The angular diameter, angular size, apparent diameter, or apparent size is an angular measurement describing how large a sphere or circle appears from a given point of view. In the vision sciences, it is called the visual angle, in optics, it is the angular aperture; the angular diameter can alternatively be thought of as the angle through which an eye or camera must rotate to look from one side of an apparent circle to the opposite side. Angular radius equals half the angular diameter; the angular diameter of a circle whose plane is perpendicular to the displacement vector between the point of view and the centre of said circle can be calculated using the formula δ = 2 arctan , in which δ is the angular diameter, d is the actual diameter of the object, D is the distance to the object. When D ≫ d, we have δ ≈ d / D, the result obtained is in radians. For a spherical object whose actual diameter equals d a c t, where D is the distance to the centre of the sphere, the angular diameter can be found by the formula δ = 2 arcsin The difference is due to the fact that the apparent edges of a sphere are its tangent points, which are closer to the observer than the centre of the sphere.
For practical use, the distinction is only significant for spherical objects that are close, since the small-angle approximation holds for x ≪ 1: arcsin x ≈ arctan x ≈ x. Estimates of angular diameter may be obtained by holding the hand at right angles to a extended arm, as shown in the figure. In astronomy, the sizes of celestial objects are given in terms of their angular diameter as seen from Earth, rather than their actual sizes. Since these angular diameters are small, it is common to present them in arcseconds. An arcsecond is 1/3600th of one degree, a radian is 180/ π degrees, so one radian equals 3,600*180/ π arcseconds, about 206,265 arcseconds. Therefore, the angular diameter of an object with physical diameter d at a distance D, expressed in arcseconds, is given by: δ = d / D arcseconds; these objects have an angular diameter of 1″: an object of diameter 1 cm at a distance of 2.06 km an object of diameter 725.27 km at a distance of 1 astronomical unit an object of diameter 45 866 916 km at 1 light-year an object of diameter 1 AU at a distance of 1 parsec Thus, the angular diameter of Earth's orbit around the Sun as viewed from a distance of 1 pc is 2″, as 1 AU is the mean radius of Earth's orbit.
The angular diameter of the Sun, from a distance of one light-year, is 0.03″, that of Earth 0.0003″. The angular diameter 0.03″ of the Sun given above is the same as that of a person at a distance of the diameter of Earth. This table shows the angular sizes of noteworthy celestial bodies as seen from Earth: The table shows that the angular diameter of Sun, when seen from Earth is 32′, as illustrated above, thus the angular diameter of the Sun is about 250,000 times that of Sirius. The angular diameter of the Sun is about 250,000 times that of Alpha Centauri A; the angular diameter of the Sun is about the same as that of the Moon. Though Pluto is physically larger than Ceres, when viewed from Earth Ceres has a much larger apparent size. Angular sizes measured in degrees are useful for larger patches of sky. However, much finer units are needed to measure the angular sizes of galaxies, nebulae, or other objects of the night sky. Degrees, are subdivided as follows: 360 degrees in a full circle 60 arc-minutes in one degree 60 arc-seconds in one arc-minuteTo put this in perspective, the full Moon as viewed from Earth is about 1⁄2°, or 30′.
The Moon's motion across the sky can be measured in angular size: 15° every hour, or 15″ per second. A one-mile-long line painte
Earth is the third planet from the Sun and the only astronomical object known to harbor life. According to radiometric dating and other sources of evidence, Earth formed over 4.5 billion years ago. Earth's gravity interacts with other objects in space the Sun and the Moon, Earth's only natural satellite. Earth revolves around the Sun in a period known as an Earth year. During this time, Earth rotates about its axis about 366.26 times. Earth's axis of rotation is tilted with respect to its orbital plane; the gravitational interaction between Earth and the Moon causes ocean tides, stabilizes Earth's orientation on its axis, slows its rotation. Earth is the largest of the four terrestrial planets. Earth's lithosphere is divided into several rigid tectonic plates that migrate across the surface over periods of many millions of years. About 71% of Earth's surface is covered with water by oceans; the remaining 29% is land consisting of continents and islands that together have many lakes and other sources of water that contribute to the hydrosphere.
The majority of Earth's polar regions are covered in ice, including the Antarctic ice sheet and the sea ice of the Arctic ice pack. Earth's interior remains active with a solid iron inner core, a liquid outer core that generates the Earth's magnetic field, a convecting mantle that drives plate tectonics. Within the first billion years of Earth's history, life appeared in the oceans and began to affect the Earth's atmosphere and surface, leading to the proliferation of aerobic and anaerobic organisms; some geological evidence indicates. Since the combination of Earth's distance from the Sun, physical properties, geological history have allowed life to evolve and thrive. In the history of the Earth, biodiversity has gone through long periods of expansion punctuated by mass extinction events. Over 99% of all species that lived on Earth are extinct. Estimates of the number of species on Earth today vary widely. Over 7.6 billion humans live on Earth and depend on its biosphere and natural resources for their survival.
Humans have developed diverse cultures. The modern English word Earth developed from a wide variety of Middle English forms, which derived from an Old English noun most spelled eorðe, it has cognates in every Germanic language, their proto-Germanic root has been reconstructed as *erþō. In its earliest appearances, eorðe was being used to translate the many senses of Latin terra and Greek γῆ: the ground, its soil, dry land, the human world, the surface of the world, the globe itself; as with Terra and Gaia, Earth was a personified goddess in Germanic paganism: the Angles were listed by Tacitus as among the devotees of Nerthus, Norse mythology included Jörð, a giantess given as the mother of Thor. Earth was written in lowercase, from early Middle English, its definite sense as "the globe" was expressed as the earth. By Early Modern English, many nouns were capitalized, the earth became the Earth when referenced along with other heavenly bodies. More the name is sometimes given as Earth, by analogy with the names of the other planets.
House styles now vary: Oxford spelling recognizes the lowercase form as the most common, with the capitalized form an acceptable variant. Another convention capitalizes "Earth" when appearing as a name but writes it in lowercase when preceded by the, it always appears in lowercase in colloquial expressions such as "what on earth are you doing?" The oldest material found in the Solar System is dated to 4.5672±0.0006 billion years ago. By 4.54±0.04 Bya the primordial Earth had formed. The bodies in the Solar System evolved with the Sun. In theory, a solar nebula partitions a volume out of a molecular cloud by gravitational collapse, which begins to spin and flatten into a circumstellar disk, the planets grow out of that disk with the Sun. A nebula contains gas, ice grains, dust. According to nebular theory, planetesimals formed by accretion, with the primordial Earth taking 10–20 million years to form. A subject of research is the formation of some 4.53 Bya. A leading hypothesis is that it was formed by accretion from material loosed from Earth after a Mars-sized object, named Theia, hit Earth.
In this view, the mass of Theia was 10 percent of Earth, it hit Earth with a glancing blow and some of its mass merged with Earth. Between 4.1 and 3.8 Bya, numerous asteroid impacts during the Late Heavy Bombardment caused significant changes to the greater surface environment of the Moon and, by inference, to that of Earth. Earth's atmosphere and oceans were formed by volcanic outgassing. Water vapor from these sources condensed into the oceans, augmented by water and ice from asteroids and comets. In this model, atmospheric "greenhouse gases" kept the oceans from freezing when the newly forming Sun had only 70% of its current luminosity. By 3.5 Bya, Earth's magnetic field was established, which helped prevent the atmosphere from being stripped away by the solar wind. A crust formed; the two models that explain land mass propose either a steady growth to the present-day forms or, more a rapid growth early in Earth history followed by a long-term steady continental area. Continents formed by plate tectonics
The ecliptic is the mean plane of the apparent path in the Earth's sky that the Sun follows over the course of one year. This plane of reference is coplanar with Earth's orbit around the Sun; the ecliptic is not noticeable from Earth's surface because the planet's rotation carries the observer through the daily cycles of sunrise and sunset, which obscure the Sun's apparent motion against the background of stars during the year. The motions as described above are simplifications. Due to the movement of Earth around the Earth–Moon center of mass, the apparent path of the Sun wobbles with a period of about one month. Due to further perturbations by the other planets of the Solar System, the Earth–Moon barycenter wobbles around a mean position in a complex fashion; the ecliptic is the apparent path of the Sun throughout the course of a year. Because Earth takes one year to orbit the Sun, the apparent position of the Sun takes one year to make a complete circuit of the ecliptic. With more than 365 days in one year, the Sun moves a little less than 1° eastward every day.
This small difference in the Sun's position against the stars causes any particular spot on Earth's surface to catch up with the Sun about four minutes each day than it would if Earth would not orbit. Again, this is a simplification, based on a hypothetical Earth that orbits at uniform speed around the Sun; the actual speed with which Earth orbits the Sun varies during the year, so the speed with which the Sun seems to move along the ecliptic varies. For example, the Sun is north of the celestial equator for about 185 days of each year, south of it for about 180 days; the variation of orbital speed accounts for part of the equation of time. Because Earth's rotational axis is not perpendicular to its orbital plane, Earth's equatorial plane is not coplanar with the ecliptic plane, but is inclined to it by an angle of about 23.4°, known as the obliquity of the ecliptic. If the equator is projected outward to the celestial sphere, forming the celestial equator, it crosses the ecliptic at two points known as the equinoxes.
The Sun, in its apparent motion along the ecliptic, crosses the celestial equator at these points, one from south to north, the other from north to south. The crossing from south to north is known as the vernal equinox known as the first point of Aries and the ascending node of the ecliptic on the celestial equator; the crossing from north to south is descending node. The orientation of Earth's axis and equator are not fixed in space, but rotate about the poles of the ecliptic with a period of about 26,000 years, a process known as lunisolar precession, as it is due to the gravitational effect of the Moon and Sun on Earth's equatorial bulge; the ecliptic itself is not fixed. The gravitational perturbations of the other bodies of the Solar System cause a much smaller motion of the plane of Earth's orbit, hence of the ecliptic, known as planetary precession; the combined action of these two motions is called general precession, changes the position of the equinoxes by about 50 arc seconds per year.
Once again, this is a simplification. Periodic motions of the Moon and apparent periodic motions of the Sun cause short-term small-amplitude periodic oscillations of Earth's axis, hence the celestial equator, known as nutation; this adds a periodic component to the position of the equinoxes. Obliquity of the ecliptic is the term used by astronomers for the inclination of Earth's equator with respect to the ecliptic, or of Earth's rotation axis to a perpendicular to the ecliptic, it is about 23.4° and is decreasing 0.013 degrees per hundred years due to planetary perturbations. The angular value of the obliquity is found by observation of the motions of Earth and other planets over many years. Astronomers produce new fundamental ephemerides as the accuracy of observation improves and as the understanding of the dynamics increases, from these ephemerides various astronomical values, including the obliquity, are derived; until 1983 the obliquity for any date was calculated from work of Newcomb, who analyzed positions of the planets until about 1895: ε = 23° 27′ 08″.26 − 46″.845 T − 0″.0059 T2 + 0″.00181 T3 where ε is the obliquity and T is tropical centuries from B1900.0 to the date in question.
From 1984, the Jet Propulsion Laboratory's DE series of computer-generated ephemerides took over as the fundamental ephemeris of the Astronomical Almanac. Obliquity based on DE200, which analyzed observations from 1911 to 1979, was calculated: ε = 23° 26′ 21″.45 − 46″.815 T − 0″.0006 T2 + 0″.00181 T3 where hereafter T is Julian centuries from J2000.0. JPL's fundamental ephemerides have been continually updated; the Astronomical Almanac for 2010 specifies:ε = 23° 26′ 21″.406 − 46″.836769 T − 0″.0001831 T2 + 0″.00200340 T3 − 0″.576×10−6 T4 − 4″.34×10−8 T5 These expressions for the obliquity are intended for high precision over a short time span ± several centuries. J. Laskar computed an expression to order T10 good to 0″.04/1000 years over 10,000 years. All of these expressions are for the mean obliquity, that is, without the nutation of the equator included; the true or instantaneous obliquity includes the nutation. Most of the major bodies of the Solar System o
In astronomy, axial tilt known as obliquity, is the angle between an object's rotational axis and its orbital axis, or, the angle between its equatorial plane and orbital plane. It differs from orbital inclination. At an obliquity of 0 degrees, the two axes point in the same direction. Earth's obliquity oscillates between 24.5 degrees on a 41,000-year cycle. Over the course of an orbital period, the obliquity does not change and the orientation of the axis remains the same relative to the background of stars; this causes one pole to be directed more toward the Sun on one side of the orbit, the other pole on the other side—the cause of the seasons on Earth. There are two standard methods of specifying tilt; the International Astronomical Union defines the north pole of a planet as that which lies on Earth's north side of the invariable plane of the Solar System. The IAU uses the right-hand rule to define a positive pole for the purpose of determining orientation. Using this convention, Venus is tilted 177°.
Earth's orbital plane is known as the ecliptic plane, Earth's tilt is known to astronomers as the obliquity of the ecliptic, being the angle between the ecliptic and the celestial equator on the celestial sphere. It is denoted by the Greek letter ε. Earth has an axial tilt of about 23.4°. This value remains about the same relative to a stationary orbital plane throughout the cycles of axial precession, but the ecliptic moves due to planetary perturbations, the obliquity of the ecliptic is not a fixed quantity. At present, it is decreasing at a rate of about 47″ per century. Earth's obliquity may have been reasonably measured as early as 1100 BC in India and China; the ancient Greeks had good measurements of the obliquity since about 350 BC, when Pytheas of Marseilles measured the shadow of a gnomon at the summer solstice. About 830 AD, the Caliph Al-Mamun of Baghdad directed his astronomers to measure the obliquity, the result was used in the Arab world for many years. In 1437, Ulugh Beg determined the Earth's axial tilt as 23°30′17″.
It was believed, during the Middle Ages, that both precession and Earth's obliquity oscillated around a mean value, with a period of 672 years, an idea known as trepidation of the equinoxes. The first to realize this was incorrect was Ibn al-Shatir in the fourteenth century and the first to realize that the obliquity is decreasing at a constant rate was Fracastoro in 1538; the first accurate, western observations of the obliquity were those of Tycho Brahe from Denmark, about 1584, although observations by several others, including al-Ma'mun, al-Tusi, Purbach and Walther, could have provided similar information. Earth's axis remains tilted in the same direction with reference to the background stars throughout a year; this means that one pole will be directed away from the Sun at one side of the orbit, half an orbit this pole will be directed towards the Sun. This is the cause of Earth's seasons. Summer occurs in the Northern hemisphere. Variations in Earth's axial tilt can influence the seasons and is a factor in long-term climate change.
The exact angular value of the obliquity is found by observation of the motions of Earth and planets over many years. Astronomers produce new fundamental ephemerides as the accuracy of observation improves and as the understanding of the dynamics increases, from these ephemerides various astronomical values, including the obliquity, are derived. Annual almanacs are published listing the methods of use; until 1983, the Astronomical Almanac's angular value of the mean obliquity for any date was calculated based on the work of Newcomb, who analyzed positions of the planets until about 1895: ε = 23° 27′ 8.26″ − 46.845″ T − 0.0059″ T2 + 0.00181″ T3where ε is the obliquity and T is tropical centuries from B1900.0 to the date in question. From 1984, the Jet Propulsion Laboratory's DE series of computer-generated ephemerides took over as the fundamental ephemeris of the Astronomical Almanac. Obliquity based on DE200, which analyzed observations from 1911 to 1979, was calculated: ε = 23° 26′ 21.448″ − 46.8150″ T − 0.00059″ T2 + 0.001813″ T3where hereafter T is Julian centuries from J2000.0.
JPL's fundamental ephemerides have been continually updated. For instance, the Astronomical Almanac for 2010 specifies: ε = 23° 26′ 21.406″ − 46.836769″ T − 0.0001831″ T2 + 0.00200340″ T3 − 5.76″ × 10−7 T4 − 4.34″ × 10−8 T5These expressions for the obliquity are intended for high precision over a short time span ± several centuries. J. Laskar computed an expression to order T10 good to 0.02″ over 1000 years and several arcseconds over 10,000 years. Ε = 23° 26′ 21.448″ − 4680.93″ t − 1.55″ t2 + 1999.25″ t3 − 51.38″ t4 − 249.67″ t5 − 39.05″ t6 + 7.12″ t7 + 27.87″ t8 + 5.79″ t9 + 2.45″ t10where here t is multiples of 10,000 Julian years from J2000.0. These expressions are for the so-called mean obliquity, that is, the obliquity free from short-term variations. Periodic motions of the Moon and of Earth in its orbit cause much smaller short-period oscillations of the rotation axis of Earth, known as nutation, which add a periodic component to Earth's obliquity; the true or instant
Aberration of light
The aberration of light is an astronomical phenomenon which produces an apparent motion of celestial objects about their true positions, dependent on the velocity of the observer. Aberration causes objects to appear to be displaced towards the direction of motion of the observer compared to when the observer is stationary; the change in angle is very small — of the order of v/c where c is the speed of light and v the velocity of the observer. In the case of "stellar" or "annual" aberration, the apparent position of a star to an observer on Earth varies periodically over the course of a year as the Earth's velocity changes as it revolves around the Sun, by a maximum angle of 20 arcseconds in right ascension or declination; the term aberration has been used to refer to a number of related phenomena concerning the propagation of light in moving bodies. Aberration should not be confused with parallax; the latter is a change in the apparent position of a nearby object, as measured by a moving observer, relative to more distant objects that define a reference frame.
The amount of parallax depends on the distance of the object from the observer, whereas aberration does not. Aberration is related to light-time correction and relativistic beaming, although it is considered separately from these effects. Aberration is significant because of its role in the development of the theories of light and the theory of special relativity, it was first observed in the late 1600s by astronomers searching for stellar parallax in order to confirm the heliocentric model of the Solar System. However, it was not understood at the time to be a different phenomenon. In 1727, James Bradley provided a classical explanation for it in terms of the finite speed of light relative to the motion of the Earth in its orbit around the Sun, which he used to make one of the earliest measurements of the speed of light. However, Bradley's theory was incompatible with 19th century theories of light, aberration became a major motivation for the aether drag theories of Augustin Fresnel and G. G. Stokes, for Hendrik Lorentz's aether theory of electromagnetism in 1892.
The aberration of light, together with Lorentz's elaboration of Maxwell's electrodynamics, the moving magnet and conductor problem, the negative aether drift experiments, as well as the Fizeau experiment, led Albert Einstein to develop the theory of special relativity in 1905, which presents a general form of the equation for aberration in terms of such theory. Aberration may be explained as the difference in angle of a beam of light in different inertial frames of reference. A common analogy is to consider the apparent direction of falling rain. If rain is falling vertically in the frame of reference of a person standing still to a person moving forwards the rain will appear to arrive at an angle, requiring the moving observer to tilt their umbrella forwards; the faster the observer moves, the more tilt is needed. The net effect is that light rays striking the moving observer from the sides in a stationary frame will come angled from ahead in the moving observer's frame; this effect is sometimes called the "searchlight" or "headlight" effect.
In the case of annual aberration of starlight, the direction of incoming starlight as seen in the Earth's moving frame is tilted relative to the angle observed in the Sun's frame. Since the direction of motion of the Earth changes during its orbit, the direction of this tilting changes during the course of the year, causes the apparent position of the star to differ from its true position as measured in the inertial frame of the Sun. While classical reasoning gives intuition for aberration, it leads to a number of physical paradoxes observable at the classical level; the theory of special relativity is required to account for aberration. The relativistic explanation is similar to the classical one however, in both theories aberration may be understood as a case of addition of velocities. In the Sun's frame, consider a beam of light with velocity equal to the speed of light c, with x and y velocity components u x and u y, at an angle tan = u y / u x. If the Earth is moving at velocity v in the x direction relative to the Sun by velocity addition the x component of the beam's velocity in the Earth's frame of reference is u x ′ = u x + v, the y velocity is unchanged, u y ′ = u y.
Thus the angle of the light in the Earth's frame in terms of the angle in the Sun's frame is tan = u y ′ u x ′ = u y u x + v = sin v / c + cos In the case of θ = 90
Figure of the Earth
Figure of the Earth is a term of art in geodesy that refers to the size and shape used to model Earth. The size and shape it refers to depend on context, including the precision needed for the model; the sphere is an approximation of the figure of the Earth, satisfactory for many purposes. Several models with greater accuracy have been developed so that coordinate systems can serve the precise needs of navigation, cadastre, land use, various other concerns. Earth's topographic surface is apparent with its variety of land forms and water areas; this topographic surface is the concern of topographers and geophysicists. While it is the surface on which Earth measurements are made, mathematically modeling it while taking the irregularities into account would be complicated; the Pythagorean concept of a spherical Earth offers a simple surface, easy to deal with mathematically. Many astronomical and navigational computations use a sphere to model the Earth as a close approximation. However, a more accurate figure is needed for measuring distances and areas on the scale beyond the purely local.
Better approximations can be had by modeling the entire surface as an oblate spheroid, using spherical harmonics to approximate the geoid, or modeling a region with a best-fit reference ellipsoids. For surveys of small areas, a planar model of Earth's surface suffices because the local topography overwhelms the curvature. Plane-table surveys are made for small areas without considering the size and shape of the entire Earth. A survey of a city, for example, might be conducted this way. By the late 1600s, serious effort was devoted to modeling the Earth as an ellipsoid, beginning with Jean Picard's measurement of a degree of arc along the Paris meridian. Improved maps and better measurement of distances and areas of national territories motivated these early attempts. Surveying instrumentation and techniques improved over the ensuing centuries. Models for the figure of the earth improved in step. In the mid- to late 20th century, research across the geosciences contributed to drastic improvements in the accuracy of the figure of the Earth.
The primary utility of this improved accuracy was to provide geographical and gravitational data for the inertial guidance systems of ballistic missiles. This funding drove the expansion of geoscientific disciplines, fostering the creation and growth of various geoscience departments at many universities; these developments benefited many civilian pursuits as well, such as weather and communication satellite control and GPS location-finding, which would be impossible without accurate models for the figure of the Earth. The models for the figure of the Earth vary in the way they are used, in their complexity, in the accuracy with which they represent the size and shape of the Earth; the simplest model for the shape of the entire Earth is a sphere. The Earth's radius is the distance from Earth's center to about 6,371 kilometers. While "radius" is a characteristic of perfect spheres, the Earth deviates from spherical by only a third of a percent, sufficiently close to treat it as a sphere in many contexts and justifying the term "the radius of the Earth".
The concept of a spherical Earth dates back to around the 6th century BC, but remained a matter of philosophical speculation until the 3rd century BC. The first scientific estimation of the radius of the Earth was given by Eratosthenes about 240 BC, with estimates of the accuracy of Eratosthenes’s measurement ranging from 2% to 15%; the Earth is only spherical, so no single value serves as its natural radius. Distances from points on the surface to the center range from 6,353 km to 6,384 km. Several different ways of modeling the Earth as a sphere each yield a mean radius of 6,371 kilometers. Regardless of the model, any radius falls between the polar minimum of about 6,357 km and the equatorial maximum of about 6,378 km; the difference 21 kilometers correspond to the polar radius being 0.3% shorter than the equator radius. Since the Earth is flattened at the poles and bulges at the Equator, geodesy represents the figure of the Earth as an oblate spheroid; the oblate spheroid, or oblate ellipsoid, is an ellipsoid of revolution obtained by rotating an ellipse about its shorter axis.
It is the regular geometric shape. A spheroid describing the figure of the Earth or other celestial body is called a reference ellipsoid; the reference ellipsoid for Earth is called an Earth ellipsoid. An ellipsoid of revolution is uniquely defined by two quantities. Several conventions for expressing the two quantities are used in geodesy, but they are all equivalent to and convertible with each other: Equatorial radius a, polar radius b. Eccentricity and flattening are different ways of expressing; when flattening appears as one of the defining quantities in geodesy it is expressed by its reciprocal. For example, in the WGS 84 spheroid used by today's GPS systems, the reciprocal of the flattening 1 / f is set to be 298.257223563. The difference between a sphere and a reference ellipsoid for Earth is small, only about one part in 300. Flattening was computed from grade measurements. Nowadays, geodetic networks and satellite geodesy are used. In practice, many reference ellip