Planet Earth/1c. Measuring the Size and Shape of Earth
Introduction to Geodesy
[edit | edit source]Geodesy is the science of accurately measuring and understanding the Earth’s size and shape, as well as Earth’s orientation in space, rotation, and gravity. Geodesy is important in mapping the Earth’s surface for transportation, navigation, establishing national and state borders, and in real estate, land ownership and management of resources on the Earth’s surface. Many people in industrialized nations carry an extremely accurate geodetic tool in our pocket (a smart phone or tablet), that only recently the United States Military allowed civilian use of Global Positioning Systems (GPS). GPS utilizes Earth orbiting satellites to pin-point your location on planet Earth with a high degree of accuracy. The recent advancement of GPS allows everything from tracking packages, mapping migrating animals, to designing self-driving cars. It is astonishing to consider that before the advent of civilian use of GPS Earth Orbiting Satellites in the late-1990s, all mapping, tracking and navigation was carried out with rudimentary tools. These rudimentary tools have established a fairly accurate measurement of Earth’s size and shape for two and half millennia.
The sun rises in the east and sets in the west due to the rotation of the Earth around its polar axis, resulting in each longitude having a different temporal occurrence of when the sun is highest in the sky. Scholars knew that if one possessed an accurate clock, which was set to a specific noon-time, one could calculate the difference in time when the sun was highest in the sky at any location on Earth and compare that with the standard time set on a clock. Using this difference in time you could determine your distance in Longitude from the standard, which was called a Meridian.
If you have ever traveled by airplane (or car) across time zones, and had to set your watch to the new local time on your arrival, you have experienced this effect. In principle you could determine the distance in Longitude you traveled by how many hours you have to adjust your watch. While no accurate clocks existed for these ancient scholars to determine Longitude with great accuracy, scholars attempted to determine longitude as best as they could to generate maps along a grid system of Latitude and Longitude laid over a globe.
The History of Measuring Latitude
[edit | edit source]The earliest written texts that illustrate the Earth as a spherical body date to the writings of Parmenides of Elea, who lived in Elea, a Phocian Greek colony in what is today Southern Italy. These writings, mostly as Greek Poems, described the cosmos as a spherical moon orbiting around a spherical Earth, date to around 535 BCE. Sailors of the Mediterranean Sea had likely learned of the curvature of the Earth from the observation of ships on large bodies of water. As ships on the open ocean traverse farther and farther away from an observer they appear to sink below the horizon. The Moon and its phases in the sky also alluded to the spherical nature of both the Moon and the Earth, as well as the record of solar and lunar eclipses, when the spherical Moon or Earth block the sun’s light. There is no record that these and other early maritime navigators had calculated the circumference or radius of the Earth, but likely had discovered the spherical nature of Earth when exploring on the open waters of the ocean.
Eratosthenes of Cyrene was born on the northern coast of Africa around 276 BCE and following an education in Athens, Greece, was appointed chief librarian in Alexandria, Egypt. The Library of Alexandria had been founded by Ptolemy I Soter, a companion to Alexander the Great, who served as the ruler of Egypt after its conquest. The library was the center of learning and education, and housed the great works of Greek and Egyptian writing of the day. Eratosthenes had the full benefit of being at the center of this educational center, and wrote proficiently, although sadly few of his writings survived to the present day. A textbook written by Cleomedes, a Greek Scholar, a few centuries later describes a famous experiment conducted by Eratosthenes.
On a little island called Elephantine in the middle of the Nile River, near present day Aswan, was a water well, which during the longest day of the year the sun would shine directly down the dark well onto the surface of the water. For a few moments the sun’s reflection was perfectly centered within the well. Eratosthenes was curious if the same thing could happen in Alexandria, about 524 miles north of Elephantine Island. Rather than dig a well, Eratosthenes held up a rod (or more technically a gnomon, which is a rod that casts a shadow), perfectly perpendicular with the ground, and observed the sun’s shadow on the ground as the time approached noon on the longest day of the year in Alexandria, when the sun would be at its highest ascent in the sky. The sunlight hitting the vertical gnomon or rod in Alexandria produced a shadow even at noon. Eratosthenes measured the minimum length of the shadow, noting that the difference between the sun being directly overhead in Elephantine Island to the south and slightly overhead in Alexandria in the north, was likely due to the curvature of the Earth.
Eratosthenes also realized that if the sun was very far away, and sun light traveled parallel to the Earth’s surface, he could use the length of the shadow to calculate the circumference of the Earth along the north-south axis. He knew the distance between Alexandria and Elephantine Island, was 5000 stades, a unit of measurement lost to time, but roughly equivalent to 524 miles (843 kilometers). Eratosthenes calculated that the angle from the center of the Earth was about 1/50th (7.2 degrees), suggesting a pole-to-pole or meridional circumference of 26,200 miles (42,165 kilometers), which is remarkably close to our modern calculated circumference of 24,860 miles (40,008 kilometers). Eratosthenes also realized that by measuring the lengths of shadows on sticks, one could deduce your position north or south. The farther north one traveled the longer the shadows would be. Shadow length was also dependent on the time of year, which could be corrected using solar calendars, for example Eratosthenes measured the minimum mid-day shadow in Alexandria from a standard-length gnomon or rod for each day of the year. A traveler could carry a similar standard-length gnomon or rod and measure the length of shadow and compare this with the measured shadow for that day in Alexandria. This would tell the traveler how far north or south of Alexandria the traveler was.
Eratosthenes discovered not only the size and shape of the Earth, but also this amazing method to determine Latitude. Like a climbing ladder, latitude is the north-south direction between the poles measured in degrees, with the Equator, the middle belt of the Earth equal distant from the poles at 0 degrees, and the poles at 90 degrees, north and south respectively. Eratosthenes is often credited as the originator of Geography, the study of the arrangement of places and physical features on the Earth.
Note that Elephantine Island in Egypt is very close to the Tropic of Cancer, the most northerly circle of latitude on Earth at which the Sun is directly overhead at noon of the June (Summer) solstice, or longest day of the year for the Northern Hemisphere. There is also a circle of latitude called the Tropic of Capricorn, which is the most southernly circle of latitude on Earth at which the sun can be directly overhead at noon of the December (Winter) solstice. This is because the Earth is tilted at 23.5 degrees relative to its orbital plane.
Using the technique of casting shadows did not work well on ships and boats, because of the rocking motion while on water. To determine latitude at sea sailors would use the night sky, and measure the angle above the horizon to the North Star (Polaris) and compare this with star charts for the time of year.
The innovations of the Indian Mathematician Aryabhata in India around 500 CE, who calculated the irrational nature of pi (π) unlocking the use of calculating the circumference of the Earth using trigonometry. Aryabhata mathematics was translated into Arabic, and put into use by early Muslim scholars, particular the Muhammad ibn Musa al-Khwarizmi (referred to as Algorithmi by Latin speakers) head librarian of the House of Wisdom in Baghdad. He published a number of ingenious calculations of the positions of various cities and places. To determine latitude, he used a simpler method than casting shadows. He would take measurements using a plumb line (a weight dangled from a string), and measure the angle from the top of a peak or mountain to the observed horizon in the distance. This angle would tell you the degrees between the top of the peak and the horizon point, if you knew this distance, you could more accurately calculate the Earth’s circumference. While this allowed a more precise measure of the meridional circumference of Earth, it still did not provide a way to measure the equatorial circumference of Earth. Scholars assumed that the Earth was a perfect sphere and that the equatorial and meridional circumference of the Earth would be equal, but this equatorial measurement had not been determined. It was particularly difficult to determine your location along the east-west axis. Muhammad ibn Musa al-Khwarizmi invented algebra, and a way to position sets of numbers along a x-y grid system. While determining the latitude of any city was fairly a straight forward affair by this time, determining the Longitude or the east-west direction was problematic. Ptolemy, a Greek scholar eight centuries before, had attempted to map the Mediterranean Sea, but failed to determine distances along the east-west axis and had overestimated the length of the sea. Muhammad ibn Musa al-Khwarizmi set about an attempt to determine both Latitude and Longitude of all the major cities, in his Book of the Description of the Earth published in Arabic in 833 CE.
The History of Measuring Longitude
[edit | edit source]The inaccuracy of determining Longitude resulted in one of the worst misunderstandings of geography in 1492 CE. Christopher Columbus’s expedition from Spain across the Atlantic Ocean, was a leap of faith that he would reach India or Asia on the other side of the ocean. When his expedition found land (the Island of Hispaniola), he was convinced that they had arrived in India, as he was unable to determine his position in longitude with any accuracy. In 1499, Alonso de Ojeda, a companion on one of Columbus’s expeditions, led his own voyage back across the Atlantic with Amerigo Vespucci, an Italian scholar who was onboard to attempt mapping these new lands. The expedition followed the coast line southward along the coast of present-day Venezuela and Brazil, to the mouth of the Amazon River. Along the way Vespucci took readings of the Latitude, and was amazed as he observed southern constellations in the night sky that he had only read about. His measurements of Latitude took him within 6 degrees of the equator, far more south than expected if the land was India. In desperation he attempted to measure the position of Longitude using the Moon and Mars. Vespucci had with him charts of Mars’s position in the night sky relative to the Moon back in Europe, and noted the times of the year that Mars would be obscured by the Moon. He measured the distances between the Moon and Mars during these evenings when the Moon would obscure Mars in Europe, but was visible in the night sky on board of the ship those same evenings. By measuring the angle between the distance of the Moon and Mars on those dates listed in his charts, he could estimate the Longitude of their position, and came to the realization that they were not close to India, but had discovered a large continent, that extended far to the south. In 1507, the German cartographer Martin Waldseemüller named this new continent America, in honor of Amerigo Vespucci’s discovery on the first accurate map of the world, Universalis Cosmographia.
A better estimate of Longitude was needed, especially as sailors traversed the world more frequently in the centuries between 1500 and 1700, and the early colonization of America by Europeans. Monarchies offered huge sums of money to any scientist who could accurately determine Longitude, with Robert Hooke, a founding member of the Royal Society attempted to devise a spring-loaded clock or using a pendulum to measure time, and hence Longitude. John Harrison an expert clock maker, devised the first truly accurate clocks, or marine chronometers, that could be used to determine Longitude with a great deal of accuracy by 1761.
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The marine chronometer or clock would be set to Greenwich Mean Time (GMT), with noon or 12:00 pm set at the point of time that the Royal Observatory in Greenwich, England, observed the sun at its highest point in the sky. Greenwich, England, was set as 0 degrees longitude, and hence the Prime Meridian. Sailors could easily determine their longitude by looking at the marine chronometer, when the sun was highest in the sky, and read the clock set at GMT. The time depicted indicates how far east or west you are from the Prime Meridian.
Latitude and Longitude are measured using quadrate degrees, divided into 60 minutes and 60 seconds. For example, a Latitude of 40°27′19″ North and Longitude of 109°31′43″ West. Indicating a place 40 degrees 27 minutes 19 seconds north of the equator and 109 degrees 31 minutes and 43 seconds west of the Prime Meridian in Greenwich, England.
In modern usage, Latitude and Longitude is often given in decimal format, for example 40.45552° and −109.52875°, with positive Latitude indicating the Northern hemisphere and negative Latitude indicating the Southern hemisphere, while Negative Longitude indicating West of the Prime Meridian and Positive Longitude as East of the Prime Meridian. Any place on the surface of the Earth can be described with these two simple numbers. In fact, you can copy and paste any decimal Latitude and Longitude into a WolframAlpha query box and find its location on a map.
Using the refined Latitude and Longitude, the meridional circumference of the Earth is 24,860 miles (40,007.86 kilometers), while the equatorial circumference of the Earth is 24,901 miles (40,075.017 kilometers), indicating a slight bulge around the equator of 67.157 kilometers, so not a perfect sphere, but a slight oblate spheroid.
While knowing latitude and longitude is significant, determining distances between points on the Earth is a more important concept for everyday travelers. Many techniques were developed by early navigators through the principle of triangulation. Triangulation is the process of determining a location by forming triangles of known points. In ancient Utah, and through-out the American Southwest, the Ancient Pueblo designed towers in the desert, which were lit by fires. A traveler could navigate distances by taking the angle between two points, such as lit fires observed during the night, and know with certainty the direction and distance to travel to reach a destination. With the inaccuracy of latitude and longitude, early maritime navigators used triangulation of light houses along a coast to help navigate dangerous coastlines into the safety of bays and safe harbors, when their estimates of navigation were off. In China, triangulation was used to determine distances between cities, as well as the heights of mountains.
Triangulation works by taking the angles between two points of a known distance apart, and an unknown point in the distance, or the distance from a point to the line of sight between two other points from which you can measure the angle from the line of sight and the original starting point. These are expressed using trigonometric expressions, that require you to know two angles and a single distance, to calculate the third distance. Triangulation requires lines of sight, and worked best in desert environments with few obstructions of the view. Triangulation is difficult in dense forests with abundant trees or on the open ocean with few observable objects on the horizon. Triangulation was used to map much of the interior of the continents, through a network of measurements starting often along coastline cities at sea level or important city centers which had determined accurate latitude and longitude.
The concept of triangulation would become a very important concept to determine the size and shape of Earth during the space age, when Earth orbiting satellites could be used with great accuracy measuring the latitude and longitude of any point on Earth, and measure distances and elevations with a great deal of certainty. Rather than measure angles, multilateration uses distance in three-dimensions to find a point that lies at the intersection of three spheres, where the distance of the three radii of the spheres are known.
Measuring Earth from Space
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On October 4, 1957 the Soviet Union successfully launched Sputnik I, the first human designed satellite measuring nearly 2 feet in diameter (58.5 cm) into Earth’s orbit. Sputnik I was of a simple spherical design, but emitted two radio frequencies that could be received on Earth. Based on the emitted radio frequencies the position of the satellite could be determined by the doppler effect. The doppler effect is when the frequency of a wave changes depending on the traveling direction of an object. When Sputnik was traveling toward a location, radio receivers on Earth would detect higher frequencies, while Sputnik was traveling away from a location, radio receivers would detect lower frequencies. Anyone with a radio receiver could determine when Sputnik was directly overhead, because the radio frequency would change pitch due to the doppler effect.
Detecting the radio frequencies emitted by Sputnik allowed any radio receiving station on Earth to know its location relative to an orbiting satellite emitting radio waves. This allowed for the positioning of points on the Earth with a greater degree of certainty. Over the next several decades numerous satellites were launched into space, and set into orbit around the Earth. Most of these early satellites emitted radio signals which could be received on Earth’s surface. Much like triangulation, if you have a minimum of three satellites in orbit above a location, a receiver could triangulate its location from the distance of the radio signals emitted from three or more satellites in space.
One of the amazing breakthroughs of these early satellites was that it allowed for the detailed measurement of any location on Earth relative to the center of the Earth, and hence altimetry, measuring the height of a location above the center of the Earth, rather than above sea level. This innovation allowed a more precise measurement of the topography of the Earth’s surface. Sea level varies with daily and monthly tides, up and down, making it not a very good baseline, and various mathematical models of Earth’s dimensions had been used instead.
As Sputnik first circled Earth, Gladys West, a young African-American mathematician was working in Dalhgren Virginia at a navy base involved in programing early main-frame computers to calculate rocket trajectories. With the advent of Sputnik, the United States Military quickly realized the importance of satellite data in determining missile trajectories and use of long-distance rockets. Gladys West was a proficient mathematician, and in the 1980s the Navy gave her the seemingly impossible task of determining the topography of the ocean surface using satellite data from the newly launched GEOSAT satellite. This meant a refinement of triangulations to such a precision that the altimetry of swells and tides of the ocean could be measured from any ship as it navigated the oceans. West devised a system of mathematical corrections so that the surface topography of the ocean and land could be compared to a reference ellipsoid, called a geoid. A geoid is a pure mathematical model of the Earth, without its irregular topography, with the most commonly used model being the World Geodetic System (WGS84), however two other older geoid models are frequently used on maps for the Continental United States, the North American Datum of 1927 (NAD27), and North American Datum of 1983 (NAD83), which can differ by as much as 95 to 47 meters across North America, and were based on models first developed in 1866 for use in mapping. They differ slightly because they model the equatorial bulge of the Earth differently.
The World Geodetic System (WGS84) was a much better geoid to use for global applications, and widely used as an international standard. Gladys West developed a mathematical model to eliminate error, allowing precise dynamic sea surface topography, as well as latitude and longitude to be calculated with onboard ship computers in the 1980s. This innovation led to the use of GPS navigation found in most cell phones, ships and vehicles today.
Today there are a number of Earth orbiting satellites that work not by sending radio waves and calculating the distance by using the doppler effect, but by transmitting time stamped radio waves from onboard high precision atomic clocks. Each satellite emits a radio wave transmitting its current time of broadcast, when the radio wave is received the time is compared to another onboard atomic clock, and the difference between the two times is the length of time it took the radio waves traveling at the speed of light to reach the receiver. With at least three satellites emitting signals, the location of each satellite can be determined, although to be more precise, four or more emitting satellites are used to establish their locations relative to each other for greater accuracy. Using a GPS receiver, the spherical waves of emitted radio transmissions from four or more satellites can be used to find a precise location anywhere on Earth. The more satellites that can be triangulated using a receiver the more precise the location can be. The number changes because of the Earth’s rotation moves the position on Earth relative to the number of visible satellites that are orbiting above.
The United States GPS (Global Positioning System) navigation satellites are a network or constellation of around 33 satellites in orbit above Earth, each providing real-time signals to Earth used in high precision calculation of any location on Earth. There are five other networks of satellites developed by other countries, including the GLONASS network maintained by Russia, the Galileo network maintained by the European Union, the BeiDou network maintained by China, and the planned IRNSS and QZSS maintained by India and Japan respectively.
The precision of knowing any location on Earth is now at the sub-centimeter (less than an inch) for fixed ground GPS receivers. This technological break-through allows for the measurement of the movement of Earth’s surface and crust on a millimeter scale. One of the more ambiguous projects using this technology was EarthScope which operated from 2012 to 2019 that deployed thousands of GPS receiving stations across the continental United States, to measure the movement of the ground at each location. These GPS receivers observed the movement of continental plates, showing the relative quick movement (up to around 40 mm a year) of the ground below Southern California in respect to the interior of the rest of the continental United States, such as Utah. Such GPS receivers also demonstrated a twice a day vertical shift in the ground up and down due to the gravitational pull of the moon of 55 centimeters, and 15 centimeters due to the sun’s gravitational pull. So, while you are seemingly living on a solid unmoving Earth, it is in fact dynamically moving each day up and down as the solid interior is stretched out by the passage of the moon and sun, and horizontally as tectonic continental plates shift under your feet.
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d. How to Navigate Across Earth using a Compass, Sextant, and Timepiece. |