The azimuth is the angle formed between a reference direction (in this example north) and a line from the observer to a point of interest projected on the same plane as the reference direction orthogonal to the zenith.
When used as a celestial coordinate, the azimuth is the horizontal direction of a star or other astronomical object in the sky. The star is the point of interest, the reference plane is the local area (e.g. a circular area 5 km in radius at sea level) around an observer on Earth's surface, and the reference vector points to true north. The azimuth is the angle between the north vector and the star's vector on the horizontal plane.[1]
The word azimuth is in all European languages today. It originates from medieval Arabic al-sumūt, pronounced as-sumūt in Arabic, meaning "the directions" (plural of Arabic al-samt
= "the direction"). The Arabic word entered late medieval Latin in an
astronomy context and in particular in the use of the Arabic version of
the astrolabe astronomy instrument. The word's first record in English is in the 1390s in Treatise on the Astrolabe by Geoffrey Chaucer.
The first known record in any Western language is in Spanish in the
1270s in an astronomy book that was largely derived from Arabic sources,
the Libros del saber de astronomía commissioned by King Alfonso X of Castile.[2]
In navigation
Azimuth Marker, Mount Allen (Sandstone Peak), Southern California, USA.
In land navigation, azimuth is usually denoted alpha, α, and defined as a horizontal angle measured clockwise from a north base line or meridian.[3][4]Azimuth
has also been more generally defined as a horizontal angle measured
clockwise from any fixed reference plane or easily established base
direction line.[5][6][7]
Today, the reference plane for an azimuth is typically true north, measured as a 0° azimuth, though other angular units (grad, mil)
can be used. Moving clockwise on a 360 degree circle, east has azimuth
90°, south 180°, and west 270°. There are exceptions: some navigation
systems use south as the reference vector. Any direction can be the
reference vector, as long as it is clearly defined.
Quite commonly, azimuths or compass bearings are stated in a
system in which either north or south can be the zero, and the angle may
be measured clockwise or anticlockwise from the zero. For example, a
bearing might be described as "(from) south, (turn) thirty degrees
(toward the) east" (the words in brackets are usually omitted),
abbreviated "S30°E", which is the bearing 30 degrees in the eastward
direction from south, i.e. the bearing 150 degrees clockwise from north.
The reference direction, stated first, is always north or south, and
the turning direction, stated last, is east or west. The directions are
chosen so that the angle, stated between them, is positive, between zero
and 90 degrees. If the bearing happens to be exactly in the direction
of one of the cardinal points, a different notation, e.g. "due east", is used instead.
We are standing at latitude , longitude zero; we want to find the azimuth from our viewpoint to Point 2 at latitude , longitude L (positive eastward). We can get a fair approximation by assuming the Earth is a sphere, in which case the azimuth α is given by
A better approximation assumes the Earth is a slightly-squashed sphere (an oblate spheroid); azimuth then has at least two very slightly different meanings. Normal-section azimuth is the angle measured at our viewpoint by a theodolite whose axis is perpendicular to the surface of the spheroid; geodetic azimuth is the angle between north and the geodesic;
that is, the shortest path on the surface of the spheroid from our
viewpoint to Point 2. The difference is usually immeasurably small; if
Point 2 is not more than 100 km away, the difference will not exceed
0.03 arc second.
Normal-section azimuth is simpler to calculate; Bomford says Cunningham's formula is exact for any distance.[citation needed] If f is the flattening, and e the eccentricity, for the chosen spheroid (e.g., 1⁄298.257223563 for WGS84) then
If φ1 = 0 then
To calculate the azimuth of the sun or a star given its declination
and hour angle at our location, we modify the formula for a spherical
earth. Replace φ2 with declination and longitude
difference with hour angle, and change the sign (since the hour angle is
positive westward instead of east).
In cartography
A standard Brunton Geo compass, commonly used by geologists and surveyors to measure azimuth
The cartographical azimuth or grid azimuth (in decimal degrees) can be calculated when the coordinates of 2 points are known in a flat plane (cartographical coordinates):
Remark that the reference axes are swapped relative to the (counterclockwise) mathematical polar coordinate system and that the azimuth is clockwise relative to the north.
This is the reason why the X and Y axis in the above formula are swapped.
If the azimuth becomes negative, one can always add 360°.
Note the swapped in contrast to the normal atan2 input order.
The opposite problem occurs when the coordinates (X1, Y1) of one point, the distance D, and the azimuth α to another point (X2, Y2) are known, one can calculate its coordinates:
This is typically used in triangulation and azimuth identification (AzID), especially in radar applications.
Map projections
There is a wide variety of azimuthal map projections.
They all have the property that directions (the azimuths) from a
central point are preserved. Some navigation systems use south as the
reference plane. However, any direction can serve as the plane of
reference, as long as it is clearly defined for everyone using that
system.
Comparison of some azimuthal projections centred on 90° N at the same scale, ordered by projection altitude in Earth radii. (click for detail)
In astronomy
Used in celestial navigation, an azimuth is the direction of a celestial body from the observer.[8] In astronomy, an azimuth is sometimes referred to as a bearing. In modern astronomy azimuth is nearly always measured from the north.
(The article on coordinate systems,
for example, uses a convention measuring from the south.) In former
times, it was common to refer to azimuth from the south, as it was then
zero at the same time that the hour angle of a star was zero. This assumes, however, that the star (upper) culminates in the south, which is only true if the star's declination is less than (i.e. further south than) the observer's latitude.
Related coordinates
Right ascension
If, instead of measuring from and along the horizon, the angles are measured from and along the celestial equator, the angles are called right ascension if referenced to the Vernal Equinox, or hour angle if referenced to the celestial meridian.
In mathematics, the azimuth angle of a point in cylindrical coordinates or spherical coordinates is the anticlockwise angle between the positive x-axis and the projection of the vector onto the xy-plane. The angle is the same as an angle in polar coordinates of the component of the vector in the xy-plane and is normally measured in radians rather than degrees. As well as measuring the angle differently, in mathematical applications theta, θ, is very often used to represent the azimuth rather than the representation of symbol phiφ.
Other uses
For magnetic tape drives, azimuth refers to the angle between the tape head(s) and tape.
In sound localization experiments and literature, the azimuth
refers to the angle the sound source makes compared to the imaginary
straight line that is drawn from within the head through the area
between the eyes.
The equation of time — above the axis a sundial will appear fast relative to a clock showing local mean time, and below the axis a sundial will appear slow.
This graph shows how many minutes the clock is ahead (+) or behind (-) the apparent sun. See the section "Sign of the equation of time" below.
The equation of time describes the discrepancy between two kinds of solar time. The word equation is used in the medieval sense of "reconcile a difference". The two times that differ are the apparent solar time, which directly tracks the diurnal motion of the Sun, and mean solar time, which tracks a theoretical mean Sun with uniform motion. Apparent solar time can be obtained by measurement of the current position (hour angle) of the Sun, as indicated (with limited accuracy) by a sundial. Mean
solar time, for the same place, would be the time indicated by a steady
clock set so that over the year its differences from apparent solar
time would have a mean of zero.
The equation of time is the east or west component of the analemma, a curve representing the angular offset of the Sun from its mean position on the celestial sphere as viewed from Earth. The equation of time values for each day of the year, compiled by astronomical observatories, were widely listed in almanacs and ephemerides.
The concept
Clock with auxiliary dial displaying the equation of time. Piazza Dante, Naples (1853).
During a year the equation of time varies as shown on the graph; its
change from one year to the next is slight. Apparent time, and the
sundial, can be ahead (fast) by as much as 16 min 33 s
(around 3 November), or behind (slow) by as much as 14 min 6 s (around
11 February). The equation of time has zeros near 15 April, 13 June,
1 September, and 25 December. Ignoring very slow changes in the Earth's
orbit and rotation, these events are repeated at the same times every tropical year. However, due to the non-integral number of days in a year, these dates can vary by a day or so from year to year.
The graph of the equation of time is closely approximated by the
sum of two sine curves, one with a period of a year and one with a
period of half a year. The curves reflect two astronomical effects, each
causing a different non-uniformity in the apparent daily motion of the
Sun relative to the stars:
the obliquity of the ecliptic
(the plane of the Earth's annual orbital motion around the Sun), which
is inclined by about 23.44 degrees relative to the plane of the Earth's equator; and
The equation of time is constant only for a planet with zero axial tilt and zero orbital eccentricity. On Mars
the difference between sundial time and clock time can be as much as
50 minutes, due to the considerably greater eccentricity of its orbit.
The planet Uranus,
which has an extremely large axial tilt, has an equation of time that
makes its days start and finish several hours earlier or later depending
on where it is in its orbit.
Sign of the equation of time
The United States Naval Observatory states "the Equation of Time is the difference apparent solar time minus mean solar time", i.e. if the sun is ahead of the clock the sign is positive, and if the clock is ahead of the sun the sign is negative.
The equation of time is shown in the upper graph above for a period of
slightly more than a year. The lower graph (which covers exactly one
calendar year) has the same absolute values but the sign
is reversed as it shows how far the clock is ahead of the sun.
Publications may use either format — in the English-speaking world, the
former usage is the more common, but is not always followed. Anyone who
makes use of a published table or graph should first check its sign
usage. Often, there is a note or caption which explains it. Otherwise,
the usage can be determined by knowing that, during the first three
months of each year, the clock is ahead of the sundial. The mnemonic
"NYSS" (pronounced "nice"), for "new year, sundial slow", can be
useful. Some published tables avoid the ambiguity by not using signs,
but by showing phrases such as "sundial fast" or "sundial slow" instead.
In this article, and others in English Wikipedia, a positive
value of the equation of time implies that a sundial is ahead of a
clock.
History
The phrase "equation of time" is derived from the medieval Latinaequātiō diērum, meaning "equation of days" or "difference of days".
The word aequātiō (and Middle Englishequation)
was used in medieval astronomy to tabulate the difference between an
observed value and the expected value (as in the equation of the centre,
the equation of the equinoxes, the equation of the epicycle).
Gerald J. Toomer uses the medieval term "equation" from the Latin aequātiō, for Ptolemy's difference between the mean solar time and the apparent solar time. Johannes Kepler's
definition of the equation is "the difference between the number of
degrees and minutes of the mean anomaly and the degrees and minutes of
the corrected anomaly."
The difference between apparent solar time and mean time was
recognized by astronomers since antiquity, but prior to the invention of
accurate mechanical clocks in the mid-17th century, sundials
were the only reliable timepieces, and apparent solar time was the
generally accepted standard. Mean time did not supplant apparent time in
national almanacs and ephemerides until the early 19th century.
Early astronomy
The irregular daily movement of the Sun was known to the Babylonians.
Book III of Ptolemy's Almagest (2nd century) is primarily concerned with the Sun's anomaly, and he tabulated the equation of time in his Handy Tables.
Ptolemy discusses the correction needed to convert the meridian
crossing of the Sun to mean solar time and takes into consideration the
nonuniform motion of the Sun along the ecliptic and the meridian
correction for the Sun's ecliptic longitude. He states the maximum
correction is 8+1⁄3 time-degrees or 5⁄9 of an hour (Book III, chapter 9).
However he did not consider the effect to be relevant for most
calculations since it was negligible for the slow-moving luminaries and
only applied it for the fastest-moving luminary, the Moon.
Based on Ptolemy's discussion in the Almagest, values for the equation of time (Arabic taʿdīl al-ayyām bi layālayhā) were standard for the tables (zij) in the works of medieval Islamic astronomy.
Early modern period
A description of apparent and mean time was given by Nevil Maskelyne in the Nautical Almanac
for 1767: "Apparent Time is that deduced immediately from the Sun,
whether from the Observation of his passing the Meridian, or from his
observed Rising or Setting.
This Time is different from that shewn by Clocks and Watches well
regulated at Land, which is called equated or mean Time." He went on to
say that, at sea, the apparent time found from observation of the Sun
must be corrected by the equation of time, if the observer requires the
mean time.
The right time was originally considered to be that which was
shown by a sundial. When good mechanical clocks were introduced, they
agreed with sundials only near four dates each year, so the equation of
time was used to "correct" their readings to obtain sundial time. Some
clocks, called equation clocks,
included an internal mechanism to perform this "correction". Later, as
clocks became the dominant good timepieces, uncorrected clock time,
i.e., "mean time", became the accepted standard. The readings of
sundials, when they were used, were then, and often still are, corrected
with the equation of time, used in the reverse direction from
previously, to obtain clock time. Many sundials, therefore, have tables
or graphs of the equation of time engraved on them to allow the user to
make this correction.
The equation of time was used historically to set clocks.
Between the invention of accurate clocks in 1656 and the advent of
commercial time distribution services around 1900, there were three
common land-based ways to set clocks. Firstly, in the unusual event of
having an astronomer present, the sun's transit across the meridian
(the moment the sun passed overhead) was noted; the clock was then set
to noon and offset by the number of minutes given by the equation of
time for that date. Secondly, and much more commonly, a sundial was
read, a table of the equation of time (usually engraved on the dial) was
consulted and the watch or clock set accordingly. These calculated the
mean time, albeit local to a point of longitude. The third method did not use the equation of time; instead, it used stellar observations to give sidereal time, exploiting the relationship between sidereal time and mean solar time.
The first tables to give the equation of time in an essentially correct way were published in 1665 by Christiaan Huygens.
Huygens, following the tradition of Ptolemy and medieval astronomers in
general, set his values for the equation of time so as to make all
values positive throughout the year.
Another set of tables was published in 1672–73 by John Flamsteed, who later became the first Astronomer Royal of the new Royal Greenwich Observatory.
These appear to have been the first essentially correct tables that
gave today's meaning of Mean Time (previously, as noted above, the sign
of the equation was always positive and it was set at zero when the
apparent time of sunrise was earliest relative to the clock time of
sunrise). Flamsteed adopted the convention of tabulating and naming the
correction in the sense that it was to be applied to the apparent time
to give mean time.
The equation of time, correctly based on the two major components of the Sun's irregularity of apparent motion, was not generally adopted until after Flamsteed's tables of 1672–73, published with the posthumous edition of the works of Jeremiah Horrocks.
Robert Hooke (1635–1703), who mathematically analyzed the universal joint,
was the first to note that the geometry and mathematical description of
the (non-secular) equation of time and the universal joint were
identical, and proposed the use of a universal joint in the construction
of a "mechanical sundial".
18th and early 19th centuries
The
corrections in Flamsteed's tables of 1672–1673 and 1680 gave mean time
computed essentially correctly and without need for further offset. But
the numerical values in tables of the equation of time have somewhat
changed since then, owing to three factors:
general improvements in accuracy that came from refinements in astronomical measurement techniques,
slow intrinsic changes in the equation of time, occurring as a
result of small long-term changes in the Earth's obliquity and
eccentricity (affecting, for instance, the distance and dates of perihelion), and
the inclusion of small sources of additional variation in the
apparent motion of the Sun, unknown in the 17th century, but discovered
from the 18th century onwards, including the effects of the Moon, Venus and Jupiter.
A sundial made in 1812, by Whitehurst & Son with a circular scale showing the equation of time correction. This is now on display in the Derby Museum.
From 1767 to 1833, the British Nautical Almanac and Astronomical Ephemeris
tabulated the equation of time in the sense 'add or subtract (as
directed) the number of minutes and seconds stated to or from the
apparent time to obtain the mean time'. Times in the Almanac were in
apparent solar time, because time aboard ship was most often determined
by observing the Sun. This operation would be performed in the unusual
case that the mean solar time of an observation was needed. In the
issues since 1834, all times have been in mean solar time, because by
then the time aboard ship was increasingly often determined by marine chronometers.
The instructions were consequently to add or subtract (as directed) the
number of minutes stated to or from the mean time to obtain the
apparent time. So now addition corresponded to the equation being
positive and subtraction corresponded to it being negative.
As the apparent daily movement of the Sun is one revolution per
day, that is 360° every 24 hours, and the Sun itself appears as a disc
of about 0.5° in the sky, simple sundials can be read to a maximum
accuracy of about one minute. Since the equation of time has a range of
about 33 minutes, the difference between sundial time and clock time
cannot be ignored. In addition to the equation of time, one also has to
apply corrections due to one's distance from the local time zone
meridian and summer time, if any.
The tiny increase of the mean solar day due to the slowing down of the Earth's rotation, by about 2 ms
per day per century, which currently accumulates up to about 1 second
every year, is not taken into account in traditional definitions of the
equation of time, as it is imperceptible at the accuracy level of
sundials.
Major components of the equation
Eccentricity of the Earth's orbit
Equation
of time (red solid line) and its two main components plotted
separately, the part due to the obliquity of the ecliptic (mauve dashed
line) and the part due to the Sun's varying apparent speed along the
ecliptic due to eccentricity of the Earth's orbit (dark blue dash &
dot line)
The Earth revolves around the Sun. As seen from Earth, the Sun
appears to revolve once around the Earth through the background stars in
one year. If the Earth orbited the Sun with a constant speed, in a
circular orbit in a plane perpendicular to the Earth's axis, then the
Sun would culminate
every day at exactly the same time, and be a perfect time keeper
(except for the very small effect of the slowing rotation of the Earth).
But the orbit of the Earth is an ellipse not centered on the Sun, and
its speed varies between 30.287 and 29.291 km/s, according to Kepler's laws of planetary motion, and its angular speed also varies, and thus the Sun appears to move faster (relative to the background stars) at perihelion (currently around 3 January) and slower at aphelion a half year later.
At these extreme points this effect varies the apparent solar day
by 7.9 s/day from its mean. Consequently, the smaller daily differences
on other days in speed are cumulative until these points, reflecting
how the planet accelerates and decelerates compared to the mean. As a
result, the eccentricity of the Earth's orbit contributes a periodic
variation which is (in the first-order approximation) a sine wave with an amplitude of 7.66 min and a period
of one year to the equation of time. The zero points are reached at
perihelion (at the beginning of January) and aphelion (beginning of
July); the extreme values are in early April (negative) and early
October (positive).
Obliquity of the ecliptic
Sun
and planets at local apparent noon (Ecliptic in red, Sun and Mercury in
yellow, Venus in white, Mars in red, Jupiter in yellow with red spot,
Saturn in white with rings).
Even if the Earth's orbit were circular, the perceived motion of the Sun along our celestial equator would still not be uniform. This is a consequence of the tilt of the Earth's rotational axis with respect to the plane of its orbit, or equivalently, the tilt of the ecliptic (the path the Sun appears to take in the celestial sphere) with respect to the celestial equator. The projection of this motion onto our celestial equator, along which "clock time" is measured, is a maximum at the solstices,
when the yearly movement of the Sun is parallel to the equator (causing
amplification of perceived speed) and yields mainly a change in right ascension. It is a minimum at the equinoxes, when the Sun's apparent motion is more sloped and yields more change in declination, leaving less for the component in right ascension,
which is the only component that affects the duration of the solar day.
A practical illustration of obliquity is that the daily shift of the
shadow cast by the Sun in a sundial even on the equator is smaller close
to the solstices and greater close to the equinoxes. If this effect
operated alone, then days would be up to 24 hours and 20.3 seconds long
(measured solar noon to solar noon) near the solstices, and as much as
20.3 seconds shorter than 24 hours near the equinoxes.
In the figure on the right, we can see the monthly variation of
the apparent slope of the plane of the ecliptic at solar midday as seen
from Earth. This variation is due to the apparent precession of the rotating Earth through the year, as seen from the Sun at solar midday.
In terms of the equation of time, the inclination of the ecliptic
results in the contribution of a sine wave variation with an amplitude
of 9.87 minutes and a period of a half year to the equation of time. The
zero points of this sine wave are reached at the equinoxes and
solstices, while the extrema are at the beginning of February and August
(negative) and the beginning of May and November (positive).
Secular effects
The
two above mentioned factors have different wavelengths, amplitudes and
phases, so their combined contribution is an irregular wave. At epoch 2000 these are the values (in minutes and seconds with UT dates):
Point
Value
Date
minimum
−14 min 15 s
11 February
zero
0 min 0 s
15 April
maximum
+3 min 41 s
14 May
zero
0 min 0 s
13 June
minimum
−6 min 30 s
26 July
zero
0 min 0 s
1 September
maximum
+16 min 25 s
3 November
zero
0 min 0 s
25 December
E.T. = apparent − mean. Positive means: Sun runs fast and
culminates earlier, or the sundial is ahead of mean time. A slight
yearly variation occurs due to presence of leap years, resetting itself
every 4 years. The exact shape of the equation of time curve and the
associated analemma slowly change over the centuries, due to secular variations
in both eccentricity and obliquity. At this moment both are slowly
decreasing, but they increase and decrease over a timescale of hundreds
of thousands of years.
On shorter timescales (thousands of years) the shifts in the dates of
equinox and perihelion will be more important. The former is caused by precession, and shifts the equinox backwards compared to the stars. But it can be ignored in the current discussion as our Gregorian calendar
is constructed in such a way as to keep the vernal equinox date at
20 March (at least at sufficient accuracy for our aim here). The shift
of the perihelion is forwards, about 1.7 days every century. In 1246 the
perihelion occurred on 22 December, the day of the solstice, so the two
contributing waves had common zero points and the equation of time
curve was symmetrical: in Astronomical Algorithms Meeus gives
February and November extrema of 15 m 39 s and May and July ones of
4 m 58 s. Before then the February minimum was larger than the November
maximum, and the May maximum larger than the July minimum. In fact, in
years before −1900 (1901 BCE) the May maximum was larger than the
November maximum. In the year −2000 (2001 BCE) the May maximum was
+12 minutes and a couple seconds while the November maximum was just
less than 10 minutes. The secular change is evident when one compares a
current graph of the equation of time (see below) with one from
2000 years ago, e.g., one constructed from the data of Ptolemy.
Graphical representation
Animation showing equation of time and analemma path over one year.
Practical use
If the gnomon
(the shadow-casting object) is not an edge but a point (e.g., a hole in
a plate), the shadow (or spot of light) will trace out a curve during
the course of a day. If the shadow is cast on a plane surface, this
curve will be a conic section
(usually a hyperbola), since the circle of the Sun's motion together
with the gnomon point define a cone. At the spring and fall equinoxes,
the cone degenerates into a plane and the hyperbola into a line. With a
different hyperbola for each day, hour marks can be put on each
hyperbola which include any necessary corrections. Unfortunately, each
hyperbola corresponds to two different days, one in each half of the
year, and these two days will require different corrections. A
convenient compromise is to draw the line for the "mean time" and add a
curve showing the exact position of the shadow points at noon during the
course of the year. This curve will take the form of a figure eight and
is known as an analemma.
By comparing the analemma to the mean noon line, the amount of
correction to be applied generally on that day can be determined.
The equation of time is used not only in connection with sundials and similar devices, but also for many applications of solar energy. Machines such as solar trackers and heliostats have to move in ways that are influenced by the equation of time.
Civil time is the local mean time for a meridian that often passes near the center of the time zone, and may possibly be further altered by daylight saving time.
When the apparent solar time that corresponds to a given civil time is
to be found, the difference in longitude between the site of interest
and the time zone meridian, daylight saving time, and the equation of
time must all be considered.
Calculating the equation of time
The
equation of time is obtained from a published table, or a graph. For
dates in the past such tables are produced from historical measurements,
or by calculation; for future dates, of course, tables can only be
calculated. In devices such as computer-controlled heliostats the
computer is often programmed to calculate the equation of time. The
calculation can be numerical or analytical. The former are based on numerical integration
of the differential equations of motion, including all significant
gravitational and relativistic effects. The results are accurate to
better than 1 second and are the basis for modern almanac data. The
latter are based on a solution that includes only the gravitational
interaction between the Sun and Earth, simpler than but not as accurate
as the former. Its accuracy can be improved by including small
corrections.
The following discussion describes a reasonably accurate
(agreeing with almanac data to within 3 seconds over a wide range of
years) algorithm for the equation of time that is well known to
astronomers.
It also shows how to obtain a simple approximate formula (accurate to
within 1 minute over a large time interval), that can be easily
evaluated with a calculator and provides the simple explanation of the
phenomenon that was used previously in this article.
GHA, the Greenwich Hour Angle of the apparent (actual) Sun;
GMHA = Universal Time − Offset, the Greenwich Mean Hour Angle of the mean (fictitious) Sun.
Here time and angle are quantities that are related by factors such as: 2π radians = 360° = 1 day = 24 hours. The difference, EOT, is measurable since GHA is an angle that can be measured and Universal Time, UT, is a scale for the measurement of time. The offset by π = 180° = 12 hours from UT is needed because UT is zero at mean midnight while GMHA = 0 at mean noon.
Both GHA and GMHA, like all physical angles, have a mathematical, but
not a physical discontinuity at their respective (apparent and mean)
noon. Despite the mathematical discontinuities of its components, EOT is
defined as a continuous function by adding (or subtracting) 24 hours in
the small time interval between the discontinuities in GHA and GMHA.
According to the definitions of the angles on the celestial sphere GHA = GAST − α (see hour angle) where:
GAST is the Greenwich apparent sidereal time (the angle between the apparent vernal equinox and the meridian in the plane of the equator). This is a known function of UT.
α is the right ascension of the apparent Sun (the angle between the apparent vernal equinox and the actual Sun in the plane of the equator).
On substituting into the equation of time, it is
EOT = GAST − α − UT + offset
Like the formula for GHA above, one can write GMHA = GAST − αM, where the last term is the right ascension of the mean Sun. The equation is often written in these terms as
EOT = αM − α
where αM = GAST − UT + offset. In this formulation a measurement or calculation of EOT at a certain value of time depends on a measurement or calculation of α at that time. Both α and αM
vary from 0 to 24 hours during the course of a year. The former has a
discontinuity at a time that depends on the value of UT, while the
latter has its at a slightly later time. As a consequence, when
calculated this way EOT has two, artificial, discontinuities. They can
both be removed by subtracting 24 hours from the value of EOT in the
small time interval after the discontinuity in α and before the one in αM. The resulting EOT is a continuous function of time.
Another definition, denoted E to distinguish it from EOT, is
E = GMST − α − UT + offset
Here GMST = GAST − eqeq, is the Greenwich
mean sidereal time (the angle between the mean vernal equinox and the
mean Sun in the plane of the equator). Therefore, GMST is an
approximation to GAST (and E is an approximation to EOT); eqeq is called the equation of the equinoxes and is due to the wobbling, or nutation
of the Earth's axis of rotation about its precessional motion. Since
the amplitude of the nutational motion is only about 1.2 s (18″ of
longitude) the difference between EOT and E can be ignored unless one is interested in subsecond accuracy.
A third definition, denoted Δt to distinguish it from EOT and E, and now called the Equation of Ephemeris Time (prior to the distinction that is now made between EOT, E, and Δt the latter was known as the equation of time) is
Δt = Λ − α
here Λ is the ecliptic longitude of the mean Sun (the angle from the mean vernal equinox to the mean Sun in the plane of the ecliptic).
The difference Λ − (GMST − UT + offset) is 1.3 s from 1960 to 2040. Therefore, over this restricted range of years Δt
is an approximation to EOT whose error is in the range 0.1 to 2.5 s
depending on the longitude correction in the equation of the equinoxes;
for many purposes, for example correcting a sundial, this accuracy is
more than good enough.
Right ascension calculation
The
right ascension, and hence the equation of time, can be calculated from
Newton's two-body theory of celestial motion, in which the bodies
(Earth and Sun) describe elliptical orbits about their common mass
center. Using this theory, the equation of time becomes
Δt = M + λp − α
where the new angles that appear are
M = 2π(t − tp)/tY, is the mean anomaly, the angle from the periapsis of the elliptical orbit to the mean Sun; its range is from 0 to 2π as t increases from tp to tp + tY;
tY = 365.2596358 days is the length of time in an anomalistic year: the time interval between two successive passages of the periapsis;
λp = Λ − M, is the ecliptic longitude of the periapsis;
t is dynamical time,
the independent variable in the theory. Here it is taken to be
identical with the continuous time based on UT (see above), but in more
precise calculations (of E or EOT) the small difference between them must be accounted for as well as the distinction between UT1 and UTC.
tp is the value of t at the periapsis.
To complete the calculation three additional angles are required:
E, the Sun's eccentric anomaly (note that this is different from M);
λ = ν + λp, the Sun's true longitude on the ecliptic.
The
celestial sphere and the Sun's elliptical orbit as seen by a geocentric
observer looking normal to the ecliptic showing the 6 angles (M, λp, α, ν, λ, E) needed for the calculation of the equation of time. For the sake of clarity the drawings are not to scale.
All these angles are shown in the figure on the right, which shows the celestial sphere and the Sun's elliptical orbit seen from the Earth (the same as the Earth's orbit seen from the Sun). In this figure ε is the obliquity, while e = √1 − (b/a)2 is the eccentricity of the ellipse.
Now given a value of 0 ≤ M ≤ 2π, one can calculate α(M) by means of the following well-known procedure:
Although this equation cannot be solved exactly in closed form, values of E(M) can be obtained from infinite (power or trigonometric) series, graphical, or numerical methods. Alternatively, note that for e = 0, E = M, and by iteration:
E ≈ M + e sin M.
This approximation can be improved, for small e, by iterating again,
E ≈ M + e sin M + 1/2e2 sin 2M,
and continued iteration produces successively higher order terms of the power series expansion in e. For small values of e (much less than 1) two or three terms of the series give a good approximation for E; the smaller e, the better the approximation.
Next, knowing E, calculate the true anomalyν from an elliptical orbit relation
The correct branch of the multiple valued function tan−1x to use is the one that makes ν a continuous function of E(M) starting from νE=0 = 0. Thus for 0 ≤ E < π use tan−1x = Tan−1x, and for π < E ≤ 2π use tan−1x = Tan−1x + π. At the specific value E = π for which the argument of tan is infinite, use ν = E. Here Tan−1x is the principal branch, |Tan−1x| < π/2;
the function that is returned by calculators and computer applications.
Alternatively, this function can be expressed in terms of its Taylor series in e, the first three terms of which are:
ν ≈ E + e sin E + 1/4e2 sin 2E.
For small e this approximation (or even just the first two terms) is a good one. Combining the approximation for E(M) with this one for ν(E) produces
ν ≈ M + 2e sin M + 5/4e2 sin 2M.
The relation ν(M) is called the equation of the center; the expression written here is a second-order approximation in e. For the small value of e that characterises the Earth's orbit this gives a very good approximation for ν(M).
Next, knowing ν, calculate λ from its definition:
λ = ν + λp
The value of λ varies non-linearly with M because the orbit is elliptical and not circular. From the approximation for ν:
λ ≈ M + λp + 2e sin M + 5/4e2 sin 2M.
Finally, knowing λ calculate α from a relation for the right triangle on the celestial sphere shown above
α = tan−1(cos ε tan λ)
Note that the quadrant of α is the same as that of λ, therefore reduce λ to the range 0 to 2π and write
α = Tan−1 (cos ε tan λ) + kπ,
where k is 0 if λ is in quadrant 1, it is 1 if λ is in quadrants 2 or 3 and it is 2 if λ is in quadrant 4. For the values at which tan is infinite, α = λ.
Although approximate values for α can be obtained from truncated Taylor series like those for ν, it is more efficacious to use the equation
α = λ − sin−1 [y sin (α + λ)]
where y = tan2(ε/2). Note that for ε = y = 0, α = λ and iterating twice:
α ≈ λ − y sin 2λ + 1/2y2 sin 4λ.
Equation of time
The
equation of time is obtained by substituting the result of the right
ascension calculation into an equation of time formula. Here Δt(M) = M + λp − α[λ(M)] is used; in part because small corrections (of the order of 1 second), that would justify using E, are not included, and in part because the goal is to obtain a simple analytical expression. Using two-term approximations for λ(M) and α(λ) allows Δt to be written as an explicit expression of two terms, which is designated Δtey because it is a first order approximation in e and in y.
Δtey = −2e sin M + y sin (2M + 2λp) = −7.659 sin M + 9.863 sin (2M + 3.5932) minutes
This equation was first derived by Milne, who wrote it in terms of λ = M + λp. The numerical values written here result from using the orbital parameter values, e = 0.016709, ε = 23.4393° = 0.409093 radians, and λp = 282.9381° = 4.938201 radians that correspond to the epoch 1 January 2000 at 12 noon UT1. When evaluating the numerical expression for Δtey as given above, a calculator must be in radian mode to obtain correct values because the value of 2λp − 2π in the argument of the second term is written there in radians. Higher order approximations can also be written, but they necessarily have more terms. For example, the second order approximation in both e and y consists of five terms
Δte2y2 = Δtey − 5/4e2 sin 2M + ey sin M cos (2M + 2λp) − 1/2y2 sin (4M + 4λp)
This approximation has the potential for high accuracy, however, in
order to achieve it over a wide range of years, the parameters e, ε, and λp must be allowed to vary with time. This creates additional calculational complications. Other approximations have been proposed, for example, Δtewhich uses the first order equation of the center but no other approximation to determine α, and Δte2 which uses the second order equation of the center.
The time variable, M, can be written either in terms of n, the number of days past perihelion, or D, the number of days past a specific date and time (epoch):
M = 2π/tYn days = MD + 2π/tYD days = 6.24004077 + 0.01720197D
Here MD is the value of M at the chosen date and time. For the values given here, in radians, MD is that measured for the actual Sun at the epoch, 1 January 2000 at 12 noon UT1, and D is the number of days past that epoch. At periapsis M = 2π, so solving gives D = Dp = 2.508109. This puts the periapsis on 4 January 2000 at 00:11:41 while the actual periapsis is, according to results from the Multiyear Interactive Computer Almanac
(abbreviated as MICA), on 3 January 2000 at 05:17:30. This large
discrepancy happens because the difference between the orbital radius at
the two locations is only 1 part in a million; in other words, radius
is a very weak function of time near periapsis. As a practical matter
this means that one cannot get a highly accurate result for the equation
of time by using n and adding the
actual periapsis date for a given year. However, high accuracy can be
achieved by using the formulation in terms of D.
Curves of Δt and Δtey along with symbols locating the daily values at noon (at 10-day intervals) obtained from the Multiyear Interactive Computer Almanac vs d for the year 2000
When D > Dp, M is greater than 2π and one must subtract a multiple of 2π (that depends on the year) from it to bring it into the range 0 to 2π. Likewise for years prior to 2000 one must add multiples of 2π. For example, for the year 2010, D varies from 3653 on 1 January at noon to 4017 on 31 December at noon; the corresponding M values are 69.0789468 and 75.3404748 and are reduced to the range 0 to 2π by subtracting 10 and 11 times 2π respectively. One can always write D = nY + d, where nY is the number of days from the epoch to noon on 1 January of the desired year, and 0 ≤ d ≤ 364 (365 if the calculation is for a leap year).
The result of the computations is usually given as either a set
of tabular values, or a graph of the equation of time as a function of d. A comparison of plots of Δt, Δtey, and results from MICA all for the year 2000 is shown in the figure on the right. The plot of Δtey is seen to be close to the results produced by MICA, the absolute error, Err = |Δtey − MICA2000|, is less than 1 minute throughout the year; its largest value is 43.2 seconds and occurs on day 276 (3 October). The plot of Δt is indistinguishable from the results of MICA, the largest absolute error between the two is 2.46 s on day 324 (20 November).
Remark on the continuity of the equation of time
For the choice of the appropriate branch of the arctan
relation with respect to function continuity a modified version of the
arctangent function is helpful. It brings in previous knowledge about
the expected value by a parameter. The modified arctangent function is
defined as:
arctanηx = arctan x + π round (η − arctan x/π).
It produces a value that is as close to η as possible. The function round rounds to the nearest integer.
Applying this yields:
Δt(M) = M + λp − arctan(M+λp) (cos ε tan λ).
The parameter M + λp arranges here to set Δt to the zero nearest value which is the desired one.
Secular effects
The difference between the MICA and Δt
results was checked every 5 years over the range from 1960 to 2040. In
every instance the maximum absolute error was less than 3 s; the largest
difference, 2.91 s, occurred on 22 May 1965 (day 141). However, in
order to achieve this level of accuracy over this range of years it is
necessary to account for the secular change in the orbital parameters
with time. The equations that describe this variation are:
According to these relations, in 100 years (D = 36525), λp increases by about 0.5% (1.7°), e decreases by about 0.25%, and ε decreases by about 0.05%.
As a result, the number of calculations required for any of the
higher-order approximations of the equation of time requires a computer
to complete them, if one wants to achieve their inherent accuracy over a
wide range of time. In this event it is no more difficult to evaluate Δt using a computer than any of its approximations.
In all this note that Δtey
as written above is easy to evaluate, even with a calculator, is
accurate enough (better than 1 minute over the 80-year range) for
correcting sundials, and has the nice physical explanation as the sum of
two terms, one due to obliquity and the other to eccentricity that was
used previously in the article. This is not true either for Δt considered as a function of M or for any of its higher-order approximations.
Alternative calculation
Another procedure for calculating the equation of time can be done as follows. Angles are in degrees; the conventional order of operations applies.
n = 360°/365.24 days,
where n is the Earth's mean angular orbital velocity in degrees per day, a.k.a. "the mean daily motion".
A = n × ( D + 9 )
where D is the date, counted in days starting at 1 on 1 January (i.e. the days part of the ordinal date in the year). 9 is the approximate number of days from the December solstice to 31 December. A is the angle the Earth would move on its orbit at its average speed from the December solstice to date D.
B = A + 360°/π × 0.0167 × sin [ n ( D − 3 ) ]
B is the angle the Earth moves from the solstice to date D,
including a first-order correction for the Earth's orbital
eccentricity, 0.0167 . The number 3 is the approximate number of days
from 31 December to the current date of the Earth's perihelion. This expression for B can be simplified by combining constants to:
B = A + 1.914° × sin [ n ( D − 3 ) ].
Here, C
is the difference between the angle moved at mean speed, and at the
angle at the corrected speed projected onto the equatorial plane, and
divided by 180° to get the difference in "half-turns". The value 23.44° is the tilt of the Earth's axis ("obliquity"). The subtraction gives the conventional sign to the equation of time. For any given value of x, arctanx (sometimes written as tan−1x)
has multiple values, differing from each other by integer numbers of
half turns. The value generated by a calculator or computer may not be
the appropriate one for this calculation. This may cause C
to be wrong by an integer number of half-turns. The excess half-turns
are removed in the next step of the calculation to give the equation of
time:
EOT = 720 × (C − nint(C)) minutes
The expression nint(C) means the nearest integer to C. On a computer, it can be programmed, for example, as INT(C + 0.5).
Its value is 0, 1, or 2 at different times of the year. Subtracting it
leaves a small positive or negative fractional number of half turns,
which is multiplied by 720, the number of minutes (12 hours) that the
Earth takes to rotate one half turn relative to the Sun, to get the
equation of time.
Compared with published values, this calculation has a root mean square
error of only 3.7 s. The greatest error is 6.0 s. This is much more
accurate than the approximation described above, but not as accurate as
the elaborate calculation.
The value of B
in the above calculation is an accurate value for the Sun's ecliptic
longitude (shifted by 90°), so the solar declination becomes readily
available:
Declination = − arcsin (sin 23.44° × cos B)
which is accurate to within a fraction of a degree.
listen); from Arabic اَلسُّمُوت as-sumūt, 'the directions', the plural form of the Arabic noun السَّمْت as-samt, meaning 'the direction') is an angular measurement in a spherical coordinate system. The vector from an observer (origin) to a point of interest is projected perpendicularly onto a reference plane; the angle between the projected vector and a reference vector on the reference plane is called the azimuth.
