Scale Model

Now let's make another scale model, using a basketball.
This time the ball's radius will represent the semimajor axis
of Pluto's orbit (39.5 AU). So all planets in our solar
system fit in the ball.
 What is the scale ratio of this model?
At what distance in meters would you put:
 Oort Cloud of comets: ~50,000 AU
 Proxima Centauri; distance 4.2 light years.
 The Center of the Milky Way (express your answer in kilometers,
and identify something that is roughly that distance from SFSU)
 Suppose you are the very first astronomer to observe the star
cluster "VSC" with a large new telescope. Previous studies found the
cluster to be a blurry object of apparent magnitude 8. Because of the
resolving power of your new telescope, you resolve the cluster and
discover, (to your surprise!) that the cluster contains exactly 100
stars, *all of the same brightness!* You name it the "Very Special
Cluster". What is the apparent magnitude of each of VSC's stars?
(Hint: consider each star's flux)
 In the last homework we mentioned that Earth's average
temperature is 290K. At what wavelength does Earth emit the most
light? Which part of the EM spectrum is this in?

The peak of emission from Betelgeuse lies at about 1 micron. What is the surface
temperature of this star according to Wien's Law?
 Betelgeuse is about 23,000 times more luminous than the Sun.
Given the Sun's temperature (5777K), use ratios to compute the
radius of Betelgeuse (in R) Given this radius, if Betelgeuse were
at the center of the Solar System, which of the planets
would be inside it?
 Explain how the color index can be used to assess a star's temperature.
Would you expect the BV index of a cool, red star like Proxima Centauri
to be larger or smaller than that of a hotter yellow star like the sun?
 Assume the apparent magnitude of a star in the "Visual" band
is measured (V=m_{V}), and the star's distance is known.
Explain why this NOT sufficient to determine the star's luminosity.
What else must be known?
 To simplify the Planck Formula (Eq. 3.22), define a dimensionless
wavelength x = (kT/hc) λ.
Substitute this into Eq. 3.22 and collect all constants, T, and
B_{ λ}(T)
on the left hand side of the equation, leaving only
functions of x on the right hand side.
HINT: You should get something a lot like this:
h^{4}
c^{3}
B_{ λ}(T) /[2 (kT)^{5}] = f(x)
Where f(x) is a function of x alone.
 Wien's Law is a very important result used throughout all
fields of astronomy. It tells us the wavelength at which the Planck
function peaks, and hence the kind of light a given object emits most
(visible, infrared, Xrays, etc.)
To derive Wien's Law, find the value of x for which the function you
derived above, f(x), is a maximum.
[ Hint: This value, x_{MAX}, is between 0 and 10.]
A.) Find the maximum of f(x) by plotting f(x) vs. x. You may use
a calculator or write a small computer program, or use software of
your choice (consult the instructor if you would like help doing this
using IDL). Show your graph, and identify the value of x at which
this maximum is found.
B.) OPTIONALNot Required (for those who know calculus): explain how calculus could
be used find the value of x at which f(x) is a maximum.
C.) OPTIONALNot Required (for those who *LOVE* calculus, especially
the chain rule): Attempt to derive an expression for x_{MAX},
the value of x which maximizes f(x). Do you know how to solve this
type of equation?
 Now, finally, use your value of x_{MAX} to
derive a relationship between temperature and
λ_{MAX}, in terms of physical constants,h,c,k.
Then, plug in these constants to derive Wien's Law (Eq. 3.15).
Philosophical Comment: If the last three problems
left you feeling exasperated, do not despair. In 1911
doing these problems would have earned you the Nobel Prize in Physics!!!