# A possibility to exist the trapped energetic particles zone near Mercury

- Lomonosov Moscow State University, Skobeltsyn Institute of Nuclear Physics, Russian Federation (a_lu@mail.ru)

We consider the possibility of radiation belts existence near Mercury. The study is carried out both using the Størmer 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 *B _{eq}*=192 nT and the north offset of the dipole

*dz*=0.18

*R*(where

_{M}*R*=2439 km is Mercury radius) were obtained, which were subsequently confirmed during the

_{M}*MESSENGER*spacecraft flybys in 2011–2015. In our work, we also introduced a formal magnetopause at the distance

*R*=1.4

_{mp}*R*.

_{M}Allowed zones of particle motion are described in the Størmer theory in cylindrical coordinates by the inequality:

*Q* = 1 – (*P*_{φ}/(*pρ*) – *qA _{φ}*/(

*pc*))

^{2}≥0,

where *P*_{φ} is particle generalized angular momentum, *p* is its momentum, *q* is its charge, *c* is the speed of light, and *A _{φ}* 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*, which is the sum of the potentials of pure dipole field and uniform external field

_{φ}*b*directing along the

_{e}*z*-axis and approximating magnetopause currents fields.

Further, we measure lengths in units of the Størmer radius *r _{st}*=√(

*q*

*M*/(

*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 Størmer integral of motion, arising as a result of axial symmetry, is the dimensionless parameter

*γ*=

*P*

_{φ}/(2

*pr*), which defines the shape and size (in

_{st}*r*) of allowed and forbidden zones. After their substitution, the inequality for

_{st}*Q*takes the form:

*Q* = 1 – (2*γ*/*ρ* – *ρ*/*r*^{3} – *Gρ*)^{2} ≥0,

where we introduce a coefficient *G*=*b _{e}r_{st}*

^{3}/(2

*B*

_{eq}R_{M}^{3}). In a pure dipole field for

*γ*>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

*r*. For external homogeneous field, in our calculations we used the value

_{st}*b*=50 nT, whence

_{e}*G*=–4.2717. In this case, when

*γ*>–1/(8

*G*) 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 Størmer’s theory can be connected with the classical description of particle motion through the initial pitch angle *α _{eq}* and phase angle

*χ*as follows:

_{eq}*γ* = 0.5 (1/*R* + *R* sin *α _{eq}* sin

*χ*+

_{eq}*R*

^{2}

*G*),

where *R* is the initial distance from the dipole.

On one hand, the condition of an internal allowed zone appearance from under the planet’s surface (that is possible, if *r _{st}* is greater then planet radius), which occurs when

*γ*=

*γ*

_{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

*γ*=

*γ*

_{out}, and the final disappearance of the captured trajectories occurs at

*γ*

_{fin}<

*γ*

_{out}. As a consequence, in the case of Mercury only capturing of particles with kinetic energy

*K*less than a few hundred keV can be possible. Particles within the internal allowed zone oscillate between mirror points, the positions of which depend on their initial pitch angles. Capturing of a particle possible if its mirror points is located above the planet’s surface, and the imprecision of the conservation of the 1

^{st}adiabatic invariant sin

^{2}

*α*/

*B*=const [3] does not lead to their significant displacement within the time

*T*of interest to us. The description of the capture region located in the vicinity of the plane of the magnetic equator can be pictorially expressed by the formula:

where each elementary trajectory ‘traj.’ lies inside the allowed zone, *α*_{0} is *α*_{eq} for the start position on the base of field line *Q*=1, *α*_{th1} and *α*_{th2} are the threshold values of *α*_{0} (Fig. 1), that as well as *γ*_{fin} slightly depend from *T*. Taking into account the variability of the Mercury magnetosphere, we chose prolonged *T*=20 minutes.

We considered sets of trajectories of 3∗10^{4} protons with *K*=100 keV, the starting point of which was located at different distances *R*. Their motion was examined over a time interval of 60 seconds, in two directions in time. A particle that was thus captured for more than 2 minutes, which is approximately three times larger then their period of drift motion, was considered as captured (see Table).

R (R) _{M} |
percentage of captured particles |

1.10 | 19.6 |

1.15 | 25.2 |

1.20 | 19.0 |

1.25 | 16.7 |

1.30 | 13.8 |

1.35 | 9.6 |

To describe the situation at the different points *R*, we also calculated maps showing the dependence of particles history on their initial pitch and phase angles (Fig. 2). Yellow color denotes particles colliding with the planet’s surface both in the past and in the future; red—either in the past or in the future; dark gray—capturing in the period under review; purple—arriving from outside the magnetopause and colliding with the surface, or vice versa; blue—flying by. We also drew the isolines *γ*_{in}, *γ*_{out} and *γ*_{fin} on these maps. As it can be seen, the flow of particles at a given point in the magnetosphere of Mercury does not consist solely of trapped, precipitated, or transit particles. At the same point, both passing particles and particles making less than one drift revolution before a collision with the planet surface can be detected.

[1] *Alexeev I.I., Belenkaya E.S., Bobrovnikov S.Yu. et al.* // J. Geophys. Res. Space Physics. 2008. V. 113. A12210.

[2] *Lavrukhin A.S., Alexeev I.I., Tyutin I.V.* // Ann. Geophys. 2019. V. 37. P. 535–547.

[3] *Kuznetsov S.N., Yushkov B.Yu.* // Plasma Physics Reports. 2002. V. 28, No. 4, pp. 342–350.

**How to cite:**
Lukashenko, A., Lavrukhin, A., Alexeev, I., and Belenkaya, E.: A possibility to exist the trapped energetic particles zone near Mercury, Europlanet Science Congress 2020, online, 21 September–9 Oct 2020, EPSC2020-735, https://doi.org/10.5194/epsc2020-735, 2020