Deep blue water, blue-green water, satellite image of the Bahamas where sunlight reflects off sand and reefs in the shallows, satellite image of phytoplankton bloom in the Southern Ocean, satellite image of the Pribilof Islands showing shades of color from different phytoplankton, and satellite image of the Baltic Sea with phytoplankton blooms.
Ocean color is the branch of ocean optics
that specifically studies the color of the water and information that
can be gained from looking at variations in color. The color of the ocean, while mainly blue, actually varies from blue to green or even yellow, brown or red in some cases. This field of study developed alongside water remote sensing, so it is focused mainly on how color is measured by instruments (like the sensors on satellites and airplanes).
Most of the ocean is blue in color, but in some places the ocean is blue-green, green, or even yellow to brown.
Blue ocean color is a result of several factors. First, water
preferentially absorbs red light, which means that blue light remains
and is reflected back out of the water. Red light is most easily
absorbed and thus does not reach great depths, usually to less than 50
meters (164 ft). Blue light, in comparison, can penetrate up to 200
meters (656 ft).
Second, water molecules and very tiny particles in ocean water
preferentially scatter blue light more than light of other colors. Blue
light scattering by water and tiny particles happens even in the very
clearest ocean water, and is similar to blue light scattering in the sky.
The main substances that affect the color of the ocean include dissolved organic matter, living phytoplankton with chlorophyll pigments, and non-living particles like marine snow and mineral sediments. Chlorophyll can be measured by satellite observations and serves as a proxy for ocean productivity (marine primary productivity)
in surface waters. In long term composite satellite images, regions
with high ocean productivity show up in yellow and green colors because
they contain more (green) phytoplankton, whereas areas of low productivity show up in blue.
Overview
Ocean color depends on how light interacts with the materials in the water. When light enters water, it can either be absorbed (light gets used up, the water gets "darker"), scattered (light gets bounced around in different directions, the water remains "bright"), or a combination of both. How underwater absorption and scattering vary spectrally, or across the spectrum of visible to infrared light energy (about 400 nm to 2000 nm wavelengths) determines what "color" the water will appear to a sensor.
Water types by color
Most
of the world’s oceans appear blue because the light leaving water is
brightest (has the highest reflectance value) in the blue part of the
visible light spectrum. Nearer to land, coastal waters often appear
green. Green waters appear this way because algae and dissolved
substances are absorbing light in the blue and red portions of the
spectrum.
Blue oceans
A deep blue colored wave viewed from the water surface near Encinitas, California, United States. The Pacific Ocean contains some of the most deep blue colored waters in the world.
The reason that open-ocean waters appear blue is that they are very
clear, somewhat similar to pure water, and have few materials present or
very tiny particles only. Pure water absorbs red light with depth.
As red light is absorbed, blue light remains. Large quantities of pure
water appear blue (even in a white-bottom swimming pool or white-painted
bucket).
The substances that are present in blue-colored open ocean waters are
often very tiny particles which scatter light, scattering light
especially strongly in the blue wavelengths. Light scattering in blue water is similar to the scattering in the atmosphere which makes the sky appear blue (called Rayleigh scattering). Some blue-colored clear water lakes appear blue for these same reasons, like Lake Tahoe in the United States.
Green oceans
Microscopic marine algae, called phytoplankton, absorb light in the blue and red wavelengths, due to their specific pigments like chlorophyll-a.
Accordingly, with more and more phytoplankton in the water, the color
of the water shifts toward the green part of the spectrum.
The most widespread light-absorbing substance in the oceans is
chlorophyll pigment, which phytoplankton use to produce carbon by photosynthesis.
Chlorophyll, a green pigment, makes phytoplankton preferentially absorb
the red and blue portions of the light spectrum . As blue and red light
are absorbed, green light remains. Ocean regions with high
concentrations of phytoplankton have shades of blue-to-green water
depending on the amount and type of the phytoplankton.
Green waters can also have a combination of phytoplankton,
dissolved substances, and sediments, while still appearing green. This
often happens in estuaries, coastal waters, and inland waters, which are
called "optically complex" waters because multiple different substances
are creating the green color seen by the sensor.
A surfer cuts through a green colored wave of ocean water at Strandhill, Ireland.
Yellow to brown oceans
Sentinel-2 satellite image of the confluence of the Rio Negro and the Solimões River, Brazil. The Rio Negro in the upper left part of the image is dark due to high concentrations of colored dissolved organic matter (CDOM). The Solimões River in the lower and right part of the image is brighter because of large amounts of sediments.
Ocean water appears yellow or brown when large amounts of dissolved substances, sediments, or both types of material are present.
Water can appear yellow or brown due to large amounts of dissolved substances. Dissolved matter or gelbstoff
(meaning yellow substance) appears dark yet relatively transparent,
much like tea. Dissolved substances absorb blue light more strongly than
light of other colors. Colored dissolved organic matter (CDOM) often comes from decaying plant matter on land or in marshes, or in the open ocean from marine phytoplankton exuding dissolved substances from their cells.
In coastal areas, runoff from rivers and resuspension of sand and
silt from the bottom add sediments to surface waters. More sediments
can make the waters appear more green, yellow, or brown because sediment
particles scatter light energy at all colors.
In large amounts, mineral particles like sediment cause the water to
turn brownish if there is a massive sediment loading event, appearing bright and opaque (not transparent), much like chocolate milk.
In Lake Boomanjin, Australia, the waters are strongly colored due to tannins from nearby trees.
MODIS satellite image of the Mississippi River sediment plume in the Gulf of Mexico following a series of rainstorms in February 2018 (image from March 4, 2018).
Red tide off the Scripps Institution of Oceanography Pier, La Jolla, California, United States.
Ocean water can appear red if there is a bloom of a specific kind of phytoplankton causing a discoloration of the sea surface. These events are called "Red tides."
However, not all red tides are harmful, and they are only considered
harmful algal blooms if the type of plankton involved contains hazardous
toxins. The red color comes from the pigments in the specific kinds of phytoplankton causing the bloom. Some examples are Karenia brevis in the Gulf of Mexico, Alexandrium fundyense in the Gulf of Maine, Margalefadinium polykroides and Alexandrium monilatum in the Chesapeake Bay, and Mesodinium rubrum in Long Island Sound.
Ocean color remote sensing is also referred to as ocean color radiometry.
Remote sensors on satellites, airplanes, and drones measure the
spectrum of light energy coming from the water surface. The sensors used
to measure light energy coming from the water are called radiometers
(or spectrometers or spectroradiometers).
Some radiometers are used in the field at earth’s surface on ships or
directly in the water. Other radiometers are designed specifically for
airplanes or earth-orbiting satellite missions. Using radiometers,
scientists measure the amount of light energy coming from the water at
all colors of the electromagnetic spectrum from ultraviolet to near-infrared.
From this reflected spectrum of light energy, or the apparent "color,"
researchers derive other variables to understand the physics and biology
of the oceans.
Ocean color measurements can be used to infer important
information such as phytoplankton biomass or concentrations of other
living and non-living material. The patterns of algal blooms from
satellite over time, over large regions up to the scale of the global
ocean, has been instrumental in characterizing variability of marine ecosystems.
Ocean color data is a key tool for research into how marine ecosystems
respond to climate change and anthropogenic perturbations.
One of the biggest challenges for ocean color remote sensing is atmospheric correction, or removing the color signal of the atmospheric haze and clouds to focus on the color signal of the ocean water. The signal from the water itself is less than 10% of the total signal of light leaving earth’s surface.
History
Scientists
including biologist Ellen Weaver helped to develop the first sensors to
measure ocean productivity from above, beginning with airplane-mounted
sensors.
People have written about the color of the ocean over many centuries, including ancient Greek poet Homer’s famous "wine-dark sea." Scientific measurements of the color of the ocean date back to the invention of the Secchi disk in Italy in the mid-1800s to study the transparency and clarity of the sea.
Major accomplishments were made in the 1960s and 1970s leading up to modern ocean color remote sensing campaigns. Nils Gunnar Jerlov’s book Optical Oceanography, published in 1968,
was a starting point for many researchers in the next decades. In 1970,
George Clarke published the first evidence that chlorophyll
concentration could be estimated based on green versus blue light coming
from the water, as measured from an airplane over George's Bank.
In the 1970s, scientist Howard Gordon and his graduate student George
Maul related imagery from the first Landsat mission to ocean color. Around the same time, a group of researchers, including John Arvesen, Dr. Ellen Weaver, and explorer Jacques Cousteau, began developing sensors to measure ocean productivity beginning with an airborne sensor.
Remote sensing of ocean color from space began in 1978 with the successful launch of NASA's Coastal Zone Color Scanner
(CZCS) on the Nimbus-7 satellite. Despite the fact that CZCS was an
experimental mission intended to last only one year as a proof of
concept, the sensor continued to generate a valuable time-series of data
over selected test sites until early 1986. Ten years passed before
other sources of ocean color data became available with the launch of
other sensors, and in particular the Sea-viewing Wide Field-of-view
sensor (SeaWiFS) in 1997 on board the NASA SeaStar satellite. Subsequent sensors have included NASA's Moderate-resolution Imaging Spectroradiometer (MODIS) on board the Aqua and Terra satellites, ESA's MEdium Resolution Imaging Spectrometer (MERIS) onboard its environmental satellite Envisat. Several new ocean-colour sensors have recently been launched, including the Indian Ocean Colour Monitor (OCM-2) on-board ISRO's Oceansat-2 satellite and the Korean Geostationary Ocean Color Imager (GOCI), which is the first ocean colour sensor to be launched on a geostationary satellite, and Visible Infrared Imager Radiometer Suite (VIIRS) aboard NASA's Suomi NPP . More ocean colour sensors are planned over the next decade by various space agencies, including hyperspectral imaging.
Applications
Ocean Color Radiometry and its derived products are also seen as fundamental Essential Climate Variables as defined by the Global Climate Observing System.
Ocean color datasets provide the only global synoptic perspective of
primary production in the oceans, giving insight into the role of the
world's oceans in the global carbon cycle. Ocean color data helps researchers map information relevant to society, such as water quality, hazards to human health like harmful algal blooms, bathymetry, and primary production and habitat types affecting commercially-important fisheries.
Chlorophyll as a proxy for phytoplankton
Season-long composites of ocean chlorophyll
concentrations. The purple and blue colors represent lower chlorophyll
concentrations. The oranges and reds represent higher chlorophyll
concentrations. These differences indicate areas with lesser or greater phytoplankton biomass.
The most widely used piece of information from ocean color remote
sensing is satellite-derived chlorophyll-a concentration. Researchers
calculate satellite-derived chlorophyll-a concentration from space based
on the central premise that the more phytoplankton is in the water, the
greener it is.
Phytoplankton are microscopic algae, marine primary producers
that turn sunlight into chemical energy that supports the ocean food
web. Like plants on land, phytoplankton create oxygen for other life on
earth. Ocean color remote sensing ever since the launch of SeaWiFS in
1997 has allowed scientists to map phytoplankton – and thus model primary production - throughout the world’s oceans over many decades, marking a major advance in our knowledge of the earth system.
Other applications
Suspended
sediments can be seen in satellite imagery following events when high
winds cause waves to stir up the seafloor, like in this image of the
western side of the Yucatan Peninsula. Darker brown colored water shows where sediments come from land via rivers, while lighter colored water shows where sediments come from the chalky calcium carbonate sands on the seafloor.
Beyond chlorophyll, a few examples of some of the ways that ocean color data are used include:
Harmful algal blooms
Researchers use ocean color data in conjunction with
meteorological data and field sampling to forecast the development and
movement of harmful algal blooms (commonly referred to as "red tides,"
although the two terms are not exactly the same). For example, MODIS
data has been used to map Karenia brevis blooms in the Gulf of Mexico.
Suspended sediments
Researchers use ocean color data to map the extent of river
plumes and document wind-driven resuspension of sediments from the
seafloor. For example, after hurricanes Katrina and Rita in the Gulf of
Mexico, ocean color remote sensing was used to map the effects offshore.
Sensors
Sensors
used to measure ocean color are instruments that measure light at
multiple wavelengths (multispectral) or a continuous spectrum of colors
(hyperspectral), usually spectroradiometers or optical radiometers. Ocean color sensors can either be mounted on satellites or airplanes, or used at earth’s surface.
Satellite sensors
The
sensors below are earth-orbiting satellite sensors. The same sensor can
be mounted on multiple satellites to give more coverage over time (aka
higher temporal resolution). For example, the MODIS sensor is mounted on
both Aqua and Terra satellites. Additionally, the VIIRS sensor is
mounted on both Suomi National Polar-Orbiting Partnership (Suomi-NPP or
SNPP) and Joint Polar Satellite System (JPSS-1, now known as NOAA-20)
satellites.
At earth’s surface, such as on research vessels, in the water using buoys,
or on piers and towers, ocean color sensors take measurements that are
then used to calibrate and validate satellite sensor data. Calibration
and validation are two types of "ground-truthing"
that are done independently. Calibration is the tuning of raw data from
the sensor to match known values, such as the brightness of the moon or
a known reflection value at earth’s surface. Calibration, done
throughout the lifetime of any sensor, is especially critical to the
early part of any satellite mission when the sensor is developed,
launched, and beginning its first raw data collection. Validation is the
independent comparison of measurements made in situ with measurements made from a satellite or airborne sensor. Satellite calibration and validation maintain the quality of ocean color satellite data.
There are many kinds of in situ sensors, and the different types are
often compared on dedicated field campaigns or lab experiments called
"round robins." In situ data are archived in data libraries such as the SeaBASS data archive. Some examples of in situ sensors (or networks of many sensors) used to calibrate or validate satellite data are:
Lunar theory attempts to account for the motions of the Moon. There are many small variations (or perturbations)
in the Moon's motion, and many attempts have been made to account for
them. After centuries of being problematic, lunar motion can now be
modeled to a very high degree of accuracy (see section Modern developments).
Lunar theory includes:
the background of general theory; including mathematical
techniques used to analyze the Moon's motion and to generate formulae
and algorithms for predicting its movements; and also
quantitative formulae, algorithms, and geometrical diagrams that may
be used to compute the Moon's position for a given time; often by the
help of tables based on the algorithms.
Lunar theory has a history of over 2000 years of investigation. Its
more modern developments have been used over the last three centuries
for fundamental scientific and technological purposes, and are still
being used in that way.
Applications
Applications of lunar theory have included the following:
In the eighteenth and nineteenth centuries, navigational tables based on lunar theory, initially in the Nautical Almanac, were much used for the determination of longitude at sea by the method of lunar distances.
In the very early twentieth century, comparison between lunar theory
and observation was used in another test of gravitational theory, to
test (and rule out) Simon Newcomb's suggestion that a well-known discrepancy in the motion of the perihelion of Mercury might be explained by a fractional adjustment of the power -2 in Newton's inverse square law of gravitation (the discrepancy was later successfully explained by the general theory of relativity).
In the mid-twentieth century, before the development of atomic
clocks, lunar theory and observation were used in combination to
implement an astronomical time scale (ephemeris time) free of the irregularities of mean solar time.
The
Moon has been observed for millennia. Over these ages, various levels of
care and precision have been possible, according to the techniques of
observation available at any time. There is a correspondingly long
history of lunar theories: it stretches from the times of the Babylonian
and Greek astronomers, down to modern lunar laser ranging.
Among notable astronomers and mathematicians down the ages, whose names are associated with lunar theories, are:
Other notable mathematicians and mathematical astronomers also made significant contributions.
The history can be considered to fall into three parts: from
ancient times to Newton; the period of classical (Newtonian) physics;
and modern developments.
Ancient times to Newton
Babylon
Of Babylonian astronomy, practically nothing was known to historians of science before the 1880s. Surviving ancient writings of Pliny had made bare mention of three astronomical schools in Mesopotamia – at Babylon, Uruk, and 'Hipparenum' (possibly 'Sippar'). But definite modern knowledge of any details only began when Joseph Epping
deciphered cuneiform texts on clay tablets from a Babylonian archive:
In these texts he identified an ephemeris of positions of the Moon.
Since then, knowledge of the subject, still fragmentary, has had to be
built up by painstaking analysis of deciphered texts, mainly in
numerical form, on tablets from Babylon and Uruk (no trace has yet been
found of anything from the third school mentioned by Pliny).
To the Babylonian astronomer Kidinnu
(in Greek or Latin, Kidenas or Cidenas) has been attributed the
invention (5th or 4th century BC) of what is now called "System B" for
predicting the position of the moon, taking account that the moon
continually changes its speed along its path relative to the background
of fixed stars. This system involved calculating daily stepwise changes
of lunar speed, up or down, with a minimum and a maximum approximately
each month.
The basis of these systems appears to have been arithmetical rather
than geometrical, but they did approximately account for the main lunar
inequality now known as the equation of the center.
The Babylonians kept very accurate records for hundreds of years of new moons and eclipses.
Some time between the years 500 BC and 400 BC they identified and began
to use the 19 year cyclic relation between lunar months and solar years
now known as the Metonic cycle.
This helped them build a numerical theory of the main
irregularities in the Moon's motion, reaching remarkably good estimates
for the (different) periods of the three most prominent features of the
Moon's motion:
The synodic month, i.e. the mean period for the phases of the
Moon. Now called "System B", it reckons the synodic month as 29 days and
(sexagesimally) 3,11;0,50 "time degrees", where each time degree is one degree of the apparent motion of the stars, or 4 minutes of time, and the sexagesimal values after the semicolon are fractions of a time degree. This converts to 29.530594 days = 29d 12h 44m 3.33s, to compare with a modern value (as at 1900 Jan 0) of 29.530589 days, or 29d 12h 44m 2.9s. This same value was used by Hipparchos and Ptolemy, was used throughout the Middle Ages, and still forms the basis of the Hebrew calendar.
The mean lunar velocity relative to the stars they estimated at
13° 10′ 35″ per day, giving a corresponding month of 27.321598 days, to compare with modern values of 13° 10′ 35.0275″ and 27.321582 days.
The anomalistic month, i.e. the mean period for the Moon's
approximately monthly accelerations and decelerations in its rate of
movement against the stars, had a Babylonian estimate of
27.5545833 days, to compare with a modern value 27.554551 days.
The draconitic month, i.e. the mean period with which the path of
the Moon against the stars deviates first north and then south in
ecliptic latitude by comparison with the ecliptic path of the Sun, was
indicated by a number of different parameters leading to various
estimates, e.g. of 27.212204 days, to compare with a modern value of 27.212221, but the Babylonians also had a numerical relationship that 5458 synodic months were equal to 5923 draconitic months,
which when compared with their accurate value for the synodic month
leads to practically exactly the modern figure for the draconitic month.
The Babylonian estimate for the synodic month was adopted for the
greater part of two millennia by Hipparchus, Ptolemy, and medieval
writers (and it is still in use as part of the basis for the calculated Hebrew (Jewish) calendar).
Greece and Hellenistic Egypt
Thereafter, from Hipparchus and Ptolemy in the Bithynian and Ptolemaic epochs down to the time of Newton's
work in the seventeenth century, lunar theories were composed mainly
with the help of geometrical ideas, inspired more or less directly by
long series of positional observations of the moon. Prominent in these
geometrical lunar theories were combinations of circular motions –
applications of the theory of epicycles.
Hipparchus,
whose works are mostly lost and known mainly from quotations by other
authors, assumed that the Moon moved in a circle inclined at 5° to the ecliptic,
rotating in a retrograde direction (i.e. opposite to the direction of
annual and monthly apparent movements of the Sun and Moon relative to
the fixed stars) once in 182⁄3 years. The circle acted as a deferent,
carrying an epicycle along which the Moon was assumed to move in a
retrograde direction. The center of the epicycle moved at a rate
corresponding to the mean change in Moon's longitude, while the period
of the Moon around the epicycle was an anomalistic month. This epicycle
approximately provided for what was later recognized as the elliptical
inequality, the equation of the center, and its size approximated to an equation of the center of about 5° 1'. This figure is much smaller than the modern value: but it is close to the difference between the modern coefficients of the equation of the center (1st term) and that of the evection:
the difference is accounted for by the fact that the ancient
measurements were taken at times of eclipses, and the effect of the
evection (which subtracts under those conditions from the equation of
the center) was at that time unknown and overlooked. For further information see also separate article Evection.
Ptolemy
Ptolemy's work the Almagest
had wide and long-lasting acceptance and influence for over a
millennium. He gave a geometrical lunar theory that improved on that of
Hipparchus by providing for a second inequality of the Moon's motion,
using a device that made the apparent apogee oscillate a little – prosneusis of the epicycle. This second inequality or second anomaly accounted rather approximately, not only for the equation of the center, but also for what became known (much later) as the evection.
But this theory, applied to its logical conclusion, would make the
distance (and apparent diameter) of the Moon appear to vary by a factor
of about 2, which is clearly not seen in reality. (The apparent angular diameter of the Moon does vary monthly, but only over a much narrower range of about 0.49°–0.55°.) This defect of the Ptolemaic theory led to proposed replacements by Ibn al-Shatir in the 14th century and by Copernicus in the 16th century.
Ibn al-Shatir and Copernicus
Significant advances in lunar theory were made by the Arab astronomer, Ibn al-Shatir (1304–1375).
Drawing on the observation that the distance to the Moon did not change
as drastically as required by Ptolemy's lunar model, he produced a new
lunar model that replaced Ptolemy's crank mechanism with a double
epicycle model that reduced the computed range of distances of the Moon
from the Earth. A similar lunar theory, developed some 150 years later by the Renaissance astronomer Nicolaus Copernicus, had the same advantage concerning the lunar distances.
Tycho Brahe, Johannes Kepler, and Jeremiah Horrocks
Tycho Brahe and Johannes Kepler
refined the Ptolemaic lunar theory, but did not overcome its central
defect of giving a poor account of the (mainly monthly) variations in
the Moon's distance, apparent diameter and parallax. Their work added to the lunar theory three substantial further discoveries.
The nodes and the inclination of the lunar orbital plane both appear to librate, with a monthly (according to Tycho) or semi-annual period (according to Kepler).
The lunar longitude has a twice-monthly Variation, by which the Moon moves faster than expected at new and full moon, and slower than expected at the quarters.
There is also an annual effect, by which the lunar motion slows down a little in January and speeds up a little in July: the annual equation.
The refinements of Brahe and Kepler were recognized by their
immediate successors as improvements, but their seventeenth-century
successors tried numerous alternative geometrical configurations for the
lunar motions to improve matters further. A notable success was
achieved by Jeremiah Horrocks,
who proposed a scheme involving an approximate 6 monthly libration in
the position of the lunar apogee and also in the size of the elliptical
eccentricity. This scheme had the great merit of giving a more realistic
description of the changes in distance, diameter and parallax of the
Moon.
Newton
A first gravitational period for lunar theory started with the work of Newton.
He was the first to define the problem of the perturbed motion of the
Moon in recognisably modern terms. His groundbreaking work is shown for
example in the Principia in all versions including the first edition published in 1687.
Newton's biographer, David Brewster,
reported that the complexity of Lunar Theory impacted Newton's health:
"[H]e was deprived of his appetite and sleep" during his work on the
problem in 1692-3, and told the astronomer John Machin that "his head never ached but when he was studying the subject". According to Brewster, Edmund Halley also told John Conduitt that when pressed to complete his analysis Newton "always replied that it made his head ache, and kept him awake so often, that he would think of it no more" [Emphasis in original].
Solar perturbation of lunar motion
Newton identified how to evaluate the perturbing effect on the
relative motion of the Earth and Moon, arising from their gravity
towards the Sun, in Book 1, Proposition 66, and in Book 3, Proposition 25. The starting-point for this approach is Corollary VI to the laws of motion.
This shows that if the external accelerative forces from some massive
body happens to act equally and in parallel on some different other
bodies considered, then those bodies would be affected equally, and in
that case their motions (relative to each other) would continue as if
there were no such external accelerative forces at all. It is only in
the case that the external forces (e.g. in Book 1, Prop. 66, and Book 3,
Prop. 25, the gravitational attractions towards the Sun) are different
in size or in direction in their accelerative effects on the different
bodies considered (e.g. on the Earth and Moon), that consequent effects
are appreciable on the relative motions of the latter bodies. (Newton
referred to accelerative forces or accelerative gravity
due to some external massive attractor such as the Sun. The measure he
used was the acceleration that the force tends to produce (in modern
terms, force per unit mass), rather than what we would now call the
force itself.)
Thus Newton concluded that it is only the difference between the
Sun's accelerative attraction on the Moon and the Sun's attraction on
the Earth that perturbs the motion of the Moon relative to the Earth.
Newton then in effect used vector decomposition of forces, to carry out this analysis. In Book 1, Proposition 66 and in Book 3, Proposition 25,
he showed by a geometrical construction, starting from the total
gravitational attraction of the Sun on the Earth, and of the Sun on the
Moon, the difference that represents the perturbing effect on the motion
of the Moon relative to the Earth. In summary, line LS in Newton's
diagram as shown below represents the size and direction of the
perturbing acceleration acting on the Moon in the Moon's current
position P (line LS does not pass through point P, but the text shows
that this is not intended to be significant, it is a result of the scale
factors and the way the diagram has been built up).
Newton's diagram 'to find the force of the Sun to perturb the Moon' accompanying Book 3, Proposition 25 of the Principia
Shown here is Newton's diagram from the first (1687) Latin edition of the Principia
(Book 3, Proposition 25, p. 434). Here he introduced his analysis of
perturbing accelerations on the Moon in the Sun-Earth-Moon system. Q represents the Sun, S the Earth, and P the Moon.
Parts of this diagram represent distances, other parts
gravitational accelerations (attractive forces per unit mass). In a dual
significance, SQ represents the Earth-Sun distance, and then it also
represents the size and direction of the Earth-Sun gravitational
acceleration. Other distances in the diagram are then in proportion to
distance SQ. Other attractions are in proportion to attraction SQ.
The Sun's attractions are SQ (on the Earth) and LQ (on the Moon).
The size of LQ is drawn so that the ratio of attractions LQ:SQ is the
inverse square of the ratio of distances PQ:SQ. (Newton constructs
KQ=SQ, giving an easier view of the proportions.) The Earth's attraction
on the Moon acts along direction PS. (But line PS signifies only
distance and direction so far, nothing has been defined about the scale
factor between solar and terrestrial attractions).
After showing solar attractions LQ on the Moon and SQ on the
Earth, on the same scale, Newton then makes a vector decomposition of LQ
into components LM and MQ. Then he identifies the perturbing
acceleration on the Moon as the difference of this from SQ. SQ and MQ
are parallel to each other, so SQ can be directly subtracted from MQ,
leaving MS. The resulting difference, after subtracting SQ from LQ, is
therefore the vector sum of LM and MS: these add up to a perturbing
acceleration LS.
Later Newton identified another resolution of the perturbing
acceleration LM+MS = LS, into orthogonal components: a transverse
component parallel to LE, and a radial component, effectively ES.
Alternative
depiction of solar perturbations, vectors LS1 and LS2, like LS in
Newton's diagram above, for 2 positions of the Moon P in its orbit
around the Earth S
Newton's diagrammatic scheme, since his time, has been re-presented
in other and perhaps visually clearer ways. Shown here is a vector
presentation
indicating, for two different positions, P1 and P2, of the Moon in its
orbit around the Earth, the respective vectors LS1 and LS2 for the
perturbing acceleration due to the Sun. The Moon's position at P1 is
fairly close to what it was at P in Newton's diagram; corresponding
perturbation LS1 is like Newton's LS in size and direction. At another
position P2, the Moon is farther away from the Sun than the Earth is,
the Sun's attraction LQ2 on the Moon is weaker than the Sun's attraction
SQ=SQ2 on the Earth, and then the resulting perturbation LS2 points
obliquely away from the Sun.
Solar perturbation vectors (arrows) analogous to LS at many positions of the Moon in its orbit around the Earth
Constructions like those in Newton's diagram can be repeated for many
different positions of the Moon in its orbit. For each position, the
result is a perturbation vector like LS1 or LS2 in the second diagram.
Shown here is an often-presented form of the diagram that summarises
sizes and directions of the perturbation vectors for many different
positions of the Moon in its orbit. Each small arrow is a perturbation
vector like LS, applicable to the Moon in the particular position around
the orbit from which the arrow begins. The perturbations on the Moon
when it is nearly in line along the Earth-Sun axis, i.e. near new or
full moon, point outwards, away from the Earth. When the Moon-Earth line
is 90° from the Earth-Sun axis they point inwards, towards the Earth,
with a size that is only half the maximum size of the axial (outwards)
perturbations. (Newton gave a rather good quantitative estimate for the
size of the solar perturbing force: at quadrature where it adds to the Earth's attraction he put it at 1⁄178.725
of the mean terrestrial attraction, and twice as much as that at the
new and full moons where it opposes and diminishes the Earth's
attraction.)
Newton also showed that the same pattern of perturbation applies,
not only to the Moon, in its relation to the Earth as disturbed by the
Sun, but also to other particles more generally in their relation to the
solid Earth as disturbed by the Sun (or by the Moon); for example
different portions of the tidal waters at the Earth's surface.[a]
The study of the common pattern of these perturbing accelerations grew
out of Newton's initial study of the perturbations of the Moon, which he
also applied to the forces moving tidal waters. Nowadays this common
pattern itself has become often known as a tidal force
whether it is being applied to the disturbances of the motions of the
Moon, or of the Earth's tidal waters – or of the motions of any other
object that suffers perturbations of analogous pattern.
After introducing his diagram 'to find the force of the Sun to
perturb the Moon' in Book 3, Proposition 25, Newton developed a first
approximation to the solar perturbing force, showing in further detail
how its components vary as the Moon follows its monthly path around the
Earth. He also took the first steps in investigating how the perturbing
force shows its effects by producing irregularities in the lunar
motions.
For a selected few of the lunar inequalities, Newton showed in
some quantitative detail how they arise from the solar perturbing force.
Much of this lunar work of Newton's was done in the 1680s, and
the extent and accuracy of his first steps in the gravitational analysis
was limited by several factors, including his own choice to develop and
present the work in what was, on the whole, a difficult geometrical
way, and by the limited accuracy and uncertainty of many astronomical
measurements in his time.
Classical gravitational period after Newton
The main aim of Newton's successors, from Leonhard Euler, Alexis Clairaut and Jean d'Alembert in the mid-eighteenth century, down to Ernest William Brown
in the late nineteenth and early twentieth century, was to account
completely and much more precisely for the moon's motions on the basis
of Newton's laws, i.e. the laws of motion and of universal gravitation
by attractions inversely proportional to the squares of the distances
between the attracting bodies. They also wished to put the
inverse-square law of gravitation to the test, and for a time in the
1740s it was seriously doubted, on account of what was then thought to
be a large discrepancy between the Newton-theoretical and the observed
rates in the motion of the lunar apogee. However Clairaut showed
shortly afterwards (1749–50) that at least the major cause of the
discrepancy lay not in the lunar theory based on Newton's laws, but in
excessive approximations that he and others had relied on to evaluate
it.
Most of the improvements in theory after Newton were made in
algebraic form: they involved voluminous and highly laborious amounts of
infinitesimal calculus and trigonometry. It also remained necessary,
for completing the theories of this period, to refer to observational
measurements.
Results of the theories
The
lunar theorists used (and invented) many different mathematical
approaches to analyse the gravitational problem. Not surprisingly, their
results tended to converge. From the time of the earliest gravitational
analysts among Newton's successors, Euler, Clairaut and d'Alembert,
it was recognized that nearly all of the main lunar perturbations could
be expressed in terms of just a few angular arguments and coefficients.
These can be represented by:
the mean motions or positions of the Moon and the Sun, together
with three coefficients and three angular positions, which together
define the shape and location of their apparent orbits:
the two eccentricities (, about 0.0549, and , about 0.01675) of the ellipses that approximate to the apparent orbits of the Moon and the Sun;
the angular direction of the perigees ( and ) (or their opposite points the apogees) of the two orbits; and
the angle of inclination (, mean value about 18523") between the planes of the two orbits, together with the direction () of the line of nodes in which those two planes intersect. The ascending node () is the node passed by the Moon when it is tending northwards relative to the ecliptic.
From these basic parameters, just four basic differential angular
arguments are enough to express, in their different combinations, nearly
all of the most significant perturbations of the lunar motions. They
are given here with their conventional symbols due to Delaunay; they are sometimes known as the Delaunay arguments:
the Moon's mean anomaly (angular distance of the mean longitude of the Moon from the mean longitude of its perigee );
the Sun's mean anomaly (angular distance of the mean longitude of the Sun from the mean longitude of its perigee );
the Moon's mean argument of latitude (angular distance of the mean
longitude of the Moon from the mean longitude of its ascending
(northward-bound) node );
the Moon's mean (solar) elongation (angular distance of the mean longitude of the Moon from the mean longitude of the Sun).
This work culminated into Brown's lunar theory (1897–1908) and Tables of the Motion of the Moon (1919). These were used in the American Ephemeris and Nautical Almanac until 1968, and in a modified form until 1984.
Largest or named lunar inequalities
Several
of the largest lunar perturbations in longitude (contributions to the
difference in its true ecliptic longitude relative to its mean
longitude) have been named. In terms of the differential arguments, they
can be expressed in the following way, with coefficients rounded to the
nearest second of arc ("):
The Moon's equation of the center, or elliptic inequality, was
known at least in approximation, to the ancients from the Babylonians
and Hipparchus onwards. Knowledge of more recent date is that it
corresponds to the approximate application of Kepler's
law of equal areas in an elliptical orbit, and represents the
speeding-up of the Moon as its distance from the Earth decreases while
it moves towards its perigee, and then its slowing down as its distance
from the Earth increases while it moves towards its apogee. The effect
on the Moon's longitude can be approximated by a series of terms, of
which the first three are .
The evection (or its approximation) was known to Ptolemy, but
its name and knowledge of its cause dates from the 17th century. Its
effect on the Moon's longitude has an odd-appearing period of about 31.8
days. This can be represented in a number of ways, for example as the
result of an approximate 6-monthly libration in the position of perigee,
with an accompanying 6-monthly pulsation in the size of the Moon's
orbital eccentricity. Its principal term is .
The Variation, discovered by Tycho Brahe, is a speeding-up of
the Moon as it approaches new-moon and full-moon, and a slowing-down as
it approaches first and last quarter. Its gravitational explanation with
a quantitative estimate was first given by Newton. Its principal term
is .
Annual equation
The
annual equation, also discovered by Brahe, was qualitatively explained
by Newton in terms that the Moon's orbit becomes slightly expanded in
size, and longer in period, when the Earth is at perihelion closest to
the Sun at the beginning of January, and the Sun's perturbing effect is
strongest, and then slightly contracted in size and shorter in period
when the Sun is most distant in early July, so that its perturbing
effect is weaker: the modern value for the principal term due to this
effect is .
Parallactic inequality
The
parallactic inequality, first found by Newton, makes Brahe's Variation a
little asymmetric as a result of the finite distance and non-zero
parallax of the Sun. Its effect is that the Moon is a little behind at
first quarter, and a little ahead at last quarter. Its principal term is
.
Reduction to the ecliptic
The
reduction to the ecliptic represents the geometric effect of expressing
the Moon's motion in terms of a longitude in the plane of the ecliptic,
although its motion is really taking place in a plane that is inclined
by about 5 degrees. Its principal term is .
The analysts of the mid-18th century expressed the perturbations of
the Moon's position in longitude using about 25-30 trigonometrical
terms. However, work in the nineteenth and twentieth century led to
very different formulations of the theory so these terms are no longer
current. The number of terms needed to express the Moon's position with
the accuracy sought at the beginning of the twentieth century was over
1400; and the number of terms needed to emulate the accuracy of modern
numerical integrations based on laser-ranging observations is in the
tens of thousands: there is no limit to the increase in number of terms
needed as requirements of accuracy increase.
Since the Second World War and especially since the 1960s, lunar
theory has been further developed in a somewhat different way. This has
been stimulated in two ways: on the one hand, by the use of automatic
digital computation, and on the other hand, by modern observational
data-types, with greatly increased accuracy and precision.
Wallace John Eckert, a student of Ernest William Brown and employee at IBM,
used the experimental digital computers developed there after the
Second World War for computation of astronomical ephemerides. One of the
projects was to put Brown's lunar theory into the machine and evaluate
the expressions directly. Another project was something entirely new: a numerical integration
of the equations of motion for the Sun and the four major planets. This
became feasible only after electronic digital computers became
available. Eventually this led to the Jet Propulsion Laboratory Development Ephemeris series.
In the meantime, Brown's theory was improved with better constants and the introduction of Ephemeris Time and the removal of some empirical corrections associated with this. This led to the Improved Lunar Ephemeris (ILE), which, with some minor successive improvements, was used in the astronomical almanacs from 1960 through 1983 and enabled lunar landing missions.
The most significant improvement of position observations of the Moon have been the Lunar Laser Ranging measurements, obtained using Earth-bound lasers and special retroreflectors
placed on the surface of the Moon. The time-of-flight of a pulse of
laser light to one of the retroreflectors and back gives a measure of
the Moon's distance at that time. The first of five retroreflectors that are operational today was taken to the Moon in the Apollo 11 spacecraft in July 1969 and placed in a suitable position on the Moon's surface by Buzz Aldrin. Range precision has been extended further by the Apache Point Observatory Lunar Laser-ranging Operation, established in 2005.
The
lunar theory, as developed numerically to fine precision using these
modern measures, is based on a larger range of considerations than the
classical theories: It takes account not only of gravitational forces
(with relativistic corrections) but also of many tidal and geophysical
effects and a greatly extended theory of lunar libration.
Like many other scientific fields this one has now developed so as to
be based on the work of large teams and institutions. An institution
notably taking one of the leading parts in these developments has been
the Jet Propulsion Laboratory (JPL) at California Institute of Technology;
and names particularly associated with the transition, from the early
1970s onwards, from classical lunar theories and ephemerides towards the
modern state of the science include those of J. Derral Mulholland and
J.G. Williams, and for the linked development of solar system
(planetary) ephemerides E. Myles Standish.
Since the 1970s, JPL has produced a series of numerically integrated Development Ephemerides
(numbered DExxx), incorporating Lunar Ephemerides (LExxx). Planetary
and lunar ephemerides DE200/LE200 were used in the official Astronomical
Almanac ephemerides for 1984–2002, and ephemerides DE405/LE405, of further improved accuracy and precision, have been in use as from the issue for 2003. The current ephemeris is DE440.
Analytical developments
In
parallel with these developments, a new class of analytical lunar
theory has also been developed in recent years, notably the Ephemeride Lunaire Parisienne by Jean Chapront and Michelle Chapront-Touzé from the Bureau des Longitudes.
Using computer-assisted algebra, the analytical developments have been
taken further than previously could be done by the classical analysts
working manually. Also, some of these new analytical theories (like ELP)
have been fitted to the numerical ephemerides previously developed at
JPL as mentioned above. The main aims of these recent analytical
theories, in contrast to the aims of the classical theories of past
centuries, have not been to generate improved positional data for
current dates; rather, their aims have included the study of further
aspects of the motion, such as long-term properties, which may not so
easily be apparent from the modern numerical theories themselves.
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