Answer Happy

6. (18 points) in clases and in my additional notes, we found out that the mary condition for a particle to be captured in a magnetic bottle (such as the van Alles belts formed by the earth's magnetic field) is that the magnetic field at some point is BB-/ where p is the particle's momentum. L is the particle's angulier momentum, and is the churge of the particle, all of which are constant. Let's put this formula in a form that you might find more table Page 1 (a) (3 points) We will tronk the velocity into components parallel and perpendicularis to the magnetic field. These will change with time, but the total speed, =+of is constant. Why is this the (b) (3 points) Explain why the momentum y can be rewritten in terms of the initial velocity: p===+ (e) (3 points) Explain why the angular momentum I can be rewritten in terms of the initial perpen- dieta velocity to initial radits of arhit Rand initial magnetic field Bo mar LEL- EL (d) 5 points) Combining the last two testalts (from past (b) anal (e)), we obtain a betetection condition: Given the capture condition above, which of the three velocities in the figure below is most likely to result in a capture! Explain why this is more likely than the other two 1 (0 points) in the graph below Blor, D.N.S. G. Kancol, T. I. Pulkicion, and J. B. Blake (1999). Equinoctial and solattal nerage of magnetospheric relatie electron Astrong sem modulation, Geophys. Rer. Lett. 26, 3133 3196 is shown salariation of high energy electrots detected in the outer van Allen belt. As is clear in the Spring and Fall about 3x the number of high energy electrots are detected in the outer belt that in the Winter and Summer
2.0 SAMPEX Electrons 2-6 MeV 2.5<L<6.5 1.5 Spring Normallized Electron Flux 1.0 Summer 0.5 0.0 February April May - July August - November October January Numeros factors contribute to this varintion, but one of these is the variation in the orientation of earth's magnetic poles relative to the sun's incoming radiation. The Earth's geographic pole is tilted about 23.4* from perpendicular to the plane of its orbit around the sun. The magnetic pole is on average 11.5 from the geographic pole. As a result, Earth's north pole (south magnetic pole) is chest to the sun in the Summer, and Earth's south pole (useth magnetic pole) is dorest to the sun in the Winter, as illtestrated below. As a consequence, anotae are noticeables more active in the Spring and Fall Solar Radiation Spring Summer Autumn Winter Explain this variation in moral activity in terms of the relative orientation of Earth's magnetic field to the incoming solar radiation. Cite the capture condition derived in part (b).
6. (18 points) In class and in my additional notes, we found out that the necessary condition for a particle to be captured in a magnetic bottle (such as the van Allen belts formed by the earth's magnetic field) is that the magnetic field at some point is B> B.. - /L where p is the particle's moment tim, L is the particle's angular moment, and is the charge of the particle, all of which are constant. Let's put this formula in a form that you might find more sable. Page 3 (a) points) We will break the velocity into components parallel and perpendicular to the magnetic field. These will change with time, but the total speed. + is constant. Why is this the case (b) 3 points) Explain why the momentum p can be rewritten in terms of the initial velocity: par = m*t + me (0) (3 points) Explain why the angular momentum . can be rewritten in terms of the initial perpen- dicula velocity . Initial radius of orbit Ra, and initinl magnetic field Bot L-.-.- (d) 3 points) Combining the last two results (from purt (b) and (c)). we obtain a new reflection condition: VE Given the capture condition above, which of the three velocities in the figure below is most likely to result in a capture? Explain why this is more likely than the other two. B. 1 VI (c) (5 points) In the graph below Baker, D.N., S. G. Kanekal, T. L. Pulkkinen, and J. B. Blake (1999). Equinoctial and solstitial averages of magnetospheric relatie electrons. A strong semiannual modulation, Geophys. Res. Lett. 26 3193 3196 is shown seasonal variation of high energy electrons detected in the outer van Allen helt. As is clear, in the Spring and Fall, about 3x the number of high energy electrons are detected in the outer belt than in the Winter in Summer
2.0 SAMPEX Electrons 2-6 MeV 2.5 L <6.5 1.5 Fad Spring Normailized Electron Flux 1.0 Winter Summer 0.5 TY 0.0 February April May - July August - November October January Numerous factors contribute to this variation, but one of these is the variation in the orientation of earth's magnetic poles relative to the sun's incoming radiation. The Earth's geographic pole is tilted about 23.4° from perpendicular to the plane of its orbit around the sun. The magnetic pole is on average = 11.5" from the geographie pole. As a result, Earth's north pole (south magnetic pole) is closest to the sun in the Summer, and Earth's south pole (north magnetic pole) is closest to the sun in the Winter, as illustrated below. As a consequence, aurorae are noticeably more active in the Spring and Fall. Solar Radiation Spring Summer Autumn Winter Explain this variation in auroral activity in terms of the relative orientation of Earth's magnetic field to the incoming solar radiation. Cite the capture condition derived in part (1)