Soni et al. Earth, Planets and Space (2020) 72:129 https://doi.org/10.1186/s40623-020-01264-5
FULL PAPER Open Access L‑shell and energy dependence of magnetic mirror point of charged particles trapped in Earth’s magnetosphere Pankaj K. Soni* , Bharati Kakad and Amar Kakad
Abstract In the Earth’s inner magnetosphere, there exist regions like plasmasphere, ring current, and radiation belts, where the population of charged particles trapped along the magnetic feld lines is more. These particles keep performing gyration, bounce and drift motions until they enter the loss cone and get precipitated to the neutral atmosphere. Theoretically, the mirror point latitude of a particle performing bounce motion is decided only by its equatorial pitch angle. This theoretical manifestation is based on the conservation of the frst adiabatic invariant, which assumes that the magnetic feld varies slowly relative to the gyro-period and gyro-radius. However, the efects of gyro-motion can- not be neglected when gyro-period and gyro-radius are large. In such a scenario, the theoretically estimated mirror point latitudes of electrons are likely to be in agreement with the actual trajectories due to their small gyro-radius. Nevertheless, for protons and other heavier charged particles like oxygen, the gyro-radius is relatively large, and the actual latitude of the mirror point may not be the same as estimated from the theory. In this context, we have carried out test particle simulations and found that the L-shell, energy, and gyro-phase of the particles do afect their mirror points. Our simulations demonstrate that the existing theoretical expression sometimes overestimates or underesti- mates the magnetic mirror point latitude depending on the value of L-shell, energy and gyro-phase due to underlying guiding centre approximation. For heavier particles like proton and oxygen, the location of the mirror point obtained from the simulation deviates considerably ( 10°–16°) from their theoretical values when energy and L-shell of the particle are higher. Furthermore, the simulations∼ show that the particles with lower equatorial pitch angles have their mirror points inside the high or mid-latitude ionosphere. Keywords: Earth’s inner magnetosphere, Trapped particle trajectories, Magnetic mirror points, Test particle simulation
Introduction In general, ∼ eV range particles are seen in the plasma- Some of the solar wind charged particles enter the sphere, ∼ 1 − 100 keV in the ring current region, and the Earth’s magnetosphere and get trapped along the mag- most energetic particles, > 100 keV are dominated in the netic feld lines. Protons, electrons, helium ions, and radiation belt regions (Ebihara and Miyoshi 2011; Millan ionospheric oxygen ions are commonly seen species in and Baker 2012). Northrop and Teller (1960) established the trapped regions of the Earth’s magnetosphere viz. the stability of charged particles trapped in the Earth’s plasmasphere, ring current, radiation belts and so on. magnetic feld. It was shown that in the Earth’s inner Satellite observations suggest that the energies of these magnetosphere, where the magnetic feld is assumed to trapped particles range from few ∼ eV to ∼ 100 MeV . be nearly dipolar, the charged particles undergo three quasi-periodic motions. Tese are, gyro-motion around *Correspondence: [email protected] its guiding centre, bounce motion along the magnetic Indian Institute of Geomagnetism, New Panvel, Navi Mumbai 410218, feld lines between the conjugate mirror points in the India northern and southern hemispheres, and longitudinal
© The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativeco mmons.org/licen ses/by/4.0/ . Soni et al. Earth, Planets and Space (2020) 72:129 Page 2 of 15
gradient-curvature drift motion of particle’s guiding found that the sixth-order Runge–Kutta method has sig- centre around the Earth (Williams 1971). A set of three nifcantly less numerical dissipation and can simulate the invariants associated with each of these motions defne a physically reliable trajectories of oxygen ions, protons, nearly stable drift shell encircling the Earth. and electrons of energy range 5 keV to 5 MeV at L = 3–6. Under the guiding centre approximation, the theory We have used this simulation to fnd the latitudes of the provides a set of reduced dynamical equations for the magnetic mirror point of charge particles, having a wide motion of the charged particles in a stable or slowly vary- range of energy and L-shells. It may be noted that we ing magnetic feld, which are averaged over gyro-radius have not ignored the gyro-motion of the particles. Te in space or gyro-period in time (Baumjohann and Treu- particles are allowed to gyrate, bounce, and drift, self- mann 2012; Li et al. 2011). However, often it is necessary consistently. For charged particles, diferent equatorial to carry out these calculations on much longer scales due pitch angles, L-shells, energies and gyro-phase are con- to the multi-scale nature of magnetized plasmas. It may sidered to estimate their latitudes of magnetic mirror be noted that the particle is likely to experience the vary- points through the simulation. Also, their deviation from ing magnetic feld over larger spatial scales as in the case the theoretical relation between m and αeq is quantifed. of larger gyro-radius. Hence, the theoretically estimated Besides, we have obtained the ranges of energy, L-shell, mirror point latitude obtained under the guiding centre and pitch angle of the particles, for which their mirror approximation can deviate from the actual particle tra- points exist in the high- or mid-latitude ionosphere. jectory. However, so far, there are no attempts to verify Te paper is structured as follows. Te model equa- or quantify such efects. Te information about the lati- tions and numerical schemes used in the simulation are tude of the magnetic mirror point of the particle during discussed in “Simulation model” section. In “Teoretical its bounce motion is essential because if the mirror point background” section, we have introduced the theoretical lies in the neutral atmosphere, the particle can lose its concepts for the estimates of the latitude of the magnetic energy through collision and cause heating. mirror point of trapped particles and their dependence Te theoretical expression for the mirror point lati- on the equatorial pitch angle. Te results obtained from tude ( m ) of a bouncing particle indicates that the mir- the test particle simulations are elaborated in “Results” ror point latitude is decided only by the equatorial pitch section. Finally, the present study is discussed and con- angle ( αeq ) of the particle (Tsurutani and Lakhina 1997). cluded in the sections “Discussion” and “Conclusions”, Tis theoretical expression is based on the guiding cen- respectively. tre approximation, where it is assumed that the magnetic feld varies slowly relative to the gyro-period and gyro- Simulation model radius. For electrons, as the gyro-radius is small, they Te relativistic equation of motion of a charged particle gyrate very close to the magnetic feld line such that the having charge q and mass m in the presence of magnetic ambient magnetic feld over a gyration can be assumed feld B can be written in the form of Lorentz equation, to be nearly constant. However, for protons and other dv heavier particles like helium and oxygen, gyro-radius is γ m0 = qv × B. dt (1) large, and it increases with energy and L-shells. In such a scenario, the ambient magnetic feld may not be constant Once velocity v is estimated using the above equation, over a gyration, and the guiding centre approximation the position of a charged particle can be obtained using may not be valid. Te presence of such heavier atoms in the following equation: the Earth’s inner magnetosphere is common (Daglis et al. r 1999). In this context, the dependency of magnetic mir- d v = . (2) ror point latitude on the L-shell and energy needs to be dt verifed. 2 2 −1/2 Here, γ = (1 − v /c ) is the relativistic factor, mass In the present study, we have developed a three-dimen- of particle is m = γ m0 , where m0 is the rest mass. Te sional relativistic test particle simulation model to visu- velocity vector is v =[vx, vy, vz] and its magnitude is esti- alize the trajectories of charged particles trapped in the mated from the kinetic energy, Ek of particle using the inner magnetosphere. Te magnetic feld of the Earth is following equation: assumed to be dipolar. Te relativistic equation of motion is solved using the sixth-order Runge–Kutta method in 2 2 m0c order to accomplish the numerically stable computa- v = c 1 − . (3) m c2 + E tions. We have tested the model by verifying the conser- 0 k vation of energy and all three adiabatic invariants from the fourth- and sixth-order Runge–Kutta methods. We Soni et al. Earth, Planets and Space (2020) 72:129 Page 3 of 15
In the Earth’s inner magnetosphere, the ambient mag- components [ vx , vy , vz ] and then the position components netic feld lines are closed and can be assumed to be [x, y, z] using the estimates of velocity components. nearly dipolar (Bittencourt 2011). In this region, the ter- To verify the stability of the simulation model, we have B r restrial dipolar magnetic feld dip( ) in the Cartesian checked the conservation of energy during all simulation coordinate system is expressed as follows (Grifths 2017; runs. As an example, the trajectories of protons having (i) ◦ ◦ Öztürk 2012): energy Ek = 10 MeV , at L = 4 , αeq = 15 , ψ = 90 and ◦ ◦ (ii) energy Ek = 10 MeV , at L = 4 , αeq = 50 , ψ = 90 B R 3 B (r) =− 0 e [3xzxˆ + 3yzyˆ + (2z2 − x2 − y2)zˆ]. are shown in Fig. 1a, b, respectively. Also, their respective dip r5 energies, Ek , estimated from the simulation at each time (4) step are plotted as a function of time in Fig. 1c, d, respec- Here, we have used the magnetic coordinate system, such tively. Te three-dimensional trajectories of both protons that the xy-plane represents the magnetic equator. Te in the dipolar magnetic feld are evident in Fig. 1a, b. As positive x is radially outward direction from the centre expected, its motion is helical around the magnetic feld of the Earth, and positive y and z, respectively, represent lines. In addition to this, it performs bounce and azi- the magnetic east and the magnetic north of the Earth. muthal drift motions due to the gradient and curvature of At the magnetic equator on the surface of the Earth, i.e., the magnetic feld lines. Also, due to the charge depend- r =[ = = = ] x Re, y 0, z 0 , the magnetic feld strength ency of ∇B × B drift, the proton moves westward as = × −5 is considered as B0 3.07 10 T. In this simulation shown by the black arrow. It is noticed that the latitudinal model, we have used the fourth- and sixth-order Runge– extent of the proton during its bounce motion is wider ◦ Kutta method to solve Eqs. (1) and (2). We have noticed for the proton with αeq = 15 (see Fig. 1a) as compared ◦ that the numerical dissipation is considerably less in the to the proton with αeq = 50 (see Fig. 1b). Here, m1 and sixth-order Runge–Kutta method, and it gives fairly good m2 can be considered as magnetic latitudes of two mir- numerical stability to simulation code. Te expression ror points of the proton in southern and northern hemi- for the sixth-order Runge–Kutta method is given below spheres. Te mirror point latitudes ( m1 = m2 = m ) (Luther 1968): of the two cases obtained from the simulation are men-
t+ t t t 16k1 6656k3 28561k4 9k5 2k6 vx = v + + + − + . (5) x 5 27 2565 11286 10 11
In discretized form, Eqs. (1) and (2) give a total of six tioned in the respective subplots. Hence we can conclude equations, which gives the rate of change of vx , vy , vz , x, that the equatorial pitch angle afects the mirror point y, and z with respect to time. At initial time, t = 0 , the latitude. In Fig. 1c, d, we have shown the total kinetic velocity and position components are [ v sin(αeq) cos(ψ) , energy computed from the simulation, which is constant. v sin(αeq) sin(ψ) , v cos(αeq) ] and [ x0 = L ± rL , 0, 0], Tis suggests that the chosen numerical scheme is appro- respectively. Te magnitude of velocity, v for ions and priate to analyse the locations of the magnetic mirror electrons of same kinetic energy will be diferent due to points of the trapped particles. their mass. Here, rL is gyro-radius and ψ is gyro-phase. Te initial gyro-phase (ψ ) of the particle can infu- Te gyro-phase, ψ is the angle made by perpendicular ence the mirror point of the particle (Liu and Qin 2011; velocity, v⊥ with positive x-direction. It decides the par- Shalchi 2016). To verify this aspect, we run the simula- ticle entry in the horizontal xy-plane and can vary from tion model for oxygen, proton, and electron of fxed 0 to 2π . We have taken ±rL in the initial position coor- energy (5 MeV), initial position x0 = L ( L = 4 and 5), and ◦ ◦ dinate so that particle’s (electron/ion) guiding centre in equatorial pitch angle(αeq = 30 and 90 ) with ψ vary- ◦ ◦ ◦ the equatorial plane is at x = L . For electron and ion, ing between 0 to 360 with an interval of 10 . It may be x0 = L + rL x0 = L − rL we took and , respectively.t +We t noted that we have fxed the initial position of the par- have varied L-shell from 3 − 6Re . In equation (5), vx ticles in the equatorial plane with x0 = L , as we want to is the Runge–Kutta approximation of vx at time t + t , understand the efect of gyro-phase on their mirror point t which is determined by the present value vx plus the latitudes. Te variation of magnetic mirror point lati- weighted average of six increments k1 to k6 . Te expres- tudes for ions (oxygen and proton) and electron is shown sions of these increments are given in Luther (1968). as a function of ψ in Fig. 2a, c, respectively. Te depend- Te size of time interval ( t ) is taken as ∼ 1/50 times ency of initial velocities [ vx0 , vy0 , vz0 ] on ψ infuences the the gyro-period. In a similar way of vx , we computed vy , position of particle during gyration. Terefore, the guid- vz , x, y, z. In this model, frst, we computed the velocity ing centre of the particle in the equatorial plane changes Soni et al. Earth, Planets and Space (2020) 72:129 Page 4 of 15
◦ Fig. 1 Simulated three-dimensional trajectories of proton of energy 10 MeV and L = 4 (i.e., x = L − rL ) with equatorial pitch angle a αeq = 15 and b 50◦ in the Earth’s dipolar magnetic feld for 60 s. The dipole moment is in the negative z-direction. The black arrow shows the westward motion ◦ ◦ ◦ ◦ of the proton due to ∇B × B drift. The values of m for proton having αeq = 15 and αeq = 50 are 40.9 and 18.6 , respectively. The corresponding energy ( Ek ) as a function of time is shown in c and d. Conservation of energy with time demonstrates the stability of the numerical scheme
◦ with ψ even if the initial position of particle, i.e., x0 = L is corresponding to ψ = 90 are marked by vertical black ◦ same. Tis results in a sinusoidal type variations of mag- lines. It is important to note that when ψ = 90 , ions netic mirror point latitudes with the initial ψ as seen in and electron with initial position ( x0 = L ∓ rL ) will have Fig. 2a, c. Te variation of m with ψ for the electron is maximum radial distance as ( x0 = L ± rL ), respectively. smaller as compare to the ions because the gyro-radius of Hence from Fig. 2b, d, it can be concluded that a parti- the electron is small. Also, the variation of m with ψ for cles (ion/electron) with initial position x0 = L ∓ rL has its ◦ the electron has an opposite phase to that of ions as they guiding centre at L only when ψ = 90 . Terefore, for the gyrate in the opposite direction. further comparison of simulation and theoretical results, ◦ ◦ For all further runs, we took ψ as 90 to obtain m from we are taking ψ = 90 in the simulation to get the parti- ◦ the simulation. Te reason for taking ψ = 90 can be cle guiding centre at (L, 0, 0). Te theoretical expression understood from Fig. 2b, d, which show the maximum to obtain the magnetic mirror point latitude is explained radial distance of the particle from the centre of Earth in next section. ( rmax ) as a function of initial ψ for the ions ( x0 = L − rL ) and electrons ( x0 = L + rL ), respectively. We have chosen ◦ αeq = 90 to see the gyration of the particle in xy-plane as Theoretical background in this case the particle does not perform bounce motion. If we inject a charged particle in the Earth’s dipolar Te initial position ( x0 = L ± rL ) and corresponding magnetic feld with an equatorial pitch angle αeq , it will maximum radial distance ( x0 = L ± rL ) are marked by follow the magnetic feld lines and bounce between mag- horizontal blue (oxygen), magenta (proton), and red netic mirror points. Te latitudes of the magnetic mirror (electron) lines. It can be noted that, all the particles with points can be determined by the equatorial pitch angle initial position ( x0 = L ∓ rL ) do not have their guiding using the following theoretical expression (Roederer and ◦ ◦ centre at (L, 0, 0) as ψ varies from 0 to 360 . Te results Zhang 2016), Soni et al. Earth, Planets and Space (2020) 72:129 Page 5 of 15
o E = 5 MeV, = 90o, L = 5 Ek = 5 MeV, eq = 30 k eq 5.4 42 a Oxygen at L=5 b L+r (H+) Oxygen at L=4 5.3 L 40 r L+r (O+) Proton at L=5 L L 5.2 38 Proton at L=4
5.1 rL Oxygen 36 Proton m
ma x 5 34 r 4.9 32 4.8 30 + x0=L-rL(O ) 4.7 + x0=L-rL(H ) 28 4.6 050 100 150 200 250 300 350 050 100 150 200 250 300 350
33.25 5.02 c Electron at L=5 d Electron Electron at L=4 5.015
5.01 33.2 x =L+r (e-) 5.005 0 L m
ma x 5 r 33.15 4.995 L-r (e-) rL L 4.99
33.1 4.985
4.98 050 100 150 200 250 300 350 050 100 150 200 250 300 350
Fig. 2 Magnetic mirror point latitude ( m ) as a function of initial gyro-phase (ψ ) for (a) oxygen (blue), and proton (magenta) ( c) electron (red) of ◦ energy 5 MeV, L = 4 and 5, and αeq = 30 . Maximum radial distance at equator ( rmax ) as a function of initial gyro-phase (ψ ) for (b) oxygen, and ◦ proton of energy 5 MeV initially placed at x0 = L − rL (d) electron of energy 5 MeV initially placed at x0 = L + rL with L = 5 , and αeq = 90
6 2 Beq cos m particles gyrating very close to the magnetic feld line, sin αeq = = . B 2 1 (6) this theoretical relation in Eq. (6) can be still applicable as m [1 + 3 sin m] 2 the magnetic feld over a gyration varies slowly. However, Here, Beq and Bm are the strength of the magnetic feld at if the gyro-radius of a particle is large, the solution of m the equator and the mirror point, respectively. Te above obtained by tracing guiding centre may deviate from the theoretical expression is derived under the guiding centre actual mirror point latitude of the particle. In order to approximation (Tsurutani and Lakhina 1997). Tis equa- verify this hypothesis, we have carried out the test par- tion implies that for a given αeq , the magnetic latitude of ticle simulations to obtain the magnetic mirror point O+ the mirror point is independent of the L-shell and energy. latitudes of electrons, protons, and oxygen ( ) ions. In In other words, Eq. (6) suggests that all charged particles Fig. 3, the Earth’s dipolar magnetic feld lines are shown of a given equatorial pitch angle ( αeq ) bounce back from in the yz-plane. In this case, the equation of dipolar mag- the same magnetic latitude in the dipolar magnetic feld, netic feld lines can be written as: irrespective of their radial position of guiding magnetic 2 r/Re = L cos , (7) feld line and energy (Baumjohann and Treumann 2012). As this equation is derived under the guiding centre where is the magnetic latitude (which can vary from ◦ ◦ approximation, it ignores the efect of gyration and only −90 to 90 ). As the magnetic feld line forms a closed traces the trajectory of the guiding centre along the mag- loop, it can pass through the Earth as well. Hence, in the netic feld line. However, in reality, particle gyrate, and it case of the Earth’s dipolar magnetic feld, there is an upper 2 1 can infuence the mirror points of the particles. For the limit on magnetic latitude l ( cos l = L ) or lower limit Soni et al. Earth, Planets and Space (2020) 72:129 Page 6 of 15
Fig. 3 Illustration of the dipolar magnetic feld lines of the Earth in the Geocentric Solar Ecliptic coordinate system. The Earth’s centre is at the origin [0 0 0], and the magnetic feld lines corresponding to L = 3 to 6 are shown. The solid red and magenta lines correspond to the theoretical magnetic mirror points with the equatorial pitch angle of 30◦ and 10◦ , respectively. The dotted lines correspond to the maximum deviation observed in the simulation with theoretical estimates of magnetic mirror point latitude. The dotted black circle represents the upper boundary of the ionosphere at 1000 km above the surface of the Earth
2 6 5 −1/2 on equatorial pitch angle αl ( sin αl =[4L − 3L ] ), L-shell, we obtain straight solid lines (i.e., same magnetic α which are associated with the footprint of the magnetic mirror point) corresponding to the fxed value of eq for feld line on the surface of the Earth. For a charged par- all L-shells. Besides, theoretical Eq. (6) suggests that the ticle to remain trapped in the Earth’s magnetic feld and electron, or proton or oxygen ion should have the same α not get lost in the Earth’s atmosphere, it is necessary to magnetic mirror points if eq is the same. However, the have m < l or αeq >αl . Tese theoretical limits are guiding centre approximation used in theory is not valid called loss cone boundaries. Particles, which have equa- for the protons and other heavier particles, whose gyro- torial pitch angle inside the loss cone boundary would radius is large. To demonstrate this, we have performed be lost into the dense neutral atmosphere of the Earth the simulations to get the trajectories of electron, pro- before they can reach their mirror points. ton and oxygen for diferent L-shells and energies, and Figure 3 shows the two-dimensional magnetic feld obtained their deviation from the theoretical results. As L = 3 L = 6 an example, we have shown the maximum deviation of lines from to obtained from Eq. (7). Te Earth’s centre is placed at the origin, and the Earth is m from its theoretical estimate for the proton having two shown with the solid blue circle. Te dotted black cir- diferent equatorial pitch angles in Fig. 3. Te dotted red α = 30◦ α = 10◦ cle represents the upper ionospheric boundary at 1000 ( eq ) and magenta ( eq ) lines correspond to ±δ km above the surface of the Earth. Te solid red and the maximum deviation ( max ) in the magnetic latitude magenta lines represent the latitudes of magnetic mir- of the mirror points obtained from the simulation for ror points estimated from theoretical Eq. (6) for equato- these two fxed equatorial pitch angles but having difer- ◦ ◦ rial pitch angles of 30 and 10 , respectively. Tese pitch ent L-shell and energy. Te actual mirror points of proton angle values are chosen such that they are greater than lie inside the region bounded by the dotted lines corre- the loss cone angle, αl . It is evident that as the equato- sponding to a fxed equatorial pitch angle. In the case of rial pitch angle decreases, the particle comes close to the the electron, this gap between the solid and dotted lines Earth and for lower L-shells particle’s mirror point lies decreases, and for heavier particles like oxygen, this gap inside the ionospheric boundary. Another feature is that will increase. In Fig. 3, it is evident that the magnetic mir- since the theoretical expression of m is independent of ror point latitude of the charged particle depends on its Soni et al. Earth, Planets and Space (2020) 72:129 Page 7 of 15
L-shell and energy. In the next section, we have examined Each subplot corresponds to the fxed energy of par- this feature in detail and quantifed the deviation in the ticles with diferent L-shells, which are depicted by magnetic latitude of the mirror point of charged particles diferent lines. Te theoretical relation of m and αeq from the theory by varying their L-shell and energy. obtained from Eq. (6) is plotted with the black dashed line in each subplot. It is evident from Fig. 4 that the Results mirror point latitudes estimated from the simulation In order to investigate the efects of L-shell, energy, and deviate from the corresponding theoretical curves. Te gyro-phase of the charged particles on their magnetic deviation is larger for the oxygen ions, moderate for the mirror point latitudes, we have performed the simulation protons, and the lowest for the electrons. Another evi- for electrons, protons and oxygen ions with energy 5 keV dent feature is that for the fxed energy, the deviation ◦ ◦ to 5 MeV at L = 3 to 6 and ψ = 0 to 360 (with an inter- between the theoretically estimated and the simulated ◦ val of 10 ), and tracked their trajectories. As an input to magnetic mirror point latitude shows an increase with the simulation code, we shoot a particle of kinetic energy, L-shell. Ek at the magnetic equator at position [ x0 , y0 , z0 ] with a In order to quantify the deviation in theoretical and α fxed equatorial pitch angle ( eq ). Te equatorial pitch simulated magnetictheory mirror points, we have computed the ◦ ◦ δ = − simulation angle is chosen in the range of 5 to 85 with an interval diference ( m m ) between theoretical ◦ of 5 such that, for a given L-shell it is always higher than and simulated latitudes of the magnetic mirror points. the loss cone angle (i.e., αeq >αl). Figure 5 shows the diference between the magnetic mir- δ L = 3 α From simulations, one can estimate the position (x, ror point latitudes, for –6 as a function of eq . Te y, z) and velocity ( vx , vy , vz ) of the particle at each time colour bar represents the diference between the theoret- step. We have converted the position from the Cartesian ical and simulated mirror point latitudes. For oxygen ions δ 0◦ coordinate system (x, y, z) to the spherical coordinate sys- (see Fig. 5a, d, g, j) the is found to vary between to 16◦ 0◦ tem (r, , φ ) using coordinate transformation. Te simu- , for protons (see Fig. 5b, e, h, k), it varies between 10◦ lation output of the magnetic latitude ( ) of the particle to and for electrons (see Fig. 5c, f, i, l) it is between 0° is obtained for one bounce period, and its maximum to 0.3°. Te maximum deviation for each energy is men- value is considered as the magnetic mirror point latitude. tioned in the respective subplots. Te deviation for all the First, we obtained the magnetic mirror point latitude particles increases with L-shells. Figure 5 indicates that of particles (electron/ion) for (i) fxed energy and vary- the deviation between the theoretically estimated and ing L-shells and (ii) varying energy and fxed L-shells by simulated magnetic mirror point latitudes increases with ◦ taking x0 = L ± rL and ψ = 90 . Later, we estimated the the L-shell, and this deviation becomes more evident for penetration depth of particle (electron/ion) in the Earth’s the higher energy particles. upper atmosphere by varying both L-shell and energy by Furthermore, we have investigated the dependence of ◦ taking x0 = L ± rL and ψ = 90 . At last, we quantifed the magnetic mirror point latitude on the particle’s energy. ◦ ◦ efect of gyro-phase, by varying ψ between 0 –360 and For this purpose, we have obtained the simulation ◦ ◦ αeq between 5 –85 for fxed L-shell ( L = 5 ) and energy results for electrons, protons and, oxygen ions of difer- ψ = 90◦ (5 MeV) with x0 = L . Tese three cases are analysed sep- ent energies, L-shell and fxed . We simulated L = 3 arately and presented in the following subsections. the magnetic mirror points of the particles for to 6, corresponding to four energy values. Figure 6 depicts Variation of mirror point latitude with L‑shell and energy the magnetic mirror points as a function of the equato- Here, we have performed the simulations to trace the rial pitch angle for diferent L-shells. In this fgure, each trajectories of the electron, proton and oxygen ion hav- of the subplots is obtained for the fxed L-shell and with ing four energies, 5 MeV, 500 keV, 50 keV, and 5 keV, four energy values (shown by diferent colours). Panels and L-shell L = 3 , 4, 5 and 6 with an equatorial pitch (a, d, g, j) are for oxygen ions, (b, e, h, k) are for protons, ◦ ◦ ◦ angle of 5 to 85 with an interval of 5 . Te initial posi- and (c, f, i, l) are for electrons. Te corresponding devia- tion of magnetic mirror point latitude from the theoreti- tion of electron/ion in the equatorial plane is taken as ◦ cally estimated m for each case is shown in Fig. 7. We x0 ± rL with ψ = 90 . Tus, for a given particle with a fxed value of L-shell and energy, we have a total of 17 noticed that for any fxed L-shell, as the energy of particle simulation runs corresponding to the chosen range of increases the simulated magnetic latitude of the mirror point deviates signifcantly from its theoretical estimate. αeq . From each simulation output, we have estimated Tis deviation is more signifcant for the oxygen ions the magnetic mirror point latitude, m and plotted it and protons, as compared to the electrons. Also, this as a function of αeq for diferent L-shell and energy in Fig. 4. Here, panels (a, d, g, j) are for oxygen ions, (b, deviation is more signifcant for the higher L-shell. Over- e, h, k) are for protons, and (c, f, i, l) are for electrons. all, this analysis shows that the magnetic mirror point Soni et al. Earth, Planets and Space (2020) 72:129 Page 8 of 15
L = 3 L = 4 L = 5 L = 6 Theoretical Oxygen (O+) Proton (H+) Electron (e-) a E = 5 MeV b E = 5 MeV c E = 5 MeV 50 k k k [Deg ] m
0 d E = 500 keV e E = 500 keV f E = 500 keV 50 k k k [Deg ] m
0 g E = 50 keV h E = 50 keV i E = 50 keV 50 k k k [Deg ] m
0 j E = 5 keV k E = 5 keV l E = 5 keV 50 k k k [Deg ] m
0 050900 50 90 05090 [Deg] [Deg] eq eq [Deg] eq Fig. 4 The magnetic mirror point latitude as a function of equatorial pitch angle of the particle with diferent energy. a, d, g, j are for oxygen ions, b, e, h, k are for protons, and c, f, i, l are for electrons. The dashed black line is drawn from the theoretical Eq. (6) and solid lines correspond to the ◦ simulation results of m for fxed ψ = 90 , fxed energy and diferent L-shells ( L = 3, 4, 5, 6)
latitude of a particle is indeed dependent on L-shell, surface of the Earth is shown in Fig. 8 as a function of the energy and mass, apart from its equatorial pitch angle. equatorial pitch angle for diferent energies and L-shells. In Fig. 8, panels (a, d, g, j) are for oxygen ions, panels (b, e, Minimum radial distance of particle h, k) are for protons and panels (c, f, i, l) are for electrons. We have investigated the penetration depth of the Each subplot is for fxed L-shell, and diferent curves in trapped particles in the Earth’s atmosphere. Our simula- it correspond to diferent energies. Te dashed-dotted tion model does not have any atmosphere, ionosphere, black line in each panel represents the theoretically and waves in the system to afect the particle’s motion. estimated position of Hmin for a given value of αeq . Te Hence, the particle’s penetration depth is decided by vertical dashed line indicates a loss cone angle, αl corre- the location of the latitude of the magnetic mirror point sponding to the L-shell. Te particles having an equato- above the surface of the Earth. As mentioned earlier, we rial pitch angle less than the loss cone angle get lost in used diferent L-shells, energy, and equatorial pitch angle the atmosphere, and theoretically, their mirror points for electrons, protons, and oxygen ions to estimate the should lie inside the Earth, i.e., Hmin < 0 . Here, Hmin = 0 latitudes of the magnetic mirror points. Te penetra- represents the Earth’s surface. Te horizontal dashed tion depth of the charged particle is measured in terms line in each panel starting from top, respectively, repre- of its minimum radial distance ( rmin ) during its bounce sents the upper ionospheric boundary at 1000 km above motion, which is the position, r of the particle at mirror the Earth’s surface. Te particles having rmin < Re , i.e., point latitude, m. Hmin < 0 does not exist practically because these parti- We have defned the minimum vertical height above cles have mirror points inside the Earth’s surface. We are the surface of the Earth corresponding to the magnetic getting such particles in the simulations as the expression mirror point such that Hmin = rmin − Re . Te minimum of the dipolar magnetic feld defnes the closed feld lines. vertical height of the magnetic mirror point above the Soni et al. Earth, Planets and Space (2020) 72:129 Page 9 of 15
Oxygen (O+) Proton (H+) Electron (e-) 16 10 0.4 6 a b c
5 9 14 0.35 4 L-shel l o o E =5 MeV, = 15.4o E =5 MeV, = 10.1 E =5 MeV, = 0.3 3 k max k max 8 k max 12 0.3 d e f 6 7 5 10 0.25 4 6 L-shel l o E =500 keV, = 12.1o E =500 keV, = 3.6 E =500 keV, = 0.1o 3 k max k max k max 8 5 0.2 6 g h i 5 4 6 0.15 4 L-shel l o E =50 keV, = 4.3o E =50 keV, = 1.2 E =50 keV, = 0.01o 3 k max k max 3 k max 4 0.1 6 j k 2 l 5 2 0.05 4 1 L-shel l o o E =5 keV, = 1.5 E =5 keV, = 0.4o E =5 keV, = 0.01 3 k max k max k max 0 0 0 20 40 60 80 20 40 60 80 20 40 60 80 [Deg] eq [Deg] eq [Deg] eq Fig. 5 The deviation of the theoretical and simulation results of magnetic mirror points as a function of the equatorial pitch angle for diferent L-shells. a, d, g, j are for oxygen ions, b, e, h, k are for proton, and c, f, i, l are for electrons. The magnitude of the absolute deviation is shown by the colour bar
It is noticed that as the equatorial pitch angle decreases the ionospheric plasma or neutral species present at that the mirror point of particle moves to lower altitudes in altitudes. the Earth’s upper atmosphere, and hence, the particle Generally, the particle precipitation occurs at around can come closer to the Earth. Te charged particles with 100 km. While reaching this altitude, the particle motion lower equatorial pitch angle get lost in the loss cone at gets afected through their collisions with ionized par- lower L-shells ( L = 3 and 4). Also, as the energy of the ticles and neutrals present below 1000 km. Te efect of particle decreases, their Hmin decreases, i.e., particles collisions dominates over the magnetic feld efects below with lower energy have their mirror points close to the 100 km and the particle dynamics in this region mainly Earth. Figure 8 indicates that the particles with equato- governed by the atmospheric chemistry. As we are using ◦ ◦ rial pitch angle between 5 <αeq < 10 have their mir- a static magnetic feld with no atmosphere or ionosphere, ror points inside the ionosphere, i.e., within 1000 km there is no possibility to explore the particle loss due to above the Earth’s surface. In Fig. 9, the minimum verti- wave–particle interaction, particle–particle interac- cal height above the surface of the Earth, Hmin for proton tion, and precipitation in the present model. Also, theo- (panel-a) and electron (panel-b) are shown as a function retically, magnetic feld lines form a closed loop that can ◦ ◦ of their energy for equatorial pitch angle of 10 and 15 take charged particles up to or inside the Earth’s surface. at L = 3 . Te horizontal dashed lines starting from top, However, in a real situation before reaching the Earth’s respectively, indicate the ionospheric boundary at 1000 surface, the particle will interact with the atmospheric km and 100 km above the Earth’s surface. Tis fgure or ionospheric particles and get lost. Terefore, we have demonstrates that the protons of energy 5 keV to 300 keV used a limit of Hmin between 100 to 1000 km. and electrons of energy 5 keV to 5 MeV have their mirror points inside the ionosphere if the equatorial pitch angle Gyro‑phase efect on mirror point latitude ◦ is less than or equal to 10 . Such particles entered in the In the last two subsections, we have taken an initial ◦ ionosphere and can get lost after their interaction with ψ = 90 to understand the efect of L-shell and energy of Soni et al. Earth, Planets and Space (2020) 72:129 Page 10 of 15
E = 5 MeV E = 500 keV E = 50 keV E = 5 keV k k k k Theoretical + + Oxygen (O ) Proton (H ) Electron (e-) 50 a L = 6 b L = 6 c L = 6 [Deg ] m
0 50 d L = 5 e L = 5 f L = 5 [Deg ] m
0 50 g L = 4 h L = 4 i L = 4 [Deg ] m
0 50 j L = 3 k L = 3 l L = 3 [Deg] m
0 050900 50 90 05090 eq [Deg] eq [Deg] eq [Deg] Fig. 6 The magnetic mirror point latitude as a function of the equatorial pitch angle of the particle at diferent L-shells. a, d, g, j are for oxygen ions, b, e, h, k are for protons, and c, f, i, l are for electrons. The dashed black line is obtained from the theoretical Eq. (6) and the solid lines correspond to ◦ the simulation results of m for fxed ψ = 90 , fxed L-shell and diferent energy ( Ek = 5 MeV , 500 keV, 50 keV, 5keV)
particle on its mirror point latitude. However, from Fig. 2 electron, variation in δ is opposite to that of ions as they it is clear that magnetic mirror point latitude varies with gyrate in opposite direction. initial gyro-phase (ψ ). Here, we have quantifed deviation Te dependency of mirror point latitude on initial in simulated m from its theoretical value for diferent ψ . gyro-phase can be understood from the direction of For this purpose, we perform simulation runs for oxy- initial force acting on the charged particle during its gen, proton, and electron of energy E = 5 MeV at L = 5 gyration. For example, ions (gyrating in the clockwise ◦ ◦ ◦ ◦ by varying αeq between 5 to 85 (with an interval of 5 ) direction) with initial ψ = 90 move toward the region of ◦ ◦ ◦ and ψ between 0 to 360 (with an interval of 10 ). Te a weaker magnetic feld (away from the Earth) and ions ◦ initial position of the particle is the same in all simula- with initial ψ = 270 move toward the stronger magnetic tion runs, i.e., x0 = L . Here, we chose the higher energy, feld (towards the Earth). It means the initial gyro-phase i.e., E = 5 MeV and L-shell, L = 5 because in the previ- afects the guiding centre of the particle in the equatorial ous two subsections we have seen that the deviation δ is plane such that L-shell associated with their guiding cen- large for higher L-shell and energy. We have a total of 629 tre varies with ψ . Hence, m varies with ψ . As gyration of simulation runs, and from each run, we estimated the the electron is opposite to ions, the efect of ψ on m is deviation δ in magnetic mirror point latitude for difer- opposite in case of the electron. ent combinations of αeq and ψ . In Fig. 10, the deviation in simulated magnetic mirror point latitude from its theo- Discussion retical value is shown for (a) oxygen, (b) proton and (c) Te theoretical expression of magnetic mirror point electron. We noticed that δ is maximum for the oxygen latitude suggests that diferent charged particles with ◦ ◦ (± 15 ), moderate for the proton (± 10 ) and minimum the same αeq will bounce back from the same mag- ◦ ◦ for the electron (± 1 ). For ions, δ is negative for ψ = 0 to netic latitude irrespective of their mass, energy, and ◦ ◦ ◦ 180 , and it is positive for ψ = 180 to 360 , whereas for L-shell. However, our test particle simulations suggest a Soni et al. Earth, Planets and Space (2020) 72:129 Page 11 of 15
Oxygen (O+) Proton (H+) Electron (e-) 16 10 0.4 7 a b c 6
E [eV] 9 14 0.35 10 5 o o o log L=6, = 15.4 L=6, = 10.1 8 L=6, = 0.3 4 max max max 12 0.3 7 d e 7 f 6 E [eV] 10 0.25 10 5 6 o o o log L=5, = 11.0 L=5, = 8.0 L=5, = 0.2 4 max max max 8 5 0.2 7 g h i 6 4 E [eV] 6 0.15 10 5 o o o log L=4, = 7.3 L=4, = 4.8 3 L=4, = 0.2 4 max max max 4 0.1 7 j k 2 l 6
E [eV] 2 0.05 1 10 5 o o o log L=3, = 5.8 L=3, = 3.0 L=3, = 0.07 4 max max max 0 0 0 20 40 60 80 20 40 60 80 20 40 60 80 [Deg] [Deg] eq [Deg] eq eq Fig. 7 The deviation of theoretical and simulation results of magnetic mirror points as a function of the equatorial pitch angle for diferent energies. a, d, g, j are for oxygen ions, b, e, h, k are for protons, and c, f, i, l are for electrons. The magnitude of the absolute deviation is shown by the colour bar
dependency of m on particle’s energy, L-shell, and mass. gyration is likely to vary, and the approximation used in Tis dependence can be understood from gyro-radius the theoretical formulation gets violated. Hence, as dem- ( rL = γ m0v⊥/qB ) of the trapped particle. When the par- onstrated by present simulations, the actual mirror lati- ticle is at higher L-shells, it experiences the weaker ambi- tude of particle deviates considerably from its theoretical ent magnetic feld resulting in larger gyro-radius. Also, as estimate when gyro-radius is larger. compared to the electrons, the gyro-radius of proton and In the present study, the trapped particle performs oxygen is more signifcant due to their higher masses. periodic motion so that their motion along the magnetic Likewise, if the energy of the charged particle increases, feld lines and drifts across the magnetic feld lines are its velocity increases, which results in more abundant symmetric. In such periodic motion, the three adiaba- gyro-radius. Terefore, higher energy, mass and L-shell tic invariants associated with particle gyro, bounce, and of the particle results in larger gyro-radius of the parti- drift motion are conserved. In the case of theoretical Eq. cle. It may be noted that the theoretical formula given (6) as gyration is ignored, and only guiding centre trajec- in Eq. (6) is based on the guiding centre approximation tory is considered, the frst adiabatic invariant is always and it considers the magnetic feld at one single point at conserved. J1 , J2 , and J3 are the adiabatic invariants asso- the equator ( Beq ) and mirror point ( Bm ) along the guid- ciated with gyro, bounce, and drift motion of particles, ing magnetic feld line. Te guiding centre approxima- respectively. Te frst adiabatic invariant is associated tion ignores the gyration and only traces the trajectory with gyro-motion, and it is given by J1 = 2πm0�µ�/q , of the guiding centre from one mirror point to another. where µ is the magnetic moment, the longitudinal invari- In reality, particle gyrates around the magnetic feld line, ant J2 is associated with the bounce motion and the fux and therefore one cannot ignore the gyration associ- invariant J3 is associated with the azimuthal drift motion ated efects. In the simulation, the trapped particles are (Mukherjee and Rajaram 1981). We estimated three adi- allowed to gyrate, bounce, and drift self-consistently. abatic invariants from the simulation and checked their For larger gyro-radius, the ambient magnetic feld over a conservation. Tese adiabatic invariants are shown as Soni et al. Earth, Planets and Space (2020) 72:129 Page 12 of 15
E =5 keV E =50 keV E =500 keV E =5 MeV k k k k Theoretical - Oxygen (O+) Proton (H+) Electron (e ) 3.5x104 a b c [km] mi n
H L = 6 L = 6 L = 6 0 3.5x104 d e f [km] mi n
H L = 5 L = 5 L = 5 0 2x104 g h i [km] mi n
H L = 4 L = 4 L = 4 0 4 2x10 j k l [km] mi n
H L = 3 L = 3 L = 3 0 050406080 0204060800 20 40 60 80
eq [Deg] eq [Deg] eq [Deg]
Fig. 8 The minimum vertical height, Hmin of oxygen ions (a, d, g, j), protons (b, e, h, k) and electrons (c, f, i, l) above the surface of the Earth as a function of equatorial pitch angle. Here, Hmin = rmin − Re . Each subplot is obtained for the fxed L-shell and four diferent values of energy. The dashed-dotted black and solid lines represent the theoretical and simulation results, respectively. The vertical dashed line corresponds to the loss cone angle, and the horizontal dashed line represents the upper ionospheric boundary at 1000 km. Hmin = 0 indicates the surface of the Earth
Proton (H+) Electron (e-) 4000 3000 a o o b = 10 = 15 2500 3000 2000
= 10o = 15o [km] 2000 [km] 1500 mi n mi n H H 1000 1000 500
0 0 44.5 55.5 66.5 44.5 55.5 66.5
log10Ek[eV] log10Ek[eV] Fig. 9 The minimum vertical height of a protons and b electrons above the surface of the Earth as a function of the energy at L = 3 with the equatorial pitch angle of 10◦ and 15◦ . The horizontal dashed lines correspond to the ionospheric boundaries at 1000 km and 100 km, respectively, above the surface of the Earth Soni et al. Earth, Planets and Space (2020) 72:129 Page 13 of 15
a 20 10 40 [Deg ] 0 eq 60
Oxygen,E = 5MeV, x = 5R -10 80 k 0 e b 20 10 40 [Deg ]
eq 60 0
80 Proton, E = 5MeV, x = 5R k 0 e -10 c 20 1
40 [Deg ] 0 eq 60
Electron, E = 5MeV, x = 5R 80 k 0 e -1 050100 150 200 250300 350 [Deg] Fig. 10 The deviation in simulated magnetic mirror point latitude from their respective theoretical value as a function of initial gyro-phase for diferent equatorial pitch angle. a is for oxygen ion, b is for proton, and c is for electron of energy Ek = 5MeV and L = 5 with initial position [x0 = L, y0 = 0, z0 = 0] . The magnitude of the deviation is shown by the colour bar
a function of time for electron, proton and oxygen in from the closed line integral of the canonical moment of Fig. 11. In upper panels of Fig. 11a–c the variation of the particle over one cycle of their respective motion. For µ is shown for one bounce period. As particle gyrates example, the frst adiabatic invariant J1 = 2πm0�µ�/q is and move from equatorial region (low magnetic feld) obtained by averaging it over one gyro-period, and it is to polar region (high magnetic feld) the µ varies but conserved over a gyration. average magnetic moment µ over gyration is constant as shown by the black dash-dot line in upper panels of Conclusions Fig 11. For electron the mean value of J1 , J2 , and J3 In the present study, we have performed the three- −14 −21 −12 −18 are 3.9 × 10 ± 4.3 × 10 , 4.2 × 10 ± 3.2 × 10 dimensional test particle simulations to investigate the −11 −14 and 7.1 × 10 ± 4.6 × 10 , for proton dynamics of charged particles trapped in the Earth’s −14 −18 −12 −17 3.5 × 10 ± 4.3 × 10 , 4.1 × 10 ± 1.3 × 10 inner magnetosphere. Te magnetic feld of the Earth is −11 −14 and 6.7 × 10 ± 4.8 × 10 , and for oxygen assumed to be dipolar and stationary with time. Since −13 −17 −12 −14 1.4 × 10 ± 3.8 × 10 , 6.5 × 10 ± 3.0 × 10 the particle is treated as a test particle, its motion does −11 −14 and 6.3 × 10 ± 6.3 × 10 , respectively. It may be not afect the ambient magnetic feld. We have simulated noted that the standard deviation in J1 , J2 , and J3 are the trajectories of electrons, protons and oxygen ions small, which implies that all three adiabatic invariants and converted the results from Cartesian to the spheri- remain constant with time. As conservation of adiabatic cal coordinate system. We have obtained the latitudes invariants is verifed, we can compare mirror point lati- of the magnetic mirror point through the simulation for tudes derived from the simulation with theory. One may particles of diferent energies, equatorial pitch angles at argue that if gyro-radius is large enough to afect the diferent L-shells and diferent initial gyro-phase. Te underlying approximation used in (6) then the frst adia- chosen range of L-shell is 3–6, energy is 5 keV–5 MeV, ◦ ◦ batic invariant J1 may also get violated. However, all three the equatorial pitch angle between 5 –85 and gyro- ◦ ◦ adiabatic invariants are found to conserved in the simu- phase between 0 –360 . Te theoretical expression of lation. It is because each of these invariant is obtained the latitude of the magnetic mirror point suggests its Soni et al. Earth, Planets and Space (2020) 72:129 Page 14 of 15
o Electron, E = 5MeV, L = 6, = 30o o Oxygen, E = 5MeV, L = 6, = 30 k eq Proton, Ek = 5MeV, L = 6, eq = 30 k eq 10-3 10-7 10-7 2 1.5 a b c 6 1.5 1 5 1 0.5 4 00.1 0.2 0.3 0123 0102030 10-14 10-14 10-13 6 d 3.92x10-14 4.3x10-21 e 3.5x10-14 4.3x10-18 f 1.4x10-13 3.8x10-17 2 1 1 4 1 J J 4 J
-5 -2 1 -2 = 1.1x10 % = 1.2x10 % = 2.7x10 % 2 2 00.1 0.2 0.3 0123 0102030 10-12 10-12 10-12 5 5 g 4.2x10-12 3.23x10-18 h 4.1x10-12 1.3x10-17 i 6.5x10-12 3.0x10-14 4.5 4.5 7 2 2 2 J J 4 J 4 6 -5 -4 -1 3.5 = 7.5x10 % 3.5 = 3.1x10 % = 4.6x10 % 5 0510 050100 150 200 0500 1000 10-11 10-11 10-11 8 7 j 7.17x10-11 4.6x10-14 k 6.7x10-11 4.8x10-14 l 6.3x10-11 6.3x10-14 7 3 3 3 6.5
J 7 J J
-2 6 -2 -1 = 6.5x10 % = 7.2x10 % 6 = 1.0x10 % 6 0500 1000 1500 2000 0500 1000 15002000 05000 10000 Time [sec] Time [sec] Time [sec]
Fig. 11 Variation of magnetic moment µ and three adiabatic invariants J1 , J2 and J3 estimated from the simulation for electron (a, d, g, j), proton (b, ◦ ◦ e, h, k), and oxygen (c, f, i, l) having energy 5 MeV, L = 6 , αeq = 30 , ψ = 90 . The mean value of adiabatic invariants J and their standard deviation σ are mentioned in the respective panels to demonstrate their conservation
dependence only on the equatorial pitch angle of the theoretical formulation of the estimation of m gets vio- charged particle (Tsurutani and Lakhina 1997; Baum- lated. Tus, one has to be cautious while using theoreti- johann and Treumann 2012). However, our simulation cally derived latitudes of magnetic mirror points. Te suggests that the magnetic mirror point of the charged initial gyro-phase (ψ ) afects the position of the guiding particle also depends on its L-shell and energy. It is centre of the particle so that L-shell associated with their because the theoretically derived magnetic mirror point guiding centre varies with ψ . Hence, the magnetic mirror latitude is based on the guiding centre approximation, point latitude of the particle is found to vary with initial which ignores the efect of gyration of particle. As com- gyro-phase of the particle. ◦ ◦ pare to oxygen ions or protons ( δmax ≈ 10 −16 ), the We have estimated the minimum vertical height ( Hmin ) deviation in the magnetic mirror point latitude of elec- of the trapped particles above the surface of the Earth in ◦ trons from theory is less signifcant ( δmax ≈ 0.3 ). Te order to know how deeper they can enter into the ion- deviation between simulated and theoretical magnetic osphere. We found that the magnetic mirror points of mirror point latitude increases with both energy and the trapped particles with smaller equatorial pitch angle ◦ L-shell of the particle and hence, can not be ignored. ( αeq ≤ 10 ) and smaller energy lies inside the ionosphere For electrons, the gyro-radius is small, so the guiding (< 1000 km ). Terefore, the particles with a smaller αeq centre approximation is applicable. For oxygen ions and will get lost in the ionosphere, even in the absence of any protons, the ambient magnetic feld over a gyration will wave–particle interaction phenomena. It is because they not be constant due to their large gyro-radius. In such are entering the region where sufcient charged and neu- a scenario, the underlying approximation used in the tral particles are present to absorb their energy through Soni et al. Earth, Planets and Space (2020) 72:129 Page 15 of 15
collisions. Such information will be useful in the atmos- Bittencourt J (2011) Fundamental Of plasma physics. Springer, Berlin. https:// doi.org/10.1007/978-1-4757-4030-1 pheric/ionospheric modelling to understand the parti- Daglis IA, Thorne RM, Baumjohann W, Orsini S (1999) The terrestrial ring cur- cle precipitation in the high latitudes. Te theoretical rent: origin, formation, and decay. Rev Geophys 37:407–438. https://doi. expression for the magnetic mirror point of the charged org/10.1029/1999RG900009 Ebihara Y, Miyoshi Y (2011) Dynamic inner magnetosphere: a tuto- particle is often used to understand their penetration rial and recent advances. Dyn Magnetosphere. https://doi. depth in the ionosphere. In this context, the simulation- org/10.1007/978-94-007-0501-2_9 based quantifed information provided in the present Grifths DJ (2017) Introduction to electrodynamics, vol 2. Cambridge Univer- sity Press study will be useful to the scientifc community to under- Li J, Qin H, Pu Z, Xie L, Fu S (2011) Variational symplectic algorithm for guiding stand the dynamics of the trapped particles in the Earth’s center dynamics in the inner magnetosphere. Phys Plasmas 18(052):902. inner magnetosphere. https://doi.org/10.1063/1.3589275 Liu Jian, Qin Hong (2011) Geometric phase of the gyromotion for charged par- ticles in a time-dependent magnetic feld. Phys Plasmas 18(07):072505. https://doi.org/10.1063/1.3609830 Abbreviations Luther H (1968) An explicit sixth-order Runge–Kutta formula. Math Comput : Magnetic mirror point latitude; α : Equatorial pitch angle; µ: Magnetic m eq 22:434–436. https://doi.org/10.1090/S0025-5718-68-99876-1 moment; R : Earth’s radius; E : Kinetic energy; α : Loss cone angle; r : e k l L Millan R, Baker D (2012) Acceleration of particles to high energies in Earth’s Gyro-radius. radiation belts. Space Sci Rev 173:103–131. https://doi.org/10.1007/s1121 4-012-9941 Acknowledgements Mukherjee G, Rajaram R (1981) Motion of charged particles in the magneto- The model computations were performed on the High-Performance Comput- sphere. Astrophys Space Sci 74:287–301. https://doi.org/10.1007/BF006 ing System at the Indian Institute of Geomagnetism. 56440 Northrop TG, Teller E (1960) Stability of the adiabatic motion of charged Authors’ contributions particles in the earth’s feld. Phys Rev 117:215. https://doi.org/10.1103/ PKS has developed a simulation model, conducted testing and analysed PhysRev.117.215 simulation data. BK has provided suggestions and support to examine the Öztürk MK (2012) Trajectories of charged particles trapped in Earth’s magnetic simulation results and other theoretical perspectives. AK provided continuous feld. Am J Phys. https://doi.org/10.1119/1.3684537 guidance and constructive comments on the work. All three authors read and Roederer JG, Zhang H (2016) Dynamics of magnetically trapped particles. approved the fnal manuscript. Springer, Berlin. https://doi.org/10.1007/978-3-642-41530-2 Shalchi A (2016) Gyrophase difusion of charged particles in random Funding magnetic felds. Monthly Notices R Astron Soc. https://doi.org/10.111 Not applicable. 1/j.1365-2966.2012.21690.x Tsurutani BT, Lakhina GS (1997) Some basic concepts of wave-particle interac- Data availability statement tions in collisionless plasmas. Rev Geophys 35:491–501. https://doi. Simulation codes and data can be shared on request to the authors. org/10.1029/97RG02200 Williams DJ (1971) Charged particles trapped in the earth’s magnetic feld. Competing interests AdGeo 15:137–218. https://doi.org/10.1016/S0065-2687(08)60302-7 The authors declare that they have no competing interests.
Received: 15 April 2020 Accepted: 29 August 2020 Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in pub- lished maps and institutional afliations.
References Baumjohann W, Treumann RA (2012) Basic space plasma physics. Imperial Col- lage Press. https://doi.org/10.1142/p850