Characteristics of typical radiation belt charged particles
The following table appears in [1][2][3] . The gyration numbers seem to be incorrect for the 500km altitude. Depending on the magnetic latitude and pitch angle of these "typical particles", the table values for the gyration period are between 2.3 and 5.0 too large.
|
Particle |
|
1 MeV |
10 MeV |
|
electron |
proton |
|
Range in aluminium (mm) |
2 |
0.4 |
Peak equatorial omni-directional flux (cm-2 s-1) |
4E6 |
3.4E5 |
Radial location (L) of peak flux (Earth-radii) |
4.4 |
1.7 |
Radius of gyration (km) |
||
@ 500 km |
0.6 |
50 |
@ 20000 km |
10 |
880 |
Gyration period (s) |
||
@ 500 km |
1E-5 |
7E-3 |
@ 20000 km |
2E-4 |
0.13 |
Bounce period (s) |
||
@ 500 km |
0.1 |
0.65 |
@ 20000 km |
0.3 |
1.7 |
Longitudinal drift period (min) |
||
@ 500 km |
10 |
3 |
@ 20000 km |
3.5 |
1.1 |
Constants:
B_0_ Magnetic field constant |
Tesla |
3.0037E-05 |
Unit charge |
Colombs |
1.6021E-19 |
Joules per MeV |
J/MeV |
1.6021E-13 |
Re Earth radius |
m |
6378210.00 |
C Speed of Light |
m/s |
2.9979E+08 |
Let's compute some numbers at 500km altitude:
L @ 0 degrees magnetic latitude |
|
1.078 |
L @ 90 degrees magnetic latitude |
|
infinite |
B @ 0 degrees magnetic latitude |
Tesla |
2.40E-05 |
B @ 90 degrees magnetic latitude |
Tesla |
4.79E-05 |
|
Particle |
||
electron |
proton |
||
Kinetic Energy |
MeV |
1 |
10 |
Kinetic Energy |
J |
1.6021E-13 |
1.6021E-12 |
Mass |
kg |
9.1094E-31 |
1.6726E-27 |
Mass Energy |
J |
8.1871E-14 |
1.5033E-10 |
Relativistic Momentum |
kg m/s |
7.60E-22 |
7.34E-20 |
Lorentz Factor |
|
2.957 |
1.0107 |
Velocity |
m/s |
2.82E+08 |
4.34E+07 |
"Relativistic Mass" |
kg |
2.69E-30 |
1.69E-27 |
Gyration Period/Field |
T-s |
1.06E-10 |
6.63E-08 |
Gyration Period @ 0 degrees |
s |
4.41E-06 |
2.77E-03 |
Gyration Period @ 90 degrees |
s |
2.21E-06 |
1.39E-03 |
Gyration Period from table |
s |
1E-5 |
7E-3 |
Calculations based on the equations in Pisacane [3] and the equations for relativistic momentum and energy scattered around Wikipedia.
Without the pitch angle and the L value (or the magnetic latitude), I can't guess at the bounce period (there are values that yield the above results). Plausible values of pitch angle yield gyration radii that are larger than those for 500km in the table.
The 20000km numbers ( the altitude of most nav-sats ) are plausible.
What am I missing here?
References:
[1] ECSS-E-10-04A, 21 January 2000, Table 28, page 94