CSI 662 / PHYS 660 Spring 2012
Introduction to Space Weather
Homework Assignment - Solution
Homework assignment #3
Assignment Date: Feb. 16, 2012
Due Date: Feb. 23, 2012
1. Plasma β.
(1) What is the definition of plasma β? You need to write down the formula, and explain
the terms.
(2) Calculate the β value for solar photosphere?
(3) Calculate the β value for solar corona?
(4) Calculate the β value for the solar wind in the near-Earth space?
(5) Discuss how the plasma β change from the photosphere, to corona, and into the
solar wind? What are the implications of the plasma β on the physical properties in
these regions?
Note: To find the characteristic values of density, temperature and magnetic field in
these regions, you are free to use any resources, e.g., books, papers and online
resources?
Answer: (1)

Pg
PB
where Pg is the plasma gas pressure (nKT) due to the thermal effect, and
PB is the plasma magnetic pressure (
B2
B2
in SI unit, and
in CGS unit).
2 0
8
Plasma β indicates the relative importance between the gas thermal pressure and the
magnetic pressure in determining the dynamics and magnetic structure of a plasma. In a
high-β regime, the thermal pressure dominates. The behavior of the plasma is mainly
determined by the thermal pressure, for instance, in the solar wind and in the convection
zone of the sun. The magnetic field just passively follows the velocity field, because of
the frozen-in effect. On the other hand, in a low-β regime, the magnetic pressure
dominates. The magnetic structure controls the flow of plasma, which are limited to only
along magnetic field lines, for instance, in the corona and in the magnetosphere.
(2) In the photosphere, in particular, in a sunspot, the parameters can be chosen as
n  1019 cm -3  10 25 m -3
T  6000 K
B  3000 G  3  10 -1 T (since 1 T  10 4 Gauss, or 1 G  10 -4 T)
Using SI or MKS unit
nkT (10 25 )  1.38  10  23  6000

 2 
 23.1
PB
B
(3  10 1 ) 2
2 0
2  (4  10 7 )
Pg
(3) In the corona, the parameters can be chosen as
n  10 9 cm -3  1015 m -3
T  10 6 K
B  100 G  1  10 -2 T

Pg
PB

nkT (1015 )  1.38  10  23  10 6

 3.47  10 -4
2
2 2
B
(1  10 )
2 0
2  (4  10 7 )
(4) in the solar wind in the near-Earth space, the parameters can be chosen as
n  1 cm -3  10 6 m -3
T  10 5 K
B  10 -5 G  10 -9 T

Pg
PB

nkT (10 6 )  1.38  10  23  10 5

 3.47
B2
(1  10 9 ) 2
2 0
2  (4  10 7 )
(5) Apparently, the plasma β undergoes a dramatic change, from a large value in the
photosphere (thanks to the large density), to an extremely small value in the corona
(thanks to the low density), and return to a large value in the solar wind (thanks to the
extremely low magnetic field). Therefore, plasma flows, driven by the thermal pressure,
dominates the magnetic behavior in the photosphere and solar wind. On the other hand,
the magnetic field structure determines the behavior of the corona, which explains why
the corona has highly distinct features from one region to the other region.
2. Plasma properties.
In a solar active region, the characteristic values are of temperature T=3 x 106 K, density
n = 1015 m-3, magnetic field B = 1000 Gauss, length scale L=104 km, velocity V = 10 km/s
(1) Calculate the thermal energy density in the active region?
(2) Calculate the magnetic energy density in the active region?
(3) Calculate the diffusivity in the active region? Then calculate the electric conductivity?
(4) Calculate the magnetic Reynolds number?
(5) Calculate the diffusive time in the active region?
Note: The magnetic diffusivity η of plasma can be calculated by the following formula:
1/ 2
c 2 e 2 me

(kT ) 3 / 2 ln 
3/ 2
3(2 )  0
1
ln   22.8  ln T  ln n
2
Answer: (1)
3
2
3
 th   1015  1.38  10 23  3  10 6  6.21  10 2 J m -3
2
 th  nkT
(2)
B2
B 
2 0
(1000  10 4 ) 2
B 
 3.98  10 3 J m -3
7
2  (4  10 )
Therefore, in the corona, the magnetic energy density is much higher
than the thermal energy density
(3)
1
ln   22.8  ln T  ln n
2
1
ln   22.8  ln( 3 10 6 )  ln( 1015 )  20.44
2
1
/
2
c 2 e 2 me

(kT ) 3 / 2 ln 
3(2 ) 3 / 2  0

(3.0 108 ) 2  (1.602 10 19 ) 2  (9.109 10 31 )1 / 2

3  (2  3.14) 3 / 2  8.854 10 12
(1.38110 23  3 10 6 ) 3 / 2  20.44
  0.404 m 2 s -1



1
 0
:the relationsh ip between diffisivit y and conductivt y
1
 0
1
4  10  0.404
7
 1.97  10 6 Siemens m -1
(4) Magnetic Reynolds Number
Rm 
UL

(10  10 3 )  (10 4  10 3 )
 2.475  1011
0.404
This numer is dimensionl ess
Rm 
(4) Magnetic Diffusion Time

L2

(10 4  10 3 ) 2
 2.475  1014 second
0.404
  7.85  10 6 yrs, or about eight millions years

Therefore, the nature dissipatio n of active region magnetic field through t he process of
classical diffusivit y is extremely large. Flare must be caused by some non - classical process