Mercury (planet)

From Wikipedia, the free encyclopedia

Mercury

Mercury in enhanced color, imaged by MESSENGER (2008)
Designations
Pronunciation	/ˈmɜːrkjəri/ ( listen)
Adjectives	Mercurian,[1] mercurial[2]
Orbital characteristics[5]
Epoch J2000
Aphelion	0.466 697 AU 69,816,900 km
Perihelion	0.307 499 AU 46,001,200 km
Semi-major axis	0.387 098 AU 57,909,050 km
Eccentricity	0.205 630[3]
Orbital period	87.969 1 d 0.240 846 yr 0.5 Mercury solar day
Synodic period	115.88 d[3]
Average orbital speed	47.362 km/s[3]
Mean anomaly	174.796°
Inclination	7.005° to ecliptic 3.38° to Sun's equator 6.34° to invariable plane[4]
Longitude of ascending node	48.331°
Argument of perihelion	29.124°
Satellites	None
Physical characteristics
Mean radius	2,439.7±1.0 km[6][7] 0.3829 Earths
Flattening	0[7]
Surface area	7.48×107 km2[6] 0.147 Earths
Volume	6.083×1010 km3[6] 0.056 Earths
Mass	3.3011×1023 kg[8] 0.055 Earths
Mean density	5.427 g/cm3[6]
Surface gravity	3.7 m/s2 0.38 g[6]
Moment of inertia factor	0.346±0.014[9]
Escape velocity	4.25 km/s[6]
Sidereal rotation period	58.646 d 1407.5 h[6]
Equatorial rotation velocity	10.892 km/h (3.026 m/s)
Axial tilt	2.04′ ± 0.08′ (to orbit)[9] (0.034°)[3]
North pole right ascension	18h 44m 2s 281.01°[3]
North pole declination	61.45°[3]
Albedo	0.068 (Bond)[10] 0.142 (geom.)[10]
Surface temp. min mean max 0°N, 0°W [11] 100 K 340 K 700 K 85°N, 0°W[11] 80 K 200 K 380 K
Apparent magnitude	−2.6[12] to 5.7[3][13]
Angular diameter	4.5–13″[3]
Atmosphere[14]
Surface pressure	trace (≲ 0.5 nPa)
Composition by volume	42% molecular oxygen 29.0% sodium 22.0% hydrogen 6.0% helium 0.5% potassium Trace amounts of argon, nitrogen, carbon dioxide, water vapor, xenon, krypton, and neon

Mercury is the smallest and innermost planet in the Solar System. Its orbital period around the Sun of 87.97 days is the shortest of all the planets in the Solar System. It is named after the Roman deity Mercury, the messenger of the gods.

Like Venus, Mercury orbits the Sun within Earth's orbit as an inferior planet, and never exceeds 28° away from the Sun. When viewed from Earth, this proximity to the Sun means the planet can only be seen near the western or eastern horizon during the early evening or early morning. At this time it may appear as a bright star-like object, but is often far more difficult to observe than Venus. The planet telescopically displays the complete range of phases, similar to Venus and the Moon, as it moves in its inner orbit relative to Earth, which reoccurs over the so-called synodic period approximately every 116 days.

Mercury is gravitationally locked with the Sun in a 3:2 spin-orbit resonance,^[15] and rotates in a way that is unique in the Solar System. As seen relative to the fixed stars, it rotates on its axis exactly three times for every two revolutions it makes around the Sun.^[a][16] As seen from the Sun, in a frame of reference that rotates with the orbital motion, it appears to rotate only once every two Mercurian years. An observer on Mercury would therefore see only one day every two years.

Mercury's axis has the smallest tilt of any of the Solar System's planets (about ​¹⁄₃₀ degree). Its orbital eccentricity is the largest of all known planets in the Solar System;^[b] at perihelion, Mercury's distance from the Sun is only about two-thirds (or 66%) of its distance at aphelion. Mercury's surface appears heavily cratered and is similar in appearance to the Moon's, indicating that it has been geologically inactive for billions of years. Having almost no atmosphere to retain heat, it has surface temperatures that vary diurnally more than on any other planet in the Solar System, ranging from 100 K (−173 °C; −280 °F) at night to 700 K (427 °C; 800 °F) during the day across the equatorial regions. The polar regions are constantly below 180 K (−93 °C; −136 °F). The planet has no known natural satellites.

Two spacecraft have visited Mercury: Mariner 10 flew by in 1974 and 1975; and MESSENGER, launched in 2004, orbited Mercury over 4,000 times in four years before exhausting its fuel and crashing into the planet's surface on April 30, 2015.^[17][18][19]

Physical characteristics

Internal structure

Internal structure of Mercury:

1. Crust: 100–300 km thick
2. Mantle: 600 km thick
3. Core: 1,800 km radius

Gravity anomalies on Mercury—mass concentrations (red) suggest subsurface structure and evolution

Mercury appears to have a solid silicate crust and mantle overlying a solid, iron sulfide outer core layer, a deeper liquid core layer, and possibly a solid inner core.^[20]

Mercury is one of four terrestrial planets in the Solar System, and is a rocky body like Earth. It is the smallest planet in the Solar System, with an equatorial radius of 2,439.7 kilometres (1,516.0 mi).^[3] Mercury is also smaller—albeit more massive—than the largest natural satellites in the Solar System, Ganymede and Titan. Mercury consists of approximately 70% metallic and 30% silicate material.^[21] Mercury's density is the second highest in the Solar System at 5.427 g/cm³, only slightly less than Earth's density of 5.515 g/cm³.^[3] If the effect of gravitational compression were to be factored out from both planets, the materials of which Mercury is made would be denser than those of Earth, with an uncompressed density of 5.3 g/cm³ versus Earth's 4.4 g/cm³.^[22]

Mercury's density can be used to infer details of its inner structure. Although Earth's high density results appreciably from gravitational compression, particularly at the core, Mercury is much smaller and its inner regions are not as compressed. Therefore, for it to have such a high density, its core must be large and rich in iron.^[23]

Geologists estimate that Mercury's core occupies about 55% of its volume; for Earth this proportion is 17%. Research published in 2007 suggests that Mercury has a molten core.^[24][25] Surrounding the core is a 500–700 km mantle consisting of silicates.^[26][27] Based on data from the Mariner 10 mission and Earth-based observation, Mercury's crust is estimated to be 35 km thick.^[28] One distinctive feature of Mercury's surface is the presence of numerous narrow ridges, extending up to several hundred kilometers in length. It is thought that these were formed as Mercury's core and mantle cooled and contracted at a time when the crust had already solidified.^[29]

Mercury's core has a higher iron content than that of any other major planet in the Solar System, and several theories have been proposed to explain this. The most widely accepted theory is that Mercury originally had a metal–silicate ratio similar to common chondrite meteorites, thought to be typical of the Solar System's rocky matter, and a mass approximately 2.25 times its current mass.^[30] Early in the Solar System's history, Mercury may have been struck by a planetesimal of approximately 1/6 that mass and several thousand kilometers across.^[30] The impact would have stripped away much of the original crust and mantle, leaving the core behind as a relatively major component.^[30] A similar process, known as the giant impact hypothesis, has been proposed to explain the formation of the Moon.^[30]

Alternatively, Mercury may have formed from the solar nebula before the Sun's energy output had stabilized. It would initially have had twice its present mass, but as the protosun contracted, temperatures near Mercury could have been between 2,500 and 3,500 K and possibly even as high as 10,000 K.^[31] Much of Mercury's surface rock could have been vaporized at such temperatures, forming an atmosphere of "rock vapor" that could have been carried away by the solar wind.^[31]

A third hypothesis proposes that the solar nebula caused drag on the particles from which Mercury was accreting, which meant that lighter particles were lost from the accreting material and not gathered by Mercury.^[32] Each hypothesis predicts a different surface composition, and there are two space missions set to make observations. MESSENGER, which ended in 2015, found higher-than-expected potassium and sulfur levels on the surface, suggesting that the giant impact hypothesis and vaporization of the crust and mantle did not occur because potassium and sulfur would have been driven off by the extreme heat of these events.^[33] BepiColombo, which will arrive at Mercury in 2025, will make observations to test these hypotheses.^[34] The findings so far would seem to favor the third hypothesis; however, further analysis of the data is needed.^[35]

Surface geology

Map of Mercury's northern hemisphere by the MLA instrument on MESSENGER lowest (purple) to 10 km (6.2 mi) highest (red).

Mercury's surface is similar in appearance to that of the Moon, showing extensive mare-like plains and heavy cratering, indicating that it has been geologically inactive for billions of years. Because knowledge of Mercury's geology had been based only on the 1975 Mariner 10 flyby and terrestrial observations, it is the least understood of the terrestrial planets.^[25] As data from MESSENGER orbiter are processed, this knowledge will increase. For example, an unusual crater with radiating troughs has been discovered that scientists called "the spider".^[36] It was later named Apollodorus.^[37]

Mercury's surface

Albedo features are areas of markedly different reflectivity, as seen by telescopic observation. Mercury has dorsa (also called "wrinkle-ridges"), Moon-like highlands, montes (mountains), planitiae (plains), rupes (escarpments), and valles (valleys).^[38][39]

MASCS spectrum scan of Mercury's surface by MESSENGER

Names for features on Mercury come from a variety of sources. Names coming from people are limited to the deceased. Craters are named for artists, musicians, painters, and authors who have made outstanding or fundamental contributions to their field. Ridges, or dorsa, are named for scientists who have contributed to the study of Mercury. Depressions or fossae are named for works of architecture. Montes are named for the word "hot" in a variety of languages. Plains or planitiae are named for Mercury in various languages. Escarpments or rupēs are named for ships of scientific expeditions. Valleys or valles are named for radio telescope facilities.^[40]

Mercury was heavily bombarded by comets and asteroids during and shortly following its formation 4.6 billion years ago, as well as during a possibly separate subsequent episode called the Late Heavy Bombardment that ended 3.8 billion years ago.^[41] During this period of intense crater formation, Mercury received impacts over its entire surface,^[39] facilitated by the lack of any atmosphere to slow impactors down.^[42] During this time Mercury was volcanically active; basins such as the Caloris Basin were filled by magma, producing smooth plains similar to the maria found on the Moon.^[43][44]

Data from the October 2008 flyby of MESSENGER gave researchers a greater appreciation for the jumbled nature of Mercury's surface. Mercury's surface is more heterogeneous than either Mars's or the Moon's, both of which contain significant stretches of similar geology, such as maria and plateaus.^[45]

Impact basins and craters

Perspective view of Caloris Basin – high (red); low (blue).

Craters on Mercury range in diameter from small bowl-shaped cavities to multi-ringed impact basins hundreds of kilometers across. They appear in all states of degradation, from relatively fresh rayed craters to highly degraded crater remnants. Mercurian craters differ subtly from lunar craters in that the area blanketed by their ejecta is much smaller, a consequence of Mercury's stronger surface gravity.^[46] According to IAU rules, each new crater must be named after an artist that was famous for more than fifty years, and dead for more than three years, before the date the crater is named.^[47]

Enhanced-color image of Munch, Sander and Poe craters amid volcanic plains (orange) near Caloris Basin

The largest known crater is Caloris Basin, with a diameter of 1,550 km.^[48] The impact that created the Caloris Basin was so powerful that it caused lava eruptions and left a concentric ring over 2 km tall surrounding the impact crater. At the antipode of the Caloris Basin is a large region of unusual, hilly terrain known as the "Weird Terrain". One hypothesis for its origin is that shock waves generated during the Caloris impact traveled around Mercury, converging at the basin's antipode (180 degrees away). The resulting high stresses fractured the surface.^[49] Alternatively, it has been suggested that this terrain formed as a result of the convergence of ejecta at this basin's antipode.^[50]

Overall, about 15 impact basins have been identified on the imaged part of Mercury. A notable basin is the 400 km wide, multi-ring Tolstoj Basin that has an ejecta blanket extending up to 500 km from its rim and a floor that has been filled by smooth plains materials. Beethoven Basin has a similar-sized ejecta blanket and a 625 km diameter rim.^[46] Like the Moon, the surface of Mercury has likely incurred the effects of space weathering processes, including Solar wind and micrometeorite impacts.^[51]

Interior of Abedin crater

Plains

Degas crater

There are two geologically distinct plains regions on Mercury.^[46][52] Gently rolling, hilly plains in the regions between craters are Mercury's oldest visible surfaces,^[46] predating the heavily cratered terrain. These inter-crater plains appear to have obliterated many earlier craters, and show a general paucity of smaller craters below about 30 km in diameter.^[52]

The so-called "Weird Terrain" formed at the point antipodal to the Caloris Basin impact

Smooth plains are widespread flat areas that fill depressions of various sizes and bear a strong resemblance to the lunar maria. Notably, they fill a wide ring surrounding the Caloris Basin. Unlike lunar maria, the smooth plains of Mercury have the same albedo as the older inter-crater plains. Despite a lack of unequivocally volcanic characteristics, the localisation and rounded, lobate shape of these plains strongly support volcanic origins.^[46] All the smooth plains of Mercury formed significantly later than the Caloris basin, as evidenced by appreciably smaller crater densities than on the Caloris ejecta blanket.^[46] The floor of the Caloris Basin is filled by a geologically distinct flat plain, broken up by ridges and fractures in a roughly polygonal pattern. It is not clear whether they are volcanic lavas induced by the impact, or a large sheet of impact melt.^[46]

Compressional features

One unusual feature of Mercury's surface is the numerous compression folds, or rupes, that crisscross the plains. As Mercury's interior cooled, it contracted and its surface began to deform, creating wrinkle ridges and lobate scarps associated with thrust faults. The scarps can reach lengths of 1000 km and heights of 3 km.^[53] These compressional features can be seen on top of other features, such as craters and smooth plains, indicating they are more recent.^[54] Mapping of the features has suggested a total shrinkage of Mercury's radius in the range of ~1 to 7 km.^[55] Small-scale thrust fault scarps have been found, tens of meters in height and with lengths in the range of a few km, that appear to be less than 50 million years old, indicating that compression of the interior and consequent surface geological activity continue to the present.^[53][55]

The Lunar Reconnaissance Orbiter discovered that similar small thrust faults exist on the Moon.

Volcanology

Picasso crater — the large arc-shaped pit located on the eastern side of its floor are postulated to have formed when subsurface magma subsided or drained, causing the surface to collapse into the resulting void.

Images obtained by MESSENGER have revealed evidence for pyroclastic flows on Mercury from low-profile shield volcanoes.^[56][57][58] MESSENGER data has helped identify 51 pyroclastic deposits on the surface,^[59] where 90% of them are found within impact craters.^[59] A study of the degradation state of the impact craters that host pyroclastic deposits suggests that pyroclastic activity occurred on Mercury over a prolonged interval.^[59]

A "rimless depression" inside the southwest rim of the Caloris Basin consists of at least nine overlapping volcanic vents, each individually up to 8 km in diameter. It is thus a "compound volcano".^[60] The vent floors are at a least 1 km below their brinks and they bear a closer resemblance to volcanic craters sculpted by explosive eruptions or modified by collapse into void spaces created by magma withdrawal back down into a conduit.^[60] The scientists could not quantify the age of the volcanic complex system, but reported that it could be of the order of a billion years.^[60]

Surface conditions and exosphere

Composite image of Mercury taken by MESSENGER

 

Radar image of Mercury's north pole

 

Composite of the north pole of Mercury, where NASA confirmed the discovery of a large volume of water ice, in permanently dark craters that exist there.^[61]

The surface temperature of Mercury ranges from 100 K to 700 K^[62] at the most extreme places: 0°N, 0°W, or 180°W. It never rises above 180 K at the poles,^[11] due to the absence of an atmosphere and a steep temperature gradient between the equator and the poles. The subsolar point reaches about 700 K during perihelion (0°W or 180°W), but only 550 K at aphelion (90° or 270°W).^[63] On the dark side of the planet, temperatures average 110 K.^[11][64] The intensity of sunlight on Mercury's surface ranges between 4.59 and 10.61 times the solar constant (1,370 W·m⁻²).^[65]

Although the daylight temperature at the surface of Mercury is generally extremely high, observations strongly suggest that ice (frozen water) exists on Mercury. The floors of deep craters at the poles are never exposed to direct sunlight, and temperatures there remain below 102 K; far lower than the global average.^[66] Water ice strongly reflects radar, and observations by the 70-meter Goldstone Solar System Radar and the VLA in the early 1990s revealed that there are patches of high radar reflection near the poles.^[67] Although ice was not the only possible cause of these reflective regions, astronomers think it was the most likely.^[68]

The icy regions are estimated to contain about 10¹⁴–10¹⁵ kg of ice,^[69] and may be covered by a layer of regolith that inhibits sublimation.^[70] By comparison, the Antarctic ice sheet on Earth has a mass of about 4×10¹⁸ kg, and Mars's south polar cap contains about 10¹⁶ kg of water.^[69] The origin of the ice on Mercury is not yet known, but the two most likely sources are from outgassing of water from the planet's interior or deposition by impacts of comets.^[69]

Mercury is too small and hot for its gravity to retain any significant atmosphere over long periods of time; it does have a tenuous surface-bounded exosphere^[71] containing hydrogen, helium, oxygen, sodium, calcium, potassium and others at a surface pressure of less than approximately 0.5 nPa (0.005 picobars).^[14] This exosphere is not stable—atoms are continuously lost and replenished from a variety of sources. Hydrogen atoms and helium atoms probably come from the solar wind, diffusing into Mercury's magnetosphere before later escaping back into space. Radioactive decay of elements within Mercury's crust is another source of helium, as well as sodium and potassium. MESSENGER found high proportions of calcium, helium, hydroxide, magnesium, oxygen, potassium, silicon and sodium. Water vapor is present, released by a combination of processes such as: comets striking its surface, sputtering creating water out of hydrogen from the solar wind and oxygen from rock, and sublimation from reservoirs of water ice in the permanently shadowed polar craters. The detection of high amounts of water-related ions like O⁺, OH⁻, and H₂O⁺ was a surprise.^[72][73] Because of the quantities of these ions that were detected in Mercury's space environment, scientists surmise that these molecules were blasted from the surface or exosphere by the solar wind.^[74][75]

Sodium, potassium and calcium were discovered in the atmosphere during the 1980–1990s, and are thought to result primarily from the vaporization of surface rock struck by micrometeorite impacts^[76] including presently from Comet Encke.^[77] In 2008, magnesium was discovered by MESSENGER.^[78] Studies indicate that, at times, sodium emissions are localized at points that correspond to the planet's magnetic poles. This would indicate an interaction between the magnetosphere and the planet's surface.^[79]

On November 29, 2012, NASA confirmed that images from MESSENGER had detected that craters at the north pole contained water ice. MESSENGER's principal investigator Sean Solomon is quoted in The New York Times estimating the volume of the ice to be large enough to "encase Washington, D.C., in a frozen block two and a half miles deep".^[61][c]

Magnetic field and magnetosphere

Graph showing relative strength of Mercury's magnetic field

Despite its small size and slow 59-day-long rotation, Mercury has a significant, and apparently global, magnetic field. According to measurements taken by Mariner 10, it is about 1.1% the strength of Earth's. The magnetic-field strength at Mercury's equator is about 300 nT.^[80][81] Like that of Earth, Mercury's magnetic field is dipolar.^[79] Unlike Earth's, Mercury's poles are nearly aligned with the planet's spin axis.^[82] Measurements from both the Mariner 10 and MESSENGER space probes have indicated that the strength and shape of the magnetic field are stable.^[82]

It is likely that this magnetic field is generated by a dynamo effect, in a manner similar to the magnetic field of Earth.^[83][84] This dynamo effect would result from the circulation of the planet's iron-rich liquid core. Particularly strong tidal effects caused by the planet's high orbital eccentricity would serve to keep the core in the liquid state necessary for this dynamo effect.^[85]

Mercury's magnetic field is strong enough to deflect the solar wind around the planet, creating a magnetosphere. The planet's magnetosphere, though small enough to fit within Earth,^[79] is strong enough to trap solar wind plasma. This contributes to the space weathering of the planet's surface.^[82] Observations taken by the Mariner 10 spacecraft detected this low energy plasma in the magnetosphere of the planet's nightside. Bursts of energetic particles in the planet's magnetotail indicate a dynamic quality to the planet's magnetosphere.^[79]

During its second flyby of the planet on October 6, 2008, MESSENGER discovered that Mercury's magnetic field can be extremely "leaky". The spacecraft encountered magnetic "tornadoes" – twisted bundles of magnetic fields connecting the planetary magnetic field to interplanetary space – that were up to 800 km wide or a third of the radius of the planet. These twisted magnetic flux tubes, technically known as flux transfer events, form open windows in the planet's magnetic shield through which the solar wind may enter and directly impact Mercury's surface via magnetic reconnection^[86] This also occurs in Earth's magnetic field. The MESSENGER observations showed the reconnection rate is ten times higher at Mercury, but its proximity to the Sun only accounts for about a third of the reconnection rate observed by MESSENGER.^[86]

Orbit, rotation, and longitude

Orbit of Mercury (yellow). Dates refer to 2006.

 

Animation of Mercury's and Earth's revolution around the Sun

Mercury has the most eccentric orbit of all the planets; its eccentricity is 0.21 with its distance from the Sun ranging from 46,000,000 to 70,000,000 km (29,000,000 to 43,000,000 mi). It takes 87.969 Earth days to complete an orbit. The diagram on the right illustrates the effects of the eccentricity, showing Mercury's orbit overlaid with a circular orbit having the same semi-major axis. Mercury's higher velocity when it is near perihelion is clear from the greater distance it covers in each 5-day interval. In the diagram the varying distance of Mercury to the Sun is represented by the size of the planet, which is inversely proportional to Mercury's distance from the Sun. This varying distance to the Sun leads to Mercury's surface being flexed by tidal bulges raised by the Sun that are about 17 times stronger than the Moon's on Earth.^[87] Combined with a 3:2 spin–orbit resonance of the planet's rotation around its axis, it also results in complex variations of the surface temperature.^[21] The resonance makes a single solar day on Mercury last exactly two Mercury years, or about 176 Earth days.^[88]

Mercury's orbit is inclined by 7 degrees to the plane of Earth's orbit (the ecliptic), as shown in the diagram on the right. As a result, transits of Mercury across the face of the Sun can only occur when the planet is crossing the plane of the ecliptic at the time it lies between Earth and the Sun. This occurs about every seven years on average.^[89]

Mercury's axial tilt is almost zero,^[90] with the best measured value as low as 0.027 degrees.^[91] This is significantly smaller than that of Jupiter, which has the second smallest axial tilt of all planets at 3.1 degrees. This means that to an observer at Mercury's poles, the center of the Sun never rises more than 2.1 arcminutes above the horizon.^[91]

At certain points on Mercury's surface, an observer would be able to see the Sun peek up about halfway over the horizon, then reverse and set before rising again, all within the same Mercurian day. This is because approximately four Earth days before perihelion, Mercury's angular orbital velocity equals its angular rotational velocity so that the Sun's apparent motion ceases; closer to perihelion, Mercury's angular orbital velocity then exceeds the angular rotational velocity. Thus, to a hypothetical observer on Mercury, the Sun appears to move in a retrograde direction. Four Earth days after perihelion, the Sun's normal apparent motion resumes.^[21] A similar effect would have occurred if Mercury had been in synchronous rotation: the alternating gain and loss of rotation over revolution would have caused a libration of 23.65° in longitude.^[92]

For the same reason, there are two points on Mercury's equator, 180 degrees apart in longitude, at either of which, around perihelion in alternate Mercurian years (once a Mercurian day), the Sun passes overhead, then reverses its apparent motion and passes overhead again, then reverses a second time and passes overhead a third time, taking a total of about 16 Earth-days for this entire process. In the other alternate Mercurian years, the same thing happens at the other of these two points. The amplitude of the retrograde motion is small, so the overall effect is that, for two or three weeks, the Sun is almost stationary overhead, and is at its most brilliant because Mercury is at perihelion, its closest to the Sun. This prolonged exposure to the Sun at its brightest makes these two points the hottest places on Mercury. Conversely, there are two other points on the equator, 90 degrees of longitude apart from the first ones, where the Sun passes overhead only when the planet is at aphelion in alternate years, when the apparent motion of the Sun in Mercury's sky is relatively rapid. These points, which are the ones on the equator where the apparent retrograde motion of the Sun happens when it is crossing the horizon as described in the preceding paragraph, receive much less solar heat than the first ones described above.

Mercury attains inferior conjunction (nearest approach to Earth) every 116 Earth days on average,^[3] but this interval can range from 105 days to 129 days due to the planet's eccentric orbit. Mercury can come as near as 82.2 gigametres (0.549 astronomical units; 51.1 million miles) to Earth, and that is slowly declining: The next approach to within 82.1 Gm (51.0 million miles) is in 2679, and to within 82.0 Gm (51.0 million miles) in 4487, but it will not be closer to Earth than 80 Gm (50 million miles) until 28,622.^[93] Its period of retrograde motion as seen from Earth can vary from 8 to 15 days on either side of inferior conjunction. This large range arises from the planet's high orbital eccentricity.^[21]

Longitude convention

The longitude convention for Mercury puts the zero of longitude at one of the two hottest points on the surface, as described above. However, when this area was first visited, by Mariner 10, this zero meridian was in darkness, so it was impossible to select a feature on the surface to define the exact position of the meridian. Therefore, a small crater further west was chosen, called Hun Kal, which provides the exact reference point for measuring longitude. The center of Hun Kal defines the 20° West meridian. A 1970 International Astronomical Union resolution suggests that longitudes be measured positively in the westerly direction on Mercury.^[94] The two hottest places on the equator are therefore at longitudes 0°W and 180°W, and the coolest points on the equator are at longitudes 90°W and 270°W. However, the MESSENGER project uses an east-positive convention.^[95]

Spin–orbit resonance

After one orbit, Mercury has rotated 1.5 times, so after two complete orbits the same hemisphere is again illuminated.

For many years it was thought that Mercury was synchronously tidally locked with the Sun, rotating once for each orbit and always keeping the same face directed towards the Sun, in the same way that the same side of the Moon always faces Earth. Radar observations in 1965 proved that the planet has a 3:2 spin–orbit resonance, rotating three times for every two revolutions around the Sun. The eccentricity of Mercury's orbit makes this resonance stable—at perihelion, when the solar tide is strongest, the Sun is nearly still in Mercury's sky.^[96]

The rare 3:2 resonant tidal locking is stabilized by the variance of the tidal force along Mercury's eccentric orbit, acting on a permanent dipole component of Mercury's mass distribution.^[97] In a circular orbit there is no such variance, so the only resonance stabilized in such an orbit is at 1:1 (e.g., Earth–Moon), when the tidal force, stretching a body along the "center-body" line, exerts a torque that aligns the body's axis of least inertia (the "longest" axis, and the axis of the aforementioned dipole) to always point at the center. However, with noticeable eccentricity, like that of Mercury's orbit, the tidal force has a maximum at perihelion and thus stabilizes resonances, like 3:2, enforcing that the planet points its axis of least inertia roughly at the Sun when passing through perihelion.^[97]

The original reason astronomers thought it was synchronously locked was that, whenever Mercury was best placed for observation, it was always nearly at the same point in its 3:2 resonance, hence showing the same face. This is because, coincidentally, Mercury's rotation period is almost exactly half of its synodic period with respect to Earth. Due to Mercury's 3:2 spin–orbit resonance, a solar day (the length between two meridian transits of the Sun) lasts about 176 Earth days.^[21] A sidereal day (the period of rotation) lasts about 58.7 Earth days.^[21]

Simulations indicate that the orbital eccentricity of Mercury varies chaotically from nearly zero (circular) to more than 0.45 over millions of years due to perturbations from the other planets.^[21][98] This was thought to explain Mercury's 3:2 spin–orbit resonance (rather than the more usual 1:1), because this state is more likely to arise during a period of high eccentricity.^[99] However, accurate modeling based on a realistic model of tidal response has demonstrated that Mercury was captured into the 3:2 spin–orbit state at a very early stage of its history, within 20 (more likely, 10) million years after its formation.^[100]

Numerical simulations show that a future secular orbital resonant perihelion interaction with Jupiter may cause the eccentricity of Mercury's orbit to increase to the point where there is a 1% chance that the planet may collide with Venus within the next five billion years.^[101][102]

Advance of perihelion

In 1859, the French mathematician and astronomer Urbain Le Verrier reported that the slow precession of Mercury's orbit around the Sun could not be completely explained by Newtonian mechanics and perturbations by the known planets. He suggested, among possible explanations, that another planet (or perhaps instead a series of smaller 'corpuscules') might exist in an orbit even closer to the Sun than that of Mercury, to account for this perturbation.^[103] (Other explanations considered included a slight oblateness of the Sun.) The success of the search for Neptune based on its perturbations of the orbit of Uranus led astronomers to place faith in this possible explanation, and the hypothetical planet was named Vulcan, but no such planet was ever found.^[104]
The perihelion precession of Mercury is 5,600 arcseconds (1.5556°) per century relative to Earth, or 574.10±0.65 arcseconds per century^[105] relative to the inertial ICRF. Newtonian mechanics, taking into account all the effects from the other planets, predicts a precession of 5,557 arcseconds (1.5436°) per century.^[105] In the early 20th century, Albert Einstein's general theory of relativity provided the explanation for the observed precession, by formalizing gravitation as being mediated by the curvature of spacetime. The effect is small: just 42.98 arcseconds per century for Mercury; it therefore requires a little over twelve million orbits for a full excess turn. Similar, but much smaller, effects exist for other Solar System bodies: 8.62 arcseconds per century for Venus, 3.84 for Earth, 1.35 for Mars, and 10.05 for 1566 Icarus.^[106][107]
Albert Einstein's formula for the perihelion shift per revolution is ${\displaystyle \epsilon =24\pi ^{3}{\frac {a^{2}}{T^{2}c^{2}(1-e^{2})}}}$, where ${\displaystyle e}$ is the orbital eccentricity, ${\displaystyle a}$ the semi-major axis, and ${\displaystyle T}$ the orbital period. Filling in the values gives a result of 0.1035 arcseconds per revolution or 0.4297 arcseconds per Earth year, i.e., 42.97 arcseconds per century.

Observation

Image mosaic by Mariner 10, 1974

Mercury's apparent magnitude varies between −2.6^[12] (brighter than the brightest star Sirius) and about +5.7 (approximating the theoretical limit of naked-eye visibility). The extremes occur when Mercury is close to the Sun in the sky.^[12][13] Observation of Mercury is complicated by its proximity to the Sun, as it is lost in the Sun's glare for much of the time. Mercury can be observed for only a brief period during either morning or evening twilight.^[108]

Mercury can, like several other planets and the brightest stars, be seen during a total solar eclipse.^[109]

Like the Moon and Venus, Mercury exhibits phases as seen from Earth. It is "new" at inferior conjunction and "full" at superior conjunction. The planet is rendered invisible from Earth on both of these occasions because of its being obscured by the Sun,^[108] except its new phase during a transit.

Mercury is technically brightest as seen from Earth when it is at a full phase. Although Mercury is farthest from Earth when it is full, the greater illuminated area that is visible and the opposition brightness surge more than compensates for the distance.^[12] The opposite is true for Venus, which appears brightest when it is a crescent, because it is much closer to Earth than when gibbous.^[12][110]

False-color map showing the maximum temperatures of the north polar region

Nonetheless, the brightest (full phase) appearance of Mercury is an essentially impossible time for practical observation, because of the extreme proximity of the Sun. Mercury is best observed at the first and last quarter, although they are phases of lesser brightness. The first and last quarter phases occur at greatest elongation east and west of the Sun, respectively. At both of these times Mercury's separation from the Sun ranges anywhere from 17.9° at perihelion to 27.8° at aphelion.^[111][112] At greatest western elongation, Mercury rises at its earliest before sunrise, and at greatest eastern elongation, it sets at its latest after sunset.^[113]

Mercury can be easily seen from the tropics and subtropics more than from higher latitudes. Viewed from low latitudes and at the right times of year, the ecliptic intersects the horizon at a steep angle. Mercury is 10° above the horizon when the planet appears directly above the Sun (i.e. its orbit appears vertical) and is at maximum elongation from the Sun (28°) and also when the Sun is 18° below the horizon, so the sky is just completely dark.^[d] This angle is the maximum altitude at which Mercury is visible in a completely dark sky.

False-color image of Carnegie Rupes, a tectonic landform—high terrain (red); low (blue).

At middle latitudes, Mercury is more often and easily visible from the Southern Hemisphere than from the Northern. This is because Mercury's maximum western elongation occurs only during early autumn in the Southern Hemisphere, whereas its greatest eastern elongation happens only during late winter in the Southern Hemisphere.^[113] In both of these cases, the angle at which the planet's orbit intersects the horizon is maximized, allowing it to rise several hours before sunrise in the former instance and not set until several hours after sundown in the latter from southern mid-latitudes, such as Argentina and South Africa.^[113]

An alternate method for viewing Mercury involves observing the planet during daylight hours when conditions are clear, ideally when it is at its greatest elongation. This allows the planet to be found easily, even when using telescopes with 8 cm (3.1 in) apertures. Care must be taken to ensure the instrument isn't pointed directly towards the Sun because of the risk for eye damage. This method bypasses the limitation of twilight observing when the ecliptic is located at a low elevation (e.g. on autumn evenings).

Ground-based telescope observations of Mercury reveal only an illuminated partial disk with limited detail. The first of two spacecraft to visit the planet was Mariner 10, which mapped about 45% of its surface from 1974 to 1975. The second is the MESSENGER spacecraft, which after three Mercury flybys between 2008 and 2009, attained orbit around Mercury on March 17, 2011,^[114] to study and map the rest of the planet.^[115]

The Hubble Space Telescope cannot observe Mercury at all, due to safety procedures that prevent its pointing too close to the Sun.^[116]

Because the shift of 0.15 revolutions in a year makes up a seven-year cycle (0.15 × 7 ≈ 1.0), in the seventh year Mercury follows almost exactly (earlier by 7 days) the sequence of phenomena it showed seven years before.^[111]

Observation history

Ancient astronomers

Mercury, from Liber astronomiae, 1550

The earliest known recorded observations of Mercury are from the Mul.Apin tablets. These observations were most likely made by an Assyrian astronomer around the 14th century BC.^[117] The cuneiform name used to designate Mercury on the Mul.Apin tablets is transcribed as Udu.Idim.Gu\u₄.Ud ("the jumping planet").^[e][118] Babylonian records of Mercury date back to the 1st millennium BC. The Babylonians called the planet Nabu after the messenger to the gods in their mythology.^[119]

The ancient Greeks knew the planet as Στίλβων (Stilbon), meaning "the gleaming", Ἑρμάων (Hermaon) and Ἑρμής (Hermes),^[120] a planetary name that is retained in modern Greek (Ερμής: Ermis).^[121] The Romans named the planet after the swift-footed Roman messenger god, Mercury (Latin Mercurius), which they equated with the Greek Hermes, because it moves across the sky faster than any other planet.^[122][123] The astronomical symbol for Mercury is a stylized version of Hermes' caduceus.^[124]

The Roman-Egyptian astronomer Ptolemy wrote about the possibility of planetary transits across the face of the Sun in his work Planetary Hypotheses. He suggested that no transits had been observed either because planets such as Mercury were too small to see, or because the transits were too infrequent.^[125]

Ibn al-Shatir's model for the appearances of Mercury, showing the multiplication of epicycles using the Tusi couple, thus eliminating the Ptolemaic eccentrics and equant.

In ancient China, Mercury was known as "the Hour Star" (Chen-xing 辰星). It was associated with the direction north and the phase of water in the Five Phases system of metaphysics.^[126] Modern Chinese, Korean, Japanese and Vietnamese cultures refer to the planet literally as the "water star" (水星), based on the Five elements.^{[127][128][129]} Hindu mythology used the name Budha for Mercury, and this god was thought to preside over Wednesday.^[130] The god Odin (or Woden) of Germanic paganism was associated with the planet Mercury and Wednesday.^[131] The Maya may have represented Mercury as an owl (or possibly four owls; two for the morning aspect and two for the evening) that served as a messenger to the underworld.^[132]

In medieval Islamic astronomy, the Andalusian astronomer Abū Ishāq Ibrāhīm al-Zarqālī in the 11th century described the deferent of Mercury's geocentric orbit as being oval, like an egg or a pignon, although this insight did not influence his astronomical theory or his astronomical calculations.^[133][134] In the 12th century, Ibn Bajjah observed "two planets as black spots on the face of the Sun", which was later suggested as the transit of Mercury and/or Venus by the Maragha astronomer Qotb al-Din Shirazi in the 13th century.^[135] (Note that most such medieval reports of transits were later taken as observations of sunspots.^[136])

In India, the Kerala school astronomer Nilakantha Somayaji in the 15th century developed a partially heliocentric planetary model in which Mercury orbits the Sun, which in turn orbits Earth, similar to the Tychonic system later proposed by Tycho Brahe in the late 16th century.^[137]

Ground-based telescopic research

Transit of Mercury. Mercury is visible as a black dot below and to the left of center. The dark area above the center of the solar disk is a sunspot.

 

Elongation is the angle between the Sun and the planet, with Earth as the reference point. Mercury appears close to the Sun.

The first telescopic observations of Mercury were made by Galileo in the early 17th century. Although he observed phases when he looked at Venus, his telescope was not powerful enough to see the phases of Mercury. In 1631, Pierre Gassendi made the first telescopic observations of the transit of a planet across the Sun when he saw a transit of Mercury predicted by Johannes Kepler. In 1639, Giovanni Zupi used a telescope to discover that the planet had orbital phases similar to Venus and the Moon. The observation demonstrated conclusively that Mercury orbited around the Sun.^[21]

A rare event in astronomy is the passage of one planet in front of another (occultation), as seen from Earth. Mercury and Venus occult each other every few centuries, and the event of May 28, 1737 is the only one historically observed, having been seen by John Bevis at the Royal Greenwich Observatory.^[138] The next occultation of Mercury by Venus will be on December 3, 2133.^[139]

The difficulties inherent in observing Mercury mean that it has been far less studied than the other planets. In 1800, Johann Schröter made observations of surface features, claiming to have observed 20-kilometre-high (12 mi) mountains. Friedrich Bessel used Schröter's drawings to erroneously estimate the rotation period as 24 hours and an axial tilt of 70°.^[140] In the 1880s, Giovanni Schiaparelli mapped the planet more accurately, and suggested that Mercury's rotational period was 88 days, the same as its orbital period due to tidal locking.^[141] This phenomenon is known as synchronous rotation. The effort to map the surface of Mercury was continued by Eugenios Antoniadi, who published a book in 1934 that included both maps and his own observations.^[79] Many of the planet's surface features, particularly the albedo features, take their names from Antoniadi's map.^[142]

In June 1962, Soviet scientists at the Institute of Radio-engineering and Electronics of the USSR Academy of Sciences, led by Vladimir Kotelnikov, became the first to bounce a radar signal off Mercury and receive it, starting radar observations of the planet.^{[143][144][145]} Three years later, radar observations by Americans Gordon Pettengill and R. Dyce, using the 300-meter Arecibo Observatory radio telescope in Puerto Rico, showed conclusively that the planet's rotational period was about 59 days.^[146][147] The theory that Mercury's rotation was synchronous had become widely held, and it was a surprise to astronomers when these radio observations were announced. If Mercury were tidally locked, its dark face would be extremely cold, but measurements of radio emission revealed that it was much hotter than expected. Astronomers were reluctant to drop the synchronous rotation theory and proposed alternative mechanisms such as powerful heat-distributing winds to explain the observations.^[148]

Water ice (yellow) at Mercury's north polar region

Italian astronomer Giuseppe Colombo noted that the rotation value was about two-thirds of Mercury's orbital period, and proposed that the planet's orbital and rotational periods were locked into a 3:2 rather than a 1:1 resonance.^[149] Data from Mariner 10 subsequently confirmed this view.^[150] This means that Schiaparelli's and Antoniadi's maps were not "wrong". Instead, the astronomers saw the same features during every second orbit and recorded them, but disregarded those seen in the meantime, when Mercury's other face was toward the Sun, because the orbital geometry meant that these observations were made under poor viewing conditions.^[140]

Ground-based optical observations did not shed much further light on Mercury, but radio astronomers using interferometry at microwave wavelengths, a technique that enables removal of the solar radiation, were able to discern physical and chemical characteristics of the subsurface layers to a depth of several meters.^[151][152] Not until the first space probe flew past Mercury did many of its most fundamental morphological properties become known. Moreover, recent technological advances have led to improved ground-based observations. In 2000, high-resolution lucky imaging observations were conducted by the Mount Wilson Observatory 1.5 meter Hale telescope. They provided the first views that resolved surface features on the parts of Mercury that were not imaged in the Mariner 10 mission.^[153] Most of the planet has been mapped by the Arecibo radar telescope, with 5 km (3.1 mi) resolution, including polar deposits in shadowed craters of what may be water ice.^[154]

Research with space probes

MESSENGER being prepared for launch

 

Mercury transiting the Sun as viewed by the Mars rover Curiosity (June 3, 2014).^[155]

Reaching Mercury from Earth poses significant technical challenges, because it orbits so much closer to the Sun than Earth. A Mercury-bound spacecraft launched from Earth must travel over 91 million kilometres (57 million miles) into the Sun's gravitational potential well. Mercury has an orbital speed of 48 km/s (30 mi/s), whereas Earth's orbital speed is 30 km/s (19 mi/s). Therefore, the spacecraft must make a large change in velocity (delta-v) to enter a Hohmann transfer orbit that passes near Mercury, as compared to the delta-v required for other planetary missions.^[156]

The potential energy liberated by moving down the Sun's potential well becomes kinetic energy; requiring another large delta-v change to do anything other than rapidly pass by Mercury. To land safely or enter a stable orbit the spacecraft would rely entirely on rocket motors. Aerobraking is ruled out because Mercury has a negligible atmosphere. A trip to Mercury requires more rocket fuel than that required to escape the Solar System completely. As a result, only two space probes have visited it so far.^[157] A proposed alternative approach would use a solar sail to attain a Mercury-synchronous orbit around the Sun.^[158]

Mariner 10

Mariner 10, the first probe to visit Mercury

The first spacecraft to visit Mercury was NASA's Mariner 10 (1974–1975).^[122] The spacecraft used the gravity of Venus to adjust its orbital velocity so that it could approach Mercury, making it both the first spacecraft to use this gravitational "slingshot" effect and the first NASA mission to visit multiple planets.^[156] Mariner 10 provided the first close-up images of Mercury's surface, which immediately showed its heavily cratered nature, and revealed many other types of geological features, such as the giant scarps that were later ascribed to the effect of the planet shrinking slightly as its iron core cools.^[159] Unfortunately, the same face of the planet was lit at each of Mariner 10's close approaches. This made close observation of both sides of the planet impossible,^[160] and resulted in the mapping of less than 45% of the planet's surface.^[161]

The spacecraft made three close approaches to Mercury, the closest of which took it to within 327 km (203 mi) of the surface.^[162] At the first close approach, instruments detected a magnetic field, to the great surprise of planetary geologists—Mercury's rotation was expected to be much too slow to generate a significant dynamo effect. The second close approach was primarily used for imaging, but at the third approach, extensive magnetic data were obtained. The data revealed that the planet's magnetic field is much like Earth's, which deflects the solar wind around the planet. For many years after the Mariner 10 encounters, the origin of Mercury's magnetic field remained the subject of several competing theories.^[163][164]

On March 24, 1975, just eight days after its final close approach, Mariner 10 ran out of fuel. Because its orbit could no longer be accurately controlled, mission controllers instructed the probe to shut down.^[165] Mariner 10 is thought to be still orbiting the Sun, passing close to Mercury every few months.^[166]

MESSENGER

Estimated details of the impact of MESSENGER on 30 April 2015

A second NASA mission to Mercury, named MESSENGER (MErcury Surface, Space ENvironment, GEochemistry, and Ranging), was launched on 3 August 2004. It made a fly-by of Earth in August 2005, and of Venus in October 2006 and June 2007 to place it onto the correct trajectory to reach an orbit around Mercury.^[167] A first fly-by of Mercury occurred on January 14, 2008, a second on October 6, 2008,^[168] and a third on September 29, 2009.^[169] Most of the hemisphere not imaged by Mariner 10 was mapped during these fly-bys. The probe successfully entered an elliptical orbit around the planet on March 18, 2011. The first orbital image of Mercury was obtained on March 29, 2011. The probe finished a one-year mapping mission,^[168] and then entered a one-year extended mission into 2013. In addition to continued observations and mapping of Mercury, MESSENGER observed the 2012 solar maximum.^[170]

The mission was designed to clear up six key issues: Mercury's high density, its geological history, the nature of its magnetic field, the structure of its core, whether it has ice at its poles, and where its tenuous atmosphere comes from. To this end, the probe carried imaging devices that gathered much-higher-resolution images of much more of Mercury than Mariner 10, assorted spectrometers to determine abundances of elements in the crust, and magnetometers and devices to measure velocities of charged particles. Measurements of changes in the probe's orbital velocity were expected to be used to infer details of the planet's interior structure.^[171] MESSENGER's final maneuver was on April 24, 2015, and it crashed into Mercury's surface on April 30, 2015.^{[172][173][174]} The spacecraft's impact with Mercury occurred near 3:26 PM EDT on April 30, 2015, leaving a crater estimated to be 16 m (52 ft) in diameter.^[175]

First (29 March 2011) and last (30 April 2015) images of

Mercury by MESSENGER

BepiColombo

The European Space Agency is planning a joint mission with Japan called BepiColombo, which will orbit Mercury with two probes: one to map the planet and the other to study its magnetosphere.^[176] Once launched in 2018, BepiColombo is expected to reach Mercury in 2025.^[177] It will release a magnetometer probe into an elliptical orbit, then chemical rockets will fire to deposit the mapper probe into a circular orbit. Both probes will operate for one terrestrial year.^[176] The mapper probe will carry an array of spectrometers similar to those on MESSENGER, and will study the planet at many different wavelengths including infrared, ultraviolet, X-ray and gamma ray.^[178]

Inertial frame of reference

From Wikipedia, the free encyclopedia

An inertial frame of reference in classical physics and special relativity is a frame of reference in which a body with zero net force acting upon it is not accelerating; that is, such a body is at rest or it is moving at a constant speed in a straight line.^[1] In analytical terms, it is a frame of reference that describes time and space homogeneously, isotropically, and in a time-independent manner.^[2] Conceptually, the physics of a system in an inertial frame have no causes external to the system.^[3] An inertial frame of reference may also be called an inertial reference frame, inertial frame, Galilean reference frame, or inertial space.^{[citation needed]}

All inertial frames are in a state of constant, rectilinear motion with respect to one another; an accelerometer moving with any of them would detect zero acceleration. Measurements in one inertial frame can be converted to measurements in another by a simple transformation (the Galilean transformation in Newtonian physics and the Lorentz transformation in special relativity). In general relativity, in any region small enough for the curvature of spacetime and tidal forces^[4] to be negligible, one can find a set of inertial frames that approximately describe that region.^[5][6]

In a non-inertial reference frame in classical physics and special relativity, the physics of a system vary depending on the acceleration of that frame with respect to an inertial frame, and the usual physical forces must be supplemented by fictitious forces.^[7][8] In contrast, systems in non-inertial frames in general relativity don't have external causes, because of the principle of geodesic motion.^[9] In classical physics, for example, a ball dropped towards the ground does not go exactly straight down because the Earth is rotating, which means the frame of reference of an observer on Earth is not inertial. The physics must account for the Coriolis effect—in this case thought of as a force—to predict the horizontal motion. Another example of such a fictitious force associated with rotating reference frames is the centrifugal effect, or centrifugal force.

Introduction

The motion of a body can only be described relative to something else—other bodies, observers, or a set of space-time coordinates. These are called frames of reference. If the coordinates are chosen badly, the laws of motion may be more complex than necessary. For example, suppose a free body that has no external forces acting on it is at rest at some instant. In many coordinate systems, it would begin to move at the next instant, even though there are no forces on it. However, a frame of reference can always be chosen in which it remains stationary. Similarly, if space is not described uniformly or time independently, a coordinate system could describe the simple flight of a free body in space as a complicated zig-zag in its coordinate system. Indeed, an intuitive summary of inertial frames can be given: in an inertial reference frame, the laws of mechanics take their simplest form.^[2]
In an inertial frame, Newton's first law, the law of inertia, is satisfied: Any free motion has a constant magnitude and direction.^[2] Newton's second law for a particle takes the form:

${\displaystyle \mathbf {F} =m\mathbf {a} \ ,}$

with F the net force (a vector), m the mass of a particle and a the acceleration of the particle (also a vector) which would be measured by an observer at rest in the frame. The force F is the vector sum of all "real" forces on the particle, such as electromagnetic, gravitational, nuclear and so forth. In contrast, Newton's second law in a rotating frame of reference, rotating at angular rate Ω about an axis, takes the form:

${\displaystyle \mathbf {F} '=m\mathbf {a} \ ,}$

which looks the same as in an inertial frame, but now the force F′ is the resultant of not only F, but also additional terms (the paragraph following this equation presents the main points without detailed mathematics):

${\displaystyle \mathbf {F} '=\mathbf {F} -2m\mathbf {\Omega } \times \mathbf {v} _{B}-m\mathbf {\Omega } \times (\mathbf {\Omega } \times \mathbf {x} _{B})-m{\frac {d\mathbf {\Omega } }{dt}}\times \mathbf {x} _{B}\ ,}$

where the angular rotation of the frame is expressed by the vector Ω pointing in the direction of the axis of rotation, and with magnitude equal to the angular rate of rotation Ω, symbol × denotes the vector cross product, vector x_B locates the body and vector v_B is the velocity of the body according to a rotating observer (different from the velocity seen by the inertial observer).

The extra terms in the force F′ are the "fictitious" forces for this frame, whose causes are external to the system in the frame. The first extra term is the Coriolis force, the second the centrifugal force, and the third the Euler force. These terms all have these properties: they vanish when Ω = 0; that is, they are zero for an inertial frame (which, of course, does not rotate); they take on a different magnitude and direction in every rotating frame, depending upon its particular value of Ω; they are ubiquitous in the rotating frame (affect every particle, regardless of circumstance); and they have no apparent source in identifiable physical sources, in particular, matter. Also, fictitious forces do not drop off with distance (unlike, for example, nuclear forces or electrical forces). For example, the centrifugal force that appears to emanate from the axis of rotation in a rotating frame increases with distance from the axis.

All observers agree on the real forces, F; only non-inertial observers need fictitious forces. The laws of physics in the inertial frame are simpler because unnecessary forces are not present.

In Newton's time the fixed stars were invoked as a reference frame, supposedly at rest relative to absolute space. In reference frames that were either at rest with respect to the fixed stars or in uniform translation relative to these stars, Newton's laws of motion were supposed to hold. In contrast, in frames accelerating with respect to the fixed stars, an important case being frames rotating relative to the fixed stars, the laws of motion did not hold in their simplest form, but had to be supplemented by the addition of fictitious forces, for example, the Coriolis force and the centrifugal force. Two interesting experiments were devised by Newton to demonstrate how these forces could be discovered, thereby revealing to an observer that they were not in an inertial frame: the example of the tension in the cord linking two spheres rotating about their center of gravity, and the example of the curvature of the surface of water in a rotating bucket. In both cases, application of Newton's second law would not work for the rotating observer without invoking centrifugal and Coriolis forces to account for their observations (tension in the case of the spheres; parabolic water surface in the case of the rotating bucket).

As we now know, the fixed stars are not fixed. Those that reside in the Milky Way turn with the galaxy, exhibiting proper motions. Those that are outside our galaxy (such as nebulae once mistaken to be stars) participate in their own motion as well, partly due to expansion of the universe, and partly due to peculiar velocities.^[10] The Andromeda galaxy is on collision course with the Milky Way at a speed of 117 km/s.^[11] The concept of inertial frames of reference is no longer tied to either the fixed stars or to absolute space. Rather, the identification of an inertial frame is based upon the simplicity of the laws of physics in the frame. In particular, the absence of fictitious forces is their identifying property.^[12]

In practice, although not a requirement, using a frame of reference based upon the fixed stars as though it were an inertial frame of reference introduces very little discrepancy. For example, the centrifugal acceleration of the Earth because of its rotation about the Sun is about thirty million times greater than that of the Sun about the galactic center.^[13]

To illustrate further, consider the question: "Does our Universe rotate?" To answer, we might attempt to explain the shape of the Milky Way galaxy using the laws of physics,^[14] although other observations might be more definitive, that is, provide larger discrepancies or less measurement uncertainty, like the anisotropy of the microwave background radiation or Big Bang nucleosynthesis.^[15][16] The flatness of the Milky Way depends on its rate of rotation in an inertial frame of reference. If we attribute its apparent rate of rotation entirely to rotation in an inertial frame, a different "flatness" is predicted than if we suppose part of this rotation actually is due to rotation of the universe and should not be included in the rotation of the galaxy itself. Based upon the laws of physics, a model is set up in which one parameter is the rate of rotation of the Universe. If the laws of physics agree more accurately with observations in a model with rotation than without it, we are inclined to select the best-fit value for rotation, subject to all other pertinent experimental observations. If no value of the rotation parameter is successful and theory is not within observational error, a modification of physical law is considered, for example, dark matter is invoked to explain the galactic rotation curve. So far, observations show any rotation of the universe is very slow, no faster than once every 60·10¹² years (10⁻¹³ rad/yr),^[17] and debate persists over whether there is any rotation. However, if rotation were found, interpretation of observations in a frame tied to the universe would have to be corrected for the fictitious forces inherent in such rotation in classical physics and special relativity, or interpreted as the curvature of spacetime and the motion of matter along the geodesics in general relativity.

When quantum effects are important, there are additional conceptual complications that arise in quantum reference frames.

Background

A set of frames where the laws of physics are simple

According to the first postulate of special relativity, all physical laws take their simplest form in an inertial frame, and there exist multiple inertial frames interrelated by uniform translation: ^[18]

Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K' moving in uniform translation relatively to K.

— Albert Einstein: The foundation of the general theory of relativity, Section A, §1

This simplicity manifests in that inertial frames have self-contained physics without the need for external causes, while physics in non-inertial frames have external causes.^[3] The principle of simplicity can be used within Newtonian physics as well as in special relativity; see Nagel^[19] and also Blagojević.^[20]

The laws of Newtonian mechanics do not always hold in their simplest form...If, for instance, an observer is placed on a disc rotating relative to the earth, he/she will sense a 'force' pushing him/her toward the periphery of the disc, which is not caused by any interaction with other bodies. Here, the acceleration is not the consequence of the usual force, but of the so-called inertial force. Newton's laws hold in their simplest form only in a family of reference frames, called inertial frames. This fact represents the essence of the Galilean principle of relativity:
   The laws of mechanics have the same form in all inertial frames.

— Milutin Blagojević: Gravitation and Gauge Symmetries, p. 4

In practical terms, the equivalence of inertial reference frames means that scientists within a box moving uniformly cannot determine their absolute velocity by any experiment. Otherwise, the differences would set up an absolute standard reference frame.^[21][22] According to this definition, supplemented with the constancy of the speed of light, inertial frames of reference transform among themselves according to the Poincaré group of symmetry transformations, of which the Lorentz transformations are a subgroup.^[23] In Newtonian mechanics, which can be viewed as a limiting case of special relativity in which the speed of light is infinite, inertial frames of reference are related by the Galilean group of symmetries.

Absolute space

Newton posited an absolute space considered well approximated by a frame of reference stationary relative to the fixed stars. An inertial frame was then one in uniform translation relative to absolute space. However, some scientists (called "relativists" by Mach^[24]), even at the time of Newton, felt that absolute space was a defect of the formulation, and should be replaced.

Indeed, the expression inertial frame of reference (German: Inertialsystem) was coined by Ludwig Lange in 1885, to replace Newton's definitions of "absolute space and time" by a more operational definition.^[25][26] As translated by Iro, Lange proposed the following definition:^[27]

A reference frame in which a mass point thrown from the same point in three different (non co-planar) directions follows rectilinear paths each time it is thrown, is called an inertial frame.

A discussion of Lange's proposal can be found in Mach.^[24]

The inadequacy of the notion of "absolute space" in Newtonian mechanics is spelled out by Blagojević:^[28]

• The existence of absolute space contradicts the internal logic of classical mechanics since, according to Galilean principle of relativity, none of the inertial frames can be singled out.
• Absolute space does not explain inertial forces since they are related to acceleration with respect to any one of the inertial frames.
• Absolute space acts on physical objects by inducing their resistance to acceleration but it cannot be acted upon.

— Milutin Blagojević: Gravitation and Gauge Symmetries, p. 5

The utility of operational definitions was carried much further in the special theory of relativity.^[29] Some historical background including Lange's definition is provided by DiSalle, who says in summary:^[30]

The original question, "relative to what frame of reference do the laws of motion hold?" is revealed to be wrongly posed. For the laws of motion essentially determine a class of reference frames, and (in principle) a procedure for constructing them.

— Robert DiSalle Space and Time: Inertial Frames

Newton's inertial frame of reference

Figure 1: Two frames of reference moving with relative velocity ${\displaystyle {\stackrel {\vec {v}}{}}}$. Frame S' has an arbitrary but fixed rotation with respect to frame S. They are both inertial frames provided a body not subject to forces appears to move in a straight line. If that motion is seen in one frame, it will also appear that way in the other.

Within the realm of Newtonian mechanics, an inertial frame of reference, or inertial reference frame, is one in which Newton's first law of motion is valid.^[31] However, the principle of special relativity generalizes the notion of inertial frame to include all physical laws, not simply Newton's first law.

Newton viewed the first law as valid in any reference frame that is in uniform motion relative to the fixed stars;^[32] that is, neither rotating nor accelerating relative to the stars.^[33] Today the notion of "absolute space" is abandoned, and an inertial frame in the field of classical mechanics is defined as:^[34][35]

An inertial frame of reference is one in which the motion of a particle not subject to forces is in a straight line at constant speed.

Hence, with respect to an inertial frame, an object or body accelerates only when a physical force is applied, and (following Newton's first law of motion), in the absence of a net force, a body at rest will remain at rest and a body in motion will continue to move uniformly—that is, in a straight line and at constant speed. Newtonian inertial frames transform among each other according to the Galilean group of symmetries.

If this rule is interpreted as saying that straight-line motion is an indication of zero net force, the rule does not identify inertial reference frames because straight-line motion can be observed in a variety of frames. If the rule is interpreted as defining an inertial frame, then we have to be able to determine when zero net force is applied. The problem was summarized by Einstein:^[36]

The weakness of the principle of inertia lies in this, that it involves an argument in a circle: a mass moves without acceleration if it is sufficiently far from other bodies; we know that it is sufficiently far from other bodies only by the fact that it moves without acceleration.

— Albert Einstein: The Meaning of Relativity, p. 58

There are several approaches to this issue. One approach is to argue that all real forces drop off with distance from their sources in a known manner, so we have only to be sure that a body is far enough away from all sources to ensure that no force is present.^[37] A possible issue with this approach is the historically long-lived view that the distant universe might affect matters (Mach's principle). Another approach is to identify all real sources for real forces and account for them. A possible issue with this approach is that we might miss something, or account inappropriately for their influence, perhaps, again, due to Mach's principle and an incomplete understanding of the universe. A third approach is to look at the way the forces transform when we shift reference frames. Fictitious forces, those that arise due to the acceleration of a frame, disappear in inertial frames, and have complicated rules of transformation in general cases. On the basis of universality of physical law and the request for frames where the laws are most simply expressed, inertial frames are distinguished by the absence of such fictitious forces.

Newton enunciated a principle of relativity himself in one of his corollaries to the laws of motion:^[38][39]

The motions of bodies included in a given space are the same among themselves, whether that space is at rest or moves uniformly forward in a straight line.

— Isaac Newton: Principia, Corollary V, p. 88 in Andrew Motte translation

This principle differs from the special principle in two ways: first, it is restricted to mechanics, and second, it makes no mention of simplicity. It shares with the special principle the invariance of the form of the description among mutually translating reference frames.^[40] The role of fictitious forces in classifying reference frames is pursued further below.

Separating non-inertial from inertial reference frames

Theory

Figure 2: Two spheres tied with a string and rotating at an angular rate ω. Because of the rotation, the string tying the spheres together is under tension.

Figure 3: Exploded view of rotating spheres in an inertial frame of reference showing the centripetal forces on the spheres provided by the tension in the tying string.

Inertial and non-inertial reference frames can be distinguished by the absence or presence of fictitious forces, as explained shortly.^[7][8]

The effect of this being in the noninertial frame is to require the observer to introduce a fictitious force into his calculations….

— Sidney Borowitz and Lawrence A Bornstein in A Contemporary View of Elementary Physics, p. 138

The presence of fictitious forces indicates the physical laws are not the simplest laws available so, in terms of the special principle of relativity, a frame where fictitious forces are present is not an inertial frame:^[41]

The equations of motion in a non-inertial system differ from the equations in an inertial system by additional terms called inertial forces. This allows us to detect experimentally the non-inertial nature of a system.

— V. I. Arnol'd: Mathematical Methods of Classical Mechanics Second Edition, p. 129

Bodies in non-inertial reference frames are subject to so-called fictitious forces (pseudo-forces); that is, forces that result from the acceleration of the reference frame itself and not from any physical force acting on the body. Examples of fictitious forces are the centrifugal force and the Coriolis force in rotating reference frames.

How then, are "fictitious" forces to be separated from "real" forces? It is hard to apply the Newtonian definition of an inertial frame without this separation. For example, consider a stationary object in an inertial frame. Being at rest, no net force is applied. But in a frame rotating about a fixed axis, the object appears to move in a circle, and is subject to centripetal force (which is made up of the Coriolis force and the centrifugal force). How can we decide that the rotating frame is a non-inertial frame? There are two approaches to this resolution: one approach is to look for the origin of the fictitious forces (the Coriolis force and the centrifugal force). We will find there are no sources for these forces, no associated force carriers, no originating bodies.^[42] A second approach is to look at a variety of frames of reference. For any inertial frame, the Coriolis force and the centrifugal force disappear, so application of the principle of special relativity would identify these frames where the forces disappear as sharing the same and the simplest physical laws, and hence rule that the rotating frame is not an inertial frame.

Newton examined this problem himself using rotating spheres, as shown in Figure 2 and Figure 3. He pointed out that if the spheres are not rotating, the tension in the tying string is measured as zero in every frame of reference.^[43] If the spheres only appear to rotate (that is, we are watching stationary spheres from a rotating frame), the zero tension in the string is accounted for by observing that the centripetal force is supplied by the centrifugal and Coriolis forces in combination, so no tension is needed. If the spheres really are rotating, the tension observed is exactly the centripetal force required by the circular motion. Thus, measurement of the tension in the string identifies the inertial frame: it is the one where the tension in the string provides exactly the centripetal force demanded by the motion as it is observed in that frame, and not a different value. That is, the inertial frame is the one where the fictitious forces vanish.

So much for fictitious forces due to rotation. However, for linear acceleration, Newton expressed the idea of undetectability of straight-line accelerations held in common:^[39]

If bodies, any how moved among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will continue to move among themselves, after the same manner as if they had been urged by no such forces.

— Isaac Newton: Principia Corollary VI, p. 89, in Andrew Motte translation

This principle generalizes the notion of an inertial frame. For example, an observer confined in a free-falling lift will assert that he himself is a valid inertial frame, even if he is accelerating under gravity, so long as he has no knowledge about anything outside the lift. So, strictly speaking, inertial frame is a relative concept. With this in mind, we can define inertial frames collectively as a set of frames which are stationary or moving at constant velocity with respect to each other, so that a single inertial frame is defined as an element of this set.

For these ideas to apply, everything observed in the frame has to be subject to a base-line, common acceleration shared by the frame itself. That situation would apply, for example, to the elevator example, where all objects are subject to the same gravitational acceleration, and the elevator itself accelerates at the same rate.

Applications

Inertial navigation systems used a cluster of gyroscopes and accelerometers to determine accelerations relative to inertial space. After a gyroscope is spun up in a particular orientation in inertial space, the law of conservation of angular momentum requires that it retain that orientation as long as no external forces are applied to it.^[44]:59 Three orthogonal gyroscopes establish an inertial reference frame, and the accelerators measure acceleration relative to that frame. The accelerations, along with a clock, can then be used to calculate the change in position. Thus, inertial navigation is a form of dead reckoning that requires no external input, and therefore cannot be jammed by any external or internal signal source.^[45]

A gyrocompass, employed for navigation of seagoing vessels, finds the geometric north. It does so, not by sensing the Earth's magnetic field, but by using inertial space as its reference. The outer casing of the gyrocompass device is held in such a way that it remains aligned with the local plumb line. When the gyroscope wheel inside the gyrocompass device is spun up, the way the gyroscope wheel is suspended causes the gyroscope wheel to gradually align its spinning axis with the Earth's axis. Alignment with the Earth's axis is the only direction for which the gyroscope's spinning axis can be stationary with respect to the Earth and not be required to change direction with respect to inertial space. After being spun up, a gyrocompass can reach the direction of alignment with the Earth's axis in as little as a quarter of an hour.^[46]

Newtonian mechanics

Classical theories that use the Galilean transformation postulate the equivalence of all inertial reference frames. Some theories may even postulate the existence of a privileged frame which provides absolute space and absolute time. The Galilean transformation transforms coordinates from one inertial reference frame, ${\displaystyle \mathbf {s} }$, to another, ${\displaystyle \mathbf {s} ^{\prime }}$, by simple addition or subtraction of corrdinates:

${\displaystyle \mathbf {r} ^{\prime }=\mathbf {r} -\mathbf {r} _{0}-\mathbf {v} t}$

${\displaystyle t^{\prime }=t-t_{0}}$

where r₀ and t₀ represent shifts in the origin of space and time, and v is the relative velocity of the two inertial reference frames. Under Galilean transformations, the time t₂ − t₁ between two events is the same for all reference frames and the distance between two simultaneous events (or, equivalently, the length of any object, |r₂ − r₁|) is also the same.

Special relativity

Einstein's theory of special relativity, like Newtonian mechanics, postulates the equivalence of all inertial reference frames. However, because special relativity postulates that the speed of light in free space is invariant, the transformation between inertial frames is the Lorentz transformation, not the Galilean transformation which is used in Newtonian mechanics. The invariance of the speed of light leads to counter-intuitive phenomena, such as time dilation and length contraction, and the relativity of simultaneity, which have been extensively verified experimentally.^[47] The Lorentz transformation reduces to the Galilean transformation as the speed of light approaches infinity or as the relative velocity between frames approaches zero.^[48]

General relativity

General relativity is based upon the principle of equivalence:^[49][50]

There is no experiment observers can perform to distinguish whether an acceleration arises because of a gravitational force or because their reference frame is accelerating.

— Douglas C. Giancoli, Physics for Scientists and Engineers with Modern Physics, p. 155.

This idea was introduced in Einstein's 1907 article "Principle of Relativity and Gravitation" and later developed in 1911.^[51] Support for this principle is found in the Eötvös experiment, which determines whether the ratio of inertial to gravitational mass is the same for all bodies, regardless of size or composition. To date no difference has been found to a few parts in 10¹¹.^[52] For some discussion of the subtleties of the Eötvös experiment, such as the local mass distribution around the experimental site (including a quip about the mass of Eötvös himself), see Franklin.^[53]

Einstein’s general theory modifies the distinction between nominally "inertial" and "noninertial" effects by replacing special relativity's "flat" Minkowski Space with a metric that produces non-zero curvature. In general relativity, the principle of inertia is replaced with the principle of geodesic motion, whereby objects move in a way dictated by the curvature of spacetime. As a consequence of this curvature, it is not a given in general relativity that inertial objects moving at a particular rate with respect to each other will continue to do so. This phenomenon of geodesic deviation means that inertial frames of reference do not exist globally as they do in Newtonian mechanics and special relativity.

However, the general theory reduces to the special theory over sufficiently small regions of spacetime, where curvature effects become less important and the earlier inertial frame arguments can come back into play.^[54][55] Consequently, modern special relativity is now sometimes described as only a "local theory".^[56] "Local" can encompass, for example, the entire Milky Way galaxy: The astronomer Karl Schwarzschild observed the motion of pairs of stars orbiting each other. He found that the two orbits of the stars of such a system lie in a plane, and the perihelion of the orbits of the two stars remains pointing in the same direction with respect to the solar system. Schwarzschild pointed out that that was invariably seen: the direction of the angular momentum of all observed double star systems remains fixed with respect to the direction of the angular momentum of the Solar System. These observations allowed him to conclude that inertial frames inside the galaxy do not rotate with respect to one another, and that the space of the Milky Way is approximately Galilean or Minkowskian.^[57]