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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 07-23a Synchornous Rotation [view large image] |
## Figure 07-23b Tidal Dragging |
locking can be expressed approximately as: |

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:

- 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.
- Value of the rigidity of the satellite is taken to be 3x10
^{10}Nm^{-2}and 0.4x10^{10}Nm^{-2}for rocky and icy objects respectively. This parameter is also proportion to T. - 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.
- The density of the satellite is assumed to be about 3 gm/cm
^{3}. It is inversely proportional to T.

System | D (10^{8}m) |
r (10^{6}m) |
M (10^{26}kg) |
m (10^{22}kg) |
T (years) |
---|---|---|---|---|---|

Sun-Earth (no locking) | 1500 | 6.5 | 1.8x10^{4} |
600 | 6x10^{9} |

Earth-Moon | 3.8 | 1.7 | 0.06 | 7.2 | 3x10^{6} |

Jupiter-Europa | 6.7 | 1.6 | 19 | 3.0 | 320 |

Saturn-Titan | 12 | 2.6 | 5.7 | 15 | 4x10^{4} |

Pluto-Charon | 0.22 | 0.6 | 1.2x10^{-4} |
0.2 | 4x10^{4} |

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