## Planetary Systems

### Synchronous Rotation

Synchronous rotation refers to the case where the satellite's spinning rate is equal to the revolution around the central body.
The satellite always presents the same face to the reference frame of the central body, i.e., it doesn't seem to be rotating (Figure 07-23a). This situation comes about when the satellite moves too close to a massive body. As shown in Figure 07-23b, the tidal force distorts the satellite into bulges, where the two ends experience difference force and hence difference torque, resulting in the retardation of the spinning rate. The time scale T for development of the

#### Figure 07-23b Tidal Dragging [view large image]

locking can be expressed approximately as:

T 6rD6/(mM2) years (x 3 for rocky body / x 0.4 for icy body)

where r is the radius of the satellite, D the distance to the central body, m and M the mass of the satellite and the central body respectively (all in multiplication factors as shown in Table 07-03). The original formula is much simplified by the followings assumptions:
1. The initial spinning rate is taken to be one revolution every 12 hours (the rotational periods for most asteroids vary between 2 to 24 hours). The locking time T is proportional to the spinning rate, thus a rapidly spinning satellite would take longer time to be locked in.
2. Value of the rigidity of the satellite is taken to be 3x1010 Nm-2 and 0.4x1010 Nm-2 for rocky and icy objects respectively. This parameter is also proportion to T.
3. The dissipation function of the satellite is assumed to be 100. It governs the rate at which mechanical energy is converted to heat. This parameter is again proportion to T.
4. The density of the satellite is assumed to be about 3 gm/cm3. It is inversely proportional to T.
Table 07-03 lists the locking time for some of the "satellite-central boy" systems. The Sun-Earth system is added to show that the Earth requires a very long time to be locked in with the Sun (longer than the age of the Solar system). The locking time for the Earth-Moon system may be much shorter in the order of thousand years as the separation is believed to be closer in the past. For the cases of Jupiter and Saturn, all the inner moons within a distance of 60 radius of the central body are tidally locked with the planets. For the Pluto-Charon system, Pluto is itself locked to Charon.

System D (108m) r (106m) M (1026kg) m (1022kg) T (years)
Sun-Earth (no locking) 1500 6.5 1.8x104 600 6x109
Earth-Moon 3.8 1.7 0.06 7.2 3x106
Jupiter-Europa 6.7 1.6 19 3.0 320
Saturn-Titan 12 2.6 5.7 15 4x104
Pluto-Charon 0.22 0.6 1.2x10-4 0.2 4x104

#### Table 07-03 Some Tidally Locked Systems

There is a tendency for a satellite to orient itself in the lowest energy configuration, with the heavy side facing the central body. Irregularly shaped bodies will align their long axis to point towards the planet (see the formation of spherical body). In many cases this planet-facing hemisphere is visibly different from the rest of the moon's surface.

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