Specifically Part f. Please explain your work and the steps you make, I don't want just a solution I want to learn how you got to the solution. Thank you! Great problem, part F involves Kinetic Theory application
A singly-charged, cold, (2 eV) ionospheric oxygen ion is found gyrating on a mag- netic field line threading the Earth's magnetosphere. When detected, the ion has no component of its velocity directed along the field line. You may assume that it is in the upper, collisionless portion of the ionosphere at an altitude of 1000 km. You may assume that the Earth's magnetic field is well-approximated by a centered dipole. The local magnetic field strength is ~ 37,000 nT and it points approximately downwards towards the Earth. Suddenly, a local wave field turns on with significant power across a very broad range of frequencies. (a) What frequency (expressed in Hz) of the wave spectrum will be resonant with the oxygen ion gyrofrequency? Assume that the ion gains energy through resonant interactions with the wave electric fields. Furthermore, assume that the energization occurs strictly in the direction per- pendicular to the magnetic field. After some time, the wave field turns off abruptly, leaving the ion with a new energy of 100 eV perpendicular to the magnetic field. (b) What is the oxygen ion's gyroradius (in meters) after it has been energized? (c) Qualitatively describe the subsequent motion of the ion, noting especially the most important force(s) acting upon it. You may make the simplifying assumption that drifts perpendicular to the magnetic field are negligible.) (d) If the oxygen ion is at an invariant latitude of 65 immediately after the heating event, at what maximum distance (in Earth radii) can it ever be found? (e) Compute the pitch angle of the ion when it is at this maximum distance. (f) At this maximum location, assume there exists an ambient isotropic plasma con- sisting of protons and electrons with a number density of 1 cm3, both with thermal energies of 10 keV, and possessing no bulk flow. Assume now a collection of many singly-charged oxygen ions heated continuously in the ionosphere such that they also yield a number density of 1 cm-3 at this maximum distance. Sketch the phase space density of the combined ion population (protons and oxygen ions) versus the velocity parallel to the magnetic field on the same plot. A singly-charged, cold, (2 eV) ionospheric oxygen ion is found gyrating on a mag- netic field line threading the Earth's magnetosphere. When detected, the ion has no component of its velocity directed along the field line. You may assume that it is in the upper, collisionless portion of the ionosphere at an altitude of 1000 km. You may assume that the Earth's magnetic field is well-approximated by a centered dipole. The local magnetic field strength is ~ 37,000 nT and it points approximately downwards towards the Earth. Suddenly, a local wave field turns on with significant power across a very broad range of frequencies. (a) What frequency (expressed in Hz) of the wave spectrum will be resonant with the oxygen ion gyrofrequency? Assume that the ion gains energy through resonant interactions with the wave electric fields. Furthermore, assume that the energization occurs strictly in the direction per- pendicular to the magnetic field. After some time, the wave field turns off abruptly, leaving the ion with a new energy of 100 eV perpendicular to the magnetic field. (b) What is the oxygen ion's gyroradius (in meters) after it has been energized? (c) Qualitatively describe the subsequent motion of the ion, noting especially the most important force(s) acting upon it. You may make the simplifying assumption that drifts perpendicular to the magnetic field are negligible.) (d) If the oxygen ion is at an invariant latitude of 65 immediately after the heating event, at what maximum distance (in Earth radii) can it ever be found? (e) Compute the pitch angle of the ion when it is at this maximum distance. (f) At this maximum location, assume there exists an ambient isotropic plasma con- sisting of protons and electrons with a number density of 1 cm3, both with thermal energies of 10 keV, and possessing no bulk flow. Assume now a collection of many singly-charged oxygen ions heated continuously in the ionosphere such that they also yield a number density of 1 cm-3 at this maximum distance. Sketch the phase space density of the combined ion population (protons and oxygen ions) versus the velocity parallel to the magnetic field on the same plot
