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Magnetospheres

Electricity and magnetism · level 8, lesson 3 of 6

A question to start

The solar wind pushes on Earth's magnetic field, and the field pushes back. How far out do they balance?

The idea

A magnetic field has its own pressure, \(B^2/(2\mu_0)\). A planet's field holds off the solar wind out to where the two pressures balance:

\[\frac{B^2}{2\mu_0} = \rho v^2.\]

The bubble inside is the planet's magnetosphere, and its edge is the magnetopause. For a field like a bar magnet's (a dipole), \(B\) falls with the cube of the distance: \(B = B_0 (R/r)^3\), where \(B_0\) is the field at the equator on the surface. Putting that in:

\[\frac{r}{R} = \left(\frac{B_0^2}{2\mu_0\,\rho v^2}\right)^{1/6}.\]
Holding off the solar wind Mars Earth
Planet-wide field at the equator none 30.6 µT
What meets the solar wind the upper air, with patches of crustal field the magnetopause, about 10 Earth radii out

Mars has no global field to push back. So the solar wind runs almost into its upper air. There, the wind's own field drapes round the planet, and in places the magnetized crust makes small bubbles of its own.

solar windEarth: a big bubbleMars: wind near the air

Worked examples

1. Earth's field pressure. \(B_0 = 30.6\) µT: \(B_0^2/(2\mu_0) = (3.06\times10^{-5})^2/(2 \times 4\pi\times10^{-7}) = 3.7\times10^{-4}\) Pa.

2. The balance. With the wind at 1.34 nPa: \(r/R = (3.7\times10^{-4}/1.34\times10^{-9})^{1/6} = (2.8\times10^5)^{1/6} = 8.1\).

3. The real value. At the magnetopause the squeezed field is about twice the plain dipole value, which pushes the balance out by \(2^{1/3} = 1.26\): \(8.1 \times 1.26 \approx 10\) Earth radii, about 64,000 km.

Try it

  1. What sets the size of a magnetosphere?
  2. A solar storm raises the wind's pressure 8 times. How far in is the magnetopause pushed?
  3. Why does the distance depend so weakly on the wind's pressure?
  4. Why can the solar wind come so close to Mars?
Answers
  1. The balance between the planet's field pressure and the solar wind's pressure.
  2. \(8^{1/6} = 1.41\), so to about \(10/1.41 = 7\) Earth radii.
  3. The dipole field falls as \(1/r^3\), so its pressure falls as \(1/r^6\): a big change in pressure moves the balance only a little.
  4. It has no global field to push back, only the upper air and patches of crustal field.

Common mistake

"A magnetosphere is a hard wall." Most particles are turned aside, but some leak in, especially when the Sun's field lines join up with the planet's. That is how auroras get their particles.

Where it leads