A primer to numerical simulations: The perihelion motion of Mercury

This is the script repository for this publication.

Abstract: Numerical simulations are playing an increasingly important role in modern science. In this work it is suggested to use a numerical study of the famous perihelion motion of the planet Mercury (one of the prime observables supporting Einsteins General Relativity) as a test case to teach numerical simulations to high school students. The paper includes details about the development of the code as well as a discussion of the visualization of the results. In addition a method is discussed how to estimate the size of the effect a priori. Which allows to double check the results found numerically.

In this repository, we provide example scripts and templates which can be used by teachers and students to simulate, visualize and extract the perihelion motion of Mercury caused by General Relativity. The programs are available in two flavors, plain Python scripts in py-scripts and corresponding Jupyter Notebooks in ipynb-scripts.

Table of Contents

1. Requirements
2. Installation instructions
   • Installation under Linux and MacOS
   • Installation under Windows
3. Running the scripts
4. Credits
5. License

Requirements

• Python 3.5 or later
• VPython, version 7 or later (tested with 7.3.2)
• (optional) Jupyter

Installing and running VPython

Here, we explain how to install Python and VPython on different operating systems.

For users familiar with Python, we recommend to pip install the vpython package for Python3 (Python3 version number 3.5 or later).

Linux and MacOS

Installing a Python distribution

All Unix based system come with an installation of Python. In case this is Python 2, a version of Python 3 needs to be installed. If you already have Python 3 installed, you can skip to Installing VPython.

The best way to install Python 3 is to use your distributions package manager, e.g. dpkg or apt-get on Linux, or Homebrew or MacPorts on MacOS. While Linux usually comes with a package manager preinstalled, macOS users must install them first. For this we refer to MacPorts or Homebrew. As an alternative, one can also install Anaconda. This is decribed in Installation under Windows

You can use the graphical interface of your package manager, if it has one, or use the terminal. Here, we give instructions for installing Python via the terminal. Depending on the installation, one usually needs superuser rights in order to update or install Python. Therefore one has to use sudo in the following commands and confirm them by entering the superuser password. The installations using a distribution manager are save and tested by vast communities. In the following, the successive order of commands to update Python and install the Python package manager Pip is presented for apt-get (Ubuntu) and MacPorts (MacOS). Other package managers use similar commands.

Ubuntu

sudo apt-get update
sudo apt-get install python3
sudo apt-get install python3-pip

MacOS

sudo port selfupdate
sudo port install python36
sudo port install py36-pip

Depending on the previously installed packages, further dependencies might be installed automatically (after confirmation). One should test if the installation was successful by typing python3 --version or python3.6 --version (depending on the installation name of Python3). This prints out the version of the Python installation (and should be larger than 3.5).

Next one must update the package manager to get the newest versions of Python packages. This is done by running

pip3 install --upgrade --user pip wheel

Depending on the installation name for Pip3, one might has to exchange pip3 → pip3.6 (MacPorts). This is best figured out by typing pip and pressing tab. Note that this just upgrades the packages for the current user. If one wants the systemwide installation, prepend sudo and remove the --user flag (sudo pip3 install --upgrade pip wheel).

Installing VPython

Finally, install VPython by running

pip3 install --user vpython

Again, leave our --user and add sudo to get a systemwide installation.

At this point, everything is installed and one can start the simulations. To test the VPython installation, run python3 or equivalent commands depending on the installation name (e.g., python3.6 on MacPorts) to start an interactive Python console.

Next run

from vpython import *
box()

in the Python console, which loads the VPython package and opens the default browser. Running box() should generate a white box in the browser window, which can be rotated using the mouse. Quit the program by running

quit()

or pressing Ctrl-c in the terminal.

Windows installation

Summary

In order to install VPython on Windows, one has to execute the following three steps:

1. Install Python3, e.g., via Anaconda.
2. Start the Anaconda Prompt application and run the command conda update conda.
3. Install VPython by running conda install -c vpython vpython in the Anaconda Prompt.

Installing a Python distribution

As the first step, one must install Python. There are several possibilities but we suggest Anaconda for an intuitive installation. There are two possible (free) options, Anaconda and Miniconda. Both are equally sufficient and one can install further updates after the initial installation. Download the newest Python 3.x (with x greater equal 5) installer for Windows and the right system type (32-bit or 64-bit). The system type can be found by pressing Windows+I, navigating to System → About and looking at the System type entry.

Installing one of the two executables creates a new Windows application called Anaconda Prompt. Starting this application opens a terminal which will be used to install the required Python packages. At first, one should test if the Anaconda installation was successful by running the command

>>>>>>> de7d3afb466420cc9ae56dcecf15279091a3ee12
conda list

This command should print a list of already installed packages. Also, run

conda update conda

to make sure you have the most recent installation.

Installing VPython

To install VPython, simply run

conda install -c vpython vpython

in the Anaconda Prompt. The additional flag -c vpython tells Anaconda the channel where the VPython package can be found. Running this command prints out a list of dependencies which will be automatically installed in order for VPython to work. The installation of any extra packages has to be confirmed.

To test the VPython installation, launch the Anaconda Prompt, enter python3 to start Python and enter

from vpython import *
box()

This should open the default browser and render a white box on black background. Using the mouse (right-click and drag), one should be able to rotate the box. The Internet Explorer was found to not be able to properly render the animation. You can use Firefox or any other modern browser of you choice.

Additions

In principle this is already sufficient to run VPython scripts. You could furthermore install Anacondas native navigator Anaconda Navigator by running

conda install anaconda-navigator

This will install another application, Anaconda Navigator, which can be started from the Desktop and simplifies programming in Python.

Running the scripts

In general, there are multiple ways to run Python scripts. We provide instructions for both Python script files and Jupyter notebooks.

Linux and MacOS

Python files

The first scenario is to write a Python file, e.g., test.py, which can be edited using any text editor of choice. To run the Python script, navigate to the containing folder in the terminal using the cd command and execute the script in the interpreter, e.g.,

cd ~/Desktop/
python3 test.py

Assuming the test.py is located in the directory ~/Desktop/. The command cd either takes relative paths (directories which are in the current directory), e.g., cd folder, or absolute paths beginning with a /, e.g., cd /Users/name/Desktop/. Once in the folder which contains the Python file (e.g., the file name is listed when typing ls), the file can be executed using the interpreter (command python3 filename.py). Again, this script can be ended (if it does not end automatically) by pressing Ctrl-c.

Jupyter notebooks

The second scenario to run Python scripts is using the interactive iPython (or equivalently Jupyter) notebooks. For this, you need to install the package first by running

pip3 install --user jupyter

in the console. Jupyter notebooks are automatically installed when installing the newest version of VPython.

To start an interactive session, one has to run

jupyter notebook

in the console (or equivalent, depending on the installation name. Again, this can be figured out by typing jupyter and pressing tab).

If succesfull, this command will print a link (containing "localhost") in the terminal. Usually, this command will also automatically open the default internet browser at the link-address. If not, one can simply copy and paste the printed link in the address bar of the browser of choice. Within this browser window, one can freely navigate to the directory of choice and create interactive Python and VPython sessions by clicking on existing files or creating new ones (upper right button new → vpython).

Windows

There are several possibilities for running Python scripts on Windows. Here we provide instructions based on an Anaconda installation.

• Create a file test.py, edit it with a text editor of choice and specify to open the file with Python3. To do so, right-click on the file, select open with and locate the Anaconda Python3 installation. Programs can be closed by closing the running window.
• Run python3 in the Anaconda Prompt application to start an interactive session. Programs can be closed by closing the running window. This command executes the content of a Python file, if one appends the location of the file, e.g., python3 test.py (where the current working directory contains the file test.py).
• Start the Anaconda Navigator application, open an interactive QT console and program within the newly opened QT terminal. Programs can be closed by closing the running window.
• Start the Anaconda Navigator application and click on the Jupyter Notebook button. This opens an interactive Python session in the default web-browser. Here, one can open existing notebooks or create new files (upper right button new → vpython).

Authors

• C. Körber
• I. Hammer
• J.-L. Wynen
• J. Heuer
• C. Müller
• C. Hanhart

License

MIT License

Copyright (c) 2018 Christopher Körber

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.