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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.

1

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.