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Planetary Wind Problem

Problem Set Up

We have 3 conditions to initialize: initial ambient medium (not important); the conditions in the base of the Hot Jupter (HJ) atmosphere; the conditions in the stellar wind at the orbital radius (a) of the HJ.

Initial Ambient Medium

The ambient medium is just a place holder to allow the planetary wind to expand into.

\rho_{a0} = m_h \mu n_{0a}

T_{a0}

c = [\frac{(k_mT_{a0}}{(\mu m_h)} ]^{1/2}

HJ Ex0base

\rho_{p0} = m_h \mu n_{pa}

T_{p0}

Stellar Wind at orbital radius a

\rho_{p0} = m_h \mu n_{pa} =\frac{\dot{M_s}}{(4\pi V_s(a) a^2)}

T_{s0}

V_{s0}

Fourier Transforms

Fourier Transforms

The basic equation for the fourier transform.

\Xi[f] = F(k) = \int^{+\infty}_{-\infty} f(x) e^{(-i 2 \pi k x)} dx

with the inverse

\Xi[F]^{-1} = f(x) = \int^{+\infty}_{-\infty} F(k)) e^{(i 2 \pi k x)} dk

And of course all the transform really means is this sum of intergrals.

F(k) = \int^{+\infty}_{-\infty} f(x) cos(2\pi kx) dx - i\int^{+\infty}_{-\infty} f(x) sin(2\pi kx) dx

Here we want to solve the signup function which is runs from -1 to 1 with a step at x=a

f(x) = sgn(x-a)

Use the derivative property of \Xi[f]

\Xi[f ] = \frac{1}{2\pi i k} \Xi[ \frac{df}{dx} ]

Since derivative of f is the direct delta function

\frac{df}{dx} = \delta(x-a)

and

\Xi[ \delta(x-a) ] = e^{-i2\pi k a}

Thus we have

\Xi[f ] = F(k) = \frac{1}{2\pi i k} e^{-i2\pi k a}

The next step is to define the Energy Spectral Density (ESD)

We use Parcivals Thm which tell us "energy" under the curve f(x)

\int^{+\infty}_{-\infty} | f(x)|^2 dx = \int^{+\infty}_{-\infty} | F(k)|^2 dk

Thus the ESD which we write as E(k) is

E(k) = | F(k)|^2

which for the sgn function is

E(k) = (\frac{1}{2\pi i k} e^{-i2\pi k a})^* (\frac{1}{2\pi i k} e^{-i2\pi k a})

E(k) = \frac{1}{4\pi^2 k^2}

So for a step function

E(k) \propto k^{-2}

which is exactly what you get in compressible turbulence because every shock is a sgn function.

Description of Problem

Results

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Results

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