Posted note about the now-"dead" NASA animation of the motion of the Sun across Mercury's sky during half of a Mercurian day (that is, from sunrise to sunset)

  The rotation of Mercury is very strange. For hundreds of years it was thought to rotate synchronously, so that it always kept one face to the Sun, just as the Moon always keeps one face to the Earth. Its actual rotation, however, causes it to turn exactly one and a half times each time it goes around the Sun, so that it turns one side toward the Sun in one orbit, and the other side toward the Sun in the next orbit, making the day on Mercury twice as long as the year. Stranger yet, at perihelion the motion of the planet around the Sun is faster than its rotation, so that the Sun seems to stop its normal (westward) motion, move the other way for a little while(!), then resume its westward motion.

How The Sun Moves In Mercury's Sky
  Mercury rotates once every 58.647 days, which is exactly 2/3 of its orbital period of 87.970 days, so it turns on its axis exactly 1 1/2 times during one of its years, causing the stars to move 1 1/2 times around the sky each year. During that time it moves once around the Sun, causing the Sun to appear to move backwards relative to the stars one full turn. As a result it has only half a day during one year (1 1/2 forward rotations by the stars, less 1 backward motion by the Sun), so that it takes two years (175.942 Earth days) for one day, three times as long as its rotation period. This seems very strange to us, but it is actually normal for a planet with a slow rotation to have a rotation period that is very different from its day length (refer to Rotation Period and Day Length for a detailed discussion of how rotation periods are related to day length).
  The reason that Mercury rotates this way is related to its orbital motion. On the average, Mercury moves four degrees around the Sun each (Earth) day, while rotating six degrees. The westward motion of the stars due to its rotation is opposite the eastward motion of the Sun caused by its orbital motion, so the Sun's net westward motion is only two degrees per (Earth) day, which is three times slower than the star's six degrees per (Earth) day; therefore, the day ought to be three times longer than the rotation, as it is.
  If the orbit of Mercury was nearly circular, this is all we'd need to say about its rotation; but since its orbit is quite eccentric, things work very differently at different times of the year.
  At aphelion Mercury is over 20% further from the Sun than normal, and as a result moves about 17% slower than normal in its orbit around the Sun; and being 20% further away than usual, even if it were moving at the same speed as usual it would LOOK like it was moving slower (that is, it would move at a slower angular velocity), as seen from the Sun. The net result is that the average orbital motion of four degrees per day slows to only 2/3 of that, or less than three degrees per day.
  The ROTATIONAL speed of Mercury on its axis, however, does NOT change. Rotational motions can change over very long periods of time as a result of tidal forces, but such changes are very slow, and in Mercury's case nonexistent, as its rotation is locked to its orbital period (as explained below). So at aphelion the westward motion of the stars is still six degrees per day, and the Sun's motion, which is that six degrees minus the apparent orbital motion, is a little over three degrees per day to the west, or more than 50% faster than normal.
  At perihelion, on the other hand, Mercury is over 20% closer to the Sun than normal, and as a result moves about 25% faster than normal in its orbit; and being 20% closer than usual, even if it were moving at the same speed as usual it would look like it was moving faster (the exact opposite of the situation described for aphelion), at least in terms of its angular velocity around the Sun. These two factors result in an increase of the orbital motion to six degrees per day, or 3/2 of the normal value (note that this is the reciprocal of the slowing of 2/3 of the normal value at aphelion).
  As at aphelion the rotational velocity is unchanged, so the stars still move westward at six degrees per day, and subtracting the orbital motion, which makes the Sun move eastward at six degrees per day, we are left with -- NOTHING! In other words, at perihelion, Mercury's orbital motion around the Sun is so fast that the Sun completely stops its westward motion. In fact, if the calculations are done very accurately instead of in the approximate way just described, it turns out that at perihelion the orbital motion actually EXCEEDS the rotational motion, so that the Sun slows, stops moving to the west, and starts slowly moving to the east.
  This eastward motion only lasts for a few days, but during that time the Sun moves a little more than one diameter (which, since it is three times closer to Mercury than to the Earth, is about a degree and a half). Then, as the planet moves past perihelion, and moving away from the Sun starts to experience a slowing of its orbital motion, the Sun stops its eastward movement and starts slowly moving to the west again.
  To summarize, during one of Mercury's years the AVERAGE motion of the Sun is two degrees per day to the west, or one-third of the stars' motion of six degrees per day, making the day three times longer than the rotation period; but at various times of the year, the motion can be very different. Near aphelion the orbital motion is slower and the net westward motion of the Sun is more than half again its normal angular velocity, or more than three degrees per day; but as the planet approaches perihelion the Sun slows, stops moving westward, moves a little over one diameter to the east, then turns around and starts slowly moving westward again, faster and faster as Mercury moves away from the Sun, until the Sun is going more than three degrees per day to the west at the next aphelion.
  (Note: At the same time that the Sun is moving slower and then faster, it gets larger and then smaller, because its apparent size depends upon how far away it is.)

A Very Strange Day
  Now let's suppose we were at a place on Mercury, such as the Caloris Basin or its antipodes, where the Sun is overhead at perihelion. At such a place, as shown in the animation below, we would see the Sun rise in the east as a ball a little over one degree in diameter (twice the size as seen from the Earth because the distance from the Sun to Mercury at aphelion is half our average distance), at a rate of more than three degrees per day, or a little over three diameters per (Earth) day. During the next half-orbital period (44 Earth days), the Sun would gradually grow larger and move westward more and more slowly, so that as it approached the meridian and its highest point in the sky, it would be more than a degree and a half in diameter and hardly moving at all. As it passed the meridian it would stop moving westward, slowly move eastward until it was a little over one diameter to the east of where it stopped moving westward, then slowly start moving to the west again. Over the next half-orbital period, the Sun would move downward in the sky, gradually moving faster and faster and shrinking in size, until as it set in the west, it was once again only a degree in size and moving more than three degrees per (Earth) day.
 The "image" below was a Flash animation of the apparent motion of the Sun in Mercury's sky. Unfortunately, Adobe ended all support for Flash animations on Jan 12, 2021, so they no longer work, and NASA has not only failed to provide a replacement, but continues to incorrectly state that the day on Mercury is 59 Earth days, whereas the point of this page is that the 59 days is the rotation period, and since the "year" (one reolution of Mercury around the Sun) is 88 days, it takes three rotations and two years on Mercury for the Sun to move once around the sky, making a "day" 176 Earth days long. At some point I will create my own animation to replace the extinct NASA "Flash" animation, and correct several errors made in the original (including the fact that although the Sun's motion was more or less correctly shown, the stars didn't change their positions at all). However, that will require a lot of time and effort, and for the moment, the text discussion will have to do.<

 The animation above showed the apparent motion of the Sun across the Mercurian sky at a location where the Sun is on the meridian (halfway between rising and setting) at perihelion. It was set for 45 degrees north latitude, so it does not correspond exactly to the Caloris Basin or its antipodes, which are on the Mercurian Equator, and have the Sun overhead at perihelion. Unfortunately, there are several problems with the animation, such as (1) it does not bother to show the constant westward motion of the stars, which is three times faster than the Sun's average westward motion, (2) it shows the Sun larger than its actual apparent size, so that the backward motion does not appear to be larger than its diameter, and (3) neither it nor the NASA site it came from bothered to explain the details of the various "dials" shown in the diagram. If and when I can produce my own version, I will make it more accurate, and include a detailed discussion of everything it shows.

  Now let's imagine moving a quarter of the way around the planet, until we are at a longitude where the Sun is overhead at aphelion, instead of at perihelion. Then we would see a sight unlike anything seen anywhere else in the solar system; for as the Sun rises on the eastern horizon Mercury would be at perihelion, and the degree and a half diameter disc of the Sun would take several days to slowly rise, then just as slowly set again as it stopped its westward motion, turned to the east, then just as slowly rise again as the Sun stopped its eastward motion and turned westward again.
  After this "double dawn" the Sun would rise in the sky, gradually shrinking as it rose, moving upward faster and faster until it passed the meridian with a motion of more than three degrees per day. Then it would sink in the west, gradually growing larger as it moved lower and lower, slower and slower, as Mercury approached perihelion. Finally, as it neared perihelion the Sun would very slowly set in the west, then rise again, as it turned to the east at perihelion, before finally setting for good (or at least until the next day, two Mercurian years later).

Why Does Mercury Rotate This Way?
 The Sun's back-and-forth dance in Mercury's sky is unique, as well as amazing; but the reason for it is quite common -- the tidal forces which act between any two bodies which have significant size compared to the distance between them. These forces cause not only the tides that we are familiar with, but also precession, the synchronous rotation of the Moon (and many other moons), the tidal slowing of the Earth and other planets, and the synchronous rotation of Pluto relative to its largest moon, Charon, and the synchronous rotation of Charon relative to Pluto.

 At this point I ran out of time for a detailed discussion. Once I've figured out how to show the animation (or a better one) of the Sun's apparent motion across Mercury's sky, I'll add a discussion of how the Sun's tidal force on Mercury tries to force Mercury to always keep one face to the Sun, and how, save for an interaction between its varying orbital velocity, the eccentricity of its orbit, and its rotation rate, it would have ended up with the synchronous rotation we once thought it had, very early in the history of the Solar System (probably, in well under a billion years). However, the following is a brief summary of the topics to be added when I have time:
 The key to Mercury's strange rotation is that the tidal force of the Sun on Mercury is so much stronger at perihelion than at aphelion that the rotation of the planet does not have to be synchronous to be stable. It just has to be a half-whole multiple of the orbital revolution, and the orbital angular velocity at perihelion has to be close to the same half-whole multiple of the average orbital angular velocity, so that the planet has a nearly fixed orientation relative to the Sun at the time when it is closest to the Sun. In this case the orbital period is 3/2 of the rotation period and the angular velocity at perihelion is approximately 3/2 of the average orbital angular velocity, so the criteria required are met.
 Finally, it turns out that the rotation period of the planet also happens to match the time required for Mercury to move from one side of the Sun to the other as seen from the Earth (a number related to the so-called "synodic period" of the planet's orbital motion). As a result, every time that Mercury is seen from the Earth on one side of the Sun, the same side of Mercury is facing the Sun, and every time it is seen from the Earth on the other side of the Sun, the same side is once again facing the Sun. In between, other parts of the planet face the Sun, but since whenever we can "easily" observe it from the Earth it has the same side facing the Sun, for centuries it was thought that it always kept one face to the Sun, even though that is completely wrong (though a long period of observations at the Pic du Midi observatory in France in the 1950's did lead to the suggestion that perhaps there was something wrong with the presumption that the planet always kept one face to the Sun). It was only after radar observations of the planet showed that it was rotating 3/2 faster than expected that the asynchronous rotation was discovered.