So far in this course, we have been studying circular orbits in a gravitational field. Today we will take a look at the other types of motion that a light object can undergo in the presence of another very massive object. These motions include elliptical orbits and free fall, and the concept of escape velocity. As discussed in the lecture part of the class, the fundamental physics idea we use is the conservation of energy.
The equations that govern this motion are complicated and we won't attempt to solve them in this course. In this exercise, however, we will use the CUPS Classical Mechanics software to solve the equations numerically. You will supply initial conditions to the programs and interpret the motion that they generate.
Bring up the CUPS Classical Mechanics program by connecting to RCS in the standard fashion. The link you want in the CUPS folder is called "Classical_Mechanics". When you open the link, you will have a menu of programs put before you. Choose the one that says Gravitational Orbits. This is a program called Orbiter, written by Bruce Hawkins at Smith College. The program comes up with a couple of windows which have the solar system orbits all prepared for you. Press F2 to start the planets in motion, if you like, and F3 to stop them. You will see the essentials of the circular orbits we've discussed so far.
The program has many interesting types of systems already built in. However, for this exercise we will use a simple two-body system consisting of a heavy object with the mass of the sun, and a light object with the mass of the Earth. Under the "Systems" menu, select "Create New System" and follow the menus:
Two screens appear, one showing the Sun-centered system and the other showing the CM-centered system. Of course, they look identical. You are now ready to proceed with the various cases of the exercise. Turn in your answers to the questions below on your own notebook paper. Be sure to include your name as well as your lab partner's if you didn't work alone.
Set the system in motion, as setup using the above parameters, using the "Go" command (F2 key). Describe the motion and explain it quantitatively, including what is magical about the value "6.2832" for the velocity of the Earth. (Hint: The program uses "natural units".)
You can change the default system parameters using options under the "Choices" menu. First, reset the parameters to their default values using the "Restart" option. Then, follow the windows of the "Change Parameters" option of the Earth in Polar Coordinates. In particular change the velocity of the earth so that it is smaller than 6.2832 AU/yr by something like a factor of two. (You are now done changing parameters.) Describe the motion, including a sketch. Include the orbit of Question (1) above in this sketch, showing the relative positions in the two orbits. Label the different parts of the new orbit.
What happens if you increase the velocity by about a factor of two? Try to find a velocity that is larger than 6.2832 AU/yr which keeps the orbit on the screen.
Calculate a new initial velocity for the "Earth" so that the orbit has an aphelion of 1 AU and a perihelion of 1/2 AU. Enter this velocity as in Question (2) and confirm that you got the right answer. You can do this in a number of ways, but consider Equation (1-34) in your textbook.
Go back to the default system parameters and now change the initial velocity of the Earth to zero. How long does it take the Earth to fall into the Sun? Compare to the calculated value. (See Section P5-4 of your textbook.)
Use the "Plots and Zoom" menu to Zoom Out by a factor of 5. (You probably want to do this twice, once for each of the two windows. Then you are done.) Calculate the escape velocity from the Sun at a distance of 1 AU and change the initial velocity of the Earth to that value, and the angle to zero degrees. Run the simulation and demonstrate that indeed the Earth (just barely) escapes from the Solar System.
Calculate the velocity you must give the Earth so that it goes out to 5 AU before falling back in towards the Sun? Put that value into the program and demonstrate you have the right answer.
