The Emissivity Function

In its original form the STEN program needed two arrays to describe the conditions in each shell, one giving the value of the source function (this was stored in array s) and one giving the optical depth function (stored in array kod). These were needed to be able to solve the radiative transfer equation $S_\nu e^{-\tau}$ for a line of sight passing through that shell. It has already been shown (on page ) that the source function is given by

$$S_\nu=\frac{2 h \nu^3}{c^2 \left[\frac{N_l g_u}{N_u g_l}-1\right]} \index{source function}$$ (4.22)

A similar equation is now needed for the optical depth function. Starting in the obvious place with the absorption function (as in equation 2.26)

$$\kappa_\nu=\frac{h\nu}{c}\left(\frac{dn_l}{d\nu}B_{lu}-\frac{dn_u}{d\nu}B_{ul}\right) \index{absorption function}$$ (4.23)

Column density   The $\frac{dn}{d\nu}$ terms represent the rate of change of the number of molecules with frequency. In this case they are distributed in a Gaussian form around the rest frequency of the photons emitted by the molecules transitioning from the upper to the lower level. Figure 4.13 shows a Gaussian curve centred on the rest frequency $\nu_0$. Any position that is offset from this maximum has a intensity given by $\kappa e^{-\frac{(\nu'-\nu_0)^2}{\Delta \nu^2}}$ where $\nu'-\nu_0$ is the offset from the rest frequency and $\kappa$ is the maximum value (ie. the value at the rest frequency). $\Delta \nu$ is the velocity width of the ring. The total number of molecules in a level is then given by integrating $\frac{dn}{d(\nu'-\nu_0)}$ over the entire frequency range

$$n=\int^\infty_{-\infty}\frac{dn}{d(\nu'-\nu_0)}d\nu'=\kappa \... ...)^2}{\Delta \nu^2}} d(\nu'-\nu_0)=\kappa \sqrt{\pi} \Delta \nu$$ (4.24)

where the final step here is a standard integral. So this gives

$$\kappa=\frac{n}{\sqrt{\pi}\Delta \nu}$$

This can be converted to be in terms of a rest velocity using the Doppler relationship $\Delta \nu=\frac{\Delta V}{c}\nu$ to give

$$\kappa=\frac{nc}{\sqrt{\pi}\Delta V\nu}$$

which can be substituted back into the original definition of $\frac{dn}{d(\nu-\nu_0)}$ to give

$$\frac{dn}{d\nu}=\frac{nc}{\sqrt{\pi}\Delta V\nu}e^{-\frac{(V'-V_0)^2}{\Delta V^2}}$$

Note that since the function $\frac{\nu'-\nu}{\Delta \nu}$ is dimensionless the units can be changed from frequency to velocity with no effect. These equations can now in turn be used to convert equation 4.23 into the form

$$\kappa_\nu=\frac{hc}{c\sqrt{\pi}\Delta V}\left(n_lB_{lu}-n_uB_{ul}\right)e^{-\frac{(V'-V_0)^2}{\Delta V^2}}$$

Using equation 2.23 one of the Einstein B coefficients can be removed. Also it is necessary to apply a conversion factor to the Einstein B coefficient. This is because equation 4.23 is the energy per unit frequency per unit volume but the definition of the Einstein B coefficients given in section 2.1.1 was given in terms of the radiation intensity which is power per unit frequency per unit solid angle per unit area. The conversion between these two different definitions of the Einstein B coefficient is given by $B'=\frac{4\pi}{c}B$ where $B'$ is the coefficient defined by the radiation intensity (see section 2.1.1for the 2 definitions). Applying these two corrections then gives an optical depth function of

$$\kappa_\nu=\frac{hc}{4\pi^\frac{3}{2}\Delta V}B_{ul}\left(n_l\frac{g_u}{g_l}-n_u\right)e^{-\frac{(V'-V_0)^2}{\Delta V^2}}$$ (4.25)

When the program is actually solving the radiative transfer equation it will be using the form of equation 3.15. This means that for every segment along a line of sight it multiplies the values of $S_\nu$ and $\kappa_\nu$ together. In certain circumstances this can cause problems as the signs of $S_\nu$ and $\kappa_\nu$ should always be the same, however if the molecules are distributed such that $N_ug_l \approx N_lg_u$ rounding errors in the computer may cause them to have different signs. In order to avoid this problem a function called the emissivity function has been defined^4.8 such that $E_\nu=S_\nu\kappa_\nu$. This then has the form

$$E_\nu=\frac{hc}{4\pi^\frac{3}{2}\Delta V}A_{ul}N_u$$ (4.26)

Note that the $B_{ul}$ from equation 4.25 has been converted to $A_{ul}$ using equation 2.24. This function is used in the ASTRA program in place of the $S_\nu \kappa_\nu$ functions that appear in the STEN program (it is referred to as array eef in the program). Note also that the $e^{-\frac{(V'-V_0)^2}{\Delta V^2}}$ factor present in equation 4.25 has been removed in equation 4.26. This must be done as this factor is different for each ring depending on which line of sight is being dealt with and therefore needs to be calculated separately each time. The same factor is also removed from the optical depth function given in equation 4.25 to give

$$\kappa_\nu=\frac{hc}{4\pi^\frac{3}{2}\Delta V}B_{ul}\left(n_l\frac{g_u}{g_l}-n_u\right)$$ (4.27)

This is the version stored in the array kod in the program.