8.4 Rotation Rates
THE ROTATION OF THE MOON
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 Moons tidal bulge is correspondingly larger. In Chapter 7 we saw how lunar tidal forces are causing Earths 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) Earths rotation will not become synchronous with the EarthMoon orbital period for hundreds of billions of years. In the case of the Moon, however, the process has already gone to completion. The Moons 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 Earths 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. Earths tidal force on the Moon would then have been more than three times greater than it is today and could have accounted for the Moons 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 Moons orbit.
MEASUREMENT OF MERCURY'S SPIN
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 Moonthe tidal bulge raised in Mercury by the Sun had modified the planets rotation rate until the bulge always pointed directly at the Sun. Although the surface features could not be seen clearly, the combination of Schiaparellis 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.
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 planets orbital period. Because there are exactly three rotations for every two revolutions, we say that there is a 3:2 spinorbit resonance in Mercurys motion. In this context, the term resonance just means that two characteristic timeshere Mercurys day and yearare related to each other in a simple way. An even simpler example of a spinorbit resonance is the Moons 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 Mercurys curious rotation for a hypothetical inhabitant of the planet. Mercurys solar daythe time from noon to noon, sayis 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.
EXPLANATION OF MERCURY'S ROTATION
Mercurys 3:2 spinorbit 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 Mercurys 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 Keplers second law, Mercurys orbital speed is greatest at perihelion (closest approach to the Sun) and least at aphelion (greatest distance from the Sun). (Sec. 2.3) A moments thought shows that because of these variations in the planets 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 Mercurys 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, Mercurys orbital and rotational motion are almost exactly synchronous at perihelion, so that particular rotation rate was naturally picked out by the Suns 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 Suns tidal influence also causes Mercurys rotation axis to be exactly perpendicular to its orbit plane. As a result, and because of Mercurys eccentric orbit and the spinorbit 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 coolera mere 550 K.
By contrast, the Sun is always on the horizon as seen from the planets poles, so temperatures there never reach the sizzling levels of the equatorial regions. Earth-based radar studies carried out during the 1990s suggest that Mercurys polar temperatures may be as low as 125 K and that, despite the planets scorched equator, the poles may be covered with extensive sheets of water ice. (See Section 8.5 for similar findings regarding the Moon.)
How has gravity influenced the rotation rates of the Moon and Mercury?