Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle , whereas the third Euler angle defines the rotation itself . In other words, if the axis of rotation of a body is itself rotating about a second axis, that body is said to be precessing about the second axis. A motion in which the second Euler angle changes is called nutation . In physics , there are two types of precession: torque -free and torque-induced.
143-459: In astronomy, precession refers to any of several slow changes in an astronomical body's rotational or orbital parameters. An important example is the steady change in the orientation of the axis of rotation of the Earth , known as the precession of the equinoxes . Torque-free precession implies that no external moment (torque) is applied to the body. In torque-free precession, the angular momentum
286-540: A circumstellar disk , and then the planets grow out of that disk with the Sun. A nebula contains gas, ice grains, and dust (including primordial nuclides ). According to nebular theory , planetesimals formed by accretion , with the primordial Earth being estimated as likely taking anywhere from 70 to 100 million years to form. Estimates of the age of the Moon range from 4.5 Ga to significantly younger. A leading hypothesis
429-526: A versor (normalized quaternion): q ^ = q i i + q j j + q k k + q r = [ q i q j q k q r ] {\displaystyle {\hat {\mathbf {q} }}=q_{i}\mathbf {i} +q_{j}\mathbf {j} +q_{k}\mathbf {k} +q_{r}={\begin{bmatrix}q_{i}\\q_{j}\\q_{k}\\q_{r}\end{bmatrix}}} The above definition stores
572-737: A 3-sphere onto the 3-dimensional pure-vector hyperplane. It has a discontinuity at 180° ( π radians): as any rotation vector r tends to an angle of π radians, its tangent tends to infinity. A rotation g followed by a rotation f in the Rodrigues representation has the simple rotation composition form ( g , f ) = g + f − f × g 1 − g ⋅ f . {\displaystyle (\mathbf {g} ,\mathbf {f} )={\frac {\mathbf {g} +\mathbf {f} -\mathbf {f} \times \mathbf {g} }{1-\mathbf {g} \cdot \mathbf {f} }}\,.} Today,
715-492: A common barycenter every 27.32 days relative to the background stars. When combined with the Earth–Moon system's common orbit around the Sun, the period of the synodic month , from new moon to new moon, is 29.53 days. Viewed from the celestial north pole , the motion of Earth, the Moon, and their axial rotations are all counterclockwise . Viewed from a vantage point above the Sun and Earth's north poles, Earth orbits in
858-548: A counterclockwise direction about the Sun. The orbital and axial planes are not precisely aligned: Earth's axis is tilted some 23.44 degrees from the perpendicular to the Earth–Sun plane (the ecliptic ), and the Earth-Moon plane is tilted up to ±5.1 degrees against the Earth–Sun plane. Without this tilt, there would be an eclipse every two weeks, alternating between lunar eclipses and solar eclipses . The Hill sphere , or
1001-416: A fixed origin ) is described by a single rotation about some axis. Such a rotation may be uniquely described by a minimum of three real parameters. However, for various reasons, there are several ways to represent it. Many of these representations use more than the necessary minimum of three parameters, although each of them still has only three degrees of freedom . An example where rotation representation
1144-420: A full rotation about its axis so that the Sun returns to the meridian . The orbital speed of Earth averages about 29.78 km/s (107,200 km/h; 66,600 mph), which is fast enough to travel a distance equal to Earth's diameter, about 12,742 km (7,918 mi), in seven minutes, and the distance from Earth to the Moon, 384,400 km (238,900 mi), in about 3.5 hours. The Moon and Earth orbit
1287-421: A globe-spanning mid-ocean ridge system. At Earth's polar regions , the ocean surface is covered by seasonally variable amounts of sea ice that often connects with polar land, permafrost and ice sheets , forming polar ice caps . Earth's land covers 29.2%, or 149 million km (58 million sq mi) of Earth's surface. The land surface includes many islands around the globe, but most of
1430-425: A liquid outer core that generates a magnetosphere capable of deflecting most of the destructive solar winds and cosmic radiation . Earth has a dynamic atmosphere , which sustains Earth's surface conditions and protects it from most meteoroids and UV-light at entry . It has a composition of primarily nitrogen and oxygen . Water vapor is widely present in the atmosphere, forming clouds that cover most of
1573-492: A lot of angular rotating velocity with respect to the rotation around the pivot axis, and as dm 1 is forced closer to the pivot axis of the rotation (by the wheel spinning further), because of the Coriolis effect , with respect to the vertical pivot axis, dm 1 tends to move in the direction of the top-left arrow in the diagram (shown at 45°) in the direction of rotation around the pivot axis. Section dm 2 of
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#17327908962481716-466: A reference placement in space. Rotation formalisms are focused on proper ( orientation-preserving ) motions of the Euclidean space with one fixed point , that a rotation refers to. Although physical motions with a fixed point are an important case (such as ones described in the center-of-mass frame , or motions of a joint ), this approach creates a knowledge about all motions. Any proper motion of
1859-432: A rotation formalism captures only the rotational part of a motion, that contains three degrees of freedom, and ignores the translational part, that contains another three. When representing a rotation as numbers in a computer, some people prefer the quaternion representation or the axis+angle representation, because they avoid the gimbal lock that can occur with Euler rotations. The above-mentioned triad of unit vectors
2002-556: A single rotation about some axis. The axis is the unit vector (unique except for sign) which remains unchanged by the rotation. The magnitude of the angle is also unique, with its sign being determined by the sign of the rotation axis. The axis can be represented as a three-dimensional unit vector e ^ = [ e x e y e z ] {\displaystyle {\hat {\mathbf {e} }}={\begin{bmatrix}e_{x}\\e_{y}\\e_{z}\end{bmatrix}}} and
2145-477: A small rotation vector ω dt for the short time dt ; e.g.: R new = exp ( [ ω ( R old ) ] × d t ) R old {\displaystyle {\boldsymbol {R}}_{\text{new}}=\exp \left(\left[{\boldsymbol {\omega }}\left({\boldsymbol {R}}_{\text{old}}\right)\right]_{\times }dt\right){\boldsymbol {R}}_{\text{old}}} for
2288-673: A small rotation vector v perpendicular to both ω and L , noting that E ( exp ( [ v ] × ) R ) ≈ E ( R ) + ( ω ( R ) × L ) ⋅ v {\displaystyle E\left(\exp \left(\left[{\boldsymbol {v}}\right]_{\times }\right){\boldsymbol {R}}\right)\approx E\left({\boldsymbol {R}}\right)+\left({\boldsymbol {\omega }}\left({\boldsymbol {R}}\right)\times {\boldsymbol {L}}\right)\cdot {\boldsymbol {v}}} Torque-induced precession ( gyroscopic precession )
2431-579: A three-dimensional object connected to its (fixed) surroundings by slack strings or bands, the strings or bands can be untangled after two complete turns about some fixed axis from an initial untangled state. Algebraically, the quaternion describing such a rotation changes from a scalar +1 (initially), through (scalar + pseudovector) values to scalar −1 (at one full turn), through (scalar + pseudovector) values back to scalar +1 (at two full turns). This cycle repeats every 2 turns. After 2 n turns (integer n > 0 ), without any intermediate untangling attempts,
2574-414: A very popular parametrization due to the following properties: Like rotation matrices, quaternions must sometimes be renormalized due to rounding errors, to make sure that they correspond to valid rotations. The computational cost of renormalizing a quaternion, however, is much less than for normalizing a 3 × 3 matrix. Quaternions also capture the spinorial character of rotations in three dimensions. For
2717-1894: Is C = cos γ 2 + sin γ 2 C = ( cos β 2 + sin β 2 B ) ( cos α 2 + sin α 2 A ) . {\displaystyle C=\cos {\frac {\gamma }{2}}+\sin {\frac {\gamma }{2}}\mathbf {C} =\left(\cos {\frac {\beta }{2}}+\sin {\frac {\beta }{2}}\mathbf {B} \right)\left(\cos {\frac {\alpha }{2}}+\sin {\frac {\alpha }{2}}\mathbf {A} \right).} Expand this quaternion product to cos γ 2 + sin γ 2 C = ( cos β 2 cos α 2 − sin β 2 sin α 2 B ⋅ A ) + ( sin β 2 cos α 2 B + sin α 2 cos β 2 A + sin β 2 sin α 2 B × A ) . {\displaystyle \cos {\frac {\gamma }{2}}+\sin {\frac {\gamma }{2}}\mathbf {C} =\left(\cos {\frac {\beta }{2}}\cos {\frac {\alpha }{2}}-\sin {\frac {\beta }{2}}\sin {\frac {\alpha }{2}}\mathbf {B} \cdot \mathbf {A} \right)+\left(\sin {\frac {\beta }{2}}\cos {\frac {\alpha }{2}}\mathbf {B} +\sin {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\mathbf {A} +\sin {\frac {\beta }{2}}\sin {\frac {\alpha }{2}}\mathbf {B} \times \mathbf {A} \right).} Divide both sides of this equation by
2860-514: Is rounded into an ellipsoid with a circumference of about 40,000 km. It is the densest planet in the Solar System . Of the four rocky planets , it is the largest and most massive. Earth is about eight light-minutes away from the Sun and orbits it , taking a year (about 365.25 days) to complete one revolution. Earth rotates around its own axis in slightly less than a day (in about 23 hours and 56 minutes). Earth's axis of rotation
3003-456: Is a chemically distinct silicate solid crust, which is underlain by a highly viscous solid mantle. The crust is separated from the mantle by the Mohorovičić discontinuity . The thickness of the crust varies from about 6 kilometres (3.7 mi) under the oceans to 30–50 km (19–31 mi) for the continents. The crust and the cold, rigid, top of the upper mantle are collectively known as
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#17327908962483146-409: Is a constant, but the angular velocity vector changes orientation with time. What makes this possible is a time-varying moment of inertia , or more precisely, a time-varying inertia matrix . The inertia matrix is composed of the moments of inertia of a body calculated with respect to separate coordinate axes (e.g. x , y , z ). If an object is asymmetric about its principal axis of rotation,
3289-402: Is added to rotation around a vertical axis. It is important to note that the torque around the gimbal axis arises without any delay; the response is instantaneous. In the discussion above, the setup was kept unchanging by preventing pitching around the gimbal axis. In the case of a spinning toy top, when the spinning top starts tilting, gravity exerts a torque. However, instead of rolling over,
3432-414: Is also called a basis . Specifying the coordinates ( components ) of vectors of this basis in its current (rotated) position, in terms of the reference (non-rotated) coordinate axes, will completely describe the rotation. The three unit vectors, û , v̂ and ŵ , that form the rotated basis each consist of 3 coordinates, yielding a total of 9 parameters. These parameters can be written as
3575-563: Is also the mechanism behind gyrocompasses . Precession is the change of angular velocity and angular momentum produced by a torque. The general equation that relates the torque to the rate of change of angular momentum is: τ = d L d t {\displaystyle {\boldsymbol {\tau }}={\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}} where τ {\displaystyle {\boldsymbol {\tau }}} and L {\displaystyle \mathbf {L} } are
3718-537: Is another way of stating that ( û , v̂ , ŵ ) form a 3D orthonormal basis . These statements comprise a total of 6 conditions (the cross product contains 3), leaving the rotation matrix with just 3 degrees of freedom, as required. Two successive rotations represented by matrices A 1 and A 2 are easily combined as elements of a group, A total = A 2 A 1 {\displaystyle \mathbf {A} _{\text{total}}=\mathbf {A} _{2}\mathbf {A} _{1}} (Note
3861-535: Is approximately 9.8 m/s (32 ft/s ). Local differences in topography, geology, and deeper tectonic structure cause local and broad regional differences in Earth's gravitational field, known as gravity anomalies . The main part of Earth's magnetic field is generated in the core, the site of a dynamo process that converts the kinetic energy of thermally and compositionally driven convection into electrical and magnetic field energy. The field extends outwards from
4004-529: Is contained in 3.45 billion-year-old Australian rocks showing fossils of microorganisms . During the Neoproterozoic , 1000 to 539 Ma , much of Earth might have been covered in ice. This hypothesis has been termed " Snowball Earth ", and it is of particular interest because it preceded the Cambrian explosion , when multicellular life forms significantly increased in complexity. Following
4147-427: Is created. Under these circumstances the angular velocity of precession is given by: where I s is the moment of inertia , ω s is the angular velocity of spin about the spin axis, m is the mass, g is the acceleration due to gravity, θ is the angle between the spin axis and the axis of precession and r is the distance between the center of mass and the pivot. The torque vector originates at
4290-792: Is enabled by Earth being an ocean world , the only one in the Solar System sustaining liquid surface water . Almost all of Earth's water is contained in its global ocean, covering 70.8% of Earth's crust . The remaining 29.2% of Earth's crust is land, most of which is located in the form of continental landmasses within Earth's land hemisphere . Most of Earth's land is at least somewhat humid and covered by vegetation , while large sheets of ice at Earth's polar deserts retain more water than Earth's groundwater , lakes, rivers and atmospheric water combined. Earth's crust consists of slowly moving tectonic plates , which interact to produce mountain ranges, volcanoes , and earthquakes. Earth has
4433-457: Is farthest out from its center of mass at its equatorial bulge, the summit of the volcano Chimborazo in Ecuador (6,384.4 km or 3,967.1 mi) is its farthest point out. Parallel to the rigid land topography the ocean exhibits a more dynamic topography . To measure the local variation of Earth's topography, geodesy employs an idealized Earth producing a geoid shape. Such a shape
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4576-516: Is gained if the ocean is idealized, covering Earth completely and without any perturbations such as tides and winds. The result is a smooth but irregular geoid surface, providing a mean sea level (MSL) as a reference level for topographic measurements. Earth's surface is the boundary between the atmosphere, and the solid Earth and oceans. Defined in this way, it has an area of about 510 million km (197 million sq mi). Earth can be divided into two hemispheres : by latitude into
4719-461: Is generally accepted to be the earliest known astronomer to recognize and assess the precession of the equinoxes at about 1° per century (which is not far from the actual value for antiquity, 1.38°), although there is some minor dispute about whether he was. In ancient China , the Jin-dynasty scholar-official Yu Xi ( fl. 307–345 AD) made a similar discovery centuries later, noting that
4862-426: Is liquid under normal atmospheric pressure. Differences in the amount of captured energy between geographic regions (as with the equatorial region receiving more sunlight than the polar regions) drive atmospheric and ocean currents , producing a global climate system with different climate regions , and a range of weather phenomena such as precipitation , allowing components such as nitrogen to cycle . Earth
5005-521: Is not straightforward, and in fact does not satisfy the law of vector addition, which shows that finite rotations are not really vectors at all. It is best to employ the rotation matrix or quaternion notation, calculate the product, and then convert back to Euler axis and angle. The idea behind Euler rotations is to split the complete rotation of the coordinate system into three simpler constitutive rotations, called precession , nutation , and intrinsic rotation , being each one of them an increment on one of
5148-513: Is now slightly longer than it was during the 19th century due to tidal deceleration , each day varies between 0 and 2 ms longer than the mean solar day. Earth's rotation period relative to the fixed stars , called its stellar day by the International Earth Rotation and Reference Systems Service (IERS), is 86,164.0989 seconds of mean solar time ( UT1 ), or 23 56 4.0989 . Earth's rotation period relative to
5291-622: Is rare, though the alternative spelling Gaia has become common due to the Gaia hypothesis , in which case its pronunciation is / ˈ ɡ aɪ . ə / rather than the more classical English / ˈ ɡ eɪ . ə / . There are a number of adjectives for the planet Earth. The word "earthly" is derived from "Earth". From the Latin Terra comes terran / ˈ t ɛr ə n / , terrestrial / t ə ˈ r ɛ s t r i ə l / , and (via French) terrene / t ə ˈ r iː n / , and from
5434-487: Is that it was formed by accretion from material loosed from Earth after a Mars -sized object with about 10% of Earth's mass, named Theia , collided with Earth. It hit Earth with a glancing blow and some of its mass merged with Earth. Between approximately 4.1 and 3.8 Ga , numerous asteroid impacts during the Late Heavy Bombardment caused significant changes to the greater surface environment of
5577-485: Is the moment of inertia , T s is the period of spin about the spin axis, and τ is the torque . In general, the problem is more complicated than this, however. The special and general theories of relativity give three types of corrections to the Newtonian precession, of a gyroscope near a large mass such as Earth, described above. They are: The Schwarzschild geodesics (sometimes Schwarzschild precession)
5720-453: Is the basis for the astronomical unit (AU) and is equal to roughly 8.3 light minutes or 380 times Earth's distance to the Moon . Earth orbits the Sun every 365.2564 mean solar days , or one sidereal year . With an apparent movement of the Sun in Earth's sky at a rate of about 1°/day eastward, which is one apparent Sun or Moon diameter every 12 hours. Due to this motion, on average it takes 24 hours—a solar day—for Earth to complete
5863-442: Is the phenomenon in which the axis of a spinning object (e.g., a gyroscope ) describes a cone in space when an external torque is applied to it. The phenomenon is commonly seen in a spinning toy top , but all rotating objects can undergo precession. If the speed of the rotation and the magnitude of the external torque are constant, the spin axis will move at right angles to the direction that would intuitively result from
Precession - Misplaced Pages Continue
6006-444: Is the precession rate, ω s is the spin rate about the axis of symmetry, I s is the moment of inertia about the axis of symmetry, I p is moment of inertia about either of the other two equal perpendicular principal axes, and α is the angle between the moment of inertia direction and the symmetry axis. When an object is not perfectly rigid , inelastic dissipation will tend to damp torque-free precession, and
6149-439: Is the science of quantitative description of a purely rotational motion . The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space. According to Euler's rotation theorem , the rotation of a rigid body (or three-dimensional coordinate system with
6292-590: Is therefore just as simple as using the rotation matrix. Just as two successive rotation matrices, A 1 followed by A 2 , are combined as A 3 = A 2 A 1 , {\displaystyle \mathbf {A} _{3}=\mathbf {A} _{2}\mathbf {A} _{1},} we can represent this with quaternion parameters in a similarly concise way: q 3 = q 2 ⊗ q 1 {\displaystyle \mathbf {q} _{3}=\mathbf {q} _{2}\otimes \mathbf {q} _{1}} Quaternions are
6435-457: Is tied to that of the Sun. Over the next 1.1 billion years , solar luminosity will increase by 10%, and over the next 3.5 billion years by 40%. Earth's increasing surface temperature will accelerate the inorganic carbon cycle , possibly reducing CO 2 concentration to levels lethally low for current plants ( 10 ppm for C4 photosynthesis ) in approximately 100–900 million years . A lack of vegetation would result in
6578-455: Is tilted with respect to the perpendicular to its orbital plane around the Sun, producing seasons . Earth is orbited by one permanent natural satellite , the Moon , which orbits Earth at 384,400 km (1.28 light seconds) and is roughly a quarter as wide as Earth. The Moon's gravity helps stabilize Earth's axis, causes tides and gradually slows Earth's rotation . Tidal locking has made
6721-426: Is used in the prediction of the anomalous perihelion precession of the planets, most notably for the accurate prediction of the apsidal precession of Mercury In astronomy, precession refers to any of several gravity-induced, slow and continuous changes in an astronomical body's rotational axis or orbital path. Precession of the equinoxes, perihelion precession, changes in the tilt of Earth's axis to its orbit, and
6864-405: Is used is in computer vision , where an automated observer needs to track a target. Consider a rigid body, with three orthogonal unit vectors fixed to its body (representing the three axes of the object's local coordinate system ). The basic problem is to specify the orientation of these three unit vectors , and hence the rigid body, with respect to the observer's coordinate system, regarded as
7007-3105: Is used to specify a composite rotation, is performed in the same manner as multiplication of complex numbers , except that the order of the elements must be taken into account, since multiplication is not commutative. In matrix notation we can write quaternion multiplication as q ~ ⊗ q = [ q r q k − q j q i − q k q r q i q j q j − q i q r q k − q i − q j − q k q r ] [ q ~ i q ~ j q ~ k q ~ r ] = [ q ~ r − q ~ k q ~ j q ~ i q ~ k q ~ r − q ~ i q ~ j − q ~ j q ~ i q ~ r q ~ k − q ~ i − q ~ j − q ~ k q ~ r ] [ q i q j q k q r ] {\displaystyle {\tilde {\mathbf {q} }}\otimes \mathbf {q} ={\begin{bmatrix}\;\;\,q_{r}&\;\;\,q_{k}&-q_{j}&\;\;\,q_{i}\\-q_{k}&\;\;\,q_{r}&\;\;\,q_{i}&\;\;\,q_{j}\\\;\;\,q_{j}&-q_{i}&\;\;\,q_{r}&\;\;\,q_{k}\\-q_{i}&-q_{j}&-q_{k}&\;\;\,q_{r}\end{bmatrix}}{\begin{bmatrix}{\tilde {q}}_{i}\\{\tilde {q}}_{j}\\{\tilde {q}}_{k}\\{\tilde {q}}_{r}\end{bmatrix}}={\begin{bmatrix}\;\;\,{\tilde {q}}_{r}&-{\tilde {q}}_{k}&\;\;\,{\tilde {q}}_{j}&\;\;\,{\tilde {q}}_{i}\\\;\;\,{\tilde {q}}_{k}&\;\;\,{\tilde {q}}_{r}&-{\tilde {q}}_{i}&\;\;\,{\tilde {q}}_{j}\\-{\tilde {q}}_{j}&\;\;\,{\tilde {q}}_{i}&\;\;\,{\tilde {q}}_{r}&\;\;\,{\tilde {q}}_{k}\\-{\tilde {q}}_{i}&-{\tilde {q}}_{j}&-{\tilde {q}}_{k}&\;\;\,{\tilde {q}}_{r}\end{bmatrix}}{\begin{bmatrix}q_{i}\\q_{j}\\q_{k}\\q_{r}\end{bmatrix}}} Combining two consecutive quaternion rotations
7150-410: Is useful in some contexts, as it represents a three-dimensional rotation with only three scalar values (its components), representing the three degrees of freedom. This is also true for representations based on sequences of three Euler angles (see below). If the rotation angle θ is zero, the axis is not uniquely defined. Combining two successive rotations, each represented by an Euler axis and angle,
7293-405: The z - y ′- x ″ convention, which are called heading , elevation , and bank (or synonymously, yaw , pitch , and roll ). Quaternions , which form a four-dimensional vector space , have proven very useful in representing rotations due to several advantages over the other representations mentioned in this article. A quaternion representation of rotation is written as
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#17327908962487436-487: The Euler angles . Notice that the outer matrix will represent a rotation around one of the axes of the reference frame, and the inner matrix represents a rotation around one of the moving frame axes. The middle matrix represents a rotation around an intermediate axis called line of nodes . However, the definition of Euler angles is not unique and in the literature many different conventions are used. These conventions depend on
7579-501: The Gibbs vector , with coordinates called Rodrigues parameters ) can be expressed in terms of the axis and angle of the rotation as follows: g = e ^ tan θ 2 {\displaystyle \mathbf {g} ={\hat {\mathbf {e} }}\tan {\frac {\theta }{2}}} This representation is a higher-dimensional analog of the gnomonic projection , mapping unit quaternions from
7722-583: The Milky Way and orbits about 28,000 light-years from its center. It is about 20 light-years above the galactic plane in the Orion Arm . The axial tilt of Earth is approximately 23.439281° with the axis of its orbit plane, always pointing towards the Celestial Poles . Due to Earth's axial tilt, the amount of sunlight reaching any given point on the surface varies over the course of
7865-759: The Pacific , North American , Eurasian , African , Antarctic , Indo-Australian , and South American . Other notable plates include the Arabian Plate , the Caribbean Plate , the Nazca Plate off the west coast of South America and the Scotia Plate in the southern Atlantic Ocean. The Australian Plate fused with the Indian Plate between 50 and 55 Ma . The fastest-moving plates are
8008-419: The asthenosphere , the solid but less-viscous part of the upper mantle that can flow and move along with the plates. As the tectonic plates migrate, oceanic crust is subducted under the leading edges of the plates at convergent boundaries. At the same time, the upwelling of mantle material at divergent boundaries creates mid-ocean ridges. The combination of these processes recycles the oceanic crust back into
8151-415: The celestial equator , this is equivalent to an apparent diameter of the Sun or the Moon every two minutes; from Earth's surface, the apparent sizes of the Sun and the Moon are approximately the same. Earth orbits the Sun, making Earth the third-closest planet to the Sun and part of the inner Solar System . Earth's average orbital distance is about 150 million km (93 million mi), which
8294-426: The eccentricity of its orbit over tens of thousands of years are all important parts of the astronomical theory of ice ages . (See Milankovitch cycles .) Axial precession is the movement of the rotational axis of an astronomical body, whereby the axis slowly traces out a cone. In the case of Earth, this type of precession is also known as the precession of the equinoxes , lunisolar precession , or precession of
8437-494: The equatorial bulge into the plane of the ecliptic , but instead causing it to precess. The torque exerted by the planets, particularly Jupiter , also plays a role. The orbits of planets around the Sun do not really follow an identical ellipse each time, but actually trace out a flower-petal shape because the major axis of each planet's elliptical orbit also precesses within its orbital plane, partly in response to perturbations in
8580-408: The ocean floor form the top of Earth's crust , which together with parts of the upper mantle form Earth's lithosphere . Earth's crust may be divided into oceanic and continental crust. Beneath the ocean-floor sediments, the oceanic crust is predominantly basaltic , while the continental crust may include lower density materials such as granite , sediments and metamorphic rocks. Nearly 75% of
8723-430: The precessing or moving mean March equinox (when the Sun is at 90° on the equator), is 86,164.0905 seconds of mean solar time (UT1) (23 56 4.0905 ) . Thus the sidereal day is shorter than the stellar day by about 8.4 ms. Apart from meteors within the atmosphere and low-orbiting satellites, the main apparent motion of celestial bodies in Earth's sky is to the west at a rate of 15°/h = 15'/min. For bodies near
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#17327908962488866-461: The skew-symmetric matrix [ ω ] × . The errors induced by finite time steps tend to increase the rotational kinetic energy: E ( R ) = ω ( R ) ⋅ L 2 {\displaystyle E\left({\boldsymbol {R}}\right)={\boldsymbol {\omega }}\left({\boldsymbol {R}}\right)\cdot {\frac {\boldsymbol {L}}{2}}} this unphysical tendency can be counteracted by repeatedly applying
9009-413: The sphere of gravitational influence , of Earth is about 1.5 million km (930,000 mi) in radius. This is the maximum distance at which Earth's gravitational influence is stronger than that of the more distant Sun and planets. Objects must orbit Earth within this radius, or they can become unbound by the gravitational perturbation of the Sun. Earth, along with the Solar System, is situated in
9152-464: The "last ice age", covered large parts of the continents, to the middle latitudes, in ice and ended about 11,700 years ago. Chemical reactions led to the first self-replicating molecules about four billion years ago. A half billion years later, the last common ancestor of all current life arose. The evolution of photosynthesis allowed the Sun's energy to be harvested directly by life forms. The resultant molecular oxygen ( O 2 ) accumulated in
9295-484: The Cambrian explosion, 535 Ma , there have been at least five major mass extinctions and many minor ones. Apart from the proposed current Holocene extinction event, the most recent was 66 Ma , when an asteroid impact triggered the extinction of non-avian dinosaurs and other large reptiles, but largely spared small animals such as insects, mammals , lizards and birds. Mammalian life has diversified over
9438-654: The Earth. Terra is also the name of the planet in some Romance languages , languages that evolved from Latin , like Italian and Portuguese , while in other Romance languages the word gave rise to names with slightly altered spellings, like the Spanish Tierra and the French Terre . The Latinate form Gæa or Gaea ( English: / ˈ dʒ iː . ə / ) of the Greek poetic name Gaia ( Γαῖα ; Ancient Greek : [ɡâi̯.a] or [ɡâj.ja] )
9581-473: The Euclidean space decomposes to a rotation around the origin and a translation . Whichever the order of their composition will be, the "pure" rotation component wouldn't change, uniquely determined by the complete motion. One can also understand "pure" rotations as linear maps in a vector space equipped with Euclidean structure, not as maps of points of a corresponding affine space . In other words,
9724-533: The Latin Tellus comes tellurian / t ɛ ˈ l ʊər i ə n / and telluric . The oldest material found in the Solar System is dated to 4.5682 +0.0002 −0.0004 Ga (billion years) ago. By 4.54 ± 0.04 Ga the primordial Earth had formed. The bodies in the Solar System formed and 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
9867-758: The Moon always face Earth with the same side. Earth, like most other bodies in the Solar System, formed 4.5 billion years ago from gas and dust in the early Solar System . During the first billion years of Earth's history , the ocean formed and then life developed within it. Life spread globally and has been altering Earth's atmosphere and surface, leading to the Great Oxidation Event two billion years ago. Humans emerged 300,000 years ago in Africa and have spread across every continent on Earth. Humans depend on Earth's biosphere and natural resources for their survival, but have increasingly impacted
10010-426: The Moon and, by inference, to that of Earth. Earth's atmosphere and oceans were formed by volcanic activity and outgassing . Water vapor from these sources condensed into the oceans, augmented by water and ice from asteroids, protoplanets , and comets . Sufficient water to fill the oceans may have been on Earth since it formed. In this model, atmospheric greenhouse gases kept the oceans from freezing when
10153-468: The Solar System have a much smaller eccentricity and precess at a much slower rate, making them nearly circular and nearly stationary. Discrepancies between the observed perihelion precession rate of the planet Mercury and that predicted by classical mechanics were prominent among the forms of experimental evidence leading to the acceptance of Einstein 's Theory of Relativity (in particular, his General Theory of Relativity ), which accurately predicted
10296-458: The Solar System's planetary-sized objects, Earth is the object with the highest density . Earth's mass is approximately 5.97 × 10 kg ( 5.970 Yg ). It is composed mostly of iron (32.1% by mass ), oxygen (30.1%), silicon (15.1%), magnesium (13.9%), sulfur (2.9%), nickel (1.8%), calcium (1.5%), and aluminium (1.4%), with the remaining 1.2% consisting of trace amounts of other elements. Due to gravitational separation ,
10439-403: The Sun when the star reaches its maximum radius, otherwise, with tidal effects, it may enter the Sun's atmosphere and be vaporized. Earth has a rounded shape , through hydrostatic equilibrium , with an average diameter of 12,742 kilometres (7,918 mi), making it the fifth largest planetary sized and largest terrestrial object of the Solar System . Due to Earth's rotation it has
10582-540: The angle by a scalar θ . Since the axis is normalized, it has only two degrees of freedom . The angle adds the third degree of freedom to this rotation representation. One may wish to express rotation as a rotation vector , or Euler vector , an un-normalized three-dimensional vector the direction of which specifies the axis, and the length of which is θ , r = θ e ^ . {\displaystyle \mathbf {r} =\theta {\hat {\mathbf {e} }}\,.} The rotation vector
10725-403: The anomalies. Deviating from Newton's law, Einstein's theory of gravitation predicts an extra term of A / r , which accurately gives the observed excess turning rate of 43 arcseconds every 100 years. Orbital nodes also precess over time. Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life . This
10868-425: The atmosphere and due to interaction with ultraviolet solar radiation, formed a protective ozone layer ( O 3 ) in the upper atmosphere. The incorporation of smaller cells within larger ones resulted in the development of complex cells called eukaryotes . True multicellular organisms formed as cells within colonies became increasingly specialized. Aided by the absorption of harmful ultraviolet radiation by
11011-434: The axes about which the rotations are carried out, and their sequence (since rotations on a sphere are non-commutative ). The convention being used is usually indicated by specifying the axes about which the consecutive rotations (before being composed) take place, referring to them by index (1, 2, 3) or letter (X, Y, Z) . The engineering and robotics communities typically use 3-1-3 Euler angles. Notice that after composing
11154-903: The axis is the only (nonzero) vector which remains unchanged by left-multiplying (rotating) it with the rotation matrix. The above properties are equivalent to | u ^ | = | v ^ | = | w ^ | = 1 u ^ ⋅ v ^ = 0 u ^ × v ^ = w ^ , {\displaystyle {\begin{aligned}|{\hat {\mathbf {u} }}|=|{\hat {\mathbf {v} }}|=|{\hat {\mathbf {w} }}|&=1\\{\hat {\mathbf {u} }}\cdot {\hat {\mathbf {v} }}&=0\\{\hat {\mathbf {u} }}\times {\hat {\mathbf {v} }}&={\hat {\mathbf {w} }}\,,\end{aligned}}} which
11297-630: The center of mass. Using ω = 2π / T , we find that the period of precession is given by: T p = 4 π 2 I s m g r T s = 4 π 2 I s sin ( θ ) τ T s {\displaystyle T_{\mathrm {p} }={\frac {4\pi ^{2}I_{\mathrm {s} }}{\ mgrT_{\mathrm {s} }}}={\frac {4\pi ^{2}I_{\mathrm {s} }\sin(\theta )}{\ \tau T_{\mathrm {s} }}}} Where I s
11440-450: The center, the temperature may be up to 6,000 °C (10,830 °F), and the pressure could reach 360 GPa (52 million psi ). Because much of the heat is provided by radioactive decay, scientists postulate that early in Earth's history, before isotopes with short half-lives were depleted, Earth's heat production was much higher. At approximately 3 Gyr , twice the present-day heat would have been produced, increasing
11583-620: The composition of the rotation R B with R A is the rotation R C = R B R A , with rotation axis and angle defined by the product of the quaternions, A = cos α 2 + sin α 2 A and B = cos β 2 + sin β 2 B , {\displaystyle A=\cos {\frac {\alpha }{2}}+\sin {\frac {\alpha }{2}}\mathbf {A} \quad {\text{and}}\quad B=\cos {\frac {\beta }{2}}+\sin {\frac {\beta }{2}}\mathbf {B} ,} that
11726-454: The continental crust , particularly during the early stages of Earth's history. New continental crust forms as a result of plate tectonics , a process ultimately driven by the continuous loss of heat from Earth's interior. Over the period of hundreds of millions of years, tectonic forces have caused areas of continental crust to group together to form supercontinents that have subsequently broken apart. At approximately 750 Ma , one of
11869-616: The continental surfaces are covered by sedimentary rocks, although they form about 5% of the mass of the crust. Earth's surface topography comprises both the topography of the ocean surface , and the shape of Earth's land surface. The submarine terrain of the ocean floor has an average bathymetric depth of 4 km, and is as varied as the terrain above sea level. Earth's surface is continually being shaped by internal plate tectonic processes including earthquakes and volcanism ; by weathering and erosion driven by ice, water, wind and temperature; and by biological processes including
12012-419: The core are chaotic; the magnetic poles drift and periodically change alignment. This causes secular variation of the main field and field reversals at irregular intervals averaging a few times every million years. The most recent reversal occurred approximately 700,000 years ago. The extent of Earth's magnetic field in space defines the magnetosphere . Ions and electrons of the solar wind are deflected by
12155-527: The core is primarily composed of the denser elements: iron (88.8%), with smaller amounts of nickel (5.8%), sulfur (4.5%), and less than 1% trace elements. The most common rock constituents of the crust are oxides . Over 99% of the crust is composed of various oxides of eleven elements, principally oxides containing silicon (the silicate minerals ), aluminium, iron, calcium, magnesium, potassium, or sodium. The major heat-producing isotopes within Earth are potassium-40 , uranium-238 , and thorium-232 . At
12298-463: The core, through the mantle, and up to Earth's surface, where it is, approximately, a dipole . The poles of the dipole are located close to Earth's geographic poles. At the equator of the magnetic field, the magnetic-field strength at the surface is 3.05 × 10 T , with a magnetic dipole moment of 7.79 × 10 Am at epoch 2000, decreasing nearly 6% per century (although it still remains stronger than its long time average). The convection movements in
12441-539: The days shorter. Above the Arctic Circle and below the Antarctic Circle there is no daylight at all for part of the year, causing a polar night , and this night extends for several months at the poles themselves. These same latitudes also experience a midnight sun , where the sun remains visible all day. By astronomical convention, the four seasons can be determined by the solstices—the points in
12584-611: The earliest known supercontinents, Rodinia , began to break apart. The continents later recombined to form Pannotia at 600–540 Ma , then finally Pangaea , which also began to break apart at 180 Ma . The most recent pattern of ice ages began about 40 Ma , and then intensified during the Pleistocene about 3 Ma . High- and middle-latitude regions have since undergone repeated cycles of glaciation and thaw, repeating about every 21,000, 41,000 and 100,000 years. The Last Glacial Period , colloquially called
12727-1191: The elements of a 3 × 3 matrix A , called a rotation matrix . Typically, the coordinates of each of these vectors are arranged along a column of the matrix (however, beware that an alternative definition of rotation matrix exists and is widely used, where the vectors' coordinates defined above are arranged by rows ) A = [ u ^ x v ^ x w ^ x u ^ y v ^ y w ^ y u ^ z v ^ z w ^ z ] {\displaystyle \mathbf {A} ={\begin{bmatrix}{\hat {\mathbf {u} }}_{x}&{\hat {\mathbf {v} }}_{x}&{\hat {\mathbf {w} }}_{x}\\{\hat {\mathbf {u} }}_{y}&{\hat {\mathbf {v} }}_{y}&{\hat {\mathbf {w} }}_{y}\\{\hat {\mathbf {u} }}_{z}&{\hat {\mathbf {v} }}_{z}&{\hat {\mathbf {w} }}_{z}\\\end{bmatrix}}} The elements of
12870-544: The equally large area of land under permafrost ) or deserts (33%). The pedosphere is the outermost layer of Earth's land surface and is composed of soil and subject to soil formation processes. Soil is crucial for land to be arable. Earth's total arable land is 10.7% of the land surface, with 1.3% being permanent cropland. Earth has an estimated 16.7 million km (6.4 million sq mi) of cropland and 33.5 million km (12.9 million sq mi) of pastureland. The land surface and
13013-482: The equator . Earth goes through one such complete precessional cycle in a period of approximately 26,000 years or 1° every 72 years, during which the positions of stars will slowly change in both equatorial coordinates and ecliptic longitude . Over this cycle, Earth's north axial pole moves from where it is now, within 1° of Polaris , in a circle around the ecliptic pole , with an angular radius of about 23.5°. The ancient Greek astronomer Hipparchus (c. 190–120 BC)
13156-402: The external torque. In the case of a toy top, its weight is acting downwards from its center of mass and the normal force (reaction) of the ground is pushing up on it at the point of contact with the support. These two opposite forces produce a torque which causes the top to precess. The device depicted on the right is gimbal mounted. From inside to outside there are three axes of rotation:
13299-406: The form of the changing gravitational forces exerted by other planets. This is called perihelion precession or apsidal precession . In the adjunct image, Earth's apsidal precession is illustrated. As the Earth travels around the Sun, its elliptical orbit rotates gradually over time. The eccentricity of its ellipse and the precession rate of its orbit are exaggerated for visualization. Most orbits in
13442-638: The growth and decomposition of biomass into soil . Earth's mechanically rigid outer layer of Earth's crust and upper mantle , the lithosphere , is divided into tectonic plates . These plates are rigid segments that move relative to each other at one of three boundaries types: at convergent boundaries , two plates come together; at divergent boundaries , two plates are pulled apart; and at transform boundaries , two plates slide past one another laterally. Along these plate boundaries, earthquakes, volcanic activity , mountain-building , and oceanic trench formation can occur. The tectonic plates ride on top of
13585-404: The heat in Earth is lost through plate tectonics, by mantle upwelling associated with mid-ocean ridges . The final major mode of heat loss is through conduction through the lithosphere, the majority of which occurs under the oceans. The gravity of Earth is the acceleration that is imparted to objects due to the distribution of mass within Earth. Near Earth's surface, gravitational acceleration
13728-399: The hub of the wheel, the gimbal axis, and the vertical pivot. To distinguish between the two horizontal axes, rotation around the wheel hub will be called spinning , and rotation around the gimbal axis will be called pitching . Rotation around the vertical pivot axis is called rotation . First, imagine that the entire device is rotating around the (vertical) pivot axis. Then, spinning of
13871-1338: The identity resulting from the previous one, cos γ 2 = cos β 2 cos α 2 − sin β 2 sin α 2 B ⋅ A , {\displaystyle \cos {\frac {\gamma }{2}}=\cos {\frac {\beta }{2}}\cos {\frac {\alpha }{2}}-\sin {\frac {\beta }{2}}\sin {\frac {\alpha }{2}}\mathbf {B} \cdot \mathbf {A} ,} and evaluate tan γ 2 C = tan β 2 B + tan α 2 A + tan β 2 tan α 2 B × A 1 − tan β 2 tan α 2 B ⋅ A . {\displaystyle \tan {\frac {\gamma }{2}}\mathbf {C} ={\frac {\tan {\frac {\beta }{2}}\mathbf {B} +\tan {\frac {\alpha }{2}}\mathbf {A} +\tan {\frac {\beta }{2}}\tan {\frac {\alpha }{2}}\mathbf {B} \times \mathbf {A} }{1-\tan {\frac {\beta }{2}}\tan {\frac {\alpha }{2}}\mathbf {B} \cdot \mathbf {A} }}.} This
14014-470: The independent rotations, they do not rotate about their axis anymore. The most external matrix rotates the other two, leaving the second rotation matrix over the line of nodes, and the third one in a frame comoving with the body. There are 3 × 3 × 3 = 27 possible combinations of three basic rotations but only 3 × 2 × 2 = 12 of them can be used for representing arbitrary 3D rotations as Euler angles. These 12 combinations avoid consecutive rotations around
14157-405: The instantaneous angular velocity is ω ( R ) = R I 0 − 1 R T L {\displaystyle {\boldsymbol {\omega }}\left({\boldsymbol {R}}\right)={\boldsymbol {R}}{\boldsymbol {I}}_{0}^{-1}{\boldsymbol {R}}^{T}{\boldsymbol {L}}} Precession occurs by repeatedly recalculating ω and applying
14300-406: The land surface is taken by the four continental landmasses , which are (in descending order): Africa-Eurasia , America (landmass) , Antarctica , and Australia (landmass) . These landmasses are further broken down and grouped into the continents . The terrain of the land surface varies greatly and consists of mountains, deserts , plains , plateaus , and other landforms . The elevation of
14443-513: The land surface varies from a low point of −418 m (−1,371 ft) at the Dead Sea , to a maximum altitude of 8,848 m (29,029 ft) at the top of Mount Everest . The mean height of land above sea level is about 797 m (2,615 ft). Land can be covered by surface water , snow, ice, artificial structures or vegetation. Most of Earth's land hosts vegetation, but considerable amounts of land are ice sheets (10%, not including
14586-399: The lithosphere, which is divided into independently moving tectonic plates. Beneath the lithosphere is the asthenosphere , a relatively low-viscosity layer on which the lithosphere rides. Important changes in crystal structure within the mantle occur at 410 and 660 km (250 and 410 mi) below the surface, spanning a transition zone that separates the upper and lower mantle. Beneath
14729-410: The loss of oxygen in the atmosphere, making current animal life impossible. Due to the increased luminosity, Earth's mean temperature may reach 100 °C (212 °F) in 1.5 billion years, and all ocean water will evaporate and be lost to space, which may trigger a runaway greenhouse effect , within an estimated 1.6 to 3 billion years. Even if the Sun were stable, a fraction of the water in
14872-456: The lowercase when it is preceded by "the", such as "the atmosphere of the earth". It almost always appears in lowercase in colloquial expressions such as "what on earth are you doing?" The name Terra / ˈ t ɛr ə / occasionally is used in scientific writing and especially in science fiction to distinguish humanity's inhabited planet from others, while in poetry Tellus / ˈ t ɛ l ə s / has been used to denote personification of
15015-497: The magnetosphere. During magnetic storms and substorms , charged particles can be deflected from the outer magnetosphere and especially the magnetotail, directed along field lines into Earth's ionosphere , where atmospheric atoms can be excited and ionized, causing an aurora . Earth's rotation period relative to the Sun—its mean solar day—is 86,400 seconds of mean solar time ( 86,400.0025 SI seconds ). Because Earth's solar day
15158-409: The magnetosphere; solar wind pressure compresses the day-side of the magnetosphere, to about 10 Earth radii, and extends the night-side magnetosphere into a long tail. Because the velocity of the solar wind is greater than the speed at which waves propagate through the solar wind, a supersonic bow shock precedes the day-side magnetosphere within the solar wind. Charged particles are contained within
15301-483: The magnetosphere; the plasmasphere is defined by low-energy particles that essentially follow magnetic field lines as Earth rotates. The ring current is defined by medium-energy particles that drift relative to the geomagnetic field, but with paths that are still dominated by the magnetic field, and the Van Allen radiation belts are formed by high-energy particles whose motion is essentially random, but contained in
15444-417: The mantle, an extremely low viscosity liquid outer core lies above a solid inner core . Earth's inner core may be rotating at a slightly higher angular velocity than the remainder of the planet, advancing by 0.1–0.5° per year, although both somewhat higher and much lower rates have also been proposed. The radius of the inner core is about one-fifth of that of Earth. The density increases with depth. Among
15587-612: The mantle. Due to this recycling, most of the ocean floor is less than 100 Ma old. The oldest oceanic crust is located in the Western Pacific and is estimated to be 200 Ma old. By comparison, the oldest dated continental crust is 4,030 Ma , although zircons have been found preserved as clasts within Eoarchean sedimentary rocks that give ages up to 4,400 Ma , indicating that at least some continental crust existed at that time. The seven major plates are
15730-986: The mathematical extension of the complex numbers, written as a + b i + c j + d k with a , b , c , d ∈ R {\displaystyle a+bi+cj+dk\qquad {\text{with }}a,b,c,d\in \mathbb {R} } and where { i , j , k } are the hypercomplex numbers satisfying i 2 = j 2 = k 2 = − 1 i j = − j i = k j k = − k j = i k i = − i k = j {\displaystyle {\begin{array}{ccccccc}i^{2}&=&j^{2}&=&k^{2}&=&-1\\ij&=&-ji&=&k&&\\jk&=&-kj&=&i&&\\ki&=&-ik&=&j&&\end{array}}} Quaternion multiplication, which
15873-548: The modern oceans will descend to the mantle , due to reduced steam venting from mid-ocean ridges. The Sun will evolve to become a red giant in about 5 billion years . Models predict that the Sun will expand to roughly 1 AU (150 million km; 93 million mi), about 250 times its present radius. Earth's fate is less clear. As a red giant, the Sun will lose roughly 30% of its mass, so, without tidal effects, Earth will move to an orbit 1.7 AU (250 million km; 160 million mi) from
16016-872: The moment of inertia with respect to each coordinate direction will change with time, while preserving angular momentum. The result is that the component of the angular velocities of the body about each axis will vary inversely with each axis' moment of inertia. The torque-free precession rate of an object with an axis of symmetry, such as a disk, spinning about an axis not aligned with that axis of symmetry can be calculated as follows: ω p = I s ω s I p cos ( α ) {\displaystyle {\boldsymbol {\omega }}_{\mathrm {p} }={\frac {{\boldsymbol {I}}_{\mathrm {s} }{\boldsymbol {\omega }}_{\mathrm {s} }}{{\boldsymbol {I}}_{\mathrm {p} }\cos({\boldsymbol {\alpha }})}}} where ω p
16159-619: The most straightforward way to prove this formula is in the (faithful) doublet representation , where g = n̂ tan a , etc. The combinatoric features of the Pauli matrix derivation just mentioned are also identical to the equivalent quaternion derivation below. Construct a quaternion associated with a spatial rotation R as, S = cos ϕ 2 + sin ϕ 2 S . {\displaystyle S=\cos {\frac {\phi }{2}}+\sin {\frac {\phi }{2}}\mathbf {S} .} Then
16302-399: The mother of Thor . Historically, "Earth" has been written in lowercase. Beginning with the use of Early Middle English , its definite sense as "the globe" was expressed as "the earth". By the era of Early Modern English , capitalization of nouns began to prevail , and the earth was also written the Earth , particularly when referenced along with other heavenly bodies. More recently,
16445-424: The name is sometimes simply given as Earth , by analogy with the names of the other planets , though "earth" and forms with "the earth" remain common. House styles now vary: Oxford spelling recognizes the lowercase form as the more common, with the capitalized form an acceptable variant. Another convention capitalizes "Earth" when appearing as a name, such as a description of the "Earth's atmosphere", but employs
16588-425: The newly forming Sun had only 70% of its current luminosity . By 3.5 Ga , Earth's magnetic field was established, which helped prevent the atmosphere from being stripped away by the solar wind . As the molten outer layer of Earth cooled it formed the first solid crust , which is thought to have been mafic in composition. The first continental crust , which was more felsic in composition, formed by
16731-593: The ocean may have covered Earth completely. The world ocean is commonly divided into the Pacific Ocean, Atlantic Ocean, Indian Ocean, Antarctic or Southern Ocean , and Arctic Ocean, from largest to smallest. The ocean covers Earth's oceanic crust , with the shelf seas covering the shelves of the continental crust to a lesser extent. The oceanic crust forms large oceanic basins with features like abyssal plains , seamounts , submarine volcanoes , oceanic trenches , submarine canyons , oceanic plateaus , and
16874-670: The oceanic plates, with the Cocos Plate advancing at a rate of 75 mm/a (3.0 in/year) and the Pacific Plate moving 52–69 mm/a (2.0–2.7 in/year). At the other extreme, the slowest-moving plate is the South American Plate, progressing at a typical rate of 10.6 mm/a (0.42 in/year). Earth's interior, like that of the other terrestrial planets, is divided into layers by their chemical or physical ( rheological ) properties. The outer layer
17017-599: The orbit of maximum axial tilt toward or away from the Sun—and the equinoxes , when Earth's rotational axis is aligned with its orbital axis. In the Northern Hemisphere, winter solstice currently occurs around 21 December; summer solstice is near 21 June, spring equinox is around 20 March and autumnal equinox is about 22 or 23 September. In the Southern Hemisphere, the situation is reversed, with
17160-408: The order, since the vector being rotated is multiplied from the right). The ease by which vectors can be rotated using a rotation matrix, as well as the ease of combining successive rotations, make the rotation matrix a useful and popular way to represent rotations, even though it is less concise than other representations. From Euler's rotation theorem we know that any rotation can be expressed as
17303-649: The ozone layer, life colonized Earth's surface. Among the earliest fossil evidence for life is microbial mat fossils found in 3.48 billion-year-old sandstone in Western Australia , biogenic graphite found in 3.7 billion-year-old metasedimentary rocks in Western Greenland , and remains of biotic material found in 4.1 billion-year-old rocks in Western Australia. The earliest direct evidence of life on Earth
17446-474: The partial melting of this mafic crust. The presence of grains of the mineral zircon of Hadean age in Eoarchean sedimentary rocks suggests that at least some felsic crust existed as early as 4.4 Ga , only 140 Ma after Earth's formation. There are two main models of how this initial small volume of continental crust evolved to reach its current abundance: (1) a relatively steady growth up to
17589-501: The past 66 Mys , and several million years ago, an African ape species gained the ability to stand upright. This facilitated tool use and encouraged communication that provided the nutrition and stimulation needed for a larger brain, which led to the evolution of humans . The development of agriculture , and then civilization , led to humans having an influence on Earth and the nature and quantity of other life forms that continues to this day. Earth's expected long-term future
17732-478: The planet's environment . Humanity's current impact on Earth's climate and biosphere is unsustainable , threatening the livelihood of humans and many other forms of life, and causing widespread extinctions . The Modern English word Earth developed, via Middle English , from an Old English noun most often spelled eorðe . It has cognates in every Germanic language , and their ancestral root has been reconstructed as * erþō . In its earliest attestation,
17875-400: The planet. The water vapor acts as a greenhouse gas and, together with other greenhouse gases in the atmosphere, particularly carbon dioxide (CO 2 ), creates the conditions for both liquid surface water and water vapor to persist via the capturing of energy from the Sun's light . This process maintains the current average surface temperature of 14.76 °C (58.57 °F), at which water
18018-421: The polar Northern and Southern hemispheres; or by longitude into the continental Eastern and Western hemispheres. Most of Earth's surface is ocean water: 70.8% or 361 million km (139 million sq mi). This vast pool of salty water is often called the world ocean , and makes Earth with its dynamic hydrosphere a water world or ocean world . Indeed, in Earth's early history
18161-426: The position of the Sun during the winter solstice had drifted roughly one degree over the course of fifty years relative to the position of the stars. The precession of Earth's axis was later explained by Newtonian physics . Being an oblate spheroid , Earth has a non-spherical shape, bulging outward at the equator. The gravitational tidal forces of the Moon and Sun apply torque to the equator, attempting to pull
18304-495: The present day, which is supported by the radiometric dating of continental crust globally and (2) an initial rapid growth in the volume of continental crust during the Archean , forming the bulk of the continental crust that now exists, which is supported by isotopic evidence from hafnium in zircons and neodymium in sedimentary rocks. The two models and the data that support them can be reconciled by large-scale recycling of
18447-1344: The quaternion as an array following the convention used in (Wertz 1980) and (Markley 2003). An alternative definition, used for example in (Coutsias 1999) and (Schmidt 2001), defines the "scalar" term as the first quaternion element, with the other elements shifted down one position. In terms of the Euler axis e ^ = [ e x e y e z ] {\displaystyle {\hat {\mathbf {e} }}={\begin{bmatrix}e_{x}\\e_{y}\\e_{z}\end{bmatrix}}} and angle θ this versor's components are expressed as follows: q i = e x sin θ 2 q j = e y sin θ 2 q k = e z sin θ 2 q r = cos θ 2 {\displaystyle {\begin{aligned}q_{i}&=e_{x}\sin {\frac {\theta }{2}}\\q_{j}&=e_{y}\sin {\frac {\theta }{2}}\\q_{k}&=e_{z}\sin {\frac {\theta }{2}}\\q_{r}&=\cos {\frac {\theta }{2}}\end{aligned}}} Inspection shows that
18590-399: The quaternion parametrization obeys the following constraint: q i 2 + q j 2 + q k 2 + q r 2 = 1 {\displaystyle q_{i}^{2}+q_{j}^{2}+q_{k}^{2}+q_{r}^{2}=1} The last term (in our definition) is often called the scalar term, which has its origin in quaternions when understood as
18733-493: The rates of mantle convection and plate tectonics, and allowing the production of uncommon igneous rocks such as komatiites that are rarely formed today. The mean heat loss from Earth is 87 mW m , for a global heat loss of 4.42 × 10 W . A portion of the core's thermal energy is transported toward the crust by mantle plumes , a form of convection consisting of upwellings of higher-temperature rock. These plumes can produce hotspots and flood basalts . More of
18876-429: The rotation axis will align itself with one of the inertia axes of the body. For a generic solid object without any axis of symmetry, the evolution of the object's orientation, represented (for example) by a rotation matrix R that transforms internal to external coordinates, may be numerically simulated. Given the object's fixed internal moment of inertia tensor I 0 and fixed external angular momentum L ,
19019-409: The rotation matrix are not all independent—as Euler's rotation theorem dictates, the rotation matrix has only three degrees of freedom. The rotation matrix has the following properties: The angle θ which appears in the eigenvalue expression corresponds to the angle of the Euler axis and angle representation. The eigenvector corresponding to the eigenvalue of 1 is the accompanying Euler axis, since
19162-442: The same axis (such as XXY) which would reduce the degrees of freedom that can be represented. Therefore, Euler angles are never expressed in terms of the external frame, or in terms of the co-moving rotated body frame, but in a mixture. Other conventions (e.g., rotation matrix or quaternions ) are used to avoid this problem. In aviation orientation of the aircraft is usually expressed as intrinsic Tait-Bryan angles following
19305-522: The shape of an ellipsoid , bulging at its Equator ; its diameter is 43 kilometres (27 mi) longer there than at its poles . Earth's shape also has local topographic variations; the largest local variations, like the Mariana Trench (10,925 metres or 35,843 feet below local sea level), shortens Earth's average radius by 0.17% and Mount Everest (8,848 metres or 29,029 feet above local sea level) lengthens it by 0.14%. Since Earth's surface
19448-477: The spinning top just pitches a little. This pitching motion reorients the spinning top with respect to the torque that is being exerted. The result is that the torque exerted by gravity – via the pitching motion – elicits gyroscopic precession (which in turn yields a counter torque against the gravity torque) rather than causing the spinning top to fall to its side. Precession or gyroscopic considerations have an effect on bicycle performance at high speed. Precession
19591-497: The strings/bands can be partially untangled back to the 2( n − 1) turns state with each application of the same procedure used in untangling from 2 turns to 0 turns. Applying the same procedure n times will take a 2 n -tangled object back to the untangled or 0 turn state. The untangling process also removes any rotation-generated twisting about the strings/bands themselves. Simple 3D mechanical models can be used to demonstrate these facts. The Rodrigues vector (sometimes called
19734-408: The summer and winter solstices exchanged and the spring and autumnal equinox dates swapped. Rotation representation (mathematics)#Euler axis and angle (rotation vector) In geometry , various formalisms exist to express a rotation in three dimensions as a mathematical transformation . In physics, this concept is applied to classical mechanics where rotational (or angular) kinematics
19877-405: The torque and angular momentum vectors respectively. Due to the way the torque vectors are defined, it is a vector that is perpendicular to the plane of the forces that create it. Thus it may be seen that the angular momentum vector will change perpendicular to those forces. Depending on how the forces are created, they will often rotate with the angular momentum vector, and then circular precession
20020-441: The wheel (around the wheelhub) is added. Imagine the gimbal axis to be locked, so that the wheel cannot pitch. The gimbal axis has sensors, that measure whether there is a torque around the gimbal axis. In the picture, a section of the wheel has been named dm 1 . At the depicted moment in time, section dm 1 is at the perimeter of the rotating motion around the (vertical) pivot axis. Section dm 1 , therefore, has
20163-430: The wheel is moving away from the pivot axis, and so a force (again, a Coriolis force) acts in the same direction as in the case of dm 1 . Note that both arrows point in the same direction. The same reasoning applies for the bottom half of the wheel, but there the arrows point in the opposite direction to that of the top arrows. Combined over the entire wheel, there is a torque around the gimbal axis when some spinning
20306-461: The word eorðe was used to translate the many senses of Latin terra and Greek γῆ gē : the ground, its soil , dry land, the human world, the surface of the world (including the sea), and the globe itself. As with Roman Terra /Tellūs and Greek Gaia , Earth may have been a personified goddess in Germanic paganism : late Norse mythology included Jörð ("Earth"), a giantess often given as
20449-620: The year. This causes the seasonal change in climate, with summer in the Northern Hemisphere occurring when the Tropic of Cancer is facing the Sun, and in the Southern Hemisphere when the Tropic of Capricorn faces the Sun. In each instance, winter occurs simultaneously in the opposite hemisphere. During the summer, the day lasts longer, and the Sun climbs higher in the sky. In winter, the climate becomes cooler and
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