Paris–Harrington theorem: Difference between revisions
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{{About|the pitch angle of a charged particle|Pitch angle in engineering|Pitch angle (engineering)}} | |||
The '''pitch angle''' of a [[charged particle]] is the angle between the particle's velocity vector and the local [[magnetic field]]. This is a common measurement and topic when studying the [[magnetosphere]], [[magnetic mirror]]s, [[ciconic cusp]]s and [[polywell]]s. See [[Aurora (astronomy)|Aurora]] and [[Ring current]] | |||
==Usage: particle motion== | |||
It is customary to discuss the direction a particle is heading by its pitch angle. A pitch angle of 0 degrees is a particle whose parallel motion is perfectly along the local [[magnetic field]]. In the [[northern hemisphere]] this particle would be heading down toward the Earth (and the opposite in the [[southern hemisphere]]). A pitch angle of 90 degrees is a particle that is locally mirroring (see: [[Magnetosphere particle motion]]). | |||
==Special case: equatorial pitch angle== | |||
The equatorial pitch angle of a particle is the pitch angle of the particle at the Earth's geomagnetic equator. This angle defines the loss cone of a particle. The loss cone is the set of angles where the particle will strike the atmosphere and no longer be trapped in the magnetosphere while particles with pitch angles outside the loss cone will continue to be trapped. The loss cone is defined as the probability of particle loss from the [[magnetic bottle]] which is: | |||
:<math> | |||
\begin{align} | |||
P &= \frac{\Omega}{2\pi} = \int_{0}^{\alpha_0} \sin\left(\alpha\right) \operatorname{d}\alpha = 1 - \cos\left(\alpha_0\right) \\ | |||
&= 1 - \sqrt{1 - \sin^2\left(\alpha_0\right)} \\ | |||
&= 1 - \sqrt{1 - \frac{B_0}{B_m}} | |||
\end{align} | |||
</math> | |||
Where <math>\Omega</math> is the solid angle we are concerned with, <math>\alpha_0</math> is the equatorial pitch angle of the particle, <math>B_0</math> is the equatorial magnetic field strength, and <math>B_m</math> is the maximum field strength. Notice that this is independent of charge, mass, or kinetic energy. | |||
==See also== | |||
* [[Adiabatic invariant]] | |||
* [[Magnetic mirror]] | |||
==External links== | |||
*''[http://www.oulu.fi/~spaceweb/textbook/ Oulu Space Physics Textbook]'' | |||
*''[http://pluto.space.swri.edu/IMAGE/glossary/pitch.html IMAGE mission glossary]'' | |||
[[Category:Electromagnetism]] | |||
Revision as of 14:49, 22 October 2013
29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church. The pitch angle of a charged particle is the angle between the particle's velocity vector and the local magnetic field. This is a common measurement and topic when studying the magnetosphere, magnetic mirrors, ciconic cusps and polywells. See Aurora and Ring current
Usage: particle motion
It is customary to discuss the direction a particle is heading by its pitch angle. A pitch angle of 0 degrees is a particle whose parallel motion is perfectly along the local magnetic field. In the northern hemisphere this particle would be heading down toward the Earth (and the opposite in the southern hemisphere). A pitch angle of 90 degrees is a particle that is locally mirroring (see: Magnetosphere particle motion).
Special case: equatorial pitch angle
The equatorial pitch angle of a particle is the pitch angle of the particle at the Earth's geomagnetic equator. This angle defines the loss cone of a particle. The loss cone is the set of angles where the particle will strike the atmosphere and no longer be trapped in the magnetosphere while particles with pitch angles outside the loss cone will continue to be trapped. The loss cone is defined as the probability of particle loss from the magnetic bottle which is:
Where is the solid angle we are concerned with, is the equatorial pitch angle of the particle, is the equatorial magnetic field strength, and is the maximum field strength. Notice that this is independent of charge, mass, or kinetic energy.