AST 101 - Lab5 - Jesse Brown

AST101: Our Corner of the Universe Lab 5: Simulating Planets’ Orbits (I) Name: Lab section: Group Members: 1 Objective To use a computer simulation to explore the properties of planets’ orbits, and in the process learn a little about how computers simulate things. 2 Introduction As we’ve discussed in class, the development of orbital mechanics proceeded in two steps. First, Johannes Kepler worked out the correct shapes of the orbits of the planets, but didn’t know why they moved in that way. Later, Isaac Newton discovered the laws of gravity and motion that explain why , not just how , the planets move in elliptical orbits. In this lab, you will work directly with a computer program that contains within it Newton’s laws of motion and gravity. You can use this program to simulate all kinds of orbits! In this lab, you will use this program to simulate different kinds of orbits and see how they behave. This lab will also expose you to a little computer code, written in the language Python. You do not need to know anything about computer programming beforehand. However, one step in this lab requires you to look at some computer code and think about what it does. We’re not going to require you to write code, but we want to give you a little exposure to what it looks like. Python is a language used by many scientists and web developers (and other people) for all kinds of things; it is one of the most popular programming languages today. Jesse Brown M030 Vivian and Tate

2.1 Kepler’s laws of planetary motion First, Kepler realized that if planets moved in elliptical orbits rather than circular ones, their movements fit the observations very well. He formulated three laws of orbits that describe the motion of the planets. Paraphrased, they are: 1. Planets move in elliptical orbits with the Sun at one focus 2. The line connecting a planet to the sun sweeps out equal area in equal time, so planets move faster when they are closer to the Sun, and slower when they are further away 3. The time it takes a planet to orbit the Sun depends only on the length of its semimajor axis. In particular, if the semimajor axis is multiplied by a factor A , the period P is multiplied by A 3/2 . 2.2 Newton’s laws of gravity and motion Newton realized several things about moving objects: 1. Forces cause objects to accelerate by an amount a = F / m in the direction of those forces. 2. Gravity is one such force. All objects attract each other with a force equal to F grav = Gm 1 m 2 r 2 where r is the distance between the two objects, and m 1 and m 2 are their masses. 2.3 Computer programs You won’t need to write your own code here – only modify my code to see what happens, and then look at some parts of it to see how it works. You can’t break anything; if something goes wrong, you can just click the three-lines menu in the top left and choose “Reset”. If you press “Run” and your code gives you an error (a big red bar at the bottom), either hit Ctrl-Z to undo your last few changes and then try again, or refresh your browser. Your GTA might be able to assist you as well (most of the graduate students will know some Python). Computers, at heart, do math on variables . Look at line 5: this assigns the value 1.017 to the variable start distance . A computer program is just a list of math instructions in order, along with other instructions that do things like draw on the screen and repeat some segments of math. It is important that you don’t change the names of any of the variables here: if you rename start distance to begin distance , even though those mean the same thing in English, the com- puter won’t know what is going on later. It is also important that you don’t add extra spaces before the beginning of any lines, or remove any that are there. Spaces before a line mean something special in Python, and if you remove them, the program will mean something different than it did before (and probably won’t work). Also, notice the text that’s displayed in green and follows a # sign. These are comments – the computer ignores them. They’re just in the code to give you “signposts” to describe what parts of the code do.

Try starting velocities of 1. 2.5 AU/year 2. 5 AU/year 3. The value you found during your prelab that matches Earth’s actual orbit 4. 8 AU/year 5. 10 AU/year

Question 3. What happened when you gave the Earth a large starting velocity of 10 AU/year? Is this what you expected? Do you think that the Earth would really move in this way if its velocity were suddenly increased to 10 AU/year? (3) Question 4. Remember that the focus of an ellipse is close to the center of the long axis if the ellipse is not very eccentric, but close to the edge if it is strongly eccentric. Look back at your drawings for the previous question. Is this true in the orbits you observed? (2) Question 5. Now, let’s look at Kepler’s second law, which compares the speed of a planet at different places in its orbit . By changing the starting distance from the Sun and the starting velocity, put your planet into an orbit with eccentricity at least 0.6. Does your planet move more quickly near perihelion or aphelion? In evaluating this, look at both the animation and the graph to make sure that they agree. How does your graph show this? (2) Question 6. Does your planet spend more time near perihelion or aphelion? (2) Question 7. Does your planet appear to follow Kepler’s second law? How do you know? (3) I did not expect the earth to go in a straight line. I think the earth would move in this way if its velocity weer suddenly increased like this because it would basically go out of orbit with the sun. Yes, this remains true in the orbits we observed. The planet moves more quickly near the perihelion . The graph reflects this just as the graphic does because the graph spikes up when when it gets closer to the sun. The planet spends more time near the Aphelion because it moves slower when it is farther than the sun. Yes, it does follow Kepler's law because it moves faster when the planet is nearer to the Sun.

4 Does the Mass Matter? The mass of the planet is set to the mass of Earth by default: 0.000003 times the mass of the Sun. Change the mass of the planet to the mass of Jupiter: 0.001 times the mass of the Sun. Question 10. Did the planet’s orbit change? Describe how it changed, or explain why it didn’t. (2) Question 11. Now, change the star’s mass. Does this affect the orbit? Why or why not? (2) Kepler’s description of planet orbits – where the Sun doesn’t move and the planet orbits around it in an ellipse – are based only on the motions of the planets in our solar system. They are true only if: • A much less massive object orbits a much more massive one (like Earth around the Sun) • The gravity of the larger object being orbited is the only force that causes much of an effect on the smaller object. Earlier, you saw that changing the mass of the planet orbiting the Sun doesn’t affect its orbit. But what happens if you make it very massive – almost as massive as the Sun itself? Question 12. Increase the mass of the planet to at least ten percent of the star’s mass. What happens now? Is this what you expect? (2) Question 13. Is this behavior something that matches Kepler’s description of planetary orbits? If it’s not, which of the assumptions above are no longer true? (2) Question 14. Suppose that you looked in space and saw a star wiggling back and forth. We can Only the eccentricity slightly increased. Yes, this makes the perihelion lower, makes the orbit more eccentric, and the orbital time is decreased. The star begins to loop as well, this makes sense because now that the planets mass is larger we can see the forces effect on the sun more clearly. No, it does not match Kepler's description because Kepler assumed that the Sun doesn't, move.

see stars with telescopes, but can’t see planets since they don’t make their own light. If you saw a star wiggling back and forth, what could you conclude about it? (2) Question 15. We know that many stars are binary stars – there are two stars that orbit each other closely. Since we know that gravity affects stars in the same way as it affects planets, we can use this program to simulate binary stars too. Set the mass of the “planet” to nearly the same as the mass of the “star”: both of them will now represent stars. See if you can create an orbit where the two stars orbit each other. Show your orbit to your TA when you’ve got it. 5 How’s This All Work? This program may look complicated. But most of the code has to do with making the animation and table you see! The only code that actually performs the simulation – where the actual physics is – is between line 50 and 65. All of the laws of nature that control the motion of orbits are the following: • Forces cause things to accelerate, where the acceleration is equal to the force divided by the mass of that thing (Newton’s second law of motion) • Any two objects attract each other with a force equal to F = Gm 1 m 2 r 2 , where the force is directed toward the other object (The law of gravity) • Forces come in pairs: if the star’s gravity exerts a force on the planet, the planet’s gravity exerts a force back on the star that goes (Newton’s third law of motion) in the opposite direction and has the same size • In a small interval of time, an object’s position changes by an amount equal to its velocity multiplied by the amount of time. (You know this already: if you are traveling at 50 miles per hour, and you do this for two hours, you will go 100 miles...) (Definition of velocity) • In a small interval of time, an object’s velocity changes by an amount equal to its acceleration multiplied by the amount of time. (Definition of acceleration) The stars mass must not be that large, or the mass of one of the planets around the star is very large.