# Solar wind and the Earth's magnetic field

I have again an old question from a comprehensive exam I took a couple of months ago. Lucky for me one could pick 5 out of 8 questions, because on some of the problems I didn't even know how to start. Now that classes are over I've now the time to revisit those problems I was dumbfounded by, such as this one: (Abriged version)

Life on earth would be impossible if we were constantly exposed to charged solar particles. Luckily, earth's magnetic field protects us from them. The solar particles have a typical energy spectrum of $d\Phi / dE \propto E^{-3}$ particles/$m^2/s/J$. What is the minimal field strength of the earth's magnetic field based on the anthropic principle, i.e., it couldn't be weaker or else we wouldn't live to observe it.

Well, this quantity $d\Phi/dE$ looks like a flux, so I guess the general setting is that of a scattering problem. But first, $d\Phi / dE$ isn't given completely, only a rough form of its energy dependence. And second, I'm not sure what a reasonably simple model for this entire process would be. Easiest in terms of calculation would be to assume some sort of homogeneous magnetic field aligned with the earth's magnetic axis, because I guess it's a pain to calculate the path of a particle in a dipole field...

Maybe the idea of this is to calculate the total cross section of earth's magnetic field and then demand that it should "cover" the earth? Or they want me to solve an equation of motion for incoming solar particles and show that all of them are deflected? My problem right now is that I don't even know how to interpret the $d\Phi/dE$ quantity whose energy dependence I'm given. I guess it makes more sense to someone with a background in elementary particle physics?

Right now I'm trying to write a vector potential $\vec{A} = \mu_o/(4\pi r^2) \vec{m} \cdot \vec{e}_r$ where $\vec{e}_r$ is the unit vector in $r$-direction in spherical coordinates, and then try to get equations of motion from the Hamiltonian $$H = \frac{(\vec{p} + q\vec{A})^2}{2m}$$ but I am not sure if I'll be able to solve whatever comes out of that, or if I'm completely on the wrong track with this.

EDIT

Image taken from here

Maybe it helps trying to understand this schematic, but I cannot easily see how the Lorentz force would create such a trajectory.

ANOTHER EDIT

From further searching, I know suspect that this has something to do with how a plasma current (the charged particles) interact with a magnetic field. That would mean that I have to calculate the radius of the ensuing magnetosphere and then demand, via the anthropic principle, that it should be at least of the same size (or larger) as the radius of earth. So the Lorentz force would probably not directly have anything to do with it. But I also have no training in plasma physics. (Some of the problems in the exam were specifically geared towards Astronomy students, so I guess they'd find it a breeze).

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Any chance you could get more specific than "Any hints..."? After all, we don't let the newbie posters ask that kind of question so it wouldn't quite be fair to let it go in this case ;-) Also, are you expected to be able to do this without external resources? Do they want an exact answer or a rough (perhaps order-of-magnitude) estimate? – David Z Dec 14 '11 at 17:43
I'll try to elaborate. No external resources are allowed. I guess an order of magnitude estimate is okay. – Lagerbaer Dec 14 '11 at 17:46
If an order of estimate then could we not assume $Bqv=mv^2/r$ and then make an assumption about $v\sim 10^8 m/s$. The other quantities we can assume to be for say an electron. Of course, this means the flux relationship given is a red herring! – Omar Dec 14 '11 at 18:32
@Omar Maybe one can combine this? If I use your reasoning, there exists a critical particle energy $E_c$ below which a particle gets deflected and above which it hits the earth. The total number of particles hitting the earth then scales like $\int_{E_c}^\infty \frac{1}{E^3}$ ~ $1/E_c^2$. Maybe then one can make some smart argument for what $E_c$ should be... – Lagerbaer Dec 14 '11 at 18:43
@Lagerbaer That could be an interesting approach. The problem is that it is a powerlaw flux relationship. I guess you could make some assumptions about the underlying particle powerlaw distribution and then assume they have a characteristic temperature (a little dubious for a powerlaw distribution). All that seems really complicated for a no-external-resources question! – Omar Dec 14 '11 at 19:12