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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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