What is Saturn's Mass Using Kepler's Laws and Moon Data?

[AnkhUNC]

Homework Statement

I am told to find the mass of Saturn using Kepler's Laws and the following information on Saturn's moons:

Mimas
r = 185.52 x 106
t = 8.14 x 104
Enceladus
r = 238.02 x 106
t = 11.839 x 104
Tethys
r = 294.66 x 106
t = 16.311 x 104
Dione
r = 377.4 x 106
t = 23.647 x 104
Rhea
r = 527.04 x 106
t = 39.031 x 104

Homework Equations

T^2 = (4pi^2/G*M)*r^3

The Attempt at a Solution

So plotting the information on the moons I get a slope of 2E-24x + 320.16. But I'm not really sure where to go from here. Why was it important for me to find the slope in order to find M? Can't I just find M via M = (4pi^2/GT^2)r^3?

 

[dynamicsolo]

You didn't say what variables you plotted for your graph, but you will not get a straight line if you simply plot T vs. r .

You are being asked to use the satellite data as points against which to fit a line. You will then get an experimental result for the mass of Saturn, M, from the slope you find. (You could find M from Newton's form of Kepler's Third Law, as you propose, but you will find that each moon's data gives you a slightly different value for M; you would still need to combine those results somehow. This is why the best-fit line is often used experimentally.)

You will need to plot T^2 vs. r^3 in order to get a straight line. The slope of the best-fit line through the five data points will then give you an estimate for [ 4·(pi^2) / GM ], from which you can then extract an estimate for M, since the value of G is known reliably to far more significant figures than you will need (as is pi).

 

[AnkhUNC]

So should I put t on the x and r on the y?

 

[dynamicsolo]

I believe your data gives orbital radii in meters and time in seconds. This means that your result for the slope will come out in SI and you will be able to get a value for Saturn's mass in kilograms (which you can then compare with references in texts or on the 'Net).

You will want to plot T^2 on the y-axis and r^3 on the x-axis. These will then function as effective variables Y = T^2 and X = r^3 , so that your data will fall on a line

Y = mX + b .

Given the form of Newton's extension of Kepler's Third Law (which you quoted), you should find a rather small value for b, the intercept of the line (ideally it would be zero, but you are working with real data) and a tiny value for m (on the order of 10^-15), which is your experimental estimate for [ 4·(pi^2) / GM ] .

You can make the values a bit easier to deal with by dividing your values for T^2 by 10^10 and those for r^3 by 10^25 , which will give numbers that are easier to plot or use in graphing software. This will keep you from getting extremely small output for m and b. You would then have to undo this rescaling by dividing your slope value by 10^15 before extracting the result for M. (The correct value for the line intercept would then be your result for b, multiplied by 10^10.)

EDIT: I fixed some of the exponents, as I was remembering planetary masses in grams, not kilograms. I tried out the plot I described a bit crudely and found that the slope is quite close to 1, making [ 4·(pi^2) / GM ] close to 1·10^-15 . Working carefully, you should get a rather satisfactory result for Saturn's mass in kilograms.