In most introductory books on astronomy Tycho Brahe's contributions to astronomy are presented as somewhat of a footnote to the story of the overturning of the old Ptolemaic, Earth-centered view of the cosmos, and its replacement by the new Copernican, Sun-centered view. It is one of the low points of Tycho's career, his proposal of a compromise between the two views in which most of the planets go around the Sun as in the Copernican theory, but the Moon and Sun go around the Earth as in the Ptolemaic view, that is most often discussed in introductory texts. However, Tycho's observations of the heavens did far more to establish a modern view of the heavens than is generally made clear, and as a result it is worth some time to explain how important Tycho was to the development of modern astronomy, despite the fact that in some respects he kept one foot firmly planted in an older view of things that makes him seem, in comparison with some of his contemporaries, a little behind the times.
  One thing that should be noted, although not covered here, is that Tycho was an extremely interesting and charismatic individual, a fabulously wealthy man who had little hesitation about spending huge sums of his own money to accomplish his goals, and even less hesitation about spending even larger sums of other people's money. For a discussion of his life refer to a Biography of Tycho Brahe. Here we shall only consider how he changed our view of the heavens.

1572 - 1574: Tycho Proves That The Heavens Are Not Immutable
  In ancient times and in early Renaissance times, when the ancient Greek texts which had been preserved by the Arabs during the thousands of years after the fall of ancient empires were recovered and translated into Western languages after the fall of Granada, it was presumed that the heavens were immutable and unchanging. The stars that were in the sky now were the same stars that had been there since the beginning of time; since indeed, the creation of the Universe by God (or in ancient views, by the Gods). But early on the evening of November 11, 1572, Tycho was surprised to see, nearly overhead, a brilliant star in the constellation of Cassiopeia which he had never noticed before. Summoning one of his assistants to verify that he wasn't just "seeing" things, he began a series of observations of what we now realize must have been a supernova (the explosive death of a massive star). It turned out that Tycho wasn't the first to notice the star, but his extensive observations, published in 1574, established beyond any doubt that it was a real phenomenon out in stellar space, and not simply some atmospheric or other near-Earth apparition.
  To see how Tycho established this fact, consider how the position of the Moon changes during the course of its passage across the sky. In the diagram below, the larger circle on the left represents the Earth, and the smaller one on the right the Moon. As is fairly typical of such diagrams, the two circles are drawn to approximately the correct relative size, but the diagram is not to scale; if it were, the Moon would have to be thirty Earth diameters away from the Earth, which would put it much further to the right, and far out of view. The Earth is cut in half by a vertical line and shaded differently on the left and right, to emphasize the fact that one part of the Earth, namely the part that is on the side that the Moon is on, is able to see the Moon, while the other part is not able to see the Moon. For the purposes of this discussion we assume that we are looking down on the orbit of the Moon from somewhere above the North Pole of the Moon's orbit, in which case the Earth would be rotating counter-clockwise in this diagram, and the Moon would be going around it in the same direction, so it would be moving upwards at the position shown in the diagram. In that case someone at point A on the Earth would see the Moon rising on their eastern horizon, someone at point B would see the Moon overhead, and someone at point C would see the Moon setting on the western horizon.

The lunar parallax (the angle that the Moon seems to move relative to its average motion as it crosses the sky) is equal to the apparent size of the Earth as seen from the Moon.

  When doing calculations of the Moon's motion around the Earth and its resulting apparent position among the stars, we would usually imagine that we are at the center of the Earth, so that our changing position on the Earth doesn't complicate the calculations. Doing so yields an apparent position exactly the same as the one seen by someone at point B, because they lie exactly along the line from the center of the Earth to the Moon, and would see the Moon in the same direction as an imaginary observer located at the Earth's center. However, as shown by the lines connecting points A and C to the Moon, people who are at different places on the Earth see the Moon in different places compared to the calculated position for point B. Someone at point A would see the Moon as being somewhat higher in the diagram, which means further along its path than it would appear to be as seen by someone at point B, and someone at point C would see the Moon as being somewhat lower in the diagram, which means not quite as far along its path as it would appear to be when seen by someone at point B. As a result, if the three different observers were to plot the position of the Moon at the same time, they would show it in slightly different positions against the starry background by an amount corresponding to the size of the Earth as seen from the Moon. This change in position is referred to as the lunar parallax caused by the Earth's physical size, and can easily be measured, as it is approximately 2 degrees, or four times the size of the Moon itself, which is approximately 1/2 a degree across. This result is shown in the next figure:

  The view shown in this diagram assumes that we can see where observers on different parts of the Earth would see the Moon at the same time, but of course in Tycho's day this wouldn't have been possible, as it would have taken weeks or months for notes and diagrams made in one place to travel halfway across the Earth. But you don't have to worry about that, because even if you stay in one place the rotation of the Earth will, in just one day, take you from point A to B to C, then all the way back to A again. During that time, the Moon will also move, and that has to be taken into account, but that can be done as shown in the following figure:

The overall motion is to the East, as shown by the large arrow. The parallax is to the West, as shown by the small arrow. As a result, the apparent motion motion of the Moon while the Moon is up (moving from the A on the right to C) is smaller than its apparent motion while it is down (from C to the A on the left).

  On the right side of the diagram you can see where the Moon would appear to be when it rose as seen by someone at point A on one day, and on the left side you can see where it would appear to be as seen from the same place when it rose on the next day. The difference in position, which is around 13 degrees, or 26 lunar diameters, is a fairly typical motion for the Moon during one day, as it only takes 27 1/3 days for the Moon to go all around the sky. Near the center of the diagram you can see two other positions for the Moon. The one on the left, marked M, is where the Moon would appear to be when it set if it moved uniformly across the sky during the approximately 12 hours and 25 minutes that it takes, on average, from moonrise to moonset. The one on the right, approximately 2 degrees to the right of that, marked C, is where the Moon would actually be observed to set, as a result of the parallax caused by the observer's motion across the face of the Earth from point A to point C while the Moon was going around us. As you can see there is a considerable error between the "midway" point that the Moon would be at if its motion were unaffected by our motion across the face of the Earth, compared with its actual position. This large "error" allowed Tycho to do several things, some of which were done by others, but none with as great an accuracy or as telling a result.
  The first thing to note is that the size of the "error" in the Moon's position, which is approximately 2 degrees, is about 4 times the apparent size of the Moon, which is only about 1/2 a degree. This means that as seen from the Moon, the Earth must appear 4 times larger than the Moon appears as seen from the Earth, and since we are at the same distance from each other, the Earth must be about 4 times larger than the Moon. Similarly, through a geometrical calculation based on the 2 degree angle of parallax that our motion across the face of the Earth produces in the apparent position of the Moon, we can calculate that the orbit of the Moon must be about 30 times larger than the Earth's diameter. Presuming that the actual size of the Earth were known to be about 8000 miles, this tells us that the Moon is about 2000 miles across, and nearly a quarter of a million miles away.
  More importantly, however, from the standpoint of this discussion, which is about the discovery of "Tycho's supernova", the "new" star or "stella nova" in Cassiopeia, is the fact that although Tycho made the greatest possible effort to observe changes in the position of that star as it rose, moved across the sky and set over the course of the several months that it was still visible to the naked eye, he never observed it to have any parallax due to our motion across the face of the Earth. This meant that the nova or new star could not merely be some transient and presumably unimportant local phenomenon, but had to lie far beyond the orbit of the Moon. In fact, given the considerable effort that Tycho took to observe the star's position, he estimated that it must be at least a hundred times further away than the Moon, as he would otherwise have been able to observe at least some change in its position. But that would put it tens of millions of miles out in space, at a distance so vast that it must literally lie among the other stars. This meant that one of the fundamental tenets of Ptolemaic astronomy, the unchanging and immutable nature of the stellar sky, was wrong. New stars could and did appear, and if one part of Ptolemy's astronomy was wrong, who knew what other parts might also be wrong?

1577- 1578: Tycho Proves That The Celestial Spheres Do Not Exist
  As it turned out, it did not take that long, astronomically speaking, for Tycho to show that another of the fundamental bases of Ptolemaic astronomy was also wrong. In Ptolemy's astronomy as interpreted in Tycho's time, the sphere of the stars was not just an apparent sphere, but a real, solid, crystalline sphere which actually physically carried the stars around us once each day. Similarly, the motions of the planets, although very complicated in comparison with those of the stars, were controlled by the motions of other just as real and solid crystalline spheres, whose turnings were as smooth and unchanging and uniform as could possibly be imagined. You couldn't see the spheres because they were made of absolutely transparent ethereal material, as impossible to observe as the Emperor's New Clothes. But as it turned out, the reason for that was that they were just as evanescent and unreal as the fairy-tale costume, and Tycho proved that, beyond any doubt, only a few years later, by observing the great comet of 1577, now known as Tycho's Comet, because of the wonderful results he obtained by observing it. Just as in the case of the Nova of 1572, he was able to demonstrate that the Comet had absolutely no parallax due to our daily motion around the Earth, so that it must lie well beyond the orbit of the Moon, out in the realm occupied by the stars and planets. Most tellingly, by comparing how it brightened and dimmed as it moved across the sky, he was able to estimate how it must have moved through the Solar System and to prove, beyond any doubt, that it would have had to pass right through the crystalline spheres that supposedly turned the planets around the Sun, presuming the spheres actually existed. But not only the comet, but its head and tail and everything associated with its appearance passed through the sky without any evidence of any disturbance or impediment. Thus, in one fell swoop Tycho established that comets, which had once been thought to be supernatural or atmospheric disturbances, were actually members of the celestial bestiary, and that the space between the planets, once thought to be filled with any number of invisible but otherwise solid, crystalline spheres, was actually absolutely empty space, devoid of anything save the planets and comets and other non-stellar objects which happened to pass through it, on one occasion or another.
  Having swept away the underpinnings of ancient Greek and early Renaissance astronomy, Tycho next turned to the problem of what the cosmos must actually be like. It seemed obvious to him that the Ptolemaic system had to be rejected, and the complicated motions of crystalline spheres within and around other crystalline spheres, which Ptolemy had used to explain the retrograde motions of the planets, must be complete nonsense. Instead, the Copernican explanation of retrograde motion as being caused by the motions of the other planets around the Sun while the Earth and Sun changed their relative positions, must be correct. However, that did not mean that Tycho was ready to accept Copernicus' theory in full. For two reasons, one due to incorrect philosophical considerations which would be swept away soon after he died, and one caused by a failure of imagination that was especially unfortunate given his other brilliant successes, he rejected the idea that the Earth moved, and although he accepted the idea that the "other" planets moved around the Sun, he claimed that the Moon and Sun must move around the Earth.

Tycho Fails To Observe Stellar Parallax, Or To Understand That Failure
  The philosophical reason that Tycho rejected the idea that the Earth went around the Sun was that he firmly believed in Aristotelian theories of motion, which proclaimed that if the Earth were moved, things would appear to move differently than they are observed to move. Things wouldn't necessarily fall towards the Earth, but might fly off into space. Things thrown in one direction might veer in other directions, because the Earth's motion was different from their own motion. The Moon, for instance, might be left behind, lost in one path in space, while the Earth went off in a separate direction. And perhaps equally important, the Earth would be denied the divine position of centrality fundamental to the Christianity of his day. All these philosophical and religious considerations would, however, have undoubtedly been of no importance to him if he had been able to observe stellar parallax.
  Stellar parallax is the same sort of thing as the lunar parallax described above, save for the particular mechanism involved in creating it. In the case of lunar parallax, because the Moon is fairly close to the Earth (only 30 diameters away from us), it is possible to easily observe the change in its motion during a single night, as a result of our motion around the Earth's axis of rotation. But the stars are much, much further away, as Tycho's own observations of the Nova of 1572 and the Comet of 1577 had shown. In neither case had he been able to see any change of position due to the rotation of the Earth, showing that the objects were at least dozens and perhaps hundreds of times further out in space than the Moon. However, there was a way, suggested by the Copernican theory of the cosmos, to observe stellar parallax. Not by using the motion of an observer around the Earth, which was far too small to be of any consequence in comparison to the vast size of the cosmos, but by using the motion of the Earth around the Sun, which would have to be many hundreds or thousands of times greater than the rotational motion of the Earth (and is in fact nearly twenty five thousand times greater).
  It is easy to see that this must be the case, for just as in the case of the Comet and the Nova, the planets' motions from night to night and during one night showed no measurable parallax due to the rotation of the Earth, meaning that they were dozens or hundreds of times further out in space than the Moon. Since the Moon is 30 Earth diameters away from us, the planets had to be hundreds or thousands of Earth diameters away from us, and if we orbit the Sun our orbit would be of similar size. But in that case, by observing the positions of stars from different portions of the Earth's very large orbit (in other words at different times of the year), it would be possible to increase the apparent parallax (or change of position of the stars due to our motion) by many hundreds or thousands of times, compared to the effect that would be observed in the case of the lunar parallax.
  Now if Tycho had been a sloppy observer, or a lazy one, so that his observations were not terribly accurate or terribly numerous, it might have been possible for the stars to have a measurable parallax, and yet not have him notice it. But he was not. He was an indefatigable observer who spent virtually every clear night for the best part of twenty years making the most accurate stellar and planetary observations that had ever been made. He observed the positions of over a thousand stars, and made thousands and thousands of observations of the changing positions of the planets, to an accuracy better than most people of even excellent vision could ever hope to see, let alone measure. Prior to his efforts it was thought reasonable to measure positions of celestial bodies to accuracies of a degree or so, but he routinely made measurements accurate to 1/30th of a degree accuracy, and in some cases, to even 1/50th or 1/100th of a degree accuracy. Even though his observations were all "naked eye" observations made before the invention of the telescope, it would be nearly a century and a half before observations of the positions of the stars and planets would be routinely made as accurately as his own observations. But despite this incredible accuracy, Tycho could not see any change in the positions of the stars as the Earth supposedly moved around the Sun. As the years piled up without any hint of anything resembling stellar parallax showing up in any of his observations, he became more and more convinced that the rational, logical laws of Aristotelian physics must be right. The Earth must be fixed in the center of the Universe, absolutely unmoving, while everything else went around it. Copernicus must be right as well, in some ways. The planets must move around the Sun, in order to provide the retrograde motions that they exhibited. But the Sun and Moon must move around the Earth, and even the vast sphere of the stars, the one true sphere which might actually have some real existence, must turn each day around an immovable Earth.

The Tychonic Cosmos. The Earth is fixed in the center of the Universe. The stars revolve around us every day, the Moon every month, and the Sun every year, while the other planets all orbit the Sun.

Why Did Tycho Fail To Observe Stellar Parallax, or to Properly Interpret That Failure?
  As mentioned above, Tycho had strong philosophical reasons for wanting to believe that the Earth was fixed in space. But above all he was an outstanding observer, and there is no doubt that if he had been able to detect the stellar parallax produced by the Earth's orbital motion around the Sun, he would have given up his preferred view of the cosmos and accepted the reality his observations would have forced upon him. So the question of why he didn't observe any parallax is of great importance in understanding his failure to accept, or even prove, the validity of the Copernican cosmos.
  The answer lies in the fact that the stars are unbelievably far away. Although Tycho's observations were very accurate, they were, after all, made without the aid of a telescope, and parallaxes of less than about 1/30th of a degree would have been too small for him to notice. As it turns out this means that he could have only detected stellar parallax if the stars were less than two or three thousand Astronomical Units away (an Astronomical Unit being defined as the approximate radius of the Earth's orbit, which, being the distance that the Earth would be covering during its orbital motion, is the appropriate sort of unit for comparing distances calculated by such measurements). As it happens, an Astronomical Unit is a vast distance, nearly 100 million miles, or 150 million kilometers, so the stars would have had to be several hundred billion miles or kilometers away in order for him to not notice their parallax. But as vast as these distances are, they pale to insignificance in comparison with the actual distance to the stars. The very closest stars that we know of are a thousand times further away than the greatest distances that Tycho could have detected, and most of the stars visible in the night-time sky are tens or hundreds of times further yet. So the reason he didn't detect stellar parallax was that the stars were simply too far away for him to do so. Naked-eye observations are just not accurate enough to detect any change in stellar positions, even though we move nearly 200 million miles across the Solar System, from one side of our orbit to the other. In fact, even looking through a telescope with a magnification of hundreds of times is far too crude a way of measuring stellar positions to detect stellar parallax. Even if you take a photograph with a telescope and put the photograph under a microscope, in the vast majority of cases the positions of the stars on the photographic plate will appear absolutely unchanging, despite our orbital motion. As a result, not only in Tycho's day, but for nearly 250 years after he died it remained beyond the ability of any astronomer to measure the parallax of any star, and prove just how far away the stars were. The distances were just too staggering to imagine, let alone measure.
  But there is another side to the story of Tycho's failure. After all, he had used his lack of parallax measurements for the Nova of 1572 and the Comet of 1577 to show that they must lie out in space, at vast distances from the Earth. Having done that why didn't he simply suppose that the stars were even further out, and just too far for his observations to reveal their parallax? Aside from the staggering distances involved, a problem with the appearance of stars may have caused him to reject the idea. Namely, when we look at stars, they appear to have a size that depends on their brightness. As it happens, that apparent size is an illusion caused by the way that the light falling on our eyes (or nowadays, on CCD imaging devices) "overexposes" brighter objects, making them look thousands or millions of times larger than they actually are. But Tycho thought that brighter stars looked bigger because they were large enough for him to see their actual sizes. And if they were as far away as they would have to be to have no observable parallax, then their apparent (though misleading) size would have required them to be immense -- in fact many times larger, for the brightest and apparently biggest stars, than the orbits of the outer planets. In other words, if he had accepted the unimaginable distances the stars would have had to have in order for him to be unable to measure their parallax, then he would have also had to assume that their physical bodies were just as outlandishly immense. And whether it was his preference for an Aristotelian view of physics, his inability to measure stellar parallax, or the ridiculous sizes that stars would have to have if they were as far away as that implied, in the end he decided to use a compromise that fit every observation that could be made of the stars, gave the most accurate calculations of the motions of astronomical bodies possible in his lifetime, and though now considered "wrong", was otherwise a perfectly acceptable description of the nature of the Solar System (and what we now call the stellar Universe).
 And if Kepler had not conceived his revolutionary theory of orbital motions a few years later, and Galileo developed a new way of thinking about physics in that same era, perhaps astronomy texts would still discuss Tycho and his astronomical accomplishments in detail similar to that on this page. But since his theory of the Solar System's structure was abandoned almost as soon as he died, Tycho is barely mentioned in most introductory astronomy texts, and often more disparagingly than not, which is a great injustice to a man who was one of the most important observational astronomers of all time.