2.3. Magnetic Moment and Angular Momentum
The angular momentum A of a spherical object of radius r_{surf} is
where = 2 (rotation period)^{-1} , and mass is the mass of the object. Thus for the Sun ( = 2.9 × 10^{-6} sec^{-1}) one gets A 10^{42} kg m^{2} sec^{-1}; the Sun's magnetic moment M_{surf} 3 × 10^{27} Gauss m^{3} . For Mercury ( = 1.2 × 10^{-6} sec^{-1}), A 10^{30} kg m^{2} sec^{-1}, and M_{surf} 5 × 10^{16} Gauss m^{3}.
A direct relationship is often but not always observed between the magnetic dipole moment M_{surf} of an object and its angular momentum A. This often observed relation is given approximately by
Figure 4 shows this equation (dashed), along with the observational data. Originally proposed only for the planets, this empirical law has been extended to include the Sun (e.g., Blackett 1947; Russell 1978), and big moons such as Ganymede and Io (Kivelson et al. 1996a, 1996b).
Figure 4. Observed relation between the magnetic moment and the angular momentum of moons, planets, and the Sun. The dashed line follows the equation for a dipolar dynamo, with a slope of 0.8. Many data are from Kivelson et al. (1996b), the rest from this text. Below a certain strength and a certain angular momentum (at bottom left), remanent magnetism is often found. |
Such a law has been called a "magnetic Bode's law" (Russell 1978), and was though to be a "long-sought connection between electromagnetic and gravitational phenomena" (Blackett 1947), and has also been called "an effect more along 'meteorological' lines" (e.g., chapter 18 in Parker 1979).
This relationship may now be better called a "dipolar dynamo law", for three reasons. (1) All the moons, planets and star(s) that obey so far this relation do have a dipolar dynamo. (2) All the moons and planets without a significant magnetic dynamo (Earth's Moon, Venus, Mars) do not follow this law - the observed data for Earth's Moon, Venus, and Mars fall significantly below the M_{surf} values predicted by this law (e.g., Kivelson et al. 1996b). (3) The relationship may not work for other (not dypolar) dynamo types - thus our Milky Way galaxy has a planar disk with an axisymmetric spiral (not dipolar) dynamo magnetic field, with A 3 × 10^{67} kg m^{2} sec^{-1} and M_{surf} 9 × 10^{55} Gauss m^{3}, and thus the equation above would predict only M_{surf} 4 × 10^{47} Gauss m^{3} - about 10^{8} times lower than observed.
Physically, no direct physical justification for this law has been found. Mathematically, since M_{surf} = B_{surf} ^{.} r^{3}_{surf} for a sphere, and since A ~ (mass) ^{.} r^{2}_{surf} ~ (density) ^{.} r^{5}_{surf} then it can be seen that M_{surf} and A are strong powers of r_{surf}, so the apparent correlation of these two quantities should predict M_{surf} ~ r^{3}_{surf} ~ A^{0.60} . This mathematical argument would predict that the data for all planets would follow this law - this argument does not explain why the observed data for some planets or some moons fall below the predictions of this law.