# Why do electrons drift Eastwards in the Van Allen belts?

## Context

The image below shows that one of the three particle movement patterns of radiation particles that are captured in the Van Allen belts is a longitudinal drift motion: However, I am experiencing some difficulties in understanding what causes this motion, and most sources I look into either simply state this motion exists, or go in detailed analysis on e.g. the magnitude of this motion without explaining what causes it.

## Question

Hence I would like to ask: What causes the westward drift of protons captured in the Van Allen Belts?

## Hypothesis

I thought perhaps a Lorentz force w.r.t. the Interplanetary Magnetic Field (IMF) and (semi) circular (bouncing) motion around Earth may cause this drift, but I am experiencing some difficulties in confirming/rejecting this hypothesis.

You are looking at $$\nabla \vec B$$-drift here (spoken: grad-E drift).

This type of motion can be explained in a simplified way, when imagining a 2-D plane, separated at y=0 with two different magnetic field strengths, $$B_z$$. Let us say that for y>0, $$\rm B_z$$ is large and for y<0, $$\rm B_z$$ is weak. The strong field imparts a small gyro-radius for the charged particles, while the weak field imparts a large gyro-radius. Positive and negative charges orbit in opposite directions in the field, hence along the field gradients, they tend to drift left and right. This type of drift therefore can lead to net charge separation, which results in an E-field, which again leads to $$\rm \vec E\times \vec B$$-drift, which is uniform in direction for both charges.

This is somewhat illustrated in this lecture, part 2.3. There are other types of drift motions, but the aforementioned two are usually all covered in introductory lectures to plasma physics. Another good one is here.

• I can follow your analogous explanation up to: "which results in an E-field". What do you mean with an E-field? And I was wondering, your explanation makes sense to me when there is a 2-D plane separated at y=0, however, could you perhaps indicate which magnetic fields follow that pattern in reality? (And hence/in context of: which physical/measured/"real life" magnetic fields are the main cause for this drift?) Thank you for the references with thorough analytical/mathematical explanations!
– a.t.
Commented Oct 4, 2021 at 8:54
• @a.t. Sorry, that might not have been super-clear. y>0 would point towards the planet, y<0 away from it, the x-y-plane being the Equatorial plane of the planet. The magnetic field that allows this in reality is Earth's dipole. A dipole possesses non-zero gradients, hence drift occurs. This drift is in opposite directions for pos. and neg. charges, which leads to charge separation, which leads to a large-scale E-field and hence ExB drift. Commented Oct 4, 2021 at 13:28