Correspondence principle

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Correspondence_principle

In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. In other words, it says that for large orbits and for large energies, quantum calculations must agree with classical calculations.

The principle was formulated by Niels Bohr in 1920, though he had previously made use of it as early as 1913 in developing his model of the atom.

The term codifies the idea that a new theory should reproduce under some conditions the results of older well-established theories in those domains where the old theories work. This concept is somewhat different from the requirement of a formal limit under which the new theory reduces to the older, thanks to the existence of a deformation parameter.
Classical quantities appear in quantum mechanics in the form of expected values of observables, and as such the Ehrenfest theorem (which predicts the time evolution of the expected values) lends support to the correspondence principle.

Quantum mechanics

The rules of quantum mechanics are highly successful in describing microscopic objects, atoms and elementary particles. But macroscopic systems, like springs and capacitors, are accurately described by classical theories like classical mechanics and classical electrodynamics. If quantum mechanics were to be applicable to macroscopic objects, there must be some limit in which quantum mechanics reduces to classical mechanics. Bohr's correspondence principle demands that classical physics and quantum physics give the same answer when the systems become large. A. Sommerfeld (1921) referred to the principle as "Bohrs Zauberstab" (Bohr's magic wand).

The conditions under which quantum and classical physics agree are referred to as the correspondence limit, or the classical limit. Bohr provided a rough prescription for the correspondence limit: it occurs when the quantum numbers describing the system are large. A more elaborated analysis of quantum-classical correspondence (QCC) in wavepacket spreading leads to the distinction between robust "restricted QCC" and fragile "detailed QCC". "Restricted QCC" refers to the first two moments of the probability distribution and is true even when the wave packets diffract, while "detailed QCC" requires smooth potentials which vary over scales much larger than the wavelength, which is what Bohr considered.

The post-1925 new quantum theory came in two different formulations. In matrix mechanics, the correspondence principle was built in and was used to construct the theory. In the Schrödinger approach classical behavior is not clear because the waves spread out as they move. Once the Schrödinger equation was given a probabilistic interpretation, Ehrenfest showed that Newton's laws hold on average: the quantum statistical expectation value of the position and momentum obey Newton's laws.

The correspondence principle is one of the tools available to physicists for selecting quantum theories corresponding to reality. The principles of quantum mechanics are broad: states of a physical system form a complex vector space and physical observables are identified with Hermitian operators that act on this Hilbert space. The correspondence principle limits the choices to those that reproduce classical mechanics in the correspondence limit.

Because quantum mechanics only reproduces classical mechanics in a statistical interpretation, and because the statistical interpretation only gives the probabilities of different classical outcomes, Bohr has argued that quantum physics does not reduce to classical mechanics similarly as classical mechanics emerges as an approximation of special relativity at small velocities. He argued that classical physics exists independently of quantum theory and cannot be derived from it. His position is that it is inappropriate to understand the experiences of observers using purely quantum mechanical notions such as wavefunctions because the different states of experience of an observer are defined classically, and do not have a quantum mechanical analog. The relative state interpretation of quantum mechanics is an attempt to understand the experience of observers using only quantum mechanical notions. Niels Bohr was an early opponent of such interpretations. 

Many of these conceptual problems, however, resolve in the phase-space formulation of quantum mechanics, where the same variables with the same interpretation are utilized to describe both quantum and classical mechanics. 

Other scientific theories

The term "correspondence principle" is used in a more general sense to mean the reduction of a new scientific theory to an earlier scientific theory in appropriate circumstances. This requires that the new theory explain all the phenomena under circumstances for which the preceding theory was known to be valid, the "correspondence limit". 

For example,

• Einstein's special relativity satisfies the correspondence principle, because it reduces to classical mechanics in the limit of velocities small compared to the speed of light (example below);
• General relativity reduces to Newtonian gravity in the limit of weak gravitational fields;
• Laplace's theory of celestial mechanics reduces to Kepler's when interplanetary interactions are ignored;
• Statistical mechanics reproduces thermodynamics when the number of particles is large;
• In biology, chromosome inheritance theory reproduces Mendel's laws of inheritance, in the domain that the inherited factors are protein coding genes.

In order for there to be a correspondence, the earlier theory has to have a domain of validity—it must work under some conditions. Not all theories have a domain of validity. For example, there is no limit where Newton's mechanics reduces to Aristotle's mechanics because Aristotle's mechanics, although academically dominant for 18 centuries, does not have any domain of validity (on the other hand, it can sensibly be said that the falling of objects through the air ("natural motion") constitutes a domain of validity for a part of Aristotle's mechanics).

Examples

Bohr model

If an electron in an atom is moving on an orbit with period T, classically the electromagnetic radiation will repeat itself every orbital period. If the coupling to the electromagnetic field is weak, so that the orbit doesn't decay very much in one cycle, the radiation will be emitted in a pattern which repeats every period, so that the Fourier transform will have frequencies which are only multiples of 1/T. This is the classical radiation law: the frequencies emitted are integer multiples of 1/T.

In quantum mechanics, this emission must be in quanta of light, of frequencies consisting of integer multiples of 1/T, so that classical mechanics is an approximate description at large quantum numbers. This means that the energy level corresponding to a classical orbit of period 1/T must have nearby energy levels which differ in energy by h/T, and they should be equally spaced near that level,

${\displaystyle \Delta E_{n}={h \over T(E_{n})}~.}$

Bohr worried whether the energy spacing 1/T should be best calculated with the period of the energy state ${\displaystyle E_{n}}$, or ${\displaystyle E_{n+1}}$, or some average—in hindsight, this model is only the leading semiclassical approximation. 

Bohr considered circular orbits. Classically, these orbits must decay to smaller circles when photons are emitted. The level spacing between circular orbits can be calculated with the correspondence formula. For a Hydrogen atom, the classical orbits have a period T determined by Kepler's third law to scale as r^3/2. The energy scales as 1/r, so the level spacing formula amounts to

${\displaystyle \Delta E\propto {1 \over r^{3 \over 2}}\propto E^{3 \over 2}.}$

It is possible to determine the energy levels by recursively stepping down orbit by orbit, but there is a shortcut. 

The angular momentum L of the circular orbit scales as √r. The energy in terms of the angular momentum is then

${\displaystyle E\propto {1 \over r}\propto {1 \over L^{2}}.}$

Assuming, with Bohr, that quantized values of L are equally spaced, the spacing between neighboring energies is

${\displaystyle \Delta E\propto {1 \over (L+\hbar )^{2}}-{1 \over L^{2}}\approx -{2\hbar \over L^{3}}\propto -E^{3 \over 2}.}$

This is as desired for equally spaced angular momenta. If one kept track of the constants, the spacing would be ħ, so the angular momentum should be an integer multiple of ħ,

${\displaystyle L={nh \over 2\pi }=n\hbar ~.}$

This is how Bohr arrived at his model. Since only the level spacing is determined heuristically by the correspondence principle, one could always add a small fixed offset to the quantum number— L could just as well have been (n+.338) ħ. 

Bohr used his physical intuition to decide which quantities were best to quantize. It is a testimony to his skill that he was able to get so much from what is only the leading order approximation. A less heuristic treatment accounts for needed offsets in the ground state L², cf. Wigner–Weyl transform.

One-dimensional potential

Bohr's correspondence condition can be solved for the level energies in a general one-dimensional potential. Define a quantity J(E) which is a function only of the energy, and has the property that

${\displaystyle {dJ \over dE}=T}$

This is the analog of the angular momentum in the case of the circular orbits. The orbits selected by the correspondence principle are the ones that obey J = nh for n integer, since

${\displaystyle \Delta E=E_{n+1}-E_{n}={dE \over dJ}(J_{n+1}-J_{n})={1 \over T}\,\Delta J}$

This quantity J is canonically conjugate to a variable θ which, by the Hamilton equations of motion changes with time as the gradient of energy with J. Since this is equal to the inverse period at all times, the variable θ increases steadily from 0 to 1 over one period.

The angle variable comes back to itself after 1 unit of increase, so the geometry of phase space in J,θ coordinates is that of a half-cylinder, capped off at J = 0, which is the motionless orbit at the lowest value of the energy. These coordinates are just as canonical as x,p, but the orbits are now lines of constant J instead of nested ovoids in x-p space. 

The area enclosed by an orbit is invariant under canonical transformations, so it is the same in x-p space as in J-θ. But in the J-θ coordinates, this area is the area of a cylinder of unit circumference between 0 and J, or just J. So J is equal to the area enclosed by the orbit in x-p coordinates too,

${\displaystyle J=\int _{0}^{T}p{dx \over dt}\,dt~.}$

The quantization rule is that the action variable J is an integer multiple of h. 

Multiperiodic motion: Bohr–Sommerfeld quantization

Bohr's correspondence principle provided a way to find the semiclassical quantization rule for a one degree of freedom system. It was an argument for the old quantum condition mostly independent from the one developed by Wien and Einstein, which focused on adiabatic invariance. But both pointed to the same quantity, the action. 

Bohr was reluctant to generalize the rule to systems with many degrees of freedom. This step was taken by Sommerfeld, who proposed the general quantization rule for an integrable system,

${\displaystyle J_{k}=hn_{k}.\,}$

Each action variable is a separate integer, a separate quantum number.

This condition reproduces the circular orbit condition for two dimensional motion: let r,θ be polar coordinates for a central potential. Then θ is already an angle variable, and the canonical momentum conjugate is L, the angular momentum. So the quantum condition for L reproduces Bohr's rule:

${\displaystyle \int _{0}^{2\pi }Ld\theta =2\pi L=nh.}$

This allowed Sommerfeld to generalize Bohr's theory of circular orbits to elliptical orbits, showing that the energy levels are the same. He also found some general properties of quantum angular momentum which seemed paradoxical at the time. One of these results was that the z-component of the angular momentum, the classical inclination of an orbit relative to the z-axis, could only take on discrete values, a result which seemed to contradict rotational invariance. This was called space quantization for a while, but this term fell out of favor with the new quantum mechanics since no quantization of space is involved. 

In modern quantum mechanics, the principle of superposition makes it clear that rotational invariance is not lost. It is possible to rotate objects with discrete orientations to produce superpositions of other discrete orientations, and this resolves the intuitive paradoxes of the Sommerfeld model. 

The quantum harmonic oscillator

Here is a demonstration of how large quantum numbers can give rise to classical (continuous) behavior. 

Consider the one-dimensional quantum harmonic oscillator. Quantum mechanics tells us that the total (kinetic and potential) energy of the oscillator, E, has a set of discrete values,

${\displaystyle E=(n+1/2)\hbar \omega ,\ n=0,1,2,3,\dots ~,}$

where ω is the angular frequency of the oscillator.

However, in a classical harmonic oscillator such as a lead ball attached to the end of a spring, we do not perceive any discreteness. Instead, the energy of such a macroscopic system appears to vary over a continuum of values. We can verify that our idea of macroscopic systems fall within the correspondence limit. The energy of the classical harmonic oscillator with amplitude A, is

${\displaystyle E={\frac {m\omega ^{2}A^{2}}{2}}.}$

Thus, the quantum number has the value

${\displaystyle n={\frac {E}{\hbar \cdot \omega }}-{\frac {1}{2}}={\frac {m\omega A^{2}}{2\hbar }}-{\frac {1}{2}}}$

If we apply typical "human-scale" values m = 1kg, ω = 1 rad/s, and A = 1 m, then n ≈ 4.74×10³³. This is a very large number, so the system is indeed in the correspondence limit. 

It is simple to see why we perceive a continuum of energy in this limit. With ω = 1 rad/s, the difference between each energy level is ħω ≈ 1.05 × 10⁻³⁴J, well below what we normally resolve for macroscopic systems. One then describes this system through an emergent classical limit. 

Relativistic kinetic energy

Here we show that the expression of kinetic energy from special relativity becomes arbitrarily close to the classical expression, for speeds that are much slower than the speed of light, v ≪ c. 

Einstein's mass-energy equation

${\displaystyle E={\frac {m_{0}}{\sqrt {1-v^{2}/c^{2}}}}c^{2}~,}$

where the velocity, v is the velocity of the body relative to the observer, ${\displaystyle m_{0}\ }$ is the rest mass (the observed mass of the body at zero velocity relative to the observer), and c is the speed of light. 

When the velocity v vanishes, the energy expressed above is not zero, and represents the rest energy,

${\displaystyle E_{0}=m_{0}c^{2}.\ }$

When the body is in motion relative to the observer, the total energy exceeds the rest energy by an amount that is, by definition, the kinetic energy,

${\displaystyle T=E-E_{0}={\frac {m_{0}c^{2}}{\sqrt {1-v^{2}/c^{2}}}}\ -\ m_{0}c^{2}~.}$

Using the approximation

${\displaystyle (1+x)^{n}\approx 1+nx\ }$

for ${\displaystyle |x|\ll 1\ }$

we get, when speeds are much slower than that of light, or v ≪ c,

${\displaystyle T\ }$	${\displaystyle =m_{0}c^{2}\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)\ }$
	${\displaystyle =m_{0}c^{2}\left(\left(1-v^{2}/c^{2}\right)^{-{\frac {1}{2}}}-1\right)\ }$
	${\displaystyle \approx m_{0}c^{2}\left((1-(-{\begin{matrix}{\frac {1}{2}}\end{matrix}})v^{2}/c^{2})-1\right)\ }$
	${\displaystyle =m_{0}c^{2}\left({\begin{matrix}{\frac {1}{2}}\end{matrix}}v^{2}/c^{2}\right)\ }$
	${\displaystyle ={\begin{matrix}{\frac {1}{2}}\end{matrix}}m_{0}v^{2}~,}$

which is the Newtonian expression for kinetic energy.

Earth's rotation

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Earth%27s_rotation

 

An animation of Earth's rotation around the planet's axis

 

This long-exposure photo of the northern night sky above the Nepali Himalayas shows the apparent paths of the stars as Earth rotates.

 

Earth's rotation imaged by DSCOVR EPIC on 29 May 2016, a few weeks before the solstice.

Earth's rotation is the rotation of Planet Earth around its own axis. Earth rotates eastward, in prograde motion. As viewed from the north pole star Polaris, Earth turns counterclockwise. 

The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is the point in the Northern Hemisphere where Earth's axis of rotation meets its surface. This point is distinct from Earth's North Magnetic Pole. The South Pole is the other point where Earth's axis of rotation intersects its surface, in Antarctica. 

Earth rotates once in about 24 hours with respect to the Sun, but once every 23 hours, 56 minutes, and 4 seconds with respect to other, distant, stars (see below). Earth's rotation is slowing slightly with time; thus, a day was shorter in the past. This is due to the tidal effects the Moon has on Earth's rotation. Atomic clocks show that a modern-day is longer by about 1.7 milliseconds than a century ago, slowly increasing the rate at which UTC is adjusted by leap seconds. Analysis of historical astronomical records shows a slowing trend of about 2.3 milliseconds per century since the 8th century BCE.

History

Among the ancient Greeks, several of the Pythagorean school believed in the rotation of Earth rather than the apparent diurnal rotation of the heavens. Perhaps the first was Philolaus (470–385 BCE), though his system was complicated, including a counter-earth rotating daily about a central fire.

A more conventional picture was that supported by Hicetas, Heraclides and Ecphantus in the fourth century BCE who assumed that Earth rotated but did not suggest that Earth revolved about the Sun. In the third century BCE, Aristarchus of Samos suggested the Sun's central place. 

However, Aristotle in the fourth century BCE criticized the ideas of Philolaus as being based on theory rather than observation. He established the idea of a sphere of fixed stars that rotated about Earth. This was accepted by most of those who came after, in particular Claudius Ptolemy (2nd century CE), who thought Earth would be devastated by gales if it rotated.

In 499 CE, the Indian astronomer Aryabhata wrote that the spherical Earth rotates about its axis daily, and that the apparent movement of the stars is a relative motion caused by the rotation of Earth. He provided the following analogy: "Just as a man in a boat going in one direction sees the stationary things on the bank as moving in the opposite direction, in the same way to a man at Lanka the fixed stars appear to be going westward."

In the 10th century, some Muslim astronomers accepted that Earth rotates around its axis. According to al-Biruni, Abu Sa'id al-Sijzi (d. circa 1020) invented an astrolabe called al-zūraqī based on the idea believed by some of his contemporaries "that the motion we see is due to the Earth's movement and not to that of the sky." The prevalence of this view is further confirmed by a reference from the 13th century which states: "According to the geometers [or engineers] (muhandisīn), the Earth is in constant circular motion, and what appears to be the motion of the heavens is actually due to the motion of the Earth and not the stars." Treatises were written to discuss its possibility, either as refutations or expressing doubts about Ptolemy's arguments against it. At the Maragha and Samarkand observatories, Earth's rotation was discussed by Tusi (b. 1201) and Qushji (b. 1403); the arguments and evidence they used resemble those used by Copernicus.

In medieval Europe, Thomas Aquinas accepted Aristotle's view and so, reluctantly, did John Buridan and Nicole Oresme in the fourteenth century. Not until Nicolaus Copernicus in 1543 adopted a heliocentric world system did the contemporary understanding of Earth's rotation begin to be established. Copernicus pointed out that if the movement of Earth is violent, then the movement of the stars must be very much more so. He acknowledged the contribution of the Pythagoreans and pointed to examples of relative motion. For Copernicus this was the first step in establishing the simpler pattern of planets circling a central Sun.

Tycho Brahe, who produced accurate observations on which Kepler based his laws, used Copernicus's work as the basis of a system assuming a stationary Earth. In 1600, William Gilbert strongly supported Earth's rotation in his treatise on Earth's magnetism and thereby influenced many of his contemporaries. Those like Gilbert who did not openly support or reject the motion of Earth about the Sun are often called "semi-Copernicans". A century after Copernicus, Riccioli disputed the model of a rotating Earth due to the lack of then-observable eastward deflections in falling bodies; such deflections would later be called the Coriolis effect. However, the contributions of Kepler, Galileo and Newton gathered support for the theory of the rotation of Earth.

Empirical tests

Earth's rotation implies that the Equator bulges and the geographical poles are flattened. In his Principia, Newton predicted this flattening would occur in the ratio of 1:230, and pointed to the pendulum measurements taken by Richer in 1673 as corroboration of the change in gravity, but initial measurements of meridian lengths by Picard and Cassini at the end of the 17th century suggested the opposite. However, measurements by Maupertuis and the French Geodesic Mission in the 1730s established the oblateness of Earth, thus confirming the positions of both Newton and Copernicus.

In Earth's rotating frame of reference, a freely moving body follows an apparent path that deviates from the one it would follow in a fixed frame of reference. Because of the Coriolis effect, falling bodies veer slightly eastward from the vertical plumb line below their point of release, and projectiles veer right in the Northern Hemisphere (and left in the Southern) from the direction in which they are shot. The Coriolis effect is mainly observable at a meteorological scale, where it is responsible for the opposite directions of cyclone rotation in the Northern and Southern hemispheres (anticlockwise and clockwise, respectively). 

Hooke, following a suggestion from Newton in 1679, tried unsuccessfully to verify the predicted eastward deviation of a body dropped from a height of 8.2 meters, but definitive results were obtained later, in the late 18th and early 19th century, by Giovanni Battista Guglielmini in Bologna, Johann Friedrich Benzenberg in Hamburg and Ferdinand Reich in Freiberg, using taller towers and carefully released weights. A ball dropped from a height of 158.5 m departed by 27.4 mm from the vertical compared with a calculated value of 28.1 mm.

The most celebrated test of Earth's rotation is the Foucault pendulum first built by physicist Léon Foucault in 1851, which consisted of a lead-filled brass sphere suspended 67 m from the top of the Panthéon in Paris. Because of Earth's rotation under the swinging pendulum, the pendulum's plane of oscillation appears to rotate at a rate depending on latitude. At the latitude of Paris the predicted and observed shift was about 11 degrees clockwise per hour. Foucault pendulums now swing in museums around the world.

Periods

Starry circles arc around the south celestial pole, seen overhead at ESO's La Silla Observatory.

 

True solar day

Earth's rotation period relative to the Sun (solar noon to solar noon) is its true solar day or apparent solar day. It depends on Earth's orbital motion and is thus affected by changes in the eccentricity and inclination of Earth's orbit. Both vary over thousands of years, so the annual variation of the true solar day also varies. Generally, it is longer than the mean solar day during two periods of the year and shorter during another two. The true solar day tends to be longer near perihelion when the Sun apparently moves along the ecliptic through a greater angle than usual, taking about 10 seconds longer to do so. Conversely, it is about 10 seconds shorter near aphelion. It is about 20 seconds longer near a solstice when the projection of the Sun's apparent motion along the ecliptic onto the celestial equator causes the Sun to move through a greater angle than usual. Conversely, near an equinox the projection onto the equator is shorter by about 20 seconds. Currently, the perihelion and solstice effects combine to lengthen the true solar day near 22 December by 30 mean solar seconds, but the solstice effect is partially cancelled by the aphelion effect near 19 June when it is only 13 seconds longer. The effects of the equinoxes shorten it near 26 March and 16 September by 18 seconds and 21 seconds, respectively.

Mean solar day

The average of the true solar day during the course of an entire year is the mean solar day, which contains 86,400 mean solar seconds. Currently, each of these seconds is slightly longer than an SI second because Earth's mean solar day is now slightly longer than it was during the 19th century due to tidal friction. The average length of the mean solar day since the introduction of the leap second in 1972 has been about 0 to 2 ms longer than 86,400 SI seconds. Random fluctuations due to core-mantle coupling have an amplitude of about 5 ms. The mean solar second between 1750 and 1892 was chosen in 1895 by Simon Newcomb as the independent unit of time in his Tables of the Sun. These tables were used to calculate the world's ephemerides between 1900 and 1983, so this second became known as the ephemeris second. In 1967 the SI second was made equal to the ephemeris second.

The apparent solar time is a measure of Earth's rotation and the difference between it and the mean solar time is known as the equation of time.

Stellar and sidereal day

On a prograde planet like Earth, the stellar day is shorter than the solar day. At time 1, the Sun and a certain distant star are both overhead. At time 2, the planet has rotated 360° and the distant star is overhead again but the Sun is not (1→2 = one stellar day). It is not until a little later, at time 3, that the Sun is overhead again (1→3 = one 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 86,164.098 903 691 seconds of mean solar time (UT1) (23^h 56^m 4.098 903 691^s, 0.997 269 663 237 16 mean solar days). Earth's rotation period relative to the precessing mean vernal equinox, named sidereal day, is 86,164.090 530 832 88 seconds of mean solar time (UT1) (23^h 56^m 4.090 530 832 88^s, 0.997 269 566 329 08 mean solar days). Thus, the sidereal day is shorter than the stellar day by about 8.4 ms.

Both the stellar day and the sidereal day are shorter than the mean solar day by about 3 minutes 56 seconds. The mean solar day in SI seconds is available from the IERS for the periods 1623–2005 and 1962–2005.

Recently (1999–2010) the average annual length of the mean solar day in excess of 86,400 SI seconds has varied between 0.25 ms and 1 ms, which must be added to both the stellar and sidereal days given in mean solar time above to obtain their lengths in SI seconds.

Angular speed

Plot of latitude vs tangential speed. The dashed line shows the Kennedy Space Center example. The dot-dash line denotes typical airliner cruise speed.

 

The angular speed of Earth's rotation in inertial space is 7.2921150 ± 0.0000001×10⁻⁵ radians per SI second. Multiplying by (180°/π radians) × (86,400 seconds/day) yields 360.985 6°/day, indicating that Earth rotates more than 360° relative to the fixed stars in one solar day. Earth's movement along its nearly circular orbit while it is rotating once around its axis requires that Earth rotate slightly more than once relative to the fixed stars before the mean Sun can pass overhead again, even though it rotates only once (360°) relative to the mean Sun. Multiplying the value in rad/s by Earth's equatorial radius of 6,378,137 m (WGS84 ellipsoid) (factors of 2π radians needed by both cancel) yields an equatorial speed of 465.10 metres per second (1,674.4 km/h). Some sources state that Earth's equatorial speed is slightly less, or 1,669.8 km/h. This is obtained by dividing Earth's equatorial circumference by 24 hours. However, the use of only one circumference unwittingly implies only one rotation in inertial space, so the corresponding time unit must be a sidereal hour. This is confirmed by multiplying by the number of sidereal days in one mean solar day, 1.002 737 909 350 795, which yields the equatorial speed in mean solar hours given above of 1,674.4 km/h. 

The tangential speed of Earth's rotation at a point on Earth can be approximated by multiplying the speed at the equator by the cosine of the latitude. For example, the Kennedy Space Center is located at latitude 28.59° N, which yields a speed of: cos 28.59° × 1674.4 km/h = 1470.2 km/h. 

Changes

Earth's axial tilt is about 23.4°. It oscillates between 22.1° and 24.5° on a 41,000-year cycle and is currently decreasing.

 

In rotational axis

Earth's rotation axis moves with respect to the fixed stars (inertial space); the components of this motion are precession and nutation. It also moves with respect to Earth's crust; this is called polar motion. 

Precession is a rotation of Earth's rotation axis, caused primarily by external torques from the gravity of the Sun, Moon and other bodies. The polar motion is primarily due to free core nutation and the Chandler wobble.

In rotational velocity

Tidal interactions

Over millions of years, Earth's rotation has been slowed significantly by tidal acceleration through gravitational interactions with the Moon. Thus angular momentum is slowly transferred to the Moon at a rate proportional to ${\displaystyle r^{-6}}$, where ${\displaystyle r}$ is the orbital radius of the Moon. This process has gradually increased the length of the day to its current value, and resulted in the Moon being tidally locked with Earth. 

This gradual rotational deceleration is empirically documented by estimates of day lengths obtained from observations of tidal rhythmites and stromatolites; a compilation of these measurements found that the length of the day has increased steadily from about 21 hours at 600 Myr ago to the current 24-hour value. By counting the microscopic lamina that form at higher tides, tidal frequencies (and thus day lengths) can be estimated, much like counting tree rings, though these estimates can be increasingly unreliable at older ages.

Resonant stabilization

A simulated history of Earth's day length, depicting a resonant-stabilizing event throughout the Precambrian era.

 

The current rate of tidal deceleration is anomalously high, implying Earth's rotational velocity must have decreased more slowly in the past. Empirical data tentatively shows a sharp increase in rotational deceleration about 600 Myr ago. Some models suggest that Earth maintained a constant day length of 21 hours throughout much of the Precambrian. This day length corresponds to the semidiurnal resonant period of the thermally-driven atmospheric tide; at this day length, the decelerative lunar torque could have been canceled by an accelerative torque from the atmospheric tide, resulting in no net torque and a constant rotational period. This stabilizing effect could have been broken by a sudden change in global temperature. Recent computational simulations support this hypothesis and suggest the Marinoan or Sturtian glaciations broke this stable configuration about 600 Myr ago; the simulated results agree quite closely with existing paleorotational data.

Global events

Deviation of day length from SI based day

 

Some recent large-scale events, such as the 2004 Indian Ocean earthquake, have caused the length of a day to shorten by 3 microseconds by reducing Earth's moment of inertia. Post-glacial rebound, ongoing since the last Ice age, is also changing the distribution of Earth's mass, thus affecting the moment of inertia of Earth and, by the conservation of angular momentum, Earth's rotation period.

The length of the day can also be influenced by manmade structures. For example, NASA scientists calculated that the water stored in the Three Gorges Dam has increased the length of Earth's day by 0.06 microseconds due to the shift in mass.

Measurement

The primary monitoring of Earth's rotation is performed by very-long-baseline interferometry coordinated with the Global Positioning System, satellite laser ranging, and other satellite techniques. This provides an absolute reference for the determination of universal time, precession, and nutation.

Ancient observations

There are recorded observations of solar and lunar eclipses by Babylonian and Chinese astronomers beginning in the 8th century BCE, as well as from the medieval Islamic world and elsewhere. These observations can be used to determine changes in Earth's rotation over the last 27 centuries, since the length of the day is a critical parameter in the calculation of the place and time of eclipses. A change in day length of milliseconds per century shows up as a change of hours and thousands of kilometers in eclipse observations. The ancient data are consistent with a shorter day, meaning Earth was turning faster throughout the past.

Cyclic variability

Around every 25-30 years Earth's rotation slows temporarily by a few milliseconds per day, usually lasting around 5 years. 2017 was the fourth consecutive year that Earth's rotation has slowed. The cause of this variability has not yet been determined.

Origin

An artist's rendering of the protoplanetary disk.

 

Earth's original rotation was a vestige of the original angular momentum of the cloud of dust, rocks, and gas that coalesced to form the Solar System. This primordial cloud was composed of hydrogen and helium produced in the Big Bang, as well as heavier elements ejected by supernovas. As this interstellar dust is heterogeneous, any asymmetry during gravitational accretion resulted in the angular momentum of the eventual planet.

However, if the giant-impact hypothesis for the origin of the Moon is correct, this primordial rotation rate would have been reset by the Theia impact 4.5 billion years ago. Regardless of the speed and tilt of Earth's rotation before the impact, it would have experienced a day some five hours long after the impact. Tidal effects would then have slowed this rate to its modern value.