Orbital eccentricity

From Wikipedia, the free encyclopedia

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

 

An elliptic, parabolic, and hyperbolic Kepler orbit:
  elliptic (eccentricity = 0.7)
  parabolic (eccentricity = 1)
  hyperbolic orbit (eccentricity = 1.3)

 

Elliptic orbit by eccentricity
  0.0 ·   0.2 ·   0.4 ·   0.6 ·   0.8

In astrodynamics, the orbital eccentricity of an astronomical object is a dimensionless parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values between 0 and 1 form an elliptic orbit, 1 is a parabolic escape orbit, and greater than 1 is a hyperbola. The term derives its name from the parameters of conic sections, as every Kepler orbit is a conic section. It is normally used for the isolated two-body problem, but extensions exist for objects following a rosette orbit through the galaxy.

Definition

e=0

e=0.5

Orbits in a two-body system for two values of the eccentricity, e. (NB: + is barycentre)

In a two-body problem with inverse-square-law force, every orbit is a Kepler orbit. The eccentricity of this Kepler orbit is a non-negative number that defines its shape.

The eccentricity may take the following values:

• circular orbit: e = 0
• elliptic orbit: 0 < e < 1 (see ellipse)
• parabolic trajectory: e = 1 (see parabola)
• hyperbolic trajectory: e > 1 (see hyperbola)

The eccentricity e is given by

${\displaystyle e={\sqrt {1+{\frac {2EL^{2}}{m_{\text{red}}^{3}\alpha ^{2}}}}}}$

where E is the total orbital energy, L is the angular momentum, m_red is the reduced mass, and α the coefficient of the inverse-square law central force such as gravity or electrostatics in classical physics:

${\displaystyle F={\frac {\alpha }{r^{2}}}}$

(α is negative for an attractive force, positive for a repulsive one; see also Kepler problem)

or in the case of a gravitational force:

${\displaystyle e={\sqrt {1+{\frac {2\varepsilon h^{2}}{\mu ^{2}}}}}}$

where ε is the specific orbital energy (total energy divided by the reduced mass), μ the standard gravitational parameter based on the total mass, and h the specific relative angular momentum (angular momentum divided by the reduced mass).

For values of e from 0 to 1 the orbit's shape is an increasingly elongated (or flatter) ellipse; for values of e from 1 to infinity the orbit is a hyperbola branch making a total turn of 2 arccsc e, decreasing from 180 to 0 degrees. The limit case between an ellipse and a hyperbola, when e equals 1, is parabola.

Radial trajectories are classified as elliptic, parabolic, or hyperbolic based on the energy of the orbit, not the eccentricity. Radial orbits have zero angular momentum and hence eccentricity equal to one. Keeping the energy constant and reducing the angular momentum, elliptic, parabolic, and hyperbolic orbits each tend to the corresponding type of radial trajectory while e tends to 1 (or in the parabolic case, remains 1).

For a repulsive force only the hyperbolic trajectory, including the radial version, is applicable.

For elliptical orbits, a simple proof shows that arcsin(${\displaystyle e}$⁠) yields the projection angle of a perfect circle to an ellipse of eccentricity e. For example, to view the eccentricity of the planet Mercury (e = 0.2056), one must simply calculate the inverse sine to find the projection angle of 11.86 degrees. Next, tilt any circular object (such as a coffee mug viewed from the top) by that angle and the apparent ellipse projected to your eye will be of that same eccentricity.

Etymology

The word "eccentricity" comes from Medieval Latin eccentricus, derived from Greek ἔκκεντρος ekkentros "out of the center", from ἐκ- ek-, "out of" + κέντρον kentron "center". "Eccentric" first appeared in English in 1551, with the definition "...a circle in which the earth, sun. etc. deviates from its center".^{[citation needed]} By five years later, in 1556, an adjectival form of the word had developed.

Calculation

The eccentricity of an orbit can be calculated from the orbital state vectors as the magnitude of the eccentricity vector:

${\displaystyle e=\left|\mathbf {e} \right|}$

where:

• e is the eccentricity vector ("Hamilton's vector").

For elliptical orbits it can also be calculated from the periapsis and apoapsis since ${\displaystyle \,r_{\text{p}}=a\,(1-e)\,}$ and ${\displaystyle \,r_{\text{a}}=a\,(1+e)\,,}$ where a is the length of the semi-major axis, the geometric-average and time-average distance.

${\displaystyle {\begin{aligned}e&={\frac {r_{\text{a}}-r_{\text{p}}}{r_{\text{a}}+r_{\text{p}}}}\\\,\\&={\frac {r_{\text{a}}/r_{\text{p}}-1}{r_{\text{a}}/r_{\text{p}}+1}}\\\,\\&=1-{\frac {2}{\;{\frac {r_{\text{a}}}{r_{\text{p}}}}+1\;}}\end{aligned}}}$

where:

• r_a is the radius at apoapsis (a.k.a. "apofocus", "aphelion", "apogee", i.e., the farthest distance of the orbit to the center of mass of the system, which is a focus of the ellipse).
• r_p is the radius at periapsis (a.k.a. "perifocus" etc., the closest distance).

The eccentricity of an elliptical orbit can also be used to obtain the ratio of the periapsis radius to the apoapsis radius:

${\displaystyle {\frac {r_{\text{a}}}{r_{\text{p}}}}={\frac {\,a\,(1+e)\,}{\,a\,(1-e)\,}}={\frac {1+e}{1-e}}}$

For Earth, orbital eccentricity e ≈ 0.01671 , apoapsis is aphelion, and periapsis is perihelion relative to sun.

For Earth's annual orbit path, the ratio of longest radius (r_a) / shortest radius (r_p) is ${\displaystyle {\frac {\,r_{\text{a}}\,}{r_{\text{p}}}}={\frac {\,1+e\,}{1-e}}{\text{ ≈ 1.03399 .}}}$

Examples

Gravity Simulator plot of the changing orbital eccentricity of Mercury, Venus, Earth, and Mars over the next 50000 years. The arrows indicate the different scales used, as the eccentricities of Mercury and Mars are much greater than those of Venus and Earth. The 0 point on this plot is the year 2007.

Eccentricities of Solar System bodies

Object	eccentricity
Triton	0.00002
Venus	0.0068
Neptune	0.0086
Earth	0.0167
Titan	0.0288
Uranus	0.0472
Jupiter	0.0484
Saturn	0.0541
Moon	0.0549
1 Ceres	0.0758
4 Vesta	0.0887
Mars	0.0934
10 Hygiea	0.1146
Makemake	0.1559
Haumea	0.1887
Mercury	0.2056
2 Pallas	0.2313
Pluto	0.2488
3 Juno	0.2555
324 Bamberga	0.3400
Eris	0.4407
Nereid	0.7507
Sedna	0.8549
Halley's Comet	0.9671
Comet Hale-Bopp	0.9951
Comet Ikeya-Seki	0.9999
C/1980 E1	1.057
ʻOumuamua	1.20
C/2019 Q4 (Borisov)	3.5

The eccentricity of the Earth's orbit is currently about 0.0167; the Earth's orbit is nearly circular. Venus and Neptune have even lower eccentricities. Over hundreds of thousands of years, the eccentricity of the Earth's orbit varies from nearly 0.0034 to almost 0.058 as a result of gravitational attractions among the planets.

The table lists the values for all planets and dwarf planets, and selected asteroids, comets, and moons. Mercury has the greatest orbital eccentricity of any planet in the Solar System (e = 0.2056). Such eccentricity is sufficient for Mercury to receive twice as much solar irradiation at perihelion compared to aphelion. Before its demotion from planet status in 2006, Pluto was considered to be the planet with the most eccentric orbit (e = 0.248). Other Trans-Neptunian objects have significant eccentricity, notably the dwarf planet Eris (0.44). Even further out, Sedna, has an extremely high eccentricity of 0.855 due to its estimated aphelion of 937 AU and perihelion of about 76 AU.

Most of the Solar System's asteroids have orbital eccentricities between 0 and 0.35 with an average value of 0.17. Their comparatively high eccentricities are probably due to the influence of Jupiter and to past collisions.

The Moon's value is 0.0549, the most eccentric of the large moons of the Solar System. The four Galilean moons have eccentricity < 0.01. Neptune's largest moon Triton has an eccentricity of 1.6×10⁻⁵ (0.000016), the smallest eccentricity of any known moon in the Solar System; its orbit is as close to a perfect circle as can be currently measured. However, smaller moons, particularly irregular moons, can have significant eccentricity, such as Neptune's third largest moon Nereid (0.75).

Comets have very different values of eccentricity. Periodic comets have eccentricities mostly between 0.2 and 0.7, but some of them have highly eccentric elliptical orbits with eccentricities just below 1, for example, Halley's Comet has a value of 0.967. Non-periodic comets follow near-parabolic orbits and thus have eccentricities even closer to 1. Examples include Comet Hale–Bopp with a value of 0.995^[5] and comet C/2006 P1 (McNaught) with a value of 1.000019. As Hale–Bopp's value is less than 1, its orbit is elliptical and it will return. Comet McNaught has a hyperbolic orbit while within the influence of the planets, but is still bound to the Sun with an orbital period of about 10⁵ years. Comet C/1980 E1 has the largest eccentricity of any known hyperbolic comet of solar origin with an eccentricity of 1.057, and will leave the Solar System eventually.

ʻOumuamua is the first interstellar object found passing through the Solar System. Its orbital eccentricity of 1.20 indicates that ʻOumuamua has never been gravitationally bound to our sun. It was discovered 0.2 AU (30,000,000 km; 19,000,000 mi) from Earth and is roughly 200 meters in diameter. It has an interstellar speed (velocity at infinity) of 26.33 km/s (58,900 mph).

Mean eccentricity

The mean eccentricity of an object is the average eccentricity as a result of perturbations over a given time period. Neptune currently has an instant (current epoch) eccentricity of 0.0113, but from 1800 to 2050 has a mean eccentricity of 0.00859.

Climatic effect

Orbital mechanics require that the duration of the seasons be proportional to the area of the Earth's orbit swept between the solstices and equinoxes, so when the orbital eccentricity is extreme, the seasons that occur on the far side of the orbit (aphelion) can be substantially longer in duration. Today, northern hemisphere autumn/fall and winter occur at closest approach (perihelion), when the earth is moving at its maximum velocity—while the opposite occurs in the southern hemisphere. As a result, in the northern hemisphere, autumn and winter are slightly shorter than spring and summer—but in global terms this is balanced with them being longer below the equator. In 2006, the northern hemisphere summer was 4.66 days longer than winter, and spring was 2.9 days longer than fall due to the Milankovitch cycles.

Apsidal precession also slowly changes the place in the Earth's orbit where the solstices and equinoxes occur. Note that this is a slow change in the orbit of the Earth, not the axis of rotation, which is referred to as axial precession. Over the next 10,000 years, the northern hemisphere winters will become gradually longer and summers will become shorter. However, any cooling effect in one hemisphere is balanced by warming in the other, and any overall change will be counteracted by the fact that the eccentricity of Earth's orbit will be almost halved. This will reduce the mean orbital radius and raise temperatures in both hemispheres closer to the mid-interglacial peak.

Exoplanets

Of the many exoplanets discovered, most have a higher orbital eccentricity than planets in our planetary system. Exoplanets found with low orbital eccentricity (near-circular orbits) are very close to their star and are tidally locked to the star. All eight planets in the Solar System have near-circular orbits. The exoplanets discovered show that the solar system, with its unusually low eccentricity, is rare and unique. One theory attributes this low eccentricity to the high number of planets in the Solar System; another suggests it arose because of its unique asteroid belts. A few other multiplanetary systems have been found, but none resemble the Solar System. The Solar System has unique planetesimal systems, which led the planets to have near-circular orbits. Solar planetesimal systems include the asteroid belt, Hilda family, Kuiper belt, Hills cloud, and the Oort cloud. The exoplanet systems discovered have either no planetesimal systems or one very large one. Low eccentricity is needed for habitability, especially advanced life. High multiplicity planet systems are much more likely to have habitable exoplanets. The grand tack hypothesis of the Solar System also helps understand its near-circular orbits and other unique features.

Milankovitch cycles

From Wikipedia, the free encyclopedia

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

 

Past and future Milankovitch cycles via VSOP model • Graphic shows variations in five orbital elements:   Axial tilt or obliquity (ε).   Eccentricity (e).   Longitude of perihelion ( sin(ϖ) ).   Precession index ( e sin(ϖ) ) • Precession index and obliquity control insolation at each latitude:   Daily-average insolation at top of atmosphere on summer solstice (${\displaystyle {\overline {Q}}^{\mathrm {day} }}$) at 65° N • Ocean sediment and Antarctic ice strata record ancient sea levels and temperatures:   Benthic forams (57 widespread locations)   Vostok ice core (Antarctica) • Vertical gray line shows present (2000 CE)

Milankovitch cycles describe the collective effects of changes in the Earth's movements on its climate over thousands of years. The term is named for Serbian geophysicist and astronomer Milutin Milanković. In the 1920s, he hypothesized that variations in eccentricity, axial tilt, and precession resulted in cyclical variation in the solar radiation reaching the Earth, and that this orbital forcing strongly influenced the Earth's climatic patterns.

Similar astronomical hypotheses had been advanced in the 19th century by Joseph Adhemar, James Croll and others, but verification was difficult because there was no reliably dated evidence, and because it was unclear which periods were important.

Now, materials on Earth that have been unchanged for millennia (obtained via ice, rock, and deep ocean cores) are being studied to indicate the history of Earth's climate. Though they are consistent with the Milankovitch hypothesis, there are still several observations that the hypothesis does not explain.

Earth's movements

The Earth's rotation around its axis, and revolution around the Sun, evolve over time due to gravitational interactions with other bodies in the Solar System. The variations are complex, but a few cycles are dominant.

Circular orbit, no eccentricity

 

Orbit with 0.5 eccentricity, exaggerated for illustration; Earth's orbit is only slightly eccentric

The Earth's orbit varies between nearly circular and mildly elliptical (its eccentricity varies). When the orbit is more elongated, there is more variation in the distance between the Earth and the Sun, and in the amount of solar radiation, at different times in the year. In addition, the rotational tilt of the Earth (its obliquity) changes slightly. A greater tilt makes the seasons more extreme. Finally, the direction in the fixed stars pointed to by the Earth's axis changes (axial precession), while the Earth's elliptical orbit around the Sun rotates (apsidal precession). The combined effect of precession with eccentricity is that proximity to the Sun occurs during different astronomical seasons.

Milankovitch studied changes in these movements of the Earth, which alter the amount and location of solar radiation reaching the Earth. This is known as solar forcing (an example of radiative forcing). Milankovitch emphasized the changes experienced at 65° north due to the great amount of land at that latitude. Land masses change temperature more quickly than oceans, because of the mixing of surface and deep water and the fact that soil has a lower volumetric heat capacity than water.

Orbital eccentricity

The Earth's orbit approximates an ellipse. Eccentricity measures the departure of this ellipse from circularity. The shape of the Earth's orbit varies between nearly circular (with the lowest eccentricity of 0.000055) and mildly elliptical (highest eccentricity of 0.0679). Its geometric or logarithmic mean is 0.0019. The major component of these variations occurs with a period of 413,000 years (eccentricity variation of ±0.012). Other components have 95,000-year and 125,000-year cycles (with a beat period of 400,000 years). They loosely combine into a 100,000-year cycle (variation of −0.03 to +0.02). The present eccentricity is 0.017 and decreasing.

Eccentricity varies primarily due to the gravitational pull of Jupiter and Saturn. The semi-major axis of the orbital ellipse, however, remains unchanged; according to perturbation theory, which computes the evolution of the orbit, the semi-major axis is invariant. The orbital period (the length of a sidereal year) is also invariant, because according to Kepler's third law, it is determined by the semi-major axis.

Effect on temperature

The semi-major axis is a constant. Therefore, when Earth's orbit becomes more eccentric, the semi-minor axis shortens. This increases the magnitude of seasonal changes.

The relative increase in solar irradiation at closest approach to the Sun (perihelion) compared to the irradiation at the furthest distance (aphelion) is slightly larger than four times the eccentricity. For Earth's current orbital eccentricity, incoming solar radiation varies by about 6.8%, while the distance from the Sun currently varies by only 3.4% (5.1 million km or 3.2 million mi or 0.034 au).

Perihelion presently occurs around January 3, while aphelion is around July 4. When the orbit is at its most eccentric, the amount of solar radiation at perihelion will be about 23% more than at aphelion. However, the Earth's eccentricity is always so small that the variation in solar irradiation is a minor factor in seasonal climate variation, compared to axial tilt and even compared to the relative ease of heating the larger land masses of the northern hemisphere.

Effect on lengths of seasons

Season durations

Year	Northern Hemisphere	Southern Hemisphere	Date: UTC	Season duration
2005	Winter solstice	Summer solstice	21 December 2005 18:35	88.99 days
2006	Spring equinox	Autumn equinox	20 March 2006 18:26	92.75 days
2006	Summer solstice	Winter solstice	21 June 2006 12:26	93.65 days
2006	Autumn equinox	Spring equinox	23 September 2006 4:03	89.85 days
2006	Winter solstice	Summer solstice	22 December 2006 0:22	88.99 days
2007	Spring equinox	Autumn equinox	21 March 2007 0:07	92.75 days
2007	Summer solstice	Winter solstice	21 June 2007 18:06	93.66 days
2007	Autumn equinox	Spring equinox	23 September 2007 9:51	89.85 days
2007	Winter solstice	Summer solstice	22 December 2007 06:08

The seasons are quadrants of the Earth's orbit, marked by the two solstices and the two equinoxes. Kepler's second law states that a body in orbit traces equal areas over equal times; its orbital velocity is highest around perihelion and lowest around aphelion. The Earth spends less time near perihelion and more time near aphelion. This means that the lengths of the seasons vary. Perihelion currently occurs around January 3, so the Earth's greater velocity shortens winter and autumn in the northern hemisphere. Summer in the northern hemisphere is 4.66 days longer than winter, and spring is 2.9 days longer than autumn. Greater eccentricity increases the variation in the Earth's orbital velocity. Currently, however, the Earth's orbit is becoming less eccentric (more nearly circular). This will make the seasons in the immediate future more similar in length.

22.1–24.5° range of Earth's obliquity

Axial tilt (obliquity)

The angle of the Earth's axial tilt with respect to the orbital plane (the obliquity of the ecliptic) varies between 22.1° and 24.5°, over a cycle of about 41,000 years. The current tilt is 23.44°, roughly halfway between its extreme values. The tilt last reached its maximum in 8,700 BCE. It is now in the decreasing phase of its cycle, and will reach its minimum around the year 11,800 CE. Increased tilt increases the amplitude of the seasonal cycle in insolation, providing more solar radiation in each hemisphere's summer and less in winter. However, these effects are not uniform everywhere on the Earth's surface. Increased tilt increases the total annual solar radiation at higher latitudes, and decreases the total closer to the equator.

The current trend of decreasing tilt, by itself, will promote milder seasons (warmer winters and colder summers), as well as an overall cooling trend. Because most of the planet's snow and ice lies at high latitude, decreasing tilt may encourage the termination of an interglacial period and the onset of a glacial period for two reasons: 1) there is less overall summer insolation, and, 2) there is less insolation at higher latitudes (which melts less of the previous winter's snow and ice).

Axial precession

Axial precessional movement

Axial precession is the trend in the direction of the Earth's axis of rotation relative to the fixed stars, with a period of 25,771.5 years. This motion means that eventually Polaris will no longer be the north pole star. It is caused by the tidal forces exerted by the Sun and the Moon on the solid Earth; both contribute roughly equally to this effect.

Currently, perihelion occurs during the southern hemisphere's summer. This means that solar radiation due to both the axial tilt inclining the southern hemisphere toward the Sun, and the Earth's proximity to the Sun, will reach maximum during the southern summer and reach minimum during the southern winter. These effects on heating are thus additive, which means that seasonal variation in irradiation of the southern hemisphere is more extreme. In the northern hemisphere, these two factors reach maximum at opposite times of the year: the north is tilted toward the Sun when the Earth is furthest from the Sun. The two effects work in opposite directions, resulting in less extreme variations in insolation.

In about 13,000 years, the north pole will be tilted toward the Sun when the Earth is at perihelion. Axial tilt and orbital eccentricity will both contribute their maximum increase in solar radiation during the northern hemisphere's summer. Axial precession will promote more extreme variation in irradiation of the northern hemisphere and less extreme variation in the south. When the Earth's axis is aligned such that aphelion and perihelion occur near the equinoxes, axial tilt will not be aligned with or against eccentricity.

Apsidal precession

Planets orbiting the Sun follow elliptical (oval) orbits that rotate gradually over time (apsidal precession). The eccentricity of this ellipse, as well as the rate of precession, is exaggerated for visualization.

The orbital ellipse itself precesses in space, in an irregular fashion, completing a full cycle every 112,000 years relative to the fixed stars. Apsidal precession occurs in the plane of the ecliptic and alters the orientation of the Earth's orbit relative to the ecliptic. This happens primarily as a result of interactions with Jupiter and Saturn. Smaller contributions are also made by the sun's oblateness and by the effects of general relativity that are well known for Mercury.

Apsidal precession combines with the 25,771.5-year cycle of axial precession (see above) to vary the position in the year that the Earth reaches perihelion. Apsidal precession shortens this period to 23,000 years on average (varying between 20,800 and 29,000 years).

 

Effects of precession on the seasons (using the Northern Hemisphere terms)

As the orientation of Earth's orbit changes, each season will gradually start earlier in the year. Precession means the Earth's nonuniform motion (see above) will affect different seasons. Winter, for instance, will be in a different section of the orbit. When the Earth's apsides (extremes of distance from the sun) are aligned with the equinoxes, the length of spring and summer combined will equal that of autumn and winter. When they are aligned with the solstices, the difference in the length of these seasons will be greatest.

Orbital inclination

The inclination of Earth's orbit drifts up and down relative to its present orbit. This three-dimensional movement is known as "precession of the ecliptic" or "planetary precession". Earth's current inclination relative to the invariable plane (the plane that represents the angular momentum of the Solar System—approximately the orbital plane of Jupiter) is 1.57°. Milankovitch did not study planetary precession. It was discovered more recently and measured, relative to Earth's orbit, to have a period of about 70,000 years. When measured independently of Earth's orbit, but relative to the invariable plane, however, precession has a period of about 100,000 years. This period is very similar to the 100,000-year eccentricity period. Both periods closely match the 100,000-year pattern of glacial events.

Theory constraints

Tabernas Desert, Spain: Cycles can be observed in the colouration and resistance of different sediment strata

Materials taken from the Earth have been studied to infer the cycles of past climate. Antarctic ice cores contain trapped air bubbles whose ratios of different oxygen isotopes are a reliable proxy for global temperatures around the time the ice was formed. Study of this data concluded that the climatic response documented in the ice cores was driven by northern hemisphere insolation as proposed by the Milankovitch hypothesis.

Analysis of deep-ocean cores and of lake depths, and a seminal paper by Hays, Imbrie, and Shackleton provide additional validation through physical evidence. Climate records contained in a 1,700 ft (520 m) core of rock drilled in Arizona show a pattern synchronized with Earth's eccentricity, and cores drilled in New England match it, going back 215 million years.

100,000-year issue

Of all the orbital cycles, Milankovitch believed that obliquity had the greatest effect on climate, and that it did so by varying the summer insolation in northern high latitudes. Therefore, he deduced a 41,000-year period for ice ages. However, subsequent research has shown that ice age cycles of the Quaternary glaciation over the last million years have been at a period of 100,000 years, which matches the eccentricity cycle. Various explanations for this discrepancy have been proposed, including frequency modulation or various feedbacks (from carbon dioxide, cosmic rays, or from ice sheet dynamics). Some models can reproduce the 100,000-year cycles as a result of non-linear interactions between small changes in the Earth's orbit and internal oscillations of the climate system.

Jung-Eun Lee of Brown University proposes that precession changes the amount of energy that Earth absorbs, because the southern hemisphere's greater ability to grow sea ice reflects more energy away from Earth. Moreover, Lee says, "Precession only matters when eccentricity is large. That's why we see a stronger 100,000-year pace than a 21,000-year pace." Some others have argued that the length of the climate record is insufficient to establish a statistically significant relationship between climate and eccentricity variations.

Transition changes

Variations of cycle times, curves determined from ocean sediments

 

420,000 years of ice core data from Vostok, Antarctica research station, with more recent times on the left

From 1–3 million years ago, climate cycles matched the 41,000-year cycle in obliquity. After one million years ago, the Mid-Pleistocene Transition (MPT) occurred with a switch to the 100,000-year cycle matching eccentricity. The transition problem refers to the need to explain what changed one million years ago. The MPT can now be reproduced in numerical simulations that include a decreasing trend in carbon dioxide and glacially induced removal of regolith.

Interpretation of unsplit peak variances

Even the well-dated climate records of the last million years do not exactly match the shape of the eccentricity curve. Eccentricity has component cycles of 95,000 and 125,000 years. Some researchers, however, say the records do not show these peaks, but only indicate a single cycle of 100,000 years. The split between the two eccentricity components, however, is observed at least once in a drill core from the 500 million years old Scandinavian Alum Shale.

Unsynced stage five observation

Deep-sea core samples show that the interglacial interval known as marine isotope stage 5 began 130,000 years ago. This is 10,000 years before the solar forcing that the Milankovitch hypothesis predicts. (This is also known as the causality problem, because the effect precedes the putative cause.)

Present and future conditions

Past and future estimations of daily average insolation at top of the atmosphere on the day of the summer solstice, at 65° N latitude. The green curve is with eccentricity e hypothetically set to 0. The red curve uses the actual (predicted) value of e; the blue dot indicates current conditions (2000 CE).

Since orbital variations are predictable, any model that relates orbital variations to climate can be run forward to predict future climate, with two caveats: the mechanism by which orbital forcing influences climate is not definitive; and non-orbital effects can be important (for example, the human impact on the environment principally increases greenhouse gases resulting in a warmer climate).

An often-cited 1980 orbital model by Imbrie predicted "the long-term cooling trend that began some 6,000 years ago will continue for the next 23,000 years." More recent work suggests that orbital variations should gradually increase 65° N summer insolation over the next 25,000 years. Earth's orbit will become less eccentric for about the next 100,000 years, so changes in this insolation will be dominated by changes in obliquity, and should not decline enough to permit a new glacial period in the next 50,000 years.

Other celestial bodies

Mars

Since 1972, speculation sought a relationship between the formation of Mars' alternating bright and dark layers in the polar layered deposits, and the planet’s orbital climate forcing. In 2002, Laska, Levard, and Mustard showed ice-layer radiance, as a function of depth, correlate with the insolation variations in summer at the Martian north pole, similar to palaeoclimate variations on Earth. They also showed Mars' precession had a period of about 51 kyr, obliquity had a period of about 120 kyr, and eccentricity had a period ranging between 95 and 99 kyr. In 2003, Head, Mustard, Kreslavsky, Milliken, and Marchant proposed Mars was in an interglacial period for the past 400 kyr, and in a glacial period between 400 and 2100 kyr, due to Mars' obliquity exceeding 30°. At this extreme obliquity, insolation is dominated by the regular periodicity of Mars' obliquity variation.  Fourier analysis of Mars' orbital elements, show an obliquity period of 128 kyr, and a precession index period of 73 kyr.

Mars has no moon large enough to stabilize its obliquity, which has varied from 10 to 70 degrees. This would explain recent observations of its surface compared to evidence of different conditions in its past, such as the extent of its polar caps.

Outer Solar system

Saturn's moon Titan has a cycle of approximately 60,000 years that could change the location of the methane lakes. Neptune's moon Triton has a variation similar to Titan's, which could cause its solid nitrogen deposits to migrate over long time scales.

Exoplanets

Scientists using computer models to study extreme axial tilts have concluded that high obliquity could cause extreme climate variations, and while that would probably not render a planet uninhabitable, it could pose difficulty for land-based life in affected areas. Most such planets would nevertheless allow development of both simple and more complex lifeforms. Although the obliquity they studied is more extreme than Earth ever experiences, there are scenarios 1.5 to 4.5 billion years from now, as the Moon's stabilizing effect lessens, where obliquity could leave its current range and the poles could eventually point almost directly at the Sun.

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 or Earth's spin is the rotation of planet Earth around its own axis, as well as changes in the orientation of the rotation axis in space. 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; the length of a day increased about 2.3 milliseconds per century since the 8th century BCE. Scientists reported that in 2020 Earth has started spinning faster, after consistently slowing down in the decades before. Because of that, engineers worldwide are discussing a 'negative leap second' and other possible timekeeping measures.

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 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 of planetary motion, 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 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 86400 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 86400 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 International Celestial Reference Frame, 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 4.098903691^s, 0.99726966323716 mean solar days). Earth's rotation period relative to the precessing mean vernal equinox, named sidereal day, is 86164.09053083288 seconds of mean solar time (UT1) (23^h 56^m 4.09053083288^s, 0.99726956632908 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. This is a result of the Earth turning 1 additional rotation, relative to the celestial reference frame, as it orbits the Sun (so 366.25 rotations/y). 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 86400 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 the solar day is incorrect; it must be the sidereal day, 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 or 1040.0mph.

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. Latitude is a placement consideration for spaceports.

Changes

Earth's axial tilt is about 23.4°. It oscillates between 22.1° and 24.5° on a 41000-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 speed

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 geodesy techniques. This provides an absolute reference for the determination of universal time, precession, and nutation. The absolute value of Earth rotation including UT1 and nutation can be determined using space geodetic observations, such as Very Long Baseline Interferometry and Lunar laser ranging, whereas their derivatives, denoted as Length-of-day excess and nutation rates can be derived from satellite observations, such as GPS, GLONASS, Galileo and Satellite laser ranging to geodetic satellites.

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.