8.4 Rotation Rates


Figure 8.11 Lunar Elongation The Moon is slightly elongated in shape, with its long axis perpetually pointing toward Earth. (The elongation is highly exaggerated in this diagram.)
 As mentioned above, the Moon’s rotation period is precisely equal to its period of revolution about Earth—27.3 days—so the Moon keeps the same side facing Earth at all times (see Figure 8.11). To an astronaut standing on the Moon’s near-side surface, Earth would appear almost stationary in the sky (although its daily rotation would be clearly evident). This condition, in which the spin of one body is precisely equal to (or synchronized with) its revolution around another body, is known as a synchronous orbit. The fact that the Moon is in a synchronous orbit around Earth is no accident. It is an inevitable consequence of the gravitational interaction between those two bodies.

Just as the Moon raises tides on Earth, Earth also produces a tidal bulge in the Moon. Indeed, because Earth is so much more massive, the tidal force on the Moon is about 20 times greater than that on Earth, and the Moon’s tidal bulge is correspondingly larger. In Chapter 7 we saw how lunar tidal forces are causing Earth’s spin to slow and how, as a result, Earth will eventually rotate on its axis at the same rate as the Moon revolves around Earth. (Sec. 7.6) Earth’s rotation will not become synchronous with the Earth–Moon orbital period for hundreds of billions of years. In the case of the Moon, however, the process has already gone to completion. The Moon’s much larger tidal deformation caused it to evolve into a synchronous orbit long ago, and the Moon is said to have become tidally locked to Earth. Most of the moons in the solar system are similarly locked by the tidal fields of their parent planets.

Actually, the size of the lunar bulge is too great to be produced by Earth’s present-day tidal influence. The explanation seems to be that, long ago, the distance from Earth to the Moon may have been as little as two-thirds of its current value, or about 250,000 km. Earth’s tidal force on the Moon would then have been more than three times greater than it is today and could have accounted for the Moon’s elongated shape. The resulting distortion could have “set” when the Moon solidified, thus surviving to the present day, while at the same time accelerating the synchronization of the Moon’s orbit.


 In principle, the ability to discern surface features on Mercury should allow us to measure its rotation rate simply by watching the motion of a particular region around the planet. In the mid-nineteenth century, an Italian astronomer named Giovanni Schiaparelli did just that. He concluded that Mercury always keeps one side facing the Sun, much as our Moon perpetually presents only one face to Earth. The explanation suggested for this synchronous rotation was the same as for the Moon—the tidal bulge raised in Mercury by the Sun had modified the planet’s rotation rate until the bulge always pointed directly at the Sun. Although the surface features could not be seen clearly, the combination of Schiaparelli’s observations and a plausible physical explanation was enough to convince most astronomers, and the belief that Mercury rotates synchronously with its revolution about the Sun (that is, once every 88 Earth days) persisted for almost half a century.

Figure 8.12 Planetary Radar A radar beam reflected from a rotating planet yields information about both the planet’s overall motion and its rotation rate. Compare Figure 4.17.
In 1965, astronomers making radar observations of Mercury from the Arecibo radio telescope in Puerto Rico (see Figure 5.22) discovered that this long-held view was in error. The technique they used is illustrated in Figure 8.12, which shows a radar signal reflecting from the surface of a hypothetical planet. Let’s imagine, for the purpose of this discussion, that the pulse of outgoing radiation is of a single frequency. The returning pulse bounced off the planet is very much weaker than the outgoing signal. Beyond this change, the reflected signal can be modified in two important ways. First, the signal as a whole may be redshifted or blueshifted as a consequence of the Doppler effect, depending on the overall radial velocity of the planet with respect to Earth. (Sec. 3.5) We will assume for simplicity that this velocity is zero, so that, on average, the frequency of the reflected signal is the same as the outgoing beam. Second, if the planet is rotating, the radiation reflected from the side of the planet moving toward us returns at a slightly higher frequency than the radiation reflected from the receding side. (Think of the two hemispheres as being separate sources of radiation and moving at slightly different velocities, one toward us and one away.) The effect is very similar to the rotational line broadening discussed in Chapter 4 (see Figure 4.18), except that in this case the radiation we are measuring was not emitted by the planet but only reflected from its surface. (Sec. 4.4) What we see in the reflected signal is a spread of frequencies on either side of the original frequency. By measuring the extent of that spread, we can determine the planet’s rotational speed.

In this way, the Arecibo researchers found that the rotation period of Mercury is not 88 days, as had previously been believed, but 59 days, exactly two-thirds of the planet’s orbital period. Because there are exactly three rotations for every two revolutions, we say that there is a 3:2 spin–orbit resonance in Mercury’s motion. In this context, the term resonance just means that two characteristic times—here Mercury’s day and year—are related to each other in a simple way. An even simpler example of a spin–orbit resonance is the Moon’s orbit around Earth. In that case, the rotation is synchronous with the revolution, and the resonance is said to be 1:1.

Figure 8.13 illustrates some implications of Mercury’s curious rotation for a hypothetical inhabitant of the planet. Mercury’s solar day—the time from noon to noon, say—is two Mercury years long! The Sun stays “up” in the black Mercury sky for almost three Earth months at a time, after which follows nearly three Earth months of darkness. At any given point in its orbit, Mercury presents the same face to the Sun not every time it revolves but every other time.

Figure 8.13 Mercury’s Rotation Mercury’s orbital and rotational motions combine to produce a day that is two Mercury years long. The arrow represents an observer standing on the surface of the planet. At day 0, it is noon for our observer, and the Sun is directly overhead. By the time Mercury has completed one full orbit around the Sun and moved from day 0 to day 88, it has rotated on its axis exactly 1.5 times, so that it is now midnight at the observer’s location. After another complete orbit, it is noon once again. The eccentricity of Mercury’s orbit is not shown in this simplified diagram.


Mercury’s 3:2 spin–orbit resonance did not occur by chance. What mechanism establishes and maintains it? In the case of the Moon orbiting Earth, the 1:1 resonance is explained as the result of tidal forces. In essence, the lunar rotation period, which probably started off much shorter than its present value, has lengthened so that the tidal bulge created by Earth is fixed relative to the body of the Moon. Tidal forces (this time due to the Sun) are also responsible for Mercury’s 3:2 resonance, but in a much more subtle way.

Mercury cannot settle into a 1:1 resonance because its orbit around the Sun is quite eccentric. By Kepler’s second law, Mercury’s orbital speed is greatest at perihelion (closest approach to the Sun) and least at aphelion (greatest distance from the Sun). (Sec. 2.3) A moment’s thought shows that because of these variations in the planet’s orbital speed, there is no way that the planet (rotating at a constant rate) can remain in a synchronous orbit. If its rotation were synchronous near perihelion, it would be too rapid at aphelion, while synchronism at aphelion would be too slow at perihelion.

Tidal forces always act to try to synchronize the rotation rate with the instantaneous orbital speed, but such synchronization cannot be maintained over Mercury’s entire orbit. What happens? The answer is found when we realize that tidal effects diminish very rapidly with increasing distance. The tidal forces acting on Mercury at perihelion are much greater than those at aphelion, and perihelion “won” the struggle to determine the rotation rate. In the 3:2 resonance, Mercury’s orbital and rotational motion are almost exactly synchronous at perihelion, so that particular rotation rate was naturally “picked out” by the Sun’s tidal influence on the planet. Notice that even though Mercury rotates through only 180° between one perihelion and the next (see Figure 8.13), the appearance of the tidal bulge is the same each time around.

The motion of Mercury is one of the simplest nonsynchronous resonances known in the solar system. Astronomers now believe that these intricate dynamic interactions are responsible for much of the fine detail observed in the motion of the solar system. Examples of resonances can be found in the orbits of many of the planets, their moons, their rings, and in the asteroid belt.

The Sun’s tidal influence also causes Mercury’s rotation axis to be exactly perpendicular to its orbit plane. As a result, and because of Mercury’s eccentric orbit and the spin–orbit resonance, some points on the surface get much hotter than others. In particular, the two (diametrically opposite) points on the equator where the Sun is directly overhead at perihelion get hottest of all. They are called the hot longitudes. The peak temperature of 700 K mentioned earlier occurs at noon at those two locations. At the warm longitudes, where the Sun is directly overhead at aphelion, the peak temperature is about 150 K cooler—a mere 550 K.

By contrast, the Sun is always on the horizon as seen from the planet’s poles, so temperatures there never reach the sizzling levels of the equatorial regions. Earth-based radar studies carried out during the 1990s suggest that Mercury’s polar temperatures may be as low as 125 K and that, despite the planet’s scorched equator, the poles may be covered with extensive sheets of water ice. (See Section 8.5 for similar findings regarding the Moon.)

Concept Check

How has gravity influenced the rotation rates of the Moon and Mercury?