Source coordinates

In the common usage case, the coordinates \((u^n,\,v^n)\) of each known source \(n\) in a reference system are provided. In order to obtain the spaxel coordinates \((x^n,\,y^n)\) of the source in the integral field data, it is assumed that a global affine transformation exists that can be described by

• a scaling by a factor \(\xi\) along the \(u\)-axis

• a scaling by a factor \(\nu\) along the \(v\)-axis

• a rotation around an angle \(\theta\)

• shifts \((x_{\text 0},\,y_{\text 0})\) along the \(x\)- and \(y\)-axis of the data cube

Thus, the coordinate transformation can be summarized by the following formulae:

\[\begin{split}x^n &= \xi\,\cos\theta\,u^n &- \nu\,\sin\theta\,v^n &+ x_{\text 0} \\ y^n &= \xi\,\sin\theta\,u^n &+ \nu\,\cos\theta\,v^n &+ y_{\text 0}\end{split}\]

Substituting \(A=\xi\,\cos\theta\), \(B=-\nu\,\cos\theta\), \(C=-\nu\,\sin\theta\), and \(D=\xi\,\sin\theta\), one obtains

\[\begin{split}x^n &= A\,u^n &+ C\,v^n &+ x_{\text 0} \\ y^n &= D\,v^n &- B\,u^n &+ y_{\text 0}\end{split}\]

The parameters \(A\) to \(y_{\text 0}\) of the coordinate transformation can be optimized in the analysis. Their behaviour during the extraction is controlled via the parameter

initfit|posfitinteger

It defines how the coordinate transformation is handled during the analysis: \(0\) - no optimization; \(1\) - only optimize the shifts, i.e. parameters \(x_{\text 0}\) and \(y_{\text 0}\); \(2\) - optimize all \(6\) parameters of the transformation.

Note

Older versions of PampelMuse only used \(4\) parameters to determine the coordinate transformation, which is the special case of an affine transformation where \(\xi=\nu\).