Magnetopause Standoff Distance (Pressure Balance)

The solar wind pushes on a planet and the planet's magnetic field pushes back. The magnetopause sits where the two pressures balance, and $R_{mp}$ is its distance from the planet's centre along the line toward the Sun. The wind's push is its dynamic pressure $q$, one half of the density times the speed squared. The field is a dipole squeezed against a conducting boundary, so its strength just inside is twice the dipole value, and $B_0$ is the equatorial surface field. The factor $K$ adjusts the wind's pressure: it is 1 for the simple balance and about 0.88 for the gas-dynamic pressure at the nose.

A dipole field falls as the cube of the distance, so its pressure falls as the sixth power, and the distance goes as the sixth root of the field squared over the push. That exponent of $1/6$ makes the boundary stubborn: doubling the wind's pressure shrinks the distance by only 11 per cent. For the Earth, with $R_p$ = 6371 km, $B_0$ about 30 000 nT ($3\times10^{-5}$ T) and a typical $q$ of 1 to 3 nPa, the boundary comes out at about 9 to 10 Earth radii, and in a storm it can be pushed in to 6 or less.

Worked example. The page's example runs the balance backward. Put the boundary at 62,594 km with $R_p$ = 6371 km, $B_0$ = 30 936.21 nT and $K$ = 1, and the pressure that holds it there is $q$ = 0.8468 nPa. That comes from the forward case: a wind of 5 protons per cm³ at 450 km/s has $\rho v^{2}$ = 1.69353 nPa, so $q$ = 0.846765 nPa, and an Earth dipole moment of $8.0\times10^{22}$ A·m² gives $B_0$ = 30 936.21 nT. The boundary then lies at 9.825 Earth radii.

Sources and limits. The sources are Chapman and Ferraro (1931) on magnetic storms, Spreiter, Summers and Alksne (1966) on the flow around the magnetosphere, and Shue and colleagues (1997) on how the solar wind controls the size and shape of the magnetopause. Shue's empirical fit, $r_0 = 11.4\,D_p^{-1/6.6}$ Earth radii for $B_z = 0$ with $D_p = \rho v^{2}$ in nPa, gives 10.525 Earth radii for that same wind ($D_p$ = 1.6935 nPa), against 9.825 here: 6.8 per cent apart, as a fit and a simple balance are. Treat the result as an estimate of the distance toward the Sun only.