Parallax in astronomy

Stellar parallax motion from annual parallax. Half the apex angle is the parallax angle.

The most important fundamental distance measurements in astronomy come from trigonometric parallax, as applied in the stellar parallax method. As the Earth orbits the Sun, the position of nearby stars will appear to shift slightly against the more distant background. These shifts are angles in an isosceles triangle, with 2 AU (the distance between the extreme positions of Earth's orbit around the Sun) making the base leg of the triangle and the distance to the star being the long equal-length legs. The amount of shift is quite small, even for the nearest stars, measuring 1 arcsecond for an object at 1 parsec's distance (3.26 light-years), and thereafter decreasing in angular amount as the distance increases. Astronomers usually express distances in units of parsecs (parallax arcseconds); light-years are used in popular media.

Because parallax becomes smaller for a greater stellar distance, useful distances can be measured only for stars which are near enough to have a parallax larger than a few times the precision of the measurement. In the 1990s, for example, the Hipparcos mission obtained parallaxes for over a hundred thousand stars with a precision of about a milliarcsecond,^[1] providing useful distances for stars out to a few hundred parsecs. The Hubble Space Telescope's Wide Field Camera 3 has the potential to provide a precision of 20 to 40 microarcseconds, enabling reliable distance measurements up to 5,000 parsecs (16,000 ly) for small numbers of stars.^[2]^[3] The Gaia space mission provided similarly accurate distances to most stars brighter than 15th magnitude.^[4]

Distances can be measured within 10% as far as the Galactic Center, about 30,000 light years away. Stars have a velocity relative to the Sun that causes proper motion (transverse across the sky) and radial velocity (motion toward or away from the Sun). The former is determined by plotting the changing position of the stars over many years, while the latter comes from measuring the Doppler shift of the star's spectrum caused by motion along the line of sight. For a group of stars with the same spectral class and a similar magnitude range, a mean parallax can be derived from statistical analysis of the proper motions relative to their radial velocities. This statistical parallax method is useful for measuring the distances of bright stars beyond 50 parsecs and giant variable stars, including Cepheids and the RR Lyrae variables.^[5]

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, while 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 observed stars is an additional unknown. When applied to samples of multiple stars, the uncertainty can be reduced; the uncertainty is inversely proportional to the square root of the sample size.^[8]

Moving cluster parallax is a technique where the motions of individual stars in a nearby star cluster can be used to find the distance to the cluster. Only open clusters are near enough for this technique to be useful. In particular the distance obtained for the Hyades has historically been an important step in the distance ladder.

Other individual objects can have fundamental distance estimates made for them under special circumstances. If the expansion of a gas cloud, like a supernova remnant or planetary nebula, can be observed over time, then an expansion parallax distance to that cloud can be estimated. Those measurements however suffer from uncertainties in the deviation of the object from sphericity. Binary stars which are both visual and spectroscopic binaries also can have their distance estimated by similar means, and do not suffer from the above geometric uncertainty. The common characteristic to these methods is that a measurement of angular motion is combined with a measurement of the absolute velocity (usually obtained via the Doppler effect). The distance estimate comes from computing how far the object must be to make its observed absolute velocity appear with the observed angular motion.

Expansion parallaxes in particular can give fundamental distance estimates for objects that are very far, because supernova ejecta have large expansion velocities and large sizes (compared to stars). Further, they can be observed with radio interferometers which can measure very small angular motions. These combine to provide fundamental distance estimates to supernovae in other galaxies.^[9] Though valuable, such cases are quite rare, so they serve as important consistency checks on the distance ladder rather than workhorse steps by themselves.

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Oh. Hey! Sorry, I don’t mean to be rude. I’m just trying to figure out how far away my thumb is. How? Parallax. Centuries ago, people thought the stars were holes in a huge crystal sphere, letting through heavenly light. It wasn’t clear just how big the sphere was, but it was pretty dang big. I have some sympathy for them. By eye, and for all intents and purposes, the stars are infinitely far away. If you drive down a road you’ll see trees nearby flying past you, but distant mountains moving more slowly. The Moon is so far it doesn’t seem to move at all compared to nearby objects — and it’s easy for your brain to think it’s much closer, smaller, and actually following you, which is a bit creepy. Sometimes people even think it’s a UFO tailing them. Finding the distance to something really far away is tough. It’s not like you can you can just pace off the distance. Or can you? The ancient Greeks knew the Earth was round and there are lots of ways to figure that out. For example, ships sailing over the horizon seem to disappear from the bottom up, as you’d expect as they slip around the Earth’s curve. But how big is the Earth? Over 2000 years ago, the Greek philosopher Eratosthenes figured it out. He knew that at the summer solstice, the Sun shone directly down a well in the city of Syene at noon. He also knew that at the same time, it was not shining straight down in Alexandria, and could measure that angle. There’s a legend that he paid someone to pace off the distance between the two cities so he could find the distance between them. But more likely he just used the numbers found by earlier surveying missions. Either way, knowing the distance and the angle, and applying a little geometry, he calculated the circumference of the Earth. His result, a little over 40,000 km, is actually amazingly accurate! For the very first time, humans had determined a scale to the Universe. That first step has since led to a much, much longer journey. Once you know how big the Earth is, other distances can be found. For example, when there’s a lunar eclipse, the shadow of the Earth is cast on the Moon. You can see the curve of the Earth’s edge as the shadow moves across the Moon. Knowing how big the Earth is, and doing a little more geometry, you can figure out how far away the Moon is! Also, the phases of the Moon depend on the angles and distances between the Earth, Moon, and Sun. Using the size of the Earth as a stepping stone, Aristarchus of Samos was able to calculate the distances to the Moon and the Sun as well as their sizes. That was 2200 years ago! His numbers weren’t terribly accurate, but that’s not the important part. His methods were sound, and they were used later by great thinkers like Hipparchus and Ptolemy to get more accurate sizes and distances. They actually did pretty well, and all over a thousand years before the invention of the telescope! And I think it also says a lot that these ancient thinkers were willing to accept a solar system that was at least millions of kilometers in size. But at this point things got sticky. Planets are pretty far away and look like dots. Our methods for finding distances failed for them. For a while, at least. In the 17th century, Johannes Kepler and Isaac Newton laid the mathematical groundwork of planetary orbits, and that in turn made it possible, in theory, to get the distances to the planets. Ah, but there was a catch. When you do the math, you find that measuring the distances to the other planets means you need to know the distance from the Earth to the Sun accurately. For example, it was known that Jupiter was about 5 times farther from the Sun than the Earth was, but that doesn’t tell you what it is in kilometers. So how far away is the Sun? Well, they had a rough idea using the number found by the Greeks, but to be able to truly understand the solar system, they needed a much more accurate value for it. To give you an idea of how important the distance from the Earth to the Sun is, they gave it a pretty high-falutin’ name: the astronomical unit, or AU. Mind you, not “an” astronomical unit, “the” one. That’s how fundamental it is to understanding everything! A lot of methods were attempted. Sometimes Mercury and Venus transit, or cross the face of the Sun. Timing these events accurately could then be used to plug numbers into the orbital equations and get the length of an AU. Grand expeditions were sent across the globe multiple times to measure the transits, and didn’t do too badly. But our atmosphere blurs the images of the planets, putting pretty big error bars on the timing measurements. The best they could do was to say the AU was 148,510,000 km -- plus or minus 800,000 km. That’s good, but not QUITE good enough to make astronomers happy. Finally, in the 1960s, astronomers used radio telescopes to bounce radar pulses off of Venus. Since we know the speed of light extremely accurately, the amount of time it takes for the light to get to Venus and back could be measured with amazing precision. Finally, after all these centuries, the astronomical unit was nailed down. It’s now defined to be 149,597,870.7 kilometers. So there. The Earth orbits the Sun on an ellipse, so think of that as the average distance of the Earth from the Sun. Knowing this number unlocked the solar system. It’s the fundamental meterstick of astronomy, and the scale we use to measure everything. Having this number meant we could predict the motions of the planets, moons, comets, and asteroids. Plus, it meant we could launch our probes into space and explore these strange new worlds for ourselves, see them up close, and truly understand the nature of the solar system. And it’s even better than that. Knowing the Astronomical Unit meant unlocking the stars. We have two eyes, and this gives us binocular vision. When you look at a nearby object, your left eye sees it at a slightly different angle than your right eye. Your brain puts these two images together, compares them, does the geometry, and gives you a sense of distance to that object. And you thought your teacher lied when she said math was useful in everyday life. We call this ability depth perception. You can see it for yourself by doing the thumb thing: as you blink one eye and then the other, your thumb appears to shift position relative to more distant objects. That shift is called parallax. The amount of shift depends on how far apart your eyes are, and how far away the object is. If you know the distance between your eyes — we’ll call this the baseline — then you can apply some trigonometry and figure out how far away the object is. If the object is nearby, it shifts a lot; if it’s farther away, it shift less. It works pretty well, but it does put a limit on how far away we can reasonably sense distance with just our eyes. Stars are a bit beyond that limit. If we want to measure their distance using parallax, we need a lot bigger baseline than the few centimeters between our eyes. Once astronomers figured out that the Earth went around the Sun rather than vice-versa, they realized that the Earth’s orbit made a huge baseline. If we observe a star when the Earth is at one spot, then wait six months for the Earth to go around the Sun to the opposite side of its orbit and observe the star again, then in principle we can determine the distance to the star, assuming we know the size of the Earth’s orbit. That’s why knowing the length of the astronomical unit is so important! The diameter of Earth’s orbit is about 300 million kilometers, which makes for a tremendous baseline. Hurray! Except, oops. When stars were observed, no parallax was seen. Was heliocentrism wrong? Pfft, no. It’s just that stars are really and truly far away, much farther than even the size of Earth’s orbit. The first star to have its parallax successfully measured was in 1838. The star was 61 Cygni, a bit of a dim bulb. But it was bright enough and close enough for astronomers to measure its shift in apparent position as the Earth orbited the Sun. 61 Cygni is about 720,000 astronomical units away. That’s a soul-crushing distance; well over 100 trillion kilometers! In fact, that’s so far that even the Earth’s orbit is too small to be a convenient unit. Astronomers came up with another one: The light year. That’s the distance light travels in a year. Light’s pretty fast, and covers about 10 trillion kilometers in a year. It’s a huge distance, but it makes the numbers easier on our poor ape brains. That makes 61 Cygni a much more palatable 11.4 light years away. Astronomers also use another unit called a parsec. It’s based on the angle a star shifts over the course of a year; a star one parsec away will have a parallax shift of one arcsecond—1/3600th of a degree. That distance turns out to be about 3.26 light years. As a unit of distance it’s convenient for astronomers, but it’s a terrible one if you’re doing the Kessel Run. Sorry, Han. The nearest star to the sun we know of, Proxima Centauri, is about 4.2 light years away. The farthest stars you can see with the naked eye are over a thousand light years distant, but the vast majority are within 100 light years. Space-based satellites are used now to accurately find the distance to hundreds of thousands of stars. Still, this method only works for relatively nearby stars, ones that are less than about 1000 light years away. But once we know those distances, we can use that information on more distant stars. How? Well, like gravity, the strength of light falls off with the square of the distance. If you have two stars that are the same intrinsic brightness—giving off the same amount of energy—and one is twice as far as the other, it will be ¼ as bright. Make it ten times farther away, it’ll be 1/100th as bright. So if you know how far away the nearer one is by measuring its parallax, you just have to compare its brightness to one farther away to get its distance. You have to make sure they’re the same kind of star; some are more luminous than others. But thanks to spectroscopy, we can do just that. A star’s distance is the key to nearly everything about it. Once we know how far it is, and we can measure its apparent brightness, we can figure out how luminous it is, how much light it’s actually giving off, and its spectrum tells us its temperature. With those in hand we can determine its mass and even its diameter. Once we figured out how far away stars are, we started to grasp their true physical nature. This led to even more methods of finding distances. The light given off by dying stars, exploding stars, stars that literally pulse, get brighter and dimmer over time. All of these and more can be used to figure out how many trillions of kilometers of space lie between us and them. And we see stars in other galaxies, which means we can use them to determine the actual size and scale of the Universe itself. And all of this started when some ancient Greeks were curious about how big the Earth was. Curiosity can take us a great, great distance. Today you learned that ancient Greeks were able to find the size of the Earth, and from that the distance to and the sizes of the Moon and Sun. Once the Earth/Sun distance was found, parallax was used to find the distance to nearby stars, and that was bootstrapped using brightness to determine the distances to much farther stars. Crash Course Astronomy is produced in association with PBS Digital Studios. Head over to their YouTube channel to catch even more awesome videos. This episode was written by me, Phil Plait. The script was edited by Blake de Pastino, and our consultant is Dr. Michelle Thaller. It was directed by Nicholas Jenkins, edited by Nicole Sweeney, the sound designer is Michael Aranda, and the graphics team is Thought Café.

Parsec

The parsec (symbol: pc) is a unit of length used to measure the large distances to astronomical objects outside the Solar System, approximately equal to 3.26 light-years or 206,265 astronomical units (AU), i.e. 30.9 trillion kilometres (19.2 trillion miles).^[a] The parsec unit is obtained by the use of parallax and trigonometry, and is defined as the distance at which 1 AU subtends an angle of one arcsecond^[10] (1/3600 of a degree). The nearest star, Proxima Centauri, is about 1.3 parsecs (4.2 light-years) from the Sun: from that distance, the gap between the Earth and the Sun spans slightly less than 1/3600 of one degree of view.^[11] Most stars visible to the naked eye are within a few hundred parsecs of the Sun, with the most distant at a few thousand parsecs, and the Andromeda Galaxy at over 700,000 parsecs.^[12]

The word parsec is a portmanteau of "parallax of one second" and was coined by the British astronomer Herbert Hall Turner in 1913^[13] to simplify astronomers' calculations of astronomical distances from only raw observational data. Partly for this reason, it is the unit preferred in astronomy and astrophysics, though the light-year remains prominent in popular science texts and common usage. Although parsecs are used for the shorter distances within the Milky Way, multiples of parsecs are required for the larger scales in the universe, including kiloparsecs (kpc) for the more distant objects within and around the Milky Way, megaparsecs (Mpc) for mid-distance galaxies, and gigaparsecs (Gpc) for many quasars and the most distant galaxies.

In August 2015, the International Astronomical Union (IAU) passed Resolution B2 which, as part of the definition of a standardized absolute and apparent bolometric magnitude scale, mentioned an existing explicit definition of the parsec as exactly 648000

π

au, or approximately 3.0856775814913673×10¹⁶ metres (based on the IAU 2012 definition of the astronomical unit). This corresponds to the small-angle definition of the parsec found in many astronomical references.^[14]^[15]

Stellar parallax

Stellar parallax created by the relative motion between the Earth and a star can be seen, in the Copernican model, as arising from the orbit of the Earth around the Sun: the star only appears to move relative to more distant objects in the sky. In a geostatic model, the movement of the star would have to be taken as real with the star oscillating across the sky with respect to the background stars.

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.^[16] 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 the surfaces of planets.^[17]

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 0.7687 ± 0.0003 arcsec.^[18] This angle is approximate that subtended by an object 2 centimeters in diameter located 5.3 kilometers away.

Hubble Space Telescope – Spatial scanning precisely measures distances up to 10,000 light-years away (10 April 2014).^[19]

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 gigantic distances involved seemed entirely implausible: it was one of Tycho's principal objections to Copernican heliocentrism that 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 (then the most distant known planet) and the eighth sphere (the fixed stars).^[20]

In 1989, the satellite Hipparcos was launched primarily for obtaining improved parallaxes and proper motions for over 100,000 nearby stars, increasing the reach of the method tenfold. Even so, Hipparcos was 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 European Space Agency's Gaia mission, launched in December 2013, can measure parallax angles to an accuracy of 10 microarcseconds, thus mapping nearby stars (and potentially planets) up to a distance of tens of thousands of light-years from Earth.^[21]^[22] In April 2014, NASA astronomers reported that the Hubble Space Telescope, by using spatial scanning, can precisely measure distances up to 10,000 light-years away, a ten-fold improvement over earlier measurements.^[19]

Diurnal parallax

Diurnal parallax is a parallax that varies with the rotation of the Earth or with a difference in location on the Earth. The Moon and to a smaller extent the terrestrial planets or asteroids seen from different viewing positions on the Earth (at one given moment) can appear differently placed against the background of fixed stars.^[23]^[24]

The diurnal parallax has been used by John Flamsteed in 1672 to measure the distance to Mars at its opposition and through that to estimate the astronomical unit and the size of the Solar System.^[25]

Lunar parallax

Lunar parallax (often short for lunar horizontal parallax or lunar equatorial horizontal parallax), is a special case of (diurnal) parallax: the Moon, being the nearest celestial body, has by far the largest maximum parallax of any celestial body, at times exceeding 1 degree.^[26]

The diagram for stellar parallax can illustrate lunar parallax as well if the diagram is taken to be scaled right down and slightly modified. Instead of 'near star', read 'Moon', and instead of taking the circle at the bottom of the diagram to represent the size of the Earth's orbit around the Sun, take it to be the size of the Earth's globe, and a circle around the Earth's surface. Then, the lunar (horizontal) parallax amounts to the difference in angular position, relative to the background of distant stars, of the Moon as seen from two different viewing positions on the Earth.

One of the viewing positions is the place from which the Moon can be seen directly overhead at a given moment. That is, viewed along the vertical line in the diagram. The other viewing position is a place from which the Moon can be seen on the horizon at the same moment. That is, viewed along one of the diagonal lines, from an Earth-surface position corresponding roughly to one of the blue dots on the modified diagram.

The lunar (horizontal) parallax can alternatively be defined as the angle subtended at the distance of the Moon by the radius of the Earth^[27]^[28]—equal to angle p in the diagram when scaled-down and modified as mentioned above.

The lunar horizontal parallax at any time depends on the linear distance of the Moon from the Earth. The Earth-Moon linear distance varies continuously as the Moon follows its perturbed and approximately elliptical orbit around the Earth. The range of the variation in linear distance is from about 56 to 63.7 Earth radii, corresponding to a horizontal parallax of about a degree of arc, but ranging from about 61.4' to about 54'.^[26] The Astronomical Almanac and similar publications tabulate the lunar horizontal parallax and/or the linear distance of the Moon from the Earth on a periodical e.g. daily basis for the convenience of astronomers (and of celestial navigators), and the study of how this coordinate varies with time forms part of lunar theory.

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 degrees, from which (with the solar apparent radius of 0.25 degrees) we get an Earth apparent radius of 1 degree. This yields for the Earth-Moon distance 60.27 Earth radii or 384,399 kilometres (238,854 mi) This procedure was first used by Aristarchus of Samos^[29] and Hipparchus, and later found its way into the work of Ptolemy.^[30]

The diagram at the right shows how daily lunar parallax arises on the geocentric and geostatic planetary model, in which the Earth is at the center 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 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} _{\mathrm {moon} }={\frac {\mathrm {distance} _{\mathrm {observerbase} }}{\tan(\mathrm {angle} )}}

This is the method referred to by Jules Verne in his 1865 novel From the Earth to the Moon:

Until 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 about 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

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 center 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^[31] 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 (with the 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 about 390 times farther away.^[29]

Aristarchus 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 suggests that the Sun is larger than the Earth, which could be taken to support the heliocentric model.^[32]

Although Aristarchus' 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.^[29] 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, such as Eros, pass much closer to Earth than Venus. In a favorable opposition, Eros can approach the Earth to within 22 million kilometers.^[33] During the opposition of 1900–1901, a worldwide program was launched to make parallax measurements of Eros to determine the solar parallax (or distance to the Sun), with the results published in 1910 by Arthur Hinks of Cambridge^[34] and Charles D. Perrine of the Lick Observatory, University of California.^[35]

Perrine published progress reports in 1906^[36] and 1908.^[37] He took 965 photographs with the Crossley Reflector and selected 525 for measurement.^[38] A similar program was then carried out, during a closer approach, in 1930–1931 by Harold Spencer Jones.^[39] The value of the Astronomical Unit (roughly the Earth-Sun distance) obtained by this program was considered definitive until 1968, when radar and dynamical parallax methods started producing more precise measurements.

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. The currently accepted value of solar parallax is 8.794143 arcseconds.^[40]

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.^[41]

Dynamical parallax

Dynamical parallax has sometimes also been used to determine the distance to a supernova when the optical wavefront 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.^[42]

Spatio-temporal parallax

From enhanced relativistic positioning systems, spatio-temporal parallax generalizing the usual notion of parallax in space only has been developed. Then, event fields in spacetime can be deduced directly without intermediate models of light bending by massive bodies such as the one used in the PPN formalism for instance.^[43]

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 individually 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. 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. 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.^[44]

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.^[44]^[45]

Other methods for distance measurement in astronomy

In astronomy, the term "parallax" has come to mean a method of estimating distances, not necessarily utilizing a true parallax, such as:

Notes

^ One trillion here is short scale, ie. 10¹² (one million million, or billion in long scale).

References

^ Perryman, M. A. C.; et al. (1999). "The HIPPARCOS Catalogue". Astronomy and Astrophysics. 323: L49–L52. Bibcode:1997A&A...323L..49P.
^ Harrington, J. D.; Villard, R. (10 April 2014). "NASA's Hubble Extends Stellar Tape Measure 10 Times Farther Into Space". NASA. Archived from the original on 17 February 2019. Retrieved 17 October 2014.
^ Riess, A. G.; Casertano, S.; Anderson, J.; MacKenty, J.; Filippenko, A. 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. S2CID 55928992.
^ 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.
^ B., Baidyanath (2003). An Introduction to Astrophysics. PHI Learning Private Limited. ISBN 978-81-203-1121-3.
^ "Hubble finds Universe may be expanding faster than expected". Archived from the original on 11 September 2018. Retrieved 3 June 2016.
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