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


\noindent
{\bf (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. 
\begin{itemize}
\item $3^1P_1 \rightarrow 2^1D_2$ (singlet)
\item $3^1P_1 \rightarrow 2^1S_0$ (singlet)
\item $2^1S_0 \rightarrow 2^1D_2$ (singlet)
\item $3^3P \rightarrow 2^3P$ (triplet)
\item $2^1S_0 \rightarrow 2^3P$ (singlet to triplet)
\end{itemize}    

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

\noindent
{\bf (3)} Neutral carbon atoms are present in a gas with a magnetic field of 2000 Gauss.  Describe {\em 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 Land\`{e} $g$ factor is 1.  \\ 

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{\bf (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? \\

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

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{\bf (6)} Problem 7.1 from textbook.\\

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{\bf (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 {\em 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?\\  

\noindent
{\bf (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.  


 
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