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Orbit of Satellites

Objective:

Determine the mass of a planet by observing the motion of its moons

Introduction & Theory:

Deriving Velocity from Net Gravitational Force

Given:

Goal:

Prove , assuming all masses are treated as point masses.

Strategy:

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1. Use net force set up to find

2. Gravitation and Net Force Solution

(Newtons Second Law):

Deriving Period from Acceleration

Given: Goal: Prove

Strategy: Couple formulas for period and velocity in terms of angular velocity and radius,

and .

Mathematical Solution:

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Matching Table

Fit y-variableRelation-

shipCoefficient

Units ofCoefficient

Power ofRm

Value ofCoefficient

Mass ofPlanet

(kg)

PercentageDifference

1 Acceleration, aa =

(GMp)/Rm2 GMp m

3/s

2-2

0

2 Speed, v

-.5 -.09%

3 Period, T

+1.5 -4.7%

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Data Table

Data Collection: Radii calculated as the vector magnitude of . Period takenby observing time at which a single orbit is completed.

CalculatingMethod of calculatingA is shown below for each graph. Value ofA can be found in DATA

TABLE above.

Fit Graph Axes Coefficient Rearrange for1 Acceleration vs. radius

2 Speed vs. radius

3 Period vs. radius

Analysis

Fit 2:

Fit 3:

Results: Here, concepts of gravitation and rotational kinematics were applied to calculate themass of the planet,Mp within 5% uncertainty, based on the orbit of its 4 moons.

Moon Radius (m) Period (s)

Red

24900

Green 505000Black 97000Orange 3.54 x 108 507200

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Conclusions:

In each of these methods, some quantity of motion was plotted against the radius of each

moon, and the planet mass was acquired by comparing the fit coefficient to relationships derivedfrom the law of universal gravitation.

The simplest example is with the graph of acceleration vs. radius. We first derived the

velocity as , and coupled with the acceleration formula , we found that .This can also be verified by applying Newtons Second Law ofMotion, such that , and canceling out the values for satellite (moon) mass,. The radii were calculated asthe magnitude of the vector with components of horizontal and vertical displacement with

respect to the planetary mass (taken from applet).

The plot ofa vs. rwas done using a LoggerPro 3.8 curve fit, AR^n, where n = -2.

Since A must map to , dividing the fit coefficient by the gravitational constant G gives Mp,the planets mass.