# CSC 161 Lab 2: 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.

## Materials

• IDLE Python 2.5
• Lab Partner

## 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 plans to launch 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).

### 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. 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 three 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. Report your results for the orbital period of these planets in the lab3_evaluation.txt described below.

### Table 2: Planets orbiting Gliese 581 (Mass = 6.16621 X 10^29 kg)

 Planet AU from Star Mass in kg b 0.041 9.30314 X 10^24 c 0.073 3.03776 X 10^24 d 0.25 4.55664 X 10^24

### 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 lab2_evaluation.txt. 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 text file called lab2_evaluation.txt:

1. How much of your code were you able to reuse from orbit_kepler.py when writing orbit_newton.py?
2. What are the orbital periods for planets b, c, and d orbiting Gliese 581?
3. 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?
4. What was the effect of changing the value used for Pi in orbit_newton.py?
5. 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.

• orbit_kepler.py
• orbit_newton.py
• lab2_evaluation.txt

### Turning in Your Work

To turn in your labs, navigate to the course home page within Blackboard. Click on the "Assignments" link on the lefthand side, and then click on the assignment you are submitting (in this case, "Lab 2: Kepler & Newton"). You will be presented with a page to upload your files and optionally enter some comments. The page will look something like this:

This document includes material © Mark Goadrich 2007