Behavior of Charged Particles near an X-line

The variability observed in Earth’s magnetic field due to solar activity is the source of many interesting charged particle motions. Here we investigate the behavior of protons and oxygen ions near the X-line, a magnetic field configuration formed by the process of magnetic reconnection. As will be seen, the motions near the X-line cannot be described by the usual guiding center drifts used in plasma physics, rather they are chaotic and must be characterized in other ways.

The X-line, shown in Figure 1, is formed by the process of magnetic reconnection, where stored magnetic field energy is converted to heat and kinetic energy. For this process to occur, two regions of oppositely directed field lines are pushed together by incoming magnetic flux (Figure 2) until the larmor radii \(r_L\) of the particles in each respective region becomes larger than the scale length change of the field lines. The particles are able to jump between the field lines, breaking the frozen-in nature of plasma particles in magnetic fields, and the two regions mix in the reconnection process.

The two regions of oppositely directed field lines create a current sheet between them, and thus an electric field. At the center of the X-line, there is theoretically a point of zero magnetic field strength, called the ’null point’. The current sheet, and therefore the electric field in this region are large and dominate over the magnetic field.

Since the change in the magnetic field occurs over a distance smaller than the larmor radius, the first adiabatic invariant, \(\mu\), is broken. The degree to which \(\mu\) is no longer constant can be described by the \(\kappa\) parameter, being \(\kappa = \sqrt{\frac{R_{field}}{r_L}}\) where \(R_{field}\) is the radius of curvature of the magnetic field lines. Referring back to the field line configuration (Figure 1), the radius of curvature is small in regions close to the X-line, making \(\kappa < 1\), and large farther away, making \(\kappa > 1\). Where \(\kappa < 1\), \(\mu\) will not be constant, but where \(\kappa >> 1\) it will be. In a graph of the \(\kappa\) parameter as a function of distance from Earth (Figure 3), it is seen that \(\kappa\) approaches zero at the null-point (\(100 R_e\)). On the other side going towards Earth, it starts increasing drastically, starting at about \(15 R_e\), and surpassing \(\kappa = 1\) at about \(10 R_e\).

Where \(\kappa >> 1\) we expect to see the usual guiding center motion described by the drift equation \(v_d = \frac{1}{q}\frac{\vec{F}\times\vec{B}}{B^2}\), where \(B\) is the magnetic field, \(q\) is the charge of the particle, and \(F\) is replaced by a force on the particle. There are many of these drifts, but with a field of this configuration, the relevant drifts are \(\vec{E}\times\vec{B}\) drift (due to the electric field force), the grad-B drift (from the spatial changes in the field), and the curvature drift (associated with the centrifugal force of the curved field lines). However, it turns out that the grad-B and curvature drifts are only important at closer than \(8 R_e\) to the Earth. As a result, these will be discussed no further as the areas of interest for this investigation lie further out than that.

Where \(\kappa < 1\), and specifically closer to the X-line, particles are imparted some amount of kinetic energy due to the reconnection process. We expect that particles passing closer to the null point will experience a greater gain in kinetic energy and that more chaotic behavior will be exhibited because as the magnetic field strength wains, the electric field’s influence will take over–even though the force due to the electric field is known, the direction that the particle will accelerate, dependent upon both the electric field and the instantaneous velocity of the particle when the electric field takes over, is not known.

We do not know exactly how the particle will move, but we will characterize the motion by comparing kinetic energy \(\Delta KE\) gained by the particle to the closest distance \(\Delta x_{null}\) of approach to the null point.

The side view of Earth’s magnetic field when the formation of an X-line has taken place. The Earth is located at \(x=0 R_e\), and the X-line at \(x=100 R_e.\)

κ as a function of distance from the Earth. From approximately 15Re outward, κ<1 and μ is not constant. At the null-point, κ approaches zero.