Longitude ( or ), identified by the
Greek letter lambda
(λ), is the
geographic
coordinate most commonly used in cartography and global
navigation for eastwest measurement.
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
eastwest 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 northsouth
angle between the local vertical and the plane of the
equator.
History
Amerigo Vespucci's means of
determining longitude
Historically, the measurement of longitude was important both to
cartography and to provide safe ocean
navigation. 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.
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 eastwest 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 twentythird 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.
John Harrison, a
selfeducated English clockmaker then invented the marine chronometer, a key piece in
solving the problem of accurately establishing longitude at sea,
thus revolutionizing 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), "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 is used to denote the location of
a place on Earth east or west of the prime meridian.
Each degree of longitude is subdivided into 60
minutes, each of which divided into 60
second. 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 righthanded
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.
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(o\!\varepsilon)^2}\;\! is the
first eccentricity squared,
{e^2}\;\!; and \scriptstyle{2\sin(\frac{o\!\varepsilon}{2})^2}\;\!
or \scriptstyle{1\cos(o\!\varepsilon)}\;\! is the
flattening, {f}\;\!). Utilized in creating the
integrand 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(o\!\varepsilon)^2n'(\phi)^3=\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
northsouth 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 eastwest longitude difference,
\scriptstyle{\cos(\phi)\Delta\lambda}\;\!, is about the same at the
equator as the northsouth, reducing to zero at the poles.
In the case of a spheroid, a
meridian and its antimeridian 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\;\!.
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 eastwest 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 (eastwest), N\;\! equals the equatorial radius.
The radius of curvature at a
right angle
to the equator (northsouth), 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 northsouth 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
northsouth 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 eastwest radius of curvature reaches the
axis, whereas the northsouth 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. 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. 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
GeospatialIntelligence Agency (NGA).
Latitude 
NS radius
of curvature
M\;\!

Surface distance
per 1° change
in latitude


EW 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
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 not parallel to
the plane of the celestial and terrestrial equator: it is the plane
of the ecliptic. This is inclined to the equator by the "
obliquity of the ecliptic",
currently 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 coordinate 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 agreedupon
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, nonorbiting 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 counterclockwise 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:
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 Airy0
crater.
Tidallylocked bodies have a natural
reference longitude passing through the point nearest to their
parent body. However,
libration due to
noncircular orbits or axial tilts causes this point to move around
any fixed point on the celestial body like an
analemma.
See also
Notes
 Oxford English Dictionary
 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. 7690. Page 80.
 Coordinate Conversion
 The Math Forum
 John P. Snyder, Map
Projections—A Working Manual (1987) 2425
 NIMA TR8350.2 page 31.
 IERS Conventions (2003) (Chp. 1, page 12)
 Length of degree calculator  National
GeospatialIntelligence Agency
 Where is zero degrees longitude on Mars?
 First map of extraterrestial planet.
External links
 Resources for determining your latitude and
longitude
 Worldwide Index
 Tageo.com – contains 2,700,000 coordinates of places
including US towns
 for each city it gives the satellite map location, country,
province, coordinates (dd,dms), variant names and nearby
places.
 IAU/IAG Working Group On Cartographic Coordinates and
Rotational Elements of the Planets and Satellites
 Average Latitude & Longitude of
Countries
 "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.
 How to find and convert Latitude &
Longitude