MAGNETIC FIELD PREDICTIONS & RELATIONS
Clifford E Carnicom
 Copyright Jan 06 2004
Edit Jan 17 2005
(Continued)

Further discussion:

1. An empirical model has been developed of the form:

B = a * exp^b*m * w² * cos( alpha / 4 ) 

where B is the magnetic field strength of the solar system body in teslas, m is the mass of the body in kilograms, and alpha is the axis of orientation of the magnetic field with the respect to the rotational axis.  Alpha is measured from the north pole of the rotational axis of the earth clockwise to the north pole of the magnetic axis of the body.  The coefficients a and b are to be determined based upon data available for solar system bodies.  B is measured at the equator of the body if the data is available. The angular velocity of the body is w, in radians per second.

The model has been developed and based upon the existence of data for the nine planets of the solar system, and the moon of earth and Ganymede, a moon of Jupiter for which data is also available.

Conceptually, the model indicates that the magnetic field is essentially proportional to the kinetic energy of the body, with the exception that the mass is exponentially related as opposed to linearly related.  In addition, the field is modified by a cyclic term which includes the orientation of the magnetic axis with respect to the rotational axis of the body.

The range of magnetic field strengths under consideration vary by a factor of up to 100,000.  The results of the model indicate general agreement within one order of magnitude.  It is upon this basis that the model is of value, and it is understood to be of empirical origin.  The model does inherit, however, a conceptual origin that macroscopically examines the kinetic energy, axis orientations and magnetic field strengths of the bodies with available data.  The parallels of the origin of magnetism with the electron spin of quantum theory remains a tantalizing prospect to pursue.  Those bodies of the greatest error, in particular Mars and Mercury, are of second order in magnitude.  The sun has currently been excluded due to the extreme variations within the magnetic field of that body.  A strategy in solving the coefficients of a and b involves a reduction in the errors of magnitude with an even distribution of this error, as opposed to a minimization of the sum of residual squares.

The problem posed is that of a non-linear regression solution, and there are varying approaches to the solution available.  A method of non-linear regression involving the Jacobian for the normal equations has been adopted.

A set of reasonable estimates for the coefficients of a and b are:

a = 1500 (approx.)

b = 1E-27 (approx.)

These values are subject to change depending upon various sources in input data that can be considered, and with respect to any changes in the model itself that may occur in the future.

2. Alternatively,

w = ( B / ( a * exp^bm * cos ( alpha / 4 ) ) ) ^1/2

3. Supplementally,

w = theta / (t - t₀)

where theta is the angle of rotation of the body in radians, t is the elapsed time of rotation in seconds, and t₀ is the origin of the time system in seconds.

4. Differential relationships of this model are of much interest to investigate in this case.  Several forms of derivative consideration are offered:

5. dB/dw = 2 * a * exp^bm * cos ( alpha / 4 ) * w

6. dB/dt = ( - 2 * a *exp^bm * cos (alpha / 4 ) * theta2 ) / ( t - t0 )³

7. dw/dt = a' = ( - theta² / ( t - t₀ )³ ) * ( B / ( a * exp^bm * cos ( alpha / 4 ) )^-1/2

8. da'/dB = ( theta² / ( 2 * ( t - t₀ )³ * a *exp^bm * cos (alpha / 4 ) ) * ( B / ( a * exp^bm * cos ( alpha / 4 ) )^-3/2

9. Relativistic mass-velocity effects have been considered within the analysis, and at this time are not considered to be of sufficient magnitude for inclusion within the modeling process.

10. Input data for the magnetic field strength model is as follows:

Magnetic field data is variable depending on the source and reference used. It is anticipated that significant uncertainties in magnetic field strength on various bodies exist. The magnetic field strength data, when available is taken as an equatorial value. Composite sources of information were used for the construction of this input table.^3,4

11. Input data for the magnetic field strength decline model is taken from the National Geophysical Data Center spherical harmonic magnetic models from the years 1900 to 2004 (Geomag V4.0) at the equator at a longitude of 120^oW.. The input data used is:

The data for 1940 appears to be potentially aberrant and was not included in this study.  The use of nuclear weapons during this decade may wish to be investigated for any association with magnetic field variations. The linear regression result for this data leads to a model form of dB(T)/dt(sec) = -3.866E-18*year + 6.456E-15. Integration of this equation with respect to time is used to determine the magnitude of the B field that corresponds to the current value of the earth's field.

Any revisions to this page will be made as is appropriate.

Clifford E Carnicom
Jan 06 2004

References:

1. National Institute of Standards and Technology (NIST) Techbeat, Greeting Another New Year Without a Leap Second, (http://www.nist.gov/public_affairs/techbeat/tb2003_1219.htm#greeting.), December 19, 2003.,
2. BBC News, Bentley, Molley, Earth Loses its Magnetism, (http://news.bbc.co.uk/go/pr/fr/-/2/hi/science/nature/3359555.stm), December 31, 2003.
3. Jay M. Pasachoff, Stars and Planets, (Houghton Mifflin, 1992), 534-535.
4. J. Kelly Beatty, The New Solar System, (Cambridge University Press, 1999), 42, 387-389.

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