Introduction to Astrophysics I, 29:119
Fall, 2006
Fifth Homework Set...October 6, 2006.
Due October 16, 2006 (Radio Astronomy Day!)

Show calculations and give reasons for your answers. Don't go around confused and despondent; if you do not know how to get started, ask me for help. All problem numbers refer to the first edition of the book.

(1) Look at the energy level diagram for carbon on the course webpage (in section ``Pictures and Diagrams to Accompany Lectures'', Lecture 8). Calculate the wavelengths of the following transitions. Also identify the part of the electromagnetic spectrum for each transition.

• $3^1P_1 \rightarrow 2^1D_2$ (singlet)
• $3^1P_1 \rightarrow 2^1S_0$ (singlet)
• $2^1S_0 \rightarrow 2^1D_2$ (singlet)
• $3^3P \rightarrow 2^3P$ (triplet)
• $2^1S_0 \rightarrow 2^3P$ (singlet to triplet)

(2) Which of the transitions from the previous problem are permitted, and which are forbidden? Give the reasons for your selections.

(3) Neutral carbon atoms are present in a gas with a magnetic field of 2000 Gauss. Describe quantitatively what happens to the $3^1P_1 \rightarrow 2^1D_2$ singlet transition. Sketch the spectrum with the magnetic field present, and put relevant numbers on your plot. Remember that for singlet states, the Landè $g$ factor is 1.

(4) Assume that significant collisional population of an excited state occurs when the mean thermal energy of atoms in a gas is $\simeq \frac{1}{3}$ the excitation energy of a state (i.e. the energy difference between the ground state and the excited state). Assume that the ground state of neutral carbon is triplet $2^3P$.
(a) At what temperature will significant population of the $2^1S_0$ singlet state occur?
(b) At what temperature will significant population of the $3^1P_1$ singlet state occur?

(5) Explain why, in the previous problem, it is not necessary for the mean thermal energy to equal the excitation energy of the excited state.

(6) Problem 7.1 from textbook.

(7) Assume that the most closely-spaced visual binary that can be ``split'' with an Earth-based telescope has a separation of 0.5 arcseconds.
(a) What is the closest physical separation (dimensions of length) that can be detected for a binary star 47 light years away?
(b) Assuming that the orbits of the binary components are circular, and that the total mass of the system is $1.5 M_{\odot}$, what is the period of this binary?

(8) Look at the spectrum of the Ring Nebula, M57 given in the online diagrams (reference problem #1 above).
(a) Calculate the energy by which the $^1D_2$ state in NII lies above the $^3P$ ground state.
(b) Calculate the energy difference between the $^3P_2$ and $^3P_1$ states.