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.
This is a really cool project, especially because it ended up being quite similar to mine! I am in awe that you got the vector field and TransformedField function in Mathematica to work, because I had such trouble during my own project. I should have asked you for help along the way, and maybe we both would have benefited from some overlap. I think my errors were mostly syntax problems in Mathematica, which is quite frustrating, but, as you mentioned, the learning curve for Mathematica can be a bit steep at times.
In regards to your issue about the vector field not appearing to have weaker magnetic field strength further away from the Earth in your model, I think that your model does have weaker fields farther out and that it is probably just a display problem with the VectorPlot functions in Mathematica. My model of a magnetic dipole ended up being pretty similar to yours in essence (it was based on the same spherical equation from Griffiths), but my use of contour plots explicitly shows that the magnetic field does, in fact, become weaker the farther you are from the origin. Hopefully my project posts, located under the categories Advanced Electromagnetism (341); Spring 2014; Brian, convince you of this. It seems like we both could have benefited from each other’s projects along the way after all.
It is a shame that you didn’t have enough success early on to be able to get to the part of your project plan where you wanted to model different types of planets and planetary cores, because that would have been really cool, although, admittedly, rather complicated. I suppose you could have substituted different magnetic moment in for different planetary core sizes (or ratios of core to total planetary radii) or something along those lines, and maybe done some cool manipulations that show the core size and the magnetic field changing together. From your Final Data post, that sounds somewhat like what you were aiming for. Having taken both introductory and intermediate Planetary Science in Vassar’s Astronomy Department, I have a good sense of where you would have been going with that type of modeling, and I’m glad you tried and got me thinking about it even if your project didn’t make it quite that far.