Relativity Science Calculator - Phenomenon of Stellar Parallax

Phenomenon of Stellar Parallax

( Or, how far are the stars from Earth? )

"The most beautiful thing we can experience is the mysterious. It is the source of all true art and all science. He to whom this emotion is a stranger, who can no longer pause to wonder and stand rapt in awe, is as good as dead: his eyes are closed" - Albert Einstein ( 1879 - 1955 )

Against a very distant seemingly "fixed star" background, any relatively nearby star will display an apparent movement as the Earth makes its yearly orbital transit about the Sun. This apparent movement produces an angle of perception called "angle of Parallax" which is used to measure the distance to the star from Earth. The Method of Parallax for determining astronomical distances to nearby stars is therefore a heliocentric method. This phenomenon is similar as to when an object is held at arm's length and seems to move against a more distant background as each eye is opened and closed. However at extreme distances the angle of Parallax becomes infinitesimally small and hence unusable.

Phenomenon and History of Parallaxradian measure for angle

The Mathematics of Stellar Parallax

parsec derivationsmall trignometric approximationradian measure for angle

We may therefore derive some equivalent distances based upon the known amount of 1 astronomical unit, AU, as follows:

And, the distances to some of the near - stars can now be obtained from the following simplified equations for parsec:

small trignometric approximation

Some Examples of Parsec Distances
Galaxy or Star θrad ( angle of parallax ) dpc ( distance ) light - year distance
Andromeda ( M31 ) 1.28205 x 10-6 0.780 mpc 2.544 M ly
Proxima Centauri 0.77233 1.29478 pc 4.223 ly
Sirius star 0.379 2.6385 pc 8.6059 ly
Procyon star 0.286 3.4965 pc 11.4043 ly

note: The European Space Agency's Gaia mission will conduct an ambitious 3 - dimensional census of our Milky Way galaxy beginning 2011 to map over one thousand million stars and their distances from the center of the galaxy and thence from our solar system. Albert Einstein's general relativity mathematics will also be stringently tested, although only about 1% of the Milky Way will have finally been taken into account how vast is our own galaxy in the midst of the cosmos. [ Gaia from Greek mythology: the goddess of the earth ]

A Short History of Parallax and Mars

( Or, how far is Paris from Ceyenne, French Guiana? )

source: jpl-time lapse mars surface images from Spirit Lander

More specifically why Parallax? The Earth - Sun distance AU had been a constant astronomical question since the time of Aristarchus ( 310 - 230 B.C. ), Samos - Alexandria, who provided the first estimate and who also preceded Copernicus in proposing a Sun - centered heliocentric universe.

The first actual measurement of AU is usually credited to Giovanni Domenico Cassini ( 1625 - 1712 ) in 1672 by simultaneous parallax measurements of Mars from Paris where he remained with astronomer and Jesuit priest Jean Picard ( 1620 - 1682, Mesure de la Terre, published 1671 ) in the newly built Paris Observatory for The French Academy of Sciences and by astronomer and colleague Jean Richer ( 1630 - 1696, Observations Astronomiques et Physiques Faites en L'Isle de Caïenne par M. Richer, de l'Academie Royale des Sciences, published 1679 having returned from Cayenne in 1673 to great acclaim and success ) whom he sent to the French colony at Cayenne, French Guiana on the northern coast of South America, 5° latitude north of the equator, for observations and measurements during Mars's perihelic opposition or closest perigee approach to Earth appearing as a bright red star. This latter observation was instigated by John Flamsteed ( England, 1646 - 1719 ), First Astronomer Royal or "The King's Astronomical Observator" by 1675 royal warrant, who predicted that on the first of October, 1672, there would be an historic perihelic opposition ( Mars at perigee ) alignment of Mars passing in front of the middle Psi star, one of several nearby little stars, in the constellation Aquarius and hence there'd be this extraordinarily rare straight line alignment of Sun - Earth - Mars offering a grand opportunity to calculate the solar Astronomical Unit between Earth and the Sun.

Naturally by the 1670s there had already been considerable advances in telescopic fabrications since the time of Galileo ( 1564 - 1642 ) with the advent of micrometer eyepieces with precisely adjustable cross - hairs in order to have made possible Cassini's major astronomical breakthrough in determining AU, the Astronomical Unit.

Knowing the distance between Paris and Cayenne as approximately 7,200 km ( 4,500 miles ) [ how did Cassini know the Paris - Cayenne distance? ] and then employing the relative AUs [ not absolute ] of Johannes Kepler ( 1571 - 1630 ) who in turn enlisted the earlier solar data marshaled by Tycho Brahe ( 1546 - 1601 ), Cassini determined a parallax of 9.5 arc second for AU where the correct modern version is 8.8 arc second. Hence, Cassini's indirect determination of solar AU by way of Mars's parallax was d = 1/9.5 or 0.1052631 parsec instead of true d = 1/8.8 or 0.1136363 parsec, an error of only 7%!! The actual km for AU will be derived in another panel using some of Cassini's parallax mathematics. Interestingly however, Danish Olaf (Ole) Roëmer ( 1644 - 1710 ) working with Cassini at the Paris Observatory in December, 1676, published the first actual calculation for the speed of light based upon Cassini's own AU calculation and speculation that light travels with a finite speed.


Giovanni Domenico Cassini ( 1625 - 1712 )

[ note: aberration, aberration of (star)light, astronomical aberration, or stellar aberration is an astronomical phenomenon different from the phenomenon of parallax whereby small apparent motion displacements of all fixed stars on the celestial sphere due to earth's orbital velocity mandates that terrestrial telescopes must also be adjusted to slightly different directions as the earth yearly transits the sun. Stellar aberration is totally independent of a star's distance from earth but rather depends upon the transverse velocity of an observer on earth, all of which is unlike the phenomenon of parallax. For example, vertically falling rain upon your umbrella will appear to come from in front of you the faster you walk and hence the more you will adjust the position of the umbrella to deflect the rain. Finally, the fact that earth does not drag with itself in its immediate vicinity any amount of aether helps dissuade the concept that indeed the aether exists. It does not according to Einstein's interpretation of The Michelson - Morley Experiment of 1887 while still utilizing "The FitzGerald - Lorentz Solution to the Michelson - Morley Null Result". ]