\documentclass[11pt]{article}
\usepackage{fullpage} % normal-person margins
\usepackage{amsmath} % for \boxed{}

\input units.tex
\newcommand\N{\u{N}}
\newcommand\vu{\vec u}
\newcommand\vo{\vec \Omega}
\newcommand\vr{\vec r}

% \d is already defined to something, so we have to obliterate
% it with a \def rather than a \newcommand.

\def\d{\u{d}} 

% This codes creates a 'question' environment that
% has paragraphs that are vertically separated rather
% than indented, and has all the text in emphasis mode.
% If I really wanted to go nuts, I could create a register
% and add an argument and automatically generate the
% question title ...

\newdimen\savedparindent
\savedparindent=\parindent

\newdimen\savedparskip
\savedparskip=\parskip

\newenvironment{question}[0]{\parindent=0in
\addtolength{\parskip}{\baselineskip}
\begin{rm}}{\parindent=\savedparindent
\parskip=\savedparskip
\end{rm}
\bigskip}

% End LaTeX magic.

\begin{document}


\begin{center}
{\Large{\bf Astrophysical Fluid Dynamics -- Problem Set 6}}

%{\large{\bf Due in class March 7}}

\end{center}

\vspace{0.1in}
\noindent
Readings: Shu pages 73--81; pages 24--31 of Course Reader, photocopied from Frank, King, and Raine

\begin{question}
{\large{\bf Problem 1.}} Steady as She Blows

Consider spherically symmetric in/out-flow near an object
of mass $M$. Assume the flow is isothermal at sound speed $c_s$.
Define the Mach number ${ \mathcal M} = u / c_s$ for radial velocity $u$.

(a) Show that

\begin{equation}
\left( \frac{1}{\mathcal{M}} - \mathcal{M} \right) \frac{\partial \mathcal{M}}{\partial r} = \frac{GM}{c_s^2r^2} - \frac{2}{r}
\end{equation}

where $r$ measures radial distance from the object and $G$ is the
gravitational constant.
\end{question}

\begin{question}
(b) Show that

\begin{equation}
\frac{ {\mathcal M}^2}{2} - \ln {\mathcal M} = 2\frac{r_s}{r} + 2 \ln \frac{r}{r_s} + C
\end{equation}

where $r_s \equiv GM/2c_s^2$ gives the radius of the ``sonic point''
and $C$ is a constant of integration.
\end{question}

%This follows from integration of (\ref{}).

\begin{question}
Plot several solutions of $\mathcal{M}$ versus $r$, drawing from all
mathematically possible families.
Label the solutions by their respective $C$ values. Identify those subset
of solutions that are physically realizable.
\end{question}

%The one and only one trans-sonic wind has $C = 3/2$. Please
%attach a plot (e.g., a free-standing .jpg or .eps, or embedded .eps).

\begin{question}
{\large{\bf Problem 2.}}
The Solar Wind

At the base of the solar corona at $r \approx R_{\odot}$,
outward flow speeds are still subsonic. Spectral line observations
of the coronal base
reveal temperatures of about $T \approx 2 \times 10^6 \K$, a fully
ionized plasma, and a number density of protons
of about $10^8 \cm^{-3}$.

(a) Estimate $\dot{M}$ for the solar wind, assuming the wind
is isothermal. Express in $M_{\odot} \yr^{-1}$.
\end{question}

\begin{question}
(b) Estimate the number density of solar wind protons flying past
the Earth. Express in $\cm^{-3}$.
\end{question}

\newpage

\begin{question}
{\large{\bf Problem 3.}}
Jet Collimation by Pressure Confinement from an External Medium

Adapted from Pringle and King, problem 3.4. The problem starts by considering
a de Laval nozzle, and ends by considering how astrophysical {\it jets} might behave
like such nozzles---with the tube of given cross-sectional area replaced
by an external medium of a given pressure profile.

Consider a $\gamma$-law gas moving inviscidly through a de Laval nozzle.
The gas initially is very subsonic, but eventually moves supersonically.
Take the sound speed of the gas initially to be $c_{s0}$.

(a) Solve for the gas velocity $u$ and the sound speed $c_s$ at the pinched throat of the nozzle, in terms of $c_{s0}$ and $\gamma$.

(b) Solve for the gas velocity $u$ at infinity, in terms of $c_{s0}$ and $\gamma$.

(c) Solve for the pressure $P$ at the pinched throat, in terms of the initial
$P_0$ and $\gamma$. Does the pressure decrease monotonically everywhere along the tube? Increase monotonically? Neither? Justify your answer.

(d) Now consider an astrophysical jet. How are jets collimated? One proposed mechanism is pressure confinement by an external, static medium through which the jet travels. For example, the jet could be a stellar jet (powered by accretion from a disk), and the external medium could be the remnant gas cloud from which the star formed. Or the jet could be an AGN (supermassive black hole) jet (powered by accretion from a disk), and the external medium could be gas in the core of the galaxy.

There are two ideas here. The first is that the external, static medium has a thermal pressure that decreases monotonically in the jet direction. The second is that the jet and the external medium reach pressure equilibrium. That is, the thermal pressure at any point within the jet equals the local thermal pressure of the external surrounding medium. Once in pressure equilibrium, the jet stops expanding in the transverse direction (some papers refer to the jet as being ``cocooned.'')
These two ingredients, coupled with the usual assumption
of steady inviscid flow, imply a very close analogy between de Laval nozzles
and jets. Instead of being given the tube's
cross-sectional area $A(x)$ and solving
for (say) $P(x)$, we are instead given $P(x)$ and solve instead for
the jet cross-sectional area $A(x)$.

Consider a steady jet composed of a gas having $\gamma=7/5$,
travelling through an external medium
whose pressure scales with distance as $x^{-2}$
(like that of a singular isothermal sphere).
Neglect gravity.
Very far along the jet, how does the {\it opening angle} of the jet (transverse
length divided by longitudinal length $x$) scale with $x$?
Take the jet to have a circular cross-section.
Justify whatever assumptions you make.

(The idea of jet as de Laval nozzle was introduced by Blandford and Rees 1974,
in the context of relativistic AGN jets. There also exists a
Bernoulli constant for relativistic flows, but it looks different
from our non-relativistic form $u^2/2 + \int dP/\rho + \Phi$.
After thinking about this problem, you should find
at least the first few pages of their paper to be fairly readable; you
will also appreciate their Figure 1.)

\end{question}

\begin{question}
{\large{\bf OPTIONAL Problem 4.}}
Isolated Accreting Compact Objects

(a) Estimate the accretion rate $\dot{M}$ of an isolated neutron star
of mass $M = M_{\odot}$
travelling through the interstellar medium (ISM) at a velocity
$v = 200 \km \s^{-1}$. Take the (warm component of the)
ISM to have density $n = 1 \cm^{-3}$
and temperature $T = 10^4 \K$. Express in $M_{\odot} \yr^{-1}$.
\end{question}

\begin{question}
(b) Estimate the accretion rate $\dot{M}$, in $M_{\odot} \yr^{-1}$,
near the Bondi radius\footnote{
We need to add the qualifier ``near the Bondi radius'' because only a small
fraction of the mass that flows past the Bondi radius actually makes it onto
the black hole (past the Schwarzchild radius), given how the observed
luminosity of Sgr A* is so low. Calculating this small
fraction from first principles remains an unsolved problem.}
of Sgr A*, the black hole
at the center of our Galaxy having mass $M = 3 \times 10^6 M_{\odot}$.
Chandra observations reveal that within 0.1 pc of Sgr A*, fully
ionized plasma has density $n \approx 30 \cm^{-3}$ and
$kT_e \approx 1.3$ keV (Baganoff et al.~2003).\footnote{This plasma
seems to originate from winds from nearby O stars.} Neglect any relative
bulk velocity between the plasma and Sgr A*.
\end{question}

\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?
