Definitions

parallaxis

Parallax

[par-uh-laks]

Parallax is an apparent displacement or difference of orientation 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. 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. In astronomy, parallax is the only direct method by which distances to objects (typically stars) beyond the Solar System can be measured. The Hipparcos satellite has used the technique for over 100,000 nearby stars. This provides the basis for all other distance measurements in astronomy, the cosmic distance ladder. Here, the term "parallax" is the angle or semi-angle of inclination between two sightlines to the star.

Parallax also affects optical instruments such as binoculars, microscopes, and twin-lens reflex cameras which view objects from slightly different angles. Many animals, including humans, have two eyes with overlapping visual fields to use parallax to gain depth perception; this process is known as stereopsis.

Distance measurement in astronomy

Stellar parallax

On an interstellar scale, parallax created by the different orbital positions of the Earth causes nearby stars to appear to move relative to more distant stars. By observing parallax, measuring angles and using geometry, one can determine the distance to various objects. When the object in question is a star, the effect is known as stellar parallax.

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. 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 of planets.

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 0.77233 ± 0.00242 arcsec. This angle is approximately the angle subtended by an object about 2 centimeters in diameter (roughly the size of a U.S. quarter dollar) located about 5.3 kilometers away.

In 1989, the satellite Hipparcos was launched primarily for obtaining parallaxes and proper motions of 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 our galaxy. The European Space Agency's Gaia mission, due to launch in 2011 and come online in 2012, will be able to measure parallax angles to an accuracy of 1 microarcsecond, thus mapping nearby stars (and potentially planets) up to a distance of tens of thousands of light-years from earth.

Computation

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, 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): d (mathrm{pc}) = 1 / p (mathrm{arcsec}). For example, the distance to Proxima Centauri is 1/0.772=.

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 Earth radii or 384,000 km. This procedure was first used by Aristarchus of Samos and Hipparchus, and later found its way into the work of Ptolemy. 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:

mathrm{distance}_{textrm{moon}} = frac {mathrm{distance}_{mathrm{observerbase}}} {tan (mathrm{angle})}

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

Up till 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

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 a gigantic size seemed entirely implausible.

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 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. 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. Although these 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.

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, like Eros, pass much closer to Earth than Venus. In a favourable opposition, Eros can approach the Earth to within 22 million kilometres. 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.

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.

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.

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.

Derivation

For a right triangle,

sin p = frac {1 AU} {d} ,
where p is the parallax, is approximately the average distance from the Sun to Earth, and d is the distance to the star. Using small-angle approximations (valid when the angle is small compared to 1 radian),
sin x approx xtextrm{ radians} = x cdot frac {180} {pi} textrm{ degrees} = x cdot 180 cdot frac {3600} {pi} textrm{ arcseconds} ,
so the parallax, measured in arcseconds, is
p'' approx frac {1 textrm{ AU}} {d} cdot 180 cdot frac{3600} {pi} .
If the parallax is 1", then the distance is
d = 1 textrm{ AU} cdot 180 cdot frac {3600} {pi} = 206,265 textrm{ AU} = 3.2616 textrm{ ly} equiv 1 textrm{ parsec} .
This defines the parsec, a convenient unit for measuring distance using parallax. Therefore, the distance, measured in parsecs, is simply d = 1 / p, when the parallax is given in arcseconds.

Parallax error

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

delta d = delta left({1 over p} right) =left| {partial over partial p} left({1 over p} right) right| delta p ={delta p over p^2}
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.

Visual perception

Because the eyes of humans and other highly evolved 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. Animals also use motion parallax, in which the animal (or just the head) moves 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.

Parallax and measurement instruments

If an optical instrument — e.g., a telescope, microscope, or theodolite — is imprecisely focused, its cross-hairs will appear to move with respect to the object focused on if one moves one's head horizontally in front of the eyepiece. This is why it is important, especially when performing measurements, to focus carefully in order to eliminate the parallax, and to check by moving one's head.

Also, in non-optical measurements the thickness of a ruler can create parallax in fine measurements. To avoid parallax error, one should take measurements with one's eye on a line directly perpendicular to the ruler so that the thickness of the ruler does not create error in positioning for fine measurements. A similar error can occur when reading the position of a pointer against a scale in an instrument such as a galvanometer. To help the user avoid this problem, the scale is sometimes printed above a narrow strip of mirror, and the user positions his eye so that the pointer obscures its own reflection. This guarantees that the user's line of sight is perpendicular to the mirror and therefore to the scale.

Parallax can cause a speedometer reading to appear different to a car's passenger than to the driver.

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

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.

In computer graphics

In many early graphical applications, such as video games, the scene was constructed of independent layers that were scrolled at different speeds when the player/cursor moved. Some hardware had explicit support for such layers, such as the Super Nintendo Entertainment System. This gave some layers the appearance of being farther away than others and was useful for creating an illusion of depth, but only worked when the player was moving. Now, most games are based on much more comprehensive three-dimensional graphic models, although portable game systems (DS, PSP) still often use parallax.

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 also invoked in the magnum opus of 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 which 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.

The word is also used in the title of Alan J. Pakula's 1974 movie The Parallax View, in which a reporter (Warren Beatty) investigates an assassination. The word in this case refers to a fictional corporation portrayed in the film.

References

General references

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