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\begin{center}
{\Large{\bf Astrophysical Fluid Dynamics -- Problem Set 10}}

%{\large{\bf Due in class Wednesday Apr 18}}
\end{center}

\bigskip
\noindent
Readings: Course Reader: selections from Binney \& Tremaine. Also
 Shu pages 98--101

\bigskip

\begin{question}
{\large{\bf Problem 1.}} Inner Disk Edge

A gas disk orbits an object of mass $M$. Disk self-gravity
is negligible. The inner edge of the disk is located at radius
$a$. Gas there has a sound speed $c_s$ and an orbital
angular frequency $\Omega$.

The inner edge is not perfectly sharp. At the inner edge, the gas pressure
decreases (going inward) over a radial
length scale $\delta$. Make an order-of-magnitude estimate for
the smallest $\delta$ can be before gas at the disk edge becomes
Rayleigh-unstable.

\end{question}

\begin{question}
{\large{\bf Problem 2.}} Clumping Sound Waves

Shu Problem Set 3, problem 1abc.

NB: The previous generation of 202 students complained that they did not get
much out of Shu's problem sets because they were excessively technical.
I have considered this complaint, but still feel that certain problems
are worthwhile. This one is worthwhile because it offers a situation
to study gravitational instability without having to pull the Jeans swindle,
and it also provides practice in deriving dispersion relations.

\end{question}

\begin{question}
{\large{\bf Problem 3.}} Magnetospheres

(a) Estimate the radius $r_A$ of the Earth's magnetosphere, in units
of the Earth's radius.

(b) Estimate the radius $r_A$ of an accreting T Tauri star's magnetosphere,
assuming $\delta \sim r$ (see lecture notes).

Use  $M_{\ast} \sim 1 M_{\odot}$,
$R_{\ast} \sim 3 R_{\odot}$, a dipole field of surface
strength $B_{\ast} \sim 1$ kG,
and the median measured accretion rate of
$\dot{M} \sim 10^{-8} M_{\odot} \yr^{-1}$.

Hot Jupiters are speculated to be deposited at $r_J \approx 0.6 r_A$
(at this location, the hot Jupiter is in 2:1 resonance with the disk
edge at $r_A$). Compare your estimate for $r_J$ with the observed radius
for a hot Jupiter orbit (0.05 AU).\footnote{Given the crudeness of the
theory, this comparison is not expected to yield
better than order-of-magnitude agreement.}

\end{question}


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