200C, Winter 2009, Homework #51. Derive the polar cap size (eqn. 9.6), the tail radius as function of the lobe field (eqn. 9.8) and the asymptotic tail radius balancing the solar wind thermal pressure (eqn. 9.14). Compute the typical value of:a. The polar cap flux (in Webers) assuming PC=15deg b. The asymptotic tail radius for typical solar wind (Ni=5/cc, Te=10eV, Ti=1eV) and polar cap size, PC=15degc. The asymptotic tail field tail field for the above solar wind conditions.2. Derive the reconnection rate (ui/uA) for Sweet-Parker reconnection (eqn 9.36) and show itis small; for Petschek reconnection (eqn 9.44) and from particle dynamics (eqn 9.63). Show that ui/uA ~5-10% for reasonable values of reconnection geometry (Bz/Bx~0.1). 3. For equatorially mirroring particles, derive the ring current energy formula (eqns 10.17, 10.19, 10.22 and 10.23). 4. Write a general form relating the magnetopause stand-off distance to solar wind dynamic pressure at the Earth [What is needed here is the functional form; assume a scaling factor that is to be determined later for the exact numerical relationship].(a) Discuss how you would expect solar wind dynamic pressure to vary as a function of distance from the Sun. (b) The table in the lecture notes comparing planetary magnetospheres [Table 15.1 in K&R, also included below] gives the dipole moment for the planets with respect to the Earth. Given the following planetary radii, calculate the equatorial magnetic field for each of the outer planets:1 Jovian radius = 11.2 Earth radii; 1 Saturn radius = 9.45 Earth radii; 1 Uranus radius = 4.00 Earth radii; 1 Neptune Radius = 3.88 Earth radii.(c) The table also shows that at the Earth the magnetopause stand-off distance is 11 Earth radii. Based on the relationship between the stand-off distance and dynamic pressure, andusing the stand-off distance for the Earth to determine the constant of proportionality, derive the expected stand-off distances of the magnetopause for each of the outer planets listed in the table. Compare your results to the values given in the table.Planet Distance (AU) Magnetic Moment (ME) Tilt Angle (degrees) Magnetopause Distance Km Rplanet Earth 1.0 1 10.8 0.7X105 11 Jupiter 5.2 20,000 9.7 30-70X105 45-100 Saturn 9.5 580 <1 12x105 21 Uranus 19.2 49 59 6.9X105 27 Neptune 30.1 27 47 6.3X105
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