At the conclusion of this project I have to say I came out of this project having learned a lot about organizing a project, the Van Allen Belts and how to work with Mathematica. But first I would like to discuss the conclusions that can be drawn from my final data.
Unfortunately I can’t draw any major conclusions about the Van Allen Belts and their location depending on core size as I was not able to incorporate that particular information into my models. I was however successfully able to model the vector fields of a magnetic dipole using Mathematica and an initial equation given by our textbook, a task that proved to be difficult enough on its own to figure out. In the end I did prove that the equation was valid and created a vector field that resembled the expected field of the dipole. Even with this however, I question the validity of the models that were obtained using Mathematica. As I mentioned in my “Final Data” post, the vectors do not seem to vary with respect to the radius away from the origin which doesn’t match the actual system I was attempting to model. It should be decreasing with respect to distance. In the end the project was never really able to combine the Van Allen Belt mapping and the modeling of Earth’s magnetic field into the single cohesive unit.
Furthermore the model itself had a lot of assumptions that needed to be put into place in order to make it easier to develop or simply because I could not figure out how to account for it within Mathematica. The assumptions were listed throughout my final data but they included using a constant magnetic field isolated from the sun’s solar wind to model out the Van Allen belts and assuming the Earth to be a sphere. In the long term however, these were all reasonable assumptions in astronomical scales.
If there was one thing I was somewhat disappointed about, it was the learning curve to reacquaint myself with Mathematica. It had been awhile since I had used Mathematica and it took some time to recall the best functions to use for particular tasks. Furthermore when trying to learn how to do a new task; like converting between Spherical and Cartesian coordinates with Mathematica, it was a long process of trial and error to achieve a result you were looking for. More often than not you would get something more like this:
So a great deal of the project was spent playing around with Mathematica more that playing around with the physics. I do certainly wish I had a bit more time to play around with the physics and even more time to play with the Mathematica program. After a few breakthroughs, I was quickly understanding how to make the program much more cooperative and finding relevant information in the help menus faster. With a little more time, I fixed and accounted for a few of the issues that had been plaguing my models.
Regardless, I was a bit overzealous with the scope of my project. It had a lot of components that were much more difficult than I initially expected them to be. What I ended with were two separate projects, a research summary of Van Allen Belts and a 3D model of Earth’s magnetic field with vague notions of where the Van Allen Belts would be located. Still I am proud of what was accomplished. Modeling the vector field of a dipole was a difficult task and a properly organized model was created.in the end. It was also plotted with proper constants though it needed a little tweaking to confirm that the structure was as expected. Finally from this point it could easily be picked up and completed at a later date with those willing to work on it needing only to manipulate the dynamo equation into an easier form.