Goal

Suppose you're an astronomer studying a newly discovered planet and its moons. By observing the motion of the moons, you'll determine the mass of the planet they are orbiting.

 Introduction

In Solving Gravitation Problems using Proportional Reasoning, you saw that the acceleration due to gravity at the surface of the Earth is given by a_g = GM_e/R_e². In the case of an arbitrary planet of mass, M_p, the acceleration of a planetary moon in a circular orbit at distance, R_m, from the center of the planet would be a_m = GM_p/R_m². For all moons of the planet, the value of GM_p would be a constant. Thus, a_m would be proportional to R_m^-2. Suppose you took data on acceleration and orbital radius for several moons of a particular planet and plotted a graph of acceleration vs. orbital radius. If you did a power law fit to the equation y = Axⁿ with n = -2, y = acceleration, and x = orbital radius, then you would expect the fit coefficient A to represent GM_p. Knowing the value of G, you could then calculate the mass of the planet. That's the plan of this exercise. You'll use fits for two other relationships to verify the value obtained for the mass of the planet.

 Prelab

Submit the prelab questions in WebAssign L144PL.

1. Orbital acceleration, speed, and period can all be expressed as a coefficient multiplied by a function of R. That is, each dependent variable has a power-law relationship with R of the form y = ARⁿ, where the coefficient, A, is a function of G and M. This is shown in the table below for the acceleration. Complete the rows for Speed and Period. For Units of Coefficient, reduce the units to a combination of meters and seconds only. (Use m and s.)

Fit	Dependent variable	Relationship	Coefficient, A	Units of Coefficient	Power of R
1	Acceleration, a	a = (GM)/R²	GM	<_>	-2
2	Speed, v	v = <_>	<_>	<_>	<_>
3	Period, T	T = <_>	<_>	<_>	<_>

2. Now you'll collect data from this animation. You should see five small circles around the center of the screen – those are the planet and 4 moons. The planet is the stationary blue circle at the origin. Coordinates of the moons are displayed in the Outputs panel. The grid spacing is 4 x 10⁷ m. For each moon, determine the period and radius of the orbit to 3 significant figures. Enter the data into the data table. The radius and period of the Orange moon's orbit are given to you.

Moon	Radius (m)	Period (s)
Red	<_>	<_>
Green	<_>	<_>
Black	<_>	<_>
Orange	3.54 x 108	5.07 x 105

 Analysis

You'll submit the analysis in WebAssign L144.

A review of curve-fitting methods: In L131, you used the method of re-expression to linearize the data. This is the default method that we use in this course for curve fitting, and it's also the method that you would be expected to use on the AP exam for a problem requiring data analysis. (There is often one such free-response problem on each AP exam.) In L113, you used a different method. In that lab, you fit the position vs. time data for dropped and projected marbles to a quadratic. That choice was made because of the expectation that the position of a free-falling object was quadratice in time. In the present exercise, you'll use a similar method to fit the satellite data. Rather than re-expressing a variable to obtain a linear fit for each of the dependent variables, you'll select an appropriate function for fitting the data.

4. Do the following.

   1. Examine your prelab results. Make any necessary corrections before continuing with the analysis.

   2. Start Logger Pro. Double click on the title box of the first column and rename it Orbital Radius (Radius for short) with units of m. Change the title of the second column to Period with units of s. Enter the data for the 4 moons.

   3. Create a calculated column for the Orbital Speed (Speed for short). Click on Data, New Calculated Column. Type in appropriate labels for a speed column. Use 2πR/T for the equation. Select the variables R and T from the drop-down box. Select the constant π from a drop-down box as well.

   4. Create another calculated column for Acceleration. Use v²/R for the equation.

   5. Now prepare your graphs. Begin with a graph of Acceleration vs. Radius. Title and format it appropriately. 

   6. Click on Insert, Graph. Plot Speed vs Radius for this graph.

   7. Repeat the last step for a graph of Period vs. Radius.

   8. Click on Page, Auto Arrange so that all your graphs will show without overlap.

   9. For each graph in turn, select Analyze, Curve Fit. Select the AR^n (variable power) fit, and type in the appropriate power. Apply the fit.

   10. Enter the numerical values for the fit coefficients into the table below. Don't include units. (This, together with the table in item 2 of the prelab, will serve as your matching table.)

Fit	Dependent variable	Value of Coefficient	Mass of Planet (kg)	|Percentage Difference|
1	Acceleration, a	<_>	<_>	0
2	Speed, v	<_>	<_>	<_>
3	Period, T	<_>	<_>	<_>

11. Knowing the values for the fit coefficients and the expressions for the coefficients from the prelab, calculate the mass of the planet for each relationship and enter the result in the table above.  For the acceleration relationship, for example,

Fit Coefficient, A = GM; therefore, M = A/G.

12. You calculated the mass of the planet using three different relationships. You would expect nearly the same result for each calculation. Calculate the absolute value of the percentage difference between the value of mass obtained for each of Fits 2 and 3 and that obtained for Fit 1. Agreement to within a few percent can be expected.  If your values for mass vary significantly from one another, then you may have a mistake in your data or calculations.

5. Check that your Logger Pro data table and graphs are formatted and labeled properly. Then upload your file.

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