Aurora Borealis

\(\)

If we have a magnetic field \(\vec B\), and a positive charged particle with velocity \(\vec v\)

Then the Lorenz force on that particle is $$ \vec F = q\,\left(\vec v\times\vec B\right) \nonumber $$

Decompose the velocity vector, into parallel and perpendicular to the magnetic field

We can rewrite the Lorenz force as below, taking into account that the force is zero for motion parallel to the magnetic field. $$ \newcommand{\parallelsum}{\mathbin{\!/\mkern-5mu/\!}} \begin{align*} \vec F &= q\left(\vec v_\parallelsum + \vec v_\perp\right)\times\vec B, & \left( \vec v_\parallelsum\times\vec B = 0\right) \\ &= q \left(\vec v_\perp\times\vec B\right) \end{align*} $$

Path

The Lorenz force makes the charged particle circle around with radius \(R\). The parallel component of the velocity is unaffected, and so the charge spirals along a field lines.

Remember: this radius is

$$ R = \frac{mv_\perp}{qB} \nonumber $$

The magnetic field of the Earth is not a straight line, but is curved. Charges will be trapped in a spiral orbit along magnetic field lines.

The field lines come down on earth near the magnetic poles. This increases the field strength in the direction of motion, causing a force to slow the charges. Some even reverse their direction and get trapped forming two radiation belts above the atmosphere.

Solar wind

Our sun emits a plasma. Plasma is highly ionized; electrons and protons. We call that the solar wind. Sometimes it’s strong, and sometimes it’s weak.

When the solar wind reaches the Earth, it ionizes the upper atmosphere of the Earth. The Auroras are a result of these electrons and protons from the solar wind colliding with the Nitrogen and Oxygen atoms in the upper atmosphere. Let’s assume it is electrons \(\text e\). The collision bumps these atoms \(X\) in an “excited” electronic state \(X^*\) at the expense of the kinetic energy of the electron $$ X + \rm{e} \to X^* + \text{e} \nonumber $$ The \(X^*\) relaxes to the ground state by emitting a photon, and produces light $$ X^* \to X^* + \text{h} f \nonumber $$

Color

The color frequency \(f\) follows from quantum mechanics, and depends on whether the nitrogen molecules in the atmosphere or oxygen molecules are being excited. It also depends on the height in the atmosphere where the ionization occurs.

The light is very faint, and can only be seen at night. That light is called Aurora, and here we call it Northern Lights. When the sun is very active, it can be breathtaking. The Aurora can change very fast on time scales of seconds to minutes. It is strongest near the magnetic poles.

Ring

For reasons that are not easy to understand, the maximum light comes from a ring that has a radius of about \(500\,\rm{km}\) from the magnetic pole.