Astronomy Lab Problem: How can you calculate the mass of Uranus using 3-5 of its moons?

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Astronomy Lab Problem: How can you calculate the mass of Uranus using 3-5 of its moons?

General information The planet Uranus is the seventh most distant from the sun and was discovered by William Hershey in It is a gas giant with a diameter of 50,000km. Nevertheless, its mean distance from the sun of 19.2 astronomical units meant that very little was known about the planet until the Voyager 2 fly-by in January of Up until then the planet was thought to have only 5 moons. You are given a picture of the moons of Uranus to use to calculate the mass of Uranus.

Mathematics Behind the calculation.

The formula you will be using is :R 3 =GM P 2 4Π 2 Where R=radius from the planet using a 2 +b 2 =c 2 a is the distance on the x-axis and b is the distance on the y-axis. P is the rotational period. You are given how many degrees the moon travels per second and need to find how many seconds in 360 o. G is a constant. 6.67x M or mass is going to be in kg X10 25

Helpful Hints and How to do one calculation. Measure the distance from the center of Uranus to a moon in the North or south direction using it orbital path. For my example, I got 8.4cm. Measure the distance from the center of Uranus to a moon in the east or west direction using its orbital path. For my example I have 5.9cm. R therefore = cm Let say the moon moves at 2.23 degrees per second. How many seconds does it take to get around Uranus? P therefore = sec

Plug the numbers into the formula: R 3 =GM P 2 4Π E14m 3 = 6.67E11m 3 /s 2 x MkgE E4 s 2 4Π 2 (2.76E10 m 3 /s 2 )(39.48)= 6.67E11m 3 /s 2 xMkgE25 MkgE25 =( E12 m 3 /s 2 )/ (6.67E11 m 3 /s 2 ) M= 1.64 E25 (Now do this for four moon. Get an average and then check to see how close you are to the actual mass of Uranus. Do % error only if you are off by magnitude of 10.