We end this chapter with a problem that shows how a classical mathematical model is modified so that it can be used to determine the period of rotation of a planet. The modification uses some of the principles of scientific accuracy discussed earlier in this chapter and illustrates the use of successive iterations to refine a computation, easily done by computer.
The procedure we develop has been used to determine the rotation period of Saturn more accurately than earlier estimates by using observations of variations in the planet's radio emissions made by Voyager 1. Since this planet has neither a solid surface nor any distinctive atmospheric features comparable to Jupiter's Great Red Spot, what is computed is the period of rotation of the magnetic field of the planet. Because of the complex nature of the radio emission data, we illustrate the method by computing the rotation period of Jupiter rather than Saturn.
We begin with a question that is essentially the same as the familiar "How much time elapses between successive alignments of the hands of the clock?" but that sets the stage for the actual problem we wish to solve.
PROBLEM 13. a. Jupiter rotates on its axis once every 9.92 hours, and its moon Io revolves around Jupiter once every 42.5 hours. What is the length of time between consecutive passages of Io over a particular spot on Jupiter?
Solution: Let Rj and Ri be the angular rotation and revolution rates for Jupiter and Io respectively.
Rj = 360/9.92= 36.3 degrees/hour
Ri = 360/42.5 = 8.47 degrees/hour.
In Fig. 2.7 (See next card.), Io moves from A to B while the point S on Jupiter makes a complete revolution and then goes on to S' to be under Io again. So we must find the time T such that RjT - 360 = RiT. Using the values above for Rj, and Ri, we get the following:
36.3 T - 360 = 8.47 T
36.3 T - 8.47 T= 360
(27.8) T= 360
T= 360/27.8 = 12.9 hours
To see how this classic problem might be altered, suppose we don't know Jupiter's rotation period (or that we don't know it very accurately). As it approached Jupiter, the Voyager was able to make observations of the times at which Jupiter's Great Red Spot appeared in the center of the disc as viewed from Voyager. We want to use these observations to determine Jupiter's rotation period.
b. Voyager's trajectory as it approached Jupiter is illustrated in Fig. 2.8. (See next card.)Modify the results of part (a) to find Jupiter's period if the Red Spot is observed to be in the center at time t1 = 2 h 25 min +/- 1 min, when Voyager's distance from Jupiter is D1 = 7.70 x 10^5 km, and again at time t2 = 16 h 24 min +/- 1 min, when Voyager's distance from Jupiter is D2 = 4.72 x 10^5 km and Voyager has moved through an angle Beta = 147.2° with respect to Jupiter's center between these two observations.