Stellar parallax

Stellar parallax is the apparent shift of position of any nearby star (or other object) against the background of distant objects. Created by the different orbital positions of Earth, the extremely small observed shift is largest at time intervals of about six months, when Earth arrives at exactly opposite sides of the Sun in its orbit, giving a baseline distance of about two astronomical units between observations. The parallax itself is considered to be half of this maximum, about equivalent to the observational shift that would occur due to the different positions of Earth and the Sun, a baseline of one astronomical unit (AU).

Stellar parallax is so difficult to detect that its existence was the subject of much debate in astronomy for hundreds of years. It was first observed in 1806 by Giuseppe Calandrelli who reported parallax in α-Lyrae in his work "Osservazione e riflessione sulla parallasse annua dall’alfa della Lira".[1] Then in 1838 Friedrich Bessel made the first successful parallax measurement, for the star 61 Cygni, using a Fraunhofer heliometer at Königsberg Observatory.[2][3]

Once a star's parallax is known, its distance from Earth can be computed trigonometrically. But the more distant an object is, the smaller its parallax. Even with 21st-century techniques in astrometry, the limits of accurate measurement make distances farther away than about 100 parsecs (or roughly 300 light years) too approximate to be useful when obtained by this technique. This limits the applicability of parallax as a measurement of distance to objects that are relatively close on a galactic scale. Other techniques, such as spectral red-shift, are required to measure the distance of more remote objects.

Stellar parallax measures are given in the tiny units of arcseconds, or even in thousandths of arcseconds (milliarcseconds). The distance unit parsec is defined as the length of the leg of a right triangle adjacent to the angle of one arcsecond at one vertex, where the other leg is 1 AU long. Because stellar parallaxes and distances all involve such skinny right triangles, a convenient trigonometric approximation can be used to convert parallaxes (in arcseconds) to distance (in parsecs). The approximate distance is simply the reciprocal of the parallax: For example, Proxima Centauri (the nearest star to Earth other than the Sun), whose parallax is 0.7685, is 1 / 0.7685 parsecs = 1.301 parsecs (4.24 ly) distant.[4]

ParallaxeV2
Stellar parallax motion from annual parallax
Stellarparallax parsec1
Stellar parallax is the basis for the parsec, which is the distance from the Sun to an astronomical object that has a parallax angle of one arcsecond. (1 AU and 1 parsec are not to scale, 1 parsec = ~206265 AU)

Early theory and attempts

Stellar parallax is so small (as to be unobservable until the 19th century) that its apparent absence was used as a 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 Brahe'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).[5]

James Bradley first tried to measure stellar parallaxes in 1729. The stellar movement proved too insignificant for his telescope, but he instead discovered the aberration of light[6] and the nutation of Earth's axis, and catalogued 3222 stars.

19th and 20th centuries

Koenigsberg helio
Bessel's heliometer

Stellar parallax is most often measured using annual parallax, defined as the difference in position of a star as seen from Earth and Sun, i.e. the angle subtended at a star by the mean radius of 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 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.[2][7]

Being very difficult to measure, only about 60 stellar parallaxes had been obtained by the end of the 19th century, mostly by use of the filar micrometer. Astrographs using astronomical photographic plates sped the process in the early 20th century. Automated plate-measuring machines[8] and more sophisticated computer technology of the 1960s allowed more efficient compilation of star catalogues. In the 1980s, charge-coupled devices (CCDs) replaced photographic plates and reduced optical uncertainties to one milliarcsecond.

Stellar parallax remains the standard for calibrating other measurement methods (see Cosmic distance ladder). Accurate calculations of distance based on stellar parallax require a measurement of the distance from Earth to the Sun, now known to exquisite accuracy based on radar reflection off the surfaces of planets.[9]

The angles involved in these calculations are very small and thus difficult to measure. The nearest star to the Sun (and also the star with the largest parallax), Proxima Centauri, has a parallax of 0.7685 ± 0.0002 arcsec.[4] This angle is approximately that subtended by an object 2 centimeters in diameter located 5.3 kilometers away.

Space astrometry for parallax

Hubble stretches the stellar tape measure ten times further
Hubble precision stellar distance measurement has been extended 10 times further into the Milky Way.[10]

In 1989 the satellite Hipparcos was launched primarily for obtaining parallaxes and proper motions of nearby stars, increasing the number of stellar parallaxes measured to milliarcsecond accuracy a thousandfold. 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 Hubble telescope WFC3 now has a precision of 20 to 40 microarcseconds, enabling reliable distance measurements up to 3,066 parsecs (10,000 ly) for a small number of stars.[11] This gives more accuracy to the Cosmic distance ladder and improves the knowledge of distances in the Universe, based on the dimensions of the Earth's orbit.

The European Space Agency's Gaia mission, launched 19 December 2013, is expected to measure parallax angles to an accuracy of 10 microarcseconds for all moderately bright stars, thus mapping nearby stars (and potentially planets) up to a distance of tens of thousands of light-years from Earth.[12] Data Release 2 in 2018 claims mean errors for the parallaxes of 15th magnitude and brighter stars of 20–40 microarcseconds.[13]

Other baselines

Statistical parallax

Two related techniques can determine the mean distances of stars by modelling the motions of stars. Both are referred to as statistical parallaxes, or individual called secular parallaxes and classical statistical parallaxes.

The motion of the Sun through space provides a longer baseline that will increase the accuracy of parallax measurements, known as secular parallax. For stars in the Milky Way disk, this corresponds to a mean baseline of 4 AU per year, whereas for halo stars the baseline is 40 AU per year. After several decades, the baseline can be orders of magnitude greater than the Earth–Sun baseline used for traditional parallax. However, secular parallax introduces a higher level of uncertainty because the relative velocity of other stars is an additional unknown. When applied to samples of multiple stars, the uncertainty can be reduced; the precision is inversely proportional to the square root of the sample size.[14]

The mean parallaxes and distances of a large group of stars can be estimated from their radial velocities and proper motions. This is known as a classical statistical parallax. The motions of the stars are modelled to statistically reproduce the velocity dispersion based on their distance.[14][15]

Other parallax in astronomy

Other uses of the term parallax in astronomy, with different meanings are the photometric parallax method, spectroscopic parallax, and dynamical parallax.

See also

References

  1. ^ Hockey, Thomas, ed. (2007). Biographical Encyclopedia of Astronomers. Springer-Verlag New York. ISBN 978-0-387-30400-7.
  2. ^ a b Zeilik & Gregory 1998, p. 44.
  3. ^ Hirshfeld, Alan W (1 May 2002). Parallax. ISBN 978-0-8050-7133-7. Page 259.
  4. ^ a b Brown, A. G. A.; et al. (Gaia collaboration) (August 2018). "Gaia Data Release 2: Summary of the contents and survey properties". Astronomy & Astrophysics. 616. A1. arXiv:1804.09365. Bibcode:2018A&A...616A...1G. doi:10.1051/0004-6361/201833051.
  5. ^ See p.51 in The reception of Copernicus' heliocentric theory: proceedings of a symposium organized by the Nicolas Copernicus Committee of the International Union of the History and Philosophy of Science, Torun, Poland, 1973, ed. Jerzy Dobrzycki, International Union of the History and Philosophy of Science. Nicolas Copernicus Committee; ISBN 90-277-0311-6, ISBN 978-90-277-0311-8
  6. ^ Buchheim, Robert (4 October 2007). The Sky is Your Laboratory. ISBN 978-0-387-73995-3. Page 184.
  7. ^ Bessel, FW, "Bestimmung der Entfernung des 61sten Sterns des Schwans Archived 2007-06-24 at the Wayback Machine" (1838) Astronomische Nachrichten, vol. 16, pp. 65–96.
  8. ^ CERN paper on plate measuring machine USNO StarScan
  9. ^ Zeilik & Gregory 1998, § 22-3.
  10. ^ "Hubble stretches the stellar tape measure ten times further". ESA/Hubble Images. Retrieved 12 April 2014.
  11. ^ Harrington, J.D.; Villard, Ray (10 April 2014). "NASA's Hubble Extends Stellar Tape Measure 10 Times Farther into Space". NASA. Retrieved 17 October 2014. Riess, Adam G.; Casertano, Stefano; Anderson, Jay; Mackenty, John; Filippenko, Alexei V. (2014). "Parallax Beyond a Kiloparsec from Spatially Scanning the Wide Field Camera 3 on the Hubble Space Telescope". The Astrophysical Journal. 785 (2): 161. arXiv:1401.0484. Bibcode:2014ApJ...785..161R. doi:10.1088/0004-637X/785/2/161.
  12. ^ Henney, Paul J. "ESA's Gaia Mission to study stars". Astronomy Today. Retrieved 8 March 2008.
  13. ^ Brown, A. G. A.; et al. (Gaia collaboration) (August 2018). "Gaia Data Release 2: Summary of the contents and survey properties". Astronomy & Astrophysics. 616. A1. arXiv:1804.09365. Bibcode:2018A&A...616A...1G. doi:10.1051/0004-6361/201833051.
  14. ^ a b Popowski, Piotr; Gould, Andrew (29 January 1998). "Mathematics of Statistical Parallax and the Local Distance Scale". arXiv:astro-ph/9703140. Bibcode:1997astro.ph..3140P.
  15. ^ Layden, Andrew C; Hanson, Robert B; Hawley, Suzanne L; Klemola, Arnold R; Hanley, Christopher J (1996). "The Absolute Magnitude and Kinematics of RR Lyrae Stars Via Statistical Parallax". The Astronomical Journal. 112: 2110. arXiv:astro-ph/9608108. Bibcode:1996AJ....112.2110L. doi:10.1086/118167.
  • Hirshfeld, Alan w. (2001). Parallax: The Race to Measure the Cosmos. New York: W. H. Freeman. ISBN 0-7167-3711-6.
  • Whipple, Fred L. (2007). Earth Moon and Planets. Read Books. ISBN 1-4067-6413-2..
  • Zeilik, Michael A.; Gregory, Stephan A. (1998). Introductory Astronomy & Astrophysics (4th ed.). Saunders College Publishing. ISBN 0-03-006228-4..

Further reading

Aberration of light

The aberration of light (also referred to as astronomical aberration, stellar aberration, or velocity aberration) is an astronomical phenomenon which produces an apparent motion of celestial objects about their true positions, dependent on the velocity of the observer. Aberration causes objects to appear to be displaced towards the direction of motion of the observer compared to when the observer is stationary. The change in angle is typically very small — of the order of v/c where c is the speed of light and v the velocity of the observer. In the case of "stellar" or "annual" aberration, the apparent position of a star to an observer on Earth varies periodically over the course of a year as the Earth's velocity changes as it revolves around the Sun, by a maximum angle of approximately 20 arcseconds in right ascension or declination.

The term aberration has historically been used to refer to a number of related phenomena concerning the propagation of light in moving bodies.

Aberration should not be confused with parallax. The latter is a change in the apparent position of a relatively nearby object, as measured by a moving observer, relative to more distant objects that define a reference frame. The amount of parallax depends on the distance of the object from the observer, whereas aberration does not. Aberration is also related to light-time correction and relativistic beaming, although it is often considered separately from these effects.

Aberration is historically significant because of its role in the development of the theories of light, electromagnetism and, ultimately, the theory of special relativity. It was first observed in the late 1600s by astronomers searching for stellar parallax in order to confirm the heliocentric model of the Solar System. However, it was not understood at the time to be a different phenomenon.

In 1727, James Bradley provided a classical explanation for it in terms of the finite speed of light relative to the motion of the Earth in its orbit around the Sun,

which he used to make one of the earliest measurements of the speed of light. However, Bradley's theory was incompatible with 19th century theories of light, and aberration became a major motivation for the aether drag theories of Augustin Fresnel (in 1818) and G. G. Stokes (in 1845), and for Hendrik Lorentz's aether theory of electromagnetism in 1892. The aberration of light, together with Lorentz's elaboration of Maxwell's electrodynamics, the moving magnet and conductor problem, the negative aether drift experiments, as well as the Fizeau experiment, led Albert Einstein to develop the theory of special relativity in 1905, which presents a general form of the equation for aberration in terms of such theory.

Astrometry

Astrometry is the branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies. The information obtained by astrometric measurements provides information on the kinematics and physical origin of the Solar System and our galaxy, the Milky Way.

Binocular disparity

Binocular disparity refers to the difference in image location of an object seen by the left and right eyes, resulting from the eyes’ horizontal separation (parallax). The brain uses binocular disparity to extract depth information from the two-dimensional retinal images in stereopsis. In computer vision, binocular disparity refers to the difference in coordinates of similar features within two stereo images.

A similar disparity can be used in rangefinding by a coincidence rangefinder to determine distance and/or altitude to a target. In astronomy, the disparity between different locations on the Earth can be used to determine various celestial parallax, and Earth's orbit can be used for stellar parallax.

Full-sky Astrometric Mapping Explorer

Full-sky Astrometric Mapping Explorer (or FAME) was a proposed astrometric satellite designed to determine with unprecedented accuracy the positions, distances, and motions of 40 million stars within our galactic neighborhood (distances by stellar parallax possible). This database was to allow astronomers to accurately determine the distance to all of the stars on this side of the Milky Way galaxy, detect large planets and planetary systems around stars within 1,000 light years of the Sun, and measure the amount of dark matter in the galaxy from its influence on stellar motions. It was to be a collaborative effort between the United States Naval Observatory (USNO) and several other institutions. FAME would have measured stellar positions to less than 50 microarcseconds. The NASA MIDEX mission was scheduled for launch in 2004. In January 2002, however, NASA abruptly cancelled this mission, mainly due to concerns about its price tag which grew from $160 million to $220 million.

This would have been an improvement over the High Precision Parallax Collecting Satellite (Hipparcos) which operated 1989-1993 and produced various star catalogs. Astrometric parallax measurements form part of the cosmic distance ladder, and can also be measured by other Space telescopes such as Hubble (HST) or ground-based telescopes to varying degrees of precision.

Compared to the FAME accuracy of 50 microarcseconds, the Gaia mission is planning 10 microarcseconds accuracy, for mapping stellar parallax up to a distance of tens of thousands of light-years from Earth.

HD 147513

HD 147513 (62 G. Scorpii) is a star in the southern constellation of Scorpius. It was first catalogued by Italian astronomer Piazzi in his star catalogue as "XVI 55". With an apparent magnitude of 5.38, according to the Bortle scale it is visible to the naked eye from suburban skies. Based upon stellar parallax measurements by the Hipparcos spacecraft, HD 147513 lies some 42 light years from the Sun.

Heliometer

A heliometer (from Greek ἥλιος hḗlios "sun" and measure) is an instrument originally designed for measuring the variation of the sun's diameter at different seasons of the year, but applied now to the modern form of the instrument which is capable of much wider use.The basic concept is to introduce a split element into a telescope's optical path so as to produce a double image. If one element is moved using a screw micrometer, precise angle measurements can be made. The simplest arrangement is to split the object lens in half, with one half fixed and the other attached to the micrometer screw and slid along the cut diameter. To measure the diameter of the sun, for example, the micrometer is first adjusted so that the two images of the solar disk coincide (the "zero" position where the split elements form essentially a single element). The micrometer is then adjusted so that diametrically opposite sides of the two images of the solar disk just touch each other. The difference in the two micrometer readings so obtained is the (angular) diameter of the sun. Similarly, a precise measurement of the apparent separation between two nearby stars, A and B, is made by first superimposing the two images of the stars and then adjusting the double image so that star A in one image coincides with star B in the other. The difference in the two micrometer readings so obtained is the apparent separation or angular distance between the two stars.

The first application of the divided object-glass and the employment of double images in astronomical measures is due to Servington Savary from Exeter in 1743. Pierre Bouguer, in 1748, originated the true conception of measurement by double image without the auxiliary aid of a filar micrometer, that is by changing the distance between two object-glasses of equal focus. John Dollond, in 1754, combined Savary's idea of the divided object-glass with Bouguer's method of measurement, resulting in the construction of the first really practical heliometers. As far as we can ascertain, Joseph von Fraunhofer, some time not long before 1820, constructed the first heliometer with an achromatic divided object-glass, i.e. the first heliometer of the modern type.The first successful measurements of stellar parallax (to determine the distance to a star) were made by Friedrich Bessel in 1838 for the star 61 Cygni using a Fraunhofer heliometer. This was the 6.2-inch (157.5 mm) aperture Fraunhofer heliometer at Königsberg Observatory built by Joseph von Fraunhofer's firm, though he did not live to see it delivered to Bessel. Although the heliometer was difficult to use, it had certain advantages for Bessel including a wider field of view compared to other great refractors of the period, and overcame atmospheric turbulence in measurements compared to a filar micrometer.

James Bradley

James Bradley (1693–1762) was an English astronomer and priest who served as Astronomer Royal from 1742. He is best known for two fundamental discoveries in astronomy, the aberration of light (1725–1728), and the nutation of the Earth's axis (1728–1748). These discoveries were called "the most brilliant and useful of the century" by Jean Baptiste Joseph Delambre, historian of astronomy, mathematical astronomer and director of the Paris Observatory, in his history of astronomy in the 18th century (1821), because "It is to these two discoveries by Bradley that we owe the exactness of modern astronomy. ... This double service assures to their discoverer the most distinguished place (after Hipparchus and Kepler) above the greatest astronomers of all ages and all countries."

Kappa Virginis

Kappa Virginis (κ Virginis, abbreviated Kappa Vir, κ Vir), officially named Kang , is a solitary star in the zodiac constellation of Virgo. It has an apparent visual magnitude of 4.18, which is sufficiently bright to be seen with the naked eye. Based upon stellar parallax measurements, the distance to this star is about 255 light-years.

Manuel John Johnson

Manuel John Johnson, FRS (23 May 1805 – 28 February 1859) was a British astronomer.He was born in Macao, China, the son of John William Roberts of the East India Company and was educated at Mr Styles' Classical Academy in Thames Ditton and at the Addiscombe Military Seminary for service in the East India Company (the HEIC).In 1823 he was sent by the HEIC to St Helena, where from 1826 he supervised the building of the Ladder Hill Observatory. He travelled twice to South Africa to consult with Fearon Fallows on the design of the observatory. In 1828 he was made Superintendent of the Observatory. In 1835 he published A Catalogue of 606 Principal Fixed Stars in the Southern Hemisphere... at St. Helena, for which he won the Gold Medal of the Royal Astronomical Society that same year. While comparing his results with those of Nicolas Louis de Lacaille he noted the high proper motion of Alpha Centauri and communicated these to Thomas Henderson at the Royal Observatory, Cape of Good Hope. This led to the first successful measurement of a stellar parallax, though not to the first publication thereof.On his return to the UK in 1833 he went up to Magdalen Hall, Oxford, graduating MA in 1839. He then served as director of the Radcliffe Observatory from 1839 until his death in Oxford in 1859. He introduced there self-registering meteorological instruments to continuously record variations in atmospheric pressure, temperature, humidity and atmospheric electricity using the recent invention of photography. The initial instruments were invented and installed by Francis Ronalds, Honorary Director of the Kew Observatory. The Radcliffe Observatory later became part of the network of observing stations established as part of the new Meteorological Office and coordinated from Kew.

Johnson was president of the Royal Astronomical Society in 1855–1857 and was elected Fellow of the Royal Society in 1856.

In 1850 he had married Caroline, the daughter of Dr James Ogle.

Omega Virginis

Omega Virginis (ω Vir, ω Virginis) is a solitary star in the zodiac constellation Virgo. It has an apparent visual magnitude of +5.22, which is bright enough to be faintly visible to the naked eye. Based upon an annual stellar parallax shift of 6.56 milliarcseconds, it is located about 500 light years from the Sun.

This is a red giant star with a stellar classification of M4 III. It is a semiregular variable with a brightness that varies over an amplitude of 0.m28 with periods of 30 and 275 days. After evolving away from the main sequence it has expanded to around 70 times the solar radius, and now shines with 1,515 times the luminosity of the Sun. The effective temperature of the outer atmosphere is 3,490 K.

Parallax

Parallax (from Ancient Greek παράλλαξις (parallaxis), meaning 'alternation') 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. Due to foreshortening, nearby objects show a larger parallax than farther objects when observed from different positions, so parallax can be used to determine distances.

To measure large distances, such as the distance of a planet or a star from Earth, astronomers use the principle of parallax. Here, the term parallax is the semi-angle of inclination between two sight-lines to the star, as observed when Earth is on opposite sides of the Sun in its orbit. 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.

Paul Wittich

Paul Wittich (c.1546 – 9 January 1586) was a German mathematician and astronomer whose Capellan geoheliocentric model, in which the inner planets Mercury and Venus orbit the sun but the outer planets Mars, Jupiter and Saturn orbit the Earth, may have directly inspired Tycho Brahe's more radically heliocentric geoheliocentric model in which all the 5 known primary planets orbited the Sun, which in turn orbited the stationary Earth.Wittich was born in Breslau (Wrocław), Silesia, and studied at the universities of Leipzig, Wittenberg and Frankfurt/Oder. About 1580 Wittich stayed with Tycho Brahe on his island Hven in Öresund, where he worked at his Uraniborg. He then was employed by Landgraf Wilhelm IV. of Hessen-Kassel. He died in Vienna.

Wittich may have been influenced by Valentin Naboth's book Primarum de coelo et terra in adopting the Capellan system to explain the motion of the inferior planets. It is evident from Wittich's diagram of his Capellan system that the Martian orbit does not intersect the solar orbit nor those of Mercury and Venus, and would thus be compatible with solid celestial orbs, with the Solar orb containing the orbs of Venus and of Mercury and itself in turn wholly circumscribed by a Martian orb. This was in significant contrast with Ursus's geoheliocentric model in which the orbits of Mercury and Venus intersect the Martian orbit but the Solar orbit does not, and also with the Tychonic model in which the Martian orbit also intersects the Solar orbit in addition to those of Mercury and Venus, and whereby both these models rule out solid celestial orbs that cannot interpenetrate, if not excluding interpenetrating fluid orbs.

However, Wittich's Capellan model of the Martian orbit contradicted Copernicus's model in which Mars at opposition is nearer to the Earth than the Sun is, whereby if true the Solar and Martian orbits must intersect in all geoheliocentric models. Thus the question of whether the daily parallax of Mars was ever greater than that of the Sun was crucial to whether Wittich's (and indeed also Praetorius's and Ursus's) model was observationally tenable or not. It seems Tycho Brahe eventually came to the conclusion by 1588 that Mars does come nearer to the Earth than the Sun is, albeit contradicting his earlier conclusion by 1584 that his observations of Mars at opposition in 1582-3 established it had no discernible parallax, whereas he put the Sun's parallax at 3 arcminutes. Thus Brahe's 1588 model crucially contradicted both Wittich's and also Ursus's geoheliocentric models at least in respect of the dimensions of the Martian orbit, by positing its intersection with the Solar orbit.

Having failed to find any Martian parallax greater than the Solar parallax, Tycho had no valid observational evidence for his 1588 conclusion that Mars comes nearer to the Earth than the Sun, and nor did anybody else at that time, whereby Tycho's uniquely distinctive geoheliocentric model had no valid observational support in this respect. It seems its credibility rested solely upon his aristocratic social status rather than any scientific evidence. And this failure to find any Martian parallax in effect also refuted Copernicus's heliocentric model in respect of its Martian orbit, and supported the geocentric models of Ptolemy and the Capellan geoheliocentric model of Wittich and Praetorius and also Ursus's more Tychonic model. The latter differed from Tycho's only in respect of its non-intersecting Martian and Solar orbits and its daily rotating Earth.

It seems a primary purpose of Wittich's Capellan model, evident from the drafting markings in his drawing, was to save the integrity of solid celestial orbs, and the only planetary models compatible with solid celestial orbs were the Ptolemaic, Copernican and Wittichan Capellan (including Praetorius's) planetary models. But in 1610 Galileo's novel telescopic confirmation that Venus has a full set of phases like the Moon, published in his 1613 Letters on Sunspots, refuted the Ptolemaic geocentric model, which implied they are only crescents in conjunction, just as in opposition, whereas they are gibbous or full in conjunction. This crucial novel fact was logically implied by the Heraclidean, Capellan and Tychonic geoheliocentric planetary models, according to all of which at least the orbits of Venus and Mercury are centred on the Sun rather than the Earth, as well as by the pure heliocentric model. Consequently this left only the Copernican and Wittichan Capellan models compatible with both solid orbs and the phases of Venus. But only the Wittichan system was also compatible with the failure to find any stellar parallax predicted by all heliocentric models, in addition to also being compatible with the failure to find any Martian parallax that refuted both the Copernican and Tychonic models.

Thus by 1610 it seems the only observationally tenable candidate for a planetary model with solid celestial orbs was Wittich's Capellan system. Indeed it also seems it was even the only planetary model that was generally observationally tenable, given the twin failures to find any stellar annual parallax nor any Martian daily parallax at that time. However, insofar as it was accepted that comets are superlunary and sphere-busting, whereby solid celestial orbs are impossible and thus intersecting orbits cease to be impossible, then this thereby also admitted the model of Ursus (and Origanus) as also observationally tenable, along with Wittich's Capellan system (and thus also Praetorius's), whilst the Ptolemaic model was ruled out by the phases of Venus, all heliocentric models by the perceived absence of any annual stellar parallax, and both the Copernican and Tychonic models were also refuted by the absence of any Martian daily parallax. Renowned anti-Copernican adherents of the Capellan planetary model included Francis Bacon, inter alia, and this model appealed to those who accepted Ptolemy's purely geocentric model was refuted by the phases of Venus, but were unpersuaded by Tychonic arguments that Mars, Jupiter and Saturn also orbited the Sun in addition to Mercury and Venus. Indeed even Newton's arguments for this stated in his commentary on Phenomenon 3 of Book 3 of his Principia were notably invalid.

Photometric parallax method

The photometric parallax method is a method of data analysis used in astronomy that uses the colours and apparent brightnesses of stars to infer their distances. It was used by the Sloan Digital Sky Survey to discover the Virgo super star cluster.

Unlike the stellar parallax method, photometric parallax can be used to estimate the distances of stars over 10 kpc away, at the expense of much more limited accuracy for individual measurements. Strictly speaking, it does not actually employ any measurements of parallax and can be considered a misnomer.

Assuming that a star is on the main sequence, the star's absolute magnitude can be determined based on its color. Once the absolute and apparent magnitudes are known, the distance to the star can be determined by using the distance modulus.

Psi Leonis

ψ Leonis (Latinised as Psi Leonis, abbreviated to ψ Leo or psi Leo), is a solitary star located in the zodiac constellation of Leo, to the east-northeast of Regulus. It is faintly visible to the naked eye with an apparent visual magnitude of 5.38. Based upon stellar parallax measurements, it is located around 95 light years from the Sun. At that distance, the visual magnitude of the star is diminished by an absorption factor of 0.3 due to interstellar dust.Psi Leonis is an evolved red giant star with a stellar classification of M2 IIIab. It shines with a luminosity over 900 times that of the Sun from a relatively cool outer atmosphere that has an effective temperature of 3,756. It is a suspected variable star with a measured brightness variation of 0m.018. Psi Leonis has a magnitude 11.63 visual companion at an angular separation of 281.60 arcseconds along a position angle of 139°, as of 2000.

Samuel Molyneux

Samuel Molyneux FRS (16 July 1689 – 13 April 1728) was an amateur astronomer and politician who sat in the British House of Commons between 1715 and 1728 and in the Irish House of Commons from 1727 to 1728. His work with James Bradley attempting to measure stellar parallax led to the discovery of the aberration of light. The aberration was the first definite evidence that the earth moved and that Copernicus and Kepler were correct. In addition to his astronomical works, Molyneux wrote about the natural history and other features of Ireland. He died in suspicious circumstances.

TAU (spacecraft)

TAU (Thousand Astronomical Units) was a proposed unmanned space probe that would go to a distance of one thousand astronomical units (1000 AU) from the Earth and Sun by NASA/JPL in 1987 using tested technology. One scientific purpose would be to measure the distance to other stars via stellar parallax. Studies continued into 1990, working with a launch in the 2005–2010 timeframe.

It was a proposed nuclear electric rocket spacecraft that used a 1 MW fission reactor and an ion drive (with a burn time of about 10 years) to reach a distance of 1000 AU in 50 years. The primary goal of the mission was to improve parallax measurements of the distances to stars inside and outside the Milky Way, with secondary goals being the study of the heliopause, measurements of conditions in the interstellar medium, and (via communications with Earth) tests of general relativity.One of the tasks envisioned for TAU would be a flyby of Pluto. A Pluto flyby was achieved in 2015 by the New Frontiers program mission New Horizons.

Some of the instruments proposed for the design included a 1.5-meter telescope for observations and a 1-meter telescope for laser communication with Earth.After launch it would accelerate to about 106 km/s (about 22.4 AU/year) over 10 years, using xenon as propellant and a nuclear fission reactor for power.

T Ceti

T Ceti is a semiregular variable star located in the equatorial constellation of Cetus. It varies between magnitudes 5.0 and 6.9 over 159.3 days. The stellar parallax shift measured by Hipparcos is 3.7 mas, which yields a distance estimate of roughly 900 light years. It is moving further from the Earth with a heliocentric radial velocity of +29 km/s.This is an MS-type star on the asymptotic giant branch with a spectral type of M5-6Se. It is often classified simply as an M-type star, for example with the spectral type of M5.5e − M8.8e. (The 'e' notation indicates the presence of emission lines in the spectrum.) It is a long period Mira variable with changing cycle lengths, showing a variation in its spectral features over the course of each cycle. Pulsation periods of 388, 398, and 382 days have been reported, as well as variations in the amplitude, which may indicate dual pulsation cycles that are interfering with each other. The star is losing mass at the rate of 8.2×10−8 M☉ y−1, and it is surrounded by a circumstellar dust shell consisting of crystallized, mostly iron-rich silicates.T Ceti has an estimated three times the mass of the Sun and has expanded to 275 times the Sun's radius. It is radiating 8,128 times the Sun's luminosity from its enlarged photosphere at an effective temperature of 3,396 K.

Xi Geminorum

Xi Geminorum (ξ Geminorum, abbreviated Xi Gem, ξ Gem), formally named Alzirr , is a star in the zodiac constellation of Gemini. It forms one of the four feet of the outline demarcating the Gemini twins. The star has an apparent visual magnitude of 3.35, which is bright enough for it to be seen with the naked eye. From stellar parallax measurements, its distance from the Sun can be estimated as 58.7 light-years (18.0 parsecs).

Östen Bergstrand

Carl Östen Emanuel Bergstrand (September 1, 1873 – September 27, 1948) was a Swedish astronomer.

He was Professor of Astronomy at Uppsala University from 1909 until 1938 and from where he received his Ph.D. in astronomy in 1899 under Nils Christoffer Dunér. His early work was focused on astrometrics, particularly in the examination of photographic plates to measure stellar parallax. He used the orbital motions of the moons of Uranus to measure the rotation period and equatorial flattening of the planet. He also made studies of the solar corona, using photographs from the 1914 solar eclipse expedition.

He wrote works on astronomy for the general public, including Astronomi (1925).

The crater Bergstrand on the Moon is named after him.

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