**The CME captured by the Solar Dynamics and Heliospheric Observatory in February, 2013.**

Coronal mass ejections are powerful, energetic events on the Sun which can also have serious repercussions for terrestrial life, see e.g.

http://brane-space.blogspot.com/2012/06/cme-that-you-dont-want-to-see.html

How do coronal mass ejections occur? This is believed to happen via a sudden, mass transient ejection probably via a high Mach number shock wave. It has also been found that collisionless shocks form ahead of CMEs when the latter’s velocity exceeds the Alfven velocity of the ambient plasma in the corona. Recall this Alfven velocity is given by:

V

_{A }= Ö (B_{o}/ m_{o }r_{ o})*collisionless*plasma . (Recall in the limit 1/N << 1 and 1/N

**®**0 we say a plasma is “

*collisionless*” where:N = 4p n

**l**

_{o }

^{3}**/ 3 and n**

_{ D }**is the number density with l**

_{ o }**the Debye length.**

_{ D }^{16}/m

^{3}) and a kinetic temperature T =10

^{6 }K. The plasma parameter is defined:

L

*=*n**l**_{o }^{3}_{ D}
l

**=[kT**_{D,s}_{ s}ε_{o}/ 4p Z_{ s}**n**^{2}_{ s}e^{2}]^{ }^{½}
Here: Z

_{ s}= 1 so that:
l

**=**_{D,s}
[(1.38 x 10

^{-23}(10^{6 }K) (8.85 x 10^{-12}F/m)_{ }/ (4p) (10**/m**^{16}^{3}) e^{2}]^{ }^{½}
l

**= 1.9 x 10**_{D,s}^{- 4}_{ }m
L

*=*7.3 x 10^{4}_{ }
And N = 4p n

**l**_{o }^{3}**/ 3**_{ D }
N = (4p)(7.3 x 10

^{4}_{ }) / 3 » 3.1 x 10^{5}_{ }
Therefore: 1/N = 1/
(3.1 x 10

^{5})
1/N = 3.2 x 10

^{- 6}_{ }
1/N

**®**0 , hence the plasma is “collisionless” so that this meets the condition.
Then
we may use the collisionless

*Vlasov-Boltzmann equation*for a plasma description, e.g.:
v
¶ f

_{x}/ ¶ x - q(s)/m(s) [¶ j/ ¶ x · ¶ f_{s}/ ¶ v] = 0
where
(f

_{s}) is any function of the constant of the motion.
This,
along with its collisional form, has been called “

*the most important equation in plasma physics*”.[1] The associated**Poisson equation**is:**Ñ**

_{ }**E =**4p e (ò f

_{e}d

^{3}V - òf

_{i}d

^{3}V) = 0

where
the

*e, i*–subscripts denote electrons and ions, respectively and f refers to the appropriate distribution for each.**implies that a relationship exists between the plasma frequency w and the wave number k. ) What it can do is form a large amplitude fixed wave called a “soliton”. (‘Steepening’ + dispersion yields solitons, while steepening + dissipation yields shocks). These soliton solutions ensure no turbulence (due to shocks), thereby meeting the collisionless condition. To estimate M we use (e.g. Sakai and Ohsawa, 1987)[2]:**

*dispersion relation*
B

**= (2M – 1) B**_{m}_{o}
Or
in terms of the ratio of the magnetic fields:

M
= 2 [B

**/ B**_{m}**] + 1**_{o}
To
reach an M= 80 value, say which might conceivably apply for coronal shocks, one would need:

[B

**/ B**_{m}**] » 39.5**_{o}
I.e.
the ratio of the maximum magnetic field intensity to the equilibrium value is
at least 39.5.

DN/
N = DB/ B

**[1 - DB/ 2 B]**_{o }^{ - 1}
We
already saw for the equilibrium collisionless corona:

N » 3.1 x 10

^{5}_{ }
It is instructional here to determine the trigger
or break point for the transient, or CME. This should be reached when the
massive influx of dissipating particles is so great that:

DN/
N

**®****¥**
By
inspection we see that if DB = 2B :

**¥**

**=**2B/ B

**[1 – 2B/ 2 B]**

_{o }^{ - 1}

To
achieve this one would most likely need a strong

*two-stream*instability requiring (cgs units):
M
> 1 + [8p N T

**/ B**_{e}^{2}^{ }]^{ 1/3}
This
elicits the necessary magnitude for B given our values. A typical soliton –like
plot will appear:

The
relevant equation for a soliton embodying

*a dissipative term*to evolve into a shock can be written:
(-
v

**+ c**_{ o}**+ v) ¶**_{ s}**v / ¶x’**^{ }__-____m____¶__**v /**^{2 }__¶____x__+ a ¶**’**^{2}**v / ¶x**^{3 }^{3}
Where the underlined term above is the

*dissipative term*. In a collisionless plasma such as the corona under consideration, the dissipation can be attributed to Landau damping and this may be the most common form to excite a shock associated with CMEs. For all such collisionless shocks the dissipative length scale, L << l**, the mean free path.**_{ mfp }
The preceding equation can be derived using an analog to

**Newton’s 2**, viz.^{nd}law of motion
a
v” = F(v) = a d

**v/ dt**^{2}**= -dF / dv**^{2 }
Where: F
= (c

**- v**_{ s}**) v**_{ o}**/ 2 + v**^{2 }**/ 6**^{3}
Is
the

**Sagdeev Potential**.(To be continued - with problems)

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