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Explanations of stellar parallax that I have found involve examining the apparent motion of a nearer star relative to a background of more distant stars, as the Earth moves around the sun or as the Earth rotates. We can see the apparent motion because we have telescopes. In Tycho Brahe's time, telescopes were not available, and this kind of motion couldn't be seen. The fact that stellar parallax couldn't be observed is supposed to be evidence that Tycho and others took as evidence that the Earth didn't move.

However, as I understand it, Tycho believed that the stars move together in a fixed sphere, so it seems that he would not even try to look for parallax using the near star/far star method. He would have had to look for motion of the moon or a planet relative to the stars. The moon and planets were in constant motion, however. This doesn't mean that you can't test for parallax, but doing so would be more complicated. How was this done?

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Parallax was always measured by measuring the angular distances between stars. If it changes during the year, you see the parallax due to the Earth motion. The accuracy is limited by the accuracy of your angle-measuring device. A typical such device at the time of Brahe could measure to 1' or perhaps 0.5' (1' is the average resolution of the human eye). This is not enough to detect any parallax of "fixed stars". Of course they always attempted to measure it, because this would decide whether the Earth moves or not (with respect to the stars). The negative result was actually one of the strongest arguments in favor of geocentric theories. To deal with this argument against his theory, Copernicus had to postulate really HUGE distance to the stars, the thing which did not look plausible to many.

Parallax was actually detected only in 18th century.

The whole story is told in great detail in this book:

Parallax: The Race to Measure the Cosmos, by Alan W. Hirshfeld

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    $\begingroup$ It is a strange sentence: "why would anyone measure...if they believed...". One reason why "they believed" is exactly that nobody has detected parallax. This argument is given in Ptolemy, together with other arguments. Scientists always try to test such "beliefs". This is how science works. $\endgroup$ Sep 27 '17 at 13:29
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    $\begingroup$ But the belief Tycho and Copernican were arguing about, at least as its usually presented, isn't whether the stars were on a fixed sphere, but whether that sphere was near or far. Parallax against a background doesn't resolve this question if all the objects you're observing are on that background. Both models predict the same results. $\endgroup$
    – simplicio
    Sep 27 '17 at 17:26
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    $\begingroup$ @simplicio: Your argument is wrong. Imagine a hollow sphere to which stars are attached. If you stay at the center, you observe fixed angular distances between them. But if you move (say rotate about the Sun which is in the center), the observed angular distances will change periodically with period 1 year. So if the sphere is of reasonable size (compared with the radius of the Earth orbit) you will observe a yearly change of some star distances. $\endgroup$ Jul 15 at 1:38
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    $\begingroup$ Actually Stellar paralax was first detected in the 1830s, no tin the 18th century. $\endgroup$ Oct 5 at 17:21
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    $\begingroup$ Stellar parallaxis not always measured by the angular distances between stars. That depends on one star being much closer than the nextstar. It is also possible to measure changes in the direction to the star from the observatory on Earth. Early astronomers believed that the stars were all attached to the inside of a giant hollow sphere, and thus were all the same distance from Earth. Thus they didn't believe in comparing the changing angles between nearer and farther stars, but in measuring the change in direction to the star over the course of a year. $\endgroup$ Oct 5 at 17:25
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Actually, ancient, medieval, and early modern astronomers mostly believed that the stars were lights attached to the inside of a hollow sphere centered on Earth and that the hollow sphere rotated 360 degrees every day.

Those who believed in the heliocentry theory believed that the stars were on the inside of a hollow sphere centered on the Sun. They believed that the hollow sphere that the stars were stuck to stood still and the Earth rotated 360 degrees every day.

Thus they almost all believed that all the stars were at the same distance from the center of the star sphere. So they didn't believe that some stars were nearer than other stars and didn't think that you could measure shifts in the relative poitions of the nearer stars compared to the farther stars, because they didn't believe in nearer and farther stars.

In the heliocentric theory Earth would orbit around the Sun once a year, and every six months it would be on the opposite side of the Sun from where it was 6 months before and where it would be six months later.

Nobody knew what the distance between Earth and Sun - which is called the Astronomical Unit, or AU - was in ancient times. Hellenistic philosophers calculated the size of the Earth and the relative size and distance of the Moon. Since the Moon sometiems eclipsed the Sun, they knew that the Sun was farther than the Moon, but didn't know how many times farther.

So if the orbit of the Earth was 1 AU (however long an AU was) from the Sun, Earth's position at any one moment would be two AU away from where it would be 6 months before or after that moment.

Ancient astronomers measured the directions to stars. They measured the angle of how far above the horizon a star was at a specific moment, and the angle of how far horizontally it was from a landmark at a specific moment. And they did that for hundreds and even a few thousand stars visible without telescopes. So they made maps of the celestial sphere, which they imagined was an actual physical sphere, and lists of the coordinates of stars on that sphere.

If the heliocentric theory was true, the Earth would travel to points 2 AU apart in 6 months. If the sphere of the stars was 20 AU from the Sun, ancient astronomers could have measured shifts in the positions of all the stars every six months. They didn't

If the sphere of the stars was 200 AU from the Sun, ancient astronomers culd have measured shifts in the positions of all the stars every six months. They didn't.

So ancient astronomers knew that either the heliocentric theory was wrong, or the distance to the sphere of the starswas many times as great as the distance between Earth and the Sun - and they knew that the distance between Earth and the Sun was at least as great as the distance between Earth and the Moon, which they knew was about 250,000 modern miles.

Most ancient astronomers who knew of the heliocentric theory couldn't believe that the sphere of the stars could be as far away as it would have to be if they couldn't measure any annual parallax. So they believed in the geocentric theory instead of the heliocentric theory.

The precession of the equinoxes is a long term wobble in the Earth's axis of rotation, taking about 25,000 years to make one full circle.

The ancient Greek astronomer Hipparchus (c. 190–120 BC) is generally accepted to be the earliest known astronomer to recognize and assess the precession of the equinoxes at about 1° per century (which is not far from the actual value for antiquity, 1.38°),[6] although there is some minor dispute about whether he was.[7] In ancient China, the Jin-dynasty scholar-official Yu Xi (fl. 307–345 AD) made a similar discovery centuries later, noting that the position of the Sun during the winter solstice had drifted roughly one degree over the course of fifty years relative to the position of the stars.[8]

https://en.wikipedia.org/wiki/Precession

The heliocentric theory was revived and expanded by Coppernicus in the early 1500s and became widely known. Copernicus was able to deduce the relative sizes of the planetary orbits, so that if the width of any orbit could be determined the sizes of all would be known.

In 1609 Galleleo began his telescopic observantions, making many discoveries. Johannes Kelper later discovered that the planets orbit in ellipical orbits, getting near to and farther from the Sun during each orbit, thus convincing those who believed in the heliocentric theory to give up the idea that the planets were attached to giant transparent spheres. If the planetary spheres didn't exist, maybe the sphere of the stars didn't exist.

In 1672 the parallax of Mars was observed from 2 different spots on Earth, thus providing the first reasonably accurate measurement of the AU. More and more accurate measurements hae been made in the following centuries.

Precession results in all the stars appearing to move in the same direction and distance. But there is also proper motion of stars, very slow apparent motion independent of precession.

Proper motion was suspected by early astronomers (according to Macrobius, AD 400) but a proof was not provided until 1718 by Edmund Halley, who noticed that Sirius, Arcturus and Aldebaran were over half a degree away from the positions charted by the ancient Greek astronomer Hipparchus roughly 1850 years earlier.[23]

https://en.wikipedia.org/wiki/Proper_motion#History

The Sun and the other stars are all orbiting the center of the galaxy with somewhat different speeds and directions, thus resulting in a slow change in the direction to each star as seen from another star, such as the Sun.

Once astronomers accepted proper motion, they had to give up the idea that all the stars were stuck to the inner surface of a hollow sphere, and had to accept that they were flying around in three dimensional space.

Proper motion means that if Earth didn't orbit around the Sun, a star's apparent motion across the sky over many years would appear to be a straight line. But because Earth does orbit the Sun, a star's apparent motion will actually be a sort of wobbling line acorss the sky.

In a treatise in 1755, Immanuel Kant, drawing on earlier work by Thomas Wright,[229] speculated (correctly) that the Milky Way might be a rotating body of a huge number of stars, held together by gravitational forces akin to the Solar System but on much larger scales.[230] The resulting disk of stars would be seen as a band on the sky from our perspective inside the disk. Wright and Kant also conjectured that some of the nebulae visible in the night sky might be separate "galaxies" themselves, similar to our own. Kant referred to both the Milky Way and the "extragalactic nebulae" as "island universes", a term still current up to the 1930s.[231][232][233]

https://en.wikipedia.org/wiki/Milky_Way

So by then astronomers seemed to have abandoned the idea that all stars were at the same distance.

Searching for stellar parallaxes, James Bradley discovered another tiny apparent motion of stars, caused by the abberation of light due to Earth's motion around the Sun inn1725-1727. Observing stars for 20 years from 17271747, Bradley proved that abberation existed, and also discovered another woblbe in stellar posiitons due to nutation, a wobble of the Earth over an 18.6 year period (there are other smaller amounts of nutatoion with different periods).

But the abberation of light and nutation make much larger changes in the apparent positons of stars than stellar parallax does.

Finally, in the 1830s, astronomical instruments and techniques adavanced enough for the prallaxes of stars to be measure. Despite al lthe evidence over the centuries of greater and greater minimum possible distances to the stars, the distances measured were shockingly to astronomers.

Thomas Henderson, at the Cape of Good HOpe, measured the parallax of Alpha Centauri, but couldn't believe how distant it was and didn't publish his results. Struve measured the parallax of Vega fairly accurately and published his results. And then Friderich Wilhelm Bessel measured the parallax of 61 Cygni. Since 61 cygni was much closer than Vega, Struve chickened out and published a revised parallax for Vega that was closer to the parallax of 61 Cygni and not nearly as accurate. And so Bessel is usually credited with making the first stellar parallax measurement.

So by 1840 astronomers knew that stars had different distances and were not attached to ine inside of a sphere.

In the later 19th century astronomical photography became more and more common and it became used for parallax measurements.

I believe that one technique became to air a telescope at a star believed to be many times father than the target star, and ppearing lcose to it in the sky. They would take photos of the sky around the distant star over time, and then measure how much the position of the target star moved compared to the position of the star the telescope was aimed at.

In modern times electronic light detecting devices have replaced photography for many astronomical purposes.

And at the present time many telescopes are capable of detecting very faint and distant galaxies. Those faint background galaxies can be seen all over the sky except in the plane of the Mikly Way Galaxy where all he stars and lcouds of gas and dust blot them out.

So it might be the modern technique to aim the telescope at a distant background galaxy thousands or millions of times farther than the target star to see how much the target star moves compared to the distant galaxy and measure its parallax.

The Hipparchos satellite automatically measured the positions, proper motions, and parallaxes of hundreds of thousands of stars from 1889-1993. The Gaia satallite from 2013-c.2022 has been measuring the parallaxes of millions of stars with even greater precision.

And I guess that the logical next step in parallax precision would be to develop satellites even more precise and accurate than Gaia and put one in Jupiter's L4 posiiton and one in Jupiter's L5 position, where they will always be about 10 AU apart, 5 times the maximum baseline for Earthbound observatiories, Hipparchos, and Gaia.

And the next step after that would be to put observatories in the L4 and L5 positions of Neptune, where they will always be about 60 AU apart, 30 times teh baseline on Earth.

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