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.

Consider the five galaxies in the table.

M 33

M 51

NGC 3109

NGC 5850

IC 3475

Use NED (nedwww.ipac.caltech.edu) to

i) obtain and print an image of each of these galaxies. Label the orientation of each image (which way are North & East?).

ii) obtain each system's recession velocity (in km/s).

iii) obtain each object's total, extinction corrected B_T⁰ magnitude and (B-V)_T⁰ color index, where available. Give preference to data from the Third Reference Catalog (RC3) where multiple measures are given.

iv) Use this information to compute the total luminosity of each galaxy.

v) Use the model relations between mass-to-light ratio and color of Bell & de Jong [(2001), ApJ, 550, 212] to convert luminosity to stellar mass. Briefly justify your choice of model from the range offered by Bell & de Jong, addressing the choice of star formation history, metallicity, and IMF.

A. How many arcseconds are in one radian?

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

s_B = 27.05 - 2.5 log(I_B).

Show explicitly that observable surface brightness s_B is distance independent.

C. Spiral and irregular galaxies have an azimuthally averaged radial light profile which is well approximated in many cases by the "exponential disk": I(r) = I₀ e^-r/h, where I₀ is the central surface brightness (in solar luminosities per square parsec) and h 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 s(a), where a is the radius in arcseconds. You will need to define appropriate replacements for I₀ and h.

ii) Integrate I(r) from 0 to 2 pi and r = 0 to x scale lengths (r = xh) 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).