Use the data for M45 and 47 Tuc tabulated below to answer this question.

Plot color magnitude diagrams for M45 and 47 Tuc.

i. Plot a line for the Zero Age Main Sequence (Table 3.9 of Binney & Merrifield). Fit this line by eye to the data. What is the distance modulus to each cluster?
(Ignore reddening.)

ii. Estimate the age of each cluster.

Use the Dias Open Cluster Catalog and the Harris Globular Cluster Catalog to answer this question.

A. Plot the RA & Dec of the open clusters.

i. What does this tell you about the structure of the Milky Way?

ii. What does this tell you about the orientation of the solar system within the Milky Way?

iii. What quantitative measure about the orientation can you extract from this plot?

B. Add the globular clusters to your plot.

i. What qualitatively new information does this add?

ii. What does the distribution of open and globular clusters suggest about the location of star formation in the Milky Way at different epochs?

iii. Estimate the RA & Dec of the center of the Milky Way.

Include a copy of your plot with your answers.

A. Explore some of the many wondrous electronic resources available to you:
NED | DSS | 2Mass | Lambda | SDSS.
Tell me about any good ones I've omitted.

B. UGC 303 and UGC 4663.

i) Use the Digital Sky Survey to find and examine the galaxies UGC 303 and UGC 4663.
Print the DSS image and give a brief description of the morphology of each.

ii) Find the same galaxies on the prints of the POSS plates in the reading room (in the cabinet immediately on your right as you enter CSS 1255). How do the POSS prints compare to the DSS? Do you perceive anthing extra on one or the other?

C. Successfully extract your choice of any interesting data from the Sloan Digital Sky Survey. Show me what you got and why you thought it interesting.

A. How many arcseconds are in one radian?

B. If Σ_B is the B-band surface brightness is solar luminosities per square parsec, show that the surface brightness μ_B in magnitudes per square arcsecond is

μ_B = 27.05 - 2.5 log(Σ_B).

Show explicitly that observable surface brightness μ_B is distance independent.

C. Spiral and irregular galaxies have an azimuthally averaged radial light profile that is well approximated in many cases by the "exponential disk": Σ(r) = Σ₀ e^-r/r_d, where Σ₀ is the central surface brightness (in solar luminosities per square parsec) and r_d is the scale length (in kpc) of a galaxy fit by this description.

i) Derive what this relation looks like in units of magnitudes per square arcsecond μ(θ), where θ is the radius in arcseconds. You will need to define appropriate substitutes for Σ₀ and r_d, e.g., μ₀ and α.

ii) Integrate Σ(r) from 0 to 2π and r = 0 to x scale lengths (r = xr_d) to obtain an expression for the luminosity enclosed by x scale lengths.

• Is the total luminosity finite as x ⇒ ∞ infinity? If not, what is it?
• How many scale lengths contain half the total light? (This is known as the half-light radius, r_e.)
• Plot the cumulative enclosed luminosity L(< x).

A simple spherically symmetric potential commonly used to approximate star clusters is the Plummer Potential:

A. Derive the density distribution ρ(r) that corresponds to this potential.

B. Integrate the result from (A) along the z axis to obtain the projected surface density Σ(R).

[Hint: r² = x²+y²+z² and R² = x²+y²; so R²+z² appears in the integrand.]

C. It is common to refer to the core radius of this distribution as the location where Σ drops to half its central value: Σ(R_c) = Σ(0)/2.

What is R_c in terms of b?

Plot ρ(r) and Σ(R) and mark the location of R_c.