# Astronomy 300: Homework Assignment #4

Due: Friday, March 1

Scale Model

1. 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)

2. 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)

3. 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?

4. The peak of emission from Betelgeuse lies at about 1 micron. What is the surface temperature of this star according to Wien's Law?

5. 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?

6. Explain how the color index can be used to assess a star's temperature. Would you expect the B-V index of a cool, red star like Proxima Centauri to be larger or smaller than that of a hotter yellow star like the sun?

7. Assume the apparent magnitude of a star in the "Visual" band is measured (V=mV), and the star's distance is known. Explain why this NOT sufficient to determine the star's luminosity. What else must be known?

8. 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:

h4 c3 B λ(T) /[2 (kT)5] = f(x)

Where f(x) is a function of x alone.

9. 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, X-rays, 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, xMAX, 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.) OPTIONAL-Not 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.) OPTIONAL-Not Required (for those who *LOVE* calculus, especially the chain rule): Attempt to derive an expression for xMAX, the value of x which maximizes f(x). Do you know how to solve this type of equation?

10. Now, finally, use your value of xMAX to derive a relationship between temperature and λMAX, 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!!!