Van Allen Belt


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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 .

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)