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\begin{document}

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\ni {\bf (PHY/AST 564; Blackman; Fall 2003) Problem Set 1}

\bigskip

\ni {\bf 1.}  

a) Explain why the plasma frequency (derived in class)
does not depend on temperature, and why the ion-electron equilibration
time does. You may consult Spitzer (Phys of Fully Ionized Gases (on reserve)
5.1-5.3).

b) Explain why charge neutrality can hold even over time scales  $<<$ ion-electron equilibration times.   How does this apply to the Galactic center?

c) Order the following equilibration times from shortest to longest
and explain physically the ordering:  $t_{pe}, t_{ee}, t_{ie}$.


\ni {\bf 2.} 

a) Derive the speed of sound for a monatomic adiabatic gas.
Hint: Linearize the fluid equations for a stationary unperturbed
state such that $P=P_0 + P'$, $\rho= \rho_0 + \rho'$, ${\bf v}={\bf v'}$,
where the 0 is the unperturbed steady state, and the perturbations
are time dependent.  Obtain a wave equation for $\rho'$ to determine
the sound speed.  

b) Obtain a numerical value for the Galactic center X-ray gas.

c) Derive the condition for justification to ignore pressure gradients
in a gas, and explain physically.


\ni {\bf 3.} Assume that the radiated luminosity from accretion onto
a black hole is $L= {0.1\dot M} c^2$, where $\dot M$ is the accretion
rate onto the hole.  Consider the Galactic center, and assume
the lower value $10^{-5}{M_\odot}$/yr on the accretion rate given in class 
based on stellar winds at 1pc from the Galactic center.
Given only this information, can you suggest a phenomenological possibility 
that could account for both the X-ray cigar and the low bolometric
luminosity from Sgr A?  Do one calculation to justify your suggestion.


%\ni {\bf 4.} 
%If the Galactic center were incurring 
%Bondi accretion, calculate the sonic point.

\vfill 
\eject

\ni {\bf (PHY/AST 564; Blackman; Fall 2003 Problem Set 2}

\medskip

\ni {\bf 1.}  For Massive X-ray Binaries with O,B type companion stars, 
accretion onto the neutron star usually occurs via graviational capture from a supersonic stellar wind.

a) Calculate the ratio 
${\dot M}_{ac}/{\dot M}_{wind}$, the accretion rate 
into the neturon star disk, over the spherical wind mass loss rate.
Your result should be a function of two ratios:
$M_{ns}/M_{comp}$, the ratio of the binary component masses, and 
$v^2_{rel}/v^2_w$, the ratio of the velocity (squared) 
of NS and the wind over the wind velocity (squared).

\smallskip

Hint: Think about what inequality must be satisfied for 
material from the wind to be gravitationally trapped the
NS as opposed to escaping and recall what we did for Galactic Center.

\smallskip

b)For wind velocity of 1000km/s, $M_{comp}=17 M_\odot$, $M_{NS}=M_\odot$
and orbit period 30 days, caculuate the ratio explicitly.

\ni {\bf 2.}  

a) Calculate and sketch the accretion disk spectrum
(flux as function of frequency $\nu$) 
for two optically thick accretion disks:
One with $R_{out}= $50 $R_{in}$ and one with  
$R_{out} = 5000 R_{in}$.   Explain the difference
between the two spectra.  
In each case, show the approximate 
frequency dependices of the flux for the regimes
$\nu << kT(R_{out}/h$,  $\nu >>  kT(R_{in}/h$,
and $kT(R_{out}/h << \nu << kT(R_{in}/h$.

b) Estimate the peak temperatures for WD and NS disks.
Why don't we need the actual disk outer radius to make this approximate
estimate?


\ni {\bf 3.}  Suppose that a thin accretion disk 
orbits a compact object in a binary system and the conditions
are favorable to produce emission lines.
(For example in a cataclysmic variable, optical-UV
spectra show emission lines hydrodgen and helium.)
Suppose that the system is an eclipsing binary, and is such that 
during maximum eclipse, the low mass companion covers exactly 1/2
of the disk around the star and that the binary inclination
angle (angle between normal to binary and line of sight) is 
$> 60^o$. 
Sketch how a given emission line profile would evolve
over the period (e.g. usually of order a day or so)
as the accreting white dwarf passes in and out of eclipse.

\smallskip
Hint: Assume you have the spectral resolution to resolve detailed
features of the line profile  and think about
the line profile emitted by material in a keplerian disk.
This problem can be done mathematically precisely, but a
qualitative discussion based on  physical intuition should suffice.











\vfill 
\eject

\ni {\bf (PHY/AST 564; Blackman; Fall 2001) Problem Set 3 Due}

\bigskip

\ni {\bf 1.}  

\ni {\bf 2.}  

\ni {\bf 3.}  


\vfill 
\eject



\end{document}





\ni {\bf 1.}
The solar luminosity is $3.8\ts 10^{33}$erg/sec
and the solar radius is $7 \ts 10^{10}$cm,
and the Earth-Sun distance is $1.5\ts 10^{13}$cm.

\ni a) What is the total flux at the earth?

\ni b) what is the frequency integrated (specific) intensity?
Calculate this at the Sun and at the Earth.
What is their ratio?



\ni {\bf 2.} It has been proposed that 
neutron stars (compact, gravitationally collapsed 
end states of stars which are extremely 
dense ( $10^{17}kg/m^3$), and all matter is compressed into neutrons)
which are accreting surrounding material from their
strong gravity 
are surrounded by a $10^8$cm plasma corona of ionized gas, with Thompson scattering
depth $\tau\sim 10$.
Suppose that the neutron star rotates with period 1.6 milliseconds.
Could we detect the rotational modulation of the emission from the
neutron star? 


\ni {\bf 3.}
A gamma-ray burst (GRB) is observed once per day, somewhere
on the sky. The total luminosity is about $10^{53}$erg/s
and they last for typically a few seconds.
From variability and relativity, 
we know that the size of the GRB emitting region 
is about $10^{15}$cm.  Post GRB emission is now also 
observed in the X-ray and optical bands. And is thought to result
from the deceleration of a relativistic expanding and radiating wind.
As the Hubble space
telescope (HST) can resolve a source $0.1$ arc seconds in diameter, if a GRB went 
off in Andromeda (700 kiloparsecs=$2\ts 10^{24}$cm away), would HST
 measure its optical intensity or its flux?
If the optical afterglow begins just after the GRB emission,
at what radius would the object have to be for this answer
to change?  What is the value of the measured quantity at that
distance?  Compare the value to that of the Sun.

\ni {\bf 4.}  Derive Einstein relations when the 
two-state atomic system is in equilibrium with a neutrino
field rather than a photon field. (Assume neutrinos to
be massless fermions).


\ni {\bf 5.}
Consider a 1-D slab of front back symmetric 
scatterers of total optical depth $\tau >>1$ 
illuminated normally, from above.  Show that the flux
and angle averaged intensity $J={\bar I}$ below the
layer are $a/(\tau + b)$ times those above where $a$ and $b$
are constants order 1. Use the two-stream approximation.


\ni {\bf 6.}  Rybicki and Lightman Problem 1.4 

\ni {\bf 7.}  Rybicki and Lightman Problem 1.5

\ni {\bf 8.}  Rybicki and Lightman Problem 1.10

