C_{12}} \lp
\eeq
It is evident that by multiplying the two preceding expressions together, one can obtain the ratio of $n_2$ to $n_0$.  

In the low--density limit $C\ll A$, Equation~\ref{R21} becomes
\beq
\label{Rlowdens}
R(2/1)|_{C\ll A} = \frac{C_{02}A_{10}}{(C_{01} + C_{02})A_{21�}} ~,
\eeq
so that the ratio of the two upper level populations approaches a constant value of 0.11 at low densities.  
  

From Equations~\ref{R21} and \ref{R10}, the fractional population in each of the three levels can be obtained from
\begin{equation}\label{fractions}
\begin{split}
\frac{n_2}{n_{\rm N^+}} &= (1 + R(2/1)^{-1} + R(2/1)^{-1}R(1/0)^{-1})^{-1}\\
\frac{n_1}{n_{\rm N^+}} &= (1 + R(2/1) + R(1/0)^{-1})^{-1}\\
\frac{n_0}{n_{\rm N^+}} &= (1 + R(1/0) + R(2/1)R(1/0))^{-1}
\end{split}
\end{equation}
The variation of the three fractional populations with electron density is shown in Fig.~\ref{fig:fractional_pops}, for a kinetic temperature of 8000 K using the \cite{Tayal11} collisional rate coefficients.
\begin{figure}
\begin{center}
\includegraphics[width = 0.8\textwidth]{ApJ99617fg12.pdf}
\caption{\label{fig:fractional_pops} Fractional populations of three \nplus\ fine structure levels for kinetic temperature of 8000 K, as a function of electron density.}
\end{center}
\end{figure}
In the low--density limit essentially all of the population is in the lowest level, and the fractional populations of the two excited levels are
\begin{equation}\label{lowdens_fractions}
\begin{split}
\frac{n_2}{n_{\rm N^+}} &= \frac{C_{02}}{A_{21}}\\
\frac{n_1}{n_{\rm N^+}} &= \frac{C_{01} + C_{02}}{A_{10}}\\
\end{split}
\end{equation}
This solution necessarily gives the same limiting value for the level population ratio as given in Equation~\ref{Rlowdens}.  It also makes clear that the population and hence the intensity of both fine structure transitions is linearly proportional to the electron density in the low--density limit.

\subsection{Excitation Temperature and Critical Density}

The excitation temperature of a transition is a measure of the relative population of its upper ($j$) and lower ($i$) levels, defined through
\beq
\label{tex}
R(j/i) = \frac{n_j}{n_i} = \frac{g_j}{g_i}exp[-(E_j - E_i)/kT_{{\rm ex}~ij}] \lc
 \eeq
 where the $g$'s are the statistical weights and $E$'s the energies of the levels.  Defining $T^*_{ij}$ = $(E_j - E_i)/k$, we see that
 \beq
 T_{{\rm ex}~ij} = \frac{-T^*_{ij}}{ln[\frac{g_i}{g_j}R(j/i)]} \lp
 \label{tex2}
 \eeq
The \cite{Tayal11} rate coefficients, as well as the earlier calculations, indicate that all three levels are coupled by collisions.  We can use equations~\ref{R21} and \ref{R10} to determine the excitation temperatures of the two \nplus\ fine structure lines, and these are plotted in Fig.~\ref{fig:tex}. It is evident in the figure that the excitation temperatures are very low for low electron densities  and that both become equal to the kinetic temperature (LTE) at sufficiently high electron densities ($\ge$ 10$^5$ cm$^{-3}$).  
\begin{figure}
\begin{center}
\includegraphics[width = 0.8\textwidth]{ApJ99617fg13.pdf}
\caption{\label{fig:tex} Excitation temperatures of \nplus\ fine structure transitions as a function of electron density, for kinetic temperature of 8000 K.  The solid curves are for the \citet{Tayal11} collision rate coefficients while the dashed curves have the same coefficients, but with the \thPtwo\ -- \thPzero\ rate set equal to zero.}
\end{center}
\end{figure}

We can gain some insight into this behavior by first considering collisions that connect only adjacent levels.  In this case, the excitation temperatures are decoupled, and for each transition we can write (for optically thin lines and neglecting any background)

\beq
T_{{\rm ex}~ij} = \frac{T^*_{ij}} {\frac{T^*_{ij}} {T_{\rm k}} + ln(1 + \frac{A_{ji}}{C_{ji}})} \lc
\label{tex3}
\eeq
where $T_{\rm k}$ is the kinetic temperature.

The critical density is unambiguously defined for a two level system as the density at which the collisional deexcitation rate is equal to the spontaneous decay rate.  With this definition, which gives a measure of the density at which the transition is significantly excited, the critical density can be written
\beq
n_{{\rm e~cr}~ji} = \frac{A_{ji}}{R_{ji}}\lp
\eeq
This definition can readily be extended to include the effects of radiative trapping through an ``effective decay rate'', typically written as $A_{\rm eff} = \beta A_{ji}$, where $\beta$ is the photon escape probability.

The excitation temperature in equation \ref{tex3} can be written in terms of the critical density as
\beq
T_{{\rm ex}~ij} = \frac{T^*_{ij}} {\frac{T^*_{ij}} {T_{\rm k}} + ln(1 + \frac{n_{\rm e~cr~ji}}{n_{\rm e}})} 
\eeq

The critical density has a more physical interpretation, which becomes evident by considering a two level or a decoupled mutilevel system as described by equation \ref{R21} or \ref{R10} with only adjacent levels coupled by collisions.  Rather than involving the kinetic temperature, we write the ratio the populations as obtained from the rate equation as
\beq
\frac{n_j}{n_i} = \frac{C_{ij}}{C_{ji} + A_{ji}} = \frac{g_j}{g_i}\frac{e^{-T^*_{ij}/T_{\rm k}}}{1 + A_{ji}/C_{ji}}
= \frac{g_j}{g_i}\frac{e^{-T^*_{ij}/T_{\rm k}}} { 1 + n_{{\rm e~cr}~ji}/n_{\rm e} }\lp
\eeq
We see that the ratio of the population of the upper level to that of the lower level in LTE ($C_{ji} \gg A_{ji}$ or $n_{\rm e} \gg n_{\rm e~ cr}$) is just $(g_j/g_i)e^{-T^*_{ij}/T_{\rm k}}$ and that the population ratio for $C_{ji}$ = $A_{ji}$ or equivalently $n_{\rm e}$ = $n_{\rm e~cr}$ is just one half of the LTE value.  Thus, the critical density is that which results in a collision rate that makes the ratio of upper to lower level population one half of its value in LTE at the kinetic temperature.  This definition is very meaningful inasmuch as it defines an appropriate fraction of the maximum possible level population for a given total column density of the species in question.  It also defines a useful measure of the optical depth relative to its LTE value, which is clearly of importance when analyzing absorption lines.  These attributes are independent of the issue of the ``detectability'' and the ``effective density'' of a given species, as discussed by \cite{evans99} and \cite{shirley15}.

The situation for multilevel systems can be more complex, because in general collisions couple the populations not just of adjacent levels, but of many levels with arbitrary separation.  This situation is illustrated by the rotational levels of the CO molecule, for which collisions with $\Delta J$ exceeding 5 can be significant \citep{Yang10}.

For \nplus\ we have only three levels, but they are in reality all collisionally coupled as discussed above.  The population ratios can be written in a form that emphasizes the similarity with a two level system as
\beq
\label{ind21}
R(2/1) = \frac{n_2}{n_1} = \