Partial Derivatives of the Spectral Function
For completeness, the partial derivatives used to calculate the matricies containing the simultaneous equations described in Mathematical Approach to Fitting are provided here.
The spectral function
The spectral radiance at a particurlar wavelength can be calculated as a sum of guaussians.
\[I(\lambda, T) = b(\lambda) + N \sum_{i=0}^{n_{lines}}\frac{I_{i}(T)}{\sigma_{i}\sqrt{2\pi}}\exp{\left(-\frac{(\lambda-(\lambda_i+s(\lambda)))^{2}}{2\sigma_{i}^{2}}\right)}\]
where
\[I(T) = \frac{ g_{ns}(2J+1)100 \times hcw_{if}A_{if}}{4\pi Q(T)}\exp{\left[-\frac{100 \times hcw_{upper}}{kT}\right]}\]
Partial derivative of temperature
\[\frac{\partial I}{\partial T} = N \sum_{i=0}^{n_{lines}}\frac{1}{\sigma_{i}\sqrt{2\pi}}\exp{\left(-\frac{(\lambda-(\lambda_i+s(\lambda)))^{2}}{2\sigma_{i}^{2}}\right)}\frac{\partial I_i(T)}{\partial}\]
where
\[\frac{\partial I_i(T)}{\partial T} = I_i(T) \frac{100 \times hcw_{upper}}{kT^2} - I_i(T) \frac{1}{Q(T)}\frac{\partial Q(T)}{\partial T}\]
where \({\partial Q(T)}/{\partial T}\) is given below.
Partial derivative of partition function
\[\frac{\partial Q}{\partial T} = \mbox{log}_{10} (Q) \times \frac{\partial}{\partial T} \left( \mbox{log}_{10} (Q) \right)\]
\[\frac{\partial}{\partial T} \left( \mbox{log}_{10} (Q) \right) = \sum_{n=0}^{6} na_n \left( \mbox{log}_{10} T \right)^{n-1}\frac{1}{T}\]
Partial derivative of column density
\[\frac{\partial I}{\partial N} = \sum_{i=0}^{n_{lines}}\frac{I_{i}(T)}{\sigma_{i}\sqrt{2\pi}}\exp{\left(-\frac{(\lambda-(\lambda_i+s(\lambda)))^{2}}{2\sigma_{i}^{2}}\right)}\]
Partial derivative of background
In the case of a constant background, the partial derivative is simply:
\[\frac{\partial I}{\partial b} = 1\]
However, h3ppy can also be configured to use a polynomial background function:
\[b(\lambda) = \sum_i a_n \lambda^i\]
In this case, the partial derivative of each term becomes:
\[\frac{\partial I}{\partial a_{i}} = \lambda^{i}\]
Partial dervative of the line shift
\[\frac{\partial I}{\partial s} = N \sum_{i=0}^{n_{lines}}\frac{I_{i}(T)}{\sigma_{i}\sqrt{2\pi}}\exp{\left(-\frac{(\lambda-(\lambda_i+s(\lambda)))^{2}}{2\sigma_{i}^{2}}\right)}\frac{2(\lambda-(\lambda_i+s(\lambda)))}{2\sigma_{i}^{2}}\frac{\partial s(\lambda)}{\partial b_{i}}\]
Partial dervative of the line width
The line width, \(\sigma(\lambda)\), describes the width of the line so that the full width at half maximum (FWHM) a particular wavelength is FWHM = \(2\sqrt{2\log(2)}\sigma(\lambda)\). The line width derivative is:
\[\frac{\partial I}{\partial \sigma} = N \sum_{i=0}^{n_{lines}}\frac{I_{i}(T)}{\sigma_{i}\sqrt{2\pi}}\exp{\left(-\frac{(\lambda-(\lambda_i+s(\lambda)))^{2}}{2\sigma_{i}^{2}}\right)}\frac{(\lambda-(\lambda_i+s(\lambda)))^2}{2\sigma_{i}^{2}} \frac{(-1)}{\sigma^3} \frac{\partial \sigma(\lambda)}{\partial a_{i}}\]
where
\[\frac{\partial \sigma(\lambda)}{\partial a_{i}} = \lambda^{i}\]