Solar wind

From Wikipedia, the free encyclopedia

The heliospheric current sheet results from the influence of the Sun's rotating magnetic field on the plasma in the solar wind.

The solar wind is a stream of plasma released from the upper atmosphere of the Sun. It consists of mostly electrons and protons with energies usually between 1.5 and 10 keV. The stream of particles varies in density, temperature, and speed over time and over solar longitude. These particles can escape the Sun's gravity because of their high energy, from the high temperature of the corona and magnetic, electrical and electromagnetic phenomena in it.

The solar wind flows outward supersonically to great distances, filling a region known as the heliosphere, an enormous bubble-like volume surrounded by the interstellar medium. Other related phenomena include the aurora (northern and southern lights), the plasma tails of comets that always point away from the Sun, and geomagnetic storms that can change the direction of magnetic field lines and create strong currents in power grids on Earth.

History

The existence of a continuous stream of particles flowing outward from the Sun was first suggested by British astronomer Richard C. Carrington. In 1859, Carrington and Richard Hodgson independently made the first observation of what would later be called a solar flare. This is a sudden outburst of energy from the Sun's atmosphere. On the following day, a geomagnetic storm was observed, and Carrington suspected that there might be a connection. George FitzGerald later suggested that matter was being regularly accelerated away from the Sun and was reaching the Earth after several days.^[1]

Laboratory simulation of the magnetosphere's influence on the Solar Wind; these auroral-like Birkeland currents were created in a terrella, a magnetised anode globe in an evacuated chamber.

In 1910 British astrophysicist Arthur Eddington essentially suggested the existence of the solar wind, without naming it, in a footnote to an article on Comet Morehouse.^[2] The idea never fully caught on even though Eddington had also made a similar suggestion at a Royal Institution address the previous year. In the latter case, he postulated that the ejected material consisted of electrons while in his study of Comet Morehouse he supposed them to be ions.^[2] The first person to suggest that they were both was Norwegian physicist Kristian Birkeland. His geomagnetic surveys showed that auroral activity was nearly uninterrupted. As these displays and other geomagnetic activity were being produced by particles from the Sun, he concluded that the Earth was being continually bombarded by "rays of electric corpuscles emitted by the Sun".^[1] In 1916, Birkeland proposed that, "From a physical point of view it is most probable that solar rays are neither exclusively negative nor positive rays, but of both kinds". In other words, the solar wind consists of both negative electrons and positive ions.^[3] Three years later in 1919, Frederick Lindemann also suggested that particles of both polarities, protons as well as electrons, come from the Sun.^[4]

Around the 1930s, scientists had determined that the temperature of the solar corona must be a million degrees Celsius because of the way it stood out into space (as seen during total eclipses). Later spectroscopic work confirmed this extraordinary temperature. In the mid-1950s the British mathematician Sydney Chapman calculated the properties of a gas at such a temperature and determined it was such a superb conductor of heat that it must extend way out into space, beyond the orbit of Earth. Also in the 1950s, a German scientist named Ludwig Biermann became interested in the fact that no matter whether a comet is headed towards or away from the Sun, its tail always points away from the Sun. Biermann postulated that this happens because the Sun emits a steady stream of particles that pushes the comet's tail away.^[5] Wilfried Schröder claims in his book, Who First Discovered the Solar Wind?, that the German astronomer Paul Ahnert was the first to relate solar wind to comet tail direction based on observations of the comet Whipple-Fedke (1942g).^[6]

Eugene Parker realised that the heat flowing from the Sun in Chapman's model and the comet tail blowing away from the Sun in Biermann's hypothesis had to be the result of the same phenomenon, which he termed the "solar wind".^[7][8] Parker showed in 1958 that even though the Sun's corona is strongly attracted by solar gravity, it is such a good conductor of heat that it is still very hot at large distances. Since gravity weakens as distance from the Sun increases, the outer coronal atmosphere escapes supersonically into interstellar space. Furthermore, Parker was the first person to notice that the weakening effect of the gravity has the same effect on hydrodynamic flow as a de Laval nozzle: it incites a transition from subsonic to supersonic flow.^[9]

Opposition to Parker's hypothesis on the solar wind was strong. The paper he submitted to the Astrophysical Journal in 1958 was rejected by two reviewers. It was saved by the editor Subrahmanyan Chandrasekhar (who later received the 1983 Nobel Prize in physics).

In January 1959, the Soviet satellite Luna 1 first directly observed the solar wind and measured its strength.^[10][11][12] They were detected by hemispherical ion traps. The discovery, made by Konstantin Gringauz, was verified by Luna 2, Luna 3 and by the more distant measurements of Venera 1. Three years later its measurement was performed by Americans (Neugebauer and collaborators) using the Mariner 2 spacecraft.^[13]

In the late 1990s the Ultraviolet Coronal Spectrometer (UVCS) instrument on board the SOHO spacecraft observed the acceleration region of the fast solar wind emanating from the poles of the Sun, and found that the wind accelerates much faster than can be accounted for by thermodynamic expansion alone. Parker's model predicted that the wind should make the transition to supersonic flow at an altitude of about 4 solar radii from the photosphere; but the transition (or "sonic point") now appears to be much lower, perhaps only 1 solar radius above the photosphere, suggesting that some additional mechanism accelerates the solar wind away from the Sun. The acceleration of the fast wind is still not understood and cannot be fully explained by Parker's theory. The gravitational and electromagnetic explanation for this acceleration is, however, detailed in an earlier paper by 1970 Nobel laureate for Physics, Hannes Alfvén.^[14][15]

The first numerical simulation of the solar wind in the solar corona including closed and open field lines was performed by Pneuman and Kopp in 1971. The magnetohydrodynamics equations in steady state were solved iteratively starting with an initial dipolar configuration.^[16]

In 1990, the Ulysses probe was launched to study the solar wind from high solar latitudes. All prior observations had been made at or near the Solar System's ecliptic plane.^[17]

Emission

While early models of the solar wind used primarily thermal energy to accelerate the material, by the 1960s it was clear that thermal acceleration alone cannot account for the high speed of solar wind. An additional unknown acceleration mechanism is required, and likely relates to magnetic fields in the solar atmosphere.

The Sun's corona, or extended outer layer, is a region of plasma that is heated to over a million degrees Celsius. As a result of thermal collisions, the particles within the inner corona have a range and distribution of speeds described by a Maxwellian distribution. The mean velocity of these particles is about 145 km/s, which is well below the solar escape velocity of 618 km/s. However, a few of the particles achieve energies sufficient to reach the terminal velocity of 400 km/s, which allows them to feed the solar wind. At the same temperature, electrons, due to their much smaller mass, reach escape velocity and build up an electric field that further accelerates ions - charged atoms - away from the Sun.^[18]

The total number of particles carried away from the Sun by the solar wind is about 1.3×10³⁶ per second.^[19] Thus, the total mass loss each year is about (2–3)×10⁻¹⁴ solar masses,^[20] or about one billion kilograms per second. This is equivalent to losing a mass equal to the Earth every 150 million years.^[21] However, only about 0.01% of the Sun's total mass has been lost through the solar wind.^[22] Other stars have much stronger stellar winds that result in significantly higher mass loss rates.

Components and speed

The solar wind is divided into two components, respectively termed the slow solar wind and the fast solar wind. The slow solar wind has a velocity of about 400 km/s, a temperature of 1.4–1.6×10⁶ K and a composition that is a close match to the corona. By contrast, the fast solar wind has a typical velocity of 750 km/s, a temperature of 8×10⁵ K and it nearly matches the composition of the Sun's photosphere.^[23] The slow solar wind is twice as dense and more variable in intensity than the fast solar wind. The slow wind also has a more complex structure, with turbulent regions and large-scale structures.^[19][24]

The slow solar wind appears to originate from a region around the Sun's equatorial belt that is known as the "streamer belt". Coronal streamers extend outward from this region, carrying plasma from the interior along closed magnetic loops.^[25][26] Observations of the Sun between 1996 and 2001 showed that emission of the slow solar wind occurred between latitudes of 30–35° around the equator during the solar minimum (the period of lowest solar activity), then expanded toward the poles as the minimum waned. By the time of the solar maximum, the poles were also emitting a slow solar wind.^[27]

The fast solar wind is thought to originate from coronal holes, which are funnel-like regions of open field lines in the Sun's magnetic field.^[28] Such open lines are particularly prevalent around the Sun's magnetic poles. The plasma source is small magnetic fields created by convection cells in the solar atmosphere. These fields confine the plasma and transport it into the narrow necks of the coronal funnels, which are located only 20,000 kilometers above the photosphere. The plasma is released into the funnel when these magnetic field lines reconnect.^[29]

Solar wind pressure

The wind exerts a pressure at 1 AU typically in the range of 1–6 nPa (1–6×10⁻⁹ N/m²), although it can readily vary outside that range.

The dynamic pressure is a function of wind speed and density. The formula is

P = 1.6726×10⁻⁶ * n * V²

where pressure P is in nPa (nano Pascals), n is the density in particles/cm³ and V is the speed in km/s of the solar wind.^[30]

Coronal mass ejection

Both the fast and slow solar wind can be interrupted by large, fast-moving bursts of plasma called interplanetary coronal mass ejections, or ICMEs. ICMEs are the interplanetary manifestation of solar coronal mass ejections, which are caused by release of magnetic energy at the Sun. CMEs are often called "solar storms" or "space storms" in the popular media. They are sometimes, but not always, associated with solar flares, which are another manifestation of magnetic energy release at the Sun. ICMEs cause shock waves in the thin plasma of the heliosphere, launching electromagnetic waves and accelerating particles (mostly protons and electrons) to form showers of ionizing radiation that precede the CME.
When a CME impacts the Earth's magnetosphere, it temporarily deforms the Earth's magnetic field, changing the direction of compass needles and inducing large electrical ground currents in Earth itself; this is called a geomagnetic storm and it is a global phenomenon. CME impacts can induce magnetic reconnection in Earth's magnetotail (the midnight side of the magnetosphere); this launches protons and electrons downward toward Earth's atmosphere, where they form the aurora.

ICMEs are not the only cause of space weather. Different patches on the Sun are known to give rise to slightly different speeds and densities of wind depending on local conditions. In isolation, each of these different wind streams would form a spiral with a slightly different angle, with fast-moving streams moving out more directly and slow-moving streams wrapping more around the Sun. Fast moving streams tend to overtake slower streams that originate westward of them on the Sun, forming turbulent co-rotating interaction regions that give rise to wave motions and accelerated particles, and that affect Earth's magnetosphere in the same way as, but more gently than, CMEs.

Effect on the Solar System

Over the lifetime of the Sun, the interaction of the Sun's surface layers with the escaping solar wind has significantly decreased its surface rotation rate.^[31] The wind is considered responsible for the tails of comets, along with the Sun's radiation.^[32] The solar wind contributes to fluctuations in celestial radio waves observed on the Earth, through an effect called interplanetary scintillation.^[33]

Magnetospheres

Schematic of Earth's magnetosphere. The solar wind flows from left to right.

As the solar wind approaches a planet that has a well-developed magnetic field (such as Earth, Jupiter and Saturn), the particles are deflected by the Lorentz force. This region, known as the magnetosphere, causes the particles to travel around the planet rather than bombarding the atmosphere or surface. The magnetosphere is roughly shaped like a hemisphere on the side facing the Sun, then is drawn out in a long wake on the opposite side. The boundary of this region is called the magnetopause, and some of the particles are able to penetrate the magnetosphere through this region by partial reconnection of the magnetic field lines.^[18]

Noon meridian section of magnetosphere.

The solar wind is responsible for the overall shape of Earth's magnetosphere, and fluctuations in its speed, density, direction, and entrained magnetic field strongly affect Earth's local space environment. For example, the levels of ionizing radiation and radio interference can vary by factors of hundreds to thousands; and the shape and location of the magnetopause and bow shock wave upstream of it can change by several Earth radii, exposing geosynchronous satellites to the direct solar wind. These phenomena are collectively called space weather.

From the European Space Agency’s Cluster mission, a new study has taken place that proposes it easier for the solar wind to infiltrate the magnetosphere than it had been previously believed. A group of scientists directly observed the existence of certain waves in the solar wind that were not expected. A recent publishing in the Journal of Geophysical Research shows that these waves enable incoming charged particles of solar wind to breach the magnetopause. This suggests that the magnetic bubble forms more as a filter than a continuous barrier. This latest discovery occurred through the distinctive arrangement of the four identical Cluster spacecraft, which fly in a strictly controlled configuration through near-Earth space. As they sweep from the magnetosphere into interplanetary space and back again, the fleet provides exceptional three-dimensional insights on the processes that connect the sun to Earth.

The team of scientists was able to characterize how variances in IMF formation largely influenced by Kelvin-Helmholtz waves (which occur upon the interface of two fluids) as a result of differences in thickness and numerous other characteristics of the boundary layer. Experts believe that this was the first occasion that the appearance "of Kelvin-Helmholtz waves at the magnetopause has be displayed at high latitude dawnward orientation of the IMF. These waves of being seen in unforeseeable places under solar wind conditions that were formerly believed to be undesired for their generation.The discoveries found through this mission are of great importance by ESA project scientists because it shows how Earth’s magnetosphere can be penetrated by solar particles under specific interplanetary magnetic field circumstances. The findings are also relevant to studies of magnetospheric progressions around other planetary bodies in the solar system. This study suggests that Kelvin-Helmholtz waves can be a somewhat common, and possibly constant, instrument for the entrance of solar wind into terrestrial magnetospheres under various IMF orientations.^[34]

Atmospheres

The solar wind affects the other incoming cosmic rays interacting with the atmosphere of planets. Moreover, planets with a weak or non-existent magnetosphere are subject to atmospheric stripping by the solar wind.

Venus, the nearest and most similar planet to Earth in the Solar System, has an atmosphere 100 times denser than our own, with little or no geo-magnetic field. Modern space probes have discovered a comet-like tail that extends to the orbit of the Earth.^[35]

Earth itself is largely protected from the solar wind by its magnetic field, which deflects most of the charged particles; however some of the charged particles are trapped in the Van Allen radiation belt. A smaller number of particles from the solar wind manage to travel, as though on an electromagnetic energy transmission line, to the Earth's upper atmosphere and ionosphere in the auroral zones. The only time the solar wind is observable on the Earth is when it is strong enough to produce phenomena such as the aurora and geomagnetic storms. Bright auroras strongly heat the ionosphere, causing its plasma to expand into the magnetosphere, increasing the size of the plasma geosphere, and causing escape of atmospheric matter into the solar wind. Geomagnetic storms result when the pressure of plasmas contained inside the magnetosphere is sufficiently large to inflate and thereby distort the geomagnetic field.

Mars is larger than Mercury and four times farther from the Sun, and yet even here it is thought that the solar wind has stripped away up to a third of its original atmosphere, leaving a layer 1/100th as dense as the Earth's. It is believed the mechanism for this atmospheric stripping is gas being caught in bubbles of magnetic field, which are ripped off by solar winds.^[36]

Planetary surfaces

Mercury, the nearest planet to the Sun, bears the full brunt of the solar wind, and its atmosphere is vestigial and transient, its surface bathed in radiation.

Mercury has an intrinsic magnetic field, so under normal solar wind conditions, the solar wind cannot penetrate the magnetosphere created around the planet, and particles only reach the surface in the cusp regions. During coronal mass ejections, however, the magnetopause may get pressed into the surface of the planet, and under these conditions, the solar wind may interact freely with the planetary surface.

The Earth's Moon has no atmosphere or intrinsic magnetic field, and consequently its surface is bombarded with the full solar wind. The Project Apollo missions deployed passive aluminum collectors in an attempt to sample the solar wind, and lunar soil returned for study confirmed that the lunar regolith is enriched in atomic nuclei deposited from the solar wind. There has been speculation that these elements may prove to be useful resources for future lunar colonies.^[37]

Outer limits

The solar wind "blows a bubble" in the interstellar medium (the rarefied hydrogen and helium gas that permeates the galaxy). The point where the solar wind's strength is no longer great enough to push back the interstellar medium is known as the heliopause, and is often considered to be the outer border of the Solar System. The distance to the heliopause is not precisely known, and probably varies widely depending on the current velocity of the solar wind and the local density of the interstellar medium, but it is known to lie far outside the orbit of Pluto. Scientists hope to gain more perspective on the heliopause from data acquired through the Interstellar Boundary Explorer (IBEX) mission, launched in October 2008.

Notable events

• From May 10 to May 12, 1999, NASA's Advanced Composition Explorer (ACE) and WIND spacecraft observed a 98% decrease of solar wind density. This allowed energetic electrons from the Sun to flow to Earth in narrow beams known as "strahl", which caused a highly unusual "polar rain" event, in which a visible aurora appeared over the North Pole. In addition, Earth's magnetosphere increased to between 5 and 6 times its normal size.^[38]
• See also the solar variation entry.
• On 13 December 2010, Voyager 1 determined that the velocity of the solar wind, at its location 10.8 billion miles from Earth has now slowed to zero. "We have gotten to the point where the wind from the Sun, which until now has always had an outward motion, is no longer moving outward; it is only moving sideways so that it can end up going down the tail of the heliosphere, which is a comet-shaped-like object," said Dr. Edward Stone, the Voyager project scientist.^[39][40]

Parallax

From Wikipedia, the free encyclopedia

A simplified illustration of the parallax of an object against a distant background due to a perspective shift. When viewed from "Viewpoint A", the object appears to be in front of the blue square. When the viewpoint is changed to "Viewpoint B", the object appears to have moved in front of the red square.

This animation is an example of parallax. As the viewpoint moves side to side, the objects in the distance appear to move slower than the objects close to the camera.

Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines.^[1][2] The term is derived from the Greek παράλλαξις (parallaxis), meaning "alteration". Nearby objects have a larger parallax than more distant objects when observed from different positions, so parallax can be used to determine distances.

Astronomers use the principle of parallax to measure distances to the closer stars. Here, the term "parallax" is the semi-angle of inclination between two sight-lines to the star, as observed when the Earth is on opposite sides of the sun in its orbit.^[3] These distances form the lowest rung of what is called "the cosmic distance ladder", the first in a succession of methods by which astronomers determine the distances to celestial objects, serving as a basis for other distance measurements in astronomy forming the higher rungs of the ladder.

Parallax also affects optical instruments such as rifle scopes, binoculars, microscopes, and twin-lens reflex cameras that view objects from slightly different angles. Many animals, including humans, have two eyes with overlapping visual fields that use parallax to gain depth perception; this process is known as stereopsis. In computer vision the effect is used for computer stereo vision, and there is a device called a parallax rangefinder that uses it to find range, and in some variations also altitude to a target.

A simple everyday example of parallax can be seen in the dashboard of motor vehicles that use a needle-style speedometer gauge. When viewed from directly in front, the speed may show exactly 60; but when viewed from the passenger seat the needle may appear to show a slightly different speed, due to the angle of viewing.

Visual perception

This image demonstrates parallax. The Sun is visible above the streetlight. The reflection in the water is a virtual image of the Sun and the streetlight. The location of the virtual image is below the surface of the water, offering a different vantage point of the streetlight, which appears to be shifted relative to the more distant Sun.

As the eyes of humans and other animals are in different positions on the head, they present different views simultaneously. This is the basis of stereopsis, the process by which the brain exploits the parallax due to the different views from the eye to gain depth perception and estimate distances to objects.^[4] Animals also use motion parallax, in which the animals (or just the head) move to gain different viewpoints. For example, pigeons (whose eyes do not have overlapping fields of view and thus cannot use stereopsis) bob their heads up and down to see depth.^[5]

The motion parallax is exploited also in wiggle stereoscopy, computer graphics which provide depth cues through viewpoint-shifting animation rather than through binocular vision.

Parallax in astronomy

Parallax is an angle subtended by a line on a point. In the upper diagram the earth in its orbit sweeps the parallax angle subtended on the sun. The lower diagram shows an equal angle swept by the sun in a geostatic model. A similar diagram can be drawn for a star except that the angle of parallax would be minuscule.

Parallax arises due to change in viewpoint occurring due to motion of the observer, of the observed, or of both. What is essential is relative motion. By observing parallax, measuring angles, and using geometry, one can determine distance.

Stellar parallax

Stellar parallax created by the relative motion between the Earth and a star can be seen, in the Copernican model, as arising from the orbit of the Earth around the Sun: the star only appears to move relative to more distant objects in the sky. In a geostatic model, the movement of the star would have to be taken as real with the star oscillating across the sky with respect to the background stars. Stellar parallax is most often measured using annual parallax, defined as the difference in position of a star as seen from the Earth and Sun, i. e. the angle subtended at a star by the mean radius of the Earth's orbit around the Sun. The parsec (3.26 light-years) is defined as the distance for which the annual parallax is 1 arcsecond. Annual parallax is normally measured by observing the position of a star at different times of the year as the Earth moves through its orbit. Measurement of annual parallax was the first reliable way to determine the distances to the closest stars. The first successful measurements of stellar parallax were made by Friedrich Bessel in 1838 for the star 61 Cygni using a heliometer.^[6] Stellar parallax remains the standard for calibrating other measurement methods. Accurate calculations of distance based on stellar parallax require a measurement of the distance from the Earth to the Sun, now based on radar reflection off the surfaces of planets.^[7]

The angles involved in these calculations are very small and thus difficult to measure. The nearest star to the Sun (and thus the star with the largest parallax), Proxima Centauri, has a parallax of 0.7687 ± 0.0003 arcsec.^[8] This angle is approximately that subtended by an object 2 centimeters in diameter located 5.3 kilometers away.

Hubble Space Telescope - Spatial scanning precisely measures distances up to 10,000 light-years away (10 April 2014).^[9]

The fact that stellar parallax was so small that it was unobservable at the time was used as the main scientific argument against heliocentrism during the early modern age. It is clear from Euclid's geometry that the effect would be undetectable if the stars were far enough away, but for various reasons such gigantic distances involved seemed entirely implausible: it was one of Tycho's principal objections to Copernican heliocentrism that in order for it to be compatible with the lack of observable stellar parallax, there would have to be an enormous and unlikely void between the orbit of Saturn and the eighth sphere (the fixed stars).^[10]

In 1989, the satellite Hipparcos was launched primarily for obtaining improved parallaxes and proper motions for over 100,000 nearby stars, increasing the reach of the method tenfold. Even so, Hipparcos is only able to measure parallax angles for stars up to about 1,600 light-years away, a little more than one percent of the diameter of the Milky Way Galaxy. The European Space Agency's Gaia mission, launched in December 2013, will be able to measure parallax angles to an accuracy of 10 microarcseconds, thus mapping nearby stars (and potentially planets) up to a distance of tens of thousands of light-years from Earth.^[11][12] In April 2014, NASA astronomers reported that the Hubble Space Telescope, by using spatial scanning, can now precisely measure distances up to 10,000 light-years away, a ten-fold improvement over earlier measurements.^[9] (related image)

Distance measurement

Stellar parallax motion

Distance measurement by parallax is a special case of the principle of triangulation, which states that one can solve for all the sides and angles in a network of triangles if, in addition to all the angles in the network, the length of at least one side has been measured. Thus, the careful measurement of the length of one baseline can fix the scale of an entire triangulation network. In parallax, the triangle is extremely long and narrow, and by measuring both its shortest side (the motion of the observer) and the small top angle (always less than 1 arcsecond,^[6] leaving the other two close to 90 degrees), the length of the long sides (in practice considered to be equal) can be determined.

Assuming the angle is small (see derivation below), the distance to an object (measured in parsecs) is the reciprocal of the parallax (measured in arcseconds): For example, the distance to Proxima Centauri is 1/0.7687=1.3009 parsecs (4.243 ly).^[8]

Diurnal parallax

Diurnal parallax is a parallax that varies with rotation of the Earth or with difference of location on the Earth. The Moon and to a smaller extent the terrestrial planets or asteroids seen from different viewing positions on the Earth (at one given moment) can appear differently placed against the background of fixed stars.^[13][14]

Lunar parallax

Lunar parallax (often short for lunar horizontal parallax or lunar equatorial horizontal parallax), is a special case of (diurnal) parallax: the Moon, being the nearest celestial body, has by far the largest maximum parallax of any celestial body, it can exceed 1 degree.^[15]

The diagram (above) for stellar parallax can illustrate lunar parallax as well, if the diagram is taken to be scaled right down and slightly modified. Instead of 'near star', read 'Moon', and instead of taking the circle at the bottom of the diagram to represent the size of the Earth's orbit around the Sun, take it to be the size of the Earth's globe, and of a circle around the Earth's surface. Then, the lunar (horizontal) parallax amounts to the difference in angular position, relative to the background of distant stars, of the Moon as seen from two different viewing positions on the Earth: one of the viewing positions is the place from which the Moon can be seen directly overhead at a given moment (that is, viewed along the vertical line in the diagram); and the other viewing position is a place from which the Moon can be seen on the horizon at the same moment (that is, viewed along one of the diagonal lines, from an Earth-surface position corresponding roughly to one of the blue dots on the modified diagram).

The lunar (horizontal) parallax can alternatively be defined as the angle subtended at the distance of the Moon by the radius of the Earth^[16]—equal to angle p in the diagram when scaled-down and modified as mentioned above.

The lunar horizontal parallax at any time depends on the linear distance of the Moon from the Earth. The Earth-Moon linear distance varies continuously as the Moon follows its perturbed and approximately elliptical orbit around the Earth. The range of the variation in linear distance is from about 56 to 63.7 earth-radii, corresponding to horizontal parallax of about a degree of arc, but ranging from about 61.4' to about 54'.^[15] The Astronomical Almanac and similar publications tabulate the lunar horizontal parallax and/or the linear distance of the Moon from the Earth on a periodical e.g. daily basis for the convenience of astronomers (and formerly, of navigators), and the study of the way in which this coordinate varies with time forms part of lunar theory.

Diagram of daily lunar parallax

Parallax can also be used to determine the distance to the Moon.

One way to determine the lunar parallax from one location is by using a lunar eclipse. A full shadow of the Earth on the Moon has an apparent radius of curvature equal to the difference between the apparent radii of the Earth and the Sun as seen from the Moon. This radius can be seen to be equal to 0.75 degree, from which (with the solar apparent radius 0.25 degree) we get an Earth apparent radius of 1 degree. This yields for the Earth-Moon distance 60.27 Earth radii or 384,399 kilometres (238,854 mi) This procedure was first used by Aristarchus of Samos^[17] and Hipparchus, and later found its way into the work of Ptolemy.^[18] The diagram at right shows how daily lunar parallax arises on the geocentric and geostatic planetary model in which the Earth is at the centre of the planetary system and does not rotate. It also illustrates the important point that parallax need not be caused by any motion of the observer, contrary to some definitions of parallax that say it is, but may arise purely from motion of the observed.

Another method is to take two pictures of the Moon at exactly the same time from two locations on Earth and compare the positions of the Moon relative to the stars. Using the orientation of the Earth, those two position measurements, and the distance between the two locations on the Earth, the distance to the Moon can be triangulated:

Example of lunar parallax: Occultation of Pleiades by the Moon

This is the method referred to by Jules Verne in From the Earth to the Moon:

Until then, many people had no idea how one could calculate the distance separating the Moon from the Earth. The circumstance was exploited to teach them that this distance was obtained by measuring the parallax of the Moon. If the word parallax appeared to amaze them, they were told that it was the angle subtended by two straight lines running from both ends of the Earth's radius to the Moon. If they had doubts on the perfection of this method, they were immediately shown that not only did this mean distance amount to a whole two hundred thirty-four thousand three hundred and forty-seven miles (94,330 leagues), but also that the astronomers were not in error by more than seventy miles (≈ 30 leagues).

Solar parallax

After Copernicus proposed his heliocentric system, with the Earth in revolution around the Sun, it was possible to build a model of the whole solar system without scale. To ascertain the scale, it is necessary only to measure one distance within the solar system, e.g., the mean distance from the Earth to the Sun (now called an astronomical unit, or AU). When found by triangulation, this is referred to as the solar parallax, the difference in position of the Sun as seen from the Earth's centre and a point one Earth radius away, i. e., the angle subtended at the Sun by the Earth's mean radius.
Knowing the solar parallax and the mean Earth radius allows one to calculate the AU, the first, small step on the long road of establishing the size and expansion age^[19] of the visible Universe.

A primitive way to determine the distance to the Sun in terms of the distance to the Moon was already proposed by Aristarchus of Samos in his book On the Sizes and Distances of the Sun and Moon. He noted that the Sun, Moon, and Earth form a right triangle (right angle at the Moon) at the moment of first or last quarter moon. He then estimated that the Moon, Earth, Sun angle was 87°. Using correct geometry but inaccurate observational data, Aristarchus concluded that the Sun was slightly less than 20 times farther away than the Moon. The true value of this angle is close to 89° 50', and the Sun is actually about 390 times farther away.^[17] He pointed out that the Moon and Sun have nearly equal apparent angular sizes and therefore their diameters must be in proportion to their distances from Earth. He thus concluded that the Sun was around 20 times larger than the Moon; this conclusion, although incorrect, follows logically from his incorrect data. It does suggest that the Sun is clearly larger than the Earth, which could be taken to support the heliocentric model.^[20]

Measuring Venus transit times to determine solar parallax

Although Aristarchus' results were incorrect due to observational errors, they were based on correct geometric principles of parallax, and became the basis for estimates of the size of the solar system for almost 2000 years, until the transit of Venus was correctly observed in 1761 and 1769.^[17] This method was proposed by Edmond Halley in 1716, although he did not live to see the results. The use of Venus transits was less successful than had been hoped due to the black drop effect, but the resulting estimate, 153 million kilometers, is just 2% above the currently accepted value, 149.6 million kilometers.

Much later, the Solar System was 'scaled' using the parallax of asteroids, some of which, such as Eros, pass much closer to Earth than Venus. In a favourable opposition, Eros can approach the Earth to within 22 million kilometres.^[21] Both the opposition of 1901 and that of 1930/1931 were used for this purpose, the calculations of the latter determination being completed by Astronomer Royal Sir Harold Spencer Jones.^[22]

Also radar reflections, both off Venus (1958) and off asteroids, like Icarus, have been used for solar parallax determination. Today, use of spacecraft telemetry links has solved this old problem. The currently accepted value of solar parallax is 8".794 143.^[23]

Dynamic or moving-cluster parallax

The open stellar cluster Hyades in Taurus extends over such a large part of the sky, 20 degrees, that the proper motions as derived from astrometry appear to converge with some precision to a perspective point north of Orion. Combining the observed apparent (angular) proper motion in seconds of arc with the also observed true (absolute) receding motion as witnessed by the Doppler redshift of the stellar spectral lines, allows estimation of the distance to the cluster (151 light-years) and its member stars in much the same way as using annual parallax.^[24]
Dynamic parallax has sometimes also been used to determine the distance to a supernova, when the optical wave front of the outburst is seen to propagate through the surrounding dust clouds at an apparent angular velocity, while its true propagation velocity is known to be the speed of light.^[25]

Derivation

For a right triangle,

where is the parallax, 1 AU (149,600,000 km) is approximately the average distance from the Sun to Earth, and is the distance to the star. Using small-angle approximations (valid when the angle is small compared to 1 radian),

so the parallax, measured in arcseconds, is

If the parallax is 1", then the distance is

This defines the parsec, a convenient unit for measuring distance using parallax. Therefore, the distance, measured in parsecs, is simply , when the parallax is given in arcseconds.^[26]

Parallax error in astronomy

Precise parallax measurements of distance have an associated error. However this error in the measured parallax angle does not translate directly into an error for the distance, except for relatively small errors. The reason for this is that an error toward a smaller angle results in a greater error in distance than an error toward a larger angle.

However, an approximation of the distance error can be computed by

where d is the distance and p is the parallax. The approximation is far more accurate for parallax errors that are small relative to the parallax than for relatively large errors. For meaningful results in stellar astronomy, Dutch astronomer Floor van Leeuwen recommends that the parallax error be no more than 10% of the total parallax when computing this error estimate.^[27]

Parallax error in measurement instruments

The correct line of sight needs to be used to avoid parallax error.

Measurements made by viewing the position of some marker relative to something to be measured are subject to parallax error if the marker is some distance away from the object of measurement and not viewed from the correct position. For example, if measuring the distance between two ticks on a line with a ruler marked on its top surface, the thickness of the ruler will separate its markings from the ticks. If viewed from a position not exactly perpendicular to the ruler, the apparent position will shift and the reading will be less accurate than the ruler is capable of.

A similar error occurs when reading the position of a pointer against a scale in an instrument such as an analog multimeter. To help the user avoid this problem, the scale is sometimes printed above a narrow strip of mirror, and the user's eye is positioned so that the pointer obscures its own reflection, guaranteeing that the user's line of sight is perpendicular to the mirror and therefore to the scale. The same effect alters the speed read on a car's speedometer by a driver in front of it and a passenger off to the side, values read from a graticule not in actual contact with the display on an oscilloscope, etc.

Photogrammetric parallax

Aerial picture pairs, when viewed through a stereo viewer, offer a pronounced stereo effect of landscape and buildings. High buildings appear to 'keel over' in the direction away from the centre of the photograph. Measurements of this parallax are used to deduce the height of the buildings, provided that flying height and baseline distances are known. This is a key component to the process of photogrammetry.

Parallax error in photography

Contax III rangefinder camera with macro photography setting. Because the viewfinder is on top of the lens and of the close proximity of the subject, goggles are fitted in front of the rangefinder and a dedicated viewfinder installed to compensate for parallax.

Parallax error can be seen when taking photos with many types of cameras, such as twin-lens reflex cameras and those including viewfinders (such as rangefinder cameras). In such cameras, the eye sees the subject through different optics (the viewfinder, or a second lens) than the one through which the photo is taken. As the viewfinder is often found above the lens of the camera, photos with parallax error are often slightly lower than intended, the classic example being the image of person with his or her head cropped off. This problem is addressed in single-lens reflex cameras, in which the viewfinder sees through the same lens through which the photo is taken (with the aid of a movable mirror), thus avoiding parallax error.

Parallax is also an issue in image stitching, such as for panoramas.

Parallax in sights

Parallax affects sights in many ways. On sights fitted to small arms, bows in archery, etc. the distance between the sighting mechanism and the weapon's bore or axis can introduce significant errors when firing at close range, particularly when firing at small targets. This difference is generally referred to as "sight height"^[28] and is compensated for (when needed) via calculations that also take in other variables such as bullet drop, windage, and the distance at which the target is expected to be.^[29] Sight height can be used to advantage when "sighting-in" rifles for field use. A typical hunting rifle (.222 with telescopic sights) sighted-in at 75m will be useful from 50m to 200m without further adjustment.^{[citation needed]}

Parallax in optical sights

In optical sights parallax refers to the apparent movement of the reticle in relationship to the target when the user moves his/her head laterally behind the sight (up/down or left/right),^[30] i.e. it is an error where the reticle does not stay aligned with the sight's own optical axis.
In optical instruments such as telescopes, microscopes, or in telescopic sights used on small arms and theodolites, the error occurs when the optics are not precisely focused: the reticle will appear to move with respect to the object focused on if one moves one's head sideways in front of the eyepiece. Some firearm telescopic sights are equipped with a parallax compensation mechanism which basically consists of a movable optical element that enables the optical system to project the picture of objects at varying distances and the reticle crosshairs pictures together in exactly the same optical plane. Telescopic sights may have no parallax compensation because they can perform very acceptably without refinement for parallax with the sight being permanently adjusted for the distance that best suits their intended usage. Typical standard factory parallax adjustment distances for hunting telescopic sights are 100 yd or 100 m to make them suited for hunting shots that rarely exceed 300 yd/m. Some target and military style telescopic sights without parallax compensation may be adjusted to be parallax free at ranges up to 300 yd/m to make them better suited for aiming at longer ranges.^{[citation needed]} Scopes for rimfires, shotguns, and muzzleloaders will have shorter parallax settings, commonly 50 yd/m^{[citation needed]} for rimfire scopes and 100 yd/m^{[citation needed]} for shotguns and muzzleloaders. Scopes for airguns are very often found with adjustable parallax, usually in the form of an adjustable objective, or AO. These may adjust down as far as 3 yards (2.74 m).^{[citation needed]}

Non-magnifying reflector or "reflex" sights have the ability to be theoretically "parallax free." But since these sights use parallel collimated light this is only true when the target is at infinity. At finite distances eye movement perpendicular to the device will cause parallax movement in the reticle image in exact relationship to eye position in the cylindrical column of light created by the collimating optics.^[31][32] Firearm sights, such as some red dot sights, try to correct for this via not focusing the reticle at infinity, but instead at some finite distance, a designed target range where the reticle will show very little movement due to parallax.^[31] Some manufactures market reflector sight models they call "parallax free,"^[33] but this refers to an optical system that compensates for off axis spherical aberration, an optical error induced by the spherical mirror used in the sight that can cause the reticle position to diverge off the sight's optical axis with change in eye position.^[34][35]

Artillery gunfire

Because of the positioning of field or naval artillery guns, each one has a slightly different perspective of the target relative to the location of the fire-control system itself. Therefore, when aiming its guns at the target, the fire control system must compensate for parallax in order to assure that fire from each gun converges on the target.

Parallax rangefinders

Parallax theory for finding naval distances

A coincidence rangefinder or parallax rangefinder can be used to find distance to a target.

As a metaphor

In a philosophic/geometric sense: an apparent change in the direction of an object, caused by a change in observational position that provides a new line of sight. The apparent displacement, or difference of position, of an object, as seen from two different stations, or points of view. In contemporary writing parallax can also be the same story, or a similar story from approximately the same time line, from one book told from a different perspective in another book. The word and concept feature prominently in James Joyce's 1922 novel, Ulysses. Orson Scott Card also used the term when referring to Ender's Shadow as compared to Ender's Game.

The metaphor is invoked by Slovenian philosopher Slavoj Žižek in his work The Parallax View. Žižek borrowed the concept of "parallax view" from the Japanese philosopher and literary critic Kojin Karatani. "The philosophical twist to be added (to parallax), of course, is that the observed distance is not simply subjective, since the same object that exists 'out there' is seen from two different stances, or points of view. It is rather that, as Hegel would have put it, subject and object are inherently mediated so that an 'epistemological' shift in the subject's point of view always reflects an ontological shift in the object itself. Or—to put it in Lacanese—the subject's gaze is always-already inscribed into the perceived object itself, in the guise of its 'blind spot,' that which is 'in the object more than object itself', the point from which the object itself returns the gaze. Sure the picture is in my eye, but I am also in the picture."^[36]