# Longitude

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Longitude (pronounced /ˈlɒndʒɨtju˝d/ or /ˈlɒŋɡɨtju˝d/),[1] identified by the Greek letter lambda (λ), is the geographic coordinate most commonly used in cartography and global navigation for east-west measurement. It is the angular distance measured east or west and usually expressed in degrees (or hours), minutes, and seconds, from the prime meridian, defined to be at the Royal Observatory, Greenwich, in England, to the meridian passing through another position on the earth's surface. A location's position along a meridian is given by its latitude. This is the angular distance of that place north or south of the equator, measured as an angle whose vertex is at the center of the earth.

## History

Mariners and explorers for most of history struggled to determine precise longitude. 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 labors 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.[2]

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.

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),[3][4] 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.

## 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.[5] 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, $o\!\varepsilon\,\!$ (which equals $\scriptstyle{\arccos(\frac{b}{a})}\,\!$, where $a\;\!$ and $b\;\!$ are the equatorial and polar radii; $\scriptstyle{\sin^2(o\!\varepsilon)}\;\!$ is the first eccentricity squared, ${e^2}\;\!$; and $\scriptstyle{2\sin^2(\frac{o\!\varepsilon}{2})}\;\!$ or $\scriptstyle{1-\cos(o\!\varepsilon)}\;\!$ is the flattening, ${f}\;\!$). Utilized in creating the integrands for curvature is the inverse of the principal elliptic integrand, $E'\;\!$:

$n'(\phi)=\frac{1}{E'(\phi)}=\frac{1}{\sqrt{1-\big(\sin(\phi)\sin(o\!\varepsilon)\big)^2}}\,\!$
\begin{align}M(\phi)&=a\cdot\cos^2(o\!\varepsilon)n'^3(\phi)=\frac{(ab)^2}{\Big((a\cos(\phi))^2+(b\sin(\phi))^2\Big)^{3/2}}\\ N(\phi)&=a{\cdot}n'(\phi)=\frac{a^2}{\sqrt{(a\cos(\phi))^2+(b\sin(\phi))^2}}\end{align}\,\!

## 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. The length of an arcdegree of east-west longitude difference, $\scriptstyle{\cos(\phi)\Delta\lambda}\;\!$, 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 exact expression for the length of an arcdegree of latitude is:

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

This radius of arc (or "arcradius") is in the plane of a meridian, and is known as the meridional radius of curvature, $M\;\!$.[6][7]

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

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

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, $N\;\!$.[6][7]

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 1 km less than the length of an arcdegree of longitude at the equator. The radii of curvature 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 6378137.0 m and an inverse flattening, (1/f), of 298.257223563, hence its polar radius is 6356752.3142 m and its first eccentricity squared is 0.00669437999014.[8] The more recent but little used IERS 2003 ellipsoid provides equatorial and polar radii of 6378136.6 and 6356751.9 m, respectively, and an inverse flattening of 298.25642.[9] 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).[10]

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
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

Ecliptic latitude and longitude are defined for the planets, stars, and other celestial bodies in a similar way to that in which the terrestrial counterparts are defined. The pole is the normal to the ecliptic nearest to the celestial north pole. 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.[11]

Tidally-locked bodies have a natural reference longitude passing through the point nearest to their parent body.[12] 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.

## Notes

1. Oxford English Dictionary
2. Vespucci, Amerigo. "Letter from Seville to Lorenzo di Pier Francesco de' Medici, 1500." Pohl, Frederick J. Amerigo Vespucci: Pilot Major. New York: Columbia University Press, 1945. 76-90. Page 80.
3. "λ = Longitude east of Greenwich (for longitude west of Greenwich, use a minus sign)."
John P. Snyder, Map Projections, A Working Manual, USGS Professional Paper 1395, page ix
4. Read HH, Watson Janet (1975). Introduction to Geology. New York: Halsted. 13–15.
5. The Math Forum
6. John P. Snyder, Map Projections—A Working Manual (1987) 24-25
7. NIMA TR8350.2 page 3-1.
8. IERS Conventions (2003) (Chp. 1, page 12)
9. Length of degree calculator - National Geospatial-Intelligence Agency
10. Where is zero degrees longitude on Mars?
11. First map of extraterrestial planet.