Diff for "TypicalParticles"

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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:

Magnetic field constant	Tesla	3.0037E-05	B₀
Unit charge	Colombs	1.6021E-19	q
Joules per MeV	J/MeV	1.6021E-13
Earth radius	m	6378210.00	R_E
Speed of Light	m/s	2.9979E+08	c

Let's compute some numbers at 500km altitude:

ratio		1.078		ratio = 1 + alt/R_E
L @ 0 degrees magnetic latitude		1.078		L=ratio
L @ 90 degrees magnetic latitude		infinite
B @ 0 degrees magnetic latitude	Tesla	2.40E-05		B_lat0=B₀/ratio³
B @ 90 degrees magnetic latitude	Tesla	4.79E-05		B_lat90=2B₀/ratio³
		Particle
		electron	proton
Kinetic Energy	MeV	1	10
Kinetic Energy	J	1.6021E-13	1.6021E-12	E_k
Rest Mass	kg	9.1094E-31	1.6726E-27	m₀
Mass Energy	J	8.1871E-14	1.5033E-10	E₀=m₀c²
Relativistic Momentum	kg m/s	7.60E-22	7.34E-20	p=sqrt( 2E₀E_k+E_k²)/c
Lorentz Factor		2.957	1.0107	gamma=1/sqrt(1-(v/c)²)
Velocity	m/s	2.82E+08	4.34E+07	v=pc²/(E_k+E₀)
"Relativistic Mass"	kg	2.69E-30	1.69E-27	m_r=m₀ gamma
Gyration Period/Field	T-s	1.06E-10	6.63E-08	2pi m_r / q
Gyration Period @ 0 degrees	s	4.41E-06	2.77E-03	Period = 2pi m_r / q B_lat0
Gyration Period @ 90 degrees	s	2.21E-06	1.39E-03	Period = 2pi m_r / q B_lat90
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

[2] ECSS-E-ST-10-04C, 15 November 2008, Table I-1, page 162

[3] "The Space Environment and Its Effects on Space Systems", Vincent L. Pisacane, AIAA 2008, Table 6.5, page 135

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+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.
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-||<|3>                                                ||||      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    ||
||<-3> Radius of gyration (km)                                               ||
||   @ 500 km                                         || 0.6      || 50      ||
||   @ 20000 km                                       || 10       || 880     ||
||<-3> Gyration period (s)                                                   ||
||   @ 500 km                                         || 1E-5     || 7E-3    ||
||   @ 20000 km                                       || 2E-4     || 0.13    ||
||<-3> Bounce period (s)                                                     ||                                                   
|| @ 500 km                                           || 0.1      || 0.65    || 
|| @ 20000 km                                         || 0.3      || 1.7     ||
||<-3> Longitudinal drift period (min)                                       ||
|| @ 500 km                                           || 10       || 3       ||
|| @ 20000 km                                         || 3.5      || 1.1     ||
+||<|3>                                                  ||||      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    ||
||<-3> Radius of gyration (km)                                                 ||
||     500 km                                           || 0.6      || 50      ||
||     20000 km                                         || 10       || 880     ||
||<-3> Gyration period (s)                                                     ||
||     500 km                                           || 1E-5     || 7E-3    ||
||     20000 km                                         || 2E-4     || 0.13    ||
||<-3> Bounce period (s)                                                       ||
||     500 km                                           || 0.1      || 0.65    ||
||     20000 km                                         || 0.3      || 1.7     ||
||<-3> Longitudinal drift period (min)                                         ||
||     500 km                                           || 10       || 3       ||
||     20000 km                                         || 3.5      || 1.1     ||

Constants:

|| Magnetic field constant || Tesla   || 3.0037E-05 || B,,0,, ||
|| Unit charge             || Colombs || 1.6021E-19 || q      ||
|| Joules per MeV          || J/MeV   || 1.6021E-13 ||
|| Earth radius            || m       || 6378210.00 || R,,E,, ||
|| Speed of Light          || m/s     || 2.9979E+08 || c      ||

Let's compute some numbers at 500km altitude:

|| ratio                            ||       |||| 1.078               || ratio = 1 + alt/R,,E,,      ||
|| L @  0 degrees magnetic latitude ||       |||| 1.078               || L=ratio                     ||
|| L @ 90 degrees magnetic latitude ||       |||| infinite            ||
|| B @  0 degrees magnetic latitude || Tesla |||| 2.40E-05            || B,,lat0,,=B,,0,,/ratio^3^   ||
|| B @ 90 degrees magnetic latitude || Tesla |||| 4.79E-05            || B,,lat90,,=2B,,0,,/ratio^3^ ||
||||<|2>                                  ||||        Particle        ||
||                                           electron   || proton     ||
|| Kinetic Energy               || MeV    || 1          || 10         ||
|| Kinetic Energy               || J      || 1.6021E-13 || 1.6021E-12 || E,,k,,                             ||
|| Rest Mass                    || kg     || 9.1094E-31 || 1.6726E-27 || m,,0,,                             ||
|| Mass Energy                  || J      || 8.1871E-14 || 1.5033E-10 || E,,0,,=m,,0,,c^2^                  ||
|| Relativistic Momentum        || kg m/s || 7.60E-22   || 7.34E-20   || p=sqrt( 2E,,0,,E,,k,,+E,,k,,^2^)/c ||
|| Lorentz Factor               ||        || 2.957      || 1.0107     || gamma=1/sqrt(1-(v/c)^2^)           ||
|| Velocity                     || m/s    || 2.82E+08   || 4.34E+07   || v=pc^2^/(E,,k,,+E,,0,,)            ||
|| "Relativistic Mass"          || kg     || 2.69E-30   || 1.69E-27   || m,,r,,=m,,0,, gamma                ||
|| Gyration Period/Field        || T-s    || 1.06E-10   || 6.63E-08   || 2pi m,,r,, / q                     ||
|| Gyration Period @  0 degrees || s      || 4.41E-06   || 2.77E-03   || Period = 2pi m,,r,, / q B,,lat0,,  ||
|| Gyration Period @ 90 degrees || s      || 2.21E-06   || 1.39E-03   || Period = 2pi m,,r,, / q B,,lat90,, ||
|| 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] [[ http://www.everyspec.com/ESA/download.php?spec=ECSS-E-10-04A.014020.pdf | ECSS-E-10-04A,     21 January  2000, Table 28,  page 94 ]]

[2] [[ http://www.spacewx.com/Docs/ECSS-E-ST-10-04C_15Nov2008.pdf | ECSS-E-ST-10-04C, 15 November 2008, Table I-1, page 162 ]]

[3] [[ http://books.google.com/books?id=se-QKAAACAAJ | "The Space Environment and Its Effects on Space Systems", Vincent L. Pisacane, AIAA 2008, Table 6.5, page 135 ]]