Calculate Solar Planetary Constant
By Reza Mihankhah Professor of Physics Henderson Community College Henderson, KY
Project Overview
Students will use real time data from ten satellites to (a) Calculate solar system planetary value for each satellite (b) Calculate the mass of the earth using Kepler’s third law (c) Obtain the mass of the earth from an internet site (d) Calculate the percent error between the two values
1 Student Learning Objectives
After completing this RWLO, the students will be able to:
Apply Kepler’s Third Law of Planetary Motion Collect radius and mass of the earth from a web site Collect altitude and period for each satellite from a web site Calculate planetary constant for each satellite, the mass of the earth, and percent error
2 Procedure
Time: Approximately 75 minutes
Materials: Computer, online access, printer, calculator, pencil and paper
Prerequisites: Students should have knowledge of Kepler’s third Law.
Implementation: This RWLO can be used as a laboratory project or as a homework assignment. Students can work individually or in groups of up to two.
1. Prior to class, the instructor should become familiar with the J-Track 3D and Hyper-Physics web sites. Students must be advised to turn off their pop-up blockers when using the J-Track 3D web site.
2. Introduce Kepler’s 3rd law: the square of the orbital period of a planet is directly proportional to the cube of the average distance of the planet from the Sun; that is,T2 R 3 . To simplify the derivation, we will use circular orbit for the planets. The centripetal force Fc necessary to keep the planet of mass Mp in a circular orbit is the gravitational force FG exerted on it by the sun of mass Ms. Thus, 2 Mp v GM s M p Fc = FG or = (1) R R2 2PR Where v is the speed of the planet, v = and T is the period for one T revolution. Substituting for v in equation (1), 2PR 1 GM M M ( )2 = s p p T R R2 Solving forT 2 gives 4P2 4P2 T2= R 3 , we will let, k = (2) GM s GM s Where k is the planetary constant. Thus,T2= kR 3 which is Kepler’s 3rd
law? Let Ms = M = mass of any planet at the center of a circular motion, solving for M in equation (2) will yield 4P 2 M = (3) Gk Students will use this equation to find the mass of the earth using k of each satellite.
3 Content Material
Student Directions: 1. Go to http://science.nasa.gov/RealTime 2. Click “+J-TRACK 3D” button 3. JTrack-3D window will open, click “satellite”, click “select” 4. Scroll down and locate the first satellite from the data Table II and click on the name 5. The orbital path of the satellite will be displayed 6. Close the satellite window, you are back to the +JTrack-3D window, click on view, click on satellite position 7. Enter the values for Altitude in meters and period in seconds in Table II. Close the satellite position window and repeat steps 3-6 for the remaining nine satellites 8. To complete the project, you will need: the mass of the earth, the radius of the earth, and the Gravitational Constant. These values can be found from: http://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html and http://en.wikipedia.org/wiki/Gravitational_constant Record these values in Table I 9. Find the distance each satellite is from the center of the earth and record it under the Radius column 10. Calculate the k for each satellite and record it in table II 11. Use equation (3) and find Mearth 12. Find and record the percent error between the calculated value of Mearth and the value from table I
4 Table I
R, Equatorial Radius of G, Universal Gravitational Mass of the Earth (kg) the Earth (m) Constant (N.m2/kg2)
Table II
Name of satellite Period Altitude R K of MEarth MEarth % Error (s) (m) (m) Satellite Calculated Table I AQUA
NOVA I
CANX-1
ORBCOMM FM 13
AO-7
SINAH 1
KO-25
SAUDISAT 1B
KOMPSAT
KITSAT-3
5 Referenced URLs:
You may review Kepler’s Laws of Planetary Motion at the following site: http://hyperphysics.phy-astr.gsu.edu/hbase/kepler.html
6 Assessment
Each student will submit a formal lab report using the following format:
1. Objective (5 pts.) - This section is a brief statement of what you are investigating with this experiment.
2. Theory (15 pts.) - This section explains the theory you believe explains your objective (5 points). The theory should be explained both qualitatively (in words, 5 points) and quantitatively (develop, list, and explain all formulas, 5 points).
3. Procedure (30 pts.) - This section is used to explain, in detail, step by step reconstruction of what you did. This section should be written so that anyone who reads the report can reconstruct your experiment. Three-point deduction for each missing step.
4. Data and Analysis (35 pts). - This section is where you present all your experimental results and the results of any calculations (10 points). The values should be labeled as to what they represent and include units (4 points). You will also perform one sample calculation of every type of calculation you will need to do in the experiment (5 points). Start by listing the formula in algebraic form and then substitute numbers in for the variables (3 points). Show how you solve for the unknown and how the units cancel (5 points). The sample calculations should be written directly below the data (8 points.).
5. Discussion/Conclusion (15 pts.) - This section is used to explain how accurately the theory explains/predicts the observation (8 points). If you are not 100% accurate, you need to suggest ways in which the experiment or theory might be improved (3 points). You should end this section with a brief summary of what was accomplished (4 points).
7 Links to Course Competencies
This RWLO could be applied in the following courses: College Physics I, Applied Physics I, Astronomy I, and Introduction to Physics I. Specifically, this RWLO meets the following course competencies:
To understand, analyze, summarize, and interpret a variety of reading material.
To communicate effectively using standard written English.
To explain the application of fundamental physical principles to various physical phenomena. To elaborate upon knowledge to create thoughts processes that are new to the student. To think critically and make connections in learning across the disciplines.
8 Supplementary Resources
Kepler’s Laws of Planetary Motion: http://hyperphysics.phy-astr.gsu.edu/hbase/kepler.html
Science @NASA: this site contains information about satellites: http://science.nasa.gov/temp/StationLoc.html
Minds-On Physics Planetary and satellite Motion: http://www2.glenbrook.k12.il.us/gbssci/phys/Class/circles/circtoc.html
Tutorials in Kepler’s Laws: http://home.cvc.org/science/kepler.htm http://hyperphysics.phy-astr.gsu.edu/hbase/kepler.html
9 Recommendations This project is designed for the first semester of a two semester algebra-based physics course. However, it can be used in any introductory and/or calculus- based physics and Astronomy I courses.
Back-up: If the internet is unavailable, the data provided in the Back-up-Data sheet could be used.
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