Trapped Particles can be studied through adiabatic invariants (see Adiabatic Invariants). In particular any kind of detailed study should deal with the McIlwain's L-parameter (see L-Shell Calculation) and the picth angle distribution. This last is related to the magnhetic flux that, under the specific trapping mechanism, can be considered contstant.

### Pitch Angle

The direction of particle's velocity with respect to the magnetic field line is called pitch angle \(\alpha\). This one can be calculated in every position, but when the magnetic flux is constant, it is related to the equatorial pithc angle, so the angle that the particle velocity (in its circling, bouncing and drifting motion) has with respct toi the magnetic field line at the Earth geomagnetic equator. More this value is close to \(90^{\circ}\) more the mirror point are close one another and the drift shell is not reaching the geomagnetic poles, as can be seen in Eq. \(\ref{eq:alpha0}\).

\begin{equation}\label{eq:alpha0}

\frac{\sin^{2} \alpha_{loc} }{B_{loc}}=\frac{\sin^{2} \alpha_{eq} }{B_{eq} }=\frac{1}{B_{M} }

\end{equation}

Because \(\alpha_{M}=90^{\circ}\) so \(\sin^{2} \alpha_{M}=1\). In addition to this relation we know that the magnetic field line, in the (shifted and tilted) dipole approximation follow the equation \(\ref{eq:field0}\):

\begin{equation}\label{eq:field0}

r=r_{eq}\cos^{2} \lambda_{m}

\end{equation}

where \(r\) is the position of the field line with respect to the dip{ole center and \(r_{eq}\) is the distance of the field line from the dipole center at the equator. This equation cal be also written as

\begin{equation}\label{eq:field1}

r=L\cdot\cos^{2} \lambda_{m}

\end{equation}

where \(L\) in Eq. \(\ref{eq:field1}\) is the McIlwain parameter. In the shifted-tilted dipole approximation of the Earth magnetic field, the local value of the magnetic field \(B\) can be obtained as a function of the Earth surface value \(B_{s}\) from the formula:

\begin{equation}\label{eq:field2}

B=B_{s}\cdot(\frac{R{e}}{r})^{3}\cdot\sqrt{1+3\cdot\sin^{2} \lambda_{m}}

\end{equation}

where \(R_{e}\) is the Eart

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h radii, \(r\) is the distance from the dipole center and \(\lambda_{m}\) is the magnetic latitude (see Geomagnetic Coordinates). Using this Eq. \(\ref{eq:field2}\) we can obtain:

\begin{equation}\label{eq:field3}

B_{loc/eq}=B_{s}\cdot(\frac{R{e}}{r_{loc/eq}})^{3}\cdot\sqrt{1+3\cdot\sin^{2} \lambda_{mloc/meq}}

\end{equation}

And in addition to the fact that \(\sin\lambda_{meq}=0\) and using Eq. \(\ref{eq:field0}\) with \(r=r_{loc}\)we can have:

\begin{equation}\label{eq:pitch0}

\frac{\sin^{2} \alpha_{loc}\cdot B_{eq}}{B_{loc}}=\sin^{2} \alpha_{eq}

\end{equation}

and

\begin{equation}\label{eq:pitch1}

\frac{B_{eq}}{B_{loc}}=\frac{(\frac{R{e}}{r_{eq}})^{3}} {(\frac{R{e}}{r_{loc}})^{3}\cdot\sqrt{1+3\cdot\sin^{2} \lambda_{mloc}}}=\frac{({r_{loc}})^{3}} {(r_{eq})^{3}\cdot\sqrt{1+3\cdot\sin^{2} \lambda_{mloc}}}

\end{equation}

but

\begin{equation}\label{eq:pitch2}

\frac{r_{loc}}{r_{eq}}=\cos^{2} \lambda_{mloc}

\end{equation}

so finally we can obtain:

\begin{equation}\label{eq:pitch3}

\frac{\sin^{2} \alpha_{loc}\cdot B_{eq}}{B_{loc}}=\sin^{2} \alpha_{eq}=\frac{(\cos^{2} \lambda_{mloc})^{3}\cdot\sin^{2} \alpha_{loc}} {\sqrt{1+3\cdot\sin^{2} \lambda_{mloc}}}

\end{equation}

And this can tells us that once we are able to evaluate the pitch angle at the detection point \(r_{loc}\) plus the magnetic latitude (in the shifted tilted dipole approximation, using IGRF-13) we can estimate the equatorial pitch angle in case of trapped particles (and conservation of the magnetic flux).

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**Pitch Angle Calculation**

This page enables users to calculate the Equatoria Pitch Angle for any trapped particle and for any position, using the previous formulas.

To perform the L-Shell Calculation please follow this link:

## Input Parameters

Input values are:

- Particle (in mass number so 1 for Protons, 2 for Helium nuclei, etc., up to 26 for iron)
- Particle Rigidity value in GV
- Geographical latitude (from 88 to -88 deg.)
- Geographic longitude (from 0 to 360 deg.)
- Altitude from the Earth's surface (in km)
- Time (year,month,day,hour,min,sec)

## Output Parameters

Output data are:

- Equatorial Picth Angle

## Info

For any additional informations or questions you can ask