Encyclopedia
Parallax, or more accurately
motion parallax is the change of angular position of two
stationary points relative to each other as seen by an observer,
due to the motion of an observer. Simply put, it is the apparent shift of an object against a background due to a change in
observer position.
Introduction
This parallax is often thought of as the "apparent motion" of an object against a distant background because of a perspective shift, as seen in Figure 1. When viewed from
Viewpoint A, the object appears to be closer than the blue square. When the viewpoint is changed to
Viewpoint B, the object
appears to have moved in front of the red square. It is most commonly used in astronomy.
Use in distance measurement
By observing parallax, measuring
angles, and using
geometry, one can determine the
distance to various objects. When this is in reference to
stars, the effect is known as
stellar parallax. The first successful measurements of a stellar parallax were made by
Friedrich Bessel in 1838, for the star 61 Cygni.
Distance measurement by parallax is a special case of the principle of
triangulation, where 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 only
one side has been measured. Thus, the careful measurement of the length of one baseline can fix the scale of a triangulation network covering the whole nation. In parallax, the triangle is extremely long and narrow, and by measuring both its shortest side and the small top angle , the long sides can be determined.
Parallax error
Precise parallax measurements of distance usually have an associated
error. Thus a parallax may be described as some angle ± some angle-error. However this "± angle-error" will not translate directly into a ± error for the range, 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 means of the following:
where
d is the distance and
p is the parallax. The approximation is more accurate for relatively small values of the parallax error when compared to the parallax.
Parallax and measurement instruments
If an optical instrument — telescope, microscope,
theodolite — is imprecisely focused, the 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 carefully focus in order to 'eliminate the parallax', and to check by moving one's head.
Also in non-optical measurements, e.g., the thickness of a ruler can create parallax in fine measurements. One is always cautioned in science classes to "avoid parallax." By this it is meant that one should always 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 to 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.
In photography, one also talks about the parallax of a camera viewfinder: for nearby objects, a viewfinder mounted on top of the
camera will show something different from what the lens 'sees', and people's heads may be cut off. The problem does not exist for the
single lens reflex camera, where the viewfinder looks through the same lens as is used for taking the photograph.
Photogrammetric parallax
Aerial photograph 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.
Lunar parallax
Jules Verne,
From the Earth to the Moon is a humorous science fiction [i] story written in 1865 [i] by Jules Verne [i] ...
. "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 , but also that the astronomers were not in error by more than seventy miles ."
A primitive way to determine the lunar parallax from one location is by using a lunar eclipse. The 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 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.
Another way to use parallax to determine the distance to the Moon would be to take two pictures of the Moon at exactly the same time from two locations on Earth, and compare the position of the Moon relative to the visible stars. Using the orientation of the Earth, and those two points, and a perpendicular displacement, a distance to the Moon can be triangulated.
Solar parallax
After
Johannes Kepler discovered his
Third Law, it was possible to build a scale model of the whole solar system, but without the scale. To fix the scale, it suffices to measure one distance within the solar system, e.g., the mean distance from the Earth to the
Sun or astronomical unit . When done 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 thus the minimum age — of the visible Universe.
A primitive way of determining the distance to the Sun in terms of the distance to the Moon was already proposed by
Aristarchus of Samos: if the Sun is relatively close by, the first and last quarters of the Moon will not happen in time precisely in the middle between new and full moon. Unfortunately the method becomes progressively imprecise for solar distances much larger than the distance of the Moon, and Aristarchus obtained a nonsensical result. It is, however, in essence a parallax method.
It was proposed by
Edmond Halley in 1716, that the
transit of Venus over the solar disc be used to derive the solar parallax. And so it was done in 1761 and 1769. There is the famous story of the French astronomer Guillaume Le Gentil, who travelled to
India to observe the 1761 event, but didn't reach his destination in time due to war. He stayed on for the 1769 event, but then there were clouds blocking the Sun...
The use of Venus transits was less successful than had been hoped due to the black drop effect.
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 and off asteroids, like Icarus, have been used for solar parallax
determination. Today, use of
spacecraft telemetry links has solved this old problem completely.
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 the more distant stars. However, this effect is so small it is undetectable without extremely precise measurements.
The
annual parallax is 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. Given two points on opposite ends of the orbit, the parallax is half the maximum parallactic shift evident from the star viewed from the two points. The
parsec is the distance for which the annual parallax is 1 arcsecond. A parsec equals 3.26 light years.
The distance of an object can be computed as the reciprocal of the parallax. For instance, the
Hipparcos satellite measured the parallax of the nearest star, Proxima Centauri, as .77233 seconds of arc . Therefore, the distance is 1/0.772=1.29
parsecs or about 4.22 light years .
The angles involved in these calculations are very small. For example, .772 arcseconds is roughly the angle subtended by an object about 2 centimeters in diameter located about 5.3 kilometers away.
Computation
The parallax in arc seconds
where
- au = astronomical unit = Average distance from sun
|-
...
to
earth = 1.4959 · 10
11 m
- d = distance to the star
Picking a good unit of measure will cancel the constants.
Derivation:
- right triangle
-
- small angle approximation
- arcseconds
- parallax
- If the parallax is 1", then the distance is = 206,265 au = 3.2616 lyr = 1 parsec
- The parallax arcseconds, when the distance is given in parsecs
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.
Measurements of the annual parallax as the earth goes through its orbit was the first reliable way to determine the distances to the closest
stars. This method was first successfully used by
Friedrich Wilhelm Bessel in 1838 when he measured the distance to 61 Cygni with a heliometer, and it remains the standard for calibrating other measurement methods .
In 1989, a satellite called "
Hipparcos" was launched with the main
purpose of obtaining parallaxes and proper motions of nearby stars, increasing the reach of the method ten-fold. Even so, Hipparcos is only able to measure parallax angles for stars up to about 1,600 light-years away — a little bit more than one percent of the diameter of
our galaxy.
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 proper motion in seconds of arc with the also observed true receding motion as witnessed by the Doppler redshift of the stellar spectral lines, allows us to estimate the distance of the cluster 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 was seen to propagate through the surrounding dust clouds at an apparent angular velocity, when we know its true propagation velocity to be the
speed of light.
The scale of the Universe
All these various astronomical parallax methods allow us to establish the first rungs on the cosmic scale ladder, out to a few hundred light years. Beyond that, other methods must be taken into use: e.g., "
spectroscopic parallaxes" — not really parallaxes at all. It is a prototype of a "standard candle" method, where we observe the apparent brightness of an object we know, based on some physical theory, the true brightness of. For groups of stars we have the
Hertzsprung-Russell diagram which allows us to derive a star's absolute brightness or magnitude from its spectral type. The observed brightness or magnitude being , we can then derive its parallax by
called "spectroscopic parallax".
Further methods, mostly of the standard candle variety, are the variable stars called
Cepheids — the
absolute brightness of which depends on their observed period of variation —,
supernova brightnesses,
globular cluster sizes and brightnesses, complete
galaxy brightnesses etc. These are all much more uncertain as they are not based on simple geometry. Yet, parallaxes are the basis of everything, as they allow the calibration of these more uncertain methods in the Solar neighbourhood.
A very modern method which is not a traditional parallax method but also geometric in nature, is "
gravitational lensing parallax". It depends on observing the differential time delay of brightness variations from a remote
quasar reaching us by two different paths through the gravitational field or "lens" of a foreground galaxy. If the redshifts of both the quasar and the foreground galaxy are known, one can show that the absolute distances of both are proportional to the differential delay, and can in fact be calculated given also the geometry of the gravitational lens on the celestial sphere.
All these independent techniques aim at determining
Hubble's constant, the constant describing how the
redshift of galaxies, due to the Universe's expansion, depends on these galaxies' distance from us. Knowing Hubble's constant again allows us to determine, by simply running the film of the cosmic expansion backwards, how long ago it was when all these galaxies were collected in a single point -- the
Big Bang. Current knowledge puts this at some 13.7 billion years ago, but with considerable uncertainty and dependence on various model assumptions.
Parallax in computer graphics
In many early graphical applications, such as video games, the scene would be constructed of independent layers that are scrolled at different speeds when the player/cursor moves. 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 is moving. Now, most games are based on much more comprehensive three-dimensional graphic models, although portable game systems still often use parallax.
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 a 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 of "parallax" feature prominently in
James Joyce's 1922 novel, Ulysses. Orson Scott Card also used this term when referring to
Ender's Shadow as compared to
Ender's Game.
See also
- Disparity
- Triangulation, wherein a point is calculated given its angles from other known points
- Trilateration, wherein a point is calculated given its distances from other known points
- Trigonometry
Sources
External links
-
- BBC's programme: Patrick Moore demonstrates Parallax using Cricket.
- - Illustrated diagrams of why this happens, what it does to the photograph, and how to prevent it from occurring
