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\begin{document}


\begin{center}
{\Large{\bf Astrophysical Fluid Dynamics -- Problem Set 5}}

%{\large{\bf Due in class Wednesday Feb 28}}

\end{center}

\vspace{0.1in}
\ni
Readings: Movies: Drag III, IV; skim pages 11 through 21 of the Course Reader

\ni
Note: Do not hesitate to make simplifying assumptions and rough estimates
for the problems below. But please justify your assumptions whenever
possible.

\vspace{0.1in}
\ni
{\large{\bf Problem 1.}} LOST

\ni Desmond turns the key
and your plane splits in half at an altitude of
32,000 feet above the Pacific.
You are not wearing your seatbelt and go flying out.

\ni (a) About how long does it take for you to hit the water's surface?
Credit awarded for detailed physical arguments.

\ni (b) About how deep do you penetrate the water?
Credit awarded for detailed physical arguments.

\vspace{0.1in}
\ni
{\large{\bf Problem 2.}}
Particle Drift in Protoplanetary Disks

\ni Planets form in disks of gas and dust surrounding young stars.

\ni Consider the young Sun, encircled by a disk of gas
of surface density (mass per unit face-on area)
$\Sigma \approx 10^3 \gm \cm^{-2}$ in the vicinity
of 1 AU. The temperature of disk gas is $T \approx T_0 (a/\AU)^{-q}$,
where $T_0 \approx 200 \K$,
$a$ is the disk radius, and $q \approx 1/2$.
Call the corresponding sound speed $c_s \approx \sqrt{kT/\mu}$,
where $k$ is Boltzmann's constant and $\mu$ is the mean molecular weight.

\ni (a) Consider a dust particle of size $s = 1 \mu$m
and internal density $\rho_p \approx 1 \gm \cm^{-3}$
located one scale height above the disk midplane, $z = h$ (see problem set 1).
Estimate the time the particle takes to settle {\it vertically}
to the midplane. Provide both a symbolic answer
and a numerical answer in years.

\ni (b) Does disk gas rotate at sub-Keplerian or super-Keplerian
speeds? Estimate the difference $\Delta v$
between the gas velocity
and the circular Keplerian speed $v_K \equiv \Omega a \equiv
 \sqrt{GM_{\odot}/a}$, at the midplane $z = 0$ at $a = 1$ AU.
Express $\Delta v$ symbolically
in terms of $c_s$, $\Omega$, $a$,
and whatever dimensionless numbers you need. Also provide
a numerical estimate.

\ni (c) Particles (rocks) are considered well-entrained
in disk gas---i.e., they rotate with nearly the same
velocity as disk gas---if their {\it momentum stopping time}

\begin{equation}
t_{\rm stop} \equiv \frac{mv_{\rm rel}}{F_D}
\end{equation}

\ni is short compared to the dynamical time $t_{\rm dyn} \sim 1/\Omega$
[this latter time is
the time for disk gas to {\it turn} / change its velocity (direction)
by $\sim$1 radian].
Here $m$ is the particle mass,
$v_{\rm rel}$ is the relative velocity between a particle
and disk gas, and $F_D$ is the drag force on the particle.

\ni Estimate the critical size $s_{\rm crit}$ for which
$t_{\rm stop} \sim t_{\rm dyn}$ at the midplane
$z=0$ at $a = 1$ AU. A numerical answer suffices.
Are bodies larger or smaller than $s_{\rm crit}$
well-entrained?

\ni (d) About how long does it take a critically sized particle
to drift {\it radially} inwards due to drag? The particle
starts at $z=0$ at $a = 1$ AU, and we want to know how
long it takes to reach, say, $a = 0.5$ AU. A numerical answer suffices.

\vspace{0.1in}
\ni
{\large{\bf Problem 3.}}
Adiabatic vs.~Isothermal Sound Waves in the Classroom

\ni Do sound waves in the classroom behave adiabatically
or isothermally? Present a quantitative, order-of-magnitude
proof, considering cooling by conduction alone. Recall that
conduction involves the random walk of molecules.
No more than a few lines of rough but still quantitative reasoning are required
to solve this problem.

\ni Audible sound waves have frequencies $\nu$ between
20 and 20000 Hz.

\end{document}

\vspace{0.1in}
\ni
{\large{\bf Problem 2.}}Torricelli's Tank, Part II

Torricelli's tank and the neglect of viscosity. When is it OK?
