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84-636: Pasiphae / p ə ˈ s ɪ f eɪ . iː / , formerly spelled Pasiphaë , is a retrograde irregular satellite of Jupiter . It was discovered in 1908 by Philibert Jacques Melotte and later named after the mythological Pasiphaë , wife of Minos and mother of the Minotaur from Greek legend . The moon was first spotted on a plate taken at the Royal Greenwich Observatory on the night of 28 February 1908. Inspection of previous plates found it as far back as January 27. It received

168-407: A disk galaxy 's general rotation are more likely to be found in the galactic halo than in the galactic disk . The Milky Way 's outer halo has many globular clusters with a retrograde orbit and with a retrograde or zero rotation. The structure of the halo is the topic of an ongoing debate. Several studies have claimed to find a halo consisting of two distinct components. These studies find

252-627: A unit vector u ^ {\displaystyle \mathbf {\hat {u}} } perpendicular to the plane of angular displacement, a scalar angular speed ω {\displaystyle \omega } results, where ω u ^ = ω , {\displaystyle \omega \mathbf {\hat {u}} ={\boldsymbol {\omega }},} and ω = v ⊥ r , {\displaystyle \omega ={\frac {v_{\perp }}{r}},} where v ⊥ {\displaystyle v_{\perp }}

336-515: A "dual" halo, with an inner, more metal-rich, prograde component (i.e. stars orbit the galaxy on average with the disk rotation), and a metal-poor, outer, retrograde (rotating against the disc) component. However, these findings have been challenged by other studies, arguing against such a duality. These studies demonstrate that the observational data can be explained without a duality, when employing an improved statistical analysis and accounting for measurement uncertainties. The nearby Kapteyn's Star

420-409: A complex function of the configuration of the matter about the center of rotation and the orientation of the rotation for the various bits. For a rigid body , for instance a wheel or an asteroid, the orientation of rotation is simply the position of the rotation axis versus the matter of the body. It may or may not pass through the center of mass , or it may lie completely outside of the body. For

504-564: A fast prograde rotation with a period of several hours much like most of the planets in the Solar System. Venus is close enough to the Sun to experience significant gravitational tidal dissipation , and also has a thick enough atmosphere to create thermally driven atmospheric tides that create a retrograde torque . Venus's present slow retrograde rotation is in equilibrium balance between gravitational tides trying to tidally lock Venus to

588-469: A particular axis. However, if the particle's trajectory lies in a single plane , it is sufficient to discard the vector nature of angular momentum, and treat it as a scalar (more precisely, a pseudoscalar ). Angular momentum can be considered a rotational analog of linear momentum. Thus, where linear momentum p is proportional to mass m and linear speed v , p = m v , {\displaystyle p=mv,} angular momentum L

672-410: A particular interaction is called angular impulse , sometimes twirl . Angular impulse is the angular analog of (linear) impulse . The trivial case of the angular momentum L {\displaystyle L} of a body in an orbit is given by L = 2 π M f r 2 {\displaystyle L=2\pi Mfr^{2}} where M {\displaystyle M}

756-403: A perpendicular rotation that is neither prograde nor retrograde. An object with an axial tilt between 90 degrees and 180 degrees is rotating in the opposite direction to its orbital direction. Regardless of inclination or axial tilt, the north pole of any planet or moon in the Solar System is defined as the pole that is in the same celestial hemisphere as Earth's north pole. All eight planets in

840-510: A planet's gravity, it can be captured into either a retrograde or prograde orbit depending on whether it first approaches the side of the planet that is rotating towards or away from it. This is an irregular moon . In the Solar System, many of the asteroid-sized moons have retrograde orbits, whereas all the large moons except Triton (the largest of Neptune's moons) have prograde orbits. The particles in Saturn's Phoebe ring are thought to have

924-407: A prograde orbit, because in this situation less propellant is required to reach the orbit. When a galaxy or a planetary system forms , its material takes a shape similar to that of a disk. Most of the material orbits and rotates in one direction. This uniformity of motion is due to the collapse of a gas cloud. The nature of the collapse is explained by conservation of angular momentum . In 2010

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1008-405: A retrograde orbit around the Sun. Most Kuiper belt objects have prograde orbits around the Sun. The first Kuiper belt object discovered to have a retrograde orbit was 2008 KV 42 . Other Kuiper belt objects with retrograde orbits are (471325) 2011 KT 19 , (342842) 2008 YB 3 , (468861) 2013 LU 28 and 2011 MM 4 . All of these orbits are highly tilted, with inclinations in

1092-453: A retrograde orbit because they originate from the irregular moon Phoebe . All retrograde satellites experience tidal deceleration to some degree. The only satellite in the Solar System for which this effect is non-negligible is Neptune's moon Triton. All the other retrograde satellites are on distant orbits and tidal forces between them and the planet are negligible. Within the Hill sphere ,

1176-463: Is a spectrally featureless object, consistent with the suspected asteroidal origin of the object. Pasiphae is believed to be a fragment from a captured asteroid along with other Pasiphae group satellites. In the visual spectrum the satellite appears grey ( colour indices B-V=0.74, R-V=0.38) similar to C-type asteroids . Retrograde motion In the Solar System , the orbits around

1260-430: Is also why hurricanes form spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system, but it does not uniquely determine it. The three-dimensional angular momentum for a point particle is classically represented as a pseudovector r × p , the cross product of the particle's position vector r (relative to some origin) and its momentum vector ;

1344-405: Is always equal to the total torque on the system; the sum of all internal torques of any system is always 0 (this is the rotational analogue of Newton's third law of motion ). Therefore, for a closed system (where there is no net external torque), the total torque on the system must be 0, which means that the total angular momentum of the system is constant. The change in angular momentum for

1428-659: Is always measured with respect to a fixed origin. Therefore, strictly speaking, L should be referred to as the angular momentum relative to that center . In the case of circular motion of a single particle, we can use I = r 2 m {\displaystyle I=r^{2}m} and ω = v / r {\displaystyle \omega ={v}/{r}} to expand angular momentum as L = r 2 m ⋅ v / r , {\displaystyle L=r^{2}m\cdot {v}/{r},} reducing to: L = r m v , {\displaystyle L=rmv,}

1512-482: Is approximately 120 degrees. Pluto and its moon Charon are tidally locked to each other. It is suspected that the Plutonian satellite system was created by a massive collision . If formed in the gravity field of a planet as the planet is forming, a moon will orbit the planet in the same direction as the planet is rotating and is a regular moon . If an object is formed elsewhere and later captured into orbit by

1596-406: Is approximately parallel with the plane of the Solar System. The reason for Uranus's unusual axial tilt is not known with certainty, but the usual speculation is that it was caused by a collision with an Earth-sized protoplanet during the formation of the Solar System. It is unlikely that Venus was formed with its present slow retrograde rotation, which takes 243 days. Venus probably began with

1680-894: Is desired to know what effect the moving matter has on the point—can it exert energy upon it or perform work about it? Energy , the ability to do work , can be stored in matter by setting it in motion—a combination of its inertia and its displacement. Inertia is measured by its mass , and displacement by its velocity . Their product, ( amount of inertia ) × ( amount of displacement ) = amount of (inertia⋅displacement) mass × velocity = momentum m × v = p {\displaystyle {\begin{aligned}({\text{amount of inertia}})\times ({\text{amount of displacement}})&={\text{amount of (inertia⋅displacement)}}\\{\text{mass}}\times {\text{velocity}}&={\text{momentum}}\\m\times v&=p\\\end{aligned}}}

1764-399: Is difficult to telescopically analyse the rotation of most asteroids. As of 2012, data is available for less than 200 asteroids and the different methods of determining the orientation of poles often result in large discrepancies. The asteroid spin vector catalog at Poznan Observatory avoids use of the phrases "retrograde rotation" or "prograde rotation" as it depends which reference plane

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1848-450: Is directed perpendicular to the plane of angular displacement, as indicated by the right-hand rule – so that the angular velocity is seen as counter-clockwise from the head of the vector. Conversely, the L {\displaystyle \mathbf {L} } vector defines the plane in which r {\displaystyle \mathbf {r} } and p {\displaystyle \mathbf {p} } lie. By defining

1932-427: Is in a retrograde orbit. A celestial object's axial tilt indicates whether the object's rotation is prograde or retrograde. Axial tilt is the angle between an object's rotation axis and a line perpendicular to its orbital plane passing through the object's centre. An object with an axial tilt up to 90 degrees is rotating in the same direction as its primary. An object with an axial tilt of exactly 90 degrees, has

2016-401: Is known, the angular momentum L {\displaystyle L} is given by L = 16 15 π 2 ρ f r 5 {\displaystyle L={\frac {16}{15}}\pi ^{2}\rho fr^{5}} where ρ {\displaystyle \rho } is the sphere's density , f {\displaystyle f} is

2100-490: Is meant and asteroid coordinates are usually given with respect to the ecliptic plane rather than the asteroid's orbital plane. Asteroids with satellites, also known as binary asteroids, make up about 15% of all asteroids less than 10 km in diameter in the main belt and near-Earth population and most are thought to be formed by the YORP effect causing an asteroid to spin so fast that it breaks up. As of 2012, and where

2184-449: Is opposite to that of its disk – spews jets much more powerful than those of a prograde black hole, which may have no jet at all. Scientists have produced a theoretical framework for the formation and evolution of retrograde black holes based on the gap between the inner edge of an accretion disk and the black hole. Conservation of angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum )

2268-591: Is possible. The last few giant impacts during planetary formation tend to be the main determiner of a terrestrial planet 's rotation rate. During the giant impact stage, the thickness of a protoplanetary disk is far larger than the size of planetary embryos so collisions are equally likely to come from any direction in three dimensions. This results in the axial tilt of accreted planets ranging from 0 to 180 degrees with any direction as likely as any other with both prograde and retrograde spins equally probable. Therefore, prograde spin with small axial tilt, common for

2352-445: Is proportional to moment of inertia I and angular speed ω measured in radians per second. L = I ω . {\displaystyle L=I\omega .} Unlike mass, which depends only on amount of matter, moment of inertia depends also on the position of the axis of rotation and the distribution of the matter. Unlike linear velocity, which does not depend upon the choice of origin, orbital angular velocity

2436-401: Is related to the angular velocity of the rotation. Because moment of inertia is a crucial part of the spin angular momentum, the latter necessarily includes all of the complications of the former, which is calculated by multiplying elementary bits of the mass by the squares of their distances from the center of rotation. Therefore, the total moment of inertia, and the angular momentum, is

2520-520: Is represented by the yellow segments (extending from the pericentre to the apocentre ). The outermost regular satellite Callisto is located for reference. Pasiphae is also known to be in a secular resonance with Jupiter (tying the longitude of its perijove with the longitude of perihelion of Jupiter). With a diameter estimated at 58 km, Pasiphae is the largest retrograde and third largest irregular satellite after Himalia and Elara . Spectroscopical measurements in infrared indicate that Pasiphae

2604-402: Is required to know the rate at which the position vector sweeps out angle, the direction perpendicular to the instantaneous plane of angular displacement, and the mass involved, as well as how this mass is distributed in space. By retaining this vector nature of angular momentum, the general nature of the equations is also retained, and can describe any sort of three-dimensional motion about

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2688-471: Is the angular momentum , sometimes called, as here, the moment of momentum of the particle versus that particular center point. The equation L = r m v {\displaystyle L=rmv} combines a moment (a mass m {\displaystyle m} turning moment arm r {\displaystyle r} ) with a linear (straight-line equivalent) speed v {\displaystyle v} . Linear speed referred to

2772-510: Is the cross product of the position vector r {\displaystyle \mathbf {r} } and the linear momentum p = m v {\displaystyle \mathbf {p} =m\mathbf {v} } of the particle. By the definition of the cross product, the L {\displaystyle \mathbf {L} } vector is perpendicular to both r {\displaystyle \mathbf {r} } and p {\displaystyle \mathbf {p} } . It

2856-509: Is the mass of the orbiting object, f {\displaystyle f} is the orbit's frequency and r {\displaystyle r} is the orbit's radius. The angular momentum L {\displaystyle L} of a uniform rigid sphere rotating around its axis, instead, is given by L = 4 5 π M f r 2 {\displaystyle L={\frac {4}{5}}\pi Mfr^{2}} where M {\displaystyle M}

2940-401: Is the radius of gyration , the distance from the axis at which the entire mass m {\displaystyle m} may be considered as concentrated. Similarly, for a point mass m {\displaystyle m} the moment of inertia is defined as, I = r 2 m {\displaystyle I=r^{2}m} where r {\displaystyle r}

3024-459: Is the rotational analog of linear momentum . It is an important physical quantity because it is a conserved quantity  – the total angular momentum of a closed system remains constant. Angular momentum has both a direction and a magnitude, and both are conserved. Bicycles and motorcycles , flying discs , rifled bullets , and gyroscopes owe their useful properties to conservation of angular momentum. Conservation of angular momentum

3108-483: Is the frequency of rotation and r {\displaystyle r} is the disk's radius. If instead the disk rotates about its diameter (e.g. coin toss), its angular momentum L {\displaystyle L} is given by L = 1 2 π M f r 2 {\displaystyle L={\frac {1}{2}}\pi Mfr^{2}} Just as for angular velocity , there are two special types of angular momentum of an object:

3192-438: Is the length of the moment arm , a line dropped perpendicularly from the origin onto the path of the particle. It is this definition, (length of moment arm) × (linear momentum) , to which the term moment of momentum refers. Another approach is to define angular momentum as the conjugate momentum (also called canonical momentum ) of the angular coordinate ϕ {\displaystyle \phi } expressed in

3276-407: Is the matter's momentum . Referring this momentum to a central point introduces a complication: the momentum is not applied to the point directly. For instance, a particle of matter at the outer edge of a wheel is, in effect, at the end of a lever of the same length as the wheel's radius, its momentum turning the lever about the center point. This imaginary lever is known as the moment arm . It has

3360-1002: Is the perpendicular component of the motion, as above. The two-dimensional scalar equations of the previous section can thus be given direction: L = I ω = I ω u ^ = ( r 2 m ) ω u ^ = r m v ⊥ u ^ = r ⊥ m v u ^ , {\displaystyle {\begin{aligned}\mathbf {L} &=I{\boldsymbol {\omega }}\\&=I\omega \mathbf {\hat {u}} \\&=\left(r^{2}m\right)\omega \mathbf {\hat {u}} \\&=rmv_{\perp }\mathbf {\hat {u}} \\&=r_{\perp }mv\mathbf {\hat {u}} ,\end{aligned}}} and L = r m v u ^ {\displaystyle \mathbf {L} =rmv\mathbf {\hat {u}} } for circular motion, where all of

3444-636: Is the perpendicular component of the motion. Expanding, L = r m v sin ⁡ ( θ ) , {\displaystyle L=rmv\sin(\theta ),} rearranging, L = r sin ⁡ ( θ ) m v , {\displaystyle L=r\sin(\theta )mv,} and reducing, angular momentum can also be expressed, L = r ⊥ m v , {\displaystyle L=r_{\perp }mv,} where r ⊥ = r sin ⁡ ( θ ) {\displaystyle r_{\perp }=r\sin(\theta )}

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3528-512: Is the sphere's mass, f {\displaystyle f} is the frequency of rotation and r {\displaystyle r} is the sphere's radius. Thus, for example, the orbital angular momentum of the Earth with respect to the Sun is about 2.66 × 10 kg⋅m ⋅s , while its rotational angular momentum is about 7.05 × 10 kg⋅m ⋅s . In the case of a uniform rigid sphere rotating around its axis, if, instead of its mass, its density

3612-463: Is thought to have ended up with its high-velocity retrograde orbit around the galaxy as a result of being ripped from a dwarf galaxy that merged with the Milky Way. Close-flybys and mergers of galaxies within galaxy clusters can pull material out of galaxies and create small satellite galaxies in either prograde or retrograde orbits around larger galaxies. A galaxy called Complex H, which

3696-665: The Lagrangian of the mechanical system. Consider a mechanical system with a mass m {\displaystyle m} constrained to move in a circle of radius r {\displaystyle r} in the absence of any external force field. The kinetic energy of the system is T = 1 2 m r 2 ω 2 = 1 2 m r 2 ϕ ˙ 2 . {\displaystyle T={\tfrac {1}{2}}mr^{2}\omega ^{2}={\tfrac {1}{2}}mr^{2}{\dot {\phi }}^{2}.} And

3780-490: The Pasiphae group , irregular retrograde moons orbiting Jupiter at distances ranging between 22.8 and 24.1 million km, and with inclinations ranging between 144.5° and 158.3°. The orbital elements are as of January 2000. They are continuously changing due to solar and planetary perturbations. The diagram illustrates its orbit in relation to other retrograde irregular satellites of Jupiter. The eccentricity of selected orbits

3864-473: The Solar System orbit the Sun in the direction of the Sun's rotation, which is counterclockwise when viewed from above the Sun's north pole . Six of the planets also rotate about their axis in this same direction. The exceptions – the planets with retrograde rotation – are Venus and Uranus . Venus's axial tilt is 177°, which means it is rotating almost exactly in the opposite direction to its orbit. Uranus has an axial tilt of 97.77°, so its axis of rotation

3948-490: The Sun of all planets and most other objects, except many comets , are prograde. They orbit around the Sun in the same direction as the sun rotates about its axis, which is counterclockwise when observed from above the Sun's north pole. Except for Venus and Uranus , planetary rotations around their axis are also prograde. Most natural satellites have prograde orbits around their planets. Prograde satellites of Uranus orbit in

4032-407: The Sun . The inclination of moons is measured from the equator of the planet they orbit. An object with an inclination between 0 and 90 degrees is orbiting or revolving in the same direction as the primary is rotating. An object with an inclination of exactly 90 degrees has a perpendicular orbit that is neither prograde nor retrograde. An object with an inclination between 90 degrees and 180 degrees

4116-441: The provisional designation 1908 CJ , as it was not clear whether it was an asteroid or a moon of Jupiter. The recognition of the latter case came by April 10. Pasiphae did not receive its present name until 1975; before then, it was simply known as Jupiter VIII . It was sometimes called "Poseidon" between 1955 and 1975. Pasiphae orbits Jupiter on a high eccentricity and high inclination retrograde orbit. It gives its name to

4200-435: The spin angular momentum is the angular momentum about the object's centre of mass , while the orbital angular momentum is the angular momentum about a chosen center of rotation. The Earth has an orbital angular momentum by nature of revolving around the Sun , and a spin angular momentum by nature of its daily rotation around the polar axis. The total angular momentum is the sum of the spin and orbital angular momenta. In

4284-441: The westerlies or from west to east through the trade wind easterlies. Prograde motion with respect to planetary rotation is seen in the atmospheric super-rotation of the thermosphere of Earth and in the upper troposphere of Venus . Simulations indicate that the atmosphere of Pluto should be dominated by winds retrograde to its rotation. Artificial satellites destined for low inclination orbits are usually launched in

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4368-513: The 100°–125° range. Meteoroids in a retrograde orbit around the Sun hit the Earth with a faster relative speed than prograde meteoroids and tend to burn up in the atmosphere and are more likely to hit the side of the Earth facing away from the Sun (i.e. at night) whereas the prograde meteoroids have slower closing speeds and more often land as meteorites and tend to hit the Sun-facing side of

4452-431: The Earth. Most meteoroids are prograde. The Sun's motion about the centre of mass of the Solar System is complicated by perturbations from the planets. Every few hundred years this motion switches between prograde and retrograde. Retrograde motion, or retrogression, within the Earth's atmosphere is seen in weather systems whose motion is opposite the general regional direction of airflow, i.e. from east to west against

4536-476: The Solar System are tidally locked to their host planet, so they have zero rotation relative to their host planet, but have the same type of rotation as their host planet relative to the Sun because they have prograde orbits around their host planet. That is, they all have prograde rotation relative to the Sun except those of Uranus. If there is a collision, material could be ejected in any direction and coalesce into either prograde or retrograde moons, which may be

4620-535: The Sun and atmospheric tides trying to spin Venus in a retrograde direction. In addition to maintaining this present day equilibrium, tides are also sufficient to account for evolution of Venus's rotation from a primordial fast prograde direction to its present-day slow retrograde rotation. In the past, various alternative hypotheses have been proposed to explain Venus's retrograde rotation, such as collisions or it having originally formed that way. Despite being closer to

4704-504: The Sun than Venus, Mercury is not tidally locked because it has entered a 3:2 spin–orbit resonance due to the eccentricity of its orbit. Mercury's prograde rotation is slow enough that due to its eccentricity, its angular orbital velocity exceeds its angular rotational velocity near perihelion , causing the motion of the sun in Mercury's sky to temporarily reverse. The rotations of Earth and Mars are also affected by tidal forces with

4788-455: The Sun, but they have not reached an equilibrium state like Mercury and Venus because they are further out from the Sun where tidal forces are weaker. The gas giants of the Solar System are too massive and too far from the Sun for tidal forces to slow down their rotations. All known dwarf planets and dwarf planet candidates have prograde orbits around the Sun, but some have retrograde rotation. Pluto has retrograde rotation; its axial tilt

4872-447: The case for the moons of dwarf planet Haumea , although Haumea's rotation direction is not known. Asteroids usually have a prograde orbit around the Sun. Only a few dozen asteroids in retrograde orbits are known. Some asteroids with retrograde orbits may be burnt-out comets, but some may acquire their retrograde orbit due to gravitational interactions with Jupiter . Due to their small size and their large distance from Earth it

4956-404: The case of the Earth the primary conserved quantity is the total angular momentum of the solar system because angular momentum is exchanged to a small but important extent among the planets and the Sun. The orbital angular momentum vector of a point particle is always parallel and directly proportional to its orbital angular velocity vector ω , where the constant of proportionality depends on both

5040-1031: The center of rotation – circular , linear , or otherwise. In vector notation , the orbital angular momentum of a point particle in motion about the origin can be expressed as: L = I ω , {\displaystyle \mathbf {L} =I{\boldsymbol {\omega }},} where This can be expanded, reduced, and by the rules of vector algebra , rearranged: L = ( r 2 m ) ( r × v r 2 ) = m ( r × v ) = r × m v = r × p , {\displaystyle {\begin{aligned}\mathbf {L} &=\left(r^{2}m\right)\left({\frac {\mathbf {r} \times \mathbf {v} }{r^{2}}}\right)\\&=m\left(\mathbf {r} \times \mathbf {v} \right)\\&=\mathbf {r} \times m\mathbf {v} \\&=\mathbf {r} \times \mathbf {p} ,\end{aligned}}} which

5124-568: The central point is simply the product of the distance r {\displaystyle r} and the angular speed ω {\displaystyle \omega } versus the point: v = r ω , {\displaystyle v=r\omega ,} another moment. Hence, angular momentum contains a double moment: L = r m r ω . {\displaystyle L=rmr\omega .} Simplifying slightly, L = r 2 m ω , {\displaystyle L=r^{2}m\omega ,}

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5208-413: The cluster and this can lead to disks and their resulting planets having inclined or retrograde orbits around their stars. Retrograde motion may also result from gravitational interactions with other celestial bodies in the same system (See Kozai mechanism ) or a near-collision with another planet, or it may be that the star itself flipped over early in their system's formation due to interactions between

5292-528: The coordinate ϕ {\displaystyle \phi } is defined by p ϕ = ∂ L ∂ ϕ ˙ = m r 2 ϕ ˙ = I ω = L . {\displaystyle p_{\phi }={\frac {\partial {\mathcal {L}}}{\partial {\dot {\phi }}}}=mr^{2}{\dot {\phi }}=I\omega =L.} To completely define orbital angular momentum in three dimensions , it

5376-470: The direction Uranus rotates, which is retrograde to the Sun. Nearly all regular satellites are tidally locked and thus have prograde rotation. Retrograde satellites are generally small and distant from their planets, except Neptune 's satellite Triton , which is large and close. All retrograde satellites are thought to have formed separately before being captured by their planets. Most low-inclination artificial satellites of Earth have been placed in

5460-453: The direction the star is rotating. A second such planet was announced just a day later: HAT-P-7b . In one study more than half of all the known hot Jupiters had orbits that were misaligned with the rotation axis of their parent stars, with six having backwards orbits. One proposed explanation is that hot Jupiters tend to form in dense clusters, where perturbations are more common and gravitational capture of planets by neighboring stars

5544-405: The discovery of several hot Jupiters with backward orbits called into question the theories about the formation of planetary systems. This can be explained by noting that stars and their planets do not form in isolation but in star clusters that contain molecular clouds . When a protoplanetary disk collides with or steals material from a cloud this can result in retrograde motion of a disk and

5628-888: The effect of multiplying the momentum's effort in proportion to its length, an effect known as a moment . Hence, the particle's momentum referred to a particular point, ( moment arm ) × ( amount of inertia ) × ( amount of displacement ) = moment of (inertia⋅displacement) length × mass × velocity = moment of momentum r × m × v = L {\displaystyle {\begin{aligned}({\text{moment arm}})\times ({\text{amount of inertia}})\times ({\text{amount of displacement}})&={\text{moment of (inertia⋅displacement)}}\\{\text{length}}\times {\text{mass}}\times {\text{velocity}}&={\text{moment of momentum}}\\r\times m\times v&=L\\\end{aligned}}}

5712-440: The frequency of rotation and r {\displaystyle r} is the sphere's radius. In the simplest case of a spinning disk, the angular momentum L {\displaystyle L} is given by L = π M f r 2 {\displaystyle L=\pi Mfr^{2}} where M {\displaystyle M} is the disk's mass, f {\displaystyle f}

5796-460: The latter is p = m v in Newtonian mechanics . Unlike linear momentum, angular momentum depends on where this origin is chosen, since the particle's position is measured from it. Angular momentum is an extensive quantity ; that is, the total angular momentum of any composite system is the sum of the angular momenta of its constituent parts. For a continuous   rigid body or a fluid ,

5880-455: The mass of the particle and its distance from origin. The spin angular momentum vector of a rigid body is proportional but not always parallel to the spin angular velocity vector Ω , making the constant of proportionality a second-rank tensor rather than a scalar. Angular momentum is a vector quantity (more precisely, a pseudovector ) that represents the product of a body's rotational inertia and rotational velocity (in radians/sec) about

5964-556: The motion is perpendicular to the radius r {\displaystyle r} . In the spherical coordinate system the angular momentum vector expresses as Angular momentum can be described as the rotational analog of linear momentum . Like linear momentum it involves elements of mass and displacement . Unlike linear momentum it also involves elements of position and shape . Many problems in physics involve matter in motion about some certain point in space, be it in actual rotation about it, or simply moving past it, where it

6048-574: The potential energy is U = 0. {\displaystyle U=0.} Then the Lagrangian is L ( ϕ , ϕ ˙ ) = T − U = 1 2 m r 2 ϕ ˙ 2 . {\displaystyle {\mathcal {L}}\left(\phi ,{\dot {\phi }}\right)=T-U={\tfrac {1}{2}}mr^{2}{\dot {\phi }}^{2}.} The generalized momentum "canonically conjugate to"

6132-658: The product of the radius of rotation r and the linear momentum of the particle p = m v {\displaystyle p=mv} , where v = r ω {\displaystyle v=r\omega } is the linear (tangential) speed . This simple analysis can also apply to non-circular motion if one uses the component of the motion perpendicular to the radius vector : L = r m v ⊥ , {\displaystyle L=rmv_{\perp },} where v ⊥ = v sin ⁡ ( θ ) {\displaystyle v_{\perp }=v\sin(\theta )}

6216-621: The prograde direction, since this minimizes the amount of propellant required to reach orbit by taking advantage of the Earth's rotation (an equatorial launch site is optimal for this effect). However, Israeli Ofeq satellites are launched in a westward, retrograde direction over the Mediterranean to ensure that launch debris does not fall onto populated land areas. Stars and planetary systems tend to be born in star clusters rather than forming in isolation. Protoplanetary disks can collide with or steal material from molecular clouds within

6300-439: The quantity r 2 m {\displaystyle r^{2}m} is the particle's moment of inertia , sometimes called the second moment of mass. It is a measure of rotational inertia. The above analogy of the translational momentum and rotational momentum can be expressed in vector form: The direction of momentum is related to the direction of the velocity for linear movement. The direction of angular momentum

6384-439: The region of stability for retrograde orbits at a large distance from the primary is larger than that for prograde orbits. This has been suggested as an explanation for the preponderance of retrograde moons around Jupiter. Because Saturn has a more even mix of retrograde/prograde moons, however, the underlying causes appear to be more complex. With the exception of Hyperion , all the known regular planetary natural satellites in

6468-428: The resulting planets. A celestial object's inclination indicates whether the object's orbit is prograde or retrograde. The inclination of a celestial object is the angle between its orbital plane and another reference frame such as the equatorial plane of the object's primary. In the Solar System , inclination of the planets is measured from the ecliptic plane , which is the plane of Earth 's orbit around

6552-507: The rotation is known, all satellites of asteroids orbit the asteroid in the same direction as the asteroid is rotating. Most known objects that are in orbital resonance are orbiting in the same direction as the objects they are in resonance with, however a few retrograde asteroids have been found in resonance with Jupiter and Saturn . Comets from the Oort cloud are much more likely than asteroids to be retrograde. Halley's Comet has

6636-414: The same body, angular momentum may take a different value for every possible axis about which rotation may take place. It reaches a minimum when the axis passes through the center of mass. For a collection of objects revolving about a center, for instance all of the bodies of the Solar System , the orientations may be somewhat organized, as is the Solar System, with most of the bodies' axes lying close to

6720-402: The solar system's terrestrial planets except for Venus, is not common for terrestrial planets in general. The pattern of stars appears fixed in the sky, insofar as human vision is concerned; this is because their massive distances relative to the Earth result in motion imperceptible to the naked eye. In reality, stars orbit the center of their galaxy. Stars with an orbit retrograde relative to

6804-436: The star's magnetic field and the planet-forming disk. The accretion disk of the protostar IRAS 16293-2422 has parts rotating in opposite directions. This is the first known example of a counterrotating accretion disk. If this system forms planets, the inner planets will likely orbit in the opposite direction to the outer planets. WASP-17b was the first exoplanet that was discovered to be orbiting its star opposite to

6888-515: The system's axis. Their orientations may also be completely random. In brief, the more mass and the farther it is from the center of rotation (the longer the moment arm ), the greater the moment of inertia, and therefore the greater the angular momentum for a given angular velocity . In many cases the moment of inertia , and hence the angular momentum, can be simplified by, I = k 2 m , {\displaystyle I=k^{2}m,} where k {\displaystyle k}

6972-466: The total angular momentum is the volume integral of angular momentum density (angular momentum per unit volume in the limit as volume shrinks to zero) over the entire body. Similar to conservation of linear momentum, where it is conserved if there is no external force, angular momentum is conserved if there is no external torque . Torque can be defined as the rate of change of angular momentum, analogous to force . The net external torque on any system

7056-417: Was orbiting the Milky Way in a retrograde direction relative to the Milky Way's rotation, is colliding with the Milky Way. NGC 7331 is an example of a galaxy that has a bulge that is rotating in the opposite direction to the rest of the disk, probably as a result of infalling material. The center of a spiral galaxy contains at least one supermassive black hole . A retrograde black hole – one whose spin

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