From the marvelous sight of the aurora to the intricacies of emission nebulae to coronal mass ejections and solar flares, space plasma physics provides an underlying thread. It is arguable that no one who embarks on an astrophysics career of any heft can ignore one or more aspects of the discipline, whether that be the behavior of charged particles in orbits, or the effects of nonlinear plasma waves.

In a new series of articles I'd like to introduce blog readers to this amazing field, and also offer a few problems to do at the end of each section. We will start with particle orbit theory and then finally work our way to linear and nonlinear plasma waves and instabilities.

*1.Plasma Orbit Theory – Particle dynamics:*
Here we consider a charged
particle (say of charge q) in a uniform and constant magnetic field (

m (dv/dt) = q(

The motion here is such that

v ⊥ F, implying circular motion. Thus:

dv/dt = q/ m [

Meanwhile, (v⊥ )

Where v⊥ is a positive, constant velocity denoting the speed in the plane perpendicular to the magnetic induction,

The quantity r above is none other than

r = m/ q [v⊥ / B] = v

for which one can have either the electron, or ion gyro-frequency (Ω) . These equations explain the physical basis for the origin of a preponderance of radio waves (i.e.

**B**).The governing equation of motion with F the Lorentz force, is:m (dv/dt) = q(

**v X B**) = FThe motion here is such that

*will always be***v***perpendicular*to the force acting on the particle sov ⊥ F, implying circular motion. Thus:

dv/dt = q/ m [

**v X B**]Meanwhile, (v⊥ )

^{2 }/ r = q/ m [**v****⊥****B**]Where v⊥ is a positive, constant velocity denoting the speed in the plane perpendicular to the magnetic induction,

**B**.The quantity r above is none other than

*the gyro-radius*. Solving for it one finds:r = m/ q [v⊥ / B] = v

_{⊥}/ (qB/m) =**v**_{⊥}**/**Ωfor which one can have either the electron, or ion gyro-frequency (Ω) . These equations explain the physical basis for the origin of a preponderance of radio waves (i.e.

*gyro-magnetic emission*) such as from the Sun, and other cosmic objects (e.g. quasars, pulsars).**v**=

**v**

_{||}**+**

_{ }**v**

_{⊥}

where the first term denotes the velocity

*along B*which stays constant so that d(

**v**

_{||}**)/ dt = 0.**

_{ }**v**

_{⊥}

**/ r = q/ m [**

^{2}**v**

_{⊥}·

*]*

**B**(setting the centripetal force = to the magnetic force producing it)

The quantity r is none other than the

*. Solving for it one finds:*

**gyro-radius**or Larmor radiusr = m/ q [

**v**⊥ / B] =

**v**⊥/ (qB/m) =

**v**⊥/ Ω

where the denominator denotes

*the gyro-frequency*.

Ω

**= qB/ m**

So that the

**ion gyrofrequency**will be:

*Ω*

_{i}

**= qB/ m**

_{i}

And the

**electron gyrofrequency**is:*Ω*

_{e}

**= qB/ m**

_{e}

**v**⊥ = Ö (2 m

_{m}

**B**/ m)

_{m}is the magnetic moment.

Thus, the proton gyrates at this rate provided m

_{m}is

*a constant of the motion*: viz.

m

_{m }= m(**v**⊥)^{2}/ 2B = const .
The

Bear in mind the gyration energy:

E = m

*is: T = 2 p /***gyro-period***Ω*Bear in mind the gyration energy:

E = m

_{m}B = m/2 (*)***E/B****,**^{2}**radio waves**(i.e. via

*gyro-magnetic emission*).

Note that the position of a particle at any time t can always be specified for a given coordinate system, if one knows the Larmor radius r and the gyrofrequency, Ω:

**x^**- sin Ω t

**y^**)

*Geometry for gyrofrequency in terms of position r, y*
The basics presented above ignore the fact that no general
solution exists to the equations of motion for a charged particle moving under
the influence of the Lorentz force in a dipole B-field. What happens is that an
approximation is needed, called “

*the guiding center approximation*”. This assumes the magnetic field changes are small over a gyroperiod, and gyroradius.**guiding center**, such that:

(a ) x – x

_{o}= - i v⊥ exp (i Ω t)/ W = r sin (Ω t)

and

(b) y – y

_{o}= ± v⊥ exp (i Ω t)/ W = r cos (Ω t)

The preceding equation pair (a, b) describes a circular orbit around the guiding center (x

_{o}, y

_{o}), with the direction of gyration always such that the magnetic field generated by the charged particle is opposite to the externally imposed field. (Plasma particle then tend to reduce the magnetic field and we say plasmas are "diamagnetic")

In a "crossed" E-B field (e.g.

**E X B**), for example, we would have:

The key point is that the guiding center (x

_{o}, y

_{o}) is fixed. while r is the "Larmor radius".

In the GC approximation, particle motion displays three components: 1) gyration about a field line (given by the gyrofrequency, or cyclotron frequency); 2) reflection between two mirror points (embodied by the “bounce period”) and 3) a gradual longitudinal drift, denoted by the (

**E X B**) drift. In this instance we allow for a slight inhomogeneity in

**E**- start by expanding E about the guiding center (Taylor series expansion). Without getting into the details - which the mathematically inclined reader can check for himself, viz.

**E**=

**E**(

*r*_{0 }) + [(

**r**-

**r**

_{0})

**·**

**Ñ**]

**E**

**+ ½**[(

**r**-

**r**

_{0})

**·**

**Ñ**]

^{2}

**E**

**+ ……**

We arrive at:

**d[v]/dt = q([E] + [v] X B)**

**Or:**q

**[E] + v**

_{d}

_{ }

**X B**= 0

**So that:**

**v**

_{d}= [E] X B/ B^{2}

**=**(1 +

**r**

_{¼}^{2 }

**v**⊥

^{2 })

**E**

_{o}

_{ }

**X B/ B**

^{2}

Where

With such an electric field present we will find the motion to be the sum of two motions: 1) the circular Larmor gyration and 2) a drift of the guiding center. The equation of motion will then be found to be:

m (dv/dt) = q(

**E**_{o}is the E-field at the guiding center.With such an electric field present we will find the motion to be the sum of two motions: 1) the circular Larmor gyration and 2) a drift of the guiding center. The equation of motion will then be found to be:

m (dv/dt) = q(

**E**+**v X B**)__Selected Problems__:

1) Find the ion and electron gyrofrequencies for an ion and electron in solar plasma with a magnetic induction (field strength) of

**B**= 0.0001 T.
2) If the perpendicular velocity component (

**v**⊥) is 10^{5 }m/s*for the electron*, find its Larmor radius and its gyro-period.
3) Thence or otherwise obtain the gyration energy in eV. (1.6 x 10

^{-19}J = 1 eV)
4) Find the guiding center positions for the electron referenced above (previous problems) if t = T/4.

5) What other critical information do you need to obtain the E-field at the guiding center?

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