In the following we take a closer look onto the practical use in wavelength calibration of the formalism developed so far and validated against two specific spectrographs. It is important to recall that we have started from text book physical principles, derived analytical equations and their simplifications for practical purposes with the primary goal to obtain robust procedures for improved calibration and data analysis.
The model represents adequately all effects for off-plane spectrographs including line curvature with an accuracy close to one pixel. Because it is mostly based on linear algebra (see Appendix A), it is straight forward to produce high performance code. Such code can be used to generate accurate initial solutions of the dispersion relation using the grating positions as provided by encoders. This allows to implement fully automatic wavelength calibration procedures.
In cases where no grating positions are available, or where the zero points of such readings are unstable, one has to resort to interactive line identifications. Clearly, for these cases one will want to derive simple procedures and code. An important question therefore is how to make use of the special properties of the echelle relation in order to minimize the amount of such line identifications required to bootstrap a semi-automatic wavelength calibration procedure. In order to remain of any practical use, the simplified formalism still has to provide a local accuracy of a few resolution elements.
Using the set of equations presented in Appendix A, here
with the approximation 
, and defining
the intermediate quantities 
and 
as:
one derives for an in-plane spectrograph (s=0, 
):
Because of the approximation this form for 
does not depend on the characteristics of the
cross-disperser. In the absence of detector rotation (see also Hall et
al. 1994) the product 
is constant for a given 
, i.e. a fixed column on the detector.
One can solve Eq. (26) for 
as
with the coefficients ai, i=0,..,4 involving the engineering parameters. If in addition the order curvature is negligible one can simplify further to a first order polynomial.
Usually in real instruments the alignment of the detector with the
dispersion direction is not perfect. A misalignment by only 1 degree
on a 1 K pixel detector amounts to an error of about 10 pixels at the
edges. Hence, it is necessary to correct either the positions or the
linear echelle relation for detector rotation.
In this case only three parameters will be needed to describe the initial relation. The value of this rotation angle may be determined geometrically, for example by comparing the difference in y position for a given wavelength in two adjacent orders.
This naive approximation would state that y-positions depend only on
the characteristics of the cross-disperser. However, it can readily be
seen that even under the assumptions from the start of this section,
this is an over-simplification, because the expression of the position
as it can be derived from the formalism in Appendix A involves
both, 
and 
respectively.
It is obvious that the y positions have to depend also on the
dispersion relation of the echelle grating as well, because the
combination of the two gratings produces the well known order
curvature in echellograms. Therefore one cannot expect a sufficiently
accurate determination of the rotation angle by simply comparing
y-positions for fixed wavelengths. If the order curvature is
significant one will need more identifications, and a least-square
minimization of equation 28 (click here) as presented in Appendix
B. Still this model based bootstrapping of the wavelength calibration
requires less initial idenfications than 2d-polynomial methods as
presented by e.g. Goodrich & Veilleux (1988) or
Verschueren et
al. (1997).