This is a summary of the program created by Brian Lundberg (REU student during the summer of 2002 under Dr. Robert Leamon) and the underlying mathematical model. For a more thorough explanation, please take a look at Brian's website.
Assumptions and Simplifications | Resulting Equations |
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This vector equations is solved for the magnetic field components in cylindrical coordinates with zeroth and first order Bessel functions of the first kind. |
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When R = R0, BA = 0. The first zero of the zeroth order Bessel function occurs at 2.4048; therefore, α = 2.4048/R0. |
Integrating the equations for current density, J = αB, and magnetic field, BA, over the cross sectional area gives us equations for current and flux.
I = (0.5192)*(2π/μ0)*B0R0 Φ = (0.21588)*2π*B0R02 |
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The program models the magnetic field of the cloud by fitting six parameters (and handedness = +/-1) and minimizing the error between the cloud model and WIND magnetic field data.
IDL's AMOEBA function is used to minimize the error calculating function by adjusting the six input parameters. As AMOEBA often converges on local minima, rather than the global minimum, it is sometimes necessary to try several sets of initial parameters in order to obtain the anticipated solution. With this in mind, we set about auditting the program with several sets of initial parameters hoping that although some of these sets would converge on undesirable answers (local minima) others would converge on the correct answer (global minimum). Since values of φ0, B0, R0, and t0 are always positive and errors in these parameters seem less likely to mislead AMOEBA, we took the average value of each from our cloud_fit results to use in the sets of parameters. But since the parameters θ0, Y0, and the handedness of the cloud which take both positive and negative values tend to cause problems when the sign of the initial value is incorrect, we needed to create sets of initial parameters for both positive and negative values and so we ended up with 8 sets of initial parameters. Unfortunately, these sets of average values could not always provide desirable solutions and trying additional combinations was still sometimes necessary.
Year | Start Time | Stop Time | θ0 | φ0 | B0 (nT) | R0 (km) | (χ/B0)2 | IMC (A) | ΦMC (Wb) | αMC(m-1) |
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1995 | Feb 08 03:00 | Feb 08 22:00 | -18.1 | 81.2 | 12.7 | 1.66E+07 | 0.016 | 5.47E+08 | 4.74E+12 | 1.45E-10 |
Mar 04 11:00 | Mar 05 04:00 | -54.8 | 154.4 | 11.8 | 1.42E+07 | 0.035 | 4.38E+08 | 3.26E+12 | 1.69E-10 | |
Dec 16 05:00 | Dec 16 22:00 | 39.0 | 49.6 | 11.6 | 3.42E+07 | 0.023 | 1.03E+09 | 1.85E+13 | 7.01E-11 | |
1997 | Aug 03 14:00 | Aug 04 04:00 | -2.9 | 90.0 | 16.2 | 1.27E+07 | 0.032 | 5.36E+08 | 3.56E+12 | 1.89E-10 |
1998 | Feb 17 10:00 | Feb 18 18:00 | -4.7 | 310 | 17.6 | 1.90E+07 | 0.032 | 8.73E+08 | 8.69E+12 | 1.26E-10 |
May 02 12:00 | May 03 17:00 | -8.5 | 357 | 10.5 | 3.36E+07 | 0.035 | 9.14E+08 | 1.60E+13 | 7.16E-11 | |
1999 | Feb 18 14:00 | Feb 19 12:00 | 40.9 | 58.3 | 16.8 | 3.54E+07 | 0.011 | 1.54E+09 | 2.86E+13 | 6.79E-11 |
Apr 16 20:00 | Apr 17 21:00 | 35.2 | 118 | 22.3 | 1.44E+07 | 0.018 | 8.34E+08 | 6.27E+12 | 1.67E-10 | |
Aug 09 10:00 | Aug 10 16:00 | 59.0 | 270 | 12.8 | 2.35E+07 | 0.023 | 7.81E+08 | 9.59E+12 | 1.02E-10 | |
*Sept 21 21:00 | Sept 22 05:00 | 5.6 | 90.0 | 14.7 | 7.08E+06 | 0.011 | 2.70E+08 | 9.99E+11 | 3.40E-10 | |
2000 | *Jul 15 19:00 | Jul 16 09:00 | 0.9 | 62.3 | 48.1 | 2.64E+07 | 0.015 | 3.29E+09 | 4.55E+13 | 9.11E-11 |
Jul 28 21:00 | Jul 29 11:00 | 4.9 | 319 | 14.6 | 1.64E+07 | 0.026 | 6.22E+08 | 5.33E+12 | 1.47E-10 | |
Aug 01 00:00 | Aug 01 16:00 | 18.2 | 118 | 12.3 | 4.09E+07 | 0.012 | 1.31E+09 | 2.79E+13 | 5.88E-11 | |
Aug 12 06:00 | Aug 13 05:00 | -10.9 | 81.8 | 29.1 | 2.51E+07 | 0.024 | 1.90E+09 | 2.49E+13 | 9.58E-11 | |
Oct 03 17:00 | Oct 04 15:00 | 15.8 | 338 | 26.3 | 2.69E+07 | 0.015 | 1.84E+09 | 2.58E+13 | 8.94E-11 | |
Nov 06 23:00 | Nov 07 18:00 | 23.0 | 100 | 23.8 | 2.26E+07 | 0.007 | 1.40E+09 | 1.65E+13 | 1.06E-10 | |
2001 | Mar 19 22:00 | Mar 22 02:00 | -85.6 | 69.2 | 23.3 | 2.10E+07 | 0.005 | 1.27E+09 | 1.39E+13 | 1.15E-10 |
Apr 22 00:00 | Apr 23 02:00 | -56.4 | 270 | 13.2 | 1.85E+07 | 0.027 | 6.34E+08 | 6.13E+12 | 1.30E-10 | |
May 28 11:00 | May 29 11:00 | -7.6 | 31.5 | 12.4 | 2.80E+07 | 0.015 | 9.01E+08 | 1.32E+13 | 8.59E-11 | |
Jul 10 17:00 | Jul 12 08:00 | -12.9 | 202 | 9.0 | 2.01E+07 | 0.012 | 4.70E+08 | 4.93E+12 | 1.20E-10 | |
Nov 24 15:00 | Nov 25 14:00 | 31.1 | 10.4 | 18.5 | 5.13E+07 | 0.024 | 2.46E+09 | 6.60E+13 | 4.69E-11 |
*Colors correspond with best fit, good fit, and worst fit given by the cloud_fit program.