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Unformatted text preview: 1 Chapman Model Please refer to http://farside.ph.utexas.edu/teaching/plasma/lectures/node65.html, on which these notes are based. I use slightly different notation and units relative to this web site. In particular note that the URL expresses temperature in units of electron volts instead of degrees Kelvin. In 1957, Sidney Chapman reasoned that the heat (the plasma) generated by the sun would diffuse out into the solar system, and so it should be possible to predict the temperature and pressure profiles by using the diffusion equation. Thus, he attempted to solve ∂ ∂t T = ∇ · ( κ ∇ T ) (1) where T is the temperature of the diffusing plasma (assumed to be the same for electrons and protons), and κ is the thermal conductivity , in units of power per unit length per unit temperature ( W/(m o K) in MKS units). For the thermal conductivity, Chapman used the classical model (i.e., conduction owing to Coulomb collisions) such that κ = κ T 5 / 2 (2) where κ is a constant. If we assume that at the base of the corona ( ’ . 7 × 10 8 m = R S ), the temperature is approximately 2 × 10 6 o K, and the number density n is ’ 10 14 m- 3 , then κ ’ 7 × 10- 14 . This results in a very high conductivity, about 20 times that of copper at room temperature. We will assume both steady state ∂T/∂t = 0 and spherical symmetry ( T depends only on radius). It follows that we can write (1) as 1 r 2 d dr r 2 T 5 / 2 dT dr ¶ = 0 (3) Note that because of the assumption that the system is in steady state, (3) is independent of κ . The solution to this equation can be written as T ( r ) = T ( R S ) R S r ¶ 2 / 7 (4) where we have assumed that T → 0 as r → ∞ . To solve for the pressure p we can use the equation for hydrostatic equilibrium dp dr =- ρ GM S r 2 =- ‡ pm p 2 KT · GM S r 2 =- pm p 2 KT ( R S ) ¶ r R S ¶ 2 / 7 GM S r 2 (5) 1 where we have used the equation of state p = 2 ρKT/m p . The factor of two results from the assumption that the electron and ion temperatures are equal. The solution to (5) is p ( r ) = p ( R S )exp ( C " R S r ¶ 5 / 7- 1 #) (6) where C ≡ 7 20 GM S m p K T ( R S ) R S . This solution is troublesome in two regards. First, it predicts p ( ∞ ) = p ( R S ) e- C ; that is, a finite pressure as r → ∞ that is much larger than the (experimentally determined) interstellar pressure. Second, the predicted density takes the form ρ ( r ) = ρ ( R S ) e C (( R S /r ) 5 / 7- 1) r R S ¶ 2 / 7 Thus, the Chapman model predicts a density profile that is slowly increasing as a function of r . For these reasons the model was deemed unphysical by Eugene Parker, who then argued that a dynamic theory was required. 2 The Solar Wind: Parker Model Please refer to http://farside.ph.utexas.edu/teaching/plasma/lectures/node66.html, on which these notes are based....
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This note was uploaded on 09/10/2010 for the course ECE 107 taught by Professor Fullterton during the Spring '07 term at UCSD.
- Spring '07