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Maya De la Belleza

Tierras Encarnesidas Glandestinas

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Planet Earth/1c. Measuring the Size and Shape of Earth

Introduction to Geodesy
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
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
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.

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
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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Planet Earth/1d. How to Navigate Across Earth using a Compass, Sexton, and Timepiece

How to Navigate Using a Compass
In an age where you can quickly determine your position on Earth in Latitude and Longitude with high precision, it may seem strange to learn about navigation methods using a Compass, Sextant and Timepiece, since most explorers of the Earth carry a GPS unit with them on their trips, or at least have a smart phone, or other electronic device with GPS capabilities. However, you may find yourself easily lost if that unit fails. You should prepare yourself to learn to navigate using older methods, including a compass, sextant and timepiece.

More than a thousand years ago, in China, it was discovered that rubbing an iron needle on a rock containing magnetite caused the needle to become magnetized. If the needle was placed through a cork and placed in a bowl of water, the needle would orientate itself along a specific direction as it floated on the surface of the water. Because all magnetized needles exhibit the same orientation, it became a useful tool for navigation. During the Song dynasty in China, about a thousand years ago, the compass was perfected as an orientating tool that travelers could carry with them, often with the magnetized needle balanced on a sharp point, under glass, to make it more practical than a cork in an open bowl of water. Widely regarded as one of the most important discoveries in China, a compass shows the cardinal directions (North, East, South, West), as they relate to the magnetized needle which is orientated along the Earth’s magnetic field.

The compass needle will orient itself along the north-south magnetic axis of the Earth, which differs depending on where you are, and changes with time. Typically, there is an N on the compass marking the direction of magnetic North. The needle will often have a red tip to indicate the southern direction, which you can line up with the S. When you hold the compass in your hand you can move it around until the needle lines up with the N and S, with the red tip (if there is one) pointing toward the S. The direction is now lined up with the Magnetic Poles of the Earth.

The North Magnetic Pole is a wandering point that currently is located at latitude 86.54°N and longitude 170.88°E, but wanders with an expected location in 2020 of 86.391°N 169.818°E. The oldest record of the magnetic pole’s location is in 1590 when it was found at 73.923°N 248.169°E. If the magnetic pole wanders so much, it may seem that a compass is an impractical tool for navigation. However, it is useful to get a quick bearing, as the magnetic needle often points north, unless you are high in the Arctic circle, or it can be adjusted if you know the declination of your location. A declination, sometimes called magnetic variation, is the angle between magnetic north and true north. Declination is written as degrees east or west from true north, or often shortened to positive degrees when east and negative when west. Magnetic declination changes over time and with location, so if you are using a compass for precise navigation you will need to keep it updated with your location.

The compass points along the magnetic axis, and a declination value is needed to obtain true north from a compass. Most topographic maps published by the United States Geological Survey will list the declination in the corner of the map, followed by a year. However, if you need to obtain the most current declination for your location you can look it up with the National Centers for Environmental Information, which is part of the National Oceanic and Atmospheric Administration of the Federal Government. They maintain an online Magnetic Field Calculator at (https://www.ngdc.noaa.gov/geomag/calculators/magcalc.shtml?#declination).

Once you found the current declination for your location, you will need to adjust your compass. For example, a declination 10° 21' E, you will need to have the needle point to 10° 21' E so that the N will point true north. Some compasses allow you to adjust the outer ring that marks the degrees that encircle the magnetic needle, so that you can set the declination for your location before you set out on a trip, for example by moving it right 10° 21' E, the compass needle will point true north, or N. Note that as a general rule in the United States locations west of the Mississippi River have an easterly magnetic declination, while east of the Mississippi River will have a westerly magnetic declination. A map of magnetic declinations is called an Isogonic Chart.

Compasses work because the Earth has a liquid and solid core made of mostly iron, which is magnetized because of its motion (called the Earth’s Dynamo). This solid inner core of the Earth orbits slowly over time, and results in a dynamic magnetic field that changes. Every few thousand or sometimes several millions of years, the magnetic pole will reverse directions. These reversals take several decades, and occur when the magnetic field of the Earth weakens. Several reversals have occurred in the last 1 million years, including an event 41,000 years ago, that only lasted 250 years, and another 780,000 years ago, which fixed the north and south magnetic poles at their current orientation. Magnetitic reversals are random events, likely due to the orientation of the solid and liquid iron core within the Earth. However, there have been some extremely long episodes during the age of dinosaurs in the Cretaceous Period when the magnetic field remained stable for millions of years. If the magnetic field undergoes a reversal during your life time, you will need to keep up to date with your local magnetic declinations, which may make using a magnetic compass problematic until the poles are fully reoriented.

Compass needles orient themselves relative to any magnetic field, such that if you hold a magnet close to the needle, the needle will orient to that field, causing the needle to be turned in orientation with the magnet. Some regions of the Earth have large deposits of magnetite, or iron ore that can cause compasses to not line up with the Earth’s magnetic field, such as the iron rich region north of Duluth Minnesota, or the mysterious Bangui magnetic anomaly in the Central African Republic. Also, be aware that high voltage electricity also emits a magnetic field, which can cause compass needles to not orient correctly if you are under a power line. There are also magnets found in mobile phones which also can cause the needles to not orient correctly when carried in a pocket with a mobile phone.

The Earth’s magnetic field is three-dimensional, such that the magnetic needle will not only point horizontally along the magnetic field but also tilt slightly vertical in orientation to a three-dimensional magnetic field. If you could measure this tilt on a compass needle, you could use it to determine latitude, as the needle will tilt more as you approach the magnetic poles. This technique has been used to determine ancient latitudes of Earth’s continents overtime, by measuring magnetic fields of the vertical tilt and horizontal orientation of magnetite grains buried in the rock record.

Compasses are used in navigation with a map, allowing a traveler to take a bearing. A compass bearing is the direction which you are headed, as shown by a compass and determined from a map.

Imagine that you are trying to find a cabin, and have come to a creek. You have a map showing the cabin on one side of creek, but you don’t know if you are on the same side of the creek as the cabin. If the map has a north arrow, you can orient your compass with the arrow. Lay down the compass, and turn both the compass and map, until the map’s north arrow lines up with north on the compass. Now you will be able to determine if you are on the right side of the creek, since you will be facing the same direction as depicted on the map. You can also determine the direction of the cabin from your location, and take a heading (measured on the compass in degrees, either in 360 degrees, or quadrate NE, SE, SW, NW each representing 90 degrees). For example, you might find that the line from your location by the river to the cabin is 25° degrees to the Northeast). As you walk you try to keep the compass bearing of your travel at 25° degrees Northeast looking at the compass, until you reach the cabin. Taking a compass bearing helps travelers to orient their direction of travel, so they don’t walk in circles. This is especially useful in dense forests or jungles where you are likely to get lost or turned around.

During the Song dynasty in China, compasses were paired with an odometer, which measured a distance not with steps, but using a wheel that would tick off rotations, providing very accurate distances between towns and villages. However, map making was oriented in respect to what a traveler might see as they journeyed through a land. With the rise of Kublai Khan during the Yuan dynasty and visits from European traders on the Silk Road, such as Marco Polo in the 1330s, China turned toward explorations westward, along the coasts of India, Arabia and Africa. During the Ming dynasty the explorer Zheng He produced the Mao Kun map, a map using only compass bearings and distances on an annotated scroll. The map is unique since it reads like a series of instructions for navigating an ocean voyage, much like one would play through a video game.

How to Navigate Using a Sextant
For navigation it is often useful to calculate your position in latitude and longitude and this requires using two other tools, a sextant and chronometer.

A Sextant is an instrument that measures the angle between an astronomical object, such as a star and the Earth’s horizon. It does this by using a mirror that reflects the sky against an image of the horizon, as you move the angle of the mirror. All you have to do is line up the astronomical object by changing the angle with the horizon line and read the angle measured. Often sextants have shaded glass to use if measuring the angle of the sun in the sky, so that you don’t damage your eye. One of the most important astronomical objects to measure using a sextant is the angle above the horizon to the star Polaris (the North Star). If you were to take a line through the axis of the North and South pole it would project northward toward a position near Polaris. Polaris can be found in the night sky by following the outer lip of the Big Dipper, or Ursa Major the Great Bear constellation. If you ever seen a time lapse photo of the night sky, all the stars will appear to rotate around this point in the night sky, because the Earth is rotating along this axis. This is called the Celestial North Pole. Once you have found Polaris in the night sky, you just measure the angle between the star and Earth’s horizon. The angle is equal to your latitude. In the southern hemisphere, there is no bright star near the Celestial South Pole, to find this point in the night sky, you have to draw a line from two constellations in the night sky. Where the line from the two bright stars in the Southern Cross and the two bright stars in Centaurus cross is the Celestial South Pole. Since it is important for navigation in the Southern Hemisphere, the Southern Cross is depicted on the flags of Australia, New Zealand, Papua New Guinea and Samoa, as well as Brazil, which shows both the Southern Cross and Centaurus on its flag.

Using a sextant to find your latitude is fairly straight forward when at sea, since the horizon line is easy to find, but when you are traveling in canyons, mountain valleys, and dense jungles it can be difficult to determine the horizon line. To help with finding the horizon, a leveling bubble can be used instead of matching the horizon line.

A sextant can also be used to determine latitude during the day, by Noon Sighting, or measuring the angle of the sun at its highest ascent at the local noon time. This angle needs to be subtracted from 90° and is recorded as the zenith distance. To convert zenith distance into latitude, you can look up the declination of the sun for each day of the year in a Nautical Almanac, which lists sun’s declination for the noon hour for each day. By subtracting or adding the declination from the zenith distance, you can find your latitude, depending on the position of the sun in the sky relative to the hemisphere.

Latitude = (90º – Noon Sighting angle) + declination

Latitude = Declination – (90º – Noon Sighting angle)

How to Navigate Using a Clock
If you measure the exact time of the sun’s highest ascent at the local noon time, you can compare this time with Greenwich Meridian Time (GMT), to determine longitude. If you maintain an accurate clock set to GMT, you can calculate your approximate longitude by adding 15° to each 1-hour difference observed from the local noon time if west of GMT. For example, if you record a local noon time of 5:00 pm GMT you are 75° West of the Greenwich Meridian. Note that you need to adjust for a slight difference between solar time and clock time, which differs by a few minutes depending on the time of year.

Rather than using a sextant, if you are on land, you can also use a gnomon or rod that casts a shadow and record the time set to GMT as the shadow length decreases during the approach to the local noon time, when the sun is highest in the sky. The GMT time at which the shadow is the shortest can be used to calculate the longitude, just remember to adjust for the slight difference between solar time and clock time.

Using these primitive methods, you can determine latitude and longitude within a kilometer (about a 1,000 yards), and this can be double checked with a modern-day GPS unit.

Public Land Survey System
Knowing latitude and longitude was of vital importance during times of war, and Thomas Hutchins fulfilled those duties as an Engineer mapping for the British Military in the mid-1700s. When hostilities broke out between the French and British forts in the western frontier of the American Colonies, Thomas Hutchins was called upon to map out the agreed peace treaty in 1763, which transferred land between the Appalachian Mountains and Mississippi River to British Colonial control, this region included the future states of western Pennsylvania, Ohio, Indiana, and Illinois. Hutchins also mapped parts of the Mississippi River and documented many of the native populations and mapped the various native towns along the rivers. He laid out plans for the city of Pittsburgh at the junction of the Ohio and Allegheny Rivers in western Pennsylvania, and mapped parts of Florida and Louisiana as well.

When the American Revolution broke out in 1776, he served with British forces, but was arrested for treason in sending coded messages to revolutionary forces. Taken to England, he escaped from prison, and made his way to France, where he met up with Benjamin Franklin, and returned to America as a patriot. After the war, Thomas Hutchins began mapping parts of Ohio using a new grid system, called the Public Land Survey System (PLSS). The public land survey system was a way to map a region by overlaying a grid of 36 numbered 1-square miles grids, with each set of 36 grids given a numbered township north or south and range west or east in reference to a local meridian line. For example, a location could be given as Section 10 Township 10 South Range 3 West. Indicating a square mile that was located in the 10th square of the 36-square mile grid, 10 townships to the south and 3 ranges to the west of the local meridian. As America’s first national geographer, the success of his system inspired its adoption by Thomas Jefferson, and all land west of the original American colonies, excluding Texas, would be mapped using this system over the next century.

The PLS system in the United States became the legal method to determine property lines, as lands were purchased during the Louisiana Purchase in 1803, or taken by force during the Mexican-American War in 1846. The PLS system was used by the United States to allocate land ownership, and used to push European settlement westward with the acquisition of lands from native tribes and peoples. Having an accurate grid map of a large land mass is a major advantage during conflict. Surveyors were often used by the military during this time, such as Captain John Fremont, who was able to acquire most of California without any shots fired during the Mexican-American War, due to his knowledge gained surveying the west. Accurate maps give small numbers of soldiers a huge advantage to mobilize and coordinate during times of war. These surveys also resulted in the United States government to allocate land ownership, that stripped much of the indigenous lands away from native peoples, and was used to establish reservation boundaries for native people in the Americas. The Public Land Survey System is still used as a legal grid system to designated land ownership for much of the western part of the United States of America.

Universal Transverse Mercator
A grid system does not fit perfectly on a round globe, and each section would not be a perfect square mile, since each line projecting north-south would converge as they approached the poles. The grid system worked for the latitudes in the continental United States, but was filled with many sections that were not completely square, and often corrections were added to the grid system in a haphazard fashion based on each local meridian. In other words, the PLS system was very localized to the western parts of the United States, and a system that could not be adapted to the rest of the world.

During World War II, there was a need to develop a better grid system of the Earth, projected on an accurate spherical model of the Earth. The United States Military adopted a method called the Universal Transverse Mercator coordinate system (UTM). The benefit of a global grid system is it takes a three-dimensional ellipsoid model of the Earth, such as the World Geodetic System (WGS84), and projects a grid onto a two-dimensional Cartesian coordinate system, which allows you to calculate distances very quickly from two points on a map. The UTM system divides the Earth into 60 north-south zones, each representing 6° of longitude. The zones are numbered, with zones 19 to 10 covering the continental United States, and zone 1 in the middle of the Pacific Ocean, near the International Date Line. Each zone has a central meridian, or north-south line that serves as an east-west refence point, and the equator which serves as the north-south reference point.

A location is defined by the distance the point is from these lines in meters. The distance from the central meridian of each zone is called the Easting, and the distance from the equator is called the Northing. For example, a location could be described as UTM Zone 18 585,000 m easting 4,515,500 m northing. You can convert UTM into Latitude and Longitude using an online converter (such as http://www.rcn.montana.edu/resources/converter.aspx), or convert Latitude and Longitude into UTM coordinates. On modern topographic maps, both UTM and Latitude and Longitude are indicated on the edges of the map, so that points can be easily located with either system. Using this knowledge, you can locate your position anywhere on Earth.


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Planet Earth/1e. Earth’s Motion and Spin

Earth’s Rotation Each Day
Right now, as you are reading this, your body is traveling at an incredibly fast speed through outer space. We can calculate one component of this speed by taking Earth’s circumference based on the ellipsoid model for the Earth’s dimensions, which exhibits an equatorial circumference of 24,901.46 miles (40,075.02 km). The Earth completes a rotation around its axis every day, or more precisely every 23 hours, 56 minutes, and 4 seconds. If you are located at the equator, your velocity (speed combined with a direction) can be calculated by dividing 24,901.46 miles by 23 hours, 56 minutes, and 4 seconds, which equals 1,040.45 miles per hour. Of course, this depends on your latitude, and decreases as you approach the poles.

One way to imagine this rotation is if you have ever watched an old record album spin, or a free spinning bike wheel. The central axis of the spinning album or wheel is stationary, while the outer edges of the circle are traveling the circumference of the circle with each revolution, the further you move from the center of rotation, the quicker your speed. In other words, the larger the wheel, the faster the rotation, and the more distance is covered per unit time.

Early scientists such as Galileo, were aware of this motion and were curious as to why we do not feel this motion on the surface of the Earth. If you imagine an ant crossing a spinning record album, at the edges the ant would feel the fast motion as air zoomed by, and the pull of a centrifugal force working to fling the poor ant off the spinning record album, but as the ant crawled toward the center its feeling of motion would decrease.

The same thing can be felt if you have ever been on a merry-go-round, the closer you are toward the center the less you feel the motion of your spin. However, on Earth we do not feel like we are traveling at over a 1,000 miles per hour at the Equator, and standing still near the north or south pole.

This bizarre paradox inspired Isaac Newton to study motion, and in the process, discovered gravity, and the three laws of motion that govern how all objects move in the universe. His discoveries were published in 1687, in his book Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy).

Before we can discuss why we do not feel the rotational force of the Earth, we need to define some terms.

 * Speed
 * is a measure of an objects’ distance traveled by the length of time the object took to travel that distance. For example, a car might have a velocity of 50 Miles per Hour (80.5 Kilometers per Hour).

 * Velocity
 * is speed combined with a direction in space.

 * Acceleration
 * is the rate of change of velocity per unit of time. For example, if a car is traveling at 50 Miles per Hour for 50 Miles and does not change speed, then it has 0 acceleration. A car that is stationary and not moving, also has 0 acceleration (within the respective frame of reference). This is because in both examples the velocity does not change.

Mathematically it is more difficult to calculate acceleration, one way to do it is to find the change of velocity for each unit of time. For example, a car going from 0 to 50 Miles Per Hour over a 5-hour long race course, we can find the speed at each 1-hour intervals and average them.

At the starting line the car is traveling at 0 miles per hour. At 1 hour the car is traveling at 10 miles per hour. At 2 hours the car is traveling at 20 miles per hour. At 3 hours the car is traveling at 30 miles per hour. At 4 hours the car is traveling at 40 miles per hour. At 5 hours the car is traveling at 50 miles per hour. Each hour the car increases its velocity by 10 miles per hour.

So, the average acceleration is equal to the average change in velocity divided by the average change in time, so the average of 10, 10, 10, 10, 10, in this example. The average acceleration is equal to 10 miles per hour, per hour (or hour squared).

If you know a little calculus, we can find what is called instantaneous acceleration, or the acceleration using the formula:
 * $$\mathbf{a} = \lim_{{\Delta t}\to 0} \frac{\Delta \mathbf{v}}{\Delta t} = \frac{d\mathbf{v}}{dt}$$

Basically, what this equation is stating is that acceleration is the derivative of velocity with respect to time.

What Isaac Newton, suggested as to the reason we do not feel this spinning motion on Earth is that the velocity of the Earth’s rotation is constant. Objects that are set into motion and have a constant velocity are said to exhibit inertia. These objects have zero acceleration.

Acceleration is when velocity changes over time. Isaac Newton realized that objects in motion will stay in motion, unless acted upon by another force. This is referred to as the law of inertia. In the weightless environment of outer space, an astronaut can spin a basketball and it will continue to spin at that velocity unless it hits another object, or another object acts against that motion. The reason that we do not feel the spin of the Earth is that everything is spinning at this constant velocity, or exhibiting the same inertia force.

However, as Isaac Newton realized, you should be feeling a centrifugal force due to this rotational force. A force is any interaction that causes an object to be moved in a direction.

Newton asked a simple question, why do objects, such as apples, fall to the Earth rather than get flung into outer space due to the rotation of the Earth?

He set about measuring the acceleration of falling objects. For example, a ball dropped from a tower. Just before the ball is dropped its velocity is 0 meters per second, but after 1 second, the ball is traveling at 10 meters per second. At 2 seconds the ball is traveling at 20 meters per second. At 3 seconds the ball is traveling at 30 meters per second. At 4 seconds the ball is traveling at 40 meters per second. At 5 seconds the ball is traveling at 50 meters per second. This sounds familiar. Each second the ball increases its velocity by 10 meters per second. So, the acceleration of the falling object is 10 meters per second per second (or second squared) or 10 m/sec2.

A century of experiments would show that falling objects on Earth’s surface have an acceleration of 9.8 m/sec2. All objects, no matter their mass, will fall at this rate.

(In reality objects will be hitting air (gas) as they fall, an object in motion will stay in motion until it is hit by another object, in this case “air” particles. This air adds resistance during free fall. So objects like parachutes, which are broad, wide and capture lots of air as they fall, or feathers will fall more slowly. Nevertheless the standard acceleration of 9.8 m/sec2 is still the same, but drag opposes this force.).

The force of a falling object is related to both mass and acceleration.

Force is measured by the mass (measured in kilograms) multiplied by the acceleration (measured in meters per second squared). Isaac Newton was rewarded by having a unit of measurement named after him!

One Newton unit of force is equal to 1-kilogram x 1 m/sec2.

Hence a bowling ball with a mass of 5 kilograms will exert a force of 5 kilograms x 9.8 m/s2 or 49 Newtons. A beach ball with a mass of 2 kilograms will exert a force of 2 kilograms x 9.8 m/s2 or 19.6 Newtons. An object measured in Newtons is a weight, since weights incorporate both mass and the acceleration. The unit of Pounds (lbs.) is also a unit of weight.

This common acceleration on the surface of the Earth is the acceleration due to gravity, which is 9.8 m/sec2. Isaac Newton realized that there was a force acting to keep objects against the surface of the Earth, and it was directly related to the mass of the Earth. The larger the mass an object had, the more its gravitational force would be. It was also related to the object’s proximity, the closer the object was, the more acceleration due to gravity the object will have. Using this mathematical relation, Newton proposed that the 9.8 m/sec2 acceleration of gravity could be used to find out how much mass the Earth had, using this formula.

 * $$g = \frac{GMe}{re^2}$$

g = 9.8 m/sec2 and is the acceleration of gravity on the surface of the Earth.

re = the radius of the Earth, or distance from the center of the Earth to the surface which can be found if we know the circumference of the Earth.

Me = the mass of the Earth, measured in kilograms.

G = the gravitational constant “sometimes called Big G”, a constant number, with the units of m3/kg ⋅s2.

Mass = Density x Volume. Density is the how compact a substance is and is measured relative to another substance, such as water. In other words, density is how well a substance or object floats or sinks. Volume is the cubic dimensions or space that an object occupies.

Isaac Newton did not know the value of Big G (the gravitational constant), but knew that it was a tiny number, since the mass and radius of the Earth were very large numbers, and the result of the equation had to equal 9.8 m/sec2.

The Quest to Find Big G
Newton’s work spurred a new generation of scientists trying to determine Big G, the gravitational constant. One way to determine Big G was to determine the density, volume and radius of the Earth. We can solve for Big G using this formula,

 * $$G = \frac{g r^2}{D V}$$

where g is the acceleration of gravity on Earth, r is the radius of the Earth from its center to the surface, D is density of Earth, and V is Earth’s volume.

One of Isaac Newton’s colleagues was Edmond Halley. Halley was one of the most brilliant scientists of the day, and is famous for his calculations of the periodicity of comets, in fact Halley’s Comet is named after him. However, he is less well known for his hypothesis that the Earth was hollow on the inside. He proposed that Earth’s density, and hence mass, was much smaller than if Earth was composed of a very dense solid inner core. During the late 1600s and early 1700s, scientists debated what the density of Earth was. Newton suggested an average density about 5 times more than water, while Halley suggested an average density less than water for the interior of the Earth. The problem was no one knew the value of Big G.

During the next century there was much discussion on the density of the Earth (the value for D). Expeditions into caverns and dark caves around the world were trying to find an entrance to the purported hollow center of the Earth. This debate captured the interest of a little short man named John Michell, who was the head of a church in Yorkshire, England, but dabbled in science in his spare time, and often wrote to fellow scientists of the day, including Benjamin Franklin. In his spare time, he thought of an experiment to measure Big G, by using a set of big very dense lead balls placed in close proximity to a set of smaller, but also very dense lead balls suspended from a string tied to a balancing rod. When the large lead balls are placed next to the smaller lead balls, the force of gravity will attract the two balls to each other. This attraction causes the balancing rod to shift slightly. To measure this movement or change in the balancing rods angle, a light was reflected off a mirror set on top of the balancing rod. Knowing the mass and radius of the lead balls, allowed one to solve for the gravitational constant, or Big G, which if known could be used to determine Earth’s density.

One of John Michell’s close friends was Henry Cavendish, a well born son of a wealthy scientist. Henry suffered from what would be called autism today, as he was incredibly shy, and struggled to carry on conversations with anyone not his close friend. Then at the age of 68, John Michell died, and left his experiment to Henry Cavendish to complete. In a large building, Henry reconstructed the experiment with the lead balls near his home, and calculated an accurate measure of Big G, the gravitational constant, which is 6.674×10−11 m3/kg⋅s2.

Using this number for Big G, it was demonstrated the Earth is not hollow, and that it is in fact, denser than rocks near the surface of the Earth which are about 3 g/cm³, with an average density of 5.51 g/cm3, or 5.5 times greater than water. This proved that the Earth is not hollow on the inside, but much denser than average rocks found on the Earth’s surface.

Henry Cavendish’s accurate calculation of the gravitational constant allows you to calculate any object’s acceleration of gravity given its mass and radius from its center of mass. The relationship between an object’s mass, radius, and the acceleration of gravity is a fundamental concept in understanding the motion of not only Earth, but other planets, moons, and stars. As well as the gravitational forces acting to hold astrological objects in orbit with each other. Furthermore, it explains why large objects in the universe take on spherical shapes around the center of mass. The acceleration of gravity also explains why we do not feel the Earth’s spin, and why objects and substances on Earth do not get flung into outer space. They are held against the Earth by its gravitational force.

Will the Earth ever stop spinning?
Should you worry about whether the Earth’s rotation or spin will slow down, and could there ever be a day in the future in when the Earth would stop rotating?

The length of the day is the time the Earth rotates once, with each longitude facing the sun once and only once during this daily rotation. If the Earth’s spin is slowing down over time, the length of the day will increase, resulting in longer days and longer nights. Today the Earth takes 23 hours, 56 minutes and 4.1 seconds to complete a rotation. (Note that it takes precisely 24 hours for the sun to reach its highest point in the sky each day, which is slightly longer than Earth’s spin, since the Earth moves a little relative to the sun each day).

Of course, the amount of daylight and night varies depending on your location and time of year, because the Earth rotates around a polar axis that is tilted at 23.5° in relationship to the sun. This is why people in Alaska (at a higher latitude) experience longer daylight during the month of July, and longer darkness during the month of December, than someone living near the Equator. The question to ask is, has the length of the Earth’s spin remained constant at 23 hours, 56 minutes and 4.1 seconds?

Like a spinning top, the Earth’s spin could be slowing down. Measuring the length of each rotation of the Earth requires clicking a very accurate stopwatch each day, and recording the time it takes for Earth to make one rotation. For the most part it says pretty close to 23 hours, 56 minutes and 4.1 seconds. However, the length does fluctuate by about 4 to 5 milliseconds. In other words, 0.004 to 0.005 seconds are added or subtracted from each day. These fluctuations appear to be on a decadal cycle, so the days in the 1860s were shorter by 0.006 seconds compared to days in the 1920s. These decadal fluctuations are believed to be the result of the transfer of angular momentum between the Earth’s fluid outer core and surrounding solid mantle, as well as tidal friction forces of the ocean as it slushes back and forth over the surface of the Earth while it spins. Weaker fluctuations occur over a yearly cycle, with days in June, July and August shorter by 0.001 seconds compared to days in December, January and February. These weaker fluctuations are cause by the atmosphere and ocean friction as the Earth spins, producing an oscillation called the “Chandler Wobble” after the American Scientist S. C. Chandler. In fact, the Earth is not just a solid mass of rock, we have a liquid ocean and a gaseous atmosphere that impacts the length of each day. It is like you are on a washing machine spinning around with wet clothes, and depending on where those clothes are in each spin cycle, there will be some variation in the speed of the spin itself.

Climate change also can have a rather important impact on the length of the day. If we were to compare the average day length during the last glacial period (25,000 years ago) to today, the day would be shorter. This is because of the Earth’s polar moment of inertia has decreased. As the great polar ice sheets that covered much of the polar regions started to melt, the distribution of the Earth’s mass shifted, from near the center of the spinning planet at the polar regions (as ice sheets), toward the equator (as melted ocean water). This change in inertia is the same phenomenon you observe when an ice skater brings his or hers arms out during a spin. The speed of the spin slows down. So as the Earth’s great ice sheets melted over the last 25,000 years, the Earth, like the spinning ice skater, projected more of its mass outward from its center toward the Equator when all that polar ice melted, slowing the spin.

While these fluctuations are interesting, they are small (several milliseconds), but we are interested in finding out when the Earth will stop spinning, and for that question we need a much longer record of day lengths, going back millions of years.

Fossil organisms keep records for the length of each year, month and day millions of years in Earth’s past. Fossil corals that live in the inter-tidal zone of the ocean are subjected to twice daily tides caused by the rotation of the Earth and gravitational pull of the moon, and amplified by the relative location of the sun. These changes in water depth result in a record in the growth rings, as well as cyclic sediments such as tidal rhythmites and banded iron formations. Using this information, Earth’s rotation has increased by 15.84 seconds every million years.

Isaac Newton proposed, objects in motion will stay in motion, unless acted upon by another force. So what force is slowing down Earth’s rotation?

The answer is our nearest neighbor— the moon! The moon is Earth’s only natural satellite with an equatorial circumference of 10,921 km (or 6,786 miles), about 27% the size of Earth. It rotates around Earth each lunar month of 27.32 days, in an unusual orbit called a synchronous rotation. This results in the strange fact that the Moon always keeps nearly the same face or surface pointed toward Earth. The opposite side of the moon, which you do not see from Earth in the night sky, is erroneously called the “dark side” of the moon.

Both sides are illuminated once every 29.5 Earth days, as the moon rotates around Earth, resulting in different phases of illumination of the moon by the sun. The moon’s axis of rotation is only slightly tilted to 5.14° in respect to the sun, and has been slowing down by Earth’s rotation to became “tidally” locked with the Earth.

With the moon’s slower lunar month-long rotation around Earth, it acts like a slow brake applied to Earth’s spin. The Earth will slow down to match the moon’s orbit of 27.32 days or 559.68 hours. At this point the Earth will be locked with the same spin as the orbit of the moon around the Earth.

An Earth with the length of rotation equal to the current lunar month would make the days on Earth last for 27.32 days, resulting in extreme daytime and nighttime temperatures like those experienced on the life-less surface of the moon today! Is this something for you to worry about?

Not anytime soon, Earth is slowed by the braking of the Moon by just a few seconds every million years, such that it will not be until 121 billion years in the future that Earth will become locked in this death orbit with the Moon, and by then, the Earth and Moon would likely have been engulfed by an expanding Sun!

The effect of the Moon’s orbit around the Earth can be observed with shifts in the ocean tide. When the moon is positioned directly above a position on the Earth (sublunar), the ocean at that position will be pulled closer to the moon due to the moon’s gravitational force producing a high tide along the coastline. An equal high tide will be felt on the opposite or antipodal side of the Earth as well. A low tide will be observed when the moon is not located on either the sublunar or antipodal sides of the Earth. As liquid water is more directly influenced by the attraction of the Moon’s gravity than rocks that compose the solid Earth you are likely more familiar with ocean tides, but there is also Earth tides which causes the Earth to bulge with the motion of the Moon. The sun also exerts some gravitational pull on Earth and can change the magnitude of the tides depending on the seasons. You can now explain the length of a day, and the length of a lunar month, the tides, but what causes the length of a year.

Earth’s Orbit around the Sun, The Year.

The Earth as a whole is not only spinning, but also traveling through space on an orbital path around the sun. Unlike the moon, Earth has a very dramatic tilt of its pole axis of 23.5° relative to the sun, such that during half of this voyage around the sun, the north pole faces the sun, and the south pole faces away from the sun. The tilt of Earth’s rotation of 23.5° results in longer days for the northern hemisphere when it is closer to the sun (June, July, August), and shorter days for the southern hemisphere, while the shorter days for the northern hemisphere (November, December, January) relate to longer days in the Southern Hemisphere. Because of the tilt of Earth’s axis, we have the four seasons of Summer, Fall, Winter, and Spring, which differ depending on your location in each hemisphere.

You might be surprised to learn that the orbit around the sun is not a perfect circle, as often depicted in illustrations of the solar system, but travels in an elliptical orbit around the sun. This can be demonstrated on Earth by documenting the sun’s position at noon every day of the year, which depicts a figure-8, called an analemma in the sky. The sun’s position at noon on the top of the figure 8 will happen on the day of the summer solstice, while the sun’s position at noon on the bottom of the figure 8 will happen on the day of the winter solstice, with the distance between the two points in the sky measuring Earth’s tilt of 23.5°. However, the width of the figure-8 is due to the elliptical path of the Earth around the sun. The figure-8 is not a perfect 8, but with one loop larger than the other. This is due to the fact that Sun is not positioned directly in the center of Earth’s elliptical orbit around it. During December-January, the Earth is closer to the sun, while in June-July, the Earth is farther away. The time of year when the Earth is closest to the sun is called the Perihelion, while the time of year when the Earth is farthest from the sun is called the Aphelion.

This is opposite of what you might think, as in the Northern Hemisphere, you are closer to the sun during the cold winter months, while during the hot summer months you are farther from the sun.

The distance from the sun varies from 0.9833 AU to 1.0167 AU, where AU is the Astronomical Unit, which is the average distance between the Sun and the Earth, which is defined as 150 million kilometers (93 million miles). Hence every year the distance from the Earth to the Sun differs by about 5 million kilometers (3.1 million miles).

While Earth’s orbit around the sun may seem like a bunch of numbers and facts to memorize, the discovery that the Sun and not the Earth was the center of the solar system was a major scientific discovery. The reason for this revolution in thought was that for centuries an equally valid explanation for the yearly cycle of Earth’s orbit was proposed.

Ptolemy’s Incorrect Geocentric Model of the Solar System
In the years shortly after the death of the Pharaoh Cleopatra and the fall of the city of Alexandria, Egypt to Roman annexation, an astronomer living in the city by the name of Claudius Ptolemy devised a model of the solar system. Ptolemy’s passion was mapping the stars, and he noticed that each night the path of Mars would move differently in reference to other stars in the night sky. Over the course of several years around 58 CE, he documented the path of Mars in the night sky demonstrating that Mars looped in the night sky over the course of several months. For example, Mars would move with the stars each night for several weeks, but then circle back for several weeks, before looping back around before heading off in the direction it started on. Because the path of Mars looped back, Ptolemy regarded this motion as a retrograde motion, and when Mars was progressing normally with the stars, a prograde motion. Ptolemy followed the Greek tradition of Aristotle, that the Earth was the center of the universe. So why were the planets of Mars and Venus looping in the night sky, they should be traveling in straight paths across the night sky, since they were orbiting the Earth, rather than the Sun? He devised a complex geocentric model of the solar system suggesting that the orbit of Mars, as well as other known planets like Venus followed an epicycle, an additional circular orbital path in addition to their orbits around Earth. It would be a millennium and a half before Ptolemy’s model of the solar system would be disproven.

Copernicus’s Correct Heliocentric Model of the Solar System
Nicolaus Copernicus published his alternative idea in his book De revolutionibus orbium coelestium (On the Revolutions of the Heavenly Spheres) in 1543. The Heliocentric view of the Solar System placed the Sun at the center of the Solar System rather than the Earth. In doing so, Copernicus demonstrated that the epicycle orbits were actually due to the observation from Earth of the passing of Mars in its own orbit around the Sun.

Copernicus viewed the Solar System as if the planets were racing around a circular track. Earth was on the inside track, while Mars was on an outside track. As Earth moved along its track on the inside, the view of Mars on the outside track would change. The retrograde motion comes naturally as a consequence of viewing a moving Mars from the perspective of a moving Earth. Copernicus rejected the epicycles needed to produce retrograde motion, rather the planets moved in a circular orbit around the Sun. Copernicus’s book is one of the most important books in science ever published, but still needed some modification, such as the fact that the Earth rotates around the Sun in an elliptical path, as do other planets rather than a circular orbit.

How fast are you traveling through space?
At the beginning of this module we talked about how fast you are traveling through space, and used the rotation or spin of the Earth, we can now add the component of Earth’s orbit through space around the sun. The distance Earth travels around the Sun is 940 million km (584 million mi) which it accomplishes every 365.256 days. The year is not evenly divided into days so calendars have to add an extra day every 4 years, or “leap days.” We can determine the velocity of this motion around the Sun, and determine that the Earth, and everything on its surface is traveling at a remarkably fast speed around the Sun of 66,619.94 miles per hour, or 107,230.73 kilometers per hour. Imagine, if you will, that as you sit there reading this you are traveling at this incredibly fast speed on a planet sling-shooting around a Star, at 30 times the speed of the fastest airplane. Because you are!

This astonishing fact, that you are on board a fast-moving object speeding through outer space, inspired Richard Buckminster Fuller in 1969, to coin the concept that Earth is simply a spaceship traveling through the vastness of the universe. Spaceship Earth, as he called our planet, is just a giant vessel, like a battleship sailing cross an empty ocean of space. He warned that you and all life on this “spacecraft” should be prepared for a long voyage.

Earth’s Galactic Voyage
John Michell, the short fat clergyman from Yorkshire, England, who devised the experiment that proved the Earth was not hollow, proposed in a letter written in 1784 that there might be objects in the universe that have so much mass that their gravitational force of acceleration would suck in even light rays, and he called these mysterious super massive objects, dark stars. Today, we call them Black Holes. The quest to find these mysterious super massive objects in the universe was enhanced by his suggestion that gravitational effects of these objects might be seen in nearby visible bodies. However, they remained a mathematical curiosity of simply taking Newton’s equations and extrapolating them for objects with enormous mass—millions of times more than the Sun.

In 1950, no one had observed one of these super massive objects in the universe, and Jocelyn Bell, a young girl at a boarding school in England, was struggling with the female only curriculum centered around the domestic subjects of cooking and sewing. When a science class was offered only the boys were allowed to attend. Furious, she and her parents protested, and she was allowed to attend the science class with two other female students. Jocelyn Bell loved physics most of all, and in 1965 went on to study physics at the University of Cambridge. She joined a team of researchers listening for radio waves from outer space. They had been picking up blips and squawks of radio waves from faint stars. Scientists called these signals quasi-stellar radio sources, which the American astronomer Hong-Yee Chiu simplified by calling them quasars. In the summer of 1967 Jocelyn Bell and her professor Antony Hewish were looking over the printouts of the newly constructed array of radio telescopes built to detect these quasar signals from space. She noticed a regular pattern of blips every 1.3373 seconds, while tempted to attribute these radio wave patterns to aliens, they jokingly called the regular pulse of signals little green men, but realized as others had that this radio signal was produced by a super massive object with enormous gravitational forces. When viewed through a telescope, the signal was coming from a faint star, which was recognized as a Neutron Star, an extremely massive star, which was spinning at an incredibly fast rate, with a pulse of electromagnetic radiation emitted every 1.3373 seconds. These radio signals are thought to be produced as nebulous clouds of gas are pulled into these supermassive stars forming an accretion disk, which emits powerful magnetic fields and radio waves as these gases fall through the disk into the neutron star—like colossal bolts of lightning.

Scientists realized that these super massive objects could be detected by using large arrays of radio telescopes to map these signals coming from space onto the sky.

Researchers focused their attention to the center of the one of the brightest stars in the night sky, which is actually a cluster of stars called Messier 87, also known as Virgo A, the brightest point in the Virgo Constellation. It had been recognized as a cluster of stars by Charles Messier in 1781, and classified by Edwin Hubble in 1931 as an elliptical nebula of stars. Today its known as a galaxy consisting of billions of stars.

Radio waves from Messier 87 indicated that near its center is a super massive object, representing a black hole. In 2019, the Event Horizon Telescope, a network of radio telescopes, focused on this point and imaged the signals coming from its center, producing the first image of a black hole, which resembles a ghostly dark spot surrounded by light. At the center of the dark spot is an object that is 6.5 billion times the mass of the Sun and 55 million light years away. The Event Horizon Telescope is also focused on a point in the night sky first detected by radio waves in 1974, which is thought to be the center of our own galaxy of stars, the Milky Way. In the night sky, a streak of stars appears to sweep across the night sky when viewed on an especially dark night. These stars are your closest stellar neighbors existing within your own galaxy. The Milky Way is a collection of billions of stars, including the Sun that swirl around a central point. The center of the Milky Way is located near the star Sagittarius A* (pronounced Sagittarius A-star), which is in the Sagittarius constellation. Here it has been observed that nearby stars swirl around a point, which is the location of another black hole that is 4 million times more massive than the Sun, and only 25,000 light years away. It is the nearest supermassive black hole to you. Astronomers have measured the rate of the Sun’s rotation around this point at the center of the Milky Way Galaxy, and determined that the entire Solar System takes about 240 to 230 million years to travel around this galactic orbit around this black hole. The last time our Solar System occupied this space relative to Sagittarius A*, was before dinosaurs had evolved on Earth!

However, do not assume this passage of the Solar System around this point is slow. Earth, and the entire Solar System is zipping along this path at an incredibly fast rate of travel.

Given that 1 light year is equal 5.879 x 1012 miles, and that it takes 240 million years to travel a circumference of 157,080 million light years, or a path of 9.23471 x 1017 miles in 2.1024 x 1012 hours, our Solar System is zipping around this black hole at a velocity of 439,246 miles per hour!

You are truly on a very fast spaceship, Earth’s motion in relation to its polar axis is between 0 and 1,040.45 miles per hour (1674.44 km/hr), depending on your latitude. Earth’s motion in relation to the Sun is 66,620 miles per hour (107,214.5 km/hr), and Earth’s galactic motion in relation to Sagittarius A* is 439,246 miles per hour (706,898 km/hr).