Instruct: Given a question, retrieve documents that answer the question.
Query: What is the circumference of the Earth?
Earth's circumference is the distance around the Earth, either around the equator () or around the poles ().
In modern times it has been used to define fundamental units of measurement of length – the nautical mile in the seventeenth century and the metre in the eighteenth. Earth's polar circumference is almost exactly 40,000 km because the metre was originally calibrated on this measurement (1/10-millionth of the distance between the poles and the equator), which is almost exactly 21,600 nautical miles (being defined as one minute of a degree – i.e. 360 multiplied by 60).
Posidonius calculated the Earth's circumference by reference to the position of the star Canopus. As explained by Cleomedes, Posidonius observed Canopus on but never above the horizon at Rhodes, while at Alexandria he saw it ascend as far as degrees above the horizon (the meridian arc between the latitude of the two locales is actually 5 degrees 14 minutes). Since he thought Rhodes was 5,000 stadia due north of Alexandria, and the difference in the star's elevation indicated the distance between the two locales was 1/48 of the circle, he multiplied 5,000 by 48 to arrive at a figure of 240,000 stadia for the circumference of the earth. It is generally thought that the stadion used by Posidonius was almost exactly 1/10 of a modern statute mile. Thus Posidonius's measure of 240,000 stadia translates to , not much short of the actual circumference of . Strabo noted that the distance between Rhodes and Alexandria is 3,750 stadia, and reported Posidonius's estimate of the Earth's circumference to be 180,000 stadia or . Pliny the Elder mentions Posidonius among his sources and without naming him reported his method for estimating the Earth's circumference. He noted, however, that Hipparchus had added some 26,000 stadia to Eratosthenes's estimate. The smaller value offered by Strabo and the different lengths of Greek and Roman stadia have created a persistent confusion around Posidonius's result. Ptolemy used Posidonius's lower value of 180,000 stades (about 33% too low) for the earth's circumference in his "Geography". This was the number used by Christopher Columbus in order to underestimate the distance to India as 70,000 stades.
A more accurate estimate was provided in Al-Biruni's "Codex Masudicus" (1037). In contrast to his predecessors, who measured the Earth's circumference by sighting the Sun simultaneously from two different locations, al-Biruni developed a new method of using trigonometric calculations, based on the angle between a plain and mountain top, which yielded more accurate measurements of the Earth's circumference, and made it possible for it to be measured by a single person from a single location.
Viète's formula for , the ratio of a circle's circumference to its diameter, is
Earth's circumference lines are the foundation of the coordinate system of latitude and longitude, introduced in the mid-second century BC by Hipparchus.
Given a circle, let "u" be the perimeter of an inscribed regular "n-"gon, and let "U" be the perimeter of a circumscribed regular "n-"gon. Then "u" and "U" are lower and upper bounds for the circumference of the circle that become sharper and sharper as "n" increases, and their average ("u" + "U")/2 is an especially good approximation to the circumference. To compute "u" and "U" for large "n", Archimedes derived the following doubling formulae:
The precise strength of Earth's gravity varies depending on location. The nominal "average" value at Earth's surface, known as is, by definition, 9.80665 m/s. This quantity is denoted variously as , (though this sometimes means the normal equatorial value on Earth, 9.78033 m/s), , gee, or simply (which is also used for the variable local value).
The summit of Mount Everest is higher above sea level, but the summit of Chimborazo is widely reported to be the farthest point on the surface from Earth's center, with Huascarán a very close second. The summit of the Chimborazo is the fixed point on Earth that has the utmost distance from the center – because of the oblate spheroid shape of the planet Earth, which is "thicker" around the Equator than measured around the poles. Chimborazo is one degree south of the Equator and the Earth's diameter at the Equator is greater than at the latitude of Everest ( above sea level), nearly 27.6° north, with sea level also elevated. Despite being lower in elevation above sea level, it is from the Earth's center, farther than the summit of Everest ( from the Earth's center). However, by height above sea level, Chimborazo is not the highest peak of the Andes.
Since the atmospheres, albedo and greenhouse effects of super-Earths are unknown, the surface temperatures are unknown and generally only an equilibrium temperature is given. For example, the black-body temperature of the Earth is 255.3 K (−18 °C or 0 °F ). It is the greenhouse gases that keep the Earth warmer. Venus has a black-body temperature of only 184.2 K (−89 °C or −128 °F ) even though Venus has a true temperature of 737 K (464 °C or 867 °F ). Though the atmosphere of Venus traps more heat than Earth's, NASA lists the black-body temperature of Venus based on the fact that Venus has an extremely high albedo (Bond albedo 0.90, Visual geometric albedo 0.67), giving it a lower black body temperature than the more absorbent (lower albedo) Earth.
In taxicab geometry, "p" = 1. Taxicab circles are squares with sides oriented at a 45° angle to the coordinate axes. While each side would have length formula_38 using a Euclidean metric, where "r" is the circle's radius, its length in taxicab geometry is 2"r". Thus, a circle's circumference is 8"r". Thus, the value of a geometric analog to formula_39 is 4 in this geometry. The formula for the unit circle in taxicab geometry is formula_40 in Cartesian coordinates and
Earth's magnetic field, also known as the geomagnetic field, is the magnetic field that extends from the Earth's interior out into space, where it meets the solar wind, a stream of charged particles emanating from the Sun. Its magnitude at the Earth's surface ranges from 25 to 65 microteslas (0.25 to 0.65 gauss). Approximately, it is the field of a magnetic dipole currently tilted at an angle of about 11 degrees with respect to Earth's rotational axis, as if there were a bar magnet placed at that angle at the center of the Earth. The North geomagnetic pole, located near Greenland in the northern hemisphere, is actually the south pole of the Earth's magnetic field, and the South geomagnetic pole is the north pole. The magnetic field is generated by electric currents due to the motion of convection currents of molten iron in the Earth's outer core driven by heat escaping from the core, a natural process called a geodynamo.
The formula for the area of a circle (more properly called the area enclosed by a circle or the area of a disk) is based on a similar method. Given a circle of radius , it is possible to partition the circle into sectors, as shown in the figure to the right. Each sector is approximately triangular in shape, and the sectors can be rearranged to form an approximate parallelogram. The height of this parallelogram is , and the width is half the circumference of the circle, or . Thus, the total area of the circle is :
Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors. The limit of the areas of the approximate parallelograms is exactly , which is the area of the circle.
The fact that many large celestial objects are approximately spheres makes it easier to calculate their surface gravity. The gravitational force outside a spherically symmetric body is the same as if its entire mass were concentrated in the center, as was established by Sir Isaac Newton. Therefore, the surface gravity of a planet or star with a given mass will be approximately inversely proportional to the square of its radius, and the surface gravity of a planet or star with a given average density will be approximately proportional to its radius. For example, the recently discovered planet, Gliese 581 c, has at least 5 times the mass of Earth, but is unlikely to have 5 times its surface gravity. If its mass is no more than 5 times that of the Earth, as is expected, and if it is a rocky planet with a large iron core, it should have a radius approximately 50% larger than that of Earth. Gravity on such a planet's surface would be approximately 2.2 times as strong as on Earth. If it is an icy or watery planet, its radius might be as large as twice the Earth's, in which case its surface gravity might be no more than 1.25 times as strong as the Earth's.
The earth sciences can include the study of geology, the lithosphere, and the large-scale structure of the earth's interior, as well as the atmosphere, hydrosphere, and biosphere. Typically, earth scientists use tools from geography, chronology, physics, chemistry, biology, and mathematics to build a quantitative understanding of how the earth works and evolves. Earth science affects our everyday lives. For example, meteorologists study the weather and watch for dangerous storms. Hydrologists study water and warn of floods. Seismologists study earthquakes and try to predict where they will strike. Geologists study rocks and help to locate useful minerals. Earth scientists often work in the field—perhaps climbing mountains, exploring the seabed, crawling through caves, or wading in swamps. They measure and collect samples (such as rocks or river water), then they record their findings on charts and maps.
The gravitational gradient of the Earth, that is, the change in the gravitational acceleration vector from one point on the Earth’s surface to another, is customarily measured in units of eotvos. The Earth's gravity gradient is dominated by the component due to Earth's near-spherical shape, which results in a vertical tensile gravity gradient of 3,080 E (an elevation increase of 1 m gives a decrease of gravity of about 0.3 mGal), and horizontal compressive gravity gradients of one half that, or 1,540 E. Earth’s rotation perturbs this in a direction-dependent manner by about 5 E. Gravity gradient anomalies in mountainous areas can be as large as several hundred eotvos.
Mars has an orbit with a semimajor axis of 1.524 astronomical units (228 million kilometers), and an eccentricity of 0.0934. The planet orbits the Sun in 687 days and travels 9.55 AU in doing so, making the average orbital speed 24 km/s.
Among irrational numbers are the ratio  of a circle's circumference to its diameter, Euler's number "e", the golden ratio "φ", and the square root of two; in fact all square roots of natural numbers, other than of perfect squares, are irrational.
Mercury is one of four terrestrial planets in the Solar System, and is a rocky body like Earth. It is the smallest planet in the Solar System, with an equatorial radius of . Mercury is also smaller—albeit more massive—than the largest natural satellites in the Solar System, Ganymede and Titan. Mercury consists of approximately 70% metallic and 30% silicate material. Mercury's density is the second highest in the Solar System at 5.427 g/cm, only slightly less than Earth's density of 5.515 g/cm. If the effect of gravitational compression were to be factored out from both planets, the materials of which Mercury is made would be denser than those of Earth, with an uncompressed density of 5.3 g/cm versus Earth's 4.4 g/cm.
The Moon's value is 0.0549, the most eccentric of the large moons of the Solar System. The four Galilean moons have eccentricity < 0.01. Neptune's largest moon Triton has an eccentricity of (), the smallest eccentricity of any known moon in the Solar System; its orbit is as close to a perfect circle as can be currently measured. However, smaller moons, particularly irregular moons, can have significant eccentricity, such as Neptune's third largest moon Nereid (0.75).
Ganymede (Jupiter III) is the largest and most massive moon of Jupiter and in the Solar System. The ninth largest object in the Solar System, it is the largest without a substantial atmosphere. It has a diameter of and is 8% larger than the planet Mercury, although only 45% as massive. Possessing a metallic core, it has the lowest moment of inertia factor of any solid body in the Solar System and is the only moon known to have a magnetic field. Outward from Jupiter, it is the seventh satellite and the third of the Galilean moons, the first group of objects discovered orbiting another planet. Ganymede orbits Jupiter in roughly seven days and is in a 1:2:4 orbital resonance with the moons Europa and Io, respectively.
Asia is the largest continent on Earth. It covers 9% of the Earth's total surface area (or 30% of its land area), and has the largest coastline, at . Asia is generally defined as comprising the eastern four-fifths of Eurasia. It is located to the east of the Suez Canal and the Ural Mountains, and south of the Caucasus Mountains (or the Kuma–Manych Depression) and the Caspian and Black Seas. It is bounded on the east by the Pacific Ocean, on the south by the Indian Ocean and on the north by the Arctic Ocean. Asia is subdivided into 48 countries, three of them (Russia, Kazakhstan and Turkey) having part of their land in Europe.
Eratosthenes, a Greek astronomer from Hellenistic Cyrenaica (276–194 BC), estimated Earth's circumference around 240 BC. He had heard that in Syene the Sun was directly overhead at the summer solstice whereas in Alexandria it still cast a shadow. Using the differing angles the shadows made as the basis of his trigonometric calculations he estimated a circumference of around 250,000 "stades". The length of a 'stade' is not precisely known, but Eratosthenes's figure only has an error of around five to fifteen percent. Eratosthenes used rough estimates and round numbers, but depending on the length of the stadion, his result is within a margin of between 2% and 20% of the actual meridional circumference, . Note that Eratosthenes could only measure the circumference of the Earth by assuming that the distance to the Sun is so great that the rays of sunlight are practically parallel.
1,700 years after Eratosthenes's death, while Christopher Columbus studied what Eratosthenes had written about the size of the Earth, he chose to believe, based on a map by Toscanelli, that the Earth's circumference was one-third smaller. Had Columbus set sail knowing that Eratosthenes's larger circumference value was more accurate, he would have known that the place that he made landfall was not Asia, but rather the New World.
Latitude lines, known as circles of latitude are parallel to each other; the equatorial circumference is the largest circle of latitude, and the Earth’s longest circumference distance. Other circles of latitude have smaller circumference; for example, the 60th parallel north or south is approximately half as long as the Equator.
In 1617 the Dutch scientist Willebrord Snellius assessed the circumference of the Earth at 24,630 Roman miles (24,024 statute miles). Around that time British mathematician Edmund Gunter improved navigational tools including a new quadrant to determine latitude at sea. He reasoned that the lines of latitude could be used as the basis for a unit of measurement for distance and proposed the nautical mile as one minute or one-sixtieth () of one degree of latitude. As one degree is of a circle, one minute of arc is of a circle – such that the polar circumference of the Earth would be exactly 21,600 miles. Gunter used Snell's circumference to define a nautical mile as 6,080 feet, the length of one minute of arc at 48 degrees latitude.