Longitude

For Dava Sobel's book about John Harrison, see Longitude (book). For the adaptation of Sobel's book, see Longitude (TV series).

Longitude is the angular distance of a point's meridian from the Prime (Greenwich) Meridian. It is usually expressed in degrees, minutes, and seconds. Lines of longitude are often referred to as meridians (Template:Pron-en or /ˈlɒŋɡɨtjuːd/), identified by the Greek letter lambda (λ), is the geographic coordinate most commonly used in cartography and global navigation for east-west measurement. Constant longitude is represented by lines running from north to south. The line of longitude (meridian) that passes through the Royal Observatory, Greenwich, in England, establishes the meaning of zero degrees of longitude, or the Prime Meridian. Any other longitude is identified by the east-west angle, referenced to the center of the Earth as vertex, between the intersections with the Equator of the meridian through the location in question and the Prime Meridian. A location's position along a meridian is given by its latitude, which is identified by the north-south angle between the local vertical and the plane of the Equator.

History

Main article: History of longitude

Amerigo Vespucci's means of determining longitude

The measurement of longitude is important both to cartography and to provide safe ocean navigation. Mariners and explorers for most of history struggled to determine precise longitude. Finding a method of determining exact longitude took centuries, resulting in the history of longitude recording the effort of some of the greatest scientific minds.

Latitude was calculated by observing with quadrant or astrolabe the inclination of the sun or of charted stars, but longitude presented no such manifest means of study. Amerigo Vespucci was perhaps the first to proffer a solution, after devoting a great deal of time and energy studying the problem during his sojourns in the New World:

As to longitude, I declare that I found so much difficulty in determining it that I was put to great pains to ascertain the east-west distance I had covered. The final result of my labours was that I found nothing better to do than to watch for and take observations at night of the conjunction of one planet with another, and especially of the conjunction of the moon with the other planets, because the moon is swifter in her course than any other planet. I compared my observations with an almanac. After I had made experiments many nights, one night, the twenty-third of August, 1499, there was a conjunction of the moon with Mars, which according to the almanac was to occur at midnight or a half hour before. I found that...at midnight Mars's position was three and a half degrees to the east.

By comparing the relative positions of the moon and Mars with their anticipated positions, Vespucci was able to crudely deduce his longitude. But this method had several limitations: First, it required the occurrence of a specific astronomical event (in this case, Mars passing through the same right ascension as the moon), and the observer needed to anticipate this event via an astronomical almanac. One needed also to know the precise time, which was difficult to ascertain in foreign lands. Finally, it required a stable viewing platform, rendering the technique useless on the rolling deck of a ship at sea.

In 1612, Galileo Galilei proposed that with sufficiently accurate knowledge of the orbits of the moons of Jupiter one could use their positions as a universal clock and this would make possible the determination of longitude, but the practical problems of the method he devised were severe and it was never used at sea. In 1714, motivated by a number of maritime disasters attributable to serious errors in reckoning position at sea, the British government established the Board of Longitude: prizes were to be awarded to the first person to demonstrate a practical method for determining the longitude of a ship at sea. These prizes motivated many to search for a solution.

Drawing of Earth with Longitudes

John Harrison, a self-educated English clockmaker then invented the marine chronometer, a key piece in solving the problem of accurately establishing longitude at sea, thus revolutionising and extending the possibility of safe long distance sea travel. Though the British rewarded John Harrison for his marine chronometer in 1773, chronometers remained very expensive and the lunar distance method continued to be used for decades. Finally, the combination of the availability of marine chronometers and wireless telegraph time signals put an end to the use of lunars in the 20th century.

Unlike latitude, which has the equator as a natural starting position, there is no natural starting position for longitude. Therefore, a reference meridian had to be chosen. It was a popular practice to use a nation's capital as the starting point, but other significant locations were also used. While British cartographers had long used the Greenwich meridian in London, other references were used elsewhere, including: El Hierro, Rome, Copenhagen, Jerusalem, Saint Petersburg, Pisa, Paris, Philadelphia, and Washington. In 1884, the International Meridian Conference adopted the Greenwich meridian as the universal Prime Meridian or zero point of longitude.

Noting and calculating longitude

Longitude is given as an angular measurement ranging from 0° at the Prime Meridian to +180° eastward and −180° westward. The Greek letter λ (lambda), is used to denote the location of a place on Earth east or west of the Prime Meridian.

Each degree of longitude is sub-divided into 60 minutes, each of which divided into 60 seconds. A longitude is thus specified in sexagesimal notation as 23° 27′ 30" E. For higher precision, the seconds are specified with a decimal fraction. An alternative representation uses degrees and minutes, where parts of a minute are expressed in decimal notation with a fraction, thus: 23° 27.500′ E. Degrees may also be expressed as a decimal fraction: 23.45833° E. For calculations, the angular measure may be converted to radians, so longitude may also be expressed in this manner as a signed fraction of π (pi), or an unsigned fraction of 2π.

For calculations, the West/East suffix is replaced by a negative sign in the western hemisphere. Confusingly, the convention of negative for East is also sometimes seen. The preferred convention—that East be positive—is consistent with a right-handed Cartesian coordinate system with the North Pole up. A specific longitude may then be combined with a specific latitude (usually positive in the northern hemisphere) to give a precise position on the Earth's surface.

Longitude at a point may be determined by calculating the time difference between that at its location and Coordinated Universal Time (UTC). Since there are 24 hours in a day and 360 degrees in a circle, the sun moves across the sky at a rate of 15 degrees per hour (360°/24 hours = 15° per hour). So if the time zone a person is in is three hours ahead of UTC then that person is near 45° longitude (3 hours × 15° per hour = 45°). The word near was used because the point might not be at the center of the time zone; also the time zones are defined politically, so their centers and boundaries often do not lie on meridians at multiples of 15°. In order to perform this calculation, however, a person needs to have a chronometer (watch) set to UTC and needs to determine local time by solar observation or astronomical observation. The details are more complex than described here: see the articles on Universal Time and on the equation of time for more details.

Singularity and discontinuity of longitude

Note that the longitude is singular at the Poles and calculations that are sufficiently accurate for other positions, may be inaccurate at or near the Poles. Also the discontinuity at the ±180° meridian must be handled with care in calculations. An example is a calculation of east displacement by subtracting two longitudes, which gives wrong answer if the two positions are on either side of this meridian. To avoid these complexities, consider replacing latitude and longitude with another horizontal position representation in calculations.

Plate movement and longitude

The surface layer of the Earth, the lithosphere, is broken up into several tectonic plates. Each plate moves in a different direction, at speeds of about 50 to 100 mm per year. As a result, for example, the longitudinal difference between a point on the Equator in Uganda (on the African Plate) and a point on the Equator in Ecuador (on the South American Plate) is increasing by about 0.0014 arcseconds per year.

If a global reference frame such as WGS84 is used, the longitude of a place on the surface will change from year to year. To minimize this change, when dealing exclusively with points on a single plate, a different reference frame can be used, whose coordinates are fixed to a particular plate, such as NAD83 for North America or ETRS89 for Europe.

Elliptic parameters

Because most planets (including Earth) are closer to ellipsoids of revolution, or spheroids, rather than to spheres, both the radius and the length of arc varies with latitude. This variation requires the introduction of elliptic parameters based on an ellipse's angular eccentricity, $\alpha \,\!{\big (}$which equals $\scriptstyle {\arccos \left({\frac {b}{a}}\right)}\,\!$, where $a\;\!$ and $b\;\!$ are the equatorial and polar radii; $\scriptstyle {\sin ^{2}(\alpha )}\;\!$ is the first eccentricity squared, ${e^{2}}\;\!$; and $\scriptstyle {2\sin ^{2}\left({\frac {\alpha }{2}}\right)}\;\!$ or $\scriptstyle {1-\cos(\alpha )}\;\!$ is the flattening, ${f}\;\!{\big )}$. Utilized in creating the integrands for curvature is the inverse of the principal elliptic integrand, $E'\;\!$:

$n'(\phi )={\frac {1}{E'(\phi )}}={\frac {1}{\sqrt {1-\left(\sin(\phi )\sin(\alpha )\right)^{2}}}};\,\!$

${\begin{aligned}M(\phi )&=a\cdot \cos ^{2}(\alpha )n'^{3}(\phi )={\frac {(ab)^{2}}{\left(\left(a\cos(\phi )\right)^{2}+\left(b\sin(\phi )\right)^{2}\right)^{3/2}}};\\N(\phi )&=a{\cdot }n'(\phi )={\frac {a^{2}}{\sqrt {\left(a\cos(\phi )\right)^{2}+\left(b\sin(\phi )\right)^{2}}}}.\end{aligned}}\,\!$

Degree length

The length of an arcdegree of north-south latitude difference, $\scriptstyle {\Delta \phi }\;\!$, is about 60 nautical miles, 111 kilometres or 69 statute miles at any latitude; more exactly, a degree of latitude at the pole covers about 1 percent more distance than a degree at the equator . The length of an arcdegree of east-west longitude difference, roughly $\scriptstyle {\cos(\phi )\Delta \lambda }\;\!$, is about the same at the Equator as the north-south, reducing to zero at the poles.

In the case of a spheroid, a meridian and its anti-meridian form an ellipse, from which an expression for the length of an arcdegree of latitude is:

${\frac {\pi }{180}}M(\phi ).\;\!$

This radius of arc (or "arcradius") is in the plane of a meridian, and is known as the meridional radius of curvature, $M\;\!$.

Similarly, an expression for the length of an arcdegree of longitude is:

${\frac {\pi }{180}}\cos(\phi )N(\phi ).\;\!$

The arcradius contained here is in the plane of the prime vertical, the east-west plane perpendicular (or "normal") to both the plane of the meridian and the plane tangent to the surface of the ellipsoid, and is known as the normal radius of curvature, $N\;\!$.

Along the Equator (east-west), $N\;\!$ equals the equatorial radius. The radius of curvature at a right angle to the Equator (north-south), $M\;\!$, is 43 km shorter, hence the length of an arcdegree of latitude at the Equator is about 0.7 km less than the length of an arcdegree of longitude at the Equator. $N\;\!$ and $M\;\!$ are equal at the poles, where they are about 64 km greater than the north-south equatorial radius of curvature because the polar "radius" is 21 km less than the equatorial radius. The shorter polar "radii" indicate that the northern and southern hemispheres are flatter, making their radii of curvature longer. This flattening also 'pinches' the north-south equatorial radius of curvature, making it 43 km less than the equatorial radius. Both radii of curvature are perpendicular to the plane tangent to the surface of the ellipsoid at all latitudes, directed toward a point on the polar axis in the opposite hemisphere (except at the Equator where both point toward Earth's center). The east-west radius of curvature reaches the axis, whereas the north-south radius of curvature is shorter at all latitudes except the poles.

The WGS84 ellipsoid, used by all GPS devices, uses an equatorial radius of 6378137.0m and an inverse flattening, (1/f), of 298.257223563, hence its polar radius is 6356752.3142m and its first eccentricity squared is 0.00669437999014. The more recent but little used IERS 2003 ellipsoid provides equatorial and polar radii of 6378136.6 and 6356751.9m, respectively, and an inverse flattening of 298.25642. Lengths of degrees on the WGS84 and IERS 2003 ellipsoids are the same when rounded to six significant digits. An appropriate calculator for any latitude is provided by the U.S. government's National Geospatial-Intelligence Agency (NGA).

Latitude	N-S radius of curvature $M\;\!$	Surface distance per 1° change in latitude		E-W radius of curvature $N\;\!$	Surface distance per 1° change in longitude
0°	6335.44 km	110.574 km		6378.14 km	111.320 km
15°	6339.70 km	110.649 km		6379.57 km	107.551 km
30°	6351.38 km	110.852 km		6383.48 km	96.486 km
45°	6367.38 km	111.132 km		6388.84 km	78.847 km
60°	6383.45 km	111.412 km		6394.21 km	55.800 km
75°	6395.26 km	111.618 km		6398.15 km	28.902 km
90°	6399.59 km	111.694 km		6399.59 km	0.000 km

Ecliptic latitude and longitude

Main article: Ecliptic coordinate system

Ecliptic latitude and longitude are defined for the planets, stars, and other celestial bodies in a broadly similar way to that in which terrestrial latitude and longitude are defined, but there is a special difference.

The plane of zero latitude for celestial objects is the plane of the ecliptic and is not parallel to the plane of the celestial and terrestrial equator. This is inclined to the Equator by the obliquity of the ecliptic, which currently has a value of about 23° 26'. The closest celestial counterpart to terrestrial latitude is declination, and the closest celestial counterpart to terrestrial longitude is right ascension. These celestial coordinates bear the same relationship to the celestial equator as terrestrial latitude and longitude do to the terrestrial equator, and they are also more frequently used in astronomy than celestial longitude and latitude.

The polar axis (relative to the celestial equator) is perpendicular to the plane of the Equator, and parallel to the terrestrial polar axis. But the (north) pole of the ecliptic, relevant to the definition of ecliptic latitude, is the normal to the ecliptic plane nearest to the direction of the celestial north pole of the Equator, i.e. 23° 26' away from it.

Ecliptic latitude is measured from 0° to 90° north (+) or south (−) of the ecliptic. Ecliptic longitude is measured from 0° to 360° eastward (the direction that the Sun appears to move relative to the stars), along the ecliptic from the vernal equinox. The equinox at a specific date and time is a fixed equinox, such as that in the J2000 reference frame.

However, the equinox moves because it is the intersection of two planes, both of which move. The ecliptic is relatively stationary, wobbling within a 4° diameter circle relative to the fixed stars over millions of years under the gravitational influence of the other planets. The greatest movement is a relatively rapid gyration of Earth's equatorial plane whose pole traces a 47° diameter circle caused by the Moon. This causes the equinox to precess westward along the ecliptic about 50" per year. This moving equinox is called the equinox of date. Ecliptic longitude relative to a moving equinox is used whenever the positions of the Sun, Moon, planets, or stars at dates other than that of a fixed equinox is important, as in calendars, astrology, or celestial mechanics. The 'error' of the Julian or Gregorian calendar is always relative to a moving equinox. The years, months, and days of the Chinese calendar all depend on the ecliptic longitudes of date of the Sun and Moon. The 30° zodiacal segments used in astrology are also relative to a moving equinox. Celestial mechanics (here restricted to the motion of solar system bodies) uses both a fixed and moving equinox. Sometimes in the study of Milankovitch cycles, the invariable plane of the solar system is substituted for the moving ecliptic. Longitude may be denominated from 0 to ${\begin{matrix}2\pi \end{matrix}}$ radians in either case.

Longitude on bodies other than Earth

Planetary co-ordinate systems are defined relative to their mean axis of rotation and various definitions of longitude depending on the body. The longitude systems of most of those bodies with observable rigid surfaces have been defined by references to a surface feature such as a crater. The north pole is that pole of rotation that lies on the north side of the invariable plane of the solar system (near the ecliptic). The location of the Prime Meridian as well as the position of body's north pole on the celestial sphere may vary with time due to precession of the axis of rotation of the planet (or satellite). If the position angle of the body's Prime Meridian increases with time, the body has a direct (or prograde) rotation; otherwise the rotation is said to be retrograde.

In the absence of other information, the axis of rotation is assumed to be normal to the mean orbital plane; Mercury and most of the satellites are in this category. For many of the satellites, it is assumed that the rotation rate is equal to the mean orbital period. In the case of the giant planets, since their surface features are constantly changing and moving at various rates, the rotation of their magnetic fields is used as a reference instead. In the case of the Sun, even this criterion fails (because its magnetosphere is very complex and does not really rotate in a steady fashion), and an agreed-upon value for the rotation of its equator is used instead.

For planetographic longitude, west longitudes (i.e., longitudes measured positively to the west) are used when the rotation is prograde, and east longitudes (i.e., longitudes measured positively to the east) when the rotation is retrograde. In simpler terms, imagine a distant, non-orbiting observer viewing a planet as it rotates. Also suppose that this observer is within the plane of the planet's equator. A point on the Equator that passes directly in front of this observer later in time has a higher planetographic longitude than a point that did so earlier in time.

However, planetocentric longitude is always measured positively to the east, regardless of which way the planet rotates. East is defined as the counter-clockwise direction around the planet, as seen from above its north pole, and the north pole is whichever pole more closely aligns with the Earth's north pole. Longitudes traditionally have been written using "E" or "W" instead of "+" or "−" to indicate this polarity. For example, the following all mean the same thing:

• −91°
• 91°W
• +269°
• 269°E.

The reference surfaces for some planets (such as Earth and Mars) are ellipsoids of revolution for which the equatorial radius is larger than the polar radius; in other words, they are oblate spheroids. Smaller bodies (Io, Mimas, etc.) tend to be better approximated by triaxial ellipsoids; however, triaxial ellipsoids would render many computations more complicated, especially those related to map projections. Many projections would lose their elegant and popular properties. For this reason spherical reference surfaces are frequently used in mapping programs.

The modern standard for maps of Mars (since about 2002) is to use planetocentric coordinates. The meridian of Mars is located at Airy-0 crater.

Tidally-locked bodies have a natural reference longitude passing through the point nearest to their parent body: 0° the center of the primary-facing hemisphere, 90° the center of the leading hemisphere, 180° the center of the anti-primary hemisphere, and 270° the center of the trailing hemisphere. However, libration due to non-circular orbits or axial tilts causes this point to move around any fixed point on the celestial body like an analemma.

See also

• American Practical Navigator
• Cardinal direction
• Geodesy
• Geodetic system
• Geographic coordinate system
• Geographical distance
• Geotagging
• Great-circle distance
• Horse latitudes
• Latitude
• List of cities by latitude
• List of cities by longitude
• Meridian arc
• Natural Area Code
• Navigation
• Orders of magnitude
• World Geodetic System

External links

• Resources for determining your latitude and longitude
• IAU/IAG Working Group On Cartographic Coordinates and Rotational Elements of the Planets and Satellites
• "Longitude forged": an essay exposing a hoax solution to the problem of calculating longitude, undetected in Dava Sobel's Longitude, from TLS, November 12, 2008.