5 Photogrammetric parallax

Aerial photograph pairs, when viewed through a stereo viewer, offer a spectacular stereo effect of landscape and buildings. High buildings appear to 'keel over' in the direction away from the centre of the photograph. Measuring this effect, also called parallax, allows one, if the flying height and the distance between the aircraft's exposure locations is known, to deduce the building's height.

6 Lunar parallax

Jules VerneJules Verne ( February 8, 1828 March 24, 1905) was a French writer and a pioneer of the science fiction ( scientific romance) genre. Early years Verne was born in Nantes to attorney Pierre Verne and his wife Sophie. The oldest of the family's five childre, De la Terre à la LuneFrom the Earth to the Moon (French title: De la Terre a la Lune is a humorous science fiction story written in 1865 by Jules Verne and is one of the earliest entries in that genre. It tells the story of three well-to-do members of a post- American Civil W ( 18651865 is a common year starting on Sunday. Events January 31 American Civil War: Confederate General Robert E. Lee becomes general-in-chief. February 17 American Civil War: Columbia, South Carolina burns as Confederate forces flee from advancing Union forc). "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)."

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 degrees, from which (with the solar apparent radius 0.25 degrees) we get an Earth apparent radius of 1 degree. This yields for the Earth-Moon distance 60 Earth radii or 384.000 km.

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.

• distance_moon = distance_{observer base} / tan(angle)

7 Solar parallax

After Johannes KeplerThis article is about Johannes Kepler the astronomer. For the planned planet-finding space telescope, see Kepler Space Mission. Johannes Kepler ( December 27, 1571 November 15, 1630), a key figure in the scientific revolution, was a German astronomer, mat discovered his Third LawJohannes Kepler's primary contribution to astronomy/ astrophysics were the three laws of planetary motion . Kepler derived these laws, in part, by studying the observations of Brahe. Isaac Newton would later design his laws of motion and universal gravita, 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 SunThe Sun (also called Sol is the star in our solar system. Planet Earth orbits the Sun. Other bodies that orbit the Sun include other planets, asteroids, meteoroids, comets and dust. Not all objects passing through the solar system have been orbitally capt or astronomical unitThe astronomical unit AU is a unit of distance, approximately equal to the mean distance between Earth and Sun. The currently accepted value of the AU is 149,597,870,691+-30 metres (about 150 million kilometres or 93 million miles). Earth's orbit is not a (AU). 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.

It was proposed by Edmund 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 (1958) and off asteroids, like

Icarus, have been used for solar parallax

determination. Today, use of spacecraft telemetry links has solved this old problem completely.

8 Stellar parallax

On an interstellar scale, parallax created by the different orbital positions of the Earth causes the stars to seem to move.

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 (in parsecs) can be computed as the reciprocal of the parallax. For instance, the nearest star, Alpha Centauri, has a parallax of 0.750". Therefore the distance is 1/0.750=1.33 parsecs or about 4.3 light years.

Computation

• The parallax in arc seconds
  • astronomical unit = Average distance from sun to earth = 1.4959e11 meters
  • 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 206264 au = 3.2616 lyr = 1 parsec (This defines the parsec)
• The parallax arc seconds, when distance 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 did not then occur to many people that the stars are so very much further away from us than the planets of the solar system as to render that argument useless.

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 used by Friedrich Wilhelm Bessel in 1838 when he measured the distance to 61 Cygni, and it remains the standard for calibrating other measurement methods (after the size of the orbit of the earth is measured by radar reflection on other planets). 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.

9 Dynamic or moving-cluster parallax

The open stellar cluster 'Hyades' (Rain Stars) 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 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 that of light.

10 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 (apparent) 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 14.7 billion years ago, but with considerable uncertainty and dependence on various model assumptions.

11 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.

Optics Computer vision Vision

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