T5: Kepler fitting

Before the tutorials:

Do, think about, and research the following:

• Bring your own program to the level of the example given for the last tutorial, so that you can start from a clean and working version.
• Which integrator showed the best performace in the last exercise? Lets use it from now on.
• What are the 3 Kepler laws ?
• How is a ellipse defined ?
• In our configuration, where is the focus of the orbits ?
• What do you need to calculate to prove the second law and how can you do it numerically ?

During the tutorials:

Program it!

• Take your leap-frog integrator from last time and produce some orbits.
  For simplicity, put your sun in (0,0), your planet at (1,0) and only modify the initial y velocity.
• Can you fit all orbits with ellipses ?
  Use the simplifications to obtain an easy way to fit the ellipses. Remember the exercises with gnuplot !
• Modify your integrator to test the Kepler's second law.
  What do you need to output additionally ?
• Measure the orbit time.
  Together with the fits from before, can you prove Kepler's third law ?

Maybe you can further explore the power of C++ (if not done already):

• Try write the output directly into a file instead of the standard output.
• Try to use a Makefile.

Example programs:

These are just examples.

• C/C++:
  • Current version (making extensively use of C++ features):
    example.cc (colorized), example.h (colorized), Makefile (colorized) (rename to “Makefile” or use with “make -f Makefile_c”)
• F90:
  • Simple version (this is sufficient for the moment, but it doesn’t hurt you to start improving it as suggested above):
    T5.f90 (colorized), Makefile (colorized) (rename to “Makefile” or use with “make -f Makefile_gfortran”)
  • More complex version:
    (To be added later. We will reorganize the program structure when we get to multiple particles; think about how you would do this on your own.)

Discussion of the results:

Plot your results with gnuplot and use it's ability to fit the orbits to obtain the orbital parameters. Can you plot the Kepler Laws ?

Example plots:

Programming goals for T5:

Goal of this tutorial is that you acquire proficiency in using gnuplot and learn how to use it for fiting functions to data. In addition you should become familiar with a Makefile.