How can we make an order-of-magnitude estimate of the strength of Earth's magnetic field?

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Solution 1

Try looking at the figures in a 2005 simulation by Takahashi et.al. in Science magazine, that at least show recurring reversals at http://www.sciencemag.org/cgi/content/full/309/5733/459?cookietest=yes.

Given that the viscosity, structure, and heat generation of the core are all to some degree unknown, and that the process may depend upon parametric amplification and even parametric resonance, this seems a pretty good beginning.

Solution 2

Live on earth is protected from solar wind by the earth's magnetic field. Charged particles from the sun (mostly) penetrate the earth's atmosphere with great velocity. These particles can be trapped by a magnetic field to follow circular path's around the magnetic field lines, thereby losing their energy due to collisions or bremstrahlung.

From first principles we can try to make an estimate of the strength of the magnetic field required to trap charged particles arriving with great velocity.

Starting with a lorentz force and a circular movement we have: $Bqv = m\frac{v^2}{r}$, so $B= \frac{mv}{qr}$.
$v$ is the velocity of the particle, approx. light velocity, order of magnitude $10^8$ m/s. $q$ is the charge of the particle, elementary unit order of magnitude $10^{-19}$ C. $m$ is the mass of the particle, approx. proton mass, order of magnitude $10^{-27}$ kg. $r$ is the radius wherein the particle has to be trapped, at the most 10 km (height of the atmosphere) $10^4$ m. This gives $B \sim 10^{-4}$ T $= 1$ Gauss.

We have to appreciate the intelligent design ...

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Mark Eichenlaub
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Updated on July 21, 2020

Comments

  • Mark Eichenlaub
    Mark Eichenlaub over 3 years

    The source of Earth's magnetic field is a dynamo driven by convection current in the molten core. Using some basic physics principles (Maxwell's equations, fluid mechanics equations), properties of Earth (mass, radius, composition, temperature gradient, angular velocity), and properties of materials (conductivity and viscosity of molten iron) or other relevant facts, is it possible to estimate the strength of the field to order of magnitude (about one gauss)?

    Descriptions I've seen of the geodynamo all refer to extensive numerical computation on a computer, but can we get a rough idea with simple estimation?

    • David Z
      David Z about 13 years
      Interesting question. My instinct is no, there's no simple explanation that can be made without some high-level assumptions about the speed of iron currents within the Earth's core and the magnetization of the iron etc. But I can't be sure.
    • Mark Eichenlaub
      Mark Eichenlaub about 13 years
      Well, I don't think the iron is magnetized because it's molten. Perhaps the speed of the currents could be estimated based on things like the temperature gradient, viscosity, and density, but I don't really know much fluid dynamics to try to answer that.
    • Mark C
      Mark C about 13 years
      @MarkE Perhaps it could read "How can I make an order-of-magnitude estimate of the earth's magnetic field?" As it is stated, I simply understood you to be asking for a "numerological"-type question which boils down to why the unit "Gauss" is defined as such. I almost came back here from this page to rail at how 0.30 is not "roughly one Gauss".
  • David Z
    David Z about 13 years
    It is interesting but it doesn't really answer the question - for instance, why 1 and not 11, 120, 0.000015, or something like that? Besides, it's a probabilistic statement, not a reason for why a particular number has the value it does. So I think I have to downvote this. (Sorry! It's nothing personal)
  • ralu
    ralu about 13 years
    I think that this kind of question is like question why is earth gravity 10 ms^-2 and speed of light 3*10^8 ms^-1 and h/ is 10^-34 Js and why on earth is earth pressure 1 bar.
  • David Z
    David Z about 13 years
    Yeah, and Benford's law doesn't address any of those numbers either.
  • ralu
    ralu about 13 years
    Of course it does. Check for instance Benford's law for distribution of first digits in the population of the 237 countries of the world. Benford's law apply for most numbers in nature that can be in range of multiple orders.
  • Mark Eichenlaub
    Mark Eichenlaub about 13 years
    Ralu, I apologize, but that is honestly a silly answer. I wanted to know about the order-of-magnitude strength of Earth's magnetic field, not about the first digit when measured in some arbitrary units.
  • Mark C
    Mark C about 13 years
    @MarkE Then perhaps you should re-title the question. For example, "Why is the mass of the proton rougly 1E-27 kg?"" could be the same kind of question, but, as given, they both are asking why our units have he magnitude they do.
  • Gerard
    Gerard about 13 years
    Actually it might be natural that units for new phenomena (e.g. earth magnetic field) establish themselves in such a way that they are approx. 1,.. for the most common situation. Rationalisation occurs later on.
  • Mark C
    Mark C about 13 years
    @Gerard: All we have to do is find out how Gauss measured the Earth's magnetic field 1885 (i.e. what constants and units he used) and it should be clear.
  • Mark Eichenlaub
    Mark Eichenlaub almost 13 years
    Ha! Very amusing Gerard. Order-of-magnitude estimate based on the anthropic principle! (If the field were weaker, we wouldn't be here to observe it).
  • Mark Eichenlaub
    Mark Eichenlaub almost 13 years
    Thanks for the link. I was not able to understand much of it, though. It looked like that were reporting on the methods and results of a computer simulation only.
  • Mark Eichenlaub
    Mark Eichenlaub almost 13 years
    I decided to mark this answer as accepted because it accurately points out what we know about this phenomenon. I asked Sterl Phinney, an astrophysicist who teaches an order-of-magnitude physics class at Caltech, the question in person. He said that although we can estimate several relevant factors (and went into details that I didn't fully understand), there's no known way at this time to make an order-of-magnitude estimate. The supercomputer simulations are pretty much the knowledge we have.
  • dmckee --- ex-moderator kitten
    dmckee --- ex-moderator kitten over 12 years
    @Mark, I think we can assume he started by using a compass as a detector. So the unit of detectable field was similar to the field the tool was designed to measure.