Rotation Diagrams

Where observations have been taken of more than one transition for a given molecule then a standard technique can be used to find the column density, namely the use of rotation diagrams. It can easily be shown that provided a source is optically thin

$$\int I_\nu d\nu =\frac{NfAh\nu}{4\pi}$$ (2.31)

since the total intensity detected, $\int I_\nu d\nu$, is equal to the number of molecules in the line of sight in a unit area, $N$ (ie. the column density), times the fraction of the population in the upper state of the transition, $f$, times the Einstein A coefficient, $A$, ( ie. the spontaneous decay rate) times the energy released per decay ,$h\nu$, divided by the surface area of a unit sphere, $4\pi$. $I_\nu$ is the radiation intensity and for a black body is given by

$$I_\nu=\frac{2h\nu^3}{c^2(e^{\frac{h\nu}{kT_R}}-1)}$$

which in the Rayleigh-Jeans approximation ( $\frac{h\nu}{kT_R} \ll 1$) reduces to

$$I_\nu=\frac{2kT_R\nu^2}{c^2}$$ (2.32)

The temperature $T_R$ is the radiation temperature and is defined by this equation. As such it applies only to a specific value of $\nu$ and may be different for a different value of $\nu$. Substituting equation 2.34 into equation 2.33 and changing the integration to be over $v$ (ie. use $d\nu=\frac{\nu \, dv}{c}$) yields

$$\int T_R dv=\frac{NfAhc^3}{8\pi k \nu^2}$$ (2.33)

The Einstein A coefficient,

$$A_{J \rightarrow J-1}=\frac{1}{g_u}\frac{64\pi^4\nu^3 S\mu^2_x}{3hc^3}$$

can now be substituted in along with the LTE assumption that all the $T_R$s are equal to a single rotation temperature, $T_{rot}$. The $g_u$ is the rotational degeneracy of the upper level, S is the intensity of the transition and $\mu_x$ is the appropriate dipole moment for the transition. The LTE assumption implies that

$$f=\frac{g_ug_kg_I}{Q(T_{rot})} e^{\frac{-E_u}{kT_{rot}}}$$

where Q is a function of $T_{rot}$ and is termed the partition function, $g_I$ and $g_k$ are the reduced nuclear spin and K-level degeneracies respectively. All this, when substituted into equation 2.35 leads to

$$\int T_R dv=\frac{8\pi^3\nu S\mu_x^2g_Ig_k}{3k}\frac{N}{Q(T_{rot})}e^{-\frac{E_u}{kT_{rot}}}$$ (2.34)

This can then be re-arranged and logarithms^2.7 taken to finally yield an equation of the form:

 

$$\underbrace{{\rm ln} \left( \frac{3k \int T_R \,dv}{8 \pi ^3 \nu S \mu ^2_x g_I g_k}\right)} = \underbrace{{\rm ln} \left(\frac{N}{Q(T_{rot})} \right)} - \underbrace{\frac{E_u}{k_{\,}}}.\underbrace{\frac{1}{T_{rot}}}$$ (2.35)

$$y\ \ \ \ \ \ \ \ \ \ \ = c +\ \ x\ \ .\ \ m$$

For a list of formulae for $Q(T_{rot})$ see Blake et al. [6] or Turner [33], for linear diatomic molecules the formula is simple

$$Q(T_{rot})=\frac{kT}{hB}$$

but for non-linear molecules it is rather more complex, for example, for CH$_3$OH :

$$Q(T_{rot})=2\left[ \frac{\pi (kT_{rot})^3}{h^3ABC} \right] ^{\frac{1}{2}}$$

All the values in x and y are known so a point can be plotted on a graph for each detection and a straight line fitted. The slope of the line is then $m$, from which T$_{rot}$ is then known and the y-axis intercept is $c$, from which $N$ is then known.

Those molecules with only 1 or 2 detections can have a lower limit placed on their column density via a method based on the same theory as the rotation diagrams. A rearrangement of the standard equation for a rotation diagram (equation 2.37 above) yields:

$$N_{mol}=\frac{3k\int T^*_Rdv}{8 \pi^3 \nu S \mu^2 g_I g_K}Q(T_{rot})e^{\frac{E_u}{kT_{rot}}}$$

As long as a value for T$_{rot}$ is known this can be solved. Unfortunately a value for T$_{rot}$ is often not known with any accuracy and this is the main source of error for molecules that are genuinely optically thin. As a first approximation $T_{rot}=\frac{2}{3}\frac{E_u}{k}$ is used for molecules that are either symmetric or only slightly asymmetric.

The main problem with this method is that it requires the molecule to be optically thin in all the transitions used. If this is not the case then it is impossible to be certain by any simple method of the column density. However for an optically thin cloud it provides a useful quick method of finding the column density. In a cloud where little or no previous data is available this could be used to provide a starting point for trying more sophisticated modelling.