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TitleAPhO Theoretical Problems (2000-2013)
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Problem 1

Eclipses of the Jupiter’s Satellite

A long time ago before scientists could measure the speed of light accurately, O Römer, a
Danish astronomer studied the time eclipses of the Jupiter’s satellite. He was able to
determine the speed of light from observed periods of a satellite around the planet Jupiter.
Figure 1 shows the orbit of the earth E around the sun S and one of the satellites M
around the planet Jupiter. (He observed the time spent between two successive
emergences of the satellite M from behind Jupiter).

A long series of observations of the eclipses permitted an accurate evaluation of the
period of M. The observed period T depends on the relative position of the earth with
respect to the frame of reference SJ as one of the coordinate axes. The average time of
revolution is T0 = 42h 28 m 16s and maximum observed period is ( T0 + 15)s.







Figure 1 : The orbits of the earth E around the sun and a satellite M around Jupiter J. The
average distance of the earth E to the Sun is RE = 149.6 x 106 . The maximum distance is
RE,max = 1.015 RE. The period of revolution of the earth is 365 days and of Jupiter is 11.9
years. The distance of the satellite M to the planet Jupiter RM= 422 x 103 km.


a. Use Newton’s law of gravitation to estimate the distance of Jupiter to the Sun. Determine
the relative angular velocity ω of the earth with respect to the frame of reference Sun-
Jupiter (SJ). Calculate the speed of the earth with respect to SJ.


b. Take a new frame which Jupiter is at rest with respect to the Sun. Determine the relative

angular velocity ω of the earth with respect to the frame of reference Sun-Jupiter ( SJ).
Calculate the speed of the earth with respect to SJ.


c. Suppose an observed saw M begin to emerge from the shadow when his position was at

θ k and the next emergence when he was at θ k+ 1 , k = 1,2,3,… From these observations he
got the apparent periods of revolution T ( t k ) as a function of time t k from Figure 1 and
then use an approximate expression to explain how the distance influences the observed
periods of revolution of M. Estimate the relative error of your approximate distance.

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