s navigators have generally come to accept the reference longitudes tabulated by the British. The reference longitude adopted by the British became known as the Prime Meridian and is now accepted by most nations as the starting point for all longitude measurements. The Prime Meridian of zero degrees longitude runs along the meridian passing through the Royal Observatory at Greenwich, England. Longitude is measured east and west from the Prime Meridian. To determine "longitude by chronometer," a navigator requires a chronometer set to the local time at the Prime Meridian. Local time at the Prime Meridian has historically been called Greenwich Mean Time (GMT), but now, due to international sensitivities, has been renamed as Coordinated Universal Time (UTC), and is known colloquially as "zulu time".

Noon sight for Longitude

Noon sights obtain the observer's Latitude. It is impossible to determine longitude with an accuracy better than 10 nautical miles (19 km) by means of a noon sight. A noon sight is called a Meridian Altitude.[2] While it is very easy to determine the observer's latitude at noon without knowing the exact time, longitude cannot accurately be measured at noon. At noon the sun's change of altitude is very slow, so determining the exact time that the sun is at its highest by direct observation is impossible, and therefore it is impossible to obtain an accurate longitude at the moment of culmination. However, it is possible to determine the time of culmination for longitude with a useful accuracy by performing a mean time of observation when the sun is on its ascent and descent prior to and following its moment of culmination. By taking a sextant reading within 15 to 30 minutes prior to local noon (culmination) and noting the time, then leaving the sextant set to the same angle and subsequently observing the moment in time at which the sun passes through the sight tube on its descent from its highest altitude between a half-hour and hour later, the two times can be averaged to obtain a longitude sufficiently accurate for navigation (within 2 nautical miles [3.7 km]).[3]

Corrections to the process

Unfortunately, the Earth does not make a perfect circular orbit around the Sun. Due to the elliptical nature of the Earth’s orbit around the Sun, the speed of the Sun’s apparent orbit around the Earth varies throughout the year and that causes it to appear to speed up and slow down very slightly. Consequently, noon at the Prime Meridian is rarely if ever exactly at 12:00 UTC, but rather it occurs some minutes and seconds before or after that time each day. This slight daily variation has been calculated and is listed for each day of the year in the Nautical almanac[4] under the title of Equation of time. This variation must be added to or subtracted from the UTC of local apparent noon to improve the accuracy of the calculation. Even with that, other factors, including the difficulty of determining the exact moment of local apparent noon due to the flattening of the Sun’s arc across the sky at its highest point, diminish the accuracy of determining longitude by chronometer as a method of celestial navigation. Accuracies of less than 10 nautical miles (19 km) error in position are difficult to achieve using the "longitude by chronometer" method. Other celestial navigation methods involving more extensive use of both the Nautical almanac and sight reduction tables are used by navigators to achieve accuracies of 1 nautical mile (1.9 km) or less.

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Longitude�Harrison's clocks H3 and H5

H3, 1740 - 1759

H5, 1770

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�Longitude, Dava Sobel

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Radar and satellite systems

Direction finding
Inertial systems
Hyperbolic navigation - Loran and GEE
Transit satellite system
Global Positioning System
GPS Control Stations
GPS chip - EM-406A
GPS .gpx file

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Radio direction finding

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Inertial navigation system�(early 1950s)

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Hyperbolic navigation�Loran and GEE (WWII)

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Transit satellite system (1964)

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�GPS system

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GPS Control Stations��Global network of ground facilities that track the GPS� satellites, monitor their transmissions, perform analyses� and send commands and data to the constellation

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GPS signal properties

satellites simultaneously send several ranging signals used to measure the distance to the satellite
also ephemeris data used to calculate the position of each satellite in orbit
information about the time and status of the entire satellite constellation - the almanac
a unique serial number
signal frequencies of 1277 or 1575 MHz

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EM-406A GPS Module

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GPS navigation

Two satellites to determine position in 2 dimensions
Three satellites to determine position in 3 dimensions
Four satellites to correct for errors in GPS unit's quartz crystal clock
Five satellites to estimate a margin of error
>5 satellites to give an accurate fix (say 5 to 10m)
dual frequency GPS (2022) gives an accuracy of cms
Differential GPS gives an accuracy of cms

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�Example of .gpx file

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Mapping

Theodolites
Barometers
Triangulation and The Great Arc
Mercator projection
Universal Transverse Mercator Projection (UTM)
UTM in Europe
Ordnance Survey
2 Degrees West
OpenStreetMap projection

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Theodolites

��

Leonard Digges, 1551

Great Trigonometrical Survey

of India, 1802 - 1871

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Mercury in glass�and aneroid barometers

Evangelista Torreclli (1643)

Lucien Vidi (1843)

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�Triangulation and The Great Arc

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Gerardus Mercator (1512-1594)�

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Universal Transverse� Mercator Projection

A cylinder is wrapped around one of 30 lines longitude

The projection is conformal

scale is the same in both directions

shape of small features is preserved

features in high latitudes are enlarged

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Gnomonic projection�Thales, 600BC

gnomonic projection is from the centre of a sphere to a plane tangent to the sphere

great circles transform to straight lines

lines of longitude and the equator are great circles and they are always shown as straight

distortion of the scale increases from the centre to the periphery

the projection is not conformal

displays all great circles as straight lines

any straight line segment shows the shortest route between two endpoints

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Universal Transverse Mercator

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Location references

Latitude and Longitude (degrees)

52.951963,-0.953978

OS 6-figures(metres)

SK 70375 39910

Eastings and Northings (meters)

470375,339910

Post code

NG13 8AD

What Three Words (3 meters)

notifying.skewed.husbands

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Ordnance Survey map

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Grid north, true north and �magnetic north

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OpenStreetMap with GPS track