Temperature Definitions

When dealing with a highly rarefied medium such as that found in a molecular cloud the usual methods of defining temperature here on Earth^2.2 are either not possible or not appropriate and some other method is needed. The definition of temperature that is most commonly used in radio astronomy is that of excitation temperature. If a molecule has two energy levels, an upper level and a lower level, then it is possible to define temperature by the ratio of the number of molecules in each level using

$$\frac{n_u}{n_l}=\frac{g_u}{g_l}e^{-\frac{h\nu}{kT_{\rm EX}}} \index{temperature!excitation}$$ (2.25)

Where $n_u$ and $n_l$ are respectively the number of molecules in the upper and lower levels. $g_u$ and $g_l$are the upper and lower level degeneracies and are given by $g_j=2j+1$ where $j$ is the quantum number for the level (see figure 2.1 for an explanation of the labelling convention).

Another commonly used definition is the kinetic temperature. This defines temperature in terms of the speed at which the molecules are moving (thus absolute zero is the the state where the molecules are stationary). It is defined by

$$\frac{1}{2}mv^2=\frac{3}{2}kT_k \index{temperature!kinetic}$$ (2.26)

where $m$ is the mass of a molecule, $k$ is the Boltzman constant and $v$ is the average velocity of the molecules in the gas.

In general in radio astronomy interest is in trace molecules such as CO, CN, HCO+, etc. which are present at a very much lower abundance than ${\rm H_2}$ or He (the two most abundant molecules). As described in section 2.1.1 the trace molecules change up and down between energy levels according to the radiation field and their natural decay probability. However, in addition to this the molecules may collide with one another which may knock them from one energy level to another. In the vast majority of cases the trace molecule will collide with a ${\rm H_2}$ molecule so most collision rates (the probability of a collision causing a transition from one particular level to another) are calculated for ${\rm H_2}$ collisions (although occasional they are calculated for He collisions). The upward collision rates are related to the downward collision rates by the equation of detailed balance

$$\frac{C_{lu}}{C_{ul}}=\frac{n_u}{n_l}=\frac{g_u}{g_l}e^{-\frac{h\nu}{kT_k}} \index{collision coefficients}$$ (2.27)

Where $C_{lu}$ is the transition rate from the lower to the upper level and $C_{ul}$ the transition rate from the upper to the lower level.

Thus it can be seen that if there are a sufficient number of ${\rm H_2}$molecules present in the cloud then the collisions will take place often enough to populate the upper level of a particular transition faster than the natural decay rate (as governed by the Einstein A coefficient). In other words if the collision transfers dominate over the radiation transfers then equation 2.29 defines the ratio of molecules in the upper and lower levels. However this ratio also defines the excitation temperature as given in equation 2.27 and thus when collisions dominate $T_{\rm EX}=T_k$. This is known as thermodynamic equilibrium. When this applies across the entire cloud but the kinetic temperature is not the same everywhere then the cloud is said to be in Local Thermodynamic Equilibrium (LTE). This is a useful approximation for some clouds and greatly simplifies the modelling of such clouds. However, there are many clouds where this is not the case, ie. they have non-LTE conditions and for such clouds the methods described in this thesis are needed to successfully model them.