[ kd ]
hwk04
Ast 4001, 2015 Oct 6
Homework set 4 -- Turn in answers on Thursday October 15.
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1. In an ordinary gas, we expect the average kinetic energy of a particle to be E av  1.5 k T .
This is untrue, however, for a thermal gas of photons -- i.e., the Planck distribution,
energy density u = A  / { exp ( h  / k T )  1 } .
The constant A has no effect in the following questions, and let's employ a dimensionless
parameter x = h  / k T instead of photon frequency. For question (a), the aim is to invent
an easy way to calculate it yourself, rather than merely looking it up or using some canned
software. Clear explanations are more important than the calculated quantities.
(a) Devise an easy way to calculate the peak of f ( x ) = x 3 / { e x  1 } to an accuracy
better than 5 decimal places. In other words, the value of x where f ( x ) is maximum,
and the value of that maximum.
Clearly explain your method, which will need to be iterative. List your successive
estimates of x , and quote the final values of x and f max to five decimal places.
("Newton's method" is OK. However, there's a much simpler way that requires more
iterations but each of them is very easy to calculate. With this method you don't need
to evaluate f ( x ) and f ' ( x ) for each iteration, and you don’t need to do as much
planning to set up the algorithm. It’ll need about 7 iterations. Use your imagination.)
(b) For temperature T = 6000 K (about the same as the Sun's photosphere), evaluate the
photon wavelength  which corresponds to the peak described in part (a). Express
your answer in m , nm , and Å .
(c) Explain why the average photon energy, E av , is directly related to the quantity
X =

f (x) dx
/
x f ( x ) d x .
Calculate the value of X to at least five decimal places. The two definite integrals
are not hard to find in various math references. Compare X to the special value
of x found in part (a).
(d) For a photon gas, what value does C have in E av = C k T ? Compare this to the
value of C that applies for an ordinary ideal gas of non-relativistic particles.
(Here E av refers only to the kinetic energies, excluding "internal degrees of freedom"
that atoms and molecules usually have.)
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----- problem 2 is on the next page -----
4001hwk02 - p2
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2. Concerning the Saha equation for ionization. Imagine a sample of pure hydrogen gas in
LTE, but there are no H 2 molecules. We’ll use n to denote density of particles per unit
volume:
n ( H 0 ) + n ( H + ) = n H , the total gas density;
n ( e ) = n ( H + ) , because in pure hydrogen every free electron must be supplied
by a hydrogen atom;
 = m H n H , the mass density.
Some of the formulae will be a little easier to write if we measure temperature T in units of
10000 K: define t = T / ( 10 4 K ). Then, if I H = 13.6 eV is the ionization potential
of hydrogen, the ratio I H / kT is equal to 15.78 / t . It is also convenient to denote
the ionization fraction as x ;
x = n ( H + ) / n H = n ( H + ) /{ n ( H0 ) + n ( H + ) } .
The Saha equation for hydrogen atoms can be written as
n ( H + ) n ( e ) / n ( H 0 ) = C t 1.5 exp (  15.78 / t ) .
(a) If we regard t as a dimensionless parameter, what dimensions or units does the constant
C have?
(b) Rewrite the equation so the left side is an expression containing the ionization fraction x
while n H and t appear only on the right side.
(c) Now assume that the gas has a particular fixed value of the total density n H . At this
density the gas is 50% ionized, n ( H 0 ) = n ( H + ) , at temperature T = 8000 K,
t = 0.8. (This is a reasonable value in many stellar atmospheres.) Accurately
calculate the ionization fraction at temperatures of 7000 and 9000 K.
( Don’t just plug this problem into some black-box software; personally do
and explain the calculations. Don't try to write a one-line solution, because that
would be too messy. Instead write the procedure in two steps. There'll be a
quadratic equation, and you can write its solution in a form that does not
require the difference between two comparable-sized quantities.)
(d) Next, suppose we wish to calculate the temperature that gives some particular
ionization fraction x . Invent, test, and describe a simple iterative procedure that
will do this.
(Hint: This is somewhat like problem 1a. Normally one would use Newton's method
or something similar, but that isn't worth the effort if we only need to calculate a few
values. If we're interested only in "low" temperatures, T < 30,000 K , a very simple
iteration process will converge fast enough. The trick is to identify an ultra-sensitive
factor in the equation.)
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