Introduction to Astrophysics II, 29:120
Winter, 2009
Third Homework Set...February 6, 2009. Due February 12, 2009

(1) Assume that the temperature in a molecular cloud is 6 K. Further assume that the intensity of a rotational transition is directly proportional to the population of a state, and that the statistical weights of rotational states are all unity. What is the ratio of the intensities of the $3 \rightarrow 2$ and $2 \rightarrow 1$ transitions of CO?

(2) Using insight from #1, explain how you would distinguish a cloud with $T=6$ K from one with $T= 15$ K. You do not need to carry out a calculation, but your explanation should be clear and use equations.

(3) Find parameters for the gas in a molecular cloud. Calculate the Jean's Mass and Jean's Length. How do they compare with the Jeans's mass and length in the general ISM? Be sure you state the source of the data you are using in your calculation.

(4) A Giant Molecular Cloud (GMC, assumed spherical) has a mass of $5 \times 10^4 M_{\odot}$ and a radius of 10 parsecs. What is the change in gravitational potential energy as it contracts to $r = 5$ parsecs? Assuming that it is in virial equilibrium, where does this energy go?

(5) Assuming virial equilibrium, what is the temperature of a GMC with $5 \times 10^4 M_{\odot}$ and $r = 5$ parsecs? The molecular cloud can be assumed to be purely molecular hydrogen.

(6) What will be the temperature and luminosity of the Sun in 5.3 Gyr, i.e. 9.8 Gyr after it formed? How about 7.1 Gyr from now (11.6 Gyr after formation)? Hint: this is not something you can compute with the mathematical toolkit we have developed in this course.

(7) A major consideration in exobiology is the main sequence lifetime of a star which hosts planets for life. The main sequence lifetime must be at least as long as the time necessary for complex life to develop. On Earth, that was 4 Gyr (from formation of the Earth 4.5 Gyr ago to the ``Cambrian Explosion'' 550 million years ago). For purposes of calculation, let's assume that the main sequence lifetime is the time it takes the luminosity of a star to increase to twice its initial value. What is the mass of a star that would have a main sequence lifetime barely adequate for the Cambrian Explosion to have occurred?

(8) Consider the highly evolved solar-type star in Figure 13.3. At what radial distance does the energy generation rate (units of Watts/kg) maximize. Hint: Answering this question will involve a number of (rather simple) calculations. Use of Mathematica is recommended.

(9) Calculate the Kelvin-Helmholtz timescale for the Sun, using its present luminosity.