Beyond the surface of the Earth, our magnetic field continues on. It stretches into space and interacts with the continuous flow of plasma coming from the Sun. Some of these particles enter deep into our magnetosphere, and can become trapped on our magnetic field lines. In this project I will look closer at where the particles get trapped, and how this varies with the energies of the particle.

The Earths Magnetic Field behaves at some points, very different from a standard dipole. On the day side it is compressed and deformed by the solar wind pressure, and on the night side it is stretched long and thin.

At the poles the field lines connect with those of the sun, running of into space, and can from our perspective be regarded as open field lines. It is still possible to make some easy estimations about or magnetic field, because it behaves approximately like a dipole, at least for the lower latitudes. These are the ones we will look closer at.

If we assume that the field lines behave like those of a dipole, their extension is easy to find. We use the equation:

[1]

Where r is the distance from center of the earth, θ is the angle from North pole to measured point.

θe is the angle from North Pole to point where the field line crosses the Earth's surface, and Re the radii of the Earth. Finally our value L is what correspond to a θ at 90 degrees, and is the point on the field line furthest from the Earth.

As mentioned in the intro, there are particles trapped in the EMF. These particles are primarily ions and electrons. Due to the Lorentz force, these particles will gyrate around the magnetic field lines, and move along them in their path towards the Earth. As the particles come closer, the magnetic field lines will converge and the Lorentz force will start slowing the particles before finally pushing them away from the Earth, back into space. This occurs at a particles mirror point. It is this mirror point that my project will be looking into. There are a few equations we will need to calculate the position of this point.

We know that the particles have a magnetic dipole moment, which gives it a certain velocity. If we decompose the velocity into parallel and perpendicular velocities, we get a profile of their velocity relative to the direction of the magnetic field line.

Since the EMF increase the energy of the particles, we know that at the mirror point the perpendicular velocity will be equal to the total velocity.

This gives us the following transformation:

[2]

Now we have a relationship between the magnetic field strength and the angle α between the particle motion and the magnetic field line direction. If we can find the field strength in any position, we can find the magnetic mirror point of a particle of any angle. Notice that particle total speed is irrelevant, as a higher velocity will also contribute to a stronger Lorentz force, and consequently mirror it in the same point as a lower velocity particle. Only the angle is of any relevance.

To find the field strength in any point we use the magnetic dipole model to find that:

[3]

With this it's easy to find the field strength in our reference point, but for convenience we shall choose the point at the equator so that θ = 90.

But we still don't know where the particle mirrors. We can find the field strength, but we still have both r and θ as unknowns.

To solve this we insert equation [3] into equation [2] to find that:

[4]

Since the left side uses θ = 90, r/Re becomes L.

We also use equation [1] to see that r/Re = L sinθ^2 on the rigth side. We can then scratch both Ls and are left with:

[5]

We now have a relationship between α and θm, and since α will be a given, we can find θm. We can then go back into equation [4], and find r/Re for the mirror point. We now have both the angle and distance to the mirror point, and we require only the latitudes of our magnetic field line, and the angle between particle motion and the field line in the equatorial plane. Both of these will be our inputs in the program.

Briefly about the program.

The program is divided into two different .C files. Earthdraw.C and Mirrorcalc.C

Earthdraw.C is simply a visualization tool if you wish to see how the magnetic field lines extend using equation [1]. Mirrorcalc.C will calculate the mirror points of particles, and then plot the results in a histogram and draw the approximate point on a model similar to Earthdraw.

The program is interactive and demands user input to function. You can see the “readme” found in esiproject.root for basic instructions on how to operate the program and what the different scenarios do.

We see the field lines drawn by Earthdraw.C from 60, 50, 40 and 30 degrees. To replicate this result choose latitude 60, 4 lines and 10 in difference in Earthdraw.C.

This is the plot showing the mirror points for all angles 1-90 degrees in integers. The angles for these points will be the same for all latitudes, but their distance will vary. To replicate this result choose deg 60 and option 2 in Mirrorcalc.C

Here we see the distributions of mirror-point distance for particles with a random pitch angle between 0 and 30 degrees. The latitude of the field line is again 60 degrees. Notice how more particles are mirrored at higher altitude than lower ones. The red line in the histogram indicates the border of actually mirrored particles, as particles going below this line will likely collide with atoms in the atmosphere and loose the energy required to escape. To replicate this result choose 60 degrees latitude, option 3, 25000 particles, 15 mean and 30 width in Mirrorcalc.C

Here we see the distributions of mirror-point distance for particles with a random pitch angle with a forced Gaussian distribution between 0 and 16 degrees with field line latitude 60. Notice how the trend from the previous plot is less apparent. To replicate this result choose 60 latitude, option 4, 25000 particles, 8 mean and 16 width in Mirrorcalc.C

The source code for this project can be downloaded by clicking here

DownloadThere is also a .root file in this archive which contains the readme.txt, and the two last histograms.

You are now free to test my program on your own, and if you have questions or critic please contact me by mail at Espen.Simonsen@student.uib.no