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\begin{center}
{\Large{\bf Astrophysical Fluid Dynamics -- Problem Set 11}}

%{\large{\bf Due in class Wednesday Apr 25}}
\end{center}

\bigskip
\noindent
Readings: Course Reader: (a) selections from Binney \& Tremaine; (b) passages on accretion disks from Frank, King, and Raine; (c) however much of Gammie (2001) that you like (the introduction is very readable). Also skim Shu 82--88, 90--92. 

\bigskip

\begin{question}
{\large{\bf Problem 1.}} Toomre $Q$

A gas disk orbits an object of mass $M$. The disk is in vertical hydrostatic
equilibrium.

(a) Show that $Q < 1$ at radius $r$ corresponds to the condition
$\rho > \rho_{\rm crit} \sim M / r^3$,
where $\rho$ is the gas density (mass per unit
volume) at the disk midplane. The condition you have derived
is an easy way to remember the Toomre instability condition.
The critical density
$\rho_{\rm crit}$ is sometimes called the Roche density (and is just
the mass of the central object divided into the sphere just enclosing
your position).

(b) Show that for $Q \sim 1$, the marginally unstable
mode has a radial wavelength comparable to the gas vertical
scale height $h$.

(c) Papers will occasionally state that $Q < 1$ corresponds
to $M_{\rm disk} / M > h/r$. Explain why this statement
makes (rough) sense.

\end{question}

\begin{question}
{\large{\bf Problem 2.}} Thank You Sir May I Have Another

Consider spiral disturbances
of the form $\exp i (k_rr - \omega t + m\phi)$ in a disk.
Note that $k_r < 0$ for leading waves and $k_r > 0$ for trailing waves.

This problem explores basic kinematics of waves in disks. By solving
part (c), you will see why the WKB dispersion relation that we presented
in class cannot be used to study the stability of non-axisymmetric
($m\neq 0$) waves, because that dispersion relation assumes that waves
always remain tightly wound and that $k$ is constant.

(a) Show that $\cot i = |k_rr/m|$, where $i$ is the pitch
angle between the tangent to a spiral arm and the local circle
$r =$ constant.

This formula is valid for any value of $i$. In the special
case that waves are tightly wound (permitting a WKB analysis),
$i \ll 1$ and $|kr| \gg 1$.

(b) Consider a parcel of gas on a purely circular orbit
with angular frequency $\Omega$. With what frequency
do the spiral arms wash over this parcel of gas, for $m \neq 0$?

This frequency is sometimes
called the Doppler-shifted forcing frequency in the disk
literature (``Doppler-shifted'' to remind you that everything
is in relative circular motion, and ``forcing'' to remind you
that a given gas parcel is being kicked by the spiral
disturbance at this frequency.)

(c) At the so-called ``corotation circle,'' the parcel does not get kicked
by the wave, because it is moving at just the right angular frequency
$\Omega$ to keep up with the wave (the parcel is ``corotating'' with the wave).
Every wave has its own corotation circle.

Consider a non-axisymmetric wave ($m \neq 0$)
at whose corotation circle $r = R_0$ and $\Omega = \Omega_0$.
Go into the frame that corotates with the wave.
In a differentially rotating
disk, the wavefronts get turned around by the background shear. In other
words, leading waves become trailing waves at the corotation circle.

Derive expressions for $k_r(t)$ and $k_\phi(t) = 2\pi / \lambda_\phi(t)$
(the wavenumber in the
azimuthal direction), in terms
of the initial value $k_r(0)$, $m$, $R_0$, $\Omega_0$, and $t$.
Restrict the analysis
to Kepler shear ($\Omega \propto r^{-3/2}$) and to small
radial displacements about corotation ($\Delta r \ll r$).
The latter restriction constitutes the so-called ``shearing sheet''
approximation of disk dynamics. In the shearing sheet, the flow
appears Cartesian (like the viscous
flow between two parallel plates that you explored
in a previous problem). You should find the wavefronts remaining straight
while they turn from being leading to trailing. Do waves that are initially
leading and tightly wound remain tightly wound?

\end{question}


\begin{question}
{\large{\bf Problem 3.}} Alpha Disks

Consider a disk having a dimensionless
viscosity $\alpha$. The disk accretes at a steady rate $\dot{M}$.

The disk cools radiatively.
Neglect the difference between the effective temperature
of the disk $T_{\rm eff}$ (which is nothing more than a convenient
way of stating what the emitted flux is) and the actual gas kinetic
temperature $T$. Take the gas to have sound speed
$c_s$ and angular frequency $\Omega$, both of which
vary with disk radius $r$.

(a) Find how $h/r$ scales with $r$, where $h$ is the disk vertical
scale height. Sketch how the disk looks.

(b) Find how $\sigma$ scales with $r$.

(c) Find how the disk midplane density $\rho$ scales with $r$.

(d) Find an approximate expression for
how long it takes a pressure disturbance to equilibrate away.
Call this time $t_z$, and express it as simply as possible.

(e) Find an approximate expression for
how long it takes a temperature disturbance to
equilibrate away. Call this time $t_{\rm cool}$, and express
in terms of $\Omega$ and $\alpha$.

(f) Find an approximate expression for
how long it takes a mass disturbance (say, a local
bunching of material) to viscously diffuse away. Call this
time $t_{\rm visc}$, and express in terms of 
$\Omega$, $\alpha$, and $h/r$.

Arrange $t_z$, $t_{\rm cool}$
and $t_{\rm visc}$ in increasing order.

(g) Find an expression for
the critical radius $r_{\rm crit}$ beyond which Toomre's $Q < 1$.
Alpha disks are generically unstable
at large radii, leading some to surmise that the outer peripheries
of quasar accretion disks / protostellar accretion disks
are fertile breeding grounds for starbursts / binary companion stars or
brown dwarfs.

\end{question}


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