Download 200C, Winter 2009, Homework #5 Derive the polar cap size (eqn

200C, Winter 2009, Homework #5
1. 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=15deg
c. 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 it
is 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.
Planet
Distance
(AU)
Magnetic Tilt
Magnetopause
Moment Angle
Distance
(ME)
(degrees) Km
Rplanet
Earth
Jupiter
Saturn
Uranus
Neptune
1.0
5.2
9.5
19.2
30.1
1
20,000
580
49
27
10.8
9.7
<1
59
47
0.7X105
30-70X105
12x105
6.9X105
6.3X105
11
45-100
21
27
26
(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, and
using 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.