Frequently Asked Questions
[ Question | Answer | Illustration | See Also ]
T.04 - Why are Roederer's L*
and McIlwain's L parameters different ?
In a magnetic dipole field, the parameter L
is defined as the distance from the center of the dipole to
the equatorial point (or minimum B value) of the field line.
The parameter L is given in units of Earth's radii and,
in a dipole field, a particle remains during its bounce and drift motion
on magnetic field lines having the same L.
For a non-dipolar geomagnetic field, e.g. IGRF, coordinate systems based on adiabatic invariants are used. To provide a familiar and easy-to-interpret variable, Roederer and McIlwain have related adiabatic invariants to the parameter L: McIlwain's parameter (Lm) is a function of the magnetic field intensity Bm at the mirror point, the integral invariant function I and the magnetic moment M of the dipole field while Roederer's parameter L* is a function of the third adiabatic invariant and the magnetic moment M. When the magnetic field model is fixed and frozen, a particle has constant Lm and constant L* values during its bounce and drift motion. Generally, both values of L are different, except in the case of a dipole magnetic field model where both values should be the same.
In the UNILIB libray, for historical reasons, the dipole magnetic
moment used to evaluate the parameters Lm and
L* is not equal to the magnetic moment of the
magnetic field but is fixed to M0 =
0.311653 Gauss Re-3
(see common block UC160,
Due to this feature, the two parameters are different even with
a magnetic dipole. The argument
kmflg of the
common block UC190 allows to control
the evaluation of the L parameters in the subroutines
UL240, UL242 and
UD330. When the variable
is set to 0 or 10, the magnetic moment M0 is
used, while when the variable is set to 1 or 11, the magnetic moment
of the magnetic field is used. Note that this last moment is
stored in the common block UC140
- McIlwain, C.E., Coordinates for mapping the distribution of magnetically trapped particles, JGR 66 (1961) 3681-3691
- Roederer, J.G., Dynamics of geomagnetically trapped radiation, 1970, Springer-Verlag, New York
The use of the variable
kmflgis illustrated with a sample program based on example #3 (evaluation of the third invariant). In this example, a magnetic dipole field is selected and a drift shell is defined with the help of the (Bm,Lm) parameter values: Bm = 0.19 Gauss and Lm = 2.00. The Roederer's L* parameter is then evaluated successively with the magnetic moment M0 and with the true magnetic field moment. The results are 2.06 and 2.01, respectively. The difference of 0.5% between the last result and Lm provides a measure of the error in the evaluation of the third invariant by the subroutine UD330.
PROGRAM faqt04 C (based on example3) C INCLUDE 'structure.h' C COMMON /UC190/ prop, stepx, stpmin, umsq, upsq, uk2, uk3, : epskm, epsrel, stplst, xclat, kmflg, nxstp C REAL*8 prop, stepx, stpmin REAL*8 umsq, upsq, uk2, uk3 REAL*8 epskm, epsrel, stplst, xclat INTEGER*4 kmflg, nxstp C INTEGER*4 kunit, kinit, ifail, kint, kext, noprint CHARACTER*32 lbint, lbext REAL*8 year, param(10), amjd INTEGER*4 knfl, ktyplus REAL*8 fbm0, flm0, falt, phi, star0, star1 C C initialize variables C DATA kunit, kinit, noprint/ 6, 1, -6/ DATA kint, kext/ 3, 0/ DATA year, amjd, param/ 1995.0d0, 0.0d0, 10*0.0d0/ C C initialize the library and the magnetic dipole field C CALL UT990 (noprint, kinit, ifail) IF( ifail .LT. 0 )STOP CALL UM510 (kint, year, lbint, kunit, ifail) IF( ifail .LT. 0 )STOP CALL UM520 (kext, amjd, param, : lbext, noprint, ifail) IF( ifail .LT. 0 )STOP C C trace the drift shell and evaluate the third invariant C in the STANDARD case C fbm0 = 0.19d0 flm0 = 2.0d0 falt = 0.0d0 knfl = 120 ktyplus = 3 C CALL UD310 (fbm0, flm0, falt, knfl, : ktyplus, ifail) IF( ifail .LT. 0 )STOP CALL UD330 (phi, star0, ifail) IF( ifail .LT. 0 )STOP C C trace the drift shell and evaluate the third invariant C when KMFLG is set to one C kmflg = 1 C CALL UD310 (fbm0, flm0, falt, knfl, : ktyplus, ifail) IF( ifail .LT. 0 )STOP CALL UD330 (phi, star1, ifail) IF( ifail .LT. 0 )STOP C C write the result C WRITE(*,*) star0, star1 C END
--- Geomagnetic field model --- Model ( 3): Dipolar magnetic field Epoch 1995. Order + 1 = 2 Calculation epoch = 1995.0 year Colatitude of the dipole pole = 10.70 deg Longitude of the dipole pole = -71.41 deg Earth dipole moment = 0.302077 G/Re^3 Correction for the SAA drift = 0.00 deg 2.06304946287526 2.01249780345246