Monday, June 21, 2010

LAB 9, Monday, June 21

PhysicsLab9 June 21, 2010 Name __________________
Dr Dave Menke, Instructor

I Title: Applying Kepler's Laws to Planetary Orbits

II Purpose: To determine the distance, orbital period, and velocity of the planets given, using Kepler's laws of Motion. To determine radio communication wait time in space.

Background information:
The German genius physicist, Herr Johannes Kepler (1571-1630), determined his three laws of motion in separate research, and they were published separately. However, one of his laws is P2 = k a3, where P is the period of revolution of the object; "k" is a constant, and the letter "a" represents the semi-major axis of the planet's elliptical orbit.
If the orbit were a circle, then "a" would be the radius of that circle.
The constant, k, is equal to the number 1.00, but ONLY if the period, P, is in Earth Years; AND if the semi-major axis, "a," is given in astronomical units (AU). Otherwise, the constant
k = [4p2 /G]/(M + m), where "G" is the constant of universal gravitation, [G = 6.673 x 10-11 N-m2 /kg2 .] The letter "M" refers to the mass of the heavier object (Sun) and the letter "m" refers to the mass of the lighter object (planet) in kilograms. The units of period, P, would be in seconds. The letter "N" is a unit of forced called a "Newton."

One astronomical unit, AU, equals 150,000,000 kilometers (approximately). The speed of light, "c," is equal to 300,000 km/sec (approximately).

III Equipment -- Each Lab Station Will Have:
Clerical supplies: pen, pencil, or quill; calculator, abacus, or mainframe; ruler, protractor, French curve; personal hard drive (brain); personal printer (hand at end of arm); graph paper, etc.

IV Procedure
1. Given the value of an astronomical unit, and both the perihelion and aphelion distances for several planets (in the DATA TABLE below), find for each planet given:
a. the semi-major axis (a) of its elliptical orbit;
b. the period of revolution about the Sun, P, in years;
c. the average velocity of all the planets listed, around the Sun (in this case,
assume all have circular orbits).

2. Pretend that we have traveled to Ganymede (a large moon of Jupiter). At that
instant, if you are given the distance from Earth to Ganymede, find out how long it will take for a radio signal to reach Earth. (The distance to Ganymede will be similar, but not exact, to the distance to Jupiter. Consult the Data section below).

3. With the same conditions, calculate the typical response time from an
astronaut on Ganymede to Houston's Johnson Space Flight Center.
(Roundtrip wait time).

V Data & Calculations
Below is a table of the distances that the indicated planets are from the Sun, at their closest point, perihelion, and at their most distant point, aphelion. The units are in Astronomical Units, also known as A.U.'s, where 1.0 A.U. = 149,500,000 kilometers = 1.49 x 1011 meters, which is the average distance that Earth is from the Sun.


Planet Perihelion Aphelion SemiMajor, a (AU) Period, P (years)
Mercury 0.31 0.47
Venus 0.72 0.73
Earth 0.98 1.02
Mars 1.39 1.66
Jupiter 4.95 5.46


[Perihelion + Aphelion]/2 = a = semi-major axis

2pr = circumference

velocity = 2pr/P = circumference/Period

VI Results
The purpose was / was not achieved because

VII Error Analysis
A. Quantitative Error
Find the Average Percent Error. Do this by first finding the percent error for each of the five planets, i.e., find the percent error for Mercury, then for Venus, etc., and then add all five percent errors and divide by 5 to get Average Percent Error.

Percent error = [|True – Yours| / True ] x 100% =

What is the “truth” as far as periods go? Look it up on Wikipedia.org or some other source, like an astronomy book.

B. Qualitative Error
Sources of error are enumerated as:
Personal –
Systematic –
Random --

VIII Questions
1. Find the average orbital velocity of each of the above planets, in km/sec. To do this, you must assume the orbits are circles in the first order approximation.

2. Pretend that we have traveled to Ganymede (a large moon of Jupiter). At that instant, if the distance from Earth to Ganymede is 5.38 AU, find out how long it will take for a radio signal to reach Earth.

3. With the same conditions, calculate the typical response time from an astronaut on Ganymede to Houston’s Johnson Space Flight Center. (Round trip wait time).

4. Use the Copernican equation (below) to determine the Sidereal (true) Period of revolution, P, from the, Synodic (observed) period, S, for Jupiter and for Saturn. The Synodic Period of Jupiter and Saturn are 13 months and 12.5 months respectively.

1/P=1-1/S

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