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Take Home #1 - Solar Wind

Take Home #1 - Solar Wind - Take Home Final 1 The Parker...

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Unformatted text preview: Take Home Final 1. The Parker Solar Wind Calculation (50poiuts) The objective of this problem is to trace how Parker may have arrived at the conclusion that a solar wind must exist from a consideration of hydrostatic equilibrium outside of the solar corona. The instruction Box it is to remind you to put a box around an important result. We will use the unmagnetized ideal MHD equations assuming that the plasma is isothermal, ie, p = nTg, where To = constant = solar corona temperature ~ lOOeV. In steady state, these equations are: . . A mama—a ~= ~— '7; $71 ~‘- 4/”er r M :- fwimmefi 1/?- MV: sic/arm”! 5.4742»? == 0 .. “.3 ~——> __ axes-i mm a - O The geometry is K rim Till") Use spherical coordinates. We seek a spherically symmetric solution for n(r). Assume that a solar wind, if any, is radial only, ie, 11 = u(r)rA. M is the solar mass, R is the solar radius. We will seek a solution for u(r) for r > R., ie, starting from the solar corona all the way past 1 AU. Proceed as follows: 1. As a first attempt, assume that u(r) = 0 (ie, as might be reasonable, look for a static solution only). Use the above equations to find a solution for n(r) if n(R) : no. Use the notation D = (ll2)Gl\/l/002, where 002 = TolMp, Mp is the proton mass. Box it. Calculate D and make a schematic sketch of n(r) vs I including r =R, D, and 1 AU . Note that R << D << 1 AU. Find the n(r>>D, ie, r ~ 1 AU). Box it. 2. From the above result, it was concluded that this solution cannot be correct since the calculated density at 1 AU as found above is bigger than the actual density observed at 1 AU. Parker therefore was motivated to try u(r) nonzero. Thus, assume that there is a nonzero outflux of mass at r=R, ifi n(R)u(R)411:R2 = F0 = constant. Using this in the above equations, find equations for n(r) and u(r). Then, manipulate the resulting system to find a I” order nonlinear ODE satisfied by V(r) = “(fl/co. Write this ODE in the dimensionless variable X = r/D. Box it. . Solve the ODE to find a nonlinear transcendental expression that relates V(x) to x. Leave the integration constant undetermined. Box it. This expression cannot be solved for V(x) in closed form. We will use some approximation techniques as below to get an idea. . Suppose a solution exists which is always supersonic, ie, V(x) >> 1 for all x. If so, what term can be discarded? (Use the usual rule for discarding terms). Do so and thus find V(x). Box it. Make a sketch of V(x). Observe that V(x) has a minimum and, as long as this minimum is supersonic, the discarded term is self—consistently small. State why this solution is unacceptable as a solution for a solar wind emanating from the solar surface. . Suppose a solution exists which is always subsonic, ie, V(x) << 1. If so, what term can be discarded? (Use the usual rule for discarding terms). Do so and thus find V(x). Box it. Make a sketch of V(x). Observe that V(x) has a maximum and, as long as this maximum is subsonic, the discarded term is self-consistently small. For this solution, what is the leading behavior of V(x) as x >> 1. In that case, what is n(r) for r ~ lAU? Thus, observe that this solution is unacceptable on the same grounds as the static solution found in 1 above. . Rewrite the ODE found in 2 above as an expression for the slope dV/dx. For what value of x is there an extremum in V(x)? Box it. Let this point be xM. Point out why such an extremum will always exist unless V(xM) = 1, ie, if the flow is exactly sonic at the extremum point. It is possible therefore that if V(xM) = 1, then there might exist a solution which is always monotonic, either always decreasing or always increasing. . Let us test this. First, find the integration constant uncovered in 3 above if it is true that V(XM)=1 when x=xM. . Next, perform an expansion of your ODE around the point {x=xM, V(xM) = l} as follows. Let V(x) = 1 + V(s), and x = XM+ s, with v << 1 and 3 << xM. Expand your solution for V(x) above (with the integration constant of 7) about this point. Keep corrections upto the significant order needed so as to find a relation v(s) which is nontrivial. Thus show that monotonic solutions V(S) exist which go thru the point x=xM with V(XM) = 1. Box it. By reasonable extrapolation, convince yourself that a critical solution can exist which is monotonically increasing over the entire domain and transits from sub to supersonic at r = DxM. This is the solar wind solution. . Find the behavior of n(r >> D) for this solution, using the behavior of u(r>>D). Box it. Notice now that the density at ~ lAU is smaller than the density obtained in the static solution of #1 above, thus overcoming the shortcoming of the static solution. ...
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