A possibility to exist the trapped energetic particles zone near Mercury
Authors: Lukashenko Anastasiia , Lavrukhin Alexander , Alexeev Igor , Belenkaya Elena 
Published: October 2020

We consider the possibility of radiation belts existence near Mercury. The study is carried out both using the St&#248;rmer theory for charged particles motion, in which zones of allowed and forbidden motion in an axially symmetric magnetic field are considered, and trajectories analysis.The internal magnetic field of Mercury was discovered in 1974 by the Mariner 10 spacecraft. In [1] values for the dipole field at the Mercury equator Beq=192 nT and the north offset of the dipole dz=0.18 RM (where RM=2439 km is Mercury radius) were obtained, which were subsequently confirmed during the MESSENGER spacecraft flybys in 2011&#8211;2015. In our work, we also introduced a formal magnetopause at the distance Rmp=1.4 RM.Allowed zones of particle motion are described in the St&#248;rmer theory in cylindrical coordinates by the inequality:Q = 1 &#8211; (P&#966;/(p&#961;) &#8211; qA&#966;/(pc))2 &#8805;0,where&#160;P&#966; is particle generalized angular momentum, p is its momentum, q is its charge, c is the speed of light, and&#160;A&#966; is magnetic field vector potential (for more details see [2]). Equality Q=0 defines the boundaries of allowed zones, and Q=1 is a force line equation. We consider such A&#966;, which is the sum of the potentials of pure dipole field and uniform external field be directing along the z-axis and approximating magnetopause currents fields.Further, we measure lengths in units of the St&#248;rmer radius rst=&#8730;(qM/(pc)), where M is the absolute value of dipole magnetic moment. This radius acts as a scale factor. In a pure dipole field, it determines the curvature radius of the unstable circular trajectory lying in the dipole equatorial plane. The second St&#248;rmer integral of motion, arising as a result of axial symmetry, is the dimensionless parameter &#947;=P&#966;/(2prst), which defines the shape and size (in rst) of allowed and forbidden zones. After their substitution, the inequality for Q takes the form:Q = 1 &#8211; (2&#947;/&#961; &#8211; &#961;/r3 &#8211; G&#961;)2 &#8805;0,where we introduce a coefficient G=berst3/(2BeqRM3). In a pure dipole field for &#947;>1, there are two allowed zones: the infinite external and internal that is isolated and enclosed in the inner part of the sphere with radius rst. For external homogeneous field, in our calculations we used the value be=50 nT, whence G=&#8211;4.2717. In this case, when &#947;>&#8211;1/(8G) there is also a closed in space internal allowed zone.Let us consider the motion of nonrelativistic particles, specifically protons, which trajectories start in points located at the dipole equatorial plane. Then an analysis based on St&#248;rmer&#8217;s theory can be connected with the classical description of particle motion through the initial pitch angle &#945;eq and phase angle &#967;eq as follows:&#947; = 0.5 (1/R + R sin &#945;eq sin &#967;eq + R2G),where R is the initial distance from the dipole.On one hand, the condition of an internal allowed zone appearance from under the planet&#8217;s surface (that is possible, if rst is greater then planet radius), which occurs when &#947;=&#947;in, must be fulfilled. On the other hand, the trajectories should not go beyond the magnetopause. The internal allowed zone firstly touches the magnetopause at &#947;=&#947;out, and the final disappearance of the captured trajectories occurs at &#947;fin