This problem set relates to the article discussed in class on 9/21/00: Barnard, E.E. 1919, On the Markings of the Sky with a Catalogue of 182 Such Objects, ApJ, 39, 1-23.

In 1919, E.E. Barnard and most other astronomers did not know much about the structure of the Milky Way, or the Initial Mass Function [IMF] of stars, or the Mass-Luminosity relation for main-sequence stars, but you do. In this problem set, you will use your fabulous late-twentieth century knowledge of astronomy to re-interpret Barnard's results in a more quantitative way.

1. Estimate the magnitude distribution of stars in a 1 degree square patch of sky:

a.) toward the galactic center

b.) 180 degrees away from the galactic center (the anticenter)

c.) directly overhead

Present your answer in mathematical and histogram form.

In order to answer this question, you can assume:

• you are observing from the vicinity of the Sun
• no extinction
• an IMF from the literature (e.g. Miller & Scalo 1979)
• that this IMF applies to the whole Galaxy
• all stars on the main sequence
• a very simple mass-luminosity relation
• a very simple geometry for the Galaxy and the distribution of stars within it.

3. Estimate the H I column density for the same three patches of sky you considered in Question 1. Again, make simple, but relatively realistic, assumptions about the distribution of material in the Galaxy.

4. If the gas/dust ratio gives A_v [mag] = N_H × 10^-21 [cm^-2] how much extinction is produced,

a.) per kpc,

b.) in total in the Galaxy

along the three lines of sight considered in Question 1? (State your assumptions carefully.)

5. If Barnard photographed the Taurus dark clouds with a limiting magnitude of m_v=14 mag:

a.) how many stars per square degree could he expect without the "extra" extinction produced by his "dark nebulae?"

b.) how much extinction, in magnitudes, would be associated (roughly) with Barnard's "almost black" and "totally black" regions in Taurus?

Note: The Taurus clouds are at a distance of roughly 140 pc, at. (Hint: you may want to convert that to galactic coordinates!)
 

Miller, G.E. and Scalo, J. 1979, The initial mass function and stellar birthrate in the solar neighborhood, Astrophysical Journal Supplement Series, 41, 513-547.