CSC207 - Lab 3
Kepler and Newton
Overview
In this lab you will write a program to help the user explore Kepler's Third Law of
Planetary motion, using both Kepler's original equation and Newton's reformulation. This
will give you practice writing, testing and maintaining simple Python programs that
follow the Python style guide.
Materials
Description
A first step in searching for intelligent life elsewhere in the universe is to find
other planets that may harbor life. While much time is being spent to analyze the closest
possibility of Mars, researchers have made progress in recent years toward finding
planets outside our solar system orbiting other stars. The first few of these
extrasolar planets found were as large or larger than
Jupiter with wildly eccentric orbits, however as our methods of detection become more
precise we are starting to find Earth-sized planets that may provide a good foundation for life.
In this lab we'll discuss planetary motion in general and apply our computing skills to
study one particular star system called Gliese 581.
Kepler
One astronomer who made a major advance in the understanding of our solar system and
astrophysics in general was
Johannes Kepler.
In fact, NASA launched a satellite to search for habitable planets in 2009, and has
named this mission after Kepler.
In the early 1600s he published his now-famous three laws of planetary motion.
These laws, based on years of stellar and planetary observation by
Tycho Brahe,
finally fixed any lingering anomalies in the
Copernican theory
that the Earth and other planets
revolved around the Sun. The first law states that the planets orbited the Sun in ellipses with
the Sun at one foci. The second law states that planets would travel faster the closer they are
to the sun and slower when farther away.
The third law describes the relationship Kepler observed between a planet's distance from the
sun and the time it takes to make one complete orbit around the sun.
Kepler stated that the square of a planet's orbital period in years was equal to that planet's
distance from the sun in Astronomical Units (AU) cubed, where and AU is the
average distance of the Earth to
the Sun (149 million kilometers).
Kepler's Third Law of Planetary Motion
Step 1
Write a program called orbit_kepler.py
for Kepler's Third Law
of planetary motion.
This program will ask the user for the name of the planet
and its average distance from the sun in astronomical units (AU).
It should then calculate the orbital period
of this planet in years and display the result to the user. Make sure to follow
the Python Style Guide when writing your program.
Test your code with the following values for planets orbiting the Sun:
Table 1: Planets and their Orbital Period
Planet | AU from Sun | Period |
Earth | 1 | 1 |
Saturn | 9.58201720 | 29.660974748248961 |
Mercury | 0.38709821 | 0.24084173359179098 |
Newton
In 1687, Isaac Newton followed
up on the laws of Kepler to publish his
Principia Mathematica. In this work, he explained that it was the
universal force of gravity which tied together the motion of the planets and the motion
of objects here on Earth. Kepler's third law was found to be a special case of a more general
law about the gravitational attraction between two objects in space, M1 and M2. Now, instead
of Kepler's law being tied to the Earth and the Sun, we can now calculate the orbital
period of any planet around any star as long as we know both their masses and the
average distance of the star from the planet.
Newton's Revised Law of Planetary Motion
The big G in Newton's equation is the
Gravitational Constant
from physics, and is in terms of meters cubed over kilograms times seconds squared.
Step 2
Revise your program orbit_kepler.py
into orbit_newton.py
to use Newton's reformulation of Kepler's Third Law. The user will be asked to enter the
name of the planet, the average distance from the star in AUs, the mass of the star in kilograms,
and the mass of the planet in kilograms. Since Newton's law uses meters instead of
AU, you will have to convert the user's input into the appropriate value, using the
definition of 1 AU as 149 million kilometers. Calculate the orbital period of the planet next,
and output the result to the user. Your formula will calculate the period in seconds, but
you should convert your answer and return to the user the number of days in the orbital period.
Evaluate orbit_newton.py
using the following data about the red dwarf star
Gliese 581 and the four planets
detected so far which orbit this star. Current scientific research is
focused on this star system since
planet c is close in size to Earth and is
a comparable distance from Gliese 581 (reported in April 2007), meaning it may lie in the
Habitable Zone for life, although further
research has shown that planet d is more likely to be located well. Report your
results for the orbital period of these planets in the Lab3 Evaluation Google Doc
described below.
Table 2: Planets orbiting Gliese 581 (Mass = 6.16621 X 10^29 kg)
Planet | AU from Star | Mass in kg |
e | 0.028 | 1.1645 X 10^25 |
b | 0.041 | 9.4719 X 10^25 |
c | 0.073 | 3.1892 X 10^25 |
d | 0.22 | 3.6192 X 10^25 |
Step 3
Your programs were dependent on the value used in Python for math.pi
,
namely 3.1415926535897931. This is only an estimate of Pi; others have calculated
1,000,000 digits of Pi, but this is still only an estimate of this irrational number.
Test out the sensitivity of your calculations above to different values of Pi, using 3.14 and
3.14159, and record your results in Lab3 Evaluation Google Doc
. Be sure to return your
code to use the original math.pi
before you turn in your code.
Evaluation
Answer the following questions in a file called Lab3 Evaluation Google Doc
:
- How much of your code were you able to reuse from
orbit_kepler.py
when writing orbit_newton.py
?
- What are the orbital periods for planets b, c, and d orbiting Gliese 581?
- Which do you think had a larger influence on the final calculation of the orbital period
in
orbit_newton.py
, the Mass of the planet or the distance from the star? Why?
- What was the effect of changing the value used for Pi in
orbit_newton.py
?
- What other sources for numerical error (like that from estimating Pi)
do you see in the code you wrote?
What to Hand In
You will be handing in the three files you wrote today and the lab evaluation document.
- drake.py
- orbit_kepler.py
- orbit_newton.py
- Lab3 Evaluation Google Doc (shared with Dr. Goadrich, mgoadric@centenary.edu)
Log into your account at cs.centenary.edu
.
Copy the above three files into your csc207/lab3 directory.
Make sure you run drake.py
, orbit_kepler.py
and orbit_newton.py
through the Python
style guide checking program before you turn in your work. If you worked with a
partner, you both must hand in the files you wrote, with both of your names in the header
at the top of the file.
Grading
- To earn a D on this lab, turn in a working version of
drake.py
- To earn a C, do the above and turn in a working version of
orbit_kepler.py
- To earn a B, do the above and turn in a working version of
orbit_newton.py
- To earn an A, do the above and complete the evaluation document.
- To earn a 100, do the above and follow the style guide exactly.
© Mark Goadrich, Centenary College of Louisiana