# Van Allen Belt

### Estimating Energy Stored in the Geomagnetic Field

The magnetic field of the earth is complicated, not a simple dipole, but it can be approximated that way.

B_r = - 2 B_0 \left( { R_E } \over r \right)^3 \cos \theta

B_{\theta} = - B_0 \left( { R_E } \over r \right)^3 \sin \theta

dE = { { {B_r}^2 + {B_{\theta}}^2 } \over { 2 {\mu}_0 } } ~ dV

dV = 2 \pi r^2 \sin \theta ~ dr ~ d\theta

dE = { { \left( 2 B_0 \left( R_E \over r \right)^3 \cos \theta \right)^2 + \left( B_0 \left( R_E \over r \right)^3 \sin \theta \right)^2 } \over { 2 {\mu}_0 } } ~ 2 \pi r^2 \sin \theta ~ dr ~ d\theta

dE = { { \pi {B_0}^2 {R_E}^6 } \over { { {\mu}_0 } r^4 } } ( 3 \cos^2 \theta + 1 ) \sin \theta ~ dr ~ d\theta

The total energy of the magnetic field is the double integral from \theta = 0 to \theta = \pi and from R = R_0 to infinity. Assuming the earth's mantle has a permeability of {\mu}_0 and the earth's iron core has an infinite permeability (and thus no appreciable magnetic energy storage), R_0 is the radius of the core, not the earth's surface, about 3500 km.

Since r and \theta and the scaling terms are independent, we separate the integrals:

E = \left( { \pi {B_0}^2 {R_E}^6 } \over { {\mu}_0 } \right) ~ \left( \int^{\infty}_{R_0} { dr \over { r^4 } } \right) ~ \left( \int_0^{\pi} ( 3 \cos^2 \theta + 1 ) \sin \theta ~ d\theta \right)

E = \left( { \pi {B_0}^2 {R_E}^6 } \over { {\mu}_0 } \right) ~ \left( 1 \over { 3 {R_0}^3 } \right) ~ \left( 3 \times { 2 \over 3 } + 2 \right)

E = { { 4 \pi {B_0}^2 {R_E}^6 } \over { 3 {\mu}_0 {R_0}^3 } } = 2 \left( { 1 \over { 2 {\mu}_0 } } \left( B_0 \left( {R_E} \over {R_0} \right)^3 \right)^2 \right) \left( { { 4 \pi } \over 3 } {R_0}^3 \right)

The B field at the surface of the core is B_0 \left( {R_E} / {R_0} \right)^3 . The total energy is twice the energy density at the surface of the core { B^2 / { 2 {\mu}_0 } } times the volume of the core { { 4 \pi } \over 3 } {R_0}^3 .

• B_0 = 3.12E-5 Tesla

• R_E = 6378000 meters

• R_0 = 3500000 meters

• \mu_0 = 4e-7 π meter3 Tesla2 Joule-1

Above the core at 3500km, E = 5.1E18 Joules = 160 gigawatt-years. Above the earth's surface, E = 8.4E17 Joules = 26 gigawatt-years.

The earth's magnetic field is decreasing at about 6% per century, so the energy is decreasing 0.12% per year, about 200 megawatts above the core.

In 1960, Dessler and Vestine estimated that the maximum stored energy of particles in the van Allen belt must be less than 6e15 Joules. If this amount of energy was expended filling and emptying the van Allen belt once per year, that would also be about 200 megawatts. It is interesting that these are about the same magnitude. Both are 1.6 parts per billion of the solar energy absorbed and reradiated by the earth, and about 1/3 of the power of the solar wind times the earth's area.

I would like to learn the actual numbers for the stored energy and the particle power flux of the van Allen belt.

### Gyration Frequency at the Equator

B = - B_0 \left( { R_E } \over r \right)^3

freq = B q / 2 \pi m . . . . . for v << c

freq = B q / 2 \pi { m_0 } ( 1 + { E_k } / {E_0} ) . . . . . E_k is the particle energy in electron volts (not sure about energy scaling)

The rest energy E_0 of a proton is 938 MeV and particle energies range from 100 keV to over 400 MeV. The rest energy of an electrom is 511 keV with particle energies up to 10 MeV.

 Equatorial Gyration Frequency (Hz) L Shell Electron Energy Proton Energy 0 100 keV 1 MeV 10 MeV 0 10 MeV 100 MeV 400 MeV 1.0 873K 730K 295K 42K 476 471 430 230 1.5 259K 216K 88K 13K 141 139 127 68.2 2.0 109K 91K 37K 5307 59.5 58.8 53.7 28.8 2.5 56K 47K 19K 2717 30.4 30.1 27.5 14.7 3.0 32K 27K 11K 1573 17.6 17.4 15.9 8.53 3.5 20K 17K 6889 990 11.1 11.0 10.0 5.37 4.0 14K 11K 4615 663 7.43 7.36 6.72 3.60 4.5 9584 8015 3241 466 5.22 5.17 4.72 2.53 5.0 6987 5843 2363 340 3.81 3.77 3.44 1.84 5.5 5249 4390 1775 255 2.86 2.83 2.58 1.38 6.0 4043 3382 1367 197 2.20 2.18 1.99 1.07 6.5 3180 2660 1075 155 1.73 1.71 1.57 0.84 7.0 2546 2129 861 124 1.39 1.37 1.25 0.67

VanAllenBelt (last edited 2013-12-23 18:55:46 by KeithLofstrom)