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--- un:bounce-motion [2024/12/03 11:43] – asad
+++ un:bounce-motion [2024/12/03 11:48] (current) – [2. Loss Cone] asad
@@ Line 41: / Line 41: @@
 ===== - Loss Cone =====
-Even if longitudinal invariance is maintained, not all particles may remain trapped along the magnetic field lines. If the particle's mirror point descends deep into the Earth's atmosphere, it may be absorbed due to interaction with neutral particles. To estimate this, an 'equatorial loss cone' \( \alpha_l \) is defined:
+Even if the longitudinal invariant remains constant, not all particles may remain trapped along the magnetic field lines. If a particle's mirror point descends deep into the Earth's atmosphere, it may be absorbed due to interactions with neutral particles, resulting in the ionized particle being lost. To account for this, an 'equatorial loss cone' \( \alpha_l \) is defined as:
 $$ \sin^2\alpha_l = \frac{B_{eq}}{B_E} = \frac{\cos^6\lambda_E}{\sqrt{1+3\sin^2\lambda_E}} $$
-where \( B_E \) is the magnetic field at the Earth's surface, and \( \lambda_E \) is the magnetic latitude where the field line touches the Earth's surface. Particles are absorbed approximately at 100 km altitude, but since the magnetic field strength differs only slightly between the surface and 100 km altitude, the surface value is used.
+where \( B_E \) is the magnetic field at the Earth's surface, and \( \lambda_E \) is the magnetic latitude where a field line touches the Earth's surface. Particles are absorbed approximately at an altitude of 100 km, but since the magnetic field strength at the surface and at 100 km altitude differ only slightly, the surface value is used here.
 {{:bn:un:loss-cone.webp?nolink&250|}}
-If the equatorial pitch angle is smaller than \( \alpha_l \), the particle's gyration will carry it deep enough into the atmosphere to be absorbed. The loss cone width depends on the L-value, which is the equatorial radius of a field line relative to Earth's radius. For distant field lines, the loss cone is very narrow. At geostationary orbit, approximately 6.6 Earth radii, the loss cone width is just 3°.
+If the equatorial pitch angle is smaller than \( \alpha_l \), the particle's gyration will take it so deep into the atmosphere that it will be absorbed. All particles within the solid angle \( d\Omega \) shown above will be lost. Using the equation above, the loss cone can be expressed in terms of the L-value, recalling that \( \cos^2\lambda_E = L^{-1} \):
-{{:bn:un:loss-cone-l.png?nolink|}}
+$$ \sin\alpha_l = (4L^6-3L^5)^{-1/4} $$
-Thus, the width of the loss cone depends only on the L-value, illustrating its critical role in determining particle dynamics in Earth's magnetosphere.
+This relationship is shown in the plot below.
+{{:bn:un:loss-cone-l.png?nolink|}}
+Thus, the width of the loss cone depends only on the L-value, which corresponds to the equatorial radius of a field line relative to the Earth's radius. For distant field lines, the loss cone becomes very narrow. At geostationary orbit, approximately 6.6 Earth radii, the loss cone width is only 3°.